{
 "schema_version": "1.0",
 "generated_at": "2026-08-14T00:00:00Z",
 "canonical_domains": 26,
 "canonical_records": 3359,
 "status_definitions": {
  "exact": "Canonical problem text retained as the clean formulation; no statement repair was required.",
  "corrected_verified": "Clean formulation repairs the canonical extraction using checked source evidence or an explicit mechanically checkable typo correction.",
  "reconstructed_unverified": "A useful clean reading is recorded, but available source evidence does not uniquely verify it.",
  "unrecoverable": "No defensible single clean statement can be recovered; clean_statement is null and the original is preserved."
 },
 "counts": {
  "corrected_verified": 61,
  "exact": 2886,
  "reconstructed_unverified": 385,
  "unrecoverable": 27
 },
 "clean_statement_sources": {
  "canonical_problem_field": 2886,
  "explicit_typographical_substitution": 23,
  "labeled_recovery_in_report_section_1": 115,
  "no_clean_statement": 27,
  "no_safe_clean_extraction": 308
 },
 "records": {
  "AIM-ALGEBRAIC_GEOMETRY-0001": {
   "statement_status": "exact",
   "original_statement": "How to find (algebraically, say) a mirror construction for fake projective planes (certain rigid surface of general type with same Hodge numbers as projective space, otherwise very different).\n\n\\begin{itemize}\n\\item $\\bullet$ There exist 100 of them, but only 22 have known explicit equations.\n\\item $\\bullet$ Not complete intersections\n\\end{itemize}",
   "clean_statement": "How to find (algebraically, say) a mirror construction for fake projective planes (certain rigid surface of general type with same Hodge numbers as projective space, otherwise very different).\n\n\\begin{itemize}\n\\item $\\bullet$ There exist 100 of them, but only 22 have known explicit equations.\n\\item $\\bullet$ Not complete intersections\n\\end{itemize}",
   "public_statement": "How to find (algebraically, say) a mirror construction for fake projective planes (certain rigid surface of general type with same Hodge numbers as projective space, otherwise very different).\n\n\\begin{itemize}\n\\item $\\bullet$ There exist 100 of them, but only 22 have known explicit equations.\n\\item $\\bullet$ Not complete intersections\n\\end{itemize}",
   "evidence": "The canonical record is item 1.1, “Mirror constructions,” from the AIM workshop *Syzygies and mirror symmetry*. Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0002": {
   "statement_status": "exact",
   "original_statement": "Given a singular variety in char. p, $X$, $\\mathcal{F}:=Frobenius$, then $\\mathcal{F}^{>>0}(Perf(X))=Coh(X)$ (as dg enhanced categories of complexes) [B-I-L-M-P].\n\nCan we see the mirror to this fact explicitly in case e.g. of mirrors to $X=\\{xy=0\\}\\subset \\mathbb{A}_{\\mathbb{F}_p}$? More generally, what is the symplectic mirror to Frobenius for each $p$?",
   "clean_statement": "Given a singular variety in char. p, $X$, $\\mathcal{F}:=Frobenius$, then $\\mathcal{F}^{>>0}(Perf(X))=Coh(X)$ (as dg enhanced categories of complexes) [B-I-L-M-P].\n\nCan we see the mirror to this fact explicitly in case e.g. of mirrors to $X=\\{xy=0\\}\\subset \\mathbb{A}_{\\mathbb{F}_p}$? More generally, what is the symplectic mirror to Frobenius for each $p$?",
   "public_statement": "Given a singular variety in char. p, $X$, $\\mathcal{F}:=Frobenius$, then $\\mathcal{F}^{>>0}(Perf(X))=Coh(X)$ (as dg enhanced categories of complexes) [B-I-L-M-P].\n\nCan we see the mirror to this fact explicitly in case e.g. of mirrors to $X=\\{xy=0\\}\\subset \\mathbb{A}_{\\mathbb{F}_p}$? More generally, what is the symplectic mirror to Frobenius for each $p$?",
   "evidence": "The canonical record is problem 1.2 in the AIM workshop list *Syzygies and mirror symmetry*, section “Mirror constructions.” Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0003": {
   "statement_status": "exact",
   "original_statement": "Probe what is know about symp. mirrors to algebraic geometric operations (not necessary toric) such as blow-ups?",
   "clean_statement": "Probe what is know about symp. mirrors to algebraic geometric operations (not necessary toric) such as blow-ups?",
   "public_statement": "Probe what is know about symp. mirrors to algebraic geometric operations (not necessary toric) such as blow-ups?",
   "evidence": "The canonical record is problem 1.3 in the AIM workshop *Syzygies and mirror symmetry*, section “Mirror constructions.” Its exact extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0004": {
   "statement_status": "exact",
   "original_statement": "Explore the mirror symmetry correspondence between Morse-theoretic decompositions and algebraic/algebro-geometric decompositions",
   "clean_statement": "Explore the mirror symmetry correspondence between Morse-theoretic decompositions and algebraic/algebro-geometric decompositions",
   "public_statement": "Explore the mirror symmetry correspondence between Morse-theoretic decompositions and algebraic/algebro-geometric decompositions",
   "evidence": "The exact canonical record is problem 1.4 in the AIM workshop *Syzygies and mirror symmetry*, section “Mirror constructions”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0005": {
   "statement_status": "exact",
   "original_statement": "HMS for ATVs (e.g. $x^2+y^2+z^2=0/\\mathbb{R}$)\n\nDoes the derived category of an arithmetic toric variety (i.e., one defined over a non-algebraically closed field) admits an exceptional collection?\n(e.g., $\\{x^2+y^2+z^2=0\\}$ in $\\mathbb{P}^2_{\\mathbb{R}}$)\n\\begin{itemize}\n\\item $\\bullet$ Over $\\mathbb{C}$ is just $\\mathbb{P}^1$ but has no $\\mathbb{R}$-points\n\\item $\\bullet$ has exceptional collection $$ with $End(\\mathcal{E})=\\mathbb{H}$.\n\\item $\\bullet$ has an action of $S^1$ which is the restriction of the $S^1\\subset \\mathbb{C}^*$ action on $\\mathbb{C}-points)\n\\item $\\bullet$ toric data here is roughly data of a fan + data of a $Gal(\\overline{k}/k)$ action on the fan + ...\n\\item $\\bullet$ One source: An exceptional collection where fan automorphism permutes objects; e.g. powers of toric Frobenius applied to $\\mathcal{O}_X$ (gives generating objects acted on by $Gal$, sometimes exceptional collection).\n\\end{itemize}",
   "clean_statement": "HMS for ATVs (e.g. $x^2+y^2+z^2=0/\\mathbb{R}$)\n\nDoes the derived category of an arithmetic toric variety (i.e., one defined over a non-algebraically closed field) admits an exceptional collection?\n(e.g., $\\{x^2+y^2+z^2=0\\}$ in $\\mathbb{P}^2_{\\mathbb{R}}$)\n\\begin{itemize}\n\\item $\\bullet$ Over $\\mathbb{C}$ is just $\\mathbb{P}^1$ but has no $\\mathbb{R}$-points\n\\item $\\bullet$ has exceptional collection $$ with $End(\\mathcal{E})=\\mathbb{H}$.\n\\item $\\bullet$ has an action of $S^1$ which is the restriction of the $S^1\\subset \\mathbb{C}^*$ action on $\\mathbb{C}-points)\n\\item $\\bullet$ toric data here is roughly data of a fan + data of a $Gal(\\overline{k}/k)$ action on the fan + ...\n\\item $\\bullet$ One source: An exceptional collection where fan automorphism permutes objects; e.g. powers of toric Frobenius applied to $\\mathcal{O}_X$ (gives generating objects acted on by $Gal$, sometimes exceptional collection).\n\\end{itemize}",
   "public_statement": "HMS for ATVs (e.g. $x^2+y^2+z^2=0/\\mathbb{R}$)\n\nDoes the derived category of an arithmetic toric variety (i.e., one defined over a non-algebraically closed field) admits an exceptional collection?\n(e.g., $\\{x^2+y^2+z^2=0\\}$ in $\\mathbb{P}^2_{\\mathbb{R}}$)\n\\begin{itemize}\n\\item $\\bullet$ Over $\\mathbb{C}$ is just $\\mathbb{P}^1$ but has no $\\mathbb{R}$-points\n\\item $\\bullet$ has exceptional collection $$ with $End(\\mathcal{E})=\\mathbb{H}$.\n\\item $\\bullet$ has an action of $S^1$ which is the restriction of the $S^1\\subset \\mathbb{C}^*$ action on $\\mathbb{C}-points)\n\\item $\\bullet$ toric data here is roughly data of a fan + data of a $Gal(\\overline{k}/k)$ action on the fan + ...\n\\item $\\bullet$ One source: An exceptional collection where fan automorphism permutes objects; e.g. powers of toric Frobenius applied to $\\mathcal{O}_X$ (gives generating objects acted on by $Gal$, sometimes exceptional collection).\n\\end{itemize}",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0006": {
   "statement_status": "exact",
   "original_statement": "Does every smooth toric variety admits an exceptional collection of sheaves\n\\begin{itemize}\n\\item $\\bullet$ if replace sheaves (good objects of $D^b(X)$, answer is yes)\n\\item $\\bullet$ First open case is the case of toric 3-folds\n\\item $\\bullet$ if replace sheaves with line bundles, answer is no [Etimov '14] counterexample smooth Fano toric with $dim\\geq 0$ and $Pic=3$.\n\\item $\\bullet$ For surfaces can do with line bundles, read off explicitly from fan (exposition of this in [Hacking-Keating])\n\\item $\\bullet$ More specifically, is it possible to find an exceptional collection from (push-forwards of) line bundles on toric strata? (This case can be analyzed, perhaps).\n\\end{itemize}",
   "clean_statement": "Does every smooth toric variety admits an exceptional collection of sheaves\n\\begin{itemize}\n\\item $\\bullet$ if replace sheaves (good objects of $D^b(X)$, answer is yes)\n\\item $\\bullet$ First open case is the case of toric 3-folds\n\\item $\\bullet$ if replace sheaves with line bundles, answer is no [Etimov '14] counterexample smooth Fano toric with $dim\\geq 0$ and $Pic=3$.\n\\item $\\bullet$ For surfaces can do with line bundles, read off explicitly from fan (exposition of this in [Hacking-Keating])\n\\item $\\bullet$ More specifically, is it possible to find an exceptional collection from (push-forwards of) line bundles on toric strata? (This case can be analyzed, perhaps).\n\\end{itemize}",
   "public_statement": "Does every smooth toric variety admits an exceptional collection of sheaves\n\\begin{itemize}\n\\item $\\bullet$ if replace sheaves (good objects of $D^b(X)$, answer is yes)\n\\item $\\bullet$ First open case is the case of toric 3-folds\n\\item $\\bullet$ if replace sheaves with line bundles, answer is no [Etimov '14] counterexample smooth Fano toric with $dim\\geq 0$ and $Pic=3$.\n\\item $\\bullet$ For surfaces can do with line bundles, read off explicitly from fan (exposition of this in [Hacking-Keating])\n\\item $\\bullet$ More specifically, is it possible to find an exceptional collection from (push-forwards of) line bundles on toric strata? (This case can be analyzed, perhaps).\n\\end{itemize}",
   "evidence": "The source is AIM Problem 2.2 in the “Syzygies and mirror symmetry” list, section “Exceptional collections.” It reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0007": {
   "statement_status": "exact",
   "original_statement": "Can the \"looking glass\" parallel/analogy between rational homotopy theory (= symplectic geometry of cotangent bundles) and local ring theory be generalized/ further exploited to other symplectic manifolds (e.g. Weinstein manifolds)?\n\nA)\n\\begin{itemize}\n\\item $\\bullet$ can \"attach handles\" to Weinstein manifolds\n\\item $\\bullet$ there exists a natural \"categorical compactification\" of Fukaya categories via \"stop\" or \"Lefschetz fibrations\"\n\\end{itemize}\nB)\n\\begin{itemize}\n\\item $\\bullet$ Can rational homotopy theory e.g. Sullivan minimal models can be used to e.g. construct resolutions?\n\\end{itemize}\nComment: The initial parallel comes from\n$$C_*(\\Omega_q M)=Ext_{C^{\\bullet}M}(k,k)=A$$\nwhere\n$$C_*(\\Omega_q M)\\simeq End_{Fuk(T^*M)}(T^*_q M, T^*_q M)$$\n(Here $M$ is assumed to be simply connected).",
   "clean_statement": "Can the \"looking glass\" parallel/analogy between rational homotopy theory (= symplectic geometry of cotangent bundles) and local ring theory be generalized/ further exploited to other symplectic manifolds (e.g. Weinstein manifolds)?\n\nA)\n\\begin{itemize}\n\\item $\\bullet$ can \"attach handles\" to Weinstein manifolds\n\\item $\\bullet$ there exists a natural \"categorical compactification\" of Fukaya categories via \"stop\" or \"Lefschetz fibrations\"\n\\end{itemize}\nB)\n\\begin{itemize}\n\\item $\\bullet$ Can rational homotopy theory e.g. Sullivan minimal models can be used to e.g. construct resolutions?\n\\end{itemize}\nComment: The initial parallel comes from\n$$C_*(\\Omega_q M)=Ext_{C^{\\bullet}M}(k,k)=A$$\nwhere\n$$C_*(\\Omega_q M)\\simeq End_{Fuk(T^*M)}(T^*_q M, T^*_q M)$$\n(Here $M$ is assumed to be simply connected).",
   "public_statement": "Can the \"looking glass\" parallel/analogy between rational homotopy theory (= symplectic geometry of cotangent bundles) and local ring theory be generalized/ further exploited to other symplectic manifolds (e.g. Weinstein manifolds)?\n\nA)\n\\begin{itemize}\n\\item $\\bullet$ can \"attach handles\" to Weinstein manifolds\n\\item $\\bullet$ there exists a natural \"categorical compactification\" of Fukaya categories via \"stop\" or \"Lefschetz fibrations\"\n\\end{itemize}\nB)\n\\begin{itemize}\n\\item $\\bullet$ Can rational homotopy theory e.g. Sullivan minimal models can be used to e.g. construct resolutions?\n\\end{itemize}\nComment: The initial parallel comes from\n$$C_*(\\Omega_q M)=Ext_{C^{\\bullet}M}(k,k)=A$$\nwhere\n$$C_*(\\Omega_q M)\\simeq End_{Fuk(T^*M)}(T^*_q M, T^*_q M)$$\n(Here $M$ is assumed to be simply connected).",
   "evidence": "This is Problem 3.1, “Symplectic geometry,” from the AIM workshop list *Syzygies and mirror symmetry*. The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0008": {
   "statement_status": "exact",
   "original_statement": "Compare $\\Delta$-resolutions\n\nCan we directly compare the Hanlon-Hicks-Lazarev and Brown-Erman construction of resolutions?\n\n$\\bullet$ Potential mechanism: (starting in degree 0)\n\n{vertices/points in stratification of $T^n$ arising in HHL}\nis in one to one correspondence with\n{additional generators for normalization of the semi-group ring, i.e. elements coming from saturation of semigroup}",
   "clean_statement": "Compare $\\Delta$-resolutions\n\nCan we directly compare the Hanlon-Hicks-Lazarev and Brown-Erman construction of resolutions?\n\n$\\bullet$ Potential mechanism: (starting in degree 0)\n\n{vertices/points in stratification of $T^n$ arising in HHL}\nis in one to one correspondence with\n{additional generators for normalization of the semi-group ring, i.e. elements coming from saturation of semigroup}",
   "public_statement": "Compare $\\Delta$-resolutions\n\nCan we directly compare the Hanlon-Hicks-Lazarev and Brown-Erman construction of resolutions?\n\n$\\bullet$ Potential mechanism: (starting in degree 0)\n\n{vertices/points in stratification of $T^n$ arising in HHL}\nis in one to one correspondence with\n{additional generators for normalization of the semi-group ring, i.e. elements coming from saturation of semigroup}",
   "evidence": "The canonical AIM record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 7,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_GEOMETRY-0009": {
   "statement_status": "exact",
   "original_statement": "Topology/Morse theory of cellular resolutions\n\nCan we interpret resolutions via Fukaya categories? (e.g. minimizing resolutions through \"cellular\" operations)",
   "clean_statement": "Topology/Morse theory of cellular resolutions\n\nCan we interpret resolutions via Fukaya categories? (e.g. minimizing resolutions through \"cellular\" operations)",
   "public_statement": "Topology/Morse theory of cellular resolutions\n\nCan we interpret resolutions via Fukaya categories? (e.g. minimizing resolutions through \"cellular\" operations)",
   "evidence": "The canonical record is Problem 4.2 in the “Resolutions” section of the AIM problem list *Syzygies and mirror symmetry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0010": {
   "statement_status": "exact",
   "original_statement": "Topology/Morse theory of cellular resolutions\n\nThe Morse-theoretic description of $H_*(M)$ can be refined (at e.g. level of \"framed flow categories\"). can this refined decomposition tell us something on the algebraic side?",
   "clean_statement": "Topology/Morse theory of cellular resolutions\n\nThe Morse-theoretic description of $H_*(M)$ can be refined (at e.g. level of \"framed flow categories\"). can this refined decomposition tell us something on the algebraic side?",
   "public_statement": "Topology/Morse theory of cellular resolutions\n\nThe Morse-theoretic description of $H_*(M)$ can be refined (at e.g. level of \"framed flow categories\"). can this refined decomposition tell us something on the algebraic side?",
   "evidence": "The canonical record is Problem 4.3 in the “Resolutions” section of the AIM workshop list *Syzygies and mirror symmetry*. The source page was checked directly on 2026-07-22 and agrees with the JSON record. Its title and text are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0011": {
   "statement_status": "exact",
   "original_statement": "$T^n=K(\\pi,1)$\n\nGiven a cellular resolution, is there a meaningful way to algebraically interpret topology of the cell complex beyond homology e.g. $\\pi_1$?",
   "clean_statement": "$T^n=K(\\pi,1)$\n\nGiven a cellular resolution, is there a meaningful way to algebraically interpret topology of the cell complex beyond homology e.g. $\\pi_1$?",
   "public_statement": "$T^n=K(\\pi,1)$\n\nGiven a cellular resolution, is there a meaningful way to algebraically interpret topology of the cell complex beyond homology e.g. $\\pi_1$?",
   "evidence": "The assigned record is Problem 4.4 in the AIM problem list *Syzygies and mirror symmetry*, section “Resolutions”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0012": {
   "statement_status": "exact",
   "original_statement": "Topology/Morse theory of resolutions\n\nIs there an analogue of \"Morse theory for cellular resolutions\" for virtual resolutions?",
   "clean_statement": "Topology/Morse theory of resolutions\n\nIs there an analogue of \"Morse theory for cellular resolutions\" for virtual resolutions?",
   "public_statement": "Topology/Morse theory of resolutions\n\nIs there an analogue of \"Morse theory for cellular resolutions\" for virtual resolutions?",
   "evidence": "The canonical record is Problem 4.5 in the AIM list *Syzygies and mirror symmetry*, section “Resolutions”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0013": {
   "statement_status": "exact",
   "original_statement": "Observation: The current definition of virtual resolution is so broad and it is not meaningful to parameterize, then.\n\nIs there a well-behaved moduli theory of virtual resolutions equipped with extra structure? e.g. condition under which there exists a finite set? or space/ stacks/...?\n(c.f. [Boij-Sölderberg] theory constructs a meaningful cone of Betti tables of all free complexes. Look at subcone of virtual resolutions of a given $M$)",
   "clean_statement": "Observation: The current definition of virtual resolution is so broad and it is not meaningful to parameterize, then.\n\nIs there a well-behaved moduli theory of virtual resolutions equipped with extra structure? e.g. condition under which there exists a finite set? or space/ stacks/...?\n(c.f. [Boij-Sölderberg] theory constructs a meaningful cone of Betti tables of all free complexes. Look at subcone of virtual resolutions of a given $M$)",
   "public_statement": "Observation: The current definition of virtual resolution is so broad and it is not meaningful to parameterize, then.\n\nIs there a well-behaved moduli theory of virtual resolutions equipped with extra structure? e.g. condition under which there exists a finite set? or space/ stacks/...?\n(c.f. [Boij-Sölderberg] theory constructs a meaningful cone of Betti tables of all free complexes. Look at subcone of virtual resolutions of a given $M$)",
   "evidence": "This is Problem 4.6 in the **Resolutions** section of the AIM list *Syzygies and mirror symmetry*. The canonical record and the live AIM page agree. The live page was checked on 2026-07-22 and contains no status update or attached remark. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0014": {
   "statement_status": "exact",
   "original_statement": "What does the Fourier-Mukai transform by either resolution of $\\Delta$ applied to $M$ tell us about $M$? More generally what does applying Fourier-Mukai transforms of resolutions of toric subvarieties tell us about the modules apply them to? [e.g. in the spirit of the way we obtain Eagon-Northcott-type complexes for modules]",
   "clean_statement": "What does the Fourier-Mukai transform by either resolution of $\\Delta$ applied to $M$ tell us about $M$? More generally what does applying Fourier-Mukai transforms of resolutions of toric subvarieties tell us about the modules apply them to? [e.g. in the spirit of the way we obtain Eagon-Northcott-type complexes for modules]",
   "public_statement": "What does the Fourier-Mukai transform by either resolution of $\\Delta$ applied to $M$ tell us about $M$? More generally what does applying Fourier-Mukai transforms of resolutions of toric subvarieties tell us about the modules apply them to? [e.g. in the spirit of the way we obtain Eagon-Northcott-type complexes for modules]",
   "evidence": "The exact canonical record is AIM Problem 5.1 from the 2023 workshop *Syzygies and mirror symmetry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0015": {
   "statement_status": "exact",
   "original_statement": "Given $M$ an $S$-module, $S=Cox(X)$,\n\\begin{enumerate}\n\\item can we always find another $S$-module $M'$ satisfying\n\\begin{itemize}\n\\item $\\bullet$ $\\tilde{M}=\\tilde{M}'$\n\\item $\\bullet$ $pdim(M')\\geq dim(X)$\n\\end{itemize}\nThis is known to be true if $X=\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$.\n\n\\item What is the relation of this to examples of [Chardin-D'Cruz] where adding embedded components drops regularity?\n\n\\item Potential mechanism by truncating $M$ to $(M)_{\\geq d}$? (This works for $\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$'s)\n\\item Alternatively, could there be an invariant of $\\tilde{M}$ which obstructs existence of such an $M'$?\n\\end{enumerate}",
   "clean_statement": "Given $M$ an $S$-module, $S=Cox(X)$,\n\\begin{enumerate}\n\\item can we always find another $S$-module $M'$ satisfying\n\\begin{itemize}\n\\item $\\bullet$ $\\tilde{M}=\\tilde{M}'$\n\\item $\\bullet$ $pdim(M')\\geq dim(X)$\n\\end{itemize}\nThis is known to be true if $X=\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$.\n\n\\item What is the relation of this to examples of [Chardin-D'Cruz] where adding embedded components drops regularity?\n\n\\item Potential mechanism by truncating $M$ to $(M)_{\\geq d}$? (This works for $\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$'s)\n\\item Alternatively, could there be an invariant of $\\tilde{M}$ which obstructs existence of such an $M'$?\n\\end{enumerate}",
   "public_statement": "Given $M$ an $S$-module, $S=Cox(X)$,\n\\begin{enumerate}\n\\item can we always find another $S$-module $M'$ satisfying\n\\begin{itemize}\n\\item $\\bullet$ $\\tilde{M}=\\tilde{M}'$\n\\item $\\bullet$ $pdim(M')\\geq dim(X)$\n\\end{itemize}\nThis is known to be true if $X=\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$.\n\n\\item What is the relation of this to examples of [Chardin-D'Cruz] where adding embedded components drops regularity?\n\n\\item Potential mechanism by truncating $M$ to $(M)_{\\geq d}$? (This works for $\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$'s)\n\\item Alternatively, could there be an invariant of $\\tilde{M}$ which obstructs existence of such an $M'$?\n\\end{enumerate}",
   "evidence": "The assigned record is Problem 6.1 in the “Modules over the Cox ring” section of the AIM list *Syzygies and mirror symmetry*. The canonical record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0016": {
   "statement_status": "exact",
   "original_statement": "What can be said about the Orlov spectrum of a toric variety (or any $X$)? (e.g. about gaps, sequences, etc.)\n\nKnown:\n\\begin{itemize}\n\\item $OrSpec(\\mathbb{P}^1)=\\{1,2\\}$\n\\item On a toric surface, there exists full arbitrary large consecutive sequence.\n\\item More genrally, by replacing $D^b(X)$ with $\\mathcal{C}$, there exists a notion of $OrSpec(\\mathcal{C})$ and $OrSpec(Rep (Q))$ ($Q$ is a quiver), computed when $Q$ is of A/D types.\n\\end{itemize}",
   "clean_statement": "What can be said about the Orlov spectrum of a toric variety (or any $X$)? (e.g. about gaps, sequences, etc.)\n\nKnown:\n\\begin{itemize}\n\\item $OrSpec(\\mathbb{P}^1)=\\{1,2\\}$\n\\item On a toric surface, there exists full arbitrary large consecutive sequence.\n\\item More genrally, by replacing $D^b(X)$ with $\\mathcal{C}$, there exists a notion of $OrSpec(\\mathcal{C})$ and $OrSpec(Rep (Q))$ ($Q$ is a quiver), computed when $Q$ is of A/D types.\n\\end{itemize}",
   "public_statement": "What can be said about the Orlov spectrum of a toric variety (or any $X$)? (e.g. about gaps, sequences, etc.)\n\nKnown:\n\\begin{itemize}\n\\item $OrSpec(\\mathbb{P}^1)=\\{1,2\\}$\n\\item On a toric surface, there exists full arbitrary large consecutive sequence.\n\\item More genrally, by replacing $D^b(X)$ with $\\mathcal{C}$, there exists a notion of $OrSpec(\\mathcal{C})$ and $OrSpec(Rep (Q))$ ($Q$ is a quiver), computed when $Q$ is of A/D types.\n\\end{itemize}",
   "evidence": "The canonical record is Problem 7.1 in the AIM list *Syzygies and mirror symmetry*, section \"Orlov spectrum and Rouquier dimension\":",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0017": {
   "statement_status": "exact",
   "original_statement": "Rdim of $D^b Sing$\n\nWhat about Rouquier dimension or orlov spectrum of $D^b(Sing(X))=D^b(Coh(X))/D^b(Perf(X))$?",
   "clean_statement": "Rdim of $D^b Sing$\n\nWhat about Rouquier dimension or orlov spectrum of $D^b(Sing(X))=D^b(Coh(X))/D^b(Perf(X))$?",
   "public_statement": "Rdim of $D^b Sing$\n\nWhat about Rouquier dimension or orlov spectrum of $D^b(Sing(X))=D^b(Coh(X))/D^b(Perf(X))$?",
   "evidence": "The exact canonical record is problem 7.2 in the AIM workshop *Syzygies and mirror symmetry*, section “Orlov spectrum and Rouquier dimension”; it is zero-based record 16 of `aim-algebraic-geometry-notes.json`:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0018": {
   "statement_status": "exact",
   "original_statement": "Same questions as above for $Fuk(X)$ where $X$ is compact symplectic.",
   "clean_statement": "Same questions as above for $Fuk(X)$ where $X$ is compact symplectic.",
   "public_statement": "Same questions as above for $Fuk(X)$ where $X$ is compact symplectic.",
   "evidence": "The canonical record is AIM problem 7.3 from the workshop *Syzygies and mirror symmetry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0019": {
   "statement_status": "exact",
   "original_statement": "There are two known ways to bound Rouquier dimension of a category\n\\begin{enumerate}\n\\item bound length of a resolution of a diagonal\n\\item the minimal depth of a presentation of $\\mathcal{C}$ as hocolim (cats with Rouquier dimension $Q$). Meaning if $F:I^{\\leq}\\to Cat$ diagram with $Rdim F(i)=0$, then $Rdim(hocolim F i)\\leq depth(I)$\n\\end{enumerate}\n\nCan (1) be used to bound (2) or vice-versa?",
   "clean_statement": "There are two known ways to bound Rouquier dimension of a category\n\\begin{enumerate}\n\\item bound length of a resolution of a diagonal\n\\item the minimal depth of a presentation of $\\mathcal{C}$ as hocolim (cats with Rouquier dimension $Q$). Meaning if $F:I^{\\leq}\\to Cat$ diagram with $Rdim F(i)=0$, then $Rdim(hocolim F i)\\leq depth(I)$\n\\end{enumerate}\n\nCan (1) be used to bound (2) or vice-versa?",
   "public_statement": "There are two known ways to bound Rouquier dimension of a category\n\\begin{enumerate}\n\\item bound length of a resolution of a diagonal\n\\item the minimal depth of a presentation of $\\mathcal{C}$ as hocolim (cats with Rouquier dimension $Q$). Meaning if $F:I^{\\leq}\\to Cat$ diagram with $Rdim F(i)=0$, then $Rdim(hocolim F i)\\leq depth(I)$\n\\end{enumerate}\n\nCan (1) be used to bound (2) or vice-versa?",
   "evidence": "The displayed text really does contain \\(Q\\), so this is not an extraction error. Because the very next clause says \\(\\operatorname{Rdim}F(i)=0\\), the only coherent reconstruction is that “\\(Q\\)” is a typographical substitution for “\\(0\\).” The notation \\(I^{\\leq}\\) is not defined on the page. I interpret it as a finite partially ordered set \\((I,\\leq)\\), regarded as a category. This agrees with the precise finite-poset formulation subsequently used by Bai--Côté.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0020": {
   "statement_status": "exact",
   "original_statement": "What can be said about the Rouquier dimension of non-toric varieties and/for their symplectic mirrors?\n\\begin{itemize}\n\\item Toric varieties of dim n $\\longleftrightarrow$(mirror) ($T^*T^n$,some stop)\n\\item What known symplecticly for $(T^*M,stops)$ can be find in (Bai-Cote, Hanlon-Hicks-Lazarev, Favero-Huang)\n\\end{itemize}\ne.g. is there a symplectic $(X^{2n},f)$ such that $Rdim \\ Fuk(X,f)\\lneq n$? (for instance by finding $(X,f)$ mirror to a singular $Y$ with $Rdim(D^b(Y))>dim(Y)$)",
   "clean_statement": "What can be said about the Rouquier dimension of non-toric varieties and/for their symplectic mirrors?\n\\begin{itemize}\n\\item Toric varieties of dim n $\\longleftrightarrow$(mirror) ($T^*T^n$,some stop)\n\\item What known symplecticly for $(T^*M,stops)$ can be find in (Bai-Cote, Hanlon-Hicks-Lazarev, Favero-Huang)\n\\end{itemize}\ne.g. is there a symplectic $(X^{2n},f)$ such that $Rdim \\ Fuk(X,f)\\lneq n$? (for instance by finding $(X,f)$ mirror to a singular $Y$ with $Rdim(D^b(Y))>dim(Y)$)",
   "public_statement": "What can be said about the Rouquier dimension of non-toric varieties and/for their symplectic mirrors?\n\\begin{itemize}\n\\item Toric varieties of dim n $\\longleftrightarrow$(mirror) ($T^*T^n$,some stop)\n\\item What known symplecticly for $(T^*M,stops)$ can be find in (Bai-Cote, Hanlon-Hicks-Lazarev, Favero-Huang)\n\\end{itemize}\ne.g. is there a symplectic $(X^{2n},f)$ such that $Rdim \\ Fuk(X,f)\\lneq n$? (for instance by finding $(X,f)$ mirror to a singular $Y$ with $Rdim(D^b(Y))>dim(Y)$)",
   "evidence": "This record is Problem 7.5 in the AIM workshop list *Syzygies and mirror symmetry*, section “Orlov spectrum and Rouquier dimension.” The canonical record and the live AIM page were both checked. The page really contains <code>\\lneq</code>; the apparent conflict below is therefore present in the source, rather than being introduced by JSON extraction.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0021": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "What operations on Fukaya categories have meaningful commutative algebra translations? e.g. maybe a spherical twist (via mirror symmetry Dehn twist) around a spherical object supported on irrelevant virtual resolution, or more generally induces an aut (space of virtual resolutions)?",
   "clean_statement": null,
   "public_statement": "What operations on Fukaya categories have meaningful commutative algebra translations? e.g. maybe a spherical twist (via mirror symmetry Dehn twist) around a spherical object supported on irrelevant virtual resolution, or more generally induces an aut (space of virtual resolutions)?",
   "evidence": "* Section 4 asks for Morse theory and a moduli theory for virtual resolutions; Problem 4.6 explicitly asks for a “space/stacks” of virtual resolutions. * Section 6 concerns modules over a Cox ring. * Thus “induces an aut (space of virtual resolutions)” most plausibly abbreviates “induces an automorphism of the space of virtual resolutions”. This is a reconstruction, not verified source text. * “A spherical object supported on irrelevant virtual resolution” has two possible meanings: (i) the center itself has Cox homology supported on the irrelevant locus, or (ii) a nonzero spherical object on the toric variety is represented by a virtual resolution that is allowed irrelevant higher homology. The distinction is decisive. Under (i) the center is zero after sheafification; under (ii) it can define a genuine twist, but only in the quotient by irrelevant homology.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0022": {
   "statement_status": "exact",
   "original_statement": "[Wang, Borisov-Wang]: Found a condition on underlying fan for a toric stack to admit infinitely many acyclic line bundles (meaning $\\mathcal{L}$ with $H^*(\\mathcal{L})=0$ for all $*$, e.g. $\\mathcal{O}(-1)$ on $\\mathbb{P}^n$) (e.g. in $2D$ if fan has collinear rays).\n\n\\begin{enumerate}\n\\item Understand symplectic mirror of these criteria.\n\\item Understand consequences of generalizations of these critera (e.g. if there exists a toric subvariety whose fans is contained in a linear subspace) or if generalized.\n\\end{enumerate}",
   "clean_statement": "[Wang, Borisov-Wang]: Found a condition on underlying fan for a toric stack to admit infinitely many acyclic line bundles (meaning $\\mathcal{L}$ with $H^*(\\mathcal{L})=0$ for all $*$, e.g. $\\mathcal{O}(-1)$ on $\\mathbb{P}^n$) (e.g. in $2D$ if fan has collinear rays).\n\n\\begin{enumerate}\n\\item Understand symplectic mirror of these criteria.\n\\item Understand consequences of generalizations of these critera (e.g. if there exists a toric subvariety whose fans is contained in a linear subspace) or if generalized.\n\\end{enumerate}",
   "public_statement": "[Wang, Borisov-Wang]: Found a condition on underlying fan for a toric stack to admit infinitely many acyclic line bundles (meaning $\\mathcal{L}$ with $H^*(\\mathcal{L})=0$ for all $*$, e.g. $\\mathcal{O}(-1)$ on $\\mathbb{P}^n$) (e.g. in $2D$ if fan has collinear rays).\n\n\\begin{enumerate}\n\\item Understand symplectic mirror of these criteria.\n\\item Understand consequences of generalizations of these critera (e.g. if there exists a toric subvariety whose fans is contained in a linear subspace) or if generalized.\n\\end{enumerate}",
   "evidence": "The canonical record is AIM Problem List 9.1 in the section “Line bundles over toric stacks” of *Syzygies and mirror symmetry*. The live AIM page was checked on 2026-07-22 and agrees with the record in `input.json`. Its mathematical content is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0023": {
   "statement_status": "exact",
   "original_statement": "The motivic filtration on the algebraic $K$-theory of the sphere spectrum\n\nDue to work of Rognes, Blumberg--Mandell, and others, we understand the algebraic K-theory of the sphere spectrum fairly well. Recent work of Hahn--Raksit--Wilson \\cite{arXiv:2206.11208} gives a motivic filtration on TC of $\\mathbb{E}_{\\infty}$-rings which is closely related to the motivic filtration on algebraic K-theory in motivic homotopy theory.\n\nCan one understand the motivic filtration (or motivic spectral sequence) on the $\\mathrm{TC}$ of the sphere spectrum? What about for the motivic filtration on related invariants such as $\\mathrm{THH}$, $\\mathrm{TP}$?",
   "clean_statement": "The motivic filtration on the algebraic $K$-theory of the sphere spectrum\n\nDue to work of Rognes, Blumberg--Mandell, and others, we understand the algebraic K-theory of the sphere spectrum fairly well. Recent work of Hahn--Raksit--Wilson \\cite{arXiv:2206.11208} gives a motivic filtration on TC of $\\mathbb{E}_{\\infty}$-rings which is closely related to the motivic filtration on algebraic K-theory in motivic homotopy theory.\n\nCan one understand the motivic filtration (or motivic spectral sequence) on the $\\mathrm{TC}$ of the sphere spectrum? What about for the motivic filtration on related invariants such as $\\mathrm{THH}$, $\\mathrm{TP}$?",
   "public_statement": "The motivic filtration on the algebraic $K$-theory of the sphere spectrum\n\nDue to work of Rognes, Blumberg--Mandell, and others, we understand the algebraic K-theory of the sphere spectrum fairly well. Recent work of Hahn--Raksit--Wilson \\cite{arXiv:2206.11208} gives a motivic filtration on TC of $\\mathbb{E}_{\\infty}$-rings which is closely related to the motivic filtration on algebraic K-theory in motivic homotopy theory.\n\nCan one understand the motivic filtration (or motivic spectral sequence) on the $\\mathrm{TC}$ of the sphere spectrum? What about for the motivic filtration on related invariants such as $\\mathrm{THH}$, $\\mathrm{TP}$?",
   "evidence": "This is problem 2.1 in the AIM workshop list *Equivariant techniques in stable homotopy theory*, section *Computations in algebraic K-theory*. The repository record and the live AIM page were compared on 2026-07-22. The mathematical question is uncorrupted:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0024": {
   "statement_status": "exact",
   "original_statement": "The $K$-theory of Johnson--Wilson theory\n\nThe algebraic $K$-theory of Johnson--Wilson theory $E(n)$ is closely related to the topological cyclic homology of the connective cover of the two periodic Johnson--Wilson theory $E'(n)$.\n\nCompute something about $\\mathrm{TC}$ (and related invariants such as $\\mathrm{TP}, \\mathrm{THH}$) of $\\tau_{\\geq0}E'(n)$, where $E'(n)$ is the $2$-periodic Johnson--Wilson theory.",
   "clean_statement": "The $K$-theory of Johnson--Wilson theory\n\nThe algebraic $K$-theory of Johnson--Wilson theory $E(n)$ is closely related to the topological cyclic homology of the connective cover of the two periodic Johnson--Wilson theory $E'(n)$.\n\nCompute something about $\\mathrm{TC}$ (and related invariants such as $\\mathrm{TP}, \\mathrm{THH}$) of $\\tau_{\\geq0}E'(n)$, where $E'(n)$ is the $2$-periodic Johnson--Wilson theory.",
   "public_statement": "The $K$-theory of Johnson--Wilson theory\n\nThe algebraic $K$-theory of Johnson--Wilson theory $E(n)$ is closely related to the topological cyclic homology of the connective cover of the two periodic Johnson--Wilson theory $E'(n)$.\n\nCompute something about $\\mathrm{TC}$ (and related invariants such as $\\mathrm{TP}, \\mathrm{THH}$) of $\\tau_{\\geq0}E'(n)$, where $E'(n)$ is the $2$-periodic Johnson--Wilson theory.",
   "evidence": "The canonical record is Problem 2.2 in the AIM list *Equivariant techniques in stable homotopy theory*, in the section \"Computations in algebraic K-theory.\" Its exact mathematical request is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 23,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_GEOMETRY-0025": {
   "statement_status": "exact",
   "original_statement": "Detecting Greek letter families in algebraic $K$-theory\n\nProve Conjecture 1.1 of \\cite{arXiv:1810.10088}. This is a version of redshift for the Greek letter family in the homotopy groups of spheres: it asks that the $K$-theory of an $\\mathbb{E}_{\\infty}$-ring detecting the $\\alpha^{(n)}$-family detects the $\\alpha^{(n+1)}$-family.",
   "clean_statement": "Detecting Greek letter families in algebraic $K$-theory\n\nProve Conjecture 1.1 of \\cite{arXiv:1810.10088}. This is a version of redshift for the Greek letter family in the homotopy groups of spheres: it asks that the $K$-theory of an $\\mathbb{E}_{\\infty}$-ring detecting the $\\alpha^{(n)}$-family detects the $\\alpha^{(n+1)}$-family.",
   "public_statement": "Detecting Greek letter families in algebraic $K$-theory\n\nProve Conjecture 1.1 of \\cite{arXiv:1810.10088}. This is a version of redshift for the Greek letter family in the homotopy groups of spheres: it asks that the $K$-theory of an $\\mathbb{E}_{\\infty}$-ring detecting the $\\alpha^{(n)}$-family detects the $\\alpha^{(n+1)}$-family.",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0026": {
   "statement_status": "exact",
   "original_statement": "Atiyah--Segal completion thoerem for iterated $K$-theory\n\nCan one prove an Atiyah--Segal completion theorem for the algebraic $K$-theory of topological $K$-theory?",
   "clean_statement": "Atiyah--Segal completion thoerem for iterated $K$-theory\n\nCan one prove an Atiyah--Segal completion theorem for the algebraic $K$-theory of topological $K$-theory?",
   "public_statement": "Atiyah--Segal completion thoerem for iterated $K$-theory\n\nCan one prove an Atiyah--Segal completion theorem for the algebraic $K$-theory of topological $K$-theory?",
   "evidence": "The canonical AIM record (workshop *Equivariant techniques in stable homotopy theory*, section “Computations in algebraic K-theory,” Problem 2.4) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0027": {
   "statement_status": "exact",
   "original_statement": "Motivic filtered Poitou-Tate duality\n\nIs there a duality at the level of motivic filtered spectra $\\mathrm{fil}_{\\text{mot}}\\mathrm{TC}(\\mathcal{O}_K)$ that recovers Poitou--Tate duality at the level of associated graded?",
   "clean_statement": "Motivic filtered Poitou-Tate duality\n\nIs there a duality at the level of motivic filtered spectra $\\mathrm{fil}_{\\text{mot}}\\mathrm{TC}(\\mathcal{O}_K)$ that recovers Poitou--Tate duality at the level of associated graded?",
   "public_statement": "Motivic filtered Poitou-Tate duality\n\nIs there a duality at the level of motivic filtered spectra $\\mathrm{fil}_{\\text{mot}}\\mathrm{TC}(\\mathcal{O}_K)$ that recovers Poitou--Tate duality at the level of associated graded?",
   "evidence": "The canonical record is Problem 2.5 in the AIM workshop list *Equivariant techniques in stable homotopy theory*, section “Computations in algebraic K-theory.” Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0028": {
   "statement_status": "exact",
   "original_statement": "Motivic filtration on $\\mathrm{TR}$\n\n\\cite{arXiv:2206.11208} gives a motivic filtration on $\\mathrm{THH}, \\mathrm{TC}, \\mathrm{TC^{-}}, \\mathrm{TP}$ of ring spectra generalizing previously defined motivic filtrations such as in \\cite{arXiv:1802.03261}. In remark 1.13 of loc. cit. it is mentioned that their techniques can be made to additionally define a motivic filtration on $\\mathrm{TR}$ of discrete rings.\n\nCan one define a well behaved motivic filtration on the $\\mathrm{TR}$ of ring spectra? Can one compute it in good cases?",
   "clean_statement": "Motivic filtration on $\\mathrm{TR}$\n\n\\cite{arXiv:2206.11208} gives a motivic filtration on $\\mathrm{THH}, \\mathrm{TC}, \\mathrm{TC^{-}}, \\mathrm{TP}$ of ring spectra generalizing previously defined motivic filtrations such as in \\cite{arXiv:1802.03261}. In remark 1.13 of loc. cit. it is mentioned that their techniques can be made to additionally define a motivic filtration on $\\mathrm{TR}$ of discrete rings.\n\nCan one define a well behaved motivic filtration on the $\\mathrm{TR}$ of ring spectra? Can one compute it in good cases?",
   "public_statement": "Motivic filtration on $\\mathrm{TR}$\n\n\\cite{arXiv:2206.11208} gives a motivic filtration on $\\mathrm{THH}, \\mathrm{TC}, \\mathrm{TC^{-}}, \\mathrm{TP}$ of ring spectra generalizing previously defined motivic filtrations such as in \\cite{arXiv:1802.03261}. In remark 1.13 of loc. cit. it is mentioned that their techniques can be made to additionally define a motivic filtration on $\\mathrm{TR}$ of discrete rings.\n\nCan one define a well behaved motivic filtration on the $\\mathrm{TR}$ of ring spectra? Can one compute it in good cases?",
   "evidence": "The canonical record is Problem 2.7 in the AIM workshop list *Equivariant techniques in stable homotopy theory*, section “Computations in algebraic K-theory.” It is record 27 (zero-based) of `aim-algebraic-geometry-notes.json`. Its statement is intact:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0029": {
   "statement_status": "exact",
   "original_statement": "Calculation of $\\mathrm{TP}(\\mathbb{Z})$\n\nBokstedt and Madsen computed $\\mathrm{TC}(\\mathbb{Z})$, without calculating $\\mathrm{TP}(\\mathbb{Z})$. However, Hesselholt and Madsen calculated (in \\cite{MR1317575}) $(L_{K(1)} \\mathbb{S})^{tS^1}$. Their results suggest the following:\n\nFix a prime $p$. Let $j = \\tau_{\\geq 0} L_{K(1)} \\mathbb{S}$. Then, $\\mathrm{TP}(\\mathbb{Z}_p) \\simeq j^{tS^1}$, for the trivial $S^1$-action on $j$.",
   "clean_statement": "Calculation of $\\mathrm{TP}(\\mathbb{Z})$\n\nBokstedt and Madsen computed $\\mathrm{TC}(\\mathbb{Z})$, without calculating $\\mathrm{TP}(\\mathbb{Z})$. However, Hesselholt and Madsen calculated (in \\cite{MR1317575}) $(L_{K(1)} \\mathbb{S})^{tS^1}$. Their results suggest the following:\n\nFix a prime $p$. Let $j = \\tau_{\\geq 0} L_{K(1)} \\mathbb{S}$. Then, $\\mathrm{TP}(\\mathbb{Z}_p) \\simeq j^{tS^1}$, for the trivial $S^1$-action on $j$.",
   "public_statement": "Calculation of $\\mathrm{TP}(\\mathbb{Z})$\n\nBokstedt and Madsen computed $\\mathrm{TC}(\\mathbb{Z})$, without calculating $\\mathrm{TP}(\\mathbb{Z})$. However, Hesselholt and Madsen calculated (in \\cite{MR1317575}) $(L_{K(1)} \\mathbb{S})^{tS^1}$. Their results suggest the following:\n\nFix a prime $p$. Let $j = \\tau_{\\geq 0} L_{K(1)} \\mathbb{S}$. Then, $\\mathrm{TP}(\\mathbb{Z}_p) \\simeq j^{tS^1}$, for the trivial $S^1$-action on $j$.",
   "evidence": "The record adds that this would follow, for example, from \\(\\operatorname{TR}(\\mathbb Z_p)\\simeq j_p\\). There is no visible corruption in the record. The citation `MR1317575` is Hesselholt--Madsen, *The \\(S^1\\)-Tate spectrum for \\(J\\)*.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0030": {
   "statement_status": "exact",
   "original_statement": "$L$-theory of integers\n\nThe symmetric and normal $L$-theory of the integers was computed in \\cite{arXiv:2004.06889}, and the Tate cohomology of $Q_8$ was computed by Atiyah in \\cite{MR0148722}. The normal $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $Q_8$; similarly, the symmetric $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $\\mathrm{Pin}(2)$.\n\nEquip $\\mathrm{Pin}(2)$ (resp. $Q_8$) with the canonical $\\mathbb{Z}/2$-action (resp. the action given by conjugation by $i$). Then, the $E_\\infty$-map $L^s(\\mathbb{Z})_{(2)} \\to L^n(\\mathbb{Z})$ can be identified by taking strict $\\mathbb{Z}/2$-fixed points of the map $\\mathbb{Z}_{(2)}^{t\\mathrm{Pin}(2)} \\to \\mathbb{Z}^{tQ_8}$. (This map is induced by the inclusion $Q_8\\subseteq \\mathrm{Pin}(2)$.)",
   "clean_statement": "$L$-theory of integers\n\nThe symmetric and normal $L$-theory of the integers was computed in \\cite{arXiv:2004.06889}, and the Tate cohomology of $Q_8$ was computed by Atiyah in \\cite{MR0148722}. The normal $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $Q_8$; similarly, the symmetric $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $\\mathrm{Pin}(2)$.\n\nEquip $\\mathrm{Pin}(2)$ (resp. $Q_8$) with the canonical $\\mathbb{Z}/2$-action (resp. the action given by conjugation by $i$). Then, the $E_\\infty$-map $L^s(\\mathbb{Z})_{(2)} \\to L^n(\\mathbb{Z})$ can be identified by taking strict $\\mathbb{Z}/2$-fixed points of the map $\\mathbb{Z}_{(2)}^{t\\mathrm{Pin}(2)} \\to \\mathbb{Z}^{tQ_8}$. (This map is induced by the inclusion $Q_8\\subseteq \\mathrm{Pin}(2)$.)",
   "public_statement": "$L$-theory of integers\n\nThe symmetric and normal $L$-theory of the integers was computed in \\cite{arXiv:2004.06889}, and the Tate cohomology of $Q_8$ was computed by Atiyah in \\cite{MR0148722}. The normal $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $Q_8$; similarly, the symmetric $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $\\mathrm{Pin}(2)$.\n\nEquip $\\mathrm{Pin}(2)$ (resp. $Q_8$) with the canonical $\\mathbb{Z}/2$-action (resp. the action given by conjugation by $i$). Then, the $E_\\infty$-map $L^s(\\mathbb{Z})_{(2)} \\to L^n(\\mathbb{Z})$ can be identified by taking strict $\\mathbb{Z}/2$-fixed points of the map $\\mathbb{Z}_{(2)}^{t\\mathrm{Pin}(2)} \\to \\mathbb{Z}^{tQ_8}$. (This map is induced by the inclusion $Q_8\\subseteq \\mathrm{Pin}(2)$.)",
   "evidence": "This is Conjecture 2.9, “\\(L\\)-theory of integers,” from the AIM workshop *Equivariant techniques in stable homotopy theory* (October 2022). The canonical record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0031": {
   "statement_status": "exact",
   "original_statement": "Devissage for equivariant/hermitian $K$-theory\n\nWe understand devissage for the $K$-theory of stable categories fairly well now due to work of Quillen, Barwick, and Burklund--Levy (see \\cite{arXiv:2112.14723}). See section 2.2 of \\cite{arXiv:2009.07225} for what is known for hermitian $K$-theory.\n\nWhat is the optimal devissage result for equivariant/hermitian $K$-theory?",
   "clean_statement": "Devissage for equivariant/hermitian $K$-theory\n\nWe understand devissage for the $K$-theory of stable categories fairly well now due to work of Quillen, Barwick, and Burklund--Levy (see \\cite{arXiv:2112.14723}). See section 2.2 of \\cite{arXiv:2009.07225} for what is known for hermitian $K$-theory.\n\nWhat is the optimal devissage result for equivariant/hermitian $K$-theory?",
   "public_statement": "Devissage for equivariant/hermitian $K$-theory\n\nWe understand devissage for the $K$-theory of stable categories fairly well now due to work of Quillen, Barwick, and Burklund--Levy (see \\cite{arXiv:2112.14723}). See section 2.2 of \\cite{arXiv:2009.07225} for what is known for hermitian $K$-theory.\n\nWhat is the optimal devissage result for equivariant/hermitian $K$-theory?",
   "evidence": "The source record has no remarks or supplied bibliography beyond those two identifiers. Both were checked: they are Burklund--Levy, *On the \\(K\\)-theory of regular coconnective rings*, and Calmès--Dotto--Harpaz--Hebestreit--Land--Moi--Nardin--Nikolaus--Steimle, *Hermitian \\(K\\)-theory for stable \\(\\infty\\)-categories III: Grothendieck--Witt groups of rings*. The latter was revised as arXiv v4 on 27 April 2026 and accepted by the *Annals of Mathematics*. There is no apparent corruption in the statement.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0032": {
   "statement_status": "exact",
   "original_statement": "a inverted Artin--Tate $\\mathbb{R}$-motives\n\nThe category of Artin--Tate $\\mathbb{R}$-motives has a topological model as a $1$-parameter deformation of $C_2$-equivariant homotopy theory \\cite{arXiv:2010.10325}. Let $\\Sp_{\\mathbb{R}}^{AT}$ denote this category.\n\nWhat category do you get when you invert the class $a$ in $\\Sp_{\\mathbb{R}}^{AT}$?",
   "clean_statement": "a inverted Artin--Tate $\\mathbb{R}$-motives\n\nThe category of Artin--Tate $\\mathbb{R}$-motives has a topological model as a $1$-parameter deformation of $C_2$-equivariant homotopy theory \\cite{arXiv:2010.10325}. Let $\\Sp_{\\mathbb{R}}^{AT}$ denote this category.\n\nWhat category do you get when you invert the class $a$ in $\\Sp_{\\mathbb{R}}^{AT}$?",
   "public_statement": "a inverted Artin--Tate $\\mathbb{R}$-motives\n\nThe category of Artin--Tate $\\mathbb{R}$-motives has a topological model as a $1$-parameter deformation of $C_2$-equivariant homotopy theory \\cite{arXiv:2010.10325}. Let $\\Sp_{\\mathbb{R}}^{AT}$ denote this category.\n\nWhat category do you get when you invert the class $a$ in $\\Sp_{\\mathbb{R}}^{AT}$?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0033": {
   "statement_status": "exact",
   "original_statement": "Mahowald invariants increase chromatic height\n\nGiven a $v_n$-periodic family in the stable homotopy groups of spheres, does its Mahowald invariant give rise to an (eventually) $v_{n+1}$-periodic family?",
   "clean_statement": "Mahowald invariants increase chromatic height\n\nGiven a $v_n$-periodic family in the stable homotopy groups of spheres, does its Mahowald invariant give rise to an (eventually) $v_{n+1}$-periodic family?",
   "public_statement": "Mahowald invariants increase chromatic height\n\nGiven a $v_n$-periodic family in the stable homotopy groups of spheres, does its Mahowald invariant give rise to an (eventually) $v_{n+1}$-periodic family?",
   "evidence": "The exact canonical record is Problem 3.3 from the AIM workshop list “Equivariant techniques in stable homotopy theory,” section “Motivic, equivariant, and synthetic spectra”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0034": {
   "statement_status": "exact",
   "original_statement": "Equivariant and synthetic\n\nCan one profitably mix synthetic spectra with equivariant homotopy theory?",
   "clean_statement": "Equivariant and synthetic\n\nCan one profitably mix synthetic spectra with equivariant homotopy theory?",
   "public_statement": "Equivariant and synthetic\n\nCan one profitably mix synthetic spectra with equivariant homotopy theory?",
   "evidence": "The exact canonical record is AIM Problem List 3.4 from the workshop *Equivariant techniques in stable homotopy theory*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0035": {
   "statement_status": "exact",
   "original_statement": "Odd primary analog of Wood cofibre sequences\n\nThe Wood cofibre sequence is the cofibre sequence $\\Sigma \\mathrm{KO} \\xrightarrow{\\eta} \\mathrm{KO} \\to \\mathrm{KU}$ (and similarly for the connective versions). There is also a version relating $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle^{C_2}$ to $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle$.\n\nWhat are odd primary analogs of the Wood cofibre sequence? For example, one might expect a $p$-stage filtration on $E_{p-1}$ with associated graded $E_{p-1}^{hC_p}$. Is there an analog for $E_{(p-1)p^{n-1}m}$?",
   "clean_statement": "Odd primary analog of Wood cofibre sequences\n\nThe Wood cofibre sequence is the cofibre sequence $\\Sigma \\mathrm{KO} \\xrightarrow{\\eta} \\mathrm{KO} \\to \\mathrm{KU}$ (and similarly for the connective versions). There is also a version relating $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle^{C_2}$ to $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle$.\n\nWhat are odd primary analogs of the Wood cofibre sequence? For example, one might expect a $p$-stage filtration on $E_{p-1}$ with associated graded $E_{p-1}^{hC_p}$. Is there an analog for $E_{(p-1)p^{n-1}m}$?",
   "public_statement": "Odd primary analog of Wood cofibre sequences\n\nThe Wood cofibre sequence is the cofibre sequence $\\Sigma \\mathrm{KO} \\xrightarrow{\\eta} \\mathrm{KO} \\to \\mathrm{KU}$ (and similarly for the connective versions). There is also a version relating $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle^{C_2}$ to $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle$.\n\nWhat are odd primary analogs of the Wood cofibre sequence? For example, one might expect a $p$-stage filtration on $E_{p-1}$ with associated graded $E_{p-1}^{hC_p}$. Is there an analog for $E_{(p-1)p^{n-1}m}$?",
   "evidence": "The local JSON record agrees with the AIM statement. The AIM web page timed out during this run, but there is no visible corruption or missing notation in the repository copy. I interpret “associated graded \\(E_{p-1}^{hC_p}\\)” in the standard stable sense: the graded pieces are suspensions of \\(E_{p-1}^{hC_p}\\). This convention is necessary even classically, since the two graded pieces of \\(KO\\wedge C(\\eta)\\simeq KU\\) are \\(KO\\) and \\(\\Sigma^2KO\\).",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0036": {
   "statement_status": "exact",
   "original_statement": "Motivic modular forms (mmf)\n\nIs there a connective version of motivic modular forms, which should be named $\\mathrm{mmf}$, and is a lift of $\\mathrm{tmf}$?",
   "clean_statement": "Motivic modular forms (mmf)\n\nIs there a connective version of motivic modular forms, which should be named $\\mathrm{mmf}$, and is a lift of $\\mathrm{tmf}$?",
   "public_statement": "Motivic modular forms (mmf)\n\nIs there a connective version of motivic modular forms, which should be named $\\mathrm{mmf}$, and is a lift of $\\mathrm{tmf}$?",
   "evidence": "The exact canonical record is Problem 3.5 in the section “Motivic, equivariant, and synthetic spectra” from the October 24–28, 2022 AIM workshop *Equivariant techniques in stable homotopy theory*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0037": {
   "statement_status": "exact",
   "original_statement": "The associated graded of the localized slice spectral sequence tower\n\nThe localized slice spectral sequence gives a way to break up the slice spectral sequence into pieces that see phenomena only on certain geometric fixed points. For the group $C_{2^n}$, there is a localized slice spectral sequence for each subgroup.\n\nFor the group $C_{2^n}$ one understand more easily the fibres between the localized slice spectral sequence for two adjacent subgroups $H$ and $H'$?",
   "clean_statement": "The associated graded of the localized slice spectral sequence tower\n\nThe localized slice spectral sequence gives a way to break up the slice spectral sequence into pieces that see phenomena only on certain geometric fixed points. For the group $C_{2^n}$, there is a localized slice spectral sequence for each subgroup.\n\nFor the group $C_{2^n}$ one understand more easily the fibres between the localized slice spectral sequence for two adjacent subgroups $H$ and $H'$?",
   "public_statement": "The associated graded of the localized slice spectral sequence tower\n\nThe localized slice spectral sequence gives a way to break up the slice spectral sequence into pieces that see phenomena only on certain geometric fixed points. For the group $C_{2^n}$, there is a localized slice spectral sequence for each subgroup.\n\nFor the group $C_{2^n}$ one understand more easily the fibres between the localized slice spectral sequence for two adjacent subgroups $H$ and $H'$?",
   "evidence": "The canonical record is Problem 4.1, “The associated graded of the localized slice spectral sequence tower,” in the AIM list *Equivariant techniques in stable homotopy theory*, section “Norms and the slice filtration.” The source record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0038": {
   "statement_status": "exact",
   "original_statement": "The image of $J$ in equivariant truncated Brown--Peterson spectra\n\nWhat is the image of the $J$ homomorphism in the fixed points of $\\mathrm{BP}^{((G))}\\langle m\\rangle$?",
   "clean_statement": "The image of $J$ in equivariant truncated Brown--Peterson spectra\n\nWhat is the image of the $J$ homomorphism in the fixed points of $\\mathrm{BP}^{((G))}\\langle m\\rangle$?",
   "public_statement": "The image of $J$ in equivariant truncated Brown--Peterson spectra\n\nWhat is the image of the $J$ homomorphism in the fixed points of $\\mathrm{BP}^{((G))}\\langle m\\rangle$?",
   "evidence": "The canonical record is Problem 4.2 in the AIM workshop list *Equivariant techniques in stable homotopy theory*, section “Norms and the slice filtration”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0039": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "The Hurewicz image of equivariant truncated Brown-Peterson spectra\n\nWhat is the Hurewicz image of the truncated Brown--Peterson spectrum $\\mathrm{BP}^{((G))}\\langle m \\rangle$?",
   "clean_statement": null,
   "public_statement": "The Hurewicz image of equivariant truncated Brown-Peterson spectra\n\nWhat is the Hurewicz image of the truncated Brown--Peterson spectrum $\\mathrm{BP}^{((G))}\\langle m \\rangle$?",
   "evidence": "Other plausible readings are genuinely different problems:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0040": {
   "statement_status": "exact",
   "original_statement": "Understanding $n$-slices\n\n\\cite{arXiv:1711.03472} gives a description of $n$-slices, the objects that appear in the associated graded of the slice filtration.\n\nIs there a way to get a good computational grasp of $n$-slices? For example understanding their Postnikov towers, homotopy groups etc.",
   "clean_statement": "Understanding $n$-slices\n\n\\cite{arXiv:1711.03472} gives a description of $n$-slices, the objects that appear in the associated graded of the slice filtration.\n\nIs there a way to get a good computational grasp of $n$-slices? For example understanding their Postnikov towers, homotopy groups etc.",
   "public_statement": "Understanding $n$-slices\n\n\\cite{arXiv:1711.03472} gives a description of $n$-slices, the objects that appear in the associated graded of the slice filtration.\n\nIs there a way to get a good computational grasp of $n$-slices? For example understanding their Postnikov towers, homotopy groups etc.",
   "evidence": "This agrees with the JSON record. There is no corruption to repair. The question is intentionally broad rather than a yes/no conjecture.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0041": {
   "statement_status": "exact",
   "original_statement": "Heights of slice differentials\n\nThe differentials in the slice spectral sequence for $\\mathrm{BP}^{((G))}\\langle n \\rangle$ look like they are stratified by height.\n\nIs it possible to make sense of the height of a differential in the slice spectral sequence of $\\mathrm{BP}^{((G))}\\langle n \\rangle$ so that it has a well defined height?",
   "clean_statement": "Heights of slice differentials\n\nThe differentials in the slice spectral sequence for $\\mathrm{BP}^{((G))}\\langle n \\rangle$ look like they are stratified by height.\n\nIs it possible to make sense of the height of a differential in the slice spectral sequence of $\\mathrm{BP}^{((G))}\\langle n \\rangle$ so that it has a well defined height?",
   "public_statement": "Heights of slice differentials\n\nThe differentials in the slice spectral sequence for $\\mathrm{BP}^{((G))}\\langle n \\rangle$ look like they are stratified by height.\n\nIs it possible to make sense of the height of a differential in the slice spectral sequence of $\\mathrm{BP}^{((G))}\\langle n \\rangle$ so that it has a well defined height?",
   "evidence": "The exact AIM record (workshop *Equivariant techniques in stable homotopy theory*, section “Norms and the slice filtration,” Problem 4.5) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0042": {
   "statement_status": "exact",
   "original_statement": "The $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for $\\mathrm{BP}^{((G))}\\langle n\\rangle$\n\n$\\mathrm{BP}^{((G))}\\langle n-1\\rangle$ is the quotient of $\\mathrm{BP}^{((G))}\\langle n\\rangle$ by $G\\cdot \\overline{v}_n$.\n\nCan one understand the $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for this quotient map?",
   "clean_statement": "The $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for $\\mathrm{BP}^{((G))}\\langle n\\rangle$\n\n$\\mathrm{BP}^{((G))}\\langle n-1\\rangle$ is the quotient of $\\mathrm{BP}^{((G))}\\langle n\\rangle$ by $G\\cdot \\overline{v}_n$.\n\nCan one understand the $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for this quotient map?",
   "public_statement": "The $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for $\\mathrm{BP}^{((G))}\\langle n\\rangle$\n\n$\\mathrm{BP}^{((G))}\\langle n-1\\rangle$ is the quotient of $\\mathrm{BP}^{((G))}\\langle n\\rangle$ by $G\\cdot \\overline{v}_n$.\n\nCan one understand the $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for this quotient map?",
   "evidence": "The canonical AIM record (source file `aim-algebraic-geometry-notes.json`, zero-based index 41, problem 4.6 in the section “Norms and the slice filtration”) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0043": {
   "statement_status": "exact",
   "original_statement": "Connecting slice differentials via power operations\n\nWe would like a systematic way of understanding slice differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$.\n\nAre there power operations connecting differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$",
   "clean_statement": "Connecting slice differentials via power operations\n\nWe would like a systematic way of understanding slice differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$.\n\nAre there power operations connecting differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$",
   "public_statement": "Connecting slice differentials via power operations\n\nWe would like a systematic way of understanding slice differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$.\n\nAre there power operations connecting differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$",
   "evidence": "The assigned record is problem 4.7 in the section “Norms and the slice filtration” of the AIM workshop list *Equivariant techniques in stable homotopy theory*. The exact stored text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0044": {
   "statement_status": "exact",
   "original_statement": "Relations in $\\mathrm{BP}^{((G))}$\n\nIs $8\\sigma = 0$ in the fixed points of $\\mathrm{BP}^{((G))}$?",
   "clean_statement": "Relations in $\\mathrm{BP}^{((G))}$\n\nIs $8\\sigma = 0$ in the fixed points of $\\mathrm{BP}^{((G))}$?",
   "public_statement": "Relations in $\\mathrm{BP}^{((G))}$\n\nIs $8\\sigma = 0$ in the fixed points of $\\mathrm{BP}^{((G))}$?",
   "evidence": "The exact corpus record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0045": {
   "statement_status": "exact",
   "original_statement": "Problem 1\n\nFind more integers $d$ and $n$ such that a general hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.\nSimilarly,\nfind more integers $d$ and $n$ such that every smooth hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.",
   "clean_statement": "Problem 1\n\nFind more integers $d$ and $n$ such that a general hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.\nSimilarly,\nfind more integers $d$ and $n$ such that every smooth hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.",
   "public_statement": "Problem 1\n\nFind more integers $d$ and $n$ such that a general hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.\nSimilarly,\nfind more integers $d$ and $n$ such that every smooth hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.",
   "evidence": "The canonical source is aim-algebraic-geometry-notes.json, zero-based index 44, problem 1.02 from the AIM workshop *Rationality problems in algebraic geometry*. The source record states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0046": {
   "statement_status": "exact",
   "original_statement": "Given $d$, what is the smallest $n$ such that there is some smooth unirational hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$?",
   "clean_statement": "Given $d$, what is the smallest $n$ such that there is some smooth unirational hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$?",
   "public_statement": "Given $d$, what is the smallest $n$ such that there is some smooth unirational hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$?",
   "evidence": "The canonical record is problem 1.04 from the AIM workshop *Rationality problems in algebraic geometry*, posed by Stefan Schreieder. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0047": {
   "statement_status": "exact",
   "original_statement": "Does unirationality specialize in smooth projective families?",
   "clean_statement": "Does unirationality specialize in smooth projective families?",
   "public_statement": "Does unirationality specialize in smooth projective families?",
   "evidence": "The exact AIM record (problem 1.06 in the 2019 workshop *Rationality problems in algebraic geometry*) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0048": {
   "statement_status": "exact",
   "original_statement": "Does there exist an irrational unirational variety with unirational parametrizations of coprime degrees?",
   "clean_statement": "Does there exist an irrational unirational variety with unirational parametrizations of coprime degrees?",
   "public_statement": "Does there exist an irrational unirational variety with unirational parametrizations of coprime degrees?",
   "evidence": "The canonical record is problem 1.08 from the AIM workshop *Rationality problems in algebraic geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0049": {
   "statement_status": "exact",
   "original_statement": "Does there exist a rationally connected smooth complex projective variety, not stably rational, that has an integral decomposition of the diagonal?",
   "clean_statement": "Does there exist a rationally connected smooth complex projective variety, not stably rational, that has an integral decomposition of the diagonal?",
   "public_statement": "Does there exist a rationally connected smooth complex projective variety, not stably rational, that has an integral decomposition of the diagonal?",
   "evidence": "The canonical AIM record is problem 1.1 from the workshop *Rationality problems in algebraic geometry*. Its question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0050": {
   "statement_status": "exact",
   "original_statement": "Does there exist a unirational variety $X$ of dimension at least 4 such that\n\\begin{align*}\n\\frac{H^4(X, \\mathbb Z) \\cap H^{2,2}(X)}{(H^4(X, \\mathbb Z) \\cap H^{2,2}(X))_{tors}}\n\\end{align*}\nis non-algebraic? Here, the subscript tors denotes torsion classes.\nDoes there exist such a variety of dimension exactly 4?",
   "clean_statement": "Does there exist a unirational variety $X$ of dimension at least 4 such that\n\\begin{align*}\n\\frac{H^4(X, \\mathbb Z) \\cap H^{2,2}(X)}{(H^4(X, \\mathbb Z) \\cap H^{2,2}(X))_{tors}}\n\\end{align*}\nis non-algebraic? Here, the subscript tors denotes torsion classes.\nDoes there exist such a variety of dimension exactly 4?",
   "public_statement": "Does there exist a unirational variety $X$ of dimension at least 4 such that\n\\begin{align*}\n\\frac{H^4(X, \\mathbb Z) \\cap H^{2,2}(X)}{(H^4(X, \\mathbb Z) \\cap H^{2,2}(X))_{tors}}\n\\end{align*}\nis non-algebraic? Here, the subscript tors denotes torsion classes.\nDoes there exist such a variety of dimension exactly 4?",
   "evidence": "The canonical AIM record, Problem 1.12 from the workshop *Rationality problems in algebraic geometry*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0051": {
   "statement_status": "exact",
   "original_statement": "Is there a smooth projective $3$-fold $X$ over a finite field for which $H^3_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z(2))$ is not zero?",
   "clean_statement": "Is there a smooth projective $3$-fold $X$ over a finite field for which $H^3_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z(2))$ is not zero?",
   "public_statement": "Is there a smooth projective $3$-fold $X$ over a finite field for which $H^3_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z(2))$ is not zero?",
   "evidence": "The canonical record is Problem 1.14 from the AIM workshop *Rationality problems in algebraic geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0052": {
   "statement_status": "exact",
   "original_statement": "Is there a smooth rational cubic hypersurface of odd dimension?",
   "clean_statement": "Is there a smooth rational cubic hypersurface of odd dimension?",
   "public_statement": "Is there a smooth rational cubic hypersurface of odd dimension?",
   "evidence": "The canonical AIM record is Problem 1.16 from the workshop “Rationality problems in algebraic geometry”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0053": {
   "statement_status": "exact",
   "original_statement": "Is there a smooth rational quartic hypersurface of some dimension?",
   "clean_statement": "Is there a smooth rational quartic hypersurface of some dimension?",
   "public_statement": "Is there a smooth rational quartic hypersurface of some dimension?",
   "evidence": "The canonical record is problem 1.18 from the AIM workshop “Rationality problems in algebraic geometry,” in the section “The problems.” Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0054": {
   "statement_status": "exact",
   "original_statement": "Can we use the degeneration method to obstruct stable rationality for $4$-folds fibered by $3$-folds over $\\mathbb P^1$?",
   "clean_statement": "Can we use the degeneration method to obstruct stable rationality for $4$-folds fibered by $3$-folds over $\\mathbb P^1$?",
   "public_statement": "Can we use the degeneration method to obstruct stable rationality for $4$-folds fibered by $3$-folds over $\\mathbb P^1$?",
   "evidence": "The canonical record is Problem 1.2 from the AIM workshop list “Rationality problems in algebraic geometry,” posed in the list by Asher Auel:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0055": {
   "statement_status": "exact",
   "original_statement": "If $X$ is a smooth rational variety of dimension $n$ and $p \\in X$ is a point, is there a Zariski open $U \\subset X$ with $p \\in U$ such that $U$ is isomorphic to a Zariski open subset of $\\mathbb P^n$?",
   "clean_statement": "If $X$ is a smooth rational variety of dimension $n$ and $p \\in X$ is a point, is there a Zariski open $U \\subset X$ with $p \\in U$ such that $U$ is isomorphic to a Zariski open subset of $\\mathbb P^n$?",
   "public_statement": "If $X$ is a smooth rational variety of dimension $n$ and $p \\in X$ is a point, is there a Zariski open $U \\subset X$ with $p \\in U$ such that $U$ is isomorphic to a Zariski open subset of $\\mathbb P^n$?",
   "evidence": "The canonical AIM record is Problem 1.22 from the workshop “Rationality problems in algebraic geometry”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0056": {
   "statement_status": "exact",
   "original_statement": "Let $X$ be a rationally connected $3$-fold which is a del Pezzo fibration over $\\mathbb P^1$.\nIs there a component $M$ of the space of sections of the map $X \\rightarrow \\mathbb P^1$\nsuch that the Stein factorization of\n$\\alpha: M \\rightarrow J(X)$\nis the MRC fibration of $M$?\n\nCan we find an example of such a rationally connected $3$-fold $X$ with does not admit a decomposition of the diagonal\nand with the above property for the components of spaces of rational curves on X?",
   "clean_statement": "Let $X$ be a rationally connected $3$-fold which is a del Pezzo fibration over $\\mathbb P^1$.\nIs there a component $M$ of the space of sections of the map $X \\rightarrow \\mathbb P^1$\nsuch that the Stein factorization of\n$\\alpha: M \\rightarrow J(X)$\nis the MRC fibration of $M$?\n\nCan we find an example of such a rationally connected $3$-fold $X$ with does not admit a decomposition of the diagonal\nand with the above property for the components of spaces of rational curves on X?",
   "public_statement": "Let $X$ be a rationally connected $3$-fold which is a del Pezzo fibration over $\\mathbb P^1$.\nIs there a component $M$ of the space of sections of the map $X \\rightarrow \\mathbb P^1$\nsuch that the Stein factorization of\n$\\alpha: M \\rightarrow J(X)$\nis the MRC fibration of $M$?\n\nCan we find an example of such a rationally connected $3$-fold $X$ with does not admit a decomposition of the diagonal\nand with the above property for the components of spaces of rational curves on X?",
   "evidence": "The canonical AIM record (problem 1.24 in the workshop *Rationality problems in algebraic geometry*) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0057": {
   "statement_status": "exact",
   "original_statement": "The following is question 2 on the website announcement for this conference.\n\nColliot-Thélène and Pirutka defined the notion of a $CH_0$-universally trivial resolution of singularities. As a preliminary question, is admitting a $CH_0$-universally trivial resolution\nan analytic local property for complex varieties? Assuming the answer is yes, we pose the following questions.\n\nIn low dimensions, say up to 4, can we give an analytic local classification of $CH_0$-universally trivial singularities?\n\nMore vaguely, is there a local classification of $CH_0$-universally trivial singularities in terms of MMP data?\n\nWe can ask the same questions for $L$-rational singularities (defined by Nicaise and Shinder), and for $B$-rational singularities (defined by Kontsevich and Tschinkel).",
   "clean_statement": "The following is question 2 on the website announcement for this conference.\n\nColliot-Thélène and Pirutka defined the notion of a $CH_0$-universally trivial resolution of singularities. As a preliminary question, is admitting a $CH_0$-universally trivial resolution\nan analytic local property for complex varieties? Assuming the answer is yes, we pose the following questions.\n\nIn low dimensions, say up to 4, can we give an analytic local classification of $CH_0$-universally trivial singularities?\n\nMore vaguely, is there a local classification of $CH_0$-universally trivial singularities in terms of MMP data?\n\nWe can ask the same questions for $L$-rational singularities (defined by Nicaise and Shinder), and for $B$-rational singularities (defined by Kontsevich and Tschinkel).",
   "public_statement": "The following is question 2 on the website announcement for this conference.\n\nColliot-Thélène and Pirutka defined the notion of a $CH_0$-universally trivial resolution of singularities. As a preliminary question, is admitting a $CH_0$-universally trivial resolution\nan analytic local property for complex varieties? Assuming the answer is yes, we pose the following questions.\n\nIn low dimensions, say up to 4, can we give an analytic local classification of $CH_0$-universally trivial singularities?\n\nMore vaguely, is there a local classification of $CH_0$-universally trivial singularities in terms of MMP data?\n\nWe can ask the same questions for $L$-rational singularities (defined by Nicaise and Shinder), and for $B$-rational singularities (defined by Kontsevich and Tschinkel).",
   "evidence": "The canonical record is AIM Problem List item 1.26, attributed on the live AIM page to Asher Auel. Its mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0058": {
   "statement_status": "exact",
   "original_statement": "Compare $CH_0$-universally trivial, $B$-rational, and $L$-rational singularities.",
   "clean_statement": "Compare $CH_0$-universally trivial, $B$-rational, and $L$-rational singularities.",
   "public_statement": "Compare $CH_0$-universally trivial, $B$-rational, and $L$-rational singularities.",
   "evidence": "The canonical record is AIM problem 1.28 from the workshop *Rationality problems in algebraic geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0059": {
   "statement_status": "exact",
   "original_statement": "Which birational transformations preserve the following property?\n\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$\nFor future reference, we call this property (*).",
   "clean_statement": "Which birational transformations preserve the following property?\n\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$\nFor future reference, we call this property (*).",
   "public_statement": "Which birational transformations preserve the following property?\n\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$\nFor future reference, we call this property (*).",
   "evidence": "The canonical record is problem 1.3 from the AIM workshop *Rationality problems in algebraic geometry*, stored as record 58 (zero-based) of `aim-algebraic-geometry-notes.json`. Its problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0060": {
   "statement_status": "exact",
   "original_statement": "Is there a rationally connected irrational variety $X$ with property (*)?\n\nRecall Property (*) from a previous question was defined as follows.\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$",
   "clean_statement": "Is there a rationally connected irrational variety $X$ with property (*)?\n\nRecall Property (*) from a previous question was defined as follows.\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$",
   "public_statement": "Is there a rationally connected irrational variety $X$ with property (*)?\n\nRecall Property (*) from a previous question was defined as follows.\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$",
   "evidence": "The canonical record is problem 1.32 from the 2019 AIM workshop *Rationality problems in algebraic geometry*. The preceding canonical record, problem 1.30, introduces property (*). Together they ask:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0061": {
   "statement_status": "exact",
   "original_statement": "There have been new rationality constructions of special cubic fourfolds $X$ due to Russo and Staglianò, for discriminant $26$ or $38$. Using these explicit constructions, write down an explicit K3 surface S and an equivalence $Ku(X) \\simeq D(S)$.",
   "clean_statement": "There have been new rationality constructions of special cubic fourfolds $X$ due to Russo and Staglianò, for discriminant $26$ or $38$. Using these explicit constructions, write down an explicit K3 surface S and an equivalence $Ku(X) \\simeq D(S)$.",
   "public_statement": "There have been new rationality constructions of special cubic fourfolds $X$ due to Russo and Staglianò, for discriminant $26$ or $38$. Using these explicit constructions, write down an explicit K3 surface S and an equivalence $Ku(X) \\simeq D(S)$.",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0062": {
   "statement_status": "exact",
   "original_statement": "For a very general cubic fourfold $X$, is there a surface $S$ such that the Kuznetsov component $Ku(X) \\subset D(X)$ has an embedding $Ku(X) \\hookrightarrow D(S)$?",
   "clean_statement": "For a very general cubic fourfold $X$, is there a surface $S$ such that the Kuznetsov component $Ku(X) \\subset D(X)$ has an embedding $Ku(X) \\hookrightarrow D(S)$?",
   "public_statement": "For a very general cubic fourfold $X$, is there a surface $S$ such that the Kuznetsov component $Ku(X) \\subset D(X)$ has an embedding $Ku(X) \\hookrightarrow D(S)$?",
   "evidence": "The canonical record is problem 1.36 from the AIM workshop *Rationality problems in algebraic geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0063": {
   "statement_status": "exact",
   "original_statement": "Let $X$ and $Y$ be cubic fourfolds. If $Ku(X) \\simeq Ku(Y)$, is $X$ birational to $Y$?",
   "clean_statement": "Let $X$ and $Y$ be cubic fourfolds. If $Ku(X) \\simeq Ku(Y)$, is $X$ birational to $Y$?",
   "public_statement": "Let $X$ and $Y$ be cubic fourfolds. If $Ku(X) \\simeq Ku(Y)$, is $X$ birational to $Y$?",
   "evidence": "The canonical record is AIM Problem Lists, workshop *Rationality problems in algebraic geometry*, problem 1.38, source file `aim-algebraic-geometry-notes.json`, zero-based record index 62. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0064": {
   "statement_status": "exact",
   "original_statement": "If two smooth projective varieties $X$ and $Y$ are K-equivalent, is the Grothendieck group $K_0(X)$ isomorphic to $K_0(Y)$?",
   "clean_statement": "If two smooth projective varieties $X$ and $Y$ are K-equivalent, is the Grothendieck group $K_0(X)$ isomorphic to $K_0(Y)$?",
   "public_statement": "If two smooth projective varieties $X$ and $Y$ are K-equivalent, is the Grothendieck group $K_0(X)$ isomorphic to $K_0(Y)$?",
   "evidence": "The exact source record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0065": {
   "statement_status": "exact",
   "original_statement": "Investigate $Ku(X)$ for a Fano hypersurface $X$ in a product of projective spaces. Specifically, investigate this for $X \\subset \\mathbb P^2 \\times \\mathbb P^3$ of bidegree $(d,2)$ for $d \\geq 2$.",
   "clean_statement": "Investigate $Ku(X)$ for a Fano hypersurface $X$ in a product of projective spaces. Specifically, investigate this for $X \\subset \\mathbb P^2 \\times \\mathbb P^3$ of bidegree $(d,2)$ for $d \\geq 2$.",
   "public_statement": "Investigate $Ku(X)$ for a Fano hypersurface $X$ in a product of projective spaces. Specifically, investigate this for $X \\subset \\mathbb P^2 \\times \\mathbb P^3$ of bidegree $(d,2)$ for $d \\geq 2$.",
   "evidence": "The canonical record (AIM Problem List, workshop *Rationality problems in algebraic geometry*, problem 1.42) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0066": {
   "statement_status": "exact",
   "original_statement": "Investigate $Ku(X)$ for $X$ a smooth quartic double 5 fold, that is, a double cover of $\\mathbb P^5$ branched along a quartic.\nThis variety is Calabi-Yau and it is never of the form $D(Y)$ for $Y$ a variety. Further, the Serre functor of $X$ is equal to shift by 3; in other words, the category $Ku(X)$ is 3-Calabi Yau.",
   "clean_statement": "Investigate $Ku(X)$ for $X$ a smooth quartic double 5 fold, that is, a double cover of $\\mathbb P^5$ branched along a quartic.\nThis variety is Calabi-Yau and it is never of the form $D(Y)$ for $Y$ a variety. Further, the Serre functor of $X$ is equal to shift by 3; in other words, the category $Ku(X)$ is 3-Calabi Yau.",
   "public_statement": "Investigate $Ku(X)$ for $X$ a smooth quartic double 5 fold, that is, a double cover of $\\mathbb P^5$ branched along a quartic.\nThis variety is Calabi-Yau and it is never of the form $D(Y)$ for $Y$ a variety. Further, the Serre functor of $X$ is equal to shift by 3; in other words, the category $Ku(X)$ is 3-Calabi Yau.",
   "evidence": "The canonical AIM record (problem 1.44, source index 65) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0067": {
   "statement_status": "exact",
   "original_statement": "Show that smooth quartic double 5-folds are irrational.",
   "clean_statement": "Show that smooth quartic double 5-folds are irrational.",
   "public_statement": "Show that smooth quartic double 5-folds are irrational.",
   "evidence": "The canonical AIM record (source file `aim-algebraic-geometry-notes.json`, zero-based index 66, Problem 1.46 of the workshop *Rationality problems in algebraic geometry*) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0068": {
   "statement_status": "exact",
   "original_statement": "Hassett and Tschinkel showed that a complete intersection of two quadrics in $\\mathbb P^5$ over an arbitrary field $k$ is rational if and only if it contains a line over $k$.\n\nGive criteria for other geometrically rational Fano $3$-folds to be rational over a field $k$.\n\nGive criteria for the complete intersection of two quadrics in $\\mathbb P^n$ to be rational over $k$.",
   "clean_statement": "Hassett and Tschinkel showed that a complete intersection of two quadrics in $\\mathbb P^5$ over an arbitrary field $k$ is rational if and only if it contains a line over $k$.\n\nGive criteria for other geometrically rational Fano $3$-folds to be rational over a field $k$.\n\nGive criteria for the complete intersection of two quadrics in $\\mathbb P^n$ to be rational over $k$.",
   "public_statement": "Hassett and Tschinkel showed that a complete intersection of two quadrics in $\\mathbb P^5$ over an arbitrary field $k$ is rational if and only if it contains a line over $k$.\n\nGive criteria for other geometrically rational Fano $3$-folds to be rational over a field $k$.\n\nGive criteria for the complete intersection of two quadrics in $\\mathbb P^n$ to be rational over $k$.",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0069": {
   "statement_status": "exact",
   "original_statement": "Let $X$ be a smooth projective complex variety. For a finite abelian group $M$, is unramified cohomology $H^i_{nr}(k(X)/k, M)$ always finite?",
   "clean_statement": "Let $X$ be a smooth projective complex variety. For a finite abelian group $M$, is unramified cohomology $H^i_{nr}(k(X)/k, M)$ always finite?",
   "public_statement": "Let $X$ be a smooth projective complex variety. For a finite abelian group $M$, is unramified cohomology $H^i_{nr}(k(X)/k, M)$ always finite?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0070": {
   "statement_status": "exact",
   "original_statement": "Let $X$ be a rationally connected smooth projective variety of dimension $n$.\nIs $H^n_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z) = 0$?",
   "clean_statement": "Let $X$ be a rationally connected smooth projective variety of dimension $n$.\nIs $H^n_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z) = 0$?",
   "public_statement": "Let $X$ be a rationally connected smooth projective variety of dimension $n$.\nIs $H^n_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z) = 0$?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 69,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0071": {
   "statement_status": "exact",
   "original_statement": "Let $X$ be a variety over $\\overline{\\mathbb Q}$. Is there an algorithm to determine whether\n$X$ is rational?\n\nSame question for unirationality.",
   "clean_statement": "Let $X$ be a variety over $\\overline{\\mathbb Q}$. Is there an algorithm to determine whether\n$X$ is rational?\n\nSame question for unirationality.",
   "public_statement": "Let $X$ be a variety over $\\overline{\\mathbb Q}$. Is there an algorithm to determine whether\n$X$ is rational?\n\nSame question for unirationality.",
   "evidence": "The canonical record is Problem 1.54 from the AIM workshop *Rationality problems in algebraic geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 70,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0072": {
   "statement_status": "exact",
   "original_statement": "Given a rationally connected variety which is not universally $CH_0$-trivial,\ndetermine a nonzero unramified class in some cycle module (in the sense of Rost).",
   "clean_statement": "Given a rationally connected variety which is not universally $CH_0$-trivial,\ndetermine a nonzero unramified class in some cycle module (in the sense of Rost).",
   "public_statement": "Given a rationally connected variety which is not universally $CH_0$-trivial,\ndetermine a nonzero unramified class in some cycle module (in the sense of Rost).",
   "evidence": "There is no visible corruption or ambiguity in the source text. To make the mathematical hypotheses precise, this report works with a smooth, proper, geometrically integral \\(X/k\\). The main new formulation also assumes that \\(X\\) has a zero-cycle \\(z\\) of degree one. This is automatic when \\(k\\) is algebraically closed and whenever \\(X(k)\\ne\\varnothing\\), which covers the usual geometric setting of the AIM question. In characteristic zero, a smooth projective rationally connected variety is rationally chain connected. Rational chain connectedness is used below only to ensure a finite torsion order.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0073": {
   "statement_status": "exact",
   "original_statement": "Prove that Oblomkov-Rozansky triply graded cohomology (HHH$^{\\text{geom}}$) is isomorphic to Khovanov-Rozansky triply graded cohomology (HHH).",
   "clean_statement": "Prove that Oblomkov-Rozansky triply graded cohomology (HHH$^{\\text{geom}}$) is isomorphic to Khovanov-Rozansky triply graded cohomology (HHH).",
   "public_statement": "Prove that Oblomkov-Rozansky triply graded cohomology (HHH$^{\\text{geom}}$) is isomorphic to Khovanov-Rozansky triply graded cohomology (HHH).",
   "evidence": "The canonical AIM record is Problem 1.1 in the section “Oblomkov-Rozansky link invariant” of the 2018 AIM workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 72,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0074": {
   "statement_status": "exact",
   "original_statement": "Prove that MF$((\\mathscr{X}^\\text{big})^\\text{stable},w)$ is equivalent as a monoidal category to K$^b(\\mathbb{S}\\text{Bim}_n)$, where $\\mathscr{X}^\\text{big}=\\mathfrak{g} \\times G \\times \\mathfrak{n} \\times G \\times \\mathfrak{n}$.",
   "clean_statement": "Prove that MF$((\\mathscr{X}^\\text{big})^\\text{stable},w)$ is equivalent as a monoidal category to K$^b(\\mathbb{S}\\text{Bim}_n)$, where $\\mathscr{X}^\\text{big}=\\mathfrak{g} \\times G \\times \\mathfrak{n} \\times G \\times \\mathfrak{n}$.",
   "public_statement": "Prove that MF$((\\mathscr{X}^\\text{big})^\\text{stable},w)$ is equivalent as a monoidal category to K$^b(\\mathbb{S}\\text{Bim}_n)$, where $\\mathscr{X}^\\text{big}=\\mathfrak{g} \\times G \\times \\mathfrak{n} \\times G \\times \\mathfrak{n}$.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 73,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0075": {
   "statement_status": "exact",
   "original_statement": "On HHH$^\\text{geom}$ of a braid $\\beta$ there is a mysterious 4th $\\mathbb{Z}$-grading coming from the $\\mathbb{Z}/2$-grading on matrix factorizations. Construct the analog for HHH of $\\beta$.",
   "clean_statement": "On HHH$^\\text{geom}$ of a braid $\\beta$ there is a mysterious 4th $\\mathbb{Z}$-grading coming from the $\\mathbb{Z}/2$-grading on matrix factorizations. Construct the analog for HHH of $\\beta$.",
   "public_statement": "On HHH$^\\text{geom}$ of a braid $\\beta$ there is a mysterious 4th $\\mathbb{Z}$-grading coming from the $\\mathbb{Z}/2$-grading on matrix factorizations. Construct the analog for HHH of $\\beta$.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0076": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "To what extend can computers compute HHH$^\\text{geom}$ directly from the definition?",
   "clean_statement": null,
   "public_statement": "To what extend can computers compute HHH$^\\text{geom}$ directly from the definition?",
   "evidence": "The word “extend” is preserved from the record; it is almost certainly a typographical error for “extent.” The mathematical phrase “directly from the definition” is not defined in the record. The contemporaneous Oblomkov–Rozansky definition gives the following unambiguous strict reading. For a braid \\(\\beta\\in Br_n\\), one:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0077": {
   "statement_status": "exact",
   "original_statement": "The space $\\mathscr{X}^{\\text{big}}$ and category MF($\\mathscr{X}^{\\text{big}},w$) makes sense not just in type A. What is the analog of $(\\mathscr{X}^{\\text{big}})^{\\text{stable}}$ in other types? What is the categorical analog of the map Br$_\\text{ext} \\to$ Br$_\\text{fin}$.",
   "clean_statement": "The space $\\mathscr{X}^{\\text{big}}$ and category MF($\\mathscr{X}^{\\text{big}},w$) makes sense not just in type A. What is the analog of $(\\mathscr{X}^{\\text{big}})^{\\text{stable}}$ in other types? What is the categorical analog of the map Br$_\\text{ext} \\to$ Br$_\\text{fin}$.",
   "public_statement": "The space $\\mathscr{X}^{\\text{big}}$ and category MF($\\mathscr{X}^{\\text{big}},w$) makes sense not just in type A. What is the analog of $(\\mathscr{X}^{\\text{big}})^{\\text{stable}}$ in other types? What is the categorical analog of the map Br$_\\text{ext} \\to$ Br$_\\text{fin}$.",
   "evidence": "The preceding AIM record specifies \\[ \\mathscr X^{\\mathrm{big}} =\\mathfrak g\\times G\\times\\mathfrak n\\times G\\times\\mathfrak n. \\] This agrees with the “non-reduced” two-fold space \\(\\mathcal X_2=\\mathfrak g\\times(G\\times\\mathfrak n)^2\\) in Oblomkov--Rozansky. We work over \\(\\mathbb C\\), take \\(G\\) to be a connected reductive group, \\(B\\subset G\\) a Borel subgroup, and \\(\\mathfrak n=\\operatorname{Lie}R_u(B)\\). Fixing a nondegenerate \\(G\\)-invariant bilinear form \\(\\kappa\\) on \\(\\mathfrak g\\), the potential is \\[ w(X,g_1,Y_1,g_2,Y_2) =\\kappa\\!\\left(X,\\operatorname{Ad}_{g_1}Y_1- \\operatorname{Ad}_{g_2}Y_2\\right). \\] No corruption of the source statement was found. “Stable” is interpreted through the type A construction in arXiv:1702.03569, Section 2.6.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0078": {
   "statement_status": "exact",
   "original_statement": "What is the interaction between cell theory and the Hilbert scheme? Matrix factorizations?\n\nIn the affine Steinberg setting this corresponds to the stratification by nilpotent orbits. Does this have an analog for $\\mathscr{X}^\\text{big}$?",
   "clean_statement": "What is the interaction between cell theory and the Hilbert scheme? Matrix factorizations?\n\nIn the affine Steinberg setting this corresponds to the stratification by nilpotent orbits. Does this have an analog for $\\mathscr{X}^\\text{big}$?",
   "public_statement": "What is the interaction between cell theory and the Hilbert scheme? Matrix factorizations?\n\nIn the affine Steinberg setting this corresponds to the stratification by nilpotent orbits. Does this have an analog for $\\mathscr{X}^\\text{big}$?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0079": {
   "statement_status": "exact",
   "original_statement": "Call a braid $\\beta$ parity if HHH$(\\beta)$ is parity. Find necessary and/or sufficient conditions for a braid to be parity.\n\nAre there operations on braids (for example, adding a positive number of full twists) which transform it into a parity braid or which preserve parity braids?",
   "clean_statement": "Call a braid $\\beta$ parity if HHH$(\\beta)$ is parity. Find necessary and/or sufficient conditions for a braid to be parity.\n\nAre there operations on braids (for example, adding a positive number of full twists) which transform it into a parity braid or which preserve parity braids?",
   "public_statement": "Call a braid $\\beta$ parity if HHH$(\\beta)$ is parity. Find necessary and/or sufficient conditions for a braid to be parity.\n\nAre there operations on braids (for example, adding a positive number of full twists) which transform it into a parity braid or which preserve parity braids?",
   "evidence": "The canonical record is problem 2.1, “Parity,” from the AIM workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0080": {
   "statement_status": "exact",
   "original_statement": "Are algebraic braids parity? If so find a (recursive) formula for HHH of an algebraic braid.",
   "clean_statement": "Are algebraic braids parity? If so find a (recursive) formula for HHH of an algebraic braid.",
   "public_statement": "Are algebraic braids parity? If so find a (recursive) formula for HHH of an algebraic braid.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0081": {
   "statement_status": "exact",
   "original_statement": "There is a commutative family of braids built as products of FT$_k$, the full twist on the first $k$ strands, for $k \\leq n$. Are the positive products in this family parity? If so, find a (recursive) formula for HHH.",
   "clean_statement": "There is a commutative family of braids built as products of FT$_k$, the full twist on the first $k$ strands, for $k \\leq n$. Are the positive products in this family parity? If so, find a (recursive) formula for HHH.",
   "public_statement": "There is a commutative family of braids built as products of FT$_k$, the full twist on the first $k$ strands, for $k \\leq n$. Are the positive products in this family parity? If so, find a (recursive) formula for HHH.",
   "evidence": "The archived AIM page timed out during this run, but the repository record is syntactically complete and the notation is fixed by the adjacent records. There is no apparent OCR corruption.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0082": {
   "statement_status": "exact",
   "original_statement": "In the presence of parity, there are two combinatorial formulas for HHH: one for localization of Hilbert schemes from the GNR conjecture and one from the Elias-Hogencamp-Mellit papers. How are these formulas related? How are they related to Demazure crystals?",
   "clean_statement": "In the presence of parity, there are two combinatorial formulas for HHH: one for localization of Hilbert schemes from the GNR conjecture and one from the Elias-Hogencamp-Mellit papers. How are these formulas related? How are they related to Demazure crystals?",
   "public_statement": "In the presence of parity, there are two combinatorial formulas for HHH: one for localization of Hilbert schemes from the GNR conjecture and one from the Elias-Hogencamp-Mellit papers. How are these formulas related? How are they related to Demazure crystals?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0083": {
   "statement_status": "exact",
   "original_statement": "To an algebraic link one can associate a Springer fibre and a Milnor fibre. Can one compute HHH from the Milnor fibre?",
   "clean_statement": "To an algebraic link one can associate a Springer fibre and a Milnor fibre. Can one compute HHH from the Milnor fibre?",
   "public_statement": "To an algebraic link one can associate a Springer fibre and a Milnor fibre. Can one compute HHH from the Milnor fibre?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 82,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0084": {
   "statement_status": "exact",
   "original_statement": "It is known that the Alexander polynomial can be computed from both the Springer fibre and the Milnor fibre. Understand the relationship between these two constructions.",
   "clean_statement": "It is known that the Alexander polynomial can be computed from both the Springer fibre and the Milnor fibre. Understand the relationship between these two constructions.",
   "public_statement": "It is known that the Alexander polynomial can be computed from both the Springer fibre and the Milnor fibre. Understand the relationship between these two constructions.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 83,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0085": {
   "statement_status": "exact",
   "original_statement": "Outside of type A is there still a relation between Hochschild cohomology of Rouquier complexes and affine Springer fibres?",
   "clean_statement": "Outside of type A is there still a relation between Hochschild cohomology of Rouquier complexes and affine Springer fibres?",
   "public_statement": "Outside of type A is there still a relation between Hochschild cohomology of Rouquier complexes and affine Springer fibres?",
   "evidence": "The canonical AIM record is Problem 3.3 in the “Springer fibres” section of the workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 84,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0086": {
   "statement_status": "exact",
   "original_statement": "What is the Drinfeld center of K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$? K$^b(\\mathbb{S}\\text{Bim}_{\\text{aff}})$? Are there additional complexes in K$^b(\\mathbb{S}\\text{Bim})$ which commute with $\\mathbb{S}$Bim?",
   "clean_statement": "What is the Drinfeld center of K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$? K$^b(\\mathbb{S}\\text{Bim}_{\\text{aff}})$? Are there additional complexes in K$^b(\\mathbb{S}\\text{Bim})$ which commute with $\\mathbb{S}$Bim?",
   "public_statement": "What is the Drinfeld center of K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$? K$^b(\\mathbb{S}\\text{Bim}_{\\text{aff}})$? Are there additional complexes in K$^b(\\mathbb{S}\\text{Bim})$ which commute with $\\mathbb{S}$Bim?",
   "evidence": "The canonical AIM record is Problem 4.1 in the section “Soergel bimodules” of the workshop list *Categorified Hecke algebras, link homology, and Hilbert schemes*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 85,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0087": {
   "statement_status": "exact",
   "original_statement": "Is Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{ext}})$) generated by Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ generated by images of Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ split generated by powers of FT$_n$?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ symmetric monoidal?",
   "clean_statement": "Is Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{ext}})$) generated by Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ generated by images of Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ split generated by powers of FT$_n$?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ symmetric monoidal?",
   "public_statement": "Is Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{ext}})$) generated by Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ generated by images of Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ split generated by powers of FT$_n$?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ symmetric monoidal?",
   "evidence": "The canonical record is Problem 4.2 in the “Soergel bimodules” section of the AIM workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*. The source page and the neighboring Problems 4.1 and 4.3 confirm the following reading:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 86,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0088": {
   "statement_status": "exact",
   "original_statement": "Is the flattening of the Gaitsgory complex for the standard representation equal to the tautological sheaf from the GNR conjecture?",
   "clean_statement": "Is the flattening of the Gaitsgory complex for the standard representation equal to the tautological sheaf from the GNR conjecture?",
   "public_statement": "Is the flattening of the Gaitsgory complex for the standard representation equal to the tautological sheaf from the GNR conjecture?",
   "evidence": "The AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 87,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0089": {
   "statement_status": "exact",
   "original_statement": "A theorem of Morton-Samuelson connected the Elliptic Hall algebra at $q=t$ to the Skein algebra of the torus. Categorify this result.",
   "clean_statement": "A theorem of Morton-Samuelson connected the Elliptic Hall algebra at $q=t$ to the Skein algebra of the torus. Categorify this result.",
   "public_statement": "A theorem of Morton-Samuelson connected the Elliptic Hall algebra at $q=t$ to the Skein algebra of the torus. Categorify this result.",
   "evidence": "The canonical record is Problem 5.1 in the “Others” section of the AIM workshop list *Categorified Hecke algebras, link homology, and Hilbert schemes*. The live AIM page confirms the exact text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 88,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0090": {
   "statement_status": "exact",
   "original_statement": "Are there simplifications of the above problem for the $\\mathfrak{sl}_n$-quotient invariant?",
   "clean_statement": "Are there simplifications of the above problem for the $\\mathfrak{sl}_n$-quotient invariant?",
   "public_statement": "Are there simplifications of the above problem for the $\\mathfrak{sl}_n$-quotient invariant?",
   "evidence": "The canonical record is Problem 5.2 in the “Others” section of the 2018 AIM workshop list *Categorified Hecke algebras, link homology, and Hilbert schemes*. The live AIM page confirms the exact wording:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 89,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0091": {
   "statement_status": "exact",
   "original_statement": "Is there an equivalence of categories between DAHA representations and a suitable extension of the Fukaya category of the character variety of a single punctured torus?",
   "clean_statement": "Is there an equivalence of categories between DAHA representations and a suitable extension of the Fukaya category of the character variety of a single punctured torus?",
   "public_statement": "Is there an equivalence of categories between DAHA representations and a suitable extension of the Fukaya category of the character variety of a single punctured torus?",
   "evidence": "The canonical AIM record, from the workshop *Categorified Hecke algebras, link homology, and Hilbert schemes* (AIM, 1--5 October 2018), asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 90,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0092": {
   "statement_status": "exact",
   "original_statement": "Is there a braid invariant extension of HHH which replaces bigraded vector spaces with bigraded $S_n$-modules suitably compatible with the action of $R=\\mathbb{Q}[x_1, \\dots, x_n]$ and the action of $\\mathbb{Q}[y_1, \\dots, y_n]$ coming from $y$-ification?",
   "clean_statement": "Is there a braid invariant extension of HHH which replaces bigraded vector spaces with bigraded $S_n$-modules suitably compatible with the action of $R=\\mathbb{Q}[x_1, \\dots, x_n]$ and the action of $\\mathbb{Q}[y_1, \\dots, y_n]$ coming from $y$-ification?",
   "public_statement": "Is there a braid invariant extension of HHH which replaces bigraded vector spaces with bigraded $S_n$-modules suitably compatible with the action of $R=\\mathbb{Q}[x_1, \\dots, x_n]$ and the action of $\\mathbb{Q}[y_1, \\dots, y_n]$ coming from $y$-ification?",
   "evidence": "The canonical record is problem 5.4, “Others,” from the AIM workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*. Its exact statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 91,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0093": {
   "statement_status": "exact",
   "original_statement": "Can one describe HHH$^{a>0}$ using HHH$^{a=0}$ after tensoring with a suitable central complex?",
   "clean_statement": "Can one describe HHH$^{a>0}$ using HHH$^{a=0}$ after tensoring with a suitable central complex?",
   "public_statement": "Can one describe HHH$^{a>0}$ using HHH$^{a=0}$ after tensoring with a suitable central complex?",
   "evidence": "The exact AIM record is Problem 5.5 in the “Others” section of the 2018 workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0094": {
   "statement_status": "exact",
   "original_statement": "How does one describe the Procesi bundle in the other settings, like K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$?",
   "clean_statement": "How does one describe the Procesi bundle in the other settings, like K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$?",
   "public_statement": "How does one describe the Procesi bundle in the other settings, like K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0095": {
   "statement_status": "exact",
   "original_statement": "Can we give lower bounds on the degree of irrationality / covering gonality.... of hypersurfaces in char $p$?\nHow do we make sense of these notions over countable / finite fields?",
   "clean_statement": "Can we give lower bounds on the degree of irrationality / covering gonality.... of hypersurfaces in char $p$?\nHow do we make sense of these notions over countable / finite fields?",
   "public_statement": "Can we give lower bounds on the degree of irrationality / covering gonality.... of hypersurfaces in char $p$?\nHow do we make sense of these notions over countable / finite fields?",
   "evidence": "The canonical record is AIM-ALGEBRAIC_GEOMETRY-0095, item 2.1 in the section “Degree of irrationality and covering gonality” of the AIM workshop *Rational subvarieties in positive characteristic*. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 94,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0096": {
   "statement_status": "exact",
   "original_statement": "How does rationality behave in a family over $\\text{Spec} \\mathbb{Z}$ or $\\text{Spec} \\mathbb{Z}[\\frac{1}{n}]$.\nIf $X$ is rational / unirational for almost all $p$, does it follow $X$ is rational / unirational in char $0$?",
   "clean_statement": "How does rationality behave in a family over $\\text{Spec} \\mathbb{Z}$ or $\\text{Spec} \\mathbb{Z}[\\frac{1}{n}]$.\nIf $X$ is rational / unirational for almost all $p$, does it follow $X$ is rational / unirational in char $0$?",
   "public_statement": "How does rationality behave in a family over $\\text{Spec} \\mathbb{Z}$ or $\\text{Spec} \\mathbb{Z}[\\frac{1}{n}]$.\nIf $X$ is rational / unirational for almost all $p$, does it follow $X$ is rational / unirational in char $0$?",
   "evidence": "The record is Problem 3.06 in the section “Rationality in a family” of the 2016 AIM workshop *Rational subvarieties in positive characteristic*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0097": {
   "statement_status": "corrected_verified",
   "original_statement": "Can we find a cubic fourfold which is conjectually irratioal, but (most of ) its reductions are rational?",
   "clean_statement": "Can one find a smooth cubic fourfold over a number field which is conjecturally irrational over an algebraic closure of characteristic zero, but for which “most” good reductions are rational?",
   "public_statement": "Can one find a smooth cubic fourfold over a number field which is conjecturally irrational over an algebraic closure of characteristic zero, but for which “most” good reductions are rational?",
   "evidence": "The same wording appears on the AIM source page, with no appended remark. I reconstruct only the evident typographical error “irratioal” as “irrational.” The mathematically recovered question is therefore:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 96,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0098": {
   "statement_status": "exact",
   "original_statement": "If the general member of a family of smooth cubic is rational, is the special fiber also rational?",
   "clean_statement": "If the general member of a family of smooth cubic is rational, is the special fiber also rational?",
   "public_statement": "If the general member of a family of smooth cubic is rational, is the special fiber also rational?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 97,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0099": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Let $\\mathbb{P}$ be the parametrizing space of higher surfaces of degree $d$ in $\\mathbb{P}^{n}$.\nWhat is the largest dimensional family of unirational / supersinglar hypersurfaces?",
   "clean_statement": null,
   "public_statement": "Let $\\mathbb{P}$ be the parametrizing space of higher surfaces of degree $d$ in $\\mathbb{P}^{n}$.\nWhat is the largest dimensional family of unirational / supersinglar hypersurfaces?",
   "evidence": "The live AIM page was checked on 2026-07-23 and contains the same words, with no status note. Thus the two apparent errors are in the source, rather than in the JSON extraction. I make the following explicit reconstruction.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 98,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0100": {
   "statement_status": "exact",
   "original_statement": "$\\mathcal{X}\\longrightarrow \\mathcal{B}$ be a family of smooth hypersurfaces of degree $d=n$ in $\\mathbb{P}^{n}$.\nIs there more than one rational section?\nDoes weak approximation hold?",
   "clean_statement": "$\\mathcal{X}\\longrightarrow \\mathcal{B}$ be a family of smooth hypersurfaces of degree $d=n$ in $\\mathbb{P}^{n}$.\nIs there more than one rational section?\nDoes weak approximation hold?",
   "public_statement": "$\\mathcal{X}\\longrightarrow \\mathcal{B}$ be a family of smooth hypersurfaces of degree $d=n$ in $\\mathbb{P}^{n}$.\nIs there more than one rational section?\nDoes weak approximation hold?",
   "evidence": "The canonical AIM record is Problem 3.17 in the workshop *Rational subvarieties in positive characteristic*, section “Rationality in a family.” Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 99,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0101": {
   "statement_status": "exact",
   "original_statement": "If $d^{2}\\leq n$, does weak approximation hold in char. $p$ for hypersurfaces? (it does in char. $0$)",
   "clean_statement": "If $d^{2}\\leq n$, does weak approximation hold in char. $p$ for hypersurfaces? (it does in char. $0$)",
   "public_statement": "If $d^{2}\\leq n$, does weak approximation hold in char. $p$ for hypersurfaces? (it does in char. $0$)",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 100,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0102": {
   "statement_status": "exact",
   "original_statement": "4th day\n\nDo there exist a cubic $4$-fold defined over number fields which is conjecured to be irrational but many reductions are conjectured to be ratioal?",
   "clean_statement": "4th day\n\nDo there exist a cubic $4$-fold defined over number fields which is conjecured to be irrational but many reductions are conjectured to be ratioal?",
   "public_statement": "4th day\n\nDo there exist a cubic $4$-fold defined over number fields which is conjecured to be irrational but many reductions are conjectured to be ratioal?",
   "evidence": "The canonical record is AIM Problem List 3.72 in the section “Rationality in a family” of the workshop *Rational subvarieties in positive characteristic*. The record, including its spelling, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 101,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0103": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Can irrational varieties over $\\#$ fields specialize to rational varieties in char. $p>0$?\n(Colliot-th\\'el\\`ene-Ojanguren's examples)",
   "clean_statement": null,
   "public_statement": "Can irrational varieties over $\\#$ fields specialize to rational varieties in char. $p>0$?\n(Colliot-th\\'el\\`ene-Ojanguren's examples)",
   "evidence": "It is Problem 3.78 in the section “Rationality in a family” of the AIM workshop list *Rational subvarieties in positive characteristic*. The archived AIM page itself contains the string `$\\#$ fields`; thus this is not merely a corruption introduced by the JSON extraction. The immediately preceding Problem 3.72 asks about cubic fourfolds “defined over number fields” and their reductions. On that contextual evidence, the most plausible reconstruction is: > **Plausible reconstruction.** Can an irrational variety over a **number field** have a rational specialization in characteristic \\(p>0\\)? What happens for the examples of Colliot-Thélène and Ojanguren?",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 102,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0104": {
   "statement_status": "exact",
   "original_statement": "When is $\\mathcal{M}_{g,n}$, $\\mathcal{A}_{g}[d]$ or $\\mathcal{K3}_{d}$ unirational over char. $p$?\nWhen is $\\mathcal{M}_{1,n}$ unirational for $n\\geq 11$?",
   "clean_statement": "When is $\\mathcal{M}_{g,n}$, $\\mathcal{A}_{g}[d]$ or $\\mathcal{K3}_{d}$ unirational over char. $p$?\nWhen is $\\mathcal{M}_{1,n}$ unirational for $n\\geq 11$?",
   "public_statement": "When is $\\mathcal{M}_{g,n}$, $\\mathcal{A}_{g}[d]$ or $\\mathcal{K3}_{d}$ unirational over char. $p$?\nWhen is $\\mathcal{M}_{1,n}$ unirational for $n\\geq 11$?",
   "evidence": "There is no apparent OCR corruption. There are, however, three genuine notational ambiguities:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 103,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0105": {
   "statement_status": "exact",
   "original_statement": "Find rational curves in $\\mathcal{A}_{g}$ in char. $p$, that do not lift to char. $0$.",
   "clean_statement": "Find rational curves in $\\mathcal{A}_{g}$ in char. $p$, that do not lift to char. $0$.",
   "public_statement": "Find rational curves in $\\mathcal{A}_{g}$ in char. $p$, that do not lift to char. $0$.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 104,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0106": {
   "statement_status": "exact",
   "original_statement": "What are the rationality properties of $\\mathcal{A}_{g,s.s.}$?\nYou can ask the same question for Newton polygon strata.",
   "clean_statement": "What are the rationality properties of $\\mathcal{A}_{g,s.s.}$?\nYou can ask the same question for Newton polygon strata.",
   "public_statement": "What are the rationality properties of $\\mathcal{A}_{g,s.s.}$?\nYou can ask the same question for Newton polygon strata.",
   "evidence": "The canonical AIM record is Problem 4.3 from the workshop *Rational subvarieties in positive characteristic*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 105,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0107": {
   "statement_status": "exact",
   "original_statement": "Are complete rational curves in $\\mathcal{A}_{g}$ dense in char. $p$?",
   "clean_statement": "Are complete rational curves in $\\mathcal{A}_{g}$ dense in char. $p$?",
   "public_statement": "Are complete rational curves in $\\mathcal{A}_{g}$ dense in char. $p$?",
   "evidence": "The canonical record is AIM Problem List item 4.4 from the workshop *Rational subvarieties in positive characteristic*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 106,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0108": {
   "statement_status": "exact",
   "original_statement": "In char. $p$ if $n\\geq d+2$,\nare the moduli spaces of rational curves on a general hypersurface of degree $d$ in $\\mathbb{P}^{n}$ irreducible of the expected dimension?",
   "clean_statement": "In char. $p$ if $n\\geq d+2$,\nare the moduli spaces of rational curves on a general hypersurface of degree $d$ in $\\mathbb{P}^{n}$ irreducible of the expected dimension?",
   "public_statement": "In char. $p$ if $n\\geq d+2$,\nare the moduli spaces of rational curves on a general hypersurface of degree $d$ in $\\mathbb{P}^{n}$ irreducible of the expected dimension?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 107,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0109": {
   "statement_status": "exact",
   "original_statement": "What is the dimension of rational curves in hypersurfaces of degree $d$ in $\\mathbb{P}^{n}$ in char. $p$? Does char. $0$ proof go through?",
   "clean_statement": "What is the dimension of rational curves in hypersurfaces of degree $d$ in $\\mathbb{P}^{n}$ in char. $p$? Does char. $0$ proof go through?",
   "public_statement": "What is the dimension of rational curves in hypersurfaces of degree $d$ in $\\mathbb{P}^{n}$ in char. $p$? Does char. $0$ proof go through?",
   "evidence": "The exact AIM record (workshop “Rational subvarieties in positive characteristic,” section “The moduli space of rational curves of hypersurfaces,” Problem 5.2) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 108,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0110": {
   "statement_status": "corrected_verified",
   "original_statement": "Is the very general hyperusrfae of degree $d\\geq 2 \\lceil(n+3)/3\\rceil$ not rational / stably rational / ruled in char $p>0$ for $p>>0$?",
   "clean_statement": "Fix \\(n\\geq1\\) and \\(d\\geq 2\\lceil(n+3)/3\\rceil\\). Is there a number \\(P(n,d)\\) such that, for every prime \\(p>P(n,d)\\), the very general smooth degree-\\(d\\), \\(n\\)-dimensional hypersurface \\(X_d\\subset\\mathbb P^{n+1}\\) in characteristic \\(p\\) is (i) irrational, (ii) not stably rational, and (iii) not ruled?",
   "public_statement": "Fix \\(n\\geq1\\) and \\(d\\geq 2\\lceil(n+3)/3\\rceil\\). Is there a number \\(P(n,d)\\) such that, for every prime \\(p>P(n,d)\\), the very general smooth degree-\\(d\\), \\(n\\)-dimensional hypersurface \\(X_d\\subset\\mathbb P^{n+1}\\) in characteristic \\(p\\) is (i) irrational, (ii) not stably rational, and (iii) not ruled?",
   "evidence": "This is Problem 6.1 in the section “New applications of Kollar's and Totaro's techniques” from the 2016 AIM workshop *Rational subvarieties in positive characteristic*. The source URL is <http://aimpl.org/ratsubvarpos/6/>. It timed out during this run, so the recovery below is based on the exact corpus record, the adjacent records, and matching primary literature rather than on a newly fetched copy of the old AIM page. There is one evident OCR error: “hyperusrfae” means “hypersurface.” The ambient projective space is omitted, but the numerical bound identifies the intended convention unambiguously. Kollár's theorem, quoted with exactly this bound by Totaro and by Lange--Schreieder, concerns an \\(n\\)-dimensional hypersurface \\[ X_d\\subset \\mathbb P^{n+1}. \\] Thus the recovered question is:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 109,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0111": {
   "statement_status": "exact",
   "original_statement": "Can we improve those techniques to find differential form for lower degree hypersurfaces?",
   "clean_statement": "Can we improve those techniques to find differential form for lower degree hypersurfaces?",
   "public_statement": "Can we improve those techniques to find differential form for lower degree hypersurfaces?",
   "evidence": "The canonical AIM record (workshop “Rational subvarieties in positive characteristic,” section 6, problem 6.2) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 110,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0112": {
   "statement_status": "corrected_verified",
   "original_statement": "Can we find new exmaples of unirational varieties that have non-vanishing differential forms?",
   "clean_statement": "Can we find new examples of unirational varieties that have non-vanishing differential forms?",
   "public_statement": "Can we find new examples of unirational varieties that have non-vanishing differential forms?",
   "evidence": "The evident typographical correction is “examples.” I interpret “non-vanishing differential forms” as “nonzero global regular differential forms,” not as forms that are nowhere zero. The surviving AIM record is otherwise unambiguous. The linked AIM page did not return usable additional text during this run.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 111,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0113": {
   "statement_status": "exact",
   "original_statement": "Look for new applications of Koll\\'ar / Totaro's techique.",
   "clean_statement": "Look for new applications of Koll\\'ar / Totaro's techique.",
   "public_statement": "Look for new applications of Koll\\'ar / Totaro's techique.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 112,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0114": {
   "statement_status": "exact",
   "original_statement": "Find an example of varieties which are degenerations of Fano hypersurfaces\nwith $H^{i}(\\mathcal{O}_{X})\\neq 0$ for some $i$.\nTry to construct varieties with non-trivial Brawer classes.",
   "clean_statement": "Find an example of varieties which are degenerations of Fano hypersurfaces\nwith $H^{i}(\\mathcal{O}_{X})\\neq 0$ for some $i$.\nTry to construct varieties with non-trivial Brawer classes.",
   "public_statement": "Find an example of varieties which are degenerations of Fano hypersurfaces\nwith $H^{i}(\\mathcal{O}_{X})\\neq 0$ for some $i$.\nTry to construct varieties with non-trivial Brawer classes.",
   "evidence": "The canonical AIM record, problem 6.5 in the section “New applications of Kollar's and Totaro's techniques,” reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 113,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0115": {
   "statement_status": "exact",
   "original_statement": "Do cubic $3$-folds have odd degree unirational prametrizations in char $p$?",
   "clean_statement": "Do cubic $3$-folds have odd degree unirational prametrizations in char $p$?",
   "public_statement": "Do cubic $3$-folds have odd degree unirational prametrizations in char $p$?",
   "evidence": "The canonical record is AIM Problem Lists, workshop *Rational subvarieties in positive characteristic*, §7, Problem 7.1. The source page contains the single sentence",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 114,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0116": {
   "statement_status": "exact",
   "original_statement": "Can we find a simply-connected variety with supersingular cohomology which is not rationally-connected?",
   "clean_statement": "Can we find a simply-connected variety with supersingular cohomology which is not rationally-connected?",
   "public_statement": "Can we find a simply-connected variety with supersingular cohomology which is not rationally-connected?",
   "evidence": "The canonical record is AIM Problem Lists, workshop *Rational subvarieties in positive characteristic*, section “Other problems,” item 9.05. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 115,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0117": {
   "statement_status": "exact",
   "original_statement": "Find a complete subvariety of $\\mathcal{M}_{g}$ of $\\text{codim}2g-1$.",
   "clean_statement": "Find a complete subvariety of $\\mathcal{M}_{g}$ of $\\text{codim}2g-1$.",
   "public_statement": "Find a complete subvariety of $\\mathcal{M}_{g}$ of $\\text{codim}2g-1$.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 116,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0118": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Fix $q$, let $g\\rightarrow \\infty$,\nwhat subvariety of $\\mathcal{M}_{g} / \\mathcal{A}_{g}$ contributes the most $\\#$ of rational points?",
   "clean_statement": null,
   "public_statement": "Fix $q$, let $g\\rightarrow \\infty$,\nwhat subvariety of $\\mathcal{M}_{g} / \\mathcal{A}_{g}$ contributes the most $\\#$ of rational points?",
   "evidence": "There is no standard quotient \\(\\mathcal M_g/\\mathcal A_g\\): \\(\\mathcal A_g\\) is not a group acting on \\(\\mathcal M_g\\). The natural relation is instead the Torelli morphism \\(\\mathcal M_g\\to\\mathcal A_g\\). The workshop report discusses moduli of curves and moduli of abelian varieties in parallel. The least speculative reconstruction is therefore: This reconstruction is not asserted to be uniquely intended.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 117,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0119": {
   "statement_status": "exact",
   "original_statement": "Is there a supersingular hypersurface of every degree, dimension and char. $p$?",
   "clean_statement": "Is there a supersingular hypersurface of every degree, dimension and char. $p$?",
   "public_statement": "Is there a supersingular hypersurface of every degree, dimension and char. $p$?",
   "evidence": "The canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 118,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0120": {
   "statement_status": "exact",
   "original_statement": "Find other examples of supersingular hypersurfaces.",
   "clean_statement": "Find other examples of supersingular hypersurfaces.",
   "public_statement": "Find other examples of supersingular hypersurfaces.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 119,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0121": {
   "statement_status": "exact",
   "original_statement": "Is being rationally connected = unirational in char $p$?",
   "clean_statement": "Is being rationally connected = unirational in char $p$?",
   "public_statement": "Is being rationally connected = unirational in char $p$?",
   "evidence": "The canonical record is AIM Problem List 9.3 in the workshop *Rational subvarieties in positive characteristic*. The statement is exactly",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 120,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0122": {
   "statement_status": "exact",
   "original_statement": "Can a surface with $c_{1}^{2}=3c_{2}$ contain a rational curve in char. $p$?",
   "clean_statement": "Can a surface with $c_{1}^{2}=3c_{2}$ contain a rational curve in char. $p$?",
   "public_statement": "Can a surface with $c_{1}^{2}=3c_{2}$ contain a rational curve in char. $p$?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 121,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0123": {
   "statement_status": "exact",
   "original_statement": "Does an automorphism of a separably rationally connected variety necessarily have a fixed point?",
   "clean_statement": "Does an automorphism of a separably rationally connected variety necessarily have a fixed point?",
   "public_statement": "Does an automorphism of a separably rationally connected variety necessarily have a fixed point?",
   "evidence": "The canonical AIM record is Problem 9.4 in the workshop *Rational subvarieties in positive characteristic*. Its complete text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 122,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0124": {
   "statement_status": "exact",
   "original_statement": "Is there an irrational smooth proper variety over $\\text{Spec}\\mathbb{Z}$? (Can we find such a hypersurface?)",
   "clean_statement": "Is there an irrational smooth proper variety over $\\text{Spec}\\mathbb{Z}$? (Can we find such a hypersurface?)",
   "public_statement": "Is there an irrational smooth proper variety over $\\text{Spec}\\mathbb{Z}$? (Can we find such a hypersurface?)",
   "evidence": "The canonical AIM record (workshop *Rational subvarieties in positive characteristic*, “Other problems,” 9.45) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 123,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0125": {
   "statement_status": "exact",
   "original_statement": "For every $p, d$ with $d\\geq n+1$, can we find hypersurface over $\\mathbb{F}_{p}$ of degree $d$ in $\\mathbb{P}^{n}$\nsuch that either $\\# X \\not\\equiv 1 \\text{mod} p$ or $X$ is not rationally connected?",
   "clean_statement": "For every $p, d$ with $d\\geq n+1$, can we find hypersurface over $\\mathbb{F}_{p}$ of degree $d$ in $\\mathbb{P}^{n}$\nsuch that either $\\# X \\not\\equiv 1 \\text{mod} p$ or $X$ is not rationally connected?",
   "public_statement": "For every $p, d$ with $d\\geq n+1$, can we find hypersurface over $\\mathbb{F}_{p}$ of degree $d$ in $\\mathbb{P}^{n}$\nsuch that either $\\# X \\not\\equiv 1 \\text{mod} p$ or $X$ is not rationally connected?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 124,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0126": {
   "statement_status": "exact",
   "original_statement": "Can we check Bloch's conjecture for known examples of super-singular surfaces?",
   "clean_statement": "Can we check Bloch's conjecture for known examples of super-singular surfaces?",
   "public_statement": "Can we check Bloch's conjecture for known examples of super-singular surfaces?",
   "evidence": "The canonical AIM record is problem 9.55 in the “Other problems” section of the 2016 workshop *Rational subvarieties in positive characteristic*. Its entire problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 125,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0127": {
   "statement_status": "exact",
   "original_statement": "Let $(X, D)$ snc.\nHow can rational curves in $X$ meet $D$?\nAre there any restrictions?",
   "clean_statement": "Let $(X, D)$ snc.\nHow can rational curves in $X$ meet $D$?\nAre there any restrictions?",
   "public_statement": "Let $(X, D)$ snc.\nHow can rational curves in $X$ meet $D$?\nAre there any restrictions?",
   "evidence": "The canonical AIM record is number 9.6 in the section “Other problems” of the workshop *Rational subvarieties in positive characteristic*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 126,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0128": {
   "statement_status": "exact",
   "original_statement": "Can we get alterations of coprime degrees?",
   "clean_statement": "Can we get alterations of coprime degrees?",
   "public_statement": "Can we get alterations of coprime degrees?",
   "evidence": "The canonical AIM record is number 9.65 in the section “Other problems” of the workshop list *Rational subvarieties in positive characteristic*. Its complete problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 127,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0129": {
   "statement_status": "exact",
   "original_statement": "Characterize a variety $X$ over $\\text{Spec}\\mathbb{Z}$ such that the indecomposable summands of $F_{*}\\mathcal{O}_{X}$ have universally bounded rank.",
   "clean_statement": "Characterize a variety $X$ over $\\text{Spec}\\mathbb{Z}$ such that the indecomposable summands of $F_{*}\\mathcal{O}_{X}$ have universally bounded rank.",
   "public_statement": "Characterize a variety $X$ over $\\text{Spec}\\mathbb{Z}$ such that the indecomposable summands of $F_{*}\\mathcal{O}_{X}$ have universally bounded rank.",
   "evidence": "The canonical AIM record is Problem 9.7 in the section “Other problems” of the 2016 workshop *Rational subvarieties in positive characteristic*. Its exact statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 128,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0130": {
   "statement_status": "exact",
   "original_statement": "If $X$ is separably rationally connected, there exists $N(\\text{dim} X, \\text{deg} X, m)$ such that,\nfor any $p_{1},\\cdots, p_{m}\\in X(\\mathbb{F}_{p})$, there exists a rational curve $C$ defined over $\\mathbb{F}_{p^{N}}$ containing all $p_{i}$. (due to Kollar)\nCan you do this without lifting the cardinality of field tend to $\\infty$?\nLet $X$ be a cubic surface over $\\mathbb{F}_{q}$.\nCan you find a rational curve defined over $\\mathbb{F}_{q}$ through each point?",
   "clean_statement": "If $X$ is separably rationally connected, there exists $N(\\text{dim} X, \\text{deg} X, m)$ such that,\nfor any $p_{1},\\cdots, p_{m}\\in X(\\mathbb{F}_{p})$, there exists a rational curve $C$ defined over $\\mathbb{F}_{p^{N}}$ containing all $p_{i}$. (due to Kollar)\nCan you do this without lifting the cardinality of field tend to $\\infty$?\nLet $X$ be a cubic surface over $\\mathbb{F}_{q}$.\nCan you find a rational curve defined over $\\mathbb{F}_{q}$ through each point?",
   "public_statement": "If $X$ is separably rationally connected, there exists $N(\\text{dim} X, \\text{deg} X, m)$ such that,\nfor any $p_{1},\\cdots, p_{m}\\in X(\\mathbb{F}_{p})$, there exists a rational curve $C$ defined over $\\mathbb{F}_{p^{N}}$ containing all $p_{i}$. (due to Kollar)\nCan you do this without lifting the cardinality of field tend to $\\infty$?\nLet $X$ be a cubic surface over $\\mathbb{F}_{q}$.\nCan you find a rational curve defined over $\\mathbb{F}_{q}$ through each point?",
   "evidence": "The canonical record is problem 9.75, “Other problems,” from the AIM workshop *Rational subvarieties in positive characteristic*. Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 129,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0131": {
   "statement_status": "exact",
   "original_statement": "For separably rationally connected varieties, is the Brauer-Manin obstruction the only obstruction for the Hasse principle?",
   "clean_statement": "For separably rationally connected varieties, is the Brauer-Manin obstruction the only obstruction for the Hasse principle?",
   "public_statement": "For separably rationally connected varieties, is the Brauer-Manin obstruction the only obstruction for the Hasse principle?",
   "evidence": "The canonical record is AIM Problem List problem 9.8 from the workshop *Rational subvarieties in positive characteristic*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 130,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0132": {
   "statement_status": "exact",
   "original_statement": "Effective cones of projective bundles\n\nIn \\cite{MR2836078} Fulger showed how to compute the effective cone of $k$-cycles $\\text{Eff}^k(\\mathbb{P}(\\mathcal{E})$ of a projective bundle over a smooth curve. In this case the answer can be given in terms of data coming from the Harder-Narasimhan filtration of $\\mathcal{E}$. However, even in the case where $X$ is a $\\mathbb{P}^2$-bundles over a surface $S$, it is not known how to compute $\\text{Eff}_2(X)$. It is natural to first consider the case where $S$ is the projective plane or a K3 surface.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(X)$ where $X$ is a $\\mathbb{P}^2$-bundle over $\\mathbb{P}^2$?",
   "clean_statement": "Effective cones of projective bundles\n\nIn \\cite{MR2836078} Fulger showed how to compute the effective cone of $k$-cycles $\\text{Eff}^k(\\mathbb{P}(\\mathcal{E})$ of a projective bundle over a smooth curve. In this case the answer can be given in terms of data coming from the Harder-Narasimhan filtration of $\\mathcal{E}$. However, even in the case where $X$ is a $\\mathbb{P}^2$-bundles over a surface $S$, it is not known how to compute $\\text{Eff}_2(X)$. It is natural to first consider the case where $S$ is the projective plane or a K3 surface.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(X)$ where $X$ is a $\\mathbb{P}^2$-bundle over $\\mathbb{P}^2$?",
   "public_statement": "Effective cones of projective bundles\n\nIn \\cite{MR2836078} Fulger showed how to compute the effective cone of $k$-cycles $\\text{Eff}^k(\\mathbb{P}(\\mathcal{E})$ of a projective bundle over a smooth curve. In this case the answer can be given in terms of data coming from the Harder-Narasimhan filtration of $\\mathcal{E}$. However, even in the case where $X$ is a $\\mathbb{P}^2$-bundles over a surface $S$, it is not known how to compute $\\text{Eff}_2(X)$. It is natural to first consider the case where $S$ is the projective plane or a K3 surface.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(X)$ where $X$ is a $\\mathbb{P}^2$-bundle over $\\mathbb{P}^2$?",
   "evidence": "The canonical record is Problem 1.1, “Effective cones of projective bundles,” from the 2016 AIM workshop *Positivity of cycles*. The archived AIM page gives the following context and question:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 131,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0133": {
   "statement_status": "exact",
   "original_statement": "What is the effective cone of surfaces $\\text{Eff}^2(X)$ where $X$ is a $\\mathbb{P}^2$-bundle over a K3 surface?",
   "clean_statement": "What is the effective cone of surfaces $\\text{Eff}^2(X)$ where $X$ is a $\\mathbb{P}^2$-bundle over a K3 surface?",
   "public_statement": "What is the effective cone of surfaces $\\text{Eff}^2(X)$ where $X$ is a $\\mathbb{P}^2$-bundle over a K3 surface?",
   "evidence": "The canonical AIM record is `AIM-ALGEBRAIC_GEOMETRY-0133` in `aim-algebraic-geometry-notes.json`, zero-based source index 132 (Positivity of Cycles, problem 1.2). It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 132,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0134": {
   "statement_status": "exact",
   "original_statement": "If $\\mathcal{E}\\cong \\mathcal{L}_1 \\oplus ... \\oplus \\mathcal{L}_r$ is a split vector bundle on a smooth projective surface $S$, what are the effective cones of k-cycles $\\text{Eff}_k(\\mathbb{P}(\\mathcal{E}))$?",
   "clean_statement": "If $\\mathcal{E}\\cong \\mathcal{L}_1 \\oplus ... \\oplus \\mathcal{L}_r$ is a split vector bundle on a smooth projective surface $S$, what are the effective cones of k-cycles $\\text{Eff}_k(\\mathbb{P}(\\mathcal{E}))$?",
   "public_statement": "If $\\mathcal{E}\\cong \\mathcal{L}_1 \\oplus ... \\oplus \\mathcal{L}_r$ is a split vector bundle on a smooth projective surface $S$, what are the effective cones of k-cycles $\\text{Eff}_k(\\mathbb{P}(\\mathcal{E}))$?",
   "evidence": "The canonical record is problem 1.3 in the section “Computing higher codimension effective and nef cones in explicit examples” from the 2016 AIM workshop *Positivity of cycles*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 133,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0135": {
   "statement_status": "exact",
   "original_statement": "Effective cones of hypersurfaces\n\nFor very basic varieties such as $Gr(m,n)$ or $\\mathbb{P}^n\\times \\mathbb{P}^m$ the effective cones of $k$-cycles can be computed. It then natural to ask if it is possible to compute the effective cones of very general hypersurfaces in these varieties.\n\nIf $X\\subset Gr(m,n)$ is a very general hypersurface, what is the effective cone of $k$-cycles $\\text{Eff}_k(K)$?",
   "clean_statement": "Effective cones of hypersurfaces\n\nFor very basic varieties such as $Gr(m,n)$ or $\\mathbb{P}^n\\times \\mathbb{P}^m$ the effective cones of $k$-cycles can be computed. It then natural to ask if it is possible to compute the effective cones of very general hypersurfaces in these varieties.\n\nIf $X\\subset Gr(m,n)$ is a very general hypersurface, what is the effective cone of $k$-cycles $\\text{Eff}_k(K)$?",
   "public_statement": "Effective cones of hypersurfaces\n\nFor very basic varieties such as $Gr(m,n)$ or $\\mathbb{P}^n\\times \\mathbb{P}^m$ the effective cones of $k$-cycles can be computed. It then natural to ask if it is possible to compute the effective cones of very general hypersurfaces in these varieties.\n\nIf $X\\subset Gr(m,n)$ is a very general hypersurface, what is the effective cone of $k$-cycles $\\text{Eff}_k(K)$?",
   "evidence": "The canonical record is AIM Problem List item 1.4 from the workshop *Positivity of cycles*. Its final sentence is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 134,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0136": {
   "statement_status": "exact",
   "original_statement": "If $X\\subset \\mathbb{P}^n\\times \\mathbb{P}^m$ is a very general hypersurface, what is the effective cone of $k$-cycles $\\text{Eff}_k(X)$?",
   "clean_statement": "If $X\\subset \\mathbb{P}^n\\times \\mathbb{P}^m$ is a very general hypersurface, what is the effective cone of $k$-cycles $\\text{Eff}_k(X)$?",
   "public_statement": "If $X\\subset \\mathbb{P}^n\\times \\mathbb{P}^m$ is a very general hypersurface, what is the effective cone of $k$-cycles $\\text{Eff}_k(X)$?",
   "evidence": "The canonical AIM record, problem 1.5 in the section “Computing higher codimension effective and nef cones in explicit examples,” asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 135,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0137": {
   "statement_status": "exact",
   "original_statement": "In the above cases, assuming $i:X\\hookrightarrow \\mathbb{P}^n\\times\\mathbb{P}^m$ or $i:X\\hookrightarrow Gr(k,n)$ is ample then the Lefschetz hyperplane theorems imply that $i_*:N_k(X) \\rightarrow N_k(Gr(k,n))$ and $i_*:N_k(X) \\rightarrow N_k(\\mathbb{P}^n\\times\\mathbb{P}^m)$ are isomorphisms for $k<\\text{dim}(X)/2$. It is natural to ask if there is any Lefschetz type phenomena for effective cones of very general hypersurfaces of sufficiently high degree.\n\nIf $X\\subset Y$ is a very general ample divisor of sufficiently high degree, does the pushforward map $i_*: N_k(X) \\rightarrow N_k(Y)$ induce an isomorphism of effective cones for $k$ sufficiently small?",
   "clean_statement": "In the above cases, assuming $i:X\\hookrightarrow \\mathbb{P}^n\\times\\mathbb{P}^m$ or $i:X\\hookrightarrow Gr(k,n)$ is ample then the Lefschetz hyperplane theorems imply that $i_*:N_k(X) \\rightarrow N_k(Gr(k,n))$ and $i_*:N_k(X) \\rightarrow N_k(\\mathbb{P}^n\\times\\mathbb{P}^m)$ are isomorphisms for $k<\\text{dim}(X)/2$. It is natural to ask if there is any Lefschetz type phenomena for effective cones of very general hypersurfaces of sufficiently high degree.\n\nIf $X\\subset Y$ is a very general ample divisor of sufficiently high degree, does the pushforward map $i_*: N_k(X) \\rightarrow N_k(Y)$ induce an isomorphism of effective cones for $k$ sufficiently small?",
   "public_statement": "In the above cases, assuming $i:X\\hookrightarrow \\mathbb{P}^n\\times\\mathbb{P}^m$ or $i:X\\hookrightarrow Gr(k,n)$ is ample then the Lefschetz hyperplane theorems imply that $i_*:N_k(X) \\rightarrow N_k(Gr(k,n))$ and $i_*:N_k(X) \\rightarrow N_k(\\mathbb{P}^n\\times\\mathbb{P}^m)$ are isomorphisms for $k<\\text{dim}(X)/2$. It is natural to ask if there is any Lefschetz type phenomena for effective cones of very general hypersurfaces of sufficiently high degree.\n\nIf $X\\subset Y$ is a very general ample divisor of sufficiently high degree, does the pushforward map $i_*: N_k(X) \\rightarrow N_k(Y)$ induce an isomorphism of effective cones for $k$ sufficiently small?",
   "evidence": "The canonical record is item 1.6 in the AIM workshop list *Positivity of cycles*, section “Computing higher codimension effective and nef cones in explicit examples.” The exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 136,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0138": {
   "statement_status": "exact",
   "original_statement": "Effective cones of Hilbert schemes and Moduli Spaces\n\nIt is natural to try to compute effective cones in the case of Hilbert schemes or moduli spaces. One natural Hilbert scheme to consider first is the Hilbert scheme of length $n$ subscheme of a smooth surface $S$, denoted by $S^{[n]}$. A great deal is known about these Hilbert schemes. In recent years there has been a lot of exciting work on the effective cones of divisors and curves of the Hilbert schemes $\\text{Eff}^1(S^{[n]})$ and $\\text{Eff}_1(S^{[n]})$. See for example Bayer and Macri's work on Hilbert schemes of K3 surfaces \\cite{MR3279532} or Huizenga's work on Hilbert schemes of $\\mathbb{P}^2$ \\cite{MR3419956}. It is then natural to ask how to compute the effective cone of surfaces or the codimension 2 effective cones.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(S^{[n]})$? What is the codimension 2 effective cone $\\text{Eff}^2(S^{[n]})$? What about higher dimension/codimension?",
   "clean_statement": "Effective cones of Hilbert schemes and Moduli Spaces\n\nIt is natural to try to compute effective cones in the case of Hilbert schemes or moduli spaces. One natural Hilbert scheme to consider first is the Hilbert scheme of length $n$ subscheme of a smooth surface $S$, denoted by $S^{[n]}$. A great deal is known about these Hilbert schemes. In recent years there has been a lot of exciting work on the effective cones of divisors and curves of the Hilbert schemes $\\text{Eff}^1(S^{[n]})$ and $\\text{Eff}_1(S^{[n]})$. See for example Bayer and Macri's work on Hilbert schemes of K3 surfaces \\cite{MR3279532} or Huizenga's work on Hilbert schemes of $\\mathbb{P}^2$ \\cite{MR3419956}. It is then natural to ask how to compute the effective cone of surfaces or the codimension 2 effective cones.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(S^{[n]})$? What is the codimension 2 effective cone $\\text{Eff}^2(S^{[n]})$? What about higher dimension/codimension?",
   "public_statement": "Effective cones of Hilbert schemes and Moduli Spaces\n\nIt is natural to try to compute effective cones in the case of Hilbert schemes or moduli spaces. One natural Hilbert scheme to consider first is the Hilbert scheme of length $n$ subscheme of a smooth surface $S$, denoted by $S^{[n]}$. A great deal is known about these Hilbert schemes. In recent years there has been a lot of exciting work on the effective cones of divisors and curves of the Hilbert schemes $\\text{Eff}^1(S^{[n]})$ and $\\text{Eff}_1(S^{[n]})$. See for example Bayer and Macri's work on Hilbert schemes of K3 surfaces \\cite{MR3279532} or Huizenga's work on Hilbert schemes of $\\mathbb{P}^2$ \\cite{MR3419956}. It is then natural to ask how to compute the effective cone of surfaces or the codimension 2 effective cones.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(S^{[n]})$? What is the codimension 2 effective cone $\\text{Eff}^2(S^{[n]})$? What about higher dimension/codimension?",
   "evidence": "There is no apparent OCR corruption. There are, however, two important convention issues in the question:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 137,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0139": {
   "statement_status": "exact",
   "original_statement": "Historically there has also been fervent interest in computing the effective cones of divisors and curves in the case of the moduli space of genus $g$ curves $\\overline{\\mathcal{M}}_g$ or their pointed versions $\\overline{\\mathcal{M}}_{g,n}$. In recent years for example there has been a great deal of work on the so called F-conjecture which predicts the effective cone of curves in $\\overline{\\mathcal{M}}_{0,n}$ is generated by explicit curve classes (so called \"F-curves\"), see e.g. Gibney's work \\cite{MR2551995}. It is then natural to ask about higher dimensional effective cones for small values of $g$ and $n$. In the pointed case, to make the problem even simpler one can ask about the symmetrized space $\\overline{\\mathcal{M}}_{g,n}^{\\mathfrak{S}_n}:=\\overline{\\mathcal{M}}_{g,n}/{\\mathfrak{S}_n}$.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(\\overline{\\mathcal{M}}_{0,7}^{\\mathfrak{S}_7})$?",
   "clean_statement": "Historically there has also been fervent interest in computing the effective cones of divisors and curves in the case of the moduli space of genus $g$ curves $\\overline{\\mathcal{M}}_g$ or their pointed versions $\\overline{\\mathcal{M}}_{g,n}$. In recent years for example there has been a great deal of work on the so called F-conjecture which predicts the effective cone of curves in $\\overline{\\mathcal{M}}_{0,n}$ is generated by explicit curve classes (so called \"F-curves\"), see e.g. Gibney's work \\cite{MR2551995}. It is then natural to ask about higher dimensional effective cones for small values of $g$ and $n$. In the pointed case, to make the problem even simpler one can ask about the symmetrized space $\\overline{\\mathcal{M}}_{g,n}^{\\mathfrak{S}_n}:=\\overline{\\mathcal{M}}_{g,n}/{\\mathfrak{S}_n}$.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(\\overline{\\mathcal{M}}_{0,7}^{\\mathfrak{S}_7})$?",
   "public_statement": "Historically there has also been fervent interest in computing the effective cones of divisors and curves in the case of the moduli space of genus $g$ curves $\\overline{\\mathcal{M}}_g$ or their pointed versions $\\overline{\\mathcal{M}}_{g,n}$. In recent years for example there has been a great deal of work on the so called F-conjecture which predicts the effective cone of curves in $\\overline{\\mathcal{M}}_{0,n}$ is generated by explicit curve classes (so called \"F-curves\"), see e.g. Gibney's work \\cite{MR2551995}. It is then natural to ask about higher dimensional effective cones for small values of $g$ and $n$. In the pointed case, to make the problem even simpler one can ask about the symmetrized space $\\overline{\\mathcal{M}}_{g,n}^{\\mathfrak{S}_n}:=\\overline{\\mathcal{M}}_{g,n}/{\\mathfrak{S}_n}$.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(\\overline{\\mathcal{M}}_{0,7}^{\\mathfrak{S}_7})$?",
   "evidence": "The AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 138,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0140": {
   "statement_status": "exact",
   "original_statement": "Effective cones of blowups\n\nLastly, it is natural to ask what the relationship is between effective cones of a variety and its blowup. This is a deceptively difficult question already in the case of the effective cone of curves and the blowup of 10 points in $\\mathbb{P}^2$, which is the setting of Nagata's conjecture. However, the question of computing effective cones of blowups is quite possibly more tractable when the center of the blowup is a positive dimensional subvariety of $\\mathbb{P}^n$.\n\nLet $Z\\subset \\mathbb{P}^n$ be a smooth subvariety with $1\\le \\text{dim}Z \\le n-2$. What is the effective cone of $k$-cycles $\\text{Eff}_k(X)$?",
   "clean_statement": "Effective cones of blowups\n\nLastly, it is natural to ask what the relationship is between effective cones of a variety and its blowup. This is a deceptively difficult question already in the case of the effective cone of curves and the blowup of 10 points in $\\mathbb{P}^2$, which is the setting of Nagata's conjecture. However, the question of computing effective cones of blowups is quite possibly more tractable when the center of the blowup is a positive dimensional subvariety of $\\mathbb{P}^n$.\n\nLet $Z\\subset \\mathbb{P}^n$ be a smooth subvariety with $1\\le \\text{dim}Z \\le n-2$. What is the effective cone of $k$-cycles $\\text{Eff}_k(X)$?",
   "public_statement": "Effective cones of blowups\n\nLastly, it is natural to ask what the relationship is between effective cones of a variety and its blowup. This is a deceptively difficult question already in the case of the effective cone of curves and the blowup of 10 points in $\\mathbb{P}^2$, which is the setting of Nagata's conjecture. However, the question of computing effective cones of blowups is quite possibly more tractable when the center of the blowup is a positive dimensional subvariety of $\\mathbb{P}^n$.\n\nLet $Z\\subset \\mathbb{P}^n$ be a smooth subvariety with $1\\le \\text{dim}Z \\le n-2$. What is the effective cone of $k$-cycles $\\text{Eff}_k(X)$?",
   "evidence": "The canonical record is source index 139 of `aim-algebraic-geometry-notes.json`: Problem 1.9, “Effective cones of blowups,” from the AIM workshop *Positivity of cycles*. Its mathematical prompt is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 139,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0141": {
   "statement_status": "corrected_verified",
   "original_statement": "Mobility and mobility counts\n\nLet $X$ be an $n$ dimensional variety. Given an effective integral $k$-cycle $\\alpha\\in N_k(X)_\\mathbb{Z}$, the \\textit{mobility count of} $\\alpha$, denoted $mc(\\alpha)$, is the maximum number of general points in $X$ that we can impose on an effective cycle of class $\\alpha$. For example, any two points in $\\mathbb{P}^2$ can be connected by a line $\\ell \\subset \\mathbb{P}^2,$ so we see $mc([\\ell])\\ge 2$.\n\nThe mobility count is supposed to be an analogue of $\\text{dim}(H^0(X,\\mathcal{O}(E)))$. Taking a cue from divisor theory it is natural to consider asymptotic invariants of a numerical cycle. Define the \\textit{mobility} of a numerical cycle $\\alpha \\in N_k(X)_\\mathbb{Z}$ class to be $$\\text{mob}(\\alpha):= \\frac{n! mc(m\\cdot\\alpha)}{m^{n/(n-k)}}.$$ Define the \\textit{Iitaka dimension of} $\\alpha$ of a numerical cycle to be $$K(\\alpha)=\\text{max}\\{ r \\in\\mathbb{R} | \\text{limsup}_{m\\rightarrow \\infty} \\frac{mc(m\\cdot \\alpha)}{m^r}>0 \\}.$$\n\nLet $[\\ell]\\in N_1(\\mathbb{P}^3)_\\mathbb{Z}$ be the class of a line. What is $\\text{mob}([\\alpha])$?",
   "clean_statement": "Over \\(\\mathbb C\\), let \\(\\ell\\) be the numerical class of a line in\n\\(\\mathbb P^3\\). Determine\n\\[\n\\operatorname{mob}(\\ell)\n=\\limsup_{m\\to\\infty}\\frac{6\\,mc(m\\ell)}{m^{3/2}}.\n\\]",
   "public_statement": "Over \\(\\mathbb C\\), let \\(\\ell\\) be the numerical class of a line in\n\\(\\mathbb P^3\\). Determine\n\\[\n\\operatorname{mob}(\\ell)\n=\\limsup_{m\\to\\infty}\\frac{6\\,mc(m\\ell)}{m^{3/2}}.\n\\]",
   "evidence": "There are two extraction defects and one later change of convention. 1. The displayed mobility formula in the record has no limiting operation. The standard definition in [Leh16, Definition 1.1] is \\[ \\operatorname{mob}(\\alpha) := \\limsup_{m\\to\\infty} \\frac{mc(m\\alpha)} {m^{\\,n/(n-k)}/n!} = \\limsup_{m\\to\\infty} \\frac{n!\\,mc(m\\alpha)}{m^{\\,n/(n-k)}}. \\tag{1.1} \\] The denominator \\(m^{n/(n-k)}/n!\\) is the standard normalization. 2. The last \\([\\alpha]\\) is reconstructed as \\([\\ell]\\). This is confirmed by the fuller AIM problem-list transcript, which asks for the mobility when \\(\\alpha\\) is the line class on \\(\\mathbb P^3\\). 3. The AIM record calls the unrescaled growth exponent \\[ K_{\\mathrm{src}}(\\alpha) := \\sup\\left\\{r\\geq0: \\limsup_{m\\to\\infty}\\frac{mc(m\\alpha)}{m^r}>0 \\right\\}. \\tag{1.2} \\] Lehmann's later convention [Leh19] multiplies this exponent by the codimension: \\[ \\kappa(\\a...",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 140,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0142": {
   "statement_status": "exact",
   "original_statement": "What is $K(\\alpha)$ for $\\alpha \\in N_k(\\text{Gr}(k,n))$ a Schubert cycle?",
   "clean_statement": "What is $K(\\alpha)$ for $\\alpha \\in N_k(\\text{Gr}(k,n))$ a Schubert cycle?",
   "public_statement": "What is $K(\\alpha)$ for $\\alpha \\in N_k(\\text{Gr}(k,n))$ a Schubert cycle?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 141,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0143": {
   "statement_status": "exact",
   "original_statement": "Develop better estimates for the growth rate of $mc(\\alpha)$ for $\\alpha$ a numerical class on $\\text{Gr}(2,4)$ or $\\mathbb{P}^2\\times \\mathbb{P}^2$.",
   "clean_statement": "Develop better estimates for the growth rate of $mc(\\alpha)$ for $\\alpha$ a numerical class on $\\text{Gr}(2,4)$ or $\\mathbb{P}^2\\times \\mathbb{P}^2$.",
   "public_statement": "Develop better estimates for the growth rate of $mc(\\alpha)$ for $\\alpha$ a numerical class on $\\text{Gr}(2,4)$ or $\\mathbb{P}^2\\times \\mathbb{P}^2$.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 142,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0144": {
   "statement_status": "exact",
   "original_statement": "Is $K(\\alpha)$ an integer?",
   "clean_statement": "Is $K(\\alpha)$ an integer?",
   "public_statement": "Is $K(\\alpha)$ an integer?",
   "evidence": "The exact canonical record from the 2016 AIM workshop *Positivity of cycles* asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 143,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0145": {
   "statement_status": "exact",
   "original_statement": "Other measures of positivity\n\nLet $X$ be a projective variety. Suppose that there is a class $\\alpha \\in \\text{Eff}_k(X)_{\\mathbb{Z}}$ such that for a general point $x\\in X$ and a general $k$-plane $V\\subset T_x X$ there is an irreducible subvariety $Y\\subset X$ with $[V]= \\alpha$ and $T_xY = V$.\n\nIs $\\alpha$ in the interior of the effective cone?",
   "clean_statement": "Other measures of positivity\n\nLet $X$ be a projective variety. Suppose that there is a class $\\alpha \\in \\text{Eff}_k(X)_{\\mathbb{Z}}$ such that for a general point $x\\in X$ and a general $k$-plane $V\\subset T_x X$ there is an irreducible subvariety $Y\\subset X$ with $[V]= \\alpha$ and $T_xY = V$.\n\nIs $\\alpha$ in the interior of the effective cone?",
   "public_statement": "Other measures of positivity\n\nLet $X$ be a projective variety. Suppose that there is a class $\\alpha \\in \\text{Eff}_k(X)_{\\mathbb{Z}}$ such that for a general point $x\\in X$ and a general $k$-plane $V\\subset T_x X$ there is an irreducible subvariety $Y\\subset X$ with $[V]= \\alpha$ and $T_xY = V$.\n\nIs $\\alpha$ in the interior of the effective cone?",
   "evidence": "The live AIM page, problem 2.5 in *Positivity of cycles*, attributes the question to Claire Voisin and currently reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 144,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0146": {
   "statement_status": "exact",
   "original_statement": "In \\cite{MR3406512}, Blumberg and Hill define\n $N_{\\infty}$-operads, equivariant generalizations of\n $E_{\\infty}$-operads.\n\nWhat is the motivic analogue of an\n $N_{\\infty}$-operad or in other words, what are the analogues of\n $G$-commutative ring spectra in motivic homotopy theory?",
   "clean_statement": "In \\cite{MR3406512}, Blumberg and Hill define\n $N_{\\infty}$-operads, equivariant generalizations of\n $E_{\\infty}$-operads.\n\nWhat is the motivic analogue of an\n $N_{\\infty}$-operad or in other words, what are the analogues of\n $G$-commutative ring spectra in motivic homotopy theory?",
   "public_statement": "In \\cite{MR3406512}, Blumberg and Hill define\n $N_{\\infty}$-operads, equivariant generalizations of\n $E_{\\infty}$-operads.\n\nWhat is the motivic analogue of an\n $N_{\\infty}$-operad or in other words, what are the analogues of\n $G$-commutative ring spectra in motivic homotopy theory?",
   "evidence": "The canonical record is problem 1.1 in the “Motivic Homotopy Theory” section of the AIM problem list *Equivariant derived algebraic geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 145,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0147": {
   "statement_status": "exact",
   "original_statement": "Can we construct an Eilenberg-Moore spectral sequence in motivic homotopy theory?",
   "clean_statement": "Can we construct an Eilenberg-Moore spectral sequence in motivic homotopy theory?",
   "public_statement": "Can we construct an Eilenberg-Moore spectral sequence in motivic homotopy theory?",
   "evidence": "The canonical record is problem 1.2 in the “Motivic Homotopy Theory” section of the AIM problem list *Equivariant derived algebraic geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 146,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0148": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Can we construct a Serre spectral sequence in motivic homotopy theory?",
   "clean_statement": null,
   "public_statement": "Can we construct a Serre spectral sequence in motivic homotopy theory?",
   "evidence": "The canonical record is AIM-ALGEBRAIC_GEOMETRY-0148, source file `aim-algebraic-geometry-notes.json`, zero-based source index 147, from the AIM workshop *Equivariant derived algebraic geometry*, section *Motivic Homotopy Theory*, problem 1.3. Its exact problem text is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 147,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0149": {
   "statement_status": "exact",
   "original_statement": "Long Term\n\nDo we have a motivic description akin to adding transfers?",
   "clean_statement": "Long Term\n\nDo we have a motivic description akin to adding transfers?",
   "public_statement": "Long Term\n\nDo we have a motivic description akin to adding transfers?",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 148,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0150": {
   "statement_status": "exact",
   "original_statement": "In classical homotopy theory, there are, among many others, three important notions/results:\n\n(1) The Freudenthal suspension theorem\n\n(2) The Barratt-Priddy-Quillen theorem\n\n(3) Infinite loop space theory\n\nCan we construct/prove the above notions in the context of motivic homotopy theory?",
   "clean_statement": "In classical homotopy theory, there are, among many others, three important notions/results:\n\n(1) The Freudenthal suspension theorem\n\n(2) The Barratt-Priddy-Quillen theorem\n\n(3) Infinite loop space theory\n\nCan we construct/prove the above notions in the context of motivic homotopy theory?",
   "public_statement": "In classical homotopy theory, there are, among many others, three important notions/results:\n\n(1) The Freudenthal suspension theorem\n\n(2) The Barratt-Priddy-Quillen theorem\n\n(3) Infinite loop space theory\n\nCan we construct/prove the above notions in the context of motivic homotopy theory?",
   "evidence": "The canonical record is number 1.5 in the “Motivic Homotopy Theory” section of the AIM workshop list *Equivariant derived algebraic geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 149,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0151": {
   "statement_status": "exact",
   "original_statement": "What can we say about the motivic Picard group?",
   "clean_statement": "What can we say about the motivic Picard group?",
   "public_statement": "What can we say about the motivic Picard group?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 150,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0152": {
   "statement_status": "exact",
   "original_statement": "What more can we say about the tensor triangular geometry of the motivic stable homotopy category? (See \\cite{arxiv:1608.02876} for a list of questions.)",
   "clean_statement": "What more can we say about the tensor triangular geometry of the motivic stable homotopy category? (See \\cite{arxiv:1608.02876} for a list of questions.)",
   "public_statement": "What more can we say about the tensor triangular geometry of the motivic stable homotopy category? (See \\cite{arxiv:1608.02876} for a list of questions.)",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 151,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0153": {
   "statement_status": "exact",
   "original_statement": "When $G = C_p$, any genuine $G$-spectrum can be described via gluing data of the form $(X,Y, X\\rightarrow Y^{tG})$ where $X$ is a spectrum, $Y$ is a Borel spectrum, and $Y^{tG}$ denotes the Tate spectrum. An analogous pasting construction can be used to describe genuine $G$-spectra for any finite group $G$.\n\nWhat is the description of $Comm_G(Sp^G)$ using the above model?",
   "clean_statement": "When $G = C_p$, any genuine $G$-spectrum can be described via gluing data of the form $(X,Y, X\\rightarrow Y^{tG})$ where $X$ is a spectrum, $Y$ is a Borel spectrum, and $Y^{tG}$ denotes the Tate spectrum. An analogous pasting construction can be used to describe genuine $G$-spectra for any finite group $G$.\n\nWhat is the description of $Comm_G(Sp^G)$ using the above model?",
   "public_statement": "When $G = C_p$, any genuine $G$-spectrum can be described via gluing data of the form $(X,Y, X\\rightarrow Y^{tG})$ where $X$ is a spectrum, $Y$ is a Borel spectrum, and $Y^{tG}$ denotes the Tate spectrum. An analogous pasting construction can be used to describe genuine $G$-spectra for any finite group $G$.\n\nWhat is the description of $Comm_G(Sp^G)$ using the above model?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 152,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0154": {
   "statement_status": "exact",
   "original_statement": "In \\cite{MR1410465}, Hesselholt and Madsen show\n that for any commutative ring $R$,\n\\[\\pi_0 THH(HR)^{C_{p^n}} \\cong W_{n+1}(R),\\]\nthe $p$-typical Witt vectors of length $n+1$. Furthermore, in \\cite{1401.5001v2}, Angeltveit,\n et. al. describe a construction of the cyclotomic structure on THH\n using $N^{S^1}_e$, an extension of the Hill-Hopkins-Ravenel\n multiplicative norm.\n\nIf $\\underline{R}$ is a commutative Green functor, what is a Witt-vector\nmodel for $\\pi_0(N^{S^1}_{C_p} H\\underline{R})^{C_{p^n}}$?",
   "clean_statement": "In \\cite{MR1410465}, Hesselholt and Madsen show\n that for any commutative ring $R$,\n\\[\\pi_0 THH(HR)^{C_{p^n}} \\cong W_{n+1}(R),\\]\nthe $p$-typical Witt vectors of length $n+1$. Furthermore, in \\cite{1401.5001v2}, Angeltveit,\n et. al. describe a construction of the cyclotomic structure on THH\n using $N^{S^1}_e$, an extension of the Hill-Hopkins-Ravenel\n multiplicative norm.\n\nIf $\\underline{R}$ is a commutative Green functor, what is a Witt-vector\nmodel for $\\pi_0(N^{S^1}_{C_p} H\\underline{R})^{C_{p^n}}$?",
   "public_statement": "In \\cite{MR1410465}, Hesselholt and Madsen show\n that for any commutative ring $R$,\n\\[\\pi_0 THH(HR)^{C_{p^n}} \\cong W_{n+1}(R),\\]\nthe $p$-typical Witt vectors of length $n+1$. Furthermore, in \\cite{1401.5001v2}, Angeltveit,\n et. al. describe a construction of the cyclotomic structure on THH\n using $N^{S^1}_e$, an extension of the Hill-Hopkins-Ravenel\n multiplicative norm.\n\nIf $\\underline{R}$ is a commutative Green functor, what is a Witt-vector\nmodel for $\\pi_0(N^{S^1}_{C_p} H\\underline{R})^{C_{p^n}}$?",
   "evidence": "This is Problem 2.2 in the AIM list *Equivariant derived algebraic geometry*, section “Equivariant Stable Homotopy Theory,” attributed to Lars Hesselholt. The canonical record is `aim-algebraic-geometry-notes.json`, index 153. The live AIM page was also inspected and agrees with the record (apart from harmless missing parentheses in the displayed fixed-point notation).",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 153,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0155": {
   "statement_status": "exact",
   "original_statement": "In order to solve the $3$-primary Kervaire\n invariant one problem using analogous methods to those in\n \\cite{0908.3724}, one would need a $C_3$-equivariant version of\n $MU_{\\R}$.\n\nCan we describe such a $C_3$-commutative ring spectrum $MU_{\\R}$ so that\nits underlying Bousfield type is $MU$ and $\\langle MU^{\\Phi\n C_3} \\rangle = \\langle H\\F_3 \\rangle$?",
   "clean_statement": "In order to solve the $3$-primary Kervaire\n invariant one problem using analogous methods to those in\n \\cite{0908.3724}, one would need a $C_3$-equivariant version of\n $MU_{\\R}$.\n\nCan we describe such a $C_3$-commutative ring spectrum $MU_{\\R}$ so that\nits underlying Bousfield type is $MU$ and $\\langle MU^{\\Phi\n C_3} \\rangle = \\langle H\\F_3 \\rangle$?",
   "public_statement": "In order to solve the $3$-primary Kervaire\n invariant one problem using analogous methods to those in\n \\cite{0908.3724}, one would need a $C_3$-equivariant version of\n $MU_{\\R}$.\n\nCan we describe such a $C_3$-commutative ring spectrum $MU_{\\R}$ so that\nits underlying Bousfield type is $MU$ and $\\langle MU^{\\Phi\n C_3} \\rangle = \\langle H\\F_3 \\rangle$?",
   "evidence": "This is Problem 2.3, attributed to D. Ravenel, in the AIM list *Equivariant derived algebraic geometry*, section “Equivariant Stable Homotopy Theory.” The source says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 154,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0156": {
   "statement_status": "exact",
   "original_statement": "What should the correct notion of a Real spectrum be? Is it ``more\" than a genuine $C_2$-spectrum? (see \\cite{0908.3724} Sec B12)",
   "clean_statement": "What should the correct notion of a Real spectrum be? Is it ``more\" than a genuine $C_2$-spectrum? (see \\cite{0908.3724} Sec B12)",
   "public_statement": "What should the correct notion of a Real spectrum be? Is it ``more\" than a genuine $C_2$-spectrum? (see \\cite{0908.3724} Sec B12)",
   "evidence": "The exact canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 155,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0157": {
   "statement_status": "exact",
   "original_statement": "Can we understand the thick subcategory of $Sp^G$ generated by representation spheres?",
   "clean_statement": "Can we understand the thick subcategory of $Sp^G$ generated by representation spheres?",
   "public_statement": "Can we understand the thick subcategory of $Sp^G$ generated by representation spheres?",
   "evidence": "The canonical record is AIM Problem List 2.5 in the section “Equivariant Stable Homotopy Theory” of the workshop *Equivariant derived algebraic geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 156,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0158": {
   "statement_status": "exact",
   "original_statement": "In \\cite{MR1337494}, Thomason showed that\n connective spectra can be modeled by symmetric monoidal categories.\n\nIs a similar statement true equivariantly? That is, is there an\nequivalence between $G$-symmetric monoidal categories and the homotopy\ncategory of connective $G$-spectra?",
   "clean_statement": "In \\cite{MR1337494}, Thomason showed that\n connective spectra can be modeled by symmetric monoidal categories.\n\nIs a similar statement true equivariantly? That is, is there an\nequivalence between $G$-symmetric monoidal categories and the homotopy\ncategory of connective $G$-spectra?",
   "public_statement": "In \\cite{MR1337494}, Thomason showed that\n connective spectra can be modeled by symmetric monoidal categories.\n\nIs a similar statement true equivariantly? That is, is there an\nequivalence between $G$-symmetric monoidal categories and the homotopy\ncategory of connective $G$-spectra?",
   "evidence": "The canonical record is Problem 2.6 in the “Equivariant Stable Homotopy Theory” section of the AIM workshop *Equivariant derived algebraic geometry* (June 13--17, 2016). Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 157,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0159": {
   "statement_status": "exact",
   "original_statement": "Long Term\n\nCan we construct a decompleted version of the Tate spectrum?",
   "clean_statement": "Long Term\n\nCan we construct a decompleted version of the Tate spectrum?",
   "public_statement": "Long Term\n\nCan we construct a decompleted version of the Tate spectrum?",
   "evidence": "The canonical AIM record is Problem 2.7 in the “Equivariant Stable Homotopy Theory” section of the June 2016 workshop *Equivariant derived algebraic geometry*. Its exact problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 158,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0160": {
   "statement_status": "exact",
   "original_statement": "In \\cite{MR2061856} Morel showed that Milnor-Witt K-theory of a field is isomorphic to the graded endomorphism ring of the unit object in the stable homotopy category of motivic spectra over that field.\n\nCan we prove a similar result in the setting of equivariant spectra? This would aid in understanding the tensor triangular geometry of $Sp^G$.",
   "clean_statement": "In \\cite{MR2061856} Morel showed that Milnor-Witt K-theory of a field is isomorphic to the graded endomorphism ring of the unit object in the stable homotopy category of motivic spectra over that field.\n\nCan we prove a similar result in the setting of equivariant spectra? This would aid in understanding the tensor triangular geometry of $Sp^G$.",
   "public_statement": "In \\cite{MR2061856} Morel showed that Milnor-Witt K-theory of a field is isomorphic to the graded endomorphism ring of the unit object in the stable homotopy category of motivic spectra over that field.\n\nCan we prove a similar result in the setting of equivariant spectra? This would aid in understanding the tensor triangular geometry of $Sp^G$.",
   "evidence": "This is Problem 2.9 in the “Equivariant Stable Homotopy Theory” section of the AIM workshop list *Equivariant derived algebraic geometry*. The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 159,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0161": {
   "statement_status": "exact",
   "original_statement": "Can we get a better understanding of the tensor triangular geometry of $Sp^G$? (See Problem 1.6)",
   "clean_statement": "Can we get a better understanding of the tensor triangular geometry of $Sp^G$? (See Problem 1.6)",
   "public_statement": "Can we get a better understanding of the tensor triangular geometry of $Sp^G$? (See Problem 1.6)",
   "evidence": "The canonical AIM record is Problem 2.8 from the workshop *Equivariant derived algebraic geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 160,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0162": {
   "statement_status": "exact",
   "original_statement": "What is an {\\'etale} map of Green or Tambara functors?",
   "clean_statement": "What is an {\\'etale} map of Green or Tambara functors?",
   "public_statement": "What is an {\\'etale} map of Green or Tambara functors?",
   "evidence": "The canonical record is AIM Problem List item 3.1 from the workshop *Equivariant derived algebraic geometry*, in the section “Connecting Equivariant Notions and Derived Algebraic Geometry.” Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 161,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0163": {
   "statement_status": "exact",
   "original_statement": "What are the explicit generators of $Pic(Sp^G)$?",
   "clean_statement": "What are the explicit generators of $Pic(Sp^G)$?",
   "public_statement": "What are the explicit generators of $Pic(Sp^G)$?",
   "evidence": "The canonical record is problem 3.2 in the workshop *Equivariant derived algebraic geometry*, section “Connecting Equivariant Notions and Derived Algebraic Geometry.” Its exact mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 162,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0164": {
   "statement_status": "exact",
   "original_statement": "Wall, Roberts, Damon, Gusein-Zade, and others have defined and studied the local $G$-degree for an equivariant polynomial map $f: \\R^n \\rightarrow \\R^n$ where $\\R^n$ is given an action of a group $G$. Their work is motivated by singularity theory, and they show that the invariants encode interesting information, but equivariant homotopy theory does not appear explicitly. Perhaps the notions of degree appearing in these works are connected to the degree in equivariant homotopy theory. We pose several questions along these lines.\n\n(1) Suppose we are given a linear representation of a finite group $G$ on $\\C^n$ and a $G$-equivariant polynomial function $f : \\C^n \\rightarrow \\C^n$ with an isolated zero at the origin. Then the local algebra\n\\[ Q(f) := \\C [\\![ x_1, \\dots, x_n ]\\!]/(f_1, \\dots, f_n)\\]\nof $f$ is a $\\C$-algebra with $G$-action. Is this $G$-representation equal to the permutation representation associated to the local degree in equivariant homotopy theory?\n\nRecall the local degree is an element of the Burnside ring, hence has an associated virtual permutation representation. Palamodov has shown that the rank of $Q(f)$ is the topological local degree.\n\n(2) Suppose that, in (1), we replace the complex numbers $\\C$ with the real numbers $\\R$. Then $Q(f)$ carries a distinguished G-invariant symmetric bilinear pairing $\\beta$. (See e.g. Eisenbud-Levine.)\n\nQuite generally, any finite dimensional real $G$-representation with $G$-invariant symmetric bilinear form $(V, \\beta)$ decomposes in an essentially unique manner as an orthogonal direct sum\n\\[ V = V_+ \\oplus V_- \\oplus V_0\\]\nwhere $\\beta$ is positive definite on $V_+$, negative definite on $V_-$, and zero on $V_0$. Call the virtual representation $[V_+] - [V_-]$ the $G$-signature of $V$. Is this virtual $G$-representation of $( Q(f), \\beta)$ equal to the permutation representation associated to the local degree in equivariant homotopy theory?\n\nNote: Eisenbud-Levine and Khimshiashvili have proven that the signature of $\\beta$ equals the topological local degree of $f$.\n\n(3) Suppose now that $f : \\C^2 \\rightarrow \\C^2$ is the gradient of a $G$-invariant equation for a plane curve singularity $0 \\in X \\subset \\C^2$. Then $G$ acts on the analytic branches of the singularity. Is this $G$-set equal to the local degree of $f$ in equivariant homotopy theory?",
   "clean_statement": "Wall, Roberts, Damon, Gusein-Zade, and others have defined and studied the local $G$-degree for an equivariant polynomial map $f: \\R^n \\rightarrow \\R^n$ where $\\R^n$ is given an action of a group $G$. Their work is motivated by singularity theory, and they show that the invariants encode interesting information, but equivariant homotopy theory does not appear explicitly. Perhaps the notions of degree appearing in these works are connected to the degree in equivariant homotopy theory. We pose several questions along these lines.\n\n(1) Suppose we are given a linear representation of a finite group $G$ on $\\C^n$ and a $G$-equivariant polynomial function $f : \\C^n \\rightarrow \\C^n$ with an isolated zero at the origin. Then the local algebra\n\\[ Q(f) := \\C [\\![ x_1, \\dots, x_n ]\\!]/(f_1, \\dots, f_n)\\]\nof $f$ is a $\\C$-algebra with $G$-action. Is this $G$-representation equal to the permutation representation associated to the local degree in equivariant homotopy theory?\n\nRecall the local degree is an element of the Burnside ring, hence has an associated virtual permutation representation. Palamodov has shown that the rank of $Q(f)$ is the topological local degree.\n\n(2) Suppose that, in (1), we replace the complex numbers $\\C$ with the real numbers $\\R$. Then $Q(f)$ carries a distinguished G-invariant symmetric bilinear pairing $\\beta$. (See e.g. Eisenbud-Levine.)\n\nQuite generally, any finite dimensional real $G$-representation with $G$-invariant symmetric bilinear form $(V, \\beta)$ decomposes in an essentially unique manner as an orthogonal direct sum\n\\[ V = V_+ \\oplus V_- \\oplus V_0\\]\nwhere $\\beta$ is positive definite on $V_+$, negative definite on $V_-$, and zero on $V_0$. Call the virtual representation $[V_+] - [V_-]$ the $G$-signature of $V$. Is this virtual $G$-representation of $( Q(f), \\beta)$ equal to the permutation representation associated to the local degree in equivariant homotopy theory?\n\nNote: Eisenbud-Levine and Khimshiashvili have proven that the signature of $\\beta$ equals the topological local degree of $f$.\n\n(3) Suppose now that $f : \\C^2 \\rightarrow \\C^2$ is the gradient of a $G$-invariant equation for a plane curve singularity $0 \\in X \\subset \\C^2$. Then $G$ acts on the analytic branches of the singularity. Is this $G$-set equal to the local degree of $f$ in equivariant homotopy theory?",
   "public_statement": "Wall, Roberts, Damon, Gusein-Zade, and others have defined and studied the local $G$-degree for an equivariant polynomial map $f: \\R^n \\rightarrow \\R^n$ where $\\R^n$ is given an action of a group $G$. Their work is motivated by singularity theory, and they show that the invariants encode interesting information, but equivariant homotopy theory does not appear explicitly. Perhaps the notions of degree appearing in these works are connected to the degree in equivariant homotopy theory. We pose several questions along these lines.\n\n(1) Suppose we are given a linear representation of a finite group $G$ on $\\C^n$ and a $G$-equivariant polynomial function $f : \\C^n \\rightarrow \\C^n$ with an isolated zero at the origin. Then the local algebra\n\\[ Q(f) := \\C [\\![ x_1, \\dots, x_n ]\\!]/(f_1, \\dots, f_n)\\]\nof $f$ is a $\\C$-algebra with $G$-action. Is this $G$-representation equal to the permutation representation associated to the local degree in equivariant homotopy theory?\n\nRecall the local degree is an element of the Burnside ring, hence has an associated virtual permutation representation. Palamodov has shown that the rank of $Q(f)$ is the topological local degree.\n\n(2) Suppose that, in (1), we replace the complex numbers $\\C$ with the real numbers $\\R$. Then $Q(f)$ carries a distinguished G-invariant symmetric bilinear pairing $\\beta$. (See e.g. Eisenbud-Levine.)\n\nQuite generally, any finite dimensional real $G$-representation with $G$-invariant symmetric bilinear form $(V, \\beta)$ decomposes in an essentially unique manner as an orthogonal direct sum\n\\[ V = V_+ \\oplus V_- \\oplus V_0\\]\nwhere $\\beta$ is positive definite on $V_+$, negative definite on $V_-$, and zero on $V_0$. Call the virtual representation $[V_+] - [V_-]$ the $G$-signature of $V$. Is this virtual $G$-representation of $( Q(f), \\beta)$ equal to the permutation representation associated to the local degree in equivariant homotopy theory?\n\nNote: Eisenbud-Levine and Khimshiashvili have proven that the signature of $\\beta$ equals the topological local degree of $f$.\n\n(3) Suppose now that $f : \\C^2 \\rightarrow \\C^2$ is the gradient of a $G$-invariant equation for a plane curve singularity $0 \\in X \\subset \\C^2$. Then $G$ acts on the analytic branches of the singularity. Is this $G$-set equal to the local degree of $f$ in equivariant homotopy theory?",
   "evidence": "The canonical record is problem 3.3, “Connecting Equivariant Notions and Derived Algebraic Geometry,” from the AIM workshop *Equivariant derived algebraic geometry*. The record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 163,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0165": {
   "statement_status": "exact",
   "original_statement": "Long Term\n\nHow best should we be making sense of equivariant derived algebraic geometry? (e.g. What is Spec? What site to use? What sheaf? etc.)",
   "clean_statement": "Long Term\n\nHow best should we be making sense of equivariant derived algebraic geometry? (e.g. What is Spec? What site to use? What sheaf? etc.)",
   "public_statement": "Long Term\n\nHow best should we be making sense of equivariant derived algebraic geometry? (e.g. What is Spec? What site to use? What sheaf? etc.)",
   "evidence": "This is AIM Problem 3.4 in the workshop list *Equivariant derived algebraic geometry*, section “Connecting Equivariant Notions and Derived Algebraic Geometry.” The canonical record is at zero-based index 164 of `aim-algebraic-geometry-notes.json`. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 164,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0166": {
   "statement_status": "exact",
   "original_statement": "If $G$ is the Galois group of a finite extension $L\n \\rightarrow K$ there is a functor $Sp_G \\rightarrow Sp^{Mot}_K$\n given by $\\Sigma^{\\infty} G/H_+ \\rightarrow \\Sigma^{\\infty}\n Spec(L^H)_+$. Moreover, if $K$ is a real closed field, this functor\n is fully faithful.\n\nIf $G$ is a profinite absolute Galois group of $K$, is there a theory\nof $G-spectra$ such that $Sp_{G} \\rightarrow Sp^{Mot}_K$ is ``rich''?",
   "clean_statement": "If $G$ is the Galois group of a finite extension $L\n \\rightarrow K$ there is a functor $Sp_G \\rightarrow Sp^{Mot}_K$\n given by $\\Sigma^{\\infty} G/H_+ \\rightarrow \\Sigma^{\\infty}\n Spec(L^H)_+$. Moreover, if $K$ is a real closed field, this functor\n is fully faithful.\n\nIf $G$ is a profinite absolute Galois group of $K$, is there a theory\nof $G-spectra$ such that $Sp_{G} \\rightarrow Sp^{Mot}_K$ is ``rich''?",
   "public_statement": "If $G$ is the Galois group of a finite extension $L\n \\rightarrow K$ there is a functor $Sp_G \\rightarrow Sp^{Mot}_K$\n given by $\\Sigma^{\\infty} G/H_+ \\rightarrow \\Sigma^{\\infty}\n Spec(L^H)_+$. Moreover, if $K$ is a real closed field, this functor\n is fully faithful.\n\nIf $G$ is a profinite absolute Galois group of $K$, is there a theory\nof $G-spectra$ such that $Sp_{G} \\rightarrow Sp^{Mot}_K$ is ``rich''?",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 165,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0167": {
   "statement_status": "exact",
   "original_statement": "For a profinite group $G$ (having subgroups of arbitrary index), are there Wirthm\\\"uller/Adams isomomorphisms in $Sp^G$?",
   "clean_statement": "For a profinite group $G$ (having subgroups of arbitrary index), are there Wirthm\\\"uller/Adams isomomorphisms in $Sp^G$?",
   "public_statement": "For a profinite group $G$ (having subgroups of arbitrary index), are there Wirthm\\\"uller/Adams isomomorphisms in $Sp^G$?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 166,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0168": {
   "statement_status": "exact",
   "original_statement": "In genuine $G$-spectra we have multiplicative norm and\n transfer maps when $G$ has finitely indexed subgroups. Furthermore,\n (via Ando-Morava) there are objects that look like what may arise\n from a multiplicative norm for compact Lie groups.\n\nAre there analogues of multiplicative norms or multiplicative\ntransfers for compact Lie groups?",
   "clean_statement": "In genuine $G$-spectra we have multiplicative norm and\n transfer maps when $G$ has finitely indexed subgroups. Furthermore,\n (via Ando-Morava) there are objects that look like what may arise\n from a multiplicative norm for compact Lie groups.\n\nAre there analogues of multiplicative norms or multiplicative\ntransfers for compact Lie groups?",
   "public_statement": "In genuine $G$-spectra we have multiplicative norm and\n transfer maps when $G$ has finitely indexed subgroups. Furthermore,\n (via Ando-Morava) there are objects that look like what may arise\n from a multiplicative norm for compact Lie groups.\n\nAre there analogues of multiplicative norms or multiplicative\ntransfers for compact Lie groups?",
   "evidence": "The canonical record is problem 4.3 in the AIM workshop list *Equivariant derived algebraic geometry*, section “Extensions to Profinite, Compact Lie Groups, etc.” The source page attributes the question to C. Rezk. The canonical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 167,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0169": {
   "statement_status": "exact",
   "original_statement": "Quillen's theorem for $MU$ states that the map from the Lazard ring $L$ to $\\pi_* MU$ is an isomorphism and so, in particular, the homotopy of $MU$ is concentrated in even degrees. Furthermore, when $G$ is a finite abelian group, it is known that\n the $G$-equivariant homotopy of $MU$ is also even.\n\nFor any finite or compact Lie group $G$, is it also the case that\n$\\pi^G_*(MU)$ is even?",
   "clean_statement": "Quillen's theorem for $MU$ states that the map from the Lazard ring $L$ to $\\pi_* MU$ is an isomorphism and so, in particular, the homotopy of $MU$ is concentrated in even degrees. Furthermore, when $G$ is a finite abelian group, it is known that\n the $G$-equivariant homotopy of $MU$ is also even.\n\nFor any finite or compact Lie group $G$, is it also the case that\n$\\pi^G_*(MU)$ is even?",
   "public_statement": "Quillen's theorem for $MU$ states that the map from the Lazard ring $L$ to $\\pi_* MU$ is an isomorphism and so, in particular, the homotopy of $MU$ is concentrated in even degrees. Furthermore, when $G$ is a finite abelian group, it is known that\n the $G$-equivariant homotopy of $MU$ is also even.\n\nFor any finite or compact Lie group $G$, is it also the case that\n$\\pi^G_*(MU)$ is even?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 168,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0170": {
   "statement_status": "exact",
   "original_statement": "Long Term\n\nHow should we think about equivariant stable homotopy theory in the context of infinite compact Lie groups?",
   "clean_statement": "Long Term\n\nHow should we think about equivariant stable homotopy theory in the context of infinite compact Lie groups?",
   "public_statement": "Long Term\n\nHow should we think about equivariant stable homotopy theory in the context of infinite compact Lie groups?",
   "evidence": "The canonical record is problem 4.5 in the AIM list *Equivariant derived algebraic geometry*, section “Extensions to Profinite, Compact Lie Groups, etc.” Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 169,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0171": {
   "statement_status": "exact",
   "original_statement": "What can we say about equivariant stable homotopy theory for infinite discrete groups?",
   "clean_statement": "What can we say about equivariant stable homotopy theory for infinite discrete groups?",
   "public_statement": "What can we say about equivariant stable homotopy theory for infinite discrete groups?",
   "evidence": "The canonical AIM record is problem 4.6 in the section **“Extensions to Profinite, Compact Lie Groups, etc.”** of the workshop **“Equivariant derived algebraic geometry.”** Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 170,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0172": {
   "statement_status": "exact",
   "original_statement": "For a finite group $G$, Lurie constructs equivariant\n elliptic cohomology in commutative naive $G$-spectra and in genuine\n $G$-spectra compatibly \\cite{MR2597740}.\n\nCan this construction be extended to commutative genuine $G$-spectra\nor $G$-commutative genuine $G$-spectra?",
   "clean_statement": "For a finite group $G$, Lurie constructs equivariant\n elliptic cohomology in commutative naive $G$-spectra and in genuine\n $G$-spectra compatibly \\cite{MR2597740}.\n\nCan this construction be extended to commutative genuine $G$-spectra\nor $G$-commutative genuine $G$-spectra?",
   "public_statement": "For a finite group $G$, Lurie constructs equivariant\n elliptic cohomology in commutative naive $G$-spectra and in genuine\n $G$-spectra compatibly \\cite{MR2597740}.\n\nCan this construction be extended to commutative genuine $G$-spectra\nor $G$-commutative genuine $G$-spectra?",
   "evidence": "This is Problem 5.1 in the “Chromatic Homotopy Theory” section of the AIM workshop list *Equivariant derived algebraic geometry*. The canonical record, including its line break, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 171,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0173": {
   "statement_status": "exact",
   "original_statement": "Can we find explicit ways to write $S \\in R-Alg$ as an $R$-module? What are objects that ``untwist\" when smashed together? (e.g. $KU \\simeq KO \\wedge C(\\eta)$, $H\\F_2 \\simeq ko \\wedge \\A(1)$)\n\nA slightly easier question: Is $tmf^{(n)}_0 \\rightarrow tmf^{(n)}_1$ relatively Gorenstein?",
   "clean_statement": "Can we find explicit ways to write $S \\in R-Alg$ as an $R$-module? What are objects that ``untwist\" when smashed together? (e.g. $KU \\simeq KO \\wedge C(\\eta)$, $H\\F_2 \\simeq ko \\wedge \\A(1)$)\n\nA slightly easier question: Is $tmf^{(n)}_0 \\rightarrow tmf^{(n)}_1$ relatively Gorenstein?",
   "public_statement": "Can we find explicit ways to write $S \\in R-Alg$ as an $R$-module? What are objects that ``untwist\" when smashed together? (e.g. $KU \\simeq KO \\wedge C(\\eta)$, $H\\F_2 \\simeq ko \\wedge \\A(1)$)\n\nA slightly easier question: Is $tmf^{(n)}_0 \\rightarrow tmf^{(n)}_1$ relatively Gorenstein?",
   "evidence": "The exact AIM record is Problem 5.8 in the “Chromatic Homotopy Theory” section of the 2016 workshop *Equivariant derived algebraic geometry*. It is attributed on the AIM problem page to V. Stojanoska and reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 172,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0174": {
   "statement_status": "exact",
   "original_statement": "When $G = C_{2^n}$ what is the $G$-equivariant\n Landweber exact functor theory for $MU^{(G)}$?",
   "clean_statement": "When $G = C_{2^n}$ what is the $G$-equivariant\n Landweber exact functor theory for $MU^{(G)}$?",
   "public_statement": "When $G = C_{2^n}$ what is the $G$-equivariant\n Landweber exact functor theory for $MU^{(G)}$?",
   "evidence": "The canonical AIM record is problem 5.6 in the “Chromatic Homotopy Theory” section of the 2016 workshop *Equivariant derived algebraic geometry*. It is record index 173 (zero-based) in `aim-algebraic-geometry-notes.json`:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 173,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0175": {
   "statement_status": "exact",
   "original_statement": "What is an algebraic model for $L_{K(1)}Comm(Sp_G)$ (for even small groups $G$)?",
   "clean_statement": "What is an algebraic model for $L_{K(1)}Comm(Sp_G)$ (for even small groups $G$)?",
   "public_statement": "What is an algebraic model for $L_{K(1)}Comm(Sp_G)$ (for even small groups $G$)?",
   "evidence": "The canonical record is AIM Problem Lists, workshop **Equivariant derived algebraic geometry**, section 5, **Chromatic Homotopy Theory**, Problem 5.3:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 174,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0176": {
   "statement_status": "exact",
   "original_statement": "Is there a Goerss-Hopkins obstruction theory for\n $Comm(Sp_G)$?",
   "clean_statement": "Is there a Goerss-Hopkins obstruction theory for\n $Comm(Sp_G)$?",
   "public_statement": "Is there a Goerss-Hopkins obstruction theory for\n $Comm(Sp_G)$?",
   "evidence": "The canonical record is AIM Problem 5.5 from the workshop *Equivariant derived algebraic geometry*, section “Chromatic Homotopy Theory”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 175,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0177": {
   "statement_status": "exact",
   "original_statement": "What would ``rings of integers'' in $tmf$\n with level structure be? Also, is there a TAF analogue of such\n objects with\n connections to TMF?",
   "clean_statement": "What would ``rings of integers'' in $tmf$\n with level structure be? Also, is there a TAF analogue of such\n objects with\n connections to TMF?",
   "public_statement": "What would ``rings of integers'' in $tmf$\n with level structure be? Also, is there a TAF analogue of such\n objects with\n connections to TMF?",
   "evidence": "The canonical record is AIM-ALGEBRAIC_GEOMETRY-0177, problem 5.2 in the “Chromatic Homotopy Theory” section of the AIM workshop *Equivariant derived algebraic geometry*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 176,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0178": {
   "statement_status": "exact",
   "original_statement": "Long Term\n\nCan we describe the moduli stack of formal groups in the derived setting?",
   "clean_statement": "Long Term\n\nCan we describe the moduli stack of formal groups in the derived setting?",
   "public_statement": "Long Term\n\nCan we describe the moduli stack of formal groups in the derived setting?",
   "evidence": "There is no apparent OCR corruption. The workshop and section make spectral algebraic geometry, with connective or nonconnective \\(\\mathbb E_\\infty\\)-rings as test rings, the relevant meaning of “derived.” Another possible meaning, Lurie's *formal moduli problems* (infinitesimal deformation germs based at a point), is related but is not the global moduli problem asked for here.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 177,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0179": {
   "statement_status": "exact",
   "original_statement": "We may examine how viewing the Tate spectrum of a given $G$-spectrum is related to chromatic localizations. For example, given the spectrum $KO$ with trivial $C_2$-action, we have\n\\[ (KO_{(2)})^{tC_2} \\simeq \\bigvee_{k\\in\\Z} \\Sigma^{4k} H\\Q_{(2)}\\]\n\nWhy should taking the Tate spectrum bring an object down in chromatic height (i.e. why do we see a blue shift)? More generally, we'd like to look at \"chromatic primes\" and \"equivariant primes\" in the same place and understand how they interact.\n\n[Needs editing - I'm sure Mark has a more eloquent statement]",
   "clean_statement": "We may examine how viewing the Tate spectrum of a given $G$-spectrum is related to chromatic localizations. For example, given the spectrum $KO$ with trivial $C_2$-action, we have\n\\[ (KO_{(2)})^{tC_2} \\simeq \\bigvee_{k\\in\\Z} \\Sigma^{4k} H\\Q_{(2)}\\]\n\nWhy should taking the Tate spectrum bring an object down in chromatic height (i.e. why do we see a blue shift)? More generally, we'd like to look at \"chromatic primes\" and \"equivariant primes\" in the same place and understand how they interact.\n\n[Needs editing - I'm sure Mark has a more eloquent statement]",
   "public_statement": "We may examine how viewing the Tate spectrum of a given $G$-spectrum is related to chromatic localizations. For example, given the spectrum $KO$ with trivial $C_2$-action, we have\n\\[ (KO_{(2)})^{tC_2} \\simeq \\bigvee_{k\\in\\Z} \\Sigma^{4k} H\\Q_{(2)}\\]\n\nWhy should taking the Tate spectrum bring an object down in chromatic height (i.e. why do we see a blue shift)? More generally, we'd like to look at \"chromatic primes\" and \"equivariant primes\" in the same place and understand how they interact.\n\n[Needs editing - I'm sure Mark has a more eloquent statement]",
   "evidence": "The canonical record is problem 5.4 in the “Chromatic Homotopy Theory” section of the AIM workshop page *Equivariant derived algebraic geometry*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 178,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0180": {
   "statement_status": "exact",
   "original_statement": "What can we say about modules over $S[BU_+]$?",
   "clean_statement": "What can we say about modules over $S[BU_+]$?",
   "public_statement": "What can we say about modules over $S[BU_+]$?",
   "evidence": "The canonical record is AIM-ALGEBRAIC_GEOMETRY-0180, problem 5.9 in the “Chromatic Homotopy Theory” section of the AIM workshop *Equivariant derived algebraic geometry*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 179,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0181": {
   "statement_status": "exact",
   "original_statement": "The essential variety is a 5-dimensional variety of degree 10. A point correspondence gives a linear constraint so a five point correspondence should give 10 complex solutions.\n\nOf the ten possibilities, how many of the solutions can be real?",
   "clean_statement": "The essential variety is a 5-dimensional variety of degree 10. A point correspondence gives a linear constraint so a five point correspondence should give 10 complex solutions.\n\nOf the ten possibilities, how many of the solutions can be real?",
   "public_statement": "The essential variety is a 5-dimensional variety of degree 10. A point correspondence gives a linear constraint so a five point correspondence should give 10 complex solutions.\n\nOf the ten possibilities, how many of the solutions can be real?",
   "evidence": "The exact AIM record (Algebraic Vision workshop, Reconstruction, Problem 1.05) is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 180,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0182": {
   "statement_status": "exact",
   "original_statement": "How many linear constraints does a point correspondence in three views determine on the trifocal tensor variety ?",
   "clean_statement": "How many linear constraints does a point correspondence in three views determine on the trifocal tensor variety ?",
   "public_statement": "How many linear constraints does a point correspondence in three views determine on the trifocal tensor variety ?",
   "evidence": "The exact canonical record is problem 1.1 in the “Reconstruction” section of the AIM workshop list *Algebraic vision*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 181,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0183": {
   "statement_status": "exact",
   "original_statement": "It is known that the trifocal tensor variety is dimension 18 with degree 297 inside of $\\mathbb{P}^{28}$. It is also known that a six point correspondence in three views is a minimal problem.\n\nHow do the 24 linear constraints lead a minimal problem? How many complex solutions are there?",
   "clean_statement": "It is known that the trifocal tensor variety is dimension 18 with degree 297 inside of $\\mathbb{P}^{28}$. It is also known that a six point correspondence in three views is a minimal problem.\n\nHow do the 24 linear constraints lead a minimal problem? How many complex solutions are there?",
   "public_statement": "It is known that the trifocal tensor variety is dimension 18 with degree 297 inside of $\\mathbb{P}^{28}$. It is also known that a six point correspondence in three views is a minimal problem.\n\nHow do the 24 linear constraints lead a minimal problem? How many complex solutions are there?",
   "evidence": "The canonical AIM record (Algebraic Vision, Reconstruction, Problem 1.15) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 182,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0184": {
   "statement_status": "exact",
   "original_statement": "In between the calibrated and uncalibrated cases there are partially calibrated cameras, say with both focal lengths known or with the center position known. What can we say about these situations?",
   "clean_statement": "In between the calibrated and uncalibrated cases there are partially calibrated cameras, say with both focal lengths known or with the center position known. What can we say about these situations?",
   "public_statement": "In between the calibrated and uncalibrated cases there are partially calibrated cameras, say with both focal lengths known or with the center position known. What can we say about these situations?",
   "evidence": "The canonical record is AIM-ALGEBRAIC_GEOMETRY-0184, item 1.2 in the “Reconstruction” section of the AIM *Algebraic vision* problem list. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 183,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0185": {
   "statement_status": "exact",
   "original_statement": "Given an algebraic curve in two views, when are they compatible? This is interesting when the cameras are fixed beforehand.",
   "clean_statement": "Given an algebraic curve in two views, when are they compatible? This is interesting when the cameras are fixed beforehand.",
   "public_statement": "Given an algebraic curve in two views, when are they compatible? This is interesting when the cameras are fixed beforehand.",
   "evidence": "The canonical record is AIM Problem List, *Algebraic vision*, §1 “Reconstruction,” Problem 1.25:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 184,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0186": {
   "statement_status": "exact",
   "original_statement": "What is the Hurwitz form of the essential variety?",
   "clean_statement": "What is the Hurwitz form of the essential variety?",
   "public_statement": "What is the Hurwitz form of the essential variety?",
   "evidence": "The exact AIM record is Problem 1.3 in the “Reconstruction” section of the *Algebraic vision* list:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 185,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0187": {
   "statement_status": "exact",
   "original_statement": "Can we use conic correspondence between two images to do reconstruction? Can we use point and curve correspondences to do reconstruction? Can we use cubic spline correspondences to do reconstruction?",
   "clean_statement": "Can we use conic correspondence between two images to do reconstruction? Can we use point and curve correspondences to do reconstruction? Can we use cubic spline correspondences to do reconstruction?",
   "public_statement": "Can we use conic correspondence between two images to do reconstruction? Can we use point and curve correspondences to do reconstruction? Can we use cubic spline correspondences to do reconstruction?",
   "evidence": "The canonical record is item 1.35 in the “Reconstruction” section of the AIM workshop list *Algebraic vision*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 186,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0188": {
   "statement_status": "exact",
   "original_statement": "Fix two cameras and two image curves. Is there is a curve in $\\mathbb{P}^3$ that projects onto these two curves? What if we put restriction on degrees or smoothness?",
   "clean_statement": "Fix two cameras and two image curves. Is there is a curve in $\\mathbb{P}^3$ that projects onto these two curves? What if we put restriction on degrees or smoothness?",
   "public_statement": "Fix two cameras and two image curves. Is there is a curve in $\\mathbb{P}^3$ that projects onto these two curves? What if we put restriction on degrees or smoothness?",
   "evidence": "The canonical AIM record is *Algebraic vision*, §1 “Reconstruction,” Problem 1.4:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 187,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0189": {
   "statement_status": "exact",
   "original_statement": "For a reconstruction what does it mean for all the points to be \"in front\" of the cameras? Can this easily be detected.",
   "clean_statement": "For a reconstruction what does it mean for all the points to be \"in front\" of the cameras? Can this easily be detected.",
   "public_statement": "For a reconstruction what does it mean for all the points to be \"in front\" of the cameras? Can this easily be detected.",
   "evidence": "The exact AIM record is Problem 1.45 in the “Reconstruction” section of the *Algebraic vision* list:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 188,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0190": {
   "statement_status": "exact",
   "original_statement": "The focal lengths of a pair of cameras can be written as a function of their fundamental matrix. During reconstruction fundamental matrices are constructed for which these focal lengths are complex. Is there an efficient way to detect if the this will occur before computing the fundamental matrix?",
   "clean_statement": "The focal lengths of a pair of cameras can be written as a function of their fundamental matrix. During reconstruction fundamental matrices are constructed for which these focal lengths are complex. Is there an efficient way to detect if the this will occur before computing the fundamental matrix?",
   "public_statement": "The focal lengths of a pair of cameras can be written as a function of their fundamental matrix. During reconstruction fundamental matrices are constructed for which these focal lengths are complex. Is there an efficient way to detect if the this will occur before computing the fundamental matrix?",
   "evidence": "The canonical record is AIM problem 1.5 in the “Reconstruction” section of the Algebraic Vision workshop:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 189,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0191": {
   "statement_status": "exact",
   "original_statement": "How can we account for uncertainty, given the geometric structure of these problems? What if we replace points with disks?",
   "clean_statement": "How can we account for uncertainty, given the geometric structure of these problems? What if we replace points with disks?",
   "public_statement": "How can we account for uncertainty, given the geometric structure of these problems? What if we replace points with disks?",
   "evidence": "The canonical record is AIM Problem Lists, *Algebraic vision*, section “Reconstruction,” Problem 1.55:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 190,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0192": {
   "statement_status": "exact",
   "original_statement": "For any of the standard varieties considered it might be possible to study the projective dual. What can we learn from studying these objects?",
   "clean_statement": "For any of the standard varieties considered it might be possible to study the projective dual. What can we learn from studying these objects?",
   "public_statement": "For any of the standard varieties considered it might be possible to study the projective dual. What can we learn from studying these objects?",
   "evidence": "The canonical record is AIM Problem Lists, *Algebraic vision*, section “More Varieties,” Problem 2.1:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 191,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0193": {
   "statement_status": "exact",
   "original_statement": "What can we say about the convex hull of all the varieties we have considered? Can we give a good description of them?",
   "clean_statement": "What can we say about the convex hull of all the varieties we have considered? Can we give a good description of them?",
   "public_statement": "What can we say about the convex hull of all the varieties we have considered? Can we give a good description of them?",
   "evidence": "The canonical record is problem 2.2 in the “More Varieties” section of the Algebraic Vision AIM problem list:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 192,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0194": {
   "statement_status": "exact",
   "original_statement": "What is the functor of points of multiview geometry?",
   "clean_statement": "What is the functor of points of multiview geometry?",
   "public_statement": "What is the functor of points of multiview geometry?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 193,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0195": {
   "statement_status": "exact",
   "original_statement": "What is the functor of points in the calibrated case?",
   "clean_statement": "What is the functor of points in the calibrated case?",
   "public_statement": "What is the functor of points in the calibrated case?",
   "evidence": "The canonical record is AIM Problem Lists, *Algebraic vision*, section “More Varieties,” Problem 2.4:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 194,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0196": {
   "statement_status": "exact",
   "original_statement": "How complete are these invariants? If two curves have the same signature what can we say about them?",
   "clean_statement": "How complete are these invariants? If two curves have the same signature what can we say about them?",
   "public_statement": "How complete are these invariants? If two curves have the same signature what can we say about them?",
   "evidence": "The exact canonical AIM record, problem 3.1 in the “Invariants” section of the 2016 Algebraic Vision workshop list, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 195,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0197": {
   "statement_status": "exact",
   "original_statement": "What can we say about the degree of a signature curve? How is it related to the automorphism group of the original curve?",
   "clean_statement": "What can we say about the degree of a signature curve? How is it related to the automorphism group of the original curve?",
   "public_statement": "What can we say about the degree of a signature curve? How is it related to the automorphism group of the original curve?",
   "evidence": "The exact canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 196,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0198": {
   "statement_status": "exact",
   "original_statement": "What is the signature curve of a general canonical curve?",
   "clean_statement": "What is the signature curve of a general canonical curve?",
   "public_statement": "What is the signature curve of a general canonical curve?",
   "evidence": "The exact source record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 197,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0199": {
   "statement_status": "exact",
   "original_statement": "What can we say about the signature of the signature of a curve?",
   "clean_statement": "What can we say about the signature of the signature of a curve?",
   "public_statement": "What can we say about the signature of the signature of a curve?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 198,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0200": {
   "statement_status": "exact",
   "original_statement": "Which curves occur as signatures of another curve?",
   "clean_statement": "Which curves occur as signatures of another curve?",
   "public_statement": "Which curves occur as signatures of another curve?",
   "evidence": "The exact canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 199,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0201": {
   "statement_status": "exact",
   "original_statement": "Can we use differential (integral, other…) invariants to construct a moduli space?",
   "clean_statement": "Can we use differential (integral, other…) invariants to construct a moduli space?",
   "public_statement": "Can we use differential (integral, other…) invariants to construct a moduli space?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 200,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0202": {
   "statement_status": "exact",
   "original_statement": "How can we use these signatures for object recognition?",
   "clean_statement": "How can we use these signatures for object recognition?",
   "public_statement": "How can we use these signatures for object recognition?",
   "evidence": "The exact canonical record is AIM problem 3.7 in the “Invariants” section of the 2016 *Algebraic vision* workshop list:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 201,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0203": {
   "statement_status": "exact",
   "original_statement": "Find local invariants for a natural (Lie) groupoid action. (e.g. bathroom tiles in an irregular bathroom). How is this related to quasi-invariants?",
   "clean_statement": "Find local invariants for a natural (Lie) groupoid action. (e.g. bathroom tiles in an irregular bathroom). How is this related to quasi-invariants?",
   "public_statement": "Find local invariants for a natural (Lie) groupoid action. (e.g. bathroom tiles in an irregular bathroom). How is this related to quasi-invariants?",
   "evidence": "The exact canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 202,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0204": {
   "statement_status": "exact",
   "original_statement": "Which visual event surfaces appear on visual hulls of algebraic surfaces?",
   "clean_statement": "Which visual event surfaces appear on visual hulls of algebraic surfaces?",
   "public_statement": "Which visual event surfaces appear on visual hulls of algebraic surfaces?",
   "evidence": "The canonical record is AIM Problem Lists, workshop **Algebraic vision**, section **Silhouettes**, Problem 4.1:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 203,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0205": {
   "statement_status": "exact",
   "original_statement": "Can we find equations of visual event surfaces?",
   "clean_statement": "Can we find equations of visual event surfaces?",
   "public_statement": "Can we find equations of visual event surfaces?",
   "evidence": "The canonical record is AIM problem 4.2 in the “Silhouettes” section of the 2016 AIM workshop *Algebraic vision*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 204,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0206": {
   "statement_status": "exact",
   "original_statement": "In silhouette reconstruction a duality appears between the visual hull and the convex hull. Is there an algebraic version of this?",
   "clean_statement": "In silhouette reconstruction a duality appears between the visual hull and the convex hull. Is there an algebraic version of this?",
   "public_statement": "In silhouette reconstruction a duality appears between the visual hull and the convex hull. Is there an algebraic version of this?",
   "evidence": "The exact canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 205,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0207": {
   "statement_status": "exact",
   "original_statement": "Can we find the fundamental matrix from silhouettes? If the silhouettes are bounded by algebraic curves are there algebraic constraints on the fundamental matrix?",
   "clean_statement": "Can we find the fundamental matrix from silhouettes? If the silhouettes are bounded by algebraic curves are there algebraic constraints on the fundamental matrix?",
   "public_statement": "Can we find the fundamental matrix from silhouettes? If the silhouettes are bounded by algebraic curves are there algebraic constraints on the fundamental matrix?",
   "evidence": "The canonical AIM record is item 4.4 in the “Silhouettes” section of the Algebraic Vision problem list:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 206,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0208": {
   "statement_status": "exact",
   "original_statement": "Which curves with cusps come from rims of algebraic surfaces?",
   "clean_statement": "Which curves with cusps come from rims of algebraic surfaces?",
   "public_statement": "Which curves with cusps come from rims of algebraic surfaces?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 207,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0209": {
   "statement_status": "exact",
   "original_statement": "Is there a relationship between Carlsson-Weinshall and Gale duality?",
   "clean_statement": "Is there a relationship between Carlsson-Weinshall and Gale duality?",
   "public_statement": "Is there a relationship between Carlsson-Weinshall and Gale duality?",
   "evidence": "The exact canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 208,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0210": {
   "statement_status": "exact",
   "original_statement": "Is there a Carlsson-Weinshallduality for point and line correspondence?",
   "clean_statement": "Is there a Carlsson-Weinshallduality for point and line correspondence?",
   "public_statement": "Is there a Carlsson-Weinshallduality for point and line correspondence?",
   "evidence": "The exact canonical record is AIM Problem List *Algebraic vision*, section 5 “Carlsson-Weinshall Duality,” Problem 5.2. Its text is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 209,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0211": {
   "statement_status": "exact",
   "original_statement": "What is a moduli space description of Carlsson-Weinshall duality? What moduli spaces does this suggest? A framed multiview variety?",
   "clean_statement": "What is a moduli space description of Carlsson-Weinshall duality? What moduli spaces does this suggest? A framed multiview variety?",
   "public_statement": "What is a moduli space description of Carlsson-Weinshall duality? What moduli spaces does this suggest? A framed multiview variety?",
   "evidence": "The exact canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 210,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0212": {
   "statement_status": "exact",
   "original_statement": "What can we say about Carlsson-Weinshall duality for other dimensions like $\\mathbb{P}^4 \\to \\mathbb{P}^3$?",
   "clean_statement": "What can we say about Carlsson-Weinshall duality for other dimensions like $\\mathbb{P}^4 \\to \\mathbb{P}^3$?",
   "public_statement": "What can we say about Carlsson-Weinshall duality for other dimensions like $\\mathbb{P}^4 \\to \\mathbb{P}^3$?",
   "evidence": "The canonical record is AIM Problem Lists, Algebraic Vision, section “Carlsson-Weinshall Duality,” Problem 5.4:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 211,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0213": {
   "statement_status": "exact",
   "original_statement": "Is there a differentiable structure for essential varieties or fundamental varieties over time. Is there a formal structure? Can we use deformation theory and cohomological techniques?",
   "clean_statement": "Is there a differentiable structure for essential varieties or fundamental varieties over time. Is there a formal structure? Can we use deformation theory and cohomological techniques?",
   "public_statement": "Is there a differentiable structure for essential varieties or fundamental varieties over time. Is there a formal structure? Can we use deformation theory and cohomological techniques?",
   "evidence": "The canonical record is AIM Problem Lists, *Algebraic vision*, Section “Multiview Geometry for Continuous Motion,” Problem 6.1:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 212,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0214": {
   "statement_status": "exact",
   "original_statement": "What is the proper algebraic formulation for a continuous multiview variety? $\\mathbb{P}^1 \\times \\mathbb{P}^3 \\to (\\mathbb{P}^2)^n$? $Hom(\\mathbb{A}^1, \\text{essential matrices})$? Kontsevich spaces? Or could we use points and optical flow data?",
   "clean_statement": "What is the proper algebraic formulation for a continuous multiview variety? $\\mathbb{P}^1 \\times \\mathbb{P}^3 \\to (\\mathbb{P}^2)^n$? $Hom(\\mathbb{A}^1, \\text{essential matrices})$? Kontsevich spaces? Or could we use points and optical flow data?",
   "public_statement": "What is the proper algebraic formulation for a continuous multiview variety? $\\mathbb{P}^1 \\times \\mathbb{P}^3 \\to (\\mathbb{P}^2)^n$? $Hom(\\mathbb{A}^1, \\text{essential matrices})$? Kontsevich spaces? Or could we use points and optical flow data?",
   "evidence": "The canonical record is AIM Problem List, workshop **Algebraic vision**, section **Multiview Geometry for Continuous Motion**, Problem 6.2. Its exact statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 213,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0215": {
   "statement_status": "exact",
   "original_statement": "Problem 1 \n\n(Suggested by Eric Larson) Given the data of a curve X with n marked points lying in projective space Pn, you can forget the curve and remember only the marked points. This can be interpreted as giving a map \n\nMg,n(Pr, β ) π\n\n−→ Conf( Pr, n).\n\nThe basic question I'm interested in is, what can we say about the various possible fiber dimensions or vertical tangent space dimensions? I have recently finished up a paper that studies when this map is dominant. The upshot is that if d ≥ g + r, then we can explicitly characterize when the map is dominant.",
   "clean_statement": "Problem 1\n\n(Suggested by Eric Larson) Given the data of a curve X with n marked points lying in projective space Pn, you can forget the curve and remember only the marked points. This can be interpreted as giving a map\n\nMg,n(Pr, β ) π\n\n−→ Conf( Pr, n).\n\nThe basic question I'm interested in is, what can we say about the various possible fiber dimensions or vertical tangent space dimensions? I have recently finished up a paper that studies when this map is dominant. The upshot is that if d ≥ g + r, then we can explicitly characterize when the map is dominant.",
   "public_statement": "Problem 1\n\n(Suggested by Eric Larson) Given the data of a curve X with n marked points lying in projective space Pn, you can forget the curve and remember only the marked points. This can be interpreted as giving a map\n\nMg,n(Pr, β ) π\n\n−→ Conf( Pr, n).\n\nThe basic question I'm interested in is, what can we say about the various possible fiber dimensions or vertical tangent space dimensions? I have recently finished up a paper that studies when this map is dominant. The upshot is that if d ≥ g + r, then we can explicitly characterize when the map is dominant.",
   "evidence": "The source is Problem 1, suggested by Eric Larson, in the American Institute of Mathematics problem-session document *Degenerations in Algebraic Geometry* (September 7, 2016). The displayed map is \\[ \\pi:\\mathcal M_{g,n}(\\mathbf P^r,\\beta)\\longrightarrow \\operatorname{Conf}(\\mathbf P^r,n), \\] and the question asks for the possible dimensions of its fibers and vertical tangent spaces, especially when the map is known to be dominant.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 214,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0216": {
   "statement_status": "exact",
   "original_statement": "Problem 1.2. Find examples where we can understand behavior distinct from the expected one. There is basically no hope of a general answer, so I mean this as an example-hunting suggestion. \n\nA specific application would be to extend the range in which we can say when π is dominant. If you have examples where the tangent space dimension is wrong, then you can degenerate to a curve where the tangent space dimension is wrong again.",
   "clean_statement": "Problem 1.2. Find examples where we can understand behavior distinct from the expected one. There is basically no hope of a general answer, so I mean this as an example-hunting suggestion.\n\nA specific application would be to extend the range in which we can say when π is dominant. If you have examples where the tangent space dimension is wrong, then you can degenerate to a curve where the tangent space dimension is wrong again.",
   "public_statement": "Problem 1.2. Find examples where we can understand behavior distinct from the expected one. There is basically no hope of a general answer, so I mean this as an example-hunting suggestion.\n\nA specific application would be to extend the range in which we can say when π is dominant. If you have examples where the tangent space dimension is wrong, then you can degenerate to a curve where the tangent space dimension is wrong again.",
   "evidence": "The source is the problem-session document from the 2016 AIM workshop *Degenerations in Algebraic Geometry*. Problem 1 introduces the evaluation map \\[ \\pi:\\mathcal M_{g,n}(\\mathbf P^r,\\beta)\\longrightarrow \\operatorname{Conf}(\\mathbf P^r,n) \\] which remembers only the images of the \\(n\\) marked points. The printed PDF says “a curve \\(X\\) with \\(n\\) marked points lying in \\(\\mathbf P^n\\)” in its first sentence, but the displayed map and every subsequent parameter use \\(\\mathbf P^r\\). Thus \\(\\mathbf P^n\\) is retained as a source typo and \\(\\mathbf P^r\\) is the mathematically consistent reading.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 215,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0217": {
   "statement_status": "exact",
   "original_statement": "Problem 2 \n\n(Suggested by Allen Knutson) Let me motivate this by stating a theorem of Brion. Suppose you have a variety X lying in a flag manifold G/P. We say that X is multiplicity-free if in cohomology [X] =\n\n∑ \n\n> S⊂W/WP\n\n[Xs],\n\ni.e. each zero-dimensional intersection with a general translate of a Schubert variety is empty or 1 point. \n\nTheorem 2.1 (Brion). If X is multiplicity-free, then X degenerates to the reduced union of Schubert varieties (as predicted by S ), and this union is Cohen-Macaulay. \n\nIn particular, if X is reduced and irreducible and multiplicity-free, then X is Cohen-Macaulay.",
   "clean_statement": "Problem 2\n\n(Suggested by Allen Knutson) Let me motivate this by stating a theorem of Brion. Suppose you have a variety X lying in a flag manifold G/P. We say that X is multiplicity-free if in cohomology [X] =\n\n∑\n\n> S⊂W/WP\n\n[Xs],\n\ni.e. each zero-dimensional intersection with a general translate of a Schubert variety is empty or 1 point.\n\nTheorem 2.1 (Brion). If X is multiplicity-free, then X degenerates to the reduced union of Schubert varieties (as predicted by S ), and this union is Cohen-Macaulay.\n\nIn particular, if X is reduced and irreducible and multiplicity-free, then X is Cohen-Macaulay.",
   "public_statement": "Problem 2\n\n(Suggested by Allen Knutson) Let me motivate this by stating a theorem of Brion. Suppose you have a variety X lying in a flag manifold G/P. We say that X is multiplicity-free if in cohomology [X] =\n\n∑\n\n> S⊂W/WP\n\n[Xs],\n\ni.e. each zero-dimensional intersection with a general translate of a Schubert variety is empty or 1 point.\n\nTheorem 2.1 (Brion). If X is multiplicity-free, then X degenerates to the reduced union of Schubert varieties (as predicted by S ), and this union is Cohen-Macaulay.\n\nIn particular, if X is reduced and irreducible and multiplicity-free, then X is Cohen-Macaulay.",
   "evidence": "The canonical record is item 2 in the problem-session notes for the AIM workshop *Degenerations in Algebraic Geometry*, suggested by Allen Knutson. The source is the six-page PDF dated September 7, 2016.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 216,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0218": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 2.2. Classify subsets S ⊂ W/WP that deform to irreducibles. \n\nFor example, is ⋃s∈S Xs Cohen-Macaulay su ffi cient? The first case to try is G/P \u001b\n\n(P1)n.",
   "clean_statement": null,
   "public_statement": "Problem 2.2. Classify subsets S ⊂ W/WP that deform to irreducibles.\n\nFor example, is ⋃s∈S Xs Cohen-Macaulay su ffi cient? The first case to try is G/P [U+001B]\n\n(P1)n.",
   "evidence": "This record is Problem 2.2 in the AIM workshop list *Degenerations in algebraic geometry*. The literal database extraction is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 217,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0219": {
   "statement_status": "exact",
   "original_statement": "Problem 3 \n\n(Suggested by Michel) Let X ⊂ P2 be an elliptic curve.",
   "clean_statement": "Problem 3\n\n(Suggested by Michel) Let X ⊂ P2 be an elliptic curve.",
   "public_statement": "Problem 3\n\n(Suggested by Michel) Let X ⊂ P2 be an elliptic curve.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 218,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0220": {
   "statement_status": "exact",
   "original_statement": "Problem 3.1. What is the number of degree d rational curves in P2 meeting X at exactly 1\n\npoint? \n\nThis is known for d ≤ 8 by Takashi. Even d = 9 would be new and interesting. The motivation is from local mirror symmetry. One approach is to degenerate X to 3 lines. Also, is a tropical approach possible? \n\n4",
   "clean_statement": "Problem 3.1. What is the number of degree d rational curves in P2 meeting X at exactly 1\n\npoint?\n\nThis is known for d ≤ 8 by Takashi. Even d = 9 would be new and interesting. The motivation is from local mirror symmetry. One approach is to degenerate X to 3 lines. Also, is a tropical approach possible?\n\n4",
   "public_statement": "Problem 3.1. What is the number of degree d rational curves in P2 meeting X at exactly 1\n\npoint?\n\nThis is known for d ≤ 8 by Takashi. Even d = 9 would be new and interesting. The motivation is from local mirror symmetry. One approach is to degenerate X to 3 lines. Also, is a tropical approach possible?\n\n4",
   "evidence": "The AIM record occurs in the workshop list *Degenerations in algebraic geometry*. The preceding item introduces \\(X\\subset \\mathbb P^2\\) as an elliptic curve, hence as a smooth plane cubic. The record itself reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 219,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0221": {
   "statement_status": "exact",
   "original_statement": "Problem 4 \n\n(Suggested by Brian Osserman) Consider forming a nodal curve X by gluing together two rational curves at g + 1 generic points. Is X Brill-Noether general?",
   "clean_statement": "Problem 4\n\n(Suggested by Brian Osserman) Consider forming a nodal curve X by gluing together two rational curves at g + 1 generic points. Is X Brill-Noether general?",
   "public_statement": "Problem 4\n\n(Suggested by Brian Osserman) Consider forming a nodal curve X by gluing together two rational curves at g + 1 generic points. Is X Brill-Noether general?",
   "evidence": "The canonical AIM record is Problem 4 from the workshop *Degenerations in algebraic geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 220,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0222": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 4.1. Does X have a g rd1,d2 if and only if ρ(g, r, d1 + d2) ≥ 0 (subject to some condition on d being \"relatively balanced\")? \n\n2Caporaso (\"Brill-Noether theory of binary curves\") thought about this. She proved some results for r ≤ 2, using a definition of \"relatively balanced\" which is the weakest reasonable one. Another possible definition could be to require 0 ≤ di ≤ g − 1. Caporaso's motivation is to give an alternate proof of the Brill-Noether theorem by degeneration to such curves, which would also give a new criterion for Brill-Noether gen-erality. For me, the motivation is related to my definition of limit linear series for curves of non-compact type. The answer to this question has concrete consequences on the dimension of limit linear series for pseudocompact curves. It is possible that the answers are distinct depending on the definition of \"relatively balanced.\" That would be interesting.",
   "clean_statement": null,
   "public_statement": "Problem 4.1. Does X have a g rd1,d2 if and only if ρ(g, r, d1 + d2) ≥ 0 (subject to some condition on d being \"relatively balanced\")?\n\n2Caporaso (\"Brill-Noether theory of binary curves\") thought about this. She proved some results for r ≤ 2, using a definition of \"relatively balanced\" which is the weakest reasonable one. Another possible definition could be to require 0 ≤ di ≤ g − 1. Caporaso's motivation is to give an alternate proof of the Brill-Noether theorem by degeneration to such curves, which would also give a new criterion for Brill-Noether gen-erality. For me, the motivation is related to my definition of limit linear series for curves of non-compact type. The answer to this question has concrete consequences on the dimension of limit linear series for pseudocompact curves. It is possible that the answers are distinct depending on the definition of \"relatively balanced.\" That would be interesting.",
   "evidence": "The canonical JSON record has lost superscripts and subscripts. The original AIM PDF recovers the notation and the preceding definition of \\(X\\):",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 221,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0223": {
   "statement_status": "exact",
   "original_statement": "Problem 5 \n\n(Suggested by Brian Harbourne) The Bounded Negativity Conjecture says that for X a sur-face which is either rational or over characteristic 0, there exists nX such that C2 ≥ nX for all reduced C ⊂ X.Nobody knows how to approach this. Recently people have started looking at a variant called \" h-constants.\" Let C be a singular reduced plane curve of degree d. Look at \n\nh(C): = d2 − ∑x∈C m2\n\n> x\n\n#singular points,\n\nwhere mx is the multiplicity of the singular point x. (This is essentially the self-intersection of the proper transform in the blowup.)",
   "clean_statement": "Problem 5\n\n(Suggested by Brian Harbourne) The Bounded Negativity Conjecture says that for X a sur-face which is either rational or over characteristic 0, there exists nX such that C2 ≥ nX for all reduced C ⊂ X.Nobody knows how to approach this. Recently people have started looking at a variant called \" h-constants.\" Let C be a singular reduced plane curve of degree d. Look at\n\nh(C): = d2 − ∑x∈C m2\n\n> x\n\n#singular points,\n\nwhere mx is the multiplicity of the singular point x. (This is essentially the self-intersection of the proper transform in the blowup.)",
   "public_statement": "Problem 5\n\n(Suggested by Brian Harbourne) The Bounded Negativity Conjecture says that for X a sur-face which is either rational or over characteristic 0, there exists nX such that C2 ≥ nX for all reduced C ⊂ X.Nobody knows how to approach this. Recently people have started looking at a variant called \" h-constants.\" Let C be a singular reduced plane curve of degree d. Look at\n\nh(C): = d2 − ∑x∈C m2\n\n> x\n\n#singular points,\n\nwhere mx is the multiplicity of the singular point x. (This is essentially the self-intersection of the proper transform in the blowup.)",
   "evidence": "This record is the introductory part of Problem 5 in the AIM workshop list *Degenerations in Algebraic Geometry* (September 7, 2016), suggested by Brian Harbourne. The database extraction breaks the displayed fraction across lines. Inspection of page 3 of the original PDF (PDF page index 2) recovers the text as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 222,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0224": {
   "statement_status": "exact",
   "original_statement": "Problem 5.1. Is h C bounded below uniformly? \n\nIf this is true, then the Bounded Negativity Conjecture holds for rational surfaces in characteristic \n0. (In characteristic p > 0, h is not bounded below. One could ask what the interesting statement would be in characteristic p.) It is known that if C is a union of lines, then h(C) > −4, while the best achieved example is something like −3.36. The point is to find a configuration of lines with many high-multiplicity intersections (basically, the point is that you want to avoid simple crossings).",
   "clean_statement": "Problem 5.1. Is h C bounded below uniformly?\n\nIf this is true, then the Bounded Negativity Conjecture holds for rational surfaces in characteristic\n0. (In characteristic p > 0, h is not bounded below. One could ask what the interesting statement would be in characteristic p.) It is known that if C is a union of lines, then h(C) > −4, while the best achieved example is something like −3.36. The point is to find a configuration of lines with many high-multiplicity intersections (basically, the point is that you want to avoid simple crossings).",
   "public_statement": "Problem 5.1. Is h C bounded below uniformly?\n\nIf this is true, then the Bounded Negativity Conjecture holds for rational surfaces in characteristic\n0. (In characteristic p > 0, h is not bounded below. One could ask what the interesting statement would be in characteristic p.) It is known that if C is a union of lines, then h(C) > −4, while the best achieved example is something like −3.36. The point is to find a configuration of lines with many high-multiplicity intersections (basically, the point is that you want to avoid simple crossings).",
   "evidence": "The source is Problem 5.1 from the AIM workshop list *Degenerations in algebraic geometry*. The extracted record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 223,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0225": {
   "statement_status": "exact",
   "original_statement": "Problem 5.3. For irreducible curves, the best known examples have h → − 2. Can we do better?",
   "clean_statement": "Problem 5.3. For irreducible curves, the best known examples have h → − 2. Can we do better?",
   "public_statement": "Problem 5.3. For irreducible curves, the best known examples have h → − 2. Can we do better?",
   "evidence": "The source is the six-page problem-session record from the AIM workshop *Degenerations in Algebraic Geometry* (September 7, 2016). The surrounding text first defines, for a singular reduced plane curve \\(C\\) of degree \\(d\\), \\[ h(C)=\\frac{d^2-\\sum_{x\\in \\operatorname{Sing}(C)}m_x(C)^2} {\\#\\operatorname{Sing}(C)}. \\] Here the sum is over the distinct **proper singular points in \\(\\mathbf P^2\\)**. This convention is explicit in the source: \\(m_x\\) is “the multiplicity of the singular point \\(x\\),” and the denominator is the number of singular points.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 224,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0226": {
   "statement_status": "exact",
   "original_statement": "Problem 6 \n\n(Suggested by Ravi Vakil) Here is an idea for a tropical approach to interpolation of points in \n\nP2. We want to show that given d and m1,..., ms, if the expected dimension is positive then the interpolation problem has the right number of solutions (this is the SHGH Conjecture ). By adding conditions, we may assume that the expected dimension is 1. We can try to do this by find a single isolated (i.e. non-deforming) and liftable tropical solution. \n\n7",
   "clean_statement": "Problem 6\n\n(Suggested by Ravi Vakil) Here is an idea for a tropical approach to interpolation of points in\n\nP2. We want to show that given d and m1,..., ms, if the expected dimension is positive then the interpolation problem has the right number of solutions (this is the SHGH Conjecture ). By adding conditions, we may assume that the expected dimension is 1. We can try to do this by find a single isolated (i.e. non-deforming) and liftable tropical solution.\n\n7",
   "public_statement": "Problem 6\n\n(Suggested by Ravi Vakil) Here is an idea for a tropical approach to interpolation of points in\n\nP2. We want to show that given d and m1,..., ms, if the expected dimension is positive then the interpolation problem has the right number of solutions (this is the SHGH Conjecture ). By adding conditions, we may assume that the expected dimension is 1. We can try to do this by find a single isolated (i.e. non-deforming) and liftable tropical solution.\n\n7",
   "evidence": "The source is Problem 6 in the AIM workshop list *Degenerations in algebraic geometry*, suggested by Ravi Vakil. The source PDF is <https://aimath.org/pastworkshops/degenalggeomproblems.pdf>. Comparison with the PDF shows that the final character `7` in the extracted record is the heading of the next problem, not part of Problem 6.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 225,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0227": {
   "statement_status": "exact",
   "original_statement": "Problem 7 \n\n(Suggested by Omid Amini) Here is a \"weak version\" of Nagata's conjecture.",
   "clean_statement": "Problem 7\n\n(Suggested by Omid Amini) Here is a \"weak version\" of Nagata's conjecture.",
   "public_statement": "Problem 7\n\n(Suggested by Omid Amini) Here is a \"weak version\" of Nagata's conjecture.",
   "evidence": "The assigned JSON record contains only the heading of Problem 7. Direct inspection of page 4 of the six-page AIM problem-session PDF recovers the complete block:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 226,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0228": {
   "statement_status": "exact",
   "original_statement": "Problem 7.1. When Nagata predicts h 0 = 0, show that dL − m1E1 −... − msEs is not big by using \"convex methods.\"",
   "clean_statement": "Problem 7.1. When Nagata predicts h 0 = 0, show that dL − m1E1 −... − msEs is not big by using \"convex methods.\"",
   "public_statement": "Problem 7.1. When Nagata predicts h 0 = 0, show that dL − m1E1 −... − msEs is not big by using \"convex methods.\"",
   "evidence": "The source is Problem 7.1 in the problem session for the AIM workshop *Degenerations in Algebraic Geometry* (September 7, 2016). The PDF says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 227,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0229": {
   "statement_status": "exact",
   "original_statement": "Problem 8 \n\n(Suggested by David Jensen) The Maximal Rank Conjecture predicts the Hilbert fuction of a general curve (for fixed g, r, d such that ρ(g, r, d) ≥ 0 and r ≥ 3). The Hilbert function measures the rank of the map Sym m H0(X, L) → H0(X, L⊗m). The Maximal Rank Conjecture predicts that this has maximal rank for general X and L.Concretely, this means that it is either injective or surjective. We proposed a combinatorial approach via the notion of tropical independence. How-ever, we can only solve the combinatorial problem for h(2) (the case of quadrics). The next step would be h(3), i.e. cubics.",
   "clean_statement": "Problem 8\n\n(Suggested by David Jensen) The Maximal Rank Conjecture predicts the Hilbert fuction of a general curve (for fixed g, r, d such that ρ(g, r, d) ≥ 0 and r ≥ 3). The Hilbert function measures the rank of the map Sym m H0(X, L) → H0(X, L⊗m). The Maximal Rank Conjecture predicts that this has maximal rank for general X and L.Concretely, this means that it is either injective or surjective. We proposed a combinatorial approach via the notion of tropical independence. How-ever, we can only solve the combinatorial problem for h(2) (the case of quadrics). The next step would be h(3), i.e. cubics.",
   "public_statement": "Problem 8\n\n(Suggested by David Jensen) The Maximal Rank Conjecture predicts the Hilbert fuction of a general curve (for fixed g, r, d such that ρ(g, r, d) ≥ 0 and r ≥ 3). The Hilbert function measures the rank of the map Sym m H0(X, L) → H0(X, L⊗m). The Maximal Rank Conjecture predicts that this has maximal rank for general X and L.Concretely, this means that it is either injective or surjective. We proposed a combinatorial approach via the notion of tropical independence. How-ever, we can only solve the combinatorial problem for h(2) (the case of quadrics). The next step would be h(3), i.e. cubics.",
   "evidence": "**Source.** AIM workshop problem list, *Degenerations in Algebraic Geometry*, Problem 8, suggested by David Jensen. The workshop took place in September 2016. The canonical record is zero-based record 228 of `aim-algebraic-geometry-notes.json`.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 228,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0230": {
   "statement_status": "exact",
   "original_statement": "Problem 8.1. Prove this for all m. \n\nThe current argument is sort of naïve. We make a choice of functions in Sym m H0(X, L)which we want to be independent. Thus, the argument works more naturally in the injective case because there is no choice needed. In that case, you have a bunch of functions and you want to show that (mr\n\n) \"m-wise\" sums are independent. To do that in the case m = 2, this becomes a combinatorial game. After this, one coud ask about higher syzygies (viewing the Hilbert function as describ-ing the first term of the syzygy).",
   "clean_statement": "Problem 8.1. Prove this for all m.\n\nThe current argument is sort of naïve. We make a choice of functions in Sym m H0(X, L)which we want to be independent. Thus, the argument works more naturally in the injective case because there is no choice needed. In that case, you have a bunch of functions and you want to show that (mr\n\n) \"m-wise\" sums are independent. To do that in the case m = 2, this becomes a combinatorial game. After this, one coud ask about higher syzygies (viewing the Hilbert function as describ-ing the first term of the syzygy).",
   "public_statement": "Problem 8.1. Prove this for all m.\n\nThe current argument is sort of naïve. We make a choice of functions in Sym m H0(X, L)which we want to be independent. Thus, the argument works more naturally in the injective case because there is no choice needed. In that case, you have a bunch of functions and you want to show that (mr\n\n) \"m-wise\" sums are independent. To do that in the case m = 2, this becomes a combinatorial game. After this, one coud ask about higher syzygies (viewing the Hilbert function as describ-ing the first term of the syzygy).",
   "evidence": "The record is Problem 8.1 in the AIM workshop list *Degenerations in algebraic geometry*, suggested in the preceding text by David Jensen. The JSON extraction is visibly damaged: it omits the preceding definition of the map, removes the entries of a binomial coefficient, and runs words together. Inspection of page 4 of the source PDF recovers the following context.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 229,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0231": {
   "statement_status": "exact",
   "original_statement": "Problem 8.2. Use degeneration techniques (tropical independence or limit linear series or both) to understand higher syzygies. \n\nThis is known by Green's conjectures for generic curves under the canonical embed-ding, by non-tropical techniques.",
   "clean_statement": "Problem 8.2. Use degeneration techniques (tropical independence or limit linear series or both) to understand higher syzygies.\n\nThis is known by Green's conjectures for generic curves under the canonical embed-ding, by non-tropical techniques.",
   "public_statement": "Problem 8.2. Use degeneration techniques (tropical independence or limit linear series or both) to understand higher syzygies.\n\nThis is known by Green's conjectures for generic curves under the canonical embed-ding, by non-tropical techniques.",
   "evidence": "The canonical record is Problem 8.2 from the AIM workshop *Degenerations in Algebraic Geometry*, September 7, 2016. The exact extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 230,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0232": {
   "statement_status": "exact",
   "original_statement": "Problem 8.3. Can one give a new proof of Green's theorem by tropical methods? \n\n49",
   "clean_statement": "Problem 8.3. Can one give a new proof of Green's theorem by tropical methods?\n\n49",
   "public_statement": "Problem 8.3. Can one give a new proof of Green's theorem by tropical methods?\n\n49",
   "evidence": "The canonical record is Problem 8.3 from the AIM workshop list *Degenerations in algebraic geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 231,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0233": {
   "statement_status": "exact",
   "original_statement": "Problem 9 \n\n(Suggested by Erwan) This is a question about Welschinger invariants of real rational sur-faces, which is a real analogue of genus 0 Gromov-Witten invariants. Specifically, we are interested in a signed count of real curves through d1 real points and d2 pairs of complex conjugate points (such that d1 + 2g2 = 3d − 1) in P\n2. (The sign is something like ( −1) raised to the number of isolated nodes.)",
   "clean_statement": "Problem 9\n\n(Suggested by Erwan) This is a question about Welschinger invariants of real rational sur-faces, which is a real analogue of genus 0 Gromov-Witten invariants. Specifically, we are interested in a signed count of real curves through d1 real points and d2 pairs of complex conjugate points (such that d1 + 2g2 = 3d − 1) in P\n2. (The sign is something like ( −1) raised to the number of isolated nodes.)",
   "public_statement": "Problem 9\n\n(Suggested by Erwan) This is a question about Welschinger invariants of real rational sur-faces, which is a real analogue of genus 0 Gromov-Witten invariants. Specifically, we are interested in a signed count of real curves through d1 real points and d2 pairs of complex conjugate points (such that d1 + 2g2 = 3d − 1) in P\n2. (The sign is something like ( −1) raised to the number of isolated nodes.)",
   "evidence": "The AIM source is Problem 9 from the workshop *Degenerations in algebraic geometry*. The extracted record contains two material OCR errors: `g2` should be \\(d_2\\), and `P\\n2` should be \\(\\mathbb P^2\\). The source's phrase “real rational surfaces” describes the ambient theory; in the specialization to the rational surface \\(\\mathbb P^2\\), the enumerated objects are real rational plane curves. The dimension constraint and the notation in the source make the reconstruction unambiguous.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 232,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0234": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 9.2. What about when d 1 = 2 or 3? (Conjecturally, it is (−1) (d−1)( d−2) /2+1 for d \u001d 0.)",
   "clean_statement": null,
   "public_statement": "Problem 9.2. What about when d 1 = 2 or 3? (Conjecturally, it is (−1) (d−1)( d−2) /2+1 for d\n 0.)",
   "evidence": "The assigned record is Problem 9.2 in the AIM problem list *Degenerations in Algebraic Geometry*. The surrounding Problem 9 concerns real irreducible rational plane curves of degree \\(d\\) through",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 233,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0235": {
   "statement_status": "exact",
   "original_statement": "Problem 10 \n\n(Suggested by Jake) Consider a line bundle L on P1. Suppose we want to study the Grass-mannian of grd ⊂ Γ(P1, L). We can consider conditions on grd's, e.g. as follows. Pick a point \n\nx ∈ P1, and considering the vanishing flag at x. This defines a complete flag, by stratifying by the degree of vanishing at x. You can then impose Schubert conditions on the linear series (which concretely means imposing conditions on the vanishing orders). 5These Schubert conditions are indexed by partitions. More generally, we can do this at several points x1,..., xs with several partitions λ1,..., λ s. Let S be the set of grd satisfying all these conditions, i.e. the intersection of all the Schubert conditions. \n\nTheorem 10.1 (Eisenbud-Harris '80s). We have codim S = ∑i |λi|.\n\nThis has an interesting relation to real algebraic geometry. \n\nTheorem 10.2 (Muken-Tarasov-Varchenko). If all the x i are real, and ∑i |λi| = dim {grd},then S consists of real reduced points. \n\nNow suppose that S is a curve rather than a finite set. \n\nTheorem 10.3. If all x i are real and ∑i |λi| = dim {grd} − 1, then S (R) ⊂ S smooth.",
   "clean_statement": "Problem 10\n\n(Suggested by Jake) Consider a line bundle L on P1. Suppose we want to study the Grass-mannian of grd ⊂ Γ(P1, L). We can consider conditions on grd's, e.g. as follows. Pick a point\n\nx ∈ P1, and considering the vanishing flag at x. This defines a complete flag, by stratifying by the degree of vanishing at x. You can then impose Schubert conditions on the linear series (which concretely means imposing conditions on the vanishing orders). 5These Schubert conditions are indexed by partitions. More generally, we can do this at several points x1,..., xs with several partitions λ1,..., λ s. Let S be the set of grd satisfying all these conditions, i.e. the intersection of all the Schubert conditions.\n\nTheorem 10.1 (Eisenbud-Harris '80s). We have codim S = ∑i |λi|.\n\nThis has an interesting relation to real algebraic geometry.\n\nTheorem 10.2 (Muken-Tarasov-Varchenko). If all the x i are real, and ∑i |λi| = dim {grd},then S consists of real reduced points.\n\nNow suppose that S is a curve rather than a finite set.\n\nTheorem 10.3. If all x i are real and ∑i |λi| = dim {grd} − 1, then S (R) ⊂ S smooth.",
   "public_statement": "Problem 10\n\n(Suggested by Jake) Consider a line bundle L on P1. Suppose we want to study the Grass-mannian of grd ⊂ Γ(P1, L). We can consider conditions on grd's, e.g. as follows. Pick a point\n\nx ∈ P1, and considering the vanishing flag at x. This defines a complete flag, by stratifying by the degree of vanishing at x. You can then impose Schubert conditions on the linear series (which concretely means imposing conditions on the vanishing orders). 5These Schubert conditions are indexed by partitions. More generally, we can do this at several points x1,..., xs with several partitions λ1,..., λ s. Let S be the set of grd satisfying all these conditions, i.e. the intersection of all the Schubert conditions.\n\nTheorem 10.1 (Eisenbud-Harris '80s). We have codim S = ∑i |λi|.\n\nThis has an interesting relation to real algebraic geometry.\n\nTheorem 10.2 (Muken-Tarasov-Varchenko). If all the x i are real, and ∑i |λi| = dim {grd},then S consists of real reduced points.\n\nNow suppose that S is a curve rather than a finite set.\n\nTheorem 10.3. If all x i are real and ∑i |λi| = dim {grd} − 1, then S (R) ⊂ S smooth.",
   "evidence": "The source is the AIM workshop list *Degenerations in Algebraic Geometry*, Problem 10. The extracted record has several OCR errors: `grd` means \\(g^r_d\\), \\(\\Gamma(\\mathbf P^1,L)\\) means \\(H^0(\\mathbf P^1,L)\\), and “Muken” is Mukhin. The original PDF also shows that the question “Is \\(S\\) smooth over \\(\\mathbf C\\)?” is the following record, Problem 10.4, and is not part of this record.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 234,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0236": {
   "statement_status": "exact",
   "original_statement": "Problem 10.4. Is S smooth? (i.e. are there singular points over C?) \n\nThere doesn't seem to be any reason that this should be true, but in the (admittedly few) examples that I've explored, I've found no singular examples. An idea for an approach is to use tautological bundles on M0,s to describe equations characterizing the locus in M0,s where S is singular (and perhaps show that this is away from the points where all the xi are real). \n\nExample 10.5. I've already checked the simplest case, e.g. G(2, 5) with the partitions s = 5points all being a single box. The next simplest cases would be with longer partitions, or G(2, 6). In this case, you can write the equations for the singular locus as a sum of squares (an obvious way to prove that there are no real solutions). In the example I looked at, the squares had the form ( xi − x j)2, which suggests a boundary geometric interpretation (points colliding). B. Osserman suggests:",
   "clean_statement": "Problem 10.4. Is S smooth? (i.e. are there singular points over C?)\n\nThere doesn't seem to be any reason that this should be true, but in the (admittedly few) examples that I've explored, I've found no singular examples. An idea for an approach is to use tautological bundles on M0,s to describe equations characterizing the locus in M0,s where S is singular (and perhaps show that this is away from the points where all the xi are real).\n\nExample 10.5. I've already checked the simplest case, e.g. G(2, 5) with the partitions s = 5points all being a single box. The next simplest cases would be with longer partitions, or G(2, 6). In this case, you can write the equations for the singular locus as a sum of squares (an obvious way to prove that there are no real solutions). In the example I looked at, the squares had the form ( xi − x j)2, which suggests a boundary geometric interpretation (points colliding). B. Osserman suggests:",
   "public_statement": "Problem 10.4. Is S smooth? (i.e. are there singular points over C?)\n\nThere doesn't seem to be any reason that this should be true, but in the (admittedly few) examples that I've explored, I've found no singular examples. An idea for an approach is to use tautological bundles on M0,s to describe equations characterizing the locus in M0,s where S is singular (and perhaps show that this is away from the points where all the xi are real).\n\nExample 10.5. I've already checked the simplest case, e.g. G(2, 5) with the partitions s = 5points all being a single box. The next simplest cases would be with longer partitions, or G(2, 6). In this case, you can write the equations for the singular locus as a sum of squares (an obvious way to prove that there are no real solutions). In the example I looked at, the squares had the form ( xi − x j)2, which suggests a boundary geometric interpretation (points colliding). B. Osserman suggests:",
   "evidence": "The source is Problem 10.4 in the AIM workshop list *Degenerations in algebraic geometry*. The preceding record supplies notation that is missing from the extracted problem. Let \\[ E=H^0(\\mathbb P^1,\\mathcal O_{\\mathbb P^1}(d)),\\qquad n=d+1,\\qquad k=r+1, \\] so that \\(g^r_d\\)'s are points of \\(G=G(k,E)\\), of dimension \\(N=k(n-k)\\). For \\(x\\in\\mathbb P^1\\), the filtration by order of vanishing at \\(x\\) is a complete osculating flag \\(F_\\bullet(x)\\). Given pairwise distinct points \\(x_1,\\ldots,x_s\\) and partitions \\(\\lambda_i\\subseteq k\\times(n-k)\\), form the **scheme-theoretic** intersection \\[ S=S(\\lambda_\\bullet;x_\\bullet) :=\\bigcap_{i=1}^s\\Omega_{\\lambda_i}(F_\\bullet(x_i)) \\subseteq G. \\] The case in the problem has \\[ \\sum_i|\\lambda_i|=N-1, \\tag{1.1} \\] so the Eisenbud--Harris proper-intersection theorem makes \\(S\\) a projective Cohen--Macaulay curve (when nonempty). All \\(x_i\\) are...",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 235,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0237": {
   "statement_status": "exact",
   "original_statement": "Problem 10.6. Reprove Mukhin-Tarasov-Varchenko theorem using degenerations. Extend it to cominiscule G /P. \n\nNotes by Tony Feng 6",
   "clean_statement": "Problem 10.6. Reprove Mukhin-Tarasov-Varchenko theorem using degenerations. Extend it to cominiscule G /P.\n\nNotes by Tony Feng 6",
   "public_statement": "Problem 10.6. Reprove Mukhin-Tarasov-Varchenko theorem using degenerations. Extend it to cominiscule G /P.\n\nNotes by Tony Feng 6",
   "evidence": "The exact extracted record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 236,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0238": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 1Question (Hailong Dao). Let X be the pun tur ed sp e trum on a lo al ring (R, m).What is the obstru tion the ory for splitting of ve tor bund les over su h a s heme? More sp e i\u001c ally, let X = Spec( R) \\ { m} for (R, m) lo al of dimension d. What is the obstru tion to splitting of rank d − 1 ve tor bundles on X?Related Question (Sat ya Mandal). Let X be a s heme and E be a ve tor bund le. What is the obstru tion to obtaining a surje tion E ։ OX?Related Question (Ara vind Asok). In the a\u001ene ase, one has a fair ide a of the obstru tion the ory for pr oje tive mo dules. What is the obstru tion the ory for gener al lasses of mo dules? Related Question (Hailong Dao). Is ther e an Euler lass gr oup for re\u001dexive mo d-ules over a regular lo al ring (R, m)?Related Question. What is the role of A1-homotopy the ory in the non-a\u001ene ase?",
   "clean_statement": null,
   "public_statement": "Question 1Question (Hailong Dao). Let X be the pun tur ed sp e trum on a lo al ring (R, m).What is the obstru tion the ory for splitting of ve tor bund les over su h a s heme? More sp e i\n ally, let X = Spec( R) \\ { m} for (R, m) lo al of dimension d. What is the obstru tion to splitting of rank d − 1 ve tor bundles on X?Related Question (Sat ya Mandal). Let X be a s heme and E be a ve tor bund le. What is the obstru tion to obtaining a surje tion E ։ OX?Related Question (Ara vind Asok). In the a\nne ase, one has a fair ide a of the obstru tion the ory for pr oje tive mo dules. What is the obstru tion the ory for gener al lasses of mo dules? Related Question (Hailong Dao). Is ther e an Euler lass gr oup for re\nexive mo d-ules over a regular lo al ring (R, m)?Related Question. What is the role of A1-homotopy the ory in the non-a\nne ase?",
   "evidence": "**Canonical record.** This is record `AIM-ALGEBRAIC_GEOMETRY-0238`, zero-based index 237 in `aim-algebraic-geometry-notes.json`, extracted from Question 1 of the AIM workshop problem list *Projective modules and \\(A^1\\)-homotopy theory* (May 2014). The following is an ASCII-escaped, reversibly exact JSON serialization of the source record. In particular, the OCR corruption and control character `\\u001c` have not been silently repaired.",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 237,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0239": {
   "statement_status": "exact",
   "original_statement": "Question 2Question (b y p opular demand). What is the onne tion betwe en Euler lass gr oups and Chow-Witt gr oups?",
   "clean_statement": "Question 2Question (b y p opular demand). What is the onne tion betwe en Euler lass gr oups and Chow-Witt gr oups?",
   "public_statement": "Question 2Question (b y p opular demand). What is the onne tion betwe en Euler lass gr oups and Chow-Witt gr oups?",
   "evidence": "The canonical JSON record is visibly damaged by PDF extraction. The original AIM workshop PDF, *Projective modules and \\(\\mathbb A^1\\)-homotopy theory*, gives the following text on page 2:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 238,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0240": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 3Question (Christian Haesemey er). What ar e \u0010motivi lo al systems\u0011? Can one de\u001cne motivi Atiyah lasses?",
   "clean_statement": null,
   "public_statement": "Question 3Question (Christian Haesemey er). What ar e [U+0010]motivi lo al systems[U+0011]? Can one de\nne motivi Atiyah lasses?",
   "evidence": "The canonical JSON record is damaged by PDF extraction: it contains control characters and loses several letters in “motivic local systems,” “define,” and “classes.” Inspection of the original AIM PDF recovers the wording on PDF page 2 as",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 239,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0241": {
   "statement_status": "exact",
   "original_statement": "Question 4Question (Krone ker-prop osed by Sat ya Mandal). Are urves in Ank set the or eti omplete interse tion ide als?",
   "clean_statement": "Question 4Question (Krone ker-prop osed by Sat ya Mandal). Are urves in Ank set the or eti omplete interse tion ide als?",
   "public_statement": "Question 4Question (Krone ker-prop osed by Sat ya Mandal). Are urves in Ank set the or eti omplete interse tion ide als?",
   "evidence": "The canonical record is entry 240 (zero-based) of `aim-algebraic-geometry-notes.json`, from the AIM workshop *Projective modules and \\(A^1\\)-homotopy theory*. Its `problem` field is preserved here verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 240,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0242": {
   "statement_status": "exact",
   "original_statement": "Question 5Question (Sat ya Mandal). Let I ⊆ k[X1, X 2,..., X n] = R be an ide al. Then is μ(I) = μ( I \n\n> I2)?",
   "clean_statement": "Question 5Question (Sat ya Mandal). Let I ⊆ k[X1, X 2,..., X n] = R be an ide al. Then is μ(I) = μ( I\n\n> I2)?",
   "public_statement": "Question 5Question (Sat ya Mandal). Let I ⊆ k[X1, X 2,..., X n] = R be an ide al. Then is μ(I) = μ( I\n\n> I2)?",
   "evidence": "This record is Question 5 in the problem list from the AIM workshop *Projective modules and \\(A^1\\)-homotopy theory*, held May 5--9, 2014. The available JSON has substantial OCR damage, so the statement below was checked against the workshop PDF.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 241,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0243": {
   "statement_status": "exact",
   "original_statement": "Question 6Question (Anastasia Sta vro va). Can A1-homotopy be extende d over gener al smo oth s hemes, e.g. over a dvr? Else explain the issues.",
   "clean_statement": "Question 6Question (Anastasia Sta vro va). Can A1-homotopy be extende d over gener al smo oth s hemes, e.g. over a dvr? Else explain the issues.",
   "public_statement": "Question 6Question (Anastasia Sta vro va). Can A1-homotopy be extende d over gener al smo oth s hemes, e.g. over a dvr? Else explain the issues.",
   "evidence": "The canonical JSON record is affected by OCR errors. The supplied AIM workshop PDF, *Projective modules and \\(\\mathbb A^1\\)-homotopy theory*, gives the following text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 242,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0244": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 7Question (Ara vind Asok). Fix an inte ger d. For ertain d, Mohan Kumar on-stru ts pr oje tive mo dules of rank d − 2 whi h ar e stably fr ee but not fr ee over asmo oth a\u001ene variety X of dimension d over an algebr ai al ly lose d \u001celd. Is this known for al l d? Is ther e a pattern?",
   "clean_statement": null,
   "public_statement": "Question 7Question (Ara vind Asok). Fix an inte ger d. For ertain d, Mohan Kumar on-stru ts pr oje tive mo dules of rank d − 2 whi h ar e stably fr ee but not fr ee over asmo oth a\nne variety X of dimension d over an algebr ai al ly lose d\neld. Is this known for al l d? Is ther e a pattern?",
   "evidence": "The canonical JSON record is visibly damaged by PDF extraction. Its raw `problem` field is:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 243,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0245": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 8Question (Jean Fasel). Let X be a smo oth a\u001ene k-variety of odd dimension d.Let P be a ve tor bund le of rank d. Do es cd(P ) ∈ CH d(X) dete t existen e of afr ee summand of rank 1 for P?",
   "clean_statement": "**Question 8. Question (Jean Fasel).** Let \\(X\\) be a smooth affine\n\\(k\\)-variety of odd dimension \\(d\\). Let \\(P\\) be a vector bundle of rank\n\\(d\\). Does \\(c_d(P)\\in CH^d(X)\\) detect existence of a free summand of\nrank 1 for \\(P\\)?",
   "public_statement": "Question 8Question (Jean Fasel). Let X be a smo oth a\nne k-variety of odd dimension d.Let P be a ve tor bund le of rank d. Do es cd(P ) ∈ CH d(X) dete t existen e of afr ee summand of rank 1 for P?",
   "evidence": "The JSON record has OCR damage (`a\\u001ene`, `ve tor bund le`, and missing spaces). Page 3 of the AIM workshop PDF gives the following text:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 244,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0246": {
   "statement_status": "exact",
   "original_statement": "Question 9Question (Kirsten Wi kelgren). Is ther e a Serr e sp e tr al se quen e for Mor el's HA1 \n\n> (,Z)?",
   "clean_statement": "Question 9Question (Kirsten Wi kelgren). Is ther e a Serr e sp e tr al se quen e for Mor el's HA1\n\n> (,Z)?",
   "public_statement": "Question 9Question (Kirsten Wi kelgren). Is ther e a Serr e sp e tr al se quen e for Mor el's HA1\n\n> (,Z)?",
   "evidence": "The canonical JSON record is visibly damaged by PDF extraction: names and ordinary words are split, and the homology notation has lost its subscript and argument. The original AIM problem-list PDF was therefore checked directly. Question 9 reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 245,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0247": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 10 Question (Anand Sa wan t). When ar e the ab ove questions wel l-de\u001cne d over singu-lar varieties? When have they been onsider ed and what is their status? 4 SARANG SANE",
   "clean_statement": "**Question 10. Question (Anand Sawant). When are the above questions\nwell-defined over singular varieties? When have they been considered and\nwhat is their status?**",
   "public_statement": "Question 10 Question (Anand Sa wan t). When ar e the ab ove questions wel l-de\nne d over singu-lar varieties? When have they been onsider ed and what is their status? 4 SARANG SANE",
   "evidence": "The canonical record is Question 10 from the AIM workshop list *Projective modules and \\(A^1\\)-homotopy theory*. The JSON extraction is visibly damaged by OCR. Inspection of the original PDF gives the following text:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 246,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0248": {
   "statement_status": "exact",
   "original_statement": "Question 11 Question (P aul Balmer). Using op er ations like ⊗, ∧ and taking ones, is ther ea onstru tion so that one obtains the Koszul omplex K•(s) fr om the omplex... 0 → P s \n\n> −→A→0...?Do es this gener alize to tensor triangulate d ate gories?",
   "clean_statement": "Question 11 Question (P aul Balmer). Using op er ations like ⊗, ∧ and taking ones, is ther ea onstru tion so that one obtains the Koszul omplex K•(s) fr om the omplex... 0 → P s\n\n> −→A→0...?Do es this gener alize to tensor triangulate d ate gories?",
   "public_statement": "Question 11 Question (P aul Balmer). Using op er ations like ⊗, ∧ and taking ones, is ther ea onstru tion so that one obtains the Koszul omplex K•(s) fr om the omplex... 0 → P s\n\n> −→A→0...?Do es this gener alize to tensor triangulate d ate gories?",
   "evidence": "The canonical record is Question 11 from the AIM workshop list *Projective modules and \\(A^1\\)-homotopy theory*. The repository copy has severe PDF extraction errors: in particular, “cones” became “ones,” “construction” lost several letters, and “categories” lost its initial letter. I checked the official PDF and recover the statement as follows (typographical spacing and the displayed arrow have been normalized, but no mathematical content has been changed):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 247,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0249": {
   "statement_status": "exact",
   "original_statement": "Question 12 Question (Anastasia Sta vro va). When G is a line ar redu ed/algebr ai gr oup, in alar ge numb er of ases, it is known on a ase-byase basis that ΠA1 \n\n> 0(G)\n\nis ab elian. Is this always true? Can this be obtaine d in a uniform way when it is true?",
   "clean_statement": "Question 12 Question (Anastasia Sta vro va). When G is a line ar redu ed/algebr ai gr oup, in alar ge numb er of ases, it is known on a ase-byase basis that ΠA1\n\n> 0(G)\n\nis ab elian. Is this always true? Can this be obtaine d in a uniform way when it is true?",
   "public_statement": "Question 12 Question (Anastasia Sta vro va). When G is a line ar redu ed/algebr ai gr oup, in alar ge numb er of ases, it is known on a ase-byase basis that ΠA1\n\n> 0(G)\n\nis ab elian. Is this always true? Can this be obtaine d in a uniform way when it is true?",
   "evidence": "The record is Question 12 from the AIM workshop *Projective modules and \\(A^1\\)-homotopy theory* (May 5--9, 2014). The official PDF verifies the following text (with the typography modernized but not the wording):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 248,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0250": {
   "statement_status": "exact",
   "original_statement": "Question 13 Question (Madha v Nori). Given positive inte gers d, r and X a CW omplex with dim X = d and lasses ci ∈ H2i(X, Z) for i = 1, 2,..., r, su h that the image of ci is 0 in H2i(X, Z \n\n> mZ)\n\nfor al l m, do es ther e exist a omplex ve tor bund le E with rank( E) = r and ci(E) = ci?",
   "clean_statement": "Question 13 Question (Madha v Nori). Given positive inte gers d, r and X a CW omplex with dim X = d and lasses ci ∈ H2i(X, Z) for i = 1, 2,..., r, su h that the image of ci is 0 in H2i(X, Z\n\n> mZ)\n\nfor al l m, do es ther e exist a omplex ve tor bund le E with rank( E) = r and ci(E) = ci?",
   "public_statement": "Question 13 Question (Madha v Nori). Given positive inte gers d, r and X a CW omplex with dim X = d and lasses ci ∈ H2i(X, Z) for i = 1, 2,..., r, su h that the image of ci is 0 in H2i(X, Z\n\n> mZ)\n\nfor al l m, do es ther e exist a omplex ve tor bund le E with rank( E) = r and ci(E) = ci?",
   "evidence": "The extracted record is Question 13 from the AIM problem list *Projective modules and \\(A^1\\)-homotopy theory*. The PDF was checked directly because the JSON text has severe OCR damage. The recovered statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 249,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0251": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 14 Question (Madha v Nori). Let X be a d-dimensional smo oth a\u001ene variety. Let α ∈ F d−1K0(X). Then do es ther e exist a ve tor bund le E of rank d − 1 su h that [E] = α?E-mail addr ess: sarangsanemath\bgmail. o m",
   "clean_statement": null,
   "public_statement": "Question 14 Question (Madha v Nori). Let X be a d-dimensional smo oth a\nne variety. Let α ∈ F d−1K0(X). Then do es ther e exist a ve tor bund le E of rank d − 1 su h that [E] = α?E-mail addr ess: sarangsanemath[U+0008]gmail. o m",
   "evidence": "The official problem list prints the following (Question 14, attributed to Madhav Nori):",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 250,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0252": {
   "statement_status": "exact",
   "original_statement": "Basepoint free theorem\n\nSuppose that $k = \\bar{k}$, $\\text{char } k = p > 0$. Let $X/k$ be a terminal threefold, and $A$ an ample $\\mathbb Q$-divisor on $X$. If $K_X+A$ is nef, must it be semiample?",
   "clean_statement": "Basepoint free theorem\n\nSuppose that $k = \\bar{k}$, $\\text{char } k = p > 0$. Let $X/k$ be a terminal threefold, and $A$ an ample $\\mathbb Q$-divisor on $X$. If $K_X+A$ is nef, must it be semiample?",
   "public_statement": "Basepoint free theorem\n\nSuppose that $k = \\bar{k}$, $\\text{char } k = p > 0$. Let $X/k$ be a terminal threefold, and $A$ an ample $\\mathbb Q$-divisor on $X$. If $K_X+A$ is nef, must it be semiample?",
   "evidence": "The canonical AIM record is problem 1.1 in the workshop section “Fundamental MMP theorems”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 251,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0253": {
   "statement_status": "exact",
   "original_statement": "Termination in dimension 3\n\nDo klt flips terminate in dimension 3?",
   "clean_statement": "Termination in dimension 3\n\nDo klt flips terminate in dimension 3?",
   "public_statement": "Termination in dimension 3\n\nDo klt flips terminate in dimension 3?",
   "evidence": "The canonical record is AIM Problem List 1.2 from the 2013 workshop *The minimal model program in characteristic \\(p\\)*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 252,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0254": {
   "statement_status": "exact",
   "original_statement": "Connectedness\n\nIs there a positive-characteristic analog of the connectedness lemma?",
   "clean_statement": "Connectedness\n\nIs there a positive-characteristic analog of the connectedness lemma?",
   "public_statement": "Connectedness\n\nIs there a positive-characteristic analog of the connectedness lemma?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 253,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0255": {
   "statement_status": "exact",
   "original_statement": "Terminal and Cohen-Macaulay\n\nAre (log) terminal singularities Cohen-Macaulay? Rational?",
   "clean_statement": "Terminal and Cohen-Macaulay\n\nAre (log) terminal singularities Cohen-Macaulay? Rational?",
   "public_statement": "Terminal and Cohen-Macaulay\n\nAre (log) terminal singularities Cohen-Macaulay? Rational?",
   "evidence": "The canonical record is AIM Problem 2.1 from the workshop *The minimal model program in characteristic \\(p\\)*, section “Singularities in char \\(p\\).” Its problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 254,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0256": {
   "statement_status": "exact",
   "original_statement": "Normality of plt pairs\n\nSuppose that $(X,D = S+B)$ is a plt pair. Must $S$ be normal?",
   "clean_statement": "Normality of plt pairs\n\nSuppose that $(X,D = S+B)$ is a plt pair. Must $S$ be normal?",
   "public_statement": "Normality of plt pairs\n\nSuppose that $(X,D = S+B)$ is a plt pair. Must $S$ be normal?",
   "evidence": "The canonical record is number 2.2 in the AIM workshop *The minimal model program in characteristic \\(p\\)*, section “Singularities in char \\(p\\)”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 255,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0257": {
   "statement_status": "exact",
   "original_statement": "Fano and MDS\n\nIs every Fano variety a Mori Dream Space?",
   "clean_statement": "Fano and MDS\n\nIs every Fano variety a Mori Dream Space?",
   "public_statement": "Fano and MDS\n\nIs every Fano variety a Mori Dream Space?",
   "evidence": "The canonical AIM record is from the workshop *The minimal model program in characteristic \\(p\\)*, section “Sections and section rings,” Problem 4.1. Its problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 256,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0258": {
   "statement_status": "exact",
   "original_statement": "Invariance of plurigenera\n\nWhat is the status of invariance of plurigenera?",
   "clean_statement": "Invariance of plurigenera\n\nWhat is the status of invariance of plurigenera?",
   "public_statement": "Invariance of plurigenera\n\nWhat is the status of invariance of plurigenera?",
   "evidence": "The canonical AIM record, from the 2013 workshop *The minimal model program in characteristic \\(p\\)*, says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 257,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0259": {
   "statement_status": "exact",
   "original_statement": "General type in families\n\nSuppose that $\\mathcal X \\to \\Delta$ is a family, and a special fiber is of general type. Does it follow that a general fiber is of general type?",
   "clean_statement": "General type in families\n\nSuppose that $\\mathcal X \\to \\Delta$ is a family, and a special fiber is of general type. Does it follow that a general fiber is of general type?",
   "public_statement": "General type in families\n\nSuppose that $\\mathcal X \\to \\Delta$ is a family, and a special fiber is of general type. Does it follow that a general fiber is of general type?",
   "evidence": "The canonical record is `aim-algebraic-geometry-notes.json`, zero-based index 258, Problem 4.4 in the AIM workshop *The minimal model program in characteristic \\(p\\)*. The exact problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 258,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0260": {
   "statement_status": "exact",
   "original_statement": "What if we look at $\\text{dim } S^0(mK_X+A)$ instead?",
   "clean_statement": "What if we look at $\\text{dim } S^0(mK_X+A)$ instead?",
   "public_statement": "What if we look at $\\text{dim } S^0(mK_X+A)$ instead?",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 259,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0261": {
   "statement_status": "exact",
   "original_statement": "Embedding by $S^0(mK_X)$\n\nSuppose that $X$ is of general type. Is there an effective constant $m = m(n)$ such that $S^0(mK_X)$ (the canonical linear system of Schwede) defines a birational map? What about the usual linear system $|mK_X|$?",
   "clean_statement": "Embedding by $S^0(mK_X)$\n\nSuppose that $X$ is of general type. Is there an effective constant $m = m(n)$ such that $S^0(mK_X)$ (the canonical linear system of Schwede) defines a birational map? What about the usual linear system $|mK_X|$?",
   "public_statement": "Embedding by $S^0(mK_X)$\n\nSuppose that $X$ is of general type. Is there an effective constant $m = m(n)$ such that $S^0(mK_X)$ (the canonical linear system of Schwede) defines a birational map? What about the usual linear system $|mK_X|$?",
   "evidence": "This is Problem 4.5 in the AIM workshop list *The minimal model program in characteristic \\(p\\)*, section “Sections and section rings.” The canonical JSON record states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 260,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0262": {
   "statement_status": "exact",
   "original_statement": "Effective Fujita vanishing\n\nIs there an effective Fujita vanishing result in flat families? Suppose that $f : \\mathcal X \\to S$ is flat, $\\mathcal F$ is a coherent sheaf on $\\mathcal X$, and $\\mathcal L$ is $f$-ample. Does there exist a constant $m_0 = m_0(\\mathcal X,\\mathcal F,\\mathcal L)$ such that $H^i(X_s,(\\mathcal F \\otimes \\mathcal L^m \\mathcal )\\vert_{X_s} ) = 0$ if $s$ is any point of $S$, $m \\geq m_0$ and $\\mathcal M$ is any $f$-nef line bundle on $X$?",
   "clean_statement": "Effective Fujita vanishing\n\nIs there an effective Fujita vanishing result in flat families? Suppose that $f : \\mathcal X \\to S$ is flat, $\\mathcal F$ is a coherent sheaf on $\\mathcal X$, and $\\mathcal L$ is $f$-ample. Does there exist a constant $m_0 = m_0(\\mathcal X,\\mathcal F,\\mathcal L)$ such that $H^i(X_s,(\\mathcal F \\otimes \\mathcal L^m \\mathcal )\\vert_{X_s} ) = 0$ if $s$ is any point of $S$, $m \\geq m_0$ and $\\mathcal M$ is any $f$-nef line bundle on $X$?",
   "public_statement": "Effective Fujita vanishing\n\nIs there an effective Fujita vanishing result in flat families? Suppose that $f : \\mathcal X \\to S$ is flat, $\\mathcal F$ is a coherent sheaf on $\\mathcal X$, and $\\mathcal L$ is $f$-ample. Does there exist a constant $m_0 = m_0(\\mathcal X,\\mathcal F,\\mathcal L)$ such that $H^i(X_s,(\\mathcal F \\otimes \\mathcal L^m \\mathcal )\\vert_{X_s} ) = 0$ if $s$ is any point of $S$, $m \\geq m_0$ and $\\mathcal M$ is any $f$-nef line bundle on $X$?",
   "evidence": "The canonical record is AIM Problem List 4.6, “Effective Fujita vanishing,” in the section “Sections and section rings” of the workshop *The minimal model program in characteristic \\(p\\)*. Its body reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 261,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0263": {
   "statement_status": "exact",
   "original_statement": "Sections of nef $K_X+A$\n\nSuppose that $A$ is Cartier and ample, and $K_X+A$ is nef. Must $H^0(X,K_X+A)$ be nonzero?",
   "clean_statement": "Sections of nef $K_X+A$\n\nSuppose that $A$ is Cartier and ample, and $K_X+A$ is nef. Must $H^0(X,K_X+A)$ be nonzero?",
   "public_statement": "Sections of nef $K_X+A$\n\nSuppose that $A$ is Cartier and ample, and $K_X+A$ is nef. Must $H^0(X,K_X+A)$ be nonzero?",
   "evidence": "The canonical record is Problem 4.7 of the AIM workshop *The minimal model program in characteristic \\(p\\)* (2013), in the section “Sections and section rings.” The archived page attributes it to Cascini and states, with no preceding local convention:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 262,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0264": {
   "statement_status": "exact",
   "original_statement": "$F$-regularity and finite generation\n\nSuppose that $X$ is an $F$-regular variety (maybe not $\\mathbb Q$-Gorenstein). If $D$ is a Weil divisor, must $\\bigoplus_{m \\geq 0} \\mathcal O_X(mD)$ be finitely generated?",
   "clean_statement": "$F$-regularity and finite generation\n\nSuppose that $X$ is an $F$-regular variety (maybe not $\\mathbb Q$-Gorenstein). If $D$ is a Weil divisor, must $\\bigoplus_{m \\geq 0} \\mathcal O_X(mD)$ be finitely generated?",
   "public_statement": "$F$-regularity and finite generation\n\nSuppose that $X$ is an $F$-regular variety (maybe not $\\mathbb Q$-Gorenstein). If $D$ is a Weil divisor, must $\\bigoplus_{m \\geq 0} \\mathcal O_X(mD)$ be finitely generated?",
   "evidence": "The canonical record is Problem 4.8 in the AIM workshop list *The minimal model program in characteristic \\(p\\)*, section “Sections and section rings.” The live AIM page attributes the problem to Schwede and gives the following text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 263,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_GEOMETRY-0265": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Pulling back forms to a resolution\n\nSuppose that $X$ is log canonical, and that there exists a log resolution $f : \\tilde{X} \\to X$ which is an isomorphism over the smooth locus. Let $\\omega$ be an $(n-1)$-form on $X$. Does $f^\\ast \\omega\\vert_{X_{\\text{smooth}}}$ extend to an $(n-1)$-form on $\\tilde{X}$ with log poles along the exceptional locus?",
   "clean_statement": null,
   "public_statement": "Pulling back forms to a resolution\n\nSuppose that $X$ is log canonical, and that there exists a log resolution $f : \\tilde{X} \\to X$ which is an isomorphism over the smooth locus. Let $\\omega$ be an $(n-1)$-form on $X$. Does $f^\\ast \\omega\\vert_{X_{\\text{smooth}}}$ extend to an $(n-1)$-form on $\\tilde{X}$ with log poles along the exceptional locus?",
   "evidence": "This is Problem 5.1, “Pulling back forms to a resolution,” in the section “Other questions” of the AIM workshop *The minimal model program in characteristic \\(p\\)*. The exact database text is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 264,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0266": {
   "statement_status": "exact",
   "original_statement": "Universal lower bounds for Seshadri constants\n\nSuppose that $X$ is a smooth variety over an uncountable field $k$ of positive characteristic. Does there exist a constant $c = c(n)$ such that $\\epsilon(L,x) \\geq c$ for any ample divisor $L$ and very general point $x$ of $X$?",
   "clean_statement": "Universal lower bounds for Seshadri constants\n\nSuppose that $X$ is a smooth variety over an uncountable field $k$ of positive characteristic. Does there exist a constant $c = c(n)$ such that $\\epsilon(L,x) \\geq c$ for any ample divisor $L$ and very general point $x$ of $X$?",
   "public_statement": "Universal lower bounds for Seshadri constants\n\nSuppose that $X$ is a smooth variety over an uncountable field $k$ of positive characteristic. Does there exist a constant $c = c(n)$ such that $\\epsilon(L,x) \\geq c$ for any ample divisor $L$ and very general point $x$ of $X$?",
   "evidence": "The canonical record is Problem 5.2 in the “Other questions” section of the 2013 AIM workshop *The minimal model program in characteristic \\(p\\)*. The archived page attributes the question to Mustaţă and states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 265,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0267": {
   "statement_status": "exact",
   "original_statement": "Nefness under mod p reduction\n\nSuppose that $X$ is a variety over $k$, with $\\text{char } k = 0$, and $L$ is a nef divisor on $X$. Must $L_p$ be nef on $X_p$ for infinitely many $p$? What about in the case $L= K_X$? What if ``nef'' is replaced by ``semiample''?",
   "clean_statement": "Nefness under mod p reduction\n\nSuppose that $X$ is a variety over $k$, with $\\text{char } k = 0$, and $L$ is a nef divisor on $X$. Must $L_p$ be nef on $X_p$ for infinitely many $p$? What about in the case $L= K_X$? What if ``nef'' is replaced by ``semiample''?",
   "public_statement": "Nefness under mod p reduction\n\nSuppose that $X$ is a variety over $k$, with $\\text{char } k = 0$, and $L$ is a nef divisor on $X$. Must $L_p$ be nef on $X_p$ for infinitely many $p$? What about in the case $L= K_X$? What if ``nef'' is replaced by ``semiample''?",
   "evidence": "The canonical AIM record is Problem 5.3, “Nefness under mod \\(p\\) reduction,” from the workshop *The minimal model program in characteristic \\(p\\)*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 266,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0268": {
   "statement_status": "exact",
   "original_statement": "Singularities and point-counting\n\nSuppose that $X$ is a variety over a finite field. Is there any relation between the singularities of $X$ and the number of points over $\\mathbb F_q$? For example, suppose that $X$ is Fano and $F$-regular.",
   "clean_statement": "Singularities and point-counting\n\nSuppose that $X$ is a variety over a finite field. Is there any relation between the singularities of $X$ and the number of points over $\\mathbb F_q$? For example, suppose that $X$ is Fano and $F$-regular.",
   "public_statement": "Singularities and point-counting\n\nSuppose that $X$ is a variety over a finite field. Is there any relation between the singularities of $X$ and the number of points over $\\mathbb F_q$? For example, suppose that $X$ is Fano and $F$-regular.",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 267,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_GEOMETRY-0269": {
   "statement_status": "exact",
   "original_statement": "Let us assume that the true probability distribution $q$ of a data generating process is outside the family of models $\\mathcal{M}$ that are being considered. Under what conditions does the posterior distribution converge to a distribution with smallest KL divergence to the true distribution $q$?",
   "clean_statement": "Let us assume that the true probability distribution $q$ of a data generating process is outside the family of models $\\mathcal{M}$ that are being considered. Under what conditions does the posterior distribution converge to a distribution with smallest KL divergence to the true distribution $q$?",
   "public_statement": "Let us assume that the true probability distribution $q$ of a data generating process is outside the family of models $\\mathcal{M}$ that are being considered. Under what conditions does the posterior distribution converge to a distribution with smallest KL divergence to the true distribution $q$?",
   "evidence": "The canonical record is Problem 11.1 in the section “When true distribution is outside the model” of the AIM workshop *Singular learning theory: connecting algebraic geometry and model selection in statistics*, held 12--16 December 2011. The official workshop report states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 268,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0270": {
   "statement_status": "exact",
   "original_statement": "Understand asymptotics of the stochastic complexity in the case where the true distribution is not in any model under consideration.",
   "clean_statement": "Understand asymptotics of the stochastic complexity in the case where the true distribution is not in any model under consideration.",
   "public_statement": "Understand asymptotics of the stochastic complexity in the case where the true distribution is not in any model under consideration.",
   "evidence": "The canonical record is source index 269 of `aim-algebraic-geometry-notes.json`, from the AIM workshop *Singular learning theory*, section “When true distribution is outside the model.” It states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 269,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0271": {
   "statement_status": "exact",
   "original_statement": "As a concrete example, suppose the data is generated from a log normal distribution, and suppose we try to fit two model classes -\n\n a) a normal distribution\n\n b) a gamma distribution\n\nUnderstand the behavior of asymptotics in this case.",
   "clean_statement": "As a concrete example, suppose the data is generated from a log normal distribution, and suppose we try to fit two model classes -\n\n a) a normal distribution\n\n b) a gamma distribution\n\nUnderstand the behavior of asymptotics in this case.",
   "public_statement": "As a concrete example, suppose the data is generated from a log normal distribution, and suppose we try to fit two model classes -\n\n a) a normal distribution\n\n b) a gamma distribution\n\nUnderstand the behavior of asymptotics in this case.",
   "evidence": "The canonical record is item 11.3 in the AIM workshop list *Singular learning theory*, under “When true distribution is outside the model.” Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 270,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_GEOMETRY-0272": {
   "statement_status": "exact",
   "original_statement": "As another example, let us assume the true model is a linear regression. $Y = exp(X) + \\epsilon$ where $\\epsilon \\sim N(\\mu,\\sigma^2)$. Let us assume that we try to fit a linear regression model to the data, i.e. $y=bx + \\epsilon$.",
   "clean_statement": "As another example, let us assume the true model is a linear regression. $Y = exp(X) + \\epsilon$ where $\\epsilon \\sim N(\\mu,\\sigma^2)$. Let us assume that we try to fit a linear regression model to the data, i.e. $y=bx + \\epsilon$.",
   "public_statement": "As another example, let us assume the true model is a linear regression. $Y = exp(X) + \\epsilon$ where $\\epsilon \\sim N(\\mu,\\sigma^2)$. Let us assume that we try to fit a linear regression model to the data, i.e. $y=bx + \\epsilon$.",
   "evidence": "The canonical record is Problem 11.4 in the section “When true distribution is outside the model” of the AIM workshop *Singular learning theory: connecting algebraic geometry and model selection in statistics*. Its exact extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 271,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0273": {
   "statement_status": "exact",
   "original_statement": "Given an ideal $I$ of $\\Z[x]$, the HSL numbers are the ``indices of nilpotency\" of the action of Frobenius on $H^\\text{top}_{(\\underline{x})}\\left( \\Z_p[\\underline{x}]/I_p \\right)$, for primes $p$. Is the limsup of the HSL numbers 1?",
   "clean_statement": "Given an ideal $I$ of $\\Z[x]$, the HSL numbers are the ``indices of nilpotency\" of the action of Frobenius on $H^\\text{top}_{(\\underline{x})}\\left( \\Z_p[\\underline{x}]/I_p \\right)$, for primes $p$. Is the limsup of the HSL numbers 1?",
   "public_statement": "Given an ideal $I$ of $\\Z[x]$, the HSL numbers are the ``indices of nilpotency\" of the action of Frobenius on $H^\\text{top}_{(\\underline{x})}\\left( \\Z_p[\\underline{x}]/I_p \\right)$, for primes $p$. Is the limsup of the HSL numbers 1?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 272,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0274": {
   "statement_status": "exact",
   "original_statement": "Realize effective computations of numerical $F$-invariants.",
   "clean_statement": "Realize effective computations of numerical $F$-invariants.",
   "public_statement": "Realize effective computations of numerical $F$-invariants.",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 273,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0275": {
   "statement_status": "exact",
   "original_statement": "Do Howald's nondegeneracy conditions with respect to the Newton polyhedron on the term ideal provide conditions under which $\\operatorname{fpt}(\\text{polynomial}) = \\operatorname{fpt}(\\text{associated term ideal})$?",
   "clean_statement": "Do Howald's nondegeneracy conditions with respect to the Newton polyhedron on the term ideal provide conditions under which $\\operatorname{fpt}(\\text{polynomial}) = \\operatorname{fpt}(\\text{associated term ideal})$?",
   "public_statement": "Do Howald's nondegeneracy conditions with respect to the Newton polyhedron on the term ideal provide conditions under which $\\operatorname{fpt}(\\text{polynomial}) = \\operatorname{fpt}(\\text{associated term ideal})$?",
   "evidence": "This is stored as Problem 11.15 in the canonical record from the AIM workshop *Relating test ideals and multiplier ideals*, in the section “Characteristic \\(p>0\\) invariants.” The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 274,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0276": {
   "statement_status": "exact",
   "original_statement": "For a graded sequence of ideals $I_m$ of a ring $R$, does $\\lim \\limits_{m \\to \\infty} m \\cdot \\operatorname{fpt}( I_m) $ exist?",
   "clean_statement": "For a graded sequence of ideals $I_m$ of a ring $R$, does $\\lim \\limits_{m \\to \\infty} m \\cdot \\operatorname{fpt}( I_m) $ exist?",
   "public_statement": "For a graded sequence of ideals $I_m$ of a ring $R$, does $\\lim \\limits_{m \\to \\infty} m \\cdot \\operatorname{fpt}( I_m) $ exist?",
   "evidence": "The canonical AIM record is Problem 11.2 in the workshop *Relating test ideals and multiplier ideals*, section “Characteristic \\(p>0\\) invariants”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 275,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0277": {
   "statement_status": "exact",
   "original_statement": "Investigate the existence and rationality of $F$-thresholds.",
   "clean_statement": "Investigate the existence and rationality of $F$-thresholds.",
   "public_statement": "Investigate the existence and rationality of $F$-thresholds.",
   "evidence": "The canonical record is `aim-algebraic-geometry-notes.json`, zero-based index 276, canonical number **11.25**:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 276,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0278": {
   "statement_status": "exact",
   "original_statement": "Investigate the discreteness and rationality of $F$-jumping numbers in the non-$\\Q$-Gorenstein case.",
   "clean_statement": "Investigate the discreteness and rationality of $F$-jumping numbers in the non-$\\Q$-Gorenstein case.",
   "public_statement": "Investigate the discreteness and rationality of $F$-jumping numbers in the non-$\\Q$-Gorenstein case.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 277,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0279": {
   "statement_status": "exact",
   "original_statement": "Fix $d$. Does there exist a sequence of singular algebras $\\{ R_n \\}$ over a fixed finite field of characteristic $p>0$, and of dimension $d$, such that the $e_{HK}\\left(R_n\\right)$ descend to some number $\\alpha_d$ from above?",
   "clean_statement": "Fix $d$. Does there exist a sequence of singular algebras $\\{ R_n \\}$ over a fixed finite field of characteristic $p>0$, and of dimension $d$, such that the $e_{HK}\\left(R_n\\right)$ descend to some number $\\alpha_d$ from above?",
   "public_statement": "Fix $d$. Does there exist a sequence of singular algebras $\\{ R_n \\}$ over a fixed finite field of characteristic $p>0$, and of dimension $d$, such that the $e_{HK}\\left(R_n\\right)$ descend to some number $\\alpha_d$ from above?",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 278,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0280": {
   "statement_status": "exact",
   "original_statement": "Fix $d$. What is the maximum value of the $F$-pure threshold of $\\mathfrak{m}$ of a non-$\\Q$-Gorenstein local ring $(R, \\mathfrak{m})$ of dimension $d$?",
   "clean_statement": "Fix $d$. What is the maximum value of the $F$-pure threshold of $\\mathfrak{m}$ of a non-$\\Q$-Gorenstein local ring $(R, \\mathfrak{m})$ of dimension $d$?",
   "public_statement": "Fix $d$. What is the maximum value of the $F$-pure threshold of $\\mathfrak{m}$ of a non-$\\Q$-Gorenstein local ring $(R, \\mathfrak{m})$ of dimension $d$?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 279,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0281": {
   "statement_status": "exact",
   "original_statement": "Given a determinantal ring $R$, compute $e_{HK}(R)$ and $s(R)$.",
   "clean_statement": "Given a determinantal ring $R$, compute $e_{HK}(R)$ and $s(R)$.",
   "public_statement": "Given a determinantal ring $R$, compute $e_{HK}(R)$ and $s(R)$.",
   "evidence": "The canonical record is numbered `11.45` and says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 280,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0282": {
   "statement_status": "exact",
   "original_statement": "Investigate the upper semicontinuity of the map $\\mathfrak{p} \\in \\operatorname{Spec} R \\mapsto e_{HK}\\left(R_\\mathfrak{p} \\right)$.",
   "clean_statement": "Investigate the upper semicontinuity of the map $\\mathfrak{p} \\in \\operatorname{Spec} R \\mapsto e_{HK}\\left(R_\\mathfrak{p} \\right)$.",
   "public_statement": "Investigate the upper semicontinuity of the map $\\mathfrak{p} \\in \\operatorname{Spec} R \\mapsto e_{HK}\\left(R_\\mathfrak{p} \\right)$.",
   "evidence": "The canonical record is from the AIM workshop problem list *Relating test ideals and multiplier ideals*, section “Characteristic \\(p>0\\) invariants.” It reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 281,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0283": {
   "statement_status": "exact",
   "original_statement": "Investigate the lower semicontinuity of the map $\\mathfrak{p} \\in \\operatorname{Spec} R \\mapsto s\\left(R_\\mathfrak{p}\\right)$.",
   "clean_statement": "Investigate the lower semicontinuity of the map $\\mathfrak{p} \\in \\operatorname{Spec} R \\mapsto s\\left(R_\\mathfrak{p}\\right)$.",
   "public_statement": "Investigate the lower semicontinuity of the map $\\mathfrak{p} \\in \\operatorname{Spec} R \\mapsto s\\left(R_\\mathfrak{p}\\right)$.",
   "evidence": "The canonical record, under the section “Characteristic \\(p>0\\) invariants,” asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 282,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0284": {
   "statement_status": "exact",
   "original_statement": "If $a_e$ denote of $F$-splitting numbers of a strongly $F$-regular local ring $R$ (i.e., $R^{1/p^e} \\cong R^{a_e} \\oplus M$ as $R$-modules, and $M$ contains no free $R$-module summands), then is $a_e = s(R) p^{ed} + C p^{e(d-1)} + O \\left(p^{e(d-2)}\\right) $ for some constant $C$?",
   "clean_statement": "If $a_e$ denote of $F$-splitting numbers of a strongly $F$-regular local ring $R$ (i.e., $R^{1/p^e} \\cong R^{a_e} \\oplus M$ as $R$-modules, and $M$ contains no free $R$-module summands), then is $a_e = s(R) p^{ed} + C p^{e(d-1)} + O \\left(p^{e(d-2)}\\right) $ for some constant $C$?",
   "public_statement": "If $a_e$ denote of $F$-splitting numbers of a strongly $F$-regular local ring $R$ (i.e., $R^{1/p^e} \\cong R^{a_e} \\oplus M$ as $R$-modules, and $M$ contains no free $R$-module summands), then is $a_e = s(R) p^{ed} + C p^{e(d-1)} + O \\left(p^{e(d-2)}\\right) $ for some constant $C$?",
   "evidence": "The canonical record is number 11.6 in the section “Characteristic \\(p>0\\) invariants” of the AIM workshop list *Relating test ideals and multiplier ideals*. It reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 283,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0285": {
   "statement_status": "exact",
   "original_statement": "For $(R, \\mathfrak{m})$ a local, normal domain of characteristic $p>0$ and dimension $d$, let $\\mu_e$ denote the minimal number of generators of $\\operatorname{Hom}_R (R^{1/p^e}, R)$ as an $R$-module. Does $\\lim \\limits_{e \\to \\infty} \\frac{\\mu_e}{p^{ed}}$ exist? If so, what is it?",
   "clean_statement": "For $(R, \\mathfrak{m})$ a local, normal domain of characteristic $p>0$ and dimension $d$, let $\\mu_e$ denote the minimal number of generators of $\\operatorname{Hom}_R (R^{1/p^e}, R)$ as an $R$-module. Does $\\lim \\limits_{e \\to \\infty} \\frac{\\mu_e}{p^{ed}}$ exist? If so, what is it?",
   "public_statement": "For $(R, \\mathfrak{m})$ a local, normal domain of characteristic $p>0$ and dimension $d$, let $\\mu_e$ denote the minimal number of generators of $\\operatorname{Hom}_R (R^{1/p^e}, R)$ as an $R$-module. Does $\\lim \\limits_{e \\to \\infty} \\frac{\\mu_e}{p^{ed}}$ exist? If so, what is it?",
   "evidence": "The canonical record is indexed as problem 11.65 in `aim-algebraic-geometry-notes.json`. The archived AIM page displays it as Problem 1.65, attributed to Tucker, in the section “Characteristic \\(p>0\\) invariants.” Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 284,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0286": {
   "statement_status": "exact",
   "original_statement": "For $(R, \\mathfrak{m})$ a local, normal domain of characteristic $p>0$, let $x$ be a minimal generator of $\\mathfrak{m}$, and let $v$ denote a $t^\\text{th}$ root of $x$. Relate $e_{HK}(R)$ and $e_{HK}\\left(R[v]\\right)$.",
   "clean_statement": "For $(R, \\mathfrak{m})$ a local, normal domain of characteristic $p>0$, let $x$ be a minimal generator of $\\mathfrak{m}$, and let $v$ denote a $t^\\text{th}$ root of $x$. Relate $e_{HK}(R)$ and $e_{HK}\\left(R[v]\\right)$.",
   "public_statement": "For $(R, \\mathfrak{m})$ a local, normal domain of characteristic $p>0$, let $x$ be a minimal generator of $\\mathfrak{m}$, and let $v$ denote a $t^\\text{th}$ root of $x$. Relate $e_{HK}(R)$ and $e_{HK}\\left(R[v]\\right)$.",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 285,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_GEOMETRY-0287": {
   "statement_status": "corrected_verified",
   "original_statement": "Generalize the Hara-Yoshida Theorem on restriction of (Hacon-de Fernex) multiplier ideals to test ideals to the non-$\\Q$-Gorenstein case. Can this be done in the numerically Gorenstein setting?\\label{generalizedHaraYoshida}",
   "clean_statement": "Let \\(X\\) be a normal variety in characteristic zero, let \\(\\mathfrak a\\subseteq\\mathcal O_X\\) be a nonzero ideal, and let \\(t\\in\\mathbb Q_{\\geq0}\\). After choosing a model over a finitely generated \\(\\mathbb Z\\)-algebra \\(A\\), is there a dense open \\(U\\subseteq\\operatorname{Spec}A\\) such that\n\\[\n\\mathcal J_{\\mathrm{dFH}}(X,\\mathfrak a^t)_\\mu\n  =\\tau_b(X_\\mu,\\mathfrak a_\\mu^t)\n\\]\nfor every closed point \\(\\mu\\in U\\)? Does this hold when \\(K_X\\) is numerically \\(\\mathbb Q\\)-Cartier?",
   "public_statement": "Let \\(X\\) be a normal variety in characteristic zero, let \\(\\mathfrak a\\subseteq\\mathcal O_X\\) be a nonzero ideal, and let \\(t\\in\\mathbb Q_{\\geq0}\\). After choosing a model over a finitely generated \\(\\mathbb Z\\)-algebra \\(A\\), is there a dense open \\(U\\subseteq\\operatorname{Spec}A\\) such that\n\\[\n\\mathcal J_{\\mathrm{dFH}}(X,\\mathfrak a^t)_\\mu\n  =\\tau_b(X_\\mu,\\mathfrak a_\\mu^t)\n\\]\nfor every closed point \\(\\mu\\in U\\)? Does this hold when \\(K_X\\) is numerically \\(\\mathbb Q\\)-Cartier?",
   "evidence": "The record is source index 286 in `aim-algebraic-geometry-notes.json`. The live AIM page places it in “Other and Related Problems,” labels it **Problem 2.05** (the corpus field `22.05` is an extraction artifact), and attributes it to de Fernex. The live page still says “restriction.” There is a one-word source error: **“restriction” should be “reduction.”** This is not a silent emendation. The Hara–Yoshida theorem at issue compares multiplier ideals in characteristic zero with test ideals after reduction to characteristic \\(p\\), and the paper that later answers the numerical case explicitly calls its result “reduction to positive characteristic.” No mathematically coherent “restriction ... to test ideals” theorem fits the surrounding workshop topic.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 286,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_GEOMETRY-0288": {
   "statement_status": "exact",
   "original_statement": "Investigate the subadditivity of (Hacon-de Fernex) multiplier ideals in the non-$\\Q$-Gorenstein case.",
   "clean_statement": "Investigate the subadditivity of (Hacon-de Fernex) multiplier ideals in the non-$\\Q$-Gorenstein case.",
   "public_statement": "Investigate the subadditivity of (Hacon-de Fernex) multiplier ideals in the non-$\\Q$-Gorenstein case.",
   "evidence": "The canonical record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 287,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0289": {
   "statement_status": "exact",
   "original_statement": "Define a ``test ideal\" that relates to the Mather discrepancy multiplier ideal, in the non-$\\mathbb{Q}$-Gorenstein case.",
   "clean_statement": "Define a ``test ideal\" that relates to the Mather discrepancy multiplier ideal, in the non-$\\mathbb{Q}$-Gorenstein case.",
   "public_statement": "Define a ``test ideal\" that relates to the Mather discrepancy multiplier ideal, in the non-$\\mathbb{Q}$-Gorenstein case.",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 288,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0290": {
   "statement_status": "exact",
   "original_statement": "Investigate lifting sections (for cohomology) using test ideals.",
   "clean_statement": "Investigate lifting sections (for cohomology) using test ideals.",
   "public_statement": "Investigate lifting sections (for cohomology) using test ideals.",
   "evidence": "The canonical record is source index 289 of `aim-algebraic-geometry-notes.json`. Its complete mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 289,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0291": {
   "statement_status": "exact",
   "original_statement": "Given ideals $I_1, \\ldots, I_n$ of a regular, $F$-finite ring $R$, consider the map sending an $r$-tuple of nonnegative real numbers $(\\lambda_1, \\ldots, \\lambda_r)$ to $\\tau\\left( {I_1}^{\\lambda_1} \\cdot \\ldots \\cdot{I_n}^{\\lambda_n}\\right)$, the mixed test ideal. If we bound all the $\\lambda_i$ by some fixed $M$, does there exist a rational polyhedral decomposition such that the function described is constant on the interior of each region?",
   "clean_statement": "Given ideals $I_1, \\ldots, I_n$ of a regular, $F$-finite ring $R$, consider the map sending an $r$-tuple of nonnegative real numbers $(\\lambda_1, \\ldots, \\lambda_r)$ to $\\tau\\left( {I_1}^{\\lambda_1} \\cdot \\ldots \\cdot{I_n}^{\\lambda_n}\\right)$, the mixed test ideal. If we bound all the $\\lambda_i$ by some fixed $M$, does there exist a rational polyhedral decomposition such that the function described is constant on the interior of each region?",
   "public_statement": "Given ideals $I_1, \\ldots, I_n$ of a regular, $F$-finite ring $R$, consider the map sending an $r$-tuple of nonnegative real numbers $(\\lambda_1, \\ldots, \\lambda_r)$ to $\\tau\\left( {I_1}^{\\lambda_1} \\cdot \\ldots \\cdot{I_n}^{\\lambda_n}\\right)$, the mixed test ideal. If we bound all the $\\lambda_i$ by some fixed $M$, does there exist a rational polyhedral decomposition such that the function described is constant on the interior of each region?",
   "evidence": "This is Problem 2.25 (attributed to Mustaţă) in the AIM problem list *Relating test ideals and multiplier ideals*, section “Other and Related Problems.” The canonical repository record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 290,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0292": {
   "statement_status": "exact",
   "original_statement": "Realize effective computations of test ideals.",
   "clean_statement": "Realize effective computations of test ideals.",
   "public_statement": "Realize effective computations of test ideals.",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 291,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0293": {
   "statement_status": "exact",
   "original_statement": "Direct summand conjecture\n\nProve/disprove the Direct Summand Conjecture.",
   "clean_statement": "Direct summand conjecture\n\nProve/disprove the Direct Summand Conjecture.",
   "public_statement": "Direct summand conjecture\n\nProve/disprove the Direct Summand Conjecture.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 292,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_GEOMETRY-0294": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Extend tight closure to mixed characteristic.",
   "clean_statement": null,
   "public_statement": "Extend tight closure to mixed characteristic.",
   "evidence": "The canonical record is from the AIM workshop list **Relating test ideals and multiplier ideals**, section **Other and Related Problems**:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 293,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0295": {
   "statement_status": "exact",
   "original_statement": "Suppose that $R$ is a local, equidimensional, $F$-pure ring of dimension $d$ with embedding dimension $n$. Is the multiplicity of $R$ at most $\\binom{n}{d}$?",
   "clean_statement": "Suppose that $R$ is a local, equidimensional, $F$-pure ring of dimension $d$ with embedding dimension $n$. Is the multiplicity of $R$ at most $\\binom{n}{d}$?",
   "public_statement": "Suppose that $R$ is a local, equidimensional, $F$-pure ring of dimension $d$ with embedding dimension $n$. Is the multiplicity of $R$ at most $\\binom{n}{d}$?",
   "evidence": "The exact canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 294,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0296": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Given a smooth, projective variety $X$ over a number field, does there exist a dense set of primes for which the action of Frobenius on the coherent cohomology $H^i(X_p, \\mathcal{O}_{X_p})$ is not nilpotent?",
   "clean_statement": null,
   "public_statement": "Given a smooth, projective variety $X$ over a number field, does there exist a dense set of primes for which the action of Frobenius on the coherent cohomology $H^i(X_p, \\mathcal{O}_{X_p})$ is not nilpotent?",
   "evidence": "The canonical record is from the AIM workshop *Relating test ideals and multiplier ideals*, section “Other and Related Problems,” with source URL <http://aimpl.org/testandmultiplierideals/2/>. The live page labels the item “Problem 2.5” and attributes it to Lyubeznik. The canonical number `22.5` is therefore a numbering/extraction artifact. The mathematical text on the live page agrees with the record:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 295,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0297": {
   "statement_status": "exact",
   "original_statement": "Do determinantal rings have finite $F$-representation type?",
   "clean_statement": "Do determinantal rings have finite $F$-representation type?",
   "public_statement": "Do determinantal rings have finite $F$-representation type?",
   "evidence": "The canonical record is AIM-ALGEBRAIC_GEOMETRY-0297, source file `aim-algebraic-geometry-notes.json`, zero-based index 296. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 296,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0298": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "For $p \\geq 11$, does $\\mathbb{F}_p[x,y,z]/(x^2+y^3+z^7)$ have finite $F$-representation type?",
   "clean_statement": null,
   "public_statement": "For $p \\geq 11$, does $\\mathbb{F}_p[x,y,z]/(x^2+y^3+z^7)$ have finite $F$-representation type?",
   "evidence": "The canonical record asks:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 297,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0299": {
   "statement_status": "exact",
   "original_statement": "Suppose that $R$ is a local ring and $f \\in R$ is a regular element such that $R/(f)$ is $F$-injective. Does this imply that $R$ is $F$-injective?",
   "clean_statement": "Suppose that $R$ is a local ring and $f \\in R$ is a regular element such that $R/(f)$ is $F$-injective. Does this imply that $R$ is $F$-injective?",
   "public_statement": "Suppose that $R$ is a local ring and $f \\in R$ is a regular element such that $R/(f)$ is $F$-injective. Does this imply that $R$ is $F$-injective?",
   "evidence": "The canonical AIM record (source file `aim-algebraic-geometry-notes.json`, zero-based index 298) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 298,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0300": {
   "statement_status": "exact",
   "original_statement": "Given a log pair $(X ,\\Delta)$, we know that there exists a finite map $\\phi: Y \\to X$ such that $\\operatorname{im}\\left(\\phi_{*} \\mathcal{O}_Y \\left( K_Y - \\phi^*\\left( K_X + \\Delta \\right)\\right) \\overset{\\text{trace}}{\\longrightarrow} \\mathcal{O}_X \\right) = \\tau(X, \\Delta)$. Is this statement true if further decorated by $\\mathfrak{a}^t$? What about for alterations?",
   "clean_statement": "Given a log pair $(X ,\\Delta)$, we know that there exists a finite map $\\phi: Y \\to X$ such that $\\operatorname{im}\\left(\\phi_{*} \\mathcal{O}_Y \\left( K_Y - \\phi^*\\left( K_X + \\Delta \\right)\\right) \\overset{\\text{trace}}{\\longrightarrow} \\mathcal{O}_X \\right) = \\tau(X, \\Delta)$. Is this statement true if further decorated by $\\mathfrak{a}^t$? What about for alterations?",
   "public_statement": "Given a log pair $(X ,\\Delta)$, we know that there exists a finite map $\\phi: Y \\to X$ such that $\\operatorname{im}\\left(\\phi_{*} \\mathcal{O}_Y \\left( K_Y - \\phi^*\\left( K_X + \\Delta \\right)\\right) \\overset{\\text{trace}}{\\longrightarrow} \\mathcal{O}_X \\right) = \\tau(X, \\Delta)$. Is this statement true if further decorated by $\\mathfrak{a}^t$? What about for alterations?",
   "evidence": "The canonical record, from `aim-algebraic-geometry-notes.json` at zero-based index 299, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 299,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0301": {
   "statement_status": "exact",
   "original_statement": "Identify a possible positive characteristic analog of minimal log discrepancy.",
   "clean_statement": "Identify a possible positive characteristic analog of minimal log discrepancy.",
   "public_statement": "Identify a possible positive characteristic analog of minimal log discrepancy.",
   "evidence": "The exact canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 300,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0302": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Investigate possible Bertini theorems for $F$-singularities.",
   "clean_statement": null,
   "public_statement": "Investigate possible Bertini theorems for $F$-singularities.",
   "evidence": "The canonical record says:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 301,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0303": {
   "statement_status": "exact",
   "original_statement": "Investigate the applications of the topics of the conference (e.g., test ideals) to projective geometry.",
   "clean_statement": "Investigate the applications of the topics of the conference (e.g., test ideals) to projective geometry.",
   "public_statement": "Investigate the applications of the topics of the conference (e.g., test ideals) to projective geometry.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 302,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0304": {
   "statement_status": "exact",
   "original_statement": "1. (Alexeev) Give a definition of log canonical, log terminal etc for pairs ( X, Y ) where X is a normal variety which is not necessarily Q-Gorenstein and Y is a formal sum of subschemes of X.",
   "clean_statement": "1. (Alexeev) Give a definition of log canonical, log terminal etc for pairs ( X, Y ) where X is a normal variety which is not necessarily Q-Gorenstein and Y is a formal sum of subschemes of X.",
   "public_statement": "1. (Alexeev) Give a definition of log canonical, log terminal etc for pairs ( X, Y ) where X is a normal variety which is not necessarily Q-Gorenstein and Y is a formal sum of subschemes of X.",
   "evidence": "The canonical record is Problem 1 in the AIM problem list from the workshop *Numerical invariants of singularities and higher-dimensional algebraic varieties* (31 July--4 August 2006). The PDF is legible, and the database transcription agrees with it:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 303,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0305": {
   "statement_status": "exact",
   "original_statement": "2. (Mustat ¸ˇ a) When f ∈ C[[ x1, · · ·, x n]] is giving a holomorphic function near 0, the log-canonical thresh-old lct (f ) is defined the usual way ( [1], [3] ). It is rational since it is computed by a log-resolution. In general when f is just a formal power series, results of [1] make it possi-ble to define lct (f ) as a limit of lct (truncations of f ). Is lct (f ) again a rational number?",
   "clean_statement": "2. (Mustat ¸ˇ a) When f ∈ C[[ x1, · · ·, x n]] is giving a holomorphic function near 0, the log-canonical thresh-old lct (f ) is defined the usual way ( [1], [3] ). It is rational since it is computed by a log-resolution. In general when f is just a formal power series, results of [1] make it possi-ble to define lct (f ) as a limit of lct (truncations of f ). Is lct (f ) again a rational number?",
   "public_statement": "2. (Mustat ¸ˇ a) When f ∈ C[[ x1, · · ·, x n]] is giving a holomorphic function near 0, the log-canonical thresh-old lct (f ) is defined the usual way ( [1], [3] ). It is rational since it is computed by a log-resolution. In general when f is just a formal power series, results of [1] make it possi-ble to define lct (f ) as a limit of lct (truncations of f ). Is lct (f ) again a rational number?",
   "evidence": "The canonical record is Problem 2, attributed to Mircea Mustaţă, from the AIM workshop *Numerical invariants of singularities and higher-dimensional algebraic varieties*. The stored text contains OCR damage in the author name, subscripts, and line breaks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 304,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0306": {
   "statement_status": "exact",
   "original_statement": "3. (Mustat ¸ˇ a) We have the ACC conjecture for log canonical thresholds ( \nConjecture 2.5 of [6], see also [3]). When we consider only smooth varieties, the conjecture is equivalent to the following question: Fix n. ∀c ∈ R, does there exist k = k(c) such that \n\n{f ∈ C[[ x1, · · ·, x n]] |lct (f ) ≥ c} depends on k-th truncations of f?A weaker question is: Does there exist k = k(f ) such that lct (f + g) ≥ lct (f ) for all g with ord (g) ≥ k?",
   "clean_statement": "3. (Mustat ¸ˇ a) We have the ACC conjecture for log canonical thresholds (\nConjecture 2.5 of [6], see also [3]). When we consider only smooth varieties, the conjecture is equivalent to the following question: Fix n. ∀c ∈ R, does there exist k = k(c) such that\n\n{f ∈ C[[ x1, · · ·, x n]] |lct (f ) ≥ c} depends on k-th truncations of f?A weaker question is: Does there exist k = k(f ) such that lct (f + g) ≥ lct (f ) for all g with ord (g) ≥ k?",
   "public_statement": "3. (Mustat ¸ˇ a) We have the ACC conjecture for log canonical thresholds (\nConjecture 2.5 of [6], see also [3]). When we consider only smooth varieties, the conjecture is equivalent to the following question: Fix n. ∀c ∈ R, does there exist k = k(c) such that\n\n{f ∈ C[[ x1, · · ·, x n]] |lct (f ) ≥ c} depends on k-th truncations of f?A weaker question is: Does there exist k = k(f ) such that lct (f + g) ≥ lct (f ) for all g with ord (g) ≥ k?",
   "evidence": "The canonical JSON record is an OCR extraction from the AIM problem list *Numerical invariants of singularities and higher-dimensional algebraic varieties*. It corrupts Mustaţă's name and joins two lines, but the original PDF is legible. Problem 3 on page 1 reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 305,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0307": {
   "statement_status": "exact",
   "original_statement": "4. (Mustat ¸ˇ a) Let X be a Q-Gorenstein variety. Then a conjecture of Shokurov ( \nConjecture 2.2 in [6] ) says mld (P; X) ≤ dim (X) for all P ∈ X. As a weaker form of this question, is there any universal upper bound of mld (P; X) one can prove (fixing dim (X))?",
   "clean_statement": "4. (Mustat ¸ˇ a) Let X be a Q-Gorenstein variety. Then a conjecture of Shokurov (\nConjecture 2.2 in [6] ) says mld (P; X) ≤ dim (X) for all P ∈ X. As a weaker form of this question, is there any universal upper bound of mld (P; X) one can prove (fixing dim (X))?",
   "public_statement": "4. (Mustat ¸ˇ a) Let X be a Q-Gorenstein variety. Then a conjecture of Shokurov (\nConjecture 2.2 in [6] ) says mld (P; X) ≤ dim (X) for all P ∈ X. As a weaker form of this question, is there any universal upper bound of mld (P; X) one can prove (fixing dim (X))?",
   "evidence": "The canonical record is item 4 of the AIM workshop list *Numerical invariants of singularities and higher-dimensional algebraic varieties*. The record has a minor OCR corruption in Mustaţă's name. The source PDF gives the following statement:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 306,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0308": {
   "statement_status": "exact",
   "original_statement": "5. (Schwede) Let X be a variety with log-canonical singularities. Does X have Du Bois singularities? ( See [5] for some nice historical discussion about the question and some progress, the introduction of [4] for some discussion of the question and [2],Chapter 12. )",
   "clean_statement": "5. (Schwede) Let X be a variety with log-canonical singularities. Does X have Du Bois singularities? ( See [5] for some nice historical discussion about the question and some progress, the introduction of [4] for some discussion of the question and [2],Chapter 12. )",
   "public_statement": "5. (Schwede) Let X be a variety with log-canonical singularities. Does X have Du Bois singularities? ( See [5] for some nice historical discussion about the question and some progress, the introduction of [4] for some discussion of the question and [2],Chapter 12. )",
   "evidence": "The canonical AIM record is Problem 5 from the workshop *Numerical invariants of singularities and higher-dimensional algebraic varieties*. Its exact extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 307,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0309": {
   "statement_status": "exact",
   "original_statement": "6. (Ishii) (1) What kind of exceptional divisor over X (with mild singularities) gives a minimal log discrepancy which is nonnegative for some ( X, D )? (2) What about over a surface X? Let E be an exceptional divisor of the minimal resolution \n\nX′ → X of X. Does E compute some mld ≥ 0?",
   "clean_statement": "6. (Ishii) (1) What kind of exceptional divisor over X (with mild singularities) gives a minimal log discrepancy which is nonnegative for some ( X, D )? (2) What about over a surface X? Let E be an exceptional divisor of the minimal resolution\n\nX′ → X of X. Does E compute some mld ≥ 0?",
   "public_statement": "6. (Ishii) (1) What kind of exceptional divisor over X (with mild singularities) gives a minimal log discrepancy which is nonnegative for some ( X, D )? (2) What about over a surface X? Let E be an exceptional divisor of the minimal resolution\n\nX′ → X of X. Does E compute some mld ≥ 0?",
   "evidence": "The canonical record is Problem 6, attributed to Ishii, in the AIM workshop list *Numerical invariants of singularities and higher-dimensional algebraic varieties*. The original PDF says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 308,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0310": {
   "statement_status": "exact",
   "original_statement": "7. (Alexeev, Mustat ¸ˇ a ) Give one (non-trivial) application of motivic integration or arc spaces to termination of flips. What about under the assumption that the varieties in a sequence of flips X1 → X2 → · · · \n\nare all smooth? \n1",
   "clean_statement": "7. (Alexeev, Mustat ¸ˇ a ) Give one (non-trivial) application of motivic integration or arc spaces to termination of flips. What about under the assumption that the varieties in a sequence of flips X1 → X2 → · · ·\n\nare all smooth?\n1",
   "public_statement": "7. (Alexeev, Mustat ¸ˇ a ) Give one (non-trivial) application of motivic integration or arc spaces to termination of flips. What about under the assumption that the varieties in a sequence of flips X1 → X2 → · · ·\n\nare all smooth?\n1",
   "evidence": "The canonical record is item 7 in the AIM workshop problem list *Numerical invariants of singularities and higher-dimensional algebraic varieties*. The source PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 309,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0311": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "8. (Schwede) Let Y be a smooth projective variety and X ⊂ Y an irreducible subvariety of codimension \n\nr. We assume that X is normal and Q-Gorenstein. If X is locally complete intersection (l.c.i.), then we have the equivalence: ( Y, I r) log-canonical ↔ X log-canonical (where I is the ideal sheaf of X on Y ). This is not true if X is not l.c.i. \n\nQuestion. ( Y, I (r)) log-canonical ↔ X log-canonical? ( I(r) is the r-th symbolic power of \n\nI. ) 9. ( Mustat ¸ˇ a ) Is there adjunction formula for multiplier ideals under restriction to subvarieties that are not defined by a regular sequence? 10. (de Fernex) Let X be a smooth variety and B ⊂ X a closed proper subscheme. We assume that there exists a prime divisor E over X, with center P, such that aE (X, cB ) ≤ 0 for some c > 0. Fix an integer e such that 1 ≤ e < dimX.Does there exist a smooth subvariety Y ⊂ X of codimension e ( P ⊂ Y, Y * B) and a divisor \n\nF over Y with its center cY (F ) = P such that the log discrepancy aF (Y, cB |Y − eP ) ≤ 0? \n1",
   "clean_statement": null,
   "public_statement": "8. (Schwede) Let Y be a smooth projective variety and X ⊂ Y an irreducible subvariety of codimension\n\nr. We assume that X is normal and Q-Gorenstein. If X is locally complete intersection (l.c.i.), then we have the equivalence: ( Y, I r) log-canonical ↔ X log-canonical (where I is the ideal sheaf of X on Y ). This is not true if X is not l.c.i.\n\nQuestion. ( Y, I (r)) log-canonical ↔ X log-canonical? ( I(r) is the r-th symbolic power of\n\nI. ) 9. ( Mustat ¸ˇ a ) Is there adjunction formula for multiplier ideals under restriction to subvarieties that are not defined by a regular sequence? 10. (de Fernex) Let X be a smooth variety and B ⊂ X a closed proper subscheme. We assume that there exists a prime divisor E over X, with center P, such that aE (X, cB ) ≤ 0 for some c > 0. Fix an integer e such that 1 ≤ e < dimX.Does there exist a smooth subvariety Y ⊂ X of codimension e ( P ⊂ Y, Y * B) and a divisor\n\nF over Y with its center cY (F ) = P such that the log discrepancy aF (Y, cB |Y − eP ) ≤ 0?\n1",
   "evidence": "The canonical JSON record accidentally joins three consecutive questions from the AIM list *Numerical invariants of singularities and higher-dimensional algebraic varieties*. The terminal `1` in the extracted text is the page number, not part of Problem 10. The original PDF says that varieties are over \\(\\mathbb C\\). With typography and the OCR error \\(Y*B\\) repaired from the PDF, the three questions are:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 310,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0312": {
   "statement_status": "exact",
   "original_statement": "1. (Shokurov) Fix n and consider projective varieties X with an exact canonical singularity of dimension \n\nn. Is the index of KX at the singularity bounded? What about the case dimX = 3? More generally, for a fixed minimal log discrepancy of ( X, 0), is the index of KX at any point with such an mld, bounded? \n1",
   "clean_statement": "1. (Shokurov) Fix n and consider projective varieties X with an exact canonical singularity of dimension\n\nn. Is the index of KX at the singularity bounded? What about the case dimX = 3? More generally, for a fixed minimal log discrepancy of ( X, 0), is the index of KX at any point with such an mld, bounded?\n1",
   "public_statement": "1. (Shokurov) Fix n and consider projective varieties X with an exact canonical singularity of dimension\n\nn. Is the index of KX at the singularity bounded? What about the case dimX = 3? More generally, for a fixed minimal log discrepancy of ( X, 0), is the index of KX at any point with such an mld, bounded?\n1",
   "evidence": "### Canonical record",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 311,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0313": {
   "statement_status": "exact",
   "original_statement": "2. (Shokurov) Consider projective varieties X of dimension n with log-canonical singularities which satisfy \n\nKX ≡ 0. Then is the index of KX bounded? This is open for dimension 4. The following four problems are concerned with positive characteristic issues. \n1",
   "clean_statement": "2. (Shokurov) Consider projective varieties X of dimension n with log-canonical singularities which satisfy\n\nKX ≡ 0. Then is the index of KX bounded? This is open for dimension 4. The following four problems are concerned with positive characteristic issues.\n1",
   "public_statement": "2. (Shokurov) Consider projective varieties X of dimension n with log-canonical singularities which satisfy\n\nKX ≡ 0. Then is the index of KX bounded? This is open for dimension 4. The following four problems are concerned with positive characteristic issues.\n1",
   "evidence": "The canonical input is record 312 (zero-based) of `aim-algebraic-geometry-notes.json`. Its extracted text is visibly damaged: the item number was shortened from 12 to 2, the transition to the next group of problems was appended, and a page number was retained as a final “1”.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 312,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0314": {
   "statement_status": "exact",
   "original_statement": "3. (Mustat ¸ˇ a) Let k be a field of characteristic p.(1) For f ∈ k[[ x1, · · ·, x n]], is the F-pure threshold f pt (f ) a rational number? (2) Let c = f pt (f ). Is τ (f c) a radical ideal? (3) If the zero set of τ (f c) is zero-dimensional, does there exist N > 1 such that ∀g ∈\n\nk[[ x1, · · ·, x n]] of order ≥ N, f pt (f ) = f pt (f + g)? 14. Does the set {f pt (f )|f ∈ k[[ x1, · · ·, x n]] } have the ACC property? 2\n1",
   "clean_statement": "3. (Mustat ¸ˇ a) Let k be a field of characteristic p.(1) For f ∈ k[[ x1, · · ·, x n]], is the F-pure threshold f pt (f ) a rational number? (2) Let c = f pt (f ). Is τ (f c) a radical ideal? (3) If the zero set of τ (f c) is zero-dimensional, does there exist N > 1 such that ∀g ∈\n\nk[[ x1, · · ·, x n]] of order ≥ N, f pt (f ) = f pt (f + g)? 14. Does the set {f pt (f )|f ∈ k[[ x1, · · ·, x n]] } have the ACC property? 2\n1",
   "public_statement": "3. (Mustat ¸ˇ a) Let k be a field of characteristic p.(1) For f ∈ k[[ x1, · · ·, x n]], is the F-pure threshold f pt (f ) a rational number? (2) Let c = f pt (f ). Is τ (f c) a radical ideal? (3) If the zero set of τ (f c) is zero-dimensional, does there exist N > 1 such that ∀g ∈\n\nk[[ x1, · · ·, x n]] of order ≥ N, f pt (f ) = f pt (f + g)? 14. Does the set {f pt (f )|f ∈ k[[ x1, · · ·, x n]] } have the ACC property? 2\n1",
   "evidence": "The canonical JSON record has merged two consecutive numbered items and has also retained a page-number/footer fragment. The original AIM PDF says that “the following four problems are concerned with positive characteristic issues” and gives the following text (with typography normalized but no mathematical change).",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 313,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0315": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "5. (Takagi) Let ( R, m) be a regular local ring of characteristic p. Suppose that ideals I, a and b of R\n\nsatisfy the following conditions: a ⊂ I, J (I−\u000f · as) ⊂ m and J (bt) ⊂ m.Then J (I−\u000f · as · bt) ⊂ J (I−\u000f · as) · J (bt)? \n1",
   "clean_statement": "Let \\((R,\\mathfrak m)\\) be a regular local ring of characteristic \\(p\\).\nSuppose that ideals \\(I,\\mathfrak a,\\mathfrak b\\) of \\(R\\) satisfy\n\\(\\mathfrak a\\subset I\\),\n\\(\\mathcal J(I^{-\\epsilon}\\cdot\\mathfrak a^s)\\subset\\mathfrak m\\), and\n\\(\\mathcal J(\\mathfrak b^t)\\subset\\mathfrak m\\). Then is\n\\[\n\\mathcal J(I^{-\\epsilon}\\cdot\\mathfrak a^s\\cdot\\mathfrak b^t)\n\\subset\n\\mathcal J(I^{-\\epsilon}\\cdot\\mathfrak a^s)\n\\mathcal J(\\mathfrak b^t)?\n\\]",
   "public_statement": "5. (Takagi) Let ( R, m) be a regular local ring of characteristic p. Suppose that ideals I, a and b of R\n\nsatisfy the following conditions: a ⊂ I, J (I−[U+000F] · as) ⊂ m and J (bt) ⊂ m.Then J (I−[U+000F] · as · bt) ⊂ J (I−[U+000F] · as) · J (bt)?\n1",
   "evidence": "The canonical JSON record contains an OCR control character in the exponent of \\(I\\), loses the superscript formatting on \\(s,t\\), and ends with a page number. The TeX source underlying the AIM PDF reads:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 314,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0316": {
   "statement_status": "exact",
   "original_statement": "6. (Takagi) Let ( R, m) be a regular local ring of characteristic p and p ⊂ R a prime ideal of height 2. If \n\nSpecR/ p has only log canonical singularities, then p(3) ⊂ p2?\n1",
   "clean_statement": "6. (Takagi) Let ( R, m) be a regular local ring of characteristic p and p ⊂ R a prime ideal of height 2. If\n\nSpecR/ p has only log canonical singularities, then p(3) ⊂ p2?\n1",
   "public_statement": "6. (Takagi) Let ( R, m) be a regular local ring of characteristic p and p ⊂ R a prime ideal of height 2. If\n\nSpecR/ p has only log canonical singularities, then p(3) ⊂ p2?\n1",
   "evidence": "The JSON extraction loses a leading digit, fraktur letters, superscript formatting, and a page break. The original AIM PDF gives **Problem 16 (Takagi)**, not Problem 6:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 315,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0317": {
   "statement_status": "exact",
   "original_statement": "7. (Cheltsov) Let X be a smooth Fano variety with P icX = Z.\n\nQuestion Do we have the following equivalence (*)? (*) X is birationally superrigid ←→ (1) ∀ linear system with no fixed components M ⊂| − mK X |, ( X, 1 \n\n> m\n\nM ) is canonical for all m > 0. ( Cheltsov explains that the implication from left to right follows from MMP. ) Now let's consider another condition: (2) ∀ divisor D ∈ | − mK X |, ( X, 1 \n\n> m\n\nD) is canonical (or plt) for all m > 0. Cheltsov says, if X satisfies (1) and (2), then X × P1 is \"almost birationally superrigid\" in the sense that the only Mori fiber structures on it are the two projections. More precisely, every birational map from X × P1 to another Mori fiber space is an isomorphism. In the other direction, there is the following: Theorem (Pukhlikov, [8] ) If XN ⊂ PN is sufficiently general ( N ≥ 6), then X satisfies the conditions (1) and (2). \n\nQuestion. What if N = 4 or N = 5? \n1",
   "clean_statement": "7. (Cheltsov) Let X be a smooth Fano variety with P icX = Z.\n\nQuestion Do we have the following equivalence (*)? (*) X is birationally superrigid ←→ (1) ∀ linear system with no fixed components M ⊂| − mK X |, ( X, 1\n\n> m\n\nM ) is canonical for all m > 0. ( Cheltsov explains that the implication from left to right follows from MMP. ) Now let's consider another condition: (2) ∀ divisor D ∈ | − mK X |, ( X, 1\n\n> m\n\nD) is canonical (or plt) for all m > 0. Cheltsov says, if X satisfies (1) and (2), then X × P1 is \"almost birationally superrigid\" in the sense that the only Mori fiber structures on it are the two projections. More precisely, every birational map from X × P1 to another Mori fiber space is an isomorphism. In the other direction, there is the following: Theorem (Pukhlikov, [8] ) If XN ⊂ PN is sufficiently general ( N ≥ 6), then X satisfies the conditions (1) and (2).\n\nQuestion. What if N = 4 or N = 5?\n1",
   "public_statement": "7. (Cheltsov) Let X be a smooth Fano variety with P icX = Z.\n\nQuestion Do we have the following equivalence (*)? (*) X is birationally superrigid ←→ (1) ∀ linear system with no fixed components M ⊂| − mK X |, ( X, 1\n\n> m\n\nM ) is canonical for all m > 0. ( Cheltsov explains that the implication from left to right follows from MMP. ) Now let's consider another condition: (2) ∀ divisor D ∈ | − mK X |, ( X, 1\n\n> m\n\nD) is canonical (or plt) for all m > 0. Cheltsov says, if X satisfies (1) and (2), then X × P1 is \"almost birationally superrigid\" in the sense that the only Mori fiber structures on it are the two projections. More precisely, every birational map from X × P1 to another Mori fiber space is an isomorphism. In the other direction, there is the following: Theorem (Pukhlikov, [8] ) If XN ⊂ PN is sufficiently general ( N ≥ 6), then X satisfies the conditions (1) and (2).\n\nQuestion. What if N = 4 or N = 5?\n1",
   "evidence": "The record comes from Problem 17 (Cheltsov) in the AIM workshop list *Numerical invariants of singularities and higher-dimensional algebraic varieties*. The PDF, rather than the damaged text extraction, gives the following question.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 316,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0318": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "8. (Cheltsov) Let Xn ⊂ P4 denote a hypersurface of degree n with a finite number of isolated ODP(ordinary double point)s. \n\nQuestion. When is Xn factorial ( Cl (Xn) = Z )? A theorem of Clemens says: \n\nXn is factorial ↔ SingX n imposes independent linear conditions on homogeneous forms on \n\nP4 of degree n.There is the following example of ( Cl (Xn) 6 = Z): \n\nXn given by xf n−1 + yg n−1 = 0 where fn−1 and gn−1 are generic homogeneous polynomials of degree n − 1 in x, y, z, w, v. Then ClX n = Z ⊕ Z.\n\nQuestion. If |SingX n| < (n − 1) 2, then Cl (Xn) = Z?Known facts. 1) |SingX n| ≤ 2 \n\n> 3\n\n(n − 1) 2 → Xn is factorial. 2) S ⊂ Xn is a smooth subvariety of codimension 1. → S is Cartier. \n\nQuestion. Let ϕ: P4 − − → P2 be a generic projection. Then is it true that at most \n\nk(n − 1) points of the set ϕ(SingX n) lie on a curve of degre k in P2?3\n1",
   "clean_statement": null,
   "public_statement": "8. (Cheltsov) Let Xn ⊂ P4 denote a hypersurface of degree n with a finite number of isolated ODP(ordinary double point)s.\n\nQuestion. When is Xn factorial ( Cl (Xn) = Z )? A theorem of Clemens says:\n\nXn is factorial ↔ SingX n imposes independent linear conditions on homogeneous forms on\n\nP4 of degree n.There is the following example of ( Cl (Xn) 6 = Z):\n\nXn given by xf n−1 + yg n−1 = 0 where fn−1 and gn−1 are generic homogeneous polynomials of degree n − 1 in x, y, z, w, v. Then ClX n = Z ⊕ Z.\n\nQuestion. If |SingX n| < (n − 1) 2, then Cl (Xn) = Z?Known facts. 1) |SingX n| ≤ 2\n\n> 3\n\n(n − 1) 2 → Xn is factorial. 2) S ⊂ Xn is a smooth subvariety of codimension 1. → S is Cartier.\n\nQuestion. Let ϕ: P4 − − → P2 be a generic projection. Then is it true that at most\n\nk(n − 1) points of the set ϕ(SingX n) lie on a curve of degre k in P2?3\n1",
   "evidence": "The canonical record is source index 317 of `aim-algebraic-geometry-notes.json`. It comes from Problem 8 (Cheltsov) in the AIM workshop list *Numerical invariants of singularities and higher-dimensional algebraic varieties*. The PDF extraction is damaged at a fraction, at the rational-map arrow, and at the end of the page. The original TeX gives the following mathematical content (notation modernized only by adding subscripts and spacing).",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 317,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0319": {
   "statement_status": "exact",
   "original_statement": "9. (Siu) Let π: X → ∆ be a holomorphic family of compact K¨ ahler manifolds over the open unit 1-disk ∆ with fiber Xt. Then is the dimension of H0(Xt, mK Xt ) independent of t ∈ ∆? This is the K¨ ahler case of Siu's famous invariance of plurigenera theorem for projective smooth varieties ( [7] ). One of the difficulties in using similar ideas from the algebraic case is that we cannot use an auxiliary ample line bundle as in [7].",
   "clean_statement": "9. (Siu) Let π: X → ∆ be a holomorphic family of compact K¨ ahler manifolds over the open unit 1-disk ∆ with fiber Xt. Then is the dimension of H0(Xt, mK Xt ) independent of t ∈ ∆? This is the K¨ ahler case of Siu's famous invariance of plurigenera theorem for projective smooth varieties ( [7] ). One of the difficulties in using similar ideas from the algebraic case is that we cannot use an auxiliary ample line bundle as in [7].",
   "public_statement": "9. (Siu) Let π: X → ∆ be a holomorphic family of compact K¨ ahler manifolds over the open unit 1-disk ∆ with fiber Xt. Then is the dimension of H0(Xt, mK Xt ) independent of t ∈ ∆? This is the K¨ ahler case of Siu's famous invariance of plurigenera theorem for projective smooth varieties ( [7] ). One of the difficulties in using similar ideas from the algebraic case is that we cannot use an auxiliary ample line bundle as in [7].",
   "evidence": "The canonical JSON record has two extraction defects: it gives the number as “9” rather than “19,” and it corrupts “Kähler.” The original AIM PDF and its TeX source have this item as the nineteenth problem in the list. The TeX source contains exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 318,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_GEOMETRY-0320": {
   "statement_status": "exact",
   "original_statement": "Problem 1.1. Let X → B be a family of surfaces. The condition \" ω[m] \n\n> X/B\n\ncommutes with base change\" means that for every base change φ: B′ → B and induced morphism \n\nφX: X′ = X ×B B′ → X, ( φ∗ \n\n> X\n\nωX/B )[m] ∼= φ∗ \n\n> X\n\n(ω[m] \n\n> X/B\n\n), where the superscript [ m] denotes the \n\nmth reflexive power, that is, the reflexive hull of the mth tensor power. Koll´ ar's condition \n\nis the condition that ω[m] \n\n> X/B\n\ncommutes with base change for all m > 0. Is the condition \" ωX/B is Q-Cartier\" equivalent to Koll´ ar's condition? Note that this is true for smoothings over the spectrum of a discrete valuation ring [Hac], Prop. 10.14. The conditions are not equivalent in positive characteristic, as Koll´ ar showed at the conference. This problem is implicitly due to Koll´ ar, who gave the stronger Koll´ ar condition in the paper [Kol90] after using the weaker definition with Shepherd-Barron for the moduli functor in [KSB88].",
   "clean_statement": "Problem 1.1. Let X → B be a family of surfaces. The condition \" ω[m]\n\n> X/B\n\ncommutes with base change\" means that for every base change φ: B′ → B and induced morphism\n\nφX: X′ = X ×B B′ → X, ( φ∗\n\n> X\n\nωX/B )[m] ∼= φ∗\n\n> X\n\n(ω[m]\n\n> X/B\n\n), where the superscript [ m] denotes the\n\nmth reflexive power, that is, the reflexive hull of the mth tensor power. Koll´ ar's condition\n\nis the condition that ω[m]\n\n> X/B\n\ncommutes with base change for all m > 0. Is the condition \" ωX/B is Q-Cartier\" equivalent to Koll´ ar's condition? Note that this is true for smoothings over the spectrum of a discrete valuation ring [Hac], Prop. 10.14. The conditions are not equivalent in positive characteristic, as Koll´ ar showed at the conference. This problem is implicitly due to Koll´ ar, who gave the stronger Koll´ ar condition in the paper [Kol90] after using the weaker definition with Shepherd-Barron for the moduli functor in [KSB88].",
   "public_statement": "Problem 1.1. Let X → B be a family of surfaces. The condition \" ω[m]\n\n> X/B\n\ncommutes with base change\" means that for every base change φ: B′ → B and induced morphism\n\nφX: X′ = X ×B B′ → X, ( φ∗\n\n> X\n\nωX/B )[m] ∼= φ∗\n\n> X\n\n(ω[m]\n\n> X/B\n\n), where the superscript [ m] denotes the\n\nmth reflexive power, that is, the reflexive hull of the mth tensor power. Koll´ ar's condition\n\nis the condition that ω[m]\n\n> X/B\n\ncommutes with base change for all m > 0. Is the condition \" ωX/B is Q-Cartier\" equivalent to Koll´ ar's condition? Note that this is true for smoothings over the spectrum of a discrete valuation ring [Hac], Prop. 10.14. The conditions are not equivalent in positive characteristic, as Koll´ ar showed at the conference. This problem is implicitly due to Koll´ ar, who gave the stronger Koll´ ar condition in the paper [Kol90] after using the weaker definition with Shepherd-Barron for the moduli functor in [KSB88].",
   "evidence": "The source is Problem 1.1 in the AIM workshop list *Compact moduli spaces and birational geometry*. The OCR has displaced subscripts and superscripts, but the mathematical statement is recoverable without ambiguity:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 319,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0321": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 1.2. The main purpose of the moduli space of stable surfaces is to serve as a compactification of the moduli space of canonically polarized surfaces. However, it is possible that some stable surfaces are not smoothable. Some parts of the construction of the moduli space have only been proved for smoothable varieties. Here are some problems: \n\n> 12MICHAEL A. VAN OPSTALL\n\n(1) The moduli space M sm \n\n> K2,χ\n\nof smoothable stable surfaces does not have a natural scheme structure at the boundary. This is because infinitesimal information is lost by throw-ing away components of the moduli space parameterizing only surfaces with worse than rational double points. Can more sense be made of the notion of smoothable to get a good scheme structure? (2) Probably we should keep all of the components and avoid the previous problem completely. In this case, however, the valuative criterion for properness has not been verified even in dimension two. The problem is: if X → ∆′ is a family of stable surfaces over a punctured disk satisfying Koll´ ar's condition, can X be completed, possibly after base change, to a family of stable surfaces over the disk, also satisfying Koll´ ar's condition? Alexeev alluded to these problems in one of his talks.",
   "clean_statement": null,
   "public_statement": "Problem 1.2. The main purpose of the moduli space of stable surfaces is to serve as a compactification of the moduli space of canonically polarized surfaces. However, it is possible that some stable surfaces are not smoothable. Some parts of the construction of the moduli space have only been proved for smoothable varieties. Here are some problems:\n\n> 12MICHAEL A. VAN OPSTALL\n\n(1) The moduli space M sm\n\n> K2,χ\n\nof smoothable stable surfaces does not have a natural scheme structure at the boundary. This is because infinitesimal information is lost by throw-ing away components of the moduli space parameterizing only surfaces with worse than rational double points. Can more sense be made of the notion of smoothable to get a good scheme structure? (2) Probably we should keep all of the components and avoid the previous problem completely. In this case, however, the valuative criterion for properness has not been verified even in dimension two. The problem is: if X → ∆′ is a family of stable surfaces over a punctured disk satisfying Koll´ ar's condition, can X be completed, possibly after base change, to a family of stable surfaces over the disk, also satisfying Koll´ ar's condition? Alexeev alluded to these problems in one of his talks.",
   "evidence": "This is Problem 1.2 in Michael A. van Opstall's notes for the AIM workshop *Compact moduli spaces and birational geometry*, held at AIM in Palo Alto on 6--10 December 2004. The canonical PDF extraction is damaged: the string `12MICHAEL A. VAN OPSTALL` is a page header, the displayed moduli symbol was split across lines, and the accent in Kollár's name was corrupted. The official source TeX recovers the notation as \\[ \\overline{M^{\\mathrm{sm}}_{K^2,\\chi}} \\] and the punctured disk as \\(\\Delta'\\).",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 320,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_GEOMETRY-0322": {
   "statement_status": "exact",
   "original_statement": "Problem 1.3 (Higher dimensions). In [Kar00], Karu proves that the semistable minimal model program in dimension n + 1 can be used (together with the weak semistable reduction of Abramovich and Karu) to construct the moduli space of smoothable stable n-folds. See the previous problem for problems with this approach. This is what needs to be done to avoid the MMP: (1) Prove boundedness of slc n-folds with fixed Kn (see Alexeev's problems below). The semistable MMP is not strong enough to do this for non-smoothable n-folds anyway. (2) Prove that small Q-Gorenstein deformations of slc singularities are slc. It is true that small Q-Gorenstein deformations of slt singularities are slt. Again, the MMP in one dimension higher would verify this. (3) Verify the valuative criterion for properness for non-smoothable n-folds. The MMP gives the valuative criterion in the smoothable case.",
   "clean_statement": "Problem 1.3 (Higher dimensions). In [Kar00], Karu proves that the semistable minimal model program in dimension n + 1 can be used (together with the weak semistable reduction of Abramovich and Karu) to construct the moduli space of smoothable stable n-folds. See the previous problem for problems with this approach. This is what needs to be done to avoid the MMP: (1) Prove boundedness of slc n-folds with fixed Kn (see Alexeev's problems below). The semistable MMP is not strong enough to do this for non-smoothable n-folds anyway. (2) Prove that small Q-Gorenstein deformations of slc singularities are slc. It is true that small Q-Gorenstein deformations of slt singularities are slt. Again, the MMP in one dimension higher would verify this. (3) Verify the valuative criterion for properness for non-smoothable n-folds. The MMP gives the valuative criterion in the smoothable case.",
   "public_statement": "Problem 1.3 (Higher dimensions). In [Kar00], Karu proves that the semistable minimal model program in dimension n + 1 can be used (together with the weak semistable reduction of Abramovich and Karu) to construct the moduli space of smoothable stable n-folds. See the previous problem for problems with this approach. This is what needs to be done to avoid the MMP: (1) Prove boundedness of slc n-folds with fixed Kn (see Alexeev's problems below). The semistable MMP is not strong enough to do this for non-smoothable n-folds anyway. (2) Prove that small Q-Gorenstein deformations of slc singularities are slc. It is true that small Q-Gorenstein deformations of slt singularities are slt. Again, the MMP in one dimension higher would verify this. (3) Verify the valuative criterion for properness for non-smoothable n-folds. The MMP gives the valuative criterion in the smoothable case.",
   "evidence": "The record is Problem 1.3 in the AIM workshop notes *Compact moduli spaces and birational geometry* (December 6--10, 2004). The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 321,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0323": {
   "statement_status": "exact",
   "original_statement": "Problem 1.\n4. (Alexeev) Are the following classes of varieties bounded? Here bounded means that all members of the class can be put into a family over a base of finite type. (1) Semi-log canonical varieties of general type with fixed Kn.(2) ≤-log terminal Fano varieties. 1.3. Positive characteristic issues.",
   "clean_statement": "Problem 1.\n4. (Alexeev) Are the following classes of varieties bounded? Here bounded means that all members of the class can be put into a family over a base of finite type. (1) Semi-log canonical varieties of general type with fixed Kn.(2) ≤-log terminal Fano varieties. 1.3. Positive characteristic issues.",
   "public_statement": "Problem 1.\n4. (Alexeev) Are the following classes of varieties bounded? Here bounded means that all members of the class can be put into a family over a base of finite type. (1) Semi-log canonical varieties of general type with fixed Kn.(2) ≤-log terminal Fano varieties. 1.3. Positive characteristic issues.",
   "evidence": "The canonical JSON record is visibly damaged by PDF extraction: “Problem 1.\\n4.” is the problem number \\(1.4\\), \\(K^n\\) lost its superscript, \\(\\epsilon\\) was misread, and the next section heading was appended to the problem. Inspection of the official AIM workshop PDF gives the following statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 322,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_GEOMETRY-0324": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 1.\n5. (Hassett) Prove the valuative criterion for properness for the moduli stack of stable surfaces in positive characteristic.",
   "clean_statement": null,
   "public_statement": "Problem 1.\n5. (Hassett) Prove the valuative criterion for properness for the moduli stack of stable surfaces in positive characteristic.",
   "evidence": "The extracted record reads:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 323,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0325": {
   "statement_status": "exact",
   "original_statement": "Problem 1.\n6. (Koll´ ar) Consider the moduli space of canonically polarized surfaces over Spec Z. Is the fiber over the prime p the moduli space of surfaces over Fp? A similar result for Mg may be due to Oort. What about other moduli problems? Ag? K3 surfaces? 1.4. Other moduli problems. An article of Schumacher and Tsuji [ST04] asserts that a separated moduli space of smooth polarized manifolds is always quasi-projective. By Viehweg's work this was known for canonically polarized manifolds and for polarized man-ifolds F with ωF nef. In the case of polarized uniruled manifolds the separatedness of the moduli space is frequently hard to check. Koll´ ar proposed a series of counterexamples by showing that every smooth toric variety is the moduli space for some class of polarized manifolds. There are many smooth, proper but nonprojective toric varieties.",
   "clean_statement": "Problem 1.\n6. (Koll´ ar) Consider the moduli space of canonically polarized surfaces over Spec Z. Is the fiber over the prime p the moduli space of surfaces over Fp? A similar result for Mg may be due to Oort. What about other moduli problems? Ag? K3 surfaces? 1.4. Other moduli problems. An article of Schumacher and Tsuji [ST04] asserts that a separated moduli space of smooth polarized manifolds is always quasi-projective. By Viehweg's work this was known for canonically polarized manifolds and for polarized man-ifolds F with ωF nef. In the case of polarized uniruled manifolds the separatedness of the moduli space is frequently hard to check. Koll´ ar proposed a series of counterexamples by showing that every smooth toric variety is the moduli space for some class of polarized manifolds. There are many smooth, proper but nonprojective toric varieties.",
   "public_statement": "Problem 1.\n6. (Koll´ ar) Consider the moduli space of canonically polarized surfaces over Spec Z. Is the fiber over the prime p the moduli space of surfaces over Fp? A similar result for Mg may be due to Oort. What about other moduli problems? Ag? K3 surfaces? 1.4. Other moduli problems. An article of Schumacher and Tsuji [ST04] asserts that a separated moduli space of smooth polarized manifolds is always quasi-projective. By Viehweg's work this was known for canonically polarized manifolds and for polarized man-ifolds F with ωF nef. In the case of polarized uniruled manifolds the separatedness of the moduli space is frequently hard to check. Koll´ ar proposed a series of counterexamples by showing that every smooth toric variety is the moduli space for some class of polarized manifolds. There are many smooth, proper but nonprojective toric varieties.",
   "evidence": "The canonical JSON record is an OCR fusion of a problem and the beginning of the next section. Inspection of the official AIM workshop PDF, *Open Problems in Compact Moduli Spaces and Birational Geometry*, printed page 2, recovers the problem as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 324,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0326": {
   "statement_status": "exact",
   "original_statement": "Problem 1.7. Can some result between Viehweg's result and the statement of Schumacher and Tsuji be proved about quasi-projectivity? For instance, if the canonical class is assumed effective? OPEN PROBLEMS IN COMPACT MODULI SPACES AND BIRATIONAL GEOMETRY 3\n\nKoll´ ar's class of examples comes from the observation that the blowups of points on a smooth variety X are parameterized by X modulo the automorphisms of X. Thus it could be true that every algebraic space which is a quotient of a quasi-projective variety by a proper group action is a moduli space for some class of polarized manifolds.",
   "clean_statement": "Problem 1.7. Can some result between Viehweg's result and the statement of Schumacher and Tsuji be proved about quasi-projectivity? For instance, if the canonical class is assumed effective? OPEN PROBLEMS IN COMPACT MODULI SPACES AND BIRATIONAL GEOMETRY 3\n\nKoll´ ar's class of examples comes from the observation that the blowups of points on a smooth variety X are parameterized by X modulo the automorphisms of X. Thus it could be true that every algebraic space which is a quotient of a quasi-projective variety by a proper group action is a moduli space for some class of polarized manifolds.",
   "public_statement": "Problem 1.7. Can some result between Viehweg's result and the statement of Schumacher and Tsuji be proved about quasi-projectivity? For instance, if the canonical class is assumed effective? OPEN PROBLEMS IN COMPACT MODULI SPACES AND BIRATIONAL GEOMETRY 3\n\nKoll´ ar's class of examples comes from the observation that the blowups of points on a smooth variety X are parameterized by X modulo the automorphisms of X. Thus it could be true that every algebraic space which is a quotient of a quasi-projective variety by a proper group action is a moduli space for some class of polarized manifolds.",
   "evidence": "The canonical record is Problem 1.7 from the AIM workshop list *Open Problems in Compact Moduli Spaces and Birational Geometry*. Its extracted `problem` field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 325,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0327": {
   "statement_status": "exact",
   "original_statement": "Problem 1.8. More generally, is every stack which is a quotient of a quasi-projective scheme by a group action a moduli stack for some class of polarized manifolds? 2. Moduli spaces of manifolds \n\nThe following problems are problems on the geometry of the \"interior\" of the moduli space, that is, on the moduli space of canonically polarized manifolds. They are motivated by the principle that even without local Torelli theorems, the geometry of moduli spaces should be similiar to the geometry of period domains.",
   "clean_statement": "Problem 1.8. More generally, is every stack which is a quotient of a quasi-projective scheme by a group action a moduli stack for some class of polarized manifolds? 2. Moduli spaces of manifolds\n\nThe following problems are problems on the geometry of the \"interior\" of the moduli space, that is, on the moduli space of canonically polarized manifolds. They are motivated by the principle that even without local Torelli theorems, the geometry of moduli spaces should be similiar to the geometry of period domains.",
   "public_statement": "Problem 1.8. More generally, is every stack which is a quotient of a quasi-projective scheme by a group action a moduli stack for some class of polarized manifolds? 2. Moduli spaces of manifolds\n\nThe following problems are problems on the geometry of the \"interior\" of the moduli space, that is, on the moduli space of canonically polarized manifolds. They are motivated by the principle that even without local Torelli theorems, the geometry of moduli spaces should be similiar to the geometry of period domains.",
   "evidence": "The record comes from the AIM workshop *Compact moduli spaces and birational geometry* (6--10 December 2004), Problem 1.8. The official TeX source has the following lead-in:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 326,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0328": {
   "statement_status": "exact",
   "original_statement": "Problem 2.\n1. (Viehweg). Let MH denote a moduli stack of canonically polarized manifolds. Can π1(MH ) be almost abelian (almost abelian means having an abelian subgroup of finite index)? More precisely, if U is a nonsingular variety (not a point) mapping generically finitely to the moduli stack, can π1(U ) be almost abelian. The conjecture is no. The result is known for Mg, and can be verified whenever some sort of local Torelli theorem holds. A U as above cannot be the complement in Pn of a normal crossings divisor with fewer than n components. At present this is only known if the number of components of H is strictly smaller than n (see [VZ02]). The following two questions are motivated by work of Viehweg and Zuo [VZ02] and [VZ05].",
   "clean_statement": "Problem 2.\n1. (Viehweg). Let MH denote a moduli stack of canonically polarized manifolds. Can π1(MH ) be almost abelian (almost abelian means having an abelian subgroup of finite index)? More precisely, if U is a nonsingular variety (not a point) mapping generically finitely to the moduli stack, can π1(U ) be almost abelian. The conjecture is no. The result is known for Mg, and can be verified whenever some sort of local Torelli theorem holds. A U as above cannot be the complement in Pn of a normal crossings divisor with fewer than n components. At present this is only known if the number of components of H is strictly smaller than n (see [VZ02]). The following two questions are motivated by work of Viehweg and Zuo [VZ02] and [VZ05].",
   "public_statement": "Problem 2.\n1. (Viehweg). Let MH denote a moduli stack of canonically polarized manifolds. Can π1(MH ) be almost abelian (almost abelian means having an abelian subgroup of finite index)? More precisely, if U is a nonsingular variety (not a point) mapping generically finitely to the moduli stack, can π1(U ) be almost abelian. The conjecture is no. The result is known for Mg, and can be verified whenever some sort of local Torelli theorem holds. A U as above cannot be the complement in Pn of a normal crossings divisor with fewer than n components. At present this is only known if the number of components of H is strictly smaller than n (see [VZ02]). The following two questions are motivated by work of Viehweg and Zuo [VZ02] and [VZ05].",
   "evidence": "The canonical record comes from the AIM workshop problem list *Compact moduli spaces and birational geometry*, Section 2, “Moduli spaces of manifolds.” In the source PDF the item is Problem 2.1, although the extracted record splits the number as `2.` followed by `1.`. The source reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 327,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0329": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 2.\n2. (Viehweg) Let U be a smooth variety, ´ etale over a moduli stack of polarized manifolds MH. Let Y be a log compactification of U and Γ = Y \\U.(1) Is Ω 1 \n\n> Y\n\n(log Γ) weakly positive with respect to U?(2) Is ωY (Γ) ample with respect to U?Both of these statements are known for period domains, and it is known that the second implies the first. These problems may be hard, but similar questions would be interesting and perhaps more tractable for configuration spaces.",
   "clean_statement": "**Problem 2.2 (Viehweg).** Let \\(U\\) be a smooth variety, étale over a moduli stack of polarized manifolds \\(M_H\\). Let \\(Y\\) be a log compactification of \\(U\\) and \\(\\Gamma=Y\\setminus U\\).\n\n1. Is \\(\\Omega_Y^1(\\log \\Gamma)\\) weakly positive with respect to \\(U\\)?\n2. Is \\(\\omega_Y(\\Gamma)\\) ample with respect to \\(U\\)?\n\nBoth of these statements are known for period domains, and it is known that the second implies the first. These problems may be hard, but similar questions would be interesting and perhaps more tractable for configuration spaces.",
   "public_statement": "**Problem 2.2 (Viehweg).** Let \\(U\\) be a smooth variety, étale over a moduli stack of polarized manifolds \\(M_H\\). Let \\(Y\\) be a log compactification of \\(U\\) and \\(\\Gamma=Y\\setminus U\\).\n\n1. Is \\(\\Omega_Y^1(\\log \\Gamma)\\) weakly positive with respect to \\(U\\)?\n2. Is \\(\\omega_Y(\\Gamma)\\) ample with respect to \\(U\\)?\n\nBoth of these statements are known for period domains, and it is known that the second implies the first. These problems may be hard, but similar questions would be interesting and perhaps more tractable for configuration spaces.",
   "evidence": "The corpus record is an OCR extraction of Problem 2.2, attributed to Viehweg, in the AIM problem list *Compact moduli spaces and birational geometry*. The official PDF gives the following statement: Here \\(M_H\\) is printed as \\(\\mathcal M_H\\) in the source. The corpus extraction has three visible OCR defects: “Problem 2. / 2.” duplicates the numbering, a stray `>` interrupts the logarithmic cotangent notation, and spacing is lost around \\(\\Gamma=Y\\setminus U\\). The recovered statement above follows the official PDF rather than those artifacts.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 328,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0330": {
   "statement_status": "exact",
   "original_statement": "Problem 2.3. Assume that MH is a moduli stack for abelian varieties or for a class of manifolds for which local Torelli holds. Let U, Y, and Γ be as in the previous problem. There is a variation of Hodge structures. Take an irreducible subvariation with Higgs bundle \n\nE1,0 π E0,1. Then one can show deg E1,0\n\nrk E1,0 − deg E0,1\n\nrk E0,1 ≤ deg Ω 1 \n\n> Y\n\n(log Γ) \n\ndim Y\n\nwhere ωY (Γ) is nef and big, and degree is taken with respect to this sheaf. The problem is to determine when equality holds. A conjecture in this direction is that U is a Shimura variety. This is true if Y is a curve and for moduli spaces Ag for small g.",
   "clean_statement": "Problem 2.3. Assume that MH is a moduli stack for abelian varieties or for a class of manifolds for which local Torelli holds. Let U, Y, and Γ be as in the previous problem. There is a variation of Hodge structures. Take an irreducible subvariation with Higgs bundle\n\nE1,0 π E0,1. Then one can show deg E1,0\n\nrk E1,0 − deg E0,1\n\nrk E0,1 ≤ deg Ω 1\n\n> Y\n\n(log Γ)\n\ndim Y\n\nwhere ωY (Γ) is nef and big, and degree is taken with respect to this sheaf. The problem is to determine when equality holds. A conjecture in this direction is that U is a Shimura variety. This is true if Y is a curve and for moduli spaces Ag for small g.",
   "public_statement": "Problem 2.3. Assume that MH is a moduli stack for abelian varieties or for a class of manifolds for which local Torelli holds. Let U, Y, and Γ be as in the previous problem. There is a variation of Hodge structures. Take an irreducible subvariation with Higgs bundle\n\nE1,0 π E0,1. Then one can show deg E1,0\n\nrk E1,0 − deg E0,1\n\nrk E0,1 ≤ deg Ω 1\n\n> Y\n\n(log Γ)\n\ndim Y\n\nwhere ωY (Γ) is nef and big, and degree is taken with respect to this sheaf. The problem is to determine when equality holds. A conjecture in this direction is that U is a Shimura variety. This is true if Y is a curve and for moduli spaces Ag for small g.",
   "evidence": "The canonical record comes from Problem 2.3 of the AIM workshop list *Compact moduli spaces and birational geometry*. The PDF extraction loses superscripts, a fraction bar, and the direct-sum symbol. The official TeX source defines `\\dirsum` to be `\\varoplus`; thus the extracted symbol `π` is not a map.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 329,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0331": {
   "statement_status": "exact",
   "original_statement": "Problem 2.\n4. (Viehweg) Does there exist a universal bound for the number of families of minimal manifolds with Hilbert polynomial H over a curve of genus q with s singular fibers depending only on H, q, and s, and not any more features of the geometry of the fibers? 4 MICHAEL A. VAN OPSTALL \n\n3. GIT \n\n3.1. Background. Geometric invariant theory plays an essential role in the usual construc-tion of Mg. GIT separates the points of a Hilbert scheme into three classes with respect to an action of a reductive group G: stable, semistable, and unstable. The quotient of the set of stable points by the group exists in a nice sense and is a moduli space of these GIT stable varieties. Note that GIT stability is not compatible with stability in the sense of the MMP. There are \"MMP-stable\" singularities which are asymptotically GIT-unstable. Also, some GIT semistable points are not MMP-stable. A good quotient of the semistable locus by the group G exists, but is not a moduli space, since some orbits of the group action corresponding to nonisomorphic varieties have intersection closures. 3.2. Problems.",
   "clean_statement": "Problem 2.\n4. (Viehweg) Does there exist a universal bound for the number of families of minimal manifolds with Hilbert polynomial H over a curve of genus q with s singular fibers depending only on H, q, and s, and not any more features of the geometry of the fibers? 4 MICHAEL A. VAN OPSTALL\n\n3. GIT\n\n3.1. Background. Geometric invariant theory plays an essential role in the usual construc-tion of Mg. GIT separates the points of a Hilbert scheme into three classes with respect to an action of a reductive group G: stable, semistable, and unstable. The quotient of the set of stable points by the group exists in a nice sense and is a moduli space of these GIT stable varieties. Note that GIT stability is not compatible with stability in the sense of the MMP. There are \"MMP-stable\" singularities which are asymptotically GIT-unstable. Also, some GIT semistable points are not MMP-stable. A good quotient of the semistable locus by the group G exists, but is not a moduli space, since some orbits of the group action corresponding to nonisomorphic varieties have intersection closures. 3.2. Problems.",
   "public_statement": "Problem 2.\n4. (Viehweg) Does there exist a universal bound for the number of families of minimal manifolds with Hilbert polynomial H over a curve of genus q with s singular fibers depending only on H, q, and s, and not any more features of the geometry of the fibers? 4 MICHAEL A. VAN OPSTALL\n\n3. GIT\n\n3.1. Background. Geometric invariant theory plays an essential role in the usual construc-tion of Mg. GIT separates the points of a Hilbert scheme into three classes with respect to an action of a reductive group G: stable, semistable, and unstable. The quotient of the set of stable points by the group exists in a nice sense and is a moduli space of these GIT stable varieties. Note that GIT stability is not compatible with stability in the sense of the MMP. There are \"MMP-stable\" singularities which are asymptotically GIT-unstable. Also, some GIT semistable points are not MMP-stable. A good quotient of the semistable locus by the group G exists, but is not a moduli space, since some orbits of the group action corresponding to nonisomorphic varieties have intersection closures. 3.2. Problems.",
   "evidence": "The source is the AIM workshop list *Compact moduli spaces and birational geometry*. The official PDF gives the following statement in Section 2, “Moduli spaces of manifolds”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 330,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0332": {
   "statement_status": "exact",
   "original_statement": "Problem 3.1. (from Koll´ ar's lecture) Fix a class of (pluri)canonically embedded manifolds in Pn. There is a notion of GIT semistability for Hilbert points in the closure of the Hilbert points of the class of manifolds considered. Now embed the family in another Pm using a higher power of KX and consider the GIT semistability conditions in that projective space. Is there an integer N such that the class of GIT semistable surfaces stabilizes for embeddings by M K X for M > N? This problem is fundamental, and the answer for curves is that N = 5.",
   "clean_statement": "Problem 3.1. (from Koll´ ar's lecture) Fix a class of (pluri)canonically embedded manifolds in Pn. There is a notion of GIT semistability for Hilbert points in the closure of the Hilbert points of the class of manifolds considered. Now embed the family in another Pm using a higher power of KX and consider the GIT semistability conditions in that projective space. Is there an integer N such that the class of GIT semistable surfaces stabilizes for embeddings by M K X for M > N? This problem is fundamental, and the answer for curves is that N = 5.",
   "public_statement": "Problem 3.1. (from Koll´ ar's lecture) Fix a class of (pluri)canonically embedded manifolds in Pn. There is a notion of GIT semistability for Hilbert points in the closure of the Hilbert points of the class of manifolds considered. Now embed the family in another Pm using a higher power of KX and consider the GIT semistability conditions in that projective space. Is there an integer N such that the class of GIT semistable surfaces stabilizes for embeddings by M K X for M > N? This problem is fundamental, and the answer for curves is that N = 5.",
   "evidence": "The record is Problem 3.1 in the AIM problem list *Compact moduli spaces and birational geometry*. The official PDF gives:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 331,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0333": {
   "statement_status": "exact",
   "original_statement": "Problem 3.2. Determine the GIT (semi)stable quintic surfaces, quartic threefolds, quintic threefolds. For hypersurfaces of a given dimension and degree, is there a bound on the exponents appearing in the diagonal 1-PS that need to be checked? All smaller cases have been checked by various authors.",
   "clean_statement": "Problem 3.2. Determine the GIT (semi)stable quintic surfaces, quartic threefolds, quintic threefolds. For hypersurfaces of a given dimension and degree, is there a bound on the exponents appearing in the diagonal 1-PS that need to be checked? All smaller cases have been checked by various authors.",
   "public_statement": "Problem 3.2. Determine the GIT (semi)stable quintic surfaces, quartic threefolds, quintic threefolds. For hypersurfaces of a given dimension and degree, is there a bound on the exponents appearing in the diagonal 1-PS that need to be checked? All smaller cases have been checked by various authors.",
   "evidence": "The record is Problem 3.2 from the 2004 AIM workshop *Compact moduli spaces and birational geometry*. The official AIM PDF and its TeX source agree with the canonical JSON record. The exact statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 332,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0334": {
   "statement_status": "exact",
   "original_statement": "Problem 3.\n3. (Morrison) For a variety X and ample line bundle L, can we work harder on the combinatorics of the monomial subspaces of H0(X, L m) to get easier estimates for the weights rλ,X of the Hilbert point of X with respect to the filtration given by some 1-PS λ?For example (1) Find more flexible filtrations of H0(X, L m) with stages of the type occurring in geo-metric estimates (e.g. cusps are not semistable) and compute the leading term of the resulting estimates for rλ,X.(2) Find methods for producing estimates for rλ,X which use estimates for non-nested \n\ncollections of subspaces of H0(X, L ). 4. Examples and applications",
   "clean_statement": "Problem 3.\n3. (Morrison) For a variety X and ample line bundle L, can we work harder on the combinatorics of the monomial subspaces of H0(X, L m) to get easier estimates for the weights rλ,X of the Hilbert point of X with respect to the filtration given by some 1-PS λ?For example (1) Find more flexible filtrations of H0(X, L m) with stages of the type occurring in geo-metric estimates (e.g. cusps are not semistable) and compute the leading term of the resulting estimates for rλ,X.(2) Find methods for producing estimates for rλ,X which use estimates for non-nested\n\ncollections of subspaces of H0(X, L ). 4. Examples and applications",
   "public_statement": "Problem 3.\n3. (Morrison) For a variety X and ample line bundle L, can we work harder on the combinatorics of the monomial subspaces of H0(X, L m) to get easier estimates for the weights rλ,X of the Hilbert point of X with respect to the filtration given by some 1-PS λ?For example (1) Find more flexible filtrations of H0(X, L m) with stages of the type occurring in geo-metric estimates (e.g. cusps are not semistable) and compute the leading term of the resulting estimates for rλ,X.(2) Find methods for producing estimates for rλ,X which use estimates for non-nested\n\ncollections of subspaces of H0(X, L ). 4. Examples and applications",
   "evidence": "The canonical record comes from the AIM workshop *Compact moduli spaces and birational geometry*, source file `aim-algebraic-geometry-notes.json`, record 333. The PDF source is the AIM list *Open Problems in Compact Moduli Spaces and Birational Geometry*. The extraction inserted a line break in the problem number and appended the next heading, “4. Examples and applications.” Inspection of the PDF shows that the heading is not part of the problem.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 333,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0335": {
   "statement_status": "exact",
   "original_statement": "Problem 4.1. Find applications of moduli of stable surfaces to surface theory. For example degenerations of curves are used to prove Brill-Noether type statements.",
   "clean_statement": "Problem 4.1. Find applications of moduli of stable surfaces to surface theory. For example degenerations of curves are used to prove Brill-Noether type statements.",
   "public_statement": "Problem 4.1. Find applications of moduli of stable surfaces to surface theory. For example degenerations of curves are used to prove Brill-Noether type statements.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 334,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0336": {
   "statement_status": "exact",
   "original_statement": "Problem 4.2. Understand the geometry of the moduli space of stable surfaces better. Here it is probably necessary to find some nice components and study their geometry. In general, after fixing even the differentiable structure of the underlying four-manifolds, the moduli space is disconnected, its connected components are reducible, and its irreducible components may be everywhere nonreduced. In addition, the boundary may not be a divisor, and the next best substitute is not a normal crossings divisor. Also, Vakil shows that the moduli space of canonically polarized surfaces is arbitrarily singular (in a precise sense). OPEN PROBLEMS IN COMPACT MODULI SPACES AND BIRATIONAL GEOMETRY 5\n\n4.1. Intersection theory.",
   "clean_statement": "Problem 4.2. Understand the geometry of the moduli space of stable surfaces better. Here it is probably necessary to find some nice components and study their geometry. In general, after fixing even the differentiable structure of the underlying four-manifolds, the moduli space is disconnected, its connected components are reducible, and its irreducible components may be everywhere nonreduced. In addition, the boundary may not be a divisor, and the next best substitute is not a normal crossings divisor. Also, Vakil shows that the moduli space of canonically polarized surfaces is arbitrarily singular (in a precise sense). OPEN PROBLEMS IN COMPACT MODULI SPACES AND BIRATIONAL GEOMETRY 5\n\n4.1. Intersection theory.",
   "public_statement": "Problem 4.2. Understand the geometry of the moduli space of stable surfaces better. Here it is probably necessary to find some nice components and study their geometry. In general, after fixing even the differentiable structure of the underlying four-manifolds, the moduli space is disconnected, its connected components are reducible, and its irreducible components may be everywhere nonreduced. In addition, the boundary may not be a divisor, and the next best substitute is not a normal crossings divisor. Also, Vakil shows that the moduli space of canonically polarized surfaces is arbitrarily singular (in a precise sense). OPEN PROBLEMS IN COMPACT MODULI SPACES AND BIRATIONAL GEOMETRY 5\n\n4.1. Intersection theory.",
   "evidence": "The canonical record comes from Problem 4.2 of the AIM workshop list *Compact moduli spaces and birational geometry*. Inspection of the official source TeX and the source PDF recovers the problem as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 335,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0337": {
   "statement_status": "exact",
   "original_statement": "Problem 4.\n3. (Hassett) Does the moduli space of stable surfaces have a virtual fundamental class?",
   "clean_statement": "Problem 4.\n3. (Hassett) Does the moduli space of stable surfaces have a virtual fundamental class?",
   "public_statement": "Problem 4.\n3. (Hassett) Does the moduli space of stable surfaces have a virtual fundamental class?",
   "evidence": "The canonical record is from the AIM workshop list *Compact moduli spaces and birational geometry*. Its extracted text is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 336,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0338": {
   "statement_status": "exact",
   "original_statement": "Problem 4.4. Find natural loci in moduli spaces of surfaces which would be interesting to intersect with each other.",
   "clean_statement": "Problem 4.4. Find natural loci in moduli spaces of surfaces which would be interesting to intersect with each other.",
   "public_statement": "Problem 4.4. Find natural loci in moduli spaces of surfaces which would be interesting to intersect with each other.",
   "evidence": "The canonical record is number 4.4 in the AIM workshop list *Compact moduli spaces and birational geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 337,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0339": {
   "statement_status": "exact",
   "original_statement": "Problem 4.\n5. (Alexeev) Is there a Gromov-Witten theory or quantum cohomology theory arising from stable pairs, perhaps from stable maps f: ( P2, D = D1 + · · · + Dn) → X, where \n\nD is a union of lines? 4.2. Explicit examples.",
   "clean_statement": "Problem 4.\n5. (Alexeev) Is there a Gromov-Witten theory or quantum cohomology theory arising from stable pairs, perhaps from stable maps f: ( P2, D = D1 + · · · + Dn) → X, where\n\nD is a union of lines? 4.2. Explicit examples.",
   "public_statement": "Problem 4.\n5. (Alexeev) Is there a Gromov-Witten theory or quantum cohomology theory arising from stable pairs, perhaps from stable maps f: ( P2, D = D1 + · · · + Dn) → X, where\n\nD is a union of lines? 4.2. Explicit examples.",
   "evidence": "The canonical JSON record has two extraction errors: “Problem 4.\\n5.” is Problem 4.5, and “4.2. Explicit examples” is the heading following the problem, not part of it. The official AIM TeX source and PDF give the following statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 338,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0340": {
   "statement_status": "exact",
   "original_statement": "Problem 4.6. Can one work out a geometric compactification of polarized K3 surfaces or Calabi-Yau manifolds in general using the framework of Alexeev/Koll´ ar/Shepherd-Barron? For example, let ( X, H ) be a pair where X is a K3 surface and H is a very ample divisor with H2 = 4. Can the stable pairs occurring as degenerations of these objects be classi-fied? Compare the resulting singularities with the work of Shah on GIT-semistable quartic surfaces. Also compare with Martin Olsson's work on moduli of log K3 surfaces.",
   "clean_statement": "Problem 4.6. Can one work out a geometric compactification of polarized K3 surfaces or Calabi-Yau manifolds in general using the framework of Alexeev/Koll´ ar/Shepherd-Barron? For example, let ( X, H ) be a pair where X is a K3 surface and H is a very ample divisor with H2 = 4. Can the stable pairs occurring as degenerations of these objects be classi-fied? Compare the resulting singularities with the work of Shah on GIT-semistable quartic surfaces. Also compare with Martin Olsson's work on moduli of log K3 surfaces.",
   "public_statement": "Problem 4.6. Can one work out a geometric compactification of polarized K3 surfaces or Calabi-Yau manifolds in general using the framework of Alexeev/Koll´ ar/Shepherd-Barron? For example, let ( X, H ) be a pair where X is a K3 surface and H is a very ample divisor with H2 = 4. Can the stable pairs occurring as degenerations of these objects be classi-fied? Compare the resulting singularities with the work of Shah on GIT-semistable quartic surfaces. Also compare with Martin Olsson's work on moduli of log K3 surfaces.",
   "evidence": "The source is Problem 4.6 in §4.2 (“Explicit examples”) of the AIM workshop notes *Compact moduli spaces and birational geometry*. The PDF text, with only the evident typographical/OCR repairs \\(H2\\mapsto H^2\\), “classi-fied” \\(\\mapsto\\) “classified,” and the accent in Kollár restored, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 339,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0341": {
   "statement_status": "exact",
   "original_statement": "Problem 4.7. Can one similarly give explicit examples of moduli spaces of surfaces of general type? For example, it would be interesting to understand the stable degenerations of (1) octic double planes, (2) quintic hypersurfaces, (3) bidouble covers, and other abelian covers, following Catanese, Manetti, Pardini, etc.",
   "clean_statement": "Problem 4.7. Can one similarly give explicit examples of moduli spaces of surfaces of general type? For example, it would be interesting to understand the stable degenerations of (1) octic double planes, (2) quintic hypersurfaces, (3) bidouble covers, and other abelian covers, following Catanese, Manetti, Pardini, etc.",
   "public_statement": "Problem 4.7. Can one similarly give explicit examples of moduli spaces of surfaces of general type? For example, it would be interesting to understand the stable degenerations of (1) octic double planes, (2) quintic hypersurfaces, (3) bidouble covers, and other abelian covers, following Catanese, Manetti, Pardini, etc.",
   "evidence": "The canonical source is the AIM workshop list *Compact moduli spaces and birational geometry*, Problem 4.7, in the subsection “Explicit examples.” The repository record agrees with the official PDF. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 340,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0342": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 4.8. Study the geometry of Hacking's moduli space of plane curves. Are there applications to questions about families of smooth plane curves? Similarly for configuration spaces or moduli spaces of marked del Pezzo surfaces. 5. Moduli of curves \n\n5.1. Birational geometry. Some references are: [FG03], [GKM02], [FP]. This is by no means a complete list; see the references in these papers for more details.",
   "clean_statement": "**Problem 4.8.** Study the geometry of Hacking's moduli space of plane\ncurves. Are there applications to questions about families of smooth plane\ncurves? Similarly for configuration spaces or moduli spaces of marked del\nPezzo surfaces.",
   "public_statement": "**Problem 4.8.** Study the geometry of Hacking's moduli space of plane\ncurves. Are there applications to questions about families of smooth plane\ncurves? Similarly for configuration spaces or moduli spaces of marked del\nPezzo surfaces.",
   "evidence": "The canonical record is Problem 4.8 from the AIM workshop *Compact moduli spaces and birational geometry*. The JSON extraction appends the beginning of Section 5 (\"Moduli of curves\") to the problem. Inspection of the official AIM source shows that this is extraction contamination: the problem environment ends before that section heading. The recovered statement is:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 341,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0343": {
   "statement_status": "exact",
   "original_statement": "Problem 5.1. Determine the cone of curves of M0,n.",
   "clean_statement": "Problem 5.1. Determine the cone of curves of M0,n.",
   "public_statement": "Problem 5.1. Determine the cone of curves of M0,n.",
   "evidence": "The canonical JSON record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 342,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0344": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 5.2. Determine the Kodaira dimension of Mg when g = 15 or 17 ≤ g ≤ 22. For g > 22 Mg is of general type, and in the other known cases, the Kodaira dimension is negative. 5.2. Other questions.",
   "clean_statement": "**Problem 5.2.** Determine the Kodaira dimension of\n\\(\\overline{\\mathcal M}_g\\) when \\(g=15\\) or \\(17\\leq g\\leq22\\).\nFor \\(g>22\\), \\(\\overline{\\mathcal M}_g\\) is of general type, and in\nthe other known cases, the Kodaira dimension is negative.",
   "public_statement": "**Problem 5.2.** Determine the Kodaira dimension of\n\\(\\overline{\\mathcal M}_g\\) when \\(g=15\\) or \\(17\\leq g\\leq22\\).\nFor \\(g>22\\), \\(\\overline{\\mathcal M}_g\\) is of general type, and in\nthe other known cases, the Kodaira dimension is negative.",
   "evidence": "The typography in the official AIM source resolves two extraction defects. On the printed page, a horizontal bar is drawn over both occurrences of \\(\\mathcal M_g\\); the repository's text extractor discarded those bars. Also, the final words “5.2. Other questions” are the heading of the next subsection, not part of Problem 5.2. The recovered statement is therefore: Source: AIM, *Compact moduli spaces and birational geometry*, Problem 5.2, [official PDF](https://aimath.org/WWN/birational/birational.pdf).",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 343,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0345": {
   "statement_status": "exact",
   "original_statement": "Problem 5.3. Does the coarse moduli space Mg contain a P1? An A1? It is known that the moduli stack in this case is algebraically hyperbolic, but the same is not likely to be true for the moduli space.",
   "clean_statement": "Problem 5.3. Does the coarse moduli space Mg contain a P1? An A1? It is known that the moduli stack in this case is algebraically hyperbolic, but the same is not likely to be true for the moduli space.",
   "public_statement": "Problem 5.3. Does the coarse moduli space Mg contain a P1? An A1? It is known that the moduli stack in this case is algebraically hyperbolic, but the same is not likely to be true for the moduli space.",
   "evidence": "The canonical record is Problem 5.3 from the AIM workshop notes *Compact moduli spaces and birational geometry*. The repository extraction reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 344,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0346": {
   "statement_status": "exact",
   "original_statement": "Problem 5.\n4. (Farkas) For g large, what is the minimal genus of a curve passing through a general point of Mg? This can be asked for the moduli space or the moduli stack. For low \n\ng, the answer for the moduli stack is zero. One could ask this question for moduli spaces of canonically polarized manifolds, although I think the point of making g large is to ensure that Mg is not uniruled. There is nothing even approximating such a condition to ensure that MH is not unirational. 6 MICHAEL A. VAN OPSTALL \n\n6. Moduli of abelian varieties",
   "clean_statement": "Problem 5.\n4. (Farkas) For g large, what is the minimal genus of a curve passing through a general point of Mg? This can be asked for the moduli space or the moduli stack. For low\n\ng, the answer for the moduli stack is zero. One could ask this question for moduli spaces of canonically polarized manifolds, although I think the point of making g large is to ensure that Mg is not uniruled. There is nothing even approximating such a condition to ensure that MH is not unirational. 6 MICHAEL A. VAN OPSTALL\n\n6. Moduli of abelian varieties",
   "public_statement": "Problem 5.\n4. (Farkas) For g large, what is the minimal genus of a curve passing through a general point of Mg? This can be asked for the moduli space or the moduli stack. For low\n\ng, the answer for the moduli stack is zero. One could ask this question for moduli spaces of canonically polarized manifolds, although I think the point of making g large is to ensure that Mg is not uniruled. There is nothing even approximating such a condition to ensure that MH is not unirational. 6 MICHAEL A. VAN OPSTALL\n\n6. Moduli of abelian varieties",
   "evidence": "The source is Michael A. van Opstall's AIM list *Open Problems in Compact Moduli Spaces and Birational Geometry*, in Section 5.2, “Other questions.” Inspection of the official PDF, including its underlying text operators, shows that the displayed notation is the open coarse space \\(M_g\\), not \\(\\overline M_g\\). The recovered item is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 345,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0347": {
   "statement_status": "exact",
   "original_statement": "Problem 6.\n1. (Grushevsky) (1) Construct divisors on Ag of small slope. (2) Find explicit complete subvarieties of Ag. There is a bound on the codimension of such a variety in [KS03].",
   "clean_statement": "Problem 6.\n1. (Grushevsky) (1) Construct divisors on Ag of small slope. (2) Find explicit complete subvarieties of Ag. There is a bound on the codimension of such a variety in [KS03].",
   "public_statement": "Problem 6.\n1. (Grushevsky) (1) Construct divisors on Ag of small slope. (2) Find explicit complete subvarieties of Ag. There is a bound on the codimension of such a variety in [KS03].",
   "evidence": "The canonical JSON record is a slightly flattened OCR extraction from the AIM workshop list *Compact moduli spaces and birational geometry*. The official PDF has a section headed “6. Moduli of abelian varieties” and reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 346,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_GEOMETRY-0348": {
   "statement_status": "exact",
   "original_statement": "Problem 6.\n2. (Shepherd-Barron) Questions on Ag and the first Voronoi compactification \n\nAFg.(1) Run the MMP on AFg when g ≤ 10; in particular, flip its extremal ray. If g ≥ 12, AFg\n\nis the canonical model of Ag, and when g = 11 it is a minimal model. (2) There is a family of natural inclusions AFg → AFg+1 parameterized by the j-line. Does \n\nH∗(AFg, C) stabilize? To what? (3) Let M be the bundle of weight 1 modular forms and D the boundary divisor. Com-pute the intersection numbers M a.D b and the plurigenera of AFg.(4) Is the total coordinate ring of AFg finitely generated? (5) Is the divisor 12 M − D semi-ample for g ≥ 12? (6) Find the effective cone of AFg. This is the well-known question of finding the possible slopes of Siegel cusp forms.",
   "clean_statement": "Problem 6.\n2. (Shepherd-Barron) Questions on Ag and the first Voronoi compactification\n\nAFg.(1) Run the MMP on AFg when g ≤ 10; in particular, flip its extremal ray. If g ≥ 12, AFg\n\nis the canonical model of Ag, and when g = 11 it is a minimal model. (2) There is a family of natural inclusions AFg → AFg+1 parameterized by the j-line. Does\n\nH∗(AFg, C) stabilize? To what? (3) Let M be the bundle of weight 1 modular forms and D the boundary divisor. Com-pute the intersection numbers M a.D b and the plurigenera of AFg.(4) Is the total coordinate ring of AFg finitely generated? (5) Is the divisor 12 M − D semi-ample for g ≥ 12? (6) Find the effective cone of AFg. This is the well-known question of finding the possible slopes of Siegel cusp forms.",
   "public_statement": "Problem 6.\n2. (Shepherd-Barron) Questions on Ag and the first Voronoi compactification\n\nAFg.(1) Run the MMP on AFg when g ≤ 10; in particular, flip its extremal ray. If g ≥ 12, AFg\n\nis the canonical model of Ag, and when g = 11 it is a minimal model. (2) There is a family of natural inclusions AFg → AFg+1 parameterized by the j-line. Does\n\nH∗(AFg, C) stabilize? To what? (3) Let M be the bundle of weight 1 modular forms and D the boundary divisor. Com-pute the intersection numbers M a.D b and the plurigenera of AFg.(4) Is the total coordinate ring of AFg finitely generated? (5) Is the divisor 12 M − D semi-ample for g ≥ 12? (6) Find the effective cone of AFg. This is the well-known question of finding the possible slopes of Siegel cusp forms.",
   "evidence": "The JSON record is an OCR extraction in which \\(A_g^F\\) appears as strings such as `AFg`. The statement was recovered from the official AIM TeX source `https://aimath.org/WWN/birational/birational.tex` (lines 555--572 at the time of access) and checked against the official PDF. The source reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-geometry-notes.json",
   "source_index": 347,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0001": {
   "statement_status": "exact",
   "original_statement": "The goal is to formulate a Manin-type conjecture for degree $d$ points on Fano varieties. Let $X/k$ be a Fano variety defined over a number field $k$. For a subset $U\\subset X$, an integer $d$, a real number $B>0$, and a line bundle $\\mathscr L$ on $X$, define\n$$N_{X,\\mathscr L}(U,B;d) := \\#\\left\\{x\\in U(\\overline k):h_{\\mathscr L}(x)\\le B\\text{ and }[\\kappa(x):k]=d\\right\\},$$\nwhere $h_{\\mathscr L}$ is a Weil height associated to $\\mathscr L$. For appropriately chosen $U$, one may expect that $N_{X,\\mathscr L}(U,B;d)\\sim cB^a(\\log B)^b$ as $B\\to\\infty$, for some choice of $a,b,c$.\n\nThere were several questions proposed around this, including\n\n* What should the predicted values of $a,b,c$ be?\n\n* What is the \"shape\" of the set $U$ that one should take? In the usual Manin conjecture, one expects that $U$ can be taken to be the complement of a thin set.\n\n* Can one compute asymptotics for $N_{X,\\mathscr L}(U,B;d)$ if $X=\\mathbb P^2_{\\mathbb Q}$ and $d=3$, for some choices of $\\mathscr L$?",
   "clean_statement": "The goal is to formulate a Manin-type conjecture for degree $d$ points on Fano varieties. Let $X/k$ be a Fano variety defined over a number field $k$. For a subset $U\\subset X$, an integer $d$, a real number $B>0$, and a line bundle $\\mathscr L$ on $X$, define\n$$N_{X,\\mathscr L}(U,B;d) := \\#\\left\\{x\\in U(\\overline k):h_{\\mathscr L}(x)\\le B\\text{ and }[\\kappa(x):k]=d\\right\\},$$\nwhere $h_{\\mathscr L}$ is a Weil height associated to $\\mathscr L$. For appropriately chosen $U$, one may expect that $N_{X,\\mathscr L}(U,B;d)\\sim cB^a(\\log B)^b$ as $B\\to\\infty$, for some choice of $a,b,c$.\n\nThere were several questions proposed around this, including\n\n* What should the predicted values of $a,b,c$ be?\n\n* What is the \"shape\" of the set $U$ that one should take? In the usual Manin conjecture, one expects that $U$ can be taken to be the complement of a thin set.\n\n* Can one compute asymptotics for $N_{X,\\mathscr L}(U,B;d)$ if $X=\\mathbb P^2_{\\mathbb Q}$ and $d=3$, for some choices of $\\mathscr L$?",
   "public_statement": "The goal is to formulate a Manin-type conjecture for degree $d$ points on Fano varieties. Let $X/k$ be a Fano variety defined over a number field $k$. For a subset $U\\subset X$, an integer $d$, a real number $B>0$, and a line bundle $\\mathscr L$ on $X$, define\n$$N_{X,\\mathscr L}(U,B;d) := \\#\\left\\{x\\in U(\\overline k):h_{\\mathscr L}(x)\\le B\\text{ and }[\\kappa(x):k]=d\\right\\},$$\nwhere $h_{\\mathscr L}$ is a Weil height associated to $\\mathscr L$. For appropriately chosen $U$, one may expect that $N_{X,\\mathscr L}(U,B;d)\\sim cB^a(\\log B)^b$ as $B\\to\\infty$, for some choice of $a,b,c$.\n\nThere were several questions proposed around this, including\n\n* What should the predicted values of $a,b,c$ be?\n\n* What is the \"shape\" of the set $U$ that one should take? In the usual Manin conjecture, one expects that $U$ can be taken to be the complement of a thin set.\n\n* Can one compute asymptotics for $N_{X,\\mathscr L}(U,B;d)$ if $X=\\mathbb P^2_{\\mathbb Q}$ and $d=3$, for some choices of $\\mathscr L$?",
   "evidence": "The source is AIM Problem List 1.02 from the workshop *Degree \\(d\\) points on algebraic surfaces*. It asks for a Manin-type conjecture for degree-\\(d\\) points on a Fano variety \\(X/k\\). Its displayed counting function is \\[ N_{X,\\mathscr L}(U,B;d) = \\#\\{x\\in U(\\overline k):h_{\\mathscr L}(x)\\leq B,\\ [\\kappa(x):k]=d\\}, \\] and the proposed form is \\(cB^a(\\log B)^b\\). It asks for \\(a,b,c\\), for the correct shape of \\(U\\), and particularly for \\(X=\\mathbb P^2_{\\mathbb Q}\\), \\(d=3\\).",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0002": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Is there a $\\mathbb Q$-irrational del Pezzo surface $X$ which is $\\mathbb Q$-unirational for which $\\operatorname{Sym}^dX$ is $\\mathbb Q$-rational for some $d>1$?",
   "clean_statement": null,
   "public_statement": "Is there a $\\mathbb Q$-irrational del Pezzo surface $X$ which is $\\mathbb Q$-unirational for which $\\operatorname{Sym}^dX$ is $\\mathbb Q$-rational for some $d>1$?",
   "evidence": "The exact canonical JSON record was checked at zero-based source index \\(1\\) in `aim-algebraic-number-theory-notes.json`, together with its neighboring records. The original URL `http://aimpl.org/degreedsurface/1/` no longer returned the problem page when checked on 2026-07-24 (the reachable AIM endpoint returned a 404 page), so no silent reconstruction from that page was made. The follow-up remark refers to “Problem 1.2”; in the canonical ordering this is a stale number and is almost certainly intended to refer to this problem, numbered 1.04.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0003": {
   "statement_status": "exact",
   "original_statement": "Let $X$ be a variety. Suppose one has a diagram $X\\dashleftarrow Y\\dashrightarrow\\mathbb P^n$, where $Y\\dashrightarrow\\mathbb P^n$ is finite of degree $d$. Then, as a consequence of Hilbert irreducibility, $X$ will have a Zariski dense set of degree $d$ points.\n\nCan we give a method to construct a Zariski dense set of degree $d$ points on a variety different from the one just mentioned?\n\n* Maybe one can replace $\\mathbb P^n$ with another sort of variety?\n\n* Maybe one can do something totally different and give a construction not coming from a correspondence.",
   "clean_statement": "Let $X$ be a variety. Suppose one has a diagram $X\\dashleftarrow Y\\dashrightarrow\\mathbb P^n$, where $Y\\dashrightarrow\\mathbb P^n$ is finite of degree $d$. Then, as a consequence of Hilbert irreducibility, $X$ will have a Zariski dense set of degree $d$ points.\n\nCan we give a method to construct a Zariski dense set of degree $d$ points on a variety different from the one just mentioned?\n\n* Maybe one can replace $\\mathbb P^n$ with another sort of variety?\n\n* Maybe one can do something totally different and give a construction not coming from a correspondence.",
   "public_statement": "Let $X$ be a variety. Suppose one has a diagram $X\\dashleftarrow Y\\dashrightarrow\\mathbb P^n$, where $Y\\dashrightarrow\\mathbb P^n$ is finite of degree $d$. Then, as a consequence of Hilbert irreducibility, $X$ will have a Zariski dense set of degree $d$ points.\n\nCan we give a method to construct a Zariski dense set of degree $d$ points on a variety different from the one just mentioned?\n\n* Maybe one can replace $\\mathbb P^n$ with another sort of variety?\n\n* Maybe one can do something totally different and give a construction not coming from a correspondence.",
   "evidence": "This is problem 1.06 from the AIM workshop *Degree \\(d\\) points on algebraic surfaces*, in the section “Initial Problem Session.” The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 2,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0004": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "With some hypotheses on $X$ and $X\\to\\operatorname{Alb}(X)$, study the image of $\\operatorname{Sym}^d(X)\\to\\operatorname{Alb}(X)$ for some small $d$.\n\nWhat are examples of $X$ for which the fibers of $\\operatorname{Sym}^dX\\to\\operatorname{Alb}(X)$ are \"well-understood\"? For example, when are they finite or rational?",
   "clean_statement": "With some hypotheses on \\(X\\) and\n\\(X\\to\\operatorname{Alb}(X)\\), study the image of\n\\(\\operatorname{Sym}^d(X)\\to\\operatorname{Alb}(X)\\) for some small\n\\(d\\).\n\nWhat are examples of \\(X\\) for which the fibers of\n\\(\\operatorname{Sym}^dX\\to\\operatorname{Alb}(X)\\) are\n“well-understood”? For example, when are they finite or rational?",
   "public_statement": "With some hypotheses on $X$ and $X\\to\\operatorname{Alb}(X)$, study the image of $\\operatorname{Sym}^d(X)\\to\\operatorname{Alb}(X)$ for some small $d$.\n\nWhat are examples of $X$ for which the fibers of $\\operatorname{Sym}^dX\\to\\operatorname{Alb}(X)$ are \"well-understood\"? For example, when are they finite or rational?",
   "evidence": "This is Problem 1.08 from the AIM workshop *Degree \\(d\\) points on algebraic surfaces*. The canonical record and the current AIM problem-list page agree. The recovered statement is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 3,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0005": {
   "statement_status": "exact",
   "original_statement": "Let $X/k$ be a nice variety. We define a few measures of irrationality.\n\n* Its covering gonality $\\operatorname{cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\\leftarrow\\mathcal C\\dashrightarrow X$, where $\\mathcal C\\dashrightarrow X$ is a dominant rational map and $\\mathcal C\\to B$ is a family of curves such that $\\operatorname{gon}(\\mathcal C_b)\\ge d$ for all $b\\in B$.\n\n* Its connected covering gonality $\\operatorname{conn.cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\\leftarrow\\mathcal C\\dashrightarrow X$, where $\\mathcal C\\to B$ is a family of curves with $\\operatorname{gon}(\\mathcal C_b)\\ge d$ for all $b\\in B$ and such that the induced rational map $\\mathcal C\\times_B\\mathcal C\\dashrightarrow X\\times X$ is dominant.\n\n* Finally, $\\operatorname{uni.irr}(X)$ is the minimal $d$ such that there exists a diagram $X\\dashleftarrow Y\\dashrightarrow\\mathbb P^n$ with $Y\\dashrightarrow X$ dominant and $Y\\dashrightarrow\\mathbb P^n$ finite of degree $d$.\n\nLet $X/k$ be a nice variety, and let $\\delta(X/k)=\\{d:$ degree $d$ points on $X$ are Zariski dense in $X\\}$ be its density degree set. How different can $\\min\\delta(X/k)$ be from $\\operatorname{uni.irr}(X),\\operatorname{cov.gon}(X)$, and/or $\\operatorname{conn.cov.gon}(X)$?",
   "clean_statement": "Let $X/k$ be a nice variety. We define a few measures of irrationality.\n\n* Its covering gonality $\\operatorname{cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\\leftarrow\\mathcal C\\dashrightarrow X$, where $\\mathcal C\\dashrightarrow X$ is a dominant rational map and $\\mathcal C\\to B$ is a family of curves such that $\\operatorname{gon}(\\mathcal C_b)\\ge d$ for all $b\\in B$.\n\n* Its connected covering gonality $\\operatorname{conn.cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\\leftarrow\\mathcal C\\dashrightarrow X$, where $\\mathcal C\\to B$ is a family of curves with $\\operatorname{gon}(\\mathcal C_b)\\ge d$ for all $b\\in B$ and such that the induced rational map $\\mathcal C\\times_B\\mathcal C\\dashrightarrow X\\times X$ is dominant.\n\n* Finally, $\\operatorname{uni.irr}(X)$ is the minimal $d$ such that there exists a diagram $X\\dashleftarrow Y\\dashrightarrow\\mathbb P^n$ with $Y\\dashrightarrow X$ dominant and $Y\\dashrightarrow\\mathbb P^n$ finite of degree $d$.\n\nLet $X/k$ be a nice variety, and let $\\delta(X/k)=\\{d:$ degree $d$ points on $X$ are Zariski dense in $X\\}$ be its density degree set. How different can $\\min\\delta(X/k)$ be from $\\operatorname{uni.irr}(X),\\operatorname{cov.gon}(X)$, and/or $\\operatorname{conn.cov.gon}(X)$?",
   "public_statement": "Let $X/k$ be a nice variety. We define a few measures of irrationality.\n\n* Its covering gonality $\\operatorname{cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\\leftarrow\\mathcal C\\dashrightarrow X$, where $\\mathcal C\\dashrightarrow X$ is a dominant rational map and $\\mathcal C\\to B$ is a family of curves such that $\\operatorname{gon}(\\mathcal C_b)\\ge d$ for all $b\\in B$.\n\n* Its connected covering gonality $\\operatorname{conn.cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\\leftarrow\\mathcal C\\dashrightarrow X$, where $\\mathcal C\\to B$ is a family of curves with $\\operatorname{gon}(\\mathcal C_b)\\ge d$ for all $b\\in B$ and such that the induced rational map $\\mathcal C\\times_B\\mathcal C\\dashrightarrow X\\times X$ is dominant.\n\n* Finally, $\\operatorname{uni.irr}(X)$ is the minimal $d$ such that there exists a diagram $X\\dashleftarrow Y\\dashrightarrow\\mathbb P^n$ with $Y\\dashrightarrow X$ dominant and $Y\\dashrightarrow\\mathbb P^n$ finite of degree $d$.\n\nLet $X/k$ be a nice variety, and let $\\delta(X/k)=\\{d:$ degree $d$ points on $X$ are Zariski dense in $X\\}$ be its density degree set. How different can $\\min\\delta(X/k)$ be from $\\operatorname{uni.irr}(X),\\operatorname{cov.gon}(X)$, and/or $\\operatorname{conn.cov.gon}(X)$?",
   "evidence": "The source record is Problem 1.1 from the AIM workshop *Degree \\(d\\) points on algebraic surfaces*. For a nice variety \\(X/k\\), it asks how far",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 4,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0006": {
   "statement_status": "exact",
   "original_statement": "Let $X/k$ be a nice variety. In addition to its density degree set $\\delta(X/\\mathbb Q)$, define\n$$\\mathcal D(X/\\mathbb Q):=\\{d:X\\text{ has a point of degree }d\\},$$\nand\n$$\\mathcal P(X/\\mathbb Q)=\\bigcup_{[k':k]<\\infty}\\delta(X/k').$$\n\nSkorobogatov constructed a bielliptic surface $X/\\mathbb Q$ such that $X(\\mathbb A_{\\mathbb Q})^{\\operatorname{Br}}\\neq\\emptyset$, but $X(\\mathbb Q)=\\emptyset$. It is known that $\\min\\delta(X/\\mathbb Q)\\le3$. Where do these Zariski-dense degree 3 points come from? Can one compute $\\mathcal D(X/\\mathbb Q),\\delta(X/\\mathbb Q),\\mathcal P(X/\\mathbb Q)$?",
   "clean_statement": "Let $X/k$ be a nice variety. In addition to its density degree set $\\delta(X/\\mathbb Q)$, define\n$$\\mathcal D(X/\\mathbb Q):=\\{d:X\\text{ has a point of degree }d\\},$$\nand\n$$\\mathcal P(X/\\mathbb Q)=\\bigcup_{[k':k]<\\infty}\\delta(X/k').$$\n\nSkorobogatov constructed a bielliptic surface $X/\\mathbb Q$ such that $X(\\mathbb A_{\\mathbb Q})^{\\operatorname{Br}}\\neq\\emptyset$, but $X(\\mathbb Q)=\\emptyset$. It is known that $\\min\\delta(X/\\mathbb Q)\\le3$. Where do these Zariski-dense degree 3 points come from? Can one compute $\\mathcal D(X/\\mathbb Q),\\delta(X/\\mathbb Q),\\mathcal P(X/\\mathbb Q)$?",
   "public_statement": "Let $X/k$ be a nice variety. In addition to its density degree set $\\delta(X/\\mathbb Q)$, define\n$$\\mathcal D(X/\\mathbb Q):=\\{d:X\\text{ has a point of degree }d\\},$$\nand\n$$\\mathcal P(X/\\mathbb Q)=\\bigcup_{[k':k]<\\infty}\\delta(X/k').$$\n\nSkorobogatov constructed a bielliptic surface $X/\\mathbb Q$ such that $X(\\mathbb A_{\\mathbb Q})^{\\operatorname{Br}}\\neq\\emptyset$, but $X(\\mathbb Q)=\\emptyset$. It is known that $\\min\\delta(X/\\mathbb Q)\\le3$. Where do these Zariski-dense degree 3 points come from? Can one compute $\\mathcal D(X/\\mathbb Q),\\delta(X/\\mathbb Q),\\mathcal P(X/\\mathbb Q)$?",
   "evidence": "The canonical AIM record, Problem 1.12 from the workshop *Degree \\(d\\) points on algebraic surfaces*, asks about three sets. For a nice variety \\(X/k\\),",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0007": {
   "statement_status": "exact",
   "original_statement": "Let $C,D$ be hyperelliptic curves of genus $\\ge2$ over a number field $k$. Must $2\\in\\delta(C\\times D/k)$? Must it be in $\\mathcal P(C\\times D/k)$?\n\nMore generally, let $Z\\subset C\\times D$ be the closure of the degree $d$ points. When is there a component of $Z$ which dominates both $C$ and $D$?",
   "clean_statement": "Let $C,D$ be hyperelliptic curves of genus $\\ge2$ over a number field $k$. Must $2\\in\\delta(C\\times D/k)$? Must it be in $\\mathcal P(C\\times D/k)$?\n\nMore generally, let $Z\\subset C\\times D$ be the closure of the degree $d$ points. When is there a component of $Z$ which dominates both $C$ and $D$?",
   "public_statement": "Let $C,D$ be hyperelliptic curves of genus $\\ge2$ over a number field $k$. Must $2\\in\\delta(C\\times D/k)$? Must it be in $\\mathcal P(C\\times D/k)$?\n\nMore generally, let $Z\\subset C\\times D$ be the closure of the degree $d$ points. When is there a component of $Z$ which dominates both $C$ and $D$?",
   "evidence": "This record is Problem 1.14 from the initial problem session of the AIM workshop *Degree \\(d\\) points on algebraic surfaces*. The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0008": {
   "statement_status": "exact",
   "original_statement": "Let $C$ be a curve of genus $\\ge2$ over a number field. Must $2\\in\\delta(\\operatorname{Sym}^2_C/k)$ or in $\\mathcal P(\\operatorname{Sym}^2_C/k)$? What do these sets look like?",
   "clean_statement": "Let $C$ be a curve of genus $\\ge2$ over a number field. Must $2\\in\\delta(\\operatorname{Sym}^2_C/k)$ or in $\\mathcal P(\\operatorname{Sym}^2_C/k)$? What do these sets look like?",
   "public_statement": "Let $C$ be a curve of genus $\\ge2$ over a number field. Must $2\\in\\delta(\\operatorname{Sym}^2_C/k)$ or in $\\mathcal P(\\operatorname{Sym}^2_C/k)$? What do these sets look like?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0009": {
   "statement_status": "exact",
   "original_statement": "Given a finite map $f:X\\to Y$ (over a number field), how are $\\delta(X/k)$ and $\\delta(Y/k)$ related? Are there examples showing that the obvious relations are sharp?",
   "clean_statement": "Given a finite map $f:X\\to Y$ (over a number field), how are $\\delta(X/k)$ and $\\delta(Y/k)$ related? Are there examples showing that the obvious relations are sharp?",
   "public_statement": "Given a finite map $f:X\\to Y$ (over a number field), how are $\\delta(X/k)$ and $\\delta(Y/k)$ related? Are there examples showing that the obvious relations are sharp?",
   "evidence": "The canonical record is item 1.18 of the AIM problem list from the workshop *Degree \\(d\\) points on algebraic surfaces* (*aim-algebraic-number-theory-notes.json*, zero-based index 8):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0010": {
   "statement_status": "exact",
   "original_statement": "Is $\\operatorname{Hilb}^2(C\\times D)(k)$ Zariski dense in $\\operatorname{Hilb}^2(C\\times D)$, where $C,D/k$ are hyperelliptic curves over a number field. More generally, for which varieties $X$ over a number field is there a $d$ such that $\\operatorname{Hilb}^d(X)(k)$ is Zariski dense?",
   "clean_statement": "Is $\\operatorname{Hilb}^2(C\\times D)(k)$ Zariski dense in $\\operatorname{Hilb}^2(C\\times D)$, where $C,D/k$ are hyperelliptic curves over a number field. More generally, for which varieties $X$ over a number field is there a $d$ such that $\\operatorname{Hilb}^d(X)(k)$ is Zariski dense?",
   "public_statement": "Is $\\operatorname{Hilb}^2(C\\times D)(k)$ Zariski dense in $\\operatorname{Hilb}^2(C\\times D)$, where $C,D/k$ are hyperelliptic curves over a number field. More generally, for which varieties $X$ over a number field is there a $d$ such that $\\operatorname{Hilb}^d(X)(k)$ is Zariski dense?",
   "evidence": "The canonical record is Problem 1.2 from the AIM workshop *Degree d points on algebraic surfaces*. Its exact `problem` field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0011": {
   "statement_status": "exact",
   "original_statement": "Given some specific geometric construction showing that $d\\in\\delta(X/k)$, can one use it to construct a dominant rational map $X\\dashrightarrow\\mathbb P^2$ of degree $nd$, for some small $n$?",
   "clean_statement": "Given some specific geometric construction showing that $d\\in\\delta(X/k)$, can one use it to construct a dominant rational map $X\\dashrightarrow\\mathbb P^2$ of degree $nd$, for some small $n$?",
   "public_statement": "Given some specific geometric construction showing that $d\\in\\delta(X/k)$, can one use it to construct a dominant rational map $X\\dashrightarrow\\mathbb P^2$ of degree $nd$, for some small $n$?",
   "evidence": "The canonical record is problem 1.22 from the initial problem session of the March 2024 AIM workshop *Degree d points on algebraic surfaces*. Its exact `problem` field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0012": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Given a covering family $T\\leftarrow\\mathcal C\\dashrightarrow X$ of curves, where $T(k)$ is Zariski dense, how does $\\operatorname{gon}(C)$ (for $C$ a generic member of the family) compare to $\\min\\delta(X/k)$?",
   "clean_statement": null,
   "public_statement": "Given a covering family $T\\leftarrow\\mathcal C\\dashrightarrow X$ of curves, where $T(k)$ is Zariski dense, how does $\\operatorname{gon}(C)$ (for $C$ a generic member of the family) compare to $\\min\\delta(X/k)$?",
   "evidence": "The exact canonical record is problem 1.24 from the AIM workshop *Degree \\(d\\) points on algebraic surfaces*:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0013": {
   "statement_status": "exact",
   "original_statement": "For a variety $X/F$, define its degree of irrationality $\\operatorname{irr}(X)$ to be the minimum degree of a rational map $X\\dashrightarrow\\mathbb P^2$ (defined over $F$).\n\nCompute $\\operatorname{irr}(X/F)$ for a del Pezzo surface $X$ over a general field $F$.",
   "clean_statement": "For a variety $X/F$, define its degree of irrationality $\\operatorname{irr}(X)$ to be the minimum degree of a rational map $X\\dashrightarrow\\mathbb P^2$ (defined over $F$).\n\nCompute $\\operatorname{irr}(X/F)$ for a del Pezzo surface $X$ over a general field $F$.",
   "public_statement": "For a variety $X/F$, define its degree of irrationality $\\operatorname{irr}(X)$ to be the minimum degree of a rational map $X\\dashrightarrow\\mathbb P^2$ (defined over $F$).\n\nCompute $\\operatorname{irr}(X/F)$ for a del Pezzo surface $X$ over a general field $F$.",
   "evidence": "The canonical AIM record is problem 1.26 from the initial problem session of the March 2024 workshop *Degree \\(d\\) points on algebraic surfaces*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0014": {
   "statement_status": "exact",
   "original_statement": "How do all of these invariants (e.g. $\\min\\delta(X/k),\\operatorname{irr}(X/k),\\operatorname{cov.gon}(X/k)$, etc.) behave under specialization/deformation in a smooth family? This includes the arithmetic case, where the base is something like $\\operatorname{Spec}\\mathscr O_{k,S}$.",
   "clean_statement": "How do all of these invariants (e.g. $\\min\\delta(X/k),\\operatorname{irr}(X/k),\\operatorname{cov.gon}(X/k)$, etc.) behave under specialization/deformation in a smooth family? This includes the arithmetic case, where the base is something like $\\operatorname{Spec}\\mathscr O_{k,S}$.",
   "public_statement": "How do all of these invariants (e.g. $\\min\\delta(X/k),\\operatorname{irr}(X/k),\\operatorname{cov.gon}(X/k)$, etc.) behave under specialization/deformation in a smooth family? This includes the arithmetic case, where the base is something like $\\operatorname{Spec}\\mathscr O_{k,S}$.",
   "evidence": "This is record `AIM-ALGEBRAIC_NUMBER_THEORY-0014`, problem 1.28 in the AIM workshop *Degree d points on algebraic surfaces*, source file `aim-algebraic-number-theory-notes.json`, source index 13.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0015": {
   "statement_status": "exact",
   "original_statement": "Is there a surface $X$ over a number field $k$ and a finite extension $k'/k$ such that $\\delta(X/k)\\not\\subset\\delta(X/k')$?",
   "clean_statement": "Is there a surface $X$ over a number field $k$ and a finite extension $k'/k$ such that $\\delta(X/k)\\not\\subset\\delta(X/k')$?",
   "public_statement": "Is there a surface $X$ over a number field $k$ and a finite extension $k'/k$ such that $\\delta(X/k)\\not\\subset\\delta(X/k')$?",
   "evidence": "The record occurs in *aim-algebraic-number-theory-notes.json* at zero-based index 14. There is no visible corruption. The AIM workshop summary specifies the intended class of surfaces: smooth, projective, and geometrically integral over a number field. That geometric-integrality hypothesis matters because the finite-union density argument below can fail on a disconnected base change.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0016": {
   "statement_status": "exact",
   "original_statement": "The following is asked as an analogue of a result due to Debarre and Klassen \\cite{arXiv:alg-geom/9210004}.\n\nLet $X_d\\subset\\mathbb P^3_k$ ($k$ a number field) be a smooth surface of degree $d$. Are the degree $\\le d-3$ points not Zariski dense? What is $\\min\\delta(X_d/k)$?",
   "clean_statement": "The following is asked as an analogue of a result due to Debarre and Klassen \\cite{arXiv:alg-geom/9210004}.\n\nLet $X_d\\subset\\mathbb P^3_k$ ($k$ a number field) be a smooth surface of degree $d$. Are the degree $\\le d-3$ points not Zariski dense? What is $\\min\\delta(X_d/k)$?",
   "public_statement": "The following is asked as an analogue of a result due to Debarre and Klassen \\cite{arXiv:alg-geom/9210004}.\n\nLet $X_d\\subset\\mathbb P^3_k$ ($k$ a number field) be a smooth surface of degree $d$. Are the degree $\\le d-3$ points not Zariski dense? What is $\\min\\delta(X_d/k)$?",
   "evidence": "The record is Problem 1.32 from the AIM workshop *Degree \\(d\\) points on algebraic surfaces* (source file `aim-algebraic-number-theory-notes.json`, source index 15). The source record is legible and no reconstruction is needed. Its problem field is, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0017": {
   "statement_status": "exact",
   "original_statement": "Given a dominant map $\\psi:X\\dashrightarrow\\mathbb P^{\\dim X}$ of minimal degree $d$, is the gonality of the pullback of a general line equal to $d$?",
   "clean_statement": "Given a dominant map $\\psi:X\\dashrightarrow\\mathbb P^{\\dim X}$ of minimal degree $d$, is the gonality of the pullback of a general line equal to $d$?",
   "public_statement": "Given a dominant map $\\psi:X\\dashrightarrow\\mathbb P^{\\dim X}$ of minimal degree $d$, is the gonality of the pullback of a general line equal to $d$?",
   "evidence": "The canonical record is Problem 1.34 in the initial problem session of the AIM workshop *Degree d points on algebraic surfaces*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0018": {
   "statement_status": "exact",
   "original_statement": "Given a general smooth degree $d$ hypersurface $X_d\\subset\\mathbb P^3_k$, for $d$ sufficiently large, must a general degree $d$ point arise from intersecting $X$ with a line?",
   "clean_statement": "Given a general smooth degree $d$ hypersurface $X_d\\subset\\mathbb P^3_k$, for $d$ sufficiently large, must a general degree $d$ point arise from intersecting $X$ with a line?",
   "public_statement": "Given a general smooth degree $d$ hypersurface $X_d\\subset\\mathbb P^3_k$, for $d$ sufficiently large, must a general degree $d$ point arise from intersecting $X$ with a line?",
   "evidence": "The source is AIM Problem 1.36 from the workshop *Degree \\(d\\) points on algebraic surfaces*, attributed to Shamil Asgarli. The exact record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0019": {
   "statement_status": "exact",
   "original_statement": "The following is asked in analogy with a result of Harris and Silverman showing that, for a curve $C/k$, $2\\in\\delta(C/k)\\iff C$ is a double cover of $\\mathbb P^1$ or a positive rank elliptic curve (see https://www.ams.org/journals/proc/1991-112-02/S0002-9939-1991-1055774-0/S0002-9939-1991-1055774-0.pdf).\n\nClassify surfaces $X/k$ of general type with $2\\in\\delta(X/k)$, possibly using geometric and/or dynamical constructions.",
   "clean_statement": "The following is asked in analogy with a result of Harris and Silverman showing that, for a curve $C/k$, $2\\in\\delta(C/k)\\iff C$ is a double cover of $\\mathbb P^1$ or a positive rank elliptic curve (see https://www.ams.org/journals/proc/1991-112-02/S0002-9939-1991-1055774-0/S0002-9939-1991-1055774-0.pdf).\n\nClassify surfaces $X/k$ of general type with $2\\in\\delta(X/k)$, possibly using geometric and/or dynamical constructions.",
   "public_statement": "The following is asked in analogy with a result of Harris and Silverman showing that, for a curve $C/k$, $2\\in\\delta(C/k)\\iff C$ is a double cover of $\\mathbb P^1$ or a positive rank elliptic curve (see https://www.ams.org/journals/proc/1991-112-02/S0002-9939-1991-1055774-0/S0002-9939-1991-1055774-0.pdf).\n\nClassify surfaces $X/k$ of general type with $2\\in\\delta(X/k)$, possibly using geometric and/or dynamical constructions.",
   "evidence": "The canonical record is source index 18 of `aim-algebraic-number-theory-notes.json`, from the AIM workshop *Degree d points on algebraic surfaces*, Initial Problem Session, Problem 1.38. The problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0020": {
   "statement_status": "exact",
   "original_statement": "1.) Find effective obstructions to the existence of solvable points on varieties (in any characteristic).\n\n2.) Do there exist curves (over surfaces) over $\\mathbb Q$ without solvable points?",
   "clean_statement": "1.) Find effective obstructions to the existence of solvable points on varieties (in any characteristic).\n\n2.) Do there exist curves (over surfaces) over $\\mathbb Q$ without solvable points?",
   "public_statement": "1.) Find effective obstructions to the existence of solvable points on varieties (in any characteristic).\n\n2.) Do there exist curves (over surfaces) over $\\mathbb Q$ without solvable points?",
   "evidence": "The canonical record is problem 1.4 from the AIM workshop *Degree $d$ points on algebraic surfaces*. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0021": {
   "statement_status": "exact",
   "original_statement": "Problem 1. Browning: Can we develop a version of the circle method over Q(t)? Wooley: The major arcs are difficult to understand. Browning: For example, consider the diagonal quadratic form \n\nA1x21 + · · · + Asx2 \n\n> s\n\n= 0 (1) with Aj ∈ Q(t). There are \"obvious\" local conditions, e.g., arising from discrete valuation rings for Q(t). Does the Hasse principle hold? Wooley: If Aj are linear, say Aj = cj + td j with cj, d j ∈ Q, then (1) is equivalent to the system of equations \n\n> s\n\n∑\n\n> j=1\n\ncj x2 \n\n> j\n\n=\n\n> s\n\n∑\n\n> j=1\n\ndj x2 \n\n> j\n\n= 0 (2) by Amer-Brumer, which defines a quartic del Pezzo surface over Q. Do the \"ob-vious\" local obstructions over Q(t) for (1) capture the Brauer-Manin obstruction over Q for (2)? Harari: We may ask these questions over Qp(t); see work of Harari-Szamuely.",
   "clean_statement": "Problem 1. Browning: Can we develop a version of the circle method over Q(t)? Wooley: The major arcs are difficult to understand. Browning: For example, consider the diagonal quadratic form\n\nA1x21 + · · · + Asx2\n\n> s\n\n= 0 (1) with Aj ∈ Q(t). There are \"obvious\" local conditions, e.g., arising from discrete valuation rings for Q(t). Does the Hasse principle hold? Wooley: If Aj are linear, say Aj = cj + td j with cj, d j ∈ Q, then (1) is equivalent to the system of equations\n\n> s\n\n∑\n\n> j=1\n\ncj x2\n\n> j\n\n=\n\n> s\n\n∑\n\n> j=1\n\ndj x2\n\n> j\n\n= 0 (2) by Amer-Brumer, which defines a quartic del Pezzo surface over Q. Do the \"ob-vious\" local obstructions over Q(t) for (1) capture the Brauer-Manin obstruction over Q for (2)? Harari: We may ask these questions over Qp(t); see work of Harari-Szamuely.",
   "public_statement": "Problem 1. Browning: Can we develop a version of the circle method over Q(t)? Wooley: The major arcs are difficult to understand. Browning: For example, consider the diagonal quadratic form\n\nA1x21 + · · · + Asx2\n\n> s\n\n= 0 (1) with Aj ∈ Q(t). There are \"obvious\" local conditions, e.g., arising from discrete valuation rings for Q(t). Does the Hasse principle hold? Wooley: If Aj are linear, say Aj = cj + td j with cj, d j ∈ Q, then (1) is equivalent to the system of equations\n\n> s\n\n∑\n\n> j=1\n\ncj x2\n\n> j\n\n=\n\n> s\n\n∑\n\n> j=1\n\ndj x2\n\n> j\n\n= 0 (2) by Amer-Brumer, which defines a quartic del Pezzo surface over Q. Do the \"ob-vious\" local obstructions over Q(t) for (1) capture the Brauer-Manin obstruction over Q for (2)? Harari: We may ask these questions over Qp(t); see work of Harari-Szamuely.",
   "evidence": "The record comes from Problem 1 of the AIM workshop list *Rational and integral points on higher-dimensional varieties* (May 28, 2014). The PDF gives the following statement (notation normalized, but wording preserved):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0022": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 2. Cheltsov: What is the \"right\" assumption on a variety V for consid-ering height zeta functions? Definitely smooth V with ample anticanonical sheaf \n\nω−1 \n\n> V\n\nshould be allowed. How about klt (i.e., Kawamata log terminal) V? Or V\n\nof Fano type (i.e., there is an effective Q-divisor ∆ such that ( V, ∆) is ample and \n\n−(KV + ∆) is ample)? An example for the latter: A quasi-smooth hypersurface V ⊂ P(a0,..., a n) in weighted projective space of degree deg( V ) < a 0 + · · · + an.",
   "clean_statement": null,
   "public_statement": "Problem 2. Cheltsov: What is the \"right\" assumption on a variety V for consid-ering height zeta functions? Definitely smooth V with ample anticanonical sheaf\n\nω−1\n\n> V\n\nshould be allowed. How about klt (i.e., Kawamata log terminal) V? Or V\n\nof Fano type (i.e., there is an effective Q-divisor ∆ such that ( V, ∆) is ample and\n\n−(KV + ∆) is ample)? An example for the latter: A quasi-smooth hypersurface V ⊂ P(a0,..., a n) in weighted projective space of degree deg( V ) < a 0 + · · · + an.",
   "evidence": "The phrase “\\((V,\\Delta)\\) is ample” occurs in the original PDF itself; it is not an OCR invention. Since ampleness is not a property of a pair, the standard and almost certainly intended definition is:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0023": {
   "statement_status": "exact",
   "original_statement": "Problem 3. Skorobogatov: Does Bhargava's machinery have implications for the Hasse principle for special surfaces? For example, let F, G ∈ Q[x, y ] be homogeneous polynomials of degree 3. Con-sider the cubic surface \n\nS = {F (x, y ) = G(z, w )} ⊂ P3\n\n> Q.\n\nThe defining equation is equivalent to the system of equations \n\n{u3 = tF (x, y ), v 3 = tG (z, w )}.\n\nThis defines a family of cubic twists of curves of genus 1 over the t-line. Swinnerton-Dyer has discussed how to search for t such that this system is solvable over Q in the diagonal case [Ann. Sci. ENS]. Can we extend his work beyond the diagonal case? Similarly, consider Kummer K3 surfaces defined by \n\nz2 = f (x)g(y),\n\nwhere f, g are quartic separable polynomials. This is equivalent to the family of quadratic twists of curves of genus 1 defined by \n\nu2 = tf (x), v 2 = tg (y).\n\nThe goal is to eliminate the condition in Swinnerton-Dyer's work that (the 2-primary part of) X has finite order for quadratic twists, using the recent work presented in Bhargava's talk. \n\n> Date: May 28, 2014.\n> 12AIM OPEN PROBLEM SESSION",
   "clean_statement": "Problem 3. Skorobogatov: Does Bhargava's machinery have implications for the Hasse principle for special surfaces? For example, let F, G ∈ Q[x, y ] be homogeneous polynomials of degree 3. Con-sider the cubic surface\n\nS = {F (x, y ) = G(z, w )} ⊂ P3\n\n> Q.\n\nThe defining equation is equivalent to the system of equations\n\n{u3 = tF (x, y ), v 3 = tG (z, w )}.\n\nThis defines a family of cubic twists of curves of genus 1 over the t-line. Swinnerton-Dyer has discussed how to search for t such that this system is solvable over Q in the diagonal case [Ann. Sci. ENS]. Can we extend his work beyond the diagonal case? Similarly, consider Kummer K3 surfaces defined by\n\nz2 = f (x)g(y),\n\nwhere f, g are quartic separable polynomials. This is equivalent to the family of quadratic twists of curves of genus 1 defined by\n\nu2 = tf (x), v 2 = tg (y).\n\nThe goal is to eliminate the condition in Swinnerton-Dyer's work that (the 2-primary part of) X has finite order for quadratic twists, using the recent work presented in Bhargava's talk.\n\n> Date: May 28, 2014.\n> 12AIM OPEN PROBLEM SESSION",
   "public_statement": "Problem 3. Skorobogatov: Does Bhargava's machinery have implications for the Hasse principle for special surfaces? For example, let F, G ∈ Q[x, y ] be homogeneous polynomials of degree 3. Con-sider the cubic surface\n\nS = {F (x, y ) = G(z, w )} ⊂ P3\n\n> Q.\n\nThe defining equation is equivalent to the system of equations\n\n{u3 = tF (x, y ), v 3 = tG (z, w )}.\n\nThis defines a family of cubic twists of curves of genus 1 over the t-line. Swinnerton-Dyer has discussed how to search for t such that this system is solvable over Q in the diagonal case [Ann. Sci. ENS]. Can we extend his work beyond the diagonal case? Similarly, consider Kummer K3 surfaces defined by\n\nz2 = f (x)g(y),\n\nwhere f, g are quartic separable polynomials. This is equivalent to the family of quadratic twists of curves of genus 1 defined by\n\nu2 = tf (x), v 2 = tg (y).\n\nThe goal is to eliminate the condition in Swinnerton-Dyer's work that (the 2-primary part of) X has finite order for quadratic twists, using the recent work presented in Bhargava's talk.\n\n> Date: May 28, 2014.\n> 12AIM OPEN PROBLEM SESSION",
   "evidence": "The source is Problem 3 (attributed to Alexei Skorobogatov) in the AIM open-problem notes *Rational and integral points on higher-dimensional varieties*, dated May 28, 2014. Restoring only mathematical typesetting, the problem says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0024": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 4. Viray: Let φ: X → E be a fibration over an elliptic curve of positive rank over Q whose generic fiber is smooth and geometrically irreducible. Let \n\nZ = {p ∈ E(Q) | Xp = φ−1(p) has points everywhere locally }.\n\nWhat can we say about Z? Is |Z| < ∞ with Z 6 = ∅ possible? The motivation is that work of Poonen, Skorobogatov-Harpaz and Colliot-Th´ el` ene- Pal-Skorobogatov constructs X failing the Hasse principle such that none of the known obstructions apply. All of these use a map X → C to a curve with 0 < |C(Q)| < ∞.Browning: The case where φ is a conic bundle may already be interesting.",
   "clean_statement": null,
   "public_statement": "Problem 4. Viray: Let φ: X → E be a fibration over an elliptic curve of positive rank over Q whose generic fiber is smooth and geometrically irreducible. Let\n\nZ = {p ∈ E(Q) | Xp = φ−1(p) has points everywhere locally }.\n\nWhat can we say about Z? Is |Z| < ∞ with Z 6 = ∅ possible? The motivation is that work of Poonen, Skorobogatov-Harpaz and Colliot-Th´ el` ene- Pal-Skorobogatov constructs X failing the Hasse principle such that none of the known obstructions apply. All of these use a map X → C to a curve with 0 < |C(Q)| < ∞.Browning: The case where φ is a conic bundle may already be interesting.",
   "evidence": "The record is Problem 4, attributed to Viray, in the AIM workshop list *Rational and integral points on higher-dimensional varieties* (PDF dated May 28, 2014). Direct inspection of page 2 of the PDF resolves the OCR errors \\(6=\\) and the damaged accents. With line-break hyphenation normalized, the statement is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0025": {
   "statement_status": "exact",
   "original_statement": "Problem 5. Harari: The following question is due to Borovoi: Consider weak approximation for X = SL n /G over Q, where G is a finite group scheme that is not necessarily constant. For example, is X(Q) dense in X(R)? If G is constant, this is known to be true. A variant is the following. Given X = SL n /G with a constant finite group scheme G over a number field K with r ≥ 2 real places v1,..., v r. Is X(K) dense in ∏rj=1 X(Kvr )? A formulation via non-abelian Galois cohomology is given in the case K = Q as follows: Is \n\nH1(Gal( Q/Q), G (Q)) → H1(Gal( C/R), G (C)) surjective? Lucchini Arteche: The algebraic Brauer-Manin obstruction says nothing for this problem. Skorobogatov: The result is known for abelian G, due to Borovoi.",
   "clean_statement": "Problem 5. Harari: The following question is due to Borovoi: Consider weak approximation for X = SL n /G over Q, where G is a finite group scheme that is not necessarily constant. For example, is X(Q) dense in X(R)? If G is constant, this is known to be true. A variant is the following. Given X = SL n /G with a constant finite group scheme G over a number field K with r ≥ 2 real places v1,..., v r. Is X(K) dense in ∏rj=1 X(Kvr )? A formulation via non-abelian Galois cohomology is given in the case K = Q as follows: Is\n\nH1(Gal( Q/Q), G (Q)) → H1(Gal( C/R), G (C)) surjective? Lucchini Arteche: The algebraic Brauer-Manin obstruction says nothing for this problem. Skorobogatov: The result is known for abelian G, due to Borovoi.",
   "public_statement": "Problem 5. Harari: The following question is due to Borovoi: Consider weak approximation for X = SL n /G over Q, where G is a finite group scheme that is not necessarily constant. For example, is X(Q) dense in X(R)? If G is constant, this is known to be true. A variant is the following. Given X = SL n /G with a constant finite group scheme G over a number field K with r ≥ 2 real places v1,..., v r. Is X(K) dense in ∏rj=1 X(Kvr )? A formulation via non-abelian Galois cohomology is given in the case K = Q as follows: Is\n\nH1(Gal( Q/Q), G (Q)) → H1(Gal( C/R), G (C)) surjective? Lucchini Arteche: The algebraic Brauer-Manin obstruction says nothing for this problem. Skorobogatov: The result is known for abelian G, due to Borovoi.",
   "evidence": "The record is Problem 5 in the AIM open problem session for *Rational and integral points on higher-dimensional varieties*, dated May 28, 2014. The primary PDF reads as follows (overlines and product indices restored from the typeset source):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 24,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0026": {
   "statement_status": "exact",
   "original_statement": "Problem 6. Wittenberg: Let X be a smooth variety over a number field K, let \n\nS be a finite set of places. Assume that X satisfies strong approximation outside \n\nS. Take a closed subvariety Z ⊂ X of codimension two. Does X \\ Z satisfy strong approximation outside S?Tschinkel: We can also ask for Zariski density (of S-integral points). Wittenberg: The result is known for X = An and arbitrary Z of codimension 2. Interesting cases are: \n\n• Wittenberg: affine quadric hypersurfaces X ⊂ A4, for example, defined by \n\nq(x1, x 2, x 3, x 4) = c for a quadratic form q and a constant c.\n\n• Harari: X a simply connected linear algebraic group \n\n• Wooley: is this true by the circle method for hypersurfaces of fixed degree as soon as the dimension is large enough? Heath-Brown: for example, is this true for quadrics X ⊂ A5?Harari: If X satisfies strong approximation, then X is algebraically simply con-nected. If X is algebraically simply connected, then X \\ Z is also simply connected. Hence considering π1 should not be helpful to get a counterexample to the problem. Also Brauer groups are not expected to be helpful. Heath-Brown: Can we drop the condition that Z ⊂ X has codimension ≥ 2? Check the topology. Colliot-Th´ el` ene: Can the circle method be used to prove that π1(X) is trivial? For example, the circle method handles \n\nxr1 \n\n> 1\n\n− xr2 \n\n> 2\n\n+ xr3 \n\n> 3\n\n− · · · ± xrn \n\n> n\n\n= c ∈ Z\n\nin sufficiently many variables. Can we show that π1(X) is trivial without the circle method?",
   "clean_statement": "Problem 6. Wittenberg: Let X be a smooth variety over a number field K, let\n\nS be a finite set of places. Assume that X satisfies strong approximation outside\n\nS. Take a closed subvariety Z ⊂ X of codimension two. Does X \\ Z satisfy strong approximation outside S?Tschinkel: We can also ask for Zariski density (of S-integral points). Wittenberg: The result is known for X = An and arbitrary Z of codimension 2. Interesting cases are:\n\n• Wittenberg: affine quadric hypersurfaces X ⊂ A4, for example, defined by\n\nq(x1, x 2, x 3, x 4) = c for a quadratic form q and a constant c.\n\n• Harari: X a simply connected linear algebraic group\n\n• Wooley: is this true by the circle method for hypersurfaces of fixed degree as soon as the dimension is large enough? Heath-Brown: for example, is this true for quadrics X ⊂ A5?Harari: If X satisfies strong approximation, then X is algebraically simply con-nected. If X is algebraically simply connected, then X \\ Z is also simply connected. Hence considering π1 should not be helpful to get a counterexample to the problem. Also Brauer groups are not expected to be helpful. Heath-Brown: Can we drop the condition that Z ⊂ X has codimension ≥ 2? Check the topology. Colliot-Th´ el` ene: Can the circle method be used to prove that π1(X) is trivial? For example, the circle method handles\n\nxr1\n\n> 1\n\n− xr2\n\n> 2\n\n+ xr3\n\n> 3\n\n− · · · ± xrn\n\n> n\n\n= c ∈ Z\n\nin sufficiently many variables. Can we show that π1(X) is trivial without the circle method?",
   "public_statement": "Problem 6. Wittenberg: Let X be a smooth variety over a number field K, let\n\nS be a finite set of places. Assume that X satisfies strong approximation outside\n\nS. Take a closed subvariety Z ⊂ X of codimension two. Does X \\ Z satisfy strong approximation outside S?Tschinkel: We can also ask for Zariski density (of S-integral points). Wittenberg: The result is known for X = An and arbitrary Z of codimension 2. Interesting cases are:\n\n• Wittenberg: affine quadric hypersurfaces X ⊂ A4, for example, defined by\n\nq(x1, x 2, x 3, x 4) = c for a quadratic form q and a constant c.\n\n• Harari: X a simply connected linear algebraic group\n\n• Wooley: is this true by the circle method for hypersurfaces of fixed degree as soon as the dimension is large enough? Heath-Brown: for example, is this true for quadrics X ⊂ A5?Harari: If X satisfies strong approximation, then X is algebraically simply con-nected. If X is algebraically simply connected, then X \\ Z is also simply connected. Hence considering π1 should not be helpful to get a counterexample to the problem. Also Brauer groups are not expected to be helpful. Heath-Brown: Can we drop the condition that Z ⊂ X has codimension ≥ 2? Check the topology. Colliot-Th´ el` ene: Can the circle method be used to prove that π1(X) is trivial? For example, the circle method handles\n\nxr1\n\n> 1\n\n− xr2\n\n> 2\n\n+ xr3\n\n> 3\n\n− · · · ± xrn\n\n> n\n\n= c ∈ Z\n\nin sufficiently many variables. Can we show that π1(X) is trivial without the circle method?",
   "evidence": "The canonical record is Problem 6 from the AIM workshop *Rational and integral points on higher-dimensional varieties*. I checked the PDF directly because the JSON extraction corrupts superscripts in the final equation. In unambiguous notation, it asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0027": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 7. Heath-Brown: Can you construct a sequence of smooth projective varieties Xk ⊂ Pk \n\n> Q\n\nwith Xk(Qp) 6 = ∅ for all places p but Xk(Q) = ∅ such that dim( Xk )deg( Xk )AIM OPEN PROBLEM SESSION 3\n\nis unbounded? Browning-Heath-Brown have given a sequence where dim( Xk )deg( Xk ) tends to 13 and dim( Xk) is tends to ∞.Wooley: How about removing the requirement of smoothness and considering the singular norm forms \n\nNK/ Q(x1α1 + · · · + xdαd) = ct d\n\nwhere d = [ K: Q]? Heath-Brown: What happens when Xk ⊂ Pk is a hypersurface? Is there any example of a smooth hypersurface of dimension ≥ 3 failing the Hasse principle? Colliot-Th´ el` ene: Sarnak-Wang have shown that the Bombieri-Lang conjecture would imply that there are many such examples of general type. Wooley: An analytic attack to show that there exist some such varieties could be as follows. Choose a locally soluble smooth hypersurface Y ⊂ PN of degree d \u001d N.The determinant method implies that the number of points in a large box grows slowly. Intersect with linear subspaces to maintain local solubility. Use a counting argument to find a linear section without rational points. Colliot-Th´ el` ene: Won't this just force the coefficients to be large? Harari: Does dim( Xk )deg( Xk ) → ∞ imply that Xk is geometrically rationally connected? Note that if X over Q is a geometrically rationally connected complete intersection, then the Hasse principle is hard to obstruct cohomologically. Browning: A conjecture of Hartshorne implies that if Y ⊂ PN is smooth, non-degenerate, with dim( Y ) ≥ 2 deg( Y ) + 1, then Y is a complete intersection, hence rationally connected. Therefore, it might be easier to look for examples with 13 < dim( Xk)deg( Xk) ≤ 2in Heath-Brown's original question. Tschinkel: Let X be a Fano variety over C. Can we have Br( X) = H3(X, Z)tors 6 =0 in all dimensions ≥ 4?",
   "clean_statement": null,
   "public_statement": "Problem 7. Heath-Brown: Can you construct a sequence of smooth projective varieties Xk ⊂ Pk\n\n> Q\n\nwith Xk(Qp) 6 = ∅ for all places p but Xk(Q) = ∅ such that dim( Xk )deg( Xk )AIM OPEN PROBLEM SESSION 3\n\nis unbounded? Browning-Heath-Brown have given a sequence where dim( Xk )deg( Xk ) tends to 13 and dim( Xk) is tends to ∞.Wooley: How about removing the requirement of smoothness and considering the singular norm forms\n\nNK/ Q(x1α1 + · · · + xdαd) = ct d\n\nwhere d = [ K: Q]? Heath-Brown: What happens when Xk ⊂ Pk is a hypersurface? Is there any example of a smooth hypersurface of dimension ≥ 3 failing the Hasse principle? Colliot-Th´ el` ene: Sarnak-Wang have shown that the Bombieri-Lang conjecture would imply that there are many such examples of general type. Wooley: An analytic attack to show that there exist some such varieties could be as follows. Choose a locally soluble smooth hypersurface Y ⊂ PN of degree d\n N.The determinant method implies that the number of points in a large box grows slowly. Intersect with linear subspaces to maintain local solubility. Use a counting argument to find a linear section without rational points. Colliot-Th´ el` ene: Won't this just force the coefficients to be large? Harari: Does dim( Xk )deg( Xk ) → ∞ imply that Xk is geometrically rationally connected? Note that if X over Q is a geometrically rationally connected complete intersection, then the Hasse principle is hard to obstruct cohomologically. Browning: A conjecture of Hartshorne implies that if Y ⊂ PN is smooth, non-degenerate, with dim( Y ) ≥ 2 deg( Y ) + 1, then Y is a complete intersection, hence rationally connected. Therefore, it might be easier to look for examples with 13 < dim( Xk)deg( Xk) ≤ 2in Heath-Brown's original question. Tschinkel: Let X be a Fano variety over C. Can we have Br( X) = H3(X, Z)tors 6 =0 in all dimensions ≥ 4?",
   "evidence": "The source is Problem 7 in the AIM workshop list *Rational and integral points on higher-dimensional varieties*. The canonical JSON has several OCR losses: `6 =` means \\(\\ne\\), the displayed quotients lost their fraction bars, `13` means \\(1/3\\), and the condition in Wooley's proposed analytic approach is \\(d\\gg N\\).",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0028": {
   "statement_status": "exact",
   "original_statement": "Problem 8. V´ arilly-Alvarado: Skorobogatov has asked whether a K3 surface X\n\nover Q can have odd order torsion in Br( X) obstructing the Hasse principle? Even in Br 1(X)? Skorobogatov: For example, for quartics X ⊂ P3 and α ∈ Br( X)[ u] for u odd: For each place v, there exists a zero cycle Zv over Qv of degree one such that α\n\nis orthogonal to Zv. Then a conjecture of Colliot-Th´ el` ene implies that X has zero cycles of degree one over Q. Will there be a rational point? So given a quartic surface X ⊂ P3 with (Br( X)/ Br( Q))[2] = 0, does the Hasse principle hold? Tschinkel: What about weak approximation? Skorobogatov: This will probably fail. Hassett: How about X ⊂ P4 of degree six?",
   "clean_statement": "Problem 8. V´ arilly-Alvarado: Skorobogatov has asked whether a K3 surface X\n\nover Q can have odd order torsion in Br( X) obstructing the Hasse principle? Even in Br 1(X)? Skorobogatov: For example, for quartics X ⊂ P3 and α ∈ Br( X)[ u] for u odd: For each place v, there exists a zero cycle Zv over Qv of degree one such that α\n\nis orthogonal to Zv. Then a conjecture of Colliot-Th´ el` ene implies that X has zero cycles of degree one over Q. Will there be a rational point? So given a quartic surface X ⊂ P3 with (Br( X)/ Br( Q))[2] = 0, does the Hasse principle hold? Tschinkel: What about weak approximation? Skorobogatov: This will probably fail. Hassett: How about X ⊂ P4 of degree six?",
   "public_statement": "Problem 8. V´ arilly-Alvarado: Skorobogatov has asked whether a K3 surface X\n\nover Q can have odd order torsion in Br( X) obstructing the Hasse principle? Even in Br 1(X)? Skorobogatov: For example, for quartics X ⊂ P3 and α ∈ Br( X)[ u] for u odd: For each place v, there exists a zero cycle Zv over Qv of degree one such that α\n\nis orthogonal to Zv. Then a conjecture of Colliot-Th´ el` ene implies that X has zero cycles of degree one over Q. Will there be a rational point? So given a quartic surface X ⊂ P3 with (Br( X)/ Br( Q))[2] = 0, does the Hasse principle hold? Tschinkel: What about weak approximation? Skorobogatov: This will probably fail. Hassett: How about X ⊂ P4 of degree six?",
   "evidence": "The record is Problem 8 in the AIM workshop list *Rational and integral points on higher-dimensional varieties*, dated May 28, 2014. The canonical record is index 27 of `aim-algebraic-number-theory-notes.json`. I checked the underlying four-page AIM PDF, including its embedded font encoding, rather than silently repairing the extracted text. In particular, the symbol in \\(\\alpha\\in \\operatorname{Br}(X)[u]\\) really is the letter \\(u\\), followed by “for \\(u\\) odd”; it is not an OCR substitution for another symbol. I interpret \\(\\operatorname{Br}(X)[u]\\) as the subgroup killed by the odd integer \\(u\\).",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0029": {
   "statement_status": "exact",
   "original_statement": "Problem 9. Wooley: Consider the set \n\nQk:= {Q(yk \n\n> 1,..., y ks ) ∈ Q[y1,..., y s] | Q ∈ Q[x1,..., x s] quadratic form }\n\nof certain forms of degree 2 k. Fixing k, how large must s be in order for the Hasse principle to hold? Let \n\nh(k):= inf \n\n> s∈N\n\n{s | the Hasse principle holds for all Q ∈ Qk in s variables }.\n\nChallenge: prove that log h(k) = o(k) as k → ∞.By Birch's result on forms in many variables, we know that h(k) ≤ 2k · 22k.On the other hand, for diagonal forms of degree k (i.e., Q(yk \n\n> 1,..., y ks ) with linear \n\nQ ∈ Q[x1,..., x s], we have the much stronger bound h(k) ∼ 2k(log k). 4 AIM OPEN PROBLEM SESSION \n\nBrowning: Replace k-th powers by norm forms from fixed extensions of degree \n\nk.",
   "clean_statement": "Problem 9. Wooley: Consider the set\n\nQk:= {Q(yk\n\n> 1,..., y ks ) ∈ Q[y1,..., y s] | Q ∈ Q[x1,..., x s] quadratic form }\n\nof certain forms of degree 2 k. Fixing k, how large must s be in order for the Hasse principle to hold? Let\n\nh(k):= inf\n\n> s∈N\n\n{s | the Hasse principle holds for all Q ∈ Qk in s variables }.\n\nChallenge: prove that log h(k) = o(k) as k → ∞.By Birch's result on forms in many variables, we know that h(k) ≤ 2k · 22k.On the other hand, for diagonal forms of degree k (i.e., Q(yk\n\n> 1,..., y ks ) with linear\n\nQ ∈ Q[x1,..., x s], we have the much stronger bound h(k) ∼ 2k(log k). 4 AIM OPEN PROBLEM SESSION\n\nBrowning: Replace k-th powers by norm forms from fixed extensions of degree\n\nk.",
   "public_statement": "Problem 9. Wooley: Consider the set\n\nQk:= {Q(yk\n\n> 1,..., y ks ) ∈ Q[y1,..., y s] | Q ∈ Q[x1,..., x s] quadratic form }\n\nof certain forms of degree 2 k. Fixing k, how large must s be in order for the Hasse principle to hold? Let\n\nh(k):= inf\n\n> s∈N\n\n{s | the Hasse principle holds for all Q ∈ Qk in s variables }.\n\nChallenge: prove that log h(k) = o(k) as k → ∞.By Birch's result on forms in many variables, we know that h(k) ≤ 2k · 22k.On the other hand, for diagonal forms of degree k (i.e., Q(yk\n\n> 1,..., y ks ) with linear\n\nQ ∈ Q[x1,..., x s], we have the much stronger bound h(k) ∼ 2k(log k). 4 AIM OPEN PROBLEM SESSION\n\nBrowning: Replace k-th powers by norm forms from fixed extensions of degree\n\nk.",
   "evidence": "The canonical record is Problem 9 from the AIM workshop *Rational and integral points on higher-dimensional varieties*. The OCR in the JSON record loses superscripts, an exponent on 2, and an asymptotic comparison symbol. Inspection of the workshop PDF gives the following statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0030": {
   "statement_status": "exact",
   "original_statement": "Problem 10. Peyre: Back to quartic surfaces X ⊂ P3. Let U be the complement of all rational curves over Q. Assume U (Q) 6 = ∅. There is numerical evidence that #{u ∈ U (Q) | H(u) ≤ B} ∼ c(log B)ρ(X)\n\nwhere ρ(X) is the rank of the Picard group of X and c is the product of local densities. Can such a formula hold without weak approximation being valid? Tschinkel: Given a quartic K3 surface X ⊂ P3, with x ∈ X(Q). Is there a procedure for deciding whether x lies on a rational curve over Q?Skorobogatov: What about removing elliptic curves? Browning: Is there any numerical evidence? (See van Luijk's work.) What is the Peyre freedom of rational curves of small degree on K3 surfaces? Colliot-Th´ el` ene: Given a K3 surface X over Q and x ∈ X(Q) 6 = ∅, does there exist a rational curve R ⊂ X over Q? Does there exist a rational curve R ⊂ X over \n\nQ containing x?Tschinkel: How about finite fields? Let X be a K3 surface over Fq and x ∈ X(Fq ). Does there exist a rational curve R ⊂ X defined over Fq containing x? Are there any rational curves R ⊂ X defined over Fq? Bogomolov-Tschinkel show for Kummer surfaces X that we can find rational curves over Fq for most x ∈ X.Skorobogatov / Testa: Are there K3 surfaces X over Q with infinitely may rational curves over Q and Pic( XC) ∼= Z?",
   "clean_statement": "Problem 10. Peyre: Back to quartic surfaces X ⊂ P3. Let U be the complement of all rational curves over Q. Assume U (Q) 6 = ∅. There is numerical evidence that #{u ∈ U (Q) | H(u) ≤ B} ∼ c(log B)ρ(X)\n\nwhere ρ(X) is the rank of the Picard group of X and c is the product of local densities. Can such a formula hold without weak approximation being valid? Tschinkel: Given a quartic K3 surface X ⊂ P3, with x ∈ X(Q). Is there a procedure for deciding whether x lies on a rational curve over Q?Skorobogatov: What about removing elliptic curves? Browning: Is there any numerical evidence? (See van Luijk's work.) What is the Peyre freedom of rational curves of small degree on K3 surfaces? Colliot-Th´ el` ene: Given a K3 surface X over Q and x ∈ X(Q) 6 = ∅, does there exist a rational curve R ⊂ X over Q? Does there exist a rational curve R ⊂ X over\n\nQ containing x?Tschinkel: How about finite fields? Let X be a K3 surface over Fq and x ∈ X(Fq ). Does there exist a rational curve R ⊂ X defined over Fq containing x? Are there any rational curves R ⊂ X defined over Fq? Bogomolov-Tschinkel show for Kummer surfaces X that we can find rational curves over Fq for most x ∈ X.Skorobogatov / Testa: Are there K3 surfaces X over Q with infinitely may rational curves over Q and Pic( XC) ∼= Z?",
   "public_statement": "Problem 10. Peyre: Back to quartic surfaces X ⊂ P3. Let U be the complement of all rational curves over Q. Assume U (Q) 6 = ∅. There is numerical evidence that #{u ∈ U (Q) | H(u) ≤ B} ∼ c(log B)ρ(X)\n\nwhere ρ(X) is the rank of the Picard group of X and c is the product of local densities. Can such a formula hold without weak approximation being valid? Tschinkel: Given a quartic K3 surface X ⊂ P3, with x ∈ X(Q). Is there a procedure for deciding whether x lies on a rational curve over Q?Skorobogatov: What about removing elliptic curves? Browning: Is there any numerical evidence? (See van Luijk's work.) What is the Peyre freedom of rational curves of small degree on K3 surfaces? Colliot-Th´ el` ene: Given a K3 surface X over Q and x ∈ X(Q) 6 = ∅, does there exist a rational curve R ⊂ X over Q? Does there exist a rational curve R ⊂ X over\n\nQ containing x?Tschinkel: How about finite fields? Let X be a K3 surface over Fq and x ∈ X(Fq ). Does there exist a rational curve R ⊂ X defined over Fq containing x? Are there any rational curves R ⊂ X defined over Fq? Bogomolov-Tschinkel show for Kummer surfaces X that we can find rational curves over Fq for most x ∈ X.Skorobogatov / Testa: Are there K3 surfaces X over Q with infinitely may rational curves over Q and Pic( XC) ∼= Z?",
   "evidence": "The canonical record is Problem 10 from the AIM workshop *Rational and integral points on higher-dimensional varieties*. The OCR in the JSON record loses several symbols and line breaks, so the statement was checked against both the workshop PDF and the AIM source TeX. The relevant opening assertion is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0031": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 1 (Brian Conrey). Let E/ Fq(T ). Let f vary over square free, large degree n and let P ∈ P1; does lim \n\n> n→∞\n\n#{f ∈ Fq[T ]n: rank Ef ≥ 2 and f (P ) ∈ (F∗ \n\n> q\n\n)2 − { 0}} \n\n#{f ∈ Fq[T ]n: rank Ef ≥ 2 and f (P ) 6 ∈ (F∗ \n\n> q\n\n)2} =?\n\n√\n\n#EP (Fq)˜EP (Fq)(This is conjectured for number fields based on numerical evidence, random matrix theory, moments of L-functions, and BSD.)",
   "clean_statement": null,
   "public_statement": "Question 1 (Brian Conrey). Let E/ Fq(T ). Let f vary over square free, large degree n and let P ∈ P1; does lim\n\n> n→∞\n\n#{f ∈ Fq[T ]n: rank Ef ≥ 2 and f (P ) ∈ (F∗\n\n> q\n\n)2 − { 0}}\n\n#{f ∈ Fq[T ]n: rank Ef ≥ 2 and f (P ) 6 ∈ (F∗\n\n> q\n\n)2} =?\n\n√\n\n#EP (Fq)˜EP (Fq)(This is conjectured for number fields based on numerical evidence, random matrix theory, moments of L-functions, and BSD.)",
   "evidence": "The PDF does not define \\(\\mathbb F_q[T]_n\\), does not say that \\(q\\) is odd, and does not impose rationality or good reduction at \\(P\\). For the rigorous result below, the explicit reconstruction assumptions are:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0032": {
   "statement_status": "exact",
   "original_statement": "Question 2 (Melanie Matchett Wood). Develop a version of the BKLPR heuristics with Zp\n\nreplaced by Z, counting up to height B. Use the rank 2 case to calibrate B with the height used when counting elliptic curves. Then use that calibration to make a rank 3 prediction and compare this with predictions coming from random matrix theory.",
   "clean_statement": "Question 2 (Melanie Matchett Wood). Develop a version of the BKLPR heuristics with Zp\n\nreplaced by Z, counting up to height B. Use the rank 2 case to calibrate B with the height used when counting elliptic curves. Then use that calibration to make a rank 3 prediction and compare this with predictions coming from random matrix theory.",
   "public_statement": "Question 2 (Melanie Matchett Wood). Develop a version of the BKLPR heuristics with Zp\n\nreplaced by Z, counting up to height B. Use the rank 2 case to calibrate B with the height used when counting elliptic curves. Then use that calibration to make a rank 3 prediction and compare this with predictions coming from random matrix theory.",
   "evidence": "The record is Question 2, attributed to Melanie Matchett Wood, in the AIM problem list from the workshop *Arithmetic statistics over finite fields and function fields*. The source PDF was checked directly; the intended local ring is \\(\\mathbb Z_p\\), and the extracted text is not hiding an ambiguity in the subscript.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0033": {
   "statement_status": "exact",
   "original_statement": "Question 3 (Jordan Ellenberg). Understand which ways of counting field extensions satisfy Principles (A) and (B).",
   "clean_statement": "Question 3 (Jordan Ellenberg). Understand which ways of counting field extensions satisfy Principles (A) and (B).",
   "public_statement": "Question 3 (Jordan Ellenberg). Understand which ways of counting field extensions satisfy Principles (A) and (B).",
   "evidence": "The canonical record is Question 3 from the January 2014 AIM workshop *Arithmetic statistics over finite fields and function fields*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0034": {
   "statement_status": "exact",
   "original_statement": "Question 4. Which definition(s) of automorphism groups give rise to the nicest formulas?",
   "clean_statement": "Question 4. Which definition(s) of automorphism groups give rise to the nicest formulas?",
   "public_statement": "Question 4. Which definition(s) of automorphism groups give rise to the nicest formulas?",
   "evidence": "The canonical record is Question 4 from the AIM workshop *Arithmetic statistics over finite fields and function fields* (27--31 January 2014):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0035": {
   "statement_status": "exact",
   "original_statement": "Question 5. For cyclic triple covers of P1 can we recover (A) and (B) while counting by genus?",
   "clean_statement": "Question 5. For cyclic triple covers of P1 can we recover (A) and (B) while counting by genus?",
   "public_statement": "Question 5. For cyclic triple covers of P1 can we recover (A) and (B) while counting by genus?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0036": {
   "statement_status": "exact",
   "original_statement": "Question 6 (Alina Bucur, Kiran Kedlaya). Are principles (A) and (B) easier/harder/true after conditioning on ramification information?",
   "clean_statement": "Question 6 (Alina Bucur, Kiran Kedlaya). Are principles (A) and (B) easier/harder/true after conditioning on ramification information?",
   "public_statement": "Question 6 (Alina Bucur, Kiran Kedlaya). Are principles (A) and (B) easier/harder/true after conditioning on ramification information?",
   "evidence": "The exact canonical record is Question 6 from the January 2014 AIM workshop *Arithmetic statistics over finite fields and function fields*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0037": {
   "statement_status": "exact",
   "original_statement": "Question 7. Does (A) fail for Z/4Z function field extensions, counted by genus?",
   "clean_statement": "Question 7. Does (A) fail for Z/4Z function field extensions, counted by genus?",
   "public_statement": "Question 7. Does (A) fail for Z/4Z function field extensions, counted by genus?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0038": {
   "statement_status": "exact",
   "original_statement": "Question 8 (Kiran Kedlaya). Fix q. Describe (conjecturally) the distribution of the number of Fq points on curves of genus g as g tends to ∞.",
   "clean_statement": "Question 8 (Kiran Kedlaya). Fix q. Describe (conjecturally) the distribution of the number of Fq points on curves of genus g as g tends to ∞.",
   "public_statement": "Question 8 (Kiran Kedlaya). Fix q. Describe (conjecturally) the distribution of the number of Fq points on curves of genus g as g tends to ∞.",
   "evidence": "The canonical AIM record is Question 8 from the workshop *Arithmetic statistics over finite fields and function fields*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0039": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 9 (Kiran Kedlaya). Look at principle (A) for tri-canonically embedded curves.",
   "clean_statement": "- the workshop report says that Melanie Matchett Wood asked when point counts in\n  a family of curves are sums of independent identically distributed local\n  variables [AIM14b];\n- nearby Question 8 asks for the fixed-\\(q\\), large-genus point-count\n  distribution, while Question 10 proposes complete intersections;\n- rational points impose asymptotically independent conditions in the earlier\n  plane-curve model [BDFL10], and Bucur--Kedlaya obtain an i.i.d. Bernoulli\n  model for high-degree complete intersections [BK12].",
   "public_statement": "Question 9 (Kiran Kedlaya). Look at principle (A) for tri-canonically embedded curves.",
   "evidence": "The source never defines principle (A). The full three-page PDF invokes principles (A) and (B) several times but does not assign either a statement. That omission is material and is not silently repaired here. The following facts do, however, support a conservative reconstruction: Accordingly, this report studies the explicit reconstruction",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0040": {
   "statement_status": "exact",
   "original_statement": "Question 10 (Melanie Matchett Wood). Look at complete intersections, and some examples with a moving target space.",
   "clean_statement": "Question 10 (Melanie Matchett Wood). Look at complete intersections, and some examples with a moving target space.",
   "public_statement": "Question 10 (Melanie Matchett Wood). Look at complete intersections, and some examples with a moving target space.",
   "evidence": "The canonical record is Question 10 in the AIM problem session for *Arithmetic statistics over finite fields and function fields*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0041": {
   "statement_status": "exact",
   "original_statement": "Question 11 (John Voight). When do counts get nicer if we are willing to partially com-pactify?",
   "clean_statement": "Question 11 (John Voight). When do counts get nicer if we are willing to partially com-pactify?",
   "public_statement": "Question 11 (John Voight). When do counts get nicer if we are willing to partially com-pactify?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0042": {
   "statement_status": "exact",
   "original_statement": "Question 12 (Julio Cesar Bueno de Andrade). Fix q, h(x) ∈ Fq[x], k ≥ 3. Get asymptotics for ∑\n\n> f\n\ndk(f ) · dk(f + h)where f ranges over degree n polynomials, as n tends to infinity.",
   "clean_statement": "Question 12 (Julio Cesar Bueno de Andrade). Fix q, h(x) ∈ Fq[x], k ≥ 3. Get asymptotics for ∑\n\n> f\n\ndk(f ) · dk(f + h)where f ranges over degree n polynomials, as n tends to infinity.",
   "public_statement": "Question 12 (Julio Cesar Bueno de Andrade). Fix q, h(x) ∈ Fq[x], k ≥ 3. Get asymptotics for ∑\n\n> f\n\ndk(f ) · dk(f + h)where f ranges over degree n polynomials, as n tends to infinity.",
   "evidence": "The exact canonical record, including the extraction line breaks, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0043": {
   "statement_status": "exact",
   "original_statement": "Question 13 (Pollack). What can we say (resp. what do we know) about gaps between irreducible polynomials over Fq?\n\n> 1",
   "clean_statement": "Question 13 (Pollack). What can we say (resp. what do we know) about gaps between irreducible polynomials over Fq?\n\n> 1",
   "public_statement": "Question 13 (Pollack). What can we say (resp. what do we know) about gaps between irreducible polynomials over Fq?\n\n> 1",
   "evidence": "The exact record in the AIM corpus is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0044": {
   "statement_status": "exact",
   "original_statement": "Question 14 (Pollack). Over a function field, how many coefficients of a polynomial (and in which positions) can be specified without eliminating the possibility of irreducibility?",
   "clean_statement": "Question 14 (Pollack). Over a function field, how many coefficients of a polynomial (and in which positions) can be specified without eliminating the possibility of irreducibility?",
   "public_statement": "Question 14 (Pollack). Over a function field, how many coefficients of a polynomial (and in which positions) can be specified without eliminating the possibility of irreducibility?",
   "evidence": "The wording was checked against the [original three-page AIM PDF](https://aimath.org/pastworkshops/arithstatffieldproblems.pdf), where it appears verbatim. There is no OCR corruption, but the short question does not state its quantifiers, the degree, or whether the polynomial is monic.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0045": {
   "statement_status": "exact",
   "original_statement": "Question 15 (Jordan Ellenberg). What algebro-geometric facts about elliptic surfaces ex-plain Poonen-Rains/BKLPR?",
   "clean_statement": "Question 15 (Jordan Ellenberg). What algebro-geometric facts about elliptic surfaces ex-plain Poonen-Rains/BKLPR?",
   "public_statement": "Question 15 (Jordan Ellenberg). What algebro-geometric facts about elliptic surfaces ex-plain Poonen-Rains/BKLPR?",
   "evidence": "The canonical record is Question 15 from the AIM workshop problem list “Arithmetic statistics over finite fields and function fields”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0046": {
   "statement_status": "exact",
   "original_statement": "Question 16 (Brian Conrey). Fix q, let g → ∞, and describe the distribution of discrimi-nants of L-polynomials of curves (or hyperelliptic curves).",
   "clean_statement": "Question 16 (Brian Conrey). Fix q, let g → ∞, and describe the distribution of discrimi-nants of L-polynomials of curves (or hyperelliptic curves).",
   "public_statement": "Question 16 (Brian Conrey). Fix q, let g → ∞, and describe the distribution of discrimi-nants of L-polynomials of curves (or hyperelliptic curves).",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0047": {
   "statement_status": "exact",
   "original_statement": "Question 17 (John Voight). What is the average number of points on the Jacobian of a hyperelliptic curve?",
   "clean_statement": "Question 17 (John Voight). What is the average number of points on the Jacobian of a hyperelliptic curve?",
   "public_statement": "Question 17 (John Voight). What is the average number of points on the Jacobian of a hyperelliptic curve?",
   "evidence": "The exact source record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0048": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 18. Fix q. How many different zeta functions are attached to curves of genus \n\ng \u001d q?",
   "clean_statement": null,
   "public_statement": "Question 18. Fix q. How many different zeta functions are attached to curves of genus\n\ng\n q?",
   "evidence": "The canonical JSON record contains an extraction error:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0049": {
   "statement_status": "exact",
   "original_statement": "Question 19 (Kiran Kedlaya). Fix g, fix C/K a curve of genus g, where K = Fq(X) and \n\nX is not necessarily rational. Can we compute the monodromy group of C? I.e., can we compute \n\n{im(Gal( K) → Aut( H1(C, Qp))): C ∈ Mg(K)}",
   "clean_statement": "Question 19 (Kiran Kedlaya). Fix g, fix C/K a curve of genus g, where K = Fq(X) and\n\nX is not necessarily rational. Can we compute the monodromy group of C? I.e., can we compute\n\n{im(Gal( K) → Aut( H1(C, Qp))): C ∈ Mg(K)}",
   "public_statement": "Question 19 (Kiran Kedlaya). Fix g, fix C/K a curve of genus g, where K = Fq(X) and\n\nX is not necessarily rational. Can we compute the monodromy group of C? I.e., can we compute\n\n{im(Gal( K) → Aut( H1(C, Qp))): C ∈ Mg(K)}",
   "evidence": "The extracted record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0050": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 20 (Melanie Matchett Wood). When do (A) and (B) predict an interesting average value of C(Fq)?",
   "clean_statement": null,
   "public_statement": "Question 20 (Melanie Matchett Wood). When do (A) and (B) predict an interesting average value of C(Fq)?",
   "evidence": "Everything proved below is therefore a rigorous answer to this explicitly labeled reconstruction. Recovering Wood's workshop slides or notes could change which label is attached to which principle, and possibly the exact normalization. No claim below depends on calling the two principles A or B; it depends only on the stated mass and independence hypotheses.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0051": {
   "statement_status": "exact",
   "original_statement": "Question 21 (Jordan Ellenberg). In what ways are discriminants of random trigonal curves like random polynomials?",
   "clean_statement": "Question 21 (Jordan Ellenberg). In what ways are discriminants of random trigonal curves like random polynomials?",
   "public_statement": "Question 21 (Jordan Ellenberg). In what ways are discriminants of random trigonal curves like random polynomials?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0052": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 22 (Katz). The Heilbron sums \n\n∑ \n\n> xmod p\n\nexp 2πix p(1 + pt )\n\np2 = ∑ \n\n> xmod p\n\nexp 2πix p\n\np2 · exp 2πipt p2\n\nseem to approximate (as t varies) the (push forward of) Haar measure better than expected. Why?",
   "clean_statement": null,
   "public_statement": "Question 22 (Katz). The Heilbron sums\n\n∑\n\n> xmod p\n\nexp 2πix p(1 + pt )\n\np2 = ∑\n\n> xmod p\n\nexp 2πix p\n\np2 · exp 2πipt p2\n\nseem to approximate (as t varies) the (push forward of) Haar measure better than expected. Why?",
   "evidence": "The PDF does **not** specify a range for $t$, a normalization, the compact group carrying Haar measure, or the pushforward map. The left side is periodic in $t\\pmod p$, so the effective range is $t\\in\\mathbb F_p$. The following is the minimal natural reconstruction compatible with (1.2): the right side samples one real Laurent polynomial at all $p$-th roots of unity, and the comparison measure is the pushforward of normalized Haar measure on the unit circle by that same polynomial. This reconstruction is mathematically canonical, but the omitted group and map were not verified as Katz's exact intended wording. A different random-matrix Haar target cannot be recovered from the PDF alone.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0053": {
   "statement_status": "exact",
   "original_statement": "Question 23 (Jordan Ellenberg). Describe the right hand side of variance of Λ( n) ∼ q? · n\n\n(as q → ∞ ) geometrically.",
   "clean_statement": "Question 23 (Jordan Ellenberg). Describe the right hand side of variance of Λ( n) ∼ q? · n\n\n(as q → ∞ ) geometrically.",
   "public_statement": "Question 23 (Jordan Ellenberg). Describe the right hand side of variance of Λ( n) ∼ q? · n\n\n(as q → ∞ ) geometrically.",
   "evidence": "The canonical record is Question 23 from the AIM problem session for *Arithmetic statistics over finite fields and function fields*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0054": {
   "statement_status": "exact",
   "original_statement": "Question 24 (Brian Conrey). What is the variance of \n\n∑\n\n> f\n\nΛ( f )Λ( f + h)(considered as a function of h).",
   "clean_statement": "Question 24 (Brian Conrey). What is the variance of\n\n∑\n\n> f\n\nΛ( f )Λ( f + h)(considered as a function of h).",
   "public_statement": "Question 24 (Brian Conrey). What is the variance of\n\n∑\n\n> f\n\nΛ( f )Λ( f + h)(considered as a function of h).",
   "evidence": "The extracted record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0055": {
   "statement_status": "exact",
   "original_statement": "Question 25 (Brian Conrey). Carry out (the other) Levinson's method for function fields.",
   "clean_statement": "Question 25 (Brian Conrey). Carry out (the other) Levinson's method for function fields.",
   "public_statement": "Question 25 (Brian Conrey). Carry out (the other) Levinson's method for function fields.",
   "evidence": "The canonical AIM record is Question 25 from the 2014 workshop *Arithmetic statistics over finite fields and function fields*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0056": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 26 (Brian Conrey). Do the analogue of Nymon-Barling criterion for RH over function fields (leading to a different proof of RH).",
   "clean_statement": null,
   "public_statement": "Question 26 (Brian Conrey). Do the analogue of Nymon-Barling criterion for RH over function fields (leading to a different proof of RH).",
   "evidence": "The canonical record is Question 26 from the AIM workshop *Arithmetic statistics over finite fields and function fields*:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0057": {
   "statement_status": "exact",
   "original_statement": "Question 27 (1st Tue speaker). Ramanujan sums \n\ncq(x):= ∑\n\n> a<q, (a,q )=1\n\nexp 2 πi (a/q )x\n\ncan write Λ in terms of cq(X) and other arithmetic functions too. How do these look in the function field setting? 2",
   "clean_statement": "Question 27 (1st Tue speaker). Ramanujan sums\n\ncq(x):= ∑\n\n> a<q, (a,q )=1\n\nexp 2 πi (a/q )x\n\ncan write Λ in terms of cq(X) and other arithmetic functions too. How do these look in the function field setting? 2",
   "public_statement": "Question 27 (1st Tue speaker). Ramanujan sums\n\ncq(x):= ∑\n\n> a<q, (a,q )=1\n\nexp 2 πi (a/q )x\n\ncan write Λ in terms of cq(X) and other arithmetic functions too. How do these look in the function field setting? 2",
   "evidence": "The canonical JSON record is Question 27 from the AIM workshop *Arithmetic statistics over finite fields and function fields*. Its extracted text is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0058": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 28 (Jordan Ellenbird). What is the role of wild ramification in questions a la Melanie Matchett Wood's work. (E.g. hyperelliptic curves in characteristic 2.)",
   "clean_statement": null,
   "public_statement": "Question 28 (Jordan Ellenbird). What is the role of wild ramification in questions a la Melanie Matchett Wood's work. (E.g. hyperelliptic curves in characteristic 2.)",
   "evidence": "The wording above is reproduced literally from the AIM problem-session PDF. The name “Jordan Ellenbird” is almost certainly a typographical error: the workshop page names Jordan Ellenberg as an organizer, and several other questions in the same PDF are attributed to Jordan Ellenberg. This report preserves the source text and treats the intended mathematical question as:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0059": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 29 (Chantal Davis). Local statistics for zeros for L-functions of orthogonal type.",
   "clean_statement": null,
   "public_statement": "Question 29 (Chantal Davis). Local statistics for zeros for L-functions of orthogonal type.",
   "evidence": "The original three-page AIM problem PDF was inspected directly. It literally prints “Chantal Davis,” so the surname is not an error introduced by the JSON extraction. The official workshop page and workshop summary instead name organizer **Chantal David**, a number theorist whose work includes statistics of zeros. Her Arizona Winter School notes from March 2014 discuss precisely the one-level densities for orthogonal families and the inability to distinguish $O$, $SO(\\mathrm{even})$, and $SO(\\mathrm{odd})$ with Fourier support inside $(-1,1)$. “Davis” is therefore almost certainly a source-level typo for “David.” This correction is recorded explicitly rather than made silently.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0060": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 30 (Rubenstein). Can we compute averages of the Hardy-Littlewood constants in the function field setting look like? Yes.",
   "clean_statement": "Question 30 (Rubenstein). Can we compute averages of the Hardy-Littlewood constants in the function field Rubinstein Yes.",
   "public_statement": "Question 30 (Rubenstein). Can we compute averages of the Hardy-Littlewood constants in the function field setting look like? Yes.",
   "evidence": "This reconstruction is explicit and is not asserted to be the only intended reading.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0061": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "1. (C. Hall) We know how to construct points on an elliptic curve E over Fq(T ) when the analytic rank is 1 by using (Drinfel'd) Heegner points. What if X is a hyperelliptic curve over Fq(T ) of higher genus? Is there any way of constructing the points on the Jacobian of X that ought to be there? Comments: 1) We have a lot of explicit examples when the analytic rank is",
   "clean_statement": null,
   "public_statement": "1. (C. Hall) We know how to construct points on an elliptic curve E over Fq(T ) when the analytic rank is 1 by using (Drinfel'd) Heegner points. What if X is a hyperelliptic curve over Fq(T ) of higher genus? Is there any way of constructing the points on the Jacobian of X that ought to be there? Comments: 1) We have a lot of explicit examples when the analytic rank is",
   "evidence": "The canonical JSON record (source file aim-algebraic-number-theory-notes.json, zero-based index 60) is truncated after the words “when the analytic rank is”. The original source is page 1 of *Problems from the AIM Tate Conjecture Workshop*, July 23--27, 2007, transcribed by Christopher Lyons. It gives the following complete record (with only mathematical typography normalized):",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0062": {
   "statement_status": "exact",
   "original_statement": "1. (C. Hall) 2) If we start with any surface S over Fq we can, via Lefschetz fibration, make it into a curve over Fq(T ). Typically (since the rank of H2 is even), one has two Tate classes. One of these is given by hyperplane sections; can we \"see\" or construct the other one? This question is like the one above, but without requiring C to be hyperelliptic. So, in some sense, the problem is almost as hard as the problem of treating general surfaces. (C. Schoen)",
   "clean_statement": "1. (C. Hall) 2) If we start with any surface S over Fq we can, via Lefschetz fibration, make it into a curve over Fq(T ). Typically (since the rank of H2 is even), one has two Tate classes. One of these is given by hyperplane sections; can we \"see\" or construct the other one? This question is like the one above, but without requiring C to be hyperelliptic. So, in some sense, the problem is almost as hard as the problem of treating general surfaces. (C. Schoen)",
   "public_statement": "1. (C. Hall) 2) If we start with any surface S over Fq we can, via Lefschetz fibration, make it into a curve over Fq(T ). Typically (since the rank of H2 is even), one has two Tate classes. One of these is given by hyperplane sections; can we \"see\" or construct the other one? This question is like the one above, but without requiring C to be hyperelliptic. So, in some sense, the problem is almost as hard as the problem of treating general surfaces. (C. Schoen)",
   "evidence": "The canonical record is the second comment on Question 1 in *Problems from the AIM Tate Conjecture Workshop*, July 23--27, 2007, transcribed by Christopher Lyons. The PDF gives Question 1 as Hall's question about constructing expected points on Jacobians of hyperelliptic curves over \\(\\mathbb F_q(T)\\). After Hall's first comment, the source states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0063": {
   "statement_status": "exact",
   "original_statement": "2. (D. Ramakrishnan) Let X be a Hilbert modular surface over Q. All of the Tate classes that arise from modular curves on X are defined over Qab; however, there also exist Tate classes which are algebraic, but not defined over Qab. (In fact, they are defined over dihedral extensions of Q.) We know the algebraicity of these classes, but do not know specific representatives. Comments: 1) One possible suggestion for such representatives could be non-congruence curves on \n\nX. (D. Ramakrishnan) 2) If we can find representatives, we can intersect them with the modular curves: this may be a way to find interesting non-Heegner points on the modular curves. (D. Ramakrishnan)",
   "clean_statement": "2. (D. Ramakrishnan) Let X be a Hilbert modular surface over Q. All of the Tate classes that arise from modular curves on X are defined over Qab; however, there also exist Tate classes which are algebraic, but not defined over Qab. (In fact, they are defined over dihedral extensions of Q.) We know the algebraicity of these classes, but do not know specific representatives. Comments: 1) One possible suggestion for such representatives could be non-congruence curves on\n\nX. (D. Ramakrishnan) 2) If we can find representatives, we can intersect them with the modular curves: this may be a way to find interesting non-Heegner points on the modular curves. (D. Ramakrishnan)",
   "public_statement": "2. (D. Ramakrishnan) Let X be a Hilbert modular surface over Q. All of the Tate classes that arise from modular curves on X are defined over Qab; however, there also exist Tate classes which are algebraic, but not defined over Qab. (In fact, they are defined over dihedral extensions of Q.) We know the algebraicity of these classes, but do not know specific representatives. Comments: 1) One possible suggestion for such representatives could be non-congruence curves on\n\nX. (D. Ramakrishnan) 2) If we can find representatives, we can intersect them with the modular curves: this may be a way to find interesting non-Heegner points on the modular curves. (D. Ramakrishnan)",
   "evidence": "The canonical record is Problem 2 in *Problems from the AIM Tate Conjecture Workshop* (July 23--27, 2007; transcribed by Christopher Lyons). It reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0064": {
   "statement_status": "exact",
   "original_statement": "3. (J. Getz) Let X be a Hilbert modular variety. At some level (say, that of cohomology), these look like a product of modular curves. Under what circumstances does X have Tate classes not coming from modular subvarieties? Thinking of these classes as submotives defined by automorphic forms (and thus as Galois representations), do these exotic classes come from CM automorphic forms as in \nQuestion 2?",
   "clean_statement": "3. (J. Getz) Let X be a Hilbert modular variety. At some level (say, that of cohomology), these look like a product of modular curves. Under what circumstances does X have Tate classes not coming from modular subvarieties? Thinking of these classes as submotives defined by automorphic forms (and thus as Galois representations), do these exotic classes come from CM automorphic forms as in\nQuestion 2?",
   "public_statement": "3. (J. Getz) Let X be a Hilbert modular variety. At some level (say, that of cohomology), these look like a product of modular curves. Under what circumstances does X have Tate classes not coming from modular subvarieties? Thinking of these classes as submotives defined by automorphic forms (and thus as Galois representations), do these exotic classes come from CM automorphic forms as in\nQuestion 2?",
   "evidence": "The record is Problem 3, attributed to J. Getz, in *Problems from AIM Tate Conjecture Workshop* (workshop held 23--27 July 2007). The original AIM PDF was inspected and agrees with the canonical JSON record:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0065": {
   "statement_status": "exact",
   "original_statement": "4. (K. Murty) Let F be a real quadratic field, π be an automorphic form over Q, and Π be its base change to F. Then Π contributes to a subspace H2(Π) of H2(SF ) for some Hilbert modular surface SF for GL 2(F ), while H1(π) sits inside H1(M ) for some modular curve \n\nM. As Gal( ¯Q/F )-modules, H2(Π) is H1(π) ⊗ H1(π). Take two real quadratic fields F1, F 2 in this way, along with the base changes Π 1, Π2\n\nof π and the associated surfaces SF1, S F2. Inside H4(SF1 × SF2 ) we have H2(Π 1) ⊗ H2(Π 2), which is H1(π)⊗ 4 as a Gal( ¯Q/F 1F2)-module. Hence we expect cycles of codimension 2 in \n\nSF1 × SF2 that are defined over F1F2. What are they? \n1",
   "clean_statement": "4. (K. Murty) Let F be a real quadratic field, π be an automorphic form over Q, and Π be its base change to F. Then Π contributes to a subspace H2(Π) of H2(SF ) for some Hilbert modular surface SF for GL 2(F ), while H1(π) sits inside H1(M ) for some modular curve\n\nM. As Gal( ¯Q/F )-modules, H2(Π) is H1(π) ⊗ H1(π). Take two real quadratic fields F1, F 2 in this way, along with the base changes Π 1, Π2\n\nof π and the associated surfaces SF1, S F2. Inside H4(SF1 × SF2 ) we have H2(Π 1) ⊗ H2(Π 2), which is H1(π)⊗ 4 as a Gal( ¯Q/F 1F2)-module. Hence we expect cycles of codimension 2 in\n\nSF1 × SF2 that are defined over F1F2. What are they?\n1",
   "public_statement": "4. (K. Murty) Let F be a real quadratic field, π be an automorphic form over Q, and Π be its base change to F. Then Π contributes to a subspace H2(Π) of H2(SF ) for some Hilbert modular surface SF for GL 2(F ), while H1(π) sits inside H1(M ) for some modular curve\n\nM. As Gal( ¯Q/F )-modules, H2(Π) is H1(π) ⊗ H1(π). Take two real quadratic fields F1, F 2 in this way, along with the base changes Π 1, Π2\n\nof π and the associated surfaces SF1, S F2. Inside H4(SF1 × SF2 ) we have H2(Π 1) ⊗ H2(Π 2), which is H1(π)⊗ 4 as a Gal( ¯Q/F 1F2)-module. Hence we expect cycles of codimension 2 in\n\nSF1 × SF2 that are defined over F1F2. What are they?\n1",
   "evidence": "The canonical record is Question 4 in the AIM workshop list *The Tate conjecture*. The final isolated **1** in the extracted record is the page number at the foot of the PDF; it is not part of the question. Superscripts, subscripts, and bars were also lost in extraction. The recovered statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0066": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "5. (D. Ulmer) Consider the curve Cd,a: yd = x(x − 1)( x − a) defined over C (though ¯Q or \n\nFq will also work). For which values of a does it happen that this curve has CM? (More precisely, for which values of a does the Jacobian of Cd,a have endomorphism algebra of dimension 2 g over Q, where g = g(Ca,d ) = d − 1 is the genus?) It happens (for dull reasons) when a = −1, 1 \n\n> 2, 2, ζ 6, ζ −16. For a fixed d > 7, there are only finitely many values of a for which it happens. Are there infinitely many d such that there's another value of a (besides the trivial ones listed above) for which Cd,a has CM? Comments: 1) An application is that, if Cd,a does have CM, then one can construct an elliptic curve defined over C(T ) of rank d +",
   "clean_statement": null,
   "public_statement": "5. (D. Ulmer) Consider the curve Cd,a: yd = x(x − 1)( x − a) defined over C (though ¯Q or\n\nFq will also work). For which values of a does it happen that this curve has CM? (More precisely, for which values of a does the Jacobian of Cd,a have endomorphism algebra of dimension 2 g over Q, where g = g(Ca,d ) = d − 1 is the genus?) It happens (for dull reasons) when a = −1, 1\n\n> 2, 2, ζ 6, ζ −16. For a fixed d > 7, there are only finitely many values of a for which it happens. Are there infinitely many d such that there's another value of a (besides the trivial ones listed above) for which Cd,a has CM? Comments: 1) An application is that, if Cd,a does have CM, then one can construct an elliptic curve defined over C(T ) of rank d +",
   "evidence": "The canonical JSON record is truncated and has two OCR errors: \\(1/2\\) appears as “1 > 2,” and \\(\\zeta _6^{-1}\\) appears as “ζ −16.” Inspection of the official AIM PDF recovers Problem 5 as follows.",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0067": {
   "statement_status": "exact",
   "original_statement": "3. (D. Ulmer) 2) Two relevant references for this problem are i) de Jong, Johan; Noot, Rutger. Jacobians with complex multiplication. Arith-metic algebraic geometry (Texel, 1989), 177-192, Progr. Math., 89, Birkhuser Boston, Boston, MA, 1991. and ii) deJong's course at AWS2002: http://swc.math.arizona.edu/aws/02/02Notes.html \n\n(D. Ulmer) 3) If x(x − 1)( x − a) is replaced by a cubic polynomial over C(T ) with Galois group S3\n\nand d > 1 is not a power of 3 then the jacobian of the resulting curve is not of CM-type (over an algebraic closure of C(T )). In addition, if d is a power prime then the jacobian does not contain non-zero abelian subvarieties of CM-type (over an algebraic closure of C(T )) if and only if d = 2 or d is an odd number that is not divisible by",
   "clean_statement": "3. (D. Ulmer) 2) Two relevant references for this problem are i) de Jong, Johan; Noot, Rutger. Jacobians with complex multiplication. Arith-metic algebraic geometry (Texel, 1989), 177-192, Progr. Math., 89, Birkhuser Boston, Boston, MA, 1991. and ii) deJong's course at AWS2002: http://swc.math.arizona.edu/aws/02/02Notes.html\n\n(D. Ulmer) 3) If x(x − 1)( x − a) is replaced by a cubic polynomial over C(T ) with Galois group S3\n\nand d > 1 is not a power of 3 then the jacobian of the resulting curve is not of CM-type (over an algebraic closure of C(T )). In addition, if d is a power prime then the jacobian does not contain non-zero abelian subvarieties of CM-type (over an algebraic closure of C(T )) if and only if d = 2 or d is an odd number that is not divisible by",
   "public_statement": "3. (D. Ulmer) 2) Two relevant references for this problem are i) de Jong, Johan; Noot, Rutger. Jacobians with complex multiplication. Arith-metic algebraic geometry (Texel, 1989), 177-192, Progr. Math., 89, Birkhuser Boston, Boston, MA, 1991. and ii) deJong's course at AWS2002: http://swc.math.arizona.edu/aws/02/02Notes.html\n\n(D. Ulmer) 3) If x(x − 1)( x − a) is replaced by a cubic polynomial over C(T ) with Galois group S3\n\nand d > 1 is not a power of 3 then the jacobian of the resulting curve is not of CM-type (over an algebraic closure of C(T )). In addition, if d is a power prime then the jacobian does not contain non-zero abelian subvarieties of CM-type (over an algebraic closure of C(T )) if and only if d = 2 or d is an odd number that is not divisible by",
   "evidence": "The canonical record is not a self-contained open problem. It splices comments 2 and 3 following Problem 5 in the AIM workshop list *The Tate conjecture*. The parent problem concerns CM specializations of",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 66,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0068": {
   "statement_status": "exact",
   "original_statement": "3. (Y. Zarhin) 4) People like Wolfart have studied the problem of determining which Jacobians of curves with Belyi parametrizations are CM. Does it help here to reparametrize this equation in the Belyi form, i.e., to express this equation as a cover of P1 ramified only at 0, 1, ∞?(D. Ramakrishnan)",
   "clean_statement": "3. (Y. Zarhin) 4) People like Wolfart have studied the problem of determining which Jacobians of curves with Belyi parametrizations are CM. Does it help here to reparametrize this equation in the Belyi form, i.e., to express this equation as a cover of P1 ramified only at 0, 1, ∞?(D. Ramakrishnan)",
   "public_statement": "3. (Y. Zarhin) 4) People like Wolfart have studied the problem of determining which Jacobians of curves with Belyi parametrizations are CM. Does it help here to reparametrize this equation in the Belyi form, i.e., to express this equation as a cover of P1 ramified only at 0, 1, ∞?(D. Ramakrishnan)",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0069": {
   "statement_status": "exact",
   "original_statement": "6. (J. Getz) Take a product of modular curves X = ∏ \n\n> i\n\nXi. Many smaller products embed diagonally into X. How much of the middle cohomology can be accounted for by these constructions? Comments: 1) Ribet considered the product of two modular curves X = X1 ×X2; here N S (X) is just the quotient of Hom( J(X1), J (X2)) by the pullbacks of divisors on the two factors. He shows it is necessary to consider twisting correspondences, in addition to the Hecke correspondences, to complete the Tate classes. Are there similarly interesting, yet explicit, correspondences for X when we consider higher dimensional factors Xi? (J. Getz, D. Ramakrishnan, D. Ulmer) 2) For a product of four Shimura curves, there are interesting Tate classes, which are also Hodge classes, which we don't know how to represent by cycles. (D. Ramakrishnan)",
   "clean_statement": "6. (J. Getz) Take a product of modular curves X = ∏\n\n> i\n\nXi. Many smaller products embed diagonally into X. How much of the middle cohomology can be accounted for by these constructions? Comments: 1) Ribet considered the product of two modular curves X = X1 ×X2; here N S (X) is just the quotient of Hom( J(X1), J (X2)) by the pullbacks of divisors on the two factors. He shows it is necessary to consider twisting correspondences, in addition to the Hecke correspondences, to complete the Tate classes. Are there similarly interesting, yet explicit, correspondences for X when we consider higher dimensional factors Xi? (J. Getz, D. Ramakrishnan, D. Ulmer) 2) For a product of four Shimura curves, there are interesting Tate classes, which are also Hodge classes, which we don't know how to represent by cycles. (D. Ramakrishnan)",
   "public_statement": "6. (J. Getz) Take a product of modular curves X = ∏\n\n> i\n\nXi. Many smaller products embed diagonally into X. How much of the middle cohomology can be accounted for by these constructions? Comments: 1) Ribet considered the product of two modular curves X = X1 ×X2; here N S (X) is just the quotient of Hom( J(X1), J (X2)) by the pullbacks of divisors on the two factors. He shows it is necessary to consider twisting correspondences, in addition to the Hecke correspondences, to complete the Tate classes. Are there similarly interesting, yet explicit, correspondences for X when we consider higher dimensional factors Xi? (J. Getz, D. Ramakrishnan, D. Ulmer) 2) For a product of four Shimura curves, there are interesting Tate classes, which are also Hodge classes, which we don't know how to represent by cycles. (D. Ramakrishnan)",
   "evidence": "The official AIM workshop PDF gives the following problem.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0070": {
   "statement_status": "exact",
   "original_statement": "7. (K. Murty) A Hilbert modular surface in characteristic p has interesting cycles parametriz-ing Hilbert-Blumenthal abelian varieties with specified Newton polygon behavior. Are these 2contained in the space of special cycles made up of (Hecke translates of) Hirzebruch-Zagier curves? How can we check this? Comments: 1) Are they pulled back from the Siegel threefold A2? (Not always.) Work of A. Langer is pertinent to this question. (J. Achter, C. Schoen) 2) Could the Teichm¨ uller curves of McMullen generate these exotic classes? (J. Ellen-berg) 3) Material by van der Geer and Moonen is relevant for explicit information on the characteristic p cycles above. (J. Achter) 4) Can we intersect these characteristic p cycles with the reduction mod p of the CM Tate cycles described in Murty's lecture? (J. Ellenberg) 5) How does the involution θ ∈ Gal( F/ Q) act on these cycles? (Here F is the real quadratic field for the Hilbert modular surface.) (J. Getz)",
   "clean_statement": "7. (K. Murty) A Hilbert modular surface in characteristic p has interesting cycles parametriz-ing Hilbert-Blumenthal abelian varieties with specified Newton polygon behavior. Are these 2contained in the space of special cycles made up of (Hecke translates of) Hirzebruch-Zagier curves? How can we check this? Comments: 1) Are they pulled back from the Siegel threefold A2? (Not always.) Work of A. Langer is pertinent to this question. (J. Achter, C. Schoen) 2) Could the Teichm¨ uller curves of McMullen generate these exotic classes? (J. Ellen-berg) 3) Material by van der Geer and Moonen is relevant for explicit information on the characteristic p cycles above. (J. Achter) 4) Can we intersect these characteristic p cycles with the reduction mod p of the CM Tate cycles described in Murty's lecture? (J. Ellenberg) 5) How does the involution θ ∈ Gal( F/ Q) act on these cycles? (Here F is the real quadratic field for the Hilbert modular surface.) (J. Getz)",
   "public_statement": "7. (K. Murty) A Hilbert modular surface in characteristic p has interesting cycles parametriz-ing Hilbert-Blumenthal abelian varieties with specified Newton polygon behavior. Are these 2contained in the space of special cycles made up of (Hecke translates of) Hirzebruch-Zagier curves? How can we check this? Comments: 1) Are they pulled back from the Siegel threefold A2? (Not always.) Work of A. Langer is pertinent to this question. (J. Achter, C. Schoen) 2) Could the Teichm¨ uller curves of McMullen generate these exotic classes? (J. Ellen-berg) 3) Material by van der Geer and Moonen is relevant for explicit information on the characteristic p cycles above. (J. Achter) 4) Can we intersect these characteristic p cycles with the reduction mod p of the CM Tate cycles described in Murty's lecture? (J. Ellenberg) 5) How does the involution θ ∈ Gal( F/ Q) act on these cycles? (Here F is the real quadratic field for the Hilbert modular surface.) (J. Getz)",
   "evidence": "The record is Problem 7, attributed to K. Murty, in the AIM workshop list *The Tate conjecture* (workshop held July 23--27, 2007; list transcribed by Christopher Lyons). The canonical JSON has the string “Are these 2contained”, but comparison with page 2 of the official PDF shows that the 2 is the printed page number captured at a page break. The recovered statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 69,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0071": {
   "statement_status": "exact",
   "original_statement": "8. (J. Ellenberg) In cases where we know all Tate classes are in the image of Ci ⊗ Q`, what can we say about the image of Ci ⊗ Z`? For instance, for which primes ` can we have non-surjectivity? Comments: 1) Koll´ ar proved that if we take a very general hypersurface in P4 such that d3 divides the degree of X, then the image of C1(X) is contained in dH 4(X, Z). But it is not known if this can happen for X over ¯Q. For C1 of X over Fq, the Q`-Tate Conjecture implies the Z`-Tate Conjecture for ` 6 = p. (C. Schoen) 2) Over Fq, the Q`-Tate Conjecture implies the Z`-Tate Conjecture for the Weil-´ etale cohomology, so this becomes a question of comparison between ´ etale and Weil-´ etale cohomology. (T. Geisser)",
   "clean_statement": "8. (J. Ellenberg) In cases where we know all Tate classes are in the image of Ci ⊗ Q`, what can we say about the image of Ci ⊗ Z`? For instance, for which primes ` can we have non-surjectivity? Comments: 1) Koll´ ar proved that if we take a very general hypersurface in P4 such that d3 divides the degree of X, then the image of C1(X) is contained in dH 4(X, Z). But it is not known if this can happen for X over ¯Q. For C1 of X over Fq, the Q`-Tate Conjecture implies the Z`-Tate Conjecture for ` 6 = p. (C. Schoen) 2) Over Fq, the Q`-Tate Conjecture implies the Z`-Tate Conjecture for the Weil-´ etale cohomology, so this becomes a question of comparison between ´ etale and Weil-´ etale cohomology. (T. Geisser)",
   "public_statement": "8. (J. Ellenberg) In cases where we know all Tate classes are in the image of Ci ⊗ Q`, what can we say about the image of Ci ⊗ Z`? For instance, for which primes ` can we have non-surjectivity? Comments: 1) Koll´ ar proved that if we take a very general hypersurface in P4 such that d3 divides the degree of X, then the image of C1(X) is contained in dH 4(X, Z). But it is not known if this can happen for X over ¯Q. For C1 of X over Fq, the Q`-Tate Conjecture implies the Z`-Tate Conjecture for ` 6 = p. (C. Schoen) 2) Over Fq, the Q`-Tate Conjecture implies the Z`-Tate Conjecture for the Weil-´ etale cohomology, so this becomes a question of comparison between ´ etale and Weil-´ etale cohomology. (T. Geisser)",
   "evidence": "The canonical JSON record has lost superscripts, subscripts, accents, and every occurrence of \\(\\ell\\). The official AIM workshop PDF gives the following statement (Problem 8, J. Ellenberg):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 70,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0072": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "9. (J. Milne) Let A be CM abelian variety over ¯Q and let c be an absolute Hodge class. Reduce A to A0 over Fp and take a Lefschetz class x on A0 of complementary dimension. Then x. ¯c ∈ Q`; is it in Q and independent of `?Comments: 1) The answer is yes, assuming either the Tate Conjecture or the Hodge Conjecture, or also when A0 is ordinary (since then x can be lifted to characteristic 0). (J. Milne) 2) This would imply the existence of the Q-subalgebras R∗ mentioned described in Milne's lecture. (J. Milne) 3) The question also makes sense for any abelian variety A which has good reduction over Fp. (J. Milne) 4) It may be that ( A0, x ) still admits a canonical lift, even if we relax the assumption that A0 is ordinary (see Milne's first comment). (J. Achter) \n1",
   "clean_statement": null,
   "public_statement": "9. (J. Milne) Let A be CM abelian variety over ¯Q and let c be an absolute Hodge class. Reduce A to A0 over Fp and take a Lefschetz class x on A0 of complementary dimension. Then x. ¯c ∈ Q`; is it in Q and independent of `?Comments: 1) The answer is yes, assuming either the Tate Conjecture or the Hodge Conjecture, or also when A0 is ordinary (since then x can be lifted to characteristic 0). (J. Milne) 2) This would imply the existence of the Q-subalgebras R∗ mentioned described in Milne's lecture. (J. Milne) 3) The question also makes sense for any abelian variety A which has good reduction over Fp. (J. Milne) 4) It may be that ( A0, x ) still admits a canonical lift, even if we relax the assumption that A0 is ordinary (see Milne's first comment). (J. Achter)\n1",
   "evidence": "The canonical JSON is damaged precisely at the coefficient field and at the prime. The official AIM workshop PDF gives the following statement.",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0073": {
   "statement_status": "exact",
   "original_statement": "0. (C. Schoen) Let J be a general abelian surface (over C, say). Describe the surfaces in J ×J.3\n1",
   "clean_statement": "0. (C. Schoen) Let J be a general abelian surface (over C, say). Describe the surfaces in J ×J.3\n1",
   "public_statement": "0. (C. Schoen) Let J be a general abelian surface (over C, say). Describe the surfaces in J ×J.3\n1",
   "evidence": "The canonical JSON record reads",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 72,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0074": {
   "statement_status": "exact",
   "original_statement": "1. (J. Getz) C. Simpson proved that if X is a variety and if ρX: πgeom \n\n> 1\n\n(X) → GL 2(Q`) is a representation whose algebraic enveloping contains SL 2, then ρX factors through ρY:\n\nπgeom \n\n> 1\n\n(Y ) → GL 2(Q`), where Y is either a curve or a Shimura variety; the factorization is induced from a map from X to some cover of Y. Is there a version of this result using just \n\nπ1 in place of πgeom \n\n> 1?\n1",
   "clean_statement": "1. (J. Getz) C. Simpson proved that if X is a variety and if ρX: πgeom\n\n> 1\n\n(X) → GL 2(Q`) is a representation whose algebraic enveloping contains SL 2, then ρX factors through ρY:\n\nπgeom\n\n> 1\n\n(Y ) → GL 2(Q`), where Y is either a curve or a Shimura variety; the factorization is induced from a map from X to some cover of Y. Is there a version of this result using just\n\nπ1 in place of πgeom\n\n> 1?\n1",
   "public_statement": "1. (J. Getz) C. Simpson proved that if X is a variety and if ρX: πgeom\n\n> 1\n\n(X) → GL 2(Q`) is a representation whose algebraic enveloping contains SL 2, then ρX factors through ρY:\n\nπgeom\n\n> 1\n\n(Y ) → GL 2(Q`), where Y is either a curve or a Shimura variety; the factorization is induced from a map from X to some cover of Y. Is there a version of this result using just\n\nπ1 in place of πgeom\n\n> 1?\n1",
   "evidence": "The canonical JSON record is an OCR-damaged extraction from the five-page list *Problems from the AIM Tate Conjecture Workshop* (July 23--27, 2007), transcribed by Christopher Lyons. The PDF shows that this is **Problem 11**, not Problem 1. The recovered text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 73,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0075": {
   "statement_status": "exact",
   "original_statement": "2. (W. Raskind) Take an abelian variety over a finite extension K of Qp with multiplicative reduction. Show the isomorphism End( A) ⊗ Qp ˜−→ End( Vp(A)) GK.\n\nComments: 1) This implies the analogous statement for A over K, with K global, having multiplica-tive reduction somewhere. (W. Raskind) 2) This should be doable by imitating methods of Serre for the case when A is an elliptic curve. (W. Raskind) 3) This is known for Drinfel'd modular varieties (done by Ito). (W. Raskind) \n1",
   "clean_statement": "2. (W. Raskind) Take an abelian variety over a finite extension K of Qp with multiplicative reduction. Show the isomorphism End( A) ⊗ Qp ˜−→ End( Vp(A)) GK.\n\nComments: 1) This implies the analogous statement for A over K, with K global, having multiplica-tive reduction somewhere. (W. Raskind) 2) This should be doable by imitating methods of Serre for the case when A is an elliptic curve. (W. Raskind) 3) This is known for Drinfel'd modular varieties (done by Ito). (W. Raskind)\n1",
   "public_statement": "2. (W. Raskind) Take an abelian variety over a finite extension K of Qp with multiplicative reduction. Show the isomorphism End( A) ⊗ Qp ˜−→ End( Vp(A)) GK.\n\nComments: 1) This implies the analogous statement for A over K, with K global, having multiplica-tive reduction somewhere. (W. Raskind) 2) This should be doable by imitating methods of Serre for the case when A is an elliptic curve. (W. Raskind) 3) This is known for Drinfel'd modular varieties (done by Ito). (W. Raskind)\n1",
   "evidence": "The canonical JSON record has lost the leading digit in the problem number. The official AIM workshop PDF, *Problems from the AIM Tate Conjecture Workshop* (July 23--27, 2007, transcribed by Christopher Lyons), states this as item **12**, not item 2:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0076": {
   "statement_status": "exact",
   "original_statement": "3. (K. Murty) According to the Tate Conjecture, if one takes the L-function of a variety X\n\nover K at the edge of the critical strip, and then increases K, the order of the pole should stay bounded. Can we prove this same statement without assuming the Tate Conjecture? Comments: 1) This is provable over function fields of finite fields since, by using the work of Lafforgue, we have (T3): the order of the pole equals the rank of the Tate classes. It is also true when X is an abelian variety in characteristic",
   "clean_statement": "3. (K. Murty) According to the Tate Conjecture, if one takes the L-function of a variety X\n\nover K at the edge of the critical strip, and then increases K, the order of the pole should stay bounded. Can we prove this same statement without assuming the Tate Conjecture? Comments: 1) This is provable over function fields of finite fields since, by using the work of Lafforgue, we have (T3): the order of the pole equals the rank of the Tate classes. It is also true when X is an abelian variety in characteristic",
   "public_statement": "3. (K. Murty) According to the Tate Conjecture, if one takes the L-function of a variety X\n\nover K at the edge of the critical strip, and then increases K, the order of the pole should stay bounded. Can we prove this same statement without assuming the Tate Conjecture? Comments: 1) This is provable over function fields of finite fields since, by using the work of Lafforgue, we have (T3): the order of the pole equals the rank of the Tate classes. It is also true when X is an abelian variety in characteristic",
   "evidence": "The canonical JSON record is truncated and misnumbers the item as Problem 3. The official AIM workshop PDF has it as Problem 13 and reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0077": {
   "statement_status": "exact",
   "original_statement": "4. (C. Hall) Let X(`) be a modular curve. What can we say about the splitting field of the \n\n`-torsion of the Jacobian J(X(`))? Comments: 1) This is needed to speak about the L-function of E → X(`) (mod `) ∈ F`[T ]. (C. Hall) \n1",
   "clean_statement": "4. (C. Hall) Let X(`) be a modular curve. What can we say about the splitting field of the\n\n`-torsion of the Jacobian J(X(`))? Comments: 1) This is needed to speak about the L-function of E → X(`) (mod `) ∈ F`[T ]. (C. Hall)\n1",
   "public_statement": "4. (C. Hall) Let X(`) be a modular curve. What can we say about the splitting field of the\n\n`-torsion of the Jacobian J(X(`))? Comments: 1) This is needed to speak about the L-function of E → X(`) (mod `) ∈ F`[T ]. (C. Hall)\n1",
   "evidence": "The canonical record is source index 76 of aim-algebraic-number-theory-notes.json. Its OCR text has replaced every occurrence of \\(\\ell\\) by a backtick, has collapsed a line break, and labels the question “4.” The official AIM workshop PDF shows that it is actually item 14:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0078": {
   "statement_status": "exact",
   "original_statement": "5. (J. Ellenberg, D. Ulmer) Let X be a modular curve, which paramterizes elliptic curves with some level structure, and let E be the universal family of elliptic curves over X. This defines an elliptic surface, typically over Q. It often happens that, for many primes p, the reduction modulo p of this surface acquires new Tate classes, not coming from characteristic zero. Can we find algebraic cycles (living in characteristic p), which account for such Tate classes? \n1",
   "clean_statement": "5. (J. Ellenberg, D. Ulmer) Let X be a modular curve, which paramterizes elliptic curves with some level structure, and let E be the universal family of elliptic curves over X. This defines an elliptic surface, typically over Q. It often happens that, for many primes p, the reduction modulo p of this surface acquires new Tate classes, not coming from characteristic zero. Can we find algebraic cycles (living in characteristic p), which account for such Tate classes?\n1",
   "public_statement": "5. (J. Ellenberg, D. Ulmer) Let X be a modular curve, which paramterizes elliptic curves with some level structure, and let E be the universal family of elliptic curves over X. This defines an elliptic surface, typically over Q. It often happens that, for many primes p, the reduction modulo p of this surface acquires new Tate classes, not coming from characteristic zero. Can we find algebraic cycles (living in characteristic p), which account for such Tate classes?\n1",
   "evidence": "The canonical record is source index 77 of “aim-algebraic-number-theory-notes.json.” Its OCR text labels the question as “5,” misspells “parametrizes” as “paramterizes,” and appends a stray page marker “1.” The cited AIM PDF is *Problems from the AIM Tate Conjecture Workshop* (July 23–27, 2007), transcribed by Christopher Lyons. In that PDF the record is Problem **15**, not Problem 5. Apart from the extraction artifacts, the statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0079": {
   "statement_status": "exact",
   "original_statement": "6. (D. Ramakrishnan) Let E be an elliptic curve over Q with CM by K and with associated Hecke character χ. Then χ2 is associated to a modular form of weight 3 appearing in \n\nH2(E), E → X for a modular curve X (whose level is the norm of the conductor of χ2\n\ntimes the discriminant of K). The Tate and Hodge Conjectures predict the existence of a correspondence E × E → E because the same motive appears in H2 of both sides. Can we exhibit this correspondence explicitly? Comments: 41) In Shioda's example (level 4), E is K3, and in fact the Kummer surface associated to \n\nE × E. (D. Ramakrishnan) 2) This problem is closely related to (the comments from) the previous problem because, when E has supersingular reduction, there are many extra cycles on E × E. (C. Schoen) \n1",
   "clean_statement": "6. (D. Ramakrishnan) Let E be an elliptic curve over Q with CM by K and with associated Hecke character χ. Then χ2 is associated to a modular form of weight 3 appearing in\n\nH2(E), E → X for a modular curve X (whose level is the norm of the conductor of χ2\n\ntimes the discriminant of K). The Tate and Hodge Conjectures predict the existence of a correspondence E × E → E because the same motive appears in H2 of both sides. Can we exhibit this correspondence explicitly? Comments: 41) In Shioda's example (level 4), E is K3, and in fact the Kummer surface associated to\n\nE × E. (D. Ramakrishnan) 2) This problem is closely related to (the comments from) the previous problem because, when E has supersingular reduction, there are many extra cycles on E × E. (C. Schoen)\n1",
   "public_statement": "6. (D. Ramakrishnan) Let E be an elliptic curve over Q with CM by K and with associated Hecke character χ. Then χ2 is associated to a modular form of weight 3 appearing in\n\nH2(E), E → X for a modular curve X (whose level is the norm of the conductor of χ2\n\ntimes the discriminant of K). The Tate and Hodge Conjectures predict the existence of a correspondence E × E → E because the same motive appears in H2 of both sides. Can we exhibit this correspondence explicitly? Comments: 41) In Shioda's example (level 4), E is K3, and in fact the Kummer surface associated to\n\nE × E. (D. Ramakrishnan) 2) This problem is closely related to (the comments from) the previous problem because, when E has supersingular reduction, there are many extra cycles on E × E. (C. Schoen)\n1",
   "evidence": "The canonical JSON record is damaged by PDF extraction: it labels the item as “6,” writes both the elliptic curve and the elliptic modular surface as `E`, drops superscripts in \\(\\chi^2\\) and \\(H^2\\), and reverses/obscures some layout. The official AIM TeX source identifies it as **Problem 16** and reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0080": {
   "statement_status": "exact",
   "original_statement": "7. (W. Raskind) In the situations where the Hodge Conjecture implies the Tate Conjecture, is it also true that the generalized Hodge Conjecture implies the generalized Tate Conjecture, e.g., for CM abelian varieties? \n1",
   "clean_statement": "7. (W. Raskind) In the situations where the Hodge Conjecture implies the Tate Conjecture, is it also true that the generalized Hodge Conjecture implies the generalized Tate Conjecture, e.g., for CM abelian varieties?\n1",
   "public_statement": "7. (W. Raskind) In the situations where the Hodge Conjecture implies the Tate Conjecture, is it also true that the generalized Hodge Conjecture implies the generalized Tate Conjecture, e.g., for CM abelian varieties?\n1",
   "evidence": "The canonical record is source index 79 of `aim-algebraic-number-theory-notes.json`. Its OCR text reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0081": {
   "statement_status": "exact",
   "original_statement": "8. (E. Izadi) Take a principally polarized abelian variety A and assume rank( N S (A)) = 1. What is the index of Ci(A) in H2i(A, Z(i))? What is the image? What if we just restrict ourselves to the line spanned by [ θ]i, where θ is the polarization? \n1",
   "clean_statement": "8. (E. Izadi) Take a principally polarized abelian variety A and assume rank( N S (A)) = 1. What is the index of Ci(A) in H2i(A, Z(i))? What is the image? What if we just restrict ourselves to the line spanned by [ θ]i, where θ is the polarization?\n1",
   "public_statement": "8. (E. Izadi) Take a principally polarized abelian variety A and assume rank( N S (A)) = 1. What is the index of Ci(A) in H2i(A, Z(i))? What is the image? What if we just restrict ourselves to the line spanned by [ θ]i, where θ is the polarization?\n1",
   "evidence": "The canonical JSON record is an OCR extraction from the problem list of the AIM workshop *The Tate conjecture* (July 23--27, 2007). The OCR calls this Problem 8 and suppresses superscripts. Inspection of page 5 of the official PDF shows that it is actually Problem 18 and that the target is literally all integral cohomology, not the subgroup of integral Hodge classes:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0082": {
   "statement_status": "exact",
   "original_statement": "9. (D. Ramakrishnan) The following question is due to P. Deligne. Let X be a variety defined over a (finitely-generated) field of characteristic 0. Given a Tate class on c on X, is there a prime p such that, upon reducing X mod p, the reduction of c is algebraic? \n2",
   "clean_statement": "9. (D. Ramakrishnan) The following question is due to P. Deligne. Let X be a variety defined over a (finitely-generated) field of characteristic 0. Given a Tate class on c on X, is there a prime p such that, upon reducing X mod p, the reduction of c is algebraic?\n2",
   "public_statement": "9. (D. Ramakrishnan) The following question is due to P. Deligne. Let X be a variety defined over a (finitely-generated) field of characteristic 0. Given a Tate class on c on X, is there a prime p such that, upon reducing X mod p, the reduction of c is algebraic?\n2",
   "evidence": "The canonical record is source index 81 of `aim-algebraic-number-theory-notes.json`. The official AIM PDF contains the record as item **19**, whereas the canonical extraction labels it item 9. The PDF literally reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0083": {
   "statement_status": "exact",
   "original_statement": "0. (M. Flach) Find a natural definition of the Weil-´ etale topology so that the cohomology of \n\nZ vanishes in degrees greater than 3. Comments: 1) The current definition only vanishes for odd degrees and gives infinitely generated groups in even degrees. (M. Flach) 2) It it not clear how hard this problem is. A better definition might be just around the corner or the only solution might be to truncate the cohomology complex. (M. Flach) \n2",
   "clean_statement": "0. (M. Flach) Find a natural definition of the Weil-´ etale topology so that the cohomology of\n\nZ vanishes in degrees greater than 3. Comments: 1) The current definition only vanishes for odd degrees and gives infinitely generated groups in even degrees. (M. Flach) 2) It it not clear how hard this problem is. A better definition might be just around the corner or the only solution might be to truncate the cohomology complex. (M. Flach)\n2",
   "public_statement": "0. (M. Flach) Find a natural definition of the Weil-´ etale topology so that the cohomology of\n\nZ vanishes in degrees greater than 3. Comments: 1) The current definition only vanishes for odd degrees and gives infinitely generated groups in even degrees. (M. Flach) 2) It it not clear how hard this problem is. A better definition might be just around the corner or the only solution might be to truncate the cohomology complex. (M. Flach)\n2",
   "evidence": "The canonical JSON record is source index 82 of `aim-algebraic-number-theory-notes.json`. Its OCR text loses the problem number, corrupts “étale,” and separates the final \\(Z\\) from the preceding sentence. The official AIM TeX source identifies it as Problem 20 and reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 82,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0084": {
   "statement_status": "exact",
   "original_statement": "1. (M. Flach) Using the current definition of the Weil-´ etale topology, define the Weil-´ etale topos for an arbitrary arithmetic scheme (i.e., a scheme of finite type over Spec( Z)) as a fibred product. Can one compute the cohomology of the sheaf Z on, say, the affine or projective line? 5",
   "clean_statement": "1. (M. Flach) Using the current definition of the Weil-´ etale topology, define the Weil-´ etale topos for an arbitrary arithmetic scheme (i.e., a scheme of finite type over Spec( Z)) as a fibred product. Can one compute the cohomology of the sheaf Z on, say, the affine or projective line? 5",
   "public_statement": "1. (M. Flach) Using the current definition of the Weil-´ etale topology, define the Weil-´ etale topos for an arbitrary arithmetic scheme (i.e., a scheme of finite type over Spec( Z)) as a fibred product. Can one compute the cohomology of the sheaf Z on, say, the affine or projective line? 5",
   "evidence": "The canonical JSON record is an OCR extraction from page 5 of the AIM workshop list *The Tate conjecture*. The official PDF gives the following as Problem 21 (not Problem 1):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 83,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0085": {
   "statement_status": "exact",
   "original_statement": "Question 1. Consider hypersurfaces fd ∈ Z[x0,..., x n] of degree d in Pn, n ≥ 2. Let \n\nN (H) = {fd: max coefficient of fd is ≤ H, fd has a solution in Zn+1 \\ { 0}}.\n\n(a) For d > n + 1, is it true that N (H)/N tot (H) → 0? Does this follow from Lang's conjectures? (Voloch) (b) Instead, look at those with those with points locally everywhere Nloc (Poonen). Is this a positive fraction, i.e. is Nloc (H)/N tot (H) > c?(c) As a special case, if you write down a plane cubic, how likely is it to have a rational point? (Voloch) (d) For cubic surfaces, there are examples where the Hasse principle fails, but maybe for almost all values of a parameter in a family, the Hasse principle holds. We should have asympotically that the N (H)/N tot (H) ∼ Nloc (H)/N tot (H)? (Colliot-Th´ el` ene) For d ≤ n, we hope this will hold in general.",
   "clean_statement": "Question 1. Consider hypersurfaces fd ∈ Z[x0,..., x n] of degree d in Pn, n ≥ 2. Let\n\nN (H) = {fd: max coefficient of fd is ≤ H, fd has a solution in Zn+1 \\ { 0}}.\n\n(a) For d > n + 1, is it true that N (H)/N tot (H) → 0? Does this follow from Lang's conjectures? (Voloch) (b) Instead, look at those with those with points locally everywhere Nloc (Poonen). Is this a positive fraction, i.e. is Nloc (H)/N tot (H) > c?(c) As a special case, if you write down a plane cubic, how likely is it to have a rational point? (Voloch) (d) For cubic surfaces, there are examples where the Hasse principle fails, but maybe for almost all values of a parameter in a family, the Hasse principle holds. We should have asympotically that the N (H)/N tot (H) ∼ Nloc (H)/N tot (H)? (Colliot-Th´ el` ene) For d ≤ n, we hope this will hold in general.",
   "public_statement": "Question 1. Consider hypersurfaces fd ∈ Z[x0,..., x n] of degree d in Pn, n ≥ 2. Let\n\nN (H) = {fd: max coefficient of fd is ≤ H, fd has a solution in Zn+1 \\ { 0}}.\n\n(a) For d > n + 1, is it true that N (H)/N tot (H) → 0? Does this follow from Lang's conjectures? (Voloch) (b) Instead, look at those with those with points locally everywhere Nloc (Poonen). Is this a positive fraction, i.e. is Nloc (H)/N tot (H) > c?(c) As a special case, if you write down a plane cubic, how likely is it to have a rational point? (Voloch) (d) For cubic surfaces, there are examples where the Hasse principle fails, but maybe for almost all values of a parameter in a family, the Hasse principle holds. We should have asympotically that the N (H)/N tot (H) ∼ Nloc (H)/N tot (H)? (Colliot-Th´ el` ene) For d ≤ n, we hope this will hold in general.",
   "evidence": "The source is the AIM workshop list *Rational and integral points on higher dimensional varieties*, workshop held December 11--20, 2002, version dated November 22, 2004, Problem/Question 1 on p. 44. Put",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 84,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0086": {
   "statement_status": "exact",
   "original_statement": "Question 2. If X is smooth projective geometrically rationally connected defined over a number field k, then the Brauer-Manin obstruction should be the only one to the Hasse principle. That is, is \n\nX(k) = X(Ak)Br X?46 \n\nAs special cases, this should be the case for Fano varieties of dimension ≥ 3, e.g. (smooth) complete intersections of degree ( d1,..., d r) in Pn with d1 + · · · + dr ≤ n. (Colliot-Th´ el` ene)",
   "clean_statement": "Question 2. If X is smooth projective geometrically rationally connected defined over a number field k, then the Brauer-Manin obstruction should be the only one to the Hasse principle. That is, is\n\nX(k) = X(Ak)Br X?46\n\nAs special cases, this should be the case for Fano varieties of dimension ≥ 3, e.g. (smooth) complete intersections of degree ( d1,..., d r) in Pn with d1 + · · · + dr ≤ n. (Colliot-Th´ el` ene)",
   "public_statement": "Question 2. If X is smooth projective geometrically rationally connected defined over a number field k, then the Brauer-Manin obstruction should be the only one to the Hasse principle. That is, is\n\nX(k) = X(Ak)Br X?46\n\nAs special cases, this should be the case for Fano varieties of dimension ≥ 3, e.g. (smooth) complete intersections of degree ( d1,..., d r) in Pn with d1 + · · · + dr ≤ n. (Colliot-Th´ el` ene)",
   "evidence": "The canonical record is source index 85 of `aim-algebraic-number-theory-notes.json`, from the AIM workshop *Rational and integral points on higher dimensional varieties*. Its extracted formula reads",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 85,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0087": {
   "statement_status": "exact",
   "original_statement": "Question 3. Let f1(x0,..., x n) = 0, f2(x0,..., x n) = 0 define the smooth com-plete intersection of two quadrics X2,2 ⊂ Pn over k a number field. For simplicity, assume \n\nk = Q. If n ≥ 8, X(R) 6 = ∅ implies X(Q) 6 = ∅. (Sansuc, Swinnerton-Dyer, Colliot-Th´ el` ene) (a) For n ≥ 9 (10 variables), this can be provably done by the circle method. (Heath-Brown) (b) For n = 7, there is the concrete problem: given two quadratic forms f1, f 2 as above in 8 variables over k a p-adic field, assume f1 = f2 = 0 is smooth, so that det( λf 1 +μf 2) = p(λ, μ )is separable. Does there exist ( λ, μ ) ∈ P1(k) such that λf 1 + μf 2 contains 3 hyperbolics (split off an extra xy in its decomposition). Solution to this problem would give a local-global principle for n = \n7. (Colliot-Th´ el` ene)",
   "clean_statement": "Question 3. Let f1(x0,..., x n) = 0, f2(x0,..., x n) = 0 define the smooth com-plete intersection of two quadrics X2,2 ⊂ Pn over k a number field. For simplicity, assume\n\nk = Q. If n ≥ 8, X(R) 6 = ∅ implies X(Q) 6 = ∅. (Sansuc, Swinnerton-Dyer, Colliot-Th´ el` ene) (a) For n ≥ 9 (10 variables), this can be provably done by the circle method. (Heath-Brown) (b) For n = 7, there is the concrete problem: given two quadratic forms f1, f 2 as above in 8 variables over k a p-adic field, assume f1 = f2 = 0 is smooth, so that det( λf 1 +μf 2) = p(λ, μ )is separable. Does there exist ( λ, μ ) ∈ P1(k) such that λf 1 + μf 2 contains 3 hyperbolics (split off an extra xy in its decomposition). Solution to this problem would give a local-global principle for n =\n7. (Colliot-Th´ el` ene)",
   "public_statement": "Question 3. Let f1(x0,..., x n) = 0, f2(x0,..., x n) = 0 define the smooth com-plete intersection of two quadrics X2,2 ⊂ Pn over k a number field. For simplicity, assume\n\nk = Q. If n ≥ 8, X(R) 6 = ∅ implies X(Q) 6 = ∅. (Sansuc, Swinnerton-Dyer, Colliot-Th´ el` ene) (a) For n ≥ 9 (10 variables), this can be provably done by the circle method. (Heath-Brown) (b) For n = 7, there is the concrete problem: given two quadratic forms f1, f 2 as above in 8 variables over k a p-adic field, assume f1 = f2 = 0 is smooth, so that det( λf 1 +μf 2) = p(λ, μ )is separable. Does there exist ( λ, μ ) ∈ P1(k) such that λf 1 + μf 2 contains 3 hyperbolics (split off an extra xy in its decomposition). Solution to this problem would give a local-global principle for n =\n7. (Colliot-Th´ el` ene)",
   "evidence": "The record is Question 3 in the AIM workshop problem list *Rational and integral points on higher dimensional varieties*. The canonical JSON is an OCR extraction; the original PDF was checked directly, including the surrounding remarks.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 86,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0088": {
   "statement_status": "exact",
   "original_statement": "Question 4. Consider the hypersurface given by ∑ \n\n> i\n\nxiy2 \n\n> i\n\n= 0 in P3 × P3. Take the height \n\nH(x, y ) = sup |xi|3 · sup |yi|2\n\nand throw out the set where some xi = 0, yi = 0. Can one estimate the counting function? (Peyre)",
   "clean_statement": "Question 4. Consider the hypersurface given by ∑\n\n> i\n\nxiy2\n\n> i\n\n= 0 in P3 × P3. Take the height\n\nH(x, y ) = sup |xi|3 · sup |yi|2\n\nand throw out the set where some xi = 0, yi = 0. Can one estimate the counting function? (Peyre)",
   "public_statement": "Question 4. Consider the hypersurface given by ∑\n\n> i\n\nxiy2\n\n> i\n\n= 0 in P3 × P3. Take the height\n\nH(x, y ) = sup |xi|3 · sup |yi|2\n\nand throw out the set where some xi = 0, yi = 0. Can one estimate the counting function? (Peyre)",
   "evidence": "The source is Question 4 in the AIM workshop document *Rational and integral points on higher dimensional varieties* (workshop held December 11--20, 2002; document version November 22, 2004). The OCR in the corpus has separated the summation indices from the formula. The PDF makes the intended statement unambiguous:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 87,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0089": {
   "statement_status": "exact",
   "original_statement": "Question 5. Consider xyz = t(x + y + z)2, with height function H(x, y, z, t ) = max( |x|, |y|, |z|, |t|), and restrict to gcd( x, y, z, t ) = 1. Excluding trivial solutions (on lines), can you prove a counting function which is \n\n∼ cB (log B)6 < B 1+ ≤?This is a singular cubic surface so we also expect an explicit constant. (Tschinkel)",
   "clean_statement": "Question 5. Consider xyz = t(x + y + z)2, with height function H(x, y, z, t ) = max( |x|, |y|, |z|, |t|), and restrict to gcd( x, y, z, t ) = 1. Excluding trivial solutions (on lines), can you prove a counting function which is\n\n∼ cB (log B)6 < B 1+ ≤?This is a singular cubic surface so we also expect an explicit constant. (Tschinkel)",
   "public_statement": "Question 5. Consider xyz = t(x + y + z)2, with height function H(x, y, z, t ) = max( |x|, |y|, |z|, |t|), and restrict to gcd( x, y, z, t ) = 1. Excluding trivial solutions (on lines), can you prove a counting function which is\n\n∼ cB (log B)6 < B 1+ ≤?This is a singular cubic surface so we also expect an explicit constant. (Tschinkel)",
   "evidence": "The canonical record is Question 5 from the AIM workshop *Rational and integral points on higher dimensional varieties*. Its OCR text is damaged at the exponent and comparison signs. The official AIM HTML and typeset PDF recover the question as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 88,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0090": {
   "statement_status": "exact",
   "original_statement": "Question \n6. (Swinnerton-Dyer) (a) Is there a K3 surface X over Q which has a finite nonzero number of rational points X(Q) <\n\n∞?(b) Is there a nonsingular quartic surface X with this property? (c) Find a third rational point on X41 + 2 X42 = X43 + 4 X44. (There are only 2 points with height \n\n≤ 215, and other non-public reasons to believe that there are only finitely many points.) (d) Find a smooth quartic X4 ⊂ P3 with Pic X ∼= Z and X(K) infinite.",
   "clean_statement": "Question\n6. (Swinnerton-Dyer) (a) Is there a K3 surface X over Q which has a finite nonzero number of rational points X(Q) <\n\n∞?(b) Is there a nonsingular quartic surface X with this property? (c) Find a third rational point on X41 + 2 X42 = X43 + 4 X44. (There are only 2 points with height\n\n≤ 215, and other non-public reasons to believe that there are only finitely many points.) (d) Find a smooth quartic X4 ⊂ P3 with Pic X ∼= Z and X(K) infinite.",
   "public_statement": "Question\n6. (Swinnerton-Dyer) (a) Is there a K3 surface X over Q which has a finite nonzero number of rational points X(Q) <\n\n∞?(b) Is there a nonsingular quartic surface X with this property? (c) Find a third rational point on X41 + 2 X42 = X43 + 4 X44. (There are only 2 points with height\n\n≤ 215, and other non-public reasons to believe that there are only finitely many points.) (d) Find a smooth quartic X4 ⊂ P3 with Pic X ∼= Z and X(K) infinite.",
   "evidence": "The record is Question 6 (attributed to Swinnerton-Dyer) in the AIM workshop list *Rational and integral points on higher dimensional varieties*, version dated 22 November 2004, pp. 46--47. The PDF was checked directly because the extracted JSON loses superscripts, a cardinality sign, and an overline. The recovered question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 89,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0091": {
   "statement_status": "exact",
   "original_statement": "Question 7. The distribution of rational points on Enriques surfaces has not been well studied. (Skorobogatov) Let X be an Enriques surface. (a) If X(k) 6 = ∅, is X(k) Zariski dense? (b) Is there an X which violates the Hasse principle? (c) Is there an X such that X(Ak)Br = ∅ but X(Ak) 6 = ∅?(d) Is there an X with X(k) = ∅, X(Ak)Br 6 = ∅?",
   "clean_statement": "Question 7. The distribution of rational points on Enriques surfaces has not been well studied. (Skorobogatov) Let X be an Enriques surface. (a) If X(k) 6 = ∅, is X(k) Zariski dense? (b) Is there an X which violates the Hasse principle? (c) Is there an X such that X(Ak)Br = ∅ but X(Ak) 6 = ∅?(d) Is there an X with X(k) = ∅, X(Ak)Br 6 = ∅?",
   "public_statement": "Question 7. The distribution of rational points on Enriques surfaces has not been well studied. (Skorobogatov) Let X be an Enriques surface. (a) If X(k) 6 = ∅, is X(k) Zariski dense? (b) Is there an X which violates the Hasse principle? (c) Is there an X such that X(Ak)Br = ∅ but X(Ak) 6 = ∅?(d) Is there an X with X(k) = ∅, X(Ak)Br 6 = ∅?",
   "evidence": "The canonical record is Question 7 from the AIM workshop *Rational and integral points on higher dimensional varieties*, in **aim-algebraic-number-theory-notes.json**, source index 90. I checked the record against the AIM PDF and the AIM HTML rendering [AIM]. The symbols rendered as “6 =” in the JSON are OCR substitutions for \\(\\ne\\), and the missing spacing in parts (c) and (d) does not change the formulas.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 90,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0092": {
   "statement_status": "exact",
   "original_statement": "Question 8. Let X an Enriques surface over a number field K, Y p\n\n→ X the double cover (K3 surface), X(AK )Br the Brauer-Manin set. Take mv ∈ X(AK )Br, lift it to an adelic point Pv on Y. Under what assumptions on mv will it be liftable to Pv on Y (AK )Br,or at least Y (AK )Br 1?",
   "clean_statement": "Question 8. Let X an Enriques surface over a number field K, Y p\n\n→ X the double cover (K3 surface), X(AK )Br the Brauer-Manin set. Take mv ∈ X(AK )Br, lift it to an adelic point Pv on Y. Under what assumptions on mv will it be liftable to Pv on Y (AK )Br,or at least Y (AK )Br 1?",
   "public_statement": "Question 8. Let X an Enriques surface over a number field K, Y p\n\n→ X the double cover (K3 surface), X(AK )Br the Brauer-Manin set. Take mv ∈ X(AK )Br, lift it to an adelic point Pv on Y. Under what assumptions on mv will it be liftable to Pv on Y (AK )Br,or at least Y (AK )Br 1?",
   "evidence": "This record is Question 8 from the AIM workshop *Rational and integral points on higher dimensional varieties*. The canonical JSON extraction has lost an arrow, spaces, and some superscripts. Comparison with the official AIM PDF and HTML version gives the following mathematical statement (the bracketed word only repairs the grammar of the source):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 91,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0093": {
   "statement_status": "exact",
   "original_statement": "Question 9. Use the intermediate Jacobian (when it is an abelian variety) in arithmetic? To fix ideas, dim X = 3, look at algebraic 1-cycles modulo rational equivalence. Compute this for a rigid Calabi-Yau 3-fold X (over C), i.e. KX ∼ 0, X simply con-nected, h1,0 = 0, h2,0 = 0, h2,1 = 0, h3,0 = 1. In this case, the intermediate Jacobian \n\nJ2(X) ∼= E is of dimension 1. Determine E given X, i.e. give its j-invariant.",
   "clean_statement": "Question 9. Use the intermediate Jacobian (when it is an abelian variety) in arithmetic? To fix ideas, dim X = 3, look at algebraic 1-cycles modulo rational equivalence. Compute this for a rigid Calabi-Yau 3-fold X (over C), i.e. KX ∼ 0, X simply con-nected, h1,0 = 0, h2,0 = 0, h2,1 = 0, h3,0 = 1. In this case, the intermediate Jacobian\n\nJ2(X) ∼= E is of dimension 1. Determine E given X, i.e. give its j-invariant.",
   "public_statement": "Question 9. Use the intermediate Jacobian (when it is an abelian variety) in arithmetic? To fix ideas, dim X = 3, look at algebraic 1-cycles modulo rational equivalence. Compute this for a rigid Calabi-Yau 3-fold X (over C), i.e. KX ∼ 0, X simply con-nected, h1,0 = 0, h2,0 = 0, h2,1 = 0, h3,0 = 1. In this case, the intermediate Jacobian\n\nJ2(X) ∼= E is of dimension 1. Determine E given X, i.e. give its j-invariant.",
   "evidence": "The canonical record is Question 9 in the AIM workshop list *Rational and integral points on higher dimensional varieties*. The source PDF was checked directly (version dated 22 November 2004). With superscripts and line breaks restored, the question reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0094": {
   "statement_status": "exact",
   "original_statement": "Question 10. Are there results at the level of number fields arising from the techniques of rationally connected varieties? (Colliot-Th´ el` ene) For example, recently Koll` ar got nice results over local fields: if X/k is a projective variety, we say P, Q ∈ X(k) are R-equivalent if you can link them by a chain of curves of genus zero over k; if X is smooth with k ↪ → C, then X is rationally connected if X(C)/R is a single point. Kollar proved that if k is a local field, and X/k is rationally connected, then \n\nX(k)/R is finite. (Szob´ o) Koll` ar and Szabo proved that if k is a number field, and X/k is rationally connected, then X(kv)/R is trivial for almost all v.",
   "clean_statement": "Question 10. Are there results at the level of number fields arising from the techniques of rationally connected varieties? (Colliot-Th´ el` ene) For example, recently Koll` ar got nice results over local fields: if X/k is a projective variety, we say P, Q ∈ X(k) are R-equivalent if you can link them by a chain of curves of genus zero over k; if X is smooth with k ↪ → C, then X is rationally connected if X(C)/R is a single point. Kollar proved that if k is a local field, and X/k is rationally connected, then\n\nX(k)/R is finite. (Szob´ o) Koll` ar and Szabo proved that if k is a number field, and X/k is rationally connected, then X(kv)/R is trivial for almost all v.",
   "public_statement": "Question 10. Are there results at the level of number fields arising from the techniques of rationally connected varieties? (Colliot-Th´ el` ene) For example, recently Koll` ar got nice results over local fields: if X/k is a projective variety, we say P, Q ∈ X(k) are R-equivalent if you can link them by a chain of curves of genus zero over k; if X is smooth with k ↪ → C, then X is rationally connected if X(C)/R is a single point. Kollar proved that if k is a local field, and X/k is rationally connected, then\n\nX(k)/R is finite. (Szob´ o) Koll` ar and Szabo proved that if k is a number field, and X/k is rationally connected, then X(kv)/R is trivial for almost all v.",
   "evidence": "The canonical record is Question 10 from the AIM workshop *Rational and integral points on higher dimensional varieties*. The PDF extraction contains several OCR errors: “\\(k\\hookrightarrow\\mathbb C\\)” became a malformed arrow, “Szabó” became “Szobó,” and \\(\\mathbb P^2\\) was split across two lines. The AIM HTML version and the workshop PDF support the following recovered text.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0095": {
   "statement_status": "exact",
   "original_statement": "Question 11. Let X/k be a smooth projective rationally connected variety with the cohomological dimension of k ≤ 1. Does X have a rational point? (Colliot-Th´ el` ene)",
   "clean_statement": "Question 11. Let X/k be a smooth projective rationally connected variety with the cohomological dimension of k ≤ 1. Does X have a rational point? (Colliot-Th´ el` ene)",
   "public_statement": "Question 11. Let X/k be a smooth projective rationally connected variety with the cohomological dimension of k ≤ 1. Does X have a rational point? (Colliot-Th´ el` ene)",
   "evidence": "The canonical record is Question 11 from the AIM workshop *Rational and integral points on higher dimensional varieties*. Its exact extracted problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 94,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0096": {
   "statement_status": "exact",
   "original_statement": "Problem 11, as phrased, had been settled by J. Ax in Bull. Am. Math. Soc. 71 (1965) p. 717. Ax produced a smooth hypersurface in 9-dimensional projective space, of degree 5, over a field k of cohomological dimension 1, with no rational point (that such hypersurfaces are rationally connected was proven much later). Ax'example has index 1, i.e. the g.c.d. of the degree of the finite field extensions over which the hypersurface acquires a rational point is 1. In Journal of the Inst. of Math. Jussieu (2004) 3 p. 1-16, J.-L. Colliot-Th´ el` ene and D. Madore produce a field of cohomological dimension 1 and a smooth cubic surface over that field which has index 3, thus settling negatively a question of Kato and Kuzumaki (1986). 50 \n\nProblem/",
   "clean_statement": "Problem 11, as phrased, had been settled by J. Ax in Bull. Am. Math. Soc. 71 (1965) p. 717. Ax produced a smooth hypersurface in 9-dimensional projective space, of degree 5, over a field k of cohomological dimension 1, with no rational point (that such hypersurfaces are rationally connected was proven much later). Ax'example has index 1, i.e. the g.c.d. of the degree of the finite field extensions over which the hypersurface acquires a rational point is 1. In Journal of the Inst. of Math. Jussieu (2004) 3 p. 1-16, J.-L. Colliot-Th´ el` ene and D. Madore produce a field of cohomological dimension 1 and a smooth cubic surface over that field which has index 3, thus settling negatively a question of Kato and Kuzumaki (1986). 50\n\nProblem/",
   "public_statement": "Problem 11, as phrased, had been settled by J. Ax in Bull. Am. Math. Soc. 71 (1965) p. 717. Ax produced a smooth hypersurface in 9-dimensional projective space, of degree 5, over a field k of cohomological dimension 1, with no rational point (that such hypersurfaces are rationally connected was proven much later). Ax'example has index 1, i.e. the g.c.d. of the degree of the finite field extensions over which the hypersurface acquires a rational point is 1. In Journal of the Inst. of Math. Jussieu (2004) 3 p. 1-16, J.-L. Colliot-Th´ el` ene and D. Madore produce a field of cohomological dimension 1 and a smooth cubic surface over that field which has index 3, thus settling negatively a question of Kato and Kuzumaki (1986). 50\n\nProblem/",
   "evidence": "This placement was checked in the official PDF: the paragraph is remark (iv) on printed page 50, immediately before Question 12. The isolated terminal text “\\(50\\) Problem/” in the extracted record is a page-number/header artifact. The raw OCR record is preserved verbatim in *input.json*; accents, punctuation, and line breaks above were restored from the official source.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0097": {
   "statement_status": "exact",
   "original_statement": "Question 12. Describe all pairs of sets A, B ⊂ μ∞, A = B, stable under Gal( Q/Q), where the elements of A and B are alternately placed around the unit circle. (Rodriguez-Villegas)",
   "clean_statement": "Question 12. Describe all pairs of sets A, B ⊂ μ∞, A = B, stable under Gal( Q/Q), where the elements of A and B are alternately placed around the unit circle. (Rodriguez-Villegas)",
   "public_statement": "Question 12. Describe all pairs of sets A, B ⊂ μ∞, A = B, stable under Gal( Q/Q), where the elements of A and B are alternately placed around the unit circle. (Rodriguez-Villegas)",
   "evidence": "This attempt concerns exactly record AIM-ALGEBRAIC_NUMBER_THEORY-0097, Question 12 of the AIM list *Rational and integral points on higher dimensional varieties* (Fernando Rodriguez-Villegas), stored at zero-based index 96 of aim-algebraic-number-theory-notes.json.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 96,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0098": {
   "statement_status": "exact",
   "original_statement": "Question 13. Find a smooth quintic hypersurface in P3 with Picard number 1 over F\n2. (Voloch)",
   "clean_statement": "Question 13. Find a smooth quintic hypersurface in P3 with Picard number 1 over F\n2. (Voloch)",
   "public_statement": "Question 13. Find a smooth quintic hypersurface in P3 with Picard number 1 over F\n2. (Voloch)",
   "evidence": "The canonical record is Question 13 from the AIM workshop *Rational and integral points on higher dimensional varieties*. The extracted text has line-break damage in the finite-field subscripts. The official AIM HTML and PDF give the unambiguous statement",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 97,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0099": {
   "statement_status": "exact",
   "original_statement": "Question 14. Consider cubic hypersurfaces a0X30 + a1X31 + a2X32 + a3X33 = \n0. (Swinnerton-Dyer) (a) If soluble, give upper bound for smallest solution in terms of the ai.(b) Look at |ai| < A, tabulate the size of the smallest solution and conjecture a particular growth rate in terms of A.",
   "clean_statement": "Question 14. Consider cubic hypersurfaces a0X30 + a1X31 + a2X32 + a3X33 =\n0. (Swinnerton-Dyer) (a) If soluble, give upper bound for smallest solution in terms of the ai.(b) Look at |ai| < A, tabulate the size of the smallest solution and conjecture a particular growth rate in terms of A.",
   "public_statement": "Question 14. Consider cubic hypersurfaces a0X30 + a1X31 + a2X32 + a3X33 =\n0. (Swinnerton-Dyer) (a) If soluble, give upper bound for smallest solution in terms of the ai.(b) Look at |ai| < A, tabulate the size of the smallest solution and conjecture a particular growth rate in terms of A.",
   "evidence": "The canonical record is Question 14 from the AIM workshop *Rational and integral points on higher dimensional varieties*, source file `aim-algebraic-number-theory-notes.json`, zero-based record index 98. The exact extracted `problem` field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 98,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0100": {
   "statement_status": "exact",
   "original_statement": "Question 15. Characterize the rational numbers α that can be written as \n\nα = 2\n\nx21 − 2 + 2\n\nx22 − 2 + · · · + 2\n\nx2 \n\n> n\n\n− 2for xi ∈ Q and fixed n. (Poonen) It is necessary that |α|p ≤ 1 for (2 /p ) = −1 and p = 2. Is it sufficient? 51",
   "clean_statement": "Question 15. Characterize the rational numbers α that can be written as\n\nα = 2\n\nx21 − 2 + 2\n\nx22 − 2 + · · · + 2\n\nx2\n\n> n\n\n− 2for xi ∈ Q and fixed n. (Poonen) It is necessary that |α|p ≤ 1 for (2 /p ) = −1 and p = 2. Is it sufficient? 51",
   "public_statement": "Question 15. Characterize the rational numbers α that can be written as\n\nα = 2\n\nx21 − 2 + 2\n\nx22 − 2 + · · · + 2\n\nx2\n\n> n\n\n− 2for xi ∈ Q and fixed n. (Poonen) It is necessary that |α|p ≤ 1 for (2 /p ) = −1 and p = 2. Is it sufficient? 51",
   "evidence": "The canonical JSON problem field is OCR-damaged. Reproduced verbatim, it is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 99,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0101": {
   "statement_status": "exact",
   "original_statement": "Question 16. Solve the local-global solubility problem for finding lines on a cubic hypersurface. (Wooley) (a) Find an example of a cubic hypersurface over Qp (in as many variables as possible) with no rational line. (b) Find an example of a cubic hypersurface over Q (in as many variables as possible) with no rational line.",
   "clean_statement": "Question 16. Solve the local-global solubility problem for finding lines on a cubic hypersurface. (Wooley) (a) Find an example of a cubic hypersurface over Qp (in as many variables as possible) with no rational line. (b) Find an example of a cubic hypersurface over Q (in as many variables as possible) with no rational line.",
   "public_statement": "Question 16. Solve the local-global solubility problem for finding lines on a cubic hypersurface. (Wooley) (a) Find an example of a cubic hypersurface over Qp (in as many variables as possible) with no rational line. (b) Find an example of a cubic hypersurface over Q (in as many variables as possible) with no rational line.",
   "evidence": "The question mark after \\(37\\) occurs in both official versions and is not an extraction error. The number \\(37\\) was subsequently a published theorem of Wooley. In contrast, I did not locate a published theorem supporting the workshop's \\(14\\)-variable local assertion; it is incompatible with the way the later general local bounds are presented, so it is treated below as an unverified workshop remark rather than as an established bound.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 100,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0102": {
   "statement_status": "exact",
   "original_statement": "Question 17. Draw a regular pentagon P, construct the circle through the 5 vertices C, and consider the curve E: P + λC 2 = 0. This is a quintic curve with 5 double points, and therefore has geometric genus 1. This curve has five points at ∞ (given by the slopes of the lines). Compute the 5-torsion of E. (McCallum)",
   "clean_statement": "Question 17. Draw a regular pentagon P, construct the circle through the 5 vertices C, and consider the curve E: P + λC 2 = 0. This is a quintic curve with 5 double points, and therefore has geometric genus 1. This curve has five points at ∞ (given by the slopes of the lines). Compute the 5-torsion of E. (McCallum)",
   "public_statement": "Question 17. Draw a regular pentagon P, construct the circle through the 5 vertices C, and consider the curve E: P + λC 2 = 0. This is a quintic curve with 5 double points, and therefore has geometric genus 1. This curve has five points at ∞ (given by the slopes of the lines). Compute the 5-torsion of E. (McCallum)",
   "evidence": "The canonical record is Question 17 from the AIM workshop *Rational and integral points on higher dimensional varieties*, source file aim-algebraic-number-theory-notes.json, zero-based record index 101. The extracted problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 101,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0103": {
   "statement_status": "exact",
   "original_statement": "Question 18. \n\n(a) Find a separable polynomial g(t) ∈ Q[t] such that the Jacobian of the hyperelliptic curve \n\ns2 = g(t) is isogeneous over Q to Er × B, E an elliptic curve, with r ≥ 4 or r = 3 and dim B ≤ \n1. (Silverberg) (b) Related problem: Give an example of a map P1 → E4/{± 1}, where ±1 acts diagonally, whose image does not lie in H/ {± 1} for any subgroup H ( E\n4. (Ellenberg)",
   "clean_statement": "Question 18.\n\n(a) Find a separable polynomial g(t) ∈ Q[t] such that the Jacobian of the hyperelliptic curve\n\ns2 = g(t) is isogeneous over Q to Er × B, E an elliptic curve, with r ≥ 4 or r = 3 and dim B ≤\n1. (Silverberg) (b) Related problem: Give an example of a map P1 → E4/{± 1}, where ±1 acts diagonally, whose image does not lie in H/ {± 1} for any subgroup H ( E\n4. (Ellenberg)",
   "public_statement": "Question 18.\n\n(a) Find a separable polynomial g(t) ∈ Q[t] such that the Jacobian of the hyperelliptic curve\n\ns2 = g(t) is isogeneous over Q to Er × B, E an elliptic curve, with r ≥ 4 or r = 3 and dim B ≤\n1. (Silverberg) (b) Related problem: Give an example of a map P1 → E4/{± 1}, where ±1 acts diagonally, whose image does not lie in H/ {± 1} for any subgroup H ( E\n4. (Ellenberg)",
   "evidence": "The canonical record is Question 18 from the AIM workshop *Rational and integral points on higher dimensional varieties*. The JSON extraction loses superscripts and the strict-containment symbol. Comparison with the official HTML and PDF gives the following statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 102,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0104": {
   "statement_status": "exact",
   "original_statement": "Question 19. Let X be a variety over a number field k and suppose that for every open Zariski dense U ⊂ X, the map π1(U ) → Gal( k/k ) has a splitting s (e.g. if X(k)is Zariski dense). In this case, if X(Ak) 6 = ∅, is there no Brauer-Manin obstruction to the Hasse principle for X, i.e. is X(Ak)Br 6 = ∅? (Ellenberg)",
   "clean_statement": "Question 19. Let X be a variety over a number field k and suppose that for every open Zariski dense U ⊂ X, the map π1(U ) → Gal( k/k ) has a splitting s (e.g. if X(k)is Zariski dense). In this case, if X(Ak) 6 = ∅, is there no Brauer-Manin obstruction to the Hasse principle for X, i.e. is X(Ak)Br 6 = ∅? (Ellenberg)",
   "public_statement": "Question 19. Let X be a variety over a number field k and suppose that for every open Zariski dense U ⊂ X, the map π1(U ) → Gal( k/k ) has a splitting s (e.g. if X(k)is Zariski dense). In this case, if X(Ak) 6 = ∅, is there no Brauer-Manin obstruction to the Hasse principle for X, i.e. is X(Ak)Br 6 = ∅? (Ellenberg)",
   "evidence": "The canonical JSON field is OCR-damaged. Reproduced verbatim, it reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 103,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0105": {
   "statement_status": "exact",
   "original_statement": "Question 20. Let p(t) = 3( t4 − 54 t2 − 117 t − 243). Write the system of two equations y2 = p(t)( x2 + 1), z2 = p(t)( x2 + 2); think of this as a one-parameter family X of curves of genus one; X(Q) is empty and X(AQ)Br 6 = ∅ (Skorobogatov). Is X(k) 6 = ∅ for some [ k: Q] odd? (Colliot-Th´ el` ene)",
   "clean_statement": "Question 20. Let p(t) = 3( t4 − 54 t2 − 117 t − 243). Write the system of two equations y2 = p(t)( x2 + 1), z2 = p(t)( x2 + 2); think of this as a one-parameter family X of curves of genus one; X(Q) is empty and X(AQ)Br 6 = ∅ (Skorobogatov). Is X(k) 6 = ∅ for some [ k: Q] odd? (Colliot-Th´ el` ene)",
   "public_statement": "Question 20. Let p(t) = 3( t4 − 54 t2 − 117 t − 243). Write the system of two equations y2 = p(t)( x2 + 1), z2 = p(t)( x2 + 2); think of this as a one-parameter family X of curves of genus one; X(Q) is empty and X(AQ)Br 6 = ∅ (Skorobogatov). Is X(k) 6 = ∅ for some [ k: Q] odd? (Colliot-Th´ el` ene)",
   "evidence": "Put \\[ g(T)=3(T^4-54T^2-117T-243),\\qquad p(X)=X^2+1,\\qquad q(X)=X^2+2. \\] The official AIM workshop PDF, *Rational and integral points on higher dimensional varieties*, Question 20, asks whether the surface with affine open \\[ S_{\\mathrm{AIM}}:\\qquad y^2=g(t)p(x),\\qquad z^2=g(t)q(x) \\tag{1} \\] has a point over some odd-degree number field. The extracted strings `t4`, `y2`, and `6 =` are OCR renderings of fourth powers, squares, and `\\(\\ne\\)`. The question attributes to Skorobogatov the facts that a smooth projective model \\(S/\\mathbb Q\\) has \\(S(\\mathbb Q)=\\varnothing\\) and \\(S(\\mathbb A_{\\mathbb Q})^{\\mathrm{Br}}\\ne\\varnothing\\).",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 104,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0106": {
   "statement_status": "exact",
   "original_statement": "Question 21. Let A/ Z be an algebra with rk Z A = 9 given by its multiplica-tion table. Suppose you know that A ' M3(Z); find an algorithm which gives an explicit isomorphism. (Stoll)",
   "clean_statement": "Question 21. Let A/ Z be an algebra with rk Z A = 9 given by its multiplica-tion table. Suppose you know that A ' M3(Z); find an algorithm which gives an explicit isomorphism. (Stoll)",
   "public_statement": "Question 21. Let A/ Z be an algebra with rk Z A = 9 given by its multiplica-tion table. Suppose you know that A ' M3(Z); find an algorithm which gives an explicit isomorphism. (Stoll)",
   "evidence": "The canonical record is Question 21 from the AIM workshop *Rational and integral points on higher dimensional varieties*. The JSON extraction loses mathematical typography and inserts a line-break hyphen. The official AIM HTML and PDF give the unambiguous statement:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 105,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0107": {
   "statement_status": "exact",
   "original_statement": "Question 22. Let k be a number field, k 6 ⊂ R. Construct an algebraic set S ⊂ kn\n\nfor some n such that the projection onto one of the coordinates is exactly the set of elements of k with |k| ≤ 1 for some (archimedean) absolute value | | of k. (Shlapentokh)",
   "clean_statement": "Question 22. Let k be a number field, k 6 ⊂ R. Construct an algebraic set S ⊂ kn\n\nfor some n such that the projection onto one of the coordinates is exactly the set of elements of k with |k| ≤ 1 for some (archimedean) absolute value | | of k. (Shlapentokh)",
   "public_statement": "Question 22. Let k be a number field, k 6 ⊂ R. Construct an algebraic set S ⊂ kn\n\nfor some n such that the projection onto one of the coordinates is exactly the set of elements of k with |k| ≤ 1 for some (archimedean) absolute value | | of k. (Shlapentokh)",
   "evidence": "The canonical record is Question 22 from the AIM workshop *Rational and integral points on higher dimensional varieties*. The official AIM PDF and HTML page both print:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 106,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0108": {
   "statement_status": "corrected_verified",
   "original_statement": "Question 23. Let E ⊂ P2 be an elliptic curve over Q, and suppose E(Q) ' Z.(Poonen) (a) Describe S ⊂ E(Q) where S = {(x, y ): y = a2 + b2, a, b ∈ Q}. Is S is finite? (b) More generally, π: X → E gives a subset π(X(Q)) ⊂ E(Q); what others can you build?",
   "clean_statement": "**Question 23 (Poonen).** Let \\(E\\subset \\mathbf P^2\\) be an elliptic curve over \\(\\mathbf Q\\), and suppose \\(E(\\mathbf Q)\\simeq \\mathbf Z\\).\n\n(a) Describe \\(S\\subset E(\\mathbf Q)\\), where\n\\[\nS=\\{(x,y):y=a^2+b^2,\\ a,b\\in\\mathbf Q\\}.\n\\]\nIs \\(S\\) finite?\n\n(b) More generally, \\(\\pi:X\\to E\\) gives a subset \\(\\pi(X(\\mathbf Q))\\subset E(\\mathbf Q)\\); what others can you build?",
   "public_statement": "**Question 23 (Poonen).** Let \\(E\\subset \\mathbf P^2\\) be an elliptic curve over \\(\\mathbf Q\\), and suppose \\(E(\\mathbf Q)\\simeq \\mathbf Z\\).\n\n(a) Describe \\(S\\subset E(\\mathbf Q)\\), where\n\\[\nS=\\{(x,y):y=a^2+b^2,\\ a,b\\in\\mathbf Q\\}.\n\\]\nIs \\(S\\) finite?\n\n(b) More generally, \\(\\pi:X\\to E\\) gives a subset \\(\\pi(X(\\mathbf Q))\\subset E(\\mathbf Q)\\); what others can you build?",
   "evidence": "The canonical record is Question 23 in the AIM problem list *Rational and integral points on higher dimensional varieties*. The supplied JSON has OCR losses (`P2`, `a2`, and an apostrophe in place of an isomorphism sign). The official PDF reads:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 107,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0109": {
   "statement_status": "exact",
   "original_statement": "Question 24. Given an elliptic curve E/ Q, rk E(Q) > 0, describe the set of primes p such that E(Q) is dense in E(Qp). Is this set nonempty? Is there a modular interpretation of this problem? (Takloo-Bighash)",
   "clean_statement": "Question 24. Given an elliptic curve E/ Q, rk E(Q) > 0, describe the set of primes p such that E(Q) is dense in E(Qp). Is this set nonempty? Is there a modular interpretation of this problem? (Takloo-Bighash)",
   "public_statement": "Question 24. Given an elliptic curve E/ Q, rk E(Q) > 0, describe the set of primes p such that E(Q) is dense in E(Qp). Is this set nonempty? Is there a modular interpretation of this problem? (Takloo-Bighash)",
   "evidence": "This is attempt 1 for record **AIM-ALGEBRAIC_NUMBER_THEORY-0109**, source file *aim-algebraic-number-theory-notes.json*, zero-based source index 108. The record is Question 24 in the AIM workshop document *Rational and integral points on higher dimensional varieties*. I checked the official PDF directly (page 52). Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 108,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0110": {
   "statement_status": "exact",
   "original_statement": "Question 25. Is there an algorithm to decide solubility of a system of linear equations \n\nai1x1 + · · · + ain xn = bi\n\nfor ai, b, x i ∈ Z, together with equations of the form \n\n{xiαj } < r j\n\nfor αj ∈ R \\ Q, ri ∈ Q? (Poonen)",
   "clean_statement": "Question 25. Is there an algorithm to decide solubility of a system of linear equations\n\nai1x1 + · · · + ain xn = bi\n\nfor ai, b, x i ∈ Z, together with equations of the form\n\n{xiαj } < r j\n\nfor αj ∈ R \\ Q, ri ∈ Q? (Poonen)",
   "public_statement": "Question 25. Is there an algorithm to decide solubility of a system of linear equations\n\nai1x1 + · · · + ain xn = bi\n\nfor ai, b, x i ∈ Z, together with equations of the form\n\n{xiαj } < r j\n\nfor αj ∈ R \\ Q, ri ∈ Q? (Poonen)",
   "evidence": "The canonical record is Question 25 from the AIM workshop *Rational and integral points on higher dimensional varieties*. The official HTML and PDF give the question as follows (with subscripts normalized typographically but not mathematically altered):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 109,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0111": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 26. Describe the variety of curves of low degree on the Fermat variety \n\nF: xd \n\n> 1\n\n+ xd \n\n> 2\n\n+ · · · + xd \n\n> 6\n\n= \n0. (Heath-Brown)",
   "clean_statement": null,
   "public_statement": "Question 26. Describe the variety of curves of low degree on the Fermat variety\n\nF: xd\n\n> 1\n\n+ xd\n\n> 2\n\n+ · · · + xd\n\n> 6\n\n=\n0. (Heath-Brown)",
   "evidence": "The canonical JSON record is affected by PDF extraction errors in the exponents. The official AIM workshop PDF gives the following statement (Question 26, page 53 of the PDF):",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 110,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0112": {
   "statement_status": "exact",
   "original_statement": "Question 27. Given a plane curve of degree d, the number of points on X of height at most d is ø d3. There are curves which have the number of rational points ¿ d2.Can you do better than ø d3? (Heath-Brown)",
   "clean_statement": "Question 27. Given a plane curve of degree d, the number of points on X of height at most d is ø d3. There are curves which have the number of rational points ¿ d2.Can you do better than ø d3? (Heath-Brown)",
   "public_statement": "Question 27. Given a plane curve of degree d, the number of points on X of height at most d is ø d3. There are curves which have the number of rational points ¿ d2.Can you do better than ø d3? (Heath-Brown)",
   "evidence": "Thus the repeated use of \\(d\\), both as degree and height cutoff, is genuine; it is not an extraction error. The corrupted symbols “ø” and “¿” in the canonical JSON are respectively \\(\\ll\\) and \\(\\gg\\), and the exponents are \\(3,2,5/2\\).",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 111,
   "attempt": 2
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0113": {
   "statement_status": "exact",
   "original_statement": "Question 28. Let E be an elliptic curve over Q, and P ∈ E(Q)[3], P 6 = O,such that rk E(Q) = 0. Let K/ Q be a cyclic extension of degree 3 of conductor f, and let \n\nν = ν(f ) be the number of distinct primes dividing f. It is true that 3 ν−1 | Sel 3(E, K )? (Chantal David)",
   "clean_statement": "Question 28. Let E be an elliptic curve over Q, and P ∈ E(Q)[3], P 6 = O,such that rk E(Q) = 0. Let K/ Q be a cyclic extension of degree 3 of conductor f, and let\n\nν = ν(f ) be the number of distinct primes dividing f. It is true that 3 ν−1 | Sel 3(E, K )? (Chantal David)",
   "public_statement": "Question 28. Let E be an elliptic curve over Q, and P ∈ E(Q)[3], P 6 = O,such that rk E(Q) = 0. Let K/ Q be a cyclic extension of degree 3 of conductor f, and let\n\nν = ν(f ) be the number of distinct primes dividing f. It is true that 3 ν−1 | Sel 3(E, K )? (Chantal David)",
   "evidence": "The canonical JSON record is an OCR extraction of Question 28 from the AIM workshop list *Rational and integral points on higher dimensional varieties*. The OCR lost the cardinality sign in the displayed divisibility. The official AIM HTML and PDF were checked and give the following question (with notation modernized only in the subscript/base-field placement):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 112,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0114": {
   "statement_status": "exact",
   "original_statement": "Question 29. If K is a field such that all O-acyclic varieties over K have a point, is Gal( K/K ) topologically generated by one element? \n\nProblem/",
   "clean_statement": "Question 29. If K is a field such that all O-acyclic varieties over K have a point, is Gal( K/K ) topologically generated by one element?\n\nProblem/",
   "public_statement": "Question 29. If K is a field such that all O-acyclic varieties over K have a point, is Gal( K/K ) topologically generated by one element?\n\nProblem/",
   "evidence": "The canonical record is Question 29 in the AIM workshop notes *Rational and integral points on higher dimensional varieties*. The extracted text reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 113,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0115": {
   "statement_status": "exact",
   "original_statement": "Question 30. This problem has been withdrawn. \n\nProblem/",
   "clean_statement": "Question 30. This problem has been withdrawn.\n\nProblem/",
   "public_statement": "Question 30. This problem has been withdrawn.\n\nProblem/",
   "evidence": "The exact extracted canonical field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 114,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0116": {
   "statement_status": "exact",
   "original_statement": "Question 31. Harmonic analysis and nonabelian torsors: Takloo-Bighash: Do these give you methods to find rational points? Hassett: Is there a harmonic analysis argument to count points on the quintic del Pezzo surface? Wooley: no examples known where you can combine harmonic analysis with torsors. \n\nProblem/",
   "clean_statement": "Question 31. Harmonic analysis and nonabelian torsors: Takloo-Bighash: Do these give you methods to find rational points? Hassett: Is there a harmonic analysis argument to count points on the quintic del Pezzo surface? Wooley: no examples known where you can combine harmonic analysis with torsors.\n\nProblem/",
   "public_statement": "Question 31. Harmonic analysis and nonabelian torsors: Takloo-Bighash: Do these give you methods to find rational points? Hassett: Is there a harmonic analysis argument to count points on the quintic del Pezzo surface? Wooley: no examples known where you can combine harmonic analysis with torsors.\n\nProblem/",
   "evidence": "The canonical JSON `problem` field is Question 31 in the AIM workshop notes *Rational and integral points on higher dimensional varieties*, followed by the stray text `Problem/` after a blank line. The official AIM PDF and HTML show that this suffix is a page-extraction/navigation artifact, not part of the question. The recovered question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 115,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0117": {
   "statement_status": "exact",
   "original_statement": "Question \n3\n2. (Voloch): Given a family of hypersurfaces, show that \"almost all\" members of the family satisfy the Hasse Principle in \"interesting\" circumstances. Colliot-Th´ el` ene: maybe the opposite? Poonen: the family was all hypersurfaces of degree d in Pn with coefficients of height \n\n≤ H, over some fixed number field K. The problem was: find the proportion of these that satisfy the Hasse Principle (originally, that have rational points, but we can find the local points easily). Wooley and Venkatesh (tentative): \n\nA−s#{(a1,..., a s): |ai| ≤ A, a 1xd \n\n> 1\n\n+ · · · + asxds = 0 satisfies HP }\n\nis asymptotic to the product of local densities, at least for s ≥ 3d + 1. This is progress for \n\nd large: for an individual a, one has HP (using current technology) for s >> d log d. Proof uses (of course) circle method. de Jong: for cubics in P2, what do conclusions of this type say about 3-Selmer groups of elliptic curves? 55 \n\nStoll: given local points, proportion that have rational points (i.e., satisfy HP) should be 0. Mazur: can you put an exponent on that? Stoll: wait for experimental evidence. \n\nProblem/",
   "clean_statement": "Question\n3\n2. (Voloch): Given a family of hypersurfaces, show that \"almost all\" members of the family satisfy the Hasse Principle in \"interesting\" circumstances. Colliot-Th´ el` ene: maybe the opposite? Poonen: the family was all hypersurfaces of degree d in Pn with coefficients of height\n\n≤ H, over some fixed number field K. The problem was: find the proportion of these that satisfy the Hasse Principle (originally, that have rational points, but we can find the local points easily). Wooley and Venkatesh (tentative):\n\nA−s#{(a1,..., a s): |ai| ≤ A, a 1xd\n\n> 1\n\n+ · · · + asxds = 0 satisfies HP }\n\nis asymptotic to the product of local densities, at least for s ≥ 3d + 1. This is progress for\n\nd large: for an individual a, one has HP (using current technology) for s >> d log d. Proof uses (of course) circle method. de Jong: for cubics in P2, what do conclusions of this type say about 3-Selmer groups of elliptic curves? 55\n\nStoll: given local points, proportion that have rational points (i.e., satisfy HP) should be 0. Mazur: can you put an exponent on that? Stoll: wait for experimental evidence.\n\nProblem/",
   "public_statement": "Question\n3\n2. (Voloch): Given a family of hypersurfaces, show that \"almost all\" members of the family satisfy the Hasse Principle in \"interesting\" circumstances. Colliot-Th´ el` ene: maybe the opposite? Poonen: the family was all hypersurfaces of degree d in Pn with coefficients of height\n\n≤ H, over some fixed number field K. The problem was: find the proportion of these that satisfy the Hasse Principle (originally, that have rational points, but we can find the local points easily). Wooley and Venkatesh (tentative):\n\nA−s#{(a1,..., a s): |ai| ≤ A, a 1xd\n\n> 1\n\n+ · · · + asxds = 0 satisfies HP }\n\nis asymptotic to the product of local densities, at least for s ≥ 3d + 1. This is progress for\n\nd large: for an individual a, one has HP (using current technology) for s >> d log d. Proof uses (of course) circle method. de Jong: for cubics in P2, what do conclusions of this type say about 3-Selmer groups of elliptic curves? 55\n\nStoll: given local points, proportion that have rational points (i.e., satisfy HP) should be 0. Mazur: can you put an exponent on that? Stoll: wait for experimental evidence.\n\nProblem/",
   "evidence": "The canonical JSON record is damaged by line-break/OCR errors: its number appears as `3\\n2`, one exponent is detached, and a page number is inserted into the sentence. The official AIM HTML version identifies it unambiguously as **Problem/Question 32** from the workshop *Rational and integral points on higher dimensional varieties* [AIM]. In cleaned mathematical notation, the record asks three related but distinct questions.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 116,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0118": {
   "statement_status": "exact",
   "original_statement": "Question \n3\n3. (Poonen) Instead of counting points of bounded height on a va-riety, count points of bounded height in a Diophantine set. (I.e., counting points in a base the fibre above which has a rational point.) What rates of growth can you get? Example: conic bundle over an elliptic curve. Heuristically, it appears rate of growth can be log log B,whereas for varieties it always turns out to be cB α(log B)β.\n\nProblem/",
   "clean_statement": "Question\n3\n3. (Poonen) Instead of counting points of bounded height on a va-riety, count points of bounded height in a Diophantine set. (I.e., counting points in a base the fibre above which has a rational point.) What rates of growth can you get? Example: conic bundle over an elliptic curve. Heuristically, it appears rate of growth can be log log B,whereas for varieties it always turns out to be cB α(log B)β.\n\nProblem/",
   "public_statement": "Question\n3\n3. (Poonen) Instead of counting points of bounded height on a va-riety, count points of bounded height in a Diophantine set. (I.e., counting points in a base the fibre above which has a rational point.) What rates of growth can you get? Example: conic bundle over an elliptic curve. Heuristically, it appears rate of growth can be log log B,whereas for varieties it always turns out to be cB α(log B)β.\n\nProblem/",
   "evidence": "The canonical record is zero-based index 117 of `aim-algebraic-number-theory-notes.json`. Its OCR has split the problem number as `3\\n3`, inserted the line-break hyphen `va-riety`, and left a trailing `Problem/`. The AIM PDF and HTML version both give the following recovered text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 117,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0119": {
   "statement_status": "exact",
   "original_statement": "Question \n3\n4. (Bogomolov) Replace Heath-Brown by Arakelov (et al). #{(x, y, z ) ∈ P2: |x|, |y|, |z| ≤ B, P (x, y, z ) = 0 } << B α+≤\n\nwith α ≤ 2 \n\n> d, d = deg P. Analogue for surfaces, etc. \n\nProblem/",
   "clean_statement": "Question\n3\n4. (Bogomolov) Replace Heath-Brown by Arakelov (et al). #{(x, y, z ) ∈ P2: |x|, |y|, |z| ≤ B, P (x, y, z ) = 0 } << B α+≤\n\nwith α ≤ 2\n\n> d, d = deg P. Analogue for surfaces, etc.\n\nProblem/",
   "public_statement": "Question\n3\n4. (Bogomolov) Replace Heath-Brown by Arakelov (et al). #{(x, y, z ) ∈ P2: |x|, |y|, |z| ≤ B, P (x, y, z ) = 0 } << B α+≤\n\nwith α ≤ 2\n\n> d, d = deg P. Analogue for surfaces, etc.\n\nProblem/",
   "evidence": "The canonical record is zero-based index 118 of `aim-algebraic-number-theory-notes.json`, extracted from the AIM workshop *Rational and integral points on higher dimensional varieties*. The record is visibly damaged: its number is split as `3\\n4`, exponents and inequality signs are displaced, and the final `Problem/` is HTML navigation text. The official AIM HTML and page 55 of the official PDF give the following unambiguous text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 118,
   "attempt": 1
  },
  "AIM-ALGEBRAIC_NUMBER_THEORY-0120": {
   "statement_status": "exact",
   "original_statement": "Question 35. Weak approximation for complex function fields: Weak approximation over a finite extension L of C(t), from Hassett's lecture. Colliot-Th´ el` ene: Known for X a connected linear algebraic group over L (reduce to reductive groups immediately; since field has cohomological dimension 1, one has Borel subgroup, reduce to tori; reduce to quasi-trivial tori, which are open subsets of affine space). CT: X = G/H, where H is a subgroup of G (not necessarily normal). Same should go through if H is connected (techniques of Borovoi et al.). CT: Question: Decide whether weak approximation holds for GL n/G, where G is a finite subgroup of GL n. This seems nontrivial. (de Jong thinks he can do this; Graber is unsure.) CT says de Jong claims: Let X/L be an arbitrary smooth projective, geometrically and rationally connected variety. Then weak approxmation holds. (de Jong: do this by reducing to characteristic p. Not in the dJ-H-S paper.) CT: there is an Enriques surface over C(t) that has a rational point but does not satisfy weak approximation.",
   "clean_statement": "Question 35. Weak approximation for complex function fields: Weak approximation over a finite extension L of C(t), from Hassett's lecture. Colliot-Th´ el` ene: Known for X a connected linear algebraic group over L (reduce to reductive groups immediately; since field has cohomological dimension 1, one has Borel subgroup, reduce to tori; reduce to quasi-trivial tori, which are open subsets of affine space). CT: X = G/H, where H is a subgroup of G (not necessarily normal). Same should go through if H is connected (techniques of Borovoi et al.). CT: Question: Decide whether weak approximation holds for GL n/G, where G is a finite subgroup of GL n. This seems nontrivial. (de Jong thinks he can do this; Graber is unsure.) CT says de Jong claims: Let X/L be an arbitrary smooth projective, geometrically and rationally connected variety. Then weak approxmation holds. (de Jong: do this by reducing to characteristic p. Not in the dJ-H-S paper.) CT: there is an Enriques surface over C(t) that has a rational point but does not satisfy weak approximation.",
   "public_statement": "Question 35. Weak approximation for complex function fields: Weak approximation over a finite extension L of C(t), from Hassett's lecture. Colliot-Th´ el` ene: Known for X a connected linear algebraic group over L (reduce to reductive groups immediately; since field has cohomological dimension 1, one has Borel subgroup, reduce to tori; reduce to quasi-trivial tori, which are open subsets of affine space). CT: X = G/H, where H is a subgroup of G (not necessarily normal). Same should go through if H is connected (techniques of Borovoi et al.). CT: Question: Decide whether weak approximation holds for GL n/G, where G is a finite subgroup of GL n. This seems nontrivial. (de Jong thinks he can do this; Graber is unsure.) CT says de Jong claims: Let X/L be an arbitrary smooth projective, geometrically and rationally connected variety. Then weak approxmation holds. (de Jong: do this by reducing to characteristic p. Not in the dJ-H-S paper.) CT: there is an Enriques surface over C(t) that has a rational point but does not satisfy weak approximation.",
   "evidence": "The canonical record is Question 35 from the AIM workshop *Rational and integral points on higher dimensional varieties*. Its official HTML transcription is available at https://www.aimath.org/WWN/qptsurface2/articles/html/27a/ and the workshop PDF at https://aimath.org/WWN/qptsurface2/qptsurface2.pdf.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-algebraic-number-theory-notes.json",
   "source_index": 119,
   "attempt": 1
  },
  "AIM-ANALYSIS-0001": {
   "statement_status": "exact",
   "original_statement": "The Borodin-Okounkov-Case-Geronimo identity is given by \\[\\text{det}(T_N(e^f))=(e^{N\\hat{f}(0)+\\sum k\\hat{f}(k)\\hat{f}(-k)})\\text{det}(1-k)\\] Find a useful adjustment to the right-hand side so that an identity holds in the Fisher-Hartwig case.",
   "clean_statement": "The Borodin-Okounkov-Case-Geronimo identity is given by \\[\\text{det}(T_N(e^f))=(e^{N\\hat{f}(0)+\\sum k\\hat{f}(k)\\hat{f}(-k)})\\text{det}(1-k)\\] Find a useful adjustment to the right-hand side so that an identity holds in the Fisher-Hartwig case.",
   "public_statement": "The Borodin-Okounkov-Case-Geronimo identity is given by \\[\\text{det}(T_N(e^f))=(e^{N\\hat{f}(0)+\\sum k\\hat{f}(k)\\hat{f}(-k)})\\text{det}(1-k)\\] Find a useful adjustment to the right-hand side so that an identity holds in the Fisher-Hartwig case.",
   "evidence": "The canonical record, and the current official AIM page, both contain the same malformed display:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-ANALYSIS-0002": {
   "statement_status": "exact",
   "original_statement": "Investigate the connection between the \\(\\tau\\) function from Painlev\\`{e} equations and block Toeplitz determinants. Seek explicit examples in which \\(\\tau\\) function asymptotics is known. Find the block Toeplitz determinant. What is it and what do we learn? Use Riemann-Hilbert to connect this to Fuchsian Riemann-Hilbert problems.",
   "clean_statement": "Investigate the connection between the \\(\\tau\\) function from Painlev\\`{e} equations and block Toeplitz determinants. Seek explicit examples in which \\(\\tau\\) function asymptotics is known. Find the block Toeplitz determinant. What is it and what do we learn? Use Riemann-Hilbert to connect this to Fuchsian Riemann-Hilbert problems.",
   "public_statement": "Investigate the connection between the \\(\\tau\\) function from Painlev\\`{e} equations and block Toeplitz determinants. Seek explicit examples in which \\(\\tau\\) function asymptotics is known. Find the block Toeplitz determinant. What is it and what do we learn? Use Riemann-Hilbert to connect this to Fuchsian Riemann-Hilbert problems.",
   "evidence": "The canonical record is item 1.2 in the section “Toeplitz Determinants and Toeplitz Operators” from the AIM workshop *Riemann-Hilbert problems, Toeplitz matrices, and applications* (March 4–8, 2024):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-ANALYSIS-0003": {
   "statement_status": "exact",
   "original_statement": "Study merging singularities or double scaling limits for Toeplitz determinants with two symbols. Compare \\(\\text{det}(T_n(a_{\\lambda}b_{\\lambda}))\\) and \\(\\text{det}T_n(a_{\\lambda})T_n(b_{\\lambda})\\). Under what conditions does the ratio converge?",
   "clean_statement": "Study merging singularities or double scaling limits for Toeplitz determinants with two symbols. Compare \\(\\text{det}(T_n(a_{\\lambda}b_{\\lambda}))\\) and \\(\\text{det}T_n(a_{\\lambda})T_n(b_{\\lambda})\\). Under what conditions does the ratio converge?",
   "public_statement": "Study merging singularities or double scaling limits for Toeplitz determinants with two symbols. Compare \\(\\text{det}(T_n(a_{\\lambda}b_{\\lambda}))\\) and \\(\\text{det}T_n(a_{\\lambda})T_n(b_{\\lambda})\\). Under what conditions does the ratio converge?",
   "evidence": "The canonical record is problem 1.3 in the section “Toeplitz Determinants and Toeplitz Operators” of the AIM workshop *Riemann-Hilbert problems, Toeplitz matrices, and applications*. Its exact repository text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-ANALYSIS-0004": {
   "statement_status": "exact",
   "original_statement": "Please see C.A. Berger and L.A. Coburn's 1994 article \"Heat Flow and Berlin-Toeplitz Estimates\" for background on this conjecture.\n\n(Berger-Coburr 1994): On the Fock/Bergman space, define the projection \\(\\prod:L^2(\\mathbb{C},e^{-|z|^2}dA) \\rightarrow \\mathscr{F}_{\\mathbb{C}}\\). Take \\(p: \\mathbb{C} \\rightarrow \\mathbb{C}\\) and define \\(T_p(f)=\\prod(p(z)f(z))\\). \\(T_p\\) is bounded on the Fock space iff for all \\((x,y) \\in \\mathbb{R}^2\\), \\(u_t=\\Delta u,\\) \\(u\\Big|_{t=0}=p\\), and \\(u(\\frac{1}{4},x,y)\\) are bounded.",
   "clean_statement": "Please see C.A. Berger and L.A. Coburn's 1994 article \"Heat Flow and Berlin-Toeplitz Estimates\" for background on this conjecture.\n\n(Berger-Coburr 1994): On the Fock/Bergman space, define the projection \\(\\prod:L^2(\\mathbb{C},e^{-|z|^2}dA) \\rightarrow \\mathscr{F}_{\\mathbb{C}}\\). Take \\(p: \\mathbb{C} \\rightarrow \\mathbb{C}\\) and define \\(T_p(f)=\\prod(p(z)f(z))\\). \\(T_p\\) is bounded on the Fock space iff for all \\((x,y) \\in \\mathbb{R}^2\\), \\(u_t=\\Delta u,\\) \\(u\\Big|_{t=0}=p\\), and \\(u(\\frac{1}{4},x,y)\\) are bounded.",
   "public_statement": "Please see C.A. Berger and L.A. Coburn's 1994 article \"Heat Flow and Berlin-Toeplitz Estimates\" for background on this conjecture.\n\n(Berger-Coburr 1994): On the Fock/Bergman space, define the projection \\(\\prod:L^2(\\mathbb{C},e^{-|z|^2}dA) \\rightarrow \\mathscr{F}_{\\mathbb{C}}\\). Take \\(p: \\mathbb{C} \\rightarrow \\mathbb{C}\\) and define \\(T_p(f)=\\prod(p(z)f(z))\\). \\(T_p\\) is bounded on the Fock space iff for all \\((x,y) \\in \\mathbb{R}^2\\), \\(u_t=\\Delta u,\\) \\(u\\Big|_{t=0}=p\\), and \\(u(\\frac{1}{4},x,y)\\) are bounded.",
   "evidence": "The canonical record is AIM-ANALYSIS-0004, item 1.4 in the AIM workshop section “Toeplitz Determinants and Toeplitz Operators.” It points to Berger and Coburn's 1994 paper but contains several transcription and normalization problems:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-ANALYSIS-0005": {
   "statement_status": "corrected_verified",
   "original_statement": "The determinant of Toeplitz \\(+\\) Handle is expressible in terms of the \\(4 \\times 4\\) Riemann-Hilbert problem for Szego type symbols. The obstacle to asymptotic analysis is the existence of a solution to model problems. If \\(d \\tilde{d}=1\\) on \\(\\{|z|=1\\}\\), where \\(d=\\frac{\\phi(z)}{w(z)}\\) and \\(\\tilde{d}=\\frac{\\phi(1/z)}{w(1/z)}\\), then it is solvable. What if this doesn't hold?",
   "clean_statement": "For Szegő-type nonvanishing symbols, what can be said about existence of the model Riemann--Hilbert problem when the ratio \\(d=w/\\phi\\) does not obey \\(d(z)d(z^{-1})=1\\)?",
   "public_statement": "For Szegő-type nonvanishing symbols, what can be said about existence of the model Riemann--Hilbert problem when the ratio \\(d=w/\\phi\\) does not obey \\(d(z)d(z^{-1})=1\\)?",
   "evidence": "The source is AIM workshop “Riemann-Hilbert problems, Toeplitz matrices, and applications,” Section “Toeplitz Determinants and Toeplitz Operators,” Problem 1.5. Two extraction issues can be resolved from the subsequent primary paper of Gharakhloo and Its [GI20]: 1. “Handle” is an OCR error for **Hankel**. 2. [GI20] writes the ratio as \\(d=w/\\phi\\), so that \\(w=d\\phi\\), whereas the AIM record writes its reciprocal \\(d_{\\rm AIM}=\\phi/w\\). This convention difference does not affect the condition, because \\[ d_{\\rm AIM}\\widetilde d_{\\rm AIM}=1 \\quad\\Longleftrightarrow\\quad d\\widetilde d=1. \\]",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-analysis-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-ANALYSIS-0006": {
   "statement_status": "exact",
   "original_statement": "Given \\(Q\\), produce \\(Q(z^d)\\) with multiple disjoint support sets for the droplet. Study the soft Riemann-Hilbert problem for orthogonal polynomials.",
   "clean_statement": "Given \\(Q\\), produce \\(Q(z^d)\\) with multiple disjoint support sets for the droplet. Study the soft Riemann-Hilbert problem for orthogonal polynomials.",
   "public_statement": "Given \\(Q\\), produce \\(Q(z^d)\\) with multiple disjoint support sets for the droplet. Study the soft Riemann-Hilbert problem for orthogonal polynomials.",
   "evidence": "The canonical record is item 2.1 in the section “Riemann-Hilbert Problems” of the AIM workshop list *Riemann-Hilbert problems, Toeplitz matrices, and applications*. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-ANALYSIS-0007": {
   "statement_status": "exact",
   "original_statement": "Study potential theory for \\(k \\times k\\) Riemann-Hilbert problems for \\(k>2\\) when multiple orthogonal polynomials are involved.",
   "clean_statement": "Study potential theory for \\(k \\times k\\) Riemann-Hilbert problems for \\(k>2\\) when multiple orthogonal polynomials are involved.",
   "public_statement": "Study potential theory for \\(k \\times k\\) Riemann-Hilbert problems for \\(k>2\\) when multiple orthogonal polynomials are involved.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-ANALYSIS-0008": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "In general, can you pose matrix Riemann Hilbert problems for KdV?",
   "clean_statement": null,
   "public_statement": "In general, can you pose matrix Riemann Hilbert problems for KdV?",
   "evidence": "1. **Scalar KdV, square unknown.** For \\[ q_t=6qq_x-q_{xxx},\\qquad q:\\mathbb R^2\\to\\mathbb R, \\] inverse scattering naturally gives a \\(1\\times2\\) row-vector Riemann--Hilbert problem. Does it always admit an equivalent, normalized, regular, invertible \\(2\\times2\\) matrix solution with the same jump and KdV symmetry? This is almost certainly the intended reading: it is the issue addressed explicitly by Egorova--Piorkowski--Teschl and Piorkowski--Teschl.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-ANALYSIS-0009": {
   "statement_status": "exact",
   "original_statement": "For planar orthogonal polynomials, we know that \\[\\overline{\\partial} Y=\\overline{Y} \\begin{bmatrix}\n0 & w \\\\\n0 & 0 \\\\\n\\end{bmatrix}\\] and \\[\\int p_j\\overline{p_k}wdA=h_j\\delta_{j,k}.\\] Is there an analogue of small norm theory of such \\(\\overline{\\partial}\\)-problems (i.e. \\(\\overline{\\partial} A=\\overline{A} W +AV\\)?) Or small norm theory for \\(A_{+}-A_{-}=\\overline{A_{-}}V_C+A_{-}V_n\\)?",
   "clean_statement": "For planar orthogonal polynomials, we know that \\[\\overline{\\partial} Y=\\overline{Y} \\begin{bmatrix}\n0 & w \\\\\n0 & 0 \\\\\n\\end{bmatrix}\\] and \\[\\int p_j\\overline{p_k}wdA=h_j\\delta_{j,k}.\\] Is there an analogue of small norm theory of such \\(\\overline{\\partial}\\)-problems (i.e. \\(\\overline{\\partial} A=\\overline{A} W +AV\\)?) Or small norm theory for \\(A_{+}-A_{-}=\\overline{A_{-}}V_C+A_{-}V_n\\)?",
   "public_statement": "For planar orthogonal polynomials, we know that \\[\\overline{\\partial} Y=\\overline{Y} \\begin{bmatrix}\n0 & w \\\\\n0 & 0 \\\\\n\\end{bmatrix}\\] and \\[\\int p_j\\overline{p_k}wdA=h_j\\delta_{j,k}.\\] Is there an analogue of small norm theory of such \\(\\overline{\\partial}\\)-problems (i.e. \\(\\overline{\\partial} A=\\overline{A} W +AV\\)?) Or small norm theory for \\(A_{+}-A_{-}=\\overline{A_{-}}V_C+A_{-}V_n\\)?",
   "evidence": "The canonical record is problem 2.4 in the AIM workshop list *Riemann-Hilbert problems, Toeplitz matrices, and applications*, section *Riemann-Hilbert Problems*. Its exact mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-ANALYSIS-0010": {
   "statement_status": "exact",
   "original_statement": "Analyze the condition given by \\[\\overline{\\partial}\\phi=\\overline{H_n}e^{-NQ},\\] where \\(Q=|z|^2+2t\\text{Re}(z^2)\\) and where \\(\\overline{H_n}\\) is Hermite.",
   "clean_statement": "Analyze the condition given by \\[\\overline{\\partial}\\phi=\\overline{H_n}e^{-NQ},\\] where \\(Q=|z|^2+2t\\text{Re}(z^2)\\) and where \\(\\overline{H_n}\\) is Hermite.",
   "public_statement": "Analyze the condition given by \\[\\overline{\\partial}\\phi=\\overline{H_n}e^{-NQ},\\] where \\(Q=|z|^2+2t\\text{Re}(z^2)\\) and where \\(\\overline{H_n}\\) is Hermite.",
   "evidence": "The exact canonical record is AIM-ANALYSIS-0010, source file `aim-analysis-notes.json`, zero-based source index 9, problem 2.5 in the “Riemann-Hilbert Problems” section of the AIM workshop *Riemann-Hilbert problems, Toeplitz matrices, and applications*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-ANALYSIS-0011": {
   "statement_status": "exact",
   "original_statement": "Consider \\[\\prod_{i=1}^N \\begin{bmatrix}\n1 & a_iz \\\\\nb_i & 1 \\\\\n\\end{bmatrix},\\] for constants \\(a_i, b_i\\). Suppose Wiener-Hopf factorization has no partial indices. Study the behavior as \\(N \\rightarrow \\infty\\). The asymptotics are known if the \\(a_i\\)'s and \\(b_i\\)'s are periodic. What happens in the more general case?",
   "clean_statement": "Consider \\[\\prod_{i=1}^N \\begin{bmatrix}\n1 & a_iz \\\\\nb_i & 1 \\\\\n\\end{bmatrix},\\] for constants \\(a_i, b_i\\). Suppose Wiener-Hopf factorization has no partial indices. Study the behavior as \\(N \\rightarrow \\infty\\). The asymptotics are known if the \\(a_i\\)'s and \\(b_i\\)'s are periodic. What happens in the more general case?",
   "public_statement": "Consider \\[\\prod_{i=1}^N \\begin{bmatrix}\n1 & a_iz \\\\\nb_i & 1 \\\\\n\\end{bmatrix},\\] for constants \\(a_i, b_i\\). Suppose Wiener-Hopf factorization has no partial indices. Study the behavior as \\(N \\rightarrow \\infty\\). The asymptotics are known if the \\(a_i\\)'s and \\(b_i\\)'s are periodic. What happens in the more general case?",
   "evidence": "The canonical record is AIM Problem 2.6 in the section “Riemann-Hilbert Problems,” attributed on the live AIM page to Tomas Berggren. Its exact mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-ANALYSIS-0012": {
   "statement_status": "exact",
   "original_statement": "Consider\n\\[Z_N=\\int_{\\Gamma} \\cdots \\int_{\\Gamma} \\prod_{i \\leq j < k \\leq N} |z_k-z_j|^{2\\beta} \\prod_{j=1}^N\\Big(\\frac{1}{1+|z_j|^2}\\Big)^{(N-1)\\beta+1} d|z_1| \\cdots d|z_N|.\\] Study \\(Z_N\\) as \\(N \\rightarrow \\infty\\), where \\(\\Gamma\\) is a smooth loop on the Riemann sphere.",
   "clean_statement": "Consider\n\\[Z_N=\\int_{\\Gamma} \\cdots \\int_{\\Gamma} \\prod_{i \\leq j < k \\leq N} |z_k-z_j|^{2\\beta} \\prod_{j=1}^N\\Big(\\frac{1}{1+|z_j|^2}\\Big)^{(N-1)\\beta+1} d|z_1| \\cdots d|z_N|.\\] Study \\(Z_N\\) as \\(N \\rightarrow \\infty\\), where \\(\\Gamma\\) is a smooth loop on the Riemann sphere.",
   "public_statement": "Consider\n\\[Z_N=\\int_{\\Gamma} \\cdots \\int_{\\Gamma} \\prod_{i \\leq j < k \\leq N} |z_k-z_j|^{2\\beta} \\prod_{j=1}^N\\Big(\\frac{1}{1+|z_j|^2}\\Big)^{(N-1)\\beta+1} d|z_1| \\cdots d|z_N|.\\] Study \\(Z_N\\) as \\(N \\rightarrow \\infty\\), where \\(\\Gamma\\) is a smooth loop on the Riemann sphere.",
   "evidence": "The AIM record is Problem 2.7 from the 2024 workshop *Riemann-Hilbert problems, Toeplitz matrices, and applications*. Its exact stored text is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-ANALYSIS-0013": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "The discrete Bessel process is connected to Poissonized Plancherel measure. Consider the distance of the largest particle: \\(\\mathbb{P}[Q_{\\text{max}}\\leq x]=q(x)\\), where \\(q\\) solves the cylindrical Toda equation. Take the limit to get cylindrical KdV. Is there a connection to Riemann-Hilbert Problems?",
   "clean_statement": "Express the largest-particle Fredholm determinant of the (possibly\nfinite-temperature) discrete Bessel process through a discrete\nRiemann--Hilbert problem, recover cylindrical Toda from its Lax pair,\nand identify the Riemann--Hilbert object and equation obtained under the\nsoft-edge Toda-to-KdV scaling.",
   "public_statement": "The discrete Bessel process is connected to Poissonized Plancherel measure. Consider the distance of the largest particle: \\(\\mathbb{P}[Q_{\\text{max}}\\leq x]=q(x)\\), where \\(q\\) solves the cylindrical Toda equation. Take the limit to get cylindrical KdV. Is there a connection to Riemann-Hilbert Problems?",
   "evidence": "The live AIM page was checked on 2026-07-27. It contains the same text, attributes the problem to Maksim Kosmakov, and has no status remark. Thus the word “distance” is verified source text, not an extraction error. The following corrections are nevertheless mathematically necessary and are explicit reconstructions rather than silent emendations. 1. “Distance” is almost surely intended to mean **distribution**. 2. The standard random variable is the largest particle \\(a_{\\max}\\), not \\(Q_{\\max}\\). For a partition \\(\\lambda\\), the particle configuration is \\(\\{\\lambda_i-i+\\tfrac12:i\\geq1\\}\\), hence \\(a_{\\max}=\\lambda_1-\\tfrac12\\). 3. The distribution has two variables. We write \\[ Q(L,s)=\\mathbb P_L(a_{\\max}\\leq s),\\qquad L>0,\\quad s\\in\\mathbb Z':=\\mathbb Z+\\tfrac12, \\] where \\(L^2\\) is the Poissonization parameter. A one-variable \\(q(x)\\) cannot by itself satisfy the differential--dif...",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-analysis-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-ANALYSIS-0014": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Formulate and then analyze general finite matrix symbol with the analog of zeros and jumps.",
   "clean_statement": null,
   "public_statement": "Formulate and then analyze general finite matrix symbol with the analog of zeros and jumps.",
   "evidence": "There are no remarks or references in the record. The grammar leaves three plausible readings of “finite”:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-ANALYSIS-0015": {
   "statement_status": "exact",
   "original_statement": "The \\(\\mathscr{l}^2\\)-spectrum for the Hilbert matrix \\(H_{m n}=\\frac{1}{m+n+1}\\) is known. Find the spectrum on spaces of the form \\(\\mathscr{l}_2(\\alpha)=\\{\\sum |a_n|^2(n+1)^{\\alpha}<\\infty\\}\\).",
   "clean_statement": "The \\(\\mathscr{l}^2\\)-spectrum for the Hilbert matrix \\(H_{m n}=\\frac{1}{m+n+1}\\) is known. Find the spectrum on spaces of the form \\(\\mathscr{l}_2(\\alpha)=\\{\\sum |a_n|^2(n+1)^{\\alpha}<\\infty\\}\\).",
   "public_statement": "The \\(\\mathscr{l}^2\\)-spectrum for the Hilbert matrix \\(H_{m n}=\\frac{1}{m+n+1}\\) is known. Find the spectrum on spaces of the form \\(\\mathscr{l}_2(\\alpha)=\\{\\sum |a_n|^2(n+1)^{\\alpha}<\\infty\\}\\).",
   "evidence": "The live AIM page contains the same text and attributes the problem to Alfonso Montes Rodríguez. There is no substantive corruption. We interpret the displayed set in the standard way:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-ANALYSIS-0016": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Define \\[u_{\\xi, \\alpha(z)}=e^{-\\alpha\\Big(\\frac{\\xi+z}{\\xi-z}\\Big)},\\] for \\(\\alpha >0\\) and \\(|\\xi|=1\\). Given \\(\\xi_1, \\xi_2, \\xi_3\\) and \\(\\alpha_1, \\alpha_2, \\alpha_3\\), can we approximate \\[\\sum_{m=0}^{\\infty} c_m^{(1)}u_1^m+c_m^{(2)}u_2^m+c_m^{(3)}u_3^m?\\]",
   "clean_statement": null,
   "public_statement": "Define \\[u_{\\xi, \\alpha(z)}=e^{-\\alpha\\Big(\\frac{\\xi+z}{\\xi-z}\\Big)},\\] for \\(\\alpha >0\\) and \\(|\\xi|=1\\). Given \\(\\xi_1, \\xi_2, \\xi_3\\) and \\(\\alpha_1, \\alpha_2, \\alpha_3\\), can we approximate \\[\\sum_{m=0}^{\\infty} c_m^{(1)}u_1^m+c_m^{(2)}u_2^m+c_m^{(3)}u_3^m?\\]",
   "evidence": "Two other plausible readings are treated separately: \\(H^\\infty\\)-norm approximation and literal convergence of the three infinite series. None of these readings is silently substituted for the source question.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-ANALYSIS-0017": {
   "statement_status": "exact",
   "original_statement": "Monotonicity of the spectral gap\n\nThe conjecture was originally formulated by R. Smits \\cite{MR1381604}. The following formulation can be found in a recent preprint by R. Laugesen \\cite{arXiv:1905.07658}.\n\nFor a bounded convex domain $\\Omega \\subset \\mathbb{R}^n$, the spectral gap $\\lambda_2(\\Omega, \\alpha)-\\lambda_1(\\Omega, \\alpha)$ is strictly increasing as a function of $\\alpha>0$. In particular, the Neumann gap provides a lower bound for the Dirichlet gap:\n$$\n \\mu_1(\\Omega)<\\lambda_2(\\Omega)-\\lambda_1(\\Omega).\n$$",
   "clean_statement": "Monotonicity of the spectral gap\n\nThe conjecture was originally formulated by R. Smits \\cite{MR1381604}. The following formulation can be found in a recent preprint by R. Laugesen \\cite{arXiv:1905.07658}.\n\nFor a bounded convex domain $\\Omega \\subset \\mathbb{R}^n$, the spectral gap $\\lambda_2(\\Omega, \\alpha)-\\lambda_1(\\Omega, \\alpha)$ is strictly increasing as a function of $\\alpha>0$. In particular, the Neumann gap provides a lower bound for the Dirichlet gap:\n$$\n \\mu_1(\\Omega)<\\lambda_2(\\Omega)-\\lambda_1(\\Omega).\n$$",
   "public_statement": "Monotonicity of the spectral gap\n\nThe conjecture was originally formulated by R. Smits \\cite{MR1381604}. The following formulation can be found in a recent preprint by R. Laugesen \\cite{arXiv:1905.07658}.\n\nFor a bounded convex domain $\\Omega \\subset \\mathbb{R}^n$, the spectral gap $\\lambda_2(\\Omega, \\alpha)-\\lambda_1(\\Omega, \\alpha)$ is strictly increasing as a function of $\\alpha>0$. In particular, the Neumann gap provides a lower bound for the Dirichlet gap:\n$$\n \\mu_1(\\Omega)<\\lambda_2(\\Omega)-\\lambda_1(\\Omega).\n$$",
   "evidence": "The canonical record is Conjecture 1.1 in the section *The Robin Laplacian* of the AIM workshop list *Shape optimization with surface interactions*. Its exact mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-ANALYSIS-0018": {
   "statement_status": "exact",
   "original_statement": "Bareket's conjecture\n\nIn a paper by M. Bareket \\cite{MR430552}, it was conjectured that the lowest Robin eigenvalue $\\lambda_1(\\Omega,\\alpha)$ on a bounded domain $\\Omega$ with a boundary parameter $\\alpha0$ centred at the origin. In the same paper, it was proved that when $|\\alpha|$ is small enough the disk is the unique maximizer.\n\nLet $\\lambda_1(\\Omega, \\alpha)$ be the lowest Robin eigenvalue on a bounded domain $\\Omega$ with a negative boundary parameter $\\alpha$.\n\\begin{enumerate}\n\\item In two dimensions, the disk maximizes $\\lambda_1(\\Omega, \\alpha)$ among all simply connected domains of the same area.\n\\item In three and higher dimensions, the ball maximizes $\\lambda_1(\\Omega, \\alpha)$ among all convex domains of the same volume.\n\\end{enumerate}",
   "clean_statement": "Bareket's conjecture\n\nIn a paper by M. Bareket \\cite{MR430552}, it was conjectured that the lowest Robin eigenvalue $\\lambda_1(\\Omega,\\alpha)$ on a bounded domain $\\Omega$ with a boundary parameter $\\alpha0$ centred at the origin. In the same paper, it was proved that when $|\\alpha|$ is small enough the disk is the unique maximizer.\n\nLet $\\lambda_1(\\Omega, \\alpha)$ be the lowest Robin eigenvalue on a bounded domain $\\Omega$ with a negative boundary parameter $\\alpha$.\n\\begin{enumerate}\n\\item In two dimensions, the disk maximizes $\\lambda_1(\\Omega, \\alpha)$ among all simply connected domains of the same area.\n\\item In three and higher dimensions, the ball maximizes $\\lambda_1(\\Omega, \\alpha)$ among all convex domains of the same volume.\n\\end{enumerate}",
   "public_statement": "Bareket's conjecture\n\nIn a paper by M. Bareket \\cite{MR430552}, it was conjectured that the lowest Robin eigenvalue $\\lambda_1(\\Omega,\\alpha)$ on a bounded domain $\\Omega$ with a boundary parameter $\\alpha0$ centred at the origin. In the same paper, it was proved that when $|\\alpha|$ is small enough the disk is the unique maximizer.\n\nLet $\\lambda_1(\\Omega, \\alpha)$ be the lowest Robin eigenvalue on a bounded domain $\\Omega$ with a negative boundary parameter $\\alpha$.\n\\begin{enumerate}\n\\item In two dimensions, the disk maximizes $\\lambda_1(\\Omega, \\alpha)$ among all simply connected domains of the same area.\n\\item In three and higher dimensions, the ball maximizes $\\lambda_1(\\Omega, \\alpha)$ among all convex domains of the same volume.\n\\end{enumerate}",
   "evidence": "This record is item 1.2, “Bareket's conjecture,” in the AIM list from the workshop *Shape optimization with surface interactions*. The canonical JSON has a visibly damaged sentence,",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-ANALYSIS-0019": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Discrete Bareket's conjecture\n\nIn the same spirit as Bareket's conjecture one can ask what polygon maximizes $\\lambda_1(\\Omega, \\alpha)$, $\\alpha0$ in which case the problem is to minimize the eigenvalue. The following conjecture concerns the case $N = 3$.\n\nAmong all triangles of a given area $\\lambda_1(\\Omega, \\alpha)$ is:\n\\begin{enumerate}\n\\item minimized by the equilateral triangle when $\\alpha>0$.\n\\item maximized by the equilateral triangle when $\\alpha<0$.\n\\end{enumerate}",
   "clean_statement": null,
   "public_statement": "Discrete Bareket's conjecture\n\nIn the same spirit as Bareket's conjecture one can ask what polygon maximizes $\\lambda_1(\\Omega, \\alpha)$, $\\alpha0$ in which case the problem is to minimize the eigenvalue. The following conjecture concerns the case $N = 3$.\n\nAmong all triangles of a given area $\\lambda_1(\\Omega, \\alpha)$ is:\n\\begin{enumerate}\n\\item minimized by the equilateral triangle when $\\alpha>0$.\n\\item maximized by the equilateral triangle when $\\alpha<0$.\n\\end{enumerate}",
   "evidence": "The canonical record is AIM Problem Lists entry 1.3, “Discrete Bareket's conjecture,” from the 2019 workshop *Shape optimization with surface interactions*. The record in `aim-analysis-notes.json` is visibly corrupted at",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-ANALYSIS-0020": {
   "statement_status": "exact",
   "original_statement": "Negative Robin eigenvalues on a locally deformed half-space\n\nConsider the Robin Laplacian with a negative boundary parameter $\\alpha$ on a locally perturbed half-space (its boundary is a hyperplane away from a bounded region). The essential spectrum of this operator is expected to be $[-\\alpha^2, \\infty)$. In the strong coupling limit $\\alpha\\to -\\infty$, existence of negative eigenvalues below the essential spectrum reduces to showing that the maximum\nfor the mean curvature of the boundary is positive \\cite{MR3626320}.\n\nProve or disprove that there are always negative eigenvalues below $-\\alpha^2$.",
   "clean_statement": "Negative Robin eigenvalues on a locally deformed half-space\n\nConsider the Robin Laplacian with a negative boundary parameter $\\alpha$ on a locally perturbed half-space (its boundary is a hyperplane away from a bounded region). The essential spectrum of this operator is expected to be $[-\\alpha^2, \\infty)$. In the strong coupling limit $\\alpha\\to -\\infty$, existence of negative eigenvalues below the essential spectrum reduces to showing that the maximum\nfor the mean curvature of the boundary is positive \\cite{MR3626320}.\n\nProve or disprove that there are always negative eigenvalues below $-\\alpha^2$.",
   "public_statement": "Negative Robin eigenvalues on a locally deformed half-space\n\nConsider the Robin Laplacian with a negative boundary parameter $\\alpha$ on a locally perturbed half-space (its boundary is a hyperplane away from a bounded region). The essential spectrum of this operator is expected to be $[-\\alpha^2, \\infty)$. In the strong coupling limit $\\alpha\\to -\\infty$, existence of negative eigenvalues below the essential spectrum reduces to showing that the maximum\nfor the mean curvature of the boundary is positive \\cite{MR3626320}.\n\nProve or disprove that there are always negative eigenvalues below $-\\alpha^2$.",
   "evidence": "The canonical record is Problem 1.4 in the AIM list *Shape optimization with surface interactions*, section “The Robin Laplacian.” The live AIM page was checked on 27 July 2026 and agrees with the extracted record. Its mathematical content is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-ANALYSIS-0021": {
   "statement_status": "exact",
   "original_statement": "Multiplicity of optimal Dirichlet eigenvalues\n\nLet $\\Omega_k^*$ be a minimizer of the shape optimization problem\n$$\n\t\\min\\{ \\lambda_k(\\Omega): \\Omega\\subset \\mathbb{R}^n, |\\Omega|=V_0\\}.\n$$\nProve that $\\lambda_{k-1}(\\Omega_k^*)=\\lambda_k(\\Omega_k^*)$.",
   "clean_statement": "Multiplicity of optimal Dirichlet eigenvalues\n\nLet $\\Omega_k^*$ be a minimizer of the shape optimization problem\n$$\n\t\\min\\{ \\lambda_k(\\Omega): \\Omega\\subset \\mathbb{R}^n, |\\Omega|=V_0\\}.\n$$\nProve that $\\lambda_{k-1}(\\Omega_k^*)=\\lambda_k(\\Omega_k^*)$.",
   "public_statement": "Multiplicity of optimal Dirichlet eigenvalues\n\nLet $\\Omega_k^*$ be a minimizer of the shape optimization problem\n$$\n\t\\min\\{ \\lambda_k(\\Omega): \\Omega\\subset \\mathbb{R}^n, |\\Omega|=V_0\\}.\n$$\nProve that $\\lambda_{k-1}(\\Omega_k^*)=\\lambda_k(\\Omega_k^*)$.",
   "evidence": "The canonical record is item 2.1, “Multiplicity of optimal Dirichlet eigenvalues,” from the AIM problem list associated with the workshop *Shape optimization with surface interactions*. Its exact mathematical text is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-ANALYSIS-0022": {
   "statement_status": "exact",
   "original_statement": "Mahler inequality for the principal Dirichlet eigenvalue\n\nLet $\\mathcal{K}^n_\\star$ be the class of centrally symmetric bounded convex domains in $\\mathbb{R}^n$. For a domain $K\\in\\mathcal{K}^n_\\star$ we define its polar set by\n $$\n K^\\circ := \\{x\\in \\mathbb{R}^n\\colon x \\cdot y \\leq 1 \\mbox{ for all }y \\in K\\}.\n $$\nFurther, let $\\mathsf{GL}_n$ be the family of invertible linear transformation in $\\mathbb{R}^n$.\n\nA hypercube in $\\mathbb{R}^n$ is an attainer for the maximization problem\n\\[\n\\sup\\left\\{\n\\inf_{T\\in\\mathsf{GL}_n} \\lambda_1(T(K))\\lambda_1((T(K))^\\circ)\\colon K\\in\\mathcal{K}^n_\\star\\right\\}.\n\\]",
   "clean_statement": "Mahler inequality for the principal Dirichlet eigenvalue\n\nLet $\\mathcal{K}^n_\\star$ be the class of centrally symmetric bounded convex domains in $\\mathbb{R}^n$. For a domain $K\\in\\mathcal{K}^n_\\star$ we define its polar set by\n $$\n K^\\circ := \\{x\\in \\mathbb{R}^n\\colon x \\cdot y \\leq 1 \\mbox{ for all }y \\in K\\}.\n $$\nFurther, let $\\mathsf{GL}_n$ be the family of invertible linear transformation in $\\mathbb{R}^n$.\n\nA hypercube in $\\mathbb{R}^n$ is an attainer for the maximization problem\n\\[\n\\sup\\left\\{\n\\inf_{T\\in\\mathsf{GL}_n} \\lambda_1(T(K))\\lambda_1((T(K))^\\circ)\\colon K\\in\\mathcal{K}^n_\\star\\right\\}.\n\\]",
   "public_statement": "Mahler inequality for the principal Dirichlet eigenvalue\n\nLet $\\mathcal{K}^n_\\star$ be the class of centrally symmetric bounded convex domains in $\\mathbb{R}^n$. For a domain $K\\in\\mathcal{K}^n_\\star$ we define its polar set by\n $$\n K^\\circ := \\{x\\in \\mathbb{R}^n\\colon x \\cdot y \\leq 1 \\mbox{ for all }y \\in K\\}.\n $$\nFurther, let $\\mathsf{GL}_n$ be the family of invertible linear transformation in $\\mathbb{R}^n$.\n\nA hypercube in $\\mathbb{R}^n$ is an attainer for the maximization problem\n\\[\n\\sup\\left\\{\n\\inf_{T\\in\\mathsf{GL}_n} \\lambda_1(T(K))\\lambda_1((T(K))^\\circ)\\colon K\\in\\mathcal{K}^n_\\star\\right\\}.\n\\]",
   "evidence": "Central symmetry forces the center to be the origin in this formulation. We harmlessly identify a bounded convex domain with its closure when taking the polar; its Dirichlet eigenvalue is that of its interior. No corruption of the mathematical statement was found.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-ANALYSIS-0023": {
   "statement_status": "exact",
   "original_statement": "Discrete Faber-Krahn inequality\n\nThe classical Faber-Krahn inequality states that among all sets of equal volume the lowest Dirichlet eigenvalue $\\lambda_1(\\Omega)$ is minimized by the ball. The following longstanding conjecture concerns the corresponding problem about minimization of $\\lambda_1(\\Omega)$ among polygons.\n\nAmong all polygons of a given area and having not more than $N$ edges the lowest Dirichlet eigenvalue $\\lambda_1(\\Omega)$ is minimized by the regular $N$-gon.",
   "clean_statement": "Discrete Faber-Krahn inequality\n\nThe classical Faber-Krahn inequality states that among all sets of equal volume the lowest Dirichlet eigenvalue $\\lambda_1(\\Omega)$ is minimized by the ball. The following longstanding conjecture concerns the corresponding problem about minimization of $\\lambda_1(\\Omega)$ among polygons.\n\nAmong all polygons of a given area and having not more than $N$ edges the lowest Dirichlet eigenvalue $\\lambda_1(\\Omega)$ is minimized by the regular $N$-gon.",
   "public_statement": "Discrete Faber-Krahn inequality\n\nThe classical Faber-Krahn inequality states that among all sets of equal volume the lowest Dirichlet eigenvalue $\\lambda_1(\\Omega)$ is minimized by the ball. The following longstanding conjecture concerns the corresponding problem about minimization of $\\lambda_1(\\Omega)$ among polygons.\n\nAmong all polygons of a given area and having not more than $N$ edges the lowest Dirichlet eigenvalue $\\lambda_1(\\Omega)$ is minimized by the regular $N$-gon.",
   "evidence": "The canonical record is AIM-ANALYSIS-0023, item 2.3 in the AIM workshop list *Shape optimization with surface interactions*, section “The Dirichlet Laplacian.” Its statement is uncorrupted and reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-ANALYSIS-0024": {
   "statement_status": "exact",
   "original_statement": "van den Berg's conjecture\n\nLet $\\Omega \\subset \\mathbb{R}^n$ be a convex domain and let $\\rho, D$ denote its inradius and diameter, respectively. Let $u$ denote the eigenfunction corresponding to the first eigenvalue of the Dirichlet Laplacian on $\\Omega$. According to a theorem due to G. Chiti \\cite{MR0652928} there exists a dimensional constant $C$ such that\n$$\n\\|u\\|_{L^\\infty(\\Omega)} \\leq \\frac{C}{\\rho^{n/2}}\\|u\\|_{L^2(\\Omega)}.\n$$\nThe following conjecture was posed by M. van den Berg \\cite{MR1804178}\n\nThere is a dimensional constant $C$ such that\n$$\n\t\\|u\\|_{L^\\infty(\\Omega)} \\leq \\frac{C}{\\rho^{n/2}}\\Bigl(\\frac{\\rho}{D}\\Bigr)^{1/6}\\|u\\|_{L^2(\\Omega)}.\n$$",
   "clean_statement": "van den Berg's conjecture\n\nLet $\\Omega \\subset \\mathbb{R}^n$ be a convex domain and let $\\rho, D$ denote its inradius and diameter, respectively. Let $u$ denote the eigenfunction corresponding to the first eigenvalue of the Dirichlet Laplacian on $\\Omega$. According to a theorem due to G. Chiti \\cite{MR0652928} there exists a dimensional constant $C$ such that\n$$\n\\|u\\|_{L^\\infty(\\Omega)} \\leq \\frac{C}{\\rho^{n/2}}\\|u\\|_{L^2(\\Omega)}.\n$$\nThe following conjecture was posed by M. van den Berg \\cite{MR1804178}\n\nThere is a dimensional constant $C$ such that\n$$\n\t\\|u\\|_{L^\\infty(\\Omega)} \\leq \\frac{C}{\\rho^{n/2}}\\Bigl(\\frac{\\rho}{D}\\Bigr)^{1/6}\\|u\\|_{L^2(\\Omega)}.\n$$",
   "public_statement": "van den Berg's conjecture\n\nLet $\\Omega \\subset \\mathbb{R}^n$ be a convex domain and let $\\rho, D$ denote its inradius and diameter, respectively. Let $u$ denote the eigenfunction corresponding to the first eigenvalue of the Dirichlet Laplacian on $\\Omega$. According to a theorem due to G. Chiti \\cite{MR0652928} there exists a dimensional constant $C$ such that\n$$\n\\|u\\|_{L^\\infty(\\Omega)} \\leq \\frac{C}{\\rho^{n/2}}\\|u\\|_{L^2(\\Omega)}.\n$$\nThe following conjecture was posed by M. van den Berg \\cite{MR1804178}\n\nThere is a dimensional constant $C$ such that\n$$\n\t\\|u\\|_{L^\\infty(\\Omega)} \\leq \\frac{C}{\\rho^{n/2}}\\Bigl(\\frac{\\rho}{D}\\Bigr)^{1/6}\\|u\\|_{L^2(\\Omega)}.\n$$",
   "evidence": "The canonical record is item 2.4, “van den Berg's conjecture,” in the AIM list *Shape optimization with surface interactions*, section “The Dirichlet Laplacian.” The archived AIM page gives the following assertion. If \\(\\Omega\\subset\\mathbb R^n\\) is a convex domain, \\(\\rho\\) and \\(D\\) are its inradius and diameter, and \\(u\\) is a first Dirichlet eigenfunction, then there should be a dimensional constant \\(C_n\\) such that",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-ANALYSIS-0025": {
   "statement_status": "exact",
   "original_statement": "Concavity of the principal Dirichlet eigenfunction\n\nLet $\\Omega\\subset \\mathbb{R}^n$ be a bouned convex domain and let $u$ be the principal eigenfunction of the Dirichlet Laplacian on $\\Omega$. By a classical result of H. Brascamp and E. Lieb $u$ is log-concave \\cite{MR0450480}. Moreover, it is known that $u$ is power-concave meaning that $u^\\alpha$ is concave in the usual sense for certain powers $\\alpha > 0$.\nThe best concavity exponent for a domain $\\Omega$ is defined as\n\\[\n\\alpha(\\Omega) :=\\sup\\{\\alpha \\ge 0\\colon u^\\alpha \\mbox{ is concave}\\}.\n\\]\nThe following conjecture was formulated in \\cite{MR1280948} by P. Lindqvist.\n\nThe best concavity exponent $\\alpha(\\Omega)$ is maximized by the ball.",
   "clean_statement": "Concavity of the principal Dirichlet eigenfunction\n\nLet $\\Omega\\subset \\mathbb{R}^n$ be a bouned convex domain and let $u$ be the principal eigenfunction of the Dirichlet Laplacian on $\\Omega$. By a classical result of H. Brascamp and E. Lieb $u$ is log-concave \\cite{MR0450480}. Moreover, it is known that $u$ is power-concave meaning that $u^\\alpha$ is concave in the usual sense for certain powers $\\alpha > 0$.\nThe best concavity exponent for a domain $\\Omega$ is defined as\n\\[\n\\alpha(\\Omega) :=\\sup\\{\\alpha \\ge 0\\colon u^\\alpha \\mbox{ is concave}\\}.\n\\]\nThe following conjecture was formulated in \\cite{MR1280948} by P. Lindqvist.\n\nThe best concavity exponent $\\alpha(\\Omega)$ is maximized by the ball.",
   "public_statement": "Concavity of the principal Dirichlet eigenfunction\n\nLet $\\Omega\\subset \\mathbb{R}^n$ be a bouned convex domain and let $u$ be the principal eigenfunction of the Dirichlet Laplacian on $\\Omega$. By a classical result of H. Brascamp and E. Lieb $u$ is log-concave \\cite{MR0450480}. Moreover, it is known that $u$ is power-concave meaning that $u^\\alpha$ is concave in the usual sense for certain powers $\\alpha > 0$.\nThe best concavity exponent for a domain $\\Omega$ is defined as\n\\[\n\\alpha(\\Omega) :=\\sup\\{\\alpha \\ge 0\\colon u^\\alpha \\mbox{ is concave}\\}.\n\\]\nThe following conjecture was formulated in \\cite{MR1280948} by P. Lindqvist.\n\nThe best concavity exponent $\\alpha(\\Omega)$ is maximized by the ball.",
   "evidence": "The canonical record is item 2.5, “Concavity of the principal Dirichlet eigenfunction,” in the AIM list *Shape optimization with surface interactions*, section “The Dirichlet Laplacian” (`aim-analysis-notes.json`, zero-based index 24). The source URL is <http://aimpl.org/shapesurface/2/>. The page timed out during this run, so the canonical repository record is the source used for the statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-ANALYSIS-0026": {
   "statement_status": "exact",
   "original_statement": "Optimizers for Neumann eigenvalues among convex sets of a fixed perimeter\n\nProve the existence of optimizers for\n $$\n \\sup\\{\\mu_k(\\Omega)\\colon \\Omega\\subset \\mathbb{R}^n \\mbox{ convex, }P(\\Omega)=P_0\\}\n $$\n and\n $$\n \\inf\\{\\mu_k(\\Omega)\\colon \\Omega\\subset \\mathbb{R}^n \\mbox{ convex, }P(\\Omega)=P_0\\}.\n $$",
   "clean_statement": "Optimizers for Neumann eigenvalues among convex sets of a fixed perimeter\n\nProve the existence of optimizers for\n $$\n \\sup\\{\\mu_k(\\Omega)\\colon \\Omega\\subset \\mathbb{R}^n \\mbox{ convex, }P(\\Omega)=P_0\\}\n $$\n and\n $$\n \\inf\\{\\mu_k(\\Omega)\\colon \\Omega\\subset \\mathbb{R}^n \\mbox{ convex, }P(\\Omega)=P_0\\}.\n $$",
   "public_statement": "Optimizers for Neumann eigenvalues among convex sets of a fixed perimeter\n\nProve the existence of optimizers for\n $$\n \\sup\\{\\mu_k(\\Omega)\\colon \\Omega\\subset \\mathbb{R}^n \\mbox{ convex, }P(\\Omega)=P_0\\}\n $$\n and\n $$\n \\inf\\{\\mu_k(\\Omega)\\colon \\Omega\\subset \\mathbb{R}^n \\mbox{ convex, }P(\\Omega)=P_0\\}.\n $$",
   "evidence": "The canonical record is `AIM-ANALYSIS-0026`, record 25 (zero-based) of `aim-analysis-notes.json`, from the AIM workshop *Shape optimization with surface interactions*, section “The Neumann Laplacian,” Problem 3.1. Its mathematical statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-ANALYSIS-0027": {
   "statement_status": "exact",
   "original_statement": "Maximizing the ratio of Neumann eigenvalues among convex domains\n\nFor any convex $\\Omega\\subset \\mathbb{R}^n$, it holds that\n \\begin{enumerate}[label=\\textup{(\\alph*)}]\n \\item $\\mu_2(\\Omega)/\\mu_1(\\Omega)\\leq 4$.\n \\item $\\mu_k(\\Omega)/\\mu_1(\\Omega)\\leq k^2$.\n \\end{enumerate}",
   "clean_statement": "Maximizing the ratio of Neumann eigenvalues among convex domains\n\nFor any convex $\\Omega\\subset \\mathbb{R}^n$, it holds that\n \\begin{enumerate}[label=\\textup{(\\alph*)}]\n \\item $\\mu_2(\\Omega)/\\mu_1(\\Omega)\\leq 4$.\n \\item $\\mu_k(\\Omega)/\\mu_1(\\Omega)\\leq k^2$.\n \\end{enumerate}",
   "public_statement": "Maximizing the ratio of Neumann eigenvalues among convex domains\n\nFor any convex $\\Omega\\subset \\mathbb{R}^n$, it holds that\n \\begin{enumerate}[label=\\textup{(\\alph*)}]\n \\item $\\mu_2(\\Omega)/\\mu_1(\\Omega)\\leq 4$.\n \\item $\\mu_k(\\Omega)/\\mu_1(\\Omega)\\leq k^2$.\n \\end{enumerate}",
   "evidence": "The canonical record is Conjecture 3.2 in the AIM workshop list *Shape optimization with surface interactions*, section “The Neumann Laplacian.” The archived AIM page agrees with the corpus record and attributes the conjecture to A. Henrot. With the indexing convention made explicit, the recovered statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-ANALYSIS-0028": {
   "statement_status": "exact",
   "original_statement": "Hot spots conjecture\n\nThis conjecture is often attributed to J. Rauch, who has formulated it in terms of large-time behaviuor of the heat semigroup generated by the Neumann Laplacian. On the qualitative level, this conjecture manifests as follows: for an insulated flat piece of metal with an arbitrary initial temperature distribution, given enough time, the hottest point on the metal will lie on its boundary.\n\nThe first non-trivial Neumann eigenfunction does not have global extrema in the interior of $\\Omega$ if\n \\begin{enumerate}\n \\item $\\Omega\\subset \\mathbb{R}^2$ is simply connected.\n \\item $\\Omega\\subset \\mathbb{R}^n$ ($n \\ge 3$) is convex.\n \\end{enumerate}",
   "clean_statement": "Hot spots conjecture\n\nThis conjecture is often attributed to J. Rauch, who has formulated it in terms of large-time behaviuor of the heat semigroup generated by the Neumann Laplacian. On the qualitative level, this conjecture manifests as follows: for an insulated flat piece of metal with an arbitrary initial temperature distribution, given enough time, the hottest point on the metal will lie on its boundary.\n\nThe first non-trivial Neumann eigenfunction does not have global extrema in the interior of $\\Omega$ if\n \\begin{enumerate}\n \\item $\\Omega\\subset \\mathbb{R}^2$ is simply connected.\n \\item $\\Omega\\subset \\mathbb{R}^n$ ($n \\ge 3$) is convex.\n \\end{enumerate}",
   "public_statement": "Hot spots conjecture\n\nThis conjecture is often attributed to J. Rauch, who has formulated it in terms of large-time behaviuor of the heat semigroup generated by the Neumann Laplacian. On the qualitative level, this conjecture manifests as follows: for an insulated flat piece of metal with an arbitrary initial temperature distribution, given enough time, the hottest point on the metal will lie on its boundary.\n\nThe first non-trivial Neumann eigenfunction does not have global extrema in the interior of $\\Omega$ if\n \\begin{enumerate}\n \\item $\\Omega\\subset \\mathbb{R}^2$ is simply connected.\n \\item $\\Omega\\subset \\mathbb{R}^n$ ($n \\ge 3$) is convex.\n \\end{enumerate}",
   "evidence": "The canonical record is item 3.3, tagged as a conjecture, in the AIM workshop list Shape optimization with surface interactions. Its mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-ANALYSIS-0029": {
   "statement_status": "exact",
   "original_statement": "Directional hot spots conjecture\n\nLet $\\Omega \\subset \\mathbb{R}^2$ be a centrally symmetric bounded convex domain and let $u$ denote the eigenfunction corresponding to the first non-trivial Neumann eigenvalue.\n\nCan one always find a direction $\\mathbf{e}\\in\\mathbb{R}^2$ such that\n \\begin{equation}\n \\mathbf{e}\\cdot \\nabla u >0 \\quad \\mbox{in }\\Omega?\n \\end{equation}",
   "clean_statement": "Directional hot spots conjecture\n\nLet $\\Omega \\subset \\mathbb{R}^2$ be a centrally symmetric bounded convex domain and let $u$ denote the eigenfunction corresponding to the first non-trivial Neumann eigenvalue.\n\nCan one always find a direction $\\mathbf{e}\\in\\mathbb{R}^2$ such that\n \\begin{equation}\n \\mathbf{e}\\cdot \\nabla u >0 \\quad \\mbox{in }\\Omega?\n \\end{equation}",
   "public_statement": "Directional hot spots conjecture\n\nLet $\\Omega \\subset \\mathbb{R}^2$ be a centrally symmetric bounded convex domain and let $u$ denote the eigenfunction corresponding to the first non-trivial Neumann eigenvalue.\n\nCan one always find a direction $\\mathbf{e}\\in\\mathbb{R}^2$ such that\n \\begin{equation}\n \\mathbf{e}\\cdot \\nabla u >0 \\quad \\mbox{in }\\Omega?\n \\end{equation}",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-ANALYSIS-0030": {
   "statement_status": "exact",
   "original_statement": "Non-existence of a maximizer for the first Steklov eigenvalue with perimeter constraint\n\nLet $\\sigma_1(\\Omega)$ denote the first non-trivial Steklov eigenvalue for a bounded domain $\\Omega\\subset\\mathbb{R}^n$.\n\nProve that there is no $\\Omega\\subset \\mathbb{R}^n$ which maximizes the quantity $P(\\Omega)^{1/(n-1)}\\sigma_1(\\Omega)$.",
   "clean_statement": "Non-existence of a maximizer for the first Steklov eigenvalue with perimeter constraint\n\nLet $\\sigma_1(\\Omega)$ denote the first non-trivial Steklov eigenvalue for a bounded domain $\\Omega\\subset\\mathbb{R}^n$.\n\nProve that there is no $\\Omega\\subset \\mathbb{R}^n$ which maximizes the quantity $P(\\Omega)^{1/(n-1)}\\sigma_1(\\Omega)$.",
   "public_statement": "Non-existence of a maximizer for the first Steklov eigenvalue with perimeter constraint\n\nLet $\\sigma_1(\\Omega)$ denote the first non-trivial Steklov eigenvalue for a bounded domain $\\Omega\\subset\\mathbb{R}^n$.\n\nProve that there is no $\\Omega\\subset \\mathbb{R}^n$ which maximizes the quantity $P(\\Omega)^{1/(n-1)}\\sigma_1(\\Omega)$.",
   "evidence": "This is Problem 4.1 in the “Steklov eigenvalues” section of the AIM workshop list *Shape optimization with surface interactions*. The exact extracted problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-ANALYSIS-0031": {
   "statement_status": "exact",
   "original_statement": "Faber-Krahn for Dirac operator with infinite mass boundary condition\n\nProve or disprove that among all planar domains of a given area, the disk is the unique minimizer for the smallest positive eigenvalue\n$\\nu_1(\\Omega)$ of the Dirac problem\nwith infinite mass boundary conditions\n$$\n\\left\\{\n\\begin{aligned}\n\t-\\mathsf{i}\n\\big(\\partial_1 v - \\mathsf{i}\\partial_2 v\\big) &= \\nu u\n\t&&\\mbox{in} \\quad \\Omega \\,,\n \\\\\n -\\mathsf{i}\\big(\\partial_1 u + \\mathsf{i}\\partial_2 u\\big)\n&= \\nu v\n &&\\mbox{in} \\quad \\Omega \\,,\n \\\\\n \\mathsf{i}\\big(n_1 + \\mathsf{i} n_2\\big) u\n &= v\n &&\\mbox{on} \\quad \\partial\\Omega \\,,\n \\end{aligned}\n \\right. $$\n where~$n = (n_1,n_2)^\\top$ is the outer unit normal to~$\\Omega$.",
   "clean_statement": "Faber-Krahn for Dirac operator with infinite mass boundary condition\n\nProve or disprove that among all planar domains of a given area, the disk is the unique minimizer for the smallest positive eigenvalue\n$\\nu_1(\\Omega)$ of the Dirac problem\nwith infinite mass boundary conditions\n$$\n\\left\\{\n\\begin{aligned}\n\t-\\mathsf{i}\n\\big(\\partial_1 v - \\mathsf{i}\\partial_2 v\\big) &= \\nu u\n\t&&\\mbox{in} \\quad \\Omega \\,,\n \\\\\n -\\mathsf{i}\\big(\\partial_1 u + \\mathsf{i}\\partial_2 u\\big)\n&= \\nu v\n &&\\mbox{in} \\quad \\Omega \\,,\n \\\\\n \\mathsf{i}\\big(n_1 + \\mathsf{i} n_2\\big) u\n &= v\n &&\\mbox{on} \\quad \\partial\\Omega \\,,\n \\end{aligned}\n \\right. $$\n where~$n = (n_1,n_2)^\\top$ is the outer unit normal to~$\\Omega$.",
   "public_statement": "Faber-Krahn for Dirac operator with infinite mass boundary condition\n\nProve or disprove that among all planar domains of a given area, the disk is the unique minimizer for the smallest positive eigenvalue\n$\\nu_1(\\Omega)$ of the Dirac problem\nwith infinite mass boundary conditions\n$$\n\\left\\{\n\\begin{aligned}\n\t-\\mathsf{i}\n\\big(\\partial_1 v - \\mathsf{i}\\partial_2 v\\big) &= \\nu u\n\t&&\\mbox{in} \\quad \\Omega \\,,\n \\\\\n -\\mathsf{i}\\big(\\partial_1 u + \\mathsf{i}\\partial_2 u\\big)\n&= \\nu v\n &&\\mbox{in} \\quad \\Omega \\,,\n \\\\\n \\mathsf{i}\\big(n_1 + \\mathsf{i} n_2\\big) u\n &= v\n &&\\mbox{on} \\quad \\partial\\Omega \\,,\n \\end{aligned}\n \\right. $$\n where~$n = (n_1,n_2)^\\top$ is the outer unit normal to~$\\Omega$.",
   "evidence": "The canonical record is Problem 5.1 in the AIM list *Shape optimization with surface interactions*, section “Dirac operators.” It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-ANALYSIS-0032": {
   "statement_status": "exact",
   "original_statement": "Hermite-Hadamard inequality in higher dimensions\n\nThe classical Hermite-Hadamard inequality states that if $f$ is a convex function on $(a, b)$ then\n$$\n \\frac{1}{b-a}\\int_a^b f(x)dx \\leq \\frac{f(b)+f(a)}{2}.\n$$\nWe consider two generalizations to higher dimensions. Let $\\Omega \\subset \\mathbb{R}^n$ be a convex domain and\n\nCase 1: $f\\colon \\Omega \\to [0, \\infty)$ convex, or\n\nCase 2: $f\\colon \\Omega \\to [0, \\infty)$ subharmonic.\n\nIn either case one can ask for smallest constants $c_n, \\tilde c_n$ such that\n\\begin{equation}\n \\frac{1}{|\\Omega|}\\int_\\Omega f(x)dx \\leq \\frac{c_n}{|\\partial\\Omega|}\\int_{\\partial\\Omega}f(x)dx,\n\\quad\\mbox{and}\\quad\n \\int_\\Omega f(x)dx \\leq \\tilde c_n|\\Omega|^{1/n}\\int_{\\partial\\Omega}f(x)dx.\n\t\\end{equation}\nThe second inequality follows from the first one by an application of the isoperimetric inequality with $\\tilde c_n = c_n/\\sqrt{n}$.\n\nFind the optimal constants in the inequalities.",
   "clean_statement": "Hermite-Hadamard inequality in higher dimensions\n\nThe classical Hermite-Hadamard inequality states that if $f$ is a convex function on $(a, b)$ then\n$$\n \\frac{1}{b-a}\\int_a^b f(x)dx \\leq \\frac{f(b)+f(a)}{2}.\n$$\nWe consider two generalizations to higher dimensions. Let $\\Omega \\subset \\mathbb{R}^n$ be a convex domain and\n\nCase 1: $f\\colon \\Omega \\to [0, \\infty)$ convex, or\n\nCase 2: $f\\colon \\Omega \\to [0, \\infty)$ subharmonic.\n\nIn either case one can ask for smallest constants $c_n, \\tilde c_n$ such that\n\\begin{equation}\n \\frac{1}{|\\Omega|}\\int_\\Omega f(x)dx \\leq \\frac{c_n}{|\\partial\\Omega|}\\int_{\\partial\\Omega}f(x)dx,\n\\quad\\mbox{and}\\quad\n \\int_\\Omega f(x)dx \\leq \\tilde c_n|\\Omega|^{1/n}\\int_{\\partial\\Omega}f(x)dx.\n\t\\end{equation}\nThe second inequality follows from the first one by an application of the isoperimetric inequality with $\\tilde c_n = c_n/\\sqrt{n}$.\n\nFind the optimal constants in the inequalities.",
   "public_statement": "Hermite-Hadamard inequality in higher dimensions\n\nThe classical Hermite-Hadamard inequality states that if $f$ is a convex function on $(a, b)$ then\n$$\n \\frac{1}{b-a}\\int_a^b f(x)dx \\leq \\frac{f(b)+f(a)}{2}.\n$$\nWe consider two generalizations to higher dimensions. Let $\\Omega \\subset \\mathbb{R}^n$ be a convex domain and\n\nCase 1: $f\\colon \\Omega \\to [0, \\infty)$ convex, or\n\nCase 2: $f\\colon \\Omega \\to [0, \\infty)$ subharmonic.\n\nIn either case one can ask for smallest constants $c_n, \\tilde c_n$ such that\n\\begin{equation}\n \\frac{1}{|\\Omega|}\\int_\\Omega f(x)dx \\leq \\frac{c_n}{|\\partial\\Omega|}\\int_{\\partial\\Omega}f(x)dx,\n\\quad\\mbox{and}\\quad\n \\int_\\Omega f(x)dx \\leq \\tilde c_n|\\Omega|^{1/n}\\int_{\\partial\\Omega}f(x)dx.\n\t\\end{equation}\nThe second inequality follows from the first one by an application of the isoperimetric inequality with $\\tilde c_n = c_n/\\sqrt{n}$.\n\nFind the optimal constants in the inequalities.",
   "evidence": "The canonical record is AIM Problem List item 6.1, “Hermite-Hadamard inequality in higher dimensions,” from the 2019 workshop *Shape optimization with surface interactions*. It asks the following.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-ANALYSIS-0033": {
   "statement_status": "exact",
   "original_statement": "Asymptotic partitioning problems\n\nIn several problems where one wants to find an in some sense optimal partition of a planar domain $\\Omega$ into $N$ subdomains it has been observed that as $N$ becomes large the partition converges to the hexagonal one.\n\nFind conditions for problems involving optimal $N$-partitions of a planar domain such that one observes in the limit $N\\to \\infty$ a packing of hexagons.",
   "clean_statement": "Asymptotic partitioning problems\n\nIn several problems where one wants to find an in some sense optimal partition of a planar domain $\\Omega$ into $N$ subdomains it has been observed that as $N$ becomes large the partition converges to the hexagonal one.\n\nFind conditions for problems involving optimal $N$-partitions of a planar domain such that one observes in the limit $N\\to \\infty$ a packing of hexagons.",
   "public_statement": "Asymptotic partitioning problems\n\nIn several problems where one wants to find an in some sense optimal partition of a planar domain $\\Omega$ into $N$ subdomains it has been observed that as $N$ becomes large the partition converges to the hexagonal one.\n\nFind conditions for problems involving optimal $N$-partitions of a planar domain such that one observes in the limit $N\\to \\infty$ a packing of hexagons.",
   "evidence": "The canonical record is AIM-ANALYSIS-0033, problem 6.2 in the AIM workshop list *Shape optimization with surface interactions*, section “Miscellaneous problems.” The archived AIM page attributes the question to K. Burdzy. The record states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-ANALYSIS-0034": {
   "statement_status": "exact",
   "original_statement": "The ovals of Benguria and Loss\n\nLet $\\Gamma\\subset \\mathbb{R}^2$ be a closed smooth curve of length $2\\pi$ and let $\\kappa(s)$ be its curvature as a function of arclength. Define the following self-adjoint one-dimensional differential operator\n\\[\nH_\\Gamma \\psi := -\\psi'' + \\kappa^2\\psi,\\qquad \\mathsf{dom}\\,H_\\Gamma := W^{2,2}(\\Gamma),\n\\]\nin the Hilbert space $L^2(\\Gamma)$ and let $\\lambda_1(H_\\Gamma)$ be its lowest eigenvalue.\n\nThe following inequality $\\lambda_1(H_\\Gamma) \\ge 1$ holds.",
   "clean_statement": "The ovals of Benguria and Loss\n\nLet $\\Gamma\\subset \\mathbb{R}^2$ be a closed smooth curve of length $2\\pi$ and let $\\kappa(s)$ be its curvature as a function of arclength. Define the following self-adjoint one-dimensional differential operator\n\\[\nH_\\Gamma \\psi := -\\psi'' + \\kappa^2\\psi,\\qquad \\mathsf{dom}\\,H_\\Gamma := W^{2,2}(\\Gamma),\n\\]\nin the Hilbert space $L^2(\\Gamma)$ and let $\\lambda_1(H_\\Gamma)$ be its lowest eigenvalue.\n\nThe following inequality $\\lambda_1(H_\\Gamma) \\ge 1$ holds.",
   "public_statement": "The ovals of Benguria and Loss\n\nLet $\\Gamma\\subset \\mathbb{R}^2$ be a closed smooth curve of length $2\\pi$ and let $\\kappa(s)$ be its curvature as a function of arclength. Define the following self-adjoint one-dimensional differential operator\n\\[\nH_\\Gamma \\psi := -\\psi'' + \\kappa^2\\psi,\\qquad \\mathsf{dom}\\,H_\\Gamma := W^{2,2}(\\Gamma),\n\\]\nin the Hilbert space $L^2(\\Gamma)$ and let $\\lambda_1(H_\\Gamma)$ be its lowest eigenvalue.\n\nThe following inequality $\\lambda_1(H_\\Gamma) \\ge 1$ holds.",
   "evidence": "The canonical record is problem 6.3, “The ovals of Benguria and Loss,” from the AIM workshop *Shape optimization with surface interactions* (`aim-analysis-notes.json`, zero-based record index 33). Its mathematical statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-ANALYSIS-0035": {
   "statement_status": "exact",
   "original_statement": "Faber-Krahn for the buckling problem\n\nLet $\\Omega\\subset\\mathbb{R}^2$ be a bounded domain and consider the lowest eigenvalue of the buckling problem\n \\begin{align}\n \\Delta^2 u+\\Lambda_1(\\Omega)\\Delta u &=0 \\quad \\mbox{in }\\Omega,\\\\\n u&=0 \\quad \\mbox{on }\\partial\\Omega,\\\\\n \\frac{\\partial u}{\\partial n}&=0 \\quad \\mbox{on }\\partial\\Omega.\n \\end{align}\n\nProve that the lowest buckling eigenvalue $\\Lambda_1(\\Omega)$ is minimized by the disk among all simply connected sets of equal area.",
   "clean_statement": "Faber-Krahn for the buckling problem\n\nLet $\\Omega\\subset\\mathbb{R}^2$ be a bounded domain and consider the lowest eigenvalue of the buckling problem\n \\begin{align}\n \\Delta^2 u+\\Lambda_1(\\Omega)\\Delta u &=0 \\quad \\mbox{in }\\Omega,\\\\\n u&=0 \\quad \\mbox{on }\\partial\\Omega,\\\\\n \\frac{\\partial u}{\\partial n}&=0 \\quad \\mbox{on }\\partial\\Omega.\n \\end{align}\n\nProve that the lowest buckling eigenvalue $\\Lambda_1(\\Omega)$ is minimized by the disk among all simply connected sets of equal area.",
   "public_statement": "Faber-Krahn for the buckling problem\n\nLet $\\Omega\\subset\\mathbb{R}^2$ be a bounded domain and consider the lowest eigenvalue of the buckling problem\n \\begin{align}\n \\Delta^2 u+\\Lambda_1(\\Omega)\\Delta u &=0 \\quad \\mbox{in }\\Omega,\\\\\n u&=0 \\quad \\mbox{on }\\partial\\Omega,\\\\\n \\frac{\\partial u}{\\partial n}&=0 \\quad \\mbox{on }\\partial\\Omega.\n \\end{align}\n\nProve that the lowest buckling eigenvalue $\\Lambda_1(\\Omega)$ is minimized by the disk among all simply connected sets of equal area.",
   "evidence": "The canonical record is problem 6.4, “Faber--Krahn for the buckling problem,” from the AIM workshop *Shape optimization with surface interactions*. It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-ANALYSIS-0036": {
   "statement_status": "exact",
   "original_statement": "Poincaré-Wirtinger extremal domain\n\nConsider the operator $-\\gamma^{-1}\\nabla \\gamma \\nabla$ on the weighted space $L^2(\\Omega, \\gamma)$ with Neumann boundary conditions, $\\Omega\\subset \\mathbb{R}^n$ and where $\\gamma$ is a Gaussian weight. It is known that the first non-trivial eigenvalue of this operator satisfies $\\mu_1(\\Omega, \\gamma)\\geq 1$ (this is the Poincaré-Wirtinger inequality).\n\nProve that the equality in the Poincaré-Wirtinger inequality is attained only for infinite strips.",
   "clean_statement": "Poincaré-Wirtinger extremal domain\n\nConsider the operator $-\\gamma^{-1}\\nabla \\gamma \\nabla$ on the weighted space $L^2(\\Omega, \\gamma)$ with Neumann boundary conditions, $\\Omega\\subset \\mathbb{R}^n$ and where $\\gamma$ is a Gaussian weight. It is known that the first non-trivial eigenvalue of this operator satisfies $\\mu_1(\\Omega, \\gamma)\\geq 1$ (this is the Poincaré-Wirtinger inequality).\n\nProve that the equality in the Poincaré-Wirtinger inequality is attained only for infinite strips.",
   "public_statement": "Poincaré-Wirtinger extremal domain\n\nConsider the operator $-\\gamma^{-1}\\nabla \\gamma \\nabla$ on the weighted space $L^2(\\Omega, \\gamma)$ with Neumann boundary conditions, $\\Omega\\subset \\mathbb{R}^n$ and where $\\gamma$ is a Gaussian weight. It is known that the first non-trivial eigenvalue of this operator satisfies $\\mu_1(\\Omega, \\gamma)\\geq 1$ (this is the Poincaré-Wirtinger inequality).\n\nProve that the equality in the Poincaré-Wirtinger inequality is attained only for infinite strips.",
   "evidence": "The canonical record is problem 6.5, “Poincaré-Wirtinger extremal domain,” from the AIM workshop *Shape optimization with surface interactions*. Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-ANALYSIS-0037": {
   "statement_status": "exact",
   "original_statement": "Nodal structure of the $k$-th Neumann eigenfunction for collapsing domains\n\nConsider the nodal domains of the $k$-th Neumann eigenfunction on a sequence of domains collapsing to a lower dimensional object.\n\nUnder what assumptions can one prove that the nodal domains are in a sense ordered?",
   "clean_statement": "Nodal structure of the $k$-th Neumann eigenfunction for collapsing domains\n\nConsider the nodal domains of the $k$-th Neumann eigenfunction on a sequence of domains collapsing to a lower dimensional object.\n\nUnder what assumptions can one prove that the nodal domains are in a sense ordered?",
   "public_statement": "Nodal structure of the $k$-th Neumann eigenfunction for collapsing domains\n\nConsider the nodal domains of the $k$-th Neumann eigenfunction on a sequence of domains collapsing to a lower dimensional object.\n\nUnder what assumptions can one prove that the nodal domains are in a sense ordered?",
   "evidence": "The canonical record is problem 6.6, “Nodal structure of the $k$-th Neumann eigenfunction for collapsing domains,” from the AIM workshop *Shape optimization with surface interactions* (`aim-analysis-notes.json`, zero-based index 36). The exact prompt is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-ANALYSIS-0038": {
   "statement_status": "exact",
   "original_statement": "Is there a pseudoconvex domain $\\Omega \\subset \\mathbb{C}^n$ with $0 < {\\rm dim} \\, A^2(\\Omega) < \\infty$?",
   "clean_statement": "Is there a pseudoconvex domain $\\Omega \\subset \\mathbb{C}^n$ with $0 < {\\rm dim} \\, A^2(\\Omega) < \\infty$?",
   "public_statement": "Is there a pseudoconvex domain $\\Omega \\subset \\mathbb{C}^n$ with $0 < {\\rm dim} \\, A^2(\\Omega) < \\infty$?",
   "evidence": "The canonical record is AIM Problem List problem 1.05 from the workshop *Problems on holomorphic function spaces and complex dynamics*, section “Problems on Holomorphic Function Spaces.” Its exact statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-ANALYSIS-0039": {
   "statement_status": "exact",
   "original_statement": "What is the dimension of the Bergman space of a short $\\mathbb{C}^2$? (We may assume vanishing Kobayashi metric)\n\nAs an example of short $\\mathbb{C}^2$, consider the sequence of mappings $f_n(z, w) = (a_n w + z^2, a_n z)$, with the coefficients $a_n$ satisfying $a_n \\searrow 0$ rapidly. Then let $\\Omega = \\bigcup_{n \\geq 1} ( \\, \\mbox{basin of attraction of} \\, f_n \\, {\\rm at}\\, 0 )$.",
   "clean_statement": "What is the dimension of the Bergman space of a short $\\mathbb{C}^2$? (We may assume vanishing Kobayashi metric)\n\nAs an example of short $\\mathbb{C}^2$, consider the sequence of mappings $f_n(z, w) = (a_n w + z^2, a_n z)$, with the coefficients $a_n$ satisfying $a_n \\searrow 0$ rapidly. Then let $\\Omega = \\bigcup_{n \\geq 1} ( \\, \\mbox{basin of attraction of} \\, f_n \\, {\\rm at}\\, 0 )$.",
   "public_statement": "What is the dimension of the Bergman space of a short $\\mathbb{C}^2$? (We may assume vanishing Kobayashi metric)\n\nAs an example of short $\\mathbb{C}^2$, consider the sequence of mappings $f_n(z, w) = (a_n w + z^2, a_n z)$, with the coefficients $a_n$ satisfying $a_n \\searrow 0$ rapidly. Then let $\\Omega = \\bigcup_{n \\geq 1} ( \\, \\mbox{basin of attraction of} \\, f_n \\, {\\rm at}\\, 0 )$.",
   "evidence": "The canonical record is Problem 1.1 in the section “Problems on Holomorphic Function Spaces” from the April 2019 AIM workshop *Problems on holomorphic function spaces and complex dynamics* (`aim-analysis-notes.json`, zero-based index 38). Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-ANALYSIS-0040": {
   "statement_status": "exact",
   "original_statement": "Is there a pseudoconvex domain $\\Omega$ with ${\\rm dim} \\, A^2(\\Omega) \\neq 0$ such that there is no bounded plurisubharmonic function $\\phi$ on $\\Omega$.",
   "clean_statement": "Is there a pseudoconvex domain $\\Omega$ with ${\\rm dim} \\, A^2(\\Omega) \\neq 0$ such that there is no bounded plurisubharmonic function $\\phi$ on $\\Omega$.",
   "public_statement": "Is there a pseudoconvex domain $\\Omega$ with ${\\rm dim} \\, A^2(\\Omega) \\neq 0$ such that there is no bounded plurisubharmonic function $\\phi$ on $\\Omega$.",
   "evidence": "The canonical record is problem 1.15 from the AIM workshop *Problems on holomorphic function spaces and complex dynamics*. Its exact extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-ANALYSIS-0041": {
   "statement_status": "exact",
   "original_statement": "What can we say about the geometry of a complete Reinhardt domain with trivial Bergman space?",
   "clean_statement": "What can we say about the geometry of a complete Reinhardt domain with trivial Bergman space?",
   "public_statement": "What can we say about the geometry of a complete Reinhardt domain with trivial Bergman space?",
   "evidence": "The canonical record is number 1.2 in the AIM list *Problems on holomorphic function spaces and complex dynamics*, section “Problems on Holomorphic Function Spaces”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-ANALYSIS-0042": {
   "statement_status": "exact",
   "original_statement": "Let $\\Delta$ be the unit disk in $\\mathbb{C}$ and let $\\omega > 0$ be an upper semicontinuous function on $\\Delta$. Consider the weightd Bergman space $A^2 (\\Delta, \\omega)$. Is it possible that $0 < {\\rm dim} \\, A^2(\\Delta, \\omega) < \\infty$?",
   "clean_statement": "Let $\\Delta$ be the unit disk in $\\mathbb{C}$ and let $\\omega > 0$ be an upper semicontinuous function on $\\Delta$. Consider the weightd Bergman space $A^2 (\\Delta, \\omega)$. Is it possible that $0 < {\\rm dim} \\, A^2(\\Delta, \\omega) < \\infty$?",
   "public_statement": "Let $\\Delta$ be the unit disk in $\\mathbb{C}$ and let $\\omega > 0$ be an upper semicontinuous function on $\\Delta$. Consider the weightd Bergman space $A^2 (\\Delta, \\omega)$. Is it possible that $0 < {\\rm dim} \\, A^2(\\Delta, \\omega) < \\infty$?",
   "evidence": "The canonical record is AIM-ANALYSIS-0042, Problem 1.25 in the section “Problems on Holomorphic Function Spaces” of the AIM list *Problems on holomorphic function spaces and complex dynamics*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-ANALYSIS-0043": {
   "statement_status": "exact",
   "original_statement": "Let $b \\in H^\\infty (\\Delta)$ with $|| b ||_{H^\\infty} \\leq 1$. Define $H(b):=(I-T_b T_{\\overline{b}})^{1/2} H^2(\\Delta)$, where $T_b$ denotes the multiplication operator by $b$.\n\n\\begin{enumerate}\n\n\\item When are polynomials dense in $H(b)$? (known in affirmative for non-extremal $b$ or inner $b$)\n\n\\item How to generate the $H(b)$s as reproducing kernel Hilbert spaces to polydisk or the ball in $\\mathbb{C}^n$?\n\n\\item Find integral representations of these spaces in setting beyond the unit disk $\\Delta$.\n\\end{enumerate}",
   "clean_statement": "Let $b \\in H^\\infty (\\Delta)$ with $|| b ||_{H^\\infty} \\leq 1$. Define $H(b):=(I-T_b T_{\\overline{b}})^{1/2} H^2(\\Delta)$, where $T_b$ denotes the multiplication operator by $b$.\n\n\\begin{enumerate}\n\n\\item When are polynomials dense in $H(b)$? (known in affirmative for non-extremal $b$ or inner $b$)\n\n\\item How to generate the $H(b)$s as reproducing kernel Hilbert spaces to polydisk or the ball in $\\mathbb{C}^n$?\n\n\\item Find integral representations of these spaces in setting beyond the unit disk $\\Delta$.\n\\end{enumerate}",
   "public_statement": "Let $b \\in H^\\infty (\\Delta)$ with $|| b ||_{H^\\infty} \\leq 1$. Define $H(b):=(I-T_b T_{\\overline{b}})^{1/2} H^2(\\Delta)$, where $T_b$ denotes the multiplication operator by $b$.\n\n\\begin{enumerate}\n\n\\item When are polynomials dense in $H(b)$? (known in affirmative for non-extremal $b$ or inner $b$)\n\n\\item How to generate the $H(b)$s as reproducing kernel Hilbert spaces to polydisk or the ball in $\\mathbb{C}^n$?\n\n\\item Find integral representations of these spaces in setting beyond the unit disk $\\Delta$.\n\\end{enumerate}",
   "evidence": "The canonical AIM record is problem 1.3 in the section “Problems on Holomorphic Function Spaces” of the AIM list *Problems on holomorphic function spaces and complex dynamics*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-ANALYSIS-0044": {
   "statement_status": "exact",
   "original_statement": "Given a $\\overline{\\partial}$-closed $(0,1)$-form $f \\in L^\\infty$ on a polydisk or bounded symmetric domain, is there a $u$ solving $\\overline{\\partial}u = f$ with the estimate $||u||_\\infty \\leq c ||f||_\\infty$?",
   "clean_statement": "Given a $\\overline{\\partial}$-closed $(0,1)$-form $f \\in L^\\infty$ on a polydisk or bounded symmetric domain, is there a $u$ solving $\\overline{\\partial}u = f$ with the estimate $||u||_\\infty \\leq c ||f||_\\infty$?",
   "public_statement": "Given a $\\overline{\\partial}$-closed $(0,1)$-form $f \\in L^\\infty$ on a polydisk or bounded symmetric domain, is there a $u$ solving $\\overline{\\partial}u = f$ with the estimate $||u||_\\infty \\leq c ||f||_\\infty$?",
   "evidence": "The canonical record is `aim-analysis-notes.json`, zero-based record 43, AIM workshop “Problems on holomorphic function spaces and complex dynamics,” problem 1.35. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-ANALYSIS-0045": {
   "statement_status": "exact",
   "original_statement": "Define $\\mathcal{H}_{p, \\alpha}:=\\left\\{ f \\,\\, \\mbox{entire}\\, : \\left(\\frac{\\alpha p}{2\\pi}\\right)^n \\left(\\int_{\\mathbb{C}^n} |f|^p e^{-\\alpha p ||z||^2/2} dm \\right)^{1/p} < \\infty \\right\\}$. What is the best constant for the H\\\"{o}lder inequality between $\\mathcal{H}_{p, \\alpha}$ and $\\mathcal{H}_{q, \\alpha}$?",
   "clean_statement": "Define $\\mathcal{H}_{p, \\alpha}:=\\left\\{ f \\,\\, \\mbox{entire}\\, : \\left(\\frac{\\alpha p}{2\\pi}\\right)^n \\left(\\int_{\\mathbb{C}^n} |f|^p e^{-\\alpha p ||z||^2/2} dm \\right)^{1/p} < \\infty \\right\\}$. What is the best constant for the H\\\"{o}lder inequality between $\\mathcal{H}_{p, \\alpha}$ and $\\mathcal{H}_{q, \\alpha}$?",
   "public_statement": "Define $\\mathcal{H}_{p, \\alpha}:=\\left\\{ f \\,\\, \\mbox{entire}\\, : \\left(\\frac{\\alpha p}{2\\pi}\\right)^n \\left(\\int_{\\mathbb{C}^n} |f|^p e^{-\\alpha p ||z||^2/2} dm \\right)^{1/p} < \\infty \\right\\}$. What is the best constant for the H\\\"{o}lder inequality between $\\mathcal{H}_{p, \\alpha}$ and $\\mathcal{H}_{q, \\alpha}$?",
   "evidence": "The canonical AIM record (source file `aim-analysis-notes.json`, zero-based index 44, problem 1.4 of *Problems on Holomorphic Function Spaces*) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-ANALYSIS-0046": {
   "statement_status": "exact",
   "original_statement": "Given a Fatou-Bieberbach domain $\\Omega$, must there exist a pseudoconvex domain $\\tilde{\\Omega}$ such that $\\tilde{\\Omega}$ with $\\Omega \\subsetneq \\tilde{\\Omega} \\subsetneq \\mathbb{C}^n$.",
   "clean_statement": "Given a Fatou-Bieberbach domain $\\Omega$, must there exist a pseudoconvex domain $\\tilde{\\Omega}$ such that $\\tilde{\\Omega}$ with $\\Omega \\subsetneq \\tilde{\\Omega} \\subsetneq \\mathbb{C}^n$.",
   "public_statement": "Given a Fatou-Bieberbach domain $\\Omega$, must there exist a pseudoconvex domain $\\tilde{\\Omega}$ such that $\\tilde{\\Omega}$ with $\\Omega \\subsetneq \\tilde{\\Omega} \\subsetneq \\mathbb{C}^n$.",
   "evidence": "The canonical record is Problem 1.45 in the section “Problems on Holomorphic Function Spaces” from the AIM workshop *Problems on holomorphic function spaces and complex dynamics*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-ANALYSIS-0047": {
   "statement_status": "exact",
   "original_statement": "Given a K\\\"{a}hler manifold $(M, \\omega)$. Define $H^2_\\omega:=\\{f: \\Delta \\rightarrow M : \\int_\\Delta \\log |z|^2 f^* \\omega > -\\infty \\}$. Investigate these spaces, for example: completeness, radial limits, etc... A special case to consider is when $M$ being toric.",
   "clean_statement": "Given a K\\\"{a}hler manifold $(M, \\omega)$. Define $H^2_\\omega:=\\{f: \\Delta \\rightarrow M : \\int_\\Delta \\log |z|^2 f^* \\omega > -\\infty \\}$. Investigate these spaces, for example: completeness, radial limits, etc... A special case to consider is when $M$ being toric.",
   "public_statement": "Given a K\\\"{a}hler manifold $(M, \\omega)$. Define $H^2_\\omega:=\\{f: \\Delta \\rightarrow M : \\int_\\Delta \\log |z|^2 f^* \\omega > -\\infty \\}$. Investigate these spaces, for example: completeness, radial limits, etc... A special case to consider is when $M$ being toric.",
   "evidence": "The canonical record is problem 1.5 in the AIM workshop list *Problems on Holomorphic Function Spaces*. The archived AIM page gives the following text (including its grammatical omissions):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-ANALYSIS-0048": {
   "statement_status": "exact",
   "original_statement": "Is there a Fatou-Bieberbach domain with real analytic boundary?",
   "clean_statement": "Is there a Fatou-Bieberbach domain with real analytic boundary?",
   "public_statement": "Is there a Fatou-Bieberbach domain with real analytic boundary?",
   "evidence": "The canonical AIM record (workshop *Problems on holomorphic function spaces and complex dynamics*, section *Problems on Holomorphic Function Spaces*, Problem 1.55) asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-ANALYSIS-0049": {
   "statement_status": "exact",
   "original_statement": "Can region of attractions for non-polynomial automorphisms of $\\mathbb{C}^2$ have $C^1$ boundary?",
   "clean_statement": "Can region of attractions for non-polynomial automorphisms of $\\mathbb{C}^2$ have $C^1$ boundary?",
   "public_statement": "Can region of attractions for non-polynomial automorphisms of $\\mathbb{C}^2$ have $C^1$ boundary?",
   "evidence": "The canonical AIM record (Analysis, problem 1.6, source index 48) reads verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-ANALYSIS-0050": {
   "statement_status": "exact",
   "original_statement": "Is there a Fatou-Bieberbach domain contained in $\\mathbb{C}^* \\times \\mathbb{C}^*$?",
   "clean_statement": "Is there a Fatou-Bieberbach domain contained in $\\mathbb{C}^* \\times \\mathbb{C}^*$?",
   "public_statement": "Is there a Fatou-Bieberbach domain contained in $\\mathbb{C}^* \\times \\mathbb{C}^*$?",
   "evidence": "The exact canonical record is AIM-ANALYSIS-0050, Problem 1.65 in the AIM list *Problems on Holomorphic Function Spaces*. The archived AIM page agrees with the record and reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-ANALYSIS-0051": {
   "statement_status": "exact",
   "original_statement": "For a domain $\\Omega \\subset \\mathbb{C}$, define\n\\begin{equation*}\nH^2(\\Omega):=\\{ f \\,\\, \\mbox{holomorphic} \\, \\, \\mbox{on} \\, \\, \\Omega : |f|^2 \\, \\, \\mbox{has a harmonic majorant} \\}.\n\\end{equation*}\nDoes log-capacity$(\\mathbb{C}\\setminus \\Omega) > 0$ imply ${\\rm dim}\\, H^2 (\\Omega) > 1$?",
   "clean_statement": "For a domain $\\Omega \\subset \\mathbb{C}$, define\n\\begin{equation*}\nH^2(\\Omega):=\\{ f \\,\\, \\mbox{holomorphic} \\, \\, \\mbox{on} \\, \\, \\Omega : |f|^2 \\, \\, \\mbox{has a harmonic majorant} \\}.\n\\end{equation*}\nDoes log-capacity$(\\mathbb{C}\\setminus \\Omega) > 0$ imply ${\\rm dim}\\, H^2 (\\Omega) > 1$?",
   "public_statement": "For a domain $\\Omega \\subset \\mathbb{C}$, define\n\\begin{equation*}\nH^2(\\Omega):=\\{ f \\,\\, \\mbox{holomorphic} \\, \\, \\mbox{on} \\, \\, \\Omega : |f|^2 \\, \\, \\mbox{has a harmonic majorant} \\}.\n\\end{equation*}\nDoes log-capacity$(\\mathbb{C}\\setminus \\Omega) > 0$ imply ${\\rm dim}\\, H^2 (\\Omega) > 1$?",
   "evidence": "The canonical record is AIM Problem Lists problem 1.7 from “Problems on Holomorphic Function Spaces,” stored at index 50 of `aim-analysis-notes.json`. The source record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-ANALYSIS-0052": {
   "statement_status": "exact",
   "original_statement": "Can we choose continuous multiples of surface area on the boundary of the Diederich-Fornaess worm (for example, powers of Levi-form) mitigating the irregularity of the Szeg\\\"{o} projection (defined using surface area/modified measure?)",
   "clean_statement": "Can we choose continuous multiples of surface area on the boundary of the Diederich-Fornaess worm (for example, powers of Levi-form) mitigating the irregularity of the Szeg\\\"{o} projection (defined using surface area/modified measure?)",
   "public_statement": "Can we choose continuous multiples of surface area on the boundary of the Diederich-Fornaess worm (for example, powers of Levi-form) mitigating the irregularity of the Szeg\\\"{o} projection (defined using surface area/modified measure?)",
   "evidence": "The AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-ANALYSIS-0053": {
   "statement_status": "exact",
   "original_statement": "Characterize the boundary data space for Neumann boundary problem for holomorphic functions on planar domain. Are there any representation formula for the solution?",
   "clean_statement": "Characterize the boundary data space for Neumann boundary problem for holomorphic functions on planar domain. Are there any representation formula for the solution?",
   "public_statement": "Characterize the boundary data space for Neumann boundary problem for holomorphic functions on planar domain. Are there any representation formula for the solution?",
   "evidence": "The archived AIM page was inspected. It contains exactly this sentence and no attribution, status note, definition of “Neumann,” regularity hypothesis, or choice of boundary norm. Thus the ambiguity is in the source, not an extraction error.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-ANALYSIS-0054": {
   "statement_status": "exact",
   "original_statement": "Let $F$ be a non-polynomial automorphism of $\\mathbb{C}^2$. What are the possible biholomorphic types of Fatou components of $F$?",
   "clean_statement": "Let $F$ be a non-polynomial automorphism of $\\mathbb{C}^2$. What are the possible biholomorphic types of Fatou components of $F$?",
   "public_statement": "Let $F$ be a non-polynomial automorphism of $\\mathbb{C}^2$. What are the possible biholomorphic types of Fatou components of $F$?",
   "evidence": "The canonical record is AIM-ANALYSIS-0054, problem 2.1 in the section “Problems on Complex Dynamics” of the AIM list *Problems on holomorphic function spaces and complex dynamics*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-ANALYSIS-0055": {
   "statement_status": "exact",
   "original_statement": "Investigate the dynamics of maps of the form\n\\begin{equation*}(x, y) \\mapsto (p(x)-\\delta y, x),\n\\end{equation*}\nwhere $p(x)$ is a degree $2$ rational map.",
   "clean_statement": "Investigate the dynamics of maps of the form\n\\begin{equation*}(x, y) \\mapsto (p(x)-\\delta y, x),\n\\end{equation*}\nwhere $p(x)$ is a degree $2$ rational map.",
   "public_statement": "Investigate the dynamics of maps of the form\n\\begin{equation*}(x, y) \\mapsto (p(x)-\\delta y, x),\n\\end{equation*}\nwhere $p(x)$ is a degree $2$ rational map.",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-ANALYSIS-0056": {
   "statement_status": "exact",
   "original_statement": "Let $F$ be a germ of automorphism of $\\mathbb{C}^n (n > 1)$ tangent to the identity with a degenerate characteristic direction $v$. When is there a domain of attraction tangent to the origin along $v$?",
   "clean_statement": "Let $F$ be a germ of automorphism of $\\mathbb{C}^n (n > 1)$ tangent to the identity with a degenerate characteristic direction $v$. When is there a domain of attraction tangent to the origin along $v$?",
   "public_statement": "Let $F$ be a germ of automorphism of $\\mathbb{C}^n (n > 1)$ tangent to the identity with a degenerate characteristic direction $v$. When is there a domain of attraction tangent to the origin along $v$?",
   "evidence": "The canonical record is `aim-analysis-notes.json`, record index 55, Problem 2.3 of the AIM list *Problems on holomorphic function spaces and complex dynamics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-ANALYSIS-0057": {
   "statement_status": "exact",
   "original_statement": "Study hedgehogs for gems of automorphism of $\\mathbb{C}^2$ with both elliptic eigenvalues.",
   "clean_statement": "Study hedgehogs for gems of automorphism of $\\mathbb{C}^2$ with both elliptic eigenvalues.",
   "public_statement": "Study hedgehogs for gems of automorphism of $\\mathbb{C}^2$ with both elliptic eigenvalues.",
   "evidence": "The canonical record reads, literally:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-ANALYSIS-0058": {
   "statement_status": "exact",
   "original_statement": "Let $C$ be the cubic surface defined by\n\\begin{equation*}\nx^2 + y^2 + z^2 - xyz = D,\n\\end{equation*}\nwhere $D$ is a complex parameter. Let $s_x: C \\rightarrow C$ be defined by\n\\begin{equation*}\ns_x(x, y, z) := (yz - x, y, z).\n\\end{equation*}\nThe maps $s_y, s_z$ are defined similarly.\n\nConsider, for example, the map $S:= s_x \\circ s_y \\circ s_z$. It is known that $S$ has non-zero entropy. Adapt the study of critical measure, etc from the study of H\\'{e}non maps to that of $F$.",
   "clean_statement": "Let $C$ be the cubic surface defined by\n\\begin{equation*}\nx^2 + y^2 + z^2 - xyz = D,\n\\end{equation*}\nwhere $D$ is a complex parameter. Let $s_x: C \\rightarrow C$ be defined by\n\\begin{equation*}\ns_x(x, y, z) := (yz - x, y, z).\n\\end{equation*}\nThe maps $s_y, s_z$ are defined similarly.\n\nConsider, for example, the map $S:= s_x \\circ s_y \\circ s_z$. It is known that $S$ has non-zero entropy. Adapt the study of critical measure, etc from the study of H\\'{e}non maps to that of $F$.",
   "public_statement": "Let $C$ be the cubic surface defined by\n\\begin{equation*}\nx^2 + y^2 + z^2 - xyz = D,\n\\end{equation*}\nwhere $D$ is a complex parameter. Let $s_x: C \\rightarrow C$ be defined by\n\\begin{equation*}\ns_x(x, y, z) := (yz - x, y, z).\n\\end{equation*}\nThe maps $s_y, s_z$ are defined similarly.\n\nConsider, for example, the map $S:= s_x \\circ s_y \\circ s_z$. It is known that $S$ has non-zero entropy. Adapt the study of critical measure, etc from the study of H\\'{e}non maps to that of $F$.",
   "evidence": "This is Problem 2.5 in the AIM list *Problems on holomorphic function spaces and complex dynamics*, section “Problems on Complex Dynamics.” The canonical record is `aim-analysis-notes.json`, zero-based index 57. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-ANALYSIS-0059": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "1. Monge-Amp` ere masses supported by analytic sets (a) Compact K¨ ahler case (e.g., CP n \\ H) [S. Dinew] \n\n• (A good) definition? Possible criteria: 1. If un ↓ and vn ↓, un, v n ∈ L∞ ∩ P SH and lim un = lim vn,then lim M A (un) = lim M A (vn). 2. Take u and un = max( u, −n). Try M A (u) = M A (un) (if it exists). \n\n• Solvability of M A (u) = μ, where μ = measure supported on an analytic set (not a point), Green's function, possibly with prescribed singularities. (b) Similar questions for the Dirichlet problem on Ω ⊂⊂ Cn.(c) Suppose 6 ∃ KE metric on X, with c1(X) > 0: Consider e.g. the continuity method ψt:= φt − sup φt. Show that a subsequence ψt →\n\nψ whose M A (ψ) is defined and M A (ψ) = μ, μ supported in the multiplier ideal sheaf. (e.g., X = CP 2 with one point blown-up.) (d) Real analogues (from the Toric case, for example). (e) X Fano, D smooth anti-canonical divisor. Let ω\u000f satisfy Ric( ωt) = \u000fω t + (1 − \u000f)[ D].\n\nWhat is the limit of (subsequence) ω\u000f? In particular, is its Ricci (in a suitable sense) supported on D? [H. Guenancia] 2. Are there non-product solutions of ( dd cu)n = 0 on M compact (e.g., CP 2), where u is smooth and not necessarily PSH? [Y. Rubinstein] 3. Are there solutions of ( dd cu)n = 0 on Cn, where u is PSH? [W. He] 4. Let Ω be a strongly pseudo-convex domain, ∂Ω ∈ C3,1. Consider the problem ( dd cu)n = f, f ≥ 0, f 1 \n\n> n\n\n∈ C1,1 and u|∂Ω = φ, φ ∈ C3,1(∂Ω). Find an analytic (independent from Krylov's) proof of u ∈ C1,1(Ω). 15. Same question as above but with f 1 \n\n> n−1\n\n∈ C1,1 (in this case, not covered by Krylov.) (Special case known: Ω = Bn, φ = 0: yes by Pli´ s.) 6. Can one construct a counterexample to the maximal rank question from the example of Ross-Witt-Nystrom of solutions of the HCMA without foliation? [M. Paun] 7. Solving ( π∗ω + i∂∂u )n+1 = 0 on X × A, where A is an annulus, and ω\n\nis possibly degenerate. u|X×{ t=1,e } = φ0, φ 1, φ0 and φ1 ω-psh (with some regularity) and ∫ \n\n> X\n\nωn > 0. [E. Di Nezza] 8. Find a PDE proof of Kolodziej's L∞ estimate; find optimal constant for a ball. [Z. Blocki] 9. Find X KE Fano and u ∈ Λ1 such that ∫ \n\n> X\n\nu3ωnKE 6 = 0. [H. Macbeth] 10. ( dd cu)n = 1 on Ω, ∆ u ∈ Ln(n−1) =⇒ u ∈ C∞? [T. Collins] 11. Complex version of Pogorelov's estimate. 12. Let p: X → D, X K¨ ahler and KX nef. Study solutions of Ric( ωt\u000f) = \n\n−ωt\u000f − \u000fβ t on Xt. [M. Paun] 13. Find an analytic proof of the ACC Conjecture/Theorem. [T. Collins] 2",
   "clean_statement": null,
   "public_statement": "1. Monge-Amp` ere masses supported by analytic sets (a) Compact K¨ ahler case (e.g., CP n \\ H) [S. Dinew]\n\n• (A good) definition? Possible criteria: 1. If un ↓ and vn ↓, un, v n ∈ L∞ ∩ P SH and lim un = lim vn,then lim M A (un) = lim M A (vn). 2. Take u and un = max( u, −n). Try M A (u) = M A (un) (if it exists).\n\n• Solvability of M A (u) = μ, where μ = measure supported on an analytic set (not a point), Green's function, possibly with prescribed singularities. (b) Similar questions for the Dirichlet problem on Ω ⊂⊂ Cn.(c) Suppose 6 ∃ KE metric on X, with c1(X) > 0: Consider e.g. the continuity method ψt:= φt − sup φt. Show that a subsequence ψt →\n\nψ whose M A (ψ) is defined and M A (ψ) = μ, μ supported in the multiplier ideal sheaf. (e.g., X = CP 2 with one point blown-up.) (d) Real analogues (from the Toric case, for example). (e) X Fano, D smooth anti-canonical divisor. Let ω[U+000F] satisfy Ric( ωt) = [U+000F]ω t + (1 − [U+000F])[ D].\n\nWhat is the limit of (subsequence) ω[U+000F]? In particular, is its Ricci (in a suitable sense) supported on D? [H. Guenancia] 2. Are there non-product solutions of ( dd cu)n = 0 on M compact (e.g., CP 2), where u is smooth and not necessarily PSH? [Y. Rubinstein] 3. Are there solutions of ( dd cu)n = 0 on Cn, where u is PSH? [W. He] 4. Let Ω be a strongly pseudo-convex domain, ∂Ω ∈ C3,1. Consider the problem ( dd cu)n = f, f ≥ 0, f 1\n\n> n\n\n∈ C1,1 and u|∂Ω = φ, φ ∈ C3,1(∂Ω). Find an analytic (independent from Krylov's) proof of u ∈ C1,1(Ω). 15. Same question as above but with f 1\n\n> n−1\n\n∈ C1,1 (in this case, not covered by Krylov.) (Special case known: Ω = Bn, φ = 0: yes by Pli´ s.) 6. Can one construct a counterexample to the maximal rank question from the example of Ross-Witt-Nystrom of solutions of the HCMA without foliation? [M. Paun] 7. Solving ( π∗ω + i∂∂u )n+1 = 0 on X × A, where A is an annulus, and ω\n\nis possibly degenerate. u|X×{ t=1,e } = φ0, φ 1, φ0 and φ1 ω-psh (with some regularity) and ∫\n\n> X\n\nωn > 0. [E. Di Nezza] 8. Find a PDE proof of Kolodziej's L∞ estimate; find optimal constant for a ball. [Z. Blocki] 9. Find X KE Fano and u ∈ Λ1 such that ∫\n\n> X\n\nu3ωnKE 6 = 0. [H. Macbeth] 10. ( dd cu)n = 1 on Ω, ∆ u ∈ Ln(n−1) =⇒ u ∈ C∞? [T. Collins] 11. Complex version of Pogorelov's estimate. 12. Let p: X → D, X K¨ ahler and KX nef. Study solutions of Ric( ωt[U+000F]) =\n\n−ωt[U+000F] − [U+000F]β t on Xt. [M. Paun] 13. Find an analytic proof of the ACC Conjecture/Theorem. [T. Collins] 2",
   "evidence": "The canonical JSON record is not one problem. It is the entire two-page list *Complex Monge-Ampère Equation Workshop: Open problems*, edited by M. Dellatorre and dated September 8, 2016. The original PDF and the workshop report were checked directly. The extraction merged all thirteen numbered items into record 1. It also turned item 5 into “15.” by adjoining the page-one footer, split the exponents \\(1/n\\) and \\(1/(n-1)\\), inserted a control character in place of \\(\\varepsilon\\), and appended the page number “2” to item 13.",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-ANALYSIS-0060": {
   "statement_status": "corrected_verified",
   "original_statement": "Question 1 (Thiele). Let u: R → R be a measurable function. Define the maximal operator along the planar vector field (1, u ) by \n\nMuf (x, y ):= sup \n\n> \u000f> 0\n\n∣∣∣∣\n\n12\u000f\n\n∫ \u000f\n\n> −\u000f\n\nf (x − t, y − u(x)t)dt \n\n∣∣∣∣. (0.1) \n\nDoes Mu satisfy any Lp bound for certain p < ∞?",
   "clean_statement": "Let \\(u:\\mathbb R\\to\\mathbb R\\) be a measurable function. Define the maximal operator along the planar vector field \\((1,u)\\) by\n\\[\nM_u f(x,y):=\\sup_{\\epsilon>0}\\left|\n\\frac{1}{2\\epsilon}\\int_{-\\epsilon}^{\\epsilon}\nf(x-t,y-u(x)t)\\,dt\n\\right|.\n\\]\nDoes \\(M_u\\) satisfy any \\(L^p\\) bound for certain \\(p<\\infty\\)?",
   "public_statement": "Let \\(u:\\mathbb R\\to\\mathbb R\\) be a measurable function. Define the maximal operator along the planar vector field \\((1,u)\\) by\n\\[\nM_u f(x,y):=\\sup_{\\epsilon>0}\\left|\n\\frac{1}{2\\epsilon}\\int_{-\\epsilon}^{\\epsilon}\nf(x-t,y-u(x)t)\\,dt\n\\right|.\n\\]\nDoes \\(M_u\\) satisfy any \\(L^p\\) bound for certain \\(p<\\infty\\)?",
   "evidence": "Here “the vector field \\((1,u)\\)” means \\(v(x,y)=(1,u(x))\\). The following repairs were made to the corpus OCR, and all were checked against the displayed formula in the PDF: 1. Each control character `\\u000f` is the glyph \\(\\epsilon\\). 2. The broken string `12\\u000f` is the fraction \\(1/(2\\epsilon)\\). 3. The integral limits are \\(-\\epsilon\\) and \\(\\epsilon\\). 4. The vertical bars enclose the whole signed average; the source does not put \\(|f|\\) inside the integral. 5. Fragmented line breaks and the superscripts in \\(\\mathbb R\\) and \\(L^p\\) were restored.",
   "classification_method": "source_verified_raw_character_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-analysis-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-ANALYSIS-0061": {
   "statement_status": "exact",
   "original_statement": "Question 2 (Christ). On Rd, let E ⊂ Rd with |E| = 1. Given q > 2,describe the set E that maximises the quantity ‖̂1E ‖q.\n\nCertain partial progress has been made. \n\nTheorem 0.1 (Christ [3]). 1. For any q > 2, the extremizing set exists. 2. For any dimension d, for any sufficiently large q which is also sufficiently close to 2N, a set E is a extremizer iff E is an ellipsoid. 3. If d = 1, then for any q close to 2N, a set E is a extremizer iff E is an ellipsoid. 4. If d = 2, then for any q close to 4, a set E is a extremizer iff E is an ellipsoid.",
   "clean_statement": "Question 2 (Christ). On Rd, let E ⊂ Rd with |E| = 1. Given q > 2,describe the set E that maximises the quantity ‖̂1E ‖q.\n\nCertain partial progress has been made.\n\nTheorem 0.1 (Christ [3]). 1. For any q > 2, the extremizing set exists. 2. For any dimension d, for any sufficiently large q which is also sufficiently close to 2N, a set E is a extremizer iff E is an ellipsoid. 3. If d = 1, then for any q close to 2N, a set E is a extremizer iff E is an ellipsoid. 4. If d = 2, then for any q close to 4, a set E is a extremizer iff E is an ellipsoid.",
   "public_statement": "Question 2 (Christ). On Rd, let E ⊂ Rd with |E| = 1. Given q > 2,describe the set E that maximises the quantity ‖̂1E ‖q.\n\nCertain partial progress has been made.\n\nTheorem 0.1 (Christ [3]). 1. For any q > 2, the extremizing set exists. 2. For any dimension d, for any sufficiently large q which is also sufficiently close to 2N, a set E is a extremizer iff E is an ellipsoid. 3. If d = 1, then for any q close to 2N, a set E is a extremizer iff E is an ellipsoid. 4. If d = 2, then for any q close to 4, a set E is a extremizer iff E is an ellipsoid.",
   "evidence": "This record is Question 2 from the AIM workshop problem list *Carleson theorems and multilinear operators*. The PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-ANALYSIS-0062": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 3 (Christ). Let B be the unit ball in R3. Let N be a positive integer. Let {Vj: 1 ≤ j ≤ N } be N different light cones in R3. Prove that \n\n∣∣∣∣∣∣∫\n\n> B\n\neiλx 23\n\n> N\n\n∏\n\n> j=1\n\nfj (x · vj )dx \n\n∣∣∣∣∣∣. λ−\u000fN∏\n\n> j=1\n\n‖fj ‖∞, (0.2) \n\nfor certain positive \u000f, where vj ∈ Vj. If possible, find the optimal \u000f.\n\n1So far (0.2) has only been proved for N ≤ 5, see [4].",
   "clean_statement": null,
   "public_statement": "Question 3 (Christ). Let B be the unit ball in R3. Let N be a positive integer. Let {Vj: 1 ≤ j ≤ N } be N different light cones in R3. Prove that\n\n∣∣∣∣∣∣∫\n\n> B\n\neiλx 23\n\n> N\n\n∏\n\n> j=1\n\nfj (x · vj )dx\n\n∣∣∣∣∣∣. λ−[U+000F]N∏\n\n> j=1\n\n‖fj ‖∞, (0.2)\n\nfor certain positive [U+000F], where vj ∈ Vj. If possible, find the optimal [U+000F].\n\n1So far (0.2) has only been proved for N ≤ 5, see [4].",
   "evidence": "The canonical record is Question 3 (attributed to Michael Christ) in the AIM open-problem list *Carleson theorems and multilinear operators*. The extraction damage can be repaired directly from the official PDF and its TeX source: the phase is",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-ANALYSIS-0063": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 4 (Bennett). Suppose we are in R4. Let \u000f > 0. Suppose that T1,\n\nT2 and T3 are three transversal families of δ-tubes (short sides δ and long side 1) such that for each j ∈ { 1, 2, 3}, {e(Tj ): Tj ∈ Tj } forms a δ-separated subset of S3. If q ≥ 43 and 1 \n\n> p\n\n+ 3 \n\n> q\n\n≤ 3, then there exists a constant C\u000f > 0\n\nsuch that \n\n∥∥∥∥∥∥\n\n> 3\n\n∏\n\n> j=1\n\n ∑ \n\n> Tj∈Tj\n\nχTj\n\n∥∥∥∥∥∥Lq/ 3(R4)\n\n≤ C\u000f\n\n> 3\n\n∏\n\n> j=1\n\nδ 4 \n\n> q−3\n> p′−\u000f\n\n(Tj )1/p. (0.3) \n\nHere e(T ) ∈ S3 denotes the direction of the long side of a tube T.",
   "clean_statement": "Suppose that we are in \\(\\mathbb R^4\\), and let \\(\\varepsilon>0\\). Suppose that \\(\\mathbb T_1,\\mathbb T_2,\\mathbb T_3\\) are three transversal families of \\(\\delta\\)-tubes (three short sides of length \\(\\delta\\) and one long side of length \\(1\\)) such that, for each \\(j\\in\\{1,2,3\\}\\),\n\\[\n\\{e(T):T\\in\\mathbb T_j\\}\n\\]\nis a \\(\\delta\\)-separated subset of \\(S^3\\). If\n\\[\nq\\geq \\frac43,\n\\qquad\n\\frac1p+\\frac3q\\leq 3,\n\\]\nthen is there a constant \\(C_\\varepsilon>0\\) such that\n\\[\n\\left\\|\n\\prod_{j=1}^{3}\n\\left(\\sum_{T\\in\\mathbb T_j}\\chi_T\\right)\n\\right\\|_{L^{q/3}(\\mathbb R^4)}\n\\leq C_\\varepsilon\n\\prod_{j=1}^{3}\n\\left[\n\\delta^{\\,4/q-3/p'-\\varepsilon}\n(\\#\\mathbb T_j)^{1/p}\n\\right]?\n\\tag{AIM}\n\\]\nHere \\(e(T)\\in S^3\\) is the direction of the long side of \\(T\\), and \\(p'\\) is the conjugate exponent.",
   "public_statement": "Question 4 (Bennett). Suppose we are in R4. Let [U+000F] > 0. Suppose that T1,\n\nT2 and T3 are three transversal families of δ-tubes (short sides δ and long side 1) such that for each j ∈ { 1, 2, 3}, {e(Tj ): Tj ∈ Tj } forms a δ-separated subset of S3. If q ≥ 43 and 1\n\n> p\n\n+ 3\n\n> q\n\n≤ 3, then there exists a constant C[U+000F] > 0\n\nsuch that\n\n∥∥∥∥∥∥\n\n> 3\n\n∏\n\n> j=1\n\n ∑\n\n> Tj∈Tj\n\nχTj\n\n∥∥∥∥∥∥Lq/ 3(R4)\n\n≤ C[U+000F]\n\n> 3\n\n∏\n\n> j=1\n\nδ 4\n\n> q−3\n> p′−[U+000F]\n\n(Tj )1/p. (0.3)\n\nHere e(T ) ∈ S3 denotes the direction of the long side of a tube T.",
   "evidence": "The canonical record is Question 4 (Bennett) in the AIM workshop list *Carleson theorems and multilinear operators*. The JSON extraction has several consequential OCR errors: “43” is \\(4/3\\), the control character is \\(\\varepsilon\\), and line breaks obscure both the admissibility condition and the location of the product. Inspection of page 2 of the authoritative AIM PDF gives the following statement.",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-analysis-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-ANALYSIS-0064": {
   "statement_status": "exact",
   "original_statement": "Question 5 (Di Plinio). Let N ∈ N. Given a collection of N directions \n\n{vj ∈ S1: 1 ≤ j ≤ N }. Does it hold true that \n\n‖ sup \n\n> j∈{ 1,2,...,N }\n\n|Hvj f |‖ L2,∞(R2). √log N ‖f ‖2? (0.4) \n\nHere Hvj f denotes the Hilbert transform along the given direction vj, namely \n\nHvj f (x):= \n\n∫\n\n> R\n\nf (x − tv j ) dt t. (0.5) The maximal variant of the estimate (0.4) has been proved by Katz [8]. Moreover, (0.4) has also been verified for sets of two \"extreme\" structures: the lacunary set and the Vargas set. One typical example of the Vargas set is the set of uniformly distributed directions. See Demeter [5], Demeter and Di Plinio [6].",
   "clean_statement": "Question 5 (Di Plinio). Let N ∈ N. Given a collection of N directions\n\n{vj ∈ S1: 1 ≤ j ≤ N }. Does it hold true that\n\n‖ sup\n\n> j∈{ 1,2,...,N }\n\n|Hvj f |‖ L2,∞(R2). √log N ‖f ‖2? (0.4)\n\nHere Hvj f denotes the Hilbert transform along the given direction vj, namely\n\nHvj f (x):=\n\n∫\n\n> R\n\nf (x − tv j ) dt t. (0.5) The maximal variant of the estimate (0.4) has been proved by Katz [8]. Moreover, (0.4) has also been verified for sets of two \"extreme\" structures: the lacunary set and the Vargas set. One typical example of the Vargas set is the set of uniformly distributed directions. See Demeter [5], Demeter and Di Plinio [6].",
   "public_statement": "Question 5 (Di Plinio). Let N ∈ N. Given a collection of N directions\n\n{vj ∈ S1: 1 ≤ j ≤ N }. Does it hold true that\n\n‖ sup\n\n> j∈{ 1,2,...,N }\n\n|Hvj f |‖ L2,∞(R2). √log N ‖f ‖2? (0.4)\n\nHere Hvj f denotes the Hilbert transform along the given direction vj, namely\n\nHvj f (x):=\n\n∫\n\n> R\n\nf (x − tv j ) dt t. (0.5) The maximal variant of the estimate (0.4) has been proved by Katz [8]. Moreover, (0.4) has also been verified for sets of two \"extreme\" structures: the lacunary set and the Vargas set. One typical example of the Vargas set is the set of uniformly distributed directions. See Demeter [5], Demeter and Di Plinio [6].",
   "evidence": "The canonical record is Question 5 (Di Plinio) in *Carleson theorems and multilinear operators: Open problems*, p. 2 of the AIM PDF. The authoritative display is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-ANALYSIS-0065": {
   "statement_status": "exact",
   "original_statement": "Question 6 (Street). Prove \n\n∣∣∣∣∫\n\n> R2\n\n∫\n\n> R2\n\nf (x)g(y)h(x + y) 1\n\ndet (x, y ) dxdy \n\n∣∣∣∣. ‖f ‖p‖g‖q‖h‖r, (0.6) \n\nfor certain p, q and r.\n\nThis question has a quite satisfactory answer. See Gressman et al. [7]. 2",
   "clean_statement": "Question 6 (Street). Prove\n\n∣∣∣∣∫\n\n> R2\n\n∫\n\n> R2\n\nf (x)g(y)h(x + y) 1\n\ndet (x, y ) dxdy\n\n∣∣∣∣. ‖f ‖p‖g‖q‖h‖r, (0.6)\n\nfor certain p, q and r.\n\nThis question has a quite satisfactory answer. See Gressman et al. [7]. 2",
   "public_statement": "Question 6 (Street). Prove\n\n∣∣∣∣∫\n\n> R2\n\n∫\n\n> R2\n\nf (x)g(y)h(x + y) 1\n\ndet (x, y ) dxdy\n\n∣∣∣∣. ‖f ‖p‖g‖q‖h‖r, (0.6)\n\nfor certain p, q and r.\n\nThis question has a quite satisfactory answer. See Gressman et al. [7]. 2",
   "evidence": "The canonical record is Question 6 (Street) from the AIM workshop *Carleson theorems and multilinear operators*. The official AIM TeX reads",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-ANALYSIS-0066": {
   "statement_status": "exact",
   "original_statement": "Question 7 (Krause). On the plane R2, let R: S1 → S1 be the rotation by \n\nπ/ 3. Prove \n\n∥∥∥∥ sup \n\n> t∈R+\n\n∣∣∣∣∫\n\n> S1\n\nf (x − tw )g(x − tR (w)) dσ (w)\n\n∣∣∣∣∥∥∥∥r. ‖f ‖p‖g‖q, (0.7) \n\nfor certain p, q and r.\n\nDuring the workshop, this has been shown to be equivalent to \n\n∥∥∥∥ sup \n\n> t∈R+\n\n∣∣∣∣∫\n\n> S1\n\nf (x − tw )g(x + tw )dσ (w)\n\n∣∣∣∣∥∥∥∥r. ‖f ‖p‖g‖q. (0.8)",
   "clean_statement": "Question 7 (Krause). On the plane R2, let R: S1 → S1 be the rotation by\n\nπ/ 3. Prove\n\n∥∥∥∥ sup\n\n> t∈R+\n\n∣∣∣∣∫\n\n> S1\n\nf (x − tw )g(x − tR (w)) dσ (w)\n\n∣∣∣∣∥∥∥∥r. ‖f ‖p‖g‖q, (0.7)\n\nfor certain p, q and r.\n\nDuring the workshop, this has been shown to be equivalent to\n\n∥∥∥∥ sup\n\n> t∈R+\n\n∣∣∣∣∫\n\n> S1\n\nf (x − tw )g(x + tw )dσ (w)\n\n∣∣∣∣∥∥∥∥r. ‖f ‖p‖g‖q. (0.8)",
   "public_statement": "Question 7 (Krause). On the plane R2, let R: S1 → S1 be the rotation by\n\nπ/ 3. Prove\n\n∥∥∥∥ sup\n\n> t∈R+\n\n∣∣∣∣∫\n\n> S1\n\nf (x − tw )g(x − tR (w)) dσ (w)\n\n∣∣∣∣∥∥∥∥r. ‖f ‖p‖g‖q, (0.7)\n\nfor certain p, q and r.\n\nDuring the workshop, this has been shown to be equivalent to\n\n∥∥∥∥ sup\n\n> t∈R+\n\n∣∣∣∣∫\n\n> S1\n\nf (x − tw )g(x + tw )dσ (w)\n\n∣∣∣∣∥∥∥∥r. ‖f ‖p‖g‖q. (0.8)",
   "evidence": "The record is Question 7 (attributed to Krause) from the AIM workshop *Carleson theorems and multilinear operators*. The OCR in `input.json` breaks several displayed formulas, so the statement was checked against the official AIM TeX source as well as the PDF linked in the record. The TeX source reads as follows (with only notation typeset more compactly here):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-ANALYSIS-0067": {
   "statement_status": "exact",
   "original_statement": "Question 8 (Anderson, Pierce). Generalise Stein and Wainger's polynomial Carleson's theorem to the discrete setting, namely to prove \n\n∥∥∥∥∥sup \n\n> λ\n\n∣∣∣∣∣∑\n\n> m∈Z\n\nf (n − m) eiλm 2\n\nm\n\n∣∣∣∣∣∥∥∥∥∥2. ‖f ‖2. (0.9) Let Λ ⊂ [0, 1]. Define sup \n\n> λ∈Λ\n\n∣∣∣∣∣∑\n\n> m∈Z\n\nf (n − m) eiλm 2\n\nm\n\n∣∣∣∣∣. (0.10) A sufficient condition has been given on the set Λ, to guarantee the l2\n\nboundedness of (0.10). See Krause and Lacey [9].",
   "clean_statement": "Question 8 (Anderson, Pierce). Generalise Stein and Wainger's polynomial Carleson's theorem to the discrete setting, namely to prove\n\n∥∥∥∥∥sup\n\n> λ\n\n∣∣∣∣∣∑\n\n> m∈Z\n\nf (n − m) eiλm 2\n\nm\n\n∣∣∣∣∣∥∥∥∥∥2. ‖f ‖2. (0.9) Let Λ ⊂ [0, 1]. Define sup\n\n> λ∈Λ\n\n∣∣∣∣∣∑\n\n> m∈Z\n\nf (n − m) eiλm 2\n\nm\n\n∣∣∣∣∣. (0.10) A sufficient condition has been given on the set Λ, to guarantee the l2\n\nboundedness of (0.10). See Krause and Lacey [9].",
   "public_statement": "Question 8 (Anderson, Pierce). Generalise Stein and Wainger's polynomial Carleson's theorem to the discrete setting, namely to prove\n\n∥∥∥∥∥sup\n\n> λ\n\n∣∣∣∣∣∑\n\n> m∈Z\n\nf (n − m) eiλm 2\n\nm\n\n∣∣∣∣∣∥∥∥∥∥2. ‖f ‖2. (0.9) Let Λ ⊂ [0, 1]. Define sup\n\n> λ∈Λ\n\n∣∣∣∣∣∑\n\n> m∈Z\n\nf (n − m) eiλm 2\n\nm\n\n∣∣∣∣∣. (0.10) A sufficient condition has been given on the set Λ, to guarantee the l2\n\nboundedness of (0.10). See Krause and Lacey [9].",
   "evidence": "The canonical record is Question 8 (Anderson, Pierce) in *Carleson theorems and multilinear operators: Open problems*, p. 3 of the authoritative AIM PDF. The PDF asks for the discrete analogue of Stein--Wainger's polynomial Carleson theorem:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-ANALYSIS-0068": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 9 (Li). Let d ≥ 3. For p ≥ 2( d + 1), prove \n\n‖\n\n> N\n\n∑\n\n> n=1\n\nane2πin dte2πin ·x‖Lp(T2). N 12 − d+1 \n\n> p+\u000f\n\n(\n\n> N\n\n∑\n\n> n=1\n\n|an|2)1/2. (0.11) This is related to Waring's problem.",
   "clean_statement": null,
   "public_statement": "Question 9 (Li). Let d ≥ 3. For p ≥ 2( d + 1), prove\n\n‖\n\n> N\n\n∑\n\n> n=1\n\nane2πin dte2πin ·x‖Lp(T2). N 12 − d+1\n\n> p+[U+000F]\n\n(\n\n> N\n\n∑\n\n> n=1\n\n|an|2)1/2. (0.11) This is related to Waring's problem.",
   "evidence": "The canonical JSON extraction is visibly corrupted: superscripts, the exponent of \\(N\\), the coefficient subscript, and the two torus variables have been split by PDF extraction. The original AIM workshop PDF was checked. Its Question 9 is:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-ANALYSIS-0069": {
   "statement_status": "exact",
   "original_statement": "Question 10 (Bez). Let S1, S 2 and S3 be transversal patches of the unit sphere S3 in R4. Let σ1, σ 2 and σ3 be the surface measure separately. Deter-mine the full range of exponents p, q > 0 such that the multi-linear singular convolution estimate \n\n‖g1dσ 1 ∗ g2dσ 2 ∗ g3dσ 3‖q.\n\n> 3\n\n∏\n\n> j=1\n\n‖gj ‖p (0.12) \n\nholds. \n\n3",
   "clean_statement": "Question 10 (Bez). Let S1, S 2 and S3 be transversal patches of the unit sphere S3 in R4. Let σ1, σ 2 and σ3 be the surface measure separately. Deter-mine the full range of exponents p, q > 0 such that the multi-linear singular convolution estimate\n\n‖g1dσ 1 ∗ g2dσ 2 ∗ g3dσ 3‖q.\n\n> 3\n\n∏\n\n> j=1\n\n‖gj ‖p (0.12)\n\nholds.\n\n3",
   "public_statement": "Question 10 (Bez). Let S1, S 2 and S3 be transversal patches of the unit sphere S3 in R4. Let σ1, σ 2 and σ3 be the surface measure separately. Deter-mine the full range of exponents p, q > 0 such that the multi-linear singular convolution estimate\n\n‖g1dσ 1 ∗ g2dσ 2 ∗ g3dσ 3‖q.\n\n> 3\n\n∏\n\n> j=1\n\n‖gj ‖p (0.12)\n\nholds.\n\n3",
   "evidence": "The canonical JSON record is visibly damaged by PDF extraction. I therefore checked the official AIM TeX source as well as the linked PDF. The source reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-ANALYSIS-0070": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 11 (Muscalu). Let K: R2 → R be a function such that \n\n|∂α ˆK(ξ)|. 1\n\n|ξ||α|, ∀ξ ∈ R2 \\ { 0}, (0.13) \n\nfor sufficiently many multi-indices α. Generalise Stein and Wainger's poly-nomial Carleson's theorem to the multi-linear setting. For example, to prove \n\n‖ sup \n\n> λ∈R\n\n|\n\n∫\n\n> R2\n\nf (x − t)g(x − s)K(t, s )eiλs 2t2\n\ndtds |‖ 2. ‖f ‖4‖g‖4. (0.14) The multi-parameter Carleson's theorem has been proved by Li and Mus-calu [11]: Let K be given as in (0.13). Define \n\nC2(f, g )( x):= sup \n\n> N1,N 2\n\n∣∣∣∣∫\n\n> R2\n\nˆK(ξ1 − N1, ξ 2, N 2) ˆf1(ξ1) ˆf2(ξ2)dξ 1dξ 2\n\n∣∣∣∣, (0.15) then \n\n‖C2(f1, f 2)‖2. ‖f1‖4‖f2‖4. (0.16)",
   "clean_statement": null,
   "public_statement": "Question 11 (Muscalu). Let K: R2 → R be a function such that\n\n|∂α ˆK(ξ)|. 1\n\n|ξ||α|, ∀ξ ∈ R2 \\ { 0}, (0.13)\n\nfor sufficiently many multi-indices α. Generalise Stein and Wainger's poly-nomial Carleson's theorem to the multi-linear setting. For example, to prove\n\n‖ sup\n\n> λ∈R\n\n|\n\n∫\n\n> R2\n\nf (x − t)g(x − s)K(t, s )eiλs 2t2\n\ndtds |‖ 2. ‖f ‖4‖g‖4. (0.14) The multi-parameter Carleson's theorem has been proved by Li and Mus-calu [11]: Let K be given as in (0.13). Define\n\nC2(f, g )( x):= sup\n\n> N1,N 2\n\n∣∣∣∣∫\n\n> R2\n\nˆK(ξ1 − N1, ξ 2, N 2) ˆf1(ξ1) ˆf2(ξ2)dξ 1dξ 2\n\n∣∣∣∣, (0.15) then\n\n‖C2(f1, f 2)‖2. ‖f1‖4‖f2‖4. (0.16)",
   "evidence": "The canonical record is Question 11 (Camil Muscalu) from the 2015 AIM workshop *Carleson theorems and multilinear operators*. The official AIM PDF gives the following multiplier hypothesis. With $m=\\widehat K$,",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 69,
   "attempt": 1
  },
  "AIM-ANALYSIS-0071": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 12 (Guo). To prove that there exists a universal constant C > 0\n\nsuch that ∀\u000f ∈ (0, 1/2), it holds that \n\n‖ sup \n\n> λ∈R\n\n∫\n\n> R\n\nf (x − t)eiλ |t|\u000f dt t ‖2 ≤ C‖f ‖2. (0.17)",
   "clean_statement": null,
   "public_statement": "Question 12 (Guo). To prove that there exists a universal constant C > 0\n\nsuch that ∀[U+000F] ∈ (0, 1/2), it holds that\n\n‖ sup\n\n> λ∈R\n\n∫\n\n> R\n\nf (x − t)eiλ |t|[U+000F] dt t ‖2 ≤ C‖f ‖2. (0.17)",
   "evidence": "The JSON extraction contains damaged occurrences of the exponent and loses some absolute-value and principal-value notation. Question 12 in the AIM workshop problem list, checked against the source PDF and against Guo's definition of the same operator, is the following.",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 70,
   "attempt": 1
  },
  "AIM-ANALYSIS-0072": {
   "statement_status": "exact",
   "original_statement": "Question 13 (Carbery). On Rn, it is a big open problem whether \n\n∥∥∥∥∥sup \n\n> R\n\n∣∣∣∣∣∫\n\n> |ξ|≤ R\n\nˆf (ξ)e2πixξ dξ \n\n∣∣∣∣∣∥∥∥∥∥2. ‖f ‖2. (0.18) \n\nHow about ∥∥∥∥∥sup \n\n> R\n\n∣∣∣∣∣f ∗\n\n(\n\nei|x|\n\n|x| n+1 2\n\n· χ{| x|≤ R}\n\n)∣ ∣∣∣∣∥∥∥∥∥2. ‖f ‖2? (0.19) For detailed discussions, see Carbery et al. [2]. 4",
   "clean_statement": "Question 13 (Carbery). On Rn, it is a big open problem whether\n\n∥∥∥∥∥sup\n\n> R\n\n∣∣∣∣∣∫\n\n> |ξ|≤ R\n\nˆf (ξ)e2πixξ dξ\n\n∣∣∣∣∣∥∥∥∥∥2. ‖f ‖2. (0.18)\n\nHow about ∥∥∥∥∥sup\n\n> R\n\n∣∣∣∣∣f ∗\n\n(\n\nei|x|\n\n|x| n+1 2\n\n· χ{| x|≤ R}\n\n)∣ ∣∣∣∣∥∥∥∥∥2. ‖f ‖2? (0.19) For detailed discussions, see Carbery et al. [2]. 4",
   "public_statement": "Question 13 (Carbery). On Rn, it is a big open problem whether\n\n∥∥∥∥∥sup\n\n> R\n\n∣∣∣∣∣∫\n\n> |ξ|≤ R\n\nˆf (ξ)e2πixξ dξ\n\n∣∣∣∣∣∥∥∥∥∥2. ‖f ‖2. (0.18)\n\nHow about ∥∥∥∥∥sup\n\n> R\n\n∣∣∣∣∣f ∗\n\n(\n\nei|x|\n\n|x| n+1 2\n\n· χ{| x|≤ R}\n\n)∣ ∣∣∣∣∥∥∥∥∥2. ‖f ‖2? (0.19) For detailed discussions, see Carbery et al. [2]. 4",
   "evidence": "The canonical JSON record is visibly damaged by PDF text extraction. I checked the official AIM problem-list PDF, *Carleson theorems and multilinear operators: Open problems*, page 4 of the PDF. With the Fourier convention",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-ANALYSIS-0073": {
   "statement_status": "exact",
   "original_statement": "Question 14 (Iliopoulou). In Rn, let Ti, i ∈ { 1, 2,..., n } be a collection of tubes with width one and infinity length. We know that \n\n∫  ∑\n\n> T1∈T 1\n\nχT1\n\n... \n\n( ∑\n\n> Tn∈T n\n\nχTn\n\n)\n\nw(T1) ∧... ∧ w(Tn)\n\n\n\n> 1\n> n−1\n\n≤ Cnn∏\n\n> i=1\n\n(Ti) 1 \n\n> n−1,\n\n(0.20) \n\nwhere w(Ti) is the unit vector parallel to the long side of the tube Ti. Could we prove (0.20) with Cn = 1?",
   "clean_statement": "Question 14 (Iliopoulou). In Rn, let Ti, i ∈ { 1, 2,..., n } be a collection of tubes with width one and infinity length. We know that\n\n∫  ∑\n\n> T1∈T 1\n\nχT1\n\n...\n\n( ∑\n\n> Tn∈T n\n\nχTn\n\n)\n\nw(T1) ∧... ∧ w(Tn)\n\n\n\n> 1\n> n−1\n\n≤ Cnn∏\n\n> i=1\n\n(Ti) 1\n\n> n−1,\n\n(0.20)\n\nwhere w(Ti) is the unit vector parallel to the long side of the tube Ti. Could we prove (0.20) with Cn = 1?",
   "public_statement": "Question 14 (Iliopoulou). In Rn, let Ti, i ∈ { 1, 2,..., n } be a collection of tubes with width one and infinity length. We know that\n\n∫  ∑\n\n> T1∈T 1\n\nχT1\n\n...\n\n( ∑\n\n> Tn∈T n\n\nχTn\n\n)\n\nw(T1) ∧... ∧ w(Tn)\n\n\n\n> 1\n> n−1\n\n≤ Cnn∏\n\n> i=1\n\n(Ti) 1\n\n> n−1,\n\n(0.20)\n\nwhere w(Ti) is the unit vector parallel to the long side of the tube Ti. Could we prove (0.20) with Cn = 1?",
   "evidence": "The canonical record is Question 14 (Marina Iliopoulou) from the 18--22 May 2015 AIM workshop *Carleson theorems and multilinear operators*. The official TeX says that, for collections \\(\\mathcal T_i\\) of doubly infinite tubes of width one in \\(\\mathbb R^n\\),",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 72,
   "attempt": 2
  },
  "AIM-ANALYSIS-0074": {
   "statement_status": "exact",
   "original_statement": "QUESTION 1: Does there exist a subset J of $1, ..., m$ such that\n$$\\|\\sum_{j\\in J} a_j - \\sum_{j\\in J^c} a_j\\|< O(\\epsilon^{1/8})\\ ?$$\n\nCOMMENT: The MSS method for Kadison-Singer seems to rely on small norm and rank one, but is it really about rank or is it about trace norm being small? Of course the 1/8 is just something to think about. We know $\\epsilon$ won't work, but maybe $\\sqrt \\epsilon$. This is related to the question raised by in the last section of the MSS paper.\n\nA von Neumann algebra is a weakly closed self adjoint sub algebra of $B(H)$. A factor is a von Neumann algebra with trivial center. Murray and von Neumann classified factors into types I, II, and III. Type I factors all have the form $B(H)$, and the only difference between them is the cardinality of $H$. Types II and III, on the other hand, each display uncountably many distinct variations as subalgebras of $B(H)$ for separable $H$. These have MASA's that are also of different types, and a Kadison-Singer problem arises as follows.",
   "clean_statement": "QUESTION 1: Does there exist a subset J of $1, ..., m$ such that\n$$\\|\\sum_{j\\in J} a_j - \\sum_{j\\in J^c} a_j\\|< O(\\epsilon^{1/8})\\ ?$$\n\nCOMMENT: The MSS method for Kadison-Singer seems to rely on small norm and rank one, but is it really about rank or is it about trace norm being small? Of course the 1/8 is just something to think about. We know $\\epsilon$ won't work, but maybe $\\sqrt \\epsilon$. This is related to the question raised by in the last section of the MSS paper.\n\nA von Neumann algebra is a weakly closed self adjoint sub algebra of $B(H)$. A factor is a von Neumann algebra with trivial center. Murray and von Neumann classified factors into types I, II, and III. Type I factors all have the form $B(H)$, and the only difference between them is the cardinality of $H$. Types II and III, on the other hand, each display uncountably many distinct variations as subalgebras of $B(H)$ for separable $H$. These have MASA's that are also of different types, and a Kadison-Singer problem arises as follows.",
   "public_statement": "QUESTION 1: Does there exist a subset J of $1, ..., m$ such that\n$$\\|\\sum_{j\\in J} a_j - \\sum_{j\\in J^c} a_j\\|< O(\\epsilon^{1/8})\\ ?$$\n\nCOMMENT: The MSS method for Kadison-Singer seems to rely on small norm and rank one, but is it really about rank or is it about trace norm being small? Of course the 1/8 is just something to think about. We know $\\epsilon$ won't work, but maybe $\\sqrt \\epsilon$. This is related to the question raised by in the last section of the MSS paper.\n\nA von Neumann algebra is a weakly closed self adjoint sub algebra of $B(H)$. A factor is a von Neumann algebra with trivial center. Murray and von Neumann classified factors into types I, II, and III. Type I factors all have the form $B(H)$, and the only difference between them is the cardinality of $H$. Types II and III, on the other hand, each display uncountably many distinct variations as subalgebras of $B(H)$ for separable $H$. These have MASA's that are also of different types, and a Kadison-Singer problem arises as follows.",
   "evidence": "The canonical record is source index 73 of aim-analysis-notes.json. Its displayed question omits the hypotheses that immediately precede it in the original workshop document. The original Google document gives the following setup:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 73,
   "attempt": 1
  },
  "AIM-ANALYSIS-0075": {
   "statement_status": "exact",
   "original_statement": "QUESTION 2: Given a MASA $A$ of a factor $M$, do the pure states of $A$ have unique state extension to $M$?\n\nSOME ANSWERS: MSS showed that for discrete MASAs of $B(H)$, $H$ separable, the answer is \"yes\". As Bernard pointed out in his opening talk, if the MASA has any continuous part, the answer is no (that was shown in the original Kadison-Singer paper). \n\nFor Type II factors on separable $H$ the answer is always NO, for any MASA but for non separable $H$, the answer is sometimes yes and sometimes no with no general results as yet. (Popa, Akemann, Sherman)\n\nFor type III factors there are no results at all for any factors and for any MASAs.",
   "clean_statement": "QUESTION 2: Given a MASA $A$ of a factor $M$, do the pure states of $A$ have unique state extension to $M$?\n\nSOME ANSWERS: MSS showed that for discrete MASAs of $B(H)$, $H$ separable, the answer is \"yes\". As Bernard pointed out in his opening talk, if the MASA has any continuous part, the answer is no (that was shown in the original Kadison-Singer paper).\n\nFor Type II factors on separable $H$ the answer is always NO, for any MASA but for non separable $H$, the answer is sometimes yes and sometimes no with no general results as yet. (Popa, Akemann, Sherman)\n\nFor type III factors there are no results at all for any factors and for any MASAs.",
   "public_statement": "QUESTION 2: Given a MASA $A$ of a factor $M$, do the pure states of $A$ have unique state extension to $M$?\n\nSOME ANSWERS: MSS showed that for discrete MASAs of $B(H)$, $H$ separable, the answer is \"yes\". As Bernard pointed out in his opening talk, if the MASA has any continuous part, the answer is no (that was shown in the original Kadison-Singer paper).\n\nFor Type II factors on separable $H$ the answer is always NO, for any MASA but for non separable $H$, the answer is sometimes yes and sometimes no with no general results as yet. (Popa, Akemann, Sherman)\n\nFor type III factors there are no results at all for any factors and for any MASAs.",
   "evidence": "The record was checked against the plain-text export of the linked AIM Google document. The text is intact; no OCR repair or mathematical reconstruction is needed. The source is the problem compilation for the AIM workshop *Beyond Kadison--Singer: paving and consequences*, held December 1--5, 2014. The compilation is dated November 30, 2014.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-ANALYSIS-0076": {
   "statement_status": "exact",
   "original_statement": "QUESTION 3: Let a C*-algebra $B$ be the direct sum of an ideal and a MASA, $B = I \\oplus A$. Under what circumstances do the pure states of $A$ have unique extension to states of $B$?\n\n \n* Cynthia Vinzant\n - there are existence proofs for KS, how to make them constructive? Likes stable polynomials and interlacers. How to make constructive proofs? \n - Find an algorithm for Weaver's conjecture/equivalence/theorem (or call it the vector partitioning problem), similarly for Paving Conjecture\n - Chuck started to describe a cooperative two-player game with partitioning the sets --> could we get an impossibility result via communication complexity?\n \n* Greg Knese\n - understanding intrinsic proofs of Feichtinger's conjecture\n - extend the theory of interlacing families to several variables\n - use a theorem to define interlacing in several variables\n - how do operators that preserve stability affect locations of roots (i.e., quantitative versions of MSS)\n \n* Petter Branden\n - combinatorics generally\n - better bounds, more quantitative results in Weaver's result\n - for the random MSS, there's a witness for the best bound, is there a conjecture for Weaver's conjecture? Weaver has a construction that \\sqrt\\eps is necessary but MSS has no constructive lower bound. Pete C. *does* have a constructive lower bound.\n - has a witness to a hyperbolic version of the problem\n - more generally, matching lower bounds for the different equivalences (the paving lower bound in the \\R case is 1/\\eps^2 and authors showed 6/\\eps^2)\n - Pete's example: can we make it better? It will affect all of the equivalences! It's almost a counter-example.\n - consequences to matroid theory: can we prove a continuous version of matroid theory using MSS theorem and what consequences to matroid theory does MSS have?\n - serious about hyperbolic frame theory!\n - extensions to hyperbolic polynomials --> just posted a new paper on the ArXiv\n \n* Marcin Bownik\n - interested in Feichtinger conjecture: suppose you have a Parseval frame and norms of vectors are bounded away from zero, what is the minimum number of partitions can we form that make Riesz basic sequences? Count the number of partitions? How does this number depend on \\epsilon, the lower bound (Pete C. also will show some results tomorrow on this question)? How to exhibit/construct such partitions?\n - Frames of complex of exponentials on a subset of the torus = exponential frames, derive better/best bounds for Feichtinger's conjecture for these frames.\n - exponential frame L^2([0,1]), consider measurable subset E \\subset [0,1]\n - \\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} are an exponential frame\n \n* Darrin Speegle\n - interested in Feichtinger conjecture, how large the partition has to be to get Riesz basic sequences, especially for complex exponentials, this suggests that there's something about Fourier series that we don't understand.\n - whether the number of partitions depends on |E| for exponential frames or any other properties of E? Are there any examples of two sets with same measure such that the number of partitions is different (or the constants are different)?\n - More concretely, can we find E, |E| = \\frac12 such that \n\\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} cannot be partitioned into two Riesz sequences?\n - there's a connection to synthetic sets and arithmetic progressions (Bourgain's work with Tzafriri): we now know we can partition exponential frames into synthetic sets (MSS says we can do so). How to do this (constructively?)?\n \n* Bill Johnson\n -Is paving true for $\\ell_p^n$? Specifically:\n\nSuppose $1\\le p<\\infty$. \n\n$Pav_p$: Does there exist for every $\\epsilon > 0$ a constant $k=k(\\epsilon, p)$ s.t. for every $n$ and every $n$ by $n$ zero diagonal matrix $A$ for which $\\|A\\|_p = 1$, there is a partition $\\sigma(1), \\dots, \\sigma(k) of $\\{1,\\dots,n\\}$ s.t. \n\n$$\n\\|\\sum_{j=1}^k P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p = \\max_j \n\\| P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p \\le \\epsilon .\n$$\n\nHere $\\|\\dot \\|_$ is the operator norm for operators on $\\ell_p^n$, NOT the Schatten $p$-class norm.\n\n$Pav_1$, due IIRC to Bourgain has been known for a long time and is relatively easy. It is more-or-less equivalent to $Pav_2$ for matrices that have non-negative entries, a result that is contained in\n\nBerman, Kenneth; Halpern, Herbert; Kaftal, Victor; Weiss, Gary; \nMatrix norm inequalities and the relative Dixmier property. \nIntegral Equations Operator Theory 11 (1988), no. 1, 28-48.\n\nIn their original paper, Bourgain and Tzafriri proved restricted invertibility for $\\ell_p^n$, but they got optimal estimates only for $p=2$. \n\n - Take A = n \\times n matrix that's a commutator and write it as a commutator of B,C with B diagonal of norm 1 and C with bounded norm, with some minimization. Bill's recent PNAS paper with Ozawa and Schechtmann showed something that needed KS with optimal estimates to improve bound from n^\\alpha (which uses only restricted invertibility) to poly(n), and that the existence of a bound independent of n implies KS. Can you improve the result using KS to get a bound independent of n?\n\n* Deanna Needell\n - design algorithms to get pavings, especially for stochastic optimization\n - interested in algorithms that are efficient!\n - algorithmic developments that use pavings\n - sampling and RIP constructions for sparse approximation, do these lead to constructions or insight for KS\n \n* Anna Gilbert\n - Fourier sampling questions, RIP for the DFT, Rudelson selection lemma, how to remove extra factors on the bounds for RIP for DFT\n \n* Olaf Mordhorst\n - applications to convex geometry and Banach spaces\n - Restricted invertibility: provide a concrete example of optimality which is proven via random matrices\n\n* Pierre Youssef\n - consequences in convex geometry\n - if you have an identity decamp in \\R^n, BSS approximated the identity with many fewer, albeit weighted, vectors. If you use MSS, you get an identity approximation without weights. Let's add a condition that the vectors are balanced (say they come from a convex body). \n - \\sum_j c_j x_j x_j^T with \\|x_j\\| = 1 and \\sum_j c_j x_j = 0\n - Find S \\subset [m] , |S| \\approx d so that\n C \\sum_{j \\in S} x_j x_j^T \\approx I with a balancing condition \\| \\sum x_j \\| \\leq B\n - this was done by Srivastava but got 4 + \\eps approximation of the identity\n - Given a convex body K in \\R^n, Restricted Invertibility says there exists an E \\subset \\R^n with dim(E) = (1-\\eps)n so that\n distance( K \\cap E, B_1) \\approx \\sqrt{dim(E)}\n - KS says that we can partition K into subspaces like this. Can we improve the Banach-Mazur distance of a convex body to the cube using this partition? The best bound currently is n^{5/6}.\n - Algorithmic version of KS: try the Batson-S-S approach to KS? BUT, they had convinced themselves that their approach wouldn't work. Perhaps it is worth to revisit this?\n \n* Leonid Gurvits\n - there's an algorithm: start with k vectors in n-dimnl space. Can you partition them into 5 sets so that set spans the space. Take each vector, repeat it 5 times and augment those matrices by identity to get dimension k. You'll get k semi-def-pos. matrices. Now you check each matrix for some property xxxx. This is an algorithm to carry out Rado-Horn?\n \n* Mirko Visontai\n - understand limitations of the proof, we don't have any idea of how many good partitions there are? come up with more quantitative versions of the proof.\n - can we generate new interlacing families?\n\n* Mihai Putinar\n - perturbation determinant: in interlacing families, we have quotients of such polynomials, they form rational functions in the Nevalinna class of functions that preserve the upper half plane\n - understand KS in terms of the phase shift theory, perhaps getting a proof of the KS in terms that analysts can understand\n \n* Dan Edidin\n - wants to geometrize this work! Paving projections is especially interesting. Fix r. Find the conditions on the diagonal of a projection so it can be r-paved? Conditions should be independent of dimension.\n - For KS2, the problem is static: given v, find a good partition. Wants a more dynamic version: look at all vectors together (in a manifold structure), so that you replace the discrete structure with a continuous one. Look over some parametrized space with geometric structure and this may be more helpful to characterize when there are lots of partitions (i.e., count) that satisfies KS2.\n\n* Pablo Parrilo\n - algorithmic questions!\n - To try to understand why the MSS proof works: many existence proofs use probabilistic methods or fixed point methods and they are non-constructive. But MSS proof combines different techniques. Understand how these pieces fit together more generally.\n \n* Rachel Ward\n - see the above on algorithmic!\n \n* Leonid Gurvits\n - Is there an analog of KS for zonotopes? Let z_1,\\ldots,z_k \\in \\R^n, k > n and let \\bar z_i denote the interval \\{ \\alpha z_i | 0 \\leq \\alpha \\leq 1\\}. Consider the Minkowski sum\n Zon = \\alpha_1 \\bar z_1 + \\cdots \\alpha_k \\bar z_k.\n - Conjecture: vol(Zon) = 1.\n - KS analog: symmetrize Zon and then seek a partition S \\cup T so that\n - \\sum_{i \\in S} \\bar z_i \\subset (1/2 \\pm \\eps) Zon.\n\n* Bernhard Bodmann\n - how to eliminate those polynomials?! Borgain-Tzafriri's proof did not have such polynomials, they used rank-1 perturbation theory. Can we reproduce such a proof at the level of multi-variate polynomials?\n \n* Adam Marcus\n - one of the biggest comments we get is that KS was so popular because it was ubiquitous so it would require some new tool that other people could use in lots of other areas, unfortunately, these polynomial tools are not portable to other areas. If someone *did* prove KS in your area, what would it look like? Let's construct such a tool so that you have a hammer to bang away with in your area! So, please think about what this tool would like!\n - There's a conjecture about repeated copies of Identity with zero padding that this gives you the worse bounds on roots?\n - Using interlacing polynomials but if you only want the maximum root, you don't need all these other roots to exist, can we trim our conditions so that the machinery is more usable? Is there some weaker form of interlacing if you don't care about all the roots.\n - There was something about adding rank-2 perturbations rather than rank-1 perturbations.",
   "clean_statement": "QUESTION 3: Let a C*-algebra $B$ be the direct sum of an ideal and a MASA, $B = I \\oplus A$. Under what circumstances do the pure states of $A$ have unique extension to states of $B$?\n\n\n* Cynthia Vinzant\n - there are existence proofs for KS, how to make them constructive? Likes stable polynomials and interlacers. How to make constructive proofs?\n - Find an algorithm for Weaver's conjecture/equivalence/theorem (or call it the vector partitioning problem), similarly for Paving Conjecture\n - Chuck started to describe a cooperative two-player game with partitioning the sets --> could we get an impossibility result via communication complexity?\n\n* Greg Knese\n - understanding intrinsic proofs of Feichtinger's conjecture\n - extend the theory of interlacing families to several variables\n - use a theorem to define interlacing in several variables\n - how do operators that preserve stability affect locations of roots (i.e., quantitative versions of MSS)\n\n* Petter Branden\n - combinatorics generally\n - better bounds, more quantitative results in Weaver's result\n - for the random MSS, there's a witness for the best bound, is there a conjecture for Weaver's conjecture? Weaver has a construction that \\sqrt\\eps is necessary but MSS has no constructive lower bound. Pete C. *does* have a constructive lower bound.\n - has a witness to a hyperbolic version of the problem\n - more generally, matching lower bounds for the different equivalences (the paving lower bound in the \\R case is 1/\\eps^2 and authors showed 6/\\eps^2)\n - Pete's example: can we make it better? It will affect all of the equivalences! It's almost a counter-example.\n - consequences to matroid theory: can we prove a continuous version of matroid theory using MSS theorem and what consequences to matroid theory does MSS have?\n - serious about hyperbolic frame theory!\n - extensions to hyperbolic polynomials --> just posted a new paper on the ArXiv\n\n* Marcin Bownik\n - interested in Feichtinger conjecture: suppose you have a Parseval frame and norms of vectors are bounded away from zero, what is the minimum number of partitions can we form that make Riesz basic sequences? Count the number of partitions? How does this number depend on \\epsilon, the lower bound (Pete C. also will show some results tomorrow on this question)? How to exhibit/construct such partitions?\n - Frames of complex of exponentials on a subset of the torus = exponential frames, derive better/best bounds for Feichtinger's conjecture for these frames.\n - exponential frame L^2([0,1]), consider measurable subset E \\subset [0,1]\n - \\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} are an exponential frame\n\n* Darrin Speegle\n - interested in Feichtinger conjecture, how large the partition has to be to get Riesz basic sequences, especially for complex exponentials, this suggests that there's something about Fourier series that we don't understand.\n - whether the number of partitions depends on |E| for exponential frames or any other properties of E? Are there any examples of two sets with same measure such that the number of partitions is different (or the constants are different)?\n - More concretely, can we find E, |E| = \\frac12 such that\n\\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} cannot be partitioned into two Riesz sequences?\n - there's a connection to synthetic sets and arithmetic progressions (Bourgain's work with Tzafriri): we now know we can partition exponential frames into synthetic sets (MSS says we can do so). How to do this (constructively?)?\n\n* Bill Johnson\n -Is paving true for $\\ell_p^n$? Specifically:\n\nSuppose $1\\le p<\\infty$.\n\n$Pav_p$: Does there exist for every $\\epsilon > 0$ a constant $k=k(\\epsilon, p)$ s.t. for every $n$ and every $n$ by $n$ zero diagonal matrix $A$ for which $\\|A\\|_p = 1$, there is a partition $\\sigma(1), \\dots, \\sigma(k) of $\\{1,\\dots,n\\}$ s.t.\n\n$$\n\\|\\sum_{j=1}^k P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p = \\max_j\n\\| P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p \\le \\epsilon .\n$$\n\nHere $\\|\\dot \\|_$ is the operator norm for operators on $\\ell_p^n$, NOT the Schatten $p$-class norm.\n\n$Pav_1$, due IIRC to Bourgain has been known for a long time and is relatively easy. It is more-or-less equivalent to $Pav_2$ for matrices that have non-negative entries, a result that is contained in\n\nBerman, Kenneth; Halpern, Herbert; Kaftal, Victor; Weiss, Gary;\nMatrix norm inequalities and the relative Dixmier property.\nIntegral Equations Operator Theory 11 (1988), no. 1, 28-48.\n\nIn their original paper, Bourgain and Tzafriri proved restricted invertibility for $\\ell_p^n$, but they got optimal estimates only for $p=2$.\n\n - Take A = n \\times n matrix that's a commutator and write it as a commutator of B,C with B diagonal of norm 1 and C with bounded norm, with some minimization. Bill's recent PNAS paper with Ozawa and Schechtmann showed something that needed KS with optimal estimates to improve bound from n^\\alpha (which uses only restricted invertibility) to poly(n), and that the existence of a bound independent of n implies KS. Can you improve the result using KS to get a bound independent of n?\n\n* Deanna Needell\n - design algorithms to get pavings, especially for stochastic optimization\n - interested in algorithms that are efficient!\n - algorithmic developments that use pavings\n - sampling and RIP constructions for sparse approximation, do these lead to constructions or insight for KS\n\n* Anna Gilbert\n - Fourier sampling questions, RIP for the DFT, Rudelson selection lemma, how to remove extra factors on the bounds for RIP for DFT\n\n* Olaf Mordhorst\n - applications to convex geometry and Banach spaces\n - Restricted invertibility: provide a concrete example of optimality which is proven via random matrices\n\n* Pierre Youssef\n - consequences in convex geometry\n - if you have an identity decamp in \\R^n, BSS approximated the identity with many fewer, albeit weighted, vectors. If you use MSS, you get an identity approximation without weights. Let's add a condition that the vectors are balanced (say they come from a convex body).\n - \\sum_j c_j x_j x_j^T with \\|x_j\\| = 1 and \\sum_j c_j x_j = 0\n - Find S \\subset [m] , |S| \\approx d so that\n C \\sum_{j \\in S} x_j x_j^T \\approx I with a balancing condition \\| \\sum x_j \\| \\leq B\n - this was done by Srivastava but got 4 + \\eps approximation of the identity\n - Given a convex body K in \\R^n, Restricted Invertibility says there exists an E \\subset \\R^n with dim(E) = (1-\\eps)n so that\n distance( K \\cap E, B_1) \\approx \\sqrt{dim(E)}\n - KS says that we can partition K into subspaces like this. Can we improve the Banach-Mazur distance of a convex body to the cube using this partition? The best bound currently is n^{5/6}.\n - Algorithmic version of KS: try the Batson-S-S approach to KS? BUT, they had convinced themselves that their approach wouldn't work. Perhaps it is worth to revisit this?\n\n* Leonid Gurvits\n - there's an algorithm: start with k vectors in n-dimnl space. Can you partition them into 5 sets so that set spans the space. Take each vector, repeat it 5 times and augment those matrices by identity to get dimension k. You'll get k semi-def-pos. matrices. Now you check each matrix for some property xxxx. This is an algorithm to carry out Rado-Horn?\n\n* Mirko Visontai\n - understand limitations of the proof, we don't have any idea of how many good partitions there are? come up with more quantitative versions of the proof.\n - can we generate new interlacing families?\n\n* Mihai Putinar\n - perturbation determinant: in interlacing families, we have quotients of such polynomials, they form rational functions in the Nevalinna class of functions that preserve the upper half plane\n - understand KS in terms of the phase shift theory, perhaps getting a proof of the KS in terms that analysts can understand\n\n* Dan Edidin\n - wants to geometrize this work! Paving projections is especially interesting. Fix r. Find the conditions on the diagonal of a projection so it can be r-paved? Conditions should be independent of dimension.\n - For KS2, the problem is static: given v, find a good partition. Wants a more dynamic version: look at all vectors together (in a manifold structure), so that you replace the discrete structure with a continuous one. Look over some parametrized space with geometric structure and this may be more helpful to characterize when there are lots of partitions (i.e., count) that satisfies KS2.\n\n* Pablo Parrilo\n - algorithmic questions!\n - To try to understand why the MSS proof works: many existence proofs use probabilistic methods or fixed point methods and they are non-constructive. But MSS proof combines different techniques. Understand how these pieces fit together more generally.\n\n* Rachel Ward\n - see the above on algorithmic!\n\n* Leonid Gurvits\n - Is there an analog of KS for zonotopes? Let z_1,\\ldots,z_k \\in \\R^n, k > n and let \\bar z_i denote the interval \\{ \\alpha z_i | 0 \\leq \\alpha \\leq 1\\}. Consider the Minkowski sum\n Zon = \\alpha_1 \\bar z_1 + \\cdots \\alpha_k \\bar z_k.\n - Conjecture: vol(Zon) = 1.\n - KS analog: symmetrize Zon and then seek a partition S \\cup T so that\n - \\sum_{i \\in S} \\bar z_i \\subset (1/2 \\pm \\eps) Zon.\n\n* Bernhard Bodmann\n - how to eliminate those polynomials?! Borgain-Tzafriri's proof did not have such polynomials, they used rank-1 perturbation theory. Can we reproduce such a proof at the level of multi-variate polynomials?\n\n* Adam Marcus\n - one of the biggest comments we get is that KS was so popular because it was ubiquitous so it would require some new tool that other people could use in lots of other areas, unfortunately, these polynomial tools are not portable to other areas. If someone *did* prove KS in your area, what would it look like? Let's construct such a tool so that you have a hammer to bang away with in your area! So, please think about what this tool would like!\n - There's a conjecture about repeated copies of Identity with zero padding that this gives you the worse bounds on roots?\n - Using interlacing polynomials but if you only want the maximum root, you don't need all these other roots to exist, can we trim our conditions so that the machinery is more usable? Is there some weaker form of interlacing if you don't care about all the roots.\n - There was something about adding rank-2 perturbations rather than rank-1 perturbations.",
   "public_statement": "QUESTION 3: Let a C*-algebra $B$ be the direct sum of an ideal and a MASA, $B = I \\oplus A$. Under what circumstances do the pure states of $A$ have unique extension to states of $B$?\n\n\n* Cynthia Vinzant\n - there are existence proofs for KS, how to make them constructive? Likes stable polynomials and interlacers. How to make constructive proofs?\n - Find an algorithm for Weaver's conjecture/equivalence/theorem (or call it the vector partitioning problem), similarly for Paving Conjecture\n - Chuck started to describe a cooperative two-player game with partitioning the sets --> could we get an impossibility result via communication complexity?\n\n* Greg Knese\n - understanding intrinsic proofs of Feichtinger's conjecture\n - extend the theory of interlacing families to several variables\n - use a theorem to define interlacing in several variables\n - how do operators that preserve stability affect locations of roots (i.e., quantitative versions of MSS)\n\n* Petter Branden\n - combinatorics generally\n - better bounds, more quantitative results in Weaver's result\n - for the random MSS, there's a witness for the best bound, is there a conjecture for Weaver's conjecture? Weaver has a construction that \\sqrt\\eps is necessary but MSS has no constructive lower bound. Pete C. *does* have a constructive lower bound.\n - has a witness to a hyperbolic version of the problem\n - more generally, matching lower bounds for the different equivalences (the paving lower bound in the \\R case is 1/\\eps^2 and authors showed 6/\\eps^2)\n - Pete's example: can we make it better? It will affect all of the equivalences! It's almost a counter-example.\n - consequences to matroid theory: can we prove a continuous version of matroid theory using MSS theorem and what consequences to matroid theory does MSS have?\n - serious about hyperbolic frame theory!\n - extensions to hyperbolic polynomials --> just posted a new paper on the ArXiv\n\n* Marcin Bownik\n - interested in Feichtinger conjecture: suppose you have a Parseval frame and norms of vectors are bounded away from zero, what is the minimum number of partitions can we form that make Riesz basic sequences? Count the number of partitions? How does this number depend on \\epsilon, the lower bound (Pete C. also will show some results tomorrow on this question)? How to exhibit/construct such partitions?\n - Frames of complex of exponentials on a subset of the torus = exponential frames, derive better/best bounds for Feichtinger's conjecture for these frames.\n - exponential frame L^2([0,1]), consider measurable subset E \\subset [0,1]\n - \\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} are an exponential frame\n\n* Darrin Speegle\n - interested in Feichtinger conjecture, how large the partition has to be to get Riesz basic sequences, especially for complex exponentials, this suggests that there's something about Fourier series that we don't understand.\n - whether the number of partitions depends on |E| for exponential frames or any other properties of E? Are there any examples of two sets with same measure such that the number of partitions is different (or the constants are different)?\n - More concretely, can we find E, |E| = \\frac12 such that\n\\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} cannot be partitioned into two Riesz sequences?\n - there's a connection to synthetic sets and arithmetic progressions (Bourgain's work with Tzafriri): we now know we can partition exponential frames into synthetic sets (MSS says we can do so). How to do this (constructively?)?\n\n* Bill Johnson\n -Is paving true for $\\ell_p^n$? Specifically:\n\nSuppose $1\\le p<\\infty$.\n\n$Pav_p$: Does there exist for every $\\epsilon > 0$ a constant $k=k(\\epsilon, p)$ s.t. for every $n$ and every $n$ by $n$ zero diagonal matrix $A$ for which $\\|A\\|_p = 1$, there is a partition $\\sigma(1), \\dots, \\sigma(k) of $\\{1,\\dots,n\\}$ s.t.\n\n$$\n\\|\\sum_{j=1}^k P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p = \\max_j\n\\| P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p \\le \\epsilon .\n$$\n\nHere $\\|\\dot \\|_$ is the operator norm for operators on $\\ell_p^n$, NOT the Schatten $p$-class norm.\n\n$Pav_1$, due IIRC to Bourgain has been known for a long time and is relatively easy. It is more-or-less equivalent to $Pav_2$ for matrices that have non-negative entries, a result that is contained in\n\nBerman, Kenneth; Halpern, Herbert; Kaftal, Victor; Weiss, Gary;\nMatrix norm inequalities and the relative Dixmier property.\nIntegral Equations Operator Theory 11 (1988), no. 1, 28-48.\n\nIn their original paper, Bourgain and Tzafriri proved restricted invertibility for $\\ell_p^n$, but they got optimal estimates only for $p=2$.\n\n - Take A = n \\times n matrix that's a commutator and write it as a commutator of B,C with B diagonal of norm 1 and C with bounded norm, with some minimization. Bill's recent PNAS paper with Ozawa and Schechtmann showed something that needed KS with optimal estimates to improve bound from n^\\alpha (which uses only restricted invertibility) to poly(n), and that the existence of a bound independent of n implies KS. Can you improve the result using KS to get a bound independent of n?\n\n* Deanna Needell\n - design algorithms to get pavings, especially for stochastic optimization\n - interested in algorithms that are efficient!\n - algorithmic developments that use pavings\n - sampling and RIP constructions for sparse approximation, do these lead to constructions or insight for KS\n\n* Anna Gilbert\n - Fourier sampling questions, RIP for the DFT, Rudelson selection lemma, how to remove extra factors on the bounds for RIP for DFT\n\n* Olaf Mordhorst\n - applications to convex geometry and Banach spaces\n - Restricted invertibility: provide a concrete example of optimality which is proven via random matrices\n\n* Pierre Youssef\n - consequences in convex geometry\n - if you have an identity decamp in \\R^n, BSS approximated the identity with many fewer, albeit weighted, vectors. If you use MSS, you get an identity approximation without weights. Let's add a condition that the vectors are balanced (say they come from a convex body).\n - \\sum_j c_j x_j x_j^T with \\|x_j\\| = 1 and \\sum_j c_j x_j = 0\n - Find S \\subset [m] , |S| \\approx d so that\n C \\sum_{j \\in S} x_j x_j^T \\approx I with a balancing condition \\| \\sum x_j \\| \\leq B\n - this was done by Srivastava but got 4 + \\eps approximation of the identity\n - Given a convex body K in \\R^n, Restricted Invertibility says there exists an E \\subset \\R^n with dim(E) = (1-\\eps)n so that\n distance( K \\cap E, B_1) \\approx \\sqrt{dim(E)}\n - KS says that we can partition K into subspaces like this. Can we improve the Banach-Mazur distance of a convex body to the cube using this partition? The best bound currently is n^{5/6}.\n - Algorithmic version of KS: try the Batson-S-S approach to KS? BUT, they had convinced themselves that their approach wouldn't work. Perhaps it is worth to revisit this?\n\n* Leonid Gurvits\n - there's an algorithm: start with k vectors in n-dimnl space. Can you partition them into 5 sets so that set spans the space. Take each vector, repeat it 5 times and augment those matrices by identity to get dimension k. You'll get k semi-def-pos. matrices. Now you check each matrix for some property xxxx. This is an algorithm to carry out Rado-Horn?\n\n* Mirko Visontai\n - understand limitations of the proof, we don't have any idea of how many good partitions there are? come up with more quantitative versions of the proof.\n - can we generate new interlacing families?\n\n* Mihai Putinar\n - perturbation determinant: in interlacing families, we have quotients of such polynomials, they form rational functions in the Nevalinna class of functions that preserve the upper half plane\n - understand KS in terms of the phase shift theory, perhaps getting a proof of the KS in terms that analysts can understand\n\n* Dan Edidin\n - wants to geometrize this work! Paving projections is especially interesting. Fix r. Find the conditions on the diagonal of a projection so it can be r-paved? Conditions should be independent of dimension.\n - For KS2, the problem is static: given v, find a good partition. Wants a more dynamic version: look at all vectors together (in a manifold structure), so that you replace the discrete structure with a continuous one. Look over some parametrized space with geometric structure and this may be more helpful to characterize when there are lots of partitions (i.e., count) that satisfies KS2.\n\n* Pablo Parrilo\n - algorithmic questions!\n - To try to understand why the MSS proof works: many existence proofs use probabilistic methods or fixed point methods and they are non-constructive. But MSS proof combines different techniques. Understand how these pieces fit together more generally.\n\n* Rachel Ward\n - see the above on algorithmic!\n\n* Leonid Gurvits\n - Is there an analog of KS for zonotopes? Let z_1,\\ldots,z_k \\in \\R^n, k > n and let \\bar z_i denote the interval \\{ \\alpha z_i | 0 \\leq \\alpha \\leq 1\\}. Consider the Minkowski sum\n Zon = \\alpha_1 \\bar z_1 + \\cdots \\alpha_k \\bar z_k.\n - Conjecture: vol(Zon) = 1.\n - KS analog: symmetrize Zon and then seek a partition S \\cup T so that\n - \\sum_{i \\in S} \\bar z_i \\subset (1/2 \\pm \\eps) Zon.\n\n* Bernhard Bodmann\n - how to eliminate those polynomials?! Borgain-Tzafriri's proof did not have such polynomials, they used rank-1 perturbation theory. Can we reproduce such a proof at the level of multi-variate polynomials?\n\n* Adam Marcus\n - one of the biggest comments we get is that KS was so popular because it was ubiquitous so it would require some new tool that other people could use in lots of other areas, unfortunately, these polynomial tools are not portable to other areas. If someone *did* prove KS in your area, what would it look like? Let's construct such a tool so that you have a hammer to bang away with in your area! So, please think about what this tool would like!\n - There's a conjecture about repeated copies of Identity with zero padding that this gives you the worse bounds on roots?\n - Using interlacing polynomials but if you only want the maximum root, you don't need all these other roots to exist, can we trim our conditions so that the machinery is more usable? Is there some weaker form of interlacing if you don't care about all the roots.\n - There was something about adding rank-2 perturbations rather than rank-1 perturbations.",
   "evidence": "The canonical record is number 3 in the AIM workshop document *Beyond Kadison--Singer: paving and consequences* (source file `aim-analysis-notes.json`, zero-based index 75). The mathematical question is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-ANALYSIS-0077": {
   "statement_status": "exact",
   "original_statement": "there are existence proofs for KS, how to make them constructive? Likes stable polynomials and interlacers. How to make constructive proofs?",
   "clean_statement": "there are existence proofs for KS, how to make them constructive? Likes stable polynomials and interlacers. How to make constructive proofs?",
   "public_statement": "there are existence proofs for KS, how to make them constructive? Likes stable polynomials and interlacers. How to make constructive proofs?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-ANALYSIS-0078": {
   "statement_status": "exact",
   "original_statement": "Find an algorithm for Weaver's conjecture/equivalence/theorem (or call it the vector partitioning problem), similarly for Paving Conjecture",
   "clean_statement": "Find an algorithm for Weaver's conjecture/equivalence/theorem (or call it the vector partitioning problem), similarly for Paving Conjecture",
   "public_statement": "Find an algorithm for Weaver's conjecture/equivalence/theorem (or call it the vector partitioning problem), similarly for Paving Conjecture",
   "evidence": "The canonical record is the following question from the AIM workshop *Beyond Kadison--Singer: paving and consequences* (December 2014):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-ANALYSIS-0079": {
   "statement_status": "exact",
   "original_statement": "Chuck started to describe a cooperative two-player game with partitioning the sets --> could we get an impossibility result via communication complexity?\n \n* Greg Knese",
   "clean_statement": "Chuck started to describe a cooperative two-player game with partitioning the sets --> could we get an impossibility result via communication complexity?\n\n* Greg Knese",
   "public_statement": "Chuck started to describe a cooperative two-player game with partitioning the sets --> could we get an impossibility result via communication complexity?\n\n* Greg Knese",
   "evidence": "The canonical record is item `b6` from the 2014 AIM workshop *Beyond Kadison--Singer: paving and consequences* (`aim-analysis-notes.json`, zero-based source index 78). Its extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-ANALYSIS-0080": {
   "statement_status": "exact",
   "original_statement": "extend the theory of interlacing families to several variables",
   "clean_statement": "extend the theory of interlacing families to several variables",
   "public_statement": "extend the theory of interlacing families to several variables",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-ANALYSIS-0081": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "how do operators that preserve stability affect locations of roots (i.e., quantitative versions of MSS)\n \n* Petter Branden",
   "clean_statement": null,
   "public_statement": "how do operators that preserve stability affect locations of roots (i.e., quantitative versions of MSS)\n\n* Petter Branden",
   "evidence": "The canonical record is `aim-analysis-notes.json`, zero-based index 80, from the AIM workshop *Beyond Kadison--Singer: paving and consequences*. Its extracted text is",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-ANALYSIS-0082": {
   "statement_status": "exact",
   "original_statement": "for the random MSS, there's a witness for the best bound, is there a conjecture for Weaver's conjecture? Weaver has a construction that \\sqrt\\eps is necessary but MSS has no constructive lower bound. Pete C. *does* have a constructive lower bound.",
   "clean_statement": "for the random MSS, there's a witness for the best bound, is there a conjecture for Weaver's conjecture? Weaver has a construction that \\sqrt\\eps is necessary but MSS has no constructive lower bound. Pete C. *does* have a constructive lower bound.",
   "public_statement": "for the random MSS, there's a witness for the best bound, is there a conjecture for Weaver's conjecture? Weaver has a construction that \\sqrt\\eps is necessary but MSS has no constructive lower bound. Pete C. *does* have a constructive lower bound.",
   "evidence": "### Exact canonical record",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-ANALYSIS-0083": {
   "statement_status": "unrecoverable",
   "original_statement": "Pete's example: can we make it better? It will affect all of the equivalences! It's almost a counter-example.",
   "clean_statement": null,
   "public_statement": "Pete's example: can we make it better? It will affect all of the equivalences! It's almost a counter-example.",
   "evidence": "It occurs in the notes for the AIM workshop *Beyond Kadison--Singer: paving and consequences* (December 1--5, 2014), under a list headed by Petter Brändén. The immediately preceding source bullets ask for matching quantitative lower bounds, say that the real paving lower bound has order \\(\\varepsilon^{-2}\\), and say that “Pete C.” has a constructive lower bound. The record itself does not define “Pete,” identify the example, specify a conjecture, or say what “better” means. Thus no unique mathematical statement can be recovered from the record alone.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-analysis-notes.json",
   "source_index": 82,
   "attempt": 1
  },
  "AIM-ANALYSIS-0084": {
   "statement_status": "exact",
   "original_statement": "consequences to matroid theory: can we prove a continuous version of matroid theory using MSS theorem and what consequences to matroid theory does MSS have?",
   "clean_statement": "consequences to matroid theory: can we prove a continuous version of matroid theory using MSS theorem and what consequences to matroid theory does MSS have?",
   "public_statement": "consequences to matroid theory: can we prove a continuous version of matroid theory using MSS theorem and what consequences to matroid theory does MSS have?",
   "evidence": "The canonical record is problem **b11** from the AIM workshop *Beyond Kadison--Singer: paving and consequences*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 83,
   "attempt": 1
  },
  "AIM-ANALYSIS-0085": {
   "statement_status": "exact",
   "original_statement": "interested in Feichtinger conjecture: suppose you have a Parseval frame and norms of vectors are bounded away from zero, what is the minimum number of partitions can we form that make Riesz basic sequences? Count the number of partitions? How does this number depend on \\epsilon, the lower bound (Pete C. also will show some results tomorrow on this question)? How to exhibit/construct such partitions?",
   "clean_statement": "interested in Feichtinger conjecture: suppose you have a Parseval frame and norms of vectors are bounded away from zero, what is the minimum number of partitions can we form that make Riesz basic sequences? Count the number of partitions? How does this number depend on \\epsilon, the lower bound (Pete C. also will show some results tomorrow on this question)? How to exhibit/construct such partitions?",
   "public_statement": "interested in Feichtinger conjecture: suppose you have a Parseval frame and norms of vectors are bounded away from zero, what is the minimum number of partitions can we form that make Riesz basic sequences? Count the number of partitions? How does this number depend on \\epsilon, the lower bound (Pete C. also will show some results tomorrow on this question)? How to exhibit/construct such partitions?",
   "evidence": "The corpus record is item `b12` from the AIM workshop *Beyond Kadison--Singer: paving and consequences*. Its extracted text asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 84,
   "attempt": 1
  },
  "AIM-ANALYSIS-0086": {
   "statement_status": "exact",
   "original_statement": "whether the number of partitions depends on |E| for exponential frames or any other properties of E? Are there any examples of two sets with same measure such that the number of partitions is different (or the constants are different)?",
   "clean_statement": "whether the number of partitions depends on |E| for exponential frames or any other properties of E? Are there any examples of two sets with same measure such that the number of partitions is different (or the constants are different)?",
   "public_statement": "whether the number of partitions depends on |E| for exponential frames or any other properties of E? Are there any examples of two sets with same measure such that the number of partitions is different (or the constants are different)?",
   "evidence": "The canonical record is AIM-ANALYSIS-0086, item b13 in the AIM workshop “Beyond Kadison-Singer: paving and consequences.” Its exact extracted problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 85,
   "attempt": 1
  },
  "AIM-ANALYSIS-0087": {
   "statement_status": "exact",
   "original_statement": "More concretely, can we find E, |E| = \\frac12 such that \n\\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} cannot be partitioned into two Riesz sequences?",
   "clean_statement": "More concretely, can we find E, |E| = \\frac12 such that\n\\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} cannot be partitioned into two Riesz sequences?",
   "public_statement": "More concretely, can we find E, |E| = \\frac12 such that\n\\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} cannot be partitioned into two Riesz sequences?",
   "evidence": "The canonical record is item b14 in the American Institute of Mathematics workshop list *Beyond Kadison--Singer: paving and consequences* (December 1--5, 2014). It appears under Darrin Speegle's name. The extracted formula is visibly damaged:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 86,
   "attempt": 1
  },
  "AIM-ANALYSIS-0088": {
   "statement_status": "exact",
   "original_statement": "there's a connection to synthetic sets and arithmetic progressions (Bourgain's work with Tzafriri): we now know we can partition exponential frames into synthetic sets (MSS says we can do so). How to do this (constructively?)?\n \n* Bill Johnson",
   "clean_statement": "there's a connection to synthetic sets and arithmetic progressions (Bourgain's work with Tzafriri): we now know we can partition exponential frames into synthetic sets (MSS says we can do so). How to do this (constructively?)?\n\n* Bill Johnson",
   "public_statement": "there's a connection to synthetic sets and arithmetic progressions (Bourgain's work with Tzafriri): we now know we can partition exponential frames into synthetic sets (MSS says we can do so). How to do this (constructively?)?\n\n* Bill Johnson",
   "evidence": "The canonical record in `aim-analysis-notes.json`, at zero-based index 87, literally reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 87,
   "attempt": 1
  },
  "AIM-ANALYSIS-0089": {
   "statement_status": "exact",
   "original_statement": "Is paving true for $\\ell_p^n$? Specifically:\n\nSuppose $1\\le p<\\infty$. \n\n$Pav_p$: Does there exist for every $\\epsilon > 0$ a constant $k=k(\\epsilon, p)$ s.t. for every $n$ and every $n$ by $n$ zero diagonal matrix $A$ for which $\\|A\\|_p = 1$, there is a partition $\\sigma(1), \\dots, \\sigma(k) of $\\{1,\\dots,n\\}$ s.t. \n\n$$\n\\|\\sum_{j=1}^k P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p = \\max_j \n\\| P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p \\le \\epsilon .\n$$\n\nHere $\\|\\dot \\|_$ is the operator norm for operators on $\\ell_p^n$, NOT the Schatten $p$-class norm.\n\n$Pav_1$, due IIRC to Bourgain has been known for a long time and is relatively easy. It is more-or-less equivalent to $Pav_2$ for matrices that have non-negative entries, a result that is contained in\n\nBerman, Kenneth; Halpern, Herbert; Kaftal, Victor; Weiss, Gary; \nMatrix norm inequalities and the relative Dixmier property. \nIntegral Equations Operator Theory 11 (1988), no. 1, 28-48.\n\nIn their original paper, Bourgain and Tzafriri proved restricted invertibility for $\\ell_p^n$, but they got optimal estimates only for $p=2$.",
   "clean_statement": "Is paving true for $\\ell_p^n$? Specifically:\n\nSuppose $1\\le p<\\infty$.\n\n$Pav_p$: Does there exist for every $\\epsilon > 0$ a constant $k=k(\\epsilon, p)$ s.t. for every $n$ and every $n$ by $n$ zero diagonal matrix $A$ for which $\\|A\\|_p = 1$, there is a partition $\\sigma(1), \\dots, \\sigma(k) of $\\{1,\\dots,n\\}$ s.t.\n\n$$\n\\|\\sum_{j=1}^k P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p = \\max_j\n\\| P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p \\le \\epsilon .\n$$\n\nHere $\\|\\dot \\|_$ is the operator norm for operators on $\\ell_p^n$, NOT the Schatten $p$-class norm.\n\n$Pav_1$, due IIRC to Bourgain has been known for a long time and is relatively easy. It is more-or-less equivalent to $Pav_2$ for matrices that have non-negative entries, a result that is contained in\n\nBerman, Kenneth; Halpern, Herbert; Kaftal, Victor; Weiss, Gary;\nMatrix norm inequalities and the relative Dixmier property.\nIntegral Equations Operator Theory 11 (1988), no. 1, 28-48.\n\nIn their original paper, Bourgain and Tzafriri proved restricted invertibility for $\\ell_p^n$, but they got optimal estimates only for $p=2$.",
   "public_statement": "Is paving true for $\\ell_p^n$? Specifically:\n\nSuppose $1\\le p<\\infty$.\n\n$Pav_p$: Does there exist for every $\\epsilon > 0$ a constant $k=k(\\epsilon, p)$ s.t. for every $n$ and every $n$ by $n$ zero diagonal matrix $A$ for which $\\|A\\|_p = 1$, there is a partition $\\sigma(1), \\dots, \\sigma(k) of $\\{1,\\dots,n\\}$ s.t.\n\n$$\n\\|\\sum_{j=1}^k P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p = \\max_j\n\\| P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p \\le \\epsilon .\n$$\n\nHere $\\|\\dot \\|_$ is the operator norm for operators on $\\ell_p^n$, NOT the Schatten $p$-class norm.\n\n$Pav_1$, due IIRC to Bourgain has been known for a long time and is relatively easy. It is more-or-less equivalent to $Pav_2$ for matrices that have non-negative entries, a result that is contained in\n\nBerman, Kenneth; Halpern, Herbert; Kaftal, Victor; Weiss, Gary;\nMatrix norm inequalities and the relative Dixmier property.\nIntegral Equations Operator Theory 11 (1988), no. 1, 28-48.\n\nIn their original paper, Bourgain and Tzafriri proved restricted invertibility for $\\ell_p^n$, but they got optimal estimates only for $p=2$.",
   "evidence": "The record is Bill Johnson's question from the AIM workshop *Beyond Kadison--Singer: paving and consequences*. The exact Google-document export agrees with `input.json`. It asks, for \\(1\\leq p<\\infty\\):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 88,
   "attempt": 2
  },
  "AIM-ANALYSIS-0090": {
   "statement_status": "exact",
   "original_statement": "Take A = n \\times n matrix that's a commutator and write it as a commutator of B,C with B diagonal of norm 1 and C with bounded norm, with some minimization. Bill's recent PNAS paper with Ozawa and Schechtmann showed something that needed KS with optimal estimates to improve bound from n^\\alpha (which uses only restricted invertibility) to poly(n), and that the existence of a bound independent of n implies KS. Can you improve the result using KS to get a bound independent of n?\n\n* Deanna Needell",
   "clean_statement": "Take A = n \\times n matrix that's a commutator and write it as a commutator of B,C with B diagonal of norm 1 and C with bounded norm, with some minimization. Bill's recent PNAS paper with Ozawa and Schechtmann showed something that needed KS with optimal estimates to improve bound from n^\\alpha (which uses only restricted invertibility) to poly(n), and that the existence of a bound independent of n implies KS. Can you improve the result using KS to get a bound independent of n?\n\n* Deanna Needell",
   "public_statement": "Take A = n \\times n matrix that's a commutator and write it as a commutator of B,C with B diagonal of norm 1 and C with bounded norm, with some minimization. Bill's recent PNAS paper with Ozawa and Schechtmann showed something that needed KS with optimal estimates to improve bound from n^\\alpha (which uses only restricted invertibility) to poly(n), and that the existence of a bound independent of n implies KS. Can you improve the result using KS to get a bound independent of n?\n\n* Deanna Needell",
   "evidence": "The canonical record (AIM, *Beyond Kadison--Singer: paving and consequences*, item b17, attributed to Deanna Needell) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 89,
   "attempt": 1
  },
  "AIM-ANALYSIS-0091": {
   "statement_status": "exact",
   "original_statement": "Find S \\subset [m] , |S| \\approx d so that\n C \\sum_{j \\in S} x_j x_j^T \\approx I with a balancing condition \\| \\sum x_j \\| \\leq B",
   "clean_statement": "Find S \\subset [m] , |S| \\approx d so that\n C \\sum_{j \\in S} x_j x_j^T \\approx I with a balancing condition \\| \\sum x_j \\| \\leq B",
   "public_statement": "Find S \\subset [m] , |S| \\approx d so that\n C \\sum_{j \\in S} x_j x_j^T \\approx I with a balancing condition \\| \\sum x_j \\| \\leq B",
   "evidence": "The canonical record says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 90,
   "attempt": 1
  },
  "AIM-ANALYSIS-0092": {
   "statement_status": "exact",
   "original_statement": "KS says that we can partition K into subspaces like this. Can we improve the Banach-Mazur distance of a convex body to the cube using this partition? The best bound currently is n^{5/6}.",
   "clean_statement": "KS says that we can partition K into subspaces like this. Can we improve the Banach-Mazur distance of a convex body to the cube using this partition? The best bound currently is n^{5/6}.",
   "public_statement": "KS says that we can partition K into subspaces like this. Can we improve the Banach-Mazur distance of a convex body to the cube using this partition? The best bound currently is n^{5/6}.",
   "evidence": "The canonical record is AIM-ANALYSIS-0092, source index 91 in aim-analysis-notes.json. Its exact problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 91,
   "attempt": 1
  },
  "AIM-ANALYSIS-0093": {
   "statement_status": "exact",
   "original_statement": "Algorithmic version of KS: try the Batson-S-S approach to KS? BUT, they had convinced themselves that their approach wouldn't work. Perhaps it is worth to revisit this?\n \n* Leonid Gurvits",
   "clean_statement": "Algorithmic version of KS: try the Batson-S-S approach to KS? BUT, they had convinced themselves that their approach wouldn't work. Perhaps it is worth to revisit this?\n\n* Leonid Gurvits",
   "public_statement": "Algorithmic version of KS: try the Batson-S-S approach to KS? BUT, they had convinced themselves that their approach wouldn't work. Perhaps it is worth to revisit this?\n\n* Leonid Gurvits",
   "evidence": "The canonical AIM record (Analysis, source index 92, item b20 from the 2014 workshop *Beyond Kadison--Singer: paving and consequences*) reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-ANALYSIS-0094": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "there's an algorithm: start with k vectors in n-dimnl space. Can you partition them into 5 sets so that set spans the space. Take each vector, repeat it 5 times and augment those matrices by identity to get dimension k. You'll get k semi-def-pos. matrices. Now you check each matrix for some property xxxx. This is an algorithm to carry out Rado-Horn?\n \n* Mirko Visontai",
   "clean_statement": null,
   "public_statement": "there's an algorithm: start with k vectors in n-dimnl space. Can you partition them into 5 sets so that set spans the space. Take each vector, repeat it 5 times and augment those matrices by identity to get dimension k. You'll get k semi-def-pos. matrices. Now you check each matrix for some property xxxx. This is an algorithm to carry out Rado-Horn?\n\n* Mirko Visontai",
   "evidence": "The canonical record is a workshop note from *Beyond Kadison--Singer: paving and consequences*. Its exact problem text is:",
   "classification_method": "invalid_literal_with_repaired_reading",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-ANALYSIS-0095": {
   "statement_status": "exact",
   "original_statement": "understand limitations of the proof, we don't have any idea of how many good partitions there are? come up with more quantitative versions of the proof.",
   "clean_statement": "understand limitations of the proof, we don't have any idea of how many good partitions there are? come up with more quantitative versions of the proof.",
   "public_statement": "understand limitations of the proof, we don't have any idea of how many good partitions there are? come up with more quantitative versions of the proof.",
   "evidence": "The exact canonical problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 94,
   "attempt": 1
  },
  "AIM-ANALYSIS-0096": {
   "statement_status": "exact",
   "original_statement": "can we generate new interlacing families?\n\n* Mihai Putinar",
   "clean_statement": "can we generate new interlacing families?\n\n* Mihai Putinar",
   "public_statement": "can we generate new interlacing families?\n\n* Mihai Putinar",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-ANALYSIS-0097": {
   "statement_status": "exact",
   "original_statement": "understand KS in terms of the phase shift theory, perhaps getting a proof of the KS in terms that analysts can understand\n \n* Dan Edidin",
   "clean_statement": "understand KS in terms of the phase shift theory, perhaps getting a proof of the KS in terms that analysts can understand\n\n* Dan Edidin",
   "public_statement": "understand KS in terms of the phase shift theory, perhaps getting a proof of the KS in terms that analysts can understand\n\n* Dan Edidin",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 96,
   "attempt": 1
  },
  "AIM-ANALYSIS-0098": {
   "statement_status": "exact",
   "original_statement": "wants to geometrize this work! Paving projections is especially interesting. Fix r. Find the conditions on the diagonal of a projection so it can be r-paved? Conditions should be independent of dimension.",
   "clean_statement": "wants to geometrize this work! Paving projections is especially interesting. Fix r. Find the conditions on the diagonal of a projection so it can be r-paved? Conditions should be independent of dimension.",
   "public_statement": "wants to geometrize this work! Paving projections is especially interesting. Fix r. Find the conditions on the diagonal of a projection so it can be r-paved? Conditions should be independent of dimension.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 97,
   "attempt": 1
  },
  "AIM-ANALYSIS-0099": {
   "statement_status": "exact",
   "original_statement": "Is there an analog of KS for zonotopes? Let z_1,\\ldots,z_k \\in \\R^n, k > n and let \\bar z_i denote the interval \\{ \\alpha z_i | 0 \\leq \\alpha \\leq 1\\}. Consider the Minkowski sum\n Zon = \\alpha_1 \\bar z_1 + \\cdots \\alpha_k \\bar z_k.",
   "clean_statement": "Is there an analog of KS for zonotopes? Let z_1,\\ldots,z_k \\in \\R^n, k > n and let \\bar z_i denote the interval \\{ \\alpha z_i | 0 \\leq \\alpha \\leq 1\\}. Consider the Minkowski sum\n Zon = \\alpha_1 \\bar z_1 + \\cdots \\alpha_k \\bar z_k.",
   "public_statement": "Is there an analog of KS for zonotopes? Let z_1,\\ldots,z_k \\in \\R^n, k > n and let \\bar z_i denote the interval \\{ \\alpha z_i | 0 \\leq \\alpha \\leq 1\\}. Consider the Minkowski sum\n Zon = \\alpha_1 \\bar z_1 + \\cdots \\alpha_k \\bar z_k.",
   "evidence": "The canonical record (AIM Problem Lists, workshop notes for *Beyond Kadison--Singer: paving and consequences*) is attributed in the original Google document to Leonid Gurvits. It reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 98,
   "attempt": 1
  },
  "AIM-ANALYSIS-0100": {
   "statement_status": "exact",
   "original_statement": "how to eliminate those polynomials?! Borgain-Tzafriri's proof did not have such polynomials, they used rank-1 perturbation theory. Can we reproduce such a proof at the level of multi-variate polynomials?\n \n* Adam Marcus",
   "clean_statement": "how to eliminate those polynomials?! Borgain-Tzafriri's proof did not have such polynomials, they used rank-1 perturbation theory. Can we reproduce such a proof at the level of multi-variate polynomials?\n\n* Adam Marcus",
   "public_statement": "how to eliminate those polynomials?! Borgain-Tzafriri's proof did not have such polynomials, they used rank-1 perturbation theory. Can we reproduce such a proof at the level of multi-variate polynomials?\n\n* Adam Marcus",
   "evidence": "The canonical record is item `b27` from the AIM workshop *Beyond Kadison--Singer: paving and consequences*. Its exact extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 99,
   "attempt": 1
  },
  "AIM-ANALYSIS-0101": {
   "statement_status": "exact",
   "original_statement": "one of the biggest comments we get is that KS was so popular because it was ubiquitous so it would require some new tool that other people could use in lots of other areas, unfortunately, these polynomial tools are not portable to other areas. If someone *did* prove KS in your area, what would it look like? Let's construct such a tool so that you have a hammer to bang away with in your area! So, please think about what this tool would like!",
   "clean_statement": "one of the biggest comments we get is that KS was so popular because it was ubiquitous so it would require some new tool that other people could use in lots of other areas, unfortunately, these polynomial tools are not portable to other areas. If someone *did* prove KS in your area, what would it look like? Let's construct such a tool so that you have a hammer to bang away with in your area! So, please think about what this tool would like!",
   "public_statement": "one of the biggest comments we get is that KS was so popular because it was ubiquitous so it would require some new tool that other people could use in lots of other areas, unfortunately, these polynomial tools are not portable to other areas. If someone *did* prove KS in your area, what would it look like? Let's construct such a tool so that you have a hammer to bang away with in your area! So, please think about what this tool would like!",
   "evidence": "The exact canonical record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 100,
   "attempt": 1
  },
  "AIM-ANALYSIS-0102": {
   "statement_status": "exact",
   "original_statement": "There's a conjecture about repeated copies of Identity with zero padding that this gives you the worse bounds on roots?",
   "clean_statement": "There's a conjecture about repeated copies of Identity with zero padding that this gives you the worse bounds on roots?",
   "public_statement": "There's a conjecture about repeated copies of Identity with zero padding that this gives you the worse bounds on roots?",
   "evidence": "The exact canonical record is the following question from the AIM workshop *Beyond Kadison--Singer: paving and consequences* (December 1--5, 2014):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 101,
   "attempt": 1
  },
  "AIM-ANALYSIS-0103": {
   "statement_status": "exact",
   "original_statement": "Using interlacing polynomials but if you only want the maximum root, you don't need all these other roots to exist, can we trim our conditions so that the machinery is more usable? Is there some weaker form of interlacing if you don't care about all the roots.",
   "clean_statement": "Using interlacing polynomials but if you only want the maximum root, you don't need all these other roots to exist, can we trim our conditions so that the machinery is more usable? Is there some weaker form of interlacing if you don't care about all the roots.",
   "public_statement": "Using interlacing polynomials but if you only want the maximum root, you don't need all these other roots to exist, can we trim our conditions so that the machinery is more usable? Is there some weaker form of interlacing if you don't care about all the roots.",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 102,
   "attempt": 1
  },
  "AIM-ANALYSIS-0104": {
   "statement_status": "exact",
   "original_statement": "Linear dependence of constants in bi-Lipschitz extension theorems\n\nDo bi-Lipschitz extension theorems admit linear dependence on constants?",
   "clean_statement": "Linear dependence of constants in bi-Lipschitz extension theorems\n\nDo bi-Lipschitz extension theorems admit linear dependence on constants?",
   "public_statement": "Linear dependence of constants in bi-Lipschitz extension theorems\n\nDo bi-Lipschitz extension theorems admit linear dependence on constants?",
   "evidence": "The canonical record is problem 1.1, “Extensions,” from the AIM workshop *Mapping theory in metric spaces*. Its exact problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 103,
   "attempt": 1
  },
  "AIM-ANALYSIS-0105": {
   "statement_status": "exact",
   "original_statement": "Dependence on parameters\n\nDo canonical Lipschitz extensions exist which depend nicely on parameters?",
   "clean_statement": "Dependence on parameters\n\nDo canonical Lipschitz extensions exist which depend nicely on parameters?",
   "public_statement": "Dependence on parameters\n\nDo canonical Lipschitz extensions exist which depend nicely on parameters?",
   "evidence": "The exact canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 104,
   "attempt": 1
  },
  "AIM-ANALYSIS-0106": {
   "statement_status": "exact",
   "original_statement": "Deterministic proof of the Lee--Naor Lipschitz extension theorem\n\nThe Lee--Naor theorem states that, given a doubling subset $A$ of a metric space $X$ and an $L$-Lipschitz map from $A$ to a Banach space $Y$, there exists a $C(\\log m)L$-Lipschitz extension $X\\to Y$. Here $m$ is the doubling constant of $A$.\n\nIs there a deterministic proof of the Lee--Naor Lipschitz extension theorem giving the sharp dependence on the doubling constant?",
   "clean_statement": "Deterministic proof of the Lee--Naor Lipschitz extension theorem\n\nThe Lee--Naor theorem states that, given a doubling subset $A$ of a metric space $X$ and an $L$-Lipschitz map from $A$ to a Banach space $Y$, there exists a $C(\\log m)L$-Lipschitz extension $X\\to Y$. Here $m$ is the doubling constant of $A$.\n\nIs there a deterministic proof of the Lee--Naor Lipschitz extension theorem giving the sharp dependence on the doubling constant?",
   "public_statement": "Deterministic proof of the Lee--Naor Lipschitz extension theorem\n\nThe Lee--Naor theorem states that, given a doubling subset $A$ of a metric space $X$ and an $L$-Lipschitz map from $A$ to a Banach space $Y$, there exists a $C(\\log m)L$-Lipschitz extension $X\\to Y$. Here $m$ is the doubling constant of $A$.\n\nIs there a deterministic proof of the Lee--Naor Lipschitz extension theorem giving the sharp dependence on the doubling constant?",
   "evidence": "The canonical record is problem 1.3 in the **Extensions** section of the AIM list *Mapping theory in metric spaces*. Its exact mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 105,
   "attempt": 2
  },
  "AIM-ANALYSIS-0107": {
   "statement_status": "exact",
   "original_statement": "Absolute Lipschitz retract constructions\n\n${\\mathbb Q}_Q(Y)$ denotes the space of unordered $Q$-tuples $y=[y_1,\\ldots,y_Q]$ of elements of $Y$ equipped with the metric $d(y,z) = \\min \\{ \\max \\{ d(y_j,z_{\\sigma(j)}) \\} : \\sigma \\in S_Q \\}$, where $S_Q$ denotes the symmetric group on $Q$ letters.\n\nIf $Y$ is an absolute Lipschitz retract, is the same true of the space ${\\mathbb Q}_Q(Y)$?",
   "clean_statement": "Absolute Lipschitz retract constructions\n\n${\\mathbb Q}_Q(Y)$ denotes the space of unordered $Q$-tuples $y=[y_1,\\ldots,y_Q]$ of elements of $Y$ equipped with the metric $d(y,z) = \\min \\{ \\max \\{ d(y_j,z_{\\sigma(j)}) \\} : \\sigma \\in S_Q \\}$, where $S_Q$ denotes the symmetric group on $Q$ letters.\n\nIf $Y$ is an absolute Lipschitz retract, is the same true of the space ${\\mathbb Q}_Q(Y)$?",
   "public_statement": "Absolute Lipschitz retract constructions\n\n${\\mathbb Q}_Q(Y)$ denotes the space of unordered $Q$-tuples $y=[y_1,\\ldots,y_Q]$ of elements of $Y$ equipped with the metric $d(y,z) = \\min \\{ \\max \\{ d(y_j,z_{\\sigma(j)}) \\} : \\sigma \\in S_Q \\}$, where $S_Q$ denotes the symmetric group on $Q$ letters.\n\nIf $Y$ is an absolute Lipschitz retract, is the same true of the space ${\\mathbb Q}_Q(Y)$?",
   "evidence": "This is Problem 1.4, “Absolute Lipschitz retract constructions,” in the “Extensions” section of the AIM workshop list *Mapping theory in metric spaces*. The canonical record (source file `aim-analysis-notes.json`, index 106) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 106,
   "attempt": 1
  },
  "AIM-ANALYSIS-0108": {
   "statement_status": "exact",
   "original_statement": "Banach space pairs with the Lipschitz extension property\n\nDoes the pair $(L^2,L^1)$ have the Lipschitz extension property?",
   "clean_statement": "Banach space pairs with the Lipschitz extension property\n\nDoes the pair $(L^2,L^1)$ have the Lipschitz extension property?",
   "public_statement": "Banach space pairs with the Lipschitz extension property\n\nDoes the pair $(L^2,L^1)$ have the Lipschitz extension property?",
   "evidence": "The canonical record is AIM Problem 1.5 in the “Extensions” section of the January 2012 workshop *Mapping theory in metric spaces*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 107,
   "attempt": 1
  },
  "AIM-ANALYSIS-0109": {
   "statement_status": "exact",
   "original_statement": "Absolute Lipschitz retracts\n\nAre all Hadamard manifolds absolute Lipschitz retracts?",
   "clean_statement": "Absolute Lipschitz retracts\n\nAre all Hadamard manifolds absolute Lipschitz retracts?",
   "public_statement": "Absolute Lipschitz retracts\n\nAre all Hadamard manifolds absolute Lipschitz retracts?",
   "evidence": "The canonical record is number 1.6, under “Extensions,” in the AIM list *Mapping theory in metric spaces*. Its exact mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 108,
   "attempt": 1
  },
  "AIM-ANALYSIS-0110": {
   "statement_status": "exact",
   "original_statement": "Lipschitz homotopy groups of the Heisenberg group\n\n${\\mathbb H}^n$ denotes the $n$th Heisenberg group, equipped with the Carnot-Carath\\'eodory metric. $\\pi_k^{\\scriptstyle{Lip}}(X)$ denotes the $k$th Lipschitz homotopy group of a metric space $X$. It is known that $\\pi_n^{\\scriptstyle{Lip}}({\\mathbb H}^n) \\ne 0$ (DeJarnette--Hajlasz--Lukyanenko--Tyson) \\cite{arXiv:1109.4641}.\n\nIf $\\pi_k({\\mathbb S}^n)$ is nontrivial, is the same true for $\\pi_k^{\\scriptstyle{Lip}}({\\mathbb H}^n)$? In particular, is $\\pi_3^{\\scriptstyle{Lip}}({\\mathbb H}^2) \\ne 0$?\n\\label{Heis-Lip}",
   "clean_statement": "Lipschitz homotopy groups of the Heisenberg group\n\n${\\mathbb H}^n$ denotes the $n$th Heisenberg group, equipped with the Carnot-Carath\\'eodory metric. $\\pi_k^{\\scriptstyle{Lip}}(X)$ denotes the $k$th Lipschitz homotopy group of a metric space $X$. It is known that $\\pi_n^{\\scriptstyle{Lip}}({\\mathbb H}^n) \\ne 0$ (DeJarnette--Hajlasz--Lukyanenko--Tyson) \\cite{arXiv:1109.4641}.\n\nIf $\\pi_k({\\mathbb S}^n)$ is nontrivial, is the same true for $\\pi_k^{\\scriptstyle{Lip}}({\\mathbb H}^n)$? In particular, is $\\pi_3^{\\scriptstyle{Lip}}({\\mathbb H}^2) \\ne 0$?\n\\label{Heis-Lip}",
   "public_statement": "Lipschitz homotopy groups of the Heisenberg group\n\n${\\mathbb H}^n$ denotes the $n$th Heisenberg group, equipped with the Carnot-Carath\\'eodory metric. $\\pi_k^{\\scriptstyle{Lip}}(X)$ denotes the $k$th Lipschitz homotopy group of a metric space $X$. It is known that $\\pi_n^{\\scriptstyle{Lip}}({\\mathbb H}^n) \\ne 0$ (DeJarnette--Hajlasz--Lukyanenko--Tyson) \\cite{arXiv:1109.4641}.\n\nIf $\\pi_k({\\mathbb S}^n)$ is nontrivial, is the same true for $\\pi_k^{\\scriptstyle{Lip}}({\\mathbb H}^n)$? In particular, is $\\pi_3^{\\scriptstyle{Lip}}({\\mathbb H}^2) \\ne 0$?\n\\label{Heis-Lip}",
   "evidence": "This is Problem 1.7 in the “Extensions” section of the AIM problem list from the 2012 workshop *Mapping theory in metric spaces*. The canonical source record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 109,
   "attempt": 1
  },
  "AIM-ANALYSIS-0111": {
   "statement_status": "exact",
   "original_statement": "Lipschitz homotopy groups of the Heisenberg group II\n\nDoes $\\pi_n^{\\scriptstyle{Lip}}({\\mathbb H}^n)$ contain torsion elements?",
   "clean_statement": "Lipschitz homotopy groups of the Heisenberg group II\n\nDoes $\\pi_n^{\\scriptstyle{Lip}}({\\mathbb H}^n)$ contain torsion elements?",
   "public_statement": "Lipschitz homotopy groups of the Heisenberg group II\n\nDoes $\\pi_n^{\\scriptstyle{Lip}}({\\mathbb H}^n)$ contain torsion elements?",
   "evidence": "The neighboring AIM record gives these definitions and cites DHLT for $\\pi_n^{\\mathrm{Lip}}(\\mathbb H^n)\\neq 0$. No corruption or ambiguity in the displayed torsion question was found.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 110,
   "attempt": 1
  },
  "AIM-ANALYSIS-0112": {
   "statement_status": "exact",
   "original_statement": "Deterministic proofs of bi-Lipschitz embedding theorems\n\nAssouad's embedding theorem asserts that to each doubling metric space $(X,d)$ and each $\\epsilon \\in (0,1)$, there corresponds $n$ so that the snowflake metric space $(X,d^\\epsilon)$ admits a bi-Lipschitz embedding into ${\\mathbb R}^n$. Naor--Neiman proved that $n$ can be chosen independent of the snowflake parameter $\\epsilon$, for $\\epsilon$ near one (say, $\\epsilon>\\tfrac12$). Their proof is probabilistic and nonconstructive. Can one give an explicit construction of such embedding?\n\nGive a deterministic proof of Naor--Neiman's improved Assouad embedding theorem.\\label{deterministic bi-Lipschitz}",
   "clean_statement": "Deterministic proofs of bi-Lipschitz embedding theorems\n\nAssouad's embedding theorem asserts that to each doubling metric space $(X,d)$ and each $\\epsilon \\in (0,1)$, there corresponds $n$ so that the snowflake metric space $(X,d^\\epsilon)$ admits a bi-Lipschitz embedding into ${\\mathbb R}^n$. Naor--Neiman proved that $n$ can be chosen independent of the snowflake parameter $\\epsilon$, for $\\epsilon$ near one (say, $\\epsilon>\\tfrac12$). Their proof is probabilistic and nonconstructive. Can one give an explicit construction of such embedding?\n\nGive a deterministic proof of Naor--Neiman's improved Assouad embedding theorem.\\label{deterministic bi-Lipschitz}",
   "public_statement": "Deterministic proofs of bi-Lipschitz embedding theorems\n\nAssouad's embedding theorem asserts that to each doubling metric space $(X,d)$ and each $\\epsilon \\in (0,1)$, there corresponds $n$ so that the snowflake metric space $(X,d^\\epsilon)$ admits a bi-Lipschitz embedding into ${\\mathbb R}^n$. Naor--Neiman proved that $n$ can be chosen independent of the snowflake parameter $\\epsilon$, for $\\epsilon$ near one (say, $\\epsilon>\\tfrac12$). Their proof is probabilistic and nonconstructive. Can one give an explicit construction of such embedding?\n\nGive a deterministic proof of Naor--Neiman's improved Assouad embedding theorem.\\label{deterministic bi-Lipschitz}",
   "evidence": "This is Problem 2.1 in the “Embeddings” section of the AIM problem list from the 2012 workshop *Mapping theory in metric spaces*. In the notation of the canonical record, the problem asks for a deterministic, explicit proof of the following assertion:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 111,
   "attempt": 2
  },
  "AIM-ANALYSIS-0113": {
   "statement_status": "exact",
   "original_statement": "Curvature conditions and bi-Lipschitz embeddings\n\nLet $X$ be a doubling metric space. Can one formulate a curvature-type condition which, if satisfied by $X$, ensures that $X$ admits a bi-Lipschitz embedding into some finite-dimensional Euclidean space?",
   "clean_statement": "Curvature conditions and bi-Lipschitz embeddings\n\nLet $X$ be a doubling metric space. Can one formulate a curvature-type condition which, if satisfied by $X$, ensures that $X$ admits a bi-Lipschitz embedding into some finite-dimensional Euclidean space?",
   "public_statement": "Curvature conditions and bi-Lipschitz embeddings\n\nLet $X$ be a doubling metric space. Can one formulate a curvature-type condition which, if satisfied by $X$, ensures that $X$ admits a bi-Lipschitz embedding into some finite-dimensional Euclidean space?",
   "evidence": "This is Problem 2.3 in the **Embeddings** section of the AIM workshop list *Mapping theory in metric spaces*. The AIM page attributes it to Leonid Kovalev. The canonical record and the live AIM page agree, and no reconstruction or OCR repair is needed.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 112,
   "attempt": 1
  },
  "AIM-ANALYSIS-0114": {
   "statement_status": "exact",
   "original_statement": "Bi-Lipschitz embeddings: Hilbert spaces vs. finite-dimensional Euclidean spaces\n\nSuppose that $X$ is a doubling space which admits a bi-Lipschitz embedding into Hilbert space. Does $X$ necessarily admit a bi-Lipschitz embedding into some finite-dimensional Euclidean space?",
   "clean_statement": "Bi-Lipschitz embeddings: Hilbert spaces vs. finite-dimensional Euclidean spaces\n\nSuppose that $X$ is a doubling space which admits a bi-Lipschitz embedding into Hilbert space. Does $X$ necessarily admit a bi-Lipschitz embedding into some finite-dimensional Euclidean space?",
   "public_statement": "Bi-Lipschitz embeddings: Hilbert spaces vs. finite-dimensional Euclidean spaces\n\nSuppose that $X$ is a doubling space which admits a bi-Lipschitz embedding into Hilbert space. Does $X$ necessarily admit a bi-Lipschitz embedding into some finite-dimensional Euclidean space?",
   "evidence": "The canonical record is zero-based index 113 of `aim-analysis-notes.json`, Problem 2.4 in the “Embeddings” section of the AIM workshop list *Mapping theory in metric spaces*. Its exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 113,
   "attempt": 1
  },
  "AIM-ANALYSIS-0115": {
   "statement_status": "exact",
   "original_statement": "Snowflake embeddings of the Heisenberg group into low-dimensional Euclidean spaces\n\nLet $d_{cc}$ denote the Carnot-Carath\\'eodory metric.\n\nDoes there exist $\\epsilon<1$ so that the first Heisenberg group ${\\mathbb H}^1$ equipped with the metric $d_{cc}^\\epsilon$ bi-Lipschitz embeds into ${\\mathbb R}^5$?",
   "clean_statement": "Snowflake embeddings of the Heisenberg group into low-dimensional Euclidean spaces\n\nLet $d_{cc}$ denote the Carnot-Carath\\'eodory metric.\n\nDoes there exist $\\epsilon<1$ so that the first Heisenberg group ${\\mathbb H}^1$ equipped with the metric $d_{cc}^\\epsilon$ bi-Lipschitz embeds into ${\\mathbb R}^5$?",
   "public_statement": "Snowflake embeddings of the Heisenberg group into low-dimensional Euclidean spaces\n\nLet $d_{cc}$ denote the Carnot-Carath\\'eodory metric.\n\nDoes there exist $\\epsilon<1$ so that the first Heisenberg group ${\\mathbb H}^1$ equipped with the metric $d_{cc}^\\epsilon$ bi-Lipschitz embeds into ${\\mathbb R}^5$?",
   "evidence": "The canonical record is problem 2.4 in the **Embeddings** section of the AIM workshop list *Mapping theory in metric spaces* (`aim-analysis-notes.json`, zero-based record index 114). Its question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 114,
   "attempt": 1
  },
  "AIM-ANALYSIS-0116": {
   "statement_status": "exact",
   "original_statement": "Bi-Lipschitz parameterization by Euclidean spaces\n\nAre there higher dimensional analogs of the Bonk--Lang parameterization theorem which give quantitative control on bi-Lipschitz constants?",
   "clean_statement": "Bi-Lipschitz parameterization by Euclidean spaces\n\nAre there higher dimensional analogs of the Bonk--Lang parameterization theorem which give quantitative control on bi-Lipschitz constants?",
   "public_statement": "Bi-Lipschitz parameterization by Euclidean spaces\n\nAre there higher dimensional analogs of the Bonk--Lang parameterization theorem which give quantitative control on bi-Lipschitz constants?",
   "evidence": "The canonical record is source index 115 of **aim-analysis-notes.json**. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 115,
   "attempt": 1
  },
  "AIM-ANALYSIS-0117": {
   "statement_status": "exact",
   "original_statement": "Quasisymmetric uniformization of metric $2$-spheres\n\nGive a geometric characterization of metric $2$-spheres which are quasisymmetrically equivalent to the standard $2$-sphere.",
   "clean_statement": "Quasisymmetric uniformization of metric $2$-spheres\n\nGive a geometric characterization of metric $2$-spheres which are quasisymmetrically equivalent to the standard $2$-sphere.",
   "public_statement": "Quasisymmetric uniformization of metric $2$-spheres\n\nGive a geometric characterization of metric $2$-spheres which are quasisymmetrically equivalent to the standard $2$-sphere.",
   "evidence": "The canonical record is AIM-ANALYSIS-0117, source file aim-analysis-notes.json, zero-based source index 116, from the AIM workshop *Mapping theory in metric spaces*, section *Uniformization and parameterization*, Problem 33.2. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 116,
   "attempt": 1
  },
  "AIM-ANALYSIS-0118": {
   "statement_status": "exact",
   "original_statement": "Bi-Lipschitz parameterization of metric $2$-spheres\n\nGive a geometric characterization of metric $2$-spheres which are equivalent to the standard $2$-sphere via a bi-Lipschitz map.",
   "clean_statement": "Bi-Lipschitz parameterization of metric $2$-spheres\n\nGive a geometric characterization of metric $2$-spheres which are equivalent to the standard $2$-sphere via a bi-Lipschitz map.",
   "public_statement": "Bi-Lipschitz parameterization of metric $2$-spheres\n\nGive a geometric characterization of metric $2$-spheres which are equivalent to the standard $2$-sphere via a bi-Lipschitz map.",
   "evidence": "The live [AIM page](http://aimpl.org/mappingmetric/3/) was checked on 2026-07-28 and agrees with `input.json`. There is no apparent OCR loss or notational ambiguity. Here “metric $2$-sphere” means a metric space homeomorphic to $S^2$, and “equivalent” means by a surjective bi-Lipschitz homeomorphism. The round sphere is denoted $(S^2,d_0)$.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 117,
   "attempt": 1
  },
  "AIM-ANALYSIS-0119": {
   "statement_status": "exact",
   "original_statement": "Almost everywhere partial differentiability of mappings of integrable lower metric distortion\n\nThe lower metric dilatation of a homeomorphism $f:X\\to Y$ of metric spaces is defined as\n$$\nh_f(x) = \\liminf_{r\\to 0} \\frac{\\sup \\{ d(f(x),f(y)) : d(x,y) \\le r \\}}{\\inf \\{ d(f(x),f(z)) : d(x,z) \\ge r \\}}.\n$$\n\nLet $f$ be a homeomorphism of planar domains with $h_f \\in L^1_{\\scriptstyle{loc}}$ and $h_f$ finite off a set of $\\sigma$-finite length. Does $f$ have partial derivatives almost everywhere?",
   "clean_statement": "Almost everywhere partial differentiability of mappings of integrable lower metric distortion\n\nThe lower metric dilatation of a homeomorphism $f:X\\to Y$ of metric spaces is defined as\n$$\nh_f(x) = \\liminf_{r\\to 0} \\frac{\\sup \\{ d(f(x),f(y)) : d(x,y) \\le r \\}}{\\inf \\{ d(f(x),f(z)) : d(x,z) \\ge r \\}}.\n$$\n\nLet $f$ be a homeomorphism of planar domains with $h_f \\in L^1_{\\scriptstyle{loc}}$ and $h_f$ finite off a set of $\\sigma$-finite length. Does $f$ have partial derivatives almost everywhere?",
   "public_statement": "Almost everywhere partial differentiability of mappings of integrable lower metric distortion\n\nThe lower metric dilatation of a homeomorphism $f:X\\to Y$ of metric spaces is defined as\n$$\nh_f(x) = \\liminf_{r\\to 0} \\frac{\\sup \\{ d(f(x),f(y)) : d(x,y) \\le r \\}}{\\inf \\{ d(f(x),f(z)) : d(x,z) \\ge r \\}}.\n$$\n\nLet $f$ be a homeomorphism of planar domains with $h_f \\in L^1_{\\scriptstyle{loc}}$ and $h_f$ finite off a set of $\\sigma$-finite length. Does $f$ have partial derivatives almost everywhere?",
   "evidence": "The canonical record is AIM-ANALYSIS-0119, source file aim-analysis-notes.json, zero-based source index 118. It comes from the AIM workshop list *Mapping theory in metric spaces*, section “Regularity.” The live AIM page identifies it as Problem 4.1 (attributed there to Pekka Koskela); the canonical field “44.1” appears to be an extraction artifact.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 118,
   "attempt": 1
  },
  "AIM-ANALYSIS-0120": {
   "statement_status": "exact",
   "original_statement": "Mapping properties of planar maps of exponentially integrable distortion\n\nDoes there exist a homeomorphism $f$ of the plane with exponentially integrable distortion such that $f$ fixes the real axis and sends a set of positive length onto the $\\tfrac13$ Cantor set?",
   "clean_statement": "Mapping properties of planar maps of exponentially integrable distortion\n\nDoes there exist a homeomorphism $f$ of the plane with exponentially integrable distortion such that $f$ fixes the real axis and sends a set of positive length onto the $\\tfrac13$ Cantor set?",
   "public_statement": "Mapping properties of planar maps of exponentially integrable distortion\n\nDoes there exist a homeomorphism $f$ of the plane with exponentially integrable distortion such that $f$ fixes the real axis and sends a set of positive length onto the $\\tfrac13$ Cantor set?",
   "evidence": "The canonical record is problem 4.2 in the AIM workshop list *Mapping theory in metric spaces* (source file `aim-analysis-notes.json`, zero-based record index 119). The original AIM page was also inspected through its archived page data. It attributes the problem to Pekka Koskela and gives exactly the same text, with no clarifying remark:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 119,
   "attempt": 1
  },
  "AIM-ANALYSIS-0121": {
   "statement_status": "exact",
   "original_statement": "Approximation of Sobolev homeomorphisms by diffeomorphisms\n\nLet $f$ be a homeomorphism of the plane such that $f$ and $f^{-1}$ lie in some Sobolev class, e.g., $W^{1,2}$. Can one approximate $f$ by diffeomorphisms $f_j$ so that $f_j$ and $f_j^{-1}$ converge in the Sobolev norm?",
   "clean_statement": "Approximation of Sobolev homeomorphisms by diffeomorphisms\n\nLet $f$ be a homeomorphism of the plane such that $f$ and $f^{-1}$ lie in some Sobolev class, e.g., $W^{1,2}$. Can one approximate $f$ by diffeomorphisms $f_j$ so that $f_j$ and $f_j^{-1}$ converge in the Sobolev norm?",
   "public_statement": "Approximation of Sobolev homeomorphisms by diffeomorphisms\n\nLet $f$ be a homeomorphism of the plane such that $f$ and $f^{-1}$ lie in some Sobolev class, e.g., $W^{1,2}$. Can one approximate $f$ by diffeomorphisms $f_j$ so that $f_j$ and $f_j^{-1}$ converge in the Sobolev norm?",
   "evidence": "The canonical record is AIM Problem Lists, workshop *Mapping theory in metric spaces*, section “Regularity,” Problem 4.3, source file `aim-analysis-notes.json`, record index 120. The live AIM page was checked on 28 July 2026 and still gives the following statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 120,
   "attempt": 1
  },
  "AIM-ANALYSIS-0122": {
   "statement_status": "exact",
   "original_statement": "Rigidity of $n$-harmonic functions in ${\\mathbb R}^n$\n\nIs there an elementary proof for the fact that every entire $n$-harmonic function in ${\\mathbb R}^n$ with linear growth is affine?",
   "clean_statement": "Rigidity of $n$-harmonic functions in ${\\mathbb R}^n$\n\nIs there an elementary proof for the fact that every entire $n$-harmonic function in ${\\mathbb R}^n$ with linear growth is affine?",
   "public_statement": "Rigidity of $n$-harmonic functions in ${\\mathbb R}^n$\n\nIs there an elementary proof for the fact that every entire $n$-harmonic function in ${\\mathbb R}^n$ with linear growth is affine?",
   "evidence": "The canonical record is AIM-ANALYSIS-0122, from aim-analysis-notes.json at zero-based index 121. The live AIM page *Mapping theory in metric spaces*, Section 5 (“Rigidity”), gives the same text as Problem 5.1 and attributes it to Mario Bonk:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 121,
   "attempt": 1
  },
  "AIM-ANALYSIS-0123": {
   "statement_status": "exact",
   "original_statement": "Are all $1$-quasiconformal homeomorphisms of Hilbert space similarities?\n\nLet $f$ be a homeomorphism between Hilbert spaces with $H_f = 1$ everywhere. Is $f$ a similarity?",
   "clean_statement": "Are all $1$-quasiconformal homeomorphisms of Hilbert space similarities?\n\nLet $f$ be a homeomorphism between Hilbert spaces with $H_f = 1$ everywhere. Is $f$ a similarity?",
   "public_statement": "Are all $1$-quasiconformal homeomorphisms of Hilbert space similarities?\n\nLet $f$ be a homeomorphism between Hilbert spaces with $H_f = 1$ everywhere. Is $f$ a similarity?",
   "evidence": "The canonical AIM record (Mapping theory in metric spaces, Rigidity, Problem 5.2) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 122,
   "attempt": 1
  },
  "AIM-ANALYSIS-0124": {
   "statement_status": "exact",
   "original_statement": "Quasiconformal mappings of the plane which destroy the rectifiability of uncountably many disjoint lines\n\nIs there a quasiconformal map $f$ of the plane and an uncountable set $E\\subset{\\mathbb R}$ so that $f(E\\times{\\mathbb R})$ contains no rectifiable curves?",
   "clean_statement": "Quasiconformal mappings of the plane which destroy the rectifiability of uncountably many disjoint lines\n\nIs there a quasiconformal map $f$ of the plane and an uncountable set $E\\subset{\\mathbb R}$ so that $f(E\\times{\\mathbb R})$ contains no rectifiable curves?",
   "public_statement": "Quasiconformal mappings of the plane which destroy the rectifiability of uncountably many disjoint lines\n\nIs there a quasiconformal map $f$ of the plane and an uncountable set $E\\subset{\\mathbb R}$ so that $f(E\\times{\\mathbb R})$ contains no rectifiable curves?",
   "evidence": "The canonical record is Problem 5.3 in the “Rigidity” section of the AIM list *Mapping theory in metric spaces*. The AIM page was checked on 2026-07-28. It agrees with the repository record and requires no reconstruction:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 123,
   "attempt": 2
  },
  "AIM-ANALYSIS-0125": {
   "statement_status": "exact",
   "original_statement": "Quasisymmetric maps of products of irreducible Carnot groups\n\nLet $H$ be an irreducible Carnot group, $G$ a product of several copies of $H$, and $f:G\\to G$ a quasisymmetric map. Must $f$ be a product map (after permutation of the factors)?",
   "clean_statement": "Quasisymmetric maps of products of irreducible Carnot groups\n\nLet $H$ be an irreducible Carnot group, $G$ a product of several copies of $H$, and $f:G\\to G$ a quasisymmetric map. Must $f$ be a product map (after permutation of the factors)?",
   "public_statement": "Quasisymmetric maps of products of irreducible Carnot groups\n\nLet $H$ be an irreducible Carnot group, $G$ a product of several copies of $H$, and $f:G\\to G$ a quasisymmetric map. Must $f$ be a product map (after permutation of the factors)?",
   "evidence": "The canonical record is AIM Problem List 5.4, from the 2012 workshop *Mapping theory in metric spaces*. Its question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 124,
   "attempt": 1
  },
  "AIM-ANALYSIS-0126": {
   "statement_status": "exact",
   "original_statement": "Product quasiconformal mappings of the plane\n\nLet $f$ be a quasiconformal map of the plane. Suppose at a.e.\\ point, the differential of $f$ either preserves both the $x$-direction and the $y$-direction, or switches the two directions. Does this imply $f$ is a product map?",
   "clean_statement": "Product quasiconformal mappings of the plane\n\nLet $f$ be a quasiconformal map of the plane. Suppose at a.e.\\ point, the differential of $f$ either preserves both the $x$-direction and the $y$-direction, or switches the two directions. Does this imply $f$ is a product map?",
   "public_statement": "Product quasiconformal mappings of the plane\n\nLet $f$ be a quasiconformal map of the plane. Suppose at a.e.\\ point, the differential of $f$ either preserves both the $x$-direction and the $y$-direction, or switches the two directions. Does this imply $f$ is a product map?",
   "evidence": "The canonical AIM record is Problem 5.5 in the Rigidity section of the 2012 workshop list *Mapping theory in metric spaces*. It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 125,
   "attempt": 1
  },
  "AIM-ANALYSIS-0127": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Loewner Sierpi\\'nski carpets\n\nIs the usual $\\tfrac13$ Sierpi\\'nski carpet $S_3$ quasisymmetrically equivalent to a Loewner space?",
   "clean_statement": null,
   "public_statement": "Loewner Sierpi\\'nski carpets\n\nIs the usual $\\tfrac13$ Sierpi\\'nski carpet $S_3$ quasisymmetrically equivalent to a Loewner space?",
   "evidence": "The canonical record is AIM Problem List 5.6 in the “Rigidity” section of *Mapping theory in metric spaces*. The live AIM page attributes the problem to Hrant Hakobyan and states:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 126,
   "attempt": 2
  },
  "AIM-ANALYSIS-0128": {
   "statement_status": "exact",
   "original_statement": "Bi-Lipschitz embeddings of the Sierpi\\'nski carpet into itself\n\nIs every bi-Lipschitz embedding of the $\\tfrac13$ Sierpi\\'nski carpet $S_3$ into itself the restriction of an affine mapping?",
   "clean_statement": "Bi-Lipschitz embeddings of the Sierpi\\'nski carpet into itself\n\nIs every bi-Lipschitz embedding of the $\\tfrac13$ Sierpi\\'nski carpet $S_3$ into itself the restriction of an affine mapping?",
   "public_statement": "Bi-Lipschitz embeddings of the Sierpi\\'nski carpet into itself\n\nIs every bi-Lipschitz embedding of the $\\tfrac13$ Sierpi\\'nski carpet $S_3$ into itself the restriction of an affine mapping?",
   "evidence": "The record adds that Bonk--Merenkov classify quasisymmetric maps **onto** \\(S_3\\). The word “onto” is essential. Here an embedding means an injective map \\[ f:S_3\\longrightarrow S_3 \\] for which, for some \\(L\\geq 1\\), \\[ L^{-1}|x-y|\\leq |f(x)-f(y)|\\leq L|x-y|\\qquad(x,y\\in S_3), \\] and it need not be surjective. “Restriction of an affine mapping” means \\(f=A|_{S_3}\\) for a plane affine map \\(A(x)=Mx+b\\). The source statement has no apparent OCR corruption or notational ambiguity.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 127,
   "attempt": 1
  },
  "AIM-ANALYSIS-0129": {
   "statement_status": "exact",
   "original_statement": "Quasisymmetric rigidity and non-rigidity of positive area Sierpi\\'nski carpets\n\nA set $X \\subset {\\mathbb R}^n$ is said to be \\emph{quasisymmetrically rigid} if every quasisymmetric map of $X$ onto itself coincides the restriction to $X$ of an isometry of ${\\mathbb R}^n$. Bonk and Merenkov have shown that the usual $\\tfrac13$ Sierpi\\'snki carpet is quasisymmetrically rigid.\n\nWhich positive area Sierpi\\'nski carpets are quasisymmetrically rigid? Which are nonrigid?",
   "clean_statement": "Quasisymmetric rigidity and non-rigidity of positive area Sierpi\\'nski carpets\n\nA set $X \\subset {\\mathbb R}^n$ is said to be \\emph{quasisymmetrically rigid} if every quasisymmetric map of $X$ onto itself coincides the restriction to $X$ of an isometry of ${\\mathbb R}^n$. Bonk and Merenkov have shown that the usual $\\tfrac13$ Sierpi\\'snki carpet is quasisymmetrically rigid.\n\nWhich positive area Sierpi\\'nski carpets are quasisymmetrically rigid? Which are nonrigid?",
   "public_statement": "Quasisymmetric rigidity and non-rigidity of positive area Sierpi\\'nski carpets\n\nA set $X \\subset {\\mathbb R}^n$ is said to be \\emph{quasisymmetrically rigid} if every quasisymmetric map of $X$ onto itself coincides the restriction to $X$ of an isometry of ${\\mathbb R}^n$. Bonk and Merenkov have shown that the usual $\\tfrac13$ Sierpi\\'snki carpet is quasisymmetrically rigid.\n\nWhich positive area Sierpi\\'nski carpets are quasisymmetrically rigid? Which are nonrigid?",
   "evidence": "The canonical record is AIM Problem List 5.8 from the 2012 workshop *Mapping theory in metric spaces*. Apart from the typographical spelling “Sierpi'snki” and the missing word “with” in “coincides the restriction,” its question is unambiguous:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 128,
   "attempt": 1
  },
  "AIM-ANALYSIS-0130": {
   "statement_status": "exact",
   "original_statement": "Inner variations\n\nConsider the inner variational equation for planar maps\n$$\n(f_z\\overline{f_{\\overline{z}}})_{\\overline{z}} = 0.\n$$\nLocally, this equation can be reduced to a first-order equation $f_zf_{\\overline{z}} = 1$, which may also be viewed as a differential inclusion $Df \\in M$, with $M$ a certain set of $2\\times 2$ matrices. Natural assumptions on $f$ are the finiteness of the energy and nonnegativity of the Jacobian.\n\nThe equation\n$$\n(f_z\\overline{f_{\\overline{z}}})_{\\overline{z}} = 0.\n$$\nexpresses the stationarity of the Dirichlet energy $\\int |Df|^2$ under inner variations of $f$ (precomposition with diffeomorphisms).\n\nIs $|Df|$ continuous?",
   "clean_statement": "Inner variations\n\nConsider the inner variational equation for planar maps\n$$\n(f_z\\overline{f_{\\overline{z}}})_{\\overline{z}} = 0.\n$$\nLocally, this equation can be reduced to a first-order equation $f_zf_{\\overline{z}} = 1$, which may also be viewed as a differential inclusion $Df \\in M$, with $M$ a certain set of $2\\times 2$ matrices. Natural assumptions on $f$ are the finiteness of the energy and nonnegativity of the Jacobian.\n\nThe equation\n$$\n(f_z\\overline{f_{\\overline{z}}})_{\\overline{z}} = 0.\n$$\nexpresses the stationarity of the Dirichlet energy $\\int |Df|^2$ under inner variations of $f$ (precomposition with diffeomorphisms).\n\nIs $|Df|$ continuous?",
   "public_statement": "Inner variations\n\nConsider the inner variational equation for planar maps\n$$\n(f_z\\overline{f_{\\overline{z}}})_{\\overline{z}} = 0.\n$$\nLocally, this equation can be reduced to a first-order equation $f_zf_{\\overline{z}} = 1$, which may also be viewed as a differential inclusion $Df \\in M$, with $M$ a certain set of $2\\times 2$ matrices. Natural assumptions on $f$ are the finiteness of the energy and nonnegativity of the Jacobian.\n\nThe equation\n$$\n(f_z\\overline{f_{\\overline{z}}})_{\\overline{z}} = 0.\n$$\nexpresses the stationarity of the Dirichlet energy $\\int |Df|^2$ under inner variations of $f$ (precomposition with diffeomorphisms).\n\nIs $|Df|$ continuous?",
   "evidence": "The exact AIM record asks about a planar Sobolev map \\(f\\) satisfying",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 129,
   "attempt": 1
  },
  "AIM-ANALYSIS-0131": {
   "statement_status": "exact",
   "original_statement": "Variational problems for dilatation functionals\n\nStudy extremal variational problems for dilatation functionals of mappings. For instance, find the extremal domain (and possibly also the extremal mappings) associated to the configuration of a doubly connected planar domain including an obstacle of prescribed diameter.",
   "clean_statement": "Variational problems for dilatation functionals\n\nStudy extremal variational problems for dilatation functionals of mappings. For instance, find the extremal domain (and possibly also the extremal mappings) associated to the configuration of a doubly connected planar domain including an obstacle of prescribed diameter.",
   "public_statement": "Variational problems for dilatation functionals\n\nStudy extremal variational problems for dilatation functionals of mappings. For instance, find the extremal domain (and possibly also the extremal mappings) associated to the configuration of a doubly connected planar domain including an obstacle of prescribed diameter.",
   "evidence": "The canonical record is AIM-ANALYSIS-0131, source file `aim-analysis-notes.json`, zero-based index 130. It is Problem 6.2 in the section “Variational problems” of the AIM list *Mapping theory in metric spaces*. The record states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 130,
   "attempt": 1
  },
  "AIM-ANALYSIS-0132": {
   "statement_status": "exact",
   "original_statement": "Hausdorff measures in $\\ell^p_n$\n\nLet ${\\mathcal H}^m$ denotes the Hausdorff $m$-measure corresponding to the norm in $\\ell^p_n$.\n\nLet $P$ be a polyhedral $m$-cycle in $\\ell^p_n$, $p\\ne 2$, with faces $F_1,\\ldots,F_k$. Is ${\\mathcal H}^m(F_1) \\le \\sum_{j=2}^k {\\mathcal H}^m(F_j)$?",
   "clean_statement": "Hausdorff measures in $\\ell^p_n$\n\nLet ${\\mathcal H}^m$ denotes the Hausdorff $m$-measure corresponding to the norm in $\\ell^p_n$.\n\nLet $P$ be a polyhedral $m$-cycle in $\\ell^p_n$, $p\\ne 2$, with faces $F_1,\\ldots,F_k$. Is ${\\mathcal H}^m(F_1) \\le \\sum_{j=2}^k {\\mathcal H}^m(F_j)$?",
   "public_statement": "Hausdorff measures in $\\ell^p_n$\n\nLet ${\\mathcal H}^m$ denotes the Hausdorff $m$-measure corresponding to the norm in $\\ell^p_n$.\n\nLet $P$ be a polyhedral $m$-cycle in $\\ell^p_n$, $p\\ne 2$, with faces $F_1,\\ldots,F_k$. Is ${\\mathcal H}^m(F_1) \\le \\sum_{j=2}^k {\\mathcal H}^m(F_j)$?",
   "evidence": "The canonical record is AIM Problem List 6.3 from the workshop *Mapping theory in metric spaces*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 131,
   "attempt": 1
  },
  "AIM-ANALYSIS-0133": {
   "statement_status": "exact",
   "original_statement": "Compact deformations of minimizing sets\n\nAn $m$-rectifiable closed set $S \\subset V = \\ell^\\infty_n$ is called {\\it minimizing} if for every Lipschitz map $f:V\\to V$ such that $f$ is the identity off a cube $C$ and $f(C)\\subset C$, ${\\mathcal H}^m(C\\cap S) \\le {\\mathcal H}^m(f(C\\cap S))$. Let $S$ be minimizing. Is it true that for ${\\mathcal H}^m$-almost every $x \\in S$, there exists a neighborhood $U$ of $x$ such that $S\\cap U$ is a Lipschitz graph over an $m$-dimensional subspace $\\Pi$ of $V$?",
   "clean_statement": "Compact deformations of minimizing sets\n\nAn $m$-rectifiable closed set $S \\subset V = \\ell^\\infty_n$ is called {\\it minimizing} if for every Lipschitz map $f:V\\to V$ such that $f$ is the identity off a cube $C$ and $f(C)\\subset C$, ${\\mathcal H}^m(C\\cap S) \\le {\\mathcal H}^m(f(C\\cap S))$. Let $S$ be minimizing. Is it true that for ${\\mathcal H}^m$-almost every $x \\in S$, there exists a neighborhood $U$ of $x$ such that $S\\cap U$ is a Lipschitz graph over an $m$-dimensional subspace $\\Pi$ of $V$?",
   "public_statement": "Compact deformations of minimizing sets\n\nAn $m$-rectifiable closed set $S \\subset V = \\ell^\\infty_n$ is called {\\it minimizing} if for every Lipschitz map $f:V\\to V$ such that $f$ is the identity off a cube $C$ and $f(C)\\subset C$, ${\\mathcal H}^m(C\\cap S) \\le {\\mathcal H}^m(f(C\\cap S))$. Let $S$ be minimizing. Is it true that for ${\\mathcal H}^m$-almost every $x \\in S$, there exists a neighborhood $U$ of $x$ such that $S\\cap U$ is a Lipschitz graph over an $m$-dimensional subspace $\\Pi$ of $V$?",
   "evidence": "The canonical record is Problem 6.4, “Compact deformations of minimizing sets,” in the AIM problem list *Mapping theory in metric spaces*, section “Variational problems.” The live AIM page was checked on 2026-07-28 and attributes the problem to Thierry De Pauw. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 132,
   "attempt": 1
  },
  "AIM-ANALYSIS-0134": {
   "statement_status": "exact",
   "original_statement": "Isoperimetric inequalities in metric measure spaces\n\nCan the perimeter measure $P$ on the right hand side of the relative isoperimetric inequality\n$$\n\\min \\{ \\mu (E\\cap B), \\mu (B \\setminus E) \\} \\le C r P(E,\\lambda B),\n$$\nwhere $E \\subset X$ is a Borel set and $B \\subset X$ is a ball of radius $r$, be replaced with the codimension one Hausdorff measure of the part of the measure-theoretic boundary of $E$ inside $B$ even without knowing ahead of time whether $E$ is of finite perimeter?",
   "clean_statement": "Isoperimetric inequalities in metric measure spaces\n\nCan the perimeter measure $P$ on the right hand side of the relative isoperimetric inequality\n$$\n\\min \\{ \\mu (E\\cap B), \\mu (B \\setminus E) \\} \\le C r P(E,\\lambda B),\n$$\nwhere $E \\subset X$ is a Borel set and $B \\subset X$ is a ball of radius $r$, be replaced with the codimension one Hausdorff measure of the part of the measure-theoretic boundary of $E$ inside $B$ even without knowing ahead of time whether $E$ is of finite perimeter?",
   "public_statement": "Isoperimetric inequalities in metric measure spaces\n\nCan the perimeter measure $P$ on the right hand side of the relative isoperimetric inequality\n$$\n\\min \\{ \\mu (E\\cap B), \\mu (B \\setminus E) \\} \\le C r P(E,\\lambda B),\n$$\nwhere $E \\subset X$ is a Borel set and $B \\subset X$ is a ball of radius $r$, be replaced with the codimension one Hausdorff measure of the part of the measure-theoretic boundary of $E$ inside $B$ even without knowing ahead of time whether $E$ is of finite perimeter?",
   "evidence": "The canonical record is AIM Problem List 6.5, “Isoperimetric inequalities in metric measure spaces,” attributed on the live AIM page to Nageswari Shanmugalingam. The exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 133,
   "attempt": 1
  },
  "AIM-ANALYSIS-0135": {
   "statement_status": "exact",
   "original_statement": "Let $\\{c_k\\}_{k=0}^n\\subset\\mathbb{R}$ be given. Does there exist a real rooted polynomial, $p(x)$, with zeros outside $[0,n]$ such that $c_k=p(k)$, $k=0,1,\\ldots,n$?",
   "clean_statement": "Let $\\{c_k\\}_{k=0}^n\\subset\\mathbb{R}$ be given. Does there exist a real rooted polynomial, $p(x)$, with zeros outside $[0,n]$ such that $c_k=p(k)$, $k=0,1,\\ldots,n$?",
   "public_statement": "Let $\\{c_k\\}_{k=0}^n\\subset\\mathbb{R}$ be given. Does there exist a real rooted polynomial, $p(x)$, with zeros outside $[0,n]$ such that $c_k=p(k)$, $k=0,1,\\ldots,n$?",
   "evidence": "The canonical record is problem 13.1 in the section “Multiplier sequences and CZDS” of the 2011 AIM workshop *Stability, hyperbolicity, and zero localization of functions*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 134,
   "attempt": 1
  },
  "AIM-ANALYSIS-0136": {
   "statement_status": "exact",
   "original_statement": "Let $f(z)$ be a meromorphic function. When is the sequence $\\{f(k)\\}_{k=0}^\\infty$ a multiplier sequence?",
   "clean_statement": "Let $f(z)$ be a meromorphic function. When is the sequence $\\{f(k)\\}_{k=0}^\\infty$ a multiplier sequence?",
   "public_statement": "Let $f(z)$ be a meromorphic function. When is the sequence $\\{f(k)\\}_{k=0}^\\infty$ a multiplier sequence?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 135,
   "attempt": 1
  },
  "AIM-ANALYSIS-0137": {
   "statement_status": "exact",
   "original_statement": "Given $p\\in\\mathbb{C}[x]$, $\\deg(p) = n$, find an $n\\times n$ Hermitian matrix whose inertia $(n_+, n_{-}, n_0)$ counts the zeros of $p$ in $H_+, H_{-}, H_0=\\mathbb{R}$.",
   "clean_statement": "Given $p\\in\\mathbb{C}[x]$, $\\deg(p) = n$, find an $n\\times n$ Hermitian matrix whose inertia $(n_+, n_{-}, n_0)$ counts the zeros of $p$ in $H_+, H_{-}, H_0=\\mathbb{R}$.",
   "public_statement": "Given $p\\in\\mathbb{C}[x]$, $\\deg(p) = n$, find an $n\\times n$ Hermitian matrix whose inertia $(n_+, n_{-}, n_0)$ counts the zeros of $p$ in $H_+, H_{-}, H_0=\\mathbb{R}$.",
   "evidence": "The canonical record is AIM-ANALYSIS-0137, item 2.3 in the “Matrix Theory” section of the AIM workshop list *Stability, hyperbolicity, and zero localization of functions*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 136,
   "attempt": 1
  },
  "AIM-ANALYSIS-0138": {
   "statement_status": "exact",
   "original_statement": "Study the zeros of Bessis-Moussa-Villani polynomials $t\\to\\text{tr}(A+tB)^m$.",
   "clean_statement": "Study the zeros of Bessis-Moussa-Villani polynomials $t\\to\\text{tr}(A+tB)^m$.",
   "public_statement": "Study the zeros of Bessis-Moussa-Villani polynomials $t\\to\\text{tr}(A+tB)^m$.",
   "evidence": "The canonical AIM record says, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 137,
   "attempt": 1
  },
  "AIM-ANALYSIS-0139": {
   "statement_status": "exact",
   "original_statement": "Find a simple proof of BMV conjecture for $3\\times 3$ matrices (and an explicit formula for the BMV measure for $3\\times 3$ matrices; the existence of the measure is due to recent work of Stahl).",
   "clean_statement": "Find a simple proof of BMV conjecture for $3\\times 3$ matrices (and an explicit formula for the BMV measure for $3\\times 3$ matrices; the existence of the measure is due to recent work of Stahl).",
   "public_statement": "Find a simple proof of BMV conjecture for $3\\times 3$ matrices (and an explicit formula for the BMV measure for $3\\times 3$ matrices; the existence of the measure is due to recent work of Stahl).",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 138,
   "attempt": 1
  },
  "AIM-ANALYSIS-0140": {
   "statement_status": "exact",
   "original_statement": "Any real polynomial can be written as $\\det(A-\\lambda I),$ where $A$ has the (tridiagonal) form\n\n\\[ \\left[\\begin{array}{cccccc}\n+ & + & & & & \\\\\n- & 0 & + & & &\\\\\n & - & 0 & + & &\\\\\n & & \\ddots & \\ddots & \\ddots & \\\\\n & & & - & 0 & + \\\\\n & & & & - & - \\\\\n\\end{array}\\right].\\]",
   "clean_statement": "Any real polynomial can be written as $\\det(A-\\lambda I),$ where $A$ has the (tridiagonal) form\n\n\\[ \\left[\\begin{array}{cccccc}\n+ & + & & & & \\\\\n- & 0 & + & & &\\\\\n & - & 0 & + & &\\\\\n & & \\ddots & \\ddots & \\ddots & \\\\\n & & & - & 0 & + \\\\\n & & & & - & - \\\\\n\\end{array}\\right].\\]",
   "public_statement": "Any real polynomial can be written as $\\det(A-\\lambda I),$ where $A$ has the (tridiagonal) form\n\n\\[ \\left[\\begin{array}{cccccc}\n+ & + & & & & \\\\\n- & 0 & + & & &\\\\\n & - & 0 & + & &\\\\\n & & \\ddots & \\ddots & \\ddots & \\\\\n & & & - & 0 & + \\\\\n & & & & - & - \\\\\n\\end{array}\\right].\\]",
   "evidence": "The exact canonical record is the declarative sentence",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 139,
   "attempt": 2
  },
  "AIM-ANALYSIS-0141": {
   "statement_status": "exact",
   "original_statement": "Investigate the signs of Hurwitz minors for P\\'olya frequency sequences.",
   "clean_statement": "Investigate the signs of Hurwitz minors for P\\'olya frequency sequences.",
   "public_statement": "Investigate the signs of Hurwitz minors for P\\'olya frequency sequences.",
   "evidence": "The canonical record is problem 2.5 in the “Matrix Theory” section of the AIM workshop *Stability, hyperbolicity, and zero localization of functions*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 140,
   "attempt": 1
  },
  "AIM-ANALYSIS-0142": {
   "statement_status": "exact",
   "original_statement": "The following conjecture of (McNamara et al.) was proved by P. Br\\\"and\\'en. Let $p(x):=\\sum_{k=0}^n a_k x^k$, with $a_{-1}:=a_{n+1}:=0$.\nIf $p\\in\\mathcal{L}\\text{-}\\mathcal{P}^+$, then\n\\[\\sum_{k=0}^n \\left|\\begin{array}{cc}\na_k & a_{k+1} \\\\\na_{k-1} & a_{k} \\end{array}\\right|x^k \\in\\mathcal{L}\\text{-}\\mathcal{P}.\\]\nS. Fisk generalized this statement to polynomials with coefficients formed from the determinants of $3\\times 3$ matrices in the conjecture below.\n\n\\label{fidk_conj}\nIf $p\\in\\mathcal{L}\\text{-}\\mathcal{P}^+$, then\n\\[\\sum_{k=0}^n \\left|\\begin{array}{ccc}\na_k & a_{k+1} & a_{k+2} \\\\\na_{k-1} & a_k & a_{k+1}\\\\\na_{k-2} & a_{k-1} & a_k\\end{array}\\right|x^k\\in\\mathcal{L}\\text{-}\\mathcal{P}^+.\\]",
   "clean_statement": "The following conjecture of (McNamara et al.) was proved by P. Br\\\"and\\'en. Let $p(x):=\\sum_{k=0}^n a_k x^k$, with $a_{-1}:=a_{n+1}:=0$.\nIf $p\\in\\mathcal{L}\\text{-}\\mathcal{P}^+$, then\n\\[\\sum_{k=0}^n \\left|\\begin{array}{cc}\na_k & a_{k+1} \\\\\na_{k-1} & a_{k} \\end{array}\\right|x^k \\in\\mathcal{L}\\text{-}\\mathcal{P}.\\]\nS. Fisk generalized this statement to polynomials with coefficients formed from the determinants of $3\\times 3$ matrices in the conjecture below.\n\n\\label{fidk_conj}\nIf $p\\in\\mathcal{L}\\text{-}\\mathcal{P}^+$, then\n\\[\\sum_{k=0}^n \\left|\\begin{array}{ccc}\na_k & a_{k+1} & a_{k+2} \\\\\na_{k-1} & a_k & a_{k+1}\\\\\na_{k-2} & a_{k-1} & a_k\\end{array}\\right|x^k\\in\\mathcal{L}\\text{-}\\mathcal{P}^+.\\]",
   "public_statement": "The following conjecture of (McNamara et al.) was proved by P. Br\\\"and\\'en. Let $p(x):=\\sum_{k=0}^n a_k x^k$, with $a_{-1}:=a_{n+1}:=0$.\nIf $p\\in\\mathcal{L}\\text{-}\\mathcal{P}^+$, then\n\\[\\sum_{k=0}^n \\left|\\begin{array}{cc}\na_k & a_{k+1} \\\\\na_{k-1} & a_{k} \\end{array}\\right|x^k \\in\\mathcal{L}\\text{-}\\mathcal{P}.\\]\nS. Fisk generalized this statement to polynomials with coefficients formed from the determinants of $3\\times 3$ matrices in the conjecture below.\n\n\\label{fidk_conj}\nIf $p\\in\\mathcal{L}\\text{-}\\mathcal{P}^+$, then\n\\[\\sum_{k=0}^n \\left|\\begin{array}{ccc}\na_k & a_{k+1} & a_{k+2} \\\\\na_{k-1} & a_k & a_{k+1}\\\\\na_{k-2} & a_{k-1} & a_k\\end{array}\\right|x^k\\in\\mathcal{L}\\text{-}\\mathcal{P}^+.\\]",
   "evidence": "The canonical record is Conjecture 3.2 in the \"Log Concavity\" section of the AIM problem list *Stability, hyperbolicity, and zero localization of functions*. It asks the following. Given",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 141,
   "attempt": 1
  },
  "AIM-ANALYSIS-0143": {
   "statement_status": "exact",
   "original_statement": "Let\n\\[Q_n^{(\\alpha, \\beta)}:=\\sum_{k=0}^n\\binom{n}{k}f_kf_{n-k}((x+\\alpha)_k(x+\\beta)_{n-k} - (x+\\alpha+\\beta)_k(x)_{n-k}),\\]\nwhere $\\alpha, \\beta>0$, $f_k^2\\ge f_{k-1}f_{k+1}$, and $(x)_k = x(x+1)\\cdots(x+k-1)$. Then the Maclaurin coefficients of $Q^{(\\,\\alpha,\\,\\beta)}$ are non-negative.",
   "clean_statement": "Let\n\\[Q_n^{(\\alpha, \\beta)}:=\\sum_{k=0}^n\\binom{n}{k}f_kf_{n-k}((x+\\alpha)_k(x+\\beta)_{n-k} - (x+\\alpha+\\beta)_k(x)_{n-k}),\\]\nwhere $\\alpha, \\beta>0$, $f_k^2\\ge f_{k-1}f_{k+1}$, and $(x)_k = x(x+1)\\cdots(x+k-1)$. Then the Maclaurin coefficients of $Q^{(\\,\\alpha,\\,\\beta)}$ are non-negative.",
   "public_statement": "Let\n\\[Q_n^{(\\alpha, \\beta)}:=\\sum_{k=0}^n\\binom{n}{k}f_kf_{n-k}((x+\\alpha)_k(x+\\beta)_{n-k} - (x+\\alpha+\\beta)_k(x)_{n-k}),\\]\nwhere $\\alpha, \\beta>0$, $f_k^2\\ge f_{k-1}f_{k+1}$, and $(x)_k = x(x+1)\\cdots(x+k-1)$. Then the Maclaurin coefficients of $Q^{(\\,\\alpha,\\,\\beta)}$ are non-negative.",
   "evidence": "The canonical AIM record, problem 3.1 in the “Log Concavity” section of the workshop *Stability, hyperbolicity, and zero localization of functions*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 142,
   "attempt": 1
  },
  "AIM-ANALYSIS-0144": {
   "statement_status": "exact",
   "original_statement": "Let $\\{\\phi_k(x)\\}_{k=0}^\\infty$ and $\\{f_k\\}_{k=0}^\\infty$ be log-concave. When is $\\sum f_k\\phi_k(x)$ log-concave ?",
   "clean_statement": "Let $\\{\\phi_k(x)\\}_{k=0}^\\infty$ and $\\{f_k\\}_{k=0}^\\infty$ be log-concave. When is $\\sum f_k\\phi_k(x)$ log-concave ?",
   "public_statement": "Let $\\{\\phi_k(x)\\}_{k=0}^\\infty$ and $\\{f_k\\}_{k=0}^\\infty$ be log-concave. When is $\\sum f_k\\phi_k(x)$ log-concave ?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 143,
   "attempt": 1
  },
  "AIM-ANALYSIS-0145": {
   "statement_status": "exact",
   "original_statement": "Let $p\\in\\mathbb{R}[x]$, $\\deg(p)=n>2$. There are at least $n-1$ points of extreme curvature $\\kappa(p)$ ( more generally where $\\kappa'(p)=0$ ) \\cite{MR2104693}.",
   "clean_statement": "Let $p\\in\\mathbb{R}[x]$, $\\deg(p)=n>2$. There are at least $n-1$ points of extreme curvature $\\kappa(p)$ ( more generally where $\\kappa'(p)=0$ ) \\cite{MR2104693}.",
   "public_statement": "Let $p\\in\\mathbb{R}[x]$, $\\deg(p)=n>2$. There are at least $n-1$ points of extreme curvature $\\kappa(p)$ ( more generally where $\\kappa'(p)=0$ ) \\cite{MR2104693}.",
   "evidence": "The canonical record, copied without correction, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 144,
   "attempt": 1
  },
  "AIM-ANALYSIS-0146": {
   "statement_status": "exact",
   "original_statement": "What are the zero spacing effects of higher order differential operators on functions which belong to $\\mathcal{L}\\text{-}\\mathcal{P}$?",
   "clean_statement": "What are the zero spacing effects of higher order differential operators on functions which belong to $\\mathcal{L}\\text{-}\\mathcal{P}$?",
   "public_statement": "What are the zero spacing effects of higher order differential operators on functions which belong to $\\mathcal{L}\\text{-}\\mathcal{P}$?",
   "evidence": "The exact canonical record is AIM-ANALYSIS-0146, source file `aim-analysis-notes.json`, index 145, workshop section “Zeros of derivatives,” Problem 5.2:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 145,
   "attempt": 1
  },
  "AIM-ANALYSIS-0147": {
   "statement_status": "exact",
   "original_statement": "Let $(L_n^\\alpha)$ denote the generalized Laguerre polynomials. It is known that $L_m^\\alpha$ and $L_n^\\alpha$ can have common zeros if $|m-n|>1$. Is the number of common zeros linked to arithmetic properties of $\\alpha$?\n\nThere are no common zeros between $L_m^\\alpha$ and $L_n^\\alpha$, $|m-n|>1$, for $\\alpha$ rational.",
   "clean_statement": "Let $(L_n^\\alpha)$ denote the generalized Laguerre polynomials. It is known that $L_m^\\alpha$ and $L_n^\\alpha$ can have common zeros if $|m-n|>1$. Is the number of common zeros linked to arithmetic properties of $\\alpha$?\n\nThere are no common zeros between $L_m^\\alpha$ and $L_n^\\alpha$, $|m-n|>1$, for $\\alpha$ rational.",
   "public_statement": "Let $(L_n^\\alpha)$ denote the generalized Laguerre polynomials. It is known that $L_m^\\alpha$ and $L_n^\\alpha$ can have common zeros if $|m-n|>1$. Is the number of common zeros linked to arithmetic properties of $\\alpha$?\n\nThere are no common zeros between $L_m^\\alpha$ and $L_n^\\alpha$, $|m-n|>1$, for $\\alpha$ rational.",
   "evidence": "The canonical record comes from Conjecture 6.1 in the “Orthogonal polynomials” section of the AIM problem list *Stability and hyperbolicity*. The live page was checked on 2026-07-28. It explicitly labels the following as “Conjecture 6.1” and attributes it to K. Driver; it does not display a restriction on the parameter:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 146,
   "attempt": 1
  },
  "AIM-ANALYSIS-0148": {
   "statement_status": "exact",
   "original_statement": "Consider a subspace $S$ of $\\mathbb{C}[x]$ of codimension $k$ (defined by $k$ linear constraints, explicitly given). Given a scalar product on $\\mathbb{C}[x]$, find an orthogonal basis of $S$ (ordered by degree); some degrees will be missing.\n\\begin{enumerate}\n\\item Which degrees get skipped?\n\\item Characterize these orthogonal polynomials.\n\\item Does this have anything to do with lacunary Pad\\'e approximants?\n\\end{enumerate}",
   "clean_statement": "Consider a subspace $S$ of $\\mathbb{C}[x]$ of codimension $k$ (defined by $k$ linear constraints, explicitly given). Given a scalar product on $\\mathbb{C}[x]$, find an orthogonal basis of $S$ (ordered by degree); some degrees will be missing.\n\\begin{enumerate}\n\\item Which degrees get skipped?\n\\item Characterize these orthogonal polynomials.\n\\item Does this have anything to do with lacunary Pad\\'e approximants?\n\\end{enumerate}",
   "public_statement": "Consider a subspace $S$ of $\\mathbb{C}[x]$ of codimension $k$ (defined by $k$ linear constraints, explicitly given). Given a scalar product on $\\mathbb{C}[x]$, find an orthogonal basis of $S$ (ordered by degree); some degrees will be missing.\n\\begin{enumerate}\n\\item Which degrees get skipped?\n\\item Characterize these orthogonal polynomials.\n\\item Does this have anything to do with lacunary Pad\\'e approximants?\n\\end{enumerate}",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 147,
   "attempt": 1
  },
  "AIM-ANALYSIS-0149": {
   "statement_status": "exact",
   "original_statement": "Let $P_n^{\\;\\alpha, \\,\\beta}$ denote the Jacobi polynomial of $\\deg n$ with parameters $\\alpha, \\beta> -1$.\nThen $P_n^{\\;\\alpha', \\,\\beta'}$ and $P_n^{\\;\\alpha, \\,\\beta}$ interlace for $\\alpha\\approx\\alpha'$, and $\\beta\\approx\\beta'$. Does the failure of interlacing correspond to some physical law being ``violated''?",
   "clean_statement": "Let $P_n^{\\;\\alpha, \\,\\beta}$ denote the Jacobi polynomial of $\\deg n$ with parameters $\\alpha, \\beta> -1$.\nThen $P_n^{\\;\\alpha', \\,\\beta'}$ and $P_n^{\\;\\alpha, \\,\\beta}$ interlace for $\\alpha\\approx\\alpha'$, and $\\beta\\approx\\beta'$. Does the failure of interlacing correspond to some physical law being ``violated''?",
   "public_statement": "Let $P_n^{\\;\\alpha, \\,\\beta}$ denote the Jacobi polynomial of $\\deg n$ with parameters $\\alpha, \\beta> -1$.\nThen $P_n^{\\;\\alpha', \\,\\beta'}$ and $P_n^{\\;\\alpha, \\,\\beta}$ interlace for $\\alpha\\approx\\alpha'$, and $\\beta\\approx\\beta'$. Does the failure of interlacing correspond to some physical law being ``violated''?",
   "evidence": "The canonical record is Problem 6.2 in the “Orthogonal polynomials” section of the AIM problem list *Stability, hyperbolicity, and zero localization of functions*. The live page was checked on 2026-07-28. It attributes the problem to K. Driver and reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 148,
   "attempt": 1
  },
  "AIM-ANALYSIS-0150": {
   "statement_status": "exact",
   "original_statement": "Let $p_k(z)$ be the polynomials orthogonal with respect to $\\Phi$ on $(-\\infty,\\infty)$.\nDefine\n\\[P_n(z):=\\frac{\\det\\left[p_i(z_j)\\right]_{i,j=1}^n}{\\det\\left[z_i^{j-1}\\right]_{i=1,j=1}^n}, \\qquad\\text{and}\\qquad\nQ_n(z):=\\frac{\\det\\left[p_i(z_j)\\right]_{i,j=2}^n}{\\det\\left[z_{i}^{j-2}\\right]_{i,j=2}^{n}}.\\]\n\nInvestigate the zeros of $P_n$ and $Q_n.$",
   "clean_statement": "Let $p_k(z)$ be the polynomials orthogonal with respect to $\\Phi$ on $(-\\infty,\\infty)$.\nDefine\n\\[P_n(z):=\\frac{\\det\\left[p_i(z_j)\\right]_{i,j=1}^n}{\\det\\left[z_i^{j-1}\\right]_{i=1,j=1}^n}, \\qquad\\text{and}\\qquad\nQ_n(z):=\\frac{\\det\\left[p_i(z_j)\\right]_{i,j=2}^n}{\\det\\left[z_{i}^{j-2}\\right]_{i,j=2}^{n}}.\\]\n\nInvestigate the zeros of $P_n$ and $Q_n.$",
   "public_statement": "Let $p_k(z)$ be the polynomials orthogonal with respect to $\\Phi$ on $(-\\infty,\\infty)$.\nDefine\n\\[P_n(z):=\\frac{\\det\\left[p_i(z_j)\\right]_{i,j=1}^n}{\\det\\left[z_i^{j-1}\\right]_{i=1,j=1}^n}, \\qquad\\text{and}\\qquad\nQ_n(z):=\\frac{\\det\\left[p_i(z_j)\\right]_{i,j=2}^n}{\\det\\left[z_{i}^{j-2}\\right]_{i,j=2}^{n}}.\\]\n\nInvestigate the zeros of $P_n$ and $Q_n.$",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 149,
   "attempt": 1
  },
  "AIM-ANALYSIS-0151": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Are there any non-real zeros of\n\\[\\int_{0}^\\infty \\Phi^\\alpha(t)\\cos(zt)dt \\qquad \\text{for}\\qquad \\alpha>0 ?\\]",
   "clean_statement": null,
   "public_statement": "Are there any non-real zeros of\n\\[\\int_{0}^\\infty \\Phi^\\alpha(t)\\cos(zt)dt \\qquad \\text{for}\\qquad \\alpha>0 ?\\]",
   "evidence": "The canonical record asks:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 150,
   "attempt": 1
  },
  "AIM-ANALYSIS-0152": {
   "statement_status": "exact",
   "original_statement": "\\label{nonzero_approx}\nLet be $K\\subset\\mathbb{C}$ be compact, let $\\mathbb{C}\\setminus K$ be connected, and let $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$. Is there a sequence of polynomials $\\{p_n\\}_{n=1}^\\infty$ with no zeros in $K$ such that $p_n\\to f$?",
   "clean_statement": "\\label{nonzero_approx}\nLet be $K\\subset\\mathbb{C}$ be compact, let $\\mathbb{C}\\setminus K$ be connected, and let $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$. Is there a sequence of polynomials $\\{p_n\\}_{n=1}^\\infty$ with no zeros in $K$ such that $p_n\\to f$?",
   "public_statement": "\\label{nonzero_approx}\nLet be $K\\subset\\mathbb{C}$ be compact, let $\\mathbb{C}\\setminus K$ be connected, and let $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$. Is there a sequence of polynomials $\\{p_n\\}_{n=1}^\\infty$ with no zeros in $K$ such that $p_n\\to f$?",
   "evidence": "The canonical record is Problem 9.1 in the “Miscellaneous Problems” section of the AIM problem list *Stability, hyperbolicity, and zero localization of functions*. The live page was checked on 2026-07-28. It attributes the problem to P. Gauthier and reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 151,
   "attempt": 1
  },
  "AIM-ANALYSIS-0153": {
   "statement_status": "exact",
   "original_statement": "\\label{conj_universal}\nLet be $K\\subset\\mathbb{C}$ be compact, $\\mathbb{C}\\setminus K$ connected, and $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$.\nSuppose $K\\subseteq\\{z:\\frac{1}{2} 0,\\]\nwhere\n\n\\[\\overline{d}(E):=\\limsup_{T\\to\\infty} \\frac{m(E\\cap[0,T\\,])}{T}.\\]",
   "clean_statement": "\\label{conj_universal}\nLet be $K\\subset\\mathbb{C}$ be compact, $\\mathbb{C}\\setminus K$ connected, and $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$.\nSuppose $K\\subseteq\\{z:\\frac{1}{2} 0,\\]\nwhere\n\n\\[\\overline{d}(E):=\\limsup_{T\\to\\infty} \\frac{m(E\\cap[0,T\\,])}{T}.\\]",
   "public_statement": "\\label{conj_universal}\nLet be $K\\subset\\mathbb{C}$ be compact, $\\mathbb{C}\\setminus K$ connected, and $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$.\nSuppose $K\\subseteq\\{z:\\frac{1}{2} 0,\\]\nwhere\n\n\\[\\overline{d}(E):=\\limsup_{T\\to\\infty} \\frac{m(E\\cap[0,T\\,])}{T}.\\]",
   "evidence": "The canonical record is object 152 (zero-based) of *aim-analysis-notes.json*. Its text is truncated after \\(\\frac12\\) and is not mathematically usable.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 152,
   "attempt": 1
  },
  "AIM-ANALYSIS-0154": {
   "statement_status": "exact",
   "original_statement": "Let $\\mu$ be a probability measure on $\\{0,1\\}^n$. Consider\n\\[f(\\vec{z}\\,) = \\int {\\vec{z}}^{\\;\\vec{\\alpha}} d\\mu(\\vec{\\alpha}\\,)\\]\nand\n\\[g(\\vec{\\lambda}\\,) = \\int e^{\\;\\vec{\\lambda}\\cdot\\vec{\\alpha}} d\\mu(\\vec{\\alpha}\\,).\\]\nWhat property of $g$ is equivalent to the stability of $f$?",
   "clean_statement": "Let $\\mu$ be a probability measure on $\\{0,1\\}^n$. Consider\n\\[f(\\vec{z}\\,) = \\int {\\vec{z}}^{\\;\\vec{\\alpha}} d\\mu(\\vec{\\alpha}\\,)\\]\nand\n\\[g(\\vec{\\lambda}\\,) = \\int e^{\\;\\vec{\\lambda}\\cdot\\vec{\\alpha}} d\\mu(\\vec{\\alpha}\\,).\\]\nWhat property of $g$ is equivalent to the stability of $f$?",
   "public_statement": "Let $\\mu$ be a probability measure on $\\{0,1\\}^n$. Consider\n\\[f(\\vec{z}\\,) = \\int {\\vec{z}}^{\\;\\vec{\\alpha}} d\\mu(\\vec{\\alpha}\\,)\\]\nand\n\\[g(\\vec{\\lambda}\\,) = \\int e^{\\;\\vec{\\lambda}\\cdot\\vec{\\alpha}} d\\mu(\\vec{\\alpha}\\,).\\]\nWhat property of $g$ is equivalent to the stability of $f$?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 153,
   "attempt": 1
  },
  "AIM-ANALYSIS-0155": {
   "statement_status": "exact",
   "original_statement": "Let $p(z)\\in\\mathbb{C}[z]$, $\\deg(p)=n$. Suppose $p(z)\\neq 0$ for all $|z|\\le1$. Define $p^*(z):=z^n\\overline{p\\left(\\frac{1}{\\bar{z}}\\right)}$. Then it is known that\n\n\\[p(A)p(A)^\\dagger \\ge p^*(A)(p^*(A))^\\dagger \\;\\; \\textit{for any contractive matrix } A.\\]\n\n($A$ is \\emph{contractive} means $\\sup\\limits_x \\frac{||Ax||_2}{||x||_2}<1.$) In 2D, we replace $A$ with a pair of commuting contractions and $p$ with a bivariate polynomial with no zeros in\n$\\{z:|z|\\le 1\\}\\times\\{w:|w|\\le 1\\}$. The 3D generalization fails -- when does it hold?",
   "clean_statement": "Let $p(z)\\in\\mathbb{C}[z]$, $\\deg(p)=n$. Suppose $p(z)\\neq 0$ for all $|z|\\le1$. Define $p^*(z):=z^n\\overline{p\\left(\\frac{1}{\\bar{z}}\\right)}$. Then it is known that\n\n\\[p(A)p(A)^\\dagger \\ge p^*(A)(p^*(A))^\\dagger \\;\\; \\textit{for any contractive matrix } A.\\]\n\n($A$ is \\emph{contractive} means $\\sup\\limits_x \\frac{||Ax||_2}{||x||_2}<1.$) In 2D, we replace $A$ with a pair of commuting contractions and $p$ with a bivariate polynomial with no zeros in\n$\\{z:|z|\\le 1\\}\\times\\{w:|w|\\le 1\\}$. The 3D generalization fails -- when does it hold?",
   "public_statement": "Let $p(z)\\in\\mathbb{C}[z]$, $\\deg(p)=n$. Suppose $p(z)\\neq 0$ for all $|z|\\le1$. Define $p^*(z):=z^n\\overline{p\\left(\\frac{1}{\\bar{z}}\\right)}$. Then it is known that\n\n\\[p(A)p(A)^\\dagger \\ge p^*(A)(p^*(A))^\\dagger \\;\\; \\textit{for any contractive matrix } A.\\]\n\n($A$ is \\emph{contractive} means $\\sup\\limits_x \\frac{||Ax||_2}{||x||_2}<1.$) In 2D, we replace $A$ with a pair of commuting contractions and $p$ with a bivariate polynomial with no zeros in\n$\\{z:|z|\\le 1\\}\\times\\{w:|w|\\le 1\\}$. The 3D generalization fails -- when does it hold?",
   "evidence": "The canonical record is Problem 9.4 in the AIM list *Stability, hyperbolicity, and zero localization of functions*, in the section “Miscellaneous Problems.” The live AIM page attributes the problem to G. Knese. The source record states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 154,
   "attempt": 1
  },
  "AIM-ANALYSIS-0156": {
   "statement_status": "exact",
   "original_statement": "Find more examples and attempt to characterize Markov processes preserving stability. Explicitly, let $\\psi:\\mathbb{R}_{MA}[x,y,z]\\to\\mathbb{R}_{MA}[x,y,z]$ be a linear operator. When is $e^{t\\psi}:\\mathbb{R}[x,y,z]\\to\\mathbb{R}[x,y,z]$ stability preserving for all $t\\ge0$ ?",
   "clean_statement": "Find more examples and attempt to characterize Markov processes preserving stability. Explicitly, let $\\psi:\\mathbb{R}_{MA}[x,y,z]\\to\\mathbb{R}_{MA}[x,y,z]$ be a linear operator. When is $e^{t\\psi}:\\mathbb{R}[x,y,z]\\to\\mathbb{R}[x,y,z]$ stability preserving for all $t\\ge0$ ?",
   "public_statement": "Find more examples and attempt to characterize Markov processes preserving stability. Explicitly, let $\\psi:\\mathbb{R}_{MA}[x,y,z]\\to\\mathbb{R}_{MA}[x,y,z]$ be a linear operator. When is $e^{t\\psi}:\\mathbb{R}[x,y,z]\\to\\mathbb{R}[x,y,z]$ stability preserving for all $t\\ge0$ ?",
   "evidence": "The canonical record is Problem 9.5 in the AIM list *Stability and hyperbolicity*, section “Miscellaneous Problems,” attributed to P. Brändén. The repository record and the live AIM HTML (checked 2026-07-28) agree verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analysis-notes.json",
   "source_index": 155,
   "attempt": 1
  },
  "AIM-ANALYSIS-0157": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "1. P´ olya and Related Inequalities \n\nConsider eigenvalues of the Dirichlet Laplacian on a bounded domain Ω ⊂ Rn:\n\n{\n\n−∆uj = Ej uj in Ω,\n\nuj = 0 on ∂Ω.Assume n ≥ 2.(1) (Michael Loss, Timo Weidl) The P´ olya Conjecture claims that the Weyl asymptotic formula provides a lower bound: \n\nEj ≥ (2 π)2(n/ |Sn−1|| Ω|)2/n j2/n, j = 1, 2, 3,.... \n\nThe conjecture remains open even for j = 3.The best partial result known is with a factor of n/ (n + 2) (which is less than 1) on the right hand side, as one deduces by estimating Ej ≤ EJ in the following inequality due to Li and Yau, \n\n> J\n\n∑\n\n> j=1\n\nEj ≥ nn + 2 (2 π)2(n/ |Sn−1|| Ω|)2/n J(n+2) /n, J = 1, 2, 3,.... \n\nBerezin proved in 1972 that \n\n∑\n\n> j\n\n(E − Ej )σ \n\n> +\n\n≤ |Ω|\n\n(2 π)n\n\n∫\n\n> Rn\n\n(E − | p|2)σ \n\n> +\n\ndp, σ ≥ 1, E > 0.\n\nThe cases 0 ≤ σ < 1 remain open. The P´ olya conjecture is exactly the case σ = 0. The inequality for σ = 1 implies the Li-Yau inequality via the Legendre transform. \n\n> 1Corrections and updates will be gratefully received at rlfrank@math.princeton.edu (for P´ olya and Lieb- Thirring inequalities) and Laugesen@illinois.edu (for Gap and Laplace eigenvalue inequalities).\n> 1\n\n(2) (Timo Weidl) Can one strengthen the Li-Yau result by including a correction term, perhaps involving the surface area of the boundary? This has been done for the discrete Laplacian on domains in a lattice; J. K. Freericks, E. H. Lieb, D. Ueltschi, Phase separation due to quantum mechanical correlations, Phys. Rev. Lett. 88, 106401 1-4 (2002). There is a result by Melas of Li-Yau type with corrections involving moments of iner-tia rather than surface area, see A. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc. 131 (2003), 631-636. February 2009: The result of Melas has been strengthened by inclusion of a correction term involving the surface area of the boundary; see H. Kovarik, S. Vugalter, T. Weidl, \n\nTwo-dimensional Berezin-Li-Yau inequalities with a correction term, Comm. Math. Phys. 287 (2009), 959-981. An earlier improvement for γ ≥ 3/2, involving a notion of effective boundary, is due to T. Weidl, Improved Berezin-Li-Yau inequalities with a remainder term,in: Spectral theory of differential operators, T. Suslina and D. Yafaev (eds.), Amer. Math. Soc. Transl. Ser. 2, 225 (2008). For γ < 3/2 further improvements seem possible and desirable. (3) (Timo Weidl) There are analogues of the P´ olya and Li-Yau inequalities under Neumann boundary conditions, with the inequality signs reversed. The P´ olya Conjecture remains open for Neumann boundary conditions for j ≥ 2, except it was recently proved for j =2 in two dimensions by A. Girouard et al., J. Diff. Geometry, to appear. The analogue of Li-Yau was proved by Pawel Kr¨ oger; see P. Kr¨ oger, Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space, J. Funct. Anal. 106 (1992), no. 2, 353-357, and also A. Laptev Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal. 151 (1997), 531-545. Can one strengthen the Kr¨ oger result by including a correction term? (4) (Timo Weidl) The questions raised above are meaningful in the presence of a magnetic field. For more information and some progress see Item (10) below. (5) (Evans Harrell, Joachim Stubbe) For σ ≥ 2 the mapping \n\nrσ: E 7 → E−σ−d/ 2\n\n( |Ω|\n\n(2 π)n\n\n∫\n\n> Rn\n\n(E − | p|2)σdp − ∑\n\n> j\n\n(E − Ej )σ\n\n> +\n\n)\n\nis non-increasing. This was proved in E. M. Harrell and L. Hermi, Differential inequali-ties for Riesz means and Weyl-type bounds for eigenvalues, J. Funct. Analysis 254 (2008), 3173-3191, using the trace identities of E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809, and Universal bounds and semiclassical estimates for eigen-values of abstract Schr¨ odinger operators, preprint 2008, available as arXiv:0808.1133. According to Weyl's asymptotic formula rσ(E) tends to zero as E tends to infinity and therefore rσ(E) ≥ 0, which is the Berezin-Li-Yau-inequality. Can one strengthen this \n\n> 2\n\nbound in the trace identity of Harrell-Stubbe to obtain correction terms involving the sur-face area of the boundary? For the Laplacian with periodic boundary conditions a similar monotonicity property holds (for details see E. M. Harrell and J. Stubbe, Trace identities for commutators, with applications to the distribution of eigenvalues, preprint 2009, avail-able as arXiv:0903.0563). In this case the search for the correction term is related to the famous Gauss circle problem (or lattice point problem). (6) (Evans Harrell, Joachim Stubbe) Prove monotonicity results like in Item (5) for higher order operators (e.g. clamped plate problem) and fractional powers of Laplacians (for some results on √−∆ see E. M. Harrell and S. Yıldırım Yolcu, Eigenvalue inequalities for Klein-Gordon Operators, accepted for publication in J. Funct. Analysis) leading to Berezin-Li-Yau inequalities for these operators. (7) (Evans Harrell, Joachim Stubbe) For p > 0 let \n\nMp(J):= \n\n(n + 2 pn\n\n1\n\nJ\n\n> J\n\n∑\n\n> j=1\n\nEpj\n\n) 1\n\n> p\n\n(1) and for p = 0 define \n\nM0(J):= e 2\n\n> n\n\n( J∏\n\n> j=1\n\nEj\n\n) 1\n\n> J. (2) According the the Weyl asymptotic formula, for all p ≥ 0,\n\nMp(J) ∼ (2 π)2(n/ |Sn−1|| Ω|)2/n J2/n \n\nas J → ∞. In E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809, it has been shown that \n\nM 21 (J) − M2(J) ≥ 14(EJ+1 − EJ )2(≥ 0) \n\nand \n\nM1(J) −\n\n√\n\nM 21 (J) − M2(J) ≤ EJ ≤ EJ+1 ≤ M1(J) + \n\n√\n\nM 21 (J) − M2(J).\n\nBoth inequalities are sharp in the Weyl limit. For extensions to other Mp(J) see E. M. Harrell and J. Stubbe, Universal bounds and semiclassical estimates for eigenvalues of abstract Schr¨ odinger operators, preprint 2008, available as arXiv:0808.1133. For p > 0\n\nfind an upper bound of the form \n\nM 2pp (J) − M p\n\n> 2p\n\n(J) ≤ C(p, Ω) E2p \n\n> 1\n\nJ2pκ \n\nwith κ < 2/n.\n\n> 3\n\n(8) (Evans Harrell, Joachim Stubbe) With the above notations does \n\nEJ ≤ Mp(J)\n\nhold for all J and all p ≥ 0? Can one find Ω and J such that the inequality \n\nM 21 (J) − M2(J) ≥ 14(EJ+1 − EJ )2\n\nis saturated? 2. Lieb-Thirring Inequalities \n\nWrite E1 < E 2 ≤ E3 ≤ · · · ≤ 0 for the eigenvalues of −∆ − V on L2(Rn), meaning \n\n(−∆ − V )uj = Ej uj.\n\nThe eigenfunctions uj represent bound states with energies Ej. For simplicity we assume V ≥ 0.Assume n ≥ 1.The Lieb-Thirring inequality can be written as \n\n∑\n\n> j\n\n|Ej |γ ≤ Ln,γ \n\n∫\n\n> Rn\n\nV γ+n/ 2 dx, \n\nThis inequality holds (with a constant Ln,γ independent of V ) iff the parameter γ satisfies γ ≥ 1/2\n\nif n = 1, γ > 0 if n = 2 and γ ≥ 0 if n ≥ 3. The case γ = 0 (counting eigenvalues) is the Cwikel-Lieb-Rozenblum Inequality (CLR). In other words Tr (−∆ − V )γ \n\n> −\n\n≤ Cn,γ \n\n(2 π)n\n\n∫\n\n> Rn\n\n∫\n\n> Rn\n\n(|p|2 − V (x)) γ \n\n> −\n\ndpdx, \n\nwhere \n\nCn,γ = Ln,γ \n\nLcl \n\n> n,γ\n\nand Lcl \n\n> n,γ\n\n= 1(2 π)n\n\n∫\n\n> Rn\n\n(|p|2 − 1) γ \n\n> −\n\ndp. \n\nThe constant Lcl \n\n> n,γ\n\nis called the semiclassical Lieb-Thirring constant. Note that Cn,γ ≥ 1 always, by the Weyl asymptotics, and that Cn,γ is decreasing in γ for each fixed n, by the Aizenman-Lieb monotonicity result. To start with, let us summarize some known results on the constants Cn,γ, along with conjectures about best (smallest) values of Cn,γ. 4n γ Best known Cn,γ Best constant? status last updated \n\n1 12 2 2 known \n\n(12, 32 ) 2* 2\n\n(γ−1/2\n\n> γ+1 /2\n\n)γ−1/2\n\nconjectured \n\n[32, ∞) 1 1 known \n\n2 (0, 12 )?\n\n[12, 1) 3.64 Feb. 2009 \n\n[1, 32 ) 1.82 Feb. 2009 \n\n[32, ∞) 1 1 known \n\n= 3 [0, 12 ) 6.87 8/√3 ' 4.62 conjectured \n\n[12, 1) 3.64 Feb. 2009 \n\n[1, 32 ) 1.82 1 conjectured Feb. 2009 \n\n[32, ∞) 1 1 known \n\n≥ 4 [0, 12 ) 10.34 Feb. 2009 \n\n[12, 1) 3.64 Feb. 2009 \n\n[1, 32 ) 1.82 1 conjectured Feb. 2009 \n\n[32, ∞) 1 1 known *better is known for γ ∈ [1, 32 ), e.g. C1,1 ≤ π√3 ' 1.82 via work of Eden-Foias.",
   "clean_statement": "1. P´ olya and Related Inequalities\n\nConsider eigenvalues of the Dirichlet Laplacian on a bounded domain Ω ⊂ Rn:\n\n{\n\n−∆uj = Ej uj in Ω,\n\nuj = 0 on ∂Ω.Assume n ≥ 2.(1) (Michael Loss, Timo Weidl) The P´ olya Conjecture claims that the Weyl asymptotic formula provides a lower bound:\n\nEj ≥ (2 π)2(n/ |Sn−1|| Ω|)2/n j2/n, j = 1, 2, 3,....\n\nThe conjecture remains open even for j = 3.The best partial result known is with a factor of n/ (n + 2) (which is less than 1) on the right hand side, as one deduces by estimating Ej ≤ EJ in the following inequality due to Li and Yau,\n\nd/2 J\n\n∑\n\nd/2 j=1\n\nEj ≥ nn + 2 (2 π)2(n/ |Sn−1|| Ω|)2/n J(n+2) /n, J = 1, 2, 3,....\n\nBerezin proved in 1972 that\n\n∑\n\nd/2 j\n\n(E − Ej )σ\n\nd/2 +\n\n≤ |Ω|\n\n(2 π)n\n\n∫\n\nd/2 Rn\n\n(E − | p|2)σ\n\nd/2 +\n\ndp, σ ≥ 1, E d/2 0.\n\nThe cases 0 ≤ σ < 1 remain open. The P´ olya conjecture is exactly the case σ = 0. The inequality for σ = 1 implies the Li-Yau inequality via the Legendre transform.\n\nd/2 1Corrections and updates will be gratefully received at rlfrank@math.princeton.edu (for P´ olya and Lieb- Thirring inequalities) and Laugesen@illinois.edu (for Gap and Laplace eigenvalue inequalities).\nd/2 1\n\n(2) (Timo Weidl) Can one strengthen the Li-Yau result by including a correction term, perhaps involving the surface area of the boundary? This has been done for the discrete Laplacian on domains in a lattice; J. K. Freericks, E. H. Lieb, D. Ueltschi, Phase separation due to quantum mechanical correlations, Phys. Rev. Lett. 88, 106401 1-4 (2002). There is a result by Melas of Li-Yau type with corrections involving moments of iner-tia rather than surface area, see A. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc. 131 (2003), 631-636. February 2009: The result of Melas has been strengthened by inclusion of a correction term involving the surface area of the boundary; see H. Kovarik, S. Vugalter, T. Weidl,\n\nTwo-dimensional Berezin-Li-Yau inequalities with a correction term, Comm. Math. Phys. 287 (2009), 959-981. An earlier improvement for γ ≥ 3/2, involving a notion of effective boundary, is due to T. Weidl, Improved Berezin-Li-Yau inequalities with a remainder term,in: Spectral theory of differential operators, T. Suslina and D. Yafaev (eds.), Amer. Math. Soc. Transl. Ser. 2, 225 (2008). For γ < 3/2 further improvements seem possible and desirable. (3) (Timo Weidl) There are analogues of the P´ olya and Li-Yau inequalities under Neumann boundary conditions, with the inequality signs reversed. The P´ olya Conjecture remains open for Neumann boundary conditions for j ≥ 2, except it was recently proved for j =2 in two dimensions by A. Girouard et al., J. Diff. Geometry, to appear. The analogue of Li-Yau was proved by Pawel Kr¨ oger; see P. Kr¨ oger, Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space, J. Funct. Anal. 106 (1992), no. 2, 353-357, and also A. Laptev Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal. 151 (1997), 531-545. Can one strengthen the Kr¨ oger result by including a correction term? (4) (Timo Weidl) The questions raised above are meaningful in the presence of a magnetic field. For more information and some progress see Item (10) below. (5) (Evans Harrell, Joachim Stubbe) For σ ≥ 2 the mapping\n\nrσ: E 7 → E−σ−d/ 2\n\n( |Ω|\n\n(2 π)n\n\n∫\n\nd/2 Rn\n\n(E − | p|2)σdp − ∑\n\nd/2 j\n\n(E − Ej )σ\n\nd/2 +\n\n)\n\nis non-increasing. This was proved in E. M. Harrell and L. Hermi, Differential inequali-ties for Riesz means and Weyl-type bounds for eigenvalues, J. Funct. Analysis 254 (2008), 3173-3191, using the trace identities of E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809, and Universal bounds and semiclassical estimates for eigen-values of abstract Schr¨ odinger operators, preprint 2008, available as arXiv:0808.1133. According to Weyl's asymptotic formula rσ(E) tends to zero as E tends to infinity and therefore rσ(E) ≥ 0, which is the Berezin-Li-Yau-inequality. Can one strengthen this\n\nd/2 2\n\nbound in the trace identity of Harrell-Stubbe to obtain correction terms involving the sur-face area of the boundary? For the Laplacian with periodic boundary conditions a similar monotonicity property holds (for details see E. M. Harrell and J. Stubbe, Trace identities for commutators, with applications to the distribution of eigenvalues, preprint 2009, avail-able as arXiv:0903.0563). In this case the search for the correction term is related to the famous Gauss circle problem (or lattice point problem). (6) (Evans Harrell, Joachim Stubbe) Prove monotonicity results like in Item (5) for higher order operators (e.g. clamped plate problem) and fractional powers of Laplacians (for some results on √−∆ see E. M. Harrell and S. Yıldırım Yolcu, Eigenvalue inequalities for Klein-Gordon Operators, accepted for publication in J. Funct. Analysis) leading to Berezin-Li-Yau inequalities for these operators. (7) (Evans Harrell, Joachim Stubbe) For p d/2 0 let\n\nMp(J):=\n\n(n + 2 pn\n\n1\n\nJ\n\nd/2 J\n\n∑\n\nd/2 j=1\n\nEpj\n\n) 1\n\nd/2 p\n\n(1) and for p = 0 define\n\nM0(J):= e 2\n\nd/2 n\n\n( J∏\n\nd/2 j=1\n\nEj\n\n) 1\n\nd/2 J. (2) According the the Weyl asymptotic formula, for all p ≥ 0,\n\nMp(J) ∼ (2 π)2(n/ |Sn−1|| Ω|)2/n J2/n\n\nas J → ∞. In E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809, it has been shown that\n\nM 21 (J) − M2(J) ≥ 14(EJ+1 − EJ )2(≥ 0)\n\nand\n\nM1(J) −\n\n√\n\nM 21 (J) − M2(J) ≤ EJ ≤ EJ+1 ≤ M1(J) +\n\n√\n\nM 21 (J) − M2(J).\n\nBoth inequalities are sharp in the Weyl limit. For extensions to other Mp(J) see E. M. Harrell and J. Stubbe, Universal bounds and semiclassical estimates for eigenvalues of abstract Schr¨ odinger operators, preprint 2008, available as arXiv:0808.1133. For p d/2 0\n\nfind an upper bound of the form\n\nM 2pp (J) − M p\n\nd/2 2p\n\n(J) ≤ C(p, Ω) E2p\n\nd/2 1\n\nJ2pκ\n\nwith κ < 2/n.\n\nd/2 3\n\n(8) (Evans Harrell, Joachim Stubbe) With the above notations does\n\nEJ ≤ Mp(J)\n\nhold for all J and all p ≥ 0? Can one find Ω and J such that the inequality\n\nM 21 (J) − M2(J) ≥ 14(EJ+1 − EJ )2\n\nis saturated? 2. Lieb-Thirring Inequalities\n\nWrite E1 < E 2 ≤ E3 ≤ · · · ≤ 0 for the eigenvalues of −∆ − V on L2(Rn), meaning\n\n(−∆ − V )uj = Ej uj.\n\nThe eigenfunctions uj represent bound states with energies Ej. For simplicity we assume V ≥ 0.Assume n ≥ 1.The Lieb-Thirring inequality can be written as\n\n∑\n\nd/2 j\n\n|Ej |γ ≤ Ln,γ\n\n∫\n\nd/2 Rn\n\nV γ+n/ 2 dx,\n\nThis inequality holds (with a constant Ln,γ independent of V ) iff the parameter γ satisfies γ ≥ 1/2\n\nif n = 1, γ d/2 0 if n = 2 and γ ≥ 0 if n ≥ 3. The case γ = 0 (counting eigenvalues) is the Cwikel-Lieb-Rozenblum Inequality (CLR). In other words Tr (−∆ − V )γ\n\nd/2 −\n\n≤ Cn,γ\n\n(2 π)n\n\n∫\n\nd/2 Rn\n\n∫\n\nd/2 Rn\n\n(|p|2 − V (x)) γ\n\nd/2 −\n\ndpdx,\n\nwhere\n\nCn,γ = Ln,γ\n\nLcl\n\nd/2 n,γ\n\nand Lcl\n\nd/2 n,γ\n\n= 1(2 π)n\n\n∫\n\nd/2 Rn\n\n(|p|2 − 1) γ\n\nd/2 −\n\ndp.\n\nThe constant Lcl\n\nd/2 n,γ\n\nis called the semiclassical Lieb-Thirring constant. Note that Cn,γ ≥ 1 always, by the Weyl asymptotics, and that Cn,γ is decreasing in γ for each fixed n, by the Aizenman-Lieb monotonicity result. To start with, let us summarize some known results on the constants Cn,γ, along with conjectures about best (smallest) values of Cn,γ. 4n γ Best known Cn,γ Best constant? status last updated\n\n1 12 2 2 known\n\n(12, 32 ) 2* 2\n\n(γ−1/2\n\nd/2 γ+1 /2\n\n)γ−1/2\n\nconjectured\n\n[32, ∞) 1 1 known\n\n2 (0, 12 )?\n\n[12, 1) 3.64 Feb. 2009\n\n[1, 32 ) 1.82 Feb. 2009\n\n[32, ∞) 1 1 known\n\n= 3 [0, 12 ) 6.87 8/√3 ' 4.62 conjectured\n\n[12, 1) 3.64 Feb. 2009\n\n[1, 32 ) 1.82 1 conjectured Feb. 2009\n\n[32, ∞) 1 1 known\n\n≥ 4 [0, 12 ) 10.34 Feb. 2009\n\n[12, 1) 3.64 Feb. 2009\n\n[1, 32 ) 1.82 1 conjectured Feb. 2009\n\n[32, ∞) 1 1 known *better is known for γ ∈ [1, 32 ), e.g. C1,1 ≤ π√3 ' 1.82 via work of Eden-Foias.",
   "public_statement": "1. P´ olya and Related Inequalities\n\nConsider eigenvalues of the Dirichlet Laplacian on a bounded domain Ω ⊂ Rn:\n\n{\n\n−∆uj = Ej uj in Ω,\n\nuj = 0 on ∂Ω.Assume n ≥ 2.(1) (Michael Loss, Timo Weidl) The P´ olya Conjecture claims that the Weyl asymptotic formula provides a lower bound:\n\nEj ≥ (2 π)2(n/ |Sn−1|| Ω|)2/n j2/n, j = 1, 2, 3,....\n\nThe conjecture remains open even for j = 3.The best partial result known is with a factor of n/ (n + 2) (which is less than 1) on the right hand side, as one deduces by estimating Ej ≤ EJ in the following inequality due to Li and Yau,\n\n> J\n\n∑\n\n> j=1\n\nEj ≥ nn + 2 (2 π)2(n/ |Sn−1|| Ω|)2/n J(n+2) /n, J = 1, 2, 3,....\n\nBerezin proved in 1972 that\n\n∑\n\n> j\n\n(E − Ej )σ\n\n> +\n\n≤ |Ω|\n\n(2 π)n\n\n∫\n\n> Rn\n\n(E − | p|2)σ\n\n> +\n\ndp, σ ≥ 1, E > 0.\n\nThe cases 0 ≤ σ < 1 remain open. The P´ olya conjecture is exactly the case σ = 0. The inequality for σ = 1 implies the Li-Yau inequality via the Legendre transform.\n\n> 1Corrections and updates will be gratefully received at rlfrank@math.princeton.edu (for P´ olya and Lieb- Thirring inequalities) and Laugesen@illinois.edu (for Gap and Laplace eigenvalue inequalities).\n> 1\n\n(2) (Timo Weidl) Can one strengthen the Li-Yau result by including a correction term, perhaps involving the surface area of the boundary? This has been done for the discrete Laplacian on domains in a lattice; J. K. Freericks, E. H. Lieb, D. Ueltschi, Phase separation due to quantum mechanical correlations, Phys. Rev. Lett. 88, 106401 1-4 (2002). There is a result by Melas of Li-Yau type with corrections involving moments of iner-tia rather than surface area, see A. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc. 131 (2003), 631-636. February 2009: The result of Melas has been strengthened by inclusion of a correction term involving the surface area of the boundary; see H. Kovarik, S. Vugalter, T. Weidl,\n\nTwo-dimensional Berezin-Li-Yau inequalities with a correction term, Comm. Math. Phys. 287 (2009), 959-981. An earlier improvement for γ ≥ 3/2, involving a notion of effective boundary, is due to T. Weidl, Improved Berezin-Li-Yau inequalities with a remainder term,in: Spectral theory of differential operators, T. Suslina and D. Yafaev (eds.), Amer. Math. Soc. Transl. Ser. 2, 225 (2008). For γ < 3/2 further improvements seem possible and desirable. (3) (Timo Weidl) There are analogues of the P´ olya and Li-Yau inequalities under Neumann boundary conditions, with the inequality signs reversed. The P´ olya Conjecture remains open for Neumann boundary conditions for j ≥ 2, except it was recently proved for j =2 in two dimensions by A. Girouard et al., J. Diff. Geometry, to appear. The analogue of Li-Yau was proved by Pawel Kr¨ oger; see P. Kr¨ oger, Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space, J. Funct. Anal. 106 (1992), no. 2, 353-357, and also A. Laptev Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal. 151 (1997), 531-545. Can one strengthen the Kr¨ oger result by including a correction term? (4) (Timo Weidl) The questions raised above are meaningful in the presence of a magnetic field. For more information and some progress see Item (10) below. (5) (Evans Harrell, Joachim Stubbe) For σ ≥ 2 the mapping\n\nrσ: E 7 → E−σ−d/ 2\n\n( |Ω|\n\n(2 π)n\n\n∫\n\n> Rn\n\n(E − | p|2)σdp − ∑\n\n> j\n\n(E − Ej )σ\n\n> +\n\n)\n\nis non-increasing. This was proved in E. M. Harrell and L. Hermi, Differential inequali-ties for Riesz means and Weyl-type bounds for eigenvalues, J. Funct. Analysis 254 (2008), 3173-3191, using the trace identities of E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809, and Universal bounds and semiclassical estimates for eigen-values of abstract Schr¨ odinger operators, preprint 2008, available as arXiv:0808.1133. According to Weyl's asymptotic formula rσ(E) tends to zero as E tends to infinity and therefore rσ(E) ≥ 0, which is the Berezin-Li-Yau-inequality. Can one strengthen this\n\n> 2\n\nbound in the trace identity of Harrell-Stubbe to obtain correction terms involving the sur-face area of the boundary? For the Laplacian with periodic boundary conditions a similar monotonicity property holds (for details see E. M. Harrell and J. Stubbe, Trace identities for commutators, with applications to the distribution of eigenvalues, preprint 2009, avail-able as arXiv:0903.0563). In this case the search for the correction term is related to the famous Gauss circle problem (or lattice point problem). (6) (Evans Harrell, Joachim Stubbe) Prove monotonicity results like in Item (5) for higher order operators (e.g. clamped plate problem) and fractional powers of Laplacians (for some results on √−∆ see E. M. Harrell and S. Yıldırım Yolcu, Eigenvalue inequalities for Klein-Gordon Operators, accepted for publication in J. Funct. Analysis) leading to Berezin-Li-Yau inequalities for these operators. (7) (Evans Harrell, Joachim Stubbe) For p > 0 let\n\nMp(J):=\n\n(n + 2 pn\n\n1\n\nJ\n\n> J\n\n∑\n\n> j=1\n\nEpj\n\n) 1\n\n> p\n\n(1) and for p = 0 define\n\nM0(J):= e 2\n\n> n\n\n( J∏\n\n> j=1\n\nEj\n\n) 1\n\n> J. (2) According the the Weyl asymptotic formula, for all p ≥ 0,\n\nMp(J) ∼ (2 π)2(n/ |Sn−1|| Ω|)2/n J2/n\n\nas J → ∞. In E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809, it has been shown that\n\nM 21 (J) − M2(J) ≥ 14(EJ+1 − EJ )2(≥ 0)\n\nand\n\nM1(J) −\n\n√\n\nM 21 (J) − M2(J) ≤ EJ ≤ EJ+1 ≤ M1(J) +\n\n√\n\nM 21 (J) − M2(J).\n\nBoth inequalities are sharp in the Weyl limit. For extensions to other Mp(J) see E. M. Harrell and J. Stubbe, Universal bounds and semiclassical estimates for eigenvalues of abstract Schr¨ odinger operators, preprint 2008, available as arXiv:0808.1133. For p > 0\n\nfind an upper bound of the form\n\nM 2pp (J) − M p\n\n> 2p\n\n(J) ≤ C(p, Ω) E2p\n\n> 1\n\nJ2pκ\n\nwith κ < 2/n.\n\n> 3\n\n(8) (Evans Harrell, Joachim Stubbe) With the above notations does\n\nEJ ≤ Mp(J)\n\nhold for all J and all p ≥ 0? Can one find Ω and J such that the inequality\n\nM 21 (J) − M2(J) ≥ 14(EJ+1 − EJ )2\n\nis saturated? 2. Lieb-Thirring Inequalities\n\nWrite E1 < E 2 ≤ E3 ≤ · · · ≤ 0 for the eigenvalues of −∆ − V on L2(Rn), meaning\n\n(−∆ − V )uj = Ej uj.\n\nThe eigenfunctions uj represent bound states with energies Ej. For simplicity we assume V ≥ 0.Assume n ≥ 1.The Lieb-Thirring inequality can be written as\n\n∑\n\n> j\n\n|Ej |γ ≤ Ln,γ\n\n∫\n\n> Rn\n\nV γ+n/ 2 dx,\n\nThis inequality holds (with a constant Ln,γ independent of V ) iff the parameter γ satisfies γ ≥ 1/2\n\nif n = 1, γ > 0 if n = 2 and γ ≥ 0 if n ≥ 3. The case γ = 0 (counting eigenvalues) is the Cwikel-Lieb-Rozenblum Inequality (CLR). In other words Tr (−∆ − V )γ\n\n> −\n\n≤ Cn,γ\n\n(2 π)n\n\n∫\n\n> Rn\n\n∫\n\n> Rn\n\n(|p|2 − V (x)) γ\n\n> −\n\ndpdx,\n\nwhere\n\nCn,γ = Ln,γ\n\nLcl\n\n> n,γ\n\nand Lcl\n\n> n,γ\n\n= 1(2 π)n\n\n∫\n\n> Rn\n\n(|p|2 − 1) γ\n\n> −\n\ndp.\n\nThe constant Lcl\n\n> n,γ\n\nis called the semiclassical Lieb-Thirring constant. Note that Cn,γ ≥ 1 always, by the Weyl asymptotics, and that Cn,γ is decreasing in γ for each fixed n, by the Aizenman-Lieb monotonicity result. To start with, let us summarize some known results on the constants Cn,γ, along with conjectures about best (smallest) values of Cn,γ. 4n γ Best known Cn,γ Best constant? status last updated\n\n1 12 2 2 known\n\n(12, 32 ) 2* 2\n\n(γ−1/2\n\n> γ+1 /2\n\n)γ−1/2\n\nconjectured\n\n[32, ∞) 1 1 known\n\n2 (0, 12 )?\n\n[12, 1) 3.64 Feb. 2009\n\n[1, 32 ) 1.82 Feb. 2009\n\n[32, ∞) 1 1 known\n\n= 3 [0, 12 ) 6.87 8/√3 ' 4.62 conjectured\n\n[12, 1) 3.64 Feb. 2009\n\n[1, 32 ) 1.82 1 conjectured Feb. 2009\n\n[32, ∞) 1 1 known\n\n≥ 4 [0, 12 ) 10.34 Feb. 2009\n\n[12, 1) 3.64 Feb. 2009\n\n[1, 32 ) 1.82 1 conjectured Feb. 2009\n\n[32, ∞) 1 1 known *better is known for γ ∈ [1, 32 ), e.g. C1,1 ≤ π√3 ' 1.82 via work of Eden-Foias.",
   "evidence": "1. In Item P5 the exponent in the Harrell--Stubbe normalized deficit must use the ambient dimension \\(n\\). The extracted `d/2` is inconsistent with the section's notation and the surrounding \\(\\mathbb R^n\\) integral. 2. Several superscripts and subscripts in Items P7--P8 collapse. The displayed source text layer reads like \\(M_1^2-M_2\\), which is dimensionally inconsistent with the stated root-normalized \\(M_p\\). The dimensionally consistent reconstructed quantity is \\[ M_1^2(J)-M_2^2(J), \\] and the corresponding general dispersion is read as \\[ M_p^{2p}(J)-M_{2p}^{2p}(J). \\] This is an explicit reconstruction, not a claim that the PDF's text layer itself is unambiguous. Confirming it against the original TeX or the authors is a useful editorial next step. 3. In the Ovals item the operator is \\[ H_C=-\\frac{d^2}{ds^2}+\\kappa(s)^2 \\] on \\(2\\pi\\)-periodic functions. The extracted run-on...",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-analysis-notes.json",
   "source_index": 156,
   "attempt": 1
  },
  "AIM-ANALYSIS-0158": {
   "statement_status": "corrected_verified",
   "original_statement": "Question 1: Wenbo Li Consider random polynomials in one variable (real or complex). Find the asymptotics of the norm of the largest zero as the degree of the random polynomials tends to infinity. \u0005",
   "clean_statement": "Question 1: Wenbo Li Consider random polynomials in one variable (real or complex). Find the asymptotics of the norm of the largest zero as the degree of the random polynomials tends to infinity.",
   "public_statement": "Question 1: Wenbo Li Consider random polynomials in one variable (real or complex). Find the asymptotics of the norm of the largest zero as the degree of the random polynomials tends to infinity.",
   "evidence": "The control character `U+0005` at the end of the extracted record is not mathematical content. Inspection of the original PDF shows the same end marker after every question. The PDF contains no preceding definition of a random-polynomial ensemble, and the following questions do not supply one.",
   "classification_method": "source_verified_raw_character_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-analysis-notes.json",
   "source_index": 157,
   "attempt": 1
  },
  "AIM-ANALYSIS-0159": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 2: Yan Fyodorov What is the mean density of permanental polynomials? This is unknown for random matrices of size greater than 5 × 5. \u0005",
   "clean_statement": null,
   "public_statement": "Question 2: Yan Fyodorov What is the mean density of permanental polynomials? This is unknown for random matrices of size greater than 5 × 5. [U+0005]",
   "evidence": "The canonical record comes from the American Institute of Mathematics workshop problem list *Random Analytic Functions*, compiled by Swaminathan Sethuraman and dated April 17, 2006. The original two-page PDF states, verbatim apart from its decorative end marker:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 158,
   "attempt": 1
  },
  "AIM-ANALYSIS-0160": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 3: Ashkan Nikeghbali What can one tell about the distribution of the zeros of the derivative of characteristic polynomial of random unitary matrix, especially near the boundary of the unit circle.Also what can one tell about E[|f ′|s] as s tends to zero? \u0005",
   "clean_statement": null,
   "public_statement": "Question 3: Ashkan Nikeghbali What can one tell about the distribution of the zeros of the derivative of characteristic polynomial of random unitary matrix, especially near the boundary of the unit circle.Also what can one tell about E[|f ′|s] as s tends to zero? [U+0005]",
   "evidence": "The record is Question 3 from the AIM workshop *Random analytic functions*. The AIM PDF, *Open Problems at the Random Analytic functions Workshop at AIM* (Swaminathan Sethuraman, April 17, 2006), reads, with only mathematical typesetting and spacing restored:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 159,
   "attempt": 1
  },
  "AIM-ANALYSIS-0161": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 4: Maurice Rojas Investigate the connections between random sparse polygons and Newton polytopes. This should be extended to the case of random Viro diagrams. \u0005",
   "clean_statement": null,
   "public_statement": "Question 4: Maurice Rojas Investigate the connections between random sparse polygons and Newton polytopes. This should be extended to the case of random Viro diagrams. [U+0005]",
   "evidence": "The canonical record is Question 4 from the AIM workshop *Random analytic functions* (April 2006), attributed to Maurice Rojas. Its printable text is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 160,
   "attempt": 1
  },
  "AIM-ANALYSIS-0162": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 5: Balint Virag Find a Gaussian entire function with negatively correlated zeros. We know that there exists such a function on the unit disc. This is related to the repulsion properties of random polynomials. \u0005",
   "clean_statement": null,
   "public_statement": "Question 5: Balint Virag Find a Gaussian entire function with negatively correlated zeros. We know that there exists such a function on the unit disc. This is related to the repulsion properties of random polynomials. [U+0005]",
   "evidence": "The canonical record comes from Question 5 of the AIM workshop list *Random analytic functions* (compiled by Swaminathan Sethuraman, dated April 17, 2006). The original PDF gives, with line breaks normalized:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 161,
   "attempt": 1
  },
  "AIM-ANALYSIS-0163": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 6: Bernard Shiffman Does the Fubini-Study metric on P1 minimize the expected number of critical points? There are reasons to conjecture that the answer is Yes. \u0005",
   "clean_statement": null,
   "public_statement": "Question 6: Bernard Shiffman Does the Fubini-Study metric on P1 minimize the expected number of critical points? There are reasons to conjecture that the answer is Yes. [U+0005]",
   "evidence": "The [AIM workshop PDF](https://aimath.org/WWN/randomzeros/arcc1.pdf) gives no degree, probability law, or definition of critical point. Those data can be recovered from the Douglas--Shiffman--Zelditch papers underlying this workshop question. The reading used in this report is therefore the following, explicitly labeled reconstruction.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 162,
   "attempt": 1
  },
  "AIM-ANALYSIS-0164": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 7: Steven Evans Is there a necessary and sufficient condition for a given correlation function to be the correlation function of an enire Gaussian function? \u0005",
   "clean_statement": null,
   "public_statement": "Question 7: Steven Evans Is there a necessary and sufficient condition for a given correlation function to be the correlation function of an enire Gaussian function? [U+0005]",
   "evidence": "The canonical record is Question 7 from the AIM workshop *Random analytic functions* (April 2006), attributed to Steven Evans. The original two-page PDF itself, not merely the extracted JSON, prints:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 163,
   "attempt": 1
  },
  "AIM-ANALYSIS-0165": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 8: Maurice Rojas What is the probability that a random Viro diagram contains no sphere? \u0005",
   "clean_statement": null,
   "public_statement": "Question 8: Maurice Rojas What is the probability that a random Viro diagram contains no sphere? [U+0005]",
   "evidence": "The canonical record is Question 8 in the AIM workshop list *Random analytic functions*, compiled by Swaminathan Sethuraman and dated April 17, 2006. The original PDF reads, with line breaks normalized:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 164,
   "attempt": 1
  },
  "AIM-ANALYSIS-0166": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 9: Ashkan Nikeghbali What are the natural physical examples of random functions with GUE zeros on the real line? \u0005\n\n> ∗Partially supported by NSF grants DMS-0211458, CAREER DMS-0349309, and AIM.\n\n1",
   "clean_statement": null,
   "public_statement": "Question 9: Ashkan Nikeghbali What are the natural physical examples of random functions with GUE zeros on the real line? [U+0005]\n\n> ∗Partially supported by NSF grants DMS-0211458, CAREER DMS-0349309, and AIM.\n\n1",
   "evidence": "The canonical JSON record contains",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 165,
   "attempt": 1
  },
  "AIM-ANALYSIS-0167": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 10: Scott Sheffield Consider a function who Fourier transform is white noise on the unit circle. We aim to understand the web like appearance, that is the zero level lines of the Gaussian free field. Zelditch and Schramm make the above question more precise. \u0005",
   "clean_statement": null,
   "public_statement": "Question 10: Scott Sheffield Consider a function who Fourier transform is white noise on the unit circle. We aim to understand the web like appearance, that is the zero level lines of the Gaussian free field. Zelditch and Schramm make the above question more precise. [U+0005]",
   "evidence": "The canonical record is Question 10 from the April 2006 AIM workshop *Random analytic functions*, attributed to Scott Sheffield. Direct inspection of the two-page source PDF gives the exact text:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 166,
   "attempt": 1
  },
  "AIM-ANALYSIS-0168": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 11: Yan Fyodorov Given a random entire function on order 1, real on real line with given distribution of real zeros, what is the distribution of zeros of f? \u0005\n\n2",
   "clean_statement": null,
   "public_statement": "Question 11: Yan Fyodorov Given a random entire function on order 1, real on real line with given distribution of real zeros, what is the distribution of zeros of f? [U+0005]\n\n2",
   "evidence": "The source itself has “on order 1”; this is almost certainly a grammatical error for “of order 1.” The control character in the extracted JSON is the end-of-question marker, and the final “2” is the PDF page number. Inspection of the PDF text gives no evidence for a prime on the final \\(f\\).",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analysis-notes.json",
   "source_index": 167,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0001": {
   "statement_status": "exact",
   "original_statement": "Obtain cancellations in additive twists of GL(3) Fourier coefficients\n\nLet $\\pi$ be a GL(3) Hecke--Maass cusp form with the n-th Fourier coefficient $\\lambda_\\pi(1,n)$. For $\\alpha\\in\\mathbb{R}$, prove a bound of the form\n$$ \\sum_{n\\ll N} \\lambda_\\pi(1,n) e(n\\alpha) \\ll_\\pi N^{1-\\delta}, $$\nfor some $\\delta>0$ using a delta method. Aim to improve upon Miller's bound.",
   "clean_statement": "Obtain cancellations in additive twists of GL(3) Fourier coefficients\n\nLet $\\pi$ be a GL(3) Hecke--Maass cusp form with the n-th Fourier coefficient $\\lambda_\\pi(1,n)$. For $\\alpha\\in\\mathbb{R}$, prove a bound of the form\n$$ \\sum_{n\\ll N} \\lambda_\\pi(1,n) e(n\\alpha) \\ll_\\pi N^{1-\\delta}, $$\nfor some $\\delta>0$ using a delta method. Aim to improve upon Miller's bound.",
   "public_statement": "Obtain cancellations in additive twists of GL(3) Fourier coefficients\n\nLet $\\pi$ be a GL(3) Hecke--Maass cusp form with the n-th Fourier coefficient $\\lambda_\\pi(1,n)$. For $\\alpha\\in\\mathbb{R}$, prove a bound of the form\n$$ \\sum_{n\\ll N} \\lambda_\\pi(1,n) e(n\\alpha) \\ll_\\pi N^{1-\\delta}, $$\nfor some $\\delta>0$ using a delta method. Aim to improve upon Miller's bound.",
   "evidence": "The canonical record is problem 1.02 from the AIM workshop “Delta symbols and the subconvexity problem” (October 16–20, 2023):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0002": {
   "statement_status": "exact",
   "original_statement": "Shifted convolution sum via various delta methods\n\nFor a holomorphic or a Hecke--Maass cusp form $f$ with the nth Fourier coefficient $\\lambda_f(n)$, obtain cancellations in the summation\n$$ \\sum_{n\\leq N} \\lambda_f(n)\\lambda_f(n+1), $$\nvia\n\\begin{enumerate}\n\\item Duke-Friedlander-Iwaniec's delta method\n\\item Trivial delta method\n\\item GL(2) Petersson trace formula as a delta method\n\\item GL(3) Kuznetsov trace formula as a delta method.\n\\end{enumerate}\n\nNote: We do not yet have an explicit GL(3) Kuznetsov trace formula with arbitrary level and nebentypus.",
   "clean_statement": "Shifted convolution sum via various delta methods\n\nFor a holomorphic or a Hecke--Maass cusp form $f$ with the nth Fourier coefficient $\\lambda_f(n)$, obtain cancellations in the summation\n$$ \\sum_{n\\leq N} \\lambda_f(n)\\lambda_f(n+1), $$\nvia\n\\begin{enumerate}\n\\item Duke-Friedlander-Iwaniec's delta method\n\\item Trivial delta method\n\\item GL(2) Petersson trace formula as a delta method\n\\item GL(3) Kuznetsov trace formula as a delta method.\n\\end{enumerate}\n\nNote: We do not yet have an explicit GL(3) Kuznetsov trace formula with arbitrary level and nebentypus.",
   "public_statement": "Shifted convolution sum via various delta methods\n\nFor a holomorphic or a Hecke--Maass cusp form $f$ with the nth Fourier coefficient $\\lambda_f(n)$, obtain cancellations in the summation\n$$ \\sum_{n\\leq N} \\lambda_f(n)\\lambda_f(n+1), $$\nvia\n\\begin{enumerate}\n\\item Duke-Friedlander-Iwaniec's delta method\n\\item Trivial delta method\n\\item GL(2) Petersson trace formula as a delta method\n\\item GL(3) Kuznetsov trace formula as a delta method.\n\\end{enumerate}\n\nNote: We do not yet have an explicit GL(3) Kuznetsov trace formula with arbitrary level and nebentypus.",
   "evidence": "The canonical source URL is <http://aimpl.org/deltasubconvex2/1/>. It timed out during this run, so the wording above was checked against the exact repository record rather than a newly downloaded copy. There is no visible OCR corruption or mathematical ambiguity.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0003": {
   "statement_status": "exact",
   "original_statement": "Level aspect subconvexity for GL(2)\n\nUse a delta method to obtain level aspect subconvexity for GL(2) L-functions.",
   "clean_statement": "Level aspect subconvexity for GL(2)\n\nUse a delta method to obtain level aspect subconvexity for GL(2) L-functions.",
   "public_statement": "Level aspect subconvexity for GL(2)\n\nUse a delta method to obtain level aspect subconvexity for GL(2) L-functions.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0004": {
   "statement_status": "exact",
   "original_statement": "Relationship between Trivial delta method and Nelson's Kernel method\n\nLet $f$ be a holomorphic or Hecke--Maass cusp form of level $1$ and $\\chi$ be a Dirichlet character of level $p$. For fixed $f$, a level aspect Burgess-type bound for $L(f\\times\\chi, 1/2)$ can be obtained by (among many others) these two methods:\n\\begin{enumerate}\n\\item Trivial delta method,\n\\item Using the Kernel $K(z,w) = \\sum_{\\gamma\\in\\Gamma_0(p)} \\omega(z^{-1}\\gamma w)$ for $z, x\\in \\mathbb{H}$ and $\\omega$ a smooth bump function, defined appropriately.\n\\end{enumerate}\nFind a precise relationship between the two methods.",
   "clean_statement": "Relationship between Trivial delta method and Nelson's Kernel method\n\nLet $f$ be a holomorphic or Hecke--Maass cusp form of level $1$ and $\\chi$ be a Dirichlet character of level $p$. For fixed $f$, a level aspect Burgess-type bound for $L(f\\times\\chi, 1/2)$ can be obtained by (among many others) these two methods:\n\\begin{enumerate}\n\\item Trivial delta method,\n\\item Using the Kernel $K(z,w) = \\sum_{\\gamma\\in\\Gamma_0(p)} \\omega(z^{-1}\\gamma w)$ for $z, x\\in \\mathbb{H}$ and $\\omega$ a smooth bump function, defined appropriately.\n\\end{enumerate}\nFind a precise relationship between the two methods.",
   "public_statement": "Relationship between Trivial delta method and Nelson's Kernel method\n\nLet $f$ be a holomorphic or Hecke--Maass cusp form of level $1$ and $\\chi$ be a Dirichlet character of level $p$. For fixed $f$, a level aspect Burgess-type bound for $L(f\\times\\chi, 1/2)$ can be obtained by (among many others) these two methods:\n\\begin{enumerate}\n\\item Trivial delta method,\n\\item Using the Kernel $K(z,w) = \\sum_{\\gamma\\in\\Gamma_0(p)} \\omega(z^{-1}\\gamma w)$ for $z, x\\in \\mathbb{H}$ and $\\omega$ a smooth bump function, defined appropriately.\n\\end{enumerate}\nFind a precise relationship between the two methods.",
   "evidence": "The canonical record is AIM problem 1.08 from the workshop *Delta symbols and the subconvexity problem*. Its mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0005": {
   "statement_status": "exact",
   "original_statement": "Character sums appearing in the works of Nelson, and Sharma\n\nRelate or compare the exponential sums appearing in the work of Paul Nelson [cite], and the work of Prahlad Sharma [cite] in the proof for level aspect subconvexity bound for $\\rm GL(1)$ twists of $\\rm GL(3)\\times GL(2)$ $L$-functions.",
   "clean_statement": "Character sums appearing in the works of Nelson, and Sharma\n\nRelate or compare the exponential sums appearing in the work of Paul Nelson [cite], and the work of Prahlad Sharma [cite] in the proof for level aspect subconvexity bound for $\\rm GL(1)$ twists of $\\rm GL(3)\\times GL(2)$ $L$-functions.",
   "public_statement": "Character sums appearing in the works of Nelson, and Sharma\n\nRelate or compare the exponential sums appearing in the work of Paul Nelson [cite], and the work of Prahlad Sharma [cite] in the proof for level aspect subconvexity bound for $\\rm GL(1)$ twists of $\\rm GL(3)\\times GL(2)$ $L$-functions.",
   "evidence": "The canonical record is Problem 1.1 in the AIM list *Delta symbols and the subconvexity problem*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0006": {
   "statement_status": "exact",
   "original_statement": "Subconvexity bound for trace function twists\n\nSubconvexity bounds for $\\rm GL(1)$ twists of a fixed $\\rm GL(2)\\times GL(2)$ $L$-function has been proved using a delta symbol by C. Raju [cite] in the archimedean aspect, and by A. Ghosh [cite] in the level aspect. Can we generalize these techniques to replace the $\\rm GL(1)$ twist by a trace function, or an appropriate archimedean analog?",
   "clean_statement": "Subconvexity bound for trace function twists\n\nSubconvexity bounds for $\\rm GL(1)$ twists of a fixed $\\rm GL(2)\\times GL(2)$ $L$-function has been proved using a delta symbol by C. Raju [cite] in the archimedean aspect, and by A. Ghosh [cite] in the level aspect. Can we generalize these techniques to replace the $\\rm GL(1)$ twist by a trace function, or an appropriate archimedean analog?",
   "public_statement": "Subconvexity bound for trace function twists\n\nSubconvexity bounds for $\\rm GL(1)$ twists of a fixed $\\rm GL(2)\\times GL(2)$ $L$-function has been proved using a delta symbol by C. Raju [cite] in the archimedean aspect, and by A. Ghosh [cite] in the level aspect. Can we generalize these techniques to replace the $\\rm GL(1)$ twist by a trace function, or an appropriate archimedean analog?",
   "evidence": "Canonical source metadata: `aim-analytic-number-theory-notes.json`, zero-based source index 5, attempt 1.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0007": {
   "statement_status": "exact",
   "original_statement": "Subconvexity for $\\rm GL(3)\\times GL(1)$ via averaging over a family\n\nLet $\\pi$ be a fixed Hecke-Maass cusp form for $\\rm SL_3(\\mathbb{Z})$, and $\\chi$ be a character of level $M$. Munshi [cite] used Petersson trace formula as a delta method to obtain a subconvexity bound for $L(1/2, \\pi\\times\\chi)$ as $M$ varies.\n\nCan we obtain a subconvexity bound by averaging over a family, e.g. by bounding\n$$\\sum_{\\chi\\bmod M} |L(1/2, \\chi)|^2. $$",
   "clean_statement": "Subconvexity for $\\rm GL(3)\\times GL(1)$ via averaging over a family\n\nLet $\\pi$ be a fixed Hecke-Maass cusp form for $\\rm SL_3(\\mathbb{Z})$, and $\\chi$ be a character of level $M$. Munshi [cite] used Petersson trace formula as a delta method to obtain a subconvexity bound for $L(1/2, \\pi\\times\\chi)$ as $M$ varies.\n\nCan we obtain a subconvexity bound by averaging over a family, e.g. by bounding\n$$\\sum_{\\chi\\bmod M} |L(1/2, \\chi)|^2. $$",
   "public_statement": "Subconvexity for $\\rm GL(3)\\times GL(1)$ via averaging over a family\n\nLet $\\pi$ be a fixed Hecke-Maass cusp form for $\\rm SL_3(\\mathbb{Z})$, and $\\chi$ be a character of level $M$. Munshi [cite] used Petersson trace formula as a delta method to obtain a subconvexity bound for $L(1/2, \\pi\\times\\chi)$ as $M$ varies.\n\nCan we obtain a subconvexity bound by averaging over a family, e.g. by bounding\n$$\\sum_{\\chi\\bmod M} |L(1/2, \\chi)|^2. $$",
   "evidence": "The canonical record, AIM workshop problem 1.14 from *Delta symbols and the subconvexity problem*, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0008": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Application to Quantum Unique Ergodicity\n\nHolowinsky and Soundararajan proved the Quantum Unique Ergodicity conjecture in the case of holomorphic Hecke eigenforms by combining two different approaches: By proving a non-trivial bound on certain shifted convolution sum, and by proving (weak) subconvexity bound for certain $L$-functions.\n\nObtaining a bound of the form\n$$ \\sum_{f,g \\in \\mathcal{B}_k} L(1/2, sym^2f) L(1/2, sym^2g)\\big|\\sum_{n\\leq k} \\lambda_f(n)\\lambda_g(n+h)\\big|^2 \\ll k^{4-\\delta},$$\nfor $|h|\\ll 1$ and some $\\delta>0$ would imply the result of Holowinsky and Soundararajan (conditionally on non-negativity of the central values $L(1/2, sym^2f)$).\n\nUse a delta method in order to prove the above bound.",
   "clean_statement": null,
   "public_statement": "Application to Quantum Unique Ergodicity\n\nHolowinsky and Soundararajan proved the Quantum Unique Ergodicity conjecture in the case of holomorphic Hecke eigenforms by combining two different approaches: By proving a non-trivial bound on certain shifted convolution sum, and by proving (weak) subconvexity bound for certain $L$-functions.\n\nObtaining a bound of the form\n$$ \\sum_{f,g \\in \\mathcal{B}_k} L(1/2, sym^2f) L(1/2, sym^2g)\\big|\\sum_{n\\leq k} \\lambda_f(n)\\lambda_g(n+h)\\big|^2 \\ll k^{4-\\delta},$$\nfor $|h|\\ll 1$ and some $\\delta>0$ would imply the result of Holowinsky and Soundararajan (conditionally on non-negativity of the central values $L(1/2, sym^2f)$).\n\nUse a delta method in order to prove the above bound.",
   "evidence": "This is an explicit reconstruction, not text verified on the unavailable AIM page. If \\(\\mathcal B_k\\) instead means an orthonormal Fourier basis, or if harmonic weights are intended, both the scale and the applicable trace formula change.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0009": {
   "statement_status": "exact",
   "original_statement": "$GL(3)$ Kuznetsov trace formula in the level aspect\n\nDevelop a Kuznetsov trace formula for automorphic forms for the group $\\Gamma_0(N) = \\left\\lbrace \\begin{pmatrix} \\star & \\star & \\star \\\\ \\star & \\star & \\star \\\\ 0 & 0 & \\star \\end{pmatrix} \\bmod N \\right\\rbrace$ for any integer $N\\geq1$.\n\nFor the purpose of applications, develop a Kuznetsov trace formula for automorphic forms for the group $\\Gamma_B(N) = \\left\\lbrace \\begin{pmatrix} \\star & \\star & \\star \\\\ 0 & \\star & \\star \\\\ 0 & 0 & \\star \\end{pmatrix} \\bmod N \\right\\rbrace$.",
   "clean_statement": "$GL(3)$ Kuznetsov trace formula in the level aspect\n\nDevelop a Kuznetsov trace formula for automorphic forms for the group $\\Gamma_0(N) = \\left\\lbrace \\begin{pmatrix} \\star & \\star & \\star \\\\ \\star & \\star & \\star \\\\ 0 & 0 & \\star \\end{pmatrix} \\bmod N \\right\\rbrace$ for any integer $N\\geq1$.\n\nFor the purpose of applications, develop a Kuznetsov trace formula for automorphic forms for the group $\\Gamma_B(N) = \\left\\lbrace \\begin{pmatrix} \\star & \\star & \\star \\\\ 0 & \\star & \\star \\\\ 0 & 0 & \\star \\end{pmatrix} \\bmod N \\right\\rbrace$.",
   "public_statement": "$GL(3)$ Kuznetsov trace formula in the level aspect\n\nDevelop a Kuznetsov trace formula for automorphic forms for the group $\\Gamma_0(N) = \\left\\lbrace \\begin{pmatrix} \\star & \\star & \\star \\\\ \\star & \\star & \\star \\\\ 0 & 0 & \\star \\end{pmatrix} \\bmod N \\right\\rbrace$ for any integer $N\\geq1$.\n\nFor the purpose of applications, develop a Kuznetsov trace formula for automorphic forms for the group $\\Gamma_B(N) = \\left\\lbrace \\begin{pmatrix} \\star & \\star & \\star \\\\ 0 & \\star & \\star \\\\ 0 & 0 & \\star \\end{pmatrix} \\bmod N \\right\\rbrace$.",
   "evidence": "The canonical record is AIM Problem List 1.18, “Delta symbols and the subconvexity problem.” Its exact statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0010": {
   "statement_status": "exact",
   "original_statement": "Subconvexity bound for Rankin-Selberg $L$-functions\n\nObtain a subconvexity bound for a $\\rm GL_n\\times GL_m$ Rankin--Selberg $L$-function when $n$ and $m$ are not coprime.\n\nThe simplest first cases would be obtaining a subconvexity bound (in any aspect) for a $\\rm GL_4\\times GL_2$ $L$-function or a $\\rm GL_3\\times GL_3$ $L$-function.",
   "clean_statement": "Subconvexity bound for Rankin-Selberg $L$-functions\n\nObtain a subconvexity bound for a $\\rm GL_n\\times GL_m$ Rankin--Selberg $L$-function when $n$ and $m$ are not coprime.\n\nThe simplest first cases would be obtaining a subconvexity bound (in any aspect) for a $\\rm GL_4\\times GL_2$ $L$-function or a $\\rm GL_3\\times GL_3$ $L$-function.",
   "public_statement": "Subconvexity bound for Rankin-Selberg $L$-functions\n\nObtain a subconvexity bound for a $\\rm GL_n\\times GL_m$ Rankin--Selberg $L$-function when $n$ and $m$ are not coprime.\n\nThe simplest first cases would be obtaining a subconvexity bound (in any aspect) for a $\\rm GL_4\\times GL_2$ $L$-function or a $\\rm GL_3\\times GL_3$ $L$-function.",
   "evidence": "The canonical record is problem 1.2 in the AIM list *Delta symbols and the subconvexity problem*, source file `aim-analytic-number-theory-notes.json`, zero-based index 9. The exact recorded problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0011": {
   "statement_status": "exact",
   "original_statement": "Cancellations in additive twists on average\n\nLet $\\alpha\\in\\mathbb{R}$ and $f_j$ be a Maass form for $\\rm SL(4, \\mathbb{Z})$ with eigenvalue $t_j$ and Fourier coefficients $\\{\\lambda_j(1,1,n)\\}_{n\\geq1}$. For parameters $T$ and $N$, prove a non-trivial bound for the averaged sum\n$$ \\sum_{t_j\\sim T} \\bigg|\\sum_{n\\leq N}\\lambda_j(1,1,n)e(n\\alpha) \\bigg|^2. $$\n\nIf $\\{f_j\\}$ are Maass forms for $\\rm SL(3, \\mathbb{Z})$ improve Miller's bound on average over the spectral parameter $t_j$.",
   "clean_statement": "Cancellations in additive twists on average\n\nLet $\\alpha\\in\\mathbb{R}$ and $f_j$ be a Maass form for $\\rm SL(4, \\mathbb{Z})$ with eigenvalue $t_j$ and Fourier coefficients $\\{\\lambda_j(1,1,n)\\}_{n\\geq1}$. For parameters $T$ and $N$, prove a non-trivial bound for the averaged sum\n$$ \\sum_{t_j\\sim T} \\bigg|\\sum_{n\\leq N}\\lambda_j(1,1,n)e(n\\alpha) \\bigg|^2. $$\n\nIf $\\{f_j\\}$ are Maass forms for $\\rm SL(3, \\mathbb{Z})$ improve Miller's bound on average over the spectral parameter $t_j$.",
   "public_statement": "Cancellations in additive twists on average\n\nLet $\\alpha\\in\\mathbb{R}$ and $f_j$ be a Maass form for $\\rm SL(4, \\mathbb{Z})$ with eigenvalue $t_j$ and Fourier coefficients $\\{\\lambda_j(1,1,n)\\}_{n\\geq1}$. For parameters $T$ and $N$, prove a non-trivial bound for the averaged sum\n$$ \\sum_{t_j\\sim T} \\bigg|\\sum_{n\\leq N}\\lambda_j(1,1,n)e(n\\alpha) \\bigg|^2. $$\n\nIf $\\{f_j\\}$ are Maass forms for $\\rm SL(3, \\mathbb{Z})$ improve Miller's bound on average over the spectral parameter $t_j$.",
   "evidence": "The canonical record is Problem 1.22, “Cancellations in additive twists on average,” from the AIM list *Delta symbols and the subconvexity problem*. The live AIM page was checked and agrees with the repository record. Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0012": {
   "statement_status": "exact",
   "original_statement": "Asymptotic for second moment of $\\rm GL_2\\times GL_2$ in level aspect\n\nLet $\\mathcal{B}_k(p)$ be a basis for holomorphic or Hecke-Maass forms for $\\Gamma_0(p)\\subset SL_2(\\mathbb{Z})$ of weight $k$. Obtain an asymptotic with a power saving error term for the second moment\n$$\\sum_{f\\in\\mathcal{B}_k(p)}\\sum_{g\\in\\mathcal{B}_\\ell(q)} |L(1/2, f\\times g)|^2, $$\nas the level $p$ and $q$ become larger, the weights $k$ and $\\ell$ are fixed.",
   "clean_statement": "Asymptotic for second moment of $\\rm GL_2\\times GL_2$ in level aspect\n\nLet $\\mathcal{B}_k(p)$ be a basis for holomorphic or Hecke-Maass forms for $\\Gamma_0(p)\\subset SL_2(\\mathbb{Z})$ of weight $k$. Obtain an asymptotic with a power saving error term for the second moment\n$$\\sum_{f\\in\\mathcal{B}_k(p)}\\sum_{g\\in\\mathcal{B}_\\ell(q)} |L(1/2, f\\times g)|^2, $$\nas the level $p$ and $q$ become larger, the weights $k$ and $\\ell$ are fixed.",
   "public_statement": "Asymptotic for second moment of $\\rm GL_2\\times GL_2$ in level aspect\n\nLet $\\mathcal{B}_k(p)$ be a basis for holomorphic or Hecke-Maass forms for $\\Gamma_0(p)\\subset SL_2(\\mathbb{Z})$ of weight $k$. Obtain an asymptotic with a power saving error term for the second moment\n$$\\sum_{f\\in\\mathcal{B}_k(p)}\\sum_{g\\in\\mathcal{B}_\\ell(q)} |L(1/2, f\\times g)|^2, $$\nas the level $p$ and $q$ become larger, the weights $k$ and $\\ell$ are fixed.",
   "evidence": "The canonical AIM record (problem 1.24 from the workshop *Delta symbols and the subconvexity problem*) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0013": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Subconvexity bound for $\\rm GL_3\\times GL_2$ in level aspect\n\nLet $\\pi$ be a fixed Hecke-Maass cusp form for $SL_3(\\mathbb{Z})$ and $f$ be a holomorphic for Hecke Maass cusp form for $\\Gamma_0(M)$. Obtain a subconvexity bound estimate for $L(1/2, \\pi\\times f)$ as $M$ grows.",
   "clean_statement": null,
   "public_statement": "Subconvexity bound for $\\rm GL_3\\times GL_2$ in level aspect\n\nLet $\\pi$ be a fixed Hecke-Maass cusp form for $SL_3(\\mathbb{Z})$ and $f$ be a holomorphic for Hecke Maass cusp form for $\\Gamma_0(M)$. Obtain a subconvexity bound estimate for $L(1/2, \\pi\\times f)$ as $M$ grows.",
   "evidence": "The phrase “holomorphic for Hecke Maass” is visibly corrupt. The neighboring record repeats the same phrase, while modern papers on this exact problem uniformly say “holomorphic or Hecke--Maass.” I therefore use the conservative reconstruction",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0014": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Shifted convolution sum for $\\rm GL_3\\times GL_2$ in level aspect\n\nLet $\\pi$ be a fixed Hecke-Maass cusp form for $SL_3(\\mathbb{Z})$ with Fourier coefficients $\\{\\lambda_\\pi(r,n)\\}$ and $f$ be a holomorphic for Hecke Maass cusp form for $\\Gamma_0(M)$ with Fourier coefficients $\\{\\lambda_f(n)\\}$. Obtain a non-trivial bound for the shifted sum,\n\\begin{equation*}\n\\sum_{n<N}\\lambda_\\pi(r,n)\\lambda_f(n+1),\n\\end{equation*}\nin the convexity range $N\\sim M^{3/2}$.",
   "clean_statement": "“holomorphic or Hecke--Maass.” The listed source URL timed out during this run, so this repair was not checked against a separately rendered workshop page. The record also leaves open whether $f$ is new or old, its nebentypus, weight or spectral parameter, whether $M$ is prime or general, whether $r$ is fixed, and whether the cutoff is sharp or smooth.",
   "public_statement": "Shifted convolution sum for $\\rm GL_3\\times GL_2$ in level aspect\n\nLet $\\pi$ be a fixed Hecke-Maass cusp form for $SL_3(\\mathbb{Z})$ with Fourier coefficients $\\{\\lambda_\\pi(r,n)\\}$ and $f$ be a holomorphic for Hecke Maass cusp form for $\\Gamma_0(M)$ with Fourier coefficients $\\{\\lambda_f(n)\\}$. Obtain a non-trivial bound for the shifted sum,\n\\begin{equation*}\n\\sum_{n<N}\\lambda_\\pi(r,n)\\lambda_f(n+1),\n\\end{equation*}\nin the convexity range $N\\sim M^{3/2}$.",
   "evidence": "The phrase “holomorphic for Hecke Maass” is visibly malformed. The conservative reconstruction is “holomorphic or Hecke--Maass.” The listed source URL timed out during this run, so this repair was not checked against a separately rendered workshop page. The record also leaves open whether $f$ is new or old, its nebentypus, weight or spectral parameter, whether $M$ is prime or general, whether $r$ is fixed, and whether the cutoff is sharp or smooth.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0015": {
   "statement_status": "exact",
   "original_statement": "$\\rm GL_3$ Kuznetsov trace formula as a $\\delta$-method\n\nObtain $t$-aspect subconvexity for a $\\rm GL_2$ $L$-function by using $\\rm GL_3$ Kuznetsov trace formula as a $\\delta$-method.",
   "clean_statement": "$\\rm GL_3$ Kuznetsov trace formula as a $\\delta$-method\n\nObtain $t$-aspect subconvexity for a $\\rm GL_2$ $L$-function by using $\\rm GL_3$ Kuznetsov trace formula as a $\\delta$-method.",
   "public_statement": "$\\rm GL_3$ Kuznetsov trace formula as a $\\delta$-method\n\nObtain $t$-aspect subconvexity for a $\\rm GL_2$ $L$-function by using $\\rm GL_3$ Kuznetsov trace formula as a $\\delta$-method.",
   "evidence": "The exact AIM record is Problem 1.3 from the October 16--20, 2023 workshop *Delta symbols and the subconvexity problem*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0016": {
   "statement_status": "exact",
   "original_statement": "Delta method over Number fields\n\nDesign a delta symbol method over number fields in order to generalize the classical techniques to the number fields setting.",
   "clean_statement": "Delta method over Number fields\n\nDesign a delta symbol method over number fields in order to generalize the classical techniques to the number fields setting.",
   "public_statement": "Delta method over Number fields\n\nDesign a delta symbol method over number fields in order to generalize the classical techniques to the number fields setting.",
   "evidence": "The canonical record is problem 1.32 from the AIM workshop *Delta symbols and the subconvexity problem* (October 16--20, 2023):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0017": {
   "statement_status": "exact",
   "original_statement": "Strong hybrid bounds for $L(1/2, f\\times \\chi)$\n\nLet $f$ be a holomorphic or Hecke-Maass cusp form of level $p$, and $\\chi$ be a primitive Dirchlet character of level $q$. Conrey--Iwaniec obtained a Weyl type subconvexity bound for $L(1/2, f\\times\\chi)$ when $p=q$, while Khan has obtained a Burgess-type bound when $p\\asymp q$, and Yang has proved a Burgess-type bound over number fields in a similar range.\n\nProve strong hybrid bounds for other ranges of $p$ and $q$.",
   "clean_statement": "Strong hybrid bounds for $L(1/2, f\\times \\chi)$\n\nLet $f$ be a holomorphic or Hecke-Maass cusp form of level $p$, and $\\chi$ be a primitive Dirchlet character of level $q$. Conrey--Iwaniec obtained a Weyl type subconvexity bound for $L(1/2, f\\times\\chi)$ when $p=q$, while Khan has obtained a Burgess-type bound when $p\\asymp q$, and Yang has proved a Burgess-type bound over number fields in a similar range.\n\nProve strong hybrid bounds for other ranges of $p$ and $q$.",
   "public_statement": "Strong hybrid bounds for $L(1/2, f\\times \\chi)$\n\nLet $f$ be a holomorphic or Hecke-Maass cusp form of level $p$, and $\\chi$ be a primitive Dirchlet character of level $q$. Conrey--Iwaniec obtained a Weyl type subconvexity bound for $L(1/2, f\\times\\chi)$ when $p=q$, while Khan has obtained a Burgess-type bound when $p\\asymp q$, and Yang has proved a Burgess-type bound over number fields in a similar range.\n\nProve strong hybrid bounds for other ranges of $p$ and $q$.",
   "evidence": "The canonical record is AIM Problem List item 1.34 from the workshop *Delta symbols and the subconvexity problem*. Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0018": {
   "statement_status": "exact",
   "original_statement": "Subconvexity bound $\\rm GL_2$ in the `conductor dropping' case\n\nLet $f$ be a holomorphic or Hecke-Maass cusp form for $\\rm GL_2$. Use a $\\delta$-method to prove the following subconvexity bounds for the following $L$-functions:\n\n\\begin{enumerate}\n\\item Level aspect subconvexity bound for $L(1/2, f)$ where $f$ has level $p$ prime (and may have non-trivial nebentypus).\n\n\\item Subconvexity for $L(1/2+it_f, f)$, where $f$ is a Maass form of level $1$ and spectral parameter $t_f$.\n\\end{enumerate}",
   "clean_statement": "Subconvexity bound $\\rm GL_2$ in the `conductor dropping' case\n\nLet $f$ be a holomorphic or Hecke-Maass cusp form for $\\rm GL_2$. Use a $\\delta$-method to prove the following subconvexity bounds for the following $L$-functions:\n\n\\begin{enumerate}\n\\item Level aspect subconvexity bound for $L(1/2, f)$ where $f$ has level $p$ prime (and may have non-trivial nebentypus).\n\n\\item Subconvexity for $L(1/2+it_f, f)$, where $f$ is a Maass form of level $1$ and spectral parameter $t_f$.\n\\end{enumerate}",
   "public_statement": "Subconvexity bound $\\rm GL_2$ in the `conductor dropping' case\n\nLet $f$ be a holomorphic or Hecke-Maass cusp form for $\\rm GL_2$. Use a $\\delta$-method to prove the following subconvexity bounds for the following $L$-functions:\n\n\\begin{enumerate}\n\\item Level aspect subconvexity bound for $L(1/2, f)$ where $f$ has level $p$ prime (and may have non-trivial nebentypus).\n\n\\item Subconvexity for $L(1/2+it_f, f)$, where $f$ is a Maass form of level $1$ and spectral parameter $t_f$.\n\\end{enumerate}",
   "evidence": "The canonical record is problem 1.36 in the AIM workshop list *Delta symbols and the subconvexity problem*, file *aim-analytic-number-theory-notes.json*, zero-based index 17. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0019": {
   "statement_status": "exact",
   "original_statement": "Bound for short second moment of $\\rm GL_4$ $L$-functions\n\nLet $f$ be a fixed Hecke-Maass cusp form for $\\rm GL_4(\\mathbb{Z})$, $T$ and $\\Delta$ be parameters. Prove a non-trivial bound for\n$$\\int_T^{T+\\Delta} |L(\\frac12+it, f)|^2 dt, $$\nfor $\\Delta = T^\\delta$ with some $\\delta<1$.",
   "clean_statement": "Bound for short second moment of $\\rm GL_4$ $L$-functions\n\nLet $f$ be a fixed Hecke-Maass cusp form for $\\rm GL_4(\\mathbb{Z})$, $T$ and $\\Delta$ be parameters. Prove a non-trivial bound for\n$$\\int_T^{T+\\Delta} |L(\\frac12+it, f)|^2 dt, $$\nfor $\\Delta = T^\\delta$ with some $\\delta<1$.",
   "public_statement": "Bound for short second moment of $\\rm GL_4$ $L$-functions\n\nLet $f$ be a fixed Hecke-Maass cusp form for $\\rm GL_4(\\mathbb{Z})$, $T$ and $\\Delta$ be parameters. Prove a non-trivial bound for\n$$\\int_T^{T+\\Delta} |L(\\frac12+it, f)|^2 dt, $$\nfor $\\Delta = T^\\delta$ with some $\\delta<1$.",
   "evidence": "The canonical record is Problem 1.38 in the AIM list *Delta symbols and the subconvexity problem* (`aim-analytic-number-theory-notes.json`, zero-based index 18). Its mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0020": {
   "statement_status": "exact",
   "original_statement": "Applications of trace formula towards point counting\n\nVarious circle methods and $\\delta$-methods have been successfully used to count points on varieties. Could one use Petersson or Kuznetsov trace formula as a $\\delta$-method to count point on varieties?",
   "clean_statement": "Applications of trace formula towards point counting\n\nVarious circle methods and $\\delta$-methods have been successfully used to count points on varieties. Could one use Petersson or Kuznetsov trace formula as a $\\delta$-method to count point on varieties?",
   "public_statement": "Applications of trace formula towards point counting\n\nVarious circle methods and $\\delta$-methods have been successfully used to count points on varieties. Could one use Petersson or Kuznetsov trace formula as a $\\delta$-method to count point on varieties?",
   "evidence": "The canonical record is problem 1.4, “Applications of trace formula towards point counting,” from the AIM workshop *Delta symbols and the subconvexity problem*. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0021": {
   "statement_status": "exact",
   "original_statement": "Hardy-type theorem for $\\rm GL_3$ $L$-functions\n\nUse a $\\delta$-method to prove that there are infinitely many critical zeros for a $\\rm GL_3(\\mathbb{Z})$ $L$-functions.",
   "clean_statement": "Hardy-type theorem for $\\rm GL_3$ $L$-functions\n\nUse a $\\delta$-method to prove that there are infinitely many critical zeros for a $\\rm GL_3(\\mathbb{Z})$ $L$-functions.",
   "public_statement": "Hardy-type theorem for $\\rm GL_3$ $L$-functions\n\nUse a $\\delta$-method to prove that there are infinitely many critical zeros for a $\\rm GL_3(\\mathbb{Z})$ $L$-functions.",
   "evidence": "The canonical record is Problem 1.42 in the AIM list *Delta symbols and the subconvexity problem*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0022": {
   "statement_status": "exact",
   "original_statement": "Let $h \\in \\N$ be fixed. Is there a sequence $X_k \\to \\infty$ such that\n\\begin{equation*}\n \\left| \\sum_{n \\leq X_k} \\lambda(n) \\lambda(n + h) \\right| \\to \\infty\n\\end{equation*}\nas $k \\to \\infty$? We could also consider variants for multiplicative functions $f$ with values in $\\{\\pm 1\\}$. We can handle $f$ pretentious, and can also do it assuming $4$-point Chowla or the existence of arbitrarily long strings of $+1$'s for $\\lambda$. Would a Sudoku-type argument work here? If $h = h_k$ is allowed to vary with $k$, then we can probably show this as well.",
   "clean_statement": "Let $h \\in \\N$ be fixed. Is there a sequence $X_k \\to \\infty$ such that\n\\begin{equation*}\n \\left| \\sum_{n \\leq X_k} \\lambda(n) \\lambda(n + h) \\right| \\to \\infty\n\\end{equation*}\nas $k \\to \\infty$? We could also consider variants for multiplicative functions $f$ with values in $\\{\\pm 1\\}$. We can handle $f$ pretentious, and can also do it assuming $4$-point Chowla or the existence of arbitrarily long strings of $+1$'s for $\\lambda$. Would a Sudoku-type argument work here? If $h = h_k$ is allowed to vary with $k$, then we can probably show this as well.",
   "public_statement": "Let $h \\in \\N$ be fixed. Is there a sequence $X_k \\to \\infty$ such that\n\\begin{equation*}\n \\left| \\sum_{n \\leq X_k} \\lambda(n) \\lambda(n + h) \\right| \\to \\infty\n\\end{equation*}\nas $k \\to \\infty$? We could also consider variants for multiplicative functions $f$ with values in $\\{\\pm 1\\}$. We can handle $f$ pretentious, and can also do it assuming $4$-point Chowla or the existence of arbitrarily long strings of $+1$'s for $\\lambda$. Would a Sudoku-type argument work here? If $h = h_k$ is allowed to vary with $k$, then we can probably show this as well.",
   "evidence": "The canonical record is AIM Problem List item 1.1 in the workshop list *Sarnak's conjecture*, section “Lower bounds.” It asks, for fixed \\(h\\in\\mathbb N\\), whether there are \\(X_k\\to\\infty\\) such that",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0023": {
   "statement_status": "exact",
   "original_statement": "Is there a bounded sequence $a(n)$ such that each $a(n)$ is computable in time $O(\\log^A n)$ for some fixed $A > 0$ and such that\n\n\\begin{equation*}\n \\left| \\sum_{n \\leq X} \\lambda(n) a(n) \\right| > \\epsilon X\n\\end{equation*}\nfor all $X$ and some fixed $\\epsilon > 0$? Can we say something about the Furstenberg systems of such an $a(n)$? In particular, do they have positive entropy?",
   "clean_statement": "Is there a bounded sequence $a(n)$ such that each $a(n)$ is computable in time $O(\\log^A n)$ for some fixed $A > 0$ and such that\n\n\\begin{equation*}\n \\left| \\sum_{n \\leq X} \\lambda(n) a(n) \\right| > \\epsilon X\n\\end{equation*}\nfor all $X$ and some fixed $\\epsilon > 0$? Can we say something about the Furstenberg systems of such an $a(n)$? In particular, do they have positive entropy?",
   "public_statement": "Is there a bounded sequence $a(n)$ such that each $a(n)$ is computable in time $O(\\log^A n)$ for some fixed $A > 0$ and such that\n\n\\begin{equation*}\n \\left| \\sum_{n \\leq X} \\lambda(n) a(n) \\right| > \\epsilon X\n\\end{equation*}\nfor all $X$ and some fixed $\\epsilon > 0$? Can we say something about the Furstenberg systems of such an $a(n)$? In particular, do they have positive entropy?",
   "evidence": "The canonical record is item 1.2, “Lower bounds,” from the AIM workshop problem list *Sarnak's conjecture* (December 2018). Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0024": {
   "statement_status": "exact",
   "original_statement": "Can we obtain lower bounds for\n\n\\begin{equation*}\n \\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} \\lambda(n) \\right| d x?\n\\end{equation*}\nIn particular, can we get a lower bound of the form $XH^{\\frac{1}{2}-\\epsilon}$?",
   "clean_statement": "Can we obtain lower bounds for\n\n\\begin{equation*}\n \\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} \\lambda(n) \\right| d x?\n\\end{equation*}\nIn particular, can we get a lower bound of the form $XH^{\\frac{1}{2}-\\epsilon}$?",
   "public_statement": "Can we obtain lower bounds for\n\n\\begin{equation*}\n \\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} \\lambda(n) \\right| d x?\n\\end{equation*}\nIn particular, can we get a lower bound of the form $XH^{\\frac{1}{2}-\\epsilon}$?",
   "evidence": "The canonical record is AIM Problem List item 1.3, in the section “Lower bounds” of the 2018 workshop on Sarnak's conjecture. Its question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0025": {
   "statement_status": "exact",
   "original_statement": "Can we get a reasonable omega result for\n\\begin{equation*}\n \\left| \\sum_{n \\leq X} f(n) \\right|,\n\\end{equation*}\nwhere $f: \\N \\to \\{\\pm1\\}$ is totally multiplicative? Anything better than $\\log \\log \\log \\log \\log X$ would be interesting.",
   "clean_statement": "Can we get a reasonable omega result for\n\\begin{equation*}\n \\left| \\sum_{n \\leq X} f(n) \\right|,\n\\end{equation*}\nwhere $f: \\N \\to \\{\\pm1\\}$ is totally multiplicative? Anything better than $\\log \\log \\log \\log \\log X$ would be interesting.",
   "public_statement": "Can we get a reasonable omega result for\n\\begin{equation*}\n \\left| \\sum_{n \\leq X} f(n) \\right|,\n\\end{equation*}\nwhere $f: \\N \\to \\{\\pm1\\}$ is totally multiplicative? Anything better than $\\log \\log \\log \\log \\log X$ would be interesting.",
   "evidence": "This is record `AIM-ANALYTIC_NUMBER_THEORY-0025`, zero-based index 24 in `aim-analytic-number-theory-notes.json`, from the AIM workshop list “Sarnak's conjecture,” section “Lower bounds,” Problem 1.4.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0026": {
   "statement_status": "exact",
   "original_statement": "Let $f$ be a multiplicative function with countable range. Can we find a coloring of that range with (say) two colors, such that the number of \"sign patterns\" of $f$ (in terms of the colors) grows super-linearly?",
   "clean_statement": "Let $f$ be a multiplicative function with countable range. Can we find a coloring of that range with (say) two colors, such that the number of \"sign patterns\" of $f$ (in terms of the colors) grows super-linearly?",
   "public_statement": "Let $f$ be a multiplicative function with countable range. Can we find a coloring of that range with (say) two colors, such that the number of \"sign patterns\" of $f$ (in terms of the colors) grows super-linearly?",
   "evidence": "The canonical AIM record is problem 2.1 in the “Sign patterns” section of the workshop list on Sarnak's conjecture:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0027": {
   "statement_status": "exact",
   "original_statement": "Can we show arbitrarily large patterns $++\\ldots+$ or $--\\ldots -$ in $(\\lambda(n))_{n \\in \\N}$? An approach through the entropy decrement argument (roughly) reduces this to showing the existence of arbitrarily large progressions $n, n+p, \\dots, n+kp$ with $\\lambda(n) = \\lambda(n+p) = \\dots = \\lambda(n+kp)$.",
   "clean_statement": "Can we show arbitrarily large patterns $++\\ldots+$ or $--\\ldots -$ in $(\\lambda(n))_{n \\in \\N}$? An approach through the entropy decrement argument (roughly) reduces this to showing the existence of arbitrarily large progressions $n, n+p, \\dots, n+kp$ with $\\lambda(n) = \\lambda(n+p) = \\dots = \\lambda(n+kp)$.",
   "public_statement": "Can we show arbitrarily large patterns $++\\ldots+$ or $--\\ldots -$ in $(\\lambda(n))_{n \\in \\N}$? An approach through the entropy decrement argument (roughly) reduces this to showing the existence of arbitrarily large progressions $n, n+p, \\dots, n+kp$ with $\\lambda(n) = \\lambda(n+p) = \\dots = \\lambda(n+kp)$.",
   "evidence": "The canonical record is AIM Problem List item 2.2 in the section “Sign patterns” of the 2018 workshop *Sarnak's conjecture*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0028": {
   "statement_status": "exact",
   "original_statement": "Can a Liouville system be isomorphic to $(\\mathbb{T}^2, m_{\\mathbb{T}^2}, T)$, where $T:\\mathbb{T}^2\\to \\mathbb{T}^2$ is given by $\\left(\\begin{smallmatrix} x\\\\ y \\end{smallmatrix}\\right) \\mapsto \\left(\\begin{smallmatrix} x+y\\\\ y \\end{smallmatrix}\\right)$ mod $1$?",
   "clean_statement": "Can a Liouville system be isomorphic to $(\\mathbb{T}^2, m_{\\mathbb{T}^2}, T)$, where $T:\\mathbb{T}^2\\to \\mathbb{T}^2$ is given by $\\left(\\begin{smallmatrix} x\\\\ y \\end{smallmatrix}\\right) \\mapsto \\left(\\begin{smallmatrix} x+y\\\\ y \\end{smallmatrix}\\right)$ mod $1$?",
   "public_statement": "Can a Liouville system be isomorphic to $(\\mathbb{T}^2, m_{\\mathbb{T}^2}, T)$, where $T:\\mathbb{T}^2\\to \\mathbb{T}^2$ is given by $\\left(\\begin{smallmatrix} x\\\\ y \\end{smallmatrix}\\right) \\mapsto \\left(\\begin{smallmatrix} x+y\\\\ y \\end{smallmatrix}\\right)$ mod $1$?",
   "evidence": "The canonical AIM record (Sarnak's conjecture workshop, section \"Möbius and Liouville systems,\" Problem 3.1) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0029": {
   "statement_status": "exact",
   "original_statement": "Is $-1$ in the spectrum of a Liouville system? Equivalently, can we show that\n\\begin{equation*}\n \\sum_{n\\leq X} (-1)^{n-1} \\frac{\\lambda(n+h_1) \\dots \\lambda(n+h_k)}{n} = o(\\log X)?\n\\end{equation*}",
   "clean_statement": "Is $-1$ in the spectrum of a Liouville system? Equivalently, can we show that\n\\begin{equation*}\n \\sum_{n\\leq X} (-1)^{n-1} \\frac{\\lambda(n+h_1) \\dots \\lambda(n+h_k)}{n} = o(\\log X)?\n\\end{equation*}",
   "public_statement": "Is $-1$ in the spectrum of a Liouville system? Equivalently, can we show that\n\\begin{equation*}\n \\sum_{n\\leq X} (-1)^{n-1} \\frac{\\lambda(n+h_1) \\dots \\lambda(n+h_k)}{n} = o(\\log X)?\n\\end{equation*}",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0030": {
   "statement_status": "exact",
   "original_statement": "Let $X_{\\mu^2} = \\overline{\\{T^k(\\mu^2) : k \\in \\N\\}}$ and $X_{\\mu} = \\overline{\\{T^k(\\mu) : k \\in \\N\\}}$, for $T$ the shift map. What are the pre-images of $f:X_{\\mu} \\to X_{\\mu^2}$ given by $(x_i) \\mapsto (x_i^2)$ in $X_{\\mu^2}$?",
   "clean_statement": "Let $X_{\\mu^2} = \\overline{\\{T^k(\\mu^2) : k \\in \\N\\}}$ and $X_{\\mu} = \\overline{\\{T^k(\\mu) : k \\in \\N\\}}$, for $T$ the shift map. What are the pre-images of $f:X_{\\mu} \\to X_{\\mu^2}$ given by $(x_i) \\mapsto (x_i^2)$ in $X_{\\mu^2}$?",
   "public_statement": "Let $X_{\\mu^2} = \\overline{\\{T^k(\\mu^2) : k \\in \\N\\}}$ and $X_{\\mu} = \\overline{\\{T^k(\\mu) : k \\in \\N\\}}$, for $T$ the shift map. What are the pre-images of $f:X_{\\mu} \\to X_{\\mu^2}$ given by $(x_i) \\mapsto (x_i^2)$ in $X_{\\mu^2}$?",
   "evidence": "The canonical AIM record (Sarnak's conjecture workshop, §3.3) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0031": {
   "statement_status": "exact",
   "original_statement": "How much can we lower $H$ in the local Fourier uniformity conjecture\n\\begin{equation*}\n \\int_X^{2X} \\sup_\\alpha \\left| \\sum_{x\\leq n\\leq x+H} \\lambda(n) e(n\\alpha) \\right| d x = o(HX)?\n\\end{equation*}",
   "clean_statement": "How much can we lower $H$ in the local Fourier uniformity conjecture\n\\begin{equation*}\n \\int_X^{2X} \\sup_\\alpha \\left| \\sum_{x\\leq n\\leq x+H} \\lambda(n) e(n\\alpha) \\right| d x = o(HX)?\n\\end{equation*}",
   "public_statement": "How much can we lower $H$ in the local Fourier uniformity conjecture\n\\begin{equation*}\n \\int_X^{2X} \\sup_\\alpha \\left| \\sum_{x\\leq n\\leq x+H} \\lambda(n) e(n\\alpha) \\right| d x = o(HX)?\n\\end{equation*}",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0032": {
   "statement_status": "exact",
   "original_statement": "Can we extend the local Fourier uniformity conjecture to nilsequences? Warm-up: do it for polynomial phases.",
   "clean_statement": "Can we extend the local Fourier uniformity conjecture to nilsequences? Warm-up: do it for polynomial phases.",
   "public_statement": "Can we extend the local Fourier uniformity conjecture to nilsequences? Warm-up: do it for polynomial phases.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0033": {
   "statement_status": "exact",
   "original_statement": "Can we prove the local Fourier uniformity conjecture in function fields? Would that suffice to obtain the full logarithmic Chowla conjecture?",
   "clean_statement": "Can we prove the local Fourier uniformity conjecture in function fields? Would that suffice to obtain the full logarithmic Chowla conjecture?",
   "public_statement": "Can we prove the local Fourier uniformity conjecture in function fields? Would that suffice to obtain the full logarithmic Chowla conjecture?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0034": {
   "statement_status": "exact",
   "original_statement": "Show that\n\\begin{equation*}\n \\left| \\sum_{1 \\leq n < N} \\lambda(n) \\lambda(N-n) \\right| < N-1.\n\\end{equation*}",
   "clean_statement": "Show that\n\\begin{equation*}\n \\left| \\sum_{1 \\leq n < N} \\lambda(n) \\lambda(N-n) \\right| < N-1.\n\\end{equation*}",
   "public_statement": "Show that\n\\begin{equation*}\n \\left| \\sum_{1 \\leq n < N} \\lambda(n) \\lambda(N-n) \\right| < N-1.\n\\end{equation*}",
   "evidence": "The exact extracted AIM record (workshop “Sarnak's conjecture,” section “Special correlations,” Problem 5.1) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0035": {
   "statement_status": "exact",
   "original_statement": "Show that\n\\begin{equation*}\n \\left| \\mathbb{E}_{n \\in \\mathbb{N}}^{\\log} \\lambda(n^2+1) \\right| 0$. It would even be interesting to show that both sets $\\{p : \\lambda(p-1) = 1 \\}$ and $\\{p : \\lambda(p-1) = -1 \\}$ are infinite. For the first bound, using the entropy decrement argument, wouldn't it be enough to show that $\\left| \\sum_{p \\approx P} \\sum_{n \\approx X} \\lambda(n^2 +p^2) \\right| = o\\left(X\\frac{P}{\\log P}\\right)$ with $P = \\log \\log \\log X$?",
   "clean_statement": "Show that\n\\begin{equation*}\n \\left| \\mathbb{E}_{n \\in \\mathbb{N}}^{\\log} \\lambda(n^2+1) \\right| 0$. It would even be interesting to show that both sets $\\{p : \\lambda(p-1) = 1 \\}$ and $\\{p : \\lambda(p-1) = -1 \\}$ are infinite. For the first bound, using the entropy decrement argument, wouldn't it be enough to show that $\\left| \\sum_{p \\approx P} \\sum_{n \\approx X} \\lambda(n^2 +p^2) \\right| = o\\left(X\\frac{P}{\\log P}\\right)$ with $P = \\log \\log \\log X$?",
   "public_statement": "Show that\n\\begin{equation*}\n \\left| \\mathbb{E}_{n \\in \\mathbb{N}}^{\\log} \\lambda(n^2+1) \\right| 0$. It would even be interesting to show that both sets $\\{p : \\lambda(p-1) = 1 \\}$ and $\\{p : \\lambda(p-1) = -1 \\}$ are infinite. For the first bound, using the entropy decrement argument, wouldn't it be enough to show that $\\left| \\sum_{p \\approx P} \\sum_{n \\approx X} \\lambda(n^2 +p^2) \\right| = o\\left(X\\frac{P}{\\log P}\\right)$ with $P = \\log \\log \\log X$?",
   "evidence": "The canonical JSON record is visibly truncated. Its `problem` field is preserved verbatim here:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0036": {
   "statement_status": "exact",
   "original_statement": "What can we say about correlations with itself of\n\\begin{equation*}\n \\mathbb{1}_{p^+(n) \\leq n^\\delta},\n\\end{equation*}\nwith $\\delta \\to 0$? Here $p^+(n) := \\max\\{p: p|n\\}$. Even upper bounds would be welcome. This kind of result would imply sharp mean values for Dirichlet polynomials supported on smooth numbers.",
   "clean_statement": "What can we say about correlations with itself of\n\\begin{equation*}\n \\mathbb{1}_{p^+(n) \\leq n^\\delta},\n\\end{equation*}\nwith $\\delta \\to 0$? Here $p^+(n) := \\max\\{p: p|n\\}$. Even upper bounds would be welcome. This kind of result would imply sharp mean values for Dirichlet polynomials supported on smooth numbers.",
   "public_statement": "What can we say about correlations with itself of\n\\begin{equation*}\n \\mathbb{1}_{p^+(n) \\leq n^\\delta},\n\\end{equation*}\nwith $\\delta \\to 0$? Here $p^+(n) := \\max\\{p: p|n\\}$. Even upper bounds would be welcome. This kind of result would imply sharp mean values for Dirichlet polynomials supported on smooth numbers.",
   "evidence": "The canonical AIM record is Problem 5.3 in the section “Special correlations” of the AIM list from the workshop *Sarnak's conjecture*. Its exact mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0037": {
   "statement_status": "exact",
   "original_statement": "Let $(X, \\mu, T)$ be a totally uniquely ergodic $0$-entropy system. We know from recent work of Frantzikinakis-Host that if $g:\\N \\to \\C$ is a bounded multiplicative function, then for all $f \\in C(X)$ with $\\int_X f d \\mu = 0$ and $x \\in X$ we have\n\n\\begin{equation*}\n \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} g(n) f(T^nx) = 0.\n\\end{equation*}\n\nNow, if $g: \\N \\to \\C$ is a (possibly unbounded) multiplicative function, show that for all $f \\in C(X)$, $x \\in X$ and fixed $c \\in \\C$,\n\n\\begin{equation*}\n \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} \\mathbb{1}_{g(n)=c} f(T^nx) = \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} \\mathbb{1}_{g(n)=c} \\cdot \\int_X f d \\mu.\n\\end{equation*}\nThis can be done for totally ergodic nilsystems.",
   "clean_statement": "Let $(X, \\mu, T)$ be a totally uniquely ergodic $0$-entropy system. We know from recent work of Frantzikinakis-Host that if $g:\\N \\to \\C$ is a bounded multiplicative function, then for all $f \\in C(X)$ with $\\int_X f d \\mu = 0$ and $x \\in X$ we have\n\n\\begin{equation*}\n \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} g(n) f(T^nx) = 0.\n\\end{equation*}\n\nNow, if $g: \\N \\to \\C$ is a (possibly unbounded) multiplicative function, show that for all $f \\in C(X)$, $x \\in X$ and fixed $c \\in \\C$,\n\n\\begin{equation*}\n \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} \\mathbb{1}_{g(n)=c} f(T^nx) = \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} \\mathbb{1}_{g(n)=c} \\cdot \\int_X f d \\mu.\n\\end{equation*}\nThis can be done for totally ergodic nilsystems.",
   "public_statement": "Let $(X, \\mu, T)$ be a totally uniquely ergodic $0$-entropy system. We know from recent work of Frantzikinakis-Host that if $g:\\N \\to \\C$ is a bounded multiplicative function, then for all $f \\in C(X)$ with $\\int_X f d \\mu = 0$ and $x \\in X$ we have\n\n\\begin{equation*}\n \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} g(n) f(T^nx) = 0.\n\\end{equation*}\n\nNow, if $g: \\N \\to \\C$ is a (possibly unbounded) multiplicative function, show that for all $f \\in C(X)$, $x \\in X$ and fixed $c \\in \\C$,\n\n\\begin{equation*}\n \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} \\mathbb{1}_{g(n)=c} f(T^nx) = \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} \\mathbb{1}_{g(n)=c} \\cdot \\int_X f d \\mu.\n\\end{equation*}\nThis can be done for totally ergodic nilsystems.",
   "evidence": "The canonical record is Problem 6.1 in the “Unbounded multiplicative functions” section of the AIM problem list from the workshop *Sarnak's conjecture*. The archived AIM page was checked and agrees with the record; no correction or reconstruction is needed.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0038": {
   "statement_status": "exact",
   "original_statement": "Can anything be done for correlations of unbounded functions? What is the behavior in almost all intervals of\n\\begin{equation*}\n \\sum_{x \\leq n \\leq x + H} g(n),\n\\end{equation*}\nwith $g$ an unbounded multiplicative function such that $g(p) = O(1)$, and $H = \\log^A x$ for $A>0$ fixed?",
   "clean_statement": "Can anything be done for correlations of unbounded functions? What is the behavior in almost all intervals of\n\\begin{equation*}\n \\sum_{x \\leq n \\leq x + H} g(n),\n\\end{equation*}\nwith $g$ an unbounded multiplicative function such that $g(p) = O(1)$, and $H = \\log^A x$ for $A>0$ fixed?",
   "public_statement": "Can anything be done for correlations of unbounded functions? What is the behavior in almost all intervals of\n\\begin{equation*}\n \\sum_{x \\leq n \\leq x + H} g(n),\n\\end{equation*}\nwith $g$ an unbounded multiplicative function such that $g(p) = O(1)$, and $H = \\log^A x$ for $A>0$ fixed?",
   "evidence": "The canonical record is AIM Problem List item 6.2 in the section “Unbounded multiplicative functions.” Its exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0039": {
   "statement_status": "exact",
   "original_statement": "We have the following polynomial versions of the Sarnak and Chowla conjectures, where $(X, \\mu, T)$ is a $0$-entropy system, $f \\in C(X)$, $x \\in X$ and $p(t) \\in \\Z[t]$:\n\n\\begin{itemize}\n \\item Polynomial Sarnak Conjecture 1: If $T$ is minimal, then $\\sum_{n\\leq X} \\lambda(n) f(T^{p(n)}x) = o(X)$.\n\n \\item Polynomial Sarnak Conjecture 2: If $p$ is not a constant multiple of the square of a polynomial, then $\\sum_{n \\leq X} \\lambda(p(n)) f(T^nx) = o(X)$.\n\n \\item Polynomial Chowla Conjecture: If $p$ is not a constant multiple of the square of a polynomial, then $\\sum_{n \\leq X} \\lambda(p(n)) = o(X)$.\n\\end{itemize}\n\nIs polynomial Sarnak true, and does it follow from polynomial Chowla in any of the two cases?",
   "clean_statement": "We have the following polynomial versions of the Sarnak and Chowla conjectures, where $(X, \\mu, T)$ is a $0$-entropy system, $f \\in C(X)$, $x \\in X$ and $p(t) \\in \\Z[t]$:\n\n\\begin{itemize}\n \\item Polynomial Sarnak Conjecture 1: If $T$ is minimal, then $\\sum_{n\\leq X} \\lambda(n) f(T^{p(n)}x) = o(X)$.\n\n \\item Polynomial Sarnak Conjecture 2: If $p$ is not a constant multiple of the square of a polynomial, then $\\sum_{n \\leq X} \\lambda(p(n)) f(T^nx) = o(X)$.\n\n \\item Polynomial Chowla Conjecture: If $p$ is not a constant multiple of the square of a polynomial, then $\\sum_{n \\leq X} \\lambda(p(n)) = o(X)$.\n\\end{itemize}\n\nIs polynomial Sarnak true, and does it follow from polynomial Chowla in any of the two cases?",
   "public_statement": "We have the following polynomial versions of the Sarnak and Chowla conjectures, where $(X, \\mu, T)$ is a $0$-entropy system, $f \\in C(X)$, $x \\in X$ and $p(t) \\in \\Z[t]$:\n\n\\begin{itemize}\n \\item Polynomial Sarnak Conjecture 1: If $T$ is minimal, then $\\sum_{n\\leq X} \\lambda(n) f(T^{p(n)}x) = o(X)$.\n\n \\item Polynomial Sarnak Conjecture 2: If $p$ is not a constant multiple of the square of a polynomial, then $\\sum_{n \\leq X} \\lambda(p(n)) f(T^nx) = o(X)$.\n\n \\item Polynomial Chowla Conjecture: If $p$ is not a constant multiple of the square of a polynomial, then $\\sum_{n \\leq X} \\lambda(p(n)) = o(X)$.\n\\end{itemize}\n\nIs polynomial Sarnak true, and does it follow from polynomial Chowla in any of the two cases?",
   "evidence": "The canonical record is Problem 7.05 in the “Generalizations” section of the AIM problem list from the workshop *Sarnak's conjecture*. The archived AIM page was checked. Its mathematical text agrees with the canonical record:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0040": {
   "statement_status": "exact",
   "original_statement": "Let $\\mathcal{A} = \\{a_1, a_2, \\dots\\}$ be a sequence of integers with $(a_i, a_j) = 1$ for $i \\not= j$. Let $\\pi(n):= (-1)^{\\#\\{a_j:a_j|n\\}}$. Is it true that\n\\begin{equation*}\n \\sum_{n \\leq X} \\pi(n) = o(X)?\n\\end{equation*}\nCan we find a set $\\mathcal{A}$ such that\n\\begin{equation*}\n \\sum_{n \\leq X} \\pi(n+h_1) \\dots \\pi(n+h_k) = o(X)\n\\end{equation*}\nfor all distinct $h_1, \\dots, h_k$?\n\n\\begin{itemize}\n \\item Remark: Let\n\\[ \\lambda_y(p) = \\begin{cases}\n -1, &\\text{if } p \\leq y \\\\\n +1, &\\text{if } p > y.\n \\end{cases}\n\\]\nIf $y = X^{o(1)}$ then\n\\begin{equation*}\n \\sum_{n\\leq X} \\lambda_y(n+h_1) \\cdots \\lambda_y(n+h_k) = o(X).\n\\end{equation*}\nFor $k=2$ this was done by Daboussi-Sarkar.\n\\end{itemize}",
   "clean_statement": "Let $\\mathcal{A} = \\{a_1, a_2, \\dots\\}$ be a sequence of integers with $(a_i, a_j) = 1$ for $i \\not= j$. Let $\\pi(n):= (-1)^{\\#\\{a_j:a_j|n\\}}$. Is it true that\n\\begin{equation*}\n \\sum_{n \\leq X} \\pi(n) = o(X)?\n\\end{equation*}\nCan we find a set $\\mathcal{A}$ such that\n\\begin{equation*}\n \\sum_{n \\leq X} \\pi(n+h_1) \\dots \\pi(n+h_k) = o(X)\n\\end{equation*}\nfor all distinct $h_1, \\dots, h_k$?\n\n\\begin{itemize}\n \\item Remark: Let\n\\[ \\lambda_y(p) = \\begin{cases}\n -1, &\\text{if } p \\leq y \\\\\n +1, &\\text{if } p > y.\n \\end{cases}\n\\]\nIf $y = X^{o(1)}$ then\n\\begin{equation*}\n \\sum_{n\\leq X} \\lambda_y(n+h_1) \\cdots \\lambda_y(n+h_k) = o(X).\n\\end{equation*}\nFor $k=2$ this was done by Daboussi-Sarkar.\n\\end{itemize}",
   "public_statement": "Let $\\mathcal{A} = \\{a_1, a_2, \\dots\\}$ be a sequence of integers with $(a_i, a_j) = 1$ for $i \\not= j$. Let $\\pi(n):= (-1)^{\\#\\{a_j:a_j|n\\}}$. Is it true that\n\\begin{equation*}\n \\sum_{n \\leq X} \\pi(n) = o(X)?\n\\end{equation*}\nCan we find a set $\\mathcal{A}$ such that\n\\begin{equation*}\n \\sum_{n \\leq X} \\pi(n+h_1) \\dots \\pi(n+h_k) = o(X)\n\\end{equation*}\nfor all distinct $h_1, \\dots, h_k$?\n\n\\begin{itemize}\n \\item Remark: Let\n\\[ \\lambda_y(p) = \\begin{cases}\n -1, &\\text{if } p \\leq y \\\\\n +1, &\\text{if } p > y.\n \\end{cases}\n\\]\nIf $y = X^{o(1)}$ then\n\\begin{equation*}\n \\sum_{n\\leq X} \\lambda_y(n+h_1) \\cdots \\lambda_y(n+h_k) = o(X).\n\\end{equation*}\nFor $k=2$ this was done by Daboussi-Sarkar.\n\\end{itemize}",
   "evidence": "The canonical record is Problem 7.2 in the “Generalizations” section of the AIM list from the workshop *Sarnak's conjecture*. The exact problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0041": {
   "statement_status": "exact",
   "original_statement": "Suppose that for all fixed $\\chi$ mod $q$ and $t \\in \\R$, $f:\\N \\to \\C$ is multiplicative with\n\\begin{equation*}\n \\sum_{p} \\frac{1 - \\operatorname{Re} f(p) \\overline{\\chi(p) p^{it}}}{p} = \\infty.\n\\end{equation*}\nDoes there exists a subsequence $X_k \\to \\infty$ such that\n\\begin{equation*}\n \\sum_{n \\leq X_j} f(n) \\overline{f(n+1)} = o(X_j)?\n\\end{equation*}\nAlternatively, is\n\\begin{equation*}\n \\sum_{n\\leq X} f(n) f(n+h_1) \\cdots f(n+h_k) = o(X)\n\\end{equation*}\nfor distinct $h_1, \\dots, h_k$?",
   "clean_statement": "Suppose that for all fixed $\\chi$ mod $q$ and $t \\in \\R$, $f:\\N \\to \\C$ is multiplicative with\n\\begin{equation*}\n \\sum_{p} \\frac{1 - \\operatorname{Re} f(p) \\overline{\\chi(p) p^{it}}}{p} = \\infty.\n\\end{equation*}\nDoes there exists a subsequence $X_k \\to \\infty$ such that\n\\begin{equation*}\n \\sum_{n \\leq X_j} f(n) \\overline{f(n+1)} = o(X_j)?\n\\end{equation*}\nAlternatively, is\n\\begin{equation*}\n \\sum_{n\\leq X} f(n) f(n+h_1) \\cdots f(n+h_k) = o(X)\n\\end{equation*}\nfor distinct $h_1, \\dots, h_k$?",
   "public_statement": "Suppose that for all fixed $\\chi$ mod $q$ and $t \\in \\R$, $f:\\N \\to \\C$ is multiplicative with\n\\begin{equation*}\n \\sum_{p} \\frac{1 - \\operatorname{Re} f(p) \\overline{\\chi(p) p^{it}}}{p} = \\infty.\n\\end{equation*}\nDoes there exists a subsequence $X_k \\to \\infty$ such that\n\\begin{equation*}\n \\sum_{n \\leq X_j} f(n) \\overline{f(n+1)} = o(X_j)?\n\\end{equation*}\nAlternatively, is\n\\begin{equation*}\n \\sum_{n\\leq X} f(n) f(n+h_1) \\cdots f(n+h_k) = o(X)\n\\end{equation*}\nfor distinct $h_1, \\dots, h_k$?",
   "evidence": "The canonical AIM record, item 7.4 in “Generalizations,” asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0042": {
   "statement_status": "exact",
   "original_statement": "What is the M\\\"obius disjointness variant of the Jewett-Krieger theorem?",
   "clean_statement": "What is the M\\\"obius disjointness variant of the Jewett-Krieger theorem?",
   "public_statement": "What is the M\\\"obius disjointness variant of the Jewett-Krieger theorem?",
   "evidence": "The canonical record is Problem 7.6 in the “Generalizations” section of the AIM problem list from the workshop *Sarnak's conjecture*. Its exact problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0043": {
   "statement_status": "exact",
   "original_statement": "Is is possible to show that\n\\begin{equation*}\n \\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} \\lambda(n) e(n \\alpha) \\right| d x = o(XH)\n\\end{equation*}\nwith $H > X^\\epsilon$ implies\n\\begin{equation*}\n \\sum_{n \\leq X} \\frac{\\lambda(n)\\lambda(n+1)}{n} = o(\\log X)?\n\\end{equation*}",
   "clean_statement": "Is is possible to show that\n\\begin{equation*}\n \\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} \\lambda(n) e(n \\alpha) \\right| d x = o(XH)\n\\end{equation*}\nwith $H > X^\\epsilon$ implies\n\\begin{equation*}\n \\sum_{n \\leq X} \\frac{\\lambda(n)\\lambda(n+1)}{n} = o(\\log X)?\n\\end{equation*}",
   "public_statement": "Is is possible to show that\n\\begin{equation*}\n \\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} \\lambda(n) e(n \\alpha) \\right| d x = o(XH)\n\\end{equation*}\nwith $H > X^\\epsilon$ implies\n\\begin{equation*}\n \\sum_{n \\leq X} \\frac{\\lambda(n)\\lambda(n+1)}{n} = o(\\log X)?\n\\end{equation*}",
   "evidence": "The canonical record is Problem 8.1 in the AIM workshop list *Sarnak's conjecture*, section “Proof techniques and examples.” The archived source page was checked directly and agrees with the repository record, including the duplicated word “Is” and the absence of a supremum over the frequency. Its mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0044": {
   "statement_status": "exact",
   "original_statement": "Does the Sarnak conjecture hold uniformly in $x \\in X$ in the case of horocycle flows? It is known that the uniform Sarnak conjecture follows from the Sarnak conjecture, but we don't know that it follows case by case.",
   "clean_statement": "Does the Sarnak conjecture hold uniformly in $x \\in X$ in the case of horocycle flows? It is known that the uniform Sarnak conjecture follows from the Sarnak conjecture, but we don't know that it follows case by case.",
   "public_statement": "Does the Sarnak conjecture hold uniformly in $x \\in X$ in the case of horocycle flows? It is known that the uniform Sarnak conjecture follows from the Sarnak conjecture, but we don't know that it follows case by case.",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0045": {
   "statement_status": "exact",
   "original_statement": "The Chowla conjecture implies that $\\mu$ cannot be orthogonal to all uniquely ergodic systems (allowing non-zero entropy). Can we find a concrete example of a uniquely ergodic system that correlates with $\\mu$? A way to find a uniquely ergodic system (of positive entropy) that does not correlate with $\\lambda$ is known.",
   "clean_statement": "The Chowla conjecture implies that $\\mu$ cannot be orthogonal to all uniquely ergodic systems (allowing non-zero entropy). Can we find a concrete example of a uniquely ergodic system that correlates with $\\mu$? A way to find a uniquely ergodic system (of positive entropy) that does not correlate with $\\lambda$ is known.",
   "public_statement": "The Chowla conjecture implies that $\\mu$ cannot be orthogonal to all uniquely ergodic systems (allowing non-zero entropy). Can we find a concrete example of a uniquely ergodic system that correlates with $\\mu$? A way to find a uniquely ergodic system (of positive entropy) that does not correlate with $\\lambda$ is known.",
   "evidence": "The canonical record is Problem 8.3 in the “Proof techniques and examples” section of the AIM problem list from the workshop *Sarnak's conjecture*. Its exact problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0046": {
   "statement_status": "exact",
   "original_statement": "Moments of divisor functions in short intervals\n\nEvaluate $$\\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} d_k(n) \\ - \\ \\mbox{main term}\\right|^{2\\ell}\\,dx$$ for different ranges of $H$.",
   "clean_statement": "Moments of divisor functions in short intervals\n\nEvaluate $$\\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} d_k(n) \\ - \\ \\mbox{main term}\\right|^{2\\ell}\\,dx$$ for different ranges of $H$.",
   "public_statement": "Moments of divisor functions in short intervals\n\nEvaluate $$\\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} d_k(n) \\ - \\ \\mbox{main term}\\right|^{2\\ell}\\,dx$$ for different ranges of $H$.",
   "evidence": "The archived AIM page was checked directly. Problem 1.1, attributed there to S. Lester, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0047": {
   "statement_status": "exact",
   "original_statement": "Some coefficients of the Ramanujan tau function appear as coefficients of the piecewise polynomial in the work of Keating, Rodgers, Roditty-Gershon, and Rudnick. Is this a coincidence or not?",
   "clean_statement": "Some coefficients of the Ramanujan tau function appear as coefficients of the piecewise polynomial in the work of Keating, Rodgers, Roditty-Gershon, and Rudnick. Is this a coincidence or not?",
   "public_statement": "Some coefficients of the Ramanujan tau function appear as coefficients of the piecewise polynomial in the work of Keating, Rodgers, Roditty-Gershon, and Rudnick. Is this a coincidence or not?",
   "evidence": "The canonical AIM record (workshop *Moments of zeta and correlations of divisor sums*, section “Divisor sums,” Problem 1.2) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0048": {
   "statement_status": "exact",
   "original_statement": "Analyze Alexandra Florea's formula for $V$ not equal to a square.",
   "clean_statement": "Analyze Alexandra Florea's formula for $V$ not equal to a square.",
   "public_statement": "Analyze Alexandra Florea's formula for $V$ not equal to a square.",
   "evidence": "The canonical record says, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0049": {
   "statement_status": "exact",
   "original_statement": "Evaluate moments of ratios of $L$-functions over function fields.",
   "clean_statement": "Evaluate moments of ratios of $L$-functions over function fields.",
   "public_statement": "Evaluate moments of ratios of $L$-functions over function fields.",
   "evidence": "The exact canonical record, from the 2016 AIM workshop *Moments of zeta and correlations of divisor sums*, section “Function fields,” Problem 2.5, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0050": {
   "statement_status": "exact",
   "original_statement": "Consider a family of elliptic curves with a known rational point. Evaluate moments of $L$-functions in this family at certain points. How does the fixed rational point analytically affect calculations? E.g. $y^2=x^3+ax+b^2$ with $a,b$ integers having $|a|,|b|\\leq X$ (or in $\\mathbb{F}_q[t]$).",
   "clean_statement": "Consider a family of elliptic curves with a known rational point. Evaluate moments of $L$-functions in this family at certain points. How does the fixed rational point analytically affect calculations? E.g. $y^2=x^3+ax+b^2$ with $a,b$ integers having $|a|,|b|\\leq X$ (or in $\\mathbb{F}_q[t]$).",
   "public_statement": "Consider a family of elliptic curves with a known rational point. Evaluate moments of $L$-functions in this family at certain points. How does the fixed rational point analytically affect calculations? E.g. $y^2=x^3+ax+b^2$ with $a,b$ integers having $|a|,|b|\\leq X$ (or in $\\mathbb{F}_q[t]$).",
   "evidence": "The canonical AIM record (workshop *Moments of zeta and correlations of divisor sums*, section “Function fields,” Problem 2.2) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0051": {
   "statement_status": "exact",
   "original_statement": "Develop a heuristic for moments of L-functions over function fields. Can one carry out the analogous Conrey-Keating calculation for high moments (fixed $q$, large degree $g$)?",
   "clean_statement": "Develop a heuristic for moments of L-functions over function fields. Can one carry out the analogous Conrey-Keating calculation for high moments (fixed $q$, large degree $g$)?",
   "public_statement": "Develop a heuristic for moments of L-functions over function fields. Can one carry out the analogous Conrey-Keating calculation for high moments (fixed $q$, large degree $g$)?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0052": {
   "statement_status": "exact",
   "original_statement": "Develop a heuristic for moments of L-functions of automorphic forms over function fields. What are families in this context?",
   "clean_statement": "Develop a heuristic for moments of L-functions of automorphic forms over function fields. What are families in this context?",
   "public_statement": "Develop a heuristic for moments of L-functions of automorphic forms over function fields. What are families in this context?",
   "evidence": "The canonical AIM record says, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0053": {
   "statement_status": "exact",
   "original_statement": "Can one find a version of the recipe using asymmetric approximate functional equations, giving a ``robust'' version of the calculations? Similarly, can one apply the Conrey-Keating method using two asymmetric Dirichlet polynomials\n$$\n\\sum_{m\\leq X} \\frac{\\tau_A(m)}{m^s} \\ \\ \\ \\ \\ \\ \\mbox{and} \\ \\ \\ \\ \\ \\\n\\sum_{n\\leq Y} \\frac{\\tau_B(n)}{n^{1-s}}\n$$\nto get the same answer as before, i.e. when $X=Y$?",
   "clean_statement": "Can one find a version of the recipe using asymmetric approximate functional equations, giving a ``robust'' version of the calculations? Similarly, can one apply the Conrey-Keating method using two asymmetric Dirichlet polynomials\n$$\n\\sum_{m\\leq X} \\frac{\\tau_A(m)}{m^s} \\ \\ \\ \\ \\ \\ \\mbox{and} \\ \\ \\ \\ \\ \\\n\\sum_{n\\leq Y} \\frac{\\tau_B(n)}{n^{1-s}}\n$$\nto get the same answer as before, i.e. when $X=Y$?",
   "public_statement": "Can one find a version of the recipe using asymmetric approximate functional equations, giving a ``robust'' version of the calculations? Similarly, can one apply the Conrey-Keating method using two asymmetric Dirichlet polynomials\n$$\n\\sum_{m\\leq X} \\frac{\\tau_A(m)}{m^s} \\ \\ \\ \\ \\ \\ \\mbox{and} \\ \\ \\ \\ \\ \\\n\\sum_{n\\leq Y} \\frac{\\tau_B(n)}{n^{1-s}}\n$$\nto get the same answer as before, i.e. when $X=Y$?",
   "evidence": "The canonical record is Problem 3.1 in the “Recipe” section of the AIM list *Moments of zeta and correlations of divisor sums*. The live source page was checked and attributes the problem to C. Hughes. It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0054": {
   "statement_status": "exact",
   "original_statement": "Can one prove the fourth moment formula starting from\n$$\n\\int \\prod_{j=1}^4 \\left( \\sum_{n\\leq x_j} \\frac{1}{n^{1/2+it}}+\\chi(s) \\sum_{n\\leq t/x_j} \\frac{1}{n^{1/2-it}}\\right)\\,dt \\ ?\n$$",
   "clean_statement": "Can one prove the fourth moment formula starting from\n$$\n\\int \\prod_{j=1}^4 \\left( \\sum_{n\\leq x_j} \\frac{1}{n^{1/2+it}}+\\chi(s) \\sum_{n\\leq t/x_j} \\frac{1}{n^{1/2-it}}\\right)\\,dt \\ ?\n$$",
   "public_statement": "Can one prove the fourth moment formula starting from\n$$\n\\int \\prod_{j=1}^4 \\left( \\sum_{n\\leq x_j} \\frac{1}{n^{1/2+it}}+\\chi(s) \\sum_{n\\leq t/x_j} \\frac{1}{n^{1/2-it}}\\right)\\,dt \\ ?\n$$",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0055": {
   "statement_status": "exact",
   "original_statement": "Using the ideas of Conrey and Keating, develop a heuristic for conjecturing asymptotic formulas for high moments of quadratic Dirichlet $L$-functions. What new terms contribute to the main term when $k\\geq 3$?",
   "clean_statement": "Using the ideas of Conrey and Keating, develop a heuristic for conjecturing asymptotic formulas for high moments of quadratic Dirichlet $L$-functions. What new terms contribute to the main term when $k\\geq 3$?",
   "public_statement": "Using the ideas of Conrey and Keating, develop a heuristic for conjecturing asymptotic formulas for high moments of quadratic Dirichlet $L$-functions. What new terms contribute to the main term when $k\\geq 3$?",
   "evidence": "The exact AIM record is Problem 5.1 in the section “Other families of \\(L\\)-functions” of the workshop list *Moments of zeta and correlations of divisor sums*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0056": {
   "statement_status": "exact",
   "original_statement": "Develop a heuristic for moments of $L$-functions of automorphic forms.",
   "clean_statement": "Develop a heuristic for moments of $L$-functions of automorphic forms.",
   "public_statement": "Develop a heuristic for moments of $L$-functions of automorphic forms.",
   "evidence": "The canonical AIM record (workshop *Moments of zeta and correlations of divisor sums*, section “Other families of \\(L\\)-functions,” problem 5.2) says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0057": {
   "statement_status": "exact",
   "original_statement": "Develop a heuristic for moments of imprimitive $L$-functions, or if your $L$-function factors into degree $1$ factors, e.g.\n$$\n\\int |L(\\tfrac{1}{2}+it,\\chi_1)\\cdots L(\\tfrac{1}{2}+it,\\chi_k)|^2\\,dt\n$$\n(or different powers).",
   "clean_statement": "Develop a heuristic for moments of imprimitive $L$-functions, or if your $L$-function factors into degree $1$ factors, e.g.\n$$\n\\int |L(\\tfrac{1}{2}+it,\\chi_1)\\cdots L(\\tfrac{1}{2}+it,\\chi_k)|^2\\,dt\n$$\n(or different powers).",
   "public_statement": "Develop a heuristic for moments of imprimitive $L$-functions, or if your $L$-function factors into degree $1$ factors, e.g.\n$$\n\\int |L(\\tfrac{1}{2}+it,\\chi_1)\\cdots L(\\tfrac{1}{2}+it,\\chi_k)|^2\\,dt\n$$\n(or different powers).",
   "evidence": "The canonical record is AIM Problem Lists, workshop *Moments of zeta and correlations of divisor sums*, section “Other families of \\(L\\)-functions,” Problem 5.3. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0058": {
   "statement_status": "exact",
   "original_statement": "Develop a heuristic for quadratic twists of $L$-functions",
   "clean_statement": "Develop a heuristic for quadratic twists of $L$-functions",
   "public_statement": "Develop a heuristic for quadratic twists of $L$-functions",
   "evidence": "The canonical record is AIM Problem List 5.4 from the workshop *Moments of zeta and correlations of divisor sums*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0059": {
   "statement_status": "exact",
   "original_statement": "Find an exact formula for $\\sum_{f\\in H_k}^h L(\\frac{1}{2},Sym^2f)^2$.",
   "clean_statement": "Find an exact formula for $\\sum_{f\\in H_k}^h L(\\frac{1}{2},Sym^2f)^2$.",
   "public_statement": "Find an exact formula for $\\sum_{f\\in H_k}^h L(\\frac{1}{2},Sym^2f)^2$.",
   "evidence": "The canonical record (AIM workshop “Moments of zeta and correlations of divisor sums,” section “Other families of $L$-functions,” Problem 5.7) says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0060": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "The mechanism for computing moments in families like $\\chi$ (mod $q$), $q\\leq Q$, involve different orthogonality relations than $|m-n|=$small. Yet the answers are the same. Why?",
   "clean_statement": null,
   "public_statement": "The mechanism for computing moments in families like $\\chi$ (mod $q$), $q\\leq Q$, involve different orthogonality relations than $|m-n|=$small. Yet the answers are the same. Why?",
   "evidence": "The conservative reconstruction used here is:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0061": {
   "statement_status": "exact",
   "original_statement": "Is there any way to get lower bounds for small moments (e.g. first moment) in families, even assuming GRH?",
   "clean_statement": "Is there any way to get lower bounds for small moments (e.g. first moment) in families, even assuming GRH?",
   "public_statement": "Is there any way to get lower bounds for small moments (e.g. first moment) in families, even assuming GRH?",
   "evidence": "The source record is Problem 5.6 in the section “Other families of $L$-functions” of the 2016 AIM workshop *Moments of zeta and correlations of divisor sums* (August 29--September 2, 2016). Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0062": {
   "statement_status": "exact",
   "original_statement": "Can one formulate a conjecture (e.g. Kloosterman sum estimates) that lead to the calculations of Conrey and Keating in a rigorous way? We need a better understanding (even mechanically) of these calculations.",
   "clean_statement": "Can one formulate a conjecture (e.g. Kloosterman sum estimates) that lead to the calculations of Conrey and Keating in a rigorous way? We need a better understanding (even mechanically) of these calculations.",
   "public_statement": "Can one formulate a conjecture (e.g. Kloosterman sum estimates) that lead to the calculations of Conrey and Keating in a rigorous way? We need a better understanding (even mechanically) of these calculations.",
   "evidence": "The canonical record is `aim-analytic-number-theory-notes.json`, zero-based index 61, problem 6.1 in the section **Making the method more rigorous** of the 2016 AIM workshop *Moments of zeta and correlations of divisor sums*. Its exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0063": {
   "statement_status": "exact",
   "original_statement": "Is there some version of M\\\"{o}bius randomness that would lead to similar calculations? And vice versa, can one go back to primes e.g. from analogs for ratios conjectures?",
   "clean_statement": "Is there some version of M\\\"{o}bius randomness that would lead to similar calculations? And vice versa, can one go back to primes e.g. from analogs for ratios conjectures?",
   "public_statement": "Is there some version of M\\\"{o}bius randomness that would lead to similar calculations? And vice versa, can one go back to primes e.g. from analogs for ratios conjectures?",
   "evidence": "The canonical record is problem 6.2 in the AIM list *Moments of zeta and correlations of divisor sums*, section “Making the method more rigorous”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0064": {
   "statement_status": "exact",
   "original_statement": "Is there a random model on the primes that leads to the Type II calculations of Bogomolny and Keating (e.g. use Hardy-Littlewood for the probability that both $n$ and $n+h$ are prime)?",
   "clean_statement": "Is there a random model on the primes that leads to the Type II calculations of Bogomolny and Keating (e.g. use Hardy-Littlewood for the probability that both $n$ and $n+h$ are prime)?",
   "public_statement": "Is there a random model on the primes that leads to the Type II calculations of Bogomolny and Keating (e.g. use Hardy-Littlewood for the probability that both $n$ and $n+h$ are prime)?",
   "evidence": "The exact canonical record is AIM Problem List 6.3 from the workshop *Moments of zeta and correlations of divisor sums*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0065": {
   "statement_status": "exact",
   "original_statement": "Can we make the calculations uniform in $k$?",
   "clean_statement": "Can we make the calculations uniform in $k$?",
   "public_statement": "Can we make the calculations uniform in $k$?",
   "evidence": "The exact canonical record is problem 6.4 in the AIM workshop section “Making the method more rigorous”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0066": {
   "statement_status": "exact",
   "original_statement": "Can one formulate a weighted version of moments with potentially simpler main terms?",
   "clean_statement": "Can one formulate a weighted version of moments with potentially simpler main terms?",
   "public_statement": "Can one formulate a weighted version of moments with potentially simpler main terms?",
   "evidence": "The exact canonical record is AIM problem 6.5 in the workshop *Moments of zeta and correlations of divisor sums*, under the section “Making the method more rigorous”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0067": {
   "statement_status": "exact",
   "original_statement": "We can get lower bounds for the Type I off-diagonal terms using sifted sequences. Is there any analog of this for Type II terms? Can one improve lower bounds for $F(\\alpha,T)$ in this way (as in Goldston-Gonek-Ozluk-Snyder)?",
   "clean_statement": "We can get lower bounds for the Type I off-diagonal terms using sifted sequences. Is there any analog of this for Type II terms? Can one improve lower bounds for $F(\\alpha,T)$ in this way (as in Goldston-Gonek-Ozluk-Snyder)?",
   "public_statement": "We can get lower bounds for the Type I off-diagonal terms using sifted sequences. Is there any analog of this for Type II terms? Can one improve lower bounds for $F(\\alpha,T)$ in this way (as in Goldston-Gonek-Ozluk-Snyder)?",
   "evidence": "The canonical record is AIM problem 7.1 in the list *Moments of zeta and correlations of divisor sums*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0068": {
   "statement_status": "exact",
   "original_statement": "Compute the $3$-point correlation function $R_3(x_1,x_2,x_3)$ for zeta rigorously under suitable hypotheses (e.g. Hardy-Littlewood conjecture for twin primes).",
   "clean_statement": "Compute the $3$-point correlation function $R_3(x_1,x_2,x_3)$ for zeta rigorously under suitable hypotheses (e.g. Hardy-Littlewood conjecture for twin primes).",
   "public_statement": "Compute the $3$-point correlation function $R_3(x_1,x_2,x_3)$ for zeta rigorously under suitable hypotheses (e.g. Hardy-Littlewood conjecture for twin primes).",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0069": {
   "statement_status": "exact",
   "original_statement": "Compute the $4$-point correlation function rigorously under suitable hypotheses (here, Type II terms come in).",
   "clean_statement": "Compute the $4$-point correlation function rigorously under suitable hypotheses (here, Type II terms come in).",
   "public_statement": "Compute the $4$-point correlation function rigorously under suitable hypotheses (here, Type II terms come in).",
   "evidence": "The exact canonical record is problem 7.3 in the AIM workshop list *Moments of zeta and correlations of divisor sums*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0070": {
   "statement_status": "exact",
   "original_statement": "Extend the work of Goldston and Gonek on mean value theorems for long Dirichlet polynomials by using Type II sums.",
   "clean_statement": "Extend the work of Goldston and Gonek on mean value theorems for long Dirichlet polynomials by using Type II sums.",
   "public_statement": "Extend the work of Goldston and Gonek on mean value theorems for long Dirichlet polynomials by using Type II sums.",
   "evidence": "The canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 69,
   "attempt": 4
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0071": {
   "statement_status": "exact",
   "original_statement": "Can one calculate moments of zeta closer to the $\\sigma=1$ line instead of on the $\\sigma=\\frac{1}{2}$ line? Can one also identify the lower order terms in the $\\sigma=1$ case?",
   "clean_statement": "Can one calculate moments of zeta closer to the $\\sigma=1$ line instead of on the $\\sigma=\\frac{1}{2}$ line? Can one also identify the lower order terms in the $\\sigma=1$ case?",
   "public_statement": "Can one calculate moments of zeta closer to the $\\sigma=1$ line instead of on the $\\sigma=\\frac{1}{2}$ line? Can one also identify the lower order terms in the $\\sigma=1$ case?",
   "evidence": "The canonical AIM record is Problem 8.1 in the “Miscellaneous” section of the workshop list *Moments of zeta and correlations of divisor sums*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 70,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0072": {
   "statement_status": "exact",
   "original_statement": "Develop a heuristic for twisted moments, e.g.\n$$\n\\int_0^T \\left( \\frac{m}{n}\\right)^{it} |\\zeta(\\tfrac{1}{2}+it)|^{2k}\\,dt.\n$$",
   "clean_statement": "Develop a heuristic for twisted moments, e.g.\n$$\n\\int_0^T \\left( \\frac{m}{n}\\right)^{it} |\\zeta(\\tfrac{1}{2}+it)|^{2k}\\,dt.\n$$",
   "public_statement": "Develop a heuristic for twisted moments, e.g.\n$$\n\\int_0^T \\left( \\frac{m}{n}\\right)^{it} |\\zeta(\\tfrac{1}{2}+it)|^{2k}\\,dt.\n$$",
   "evidence": "The canonical AIM record is problem 8.2 in the “Miscellaneous” section of the workshop *Moments of zeta and correlations of divisor sums*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0073": {
   "statement_status": "exact",
   "original_statement": "Develop a heuristic for\n$$\n\\int_T^{2T} \\left| \\sum_{p\\leq X} \\frac{1}{p^{1/2+it}}\\right|^{2k} \\,dt.\n$$\nFind connections with the work of Bogomolny-Keating and with the Ratios Conjectures.",
   "clean_statement": "Develop a heuristic for\n$$\n\\int_T^{2T} \\left| \\sum_{p\\leq X} \\frac{1}{p^{1/2+it}}\\right|^{2k} \\,dt.\n$$\nFind connections with the work of Bogomolny-Keating and with the Ratios Conjectures.",
   "public_statement": "Develop a heuristic for\n$$\n\\int_T^{2T} \\left| \\sum_{p\\leq X} \\frac{1}{p^{1/2+it}}\\right|^{2k} \\,dt.\n$$\nFind connections with the work of Bogomolny-Keating and with the Ratios Conjectures.",
   "evidence": "The canonical AIM record (workshop *Moments of zeta and correlations of divisor sums*, section “Miscellaneous,” problem 8.3) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 72,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0074": {
   "statement_status": "exact",
   "original_statement": "Evaluate $$\\int_0^T f(|\\zeta(\\tfrac{1}{2}+it)|)\\,dt,$$ where $f=J_{2k}$, the Bessel function, or $f=T_{2k}$, the Chebyshev polynomial.",
   "clean_statement": "Evaluate $$\\int_0^T f(|\\zeta(\\tfrac{1}{2}+it)|)\\,dt,$$ where $f=J_{2k}$, the Bessel function, or $f=T_{2k}$, the Chebyshev polynomial.",
   "public_statement": "Evaluate $$\\int_0^T f(|\\zeta(\\tfrac{1}{2}+it)|)\\,dt,$$ where $f=J_{2k}$, the Bessel function, or $f=T_{2k}$, the Chebyshev polynomial.",
   "evidence": "The canonical AIM record (source file `aim-analytic-number-theory-notes.json`, zero-based index 73) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 73,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0075": {
   "statement_status": "exact",
   "original_statement": "Evaluate moments of zeta over sets other than $[0,T]$ or $[T,2T]$.",
   "clean_statement": "Evaluate moments of zeta over sets other than $[0,T]$ or $[T,2T]$.",
   "public_statement": "Evaluate moments of zeta over sets other than $[0,T]$ or $[T,2T]$.",
   "evidence": "The canonical AIM record is problem 8.4 in the “Miscellaneous” section of the workshop *Moments of zeta and correlations of divisor sums*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0076": {
   "statement_status": "exact",
   "original_statement": "Evaluate fractional moments of zeta.",
   "clean_statement": "Evaluate fractional moments of zeta.",
   "public_statement": "Evaluate fractional moments of zeta.",
   "evidence": "The canonical AIM record says, exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0077": {
   "statement_status": "exact",
   "original_statement": "Evaluate $$ \\sum \\int_{t_j}^{t_j+T^{7/8}} |\\zeta(\\tfrac{1}{2}+it)|^4 \\,dt, $$ where the sum is over a set of $t_j$'s with modulus $\\leq T$ and spaced more than $T^{7/8}$ apart from each other (related to work of Heath-Brown and of Zavorotnyi)",
   "clean_statement": "Evaluate $$ \\sum \\int_{t_j}^{t_j+T^{7/8}} |\\zeta(\\tfrac{1}{2}+it)|^4 \\,dt, $$ where the sum is over a set of $t_j$'s with modulus $\\leq T$ and spaced more than $T^{7/8}$ apart from each other (related to work of Heath-Brown and of Zavorotnyi)",
   "public_statement": "Evaluate $$ \\sum \\int_{t_j}^{t_j+T^{7/8}} |\\zeta(\\tfrac{1}{2}+it)|^4 \\,dt, $$ where the sum is over a set of $t_j$'s with modulus $\\leq T$ and spaced more than $T^{7/8}$ apart from each other (related to work of Heath-Brown and of Zavorotnyi)",
   "evidence": "The canonical AIM record (workshop *Moments of zeta and correlations of divisor sums*, section “Miscellaneous,” item 8.6) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0078": {
   "statement_status": "exact",
   "original_statement": "Does the holomorphic anomaly equation \"integrate\" over elliptic fibrations?",
   "clean_statement": "Does the holomorphic anomaly equation \"integrate\" over elliptic fibrations?",
   "public_statement": "Does the holomorphic anomaly equation \"integrate\" over elliptic fibrations?",
   "evidence": "The canonical AIM record (source file `aim-analytic-number-theory-notes.json`, zero-based index 77) asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0079": {
   "statement_status": "exact",
   "original_statement": "Eta-products and root systems and holomorphic anomaly equation\n\nAs it turns out mock modular forms (such as eta-products) do not satisfy the holomorphic anaomaly equation.\n\nDo they satisfy other differential equations?",
   "clean_statement": "Eta-products and root systems and holomorphic anomaly equation\n\nAs it turns out mock modular forms (such as eta-products) do not satisfy the holomorphic anaomaly equation.\n\nDo they satisfy other differential equations?",
   "public_statement": "Eta-products and root systems and holomorphic anomaly equation\n\nAs it turns out mock modular forms (such as eta-products) do not satisfy the holomorphic anaomaly equation.\n\nDo they satisfy other differential equations?",
   "evidence": "The canonical record is problem 1.2 in the section “Holomorphic anomaly equation” of the April 2013 AIM workshop *Gromov--Witten invariants and number theory*. It asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0080": {
   "statement_status": "exact",
   "original_statement": "Investigate the Holomorphic anomaly equation (HAE) in fibered Calabi-Yau 3-folds and BPS counting. \\label{hae-cy}",
   "clean_statement": "Investigate the Holomorphic anomaly equation (HAE) in fibered Calabi-Yau 3-folds and BPS counting. \\label{hae-cy}",
   "public_statement": "Investigate the Holomorphic anomaly equation (HAE) in fibered Calabi-Yau 3-folds and BPS counting. \\label{hae-cy}",
   "evidence": "The canonical AIM record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0081": {
   "statement_status": "exact",
   "original_statement": "When can the elliptic genus be defined for noncompact Calabi-Yau manifolds?",
   "clean_statement": "When can the elliptic genus be defined for noncompact Calabi-Yau manifolds?",
   "public_statement": "When can the elliptic genus be defined for noncompact Calabi-Yau manifolds?",
   "evidence": "The record has no remarks or literature field. The neighboring records ask for an index-theoretic interpretation and for links with modular forms, so the natural reading is deliberately broad: it asks both for hypotheses under which a geometric/index definition exists and for an explanation of the extra choices required by noncompactness. There is no visible OCR corruption. The canonical repository record and neighboring records were checked. The historical AIM URL was requested, but the live page could not be retrieved through the available browser, so the wording above is verified from the canonical local record rather than independently from the live page.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0082": {
   "statement_status": "exact",
   "original_statement": "When can modular forms on Calabi Yau 3-folds moduli spaces be reduced to lower dimensional forms?",
   "clean_statement": "When can modular forms on Calabi Yau 3-folds moduli spaces be reduced to lower dimensional forms?",
   "public_statement": "When can modular forms on Calabi Yau 3-folds moduli spaces be reduced to lower dimensional forms?",
   "evidence": "The canonical record is problem 3.3 in the “Calabi--Yau manifolds” section of the 2013 AIM workshop *Gromov--Witten invariants and number theory*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0083": {
   "statement_status": "exact",
   "original_statement": "Is there an index theoretic interpretation of elliptic genus? When it can, how do we compute it and what is the geometric interpretation of the coefficients? Check that this matches CFT calculations.",
   "clean_statement": "Is there an index theoretic interpretation of elliptic genus? When it can, how do we compute it and what is the geometric interpretation of the coefficients? Check that this matches CFT calculations.",
   "public_statement": "Is there an index theoretic interpretation of elliptic genus? When it can, how do we compute it and what is the geometric interpretation of the coefficients? Check that this matches CFT calculations.",
   "evidence": "The exact JSON record and the neighboring records were inspected. There is no visible OCR corruption. The original HTTP source URL was also requested, but was unavailable through the browser used in this run. The local canonical record is internally complete. The only substantive ambiguity is that “elliptic genus” has several meanings. The Calabi--Yau and CFT context strongly indicates the standard two-variable holomorphic elliptic genus of a compact complex manifold, rather than only the one-variable Ochanine or Witten genus. That is the convention used below. We separately flag where compactness and the Calabi--Yau condition enter.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 82,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0084": {
   "statement_status": "exact",
   "original_statement": "Investigate the role of paramodular groups in counting problems in Calabi Yau 3-folds",
   "clean_statement": "Investigate the role of paramodular groups in counting problems in Calabi Yau 3-folds",
   "public_statement": "Investigate the role of paramodular groups in counting problems in Calabi Yau 3-folds",
   "evidence": "The canonical record is AIM-ANALYTIC_NUMBER_THEORY-0084, source file aim-analytic-number-theory-notes.json, zero-based source index 83, from the AIM workshop *Gromov-Witten invariants and number theory*, section *Calabi-Yau manifolds*, problem 3.4. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 83,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0085": {
   "statement_status": "exact",
   "original_statement": "Can we calculate the leading terms of $F_g^B(q)$ at the orbifold point $z=0$ of the quintic\n\\[\n(\\sum x_i^5 + z \\prod x_i =0 )/ \\mathbb{Z}_5^3?\n\\]",
   "clean_statement": "Can we calculate the leading terms of $F_g^B(q)$ at the orbifold point $z=0$ of the quintic\n\\[\n(\\sum x_i^5 + z \\prod x_i =0 )/ \\mathbb{Z}_5^3?\n\\]",
   "public_statement": "Can we calculate the leading terms of $F_g^B(q)$ at the orbifold point $z=0$ of the quintic\n\\[\n(\\sum x_i^5 + z \\prod x_i =0 )/ \\mathbb{Z}_5^3?\n\\]",
   "evidence": "The exact canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 84,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0086": {
   "statement_status": "exact",
   "original_statement": "Here $h_m$ is the $m$-th Hurwitz class number, i.e. the number of equivalence classes of positive definite binary quadratic forms of discriminant $-m$ with the class containing $x^2+y^2$ weighted by 1/2 and the class containing $x^2 + xy + y^2$ weighted by 1/3. Moreover by convention $h_0=-1/12.$\n\nFor more information in regards to the connection between physics and number theory, see the following work of\n\n Kathrin Bringmann and Ben Kane\nhttp://arxiv.org/pdf/1305.0112v1.pdf\n\nKatrin Bringmann and Sameer Murthy\nhttp://arxiv.org/pdf/1208.3476v2.pdf\n\nKatrin Bringmann and Jan Manschot\nhttp://arxiv.org/pdf/1304.7208v1.pdf\n\nDo the coefficients of $h_m$ have enumerative significance? Can we find $h_m$ for $m \\geq 9$?",
   "clean_statement": "Here $h_m$ is the $m$-th Hurwitz class number, i.e. the number of equivalence classes of positive definite binary quadratic forms of discriminant $-m$ with the class containing $x^2+y^2$ weighted by 1/2 and the class containing $x^2 + xy + y^2$ weighted by 1/3. Moreover by convention $h_0=-1/12.$\n\nFor more information in regards to the connection between physics and number theory, see the following work of\n\n Kathrin Bringmann and Ben Kane\nhttp://arxiv.org/pdf/1305.0112v1.pdf\n\nKatrin Bringmann and Sameer Murthy\nhttp://arxiv.org/pdf/1208.3476v2.pdf\n\nKatrin Bringmann and Jan Manschot\nhttp://arxiv.org/pdf/1304.7208v1.pdf\n\nDo the coefficients of $h_m$ have enumerative significance? Can we find $h_m$ for $m \\geq 9$?",
   "public_statement": "Here $h_m$ is the $m$-th Hurwitz class number, i.e. the number of equivalence classes of positive definite binary quadratic forms of discriminant $-m$ with the class containing $x^2+y^2$ weighted by 1/2 and the class containing $x^2 + xy + y^2$ weighted by 1/3. Moreover by convention $h_0=-1/12.$\n\nFor more information in regards to the connection between physics and number theory, see the following work of\n\n Kathrin Bringmann and Ben Kane\nhttp://arxiv.org/pdf/1305.0112v1.pdf\n\nKatrin Bringmann and Sameer Murthy\nhttp://arxiv.org/pdf/1208.3476v2.pdf\n\nKatrin Bringmann and Jan Manschot\nhttp://arxiv.org/pdf/1304.7208v1.pdf\n\nDo the coefficients of $h_m$ have enumerative significance? Can we find $h_m$ for $m \\geq 9$?",
   "evidence": "The canonical record is item 4.2, “Specific functions,” from the 2013 AIM workshop *Gromov–Witten invariants and number theory*. Its mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 85,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0087": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Does $F= \\sum f_g(\\tau)\\lambda^{2g-2}$ have transformation properties with respect to $\\lambda$?",
   "clean_statement": null,
   "public_statement": "Does $F= \\sum f_g(\\tau)\\lambda^{2g-2}$ have transformation properties with respect to $\\lambda$?",
   "evidence": "The canonical record is problem 4.3, in the “Specific functions” section of the 2013 AIM workshop *Gromov-Witten invariants and number theory*. Its entire mathematical text is",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 86,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0088": {
   "statement_status": "exact",
   "original_statement": "Is $\\sum \\frac{(5n)!}{(n!)^5}z^n$ related to (non-holomorphic) forms? Is it related to automorphic objects?\nDo other solutions to the Picard-Fuchs equation have modular properties?",
   "clean_statement": "Is $\\sum \\frac{(5n)!}{(n!)^5}z^n$ related to (non-holomorphic) forms? Is it related to automorphic objects?\nDo other solutions to the Picard-Fuchs equation have modular properties?",
   "public_statement": "Is $\\sum \\frac{(5n)!}{(n!)^5}z^n$ related to (non-holomorphic) forms? Is it related to automorphic objects?\nDo other solutions to the Picard-Fuchs equation have modular properties?",
   "evidence": "The canonical AIM record (workshop *Gromov--Witten invariants and number theory*, section “Specific functions,” problem 4.4) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 87,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0089": {
   "statement_status": "exact",
   "original_statement": "Do the L-series of mixed mock modular forms have interesting properties?",
   "clean_statement": "Do the L-series of mixed mock modular forms have interesting properties?",
   "public_statement": "Do the L-series of mixed mock modular forms have interesting properties?",
   "evidence": "The exact canonical record is problem 5.1 in the section “Other problems” of the AIM workshop *Gromov–Witten invariants and number theory*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 88,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0090": {
   "statement_status": "exact",
   "original_statement": "Do mixed mock modular forms satisfy differential equations with respect to a modular function?",
   "clean_statement": "Do mixed mock modular forms satisfy differential equations with respect to a modular function?",
   "public_statement": "Do mixed mock modular forms satisfy differential equations with respect to a modular function?",
   "evidence": "The canonical record is problem 5.2 in the AIM workshop list *Gromov--Witten invariants and number theory*, section “Other problems”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 89,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0091": {
   "statement_status": "exact",
   "original_statement": "Determine the connection between Mock modular forms and geometric invariants like Gromov-Witten, Donaldson, and OSV conjecture.",
   "clean_statement": "Determine the connection between Mock modular forms and geometric invariants like Gromov-Witten, Donaldson, and OSV conjecture.",
   "public_statement": "Determine the connection between Mock modular forms and geometric invariants like Gromov-Witten, Donaldson, and OSV conjecture.",
   "evidence": "The canonical record is problem 5.3 in the AIM workshop list *Gromov--Witten invariants and number theory*, section “Other problems.” Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 90,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0092": {
   "statement_status": "exact",
   "original_statement": "Problem 1.1. Given a q-series, determine fast methods to find (heuristically) if it is modular. \n\nThe method should require knowledge of the weight, but not require knowledge of the level or group.",
   "clean_statement": "Problem 1.1. Given a q-series, determine fast methods to find (heuristically) if it is modular.\n\nThe method should require knowledge of the weight, but not require knowledge of the level or group.",
   "public_statement": "Problem 1.1. Given a q-series, determine fast methods to find (heuristically) if it is modular.\n\nThe method should require knowledge of the weight, but not require knowledge of the level or group.",
   "evidence": "The source is the two-page AIM workshop problem list *Mock Modular Forms*, edited by Sharon Anne Garthwaite after the workshop “Mock modular forms in combinatorics and arithmetic geometry,” March 8–12, 2010. The PDF was inspected directly. Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 91,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0093": {
   "statement_status": "exact",
   "original_statement": "Problem 1.2. Find a method to prove modularity directly from the sum ( q-hypergeometric) expansion. \n\nFor motivation, consider the Rogers-Ramanujan identities \n\nG(q) =\n\n> ∞\n\n∑\n\n> n=0\n\nqn2\n\n(q; q)n\n\n= 1\n\n(q; q5)∞(q4; q5)∞,\n\nH(q) =\n\n> ∞\n\n∑\n\n> n=0\n\nqn2+n\n\n(q; q)n\n\n= 1\n\n(q2; q5)∞(q3; q5)∞;\n\nit is apparent from the right hand side that q−1/60 G(q) and q11 /60 H(q) are modular. How can we see this from the left hand side?",
   "clean_statement": "Problem 1.2. Find a method to prove modularity directly from the sum ( q-hypergeometric) expansion.\n\nFor motivation, consider the Rogers-Ramanujan identities\n\nG(q) =\n\n> ∞\n\n∑\n\n> n=0\n\nqn2\n\n(q; q)n\n\n= 1\n\n(q; q5)∞(q4; q5)∞,\n\nH(q) =\n\n> ∞\n\n∑\n\n> n=0\n\nqn2+n\n\n(q; q)n\n\n= 1\n\n(q2; q5)∞(q3; q5)∞;\n\nit is apparent from the right hand side that q−1/60 G(q) and q11 /60 H(q) are modular. How can we see this from the left hand side?",
   "public_statement": "Problem 1.2. Find a method to prove modularity directly from the sum ( q-hypergeometric) expansion.\n\nFor motivation, consider the Rogers-Ramanujan identities\n\nG(q) =\n\n> ∞\n\n∑\n\n> n=0\n\nqn2\n\n(q; q)n\n\n= 1\n\n(q; q5)∞(q4; q5)∞,\n\nH(q) =\n\n> ∞\n\n∑\n\n> n=0\n\nqn2+n\n\n(q; q)n\n\n= 1\n\n(q2; q5)∞(q3; q5)∞;\n\nit is apparent from the right hand side that q−1/60 G(q) and q11 /60 H(q) are modular. How can we see this from the left hand side?",
   "evidence": "The source is the AIM workshop problem list *Mock modular forms in combinatorics and arithmetic geometry* (March 8--12, 2010), Problem 1.2. With",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0094": {
   "statement_status": "exact",
   "original_statement": "Problem 1.3. Is there any kind of relations between the modularity of a series and the expansion in terms of q-orthogonal polynomials?",
   "clean_statement": "Problem 1.3. Is there any kind of relations between the modularity of a series and the expansion in terms of q-orthogonal polynomials?",
   "public_statement": "Problem 1.3. Is there any kind of relations between the modularity of a series and the expansion in terms of q-orthogonal polynomials?",
   "evidence": "The canonical record is Problem 1.3 from the AIM workshop *Mock modular forms in combinatorics and arithmetic geometry* (March 8--12, 2010):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0095": {
   "statement_status": "exact",
   "original_statement": "Problem 1.4. For vector-valued modular forms, do the matrices involved have interesting digi-talization? \n\nFor example, in the Rogers-Ramanujan example, the transformation z 7 → z+1 leads to a diagonal matrix, where as z 7 → − 1/z does not. Is the diagonalization of this interesting? \n\n> 12EDITED BY SHARON ANNE GARTHWAITE\n\n2. B LOCH GROUP METHOD PROBLEMS \n\nWork by Werner Nahm gives us a way to attack modularity questions with Algebraic K-theory. Let B(F) denote the Bloch group for a field F. For more about the Bloch group, see Nahm's article \n\nConformal Field Theory and Torsion Elements of the Bloch Group, which is available on the arXiv. Let A ∈ Mr×r(Q) be a positive definite symmetric matrix. Let B ∈ Qr and C ∈ Q; define \n\nfA,B,C (τ): =\n\n∑ \n\n> n!,..., nr≥0\n\nq1 \n\n> 2\n> −→n A −→nt+B−→n+C\n\n(q)n1 · · · (q)nr,\n\nwhere q:= e2πiτ and (q)m:= (1 − q)(1 − q2) · · · (1 − qm), taking (q)0:= 1.In the case r = 1, Zagier proved that this function is only modular for seven choices of A, B, C,such as (2, 0, −1/60) and (2, 1, 11 /60); the proof involves looking at the behavior of the function as q tends to one. Nahm conjectures that for general r, there is an A for which fA,B,C (τ) is modular if and only if the image of A in the Bloch group is torsion. If such an A exists, there may be multiple choices of \n\nB and C for which fA,B,C (τ) is indeed modular. Motivation for this idea comes from the study of the dilogarithm function and other related functions. For more on this, see Don Zagier's chapter ' The Dilogarithm Function in \"Frontiers in number theory, physics, and geometry II.\" The problems throughout this section assume the notation above.",
   "clean_statement": "Problem 1.4. For vector-valued modular forms, do the matrices involved have interesting digi-talization?\n\nFor example, in the Rogers-Ramanujan example, the transformation z 7 → z+1 leads to a diagonal matrix, where as z 7 → − 1/z does not. Is the diagonalization of this interesting?\n\n> 12EDITED BY SHARON ANNE GARTHWAITE\n\n2. B LOCH GROUP METHOD PROBLEMS\n\nWork by Werner Nahm gives us a way to attack modularity questions with Algebraic K-theory. Let B(F) denote the Bloch group for a field F. For more about the Bloch group, see Nahm's article\n\nConformal Field Theory and Torsion Elements of the Bloch Group, which is available on the arXiv. Let A ∈ Mr×r(Q) be a positive definite symmetric matrix. Let B ∈ Qr and C ∈ Q; define\n\nfA,B,C (τ): =\n\n∑\n\n> n!,..., nr≥0\n\nq1\n\n> 2\n> −→n A −→nt+B−→n+C\n\n(q)n1 · · · (q)nr,\n\nwhere q:= e2πiτ and (q)m:= (1 − q)(1 − q2) · · · (1 − qm), taking (q)0:= 1.In the case r = 1, Zagier proved that this function is only modular for seven choices of A, B, C,such as (2, 0, −1/60) and (2, 1, 11 /60); the proof involves looking at the behavior of the function as q tends to one. Nahm conjectures that for general r, there is an A for which fA,B,C (τ) is modular if and only if the image of A in the Bloch group is torsion. If such an A exists, there may be multiple choices of\n\nB and C for which fA,B,C (τ) is indeed modular. Motivation for this idea comes from the study of the dilogarithm function and other related functions. For more on this, see Don Zagier's chapter ' The Dilogarithm Function in \"Frontiers in number theory, physics, and geometry II.\" The problems throughout this section assume the notation above.",
   "public_statement": "Problem 1.4. For vector-valued modular forms, do the matrices involved have interesting digi-talization?\n\nFor example, in the Rogers-Ramanujan example, the transformation z 7 → z+1 leads to a diagonal matrix, where as z 7 → − 1/z does not. Is the diagonalization of this interesting?\n\n> 12EDITED BY SHARON ANNE GARTHWAITE\n\n2. B LOCH GROUP METHOD PROBLEMS\n\nWork by Werner Nahm gives us a way to attack modularity questions with Algebraic K-theory. Let B(F) denote the Bloch group for a field F. For more about the Bloch group, see Nahm's article\n\nConformal Field Theory and Torsion Elements of the Bloch Group, which is available on the arXiv. Let A ∈ Mr×r(Q) be a positive definite symmetric matrix. Let B ∈ Qr and C ∈ Q; define\n\nfA,B,C (τ): =\n\n∑\n\n> n!,..., nr≥0\n\nq1\n\n> 2\n> −→n A −→nt+B−→n+C\n\n(q)n1 · · · (q)nr,\n\nwhere q:= e2πiτ and (q)m:= (1 − q)(1 − q2) · · · (1 − qm), taking (q)0:= 1.In the case r = 1, Zagier proved that this function is only modular for seven choices of A, B, C,such as (2, 0, −1/60) and (2, 1, 11 /60); the proof involves looking at the behavior of the function as q tends to one. Nahm conjectures that for general r, there is an A for which fA,B,C (τ) is modular if and only if the image of A in the Bloch group is torsion. If such an A exists, there may be multiple choices of\n\nB and C for which fA,B,C (τ) is indeed modular. Motivation for this idea comes from the study of the dilogarithm function and other related functions. For more on this, see Don Zagier's chapter ' The Dilogarithm Function in \"Frontiers in number theory, physics, and geometry II.\" The problems throughout this section assume the notation above.",
   "evidence": "The source is the American Institute of Mathematics problem list from the March 8--12, 2010 workshop *Mock modular forms in combinatorics and arithmetic geometry*, Problem 1.4 on PDF page 1. The first sentence in the PDF is literally:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 94,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0096": {
   "statement_status": "exact",
   "original_statement": "Problem 2.1. Given a particular A, can we bound the number of possible B? Similarly, given a particular A, can we bound the dimension of the vector space determined by the B?\n\nThe bounds may or may not be effective.",
   "clean_statement": "Problem 2.1. Given a particular A, can we bound the number of possible B? Similarly, given a particular A, can we bound the dimension of the vector space determined by the B?\n\nThe bounds may or may not be effective.",
   "public_statement": "Problem 2.1. Given a particular A, can we bound the number of possible B? Similarly, given a particular A, can we bound the dimension of the vector space determined by the B?\n\nThe bounds may or may not be effective.",
   "evidence": "The exact canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0097": {
   "statement_status": "exact",
   "original_statement": "Problem 2.2. Consider the Laplace transform of the q-hypergeometric series above. Note that the Mellin transform will not converge. Look at the properties, in particular the \"jumps;\" what can we say about the dependence on A, B, and C?",
   "clean_statement": "Problem 2.2. Consider the Laplace transform of the q-hypergeometric series above. Note that the Mellin transform will not converge. Look at the properties, in particular the \"jumps;\" what can we say about the dependence on A, B, and C?",
   "public_statement": "Problem 2.2. Consider the Laplace transform of the q-hypergeometric series above. Note that the Mellin transform will not converge. Look at the properties, in particular the \"jumps;\" what can we say about the dependence on A, B, and C?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 96,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0098": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 2.3. Look at explicit examples where r ≥ 2 and fA,B,C (τ) is modular. Can we write these in terms of natural objects? Similarly, can we related them to combinatorial identities; in the case r = 1, the two triples given relate to the Rogers-Ramanujan identities.",
   "clean_statement": null,
   "public_statement": "Problem 2.3. Look at explicit examples where r ≥ 2 and fA,B,C (τ) is modular. Can we write these in terms of natural objects? Similarly, can we related them to combinatorial identities; in the case r = 1, the two triples given relate to the Rogers-Ramanujan identities.",
   "evidence": "The canonical record is Problem 2.3 in the two-page problem list from the AIM workshop *Mock modular forms in combinatorics and arithmetic geometry* (March 8--12, 2010). The record reads, exactly:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 97,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0099": {
   "statement_status": "exact",
   "original_statement": "Problem 2.4. Relate Bailey pair techniques with Bloch group techniques. In particular, develop a dictionary. Perhaps this might might help with the computation of torsion. Similarly, connect orthogonal polynomials to Bloch group techniques.",
   "clean_statement": "Problem 2.4. Relate Bailey pair techniques with Bloch group techniques. In particular, develop a dictionary. Perhaps this might might help with the computation of torsion. Similarly, connect orthogonal polynomials to Bloch group techniques.",
   "public_statement": "Problem 2.4. Relate Bailey pair techniques with Bloch group techniques. In particular, develop a dictionary. Perhaps this might might help with the computation of torsion. Similarly, connect orthogonal polynomials to Bloch group techniques.",
   "evidence": "The canonical record is Problem 2.4 from the AIM workshop list *Mock modular forms in combinatorics and arithmetic geometry*. The source file is `aim-analytic-number-theory-notes.json`, record index 98. The exact extracted statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 98,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0100": {
   "statement_status": "exact",
   "original_statement": "Problem 2.5. Determine how mock modular forms fit into this theory.",
   "clean_statement": "Problem 2.5. Determine how mock modular forms fit into this theory.",
   "public_statement": "Problem 2.5. Determine how mock modular forms fit into this theory.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 99,
   "attempt": 2
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0101": {
   "statement_status": "exact",
   "original_statement": "Problem 2.6. How do q-analogues of the dilogarithm figure into the theory?",
   "clean_statement": "Problem 2.6. How do q-analogues of the dilogarithm figure into the theory?",
   "public_statement": "Problem 2.6. How do q-analogues of the dilogarithm figure into the theory?",
   "evidence": "The exact source question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 100,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0102": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "1. Monday problem session 11.1. Unknown cases of (sub)convexity 11.2. Universality of convexity breaking exponents 32. Tuesday problem session 42.1. General definition and interesting examples of periods 42.2. A convexity bound for periods? 52.3. List of problems to be discussed into small groups 53. Wednesday problem session 54. Thursday problem session 65. Friday problem session 71. M ONDAY PROBLEM SESSION \n\n1.1. Unknown cases of (sub)convexity. \n\n• This point was mentioned by Reznikov. Let π be an automorphic cuspi-dal representation of GL m for m Ê 1 and π′ be an automorphic cuspidal representation of GL n for n Ê 1. Do we know the convexity bound for \n\nL(π×π′, s)? It absolutely converges on ℜs > 1 and thus we know the con-vexity bound in the s-aspect. Do we know the convexity bound in any aspect for any m Ê 1 and any n Ê 1? No! For instance, when m = n = 2, it is known via L2-theory of automorphic forms. Is there a geometric method which could give the result for any m Ê 1 and any n Ê 1? \n\n• These subconvexity problems were mainly suggested by Michel. Let f\n\nbe a Hecke Maass cusp form of level 1 and Laplacian eigenvalue λf:=\n\n1/4 + i t 2 \n\n> f. We want to break the convexity bound for L( f, 1/2 + i t f ) in the spectral aspect namely to find δ > 0 (even microcospic) such that (1.1) L( f, 1/2 + i t f ) ¿ε t 1/4 −δ+ε\n\n> f\n\nfor any ε > 0. About this problem, two questions arose during the dis-cussion. \n\n- Is there a work on this from Luo? \n\n> Date: Version of December 8, 2006. Guillaume.Ricotta@math.u-bordeaux1.fr.\n> 12G. RICOTTA\n\n- Does somebody know an arithmetic application of such unkown convexity bound? Note that this is a case for which the analytic conductor drops since \n\nQ( f, 1/2 + i t f ) = (1 + ∣∣1/2 + i t f − i t f\n\n∣∣) ( 1 + ∣∣1/2 + i t f + i t f\n\n∣∣) ≈ t f.Thus, previous experience suggests that it should be difficult to prove. You can also think of the method of moments to be convinced: you will have to estimate an higher moment if you want to break convexity. An-other example in which the analytic conductor drops is given by (1.2) L( f × g, 1/2 + i t f ) ¿g,ε t 1/2 −δ+ε\n\n> f\n\nfor any ε > 0. Here, g is a fixed Hecke Maass cusp form. Note that if \n\ng is an holomorphic Hecke cusp form of weight 4 and level q, such L-functions already appear in Phillips-Sarnak's deformation theory even if people are more interested in non-vanishing results in this context. Roughly speaking, this theory deals with the deformation of Γ0(q) in the direction given by g. The authors proved that a positive proportion (in a suitable sense) of L( f × g, 1/2 + i t f ) does not vanish which entails that a positive proportion (in the same suitable sense) of f 's is annihilated by such deformation. \n\n• Let f and g be two fixed Hecke Maass forms. We want to break the con-vexity bound for L( f × g, 1/2 + i t ) in the s-aspect namely to find δ > 0such that (1.3) L( f × g, 1/2 + i t ) ¿f,g,ε t 1−δ+ε\n\nfor any ε > 0. Jutila suggested an extra-average over the spectral param-eter t f of f to produce some saving. For instance, he proved with Moto-hashi that \n\nL( f × g, 1/2 + i t ) ¿g,ε\n\n{t 1+ε /√ t f if t 3/2 \n\n> f\n\n¿ t ¿ε t 2−ε \n\n> f,\n\n(t + t f\n\n)2/3 +ε if t ¿ t 3/2 \n\n> f.How can we extend this range? \n\n• Let us talk about the symmetric-square L-function L(Sym 2 f, s) for any Hecke Maass cusp form of level q f, spectral parameter t f and nebenty-pus χf. This L-function is of degree 3. Thus, the subconvexity problem in the s-aspect is given by (1.4) L(Sym 2 f, 1/2 + i t ) ¿f,ε t 3/4 −δ+ε\n\nfor any ε > 0 and for some δ > 0. About the three spectral parameters at infinity of L(Sym 2 f, s), one is of constant size and the two others are of size t f. Thus, the subconvexity problem in the spectral aspect is given by (1.5) L(Sym 2 f, 1/2 + i t ) ¿q f,t,ε t 1/2 −δ+ε\n\n> f\n\nfor any ε > 0 and for some δ > 0. The subconvexity problem in the level aspect is given by (1.6) L(Sym 2 f, 1/2 + i t ) ¿q f,t,ε qε ×\n\n{q1/2 −δ \n\n> f\n\nif χf trivial or non-quadratic, \n\nq1/4 −δ \n\n> f\n\notherwise PROBLEM SESSIONS: \"SUBCONVEXITY BOUNDS FOR L-FUNCTIONS\" 3\n\nfor some δ > 0 and for any ε > 0 since \n\nL( f × f, s) = L(χf, s)L(Sym 2 f, s). Michel said that if you know a subconvexity bound for L(Sym 2 f, s) in the \n\ns-aspect then you know (by Cauchy-Schwarz) a subconvexity bound for \n\nL( f × g, s) when f 6 = g and g is fixed in the s-aspect. He also said that the subconvexity problem for symmetric square L-functions could have some link with metaplectic tools on ˜GL 2 via an integral representation of this L-function in which an Eisenstein series on ˜GL 2 occurs. \n\n• Let us talk about triple L-functions. Let f, g, h some Hecke Maass forms, \n\nf being of level q f, spectral parameter t f, weight k f and the same nota-tions for the two others. We could be interested in the following subcon-vexity problems (1.7) L( f × g × h, 1/2 + i t ) ¿f,g,t,qh k2−δ+ε\n\nwhen h is holomorphic. (1.8) L( f × g × h, 1/2 + i t ) ¿f,g,h,ε t 2−δ+ε.(1.9) L( f × f × h, 1/2) ¿q f,h,ε t 1−δ+ε \n\n> f.This last case is again an example of situation in which the conductor drops. Reznikov said that it is easier and doable to prove (1.10) L( f × g × h, 1/2) ¿ε t 2−δ+ε\n\n> f\n\nwhen f, g and h have some comparable but not equal spectral parame-ters at infinity such that the conductor does not drop. 1.2. Universality of convexity breaking exponents. \n\n• One aims at explaining the apparently unrelated occurences of Weyl's subconvexity exponent given by 1/4(1 − 1/3) and Burgess' subconvexity exponent given by 1/4(1 − 1/4). Remember that Weyl's subconvexity ex-ponent appears \n\n- in the subconvexity problem for GL 2 L-functions L( f, s) in the s and spectral aspect, \n\n- in the subconvexity problem for twisted L-functions L( f × χ, s) in the level aspect ( f and χ are of same level), \n\n- in the subconvexity problem for Rankin-Selberg L-functions L( f ×\n\ng, s) in the t f aspect whereas Burgess' subconvexity exponent appears \n\n- in the subconvexity problem for Dirichlet L-functions L(χ, s) in the level aspect, \n\n- in the subconvexity problem for twisted L-functions L( f × χ, s) in the conductor aspect of the character. \n\n• A natural problem is to find particular L-functions for which we know how to prove better exponents than Weyl and Burgess' ones. Soundarara-jan suggested to look at L(χ, s) when χ is of conductor say 3 n and n goes to infinity by taking advantage of Vinogradov's method. We can talk about subconvexity in the depth aspect in such context. A similar ex-ample is L( f × χ, s) by taking advantage of Graham-Ringrose's method when the modulus of χ is a higly divisible square-free integer. 4 G. RICOTTA \n\n• It is also natural to wonder if there exists some applications which need a better subconvexity exponent than Weyl or Burgess' one. Michel sug-gested an application to André Oort conjecture discovered by Edixhoven. This conjecture asserts that a curve contained in X0(1) × X0(1) which does not project to X0(1) itself and contains infinitely many CM points is the modular curve itself (embedded in the product as a graph of Hecke correspondence). Assuming GRH for quadratic imaginary fields, Edix-hoven \"proved\" this conjecture. The main hole for an unconditional proof being that he needs to know that the number of primes less than log 2 (|d|) log 22 (|d|) which are split in the quadratic imaginary field of dis-criminant d tends to infinity with d. Fouvry, applying a result of Linnik and Vinogradov, noticed that the number of primes less than |d|1/4 +ε\n\nwhich are split in the quadratic imaginary field of discriminant d tends to infinity with d. Improving Burgess' bound for character sums will im-prove the previous range. Studying these small split primes is a challeng-ing problem because it may occur when one wants to build an efficient amplifier. For instance, Duke-Friedlander-Iwaniec faced this problem when they tried to prove subconvexity bound for class group L-functions without appealing to the spectral theory of automorphic forms. 2. T UESDAY PROBLEM SESSION \n\n2.1. General definition and interesting examples of periods. Lindenstrauss gave the following general definition of period. Let G be a group and H be a subgroup of G. The general space is given by \n\nX:= G(Q) ∖G(AQ). To any automorphic form f on X (a smooth function on X which belongs to the space of an automorphic representation), we define some periods by \n\n∫ \n\n> H(Q)∖H(AQ)\n\nf (h)g (h)d h\n\nfor any automorphic form g on H (Q) ∖H (AQ). Then, people gave fundamental examples of periods. \n\nExample 1: Fourier coefficients of cusp forms on GL 2\n\nHere, G = GL 2 and H is the unipotent subgroup of G namely \n\nH:=\n\n{( 1 x\n\n0 1\n\n), x ∈ R\n\n}.For any g in GL 2(Q) ∖GL 2(AQ), the period \n\n∫\n\n> R\n\nf\n\n(\n\ng\n\n(1 t\n\n0 1\n\n)) \n\ne(−nt )d t\n\nis directly linked to the n-th Fourier coefficient of f for any n ∈ Z.\n\nExample 2: Special values of GL 2 L-functions Here, F is a number field, G = GL 2 and H is the torus subgroup of G namely \n\nH:=\n\n{( y 00 1\n\n), y > 0\n\n}.PROBLEM SESSIONS: \"SUBCONVEXITY BOUNDS FOR L-FUNCTIONS\" 5\n\nThe period ∫ \n\n> F×∖A×\n> F\n\nf\n\n(( y 00 1\n\n)) \n\nd× y\n\nis directly linked to the special value L( f, 1/2) up to Γ-factors. \n\nExample 3: Triple product formula Here, G = GL 2 × GL 2 and H = GL 2 is a subgroup of G via the diagonal embed-ding.Thus, an automorphic form F on G is a pair of automorphic forms f1 and \n\nf2 on GL 2. If h ∈ H then F (h) = f1(h) f2(h). The period \n\n∫\n\n> GL 2(Q)∖GL 2(AQ)\n\nf1(h) f2(h) f3(h)d h\n\nis linked (up to some factors) to the special value L( f1 × f2 × f3, 1/2). 2.2. A convexity bound for periods? The discussion was about understanding what could be a general bound for period which specializes to (sub)convex bounds for L-functions. Lindenstrauss said that a convex bound for period should be a bound that comes from general harmonic analysis results (it is the case for L-functions). Then, Miller defined an automorphic period which is our previous geometric period up to some factors (which are sometimes special values of L-functions). Thus, bounding these factors turns out to bounding both (automor-phic and geometric) periods. Another problem was trying to understand if there is a canonical way to define a period such that it does not depend on the choice of test vectors in the space of representations that occur. 2.3. List of problems to be discussed into small groups. \n\n• Subconvexity problems when the analytic conductor drops (in particu-lar for the symmetric square L-function). \n\n• Improving Weyl's exponent in various examples. In particular, investi-gate Soundararajan's idea about Dirichlet characters of highly compos-ite moduli or try to prove some explicit spectral decomposition of shifted convolution sums (Harcos' suggestion). \n\n• Develop explicit Good-Motoashi type identities. \n\n• Develop associativity type identities in the conductor aspect. \n\n• Formulate clearly period problems in relation to L-functions and repre-sentation theory. \n\n• Quantitative equidistribution results to prove subconvexity bounds. 3. W EDNESDAY PROBLEM SESSION \n\nHere is an incomplete list of the problems which could be understood in a close future. \n\n• Silberman suggested to try to prove a strong hybrid subconvexity bound for standard L-functions on GL n namely try to find δ > 0 such that \n\nL(π, 1/2 + i t ) ¿ε Q(π, 1/2 + i t )1/4 −δ+ε\n\nfor any ε > 0. According to Garett, the Diaconu-Garrett-Goldfeld ex-tension of Good's method to GL n xGL n−1 may have something to offer in this direction.Venkatesh also suggested to try to find some applica-tions of such subconvexity bound before proving them. For instance, 6 G. RICOTTA \n\nit can be interesting to clarify the links between subconvexity problems and equidistribution problems in higher rank for classical groups. Duke also had in mind to extract explicit information from higher rank Artin \n\nL-functions. \n\n• About periods, a very important point is to find an heuristic way of pre-dicting what is an analogue of convexity bounds and Lindelöf hypoth-esis for periods. Also, prove some bounds for periods which special-ize to subconvexity bounds for L-functions. Silberman mentioned the problem of predicting what could be the expected bound for the infinite norm of general automorphic forms when the spectral parameters go to infinity. \n\n• Jutila suggested to prove some Ω-results about the error term which oc-curs in some moments of families of L-functions. For instance, \n\n∫ \n\n> tÉT\n\n∣∣L( f, 1/2 + i t )∣∣2 dt = Main( T ) + Error( T )where Error( T ) = Ω(pT ) is expected to hold for any GL 2-automorphic form. What could be some applications of such Ω-results? \n\n• Venkatesh suggested to find a way to guess when it is possible to prove some asymptotic formula for some moment given by \n\n∑ \n\n> f∈F\n\nL( f, 1/2) where as usual the conductor of each L-function of F is of size almost constant say Q(F ) in the logarithmic scale and Q(F ) → +∞. If 4 log |F | > log Q(F )then we generally can prove an asymptotic formula. At the moment, the record is 6 log |F | = log Q(F )in the work of Conrey and Iwaniec on the cubic moment of automorphic \n\nL-functions. Can we do better? \n\n• Soundararajan asked if it is possible to prove some subconvexity bound in the critical strip but outside the critical line and eventually near the edge of the critical strip. One known instance is the ζ function in the \n\ns-aspect near ℜs = 1 via Vinogradov. \n\n• Michel asked about an analogy of (sub)convexity for p-adic L-functions. The answer could be some integrality property (Mazur, Prasad,...). 4. T HURSDAY PROBLEM SESSION \n\n• Venkatesh convinced us that many equidistribution results are useful to prove asymptotic formula for moments of L-functions with a power sav-ing in the error term if the suitable equidistribution results are quantita-tive ones. He illustrated this by two GL 2-examples namely \n\n∫ +T\n\n> −T\n\n∣∣L( f, 1/2 + i t )∣∣2 dt\n\nand ×∑ \n\n> χmod ( q)\n\n∫\n\n> R\n\n∣∣L( f × χ, 1/2 + i t )∣∣2 dt.PROBLEM SESSIONS: \"SUBCONVEXITY BOUNDS FOR L-FUNCTIONS\" 7\n\n• In addition, he mentioned what could be the obstacles to do the same in higher rank cases. On one hand, it is very hard to prove an inte-gral representation (period) of L-functions which occured in higher rank since the archimedean computation may be very far away from obvious. According to Garett, the Diaconu-Garrett-Goldfeld extension of Good's spectral-theory-based method to GL n xGL n−1 illustrates the complica-tions at archimedean places. On the other hand, it is necessary to anal-yse such integral representation via ergodic theory or spectral methods. Two difficult instances are given by On × On−1 and GL n ×GL n−1.\n\n• An application of subconvexity bounds for GL n may be some informa-tion about the 2 n-moment of the Riemann ζ function. For instance, some Motohashi type formula make the link between GL 3 and |ζ|6. We also have to mention the work of Conrey and Iwaniec. 5. F RIDAY PROBLEM SESSION \n\n• There will be a website dedicated to subconvexity for L-functions. Peo-ple agreed that it should contain references to important results, a list of people with their current attempts and previous results also. \n\n• Venkatesh put the stress on the subconvexity problem in higher-rank cases. Few methods are known which have bearing on it. After Venkatesh-Lindenstrauss and Bernstein-Reznikoff treatments of triple products, the exceptions are works in progress mentionned by Garett: Venkatesh's ap-plications of ergodic theoretic ideas coming from Ratner and Clozel, and Diaconu-Garrett-Goldfeld's GL n version of an old method of Good. Venkatesh suggested that a first higher rank example to undertake should be GL 3 × GL 2 with f3 on GL 3 is fixed and f2 is varying. In order to get some insight into that, it would be profitable for everybody that classi-cal analytic number theorists try to understand the case when f3 is an Eisenstein series namely \n\n∑ \n\n> fHecke-Maass of level 1 and eigenvalue 1/4 +t2\n> f\n> tfvT\n\n∫ \n\n> tvT\n\n∣∣L( f, 1/2 + i t )∣∣6 dt.Note that the size of the family is about T 3 whereas the size of the an-alytic conductor is about T 12 which reveals the level of difficulty. Also, people should understand where the GL 3-theory occurs in the analytic analysis.",
   "clean_statement": null,
   "public_statement": "1. Monday problem session 11.1. Unknown cases of (sub)convexity 11.2. Universality of convexity breaking exponents 32. Tuesday problem session 42.1. General definition and interesting examples of periods 42.2. A convexity bound for periods? 52.3. List of problems to be discussed into small groups 53. Wednesday problem session 54. Thursday problem session 65. Friday problem session 71. M ONDAY PROBLEM SESSION\n\n1.1. Unknown cases of (sub)convexity.\n\n• This point was mentioned by Reznikov. Let π be an automorphic cuspi-dal representation of GL m for m Ê 1 and π′ be an automorphic cuspidal representation of GL n for n Ê 1. Do we know the convexity bound for\n\nL(π×π′, s)? It absolutely converges on ℜs > 1 and thus we know the con-vexity bound in the s-aspect. Do we know the convexity bound in any aspect for any m Ê 1 and any n Ê 1? No! For instance, when m = n = 2, it is known via L2-theory of automorphic forms. Is there a geometric method which could give the result for any m Ê 1 and any n Ê 1?\n\n• These subconvexity problems were mainly suggested by Michel. Let f\n\nbe a Hecke Maass cusp form of level 1 and Laplacian eigenvalue λf:=\n\n1/4 + i t 2\n\n> f. We want to break the convexity bound for L( f, 1/2 + i t f ) in the spectral aspect namely to find δ > 0 (even microcospic) such that (1.1) L( f, 1/2 + i t f ) ¿ε t 1/4 −δ+ε\n\n> f\n\nfor any ε > 0. About this problem, two questions arose during the dis-cussion.\n\n- Is there a work on this from Luo?\n\n> Date: Version of December 8, 2006. Guillaume.Ricotta@math.u-bordeaux1.fr.\n> 12G. RICOTTA\n\n- Does somebody know an arithmetic application of such unkown convexity bound? Note that this is a case for which the analytic conductor drops since\n\nQ( f, 1/2 + i t f ) = (1 + ∣∣1/2 + i t f − i t f\n\n∣∣) ( 1 + ∣∣1/2 + i t f + i t f\n\n∣∣) ≈ t f.Thus, previous experience suggests that it should be difficult to prove. You can also think of the method of moments to be convinced: you will have to estimate an higher moment if you want to break convexity. An-other example in which the analytic conductor drops is given by (1.2) L( f × g, 1/2 + i t f ) ¿g,ε t 1/2 −δ+ε\n\n> f\n\nfor any ε > 0. Here, g is a fixed Hecke Maass cusp form. Note that if\n\ng is an holomorphic Hecke cusp form of weight 4 and level q, such L-functions already appear in Phillips-Sarnak's deformation theory even if people are more interested in non-vanishing results in this context. Roughly speaking, this theory deals with the deformation of Γ0(q) in the direction given by g. The authors proved that a positive proportion (in a suitable sense) of L( f × g, 1/2 + i t f ) does not vanish which entails that a positive proportion (in the same suitable sense) of f 's is annihilated by such deformation.\n\n• Let f and g be two fixed Hecke Maass forms. We want to break the con-vexity bound for L( f × g, 1/2 + i t ) in the s-aspect namely to find δ > 0such that (1.3) L( f × g, 1/2 + i t ) ¿f,g,ε t 1−δ+ε\n\nfor any ε > 0. Jutila suggested an extra-average over the spectral param-eter t f of f to produce some saving. For instance, he proved with Moto-hashi that\n\nL( f × g, 1/2 + i t ) ¿g,ε\n\n{t 1+ε /√ t f if t 3/2\n\n> f\n\n¿ t ¿ε t 2−ε\n\n> f,\n\n(t + t f\n\n)2/3 +ε if t ¿ t 3/2\n\n> f.How can we extend this range?\n\n• Let us talk about the symmetric-square L-function L(Sym 2 f, s) for any Hecke Maass cusp form of level q f, spectral parameter t f and nebenty-pus χf. This L-function is of degree 3. Thus, the subconvexity problem in the s-aspect is given by (1.4) L(Sym 2 f, 1/2 + i t ) ¿f,ε t 3/4 −δ+ε\n\nfor any ε > 0 and for some δ > 0. About the three spectral parameters at infinity of L(Sym 2 f, s), one is of constant size and the two others are of size t f. Thus, the subconvexity problem in the spectral aspect is given by (1.5) L(Sym 2 f, 1/2 + i t ) ¿q f,t,ε t 1/2 −δ+ε\n\n> f\n\nfor any ε > 0 and for some δ > 0. The subconvexity problem in the level aspect is given by (1.6) L(Sym 2 f, 1/2 + i t ) ¿q f,t,ε qε ×\n\n{q1/2 −δ\n\n> f\n\nif χf trivial or non-quadratic,\n\nq1/4 −δ\n\n> f\n\notherwise PROBLEM SESSIONS: \"SUBCONVEXITY BOUNDS FOR L-FUNCTIONS\" 3\n\nfor some δ > 0 and for any ε > 0 since\n\nL( f × f, s) = L(χf, s)L(Sym 2 f, s). Michel said that if you know a subconvexity bound for L(Sym 2 f, s) in the\n\ns-aspect then you know (by Cauchy-Schwarz) a subconvexity bound for\n\nL( f × g, s) when f 6 = g and g is fixed in the s-aspect. He also said that the subconvexity problem for symmetric square L-functions could have some link with metaplectic tools on ˜GL 2 via an integral representation of this L-function in which an Eisenstein series on ˜GL 2 occurs.\n\n• Let us talk about triple L-functions. Let f, g, h some Hecke Maass forms,\n\nf being of level q f, spectral parameter t f, weight k f and the same nota-tions for the two others. We could be interested in the following subcon-vexity problems (1.7) L( f × g × h, 1/2 + i t ) ¿f,g,t,qh k2−δ+ε\n\nwhen h is holomorphic. (1.8) L( f × g × h, 1/2 + i t ) ¿f,g,h,ε t 2−δ+ε.(1.9) L( f × f × h, 1/2) ¿q f,h,ε t 1−δ+ε\n\n> f.This last case is again an example of situation in which the conductor drops. Reznikov said that it is easier and doable to prove (1.10) L( f × g × h, 1/2) ¿ε t 2−δ+ε\n\n> f\n\nwhen f, g and h have some comparable but not equal spectral parame-ters at infinity such that the conductor does not drop. 1.2. Universality of convexity breaking exponents.\n\n• One aims at explaining the apparently unrelated occurences of Weyl's subconvexity exponent given by 1/4(1 − 1/3) and Burgess' subconvexity exponent given by 1/4(1 − 1/4). Remember that Weyl's subconvexity ex-ponent appears\n\n- in the subconvexity problem for GL 2 L-functions L( f, s) in the s and spectral aspect,\n\n- in the subconvexity problem for twisted L-functions L( f × χ, s) in the level aspect ( f and χ are of same level),\n\n- in the subconvexity problem for Rankin-Selberg L-functions L( f ×\n\ng, s) in the t f aspect whereas Burgess' subconvexity exponent appears\n\n- in the subconvexity problem for Dirichlet L-functions L(χ, s) in the level aspect,\n\n- in the subconvexity problem for twisted L-functions L( f × χ, s) in the conductor aspect of the character.\n\n• A natural problem is to find particular L-functions for which we know how to prove better exponents than Weyl and Burgess' ones. Soundarara-jan suggested to look at L(χ, s) when χ is of conductor say 3 n and n goes to infinity by taking advantage of Vinogradov's method. We can talk about subconvexity in the depth aspect in such context. A similar ex-ample is L( f × χ, s) by taking advantage of Graham-Ringrose's method when the modulus of χ is a higly divisible square-free integer. 4 G. RICOTTA\n\n• It is also natural to wonder if there exists some applications which need a better subconvexity exponent than Weyl or Burgess' one. Michel sug-gested an application to André Oort conjecture discovered by Edixhoven. This conjecture asserts that a curve contained in X0(1) × X0(1) which does not project to X0(1) itself and contains infinitely many CM points is the modular curve itself (embedded in the product as a graph of Hecke correspondence). Assuming GRH for quadratic imaginary fields, Edix-hoven \"proved\" this conjecture. The main hole for an unconditional proof being that he needs to know that the number of primes less than log 2 (|d|) log 22 (|d|) which are split in the quadratic imaginary field of dis-criminant d tends to infinity with d. Fouvry, applying a result of Linnik and Vinogradov, noticed that the number of primes less than |d|1/4 +ε\n\nwhich are split in the quadratic imaginary field of discriminant d tends to infinity with d. Improving Burgess' bound for character sums will im-prove the previous range. Studying these small split primes is a challeng-ing problem because it may occur when one wants to build an efficient amplifier. For instance, Duke-Friedlander-Iwaniec faced this problem when they tried to prove subconvexity bound for class group L-functions without appealing to the spectral theory of automorphic forms. 2. T UESDAY PROBLEM SESSION\n\n2.1. General definition and interesting examples of periods. Lindenstrauss gave the following general definition of period. Let G be a group and H be a subgroup of G. The general space is given by\n\nX:= G(Q) ∖G(AQ). To any automorphic form f on X (a smooth function on X which belongs to the space of an automorphic representation), we define some periods by\n\n∫\n\n> H(Q)∖H(AQ)\n\nf (h)g (h)d h\n\nfor any automorphic form g on H (Q) ∖H (AQ). Then, people gave fundamental examples of periods.\n\nExample 1: Fourier coefficients of cusp forms on GL 2\n\nHere, G = GL 2 and H is the unipotent subgroup of G namely\n\nH:=\n\n{( 1 x\n\n0 1\n\n), x ∈ R\n\n}.For any g in GL 2(Q) ∖GL 2(AQ), the period\n\n∫\n\n> R\n\nf\n\n(\n\ng\n\n(1 t\n\n0 1\n\n))\n\ne(−nt )d t\n\nis directly linked to the n-th Fourier coefficient of f for any n ∈ Z.\n\nExample 2: Special values of GL 2 L-functions Here, F is a number field, G = GL 2 and H is the torus subgroup of G namely\n\nH:=\n\n{( y 00 1\n\n), y > 0\n\n}.PROBLEM SESSIONS: \"SUBCONVEXITY BOUNDS FOR L-FUNCTIONS\" 5\n\nThe period ∫\n\n> F×∖A×\n> F\n\nf\n\n(( y 00 1\n\n))\n\nd× y\n\nis directly linked to the special value L( f, 1/2) up to Γ-factors.\n\nExample 3: Triple product formula Here, G = GL 2 × GL 2 and H = GL 2 is a subgroup of G via the diagonal embed-ding.Thus, an automorphic form F on G is a pair of automorphic forms f1 and\n\nf2 on GL 2. If h ∈ H then F (h) = f1(h) f2(h). The period\n\n∫\n\n> GL 2(Q)∖GL 2(AQ)\n\nf1(h) f2(h) f3(h)d h\n\nis linked (up to some factors) to the special value L( f1 × f2 × f3, 1/2). 2.2. A convexity bound for periods? The discussion was about understanding what could be a general bound for period which specializes to (sub)convex bounds for L-functions. Lindenstrauss said that a convex bound for period should be a bound that comes from general harmonic analysis results (it is the case for L-functions). Then, Miller defined an automorphic period which is our previous geometric period up to some factors (which are sometimes special values of L-functions). Thus, bounding these factors turns out to bounding both (automor-phic and geometric) periods. Another problem was trying to understand if there is a canonical way to define a period such that it does not depend on the choice of test vectors in the space of representations that occur. 2.3. List of problems to be discussed into small groups.\n\n• Subconvexity problems when the analytic conductor drops (in particu-lar for the symmetric square L-function).\n\n• Improving Weyl's exponent in various examples. In particular, investi-gate Soundararajan's idea about Dirichlet characters of highly compos-ite moduli or try to prove some explicit spectral decomposition of shifted convolution sums (Harcos' suggestion).\n\n• Develop explicit Good-Motoashi type identities.\n\n• Develop associativity type identities in the conductor aspect.\n\n• Formulate clearly period problems in relation to L-functions and repre-sentation theory.\n\n• Quantitative equidistribution results to prove subconvexity bounds. 3. W EDNESDAY PROBLEM SESSION\n\nHere is an incomplete list of the problems which could be understood in a close future.\n\n• Silberman suggested to try to prove a strong hybrid subconvexity bound for standard L-functions on GL n namely try to find δ > 0 such that\n\nL(π, 1/2 + i t ) ¿ε Q(π, 1/2 + i t )1/4 −δ+ε\n\nfor any ε > 0. According to Garett, the Diaconu-Garrett-Goldfeld ex-tension of Good's method to GL n xGL n−1 may have something to offer in this direction.Venkatesh also suggested to try to find some applica-tions of such subconvexity bound before proving them. For instance, 6 G. RICOTTA\n\nit can be interesting to clarify the links between subconvexity problems and equidistribution problems in higher rank for classical groups. Duke also had in mind to extract explicit information from higher rank Artin\n\nL-functions.\n\n• About periods, a very important point is to find an heuristic way of pre-dicting what is an analogue of convexity bounds and Lindelöf hypoth-esis for periods. Also, prove some bounds for periods which special-ize to subconvexity bounds for L-functions. Silberman mentioned the problem of predicting what could be the expected bound for the infinite norm of general automorphic forms when the spectral parameters go to infinity.\n\n• Jutila suggested to prove some Ω-results about the error term which oc-curs in some moments of families of L-functions. For instance,\n\n∫\n\n> tÉT\n\n∣∣L( f, 1/2 + i t )∣∣2 dt = Main( T ) + Error( T )where Error( T ) = Ω(pT ) is expected to hold for any GL 2-automorphic form. What could be some applications of such Ω-results?\n\n• Venkatesh suggested to find a way to guess when it is possible to prove some asymptotic formula for some moment given by\n\n∑\n\n> f∈F\n\nL( f, 1/2) where as usual the conductor of each L-function of F is of size almost constant say Q(F ) in the logarithmic scale and Q(F ) → +∞. If 4 log |F | > log Q(F )then we generally can prove an asymptotic formula. At the moment, the record is 6 log |F | = log Q(F )in the work of Conrey and Iwaniec on the cubic moment of automorphic\n\nL-functions. Can we do better?\n\n• Soundararajan asked if it is possible to prove some subconvexity bound in the critical strip but outside the critical line and eventually near the edge of the critical strip. One known instance is the ζ function in the\n\ns-aspect near ℜs = 1 via Vinogradov.\n\n• Michel asked about an analogy of (sub)convexity for p-adic L-functions. The answer could be some integrality property (Mazur, Prasad,...). 4. T HURSDAY PROBLEM SESSION\n\n• Venkatesh convinced us that many equidistribution results are useful to prove asymptotic formula for moments of L-functions with a power sav-ing in the error term if the suitable equidistribution results are quantita-tive ones. He illustrated this by two GL 2-examples namely\n\n∫ +T\n\n> −T\n\n∣∣L( f, 1/2 + i t )∣∣2 dt\n\nand ×∑\n\n> χmod ( q)\n\n∫\n\n> R\n\n∣∣L( f × χ, 1/2 + i t )∣∣2 dt.PROBLEM SESSIONS: \"SUBCONVEXITY BOUNDS FOR L-FUNCTIONS\" 7\n\n• In addition, he mentioned what could be the obstacles to do the same in higher rank cases. On one hand, it is very hard to prove an inte-gral representation (period) of L-functions which occured in higher rank since the archimedean computation may be very far away from obvious. According to Garett, the Diaconu-Garrett-Goldfeld extension of Good's spectral-theory-based method to GL n xGL n−1 illustrates the complica-tions at archimedean places. On the other hand, it is necessary to anal-yse such integral representation via ergodic theory or spectral methods. Two difficult instances are given by On × On−1 and GL n ×GL n−1.\n\n• An application of subconvexity bounds for GL n may be some informa-tion about the 2 n-moment of the Riemann ζ function. For instance, some Motohashi type formula make the link between GL 3 and |ζ|6. We also have to mention the work of Conrey and Iwaniec. 5. F RIDAY PROBLEM SESSION\n\n• There will be a website dedicated to subconvexity for L-functions. Peo-ple agreed that it should contain references to important results, a list of people with their current attempts and previous results also.\n\n• Venkatesh put the stress on the subconvexity problem in higher-rank cases. Few methods are known which have bearing on it. After Venkatesh-Lindenstrauss and Bernstein-Reznikoff treatments of triple products, the exceptions are works in progress mentionned by Garett: Venkatesh's ap-plications of ergodic theoretic ideas coming from Ratner and Clozel, and Diaconu-Garrett-Goldfeld's GL n version of an old method of Good. Venkatesh suggested that a first higher rank example to undertake should be GL 3 × GL 2 with f3 on GL 3 is fixed and f2 is varying. In order to get some insight into that, it would be profitable for everybody that classi-cal analytic number theorists try to understand the case when f3 is an Eisenstein series namely\n\n∑\n\n> fHecke-Maass of level 1 and eigenvalue 1/4 +t2\n> f\n> tfvT\n\n∫\n\n> tvT\n\n∣∣L( f, 1/2 + i t )∣∣6 dt.Note that the size of the family is about T 3 whereas the size of the an-alytic conductor is about T 12 which reveals the level of difficulty. Also, people should understand where the GL 3-theory occurs in the analytic analysis.",
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   "original_statement": "1. Evidence in support of the Elliott-Halberstam conjecture. \n\n• J.B. Conrey asked what evidence exists in support of the Elliott-Halberstam conjecture (EH). He also suggested that the search for further such evidence may shed some light on the conjecture or even lead to the resolution of an \"easy\" case, which in turn would su ffi ce to establish the existence of infinitely many bounded gaps between primes by the GPY method. Similar comments were also made by G. Harcos in his contributed statement. \n\n• R.C. Vaughan observed that the Barban-Davenport-Halberstam theorem is consistent with EH. \n\n• A. Granville noted that the Cram´ er probabilistic model also supports EH. \n\n• In an earlier lecture on the Bombieri-Vinogradov theorem and the large sieve, R.C. Vaughan had remarked that progress toward EH would most likely exploit cancellation among the values of ¯ χ(a) in the formula \n\nψ(x; q, a) = 1\n\nφ(q)\n\n∑ \n\n> χmod q\n\n¯χ(a)ψ(x, χ ). (1) J.B. Conrey asked whether known results on the asymptotic of the sum \n\n∑∗ \n\n> χmod q\n\nχ(a)|L(1 /2, χ )|2\n\nsuggest that the desired cancellation in (1) may be nonexistent. \n\n• K. Soundararajan noted that the results mentioned by Conrey can be explained via the Euler product of L(s, χ )and that there is no such phenomenon in the case of ψ(x, χ ). Therefore, it seems that Vaughan's remark remains valid until further notice.",
   "clean_statement": "1. Evidence in support of the Elliott-Halberstam conjecture.\n\n• J.B. Conrey asked what evidence exists in support of the Elliott-Halberstam conjecture (EH). He also suggested that the search for further such evidence may shed some light on the conjecture or even lead to the resolution of an \"easy\" case, which in turn would su ffi ce to establish the existence of infinitely many bounded gaps between primes by the GPY method. Similar comments were also made by G. Harcos in his contributed statement.\n\n• R.C. Vaughan observed that the Barban-Davenport-Halberstam theorem is consistent with EH.\n\n• A. Granville noted that the Cram´ er probabilistic model also supports EH.\n\n• In an earlier lecture on the Bombieri-Vinogradov theorem and the large sieve, R.C. Vaughan had remarked that progress toward EH would most likely exploit cancellation among the values of ¯ χ(a) in the formula\n\nψ(x; q, a) = 1\n\nφ(q)\n\n∑\n\n> χmod q\n\n¯χ(a)ψ(x, χ ). (1) J.B. Conrey asked whether known results on the asymptotic of the sum\n\n∑∗\n\n> χmod q\n\nχ(a)|L(1 /2, χ )|2\n\nsuggest that the desired cancellation in (1) may be nonexistent.\n\n• K. Soundararajan noted that the results mentioned by Conrey can be explained via the Euler product of L(s, χ )and that there is no such phenomenon in the case of ψ(x, χ ). Therefore, it seems that Vaughan's remark remains valid until further notice.",
   "public_statement": "1. Evidence in support of the Elliott-Halberstam conjecture.\n\n• J.B. Conrey asked what evidence exists in support of the Elliott-Halberstam conjecture (EH). He also suggested that the search for further such evidence may shed some light on the conjecture or even lead to the resolution of an \"easy\" case, which in turn would su ffi ce to establish the existence of infinitely many bounded gaps between primes by the GPY method. Similar comments were also made by G. Harcos in his contributed statement.\n\n• R.C. Vaughan observed that the Barban-Davenport-Halberstam theorem is consistent with EH.\n\n• A. Granville noted that the Cram´ er probabilistic model also supports EH.\n\n• In an earlier lecture on the Bombieri-Vinogradov theorem and the large sieve, R.C. Vaughan had remarked that progress toward EH would most likely exploit cancellation among the values of ¯ χ(a) in the formula\n\nψ(x; q, a) = 1\n\nφ(q)\n\n∑\n\n> χmod q\n\n¯χ(a)ψ(x, χ ). (1) J.B. Conrey asked whether known results on the asymptotic of the sum\n\n∑∗\n\n> χmod q\n\nχ(a)|L(1 /2, χ )|2\n\nsuggest that the desired cancellation in (1) may be nonexistent.\n\n• K. Soundararajan noted that the results mentioned by Conrey can be explained via the Euler product of L(s, χ )and that there is no such phenomenon in the case of ψ(x, χ ). Therefore, it seems that Vaughan's remark remains valid until further notice.",
   "evidence": "The source is Problem 1 in Angel Kumchev's notes from the December 2005 ARCC workshop *Gaps Between Primes*. The original PDF was inspected directly. The extracted record is faithful in substance, but its PDF-to-text conversion split ligatures and displayed sums vertically. With only those typographical defects repaired, the key passage is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 102,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0104": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "2. On the large sieve, I. \n\n• H. Helfgott asked whether the standard large sieve inequality for the exponential sum \n\nS (α) =\n\n> M+N\n\n∑\n\n> n=M+1\n\nane(αn)is best possible for coe ffi cient sequences with an ∈ { 0, 1}. He noted that the usual proof that the large sieve is essentially best possible uses a sequence that does not belong to this class. \n\n• R.C. Vaughan said that the answer to this question is most likely in the a ffi rmative.",
   "clean_statement": "2. On the large sieve, I.\n\n• H. Helfgott asked whether the standard large sieve inequality for the exponential sum\n\nS (α) =\nM+N\n\n∑\nn=M+1\n\n, and the split best possible for coe ffi cient sequences with an ∈ { 0, 1}. He noted that the usual proof that the large sieve is essentially best possible uses a sequence that does not belong to this class.\n\n• R.C. Vaughan said that the answer to this question is most likely in the a ffi rmative.",
   "public_statement": "2. On the large sieve, I.\n\n• H. Helfgott asked whether the standard large sieve inequality for the exponential sum\n\nS (α) =\n\n> M+N\n\n∑\n\n> n=M+1\n\nane(αn)is best possible for coe ffi cient sequences with an ∈ { 0, 1}. He noted that the usual proof that the large sieve is essentially best possible uses a sequence that does not belong to this class.\n\n• R.C. Vaughan said that the answer to this question is most likely in the a ffi rmative.",
   "evidence": "A second plausible reading is the additive arithmetic/Farey form \\[ \\mathcal L_Q(a):= \\sum_{q\\leq Q}\\ \\sum_{\\substack{1\\leq h\\leq q\\\\(h,q)=1}} \\left|S\\!\\left(\\frac hq\\right)\\right|^2 \\leq (N-1+Q^2)\\sum_n|a_n|^2, \\tag{FLS} \\] where \\(h/q\\) is viewed modulo one and \\(q=1,h=1\\) supplies the point \\(0\\). This follows from (LS), since two distinct reduced fractions of denominator at most \\(Q\\) are at circular distance at least \\(Q^{-2}\\). It is often written with \\(N+Q^2\\). Both readings are treated below; their sharpness questions are not identical.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 103,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0105": {
   "statement_status": "unrecoverable",
   "original_statement": "3. On the large sieve, II. Y. Motohashi proposed to try to improve the large sieve by restricting the moduli. He mentioned that D. Wolke has results in that direction for prime moduli. 14. |tζ(1 + it )| ≥ 1.\n\n• J. Pintz asked whether it is true that \n\n|tζ(1 + it )| ≥ 1 for all t ∈ R.\n\nHe noted that this inequality holds when |t| ≥ t0 and when |t| ≤ δ. A complete proof would simplify the GPY method at certain points. \n\n• R.C. Vaughan recalled being asked a similar question in the past and that he was able to answer it in the affi rmative. He said that he thinks that he has the solution written somewhere in his files.",
   "clean_statement": null,
   "public_statement": "3. On the large sieve, II. Y. Motohashi proposed to try to improve the large sieve by restricting the moduli. He mentioned that D. Wolke has results in that direction for prime moduli. 14. |tζ(1 + it )| ≥ 1.\n\n• J. Pintz asked whether it is true that\n\n|tζ(1 + it )| ≥ 1 for all t ∈ R.\n\nHe noted that this inequality holds when |t| ≥ t0 and when |t| ≤ δ. A complete proof would simplify the GPY method at certain points.\n\n• R.C. Vaughan recalled being asked a similar question in the past and that he was able to answer it in the affi rmative. He said that he thinks that he has the solution written somewhere in his files.",
   "evidence": "The canonical record is not one mathematical problem. It joins two consecutive items from the AIM workshop list *Gaps between primes*:",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 104,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0106": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "5. The multiplicative twin-prime problem of Elliott. \n\n• In 2003 P.D.T.A. Elliott proved that for every fixed integer a > 1, the equation \n\nak = p + 1\n\nq + 1 (2) has infinitely many solutions in primes p, q and integers k with 2 ≤ k ≤ log q. K.B. Ford suggested to try to adapt the GPY method to reduce the range for k in Elliott's result. \n\n• J. Pintz remarked that an upper bound of the form k \u001c (log q)1/2+\u000f seemed a reasonable goal at that stage of our understanding of the GPY method. \n\n• Several participants noted that the adaptation of the GPY method to this problem would require an analogue of Gallagher's result on the average size of the singular series for prime k-tuples. A. Granville noted that the problem of understanding the singular series of (2) may be of independent interest.",
   "clean_statement": null,
   "public_statement": "5. The multiplicative twin-prime problem of Elliott.\n\n• In 2003 P.D.T.A. Elliott proved that for every fixed integer a > 1, the equation\n\nak = p + 1\n\nq + 1 (2) has infinitely many solutions in primes p, q and integers k with 2 ≤ k ≤ log q. K.B. Ford suggested to try to adapt the GPY method to reduce the range for k in Elliott's result.\n\n• J. Pintz remarked that an upper bound of the form k\n (log q)1/2+[U+000F] seemed a reasonable goal at that stage of our understanding of the GPY method.\n\n• Several participants noted that the adaptation of the GPY method to this problem would require an analogue of Gallagher's result on the average size of the singular series for prime k-tuples. A. Granville noted that the problem of understanding the singular series of (2) may be of independent interest.",
   "evidence": "The canonical JSON record is visibly damaged by vertical-layout extraction and two control characters. In particular, it contains the lines",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 105,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0107": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "6. Long chains of primes. \n\n• K.B. Ford proposed the following problem: Let x be large and consider a sequence of primes p1,..., pk ≤ x\n\nsuch that for all j = 2,..., k,\n\np j = m j p j−1 + 1 for some m j ∈ N.\n\nObviously, k = O(log x). Prove or disprove that k = o(log x). \n\n• H.L. Montgomery noted that this question reminded him of a problem of Erd ¨ os: Show that there are finitely many n such that n − 2k is prime for all integers k with 2 k < n.\n\n• R.C. Vaughan recalled that Erd¨ os thought that n = 105 is the largest such n.",
   "clean_statement": null,
   "public_statement": "6. Long chains of primes.\n\n• K.B. Ford proposed the following problem: Let x be large and consider a sequence of primes p1,..., pk ≤ x\n\nsuch that for all j = 2,..., k,\n\np j = m j p j−1 + 1 for some m j ∈ N.\n\nObviously, k = O(log x). Prove or disprove that k = o(log x).\n\n• H.L. Montgomery noted that this question reminded him of a problem of Erd ¨ os: Show that there are finitely many n such that n − 2k is prime for all integers k with 2 k < n.\n\n• R.C. Vaughan recalled that Erd¨ os thought that n = 105 is the largest such n.",
   "evidence": "The canonical input is record 106 (zero-based) of `aim-analytic-number-theory-notes.json`. Its primary source is Angel Kumchev's notes from the December 2005 ARCC workshop *Gaps Between Primes* [AIM2005, p. 2 of the PDF]. The source was inspected directly, rather than relying only on the extracted record.",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 106,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0108": {
   "statement_status": "unrecoverable",
   "original_statement": "7. Small gaps between primes in thin sequences. \n\n• A. Kantorovich asked whether it is possible to prove the existence of small gaps between primes from the sequence \n\n{p ≤ x | p = [n log n] for some n ∈ N}.\n\nThis is a \"thin\" set of primes: #{p ≤ x | p = [n log n] for some n ∈ N} ∼ x(log x)−2.\n\n• Several participants noted that if one is interested in this question, then one may also investigate the analogous question for Piatetski-Shapiro primes. 28. Mersenne composites. \n\n• A. Granville asked whether it is possible to use the GPY method to prove (under the assumption of EH, if necessary) that there are infinitely many composites of the form 2 p − 1 (Mersenne composites). \n\n• C. Elsholtz noted that there may exist related work by M.R. Murty under the assumption of Artin's primitive root conjecture.",
   "clean_statement": null,
   "public_statement": "7. Small gaps between primes in thin sequences.\n\n• A. Kantorovich asked whether it is possible to prove the existence of small gaps between primes from the sequence\n\n{p ≤ x | p = [n log n] for some n ∈ N}.\n\nThis is a \"thin\" set of primes: #{p ≤ x | p = [n log n] for some n ∈ N} ∼ x(log x)−2.\n\n• Several participants noted that if one is interested in this question, then one may also investigate the analogous question for Piatetski-Shapiro primes. 28. Mersenne composites.\n\n• A. Granville asked whether it is possible to use the GPY method to prove (under the assumption of EH, if necessary) that there are infinitely many composites of the form 2 p − 1 (Mersenne composites).\n\n• C. Elsholtz noted that there may exist related work by M.R. Murty under the assumption of Artin's primitive root conjecture.",
   "evidence": "The canonical record is not one mathematical problem. It is an extraction splice from Angel Kumchev's notes for the December 2005 ARCC workshop *Gaps Between Primes*. The primary five-page PDF and the neighboring canonical records were checked directly.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 107,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0109": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "9. Second order di ff erences between primes. \n\n• T.D. Wooley proposed the following problem: How often are the second di ff erences \n\npn+2 − 2pn+1 + pn\n\nsmall (large)? He noted that Erd¨ os has proved that \n\n|pn+2 − 2pn+1 + pn| ≥ (1 + δ) log pn (3) infinitely often. \n\n• A. Balog added that he had thought about the problem in the past and had convinced himself that the left side of (3) is infinitely often as large as the largest known gap between consecutive primes.",
   "clean_statement": null,
   "public_statement": "9. Second order di ff erences between primes.\n\n• T.D. Wooley proposed the following problem: How often are the second di ff erences\n\npn+2 − 2pn+1 + pn\n\nsmall (large)? He noted that Erd¨ os has proved that\n\n|pn+2 − 2pn+1 + pn| ≥ (1 + δ) log pn (3) infinitely often.\n\n• A. Balog added that he had thought about the problem in the past and had convinced himself that the left side of (3) is infinitely often as large as the largest known gap between consecutive primes.",
   "evidence": "This record is Problem 9 in the AIM workshop notes *Gaps Between Primes* (December 2005), recorded by Angel Kumchev. The source PDF gives the following question (typographical spacing and OCR errors have been repaired, but the mathematical wording has not been strengthened):",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 108,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0110": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "10. Di ff erences between primes and pair correlation. \n\n• Let F(α) be Montgomery's pair correlation function. In 1982 D.R. Heath-Brown proved that, under RH and Montgomery's Pair Correlation Conjecture, lim inf \n\n> n→∞\n\npn+1 − pn\n\nlog pn\n\n= 0.\n\nT.H. Chan asked whether it is possible to prove a converse result (which, of course, would be unconditional post-GPY). \n\n• J. Pintz expressed serious doubt. \n\n• In the same paper, Heath-Brown proved also (under the same assumptions) that \n\npn+1 − pn \u001c √pn log pn.\n\nJ.B. Conrey asked whether it is possible to use random matrix theory (RMT) to improve further on this result. \n\n• K. Soundararajan felt quite strongly that RMT should not help in this problem.",
   "clean_statement": null,
   "public_statement": "10. Di ff erences between primes and pair correlation.\n\n• Let F(α) be Montgomery's pair correlation function. In 1982 D.R. Heath-Brown proved that, under RH and Montgomery's Pair Correlation Conjecture, lim inf\n\n> n→∞\n\npn+1 − pn\n\nlog pn\n\n= 0.\n\nT.H. Chan asked whether it is possible to prove a converse result (which, of course, would be unconditional post-GPY).\n\n• J. Pintz expressed serious doubt.\n\n• In the same paper, Heath-Brown proved also (under the same assumptions) that\n\npn+1 − pn\n √pn log pn.\n\nJ.B. Conrey asked whether it is possible to use random matrix theory (RMT) to improve further on this result.\n\n• K. Soundararajan felt quite strongly that RMT should not help in this problem.",
   "evidence": "The record is Problem 10 in Angel Kumchev's notes from the December 2005 ARCC workshop *Gaps Between Primes*. The official AIM PDF was checked, as was the primary paper of Heath-Brown. The mathematically unambiguous repaired statement is as follows. Write",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 109,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0111": {
   "statement_status": "corrected_verified",
   "original_statement": "11. Di ff erences between primes and ζ(s). Traditionally, analytic information about the zeta-function has been used to derive upper bounds for the di ff erences between consecutive primes. Y. Motohashi asked whether this process can be reversed. For example, what (if anything) can be said about ζ(s) in the critical strip under the assumption that \n\npn+1 − pn \u001c\u000f p\u000f\n\n> n?",
   "clean_statement": "Traditionally, analytic information about the zeta-function has been\nused to derive upper bounds for differences between consecutive primes.\nY. Motohashi asked whether this process can be reversed.  For example,\nwhat (if anything) can be said about \\(\\zeta(s)\\) in the critical strip\nunder (H)?",
   "public_statement": "Traditionally, analytic information about the zeta-function has been\nused to derive upper bounds for differences between consecutive primes.\nY. Motohashi asked whether this process can be reversed.  For example,\nwhat (if anything) can be said about \\(\\zeta(s)\\) in the critical strip\nunder (H)?",
   "evidence": "The canonical record is problem 11 from the December 2005 ARCC workshop *Gaps Between Primes*, in notes prepared by Angel Kumchev. The repository extraction ends with corrupted control characters: The official AIM PDF was checked at the displayed formula on its third page. The formula is unambiguously",
   "classification_method": "source_verified_raw_character_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 110,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0112": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "12. Limits of the GPY method. \n\n• H. Helfgott noted that it seems that the GPY method is by nature a lower bound method and will most likely never yield an asymptotic formula for the number of primes with a given spacing. He asked those who were well-versed in the GPY method whether this is indeed so and also whether the GPY method is capable of producing a right-order lower bound. \n\n• J. Pintz replied that an asymptotic formula of any kind appears to be out of the reach of the method. He also noted that the method could potentially yield a \"right-order lower bound\", but that that will involve considerable technical di ffi culties. Some ideas in that direction were sketched in C. Yildirim's talk. It should be noted that in this context, a \"right-order lower bound\" means that one can show that a positive proportion of the k-tuples we consider contain at least two primes. 313. Distances between Gaussian primes. Several members of the audience asked whether the GPY method can be adapted to study small distances between Gaussian primes. T.D. Wooley remarked that such an adaptation should be straightforward.",
   "clean_statement": null,
   "public_statement": "12. Limits of the GPY method.\n\n• H. Helfgott noted that it seems that the GPY method is by nature a lower bound method and will most likely never yield an asymptotic formula for the number of primes with a given spacing. He asked those who were well-versed in the GPY method whether this is indeed so and also whether the GPY method is capable of producing a right-order lower bound.\n\n• J. Pintz replied that an asymptotic formula of any kind appears to be out of the reach of the method. He also noted that the method could potentially yield a \"right-order lower bound\", but that that will involve considerable technical di ffi culties. Some ideas in that direction were sketched in C. Yildirim's talk. It should be noted that in this context, a \"right-order lower bound\" means that one can show that a positive proportion of the k-tuples we consider contain at least two primes. 313. Distances between Gaussian primes. Several members of the audience asked whether the GPY method can be adapted to study small distances between Gaussian primes. T.D. Wooley remarked that such an adaptation should be straightforward.",
   "evidence": "The canonical problem field is preserved here verbatim:",
   "classification_method": "invalid_literal_with_repaired_reading",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 111,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0113": {
   "statement_status": "exact",
   "original_statement": "14. Small values of linear forms at prime arguments. A. Balog proposed the following problem: Show that the di ff erence |ap − q|, where p, q are primes and a > 1 is a fixed integer, can be small. More generally, consider the case when a is real, a, 1.",
   "clean_statement": "14. Small values of linear forms at prime arguments. A. Balog proposed the following problem: Show that the di ff erence |ap − q|, where p, q are primes and a > 1 is a fixed integer, can be small. More generally, consider the case when a is real, a, 1.",
   "public_statement": "14. Small values of linear forms at prime arguments. A. Balog proposed the following problem: Show that the di ff erence |ap − q|, where p, q are primes and a > 1 is a fixed integer, can be small. More generally, consider the case when a is real, a, 1.",
   "evidence": "The canonical record is item 14 of the AIM workshop list *Gaps between primes*. Its OCR text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 112,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0114": {
   "statement_status": "exact",
   "original_statement": "15. Alternative weights in the GPY method. \n\n• The GPY method uses the weights \n\nΛR(n; H) = ∑\n\n> d|P(n;H)\n\nλd, λd = μ(d) (log +(R/d)) k+l\n\n(k + l)!,\n\nwhere P(n; H) = (n + h1) · · · (n + hk) and log + x = max(0, log x). These λd's are Selberg-type sieve weights. J.B. Conrey asked whether it is possible to strenghten the method quantitatively by using a di ff erent choice of the \n\nλd's. For the duration of the workshop, there was an ongoing discussion of this question. Much of it focused on weights of the form \n\nλd = μ(d)\n\n{ ∏\n\n> p|d\n\nf\n\n( log p\n\nlog R\n\n) } \n\nQ( log +(R/d)),\n\nwhere f and Q are smooth functions at our disposal. \n\n• R.C. Vaughan noted that it may be worth investigating how the method is a ff ected by changing Selberg's weights to other traditional sieve weights. \n\n• Another suggestion was to try to combine upper and lower sieves a la Chen. K. Soundararajan commented that there will be an issue calculating the negative contribution.",
   "clean_statement": "15. Alternative weights in the GPY method.\n\n• The GPY method uses the weights\n\nΛR(n; H) = ∑\n\n> d|P(n;H)\n\nλd, λd = μ(d) (log +(R/d)) k+l\n\n(k + l)!,\n\nwhere P(n; H) = (n + h1) · · · (n + hk) and log + x = max(0, log x). These λd's are Selberg-type sieve weights. J.B. Conrey asked whether it is possible to strenghten the method quantitatively by using a di ff erent choice of the\n\nλd's. For the duration of the workshop, there was an ongoing discussion of this question. Much of it focused on weights of the form\n\nλd = μ(d)\n\n{ ∏\n\n> p|d\n\nf\n\n( log p\n\nlog R\n\n) }\n\nQ( log +(R/d)),\n\nwhere f and Q are smooth functions at our disposal.\n\n• R.C. Vaughan noted that it may be worth investigating how the method is a ff ected by changing Selberg's weights to other traditional sieve weights.\n\n• Another suggestion was to try to combine upper and lower sieves a la Chen. K. Soundararajan commented that there will be an issue calculating the negative contribution.",
   "public_statement": "15. Alternative weights in the GPY method.\n\n• The GPY method uses the weights\n\nΛR(n; H) = ∑\n\n> d|P(n;H)\n\nλd, λd = μ(d) (log +(R/d)) k+l\n\n(k + l)!,\n\nwhere P(n; H) = (n + h1) · · · (n + hk) and log + x = max(0, log x). These λd's are Selberg-type sieve weights. J.B. Conrey asked whether it is possible to strenghten the method quantitatively by using a di ff erent choice of the\n\nλd's. For the duration of the workshop, there was an ongoing discussion of this question. Much of it focused on weights of the form\n\nλd = μ(d)\n\n{ ∏\n\n> p|d\n\nf\n\n( log p\n\nlog R\n\n) }\n\nQ( log +(R/d)),\n\nwhere f and Q are smooth functions at our disposal.\n\n• R.C. Vaughan noted that it may be worth investigating how the method is a ff ected by changing Selberg's weights to other traditional sieve weights.\n\n• Another suggestion was to try to combine upper and lower sieves a la Chen. K. Soundararajan commented that there will be an issue calculating the negative contribution.",
   "evidence": "This is Problem 15 in the AIM workshop list *Gaps between primes*. The official PDF is the authority for the reconstruction below. The JSON extraction lost fraction layout and line breaks but did not change the mathematical content.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 113,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0115": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "16. Possible improvements on the Bombieri-Vinogradov theorem. \n\n• J.B. Friedlander posed several questions in his talk: \n\n◦ Show that for some fixed θ > 1/2 and D = xθ, one has \n\n∑\n\n> d≤D\n> (d,a)=1\n\nμ(d)\n\n(\n\nψ(x; d, a) − x\n\nφ(d)\n\n)\n\n\u001c x(log x)−A.\n\n◦ Prove anything beyond the Bombieri-Vinogradov theorem for the sum \n\n∑\n\n> d≤D\n\nmax \n\n> (a,d)=1\n> |a|<(log x)2005\n\n∣∣∣∣ψ(x; d, a) − x\n\nφ(d)\n\n∣∣∣∣.\n\n◦ For any \"reasonable\" weights λd, show, for some fixed θ > 1/2 and D = xθ, that \n\n∑\n\n> d≤D\n> (d,a)=1\n\nλd\n\n(\n\nψ(x; d, ¯a) − x\n\nφ(d)\n\n)\n\n\u001c x(log x)−A.\n\nHere, ¯ a is defined modulo d by a¯a ≡ 1 (mod d). All of these were motivated by the limitations of the Bombieri-Friedlander-Iwaniec method for primes in arith-metic progressions to large moduli. 4• Several participants proposed to investigate theorems of the Bombieri-Vinogradov type for averages of arith-metic functions other than Λ(n) (cf. known results for the divisor functions d(n) and d3(n)). \n\n• A. Granville asked whether it is possible to formulate a reasonable conjecture about zeros of L-functions that would yield EH( θ) with a fixed θ > 1/2.",
   "clean_statement": null,
   "public_statement": "16. Possible improvements on the Bombieri-Vinogradov theorem.\n\n• J.B. Friedlander posed several questions in his talk:\n\n◦ Show that for some fixed θ > 1/2 and D = xθ, one has\n\n∑\n\n> d≤D\n> (d,a)=1\n\nμ(d)\n\n(\n\nψ(x; d, a) − x\n\nφ(d)\n\n)\n\n\n x(log x)−A.\n\n◦ Prove anything beyond the Bombieri-Vinogradov theorem for the sum\n\n∑\n\n> d≤D\n\nmax\n\n> (a,d)=1\n> |a|<(log x)2005\n\n∣∣∣∣ψ(x; d, a) − x\n\nφ(d)\n\n∣∣∣∣.\n\n◦ For any \"reasonable\" weights λd, show, for some fixed θ > 1/2 and D = xθ, that\n\n∑\n\n> d≤D\n> (d,a)=1\n\nλd\n\n(\n\nψ(x; d, ¯a) − x\n\nφ(d)\n\n)\n\n\n x(log x)−A.\n\nHere, ¯ a is defined modulo d by a¯a ≡ 1 (mod d). All of these were motivated by the limitations of the Bombieri-Friedlander-Iwaniec method for primes in arith-metic progressions to large moduli. 4• Several participants proposed to investigate theorems of the Bombieri-Vinogradov type for averages of arith-metic functions other than Λ(n) (cf. known results for the divisor functions d(n) and d3(n)).\n\n• A. Granville asked whether it is possible to formulate a reasonable conjecture about zeros of L-functions that would yield EH( θ) with a fixed θ > 1/2.",
   "evidence": "The record is Problem 16 in the AIM workshop list *Gaps between primes* (2005). I checked the typeset workshop PDF rather than relying on the damaged extracted text. With \\[ \\psi(x;q,a)=\\sum_{\\substack{n\\le x\\\\n\\equiv a\\pmod q}}\\Lambda(n), \\] the three questions attributed to J. B. Friedlander are:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 114,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0116": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "17. Elliott-Halberstam from bounded gaps. \n\n• During the workshop, much attention focused on how to prove EH( θ) with θ > 1/2, so we can deduce the existence of bounded gaps between primes. V. Blomer asked whether it is possible to go the opposite way and derive an improvement on the Bombieri-Vinogradov theorem from a \"strong\" quantitative result on bounded gaps between primes. \n\n• R.C. Vaughan commented that such an improvement would encode so much information about L-functions that the hypothesis would have to be extremely strong. \n\n• K. Soundararajan added that the \"strong quantitative result\" could be as strong as the Hardy-Littlewood k-tuple conjecture with a sharp error term, say O(N1/2+\u000f ).",
   "clean_statement": null,
   "public_statement": "17. Elliott-Halberstam from bounded gaps.\n\n• During the workshop, much attention focused on how to prove EH( θ) with θ > 1/2, so we can deduce the existence of bounded gaps between primes. V. Blomer asked whether it is possible to go the opposite way and derive an improvement on the Bombieri-Vinogradov theorem from a \"strong\" quantitative result on bounded gaps between primes.\n\n• R.C. Vaughan commented that such an improvement would encode so much information about L-functions that the hypothesis would have to be extremely strong.\n\n• K. Soundararajan added that the \"strong quantitative result\" could be as strong as the Hardy-Littlewood k-tuple conjecture with a sharp error term, say O(N1/2+[U+000F] ).",
   "evidence": "This is Problem 17 in the AIM workshop list *Gaps between primes* (December 2005), notes by Angel Kumchev. The exact mathematical prompt is:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 115,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0117": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "18. From k-tuples to (k + 1) -tuples. \n\n• J.B. Conrey asked whether it is possible to deduce the existence of prime ( k + 1)-tuples from a version of the Elliott-Halberstam conjecture for prime k-tuples. \n\n• T.D. Wooley and several others pointed out that it is not quite clear what the proper statement of EH for prime \n\nk-tuples is. It was mentioned that there are results by Balog, Kawada, and Mikawa of the form \n\n∑\n\n> q≤Q\n\nmax \n\n> (a,q)=1\n\n∑\n\n> 1≤b≤q\n> (b,q)=1\n\n∣∣∣∣∑\n\n> n≤x/q\n\n$(qn + a)$(qn + b) − MT( x, q; a, b)\n\n∣∣∣∣ \u001c x2(log x)−A.\n\nHere, $(·) is the characteristic function of the primes and MT( x, q; a, b) is a main term. \n\n• R.C. Vaughan suggested that the following estimate is another possible candidate: \n\n∑\n\n> q≤Q\n\nmax \n\n> (a,q)=1\n\n∣∣∣∣∑\n\n> 1≤b≤B\n> (b,q)=1\n\n( ∑\n\n> n≤x/q\n\n$(qn + a)$(qn + b) − MT( x, q; a, b)\n\n)∣ ∣∣∣ \u001c Bx (log x)−A.\n\n• J. Pintz proposed to try to use EH for twin primes to obtain two pairs of twin primes that are close to each other.",
   "clean_statement": null,
   "public_statement": "18. From k-tuples to (k + 1) -tuples.\n\n• J.B. Conrey asked whether it is possible to deduce the existence of prime ( k + 1)-tuples from a version of the Elliott-Halberstam conjecture for prime k-tuples.\n\n• T.D. Wooley and several others pointed out that it is not quite clear what the proper statement of EH for prime\n\nk-tuples is. It was mentioned that there are results by Balog, Kawada, and Mikawa of the form\n\n∑\n\n> q≤Q\n\nmax\n\n> (a,q)=1\n\n∑\n\n> 1≤b≤q\n> (b,q)=1\n\n∣∣∣∣∑\n\n> n≤x/q\n\n$(qn + a)$(qn + b) − MT( x, q; a, b)\n\n∣∣∣∣\n x2(log x)−A.\n\nHere, $(·) is the characteristic function of the primes and MT( x, q; a, b) is a main term.\n\n• R.C. Vaughan suggested that the following estimate is another possible candidate:\n\n∑\n\n> q≤Q\n\nmax\n\n> (a,q)=1\n\n∣∣∣∣∑\n\n> 1≤b≤B\n> (b,q)=1\n\n( ∑\n\n> n≤x/q\n\n$(qn + a)$(qn + b) − MT( x, q; a, b)\n\n)∣ ∣∣∣\n Bx (log x)−A.\n\n• J. Pintz proposed to try to use EH for twin primes to obtain two pairs of twin primes that are close to each other.",
   "evidence": "The source is page 5 of Angel Kumchev's notes from the December 2005 ARCC workshop *Gaps Between Primes*. The heading and first question are:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 116,
   "attempt": 1
  },
  "AIM-ANALYTIC_NUMBER_THEORY-0118": {
   "statement_status": "exact",
   "original_statement": "19. Triples in prime-like sequences. \n\n• C. Elsholtz asked whether it is possible to find infinitely many triples sn, sn+1, sn+2, with sn+2 − sn bounded, in sequences that are commonly considered to be similar to but more manageable than the primes (e.g., integers that are sums of two squares). \n\n• K.B. Ford noted that for sums of two squares the problem is trivial, because of triples of the form n2, n2 +1, n2 +4, for example. \n\n• J.B. Friedlander suggested that such trivialities can be avoided by requiring that the two squares be of compara-ble sizes. S.W. Graham and T.D. Wooley proposed another option: produce \"many\" gaps. \n\n• R.C. Vaughan proposed considering the more general question of triples of values of norm-forms. 5",
   "clean_statement": "19. Triples in prime-like sequences.\n\n• C. Elsholtz asked whether it is possible to find infinitely many triples sn, sn+1, sn+2, with sn+2 − sn bounded, in sequences that are commonly considered to be similar to but more manageable than the primes (e.g., integers that are sums of two squares).\n\n• K.B. Ford noted that for sums of two squares the problem is trivial, because of triples of the form n2, n2 +1, n2 +4, for example.\n\n• J.B. Friedlander suggested that such trivialities can be avoided by requiring that the two squares be of compara-ble sizes. S.W. Graham and T.D. Wooley proposed another option: produce \"many\" gaps.\n\n• R.C. Vaughan proposed considering the more general question of triples of values of norm-forms. 5",
   "public_statement": "19. Triples in prime-like sequences.\n\n• C. Elsholtz asked whether it is possible to find infinitely many triples sn, sn+1, sn+2, with sn+2 − sn bounded, in sequences that are commonly considered to be similar to but more manageable than the primes (e.g., integers that are sums of two squares).\n\n• K.B. Ford noted that for sums of two squares the problem is trivial, because of triples of the form n2, n2 +1, n2 +4, for example.\n\n• J.B. Friedlander suggested that such trivialities can be avoided by requiring that the two squares be of compara-ble sizes. S.W. Graham and T.D. Wooley proposed another option: produce \"many\" gaps.\n\n• R.C. Vaughan proposed considering the more general question of triples of values of norm-forms. 5",
   "evidence": "The source is Angel Kumchev's notes from the December 2005 ARCC workshop *Gaps Between Primes*, Problem 19, p. 5. The original PDF was inspected directly. Apart from line wrapping (including the split word “compara-ble”) and the extracted page number, the repository record is accurate. The PDF says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-analytic-number-theory-notes.json",
   "source_index": 117,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0001": {
   "statement_status": "exact",
   "original_statement": "Is it possible to define a generating series of special cycles over Shimura varieties related to automorphic representations of higher weight? Do we have similar geometric/arithmetic modularity and Siegel-Weil formulas as in the Kudla program?",
   "clean_statement": "Is it possible to define a generating series of special cycles over Shimura varieties related to automorphic representations of higher weight? Do we have similar geometric/arithmetic modularity and Siegel-Weil formulas as in the Kudla program?",
   "public_statement": "Is it possible to define a generating series of special cycles over Shimura varieties related to automorphic representations of higher weight? Do we have similar geometric/arithmetic modularity and Siegel-Weil formulas as in the Kudla program?",
   "evidence": "The canonical record is Problem 1.1 in the section **“Special cycles, modularity and arithmetic intersection”** of the AIM workshop *Arithmetic intersection theory on Shimura varieties* (January 8--12, 2024). The exact recorded question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 0,
   "attempt": 2
  },
  "AIM-ARITHMETIC_GEOMETRY-0002": {
   "statement_status": "exact",
   "original_statement": "In recent proof of Kudla-Rapoport conjectures by Li--Zhang, numerical local modularity results for local Kudla--Rapoport divisors have been discovered and proved. Is it possible to formulate local modularity geometrically e.g. using moduli of p-adic Shtukas? Could the method of Feng-Yun-Zhang for global modularity be adapted to prove geometric local modularity?",
   "clean_statement": "In recent proof of Kudla-Rapoport conjectures by Li--Zhang, numerical local modularity results for local Kudla--Rapoport divisors have been discovered and proved. Is it possible to formulate local modularity geometrically e.g. using moduli of p-adic Shtukas? Could the method of Feng-Yun-Zhang for global modularity be adapted to prove geometric local modularity?",
   "public_statement": "In recent proof of Kudla-Rapoport conjectures by Li--Zhang, numerical local modularity results for local Kudla--Rapoport divisors have been discovered and proved. Is it possible to formulate local modularity geometrically e.g. using moduli of p-adic Shtukas? Could the method of Feng-Yun-Zhang for global modularity be adapted to prove geometric local modularity?",
   "evidence": "The canonical record is Problem 1.3 in the section “Special cycles, modularity and arithmetic intersection” of the AIM list *Arithmetic intersection theory on Shimura varieties*. The live AIM page agrees with the repository record and attributes the question to Q. He and Z. Zhang. The exact statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 1,
   "attempt": 2
  },
  "AIM-ARITHMETIC_GEOMETRY-0003": {
   "statement_status": "exact",
   "original_statement": "In the formulation of the arithmetic fundamental lemma, arithmetic intersection numbers are only formulated for regular semi-simple orbits because of proper intersections of cycles. Is it possible to define arithmetic intersection numbers in case of improper intersections of cycles (and formulate a similar analytic side)?",
   "clean_statement": "In the formulation of the arithmetic fundamental lemma, arithmetic intersection numbers are only formulated for regular semi-simple orbits because of proper intersections of cycles. Is it possible to define arithmetic intersection numbers in case of improper intersections of cycles (and formulate a similar analytic side)?",
   "public_statement": "In the formulation of the arithmetic fundamental lemma, arithmetic intersection numbers are only formulated for regular semi-simple orbits because of proper intersections of cycles. Is it possible to define arithmetic intersection numbers in case of improper intersections of cycles (and formulate a similar analytic side)?",
   "evidence": "The canonical record is item 1.4 in the section **“Special cycles, modularity and arithmetic intersection”** of the AIM workshop list **“Arithmetic intersection theory on Shimura varieties.”** The exact corpus statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 2,
   "attempt": 2
  },
  "AIM-ARITHMETIC_GEOMETRY-0004": {
   "statement_status": "exact",
   "original_statement": "Derived special cycles have been defined on integral models of certain Shimura varieties (resp. moduli of Shtukas) in recent works of Madapusi (resp. Feng-Yun-Zhang) in the good reduction case. Is it possible to extend their method and define derived special cycles in the bad reduction cases e.g. for some specific parahoric levels?",
   "clean_statement": "Derived special cycles have been defined on integral models of certain Shimura varieties (resp. moduli of Shtukas) in recent works of Madapusi (resp. Feng-Yun-Zhang) in the good reduction case. Is it possible to extend their method and define derived special cycles in the bad reduction cases e.g. for some specific parahoric levels?",
   "public_statement": "Derived special cycles have been defined on integral models of certain Shimura varieties (resp. moduli of Shtukas) in recent works of Madapusi (resp. Feng-Yun-Zhang) in the good reduction case. Is it possible to extend their method and define derived special cycles in the bad reduction cases e.g. for some specific parahoric levels?",
   "evidence": "The canonical record is problem 1.2 in the section “Special cycles, modularity and arithmetic intersection” of the AIM workshop list *Arithmetic intersection theory on Shimura varieties*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0005": {
   "statement_status": "exact",
   "original_statement": "In any arithmetic transfer conjectures, could we find perfect test transfer functions for (derived) orbital integrals to match arithmetic intersection numbers with no correction factors?",
   "clean_statement": "In any arithmetic transfer conjectures, could we find perfect test transfer functions for (derived) orbital integrals to match arithmetic intersection numbers with no correction factors?",
   "public_statement": "In any arithmetic transfer conjectures, could we find perfect test transfer functions for (derived) orbital integrals to match arithmetic intersection numbers with no correction factors?",
   "evidence": "The canonical record is problem 1.5 in the section “Special cycles, modularity and arithmetic intersection” of the AIM workshop list *Arithmetic intersection theory on Shimura varieties*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0006": {
   "statement_status": "exact",
   "original_statement": "In the recent work of Li--Rapoport--Zhang on arithmetic fundamental lemma conjecture for spherical Hecke algebras, there is a conjecture on commutativity of certain Hecke operators on Grothendieck groups of coherent sheaves on unitary Rapoport--Zink spaces. How to prove this conjecture?",
   "clean_statement": "In the recent work of Li--Rapoport--Zhang on arithmetic fundamental lemma conjecture for spherical Hecke algebras, there is a conjecture on commutativity of certain Hecke operators on Grothendieck groups of coherent sheaves on unitary Rapoport--Zink spaces. How to prove this conjecture?",
   "public_statement": "In the recent work of Li--Rapoport--Zhang on arithmetic fundamental lemma conjecture for spherical Hecke algebras, there is a conjecture on commutativity of certain Hecke operators on Grothendieck groups of coherent sheaves on unitary Rapoport--Zink spaces. How to prove this conjecture?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0007": {
   "statement_status": "exact",
   "original_statement": "Similar to Kudla program, we could also define a generating series of (derived) complex multiplication cycles on Shimura varieties and conjecture its modularity. Is it possible to compute arithmetic intersection of the generating series with special divisors / Hodge bundle and prove its numerical modularity when the Shimura variety has low dimension e.g. on Siegel threefold or Hilbert modular surface?",
   "clean_statement": "Similar to Kudla program, we could also define a generating series of (derived) complex multiplication cycles on Shimura varieties and conjecture its modularity. Is it possible to compute arithmetic intersection of the generating series with special divisors / Hodge bundle and prove its numerical modularity when the Shimura variety has low dimension e.g. on Siegel threefold or Hilbert modular surface?",
   "public_statement": "Similar to Kudla program, we could also define a generating series of (derived) complex multiplication cycles on Shimura varieties and conjecture its modularity. Is it possible to compute arithmetic intersection of the generating series with special divisors / Hodge bundle and prove its numerical modularity when the Shimura variety has low dimension e.g. on Siegel threefold or Hilbert modular surface?",
   "evidence": "The canonical record is Problem 2.1 in the AIM list *Arithmetic intersection theory on Shimura varieties*, section “Arithmetic intersection theory beyond classical unitary Shimura varieties.” The live AIM page was checked on 2026-07-29. It attributes the question to A. Mihatsch and contains no status update or remark. The exact statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0008": {
   "statement_status": "exact",
   "original_statement": "Could we formulate and prove arithmetic Siegel--Weil formulas of special cycles on unitary Shimura varieties for U(a,b) (a, b>1) where there are no special divisors?",
   "clean_statement": "Could we formulate and prove arithmetic Siegel--Weil formulas of special cycles on unitary Shimura varieties for U(a,b) (a, b>1) where there are no special divisors?",
   "public_statement": "Could we formulate and prove arithmetic Siegel--Weil formulas of special cycles on unitary Shimura varieties for U(a,b) (a, b>1) where there are no special divisors?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 7,
   "attempt": 3
  },
  "AIM-ARITHMETIC_GEOMETRY-0009": {
   "statement_status": "exact",
   "original_statement": "Could we formulate and prove arithmetic (resp. higher) Siegel--Weil formulas on Shimura varieties (resp. moduli of Shtukas) for exceptional groups?",
   "clean_statement": "Could we formulate and prove arithmetic (resp. higher) Siegel--Weil formulas on Shimura varieties (resp. moduli of Shtukas) for exceptional groups?",
   "public_statement": "Could we formulate and prove arithmetic (resp. higher) Siegel--Weil formulas on Shimura varieties (resp. moduli of Shtukas) for exceptional groups?",
   "evidence": "The canonical record is item 2.3 in the section “Arithmetic intersection theory beyond classical unitary Shimura varieties” of the AIM problem list attached to the January 8--12, 2024 workshop *Arithmetic intersection theory on Shimura varieties*. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0010": {
   "statement_status": "exact",
   "original_statement": "Could we describe the reduced locus of Rapoport--Zink spaces (integral local Shimura varieties) for inner forms of general linear groups? It will be useful for studying arithmetic transfer conjectures in the context of linear arithmetic fundamental lemmas.",
   "clean_statement": "Could we describe the reduced locus of Rapoport--Zink spaces (integral local Shimura varieties) for inner forms of general linear groups? It will be useful for studying arithmetic transfer conjectures in the context of linear arithmetic fundamental lemmas.",
   "public_statement": "Could we describe the reduced locus of Rapoport--Zink spaces (integral local Shimura varieties) for inner forms of general linear groups? It will be useful for studying arithmetic transfer conjectures in the context of linear arithmetic fundamental lemmas.",
   "evidence": "The canonical source is aim-arithmetic-geometry-notes.json, zero-based index 9, problem 2.4 from the AIM workshop *Arithmetic intersection theory on Shimura varieties*. The exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 9,
   "attempt": 2
  },
  "AIM-ARITHMETIC_GEOMETRY-0011": {
   "statement_status": "exact",
   "original_statement": "Study non-algebraic locally symmetric spaces for $O(4,4)$ and natural special cycles on them. What kind of structures do we expect?",
   "clean_statement": "Study non-algebraic locally symmetric spaces for $O(4,4)$ and natural special cycles on them. What kind of structures do we expect?",
   "public_statement": "Study non-algebraic locally symmetric spaces for $O(4,4)$ and natural special cycles on them. What kind of structures do we expect?",
   "evidence": "The record is `AIM-ARITHMETIC_GEOMETRY-0011`, source file `aim-arithmetic-geometry-notes.json`, zero-based source index 10. The neighboring records confirm that the intended setting is an extension of Kudla-style cycle and intersection structures beyond algebraic Shimura varieties. There is no visible OCR corruption. The record does not specify a rational quadratic space, level, connected component, or cohomology theory, so those choices must be made explicitly.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 10,
   "attempt": 2
  },
  "AIM-ARITHMETIC_GEOMETRY-0012": {
   "statement_status": "exact",
   "original_statement": "Is there an algorithm to compute derived equivalence for K3 surfaces over a finitely generated field?",
   "clean_statement": "Is there an algorithm to compute derived equivalence for K3 surfaces over a finitely generated field?",
   "public_statement": "Is there an algorithm to compute derived equivalence for K3 surfaces over a finitely generated field?",
   "evidence": "The canonical record is AIM problem 1.05 in the section “K3 surfaces” of the workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 11,
   "attempt": 2
  },
  "AIM-ARITHMETIC_GEOMETRY-0013": {
   "statement_status": "exact",
   "original_statement": "If you fix a degree 2 K3 surface over $\\mathbb{Q}$ with a 2-torsion Brauer class $\\alpha$ (over $\\overline{\\mathbb{Q}}$), van Geeman tells us that we can find a variety $Y$ such that $X$ is a moduli space of sheaves on $Y$ and such that $\\alpha$ is the obstruction for the moduli space to be fine.\n\nIs $Y$ unique? How many such $Y$ are there? If $X$ is defined over $\\mathbb{Q}$ and $Y$ is defined over some nontrivial extension of $\\mathbb{Q}$, is it possible that $\\alpha$ is defined over $\\mathbb{Q}$, i.e.~in the image of $\\mathrm{Br}(X) \\to \\mathrm{Br}(\\overline{X})$?",
   "clean_statement": "If you fix a degree 2 K3 surface over $\\mathbb{Q}$ with a 2-torsion Brauer class $\\alpha$ (over $\\overline{\\mathbb{Q}}$), van Geeman tells us that we can find a variety $Y$ such that $X$ is a moduli space of sheaves on $Y$ and such that $\\alpha$ is the obstruction for the moduli space to be fine.\n\nIs $Y$ unique? How many such $Y$ are there? If $X$ is defined over $\\mathbb{Q}$ and $Y$ is defined over some nontrivial extension of $\\mathbb{Q}$, is it possible that $\\alpha$ is defined over $\\mathbb{Q}$, i.e.~in the image of $\\mathrm{Br}(X) \\to \\mathrm{Br}(\\overline{X})$?",
   "public_statement": "If you fix a degree 2 K3 surface over $\\mathbb{Q}$ with a 2-torsion Brauer class $\\alpha$ (over $\\overline{\\mathbb{Q}}$), van Geeman tells us that we can find a variety $Y$ such that $X$ is a moduli space of sheaves on $Y$ and such that $\\alpha$ is the obstruction for the moduli space to be fine.\n\nIs $Y$ unique? How many such $Y$ are there? If $X$ is defined over $\\mathbb{Q}$ and $Y$ is defined over some nontrivial extension of $\\mathbb{Q}$, is it possible that $\\alpha$ is defined over $\\mathbb{Q}$, i.e.~in the image of $\\mathrm{Br}(X) \\to \\mathrm{Br}(\\overline{X})$?",
   "evidence": "The AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0014": {
   "statement_status": "exact",
   "original_statement": "Does there exist a K3 surface $X$ and a Brauer class $\\alpha$ of odd order $n$ defined over a number field $k$ such that $\\alpha$ gives an obstruction to the existence of rational points?\n\nWhat about an example with $n>2$?\n\nWhat about a transcendental example?",
   "clean_statement": "Does there exist a K3 surface $X$ and a Brauer class $\\alpha$ of odd order $n$ defined over a number field $k$ such that $\\alpha$ gives an obstruction to the existence of rational points?\n\nWhat about an example with $n>2$?\n\nWhat about a transcendental example?",
   "public_statement": "Does there exist a K3 surface $X$ and a Brauer class $\\alpha$ of odd order $n$ defined over a number field $k$ such that $\\alpha$ gives an obstruction to the existence of rational points?\n\nWhat about an example with $n>2$?\n\nWhat about a transcendental example?",
   "evidence": "The canonical record is item 1.2 in the K3-surfaces section of the AIM workshop list *Brauer groups and obstruction problems: moduli spaces and arithmetic*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0015": {
   "statement_status": "exact",
   "original_statement": "Say we have a K3 surface over a number field. Are the places of bad reduction of the surface determined by the derived category? What about over $\\mathbb{C}((t))$?",
   "clean_statement": "Say we have a K3 surface over a number field. Are the places of bad reduction of the surface determined by the derived category? What about over $\\mathbb{C}((t))$?",
   "public_statement": "Say we have a K3 surface over a number field. Are the places of bad reduction of the surface determined by the derived category? What about over $\\mathbb{C}((t))$?",
   "evidence": "The canonical AIM record (workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*, section “K3 surfaces”, Problem 1.25) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0016": {
   "statement_status": "exact",
   "original_statement": "Can we identify 3-torsion elements of the Brauer group of a degree 2 K3 surface as obstructions to the fineness of some moduli space?\n\nCan we do this for K3 surfaces of other degrees? What about degree 4?",
   "clean_statement": "Can we identify 3-torsion elements of the Brauer group of a degree 2 K3 surface as obstructions to the fineness of some moduli space?\n\nCan we do this for K3 surfaces of other degrees? What about degree 4?",
   "public_statement": "Can we identify 3-torsion elements of the Brauer group of a degree 2 K3 surface as obstructions to the fineness of some moduli space?\n\nCan we do this for K3 surfaces of other degrees? What about degree 4?",
   "evidence": "The canonical record is Problem 1.3 in the K3-surfaces section of the AIM workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0017": {
   "statement_status": "exact",
   "original_statement": "If $X$ is a non-fine moduli space of sheaves on a K3 surface $Y$, and the dimension of $X$ is greater than 2, can the corresponding Brauer class obstruct the Hasse principle?",
   "clean_statement": "If $X$ is a non-fine moduli space of sheaves on a K3 surface $Y$, and the dimension of $X$ is greater than 2, can the corresponding Brauer class obstruct the Hasse principle?",
   "public_statement": "If $X$ is a non-fine moduli space of sheaves on a K3 surface $Y$, and the dimension of $X$ is greater than 2, can the corresponding Brauer class obstruct the Hasse principle?",
   "evidence": "The archived AIM URL in the record is <http://aimpl.org/brauermoduli/1/>. It returned an HTTP 502 error during this run, so the wording above is verified from the canonical JSON record and its neighbouring K3 questions, not from a currently accessible copy of the web page. There is no visible OCR corruption.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0018": {
   "statement_status": "exact",
   "original_statement": "Given a K3 surface $X$ with a Brauer class, is that Brauer class the obstruction to fineness as a moduli space of sheaves?",
   "clean_statement": "Given a K3 surface $X$ with a Brauer class, is that Brauer class the obstruction to fineness as a moduli space of sheaves?",
   "public_statement": "Given a K3 surface $X$ with a Brauer class, is that Brauer class the obstruction to fineness as a moduli space of sheaves?",
   "evidence": "The canonical record is AIM Problem List item 1.4 in the section “K3 surfaces” of the 2013 workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0019": {
   "statement_status": "exact",
   "original_statement": "Let $X$ be a K3 surface of degree $2d$. Consider the Hilbert scheme $\\mathscr{H}$ of length $d+1$ zero-dimensional subschemes of $X$. $\\mathrm{Br}(X)=\\mathrm{Br}(\\mathscr{H})$. There is a map $\\mathscr{H} \\to \\mathbb{P}^{d+1}$ with generic fiber an abelian variety. Does every $\\alpha \\in \\mathrm{Br}(X)$ be correspond to a torsor for the relative Jacobian of this vibration in such a way that the order of the torsor associated to $\\alpha$ times the order of $\\alpha$ is the order of the torsor $\\mathscr{H}$? In this question we are using the same correspondence between principal homogeneous spaces and Brauer classes as in Bianca's talk.\n\nIf we consider the moduli space of $\\alpha$-twisted sheaves on $X$ of dimension 0 and length $d+1$, does this realize the torsors of the previous part of this question?",
   "clean_statement": "Let $X$ be a K3 surface of degree $2d$. Consider the Hilbert scheme $\\mathscr{H}$ of length $d+1$ zero-dimensional subschemes of $X$. $\\mathrm{Br}(X)=\\mathrm{Br}(\\mathscr{H})$. There is a map $\\mathscr{H} \\to \\mathbb{P}^{d+1}$ with generic fiber an abelian variety. Does every $\\alpha \\in \\mathrm{Br}(X)$ be correspond to a torsor for the relative Jacobian of this vibration in such a way that the order of the torsor associated to $\\alpha$ times the order of $\\alpha$ is the order of the torsor $\\mathscr{H}$? In this question we are using the same correspondence between principal homogeneous spaces and Brauer classes as in Bianca's talk.\n\nIf we consider the moduli space of $\\alpha$-twisted sheaves on $X$ of dimension 0 and length $d+1$, does this realize the torsors of the previous part of this question?",
   "public_statement": "Let $X$ be a K3 surface of degree $2d$. Consider the Hilbert scheme $\\mathscr{H}$ of length $d+1$ zero-dimensional subschemes of $X$. $\\mathrm{Br}(X)=\\mathrm{Br}(\\mathscr{H})$. There is a map $\\mathscr{H} \\to \\mathbb{P}^{d+1}$ with generic fiber an abelian variety. Does every $\\alpha \\in \\mathrm{Br}(X)$ be correspond to a torsor for the relative Jacobian of this vibration in such a way that the order of the torsor associated to $\\alpha$ times the order of $\\alpha$ is the order of the torsor $\\mathscr{H}$? In this question we are using the same correspondence between principal homogeneous spaces and Brauer classes as in Bianca's talk.\n\nIf we consider the moduli space of $\\alpha$-twisted sheaves on $X$ of dimension 0 and length $d+1$, does this realize the torsors of the previous part of this question?",
   "evidence": "The canonical record (AIM Problem Lists, workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*, item 1.45) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0020": {
   "statement_status": "exact",
   "original_statement": "Fix a number field and a degree. Look at all K3 surfaces over the number field of degree $2d$. Look at the Brauer group of the K3 surface modulo the Brauer group of the number field. Is there a uniform bound on the size of the quotient? What about the algebraic part? The transcendental part of the Brauer group? Note: the first two groups are known to be finite.",
   "clean_statement": "Fix a number field and a degree. Look at all K3 surfaces over the number field of degree $2d$. Look at the Brauer group of the K3 surface modulo the Brauer group of the number field. Is there a uniform bound on the size of the quotient? What about the algebraic part? The transcendental part of the Brauer group? Note: the first two groups are known to be finite.",
   "public_statement": "Fix a number field and a degree. Look at all K3 surfaces over the number field of degree $2d$. Look at the Brauer group of the K3 surface modulo the Brauer group of the number field. Is there a uniform bound on the size of the quotient? What about the algebraic part? The transcendental part of the Brauer group? Note: the first two groups are known to be finite.",
   "evidence": "The canonical AIM record (source file `aim-arithmetic-geometry-notes.json`, zero-based index 19) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0021": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "If we have a K3 surface $X$ defined over some number field $k$, what can we say about the fields of definition of its Fourier-Mukai partners (partners over $\\mathbb{C}$)? What about twisted K3 surfaces? What about Abelian varieties? What about other varieties?\n\nIf we look at real quadratic extensions of $\\mathbb{Q}$, can the order of the odd part of the class group go to $\\infty$? Does this answer the previous question? How does this relate to the previous problem?",
   "clean_statement": null,
   "public_statement": "If we have a K3 surface $X$ defined over some number field $k$, what can we say about the fields of definition of its Fourier-Mukai partners (partners over $\\mathbb{C}$)? What about twisted K3 surfaces? What about Abelian varieties? What about other varieties?\n\nIf we look at real quadratic extensions of $\\mathbb{Q}$, can the order of the odd part of the class group go to $\\infty$? Does this answer the previous question? How does this relate to the previous problem?",
   "evidence": "where $h^+(p)$ is the narrow class number. Thus the wording “real quadratic extensions” has two plausible readings:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0022": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "How does the group structure of the Brauer group relate to Brauer classes as obstructions to fineness of moduli spaces? Given a K3 surface $X$ and two K3 surfaces $S_1$ and $S_2$ such that $X$ is a moduli space of sheaves on $S_1$ and $S_1$, can we explicitly construct a third K3 surface or variety $Y$ such that $X$ is a moduli space of sheaves on $Y$ and the obstruction class to fineness of this moduli space is the sum of the obstruction classes coming from $S_1$ and $S_2$?",
   "clean_statement": null,
   "public_statement": "How does the group structure of the Brauer group relate to Brauer classes as obstructions to fineness of moduli spaces? Given a K3 surface $X$ and two K3 surfaces $S_1$ and $S_2$ such that $X$ is a moduli space of sheaves on $S_1$ and $S_1$, can we explicitly construct a third K3 surface or variety $Y$ such that $X$ is a moduli space of sheaves on $Y$ and the obstruction class to fineness of this moduli space is the sum of the obstruction classes coming from $S_1$ and $S_2$?",
   "evidence": "This report treats that sentence as an explicit reconstruction, not as verified source text. The original record also omits a base field and stability data. We work over $\\mathbb C$, use primitive isotropic Mukai vectors and a generic polarization, and keep the chosen identifications with $X$ as part of the data. These hypotheses are essential: the two obstruction classes live in the same group only after transporting them to $\\operatorname{Br}(X)$.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0023": {
   "statement_status": "exact",
   "original_statement": "Pick three quadrics, $Q_0$, $Q_1$, and $Q_2$. Let $X=V(x_0Q_0+x_1Q_1+x_2Q_2)$. This is a cubic fourfold which contains the plane $V(x_0,x_1,x_2)$. The net of quadrics generated by the $Q_i$ gives us a family of quadric fourfolds over $\\mathbb{P}^2$. Let $\\mathcal{V}$ be the locus of maximal isotropic subspaces of the fibers. If we take the Stein factorization of the map $\\mathcal{V} \\to \\mathbb{P}^2$, this gives a bundle of projective spaces of dimension 3 over a double cover of $\\mathbb{P}^2$. The double cover will be a K3 surface. This give a Brauer class on the K3 surface.\n\nIf we look at 3-planes in $\\mathbb{P}^5$ containing the plane, they will be parametrized by $\\mathbb{P}^2$, and we will get a family of quadric surfaces over the $\\mathbb{P}^2$, namely the residual intersection of $X$ with the 3-plane. We can do the same construction as before with the space of maximal isotropic subspaces of the fibers to get another projective bundle over a K3 surface.\n\nIs there an intrinsic description of the relation between these twisted K3 surfaces?",
   "clean_statement": "Pick three quadrics, $Q_0$, $Q_1$, and $Q_2$. Let $X=V(x_0Q_0+x_1Q_1+x_2Q_2)$. This is a cubic fourfold which contains the plane $V(x_0,x_1,x_2)$. The net of quadrics generated by the $Q_i$ gives us a family of quadric fourfolds over $\\mathbb{P}^2$. Let $\\mathcal{V}$ be the locus of maximal isotropic subspaces of the fibers. If we take the Stein factorization of the map $\\mathcal{V} \\to \\mathbb{P}^2$, this gives a bundle of projective spaces of dimension 3 over a double cover of $\\mathbb{P}^2$. The double cover will be a K3 surface. This give a Brauer class on the K3 surface.\n\nIf we look at 3-planes in $\\mathbb{P}^5$ containing the plane, they will be parametrized by $\\mathbb{P}^2$, and we will get a family of quadric surfaces over the $\\mathbb{P}^2$, namely the residual intersection of $X$ with the 3-plane. We can do the same construction as before with the space of maximal isotropic subspaces of the fibers to get another projective bundle over a K3 surface.\n\nIs there an intrinsic description of the relation between these twisted K3 surfaces?",
   "public_statement": "Pick three quadrics, $Q_0$, $Q_1$, and $Q_2$. Let $X=V(x_0Q_0+x_1Q_1+x_2Q_2)$. This is a cubic fourfold which contains the plane $V(x_0,x_1,x_2)$. The net of quadrics generated by the $Q_i$ gives us a family of quadric fourfolds over $\\mathbb{P}^2$. Let $\\mathcal{V}$ be the locus of maximal isotropic subspaces of the fibers. If we take the Stein factorization of the map $\\mathcal{V} \\to \\mathbb{P}^2$, this gives a bundle of projective spaces of dimension 3 over a double cover of $\\mathbb{P}^2$. The double cover will be a K3 surface. This give a Brauer class on the K3 surface.\n\nIf we look at 3-planes in $\\mathbb{P}^5$ containing the plane, they will be parametrized by $\\mathbb{P}^2$, and we will get a family of quadric surfaces over the $\\mathbb{P}^2$, namely the residual intersection of $X$ with the 3-plane. We can do the same construction as before with the space of maximal isotropic subspaces of the fibers to get another projective bundle over a K3 surface.\n\nIs there an intrinsic description of the relation between these twisted K3 surfaces?",
   "evidence": "This is Problem 1.1 in the “K3 surfaces” section of the AIM workshop list *Brauer groups and obstruction problems: moduli spaces and arithmetic* (25 February--1 March 2013). The exact source record asks the following.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0024": {
   "statement_status": "exact",
   "original_statement": "Consider $D^b(\\mathbb{P}^n)$. Take an admissible thick subcategory, $\\mathcal{A}$, i.e. a triangulated subcategory closed under taking summands and such that the inclusion has a left (equivalently right) adjoint. If $K_0(\\mathcal{A})=0$, is $\\mathcal{A}=0$? Such an $\\mathcal{A}$ is called a phantom. When $n=1$, it is known that there is no such phantom subcategory. This is related to Caldararu's conjecture on the existence of exceptional collections on projective homogeneous varieties. The answer is no if we replace $\\mathbb{P}^n$ by an arbitrary variety (there is a surface counterexample).\n\nIf we replace $\\mathbb{P}^n$ by another regular surface, is the condition of not having phantom subcategories equivalent to rationality? What if we take $K_0(X) \\otimes \\mathbb{Q}$?",
   "clean_statement": "Consider $D^b(\\mathbb{P}^n)$. Take an admissible thick subcategory, $\\mathcal{A}$, i.e. a triangulated subcategory closed under taking summands and such that the inclusion has a left (equivalently right) adjoint. If $K_0(\\mathcal{A})=0$, is $\\mathcal{A}=0$? Such an $\\mathcal{A}$ is called a phantom. When $n=1$, it is known that there is no such phantom subcategory. This is related to Caldararu's conjecture on the existence of exceptional collections on projective homogeneous varieties. The answer is no if we replace $\\mathbb{P}^n$ by an arbitrary variety (there is a surface counterexample).\n\nIf we replace $\\mathbb{P}^n$ by another regular surface, is the condition of not having phantom subcategories equivalent to rationality? What if we take $K_0(X) \\otimes \\mathbb{Q}$?",
   "public_statement": "Consider $D^b(\\mathbb{P}^n)$. Take an admissible thick subcategory, $\\mathcal{A}$, i.e. a triangulated subcategory closed under taking summands and such that the inclusion has a left (equivalently right) adjoint. If $K_0(\\mathcal{A})=0$, is $\\mathcal{A}=0$? Such an $\\mathcal{A}$ is called a phantom. When $n=1$, it is known that there is no such phantom subcategory. This is related to Caldararu's conjecture on the existence of exceptional collections on projective homogeneous varieties. The answer is no if we replace $\\mathbb{P}^n$ by an arbitrary variety (there is a surface counterexample).\n\nIf we replace $\\mathbb{P}^n$ by another regular surface, is the condition of not having phantom subcategories equivalent to rationality? What if we take $K_0(X) \\otimes \\mathbb{Q}$?",
   "evidence": "The canonical record is problem 2.1, “Other,” from the 2013 AIM workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0025": {
   "statement_status": "exact",
   "original_statement": "Suppose we have two genus one curves $C$ and $C'$ over a field $F$ which are derived-equivalent. Is $C(F) \\neq \\emptyset$ equivalent to $C'(F) \\neq \\emptyset$? Are $C$ and $C'$ isomorphic over $F$?",
   "clean_statement": "Suppose we have two genus one curves $C$ and $C'$ over a field $F$ which are derived-equivalent. Is $C(F) \\neq \\emptyset$ equivalent to $C'(F) \\neq \\emptyset$? Are $C$ and $C'$ isomorphic over $F$?",
   "public_statement": "Suppose we have two genus one curves $C$ and $C'$ over a field $F$ which are derived-equivalent. Is $C(F) \\neq \\emptyset$ equivalent to $C'(F) \\neq \\emptyset$? Are $C$ and $C'$ isomorphic over $F$?",
   "evidence": "There is no visible corruption or missing notation. Following the paper that arose from the same AIM workshop, “derived-equivalent” is interpreted as an \\(F\\)-linear exact equivalence \\[ D^b(\\operatorname{Coh} C)\\simeq D^b(\\operatorname{Coh} C') \\] of triangulated categories. A genus-one curve means a smooth, projective, geometrically connected curve of genus one. No perfectness hypothesis is imposed: the classification used below is stated over an arbitrary field.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0026": {
   "statement_status": "exact",
   "original_statement": "Consider $\\mathbb{C}(u,v)=\\mathbb{C}(\\mathbb{P}^2)$. Look at the cyclic algebra $A=(u,v)_n$, defined to be $\\mathbb{C}(u,v)\\langle x,y \\rangle/(x^n-u, y^n-v,xy-\\zeta yx)$ with $\\zeta$ a primitive $n$th root of unity. Does there exist a transcendence degree 1 subfield of $A$ (also a $\\mathbb{C}$-subalgebra) which is not rational? Geometrically, can we find a branched cover $S$ of $\\mathbb{P}^2$ corresponding to a maximal subfield of the cyclic algebra, which has a rational map to an irrational curve?",
   "clean_statement": "Consider $\\mathbb{C}(u,v)=\\mathbb{C}(\\mathbb{P}^2)$. Look at the cyclic algebra $A=(u,v)_n$, defined to be $\\mathbb{C}(u,v)\\langle x,y \\rangle/(x^n-u, y^n-v,xy-\\zeta yx)$ with $\\zeta$ a primitive $n$th root of unity. Does there exist a transcendence degree 1 subfield of $A$ (also a $\\mathbb{C}$-subalgebra) which is not rational? Geometrically, can we find a branched cover $S$ of $\\mathbb{P}^2$ corresponding to a maximal subfield of the cyclic algebra, which has a rational map to an irrational curve?",
   "public_statement": "Consider $\\mathbb{C}(u,v)=\\mathbb{C}(\\mathbb{P}^2)$. Look at the cyclic algebra $A=(u,v)_n$, defined to be $\\mathbb{C}(u,v)\\langle x,y \\rangle/(x^n-u, y^n-v,xy-\\zeta yx)$ with $\\zeta$ a primitive $n$th root of unity. Does there exist a transcendence degree 1 subfield of $A$ (also a $\\mathbb{C}$-subalgebra) which is not rational? Geometrically, can we find a branched cover $S$ of $\\mathbb{P}^2$ corresponding to a maximal subfield of the cyclic algebra, which has a rational map to an irrational curve?",
   "evidence": "The canonical record is problem 2.3 in the “Other” section of the 2013 AIM workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0027": {
   "statement_status": "exact",
   "original_statement": "Does any Abelian variety have a transcendental Brauer class which obstructs the existence of rational points?\n\nA theorem of Manin says that if an Abelian variety has a transcendental Brauer class, and $\\mathrm{Sha}(A)$ is finite, $E$ a torsor for $A$, and $E(\\mathbb{A})^{\\mathrm{Br}_{\\mathrm{alg}}} \\neq \\emptyset$, then $E(K) \\neq \\emptyset$. Can we show that the existence of a transcendental obstruction implies the existence of algebraic obstructions which rule out the same adelic points as coming from global points?",
   "clean_statement": "Does any Abelian variety have a transcendental Brauer class which obstructs the existence of rational points?\n\nA theorem of Manin says that if an Abelian variety has a transcendental Brauer class, and $\\mathrm{Sha}(A)$ is finite, $E$ a torsor for $A$, and $E(\\mathbb{A})^{\\mathrm{Br}_{\\mathrm{alg}}} \\neq \\emptyset$, then $E(K) \\neq \\emptyset$. Can we show that the existence of a transcendental obstruction implies the existence of algebraic obstructions which rule out the same adelic points as coming from global points?",
   "public_statement": "Does any Abelian variety have a transcendental Brauer class which obstructs the existence of rational points?\n\nA theorem of Manin says that if an Abelian variety has a transcendental Brauer class, and $\\mathrm{Sha}(A)$ is finite, $E$ a torsor for $A$, and $E(\\mathbb{A})^{\\mathrm{Br}_{\\mathrm{alg}}} \\neq \\emptyset$, then $E(K) \\neq \\emptyset$. Can we show that the existence of a transcendental obstruction implies the existence of algebraic obstructions which rule out the same adelic points as coming from global points?",
   "evidence": "The canonical AIM record (workshop “Brauer groups and obstruction problems: moduli spaces and arithmetic,” section “Other,” Problem 2.4) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0028": {
   "statement_status": "corrected_verified",
   "original_statement": "Let $X$ be a smooth projective complex threefold and a class $\\gamma \\in H^2(X,\\Z/2)$. Consider the integral Bockstein of its square, $\\beta(\\gamma^2) \\in H^5(X,\\Z)$ (coming from the short exact sequence $0 \\to \\Z \\to \\Z \\to \\Z/2 \\to 0$). Take $H^5(X,\\Z)/(H^2(X,\\Z) \\mathbb{C}up \\beta(\\gamma))$. If $\\beta(\\gamma^2) \\neq 0$ in this quotient, then the period index conjecture would be false, since we would have a Brauer class of period 2, and index at least 8. Is there a threefold satisfying these conditions? The period-index conjecture says that the index of $\\alpha$ divides the period of $\\alpha$ to the $(d-1)$-th power, where $d$ is the dimension of the variety.",
   "clean_statement": "Does there exist a smooth projective complex threefold \\(X\\) and\n\\(\\gamma\\in H^2(X,\\mathbb Z/2)\\) for which\n\\([\\beta_2(\\gamma^2)]\\ne0\\) in\n\\(\\mathcal Q_{\\beta_2(\\gamma)}(X)\\)?",
   "public_statement": "Does there exist a smooth projective complex threefold \\(X\\) and\n\\(\\gamma\\in H^2(X,\\mathbb Z/2)\\) for which\n\\([\\beta_2(\\gamma^2)]\\ne0\\) in\n\\(\\mathcal Q_{\\beta_2(\\gamma)}(X)\\)?",
   "evidence": "The source has two corruptions. 1. The integral Bockstein is the connecting map for \\[ 0\\longrightarrow\\mathbb Z\\xrightarrow{\\times2}\\mathbb Z \\longrightarrow\\mathbb Z/2\\longrightarrow0. \\] The multiplication-by-\\(2\\) label is missing from the AIM text. 2. Comparison with Antieau--Williams, *The topological period-index problem over 6-complexes*, identifies the string “\\(\\mathbb{C}up\\)” as a corrupted cup product.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0029": {
   "statement_status": "exact",
   "original_statement": "Fix a variety $X$ and a Brauer class $\\alpha$. Is the set of $\\beta$ such that the $D^b(X,\\alpha)$ are equivalent finite?",
   "clean_statement": "Fix a variety $X$ and a Brauer class $\\alpha$. Is the set of $\\beta$ such that the $D^b(X,\\alpha)$ are equivalent finite?",
   "public_statement": "Fix a variety $X$ and a Brauer class $\\alpha$. Is the set of $\\beta$ such that the $D^b(X,\\alpha)$ are equivalent finite?",
   "evidence": "The canonical record (AIM workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*, item 2.6) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0030": {
   "statement_status": "exact",
   "original_statement": "Let $A$ be an abelian variety over a number field $K$. Does there exist a finite extension $K'/K$ such that for a set of primes $v$ of $K'$ of density $1$, $A/\\F_v$ is ordinary?",
   "clean_statement": "Let $A$ be an abelian variety over a number field $K$. Does there exist a finite extension $K'/K$ such that for a set of primes $v$ of $K'$ of density $1$, $A/\\F_v$ is ordinary?",
   "public_statement": "Let $A$ be an abelian variety over a number field $K$. Does there exist a finite extension $K'/K$ such that for a set of primes $v$ of $K'$ of density $1$, $A/\\F_v$ is ordinary?",
   "evidence": "The repository record has no visible OCR corruption. Its cited source URL, <http://aimpl.org/cohomabelian/1/>, timed out during this run, so it could not be compared directly with the live page. The same formulation appears at the beginning of Section 7 of Pink [Pin98] and in Section 1.1 of Cantoral Farfán--Li--Mantovan--Pries--Tang [CFLMPT25]. There is therefore no substantive ambiguity in the recovered question.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0031": {
   "statement_status": "exact",
   "original_statement": "Let $X/\\mathbb C$ be a surface of general type. Then Lang's conjecture implies that there exists a constant $N$, depending only on $X$, such that if $C \\subseteq X$ is rational, then $K_X \\cdot C \\leq N$.\n\nWhat happens if $X$ is an elliptic surface? Is there a bound on $K_X \\cdot C$ as $C$ varies through rational curves?",
   "clean_statement": "Let $X/\\mathbb C$ be a surface of general type. Then Lang's conjecture implies that there exists a constant $N$, depending only on $X$, such that if $C \\subseteq X$ is rational, then $K_X \\cdot C \\leq N$.\n\nWhat happens if $X$ is an elliptic surface? Is there a bound on $K_X \\cdot C$ as $C$ varies through rational curves?",
   "public_statement": "Let $X/\\mathbb C$ be a surface of general type. Then Lang's conjecture implies that there exists a constant $N$, depending only on $X$, such that if $C \\subseteq X$ is rational, then $K_X \\cdot C \\leq N$.\n\nWhat happens if $X$ is an elliptic surface? Is there a bound on $K_X \\cdot C$ as $C$ varies through rational curves?",
   "evidence": "The canonical AIM record (source file `aim-arithmetic-geometry-notes.json`, zero-based index 30) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0032": {
   "statement_status": "exact",
   "original_statement": "Let $K$ be a number field, and let $A/K$ be an abelian variety. Do there exist infinitely many places $v$ of $K$ such that the reduction $A/\\F_v$ is not ordinary? More strongly, do there exist infinitely many places $v$ of $K$ such that the reduction $A/\\F_v$ is supersingular?",
   "clean_statement": "Let $K$ be a number field, and let $A/K$ be an abelian variety. Do there exist infinitely many places $v$ of $K$ such that the reduction $A/\\F_v$ is not ordinary? More strongly, do there exist infinitely many places $v$ of $K$ such that the reduction $A/\\F_v$ is supersingular?",
   "public_statement": "Let $K$ be a number field, and let $A/K$ be an abelian variety. Do there exist infinitely many places $v$ of $K$ such that the reduction $A/\\F_v$ is not ordinary? More strongly, do there exist infinitely many places $v$ of $K$ such that the reduction $A/\\F_v$ is supersingular?",
   "evidence": "Here and below, $v$ means a finite place of good reduction and $\\mathbb F_v=\\kappa(v)$ is its residue field. This restriction is implicit in the phrase \"the reduction $A/\\mathbb F_v$.\" The local canonical JSON record is internally consistent with the adjacent workshop problems 1.1, 1.2, and 1.4. The original AIM URL timed out during this run, so the wording above is verified from the canonical repository copy rather than a fresh rendering of the web page. There is no apparent OCR corruption.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0033": {
   "statement_status": "exact",
   "original_statement": "(a) Let $K$ be an algebraically closed field, $X/K$ an abelian variety. The Voevodsky motive $F$ is given as a sheaf of groups $F(S) = \\textrm{Mor}_K(S,X)$. Is $F$ geometric (i.e. in the triangulated subcategory generated by smooth projective varieties)?\n\n(b) (This is a stronger variant of (a)): Is every principally polarized abelian variety a direct summand of a Jacobian integrally (as an abelian variety)?",
   "clean_statement": "(a) Let $K$ be an algebraically closed field, $X/K$ an abelian variety. The Voevodsky motive $F$ is given as a sheaf of groups $F(S) = \\textrm{Mor}_K(S,X)$. Is $F$ geometric (i.e. in the triangulated subcategory generated by smooth projective varieties)?\n\n(b) (This is a stronger variant of (a)): Is every principally polarized abelian variety a direct summand of a Jacobian integrally (as an abelian variety)?",
   "public_statement": "(a) Let $K$ be an algebraically closed field, $X/K$ an abelian variety. The Voevodsky motive $F$ is given as a sheaf of groups $F(S) = \\textrm{Mor}_K(S,X)$. Is $F$ geometric (i.e. in the triangulated subcategory generated by smooth projective varieties)?\n\n(b) (This is a stronger variant of (a)): Is every principally polarized abelian variety a direct summand of a Jacobian integrally (as an abelian variety)?",
   "evidence": "The record has no apparent OCR corruption. It does, however, suppress three choices which affect the answer to (a): Nisnevich versus étale motives, the coefficient ring, and the placement of the sheaf as a complex. The standard reconstruction is that \\(F_X\\) is the homotopy-invariant sheaf with transfers represented by the commutative group scheme \\(X\\), placed in cohomological degree zero in \\(DM^{\\mathrm{eff}}_-(K,\\mathbb Z)\\). The transfer along a finite correspondence is induced by the sum/trace map on the abelian variety.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0034": {
   "statement_status": "exact",
   "original_statement": "Let $p$ be a prime number. For a fixed dimension $g$ and a fixed $p$-rank $f$, what is the smallest field of definition of a simple abelian variety over $\\overline{\\F_p}$ with dimension $g$ and $p$-rank $f$?",
   "clean_statement": "Let $p$ be a prime number. For a fixed dimension $g$ and a fixed $p$-rank $f$, what is the smallest field of definition of a simple abelian variety over $\\overline{\\F_p}$ with dimension $g$ and $p$-rank $f$?",
   "public_statement": "Let $p$ be a prime number. For a fixed dimension $g$ and a fixed $p$-rank $f$, what is the smallest field of definition of a simple abelian variety over $\\overline{\\F_p}$ with dimension $g$ and $p$-rank $f$?",
   "evidence": "The canonical record (AIM workshop *Cohomological methods in abelian varieties*, Problems 1.5) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0035": {
   "statement_status": "exact",
   "original_statement": "Is every abelian variety the quotient of $\\textrm{Jac}(C)$, where $C$ is a smooth plane curve?",
   "clean_statement": "Is every abelian variety the quotient of $\\textrm{Jac}(C)$, where $C$ is a smooth plane curve?",
   "public_statement": "Is every abelian variety the quotient of $\\textrm{Jac}(C)$, where $C$ is a smooth plane curve?",
   "evidence": "The canonical AIM record (workshop *Cohomological methods in abelian varieties*, Problems 1.6) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0036": {
   "statement_status": "exact",
   "original_statement": "Let $X$ be a principally polarized abelian variety of dimension $\\geq 4$. Is $\\theta^2 \\in CH^2(X)/\\textrm{tors}$ divisible by $2$?",
   "clean_statement": "Let $X$ be a principally polarized abelian variety of dimension $\\geq 4$. Is $\\theta^2 \\in CH^2(X)/\\textrm{tors}$ divisible by $2$?",
   "public_statement": "Let $X$ be a principally polarized abelian variety of dimension $\\geq 4$. Is $\\theta^2 \\in CH^2(X)/\\textrm{tors}$ divisible by $2$?",
   "evidence": "The canonical AIM record (workshop *Cohomological methods in abelian varieties*, Problems 1.7) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0037": {
   "statement_status": "exact",
   "original_statement": "Let $A$ be an abelian variety over a finite field $K$, and $u: A \\to A$ a fixed automorphism. Then $u$ induces $u^*: H^*(\\overline{A}, \\Q_l) \\to H^*(\\overline{A}, \\Q_l)$. Let $\\alpha_i$ be its eigenvalues, and let $|\\alpha_i|$ denote the archimedean size of $\\alpha_i$. Take the subring in $H^*(\\overline{A}, \\Q_l)$ generated by a polarization and stabilized by $u$. Is the maximum archimedean size of the eigenvalues of the subring equal to the maximum archimedean size of the eigenvalues of $H^*(\\overline{A}, \\Q_l)$?",
   "clean_statement": "Let $A$ be an abelian variety over a finite field $K$, and $u: A \\to A$ a fixed automorphism. Then $u$ induces $u^*: H^*(\\overline{A}, \\Q_l) \\to H^*(\\overline{A}, \\Q_l)$. Let $\\alpha_i$ be its eigenvalues, and let $|\\alpha_i|$ denote the archimedean size of $\\alpha_i$. Take the subring in $H^*(\\overline{A}, \\Q_l)$ generated by a polarization and stabilized by $u$. Is the maximum archimedean size of the eigenvalues of the subring equal to the maximum archimedean size of the eigenvalues of $H^*(\\overline{A}, \\Q_l)$?",
   "public_statement": "Let $A$ be an abelian variety over a finite field $K$, and $u: A \\to A$ a fixed automorphism. Then $u$ induces $u^*: H^*(\\overline{A}, \\Q_l) \\to H^*(\\overline{A}, \\Q_l)$. Let $\\alpha_i$ be its eigenvalues, and let $|\\alpha_i|$ denote the archimedean size of $\\alpha_i$. Take the subring in $H^*(\\overline{A}, \\Q_l)$ generated by a polarization and stabilized by $u$. Is the maximum archimedean size of the eigenvalues of the subring equal to the maximum archimedean size of the eigenvalues of $H^*(\\overline{A}, \\Q_l)$?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 36,
   "attempt": 2
  },
  "AIM-ARITHMETIC_GEOMETRY-0038": {
   "statement_status": "exact",
   "original_statement": "If $u(F) < \\infty$, does there exist $B \\in \\N$ such that $u(F') <\n \\infty$ for all $F'/F$ finite? In particular, is $B = 2u(F)$\n sufficient?",
   "clean_statement": "If $u(F) < \\infty$, does there exist $B \\in \\N$ such that $u(F') <\n \\infty$ for all $F'/F$ finite? In particular, is $B = 2u(F)$\n sufficient?",
   "public_statement": "If $u(F) < \\infty$, does there exist $B \\in \\N$ such that $u(F') <\n \\infty$ for all $F'/F$ finite? In particular, is $B = 2u(F)$\n sufficient?",
   "evidence": "The exact canonical record in `aim-arithmetic-geometry-notes.json`, source index 37, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 37,
   "attempt": 2
  },
  "AIM-ARITHMETIC_GEOMETRY-0039": {
   "statement_status": "exact",
   "original_statement": "Find interesting classes $\\mathcal F$ of fields such that for $F \\in\n \\mathcal F$,\n\\[u(F) < \\infty \\Longrightarrow u(F(t)) < \\infty.\\]",
   "clean_statement": "Find interesting classes $\\mathcal F$ of fields such that for $F \\in\n \\mathcal F$,\n\\[u(F) < \\infty \\Longrightarrow u(F(t)) < \\infty.\\]",
   "public_statement": "Find interesting classes $\\mathcal F$ of fields such that for $F \\in\n \\mathcal F$,\n\\[u(F) < \\infty \\Longrightarrow u(F(t)) < \\infty.\\]",
   "evidence": "The canonical AIM record (workshop section “The $u$-invariant problem,” Problem 1.2) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0040": {
   "statement_status": "exact",
   "original_statement": "Define \\[u_k(F) = \\max \\{ \\dim (q)\\ \\ |\\ \\ q \\in I^k \\}.\\]\nThen $u_0 = u$.\n\nCan we compute $u_k(F)$? How about $u_3(F)$?",
   "clean_statement": "Define \\[u_k(F) = \\max \\{ \\dim (q)\\ \\ |\\ \\ q \\in I^k \\}.\\]\nThen $u_0 = u$.\n\nCan we compute $u_k(F)$? How about $u_3(F)$?",
   "public_statement": "Define \\[u_k(F) = \\max \\{ \\dim (q)\\ \\ |\\ \\ q \\in I^k \\}.\\]\nThen $u_0 = u$.\n\nCan we compute $u_k(F)$? How about $u_3(F)$?",
   "evidence": "The canonical repository record (source index 39 in `aim-arithmetic-geometry-notes.json`) transcribes the 2011 AIM problem as",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0041": {
   "statement_status": "exact",
   "original_statement": "Let \\[\\nu(F) = \\max \\{n\\ |\\ I^n \\neq 0 \\}. \\]\n\nIf $\\nu(F) < \\infty$, is it true that $\\nu(F') < \\nu(F) +1$ for all\n$F'/F$ finite?",
   "clean_statement": "Let \\[\\nu(F) = \\max \\{n\\ |\\ I^n \\neq 0 \\}. \\]\n\nIf $\\nu(F) < \\infty$, is it true that $\\nu(F') < \\nu(F) +1$ for all\n$F'/F$ finite?",
   "public_statement": "Let \\[\\nu(F) = \\max \\{n\\ |\\ I^n \\neq 0 \\}. \\]\n\nIf $\\nu(F) < \\infty$, is it true that $\\nu(F') < \\nu(F) +1$ for all\n$F'/F$ finite?",
   "evidence": "The canonical record at index 40 of `aim-arithmetic-geometry-notes.json`, from the 2011 AIM workshop *Deformation theory, patching, quadratic forms, and the Brauer group*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0042": {
   "statement_status": "exact",
   "original_statement": "For a quadratic form $q$ over $F$, we define\n\\[\\textrm{splitting degree} = \\min\\{[L:F]\\ :\\ q_{L}\\ \\textrm{is a\n direct sum of hyperbolics}\\}.\\]\nThe \\emph{torsion index} of $F$, denoted $\\tau_F$ is the maximum\nsplitting degree, taken over all even dimensional $q$.\n\nIs $\\tau_F < 2^{(\\frac{u(F)}{2} -1)}$?",
   "clean_statement": "For a quadratic form $q$ over $F$, we define\n\\[\\textrm{splitting degree} = \\min\\{[L:F]\\ :\\ q_{L}\\ \\textrm{is a\n direct sum of hyperbolics}\\}.\\]\nThe \\emph{torsion index} of $F$, denoted $\\tau_F$ is the maximum\nsplitting degree, taken over all even dimensional $q$.\n\nIs $\\tau_F < 2^{(\\frac{u(F)}{2} -1)}$?",
   "public_statement": "For a quadratic form $q$ over $F$, we define\n\\[\\textrm{splitting degree} = \\min\\{[L:F]\\ :\\ q_{L}\\ \\textrm{is a\n direct sum of hyperbolics}\\}.\\]\nThe \\emph{torsion index} of $F$, denoted $\\tau_F$ is the maximum\nsplitting degree, taken over all even dimensional $q$.\n\nIs $\\tau_F < 2^{(\\frac{u(F)}{2} -1)}$?",
   "evidence": "The canonical record is problem 1.5 in the AIM list for the workshop *Deformation theory, patching, quadratic forms, and the Brauer group*. It says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0043": {
   "statement_status": "exact",
   "original_statement": "When is $\\tau_F$ finite?",
   "clean_statement": "When is $\\tau_F$ finite?",
   "public_statement": "When is $\\tau_F$ finite?",
   "evidence": "The canonical record is Problem 1.6 in the AIM workshop section *The $u$-invariant problem*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0044": {
   "statement_status": "exact",
   "original_statement": "When $F = \\R$ , we have $u(F) = \\infty$ but $\\tau_F < \\infty$. Is there\n an example where this holds for $F$ not formally real?",
   "clean_statement": "When $F = \\R$ , we have $u(F) = \\infty$ but $\\tau_F < \\infty$. Is there\n an example where this holds for $F$ not formally real?",
   "public_statement": "When $F = \\R$ , we have $u(F) = \\infty$ but $\\tau_F < \\infty$. Is there\n an example where this holds for $F$ not formally real?",
   "evidence": "The canonical record in `aim-arithmetic-geometry-notes.json`, source index 43, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0045": {
   "statement_status": "exact",
   "original_statement": "Find reasonable classes $\\mathcal F$ of fields such that $u(F)$ is a\n power of 2 for all $F \\in \\mathcal F$. For these fields, assume $k$ is the\n period-index bound for $l=2$. Does this imply $u(F) = 2^{k+1}$?",
   "clean_statement": "Find reasonable classes $\\mathcal F$ of fields such that $u(F)$ is a\n power of 2 for all $F \\in \\mathcal F$. For these fields, assume $k$ is the\n period-index bound for $l=2$. Does this imply $u(F) = 2^{k+1}$?",
   "public_statement": "Find reasonable classes $\\mathcal F$ of fields such that $u(F)$ is a\n power of 2 for all $F \\in \\mathcal F$. For these fields, assume $k$ is the\n period-index bound for $l=2$. Does this imply $u(F) = 2^{k+1}$?",
   "evidence": "The record is Problem 1.8 in the section *The \\(u\\)-invariant problem* from the AIM workshop *Deformation theory, patching, quadratic forms, and the Brauer group*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 44,
   "attempt": 2
  },
  "AIM-ARITHMETIC_GEOMETRY-0046": {
   "statement_status": "exact",
   "original_statement": "Name of the problem block\n\nIntro to problem block\n\nThis is the problem statement.",
   "clean_statement": "Name of the problem block\n\nIntro to problem block\n\nThis is the problem statement.",
   "public_statement": "Name of the problem block\n\nIntro to problem block\n\nThis is the problem statement.",
   "evidence": "The exact canonical record is labeled as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0047": {
   "statement_status": "exact",
   "original_statement": "Test introduction\n\nthe statement of a throw-away problem",
   "clean_statement": "Test introduction\n\nthe statement of a throw-away problem",
   "public_statement": "Test introduction\n\nthe statement of a throw-away problem",
   "evidence": "The exact canonical metadata are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0048": {
   "statement_status": "exact",
   "original_statement": "Problem 1. Describe the singularities of the smoothable component of Hilb d(An). To be more specific, \n\n• How large can the dimension of the Zariski tangent space to this component get? \n\n• Does the maximum occur in the intersection of components? Does it occur in the smoothable component?",
   "clean_statement": "Problem 1. Describe the singularities of the smoothable component of Hilb d(An). To be more specific,\n\n• How large can the dimension of the Zariski tangent space to this component get?\n\n• Does the maximum occur in the intersection of components? Does it occur in the smoothable component?",
   "public_statement": "Problem 1. Describe the singularities of the smoothable component of Hilb d(An). To be more specific,\n\n• How large can the dimension of the Zariski tangent space to this component get?\n\n• Does the maximum occur in the intersection of components? Does it occur in the smoothable component?",
   "evidence": "The final sentence is verified verbatim from the PDF; it is not an extraction error. Read literally, its last question is redundant, since the tangent space “to this component” is based at a point of the component. A plausible intended contrast is between points lying on several components and points lying only on the smoothable component, or between the ambient Hilbert-scheme tangent space along the smoothable component and the tangent space of the reduced component itself. The source does not settle that ambiguity. This report therefore distinguishes \\[ T_{[Z]}\\operatorname{Hilb}^{D}(\\mathbb A^n) \\quad\\text{from}\\quad T_{[Z]}\\mathcal R_D^n, \\] where \\(\\mathcal R_D^n\\) is the reduced smoothable component, and constructs points at which these spaces are equal. It separately determines when the constructed point is known to lie at an intersection of components.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0049": {
   "statement_status": "exact",
   "original_statement": "Problem 2. Can you describe the Zariski tangent space to the smoothable component of Hilb d(An)?",
   "clean_statement": "Problem 2. Can you describe the Zariski tangent space to the smoothable component of Hilb d(An)?",
   "public_statement": "Problem 2. Can you describe the Zariski tangent space to the smoothable component of Hilb d(An)?",
   "evidence": "The exact canonical extraction is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 48,
   "attempt": 3
  },
  "AIM-ARITHMETIC_GEOMETRY-0050": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 3. Fix d, g, r, e > 0. Let Hilb sm d,g (Pr) be the open subscheme of the Hilbert scheme \n\nHilb d,g (Pr) that parameterizes smooth curves. For each point [C] ∈ Hilb sm d,g (Pr), we define the Gauss map C → Gr(1, r ) sending a point of C to the tangent line at that point. Define \n\nZe:= {[C] ∈ Hilb sm d,g (Pr)| the Gauss map of the curve C is inseparable of degree pe}.\n\nThen ∪ \n\n> e≥0\n\nZe = Hilb sm d,g (Pr).What can we say about the set Ze? Can we construct exotic components (i.e. components that only exist in characteristic p) using this stratification? Study the action by the Galois group \n\nGal( Fp/Fp).",
   "clean_statement": null,
   "public_statement": "Problem 3. Fix d, g, r, e > 0. Let Hilb sm d,g (Pr) be the open subscheme of the Hilbert scheme\n\nHilb d,g (Pr) that parameterizes smooth curves. For each point [C] ∈ Hilb sm d,g (Pr), we define the Gauss map C → Gr(1, r ) sending a point of C to the tangent line at that point. Define\n\nZe:= {[C] ∈ Hilb sm d,g (Pr)| the Gauss map of the curve C is inseparable of degree pe}.\n\nThen ∪\n\n> e≥0\n\nZe = Hilb sm d,g (Pr).What can we say about the set Ze? Can we construct exotic components (i.e. components that only exist in characteristic p) using this stratification? Study the action by the Galois group\n\nGal( Fp/Fp).",
   "evidence": "The primary source is the four-page problem list *Components of Hilbert Schemes*, recorded by Izzet Coskun and edited by Li Li after the AIM workshop of July 19--23, 2010. The PDF typography restores the corrupted extraction as follows.",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 49,
   "attempt": 2
  },
  "AIM-ARITHMETIC_GEOMETRY-0051": {
   "statement_status": "exact",
   "original_statement": "Problem 4. (1) Is there a component of Hilb d(An) that exists only in characteristic p for some p?(2) Same question for the Hilbert schemes of curves in P3.",
   "clean_statement": "Problem 4. (1) Is there a component of Hilb d(An) that exists only in characteristic p for some p?(2) Same question for the Hilbert schemes of curves in P3.",
   "public_statement": "Problem 4. (1) Is there a component of Hilb d(An) that exists only in characteristic p for some p?(2) Same question for the Hilbert schemes of curves in P3.",
   "evidence": "The AIM workshop list *Components of Hilbert Schemes* asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0052": {
   "statement_status": "exact",
   "original_statement": "Problem 5. Is there a nonreduced component of Hilb d(An)? If so, find it.",
   "clean_statement": "Problem 5. Is there a nonreduced component of Hilb d(An)? If so, find it.",
   "public_statement": "Problem 5. Is there a nonreduced component of Hilb d(An)? If so, find it.",
   "evidence": "The canonical record is Problem 5 from the 2010 AIM workshop list *Components of Hilbert schemes*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0053": {
   "statement_status": "exact",
   "original_statement": "Problem 6. (1) Give an explicit example (or show it does not happen) of a geometrically irreducible component of Hilb d(An), which is not fixed under the action of Gal( Q/Q).(2) Same question for the Hilbert schemes of curves in P3.",
   "clean_statement": "Problem 6. (1) Give an explicit example (or show it does not happen) of a geometrically irreducible component of Hilb d(An), which is not fixed under the action of Gal( Q/Q).(2) Same question for the Hilbert schemes of curves in P3.",
   "public_statement": "Problem 6. (1) Give an explicit example (or show it does not happen) of a geometrically irreducible component of Hilb d(An), which is not fixed under the action of Gal( Q/Q).(2) Same question for the Hilbert schemes of curves in P3.",
   "evidence": "The typeset AIM source reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0054": {
   "statement_status": "exact",
   "original_statement": "Problem 7. Does there exist a nonrational component of Hilb d(An)? \n\n> 12RECORDED ON WHITEBOARD BY IZZET COSKUN, AND EDITED BY LI LI",
   "clean_statement": "Problem 7. Does there exist a nonrational component of Hilb d(An)?\n\n> 12RECORDED ON WHITEBOARD BY IZZET COSKUN, AND EDITED BY LI LI",
   "public_statement": "Problem 7. Does there exist a nonrational component of Hilb d(An)?\n\n> 12RECORDED ON WHITEBOARD BY IZZET COSKUN, AND EDITED BY LI LI",
   "evidence": "The primary source is the four-page AIM problem list *Components of Hilbert Schemes*, recorded by Izzet Coskun and edited by Li Li after the workshop of 19--23 July 2010. Its notation paragraph says that \\(\\operatorname{Hilb}^{d}(\\mathbb A^{n})\\) denotes the Hilbert scheme of \\(d\\) points in affine \\(n\\)-space and that “component” means irreducible component. Page 1 gives the exact problem:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 53,
   "attempt": 2
  },
  "AIM-ARITHMETIC_GEOMETRY-0055": {
   "statement_status": "exact",
   "original_statement": "Problem 8. If a component of the Hilbert scheme contains a smooth Borel fixed point, does the component have to be rational?",
   "clean_statement": "Problem 8. If a component of the Hilbert scheme contains a smooth Borel fixed point, does the component have to be rational?",
   "public_statement": "Problem 8. If a component of the Hilbert scheme contains a smooth Borel fixed point, does the component have to be rational?",
   "evidence": "The exact 2010 AIM question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0056": {
   "statement_status": "exact",
   "original_statement": "Problem 9. Is the Hilbert scheme of local Cohen-Macaulay curves in P3 connected?",
   "clean_statement": "Problem 9. Is the Hilbert scheme of local Cohen-Macaulay curves in P3 connected?",
   "public_statement": "Problem 9. Is the Hilbert scheme of local Cohen-Macaulay curves in P3 connected?",
   "evidence": "The primary AIM workshop PDF (page 2 of the PDF, numbered page 2) literally prints:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0057": {
   "statement_status": "exact",
   "original_statement": "Problem 10. Let ρ = ( p, q, r, 0, 0,... ) and define Ep,q,r:= Eρ to be the moduli space of finite length graded modules of function ρ. For which p, q, r is Ep,q,r irreducible?",
   "clean_statement": "Problem 10. Let ρ = ( p, q, r, 0, 0,... ) and define Ep,q,r:= Eρ to be the moduli space of finite length graded modules of function ρ. For which p, q, r is Ep,q,r irreducible?",
   "public_statement": "Problem 10. Let ρ = ( p, q, r, 0, 0,... ) and define Ep,q,r:= Eρ to be the moduli space of finite length graded modules of function ρ. For which p, q, r is Ep,q,r irreducible?",
   "evidence": "The AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0058": {
   "statement_status": "exact",
   "original_statement": "Problem 11. Describe the irreducible component of Eρ, the moduli of finite length graded modules of function ρ.",
   "clean_statement": "Problem 11. Describe the irreducible component of Eρ, the moduli of finite length graded modules of function ρ.",
   "public_statement": "Problem 11. Describe the irreducible component of Eρ, the moduli of finite length graded modules of function ρ.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0059": {
   "statement_status": "exact",
   "original_statement": "Problem 12. What do properties of the Rao modules imply about C? For example, if MC is Gorenstein or annihilated by a linear form, does C have any nice properties?",
   "clean_statement": "Problem 12. What do properties of the Rao modules imply about C? For example, if MC is Gorenstein or annihilated by a linear form, does C have any nice properties?",
   "public_statement": "Problem 12. What do properties of the Rao modules imply about C? For example, if MC is Gorenstein or annihilated by a linear form, does C have any nice properties?",
   "evidence": "The primary AIM workshop PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0060": {
   "statement_status": "exact",
   "original_statement": "Problem 13. Let Ct be a family of curves in P3 such that a general curve in this family is a smooth complete intersection, and the special curve C0 is smooth. Does it imply that C0 is also a complete intersection, assuming that the characteristic is 0?",
   "clean_statement": "Problem 13. Let Ct be a family of curves in P3 such that a general curve in this family is a smooth complete intersection, and the special curve C0 is smooth. Does it imply that C0 is also a complete intersection, assuming that the characteristic is 0?",
   "public_statement": "Problem 13. Let Ct be a family of curves in P3 such that a general curve in this family is a smooth complete intersection, and the special curve C0 is smooth. Does it imply that C0 is also a complete intersection, assuming that the characteristic is 0?",
   "evidence": "The AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0061": {
   "statement_status": "exact",
   "original_statement": "Problem 14. Give a geometric algebraic description of generic points of irreducible components of Hilb d(An).",
   "clean_statement": "Problem 14. Give a geometric algebraic description of generic points of irreducible components of Hilb d(An).",
   "public_statement": "Problem 14. Give a geometric algebraic description of generic points of irreducible components of Hilb d(An).",
   "evidence": "The exact extracted record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0062": {
   "statement_status": "exact",
   "original_statement": "Problem 15. Is the Gr¨ obner fan a discrete invariant that distinguishes the components of Hilb d(An)?",
   "clean_statement": "Problem 15. Is the Gr¨ obner fan a discrete invariant that distinguishes the components of Hilb d(An)?",
   "public_statement": "Problem 15. Is the Gr¨ obner fan a discrete invariant that distinguishes the components of Hilb d(An)?",
   "evidence": "The source is Problem 15 in the AIM workshop list *Components of Hilbert Schemes* (2010). The OCR record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0063": {
   "statement_status": "exact",
   "original_statement": "Problem 16. Does the set of monomial ideals contained in a component of Hilb d(An) determine that component? (Or more generally for Hilb P (Pn) for an arbitrary Hilbert polynomial P.)",
   "clean_statement": "Problem 16. Does the set of monomial ideals contained in a component of Hilb d(An) determine that component? (Or more generally for Hilb P (Pn) for an arbitrary Hilbert polynomial P.)",
   "public_statement": "Problem 16. Does the set of monomial ideals contained in a component of Hilb d(An) determine that component? (Or more generally for Hilb P (Pn) for an arbitrary Hilbert polynomial P.)",
   "evidence": "The AIM record (Problem 16 in *Components of Hilbert schemes*) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 62,
   "attempt": 2
  },
  "AIM-ARITHMETIC_GEOMETRY-0064": {
   "statement_status": "exact",
   "original_statement": "Problem 17. Let S ⊂ P3 be an integral surface of degree d with a double curve D of degree e, and triple points, pinch point,.... Conjecture: there exists n0 ∈ Z such that for any smooth curve C ⊂ S of degree n ≥ n0, let HC\n\nbe the irreducible component of the Hilbert scheme containing C, then a general C′ ∈ HC is also contained in a surface S′ ⊂ P3 of degree d and a double curve D′ of degree e.",
   "clean_statement": "Problem 17. Let S ⊂ P3 be an integral surface of degree d with a double curve D of degree e, and triple points, pinch point,.... Conjecture: there exists n0 ∈ Z such that for any smooth curve C ⊂ S of degree n ≥ n0, let HC\n\nbe the irreducible component of the Hilbert scheme containing C, then a general C′ ∈ HC is also contained in a surface S′ ⊂ P3 of degree d and a double curve D′ of degree e.",
   "public_statement": "Problem 17. Let S ⊂ P3 be an integral surface of degree d with a double curve D of degree e, and triple points, pinch point,.... Conjecture: there exists n0 ∈ Z such that for any smooth curve C ⊂ S of degree n ≥ n0, let HC\n\nbe the irreducible component of the Hilbert scheme containing C, then a general C′ ∈ HC is also contained in a surface S′ ⊂ P3 of degree d and a double curve D′ of degree e.",
   "evidence": "The exact extracted record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 63,
   "attempt": 2
  },
  "AIM-ARITHMETIC_GEOMETRY-0065": {
   "statement_status": "exact",
   "original_statement": "Problem 18. Is it possible to define analogues of the Nakajima operators for the cohomology of a desingularization of the smoothable component of Hilb d(An)?",
   "clean_statement": "Problem 18. Is it possible to define analogues of the Nakajima operators for the cohomology of a desingularization of the smoothable component of Hilb d(An)?",
   "public_statement": "Problem 18. Is it possible to define analogues of the Nakajima operators for the cohomology of a desingularization of the smoothable component of Hilb d(An)?",
   "evidence": "The AIM source asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0066": {
   "statement_status": "exact",
   "original_statement": "Problem 19. What is the geography of locally Cohen-Macaulay surfaces in P4? To be more precise, what numerical invariants (degree, sectional genus,... ) occur?",
   "clean_statement": "Problem 19. What is the geography of locally Cohen-Macaulay surfaces in P4? To be more precise, what numerical invariants (degree, sectional genus,... ) occur?",
   "public_statement": "Problem 19. What is the geography of locally Cohen-Macaulay surfaces in P4? To be more precise, what numerical invariants (degree, sectional genus,... ) occur?",
   "evidence": "The AIM workshop list *Components of Hilbert schemes* asks in Problem 19:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0067": {
   "statement_status": "exact",
   "original_statement": "Problem 20. Let C be a curve, can Hilb d(C) have a component of dim < d?",
   "clean_statement": "Problem 20. Let C be a curve, can Hilb d(C) have a component of dim < d?",
   "public_statement": "Problem 20. Let C be a curve, can Hilb d(C) have a component of dim < d?",
   "evidence": "The canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0068": {
   "statement_status": "exact",
   "original_statement": "Problem 21. Is there a multigraded Hilbert scheme with a connected component isomorphic to a fat point?",
   "clean_statement": "Problem 21. Is there a multigraded Hilbert scheme with a connected component isomorphic to a fat point?",
   "public_statement": "Problem 21. Is there a multigraded Hilbert scheme with a connected component isomorphic to a fat point?",
   "evidence": "The exact AIM workshop question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0069": {
   "statement_status": "exact",
   "original_statement": "Problem 22. Fix d, g, n. Find a good/sharp lower bound for the dimension of components of \n\nHilb d,g (Pn).(For P3 a lower bound is 4d; for P4 the lower bound is 5d + 1 − g which is obviously not a good bound for d fixed and g sufficiently large.)",
   "clean_statement": "Problem 22. Fix d, g, n. Find a good/sharp lower bound for the dimension of components of\n\nHilb d,g (Pn).(For P3 a lower bound is 4d; for P4 the lower bound is 5d + 1 − g which is obviously not a good bound for d fixed and g sufficiently large.)",
   "public_statement": "Problem 22. Fix d, g, n. Find a good/sharp lower bound for the dimension of components of\n\nHilb d,g (Pn).(For P3 a lower bound is 4d; for P4 the lower bound is 5d + 1 − g which is obviously not a good bound for d fixed and g sufficiently large.)",
   "evidence": "The canonical record is Problem 22 from the AIM workshop *Components of Hilbert Schemes* (19--23 July 2010). Its OCR text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0070": {
   "statement_status": "exact",
   "original_statement": "Problem \n2\n3. (An old question of Joe Harris) Does there exist a nondegenerate rigid curve in Pn\n\nother than the rational normal curve? Here rigid means the only deformations of the curve are those induced by automorphisms of Pn.",
   "clean_statement": "Problem\n2\n3. (An old question of Joe Harris) Does there exist a nondegenerate rigid curve in Pn\n\nother than the rational normal curve? Here rigid means the only deformations of the curve are those induced by automorphisms of Pn.",
   "public_statement": "Problem\n2\n3. (An old question of Joe Harris) Does there exist a nondegenerate rigid curve in Pn\n\nother than the rational normal curve? Here rigid means the only deformations of the curve are those induced by automorphisms of Pn.",
   "evidence": "The machine-extracted record breaks the problem number and superscripts across lines. Page 2 of the original AIM workshop PDF gives the following unambiguous statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 69,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0071": {
   "statement_status": "exact",
   "original_statement": "Problem 24. Is Hilb 8(A4) reduced? More generally, develop techniques to prove the reducedness of Hilbert schemes.",
   "clean_statement": "Problem 24. Is Hilb 8(A4) reduced? More generally, develop techniques to prove the reducedness of Hilbert schemes.",
   "public_statement": "Problem 24. Is Hilb 8(A4) reduced? More generally, develop techniques to prove the reducedness of Hilbert schemes.",
   "evidence": "The canonical record is Problem 24 in the AIM workshop list *Components of Hilbert Schemes* (July 2010):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 70,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0072": {
   "statement_status": "exact",
   "original_statement": "Problem 25. For R = k[x1, x 2, x 3]. Is every multigraded Hilbert scheme connected?",
   "clean_statement": "Problem 25. For R = k[x1, x 2, x 3]. Is every multigraded Hilbert scheme connected?",
   "public_statement": "Problem 25. For R = k[x1, x 2, x 3]. Is every multigraded Hilbert scheme connected?",
   "evidence": "The exact AIM workshop record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0073": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 26. (i) Let C be of bidegree (3, 7) on a nonsingular quadric surface in P3. Can C be connected to an extremal curve in Hilb 10,12 (P3)?(ii) Given 4 skew lines C1 on a nonsingular quadric Q1 in P3. Does there exists a family \n\nQt 2H, and a family Ct ⊂ Qt such that C0 ⊂ Q0 = 2 H is locally Cohen-Macaulay? 4 RECORDED ON WHITEBOARD BY IZZET COSKUN, AND EDITED BY LI LI",
   "clean_statement": "**Problem 26.** (i) Let \\(C\\) be of bidegree \\((3,7)\\) on a nonsingular\nquadric surface in \\(\\mathbf P^3\\).  Can \\(C\\) be connected to an extremal\ncurve in \\(\\operatorname{Hilb}_{10,12}(\\mathbf P^3)\\)?\n\n(ii) Given four skew lines \\(C_1\\) on a nonsingular quadric \\(Q_1\\), does\nthere exist a family \\(Q_t\\rightsquigarrow 2H\\), and a family\n\\(C_t\\subset Q_t\\), such that \\(C_0\\subset Q_0=2H\\) is locally\nCohen--Macaulay?\n\n**Remark.** (ii) implies (i).",
   "public_statement": "Problem 26. (i) Let C be of bidegree (3, 7) on a nonsingular quadric surface in P3. Can C be connected to an extremal curve in Hilb 10,12 (P3)?(ii) Given 4 skew lines C1 on a nonsingular quadric Q1 in P3. Does there exists a family\n\nQt 2H, and a family Ct ⊂ Qt such that C0 ⊂ Q0 = 2 H is locally Cohen-Macaulay? 4 RECORDED ON WHITEBOARD BY IZZET COSKUN, AND EDITED BY LI LI",
   "evidence": "The canonical JSON record preserves OCR and page-footer damage. Its problem text is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 72,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0074": {
   "statement_status": "exact",
   "original_statement": "Problem 27. Are there necessary or sufficient conditions on Borel fixed monomial ideals such that they are the generic initial ideals of local Cohen-Macaulay curves?",
   "clean_statement": "Problem 27. Are there necessary or sufficient conditions on Borel fixed monomial ideals such that they are the generic initial ideals of local Cohen-Macaulay curves?",
   "public_statement": "Problem 27. Are there necessary or sufficient conditions on Borel fixed monomial ideals such that they are the generic initial ideals of local Cohen-Macaulay curves?",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 73,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0075": {
   "statement_status": "exact",
   "original_statement": "Problem 28. What is the smallest d such that Hilb d(A3) is reducible?",
   "clean_statement": "Problem 28. What is the smallest d such that Hilb d(A3) is reducible?",
   "public_statement": "Problem 28. What is the smallest d such that Hilb d(A3) is reducible?",
   "evidence": "The AIM record (Components of Hilbert Schemes, Problem 28) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0076": {
   "statement_status": "exact",
   "original_statement": "Problem 29. Fix a Hilbert polynomial P, consider the moduli space BrV P Pn of branchvarieties that are equidimensional and connected in codimension 1. Is BrV P (Pn) connected when non-empty?",
   "clean_statement": "Problem 29. Fix a Hilbert polynomial P, consider the moduli space BrV P Pn of branchvarieties that are equidimensional and connected in codimension 1. Is BrV P (Pn) connected when non-empty?",
   "public_statement": "Problem 29. Fix a Hilbert polynomial P, consider the moduli space BrV P Pn of branchvarieties that are equidimensional and connected in codimension 1. Is BrV P (Pn) connected when non-empty?",
   "evidence": "The canonical record is an OCR rendering of AIM Problem 29:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0077": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 30. Is there a rigid local Artinian algebra besides kn?",
   "clean_statement": "**Recovered local question.** Over an algebraically closed field \\(k\\), is\nthere a nontrivial finite local \\(k\\)-algebra \\(A\\) with\n\\(T^1_{A/k}=0\\)?",
   "public_statement": "**Recovered local question.** Over an algebraically closed field \\(k\\), is\nthere a nontrivial finite local \\(k\\)-algebra \\(A\\) with\n\\(T^1_{A/k}=0\\)?",
   "evidence": "The record comes from Problem 30 in the AIM workshop list *Components of Hilbert Schemes*. Direct inspection of the typeset PDF, including its decompressed page-content stream, shows that the last expression is \\(k^n\\); the string “kn” is an OCR loss of the superscript. Thus the literal source statement is: There is an internal inconsistency: \\(k^n\\) is a product of fields and is not local when \\(n>1\\). The natural local baseline is \\(k\\), whereas \\(k^n\\) is the familiar finite étale baseline when locality is dropped. Jelisiejew’s 2026 survey corroborates this repair. Its Problem XXIII asks whether there is an irreducible finite \\(k\\)-scheme \\(\\Gamma\\), other than \\(\\operatorname{Spec} k\\), such that \\(T^1_\\Gamma=0\\), with characteristic zero assumed for safety.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 76,
   "attempt": 2
  },
  "AIM-ARITHMETIC_GEOMETRY-0078": {
   "statement_status": "exact",
   "original_statement": "Problem 31. Consider 1, 4, 10, a where 6 ≤ a ≤ 10. The general Artinian algebra with this Hilbert function is nonsmoothable. What is the generic point of the component it lies on?",
   "clean_statement": "Problem 31. Consider 1, 4, 10, a where 6 ≤ a ≤ 10. The general Artinian algebra with this Hilbert function is nonsmoothable. What is the generic point of the component it lies on?",
   "public_statement": "Problem 31. Consider 1, 4, 10, a where 6 ≤ a ≤ 10. The general Artinian algebra with this Hilbert function is nonsmoothable. What is the generic point of the component it lies on?",
   "evidence": "The AIM record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0079": {
   "statement_status": "exact",
   "original_statement": "Problem 32. Fix d, g. Let H ⊂ Hilb d,g (P3) be an irreducible component. What is the largest number of points in general position you can make these curves pass through?",
   "clean_statement": "Problem 32. Fix d, g. Let H ⊂ Hilb d,g (P3) be an irreducible component. What is the largest number of points in general position you can make these curves pass through?",
   "public_statement": "Problem 32. Fix d, g. Let H ⊂ Hilb d,g (P3) be an irreducible component. What is the largest number of points in general position you can make these curves pass through?",
   "evidence": "The source states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0080": {
   "statement_status": "exact",
   "original_statement": "Problem 33. What is the best pair (d, g ) (in the vague sense that d is as small as possible and g\n\nis close to 0 as possible) such that there is a component of Hilb d,g whose general member has an embedded point.",
   "clean_statement": "Problem 33. What is the best pair (d, g ) (in the vague sense that d is as small as possible and g\n\nis close to 0 as possible) such that there is a component of Hilb d,g whose general member has an embedded point.",
   "public_statement": "Problem 33. What is the best pair (d, g ) (in the vague sense that d is as small as possible and g\n\nis close to 0 as possible) such that there is a component of Hilb d,g whose general member has an embedded point.",
   "evidence": "The canonical record is Problem 33 from the AIM workshop *Components of Hilbert Schemes*. Its exact OCR text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-ARITHMETIC_GEOMETRY-0081": {
   "statement_status": "unrecoverable",
   "original_statement": "Conjecture 1, p. 4). For that matter, prove Morita's conjecture! Perhaps this is the \"right\" way to prove Faber's intersection number conjecture. (10) (Mondello) For M≤krational components \n\n> g,n, is Rg−1+ k ∼= Q? What is Rg−1+ k generated by? When should we expect a 1-dimensional socle, and what should we expect for R(M≤kg,n )? 4 A LIST OF OPEN PROBLEMS AND QUESTIONS ON THE MODULI SPACE OF CURVES \n\n(11) (Igusa) Are there operations which relate the stable classes on BΓ∞? We have H∗\n\n> spec\n\n(CP )∞−1 ∼=\n\nZ[c1] · u, where u is the Thom class in degree −2. How is this reflected at the level of infinite loop spaces? What are the stable maps CP ∞−1 → CP ∞−1?(12) (Baldwin) Has anyone computed the intersection cohomology of Mg,n (Pr, d )? These can be arbitrarily singular, but this is what intersection cohomology is designed for. Is there a good notion of the tautological ring here? Perhaps the virtual fundamental class plays the usual role of the fundamental class. (13) (Faber) From Ekedahl and van der Geer, λg is 0 on Ag rationally but not integrally. The order in integral cohomology has been computed up to a factor of two. What is it? (Compare to 5 above.) (14) (Bertram) When will Getzler's paper on M1 appear (even just as a preprint)? Conjecture: \n\nt → ∞. Getzler comments that he does not like how this question is phrased. (15) (Ellenberg) Consider Hurwitz space (genus g, degree d). Could the cohomology stabilize as \n\ng → ∞ with d fixed? The reason behind this question is that point counting over finite fields gives exactly the behavior we would expect if we had Harer stability in degree 2. So, could some sort of Harer stability hold for some sort of Hurwitz schemes? Motivation for this question comes from work on number fields/function fields done in the '80s by Darskovksy and Wright. The general philosophy is this: suppose we have a nice sequence of varieties {Xn}n∈N,and lim n→∞ \n\npoints on Xn(Fq )\n\nqdim Xn\n\nexists for all q. Is this because of some version of Harer stability at play here? (16) (Tseng) Same question for C → BG, with G a finite group. Tillmann says \"yes\" for G = S1.More precisely, consider \n\nEDiff( Fg, 1) ×Diff( Fg, 1) Map ∂ (Fg, 1, BG )for G connected, such as S1. This stabilizes by gluing in tori and the induced map on homology is an isomorphism in some range. Is this related? (17) (Sullivan) Fix a curve C and look at all unbranched covers of it. This gives points in Mg.Do these become uniformly dense for any C (with respect to the Teichmuller metric) in universally defined regions Ug of Mg for g large? More precisely, given \u000f > 0, can we find \n\ng0 such that, for g > g 0 every point of Ug is within distance \u000f of an unbranched cover of C\n\nof genus g? This would imply that, given curves C1, C 2, one could find covers ˜C1, ˜C2 which are arbitrarily close in the moduli space, the Siegel-Ehrenpreis problem. (18) (Morita) There are many numerical invariants that can be associated to the moduli space. (a) One may compute the signature of the cohomology ring of Mg,n.(b) Since Mg,n is a rational cohomology manifold, there are Thom's rational Pontrjagin classes, and one may compute the rational L-genus with respect to these. (c) Lastly, Mg,n is an orbifold of a complex manifold, so there are orbifold Chern classes, and hence orbifold Pontrjagin classes. Thus one may talk about the orbifold L-genus. What are these numbers? Do they agree? Probably not, but their disagreement would tell us interesting information about the types of singularities in the moduli space. The differ-ence between the signature and the rational L-genus detects geometric singularities. The difference between the rational L-genus and the orbifold L-genus detects complex analytic singularities. One may similarly ask questions about the signature of the tautological ring, and many other variations on this theme. A LIST OF OPEN PROBLEMS AND QUESTIONS ON THE MODULI SPACE OF CURVES 5\n\n(19) (Sullivan) This question regards the algebraic structure on the homology of the free loop space of a manifold. There are maps \n\nH∗LM ∆\n\n−→ H∗LM ⊗ H∗LM \n\nand \n\nH∗LM ⊗ H∗LM μ\n\n−→ H∗LM \n\nFirst the naive question: what is the algebraic structure here? The spectral sequence converging to H∗LM has E2 term a tensor product of a Hopf algebra (coming from the base \n\nM ) and a Frobenius algebra (coming from the fibre). But the differential does not respect these structures. A perhaps better question is: can we illuminate the situation be reformulating in terms of the category of spaces over M? We have \n\nLM LM ×M LM LM × LM M M M × M\n\n\u000f\n\n\u000f\n\n/\n\n/\n\n\u000f\n\n\u000f\n\n/\n\n/\n\n\u000f\n\n\u000f// \n\n> id\n\n/ / ∆\n\nThe left square corresponds to the Frobenius algebra part, and the right square corresponds to the Hopf algebra part. So, what is the full algebraic structure here?",
   "clean_statement": null,
   "public_statement": "Conjecture 1, p. 4). For that matter, prove Morita's conjecture! Perhaps this is the \"right\" way to prove Faber's intersection number conjecture. (10) (Mondello) For M≤krational components\n\n> g,n, is Rg−1+ k ∼= Q? What is Rg−1+ k generated by? When should we expect a 1-dimensional socle, and what should we expect for R(M≤kg,n )? 4 A LIST OF OPEN PROBLEMS AND QUESTIONS ON THE MODULI SPACE OF CURVES\n\n(11) (Igusa) Are there operations which relate the stable classes on BΓ∞? We have H∗\n\n> spec\n\n(CP )∞−1 ∼=\n\nZ[c1] · u, where u is the Thom class in degree −2. How is this reflected at the level of infinite loop spaces? What are the stable maps CP ∞−1 → CP ∞−1?(12) (Baldwin) Has anyone computed the intersection cohomology of Mg,n (Pr, d )? These can be arbitrarily singular, but this is what intersection cohomology is designed for. Is there a good notion of the tautological ring here? Perhaps the virtual fundamental class plays the usual role of the fundamental class. (13) (Faber) From Ekedahl and van der Geer, λg is 0 on Ag rationally but not integrally. The order in integral cohomology has been computed up to a factor of two. What is it? (Compare to 5 above.) (14) (Bertram) When will Getzler's paper on M1 appear (even just as a preprint)? Conjecture:\n\nt → ∞. Getzler comments that he does not like how this question is phrased. (15) (Ellenberg) Consider Hurwitz space (genus g, degree d). Could the cohomology stabilize as\n\ng → ∞ with d fixed? The reason behind this question is that point counting over finite fields gives exactly the behavior we would expect if we had Harer stability in degree 2. So, could some sort of Harer stability hold for some sort of Hurwitz schemes? Motivation for this question comes from work on number fields/function fields done in the '80s by Darskovksy and Wright. The general philosophy is this: suppose we have a nice sequence of varieties {Xn}n∈N,and lim n→∞\n\npoints on Xn(Fq )\n\nqdim Xn\n\nexists for all q. Is this because of some version of Harer stability at play here? (16) (Tseng) Same question for C → BG, with G a finite group. Tillmann says \"yes\" for G = S1.More precisely, consider\n\nEDiff( Fg, 1) ×Diff( Fg, 1) Map ∂ (Fg, 1, BG )for G connected, such as S1. This stabilizes by gluing in tori and the induced map on homology is an isomorphism in some range. Is this related? (17) (Sullivan) Fix a curve C and look at all unbranched covers of it. This gives points in Mg.Do these become uniformly dense for any C (with respect to the Teichmuller metric) in universally defined regions Ug of Mg for g large? More precisely, given [U+000F] > 0, can we find\n\ng0 such that, for g > g 0 every point of Ug is within distance [U+000F] of an unbranched cover of C\n\nof genus g? This would imply that, given curves C1, C 2, one could find covers ˜C1, ˜C2 which are arbitrarily close in the moduli space, the Siegel-Ehrenpreis problem. (18) (Morita) There are many numerical invariants that can be associated to the moduli space. (a) One may compute the signature of the cohomology ring of Mg,n.(b) Since Mg,n is a rational cohomology manifold, there are Thom's rational Pontrjagin classes, and one may compute the rational L-genus with respect to these. (c) Lastly, Mg,n is an orbifold of a complex manifold, so there are orbifold Chern classes, and hence orbifold Pontrjagin classes. Thus one may talk about the orbifold L-genus. What are these numbers? Do they agree? Probably not, but their disagreement would tell us interesting information about the types of singularities in the moduli space. The differ-ence between the signature and the rational L-genus detects geometric singularities. The difference between the rational L-genus and the orbifold L-genus detects complex analytic singularities. One may similarly ask questions about the signature of the tautological ring, and many other variations on this theme. A LIST OF OPEN PROBLEMS AND QUESTIONS ON THE MODULI SPACE OF CURVES 5\n\n(19) (Sullivan) This question regards the algebraic structure on the homology of the free loop space of a manifold. There are maps\n\nH∗LM ∆\n\n−→ H∗LM ⊗ H∗LM\n\nand\n\nH∗LM ⊗ H∗LM μ\n\n−→ H∗LM\n\nFirst the naive question: what is the algebraic structure here? The spectral sequence converging to H∗LM has E2 term a tensor product of a Hopf algebra (coming from the base\n\nM ) and a Frobenius algebra (coming from the fibre). But the differential does not respect these structures. A perhaps better question is: can we illuminate the situation be reformulating in terms of the category of spaces over M? We have\n\nLM LM ×M LM LM × LM M M M × M\n\n[U+000F]\n\n[U+000F]\n\n/\n\n/\n\n[U+000F]\n\n[U+000F]\n\n/\n\n/\n\n[U+000F]\n\n[U+000F]//\n\n> id\n\n/ / ∆\n\nThe left square corresponds to the Frobenius algebra part, and the right square corresponds to the Hopf algebra part. So, what is the full algebraic structure here?",
   "evidence": "The canonical JSON object is not one problem. Comparison with the five-page AIM source shows that it starts in the middle of numbered Problem (9), at the parenthetical citation ``[Mor05] (Conjecture 1, p. 4),'' and then concatenates the end of Problem (9) with all of Problems (10)--(19). The JSON fields `number: \"1\"` and `tag: \"conjecture\"` were evidently inferred from the words ``Conjecture 1'' inside that citation, not from a heading for a new problem. Page headers, broken formulae, and control characters were also incorporated into the object. Consequently there is no single proposition whose truth could constitute a solution of the record as stored.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-arithmetic-geometry-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-BIOLOGY-0001": {
   "statement_status": "exact",
   "original_statement": "How does vasculature in eye change with diabetes?",
   "clean_statement": "How does vasculature in eye change with diabetes?",
   "public_statement": "How does vasculature in eye change with diabetes?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-BIOLOGY-0002": {
   "statement_status": "exact",
   "original_statement": "How does change in vasculature can change oxygenation in the eye?",
   "clean_statement": "How does change in vasculature can change oxygenation in the eye?",
   "public_statement": "How does change in vasculature can change oxygenation in the eye?",
   "evidence": "The exact canonical record is AIM Problem Lists workshop *Modeling the eye as a window on the body*, section “Diabetes,” problem 1.2:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-BIOLOGY-0003": {
   "statement_status": "exact",
   "original_statement": "Can we identify any early warning signs by monitoring microvasculature?",
   "clean_statement": "Can we identify any early warning signs by monitoring microvasculature?",
   "public_statement": "Can we identify any early warning signs by monitoring microvasculature?",
   "evidence": "The exact canonical question is: “Can we identify any early warning signs by monitoring microvasculature?” It appears as Diabetes Problem 1.3 in the AIM workshop “Modeling the eye as a window on the body.” The official workshop summary explains that ocular vascular and structural changes can be measured noninvasively, that one problem group studied early imaging warnings and diabetes progression, and that the group proposed following vascular measurements over a long period. The summary specifically mentions color-Doppler waveform parameters in retrobulbar vessels; those are upstream flow measurements rather than direct retinal capillary measurements. The present record itself says “microvasculature,” so OCT angiography (OCTA) is the most direct candidate modality, with Doppler indices treated as optional auxiliary observations.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-BIOLOGY-0004": {
   "statement_status": "exact",
   "original_statement": "Can we characterize and quantify the leakage factors from a clinical and modeling perspective in diabetes?",
   "clean_statement": "Can we characterize and quantify the leakage factors from a clinical and modeling perspective in diabetes?",
   "public_statement": "Can we characterize and quantify the leakage factors from a clinical and modeling perspective in diabetes?",
   "evidence": "The canonical source is the AIM workshop *Modeling the eye as a window on the body*, section “Diabetes,” problem 1.4. The exact record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-BIOLOGY-0005": {
   "statement_status": "exact",
   "original_statement": "How can we develop a predictive model for early imaging related warnings and diabetes progression?",
   "clean_statement": "How can we develop a predictive model for early imaging related warnings and diabetes progression?",
   "public_statement": "How can we develop a predictive model for early imaging related warnings and diabetes progression?",
   "evidence": "The canonical AIM record (Biology, workshop *Modeling the eye as a window on the body*, Diabetes problem 1.5) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 4,
   "attempt": 2
  },
  "AIM-BIOLOGY-0006": {
   "statement_status": "exact",
   "original_statement": "How does aging influence intraocular pressure(IOP)of an eye? How does aging influence the structure of the eye?",
   "clean_statement": "How does aging influence intraocular pressure(IOP)of an eye? How does aging influence the structure of the eye?",
   "public_statement": "How does aging influence intraocular pressure(IOP)of an eye? How does aging influence the structure of the eye?",
   "evidence": "The canonical AIM record, from the workshop *Modeling the eye as a window on the body*, section “Aging” (source URL: http://aimpl.org/eyewindow/2/), asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-BIOLOGY-0007": {
   "statement_status": "exact",
   "original_statement": "How does velocity profile change in young vs old healthy population?",
   "clean_statement": "How does velocity profile change in young vs old healthy population?",
   "public_statement": "How does velocity profile change in young vs old healthy population?",
   "evidence": "The exact canonical statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-BIOLOGY-0008": {
   "statement_status": "exact",
   "original_statement": "Can we quantify and characterize the factors that determine the oxygenation in the eye? (e.g. Light + dark ).",
   "clean_statement": "Can we quantify and characterize the factors that determine the oxygenation in the eye? (e.g. Light + dark ).",
   "public_statement": "Can we quantify and characterize the factors that determine the oxygenation in the eye? (e.g. Light + dark ).",
   "evidence": "The canonical AIM record, in the “Basic Physiology” section of the workshop *Modeling the eye as a window on the body*, asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-BIOLOGY-0009": {
   "statement_status": "exact",
   "original_statement": "Can we develop kidney inspired models for aqueous humor production?",
   "clean_statement": "Can we develop kidney inspired models for aqueous humor production?",
   "public_statement": "Can we develop kidney inspired models for aqueous humor production?",
   "evidence": "The canonical AIM record is from the workshop *Modeling the eye as a window on the body*, section “Aqueous Humor Formation,” item 4.2. It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-BIOLOGY-0010": {
   "statement_status": "exact",
   "original_statement": "Characterizing and modeling tumor heterogeneity",
   "clean_statement": "Characterizing and modeling tumor heterogeneity",
   "public_statement": "Characterizing and modeling tumor heterogeneity",
   "evidence": "The canonical record is problem 1.1, “Characterizing and modeling tumor heterogeneity,” from the AIM workshop *Systems approaches to drug discovery and development in oncology*. Its main questions are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 9,
   "attempt": 2
  },
  "AIM-BIOLOGY-0011": {
   "statement_status": "exact",
   "original_statement": "Systems approaches to drug resistance",
   "clean_statement": "Systems approaches to drug resistance",
   "public_statement": "Systems approaches to drug resistance",
   "evidence": "The canonical AIM record is problem 1.2, **“Systems approaches to drug resistance,”** from the workshop *Systems approaches to drug discovery and development in oncology*. Its accompanying questions are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 10,
   "attempt": 2
  },
  "AIM-BIOLOGY-0012": {
   "statement_status": "exact",
   "original_statement": "Linking signaling models to phenotype (e.g., tumor growth)",
   "clean_statement": "Linking signaling models to phenotype (e.g., tumor growth)",
   "public_statement": "Linking signaling models to phenotype (e.g., tumor growth)",
   "evidence": "The exact canonical problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-BIOLOGY-0013": {
   "statement_status": "exact",
   "original_statement": "Translating pre-clinical models to human",
   "clean_statement": "Translating pre-clinical models to human",
   "public_statement": "Translating pre-clinical models to human",
   "evidence": "The canonical AIM record is problem 1.4, **“Translating pre-clinical models to human,”** in the workshop *Systems approaches to drug discovery and development in oncology*. Its accompanying questions are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-BIOLOGY-0014": {
   "statement_status": "exact",
   "original_statement": "The role of cellular metabolism in cancer",
   "clean_statement": "The role of cellular metabolism in cancer",
   "public_statement": "The role of cellular metabolism in cancer",
   "evidence": "The canonical AIM record is problem 2.1, **“The role of cellular metabolism in cancer,”** from the workshop *Systems approaches to drug discovery and development in oncology*. The record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 13,
   "attempt": 2
  },
  "AIM-BIOLOGY-0015": {
   "statement_status": "exact",
   "original_statement": "Spatial effects on cancer cells",
   "clean_statement": "Spatial effects on cancer cells",
   "public_statement": "Spatial effects on cancer cells",
   "evidence": "The canonical AIM record is “Spatial effects on cancer cells,” problem 2.2 in the “Other biology-centric problem areas” from the workshop *Systems approaches to drug discovery and development in oncology*. Its exact main questions are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-BIOLOGY-0016": {
   "statement_status": "exact",
   "original_statement": "Cell cycle-dependent variation in tumor cells.",
   "clean_statement": "Cell cycle-dependent variation in tumor cells.",
   "public_statement": "Cell cycle-dependent variation in tumor cells.",
   "evidence": "The canonical AIM record is problem 2.3, **“Cell cycle-dependent variation in tumor cells.”** Its full accompanying text asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-BIOLOGY-0017": {
   "statement_status": "exact",
   "original_statement": "Accounting for and modeling metastasis",
   "clean_statement": "Accounting for and modeling metastasis",
   "public_statement": "Accounting for and modeling metastasis",
   "evidence": "The canonical record is AIM-BIOLOGY-0017, source file `aim-biology-notes.json`, zero-based source index 16, from the AIM workshop *Systems approaches to drug discovery and development in oncology*. Its short problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-BIOLOGY-0018": {
   "statement_status": "exact",
   "original_statement": "Uncertainty in model structure",
   "clean_statement": "Uncertainty in model structure",
   "public_statement": "Uncertainty in model structure",
   "evidence": "The record points to Ciaccio et al. (2010) and Morris et al. (2011) as examples of methods for exploring structures. The first reference can be identified unambiguously as the microwestern-array study of EGF-receptor signaling [Ciaccio2010]. The year, topic, and workshop context identify the second as the constrained-fuzzy-logic study of inflammatory signaling [Morris2011]. No corruption of the short source statement was found.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-BIOLOGY-0019": {
   "statement_status": "exact",
   "original_statement": "Uncertainty in model parameters",
   "clean_statement": "Uncertainty in model parameters",
   "public_statement": "Uncertainty in model parameters",
   "evidence": "The canonical AIM record comes from the workshop *Systems approaches to drug discovery and development in oncology*, section “Other modeling-centric problem areas,” problem 3.3. Its title is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-BIOLOGY-0020": {
   "statement_status": "exact",
   "original_statement": "Challenges: Time lines, focus, intellectual property",
   "clean_statement": "Challenges: Time lines, focus, intellectual property",
   "public_statement": "Challenges: Time lines, focus, intellectual property",
   "evidence": "The canonical record is AIM-BIOLOGY-0020, source file aim-biology-notes.json, zero-based source index 19, from the AIM workshop *Systems approaches to drug discovery and development in oncology*. Its complete problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 19,
   "attempt": 2
  },
  "AIM-BIOLOGY-0021": {
   "statement_status": "exact",
   "original_statement": "Opportunities: Student clinics, Research Experience for Undergraduates (REU) programs, internships",
   "clean_statement": "Opportunities: Student clinics, Research Experience for Undergraduates (REU) programs, internships",
   "public_statement": "Opportunities: Student clinics, Research Experience for Undergraduates (REU) programs, internships",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-BIOLOGY-0022": {
   "statement_status": "exact",
   "original_statement": "Funding for industry/academic interface: US National Institutes of Health (1) R01 \"Interface of the Life and Physical Sciences\" grants (PAR-10-141, PAR-10-142), (2) Small Business Innovation research (SBIR) and Small Business Technology Transfer (STTR) grants",
   "clean_statement": "Funding for industry/academic interface: US National Institutes of Health (1) R01 \"Interface of the Life and Physical Sciences\" grants (PAR-10-141, PAR-10-142), (2) Small Business Innovation research (SBIR) and Small Business Technology Transfer (STTR) grants",
   "public_statement": "Funding for industry/academic interface: US National Institutes of Health (1) R01 \"Interface of the Life and Physical Sciences\" grants (PAR-10-141, PAR-10-142), (2) Small Business Innovation research (SBIR) and Small Business Technology Transfer (STTR) grants",
   "evidence": "The exact corpus record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-BIOLOGY-0023": {
   "statement_status": "exact",
   "original_statement": "Chapter A: How does human vision make good perceptual guesses about objects? \n\nDaniel Kersten Talk Summary \n\nThe speaker provided an explanation of how human vision makes good perceptual guesses about objects using bayesian influence graphs. To this end, he decomposed an object S and its image I into world/object properties and features.The resulting probability distribution was then represented using a graph where nodes represent random variables and links represent the influences. He then illustrated this approach using examples of discounting and cue integration. In the second half of his talk the speaker concentrated on V1 and LOC mechanisms involved in perception. His experiments revealed that perceptual organization correlated with reduced V1 activity and increases LOC activity. He showed that V1 activity can predict percept on the time-scale of behaviour. The decrease in V1 activity could mean two things 1) Predictive Coding 2) Sparsification. In predictive Coding high-level object models project back predictions of the incoming data. In this case a good fit implies a low activity at the lower areas due to subtraction. (The \"shut-up\" theory). In sparsification, a good high-level fit tells the lower areas to 'stop gossiping\". This essentially amplifies the activity for features belong to the object and suppress the rest. Since predictive coding and sparsification have the same empirical fMRI observation, the experiments were inconclusive in deciding which of the two mechanisms (shut-up or stop gossiping) reduces the V1 activity when a higher level (LOC) has a good explanation for an object. \n\nQuestions During the Presentation: \n\n• Bill: Could you explain the term Discounting? \n\n• Ans: Discounting is throwing away information that is not required for perception. For eg: lighting \n\n• Jeff: (Regarding shape cues explaining away pinkishness as white paper and pink light)Are these things learnt? \n\n• Ans: Don't know yet. We are in the process of esigning and experiment to find that out. \n\nDiscussion after the Presentation: \n\n• Grossberg: In non-stationary environments priors don't exist. Also the bayesian framework breaks apart for some Kanisza illusions. \n\n• Bruno: Nothing in the Bayesian framework is inconsistent with Kanizsa. Bayesian framework is taking out the Kanizsa illusions and explaining those. The correct choice of priors is the trick. \n\n• Dan: For eg: one prior for natural vision is that motion tends to be slow. \n\n• Bill Softky: You showed us data showing suppression of activity in V1. Is there any data for suppression of thalamus? \n\n• Dan: In an fMRI experiment, localizing the thalamus and especially the LGN is not easy. Hence we don't have any data for that. \n\n• This was followed by Steve explaining a Kanizsa illusion. 4",
   "clean_statement": "Chapter A: How does human vision make good perceptual guesses about objects?\n\nDaniel Kersten Talk Summary\n\nThe speaker provided an explanation of how human vision makes good perceptual guesses about objects using bayesian influence graphs. To this end, he decomposed an object S and its image I into world/object properties and features.The resulting probability distribution was then represented using a graph where nodes represent random variables and links represent the influences. He then illustrated this approach using examples of discounting and cue integration. In the second half of his talk the speaker concentrated on V1 and LOC mechanisms involved in perception. His experiments revealed that perceptual organization correlated with reduced V1 activity and increases LOC activity. He showed that V1 activity can predict percept on the time-scale of behaviour. The decrease in V1 activity could mean two things 1) Predictive Coding 2) Sparsification. In predictive Coding high-level object models project back predictions of the incoming data. In this case a good fit implies a low activity at the lower areas due to subtraction. (The \"shut-up\" theory). In sparsification, a good high-level fit tells the lower areas to 'stop gossiping\". This essentially amplifies the activity for features belong to the object and suppress the rest. Since predictive coding and sparsification have the same empirical fMRI observation, the experiments were inconclusive in deciding which of the two mechanisms (shut-up or stop gossiping) reduces the V1 activity when a higher level (LOC) has a good explanation for an object.\n\nQuestions During the Presentation:\n\n• Bill: Could you explain the term Discounting?\n\n• Ans: Discounting is throwing away information that is not required for perception. For eg: lighting\n\n• Jeff: (Regarding shape cues explaining away pinkishness as white paper and pink light)Are these things learnt?\n\n• Ans: Don't know yet. We are in the process of esigning and experiment to find that out.\n\nDiscussion after the Presentation:\n\n• Grossberg: In non-stationary environments priors don't exist. Also the bayesian framework breaks apart for some Kanisza illusions.\n\n• Bruno: Nothing in the Bayesian framework is inconsistent with Kanizsa. Bayesian framework is taking out the Kanizsa illusions and explaining those. The correct choice of priors is the trick.\n\n• Dan: For eg: one prior for natural vision is that motion tends to be slow.\n\n• Bill Softky: You showed us data showing suppression of activity in V1. Is there any data for suppression of thalamus?\n\n• Dan: In an fMRI experiment, localizing the thalamus and especially the LGN is not easy. Hence we don't have any data for that.\n\n• This was followed by Steve explaining a Kanizsa illusion. 4",
   "public_statement": "Chapter A: How does human vision make good perceptual guesses about objects?\n\nDaniel Kersten Talk Summary\n\nThe speaker provided an explanation of how human vision makes good perceptual guesses about objects using bayesian influence graphs. To this end, he decomposed an object S and its image I into world/object properties and features.The resulting probability distribution was then represented using a graph where nodes represent random variables and links represent the influences. He then illustrated this approach using examples of discounting and cue integration. In the second half of his talk the speaker concentrated on V1 and LOC mechanisms involved in perception. His experiments revealed that perceptual organization correlated with reduced V1 activity and increases LOC activity. He showed that V1 activity can predict percept on the time-scale of behaviour. The decrease in V1 activity could mean two things 1) Predictive Coding 2) Sparsification. In predictive Coding high-level object models project back predictions of the incoming data. In this case a good fit implies a low activity at the lower areas due to subtraction. (The \"shut-up\" theory). In sparsification, a good high-level fit tells the lower areas to 'stop gossiping\". This essentially amplifies the activity for features belong to the object and suppress the rest. Since predictive coding and sparsification have the same empirical fMRI observation, the experiments were inconclusive in deciding which of the two mechanisms (shut-up or stop gossiping) reduces the V1 activity when a higher level (LOC) has a good explanation for an object.\n\nQuestions During the Presentation:\n\n• Bill: Could you explain the term Discounting?\n\n• Ans: Discounting is throwing away information that is not required for perception. For eg: lighting\n\n• Jeff: (Regarding shape cues explaining away pinkishness as white paper and pink light)Are these things learnt?\n\n• Ans: Don't know yet. We are in the process of esigning and experiment to find that out.\n\nDiscussion after the Presentation:\n\n• Grossberg: In non-stationary environments priors don't exist. Also the bayesian framework breaks apart for some Kanisza illusions.\n\n• Bruno: Nothing in the Bayesian framework is inconsistent with Kanizsa. Bayesian framework is taking out the Kanizsa illusions and explaining those. The correct choice of priors is the trick.\n\n• Dan: For eg: one prior for natural vision is that motion tends to be slow.\n\n• Bill Softky: You showed us data showing suppression of activity in V1. Is there any data for suppression of thalamus?\n\n• Dan: In an fMRI experiment, localizing the thalamus and especially the LGN is not easy. Hence we don't have any data for that.\n\n• This was followed by Steve explaining a Kanizsa illusion. 4",
   "evidence": "The canonical record is Chapter A of the 2003 AIM workshop notes *Inference and Prediction in Neocortical Circuits*, a talk summary for Daniel Kersten [AIM2003]. The official PDF confirms the following central passage:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-BIOLOGY-0024": {
   "statement_status": "exact",
   "original_statement": "Chapter B: Anatomical Substrates for Functional Responses of Neurons in the Primate Visual Cerebral Cortex \n\nAlessandra Angelucci \n\nAlessandra's talk addressed specific brain circuits and perception. Specifically, it re-ported results on the study \"How does the spatial scale of classical receptive fields and and surround modulation compare to the spatial scale of horizontal, feedforward and feedback connections? The study consisted of injecting tracers and correlating the connection spread to the receptive field size at various levels (V1,..., ). The speaker noted that receptive field sizes depend on the measurement technique. Typically encountered measurements of receptive fields are (1) the minimum response field is denoted by mRF, (2) the summation receptive field at high contract denoted by hsRF and (3) the summation receptive field at low contract denoted by hsRF. Her study showed that horizontal connections are commensurate with the low contrast sRF size of V1 cells and might mediate expansion of the sRF at low contrast and collinear facilitation effects. Moreover, the extent of feedback connections were found to be commen-surate with the whole range of V1 cells center surround field sizes.These connections might mediate center-surround interactions and global-to-local signal integration. She also reported the following; (1)Feedback connections are 10 times faster than hor-izontal connections, and as fast as feedorward connections. (2)Feedback connections are patchy (3)Feedback connections to V1 are specific (4)Horizontal and feedback connections arise from excitatory neurons (5)There is no such thing as \"long-range inhibitory connec-tions\" in visual cortex. (6)Horizontal and feedback axons contact excitatory (approx 80%) as well as inhibitory (approx 20%) neurons \n\nDiscussion:Question: Our impression was that feedback is diffused and not orientation specific? Ans: The results presented here say the opposite. It says that feeback is patchy and orientation specific. Qn: Connections from V1 to LGN? Ans: We don't have any data on that. Qn: Did you say that 98% of the feedback connections go to excitatory neurons? Ans: No, the division is 80/20",
   "clean_statement": "Chapter B: Anatomical Substrates for Functional Responses of Neurons in the Primate Visual Cerebral Cortex\n\nAlessandra Angelucci\n\nAlessandra's talk addressed specific brain circuits and perception. Specifically, it re-ported results on the study \"How does the spatial scale of classical receptive fields and and surround modulation compare to the spatial scale of horizontal, feedforward and feedback connections? The study consisted of injecting tracers and correlating the connection spread to the receptive field size at various levels (V1,..., ). The speaker noted that receptive field sizes depend on the measurement technique. Typically encountered measurements of receptive fields are (1) the minimum response field is denoted by mRF, (2) the summation receptive field at high contract denoted by hsRF and (3) the summation receptive field at low contract denoted by hsRF. Her study showed that horizontal connections are commensurate with the low contrast sRF size of V1 cells and might mediate expansion of the sRF at low contrast and collinear facilitation effects. Moreover, the extent of feedback connections were found to be commen-surate with the whole range of V1 cells center surround field sizes.These connections might mediate center-surround interactions and global-to-local signal integration. She also reported the following; (1)Feedback connections are 10 times faster than hor-izontal connections, and as fast as feedorward connections. (2)Feedback connections are patchy (3)Feedback connections to V1 are specific (4)Horizontal and feedback connections arise from excitatory neurons (5)There is no such thing as \"long-range inhibitory connec-tions\" in visual cortex. (6)Horizontal and feedback axons contact excitatory (approx 80%) as well as inhibitory (approx 20%) neurons\n\nDiscussion:Question: Our impression was that feedback is diffused and not orientation specific? Ans: The results presented here say the opposite. It says that feeback is patchy and orientation specific. Qn: Connections from V1 to LGN? Ans: We don't have any data on that. Qn: Did you say that 98% of the feedback connections go to excitatory neurons? Ans: No, the division is 80/20",
   "public_statement": "Chapter B: Anatomical Substrates for Functional Responses of Neurons in the Primate Visual Cerebral Cortex\n\nAlessandra Angelucci\n\nAlessandra's talk addressed specific brain circuits and perception. Specifically, it re-ported results on the study \"How does the spatial scale of classical receptive fields and and surround modulation compare to the spatial scale of horizontal, feedforward and feedback connections? The study consisted of injecting tracers and correlating the connection spread to the receptive field size at various levels (V1,..., ). The speaker noted that receptive field sizes depend on the measurement technique. Typically encountered measurements of receptive fields are (1) the minimum response field is denoted by mRF, (2) the summation receptive field at high contract denoted by hsRF and (3) the summation receptive field at low contract denoted by hsRF. Her study showed that horizontal connections are commensurate with the low contrast sRF size of V1 cells and might mediate expansion of the sRF at low contrast and collinear facilitation effects. Moreover, the extent of feedback connections were found to be commen-surate with the whole range of V1 cells center surround field sizes.These connections might mediate center-surround interactions and global-to-local signal integration. She also reported the following; (1)Feedback connections are 10 times faster than hor-izontal connections, and as fast as feedorward connections. (2)Feedback connections are patchy (3)Feedback connections to V1 are specific (4)Horizontal and feedback connections arise from excitatory neurons (5)There is no such thing as \"long-range inhibitory connec-tions\" in visual cortex. (6)Horizontal and feedback axons contact excitatory (approx 80%) as well as inhibitory (approx 20%) neurons\n\nDiscussion:Question: Our impression was that feedback is diffused and not orientation specific? Ans: The results presented here say the opposite. It says that feeback is patchy and orientation specific. Qn: Connections from V1 to LGN? Ans: We don't have any data on that. Qn: Did you say that 98% of the feedback connections go to excitatory neurons? Ans: No, the division is 80/20",
   "evidence": "This canonical record is tagged `section`, but it contains a genuine research question rather than only a heading. Chapter B of the AIM workshop notes summarizes Alessandra Angelucci's question:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-BIOLOGY-0025": {
   "statement_status": "exact",
   "original_statement": "Chapter C: Breakthroughs in Brain Computing \n\nSteve Grossberg Discussion Following the Talk \n\nQ1: Why do you choose very specific kind of laminar circuits? What is the particular reason for ur circuits? Any predictions for us? Answer: I would like you to refer to my paper with Razada. There we stimulate in layer 6 and observe what happens in layer 4. The reason for Laminar architecture and how we came to it: We started working on grouping. What are the units of perception? To get at the grouping properties and analog coherence required certain ordering of mechanisms or else the data would collapse. Needed something which is robust and not parameter sensitive. We could see by analyzing the anatomy that it provided this required ordering of mechanisms in a compact way. We 5\n\ntried knocking down parts of it to see whether this breaks down and we found that the entire thing was necessary. What we basically have is a minimal mechanis for the required ordering property. Qn:Do you insist that the lateral connections add superlinearly? Ans: Not superlinearly but faster. Comment: The normalization property with just one inhibitory neron might not be sufficient for all. By variations of the simple circuit you can get a variety of effects from bipole to modulatory. Qn: What is your prediction for an experiment where there is a stimulus in the center and the flanks are detected? Qn: Does your model need spikes? Answer: No you don't need spikes. Spikes are important in some situations (self syncrhonizing nets, order preserving limit cycles) but not required for other. Qn: Why are spikes there? Ans: So that we have non attenuated signal transmission over long distance. Also with spikes, we can compensate for axonal delay by increasing the diameter of axons. Also I talked about a balance between excitation and inhibition. If these balances doesn't occur due to development then the system becomes to unstable. If the system is designed to 'stabilize' using spikes, then spikes become important. These systems show that they love to resynchronize these spikes for stability reasons. Qn: Does your model do computer vision tasks that are not done yet? For this the speaker answered with a list of computer vision tasks his algorithms have been employed on. Qn: The neurophysiological detail is that there are spikes. A: Why don't u put in all the channels? Jeff (Comment): Analogy between computers and quantum mechanics. Although the transistors operate based on principles of quantum mechanics, an understanding of quantum mechanics is not required to understand the working of computers. Speaker(Comment): My equations are mean values of stochastic differential equations. Speaker(Comment): An example where spikes are not important: Phonemic Restora-tion. BB noise colored by eel is on the wheel wagon peel orange heel is on the shoes. Pentti(Comment): Syncrhonous computing might not need spikes. Need not be the case for asynchronous operations.",
   "clean_statement": "Chapter C: Breakthroughs in Brain Computing\n\nSteve Grossberg Discussion Following the Talk\n\nQ1: Why do you choose very specific kind of laminar circuits? What is the particular reason for ur circuits? Any predictions for us? Answer: I would like you to refer to my paper with Razada. There we stimulate in layer 6 and observe what happens in layer 4. The reason for Laminar architecture and how we came to it: We started working on grouping. What are the units of perception? To get at the grouping properties and analog coherence required certain ordering of mechanisms or else the data would collapse. Needed something which is robust and not parameter sensitive. We could see by analyzing the anatomy that it provided this required ordering of mechanisms in a compact way. We 5\n\ntried knocking down parts of it to see whether this breaks down and we found that the entire thing was necessary. What we basically have is a minimal mechanis for the required ordering property. Qn:Do you insist that the lateral connections add superlinearly? Ans: Not superlinearly but faster. Comment: The normalization property with just one inhibitory neron might not be sufficient for all. By variations of the simple circuit you can get a variety of effects from bipole to modulatory. Qn: What is your prediction for an experiment where there is a stimulus in the center and the flanks are detected? Qn: Does your model need spikes? Answer: No you don't need spikes. Spikes are important in some situations (self syncrhonizing nets, order preserving limit cycles) but not required for other. Qn: Why are spikes there? Ans: So that we have non attenuated signal transmission over long distance. Also with spikes, we can compensate for axonal delay by increasing the diameter of axons. Also I talked about a balance between excitation and inhibition. If these balances doesn't occur due to development then the system becomes to unstable. If the system is designed to 'stabilize' using spikes, then spikes become important. These systems show that they love to resynchronize these spikes for stability reasons. Qn: Does your model do computer vision tasks that are not done yet? For this the speaker answered with a list of computer vision tasks his algorithms have been employed on. Qn: The neurophysiological detail is that there are spikes. A: Why don't u put in all the channels? Jeff (Comment): Analogy between computers and quantum mechanics. Although the transistors operate based on principles of quantum mechanics, an understanding of quantum mechanics is not required to understand the working of computers. Speaker(Comment): My equations are mean values of stochastic differential equations. Speaker(Comment): An example where spikes are not important: Phonemic Restora-tion. BB noise colored by eel is on the wheel wagon peel orange heel is on the shoes. Pentti(Comment): Syncrhonous computing might not need spikes. Need not be the case for asynchronous operations.",
   "public_statement": "Chapter C: Breakthroughs in Brain Computing\n\nSteve Grossberg Discussion Following the Talk\n\nQ1: Why do you choose very specific kind of laminar circuits? What is the particular reason for ur circuits? Any predictions for us? Answer: I would like you to refer to my paper with Razada. There we stimulate in layer 6 and observe what happens in layer 4. The reason for Laminar architecture and how we came to it: We started working on grouping. What are the units of perception? To get at the grouping properties and analog coherence required certain ordering of mechanisms or else the data would collapse. Needed something which is robust and not parameter sensitive. We could see by analyzing the anatomy that it provided this required ordering of mechanisms in a compact way. We 5\n\ntried knocking down parts of it to see whether this breaks down and we found that the entire thing was necessary. What we basically have is a minimal mechanis for the required ordering property. Qn:Do you insist that the lateral connections add superlinearly? Ans: Not superlinearly but faster. Comment: The normalization property with just one inhibitory neron might not be sufficient for all. By variations of the simple circuit you can get a variety of effects from bipole to modulatory. Qn: What is your prediction for an experiment where there is a stimulus in the center and the flanks are detected? Qn: Does your model need spikes? Answer: No you don't need spikes. Spikes are important in some situations (self syncrhonizing nets, order preserving limit cycles) but not required for other. Qn: Why are spikes there? Ans: So that we have non attenuated signal transmission over long distance. Also with spikes, we can compensate for axonal delay by increasing the diameter of axons. Also I talked about a balance between excitation and inhibition. If these balances doesn't occur due to development then the system becomes to unstable. If the system is designed to 'stabilize' using spikes, then spikes become important. These systems show that they love to resynchronize these spikes for stability reasons. Qn: Does your model do computer vision tasks that are not done yet? For this the speaker answered with a list of computer vision tasks his algorithms have been employed on. Qn: The neurophysiological detail is that there are spikes. A: Why don't u put in all the channels? Jeff (Comment): Analogy between computers and quantum mechanics. Although the transistors operate based on principles of quantum mechanics, an understanding of quantum mechanics is not required to understand the working of computers. Speaker(Comment): My equations are mean values of stochastic differential equations. Speaker(Comment): An example where spikes are not important: Phonemic Restora-tion. BB noise colored by eel is on the wheel wagon peel orange heel is on the shoes. Pentti(Comment): Syncrhonous computing might not need spikes. Need not be the case for asynchronous operations.",
   "evidence": "The canonical record is AIM-BIOLOGY-0025, zero-based index 24 of aim-biology-notes.json. It is tagged section and headed “Chapter C: Breakthroughs in Brain Computing.” The official source is the report of the AIM workshop *Inference and Prediction in Neocortical Circuits*, held 21--24 September 2003 [AIM2003]. The passage contains discussion after a talk by Steve Grossberg, not a stated open problem.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-BIOLOGY-0026": {
   "statement_status": "exact",
   "original_statement": "Chapter D: A saliency map in primary visual cortex \n\nLi Zhaoping University College London Talk Summary: The speaker presented a model of V1 that produces a saliency mapl. This model transforms contrast inputs to saliencies using contextual influences. The con-textual influences are implemented in a recurrent network with intra-cortical connections. 6\n\nSaliencies are signalled using firing rates of the neurons.The speaker compared the perfor-mance of her model with known psyho-physical results. \n\nDiscussion And Questions Following the Talk \n\nQ(Bill): Do all V1 cells signal saliency or a few of them. A: Yes all cells. Q: What is the role of other visual areas in this model? A: No role. Speakers' Comment: V1 cells firing rates signal saliencies, despite their feature tuning. Strongest response to any visual location signals saliency. Also, this theory is only bottom up. V1's output as saliency map is viewed under the idealziation of the top-down feedback to V1 being disabled. eg. shortly after visual exposure or under anesthesia. (Saliency from bottom up factors only) Q: Are these results after the system have settled? A: After one time constant. Q: Long range excitatory connections are not isotropic. In your figure it seems as if they are not. Comment: The speaker then explained how they are not isotropic in her model. Bill: What about color, motion etc. how do u combine in the model? Ans: These are not included in the current model. Q: Did you do any of these on natural images? A: No the limitation is the processing power. Q: Do you need any endstopping? A: Steve (Comment): I am worried about about the use of the word \"saliency\" map for the visual search tasks you mentioned. For eg we have found that 3D properties (surfaces) can affect pop-out mechanisms and the word 'saliency' is usually used in a more wider context. Have you done any experiments using surface properties? Answer: I just compared with visual search tasks. This model does not explaining 3D surface properties. Dana(Comment): The same data can be explained using Signal to Noise Ratio. (Tries-mann data) Some of the experimental data could be false because they were done in blocks. Q:What about multiple scales? A: Same principle will apply at all scales. Q: Regarding the disynaptic inhibitory connection, is there a time delay that is re-quired? A: Yes. To get rid of the symmetry breaking behaviour we need the disynaptic be-haviour.",
   "clean_statement": "Chapter D: A saliency map in primary visual cortex\n\nLi Zhaoping University College London Talk Summary: The speaker presented a model of V1 that produces a saliency mapl. This model transforms contrast inputs to saliencies using contextual influences. The con-textual influences are implemented in a recurrent network with intra-cortical connections. 6\n\nSaliencies are signalled using firing rates of the neurons.The speaker compared the perfor-mance of her model with known psyho-physical results.\n\nDiscussion And Questions Following the Talk\n\nQ(Bill): Do all V1 cells signal saliency or a few of them. A: Yes all cells. Q: What is the role of other visual areas in this model? A: No role. Speakers' Comment: V1 cells firing rates signal saliencies, despite their feature tuning. Strongest response to any visual location signals saliency. Also, this theory is only bottom up. V1's output as saliency map is viewed under the idealziation of the top-down feedback to V1 being disabled. eg. shortly after visual exposure or under anesthesia. (Saliency from bottom up factors only) Q: Are these results after the system have settled? A: After one time constant. Q: Long range excitatory connections are not isotropic. In your figure it seems as if they are not. Comment: The speaker then explained how they are not isotropic in her model. Bill: What about color, motion etc. how do u combine in the model? Ans: These are not included in the current model. Q: Did you do any of these on natural images? A: No the limitation is the processing power. Q: Do you need any endstopping? A: Steve (Comment): I am worried about about the use of the word \"saliency\" map for the visual search tasks you mentioned. For eg we have found that 3D properties (surfaces) can affect pop-out mechanisms and the word 'saliency' is usually used in a more wider context. Have you done any experiments using surface properties? Answer: I just compared with visual search tasks. This model does not explaining 3D surface properties. Dana(Comment): The same data can be explained using Signal to Noise Ratio. (Tries-mann data) Some of the experimental data could be false because they were done in blocks. Q:What about multiple scales? A: Same principle will apply at all scales. Q: Regarding the disynaptic inhibitory connection, is there a time delay that is re-quired? A: Yes. To get rid of the symmetry breaking behaviour we need the disynaptic be-haviour.",
   "public_statement": "Chapter D: A saliency map in primary visual cortex\n\nLi Zhaoping University College London Talk Summary: The speaker presented a model of V1 that produces a saliency mapl. This model transforms contrast inputs to saliencies using contextual influences. The con-textual influences are implemented in a recurrent network with intra-cortical connections. 6\n\nSaliencies are signalled using firing rates of the neurons.The speaker compared the perfor-mance of her model with known psyho-physical results.\n\nDiscussion And Questions Following the Talk\n\nQ(Bill): Do all V1 cells signal saliency or a few of them. A: Yes all cells. Q: What is the role of other visual areas in this model? A: No role. Speakers' Comment: V1 cells firing rates signal saliencies, despite their feature tuning. Strongest response to any visual location signals saliency. Also, this theory is only bottom up. V1's output as saliency map is viewed under the idealziation of the top-down feedback to V1 being disabled. eg. shortly after visual exposure or under anesthesia. (Saliency from bottom up factors only) Q: Are these results after the system have settled? A: After one time constant. Q: Long range excitatory connections are not isotropic. In your figure it seems as if they are not. Comment: The speaker then explained how they are not isotropic in her model. Bill: What about color, motion etc. how do u combine in the model? Ans: These are not included in the current model. Q: Did you do any of these on natural images? A: No the limitation is the processing power. Q: Do you need any endstopping? A: Steve (Comment): I am worried about about the use of the word \"saliency\" map for the visual search tasks you mentioned. For eg we have found that 3D properties (surfaces) can affect pop-out mechanisms and the word 'saliency' is usually used in a more wider context. Have you done any experiments using surface properties? Answer: I just compared with visual search tasks. This model does not explaining 3D surface properties. Dana(Comment): The same data can be explained using Signal to Noise Ratio. (Tries-mann data) Some of the experimental data could be false because they were done in blocks. Q:What about multiple scales? A: Same principle will apply at all scales. Q: Regarding the disynaptic inhibitory connection, is there a time delay that is re-quired? A: Yes. To get rid of the symmetry breaking behaviour we need the disynaptic be-haviour.",
   "evidence": "This record is Chapter D of the American Institute of Mathematics workshop notes *Inference and Prediction in Neocortical Circuits*. The official PDF identifies itself as a hard-copy version of an AIM web page and is dated October 24, 2003. The recovered chapter is:",
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   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
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  },
  "AIM-BIOLOGY-0027": {
   "statement_status": "corrected_verified",
   "original_statement": "Chapter E: Different Functional Roles of Feedback and Horizontal Connnections \n\nVirginia de Sa Q: (Regarding the processing of lip movement patterns). Why did you use the motion and not the position. 7\n\nA: Expediency. That system was already working when I started my work. Q: Do you have any sense of how this scales for more categories. A: No we haven't done the experiment on more classes. If we have more modalities the task is clearly easier. Comment: Feedback and horizontal connections both bring in greater context but feeback conxsn are unique in bringing back information from other sensory modalities. Hor-izontal conections tend to connect like recpetive fields. Comment(Regarding the slice experiments on rat): There aren't many lateral connec-tions in rat. So her experiment was on short-range connections. Q: (Regarding the slice experiment) Where is your recording electrode? A: In layer 2-3. Q: Is maximising agreement different from minimizing disagreement? A: In the way it is defined here, both turn out to be the same. Comment: Isn't it the independence of the modalities that helped the inter-training of the modalities? Q: Does the association come about because lip-movement and the associated phoneme have the same physical cause? A: Yes.",
   "clean_statement": "Chapter E: Different Functional Roles of Feedback and Horizontal Connnections\n\nVirginia de Sa Q: (Regarding the processing of lip movement patterns). Why did you use the motion and not the position. Connnections,\n\nA: Expediency. That system was already working when I started my work. Q: Do you have any sense of how this scales for more categories. A: No we haven't done the experiment on more classes. If we have more modalities the task is clearly easier. Comment: Feedback and horizontal connections both bring in greater context but feeback conxsn are unique in bringing back information from other sensory modalities. Hor-izontal conections tend to connect like recpetive fields. Comment(Regarding the slice experiments on rat): There aren't many lateral connec-tions in rat. So her experiment was on short-range connections. Q: (Regarding the slice experiment) Where is your recording electrode? A: In layer 2-3. Q: Is maximising agreement different from minimizing disagreement? A: In the way it is defined here, both turn out to be the same. Comment: Isn't it the independence of the modalities that helped the inter-training of the modalities? Q: Does the association come about because lip-movement and the associated phoneme have the same physical cause? A: Yes.",
   "public_statement": "Chapter E: Different Functional Roles of Feedback and Horizontal Connnections\n\nVirginia de Sa Q: (Regarding the processing of lip movement patterns). Why did you use the motion and not the position. Connnections,\n\nA: Expediency. That system was already working when I started my work. Q: Do you have any sense of how this scales for more categories. A: No we haven't done the experiment on more classes. If we have more modalities the task is clearly easier. Comment: Feedback and horizontal connections both bring in greater context but feeback conxsn are unique in bringing back information from other sensory modalities. Hor-izontal conections tend to connect like recpetive fields. Comment(Regarding the slice experiments on rat): There aren't many lateral connec-tions in rat. So her experiment was on short-range connections. Q: (Regarding the slice experiment) Where is your recording electrode? A: In layer 2-3. Q: Is maximising agreement different from minimizing disagreement? A: In the way it is defined here, both turn out to be the same. Comment: Isn't it the independence of the modalities that helped the inter-training of the modalities? Q: Does the association come about because lip-movement and the associated phoneme have the same physical cause? A: Yes.",
   "evidence": "This record is Chapter E of the American Institute of Mathematics workshop notes *Inference and Prediction in Neocortical Circuits*, version October 24, 2003. It is a discussion transcript, not an AIM problem stated as a conjecture. The official PDF gives the following content, with only line-break hyphenation and obvious typographical defects repaired here: The following defects were checked against the PDF:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-biology-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-BIOLOGY-0028": {
   "statement_status": "corrected_verified",
   "original_statement": "Chapter F: Distributed Syncrhony \n\nZhohua Zhang, Dana Ballard. (Speaker) Talk Summary: The speaker presented a spike-timing based communication mechanism for inter-neuron communication. He suggested reasons why a rate-coding based approach will not work. The proposed model includes a clock of period 20 msec. A signal P to be transmitted is encoded as the multiplication of a probability of firing sqrt(P) and a phase offset of sqrt(P) with respect to the clock. This model can reproduce observed PSTHs and receptive field characteristics. The 'Distributed Synchrony Model' was motivated by three criteria (1) Fast, reliable inter-cortical communication, (2) the need for a cell to multiplex and (3) the need to reproduce observed cell responses. Discussion and Questions Q: Is the cortex trying to reconstruct the image or trying to extract some features from the image? A: In this case yes..trying to reconstruct. Q: (Regarding the PSTHs). Your model will reproduce the PSTHs only when the stimulus is presented in synchronization with the internal clock. A: That is true. But we assume that the cortex picks up the signal from the retina in a syncrhonized fashion. Q: (Regarding) Coding cost of different strategies. Does the cost come from the learning part or the prediction part? A: We don't make that distinction. Q: Does your feedback rule have any relation with EM? A: The feedback rule is motivated by the EM rule. 8\n\nQ: (Regarding the synaptic weights used for receptive field simulations) Are the synap-tic weights pretty similar or are they different. A: Depending on how we pick the weights we can get different receptive fields (Steve): You have a shut up model. But in my talk I gave an example where top-down feedback is excitatory. Could you comment on all these experiments that say that feedback is not shut up. A: In this model feedback is indeed shutup. Comment(Alessandra): Anatomical studies show that feedback is excitatory in center and inhibitory in surround. Comment(Steve): I thinkt that end stopping is caused by feedback from V2 is false. Q: It seems that phase has done two things with the clock. Multiplexing and Coding. A: That is right. The width of the 20 ms is related to the number of 'threads' that you have. Q(Bill): At the beginning of the clock..how does the neuron forget everything? A: We assume that there is some magical swap space available for storing all the information and then swapping it back in. Q(Bill): Also how does the cell generate the random number? A: There are some known mechanisms. Q: Are spikes are essential? A: I think spikes are an innovation by biology to send signals reliably over long distance to Q: If we have a package of spikes....what is the necessity of clock? A: So that we can do feedback. If there is no clock, there is no way to align the feedback signal. Q: How neurons syncrhonize critically depend on the neuron model. For eg the Aertson model A: He has an integrate and fire in combination with a random number generator.",
   "clean_statement": "Chapter F: Distributed Syncrhony\n\nsynap-tic, Dana Ballard. (Speaker) Talk Summary: The speaker presented a spike-timing based communication mechanism for inter-neuron communication. He suggested reasons why a rate-coding based approach will not work. The proposed model includes a clock of period 20 msec. A signal P to be transmitted is encoded as the multiplication of a probability of firing sqrt(P) and a phase offset of sqrt(P) with respect to the clock. This model can reproduce observed PSTHs and receptive field characteristics. The 'Distributed Synchrony Model' was motivated by three criteria (1) Fast, reliable inter-cortical communication, (2) the need for a cell to multiplex and (3) the need to reproduce observed cell responses. Discussion and Questions Q: Is the cortex trying to reconstruct the image or trying to extract some features from the image? A: In this case yes..trying to reconstruct. Q: (Regarding the PSTHs). Your model will reproduce the PSTHs only when the stimulus is presented in synchronization with the internal clock. A: That is true. But we assume that the cortex picks up the signal from the retina in a syncrhonized fashion. Q: (Regarding) Coding cost of different strategies. Does the cost come from the learning part or the prediction part? A: We don't make that distinction. Q: Does your feedback rule have any relation with EM? A: The feedback rule is motivated by the EM rule. 8\n\nQ: (Regarding the synaptic weights used for receptive field simulations) Are the synap-tic weights pretty similar or are they different. A: Depending on how we pick the weights we can get different receptive fields (Steve): You have a shut up model. But in my talk I gave an example where top-down feedback is excitatory. Could you comment on all these experiments that say that feedback is not shut up. A: In this model feedback is indeed shutup. Comment(Alessandra): Anatomical studies show that feedback is excitatory in center and inhibitory in surround. Comment(Steve): I thinkt that end stopping is caused by feedback from V2 is false. Q: It seems that phase has done two things with the clock. Multiplexing and Coding. A: That is right. The width of the 20 ms is related to the number of 'threads' that you have. Q(Bill): At the beginning of the clock..how does the neuron forget everything? A: We assume that there is some magical swap space available for storing all the information and then swapping it back in. Q(Bill): Also how does the cell generate the random number? A: There are some known mechanisms. Q: Are spikes are essential? A: I think spikes are an innovation by biology to send signals reliably over long distance to Q: If we have a package of spikes....what is the necessity of clock? A: So that we can do feedback. If there is no clock, there is no way to align the feedback signal. Q: How neurons syncrhonize critically depend on the neuron model. For eg the Aertson model A: He has an integrate and fire in combination with a random number generator.",
   "public_statement": "Chapter F: Distributed Syncrhony\n\nsynap-tic, Dana Ballard. (Speaker) Talk Summary: The speaker presented a spike-timing based communication mechanism for inter-neuron communication. He suggested reasons why a rate-coding based approach will not work. The proposed model includes a clock of period 20 msec. A signal P to be transmitted is encoded as the multiplication of a probability of firing sqrt(P) and a phase offset of sqrt(P) with respect to the clock. This model can reproduce observed PSTHs and receptive field characteristics. The 'Distributed Synchrony Model' was motivated by three criteria (1) Fast, reliable inter-cortical communication, (2) the need for a cell to multiplex and (3) the need to reproduce observed cell responses. Discussion and Questions Q: Is the cortex trying to reconstruct the image or trying to extract some features from the image? A: In this case yes..trying to reconstruct. Q: (Regarding the PSTHs). Your model will reproduce the PSTHs only when the stimulus is presented in synchronization with the internal clock. A: That is true. But we assume that the cortex picks up the signal from the retina in a syncrhonized fashion. Q: (Regarding) Coding cost of different strategies. Does the cost come from the learning part or the prediction part? A: We don't make that distinction. Q: Does your feedback rule have any relation with EM? A: The feedback rule is motivated by the EM rule. 8\n\nQ: (Regarding the synaptic weights used for receptive field simulations) Are the synap-tic weights pretty similar or are they different. A: Depending on how we pick the weights we can get different receptive fields (Steve): You have a shut up model. But in my talk I gave an example where top-down feedback is excitatory. Could you comment on all these experiments that say that feedback is not shut up. A: In this model feedback is indeed shutup. Comment(Alessandra): Anatomical studies show that feedback is excitatory in center and inhibitory in surround. Comment(Steve): I thinkt that end stopping is caused by feedback from V2 is false. Q: It seems that phase has done two things with the clock. Multiplexing and Coding. A: That is right. The width of the 20 ms is related to the number of 'threads' that you have. Q(Bill): At the beginning of the clock..how does the neuron forget everything? A: We assume that there is some magical swap space available for storing all the information and then swapping it back in. Q(Bill): Also how does the cell generate the random number? A: There are some known mechanisms. Q: Are spikes are essential? A: I think spikes are an innovation by biology to send signals reliably over long distance to Q: If we have a package of spikes....what is the necessity of clock? A: So that we can do feedback. If there is no clock, there is no way to align the feedback signal. Q: How neurons syncrhonize critically depend on the neuron model. For eg the Aertson model A: He has an integrate and fire in combination with a random number generator.",
   "evidence": "This record is Chapter F of the AIM workshop notes *Inference and Prediction in Neocortical Circuits*. It is a talk summary and discussion, tagged “section,” rather than a stated open problem. The official AIM PDF verifies the central sentence (including its informal notation): The official PDF contains several source/OCR defects. “Distributed Syncrhony” and “syncrhonized” are misspellings; the author printed as “Zhohua Zhang” is identifiable from the primary publications as **Zuohua Zhang**; the isolated 8 after the EM discussion is a page number; and “synap-tic” is a line-break artifact. The broken sentence after “send signals reliably over long distance” cannot safely be completed and is not used here.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-biology-notes.json",
   "source_index": 27,
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  },
  "AIM-BIOLOGY-0029": {
   "statement_status": "exact",
   "original_statement": "Chapter G: Helmholtz Inference in Early Vision Areas \n\nKen Nakayama \n\nDiscussion/Questions After the Talk. Dana: If surfaces are creating the percept of an image space, it seems what you are calling generic views are quite in line with Bayes. A: When I talk about Bayes, the priors are stochastic and non ergodic. Also the Kanizsa diagrams show that we are not using the priors. Bayesian formulation is far too unconstrained and general. Comment(Jeff?): I see most of the figures as ambiguous. Q: Isn't the generic view principle a prior? (Comment): Bill Freeman has a paper which said the generic view principle in terms of Bayes. Q: Most of the things can be explained because you see motor experience and not just image? 9\n\nComment: We see the impossible triangle through bottom up process even though we know at the top that its not possible. Q (Jeff): You are confusing with models and theory. Do you think there is a neurobi-ological explanation for all this. Answer: Reductionism is metaphysical. Comment(Dana): Through development we get the the kinetic depth and stereo vision. We might not be able to get this figured out until we track the developmental trajectory. Also perception and motor systems do interact. Comment(Jeff): All the speakers yesterday were in the optimistic camp and today we have the speakers in the pessimistic camp",
   "clean_statement": "Chapter G: Helmholtz Inference in Early Vision Areas\n\nKen Nakayama\n\nDiscussion/Questions After the Talk. Dana: If surfaces are creating the percept of an image space, it seems what you are calling generic views are quite in line with Bayes. A: When I talk about Bayes, the priors are stochastic and non ergodic. Also the Kanizsa diagrams show that we are not using the priors. Bayesian formulation is far too unconstrained and general. Comment(Jeff?): I see most of the figures as ambiguous. Q: Isn't the generic view principle a prior? (Comment): Bill Freeman has a paper which said the generic view principle in terms of Bayes. Q: Most of the things can be explained because you see motor experience and not just image? 9\n\nComment: We see the impossible triangle through bottom up process even though we know at the top that its not possible. Q (Jeff): You are confusing with models and theory. Do you think there is a neurobi-ological explanation for all this. Answer: Reductionism is metaphysical. Comment(Dana): Through development we get the the kinetic depth and stereo vision. We might not be able to get this figured out until we track the developmental trajectory. Also perception and motor systems do interact. Comment(Jeff): All the speakers yesterday were in the optimistic camp and today we have the speakers in the pessimistic camp",
   "public_statement": "Chapter G: Helmholtz Inference in Early Vision Areas\n\nKen Nakayama\n\nDiscussion/Questions After the Talk. Dana: If surfaces are creating the percept of an image space, it seems what you are calling generic views are quite in line with Bayes. A: When I talk about Bayes, the priors are stochastic and non ergodic. Also the Kanizsa diagrams show that we are not using the priors. Bayesian formulation is far too unconstrained and general. Comment(Jeff?): I see most of the figures as ambiguous. Q: Isn't the generic view principle a prior? (Comment): Bill Freeman has a paper which said the generic view principle in terms of Bayes. Q: Most of the things can be explained because you see motor experience and not just image? 9\n\nComment: We see the impossible triangle through bottom up process even though we know at the top that its not possible. Q (Jeff): You are confusing with models and theory. Do you think there is a neurobi-ological explanation for all this. Answer: Reductionism is metaphysical. Comment(Dana): Through development we get the the kinetic depth and stereo vision. We might not be able to get this figured out until we track the developmental trajectory. Also perception and motor systems do interact. Comment(Jeff): All the speakers yesterday were in the optimistic camp and today we have the speakers in the pessimistic camp",
   "evidence": "This canonical record is Chapter G of the American Institute of Mathematics workshop notes *Inference and Prediction in Neocortical Circuits*. The official PDF describes itself as a hard-copy version of an AIM web page and is dated October 24, 2003. Chapter G begins on PDF page index 7 (printed page 8) and continues on PDF page index 8 (printed page 9). It contains no talk summary, equations, or formal conjecture, only this discussion after Ken Nakayama's talk:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-BIOLOGY-0030": {
   "statement_status": "exact",
   "original_statement": "Chapter H: Neural Mechanisms of Perceptual Inference \n\nRudiger vonder Heydt \n\nQ: What percentage of the recorded cells were in V1? A: 50 Q: Are these unbiased selection of random cells? A: Yes, as we physiologists do. Q: How is the response defined? A: Its the total number of spikes in that window. (800ms). Q: What is the largest extent of the visual field you tried? A: 8 degrees. Q:How do you measure the intitial receptive field of a cell? A: We use an optimal bar. We move the bar out and figure out the point where this cell does not respond any more. The minimum region outside which you don't get a response. Comment: Other techniques can give different receptive field sizes. Q: Figure ground and ownership? A: Its just different words for the same thing. Q: Didn't understand the comment about influence of IT. A: We have the half-strength of the ownership signal at 70ms. IT, upto 100ms is silent. Q: What about the illlusory contours? ARe they more local? A: Q: Does that mean that the illusory contour is withing the summation field of the neuron? A: What we know as the summation field is quite controversial. Summation os not an explantion of the illusory contours. Q: What about natural scenes? A: I suspect that some of this would hold up.",
   "clean_statement": "Chapter H: Neural Mechanisms of Perceptual Inference\n\nRudiger vonder Heydt\n\nQ: What percentage of the recorded cells were in V1? A: 50 Q: Are these unbiased selection of random cells? A: Yes, as we physiologists do. Q: How is the response defined? A: Its the total number of spikes in that window. (800ms). Q: What is the largest extent of the visual field you tried? A: 8 degrees. Q:How do you measure the intitial receptive field of a cell? A: We use an optimal bar. We move the bar out and figure out the point where this cell does not respond any more. The minimum region outside which you don't get a response. Comment: Other techniques can give different receptive field sizes. Q: Figure ground and ownership? A: Its just different words for the same thing. Q: Didn't understand the comment about influence of IT. A: We have the half-strength of the ownership signal at 70ms. IT, upto 100ms is silent. Q: What about the illlusory contours? ARe they more local? A: Q: Does that mean that the illusory contour is withing the summation field of the neuron? A: What we know as the summation field is quite controversial. Summation os not an explantion of the illusory contours. Q: What about natural scenes? A: I suspect that some of this would hold up.",
   "public_statement": "Chapter H: Neural Mechanisms of Perceptual Inference\n\nRudiger vonder Heydt\n\nQ: What percentage of the recorded cells were in V1? A: 50 Q: Are these unbiased selection of random cells? A: Yes, as we physiologists do. Q: How is the response defined? A: Its the total number of spikes in that window. (800ms). Q: What is the largest extent of the visual field you tried? A: 8 degrees. Q:How do you measure the intitial receptive field of a cell? A: We use an optimal bar. We move the bar out and figure out the point where this cell does not respond any more. The minimum region outside which you don't get a response. Comment: Other techniques can give different receptive field sizes. Q: Figure ground and ownership? A: Its just different words for the same thing. Q: Didn't understand the comment about influence of IT. A: We have the half-strength of the ownership signal at 70ms. IT, upto 100ms is silent. Q: What about the illlusory contours? ARe they more local? A: Q: Does that mean that the illusory contour is withing the summation field of the neuron? A: What we know as the summation field is quite controversial. Summation os not an explantion of the illusory contours. Q: What about natural scenes? A: I suspect that some of this would hold up.",
   "evidence": "This canonical record is **not an open problem**. It is Chapter H of the AIM workshop notes *Inference and Prediction in Neocortical Circuits* (AIM PDF version dated 24 October 2003), headed “Neural Mechanisms of Perceptual Inference” and attributed to Rüdiger von der Heydt. It records questions and short answers after a talk.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 29,
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  },
  "AIM-BIOLOGY-0031": {
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   "original_statement": "Chapter I: Synaptic Integration in the Early Visual Pathway \n\nJudith Hirsch \n\nDiscussion after the presentation. Q: Isn't there a push-pull mechanism in the retina? 10 \n\nA: Yes I think there is. But it has to be recreated in the thalamus because the connec-tions from retina to LGN are all excitatory. Q: What is the proportion of cells? A: Don't know. Q: How do you select the response window? A: Centered around the peak of the response. Q: (About Laminar Distribution of Simple Receptive Fields). Will this hold in monkey as well? A: No. I expect it to be different. The circuits that form the simple cells in monkeys are very different from than in cats. Comment: Excitatory and inhibitory cells look differnt. Excitatory cells are spiny. Inhibitory are smooth. Q: Orientation selectivity is coming from?? A: The pull can contribute to orientation selectivity. Q: What do the complex cells look like? A: They don't respond well to the \"sparsely dynamic\" stimulii. You need just the correct data set to come into a cell to make it fire.(?) Q: Does anyone try to knock out LGN and try to record from V1? Q: Do inhibitory cells make contact with other inhibitory cells? A: Yes I believe they do. It seems that the orientation selective inhibitory cells have the same inhibition pattern as the excitatory cells. Q: Did you look for evidence of syncrhony now that you have intra-cellular recording. A: There are lot of blips Q: What was the relative receptive field sizes of the complex interneurons and the simple excitatory neurons? A: Roughly the same(?). We don't have an accurate measure.",
   "clean_statement": "Chapter I: Synaptic Integration in the Early Visual Pathway\n\nJudith Hirsch\n\nDiscussion after the presentation. Q: Isn't there a push-pull mechanism in the retina? 10\n\nA: Yes I think there is. But it has to be recreated in the thalamus because the connec-tions from retina to LGN are all excitatory. Q: What is the proportion of cells? A: Don't know. Q: How do you select the response window? A: Centered around the peak of the response. Q: (About Laminar Distribution of Simple Receptive Fields). Will this hold in monkey as well? A: No. I expect it to be different. The circuits that form the simple cells in monkeys are very different from than in cats. Comment: Excitatory and inhibitory cells look differnt. Excitatory cells are spiny. Inhibitory are smooth. Q: Orientation selectivity is coming from?? A: The pull can contribute to orientation selectivity. Q: What do the complex cells look like? A: They don't respond well to the \"sparsely dynamic\" stimulii. You need just the correct data set to come into a cell to make it fire.(?) Q: Does anyone try to knock out LGN and try to record from V1? Q: Do inhibitory cells make contact with other inhibitory cells? A: Yes I believe they do. It seems that the orientation selective inhibitory cells have the same inhibition pattern as the excitatory cells. Q: Did you look for evidence of syncrhony now that you have intra-cellular recording. A: There are lot of blips Q: What was the relative receptive field sizes of the complex interneurons and the simple excitatory neurons? A: Roughly the same(?). We don't have an accurate measure.",
   "public_statement": "Chapter I: Synaptic Integration in the Early Visual Pathway\n\nJudith Hirsch\n\nDiscussion after the presentation. Q: Isn't there a push-pull mechanism in the retina? 10\n\nA: Yes I think there is. But it has to be recreated in the thalamus because the connec-tions from retina to LGN are all excitatory. Q: What is the proportion of cells? A: Don't know. Q: How do you select the response window? A: Centered around the peak of the response. Q: (About Laminar Distribution of Simple Receptive Fields). Will this hold in monkey as well? A: No. I expect it to be different. The circuits that form the simple cells in monkeys are very different from than in cats. Comment: Excitatory and inhibitory cells look differnt. Excitatory cells are spiny. Inhibitory are smooth. Q: Orientation selectivity is coming from?? A: The pull can contribute to orientation selectivity. Q: What do the complex cells look like? A: They don't respond well to the \"sparsely dynamic\" stimulii. You need just the correct data set to come into a cell to make it fire.(?) Q: Does anyone try to knock out LGN and try to record from V1? Q: Do inhibitory cells make contact with other inhibitory cells? A: Yes I believe they do. It seems that the orientation selective inhibitory cells have the same inhibition pattern as the excitatory cells. Q: Did you look for evidence of syncrhony now that you have intra-cellular recording. A: There are lot of blips Q: What was the relative receptive field sizes of the complex interneurons and the simple excitatory neurons? A: Roughly the same(?). We don't have an accurate measure.",
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   "original_statement": "Chapter J: Resonance Prediction and Priors \n\nTai Sing Lee \n\nQ: Can you tell us anything about the time course of the response between V1 and V2? A: This is hard to detect in our experiments. Q: Is there a connection between predictive ideas and pop out? A: Here top down predictions are more like a prior Qn: What is the attention effect in V2 neurons? A: The V2 neurons don't get attenuated even when you are attending to other locations. Q: Did you try turning off the target? If you turn the target off after 100ms. then probably you would see something interesting. A: We didn't try that. But this looks like an interesting experiment to do. Q: Did you do experiments where the monkey doesnt know where the pop out ball is? Q: The latency of the response to the popout seems to be very long to me comapred to the poput of bars. A: Here the pop out is working on 3D surfaces. Infering this 3D takes longer time. 11",
   "clean_statement": "Chapter J: Resonance Prediction and Priors\n\nTai Sing Lee\n\nQ: Can you tell us anything about the time course of the response between V1 and V2? A: This is hard to detect in our experiments. Q: Is there a connection between predictive ideas and pop out? A: Here top down predictions are more like a prior Qn: What is the attention effect in V2 neurons? A: The V2 neurons don't get attenuated even when you are attending to other locations. Q: Did you try turning off the target? If you turn the target off after 100ms. then probably you would see something interesting. A: We didn't try that. But this looks like an interesting experiment to do. Q: Did you do experiments where the monkey doesnt know where the pop out ball is? Q: The latency of the response to the popout seems to be very long to me comapred to the poput of bars. A: Here the pop out is working on 3D surfaces. Infering this 3D takes longer time. 11",
   "public_statement": "Chapter J: Resonance Prediction and Priors\n\nTai Sing Lee\n\nQ: Can you tell us anything about the time course of the response between V1 and V2? A: This is hard to detect in our experiments. Q: Is there a connection between predictive ideas and pop out? A: Here top down predictions are more like a prior Qn: What is the attention effect in V2 neurons? A: The V2 neurons don't get attenuated even when you are attending to other locations. Q: Did you try turning off the target? If you turn the target off after 100ms. then probably you would see something interesting. A: We didn't try that. But this looks like an interesting experiment to do. Q: Did you do experiments where the monkey doesnt know where the pop out ball is? Q: The latency of the response to the popout seems to be very long to me comapred to the poput of bars. A: Here the pop out is working on 3D surfaces. Infering this 3D takes longer time. 11",
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   "original_statement": "Chapter K: Notes from Breakout Session I \n\nOn the second day of the workshop the participants broke out into four different groups based on their research oreintation. The groups were (1) Pyschology and Psychophysics, (2) Anatomists, and (3) Theorists. Due to the large number of theorists, the theory group was split into two (Theory 1 and Theory 2). The groups met separately for 45 minutes. Each group was supposed to come up with a list of three things that they want from the other groups. At the end of 45 minutes all the groups met at one place and a discussion ensued on the requirements. What follows is a compilation of suggestions from different groups. \n\nSuggestions From Psychology/Psychophysics group \n\nFor theorists \n\n• models that can act on real stimuli real images, moving images \n\n• Are there associative memory models running on the hardware that we have? (spiking models) \n\n• Theory of mid-level vision that is understandable to us. For anatomists \n\n• more quantitative descriptions organized into ciruits. proportions of neurons in one area compared to another area are they myelinated or not \n\n• More oranized quantitative information. \n\n• Timing \n\n• Thalamus \n\n• Anatomists comments on theory slides. \n\nFrom Theory Group I \n\nFor Anatomists \n\n• Naturalize Stimulii \n\n• Make the raw data available. \n\n• Fast-forward development of multi-electrode recordings. \n\n• Specify data on excitatory/inhibitory..which layer they are in. For Psychologists \n\n• Is the BOLD signal reactive or predictive? Need explanation. \n\n• Request for more mathematical/theoretical training. \n\n• Theoretically driven experiments. \n\n• Theories posited before data. \n\nFrom Anatomists \n\nFor Theorists \n\n• Falsifiable predictions crucial tests. - test on how bayesian inference occurs in brain. 12 \n\n• Rigorous comparisons between models. - surface representation (biologically plausible) - Where does the model fail? \n\n• Different feedback systems. - predictions from these models. To Psychologists \n\n• Ken: What approach, what end point? \n\n• More natural stimulii. \n\nFrom Theory group II \n\nFor Both Anatomists and Psychologists \n\n• Better communication - theoretical training (academic curriculum) - tutorials or invited talks @ conferences. \n\n• Better Data A) centralized database, systematic connectivity diagram B) Reporting (availability/completeness) C) Natural stimulii/ multi-electrode recordings. \n\n• Hire us!!",
   "clean_statement": "Chapter K: Notes from Breakout Session I\n\nOn the second day of the workshop the participants broke out into four different groups based on their research oreintation. The groups were (1) Pyschology and Psychophysics, (2) Anatomists, and (3) Theorists. Due to the large number of theorists, the theory group was split into two (Theory 1 and Theory 2). The groups met separately for 45 minutes. Each group was supposed to come up with a list of three things that they want from the other groups. At the end of 45 minutes all the groups met at one place and a discussion ensued on the requirements. What follows is a compilation of suggestions from different groups.\n\nSuggestions From Psychology/Psychophysics group\n\nFor theorists\n\n• models that can act on real stimuli real images, moving images\n\n• Are there associative memory models running on the hardware that we have? (spiking models)\n\n• Theory of mid-level vision that is understandable to us. For anatomists\n\n• more quantitative descriptions organized into ciruits. proportions of neurons in one area compared to another area are they myelinated or not\n\n• More oranized quantitative information.\n\n• Timing\n\n• Thalamus\n\n• Anatomists comments on theory slides.\n\nFrom Theory Group I\n\nFor Anatomists\n\n• Naturalize Stimulii\n\n• Make the raw data available.\n\n• Fast-forward development of multi-electrode recordings.\n\n• Specify data on excitatory/inhibitory..which layer they are in. For Psychologists\n\n• Is the BOLD signal reactive or predictive? Need explanation.\n\n• Request for more mathematical/theoretical training.\n\n• Theoretically driven experiments.\n\n• Theories posited before data.\n\nFrom Anatomists\n\nFor Theorists\n\n• Falsifiable predictions crucial tests. - test on how bayesian inference occurs in brain. 12\n\n• Rigorous comparisons between models. - surface representation (biologically plausible) - Where does the model fail?\n\n• Different feedback systems. - predictions from these models. To Psychologists\n\n• Ken: What approach, what end point?\n\n• More natural stimulii.\n\nFrom Theory group II\n\nFor Both Anatomists and Psychologists\n\n• Better communication - theoretical training (academic curriculum) - tutorials or invited talks @ conferences.\n\n• Better Data A) centralized database, systematic connectivity diagram B) Reporting (availability/completeness) C) Natural stimulii/ multi-electrode recordings.\n\n• Hire us!!",
   "public_statement": "Chapter K: Notes from Breakout Session I\n\nOn the second day of the workshop the participants broke out into four different groups based on their research oreintation. The groups were (1) Pyschology and Psychophysics, (2) Anatomists, and (3) Theorists. Due to the large number of theorists, the theory group was split into two (Theory 1 and Theory 2). The groups met separately for 45 minutes. Each group was supposed to come up with a list of three things that they want from the other groups. At the end of 45 minutes all the groups met at one place and a discussion ensued on the requirements. What follows is a compilation of suggestions from different groups.\n\nSuggestions From Psychology/Psychophysics group\n\nFor theorists\n\n• models that can act on real stimuli real images, moving images\n\n• Are there associative memory models running on the hardware that we have? (spiking models)\n\n• Theory of mid-level vision that is understandable to us. For anatomists\n\n• more quantitative descriptions organized into ciruits. proportions of neurons in one area compared to another area are they myelinated or not\n\n• More oranized quantitative information.\n\n• Timing\n\n• Thalamus\n\n• Anatomists comments on theory slides.\n\nFrom Theory Group I\n\nFor Anatomists\n\n• Naturalize Stimulii\n\n• Make the raw data available.\n\n• Fast-forward development of multi-electrode recordings.\n\n• Specify data on excitatory/inhibitory..which layer they are in. For Psychologists\n\n• Is the BOLD signal reactive or predictive? Need explanation.\n\n• Request for more mathematical/theoretical training.\n\n• Theoretically driven experiments.\n\n• Theories posited before data.\n\nFrom Anatomists\n\nFor Theorists\n\n• Falsifiable predictions crucial tests. - test on how bayesian inference occurs in brain. 12\n\n• Rigorous comparisons between models. - surface representation (biologically plausible) - Where does the model fail?\n\n• Different feedback systems. - predictions from these models. To Psychologists\n\n• Ken: What approach, what end point?\n\n• More natural stimulii.\n\nFrom Theory group II\n\nFor Both Anatomists and Psychologists\n\n• Better communication - theoretical training (academic curriculum) - tutorials or invited talks @ conferences.\n\n• Better Data A) centralized database, systematic connectivity diagram B) Reporting (availability/completeness) C) Natural stimulii/ multi-electrode recordings.\n\n• Hire us!!",
   "evidence": "This record is Chapter K, “Notes from Breakout Session I,” in the official AIM workshop notes *Inference and Prediction in Neocortical Circuits*, version dated 24 October 2003. It is a compilation of requests exchanged by workshop groups, not a single open problem.",
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   "original_statement": "Chapter L: Notes from Breakout Session II \n\nThe four groups met separately to prepare responses to the comments from the previous day. \n\nReply From Psychologists \n\nThe theorists wanted to be hired. The primate experiments would go much better if they did psychophysical experiments. Giving away raw data looks like giving away ones own work. Alessandra: Once something is published it is public domain and so the data should me bade public. Judith: It could be that you have one sort of analysis and more analysis might be waiting. you just don't want to give away the data. We should be able to publish the data \n\nFrom Theory group I. \n\n• Are there associative spiking models with spiking neurons? (Yes). Fritz will give a presentation on that. \n\n• Theoreticians will become aware of training in experimentation. This will give us more intuition. But we need more help in doing that. We would also like the data from you! \n\n• Russ: Rigorous model comparisons between different models. One can apply the same inputs to models and get the results. Should document when the models fail. \n\n• Falsifiable predictions?: We alread do. \n\n• The job of theorists is not just to make testable hypothesis. 13 \n\nFrom Anatomists \n\n• Now more labs are producing less descriptive and more quantitative data. There is some kind of questions that take too much time to address. (For eg: number of synapses). How can computer science help us for this? \n\n• There are computer vision techniques and tools available that could be used. Multi-electrode recording? Its a lot of effort. If the cells cannot be activated at the same time, it amounts to recording one after another. \n\n• Natural stimuli. We have acquired multichannel recording systems and we are racing to do multi electrode recording. \n\nFrom Theory II \n\n• Bayesian Framework.: Bayesian framework is a principled way of looking at the computational problems faced by the brian given the ambiguous Priors can be implemented as lateral connectivity or synaptic weights or top-down predictions. Predictions for testing the bayesian framework: Rajesh Rao's work \n\n• Different forms of feedback: We really don't know how these forms of feedback are. The diffused feedback that is topographic could be doing an spatial priming and the patchy ones are doing feature priming. \n\n• Tony: Natural stimulii. Adaptive models: Natural stimulii. Yes. In the case of hardwired models: it will give tighter prediction to the experimentalists.",
   "clean_statement": "Chapter L: Notes from Breakout Session II\n\nThe four groups met separately to prepare responses to the comments from the previous day.\n\nReply From Psychologists\n\nThe theorists wanted to be hired. The primate experiments would go much better if they did psychophysical experiments. Giving away raw data looks like giving away ones own work. Alessandra: Once something is published it is public domain and so the data should me bade public. Judith: It could be that you have one sort of analysis and more analysis might be waiting. you just don't want to give away the data. We should be able to publish the data\n\nFrom Theory group I.\n\n• Are there associative spiking models with spiking neurons? (Yes). Fritz will give a presentation on that.\n\n• Theoreticians will become aware of training in experimentation. This will give us more intuition. But we need more help in doing that. We would also like the data from you!\n\n• Russ: Rigorous model comparisons between different models. One can apply the same inputs to models and get the results. Should document when the models fail.\n\n• Falsifiable predictions?: We alread do.\n\n• The job of theorists is not just to make testable hypothesis. 13\n\nFrom Anatomists\n\n• Now more labs are producing less descriptive and more quantitative data. There is some kind of questions that take too much time to address. (For eg: number of synapses). How can computer science help us for this?\n\n• There are computer vision techniques and tools available that could be used. Multi-electrode recording? Its a lot of effort. If the cells cannot be activated at the same time, it amounts to recording one after another.\n\n• Natural stimuli. We have acquired multichannel recording systems and we are racing to do multi electrode recording.\n\nFrom Theory II\n\n• Bayesian Framework.: Bayesian framework is a principled way of looking at the computational problems faced by the brian given the ambiguous Priors can be implemented as lateral connectivity or synaptic weights or top-down predictions. Predictions for testing the bayesian framework: Rajesh Rao's work\n\n• Different forms of feedback: We really don't know how these forms of feedback are. The diffused feedback that is topographic could be doing an spatial priming and the patchy ones are doing feature priming.\n\n• Tony: Natural stimulii. Adaptive models: Natural stimulii. Yes. In the case of hardwired models: it will give tighter prediction to the experimentalists.",
   "public_statement": "Chapter L: Notes from Breakout Session II\n\nThe four groups met separately to prepare responses to the comments from the previous day.\n\nReply From Psychologists\n\nThe theorists wanted to be hired. The primate experiments would go much better if they did psychophysical experiments. Giving away raw data looks like giving away ones own work. Alessandra: Once something is published it is public domain and so the data should me bade public. Judith: It could be that you have one sort of analysis and more analysis might be waiting. you just don't want to give away the data. We should be able to publish the data\n\nFrom Theory group I.\n\n• Are there associative spiking models with spiking neurons? (Yes). Fritz will give a presentation on that.\n\n• Theoreticians will become aware of training in experimentation. This will give us more intuition. But we need more help in doing that. We would also like the data from you!\n\n• Russ: Rigorous model comparisons between different models. One can apply the same inputs to models and get the results. Should document when the models fail.\n\n• Falsifiable predictions?: We alread do.\n\n• The job of theorists is not just to make testable hypothesis. 13\n\nFrom Anatomists\n\n• Now more labs are producing less descriptive and more quantitative data. There is some kind of questions that take too much time to address. (For eg: number of synapses). How can computer science help us for this?\n\n• There are computer vision techniques and tools available that could be used. Multi-electrode recording? Its a lot of effort. If the cells cannot be activated at the same time, it amounts to recording one after another.\n\n• Natural stimuli. We have acquired multichannel recording systems and we are racing to do multi electrode recording.\n\nFrom Theory II\n\n• Bayesian Framework.: Bayesian framework is a principled way of looking at the computational problems faced by the brian given the ambiguous Priors can be implemented as lateral connectivity or synaptic weights or top-down predictions. Predictions for testing the bayesian framework: Rajesh Rao's work\n\n• Different forms of feedback: We really don't know how these forms of feedback are. The diffused feedback that is topographic could be doing an spatial priming and the patchy ones are doing feature priming.\n\n• Tony: Natural stimulii. Adaptive models: Natural stimulii. Yes. In the case of hardwired models: it will give tighter prediction to the experimentalists.",
   "evidence": "This canonical record is Chapter L, “Notes from Breakout Session II,” in the American Institute of Mathematics workshop notes *Inference and Prediction in Neocortical Circuits* (version dated 24 October 2003). It is a discussion summary, tagged `section`, rather than a formal mathematical problem.",
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   "original_statement": "A.1 Introduction to Discussion Session Goals \n\nThis document presents an outline of some of content of the afternoon discussions. In the interest of producing these notes as quickly as possible, no attempt has been made to track down every reference or attribute every comment to a participant. Also, as the scribe, I summarized what I understood from the discussion, so attributed comments are, in some cases, paraphrases. (Thus, you may wish to check with the attributed before making any consequent attribution.) Each afternoon session was centered around some theme. The purpose of the ses-sions was to try to identify problems that people might want to pursue, and identify which problems are most important to work on. Also, with participants from many different aca-demic communities (biology, combinatorics, topology, statistics), it was important for each community to hear perspectives from the others.",
   "clean_statement": "A.1 Introduction to Discussion Session Goals\n\nThis document presents an outline of some of content of the afternoon discussions. In the interest of producing these notes as quickly as possible, no attempt has been made to track down every reference or attribute every comment to a participant. Also, as the scribe, I summarized what I understood from the discussion, so attributed comments are, in some cases, paraphrases. (Thus, you may wish to check with the attributed before making any consequent attribution.) Each afternoon session was centered around some theme. The purpose of the ses-sions was to try to identify problems that people might want to pursue, and identify which problems are most important to work on. Also, with participants from many different aca-demic communities (biology, combinatorics, topology, statistics), it was important for each community to hear perspectives from the others.",
   "public_statement": "A.1 Introduction to Discussion Session Goals\n\nThis document presents an outline of some of content of the afternoon discussions. In the interest of producing these notes as quickly as possible, no attempt has been made to track down every reference or attribute every comment to a participant. Also, as the scribe, I summarized what I understood from the discussion, so attributed comments are, in some cases, paraphrases. (Thus, you may wish to check with the attributed before making any consequent attribution.) Each afternoon session was centered around some theme. The purpose of the ses-sions was to try to identify problems that people might want to pursue, and identify which problems are most important to work on. Also, with participants from many different aca-demic communities (biology, combinatorics, topology, statistics), it was important for each community to hear perspectives from the others.",
   "evidence": "This canonical record is item A.1, tagged as a section rather than as a question, in the AIM workshop notes *Geometric models of biological phenomena*. The source record reads:",
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   "original_statement": "A.2 Sunday Discussion on Biological Issues \n\nThe discussion was moderated by John Luecke. Susan Holmes opened the discussion by suggesting that we might ask the biologists: what mathematical or biological questions related to phylogenetic trees are most important to biologists? She also invited clarifications about things that have been addressed in talks earlier in the day. Q. What properties of phylogenetic trees make them different from random trees? Is there some kind of structure that makes them different? (Epstein) \n\n• Phylogenetic trees don't look like trees generated by random branching processes. (Huelsenbeck) Q. What are the characteristics that a \"distance between trees\" should have that biologists would want? (Luecke) \n\n• Luecke: For instance, the Billara-Holmes-Vogtmann (BHV) metric measures dis-tances along geodesics in tree space, and the trees change as you move along the geodesic, but does this have a reasonable biological intepretation? \n\n• Sober: that does not have biological meaning. Biologists just want to know how to tell if trees are similar or not. \n\n• Felsenstein: you are assuming we (biologists) want a notion of distance between trees? (Scribe: this appears to have been in contrast to just a notion of similarity. This elicited remarks from mathematicians about the fruitfulness of using a metric to tell how similar two trees are.) \n\n• Huelsenbeck pointed out several ways to measure how different trees are, e.g., involv-ing the contraction/expansion of edges to get from one to the other, or differences in length of edges, or squared branch length, etc. Some discussion of the Robinson-Foulds distance ensued. \n\n• Luecke: so computational simplicity would be one important consideration for a distance. \n\n• Huelsenbeck: Any distance notion on phylogenetic trees should have a good theoret-ical foundation. 4\n\nQ. Is there a difference between some real distribution of trees on tree space rather than some random distribution? (Holmes) \n\n• Evans: this will help us to construct meaningful confidence sets on trees. \n\n• Felsenstein: Tree spaces are weird. For a cloud of trees, we want to characterize where in this space those trees are. Q. Problem: make notions of distances accessible to biologists, so they might use them. (Huelsenbeck) \n\n• Some comments were made about the use of methods of Kuhner-Felsenstein, weighted Robinson-Foulds, in biology. \n\n• Diaconis pointed out that in his work on non-standard data structures, the notions of distances are very useful and basic. In doing exploratory analyses, they can be useful in statistical tasks. So, what do biologists think of these distances? \n\n• Holmes pointed out that how fast MCMC methods converge depends on the geometry of the space, and the metric chosen. \n\n• Huelsenbeck suggested that a useful task for mathematicians would be to make some sense of the uses of these distances to biologists? (such as in an article pitched to biologists). More comments about tree space topology and branch lengths: \n\n• Huelsenbeck: are there versions of tree space that biologists might not be interested in, due to technical conditions? For instance, if you give a program an alignment, the branches have to be short enough so that you can align them. \n\n• Felsenstein: In many cases in biology we are interested in grappling with, such as origins of mammals, many of the orders seem to have popped up rather quickly. We could be interested in (a) the branch length and (b) the topology. When the orders diverged, important things were happening very quickly, such as morphological changes. But the molecules you are studying may not have been involved. If you are interested in those morphological changes, it is important to know what order the branching occurred. On the other hand, in some other questions, topology many not be as important as branch length. It really depends on the question you are asking. \n\n• Sober: A third issue is important: the character states of the interior nodes of trees. You might want to know what is the sequence of changes that occurred on some branch? for instance in some calculation, you may want to integrate over all possible states of interior nodes. Felsenstein pointed out that these states on interior nodes are sometimes overinterpreted- they are often viewed as actual states, rather than estimates. \n\n• Penny: in biology, knowing the \"true\" tree may only be a starting point of the real investigation- some other aspect that the biologist is truly interested in. Huelsen-beck gave an example of an evolution biologist interested in sexual selection. Al-though she made a phylogeny (what she thought was the best tree), she wasn't primarily interested in the phylogeny, but in further questions about sexual selec-tion. Q. How to pick a tree (or tree average) from a set of trees resulting from data? 5\n\n• Epstein: in bacteria dna fragments, each one gives a tree. Which one to use? Need a tree distance to analyze the resulting 22 different trees. The distance he used was the BHV metric, but he suspects one obtains much the same answer with many different tree metrics. \n\n• Evans: this is similar to Mallow's model in statistics, where you have a distribution on permutations, centered on a permutation, dropping off in some radial sense. Here, you have a central tree, and then the probability that you observe some other tree dies off as you move farther away. \n\n• Penny: one approach to identify \"bad\" portions of a tree is to try to \"identify the guilty taxon\", by succedssively removing taxa and then see whether this stabilizes the tree significantly. Q. How to understand or deal with residuals, such as non-tree like data? \n\n• Holmes: gave an example in which a friend has a reference tree, and 8 different plumage trees. In tree space, if the plumage tree differs from the reference dna tree in some direction, is that present in the way these things actually evolved? In statistics, when doing regression, you compare data points to a fitted line, and ignore the ones that are way off. Here, in tree space, there's a similar question for non-tree like data: how much did I have to bend the data to make it into a tree? \n\n• Sober: how do you decide how much off is too much, before you are worried? (Scribe's note: this seems to require a notion of distance not just on tree space, but some larger space- such as on the space of DNA sequences- in which tree space is embedded. We discussed embedding questions on Wednesday.) More comments on distances: \n\n• Diaconis told a story where distances fit data remarkably well. Perception psycholo-gist Roger Shepard studied the visual system, by showing subjects a configuration of blocks, then some other configuration, and then asked them: are they are the same? He found that the time it took people to decide was the geodesic in the three di-mensional rotation group. The data is quite remarkable, and gave straight line plots. We know what it means to measure the distance in the rotation group (but why the brain should know about this is some fascinating subject in itself.) \n\n• Similarly, does a distance between trees have to have some interpretation in terms of evolution, in order to be the \"right\" notion of distance? \n\n• Epstein: For instance, given dna or amino acid sequences, one for each taxon, this data produces a tree. As sequences change or evolve one nucleotide at a time, the trees will change. What path does this trace out in tree space? (Scribe's note: this appears to require a space of trees that includes trees with many different numbers of leaves, so that one can speak of how a tree evolves as species split off from one another.) \n\n• In response to a question, Felsenstein mentioned a few sources of variation in trees: (a) statistical error. (b) coalescence: take a gene copy in three species, and think of copies ancestral to these. The copies do not come together instantly, but have some stochastic chance of mixing, and they may come in some random order that conflicts with a species tree. (c) horizontal gene transfer. (Scribe's note: a more detailed discussion of this topic can be found in Tuesday's discussion.) 6\n\nQ. Which is better: concatenating DNA sequences first or averaging trees later? \n\n• Holmes: empirical studies show that if you take all the data and compute one tree, you generally do less well at estimating the tree than than if you take the fragments, use them, then average them. This comes from the CAT(0) property. The intuition behind it is if you average in a negatively curved space, you converge much faster than you should. \n\n• St. John: pointed out an example with 20 simulated DNA sequences: if a is one part of the genome, and b is another part, then the trees obtained from using a combined with b do not overlap, and are somewhat in between, the trees obtained from a and from b alone. Felsenstein commented that the stochastic effects pushed a and b in different directions in tree space. She also noted that in studying hybridization (e.g., sunflowers), she was surprised that she didn't get more overlap, and often the right answer was with part of the data, not all the data. \n\n• Felsenstein: Biologists are in disagreement about whether concatenating or averaging is better. Statistically, which is better? \n\n• It was pointed out that a paper by Cunningham (1997) does comparisons. Also, work by Amit, and statistical literature on boosting and bagging. \n\n• Billera: there's a notion of equivalent metrics in topology, and any of these will do. (In doing geometry, where issues like curvature come into play, the choice of metrics is important.) It may be the case that even though certain metrics are good enough for some purposes, it doesn't mean that other metrics aren't valid. Q. What is the right notion of an \"average\" of trees? \n\n• The biologists present agreed that this was a very interesting question for biologists. \n\n• Vert: Two important ideas emerge when working with decision trees: (1) average decrease some variance, (2) averaging can leave the space. Boosting will leave the space, if we don't leave the space it's really not boosting. The averaging we are discussing here doesn't leave the space of trees. \n\n• More discussion on averaging took place here. Holmes made a comment about the \"non-associativity\" of trees (related to considering trees of trees, when building trees one character at a time). Billera made some remarks about how averaging may be better than concatenation, but we have no guarantee that it is any good. \n\n• Penny discussed the notion of a median tree: the tree that is closest on average to all the others. \n\n• Felsenstein: other examples are the consensus tree, and majority rule consensus tree. Q. What are good properties of averaging? (Luecke) \n\n• Penny: would like a fully resolved binary tree, though he points out that some others might sacrifice that in favor of other features. \n\n• Felsenstein posed a problem about averaging properties: given a bunch of trees from different genes, to study a question like: are chimps closer to humans than gorillas? Say 2/3 of the trees trees show humans with chimps, others show chimps with gorillas. Suppose in trees that group humans with chimps, the average branch length is 1, but the others don't have it. When you average... do you want it to be length.66 or length 1? 7\n\nThere was a question about how different are trees that result from diffrent averaging methods. It was pointed out (Huelsenbeck) that differences between methods for a single gene is much smaller than the differences across genes. Q. What would be the distribution on trees that gives majority rule consensus as its average? (Holmes) \n\n• Holmes: would like any tree average to be an expected value with respect to some distribution on tree space.",
   "clean_statement": "A.2 Sunday Discussion on Biological Issues\n\nThe discussion was moderated by John Luecke. Susan Holmes opened the discussion by suggesting that we might ask the biologists: what mathematical or biological questions related to phylogenetic trees are most important to biologists? She also invited clarifications about things that have been addressed in talks earlier in the day. Q. What properties of phylogenetic trees make them different from random trees? Is there some kind of structure that makes them different? (Epstein)\n\n• Phylogenetic trees don't look like trees generated by random branching processes. (Huelsenbeck) Q. What are the characteristics that a \"distance between trees\" should have that biologists would want? (Luecke)\n\n• Luecke: For instance, the Billara-Holmes-Vogtmann (BHV) metric measures dis-tances along geodesics in tree space, and the trees change as you move along the geodesic, but does this have a reasonable biological intepretation?\n\n• Sober: that does not have biological meaning. Biologists just want to know how to tell if trees are similar or not.\n\n• Felsenstein: you are assuming we (biologists) want a notion of distance between trees? (Scribe: this appears to have been in contrast to just a notion of similarity. This elicited remarks from mathematicians about the fruitfulness of using a metric to tell how similar two trees are.)\n\n• Huelsenbeck pointed out several ways to measure how different trees are, e.g., involv-ing the contraction/expansion of edges to get from one to the other, or differences in length of edges, or squared branch length, etc. Some discussion of the Robinson-Foulds distance ensued.\n\n• Luecke: so computational simplicity would be one important consideration for a distance.\n\n• Huelsenbeck: Any distance notion on phylogenetic trees should have a good theoret-ical foundation. 4\n\nQ. Is there a difference between some real distribution of trees on tree space rather than some random distribution? (Holmes)\n\n• Evans: this will help us to construct meaningful confidence sets on trees.\n\n• Felsenstein: Tree spaces are weird. For a cloud of trees, we want to characterize where in this space those trees are. Q. Problem: make notions of distances accessible to biologists, so they might use them. (Huelsenbeck)\n\n• Some comments were made about the use of methods of Kuhner-Felsenstein, weighted Robinson-Foulds, in biology.\n\n• Diaconis pointed out that in his work on non-standard data structures, the notions of distances are very useful and basic. In doing exploratory analyses, they can be useful in statistical tasks. So, what do biologists think of these distances?\n\n• Holmes pointed out that how fast MCMC methods converge depends on the geometry of the space, and the metric chosen.\n\n• Huelsenbeck suggested that a useful task for mathematicians would be to make some sense of the uses of these distances to biologists? (such as in an article pitched to biologists). More comments about tree space topology and branch lengths:\n\n• Huelsenbeck: are there versions of tree space that biologists might not be interested in, due to technical conditions? For instance, if you give a program an alignment, the branches have to be short enough so that you can align them.\n\n• Felsenstein: In many cases in biology we are interested in grappling with, such as origins of mammals, many of the orders seem to have popped up rather quickly. We could be interested in (a) the branch length and (b) the topology. When the orders diverged, important things were happening very quickly, such as morphological changes. But the molecules you are studying may not have been involved. If you are interested in those morphological changes, it is important to know what order the branching occurred. On the other hand, in some other questions, topology many not be as important as branch length. It really depends on the question you are asking.\n\n• Sober: A third issue is important: the character states of the interior nodes of trees. You might want to know what is the sequence of changes that occurred on some branch? for instance in some calculation, you may want to integrate over all possible states of interior nodes. Felsenstein pointed out that these states on interior nodes are sometimes overinterpreted- they are often viewed as actual states, rather than estimates.\n\n• Penny: in biology, knowing the \"true\" tree may only be a starting point of the real investigation- some other aspect that the biologist is truly interested in. Huelsen-beck gave an example of an evolution biologist interested in sexual selection. Al-though she made a phylogeny (what she thought was the best tree), she wasn't primarily interested in the phylogeny, but in further questions about sexual selec-tion. Q. How to pick a tree (or tree average) from a set of trees resulting from data? 5\n\n• Epstein: in bacteria dna fragments, each one gives a tree. Which one to use? Need a tree distance to analyze the resulting 22 different trees. The distance he used was the BHV metric, but he suspects one obtains much the same answer with many different tree metrics.\n\n• Evans: this is similar to Mallow's model in statistics, where you have a distribution on permutations, centered on a permutation, dropping off in some radial sense. Here, you have a central tree, and then the probability that you observe some other tree dies off as you move farther away.\n\n• Penny: one approach to identify \"bad\" portions of a tree is to try to \"identify the guilty taxon\", by succedssively removing taxa and then see whether this stabilizes the tree significantly. Q. How to understand or deal with residuals, such as non-tree like data?\n\n• Holmes: gave an example in which a friend has a reference tree, and 8 different plumage trees. In tree space, if the plumage tree differs from the reference dna tree in some direction, is that present in the way these things actually evolved? In statistics, when doing regression, you compare data points to a fitted line, and ignore the ones that are way off. Here, in tree space, there's a similar question for non-tree like data: how much did I have to bend the data to make it into a tree?\n\n• Sober: how do you decide how much off is too much, before you are worried? (Scribe's note: this seems to require a notion of distance not just on tree space, but some larger space- such as on the space of DNA sequences- in which tree space is embedded. We discussed embedding questions on Wednesday.) More comments on distances:\n\n• Diaconis told a story where distances fit data remarkably well. Perception psycholo-gist Roger Shepard studied the visual system, by showing subjects a configuration of blocks, then some other configuration, and then asked them: are they are the same? He found that the time it took people to decide was the geodesic in the three di-mensional rotation group. The data is quite remarkable, and gave straight line plots. We know what it means to measure the distance in the rotation group (but why the brain should know about this is some fascinating subject in itself.)\n\n• Similarly, does a distance between trees have to have some interpretation in terms of evolution, in order to be the \"right\" notion of distance?\n\n• Epstein: For instance, given dna or amino acid sequences, one for each taxon, this data produces a tree. As sequences change or evolve one nucleotide at a time, the trees will change. What path does this trace out in tree space? (Scribe's note: this appears to require a space of trees that includes trees with many different numbers of leaves, so that one can speak of how a tree evolves as species split off from one another.)\n\n• In response to a question, Felsenstein mentioned a few sources of variation in trees: (a) statistical error. (b) coalescence: take a gene copy in three species, and think of copies ancestral to these. The copies do not come together instantly, but have some stochastic chance of mixing, and they may come in some random order that conflicts with a species tree. (c) horizontal gene transfer. (Scribe's note: a more detailed discussion of this topic can be found in Tuesday's discussion.) 6\n\nQ. Which is better: concatenating DNA sequences first or averaging trees later?\n\n• Holmes: empirical studies show that if you take all the data and compute one tree, you generally do less well at estimating the tree than than if you take the fragments, use them, then average them. This comes from the CAT(0) property. The intuition behind it is if you average in a negatively curved space, you converge much faster than you should.\n\n• St. John: pointed out an example with 20 simulated DNA sequences: if a is one part of the genome, and b is another part, then the trees obtained from using a combined with b do not overlap, and are somewhat in between, the trees obtained from a and from b alone. Felsenstein commented that the stochastic effects pushed a and b in different directions in tree space. She also noted that in studying hybridization (e.g., sunflowers), she was surprised that she didn't get more overlap, and often the right answer was with part of the data, not all the data.\n\n• Felsenstein: Biologists are in disagreement about whether concatenating or averaging is better. Statistically, which is better?\n\n• It was pointed out that a paper by Cunningham (1997) does comparisons. Also, work by Amit, and statistical literature on boosting and bagging.\n\n• Billera: there's a notion of equivalent metrics in topology, and any of these will do. (In doing geometry, where issues like curvature come into play, the choice of metrics is important.) It may be the case that even though certain metrics are good enough for some purposes, it doesn't mean that other metrics aren't valid. Q. What is the right notion of an \"average\" of trees?\n\n• The biologists present agreed that this was a very interesting question for biologists.\n\n• Vert: Two important ideas emerge when working with decision trees: (1) average decrease some variance, (2) averaging can leave the space. Boosting will leave the space, if we don't leave the space it's really not boosting. The averaging we are discussing here doesn't leave the space of trees.\n\n• More discussion on averaging took place here. Holmes made a comment about the \"non-associativity\" of trees (related to considering trees of trees, when building trees one character at a time). Billera made some remarks about how averaging may be better than concatenation, but we have no guarantee that it is any good.\n\n• Penny discussed the notion of a median tree: the tree that is closest on average to all the others.\n\n• Felsenstein: other examples are the consensus tree, and majority rule consensus tree. Q. What are good properties of averaging? (Luecke)\n\n• Penny: would like a fully resolved binary tree, though he points out that some others might sacrifice that in favor of other features.\n\n• Felsenstein posed a problem about averaging properties: given a bunch of trees from different genes, to study a question like: are chimps closer to humans than gorillas? Say 2/3 of the trees trees show humans with chimps, others show chimps with gorillas. Suppose in trees that group humans with chimps, the average branch length is 1, but the others don't have it. When you average... do you want it to be length.66 or length 1? 7\n\nThere was a question about how different are trees that result from diffrent averaging methods. It was pointed out (Huelsenbeck) that differences between methods for a single gene is much smaller than the differences across genes. Q. What would be the distribution on trees that gives majority rule consensus as its average? (Holmes)\n\n• Holmes: would like any tree average to be an expected value with respect to some distribution on tree space.",
   "public_statement": "A.2 Sunday Discussion on Biological Issues\n\nThe discussion was moderated by John Luecke. Susan Holmes opened the discussion by suggesting that we might ask the biologists: what mathematical or biological questions related to phylogenetic trees are most important to biologists? She also invited clarifications about things that have been addressed in talks earlier in the day. Q. What properties of phylogenetic trees make them different from random trees? Is there some kind of structure that makes them different? (Epstein)\n\n• Phylogenetic trees don't look like trees generated by random branching processes. (Huelsenbeck) Q. What are the characteristics that a \"distance between trees\" should have that biologists would want? (Luecke)\n\n• Luecke: For instance, the Billara-Holmes-Vogtmann (BHV) metric measures dis-tances along geodesics in tree space, and the trees change as you move along the geodesic, but does this have a reasonable biological intepretation?\n\n• Sober: that does not have biological meaning. Biologists just want to know how to tell if trees are similar or not.\n\n• Felsenstein: you are assuming we (biologists) want a notion of distance between trees? (Scribe: this appears to have been in contrast to just a notion of similarity. This elicited remarks from mathematicians about the fruitfulness of using a metric to tell how similar two trees are.)\n\n• Huelsenbeck pointed out several ways to measure how different trees are, e.g., involv-ing the contraction/expansion of edges to get from one to the other, or differences in length of edges, or squared branch length, etc. Some discussion of the Robinson-Foulds distance ensued.\n\n• Luecke: so computational simplicity would be one important consideration for a distance.\n\n• Huelsenbeck: Any distance notion on phylogenetic trees should have a good theoret-ical foundation. 4\n\nQ. Is there a difference between some real distribution of trees on tree space rather than some random distribution? (Holmes)\n\n• Evans: this will help us to construct meaningful confidence sets on trees.\n\n• Felsenstein: Tree spaces are weird. For a cloud of trees, we want to characterize where in this space those trees are. Q. Problem: make notions of distances accessible to biologists, so they might use them. (Huelsenbeck)\n\n• Some comments were made about the use of methods of Kuhner-Felsenstein, weighted Robinson-Foulds, in biology.\n\n• Diaconis pointed out that in his work on non-standard data structures, the notions of distances are very useful and basic. In doing exploratory analyses, they can be useful in statistical tasks. So, what do biologists think of these distances?\n\n• Holmes pointed out that how fast MCMC methods converge depends on the geometry of the space, and the metric chosen.\n\n• Huelsenbeck suggested that a useful task for mathematicians would be to make some sense of the uses of these distances to biologists? (such as in an article pitched to biologists). More comments about tree space topology and branch lengths:\n\n• Huelsenbeck: are there versions of tree space that biologists might not be interested in, due to technical conditions? For instance, if you give a program an alignment, the branches have to be short enough so that you can align them.\n\n• Felsenstein: In many cases in biology we are interested in grappling with, such as origins of mammals, many of the orders seem to have popped up rather quickly. We could be interested in (a) the branch length and (b) the topology. When the orders diverged, important things were happening very quickly, such as morphological changes. But the molecules you are studying may not have been involved. If you are interested in those morphological changes, it is important to know what order the branching occurred. On the other hand, in some other questions, topology many not be as important as branch length. It really depends on the question you are asking.\n\n• Sober: A third issue is important: the character states of the interior nodes of trees. You might want to know what is the sequence of changes that occurred on some branch? for instance in some calculation, you may want to integrate over all possible states of interior nodes. Felsenstein pointed out that these states on interior nodes are sometimes overinterpreted- they are often viewed as actual states, rather than estimates.\n\n• Penny: in biology, knowing the \"true\" tree may only be a starting point of the real investigation- some other aspect that the biologist is truly interested in. Huelsen-beck gave an example of an evolution biologist interested in sexual selection. Al-though she made a phylogeny (what she thought was the best tree), she wasn't primarily interested in the phylogeny, but in further questions about sexual selec-tion. Q. How to pick a tree (or tree average) from a set of trees resulting from data? 5\n\n• Epstein: in bacteria dna fragments, each one gives a tree. Which one to use? Need a tree distance to analyze the resulting 22 different trees. The distance he used was the BHV metric, but he suspects one obtains much the same answer with many different tree metrics.\n\n• Evans: this is similar to Mallow's model in statistics, where you have a distribution on permutations, centered on a permutation, dropping off in some radial sense. Here, you have a central tree, and then the probability that you observe some other tree dies off as you move farther away.\n\n• Penny: one approach to identify \"bad\" portions of a tree is to try to \"identify the guilty taxon\", by succedssively removing taxa and then see whether this stabilizes the tree significantly. Q. How to understand or deal with residuals, such as non-tree like data?\n\n• Holmes: gave an example in which a friend has a reference tree, and 8 different plumage trees. In tree space, if the plumage tree differs from the reference dna tree in some direction, is that present in the way these things actually evolved? In statistics, when doing regression, you compare data points to a fitted line, and ignore the ones that are way off. Here, in tree space, there's a similar question for non-tree like data: how much did I have to bend the data to make it into a tree?\n\n• Sober: how do you decide how much off is too much, before you are worried? (Scribe's note: this seems to require a notion of distance not just on tree space, but some larger space- such as on the space of DNA sequences- in which tree space is embedded. We discussed embedding questions on Wednesday.) More comments on distances:\n\n• Diaconis told a story where distances fit data remarkably well. Perception psycholo-gist Roger Shepard studied the visual system, by showing subjects a configuration of blocks, then some other configuration, and then asked them: are they are the same? He found that the time it took people to decide was the geodesic in the three di-mensional rotation group. The data is quite remarkable, and gave straight line plots. We know what it means to measure the distance in the rotation group (but why the brain should know about this is some fascinating subject in itself.)\n\n• Similarly, does a distance between trees have to have some interpretation in terms of evolution, in order to be the \"right\" notion of distance?\n\n• Epstein: For instance, given dna or amino acid sequences, one for each taxon, this data produces a tree. As sequences change or evolve one nucleotide at a time, the trees will change. What path does this trace out in tree space? (Scribe's note: this appears to require a space of trees that includes trees with many different numbers of leaves, so that one can speak of how a tree evolves as species split off from one another.)\n\n• In response to a question, Felsenstein mentioned a few sources of variation in trees: (a) statistical error. (b) coalescence: take a gene copy in three species, and think of copies ancestral to these. The copies do not come together instantly, but have some stochastic chance of mixing, and they may come in some random order that conflicts with a species tree. (c) horizontal gene transfer. (Scribe's note: a more detailed discussion of this topic can be found in Tuesday's discussion.) 6\n\nQ. Which is better: concatenating DNA sequences first or averaging trees later?\n\n• Holmes: empirical studies show that if you take all the data and compute one tree, you generally do less well at estimating the tree than than if you take the fragments, use them, then average them. This comes from the CAT(0) property. The intuition behind it is if you average in a negatively curved space, you converge much faster than you should.\n\n• St. John: pointed out an example with 20 simulated DNA sequences: if a is one part of the genome, and b is another part, then the trees obtained from using a combined with b do not overlap, and are somewhat in between, the trees obtained from a and from b alone. Felsenstein commented that the stochastic effects pushed a and b in different directions in tree space. She also noted that in studying hybridization (e.g., sunflowers), she was surprised that she didn't get more overlap, and often the right answer was with part of the data, not all the data.\n\n• Felsenstein: Biologists are in disagreement about whether concatenating or averaging is better. Statistically, which is better?\n\n• It was pointed out that a paper by Cunningham (1997) does comparisons. Also, work by Amit, and statistical literature on boosting and bagging.\n\n• Billera: there's a notion of equivalent metrics in topology, and any of these will do. (In doing geometry, where issues like curvature come into play, the choice of metrics is important.) It may be the case that even though certain metrics are good enough for some purposes, it doesn't mean that other metrics aren't valid. Q. What is the right notion of an \"average\" of trees?\n\n• The biologists present agreed that this was a very interesting question for biologists.\n\n• Vert: Two important ideas emerge when working with decision trees: (1) average decrease some variance, (2) averaging can leave the space. Boosting will leave the space, if we don't leave the space it's really not boosting. The averaging we are discussing here doesn't leave the space of trees.\n\n• More discussion on averaging took place here. Holmes made a comment about the \"non-associativity\" of trees (related to considering trees of trees, when building trees one character at a time). Billera made some remarks about how averaging may be better than concatenation, but we have no guarantee that it is any good.\n\n• Penny discussed the notion of a median tree: the tree that is closest on average to all the others.\n\n• Felsenstein: other examples are the consensus tree, and majority rule consensus tree. Q. What are good properties of averaging? (Luecke)\n\n• Penny: would like a fully resolved binary tree, though he points out that some others might sacrifice that in favor of other features.\n\n• Felsenstein posed a problem about averaging properties: given a bunch of trees from different genes, to study a question like: are chimps closer to humans than gorillas? Say 2/3 of the trees trees show humans with chimps, others show chimps with gorillas. Suppose in trees that group humans with chimps, the average branch length is 1, but the others don't have it. When you average... do you want it to be length.66 or length 1? 7\n\nThere was a question about how different are trees that result from diffrent averaging methods. It was pointed out (Huelsenbeck) that differences between methods for a single gene is much smaller than the differences across genes. Q. What would be the distribution on trees that gives majority rule consensus as its average? (Holmes)\n\n• Holmes: would like any tree average to be an expected value with respect to some distribution on tree space.",
   "evidence": "This record is Section A.2, “Sunday Discussion on Biological Issues,” in the official AIM workshop notes *Geometric Models of Biological Phenomena*, version dated 18 June 2003. It is a moderated, multi-question discussion rather than one narrowly stated conjecture. It nevertheless contains genuine research questions about biologically meaningful tree metrics, confidence sets and distributions on tree space, topology versus branch length and ancestral states, non-tree-like residuals, concatenating sequence data versus combining gene trees, and the correct notion of a tree average.",
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   "original_statement": "A.3 Monday Discussion on Combinatorial Issues \n\nThe disussion was moderated by Michelle Wachs. She started off the discussion by asking: Q. What combinatorial questions arise in biology that might be of interest to biologists? Or, what problems motivated by biology would be of interest to mathematicians, to give us some interesting problems? \n\n• Huelsenbeck: In putting priors on the set of all possible trees, we sometimes need to count trees with particular characteristics, such as ones with a particular kind of edge (split, bipartition), or a list of either this or that kind of edge. The constraints may be multiple or conflicting. The count is needed so that we can sum probabilities over these trees. \n\n• Holmes pointed out that inclusion/exclusion issues are involved here. There was some discussion about NP-hard problems, but Billera pointed out that complexity issues are not that relevant for the basic question of how to count these things. Q. Given a set S of k edges, and a subset J of S, how many leaf labeled binary trees have some subset J of S but not the edges in S − J? Do this for all J.\n\n• Evans asked why we can't just use MCMC to solve this problem? Holmes pointed out that the combinatorics may give some insight. \n\n• Another example of a question that arises: the number of trees that are distance d\n\nfrom a particular set of trees (e.g., using a distance such as nearest-neighbor inter-change)? Q. What are good codings for trees? \n\n• Wachs described a bijection between phylogenetic trees and type B permutations whose left-to-right minimum is unbarred. (Type B permutations are permutations in which each number can be barred or unbarred.) This can be used as a coding for trees. \n\n• There was a question about Penny's method for coding trees (in which each time you add an edge there are 2 n − 1 places to add it, so that you get (2 n − 1)!! of them). Billera said that this is the same as what was on the board. \n\n• Penny mentioned the existence of another scheme for coding each tree by a unique number. Holmes pointed out that the biologists use a Newick notation (New Hamp-shire format) that uses lots of parentheses. Another coding is a matrix where the columns are edges, and the rows are vertices, and you put a 1 if the edge is an ancestor of the vertex. (Brooks used it in biology, and Graham-Winkler used for addressing a network). 8\n\n• Holmes described a matching representation is one due to Diaconis-Holmes. When \n\nn = 2 there is one matching, when n = 4 there are 3 matchings, when n = 6 there are 15 matchings, corresponding to 4 leaf trees. The matchings are the sibling pairs. Example: n = 6. Call the numbers from 1 to 4 available numbers, and the other numbers unavailable. Given a matching 16 − 42 − 53, pick the first available pair of numbers. By pigeonhole argument, there is some pair, that's a sibling pair in a tree. Then the smallest non-available number (5, in this case) is the parent, and the other member of the pair (3) is the sibling of 5. Continue. \n\n• Diaconis pointed out their matching method is related to Billera's method using Prufer codes. Q. Is there an efficient representation for coding trees? (Billera) Q. Is there a nice neat notation for trees that is continuous when doing a random walk on trees? In other words, do small changes in notation correspond to close trees? (Diaconis) \n\n• The type B permutation coding might be very amenable to doing a random walk. Bridson pointed out that this representation is related to subsets of the Cayley graph of the Weyl group. Q. Are there questions that combinatorialists might have for biologists? Combinatorial structures that biologists might be interesting? (Wachs) Q. If we took trees and instead of putting real numbers of them, putting discrete values, like 0-1, would that be of interest in biologists? (Diaconis) \n\n• Huelsenbeck: this maybe useful in character-mapping, mapping characters that are on or off. Perhaps some substitutions are more likely when characters are on. \n\n• Epstein: it certainly seems reasonable to associate to each edge a vector, rather than a real number. \n\n• Evans: as in Penny's talk, characters of each of the vertices can be thought of as elements of the Klein 4-group... the edges can represent differences of elements at the vertices. \n\n• Wachs: what about k-ary trees? \n\n• Penny: people find cycles useful for representing uncertainty. Referred to some who tried to define how tree-like the data was, something tree-like would have only small cycles. \n\n• Holmes referred to a program called splitstree that makes such diagrams. \n\n• Penny: biologist would like structures that represents distances accurately. \n\n• Shareshian: described a space that arose in his work, a space in which ( n − 2)! cycles correspond to associahedra.",
   "clean_statement": "A.3 Monday Discussion on Combinatorial Issues\n\nThe disussion was moderated by Michelle Wachs. She started off the discussion by asking: Q. What combinatorial questions arise in biology that might be of interest to biologists? Or, what problems motivated by biology would be of interest to mathematicians, to give us some interesting problems?\n\n• Huelsenbeck: In putting priors on the set of all possible trees, we sometimes need to count trees with particular characteristics, such as ones with a particular kind of edge (split, bipartition), or a list of either this or that kind of edge. The constraints may be multiple or conflicting. The count is needed so that we can sum probabilities over these trees.\n\n• Holmes pointed out that inclusion/exclusion issues are involved here. There was some discussion about NP-hard problems, but Billera pointed out that complexity issues are not that relevant for the basic question of how to count these things. Q. Given a set S of k edges, and a subset J of S, how many leaf labeled binary trees have some subset J of S but not the edges in S − J? Do this for all J.\n\n• Evans asked why we can't just use MCMC to solve this problem? Holmes pointed out that the combinatorics may give some insight.\n\n• Another example of a question that arises: the number of trees that are distance d\n\nfrom a particular set of trees (e.g., using a distance such as nearest-neighbor inter-change)? Q. What are good codings for trees?\n\n• Wachs described a bijection between phylogenetic trees and type B permutations whose left-to-right minimum is unbarred. (Type B permutations are permutations in which each number can be barred or unbarred.) This can be used as a coding for trees.\n\n• There was a question about Penny's method for coding trees (in which each time you add an edge there are 2 n − 1 places to add it, so that you get (2 n − 1)!! of them). Billera said that this is the same as what was on the board.\n\n• Penny mentioned the existence of another scheme for coding each tree by a unique number. Holmes pointed out that the biologists use a Newick notation (New Hamp-shire format) that uses lots of parentheses. Another coding is a matrix where the columns are edges, and the rows are vertices, and you put a 1 if the edge is an ancestor of the vertex. (Brooks used it in biology, and Graham-Winkler used for addressing a network). 8\n\n• Holmes described a matching representation is one due to Diaconis-Holmes. When\n\nn = 2 there is one matching, when n = 4 there are 3 matchings, when n = 6 there are 15 matchings, corresponding to 4 leaf trees. The matchings are the sibling pairs. Example: n = 6. Call the numbers from 1 to 4 available numbers, and the other numbers unavailable. Given a matching 16 − 42 − 53, pick the first available pair of numbers. By pigeonhole argument, there is some pair, that's a sibling pair in a tree. Then the smallest non-available number (5, in this case) is the parent, and the other member of the pair (3) is the sibling of 5. Continue.\n\n• Diaconis pointed out their matching method is related to Billera's method using Prufer codes. Q. Is there an efficient representation for coding trees? (Billera) Q. Is there a nice neat notation for trees that is continuous when doing a random walk on trees? In other words, do small changes in notation correspond to close trees? (Diaconis)\n\n• The type B permutation coding might be very amenable to doing a random walk. Bridson pointed out that this representation is related to subsets of the Cayley graph of the Weyl group. Q. Are there questions that combinatorialists might have for biologists? Combinatorial structures that biologists might be interesting? (Wachs) Q. If we took trees and instead of putting real numbers of them, putting discrete values, like 0-1, would that be of interest in biologists? (Diaconis)\n\n• Huelsenbeck: this maybe useful in character-mapping, mapping characters that are on or off. Perhaps some substitutions are more likely when characters are on.\n\n• Epstein: it certainly seems reasonable to associate to each edge a vector, rather than a real number.\n\n• Evans: as in Penny's talk, characters of each of the vertices can be thought of as elements of the Klein 4-group... the edges can represent differences of elements at the vertices.\n\n• Wachs: what about k-ary trees?\n\n• Penny: people find cycles useful for representing uncertainty. Referred to some who tried to define how tree-like the data was, something tree-like would have only small cycles.\n\n• Holmes referred to a program called splitstree that makes such diagrams.\n\n• Penny: biologist would like structures that represents distances accurately.\n\n• Shareshian: described a space that arose in his work, a space in which ( n − 2)! cycles correspond to associahedra.",
   "public_statement": "A.3 Monday Discussion on Combinatorial Issues\n\nThe disussion was moderated by Michelle Wachs. She started off the discussion by asking: Q. What combinatorial questions arise in biology that might be of interest to biologists? Or, what problems motivated by biology would be of interest to mathematicians, to give us some interesting problems?\n\n• Huelsenbeck: In putting priors on the set of all possible trees, we sometimes need to count trees with particular characteristics, such as ones with a particular kind of edge (split, bipartition), or a list of either this or that kind of edge. The constraints may be multiple or conflicting. The count is needed so that we can sum probabilities over these trees.\n\n• Holmes pointed out that inclusion/exclusion issues are involved here. There was some discussion about NP-hard problems, but Billera pointed out that complexity issues are not that relevant for the basic question of how to count these things. Q. Given a set S of k edges, and a subset J of S, how many leaf labeled binary trees have some subset J of S but not the edges in S − J? Do this for all J.\n\n• Evans asked why we can't just use MCMC to solve this problem? Holmes pointed out that the combinatorics may give some insight.\n\n• Another example of a question that arises: the number of trees that are distance d\n\nfrom a particular set of trees (e.g., using a distance such as nearest-neighbor inter-change)? Q. What are good codings for trees?\n\n• Wachs described a bijection between phylogenetic trees and type B permutations whose left-to-right minimum is unbarred. (Type B permutations are permutations in which each number can be barred or unbarred.) This can be used as a coding for trees.\n\n• There was a question about Penny's method for coding trees (in which each time you add an edge there are 2 n − 1 places to add it, so that you get (2 n − 1)!! of them). Billera said that this is the same as what was on the board.\n\n• Penny mentioned the existence of another scheme for coding each tree by a unique number. Holmes pointed out that the biologists use a Newick notation (New Hamp-shire format) that uses lots of parentheses. Another coding is a matrix where the columns are edges, and the rows are vertices, and you put a 1 if the edge is an ancestor of the vertex. (Brooks used it in biology, and Graham-Winkler used for addressing a network). 8\n\n• Holmes described a matching representation is one due to Diaconis-Holmes. When\n\nn = 2 there is one matching, when n = 4 there are 3 matchings, when n = 6 there are 15 matchings, corresponding to 4 leaf trees. The matchings are the sibling pairs. Example: n = 6. Call the numbers from 1 to 4 available numbers, and the other numbers unavailable. Given a matching 16 − 42 − 53, pick the first available pair of numbers. By pigeonhole argument, there is some pair, that's a sibling pair in a tree. Then the smallest non-available number (5, in this case) is the parent, and the other member of the pair (3) is the sibling of 5. Continue.\n\n• Diaconis pointed out their matching method is related to Billera's method using Prufer codes. Q. Is there an efficient representation for coding trees? (Billera) Q. Is there a nice neat notation for trees that is continuous when doing a random walk on trees? In other words, do small changes in notation correspond to close trees? (Diaconis)\n\n• The type B permutation coding might be very amenable to doing a random walk. Bridson pointed out that this representation is related to subsets of the Cayley graph of the Weyl group. Q. Are there questions that combinatorialists might have for biologists? Combinatorial structures that biologists might be interesting? (Wachs) Q. If we took trees and instead of putting real numbers of them, putting discrete values, like 0-1, would that be of interest in biologists? (Diaconis)\n\n• Huelsenbeck: this maybe useful in character-mapping, mapping characters that are on or off. Perhaps some substitutions are more likely when characters are on.\n\n• Epstein: it certainly seems reasonable to associate to each edge a vector, rather than a real number.\n\n• Evans: as in Penny's talk, characters of each of the vertices can be thought of as elements of the Klein 4-group... the edges can represent differences of elements at the vertices.\n\n• Wachs: what about k-ary trees?\n\n• Penny: people find cycles useful for representing uncertainty. Referred to some who tried to define how tree-like the data was, something tree-like would have only small cycles.\n\n• Holmes referred to a program called splitstree that makes such diagrams.\n\n• Penny: biologist would like structures that represents distances accurately.\n\n• Shareshian: described a space that arose in his work, a space in which ( n − 2)! cycles correspond to associahedra.",
   "evidence": "This record is Section A.3, “Monday Discussion on Combinatorial Issues,” in the American Institute of Mathematics workshop notes *Geometric Models of Biological Phenomena*. Unlike a purely contextual section, it contains a definite counting problem:",
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   "original_statement": "A.4 Tuesday Discussion on Statistical Issues \n\nThis discussion was moderated by Ruth Charney. Q. Diaconis suggested making a list of the kinds of noise, or sources of variation, that are implicit in estimating trees? Participant discussion led to the following list: \n\n• alignment \n\n• errors in sequencing, (though Epstein pointed out that the quality is steadily improv-ing) 9\n\n• misspecification of the model (process of going from the data to a tree): mutation rates, independence between sites, change of nucleotide composition \n\n• variation within species (1 in 1000 genes) \n\n• bias in corrections for distances (for distance based models) \n\n• variation between fragments of same DNA (bias created by choice of fragments) \n\n• selection varies in different parts of the genome \n\n• gene identification (problems created by gene duplication and gene loss) \n\n• hybridization (branching comes back together, structure is not a tree) and horizontal gene transfer \n\n• optimization \n\n• not enough data- length of sequences, number of taxa Q. are these problems worth working on incrementally or all at once? Which ones are most important? (Diaconis) \n\n• Penny: differences in nucleotide composition? Q. Can we define the geometry (distance) on tree space to behave well with respect to a particular model? \n\n• Vert: what are the implications of the noise in the definition of the tree space? Instead of defining the geometry of tree space beforehand, and then asking what properties it satisfies, would it be possible to define a tree space in terms in such a way that it has good properties (is stable, averages are at least as good as the trees themselves, etc.) \n\n• Diaconis: this is perhaps related to statistical geometry or information geometry- using the Fisher information to define a Riemannian metric on the space of distribu-tions. \n\n• Epstein: an average is a summary statistic, but doesn't summarize everything we might want. \n\n• Evans: information geometry example: using the upper half plane to parametrize normals on a line, the geometry that best reflects closeness of normals is hyperbolic geometry. Q. What are residuals for trees, and how do we estimate the residuals? (Residuals measure how far each data point is from fitted tree.) Are there graphical methods for detecting departure from the model? (Penny) \n\n• Penny also asked why there are so many definitions of maximum likelihood? \n\n• Diaconis suggested that one topic probabalists and biologists may benefit from is work of Aldous and students, on analyzing rates of convergence of random walks on phylogenetic trees. Aldous responded by noting that the case they can analyze is not necessarily so useful to biologists (one where leaf is taken off and pushed somewhere else). For an n-leaf tree, the random walk needs n2 random steps. A more realistic example is to cut somewhere, and attach whole subtree in a different place. It is believed that about n3/2 steps is needed to mix this kind of chain. This could be useful in MCMC algorithms. Q. Random walks on sets of trees (using tree rotations)- how fast does it converge? \n\n• Once again, the question of how to measure distance in the space of trees emerged. Q. What are appropriate meeasures on tree space? 10 \n\n• Glenn asked if there a need for a non-parametric likelihood on trees? Diaconis said that empirical likelihood related to the boostrap, so it could contribute. Q. If exact distances aren't easy to compute, can we obtain any upper and lower bounds for distances (such as the BHV metric)? (Su)",
   "clean_statement": "A.4 Tuesday Discussion on Statistical Issues\n\nThis discussion was moderated by Ruth Charney. Q. Diaconis suggested making a list of the kinds of noise, or sources of variation, that are implicit in estimating trees? Participant discussion led to the following list:\n\n• alignment\n\n• errors in sequencing, (though Epstein pointed out that the quality is steadily improv-ing) 9\n\n• misspecification of the model (process of going from the data to a tree): mutation rates, independence between sites, change of nucleotide composition\n\n• variation within species (1 in 1000 genes)\n\n• bias in corrections for distances (for distance based models)\n\n• variation between fragments of same DNA (bias created by choice of fragments)\n\n• selection varies in different parts of the genome\n\n• gene identification (problems created by gene duplication and gene loss)\n\n• hybridization (branching comes back together, structure is not a tree) and horizontal gene transfer\n\n• optimization\n\n• not enough data- length of sequences, number of taxa Q. are these problems worth working on incrementally or all at once? Which ones are most important? (Diaconis)\n\n• Penny: differences in nucleotide composition? Q. Can we define the geometry (distance) on tree space to behave well with respect to a particular model?\n\n• Vert: what are the implications of the noise in the definition of the tree space? Instead of defining the geometry of tree space beforehand, and then asking what properties it satisfies, would it be possible to define a tree space in terms in such a way that it has good properties (is stable, averages are at least as good as the trees themselves, etc.)\n\n• Diaconis: this is perhaps related to statistical geometry or information geometry- using the Fisher information to define a Riemannian metric on the space of distribu-tions.\n\n• Epstein: an average is a summary statistic, but doesn't summarize everything we might want.\n\n• Evans: information geometry example: using the upper half plane to parametrize normals on a line, the geometry that best reflects closeness of normals is hyperbolic geometry. Q. What are residuals for trees, and how do we estimate the residuals? (Residuals measure how far each data point is from fitted tree.) Are there graphical methods for detecting departure from the model? (Penny)\n\n• Penny also asked why there are so many definitions of maximum likelihood?\n\n• Diaconis suggested that one topic probabalists and biologists may benefit from is work of Aldous and students, on analyzing rates of convergence of random walks on phylogenetic trees. Aldous responded by noting that the case they can analyze is not necessarily so useful to biologists (one where leaf is taken off and pushed somewhere else). For an n-leaf tree, the random walk needs n2 random steps. A more realistic example is to cut somewhere, and attach whole subtree in a different place. It is believed that about n3/2 steps is needed to mix this kind of chain. This could be useful in MCMC algorithms. Q. Random walks on sets of trees (using tree rotations)- how fast does it converge?\n\n• Once again, the question of how to measure distance in the space of trees emerged. Q. What are appropriate meeasures on tree space? 10\n\n• Glenn asked if there a need for a non-parametric likelihood on trees? Diaconis said that empirical likelihood related to the boostrap, so it could contribute. Q. If exact distances aren't easy to compute, can we obtain any upper and lower bounds for distances (such as the BHV metric)? (Su)",
   "public_statement": "A.4 Tuesday Discussion on Statistical Issues\n\nThis discussion was moderated by Ruth Charney. Q. Diaconis suggested making a list of the kinds of noise, or sources of variation, that are implicit in estimating trees? Participant discussion led to the following list:\n\n• alignment\n\n• errors in sequencing, (though Epstein pointed out that the quality is steadily improv-ing) 9\n\n• misspecification of the model (process of going from the data to a tree): mutation rates, independence between sites, change of nucleotide composition\n\n• variation within species (1 in 1000 genes)\n\n• bias in corrections for distances (for distance based models)\n\n• variation between fragments of same DNA (bias created by choice of fragments)\n\n• selection varies in different parts of the genome\n\n• gene identification (problems created by gene duplication and gene loss)\n\n• hybridization (branching comes back together, structure is not a tree) and horizontal gene transfer\n\n• optimization\n\n• not enough data- length of sequences, number of taxa Q. are these problems worth working on incrementally or all at once? Which ones are most important? (Diaconis)\n\n• Penny: differences in nucleotide composition? Q. Can we define the geometry (distance) on tree space to behave well with respect to a particular model?\n\n• Vert: what are the implications of the noise in the definition of the tree space? Instead of defining the geometry of tree space beforehand, and then asking what properties it satisfies, would it be possible to define a tree space in terms in such a way that it has good properties (is stable, averages are at least as good as the trees themselves, etc.)\n\n• Diaconis: this is perhaps related to statistical geometry or information geometry- using the Fisher information to define a Riemannian metric on the space of distribu-tions.\n\n• Epstein: an average is a summary statistic, but doesn't summarize everything we might want.\n\n• Evans: information geometry example: using the upper half plane to parametrize normals on a line, the geometry that best reflects closeness of normals is hyperbolic geometry. Q. What are residuals for trees, and how do we estimate the residuals? (Residuals measure how far each data point is from fitted tree.) Are there graphical methods for detecting departure from the model? (Penny)\n\n• Penny also asked why there are so many definitions of maximum likelihood?\n\n• Diaconis suggested that one topic probabalists and biologists may benefit from is work of Aldous and students, on analyzing rates of convergence of random walks on phylogenetic trees. Aldous responded by noting that the case they can analyze is not necessarily so useful to biologists (one where leaf is taken off and pushed somewhere else). For an n-leaf tree, the random walk needs n2 random steps. A more realistic example is to cut somewhere, and attach whole subtree in a different place. It is believed that about n3/2 steps is needed to mix this kind of chain. This could be useful in MCMC algorithms. Q. Random walks on sets of trees (using tree rotations)- how fast does it converge?\n\n• Once again, the question of how to measure distance in the space of trees emerged. Q. What are appropriate meeasures on tree space? 10\n\n• Glenn asked if there a need for a non-parametric likelihood on trees? Diaconis said that empirical likelihood related to the boostrap, so it could contribute. Q. If exact distances aren't easy to compute, can we obtain any upper and lower bounds for distances (such as the BHV metric)? (Su)",
   "evidence": "The canonical record is section A.4, “Tuesday Discussion on Statistical Issues,” in the American Institute of Mathematics workshop notes *Geometric models of biological phenomena*. The official PDF was inspected directly. It identifies the American Institute of Mathematics, gives version time “Wed Jun 18 16:51:55 2003,” and credits the discussion-session notes to Francis Su at the chapter level. A.4 occupies PDF pages 8–10 (printed pages 9–10) and says the session was moderated by Ruth Charney.",
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   "statement_status": "exact",
   "original_statement": "A.5 Wednesday Discussion on Geometric Issues \n\nThis session was moderated by John Shareshian. The discussion started off with some comments about Riemannian metrics and whether the set of metrics on tree space would be useful to study. \n\n• Vogtmann asked if this space may be too large to study? \n\n• Forman noted that when geometers look at the space of all Riemannian metrics on a given space, they do it to try to find the \"best\" metric in some sense (e.g., such as one with constant curvature). Q. Are there many useful or interesting metrics on tree space? Q. Is the BHV metric the only one (up to scalars) of non-positive curvature? (Bridson) \n\n• Charney: metrics are slightly different, but they should all be similar. If there is no reason that biologists suggest that one is better than another, then it makes sense to choose one that has useful properties, such as metrics of nonpositive curvature. \n\n• Billera: for biologists, what metrics do you want to have on the orthants? Q. What metrics \"should\" be used on the subspace obtained by fixing the tree type? \n\n• St. John: uses the L1 metric, though Felsenstein uses L2.\n\n• Penny: we use L1 because then lengths scale with time. \n\n• Charney: since the data is not completely in tree space, maybe we should be consid-ering non-intrinsic metrics. \n\n• Forman: want to choose metrics so that the statistical methods we are using are continuous. \n\n• Flath: perhaps some edges are more important than others? \n\n• Penny: we are not surprised when we get edges near leaves are accurate, but we are really interested in getting deep internal edges right. \n\n• Penny: taxonomic studies, just care about the branch order (weight 1 on edges), but when we consider time studies, we do want the lengths. \n\n• Forman: measuring the residuals seem to imply that embedding tree space in some larger space and considering an extrinsic metric would be important. \n\n• Vogtmann noted that Felsenstein was talking about embedding tree space in Eu-clidean space (using the Robinson-Foulds metric). Q. How does the Robinson-Foulds metric compare with the BHV metric? (Diaconis) What's the Lipshitz constant? (Bridson) \n\n• Vert: Another distance (referred to by Felsenstein in his talk) is Kullback-Leibler distance (relative entropy) between probabilities. Fix a model. Each tree defines a probability on the set of assignments of letters (A,G,C,T) to leaves. Maximum likelihood corresponds to projection of the empirical measure of a data point to tree space according to this distance. 11 \n\nThere was some explanation and discussion of this embedding. Diaconis noted that while there are many metrics on probability distributions to consider here, the Kullback-Liebler separation is good for maximum likelihood. Q. Should the metric on tree space come from an intrinsic metric, or an extrinsic metric on a larger space in which tree space is embedded? Which is a more natural way to view tree space? \n\n• Forman: studying the residuals seems to be very important. This larger space is part of what you are given in the data, and should be useful. Charney rephrased the question as: what is the right ambient space to embed tree space? While Bridson remarked that he viewed tree space as God-given, Forman asserted that he views the data as God-given. \n\n• Vert remarked that the \"average\" of trees with high likelihood may not be high likelihood with the intrinsic metric. Q. How do you deliver geometry to biologists? (Billera) \n\n• Billera pointed out that interesting mathematics in biology seems to happen by serendipity- cases in which someone happened to know some mathematics or knew someone who did. \n\n• Penny: notes that there is a role for mathematics to play. Also new majors in mathematical biology (at some schools) have arisen that will train students to think in both disciplines. \n\n• Diaconis notes that as a result of this conference, and ensuing discussions, people might write an article for Science. This may bring biologists to mathematics. \n\n• Snel suggests advocating a notion of average, and communicate why it is better. For instance, articles on concatenated alignments have appeared in Science. \n\n• Huelsenbeck: write review articles that explain things clearly. Write for Systematics Biology - that's where stuff on trees would appear. \n\n• Billera asked again how one spreads mathematical knowledge in the biological com-munity, noting that courses offered at most institutions are either at too low or too high a level for biologists.",
   "clean_statement": "A.5 Wednesday Discussion on Geometric Issues\n\nThis session was moderated by John Shareshian. The discussion started off with some comments about Riemannian metrics and whether the set of metrics on tree space would be useful to study.\n\n• Vogtmann asked if this space may be too large to study?\n\n• Forman noted that when geometers look at the space of all Riemannian metrics on a given space, they do it to try to find the \"best\" metric in some sense (e.g., such as one with constant curvature). Q. Are there many useful or interesting metrics on tree space? Q. Is the BHV metric the only one (up to scalars) of non-positive curvature? (Bridson)\n\n• Charney: metrics are slightly different, but they should all be similar. If there is no reason that biologists suggest that one is better than another, then it makes sense to choose one that has useful properties, such as metrics of nonpositive curvature.\n\n• Billera: for biologists, what metrics do you want to have on the orthants? Q. What metrics \"should\" be used on the subspace obtained by fixing the tree type?\n\n• St. John: uses the L1 metric, though Felsenstein uses L2.\n\n• Penny: we use L1 because then lengths scale with time.\n\n• Charney: since the data is not completely in tree space, maybe we should be consid-ering non-intrinsic metrics.\n\n• Forman: want to choose metrics so that the statistical methods we are using are continuous.\n\n• Flath: perhaps some edges are more important than others?\n\n• Penny: we are not surprised when we get edges near leaves are accurate, but we are really interested in getting deep internal edges right.\n\n• Penny: taxonomic studies, just care about the branch order (weight 1 on edges), but when we consider time studies, we do want the lengths.\n\n• Forman: measuring the residuals seem to imply that embedding tree space in some larger space and considering an extrinsic metric would be important.\n\n• Vogtmann noted that Felsenstein was talking about embedding tree space in Eu-clidean space (using the Robinson-Foulds metric). Q. How does the Robinson-Foulds metric compare with the BHV metric? (Diaconis) What's the Lipshitz constant? (Bridson)\n\n• Vert: Another distance (referred to by Felsenstein in his talk) is Kullback-Leibler distance (relative entropy) between probabilities. Fix a model. Each tree defines a probability on the set of assignments of letters (A,G,C,T) to leaves. Maximum likelihood corresponds to projection of the empirical measure of a data point to tree space according to this distance. 11\n\nThere was some explanation and discussion of this embedding. Diaconis noted that while there are many metrics on probability distributions to consider here, the Kullback-Liebler separation is good for maximum likelihood. Q. Should the metric on tree space come from an intrinsic metric, or an extrinsic metric on a larger space in which tree space is embedded? Which is a more natural way to view tree space?\n\n• Forman: studying the residuals seems to be very important. This larger space is part of what you are given in the data, and should be useful. Charney rephrased the question as: what is the right ambient space to embed tree space? While Bridson remarked that he viewed tree space as God-given, Forman asserted that he views the data as God-given.\n\n• Vert remarked that the \"average\" of trees with high likelihood may not be high likelihood with the intrinsic metric. Q. How do you deliver geometry to biologists? (Billera)\n\n• Billera pointed out that interesting mathematics in biology seems to happen by serendipity- cases in which someone happened to know some mathematics or knew someone who did.\n\n• Penny: notes that there is a role for mathematics to play. Also new majors in mathematical biology (at some schools) have arisen that will train students to think in both disciplines.\n\n• Diaconis notes that as a result of this conference, and ensuing discussions, people might write an article for Science. This may bring biologists to mathematics.\n\n• Snel suggests advocating a notion of average, and communicate why it is better. For instance, articles on concatenated alignments have appeared in Science.\n\n• Huelsenbeck: write review articles that explain things clearly. Write for Systematics Biology - that's where stuff on trees would appear.\n\n• Billera asked again how one spreads mathematical knowledge in the biological com-munity, noting that courses offered at most institutions are either at too low or too high a level for biologists.",
   "public_statement": "A.5 Wednesday Discussion on Geometric Issues\n\nThis session was moderated by John Shareshian. The discussion started off with some comments about Riemannian metrics and whether the set of metrics on tree space would be useful to study.\n\n• Vogtmann asked if this space may be too large to study?\n\n• Forman noted that when geometers look at the space of all Riemannian metrics on a given space, they do it to try to find the \"best\" metric in some sense (e.g., such as one with constant curvature). Q. Are there many useful or interesting metrics on tree space? Q. Is the BHV metric the only one (up to scalars) of non-positive curvature? (Bridson)\n\n• Charney: metrics are slightly different, but they should all be similar. If there is no reason that biologists suggest that one is better than another, then it makes sense to choose one that has useful properties, such as metrics of nonpositive curvature.\n\n• Billera: for biologists, what metrics do you want to have on the orthants? Q. What metrics \"should\" be used on the subspace obtained by fixing the tree type?\n\n• St. John: uses the L1 metric, though Felsenstein uses L2.\n\n• Penny: we use L1 because then lengths scale with time.\n\n• Charney: since the data is not completely in tree space, maybe we should be consid-ering non-intrinsic metrics.\n\n• Forman: want to choose metrics so that the statistical methods we are using are continuous.\n\n• Flath: perhaps some edges are more important than others?\n\n• Penny: we are not surprised when we get edges near leaves are accurate, but we are really interested in getting deep internal edges right.\n\n• Penny: taxonomic studies, just care about the branch order (weight 1 on edges), but when we consider time studies, we do want the lengths.\n\n• Forman: measuring the residuals seem to imply that embedding tree space in some larger space and considering an extrinsic metric would be important.\n\n• Vogtmann noted that Felsenstein was talking about embedding tree space in Eu-clidean space (using the Robinson-Foulds metric). Q. How does the Robinson-Foulds metric compare with the BHV metric? (Diaconis) What's the Lipshitz constant? (Bridson)\n\n• Vert: Another distance (referred to by Felsenstein in his talk) is Kullback-Leibler distance (relative entropy) between probabilities. Fix a model. Each tree defines a probability on the set of assignments of letters (A,G,C,T) to leaves. Maximum likelihood corresponds to projection of the empirical measure of a data point to tree space according to this distance. 11\n\nThere was some explanation and discussion of this embedding. Diaconis noted that while there are many metrics on probability distributions to consider here, the Kullback-Liebler separation is good for maximum likelihood. Q. Should the metric on tree space come from an intrinsic metric, or an extrinsic metric on a larger space in which tree space is embedded? Which is a more natural way to view tree space?\n\n• Forman: studying the residuals seems to be very important. This larger space is part of what you are given in the data, and should be useful. Charney rephrased the question as: what is the right ambient space to embed tree space? While Bridson remarked that he viewed tree space as God-given, Forman asserted that he views the data as God-given.\n\n• Vert remarked that the \"average\" of trees with high likelihood may not be high likelihood with the intrinsic metric. Q. How do you deliver geometry to biologists? (Billera)\n\n• Billera pointed out that interesting mathematics in biology seems to happen by serendipity- cases in which someone happened to know some mathematics or knew someone who did.\n\n• Penny: notes that there is a role for mathematics to play. Also new majors in mathematical biology (at some schools) have arisen that will train students to think in both disciplines.\n\n• Diaconis notes that as a result of this conference, and ensuing discussions, people might write an article for Science. This may bring biologists to mathematics.\n\n• Snel suggests advocating a notion of average, and communicate why it is better. For instance, articles on concatenated alignments have appeared in Science.\n\n• Huelsenbeck: write review articles that explain things clearly. Write for Systematics Biology - that's where stuff on trees would appear.\n\n• Billera asked again how one spreads mathematical knowledge in the biological com-munity, noting that courses offered at most institutions are either at too low or too high a level for biologists.",
   "evidence": "This record is Section A.5, “Wednesday Discussion on Geometric Issues,” in the official AIM workshop notes *Geometric Models of Biological Phenomena*, version dated 18 June 2003. John Shareshian moderated the session. The record contains genuine questions:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-BIOLOGY-0040": {
   "statement_status": "exact",
   "original_statement": "A.6 Thursday Open Discussion \n\nThis discussion was moderated by Lou Billera. Billera noted that, as it was the final discussion session, anything is a valid topic today. The discussion started off with some comments about the usefulness of the workshop: \n\n• Billera began by remarking that the organizers worried that having 4 different groups of people at this workshop from different backgrounds could have been disastrous. But he noted that it didn't appear to be... and had found it interesting, and was pleased with how much discussion had taken place. \n\n• Diaconis emphasized the usefulness of this workshop for generating problems to work on, and cited an example of residuals for discrete data analysis, as something worth following up, as well as the discussion in the geometry session about what are natural measures on tree space. He noted that the workshop was valuable because he couldn't think of another setting in which all these ideas from different arenas could all come together. 12 \n\nQ. Concatenation as \"averaging\"? \n\n• Billera noted that in the discussion about averaging, he was struck with the idea that concatenation was an average in sequence space, an idea that he hadn't appreciated until this workshop. One question He noted that the conference had touched on many ways to go from sequences to trees, and one of the big questions was whether these methods were coherent with concatenation-averaging. \n\n• Penny: it is worrisome that there are many different ways of concatenating, and how to handle the number of parameters? Q. Likelihood contours? \n\n• There was further discussion about confidence sets and likelihood contours, and how these might intersect transition points where the tree space branches in several di-rections. For instance, if a maximum likelihood process produces a tree with very short edge, then it would be near a transition. There was also discussion of whether the notion of curvature on tree space would be helpful. Q. Multivariate \"metrics\"? \n\n• Holmes: in the date that arises for trees, you get multivariate distances. We may want to speak of directions? is there anything like that in metric topology? \n\n• Diaconis: there are some distances that take values in partially ordered sets, etc., distances as a vector, etc. Q. Alternate representations (not trees)? \n\n• Epstein asked if are there other notions besides trees that would be helpful? \n\n• St. John: Networks? Trees with several non-tree portions. Q. Random walks on trees (generate letters, prodcue new trees) \n\n• Vert: One can imagine a random walks in this space of trees, by starting a tree, generate data, estimate a tree (by maximum likelihood). Do this number of times, get a random walk? Might be interesting to study this walk? Q. Would it be useful to study 2-colored trees to model orthology and paralogy? (Shareshian) During the session, Diaconis suggested that, as a follow up to this conference, people can now get together in small groups and communicate and start to work on a problem, whereas this wasn't true before the beginning of the week. AIM could facilitate such follow-up meetings. There was some brief discussion about holding a follow-up conference to this one, since all the participants now have a solid foundation on which to try to work on some joint problems. ARCC and the organizers (Lou Billera, Susan Holmes, Karen Vogtmann) were all thanked enthusiastically for a stimulating week of talks and discussions. The workshop was concluded over evening refreshments.",
   "clean_statement": "A.6 Thursday Open Discussion\n\nThis discussion was moderated by Lou Billera. Billera noted that, as it was the final discussion session, anything is a valid topic today. The discussion started off with some comments about the usefulness of the workshop:\n\n• Billera began by remarking that the organizers worried that having 4 different groups of people at this workshop from different backgrounds could have been disastrous. But he noted that it didn't appear to be... and had found it interesting, and was pleased with how much discussion had taken place.\n\n• Diaconis emphasized the usefulness of this workshop for generating problems to work on, and cited an example of residuals for discrete data analysis, as something worth following up, as well as the discussion in the geometry session about what are natural measures on tree space. He noted that the workshop was valuable because he couldn't think of another setting in which all these ideas from different arenas could all come together. 12\n\nQ. Concatenation as \"averaging\"?\n\n• Billera noted that in the discussion about averaging, he was struck with the idea that concatenation was an average in sequence space, an idea that he hadn't appreciated until this workshop. One question He noted that the conference had touched on many ways to go from sequences to trees, and one of the big questions was whether these methods were coherent with concatenation-averaging.\n\n• Penny: it is worrisome that there are many different ways of concatenating, and how to handle the number of parameters? Q. Likelihood contours?\n\n• There was further discussion about confidence sets and likelihood contours, and how these might intersect transition points where the tree space branches in several di-rections. For instance, if a maximum likelihood process produces a tree with very short edge, then it would be near a transition. There was also discussion of whether the notion of curvature on tree space would be helpful. Q. Multivariate \"metrics\"?\n\n• Holmes: in the date that arises for trees, you get multivariate distances. We may want to speak of directions? is there anything like that in metric topology?\n\n• Diaconis: there are some distances that take values in partially ordered sets, etc., distances as a vector, etc. Q. Alternate representations (not trees)?\n\n• Epstein asked if are there other notions besides trees that would be helpful?\n\n• St. John: Networks? Trees with several non-tree portions. Q. Random walks on trees (generate letters, prodcue new trees)\n\n• Vert: One can imagine a random walks in this space of trees, by starting a tree, generate data, estimate a tree (by maximum likelihood). Do this number of times, get a random walk? Might be interesting to study this walk? Q. Would it be useful to study 2-colored trees to model orthology and paralogy? (Shareshian) During the session, Diaconis suggested that, as a follow up to this conference, people can now get together in small groups and communicate and start to work on a problem, whereas this wasn't true before the beginning of the week. AIM could facilitate such follow-up meetings. There was some brief discussion about holding a follow-up conference to this one, since all the participants now have a solid foundation on which to try to work on some joint problems. ARCC and the organizers (Lou Billera, Susan Holmes, Karen Vogtmann) were all thanked enthusiastically for a stimulating week of talks and discussions. The workshop was concluded over evening refreshments.",
   "public_statement": "A.6 Thursday Open Discussion\n\nThis discussion was moderated by Lou Billera. Billera noted that, as it was the final discussion session, anything is a valid topic today. The discussion started off with some comments about the usefulness of the workshop:\n\n• Billera began by remarking that the organizers worried that having 4 different groups of people at this workshop from different backgrounds could have been disastrous. But he noted that it didn't appear to be... and had found it interesting, and was pleased with how much discussion had taken place.\n\n• Diaconis emphasized the usefulness of this workshop for generating problems to work on, and cited an example of residuals for discrete data analysis, as something worth following up, as well as the discussion in the geometry session about what are natural measures on tree space. He noted that the workshop was valuable because he couldn't think of another setting in which all these ideas from different arenas could all come together. 12\n\nQ. Concatenation as \"averaging\"?\n\n• Billera noted that in the discussion about averaging, he was struck with the idea that concatenation was an average in sequence space, an idea that he hadn't appreciated until this workshop. One question He noted that the conference had touched on many ways to go from sequences to trees, and one of the big questions was whether these methods were coherent with concatenation-averaging.\n\n• Penny: it is worrisome that there are many different ways of concatenating, and how to handle the number of parameters? Q. Likelihood contours?\n\n• There was further discussion about confidence sets and likelihood contours, and how these might intersect transition points where the tree space branches in several di-rections. For instance, if a maximum likelihood process produces a tree with very short edge, then it would be near a transition. There was also discussion of whether the notion of curvature on tree space would be helpful. Q. Multivariate \"metrics\"?\n\n• Holmes: in the date that arises for trees, you get multivariate distances. We may want to speak of directions? is there anything like that in metric topology?\n\n• Diaconis: there are some distances that take values in partially ordered sets, etc., distances as a vector, etc. Q. Alternate representations (not trees)?\n\n• Epstein asked if are there other notions besides trees that would be helpful?\n\n• St. John: Networks? Trees with several non-tree portions. Q. Random walks on trees (generate letters, prodcue new trees)\n\n• Vert: One can imagine a random walks in this space of trees, by starting a tree, generate data, estimate a tree (by maximum likelihood). Do this number of times, get a random walk? Might be interesting to study this walk? Q. Would it be useful to study 2-colored trees to model orthology and paralogy? (Shareshian) During the session, Diaconis suggested that, as a follow up to this conference, people can now get together in small groups and communicate and start to work on a problem, whereas this wasn't true before the beginning of the week. AIM could facilitate such follow-up meetings. There was some brief discussion about holding a follow-up conference to this one, since all the participants now have a solid foundation on which to try to work on some joint problems. ARCC and the organizers (Lou Billera, Susan Holmes, Karen Vogtmann) were all thanked enthusiastically for a stimulating week of talks and discussions. The workshop was concluded over evening refreshments.",
   "evidence": "This record is section A.6, “Thursday Open Discussion,” in the AIM workshop report *Geometric models of biological phenomena*. It is a collection of discussion prompts, not a single formal open problem. The source PDF was checked against the extracted JSON. The isolated “12” in the JSON is the printed page number. The phrases “One question He noted,” “date,” and “prodcue” occur in the PDF itself and are retained as source defects; the mathematics below does not depend on reconstructing them.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-biology-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0001": {
   "statement_status": "exact",
   "original_statement": "Entropy Freiman-Ruzsa lower bounds\n\nTheorem. Let $X$ be any $\\mathbb{F}_2^n$-valued random variable such that $d[X,X]\\leq \\log K$, where $d$ is the entropic Ruzsa distance. Then there is a subspace $H\\subset \\mathbb{F}_2^n$ such that $d[X,U_H]\\leq 11\\log K$, where $U_H$ is uniform on $H$.\n\nCan we give lower bounds for $d[X,U_H]$?",
   "clean_statement": "Entropy Freiman-Ruzsa lower bounds\n\nTheorem. Let $X$ be any $\\mathbb{F}_2^n$-valued random variable such that $d[X,X]\\leq \\log K$, where $d$ is the entropic Ruzsa distance. Then there is a subspace $H\\subset \\mathbb{F}_2^n$ such that $d[X,U_H]\\leq 11\\log K$, where $U_H$ is uniform on $H$.\n\nCan we give lower bounds for $d[X,U_H]$?",
   "public_statement": "Entropy Freiman-Ruzsa lower bounds\n\nTheorem. Let $X$ be any $\\mathbb{F}_2^n$-valued random variable such that $d[X,X]\\leq \\log K$, where $d$ is the entropic Ruzsa distance. Then there is a subspace $H\\subset \\mathbb{F}_2^n$ such that $d[X,U_H]\\leq 11\\log K$, where $U_H$ is uniform on $H$.\n\nCan we give lower bounds for $d[X,U_H]$?",
   "evidence": "The canonical record is problem 1.1, “Entropy Freiman-Ruzsa lower bounds,” from the AIM workshop *High-dimensional phenomena in discrete analysis*, section “Entropic methods.” Its mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0002": {
   "statement_status": "exact",
   "original_statement": "Entropy proof of Sidorenko\n\nThere is a proof system for graph homomorphism inequalities using flag algebras.\n\nDefine a proof system for graph homomorphism inequalities using entropy.",
   "clean_statement": "Entropy proof of Sidorenko\n\nThere is a proof system for graph homomorphism inequalities using flag algebras.\n\nDefine a proof system for graph homomorphism inequalities using entropy.",
   "public_statement": "Entropy proof of Sidorenko\n\nThere is a proof system for graph homomorphism inequalities using flag algebras.\n\nDefine a proof system for graph homomorphism inequalities using entropy.",
   "evidence": "The canonical AIM record is workshop *High-dimensional phenomena in discrete analysis*, section “Entropic methods,” Problem 1.2. Its problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0003": {
   "statement_status": "exact",
   "original_statement": "Inverse theorems with polynomial bounds\n\nGiven $f:\\mathbb{F}_2^n\\to \\mathbb{F}_2^m$ with $\\mathbb{P}_{x+y=z+w}(f(x)+f(y)=f(z)+f(w))\\geq \\epsilon$, then Polynomial Freiman-Ruzsa implies that $f$ correlates with a linear function.\n\nSet $D_hf(x)=f(x+h)-f(x)$. If $\\mathbb{P}_{x,h_1,h_2,h_3}(D_{h_1}D_{h_2}D_{h_3}f(x)=0)\\geq \\epsilon$, then PFR implies that $f$ correlates with a quadratic function.\n\nIs there a direct entropic proof of the above results with polynomial bounds?",
   "clean_statement": "Inverse theorems with polynomial bounds\n\nGiven $f:\\mathbb{F}_2^n\\to \\mathbb{F}_2^m$ with $\\mathbb{P}_{x+y=z+w}(f(x)+f(y)=f(z)+f(w))\\geq \\epsilon$, then Polynomial Freiman-Ruzsa implies that $f$ correlates with a linear function.\n\nSet $D_hf(x)=f(x+h)-f(x)$. If $\\mathbb{P}_{x,h_1,h_2,h_3}(D_{h_1}D_{h_2}D_{h_3}f(x)=0)\\geq \\epsilon$, then PFR implies that $f$ correlates with a quadratic function.\n\nIs there a direct entropic proof of the above results with polynomial bounds?",
   "public_statement": "Inverse theorems with polynomial bounds\n\nGiven $f:\\mathbb{F}_2^n\\to \\mathbb{F}_2^m$ with $\\mathbb{P}_{x+y=z+w}(f(x)+f(y)=f(z)+f(w))\\geq \\epsilon$, then Polynomial Freiman-Ruzsa implies that $f$ correlates with a linear function.\n\nSet $D_hf(x)=f(x+h)-f(x)$. If $\\mathbb{P}_{x,h_1,h_2,h_3}(D_{h_1}D_{h_2}D_{h_3}f(x)=0)\\geq \\epsilon$, then PFR implies that $f$ correlates with a quadratic function.\n\nIs there a direct entropic proof of the above results with polynomial bounds?",
   "evidence": "This is Problem 1.3, “Inverse theorems with polynomial bounds,” from the May 2024 AIM workshop *High-dimensional phenomena in discrete analysis*, in the section “Entropic methods.” The live AimPL problem URL returned an access error during this run, so the exact canonical JSON record and the neighboring record 1.1 were used. Neighbor 1.1 states the entropic Polynomial Freiman–Ruzsa theorem with constant 11, confirming that Problem 1.3 asks for a more direct entropic derivation of inverse consequences, not for a proof of PFR itself.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0004": {
   "statement_status": "exact",
   "original_statement": "Non-linear Roth\n\nLet $P$ be any (non-linear) polynomial, $A\\subset \\mathbb{F}_p$ with $|A|>p^{0.99}$. Does there always exist distinct $x,y$ such that\n$$x,y,y+P(x)-P(y)\\in A?$$",
   "clean_statement": "Non-linear Roth\n\nLet $P$ be any (non-linear) polynomial, $A\\subset \\mathbb{F}_p$ with $|A|>p^{0.99}$. Does there always exist distinct $x,y$ such that\n$$x,y,y+P(x)-P(y)\\in A?$$",
   "public_statement": "Non-linear Roth\n\nLet $P$ be any (non-linear) polynomial, $A\\subset \\mathbb{F}_p$ with $|A|>p^{0.99}$. Does there always exist distinct $x,y$ such that\n$$x,y,y+P(x)-P(y)\\in A?$$",
   "evidence": "The canonical record is problem 2.1, “Non-linear Roth,” from the additive-combinatorics section of the May 2024 AIM workshop *High-dimensional phenomena in discrete analysis*. Its exact mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0005": {
   "statement_status": "exact",
   "original_statement": "Roth with common difference $p-1$\n\nFor all $C>0$ and sufficiently large $N$, if $A\\subset [N]$ with $|A|\\geq N/\\log^C N$, then there exists $x$ and $p$ prime such that\n$$x,x+p-1,x+2(p-1)\\in A.$$",
   "clean_statement": "Roth with common difference $p-1$\n\nFor all $C>0$ and sufficiently large $N$, if $A\\subset [N]$ with $|A|\\geq N/\\log^C N$, then there exists $x$ and $p$ prime such that\n$$x,x+p-1,x+2(p-1)\\in A.$$",
   "public_statement": "Roth with common difference $p-1$\n\nFor all $C>0$ and sufficiently large $N$, if $A\\subset [N]$ with $|A|\\geq N/\\log^C N$, then there exists $x$ and $p$ prime such that\n$$x,x+p-1,x+2(p-1)\\in A.$$",
   "evidence": "The canonical AIM record is workshop *High-dimensional phenomena in discrete analysis*, section “Additive combinatorics,” Conjecture 2.2. Its exact main statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0006": {
   "statement_status": "exact",
   "original_statement": "Roth with constrained differences\n\nFor which $0\\in D\\subseteq \\mathbb{F}_p$ is it true that any $A\\subset \\mathbb{F}_p^n$ avoiding\n$$x,x+y,x+2y\\in A\\quad \\text{with } y\\in D^n\\setminus\\{0\\}$$\nhas size $\\leq C^n$ for some $C < p$?",
   "clean_statement": "Roth with constrained differences\n\nFor which $0\\in D\\subseteq \\mathbb{F}_p$ is it true that any $A\\subset \\mathbb{F}_p^n$ avoiding\n$$x,x+y,x+2y\\in A\\quad \\text{with } y\\in D^n\\setminus\\{0\\}$$\nhas size $\\leq C^n$ for some $C < p$?",
   "public_statement": "Roth with constrained differences\n\nFor which $0\\in D\\subseteq \\mathbb{F}_p$ is it true that any $A\\subset \\mathbb{F}_p^n$ avoiding\n$$x,x+y,x+2y\\in A\\quad \\text{with } y\\in D^n\\setminus\\{0\\}$$\nhas size $\\leq C^n$ for some $C < p$?",
   "evidence": "The source is Problem 2.3 in the “Additive combinatorics” section of the AIM workshop list *High-dimensional phenomena in discrete analysis*. The exact mathematical question in the supplied record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0007": {
   "statement_status": "exact",
   "original_statement": "Inverse theorem for equilateral triangles on the sphere\n\nDenote by $S^n\\subset \\mathbb{R}^{n+1}$ the unit sphere. For a function $f:S^n\\to \\mathbb{R}$, define the quantity\n$$Tf:= \\mathbb{E}_{x,y,z\\in S^n,x+y+z=0} f(x)f(y)f(z).$$\n\nIf $f$ is 1-bounded and $|Tf|>\\delta$, does $f$ correlate with a function that depends on $O_{\\delta}(1)$ coordinates (with respect to some basis)?",
   "clean_statement": "Inverse theorem for equilateral triangles on the sphere\n\nDenote by $S^n\\subset \\mathbb{R}^{n+1}$ the unit sphere. For a function $f:S^n\\to \\mathbb{R}$, define the quantity\n$$Tf:= \\mathbb{E}_{x,y,z\\in S^n,x+y+z=0} f(x)f(y)f(z).$$\n\nIf $f$ is 1-bounded and $|Tf|>\\delta$, does $f$ correlate with a function that depends on $O_{\\delta}(1)$ coordinates (with respect to some basis)?",
   "public_statement": "Inverse theorem for equilateral triangles on the sphere\n\nDenote by $S^n\\subset \\mathbb{R}^{n+1}$ the unit sphere. For a function $f:S^n\\to \\mathbb{R}$, define the quantity\n$$Tf:= \\mathbb{E}_{x,y,z\\in S^n,x+y+z=0} f(x)f(y)f(z).$$\n\nIf $f$ is 1-bounded and $|Tf|>\\delta$, does $f$ correlate with a function that depends on $O_{\\delta}(1)$ coordinates (with respect to some basis)?",
   "evidence": "The source record asks the following.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0008": {
   "statement_status": "exact",
   "original_statement": "Easier arithmetic Kakeya\n\nLet $S\\subset\\mathbb{Z}^n$ be such that for all $(d_1,\\ldots,d_r)\\in ([N]^n)^r$, $S$ contains the GAP (for some translate $a_0$)\n$$\\{a_0+i_1d_1+\\cdots +i_rd_r : i_1,\\ldots,i_r\\in \\{0,1,\\ldots,k-1\\}\\}.$$\nThen $|S|\\geq N^{c_kn+o_{k\\to\\infty}(1)}$.",
   "clean_statement": "Easier arithmetic Kakeya\n\nLet $S\\subset\\mathbb{Z}^n$ be such that for all $(d_1,\\ldots,d_r)\\in ([N]^n)^r$, $S$ contains the GAP (for some translate $a_0$)\n$$\\{a_0+i_1d_1+\\cdots +i_rd_r : i_1,\\ldots,i_r\\in \\{0,1,\\ldots,k-1\\}\\}.$$\nThen $|S|\\geq N^{c_kn+o_{k\\to\\infty}(1)}$.",
   "public_statement": "Easier arithmetic Kakeya\n\nLet $S\\subset\\mathbb{Z}^n$ be such that for all $(d_1,\\ldots,d_r)\\in ([N]^n)^r$, $S$ contains the GAP (for some translate $a_0$)\n$$\\{a_0+i_1d_1+\\cdots +i_rd_r : i_1,\\ldots,i_r\\in \\{0,1,\\ldots,k-1\\}\\}.$$\nThen $|S|\\geq N^{c_kn+o_{k\\to\\infty}(1)}$.",
   "evidence": "The supplied AIM record, Problem 2.5 in the “Additive combinatorics” section of *High-dimensional phenomena in discrete analysis*, states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0009": {
   "statement_status": "exact",
   "original_statement": "Popular Furstenberg-Sárközy\n\n(Furstenberg-Sárközy) For all $\\epsilon>0$, there exists $N_0=N_0(\\epsilon)$ such that for all $N\\geq N_0$, $A\\subseteq [N]$ with $|A|\\geq \\delta N$, there exists $y\\neq 0$ such that\n$$|\\{x : x,x+y^2\\in A\\}|\\geq (\\delta^2-\\epsilon)N,$$\nwith $N_0(\\epsilon)\\leq e^{1/\\epsilon^c}$.\n\n(1) Prove lower bounds of the form $N_0(\\epsilon)>superpoly(1/\\epsilon)$, or even just $N_0(\\epsilon)>1/\\epsilon^{10}$.\n\n(2) What happens in the function field case $\\mathbb{F}_p[t]_{< n}$? Can we prove upper/lower bounds?",
   "clean_statement": "Popular Furstenberg-Sárközy\n\n(Furstenberg-Sárközy) For all $\\epsilon>0$, there exists $N_0=N_0(\\epsilon)$ such that for all $N\\geq N_0$, $A\\subseteq [N]$ with $|A|\\geq \\delta N$, there exists $y\\neq 0$ such that\n$$|\\{x : x,x+y^2\\in A\\}|\\geq (\\delta^2-\\epsilon)N,$$\nwith $N_0(\\epsilon)\\leq e^{1/\\epsilon^c}$.\n\n(1) Prove lower bounds of the form $N_0(\\epsilon)>superpoly(1/\\epsilon)$, or even just $N_0(\\epsilon)>1/\\epsilon^{10}$.\n\n(2) What happens in the function field case $\\mathbb{F}_p[t]_{< n}$? Can we prove upper/lower bounds?",
   "public_statement": "Popular Furstenberg-Sárközy\n\n(Furstenberg-Sárközy) For all $\\epsilon>0$, there exists $N_0=N_0(\\epsilon)$ such that for all $N\\geq N_0$, $A\\subseteq [N]$ with $|A|\\geq \\delta N$, there exists $y\\neq 0$ such that\n$$|\\{x : x,x+y^2\\in A\\}|\\geq (\\delta^2-\\epsilon)N,$$\nwith $N_0(\\epsilon)\\leq e^{1/\\epsilon^c}$.\n\n(1) Prove lower bounds of the form $N_0(\\epsilon)>superpoly(1/\\epsilon)$, or even just $N_0(\\epsilon)>1/\\epsilon^{10}$.\n\n(2) What happens in the function field case $\\mathbb{F}_p[t]_{< n}$? Can we prove upper/lower bounds?",
   "evidence": "The canonical record is Problem 2.6 in the additive-combinatorics section of the AIM workshop *High-dimensional phenomena in discrete analysis*. Its main paragraph reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0010": {
   "statement_status": "exact",
   "original_statement": "How badly can Sidorenko fail for hypergraphs\n\nLet $H$ be a 3-uniform, 3-partite hypergraph. Then any $n$-vertex 3-uniform hypergraph with $pn^3$ edges contains at least $p^{Ce(H)}n^{V(H)}$ copies of $H$.",
   "clean_statement": "How badly can Sidorenko fail for hypergraphs\n\nLet $H$ be a 3-uniform, 3-partite hypergraph. Then any $n$-vertex 3-uniform hypergraph with $pn^3$ edges contains at least $p^{Ce(H)}n^{V(H)}$ copies of $H$.",
   "public_statement": "How badly can Sidorenko fail for hypergraphs\n\nLet $H$ be a 3-uniform, 3-partite hypergraph. Then any $n$-vertex 3-uniform hypergraph with $pn^3$ edges contains at least $p^{Ce(H)}n^{V(H)}$ copies of $H$.",
   "evidence": "The AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0011": {
   "statement_status": "exact",
   "original_statement": "Expansion of non-Sidorenko graphs\n\nGiven a $k$-graph $F$, define the $r$-expansion of $F$ to be the $r$-graph obtained by enlarging each $k$-edge of $F$ with a set of $r-k$ vertices of degree one.\n\nDoes there exist a non-Sidorenko (hyper)graph $H$ whose $r$-expansion is Sidorenko?",
   "clean_statement": "Expansion of non-Sidorenko graphs\n\nGiven a $k$-graph $F$, define the $r$-expansion of $F$ to be the $r$-graph obtained by enlarging each $k$-edge of $F$ with a set of $r-k$ vertices of degree one.\n\nDoes there exist a non-Sidorenko (hyper)graph $H$ whose $r$-expansion is Sidorenko?",
   "public_statement": "Expansion of non-Sidorenko graphs\n\nGiven a $k$-graph $F$, define the $r$-expansion of $F$ to be the $r$-graph obtained by enlarging each $k$-edge of $F$ with a set of $r-k$ vertices of degree one.\n\nDoes there exist a non-Sidorenko (hyper)graph $H$ whose $r$-expansion is Sidorenko?",
   "evidence": "The record is problem 3.2, “Expansion of non-Sidorenko graphs,” from the AIM workshop *High-dimensional phenomena in discrete analysis*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0012": {
   "statement_status": "exact",
   "original_statement": "Induced packing and covering\n\nLet $H$ be a fixed graph, $G$ an $n$-vertex graph. If the minimum number of edges needed to add or remove from $G$ in order to remove all induced copies of $H$ is $o(n^2)$, then the regularity lemma implies that the maximum number of induced copies of $H$, with each pair intersecting in at most one vertex, is $o(n^2)$.\n\nCan we get better bounds, even for the case $H=C_5$?",
   "clean_statement": "Induced packing and covering\n\nLet $H$ be a fixed graph, $G$ an $n$-vertex graph. If the minimum number of edges needed to add or remove from $G$ in order to remove all induced copies of $H$ is $o(n^2)$, then the regularity lemma implies that the maximum number of induced copies of $H$, with each pair intersecting in at most one vertex, is $o(n^2)$.\n\nCan we get better bounds, even for the case $H=C_5$?",
   "public_statement": "Induced packing and covering\n\nLet $H$ be a fixed graph, $G$ an $n$-vertex graph. If the minimum number of edges needed to add or remove from $G$ in order to remove all induced copies of $H$ is $o(n^2)$, then the regularity lemma implies that the maximum number of induced copies of $H$, with each pair intersecting in at most one vertex, is $o(n^2)$.\n\nCan we get better bounds, even for the case $H=C_5$?",
   "evidence": "The canonical record is Problem 3.3 in the extremal-graph-theory section of the AIM workshop *High-dimensional phenomena in discrete analysis*. The live AIM page, which attributes the problem to Jacob Fox, has exactly the following introduction and question:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0013": {
   "statement_status": "exact",
   "original_statement": "Turan exponents of hypergraphs\n\nThere exists $c>0$ such that the following holds. If $H$ is an $n$-vertex, 3-uniform hypergraph with at least $n^{3-c/d}$ edges, then $H$ contains some/every 3-uniform, 3-partite linear hypergraph on $d$ vertices with at least $d^2/100$ edges.",
   "clean_statement": "Turan exponents of hypergraphs\n\nThere exists $c>0$ such that the following holds. If $H$ is an $n$-vertex, 3-uniform hypergraph with at least $n^{3-c/d}$ edges, then $H$ contains some/every 3-uniform, 3-partite linear hypergraph on $d$ vertices with at least $d^2/100$ edges.",
   "public_statement": "Turan exponents of hypergraphs\n\nThere exists $c>0$ such that the following holds. If $H$ is an $n$-vertex, 3-uniform hypergraph with at least $n^{3-c/d}$ edges, then $H$ contains some/every 3-uniform, 3-partite linear hypergraph on $d$ vertices with at least $d^2/100$ edges.",
   "evidence": "The AIM record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0014": {
   "statement_status": "exact",
   "original_statement": "Equi-isoclinic subspaces\n\nGiven $k$-dimensional linear subspaces $U,V\\leq \\mathbb{R}^d$, we say that $U,V$ are equi-isoclinic with angle $\\alpha$ if every unit vector in $U$ projects down onto $V$ a vector of length $\\alpha$.\n\nLet $N_\\alpha^k(d)$ be the maximum number of $k$-dimensional subspaces of $\\mathbb{R}^d$, every pair of which is equi-isoclinic with angle $\\alpha$. Then\n\n(1) $N_{\\alpha}^k(d)=O_{\\alpha,k}(d)$.\n\n(2) $N_{\\alpha}^k(d)=O_{k}(d)+o_{\\alpha,k}(d)$.",
   "clean_statement": "Equi-isoclinic subspaces\n\nGiven $k$-dimensional linear subspaces $U,V\\leq \\mathbb{R}^d$, we say that $U,V$ are equi-isoclinic with angle $\\alpha$ if every unit vector in $U$ projects down onto $V$ a vector of length $\\alpha$.\n\nLet $N_\\alpha^k(d)$ be the maximum number of $k$-dimensional subspaces of $\\mathbb{R}^d$, every pair of which is equi-isoclinic with angle $\\alpha$. Then\n\n(1) $N_{\\alpha}^k(d)=O_{\\alpha,k}(d)$.\n\n(2) $N_{\\alpha}^k(d)=O_{k}(d)+o_{\\alpha,k}(d)$.",
   "public_statement": "Equi-isoclinic subspaces\n\nGiven $k$-dimensional linear subspaces $U,V\\leq \\mathbb{R}^d$, we say that $U,V$ are equi-isoclinic with angle $\\alpha$ if every unit vector in $U$ projects down onto $V$ a vector of length $\\alpha$.\n\nLet $N_\\alpha^k(d)$ be the maximum number of $k$-dimensional subspaces of $\\mathbb{R}^d$, every pair of which is equi-isoclinic with angle $\\alpha$. Then\n\n(1) $N_{\\alpha}^k(d)=O_{\\alpha,k}(d)$.\n\n(2) $N_{\\alpha}^k(d)=O_{k}(d)+o_{\\alpha,k}(d)$.",
   "evidence": "The AIM record, problem 4.1 from the workshop *High-dimensional phenomena in discrete analysis*, states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0015": {
   "statement_status": "exact",
   "original_statement": "Distinct distances and related problems\n\n(1) Let $S\\subset\\mathbb{R}^2$ with no three points collinear. Then $S$ determines at least $|S|/2$ distinct distances.\n\n(2) Given $n$ points in $\\mathbb{R}^2$ with no three collinear. Then one can remove $O(1)$-many points so that the number of isosceles triangles is at most $1.99\\binom{n}{2}$.\n\n(3) Given a set $S$ of $n$ points in $\\mathbb{R}^2$ with no three collinear. Then $S$ determines $\\Omega(n^2)$ perpendicular bisectors containing at most 1 point of $S$ each, unless $S$ is the vertices of a regular polygon together with its center.",
   "clean_statement": "Distinct distances and related problems\n\n(1) Let $S\\subset\\mathbb{R}^2$ with no three points collinear. Then $S$ determines at least $|S|/2$ distinct distances.\n\n(2) Given $n$ points in $\\mathbb{R}^2$ with no three collinear. Then one can remove $O(1)$-many points so that the number of isosceles triangles is at most $1.99\\binom{n}{2}$.\n\n(3) Given a set $S$ of $n$ points in $\\mathbb{R}^2$ with no three collinear. Then $S$ determines $\\Omega(n^2)$ perpendicular bisectors containing at most 1 point of $S$ each, unless $S$ is the vertices of a regular polygon together with its center.",
   "public_statement": "Distinct distances and related problems\n\n(1) Let $S\\subset\\mathbb{R}^2$ with no three points collinear. Then $S$ determines at least $|S|/2$ distinct distances.\n\n(2) Given $n$ points in $\\mathbb{R}^2$ with no three collinear. Then one can remove $O(1)$-many points so that the number of isosceles triangles is at most $1.99\\binom{n}{2}$.\n\n(3) Given a set $S$ of $n$ points in $\\mathbb{R}^2$ with no three collinear. Then $S$ determines $\\Omega(n^2)$ perpendicular bisectors containing at most 1 point of $S$ each, unless $S$ is the vertices of a regular polygon together with its center.",
   "evidence": "The canonical record is Conjecture 4.2, attributed to Cosmin Pohoata, in the AIM list *High-dimensional phenomena in discrete analysis*. It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0016": {
   "statement_status": "exact",
   "original_statement": "Kupitz conjecture in three dimensions\n\n(Kupitz conjecture) Given $n$ points in $\\mathbb{R}^2$, is there always a line through at least two points, such that the number of points on each side differ by $O(1)$?\n\nGiven $n$ points in $\\mathbb{R}^3$, is there always a plane through at least two points, such that the number of points on each side differ by $O(1)$?",
   "clean_statement": "Kupitz conjecture in three dimensions\n\n(Kupitz conjecture) Given $n$ points in $\\mathbb{R}^2$, is there always a line through at least two points, such that the number of points on each side differ by $O(1)$?\n\nGiven $n$ points in $\\mathbb{R}^3$, is there always a plane through at least two points, such that the number of points on each side differ by $O(1)$?",
   "public_statement": "Kupitz conjecture in three dimensions\n\n(Kupitz conjecture) Given $n$ points in $\\mathbb{R}^2$, is there always a line through at least two points, such that the number of points on each side differ by $O(1)$?\n\nGiven $n$ points in $\\mathbb{R}^3$, is there always a plane through at least two points, such that the number of points on each side differ by $O(1)$?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0017": {
   "statement_status": "exact",
   "original_statement": "Rudin's problem for $\\mathbb{F}_p[x], p>2$\n\nSuppose $A\\subset \\mathbb{F}_p[x]$ is of finite size, and every element of $A$ is a square. Show that $|A+A|\\geq |A|^{1+\\epsilon}$ for some constant $\\epsilon$.",
   "clean_statement": "Rudin's problem for $\\mathbb{F}_p[x], p>2$\n\nSuppose $A\\subset \\mathbb{F}_p[x]$ is of finite size, and every element of $A$ is a square. Show that $|A+A|\\geq |A|^{1+\\epsilon}$ for some constant $\\epsilon$.",
   "public_statement": "Rudin's problem for $\\mathbb{F}_p[x], p>2$\n\nSuppose $A\\subset \\mathbb{F}_p[x]$ is of finite size, and every element of $A$ is a square. Show that $|A+A|\\geq |A|^{1+\\epsilon}$ for some constant $\\epsilon$.",
   "evidence": "The canonical record is from the 2018 AIM workshop *Additive combinatorics and its applications*, section “Polynomial method,” problem 1.1. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0018": {
   "statement_status": "exact",
   "original_statement": "3APs in sets of small growth\n\nObtain a proof of the following theorem using the polynomial method.\n\nTheorem. There is an absolute constant $c>0$ that the following holds. If $A\\subset \\mathbb{F}_p^n$ and $|A+A|\\leq |A|^{1+c}$ then $A$ contains a 3AP.",
   "clean_statement": "3APs in sets of small growth\n\nObtain a proof of the following theorem using the polynomial method.\n\nTheorem. There is an absolute constant $c>0$ that the following holds. If $A\\subset \\mathbb{F}_p^n$ and $|A+A|\\leq |A|^{1+c}$ then $A$ contains a 3AP.",
   "public_statement": "3APs in sets of small growth\n\nObtain a proof of the following theorem using the polynomial method.\n\nTheorem. There is an absolute constant $c>0$ that the following holds. If $A\\subset \\mathbb{F}_p^n$ and $|A+A|\\leq |A|^{1+c}$ then $A$ contains a 3AP.",
   "evidence": "The AIM record, from the workshop “Additive combinatorics and its applications,” Polynomial method, Problem 1.2, says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0019": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Polynomial method via tensorizing\n\nThe purpose of this problem is to extend the current polynomial method techniques to handle 4APs and more complicated patterns. It's about suggesting a notion of tensor rank that tensorizes. Let $\\mathbb{F}$ be a field and $A\\in \\mathbb{F}^{n\\times \\cdots \\times n}$ be a $d$-dimensional tensor. We can think of $A$ as a multilinear polynomial $f:(\\mathbb{F}^n)^d\\rightarrow \\mathbb{F}$ on variables $X_1,\\cdots,X_d \\in \\mathbb{F}^n$ where $A$ specifies the coefficients of the monomials. In the following we introduce several notions of rank.\nLet $rank(A)$ be the minimum number of rank-one tensors that sum up to $A$. We need to specify the definition of a rank-one tensor. There are several definitions of a rank-1 tensor $f$.\n\n$rank$-1:\n$$f(X_1,\\cdots,X_d) = g_1(X_1)\\cdots g_d(X_d)$$\nwhere each $g_i:\\mathbb{F}^n\\rightarrow\\mathbb{F}$ is linear\n\n$srank$-1: There is some $i\\in [d]$ so that\n$$f(X_1,\\cdots,X_d) = g(X_i)f'(X_1,\\cdots,X_d)$$\nwhere $g$ is linear and $f'$ does not depend on $X_i$.\n\n$prank$-1: There is a subset $\\phi\\neq S\\subset[d]$ so that\n$$f(X_1,\\cdots,X_d) = g(X_i: i\\in S) h(X_i: i\\notin S)$$\nNote that $prank(\\cdot)\\leq srank(\\cdot)\\leq rank(\\cdot)$.\nWe introduce another notion of rank (only for $\\mathbb{F}_p$, where $p$ is prime) called analytic rank.\nLet $$Bias (f) = \\mathbb{E}_{X_1,\\cdots,X_d} \\omega^{f(X_1,\\cdots,X_d)}$$\nwhere $\\omega$ is a $p$th root of unity. Also note that $Bias(f)\\in [0,1]$.\nDefine $$arank(f) = -\\log Bias(f).$$\nWe have that $arank(\\cdot)\\leq prank(\\cdot)$.\n\nNow, let's see how these notions of rank are used to solve, say the capset problem. The idea is that the identity tensor has high rank for all definitions of $arank,srank,prank$, but the capset tensor has low rank. The capset tensor is $C\\in \\mathbb{F}_3^{n\\times n \\times n}$ where $n= 3^k$ for some $k$. Given $x,y,z\\in \\mathbb{F}_3^k$ we have $C(x,y,z)= 0 $ if $x+y+z\\neq 0$ and is 1 otherwise.\nIf $S\\subset \\mathbb{F}_3^k$ is $AP$-free, then $C_{|S}$ ($C$ restricted to the set $S$) is identity.\n\nLet $\\mathbf{r}$ be either $arank$ or $prank$. Let $A$ be a tensor, and $Id$ is the identity tensor on the same space as $A$. Suppose $\\mathbf{r}(A)< \\mathbf{r}(Id)$. Is it true that $\\mathbf{r}(A^{\\otimes m}) < \\mathbf{r}(Id)^m c^m$ for some constant $c<1$?",
   "clean_statement": null,
   "public_statement": "Polynomial method via tensorizing\n\nThe purpose of this problem is to extend the current polynomial method techniques to handle 4APs and more complicated patterns. It's about suggesting a notion of tensor rank that tensorizes. Let $\\mathbb{F}$ be a field and $A\\in \\mathbb{F}^{n\\times \\cdots \\times n}$ be a $d$-dimensional tensor. We can think of $A$ as a multilinear polynomial $f:(\\mathbb{F}^n)^d\\rightarrow \\mathbb{F}$ on variables $X_1,\\cdots,X_d \\in \\mathbb{F}^n$ where $A$ specifies the coefficients of the monomials. In the following we introduce several notions of rank.\nLet $rank(A)$ be the minimum number of rank-one tensors that sum up to $A$. We need to specify the definition of a rank-one tensor. There are several definitions of a rank-1 tensor $f$.\n\n$rank$-1:\n$$f(X_1,\\cdots,X_d) = g_1(X_1)\\cdots g_d(X_d)$$\nwhere each $g_i:\\mathbb{F}^n\\rightarrow\\mathbb{F}$ is linear\n\n$srank$-1: There is some $i\\in [d]$ so that\n$$f(X_1,\\cdots,X_d) = g(X_i)f'(X_1,\\cdots,X_d)$$\nwhere $g$ is linear and $f'$ does not depend on $X_i$.\n\n$prank$-1: There is a subset $\\phi\\neq S\\subset[d]$ so that\n$$f(X_1,\\cdots,X_d) = g(X_i: i\\in S) h(X_i: i\\notin S)$$\nNote that $prank(\\cdot)\\leq srank(\\cdot)\\leq rank(\\cdot)$.\nWe introduce another notion of rank (only for $\\mathbb{F}_p$, where $p$ is prime) called analytic rank.\nLet $$Bias (f) = \\mathbb{E}_{X_1,\\cdots,X_d} \\omega^{f(X_1,\\cdots,X_d)}$$\nwhere $\\omega$ is a $p$th root of unity. Also note that $Bias(f)\\in [0,1]$.\nDefine $$arank(f) = -\\log Bias(f).$$\nWe have that $arank(\\cdot)\\leq prank(\\cdot)$.\n\nNow, let's see how these notions of rank are used to solve, say the capset problem. The idea is that the identity tensor has high rank for all definitions of $arank,srank,prank$, but the capset tensor has low rank. The capset tensor is $C\\in \\mathbb{F}_3^{n\\times n \\times n}$ where $n= 3^k$ for some $k$. Given $x,y,z\\in \\mathbb{F}_3^k$ we have $C(x,y,z)= 0 $ if $x+y+z\\neq 0$ and is 1 otherwise.\nIf $S\\subset \\mathbb{F}_3^k$ is $AP$-free, then $C_{|S}$ ($C$ restricted to the set $S$) is identity.\n\nLet $\\mathbf{r}$ be either $arank$ or $prank$. Let $A$ be a tensor, and $Id$ is the identity tensor on the same space as $A$. Suppose $\\mathbf{r}(A)< \\mathbf{r}(Id)$. Is it true that $\\mathbf{r}(A^{\\otimes m}) < \\mathbf{r}(Id)^m c^m$ for some constant $c<1$?",
   "evidence": "1. The printed `\\(\\phi\\neq S\\subset[d]\\)` is almost certainly an extraction/OCR error for the standard condition \\(\\varnothing\\ne S\\subsetneq[d]\\). Both factors in a partition-rank-one decomposition must use nonempty complementary sets of modes. 2. \\(\\omega\\) must be a **nontrivial** additive character, i.e. a primitive \\(p\\)-th root in the prime-field notation. If \\(\\omega=1\\), every bias is one. 3. Standard analytic rank is \\[ \\operatorname{arank}_p(T)=-\\log_p\\operatorname{bias}(T). \\] The missing logarithm base matters critically when a numerical rank is raised to the \\(m\\)-th power. Base \\(p\\) is also the normalization for which matrix analytic rank equals matrix rank and Lovett's inequality \\(\\operatorname{arank}\\le\\operatorname{prank}\\) has constant one. 4. The identity must be the order-\\(d\\), side-\\(n\\) diagonal tensor \\[ I_{n,d}=\\sum_{i=1}^n e_i^*\\otimes\\cdots\\otimes e_i^*. \\...",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0020": {
   "statement_status": "exact",
   "original_statement": "Arithmetic removal lemma for cyclic groups\n\nThe arithmetic removal lemma is the following. $\\forall\\varepsilon>0, \\exists\\delta>0$ such that if $G$ is an abelian group and $A,B,C\\subset G$, and suppose there are at least $\\delta|G|^2$ solutions to $a+b+c=0, a\\in A, b\\in B, c\\in C$. Then you can remove $\\varepsilon|G|$ elements from each of the sets $A,B,C$ so that there are no solutions to $a+b+c=0$. In the case of $G = \\mathbb{F}_p^n$ we know $\\delta = \\varepsilon^{c_p+o(1)}$ where $c_p$ is a known absolute constant and this is sharp. In the case of cyclic groups, we know that because of Behrend's construction $\\delta$ can not be a polynomial in $\\varepsilon$. The known bound is however is of tower type.\n\nObtain improved bound for arithmetic regularity lemma for cyclic groups.",
   "clean_statement": "Arithmetic removal lemma for cyclic groups\n\nThe arithmetic removal lemma is the following. $\\forall\\varepsilon>0, \\exists\\delta>0$ such that if $G$ is an abelian group and $A,B,C\\subset G$, and suppose there are at least $\\delta|G|^2$ solutions to $a+b+c=0, a\\in A, b\\in B, c\\in C$. Then you can remove $\\varepsilon|G|$ elements from each of the sets $A,B,C$ so that there are no solutions to $a+b+c=0$. In the case of $G = \\mathbb{F}_p^n$ we know $\\delta = \\varepsilon^{c_p+o(1)}$ where $c_p$ is a known absolute constant and this is sharp. In the case of cyclic groups, we know that because of Behrend's construction $\\delta$ can not be a polynomial in $\\varepsilon$. The known bound is however is of tower type.\n\nObtain improved bound for arithmetic regularity lemma for cyclic groups.",
   "public_statement": "Arithmetic removal lemma for cyclic groups\n\nThe arithmetic removal lemma is the following. $\\forall\\varepsilon>0, \\exists\\delta>0$ such that if $G$ is an abelian group and $A,B,C\\subset G$, and suppose there are at least $\\delta|G|^2$ solutions to $a+b+c=0, a\\in A, b\\in B, c\\in C$. Then you can remove $\\varepsilon|G|$ elements from each of the sets $A,B,C$ so that there are no solutions to $a+b+c=0$. In the case of $G = \\mathbb{F}_p^n$ we know $\\delta = \\varepsilon^{c_p+o(1)}$ where $c_p$ is a known absolute constant and this is sharp. In the case of cyclic groups, we know that because of Behrend's construction $\\delta$ can not be a polynomial in $\\varepsilon$. The known bound is however is of tower type.\n\nObtain improved bound for arithmetic regularity lemma for cyclic groups.",
   "evidence": "The AIM record is titled **“Arithmetic removal lemma for cyclic groups.”** Its displayed formulation says that if there are *at least* \\(\\delta |G|^2\\) solutions of \\[ a+b+c=0,\\qquad (a,b,c)\\in A\\times B\\times C, \\] then one may remove \\(\\varepsilon |G|\\) elements from each set and destroy all solutions. It then contrasts the sharp finite-vector-space bound with a tower-type bound for cyclic groups, mentions Behrend's obstruction to polynomial dependence, and ends with “Obtain improved bound for arithmetic regularity lemma for cyclic groups.” The accompanying literature note points to tri-colored sum-free sets.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0021": {
   "statement_status": "exact",
   "original_statement": "A polynomial method proof of Frankl-Rodl theorem\n\nThe Frankl-Rodl theorem is the following: let $A\\subset \\mathbb{F}_2^n$ be so that $A+A$ does not contain any element with hamming weight $k$ for some $k$ satisfying $C\\sqrt[]{\\log \\alpha^{-1}n}\\leq k\\leq n-C\\sqrt[]{\\log \\alpha^{-1}n}$. Then $A$ has density at most $\\alpha$.\n\nFind a proof of Frankl-Rodl theorem using the polynomial method, in particular Croot-Lev-Pach method.",
   "clean_statement": "A polynomial method proof of Frankl-Rodl theorem\n\nThe Frankl-Rodl theorem is the following: let $A\\subset \\mathbb{F}_2^n$ be so that $A+A$ does not contain any element with hamming weight $k$ for some $k$ satisfying $C\\sqrt[]{\\log \\alpha^{-1}n}\\leq k\\leq n-C\\sqrt[]{\\log \\alpha^{-1}n}$. Then $A$ has density at most $\\alpha$.\n\nFind a proof of Frankl-Rodl theorem using the polynomial method, in particular Croot-Lev-Pach method.",
   "public_statement": "A polynomial method proof of Frankl-Rodl theorem\n\nThe Frankl-Rodl theorem is the following: let $A\\subset \\mathbb{F}_2^n$ be so that $A+A$ does not contain any element with hamming weight $k$ for some $k$ satisfying $C\\sqrt[]{\\log \\alpha^{-1}n}\\leq k\\leq n-C\\sqrt[]{\\log \\alpha^{-1}n}$. Then $A$ has density at most $\\alpha$.\n\nFind a proof of Frankl-Rodl theorem using the polynomial method, in particular Croot-Lev-Pach method.",
   "evidence": "The canonical `problem` field is reproduced verbatim below. This preserves the missing parity hypothesis and the ambiguous expression under the square root.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0022": {
   "statement_status": "exact",
   "original_statement": "Doubling of generic set of points\n\nThere is a result of Stancescu as follows: If $A\\subset \\mathbb{R}^2$ is a finite set with no three points on a line, then $$|A+A|\\geq |A|(\\log |A|)^c $$ for some $c>0$.\nAlso we know that the lower bound of $|A|^{1-\\varepsilon}$ is false because of the Behrend construction. We would like to generalize this statement to higher dimensions.\n\nLet $A\\subset \\mathbb{R}^3$ be a finite set with no five points on a plane. What is the best inequality $$|A+A|\\geq |A| f(A)$$\nthat we can have?",
   "clean_statement": "Doubling of generic set of points\n\nThere is a result of Stancescu as follows: If $A\\subset \\mathbb{R}^2$ is a finite set with no three points on a line, then $$|A+A|\\geq |A|(\\log |A|)^c $$ for some $c>0$.\nAlso we know that the lower bound of $|A|^{1-\\varepsilon}$ is false because of the Behrend construction. We would like to generalize this statement to higher dimensions.\n\nLet $A\\subset \\mathbb{R}^3$ be a finite set with no five points on a plane. What is the best inequality $$|A+A|\\geq |A| f(A)$$\nthat we can have?",
   "public_statement": "Doubling of generic set of points\n\nThere is a result of Stancescu as follows: If $A\\subset \\mathbb{R}^2$ is a finite set with no three points on a line, then $$|A+A|\\geq |A|(\\log |A|)^c $$ for some $c>0$.\nAlso we know that the lower bound of $|A|^{1-\\varepsilon}$ is false because of the Behrend construction. We would like to generalize this statement to higher dimensions.\n\nLet $A\\subset \\mathbb{R}^3$ be a finite set with no five points on a plane. What is the best inequality $$|A+A|\\geq |A| f(A)$$\nthat we can have?",
   "evidence": "The exact AIM record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0023": {
   "statement_status": "exact",
   "original_statement": "A question about 2-coloring of integers\n\nLet $W(k,\\ell)$ be the minimum $n$ so that any red/blue coloring of $\\{1,\\cdots, n\\}$ has either a red $k$-AP or a blue $\\ell$-AP.\nThe following conjecture is supported by numerical evidence.\n\n$W(3,\\ell)= \\ell^{2+o(1)}$.",
   "clean_statement": "A question about 2-coloring of integers\n\nLet $W(k,\\ell)$ be the minimum $n$ so that any red/blue coloring of $\\{1,\\cdots, n\\}$ has either a red $k$-AP or a blue $\\ell$-AP.\nThe following conjecture is supported by numerical evidence.\n\n$W(3,\\ell)= \\ell^{2+o(1)}$.",
   "public_statement": "A question about 2-coloring of integers\n\nLet $W(k,\\ell)$ be the minimum $n$ so that any red/blue coloring of $\\{1,\\cdots, n\\}$ has either a red $k$-AP or a blue $\\ell$-AP.\nThe following conjecture is supported by numerical evidence.\n\n$W(3,\\ell)= \\ell^{2+o(1)}$.",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0024": {
   "statement_status": "exact",
   "original_statement": "A stronger BSG theorem\n\nThe standard BSG theorem is as follows: let $E(A) = |\\{(a,b,c,d)\\in A^4: a+b=c+d\\}|$. If $\\frac{E(A)}{|A|^3}\\geq \\frac{1}{K}$ then there is a subset $A'\\subset A$ so that $|A'|\\geq \\frac{|A|}{K^{100}}$ and $|A'+A'|\\leq K^{100}|A'|$.\nOne of the drawbacks of BSG theorem is that we lose control on $E(A')$. Can we have a stronger BSG theorem as follows?\n\nLet $A\\subset G$, and suppose $E(A)\\geq \\frac{|A|^3}{K}$. Then there is a set $A'\\subset A$ so that $$|A'|\\geq \\frac{|A|}{K^{100}},$$\n$$|A'+A'|\\leq K^{100}|A'|,$$\n$$\\frac{E(A')}{|A'|^3}\\geq \\frac{E(A)}{|A|^3}$$\nCan we moreover obtain the following?\n$$\\frac{E(A')}{|A'|^3}\\geq \\frac{E(A)}{|A|^3} \\left(\\frac{|A|}{|A'|}\\right)^{\\frac{1}{100}}$$",
   "clean_statement": "A stronger BSG theorem\n\nThe standard BSG theorem is as follows: let $E(A) = |\\{(a,b,c,d)\\in A^4: a+b=c+d\\}|$. If $\\frac{E(A)}{|A|^3}\\geq \\frac{1}{K}$ then there is a subset $A'\\subset A$ so that $|A'|\\geq \\frac{|A|}{K^{100}}$ and $|A'+A'|\\leq K^{100}|A'|$.\nOne of the drawbacks of BSG theorem is that we lose control on $E(A')$. Can we have a stronger BSG theorem as follows?\n\nLet $A\\subset G$, and suppose $E(A)\\geq \\frac{|A|^3}{K}$. Then there is a set $A'\\subset A$ so that $$|A'|\\geq \\frac{|A|}{K^{100}},$$\n$$|A'+A'|\\leq K^{100}|A'|,$$\n$$\\frac{E(A')}{|A'|^3}\\geq \\frac{E(A)}{|A|^3}$$\nCan we moreover obtain the following?\n$$\\frac{E(A')}{|A'|^3}\\geq \\frac{E(A)}{|A|^3} \\left(\\frac{|A|}{|A'|}\\right)^{\\frac{1}{100}}$$",
   "public_statement": "A stronger BSG theorem\n\nThe standard BSG theorem is as follows: let $E(A) = |\\{(a,b,c,d)\\in A^4: a+b=c+d\\}|$. If $\\frac{E(A)}{|A|^3}\\geq \\frac{1}{K}$ then there is a subset $A'\\subset A$ so that $|A'|\\geq \\frac{|A|}{K^{100}}$ and $|A'+A'|\\leq K^{100}|A'|$.\nOne of the drawbacks of BSG theorem is that we lose control on $E(A')$. Can we have a stronger BSG theorem as follows?\n\nLet $A\\subset G$, and suppose $E(A)\\geq \\frac{|A|^3}{K}$. Then there is a set $A'\\subset A$ so that $$|A'|\\geq \\frac{|A|}{K^{100}},$$\n$$|A'+A'|\\leq K^{100}|A'|,$$\n$$\\frac{E(A')}{|A'|^3}\\geq \\frac{E(A)}{|A|^3}$$\nCan we moreover obtain the following?\n$$\\frac{E(A')}{|A'|^3}\\geq \\frac{E(A)}{|A|^3} \\left(\\frac{|A|}{|A'|}\\right)^{\\frac{1}{100}}$$",
   "evidence": "The canonical record is Problem 2.15, “A stronger BSG theorem,” from the AIM list *Additive combinatorics and its applications*, section “Additive combinatorics.” The stored source record is uncorrupted and asks the following.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0025": {
   "statement_status": "exact",
   "original_statement": "Finding examples.\n\nFind counter examples to the polynomial Bogolyubov Ruzsa theorem.\n\nFind interesting examples of subset $A\\subset \\mathbb{F}_2^n$ so that $A+A$ is far from all of $\\mathbb{F}_2^n$.\n\nFind interesting examples of $A\\subset [N]$ of density $\\alpha$ so that $4A$ does not contain long APs.",
   "clean_statement": "Finding examples.\n\nFind counter examples to the polynomial Bogolyubov Ruzsa theorem.\n\nFind interesting examples of subset $A\\subset \\mathbb{F}_2^n$ so that $A+A$ is far from all of $\\mathbb{F}_2^n$.\n\nFind interesting examples of $A\\subset [N]$ of density $\\alpha$ so that $4A$ does not contain long APs.",
   "public_statement": "Finding examples.\n\nFind counter examples to the polynomial Bogolyubov Ruzsa theorem.\n\nFind interesting examples of subset $A\\subset \\mathbb{F}_2^n$ so that $A+A$ is far from all of $\\mathbb{F}_2^n$.\n\nFind interesting examples of $A\\subset [N]$ of density $\\alpha$ so that $4A$ does not contain long APs.",
   "evidence": "The AIM record, under the heading “Finding examples,” contains three requests:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0026": {
   "statement_status": "exact",
   "original_statement": "Approximate Caratheodory theorem\n\nThere is the following important theorem:\nLet $A\\subset \\mathbb{R}^d$ be a finite set of points. Also assume that $A$ is contained in the $L^p$ unit Hamming ball for $p\\geq 2$.\nThen any arbitrary point $a$ in the convex hull of $A$ can be approximated by a convex combination of only $O(p\\varepsilon^{-2})$ many points of $A$.\n\nQ. First, assuming some assumptions on the point set $A$, find a better dependence (sub-linear) on $p$.\n\nQ. Characterize configurations for which the bound $O(p\\varepsilon^{-2})$ is tight.\nstatus",
   "clean_statement": "Approximate Caratheodory theorem\n\nThere is the following important theorem:\nLet $A\\subset \\mathbb{R}^d$ be a finite set of points. Also assume that $A$ is contained in the $L^p$ unit Hamming ball for $p\\geq 2$.\nThen any arbitrary point $a$ in the convex hull of $A$ can be approximated by a convex combination of only $O(p\\varepsilon^{-2})$ many points of $A$.\n\nQ. First, assuming some assumptions on the point set $A$, find a better dependence (sub-linear) on $p$.\n\nQ. Characterize configurations for which the bound $O(p\\varepsilon^{-2})$ is tight.\nstatus",
   "public_statement": "Approximate Caratheodory theorem\n\nThere is the following important theorem:\nLet $A\\subset \\mathbb{R}^d$ be a finite set of points. Also assume that $A$ is contained in the $L^p$ unit Hamming ball for $p\\geq 2$.\nThen any arbitrary point $a$ in the convex hull of $A$ can be approximated by a convex combination of only $O(p\\varepsilon^{-2})$ many points of $A$.\n\nQ. First, assuming some assumptions on the point set $A$, find a better dependence (sub-linear) on $p$.\n\nQ. Characterize configurations for which the bound $O(p\\varepsilon^{-2})$ is tight.\nstatus",
   "evidence": "The canonical `problem` field is reproduced verbatim below. In particular, this preserves the phrases “unit Hamming ball,” the omitted approximation norm, and the trailing word `status`.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0027": {
   "statement_status": "exact",
   "original_statement": "Finding good modeling\n\nLet $A,B$ be subset of abelian groups $G, H$ respectively. A map $\\phi: A\\rightarrow B$ is called a Freiman isomorphism if\n$$a+b=c+d \\iff \\phi(a)+\\phi(b)=\\phi(c)+\\phi(d).$$\nWe have the following modeling lemmas.\n\n1. Let $A\\subset \\mathbb{F}_2^n$ and $|A+A|\\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of $\\mathbb{F}_2^m$ where $|\\mathbb{F}_2^m|\\leq K^{O(1)}|A|$.\n\n2. Let $A\\subset \\mathbb{Z}^d$ and $|A+A|\\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of a group $H$ where $|H|\\leq K^{O(1)}|A|$.\n\nMoreover, this is false for arbitrary groups instead of $\\mathbb{Z}^d$ or $\\mathbb{F}_2^n$.\n\nSuppose $A\\subset G$, and $|A+A|\\leq K |A|$. Can we find a subset $A'\\subset A$ with $|A'|\\geq K^{-O(1)}|A|$ and a group $H$ so that $A'$ is Freiman isomorphic to a subset of $H$ and $|H|\\leq K^{O(1)}|A|$?",
   "clean_statement": "Finding good modeling\n\nLet $A,B$ be subset of abelian groups $G, H$ respectively. A map $\\phi: A\\rightarrow B$ is called a Freiman isomorphism if\n$$a+b=c+d \\iff \\phi(a)+\\phi(b)=\\phi(c)+\\phi(d).$$\nWe have the following modeling lemmas.\n\n1. Let $A\\subset \\mathbb{F}_2^n$ and $|A+A|\\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of $\\mathbb{F}_2^m$ where $|\\mathbb{F}_2^m|\\leq K^{O(1)}|A|$.\n\n2. Let $A\\subset \\mathbb{Z}^d$ and $|A+A|\\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of a group $H$ where $|H|\\leq K^{O(1)}|A|$.\n\nMoreover, this is false for arbitrary groups instead of $\\mathbb{Z}^d$ or $\\mathbb{F}_2^n$.\n\nSuppose $A\\subset G$, and $|A+A|\\leq K |A|$. Can we find a subset $A'\\subset A$ with $|A'|\\geq K^{-O(1)}|A|$ and a group $H$ so that $A'$ is Freiman isomorphic to a subset of $H$ and $|H|\\leq K^{O(1)}|A|$?",
   "public_statement": "Finding good modeling\n\nLet $A,B$ be subset of abelian groups $G, H$ respectively. A map $\\phi: A\\rightarrow B$ is called a Freiman isomorphism if\n$$a+b=c+d \\iff \\phi(a)+\\phi(b)=\\phi(c)+\\phi(d).$$\nWe have the following modeling lemmas.\n\n1. Let $A\\subset \\mathbb{F}_2^n$ and $|A+A|\\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of $\\mathbb{F}_2^m$ where $|\\mathbb{F}_2^m|\\leq K^{O(1)}|A|$.\n\n2. Let $A\\subset \\mathbb{Z}^d$ and $|A+A|\\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of a group $H$ where $|H|\\leq K^{O(1)}|A|$.\n\nMoreover, this is false for arbitrary groups instead of $\\mathbb{Z}^d$ or $\\mathbb{F}_2^n$.\n\nSuppose $A\\subset G$, and $|A+A|\\leq K |A|$. Can we find a subset $A'\\subset A$ with $|A'|\\geq K^{-O(1)}|A|$ and a group $H$ so that $A'$ is Freiman isomorphic to a subset of $H$ and $|H|\\leq K^{O(1)}|A|$?",
   "evidence": "The canonical record (AIM workshop *Additive combinatorics and its applications*, problem 2.3) asks the following. If \\(A\\) is a finite subset of an abelian group \\(G\\) and",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0028": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Product sets in iterated sumsets\n\nObserve that the interval $[N]$ contains $[N^c][N^c]$ for $c=0.5$.\n\nThere exists $n,\\varepsilon,\\delta$ for which the following holds.\nLet $A\\subset \\mathbb{F}_p$ and suppose $|A+A|\\leq |A|^{1+\\varepsilon}$. Then $nA-nA$ contains a product set $XY$ where $|X|,|Y|\\geq |A|^{1+\\varepsilon}$.\n\nIf $X$ is polynomially large, how big can you make $Y$?",
   "clean_statement": null,
   "public_statement": "Product sets in iterated sumsets\n\nObserve that the interval $[N]$ contains $[N^c][N^c]$ for $c=0.5$.\n\nThere exists $n,\\varepsilon,\\delta$ for which the following holds.\nLet $A\\subset \\mathbb{F}_p$ and suppose $|A+A|\\leq |A|^{1+\\varepsilon}$. Then $nA-nA$ contains a product set $XY$ where $|X|,|Y|\\geq |A|^{1+\\varepsilon}$.\n\nIf $X$ is polynomially large, how big can you make $Y$?",
   "evidence": "The most plausible repair is to replace both occurrences of \\(|A|^{1+\\varepsilon}\\) in the conclusion by \\(|A|^\\delta\\), or more generally to ask for \\[ |X|\\ge |A|^\\alpha,\\qquad |Y|\\ge |A|^\\beta, \\tag{1} \\] under an explicit density window such as \\(|A|\\le p^{1-\\eta}\\), with \\(n,\\varepsilon,\\alpha,\\beta,\\eta\\) absolute. This is a reconstruction, not verified source text.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0029": {
   "statement_status": "exact",
   "original_statement": "Entropy of sum of random variables.\n\nLet $G$ be a torsion free abelian group. Suppose $X$ is a random variable on $G$ and $S_n$ is sum of $n$ many $i.i.d$ copies of $X$. Let $H(\\cdot)$ be the discrete entropy function.\n\nWe have the following theorem by T. Tao:\n$$H(S_2)\\geq H(S_1)+\\log_2 \\sqrt[]{2}-o(1) $$ where the $o(1)$ term goes to zero as $H(X)$ grows to infinity.\n\nShow $$H(S_n)\\geq H(S_{n-1})+\\log_2 \\sqrt[]{\\frac{n}{n-1}} - o(1).$$",
   "clean_statement": "Entropy of sum of random variables.\n\nLet $G$ be a torsion free abelian group. Suppose $X$ is a random variable on $G$ and $S_n$ is sum of $n$ many $i.i.d$ copies of $X$. Let $H(\\cdot)$ be the discrete entropy function.\n\nWe have the following theorem by T. Tao:\n$$H(S_2)\\geq H(S_1)+\\log_2 \\sqrt[]{2}-o(1) $$ where the $o(1)$ term goes to zero as $H(X)$ grows to infinity.\n\nShow $$H(S_n)\\geq H(S_{n-1})+\\log_2 \\sqrt[]{\\frac{n}{n-1}} - o(1).$$",
   "public_statement": "Entropy of sum of random variables.\n\nLet $G$ be a torsion free abelian group. Suppose $X$ is a random variable on $G$ and $S_n$ is sum of $n$ many $i.i.d$ copies of $X$. Let $H(\\cdot)$ be the discrete entropy function.\n\nWe have the following theorem by T. Tao:\n$$H(S_2)\\geq H(S_1)+\\log_2 \\sqrt[]{2}-o(1) $$ where the $o(1)$ term goes to zero as $H(X)$ grows to infinity.\n\nShow $$H(S_n)\\geq H(S_{n-1})+\\log_2 \\sqrt[]{\\frac{n}{n-1}} - o(1).$$",
   "evidence": "The canonical AIM record (workshop *Additive combinatorics and its applications*, section 2.4) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0030": {
   "statement_status": "exact",
   "original_statement": "Polynomial Freiman Ruzsa conjecture implies Polynomial Bogolyubov Ruzsa conjecture.\n\nProve that polynomial Freiman Ruzsa conjecture implies Polynomial Bogolyubov Ruzsa conjecture.",
   "clean_statement": "Polynomial Freiman Ruzsa conjecture implies Polynomial Bogolyubov Ruzsa conjecture.\n\nProve that polynomial Freiman Ruzsa conjecture implies Polynomial Bogolyubov Ruzsa conjecture.",
   "public_statement": "Polynomial Freiman Ruzsa conjecture implies Polynomial Bogolyubov Ruzsa conjecture.\n\nProve that polynomial Freiman Ruzsa conjecture implies Polynomial Bogolyubov Ruzsa conjecture.",
   "evidence": "The canonical record (AIM problem 2.45, workshop *Additive combinatorics and its applications*) says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0031": {
   "statement_status": "exact",
   "original_statement": "Finding almost subspace inside A+A\n\nLet $A\\subset \\mathbb{F}_2^n$ be a set of density $\\alpha$. Then there is a subspace $V$ of codimension $\\log^{O(1)}(\\varepsilon^{-1}\\alpha^{-1})$ so that $$|V\\cap (A+A)|\\geq (1-\\varepsilon)|V|$$",
   "clean_statement": "Finding almost subspace inside A+A\n\nLet $A\\subset \\mathbb{F}_2^n$ be a set of density $\\alpha$. Then there is a subspace $V$ of codimension $\\log^{O(1)}(\\varepsilon^{-1}\\alpha^{-1})$ so that $$|V\\cap (A+A)|\\geq (1-\\varepsilon)|V|$$",
   "public_statement": "Finding almost subspace inside A+A\n\nLet $A\\subset \\mathbb{F}_2^n$ be a set of density $\\alpha$. Then there is a subspace $V$ of codimension $\\log^{O(1)}(\\varepsilon^{-1}\\alpha^{-1})$ so that $$|V\\cap (A+A)|\\geq (1-\\varepsilon)|V|$$",
   "evidence": "The canonical record is AIM Problem Lists, workshop *Additive combinatorics and its applications*, section *Additive combinatorics*, Problem 2.5, “Finding almost subspace inside \\(A+A\\).” Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0032": {
   "statement_status": "exact",
   "original_statement": "Finding structure inside $A+A$\n\nLet $A\\subset \\mathbb{F}_2^n$ be a subset of density $0.01$. Then there is a subset $B\\subset \\mathbb{F}_2^n$ with density absolute constant, so that $$B+B\\subset A+A$$ and $$B = S_1+\\cdots+S_\\ell$$ for some sets $S_i$'s with $|S_i|\\leq poly(n)$ and $\\ell\\leq poly(n)$.",
   "clean_statement": "Finding structure inside $A+A$\n\nLet $A\\subset \\mathbb{F}_2^n$ be a subset of density $0.01$. Then there is a subset $B\\subset \\mathbb{F}_2^n$ with density absolute constant, so that $$B+B\\subset A+A$$ and $$B = S_1+\\cdots+S_\\ell$$ for some sets $S_i$'s with $|S_i|\\leq poly(n)$ and $\\ell\\leq poly(n)$.",
   "public_statement": "Finding structure inside $A+A$\n\nLet $A\\subset \\mathbb{F}_2^n$ be a subset of density $0.01$. Then there is a subset $B\\subset \\mathbb{F}_2^n$ with density absolute constant, so that $$B+B\\subset A+A$$ and $$B = S_1+\\cdots+S_\\ell$$ for some sets $S_i$'s with $|S_i|\\leq poly(n)$ and $\\ell\\leq poly(n)$.",
   "evidence": "The canonical record is AIM Problem 2.55, “Finding structure inside \\(A+A\\),” attributed on the live AIM page to Kaave Hosseini. The source says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0033": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A conjecture about generalized additive energy\n\nLet $A\\subset G$ and define $$E_{2n}(A) = |\\{(a_1,\\cdots,a_n)\\in A^{2n}: a_1+\\cdots+a_n = a_{n+1}+\\cdots+a_{2n}\\}|.$$\nTrivial bounds on additive energy are $$|A|^n\\leq E_{2n}(A)\\leq |A|^{2n-1}.$$\nDefine $$\\rho_{2n}(A)= \\frac{E_{2n}(A)}{|A|^n}.$$\n\nGiven this, one can compare various $\\rho_k$'s. For example by an applications of Holder's inequality one can get\n$$\\rho_8(A) \\geq \\rho_4^3(A).$$\nMoreover, there is the following theorem.\n\nTheorem. Fix some $0<\\varepsilon<1$. Suppose $\\rho_8(A)\\leq K\\rho_4^3(A)$. then there is a subset $H\\subset A$, so that\n$$|H|\\geq \\rho_4(A)/K^c$$ and\n$$E_4(H)\\geq \\frac{|H|^3}{|A|^{\\varepsilon}K^{c'}}.$$\n\nNow, using Holder's inequality one can show\n$$\\rho_{2n}\\leq \\rho_{2n+2}^{\\frac{n-1}{n}}.$$\n\nSuppose $$\\rho_{2n+2}^{\\frac{n-1}{n}}(A)< K\\rho_{2n}(A)$$. Then there is $H\\subset (n-1)A, |H|\\geq \\frac{\\rho_{2n}^{\\frac{1}{n-1}}}{K^c}$ so that $$E_4(H)\\geq \\frac{|H|^3}{|A|^{\\varepsilon}K^{c'}}.$$",
   "clean_statement": "**Recovered conjecture.** For every integer \\(n\\ge2\\) and every \\(0<\\varepsilon<1\\), there exist \\(c=c(n,\\varepsilon)>0\\) and \\(c'=c'(n,\\varepsilon)>0\\) such that the following holds uniformly for every finite abelian group \\(G\\), every nonempty \\(A\\subseteq G\\), and every \\(K>1\\). If\n\\[\n\\rho_{2n+2}(A)^{(n-1)/n}\\le K\\rho_{2n}(A),\n\\]\nthen there is a subset \\(H\\subseteq(n-1)A\\) such that\n\\[\n|H|\\ge \\rho_{2n}(A)^{1/(n-1)}K^{-c},\n\\qquad\nE_4(H)\\ge\n\\frac{|H|^3}{|A|^\\varepsilon K^{c'}}.\n\\]",
   "public_statement": "A conjecture about generalized additive energy\n\nLet $A\\subset G$ and define $$E_{2n}(A) = |\\{(a_1,\\cdots,a_n)\\in A^{2n}: a_1+\\cdots+a_n = a_{n+1}+\\cdots+a_{2n}\\}|.$$\nTrivial bounds on additive energy are $$|A|^n\\leq E_{2n}(A)\\leq |A|^{2n-1}.$$\nDefine $$\\rho_{2n}(A)= \\frac{E_{2n}(A)}{|A|^n}.$$\n\nGiven this, one can compare various $\\rho_k$'s. For example by an applications of Holder's inequality one can get\n$$\\rho_8(A) \\geq \\rho_4^3(A).$$\nMoreover, there is the following theorem.\n\nTheorem. Fix some $0<\\varepsilon<1$. Suppose $\\rho_8(A)\\leq K\\rho_4^3(A)$. then there is a subset $H\\subset A$, so that\n$$|H|\\geq \\rho_4(A)/K^c$$ and\n$$E_4(H)\\geq \\frac{|H|^3}{|A|^{\\varepsilon}K^{c'}}.$$\n\nNow, using Holder's inequality one can show\n$$\\rho_{2n}\\leq \\rho_{2n+2}^{\\frac{n-1}{n}}.$$\n\nSuppose $$\\rho_{2n+2}^{\\frac{n-1}{n}}(A)< K\\rho_{2n}(A)$$. Then there is $H\\subset (n-1)A, |H|\\geq \\frac{\\rho_{2n}^{\\frac{1}{n-1}}}{K^c}$ so that $$E_4(H)\\geq \\frac{|H|^3}{|A|^{\\varepsilon}K^{c'}}.$$",
   "evidence": "The higher-energy conjecture must quantify \\(n\\). A conservative reconstruction is:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0034": {
   "statement_status": "exact",
   "original_statement": "Koml\\'{o}s conjecture\n\nThere is a universal constant $c>0$ s.t. for every $v_1,v_2,...,v_n \\in \\mathbb{B_2}^d$, there exist signs $\\epsilon_1,...,\\epsilon_n\\in \\{-1,1\\}$ s.t. $|| \\sum_{i=1}^n \\epsilon_i v_i||_{\\infty} \\le c$",
   "clean_statement": "Koml\\'{o}s conjecture\n\nThere is a universal constant $c>0$ s.t. for every $v_1,v_2,...,v_n \\in \\mathbb{B_2}^d$, there exist signs $\\epsilon_1,...,\\epsilon_n\\in \\{-1,1\\}$ s.t. $|| \\sum_{i=1}^n \\epsilon_i v_i||_{\\infty} \\le c$",
   "public_statement": "Koml\\'{o}s conjecture\n\nThere is a universal constant $c>0$ s.t. for every $v_1,v_2,...,v_n \\in \\mathbb{B_2}^d$, there exist signs $\\epsilon_1,...,\\epsilon_n\\in \\{-1,1\\}$ s.t. $|| \\sum_{i=1}^n \\epsilon_i v_i||_{\\infty} \\le c$",
   "evidence": "The canonical record is AIM Problem Lists, workshop *Hereditary discrepancy and factorization norms*, section *Open problems*, Problem 1.05, “Komlós conjecture.” Its exact mathematical content is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0035": {
   "statement_status": "exact",
   "original_statement": "Constructive Koml\\'{o}s\n\nFind a polynomial time algorithm to find the signs above.",
   "clean_statement": "Constructive Koml\\'{o}s\n\nFind a polynomial time algorithm to find the signs above.",
   "public_statement": "Constructive Koml\\'{o}s\n\nFind a polynomial time algorithm to find the signs above.",
   "evidence": "The exact extracted record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0036": {
   "statement_status": "exact",
   "original_statement": "Beck-Fiala conjecture and making it constructive\n\nGiven a set system $(V,\\mathcal{S})$ with $V=[n]$, $\\mathcal{S}=S_1,...,S_m$. Each $i\\in V$ occurs in at most $t$ sets from $\\mathcal{S}$. Then there is a coloring to the elements of $V$ achieving a discrepancy of $O(\\sqrt{t})$. Koml\\'{o}s conjecture implies this by treating the incidence vector of each $i\\in V$ as a vector and scaling it down by $\\sqrt{t}$.",
   "clean_statement": "Beck-Fiala conjecture and making it constructive\n\nGiven a set system $(V,\\mathcal{S})$ with $V=[n]$, $\\mathcal{S}=S_1,...,S_m$. Each $i\\in V$ occurs in at most $t$ sets from $\\mathcal{S}$. Then there is a coloring to the elements of $V$ achieving a discrepancy of $O(\\sqrt{t})$. Koml\\'{o}s conjecture implies this by treating the incidence vector of each $i\\in V$ as a vector and scaling it down by $\\sqrt{t}$.",
   "public_statement": "Beck-Fiala conjecture and making it constructive\n\nGiven a set system $(V,\\mathcal{S})$ with $V=[n]$, $\\mathcal{S}=S_1,...,S_m$. Each $i\\in V$ occurs in at most $t$ sets from $\\mathcal{S}$. Then there is a coloring to the elements of $V$ achieving a discrepancy of $O(\\sqrt{t})$. Koml\\'{o}s conjecture implies this by treating the incidence vector of each $i\\in V$ as a vector and scaling it down by $\\sqrt{t}$.",
   "evidence": "The canonical record is problem 1.15, “Beck-Fiala conjecture and making it constructive,” from the AIM workshop *Hereditary discrepancy and factorization norms*. Its mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0037": {
   "statement_status": "exact",
   "original_statement": "Beck-Fiala for random set system\n\nEach element $i\\in [n]$ lies in $t$ randomly selected sets from $\\mathcal{S}$.",
   "clean_statement": "Beck-Fiala for random set system\n\nEach element $i\\in [n]$ lies in $t$ randomly selected sets from $\\mathcal{S}$.",
   "public_statement": "Beck-Fiala for random set system\n\nEach element $i\\in [n]$ lies in $t$ randomly selected sets from $\\mathcal{S}$.",
   "evidence": "The exact corpus record is problem 1.2 from the AIM workshop *Hereditary discrepancy and factorization norms*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0038": {
   "statement_status": "exact",
   "original_statement": "Hypercube set system\n\nGiven a set system where the elements are the vertices of the hypercube $\\{0,1\\}^n$. Corresponding to each vertex $v$, there is a set $S_v$ which is generated by picking a random subset of the neighbours of $v$ in the hypercube. What is the discrepancy of this set system?",
   "clean_statement": "Hypercube set system\n\nGiven a set system where the elements are the vertices of the hypercube $\\{0,1\\}^n$. Corresponding to each vertex $v$, there is a set $S_v$ which is generated by picking a random subset of the neighbours of $v$ in the hypercube. What is the discrepancy of this set system?",
   "public_statement": "Hypercube set system\n\nGiven a set system where the elements are the vertices of the hypercube $\\{0,1\\}^n$. Corresponding to each vertex $v$, there is a set $S_v$ which is generated by picking a random subset of the neighbours of $v$ in the hypercube. What is the discrepancy of this set system?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0039": {
   "statement_status": "exact",
   "original_statement": "Discrepancy of permutations\n\na) discrepancy of k-permutations\n\nFor $k=3$, $[NNN'12]$ showed a lower bound of $\\Omega(\\log n)$ where $n$ is the number of elements in the ground set. For general $k$, we know that the answer lies in $[\\sqrt{k}+\\log n,\\sqrt{k}\\log n]$. An open question is to clsoe this gap.\n\nb) discrepancy of k-random permutations\n\nA lower bound of $\\Omega(\\sqrt{k})$ is known.",
   "clean_statement": "Discrepancy of permutations\n\na) discrepancy of k-permutations\n\nFor $k=3$, $[NNN'12]$ showed a lower bound of $\\Omega(\\log n)$ where $n$ is the number of elements in the ground set. For general $k$, we know that the answer lies in $[\\sqrt{k}+\\log n,\\sqrt{k}\\log n]$. An open question is to clsoe this gap.\n\nb) discrepancy of k-random permutations\n\nA lower bound of $\\Omega(\\sqrt{k})$ is known.",
   "public_statement": "Discrepancy of permutations\n\na) discrepancy of k-permutations\n\nFor $k=3$, $[NNN'12]$ showed a lower bound of $\\Omega(\\log n)$ where $n$ is the number of elements in the ground set. For general $k$, we know that the answer lies in $[\\sqrt{k}+\\log n,\\sqrt{k}\\log n]$. An open question is to clsoe this gap.\n\nb) discrepancy of k-random permutations\n\nA lower bound of $\\Omega(\\sqrt{k})$ is known.",
   "evidence": "The canonical AIM record (workshop *Hereditary discrepancy and factorization norms*, Open Problem 1.3) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0040": {
   "statement_status": "exact",
   "original_statement": "Matrix Spencer\n\nGiven matrices $A_1,...,A_n \\in\\mathbb{R}^{n\\times n}$ satisfying $||A_i||_{op} \\le 1$ where $||.||_{op}$ is the operator norm of matrices. Do there exists signs $\\epsilon_1,...,\\epsilon_n\\in\\{-1,1\\}$ s.t. $||\\sum_{i=1}^n\\epsilon_i A_i || =O(\\sqrt{n})$.",
   "clean_statement": "Matrix Spencer\n\nGiven matrices $A_1,...,A_n \\in\\mathbb{R}^{n\\times n}$ satisfying $||A_i||_{op} \\le 1$ where $||.||_{op}$ is the operator norm of matrices. Do there exists signs $\\epsilon_1,...,\\epsilon_n\\in\\{-1,1\\}$ s.t. $||\\sum_{i=1}^n\\epsilon_i A_i || =O(\\sqrt{n})$.",
   "public_statement": "Matrix Spencer\n\nGiven matrices $A_1,...,A_n \\in\\mathbb{R}^{n\\times n}$ satisfying $||A_i||_{op} \\le 1$ where $||.||_{op}$ is the operator norm of matrices. Do there exists signs $\\epsilon_1,...,\\epsilon_n\\in\\{-1,1\\}$ s.t. $||\\sum_{i=1}^n\\epsilon_i A_i || =O(\\sqrt{n})$.",
   "evidence": "Source metadata: `aim-combinatorics-notes.json`, zero-based record index 39, AIM problem ID `AIM-COMBINATORICS-0040`, workshop *Hereditary discrepancy and factorization norms*, source URL `http://aimpl.org/hereddiscrep/1/`.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0041": {
   "statement_status": "exact",
   "original_statement": "Steinitz conjecture\n\nGiven vectors $v_1,...,v_n\\in\\mathbb{R}^d$ and a norm $X$ satisfying $\\sum_{i=1}^n v_i=0$ and $||v_i||_X\\le 1$ for all $i$. Define $S_n(X,d)=min_{\\pi\\in S^n}max_{k=1,...,n}|| \\sum_{i=1}^k v_{\\pi(i)}||_X$ where $S^n$ is the set of all permutations of $[n]$. Let $S(X,d)=sup_n S_n(X,d)$. Then, $S(X,d) =O(\\sqrt{d})$.",
   "clean_statement": "Steinitz conjecture\n\nGiven vectors $v_1,...,v_n\\in\\mathbb{R}^d$ and a norm $X$ satisfying $\\sum_{i=1}^n v_i=0$ and $||v_i||_X\\le 1$ for all $i$. Define $S_n(X,d)=min_{\\pi\\in S^n}max_{k=1,...,n}|| \\sum_{i=1}^k v_{\\pi(i)}||_X$ where $S^n$ is the set of all permutations of $[n]$. Let $S(X,d)=sup_n S_n(X,d)$. Then, $S(X,d) =O(\\sqrt{d})$.",
   "public_statement": "Steinitz conjecture\n\nGiven vectors $v_1,...,v_n\\in\\mathbb{R}^d$ and a norm $X$ satisfying $\\sum_{i=1}^n v_i=0$ and $||v_i||_X\\le 1$ for all $i$. Define $S_n(X,d)=min_{\\pi\\in S^n}max_{k=1,...,n}|| \\sum_{i=1}^k v_{\\pi(i)}||_X$ where $S^n$ is the set of all permutations of $[n]$. Let $S(X,d)=sup_n S_n(X,d)$. Then, $S(X,d) =O(\\sqrt{d})$.",
   "evidence": "The AIM record is Problem 1.4, “Steinitz conjecture,” from the workshop list *Hereditary discrepancy and factorization norms*. Its displayed text says that, for zero-sum vectors \\(v_1,\\ldots,v_n\\in\\mathbb R^d\\) of norm at most one in a norm \\(X\\), \\[ \\min_{\\pi}\\max_{1\\le k\\le n} \\left\\|\\sum_{i=1}^k v_{\\pi(i)}\\right\\|_X =O(\\sqrt d), \\] after taking a supremum over \\(n\\). As written, this is not a consistent definition or conjecture:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0042": {
   "statement_status": "exact",
   "original_statement": "Vector Balancing\n\nLet $X,Y$ be norms on $\\mathbb{R}^d$. Given vectors $v_1,...,v_n\\in \\mathbb{R}^d$ satisfying $||v_i||_X\\le 1$ for all $i$. We want to find signs $\\epsilon_i \\in \\{-1,1\\}$ to minimize $|| \\sum_{i=1}^n\\epsilon_iv_i ||_Y$. Call this the discrepancy of the vectors. Define $\\alpha_n(X,Y,d)$ as the maximum of this discrepancy over all choice of $n$ vectors satisfying $||v_i||_X\\le 1$ and $\\alpha(X,Y,d) =sup_n \\alpha_n(X,Y,d)$. Banaszczyk proved $\\alpha(X,Y,d) \\ge \\sqrt{d}(\\frac{vol(B_X)}{vol(B_Y)})^{1/d}$ where $B_X,B_Y$ are the unit balls of the corresponding norms.\n\nIs this lower bound also an upper bound for $\\alpha(X,Y,d)$ upto $\\log d$ factors?",
   "clean_statement": "Vector Balancing\n\nLet $X,Y$ be norms on $\\mathbb{R}^d$. Given vectors $v_1,...,v_n\\in \\mathbb{R}^d$ satisfying $||v_i||_X\\le 1$ for all $i$. We want to find signs $\\epsilon_i \\in \\{-1,1\\}$ to minimize $|| \\sum_{i=1}^n\\epsilon_iv_i ||_Y$. Call this the discrepancy of the vectors. Define $\\alpha_n(X,Y,d)$ as the maximum of this discrepancy over all choice of $n$ vectors satisfying $||v_i||_X\\le 1$ and $\\alpha(X,Y,d) =sup_n \\alpha_n(X,Y,d)$. Banaszczyk proved $\\alpha(X,Y,d) \\ge \\sqrt{d}(\\frac{vol(B_X)}{vol(B_Y)})^{1/d}$ where $B_X,B_Y$ are the unit balls of the corresponding norms.\n\nIs this lower bound also an upper bound for $\\alpha(X,Y,d)$ upto $\\log d$ factors?",
   "public_statement": "Vector Balancing\n\nLet $X,Y$ be norms on $\\mathbb{R}^d$. Given vectors $v_1,...,v_n\\in \\mathbb{R}^d$ satisfying $||v_i||_X\\le 1$ for all $i$. We want to find signs $\\epsilon_i \\in \\{-1,1\\}$ to minimize $|| \\sum_{i=1}^n\\epsilon_iv_i ||_Y$. Call this the discrepancy of the vectors. Define $\\alpha_n(X,Y,d)$ as the maximum of this discrepancy over all choice of $n$ vectors satisfying $||v_i||_X\\le 1$ and $\\alpha(X,Y,d) =sup_n \\alpha_n(X,Y,d)$. Banaszczyk proved $\\alpha(X,Y,d) \\ge \\sqrt{d}(\\frac{vol(B_X)}{vol(B_Y)})^{1/d}$ where $B_X,B_Y$ are the unit balls of the corresponding norms.\n\nIs this lower bound also an upper bound for $\\alpha(X,Y,d)$ upto $\\log d$ factors?",
   "evidence": "Let $X,Y$ be norms on $\\mathbb R^d$, with unit balls $C=B_X$ and $K=B_Y$. For $v_1,\\ldots,v_N\\in C$, the source defines \\[ \\operatorname{disc}_Y(v_1,\\ldots,v_N) =\\min_{\\varepsilon\\in\\{-1,1\\}^N} \\left\\|\\sum_{i=1}^N\\varepsilon_i v_i\\right\\|_Y, \\] then takes the maximum over all such $N$-tuples to obtain $\\alpha_N(X,Y,d)$ and finally $\\alpha(X,Y,d)=\\sup_N\\alpha_N(X,Y,d)$.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0043": {
   "statement_status": "exact",
   "original_statement": "Tusnady's problem\n\nLet $d_n$ be the discrepancy of $n$ points in $\\mathbb{R}^2$ wrt all alix-aligned rectangles i.e. the set system comprises of all axis-aligned rectangles in $\\mathbb{R}^2$ as sets, maximised over all choices of $n$ points. Tusnady's problem then asks for the exact value of $d_n$.\n\nWe can also ask for the same question in $\\mathbb{R}^d$.",
   "clean_statement": "Tusnady's problem\n\nLet $d_n$ be the discrepancy of $n$ points in $\\mathbb{R}^2$ wrt all alix-aligned rectangles i.e. the set system comprises of all axis-aligned rectangles in $\\mathbb{R}^2$ as sets, maximised over all choices of $n$ points. Tusnady's problem then asks for the exact value of $d_n$.\n\nWe can also ask for the same question in $\\mathbb{R}^d$.",
   "public_statement": "Tusnady's problem\n\nLet $d_n$ be the discrepancy of $n$ points in $\\mathbb{R}^2$ wrt all alix-aligned rectangles i.e. the set system comprises of all axis-aligned rectangles in $\\mathbb{R}^2$ as sets, maximised over all choices of $n$ points. Tusnady's problem then asks for the exact value of $d_n$.\n\nWe can also ask for the same question in $\\mathbb{R}^d$.",
   "evidence": "Source metadata: `aim-combinatorics-notes.json`, zero-based index 42, problem ID `AIM-COMBINATORICS-0043`, workshop *Hereditary discrepancy and factorization norms*, problem number 1.65, source URL `http://aimpl.org/hereddiscrep/1/`.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0044": {
   "statement_status": "exact",
   "original_statement": "Beating LLL\n\nConsider a matrix where both columns are rows are t-sparse. LLL gives a discrepancy bound of $O(\\sqrt{t\\log t})$ here.\nIf we assume the stronger property that every pair of rows intersect in at most one element, can we get a $O(\\sqrt{t})$?",
   "clean_statement": "Beating LLL\n\nConsider a matrix where both columns are rows are t-sparse. LLL gives a discrepancy bound of $O(\\sqrt{t\\log t})$ here.\nIf we assume the stronger property that every pair of rows intersect in at most one element, can we get a $O(\\sqrt{t})$?",
   "public_statement": "Beating LLL\n\nConsider a matrix where both columns are rows are t-sparse. LLL gives a discrepancy bound of $O(\\sqrt{t\\log t})$ here.\nIf we assume the stronger property that every pair of rows intersect in at most one element, can we get a $O(\\sqrt{t})$?",
   "evidence": "The AIM entry is Problem 1.7, “Beating LLL,” from the workshop *Hereditary discrepancy and factorization norms*. The stored text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0045": {
   "statement_status": "exact",
   "original_statement": "Hardness of Komlos\n\nCan we prove some hardness statement about finding a coloring achieving Komlos guarantee or the Banaszczyk discrepancy guarantee?",
   "clean_statement": "Hardness of Komlos\n\nCan we prove some hardness statement about finding a coloring achieving Komlos guarantee or the Banaszczyk discrepancy guarantee?",
   "public_statement": "Hardness of Komlos\n\nCan we prove some hardness statement about finding a coloring achieving Komlos guarantee or the Banaszczyk discrepancy guarantee?",
   "evidence": "The canonical record is problem 1.75, “Hardness of Komlos,” from the AIM workshop *Hereditary discrepancy and factorization norms*. Its complete question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0046": {
   "statement_status": "exact",
   "original_statement": "Reverse Banaszczyk-type problems\n\nSay that a convex body $K$ has property B if for any set of vectors of length no more than $c$ (some small constant), there exists a signing of them s.t. this signed sum lies inside $K$.\n\nWhat is the convex body with the smallest gaussian volume which has property $B$?",
   "clean_statement": "Reverse Banaszczyk-type problems\n\nSay that a convex body $K$ has property B if for any set of vectors of length no more than $c$ (some small constant), there exists a signing of them s.t. this signed sum lies inside $K$.\n\nWhat is the convex body with the smallest gaussian volume which has property $B$?",
   "public_statement": "Reverse Banaszczyk-type problems\n\nSay that a convex body $K$ has property B if for any set of vectors of length no more than $c$ (some small constant), there exists a signing of them s.t. this signed sum lies inside $K$.\n\nWhat is the convex body with the smallest gaussian volume which has property $B$?",
   "evidence": "The canonical AIM record is Problem 1.8, “Reverse Banaszczyk-type problems,” from the workshop *Hereditary discrepancy and factorization norms*. Its stored text says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0047": {
   "statement_status": "exact",
   "original_statement": "Polysize list of colorings\n\nIs there a list of colorings of polynomial size such that for any subset of the columns, one of these colorings gives it a discrepancy of O(h), where $h$ is the hereditary discrepancy?",
   "clean_statement": "Polysize list of colorings\n\nIs there a list of colorings of polynomial size such that for any subset of the columns, one of these colorings gives it a discrepancy of O(h), where $h$ is the hereditary discrepancy?",
   "public_statement": "Polysize list of colorings\n\nIs there a list of colorings of polynomial size such that for any subset of the columns, one of these colorings gives it a discrepancy of O(h), where $h$ is the hereditary discrepancy?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0048": {
   "statement_status": "exact",
   "original_statement": "Tight paths versus cliques\n\nDhruv Mubayi\n\nLet $P_s$ denote a tight path of length $s$ (meaning with $s$ edges),\ni.e.,\n\\[\nP_s = \\{123, 234,345,\\dots, s\\ s+1\\ s+2\\}.\n\\]\n\nE.g., $P_3 = \\{123,234,345\\}$\n\nKnown: $\\Omega(t^2/\\log t) \\le r_3(P_3, K_t) \\le O(t^2)$\n\nProblem: Improve the bounds. Is it $o(t^2)$? Perhaps the lower bound is the truth.\n\nRelated results:\n\n(Phelps--R\\\"odl) $r_3(P_2, K_t) = \\Theta(t^2/\\log t)$. This is the\nsame as giving the minimum independence number of a linear triple system on $n$ points.\n\n(Mubayi--Cooper) $r_3(P_s,K_t) = \\Theta(t^2)$ for fixed $s \\geq 4$\n\nHere is the construction giving the lower bound for $r_3(P_s,K_4)$ for\n$s \\geq 4$. Consider the 3-uniform $t^2$-vertex hypergraph with the vertex\nset being points of a $t \\times t$ grid, i.e., $V = [t] \\times [t]$, and\nthree points form an edge if and only if they form an L-shape, i.e.,\n\\[\n\\{(x_1,y_2),(x_1,y_1),(x_2,y_1)\\} \\quad \\text{for all } 1 \\le x_1 < x_2 \\le t\n\\text{ and } 1 \\le y_1 < y_2 \\le t.\n\\]\nThis hypergraph has independence number $2t -1$ and is $P_4$-free.\n\nComments:\n\nConlon: for what $H$ is $r_3(H, K_t)$ polynomial in $t$? E.g., is it\nso when $H$ is the Fano plane?\n\nMubayi: (Samotij--Mubayi) If a $3$-uniform hypergraph does not contain\na copy of the Fano plane, then one can find a subset $S$ of vertices\nof polynomial size (i.e., $\\geq |V|^\\epsilon$) such that there is a\npartition $S = S_1 \\cup S_2$ with $|S_1| = |S_2|$ where no triple has\nat least one vertex in both $S_1$ and $S_2$.\n\nExtensions to hypergraphs:\n\nFix uniformity $k$.\nLet $P_s^{(k)}$ denote the tight $k$-uniform path of length $s$. So previously $P_s$ meant $P_s^{(3)}$.\n\nKnown: $\\Omega(t^{k-1} / \\log t) \\le r_k(P_s^{(k)},K_t^{(k)}) \\le O(t^{k-1})$.",
   "clean_statement": "Tight paths versus cliques\n\nDhruv Mubayi\n\nLet $P_s$ denote a tight path of length $s$ (meaning with $s$ edges),\ni.e.,\n\\[\nP_s = \\{123, 234,345,\\dots, s\\ s+1\\ s+2\\}.\n\\]\n\nE.g., $P_3 = \\{123,234,345\\}$\n\nKnown: $\\Omega(t^2/\\log t) \\le r_3(P_3, K_t) \\le O(t^2)$\n\nProblem: Improve the bounds. Is it $o(t^2)$? Perhaps the lower bound is the truth.\n\nRelated results:\n\n(Phelps--R\\\"odl) $r_3(P_2, K_t) = \\Theta(t^2/\\log t)$. This is the\nsame as giving the minimum independence number of a linear triple system on $n$ points.\n\n(Mubayi--Cooper) $r_3(P_s,K_t) = \\Theta(t^2)$ for fixed $s \\geq 4$\n\nHere is the construction giving the lower bound for $r_3(P_s,K_4)$ for\n$s \\geq 4$. Consider the 3-uniform $t^2$-vertex hypergraph with the vertex\nset being points of a $t \\times t$ grid, i.e., $V = [t] \\times [t]$, and\nthree points form an edge if and only if they form an L-shape, i.e.,\n\\[\n\\{(x_1,y_2),(x_1,y_1),(x_2,y_1)\\} \\quad \\text{for all } 1 \\le x_1 < x_2 \\le t\n\\text{ and } 1 \\le y_1 < y_2 \\le t.\n\\]\nThis hypergraph has independence number $2t -1$ and is $P_4$-free.\n\nComments:\n\nConlon: for what $H$ is $r_3(H, K_t)$ polynomial in $t$? E.g., is it\nso when $H$ is the Fano plane?\n\nMubayi: (Samotij--Mubayi) If a $3$-uniform hypergraph does not contain\na copy of the Fano plane, then one can find a subset $S$ of vertices\nof polynomial size (i.e., $\\geq |V|^\\epsilon$) such that there is a\npartition $S = S_1 \\cup S_2$ with $|S_1| = |S_2|$ where no triple has\nat least one vertex in both $S_1$ and $S_2$.\n\nExtensions to hypergraphs:\n\nFix uniformity $k$.\nLet $P_s^{(k)}$ denote the tight $k$-uniform path of length $s$. So previously $P_s$ meant $P_s^{(3)}$.\n\nKnown: $\\Omega(t^{k-1} / \\log t) \\le r_k(P_s^{(k)},K_t^{(k)}) \\le O(t^{k-1})$.",
   "public_statement": "Tight paths versus cliques\n\nDhruv Mubayi\n\nLet $P_s$ denote a tight path of length $s$ (meaning with $s$ edges),\ni.e.,\n\\[\nP_s = \\{123, 234,345,\\dots, s\\ s+1\\ s+2\\}.\n\\]\n\nE.g., $P_3 = \\{123,234,345\\}$\n\nKnown: $\\Omega(t^2/\\log t) \\le r_3(P_3, K_t) \\le O(t^2)$\n\nProblem: Improve the bounds. Is it $o(t^2)$? Perhaps the lower bound is the truth.\n\nRelated results:\n\n(Phelps--R\\\"odl) $r_3(P_2, K_t) = \\Theta(t^2/\\log t)$. This is the\nsame as giving the minimum independence number of a linear triple system on $n$ points.\n\n(Mubayi--Cooper) $r_3(P_s,K_t) = \\Theta(t^2)$ for fixed $s \\geq 4$\n\nHere is the construction giving the lower bound for $r_3(P_s,K_4)$ for\n$s \\geq 4$. Consider the 3-uniform $t^2$-vertex hypergraph with the vertex\nset being points of a $t \\times t$ grid, i.e., $V = [t] \\times [t]$, and\nthree points form an edge if and only if they form an L-shape, i.e.,\n\\[\n\\{(x_1,y_2),(x_1,y_1),(x_2,y_1)\\} \\quad \\text{for all } 1 \\le x_1 < x_2 \\le t\n\\text{ and } 1 \\le y_1 < y_2 \\le t.\n\\]\nThis hypergraph has independence number $2t -1$ and is $P_4$-free.\n\nComments:\n\nConlon: for what $H$ is $r_3(H, K_t)$ polynomial in $t$? E.g., is it\nso when $H$ is the Fano plane?\n\nMubayi: (Samotij--Mubayi) If a $3$-uniform hypergraph does not contain\na copy of the Fano plane, then one can find a subset $S$ of vertices\nof polynomial size (i.e., $\\geq |V|^\\epsilon$) such that there is a\npartition $S = S_1 \\cup S_2$ with $|S_1| = |S_2|$ where no triple has\nat least one vertex in both $S_1$ and $S_2$.\n\nExtensions to hypergraphs:\n\nFix uniformity $k$.\nLet $P_s^{(k)}$ denote the tight $k$-uniform path of length $s$. So previously $P_s$ meant $P_s^{(3)}$.\n\nKnown: $\\Omega(t^{k-1} / \\log t) \\le r_k(P_s^{(k)},K_t^{(k)}) \\le O(t^{k-1})$.",
   "evidence": "The canonical AIM record (Graph Ramsey theory workshop, section “Tight paths versus cliques,” problem 1, attributed to Dhruv Mubayi) uses **edge count** for the path parameter:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0049": {
   "statement_status": "exact",
   "original_statement": "Question: Is it true that for fixed $k$, and $s$ sufficiently large (in terms of $k$), one has $r_k(P_s^{(k)},K_t^{(k)}) = \\Theta(t^{k-1})$.",
   "clean_statement": "Question: Is it true that for fixed $k$, and $s$ sufficiently large (in terms of $k$), one has $r_k(P_s^{(k)},K_t^{(k)}) = \\Theta(t^{k-1})$.",
   "public_statement": "Question: Is it true that for fixed $k$, and $s$ sufficiently large (in terms of $k$), one has $r_k(P_s^{(k)},K_t^{(k)}) = \\Theta(t^{k-1})$.",
   "evidence": "The canonical record (AIM Graph Ramsey theory workshop, section “Tight paths versus cliques,” question 2) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0050": {
   "statement_status": "exact",
   "original_statement": "Paths in 3-colorings of 6-chromatic graphs\n\nAndr\\'as Gy\\'arf\\'as\n\n$P_k$ denotes a path of length $k$, i.e., with $k$ edges\n\nFact: $r(P_3,P_3,P_3) = 6$\n\nQuestion: Is it true that in any 3-edge-coloring of a graph\nof chromatic number at least 6, there is a monochromatic path of\nlength 3?\n\nKnown: (Garrison) For $k \\ne 3$, if we $k$-edge-color an\n$\\ell$-chromatic graph and $\\ell \\ge r(\\underbrace{P_3,\\dots,P_3}_{k\n \\text{ times}})$, then we get a monochromatic $P_3$. (That is, $k=3$\nis the only unknown case.)\n\nComment: For $k\\ne 3$, this follows from\n$\\operatorname{ex}(n,P_3)\\le n$ (look at the majority color).",
   "clean_statement": "Paths in 3-colorings of 6-chromatic graphs\n\nAndr\\'as Gy\\'arf\\'as\n\n$P_k$ denotes a path of length $k$, i.e., with $k$ edges\n\nFact: $r(P_3,P_3,P_3) = 6$\n\nQuestion: Is it true that in any 3-edge-coloring of a graph\nof chromatic number at least 6, there is a monochromatic path of\nlength 3?\n\nKnown: (Garrison) For $k \\ne 3$, if we $k$-edge-color an\n$\\ell$-chromatic graph and $\\ell \\ge r(\\underbrace{P_3,\\dots,P_3}_{k\n \\text{ times}})$, then we get a monochromatic $P_3$. (That is, $k=3$\nis the only unknown case.)\n\nComment: For $k\\ne 3$, this follows from\n$\\operatorname{ex}(n,P_3)\\le n$ (look at the majority color).",
   "public_statement": "Paths in 3-colorings of 6-chromatic graphs\n\nAndr\\'as Gy\\'arf\\'as\n\n$P_k$ denotes a path of length $k$, i.e., with $k$ edges\n\nFact: $r(P_3,P_3,P_3) = 6$\n\nQuestion: Is it true that in any 3-edge-coloring of a graph\nof chromatic number at least 6, there is a monochromatic path of\nlength 3?\n\nKnown: (Garrison) For $k \\ne 3$, if we $k$-edge-color an\n$\\ell$-chromatic graph and $\\ell \\ge r(\\underbrace{P_3,\\dots,P_3}_{k\n \\text{ times}})$, then we get a monochromatic $P_3$. (That is, $k=3$\nis the only unknown case.)\n\nComment: For $k\\ne 3$, this follows from\n$\\operatorname{ex}(n,P_3)\\le n$ (look at the majority color).",
   "evidence": "The source record states that \\(P_k\\) means a path of **length** \\(k\\), hence with \\(k\\) edges. It records \\[ r(P_3,P_3,P_3)=6 \\] and asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0051": {
   "statement_status": "exact",
   "original_statement": "4-color Ramsey number of triangles\n\nFan Chung\n\nProblem: Find $r(3,3,3,3)$. Is it 51?\n\nKnown: $51 \\le r(3,3,3,3) \\le 62$. Lower bound due to (Chung 1973),\nand upper bound due to (Fettes--Kramer--Radziszowski 2004).",
   "clean_statement": "4-color Ramsey number of triangles\n\nFan Chung\n\nProblem: Find $r(3,3,3,3)$. Is it 51?\n\nKnown: $51 \\le r(3,3,3,3) \\le 62$. Lower bound due to (Chung 1973),\nand upper bound due to (Fettes--Kramer--Radziszowski 2004).",
   "public_statement": "4-color Ramsey number of triangles\n\nFan Chung\n\nProblem: Find $r(3,3,3,3)$. Is it 51?\n\nKnown: $51 \\le r(3,3,3,3) \\le 62$. Lower bound due to (Chung 1973),\nand upper bound due to (Fettes--Kramer--Radziszowski 2004).",
   "evidence": "The canonical AIM record (Graph Ramsey theory workshop, section “4-color Ramsey number of triangles,” problem 4, attributed to Fan Chung) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0052": {
   "statement_status": "exact",
   "original_statement": "Hypergraph size Ramsey numbers\n\nAndrzej Dudek\n\nThe size Ramsey number $\\hat r(n,n)$ is the minimum number $m$\nfor which there exists a graph with $m$ edges such that every\n2-edge-coloring of it produces a monochromatic $K_n$. Similarly define\n$\\hat r_3(n,n)$ for the corresponding 3-uniform size Ramsey number.\n\nFor graphs, $\\hat r (n,n) = \\binom{r(n,n)}{2}$.\n\nProblem: (Dudek--R\\\"odl) Is $\\hat r_3(n,n) = \\binom{r_3(n,n)}{3}$?\n\nIn particular,",
   "clean_statement": "Hypergraph size Ramsey numbers\n\nAndrzej Dudek\n\nThe size Ramsey number $\\hat r(n,n)$ is the minimum number $m$\nfor which there exists a graph with $m$ edges such that every\n2-edge-coloring of it produces a monochromatic $K_n$. Similarly define\n$\\hat r_3(n,n)$ for the corresponding 3-uniform size Ramsey number.\n\nFor graphs, $\\hat r (n,n) = \\binom{r(n,n)}{2}$.\n\nProblem: (Dudek--R\\\"odl) Is $\\hat r_3(n,n) = \\binom{r_3(n,n)}{3}$?\n\nIn particular,",
   "public_statement": "Hypergraph size Ramsey numbers\n\nAndrzej Dudek\n\nThe size Ramsey number $\\hat r(n,n)$ is the minimum number $m$\nfor which there exists a graph with $m$ edges such that every\n2-edge-coloring of it produces a monochromatic $K_n$. Similarly define\n$\\hat r_3(n,n)$ for the corresponding 3-uniform size Ramsey number.\n\nFor graphs, $\\hat r (n,n) = \\binom{r(n,n)}{2}$.\n\nProblem: (Dudek--R\\\"odl) Is $\\hat r_3(n,n) = \\binom{r_3(n,n)}{3}$?\n\nIn particular,",
   "evidence": "The canonical AIM Graph Ramsey theory workshop record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0053": {
   "statement_status": "exact",
   "original_statement": "Problem: (Dudek--R\\\"odl) Let $N = r_3(n,n)$. Is it true that every\n2-edge-coloring of $K_N^{(3)}-e$ contains a monochromatic $K_n^{(3)}$? Here $K_N^{(3)} - e$ denotes $K_N^{(3)}$ with one edge deleted.\n\nLower bound: (Dudek--La Fleur--Mubayi--R\\\"odl) $\\hat r_3(n,n) \\ge\n\\text{poly}(n) r_3(n,n)^2$\n\nComment: $r_3(4,4) = 13$ ($\\ge$ Seymour; $\\le$\nRadziszowski). So does it work for $n=4$? This is a matter of computational verification.",
   "clean_statement": "Problem: (Dudek--R\\\"odl) Let $N = r_3(n,n)$. Is it true that every\n2-edge-coloring of $K_N^{(3)}-e$ contains a monochromatic $K_n^{(3)}$? Here $K_N^{(3)} - e$ denotes $K_N^{(3)}$ with one edge deleted.\n\nLower bound: (Dudek--La Fleur--Mubayi--R\\\"odl) $\\hat r_3(n,n) \\ge\n\\text{poly}(n) r_3(n,n)^2$\n\nComment: $r_3(4,4) = 13$ ($\\ge$ Seymour; $\\le$\nRadziszowski). So does it work for $n=4$? This is a matter of computational verification.",
   "public_statement": "Problem: (Dudek--R\\\"odl) Let $N = r_3(n,n)$. Is it true that every\n2-edge-coloring of $K_N^{(3)}-e$ contains a monochromatic $K_n^{(3)}$? Here $K_N^{(3)} - e$ denotes $K_N^{(3)}$ with one edge deleted.\n\nLower bound: (Dudek--La Fleur--Mubayi--R\\\"odl) $\\hat r_3(n,n) \\ge\n\\text{poly}(n) r_3(n,n)^2$\n\nComment: $r_3(4,4) = 13$ ($\\ge$ Seymour; $\\le$\nRadziszowski). So does it work for $n=4$? This is a matter of computational verification.",
   "evidence": "The source record (AIM problem list, workshop *Graph Ramsey theory*, section *Hypergraph size Ramsey numbers*) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0054": {
   "statement_status": "exact",
   "original_statement": "Online Ramsey numbers\n\nJacob Fox\n\nThe online Ramsey number $\\tilde r(s,n)$ is defined via a game as\nfollows. Initially there are infinitely many vertices and no\nedges. Each turn, the builder adds an edge to the graph, and\nthe painter colors the edge blue or red. We define $\\tilde\nr(s,n)$ to be the minimum $m$ such that the builder can force a win in\n$m$ moves, i.e., force a red $K_s$ or a blue $K_n$.\n\nNote: $\\tilde r(s,n) \\le \\hat r (s,n) = \\binom{r(s,n)}{2}$.\n\n(Conlon) For infinitely many $n$, $\\tilde r(n,n) \\le (0.99)^n\n\\binom{r(n,n)}{2}$. The constant $0.99$ can probably be made much\nsmaller.\n\nNote that if $\\tilde r(n,n) \\ge (1.999)^n$ for all $n$, then $r(n,n)\n\\ge (\\sqrt 2 + \\delta)^n$ for some $\\delta > 0$. This is a possible\nstrategy for improving lower bounds on diagonal Ramsey numbers. More\nmodestly, we can ask:\n\nProblem: Show that $\\tilde r(n,n) \\ge (\\sqrt 2 + \\epsilon)^n$\nfor some $\\epsilon > 0$.\n\nRelated: (Conlon--Fox--Grinshpun) $\\tilde r(3,n) = \\tilde \\Theta(n^3)$, where\n$\\tilde\\Theta$ means up to a $\\text{poly\\,log}(n)$ factor. The lower bound is\nproved via a lopsided local lemma.\n\n\\section{Erd\\H{o}s--Hajnal for tournaments}\n\nMaria Chudnovsky\n\nLet $T^*$ denote the following tournament on $6$ vertices labeled by\n$\\{1, 2, \\dots, 6\\}$: orient $6 \\to 1, 6 \\to 3, 5 \\to 2, 4 \\to 1$, and\norient all remaining pairs $i \\to j$ where $i < j$.",
   "clean_statement": "Online Ramsey numbers\n\nJacob Fox\n\nThe online Ramsey number $\\tilde r(s,n)$ is defined via a game as\nfollows. Initially there are infinitely many vertices and no\nedges. Each turn, the builder adds an edge to the graph, and\nthe painter colors the edge blue or red. We define $\\tilde\nr(s,n)$ to be the minimum $m$ such that the builder can force a win in\n$m$ moves, i.e., force a red $K_s$ or a blue $K_n$.\n\nNote: $\\tilde r(s,n) \\le \\hat r (s,n) = \\binom{r(s,n)}{2}$.\n\n(Conlon) For infinitely many $n$, $\\tilde r(n,n) \\le (0.99)^n\n\\binom{r(n,n)}{2}$. The constant $0.99$ can probably be made much\nsmaller.\n\nNote that if $\\tilde r(n,n) \\ge (1.999)^n$ for all $n$, then $r(n,n)\n\\ge (\\sqrt 2 + \\delta)^n$ for some $\\delta > 0$. This is a possible\nstrategy for improving lower bounds on diagonal Ramsey numbers. More\nmodestly, we can ask:\n\nProblem: Show that $\\tilde r(n,n) \\ge (\\sqrt 2 + \\epsilon)^n$\nfor some $\\epsilon > 0$.\n\nRelated: (Conlon--Fox--Grinshpun) $\\tilde r(3,n) = \\tilde \\Theta(n^3)$, where\n$\\tilde\\Theta$ means up to a $\\text{poly\\,log}(n)$ factor. The lower bound is\nproved via a lopsided local lemma.\n\n\\section{Erd\\H{o}s--Hajnal for tournaments}\n\nMaria Chudnovsky\n\nLet $T^*$ denote the following tournament on $6$ vertices labeled by\n$\\{1, 2, \\dots, 6\\}$: orient $6 \\to 1, 6 \\to 3, 5 \\to 2, 4 \\to 1$, and\norient all remaining pairs $i \\to j$ where $i < j$.",
   "public_statement": "Online Ramsey numbers\n\nJacob Fox\n\nThe online Ramsey number $\\tilde r(s,n)$ is defined via a game as\nfollows. Initially there are infinitely many vertices and no\nedges. Each turn, the builder adds an edge to the graph, and\nthe painter colors the edge blue or red. We define $\\tilde\nr(s,n)$ to be the minimum $m$ such that the builder can force a win in\n$m$ moves, i.e., force a red $K_s$ or a blue $K_n$.\n\nNote: $\\tilde r(s,n) \\le \\hat r (s,n) = \\binom{r(s,n)}{2}$.\n\n(Conlon) For infinitely many $n$, $\\tilde r(n,n) \\le (0.99)^n\n\\binom{r(n,n)}{2}$. The constant $0.99$ can probably be made much\nsmaller.\n\nNote that if $\\tilde r(n,n) \\ge (1.999)^n$ for all $n$, then $r(n,n)\n\\ge (\\sqrt 2 + \\delta)^n$ for some $\\delta > 0$. This is a possible\nstrategy for improving lower bounds on diagonal Ramsey numbers. More\nmodestly, we can ask:\n\nProblem: Show that $\\tilde r(n,n) \\ge (\\sqrt 2 + \\epsilon)^n$\nfor some $\\epsilon > 0$.\n\nRelated: (Conlon--Fox--Grinshpun) $\\tilde r(3,n) = \\tilde \\Theta(n^3)$, where\n$\\tilde\\Theta$ means up to a $\\text{poly\\,log}(n)$ factor. The lower bound is\nproved via a lopsided local lemma.\n\n\\section{Erd\\H{o}s--Hajnal for tournaments}\n\nMaria Chudnovsky\n\nLet $T^*$ denote the following tournament on $6$ vertices labeled by\n$\\{1, 2, \\dots, 6\\}$: orient $6 \\to 1, 6 \\to 3, 5 \\to 2, 4 \\to 1$, and\norient all remaining pairs $i \\to j$ where $i < j$.",
   "evidence": "The online-Ramsey part of the verified AIM record defines the game on infinitely many initially isolated vertices: in each move Builder exposes one edge and Painter immediately colors it red or blue. It then asks",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0055": {
   "statement_status": "exact",
   "original_statement": "Question: Does there exist $\\epsilon > 0$ such that if $T_n$\nis an $n$-vertex tournament not containing $T^*$ as a subtournament,\nthen $T_n$ contains a transitive tournament with $\\ge n^\\epsilon$\nvertices?\n\nRemark: (Berger--Choromanski--Chudnovsky) This is the smallest\ntournament for which this conclusion is not known.\n\nRemark: Such a statement for all tournaments is equivalent to\nthe Erd\\H os--Hajnal conjecture.",
   "clean_statement": "Question: Does there exist $\\epsilon > 0$ such that if $T_n$\nis an $n$-vertex tournament not containing $T^*$ as a subtournament,\nthen $T_n$ contains a transitive tournament with $\\ge n^\\epsilon$\nvertices?\n\nRemark: (Berger--Choromanski--Chudnovsky) This is the smallest\ntournament for which this conclusion is not known.\n\nRemark: Such a statement for all tournaments is equivalent to\nthe Erd\\H os--Hajnal conjecture.",
   "public_statement": "Question: Does there exist $\\epsilon > 0$ such that if $T_n$\nis an $n$-vertex tournament not containing $T^*$ as a subtournament,\nthen $T_n$ contains a transitive tournament with $\\ge n^\\epsilon$\nvertices?\n\nRemark: (Berger--Choromanski--Chudnovsky) This is the smallest\ntournament for which this conclusion is not known.\n\nRemark: Such a statement for all tournaments is equivalent to\nthe Erd\\H os--Hajnal conjecture.",
   "evidence": "The canonical record asks whether there is an \\(\\epsilon>0\\) such that every \\(n\\)-vertex tournament with no subtournament isomorphic to \\(T^*\\) contains a transitive subtournament on at least \\(n^\\epsilon\\) vertices. Its historical remark says that Berger--Choromanski--Chudnovsky had reduced the six-vertex case to this one tournament.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0056": {
   "statement_status": "exact",
   "original_statement": "Local trees of triangles\n\nVojt\\v{e}ch R\\\"odl\n\nA tree of triangles is any graph built recursively by starting with a triangle, and then adding two edges at a time that forms a single new triangle with an existing edge.\n\nKnown: Given any tree $T$ of triangles, there exists $k$ such\nthat if $G \\xrightarrow{k} K_3$ (the $k$ above the arrow means with $k$\ncolors), then $G$ contains $T$.\n\nQuestion: For every $\\ell$, does there exist a graph $G\n\\xrightarrow{2} K_3$ which on every $\\ell$ vertices induces a subgraph\nof a tree of triangles.\n\nThe property in the conclusion of the question will be referred to as\nlocally a tree of triangles.",
   "clean_statement": "Local trees of triangles\n\nVojt\\v{e}ch R\\\"odl\n\nA tree of triangles is any graph built recursively by starting with a triangle, and then adding two edges at a time that forms a single new triangle with an existing edge.\n\nKnown: Given any tree $T$ of triangles, there exists $k$ such\nthat if $G \\xrightarrow{k} K_3$ (the $k$ above the arrow means with $k$\ncolors), then $G$ contains $T$.\n\nQuestion: For every $\\ell$, does there exist a graph $G\n\\xrightarrow{2} K_3$ which on every $\\ell$ vertices induces a subgraph\nof a tree of triangles.\n\nThe property in the conclusion of the question will be referred to as\nlocally a tree of triangles.",
   "public_statement": "Local trees of triangles\n\nVojt\\v{e}ch R\\\"odl\n\nA tree of triangles is any graph built recursively by starting with a triangle, and then adding two edges at a time that forms a single new triangle with an existing edge.\n\nKnown: Given any tree $T$ of triangles, there exists $k$ such\nthat if $G \\xrightarrow{k} K_3$ (the $k$ above the arrow means with $k$\ncolors), then $G$ contains $T$.\n\nQuestion: For every $\\ell$, does there exist a graph $G\n\\xrightarrow{2} K_3$ which on every $\\ell$ vertices induces a subgraph\nof a tree of triangles.\n\nThe property in the conclusion of the question will be referred to as\nlocally a tree of triangles.",
   "evidence": "The record comes from the January 2015 AIM workshop *Graph Ramsey theory* and attributes the question to Vojtěch Rödl. In normalized notation it asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0057": {
   "statement_status": "exact",
   "original_statement": "Question Does there exist any finite graph where every edge\nis in at least two triangles and the graph is locally a tree of triangles? (Solved by J. Verstraete)\n\n(Ne\\v{s}et\\v{r}il--R\\\"odl) Partial results",
   "clean_statement": "Question Does there exist any finite graph where every edge\nis in at least two triangles and the graph is locally a tree of triangles? (Solved by J. Verstraete)\n\n(Ne\\v{s}et\\v{r}il--R\\\"odl) Partial results",
   "public_statement": "Question Does there exist any finite graph where every edge\nis in at least two triangles and the graph is locally a tree of triangles? (Solved by J. Verstraete)\n\n(Ne\\v{s}et\\v{r}il--R\\\"odl) Partial results",
   "evidence": "The extracted source record reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 56,
   "attempt": 3
  },
  "AIM-COMBINATORICS-0058": {
   "statement_status": "exact",
   "original_statement": "Ramsey minimal graphs\n\nTibor Szab\\'o\n\nLet $G \\xrightarrow{r} K_k$ be minimal, i.e., $G - e\n\\not\\xrightarrow{r} K_3$ for any $e \\in E(G)$.\nLet $\\mathcal{M}_r(k)$ denote the family of such graphs $G$. Let\n\\[\ns_r(k) = \\min\\{\\delta(G) : G \\in \\mathcal{M}_r(k)\\}\n\\]\nwhere $\\delta(G)$ is the minimum degree of $G$.\n\nConjecture: (Fox--Grinshpun--Liebenau--Person--Szab\\'o)\n$s_r(k) \\le s_r(k+1)$ for all $k$.\n\nResults: $s_2(k) = (k-1)^2$ (Erd\\H os--Burr--Lov\\'asz 1976)\n\n$\\Omega(r^2 \\log r) \\le s_r(3) \\le O(r^2(\\log r)^2 )$\n\n$\\Omega\\left(\\frac{r^2 \\log r }{ \\log\\log r}\\right) \\le s_r(k) \\le\nO\\left(r^2 (\\log r)^{8(k-1)^2}\\right)$ for $k \\ge 4$",
   "clean_statement": "Ramsey minimal graphs\n\nTibor Szab\\'o\n\nLet $G \\xrightarrow{r} K_k$ be minimal, i.e., $G - e\n\\not\\xrightarrow{r} K_3$ for any $e \\in E(G)$.\nLet $\\mathcal{M}_r(k)$ denote the family of such graphs $G$. Let\n\\[\ns_r(k) = \\min\\{\\delta(G) : G \\in \\mathcal{M}_r(k)\\}\n\\]\nwhere $\\delta(G)$ is the minimum degree of $G$.\n\nConjecture: (Fox--Grinshpun--Liebenau--Person--Szab\\'o)\n$s_r(k) \\le s_r(k+1)$ for all $k$.\n\nResults: $s_2(k) = (k-1)^2$ (Erd\\H os--Burr--Lov\\'asz 1976)\n\n$\\Omega(r^2 \\log r) \\le s_r(3) \\le O(r^2(\\log r)^2 )$\n\n$\\Omega\\left(\\frac{r^2 \\log r }{ \\log\\log r}\\right) \\le s_r(k) \\le\nO\\left(r^2 (\\log r)^{8(k-1)^2}\\right)$ for $k \\ge 4$",
   "public_statement": "Ramsey minimal graphs\n\nTibor Szab\\'o\n\nLet $G \\xrightarrow{r} K_k$ be minimal, i.e., $G - e\n\\not\\xrightarrow{r} K_3$ for any $e \\in E(G)$.\nLet $\\mathcal{M}_r(k)$ denote the family of such graphs $G$. Let\n\\[\ns_r(k) = \\min\\{\\delta(G) : G \\in \\mathcal{M}_r(k)\\}\n\\]\nwhere $\\delta(G)$ is the minimum degree of $G$.\n\nConjecture: (Fox--Grinshpun--Liebenau--Person--Szab\\'o)\n$s_r(k) \\le s_r(k+1)$ for all $k$.\n\nResults: $s_2(k) = (k-1)^2$ (Erd\\H os--Burr--Lov\\'asz 1976)\n\n$\\Omega(r^2 \\log r) \\le s_r(3) \\le O(r^2(\\log r)^2 )$\n\n$\\Omega\\left(\\frac{r^2 \\log r }{ \\log\\log r}\\right) \\le s_r(k) \\le\nO\\left(r^2 (\\log r)^{8(k-1)^2}\\right)$ for $k \\ge 4$",
   "evidence": "The source record is from the AIM workshop *Graph Ramsey theory*, section “Ramsey minimal graphs,” problem 11, attributed to Tibor Szabó. The displayed source text contains",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0059": {
   "statement_status": "exact",
   "original_statement": "Conjecture: (Fox--Grinshpun--Liebenau--Person--Szab\\'o) $s_r(k)\n= \\Theta(r \\cdot f_{k,k+1}(r)^2)$ where $f_{k,k+1}$ is the\nErd\\H{o}s-Rogers function, to be discussed in Dudek's talk.",
   "clean_statement": "Conjecture: (Fox--Grinshpun--Liebenau--Person--Szab\\'o) $s_r(k)\n= \\Theta(r \\cdot f_{k,k+1}(r)^2)$ where $f_{k,k+1}$ is the\nErd\\H{o}s-Rogers function, to be discussed in Dudek's talk.",
   "public_statement": "Conjecture: (Fox--Grinshpun--Liebenau--Person--Szab\\'o) $s_r(k)\n= \\Theta(r \\cdot f_{k,k+1}(r)^2)$ where $f_{k,k+1}$ is the\nErd\\H{o}s-Rogers function, to be discussed in Dudek's talk.",
   "evidence": "The exact AIM record (Graph Ramsey Theory workshop, section \"Ramsey minimal graphs\", problem 12) says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0060": {
   "statement_status": "exact",
   "original_statement": "Conjecture: $s_r(3) = \\Theta(r^2 \\log r)$\n\nRemark: This would follow from the extension of the triangle-free\nprocess to $r$ colors.",
   "clean_statement": "Conjecture: $s_r(3) = \\Theta(r^2 \\log r)$\n\nRemark: This would follow from the extension of the triangle-free\nprocess to $r$ colors.",
   "public_statement": "Conjecture: $s_r(3) = \\Theta(r^2 \\log r)$\n\nRemark: This would follow from the extension of the triangle-free\nprocess to $r$ colors.",
   "evidence": "The exact AIM record (Graph Ramsey Theory workshop, section \"Ramsey minimal graphs\", problem 13) states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0061": {
   "statement_status": "exact",
   "original_statement": "Ramsey numbers of knots\n\nDavid Conlon\n\nTheorem. (Negami) Fix a knot $K$. If $n$ if large enough, any\nlinear spatial embedding of $K_n$ contains $K$.\n\nLinear spatial embedding of a graph means placing the vertices of the graph in $\\mathbb{R}^3$ and drawing edges as straight lines between the vertices.\n\n``Contains'' would mean either \n\n(i) a cycle homeomorphic to $K$, or \n\n(ii) a cycle isotopic to $K$ \n\nThe distinction is that (ii) does not allow reflections.\n\nKnown: Any\nlinear spatial embedding of $K_7$ contains a homeomorphic copy of a\ntrefoil knot.\n\nQuestion: If $K$ has $k$ crossings, how big does $n = n(k)$\nhave to be?\n\nKnown:\n(Negami) at most 5-fold exponential\n\n(Conlon--Fox) $n(k) \\le 2^{2^{ck}}$",
   "clean_statement": "Ramsey numbers of knots\n\nDavid Conlon\n\nTheorem. (Negami) Fix a knot $K$. If $n$ if large enough, any\nlinear spatial embedding of $K_n$ contains $K$.\n\nLinear spatial embedding of a graph means placing the vertices of the graph in $\\mathbb{R}^3$ and drawing edges as straight lines between the vertices.\n\n``Contains'' would mean either\n\n(i) a cycle homeomorphic to $K$, or\n\n(ii) a cycle isotopic to $K$\n\nThe distinction is that (ii) does not allow reflections.\n\nKnown: Any\nlinear spatial embedding of $K_7$ contains a homeomorphic copy of a\ntrefoil knot.\n\nQuestion: If $K$ has $k$ crossings, how big does $n = n(k)$\nhave to be?\n\nKnown:\n(Negami) at most 5-fold exponential\n\n(Conlon--Fox) $n(k) \\le 2^{2^{ck}}$",
   "public_statement": "Ramsey numbers of knots\n\nDavid Conlon\n\nTheorem. (Negami) Fix a knot $K$. If $n$ if large enough, any\nlinear spatial embedding of $K_n$ contains $K$.\n\nLinear spatial embedding of a graph means placing the vertices of the graph in $\\mathbb{R}^3$ and drawing edges as straight lines between the vertices.\n\n``Contains'' would mean either\n\n(i) a cycle homeomorphic to $K$, or\n\n(ii) a cycle isotopic to $K$\n\nThe distinction is that (ii) does not allow reflections.\n\nKnown: Any\nlinear spatial embedding of $K_7$ contains a homeomorphic copy of a\ntrefoil knot.\n\nQuestion: If $K$ has $k$ crossings, how big does $n = n(k)$\nhave to be?\n\nKnown:\n(Negami) at most 5-fold exponential\n\n(Conlon--Fox) $n(k) \\le 2^{2^{ck}}$",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0062": {
   "statement_status": "exact",
   "original_statement": "Question:\nIs it polynomial in $k$?\n\nLower bound: $n(K) \\ge \\sqrt{\\nu(K)}$, where $\\nu(K)$ is the crossing\nnumber of a knot.\n\nWe can also ask the same question for links.\nFor the simplest nontrivial link of two unknots (i.e., Hopf link), the answer is 6.",
   "clean_statement": "Question:\nIs it polynomial in $k$?\n\nLower bound: $n(K) \\ge \\sqrt{\\nu(K)}$, where $\\nu(K)$ is the crossing\nnumber of a knot.\n\nWe can also ask the same question for links.\nFor the simplest nontrivial link of two unknots (i.e., Hopf link), the answer is 6.",
   "public_statement": "Question:\nIs it polynomial in $k$?\n\nLower bound: $n(K) \\ge \\sqrt{\\nu(K)}$, where $\\nu(K)$ is the crossing\nnumber of a knot.\n\nWe can also ask the same question for links.\nFor the simplest nontrivial link of two unknots (i.e., Hopf link), the answer is 6.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0063": {
   "statement_status": "exact",
   "original_statement": "Semi-algebraic Ramsey numbers\n\nAndrew Suk\n\nSuppose that that for each $n$ we have a $k$-uniform hypergraph $H_n$\nwhere $V(H_n)$ is a set of $n$ points in the plane, say $V(H_n) =\n\\{v_1, \\dots, v_n\\} \\subseteq \\mathbb{R}^2$, and the edges are defined\nby a $2k$-variable polynomial $f$, namely\n\\[\n\\{v_{i_1}, \\dots, v_{i_k}\\} \\in E(H_n) \\Leftrightarrow\nf(v_{i_1},\\dots, v_{i_k}) \\ge 0, \\qquad i_1 < i_2 < \\dots < i_k.\n\\]\n\nQuestion: What is the size of the largest\nhomogeneous set (i.e., clique or independent set) that is guaranteed to\nexist in every such\n$H_n$?\n\nLet $r_f(n) = \\min_{H_n} \\max\\{\\alpha(H_n),\n\\omega(H_n)\\}$ denote the above quantity.\n\nKnown: From bounds on Ramsey numbers we have $r_f(n) \\ge c\n\\underbrace{\\log \\dots \\log}_{k-1 \\text{ times}} n$.",
   "clean_statement": "Semi-algebraic Ramsey numbers\n\nAndrew Suk\n\nSuppose that that for each $n$ we have a $k$-uniform hypergraph $H_n$\nwhere $V(H_n)$ is a set of $n$ points in the plane, say $V(H_n) =\n\\{v_1, \\dots, v_n\\} \\subseteq \\mathbb{R}^2$, and the edges are defined\nby a $2k$-variable polynomial $f$, namely\n\\[\n\\{v_{i_1}, \\dots, v_{i_k}\\} \\in E(H_n) \\Leftrightarrow\nf(v_{i_1},\\dots, v_{i_k}) \\ge 0, \\qquad i_1 < i_2 < \\dots < i_k.\n\\]\n\nQuestion: What is the size of the largest\nhomogeneous set (i.e., clique or independent set) that is guaranteed to\nexist in every such\n$H_n$?\n\nLet $r_f(n) = \\min_{H_n} \\max\\{\\alpha(H_n),\n\\omega(H_n)\\}$ denote the above quantity.\n\nKnown: From bounds on Ramsey numbers we have $r_f(n) \\ge c\n\\underbrace{\\log \\dots \\log}_{k-1 \\text{ times}} n$.",
   "public_statement": "Semi-algebraic Ramsey numbers\n\nAndrew Suk\n\nSuppose that that for each $n$ we have a $k$-uniform hypergraph $H_n$\nwhere $V(H_n)$ is a set of $n$ points in the plane, say $V(H_n) =\n\\{v_1, \\dots, v_n\\} \\subseteq \\mathbb{R}^2$, and the edges are defined\nby a $2k$-variable polynomial $f$, namely\n\\[\n\\{v_{i_1}, \\dots, v_{i_k}\\} \\in E(H_n) \\Leftrightarrow\nf(v_{i_1},\\dots, v_{i_k}) \\ge 0, \\qquad i_1 < i_2 < \\dots < i_k.\n\\]\n\nQuestion: What is the size of the largest\nhomogeneous set (i.e., clique or independent set) that is guaranteed to\nexist in every such\n$H_n$?\n\nLet $r_f(n) = \\min_{H_n} \\max\\{\\alpha(H_n),\n\\omega(H_n)\\}$ denote the above quantity.\n\nKnown: From bounds on Ramsey numbers we have $r_f(n) \\ge c\n\\underbrace{\\log \\dots \\log}_{k-1 \\text{ times}} n$.",
   "evidence": "The source asks the following question, attributed to Andrew Suk. For each \\(n\\), let \\(H_n\\) be a \\(k\\)-uniform hypergraph whose vertices are labelled points \\[ P=(v_1,\\ldots,v_n)\\in(\\mathbb R^2)^n, \\] and, for \\(i_1<\\cdots<i_k\\), put \\[ \\{v_{i_1},\\ldots,v_{i_k}\\}\\in E(H_n) \\quad\\Longleftrightarrow\\quad f(v_{i_1},\\ldots,v_{i_k})\\geq 0, \\] where \\(f\\) is one polynomial in the \\(2k\\) coordinates. What order of magnitude can be guaranteed for the largest clique or independent set? The source denotes the worst guarantee by \\[ r_f(n)=\\min_{H_n}\\max\\{\\alpha(H_n),\\omega(H_n)\\}. \\]",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0064": {
   "statement_status": "exact",
   "original_statement": "Question: (Bukh--Matou\\v{s}ek) Can this be improved to a\nconstant number of logs?\n\nKnown: (Bukh--Matou\\v{s}ek) In dimension 1, two logs suffice,\nand this is tight.\n\n(Conlon--Fox--Pach--Sudakov--Suk) $r_f(n) \\ge c \\underbrace{\\log \\dots\n \\log}_{k-2 \\text{ times}} n$.\n\nIf the dimension is high enough, this is sharp (i.e., in\n$\\mathbb{R}^{c(k)}$).\n\nVariant: Unbalanced version $R_f^{(k)}(s,n)$, where the\nsuperscript means $k$-uniform.",
   "clean_statement": "Question: (Bukh--Matou\\v{s}ek) Can this be improved to a\nconstant number of logs?\n\nKnown: (Bukh--Matou\\v{s}ek) In dimension 1, two logs suffice,\nand this is tight.\n\n(Conlon--Fox--Pach--Sudakov--Suk) $r_f(n) \\ge c \\underbrace{\\log \\dots\n \\log}_{k-2 \\text{ times}} n$.\n\nIf the dimension is high enough, this is sharp (i.e., in\n$\\mathbb{R}^{c(k)}$).\n\nVariant: Unbalanced version $R_f^{(k)}(s,n)$, where the\nsuperscript means $k$-uniform.",
   "public_statement": "Question: (Bukh--Matou\\v{s}ek) Can this be improved to a\nconstant number of logs?\n\nKnown: (Bukh--Matou\\v{s}ek) In dimension 1, two logs suffice,\nand this is tight.\n\n(Conlon--Fox--Pach--Sudakov--Suk) $r_f(n) \\ge c \\underbrace{\\log \\dots\n \\log}_{k-2 \\text{ times}} n$.\n\nIf the dimension is high enough, this is sharp (i.e., in\n$\\mathbb{R}^{c(k)}$).\n\nVariant: Unbalanced version $R_f^{(k)}(s,n)$, where the\nsuperscript means $k$-uniform.",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0065": {
   "statement_status": "exact",
   "original_statement": "Problem: (Conlon--Fox--Pach--Sudakov--Suk) Is\n$R_f^{(3)}(4,n)$ polynomial in $n$?\n\nResults: (Suk) $R_f^{(3)}(4,n) = e^{e^{O(\\sqrt{\\log\n n})}}$. (Clearly $R_f^{(3)}(4,n) = 2^{O(n^2\\log n)}$ from Ramsey number bounds).\n\nMotivation: $R_f^{(4)}(5,n)$ connected to Erd\\H{o}s-Szekeres\nproblem on $n$ points in convex position.",
   "clean_statement": "Problem: (Conlon--Fox--Pach--Sudakov--Suk) Is\n$R_f^{(3)}(4,n)$ polynomial in $n$?\n\nResults: (Suk) $R_f^{(3)}(4,n) = e^{e^{O(\\sqrt{\\log\n n})}}$. (Clearly $R_f^{(3)}(4,n) = 2^{O(n^2\\log n)}$ from Ramsey number bounds).\n\nMotivation: $R_f^{(4)}(5,n)$ connected to Erd\\H{o}s-Szekeres\nproblem on $n$ points in convex position.",
   "public_statement": "Problem: (Conlon--Fox--Pach--Sudakov--Suk) Is\n$R_f^{(3)}(4,n)$ polynomial in $n$?\n\nResults: (Suk) $R_f^{(3)}(4,n) = e^{e^{O(\\sqrt{\\log\n n})}}$. (Clearly $R_f^{(3)}(4,n) = 2^{O(n^2\\log n)}$ from Ramsey number bounds).\n\nMotivation: $R_f^{(4)}(5,n)$ connected to Erd\\H{o}s-Szekeres\nproblem on $n$ points in convex position.",
   "evidence": "The source record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0066": {
   "statement_status": "exact",
   "original_statement": "Question: (Fox) Is $R_f^{(2)}(3,n) = n^{1 + o(1)}$?",
   "clean_statement": "Question: (Fox) Is $R_f^{(2)}(3,n) = n^{1 + o(1)}$?",
   "public_statement": "Question: (Fox) Is $R_f^{(2)}(3,n) = n^{1 + o(1)}$?",
   "evidence": "The complete extracted record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0067": {
   "statement_status": "exact",
   "original_statement": "Large cliques in straight edge intersection graphs\n\nJ\\'anos Pach\n\nQuestion: If $K_n$ is drawn in the plane with straight line\nedges, can we always find $\\Omega(n)$ pairwise crossing edges?\n\nKnown: (Aronov--Erd\\H{os}--Kleitman--Pach) Can get\n$\\Omega(\\sqrt{n})$.\n\nResults: (Csaba--Fox--Pach) For some $\\epsilon > 0$, if edges\nare curves and each pair crosses $O(1)$ times, then we get\n$\\Omega(n^{\\epsilon})$ pairwise crossing curves.",
   "clean_statement": "Large cliques in straight edge intersection graphs\n\nJ\\'anos Pach\n\nQuestion: If $K_n$ is drawn in the plane with straight line\nedges, can we always find $\\Omega(n)$ pairwise crossing edges?\n\nKnown: (Aronov--Erd\\H{os}--Kleitman--Pach) Can get\n$\\Omega(\\sqrt{n})$.\n\nResults: (Csaba--Fox--Pach) For some $\\epsilon > 0$, if edges\nare curves and each pair crosses $O(1)$ times, then we get\n$\\Omega(n^{\\epsilon})$ pairwise crossing curves.",
   "public_statement": "Large cliques in straight edge intersection graphs\n\nJ\\'anos Pach\n\nQuestion: If $K_n$ is drawn in the plane with straight line\nedges, can we always find $\\Omega(n)$ pairwise crossing edges?\n\nKnown: (Aronov--Erd\\H{os}--Kleitman--Pach) Can get\n$\\Omega(\\sqrt{n})$.\n\nResults: (Csaba--Fox--Pach) For some $\\epsilon > 0$, if edges\nare curves and each pair crosses $O(1)$ times, then we get\n$\\Omega(n^{\\epsilon})$ pairwise crossing curves.",
   "evidence": "The canonical record is item 20 of the AIM workshop list “Graph Ramsey theory,” attributed there to János Pach:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0068": {
   "statement_status": "exact",
   "original_statement": "Chromatic number of graphs with everywhere high independence number\n\nJacque Verstraete\n\nQuestion: Is there an $\\epsilon > 0$ so that $\\chi(G) \\le\nd^{1-\\epsilon}$ for every graph $G$ of maximum degree $d$ such that\n$\\alpha(H) \\geq \\frac13 |V(H)|$ for every subgraph $H \\subseteq G$?\n\nMotivation: Kneser graphs $KG_{3n,n}$ satisfies the constraint\nand has chromatic number $\\Theta(\\log d)$.\n\nNote that any graph satisfying the constraint is $K_4$-free, and it is\nknown (Shearer) that $\\alpha(G) \\ge \\frac{cn\\log d}{d \\log\\log d}$ if\n$G$ is $K_4$-free, and hence $\\chi(G) \\le \\frac{n}{\\alpha(G)} \\le\n\\frac{d \\log\\log d}{c \\log d}$.\n\n\\section{Burr--Erd\\H{o}s for hypergraphs}\n\nJacob Fox\n\nA graph $G$ is $d$-degenerate if every $H \\subseteq G$ has\n$\\delta(H)\\le d$.",
   "clean_statement": "Chromatic number of graphs with everywhere high independence number\n\nJacque Verstraete\n\nQuestion: Is there an $\\epsilon > 0$ so that $\\chi(G) \\le\nd^{1-\\epsilon}$ for every graph $G$ of maximum degree $d$ such that\n$\\alpha(H) \\geq \\frac13 |V(H)|$ for every subgraph $H \\subseteq G$?\n\nMotivation: Kneser graphs $KG_{3n,n}$ satisfies the constraint\nand has chromatic number $\\Theta(\\log d)$.\n\nNote that any graph satisfying the constraint is $K_4$-free, and it is\nknown (Shearer) that $\\alpha(G) \\ge \\frac{cn\\log d}{d \\log\\log d}$ if\n$G$ is $K_4$-free, and hence $\\chi(G) \\le \\frac{n}{\\alpha(G)} \\le\n\\frac{d \\log\\log d}{c \\log d}$.\n\n\\section{Burr--Erd\\H{o}s for hypergraphs}\n\nJacob Fox\n\nA graph $G$ is $d$-degenerate if every $H \\subseteq G$ has\n$\\delta(H)\\le d$.",
   "public_statement": "Chromatic number of graphs with everywhere high independence number\n\nJacque Verstraete\n\nQuestion: Is there an $\\epsilon > 0$ so that $\\chi(G) \\le\nd^{1-\\epsilon}$ for every graph $G$ of maximum degree $d$ such that\n$\\alpha(H) \\geq \\frac13 |V(H)|$ for every subgraph $H \\subseteq G$?\n\nMotivation: Kneser graphs $KG_{3n,n}$ satisfies the constraint\nand has chromatic number $\\Theta(\\log d)$.\n\nNote that any graph satisfying the constraint is $K_4$-free, and it is\nknown (Shearer) that $\\alpha(G) \\ge \\frac{cn\\log d}{d \\log\\log d}$ if\n$G$ is $K_4$-free, and hence $\\chi(G) \\le \\frac{n}{\\alpha(G)} \\le\n\\frac{d \\log\\log d}{c \\log d}$.\n\n\\section{Burr--Erd\\H{o}s for hypergraphs}\n\nJacob Fox\n\nA graph $G$ is $d$-degenerate if every $H \\subseteq G$ has\n$\\delta(H)\\le d$.",
   "evidence": "The mathematical question in the source record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0069": {
   "statement_status": "exact",
   "original_statement": "Conjecture: (Burr--Erd\\H{o}s) $r(G,G) \\le c(d) n$ if $G$ is\nan $n$-vertex $d$-degenerate graph.\n\nKnown: (Kostochka--Sudakov) $r(G,G) \\le n^{1+o_d(1)}$.\n\nIf $G$ is a 3-uniform $d$-degenerate hypergraph, the statement can be\nfalse. There is a 1-degenerate 4-uniform $G$ such that $r_4(G,G) \\ge\n2^{\\Omega(n^{1/3})}$.\n\nPerhaps we can redefine degeneracy for $k$-uniform hypergraphs: call a\n$k$-uniform hypergraph $d$-degenerate if the 2-graph obtained by\nreplacing every edge by a graph clique $K_k^{(2)}$ is $d$-degenerate\nas a graph.",
   "clean_statement": "Conjecture: (Burr--Erd\\H{o}s) $r(G,G) \\le c(d) n$ if $G$ is\nan $n$-vertex $d$-degenerate graph.\n\nKnown: (Kostochka--Sudakov) $r(G,G) \\le n^{1+o_d(1)}$.\n\nIf $G$ is a 3-uniform $d$-degenerate hypergraph, the statement can be\nfalse. There is a 1-degenerate 4-uniform $G$ such that $r_4(G,G) \\ge\n2^{\\Omega(n^{1/3})}$.\n\nPerhaps we can redefine degeneracy for $k$-uniform hypergraphs: call a\n$k$-uniform hypergraph $d$-degenerate if the 2-graph obtained by\nreplacing every edge by a graph clique $K_k^{(2)}$ is $d$-degenerate\nas a graph.",
   "public_statement": "Conjecture: (Burr--Erd\\H{o}s) $r(G,G) \\le c(d) n$ if $G$ is\nan $n$-vertex $d$-degenerate graph.\n\nKnown: (Kostochka--Sudakov) $r(G,G) \\le n^{1+o_d(1)}$.\n\nIf $G$ is a 3-uniform $d$-degenerate hypergraph, the statement can be\nfalse. There is a 1-degenerate 4-uniform $G$ such that $r_4(G,G) \\ge\n2^{\\Omega(n^{1/3})}$.\n\nPerhaps we can redefine degeneracy for $k$-uniform hypergraphs: call a\n$k$-uniform hypergraph $d$-degenerate if the 2-graph obtained by\nreplacing every edge by a graph clique $K_k^{(2)}$ is $d$-degenerate\nas a graph.",
   "evidence": "The canonical AIM record (Graph Ramsey theory workshop, item 22, zero-based source index 68) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0070": {
   "statement_status": "exact",
   "original_statement": "Question: (Conlon--Fox--Sudakov) Is $r_f(H,H) \\le c_k(d) n$\nfor all $n$-vertex $d$-degenerate (in the above sense) $k$-uniform\nhypergraph $H$?",
   "clean_statement": "Question: (Conlon--Fox--Sudakov) Is $r_f(H,H) \\le c_k(d) n$\nfor all $n$-vertex $d$-degenerate (in the above sense) $k$-uniform\nhypergraph $H$?",
   "public_statement": "Question: (Conlon--Fox--Sudakov) Is $r_f(H,H) \\le c_k(d) n$\nfor all $n$-vertex $d$-degenerate (in the above sense) $k$-uniform\nhypergraph $H$?",
   "evidence": "The canonical AIM record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 69,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0071": {
   "statement_status": "exact",
   "original_statement": "Cycle versus cliques\n\nJozef Skokan\n\nEasy: $r(C_m, K_n) \\ge (m-1)(n-1) + 1$.\n\nConjecture: (Erd\\H{o}s) $r(C_m, K_n) = (m-1)(n-1) + 1$ if $m\n\\ge n$.\n\nKnown: (Bondy--Erd\\H{o}s 1973) True if $m \\ge\nn^2$. (Nikiforov) True if $m \\ge 4n + 6$.\n\nWhy should $m \\ge n$?\n\nEasy: $r(P_m,K_n) = (m-1)(n-1) + 1$ for all $m, n$.",
   "clean_statement": "Cycle versus cliques\n\nJozef Skokan\n\nEasy: $r(C_m, K_n) \\ge (m-1)(n-1) + 1$.\n\nConjecture: (Erd\\H{o}s) $r(C_m, K_n) = (m-1)(n-1) + 1$ if $m\n\\ge n$.\n\nKnown: (Bondy--Erd\\H{o}s 1973) True if $m \\ge\nn^2$. (Nikiforov) True if $m \\ge 4n + 6$.\n\nWhy should $m \\ge n$?\n\nEasy: $r(P_m,K_n) = (m-1)(n-1) + 1$ for all $m, n$.",
   "public_statement": "Cycle versus cliques\n\nJozef Skokan\n\nEasy: $r(C_m, K_n) \\ge (m-1)(n-1) + 1$.\n\nConjecture: (Erd\\H{o}s) $r(C_m, K_n) = (m-1)(n-1) + 1$ if $m\n\\ge n$.\n\nKnown: (Bondy--Erd\\H{o}s 1973) True if $m \\ge\nn^2$. (Nikiforov) True if $m \\ge 4n + 6$.\n\nWhy should $m \\ge n$?\n\nEasy: $r(P_m,K_n) = (m-1)(n-1) + 1$ for all $m, n$.",
   "evidence": "The canonical AIM record, item 24 of the Graph Ramsey theory workshop list, says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 70,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0072": {
   "statement_status": "exact",
   "original_statement": "Local conditions for distinct distances\n\nAndrew Suk\n\nQuestion: (Erd\\H{o}s 1986) What is the least possible number\nof distinct distances among $n$ points in the plane if every 5 of them\nspan at least 9 distinct distances?\n\nConjecture: $\\Omega(n^2)$\n\nBest known: $\\Omega(n)$.\n\nRelated to $F(r,s,9)$.",
   "clean_statement": "Local conditions for distinct distances\n\nAndrew Suk\n\nQuestion: (Erd\\H{o}s 1986) What is the least possible number\nof distinct distances among $n$ points in the plane if every 5 of them\nspan at least 9 distinct distances?\n\nConjecture: $\\Omega(n^2)$\n\nBest known: $\\Omega(n)$.\n\nRelated to $F(r,s,9)$.",
   "public_statement": "Local conditions for distinct distances\n\nAndrew Suk\n\nQuestion: (Erd\\H{o}s 1986) What is the least possible number\nof distinct distances among $n$ points in the plane if every 5 of them\nspan at least 9 distinct distances?\n\nConjecture: $\\Omega(n^2)$\n\nBest known: $\\Omega(n)$.\n\nRelated to $F(r,s,9)$.",
   "evidence": "The canonical AIM record is Question 25 in the Graph Ramsey theory workshop list, under “Local conditions for distinct distances,” attributed to Andrew Suk. Its mathematical question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0073": {
   "statement_status": "exact",
   "original_statement": "Ramsey number of simplices\n\nJacob Fox\n\nQuestion: Estimate $r_k(k+1) := r_k(k+1,k+1)$.\n\nBounds: $2^{\\Omega(k)} \\le r_k(k+1) \\le\n2^{2^{.^{.^{.^{2^C}}}}}$, where the tower of $2$'s has height\n$\\approx k$.\n\nC.f. (Duffus--Lefmann--R\\\"odl) for $r_k(k+2)$.",
   "clean_statement": "Ramsey number of simplices\n\nJacob Fox\n\nQuestion: Estimate $r_k(k+1) := r_k(k+1,k+1)$.\n\nBounds: $2^{\\Omega(k)} \\le r_k(k+1) \\le\n2^{2^{.^{.^{.^{2^C}}}}}$, where the tower of $2$'s has height\n$\\approx k$.\n\nC.f. (Duffus--Lefmann--R\\\"odl) for $r_k(k+2)$.",
   "public_statement": "Ramsey number of simplices\n\nJacob Fox\n\nQuestion: Estimate $r_k(k+1) := r_k(k+1,k+1)$.\n\nBounds: $2^{\\Omega(k)} \\le r_k(k+1) \\le\n2^{2^{.^{.^{.^{2^C}}}}}$, where the tower of $2$'s has height\n$\\approx k$.\n\nC.f. (Duffus--Lefmann--R\\\"odl) for $r_k(k+2)$.",
   "evidence": "The canonical AIM record, attributed to Jacob Fox, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 72,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0074": {
   "statement_status": "exact",
   "original_statement": "Hypergraph triangles versus cliques\n\nJacque Verstraete\n\nKnown: (Kim, Ajtai--Koml\\'os--Szemer\\'edi) $r(3,t) =\n\\Theta(t^2/\\log t)$.\n\nLet $\\triangle$ denote the 3-uniform hypergraph with vertex set $[6]$\nand edges $\\{123,345,561\\}$.\n\nKnown: $\\Omega\\left(\\frac{t^{3/2}}{(\\log t)^{3/4}}\\right) =\nr_3(\\triangle, K_t^{(3)}) = O(t^{3/2})$\n\nConjecture (Kostochka--Mubayi--Verstraete) $r_3(\\triangle,\nK_t^{(3)}) = o(t^{3/2})$.",
   "clean_statement": "Hypergraph triangles versus cliques\n\nJacque Verstraete\n\nKnown: (Kim, Ajtai--Koml\\'os--Szemer\\'edi) $r(3,t) =\n\\Theta(t^2/\\log t)$.\n\nLet $\\triangle$ denote the 3-uniform hypergraph with vertex set $[6]$\nand edges $\\{123,345,561\\}$.\n\nKnown: $\\Omega\\left(\\frac{t^{3/2}}{(\\log t)^{3/4}}\\right) =\nr_3(\\triangle, K_t^{(3)}) = O(t^{3/2})$\n\nConjecture (Kostochka--Mubayi--Verstraete) $r_3(\\triangle,\nK_t^{(3)}) = o(t^{3/2})$.",
   "public_statement": "Hypergraph triangles versus cliques\n\nJacque Verstraete\n\nKnown: (Kim, Ajtai--Koml\\'os--Szemer\\'edi) $r(3,t) =\n\\Theta(t^2/\\log t)$.\n\nLet $\\triangle$ denote the 3-uniform hypergraph with vertex set $[6]$\nand edges $\\{123,345,561\\}$.\n\nKnown: $\\Omega\\left(\\frac{t^{3/2}}{(\\log t)^{3/4}}\\right) =\nr_3(\\triangle, K_t^{(3)}) = O(t^{3/2})$\n\nConjecture (Kostochka--Mubayi--Verstraete) $r_3(\\triangle,\nK_t^{(3)}) = o(t^{3/2})$.",
   "evidence": "The canonical Graph Ramsey theory workshop record, problem 27, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 73,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0075": {
   "statement_status": "exact",
   "original_statement": "Ramsey goodness for hypergraphs\n\nDavid Conlon\n\nA graph $G$ is $t$-good if\n\\[\nr(G,K_t) = (t-1)(v(G) - 1) + 1.\n\\]\n$G$ is $H$-good is\n\\[\nr(G, H) = (\\chi(H)-1)(v(G) - 1) + \\sigma(H)\n\\]\nwhere $\\sigma(H)$ is the size of the smalles color class in any\n$\\chi(H)$-coloring of $H$.\n\nIntuition: graphs that are $H$-good tend to be poor expanders.\n\n\\medskip\n\nLet $F$ denote the Fano plane. A 3-uniform hypergraph $H$ is $F$-good if\n\\[\nr(H,F) = 2(v(H)-1) + 1.\n\\]\nConstruction giving $\\ge$: $v(H)-1$ vertices on the left,\n$v(H)-1$ vertice on the right, color an edge red if it stays in one\npart, and blue if it contains vertices in both parts.\n\nQuestion: Which $H$ are $F$-good?",
   "clean_statement": "Ramsey goodness for hypergraphs\n\nDavid Conlon\n\nA graph $G$ is $t$-good if\n\\[\nr(G,K_t) = (t-1)(v(G) - 1) + 1.\n\\]\n$G$ is $H$-good is\n\\[\nr(G, H) = (\\chi(H)-1)(v(G) - 1) + \\sigma(H)\n\\]\nwhere $\\sigma(H)$ is the size of the smalles color class in any\n$\\chi(H)$-coloring of $H$.\n\nIntuition: graphs that are $H$-good tend to be poor expanders.\n\n\\medskip\n\nLet $F$ denote the Fano plane. A 3-uniform hypergraph $H$ is $F$-good if\n\\[\nr(H,F) = 2(v(H)-1) + 1.\n\\]\nConstruction giving $\\ge$: $v(H)-1$ vertices on the left,\n$v(H)-1$ vertice on the right, color an edge red if it stays in one\npart, and blue if it contains vertices in both parts.\n\nQuestion: Which $H$ are $F$-good?",
   "public_statement": "Ramsey goodness for hypergraphs\n\nDavid Conlon\n\nA graph $G$ is $t$-good if\n\\[\nr(G,K_t) = (t-1)(v(G) - 1) + 1.\n\\]\n$G$ is $H$-good is\n\\[\nr(G, H) = (\\chi(H)-1)(v(G) - 1) + \\sigma(H)\n\\]\nwhere $\\sigma(H)$ is the size of the smalles color class in any\n$\\chi(H)$-coloring of $H$.\n\nIntuition: graphs that are $H$-good tend to be poor expanders.\n\n\\medskip\n\nLet $F$ denote the Fano plane. A 3-uniform hypergraph $H$ is $F$-good if\n\\[\nr(H,F) = 2(v(H)-1) + 1.\n\\]\nConstruction giving $\\ge$: $v(H)-1$ vertices on the left,\n$v(H)-1$ vertice on the right, color an edge red if it stays in one\npart, and blue if it contains vertices in both parts.\n\nQuestion: Which $H$ are $F$-good?",
   "evidence": "The AIM record, attributed to David Conlon, asks which 3-uniform hypergraphs are Ramsey-good with respect to the Fano plane. The record defines \\[ r(H,F)=2(v(H)-1)+1 \\] as the equality of interest and describes a two-part coloring intended to prove the corresponding lower bound.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0076": {
   "statement_status": "exact",
   "original_statement": "Increasing paths in edge labelings\n\nRon Graham\n\nLabel the edges of $K_n$ with numbers $1, 2, \\dots,\n\\binom{n}{2}$. For each $v \\in V(K_n)$, let $t(v)$ be the length of\nthe longest increasing path from $v$.\n\nQuestion: Is it true that $\\max_v t(v) \\ge cn$ for some $c > 0$?\n\nFurthermore,\nis it true that for any graph $G$, edges labeled $1, 2, \\dots,\n|E(G)|$, we always have $\\sum_{v \\in V(G)} t(v) \\ge |E(G)|$?\n\nKnown: (Calderbank--Chung--Sturtevant 1984).\nThere exists a labeling with $\\max_v t(v) \\le n/2$.\n\nKnown: (Graham--Kleitman 1973) $\\max_v t(v) \\ge (1+o(1))\n\\sqrt{n}$ for $K_n$\n\nRecently claimed to be improved to $\\ge (\\sqrt{2} + o(1)) \\sqrt{n}$ by a\ngroup of graduate students.",
   "clean_statement": "Increasing paths in edge labelings\n\nRon Graham\n\nLabel the edges of $K_n$ with numbers $1, 2, \\dots,\n\\binom{n}{2}$. For each $v \\in V(K_n)$, let $t(v)$ be the length of\nthe longest increasing path from $v$.\n\nQuestion: Is it true that $\\max_v t(v) \\ge cn$ for some $c > 0$?\n\nFurthermore,\nis it true that for any graph $G$, edges labeled $1, 2, \\dots,\n|E(G)|$, we always have $\\sum_{v \\in V(G)} t(v) \\ge |E(G)|$?\n\nKnown: (Calderbank--Chung--Sturtevant 1984).\nThere exists a labeling with $\\max_v t(v) \\le n/2$.\n\nKnown: (Graham--Kleitman 1973) $\\max_v t(v) \\ge (1+o(1))\n\\sqrt{n}$ for $K_n$\n\nRecently claimed to be improved to $\\ge (\\sqrt{2} + o(1)) \\sqrt{n}$ by a\ngroup of graduate students.",
   "public_statement": "Increasing paths in edge labelings\n\nRon Graham\n\nLabel the edges of $K_n$ with numbers $1, 2, \\dots,\n\\binom{n}{2}$. For each $v \\in V(K_n)$, let $t(v)$ be the length of\nthe longest increasing path from $v$.\n\nQuestion: Is it true that $\\max_v t(v) \\ge cn$ for some $c > 0$?\n\nFurthermore,\nis it true that for any graph $G$, edges labeled $1, 2, \\dots,\n|E(G)|$, we always have $\\sum_{v \\in V(G)} t(v) \\ge |E(G)|$?\n\nKnown: (Calderbank--Chung--Sturtevant 1984).\nThere exists a labeling with $\\max_v t(v) \\le n/2$.\n\nKnown: (Graham--Kleitman 1973) $\\max_v t(v) \\ge (1+o(1))\n\\sqrt{n}$ for $K_n$\n\nRecently claimed to be improved to $\\ge (\\sqrt{2} + o(1)) \\sqrt{n}$ by a\ngroup of graduate students.",
   "evidence": "The AIM record, attributed to Ron Graham, gives every edge of a graph a distinct label. For the complete graph it asks whether there is an absolute constant \\(c>0\\) such that \\[ \\max_{v\\in V(K_n)}t(v)\\ge cn, \\] where \\(t(v)\\) is the maximum length of an increasing path starting at \\(v\\). It then asks, for every edge-ordered graph \\(G\\), whether \\[ \\sum_{v\\in V(G)}t(v)\\ge |E(G)|. \\tag{1.1} \\]",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0077": {
   "statement_status": "exact",
   "original_statement": "Directed paths in Eulerian digraphs\n\nJacques Verstraete\n\nThe following is a special case of a conjecture of Bollob\\'{a}s and Scott on weighted paths in directed graphs (see: A proof of a conjecture of Bondy concerning paths in weighted digraphs. J.\\ Combin.\\ Theory Ser.\\ B 66 (1996), no. 2, 283--292):\n\nQuestion: Does every Eulerian digraph of average degree $d$\nhave a directed path of length $\\Omega(d)$?\n\nGetting $\\Omega(\\sqrt{d})$ is not too hard.",
   "clean_statement": "Directed paths in Eulerian digraphs\n\nJacques Verstraete\n\nThe following is a special case of a conjecture of Bollob\\'{a}s and Scott on weighted paths in directed graphs (see: A proof of a conjecture of Bondy concerning paths in weighted digraphs. J.\\ Combin.\\ Theory Ser.\\ B 66 (1996), no. 2, 283--292):\n\nQuestion: Does every Eulerian digraph of average degree $d$\nhave a directed path of length $\\Omega(d)$?\n\nGetting $\\Omega(\\sqrt{d})$ is not too hard.",
   "public_statement": "Directed paths in Eulerian digraphs\n\nJacques Verstraete\n\nThe following is a special case of a conjecture of Bollob\\'{a}s and Scott on weighted paths in directed graphs (see: A proof of a conjecture of Bondy concerning paths in weighted digraphs. J.\\ Combin.\\ Theory Ser.\\ B 66 (1996), no. 2, 283--292):\n\nQuestion: Does every Eulerian digraph of average degree $d$\nhave a directed path of length $\\Omega(d)$?\n\nGetting $\\Omega(\\sqrt{d})$ is not too hard.",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0078": {
   "statement_status": "exact",
   "original_statement": "Ordered Ramsey numbers\n\nDavid Conlon, Jacob Fox, Choongbum Lee, and Benny Sudakov\n\nAn ordered graph on $N$ vertices is a graph whose vertices have been labeled with $\\{1,\\dots,N\\}$. We say that an ordered graph $G$ on $\\{1,\\dots,N\\}$ contains another ordered graph $H$ on $\\{1,\\dots,n\\}$ if $H$ occurs as a subgraph of $G$ with vertices appearing in the correct order.\n\nLet $r_<(H)$ denote the minimum $N$ such that every 2-edge coloring of $K_N$ contains a monochromatic ordered copy of $H$. Similarly for $r_<(H_1, H_2)$.\n\nQuestion: Does there exist $\\epsilon > 0$ such that $r_<(K_3,M) = O(n^{2-\\epsilon})$ for every ordered matching $M$ on $n$ vertices.\n\nKnown: $r_<(K_3, M) \\le r_< (K_3, K_n) = \\Theta(n^2/\\log n)$.\n\nKnown: There exists an ordered matching $M$ on $n$ vertices such that $r_<(K_3, M) = n^{4/3 + o(1)}$.",
   "clean_statement": "Ordered Ramsey numbers\n\nDavid Conlon, Jacob Fox, Choongbum Lee, and Benny Sudakov\n\nAn ordered graph on $N$ vertices is a graph whose vertices have been labeled with $\\{1,\\dots,N\\}$. We say that an ordered graph $G$ on $\\{1,\\dots,N\\}$ contains another ordered graph $H$ on $\\{1,\\dots,n\\}$ if $H$ occurs as a subgraph of $G$ with vertices appearing in the correct order.\n\nLet $r_<(H)$ denote the minimum $N$ such that every 2-edge coloring of $K_N$ contains a monochromatic ordered copy of $H$. Similarly for $r_<(H_1, H_2)$.\n\nQuestion: Does there exist $\\epsilon > 0$ such that $r_<(K_3,M) = O(n^{2-\\epsilon})$ for every ordered matching $M$ on $n$ vertices.\n\nKnown: $r_<(K_3, M) \\le r_< (K_3, K_n) = \\Theta(n^2/\\log n)$.\n\nKnown: There exists an ordered matching $M$ on $n$ vertices such that $r_<(K_3, M) = n^{4/3 + o(1)}$.",
   "public_statement": "Ordered Ramsey numbers\n\nDavid Conlon, Jacob Fox, Choongbum Lee, and Benny Sudakov\n\nAn ordered graph on $N$ vertices is a graph whose vertices have been labeled with $\\{1,\\dots,N\\}$. We say that an ordered graph $G$ on $\\{1,\\dots,N\\}$ contains another ordered graph $H$ on $\\{1,\\dots,n\\}$ if $H$ occurs as a subgraph of $G$ with vertices appearing in the correct order.\n\nLet $r_<(H)$ denote the minimum $N$ such that every 2-edge coloring of $K_N$ contains a monochromatic ordered copy of $H$. Similarly for $r_<(H_1, H_2)$.\n\nQuestion: Does there exist $\\epsilon > 0$ such that $r_<(K_3,M) = O(n^{2-\\epsilon})$ for every ordered matching $M$ on $n$ vertices.\n\nKnown: $r_<(K_3, M) \\le r_< (K_3, K_n) = \\Theta(n^2/\\log n)$.\n\nKnown: There exists an ordered matching $M$ on $n$ vertices such that $r_<(K_3, M) = n^{4/3 + o(1)}$.",
   "evidence": "The canonical record is Question 31 in the AIM workshop list *Graph Ramsey theory*, section “Ordered Ramsey numbers,” attributed to David Conlon, Jacob Fox, Choongbum Lee, and Benny Sudakov. It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0079": {
   "statement_status": "exact",
   "original_statement": "Question: Does there exist a constant $c > 0$ such that for every sufficiently large $n$ there exists a 100-regular graph $H$ on $n$ vertices such that $r_<(H) \\ge n^{1 + c}$ for every ordering of $H$?\n\n\\section{Ramsey meets Erd\\H{o}s--Szekeres}\n\nDhruv Mubayi\n\nLet $g(s,n)$ be the minimum $N$ such that for every set of $N$ points in general position in the plane, and every red/blue coloring of the pairs, there are $s$ points in convex position, with all pairs red, or $n$ points in convex position with all pairs blue.\n\nKnown: $4^n < g(n,n) < 2^{O( n^2 \\log n)}$\n\n$2^{n-1} < g(s,n) < r(4^s, 4^n) < (4^n)^{4^s}$",
   "clean_statement": "Question: Does there exist a constant $c > 0$ such that for every sufficiently large $n$ there exists a 100-regular graph $H$ on $n$ vertices such that $r_<(H) \\ge n^{1 + c}$ for every ordering of $H$?\n\n\\section{Ramsey meets Erd\\H{o}s--Szekeres}\n\nDhruv Mubayi\n\nLet $g(s,n)$ be the minimum $N$ such that for every set of $N$ points in general position in the plane, and every red/blue coloring of the pairs, there are $s$ points in convex position, with all pairs red, or $n$ points in convex position with all pairs blue.\n\nKnown: $4^n < g(n,n) < 2^{O( n^2 \\log n)}$\n\n$2^{n-1} < g(s,n) < r(4^s, 4^n) < (4^n)^{4^s}$",
   "public_statement": "Question: Does there exist a constant $c > 0$ such that for every sufficiently large $n$ there exists a 100-regular graph $H$ on $n$ vertices such that $r_<(H) \\ge n^{1 + c}$ for every ordering of $H$?\n\n\\section{Ramsey meets Erd\\H{o}s--Szekeres}\n\nDhruv Mubayi\n\nLet $g(s,n)$ be the minimum $N$ such that for every set of $N$ points in general position in the plane, and every red/blue coloring of the pairs, there are $s$ points in convex position, with all pairs red, or $n$ points in convex position with all pairs blue.\n\nKnown: $4^n < g(n,n) < 2^{O( n^2 \\log n)}$\n\n$2^{n-1} < g(s,n) < r(4^s, 4^n) < (4^n)^{4^s}$",
   "evidence": "The canonical record begins in the section “Ordered Ramsey numbers” and asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0080": {
   "statement_status": "exact",
   "original_statement": "Problem: Improve these bounds.",
   "clean_statement": "Problem: Improve these bounds.",
   "public_statement": "Problem: Improve these bounds.",
   "evidence": "The canonical record consists only of",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0081": {
   "statement_status": "exact",
   "original_statement": "Euclidean Ramsey sets\n\nRon Graham\n\nLet $X$ be a finite set in $\\mathbb{E}^n$ ($n$-dimensional Euclidean\nspace). We say that $X$ is Ramsey if for all positive integers\n$r$ there exists $N = N(X,r)$ such that if one colors the points of\n$\\mathbb{E}^N$ with $r$ colors, then there exists a monochromatic\nset congruent to $X$.\n\nFor example, if $X$ consist of two points,\nthen $X$ is Ramsey. (Consider an $r$-coloring of the $2r+1$ vertices of\na unit simplex simplex in $\\mathbb{E}^{2r}$).\n\nA set is spherical if it lies on some sphere.\n\nTheorem. (Erd\\H{o}s--Graham--Montgomery--Rothschild--Spencer--Straus). Every Ramsey set is spherical.\n\nConjecture. (\\$1000) Every finite spherical set is Ramsey.\n\nA ``warm up'':",
   "clean_statement": "Euclidean Ramsey sets\n\nRon Graham\n\nLet $X$ be a finite set in $\\mathbb{E}^n$ ($n$-dimensional Euclidean\nspace). We say that $X$ is Ramsey if for all positive integers\n$r$ there exists $N = N(X,r)$ such that if one colors the points of\n$\\mathbb{E}^N$ with $r$ colors, then there exists a monochromatic\nset congruent to $X$.\n\nFor example, if $X$ consist of two points,\nthen $X$ is Ramsey. (Consider an $r$-coloring of the $2r+1$ vertices of\na unit simplex simplex in $\\mathbb{E}^{2r}$).\n\nA set is spherical if it lies on some sphere.\n\nTheorem. (Erd\\H{o}s--Graham--Montgomery--Rothschild--Spencer--Straus). Every Ramsey set is spherical.\n\nConjecture. (\\$1000) Every finite spherical set is Ramsey.\n\nA ``warm up'':",
   "public_statement": "Euclidean Ramsey sets\n\nRon Graham\n\nLet $X$ be a finite set in $\\mathbb{E}^n$ ($n$-dimensional Euclidean\nspace). We say that $X$ is Ramsey if for all positive integers\n$r$ there exists $N = N(X,r)$ such that if one colors the points of\n$\\mathbb{E}^N$ with $r$ colors, then there exists a monochromatic\nset congruent to $X$.\n\nFor example, if $X$ consist of two points,\nthen $X$ is Ramsey. (Consider an $r$-coloring of the $2r+1$ vertices of\na unit simplex simplex in $\\mathbb{E}^{2r}$).\n\nA set is spherical if it lies on some sphere.\n\nTheorem. (Erd\\H{o}s--Graham--Montgomery--Rothschild--Spencer--Straus). Every Ramsey set is spherical.\n\nConjecture. (\\$1000) Every finite spherical set is Ramsey.\n\nA ``warm up'':",
   "evidence": "The canonical record is Conjecture 34 in the AIM workshop list *Graph Ramsey theory*, section “Euclidean Ramsey sets,” attributed to Ron Graham. Its exact mathematical claim is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0082": {
   "statement_status": "exact",
   "original_statement": "Conjecture. (\\$100) Every 4-point subset of a circle is\nRamsey.\n\nA rival conjecture:",
   "clean_statement": "Conjecture. (\\$100) Every 4-point subset of a circle is\nRamsey.\n\nA rival conjecture:",
   "public_statement": "Conjecture. (\\$100) Every 4-point subset of a circle is\nRamsey.\n\nA rival conjecture:",
   "evidence": "The canonical record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0083": {
   "statement_status": "exact",
   "original_statement": "Conjecture. (Leader--Russell--Walters) A finite set is Ramsey\nif and only if it is a subset of some set with a transitive symmetry group.\n\nFor 3-point sets:",
   "clean_statement": "Conjecture. (Leader--Russell--Walters) A finite set is Ramsey\nif and only if it is a subset of some set with a transitive symmetry group.\n\nFor 3-point sets:",
   "public_statement": "Conjecture. (Leader--Russell--Walters) A finite set is Ramsey\nif and only if it is a subset of some set with a transitive symmetry group.\n\nFor 3-point sets:",
   "evidence": "The canonical record is number 36 in the section “Euclidean Ramsey sets” of the AIM workshop list “Graph Ramsey theory.” Its mathematical content is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 82,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0084": {
   "statement_status": "exact",
   "original_statement": "Conjecture. Let $T$ be a set of three points in\n$\\mathbb{E}^2$. Then there exists a 3-coloring of $\\mathbb{E}^2$ with no\nmonochromatic copy of $T$.",
   "clean_statement": "Conjecture. Let $T$ be a set of three points in\n$\\mathbb{E}^2$. Then there exists a 3-coloring of $\\mathbb{E}^2$ with no\nmonochromatic copy of $T$.",
   "public_statement": "Conjecture. Let $T$ be a set of three points in\n$\\mathbb{E}^2$. Then there exists a 3-coloring of $\\mathbb{E}^2$ with no\nmonochromatic copy of $T$.",
   "evidence": "The exact canonical record is Conjecture 37 in the AIM workshop list *Graph Ramsey theory*, section “Euclidean Ramsey sets”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 83,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0085": {
   "statement_status": "exact",
   "original_statement": "Conjecture. In any 2-coloring of $\\mathbb{E}^2$, every\n3-point set $X$ occurs monochromatically (as some congruent copy), except\npossibly for the three points that form some specific equilateral triangle.\n\n(Example coloring: half-open horizontal stripes of height $\\sqrt{3}/2$ in\nalternating colors will stop the three vertices of a unit equilateral triangle\nfrom occurring monochromatically).\n\nKnown: If $T$ is a set of 3 collinear points, then there exists\na 16-coloring of $\\mathbb{E}^N$ with no monochromatic copy of $T$.",
   "clean_statement": "Conjecture. In any 2-coloring of $\\mathbb{E}^2$, every\n3-point set $X$ occurs monochromatically (as some congruent copy), except\npossibly for the three points that form some specific equilateral triangle.\n\n(Example coloring: half-open horizontal stripes of height $\\sqrt{3}/2$ in\nalternating colors will stop the three vertices of a unit equilateral triangle\nfrom occurring monochromatically).\n\nKnown: If $T$ is a set of 3 collinear points, then there exists\na 16-coloring of $\\mathbb{E}^N$ with no monochromatic copy of $T$.",
   "public_statement": "Conjecture. In any 2-coloring of $\\mathbb{E}^2$, every\n3-point set $X$ occurs monochromatically (as some congruent copy), except\npossibly for the three points that form some specific equilateral triangle.\n\n(Example coloring: half-open horizontal stripes of height $\\sqrt{3}/2$ in\nalternating colors will stop the three vertices of a unit equilateral triangle\nfrom occurring monochromatically).\n\nKnown: If $T$ is a set of 3 collinear points, then there exists\na 16-coloring of $\\mathbb{E}^N$ with no monochromatic copy of $T$.",
   "evidence": "The exact source record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 84,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0086": {
   "statement_status": "exact",
   "original_statement": "Question: What is the minimum number of colors needed above?",
   "clean_statement": "Question: What is the minimum number of colors needed above?",
   "public_statement": "Question: What is the minimum number of colors needed above?",
   "evidence": "The canonical record says only:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 85,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0087": {
   "statement_status": "exact",
   "original_statement": "Folkman graphs\n\nRon Graham\n\nA Folkman graph is a $K_4$-free graph such that every\n2-edge-coloring contains a monochromatic triangle.\n\nFolkman showed that such a graph exists, but he required an enormous\nnumber of vertices. Proofs of existence of smaller Folkman graphs were\ngiven by Frankl--R\\\"odl, Spencer, Lu, Dudek--R\\\"odl,\nLange--Radziszowski--Xu.\n\nChallenge. (\\$100) Find a Folkman graph with fewer than 100\nvertices.\n\nCandidate construction: (Exoo) take $\\mathbb{Z}_{127}$ as vertices, and join\n$i$ with $j$ if $i-j$ is a cubic residue mod 127, and then delete 33\nvertices (three independent sets of size 11).",
   "clean_statement": "Folkman graphs\n\nRon Graham\n\nA Folkman graph is a $K_4$-free graph such that every\n2-edge-coloring contains a monochromatic triangle.\n\nFolkman showed that such a graph exists, but he required an enormous\nnumber of vertices. Proofs of existence of smaller Folkman graphs were\ngiven by Frankl--R\\\"odl, Spencer, Lu, Dudek--R\\\"odl,\nLange--Radziszowski--Xu.\n\nChallenge. (\\$100) Find a Folkman graph with fewer than 100\nvertices.\n\nCandidate construction: (Exoo) take $\\mathbb{Z}_{127}$ as vertices, and join\n$i$ with $j$ if $i-j$ is a cubic residue mod 127, and then delete 33\nvertices (three independent sets of size 11).",
   "public_statement": "Folkman graphs\n\nRon Graham\n\nA Folkman graph is a $K_4$-free graph such that every\n2-edge-coloring contains a monochromatic triangle.\n\nFolkman showed that such a graph exists, but he required an enormous\nnumber of vertices. Proofs of existence of smaller Folkman graphs were\ngiven by Frankl--R\\\"odl, Spencer, Lu, Dudek--R\\\"odl,\nLange--Radziszowski--Xu.\n\nChallenge. (\\$100) Find a Folkman graph with fewer than 100\nvertices.\n\nCandidate construction: (Exoo) take $\\mathbb{Z}_{127}$ as vertices, and join\n$i$ with $j$ if $i-j$ is a cubic residue mod 127, and then delete 33\nvertices (three independent sets of size 11).",
   "evidence": "The source record is challenge 40 from the AIM workshop section “Folkman graphs”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 86,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0088": {
   "statement_status": "exact",
   "original_statement": "van der Waerden numbers\n\nRon Graham\n\nLet $W(n,m)$ denote the least $N$ so that any red/blue coloring of $\\{1,\n\\dots, N\\}$ contains either a red $n$-term arithmetic progression or a\nblue $m$-term arithmetic progression.\n\nLet $W(n) = W(n,n)$.\n\nSome data:\n{c|ccccc}\n $n$ & 2 & 3 & 4 & 5 & 6\n \n\n \\hline\n $W(n)$ & 3 & 9 & 35 & 178 & 1132\n\nKnown: (Berlekamp) $W(n+1) \\ge n 2^n$ for $n$ prime.\n\n(Gowers) $W(n) \\le 2^{2^{2^{2^{2^{n+9}}}}}$.\n\nConjecture: (\\$1000) $W(n) \\le 2^{n^2}$.\n\nLet $W^*(n)$ denote the size of the smallest subset $S$ of\n$\\mathbb{N}$ such that any 2-coloring of $S$ has a monochromatic\n$n$-term arithmetic progression.\n\nKnown: (Elkies) $W^*(3) = W(3).$\n\n$W^*(4) \\le 27 < W(4) = 35.$",
   "clean_statement": "van der Waerden numbers\n\nRon Graham\n\nLet $W(n,m)$ denote the least $N$ so that any red/blue coloring of $\\{1,\n\\dots, N\\}$ contains either a red $n$-term arithmetic progression or a\nblue $m$-term arithmetic progression.\n\nLet $W(n) = W(n,n)$.\n\nSome data:\n{c|ccccc}\n $n$ & 2 & 3 & 4 & 5 & 6\n\n\n \\hline\n $W(n)$ & 3 & 9 & 35 & 178 & 1132\n\nKnown: (Berlekamp) $W(n+1) \\ge n 2^n$ for $n$ prime.\n\n(Gowers) $W(n) \\le 2^{2^{2^{2^{2^{n+9}}}}}$.\n\nConjecture: (\\$1000) $W(n) \\le 2^{n^2}$.\n\nLet $W^*(n)$ denote the size of the smallest subset $S$ of\n$\\mathbb{N}$ such that any 2-coloring of $S$ has a monochromatic\n$n$-term arithmetic progression.\n\nKnown: (Elkies) $W^*(3) = W(3).$\n\n$W^*(4) \\le 27 < W(4) = 35.$",
   "public_statement": "van der Waerden numbers\n\nRon Graham\n\nLet $W(n,m)$ denote the least $N$ so that any red/blue coloring of $\\{1,\n\\dots, N\\}$ contains either a red $n$-term arithmetic progression or a\nblue $m$-term arithmetic progression.\n\nLet $W(n) = W(n,n)$.\n\nSome data:\n{c|ccccc}\n $n$ & 2 & 3 & 4 & 5 & 6\n\n\n \\hline\n $W(n)$ & 3 & 9 & 35 & 178 & 1132\n\nKnown: (Berlekamp) $W(n+1) \\ge n 2^n$ for $n$ prime.\n\n(Gowers) $W(n) \\le 2^{2^{2^{2^{2^{n+9}}}}}$.\n\nConjecture: (\\$1000) $W(n) \\le 2^{n^2}$.\n\nLet $W^*(n)$ denote the size of the smallest subset $S$ of\n$\\mathbb{N}$ such that any 2-coloring of $S$ has a monochromatic\n$n$-term arithmetic progression.\n\nKnown: (Elkies) $W^*(3) = W(3).$\n\n$W^*(4) \\le 27 < W(4) = 35.$",
   "evidence": "The source record, attributed to Ron Graham, defines \\(W(n,m)\\) to be the least \\(N\\) such that every red/blue coloring of \\([N]=\\{1,\\ldots,N\\}\\) contains either a red \\(n\\)-term arithmetic progression or a blue \\(m\\)-term arithmetic progression, and puts \\(W(n)=W(n,n)\\). It records",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 87,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0089": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question: Does $W(n) - W^*(n) \\to \\infty$?\n\n\\medskip\n\nOff diagonal van der Waerden numbers $W(k,3)$\n\nFrom computed values, $W(k,3)$ seems to grow quadratically in $k$.\n\nKnown: $k^{2-o(1)} < W(k) < e^{O(k \\log^5 k)}$.",
   "clean_statement": "Question: Does $W(n)-W^*(n)\\to\\infty$?\n\nOff diagonal van der Waerden numbers $W(k,3)$\n\nFrom computed values, $W(k,3)$ seems to grow quadratically in $k$.\n\nKnown: $k^{2-o(1)}<W(k)<e^{O(k\\log^5 k)}$.",
   "public_statement": "Question: Does $W(n) - W^*(n) \\to \\infty$?\n\n\\medskip\n\nOff diagonal van der Waerden numbers $W(k,3)$\n\nFrom computed values, $W(k,3)$ seems to grow quadratically in $k$.\n\nKnown: $k^{2-o(1)} < W(k) < e^{O(k \\log^5 k)}$.",
   "evidence": "In the last displayed bound of the record, the occurrence of $W(k)$ is almost certainly a transcription error for $W(k,3)$. The heading immediately before it says $W(k,3)$, Brown--Landman--Robertson proved the corresponding $k^{2-o(1)}$ lower bound for the off-diagonal number, and the next corpus record separately asks about polynomial growth of the diagonal $W(k)$. All conclusions below use this explicitly labelled reconstruction.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 88,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0090": {
   "statement_status": "exact",
   "original_statement": "Question: Does $W(k)$ grow polynomially in $k$?",
   "clean_statement": "Question: Does $W(k)$ grow polynomially in $k$?",
   "public_statement": "Question: Does $W(k)$ grow polynomially in $k$?",
   "evidence": "The exact source record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 89,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0091": {
   "statement_status": "exact",
   "original_statement": "(1) Spencer Backman: \"Is there a generalization of chip-firing to k-uniform hypergraphs, where chips are represented by kth roots of unity instead of ±1? I have the general set-up. You need some additional data: a cyclic ordering of each edge. When a vertex fires, it sends the appropriate root of unity according to this order to its adjacent vertices. There are some encouraging signs: the complex Laplacian matrix is positive-semidefinite and the process is invariant under which total ordering of the vertices you take. I wrote down my thoughts about this problem in my statement of interest on the workshop's website. 1\"(a) Andrea Sportiello: \"Perhaps we need only that the vectors sum to 0, not necessarily that they be roots of unity.\" (b) Spencer Backman: \"When is a chip value 'non-negative'? I be-lieve it is when the argument lies in some range, say between ω\n\nand ωn−1. But what is the kernel of the Laplacian in this case? What is a good notion of reduced divisor, recurrent configura-tion, and so on?\"",
   "clean_statement": "(1) Spencer Backman: \"Is there a generalization of chip-firing to k-uniform hypergraphs, where chips are represented by kth roots of unity instead of ±1? I have the general set-up. You need some additional data: a cyclic ordering of each edge. When a vertex fires, it sends the appropriate root of unity according to this order to its adjacent vertices. There are some encouraging signs: the complex Laplacian matrix is positive-semidefinite and the process is invariant under which total ordering of the vertices you take. I wrote down my thoughts about this problem in my statement of interest on the workshop's website. 1\"(a) Andrea Sportiello: \"Perhaps we need only that the vectors sum to 0, not necessarily that they be roots of unity.\" (b) Spencer Backman: \"When is a chip value 'non-negative'? I be-lieve it is when the argument lies in some range, say between ω\n\nand ωn−1. But what is the kernel of the Laplacian in this case? What is a good notion of reduced divisor, recurrent configura-tion, and so on?\"",
   "public_statement": "(1) Spencer Backman: \"Is there a generalization of chip-firing to k-uniform hypergraphs, where chips are represented by kth roots of unity instead of ±1? I have the general set-up. You need some additional data: a cyclic ordering of each edge. When a vertex fires, it sends the appropriate root of unity according to this order to its adjacent vertices. There are some encouraging signs: the complex Laplacian matrix is positive-semidefinite and the process is invariant under which total ordering of the vertices you take. I wrote down my thoughts about this problem in my statement of interest on the workshop's website. 1\"(a) Andrea Sportiello: \"Perhaps we need only that the vectors sum to 0, not necessarily that they be roots of unity.\" (b) Spencer Backman: \"When is a chip value 'non-negative'? I be-lieve it is when the argument lies in some range, say between ω\n\nand ωn−1. But what is the kernel of the Laplacian in this case? What is a good notion of reduced divisor, recurrent configura-tion, and so on?\"",
   "evidence": "The source is the first problem in *Problems from the AIM Chip-Firing Workshop*, recorded by Sam Hopkins after the July 8--12, 2013 AIM workshop “Generalizations of chip-firing and the critical group.” The PDF explicitly warns that the quotations are summaries rather than direct quotations.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 90,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0092": {
   "statement_status": "exact",
   "original_statement": "(2) Jeremy Martin: \"We can efficiently sample random spanning trees of a graph in a uniform way by sampling the Jacobian group randomly. Can this approach be made to work when I change the graph to something more general like a simplicial complex?\" (a) Farbod Shokrieh: \"The class of regular matroids is a good place to start looking for such a generalization. In this case we can define the Jacobian and we know the size is correct (that is, is equal to the size of a basis of the matroid). The problem is finding an efficient bijection. We have no analog of Dhar's burning algorithm at this level.\" \n\n> 1The wokshop's website is aimath.org/WWN/chipfiring/.\n> 12PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\n\n(b) Jeremy Martin: \"I can state the problem more concretely as follows. Let D be an n × m matrix over Z of rank n, and let L: Zn → Zn be given by L:= DD t. Define \n\nJ(D):= Zn/im L = coker( L).\n\nWe can uniformly sample from J(D). But can we sample uni-formly from the column bases of D?\" (c) Matt Baker: \"If D is unimodular, then the usual Cauchy-Binet proof of Matrix-Tree theorem works and shows that J(D) is in bijection with a column basis of D. We are really asking: is this bijection efficiently computable?\" (d) Jeremy Martin: \"Actually, I believe you do not need unimodu-larity; you just need the determinants of all the maximal minors to be ±1.\"",
   "clean_statement": "(2) Jeremy Martin: \"We can efficiently sample random spanning trees of a graph in a uniform way by sampling the Jacobian group randomly. Can this approach be made to work when I change the graph to something more general like a simplicial complex?\" (a) Farbod Shokrieh: \"The class of regular matroids is a good place to start looking for such a generalization. In this case we can define the Jacobian and we know the size is correct (that is, is equal to the size of a basis of the matroid). The problem is finding an efficient bijection. We have no analog of Dhar's burning algorithm at this level.\"\n\n> 1The wokshop's website is aimath.org/WWN/chipfiring/.\n> 12PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\n\n(b) Jeremy Martin: \"I can state the problem more concretely as follows. Let D be an n × m matrix over Z of rank n, and let L: Zn → Zn be given by L:= DD t. Define\n\nJ(D):= Zn/im L = coker( L).\n\nWe can uniformly sample from J(D). But can we sample uni-formly from the column bases of D?\" (c) Matt Baker: \"If D is unimodular, then the usual Cauchy-Binet proof of Matrix-Tree theorem works and shows that J(D) is in bijection with a column basis of D. We are really asking: is this bijection efficiently computable?\" (d) Jeremy Martin: \"Actually, I believe you do not need unimodu-larity; you just need the determinants of all the maximal minors to be ±1.\"",
   "public_statement": "(2) Jeremy Martin: \"We can efficiently sample random spanning trees of a graph in a uniform way by sampling the Jacobian group randomly. Can this approach be made to work when I change the graph to something more general like a simplicial complex?\" (a) Farbod Shokrieh: \"The class of regular matroids is a good place to start looking for such a generalization. In this case we can define the Jacobian and we know the size is correct (that is, is equal to the size of a basis of the matroid). The problem is finding an efficient bijection. We have no analog of Dhar's burning algorithm at this level.\"\n\n> 1The wokshop's website is aimath.org/WWN/chipfiring/.\n> 12PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\n\n(b) Jeremy Martin: \"I can state the problem more concretely as follows. Let D be an n × m matrix over Z of rank n, and let L: Zn → Zn be given by L:= DD t. Define\n\nJ(D):= Zn/im L = coker( L).\n\nWe can uniformly sample from J(D). But can we sample uni-formly from the column bases of D?\" (c) Matt Baker: \"If D is unimodular, then the usual Cauchy-Binet proof of Matrix-Tree theorem works and shows that J(D) is in bijection with a column basis of D. We are really asking: is this bijection efficiently computable?\" (d) Jeremy Martin: \"Actually, I believe you do not need unimodu-larity; you just need the determinants of all the maximal minors to be ±1.\"",
   "evidence": "This is Problem 2 in the problem list from the July 8--12, 2013 AIM workshop *Generalizations of chip-firing and the critical group*. The source PDF is the authoritative text [AIM13]. The corpus extraction contains three OCR/layout errors that matter:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 91,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0093": {
   "statement_status": "exact",
   "original_statement": "(3) Persi Diaconis: \"Say you play the explorers game [16] on a circle with n vertices labeled cyclically 0 through n − 1 and you repeatedly explore from 0. The probability that the last point occupied before the circle is filled is j is given by A(n, j )/n!, where A(n, j ) are the Eulerian numbers. 2 This fact was shown to me by Andrea Sportiello during the lunch break. What is interesting is that the situation is very different from a random walk, in which the distribution is uniform. It may be worth looking at the probability distribution of last occupied spot in the explorers game for other sets (for example in higher dimensions or on circulant graphs).\" (a) Lionel Levine: \"Jim Propp made this observation about the Eulerian numbers a while ago.\" (b) James Propp: \"There are connections here to other models such as P´ olya's urn and Bernard Friedman's urn. Are these connec-tions useful?\" (c) Andrea Sportiello: \"If you change the graph to something like Z2,I would guess that things become ugly very fast.\" (d) Lionel Levine: \"It is not clear how to generalize to higher di-mensions because so much is going on, but sticking to the one-dimensional case and putting weights on the edges may give some new q-analog.\"",
   "clean_statement": "(3) Persi Diaconis: \"Say you play the explorers game [16] on a circle with n vertices labeled cyclically 0 through n − 1 and you repeatedly explore from 0. The probability that the last point occupied before the circle is filled is j is given by A(n, j )/n!, where A(n, j ) are the Eulerian numbers. 2 This fact was shown to me by Andrea Sportiello during the lunch break. What is interesting is that the situation is very different from a random walk, in which the distribution is uniform. It may be worth looking at the probability distribution of last occupied spot in the explorers game for other sets (for example in higher dimensions or on circulant graphs).\" (a) Lionel Levine: \"Jim Propp made this observation about the Eulerian numbers a while ago.\" (b) James Propp: \"There are connections here to other models such as P´ olya's urn and Bernard Friedman's urn. Are these connec-tions useful?\" (c) Andrea Sportiello: \"If you change the graph to something like Z2,I would guess that things become ugly very fast.\" (d) Lionel Levine: \"It is not clear how to generalize to higher di-mensions because so much is going on, but sticking to the one-dimensional case and putting weights on the edges may give some new q-analog.\"",
   "public_statement": "(3) Persi Diaconis: \"Say you play the explorers game [16] on a circle with n vertices labeled cyclically 0 through n − 1 and you repeatedly explore from 0. The probability that the last point occupied before the circle is filled is j is given by A(n, j )/n!, where A(n, j ) are the Eulerian numbers. 2 This fact was shown to me by Andrea Sportiello during the lunch break. What is interesting is that the situation is very different from a random walk, in which the distribution is uniform. It may be worth looking at the probability distribution of last occupied spot in the explorers game for other sets (for example in higher dimensions or on circulant graphs).\" (a) Lionel Levine: \"Jim Propp made this observation about the Eulerian numbers a while ago.\" (b) James Propp: \"There are connections here to other models such as P´ olya's urn and Bernard Friedman's urn. Are these connec-tions useful?\" (c) Andrea Sportiello: \"If you change the graph to something like Z2,I would guess that things become ugly very fast.\" (d) Lionel Levine: \"It is not clear how to generalize to higher di-mensions because so much is going on, but sticking to the one-dimensional case and putting weights on the edges may give some new q-analog.\"",
   "evidence": "The official AIM PDF is the problem-session record from the July 2013 workshop “Generalizations of chip-firing and the critical group.” Its introduction warns that the quotations are summaries rather than direct transcriptions.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0094": {
   "statement_status": "exact",
   "original_statement": "(4) Sam Hopkins: \"This question concerns a recent preprint of Venkatesh and Viswanath [34], which shows that for a finite graph G on n ver-tices, the dimension of the root space associated to the root that is the sum of all simple roots in the Kac-Moody algebra having Cartan matrix 2 In − A(G) is the number of maximal G-parking functions. (Here A(G) is the adjacency matrix of G.) Can this be exploited further? Are all G-parking functions present somewhere? Does the sandpile group act on some such root space?\" \n\n> 2For a definition of Eulerian numbers, see en.wikipedia.org/wiki/Eulerian number. PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 3\n\n(a) Sam Hopkins: \"I should also mention that there is a natural labeling of a basis of this root space by maximal parking func-tions.\" (b) Anton Dochtermann: \"The number of maximal parking func-tions is also the number of last syzygies in a minimal free reso-lution of the toppling ideal, and also the number of chambers of a certain slice of the graphic arrangement. Can we connect the Lie theory to commutative algebra or hyperlane arrangements?\"",
   "clean_statement": "(4) Sam Hopkins: \"This question concerns a recent preprint of Venkatesh and Viswanath [34], which shows that for a finite graph G on n ver-tices, the dimension of the root space associated to the root that is the sum of all simple roots in the Kac-Moody algebra having Cartan matrix 2 In − A(G) is the number of maximal G-parking functions. (Here A(G) is the adjacency matrix of G.) Can this be exploited further? Are all G-parking functions present somewhere? Does the sandpile group act on some such root space?\"\n\n> 2For a definition of Eulerian numbers, see en.wikipedia.org/wiki/Eulerian number. PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 3\n\n(a) Sam Hopkins: \"I should also mention that there is a natural labeling of a basis of this root space by maximal parking func-tions.\" (b) Anton Dochtermann: \"The number of maximal parking func-tions is also the number of last syzygies in a minimal free reso-lution of the toppling ideal, and also the number of chambers of a certain slice of the graphic arrangement. Can we connect the Lie theory to commutative algebra or hyperlane arrangements?\"",
   "public_statement": "(4) Sam Hopkins: \"This question concerns a recent preprint of Venkatesh and Viswanath [34], which shows that for a finite graph G on n ver-tices, the dimension of the root space associated to the root that is the sum of all simple roots in the Kac-Moody algebra having Cartan matrix 2 In − A(G) is the number of maximal G-parking functions. (Here A(G) is the adjacency matrix of G.) Can this be exploited further? Are all G-parking functions present somewhere? Does the sandpile group act on some such root space?\"\n\n> 2For a definition of Eulerian numbers, see en.wikipedia.org/wiki/Eulerian number. PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 3\n\n(a) Sam Hopkins: \"I should also mention that there is a natural labeling of a basis of this root space by maximal parking func-tions.\" (b) Anton Dochtermann: \"The number of maximal parking func-tions is also the number of last syzygies in a minimal free reso-lution of the toppling ideal, and also the number of chambers of a certain slice of the graphic arrangement. Can we connect the Lie theory to commutative algebra or hyperlane arrangements?\"",
   "evidence": "The canonical JSON record is an OCR extraction of problem (4) from the AIM workshop list *Generalizations of chip-firing and the critical group*. The extraction splits words at line endings, renders the identity matrix incorrectly, misspells “hyperplane,” and inserts a page header and an Eulerian-number footnote belonging to the preceding problem. Inspection of pages 2–3 of the original PDF gives the following recovered text (with only mathematical typesetting normalized):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0095": {
   "statement_status": "exact",
   "original_statement": "(5) Lionel Levine: \"How can we uniformly and efficiently sample from the set of maximal G-parking functions?\" (a) Persi Diacons: \"There may be a connection to my paper with Christos Athanasiadis [5].\"",
   "clean_statement": "(5) Lionel Levine: \"How can we uniformly and efficiently sample from the set of maximal G-parking functions?\" (a) Persi Diacons: \"There may be a connection to my paper with Christos Athanasiadis [5].\"",
   "public_statement": "(5) Lionel Levine: \"How can we uniformly and efficiently sample from the set of maximal G-parking functions?\" (a) Persi Diacons: \"There may be a connection to my paper with Christos Athanasiadis [5].\"",
   "evidence": "The source is the AIM workshop problem list *Generalizations of chip-firing and the critical group*, compiled after the workshop of 8--12 July 2013. The printed item is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 94,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0096": {
   "statement_status": "exact",
   "original_statement": "(6) Persi Diaconis: \"When we play the explorers game and repeatedly explore from the origin on a square lattice in any dimension, the lim-iting shape is round. In dimensions d ≥ 3, the fluctuations after n\n\nsteps from this ball of radius n1/d are tiny, on the order of √log( n). For d = 2, the fluctuations are of size log( n), and for d = 1 the fluctuations are of size √n. These results in the higher dimensional cases have only been obtained recently; see [22], [4], and [21]. Are there other growth models where the fluctuations take on some in-termediate value, say between √n and log( n)?\" (a) Andrea Sportiello: \"If we consider a wedge of the two dimen-sional lattice Z2, is it clear that the limiting shape is the sector of the circle and does not depend on the angles?\" (b) Charles Smart: \"Away from the border we should be fine.\" (c) Andrea Sportiello: \"Then if you take the part of the lattice inside a parabola of exponent γ (that is, where |y| ≤ x1/γ ), which is a further generalization of a wedge, this interpolates between one and two dimensions and is thus a candidate for intermediate growth of fluctuations.\" (d) Igor Pak: \"Yes, in general if we consider the part of the lattice given by |y| ≤ f (x), we should be able to interpolate however we want between √n and log( n) by choosing the appropriate function f.\"",
   "clean_statement": "(6) Persi Diaconis: \"When we play the explorers game and repeatedly explore from the origin on a square lattice in any dimension, the lim-iting shape is round. In dimensions d ≥ 3, the fluctuations after n\n\nsteps from this ball of radius n1/d are tiny, on the order of √log( n). For d = 2, the fluctuations are of size log( n), and for d = 1 the fluctuations are of size √n. These results in the higher dimensional cases have only been obtained recently; see [22], [4], and [21]. Are there other growth models where the fluctuations take on some in-termediate value, say between √n and log( n)?\" (a) Andrea Sportiello: \"If we consider a wedge of the two dimen-sional lattice Z2, is it clear that the limiting shape is the sector of the circle and does not depend on the angles?\" (b) Charles Smart: \"Away from the border we should be fine.\" (c) Andrea Sportiello: \"Then if you take the part of the lattice inside a parabola of exponent γ (that is, where |y| ≤ x1/γ ), which is a further generalization of a wedge, this interpolates between one and two dimensions and is thus a candidate for intermediate growth of fluctuations.\" (d) Igor Pak: \"Yes, in general if we consider the part of the lattice given by |y| ≤ f (x), we should be able to interpolate however we want between √n and log( n) by choosing the appropriate function f.\"",
   "public_statement": "(6) Persi Diaconis: \"When we play the explorers game and repeatedly explore from the origin on a square lattice in any dimension, the lim-iting shape is round. In dimensions d ≥ 3, the fluctuations after n\n\nsteps from this ball of radius n1/d are tiny, on the order of √log( n). For d = 2, the fluctuations are of size log( n), and for d = 1 the fluctuations are of size √n. These results in the higher dimensional cases have only been obtained recently; see [22], [4], and [21]. Are there other growth models where the fluctuations take on some in-termediate value, say between √n and log( n)?\" (a) Andrea Sportiello: \"If we consider a wedge of the two dimen-sional lattice Z2, is it clear that the limiting shape is the sector of the circle and does not depend on the angles?\" (b) Charles Smart: \"Away from the border we should be fine.\" (c) Andrea Sportiello: \"Then if you take the part of the lattice inside a parabola of exponent γ (that is, where |y| ≤ x1/γ ), which is a further generalization of a wedge, this interpolates between one and two dimensions and is thus a candidate for intermediate growth of fluctuations.\" (d) Igor Pak: \"Yes, in general if we consider the part of the lattice given by |y| ≤ f (x), we should be able to interpolate however we want between √n and log( n) by choosing the appropriate function f.\"",
   "evidence": "This is Problem 6 in the AIM workshop problem list *Generalizations of chip-firing and the critical group* (workshop held 8--12 July 2013). The introduction to the list says that the displayed comments are recorder's summaries rather than necessarily verbatim quotations. The source is the official AIM PDF [1].",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0097": {
   "statement_status": "exact",
   "original_statement": "(7) Matt Macauley: \"Consider the equivalence relation on the set of all acyclic orientations of a graph G of converting sources to sinks. This relation encodes the chip-firing game on G among a special class of divisors. In this special case, there is a nice geometric way to view chip-firing via passing from the graphic arrangement (whose regions are in bijection with all acyclic orientations) to the toric graphic arrangement. Can we find a similar geometric way of viewing chip-firing equivalence for other (larger) classes of divisors?\" (a) Andrea Sportiello: \"I think we need to look at multi-toppling, or what has also been called hereditary chip-firing, to accomplish this.\" 4 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP \n\n(b) James Propp: \"Does multi-toppling have the abelian property?\" (c) Andrea Sportiello: \"It has the abelian property precisely when it is hereditary, when we fire clusters and then sub-clusters and so on.\" (d) Lionel Levine: \"A good source for hereditary chip-firing is a paper of Spencer Backman [6].\"",
   "clean_statement": "(7) Matt Macauley: \"Consider the equivalence relation on the set of all acyclic orientations of a graph G of converting sources to sinks. This relation encodes the chip-firing game on G among a special class of divisors. In this special case, there is a nice geometric way to view chip-firing via passing from the graphic arrangement (whose regions are in bijection with all acyclic orientations) to the toric graphic arrangement. Can we find a similar geometric way of viewing chip-firing equivalence for other (larger) classes of divisors?\" (a) Andrea Sportiello: \"I think we need to look at multi-toppling, or what has also been called hereditary chip-firing, to accomplish this.\" 4 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\n\n(b) James Propp: \"Does multi-toppling have the abelian property?\" (c) Andrea Sportiello: \"It has the abelian property precisely when it is hereditary, when we fire clusters and then sub-clusters and so on.\" (d) Lionel Levine: \"A good source for hereditary chip-firing is a paper of Spencer Backman [6].\"",
   "public_statement": "(7) Matt Macauley: \"Consider the equivalence relation on the set of all acyclic orientations of a graph G of converting sources to sinks. This relation encodes the chip-firing game on G among a special class of divisors. In this special case, there is a nice geometric way to view chip-firing via passing from the graphic arrangement (whose regions are in bijection with all acyclic orientations) to the toric graphic arrangement. Can we find a similar geometric way of viewing chip-firing equivalence for other (larger) classes of divisors?\" (a) Andrea Sportiello: \"I think we need to look at multi-toppling, or what has also been called hereditary chip-firing, to accomplish this.\" 4 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\n\n(b) James Propp: \"Does multi-toppling have the abelian property?\" (c) Andrea Sportiello: \"It has the abelian property precisely when it is hereditary, when we fire clusters and then sub-clusters and so on.\" (d) Lionel Levine: \"A good source for hereditary chip-firing is a paper of Spencer Backman [6].\"",
   "evidence": "This is Problem 7 in the official AIM list *Generalizations of chip-firing and the critical group*, produced after the workshop of 8--12 July 2013 [1]. The PDF introduction warns that the displayed comments are recorder's summaries and not necessarily verbatim quotations.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 96,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0098": {
   "statement_status": "exact",
   "original_statement": "(8) Jordan Ellenberg: \"By the Matrix-Tree theorem, the number of spanning trees of a graph G equals the size of the critical group of G. When are the spanning trees a torsor for the critical group? Making G into a ribbon graph, and given choice of sink, we do get a torsor structure from the rotor-router model [20]. But under what conditions can we get a natural torsor action without this additional data?\" (a) Jordan Ellenberg: \"If you do not know what a torsor is, I am asking when there is a simply-transitive action of the critical group on the set of spanning trees. Simply-transitive means that for any pair of spanning trees T and T ′, there is a unique group element taking T to T ′. When I say I want this action to be 'natural', I mean at least that I want it to respect the automorphism group of G.\" (b) Farbod Shokrieh: \"David Wagner has an argument (although not a proof) that we cannot expect to get a torsor in general.\" (c) Matt Baker: \"A recent paper of mine and several coauthors [2] almost achieves this through a polyhedral decomposition of the real g-dimensional torus (where g is the genus of the graph). The cells of this decomposition are canonically in bijection with the spanning trees, the vertices are canonically in bijection with Pic g(G), and Pic g(G) is canonically a torsor for the critical group. The only thing that is not canonical here is the bijection between vertices and cells of the decomposition.\" (d) Lionel Levine: \"Why is the genus special here?\" (e) Matt Baker: \"The set of reduced divisors depends on the choice of sink vertex q. It turns out that there is another nice set of coset representatives (well not really coset representatives because they do not form a group, but representatives of the set of degree g divisors modulo chip-firing equivalence) that can be defined in a canonical way.\" (f) Melody Chan: \"In my recent paper with Church and Gro-chow [14], we look at when the choice of sink is needed in the rotor-router model. We show that the resulting torsor is in-dependent of the choice of sink if and only if the graph G is planar.\"",
   "clean_statement": "(8) Jordan Ellenberg: \"By the Matrix-Tree theorem, the number of spanning trees of a graph G equals the size of the critical group of G. When are the spanning trees a torsor for the critical group? Making G into a ribbon graph, and given choice of sink, we do get a torsor structure from the rotor-router model [20]. But under what conditions can we get a natural torsor action without this additional data?\" (a) Jordan Ellenberg: \"If you do not know what a torsor is, I am asking when there is a simply-transitive action of the critical group on the set of spanning trees. Simply-transitive means that for any pair of spanning trees T and T ′, there is a unique group element taking T to T ′. When I say I want this action to be 'natural', I mean at least that I want it to respect the automorphism group of G.\" (b) Farbod Shokrieh: \"David Wagner has an argument (although not a proof) that we cannot expect to get a torsor in general.\" (c) Matt Baker: \"A recent paper of mine and several coauthors [2] almost achieves this through a polyhedral decomposition of the real g-dimensional torus (where g is the genus of the graph). The cells of this decomposition are canonically in bijection with the spanning trees, the vertices are canonically in bijection with Pic g(G), and Pic g(G) is canonically a torsor for the critical group. The only thing that is not canonical here is the bijection between vertices and cells of the decomposition.\" (d) Lionel Levine: \"Why is the genus special here?\" (e) Matt Baker: \"The set of reduced divisors depends on the choice of sink vertex q. It turns out that there is another nice set of coset representatives (well not really coset representatives because they do not form a group, but representatives of the set of degree g divisors modulo chip-firing equivalence) that can be defined in a canonical way.\" (f) Melody Chan: \"In my recent paper with Church and Gro-chow [14], we look at when the choice of sink is needed in the rotor-router model. We show that the resulting torsor is in-dependent of the choice of sink if and only if the graph G is planar.\"",
   "public_statement": "(8) Jordan Ellenberg: \"By the Matrix-Tree theorem, the number of spanning trees of a graph G equals the size of the critical group of G. When are the spanning trees a torsor for the critical group? Making G into a ribbon graph, and given choice of sink, we do get a torsor structure from the rotor-router model [20]. But under what conditions can we get a natural torsor action without this additional data?\" (a) Jordan Ellenberg: \"If you do not know what a torsor is, I am asking when there is a simply-transitive action of the critical group on the set of spanning trees. Simply-transitive means that for any pair of spanning trees T and T ′, there is a unique group element taking T to T ′. When I say I want this action to be 'natural', I mean at least that I want it to respect the automorphism group of G.\" (b) Farbod Shokrieh: \"David Wagner has an argument (although not a proof) that we cannot expect to get a torsor in general.\" (c) Matt Baker: \"A recent paper of mine and several coauthors [2] almost achieves this through a polyhedral decomposition of the real g-dimensional torus (where g is the genus of the graph). The cells of this decomposition are canonically in bijection with the spanning trees, the vertices are canonically in bijection with Pic g(G), and Pic g(G) is canonically a torsor for the critical group. The only thing that is not canonical here is the bijection between vertices and cells of the decomposition.\" (d) Lionel Levine: \"Why is the genus special here?\" (e) Matt Baker: \"The set of reduced divisors depends on the choice of sink vertex q. It turns out that there is another nice set of coset representatives (well not really coset representatives because they do not form a group, but representatives of the set of degree g divisors modulo chip-firing equivalence) that can be defined in a canonical way.\" (f) Melody Chan: \"In my recent paper with Church and Gro-chow [14], we look at when the choice of sink is needed in the rotor-router model. We show that the resulting torsor is in-dependent of the choice of sink if and only if the graph G is planar.\"",
   "evidence": "This is Problem 8 from the AIM workshop list *Generalizations of chip-firing and the critical group*. Jordan Ellenberg asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 97,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0099": {
   "statement_status": "exact",
   "original_statement": "(9) Lionel Levine: \"Is there an efficient way to compute the rank (in the sense of Baker and Norine [7]) of a divisor?\" PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 5",
   "clean_statement": "(9) Lionel Levine: \"Is there an efficient way to compute the rank (in the sense of Baker and Norine [7]) of a divisor?\" PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 5",
   "public_statement": "(9) Lionel Levine: \"Is there an efficient way to compute the rank (in the sense of Baker and Norine [7]) of a divisor?\" PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 5",
   "evidence": "The canonical JSON record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 98,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0100": {
   "statement_status": "exact",
   "original_statement": "(10) Lionel Levine: \"This question is inspired by Criel Merino's work [25] relating the Tutte polynomial to chip-firing. The Tutte polynomial is the most general invariant of a graph that has a deletion-contraction recurrence. Is there a deletion-contraction recurrence of some sort for critical groups? It would have to be at the level of group algebras rather than groups themselves.\" (a) Jeremy Martin: \"Start by considering just a cycle graph.\" (b) Lionel Levine: \"For hyperplane arrangement, there is a nice deletion-contraction rule that gives an exact sequence of alge-bras; is there a corresponding exact sequence for critical groups?\"",
   "clean_statement": "(10) Lionel Levine: \"This question is inspired by Criel Merino's work [25] relating the Tutte polynomial to chip-firing. The Tutte polynomial is the most general invariant of a graph that has a deletion-contraction recurrence. Is there a deletion-contraction recurrence of some sort for critical groups? It would have to be at the level of group algebras rather than groups themselves.\" (a) Jeremy Martin: \"Start by considering just a cycle graph.\" (b) Lionel Levine: \"For hyperplane arrangement, there is a nice deletion-contraction rule that gives an exact sequence of alge-bras; is there a corresponding exact sequence for critical groups?\"",
   "public_statement": "(10) Lionel Levine: \"This question is inspired by Criel Merino's work [25] relating the Tutte polynomial to chip-firing. The Tutte polynomial is the most general invariant of a graph that has a deletion-contraction recurrence. Is there a deletion-contraction recurrence of some sort for critical groups? It would have to be at the level of group algebras rather than groups themselves.\" (a) Jeremy Martin: \"Start by considering just a cycle graph.\" (b) Lionel Levine: \"For hyperplane arrangement, there is a nice deletion-contraction rule that gives an exact sequence of alge-bras; is there a corresponding exact sequence for critical groups?\"",
   "evidence": "The canonical record is problem 10 from the AIM workshop *Generalizations of chip-firing and the critical group*. The official PDF was checked directly (page 5 of the printed report, PDF page 4). Apart from the line-break OCR error “alge-bras,” the corpus record is accurate:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 99,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0101": {
   "statement_status": "exact",
   "original_statement": "(11) Lionel Levine: \"Suppose I give you a stable, periodic sandpile con-figuration c on the square grid Z2. Is it decidable whether there is an integer n such that c + nδ 0 (i.e. the addition of n grains of sand to the origin) is not stabilizable?\" (a) Matt Baker: \"What do the sets of n such that c + nδ 0 is not stabilizable look like when there is such an n?\" (b) Lionel Levine: \"In that case there is a minimal such n.\" (c) Andrea Sportiello: \"This minimal n must depend on the period; what else could it depend on?\" (d) Lionel Levine: \"Well that would certainly make the problem decidable.\"",
   "clean_statement": "(11) Lionel Levine: \"Suppose I give you a stable, periodic sandpile con-figuration c on the square grid Z2. Is it decidable whether there is an integer n such that c + nδ 0 (i.e. the addition of n grains of sand to the origin) is not stabilizable?\" (a) Matt Baker: \"What do the sets of n such that c + nδ 0 is not stabilizable look like when there is such an n?\" (b) Lionel Levine: \"In that case there is a minimal such n.\" (c) Andrea Sportiello: \"This minimal n must depend on the period; what else could it depend on?\" (d) Lionel Levine: \"Well that would certainly make the problem decidable.\"",
   "public_statement": "(11) Lionel Levine: \"Suppose I give you a stable, periodic sandpile con-figuration c on the square grid Z2. Is it decidable whether there is an integer n such that c + nδ 0 (i.e. the addition of n grains of sand to the origin) is not stabilizable?\" (a) Matt Baker: \"What do the sets of n such that c + nδ 0 is not stabilizable look like when there is such an n?\" (b) Lionel Levine: \"In that case there is a minimal such n.\" (c) Andrea Sportiello: \"This minimal n must depend on the period; what else could it depend on?\" (d) Lionel Levine: \"Well that would certainly make the problem decidable.\"",
   "evidence": "The original AIM PDF, *Problems from the AIM Chip-Firing Workshop*, was checked directly. Problem 11 reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 100,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0102": {
   "statement_status": "exact",
   "original_statement": "(12) Persi Diaconis: \"Is there an abelian growth model for non-overlapping discs growing in the plane?\" (a) James Propp: \"There is such an abelian model in the case of \n\nnon-nesting discs; firing means if you have a disc that is totally contained in another one, you move it in some fixed way.\" (b) Anton Dochtermann: \"We could even ask if there is an abelian model of the one-dimensional case of non-overlapping intervals on a line.\"",
   "clean_statement": "(12) Persi Diaconis: \"Is there an abelian growth model for non-overlapping discs growing in the plane?\" (a) James Propp: \"There is such an abelian model in the case of\n\nnon-nesting discs; firing means if you have a disc that is totally contained in another one, you move it in some fixed way.\" (b) Anton Dochtermann: \"We could even ask if there is an abelian model of the one-dimensional case of non-overlapping intervals on a line.\"",
   "public_statement": "(12) Persi Diaconis: \"Is there an abelian growth model for non-overlapping discs growing in the plane?\" (a) James Propp: \"There is such an abelian model in the case of\n\nnon-nesting discs; firing means if you have a disc that is totally contained in another one, you move it in some fixed way.\" (b) Anton Dochtermann: \"We could even ask if there is an abelian model of the one-dimensional case of non-overlapping intervals on a line.\"",
   "evidence": "The official AIM problem list states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 101,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0103": {
   "statement_status": "exact",
   "original_statement": "(13) Lionel Levine: \"What is the right definition of an abelian network with shared memory?\" (a) Lionel Levine: \"I have unified some subset of the class of models that behave like chip-firing does, but a lot of models with an abelian property still fall outside this definition and seem to re-quire some kind of shared memory. An example of shared mem-ory is cluster-firing, as investigated by Spencer Backman [6], who in particular studied when the abelian property is pre-served while allowing firing of multiple vertices at once.\" (b) Jordan Ellenberg: \"This seems formally similar to Persi Diaco-nis's question on non-overlapping discs.\"",
   "clean_statement": "(13) Lionel Levine: \"What is the right definition of an abelian network with shared memory?\" (a) Lionel Levine: \"I have unified some subset of the class of models that behave like chip-firing does, but a lot of models with an abelian property still fall outside this definition and seem to re-quire some kind of shared memory. An example of shared mem-ory is cluster-firing, as investigated by Spencer Backman [6], who in particular studied when the abelian property is pre-served while allowing firing of multiple vertices at once.\" (b) Jordan Ellenberg: \"This seems formally similar to Persi Diaco-nis's question on non-overlapping discs.\"",
   "public_statement": "(13) Lionel Levine: \"What is the right definition of an abelian network with shared memory?\" (a) Lionel Levine: \"I have unified some subset of the class of models that behave like chip-firing does, but a lot of models with an abelian property still fall outside this definition and seem to re-quire some kind of shared memory. An example of shared mem-ory is cluster-firing, as investigated by Spencer Backman [6], who in particular studied when the abelian property is pre-served while allowing firing of multiple vertices at once.\" (b) Jordan Ellenberg: \"This seems formally similar to Persi Diaco-nis's question on non-overlapping discs.\"",
   "evidence": "The canonical record is Problem 13 from the AIM workshop *Generalizations of chip-firing and the critical group*. Its extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 102,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0104": {
   "statement_status": "exact",
   "original_statement": "(14) David Perkinson: \"This question is due to Richard Stanley; it is problem 4 of chapter 6 in his notes on hyperplane arrangements [32]. Is there a natural bijection φ between labeled trees T on {0, 1, 2,..., n }6 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP \n\nrooted at 0 and parking functions of length n such that deg( φ(T )) = g − inv( T ),\n\nwhere g is the genus of the complete graph Kn+1 and inv( T ) is the number of inversions of T? I believe such a bijection would extend to an arbitrary graph by replacing the inversion statistic with the κ\n\nstatistic of Ira Gessel [19]. No bijection I have tried works (and there are many bijections between spanning trees and parking functions in the literature).\" (a) Recorder's note: this problem was resolved in [30].",
   "clean_statement": "(14) David Perkinson: \"This question is due to Richard Stanley; it is problem 4 of chapter 6 in his notes on hyperplane arrangements [32]. Is there a natural bijection φ between labeled trees T on {0, 1, 2,..., n }6 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\n\nrooted at 0 and parking functions of length n such that deg( φ(T )) = g − inv( T ),\n\nwhere g is the genus of the complete graph Kn+1 and inv( T ) is the number of inversions of T? I believe such a bijection would extend to an arbitrary graph by replacing the inversion statistic with the κ\n\nstatistic of Ira Gessel [19]. No bijection I have tried works (and there are many bijections between spanning trees and parking functions in the literature).\" (a) Recorder's note: this problem was resolved in [30].",
   "public_statement": "(14) David Perkinson: \"This question is due to Richard Stanley; it is problem 4 of chapter 6 in his notes on hyperplane arrangements [32]. Is there a natural bijection φ between labeled trees T on {0, 1, 2,..., n }6 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\n\nrooted at 0 and parking functions of length n such that deg( φ(T )) = g − inv( T ),\n\nwhere g is the genus of the complete graph Kn+1 and inv( T ) is the number of inversions of T? I believe such a bijection would extend to an arbitrary graph by replacing the inversion statistic with the κ\n\nstatistic of Ira Gessel [19]. No bijection I have tried works (and there are many bijections between spanning trees and parking functions in the literature).\" (a) Recorder's note: this problem was resolved in [30].",
   "evidence": "The canonical record is David Perkinson's Problem (14) in the AIM chip-firing problem list. The extracted text has only a page-break OCR artifact. Comparison with the linked AIM PDF recovers the question as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 103,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0105": {
   "statement_status": "exact",
   "original_statement": "(15) Jeremy Martin: \"What is the average value of the scaling limit of the identity element of the sandpile group on Z2?\" (a) Wesley Pegden: \"It might be possible to explicitly construct the limit of the identity in the case of the square lattice, and then compute this average and other numbers as well such as the side length of the square of constant value 2 that appears in the middle. We do not know how to do this yet.\"",
   "clean_statement": "(15) Jeremy Martin: \"What is the average value of the scaling limit of the identity element of the sandpile group on Z2?\" (a) Wesley Pegden: \"It might be possible to explicitly construct the limit of the identity in the case of the square lattice, and then compute this average and other numbers as well such as the side length of the square of constant value 2 that appears in the middle. We do not know how to do this yet.\"",
   "public_statement": "(15) Jeremy Martin: \"What is the average value of the scaling limit of the identity element of the sandpile group on Z2?\" (a) Wesley Pegden: \"It might be possible to explicitly construct the limit of the identity in the case of the square lattice, and then compute this average and other numbers as well such as the side length of the square of constant value 2 that appears in the middle. We do not know how to do this yet.\"",
   "evidence": "The AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 104,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0106": {
   "statement_status": "exact",
   "original_statement": "(16) Wesley Pegden: \"Are all the ways to hexagonally tile the plane with polyominoes that are 90 ◦ rotationally symmetric given by the tilings (up to dilations, say) considered in my recent paper with Lionel Levine and Charles Smart [23] on the Apollonian circle-packing pat-terns that emerge in the scaling limit of the sandpile model? We believe this conjecture (and in fact state a stronger conjecture in the paper).\"",
   "clean_statement": "(16) Wesley Pegden: \"Are all the ways to hexagonally tile the plane with polyominoes that are 90 ◦ rotationally symmetric given by the tilings (up to dilations, say) considered in my recent paper with Lionel Levine and Charles Smart [23] on the Apollonian circle-packing pat-terns that emerge in the scaling limit of the sandpile model? We believe this conjecture (and in fact state a stronger conjecture in the paper).\"",
   "public_statement": "(16) Wesley Pegden: \"Are all the ways to hexagonally tile the plane with polyominoes that are 90 ◦ rotationally symmetric given by the tilings (up to dilations, say) considered in my recent paper with Lionel Levine and Charles Smart [23] on the Apollonian circle-packing pat-terns that emerge in the scaling limit of the sandpile model? We believe this conjecture (and in fact state a stronger conjecture in the paper).\"",
   "evidence": "The canonical record is Problem 16 from the AIM workshop *Generalizations of chip-firing and the critical group*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 105,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0107": {
   "statement_status": "exact",
   "original_statement": "(17) Caroline Klivans: \"What are higher dimensional (in the sense of [18] and [17]) critical configurations?\" (a) Sam Hopkins: \"Are you looking specifically for analogs of stable recurrent configurations? Or do analogs of parking functions or superstable elements also interest you?\" (b) Caroline Klivans: \"We would like to generalize any of these notions from the graph case to the simplicial complex case.\"",
   "clean_statement": "(17) Caroline Klivans: \"What are higher dimensional (in the sense of [18] and [17]) critical configurations?\" (a) Sam Hopkins: \"Are you looking specifically for analogs of stable recurrent configurations? Or do analogs of parking functions or superstable elements also interest you?\" (b) Caroline Klivans: \"We would like to generalize any of these notions from the graph case to the simplicial complex case.\"",
   "public_statement": "(17) Caroline Klivans: \"What are higher dimensional (in the sense of [18] and [17]) critical configurations?\" (a) Sam Hopkins: \"Are you looking specifically for analogs of stable recurrent configurations? Or do analogs of parking functions or superstable elements also interest you?\" (b) Caroline Klivans: \"We would like to generalize any of these notions from the graph case to the simplicial complex case.\"",
   "evidence": "The canonical AIM record is Problem (17), attributed to Caroline Klivans, from the workshop *Generalizations of chip-firing and the critical group*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 106,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0108": {
   "statement_status": "exact",
   "original_statement": "(18) Caroline Klivans: \"How can we extend to higher dimensions Merino's result [25] that the generating function of the critical configurations (by degrees) is equal to an evaluation of the Tutte polynomial of G\n\n(in particular, is equal to T (1, y ))?\" (a) David Perkinson: \"So you have a notion of external activity in this case?\" (b) Caroline Klivans: \"We can look at the cellular matroid corre-sponding to the simplicial complex. However, the size of our critical group is a weighted sum of spanning trees, so we will need to look at the arithmetic Tutte polynomial instead of the normal Tutte polynomial.\" (c) Igor Pak: \"The degree should be the same across each coset, and in this case we can define the generating function formally without choosing representatives. So we may be able to answer PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 7\n\nthis question without first answering the question about what the right notion of critical configurations is.\"",
   "clean_statement": "(18) Caroline Klivans: \"How can we extend to higher dimensions Merino's result [25] that the generating function of the critical configurations (by degrees) is equal to an evaluation of the Tutte polynomial of G\n\n(in particular, is equal to T (1, y ))?\" (a) David Perkinson: \"So you have a notion of external activity in this case?\" (b) Caroline Klivans: \"We can look at the cellular matroid corre-sponding to the simplicial complex. However, the size of our critical group is a weighted sum of spanning trees, so we will need to look at the arithmetic Tutte polynomial instead of the normal Tutte polynomial.\" (c) Igor Pak: \"The degree should be the same across each coset, and in this case we can define the generating function formally without choosing representatives. So we may be able to answer PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 7\n\nthis question without first answering the question about what the right notion of critical configurations is.\"",
   "public_statement": "(18) Caroline Klivans: \"How can we extend to higher dimensions Merino's result [25] that the generating function of the critical configurations (by degrees) is equal to an evaluation of the Tutte polynomial of G\n\n(in particular, is equal to T (1, y ))?\" (a) David Perkinson: \"So you have a notion of external activity in this case?\" (b) Caroline Klivans: \"We can look at the cellular matroid corre-sponding to the simplicial complex. However, the size of our critical group is a weighted sum of spanning trees, so we will need to look at the arithmetic Tutte polynomial instead of the normal Tutte polynomial.\" (c) Igor Pak: \"The degree should be the same across each coset, and in this case we can define the generating function formally without choosing representatives. So we may be able to answer PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 7\n\nthis question without first answering the question about what the right notion of critical configurations is.\"",
   "evidence": "The canonical record is problem 18 in `aim-combinatorics-notes.json`, source index 107, from the workshop *Generalizations of chip-firing and the critical group*. It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 107,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0109": {
   "statement_status": "exact",
   "original_statement": "(19) Andrea Sportiello and Jordan Ellenberg: \"Can we develop a chip-firing model for cell complexes that includes chips on cells of different dimensions and features interaction between dimensions? Probably, chip configurations and firing in lower dimensions influence what can happen in higher dimensions.\" (a) Gregg Musiker: \"Is there a Chow ring lurking here?\"",
   "clean_statement": "(19) Andrea Sportiello and Jordan Ellenberg: \"Can we develop a chip-firing model for cell complexes that includes chips on cells of different dimensions and features interaction between dimensions? Probably, chip configurations and firing in lower dimensions influence what can happen in higher dimensions.\" (a) Gregg Musiker: \"Is there a Chow ring lurking here?\"",
   "public_statement": "(19) Andrea Sportiello and Jordan Ellenberg: \"Can we develop a chip-firing model for cell complexes that includes chips on cells of different dimensions and features interaction between dimensions? Probably, chip configurations and firing in lower dimensions influence what can happen in higher dimensions.\" (a) Gregg Musiker: \"Is there a Chow ring lurking here?\"",
   "evidence": "This is Problem (19) from the 2013 AIM workshop *Generalizations of chip-firing and the critical group*. The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 108,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0110": {
   "statement_status": "exact",
   "original_statement": "(20) Charles Smart: \"Let Γ be the set of 2 × 2 real symmetric matrices A\n\nsuch that there is a function u: Z2 → Z with \n\nu(x) = 12 x · Ax + o(1 + |x|2)and ∆ u(x) ≤ 0 for all x. When we consider the Laplacian given by the standard lattice Z2, we find that Γ exhibits Apollonian circle-packing phenomenon. However, with the F -lattice instead (which has two edges oriented in and two edges oriented out at each vertex), we appear to get a very simple set for Γ. Is this really the case?\"",
   "clean_statement": "(20) Charles Smart: \"Let Γ be the set of 2 × 2 real symmetric matrices A\n\nsuch that there is a function u: Z2 → Z with\n\nu(x) = 12 x · Ax + o(1 + |x|2)and ∆ u(x) ≤ 0 for all x. When we consider the Laplacian given by the standard lattice Z2, we find that Γ exhibits Apollonian circle-packing phenomenon. However, with the F -lattice instead (which has two edges oriented in and two edges oriented out at each vertex), we appear to get a very simple set for Γ. Is this really the case?\"",
   "public_statement": "(20) Charles Smart: \"Let Γ be the set of 2 × 2 real symmetric matrices A\n\nsuch that there is a function u: Z2 → Z with\n\nu(x) = 12 x · Ax + o(1 + |x|2)and ∆ u(x) ≤ 0 for all x. When we consider the Laplacian given by the standard lattice Z2, we find that Γ exhibits Apollonian circle-packing phenomenon. However, with the F -lattice instead (which has two edges oriented in and two edges oriented out at each vertex), we appear to get a very simple set for Γ. Is this really the case?\"",
   "evidence": "The canonical record is Problem (20), attributed to Charles Smart, from the AIM workshop *Generalizations of chip-firing and the critical group*. The extraction has lost superscripts and a fraction bar. The mathematically consistent reading, confirmed by Smart's 2013 F-lattice handout and the later paper of Bou-Rabee, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 109,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0111": {
   "statement_status": "exact",
   "original_statement": "(21) Many participants: \"How can we generalize Baker and Norine's graphical Riemann-Roch theorem [7] from graphs to cell complexes? Solving this problem may require a good combinatorial definition of critical configurations or superstables in arbitrary dimension, and/or understanding higher-dimensional generalizations of the Riemann-Roch theorem in algebraic geometry.\"",
   "clean_statement": "(21) Many participants: \"How can we generalize Baker and Norine's graphical Riemann-Roch theorem [7] from graphs to cell complexes? Solving this problem may require a good combinatorial definition of critical configurations or superstables in arbitrary dimension, and/or understanding higher-dimensional generalizations of the Riemann-Roch theorem in algebraic geometry.\"",
   "public_statement": "(21) Many participants: \"How can we generalize Baker and Norine's graphical Riemann-Roch theorem [7] from graphs to cell complexes? Solving this problem may require a good combinatorial definition of critical configurations or superstables in arbitrary dimension, and/or understanding higher-dimensional generalizations of the Riemann-Roch theorem in algebraic geometry.\"",
   "evidence": "This is Problem 21 from the AIM workshop *Generalizations of chip-firing and the critical group*. The exact source record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 110,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0112": {
   "statement_status": "exact",
   "original_statement": "(22) Caroline Klivans: \"Can we develop a chip-firing model for directed rooted forests (as defined, for example, by Olivier Bernardi [10]) in higher dimension?\"",
   "clean_statement": "(22) Caroline Klivans: \"Can we develop a chip-firing model for directed rooted forests (as defined, for example, by Olivier Bernardi [10]) in higher dimension?\"",
   "public_statement": "(22) Caroline Klivans: \"Can we develop a chip-firing model for directed rooted forests (as defined, for example, by Olivier Bernardi [10]) in higher dimension?\"",
   "evidence": "The exact canonical statement is Problem (22) from the 2013 AIM workshop *Generalizations of chip-firing and the critical group*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 111,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0113": {
   "statement_status": "exact",
   "original_statement": "(23) Farbod Shokrieh: \"What is the full Gr¨ obner fan of the toppling ideal (as in [26])?\"",
   "clean_statement": "(23) Farbod Shokrieh: \"What is the full Gr¨ obner fan of the toppling ideal (as in [26])?\"",
   "public_statement": "(23) Farbod Shokrieh: \"What is the full Gr¨ obner fan of the toppling ideal (as in [26])?\"",
   "evidence": "The source record is problem 23 from the AIM workshop *Generalizations of chip-firing and the critical group*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 112,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0114": {
   "statement_status": "exact",
   "original_statement": "(24) Art Duval: \"Can we figure out the abelian sandpile dynamics in higher dimensions?\" (a) Matt Baker: \"Do we have pretty three-dimensional pictures?\" (b) Caroline Klivans: \"I have code for this.\"",
   "clean_statement": "(24) Art Duval: \"Can we figure out the abelian sandpile dynamics in higher dimensions?\" (a) Matt Baker: \"Do we have pretty three-dimensional pictures?\" (b) Caroline Klivans: \"I have code for this.\"",
   "public_statement": "(24) Art Duval: \"Can we figure out the abelian sandpile dynamics in higher dimensions?\" (a) Matt Baker: \"Do we have pretty three-dimensional pictures?\" (b) Caroline Klivans: \"I have code for this.\"",
   "evidence": "The source is Problem 24 from the AIM workshop *Generalizations of chip-firing and the critical group*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 113,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0115": {
   "statement_status": "exact",
   "original_statement": "(25) Andrea Sportiello: \"What is the higher-dimensional version of the explorer model [16]? Is this model more amenable to generalization the abelian sandpile model?\"",
   "clean_statement": "(25) Andrea Sportiello: \"What is the higher-dimensional version of the explorer model [16]? Is this model more amenable to generalization the abelian sandpile model?\"",
   "public_statement": "(25) Andrea Sportiello: \"What is the higher-dimensional version of the explorer model [16]? Is this model more amenable to generalization the abelian sandpile model?\"",
   "evidence": "The exact corpus record is Andrea Sportiello's Problem (25) from the 2013 AIM workshop *Generalizations of chip-firing and the critical group*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 114,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0116": {
   "statement_status": "exact",
   "original_statement": "(26) Caroline Klivans: \"Is there an efficient algorithm to generate a higher dimensional spanning tree uniformly at random?\"",
   "clean_statement": "(26) Caroline Klivans: \"Is there an efficient algorithm to generate a higher dimensional spanning tree uniformly at random?\"",
   "public_statement": "(26) Caroline Klivans: \"Is there an efficient algorithm to generate a higher dimensional spanning tree uniformly at random?\"",
   "evidence": "This is Problem 26 from the AIM workshop *Generalizations of chip-firing and the critical group*. The exact source record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 115,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0117": {
   "statement_status": "exact",
   "original_statement": "(27) Adrien Kassel: \"How does chip-firing on planar metric graphs relate to questions in analysis?\"",
   "clean_statement": "(27) Adrien Kassel: \"How does chip-firing on planar metric graphs relate to questions in analysis?\"",
   "public_statement": "(27) Adrien Kassel: \"How does chip-firing on planar metric graphs relate to questions in analysis?\"",
   "evidence": "The canonical AIM record is the following broad question from the workshop *Generalizations of chip-firing and the critical group*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 116,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0118": {
   "statement_status": "exact",
   "original_statement": "(28) Vic Reiner: \"We know results about the homomorphism between critical groups for two graphs that are related by a Galois covering map (see, e.g., [31]). How do these results relate to class field theory, when the Galois group of the covering is abelian?\" 8 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP",
   "clean_statement": "(28) Vic Reiner: \"We know results about the homomorphism between critical groups for two graphs that are related by a Galois covering map (see, e.g., [31]). How do these results relate to class field theory, when the Galois group of the covering is abelian?\" 8 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP",
   "public_statement": "(28) Vic Reiner: \"We know results about the homomorphism between critical groups for two graphs that are related by a Galois covering map (see, e.g., [31]). How do these results relate to class field theory, when the Galois group of the covering is abelian?\" 8 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP",
   "evidence": "The canonical record is problem (28), attributed to Vic Reiner, from the AIM workshop *Generalizations of chip-firing and the critical group*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 117,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0119": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "(29) Matt Baker: \"Is there a relation between the full Tutte polynomial (as opposed to just TG(1, y ) from Merino's theorem [25]) and chip-firing? We made to use some additional geometric structure.\"",
   "clean_statement": null,
   "public_statement": "(29) Matt Baker: \"Is there a relation between the full Tutte polynomial (as opposed to just TG(1, y ) from Merino's theorem [25]) and chip-firing? We made to use some additional geometric structure.\"",
   "evidence": "The last sentence is ungrammatical. A highly plausible reconstruction is",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 118,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0120": {
   "statement_status": "exact",
   "original_statement": "(30) Persi Diaconis: \"What does a limiting shape theory for generalized internal DLA (see [22], [4], and [21]) look like?\"",
   "clean_statement": "(30) Persi Diaconis: \"What does a limiting shape theory for generalized internal DLA (see [22], [4], and [21]) look like?\"",
   "public_statement": "(30) Persi Diaconis: \"What does a limiting shape theory for generalized internal DLA (see [22], [4], and [21]) look like?\"",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 119,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0121": {
   "statement_status": "exact",
   "original_statement": "(31) Dustin Cartwright: \"What is the (tropical) Picard group (in the sense of [13]) of the product of graphs? I have a guess if one of the graphs is a tree.\"",
   "clean_statement": "(31) Dustin Cartwright: \"What is the (tropical) Picard group (in the sense of [13]) of the product of graphs? I have a guess if one of the graphs is a tree.\"",
   "public_statement": "(31) Dustin Cartwright: \"What is the (tropical) Picard group (in the sense of [13]) of the product of graphs? I have a guess if one of the graphs is a tree.\"",
   "evidence": "The canonical record is AIM problem 31 from the workshop *Generalizations of chip-firing and the critical group*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 120,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0122": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "(32) Jim Propp: \"Is Andrea Sportiello's model of inverse topplings [12] the same as the particle creation and deletion model?\"",
   "clean_statement": null,
   "public_statement": "(32) Jim Propp: \"Is Andrea Sportiello's model of inverse topplings [12] the same as the particle creation and deletion model?\"",
   "evidence": "There is a second plausible reading. CPS explicitly identify one of their Markov chains with the Karmakar--Manna particle--hole protocol. On that reading the answer is tautologically yes, but it does not answer the rotor-routing question described by the workshop summary. Both readings are separated below.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 121,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0123": {
   "statement_status": "exact",
   "original_statement": "(33) Lionel Levine: \"What is the infinite volume limit of Andrea's prop-agator model [12] with a finite density of propagators?\"",
   "clean_statement": "(33) Lionel Levine: \"What is the infinite volume limit of Andrea's prop-agator model [12] with a finite density of propagators?\"",
   "public_statement": "(33) Lionel Levine: \"What is the infinite volume limit of Andrea's prop-agator model [12] with a finite density of propagators?\"",
   "evidence": "The AIM list asks, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 122,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0124": {
   "statement_status": "exact",
   "original_statement": "(34) Jordan Ellenberg: \"What does a stochastic sandpile model that in-terpolates between DLA [16] and the sandpile model look like?\"",
   "clean_statement": "(34) Jordan Ellenberg: \"What does a stochastic sandpile model that in-terpolates between DLA [16] and the sandpile model look like?\"",
   "public_statement": "(34) Jordan Ellenberg: \"What does a stochastic sandpile model that in-terpolates between DLA [16] and the sandpile model look like?\"",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 123,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0125": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "(35) Leonardo Rolla: \"If you start with an i.i.d. number of chips with some small density μ in Zd, is this configuration stabilizable with high probability?\"",
   "clean_statement": "the **ordinary fixed-energy abelian sandpile**. A configuration is \\(\\eta:\\mathbb Z^d\\to\\mathbb Z_{\\ge0}\\); a site with at least \\(2d\\) chips may topple, losing \\(2d\\) chips and sending one to each nearest neighbour. Randomness is only in the i.i.d. initial heights. The configuration is stabilizable if a legal toppling procedure reaches heights at most \\(2d-1\\) while toppling every site only finitely often. This matches the terminology, workshop setting, and the precise infinite-volume problem studied by Fey--Meester--Redig.",
   "public_statement": "(35) Leonardo Rolla: \"If you start with an i.i.d. number of chips with some small density μ in Zd, is this configuration stabilizable with high probability?\"",
   "evidence": "Throughout this report, \\(\\mu=\\mathbb E\\eta(0)\\) denotes mean chip density. For a product law, the stabilization event is translation invariant and the law is ergodic. Its probability is therefore exactly zero or one. Thus “with high probability” in infinite volume should be read as “almost surely.” On a finite wired box every configuration stabilizes, so a literal finite-volume stabilization probability would be identically one and would not express the intended question.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 124,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0126": {
   "statement_status": "exact",
   "original_statement": "(36) Lionel Levine: \"The graph Laplacian is the generator of a random walk on a graph. Does the Laplacian of a simplicial complex generate some kind of stochastic process on this higher-dimensional object?\" (a) Caroline Klivans: \"Look up the work of Russel Lyons [24].\" (b) Lionel Levine: \"I have and it is interesting but not what I want.\" (c) Andrea Sportiello: \"Dustin Cartwright's model [13] looks more like a random walk with the way it assigns weights.\" (d) Matt Baker: \"So the concrete question is: does Dustin's model generate some stochastic process?\"",
   "clean_statement": "(36) Lionel Levine: \"The graph Laplacian is the generator of a random walk on a graph. Does the Laplacian of a simplicial complex generate some kind of stochastic process on this higher-dimensional object?\" (a) Caroline Klivans: \"Look up the work of Russel Lyons [24].\" (b) Lionel Levine: \"I have and it is interesting but not what I want.\" (c) Andrea Sportiello: \"Dustin Cartwright's model [13] looks more like a random walk with the way it assigns weights.\" (d) Matt Baker: \"So the concrete question is: does Dustin's model generate some stochastic process?\"",
   "public_statement": "(36) Lionel Levine: \"The graph Laplacian is the generator of a random walk on a graph. Does the Laplacian of a simplicial complex generate some kind of stochastic process on this higher-dimensional object?\" (a) Caroline Klivans: \"Look up the work of Russel Lyons [24].\" (b) Lionel Levine: \"I have and it is interesting but not what I want.\" (c) Andrea Sportiello: \"Dustin Cartwright's model [13] looks more like a random walk with the way it assigns weights.\" (d) Matt Baker: \"So the concrete question is: does Dustin's model generate some stochastic process?\"",
   "evidence": "The canonical AIM record is problem 36 from the workshop *Generalizations of chip-firing and the critical group*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 125,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0127": {
   "statement_status": "exact",
   "original_statement": "(37) James Propp: \"What is known about harmonic maps (in the sense of Baker and Norine [8], Cory [15], etc.) between Markov chains?\" (a) Matt Baker: \"The paper of Urakawa's [33] that we cite in our original article already is aware of this connection to random walks. I have not followed it up since then. We also have a notion of harmonic maps for metric graphs.\"",
   "clean_statement": "(37) James Propp: \"What is known about harmonic maps (in the sense of Baker and Norine [8], Cory [15], etc.) between Markov chains?\" (a) Matt Baker: \"The paper of Urakawa's [33] that we cite in our original article already is aware of this connection to random walks. I have not followed it up since then. We also have a notion of harmonic maps for metric graphs.\"",
   "public_statement": "(37) James Propp: \"What is known about harmonic maps (in the sense of Baker and Norine [8], Cory [15], etc.) between Markov chains?\" (a) Matt Baker: \"The paper of Urakawa's [33] that we cite in our original article already is aware of this connection to random walks. I have not followed it up since then. We also have a notion of harmonic maps for metric graphs.\"",
   "evidence": "The canonical AIM record is Problem 37 from the workshop *Generalizations of chip-firing and the critical group*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 126,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0128": {
   "statement_status": "exact",
   "original_statement": "(38) David Perkinson: \"If you have an n × n integer matrix of full rank, is that the Laplacian of a graph? Well, of course not in general, but if you are allowed to do row operations you can always get a graph (in fact an infinite number of them) with this property. This is due to an algorithm of Wilmes [29]. But to accomplish this, you need to allow the resulting graph to be directed. (a) Spencer Backman: \"The question is related to trying to under-stand when a directed graph has the Riemann-Roch property. Amini and Manjunath [1] studied when orthogonal lattices have the Riemann-Roch property, and via the algorithm of Wilmes, Arash Asadi and I [3] were able to reduce this to a question of directed graphs. So a solution to Dave's question would help PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 9\n\nwith understanding Riemann-Roch on directed graphs (which appears to be a very delicate question).\"",
   "clean_statement": "(38) David Perkinson: \"If you have an n × n integer matrix of full rank, is that the Laplacian of a graph? Well, of course not in general, but if you are allowed to do row operations you can always get a graph (in fact an infinite number of them) with this property. This is due to an algorithm of Wilmes [29]. But to accomplish this, you need to allow the resulting graph to be directed. (a) Spencer Backman: \"The question is related to trying to under-stand when a directed graph has the Riemann-Roch property. Amini and Manjunath [1] studied when orthogonal lattices have the Riemann-Roch property, and via the algorithm of Wilmes, Arash Asadi and I [3] were able to reduce this to a question of directed graphs. So a solution to Dave's question would help PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 9\n\nwith understanding Riemann-Roch on directed graphs (which appears to be a very delicate question).\"",
   "public_statement": "(38) David Perkinson: \"If you have an n × n integer matrix of full rank, is that the Laplacian of a graph? Well, of course not in general, but if you are allowed to do row operations you can always get a graph (in fact an infinite number of them) with this property. This is due to an algorithm of Wilmes [29]. But to accomplish this, you need to allow the resulting graph to be directed. (a) Spencer Backman: \"The question is related to trying to under-stand when a directed graph has the Riemann-Roch property. Amini and Manjunath [1] studied when orthogonal lattices have the Riemann-Roch property, and via the algorithm of Wilmes, Arash Asadi and I [3] were able to reduce this to a question of directed graphs. So a solution to Dave's question would help PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 9\n\nwith understanding Riemann-Roch on directed graphs (which appears to be a very delicate question).\"",
   "evidence": "The AIM workshop list records David Perkinson's setup:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 127,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0129": {
   "statement_status": "exact",
   "original_statement": "(39) Gregg Musiker: \"Can we characterize which genus g graphs have a maximal harmonic group (in the sense of Corry [15]) of size 6( g−1)?\" (a) Scott Corry: \"If you know the maximal graph group, I can make a graph on which it acts harmonically by a variation of a Cayley graph construction.\" (b) Gregg Musiker: \"But if I hand you a graph, can you tell me if it has a group of size 6( g − 1) that acts harmonically on it?\"",
   "clean_statement": "(39) Gregg Musiker: \"Can we characterize which genus g graphs have a maximal harmonic group (in the sense of Corry [15]) of size 6( g−1)?\" (a) Scott Corry: \"If you know the maximal graph group, I can make a graph on which it acts harmonically by a variation of a Cayley graph construction.\" (b) Gregg Musiker: \"But if I hand you a graph, can you tell me if it has a group of size 6( g − 1) that acts harmonically on it?\"",
   "public_statement": "(39) Gregg Musiker: \"Can we characterize which genus g graphs have a maximal harmonic group (in the sense of Corry [15]) of size 6( g−1)?\" (a) Scott Corry: \"If you know the maximal graph group, I can make a graph on which it acts harmonically by a variation of a Cayley graph construction.\" (b) Gregg Musiker: \"But if I hand you a graph, can you tell me if it has a group of size 6( g − 1) that acts harmonically on it?\"",
   "evidence": "The canonical AIM record is Problem 39 from the 2013 workshop *Generalizations of chip-firing and the critical group*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 128,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0130": {
   "statement_status": "exact",
   "original_statement": "(40) Jordan Ellenberg: \"Is there a Deligne-Mumford stack on graphs?\" (a) Matt Baker: \"We have a paper on metric graphs that accom-plishes some of this.\" (b) Jordan: \"But there is still a question of what are harmonic group actions on metric graphs?\"",
   "clean_statement": "(40) Jordan Ellenberg: \"Is there a Deligne-Mumford stack on graphs?\" (a) Matt Baker: \"We have a paper on metric graphs that accom-plishes some of this.\" (b) Jordan: \"But there is still a question of what are harmonic group actions on metric graphs?\"",
   "public_statement": "(40) Jordan Ellenberg: \"Is there a Deligne-Mumford stack on graphs?\" (a) Matt Baker: \"We have a paper on metric graphs that accom-plishes some of this.\" (b) Jordan: \"But there is still a question of what are harmonic group actions on metric graphs?\"",
   "evidence": "The exact canonical record is Problem 40 from the 2013 AIM workshop *Generalizations of chip-firing and the critical group*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 129,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0131": {
   "statement_status": "exact",
   "original_statement": "(41) Lionel Levine: \"If you have a graph with a harmonic action, then its critical group comes with some extra structure, namely a group of automorphisms of the graph. That extra object might be worth studying.\"",
   "clean_statement": "(41) Lionel Levine: \"If you have a graph with a harmonic action, then its critical group comes with some extra structure, namely a group of automorphisms of the graph. That extra object might be worth studying.\"",
   "public_statement": "(41) Lionel Levine: \"If you have a graph with a harmonic action, then its critical group comes with some extra structure, namely a group of automorphisms of the graph. That extra object might be worth studying.\"",
   "evidence": "This is item (41), attributed to Lionel Levine, in the AIM workshop list *Generalizations of chip-firing and the critical group*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 130,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0132": {
   "statement_status": "exact",
   "original_statement": "(42) Matt Baker: \"Peter Winkler [35] gives an interesting solution to a famous math Olympiad problem involving numbers on a pentagon in his book on math puzzles. His solution takes infinite covers of the pentagon. I have not tried to generalize this proof to the usual chip-firing game; Farbod and I [9] have a solution that uses potential theory. But maybe we could apply Winkler's method to general graphs?\" (a) Matt Macauley: \"Is this pentagon game the same as numbers game for Coxeter groups?\" (b) Matt Baker \"In fact it is a special case. The numbers game is a different generalization of chip-firing on a cycle than chip-firing on a general graph is.\"",
   "clean_statement": "(42) Matt Baker: \"Peter Winkler [35] gives an interesting solution to a famous math Olympiad problem involving numbers on a pentagon in his book on math puzzles. His solution takes infinite covers of the pentagon. I have not tried to generalize this proof to the usual chip-firing game; Farbod and I [9] have a solution that uses potential theory. But maybe we could apply Winkler's method to general graphs?\" (a) Matt Macauley: \"Is this pentagon game the same as numbers game for Coxeter groups?\" (b) Matt Baker \"In fact it is a special case. The numbers game is a different generalization of chip-firing on a cycle than chip-firing on a general graph is.\"",
   "public_statement": "(42) Matt Baker: \"Peter Winkler [35] gives an interesting solution to a famous math Olympiad problem involving numbers on a pentagon in his book on math puzzles. His solution takes infinite covers of the pentagon. I have not tried to generalize this proof to the usual chip-firing game; Farbod and I [9] have a solution that uses potential theory. But maybe we could apply Winkler's method to general graphs?\" (a) Matt Macauley: \"Is this pentagon game the same as numbers game for Coxeter groups?\" (b) Matt Baker \"In fact it is a special case. The numbers game is a different generalization of chip-firing on a cycle than chip-firing on a general graph is.\"",
   "evidence": "The AIM workshop record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 131,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0133": {
   "statement_status": "exact",
   "original_statement": "A more complete introduction to this problem is given in V. Ripoll's paper \\htmladdnormallink{Lyashoko-Looijenga Morphisms and Submaximal Factorizations of a Coxeter Element}{http://arxiv.org/pdf/1012.3825v3.pdf} \\cite{arXiv:1012.3825}.\n\nLet $W$ be a well-generated irreducible complex reflection group. One of the major open problems in Catalan combinatorics that V. Reiner outlined was to give a uniform proof that noncrossing objects are counted by $$Cat^p(W) = \\prod_{i=1}^n \\frac{ph+d_i}{d_i}.$$\n\nThis is a suggestion for how to uniformly prove this formula for such $W$ (note that in the non-well-generated case, $Cat^p(W)$ is not even an integer). Such a uniform proof would be new, even for $p=1$.\n\nWe know case-by-case that $Cat^p(W)$ counts multichains $w_1 \\leq w_2 \\leq \\cdots \\leq w_p \\leq c$ of length $p$ in $NC(W,c)$. Using standard zeta polynomial computations, we can relate multichains in $[1,c]_T$ to strictly increasing chains in $[1,c]_T$, which we then interpret as reduced factorizations of the Coxeter element into nontrivial factors. It is interesting that even though the formula for multichains has a beautiful uniform expression, the formula for strict chains does not seem nice in general.\n\nA $k$-block factorization of $c$ is a factorization of c into $k$ factors $$c=u_1 u_2 \\cdots u_k,$$ such that $\\sum_{i=1}^k \\ell_T(u_i) = \\ell_T(c) = n$ and $u_i \\neq 1$.\n\nLet $Fact_k(c)$ denote the number of $k$-block factorizations of $c$.\n\nUsing the relation between multichains and strict chains, we obtain the identity $$\\prod_{i=1}^n \\frac{ph+d_i}{d_i} = \\sum_{k=1}^n \\binom{p+1}{k} Fact_k (c).$$\n\nDespite the fact that $Fact_k (c)$ does not seem to have a nice uniform formula in general, we do understand it geometrically---$Fact_k (c)$ counts the cardinality of fibers of the Lyashko-Looijenga map. In particular, D. Bessis showed case-by-case that the generic fibers of this map correspond to $n$-block factorizations, or reduced decompositions in the reflections $T$ \\cite{arXiv:math/0610777}. Even though it was done case-by-case, we have the uniform formula $$Fact_n(c) = \\frac{n! h^n}{|W|} = \\frac{(h)(2h)(3h)\\cdots(nh)}{d_1 d_2 \\cdots d_n}.$$ V. Ripoll gave a similar formula for the almost generic fibers.\n\nUniformly prove $$\\prod_{i=1}^n \\frac{ph+d_i}{d_i} = \\sum_{k=1}^n \\binom{p+1}{k} Fact_k (c)$$ by interpreting the right-hand side as fibers of the Lyashko-Looijenga map.",
   "clean_statement": "A more complete introduction to this problem is given in V. Ripoll's paper \\htmladdnormallink{Lyashoko-Looijenga Morphisms and Submaximal Factorizations of a Coxeter Element}{http://arxiv.org/pdf/1012.3825v3.pdf} \\cite{arXiv:1012.3825}.\n\nLet $W$ be a well-generated irreducible complex reflection group. One of the major open problems in Catalan combinatorics that V. Reiner outlined was to give a uniform proof that noncrossing objects are counted by $$Cat^p(W) = \\prod_{i=1}^n \\frac{ph+d_i}{d_i}.$$\n\nThis is a suggestion for how to uniformly prove this formula for such $W$ (note that in the non-well-generated case, $Cat^p(W)$ is not even an integer). Such a uniform proof would be new, even for $p=1$.\n\nWe know case-by-case that $Cat^p(W)$ counts multichains $w_1 \\leq w_2 \\leq \\cdots \\leq w_p \\leq c$ of length $p$ in $NC(W,c)$. Using standard zeta polynomial computations, we can relate multichains in $[1,c]_T$ to strictly increasing chains in $[1,c]_T$, which we then interpret as reduced factorizations of the Coxeter element into nontrivial factors. It is interesting that even though the formula for multichains has a beautiful uniform expression, the formula for strict chains does not seem nice in general.\n\nA $k$-block factorization of $c$ is a factorization of c into $k$ factors $$c=u_1 u_2 \\cdots u_k,$$ such that $\\sum_{i=1}^k \\ell_T(u_i) = \\ell_T(c) = n$ and $u_i \\neq 1$.\n\nLet $Fact_k(c)$ denote the number of $k$-block factorizations of $c$.\n\nUsing the relation between multichains and strict chains, we obtain the identity $$\\prod_{i=1}^n \\frac{ph+d_i}{d_i} = \\sum_{k=1}^n \\binom{p+1}{k} Fact_k (c).$$\n\nDespite the fact that $Fact_k (c)$ does not seem to have a nice uniform formula in general, we do understand it geometrically---$Fact_k (c)$ counts the cardinality of fibers of the Lyashko-Looijenga map. In particular, D. Bessis showed case-by-case that the generic fibers of this map correspond to $n$-block factorizations, or reduced decompositions in the reflections $T$ \\cite{arXiv:math/0610777}. Even though it was done case-by-case, we have the uniform formula $$Fact_n(c) = \\frac{n! h^n}{|W|} = \\frac{(h)(2h)(3h)\\cdots(nh)}{d_1 d_2 \\cdots d_n}.$$ V. Ripoll gave a similar formula for the almost generic fibers.\n\nUniformly prove $$\\prod_{i=1}^n \\frac{ph+d_i}{d_i} = \\sum_{k=1}^n \\binom{p+1}{k} Fact_k (c)$$ by interpreting the right-hand side as fibers of the Lyashko-Looijenga map.",
   "public_statement": "A more complete introduction to this problem is given in V. Ripoll's paper \\htmladdnormallink{Lyashoko-Looijenga Morphisms and Submaximal Factorizations of a Coxeter Element}{http://arxiv.org/pdf/1012.3825v3.pdf} \\cite{arXiv:1012.3825}.\n\nLet $W$ be a well-generated irreducible complex reflection group. One of the major open problems in Catalan combinatorics that V. Reiner outlined was to give a uniform proof that noncrossing objects are counted by $$Cat^p(W) = \\prod_{i=1}^n \\frac{ph+d_i}{d_i}.$$\n\nThis is a suggestion for how to uniformly prove this formula for such $W$ (note that in the non-well-generated case, $Cat^p(W)$ is not even an integer). Such a uniform proof would be new, even for $p=1$.\n\nWe know case-by-case that $Cat^p(W)$ counts multichains $w_1 \\leq w_2 \\leq \\cdots \\leq w_p \\leq c$ of length $p$ in $NC(W,c)$. Using standard zeta polynomial computations, we can relate multichains in $[1,c]_T$ to strictly increasing chains in $[1,c]_T$, which we then interpret as reduced factorizations of the Coxeter element into nontrivial factors. It is interesting that even though the formula for multichains has a beautiful uniform expression, the formula for strict chains does not seem nice in general.\n\nA $k$-block factorization of $c$ is a factorization of c into $k$ factors $$c=u_1 u_2 \\cdots u_k,$$ such that $\\sum_{i=1}^k \\ell_T(u_i) = \\ell_T(c) = n$ and $u_i \\neq 1$.\n\nLet $Fact_k(c)$ denote the number of $k$-block factorizations of $c$.\n\nUsing the relation between multichains and strict chains, we obtain the identity $$\\prod_{i=1}^n \\frac{ph+d_i}{d_i} = \\sum_{k=1}^n \\binom{p+1}{k} Fact_k (c).$$\n\nDespite the fact that $Fact_k (c)$ does not seem to have a nice uniform formula in general, we do understand it geometrically---$Fact_k (c)$ counts the cardinality of fibers of the Lyashko-Looijenga map. In particular, D. Bessis showed case-by-case that the generic fibers of this map correspond to $n$-block factorizations, or reduced decompositions in the reflections $T$ \\cite{arXiv:math/0610777}. Even though it was done case-by-case, we have the uniform formula $$Fact_n(c) = \\frac{n! h^n}{|W|} = \\frac{(h)(2h)(3h)\\cdots(nh)}{d_1 d_2 \\cdots d_n}.$$ V. Ripoll gave a similar formula for the almost generic fibers.\n\nUniformly prove $$\\prod_{i=1}^n \\frac{ph+d_i}{d_i} = \\sum_{k=1}^n \\binom{p+1}{k} Fact_k (c)$$ by interpreting the right-hand side as fibers of the Lyashko-Looijenga map.",
   "evidence": "The canonical record is `aim-combinatorics-notes.json`, zero-based index 132, from the AIM workshop *Rational Catalan combinatorics*, section “Catalan Combinatorics and Reflection Groups,” Problem 1.1. The supplied source URL is <http://aimpl.org/rationalcatalan/1/>. It returned an HTTP 502 error during this run (30 July 2026), so the text below is preserved exactly from `input.json`. In particular, the misspelling “Lyashoko-Looijenga” is in the source; the standard spelling is “Lyashko--Looijenga.”",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 132,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0134": {
   "statement_status": "unrecoverable",
   "original_statement": "This is just a test!",
   "clean_statement": null,
   "public_statement": "This is just a test!",
   "evidence": "**Recovered statement.** No mathematical statement is recoverable at item level from the canonical record. The record is an AIMPL test/placeholder entry, or the residue of an overwritten entry; the available evidence does not distinguish those possibilities. The status used here is `invalid_statement`, with the underlying record classified as `not_a_problem`.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 133,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0135": {
   "statement_status": "exact",
   "original_statement": "How do we measure the curvature of real networks?",
   "clean_statement": "How do we measure the curvature of real networks?",
   "public_statement": "How do we measure the curvature of real networks?",
   "evidence": "The exact AIM Problem List record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 134,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0136": {
   "statement_status": "exact",
   "original_statement": "Need ways to compute geodesics on spaces where curvature is variable (using embedded graphs).",
   "clean_statement": "Need ways to compute geodesics on spaces where curvature is variable (using embedded graphs).",
   "public_statement": "Need ways to compute geodesics on spaces where curvature is variable (using embedded graphs).",
   "evidence": "The canonical repository record is AIM-COMBINATORICS-0136, item 11.1 in the “Problem session” of the AIM workshop *Geometry of large networks*. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 135,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0137": {
   "statement_status": "exact",
   "original_statement": "Why measure it? (What are the real-world applications?)",
   "clean_statement": "Why measure it? (What are the real-world applications?)",
   "public_statement": "Why measure it? (What are the real-world applications?)",
   "evidence": "The exact canonical record, zero-based index 136 of `aim-combinatorics-notes.json`, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 136,
   "attempt": 2
  },
  "AIM-COMBINATORICS-0138": {
   "statement_status": "exact",
   "original_statement": "Find (different) notions of distance on the space of graphs, or perhaps measures on this space. In particular: how far is a given graph from being a tree?",
   "clean_statement": "Find (different) notions of distance on the space of graphs, or perhaps measures on this space. In particular: how far is a given graph from being a tree?",
   "public_statement": "Find (different) notions of distance on the space of graphs, or perhaps measures on this space. In particular: how far is a given graph from being a tree?",
   "evidence": "The canonical record is item 11.2 in the problem session of the 2011 AIM workshop *Geometry of large networks*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 137,
   "attempt": 2
  },
  "AIM-COMBINATORICS-0139": {
   "statement_status": "exact",
   "original_statement": "Is curvature related to clustering?",
   "clean_statement": "Is curvature related to clustering?",
   "public_statement": "Is curvature related to clustering?",
   "evidence": "The canonical record is problem 11.25 from the AIM workshop **Geometry of large networks**:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 138,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0140": {
   "statement_status": "exact",
   "original_statement": "What is the curvature of a network? How to define it?",
   "clean_statement": "What is the curvature of a network? How to define it?",
   "public_statement": "What is the curvature of a network? How to define it?",
   "evidence": "The exact canonical AIM Problem Lists record, zero-based index 139 of `aim-combinatorics-notes.json`, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 139,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0141": {
   "statement_status": "exact",
   "original_statement": "For random graphs, is there an idea for a degree of heterogeneity that could indicate hyperbolicity",
   "clean_statement": "For random graphs, is there an idea for a degree of heterogeneity that could indicate hyperbolicity",
   "public_statement": "For random graphs, is there an idea for a degree of heterogeneity that could indicate hyperbolicity",
   "evidence": "The source URL in the record did not yield a browsable copy during this run. The statement above is therefore reproduced from the exact canonical repository record; it contains no apparent OCR corruption.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 140,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0142": {
   "statement_status": "exact",
   "original_statement": "What would be meant by a graph having *positive* curvature? (Grids don't have it; trees don't; how about spheres?)",
   "clean_statement": "What would be meant by a graph having *positive* curvature? (Grids don't have it; trees don't; how about spheres?)",
   "public_statement": "What would be meant by a graph having *positive* curvature? (Grids don't have it; trees don't; how about spheres?)",
   "evidence": "The canonical record is problem 11.4 from the AIM workshop *Geometry of large networks*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 141,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0143": {
   "statement_status": "exact",
   "original_statement": "What precise notions of negative curvature imply congestion?",
   "clean_statement": "What precise notions of negative curvature imply congestion?",
   "public_statement": "What precise notions of negative curvature imply congestion?",
   "evidence": "The exact AIM record, from the workshop *Geometry of large networks*, Problem 11.45, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 142,
   "attempt": 2
  },
  "AIM-COMBINATORICS-0144": {
   "statement_status": "exact",
   "original_statement": "What are the data required to define congestion?",
   "clean_statement": "What are the data required to define congestion?",
   "public_statement": "What are the data required to define congestion?",
   "evidence": "The canonical AIM record is problem 11.5 from the problem session of the workshop *Geometry of large networks*. It is record AIM-COMBINATORICS-0144 at zero-based index 143 of aim-combinatorics-notes.json. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 143,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0145": {
   "statement_status": "exact",
   "original_statement": "Is there a scale at which the structure of a (real) network ``looks like'' a manifold? (e.g., Krioukov's result)",
   "clean_statement": "Is there a scale at which the structure of a (real) network ``looks like'' a manifold? (e.g., Krioukov's result)",
   "public_statement": "Is there a scale at which the structure of a (real) network ``looks like'' a manifold? (e.g., Krioukov's result)",
   "evidence": "The exact canonical record, item 11.55 in the problem session of the AIM workshop *Geometry of large networks*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 144,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0146": {
   "statement_status": "exact",
   "original_statement": "$\\pi(K_4^3)=\\frac59$.\\label{K34}",
   "clean_statement": "$\\pi(K_4^3)=\\frac59$.\\label{K34}",
   "public_statement": "$\\pi(K_4^3)=\\frac59$.\\label{K34}",
   "evidence": "The canonical record is Conjecture 11.1 in the AIM list *Hypergraph Turán problem*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 145,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0147": {
   "statement_status": "exact",
   "original_statement": "$\\pi(K_m^3)=1-\\left(\\frac{2}{m-1}\\right)^2$.\n\\label{Km3}",
   "clean_statement": "$\\pi(K_m^3)=1-\\left(\\frac{2}{m-1}\\right)^2$.\n\\label{Km3}",
   "public_statement": "$\\pi(K_m^3)=1-\\left(\\frac{2}{m-1}\\right)^2$.\n\\label{Km3}",
   "evidence": "This is attempt 1 for canonical record AIM-COMBINATORICS-0147, zero-based record 146 of aim-combinatorics-notes.json. Its exact mathematical assertion is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 146,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0148": {
   "statement_status": "unrecoverable",
   "original_statement": "Intro to this problem ...\n\nThis problem is just a placeholder. Replace it bu a real problem,\nor we can jsut delete it later.",
   "clean_statement": null,
   "public_statement": "Intro to this problem ...\n\nThis problem is just a placeholder. Replace it bu a real problem,\nor we can jsut delete it later.",
   "evidence": "**Recovered statement:** no mathematical statement is recoverable from this record. The text is an editorial instruction to replace or delete a stub.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 147,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0149": {
   "statement_status": "exact",
   "original_statement": "Let us mention here another very interesting question for whose solution de\nCaen~\\cite[Page~190]{decaen:94} offered 500 Canadian dollars.\n\nDoes $k(1-\\pi(K_{k+1}^k))$ tend to $\\infty$\nas $k\\to\\infty$?",
   "clean_statement": "Let us mention here another very interesting question for whose solution de\nCaen~\\cite[Page~190]{decaen:94} offered 500 Canadian dollars.\n\nDoes $k(1-\\pi(K_{k+1}^k))$ tend to $\\infty$\nas $k\\to\\infty$?",
   "public_statement": "Let us mention here another very interesting question for whose solution de\nCaen~\\cite[Page~190]{decaen:94} offered 500 Canadian dollars.\n\nDoes $k(1-\\pi(K_{k+1}^k))$ tend to $\\infty$\nas $k\\to\\infty$?",
   "evidence": "The AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 148,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0150": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "$K_4^3$ minus an edge\n\nLet $K_4^-$ be obtained from $K_4^3$ by removing one edge.\n\n$\\pi(K_4^-)=\\frac27$",
   "clean_statement": null,
   "public_statement": "$K_4^3$ minus an edge\n\nLet $K_4^-$ be obtained from $K_4^3$ by removing one edge.\n\n$\\pi(K_4^-)=\\frac27$",
   "evidence": "The canonical record is `aim-combinatorics-notes.json`, zero-based index 149, from the AIM workshop “Hypergraph Turan problem,” section “Complete hypergraphs,” number 11.4. Its statement is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 149,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0151": {
   "statement_status": "exact",
   "original_statement": "Frankl and F\\\"uredi~\\cite{frankl+furedi:89} determined\n$\\pi(B_{k,2})$ for $k=5,6$; in both cases the lower bounds comes by\nblowing up a small design. The following question is still open (see\nFrankl and F\\\"uredi~\\cite[Conjecture~1.5]{frankl+furedi:89}):\n\nDetermine $\\operatorname{ex}(n,B_{5,2})$ and $\\operatorname{ex}(n,B_{6,2})$ exactly for all\nlarge $n$.",
   "clean_statement": "Frankl and F\\\"uredi~\\cite{frankl+furedi:89} determined\n$\\pi(B_{k,2})$ for $k=5,6$; in both cases the lower bounds comes by\nblowing up a small design. The following question is still open (see\nFrankl and F\\\"uredi~\\cite[Conjecture~1.5]{frankl+furedi:89}):\n\nDetermine $\\operatorname{ex}(n,B_{5,2})$ and $\\operatorname{ex}(n,B_{6,2})$ exactly for all\nlarge $n$.",
   "public_statement": "Frankl and F\\\"uredi~\\cite{frankl+furedi:89} determined\n$\\pi(B_{k,2})$ for $k=5,6$; in both cases the lower bounds comes by\nblowing up a small design. The following question is still open (see\nFrankl and F\\\"uredi~\\cite[Conjecture~1.5]{frankl+furedi:89}):\n\nDetermine $\\operatorname{ex}(n,B_{5,2})$ and $\\operatorname{ex}(n,B_{6,2})$ exactly for all\nlarge $n$.",
   "evidence": "The canonical record is AIM-COMBINATORICS-0151, zero-based record 150 of aim-combinatorics-notes.json, attempt 1. It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 150,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0152": {
   "statement_status": "exact",
   "original_statement": "\\[\n \\pi(B_{5,5})=\\frac{40}{81},\n\\]\nand\n\\[\n \\pi(B_{6,6})=\\frac12.\n\\]",
   "clean_statement": "\\[\n \\pi(B_{5,5})=\\frac{40}{81},\n\\]\nand\n\\[\n \\pi(B_{6,6})=\\frac12.\n\\]",
   "public_statement": "\\[\n \\pi(B_{5,5})=\\frac{40}{81},\n\\]\nand\n\\[\n \\pi(B_{6,6})=\\frac12.\n\\]",
   "evidence": "The AIM record, from the workshop list “Hypergraph Turán problem,” section “Turán functions for books,” conjectures",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 151,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0153": {
   "statement_status": "exact",
   "original_statement": "Mubayi and R\\\"odl~\\cite{mubayi+rodl:02} have given bounds on\n$\\pi(C_5^3)$, where $C_5^3$ is the tight $3$-graph $5$-cycle:\n $$\n C_5^3=\\{123,234,345,451,512\\}.\n $$\n In particular, the lower bound $\\pi(C_5^3)\\ge 2\\sqrt3-3$ comes from the\nfollowing construction: partition the vertex set into two parts $A$ and $B$, take all triples that intersect $A$ precisely in $2$ vertices, and\nrecursively repeat this construction within $B$. Finding the optimal ratio\nbetween $|A|$ and $|B|$ gives the required.\nRazborov's~\\cite{razborov:10} flag algebra computations showed that\n$\\pi(C_5^3)< 0.4683$ (note that $2\\sqrt3-3=0.4641...$). This makes the following\nconjecture plausible.\n\n$\\pi(C_5^3)=2\\sqrt3-3$",
   "clean_statement": "Mubayi and R\\\"odl~\\cite{mubayi+rodl:02} have given bounds on\n$\\pi(C_5^3)$, where $C_5^3$ is the tight $3$-graph $5$-cycle:\n $$\n C_5^3=\\{123,234,345,451,512\\}.\n $$\n In particular, the lower bound $\\pi(C_5^3)\\ge 2\\sqrt3-3$ comes from the\nfollowing construction: partition the vertex set into two parts $A$ and $B$, take all triples that intersect $A$ precisely in $2$ vertices, and\nrecursively repeat this construction within $B$. Finding the optimal ratio\nbetween $|A|$ and $|B|$ gives the required.\nRazborov's~\\cite{razborov:10} flag algebra computations showed that\n$\\pi(C_5^3)< 0.4683$ (note that $2\\sqrt3-3=0.4641...$). This makes the following\nconjecture plausible.\n\n$\\pi(C_5^3)=2\\sqrt3-3$",
   "public_statement": "Mubayi and R\\\"odl~\\cite{mubayi+rodl:02} have given bounds on\n$\\pi(C_5^3)$, where $C_5^3$ is the tight $3$-graph $5$-cycle:\n $$\n C_5^3=\\{123,234,345,451,512\\}.\n $$\n In particular, the lower bound $\\pi(C_5^3)\\ge 2\\sqrt3-3$ comes from the\nfollowing construction: partition the vertex set into two parts $A$ and $B$, take all triples that intersect $A$ precisely in $2$ vertices, and\nrecursively repeat this construction within $B$. Finding the optimal ratio\nbetween $|A|$ and $|B|$ gives the required.\nRazborov's~\\cite{razborov:10} flag algebra computations showed that\n$\\pi(C_5^3)< 0.4683$ (note that $2\\sqrt3-3=0.4641...$). This makes the following\nconjecture plausible.\n\n$\\pi(C_5^3)=2\\sqrt3-3$",
   "evidence": "The canonical AIM record is problem 33.1 in the workshop list *Hypergraph Turán problem*, section “Tight \\(5\\)-Cycle,” at http://aimpl.org/hypergraphturan/3/ . It defines \\[ C_5^3=\\{123,234,345,451,512\\} \\] and asks for the classical edge-density Turán value. Its conjecture is \\[ \\boxed{\\pi(C_5^3)=2\\sqrt3-3.} \\] Here \\[ \\pi(F)=\\lim_{n\\to\\infty}\\frac{\\operatorname{ex}(n,F)}{\\binom n3}, \\] where copies are ordinary (not necessarily induced) injective copies of a 3-uniform hypergraph.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 152,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0154": {
   "statement_status": "exact",
   "original_statement": "Tight $5$-Cycle Minus an Edge\n\nLet the $3$-graph $C_5^-$ be obtained from $C_5^3$ by removing one\nedge. An example of a $C_5^-$-free $3$-graph can be obtained by taking\na complete $3$-partite $3$-graph and repeating this construction\nrecursively within each of the three parts. This gives density $1/4$\nin the limit.\n\n$\\pi(C_5^-)=1/4$.",
   "clean_statement": "Tight $5$-Cycle Minus an Edge\n\nLet the $3$-graph $C_5^-$ be obtained from $C_5^3$ by removing one\nedge. An example of a $C_5^-$-free $3$-graph can be obtained by taking\na complete $3$-partite $3$-graph and repeating this construction\nrecursively within each of the three parts. This gives density $1/4$\nin the limit.\n\n$\\pi(C_5^-)=1/4$.",
   "public_statement": "Tight $5$-Cycle Minus an Edge\n\nLet the $3$-graph $C_5^-$ be obtained from $C_5^3$ by removing one\nedge. An example of a $C_5^-$-free $3$-graph can be obtained by taking\na complete $3$-partite $3$-graph and repeating this construction\nrecursively within each of the three parts. This gives density $1/4$\nin the limit.\n\n$\\pi(C_5^-)=1/4$.",
   "evidence": "The canonical record is zero-based index 153 of aim-combinatorics-notes.json, problem 33.2 in the AIM workshop list *Hypergraph Turan problem*, section “Tight \\(5\\)-Cycle.” It states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 153,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0155": {
   "statement_status": "exact",
   "original_statement": "For any $r\\ge 3$ and $s\\ge 4$ we have\n $$\n f^r(n,s(r-2)+3,s)=o(n^2).\n $$",
   "clean_statement": "For any $r\\ge 3$ and $s\\ge 4$ we have\n $$\n f^r(n,s(r-2)+3,s)=o(n^2).\n $$",
   "public_statement": "For any $r\\ge 3$ and $s\\ge 4$ we have\n $$\n f^r(n,s(r-2)+3,s)=o(n^2).\n $$",
   "evidence": "The canonical AIM record (Hypergraph Turan problem workshop, Section 44.1) states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 154,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0156": {
   "statement_status": "exact",
   "original_statement": "A related question is as follows. Let $A$, $B$, and $C$ be disjoint\nsets each of size $n$. Let $M_1,\\dots ,M_l$, be matchings, where each\nedge of $M_i$ has one point in each of $A$, $B$, and $C$. The\nforbidden configuration is: three edges $abc$, $a'b'c'$, $a''b''c''$\nall in some $M_i$ and one edge of the form $ab'c''$ in some other\n$M_j$ (that is, the edge from $M_j$ crosses the three edges of $M_i$).\nAdditionally, we require that the union of all matchings $M_i$ makes a\nsimple (linear) $3$-graph, call it $M$.\n\n$$\n |M| = o(n^2).\n $$",
   "clean_statement": "A related question is as follows. Let $A$, $B$, and $C$ be disjoint\nsets each of size $n$. Let $M_1,\\dots ,M_l$, be matchings, where each\nedge of $M_i$ has one point in each of $A$, $B$, and $C$. The\nforbidden configuration is: three edges $abc$, $a'b'c'$, $a''b''c''$\nall in some $M_i$ and one edge of the form $ab'c''$ in some other\n$M_j$ (that is, the edge from $M_j$ crosses the three edges of $M_i$).\nAdditionally, we require that the union of all matchings $M_i$ makes a\nsimple (linear) $3$-graph, call it $M$.\n\n$$\n |M| = o(n^2).\n $$",
   "public_statement": "A related question is as follows. Let $A$, $B$, and $C$ be disjoint\nsets each of size $n$. Let $M_1,\\dots ,M_l$, be matchings, where each\nedge of $M_i$ has one point in each of $A$, $B$, and $C$. The\nforbidden configuration is: three edges $abc$, $a'b'c'$, $a''b''c''$\nall in some $M_i$ and one edge of the form $ab'c''$ in some other\n$M_j$ (that is, the edge from $M_j$ crosses the three edges of $M_i$).\nAdditionally, we require that the union of all matchings $M_i$ makes a\nsimple (linear) $3$-graph, call it $M$.\n\n$$\n |M| = o(n^2).\n $$",
   "evidence": "The exact AIM record is Conjecture 10 in the 2011 workshop list, in the section “Ruzsa–Szemerédi Theorem and Relatives.” It takes three disjoint sets \\(A,B,C\\), each of size \\(n\\), and matchings \\(M_1,\\ldots,M_\\ell\\) of triples with one vertex in each part. The forbidden configuration consists of three edges",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 155,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0157": {
   "statement_status": "exact",
   "original_statement": "The following conjecture seems to be related.\n\nLet $F$ be a graph and\n$\\alpha>1$ be such that $\\operatorname{ex}(n,F)=\\Omega(n^{\\alpha}$). Then for any\n$\\epsilon > 0$ there is $n_0$ so that if $n > n_0$ and a\ngraph $H$ is the edge-disjoint union of $m=\\lceil \\epsilon\nn^{\\alpha}\\rceil$ copies of $F$, then $H$ contains another copy of $F$\n(i.e.\\ has at least $m+1$ copies of $F$).",
   "clean_statement": "The following conjecture seems to be related.\n\nLet $F$ be a graph and\n$\\alpha>1$ be such that $\\operatorname{ex}(n,F)=\\Omega(n^{\\alpha}$). Then for any\n$\\epsilon > 0$ there is $n_0$ so that if $n > n_0$ and a\ngraph $H$ is the edge-disjoint union of $m=\\lceil \\epsilon\nn^{\\alpha}\\rceil$ copies of $F$, then $H$ contains another copy of $F$\n(i.e.\\ has at least $m+1$ copies of $F$).",
   "public_statement": "The following conjecture seems to be related.\n\nLet $F$ be a graph and\n$\\alpha>1$ be such that $\\operatorname{ex}(n,F)=\\Omega(n^{\\alpha}$). Then for any\n$\\epsilon > 0$ there is $n_0$ so that if $n > n_0$ and a\ngraph $H$ is the edge-disjoint union of $m=\\lceil \\epsilon\nn^{\\alpha}\\rceil$ copies of $F$, then $H$ contains another copy of $F$\n(i.e.\\ has at least $m+1$ copies of $F$).",
   "evidence": "The AIM record (Hypergraph Turan problem workshop, Section “Ruzsa--Szemerédi Theorem and Relatives,” item 44.3) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 156,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0158": {
   "statement_status": "exact",
   "original_statement": "Conjecture 2.\n1. (L. Caccetta, R. H¨ aggkvist [5]) Every simple n-vertex digraph with minimum out-degree at least r has a cycle with length at most d n \n\n> r\n\ne.\n\nThis can be restated in the following way: Let A be an n × n 0-1 matrix such that aij = 1 implies \n\naji 6 = 1 for all i 6 = j. Let aii = 1 for all i. If the sum of every row in A is at least r + 1, Adn/r e\n\nhas trace greater than n.\n\n#2.1 Partial Results \n\nThe C-H conjecture has been proved for: \n\n• r = 2 by Caccetta and H¨ aggkvist [5] \n\n• r = 3 by Hamidoune [17] \n\n• r = 4 and r = 5 by Ho´ ang and Reed [19] \n\n• r ≤ √n/ 2 by Shen [30]. For the exact statement of his result, see Theorem 5.1. This shows that for any given r, the number of counterexamples to the conjecture (if any) is finite. \n\n• Cayley graphs (which implies all vertex transitive graphs using coset representations) by Hamidoune [15]. This proof uses a lemma of Kemperman [21] (Lemma 5.9). Also, Shen [32] proved that if deg +(u) + deg +(v) ≥ 4 for all ( u, v ) ∈ E(G), then g ≤ d n/ 2e,where g denotes the girth of G. This is an average local outdegree version for the r = 2 case of the Caccetta-H¨ aggkvist conjecture. \n\n#2.2 Approximate Results I - Additive Constant \n\nAnother approach is to show that if δ+ \n\n> G\n\n≥ r, then there is a cycle of length at most n \n\n> r\n\n+ c for some small c. This has been proved for some values of c, as follows: \n\n• c = 2500 by Chv´ atal and Szemer´ edi [9] \n\n• c = 304 by Nishimura [27] \n\n• c = 73 by Shen [31]. 22.3 Approximate Results II - Special Case n/3 \n\nThe case r = n/ 2 is trivial, but r = n/ 3 has received much attention. Research has sought the minimum constant c such that δ+ \n\n> G\n\n≥ cn in an n-vertex simple digraph G forces a directed cycle of length at most 3. The conjecture is that c = 1 /3, and the current results are: \n\n• c ≤ (3 − √5) /2 = 0.382 by Caccetta and H¨ aggkvist [5] \n\n• c ≤ (2 √6 − 3) /5 = 0.3797 by Bondy in a neat subgraph counting argument [4] \n\n• c ≤ 3 − √7 = 0.3542 by [29] Similarly, Seymour, Graaf, and Schrijver [12] asked for the minimum value of β so that when the minimum inand out-degrees of G are at least βn, G has a directed cycle of length at most 3. They proved that β ≤ 0.3487 and gave a formula relating β and c. Shen applied this formula to his 1998 result to get a slight improvement to β ≤ 0.3477 [29]. \n\n3 Seymour's Second Neighborhood Conjecture \n\nThis conjecture implies the special case of Caccetta-H¨ aggkvist when both inand out-degrees are at least n/ 3, and has received much attention of its own.",
   "clean_statement": "Conjecture 2.\n1. (L. Caccetta, R. H¨ aggkvist [5]) Every simple n-vertex digraph with minimum out-degree at least r has a cycle with length at most d n\n\n> r\n\ne.\n\nThis can be restated in the following way: Let A be an n × n 0-1 matrix such that aij = 1 implies\n\naji 6 = 1 for all i 6 = j. Let aii = 1 for all i. If the sum of every row in A is at least r + 1, Adn/r e\n\nhas trace greater than n.\n\n#2.1 Partial Results\n\nThe C-H conjecture has been proved for:\n\n• r = 2 by Caccetta and H¨ aggkvist [5]\n\n• r = 3 by Hamidoune [17]\n\n• r = 4 and r = 5 by Ho´ ang and Reed [19]\n\n• r ≤ √n/ 2 by Shen [30]. For the exact statement of his result, see Theorem 5.1. This shows that for any given r, the number of counterexamples to the conjecture (if any) is finite.\n\n• Cayley graphs (which implies all vertex transitive graphs using coset representations) by Hamidoune [15]. This proof uses a lemma of Kemperman [21] (Lemma 5.9). Also, Shen [32] proved that if deg +(u) + deg +(v) ≥ 4 for all ( u, v ) ∈ E(G), then g ≤ d n/ 2e,where g denotes the girth of G. This is an average local outdegree version for the r = 2 case of the Caccetta-H¨ aggkvist conjecture.\n\n#2.2 Approximate Results I - Additive Constant\n\nAnother approach is to show that if δ+\n\n> G\n\n≥ r, then there is a cycle of length at most n\n\n> r\n\n+ c for some small c. This has been proved for some values of c, as follows:\n\n• c = 2500 by Chv´ atal and Szemer´ edi [9]\n\n• c = 304 by Nishimura [27]\n\n• c = 73 by Shen [31]. 22.3 Approximate Results II - Special Case n/3\n\nThe case r = n/ 2 is trivial, but r = n/ 3 has received much attention. Research has sought the minimum constant c such that δ+\n\n> G\n\n≥ cn in an n-vertex simple digraph G forces a directed cycle of length at most 3. The conjecture is that c = 1 /3, and the current results are:\n\n• c ≤ (3 − √5) /2 = 0.382 by Caccetta and H¨ aggkvist [5]\n\n• c ≤ (2 √6 − 3) /5 = 0.3797 by Bondy in a neat subgraph counting argument [4]\n\n• c ≤ 3 − √7 = 0.3542 by [29] Similarly, Seymour, Graaf, and Schrijver [12] asked for the minimum value of β so that when the minimum inand out-degrees of G are at least βn, G has a directed cycle of length at most 3. They proved that β ≤ 0.3487 and gave a formula relating β and c. Shen applied this formula to his 1998 result to get a slight improvement to β ≤ 0.3477 [29].\n\n3 Seymour's Second Neighborhood Conjecture\n\nThis conjecture implies the special case of Caccetta-H¨ aggkvist when both inand out-degrees are at least n/ 3, and has received much attention of its own.",
   "public_statement": "Conjecture 2.\n1. (L. Caccetta, R. H¨ aggkvist [5]) Every simple n-vertex digraph with minimum out-degree at least r has a cycle with length at most d n\n\n> r\n\ne.\n\nThis can be restated in the following way: Let A be an n × n 0-1 matrix such that aij = 1 implies\n\naji 6 = 1 for all i 6 = j. Let aii = 1 for all i. If the sum of every row in A is at least r + 1, Adn/r e\n\nhas trace greater than n.\n\n#2.1 Partial Results\n\nThe C-H conjecture has been proved for:\n\n• r = 2 by Caccetta and H¨ aggkvist [5]\n\n• r = 3 by Hamidoune [17]\n\n• r = 4 and r = 5 by Ho´ ang and Reed [19]\n\n• r ≤ √n/ 2 by Shen [30]. For the exact statement of his result, see Theorem 5.1. This shows that for any given r, the number of counterexamples to the conjecture (if any) is finite.\n\n• Cayley graphs (which implies all vertex transitive graphs using coset representations) by Hamidoune [15]. This proof uses a lemma of Kemperman [21] (Lemma 5.9). Also, Shen [32] proved that if deg +(u) + deg +(v) ≥ 4 for all ( u, v ) ∈ E(G), then g ≤ d n/ 2e,where g denotes the girth of G. This is an average local outdegree version for the r = 2 case of the Caccetta-H¨ aggkvist conjecture.\n\n#2.2 Approximate Results I - Additive Constant\n\nAnother approach is to show that if δ+\n\n> G\n\n≥ r, then there is a cycle of length at most n\n\n> r\n\n+ c for some small c. This has been proved for some values of c, as follows:\n\n• c = 2500 by Chv´ atal and Szemer´ edi [9]\n\n• c = 304 by Nishimura [27]\n\n• c = 73 by Shen [31]. 22.3 Approximate Results II - Special Case n/3\n\nThe case r = n/ 2 is trivial, but r = n/ 3 has received much attention. Research has sought the minimum constant c such that δ+\n\n> G\n\n≥ cn in an n-vertex simple digraph G forces a directed cycle of length at most 3. The conjecture is that c = 1 /3, and the current results are:\n\n• c ≤ (3 − √5) /2 = 0.382 by Caccetta and H¨ aggkvist [5]\n\n• c ≤ (2 √6 − 3) /5 = 0.3797 by Bondy in a neat subgraph counting argument [4]\n\n• c ≤ 3 − √7 = 0.3542 by [29] Similarly, Seymour, Graaf, and Schrijver [12] asked for the minimum value of β so that when the minimum inand out-degrees of G are at least βn, G has a directed cycle of length at most 3. They proved that β ≤ 0.3487 and gave a formula relating β and c. Shen applied this formula to his 1998 result to get a slight improvement to β ≤ 0.3477 [29].\n\n3 Seymour's Second Neighborhood Conjecture\n\nThis conjecture implies the special case of Caccetta-H¨ aggkvist when both inand out-degrees are at least n/ 3, and has received much attention of its own.",
   "evidence": "The source is Blair D. Sullivan's 14 April 2006 AIM workshop survey, *A Summary of Results and Problems Related to the Caccetta--Häggkvist Conjecture*. The canonical `input.json` preserves the full OCR extraction. Inspection of the original PDF recovers Conjecture 2.1 as follows.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 157,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0159": {
   "statement_status": "exact",
   "original_statement": "Conjecture 3.\n1. (Seymour) Any simple digraph with no loops or digons has a vertex v whose second neighborhood is at least as big as its first neighborhood, i.e. |N +2 (v)| ≥ | N +(v)|.\n\nThe following is known for Seymour's second neighborhood conjecture: \n\n• When G is a tournament, this is Dean's conjecture, and was proved by Fisher [13] using probabilistic methods. There is also a combinatorial proof by Havet and Thomass´ e [18]. \n\n• It was proved for digraphs with minimum outdegree ≤ 6 by Kaneko and Locke [20]. \n\n• There is a vertex v where |N +2 (v)| ≥ γ|N +(v)| and γ = 0.657298... is the unique real root of 2 x3 + x2 − 1 = \n0. (Note the conjecture is that γ = 1) by Chen, Shen, and Yuster [7]. They also claim a slight improvement to γ = 0.67815 (proof unpublished). \n\n• Godbole, Cole, and Wright [14] showed that the conjecture holds for almost all digraphs. \n\n4 r-Regular Digraphs \n\nA digraph G is r-regular if every vertex v has δ+ \n\n> G\n\n(v) = δ− \n\n> G\n\n(v) = r.",
   "clean_statement": "Conjecture 3.\n1. (Seymour) Any simple digraph with no loops or digons has a vertex v whose second neighborhood is at least as big as its first neighborhood, i.e. |N +2 (v)| ≥ | N +(v)|.\n\nThe following is known for Seymour's second neighborhood conjecture:\n\n• When G is a tournament, this is Dean's conjecture, and was proved by Fisher [13] using probabilistic methods. There is also a combinatorial proof by Havet and Thomass´ e [18].\n\n• It was proved for digraphs with minimum outdegree ≤ 6 by Kaneko and Locke [20].\n\n• There is a vertex v where |N +2 (v)| ≥ γ|N +(v)| and γ = 0.657298... is the unique real root of 2 x3 + x2 − 1 =\n0. (Note the conjecture is that γ = 1) by Chen, Shen, and Yuster [7]. They also claim a slight improvement to γ = 0.67815 (proof unpublished).\n\n• Godbole, Cole, and Wright [14] showed that the conjecture holds for almost all digraphs.\n\n4 r-Regular Digraphs\n\nA digraph G is r-regular if every vertex v has δ+\n\n> G\n\n(v) = δ−\n\n> G\n\n(v) = r.",
   "public_statement": "Conjecture 3.\n1. (Seymour) Any simple digraph with no loops or digons has a vertex v whose second neighborhood is at least as big as its first neighborhood, i.e. |N +2 (v)| ≥ | N +(v)|.\n\nThe following is known for Seymour's second neighborhood conjecture:\n\n• When G is a tournament, this is Dean's conjecture, and was proved by Fisher [13] using probabilistic methods. There is also a combinatorial proof by Havet and Thomass´ e [18].\n\n• It was proved for digraphs with minimum outdegree ≤ 6 by Kaneko and Locke [20].\n\n• There is a vertex v where |N +2 (v)| ≥ γ|N +(v)| and γ = 0.657298... is the unique real root of 2 x3 + x2 − 1 =\n0. (Note the conjecture is that γ = 1) by Chen, Shen, and Yuster [7]. They also claim a slight improvement to γ = 0.67815 (proof unpublished).\n\n• Godbole, Cole, and Wright [14] showed that the conjecture holds for almost all digraphs.\n\n4 r-Regular Digraphs\n\nA digraph G is r-regular if every vertex v has δ+\n\n> G\n\n(v) = δ−\n\n> G\n\n(v) = r.",
   "evidence": "The source is the AIM workshop problem list *The Caccetta--Haggkvist conjecture*, Conjecture 3.1 (Seymour). After repairing the PDF extraction, the mathematical statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 158,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0160": {
   "statement_status": "exact",
   "original_statement": "Conjecture 4.\n1. (Behzad, Chartrand, Wall [2]) The minimum number of vertices in an r-regular digraph G with girth g is r(g − 1) + 1.\n\nBehzad, Chartrand and Wall give an example achieving this by placing r(g − 1) + 1 vertices on a circle with each vertex having edges to the next r vertices in clockwise order. The Caccetta-H¨ aggkvist Conjecture is a generalization of this earlier conjecture. The Behzad-Chartrand-Wall conjecture was proved for the following special cases: 3• r = 2 by Behzad [1] \n\n• r = 3 by Bermond [3] \n\n• Vertex-transitive graphs by Hamidoune [16] \n\n• If δ+ \n\n> G\n\n≥ r, then g ≤ 3d n \n\n> r\n\nln( 2+ √7 \n\n> 3\n\n)e ≈ 1.312 n \n\n> r\n\nby Shen [31]. \n\n5 Related Results \n\nTheorem 5.\n1. (Shen [30]) For a digraph G on n vertices, if δ+ \n\n> G\n\n≥ r, and n ≥ 2r2 − 3r + 1,then G has a cycle of length at most d n \n\n> r\n\ne.\n\nIn a graph G, for u, v ∈ V (G), κ(u, v ) denotes the maximum number of internally disjoint paths between u and v. If G is a digraph, κ counts the maximum number of internally disjoint directed paths from u to v.In a graph G, for u, v ∈ V (G), λ(u, v ) denotes the maximum number of edge-disjoint paths between u and v. If G is a digraph, λ counts the maximum number of edge-disjoint directed paths from u to v.\n\nTheorem 5.\n2. (Thomassen [34]) For all positive integers r, there is a digraph D without digons with δ+ \n\n> D\n\n≥ r and δ− \n\n> D\n\n≥ r such that: i. no vertex v ∈ V (D) is contained in three openly disjoint circuits (that is, three circuits which pairwise share only v)ii. no edge (x, y ) ∈ E(D) has κ(y, x ) ≥ 3.\n\nTheorem 5.\n3. (Mader [25]) For every integer k ≥ 0, If G has |V (G)| > k 2(k + 1) and at most \n\nk2(k + 1) vertices of G have out-degree at most k3(k + 1), then there are vertices x 6 = y so that \n\nκ(x, y ) > k.\n\n#5.1 Undirected Graph Theorems \n\nTheorem 5.\n4. (Mader [22]) Every graph G with δG ≥ r contains vertices x, y with κ(x, y ) ≥ r\n\nwhen r ≥ 1.\n\nTheorem 5.\n5. (Mader [23]) Every graph G with δG ≥ r contains r + 1 vertices v1,..., v r+1 \n\nwith λ(vi, v j ) ≥ r for all i 6 = j.\n\n#5.2 Additive Number Theory Results \n\nGiven an additive group Γ, and sets A, B ⊆ Γ, let A + B:= {a + b | a ∈ A, b ∈ B}, and A ˆ+B:= {a + b | a ∈ A, b ∈ B, a 6 = b}. Finally, for a positive integer r, let rB:= {b1 + · · · + bh| all \n\nbi ∈ B, not necessarily distinct }.\n\nTheorem 5.\n6. (Cauchy [6] and Davenport [10], [11]) Let p be a prime, and A, B ⊆ Z/p Z be nonempty. Then |A + B| ≥ min( p, |A| + |B| − 1).\n\n4Theorem 5.\n7. (I. Chowla [8]) Let m be a positive integer, and A, B ⊆ Z/m Z such that 0 ∈ B\n\nand gcd (b, m ) = 1 for all nonzero b ∈ B. Then |A + B| ≥ min( m, |A| + |B| − 1).\n\nTheorem 5.\n8. (Dias de Silva and Hamidoune [33]) The Erd¨ os-Heilbronn Conjecture: Let \n\nA ⊆ Z/p Z, with p prime. Then |A ˆ+A| ≥ min(2 |A| − 3, p ).\n\nFor a multiplicative group Γ and sets A, B ⊆ Γ, let AB:= {ab | a ∈ A, b ∈ B}.\n\nLemma 5.\n9. (Kemperman [21]) Given a group Γ and finite non-empty subsets A, B ⊆ Γ, if \n\n1 ∈ A, B but (1, 1) is the only pair (a, b ) with a ∈ A, b ∈ B such that ab = 1, then |AB | ≥ |A| + |B| − 1.\n\nWe say G is a layered digraph if G is a digraph with V (G) = ∪hi=0 Vi with Vi 6 = ∅, and Vi ∩ Vj = ∅\n\nfor all i 6 = j, and ( u, v ) ∈ E(G) implies u ∈ Vi−1, v ∈ Vi for some i ∈ { 1,..., h }.A Pl¨ unnecke graph is a layered digraph G with the following two properties: 1. If u, v, w 1,..., w k are vertices of G with ( u, v ), (v, w 1),..., (v, w k) ∈ E(G), then there are distinct vertices v1,..., v k so that ( u, v i), (vi, w i) ∈ E(G) for i = 1,..., k.2. If v, w, u 1,..., u k are vertices of G with ( v, w ), (u1, v ),..., (uk, v ) ∈ E(G), then there are distinct vertices v1,..., v k so that ( ui, v i), (vi, w ) ∈ E(G) for i = 1,..., k.Let G be a digraph, and X, Y nonempty subsets of V (G). Then \n\nIm(X,Y):= {y ∈ Y | there is a directed path from X to y}. The magnification ratio D(X,Y) is \n\nD(X, Y ):= min \n\n> Z⊆X,Z 6=∅\n\n{|Im (Z, Y )|\n\n|Z|\n\n}.\n\nTheorem 5.\n1\n0. (Pl¨ unnecke [28]) In a Pl¨ unnecke graph, let Di = D(V0, V i). Then \n\nD1 ≥ D 1 \n\n> 2\n> 2\n\n≥ · · · ≥ D 1 \n\n> h\n> h.\n\nThe following are consequences of applying Pl¨ unnecke's Inequalities to a special graph created from subsets A, B of a group: \n\nTheorem 5.11. For sets A, B ⊂ Γ:1. |iB | 1 \n\n> i\n\n≥ | hB | 1 \n\n> h\n\nfor all 0 ≤ i < h.2. If |B| = k, and |B + B| ≤ ck, then |hB | ≤ chk.3. If |A| = n, and |A + B| < cn then for all k, ` ∈ Z+, we have that |kB − `B | ≤ ck+`n where \n\nkB − `B denotes the set of all elements expressable as (b1 + · · · + bk) − (b′ \n\n> 1\n\n+ · · · + b′\n\n> `\n\n) where all bi, b ′ \n\n> i\n\nare in B.\n\nTheorem 5.12. (generalization of Erd¨ os-Heilbronn (Thm. 5.8)) Let A, B ⊆ Z/p Z, with p prime and |A| 6 = |B|. Let C = A ˆ+B. Then |C| ≥ min( |A| + |B| − 2, p ).\n\n56 Open Problems and Conjectures \n\n#6.1 Rainbow Conjectures \n\n6.1.1 A Colored Generalization of Seymour's Second Neighborhood \n\nGiven a digraph G = ( V, E ) with each edge e ∈ E having a set Se of labels in {1, 2,..., k G}, a \n\nrainbow structure H in G (such as a path or cycle) means one in which there is a way to assign each edge e ∈ E(H) a label `(e) ∈ Se so that ell (e) 6 = `(f ) for all edges e 6 = f in E(H).",
   "clean_statement": "Conjecture 4.\n1. (Behzad, Chartrand, Wall [2]) The minimum number of vertices in an r-regular digraph G with girth g is r(g − 1) + 1.\n\nBehzad, Chartrand and Wall give an example achieving this by placing r(g − 1) + 1 vertices on a circle with each vertex having edges to the next r vertices in clockwise order. The Caccetta-H¨ aggkvist Conjecture is a generalization of this earlier conjecture. The Behzad-Chartrand-Wall conjecture was proved for the following special cases: 3• r = 2 by Behzad [1]\n\n• r = 3 by Bermond [3]\n\n• Vertex-transitive graphs by Hamidoune [16]\n\n• If δ+\n\n> G\n\n≥ r, then g ≤ 3d n\n\n> r\n\nln( 2+ √7\n\n> 3\n\n)e ≈ 1.312 n\n\n> r\n\nby Shen [31].\n\n5 Related Results\n\nTheorem 5.\n1. (Shen [30]) For a digraph G on n vertices, if δ+\n\n> G\n\n≥ r, and n ≥ 2r2 − 3r + 1,then G has a cycle of length at most d n\n\n> r\n\ne.\n\nIn a graph G, for u, v ∈ V (G), κ(u, v ) denotes the maximum number of internally disjoint paths between u and v. If G is a digraph, κ counts the maximum number of internally disjoint directed paths from u to v.In a graph G, for u, v ∈ V (G), λ(u, v ) denotes the maximum number of edge-disjoint paths between u and v. If G is a digraph, λ counts the maximum number of edge-disjoint directed paths from u to v.\n\nTheorem 5.\n2. (Thomassen [34]) For all positive integers r, there is a digraph D without digons with δ+\n\n> D\n\n≥ r and δ−\n\n> D\n\n≥ r such that: i. no vertex v ∈ V (D) is contained in three openly disjoint circuits (that is, three circuits which pairwise share only v)ii. no edge (x, y ) ∈ E(D) has κ(y, x ) ≥ 3.\n\nTheorem 5.\n3. (Mader [25]) For every integer k ≥ 0, If G has |V (G)| > k 2(k + 1) and at most\n\nk2(k + 1) vertices of G have out-degree at most k3(k + 1), then there are vertices x 6 = y so that\n\nκ(x, y ) > k.\n\n#5.1 Undirected Graph Theorems\n\nTheorem 5.\n4. (Mader [22]) Every graph G with δG ≥ r contains vertices x, y with κ(x, y ) ≥ r\n\nwhen r ≥ 1.\n\nTheorem 5.\n5. (Mader [23]) Every graph G with δG ≥ r contains r + 1 vertices v1,..., v r+1\n\nwith λ(vi, v j ) ≥ r for all i 6 = j.\n\n#5.2 Additive Number Theory Results\n\nGiven an additive group Γ, and sets A, B ⊆ Γ, let A + B:= {a + b | a ∈ A, b ∈ B}, and A ˆ+B:= {a + b | a ∈ A, b ∈ B, a 6 = b}. Finally, for a positive integer r, let rB:= {b1 + · · · + bh| all\n\nbi ∈ B, not necessarily distinct }.\n\nTheorem 5.\n6. (Cauchy [6] and Davenport [10], [11]) Let p be a prime, and A, B ⊆ Z/p Z be nonempty. Then |A + B| ≥ min( p, |A| + |B| − 1).\n\n4Theorem 5.\n7. (I. Chowla [8]) Let m be a positive integer, and A, B ⊆ Z/m Z such that 0 ∈ B\n\nand gcd (b, m ) = 1 for all nonzero b ∈ B. Then |A + B| ≥ min( m, |A| + |B| − 1).\n\nTheorem 5.\n8. (Dias de Silva and Hamidoune [33]) The Erd¨ os-Heilbronn Conjecture: Let\n\nA ⊆ Z/p Z, with p prime. Then |A ˆ+A| ≥ min(2 |A| − 3, p ).\n\nFor a multiplicative group Γ and sets A, B ⊆ Γ, let AB:= {ab | a ∈ A, b ∈ B}.\n\nLemma 5.\n9. (Kemperman [21]) Given a group Γ and finite non-empty subsets A, B ⊆ Γ, if\n\n1 ∈ A, B but (1, 1) is the only pair (a, b ) with a ∈ A, b ∈ B such that ab = 1, then |AB | ≥ |A| + |B| − 1.\n\nWe say G is a layered digraph if G is a digraph with V (G) = ∪hi=0 Vi with Vi 6 = ∅, and Vi ∩ Vj = ∅\n\nfor all i 6 = j, and ( u, v ) ∈ E(G) implies u ∈ Vi−1, v ∈ Vi for some i ∈ { 1,..., h }.A Pl¨ unnecke graph is a layered digraph G with the following two properties: 1. If u, v, w 1,..., w k are vertices of G with ( u, v ), (v, w 1),..., (v, w k) ∈ E(G), then there are distinct vertices v1,..., v k so that ( u, v i), (vi, w i) ∈ E(G) for i = 1,..., k.2. If v, w, u 1,..., u k are vertices of G with ( v, w ), (u1, v ),..., (uk, v ) ∈ E(G), then there are distinct vertices v1,..., v k so that ( ui, v i), (vi, w ) ∈ E(G) for i = 1,..., k.Let G be a digraph, and X, Y nonempty subsets of V (G). Then\n\nIm(X,Y):= {y ∈ Y | there is a directed path from X to y}. The magnification ratio D(X,Y) is\n\nD(X, Y ):= min\n\n> Z⊆X,Z 6=∅\n\n{|Im (Z, Y )|\n\n|Z|\n\n}.\n\nTheorem 5.\n1\n0. (Pl¨ unnecke [28]) In a Pl¨ unnecke graph, let Di = D(V0, V i). Then\n\nD1 ≥ D 1\n\n> 2\n> 2\n\n≥ · · · ≥ D 1\n\n> h\n> h.\n\nThe following are consequences of applying Pl¨ unnecke's Inequalities to a special graph created from subsets A, B of a group:\n\nTheorem 5.11. For sets A, B ⊂ Γ:1. |iB | 1\n\n> i\n\n≥ | hB | 1\n\n> h\n\nfor all 0 ≤ i < h.2. If |B| = k, and |B + B| ≤ ck, then |hB | ≤ chk.3. If |A| = n, and |A + B| < cn then for all k, ` ∈ Z+, we have that |kB − `B | ≤ ck+`n where\n\nkB − `B denotes the set of all elements expressable as (b1 + · · · + bk) − (b′\n\n> 1\n\n+ · · · + b′\n\n> `\n\n) where all bi, b ′\n\n> i\n\nare in B.\n\nTheorem 5.12. (generalization of Erd¨ os-Heilbronn (Thm. 5.8)) Let A, B ⊆ Z/p Z, with p prime and |A| 6 = |B|. Let C = A ˆ+B. Then |C| ≥ min( |A| + |B| − 2, p ).\n\n56 Open Problems and Conjectures\n\n#6.1 Rainbow Conjectures\n\n6.1.1 A Colored Generalization of Seymour's Second Neighborhood\n\nGiven a digraph G = ( V, E ) with each edge e ∈ E having a set Se of labels in {1, 2,..., k G}, a\n\nrainbow structure H in G (such as a path or cycle) means one in which there is a way to assign each edge e ∈ E(H) a label `(e) ∈ Se so that ell (e) 6 = `(f ) for all edges e 6 = f in E(H).",
   "public_statement": "Conjecture 4.\n1. (Behzad, Chartrand, Wall [2]) The minimum number of vertices in an r-regular digraph G with girth g is r(g − 1) + 1.\n\nBehzad, Chartrand and Wall give an example achieving this by placing r(g − 1) + 1 vertices on a circle with each vertex having edges to the next r vertices in clockwise order. The Caccetta-H¨ aggkvist Conjecture is a generalization of this earlier conjecture. The Behzad-Chartrand-Wall conjecture was proved for the following special cases: 3• r = 2 by Behzad [1]\n\n• r = 3 by Bermond [3]\n\n• Vertex-transitive graphs by Hamidoune [16]\n\n• If δ+\n\n> G\n\n≥ r, then g ≤ 3d n\n\n> r\n\nln( 2+ √7\n\n> 3\n\n)e ≈ 1.312 n\n\n> r\n\nby Shen [31].\n\n5 Related Results\n\nTheorem 5.\n1. (Shen [30]) For a digraph G on n vertices, if δ+\n\n> G\n\n≥ r, and n ≥ 2r2 − 3r + 1,then G has a cycle of length at most d n\n\n> r\n\ne.\n\nIn a graph G, for u, v ∈ V (G), κ(u, v ) denotes the maximum number of internally disjoint paths between u and v. If G is a digraph, κ counts the maximum number of internally disjoint directed paths from u to v.In a graph G, for u, v ∈ V (G), λ(u, v ) denotes the maximum number of edge-disjoint paths between u and v. If G is a digraph, λ counts the maximum number of edge-disjoint directed paths from u to v.\n\nTheorem 5.\n2. (Thomassen [34]) For all positive integers r, there is a digraph D without digons with δ+\n\n> D\n\n≥ r and δ−\n\n> D\n\n≥ r such that: i. no vertex v ∈ V (D) is contained in three openly disjoint circuits (that is, three circuits which pairwise share only v)ii. no edge (x, y ) ∈ E(D) has κ(y, x ) ≥ 3.\n\nTheorem 5.\n3. (Mader [25]) For every integer k ≥ 0, If G has |V (G)| > k 2(k + 1) and at most\n\nk2(k + 1) vertices of G have out-degree at most k3(k + 1), then there are vertices x 6 = y so that\n\nκ(x, y ) > k.\n\n#5.1 Undirected Graph Theorems\n\nTheorem 5.\n4. (Mader [22]) Every graph G with δG ≥ r contains vertices x, y with κ(x, y ) ≥ r\n\nwhen r ≥ 1.\n\nTheorem 5.\n5. (Mader [23]) Every graph G with δG ≥ r contains r + 1 vertices v1,..., v r+1\n\nwith λ(vi, v j ) ≥ r for all i 6 = j.\n\n#5.2 Additive Number Theory Results\n\nGiven an additive group Γ, and sets A, B ⊆ Γ, let A + B:= {a + b | a ∈ A, b ∈ B}, and A ˆ+B:= {a + b | a ∈ A, b ∈ B, a 6 = b}. Finally, for a positive integer r, let rB:= {b1 + · · · + bh| all\n\nbi ∈ B, not necessarily distinct }.\n\nTheorem 5.\n6. (Cauchy [6] and Davenport [10], [11]) Let p be a prime, and A, B ⊆ Z/p Z be nonempty. Then |A + B| ≥ min( p, |A| + |B| − 1).\n\n4Theorem 5.\n7. (I. Chowla [8]) Let m be a positive integer, and A, B ⊆ Z/m Z such that 0 ∈ B\n\nand gcd (b, m ) = 1 for all nonzero b ∈ B. Then |A + B| ≥ min( m, |A| + |B| − 1).\n\nTheorem 5.\n8. (Dias de Silva and Hamidoune [33]) The Erd¨ os-Heilbronn Conjecture: Let\n\nA ⊆ Z/p Z, with p prime. Then |A ˆ+A| ≥ min(2 |A| − 3, p ).\n\nFor a multiplicative group Γ and sets A, B ⊆ Γ, let AB:= {ab | a ∈ A, b ∈ B}.\n\nLemma 5.\n9. (Kemperman [21]) Given a group Γ and finite non-empty subsets A, B ⊆ Γ, if\n\n1 ∈ A, B but (1, 1) is the only pair (a, b ) with a ∈ A, b ∈ B such that ab = 1, then |AB | ≥ |A| + |B| − 1.\n\nWe say G is a layered digraph if G is a digraph with V (G) = ∪hi=0 Vi with Vi 6 = ∅, and Vi ∩ Vj = ∅\n\nfor all i 6 = j, and ( u, v ) ∈ E(G) implies u ∈ Vi−1, v ∈ Vi for some i ∈ { 1,..., h }.A Pl¨ unnecke graph is a layered digraph G with the following two properties: 1. If u, v, w 1,..., w k are vertices of G with ( u, v ), (v, w 1),..., (v, w k) ∈ E(G), then there are distinct vertices v1,..., v k so that ( u, v i), (vi, w i) ∈ E(G) for i = 1,..., k.2. If v, w, u 1,..., u k are vertices of G with ( v, w ), (u1, v ),..., (uk, v ) ∈ E(G), then there are distinct vertices v1,..., v k so that ( ui, v i), (vi, w ) ∈ E(G) for i = 1,..., k.Let G be a digraph, and X, Y nonempty subsets of V (G). Then\n\nIm(X,Y):= {y ∈ Y | there is a directed path from X to y}. The magnification ratio D(X,Y) is\n\nD(X, Y ):= min\n\n> Z⊆X,Z 6=∅\n\n{|Im (Z, Y )|\n\n|Z|\n\n}.\n\nTheorem 5.\n1\n0. (Pl¨ unnecke [28]) In a Pl¨ unnecke graph, let Di = D(V0, V i). Then\n\nD1 ≥ D 1\n\n> 2\n> 2\n\n≥ · · · ≥ D 1\n\n> h\n> h.\n\nThe following are consequences of applying Pl¨ unnecke's Inequalities to a special graph created from subsets A, B of a group:\n\nTheorem 5.11. For sets A, B ⊂ Γ:1. |iB | 1\n\n> i\n\n≥ | hB | 1\n\n> h\n\nfor all 0 ≤ i < h.2. If |B| = k, and |B + B| ≤ ck, then |hB | ≤ chk.3. If |A| = n, and |A + B| < cn then for all k, ` ∈ Z+, we have that |kB − `B | ≤ ck+`n where\n\nkB − `B denotes the set of all elements expressable as (b1 + · · · + bk) − (b′\n\n> 1\n\n+ · · · + b′\n\n> `\n\n) where all bi, b ′\n\n> i\n\nare in B.\n\nTheorem 5.12. (generalization of Erd¨ os-Heilbronn (Thm. 5.8)) Let A, B ⊆ Z/p Z, with p prime and |A| 6 = |B|. Let C = A ˆ+B. Then |C| ≥ min( |A| + |B| − 2, p ).\n\n56 Open Problems and Conjectures\n\n#6.1 Rainbow Conjectures\n\n6.1.1 A Colored Generalization of Seymour's Second Neighborhood\n\nGiven a digraph G = ( V, E ) with each edge e ∈ E having a set Se of labels in {1, 2,..., k G}, a\n\nrainbow structure H in G (such as a path or cycle) means one in which there is a way to assign each edge e ∈ E(H) a label `(e) ∈ Se so that ell (e) 6 = `(f ) for all edges e 6 = f in E(H).",
   "evidence": "The source is Blair D. Sullivan's 2006 AIM workshop survey *A Summary of Results and Problems Related to the Caccetta--Häggkvist Conjecture*. Section 4 defines an \\(r\\)-regular digraph by \\[ d^+(v)=d^-(v)=r\\qquad\\text{for every vertex }v. \\] After repairing the extraction, Conjecture 4.1 reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 159,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0161": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n1. (Seymour, Sullivan) Let G be a simple digraph on the vertex set V, and \n\nE1,..., E k ⊆ E(G). Say an edge e ∈ E(G) has label set Se ⊆ { 1,..., k } where i ∈ Se if and only if e ∈ Ei. Finally, let Gi = ( V, E i).i. There exists a rainbow (di)cycle in G or ii. There exists a vertex v such that |{ w | there exists a rainbow path from v to w}| ≥ ∑ki=1 δ+ \n\n> Gi\n\n(v).Notes: This is false if you require that the colors appear in an increasing order (cyclic on the cycle). We have been able to show that this conjecture holds when G1,..., G k are Cayley graphs on a common group Γ (using induction on Lemma 5.9), and when δ+ \n\n> Gi\n\n(v) ≤ 1 for all v and all i\n\nexcept i = 1, where we allow the outdegrees to be unbounded (but finite). \n\n6.1.2 Implications of",
   "clean_statement": "Conjecture 6.\n1. (Seymour, Sullivan) Let G be a simple digraph on the vertex set V, and\n\nE1,..., E k ⊆ E(G). Say an edge e ∈ E(G) has label set Se ⊆ { 1,..., k } where i ∈ Se if and only if e ∈ Ei. Finally, let Gi = ( V, E i).i. There exists a rainbow (di)cycle in G or ii. There exists a vertex v such that |{ w | there exists a rainbow path from v to w}| ≥ ∑ki=1 δ+\n\n> Gi\n\n(v).Notes: This is false if you require that the colors appear in an increasing order (cyclic on the cycle). We have been able to show that this conjecture holds when G1,..., G k are Cayley graphs on a common group Γ (using induction on Lemma 5.9), and when δ+\n\n> Gi\n\n(v) ≤ 1 for all v and all i\n\nexcept i = 1, where we allow the outdegrees to be unbounded (but finite).\n\n6.1.2 Implications of",
   "public_statement": "Conjecture 6.\n1. (Seymour, Sullivan) Let G be a simple digraph on the vertex set V, and\n\nE1,..., E k ⊆ E(G). Say an edge e ∈ E(G) has label set Se ⊆ { 1,..., k } where i ∈ Se if and only if e ∈ Ei. Finally, let Gi = ( V, E i).i. There exists a rainbow (di)cycle in G or ii. There exists a vertex v such that |{ w | there exists a rainbow path from v to w}| ≥ ∑ki=1 δ+\n\n> Gi\n\n(v).Notes: This is false if you require that the colors appear in an increasing order (cyclic on the cycle). We have been able to show that this conjecture holds when G1,..., G k are Cayley graphs on a common group Γ (using induction on Lemma 5.9), and when δ+\n\n> Gi\n\n(v) ≤ 1 for all v and all i\n\nexcept i = 1, where we allow the outdegrees to be unbounded (but finite).\n\n6.1.2 Implications of",
   "evidence": "The record comes from Blair D. Sullivan's 2006 AIM workshop summary *A Summary of Problems and Results Related to the Caccetta--Häggkvist Conjecture*, Section 6.1.1, “A Colored Generalization of Seymour's Second Neighborhood.” The corpus extraction runs into the next heading (“6.1.2 Implications of”); that phrase is not part of the conjecture.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 160,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0162": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.2. Seymour's Second Neighborhood Conjecture Notes: To see this, for a digraph H, let k = 2, G = H, and E1 = E2 = E(H).",
   "clean_statement": "Conjecture 6.2. Seymour's Second Neighborhood Conjecture Notes: To see this, for a digraph H, let k = 2, G = H, and E1 = E2 = E(H).",
   "public_statement": "Conjecture 6.2. Seymour's Second Neighborhood Conjecture Notes: To see this, for a digraph H, let k = 2, G = H, and E1 = E2 = E(H).",
   "evidence": "The canonical record is the following extracted text from Blair D. Sullivan's AIM survey:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 161,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0163": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.3. Caccetta-H¨ aggkvist Conjecture (general case) Notes: For a digraph H, take G = H, k = d n\n\n> δ+\n> H\n\ne, and E1 = · · · = Ek = E(H). We must get a rainbow cycle because the sum of the outdegrees at each vertex is ≥ n. This corresponds to a dicycle of length at most d n\n\n> δ+\n> H\n\ne in H, as desired.",
   "clean_statement": "Conjecture 6.3. Caccetta-H¨ aggkvist Conjecture (general case) Notes: For a digraph H, take G = H, k = d n\n\n> δ+\n> H\n\ne, and E1 = · · · = Ek = E(H). We must get a rainbow cycle because the sum of the outdegrees at each vertex is ≥ n. This corresponds to a dicycle of length at most d n\n\n> δ+\n> H\n\ne in H, as desired.",
   "public_statement": "Conjecture 6.3. Caccetta-H¨ aggkvist Conjecture (general case) Notes: For a digraph H, take G = H, k = d n\n\n> δ+\n> H\n\ne, and E1 = · · · = Ek = E(H). We must get a rainbow cycle because the sum of the outdegrees at each vertex is ≥ n. This corresponds to a dicycle of length at most d n\n\n> δ+\n> H\n\ne in H, as desired.",
   "evidence": "The source record is Conjecture 6.3 in the AIM notes from the 2006 workshop “The Caccetta--Haggkvist conjecture.” Its displayed text is an implication from the preceding Seymour--Sullivan rainbow conjecture, rather than a new independent formulation:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 162,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0164": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.4. Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| +\n\n|N+(v)| ≥ 2|N −(v)|.Notes: Recall, for comparison, SSN can be written as |N +2 (v)| + |N +(v)| ≥ 2|N +(v)|. To see how",
   "clean_statement": "Conjecture 6.4. Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| +\n\n|N+(v)| ≥ 2|N −(v)|.Notes: Recall, for comparison, SSN can be written as |N +2 (v)| + |N +(v)| ≥ 2|N +(v)|. To see how",
   "public_statement": "Conjecture 6.4. Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| +\n\n|N+(v)| ≥ 2|N −(v)|.Notes: Recall, for comparison, SSN can be written as |N +2 (v)| + |N +(v)| ≥ 2|N +(v)|. To see how",
   "evidence": "The canonical record is visibly truncated after “To see how.” Its exact extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 163,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0165": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.1 implies 6.4, take a simple digraph H with no loops or digons, and set G = H and \n\nE1 = E2 = E(H). G cannot have a rainbow cycle by definition of H. Define N +∗ \n\n> G\n\n(u) = {vertices you can reach by a rainbow path in G from u}, and N −∗ \n\n> G\n\n(u) = {vertices that have a rainbow path in G to u}. Let E3 be the edges {(u, v ) | v is not in the set N −∗ \n\n> G\n\n(u)}. Let G′ = H, and have subsets E1, E 2, E 3 ⊆ E(G′) giving rise to label sets S′ \n\n> e\n\n⊆ { 1, 2, 3} for e ∈ G′. We can see from these definitions that for any u in V (G′), \n\n> 3\n\n∑\n\n> i=1\n\nδ+ \n\n> Gi\n\n(u) = 2 |N +(u)| + (( n − 1) − | N −(u)| − | N − \n\n> 2\n\n(u)|),\n\n6where all neighborhoods referenced on the RHS are in H. We can rewrite this as: \n\n> 3\n\n∑\n\n> i=1\n\nδ+ \n\n> Gi\n\n(u) = (( n − 1) − (|N −(u)| + |N − \n\n> 2\n\n(u)| − 2|N +(u)|)).\n\nThen ∑3 \n\n> i=1\n\nδ+ \n\n> Gi\n\n(u) ≥ n whenever |N −(u)| + |N − \n\n> 2\n\n(u)| < 2|N +(u)|. If this were true for all vertices \n\nu, then no vertex could have |N +∗ \n\n> G′\n\n(u)| ≥ ∑3 \n\n> i=1\n\nδ+ \n\n> Gi\n\n(u) ≥ n, so we must have a rainbow cycle in G′,by",
   "clean_statement": "Conjecture 6.1 implies 6.4, take a simple digraph H with no loops or digons, and set G = H and\n\nE1 = E2 = E(H). G cannot have a rainbow cycle by definition of H. Define N +∗\n\n> G\n\n(u) = {vertices you can reach by a rainbow path in G from u}, and N −∗\n\n> G\n\n(u) = {vertices that have a rainbow path in G to u}. Let E3 be the edges {(u, v ) | v is not in the set N −∗\n\n> G\n\n(u)}. Let G′ = H, and have subsets E1, E 2, E 3 ⊆ E(G′) giving rise to label sets S′\n\n> e\n\n⊆ { 1, 2, 3} for e ∈ G′. We can see from these definitions that for any u in V (G′),\n\n> 3\n\n∑\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) = 2 |N +(u)| + (( n − 1) − | N −(u)| − | N −\n\n> 2\n\n(u)|),\n\n6where all neighborhoods referenced on the RHS are in H. We can rewrite this as:\n\n> 3\n\n∑\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) = (( n − 1) − (|N −(u)| + |N −\n\n> 2\n\n(u)| − 2|N +(u)|)).\n\nThen ∑3\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) ≥ n whenever |N −(u)| + |N −\n\n> 2\n\n(u)| < 2|N +(u)|. If this were true for all vertices\n\nu, then no vertex could have |N +∗\n\n> G′\n\n(u)| ≥ ∑3\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) ≥ n, so we must have a rainbow cycle in G′,by",
   "public_statement": "Conjecture 6.1 implies 6.4, take a simple digraph H with no loops or digons, and set G = H and\n\nE1 = E2 = E(H). G cannot have a rainbow cycle by definition of H. Define N +∗\n\n> G\n\n(u) = {vertices you can reach by a rainbow path in G from u}, and N −∗\n\n> G\n\n(u) = {vertices that have a rainbow path in G to u}. Let E3 be the edges {(u, v ) | v is not in the set N −∗\n\n> G\n\n(u)}. Let G′ = H, and have subsets E1, E 2, E 3 ⊆ E(G′) giving rise to label sets S′\n\n> e\n\n⊆ { 1, 2, 3} for e ∈ G′. We can see from these definitions that for any u in V (G′),\n\n> 3\n\n∑\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) = 2 |N +(u)| + (( n − 1) − | N −(u)| − | N −\n\n> 2\n\n(u)|),\n\n6where all neighborhoods referenced on the RHS are in H. We can rewrite this as:\n\n> 3\n\n∑\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) = (( n − 1) − (|N −(u)| + |N −\n\n> 2\n\n(u)| − 2|N +(u)|)).\n\nThen ∑3\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) ≥ n whenever |N −(u)| + |N −\n\n> 2\n\n(u)| < 2|N +(u)|. If this were true for all vertices\n\nu, then no vertex could have |N +∗\n\n> G′\n\n(u)| ≥ ∑3\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) ≥ n, so we must have a rainbow cycle in G′,by",
   "evidence": "This canonical record is not an independent conjecture. It is the middle of the proof, split across records AIM-COMBINATORICS-0164 through AIM-COMBINATORICS-0166, that the rainbow-reachability Conjecture 6.1 of Seymour and Sullivan implies Sullivan's neighborhood Conjecture 6.4. The exact source is Blair D. Sullivan's 2006 AIM workshop summary *A Summary of Problems and Results Related to the Caccetta--Häggkvist Conjecture*, pages 5--6. The original arXiv TeX source was also inspected.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 164,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0166": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.1. However, by construction, since G has no rainbow cycle, G′ has no rainbow cycle. Thus there is a vertex v ∈ V (H) so that |N −(u)| + |N − \n\n> 2\n\n(u)| ≥ 2|N +(u)|. If we reverse all edges in H, this gives |N +(u)| + |N +2 (u)| ≥ 2|N −(u)|, as claimed.",
   "clean_statement": "Conjecture 6.1. However, by construction, since G has no rainbow cycle, G′ has no rainbow cycle. Thus there is a vertex v ∈ V (H) so that |N −(u)| + |N −\n\n> 2\n\n(u)| ≥ 2|N +(u)|. If we reverse all edges in H, this gives |N +(u)| + |N +2 (u)| ≥ 2|N −(u)|, as claimed.",
   "public_statement": "Conjecture 6.1. However, by construction, since G has no rainbow cycle, G′ has no rainbow cycle. Thus there is a vertex v ∈ V (H) so that |N −(u)| + |N −\n\n> 2\n\n(u)| ≥ 2|N +(u)|. If we reverse all edges in H, this gives |N +(u)| + |N +2 (u)| ≥ 2|N −(u)|, as claimed.",
   "evidence": "The assigned record is not an independent conjecture. It is the final OCR-split fragment of the notes following Conjecture 6.4 in Blair D. Sullivan's 2006 AIM workshop summary *A Summary of Problems and Results Related to the Caccetta--Häggkvist Conjecture*. The exact PDF and original arXiv TeX were inspected. The source passage reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 165,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0167": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n5. (Seymour) Under the hypotheses of",
   "clean_statement": "Conjecture 6.\n5. (Seymour) Under the hypotheses of",
   "public_statement": "Conjecture 6.\n5. (Seymour) Under the hypotheses of",
   "evidence": "The canonical record is severely truncated:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 166,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0168": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 6.1, if |V | = d and ∑ki=1 δGi (v) ≥\n\nd for all v, G must have a rainbow cycle. Note: This conjecture is false if |V | = d is replaced by |V | = d + 1. \n\n6.1.3 Other Conjectures Inspired by (or related to)",
   "clean_statement": null,
   "public_statement": "Conjecture 6.1, if |V | = d and ∑ki=1 δGi (v) ≥\n\nd for all v, G must have a rainbow cycle. Note: This conjecture is false if |V | = d is replaced by |V | = d + 1.\n\n6.1.3 Other Conjectures Inspired by (or related to)",
   "evidence": "The canonical record is the second half of Conjecture 6.5 in Blair Sullivan's summary of the 2006 AIM workshop *The Caccetta--Haggkvist conjecture*. The preceding canonical record, AIM-COMBINATORICS-0167, contains only the truncated prefix “Conjecture 6.5. (Seymour) Under the hypotheses of”. Reading the two records against page 6 of the source PDF recovers the complete statement:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 167,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0169": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 6.1 \n\nIf we believe Seymour's second neighborhood conjecture and",
   "clean_statement": null,
   "public_statement": "Conjecture 6.1\n\nIf we believe Seymour's second neighborhood conjecture and",
   "evidence": "The assigned record is an extremely truncated transition, not a conjecture by itself. Comparison with the original arXiv TeX and AIM PDF recovers the complete sentence:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 168,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0170": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.1 (specifically 6.4), we might be led to ask if the following holds:",
   "clean_statement": "Conjecture 6.1 (specifically 6.4), we might be led to ask if the following holds:",
   "public_statement": "Conjecture 6.1 (specifically 6.4), we might be led to ask if the following holds:",
   "evidence": "The canonical record is not a complete conjecture. It is the second half of a transition sentence split across two extracted records:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 169,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0171": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.6. (\"Compromise Conjecture\") Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| ≥ | N −(v)|.",
   "clean_statement": "Conjecture 6.6. (\"Compromise Conjecture\") Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| ≥ | N −(v)|.",
   "public_statement": "Conjecture 6.6. (\"Compromise Conjecture\") Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| ≥ | N −(v)|.",
   "evidence": "The exact corpus record is Conjecture 6.6 from the AIM workshop list on the Caccetta--Haggkvist conjecture:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 170,
   "attempt": 2
  },
  "AIM-COMBINATORICS-0172": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.7. Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| +\n\n|N+(v)| ≥ 2 min( |N −(v)|, |N +(v)|).",
   "clean_statement": "Conjecture 6.7. Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| +\n\n|N+(v)| ≥ 2 min( |N −(v)|, |N +(v)|).",
   "public_statement": "Conjecture 6.7. Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| +\n\n|N+(v)| ≥ 2 min( |N −(v)|, |N +(v)|).",
   "evidence": "The original arXiv TeX and AIM PDF give the following unambiguous statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 171,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0173": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.8. Under the hypotheses of",
   "clean_statement": "Conjecture 6.8. Under the hypotheses of",
   "public_statement": "Conjecture 6.8. Under the hypotheses of",
   "evidence": "The canonical record is split across two consecutive extraction records. The exact text in the assigned record is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 172,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0174": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.1, if δ+ \n\n> Gi\n\n≥ ri, and ∑ti=1 ri ≥ | V |, there is a rainbow cycle in G.",
   "clean_statement": "Conjecture 6.1, if δ+\n\n> Gi\n\n≥ ri, and ∑ti=1 ri ≥ | V |, there is a rainbow cycle in G.",
   "public_statement": "Conjecture 6.1, if δ+\n\n> Gi\n\n≥ ri, and ∑ti=1 ri ≥ | V |, there is a rainbow cycle in G.",
   "evidence": "The canonical record is the second half of a sentence split across two extraction records. Its literal text is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 173,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0175": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.9. Under the hypotheses of",
   "clean_statement": "Conjecture 6.9. Under the hypotheses of",
   "public_statement": "Conjecture 6.9. Under the hypotheses of",
   "evidence": "The assigned canonical record is an extraction fragment:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 174,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0176": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.1, if ∑ti=1 δ+ \n\n> Gi\n\n(v) ≥ | V | for all vertices \n\nv, there is a rainbow cycle in G.",
   "clean_statement": "Conjecture 6.1, if ∑ti=1 δ+\n\n> Gi\n\n(v) ≥ | V | for all vertices\n\nv, there is a rainbow cycle in G.",
   "public_statement": "Conjecture 6.1, if ∑ti=1 δ+\n\n> Gi\n\n(v) ≥ | V | for all vertices\n\nv, there is a rainbow cycle in G.",
   "evidence": "The canonical record is an OCR fragment:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 175,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0177": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n1\n0. (Devos) Under the hypotheses of",
   "clean_statement": "Conjecture 6.\n1\n0. (Devos) Under the hypotheses of",
   "public_statement": "Conjecture 6.\n1\n0. (Devos) Under the hypotheses of",
   "evidence": "The assigned canonical record is only the first fragment",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 176,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0178": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.1, if there is no rainbow cycle in G with strictly increasing edge labels, then the average number of vertices reachable from a fixed vertex v by label-increasing (possibly trivial) paths is at least 1 + ∑ki=1 δ+ \n\n> Gi.Note: This can be proved when Gi = Cayley(Γ, A i) for some group Γ, using Lemma 5.9 \n\n#6.2 Second & Kth Neighborhood Conjectures",
   "clean_statement": "Conjecture 6.1, if there is no rainbow cycle in G with strictly increasing edge labels, then the average number of vertices reachable from a fixed vertex v by label-increasing (possibly trivial) paths is at least 1 + ∑ki=1 δ+\n\n> Gi.Note: This can be proved when Gi = Cayley(Γ, A i) for some group Γ, using Lemma 5.9\n\n#6.2 Second & Kth Neighborhood Conjectures",
   "public_statement": "Conjecture 6.1, if there is no rainbow cycle in G with strictly increasing edge labels, then the average number of vertices reachable from a fixed vertex v by label-increasing (possibly trivial) paths is at least 1 + ∑ki=1 δ+\n\n> Gi.Note: This can be proved when Gi = Cayley(Γ, A i) for some group Γ, using Lemma 5.9\n\n#6.2 Second & Kth Neighborhood Conjectures",
   "evidence": "The assigned record is the second part of a split extraction. The preceding source fragment supplies only the heading “Conjecture 6.10. (Devos) Under the hypotheses of”; the assigned record and the original PDF/TeX supply the rest. The exact TeX source reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 177,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0179": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.11. Is Seymour's Second Neighborhood true for locally finite digraphs? What if we just require the outdegrees to be finite?",
   "clean_statement": "Conjecture 6.11. Is Seymour's Second Neighborhood true for locally finite digraphs? What if we just require the outdegrees to be finite?",
   "public_statement": "Conjecture 6.11. Is Seymour's Second Neighborhood true for locally finite digraphs? What if we just require the outdegrees to be finite?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 178,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0180": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n1\n2. (Thomass´ e) Let G be a digraph with no directed cycle of length at most three. Then there is a vertex v ∈ V (G) with δ+(v) at most the number of non-neighbors of v.\n\nThe following generalization of second neighborhood to kth neighborhood was taken from a (unpublished) paper of Serge Burckel: 7",
   "clean_statement": "Conjecture 6.\n1\n2. (Thomass´ e) Let G be a digraph with no directed cycle of length at most three. Then there is a vertex v ∈ V (G) with δ+(v) at most the number of non-neighbors of v.\n\nThe following generalization of second neighborhood to kth neighborhood was taken from a (unpublished) paper of Serge Burckel: 7",
   "public_statement": "Conjecture 6.\n1\n2. (Thomass´ e) Let G be a digraph with no directed cycle of length at most three. Then there is a vertex v ∈ V (G) with δ+(v) at most the number of non-neighbors of v.\n\nThe following generalization of second neighborhood to kth neighborhood was taken from a (unpublished) paper of Serge Burckel: 7",
   "evidence": "The canonical JSON has two extraction defects: the displayed number is split as `6.\\n1\\n2`, and the conjecture is followed by the opening sentence of the next item. Inspection of the TeX source behind the AIM survey recovers the item as **Conjecture 6.12** and gives the statement",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 179,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0181": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.13. Any simple digraph with no directed cycles of length at most k has a vertex \n\nv such that |N + \n\n> k\n\n(v)| ≥ | N +\n\n> k−1\n\n(v)|.\n\nSerge Burckel also asked the following structural question:",
   "clean_statement": "Conjecture 6.13. Any simple digraph with no directed cycles of length at most k has a vertex\n\nv such that |N +\n\n> k\n\n(v)| ≥ | N +\n\n> k−1\n\n(v)|.\n\nSerge Burckel also asked the following structural question:",
   "public_statement": "Conjecture 6.13. Any simple digraph with no directed cycles of length at most k has a vertex\n\nv such that |N +\n\n> k\n\n(v)| ≥ | N +\n\n> k−1\n\n(v)|.\n\nSerge Burckel also asked the following structural question:",
   "evidence": "The canonical record is visibly damaged by line-oriented PDF extraction:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 180,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0182": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.14. For any k and any digraph G, define G∗ to be those vertices with |N +2 (v)| ≥ |N+(v)|. Then any vertex of out-degree k is at distance at most k from a vertex in G∗.Notes: He motivates this with the following remark: \"If a vertex x has one successor y, then if y has no successors, y ∈ G∗, otherwise x ∈ G∗. This property seems to generalize for any out-degree, and if it is true, is optimal considering 'pyramids' where any vertex of out-degree k\n\nis at distance exactly k from the (unique) solution.\" His 'pyramids' are formed by placing one vertex, then two in the row beneath it, and so forth ( i vertices in row i) to form a triangle. The vertices in row i are then completely joined to all vertices in row i − 1 for i ≥ 2.",
   "clean_statement": "Conjecture 6.14. For any k and any digraph G, define G∗ to be those vertices with |N +2 (v)| ≥ |N+(v)|. Then any vertex of out-degree k is at distance at most k from a vertex in G∗.Notes: He motivates this with the following remark: \"If a vertex x has one successor y, then if y has no successors, y ∈ G∗, otherwise x ∈ G∗. This property seems to generalize for any out-degree, and if it is true, is optimal considering 'pyramids' where any vertex of out-degree k\n\nis at distance exactly k from the (unique) solution.\" His 'pyramids' are formed by placing one vertex, then two in the row beneath it, and so forth ( i vertices in row i) to form a triangle. The vertices in row i are then completely joined to all vertices in row i − 1 for i ≥ 2.",
   "public_statement": "Conjecture 6.14. For any k and any digraph G, define G∗ to be those vertices with |N +2 (v)| ≥ |N+(v)|. Then any vertex of out-degree k is at distance at most k from a vertex in G∗.Notes: He motivates this with the following remark: \"If a vertex x has one successor y, then if y has no successors, y ∈ G∗, otherwise x ∈ G∗. This property seems to generalize for any out-degree, and if it is true, is optimal considering 'pyramids' where any vertex of out-degree k\n\nis at distance exactly k from the (unique) solution.\" His 'pyramids' are formed by placing one vertex, then two in the row beneath it, and so forth ( i vertices in row i) to form a triangle. The vertices in row i are then completely joined to all vertices in row i − 1 for i ≥ 2.",
   "evidence": "The source is Conjecture 6.14 in the AIM workshop problem list *The Caccetta--Häggkvist conjecture*. The source list declares at the outset that digraphs are finite unless explicitly stated otherwise, and defines \\(N_j^+(v)\\) to be the vertices at directed distance exactly \\(j\\) from \\(v\\). With \\[ G^*=\\{v\\in V(G): |N_2^+(v)|\\geq |N^+(v)|\\}, \\] the displayed conjecture says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 181,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0183": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n1\n5. (Seymour &/or Jackson) If G is an Eulerian digraph with no loops or digons, then ∑ \n\n> v∈V(G)\n\n|N+2 (v)| ≥ ∑ \n\n> v∈V(G)\n\n|N+(v)|.",
   "clean_statement": "Conjecture 6.\n1\n5. (Seymour &/or Jackson) If G is an Eulerian digraph with no loops or digons, then ∑\n\n> v∈V(G)\n\n|N+2 (v)| ≥ ∑\n\n> v∈V(G)\n\n|N+(v)|.",
   "public_statement": "Conjecture 6.\n1\n5. (Seymour &/or Jackson) If G is an Eulerian digraph with no loops or digons, then ∑\n\n> v∈V(G)\n\n|N+2 (v)| ≥ ∑\n\n> v∈V(G)\n\n|N+(v)|.",
   "evidence": "The canonical JSON breaks the conjecture number across lines as “6. 1 5” and inserts `>` extraction debris before the summation indices. The primary AIM PDF gives the unambiguous statement:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 182,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0184": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n1\n6. (Thomass´ e & Kral) Let G be an Eulerian digraph on n vertices, so that \n\n|E(G)| ≥ n2/3. Then G has a directed cycle of length at most three. \n\n#6.3 Matrices",
   "clean_statement": "Conjecture 6.\n1\n6. (Thomass´ e & Kral) Let G be an Eulerian digraph on n vertices, so that\n\n|E(G)| ≥ n2/3. Then G has a directed cycle of length at most three.\n\n#6.3 Matrices",
   "public_statement": "Conjecture 6.\n1\n6. (Thomass´ e & Kral) Let G be an Eulerian digraph on n vertices, so that\n\n|E(G)| ≥ n2/3. Then G has a directed cycle of length at most three.\n\n#6.3 Matrices",
   "evidence": "The canonical JSON record is damaged by line breaks and OCR. The source PDF gives the following statement on page 7:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 183,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0185": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.17. The following were all presented by Seymour, with no other attributions given: For the following questions, matrices are assumed to be n × n 0-1 matrices such that aij = 1 \n\nimplies aji 6 = 1, and all diagonal elements equal to 1.1. Let A1, A 2,..., A dn/r e be matrices (not necessarily distinct) so that the row sums of Ai are at least r + 1 for all i. Does A1A2 · · · Adn/r e have trace > n? This is a special case of",
   "clean_statement": "Conjecture 6.17. The following were all presented by Seymour, with no other attributions given: For the following questions, matrices are assumed to be n × n 0-1 matrices such that aij = 1\n\nimplies aji 6 = 1, and all diagonal elements equal to 1.1. Let A1, A 2,..., A dn/r e be matrices (not necessarily distinct) so that the row sums of Ai are at least r + 1 for all i. Does A1A2 · · · Adn/r e have trace > n? This is a special case of",
   "public_statement": "Conjecture 6.17. The following were all presented by Seymour, with no other attributions given: For the following questions, matrices are assumed to be n × n 0-1 matrices such that aij = 1\n\nimplies aji 6 = 1, and all diagonal elements equal to 1.1. Let A1, A 2,..., A dn/r e be matrices (not necessarily distinct) so that the row sums of Ai are at least r + 1 for all i. Does A1A2 · · · Adn/r e have trace > n? This is a special case of",
   "evidence": "The canonical record is an OCR-truncated extraction of item 1 in AIM Conjecture 6.17. It renders a ceiling as “dn/r e,” omits the off-diagonal qualification from the asymmetry condition, joins the item number to the preceding sentence, and stops after “This is a special case of.”",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 184,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0186": {
   "statement_status": "corrected_verified",
   "original_statement": "Conjecture 6.5. 2. Let A1, A 2,..., A t be matrices (not necessarily distinct) so Ai has row sums at least ri + 1 \n\nand ∑ti=1 ri ≥ n. Does A1A2 · · · At have trace > n? This is equivalent to",
   "clean_statement": "The matrices \\(A_1,\\ldots,A_t\\) are \\(n\\times n\\) \\(0\\)-\\(1\\) matrices, \\(a^{(i)}_{uv}=1\\) implies \\(a^{(i)}_{vu}\\ne1\\) for \\(u\\ne v\\), and all diagonal entries equal \\(1\\). They need not be distinct. If every row sum of \\(A_i\\) is at least \\(r_i+1\\) and\n\\[\n\\sum_{i=1}^t r_i\\ge n,\n\\]\nmust ordinary matrix multiplication satisfy\n\\[\n\\operatorname{tr}(A_1A_2\\cdots A_t)>n?\n\\]",
   "public_statement": "The matrices \\(A_1,\\ldots,A_t\\) are \\(n\\times n\\) \\(0\\)-\\(1\\) matrices, \\(a^{(i)}_{uv}=1\\) implies \\(a^{(i)}_{vu}\\ne1\\) for \\(u\\ne v\\), and all diagonal entries equal \\(1\\). They need not be distinct. If every row sum of \\(A_i\\) is at least \\(r_i+1\\) and\n\\[\n\\sum_{i=1}^t r_i\\ge n,\n\\]\nmust ordinary matrix multiplication satisfy\n\\[\n\\operatorname{tr}(A_1A_2\\cdots A_t)>n?\n\\]",
   "evidence": "This fragment was split in the middle of an item. Inspection of page 7 of the primary source recovers it as item 2 under **Conjecture 6.17**, not as a new item numbered “Conjecture 6.5.2.” The shared preamble and continuation give the complete recovered statement: The missing words after the owned fragment are “Conjecture 6.5.” Items 3 and 4, printed immediately afterward, give the intended layered-path and increasing-color-cycle pictures. In item 3 there are \\(t+1\\) copies of the vertex set; an \\(A_i\\)-arc goes from layer \\(i-1\\) to layer \\(i\\), and diagonal entries are the horizontal waiting edges. The question is whether some vertex has a non-horizontal path back to its copy in the last layer. Item 4 identifies this with a nontrivial rainbow directed cycle whose colors occur in increasing order.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 185,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0187": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.5. 3. Form a digraph from the matrices A1,..., A t by putting t + 1 copies of the vertex set V in a row, connecting copy k of v to copy k + 1 of v with a horizontal edge for all vertices v and all k, and then from copy k to copy k + 1 put in the edge from copy k of vi to copy k + 1 of \n\nvj precisely when aij = 1 in matrix Ak for k = 1,..., t. The question now is whether there exists a vertex u so that there is a non-trivial (i.e. not all horizontal edges) path from copy \n\n1 of u to copy t + 1 of u.4. If we \"squish\" all the bipartite graphs from the previous items so they live on a single copy of V, marking edges from copy i to copy i + 1 with color i, then we have the sum of the (colored) outdegrees at each vertex is at least n, and we're asking for a non-trivial rainbow cycle which has the colors appearing in increasing order. \n\nFor a matrix A, the spectral radius of A is defined to be max {| λ|: λ an eigenvalue of A}.8",
   "clean_statement": "Conjecture 6.5. 3. Form a digraph from the matrices A1,..., A t by putting t + 1 copies of the vertex set V in a row, connecting copy k of v to copy k + 1 of v with a horizontal edge for all vertices v and all k, and then from copy k to copy k + 1 put in the edge from copy k of vi to copy k + 1 of\n\nvj precisely when aij = 1 in matrix Ak for k = 1,..., t. The question now is whether there exists a vertex u so that there is a non-trivial (i.e. not all horizontal edges) path from copy\n\n1 of u to copy t + 1 of u.4. If we \"squish\" all the bipartite graphs from the previous items so they live on a single copy of V, marking edges from copy i to copy i + 1 with color i, then we have the sum of the (colored) outdegrees at each vertex is at least n, and we're asking for a non-trivial rainbow cycle which has the colors appearing in increasing order.\n\nFor a matrix A, the spectral radius of A is defined to be max {| λ|: λ an eigenvalue of A}.8",
   "public_statement": "Conjecture 6.5. 3. Form a digraph from the matrices A1,..., A t by putting t + 1 copies of the vertex set V in a row, connecting copy k of v to copy k + 1 of v with a horizontal edge for all vertices v and all k, and then from copy k to copy k + 1 put in the edge from copy k of vi to copy k + 1 of\n\nvj precisely when aij = 1 in matrix Ak for k = 1,..., t. The question now is whether there exists a vertex u so that there is a non-trivial (i.e. not all horizontal edges) path from copy\n\n1 of u to copy t + 1 of u.4. If we \"squish\" all the bipartite graphs from the previous items so they live on a single copy of V, marking edges from copy i to copy i + 1 with color i, then we have the sum of the (colored) outdegrees at each vertex is at least n, and we're asking for a non-trivial rainbow cycle which has the colors appearing in increasing order.\n\nFor a matrix A, the spectral radius of A is defined to be max {| λ|: λ an eigenvalue of A}.8",
   "evidence": "The canonical JSON record is an OCR-split fragment labeled “Conjecture 6.5,” beginning with item 3 and ending with item 4 plus the first sentence of the next subsection. The original AIM PDF verifies that the fragment is actually **items 3 and 4 of Conjecture 6.17**, in Section 6.3, “Matrices.” The stray definition of spectral radius and terminal “8” belong to the transition to Conjecture 6.18, not to this problem.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 186,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0188": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n1\n8. (Charbit) Let A be the adjacency matrix of a digraph G with zero on the diagonal and aij = 1 if and only if the edge (i, j ) ∈ E(G). If the spectral radius of A is at least \n\nn/k, then G has a cycle of length ≤ k.\n\n#6.4 Disjoint Cycles",
   "clean_statement": "Conjecture 6.\n1\n8. (Charbit) Let A be the adjacency matrix of a digraph G with zero on the diagonal and aij = 1 if and only if the edge (i, j ) ∈ E(G). If the spectral radius of A is at least\n\nn/k, then G has a cycle of length ≤ k.\n\n#6.4 Disjoint Cycles",
   "public_statement": "Conjecture 6.\n1\n8. (Charbit) Let A be the adjacency matrix of a digraph G with zero on the diagonal and aij = 1 if and only if the edge (i, j ) ∈ E(G). If the spectral radius of A is at least\n\nn/k, then G has a cycle of length ≤ k.\n\n#6.4 Disjoint Cycles",
   "evidence": "The canonical record is an OCR extraction of Conjecture 6.18 in Blair Sullivan's report from the 2006 AIM workshop on the Caccetta--Haggkvist conjecture. The split number “6. 1 8” is an OCR artifact, and the attached heading “6.4 Disjoint Cycles” begins the next section. The recovered statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 187,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0189": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n1\n9. (Bermond-Thomassen) In a digraph D with δ+ \n\n> D\n\n≥ 2k − 1, there are k vertex disjoint cycles. Notes: Open for k ≥ 3, and the proof for k = 2 by Thomassen is not intuitive.",
   "clean_statement": "Conjecture 6.\n1\n9. (Bermond-Thomassen) In a digraph D with δ+\n\n> D\n\n≥ 2k − 1, there are k vertex disjoint cycles. Notes: Open for k ≥ 3, and the proof for k = 2 by Thomassen is not intuitive.",
   "public_statement": "Conjecture 6.\n1\n9. (Bermond-Thomassen) In a digraph D with δ+\n\n> D\n\n≥ 2k − 1, there are k vertex disjoint cycles. Notes: Open for k ≥ 3, and the proof for k = 2 by Thomassen is not intuitive.",
   "evidence": "The corpus record is Conjecture 6.19 in Blair D. Sullivan's report from the 2006 AIM workshop on the Caccetta--Haggkvist conjecture. After repairing only the PDF line breaks, its statement is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 188,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0190": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n2\n0. (Ho´ ang & Reed) If G is a digraph with minimum outdegree r, then there are directed cycles C1,..., C r such that for all `,\n\n|V (C`) ∩ (∪`−1 \n\n> i=1\n\nV (Ci)) | ≤ 1.\n\nNotes: The related conjecture that given minimum outdegree at least r, there should be a vertex \n\nv with r cycles through v which are otherwise vertex-disjoint is false. A counterexample for r = 3 was given by Thomassen in 1985 (see Theorem 5.2 for his complete result). Adding the condition that the minimum indegree is also at least r does not improve the veracity of the conjecture, though it is still open when indegree and outdegree are identically r everywhere. \n\n#6.5 Connectivity",
   "clean_statement": "Conjecture 6.\n2\n0. (Ho´ ang & Reed) If G is a digraph with minimum outdegree r, then there are directed cycles C1,..., C r such that for all `,\n\n|V (C`) ∩ (∪`−1\n\n> i=1\n\nV (Ci)) | ≤ 1.\n\nNotes: The related conjecture that given minimum outdegree at least r, there should be a vertex\n\nv with r cycles through v which are otherwise vertex-disjoint is false. A counterexample for r = 3 was given by Thomassen in 1985 (see Theorem 5.2 for his complete result). Adding the condition that the minimum indegree is also at least r does not improve the veracity of the conjecture, though it is still open when indegree and outdegree are identically r everywhere.\n\n#6.5 Connectivity",
   "public_statement": "Conjecture 6.\n2\n0. (Ho´ ang & Reed) If G is a digraph with minimum outdegree r, then there are directed cycles C1,..., C r such that for all `,\n\n|V (C`) ∩ (∪`−1\n\n> i=1\n\nV (Ci)) | ≤ 1.\n\nNotes: The related conjecture that given minimum outdegree at least r, there should be a vertex\n\nv with r cycles through v which are otherwise vertex-disjoint is false. A counterexample for r = 3 was given by Thomassen in 1985 (see Theorem 5.2 for his complete result). Adding the condition that the minimum indegree is also at least r does not improve the veracity of the conjecture, though it is still open when indegree and outdegree are identically r everywhere.\n\n#6.5 Connectivity",
   "evidence": "The canonical JSON record is an OCR-damaged extraction of Conjecture 6.20 in Blair D. Sullivan's 2006 AIM survey. The PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 189,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0191": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n2\n1. (Hamidoune, 1981) Let D be a digraph with δ+ \n\n> D\n\n≥ r, δ− \n\n> D\n\n≥ r, and r ≥ 1.Then there is an edge (x, y ) such that κ(y, x ) ≥ r.\n\nNote that a counterexample to the above conjecture for all r ≥ 3 is given by Thomassen in Theorem 5.2.",
   "clean_statement": "Conjecture 6.\n2\n1. (Hamidoune, 1981) Let D be a digraph with δ+\n\n> D\n\n≥ r, δ−\n\n> D\n\n≥ r, and r ≥ 1.Then there is an edge (x, y ) such that κ(y, x ) ≥ r.\n\nNote that a counterexample to the above conjecture for all r ≥ 3 is given by Thomassen in Theorem 5.2.",
   "public_statement": "Conjecture 6.\n2\n1. (Hamidoune, 1981) Let D be a digraph with δ+\n\n> D\n\n≥ r, δ−\n\n> D\n\n≥ r, and r ≥ 1.Then there is an edge (x, y ) such that κ(y, x ) ≥ r.\n\nNote that a counterexample to the above conjecture for all r ≥ 3 is given by Thomassen in Theorem 5.2.",
   "evidence": "The OCR in `input.json` breaks the subscripts and the conjecture number. The AIM source gives the following statement as Conjecture 6.21:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 190,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0192": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n2\n2. (Mader) If a digraph D has δ+ \n\n> D\n\n≥ r, then there are vertices x 6 = y such that \n\nλ(x, y ) ≥ r. Also, there is an edge (x, y ) so that λ(x, y ) ≥ r.\n\nThe first part of this conjecture was proven by Mader [24] for λ(x, y ) ≥ r − 1. \n\n#6.6 Weighted Versions",
   "clean_statement": "Conjecture 6.\n2\n2. (Mader) If a digraph D has δ+\n\n> D\n\n≥ r, then there are vertices x 6 = y such that\n\nλ(x, y ) ≥ r. Also, there is an edge (x, y ) so that λ(x, y ) ≥ r.\n\nThe first part of this conjecture was proven by Mader [24] for λ(x, y ) ≥ r − 1.\n\n#6.6 Weighted Versions",
   "public_statement": "Conjecture 6.\n2\n2. (Mader) If a digraph D has δ+\n\n> D\n\n≥ r, then there are vertices x 6 = y such that\n\nλ(x, y ) ≥ r. Also, there is an edge (x, y ) so that λ(x, y ) ≥ r.\n\nThe first part of this conjecture was proven by Mader [24] for λ(x, y ) ≥ r − 1.\n\n#6.6 Weighted Versions",
   "evidence": "The corpus record is Conjecture 6.22 in Section 6.5 (“Connectivity”) of Blair D. Sullivan's report from the 2006 AIM workshop on the Caccetta--Haggkvist conjecture. The primary PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 191,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0193": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n2\n3. (Bollob´ as & Scott) Let p: E(G) → [0, 1]. If ∑ \n\n> v∈N+\n> G(u)\n\np(uv ) ≥ 1 and \n\n∑ \n\n> v∈N−\n> G(u)\n\np(vu ) ≥ 1 for all u ∈ V (G), there is a directed cycle in G of total weight ≥ 1.Note: There is a nice proof that there is a dipath of total weight at least 1.",
   "clean_statement": "Conjecture 6.\n2\n3. (Bollob´ as & Scott) Let p: E(G) → [0, 1]. If ∑\n\n> v∈N+\n> G(u)\n\np(uv ) ≥ 1 and\n\n∑\n\n> v∈N−\n> G(u)\n\np(vu ) ≥ 1 for all u ∈ V (G), there is a directed cycle in G of total weight ≥ 1.Note: There is a nice proof that there is a dipath of total weight at least 1.",
   "public_statement": "Conjecture 6.\n2\n3. (Bollob´ as & Scott) Let p: E(G) → [0, 1]. If ∑\n\n> v∈N+\n> G(u)\n\np(uv ) ≥ 1 and\n\n∑\n\n> v∈N−\n> G(u)\n\np(vu ) ≥ 1 for all u ∈ V (G), there is a directed cycle in G of total weight ≥ 1.Note: There is a nice proof that there is a dipath of total weight at least 1.",
   "evidence": "The assigned record is Conjecture 6.23 in Section 6.6 (“Weighted Versions”) of Blair D. Sullivan's 2006 AIM report on the Caccetta--Haggkvist conjecture. The primary PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 192,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0194": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n2\n4. (Zhang) Let G be a digraph on n vertices, and f: E(G) → { 0, 1,... }. If ∑ \n\n> e∈E+(v)\n\nf (e) ≥ n/k for all v ∈ V (G), then there is a directed cycle C such that ∑ \n\n> e∈C\n> 1\n> f(e)\n\n≤ k.Counterexample: (Charbit) Take the directed Cayley graph G with group G = Z8, and generators \n\n{1, 2}. Let the weight function f be 1 on 2-edges, and 2 on 1-edges (where r-edges are those coming from the generator r). Now let k = 3. We can calculate ∑ \n\n> e∈E+(v)\n\nf (e) = 4 ≥ 8/3 for all vertices v, but there is no directed cycle with ∑ \n\n> e∈C\n> 1\n> f(e)\n\n≤ 3. 96.7 Averaged Outdegree Conditions",
   "clean_statement": "Conjecture 6.\n2\n4. (Zhang) Let G be a digraph on n vertices, and f: E(G) → { 0, 1,... }. If ∑\n\n> e∈E+(v)\n\nf (e) ≥ n/k for all v ∈ V (G), then there is a directed cycle C such that ∑\n\n> e∈C\n> 1\n> f(e)\n\n≤ k.Counterexample: (Charbit) Take the directed Cayley graph G with group G = Z8, and generators\n\n{1, 2}. Let the weight function f be 1 on 2-edges, and 2 on 1-edges (where r-edges are those coming from the generator r). Now let k = 3. We can calculate ∑\n\n> e∈E+(v)\n\nf (e) = 4 ≥ 8/3 for all vertices v, but there is no directed cycle with ∑\n\n> e∈C\n> 1\n> f(e)\n\n≤ 3. 96.7 Averaged Outdegree Conditions",
   "public_statement": "Conjecture 6.\n2\n4. (Zhang) Let G be a digraph on n vertices, and f: E(G) → { 0, 1,... }. If ∑\n\n> e∈E+(v)\n\nf (e) ≥ n/k for all v ∈ V (G), then there is a directed cycle C such that ∑\n\n> e∈C\n> 1\n> f(e)\n\n≤ k.Counterexample: (Charbit) Take the directed Cayley graph G with group G = Z8, and generators\n\n{1, 2}. Let the weight function f be 1 on 2-edges, and 2 on 1-edges (where r-edges are those coming from the generator r). Now let k = 3. We can calculate ∑\n\n> e∈E+(v)\n\nf (e) = 4 ≥ 8/3 for all vertices v, but there is no directed cycle with ∑\n\n> e∈C\n> 1\n> f(e)\n\n≤ 3. 96.7 Averaged Outdegree Conditions",
   "evidence": "The OCR in `input.json` splits the conjecture number and appends the next page number and section heading. The primary PDF gives the following display as Conjecture 6.24:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 193,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0195": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.25. If D is a digraph on n vertices with \n\n∑ \n\n> v∈V(D)\n\nlog(1 + 1\n\nδ+ \n\n> D\n\n(v)) ≥ n log(1 + 1\n\nr ),\n\nthen D has a cycle of length at most dn/r e.Counterexample: Take a transitive tournament of size n − 1, and replace the edge from the vertex of out-degree n − 2 to the vertex of out-degree zero with a path of length 2 in the opposite direction (thus increasing the number of vertices to n).",
   "clean_statement": "Conjecture 6.25. If D is a digraph on n vertices with\n\n∑\n\n> v∈V(D)\n\nlog(1 + 1\n\nδ+\n\n> D\n\n(v)) ≥ n log(1 + 1\n\nr ),\n\nthen D has a cycle of length at most dn/r e.Counterexample: Take a transitive tournament of size n − 1, and replace the edge from the vertex of out-degree n − 2 to the vertex of out-degree zero with a path of length 2 in the opposite direction (thus increasing the number of vertices to n).",
   "public_statement": "Conjecture 6.25. If D is a digraph on n vertices with\n\n∑\n\n> v∈V(D)\n\nlog(1 + 1\n\nδ+\n\n> D\n\n(v)) ≥ n log(1 + 1\n\nr ),\n\nthen D has a cycle of length at most dn/r e.Counterexample: Take a transitive tournament of size n − 1, and replace the edge from the vertex of out-degree n − 2 to the vertex of out-degree zero with a path of length 2 in the opposite direction (thus increasing the number of vertices to n).",
   "evidence": "The canonical record comes from Conjecture 6.25 in Blair D. Sullivan's 2006 AIM survey. Inspection of page 10 of the PDF confirms that the final bound uses a ceiling, not a floor. In modern notation, the displayed statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 194,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0196": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n2\n6. (Shen [32]) Let G be a digraph with n vertices and minimum outdegree at least one. If δ+ \n\n> G\n\n(v) + δ+ \n\n> G\n\n(u) ≥ 2r for every edge (u, v ) in G, then the girth g of G is at most \n\ndn/r e.Note: This was proved by Shen for r = 2 in [32]. \n\n#6.8 UnCategorized \n\nThe following conjecture would imply that in a counterexample for Caccetta-H¨ aggkvist for r =\n\nn/ 3, one can order the vertices so that at least 75% of the edges go from left to right:",
   "clean_statement": "Conjecture 6.\n2\n6. (Shen [32]) Let G be a digraph with n vertices and minimum outdegree at least one. If δ+\n\n> G\n\n(v) + δ+\n\n> G\n\n(u) ≥ 2r for every edge (u, v ) in G, then the girth g of G is at most\n\ndn/r e.Note: This was proved by Shen for r = 2 in [32].\n\n#6.8 UnCategorized\n\nThe following conjecture would imply that in a counterexample for Caccetta-H¨ aggkvist for r =\n\nn/ 3, one can order the vertices so that at least 75% of the edges go from left to right:",
   "public_statement": "Conjecture 6.\n2\n6. (Shen [32]) Let G be a digraph with n vertices and minimum outdegree at least one. If δ+\n\n> G\n\n(v) + δ+\n\n> G\n\n(u) ≥ 2r for every edge (u, v ) in G, then the girth g of G is at most\n\ndn/r e.Note: This was proved by Shen for r = 2 in [32].\n\n#6.8 UnCategorized\n\nThe following conjecture would imply that in a counterexample for Caccetta-H¨ aggkvist for r =\n\nn/ 3, one can order the vertices so that at least 75% of the edges go from left to right:",
   "evidence": "The primary AIM PDF gives the following statement. Its bracket glyphs are a ceiling, not a floor:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 195,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0197": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n2\n7. (Chudnovsky, Seymour, Sullivan) Let G be a simple digraph with k non-edges (unordered pairs {u, v } where both uv and vu are not in E(G)). If G has no directed cycle of length at most 3, one can delete at most k/ 2 edges from G and obtain a graph with no directed cycle. Notes: We know that there are tight examples for this conjecture (transitive tournaments, C4,and products of these). It is also known that a minimal counterexample has no source or sink vertex, and no directed cut. Kostochka recently proved all vertices in a minimal counterexample have at least 3 and at most ( n − 1) /2 non-neighbors. He has an argument using these facts to show the conjecture for all k ≤ 14.",
   "clean_statement": "Conjecture 6.\n2\n7. (Chudnovsky, Seymour, Sullivan) Let G be a simple digraph with k non-edges (unordered pairs {u, v } where both uv and vu are not in E(G)). If G has no directed cycle of length at most 3, one can delete at most k/ 2 edges from G and obtain a graph with no directed cycle. Notes: We know that there are tight examples for this conjecture (transitive tournaments, C4,and products of these). It is also known that a minimal counterexample has no source or sink vertex, and no directed cut. Kostochka recently proved all vertices in a minimal counterexample have at least 3 and at most ( n − 1) /2 non-neighbors. He has an argument using these facts to show the conjecture for all k ≤ 14.",
   "public_statement": "Conjecture 6.\n2\n7. (Chudnovsky, Seymour, Sullivan) Let G be a simple digraph with k non-edges (unordered pairs {u, v } where both uv and vu are not in E(G)). If G has no directed cycle of length at most 3, one can delete at most k/ 2 edges from G and obtain a graph with no directed cycle. Notes: We know that there are tight examples for this conjecture (transitive tournaments, C4,and products of these). It is also known that a minimal counterexample has no source or sink vertex, and no directed cut. Kostochka recently proved all vertices in a minimal counterexample have at least 3 and at most ( n − 1) /2 non-neighbors. He has an argument using these facts to show the conjecture for all k ≤ 14.",
   "evidence": "The canonical record is Conjecture 6.27 in Blair D. Sullivan's 2006 AIM survey. The primary PDF gives the following statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 196,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0198": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n2\n8. (Devos) For any digraph G with no directed cycles of length at most three, there is a probability distribution p on V (G) such that at every vertex v, p(N +(v)) ≥ p(N −(v)) \n\nand p(N − \n\n> 2\n\n(v)) ≥ p(N −(v)), where p(S):= ∑ \n\n> s∈S\n\np(s) for a set S ⊆ V (G).Notes: For tournaments, such a distribution exists, and is unique. Its existence for a general digraph would imply Seymour's Second Neighborhood Conjecture, and actually it would suffice to just have a probability distribution p so that p(N − \n\n> 2\n\n(v)) ≥ p(N −(v)) on average in G.Given a digraph G = ( V, E ), we say F ⊆ E(G) is a feedback arc set if the digraph G′ = ( V, E −F )has no directed cycles.",
   "clean_statement": "Conjecture 6.\n2\n8. (Devos) For any digraph G with no directed cycles of length at most three, there is a probability distribution p on V (G) such that at every vertex v, p(N +(v)) ≥ p(N −(v))\n\nand p(N −\n\n> 2\n\n(v)) ≥ p(N −(v)), where p(S):= ∑\n\n> s∈S\n\np(s) for a set S ⊆ V (G).Notes: For tournaments, such a distribution exists, and is unique. Its existence for a general digraph would imply Seymour's Second Neighborhood Conjecture, and actually it would suffice to just have a probability distribution p so that p(N −\n\n> 2\n\n(v)) ≥ p(N −(v)) on average in G.Given a digraph G = ( V, E ), we say F ⊆ E(G) is a feedback arc set if the digraph G′ = ( V, E −F )has no directed cycles.",
   "public_statement": "Conjecture 6.\n2\n8. (Devos) For any digraph G with no directed cycles of length at most three, there is a probability distribution p on V (G) such that at every vertex v, p(N +(v)) ≥ p(N −(v))\n\nand p(N −\n\n> 2\n\n(v)) ≥ p(N −(v)), where p(S):= ∑\n\n> s∈S\n\np(s) for a set S ⊆ V (G).Notes: For tournaments, such a distribution exists, and is unique. Its existence for a general digraph would imply Seymour's Second Neighborhood Conjecture, and actually it would suffice to just have a probability distribution p so that p(N −\n\n> 2\n\n(v)) ≥ p(N −(v)) on average in G.Given a digraph G = ( V, E ), we say F ⊆ E(G) is a feedback arc set if the digraph G′ = ( V, E −F )has no directed cycles.",
   "evidence": "The primary source is Sullivan's AIM problem-list article on the Caccetta--Häggkvist conjecture. The corpus extraction split the subscript in the second inequality and appended text from the next problem. Using the definitions earlier in that source, the recovered statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 197,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0199": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n2\n9. (Lichiardopol's Conjecture) Every digraph D has some minimal feedback arc set F which contains a path of length δ+\n\n> D.\n\n10 Notes: This implies Ho´ ang & Reed (6.20), Caccetta-H¨ aggkvist (2.1), Bermond-Thomassen (6.19), and Thomass´ e (6.34).",
   "clean_statement": "Conjecture 6.\n2\n9. (Lichiardopol's Conjecture) Every digraph D has some minimal feedback arc set F which contains a path of length δ+\n\n> D.\n\n10 Notes: This implies Ho´ ang & Reed (6.20), Caccetta-H¨ aggkvist (2.1), Bermond-Thomassen (6.19), and Thomass´ e (6.34).",
   "public_statement": "Conjecture 6.\n2\n9. (Lichiardopol's Conjecture) Every digraph D has some minimal feedback arc set F which contains a path of length δ+\n\n> D.\n\n10 Notes: This implies Ho´ ang & Reed (6.20), Caccetta-H¨ aggkvist (2.1), Bermond-Thomassen (6.19), and Thomass´ e (6.34).",
   "evidence": "The primary AIM PDF defines a feedback arc set immediately before the conjecture: if \\(D=(V,E)\\), then \\(F\\subseteq E\\) is a feedback arc set (FAS) when \\((V,E\\setminus F)\\) has no directed cycle. The exact statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 198,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0200": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n3\n0. (Mader, 1985) For all k ∈ Z+, there exists r ∈ Z+ such that every r-out-regular digraph contains a subdivision of the transitive tournament on k vertices. Notes: This conjecture is known for k = 3 (where r = 2), and k = 4 ( r = 3 proven by Mader in 1996 [26]). The existence of r for k = 5 is still not known, though r = 6 has been conjectured.",
   "clean_statement": "Conjecture 6.\n3\n0. (Mader, 1985) For all k ∈ Z+, there exists r ∈ Z+ such that every r-out-regular digraph contains a subdivision of the transitive tournament on k vertices. Notes: This conjecture is known for k = 3 (where r = 2), and k = 4 ( r = 3 proven by Mader in 1996 [26]). The existence of r for k = 5 is still not known, though r = 6 has been conjectured.",
   "public_statement": "Conjecture 6.\n3\n0. (Mader, 1985) For all k ∈ Z+, there exists r ∈ Z+ such that every r-out-regular digraph contains a subdivision of the transitive tournament on k vertices. Notes: This conjecture is known for k = 3 (where r = 2), and k = 4 ( r = 3 proven by Mader in 1996 [26]). The existence of r for k = 5 is still not known, though r = 6 has been conjectured.",
   "evidence": "The canonical record is Conjecture 6.30 from the AIM workshop list *The Caccetta--Haggkvist conjecture*. The OCR has split the number “30” across lines, but the mathematical text recovers unambiguously as follows.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 199,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0201": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n3\n1. (Shen) For a digraph G on n vertices of girth g, define \n\nt(G, r ) = ∑ \n\n> u:δ+\n> G(u)<r\n\n(r − δ+ \n\n> G\n\n(u)).\n\nIf δ+ \n\n> G\n\n≥ 1, then n ≥ r(g − 1) + 1 − t(G, r ).",
   "clean_statement": "Conjecture 6.\n3\n1. (Shen) For a digraph G on n vertices of girth g, define\n\nt(G, r ) = ∑\n\n> u:δ+\n> G(u)<r\n\n(r − δ+\n\n> G\n\n(u)).\n\nIf δ+\n\n> G\n\n≥ 1, then n ≥ r(g − 1) + 1 − t(G, r ).",
   "public_statement": "Conjecture 6.\n3\n1. (Shen) For a digraph G on n vertices of girth g, define\n\nt(G, r ) = ∑\n\n> u:δ+\n> G(u)<r\n\n(r − δ+\n\n> G\n\n(u)).\n\nIf δ+\n\n> G\n\n≥ 1, then n ≥ r(g − 1) + 1 − t(G, r ).",
   "evidence": "The primary AIM PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 200,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0202": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n3\n2. (Thomass´ e) If G is a loopless digon-free digraph the maximum number of induced directed 2-edge paths is n3/15 + O(n2).Notes: First, note that n3/15 + O(n2) can be obtained by substituting C4's inside C4's. Next, given a digraph G, let τ be the number of induced directed 2-paths, η the number of induced edges, and θ the number of cyclic triangles. Then if |V (G)| = n,\n\nτ + η/ 3 + 2 θ = n3/12 + O(n2) − 1\n\n6\n\n∑ \n\n> v∈V(G)\n\n(\n\n(δ+(v) − δ−(v)) 2 + ( n\n\n2 − δ+(v)) 2 + ( n\n\n2 − δ−(v)) 2\n\n).\n\nF¨ uredi rewrote the terms on the right hand side in terms of τ, η, and θ, and showed \n\nτ ≤ n3\n\n12 + O(n2).\n\nBondy has a slight improvement of this result, proving that \n\nτ ≤ 2n3\n\n25.\n\nDefine an α/β-digraph to be a digraph D on vertex set V and edges defined by σ1,..., σ β\n\npermutations (linear orders) on V where the edge ( i, j ) ∈ E ⇐⇒ i < j in at least α of the σi.We say G is a majority digraph if α \n\n> β\n\n> 2/3.",
   "clean_statement": "Conjecture 6.\n3\n2. (Thomass´ e) If G is a loopless digon-free digraph the maximum number of induced directed 2-edge paths is n3/15 + O(n2).Notes: First, note that n3/15 + O(n2) can be obtained by substituting C4's inside C4's. Next, given a digraph G, let τ be the number of induced directed 2-paths, η the number of induced edges, and θ the number of cyclic triangles. Then if |V (G)| = n,\n\nτ + η/ 3 + 2 θ = n3/12 + O(n2) − 1\n\n6\n\n∑\n\n> v∈V(G)\n\n(\n\n(δ+(v) − δ−(v)) 2 + ( n\n\n2 − δ+(v)) 2 + ( n\n\n2 − δ−(v)) 2\n\n).\n\nF¨ uredi rewrote the terms on the right hand side in terms of τ, η, and θ, and showed\n\nτ ≤ n3\n\n12 + O(n2).\n\nBondy has a slight improvement of this result, proving that\n\nτ ≤ 2n3\n\n25.\n\nDefine an α/β-digraph to be a digraph D on vertex set V and edges defined by σ1,..., σ β\n\npermutations (linear orders) on V where the edge ( i, j ) ∈ E ⇐⇒ i < j in at least α of the σi.We say G is a majority digraph if α\n\n> β\n\n> 2/3.",
   "public_statement": "Conjecture 6.\n3\n2. (Thomass´ e) If G is a loopless digon-free digraph the maximum number of induced directed 2-edge paths is n3/15 + O(n2).Notes: First, note that n3/15 + O(n2) can be obtained by substituting C4's inside C4's. Next, given a digraph G, let τ be the number of induced directed 2-paths, η the number of induced edges, and θ the number of cyclic triangles. Then if |V (G)| = n,\n\nτ + η/ 3 + 2 θ = n3/12 + O(n2) − 1\n\n6\n\n∑\n\n> v∈V(G)\n\n(\n\n(δ+(v) − δ−(v)) 2 + ( n\n\n2 − δ+(v)) 2 + ( n\n\n2 − δ−(v)) 2\n\n).\n\nF¨ uredi rewrote the terms on the right hand side in terms of τ, η, and θ, and showed\n\nτ ≤ n3\n\n12 + O(n2).\n\nBondy has a slight improvement of this result, proving that\n\nτ ≤ 2n3\n\n25.\n\nDefine an α/β-digraph to be a digraph D on vertex set V and edges defined by σ1,..., σ β\n\npermutations (linear orders) on V where the edge ( i, j ) ∈ E ⇐⇒ i < j in at least α of the σi.We say G is a majority digraph if α\n\n> β\n\n> 2/3.",
   "evidence": "The primary AIM PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 201,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0203": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n3\n3. (Thomass´ e & Charbit) Caccetta-H¨ aggkvist holds for 3/4-digraphs. Notes: Majority digraphs have no cycles of length at most three. The class of 3/4-digraphs is stable under substitution, and contains the circular interval graph on 3 k + 1 vertices with outdegree k going clockwise. The class includes all known extremal examples for C-H, yet this class of 3/4-digraphs seems manageably small. It is still open whether or not 3/4 digraphs must have vertices x of outdegree less than n/ 3. One can more generally ask if Caccetta-H¨ aggkvist holds for larger classes of majority digraphs. 11",
   "clean_statement": "Conjecture 6.\n3\n3. (Thomass´ e & Charbit) Caccetta-H¨ aggkvist holds for 3/4-digraphs. Notes: Majority digraphs have no cycles of length at most three. The class of 3/4-digraphs is stable under substitution, and contains the circular interval graph on 3 k + 1 vertices with outdegree k going clockwise. The class includes all known extremal examples for C-H, yet this class of 3/4-digraphs seems manageably small. It is still open whether or not 3/4 digraphs must have vertices x of outdegree less than n/ 3. One can more generally ask if Caccetta-H¨ aggkvist holds for larger classes of majority digraphs. 11",
   "public_statement": "Conjecture 6.\n3\n3. (Thomass´ e & Charbit) Caccetta-H¨ aggkvist holds for 3/4-digraphs. Notes: Majority digraphs have no cycles of length at most three. The class of 3/4-digraphs is stable under substitution, and contains the circular interval graph on 3 k + 1 vertices with outdegree k going clockwise. The class includes all known extremal examples for C-H, yet this class of 3/4-digraphs seems manageably small. It is still open whether or not 3/4 digraphs must have vertices x of outdegree less than n/ 3. One can more generally ask if Caccetta-H¨ aggkvist holds for larger classes of majority digraphs. 11",
   "evidence": "The canonical AIM record reads, after repairing line-break and accent OCR:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 202,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0204": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n3\n4. (Thomass´ e) Every digraph D has a path of length δ+\n\n> D\n\n(g − 1), where g is the girth of D.Note: This is open for g = 3, and implies Caccetta-H¨ aggkvist.",
   "clean_statement": "Conjecture 6.\n3\n4. (Thomass´ e) Every digraph D has a path of length δ+\n\n> D\n\n(g − 1), where g is the girth of D.Note: This is open for g = 3, and implies Caccetta-H¨ aggkvist.",
   "public_statement": "Conjecture 6.\n3\n4. (Thomass´ e) Every digraph D has a path of length δ+\n\n> D\n\n(g − 1), where g is the girth of D.Note: This is open for g = 3, and implies Caccetta-H¨ aggkvist.",
   "evidence": "The canonical JSON is visibly damaged by PDF extraction: the conjecture number is split across lines as 6.34, and the superscript and subscript in the degree symbol are detached. Page 11 of the primary AIM PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 203,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0205": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n3\n5. (Thomass´ e) In a digraph D,\n\n∑ \n\n> v∈V(D)\n\n|δ+\n\n> D\n\n(v) − δ− \n\n> D\n\n(v)| + |{ (u, v )|d(u, v ) ≤ 2}| ≥ 2|E(D)| + |{ v|δ+\n\n> D\n\n(v) > δ − \n\n> D\n\n(v)}|.\n\nNote: This is exact for transitive tournaments.",
   "clean_statement": "Conjecture 6.\n3\n5. (Thomass´ e) In a digraph D,\n\n∑\n\n> v∈V(D)\n\n|δ+\n\n> D\n\n(v) − δ−\n\n> D\n\n(v)| + |{ (u, v )|d(u, v ) ≤ 2}| ≥ 2|E(D)| + |{ v|δ+\n\n> D\n\n(v) > δ −\n\n> D\n\n(v)}|.\n\nNote: This is exact for transitive tournaments.",
   "public_statement": "Conjecture 6.\n3\n5. (Thomass´ e) In a digraph D,\n\n∑\n\n> v∈V(D)\n\n|δ+\n\n> D\n\n(v) − δ−\n\n> D\n\n(v)| + |{ (u, v )|d(u, v ) ≤ 2}| ≥ 2|E(D)| + |{ v|δ+\n\n> D\n\n(v) > δ −\n\n> D\n\n(v)}|.\n\nNote: This is exact for transitive tournaments.",
   "evidence": "The primary AIM PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 204,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0206": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.\n3\n6. (Thomass´ e) Let G be a digraph on n vertices with minimum outdegree at least 4n/ 15 so that G is maximal with no cycles of length at most three, and has no homogeneous set (in other words, G cannot be obtained by substitution). Then G is a Cayley graph on 3k + 1 \n\nvertices with 1,..., k as generators for some k (i.e. the circular interval graph with everyone joined to next k clockwise). Note: Z/15 Z with generators S = {1, 2, 4, 8} gives exactly 4 n/ 15. \n\n#Acknowledgements \n\nThis research was performed while on appointment as a U.S. Department of Homeland Security (DHS) Fellow under the DHS Scholarship and Fellowship Program, a program administered by the Oak Ridge Institute for Science and Education (ORISE) for DHS through an interagency agreement with the U.S Department of Energy (DOE). ORISE is managed by Oak Ridge Asso-ciated Universities under DOE contract number DE-AC05-06OR23100. All opinions expressed in this paper are the author's and do not necessarily reflect the policies and views of DHS, DOE, or ORISE.",
   "clean_statement": "Conjecture 6.\n3\n6. (Thomass´ e) Let G be a digraph on n vertices with minimum outdegree at least 4n/ 15 so that G is maximal with no cycles of length at most three, and has no homogeneous set (in other words, G cannot be obtained by substitution). Then G is a Cayley graph on 3k + 1\n\nvertices with 1,..., k as generators for some k (i.e. the circular interval graph with everyone joined to next k clockwise). Note: Z/15 Z with generators S = {1, 2, 4, 8} gives exactly 4 n/ 15.\n\n#Acknowledgements\n\nThis research was performed while on appointment as a U.S. Department of Homeland Security (DHS) Fellow under the DHS Scholarship and Fellowship Program, a program administered by the Oak Ridge Institute for Science and Education (ORISE) for DHS through an interagency agreement with the U.S Department of Energy (DOE). ORISE is managed by Oak Ridge Asso-ciated Universities under DOE contract number DE-AC05-06OR23100. All opinions expressed in this paper are the author's and do not necessarily reflect the policies and views of DHS, DOE, or ORISE.",
   "public_statement": "Conjecture 6.\n3\n6. (Thomass´ e) Let G be a digraph on n vertices with minimum outdegree at least 4n/ 15 so that G is maximal with no cycles of length at most three, and has no homogeneous set (in other words, G cannot be obtained by substitution). Then G is a Cayley graph on 3k + 1\n\nvertices with 1,..., k as generators for some k (i.e. the circular interval graph with everyone joined to next k clockwise). Note: Z/15 Z with generators S = {1, 2, 4, 8} gives exactly 4 n/ 15.\n\n#Acknowledgements\n\nThis research was performed while on appointment as a U.S. Department of Homeland Security (DHS) Fellow under the DHS Scholarship and Fellowship Program, a program administered by the Oak Ridge Institute for Science and Education (ORISE) for DHS through an interagency agreement with the U.S Department of Energy (DOE). ORISE is managed by Oak Ridge Asso-ciated Universities under DOE contract number DE-AC05-06OR23100. All opinions expressed in this paper are the author's and do not necessarily reflect the policies and views of DHS, DOE, or ORISE.",
   "evidence": "The primary source is page 11 of Blair D. Sullivan's 2006 AIM report. It prints the following statement:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 205,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0207": {
   "statement_status": "exact",
   "original_statement": "Problem 1.1 (Y. Katznelson). Given a constant a > 0, does there exist d0 = d0(a)such that for any integer d > d 0 and any closed subset A of the d-dimensional torus group with the measure μ(A) > a, the difference set A − A contains a subgroup?",
   "clean_statement": "Problem 1.1 (Y. Katznelson). Given a constant a > 0, does there exist d0 = d0(a)such that for any integer d > d 0 and any closed subset A of the d-dimensional torus group with the measure μ(A) > a, the difference set A − A contains a subgroup?",
   "public_statement": "Problem 1.1 (Y. Katznelson). Given a constant a > 0, does there exist d0 = d0(a)such that for any integer d > d 0 and any closed subset A of the d-dimensional torus group with the measure μ(A) > a, the difference set A − A contains a subgroup?",
   "evidence": "The source record is Katznelson's Problem 1.1 from the AIM workshop notes on additive combinatorics:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 206,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0208": {
   "statement_status": "exact",
   "original_statement": "Problem 1.2 (T. Tao). Is there a hypergraph regularity lemma for subsets of pseu-dorandom sparse hypergraphs of large density? If so, it would give a new proof that there are arbitrarily long arithmetic progressions among the primes, which may possi-bly extend to a more general situation. The following analogue for graphs is known: If \n\n|A| = |B| = N, G0 ⊆ A × B is \"sparsely c(≤, δ )-quasirandom\", G ⊂ G0, |G| > δ |G0|,then there exist equitable partitions \n\nA = A1 ∪ · · · ∪ As, B = B1 ∪ · · · ∪ Bt,\n\nwhere s, t < C (≤, δ ), such that for (1 − ≤)st of the pairs ( i, j ) the restriction of G to \n\nAi × Bj is ≤-regular relative to G0. \n\n> 12COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV\n\nThe presenter remarks that despite being a generalization of the already rather diffi-cult hypergraph regularity lemmas, if done correctly the proof of such a result may be \n\neasier than that of the existing regularity lemmas. This is because the induction used to prove such lemmas may be cleaner.",
   "clean_statement": "Problem 1.2 (T. Tao). Is there a hypergraph regularity lemma for subsets of pseu-dorandom sparse hypergraphs of large density? If so, it would give a new proof that there are arbitrarily long arithmetic progressions among the primes, which may possi-bly extend to a more general situation. The following analogue for graphs is known: If\n\n|A| = |B| = N, G0 ⊆ A × B is \"sparsely c(≤, δ )-quasirandom\", G ⊂ G0, |G| > δ |G0|,then there exist equitable partitions\n\nA = A1 ∪ · · · ∪ As, B = B1 ∪ · · · ∪ Bt,\n\nwhere s, t < C (≤, δ ), such that for (1 − ≤)st of the pairs ( i, j ) the restriction of G to\n\nAi × Bj is ≤-regular relative to G0.\n\n> 12COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV\n\nThe presenter remarks that despite being a generalization of the already rather diffi-cult hypergraph regularity lemmas, if done correctly the proof of such a result may be\n\neasier than that of the existing regularity lemmas. This is because the induction used to prove such lemmas may be cleaner.",
   "public_statement": "Problem 1.2 (T. Tao). Is there a hypergraph regularity lemma for subsets of pseu-dorandom sparse hypergraphs of large density? If so, it would give a new proof that there are arbitrarily long arithmetic progressions among the primes, which may possi-bly extend to a more general situation. The following analogue for graphs is known: If\n\n|A| = |B| = N, G0 ⊆ A × B is \"sparsely c(≤, δ )-quasirandom\", G ⊂ G0, |G| > δ |G0|,then there exist equitable partitions\n\nA = A1 ∪ · · · ∪ As, B = B1 ∪ · · · ∪ Bt,\n\nwhere s, t < C (≤, δ ), such that for (1 − ≤)st of the pairs ( i, j ) the restriction of G to\n\nAi × Bj is ≤-regular relative to G0.\n\n> 12COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV\n\nThe presenter remarks that despite being a generalization of the already rather diffi-cult hypergraph regularity lemmas, if done correctly the proof of such a result may be\n\neasier than that of the existing regularity lemmas. This is because the induction used to prove such lemmas may be cleaner.",
   "evidence": "This is Problem 1.2, attributed to T. Tao, in the AIM workshop list *Recent trends in additive combinatorics* (collected by Ernie Croot and Vsevolod F. Lev). It asks whether a subset of large relative density in a pseudorandom sparse hypergraph satisfies a hypergraph regularity lemma, and observes that such a result could reprove the existence of arbitrarily long arithmetic progressions in the primes.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 207,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0209": {
   "statement_status": "exact",
   "original_statement": "Problem 1.3 (G. Freiman). Fix an integer s ≥ 3 and suppose that A = ( a1, a 2,... ) is a strictly increasing sequence of integers such that no segment of this sequence of the form ( ai+1, a i+2,..., a i+s) ( i = 0, 1,... ) contains a three-term arithmetic progression. How large can the density of A be under this assumption? Describe all extremal sets. Examples: 1) For s = 4 one can take A = (0, 1, 3, 4, 6, 7, 9, 10,... ) (all non-negative integers congruent to any of 0, 1, 3, and 4 modulo six) with the density 2 /3; 2) for s = 8 one can take A = (0, 1, 3, 4, 9, 10, 12, 13, 18, 19, 21,... ) (all non-negative integers congruent to any of 0, 1, 3, 4, 9, 10, 12, and 13 modulo eighteen) with the density 4 /9.",
   "clean_statement": "Problem 1.3 (G. Freiman). Fix an integer s ≥ 3 and suppose that A = ( a1, a 2,... ) is a strictly increasing sequence of integers such that no segment of this sequence of the form ( ai+1, a i+2,..., a i+s) ( i = 0, 1,... ) contains a three-term arithmetic progression. How large can the density of A be under this assumption? Describe all extremal sets. Examples: 1) For s = 4 one can take A = (0, 1, 3, 4, 6, 7, 9, 10,... ) (all non-negative integers congruent to any of 0, 1, 3, and 4 modulo six) with the density 2 /3; 2) for s = 8 one can take A = (0, 1, 3, 4, 9, 10, 12, 13, 18, 19, 21,... ) (all non-negative integers congruent to any of 0, 1, 3, 4, 9, 10, 12, and 13 modulo eighteen) with the density 4 /9.",
   "public_statement": "Problem 1.3 (G. Freiman). Fix an integer s ≥ 3 and suppose that A = ( a1, a 2,... ) is a strictly increasing sequence of integers such that no segment of this sequence of the form ( ai+1, a i+2,..., a i+s) ( i = 0, 1,... ) contains a three-term arithmetic progression. How large can the density of A be under this assumption? Describe all extremal sets. Examples: 1) For s = 4 one can take A = (0, 1, 3, 4, 6, 7, 9, 10,... ) (all non-negative integers congruent to any of 0, 1, 3, and 4 modulo six) with the density 2 /3; 2) for s = 8 one can take A = (0, 1, 3, 4, 9, 10, 12, 13, 18, 19, 21,... ) (all non-negative integers congruent to any of 0, 1, 3, 4, 9, 10, 12, and 13 modulo eighteen) with the density 4 /9.",
   "evidence": "The primary AIM workshop PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 208,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0210": {
   "statement_status": "exact",
   "original_statement": "Problem 1.4 (B. Green). For a prime p, what is the least number of three-term arithmetic progressions that a subset A ⊂ Fp with |A| = ( p − 1) /2 can have? What happens if |A| = δp where δ < 0.5?",
   "clean_statement": "Problem 1.4 (B. Green). For a prime p, what is the least number of three-term arithmetic progressions that a subset A ⊂ Fp with |A| = ( p − 1) /2 can have? What happens if |A| = δp where δ < 0.5?",
   "public_statement": "Problem 1.4 (B. Green). For a prime p, what is the least number of three-term arithmetic progressions that a subset A ⊂ Fp with |A| = ( p − 1) /2 can have? What happens if |A| = δp where δ < 0.5?",
   "evidence": "The AIM list *Recent trends in additive combinatorics*, Problem 1.4 (attributed to B. Green), asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 209,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0211": {
   "statement_status": "exact",
   "original_statement": "Problem 1.5 (N. Katz). Fix integer k ≥ 3 and real C ≥ 2. Given that A is a finite set of integers with |A + A| < C |A|, is it true that the number of k-term arithmetic progressions in A is at least c|A|2 with a positive constant c depending on k and C\n\nonly? B. Green comments: The answer to this is surely \"yes\". A (or a large subset of it) is Freiman isomorphic to a dense subset of Z/p Z by a lemma of Ruzsa. Now apply Szemeredi's theorem. J. Solymosi mentions that Katz's question was part of an induction step in his alter-native proof of the Balog-Szemeredi theorem.",
   "clean_statement": "Problem 1.5 (N. Katz). Fix integer k ≥ 3 and real C ≥ 2. Given that A is a finite set of integers with |A + A| < C |A|, is it true that the number of k-term arithmetic progressions in A is at least c|A|2 with a positive constant c depending on k and C\n\nonly? B. Green comments: The answer to this is surely \"yes\". A (or a large subset of it) is Freiman isomorphic to a dense subset of Z/p Z by a lemma of Ruzsa. Now apply Szemeredi's theorem. J. Solymosi mentions that Katz's question was part of an induction step in his alter-native proof of the Balog-Szemeredi theorem.",
   "public_statement": "Problem 1.5 (N. Katz). Fix integer k ≥ 3 and real C ≥ 2. Given that A is a finite set of integers with |A + A| < C |A|, is it true that the number of k-term arithmetic progressions in A is at least c|A|2 with a positive constant c depending on k and C\n\nonly? B. Green comments: The answer to this is surely \"yes\". A (or a large subset of it) is Freiman isomorphic to a dense subset of Z/p Z by a lemma of Ruzsa. Now apply Szemeredi's theorem. J. Solymosi mentions that Katz's question was part of an induction step in his alter-native proof of the Balog-Szemeredi theorem.",
   "evidence": "The primary AIM workshop sheet states (with its mathematical typography restored):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 210,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0212": {
   "statement_status": "exact",
   "original_statement": "Problem 1.6 (G. Freiman). Given that A is an n-element integer set free of three-term arithmetic progressions, how small can |2A| be?",
   "clean_statement": "Problem 1.6 (G. Freiman). Given that A is an n-element integer set free of three-term arithmetic progressions, how small can |2A| be?",
   "public_statement": "Problem 1.6 (G. Freiman). Given that A is an n-element integer set free of three-term arithmetic progressions, how small can |2A| be?",
   "evidence": "Problem 1.6 of the AIM list *Recent trends in additive combinatorics*, attributed to G. Freiman, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 211,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0213": {
   "statement_status": "exact",
   "original_statement": "Problem 1.7 (Brought by T. Tao). For a positive integer r, what is the largest possible size of a subset A ⊆ Fr \n\n> 3\n\ncontaining no three points on a line?",
   "clean_statement": "Problem 1.7 (Brought by T. Tao). For a positive integer r, what is the largest possible size of a subset A ⊆ Fr\n\n> 3\n\ncontaining no three points on a line?",
   "public_statement": "Problem 1.7 (Brought by T. Tao). For a positive integer r, what is the largest possible size of a subset A ⊆ Fr\n\n> 3\n\ncontaining no three points on a line?",
   "evidence": "The AIM workshop list *Recent trend in additive combinatorics*, Problem 1.7 (brought by T. Tao), asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 212,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0214": {
   "statement_status": "exact",
   "original_statement": "Problem 1.8 (R. Graham). Define W (k) to be the least integer such that in any 2-coloring of the integers {1, 2,..., W (k)} there must always exist a monochromatic k-term arithmetic progression. What is the true order of growth of W (k)? The presenter offers $1000 for the proof that W (k) ≤ 2k2.",
   "clean_statement": "Problem 1.8 (R. Graham). Define W (k) to be the least integer such that in any 2-coloring of the integers {1, 2,..., W (k)} there must always exist a monochromatic k-term arithmetic progression. What is the true order of growth of W (k)? The presenter offers $1000 for the proof that W (k) ≤ 2k2.",
   "public_statement": "Problem 1.8 (R. Graham). Define W (k) to be the least integer such that in any 2-coloring of the integers {1, 2,..., W (k)} there must always exist a monochromatic k-term arithmetic progression. What is the true order of growth of W (k)? The presenter offers $1000 for the proof that W (k) ≤ 2k2.",
   "evidence": "The exact canonical record assigned to this attempt is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 213,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0215": {
   "statement_status": "exact",
   "original_statement": "Problem 1.9 (R. Graham). Define W ∗(k) to be the size of the smallest set X ⊆ Z such that any 2-coloring of X always has a monochromatic k-term arithmetic progression. Then, W ∗(k) ≤ W (k). For $100: Is W (k) − W ∗(k) unbounded as k → ∞? Does lim \n\n> k→∞\n\nW ∗(k)\n\nW (k) = 1?",
   "clean_statement": "Problem 1.9 (R. Graham). Define W ∗(k) to be the size of the smallest set X ⊆ Z such that any 2-coloring of X always has a monochromatic k-term arithmetic progression. Then, W ∗(k) ≤ W (k). For $100: Is W (k) − W ∗(k) unbounded as k → ∞? Does lim\n\n> k→∞\n\nW ∗(k)\n\nW (k) = 1?",
   "public_statement": "Problem 1.9 (R. Graham). Define W ∗(k) to be the size of the smallest set X ⊆ Z such that any 2-coloring of X always has a monochromatic k-term arithmetic progression. Then, W ∗(k) ≤ W (k). For $100: Is W (k) − W ∗(k) unbounded as k → ∞? Does lim\n\n> k→∞\n\nW ∗(k)\n\nW (k) = 1?",
   "evidence": "The stored `problem` field is reproduced verbatim below. It is visibly damaged by PDF extraction around the displayed limit.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 214,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0216": {
   "statement_status": "exact",
   "original_statement": "Problem 1.10 (R. Graham). Let A ⊆ Z × Z satisfy \n\n∑\n\n> (x,y )∈A\n\n1\n\nx2 + y2 = ∞.\n\nConjecture ($1000): A contains the four vertices of a square, i.e. four points of the form (x, y ), (x + a, y ), (x, y + a), and ( x + a, y + a). (More generally, it should be true that \n\nA contains a k × k square grid.)",
   "clean_statement": "Problem 1.10 (R. Graham). Let A ⊆ Z × Z satisfy\n\n∑\n\n> (x,y )∈A\n\n1\n\nx2 + y2 = ∞.\n\nConjecture ($1000): A contains the four vertices of a square, i.e. four points of the form (x, y ), (x + a, y ), (x, y + a), and ( x + a, y + a). (More generally, it should be true that\n\nA contains a k × k square grid.)",
   "public_statement": "Problem 1.10 (R. Graham). Let A ⊆ Z × Z satisfy\n\n∑\n\n> (x,y )∈A\n\n1\n\nx2 + y2 = ∞.\n\nConjecture ($1000): A contains the four vertices of a square, i.e. four points of the form (x, y ), (x + a, y ), (x, y + a), and ( x + a, y + a). (More generally, it should be true that\n\nA contains a k × k square grid.)",
   "evidence": "The original AIM workshop PDF was checked directly. Problem 1.10, presented by R. Graham, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 215,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0217": {
   "statement_status": "exact",
   "original_statement": "Problem 1.11 (R. Graham). Given real α ∈ (0, 1) and integer k ≥ 3, estimate the size of the smallest set Sα,k with the properties that 1) If X ⊆ Sα,k satisfies |X| ≥ α|Sα,k |, then X has a k-term arithmetic progression; 2) Sα,k has no ( k + 1)-term arithmetic progression.",
   "clean_statement": "Problem 1.11 (R. Graham). Given real α ∈ (0, 1) and integer k ≥ 3, estimate the size of the smallest set Sα,k with the properties that 1) If X ⊆ Sα,k satisfies |X| ≥ α|Sα,k |, then X has a k-term arithmetic progression; 2) Sα,k has no ( k + 1)-term arithmetic progression.",
   "public_statement": "Problem 1.11 (R. Graham). Given real α ∈ (0, 1) and integer k ≥ 3, estimate the size of the smallest set Sα,k with the properties that 1) If X ⊆ Sα,k satisfies |X| ≥ α|Sα,k |, then X has a k-term arithmetic progression; 2) Sα,k has no ( k + 1)-term arithmetic progression.",
   "evidence": "The exact canonical record assigned to this attempt is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 216,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0218": {
   "statement_status": "exact",
   "original_statement": "Problem 1.12 (R. Graham). Let S be a set of homogeneous linear equations which is partition regular, i.e. in any r-coloring of Z there is always a non-trivial monochromatic solution to S. What is the minimum number of monochromatic solutions to S which can occur in some r-coloring of {1, 2,..., n } as a function of n and r?For example, with r = 2 and S is the single equation x + y = z, the correct answer is n2(1 + o(1)) /22 (Schoen, Roberts-Zeilberger). A random 2-coloring of {1, 2,..., n }\n\nwould give ∼ n2/16 such solutions. What happens for the equation x+y = 2 z? Are random 2-colorings best in this case? 4 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV",
   "clean_statement": "Problem 1.12 (R. Graham). Let S be a set of homogeneous linear equations which is partition regular, i.e. in any r-coloring of Z there is always a non-trivial monochromatic solution to S. What is the minimum number of monochromatic solutions to S which can occur in some r-coloring of {1, 2,..., n } as a function of n and r?For example, with r = 2 and S is the single equation x + y = z, the correct answer is n2(1 + o(1)) /22 (Schoen, Roberts-Zeilberger). A random 2-coloring of {1, 2,..., n }\n\nwould give ∼ n2/16 such solutions. What happens for the equation x+y = 2 z? Are random 2-colorings best in this case? 4 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV",
   "public_statement": "Problem 1.12 (R. Graham). Let S be a set of homogeneous linear equations which is partition regular, i.e. in any r-coloring of Z there is always a non-trivial monochromatic solution to S. What is the minimum number of monochromatic solutions to S which can occur in some r-coloring of {1, 2,..., n } as a function of n and r?For example, with r = 2 and S is the single equation x + y = z, the correct answer is n2(1 + o(1)) /22 (Schoen, Roberts-Zeilberger). A random 2-coloring of {1, 2,..., n }\n\nwould give ∼ n2/16 such solutions. What happens for the equation x+y = 2 z? Are random 2-colorings best in this case? 4 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV",
   "evidence": "The stored record is OCR text from Problem 1.12 of the AIM list *Recent trends in additive combinatorics*. Inspection of the official PDF confirms that the lost superscripts and fractions are \\[ \\frac{n^2(1+o(1))}{22}\\qquad\\hbox{and}\\qquad \\frac{n^2}{16}. \\] The words beginning “COLLECTED BY” are a page footer, not part of the problem. The source also says “Roberts-Zeilberger”; the cited paper is by Aaron Robertson and Doron Zeilberger.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 217,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0219": {
   "statement_status": "exact",
   "original_statement": "Problem 1.13 (R. Graham). Obtain \"reasonable\" bounds for the Hales-Jewett theo-rem and for the density version of it.",
   "clean_statement": "Problem 1.13 (R. Graham). Obtain \"reasonable\" bounds for the Hales-Jewett theo-rem and for the density version of it.",
   "public_statement": "Problem 1.13 (R. Graham). Obtain \"reasonable\" bounds for the Hales-Jewett theo-rem and for the density version of it.",
   "evidence": "The exact extracted AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 218,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0220": {
   "statement_status": "exact",
   "original_statement": "Problem 1.14 (R. Graham). Instead of k-term arithmetic progressions, one could consider a more flexible structure, namely weak k-term arithmetic progressions, which are sets of the form \n\n{b nα + βc: 1 ≤ n ≤ k} (for some α ≥ 1 and β).\n\nSince there are substantially more weak k-term arithmetic progressions than k-term arithmetic progressions, some of the standard problems and results might be easier to attack.",
   "clean_statement": "Problem 1.14 (R. Graham). Instead of k-term arithmetic progressions, one could consider a more flexible structure, namely weak k-term arithmetic progressions, which are sets of the form\n\n{b nα + βc: 1 ≤ n ≤ k} (for some α ≥ 1 and β).\n\nSince there are substantially more weak k-term arithmetic progressions than k-term arithmetic progressions, some of the standard problems and results might be easier to attack.",
   "public_statement": "Problem 1.14 (R. Graham). Instead of k-term arithmetic progressions, one could consider a more flexible structure, namely weak k-term arithmetic progressions, which are sets of the form\n\n{b nα + βc: 1 ≤ n ≤ k} (for some α ≥ 1 and β).\n\nSince there are substantially more weak k-term arithmetic progressions than k-term arithmetic progressions, some of the standard problems and results might be easier to attack.",
   "evidence": "The stored corpus record reads verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 219,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0221": {
   "statement_status": "exact",
   "original_statement": "Problem 1.15 (E. Croot). Let f (S) denote the length of the longest arithmetic pro-gression in a set of integers S. Given a real θ ∈ (0, 1] and an integer N ≥ 1, among all subsets A ⊆ [N ] satisfying |A| ≥ N θ, how small can f (A + A) be?",
   "clean_statement": "Problem 1.15 (E. Croot). Let f (S) denote the length of the longest arithmetic pro-gression in a set of integers S. Given a real θ ∈ (0, 1] and an integer N ≥ 1, among all subsets A ⊆ [N ] satisfying |A| ≥ N θ, how small can f (A + A) be?",
   "public_statement": "Problem 1.15 (E. Croot). Let f (S) denote the length of the longest arithmetic pro-gression in a set of integers S. Given a real θ ∈ (0, 1] and an integer N ≥ 1, among all subsets A ⊆ [N ] satisfying |A| ≥ N θ, how small can f (A + A) be?",
   "evidence": "The exact stored record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 220,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0222": {
   "statement_status": "exact",
   "original_statement": "Problem 1.16 (T. Wooley). Can one generalize Behrend's construction to produce large subsets S ⊂ [N ] such that S does not admit solutions to \n\n> s\n\n∑\n\n> i=1\n\naixi = 0, where \n\n> s\n\n∑\n\n> i=1\n\nai = 0 and |ai| < A?What about the same question, but with \n\n> t\n\n∑\n\n> i=1\n\naix2 \n\n> i\n\n= 0? N. Alon comments: the answer to the question is \"No\", if, for example, a1 = a2 = 1 and a3 = a4 = −1, then S cannot have size bigger than (1 + o(1)) √n (it is a Sidon set). For the quadratic version the answer is also \"no\"; if say, \n\ns = 100, a 1 = a2 = · · · = a50 = 1, a 51 = −1,..., a 100 = −1,\n\nthen by the pigeonhole principle S cannot of size bigger than O(n1/25 ). For the linear case the Behrend construction easily generalizes if only one ai is positive and the others are negative (even simultaneously for all such sets ai). There are also extensions to some other cases that appear in papers of Ruzsa in Acta Arithmetica in 93 or so.",
   "clean_statement": "Problem 1.16 (T. Wooley). Can one generalize Behrend's construction to produce large subsets S ⊂ [N ] such that S does not admit solutions to\n\n> s\n\n∑\n\n> i=1\n\naixi = 0, where\n\n> s\n\n∑\n\n> i=1\n\nai = 0 and |ai| < A?What about the same question, but with\n\n> t\n\n∑\n\n> i=1\n\naix2\n\n> i\n\n= 0? N. Alon comments: the answer to the question is \"No\", if, for example, a1 = a2 = 1 and a3 = a4 = −1, then S cannot have size bigger than (1 + o(1)) √n (it is a Sidon set). For the quadratic version the answer is also \"no\"; if say,\n\ns = 100, a 1 = a2 = · · · = a50 = 1, a 51 = −1,..., a 100 = −1,\n\nthen by the pigeonhole principle S cannot of size bigger than O(n1/25 ). For the linear case the Behrend construction easily generalizes if only one ai is positive and the others are negative (even simultaneously for all such sets ai). There are also extensions to some other cases that appear in papers of Ruzsa in Acta Arithmetica in 93 or so.",
   "public_statement": "Problem 1.16 (T. Wooley). Can one generalize Behrend's construction to produce large subsets S ⊂ [N ] such that S does not admit solutions to\n\n> s\n\n∑\n\n> i=1\n\naixi = 0, where\n\n> s\n\n∑\n\n> i=1\n\nai = 0 and |ai| < A?What about the same question, but with\n\n> t\n\n∑\n\n> i=1\n\naix2\n\n> i\n\n= 0? N. Alon comments: the answer to the question is \"No\", if, for example, a1 = a2 = 1 and a3 = a4 = −1, then S cannot have size bigger than (1 + o(1)) √n (it is a Sidon set). For the quadratic version the answer is also \"no\"; if say,\n\ns = 100, a 1 = a2 = · · · = a50 = 1, a 51 = −1,..., a 100 = −1,\n\nthen by the pigeonhole principle S cannot of size bigger than O(n1/25 ). For the linear case the Behrend construction easily generalizes if only one ai is positive and the others are negative (even simultaneously for all such sets ai). There are also extensions to some other cases that appear in papers of Ruzsa in Acta Arithmetica in 93 or so.",
   "evidence": "The official AIM workshop PDF, *Recent trends in additive combinatorics*, Problem 1.16 (T. Wooley), asks whether Behrend's construction can be generalized to give large sets \\(S\\subset [N]\\) having no solutions of \\[ \\sum_{i=1}^{s}a_i x_i=0, \\qquad \\sum_{i=1}^{s}a_i=0, \\qquad |a_i|<A, \\] and asks the analogous question for \\[ \\sum_{i=1}^{t}a_i x_i^2=0. \\] The PDF then records N. Alon's two negative examples: coefficients \\(1,1,-1,-1\\) force a Sidon-type \\(O(\\sqrt N)\\) bound, while fifty \\(+1\\)'s and fifty \\(-1\\)'s in the quadratic equation force \\(O(N^{1/25})\\). It also records that the linear construction works when only one coefficient has one sign, and points to Ruzsa's Acta Arithmetica papers.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 221,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0223": {
   "statement_status": "exact",
   "original_statement": "Problem 1.17 (A. Granville). Given a set in a finite field, how to determine (in rea-sonable time) whether it is a sumset of yet another set?",
   "clean_statement": "Problem 1.17 (A. Granville). Given a set in a finite field, how to determine (in rea-sonable time) whether it is a sumset of yet another set?",
   "public_statement": "Problem 1.17 (A. Granville). Given a set in a finite field, how to determine (in rea-sonable time) whether it is a sumset of yet another set?",
   "evidence": "The stored record reads verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 222,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0224": {
   "statement_status": "exact",
   "original_statement": "Problem 1.18 (V. Lev). Let A be a finite non-empty subset of an abelian group G, and write D:= A−A. If any d ∈ D has strictly more than |A|/2 representations of the form \n\nd = a′ − a′′ with a′, a ′′ ∈ A, then D is a subgroup: indeed, by the pigeonhole principle for any d1, d 2 ∈ D there exists a pair of representations d1 = a′ \n\n> 1\n\n− a′′ \n\n> 1, d 2 = a′ \n\n> 2\n\n− a′′ \n\n> 2\n\nsuch that a′′ \n\n> 1\n\n= a′′ \n\n> 2, and it follows that d1 − d2 = a′ \n\n> 1\n\n− a′ \n\n> 2\n\n∈ D.Assume now that any d ∈ D is only guaranteed to have at least |A|/2 representations as d = a′ −a′′ with a′, a ′′ ∈ A. In this case the argument above doesn't work, and in fact, the conclusion is not true either. To see this, consider the set A:= H ∪ (g + H), where PALO ALTO PROBLEMS 5\n\nH < G is a finite subgroup and g ∈ G is so chosen that the order of g in the quotient group G/H is at least five. Then D = ( −g + H) ∪ H ∪ (g + H) is not a subgroup, but a union of three cosets. At the same time, it is easily seen that any d ∈ D has at least \n\n|H| = |A|/2 representations of the form d = a′ − a′′.Is this example unique? In other words, given that any d ∈ D has at least |A|/2representations as d = a′ − a′′, is it necessarily true that D is either a subgroup or a union of three cosets? For practical applications one should go somewhat beyond the |A|/2 bound. Problem: assuming that any d ∈ D:= A − A has more than |A|/3representations of the form d = a′ − a′′ with a′, a ′′ ∈ A, is it necessarily true that D is either a subgroup or a union of three cosets?",
   "clean_statement": "Problem 1.18 (V. Lev). Let A be a finite non-empty subset of an abelian group G, and write D:= A−A. If any d ∈ D has strictly more than |A|/2 representations of the form\n\nd = a′ − a′′ with a′, a ′′ ∈ A, then D is a subgroup: indeed, by the pigeonhole principle for any d1, d 2 ∈ D there exists a pair of representations d1 = a′\n\n> 1\n\n− a′′\n\n> 1, d 2 = a′\n\n> 2\n\n− a′′\n\n> 2\n\nsuch that a′′\n\n> 1\n\n= a′′\n\n> 2, and it follows that d1 − d2 = a′\n\n> 1\n\n− a′\n\n> 2\n\n∈ D.Assume now that any d ∈ D is only guaranteed to have at least |A|/2 representations as d = a′ −a′′ with a′, a ′′ ∈ A. In this case the argument above doesn't work, and in fact, the conclusion is not true either. To see this, consider the set A:= H ∪ (g + H), where PALO ALTO PROBLEMS 5\n\nH < G is a finite subgroup and g ∈ G is so chosen that the order of g in the quotient group G/H is at least five. Then D = ( −g + H) ∪ H ∪ (g + H) is not a subgroup, but a union of three cosets. At the same time, it is easily seen that any d ∈ D has at least\n\n|H| = |A|/2 representations of the form d = a′ − a′′.Is this example unique? In other words, given that any d ∈ D has at least |A|/2representations as d = a′ − a′′, is it necessarily true that D is either a subgroup or a union of three cosets? For practical applications one should go somewhat beyond the |A|/2 bound. Problem: assuming that any d ∈ D:= A − A has more than |A|/3representations of the form d = a′ − a′′ with a′, a ′′ ∈ A, is it necessarily true that D is either a subgroup or a union of three cosets?",
   "public_statement": "Problem 1.18 (V. Lev). Let A be a finite non-empty subset of an abelian group G, and write D:= A−A. If any d ∈ D has strictly more than |A|/2 representations of the form\n\nd = a′ − a′′ with a′, a ′′ ∈ A, then D is a subgroup: indeed, by the pigeonhole principle for any d1, d 2 ∈ D there exists a pair of representations d1 = a′\n\n> 1\n\n− a′′\n\n> 1, d 2 = a′\n\n> 2\n\n− a′′\n\n> 2\n\nsuch that a′′\n\n> 1\n\n= a′′\n\n> 2, and it follows that d1 − d2 = a′\n\n> 1\n\n− a′\n\n> 2\n\n∈ D.Assume now that any d ∈ D is only guaranteed to have at least |A|/2 representations as d = a′ −a′′ with a′, a ′′ ∈ A. In this case the argument above doesn't work, and in fact, the conclusion is not true either. To see this, consider the set A:= H ∪ (g + H), where PALO ALTO PROBLEMS 5\n\nH < G is a finite subgroup and g ∈ G is so chosen that the order of g in the quotient group G/H is at least five. Then D = ( −g + H) ∪ H ∪ (g + H) is not a subgroup, but a union of three cosets. At the same time, it is easily seen that any d ∈ D has at least\n\n|H| = |A|/2 representations of the form d = a′ − a′′.Is this example unique? In other words, given that any d ∈ D has at least |A|/2representations as d = a′ − a′′, is it necessarily true that D is either a subgroup or a union of three cosets? For practical applications one should go somewhat beyond the |A|/2 bound. Problem: assuming that any d ∈ D:= A − A has more than |A|/3representations of the form d = a′ − a′′ with a′, a ′′ ∈ A, is it necessarily true that D is either a subgroup or a union of three cosets?",
   "evidence": "The canonical input remains unchanged in input.json. Its problem field is reproduced verbatim here, including extraction line breaks and the page-footer intrusion:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 223,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0225": {
   "statement_status": "exact",
   "original_statement": "Problem 1.19 (M.-C. Chang). Is it true that for any ≤ > 0 there exists ≤′ > 0 with the following property: if A ⊆ Z/q Z (with a sufficiently large integer q) satisfies |A +\n\nA| + |A · A| < q ≤, then either |A| > q 1−≤′\n\nor there exists q1 | q, q 1 > 1 such that the canonical image of A in Z/q 1Z has at most q≤′\n\nelements? 2. Sumsets Problem Session",
   "clean_statement": "Problem 1.19 (M.-C. Chang). Is it true that for any ≤ > 0 there exists ≤′ > 0 with the following property: if A ⊆ Z/q Z (with a sufficiently large integer q) satisfies |A +\n\nA| + |A · A| < q ≤, then either |A| > q 1−≤′\n\nor there exists q1 | q, q 1 > 1 such that the canonical image of A in Z/q 1Z has at most q≤′\n\nelements? 2. Sumsets Problem Session",
   "public_statement": "Problem 1.19 (M.-C. Chang). Is it true that for any ≤ > 0 there exists ≤′ > 0 with the following property: if A ⊆ Z/q Z (with a sufficiently large integer q) satisfies |A +\n\nA| + |A · A| < q ≤, then either |A| > q 1−≤′\n\nor there exists q1 | q, q 1 > 1 such that the canonical image of A in Z/q 1Z has at most q≤′\n\nelements? 2. Sumsets Problem Session",
   "evidence": "The exact stored record is visibly corrupted by OCR:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 224,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0226": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 2.1 (V. Lev). Solving a problem by Leo Moser, Peter Scherk proved in 1955 that if A and B are finite subsets of an abelian group such that A ∩ (−B) = {0}, then \n\n|A + B| ≥ | A| + |B| − \n1. (The condition A ∩ (−B) = {0} means that both A and B\n\ncontain zero and, moreover, the only representation of zero as 0 = a + b with a ∈ A and \n\nb ∈ B is that with a = b = 0). The estimate of Scherk's theorem is best possible: the bound is attained, for instance, if A = {0, d,..., (m − 1) d} and B = {0, d,..., (n − 1) d},where m and n are positive integers and d is a group element of order at least m + n − 1. Is there an analog of Scherk's theorem for the restricted sumset A ˙+B (the set of all sums a + b with a ∈ A, b ∈ B and a 6 = b)? Conjecture: if A and B are finite subsets of an abelian group such that A ∩ (−B) = {0}, then |A ˙+B| ≥ | A| + |B| − 3.",
   "clean_statement": "Lev's restricted Scherk conjecture.** Let \\(A,B\\) be finite subsets of an abelian group \\(G\\), with\n\\[\n A\\cap(-B)=\\{0\\}.\n\\]\nFor\n\\[\n A\\mathbin{\\dot+}B:=\\{a+b:a\\in A,\\ b\\in B,\\ a\\ne b\\},\n\\]\nprove or disprove\n\\[\n |A\\mathbin{\\dot+}B|\\ge |A|+|B|-3. \\tag{1}\n\\]",
   "public_statement": "Problem 2.1 (V. Lev). Solving a problem by Leo Moser, Peter Scherk proved in 1955 that if A and B are finite subsets of an abelian group such that A ∩ (−B) = {0}, then\n\n|A + B| ≥ | A| + |B| −\n1. (The condition A ∩ (−B) = {0} means that both A and B\n\ncontain zero and, moreover, the only representation of zero as 0 = a + b with a ∈ A and\n\nb ∈ B is that with a = b = 0). The estimate of Scherk's theorem is best possible: the bound is attained, for instance, if A = {0, d,..., (m − 1) d} and B = {0, d,..., (n − 1) d},where m and n are positive integers and d is a group element of order at least m + n − 1. Is there an analog of Scherk's theorem for the restricted sumset A ˙+B (the set of all sums a + b with a ∈ A, b ∈ B and a 6 = b)? Conjecture: if A and B are finite subsets of an abelian group such that A ∩ (−B) = {0}, then |A ˙+B| ≥ | A| + |B| − 3.",
   "evidence": "The stored AIM record reads (including extraction artifacts):",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 225,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0227": {
   "statement_status": "exact",
   "original_statement": "Problem 2.2 (V. Lev). Given two finite integer sets A and B, write \n\nν(n):= {(a, b ) ∈ A × B: a + b = n}; n ∈ Z.\n\nThe spectrum of ν defines a partition of the integer |A|| B| which can be visualized using a Ferrers diagram; that is, an arrangement of |A|| B| square boxes in bottom-aligned columns such that the height of the leftmost column is the largest value attained by ν,the height of next column is the second largest value of ν, and so on. It is not difficult to show that if rk denotes the height of the kth column of the diagram (that is, the kth largest value attained by ν), then \n\nr2 \n\n> k\n\n≤ rk + rk+1 + rk+2 + · · · (∗)6 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV \n\nfor any k ≥ 1. Problem: what are the general properties shared by the functions ν for all finite sets A, B ⊆ Z, other than that reflected by ( ∗)? Notice that for any t ∈ N, the length of the tth row of the above described diagram (counting the rows from the bottom) is Nt:= {n: ν(n) ≥ t}. From a well-known result of Pollard it follows that N1 + · · · + Nt ≥ t(|A| + |B| − t) for any t ≤ min {| A|, |B|}, and this can be derived also as a corollary of ( ∗).",
   "clean_statement": "Problem 2.2 (V. Lev). Given two finite integer sets A and B, write\n\nν(n):= {(a, b ) ∈ A × B: a + b = n}; n ∈ Z.\n\nThe spectrum of ν defines a partition of the integer |A|| B| which can be visualized using a Ferrers diagram; that is, an arrangement of |A|| B| square boxes in bottom-aligned columns such that the height of the leftmost column is the largest value attained by ν,the height of next column is the second largest value of ν, and so on. It is not difficult to show that if rk denotes the height of the kth column of the diagram (that is, the kth largest value attained by ν), then\n\nr2\n\n> k\n\n≤ rk + rk+1 + rk+2 + · · · (∗)6 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV\n\nfor any k ≥ 1. Problem: what are the general properties shared by the functions ν for all finite sets A, B ⊆ Z, other than that reflected by ( ∗)? Notice that for any t ∈ N, the length of the tth row of the above described diagram (counting the rows from the bottom) is Nt:= {n: ν(n) ≥ t}. From a well-known result of Pollard it follows that N1 + · · · + Nt ≥ t(|A| + |B| − t) for any t ≤ min {| A|, |B|}, and this can be derived also as a corollary of ( ∗).",
   "public_statement": "Problem 2.2 (V. Lev). Given two finite integer sets A and B, write\n\nν(n):= {(a, b ) ∈ A × B: a + b = n}; n ∈ Z.\n\nThe spectrum of ν defines a partition of the integer |A|| B| which can be visualized using a Ferrers diagram; that is, an arrangement of |A|| B| square boxes in bottom-aligned columns such that the height of the leftmost column is the largest value attained by ν,the height of next column is the second largest value of ν, and so on. It is not difficult to show that if rk denotes the height of the kth column of the diagram (that is, the kth largest value attained by ν), then\n\nr2\n\n> k\n\n≤ rk + rk+1 + rk+2 + · · · (∗)6 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV\n\nfor any k ≥ 1. Problem: what are the general properties shared by the functions ν for all finite sets A, B ⊆ Z, other than that reflected by ( ∗)? Notice that for any t ∈ N, the length of the tth row of the above described diagram (counting the rows from the bottom) is Nt:= {n: ν(n) ≥ t}. From a well-known result of Pollard it follows that N1 + · · · + Nt ≥ t(|A| + |B| − t) for any t ≤ min {| A|, |B|}, and this can be derived also as a corollary of ( ∗).",
   "evidence": "The raw corpus record defines, for finite integer sets \\(A,B\\), \\[ \\nu(n):=\\#\\{(a,b)\\in A\\times B:a+b=n\\},\\qquad n\\in\\mathbb Z, \\] then sorts the positive values of \\(\\nu\\) as column heights \\(r_1\\ge r_2\\ge\\cdots\\) of a Ferrers diagram. The extracted display is corrupted across a page boundary as `r2 > k <= rk + rk+1 + ...`.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 226,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0228": {
   "statement_status": "exact",
   "original_statement": "Problem 2.3 (G. Freiman). Suppose A ⊆ Z2 is finite set no three points of which are on a line. What is the smallest size of 2 A, given that |A| = n?",
   "clean_statement": "Problem 2.3 (G. Freiman). Suppose A ⊆ Z2 is finite set no three points of which are on a line. What is the smallest size of 2 A, given that |A| = n?",
   "public_statement": "Problem 2.3 (G. Freiman). Suppose A ⊆ Z2 is finite set no three points of which are on a line. What is the smallest size of 2 A, given that |A| = n?",
   "evidence": "The canonical JSON lost superscripts and comparison symbols. The AIM workshop PDF gives the following problem.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 227,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0229": {
   "statement_status": "exact",
   "original_statement": "Problem 2.4 (T. Tao). Suppose that A and B are finite sets of integers with |A| = m\n\nand |B| = n, where m > n, and suppose that G is a subset of A × B having size at least δmn. Further, suppose that \n\n|{ a + b: a ∈ A, b ∈ B, (a, b ) ∈ G}| < Cm. \n\nDoes this imply anything about the structure of A and B? In particular, must there exist A′ ⊂ A, B′ ⊂ B, |A′| ≥ cm, |B′| ≥ cn such that |A′ + B′| ≤ Km, where c and K\n\ndepend on δ and C?",
   "clean_statement": "Problem 2.4 (T. Tao). Suppose that A and B are finite sets of integers with |A| = m\n\nand |B| = n, where m > n, and suppose that G is a subset of A × B having size at least δmn. Further, suppose that\n\n|{ a + b: a ∈ A, b ∈ B, (a, b ) ∈ G}| < Cm.\n\nDoes this imply anything about the structure of A and B? In particular, must there exist A′ ⊂ A, B′ ⊂ B, |A′| ≥ cm, |B′| ≥ cn such that |A′ + B′| ≤ Km, where c and K\n\ndepend on δ and C?",
   "public_statement": "Problem 2.4 (T. Tao). Suppose that A and B are finite sets of integers with |A| = m\n\nand |B| = n, where m > n, and suppose that G is a subset of A × B having size at least δmn. Further, suppose that\n\n|{ a + b: a ∈ A, b ∈ B, (a, b ) ∈ G}| < Cm.\n\nDoes this imply anything about the structure of A and B? In particular, must there exist A′ ⊂ A, B′ ⊂ B, |A′| ≥ cm, |B′| ≥ cn such that |A′ + B′| ≤ Km, where c and K\n\ndepend on δ and C?",
   "evidence": "No corruption of the mathematical statement was found. The line breaks in the canonical JSON are extraction artifacts only.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 228,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0230": {
   "statement_status": "exact",
   "original_statement": "Problem 2.5 (T. Tao). Suppose that A and B are finite sets of integers with |A| = m\n\nand |B| = n, m > n, satisfying |A+B| < Km. Must there exist a generalized arithmetic progression P of rank c(K) containing B such that A ⊂ P + X and |P + X| ≤ c(K)|A|?",
   "clean_statement": "Problem 2.5 (T. Tao). Suppose that A and B are finite sets of integers with |A| = m\n\nand |B| = n, m > n, satisfying |A+B| < Km. Must there exist a generalized arithmetic progression P of rank c(K) containing B such that A ⊂ P + X and |P + X| ≤ c(K)|A|?",
   "public_statement": "Problem 2.5 (T. Tao). Suppose that A and B are finite sets of integers with |A| = m\n\nand |B| = n, m > n, satisfying |A+B| < Km. Must there exist a generalized arithmetic progression P of rank c(K) containing B such that A ⊂ P + X and |P + X| ≤ c(K)|A|?",
   "evidence": "The canonical record asks the following question of T. Tao. Let \\(A,B\\subseteq\\mathbb Z\\) be finite, with \\[ |A|=m>|B|=n, \\qquad |A+B|<Km. \\] Must there be a generalized arithmetic progression \\(P\\) of rank \\(c(K)\\), containing \\(B\\), and a set \\(X\\subseteq\\mathbb Z\\) such that \\[ A\\subseteq P+X, \\qquad |P+X|\\le c(K)|A|? \\tag{Q} \\]",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 229,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0231": {
   "statement_status": "exact",
   "original_statement": "Problem 2.6 (Y. Stanchescu). Suppose that A is a finite subset of Zd, not contained in a hyperplane of dimension smaller than d. Determine the smallest possible value of \n\n|A − A| as a function of |A|.",
   "clean_statement": "Problem 2.6 (Y. Stanchescu). Suppose that A is a finite subset of Zd, not contained in a hyperplane of dimension smaller than d. Determine the smallest possible value of\n\n|A − A| as a function of |A|.",
   "public_statement": "Problem 2.6 (Y. Stanchescu). Suppose that A is a finite subset of Zd, not contained in a hyperplane of dimension smaller than d. Determine the smallest possible value of\n\n|A − A| as a function of |A|.",
   "evidence": "The canonical JSON contains line-break and fraction damage. The official AIM workshop PDF gives:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 230,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0232": {
   "statement_status": "exact",
   "original_statement": "Problem 2.7 (B. Green). What is the size of the largest subset of Fp which is not a sumset B + B?PALO ALTO PROBLEMS 7",
   "clean_statement": "Problem 2.7 (B. Green). What is the size of the largest subset of Fp which is not a sumset B + B?PALO ALTO PROBLEMS 7",
   "public_statement": "Problem 2.7 (B. Green). What is the size of the largest subset of Fp which is not a sumset B + B?PALO ALTO PROBLEMS 7",
   "evidence": "The official AIM PDF gives the following question.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 231,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0233": {
   "statement_status": "exact",
   "original_statement": "Problem 2.8 (T. Wooley). Suppose A is a subset of the naturals. We say that A is an additive basis of order h for a polynomial sequence {f (n): n = 1, 2,... } if hA contains this sequence. If f is linear, and A is any order h basis for f (n), then |A∩[n]| ≥ n1/h. If d =deg( f ) ≥\n\n2, then one can trivially deduce that |A ∩ [n]| ≥ n1/hd. Can one get a substantially sharper lower bound in the case d ≥ 2?",
   "clean_statement": "Problem 2.8 (T. Wooley). Suppose A is a subset of the naturals. We say that A is an additive basis of order h for a polynomial sequence {f (n): n = 1, 2,... } if hA contains this sequence. If f is linear, and A is any order h basis for f (n), then |A∩[n]| ≥ n1/h. If d =deg( f ) ≥\n\n2, then one can trivially deduce that |A ∩ [n]| ≥ n1/hd. Can one get a substantially sharper lower bound in the case d ≥ 2?",
   "public_statement": "Problem 2.8 (T. Wooley). Suppose A is a subset of the naturals. We say that A is an additive basis of order h for a polynomial sequence {f (n): n = 1, 2,... } if hA contains this sequence. If f is linear, and A is any order h basis for f (n), then |A∩[n]| ≥ n1/h. If d =deg( f ) ≥\n\n2, then one can trivially deduce that |A ∩ [n]| ≥ n1/hd. Can one get a substantially sharper lower bound in the case d ≥ 2?",
   "evidence": "This is Problem 2.8, attributed to T. Wooley, in the AIM workshop list *Recent trends in additive combinatorics*. The repository extraction has lost superscript formatting: its strings `n1/h` and `n1/hd` mean, respectively, $n^{1/h}$ and $n^{1/(hd)}$. This was checked against the source PDF.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 232,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0234": {
   "statement_status": "exact",
   "original_statement": "Problem 2.9 (T. Gowers). Suppose A ⊆ Z, |A| = n. Let \n\nS = {x + a + b + c: x, x + a, x + b, x + c, x + a + b, x + b + c, x + a + c ∈ A}.\n\nIf |S| < cn, then can one deduce anything about the structure of A?B. Green comments: The answer to this is \"no\" as it stands. For example S could be a dissociated set. Tim, Terry and I [Green] tried to formulate a decent question along these lines but couldn't come up with anything we liked.",
   "clean_statement": "Problem 2.9 (T. Gowers). Suppose A ⊆ Z, |A| = n. Let\n\nS = {x + a + b + c: x, x + a, x + b, x + c, x + a + b, x + b + c, x + a + c ∈ A}.\n\nIf |S| < cn, then can one deduce anything about the structure of A?B. Green comments: The answer to this is \"no\" as it stands. For example S could be a dissociated set. Tim, Terry and I [Green] tried to formulate a decent question along these lines but couldn't come up with anything we liked.",
   "public_statement": "Problem 2.9 (T. Gowers). Suppose A ⊆ Z, |A| = n. Let\n\nS = {x + a + b + c: x, x + a, x + b, x + c, x + a + b, x + b + c, x + a + c ∈ A}.\n\nIf |S| < cn, then can one deduce anything about the structure of A?B. Green comments: The answer to this is \"no\" as it stands. For example S could be a dissociated set. Tim, Terry and I [Green] tried to formulate a decent question along these lines but couldn't come up with anything we liked.",
   "evidence": "The canonical record is Problem 2.9 from the AIM workshop problem list *Recent Trends in Additive Combinatorics*. The official PDF was checked directly. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 233,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0235": {
   "statement_status": "exact",
   "original_statement": "Problem 2.10. Suppose A is a subset of the naturals, and is an additive basis of N of order 2. Does there exist a proper subset B of A, where B is an additive basis of the naturals of order 2? What is the slowest growing \n\nB(x):= |{ b ∈ B: b ≤ x}|?",
   "clean_statement": "Problem 2.10. Suppose A is a subset of the naturals, and is an additive basis of N of order 2. Does there exist a proper subset B of A, where B is an additive basis of the naturals of order 2? What is the slowest growing\n\nB(x):= |{ b ∈ B: b ≤ x}|?",
   "public_statement": "Problem 2.10. Suppose A is a subset of the naturals, and is an additive basis of N of order 2. Does there exist a proper subset B of A, where B is an additive basis of the naturals of order 2? What is the slowest growing\n\nB(x):= |{ b ∈ B: b ≤ x}|?",
   "evidence": "The record is Problem 2.10 in the AIM workshop list *Recent trends in additive combinatorics*. The official PDF and the corpus record agree on the wording:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 234,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0236": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 2.11 (Brought by V. Vu, originally stated by Erd˝ os and Turan). Suppose \n\nA ⊆ N is an additive basis of order n. Let r(m) be the number of pairs ( a1, a 2) ∈ A × A\n\nsuch that m = a1 + a2. Must lim sup m→∞ r(m) = ∞?",
   "clean_statement": null,
   "public_statement": "Problem 2.11 (Brought by V. Vu, originally stated by Erd˝ os and Turan). Suppose\n\nA ⊆ N is an additive basis of order n. Let r(m) be the number of pairs ( a1, a 2) ∈ A × A\n\nsuch that m = a1 + a2. Must lim sup m→∞ r(m) = ∞?",
   "evidence": "The official AIM PDF prints:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 235,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0237": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 2.12 (Y. Stanchescu). Fix an integer t ≥ 1 and suppose that A ⊆ [N ] is a set such that none of the t2 equations mx + ny = ( m + n)z with 1 ≤ m, n ≤ t\n\nhas a non-trivial solution in the variables x, y, z ∈ A. How large can A be under this assumption?",
   "clean_statement": null,
   "public_statement": "Problem 2.12 (Y. Stanchescu). Fix an integer t ≥ 1 and suppose that A ⊆ [N ] is a set such that none of the t2 equations mx + ny = ( m + n)z with 1 ≤ m, n ≤ t\n\nhas a non-trivial solution in the variables x, y, z ∈ A. How large can A be under this assumption?",
   "evidence": "The canonical record comes from Problem 2.12 of the AIM workshop list *Recent trends in additive combinatorics*. Direct inspection of the official PDF gives the following statement.",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 236,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0238": {
   "statement_status": "exact",
   "original_statement": "Problem 3.1 (J. Bourgain). Find explicitly a function f: Fp × Fp → Fp such that for every A, B ⊆ Fp with |A|, |B| ∼ p1/2 we have \n\n|f (A × B)| ≥ p1/2+ ≤.8 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV",
   "clean_statement": "Problem 3.1 (J. Bourgain). Find explicitly a function f: Fp × Fp → Fp such that for every A, B ⊆ Fp with |A|, |B| ∼ p1/2 we have\n\n|f (A × B)| ≥ p1/2+ ≤.8 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV",
   "public_statement": "Problem 3.1 (J. Bourgain). Find explicitly a function f: Fp × Fp → Fp such that for every A, B ⊆ Fp with |A|, |B| ∼ p1/2 we have\n\n|f (A × B)| ≥ p1/2+ ≤.8 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV",
   "evidence": "The canonical record is Problem 3.1 in the AIM workshop problem list *Recent Trends in Additive Combinatorics*. The official PDF was checked directly. It states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 237,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0239": {
   "statement_status": "exact",
   "original_statement": "Problem 3.2 (J. Bourgain). Given that H ≤ F∗ \n\n> p\n\nand |H| > p δ, what is the smallest k\n\nwhich is guaranteed to satisfy kH (= H + · · · + H) = Fp? It is known that this holds provided that log k > (1 /δ )C, but can one do better?",
   "clean_statement": "Problem 3.2 (J. Bourgain). Given that H ≤ F∗\n\n> p\n\nand |H| > p δ, what is the smallest k\n\nwhich is guaranteed to satisfy kH (= H + · · · + H) = Fp? It is known that this holds provided that log k > (1 /δ )C, but can one do better?",
   "public_statement": "Problem 3.2 (J. Bourgain). Given that H ≤ F∗\n\n> p\n\nand |H| > p δ, what is the smallest k\n\nwhich is guaranteed to satisfy kH (= H + · · · + H) = Fp? It is known that this holds provided that log k > (1 /δ )C, but can one do better?",
   "evidence": "The corpus record is visibly damaged by PDF extraction: the subgroup notation, the exponent on \\(p\\), and the exponent on \\(1/\\delta\\) were separated from their surrounding text. The AIM workshop PDF and the expanded problem list of Croot--Lev give the following unambiguous statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 238,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0240": {
   "statement_status": "exact",
   "original_statement": "Problem 3.3 (J. Bourgain). Suppose that H ≤ F∗\n\n> p. How large must H be in order that ∣∣∣∣∣∑\n\n> x∈H\n\nep(ax )\n\n∣∣∣∣∣ = o(|H|)to hold for all a 6 = 0?",
   "clean_statement": "Problem 3.3 (J. Bourgain). Suppose that H ≤ F∗\n\n> p. How large must H be in order that ∣∣∣∣∣∑\n\n> x∈H\n\nep(ax )\n\n∣∣∣∣∣ = o(|H|)to hold for all a 6 = 0?",
   "public_statement": "Problem 3.3 (J. Bourgain). Suppose that H ≤ F∗\n\n> p. How large must H be in order that ∣∣∣∣∣∑\n\n> x∈H\n\nep(ax )\n\n∣∣∣∣∣ = o(|H|)to hold for all a 6 = 0?",
   "evidence": "The repository record is visibly damaged by PDF extraction: subscripts, superscripts, the absolute-value bars, and the not-equal sign have been split across lines. The official AIM PDF, page 8 of the printed document (PDF page 7), displays Problem 3.3 as follows.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 239,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0241": {
   "statement_status": "exact",
   "original_statement": "Problem 3.4 (Brought by B. Green, originally stated by A. Venkatesh). Let A ⊆\n\nSL 2(Fp) satisfy |A| ∼ p5/2. Does it follow that \n\n|A · A| > p5/2+ δ?",
   "clean_statement": "Problem 3.4 (Brought by B. Green, originally stated by A. Venkatesh). Let A ⊆\n\nSL 2(Fp) satisfy |A| ∼ p5/2. Does it follow that\n\n|A · A| > p5/2+ δ?",
   "public_statement": "Problem 3.4 (Brought by B. Green, originally stated by A. Venkatesh). Let A ⊆\n\nSL 2(Fp) satisfy |A| ∼ p5/2. Does it follow that\n\n|A · A| > p5/2+ δ?",
   "evidence": "The canonical record is Problem 3.4 in the AIM workshop list *Recent Trends in Additive Combinatorics*. The official PDF was inspected directly. Its mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 240,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0242": {
   "statement_status": "exact",
   "original_statement": "Problem 3.5 (T. Tao). Let A be a finite subset of a (not necessarily abelian) group. Given that A · A is small, is A · A · A necessarily small, too? T. Tao comments: it was pointed out to me by Ben Green and Mei-Chu Chang that the problem as stated is false, as it follows by considering A = H ∪ { x} where H is a non-normal subgroup and x is not in the normalizer of H. However, what does appear to be true is that there is a large subset A′ of A such that A′A′A′ is small. A more ambitious problem would be to attempt an inverse theorem; for instance, if |AA | < 2|A|,what can one say about A?",
   "clean_statement": "Problem 3.5 (T. Tao). Let A be a finite subset of a (not necessarily abelian) group. Given that A · A is small, is A · A · A necessarily small, too? T. Tao comments: it was pointed out to me by Ben Green and Mei-Chu Chang that the problem as stated is false, as it follows by considering A = H ∪ { x} where H is a non-normal subgroup and x is not in the normalizer of H. However, what does appear to be true is that there is a large subset A′ of A such that A′A′A′ is small. A more ambitious problem would be to attempt an inverse theorem; for instance, if |AA | < 2|A|,what can one say about A?",
   "public_statement": "Problem 3.5 (T. Tao). Let A be a finite subset of a (not necessarily abelian) group. Given that A · A is small, is A · A · A necessarily small, too? T. Tao comments: it was pointed out to me by Ben Green and Mei-Chu Chang that the problem as stated is false, as it follows by considering A = H ∪ { x} where H is a non-normal subgroup and x is not in the normalizer of H. However, what does appear to be true is that there is a large subset A′ of A such that A′A′A′ is small. A more ambitious problem would be to attempt an inverse theorem; for instance, if |AA | < 2|A|,what can one say about A?",
   "evidence": "The canonical record is Problem 3.5 in *Problems Presented at the Workshop on Recent Trend in Additive Combinatorics*, collected by Ernie Croot and Vsevolod F. Lev. The official AIM PDF, page 8 (PDF page index 7), says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 241,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0243": {
   "statement_status": "exact",
   "original_statement": "Problem 3.6 (Brought by V. Lev). Given a prime p, how large can a set A ⊆ Fp be given that the difference between any two elements of A is a quadratic residue modulo \n\np?",
   "clean_statement": "Problem 3.6 (Brought by V. Lev). Given a prime p, how large can a set A ⊆ Fp be given that the difference between any two elements of A is a quadratic residue modulo\n\np?",
   "public_statement": "Problem 3.6 (Brought by V. Lev). Given a prime p, how large can a set A ⊆ Fp be given that the difference between any two elements of A is a quadratic residue modulo\n\np?",
   "evidence": "The AIM workshop PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 242,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0244": {
   "statement_status": "exact",
   "original_statement": "Problem 3.7 (Brought by B. Green and T. Tao). Take a finite subset of the squares of integers, |A| = N. What lower bound can you get on A + A? One can get better than cN via Freiman's theorem. B. Green comments: I am not sure who posed this. It is certainly implicit in a paper of Chang on Rudin's problem (are the squares a Λ( p) set?)",
   "clean_statement": "Problem 3.7 (Brought by B. Green and T. Tao). Take a finite subset of the squares of integers, |A| = N. What lower bound can you get on A + A? One can get better than cN via Freiman's theorem. B. Green comments: I am not sure who posed this. It is certainly implicit in a paper of Chang on Rudin's problem (are the squares a Λ( p) set?)",
   "public_statement": "Problem 3.7 (Brought by B. Green and T. Tao). Take a finite subset of the squares of integers, |A| = N. What lower bound can you get on A + A? One can get better than cN via Freiman's theorem. B. Green comments: I am not sure who posed this. It is certainly implicit in a paper of Chang on Rudin's problem (are the squares a Λ( p) set?)",
   "evidence": "The canonical record is Problem 3.7 from the AIM workshop *Recent trends in additive combinatorics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 243,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0245": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 3.8 (T. Tao). Take F to be a finite field, and suppose that E ⊆ F × F × F,where E is a Besicovich set; i.e. E contains a line in every direction. It is known from the work of Wolf that |E| ≥ | F|5/2; prove the better lower bound |E| ≥ | F|5/2+ ≤.",
   "clean_statement": null,
   "public_statement": "Problem 3.8 (T. Tao). Take F to be a finite field, and suppose that E ⊆ F × F × F,where E is a Besicovich set; i.e. E contains a line in every direction. It is known from the work of Wolf that |E| ≥ | F|5/2; prove the better lower bound |E| ≥ | F|5/2+ ≤.",
   "evidence": "The canonical record is Problem 3.8 in *Problems Presented at the Workshop on Recent Trend in Additive Combinatorics*, collected by Ernie Croot and Vsevolod F. Lev. The official AIM PDF, page 8 (PDF page index 7), reads, after repairing its text layer:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 244,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0246": {
   "statement_status": "exact",
   "original_statement": "Problem 3.9 (A. Granville). Given a finite field F and an integer n ≥ 1, find the smallest size of a subset E ⊂ F n which determines all directions in Fn.PALO ALTO PROBLEMS 9",
   "clean_statement": "Problem 3.9 (A. Granville). Given a finite field F and an integer n ≥ 1, find the smallest size of a subset E ⊂ F n which determines all directions in Fn.PALO ALTO PROBLEMS 9",
   "public_statement": "Problem 3.9 (A. Granville). Given a finite field F and an integer n ≥ 1, find the smallest size of a subset E ⊂ F n which determines all directions in Fn.PALO ALTO PROBLEMS 9",
   "evidence": "The canonical record is an OCR extraction of Problem 3.9 from the AIM workshop list *Problems presented at the workshop: Additive Combinatorics*. The PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 245,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0247": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 3.10 (T. Tao). Find an analogue for the Szemer´ edi-Trotter theorem for \n\nFp × Fp. More precisely, suppose we have a system of n points and l lines in Fp × Fp.Does the number i of point-line incidences necessarily satisfy \n\ni ø (nl )2/3 + n + l?In particular, if both n and l are about log p, is it true that i = O(( nl )2/3)?",
   "clean_statement": null,
   "public_statement": "Problem 3.10 (T. Tao). Find an analogue for the Szemer´ edi-Trotter theorem for\n\nFp × Fp. More precisely, suppose we have a system of n points and l lines in Fp × Fp.Does the number i of point-line incidences necessarily satisfy\n\ni ø (nl )2/3 + n + l?In particular, if both n and l are about log p, is it true that i = O(( nl )2/3)?",
   "evidence": "The record is Problem 3.10, attributed to T. Tao, in the AIM workshop list *Recent trends in additive combinatorics*. Direct inspection of the official PDF gives the following statement (notation normalized but not changed mathematically):",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 246,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0248": {
   "statement_status": "exact",
   "original_statement": "Problem 3.11 (J. Solymosi). Do there exist sets A, B, C ⊆ Fp with |A|, |B|, |C| ≈ √p\n\nand |A + B| + |AC | < p?",
   "clean_statement": "Problem 3.11 (J. Solymosi). Do there exist sets A, B, C ⊆ Fp with |A|, |B|, |C| ≈ √p\n\nand |A + B| + |AC | < p?",
   "public_statement": "Problem 3.11 (J. Solymosi). Do there exist sets A, B, C ⊆ Fp with |A|, |B|, |C| ≈ √p\n\nand |A + B| + |AC | < p?",
   "evidence": "The canonical record is Problem 3.11, attributed to J. Solymosi, from the AIM workshop list *Problems Presented at the Workshop on Recent Trend in Additive Combinatorics*. Its literal question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 247,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0249": {
   "statement_status": "exact",
   "original_statement": "Problem 3.12 (J. Bourgain). Consider the Szemer´ edi-Trotter theorem in R3 with n2\n\nlines, each containing n points from a given set S. Assume no n lines are coplanar. Find a lower bound for S (e.g. |S| ≥ n3−≤).",
   "clean_statement": "Problem 3.12 (J. Bourgain). Consider the Szemer´ edi-Trotter theorem in R3 with n2\n\nlines, each containing n points from a given set S. Assume no n lines are coplanar. Find a lower bound for S (e.g. |S| ≥ n3−≤).",
   "public_statement": "Problem 3.12 (J. Bourgain). Consider the Szemer´ edi-Trotter theorem in R3 with n2\n\nlines, each containing n points from a given set S. Assume no n lines are coplanar. Find a lower bound for S (e.g. |S| ≥ n3−≤).",
   "evidence": "The canonical JSON record is an OCR extraction of Problem 3.12 in the AIM workshop list *Recent trends in additive combinatorics*. The official source PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 248,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0250": {
   "statement_status": "exact",
   "original_statement": "Problem 3.13 (T. Tao). Take n lines in R3. Define a joint to be a point with three lines passing through it that are not coplanar. How many joints can there be?",
   "clean_statement": "Problem 3.13 (T. Tao). Take n lines in R3. Define a joint to be a point with three lines passing through it that are not coplanar. How many joints can there be?",
   "public_statement": "Problem 3.13 (T. Tao). Take n lines in R3. Define a joint to be a point with three lines passing through it that are not coplanar. How many joints can there be?",
   "evidence": "The record is Problem 3.13, attributed to T. Tao, in the AIM workshop list *Recent trends in additive combinatorics*. Direct inspection of the official PDF recovers the statement as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 249,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0251": {
   "statement_status": "exact",
   "original_statement": "Problem 3.14 (T. Tao). Same problem, but over finite fields.",
   "clean_statement": "Problem 3.14 (T. Tao). Same problem, but over finite fields.",
   "public_statement": "Problem 3.14 (T. Tao). Same problem, but over finite fields.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 250,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0252": {
   "statement_status": "exact",
   "original_statement": "Problem 3.15 (Brought by T. Tao, originally stated by I. Ruzsa). Choose A ⊆ Fn\n\n> 2\n\nsuch that \n\n|A + A| ≤ k|A|.\n\nDoes there exist a subspace V ⊆ Fn \n\n> 2\n\nsuch that |V | < k c|A|, and \n\n|A ∩ V | ≥ k−c|A|?",
   "clean_statement": "Problem 3.15 (Brought by T. Tao, originally stated by I. Ruzsa). Choose A ⊆ Fn\n\n> 2\n\nsuch that\n\n|A + A| ≤ k|A|.\n\nDoes there exist a subspace V ⊆ Fn\n\n> 2\n\nsuch that |V | < k c|A|, and\n\n|A ∩ V | ≥ k−c|A|?",
   "public_statement": "Problem 3.15 (Brought by T. Tao, originally stated by I. Ruzsa). Choose A ⊆ Fn\n\n> 2\n\nsuch that\n\n|A + A| ≤ k|A|.\n\nDoes there exist a subspace V ⊆ Fn\n\n> 2\n\nsuch that |V | < k c|A|, and\n\n|A ∩ V | ≥ k−c|A|?",
   "evidence": "The official AIM PDF gives the following statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 251,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0253": {
   "statement_status": "exact",
   "original_statement": "Problem 3.16 (T. Tao). Suppose A ⊆ Z. If |2A| < k |A|, describe the sumset |nA | as \n\nn tends to infinity. Find f (n, k ), which is the smallest number such that \n\n|nA | ≤ f (n, k )|A|\n\nfor all A such that |2A| < k |A|.",
   "clean_statement": "Problem 3.16 (T. Tao). Suppose A ⊆ Z. If |2A| < k |A|, describe the sumset |nA | as\n\nn tends to infinity. Find f (n, k ), which is the smallest number such that\n\n|nA | ≤ f (n, k )|A|\n\nfor all A such that |2A| < k |A|.",
   "public_statement": "Problem 3.16 (T. Tao). Suppose A ⊆ Z. If |2A| < k |A|, describe the sumset |nA | as\n\nn tends to infinity. Find f (n, k ), which is the smallest number such that\n\n|nA | ≤ f (n, k )|A|\n\nfor all A such that |2A| < k |A|.",
   "evidence": "The canonical record is Problem 3.16 from the AIM workshop *Recent trends in additive combinatorics*, attributed to T. Tao:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 252,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0254": {
   "statement_status": "exact",
   "original_statement": "Problem 3.17 (N. Katz). Does there exist a finite subset A ⊂ Z, |2A| = k|A|, so that for the function f (n, k ) defined in the previous problem every proper A′ ⊂ A satisfies \n\n|nA ′| ≥ f (n, k )|A′|?",
   "clean_statement": "Problem 3.17 (N. Katz). Does there exist a finite subset A ⊂ Z, |2A| = k|A|, so that for the function f (n, k ) defined in the previous problem every proper A′ ⊂ A satisfies\n\n|nA ′| ≥ f (n, k )|A′|?",
   "public_statement": "Problem 3.17 (N. Katz). Does there exist a finite subset A ⊂ Z, |2A| = k|A|, so that for the function f (n, k ) defined in the previous problem every proper A′ ⊂ A satisfies\n\n|nA ′| ≥ f (n, k )|A′|?",
   "evidence": "The official AIM PDF confirms all three potentially surprising features: Problem 3.16 uses the strict inequality \\(<\\), Problem 3.17 puts \\(A\\) on the equality boundary \\(=\\), and the required inequality for every proper subset is \\(\\geq\\). These are not extraction errors.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 253,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0255": {
   "statement_status": "exact",
   "original_statement": "Problem 3.18 (J. Bourgain). Find a good upper bound for the absolute value of the exponential sum ∑\n\n> x∈Fp\n\nep(aθ x + bθ x2\n\n),\n\nwhere θ is a generator for F∗\n\n> p.",
   "clean_statement": "Problem 3.18 (J. Bourgain). Find a good upper bound for the absolute value of the exponential sum ∑\n\n> x∈Fp\n\nep(aθ x + bθ x2\n\n),\n\nwhere θ is a generator for F∗\n\n> p.",
   "public_statement": "Problem 3.18 (J. Bourgain). Find a good upper bound for the absolute value of the exponential sum ∑\n\n> x∈Fp\n\nep(aθ x + bθ x2\n\n),\n\nwhere θ is a generator for F∗\n\n> p.",
   "evidence": "The corpus OCR reads",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 254,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0256": {
   "statement_status": "exact",
   "original_statement": "Problem 3.19 (Brought by V. Lev, originally stated by Konyagin and the presenter).\n\nFor any integer n ≥ 2 the set {0, 1, 2, 4,..., 2n−2} is \"linear\" (has Freiman rank one) and not contained in an arithmetic progression with difference larger than one, hence it is not isomorphic to a set of integers of length smaller than 2 n−2. Is this the extremal case? That is, is it true that in any class of isomorphic n-element sets there is a set of length at most 2 n−2? Conjecture: for n ≥ 7 any n-element set of integers is isomorphic (in Freiman's sense) to a subset of [0, 2n−2].",
   "clean_statement": "Problem 3.19 (Brought by V. Lev, originally stated by Konyagin and the presenter).\n\nFor any integer n ≥ 2 the set {0, 1, 2, 4,..., 2n−2} is \"linear\" (has Freiman rank one) and not contained in an arithmetic progression with difference larger than one, hence it is not isomorphic to a set of integers of length smaller than 2 n−2. Is this the extremal case? That is, is it true that in any class of isomorphic n-element sets there is a set of length at most 2 n−2? Conjecture: for n ≥ 7 any n-element set of integers is isomorphic (in Freiman's sense) to a subset of [0, 2n−2].",
   "public_statement": "Problem 3.19 (Brought by V. Lev, originally stated by Konyagin and the presenter).\n\nFor any integer n ≥ 2 the set {0, 1, 2, 4,..., 2n−2} is \"linear\" (has Freiman rank one) and not contained in an arithmetic progression with difference larger than one, hence it is not isomorphic to a set of integers of length smaller than 2 n−2. Is this the extremal case? That is, is it true that in any class of isomorphic n-element sets there is a set of length at most 2 n−2? Conjecture: for n ≥ 7 any n-element set of integers is isomorphic (in Freiman's sense) to a subset of [0, 2n−2].",
   "evidence": "The canonical JSON has lost superscripts. Inspection of the official AIM PDF and the original Konyagin--Lev formulation recovers the statement as follows.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 255,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0257": {
   "statement_status": "exact",
   "original_statement": "Problem 4.1 (T. Tao). Suppose that S ⊆ F2 \n\n> p\n\nwith |S| = p. How many pairs of points of distance one apart can there be? That is, how large can #{(x, y ) ∈ S × S: ( x1 − x2)2 + ( y1 − y2)2 = 1 }\n\nbe?",
   "clean_statement": "Problem 4.1 (T. Tao). Suppose that S ⊆ F2\n\n> p\n\nwith |S| = p. How many pairs of points of distance one apart can there be? That is, how large can #{(x, y ) ∈ S × S: ( x1 − x2)2 + ( y1 − y2)2 = 1 }\n\nbe?",
   "public_statement": "Problem 4.1 (T. Tao). Suppose that S ⊆ F2\n\n> p\n\nwith |S| = p. How many pairs of points of distance one apart can there be? That is, how large can #{(x, y ) ∈ S × S: ( x1 − x2)2 + ( y1 − y2)2 = 1 }\n\nbe?",
   "evidence": "Inspection of the original AIM PDF recovers the statement as follows.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 256,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0258": {
   "statement_status": "exact",
   "original_statement": "Problem 4.2 (V. S´ os). Let A be strictly increasing infinite sequence of integers, and denote by An the initial n-element segment of A. Suppose that \n\n|An + An| < Cn. \n\nWhat can be said about the structure of A?PALO ALTO PROBLEMS 11",
   "clean_statement": "Problem 4.2 (V. S´ os). Let A be strictly increasing infinite sequence of integers, and denote by An the initial n-element segment of A. Suppose that\n\n|An + An| < Cn.\n\nWhat can be said about the structure of A?PALO ALTO PROBLEMS 11",
   "public_statement": "Problem 4.2 (V. S´ os). Let A be strictly increasing infinite sequence of integers, and denote by An the initial n-element segment of A. Suppose that\n\n|An + An| < Cn.\n\nWhat can be said about the structure of A?PALO ALTO PROBLEMS 11",
   "evidence": "The AIM PDF gives the following problem (subscripts and the accent in the presenter's name have been restored from the typeset PDF):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 257,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0259": {
   "statement_status": "exact",
   "original_statement": "Problem 4.3 (V. S´ os). Let P be a product-free subset of an abelian group G with \n\n|G| = n. How large can P be? For abelian groups one has |P | ≥ 2n/ 7 (Alon-Kleitman), which is sharp. The case G = An is already interesting, and Green conjectures that in this case |P | = o(|An|).",
   "clean_statement": "Problem 4.3 (V. S´ os). Let P be a product-free subset of an abelian group G with\n\n|G| = n. How large can P be? For abelian groups one has |P | ≥ 2n/ 7 (Alon-Kleitman), which is sharp. The case G = An is already interesting, and Green conjectures that in this case |P | = o(|An|).",
   "public_statement": "Problem 4.3 (V. S´ os). Let P be a product-free subset of an abelian group G with\n\n|G| = n. How large can P be? For abelian groups one has |P | ≥ 2n/ 7 (Alon-Kleitman), which is sharp. The case G = An is already interesting, and Green conjectures that in this case |P | = o(|An|).",
   "evidence": "The canonical record is Problem 4.3 in *Problems Presented at the Workshop on Recent Trend in Additive Combinatorics*, collected by Ernie Croot and Vsevolod F. Lev. The official PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 258,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0260": {
   "statement_status": "exact",
   "original_statement": "Problem 4.4 (G. Freiman). Let A ⊆ Z with |A| = n and let G be a subset of A × A.Write \n\nG1 = {ai + aj: (ai, a j ) ∈ G}\n\nand \n\nG2 = {ai − aj: (ai, a j ) ∈ G}.\n\nGiven |G1|, estimate |G2| from above and describe those sets A with largest possible \n\n|G2|.Examples: 1) If |G1| = 1, then |G2| = n;2) If |G1| = 2, then |G2| = 2 n − 1, and A is an arithmetic progression; 3) If |G1| = 4, then |G2| = 4 n − c√n, and A is isomorphic to the set of interior points of some convex set; 4) If |G1| = 8, then |G2| = 8 n − cn 2/3, and A is near a three-dimensional convex body.",
   "clean_statement": "Problem 4.4 (G. Freiman). Let A ⊆ Z with |A| = n and let G be a subset of A × A.Write\n\nG1 = {ai + aj: (ai, a j ) ∈ G}\n\nand\n\nG2 = {ai − aj: (ai, a j ) ∈ G}.\n\nGiven |G1|, estimate |G2| from above and describe those sets A with largest possible\n\n|G2|.Examples: 1) If |G1| = 1, then |G2| = n;2) If |G1| = 2, then |G2| = 2 n − 1, and A is an arithmetic progression; 3) If |G1| = 4, then |G2| = 4 n − c√n, and A is isomorphic to the set of interior points of some convex set; 4) If |G1| = 8, then |G2| = 8 n − cn 2/3, and A is near a three-dimensional convex body.",
   "public_statement": "Problem 4.4 (G. Freiman). Let A ⊆ Z with |A| = n and let G be a subset of A × A.Write\n\nG1 = {ai + aj: (ai, a j ) ∈ G}\n\nand\n\nG2 = {ai − aj: (ai, a j ) ∈ G}.\n\nGiven |G1|, estimate |G2| from above and describe those sets A with largest possible\n\n|G2|.Examples: 1) If |G1| = 1, then |G2| = n;2) If |G1| = 2, then |G2| = 2 n − 1, and A is an arithmetic progression; 3) If |G1| = 4, then |G2| = 4 n − c√n, and A is isomorphic to the set of interior points of some convex set; 4) If |G1| = 8, then |G2| = 8 n − cn 2/3, and A is near a three-dimensional convex body.",
   "evidence": "Inspection of the original AIM PDF gives the following exact typography.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 259,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0261": {
   "statement_status": "exact",
   "original_statement": "Problem 4.5 (N. Katz). Let C be a triadic Cantor set. Does there exist E ⊂ R2 with the following properties: 1) E is the union of a 1-D family of unit line segments whose slopes are in C;2) the Lebesgue measure of E is 0. 3) the union of the doubles of the above line segments has positive Lebesgue mea-sure.",
   "clean_statement": "Problem 4.5 (N. Katz). Let C be a triadic Cantor set. Does there exist E ⊂ R2 with the following properties: 1) E is the union of a 1-D family of unit line segments whose slopes are in C;2) the Lebesgue measure of E is 0. 3) the union of the doubles of the above line segments has positive Lebesgue mea-sure.",
   "public_statement": "Problem 4.5 (N. Katz). Let C be a triadic Cantor set. Does there exist E ⊂ R2 with the following properties: 1) E is the union of a 1-D family of unit line segments whose slopes are in C;2) the Lebesgue measure of E is 0. 3) the union of the doubles of the above line segments has positive Lebesgue mea-sure.",
   "evidence": "The canonical record is Problem 4.5, attributed to N. Katz, in the AIM workshop list *Recent trends in additive combinatorics*. The official PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 260,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0262": {
   "statement_status": "exact",
   "original_statement": "Problem 4.6 (T. Tao). Given n lines and n points in R2, the number of point-line incidences is O(n4/3). Suppose that this number of incidences is indeed of this order. What can be said about the structure of our configuration of points and lines?",
   "clean_statement": "Problem 4.6 (T. Tao). Given n lines and n points in R2, the number of point-line incidences is O(n4/3). Suppose that this number of incidences is indeed of this order. What can be said about the structure of our configuration of points and lines?",
   "public_statement": "Problem 4.6 (T. Tao). Given n lines and n points in R2, the number of point-line incidences is O(n4/3). Suppose that this number of incidences is indeed of this order. What can be said about the structure of our configuration of points and lines?",
   "evidence": "The canonical record is Problem 4.6, attributed to T. Tao, from the AIM workshop list Recent trends in additive combinatorics. The extracted text reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 261,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0263": {
   "statement_status": "exact",
   "original_statement": "Problem 4.7 (A. Granville and T. Tao). Suppose that G is a proper subgroup of the multiplicative group of Fp. Let A ⊂ Fp such that A + A = G or A + A is slightly larger than G. Do such A exist, and do they necessarily have structure? Another version of this question is as follows: given that |A| > p ≤, can A + A be contained in a proper subgroup of F∗\n\n> p?",
   "clean_statement": "Problem 4.7 (A. Granville and T. Tao). Suppose that G is a proper subgroup of the multiplicative group of Fp. Let A ⊂ Fp such that A + A = G or A + A is slightly larger than G. Do such A exist, and do they necessarily have structure? Another version of this question is as follows: given that |A| > p ≤, can A + A be contained in a proper subgroup of F∗\n\n> p?",
   "public_statement": "Problem 4.7 (A. Granville and T. Tao). Suppose that G is a proper subgroup of the multiplicative group of Fp. Let A ⊂ Fp such that A + A = G or A + A is slightly larger than G. Do such A exist, and do they necessarily have structure? Another version of this question is as follows: given that |A| > p ≤, can A + A be contained in a proper subgroup of F∗\n\n> p?",
   "evidence": "The canonical JSON extraction is visibly damaged: it prints `p ≤` where an exponent should occur, loses the subscript and star in \\(\\mathbb F_p^*\\), and ends the remark after “very close to”. The official AIM workshop PDF restores the text as follows (notation normalized only from the PDF typography):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 262,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0264": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 4.8 (I. √ Laba). Suppose that α is transcendental, and |A| = n. What is the best lower bound for |A + αA |?12 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV",
   "clean_statement": "**Problem 4.8 (I. Łaba).** Suppose that \\(\\alpha\\) is transcendental, and \\(|A|=n\\). What is the best lower bound for \\(|A+\\alpha A|\\)?\n\n**Remark(s).** Konyagin and Łaba have shown that this cardinality is\n\\[\n\\gg \\frac{n\\log n}{\\log\\log n}.\n\\]\nThe best example (lowest known cardinality) is\n\\[\nn e^{c\\sqrt{\\log n}}.\n\\]",
   "public_statement": "**Problem 4.8 (I. Łaba).** Suppose that \\(\\alpha\\) is transcendental, and \\(|A|=n\\). What is the best lower bound for \\(|A+\\alpha A|\\)?\n\n**Remark(s).** Konyagin and Łaba have shown that this cardinality is\n\\[\n\\gg \\frac{n\\log n}{\\log\\log n}.\n\\]\nThe best example (lowest known cardinality) is\n\\[\nn e^{c\\sqrt{\\log n}}.\n\\]",
   "evidence": "The canonical JSON has three OCR defects: the author is Izabella Łaba, not “I. √ Laba”; a page footer was inserted into the problem; and the exponent in the example was flattened. The official AIM PDF places the problem at the bottom of printed page 11 and the remark at the top of printed page 12. With mathematical typography restored, it reads:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 263,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0265": {
   "statement_status": "unrecoverable",
   "original_statement": "Problem 4.9 (N. Katz).\n\nSD( r1,..., r n; α),\n\nfor some r1,..., r n ∈ R.",
   "clean_statement": null,
   "public_statement": "Problem 4.9 (N. Katz).\n\nSD( r1,..., r n; α),\n\nfor some r1,..., r n ∈ R.",
   "evidence": "This is not merely an OCR truncation. The official PDF, page 12, contains exactly the same two-line fragment, and the official TeX source reads: \\[ {\\rm SD}(r_1,\\ldots,r_n;\\alpha), \\qquad \\text{for some }r_1,\\ldots,r_n\\in\\mathbb R. \\] There is no verb, no quantifier on \\(\\alpha\\), and no definition of \\(SD\\) anywhere in that source. Therefore the exact problem cannot be recovered as a well-formed mathematical statement.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 264,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0266": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 4.10 (T. Tao). What is the best ≤ for which there exist real numbers r1,..., r n\n\nwith the following property: given any two random values x, y taking finitely many real values, and obeying the entropy bound H(x + rj y) < log N for all j = 1,..., N, one necessarily has H(x − y) < (1 + ≤) log N.",
   "clean_statement": "**Problem 4.10 (T. Tao).** What is the best \\(\\varepsilon\\) for which there\nexist real numbers \\(r_1,\\ldots,r_n\\) with the following property: given any\ntwo random values \\(x,y\\) taking finitely many real values, and obeying the\nentropy bound \\(H(x+r_jy)<\\log N\\) for all \\(j=1,\\ldots,N\\), one necessarily\nhas \\(H(x-y)<(1+\\varepsilon)\\log N\\).",
   "public_statement": "**Problem 4.10 (T. Tao).** What is the best \\(\\varepsilon\\) for which there\nexist real numbers \\(r_1,\\ldots,r_n\\) with the following property: given any\ntwo random values \\(x,y\\) taking finitely many real values, and obeying the\nentropy bound \\(H(x+r_jy)<\\log N\\) for all \\(j=1,\\ldots,N\\), one necessarily\nhas \\(H(x-y)<(1+\\varepsilon)\\log N\\).",
   "evidence": "The record comes from Problem 4.10 of the AIM workshop *Recent Trends in Additive Combinatorics* (September 9--12, 2004). Inspection of the official PDF, rather than the OCR record alone, recovers the missing symbol as \\(\\varepsilon\\). The PDF literally states: There are genuine defects in the printed statement, not merely OCR defects.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 265,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0267": {
   "statement_status": "exact",
   "original_statement": "Problem 4.11 (Brought by B. Green, implicit in a paper of Solomyak and Peres).\n\nGive an estimate for the size of the δ-thickened Kahane Besicovich set C4 × C 4, where \n\nC4 =\n\n{ ∞∑\n\n> n=0\n\nan\n\n4n: an ∈ { 0, 1}\n\n}.",
   "clean_statement": "Problem 4.11 (Brought by B. Green, implicit in a paper of Solomyak and Peres).\n\nGive an estimate for the size of the δ-thickened Kahane Besicovich set C4 × C 4, where\n\nC4 =\n\n{ ∞∑\n\n> n=0\n\nan\n\n4n: an ∈ { 0, 1}\n\n}.",
   "public_statement": "Problem 4.11 (Brought by B. Green, implicit in a paper of Solomyak and Peres).\n\nGive an estimate for the size of the δ-thickened Kahane Besicovich set C4 × C 4, where\n\nC4 =\n\n{ ∞∑\n\n> n=0\n\nan\n\n4n: an ∈ { 0, 1}\n\n}.",
   "evidence": "The official AIM workshop PDF, on its final page and in the section headed “Erdős Distance and Kakea Problem Session,” prints:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 266,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0268": {
   "statement_status": "exact",
   "original_statement": "Problem 4.12 (T. Tao). For which real numbers α the set C4 + αC4 has zero Lebesgue measure, where C4 is as in the previous problem?",
   "clean_statement": "Problem 4.12 (T. Tao). For which real numbers α the set C4 + αC4 has zero Lebesgue measure, where C4 is as in the previous problem?",
   "public_statement": "Problem 4.12 (T. Tao). For which real numbers α the set C4 + αC4 has zero Lebesgue measure, where C4 is as in the previous problem?",
   "evidence": "The official AIM TeX source gives Problem 4.11 as \\[ \\mathcal C_4= \\left\\{ \\sum_{n=0}^{\\infty}\\frac{a_n}{4^n}:a_n\\in\\{0,1\\} \\right\\}, \\] and Problem 4.12, attributed to T. Tao, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 267,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0269": {
   "statement_status": "exact",
   "original_statement": "Problem 4.13 (Brought by V. Lev, originally stated by Konyagin and the presenter).\n\nSuppose we are given r ≥ 1 points z1,..., z r on the unit circle and corresponding non-negative weights p1,..., p r, normalized by the condition p1 + · · · + pr = r. We want to find yet another point z on the circle which should be as far as possible from all points \n\nzj in the sense that the product ∏rj=1 |z − zj |pj is to be maximized. Conjecture: for any points zj and weights pj as above, there exists z such that \n\n> r\n\n∏\n\n> j=1\n\n|z − zj |pj ≥ 2.",
   "clean_statement": "Problem 4.13 (Brought by V. Lev, originally stated by Konyagin and the presenter).\n\nSuppose we are given r ≥ 1 points z1,..., z r on the unit circle and corresponding non-negative weights p1,..., p r, normalized by the condition p1 + · · · + pr = r. We want to find yet another point z on the circle which should be as far as possible from all points\n\nzj in the sense that the product ∏rj=1 |z − zj |pj is to be maximized. Conjecture: for any points zj and weights pj as above, there exists z such that\n\n> r\n\n∏\n\n> j=1\n\n|z − zj |pj ≥ 2.",
   "public_statement": "Problem 4.13 (Brought by V. Lev, originally stated by Konyagin and the presenter).\n\nSuppose we are given r ≥ 1 points z1,..., z r on the unit circle and corresponding non-negative weights p1,..., p r, normalized by the condition p1 + · · · + pr = r. We want to find yet another point z on the circle which should be as far as possible from all points\n\nzj in the sense that the product ∏rj=1 |z − zj |pj is to be maximized. Conjecture: for any points zj and weights pj as above, there exists z such that\n\n> r\n\n∏\n\n> j=1\n\n|z − zj |pj ≥ 2.",
   "evidence": "The source is Problem 4.13 in the AIM workshop list *Recent trends in additive combinatorics*. Its mathematical statement, with the typography restored but without changing its content, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 268,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0270": {
   "statement_status": "unrecoverable",
   "original_statement": "Problem 4. NP Characterization of Perfect Graphs 5. Recognition Algorithm Given the List of Maximal Cliques a. Berge Graphs with Poly-bounded Number of Max Cliques 6. TDI Matrices 7. Fixed Parameter Algorithms 8. Clique Joins 9. Polynomial Size Decomposition Tree B. Structural Characterization of Perfect Graphs............... 6C. Coloring Perfect Graphs......................... 71. Uniquely colorable perfect graphs D. Optimization on Perfect Graphs..................... 71. New Optimization Problems on Perfect Graphs a. A Possible New Problem E. Skew-Partitions............................ 71. Extending a Skew -Partition 2. Graphs Without Skew-Partitions 3. Graphs Without Star Cutsets 4. Finding Skew-Partitions in Berge Graphs 5. Interaction Between Different Skew-Partitions in a Graph 6. Skew -Partitions of Balanced Size 7. Recognizing Balanced Skew-Partitions 8. Even-Pair Skew-Partition F. Even Pairs in Berge Graphs....................... 91. Coloring Berge Graphs Using Even Pairs 2. Recognizing Even Pairs 3. Quasi-Parity and Strict Quasi-Parity Graphs a. Forbidden Subgraphs for The Class of Strict Quasi-Parity Graphs b. Recognition of Quasi-Parity and Strict Quasi-Parity Graphs 4. Perfectly Contractile Graphs a. Perfectly Contractile Graphs and the Decomposition Method 5. Possible Structure Theorem for Berge Graphs 6. Odd holes and odd walks G. Forbidding Holes and Antiholes.................... 12 1. 2-divisible Graphs 2. Clique Coloring of Perfect Graphs 3. Recognition of Odd-Hole-Free Graphs 4. Even-Hole-Free Graphs 5. Even-hole-free circulants, 6. beta-perfect graphs H. Partitionable Graphs......................... 14 1. Perfect, Partitionable, and Kernel-Solvable Graphs 3\n\n2. Partitionable graphs and odd holes 3. A Property of Partitionable Graphs 4. Small Transversals in Partitionable Graphs I. The Imperfection Ratio........................ 16 J. Integer Programming......................... 18 1. Partitionable Graphs as Cutting Planes for Packing Problems? 2. Feasibility/Membership Problem For the Theta Body K. Balanced Graphs........................... 18 1. Balanced circulants L. P4-structure and Its Relatives..................... 19 4\n\nChapter A: Recognition of Perfect Graphs \n\nCan one decide in polynomial time if a graph is perfect? \n\nA.1 Polynomial Recognition Algorithm Found \n\nA polynomial algorithm to test whether a graph is Berge was found in November 2002. A paper summarizing the work of two groups- Chudnovsky and Seymour and Cornuejols, Liu and Vuscovic- is due to appear in Combinatorica. The algorithm is independent of the proof of the strong perfect graph conjecture. \n\nA.2 Interaction Between Skew-Partitions and 2-joins \n\nOne can think of algorithms for testing for odd holes if you use only 2-joins to decom-pose a graph, or if you use only skew partitions. However, the interaction between skew partition steps and 2-join steps adds another level of difficulty. Can one argue that this can be reduced only to testing for holes as you decompose using skew partitions only and testing for holes using 2-joins only? Contributed by Jeremy Spinrad Similar approach worked in algorithms for recognizing even-hole-free graphs: first the graph is decomposed via vertex cut-sets, and only then via 2-joins. The 2-join decomposition blocks are defined in such a way that no new vertex cutset is introduced. Contributed by Kristina Vuskoic \n\nA.3 The Perfect-Graph Robust Algorithm Problem \n\nAn algorithm, which for any easily recognizable input, A, finds either an easily rec-ognizable B or an easily recognizable C, is sometimes called a \"robust algorithm\" - to be distinguished from a non-robust algorithm, which, for any A without a B, finds a C. Either provides a proof of the \"existentially polytime (EP) theorem\": For any A, there exists a B or a C. In other words: For any A without a B, there is a C. In [92i:68043 ], Jack Edmonds and Kathie Cameron advocated seeking a robust al-gorithm which, for any graph G, finds either a clique and colouring the same size or else finds an easily recognizable combinatorial obstruction to G being perfect. The obstruction might be specified to be a \"alpha-omega partitioned subgraph\", or it might be specified more particularly to be an odd hole or odd antihole. Such an algorithm might be simpler than an algorithm for recognizing whether or not a graph is perfect, in view of precedents, and since what it would do is incomparable with perfect-graph recognition. Such an algorithm could end up giving a clique and colouring the same size in a non-perfect graph. Here are two examples of similar problems which have been solved. Edmonds has given a simple robust algorithm which, for any graph G, either finds an odd cycle > 3 with at most one chord (a defining obstruction to G being Meyniel) or else finds a clique and colouring the same size. This is an improvement on the non-robust algorithms of Hoang and Hertz which, assuming a graph is Meyniel, find a clique and colouring the same size. Edmonds' algorithm is much simpler than the Burlet-Fonlupt decomposition algorithm for recognizing Meyniel graphs, which was motivated by an interest in optimizing in Meyniel graphs, and which is used by Hoang and Hertz. 5\n\nConforti and Cornuejols give a complicated decomposition algorithm for recognizing whether or not a matrix is balanced. At about the same time, to motivate the advocacy of a robust algorithm for either node-colouring a graph or recognizing it to be not perfect, Cameron and Edmonds presented a simple algorithm which, for any 0-1 matrix M, either finds, where x is the largest number of ones in any row, an x-colouring of the columns so that the 1's of any row are in different coloured columns, or else finds \"an odd hole\" in M (the defining obstruction to M being balanced). This introduced the \"EP - robust algorithm\" paradigm which is followed in Edmonds' Meyniel-related algorithm, and is related to the Conforti-Cornuejols-Rao treatment of balanced matrices in the same way that Edmonds' is related to the Burlet-Fonlupt treatment of Meyniel graphs. Following the same paradigm we expect there to be a robust algorithm proving the SPCG, related in the same way to the Chudnovsky-Robertson-Seymour-Thomas decomposition of Berge graphs. In conclusion, we know the following EP Theorem 1: For any graph, there is either a clique and a colouring of the same size, or there is an alpha-omega partitioned subgraph (or both). EP Theorem 2 (SPGT): For any graph, there is either a clique and a colouring of the same size, or there is a odd hole or odd antihole (or both). So: Give a combinatorial polytime algorithm to find what the EP theorem asserts to exist. Contributed by Kathie Cameron and Jack Edmonds \n\nA.4 NP Description of Perfect Graphs \n\nGive an NP description of perfect graphs. Contributed by Jack Edmonds. \n\nA.5 Recognition Algorithm Given the List of Maximal Cliques \n\nIs there a polytime recognition algorithm for perfect graphs where the input is the list of all maximal cliques in the graph? This probelm has been resolved, since a perfect graph can itself now be recognized. Contributed by Bruce Shepherd \n\nA.5.a Berge Graphs with Poly-bounded Number of Max Cliques. Give a poly-nomial time recognition algorithm for Berge graphs with polynomially bounded number of maximal cliques. Contributed by Jeremy Spinrad. \n\nA.6 TDI Matrices \n\n1. Given an m × n 0 − 1 matrix A and an m-dimensional vector b, decide whether the system \n\nAx ≤ b is totally dual integral (TDI), that is, is there an integer dual solution for every objective function for which the dual optimum exists. \n\nThis is the 0 − 1 special case of the well-known problem of TDI system recognition. Let \n\nP:= Ax ≤ b, x ≥ 0. Let A(u) be the matrix whose rows are the normal vectors of facets of \n\nP containing u, each row is integer and the gcd of its entries is 1. There are several known relations between TDI and unimodular systems. As Serkan Ho¸ sten pointed out, 'nondegenerate' TDI matrices are exactly those in which A(u) is an \n\nn × n matrix having determinant 1 for every vertex u.The following problem involves unimodularity in perfectness test: 6\n\nINPUT: m × n 0-1 matrix A and an m-dimensional positive vector b,QUESTION: Is the matrix A(u) for every vertex u of P a square matrix of determinant 1? \n\n2. Can this problem be solved in polynomial time? \n\nFor reducing the test for the perfectness of matrix A to this problem define b with a 'lexicographic perturbation' from the all 1 objective function. Contributed by Andr´ as Seb˝ o\n\nA.7 Fixed Parameter Algorithms \n\nIn the context of fixed parameter algorithms, which was recently introduced by Downey and Fellows, it would be interesting to design an algorithm with running time O(f (k)|V |c)where k is the size of the maximum clique, c is a small constant independent of k and f (k)is a (exponential) function of k. Such an algorithm can work for small k even for large n.Contributed by Mohammad Taghi Hajiaghayi \n\nA.8 Clique Joins \n\nA k-clique-join of G = ( V, E ) is a set of pairs {(A0, B 0), (A1, B 1),..., (Ak, B k)}, where \n\n{A0, B 0} is a partition of V, both A0 and B0 contain at least one ω-clique, and Ai ⊆ A0,\n\nBi ⊆ B0 (i = 1,..., k ) (not necessarily disjoint), moreover (i) If x ∈ Ai and y ∈ Bi, then xy ∈ E\n\n(ii) If K is an ω-clique of G that meets both A0 and B0, then there exists i so that \n\nK ⊆ Ai ∪ Bi.A partitionable graph does not contain a k-clique-join for k < 2( ω − 1), on the other hand odd holes, odd antiholes do all contain 2( ω − 1)-clique-joins. \n\nCould the minimum of k for which a k-clique-join exists be computed (or well-characterized)? For Berge-graphs? Is there a variant of this operation that would allow to compose perfect graphs and keep perfectness? \n\nContributed by Andr´ as Seb˝ o\n\nA.9 Polynomial Size Decomposition Tree \n\nThe problem with using skew-paritions (or star cutsets) for recognition algorithms is that the decomposition tree they induce does not have polynomial size. The following questions have been suggested during the workshop: 1. Suggest different endblocks of the decompostition (other than basic perfect graphs), that can be recognized in polynomial time and yet make the decomposition tree polynomial (Bruce Reed) 2. Normally every skew-partition has 4 decomposition blocks. What if we could prove that it is enough to consider only two blocks for each skew-partition. Would that imply a polynomial size decomposition tree? Possibly introducing new endblocks or using cleaning? (Kristina Vuskovic). 3.Does decomposition of C4-free graphs via star-cutsets induce a polynomial size de-composition tree? Possibly using cleaning? (Kristina Vuskovic). 7\n\nChapter B: Structural Characterization of Perfect Graphs \n\nPossible structural characterization of perfect graphs. Give explicit constructions for subclasses of Berge graphs. Contributed by Paul Seymour \n\nChapter C: Coloring Perfect Graphs \n\nCan one find an efficient algorithms to color a perfect graph? \n\nC.1 Uniquely colorable perfect graphs \n\nUniquely colorable perfect graphs (in which there is a unique partition into ω stable-sets) are closely related to minimal imperfect graphs: according to a result of Padberg (Perfect zero-one matrices, Math Programming, 6, (1974)) if G is minimal imperfect then for all of its vertices v, the graph G − v is uniquely colorable. There is also a combinatorial good characterization theorem for unique colorability of perfect graphs, and a polynomial algorithm for testing the property using the ellipsoid method (IPCO 1, Kannan, Pulleyblank eds, Waterloo Univ. Press, 1990). In other words \n\nUNIQUE COLORABILITY is a tractable property for perfect graphs, closely related to min-imal imperfect graphs. Yet, some simple conjectures related to the SPGT, resist through the years. The following one arises both by specializing more general conjectures occurring in various papers, and does not seem to trivially follow from the SPGT: \n\nIf G is perfect and uniquely colorable, does there exist two ω-cliques which meet in ω −1\n\npoints? \n\nLet us call the two vertices in the symmetric difference of two such cliques forced.\n\nIs it true that every known uniquely colorable perfect graph collapses to an ω-clique by successive identification of forced vertices? \n\nIt can be simply proved that a minimal imperfect graph with three forced vertices in particular positions is an odd hole or an odd antihole. A simpler proof of the following statement would shortcut the proof of the SPGT: \n\nIf G is minimal imperfect, there is a vertex v so that N (v) is uniquely colorable. \n\nContributed by Jean Fonlupt and Andr´ as Seb˝ o\n\nChapter D: Optimization on Perfect Graphs \n\nOptimization on perfect graphs without using the ellipsoid method. \n\nD.1 New Optimization Problems on Perfect Graphs \n\nAre there any new optimization problems (other that coloring and finding the size of the max clique) that are easier to solve for a perfect graph that for a general graph? Can we use any of the existing (future) recognition algorithms in order to do that? Contributed by Mohammad Hajiaghayi \n\nD.1.a A Possible New Problem. Solve in a perfect graph: do two given vertices belong to an induced hole? Contributed by Bruce Reed 8\n\nChapter E: Skew-Partitions \n\nE.1 Extending a Skew -Partition \n\nWhen can a skew partition of an induced subgraph be extended to a skew partition of G? Algorithmically this is answered by the algorithm of de Figueiredo, Klein, Kohayakawa and Reed [2001j:05114], but what about a theorem? Is there some theorem that says \"either the skew partition is extendable, or there is a reason why not (an obstruction)\"? Contributed by Paul Seymour \n\nE.2 Graphs Without Skew-Partitions \n\nIs there a structure theorem for such graphs? Can they all be constructed somehow? Maybe by starting with a small one, and adding little bits so that at each stage there is no skew partition? Contributed by Paul Seymour \n\nE.3 Graphs Without Star Cutsets \n\nIs there a structure theorem for such graphs? Can they all be constructed somehow? Maybe by starting with a small one, and adding little bits so that at each stage there is no star cutset? Contributed by Bruce Reed \n\nConjecture If neither G nor Gc has a star cutset then the disk-structure of G is connected (A disk is a hole or an antihole. Two disks are adjacent in the disk structure if they share at least 2 vertices). Contributed by Ryan Hayward \n\nE.4 Finding Skew-Partitions in Berge Graphs \n\nIs it easier to detect skew partitions in Berge graphs than in general ones? Contributed by Paul Seymour \n\nE.5 Interaction Between Different Skew-Partitions in a Graph \n\nFor 1 ≤ i ≤ n, let ( Ai, B i, C i, D i) be a skew partition of G, where there are no edges between Ai and Bi, and Ci is complete to Di. For each i, choose one of Ai, B i, C i, D i, say \n\nXi, and let X be the union of all these Xi. Call G \\ X a chunk. If there is an odd hole or antihole in G, then it belongs to the chunk (for some choice of the Xi's), so to check Bergeness of G, it is enough to check Bergeness of all the chunks. But even if n is linear in the size of G, the number of chunks can be exponential. Maybe there is a way around this. With decomposition theorems that come up from excluded minors, using separations instead of skew partitions, the same exponential blowup happens, but it can be avoided by using separations that are pairwise noncrossing - then you only get linearly many pieces. Is there an analogous nice way for skew partitions to fit together, so that we only get linearly many (or polynomially many) chunks? Contributed by Paul Seymour 9\n\nE.6 Skew -Partitions of Balanced Size \n\nThe problem with recursive skew decomposition is that at least at first glance, you get exponential behavior. This would not occur if you were always able to find a decomposition in which each of A,B,C,D have at least n/c vertices for some c. Call this a skew partition of balanced size. a) Can you find a skew partition of balanced in polynomial time, if one exists? b) Will always looking for a balanced skew partition if possible lead to a polynomial size decomposition tree for perfect graphs (ie always use the skew partition which maximizes the size of the smallest set) Contributed by Jeremy Spinrad Answer: There exists a graph admitting no skew-partition of balanced size: take a clique and for some edges e1,..., e k of it add a vertices v1,..., v k s.t. each vi has degree 2 and is adjacent to both ends of ei.\n\nE.7 Recognizing Balanced Skew-Partitions \n\nGiven a skew-partition, can one check in polynomial time whether it is balanced. Contributed by Jeremy Spinrad \n\nE.8 Even-Pair Skew-Partition \n\nAn even-pair skew-partition is a partition of the vertex set of a graph G into four sets \n\nA, B, C, D s.t. A is complete to B and C is anti-complete to D, and any two non-adjacent vertices in A or B are an even pair.",
   "clean_statement": null,
   "public_statement": "Problem 4. NP Characterization of Perfect Graphs 5. Recognition Algorithm Given the List of Maximal Cliques a. Berge Graphs with Poly-bounded Number of Max Cliques 6. TDI Matrices 7. Fixed Parameter Algorithms 8. Clique Joins 9. Polynomial Size Decomposition Tree B. Structural Characterization of Perfect Graphs............... 6C. Coloring Perfect Graphs......................... 71. Uniquely colorable perfect graphs D. Optimization on Perfect Graphs..................... 71. New Optimization Problems on Perfect Graphs a. A Possible New Problem E. Skew-Partitions............................ 71. Extending a Skew -Partition 2. Graphs Without Skew-Partitions 3. Graphs Without Star Cutsets 4. Finding Skew-Partitions in Berge Graphs 5. Interaction Between Different Skew-Partitions in a Graph 6. Skew -Partitions of Balanced Size 7. Recognizing Balanced Skew-Partitions 8. Even-Pair Skew-Partition F. Even Pairs in Berge Graphs....................... 91. Coloring Berge Graphs Using Even Pairs 2. Recognizing Even Pairs 3. Quasi-Parity and Strict Quasi-Parity Graphs a. Forbidden Subgraphs for The Class of Strict Quasi-Parity Graphs b. Recognition of Quasi-Parity and Strict Quasi-Parity Graphs 4. Perfectly Contractile Graphs a. Perfectly Contractile Graphs and the Decomposition Method 5. Possible Structure Theorem for Berge Graphs 6. Odd holes and odd walks G. Forbidding Holes and Antiholes.................... 12 1. 2-divisible Graphs 2. Clique Coloring of Perfect Graphs 3. Recognition of Odd-Hole-Free Graphs 4. Even-Hole-Free Graphs 5. Even-hole-free circulants, 6. beta-perfect graphs H. Partitionable Graphs......................... 14 1. Perfect, Partitionable, and Kernel-Solvable Graphs 3\n\n2. Partitionable graphs and odd holes 3. A Property of Partitionable Graphs 4. Small Transversals in Partitionable Graphs I. The Imperfection Ratio........................ 16 J. Integer Programming......................... 18 1. Partitionable Graphs as Cutting Planes for Packing Problems? 2. Feasibility/Membership Problem For the Theta Body K. Balanced Graphs........................... 18 1. Balanced circulants L. P4-structure and Its Relatives..................... 19 4\n\nChapter A: Recognition of Perfect Graphs\n\nCan one decide in polynomial time if a graph is perfect?\n\nA.1 Polynomial Recognition Algorithm Found\n\nA polynomial algorithm to test whether a graph is Berge was found in November 2002. A paper summarizing the work of two groups- Chudnovsky and Seymour and Cornuejols, Liu and Vuscovic- is due to appear in Combinatorica. The algorithm is independent of the proof of the strong perfect graph conjecture.\n\nA.2 Interaction Between Skew-Partitions and 2-joins\n\nOne can think of algorithms for testing for odd holes if you use only 2-joins to decom-pose a graph, or if you use only skew partitions. However, the interaction between skew partition steps and 2-join steps adds another level of difficulty. Can one argue that this can be reduced only to testing for holes as you decompose using skew partitions only and testing for holes using 2-joins only? Contributed by Jeremy Spinrad Similar approach worked in algorithms for recognizing even-hole-free graphs: first the graph is decomposed via vertex cut-sets, and only then via 2-joins. The 2-join decomposition blocks are defined in such a way that no new vertex cutset is introduced. Contributed by Kristina Vuskoic\n\nA.3 The Perfect-Graph Robust Algorithm Problem\n\nAn algorithm, which for any easily recognizable input, A, finds either an easily rec-ognizable B or an easily recognizable C, is sometimes called a \"robust algorithm\" - to be distinguished from a non-robust algorithm, which, for any A without a B, finds a C. Either provides a proof of the \"existentially polytime (EP) theorem\": For any A, there exists a B or a C. In other words: For any A without a B, there is a C. In [92i:68043 ], Jack Edmonds and Kathie Cameron advocated seeking a robust al-gorithm which, for any graph G, finds either a clique and colouring the same size or else finds an easily recognizable combinatorial obstruction to G being perfect. The obstruction might be specified to be a \"alpha-omega partitioned subgraph\", or it might be specified more particularly to be an odd hole or odd antihole. Such an algorithm might be simpler than an algorithm for recognizing whether or not a graph is perfect, in view of precedents, and since what it would do is incomparable with perfect-graph recognition. Such an algorithm could end up giving a clique and colouring the same size in a non-perfect graph. Here are two examples of similar problems which have been solved. Edmonds has given a simple robust algorithm which, for any graph G, either finds an odd cycle > 3 with at most one chord (a defining obstruction to G being Meyniel) or else finds a clique and colouring the same size. This is an improvement on the non-robust algorithms of Hoang and Hertz which, assuming a graph is Meyniel, find a clique and colouring the same size. Edmonds' algorithm is much simpler than the Burlet-Fonlupt decomposition algorithm for recognizing Meyniel graphs, which was motivated by an interest in optimizing in Meyniel graphs, and which is used by Hoang and Hertz. 5\n\nConforti and Cornuejols give a complicated decomposition algorithm for recognizing whether or not a matrix is balanced. At about the same time, to motivate the advocacy of a robust algorithm for either node-colouring a graph or recognizing it to be not perfect, Cameron and Edmonds presented a simple algorithm which, for any 0-1 matrix M, either finds, where x is the largest number of ones in any row, an x-colouring of the columns so that the 1's of any row are in different coloured columns, or else finds \"an odd hole\" in M (the defining obstruction to M being balanced). This introduced the \"EP - robust algorithm\" paradigm which is followed in Edmonds' Meyniel-related algorithm, and is related to the Conforti-Cornuejols-Rao treatment of balanced matrices in the same way that Edmonds' is related to the Burlet-Fonlupt treatment of Meyniel graphs. Following the same paradigm we expect there to be a robust algorithm proving the SPCG, related in the same way to the Chudnovsky-Robertson-Seymour-Thomas decomposition of Berge graphs. In conclusion, we know the following EP Theorem 1: For any graph, there is either a clique and a colouring of the same size, or there is an alpha-omega partitioned subgraph (or both). EP Theorem 2 (SPGT): For any graph, there is either a clique and a colouring of the same size, or there is a odd hole or odd antihole (or both). So: Give a combinatorial polytime algorithm to find what the EP theorem asserts to exist. Contributed by Kathie Cameron and Jack Edmonds\n\nA.4 NP Description of Perfect Graphs\n\nGive an NP description of perfect graphs. Contributed by Jack Edmonds.\n\nA.5 Recognition Algorithm Given the List of Maximal Cliques\n\nIs there a polytime recognition algorithm for perfect graphs where the input is the list of all maximal cliques in the graph? This probelm has been resolved, since a perfect graph can itself now be recognized. Contributed by Bruce Shepherd\n\nA.5.a Berge Graphs with Poly-bounded Number of Max Cliques. Give a poly-nomial time recognition algorithm for Berge graphs with polynomially bounded number of maximal cliques. Contributed by Jeremy Spinrad.\n\nA.6 TDI Matrices\n\n1. Given an m × n 0 − 1 matrix A and an m-dimensional vector b, decide whether the system\n\nAx ≤ b is totally dual integral (TDI), that is, is there an integer dual solution for every objective function for which the dual optimum exists.\n\nThis is the 0 − 1 special case of the well-known problem of TDI system recognition. Let\n\nP:= Ax ≤ b, x ≥ 0. Let A(u) be the matrix whose rows are the normal vectors of facets of\n\nP containing u, each row is integer and the gcd of its entries is 1. There are several known relations between TDI and unimodular systems. As Serkan Ho¸ sten pointed out, 'nondegenerate' TDI matrices are exactly those in which A(u) is an\n\nn × n matrix having determinant 1 for every vertex u.The following problem involves unimodularity in perfectness test: 6\n\nINPUT: m × n 0-1 matrix A and an m-dimensional positive vector b,QUESTION: Is the matrix A(u) for every vertex u of P a square matrix of determinant 1?\n\n2. Can this problem be solved in polynomial time?\n\nFor reducing the test for the perfectness of matrix A to this problem define b with a 'lexicographic perturbation' from the all 1 objective function. Contributed by Andr´ as Seb˝ o\n\nA.7 Fixed Parameter Algorithms\n\nIn the context of fixed parameter algorithms, which was recently introduced by Downey and Fellows, it would be interesting to design an algorithm with running time O(f (k)|V |c)where k is the size of the maximum clique, c is a small constant independent of k and f (k)is a (exponential) function of k. Such an algorithm can work for small k even for large n.Contributed by Mohammad Taghi Hajiaghayi\n\nA.8 Clique Joins\n\nA k-clique-join of G = ( V, E ) is a set of pairs {(A0, B 0), (A1, B 1),..., (Ak, B k)}, where\n\n{A0, B 0} is a partition of V, both A0 and B0 contain at least one ω-clique, and Ai ⊆ A0,\n\nBi ⊆ B0 (i = 1,..., k ) (not necessarily disjoint), moreover (i) If x ∈ Ai and y ∈ Bi, then xy ∈ E\n\n(ii) If K is an ω-clique of G that meets both A0 and B0, then there exists i so that\n\nK ⊆ Ai ∪ Bi.A partitionable graph does not contain a k-clique-join for k < 2( ω − 1), on the other hand odd holes, odd antiholes do all contain 2( ω − 1)-clique-joins.\n\nCould the minimum of k for which a k-clique-join exists be computed (or well-characterized)? For Berge-graphs? Is there a variant of this operation that would allow to compose perfect graphs and keep perfectness?\n\nContributed by Andr´ as Seb˝ o\n\nA.9 Polynomial Size Decomposition Tree\n\nThe problem with using skew-paritions (or star cutsets) for recognition algorithms is that the decomposition tree they induce does not have polynomial size. The following questions have been suggested during the workshop: 1. Suggest different endblocks of the decompostition (other than basic perfect graphs), that can be recognized in polynomial time and yet make the decomposition tree polynomial (Bruce Reed) 2. Normally every skew-partition has 4 decomposition blocks. What if we could prove that it is enough to consider only two blocks for each skew-partition. Would that imply a polynomial size decomposition tree? Possibly introducing new endblocks or using cleaning? (Kristina Vuskovic). 3.Does decomposition of C4-free graphs via star-cutsets induce a polynomial size de-composition tree? Possibly using cleaning? (Kristina Vuskovic). 7\n\nChapter B: Structural Characterization of Perfect Graphs\n\nPossible structural characterization of perfect graphs. Give explicit constructions for subclasses of Berge graphs. Contributed by Paul Seymour\n\nChapter C: Coloring Perfect Graphs\n\nCan one find an efficient algorithms to color a perfect graph?\n\nC.1 Uniquely colorable perfect graphs\n\nUniquely colorable perfect graphs (in which there is a unique partition into ω stable-sets) are closely related to minimal imperfect graphs: according to a result of Padberg (Perfect zero-one matrices, Math Programming, 6, (1974)) if G is minimal imperfect then for all of its vertices v, the graph G − v is uniquely colorable. There is also a combinatorial good characterization theorem for unique colorability of perfect graphs, and a polynomial algorithm for testing the property using the ellipsoid method (IPCO 1, Kannan, Pulleyblank eds, Waterloo Univ. Press, 1990). In other words\n\nUNIQUE COLORABILITY is a tractable property for perfect graphs, closely related to min-imal imperfect graphs. Yet, some simple conjectures related to the SPGT, resist through the years. The following one arises both by specializing more general conjectures occurring in various papers, and does not seem to trivially follow from the SPGT:\n\nIf G is perfect and uniquely colorable, does there exist two ω-cliques which meet in ω −1\n\npoints?\n\nLet us call the two vertices in the symmetric difference of two such cliques forced.\n\nIs it true that every known uniquely colorable perfect graph collapses to an ω-clique by successive identification of forced vertices?\n\nIt can be simply proved that a minimal imperfect graph with three forced vertices in particular positions is an odd hole or an odd antihole. A simpler proof of the following statement would shortcut the proof of the SPGT:\n\nIf G is minimal imperfect, there is a vertex v so that N (v) is uniquely colorable.\n\nContributed by Jean Fonlupt and Andr´ as Seb˝ o\n\nChapter D: Optimization on Perfect Graphs\n\nOptimization on perfect graphs without using the ellipsoid method.\n\nD.1 New Optimization Problems on Perfect Graphs\n\nAre there any new optimization problems (other that coloring and finding the size of the max clique) that are easier to solve for a perfect graph that for a general graph? Can we use any of the existing (future) recognition algorithms in order to do that? Contributed by Mohammad Hajiaghayi\n\nD.1.a A Possible New Problem. Solve in a perfect graph: do two given vertices belong to an induced hole? Contributed by Bruce Reed 8\n\nChapter E: Skew-Partitions\n\nE.1 Extending a Skew -Partition\n\nWhen can a skew partition of an induced subgraph be extended to a skew partition of G? Algorithmically this is answered by the algorithm of de Figueiredo, Klein, Kohayakawa and Reed [2001j:05114], but what about a theorem? Is there some theorem that says \"either the skew partition is extendable, or there is a reason why not (an obstruction)\"? Contributed by Paul Seymour\n\nE.2 Graphs Without Skew-Partitions\n\nIs there a structure theorem for such graphs? Can they all be constructed somehow? Maybe by starting with a small one, and adding little bits so that at each stage there is no skew partition? Contributed by Paul Seymour\n\nE.3 Graphs Without Star Cutsets\n\nIs there a structure theorem for such graphs? Can they all be constructed somehow? Maybe by starting with a small one, and adding little bits so that at each stage there is no star cutset? Contributed by Bruce Reed\n\nConjecture If neither G nor Gc has a star cutset then the disk-structure of G is connected (A disk is a hole or an antihole. Two disks are adjacent in the disk structure if they share at least 2 vertices). Contributed by Ryan Hayward\n\nE.4 Finding Skew-Partitions in Berge Graphs\n\nIs it easier to detect skew partitions in Berge graphs than in general ones? Contributed by Paul Seymour\n\nE.5 Interaction Between Different Skew-Partitions in a Graph\n\nFor 1 ≤ i ≤ n, let ( Ai, B i, C i, D i) be a skew partition of G, where there are no edges between Ai and Bi, and Ci is complete to Di. For each i, choose one of Ai, B i, C i, D i, say\n\nXi, and let X be the union of all these Xi. Call G \\ X a chunk. If there is an odd hole or antihole in G, then it belongs to the chunk (for some choice of the Xi's), so to check Bergeness of G, it is enough to check Bergeness of all the chunks. But even if n is linear in the size of G, the number of chunks can be exponential. Maybe there is a way around this. With decomposition theorems that come up from excluded minors, using separations instead of skew partitions, the same exponential blowup happens, but it can be avoided by using separations that are pairwise noncrossing - then you only get linearly many pieces. Is there an analogous nice way for skew partitions to fit together, so that we only get linearly many (or polynomially many) chunks? Contributed by Paul Seymour 9\n\nE.6 Skew -Partitions of Balanced Size\n\nThe problem with recursive skew decomposition is that at least at first glance, you get exponential behavior. This would not occur if you were always able to find a decomposition in which each of A,B,C,D have at least n/c vertices for some c. Call this a skew partition of balanced size. a) Can you find a skew partition of balanced in polynomial time, if one exists? b) Will always looking for a balanced skew partition if possible lead to a polynomial size decomposition tree for perfect graphs (ie always use the skew partition which maximizes the size of the smallest set) Contributed by Jeremy Spinrad Answer: There exists a graph admitting no skew-partition of balanced size: take a clique and for some edges e1,..., e k of it add a vertices v1,..., v k s.t. each vi has degree 2 and is adjacent to both ends of ei.\n\nE.7 Recognizing Balanced Skew-Partitions\n\nGiven a skew-partition, can one check in polynomial time whether it is balanced. Contributed by Jeremy Spinrad\n\nE.8 Even-Pair Skew-Partition\n\nAn even-pair skew-partition is a partition of the vertex set of a graph G into four sets\n\nA, B, C, D s.t. A is complete to B and C is anti-complete to D, and any two non-adjacent vertices in A or B are an even pair.",
   "evidence": "The canonical input is not one mathematical problem. It is a malformed extraction from the 21-page AIM workshop document *Perfect Graphs* (version dated 24 August 2004). The input begins",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 269,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0271": {
   "statement_status": "exact",
   "original_statement": "Question 1 Is it true that every Berge graph is either basic or has a 2-join or has an even-pair skew-partition?",
   "clean_statement": "Question 1 Is it true that every Berge graph is either basic or has a 2-join or has an even-pair skew-partition?",
   "public_statement": "Question 1 Is it true that every Berge graph is either basic or has a 2-join or has an even-pair skew-partition?",
   "evidence": "The canonical record comes from the AIM workshop *The Perfect Graph Conjecture* (official PDF version dated 24 August 2004). The source defines an **even-pair skew-partition** as a partition of \\(V(G)\\) into four sets \\(A,B,C,D\\) such that \\(A\\) is complete to \\(B\\), \\(C\\) is anticomplete to \\(D\\), and every two nonadjacent vertices lying together in \\(A\\) or together in \\(B\\) form an even pair. As usual for a split of a skew partition, all four sets are nonempty. An even pair is a nonadjacent pair for which every induced path between its vertices has even length.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 270,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0272": {
   "statement_status": "unrecoverable",
   "original_statement": "Question 2 Is even-pair skew-partition a composition? Contributed by Bruce Reed \n\nChapter F: Even Pairs in Berge Graphs \n\nAn even pair is a pair of vertices such that each chordless path between them has even length. Results of Fonlupt and Uhry, Meyniel, and also Bertschi and Reed imply that no minimal imperfect graph contains an even pair. A graph G is called quasi-parity (QP) if, for every induced subgraph H of G on at least two vertices, either H or its complement has an even pair. A graph G is called strict quasi-parity (SQP) if every induced subgraph H of G either H is aclique or has an even pair. In the past 20 years many classical families of perfect graphs were proven to be SQP, which shows the interest of this class. However there are perfect graphs with no even pairs, e.g., all the line-graphs of 3-connected bipartite graphs. None of the following questions or conjectures has been settled in full. Solutions are known only for special subclasses of graphs, e.g., planar graphs, claw-free graphs, bull-free graphs, etc. Contributed by Fr´ ed´ eric Maffray 10 \n\nF.1 Coloring Berge Graphs Using Even Pairs \n\nIt is known (from Fonlupt and Uhry) that contracting an even pair in a perfect graph yields a perfect graph with the same chromaticnumber. This idea can be used as the basis for a conceptually simple coloring algorithm. Contributed by Fr´ ed´ eric Maffray \n\nF.2 Recognizing Even Pairs \n\nCan one decide in polynomial time if a given Berge graph has an even pair? (The general problem, i.e., not restricted to Berge graphs, is known to be co-NP-complete.) Contributed by Fr´ ed´ eric Maffray Can one find even pairs using balanced skew-partitions? (Is there always an even pair in the cutset of a balanced skew-partition?) Contributed by Bruce Ree \n\nF.3 Quasi-Parity and Strict Quasi-Parity Graphs \n\nA graph G is called quasi-parity (QP) if, for every induced subgraph H of G on at least two vertices, either H or its complement has an even pair. A graph G is called strict quasi-parity (SQP) if every induced subgraph H of G either H is a clique or has an even pair. In the past 20 years many classical families of perfect graphs were proven to be SQP, which shows the interest of this class. Contributed by Fr´ ed´ eric Maffray \n\nF.3.a Forbidden Subgraphs for The Class of Strict Quasi-Parity Graphs. Hougardy conjectured that the minimal forbidden induced subgraphs for the class SQP are odd hole, antiholes, and some line-graphs of bipartite graphs (not determined explicitly). Contributed by Fr´ ed´ eric Maffray \n\nF.3.b Recognition of Quasi-Parity and Strict Quasi-Parity Graphs. Can one decide in polynomial time if a given Berge graph is in the class QP, or SQP? Contributed by Fr´ ed´ eric Maffray \n\nF.4 Perfectly Contractile Graphs \n\nBertschi called a graph G even-contractile if there exists a sequence of even-pair con-tractions that turn G into a clique, and he called a graph G perfectly contractile (PC) if every induced subgraph of G is even-contractile. Many classical families of graphs (Meyniel graphs, weakly chordal graphs, perfectly orderable graphs, etc) are perfectly contractile, and for some of them (Meyniel graphs, weakly chordal graphs) the coloring algorithm based on even-pair contractions is the most efficient that is known so far. Everett and Reed conjectured that a graph is PC if and only if it contains no odd hole, no antihole, and no odd prism (two disjoint triangles with three disjoint chordless odd paths between them). Maffray and Trotignon proved a weaker form of this conjecture, also due to Everett and Reed: if a graph contains no odd hole, no antihole, and no prism, and it is not a clique, then the graph admits an even pair whose contraction yields a graph with no odd hole, no 11 \n\nantihole, and no prism. The proof is an algorithm to find such a pair. Here is a link to a preprint 1.The same authors found a polynomial-time algorithm to decide if a graph belongs to that class (the class of graphs with no odd hole, no antihole, and no prism). Here is a link to a preprint 2.Contributed by Fr´ ed´ eric Maffray \n\nF.4.a Perfectly Contractile Graphs and the Decomposition Method. The following conjecture, due to Everett and Reed attempts to characterize perfectly contractile graphs. \n\nPerfectly Contractile Graph Conjecture (PCGC) A graph is perfectly contractile if and only if it does not contain an odd hole, an antihole nor an odd prism. \n\nWe propose to investigate the PCGC and a possible construction of a polynomial-time recognition algorithm for perfectly contractile graphs, through the decomposition method. The decomposition method is based on a decomposition theorem of the following form, for the class of graphs C we want to analyse. \n\nDecomposition Theorem If G ∈ C, then G is either basic or it contains certain types of cutsets. \n\nBasic stands for a certain \"simple\" subclass of C.The idea of a decomposition based recognition algorithm for the class C is as follows. In a connected graph G, a node set (or an edge set or a combination of the two) is a \n\ncutset if its removal disconnects G into two or more connected components. From these components blocks of decomposition are constructed by adding some more nodes and edges. A decomposition is C-preserving if it satisfies the following: G belongs to C if and only if all the blocks of decomposition belong to C. A decomposition based recognition algorithm takes an input graph G and decomposes it using C-preserving decompositions into a polynomial number of basic blocks, which are then checked, in polynomial time, whether they belong to \n\nC.Such a construction of blocks works nicely for clique cutsets. A node set S is a star cutset of a graph G if its removal disconnects G and S contains a node that is adjacent to all the other nodes of S. With the usual construction of blocks for the node cutsets, the star cutset decomposition is not preserving for the class of perfectly contractile graphs. A generalization of star cutsets is obtained as follows. 1-Amalgams are defined and used in for the construction of a recognition algorithm for Meyniel graphs. A graph G has a \n\n1-amalgam if its vertex set can be partitioned into sets V1, V2 and K (where K is possibly empty) in such a way that: \n\n• for i = 1, 2, |Vi| ≥ 2 and Vi contains a nonempty set Ai;\n\n• every node of A1 is adjacent to every node of A2 and these are the only adjacencies between the nodes of V1 and the nodes of V2; and \n\n• if K 6 = ∅, then it induces a clique, and every node of K is adjacent to every node of \n\nA1 ∪ A2.A graph G has a 2-join if its node set can be partitioned into sets V1 and V2 so that for i = 1, 2, Vi contains disjoint nonempty sets Ai and Bi, and the following properties hold: \n\n> 1http://www-leibniz.imag.fr/LesCahiers/2002/Cahier67/ResumCahier67.html\n> 2http://www-leibniz.imag.fr/NEWLEIBNIZ/LesCahiers/Cahier106/ResumCahier106.html 12\n\n• every node of A1 (resp. B1) is adjacent to every node of A2 (resp. B2), and these are the only adjacencies between the nodes of V1 and the nodes of V2;\n\n• for i = 1, 2, let Pi be the set of all chordless paths in G[Vi] with one endnode in Ai,the other endnode in Bi, and no intermediate node in Ai ∪ Bi. For i = 1, 2, Pi 6 = ∅\n\nand G[Vi] is not isomorphic to a path in Pi.Let Cpc denote the class of perfectly contractile graphs. We have the following conjectures.",
   "clean_statement": null,
   "public_statement": "Question 2 Is even-pair skew-partition a composition? Contributed by Bruce Reed\n\nChapter F: Even Pairs in Berge Graphs\n\nAn even pair is a pair of vertices such that each chordless path between them has even length. Results of Fonlupt and Uhry, Meyniel, and also Bertschi and Reed imply that no minimal imperfect graph contains an even pair. A graph G is called quasi-parity (QP) if, for every induced subgraph H of G on at least two vertices, either H or its complement has an even pair. A graph G is called strict quasi-parity (SQP) if every induced subgraph H of G either H is aclique or has an even pair. In the past 20 years many classical families of perfect graphs were proven to be SQP, which shows the interest of this class. However there are perfect graphs with no even pairs, e.g., all the line-graphs of 3-connected bipartite graphs. None of the following questions or conjectures has been settled in full. Solutions are known only for special subclasses of graphs, e.g., planar graphs, claw-free graphs, bull-free graphs, etc. Contributed by Fr´ ed´ eric Maffray 10\n\nF.1 Coloring Berge Graphs Using Even Pairs\n\nIt is known (from Fonlupt and Uhry) that contracting an even pair in a perfect graph yields a perfect graph with the same chromaticnumber. This idea can be used as the basis for a conceptually simple coloring algorithm. Contributed by Fr´ ed´ eric Maffray\n\nF.2 Recognizing Even Pairs\n\nCan one decide in polynomial time if a given Berge graph has an even pair? (The general problem, i.e., not restricted to Berge graphs, is known to be co-NP-complete.) Contributed by Fr´ ed´ eric Maffray Can one find even pairs using balanced skew-partitions? (Is there always an even pair in the cutset of a balanced skew-partition?) Contributed by Bruce Ree\n\nF.3 Quasi-Parity and Strict Quasi-Parity Graphs\n\nA graph G is called quasi-parity (QP) if, for every induced subgraph H of G on at least two vertices, either H or its complement has an even pair. A graph G is called strict quasi-parity (SQP) if every induced subgraph H of G either H is a clique or has an even pair. In the past 20 years many classical families of perfect graphs were proven to be SQP, which shows the interest of this class. Contributed by Fr´ ed´ eric Maffray\n\nF.3.a Forbidden Subgraphs for The Class of Strict Quasi-Parity Graphs. Hougardy conjectured that the minimal forbidden induced subgraphs for the class SQP are odd hole, antiholes, and some line-graphs of bipartite graphs (not determined explicitly). Contributed by Fr´ ed´ eric Maffray\n\nF.3.b Recognition of Quasi-Parity and Strict Quasi-Parity Graphs. Can one decide in polynomial time if a given Berge graph is in the class QP, or SQP? Contributed by Fr´ ed´ eric Maffray\n\nF.4 Perfectly Contractile Graphs\n\nBertschi called a graph G even-contractile if there exists a sequence of even-pair con-tractions that turn G into a clique, and he called a graph G perfectly contractile (PC) if every induced subgraph of G is even-contractile. Many classical families of graphs (Meyniel graphs, weakly chordal graphs, perfectly orderable graphs, etc) are perfectly contractile, and for some of them (Meyniel graphs, weakly chordal graphs) the coloring algorithm based on even-pair contractions is the most efficient that is known so far. Everett and Reed conjectured that a graph is PC if and only if it contains no odd hole, no antihole, and no odd prism (two disjoint triangles with three disjoint chordless odd paths between them). Maffray and Trotignon proved a weaker form of this conjecture, also due to Everett and Reed: if a graph contains no odd hole, no antihole, and no prism, and it is not a clique, then the graph admits an even pair whose contraction yields a graph with no odd hole, no 11\n\nantihole, and no prism. The proof is an algorithm to find such a pair. Here is a link to a preprint 1.The same authors found a polynomial-time algorithm to decide if a graph belongs to that class (the class of graphs with no odd hole, no antihole, and no prism). Here is a link to a preprint 2.Contributed by Fr´ ed´ eric Maffray\n\nF.4.a Perfectly Contractile Graphs and the Decomposition Method. The following conjecture, due to Everett and Reed attempts to characterize perfectly contractile graphs.\n\nPerfectly Contractile Graph Conjecture (PCGC) A graph is perfectly contractile if and only if it does not contain an odd hole, an antihole nor an odd prism.\n\nWe propose to investigate the PCGC and a possible construction of a polynomial-time recognition algorithm for perfectly contractile graphs, through the decomposition method. The decomposition method is based on a decomposition theorem of the following form, for the class of graphs C we want to analyse.\n\nDecomposition Theorem If G ∈ C, then G is either basic or it contains certain types of cutsets.\n\nBasic stands for a certain \"simple\" subclass of C.The idea of a decomposition based recognition algorithm for the class C is as follows. In a connected graph G, a node set (or an edge set or a combination of the two) is a\n\ncutset if its removal disconnects G into two or more connected components. From these components blocks of decomposition are constructed by adding some more nodes and edges. A decomposition is C-preserving if it satisfies the following: G belongs to C if and only if all the blocks of decomposition belong to C. A decomposition based recognition algorithm takes an input graph G and decomposes it using C-preserving decompositions into a polynomial number of basic blocks, which are then checked, in polynomial time, whether they belong to\n\nC.Such a construction of blocks works nicely for clique cutsets. A node set S is a star cutset of a graph G if its removal disconnects G and S contains a node that is adjacent to all the other nodes of S. With the usual construction of blocks for the node cutsets, the star cutset decomposition is not preserving for the class of perfectly contractile graphs. A generalization of star cutsets is obtained as follows. 1-Amalgams are defined and used in for the construction of a recognition algorithm for Meyniel graphs. A graph G has a\n\n1-amalgam if its vertex set can be partitioned into sets V1, V2 and K (where K is possibly empty) in such a way that:\n\n• for i = 1, 2, |Vi| ≥ 2 and Vi contains a nonempty set Ai;\n\n• every node of A1 is adjacent to every node of A2 and these are the only adjacencies between the nodes of V1 and the nodes of V2; and\n\n• if K 6 = ∅, then it induces a clique, and every node of K is adjacent to every node of\n\nA1 ∪ A2.A graph G has a 2-join if its node set can be partitioned into sets V1 and V2 so that for i = 1, 2, Vi contains disjoint nonempty sets Ai and Bi, and the following properties hold:\n\n> 1http://www-leibniz.imag.fr/LesCahiers/2002/Cahier67/ResumCahier67.html\n> 2http://www-leibniz.imag.fr/NEWLEIBNIZ/LesCahiers/Cahier106/ResumCahier106.html 12\n\n• every node of A1 (resp. B1) is adjacent to every node of A2 (resp. B2), and these are the only adjacencies between the nodes of V1 and the nodes of V2;\n\n• for i = 1, 2, let Pi be the set of all chordless paths in G[Vi] with one endnode in Ai,the other endnode in Bi, and no intermediate node in Ai ∪ Bi. For i = 1, 2, Pi 6 = ∅\n\nand G[Vi] is not isomorphic to a path in Pi.Let Cpc denote the class of perfectly contractile graphs. We have the following conjectures.",
   "evidence": "The canonical input is another extraction mega-record, not one mathematical question. In the official 21-page AIM document *Perfect Graphs* (version 24 August 2004), the record begins at the second question of Section E.8 on PDF page index 8:",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 271,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0273": {
   "statement_status": "exact",
   "original_statement": "Conjecture 1-Amalgam decomposition is Cpc -preserving.",
   "clean_statement": "Conjecture 1-Amalgam decomposition is Cpc -preserving.",
   "public_statement": "Conjecture 1-Amalgam decomposition is Cpc -preserving.",
   "evidence": "The canonical record contains only:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 272,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0274": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 2-Join decomposition is Cpc -preserving. \n\nConjecture No minimal non perfectly contractile graph has a star cutset. Note that the PCGC implies all three of these conjectures. Contributed by Claudia Linhares Sales. \n\nF.5 Possible Structure Theorem for Berge Graphs \n\nConjecture For every even-pairfree Berge graph, either it or its complement is the line graph of a bipartite graph, or has a 2-join. A direct proof (if there is one) might give a shorter proof of the SPGC. Contributed by Robin Thomas A general even pair is not a compositiona non-Berge graph may become Berge by contracting an even pair. How much more do we need in order to be able to find a construction for Berge graphs using even pairs? \n\nQuestion Give a sufficient condition such that if x, y is an even pair satisfying the condition then G is Berge if and only if the graph obtained from G by contracting x, y is Berge. Contributed by Paul Seymour \n\nF.6 Odd holes and odd walks \n\nGiven a graph G = ( V, E ) and two vertices a, b can the following problem be solved in polynomial time? \n\nFind a triangle-free odd (a, b )-walk in G, where a walk can also contain repetitions of edges, and triangle-free means that the vertex-set of the walk does not contain any triangle (but can contain an odd hole). \n\nA polynomial algorithm for this problem would specialize to a polynomial algorithm for finding odd holes (and even pairs in odd-hole-free graphs). Bienstock proved that it is NP-hard to find odd holes containing a given a ∈ V. However, a triangle-free odd ( a, a )-walk exists in a 2-connected graph if and only if there exists an odd hole in G (not necessarily containing a). Contributed by Andr´ as Seb˝ o and Nicolas Trotignon 13 \n\nChapter G: Forbidding Holes and Antiholes \n\nG.1 2-divisible Graphs \n\nA 2 division of a graph is a partition of its vertex set into two parts neither of which contains a maximum clique. Hoang and McDiarmid call a graph 2-divisible if all of its induced subgraphs permit 2-division. Every perfect graph is 2-divisible, and an odd hole has no 2-division. Thus 2-divisible graphs are odd-hole-free. Hoang and McDiarmid made the following conjectures: (1) G is 2-divisible iff G is odd-hole-free Contributed by Bruce Reed \n\nG.2 Clique Coloring of Perfect Graphs \n\nIt has been asked by Duffus et al [92e:06009] whether the clique hypergraph of perfect graphs is colorable with a constant number of colors. Several results followed showing that for some classes of perfect graphs this constant is actually 2 or 3: They proved it for comparability and cocomparability graphs, Bacs´ o et al proved the same for some more perfect graphs, and noticed that 'almost all' perfect graphs are 3-clique-colorable (applying Pr¨ omel and Steger's result (Probability and Computation, 1, 1992)); they ask whether this bound holds for all perfect graphs: \n\nCan the vertex-set of any perfect graph be partitioned into three classes, so that no (inclusionwise) maximal clique (of size > 1) is included in any of these? Is the same true already for odd hole free graphs? \n\nIf the clique-hypergraph of a graph and of all of its subgraphs can be colored with k\n\ncolors, that is, there exists a partition of the vertex-set into k parts so that none of them contains an (inclusionwise) maximal clique, then Ho` ang and McDiarmid [2002j:05110] say the graph is strongly k-divisible. This is indeed a sharpening of k-divisibility where 'maximal' is replaced by 'maximum' (cardinality). In these terms the above conjecture states that perfect graphs are strongly 3-divisible. We formulate another problem in this language: \n\nCan strong 2-divisibility be decided in polytime? \n\nA major difficulty with the coloration of the maximal clique hypergraph is that it is NP-hard to decide whether a partition of the vertices is a clique-coloration, and even in very particular classes of perfect graphs. Contributed by Myriam Preissmann and Andr´ as Seb˝ o\n\nG.3 Recognition of Odd-Hole-Free Graphs \n\nFind a polynomial algorithm to recognize odd-hole-free graphs. Contributed by Chinh Hoang \n\nG.4 Even-Hole-Free Graphs \n\nA k-division of a graph G is a partition of its vertex-set into sets V1,..., V k such that no Vi contains a largest clique of G. A graph is k-divisible if each of its induced subgraphs with at least one edge has a k-division. Conforti, Cornu´ ejols, Kapoor and Vuˇ skovi´ c designed a polynomial algorithm to recog-nize even-hole-free graphs. 14",
   "clean_statement": null,
   "public_statement": "Conjecture 2-Join decomposition is Cpc -preserving.\n\nConjecture No minimal non perfectly contractile graph has a star cutset. Note that the PCGC implies all three of these conjectures. Contributed by Claudia Linhares Sales.\n\nF.5 Possible Structure Theorem for Berge Graphs\n\nConjecture For every even-pairfree Berge graph, either it or its complement is the line graph of a bipartite graph, or has a 2-join. A direct proof (if there is one) might give a shorter proof of the SPGC. Contributed by Robin Thomas A general even pair is not a compositiona non-Berge graph may become Berge by contracting an even pair. How much more do we need in order to be able to find a construction for Berge graphs using even pairs?\n\nQuestion Give a sufficient condition such that if x, y is an even pair satisfying the condition then G is Berge if and only if the graph obtained from G by contracting x, y is Berge. Contributed by Paul Seymour\n\nF.6 Odd holes and odd walks\n\nGiven a graph G = ( V, E ) and two vertices a, b can the following problem be solved in polynomial time?\n\nFind a triangle-free odd (a, b )-walk in G, where a walk can also contain repetitions of edges, and triangle-free means that the vertex-set of the walk does not contain any triangle (but can contain an odd hole).\n\nA polynomial algorithm for this problem would specialize to a polynomial algorithm for finding odd holes (and even pairs in odd-hole-free graphs). Bienstock proved that it is NP-hard to find odd holes containing a given a ∈ V. However, a triangle-free odd ( a, a )-walk exists in a 2-connected graph if and only if there exists an odd hole in G (not necessarily containing a). Contributed by Andr´ as Seb˝ o and Nicolas Trotignon 13\n\nChapter G: Forbidding Holes and Antiholes\n\nG.1 2-divisible Graphs\n\nA 2 division of a graph is a partition of its vertex set into two parts neither of which contains a maximum clique. Hoang and McDiarmid call a graph 2-divisible if all of its induced subgraphs permit 2-division. Every perfect graph is 2-divisible, and an odd hole has no 2-division. Thus 2-divisible graphs are odd-hole-free. Hoang and McDiarmid made the following conjectures: (1) G is 2-divisible iff G is odd-hole-free Contributed by Bruce Reed\n\nG.2 Clique Coloring of Perfect Graphs\n\nIt has been asked by Duffus et al [92e:06009] whether the clique hypergraph of perfect graphs is colorable with a constant number of colors. Several results followed showing that for some classes of perfect graphs this constant is actually 2 or 3: They proved it for comparability and cocomparability graphs, Bacs´ o et al proved the same for some more perfect graphs, and noticed that 'almost all' perfect graphs are 3-clique-colorable (applying Pr¨ omel and Steger's result (Probability and Computation, 1, 1992)); they ask whether this bound holds for all perfect graphs:\n\nCan the vertex-set of any perfect graph be partitioned into three classes, so that no (inclusionwise) maximal clique (of size > 1) is included in any of these? Is the same true already for odd hole free graphs?\n\nIf the clique-hypergraph of a graph and of all of its subgraphs can be colored with k\n\ncolors, that is, there exists a partition of the vertex-set into k parts so that none of them contains an (inclusionwise) maximal clique, then Ho` ang and McDiarmid [2002j:05110] say the graph is strongly k-divisible. This is indeed a sharpening of k-divisibility where 'maximal' is replaced by 'maximum' (cardinality). In these terms the above conjecture states that perfect graphs are strongly 3-divisible. We formulate another problem in this language:\n\nCan strong 2-divisibility be decided in polytime?\n\nA major difficulty with the coloration of the maximal clique hypergraph is that it is NP-hard to decide whether a partition of the vertices is a clique-coloration, and even in very particular classes of perfect graphs. Contributed by Myriam Preissmann and Andr´ as Seb˝ o\n\nG.3 Recognition of Odd-Hole-Free Graphs\n\nFind a polynomial algorithm to recognize odd-hole-free graphs. Contributed by Chinh Hoang\n\nG.4 Even-Hole-Free Graphs\n\nA k-division of a graph G is a partition of its vertex-set into sets V1,..., V k such that no Vi contains a largest clique of G. A graph is k-divisible if each of its induced subgraphs with at least one edge has a k-division. Conforti, Cornu´ ejols, Kapoor and Vuˇ skovi´ c designed a polynomial algorithm to recog-nize even-hole-free graphs. 14",
   "evidence": "The canonical JSON record begins with",
   "classification_method": "invalid_literal_with_repaired_reading",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 273,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0275": {
   "statement_status": "exact",
   "original_statement": "Conjecture 2 (Ho` ang). Every even-hole-free graph is 3-divisible.",
   "clean_statement": "Conjecture 2 (Ho` ang). Every even-hole-free graph is 3-divisible.",
   "public_statement": "Conjecture 2 (Ho` ang). Every even-hole-free graph is 3-divisible.",
   "evidence": "The canonical record is in Chapter G.4, “Even-Hole-Free Graphs,” of the AIM workshop list *The Perfect Graph Conjecture* (official PDF version dated 24 August 2004). The nearby source text gives the needed definition:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 274,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0276": {
   "statement_status": "exact",
   "original_statement": "Conjecture 3 (Ho` ang). If G is an even-hole-free graph then χ(G) ≤ 2ω(G) − 1. Hayward and Reed proposed the following.",
   "clean_statement": "Conjecture 3 (Ho` ang). If G is an even-hole-free graph then χ(G) ≤ 2ω(G) − 1. Hayward and Reed proposed the following.",
   "public_statement": "Conjecture 3 (Ho` ang). If G is an even-hole-free graph then χ(G) ≤ 2ω(G) − 1. Hayward and Reed proposed the following.",
   "evidence": "The canonical record is in Chapter G.4, “Even-Hole-Free Graphs,” of the AIM workshop list *The Perfect Graph Conjecture* (official PDF version dated 24 August 2004). The exact record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 275,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0277": {
   "statement_status": "exact",
   "original_statement": "Conjecture 4 (Hayward and Reed). An even-hole-free graph contains a vertex whose neighbourhood can be partitioned into two cliques.",
   "clean_statement": "Conjecture 4 (Hayward and Reed). An even-hole-free graph contains a vertex whose neighbourhood can be partitioned into two cliques.",
   "public_statement": "Conjecture 4 (Hayward and Reed). An even-hole-free graph contains a vertex whose neighbourhood can be partitioned into two cliques.",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 276,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0278": {
   "statement_status": "exact",
   "original_statement": "Conjecture 3 which in turn implies",
   "clean_statement": "Conjecture 3 which in turn implies",
   "public_statement": "Conjecture 3 which in turn implies",
   "evidence": "The exact canonical record is the five-word fragment",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 277,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0279": {
   "statement_status": "exact",
   "original_statement": "Conjecture 2. Let G be an even-hole-free graph.",
   "clean_statement": "Conjecture 2. Let G be an even-hole-free graph.",
   "public_statement": "Conjecture 2. Let G be an even-hole-free graph.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 278,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0280": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 4 implies that each induced subgraph H of G has a vertex of degree at most 2 ω(H) − 2, and therefore χ(G) ≤ 2ω(G) − 1. Since any graph F is \n\n> χ(F)\n> ω(F)−1\n\n-divisible, G is 3-divisible. Contributed by Chinh Hoang. \n\nG.5 Even-hole-free circulants, \n\nGiven interer k ≥ 1 and m ≥ 0, let us introduce a graph G(k, m ) = ( V, E ) with circular symmetry as follows: V = Zn = {1,..., n }, where n = k(2 m + 1), and ( i, j ) ∈ E iff \n\ni − j + t (2 m + 1) = 0 or + 1 or − 1 ( mod n )for some integer t. (For convenience, the loops i = j are included.) E.g. if k = 5, m = 2 then n = 25 and ( i, j ) ∈ E iff i − j(mod 25) ∈ { 4, 5, 6; 9, 10, 11; 14, 15, 16; 19, 20, 21; 24, 0, 1}.\n\nIt is not difficult to check that G(k, m ) has no even holes, (in fact, it can only have holes of length 2 m + 1); furthermore, \n\nω(G(k, m )) = 2 k, 2k + dk/m e ≤ χ(G(k, m )) ≤ 2k + dk/m e + 1, and G(k, m ) satisfies Conjectures 2,3,4 from the section \"Even-Hole-Free Graphs\". \n\nConjecture. Every non-empty even-hole-free circulant is isomorphic to a G(k, m ). Contributed by Diogo Andrade, Endre Boros, and Vladimir Gurvich \n\nG.6 beta-perfect graphs \n\nDefinition β(G) = max G′⊆G(mindeg (G′) + 1) (the maximum is taken over all induced subgraphs). Note that β(G) ≥ χ(G). A graph G is called β-perfect if β(G′) = χ(G′) for all induced subgraphs G′ of G.\n\nQuestion Characterize β-perfect graphs. Even holes and graphs obtained from odd holes by replacing every vertex by two adjacent vertices preserving the adjacencies in the hole (so every edge is replaced by a K4)are known not to be β-perfect. So a β-perfect graph has no induced subgraph of those types. Contributed by Bruce Reed \n\nChapter H: Partitionable Graphs \n\nH.1 Perfect, Partitionable, and Kernel-Solvable Graphs \n\nGiven a graph G = (V,E), assign to every its edge e = (u,v) either the directed arc [u,v), or [v,u), or both. The obtained directed multi-graph D = (V,A) is called an orientation of G. 15 \n\nA vertex-subset K of V is called a KERNEL if K is (i) independent and (ii) absorbant, that is for each u from V K there is an arc [u,v) in A such that v in K. Orientation D is called clique acyclic if every clique of G has a kernel in D. Orientation \n\nD is called kernel-less if it has no kernel. Graph G is called kernel-solvable if every its clique-acyclic orientation has a kernel. Berge and Duchet (1983) conjectured that (BD1) Perfect graphs are kernel-solvable, and (BD2) Kernel-solvable graphs are perfect. BD1 was proved by Boros and Gurvich (1996) and by Holzman and Aharoni (1998), BD2 follows from the SPGC but no independent proof is known. An orientation D of a PARTITIONABLE graph G is called UNIFORM if D is (0) kernel-less and clique acyclic; (a) for each maximum stable set S there exists a unique unabsorbed vertex v(S); (b) v(S) belongs to the vis-a-vis clique C(S) of S; (c) for each vertex v there exists a unique maximal stable set S(v) which does not absorb v. Sebo (1998) proved that every kernel-less and clique-acyclic orientation of a minimal imperfect graph is uniform. Conjecture. Each partitionable graph has a uniform orientation. This, if true, implies BD2 Contributed by Boros and Gurvich \n\nH.2 Partitionable graphs and odd holes \n\nLet G = ( V, E ) be a graph, and α, ω arbitrary natural numbers. Assume that a, v, b ∈\n\nV, av ∈ E, vb / ∈ E are such that G − a, G − v have a partition of size α into ω-cliques, and \n\nG − v, G − b have a partition of size ω into α-stable sets. It is easy to show then that G is not perfect. \n\nGiven the four partitions, find an odd hole or an odd antihole. \n\nThis would imply SPGC. This contains the following: \n\nGiven a partitionable graph (with all the partitions), find an odd hole or an odd antihole. Does the fact that the partitions are given make the task easier? \n\nContributed by Andr´ as Seb˝ o\n\nH.3 A Property of Partitionable Graphs \n\nWe say that a graph satisfies the \"no-week-pair\" property if each pair of vertices of a graph is either in a maximum clique or in a maximum stable set Conjecture: If a partitionable graph satisfies the \"no-week-pair\" property then the graph is an odd hole or an odd anti-hole. The following graph is a counter-example to this conjecture: take a 17-gon and and add all 34- and 5-chords. (This 17-vertex graph is the only known partitionable graph without a small transver-sal.) It's still interesting if there are other such graphs (i.e. partitionable graph satisfying \"no-week-pair\" property). If there are then it would be nice to characterize them. 16 \n\nContributed by Ara Markosian \n\nH.4 Small Transversals in Partitionable Graphs \n\nFollowing Bland, Huang, and Trotter [80g:05034];[86e:05075] a graph is called parti-tionable if, for some r and s, it has rs + 1 vertices and, no matter which vertex is removed, the set of the remaining rs vertices can be partitioned into r pairwise disjoint cliques of size \n\ns and also into s pairwise disjoint stable sets of size r. Odd holes and odd antiholes are partitionable; many additional partitionable graphs have been constructed by V. Chvtal, R. L. Graham, A. F. Perold, and S. H. Whitesides [81b:05044]. A small transversal in a graph G is a set of α(G) + ω(G) − 1 vertices which meets all cliques of size ω(G) and all stable sets of size α(G). The following problem is an easier variation on a conjecture contributed to the 1993 workshop on perfect graphs 3 by Gurvich and Temkin and on two conjectures proposed by Bacso, Boros, Gurvich, Maffray, and Preissmann [2000h:05116]. \n\nConjecture. Every partitionable graph G with α(G) > 2 and ω(G) > 2 has a small transversal or else contains a hole of length five. One of the milestones in the development of our understanding of perfect graphs was the theorem of Lovasz [46 #8885], asserting that every minimal imperfect graph G has precisely α(G)ω(G) +1 vertices. This theorem implies that every minimal imperfect graph is partitionable and that - as pointed out by Chvatal [86h:05091] - no minimal imperfect graph contains a small transversal. It follows that a proof of the conjecture would provide another proof of the Strong Perfect Graph Theorem. A partitionable graph without a small transversal has been constructed by by Chvatal, Graham, Perold, and Whitesides (op.cit.). Its vertices are 0, 1,..., 16; vertices i and j are adjacent if and only if |i − j| mod 17 is one of 1, 3, 4, 5, 12, 13, 14, 16. Ara Markosian claims here 4 that this is the only known partitionable graph without a small transversal. One of the many holes of length five in this graph is 1 − 4 − 8 − 12 − 15 − 1Additional information on related results and problems can be found here 5\n\nContributed by Vasek Chvatal \n\nChapter I: The Imperfection Ratio \n\nA demand vector for a graph G with node set V is a non-negative vector of integers indexed by nodes of G. Given a graph G and a demand vector x = ( xv: v ∈ V (G)) a coloring of the pair ( G.x ) is an assignment of a set of xv colors to each node v of G such that two adjacet nodes receive disjoint sets of colors. Coloring the pair ( G, x ) corresponds exactly to usual proper coloring of the replicated graph Gx. Let G be a graph.Define the \n\nimperfection ratio of G by setting \n\nimp (G) = max x{χf (Gx)\n\nω(Gx) }\n\n> 3http://dimacs.rutgers.edu/\"\n> 4http://www.aimath.org/WWN/perfectgraph/articles/html/46a/\n> 5http://www.cs.rutgers.edu/ ∼chvatal/perfect/problems.html#partitionable 17\n\nwhere the maximum is over all non-zero integral demand vectors x and χF (Gx) and \n\nω(Gx) are the fractional chromatic number and the clique number of the replicated graph Gx\n\nrespectively. (The ratios on the right-hand side above do indeed attain a maximum value). Observe that imp (G) ≥ 1. The next result establishes the connection of the imperfection ratio with perfection. \n\nProposition For any graph G, imp (G) = 1 iff G is perfect. One of the motivations for studying graph imperfection is its connection to frequency assignment. With this motivation in mind one is particularly interested in bounding the imperfection ratio for graphs of relevant graph classes. One relevant graph class are for example unit disk graph, that is graphs the node set of which can be represented by unit size disks such that two nodes are adjacent if and only if the corresponding disks intersect. It is known that imp (G) ≤ 2.155 for any unit disk graph G, and that there exists a unit disk graph G with imp (G) arbitrarily close to 3/2. \n\nConjecture: For any unit disk graph G, imp (G) ≤ 3/2. A subclass of unit disk graphs are the induced subgraphs of the triangular lattice. These graphs are of importance for channel assignment, since a pattern of omni-directional transmitters in two dimensions laid out like nodes of the triangular lattice in the plane give good coverage. Let us now consider an induced subgraph G of the triangular lattice T. Such a graph has a natural 3-coloring. It is possible to have ω(Gx) = 3 and χ(Gx)=4 for such a graph G.There is a polynomial-time coloring algorithm (by McDiarmid and Reed) which shows that for such a graph G we always have \n\nχ(Gx) ≤ 4ω(Gx) + 1 \n\n3Thus imp (G) ≤ 4 \n\n> 3\n\nfor any finite induced subgraph G of the triangular lattice T. The 9-cycle C9 is an induced subgraph of the triangular lattice T. For any integer k the graph obtained from C9 by replicating each of its nodes k times has clique number 2 k and chromatic number d 9k \n\n> 4\n\ne Is the ratio 9 \n\n> 8\n\nof chromatic number to clique number asymptotically the worst (greatest) possible with large demands? This questions may be rephrased in terms of imp (G)as follows: \n\nConjecture For any induced subgraph G of the triangular lattice T, we have imp (G) ≤\n\n> 9\n> 8\n\nThis would imply the following weaker and perhaps more tractable conjecture: \n\nConjecture If G is a trianlgle-free induced subgraph of the triangular lattice then \n\n|V (G)| ≤ 9 \n\n> 4\n\nα(G) (where α(G) is the size of the maximum stable set in G), and indeed \n\nχf (G) ≤ 9\n\n> 4\n\nTo get a feeling for the behavior of the imperfection ratio, we mention the following elementary decomposition result: if G is composed of two parts G1 and G2 that are either disjoint or overlap in a clique, then \n\nimp (G) = max {imp (G1), imp (G2)}\n\nThe following is another property of imperfection which is desirable for any graph invariant related to perfection: 18 \n\nProposition For any graph G imp (G) = imp (Gc) where Gc denotes the completemt of G.Another graph class of interest are planar graphs. It follows from the 4-colour theorem that imp (G) ≤ 2 for any planar graph G. It is known that one can improve a little on this but we conjecture that the true value is 3 /2.\n\nConjecture: For any planar graph G, imp (G) ≤ 3/2. Another area of interest are complexity issues concerning imperfection. It is known that it is NP hard to determine the imperfection ratio. One open question is whether for a fixed k one can determine in polynomial time whether a graph G satisfies imp (G) ≤ k. For the special case of k = 1, this is the recognition problem for perfect graphs. Another open question is how hard is it to approximate the imperfection ratio of a graph. Initial contribution by Bruce Reed, extended by Stefanie Gerke \n\nChapter J: Integer Programming \n\nJ.1 Partitionable Graphs as Cutting Planes for Packing Problems? \n\nAs is well known, the strong perfect graph conjecture has been of interest to the integer programming community as well as the combinatorics community. Now that the SPGC has been established I wanted to mention another problem which may shed light on cutting plane approaches to packing problems. Sewell (and later Bram Verweij and Aardel) gave successful cutting plane codes for solving maximum stable set problems in sparse graphs by adding odd hole inequalities as they are violated by fractional solutions. Moura studied this approach for some problems arising in design theory, but met with much less success since the graph instances were much more dense (and hence odd hole inequalities were unlikely to be violated). Can we extend the class of odd cycle inequalities to the class of partitionable graph inequalities \n\n∑\n\n> v∈I\n\nxv ≤ α(I)for each partitionable subgraph. I.e., can we develop algorithms to solve the separa-tion problem for this class of inequalities. One positive result is that partitionable graphs themselves can be recognized in polynomial time (another problem is to find a combinatorial algorithm to recognize partitionable graphs). Contributed by Bruce Shepherd. \n\nJ.2 Feasibility/Membership Problem For the Theta Body \n\nFind a polynomial time algorithm to solve the (exact) feasibility/membership problem for the theta body. Contributed by Bruce Shepherd \n\nChapter K: Balanced Graphs \n\nDefinition A graph is balanced if every induced cycle has length 0( mod 4). Clearly balanced graphs are bipartite. 19 \n\nA balanced graph is basic if all its vertices on one side of the bipartition have degree at most 2 or G contains a hole H such that the vertices of G \\ H induce a complete bipartite graph. Here are two conjectures concerning balanced graphs.",
   "clean_statement": null,
   "public_statement": "Conjecture 4 implies that each induced subgraph H of G has a vertex of degree at most 2 ω(H) − 2, and therefore χ(G) ≤ 2ω(G) − 1. Since any graph F is\n\n> χ(F)\n> ω(F)−1\n\n-divisible, G is 3-divisible. Contributed by Chinh Hoang.\n\nG.5 Even-hole-free circulants,\n\nGiven interer k ≥ 1 and m ≥ 0, let us introduce a graph G(k, m ) = ( V, E ) with circular symmetry as follows: V = Zn = {1,..., n }, where n = k(2 m + 1), and ( i, j ) ∈ E iff\n\ni − j + t (2 m + 1) = 0 or + 1 or − 1 ( mod n )for some integer t. (For convenience, the loops i = j are included.) E.g. if k = 5, m = 2 then n = 25 and ( i, j ) ∈ E iff i − j(mod 25) ∈ { 4, 5, 6; 9, 10, 11; 14, 15, 16; 19, 20, 21; 24, 0, 1}.\n\nIt is not difficult to check that G(k, m ) has no even holes, (in fact, it can only have holes of length 2 m + 1); furthermore,\n\nω(G(k, m )) = 2 k, 2k + dk/m e ≤ χ(G(k, m )) ≤ 2k + dk/m e + 1, and G(k, m ) satisfies Conjectures 2,3,4 from the section \"Even-Hole-Free Graphs\".\n\nConjecture. Every non-empty even-hole-free circulant is isomorphic to a G(k, m ). Contributed by Diogo Andrade, Endre Boros, and Vladimir Gurvich\n\nG.6 beta-perfect graphs\n\nDefinition β(G) = max G′⊆G(mindeg (G′) + 1) (the maximum is taken over all induced subgraphs). Note that β(G) ≥ χ(G). A graph G is called β-perfect if β(G′) = χ(G′) for all induced subgraphs G′ of G.\n\nQuestion Characterize β-perfect graphs. Even holes and graphs obtained from odd holes by replacing every vertex by two adjacent vertices preserving the adjacencies in the hole (so every edge is replaced by a K4)are known not to be β-perfect. So a β-perfect graph has no induced subgraph of those types. Contributed by Bruce Reed\n\nChapter H: Partitionable Graphs\n\nH.1 Perfect, Partitionable, and Kernel-Solvable Graphs\n\nGiven a graph G = (V,E), assign to every its edge e = (u,v) either the directed arc [u,v), or [v,u), or both. The obtained directed multi-graph D = (V,A) is called an orientation of G. 15\n\nA vertex-subset K of V is called a KERNEL if K is (i) independent and (ii) absorbant, that is for each u from V K there is an arc [u,v) in A such that v in K. Orientation D is called clique acyclic if every clique of G has a kernel in D. Orientation\n\nD is called kernel-less if it has no kernel. Graph G is called kernel-solvable if every its clique-acyclic orientation has a kernel. Berge and Duchet (1983) conjectured that (BD1) Perfect graphs are kernel-solvable, and (BD2) Kernel-solvable graphs are perfect. BD1 was proved by Boros and Gurvich (1996) and by Holzman and Aharoni (1998), BD2 follows from the SPGC but no independent proof is known. An orientation D of a PARTITIONABLE graph G is called UNIFORM if D is (0) kernel-less and clique acyclic; (a) for each maximum stable set S there exists a unique unabsorbed vertex v(S); (b) v(S) belongs to the vis-a-vis clique C(S) of S; (c) for each vertex v there exists a unique maximal stable set S(v) which does not absorb v. Sebo (1998) proved that every kernel-less and clique-acyclic orientation of a minimal imperfect graph is uniform. Conjecture. Each partitionable graph has a uniform orientation. This, if true, implies BD2 Contributed by Boros and Gurvich\n\nH.2 Partitionable graphs and odd holes\n\nLet G = ( V, E ) be a graph, and α, ω arbitrary natural numbers. Assume that a, v, b ∈\n\nV, av ∈ E, vb / ∈ E are such that G − a, G − v have a partition of size α into ω-cliques, and\n\nG − v, G − b have a partition of size ω into α-stable sets. It is easy to show then that G is not perfect.\n\nGiven the four partitions, find an odd hole or an odd antihole.\n\nThis would imply SPGC. This contains the following:\n\nGiven a partitionable graph (with all the partitions), find an odd hole or an odd antihole. Does the fact that the partitions are given make the task easier?\n\nContributed by Andr´ as Seb˝ o\n\nH.3 A Property of Partitionable Graphs\n\nWe say that a graph satisfies the \"no-week-pair\" property if each pair of vertices of a graph is either in a maximum clique or in a maximum stable set Conjecture: If a partitionable graph satisfies the \"no-week-pair\" property then the graph is an odd hole or an odd anti-hole. The following graph is a counter-example to this conjecture: take a 17-gon and and add all 34- and 5-chords. (This 17-vertex graph is the only known partitionable graph without a small transver-sal.) It's still interesting if there are other such graphs (i.e. partitionable graph satisfying \"no-week-pair\" property). If there are then it would be nice to characterize them. 16\n\nContributed by Ara Markosian\n\nH.4 Small Transversals in Partitionable Graphs\n\nFollowing Bland, Huang, and Trotter [80g:05034];[86e:05075] a graph is called parti-tionable if, for some r and s, it has rs + 1 vertices and, no matter which vertex is removed, the set of the remaining rs vertices can be partitioned into r pairwise disjoint cliques of size\n\ns and also into s pairwise disjoint stable sets of size r. Odd holes and odd antiholes are partitionable; many additional partitionable graphs have been constructed by V. Chvtal, R. L. Graham, A. F. Perold, and S. H. Whitesides [81b:05044]. A small transversal in a graph G is a set of α(G) + ω(G) − 1 vertices which meets all cliques of size ω(G) and all stable sets of size α(G). The following problem is an easier variation on a conjecture contributed to the 1993 workshop on perfect graphs 3 by Gurvich and Temkin and on two conjectures proposed by Bacso, Boros, Gurvich, Maffray, and Preissmann [2000h:05116].\n\nConjecture. Every partitionable graph G with α(G) > 2 and ω(G) > 2 has a small transversal or else contains a hole of length five. One of the milestones in the development of our understanding of perfect graphs was the theorem of Lovasz [46 #8885], asserting that every minimal imperfect graph G has precisely α(G)ω(G) +1 vertices. This theorem implies that every minimal imperfect graph is partitionable and that - as pointed out by Chvatal [86h:05091] - no minimal imperfect graph contains a small transversal. It follows that a proof of the conjecture would provide another proof of the Strong Perfect Graph Theorem. A partitionable graph without a small transversal has been constructed by by Chvatal, Graham, Perold, and Whitesides (op.cit.). Its vertices are 0, 1,..., 16; vertices i and j are adjacent if and only if |i − j| mod 17 is one of 1, 3, 4, 5, 12, 13, 14, 16. Ara Markosian claims here 4 that this is the only known partitionable graph without a small transversal. One of the many holes of length five in this graph is 1 − 4 − 8 − 12 − 15 − 1Additional information on related results and problems can be found here 5\n\nContributed by Vasek Chvatal\n\nChapter I: The Imperfection Ratio\n\nA demand vector for a graph G with node set V is a non-negative vector of integers indexed by nodes of G. Given a graph G and a demand vector x = ( xv: v ∈ V (G)) a coloring of the pair ( G.x ) is an assignment of a set of xv colors to each node v of G such that two adjacet nodes receive disjoint sets of colors. Coloring the pair ( G, x ) corresponds exactly to usual proper coloring of the replicated graph Gx. Let G be a graph.Define the\n\nimperfection ratio of G by setting\n\nimp (G) = max x{χf (Gx)\n\nω(Gx) }\n\n> 3http://dimacs.rutgers.edu/\"\n> 4http://www.aimath.org/WWN/perfectgraph/articles/html/46a/\n> 5http://www.cs.rutgers.edu/ ∼chvatal/perfect/problems.html#partitionable 17\n\nwhere the maximum is over all non-zero integral demand vectors x and χF (Gx) and\n\nω(Gx) are the fractional chromatic number and the clique number of the replicated graph Gx\n\nrespectively. (The ratios on the right-hand side above do indeed attain a maximum value). Observe that imp (G) ≥ 1. The next result establishes the connection of the imperfection ratio with perfection.\n\nProposition For any graph G, imp (G) = 1 iff G is perfect. One of the motivations for studying graph imperfection is its connection to frequency assignment. With this motivation in mind one is particularly interested in bounding the imperfection ratio for graphs of relevant graph classes. One relevant graph class are for example unit disk graph, that is graphs the node set of which can be represented by unit size disks such that two nodes are adjacent if and only if the corresponding disks intersect. It is known that imp (G) ≤ 2.155 for any unit disk graph G, and that there exists a unit disk graph G with imp (G) arbitrarily close to 3/2.\n\nConjecture: For any unit disk graph G, imp (G) ≤ 3/2. A subclass of unit disk graphs are the induced subgraphs of the triangular lattice. These graphs are of importance for channel assignment, since a pattern of omni-directional transmitters in two dimensions laid out like nodes of the triangular lattice in the plane give good coverage. Let us now consider an induced subgraph G of the triangular lattice T. Such a graph has a natural 3-coloring. It is possible to have ω(Gx) = 3 and χ(Gx)=4 for such a graph G.There is a polynomial-time coloring algorithm (by McDiarmid and Reed) which shows that for such a graph G we always have\n\nχ(Gx) ≤ 4ω(Gx) + 1\n\n3Thus imp (G) ≤ 4\n\n> 3\n\nfor any finite induced subgraph G of the triangular lattice T. The 9-cycle C9 is an induced subgraph of the triangular lattice T. For any integer k the graph obtained from C9 by replicating each of its nodes k times has clique number 2 k and chromatic number d 9k\n\n> 4\n\ne Is the ratio 9\n\n> 8\n\nof chromatic number to clique number asymptotically the worst (greatest) possible with large demands? This questions may be rephrased in terms of imp (G)as follows:\n\nConjecture For any induced subgraph G of the triangular lattice T, we have imp (G) ≤\n\n> 9\n> 8\n\nThis would imply the following weaker and perhaps more tractable conjecture:\n\nConjecture If G is a trianlgle-free induced subgraph of the triangular lattice then\n\n|V (G)| ≤ 9\n\n> 4\n\nα(G) (where α(G) is the size of the maximum stable set in G), and indeed\n\nχf (G) ≤ 9\n\n> 4\n\nTo get a feeling for the behavior of the imperfection ratio, we mention the following elementary decomposition result: if G is composed of two parts G1 and G2 that are either disjoint or overlap in a clique, then\n\nimp (G) = max {imp (G1), imp (G2)}\n\nThe following is another property of imperfection which is desirable for any graph invariant related to perfection: 18\n\nProposition For any graph G imp (G) = imp (Gc) where Gc denotes the completemt of G.Another graph class of interest are planar graphs. It follows from the 4-colour theorem that imp (G) ≤ 2 for any planar graph G. It is known that one can improve a little on this but we conjecture that the true value is 3 /2.\n\nConjecture: For any planar graph G, imp (G) ≤ 3/2. Another area of interest are complexity issues concerning imperfection. It is known that it is NP hard to determine the imperfection ratio. One open question is whether for a fixed k one can determine in polynomial time whether a graph G satisfies imp (G) ≤ k. For the special case of k = 1, this is the recognition problem for perfect graphs. Another open question is how hard is it to approximate the imperfection ratio of a graph. Initial contribution by Bruce Reed, extended by Stefanie Gerke\n\nChapter J: Integer Programming\n\nJ.1 Partitionable Graphs as Cutting Planes for Packing Problems?\n\nAs is well known, the strong perfect graph conjecture has been of interest to the integer programming community as well as the combinatorics community. Now that the SPGC has been established I wanted to mention another problem which may shed light on cutting plane approaches to packing problems. Sewell (and later Bram Verweij and Aardel) gave successful cutting plane codes for solving maximum stable set problems in sparse graphs by adding odd hole inequalities as they are violated by fractional solutions. Moura studied this approach for some problems arising in design theory, but met with much less success since the graph instances were much more dense (and hence odd hole inequalities were unlikely to be violated). Can we extend the class of odd cycle inequalities to the class of partitionable graph inequalities\n\n∑\n\n> v∈I\n\nxv ≤ α(I)for each partitionable subgraph. I.e., can we develop algorithms to solve the separa-tion problem for this class of inequalities. One positive result is that partitionable graphs themselves can be recognized in polynomial time (another problem is to find a combinatorial algorithm to recognize partitionable graphs). Contributed by Bruce Shepherd.\n\nJ.2 Feasibility/Membership Problem For the Theta Body\n\nFind a polynomial time algorithm to solve the (exact) feasibility/membership problem for the theta body. Contributed by Bruce Shepherd\n\nChapter K: Balanced Graphs\n\nDefinition A graph is balanced if every induced cycle has length 0( mod 4). Clearly balanced graphs are bipartite. 19\n\nA balanced graph is basic if all its vertices on one side of the bipartition have degree at most 2 or G contains a hole H such that the vertices of G \\ H induce a complete bipartite graph. Here are two conjectures concerning balanced graphs.",
   "evidence": "The canonical **problem** field is not one mathematical record. It begins in the middle of the proof of the implications among Conjectures 2, 3, and 4 in Section G.4, contains all of Section G.5, and then absorbs G.6, Chapters H, I, and J, and the beginning of Chapter K. The exact 13,034-character field is preserved in **input.json**. Thus the canonical record, as a record, has status `invalid_statement`: it has no single set of hypotheses or conclusion.",
   "classification_method": "invalid_literal_with_repaired_reading",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 279,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0281": {
   "statement_status": "exact",
   "original_statement": "Conjecture 1 (Conforti, Cornuej´ ols,Rao) Every balanced graph is either basic or has a 2-join or a skew-partition.",
   "clean_statement": "Conjecture 1 (Conforti, Cornuej´ ols,Rao) Every balanced graph is either basic or has a 2-join or a skew-partition.",
   "public_statement": "Conjecture 1 (Conforti, Cornuej´ ols,Rao) Every balanced graph is either basic or has a 2-join or a skew-partition.",
   "evidence": "The canonical record states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 280,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0282": {
   "statement_status": "exact",
   "original_statement": "Conjecture 2 (Conforti, Rao) In every balanced graph, there exists an edge that can be deleted, so that the resulting graph remains balanced. Contributed by Gerard Cornuej` ols \n\nK.1 Balanced circulants \n\nGiven interer k ≥ 1 and m ≥ 1, let us introduce a graph G(k, m ) = ( V, E ) with circular symmetry as follows: V = Zn = {0, 1,..., n − 1}, where n = 4 km, and ( i, j ) ∈ E iff \n\ni − j + 4 mt = +1 or − 1 ( mod n )for some integer t. E.g. if k = 5, m = 2 then n = 40 and ( i, j ) ∈ E iff i − j(mod 40) ∈{7, 9; 15, 17; 23, 25; 31, 33; 39, 1}.\n\nIt is not difficult to check that G(k, m ) is balanced, i.e. it has no odd cycles, nor (4 i + 2)-holes. In fact, it may only contain holes of length 4 and 4 m. Moreover, every (4 i + 2)-cycle has at least two chords. Hence, G(k, m ) − e is still balanced for any edge e, in agreement with",
   "clean_statement": "Conjecture 2 (Conforti, Rao) In every balanced graph, there exists an edge that can be deleted, so that the resulting graph remains balanced. Contributed by Gerard Cornuej` ols\n\nK.1 Balanced circulants\n\nGiven interer k ≥ 1 and m ≥ 1, let us introduce a graph G(k, m ) = ( V, E ) with circular symmetry as follows: V = Zn = {0, 1,..., n − 1}, where n = 4 km, and ( i, j ) ∈ E iff\n\ni − j + 4 mt = +1 or − 1 ( mod n )for some integer t. E.g. if k = 5, m = 2 then n = 40 and ( i, j ) ∈ E iff i − j(mod 40) ∈{7, 9; 15, 17; 23, 25; 31, 33; 39, 1}.\n\nIt is not difficult to check that G(k, m ) is balanced, i.e. it has no odd cycles, nor (4 i + 2)-holes. In fact, it may only contain holes of length 4 and 4 m. Moreover, every (4 i + 2)-cycle has at least two chords. Hence, G(k, m ) − e is still balanced for any edge e, in agreement with",
   "public_statement": "Conjecture 2 (Conforti, Rao) In every balanced graph, there exists an edge that can be deleted, so that the resulting graph remains balanced. Contributed by Gerard Cornuej` ols\n\nK.1 Balanced circulants\n\nGiven interer k ≥ 1 and m ≥ 1, let us introduce a graph G(k, m ) = ( V, E ) with circular symmetry as follows: V = Zn = {0, 1,..., n − 1}, where n = 4 km, and ( i, j ) ∈ E iff\n\ni − j + 4 mt = +1 or − 1 ( mod n )for some integer t. E.g. if k = 5, m = 2 then n = 40 and ( i, j ) ∈ E iff i − j(mod 40) ∈{7, 9; 15, 17; 23, 25; 31, 33; 39, 1}.\n\nIt is not difficult to check that G(k, m ) is balanced, i.e. it has no odd cycles, nor (4 i + 2)-holes. In fact, it may only contain holes of length 4 and 4 m. Moreover, every (4 i + 2)-cycle has at least two chords. Hence, G(k, m ) − e is still balanced for any edge e, in agreement with",
   "evidence": "The exact leading conjecture in the canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 281,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0283": {
   "statement_status": "exact",
   "original_statement": "Conjecture 2 from the section \"Balanced Graphs\".",
   "clean_statement": "Conjecture 2 from the section \"Balanced Graphs\".",
   "public_statement": "Conjecture 2 from the section \"Balanced Graphs\".",
   "evidence": "The exact canonical record is only the fragment",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 282,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0284": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 1 of the same section also holds for G(k, m ). Indeed, if k > 1 then S = {0; 4 mi + 1, 4mi − 1 | i = 1,..., k }\n\nis a star cutset: 0 is an isolated vertex in ¯G[S], while 4 mj is an isolated vertex in G[V \\ S]for every j = 1,..., k; and if k = 1 then G(k, m ) is 4 m-cycle, that is a basic graph CONJECTURE. Every non-empty balanced circulant is isomorphic to a G(k, m ). Contributed by Diogo Andrade, Endre Boros, and Vladimir Gurvich \n\nChapter L: P4-structure and Its Relatives \n\nThe P4-structure of a graph G is the 4-uniform hypergraph whose vertex-set V is the vertex-set of G and whose hyperedges are the subsets of V that induce P4's (chordless paths on four vertices) in G. The graph property of being Berge can be formulated directly in terms of P4-structure: a graph is Berge if and only if its P4-structure contains no induced \n\nodd ring, meaning a 4-uniform hypergraph with vertices \n\nu0, u 1,..., u k−1,\n\nwhere k is odd and at least five, and with the k hyperedges \n\n{ui+1, u i+2, u i+3, u i+4 },\n\nwhere the subscripts are taken modulo k. (The \"if\" part is trivial and the \"only if\" part is easy, even though a little tedious; see [MR 86j:05119] for a sketch of the argument.) Can the Chudnovsky-Robertson-Seymour-Thomas decomposition theorem for Berge graphs be reformulated directly in terms of P4-structure? The five classes of basic graphs featured in the theorem lend themselves nicely to such reformulations: there are classes C of 4-uniform hypergraphs such that 20 \n\n• the P4-structure of every basic graph belongs to C,\n\n• no 4-uniform hypergraph in C contains an induced odd ring, \n\n• membership in C can be tested in polynomial time. One such class is defined in terms of a certain directed graph, D6(H), associated with every 4-uniform hypergraph H: C consists of all 4-uniform hypergraphs H such that \n\n• H contains no induced ring with five vertices and \n\n• all strongly connected components of D6(H) are bipartite. The vertices of D6(H) are all the ordered 6-tuples (u1, u 2, u 3, u 4, u 5, u 6)of distinct vertices of H such that the sub-hypergraph of H induced by the set \n\n{u1, u 2, u 3, u 4, u 5, u 6}\n\nconsists of the three hyperedges \n\n{u1, u 2, u 3, u 4}, {u2, u 3, u 4, u 5}, {u3, u 4, u 5, u 6};there is a directed edge from vertex (u1, u 2, u 3, u 4, u 5, u 6)of D6(H) to vertex (v1, v 2, v 3, v 4, v 5, v 6)of D6(H) if and only if \n\nv1 = u2, v 2 = u3, v 3 = u4, v 4 = u5, v 5 = u6.\n\nTrivially, membership in C can be tested in polynomial time; trivially, no 4-uniform hyper-graph in C contains an induced odd ring; a proof that the P4-structure of every basic graph belongs to C is easy, even though a little tedious (here 6 is a sketch of the argument). The four kinds of structural faults featured in the decomposition theorem suggest the following three problems.",
   "clean_statement": null,
   "public_statement": "Conjecture 1 of the same section also holds for G(k, m ). Indeed, if k > 1 then S = {0; 4 mi + 1, 4mi − 1 | i = 1,..., k }\n\nis a star cutset: 0 is an isolated vertex in ¯G[S], while 4 mj is an isolated vertex in G[V \\ S]for every j = 1,..., k; and if k = 1 then G(k, m ) is 4 m-cycle, that is a basic graph CONJECTURE. Every non-empty balanced circulant is isomorphic to a G(k, m ). Contributed by Diogo Andrade, Endre Boros, and Vladimir Gurvich\n\nChapter L: P4-structure and Its Relatives\n\nThe P4-structure of a graph G is the 4-uniform hypergraph whose vertex-set V is the vertex-set of G and whose hyperedges are the subsets of V that induce P4's (chordless paths on four vertices) in G. The graph property of being Berge can be formulated directly in terms of P4-structure: a graph is Berge if and only if its P4-structure contains no induced\n\nodd ring, meaning a 4-uniform hypergraph with vertices\n\nu0, u 1,..., u k−1,\n\nwhere k is odd and at least five, and with the k hyperedges\n\n{ui+1, u i+2, u i+3, u i+4 },\n\nwhere the subscripts are taken modulo k. (The \"if\" part is trivial and the \"only if\" part is easy, even though a little tedious; see [MR 86j:05119] for a sketch of the argument.) Can the Chudnovsky-Robertson-Seymour-Thomas decomposition theorem for Berge graphs be reformulated directly in terms of P4-structure? The five classes of basic graphs featured in the theorem lend themselves nicely to such reformulations: there are classes C of 4-uniform hypergraphs such that 20\n\n• the P4-structure of every basic graph belongs to C,\n\n• no 4-uniform hypergraph in C contains an induced odd ring,\n\n• membership in C can be tested in polynomial time. One such class is defined in terms of a certain directed graph, D6(H), associated with every 4-uniform hypergraph H: C consists of all 4-uniform hypergraphs H such that\n\n• H contains no induced ring with five vertices and\n\n• all strongly connected components of D6(H) are bipartite. The vertices of D6(H) are all the ordered 6-tuples (u1, u 2, u 3, u 4, u 5, u 6)of distinct vertices of H such that the sub-hypergraph of H induced by the set\n\n{u1, u 2, u 3, u 4, u 5, u 6}\n\nconsists of the three hyperedges\n\n{u1, u 2, u 3, u 4}, {u2, u 3, u 4, u 5}, {u3, u 4, u 5, u 6};there is a directed edge from vertex (u1, u 2, u 3, u 4, u 5, u 6)of D6(H) to vertex (v1, v 2, v 3, v 4, v 5, v 6)of D6(H) if and only if\n\nv1 = u2, v 2 = u3, v 3 = u4, v 4 = u5, v 5 = u6.\n\nTrivially, membership in C can be tested in polynomial time; trivially, no 4-uniform hyper-graph in C contains an induced odd ring; a proof that the P4-structure of every basic graph belongs to C is easy, even though a little tedious (here 6 is a sketch of the argument). The four kinds of structural faults featured in the decomposition theorem suggest the following three problems.",
   "evidence": "The canonical record is not one coherent problem. It begins in the middle of the balanced-circulant item K.1, gives a conjecture, and then runs through the heading and introductory material of Chapter L on P4-structure. The latter material sets up later, separately indexed problems and is not part of the K.1 conjecture. Consequently the record is classified as `invalid_statement` as a unit.",
   "classification_method": "invalid_literal_with_repaired_reading",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 283,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0285": {
   "statement_status": "exact",
   "original_statement": "Problem 1: Find a class C1 of 4-uniform hypergraphs such that \n\n• the P4-structure of every graph with a 2-join belongs to C1,\n\n• no odd ring belongs to C1,\n\n• C1 belongs to NP.",
   "clean_statement": "Problem 1: Find a class C1 of 4-uniform hypergraphs such that\n\n• the P4-structure of every graph with a 2-join belongs to C1,\n\n• no odd ring belongs to C1,\n\n• C1 belongs to NP.",
   "public_statement": "Problem 1: Find a class C1 of 4-uniform hypergraphs such that\n\n• the P4-structure of every graph with a 2-join belongs to C1,\n\n• no odd ring belongs to C1,\n\n• C1 belongs to NP.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 284,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0286": {
   "statement_status": "exact",
   "original_statement": "Problem 2: Find a class C2 of 4-uniform hypergraphs such that \n\n• the P4-structure of every graph with an M-join belongs to C2,\n\n• no odd ring belongs to C2,\n\n• C2 belongs to NP.",
   "clean_statement": "Problem 2: Find a class C2 of 4-uniform hypergraphs such that\n\n• the P4-structure of every graph with an M-join belongs to C2,\n\n• no odd ring belongs to C2,\n\n• C2 belongs to NP.",
   "public_statement": "Problem 2: Find a class C2 of 4-uniform hypergraphs such that\n\n• the P4-structure of every graph with an M-join belongs to C2,\n\n• no odd ring belongs to C2,\n\n• C2 belongs to NP.",
   "evidence": "The official AIM PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 285,
   "attempt": 1
  },
  "AIM-COMBINATORICS-0287": {
   "statement_status": "exact",
   "original_statement": "Problem 3: Find a class C3 of 4-uniform hypergraphs such that \n\n• the P4-structure of every graph with a balanced skew partition belongs to C3,\n\n• no odd ring belongs to C3,\n\n• C3 belongs to NP.\n\n> 6http://www.cs.rutgers.edu/ ∼chvatal/perfect/problems.html#basic 21\n\nCh´ ınh Ho` ang introduced additional graph functions that are invariant under comple-mentation and determine whether or not a graph is Berge: these are the co-paw-structure \n\n([MR 2001a:05065]), the co-C4-structure (Discrete Math. 252 (2002), 141-159), and the \n\nco-P3-structure (to appear in SIAM J. Discrete Math.). Can the decomposition theorem be reformulated directly in terms of one of Ho` ang's invariants? Contributed by Vaˇ sek Chv´ atal",
   "clean_statement": "Problem 3: Find a class C3 of 4-uniform hypergraphs such that\n\n• the P4-structure of every graph with a balanced skew partition belongs to C3,\n\n• no odd ring belongs to C3,\n\n• C3 belongs to NP.\n\n> 6http://www.cs.rutgers.edu/ ∼chvatal/perfect/problems.html#basic 21\n\nCh´ ınh Ho` ang introduced additional graph functions that are invariant under comple-mentation and determine whether or not a graph is Berge: these are the co-paw-structure\n\n([MR 2001a:05065]), the co-C4-structure (Discrete Math. 252 (2002), 141-159), and the\n\nco-P3-structure (to appear in SIAM J. Discrete Math.). Can the decomposition theorem be reformulated directly in terms of one of Ho` ang's invariants? Contributed by Vaˇ sek Chv´ atal",
   "public_statement": "Problem 3: Find a class C3 of 4-uniform hypergraphs such that\n\n• the P4-structure of every graph with a balanced skew partition belongs to C3,\n\n• no odd ring belongs to C3,\n\n• C3 belongs to NP.\n\n> 6http://www.cs.rutgers.edu/ ∼chvatal/perfect/problems.html#basic 21\n\nCh´ ınh Ho` ang introduced additional graph functions that are invariant under comple-mentation and determine whether or not a graph is Berge: these are the co-paw-structure\n\n([MR 2001a:05065]), the co-C4-structure (Discrete Math. 252 (2002), 141-159), and the\n\nco-P3-structure (to appear in SIAM J. Discrete Math.). Can the decomposition theorem be reformulated directly in terms of one of Ho` ang's invariants? Contributed by Vaˇ sek Chv´ atal",
   "evidence": "The official AIM page states the numbered item as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-combinatorics-notes.json",
   "source_index": 286,
   "attempt": 1
  },
  "AIM-COMPUTATION-0001": {
   "statement_status": "unrecoverable",
   "original_statement": "1. Is there a deterministic algorithm that is polynomial time in n and log( q)? State of the art algorithm should be seen on May 16th as a talk. 2. How quickly can we factor x2 −a without hypotheses (Qi Cheng's question) State of the art algorithm Burgess O(p 1 \n\n> 2e\n\n)\n\n1.2 Open Question \n\nWhat is the exponent of the probabilistic complexity (for q = 2)? \n\n1.3 Open Question \n\nGiven a set of polynomials with very low degree (2 or 3), decide if all of them factor into linear factors. Is it faster to do this by factoring their product or each one individually? (Tanja Lange's question) \n\n2 Sparse Polynomials in Fq[x]\n\n2.1 Open Question \n\nDecide whether a trinomial xα + ax β + b with 0 < β < α ≤ q − 1, a, b ∈ Fq has a root in Fq, in polynomial time (in log( q)). (Erich Kaltofen's Question) \n\n2.2 Open Questions",
   "clean_statement": null,
   "public_statement": "1. Is there a deterministic algorithm that is polynomial time in n and log( q)? State of the art algorithm should be seen on May 16th as a talk. 2. How quickly can we factor x2 −a without hypotheses (Qi Cheng's question) State of the art algorithm Burgess O(p 1\n\n> 2e\n\n)\n\n1.2 Open Question\n\nWhat is the exponent of the probabilistic complexity (for q = 2)?\n\n1.3 Open Question\n\nGiven a set of polynomials with very low degree (2 or 3), decide if all of them factor into linear factors. Is it faster to do this by factoring their product or each one individually? (Tanja Lange's question)\n\n2 Sparse Polynomials in Fq[x]\n\n2.1 Open Question\n\nDecide whether a trinomial xα + ax β + b with 0 < β < α ≤ q − 1, a, b ∈ Fq has a root in Fq, in polynomial time (in log( q)). (Erich Kaltofen's Question)\n\n2.2 Open Questions",
   "evidence": "The canonical record is not one mathematical problem. The official three-page AIM PDF verifies that extraction joined the following distinct items from page 1:",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-computation-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-COMPUTATION-0002": {
   "statement_status": "unrecoverable",
   "original_statement": "1. Finding better certificates for irreducibility (a) Are there certificates for irreducibility that one can verify faster than the existing irreducibility tests? (Victor Miller's Question) (b) Can you exhibit families of sparse polynomials for which you can find shorter certificates? 2. Find a polynomial-time algorithm in log( q), where q = pn, that solves \n\n∑n−1 \n\n> i,j =0\n\nai,j xpi +pj\n\n+ ∑n−1 \n\n> i=0\n\nbixpi\n\n+ c in Fq (or shows no solutions) and works for 1 \n\n> poly\n\nof the inputs, and never lies. (We know that this is NP-complete.) (Jintai Ding's Question) 13 Factoring over Q[x]\n\n3.1 Open \nQuestion",
   "clean_statement": null,
   "public_statement": "1. Finding better certificates for irreducibility (a) Are there certificates for irreducibility that one can verify faster than the existing irreducibility tests? (Victor Miller's Question) (b) Can you exhibit families of sparse polynomials for which you can find shorter certificates? 2. Find a polynomial-time algorithm in log( q), where q = pn, that solves\n\n∑n−1\n\n> i,j =0\n\nai,j xpi +pj\n\n+ ∑n−1\n\n> i=0\n\nbixpi\n\n+ c in Fq (or shows no solutions) and works for 1\n\n> poly\n\nof the inputs, and never lies. (We know that this is NP-complete.) (Jintai Ding's Question) 13 Factoring over Q[x]\n\n3.1 Open\nQuestion",
   "evidence": "The canonical JSON record is not one mathematical problem. It is a damaged extraction of page 1 of the AIM workshop notes *The computational complexity of polynomial factorization*. Comparison with the source PDF recovers the following text in Section 2.2, “Open Questions” (line breaks normalized but wording retained):",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-computation-notes.json",
   "source_index": 1,
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  },
  "AIM-COMPUTATION-0003": {
   "statement_status": "unrecoverable",
   "original_statement": "1. How long does the gradual feeding (or some modification) algorithm actu-ally take to solve one approximate linear equation (with the ci's bounded by kn bits)? (Mark van Hoeij's Question) 2. (a) Can we do lattice basis reduction in essentially linear time in the total number of digits of the input? (Dan Bernstein's Question) (b) Is there a softly linear time reduction from linear system solving to basis reduction in lattices? (Joachim von zur Gathen's question) \n\n3.2 Open Question \n\nCan we use Niederreiter's equation to find a faster algorithm to factor dense polynomials in Q[x]? (Shuhong Gao's Question) The main possible advantage of such a technique would be to use linear algebra over Fp instead of lattice basis reduction. \n\n4 Bivariate \n\n4.1 Open Question \n\nFor Dense Bivariate polynomials over Fp with total degree, t: Can we improve on the probabilistic complexity ˜O( d3) + Factorization Complexity of Univariate, degree d, polynomial. Lecerf believes ˜O( dω ) is possible. The bottleneck is solving d2 equations in d variables. \n\n4.2 Open Question \n\nCan we find low-degree ( ≤ n) factors of a polynomial with t terms and degree \n\nd (t \u001c d2, n \u001c d)\n\n4.3 Open Question \n\nFind an Absolute Factorization (or test for irreducibility) over fields with small (p < d 2) characteristic, in ˜O( d3). \n\n5 Intermezzo \n\nGiven f ∈ Q[x] irreducible, and f (α) = 0. Factor f over Q[α]. State of Art: Belabas 26 Numerical \n\n6.1 Open Question \n\nChallenge Problem #1 in Kaltofen's JSC 2000 paper. \n\n6.2 Open Question \n\nGiven f ∈ C[x, y ] find the nearest polynomial that factors into linear factors in \n\nC[x, y ]. Find/Bound the distance to the nearest polynomial that factors. \n\n6.3 Open Question \n\nGiven f ∈ Q[x, y, z ], for which values of z ∈ Q does f factor (completely) in \n\nC[x, y ]? What is the degree of a given z?\n\n6.4 Open Question \n\nHow to compute/approximate the structured condition number of a Ruppert Matrix. 3",
   "clean_statement": null,
   "public_statement": "1. How long does the gradual feeding (or some modification) algorithm actu-ally take to solve one approximate linear equation (with the ci's bounded by kn bits)? (Mark van Hoeij's Question) 2. (a) Can we do lattice basis reduction in essentially linear time in the total number of digits of the input? (Dan Bernstein's Question) (b) Is there a softly linear time reduction from linear system solving to basis reduction in lattices? (Joachim von zur Gathen's question)\n\n3.2 Open Question\n\nCan we use Niederreiter's equation to find a faster algorithm to factor dense polynomials in Q[x]? (Shuhong Gao's Question) The main possible advantage of such a technique would be to use linear algebra over Fp instead of lattice basis reduction.\n\n4 Bivariate\n\n4.1 Open Question\n\nFor Dense Bivariate polynomials over Fp with total degree, t: Can we improve on the probabilistic complexity ˜O( d3) + Factorization Complexity of Univariate, degree d, polynomial. Lecerf believes ˜O( dω ) is possible. The bottleneck is solving d2 equations in d variables.\n\n4.2 Open Question\n\nCan we find low-degree ( ≤ n) factors of a polynomial with t terms and degree\n\nd (t\n d2, n\n d)\n\n4.3 Open Question\n\nFind an Absolute Factorization (or test for irreducibility) over fields with small (p < d 2) characteristic, in ˜O( d3).\n\n5 Intermezzo\n\nGiven f ∈ Q[x] irreducible, and f (α) = 0. Factor f over Q[α]. State of Art: Belabas 26 Numerical\n\n6.1 Open Question\n\nChallenge Problem #1 in Kaltofen's JSC 2000 paper.\n\n6.2 Open Question\n\nGiven f ∈ C[x, y ] find the nearest polynomial that factors into linear factors in\n\nC[x, y ]. Find/Bound the distance to the nearest polynomial that factors.\n\n6.3 Open Question\n\nGiven f ∈ Q[x, y, z ], for which values of z ∈ Q does f factor (completely) in\n\nC[x, y ]? What is the degree of a given z?\n\n6.4 Open Question\n\nHow to compute/approximate the structured condition number of a Ruppert Matrix. 3",
   "evidence": "The canonical `problem` field is preserved verbatim in `input.json`. It is not a single mathematical question. Comparison with the official three-page AIM PDF shows that the extraction joined almost all of pages 2 and 3 of a workshop problem list into one record. More precisely, the record contains:",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-computation-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-COMPUTATION-0004": {
   "statement_status": "exact",
   "original_statement": "A.1 Arisawa, Mariko \n\nWe would like to introduce a viscosity solution's approach to a class of second-order partial-integro-differential equations (PIDE, in short) arising in the mathematical finance. The usual brownian models for the processes of the asset prices are replaced by the jump-diffusion processes. The derived PIDEs contain in the integral term, various Le'vy measures which correspond to the frequencies and the sizes of the jumps. In addition, according to the financial settings such as the American option, the look back option, the problem with transaction costs, a variety of nonlinear PIDEs are engendered. We shall introduce a new definition of the viscosity solution for such problems, and shall give the comparison, the existence, and the regularity of the solutions of viscosity solutions in this framework. As far as the high-dimensional numerical analysis, a very few is known at this stage, while from the practical view point a lot of important problems should be explored, such as the modelization of the high-dimensional Le'vy measures arising in the mathematical finances. This is the problem which I would like to discuss with the participants of the workshop.",
   "clean_statement": "A.1 Arisawa, Mariko\n\nWe would like to introduce a viscosity solution's approach to a class of second-order partial-integro-differential equations (PIDE, in short) arising in the mathematical finance. The usual brownian models for the processes of the asset prices are replaced by the jump-diffusion processes. The derived PIDEs contain in the integral term, various Le'vy measures which correspond to the frequencies and the sizes of the jumps. In addition, according to the financial settings such as the American option, the look back option, the problem with transaction costs, a variety of nonlinear PIDEs are engendered. We shall introduce a new definition of the viscosity solution for such problems, and shall give the comparison, the existence, and the regularity of the solutions of viscosity solutions in this framework. As far as the high-dimensional numerical analysis, a very few is known at this stage, while from the practical view point a lot of important problems should be explored, such as the modelization of the high-dimensional Le'vy measures arising in the mathematical finances. This is the problem which I would like to discuss with the participants of the workshop.",
   "public_statement": "A.1 Arisawa, Mariko\n\nWe would like to introduce a viscosity solution's approach to a class of second-order partial-integro-differential equations (PIDE, in short) arising in the mathematical finance. The usual brownian models for the processes of the asset prices are replaced by the jump-diffusion processes. The derived PIDEs contain in the integral term, various Le'vy measures which correspond to the frequencies and the sizes of the jumps. In addition, according to the financial settings such as the American option, the look back option, the problem with transaction costs, a variety of nonlinear PIDEs are engendered. We shall introduce a new definition of the viscosity solution for such problems, and shall give the comparison, the existence, and the regularity of the solutions of viscosity solutions in this framework. As far as the high-dimensional numerical analysis, a very few is known at this stage, while from the practical view point a lot of important problems should be explored, such as the modelization of the high-dimensional Le'vy measures arising in the mathematical finances. This is the problem which I would like to discuss with the participants of the workshop.",
   "evidence": "The canonical item is Section A.1, “Arisawa, Mariko,” in the participant contributions to the AIM workshop *Numerical methods for optimal control in high dimensions*. The official PDF and the extracted record agree. The entry says that second-order partial integro-differential equations (PIDEs) arising from jump-diffusion models in mathematical finance involve a variety of Lévy measures and nonlinearities associated with American options, lookback options, and transaction costs. It proposes a viscosity-solution framework with comparison, existence, and regularity results. With the PDF's malformed accent in “Lévy” normalized, the entry closes with:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 3,
   "attempt": 2
  },
  "AIM-COMPUTATION-0005": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A.2 Capobianco, Enrico \n\nPart 1 \n\nSystems Biology: On Dimensionality Reduction and Feature Se-lection in Genomics (work carried out at Boston University, Biomedical Engineering Department) Gene networks offer a wealth of data; this is mainly due to the genomic dimensionality rather than the samples, as the latter usually come from measurements obtained under only a few experimental conditions or time points. It is therefore a challenging task to design suitable statistical models and to develop effective reverse engineering algorithms. Network inference and reverse engineering deal basically with an inverse problem that is stimulating a great deal of biomedical research in systems biology: reconstructing gene-gene interactions from measurements obtained under specific experimental conditions. Some of the open problems calling for solutions are now listed. First, the signature of noise is pervasive in genetic networks. For instance, in pertur-bation experiments only a few genes change expression value, while most genes show either noisy or constant patterns. The measurements are usually taken at an initial time and then regularly recorded until the genes reach a steady state. During this time interval, the gene temporal patterns are subject to several kinds of fluctuations, and only some of them are induced by the perturbations, directly or indirectly as a cascade effect. Noise is thus a strong conditioning factor for the gene temporal patterns; the ones that appear smooth are usually of interest compared to those characterized by a random signature. Consequently, sharp time-localized fluctuations lead to discard many patterns and focus just on subsets of genes. An important tool is thresholding, which deals with the separation of signal from noise. It identifies gene groups by selecting the genes that are differentially expressed with regard to 4\n\nnoise-dependent patterns. Furthermore, a gene can be considered outlying if its expression value is significantly different (overor under-expressed) compared to the average expression value computed for all the genes. \n\nMethodological aspects \n\nStatistically speaking, if we are able to isolate a certain number of modes or components which represent in a compact way the systems dynamics underlying the gene network under examination, we might also try to compute statistics based on these modes through the gene sets that have been identified by them. Due to the inherent biological characterization of the estimated components, the thresh-olding step can validate or contradict the hypothesis that the observed expression values of the genes are significantly different from specific test statistics. The idea is that for each available condition (a sample point where a measurement is taken or an experiment is conducted), a genomic profile X is considered a mixture (via a certain matrix A) of unknown biological influences (modes or components) S that may have more or less impact over each specific condition. Then, feature sets are identified by linear combinations of genes which are delivered by each identified and estimated component. More formally, consider at time t an ≤− noise genomic system such as: \n\nX = AS + ≤ (1) The dimensions of the model (for k = 1) are S ∈ Rm, X ∈ Rn, and A ∈ Rn×m, and the gene expression matrix X has rows representing the different genes involved in the regulatory network under exam, while the columns list the measurements at successive conditions. The mixture signals in X represent k−dimensional vectors xi, i = 1,..., n, and the components S represent k−dimensional vectors sj, j = 1,..., m. The latter are mixed up linearly (or non-linearly) through the mixing matrix A. Both S and A are unknown variables and must be estimated from the only available information set X.This gene selection goal implies that a genetic regulatory network is a redundant system because inherently noisy, but also owing to the high-dimensionality and dependence between genes, only in part justified by biological reasons. Equivalently, the underlying system of gene dynamics is sparse; in other words, the gene-gene interaction matrix is only partially active, as some of the supposed links among genes are not supported by significant evidence. Exploiting sparsity when designing statistical inference procedures for gene network analysis as a key goal is to target just subsets of genes when measuring the effects of the experiments. And sparsity refers to dimensionality too, through the possibility of approxi-mating the truly intrinsic dimension of the input space. In turn, this may bring some valuable experimental insight too, by suggesting for instance how to calibrate the perturbations and selecting the genes expected to become targets. In order to explore these two aspects, redundancy and sparsity, clustering has been widely employed by computational biologists, but several limitations have appeared; there-fore, independent and principal component analysis (ICA and PCA, respectively) can also 5\n\nbe proposed (and compared) as examples of flexible approximation tools targeted to dimen-sionality reduction and gene feature selection. \n\nPart 2 \n\nSemi-Parametric Estimation in In Vivo MR Spectroscopy (ongoing joint work with: H. Ratiney, C. Cudalbu, S. Cavassila, D. Graveron-Demilly RMN, CNRS UMR 5012, Univ. Claude Bernard LYON I-CPE, France R. de Beer, D. van Ormondt, Applied Physics, TU Delft, The Netherlands). Magnetic Resonance Spectroscopy (MRS) is a unique tool for non-invasive in vivo detection and quantitation of metabolites. This makes MRS an indispensable tool for com-bating major diseases. Although modern MRS-methods are increasingly capable of detecting and quantifying metabolites, perturbation by signals from so-called macromolecules and un-wanted metabolites poses problems. Three procedures for alleviating the problem exist: 1) Tuning of the scanner to a metabolite of interest - 'spectral editing' - can clean up the MRS signal significantly. 2) Separate (approximate) measurement of the macromolecule signal and subsequent subtraction from the MRS signal. 3) Semi-parametric estimation of the model parameters of interest from the MRS signal. Our strategy is to process directly in the measurement domain; in MRS, this is the time domain. Alternatively, one can process in the frequency or spectral domain. A main disadvantage of the latter is that it starts with estimation of the spectrum which is not trivial in the case of missing samples or non-Cartesian sampling. \n\nMethodological aspects \n\nIn semi-parametric estimation, the task is to disentangle the parametric and non-parametric parts of a signal. In MRS, the parametric part pertains to the metabolites, the non-parametric part to the macromolecules. In the measurement domain, the metabolite signal persists over time much longer than the macromolecule signal. On the other hand, during the initial period, the latter strongly dominates the former. This property is exploited to bring about the disentanglement. Through simulations, we have successfully investigated: 1) Iteration of the disentanglement procedure. 2) Derivation of Cramer-Rao Bounds (CRB) for the metabolite concentrations, taking the non-parametric macromolecule signal into account. This contributes to experimental design. Furthermore: 1) Our previous contributions did not report on iteration of the disentanglement pro-cedure. Now, we include cases where such iteration clearly helps. 2) We have devised a novel way (in MRS, at least) to estimate CRBs for the parametric part, taking the non-parametric part into account. 3) Only information about the point of time where the non-parametric part 'decays into the noise' is needed. We thus have shown that through semiparametric models it is possible to augment the applicability of MRS to Medicine, while in order to have the best possible impact we need to optimize the accuracy of the estimates for the parameters of interest. 6\n\nIn particular, one seeks an improved reliability of the lower bounds computed on pa-rameters to be estimated with a planned scan, from which it depends in turn an improved reliability of the 'experimental design', or, in other words, a better prediction of feasibility of expensive MRS-scans in the clinic. Then, one would like to address: Awhat is the predicted minimum detectable concentration of each of the 40 or so metabolites of interest with a given scanner measurement protocol? BIs the try justifiable?",
   "clean_statement": null,
   "public_statement": "A.2 Capobianco, Enrico\n\nPart 1\n\nSystems Biology: On Dimensionality Reduction and Feature Se-lection in Genomics (work carried out at Boston University, Biomedical Engineering Department) Gene networks offer a wealth of data; this is mainly due to the genomic dimensionality rather than the samples, as the latter usually come from measurements obtained under only a few experimental conditions or time points. It is therefore a challenging task to design suitable statistical models and to develop effective reverse engineering algorithms. Network inference and reverse engineering deal basically with an inverse problem that is stimulating a great deal of biomedical research in systems biology: reconstructing gene-gene interactions from measurements obtained under specific experimental conditions. Some of the open problems calling for solutions are now listed. First, the signature of noise is pervasive in genetic networks. For instance, in pertur-bation experiments only a few genes change expression value, while most genes show either noisy or constant patterns. The measurements are usually taken at an initial time and then regularly recorded until the genes reach a steady state. During this time interval, the gene temporal patterns are subject to several kinds of fluctuations, and only some of them are induced by the perturbations, directly or indirectly as a cascade effect. Noise is thus a strong conditioning factor for the gene temporal patterns; the ones that appear smooth are usually of interest compared to those characterized by a random signature. Consequently, sharp time-localized fluctuations lead to discard many patterns and focus just on subsets of genes. An important tool is thresholding, which deals with the separation of signal from noise. It identifies gene groups by selecting the genes that are differentially expressed with regard to 4\n\nnoise-dependent patterns. Furthermore, a gene can be considered outlying if its expression value is significantly different (overor under-expressed) compared to the average expression value computed for all the genes.\n\nMethodological aspects\n\nStatistically speaking, if we are able to isolate a certain number of modes or components which represent in a compact way the systems dynamics underlying the gene network under examination, we might also try to compute statistics based on these modes through the gene sets that have been identified by them. Due to the inherent biological characterization of the estimated components, the thresh-olding step can validate or contradict the hypothesis that the observed expression values of the genes are significantly different from specific test statistics. The idea is that for each available condition (a sample point where a measurement is taken or an experiment is conducted), a genomic profile X is considered a mixture (via a certain matrix A) of unknown biological influences (modes or components) S that may have more or less impact over each specific condition. Then, feature sets are identified by linear combinations of genes which are delivered by each identified and estimated component. More formally, consider at time t an ≤− noise genomic system such as:\n\nX = AS + ≤ (1) The dimensions of the model (for k = 1) are S ∈ Rm, X ∈ Rn, and A ∈ Rn×m, and the gene expression matrix X has rows representing the different genes involved in the regulatory network under exam, while the columns list the measurements at successive conditions. The mixture signals in X represent k−dimensional vectors xi, i = 1,..., n, and the components S represent k−dimensional vectors sj, j = 1,..., m. The latter are mixed up linearly (or non-linearly) through the mixing matrix A. Both S and A are unknown variables and must be estimated from the only available information set X.This gene selection goal implies that a genetic regulatory network is a redundant system because inherently noisy, but also owing to the high-dimensionality and dependence between genes, only in part justified by biological reasons. Equivalently, the underlying system of gene dynamics is sparse; in other words, the gene-gene interaction matrix is only partially active, as some of the supposed links among genes are not supported by significant evidence. Exploiting sparsity when designing statistical inference procedures for gene network analysis as a key goal is to target just subsets of genes when measuring the effects of the experiments. And sparsity refers to dimensionality too, through the possibility of approxi-mating the truly intrinsic dimension of the input space. In turn, this may bring some valuable experimental insight too, by suggesting for instance how to calibrate the perturbations and selecting the genes expected to become targets. In order to explore these two aspects, redundancy and sparsity, clustering has been widely employed by computational biologists, but several limitations have appeared; there-fore, independent and principal component analysis (ICA and PCA, respectively) can also 5\n\nbe proposed (and compared) as examples of flexible approximation tools targeted to dimen-sionality reduction and gene feature selection.\n\nPart 2\n\nSemi-Parametric Estimation in In Vivo MR Spectroscopy (ongoing joint work with: H. Ratiney, C. Cudalbu, S. Cavassila, D. Graveron-Demilly RMN, CNRS UMR 5012, Univ. Claude Bernard LYON I-CPE, France R. de Beer, D. van Ormondt, Applied Physics, TU Delft, The Netherlands). Magnetic Resonance Spectroscopy (MRS) is a unique tool for non-invasive in vivo detection and quantitation of metabolites. This makes MRS an indispensable tool for com-bating major diseases. Although modern MRS-methods are increasingly capable of detecting and quantifying metabolites, perturbation by signals from so-called macromolecules and un-wanted metabolites poses problems. Three procedures for alleviating the problem exist: 1) Tuning of the scanner to a metabolite of interest - 'spectral editing' - can clean up the MRS signal significantly. 2) Separate (approximate) measurement of the macromolecule signal and subsequent subtraction from the MRS signal. 3) Semi-parametric estimation of the model parameters of interest from the MRS signal. Our strategy is to process directly in the measurement domain; in MRS, this is the time domain. Alternatively, one can process in the frequency or spectral domain. A main disadvantage of the latter is that it starts with estimation of the spectrum which is not trivial in the case of missing samples or non-Cartesian sampling.\n\nMethodological aspects\n\nIn semi-parametric estimation, the task is to disentangle the parametric and non-parametric parts of a signal. In MRS, the parametric part pertains to the metabolites, the non-parametric part to the macromolecules. In the measurement domain, the metabolite signal persists over time much longer than the macromolecule signal. On the other hand, during the initial period, the latter strongly dominates the former. This property is exploited to bring about the disentanglement. Through simulations, we have successfully investigated: 1) Iteration of the disentanglement procedure. 2) Derivation of Cramer-Rao Bounds (CRB) for the metabolite concentrations, taking the non-parametric macromolecule signal into account. This contributes to experimental design. Furthermore: 1) Our previous contributions did not report on iteration of the disentanglement pro-cedure. Now, we include cases where such iteration clearly helps. 2) We have devised a novel way (in MRS, at least) to estimate CRBs for the parametric part, taking the non-parametric part into account. 3) Only information about the point of time where the non-parametric part 'decays into the noise' is needed. We thus have shown that through semiparametric models it is possible to augment the applicability of MRS to Medicine, while in order to have the best possible impact we need to optimize the accuracy of the estimates for the parameters of interest. 6\n\nIn particular, one seeks an improved reliability of the lower bounds computed on pa-rameters to be estimated with a planned scan, from which it depends in turn an improved reliability of the 'experimental design', or, in other words, a better prediction of feasibility of expensive MRS-scans in the clinic. Then, one would like to address: Awhat is the predicted minimum detectable concentration of each of the 40 or so metabolites of interest with a given scanner measurement protocol? BIs the try justifiable?",
   "evidence": "The source is Section A.2, “Capobianco, Enrico,” of the AIM workshop document *Numerical Methods for Optimal Control in High Dimensions*, version dated 19 August 2005. The table of contents identifies Appendix A as “Participant Contributions.” Thus this canonical `tag: section` record is an agenda contribution, not one theorem-like open problem.",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-computation-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-COMPUTATION-0006": {
   "statement_status": "exact",
   "original_statement": "A.3 de Farias, Daniela \n\nIn this workshop, I am interested in learning more about the main open issues and approaches across the different subfields related to high-dimensional optimal control. In addition, I also see the workshop as an opportunity to discuss some of the open issues and explore potential directions in my own subfield. My current research interests lie in approximate methods for large-scale stochastic dynamic programming, with particular focus on value function approximation. I would hope to see at least some activities and discussions pertaining to that topic. Some of the directions in approximate dynamic programming that I consider worth exploring are as follows: \n\n• Approaches for validation of approximate DP methods. Interest in approximate DP has been fuelled by successful application in a variety of areas ranging from finance to games and helicopter control. Yet, it is still difficult to know a priori when, how and why a given method will be successful in tackling a particular problem. In my view, there is a need for more systematic approaches to validation of algorithms beyond illustration via application to one or two specific problems. These systematic approaches could be of many different kinds, which could be discussed in more or less technical detail at the workshop. For instance, one could pursue the development theoretical guarantees of performance and errors, or a more systematic empirical comparison based on a benchmark of large-scale DP problems. \n\n• Comparison and unification of different algorithms. Approximate DP is currently represented by a conglomeration of ideas and algorithms, and it is often difficult to know how they relate to one another and which is most appropriate for any class of applications. A streamlined comparison among the approaches and potentially unifying theory would be useful developments to the field. \n\n• Systematic and/or adaptive design of approximation architectures. A central issue in approximate dynamic programming that remains largely open is how to choose the approximation architecture, which remains a problem-specific task, and approximate DP algorithms typically start with the premise that an approximation architecture has been pre-specified. More streamlined methods of approximation architecture selection together with theory characterizing optimality and convergence rates would be desirable. Alternatively, it may be worth trying to explore whether a canonical approximation architecture could be developed at least for some specific class of problems. \n\n• Performance-Oriented Approximate DP. The design, analysis and comparison of value function approximation algorithms is often based on approximation or Bellman er-rors. Both situations entail the choice of a norm whereby to measure and trade off approximation or Bellman errors across states. Little work has been done to develop 7\n\na better understanding of how different criteria capture the quality of the policies being generated. Theoretical analysis is required to identify which criteria - Bell-man or approximation error, which norms? - best reflect policy performance; at the same time, algorithms that take such criteria into account must be designed. \n\n• Alternative paradigms and connections with stochastic programming. As a more open-ended direction, one could consider whether extensions of the Markov deci-sion process paradigm, possibly combining it with stochastic programming, and/or a completely different approach could be more suitable for tackling certain classes of real-world problems without running into the curse of dimensionality.",
   "clean_statement": "A.3 de Farias, Daniela\n\nIn this workshop, I am interested in learning more about the main open issues and approaches across the different subfields related to high-dimensional optimal control. In addition, I also see the workshop as an opportunity to discuss some of the open issues and explore potential directions in my own subfield. My current research interests lie in approximate methods for large-scale stochastic dynamic programming, with particular focus on value function approximation. I would hope to see at least some activities and discussions pertaining to that topic. Some of the directions in approximate dynamic programming that I consider worth exploring are as follows:\n\n• Approaches for validation of approximate DP methods. Interest in approximate DP has been fuelled by successful application in a variety of areas ranging from finance to games and helicopter control. Yet, it is still difficult to know a priori when, how and why a given method will be successful in tackling a particular problem. In my view, there is a need for more systematic approaches to validation of algorithms beyond illustration via application to one or two specific problems. These systematic approaches could be of many different kinds, which could be discussed in more or less technical detail at the workshop. For instance, one could pursue the development theoretical guarantees of performance and errors, or a more systematic empirical comparison based on a benchmark of large-scale DP problems.\n\n• Comparison and unification of different algorithms. Approximate DP is currently represented by a conglomeration of ideas and algorithms, and it is often difficult to know how they relate to one another and which is most appropriate for any class of applications. A streamlined comparison among the approaches and potentially unifying theory would be useful developments to the field.\n\n• Systematic and/or adaptive design of approximation architectures. A central issue in approximate dynamic programming that remains largely open is how to choose the approximation architecture, which remains a problem-specific task, and approximate DP algorithms typically start with the premise that an approximation architecture has been pre-specified. More streamlined methods of approximation architecture selection together with theory characterizing optimality and convergence rates would be desirable. Alternatively, it may be worth trying to explore whether a canonical approximation architecture could be developed at least for some specific class of problems.\n\n• Performance-Oriented Approximate DP. The design, analysis and comparison of value function approximation algorithms is often based on approximation or Bellman er-rors. Both situations entail the choice of a norm whereby to measure and trade off approximation or Bellman errors across states. Little work has been done to develop 7\n\na better understanding of how different criteria capture the quality of the policies being generated. Theoretical analysis is required to identify which criteria - Bell-man or approximation error, which norms? - best reflect policy performance; at the same time, algorithms that take such criteria into account must be designed.\n\n• Alternative paradigms and connections with stochastic programming. As a more open-ended direction, one could consider whether extensions of the Markov deci-sion process paradigm, possibly combining it with stochastic programming, and/or a completely different approach could be more suitable for tackling certain classes of real-world problems without running into the curse of dimensionality.",
   "public_statement": "A.3 de Farias, Daniela\n\nIn this workshop, I am interested in learning more about the main open issues and approaches across the different subfields related to high-dimensional optimal control. In addition, I also see the workshop as an opportunity to discuss some of the open issues and explore potential directions in my own subfield. My current research interests lie in approximate methods for large-scale stochastic dynamic programming, with particular focus on value function approximation. I would hope to see at least some activities and discussions pertaining to that topic. Some of the directions in approximate dynamic programming that I consider worth exploring are as follows:\n\n• Approaches for validation of approximate DP methods. Interest in approximate DP has been fuelled by successful application in a variety of areas ranging from finance to games and helicopter control. Yet, it is still difficult to know a priori when, how and why a given method will be successful in tackling a particular problem. In my view, there is a need for more systematic approaches to validation of algorithms beyond illustration via application to one or two specific problems. These systematic approaches could be of many different kinds, which could be discussed in more or less technical detail at the workshop. For instance, one could pursue the development theoretical guarantees of performance and errors, or a more systematic empirical comparison based on a benchmark of large-scale DP problems.\n\n• Comparison and unification of different algorithms. Approximate DP is currently represented by a conglomeration of ideas and algorithms, and it is often difficult to know how they relate to one another and which is most appropriate for any class of applications. A streamlined comparison among the approaches and potentially unifying theory would be useful developments to the field.\n\n• Systematic and/or adaptive design of approximation architectures. A central issue in approximate dynamic programming that remains largely open is how to choose the approximation architecture, which remains a problem-specific task, and approximate DP algorithms typically start with the premise that an approximation architecture has been pre-specified. More streamlined methods of approximation architecture selection together with theory characterizing optimality and convergence rates would be desirable. Alternatively, it may be worth trying to explore whether a canonical approximation architecture could be developed at least for some specific class of problems.\n\n• Performance-Oriented Approximate DP. The design, analysis and comparison of value function approximation algorithms is often based on approximation or Bellman er-rors. Both situations entail the choice of a norm whereby to measure and trade off approximation or Bellman errors across states. Little work has been done to develop 7\n\na better understanding of how different criteria capture the quality of the policies being generated. Theoretical analysis is required to identify which criteria - Bell-man or approximation error, which norms? - best reflect policy performance; at the same time, algorithms that take such criteria into account must be designed.\n\n• Alternative paradigms and connections with stochastic programming. As a more open-ended direction, one could consider whether extensions of the Markov deci-sion process paradigm, possibly combining it with stochastic programming, and/or a completely different approach could be more suitable for tackling certain classes of real-world problems without running into the curse of dimensionality.",
   "evidence": "This record is item A.3, Daniela de Farias's participant contribution to the AIM workshop *Numerical methods for optimal control in high dimensions* (August 29--September 2, 2005). It is a research agenda rather than a single quantified open problem. The contribution asks for progress on five directions in approximate dynamic programming (ADP):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-COMPUTATION-0007": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A.4 Haykin, Simon \n\nFor my talk, I will do the following: 1. Provide a brief background on the classical Kalman filtering algorithm, emphasizing its virtues and limitations. 2. Describe particle filtering, rooted in Bayesian estimation and Monte Carlo simula-tion. Here again, I will highlight the virtues and limitations of this second approach. 3. The background would then be set for describing a new approach that has the potential for making a significant difference to the literature. In particular, I will pose a nonlinear recursive problem that is currently the stumbling block. As such, to progress further with this approach, we have to solve this problem. It could be that the solution I am looking for will pop up in the supplemental technical breakout session following my talk on the second day.",
   "clean_statement": null,
   "public_statement": "A.4 Haykin, Simon\n\nFor my talk, I will do the following: 1. Provide a brief background on the classical Kalman filtering algorithm, emphasizing its virtues and limitations. 2. Describe particle filtering, rooted in Bayesian estimation and Monte Carlo simula-tion. Here again, I will highlight the virtues and limitations of this second approach. 3. The background would then be set for describing a new approach that has the potential for making a significant difference to the literature. In particular, I will pose a nonlinear recursive problem that is currently the stumbling block. As such, to progress further with this approach, we have to solve this problem. It could be that the solution I am looking for will pop up in the supplemental technical breakout session following my talk on the second day.",
   "evidence": "The source is Appendix A.4, “Haykin, Simon,” in the AIM document *Numerical Methods for Optimal Control in High Dimensions*, version dated 19 August 2005. Appendix A is explicitly a collection of participant contributions. The complete A.4 text, with line wrapping normalized, is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-computation-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-COMPUTATION-0008": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A.5 Kurzhanski, Alexander \n\nAmong the central topics of modern control theory are problems of control synthesis for complex systems. These include problems of control under uncertainty and conflict as well as those under complex constraints. The solutions to such problems are well formalizable within Hamiltonian techniques and Dynamic Programming ideas, being as a rule reduced to the solution of HJB and HJBI partial differential equations or/and variational inequalities. An example where such methods are successfully applied are problems of forward and backward reachability for uncertain systems with further passage to control synthesis, safety verification, measurement feedback and related issues. The advent of HJB methods for continuous systems is due to the introduction of appropriate theories for generalized viscosity-type solutions and their equivalents. Another important topic is the solution to problems in dynamics and control for set-valued systems in terms of Hamiltonian formalism and Dynamic Programming. At present there is a sharp necessity for a breakthrough in numerical methods for solving equations and variational inequalities of the HJB-HJBI types motivated by problems of the above. There is also an interest in numerical methods for the comparison principle in HJB theory, which allows to calculate upper and lower bounds to exact solutions of HJB equations. Another perspective is the calculation of set-valued solutions to problems in evolution dynamics, estimation and control. Together with my colleagues we have developed an ellipsoidal calculus aimed at such problems and closely connected to HJB theory. The calculus is applicable to systems with original linear structure, but its methods allow effective calculations in high dimensions as well as computer animation in high dimensions through computer windows. For nonlinear systems some comparison methods were indicated. 8\n\nThe discussion of such methods as well as those for stochastic dynamics are within my primary interests at the AIM workshop.",
   "clean_statement": null,
   "public_statement": "A.5 Kurzhanski, Alexander\n\nAmong the central topics of modern control theory are problems of control synthesis for complex systems. These include problems of control under uncertainty and conflict as well as those under complex constraints. The solutions to such problems are well formalizable within Hamiltonian techniques and Dynamic Programming ideas, being as a rule reduced to the solution of HJB and HJBI partial differential equations or/and variational inequalities. An example where such methods are successfully applied are problems of forward and backward reachability for uncertain systems with further passage to control synthesis, safety verification, measurement feedback and related issues. The advent of HJB methods for continuous systems is due to the introduction of appropriate theories for generalized viscosity-type solutions and their equivalents. Another important topic is the solution to problems in dynamics and control for set-valued systems in terms of Hamiltonian formalism and Dynamic Programming. At present there is a sharp necessity for a breakthrough in numerical methods for solving equations and variational inequalities of the HJB-HJBI types motivated by problems of the above. There is also an interest in numerical methods for the comparison principle in HJB theory, which allows to calculate upper and lower bounds to exact solutions of HJB equations. Another perspective is the calculation of set-valued solutions to problems in evolution dynamics, estimation and control. Together with my colleagues we have developed an ellipsoidal calculus aimed at such problems and closely connected to HJB theory. The calculus is applicable to systems with original linear structure, but its methods allow effective calculations in high dimensions as well as computer animation in high dimensions through computer windows. For nonlinear systems some comparison methods were indicated. 8\n\nThe discussion of such methods as well as those for stochastic dynamics are within my primary interests at the AIM workshop.",
   "evidence": "The canonical record is Section A.5, “Kurzhanski, Alexander,” in the participant contributions to the AIM workshop *Numerical methods for optimal control in high dimensions*. It is a research agenda, not a numbered conjecture. It identifies:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-computation-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-COMPUTATION-0009": {
   "statement_status": "exact",
   "original_statement": "A.6 Kushner, Harold \n\nI have been interested in numerical methods in stochastic (and deterministic) control for several decades, and have developed the current methods of choice for the stochastic problem (which is often the best method for determinstic problems as well, as seen in applications to complex variational problems in out book). These cover virtually all of the cost functions and systems of current interest. But, in applications, there is a serious dimensioanlity problem, beyond 4 dimensions. Various approximation schemes (say, Q-learning or neural nets) have been hyped, but with llittle to show, outside of some very special cases. Additionally, solving the Bellman or HJ-Bellman equation is not enough. One wants the control, and to investigate the effects of implemetation of reasonable approximations. The visualization problem, even in 3 dimensions, is formidable.",
   "clean_statement": "A.6 Kushner, Harold\n\nI have been interested in numerical methods in stochastic (and deterministic) control for several decades, and have developed the current methods of choice for the stochastic problem (which is often the best method for determinstic problems as well, as seen in applications to complex variational problems in out book). These cover virtually all of the cost functions and systems of current interest. But, in applications, there is a serious dimensioanlity problem, beyond 4 dimensions. Various approximation schemes (say, Q-learning or neural nets) have been hyped, but with llittle to show, outside of some very special cases. Additionally, solving the Bellman or HJ-Bellman equation is not enough. One wants the control, and to investigate the effects of implemetation of reasonable approximations. The visualization problem, even in 3 dimensions, is formidable.",
   "public_statement": "A.6 Kushner, Harold\n\nI have been interested in numerical methods in stochastic (and deterministic) control for several decades, and have developed the current methods of choice for the stochastic problem (which is often the best method for determinstic problems as well, as seen in applications to complex variational problems in out book). These cover virtually all of the cost functions and systems of current interest. But, in applications, there is a serious dimensioanlity problem, beyond 4 dimensions. Various approximation schemes (say, Q-learning or neural nets) have been hyped, but with llittle to show, outside of some very special cases. Additionally, solving the Bellman or HJ-Bellman equation is not enough. One wants the control, and to investigate the effects of implemetation of reasonable approximations. The visualization problem, even in 3 dimensions, is formidable.",
   "evidence": "This record is item A.6, Harold Kushner's participant contribution to the AIM workshop *Numerical methods for optimal control in high dimensions* (August 29--September 2, 2005). The official PDF, version dated August 19, 2005, contains a short position statement rather than a single quantified conjecture. Kushner says that established numerical methods cover broad stochastic and deterministic control classes but face a severe dimensionality barrier beyond four dimensions; he is skeptical of then-current claims for Q-learning and neural-network approximations; and he stresses that solving a Bellman or Hamilton--Jacobi--Bellman (HJB) equation is insufficient because one must extract a control and understand what happens when an approximation is implemented. He also flags visualization in three dimensions as formidable.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-COMPUTATION-0010": {
   "statement_status": "exact",
   "original_statement": "A.7 Mitchell, Ian \n\nThe solution to the Hamilton-Jacobi(-Bellman)(-Isaacs) partial differential equation (HJ PDE) provides the optimal value function (eg cost to go) for bounded input optimal control or zero-sum differential game problems respectively. These equations arise in many disparate fields, and my personal interest lies in how they can be used to approximate the reachable sets of continuous dynamic systems in order to verify and/or synthesize provably safe control policies. For these purposes, it may be possible to relax the goal of a convergent approximation of the solution of the PDE. Instead, we seek guaranteed under or over approximations of the solutions; for example, an underapproximation of the solution might overapproximate the unsafe reachable set, which might thereby generate false negative verifications but no false positives. With or without such relaxations, is it possible to take advantage of structure in the system dynamics to reduce the typically exponential growth in computational cost with system dimension? For example, is it possible to take advantage of weak or linear coupling between certain components to reduce the degrees of freedom needed by an approximation, perhaps through partial parameterization? Or another approach: Can sampling methods be applied to produce quantifiable probabilistic verifications? In addition to these questions related to the application area of verification, I am inter-ested in the class of algorithms designed to approximate solutions of HJ PDEs - often broadly called \"level set methods.\" This interest includes both the time-dependent and stationary (or static) versions of the equation, as well as issues related to the use of such numerical approximations in the control of robotic systems.",
   "clean_statement": "A.7 Mitchell, Ian\n\nThe solution to the Hamilton-Jacobi(-Bellman)(-Isaacs) partial differential equation (HJ PDE) provides the optimal value function (eg cost to go) for bounded input optimal control or zero-sum differential game problems respectively. These equations arise in many disparate fields, and my personal interest lies in how they can be used to approximate the reachable sets of continuous dynamic systems in order to verify and/or synthesize provably safe control policies. For these purposes, it may be possible to relax the goal of a convergent approximation of the solution of the PDE. Instead, we seek guaranteed under or over approximations of the solutions; for example, an underapproximation of the solution might overapproximate the unsafe reachable set, which might thereby generate false negative verifications but no false positives. With or without such relaxations, is it possible to take advantage of structure in the system dynamics to reduce the typically exponential growth in computational cost with system dimension? For example, is it possible to take advantage of weak or linear coupling between certain components to reduce the degrees of freedom needed by an approximation, perhaps through partial parameterization? Or another approach: Can sampling methods be applied to produce quantifiable probabilistic verifications? In addition to these questions related to the application area of verification, I am inter-ested in the class of algorithms designed to approximate solutions of HJ PDEs - often broadly called \"level set methods.\" This interest includes both the time-dependent and stationary (or static) versions of the equation, as well as issues related to the use of such numerical approximations in the control of robotic systems.",
   "public_statement": "A.7 Mitchell, Ian\n\nThe solution to the Hamilton-Jacobi(-Bellman)(-Isaacs) partial differential equation (HJ PDE) provides the optimal value function (eg cost to go) for bounded input optimal control or zero-sum differential game problems respectively. These equations arise in many disparate fields, and my personal interest lies in how they can be used to approximate the reachable sets of continuous dynamic systems in order to verify and/or synthesize provably safe control policies. For these purposes, it may be possible to relax the goal of a convergent approximation of the solution of the PDE. Instead, we seek guaranteed under or over approximations of the solutions; for example, an underapproximation of the solution might overapproximate the unsafe reachable set, which might thereby generate false negative verifications but no false positives. With or without such relaxations, is it possible to take advantage of structure in the system dynamics to reduce the typically exponential growth in computational cost with system dimension? For example, is it possible to take advantage of weak or linear coupling between certain components to reduce the degrees of freedom needed by an approximation, perhaps through partial parameterization? Or another approach: Can sampling methods be applied to produce quantifiable probabilistic verifications? In addition to these questions related to the application area of verification, I am inter-ested in the class of algorithms designed to approximate solutions of HJ PDEs - often broadly called \"level set methods.\" This interest includes both the time-dependent and stationary (or static) versions of the equation, as well as issues related to the use of such numerical approximations in the control of robotic systems.",
   "evidence": "This record is item A.7, attributed to Ian Mitchell, in the 2005 AIM workshop list *Numerical methods for optimal control in high dimensions*. The source is a research agenda rather than a single formally quantified problem. The official PDF was checked directly; A.7 occupies page 7 and ends immediately before A.8. The JSON's `inter-ested` is only line-break hyphenation. The source's `eg` is retained in the extraction and is naturally read as “e.g.”",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-COMPUTATION-0011": {
   "statement_status": "exact",
   "original_statement": "A.8 Oberman, Adam \n\nI am interested in finite difference methods for high dimensional control and stochastic control problems. The types of general questions I am interested in are in 2 groups: - we know in theory that we can solve the PDE for the value function. How can we do this in practice for medium dimensional (3-4-5) problems? - Given an approximate numerical solution of the PDE for 9\n\nthe value function, it isn't trivial to implement a near-optimal control strategy. How do we best do this and can we understand how far from optimal these approximations are? I am curious about how much of a gap there is between theory and practice for nonlinear control problems. Given a control problem, we can write down the PDE for the value function, and we know that this PDE has unique viscosity solutions. From solutions of the PDE, we can then obtain the control. What I would like to know is the following: Suppose we have an approximate solution of the PDE, and we use this approximation to derive a control. How far from optimal will the resulting solution be? Also: suppose the optimal solution swtiches strategy infinitely many times. (This happens even in well-known examples, like stabilizing the linear/nonliner pendulum). Even supposing we had the exact solution, how far off are we from optimal if we approximate the control by one which only switches finitely many times, (which is all that we can do in practice).",
   "clean_statement": "A.8 Oberman, Adam\n\nI am interested in finite difference methods for high dimensional control and stochastic control problems. The types of general questions I am interested in are in 2 groups: - we know in theory that we can solve the PDE for the value function. How can we do this in practice for medium dimensional (3-4-5) problems? - Given an approximate numerical solution of the PDE for 9\n\nthe value function, it isn't trivial to implement a near-optimal control strategy. How do we best do this and can we understand how far from optimal these approximations are? I am curious about how much of a gap there is between theory and practice for nonlinear control problems. Given a control problem, we can write down the PDE for the value function, and we know that this PDE has unique viscosity solutions. From solutions of the PDE, we can then obtain the control. What I would like to know is the following: Suppose we have an approximate solution of the PDE, and we use this approximation to derive a control. How far from optimal will the resulting solution be? Also: suppose the optimal solution swtiches strategy infinitely many times. (This happens even in well-known examples, like stabilizing the linear/nonliner pendulum). Even supposing we had the exact solution, how far off are we from optimal if we approximate the control by one which only switches finitely many times, (which is all that we can do in practice).",
   "public_statement": "A.8 Oberman, Adam\n\nI am interested in finite difference methods for high dimensional control and stochastic control problems. The types of general questions I am interested in are in 2 groups: - we know in theory that we can solve the PDE for the value function. How can we do this in practice for medium dimensional (3-4-5) problems? - Given an approximate numerical solution of the PDE for 9\n\nthe value function, it isn't trivial to implement a near-optimal control strategy. How do we best do this and can we understand how far from optimal these approximations are? I am curious about how much of a gap there is between theory and practice for nonlinear control problems. Given a control problem, we can write down the PDE for the value function, and we know that this PDE has unique viscosity solutions. From solutions of the PDE, we can then obtain the control. What I would like to know is the following: Suppose we have an approximate solution of the PDE, and we use this approximation to derive a control. How far from optimal will the resulting solution be? Also: suppose the optimal solution swtiches strategy infinitely many times. (This happens even in well-known examples, like stabilizing the linear/nonliner pendulum). Even supposing we had the exact solution, how far off are we from optimal if we approximate the control by one which only switches finitely many times, (which is all that we can do in practice).",
   "evidence": "This is item A.8, contributed by Adam Oberman to the AIM workshop *Numerical methods for optimal control in high dimensions*. The record asks three broad questions rather than posing a single theorem:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-COMPUTATION-0012": {
   "statement_status": "exact",
   "original_statement": "A.9 Osher, Stanley \n\nI am interested in solving high dimensional Hamilton-Jacobi equations numerically, especially those arising in control theory and involving the level set method.",
   "clean_statement": "A.9 Osher, Stanley\n\nI am interested in solving high dimensional Hamilton-Jacobi equations numerically, especially those arising in control theory and involving the level set method.",
   "public_statement": "A.9 Osher, Stanley\n\nI am interested in solving high dimensional Hamilton-Jacobi equations numerically, especially those arising in control theory and involving the level set method.",
   "evidence": "The canonical record is item A.9 of the AIM workshop notes *Numerical methods for optimal control in high dimensions* (Stanley Osher):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-COMPUTATION-0013": {
   "statement_status": "exact",
   "original_statement": "A.10 Ostrov, Daniel \n\nMy primary interest in optimal control in multidimensions has come from applications in financial mathematics. For example, I have recently been working (with Jonathan Good-man) on how to determine the best strategy for trading a number of different stocks so that the expected \"utility\" of the stocks is maximized. (The term \"utility\" refers to a specified balance between the desires for a higher return from the stocks and a low total variance.) It is known that in the presence of transactions costs (that is, a charge for buying or selling stocks), there is a \"hold region\" in the multidimensional space of stock prices, and within this region, one should not trade stocks, but on the boundary of the hold region, one trades so that the stocks do not leave this hold region. It is believed, but not known for certain, that in the asymptotic limit as the transaction costs get small, the hold region becomes a multidimensional parallelogram. However, even assuming that this is true, it is not clear what the orientation of the parallelogram should be.",
   "clean_statement": "A.10 Ostrov, Daniel\n\nMy primary interest in optimal control in multidimensions has come from applications in financial mathematics. For example, I have recently been working (with Jonathan Good-man) on how to determine the best strategy for trading a number of different stocks so that the expected \"utility\" of the stocks is maximized. (The term \"utility\" refers to a specified balance between the desires for a higher return from the stocks and a low total variance.) It is known that in the presence of transactions costs (that is, a charge for buying or selling stocks), there is a \"hold region\" in the multidimensional space of stock prices, and within this region, one should not trade stocks, but on the boundary of the hold region, one trades so that the stocks do not leave this hold region. It is believed, but not known for certain, that in the asymptotic limit as the transaction costs get small, the hold region becomes a multidimensional parallelogram. However, even assuming that this is true, it is not clear what the orientation of the parallelogram should be.",
   "public_statement": "A.10 Ostrov, Daniel\n\nMy primary interest in optimal control in multidimensions has come from applications in financial mathematics. For example, I have recently been working (with Jonathan Good-man) on how to determine the best strategy for trading a number of different stocks so that the expected \"utility\" of the stocks is maximized. (The term \"utility\" refers to a specified balance between the desires for a higher return from the stocks and a low total variance.) It is known that in the presence of transactions costs (that is, a charge for buying or selling stocks), there is a \"hold region\" in the multidimensional space of stock prices, and within this region, one should not trade stocks, but on the boundary of the hold region, one trades so that the stocks do not leave this hold region. It is believed, but not known for certain, that in the asymptotic limit as the transaction costs get small, the hold region becomes a multidimensional parallelogram. However, even assuming that this is true, it is not clear what the orientation of the parallelogram should be.",
   "evidence": "This record is item A.10, attributed to Daniel Ostrov, in the AIM workshop list *Numerical methods for optimal control in high dimensions*. The official nine-page PDF was checked directly. A.10 occupies page 9 of the displayed document (PDF page index 8), lines 274--285 in the extracted text. The name printed across a line break as `Good-man` is Jonathan **Goodman**; the hyphen is not part of his surname.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-COMPUTATION-0014": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "1. Spectral efficiency AP: One can consider spectral efficiency (bps/Hz) for (a) a single user (b) multiple users within the same cell (c) multiple cells AP: HDB cannot improve capacity. (This fact has not been proven for the multi-user case, although it is believed to be true.) BH: In the multi-user case, HDB provides frequency diversity. 2. Coverage (SNR + fade margins) AP: SF does not buy coverage. HDB only buys coverage through di-versity. GP: There is a trade-off between coverage and transmission rate. 3. Reliability 14 AP: Reliability is a measure on the statistics of the link and HDB helps, but there is no benefit from SF. AP: Tx processing does not gain over Rx processing. CT: The capacity of a SIMO channel is the same as the capacity of a MISO channel (if CSI is available at Tx), but the SIMO channel does not have SF. 4. Channel estimation AP: For the same amount of power, you have to estimate a lot more parameters in a HDB channel than in a non-HDB channel. This is problematic, especially for the weaker taps. AP: What is the interaction with SF? This relates to iterative TR. 5. Signaling overhead AP: Current systems have about 25-30 % signaling overhead, which eats up spectral efficiency. In these scenarios, iterative TR would not be worth its cost in delay. However, channel estimation is especially important if there is fading. 6. Low probability of intercept AP: In this case, SF is very important. In the context of TR, it can be achieved even with 1 transmit antenna. CDMA technology is not a competitive contender in this regard, because pseudo-random sequences are well-known. LPI applications are probably prepared to throw away bandwidth. In UWB applications, e.g., cable replacement applications, one is not interested in spectral efficiency. Also, the transmit power is limited. GP: In each LPI transmission, the power delivered can be very low, but the power from several transmissions will add up. GP: Are there commercial applications for LPI? Probably not, because they wouldn't tolerate the high rate back-off. In these cases, se-curity is achieved with higher layer mechanisms. BH: You don't have to do a complete rate back-off, but only to the point which the equalizer can handle the channel. 15 VI. Further Discussion on Topics from Open Discussion Session on Thursday",
   "clean_statement": null,
   "public_statement": "1. Spectral efficiency AP: One can consider spectral efficiency (bps/Hz) for (a) a single user (b) multiple users within the same cell (c) multiple cells AP: HDB cannot improve capacity. (This fact has not been proven for the multi-user case, although it is believed to be true.) BH: In the multi-user case, HDB provides frequency diversity. 2. Coverage (SNR + fade margins) AP: SF does not buy coverage. HDB only buys coverage through di-versity. GP: There is a trade-off between coverage and transmission rate. 3. Reliability 14 AP: Reliability is a measure on the statistics of the link and HDB helps, but there is no benefit from SF. AP: Tx processing does not gain over Rx processing. CT: The capacity of a SIMO channel is the same as the capacity of a MISO channel (if CSI is available at Tx), but the SIMO channel does not have SF. 4. Channel estimation AP: For the same amount of power, you have to estimate a lot more parameters in a HDB channel than in a non-HDB channel. This is problematic, especially for the weaker taps. AP: What is the interaction with SF? This relates to iterative TR. 5. Signaling overhead AP: Current systems have about 25-30 % signaling overhead, which eats up spectral efficiency. In these scenarios, iterative TR would not be worth its cost in delay. However, channel estimation is especially important if there is fading. 6. Low probability of intercept AP: In this case, SF is very important. In the context of TR, it can be achieved even with 1 transmit antenna. CDMA technology is not a competitive contender in this regard, because pseudo-random sequences are well-known. LPI applications are probably prepared to throw away bandwidth. In UWB applications, e.g., cable replacement applications, one is not interested in spectral efficiency. Also, the transmit power is limited. GP: In each LPI transmission, the power delivered can be very low, but the power from several transmissions will add up. GP: Are there commercial applications for LPI? Probably not, because they wouldn't tolerate the high rate back-off. In these cases, se-curity is achieved with higher layer mechanisms. BH: You don't have to do a complete rate back-off, but only to the point which the equalizer can handle the channel. 15 VI. Further Discussion on Topics from Open Discussion Session on Thursday",
   "evidence": "The source is not a single numbered theorem. It is Section V, “Open Discussion,” in the Friday session of the 2004 AIM workshop *Time Reversal Communications in Richly Scattering Environments*. The extraction merged all six discussion headings into one record:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-computation-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-COMPUTATION-0015": {
   "statement_status": "exact",
   "original_statement": "1. Equalization complexity AP: MC modulation, e.g., OFDM, can handle the equalization task easily. However, for HDB channels, one would need long symbols, and problems with channel variability and channel estimation arise. Equalization complexity in MC modulation is O(N log N ), whereas in SC modulation, the complexity is exponential in N (N\n\nis the channel length). AP: Linear vs. nonlinear equalization Linear equalization algorithms are not efficient. It appears that nonlinear techniques, e.g., THP, DPC, perform better, but their interactions with SF have not been investigated. AP: Special equalization techniques The problem frequently reduces to that of equalizing a long but sparse channel. These channels are encountered in TV systems, and are handled in DVB-T with OFDM-like techniques. 2. OFDM AP: OFDM is the system of choice for 3.5G, 4G and Wi-Fi systems. The delay spread and the Doppler define the cyclic prefix, symbol length and FFT size for an OFDM system. BH: Coding in OFDM with fine frequency spacing, such as in HDB channels that are very frequency selective, becomes more complex. AP: Spatial focusing OFDM and SF have not been studied. We expect some perfor-mance and complexity trade-offs. 3. Waveform Selection TS: The waveform selection problem is not only relevant to HDB chan-nels. AP: In UWB systems, OFDM and PPM are contenders. 4. Open vs. closed loop techniques 16 AP: What happens when you don't know the channel exactly? What happens when you estimate the channel on the uplink and use this knowledge on the downlink? 17 A Acronyms Acronyms \n\nCDMA Code Division Multiple Access CSI Channel State Information DFE Decision Feedback Equalizer DPC Dirty Paper Coding DVB-T Digital Video Broadcasting - Terrestrial FDD Frequency Division Duplexing FFT Fast Fourier Transform FIR Finite Impulse Response HDB High Delay spread Bandwidth HSDPA High Speed Data Packet Access IIR Infinite Impulse Response ISI Inter Symbol Interference ITU International Telecommunication Union LE Linear Equalizer LMS Least Mean Square LPI Low Probability of Intercept MC Multi Carrier MIMO Multiple Input Multiple Output MISO Multiple Input Single Output ML Maximum Likelihood 18 OFDM Orthogonal Frequency Division Multiplexing PDP Power Delay Profile PPM Pulse Position Modulation QAM Quadrature Amplitude Modulation Rx Receiver SC Single Carrier SDMA Space Division Multiple Access SF Spatial Focusing SIMO Single Input Multiple Output SINR Signal to Interference plus Noise Ratio SNR Signal to Noise Ratio SONAR Sonic Radar TDD Time Division Duplexing THP Tomlinson Harashima Precoding TR Time Reversal TV Television Tx Transmitter UWB Ultra Wideband 19 B Participants Abbreviations of Names of Participants \n\nAA Agissilaos Athanassoulis AF Albert Fannjiang AK Arnold Kim AP Arogyaswami Paulraj BH Babak Hassibi CT Chrysoula Tsogka DB Deborah Berebichez FL Frederick Lee ES Erik Stauffer GD Gregoire Derveaux GL Gabriel Lerosey GP George Papanicolaou HB Harry Bims HS Heechun Song JF Jean-Pierre Fouque JH Jan Hansen JS James Frederick Sifferlen KS Knut Solna LB Liliana Borcea MF Mathias Fink 20 MV Mai Vu MC Mohamad Charafeddine ME Majid Emami OO Ozgur Oyman PK Persefoni Kyritsi RC Robert Calderbank RD Robert Clark Daniels RH Robert Heath RM Raghuraman Mudumbai RN Rohit Nabar ST Stephan ten Brink TS Thomas Strohmer UM Upamanyu Madhow 21",
   "clean_statement": "1. Equalization complexity AP: MC modulation, e.g., OFDM, can handle the equalization task easily. However, for HDB channels, one would need long symbols, and problems with channel variability and channel estimation arise. Equalization complexity in MC modulation is O(N log N ), whereas in SC modulation, the complexity is exponential in N (N\n\nis the channel length). AP: Linear vs. nonlinear equalization Linear equalization algorithms are not efficient. It appears that nonlinear techniques, e.g., THP, DPC, perform better, but their interactions with SF have not been investigated. AP: Special equalization techniques The problem frequently reduces to that of equalizing a long but sparse channel. These channels are encountered in TV systems, and are handled in DVB-T with OFDM-like techniques. 2. OFDM AP: OFDM is the system of choice for 3.5G, 4G and Wi-Fi systems. The delay spread and the Doppler define the cyclic prefix, symbol length and FFT size for an OFDM system. BH: Coding in OFDM with fine frequency spacing, such as in HDB channels that are very frequency selective, becomes more complex. AP: Spatial focusing OFDM and SF have not been studied. We expect some perfor-mance and complexity trade-offs. 3. Waveform Selection TS: The waveform selection problem is not only relevant to HDB chan-nels. AP: In UWB systems, OFDM and PPM are contenders. 4. Open vs. closed loop techniques 16 AP: What happens when you don't know the channel exactly? What happens when you estimate the channel on the uplink and use this knowledge on the downlink? 17 A Acronyms Acronyms\n\nCDMA Code Division Multiple Access CSI Channel State Information DFE Decision Feedback Equalizer DPC Dirty Paper Coding DVB-T Digital Video Broadcasting - Terrestrial FDD Frequency Division Duplexing FFT Fast Fourier Transform FIR Finite Impulse Response HDB High Delay spread Bandwidth HSDPA High Speed Data Packet Access IIR Infinite Impulse Response ISI Inter Symbol Interference ITU International Telecommunication Union LE Linear Equalizer LMS Least Mean Square LPI Low Probability of Intercept MC Multi Carrier MIMO Multiple Input Multiple Output MISO Multiple Input Single Output ML Maximum Likelihood 18 OFDM Orthogonal Frequency Division Multiplexing PDP Power Delay Profile PPM Pulse Position Modulation QAM Quadrature Amplitude Modulation Rx Receiver SC Single Carrier SDMA Space Division Multiple Access SF Spatial Focusing SIMO Single Input Multiple Output SINR Signal to Interference plus Noise Ratio SNR Signal to Noise Ratio SONAR Sonic Radar TDD Time Division Duplexing THP Tomlinson Harashima Precoding TR Time Reversal TV Television Tx Transmitter UWB Ultra Wideband 19 B Participants Abbreviations of Names of Participants\n\nAA Agissilaos Athanassoulis AF Albert Fannjiang AK Arnold Kim AP Arogyaswami Paulraj BH Babak Hassibi CT Chrysoula Tsogka DB Deborah Berebichez FL Frederick Lee ES Erik Stauffer GD Gregoire Derveaux GL Gabriel Lerosey GP George Papanicolaou HB Harry Bims HS Heechun Song JF Jean-Pierre Fouque JH Jan Hansen JS James Frederick Sifferlen KS Knut Solna LB Liliana Borcea MF Mathias Fink 20 MV Mai Vu MC Mohamad Charafeddine ME Majid Emami OO Ozgur Oyman PK Persefoni Kyritsi RC Robert Calderbank RD Robert Clark Daniels RH Robert Heath RM Raghuraman Mudumbai RN Rohit Nabar ST Stephan ten Brink TS Thomas Strohmer UM Upamanyu Madhow 21",
   "public_statement": "1. Equalization complexity AP: MC modulation, e.g., OFDM, can handle the equalization task easily. However, for HDB channels, one would need long symbols, and problems with channel variability and channel estimation arise. Equalization complexity in MC modulation is O(N log N ), whereas in SC modulation, the complexity is exponential in N (N\n\nis the channel length). AP: Linear vs. nonlinear equalization Linear equalization algorithms are not efficient. It appears that nonlinear techniques, e.g., THP, DPC, perform better, but their interactions with SF have not been investigated. AP: Special equalization techniques The problem frequently reduces to that of equalizing a long but sparse channel. These channels are encountered in TV systems, and are handled in DVB-T with OFDM-like techniques. 2. OFDM AP: OFDM is the system of choice for 3.5G, 4G and Wi-Fi systems. The delay spread and the Doppler define the cyclic prefix, symbol length and FFT size for an OFDM system. BH: Coding in OFDM with fine frequency spacing, such as in HDB channels that are very frequency selective, becomes more complex. AP: Spatial focusing OFDM and SF have not been studied. We expect some perfor-mance and complexity trade-offs. 3. Waveform Selection TS: The waveform selection problem is not only relevant to HDB chan-nels. AP: In UWB systems, OFDM and PPM are contenders. 4. Open vs. closed loop techniques 16 AP: What happens when you don't know the channel exactly? What happens when you estimate the channel on the uplink and use this knowledge on the downlink? 17 A Acronyms Acronyms\n\nCDMA Code Division Multiple Access CSI Channel State Information DFE Decision Feedback Equalizer DPC Dirty Paper Coding DVB-T Digital Video Broadcasting - Terrestrial FDD Frequency Division Duplexing FFT Fast Fourier Transform FIR Finite Impulse Response HDB High Delay spread Bandwidth HSDPA High Speed Data Packet Access IIR Infinite Impulse Response ISI Inter Symbol Interference ITU International Telecommunication Union LE Linear Equalizer LMS Least Mean Square LPI Low Probability of Intercept MC Multi Carrier MIMO Multiple Input Multiple Output MISO Multiple Input Single Output ML Maximum Likelihood 18 OFDM Orthogonal Frequency Division Multiplexing PDP Power Delay Profile PPM Pulse Position Modulation QAM Quadrature Amplitude Modulation Rx Receiver SC Single Carrier SDMA Space Division Multiple Access SF Spatial Focusing SIMO Single Input Multiple Output SINR Signal to Interference plus Noise Ratio SNR Signal to Noise Ratio SONAR Sonic Radar TDD Time Division Duplexing THP Tomlinson Harashima Precoding TR Time Reversal TV Television Tx Transmitter UWB Ultra Wideband 19 B Participants Abbreviations of Names of Participants\n\nAA Agissilaos Athanassoulis AF Albert Fannjiang AK Arnold Kim AP Arogyaswami Paulraj BH Babak Hassibi CT Chrysoula Tsogka DB Deborah Berebichez FL Frederick Lee ES Erik Stauffer GD Gregoire Derveaux GL Gabriel Lerosey GP George Papanicolaou HB Harry Bims HS Heechun Song JF Jean-Pierre Fouque JH Jan Hansen JS James Frederick Sifferlen KS Knut Solna LB Liliana Borcea MF Mathias Fink 20 MV Mai Vu MC Mohamad Charafeddine ME Majid Emami OO Ozgur Oyman PK Persefoni Kyritsi RC Robert Calderbank RD Robert Clark Daniels RH Robert Heath RM Raghuraman Mudumbai RN Rohit Nabar ST Stephan ten Brink TS Thomas Strohmer UM Upamanyu Madhow 21",
   "evidence": "The canonical record comes from the final open-discussion section of the AIM workshop notes *Time-reversal communications in richly scattering environments* (18--22 October 2004). The official PDF was inspected directly. The mathematical discussion occupies PDF pages 16--17 (PDF indices 15--16), and has four branches:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-COMPUTATION-0016": {
   "statement_status": "exact",
   "original_statement": "Problem 1. Explicitly describe a Markov basis for tables with fixed first-order summary. That is, describe a set of moves which will connect any two surveys that have the same first order summary. In a purely mathematical language: describe a generating set for the toric ideal given by the p2 × p! matrix whose columns are the p! p × p permutation matrices. \n\nContingency tables with quadratic statistics.",
   "clean_statement": "Problem 1. Explicitly describe a Markov basis for tables with fixed first-order summary. That is, describe a set of moves which will connect any two surveys that have the same first order summary. In a purely mathematical language: describe a generating set for the toric ideal given by the p2 × p! matrix whose columns are the p! p × p permutation matrices.\n\nContingency tables with quadratic statistics.",
   "public_statement": "Problem 1. Explicitly describe a Markov basis for tables with fixed first-order summary. That is, describe a set of moves which will connect any two surveys that have the same first order summary. In a purely mathematical language: describe a generating set for the toric ideal given by the p2 × p! matrix whose columns are the p! p × p permutation matrices.\n\nContingency tables with quadratic statistics.",
   "evidence": "The official AIM PDF *Computational Algebraic Statistics*, version 27 March 2004, was checked directly. On displayed page 3 (PDF page index 2), under Persi Diaconis's heading “Fixed First-Order Summaries,” it defines a survey as a frequency function $f:S_p\\to\\mathbb N$. Its first-order summary is the $p\\times p$ table whose $(i,j)$ entry counts voters who put candidate $i$ in position $j$. The problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-COMPUTATION-0017": {
   "statement_status": "exact",
   "original_statement": "Problem 2. Find a Markov basis for studying the set X of 2-way tables with a fixed quadratic statistic: \n\nX = {T | ∑\n\n> j\n\nTij = Ri, ∑\n\n> i\n\nTij = Cj and ∑\n\n> i,j\n\n(i − 2)( j − 2) Tij = Q}.\n\nFor 3 × 4 tables, the matrix that computes the sufficient statistics (i.e. the matrix which represents the log-linear model) is the following: \n\n\n\n1 1 1 1 0 0 0 0 0 0 0 00 0 0 0 1 1 1 1 0 0 0 00 0 0 0 0 0 0 0 1 1 1 11 0 0 0 1 0 0 0 1 0 0 00 1 0 0 0 1 0 0 0 1 0 00 0 1 0 0 0 1 0 0 0 1 00 0 0 1 0 0 0 1 0 0 0 11 0 −1 −2 0 0 0 0 −1 0 1 2\n\n.\n\nPolyhedral Cones and Testing. \n\nObserve \n\nXij = {10|P (Xij |Xij ) = Z−1(θ)i)exij θj }4\n\nTest θi = θ∀i against A · θ ≥ 0 with A fixed. \n\nExample 3. Possible conditions enforced by A · · · θ ≥ 0 are \n\nθ1 ≥ θ2 ≥ · · · ≥ θI\n\nor \n\nθi ≥ θI ∀i. \n\nThere should be a nice interaction between statistics and polyhedral combinatorics in this problem. \n\nCompare \"Competing\" Techniques. \n\nA. Classical Asymptotics B. Edgeworth Corrections C. StatXact D. Sequential Importance Sampling E. Others How do these different techniques for estimating p-values relate to each other? With respect to sequential importance sampling: what are possible multi-way gener-alizations of the Gale-Ryser theorem which gives necessary and sufficient conditions for the existence of a 0 /1 matrix with fixed row and column sums? \n\nStephen Fienberg \n\nQuestions about Odds-Ratios.",
   "clean_statement": "Problem 2. Find a Markov basis for studying the set X of 2-way tables with a fixed quadratic statistic:\n\nX = {T | ∑\n\n> j\n\nTij = Ri, ∑\n\n> i\n\nTij = Cj and ∑\n\n> i,j\n\n(i − 2)( j − 2) Tij = Q}.\n\nFor 3 × 4 tables, the matrix that computes the sufficient statistics (i.e. the matrix which represents the log-linear model) is the following:\n\n\n\n1 1 1 1 0 0 0 0 0 0 0 00 0 0 0 1 1 1 1 0 0 0 00 0 0 0 0 0 0 0 1 1 1 11 0 0 0 1 0 0 0 1 0 0 00 1 0 0 0 1 0 0 0 1 0 00 0 1 0 0 0 1 0 0 0 1 00 0 0 1 0 0 0 1 0 0 0 11 0 −1 −2 0 0 0 0 −1 0 1 2\n\n.\n\nPolyhedral Cones and Testing.\n\nObserve\n\nXij = {10|P (Xij |Xij ) = Z−1(θ)i)exij θj }4\n\nTest θi = θ∀i against A · θ ≥ 0 with A fixed.\n\nExample 3. Possible conditions enforced by A · · · θ ≥ 0 are\n\nθ1 ≥ θ2 ≥ · · · ≥ θI\n\nor\n\nθi ≥ θI ∀i.\n\nThere should be a nice interaction between statistics and polyhedral combinatorics in this problem.\n\nCompare \"Competing\" Techniques.\n\nA. Classical Asymptotics B. Edgeworth Corrections C. StatXact D. Sequential Importance Sampling E. Others How do these different techniques for estimating p-values relate to each other? With respect to sequential importance sampling: what are possible multi-way gener-alizations of the Gale-Ryser theorem which gives necessary and sufficient conditions for the existence of a 0 /1 matrix with fixed row and column sums?\n\nStephen Fienberg\n\nQuestions about Odds-Ratios.",
   "public_statement": "Problem 2. Find a Markov basis for studying the set X of 2-way tables with a fixed quadratic statistic:\n\nX = {T | ∑\n\n> j\n\nTij = Ri, ∑\n\n> i\n\nTij = Cj and ∑\n\n> i,j\n\n(i − 2)( j − 2) Tij = Q}.\n\nFor 3 × 4 tables, the matrix that computes the sufficient statistics (i.e. the matrix which represents the log-linear model) is the following:\n\n\n\n1 1 1 1 0 0 0 0 0 0 0 00 0 0 0 1 1 1 1 0 0 0 00 0 0 0 0 0 0 0 1 1 1 11 0 0 0 1 0 0 0 1 0 0 00 1 0 0 0 1 0 0 0 1 0 00 0 1 0 0 0 1 0 0 0 1 00 0 0 1 0 0 0 1 0 0 0 11 0 −1 −2 0 0 0 0 −1 0 1 2\n\n.\n\nPolyhedral Cones and Testing.\n\nObserve\n\nXij = {10|P (Xij |Xij ) = Z−1(θ)i)exij θj }4\n\nTest θi = θ∀i against A · θ ≥ 0 with A fixed.\n\nExample 3. Possible conditions enforced by A · · · θ ≥ 0 are\n\nθ1 ≥ θ2 ≥ · · · ≥ θI\n\nor\n\nθi ≥ θI ∀i.\n\nThere should be a nice interaction between statistics and polyhedral combinatorics in this problem.\n\nCompare \"Competing\" Techniques.\n\nA. Classical Asymptotics B. Edgeworth Corrections C. StatXact D. Sequential Importance Sampling E. Others How do these different techniques for estimating p-values relate to each other? With respect to sequential importance sampling: what are possible multi-way gener-alizations of the Gale-Ryser theorem which gives necessary and sufficient conditions for the existence of a 0 /1 matrix with fixed row and column sums?\n\nStephen Fienberg\n\nQuestions about Odds-Ratios.",
   "evidence": "The canonical JSON record contains extraction spillover. The authoritative AIM PDF, *Computational Algebraic Statistics* (version dated 27 March 2004), gives the following problem and then ends it immediately after the displayed matrix:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-COMPUTATION-0018": {
   "statement_status": "exact",
   "original_statement": "Question 4. For a 2 × 2 table of probabilities \n\n( p00 p01 \n\np10 p11 \n\n): does it makes sense to fix values of two of the odds ratios α = p00 p11 \n\n> p01 p10\n\nand α∗ = p00 p01 \n\n> p11 p10\n\nand look at the locus of possible values for the third odds-ratio α∗∗ = p00 p10 \n\n> p01 p11?Describe the curve (in the probability simplex) obtained by intersecting two of these quadric surfaces.",
   "clean_statement": "Question 4. For a 2 × 2 table of probabilities\n\n( p00 p01\n\np10 p11\n\n): does it makes sense to fix values of two of the odds ratios α = p00 p11\n\n> p01 p10\n\nand α∗ = p00 p01\n\n> p11 p10\n\nand look at the locus of possible values for the third odds-ratio α∗∗ = p00 p10\n\n> p01 p11?Describe the curve (in the probability simplex) obtained by intersecting two of these quadric surfaces.",
   "public_statement": "Question 4. For a 2 × 2 table of probabilities\n\n( p00 p01\n\np10 p11\n\n): does it makes sense to fix values of two of the odds ratios α = p00 p11\n\n> p01 p10\n\nand α∗ = p00 p01\n\n> p11 p10\n\nand look at the locus of possible values for the third odds-ratio α∗∗ = p00 p10\n\n> p01 p11?Describe the curve (in the probability simplex) obtained by intersecting two of these quadric surfaces.",
   "evidence": "Write",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-COMPUTATION-0019": {
   "statement_status": "exact",
   "original_statement": "Question 5. What is the algebro-geometric structure of passing from local to global odds ratios? Note that local odds ratio is obtained by looking at a 2 k subtable of a given table whereas a global odds ratio is obtained by looking at the odds-ratio in a 2 k collapsing of a given table.",
   "clean_statement": "Question 5. What is the algebro-geometric structure of passing from local to global odds ratios? Note that local odds ratio is obtained by looking at a 2 k subtable of a given table whereas a global odds ratio is obtained by looking at the odds-ratio in a 2 k collapsing of a given table.",
   "public_statement": "Question 5. What is the algebro-geometric structure of passing from local to global odds ratios? Note that local odds ratio is obtained by looking at a 2 k subtable of a given table whereas a global odds ratio is obtained by looking at the odds-ratio in a 2 k collapsing of a given table.",
   "evidence": "The canonical record reads “\\(2 k\\)” twice because superscripts were lost during extraction. The official AIM HTML source preserves the mathematical alternative text and resolves both occurrences as \\(2^k\\). The verified statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-COMPUTATION-0020": {
   "statement_status": "exact",
   "original_statement": "Question 6. For multi-way tables, what are the complete and partial specifications that come from specifying sets of odds-ratios? \n\nExistence of Maximum Likelihood Estimates.",
   "clean_statement": "Question 6. For multi-way tables, what are the complete and partial specifications that come from specifying sets of odds-ratios?\n\nExistence of Maximum Likelihood Estimates.",
   "public_statement": "Question 6. For multi-way tables, what are the complete and partial specifications that come from specifying sets of odds-ratios?\n\nExistence of Maximum Likelihood Estimates.",
   "evidence": "The official AIM PDF *Computational Algebraic Statistics*, version 27 March 2004, was inspected directly. On displayed page 4 (PDF page index 3), Stephen Fienberg's subsection is headed “Questions about Odds-Ratios.” It first asks about three multiplicative contrasts in a positive \\(2\\times2\\) probability table (Question 4), then distinguishes local odds ratios on \\(2^k\\) subtables from global odds ratios formed after a \\(2^k\\) collapsing (Question 5). The assigned statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-COMPUTATION-0021": {
   "statement_status": "exact",
   "original_statement": "Question 7. What conditions on the structures of zeros in a table of counts will guarantee that the maximum likelihood estimates do not exist? In this setting, a maximum likelihood estimate is required to have all strictly positive probabilities. 5\n\nAs Alessandro Rinaldo, later pointed out in his talk, this is equivalent to giving a combinatorial description of the facets of the cone of all feasible margins. This polyhedral cone is obtained by taking of the positive hull of the columns of the matrices AG which appear in the working questions on graphical models by Chris Meek and Serkan Ho¸ sten.",
   "clean_statement": "Question 7. What conditions on the structures of zeros in a table of counts will guarantee that the maximum likelihood estimates do not exist? In this setting, a maximum likelihood estimate is required to have all strictly positive probabilities. 5\n\nAs Alessandro Rinaldo, later pointed out in his talk, this is equivalent to giving a combinatorial description of the facets of the cone of all feasible margins. This polyhedral cone is obtained by taking of the positive hull of the columns of the matrices AG which appear in the working questions on graphical models by Chris Meek and Serkan Ho¸ sten.",
   "public_statement": "Question 7. What conditions on the structures of zeros in a table of counts will guarantee that the maximum likelihood estimates do not exist? In this setting, a maximum likelihood estimate is required to have all strictly positive probabilities. 5\n\nAs Alessandro Rinaldo, later pointed out in his talk, this is equivalent to giving a combinatorial description of the facets of the cone of all feasible margins. This polyhedral cone is obtained by taking of the positive hull of the columns of the matrices AG which appear in the working questions on graphical models by Chris Meek and Serkan Ho¸ sten.",
   "evidence": "The source is Question 7 in the AIM workshop notes *Computational algebraic statistics* (version dated 27 March 2004). The extracted record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-COMPUTATION-0022": {
   "statement_status": "exact",
   "original_statement": "Conjecture 8. Maximum likelihood estimates for an I × J × K table under the no 3-way interaction model do not exist if and only if they do not exist for all collapsings of the table to a 2 × 2 × 2 table. \n\nLater in the week, Nicholas Eriksson showed that this conjecture is false already for 3 × 3 × 3 tables. Mathias Drton asked \"How can we incorporate structural zeroes into this problem?\" Stephen Roehrig asked \"Is it possible to relate the MLE solutions for decomposable models to decomposable network flow problems?\" \n\nAkimichi Takemura",
   "clean_statement": "Conjecture 8. Maximum likelihood estimates for an I × J × K table under the no 3-way interaction model do not exist if and only if they do not exist for all collapsings of the table to a 2 × 2 × 2 table.\n\nLater in the week, Nicholas Eriksson showed that this conjecture is false already for 3 × 3 × 3 tables. Mathias Drton asked \"How can we incorporate structural zeroes into this problem?\" Stephen Roehrig asked \"Is it possible to relate the MLE solutions for decomposable models to decomposable network flow problems?\"\n\nAkimichi Takemura",
   "public_statement": "Conjecture 8. Maximum likelihood estimates for an I × J × K table under the no 3-way interaction model do not exist if and only if they do not exist for all collapsings of the table to a 2 × 2 × 2 table.\n\nLater in the week, Nicholas Eriksson showed that this conjecture is false already for 3 × 3 × 3 tables. Mathias Drton asked \"How can we incorporate structural zeroes into this problem?\" Stephen Roehrig asked \"Is it possible to relate the MLE solutions for decomposable models to decomposable network flow problems?\"\n\nAkimichi Takemura",
   "evidence": "The source is Conjecture 8 in the AIM workshop notes *Computational algebraic statistics* (version dated 27 March 2004). The official PDF says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-COMPUTATION-0023": {
   "statement_status": "exact",
   "original_statement": "Question 9. Do I × J × K tables under the no 3-way interaction model always have a unique minimal Markov basis? Bernd Sturmfels asked, \"What about problems for 3-way tables that have structural zeros coming from a very regular design?\" Henry Wynn asked, \"Does the group structure of circuits play a role in the solution of this problem?\" \n\nEmily Gamundi",
   "clean_statement": "Question 9. Do I × J × K tables under the no 3-way interaction model always have a unique minimal Markov basis? Bernd Sturmfels asked, \"What about problems for 3-way tables that have structural zeros coming from a very regular design?\" Henry Wynn asked, \"Does the group structure of circuits play a role in the solution of this problem?\"\n\nEmily Gamundi",
   "public_statement": "Question 9. Do I × J × K tables under the no 3-way interaction model always have a unique minimal Markov basis? Bernd Sturmfels asked, \"What about problems for 3-way tables that have structural zeros coming from a very regular design?\" Henry Wynn asked, \"Does the group structure of circuits play a role in the solution of this problem?\"\n\nEmily Gamundi",
   "evidence": "Question 9 in the AIM workshop notes *Computational algebraic statistics* (version dated 27 March 2004) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-COMPUTATION-0024": {
   "statement_status": "exact",
   "original_statement": "Question 10. Are there always only 0, 1, or 2 statistically interesting solutions to the blocking probability equations in network tomography? \n\nRuriko Yoshida \n\nRuriko Yoshida discussed short encodings of the set of points in polytopes by means of rational generating functions, and the applications of rational generating functions for giving short encodings of Markov bases. Ian Dinwoodie asked, \"Is there a way to extract (random) monomials from the gener-ating functions?\" Akimichi Takemura asked, \"Is there a way to extract only the absolutely essential Markov basis elements that are needed to connect a given fiber of the form {Ax = b|x ∈ Nd}?Ruriko answered,\"We are still attempting to solve these problems.\" Henry Wynn asked, \" Can these techniques be used to approximate the number of integral points inside an arbitrary convex set?\" \n\nMathias Drton \n\nMathias Drton discussed multimodality of the likelihood function for the model of seemingly unrelated regression (SUR). These can be thought of as conditional independence models induced by the chain graph with two sets of nodes, ( X1,..., X n) and ( Y1,..., Y n)and directed edges from Xi → Yi and undirected edges between any pair Yi − Yj.6",
   "clean_statement": "Question 10. Are there always only 0, 1, or 2 statistically interesting solutions to the blocking probability equations in network tomography?\n\nRuriko Yoshida\n\nRuriko Yoshida discussed short encodings of the set of points in polytopes by means of rational generating functions, and the applications of rational generating functions for giving short encodings of Markov bases. Ian Dinwoodie asked, \"Is there a way to extract (random) monomials from the gener-ating functions?\" Akimichi Takemura asked, \"Is there a way to extract only the absolutely essential Markov basis elements that are needed to connect a given fiber of the form {Ax = b|x ∈ Nd}?Ruriko answered,\"We are still attempting to solve these problems.\" Henry Wynn asked, \" Can these techniques be used to approximate the number of integral points inside an arbitrary convex set?\"\n\nMathias Drton\n\nMathias Drton discussed multimodality of the likelihood function for the model of seemingly unrelated regression (SUR). These can be thought of as conditional independence models induced by the chain graph with two sets of nodes, ( X1,..., X n) and ( Y1,..., Y n)and directed edges from Xi → Yi and undirected edges between any pair Yi − Yj.6",
   "public_statement": "Question 10. Are there always only 0, 1, or 2 statistically interesting solutions to the blocking probability equations in network tomography?\n\nRuriko Yoshida\n\nRuriko Yoshida discussed short encodings of the set of points in polytopes by means of rational generating functions, and the applications of rational generating functions for giving short encodings of Markov bases. Ian Dinwoodie asked, \"Is there a way to extract (random) monomials from the gener-ating functions?\" Akimichi Takemura asked, \"Is there a way to extract only the absolutely essential Markov basis elements that are needed to connect a given fiber of the form {Ax = b|x ∈ Nd}?Ruriko answered,\"We are still attempting to solve these problems.\" Henry Wynn asked, \" Can these techniques be used to approximate the number of integral points inside an arbitrary convex set?\"\n\nMathias Drton\n\nMathias Drton discussed multimodality of the likelihood function for the model of seemingly unrelated regression (SUR). These can be thought of as conditional independence models induced by the chain graph with two sets of nodes, ( X1,..., X n) and ( Y1,..., Y n)and directed edges from Xi → Yi and undirected edges between any pair Yi − Yj.6",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-COMPUTATION-0025": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 11. Can algebraic techniques be used to see the unimodality of the likelihood function in a MANOVA model. These models correspond to the chain graphs as described above but with the extra edges Xi → Yj for each i and j.Akimichi Takemura asked,\"As the number of samples grows, how quickly does the likelihood converge to a unimodal function in the bivariate SUR model?\" \n\nAleksandra Slavkovic \n\nAleksandra Slavkovic spoke about disclosure limitation problems associated with the releasse of marginals and conditionals. Since fixing conditionals is a linear constraint, the theory of Markov bases can be applied to the problem of deciding how many tables have the given fixed conditionals.",
   "clean_statement": null,
   "public_statement": "Question 11. Can algebraic techniques be used to see the unimodality of the likelihood function in a MANOVA model. These models correspond to the chain graphs as described above but with the extra edges Xi → Yj for each i and j.Akimichi Takemura asked,\"As the number of samples grows, how quickly does the likelihood converge to a unimodal function in the bivariate SUR model?\"\n\nAleksandra Slavkovic\n\nAleksandra Slavkovic spoke about disclosure limitation problems associated with the releasse of marginals and conditionals. Since fixing conditionals is a linear constraint, the theory of Markov bases can be applied to the problem of deciding how many tables have the given fixed conditionals.",
   "evidence": "The source is Question 11 in the AIM workshop notes *Computational algebraic statistics*. The relevant text in the official PDF is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-computation-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-COMPUTATION-0026": {
   "statement_status": "exact",
   "original_statement": "Question 12. What can be done with Markov bases when their are floating point approxi-mations made in marginals or conditionals?",
   "clean_statement": "Question 12. What can be done with Markov bases when their are floating point approxi-mations made in marginals or conditionals?",
   "public_statement": "Question 12. What can be done with Markov bases when their are floating point approxi-mations made in marginals or conditionals?",
   "evidence": "The exact extracted record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-COMPUTATION-0027": {
   "statement_status": "exact",
   "original_statement": "Question 13. What is the family of distributions that has margins and conditionals as sufficient statistics? What should the stationary distribution be for MCMC? Jon Forster asked, \"Is it safe to release full conditionals that have been perturbed?\" Aleksandra replied, \"Probably not.\" \n\nLuis Garcia \n\nLuis Garcia spoke about the algebraic structure of the probability distributions which arise from graphical models with hidden variables. Chris Meek asked, \"What is the algebraic characterization of the Verma constraints? (non-independence constraints)\" Mathias Drton asked, \"Are these the same constraints described by Richardson and Spirtes (Annals of Statistics 2002)?\" Henry Wynn asked, \"How do inequalities play a role in this problem?\" \n\nHenry Wynn \n\nHenry Wynn spoke about formulae relating cumulants to moments.",
   "clean_statement": "Question 13. What is the family of distributions that has margins and conditionals as sufficient statistics? What should the stationary distribution be for MCMC? Jon Forster asked, \"Is it safe to release full conditionals that have been perturbed?\" Aleksandra replied, \"Probably not.\"\n\nLuis Garcia\n\nLuis Garcia spoke about the algebraic structure of the probability distributions which arise from graphical models with hidden variables. Chris Meek asked, \"What is the algebraic characterization of the Verma constraints? (non-independence constraints)\" Mathias Drton asked, \"Are these the same constraints described by Richardson and Spirtes (Annals of Statistics 2002)?\" Henry Wynn asked, \"How do inequalities play a role in this problem?\"\n\nHenry Wynn\n\nHenry Wynn spoke about formulae relating cumulants to moments.",
   "public_statement": "Question 13. What is the family of distributions that has margins and conditionals as sufficient statistics? What should the stationary distribution be for MCMC? Jon Forster asked, \"Is it safe to release full conditionals that have been perturbed?\" Aleksandra replied, \"Probably not.\"\n\nLuis Garcia\n\nLuis Garcia spoke about the algebraic structure of the probability distributions which arise from graphical models with hidden variables. Chris Meek asked, \"What is the algebraic characterization of the Verma constraints? (non-independence constraints)\" Mathias Drton asked, \"Are these the same constraints described by Richardson and Spirtes (Annals of Statistics 2002)?\" Henry Wynn asked, \"How do inequalities play a role in this problem?\"\n\nHenry Wynn\n\nHenry Wynn spoke about formulae relating cumulants to moments.",
   "evidence": "This record comes from the AIM workshop *Computational algebraic statistics*, Question 13. The canonical JSON extraction contains later speaker material, but inspection of the workshop PDF shows that the question ends before the heading “Luis Garcia.” The recovered statement is therefore:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-COMPUTATION-0028": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 14. Use the transformations from probabilities to moments to cumulants to de-scribe independence models/ hierarchical models in terms of polynomial functions in the cumulants.",
   "clean_statement": null,
   "public_statement": "Problem 14. Use the transformations from probabilities to moments to cumulants to de-scribe independence models/ hierarchical models in terms of polynomial functions in the cumulants.",
   "evidence": "The exact extracted record is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-computation-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-COMPUTATION-0029": {
   "statement_status": "exact",
   "original_statement": "Question 15. What is the cumulant ideal of the binary four-cycle model? \n\nRussell Steele \n\nRussell Steele spoke about mixture models, model selection, and the Bayesian Infor-mation Criterion (BIC).",
   "clean_statement": "Question 15. What is the cumulant ideal of the binary four-cycle model?\n\nRussell Steele\n\nRussell Steele spoke about mixture models, model selection, and the Bayesian Infor-mation Criterion (BIC).",
   "public_statement": "Question 15. What is the cumulant ideal of the binary four-cycle model?\n\nRussell Steele\n\nRussell Steele spoke about mixture models, model selection, and the Bayesian Infor-mation Criterion (BIC).",
   "evidence": "The source is the AIM workshop list *Computational Algebraic Statistics* (PDF version dated 27 March 2004). The recovered question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-COMPUTATION-0030": {
   "statement_status": "exact",
   "original_statement": "Question 16. Computing normalization constants and posterior distributions for continuous random variables involves the computation of integrals involving rational and transcendental funtions. Can algebraic techniques be used to give exact expressions for these integrals or help give approximations for them? 7",
   "clean_statement": "Question 16. Computing normalization constants and posterior distributions for continuous random variables involves the computation of integrals involving rational and transcendental funtions. Can algebraic techniques be used to give exact expressions for these integrals or help give approximations for them? 7",
   "public_statement": "Question 16. Computing normalization constants and posterior distributions for continuous random variables involves the computation of integrals involving rational and transcendental funtions. Can algebraic techniques be used to give exact expressions for these integrals or help give approximations for them? 7",
   "evidence": "The record is Question 16 in the AIM workshop problem list *Computational algebraic statistics* (version dated March 27, 2004). It occurs under the speaker heading “Russell Steele,” after a sentence about mixture models, model selection, and BIC. Inspection of the official PDF gives the source text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-COMPUTATION-0031": {
   "statement_status": "exact",
   "original_statement": "Question 17. Why does the BIC work for model fitting with mixture models? What is being approximated (and why is it good to approximate these things?) This question raised some controversy during the discussion following the talk which was later resolved. The main issue that is interesting in this situation is that the likelihood function for mixture models does not satisfy the necessary regularity conditions to apply the classical results about the BIC.",
   "clean_statement": "Question 17. Why does the BIC work for model fitting with mixture models? What is being approximated (and why is it good to approximate these things?) This question raised some controversy during the discussion following the talk which was later resolved. The main issue that is interesting in this situation is that the likelihood function for mixture models does not satisfy the necessary regularity conditions to apply the classical results about the BIC.",
   "public_statement": "Question 17. Why does the BIC work for model fitting with mixture models? What is being approximated (and why is it good to approximate these things?) This question raised some controversy during the discussion following the talk which was later resolved. The main issue that is interesting in this situation is that the likelihood function for mixture models does not satisfy the necessary regularity conditions to apply the classical results about the BIC.",
   "evidence": "The canonical record is Question 17 from the AIM workshop *Computational algebraic statistics*. The source PDF was inspected directly. The preceding heading says that Russell Steele spoke about mixture models, model selection, and the Bayesian Information Criterion (BIC). The exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-COMPUTATION-0032": {
   "statement_status": "exact",
   "original_statement": "Question 18. Can algebra be used to better approximate the location of the modes in a mixture?",
   "clean_statement": "Question 18. Can algebra be used to better approximate the location of the modes in a mixture?",
   "public_statement": "Question 18. Can algebra be used to better approximate the location of the modes in a mixture?",
   "evidence": "The exact AIM question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-COMPUTATION-0033": {
   "statement_status": "exact",
   "original_statement": "Question 19. Can algebraic techniques be used to better imputing missing categorical variable?",
   "clean_statement": "Question 19. Can algebraic techniques be used to better imputing missing categorical variable?",
   "public_statement": "Question 19. Can algebraic techniques be used to better imputing missing categorical variable?",
   "evidence": "The record is Question 19 in the AIM workshop problem list *Computational algebraic statistics*, version dated March 27, 2004. Inspection of the official PDF places it under Russell Steele, immediately after Question 18 about locating modes in a mixture and immediately before Question 20 about collapsing levels of categorical variables for multiple imputation. The source says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-COMPUTATION-0034": {
   "statement_status": "exact",
   "original_statement": "Question 20. Can algebraic techniques be used to collapse levels of categorical variables in order to improve the quality of multiple imputation without destroying the quality of inference? \n\nFrantisek Matus \n\nFrantisek Matus spoke about the representability of conditional independence (CI) structures.",
   "clean_statement": "Question 20. Can algebraic techniques be used to collapse levels of categorical variables in order to improve the quality of multiple imputation without destroying the quality of inference?\n\nFrantisek Matus\n\nFrantisek Matus spoke about the representability of conditional independence (CI) structures.",
   "public_statement": "Question 20. Can algebraic techniques be used to collapse levels of categorical variables in order to improve the quality of multiple imputation without destroying the quality of inference?\n\nFrantisek Matus\n\nFrantisek Matus spoke about the representability of conditional independence (CI) structures.",
   "evidence": "The canonical record is Question 20 from the AIM workshop list *Computational algebraic statistics*. Direct inspection of the official PDF and its neighboring text shows the following layout:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-COMPUTATION-0035": {
   "statement_status": "exact",
   "original_statement": "Conjecture 21. Every p-representable CI-structure can be p-represented by a random vector living on a finite probability space endowed with the uniform distribution.",
   "clean_statement": "Conjecture 21. Every p-representable CI-structure can be p-represented by a random vector living on a finite probability space endowed with the uniform distribution.",
   "public_statement": "Conjecture 21. Every p-representable CI-structure can be p-represented by a random vector living on a finite probability space endowed with the uniform distribution.",
   "evidence": "The canonical record is Conjecture 21 from the AIM workshop list *Computational algebraic statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-COMPUTATION-0036": {
   "statement_status": "exact",
   "original_statement": "Conjecture 22. There exists a connected matroid whose CI-structure is p-representable but not multilinear. \n\nSuppose independence vatieties are properly defined, also for the CI constraints involv-ing functional independences, over the field of complex numbers.",
   "clean_statement": "Conjecture 22. There exists a connected matroid whose CI-structure is p-representable but not multilinear.\n\nSuppose independence vatieties are properly defined, also for the CI constraints involv-ing functional independences, over the field of complex numbers.",
   "public_statement": "Conjecture 22. There exists a connected matroid whose CI-structure is p-representable but not multilinear.\n\nSuppose independence vatieties are properly defined, also for the CI constraints involv-ing functional independences, over the field of complex numbers.",
   "evidence": "The exact conjecture in the official AIM *Computational Algebraic Statistics* workshop report is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-COMPUTATION-0037": {
   "statement_status": "exact",
   "original_statement": "Conjecture 23. When P belongs to such a zero dimensional independence variety then P\n\nis the distribution of a random vector with its CI-structure equal to the CI-structure of a matroid. \n\nDonald Richards \n\nDonald Richards spoke on the problem of determining the number of solutions to generic maximum likelihood problems and the use of computational algebra, homotopy con-tinuation, and mixed volume in these calculations.",
   "clean_statement": "Conjecture 23. When P belongs to such a zero dimensional independence variety then P\n\nis the distribution of a random vector with its CI-structure equal to the CI-structure of a matroid.\n\nDonald Richards\n\nDonald Richards spoke on the problem of determining the number of solutions to generic maximum likelihood problems and the use of computational algebra, homotopy con-tinuation, and mixed volume in these calculations.",
   "public_statement": "Conjecture 23. When P belongs to such a zero dimensional independence variety then P\n\nis the distribution of a random vector with its CI-structure equal to the CI-structure of a matroid.\n\nDonald Richards\n\nDonald Richards spoke on the problem of determining the number of solutions to generic maximum likelihood problems and the use of computational algebra, homotopy con-tinuation, and mixed volume in these calculations.",
   "evidence": "The canonical record contains:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-COMPUTATION-0038": {
   "statement_status": "exact",
   "original_statement": "Problem 24 (Behrens-Fischer Problem). Solve the maximum likelihood equations for the mixture of two multivariate Gaussians with the same mean and different covariance matrices.",
   "clean_statement": "Problem 24 (Behrens-Fischer Problem). Solve the maximum likelihood equations for the mixture of two multivariate Gaussians with the same mean and different covariance matrices.",
   "public_statement": "Problem 24 (Behrens-Fischer Problem). Solve the maximum likelihood equations for the mixture of two multivariate Gaussians with the same mean and different covariance matrices.",
   "evidence": "The canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-COMPUTATION-0039": {
   "statement_status": "exact",
   "original_statement": "Problem 25. Solve the maximum likelihood equations for a multivariate guassian with arbitrary missing data patterns.",
   "clean_statement": "Problem 25. Solve the maximum likelihood equations for a multivariate guassian with arbitrary missing data patterns.",
   "public_statement": "Problem 25. Solve the maximum likelihood equations for a multivariate guassian with arbitrary missing data patterns.",
   "evidence": "The canonical record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-COMPUTATION-0040": {
   "statement_status": "exact",
   "original_statement": "Question 26. What are the possible implications of semi-algebraic geometry to maximum likelihood estimation? \n\nSeth Sullivant \n\nSeth Sullivant spoke about Markov bases for hierarchical models and theoretical results about the structure of minimal Markov basis elements. 8",
   "clean_statement": "Question 26. What are the possible implications of semi-algebraic geometry to maximum likelihood estimation?\n\nSeth Sullivant\n\nSeth Sullivant spoke about Markov bases for hierarchical models and theoretical results about the structure of minimal Markov basis elements. 8",
   "public_statement": "Question 26. What are the possible implications of semi-algebraic geometry to maximum likelihood estimation?\n\nSeth Sullivant\n\nSeth Sullivant spoke about Markov bases for hierarchical models and theoretical results about the structure of minimal Markov basis elements. 8",
   "evidence": "The canonical record contains Question 26 followed by material from the next speaker entry. The official AIM workshop PDF shows that the recovered statement is exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 39,
   "attempt": 2
  },
  "AIM-COMPUTATION-0041": {
   "statement_status": "exact",
   "original_statement": "Question 27. What types of \"finiteness\" statements can be made about Markov bases for hierarchical models as more than one set of levels is allowed to vary?",
   "clean_statement": "Question 27. What types of \"finiteness\" statements can be made about Markov bases for hierarchical models as more than one set of levels is allowed to vary?",
   "public_statement": "Question 27. What types of \"finiteness\" statements can be made about Markov bases for hierarchical models as more than one set of levels is allowed to vary?",
   "evidence": "The canonical record is Question 27 from the AIM workshop list *Computational Algebraic Statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-COMPUTATION-0042": {
   "statement_status": "exact",
   "original_statement": "Problem 28. Characterize the binary hierarchical models which have Markov bases con-sisting of moves of degree 4 or less. \n\nElizabeth Allman",
   "clean_statement": "Problem 28. Characterize the binary hierarchical models which have Markov bases con-sisting of moves of degree 4 or less.\n\nElizabeth Allman",
   "public_statement": "Problem 28. Characterize the binary hierarchical models which have Markov bases con-sisting of moves of degree 4 or less.\n\nElizabeth Allman",
   "evidence": "The canonical record is Problem 28 from the AIM workshop page *Computational Algebraic Statistics*. The extracted text is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-COMPUTATION-0043": {
   "statement_status": "exact",
   "original_statement": "Problem 29. Compute the ideal of the secant varieties Sec k(Pk−1 × Pk−1 × Pk−1) for k ≥ 4.",
   "clean_statement": "Problem 29. Compute the ideal of the secant varieties Sec k(Pk−1 × Pk−1 × Pk−1) for k ≥ 4.",
   "public_statement": "Problem 29. Compute the ideal of the secant varieties Sec k(Pk−1 × Pk−1 × Pk−1) for k ≥ 4.",
   "evidence": "The exact extracted record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-COMPUTATION-0044": {
   "statement_status": "exact",
   "original_statement": "Question 30. Is the ideal obtained by taking the invariants arising from all 3-dimensional flattenings equal to the ideal of all invariants for a phylogenetic tree?",
   "clean_statement": "Question 30. Is the ideal obtained by taking the invariants arising from all 3-dimensional flattenings equal to the ideal of all invariants for a phylogenetic tree?",
   "public_statement": "Question 30. Is the ideal obtained by taking the invariants arising from all 3-dimensional flattenings equal to the ideal of all invariants for a phylogenetic tree?",
   "evidence": "The canonical record is Question 30 from the AIM workshop page *Computational Algebraic Statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-COMPUTATION-0045": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 31. Determine the image of the stochastic parameterization (as opposed to the complex parameterization).",
   "clean_statement": null,
   "public_statement": "Problem 31. Determine the image of the stochastic parameterization (as opposed to the complex parameterization).",
   "evidence": "Three historically plausible readings must be distinguished.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-computation-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-COMPUTATION-0046": {
   "statement_status": "exact",
   "original_statement": "Problem 32. \"Finding good invariants\": Do some of the invariants have more of a statis-tical/biological significance than others?",
   "clean_statement": "Problem 32. \"Finding good invariants\": Do some of the invariants have more of a statis-tical/biological significance than others?",
   "public_statement": "Problem 32. \"Finding good invariants\": Do some of the invariants have more of a statis-tical/biological significance than others?",
   "evidence": "This is Problem 32 in Elizabeth Allman's contribution to the AIM workshop *Computational algebraic statistics*. The exact canonical record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 45,
   "attempt": 2
  },
  "AIM-COMPUTATION-0047": {
   "statement_status": "exact",
   "original_statement": "Question 33. How can we efficiently decide which trees fit the data best? Which model of mutation is most accurate?",
   "clean_statement": "Question 33. How can we efficiently decide which trees fit the data best? Which model of mutation is most accurate?",
   "public_statement": "Question 33. How can we efficiently decide which trees fit the data best? Which model of mutation is most accurate?",
   "evidence": "The canonical record is Question 33 in the American Institute of Mathematics list *Computational Algebraic Statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-COMPUTATION-0048": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 34. Find invariants for other models with rate variation among sites: secant varieties of the phylogenetic variaties.",
   "clean_statement": null,
   "public_statement": "Problem 34. Find invariants for other models with rate variation among sites: secant varieties of the phylogenetic variaties.",
   "evidence": "This is Problem 34 in the Elizabeth Allman portion of the AIM workshop list *Computational algebraic statistics*. The exact extracted record, including its typographical error, is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-computation-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-COMPUTATION-0049": {
   "statement_status": "exact",
   "original_statement": "Problem 35. Develop techniques to use invariants in combination with quartet methods.",
   "clean_statement": "Problem 35. Develop techniques to use invariants in combination with quartet methods.",
   "public_statement": "Problem 35. Develop techniques to use invariants in combination with quartet methods.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-COMPUTATION-0050": {
   "statement_status": "exact",
   "original_statement": "Question 36. Can we develop algebraic invariant techniques for identifying clades? \n\nShmuel Onn \n\nShmuel Onn spoke about the spectra of hierarchical models. The spectra is defined to be the set of all sets of possible cell entries in for a multiway nonnegative integral table with fixed marginal totals.",
   "clean_statement": "Question 36. Can we develop algebraic invariant techniques for identifying clades?\n\nShmuel Onn\n\nShmuel Onn spoke about the spectra of hierarchical models. The spectra is defined to be the set of all sets of possible cell entries in for a multiway nonnegative integral table with fixed marginal totals.",
   "public_statement": "Question 36. Can we develop algebraic invariant techniques for identifying clades?\n\nShmuel Onn\n\nShmuel Onn spoke about the spectra of hierarchical models. The spectra is defined to be the set of all sets of possible cell entries in for a multiway nonnegative integral table with fixed marginal totals.",
   "evidence": "The official AIM PDF resolves the apparent topic change. On PDF page 7, Questions 29--36 are listed under the heading **Elizabeth Allman**. Immediately after Question 36, **Shmuel Onn** appears as a new speaker heading; the paragraph about spectra follows it, and Problems 37--38 concern those spectra. Thus the last two paragraphs in the canonical record are extraction bleed from the next speaker section. They are preserved verbatim in `input.json`, but they are not part of the phylogenetic question and are not used below.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-COMPUTATION-0051": {
   "statement_status": "exact",
   "original_statement": "Problem 37. Find the spectra of various models and classes of models.",
   "clean_statement": "Problem 37. Find the spectra of various models and classes of models.",
   "public_statement": "Problem 37. Find the spectra of various models and classes of models.",
   "evidence": "The canonical record is Problem 37 from the 2004 AIM workshop list *Computational algebraic statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-COMPUTATION-0052": {
   "statement_status": "exact",
   "original_statement": "Question 38. With respect to some distribution is the set of tables whose range of cell entries are intervals dense among all tables? \n\nJoseph Landsberg Ilias Kotsireas \n\nChallenge problem in Gr¨ obner bases of polynomial ideals. \n\nThe system of polynomial equations given in 3 different formats at http://www.cargo.wlu.ca/hi/had7.cocoa http://www.cargo.wlu.ca/hi/had7.maple http://www.cargo.wlu.ca/hi/had7.reduce has 41 equations in the 27 variables a1,..., a27.Prove that this system of polynomial equations, does not have any solutions. This would constitute an algebraic proof of the combinatorial result of L. D. Baumert: \n\nthere are no Hadamard matrices of order 28 with one circulant core.9\n\nMoreover, it is of interest to write down the effective Nullstellensatz for this ideal, i.e. write down explicitly a linear combination of (some of) the generators of the ideal, which is equal to 1. Similar algebraic proofs could then be devised for Hadamard matrices with one circulant core of bigger order. Reference: Ilias Kotsireas, Christos Koukouvinos, Jennifer Seberry, Hadamard ideals and Hadamard matrices with one circulant core. Submitted, November 2003. Ideas that were suggested during the workshop: \n\n• (Bernd Sturmfels) Devise some kind of incremental process to write down the effective Nullstellensatz. e.g. break the system into pieces, write the effective Nullstellensatz for each piece separately and then combine the results to obtain the effective Null-stellensatz for the whole system; \n\n• (Beth Arnold) Use CoCoA and her own mod p Gr¨ obner implementation, to see if the Gr¨ obner basis can be computed. \n\n• (Nick Eriksson) Use 4ti2, to see if the Grobner basis can be computed. \n\nDesign efficient heuristics in binary trees. \n\nGiven sets of binary vectors in a set of binary trees with given depths, design efficient heuristics to predict the positions of the analogous vectors in binary trees of bigger depth. The adjective \"analogous\" can take on various meanings. In our context, it designates solu-tions of the associated polynomial system. Ideas that were suggested during the workshop: (Persi Diaconis) first of all obtain graphical visualizations and then analyze the data with DFT and other transforms. Work in progress. \n\nIndicator function approach for Hadamard Equivalence. \n\nCheck data sets of Hadamard matrices of big orders to identify the inequivalent Hadamard matrices. Ideas that were suggested during the workshop: (Maria Piera Rogantin) A JSPI paper by Fontana-Pistone-Rogantin contains a description of the indicator function approach to check Hadamard equivalence. In general, one needs to compute the indicator function which has exponentially many monomials. However, this approach is especially efficient for Hadamard matrices with some structure, e.g. with one circulant core. The main reason for this is that the exponentially many monomials fall naturally in a few categories and we only need to check equality of the coefficients of one representative from these categories. Work in progress. \n\nFirst Open Problem Session",
   "clean_statement": "Question 38. With respect to some distribution is the set of tables whose range of cell entries are intervals dense among all tables?\n\nJoseph Landsberg Ilias Kotsireas\n\nChallenge problem in Gr¨ obner bases of polynomial ideals.\n\nThe system of polynomial equations given in 3 different formats at http://www.cargo.wlu.ca/hi/had7.cocoa http://www.cargo.wlu.ca/hi/had7.maple http://www.cargo.wlu.ca/hi/had7.reduce has 41 equations in the 27 variables a1,..., a27.Prove that this system of polynomial equations, does not have any solutions. This would constitute an algebraic proof of the combinatorial result of L. D. Baumert:\n\nthere are no Hadamard matrices of order 28 with one circulant core.9\n\nMoreover, it is of interest to write down the effective Nullstellensatz for this ideal, i.e. write down explicitly a linear combination of (some of) the generators of the ideal, which is equal to 1. Similar algebraic proofs could then be devised for Hadamard matrices with one circulant core of bigger order. Reference: Ilias Kotsireas, Christos Koukouvinos, Jennifer Seberry, Hadamard ideals and Hadamard matrices with one circulant core. Submitted, November 2003. Ideas that were suggested during the workshop:\n\n• (Bernd Sturmfels) Devise some kind of incremental process to write down the effective Nullstellensatz. e.g. break the system into pieces, write the effective Nullstellensatz for each piece separately and then combine the results to obtain the effective Null-stellensatz for the whole system;\n\n• (Beth Arnold) Use CoCoA and her own mod p Gr¨ obner implementation, to see if the Gr¨ obner basis can be computed.\n\n• (Nick Eriksson) Use 4ti2, to see if the Grobner basis can be computed.\n\nDesign efficient heuristics in binary trees.\n\nGiven sets of binary vectors in a set of binary trees with given depths, design efficient heuristics to predict the positions of the analogous vectors in binary trees of bigger depth. The adjective \"analogous\" can take on various meanings. In our context, it designates solu-tions of the associated polynomial system. Ideas that were suggested during the workshop: (Persi Diaconis) first of all obtain graphical visualizations and then analyze the data with DFT and other transforms. Work in progress.\n\nIndicator function approach for Hadamard Equivalence.\n\nCheck data sets of Hadamard matrices of big orders to identify the inequivalent Hadamard matrices. Ideas that were suggested during the workshop: (Maria Piera Rogantin) A JSPI paper by Fontana-Pistone-Rogantin contains a description of the indicator function approach to check Hadamard equivalence. In general, one needs to compute the indicator function which has exponentially many monomials. However, this approach is especially efficient for Hadamard matrices with some structure, e.g. with one circulant core. The main reason for this is that the exponentially many monomials fall naturally in a few categories and we only need to check equality of the coefficients of one representative from these categories. Work in progress.\n\nFirst Open Problem Session",
   "public_statement": "Question 38. With respect to some distribution is the set of tables whose range of cell entries are intervals dense among all tables?\n\nJoseph Landsberg Ilias Kotsireas\n\nChallenge problem in Gr¨ obner bases of polynomial ideals.\n\nThe system of polynomial equations given in 3 different formats at http://www.cargo.wlu.ca/hi/had7.cocoa http://www.cargo.wlu.ca/hi/had7.maple http://www.cargo.wlu.ca/hi/had7.reduce has 41 equations in the 27 variables a1,..., a27.Prove that this system of polynomial equations, does not have any solutions. This would constitute an algebraic proof of the combinatorial result of L. D. Baumert:\n\nthere are no Hadamard matrices of order 28 with one circulant core.9\n\nMoreover, it is of interest to write down the effective Nullstellensatz for this ideal, i.e. write down explicitly a linear combination of (some of) the generators of the ideal, which is equal to 1. Similar algebraic proofs could then be devised for Hadamard matrices with one circulant core of bigger order. Reference: Ilias Kotsireas, Christos Koukouvinos, Jennifer Seberry, Hadamard ideals and Hadamard matrices with one circulant core. Submitted, November 2003. Ideas that were suggested during the workshop:\n\n• (Bernd Sturmfels) Devise some kind of incremental process to write down the effective Nullstellensatz. e.g. break the system into pieces, write the effective Nullstellensatz for each piece separately and then combine the results to obtain the effective Null-stellensatz for the whole system;\n\n• (Beth Arnold) Use CoCoA and her own mod p Gr¨ obner implementation, to see if the Gr¨ obner basis can be computed.\n\n• (Nick Eriksson) Use 4ti2, to see if the Grobner basis can be computed.\n\nDesign efficient heuristics in binary trees.\n\nGiven sets of binary vectors in a set of binary trees with given depths, design efficient heuristics to predict the positions of the analogous vectors in binary trees of bigger depth. The adjective \"analogous\" can take on various meanings. In our context, it designates solu-tions of the associated polynomial system. Ideas that were suggested during the workshop: (Persi Diaconis) first of all obtain graphical visualizations and then analyze the data with DFT and other transforms. Work in progress.\n\nIndicator function approach for Hadamard Equivalence.\n\nCheck data sets of Hadamard matrices of big orders to identify the inequivalent Hadamard matrices. Ideas that were suggested during the workshop: (Maria Piera Rogantin) A JSPI paper by Fontana-Pistone-Rogantin contains a description of the indicator function approach to check Hadamard equivalence. In general, one needs to compute the indicator function which has exponentially many monomials. However, this approach is especially efficient for Hadamard matrices with some structure, e.g. with one circulant core. The main reason for this is that the exponentially many monomials fall naturally in a few categories and we only need to check equality of the coefficients of one representative from these categories. Work in progress.\n\nFirst Open Problem Session",
   "evidence": "The canonical JSON record contains a genuine question followed by unrelated text about Hadamard matrices. The official AIM workshop page separates these items by speaker. Under **Shmuel Onn** it says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-COMPUTATION-0053": {
   "statement_status": "exact",
   "original_statement": "Question 39 (Sturmfels). In a graphical model with hidden variables how many (nonneg-ative) real modes might exist in solutions to the likelihood equations?",
   "clean_statement": "Question 39 (Sturmfels). In a graphical model with hidden variables how many (nonneg-ative) real modes might exist in solutions to the likelihood equations?",
   "public_statement": "Question 39 (Sturmfels). In a graphical model with hidden variables how many (nonneg-ative) real modes might exist in solutions to the likelihood equations?",
   "evidence": "The canonical record is Question 39 from the AIM workshop notes *Computational algebraic statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-COMPUTATION-0054": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 40 (Wynn). Which nodes being hidden imply the multimodality/ unimodality of the likelihood? Given a particular graph, what nodes must be observed to ensure unimodal-ity?",
   "clean_statement": null,
   "public_statement": "Question 40 (Wynn). Which nodes being hidden imply the multimodality/ unimodality of the likelihood? Given a particular graph, what nodes must be observed to ensure unimodal-ity?",
   "evidence": "The canonical record is Question 40 from the 2004 AIM workshop list *Computational algebraic statistics*:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-computation-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-COMPUTATION-0055": {
   "statement_status": "exact",
   "original_statement": "Question 41 (Ho¸ sten). How many complex roots to the likelihood equations can there be? What is the statistical significance of complex roots/ negative real roots or other roots that do not yield a valid statistical model (e.g. covariance matrices that are not positive definite)?",
   "clean_statement": "Question 41 (Ho¸ sten). How many complex roots to the likelihood equations can there be? What is the statistical significance of complex roots/ negative real roots or other roots that do not yield a valid statistical model (e.g. covariance matrices that are not positive definite)?",
   "public_statement": "Question 41 (Ho¸ sten). How many complex roots to the likelihood equations can there be? What is the statistical significance of complex roots/ negative real roots or other roots that do not yield a valid statistical model (e.g. covariance matrices that are not positive definite)?",
   "evidence": "The official AIM workshop page gives the following standalone item in the first open-problem session:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-COMPUTATION-0056": {
   "statement_status": "exact",
   "original_statement": "Question 42 (Richards). What can be said about the number and structure of solutions to the likelihood equations which arise in the Behrens-Fischer problem? 10",
   "clean_statement": "Question 42 (Richards). What can be said about the number and structure of solutions to the likelihood equations which arise in the Behrens-Fischer problem? 10",
   "public_statement": "Question 42 (Richards). What can be said about the number and structure of solutions to the likelihood equations which arise in the Behrens-Fischer problem? 10",
   "evidence": "The canonical record quotes Question 42 from the AIM workshop notes *Computational algebraic statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-COMPUTATION-0057": {
   "statement_status": "exact",
   "original_statement": "Question 43 (Sturmfels). Which is more important to solve: fixing the model and increasing the number of levels in the model, or fixing binary random variables and letting the number of random variables get large? Meek: It depends on the type of problems you are interested in studying. Roehrig: You can exchange a solution to problems on binary random variables to a problem on nonbinary random variables.",
   "clean_statement": "Question 43 (Sturmfels). Which is more important to solve: fixing the model and increasing the number of levels in the model, or fixing binary random variables and letting the number of random variables get large? Meek: It depends on the type of problems you are interested in studying. Roehrig: You can exchange a solution to problems on binary random variables to a problem on nonbinary random variables.",
   "public_statement": "Question 43 (Sturmfels). Which is more important to solve: fixing the model and increasing the number of levels in the model, or fixing binary random variables and letting the number of random variables get large? Meek: It depends on the type of problems you are interested in studying. Roehrig: You can exchange a solution to problems on binary random variables to a problem on nonbinary random variables.",
   "evidence": "The canonical record is Question 43 from the 2004 AIM workshop list *Computational algebraic statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-COMPUTATION-0058": {
   "statement_status": "exact",
   "original_statement": "Question 44 (Landsberg). Why is flattening used so frequently in statistics? Might other linear transformations on categorical data be more useful? Why not project onto the irre-ducible representations of the symmetric group? What about other forms of collapsing data that takes into account group actions?",
   "clean_statement": "Question 44 (Landsberg). Why is flattening used so frequently in statistics? Might other linear transformations on categorical data be more useful? Why not project onto the irre-ducible representations of the symmetric group? What about other forms of collapsing data that takes into account group actions?",
   "public_statement": "Question 44 (Landsberg). Why is flattening used so frequently in statistics? Might other linear transformations on categorical data be more useful? Why not project onto the irre-ducible representations of the symmetric group? What about other forms of collapsing data that takes into account group actions?",
   "evidence": "The official AIM workshop page for *Computational algebraic statistics* gives the following question:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-COMPUTATION-0059": {
   "statement_status": "exact",
   "original_statement": "Problem 45 (Fienberg). Computing the full distributions for observables using Bayesian extensions. Predictive distributions. Dinwoodie: How might you use algebra to pursue Bayesian directions in model fit criteria? Sturmfels: How to you compute this in Macaulay 2? What is Bayesian algebraic geometry? Fienberg: Taking mixtures of things the different models we have discussed at this conference. Meek: Finding the modes of the posterior distributions could be an interesting research problem. Pistone: You could use moment generating functions and cumulant generating func-tions for these problems. Fienberg: Identifiability is a big issue in the Bayesian context. Wynn: How may Edgeworth corrections play a role in this problem?",
   "clean_statement": "Problem 45 (Fienberg). Computing the full distributions for observables using Bayesian extensions. Predictive distributions. Dinwoodie: How might you use algebra to pursue Bayesian directions in model fit criteria? Sturmfels: How to you compute this in Macaulay 2? What is Bayesian algebraic geometry? Fienberg: Taking mixtures of things the different models we have discussed at this conference. Meek: Finding the modes of the posterior distributions could be an interesting research problem. Pistone: You could use moment generating functions and cumulant generating func-tions for these problems. Fienberg: Identifiability is a big issue in the Bayesian context. Wynn: How may Edgeworth corrections play a role in this problem?",
   "public_statement": "Problem 45 (Fienberg). Computing the full distributions for observables using Bayesian extensions. Predictive distributions. Dinwoodie: How might you use algebra to pursue Bayesian directions in model fit criteria? Sturmfels: How to you compute this in Macaulay 2? What is Bayesian algebraic geometry? Fienberg: Taking mixtures of things the different models we have discussed at this conference. Meek: Finding the modes of the posterior distributions could be an interesting research problem. Pistone: You could use moment generating functions and cumulant generating func-tions for these problems. Fienberg: Identifiability is a big issue in the Bayesian context. Wynn: How may Edgeworth corrections play a role in this problem?",
   "evidence": "The source is Problem 45 from the AIM workshop list *Computational algebraic statistics*. The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-COMPUTATION-0060": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 46 (Steele). What might be the application of these techniques to nonparametric models?",
   "clean_statement": null,
   "public_statement": "Question 46 (Steele). What might be the application of these techniques to nonparametric models?",
   "evidence": "That reading is faithful to the surrounding discussion of mixtures and likelihood, but it is not asserted to be the only intended reading.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-computation-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-COMPUTATION-0061": {
   "statement_status": "exact",
   "original_statement": "Question 47. [Pistone] Take a design of N distinct rational vectors D = {v1,..., v N } ⊆ Qd\n\nand xα1,..., x αN are monomials in Q[x1,..., x n] which are linearly independent modulo the design ideal I(D). Further suppose that these monomials are equal to the set of standard monomials of some monomial ideal M.Suppose that M = 〈m1,..., m r〉. Since the monomials xα1,..., x αN are a basis for a quotient space, there is a unique representation of the mi as linear combinations of the xαj.This produces r polynomials f1,..., f r which belong to I(D). Question: are the only points which vanish on these polynomials the design points; that is, does V (f1,..., f r) = D?",
   "clean_statement": "Question 47. [Pistone] Take a design of N distinct rational vectors D = {v1,..., v N } ⊆ Qd\n\nand xα1,..., x αN are monomials in Q[x1,..., x n] which are linearly independent modulo the design ideal I(D). Further suppose that these monomials are equal to the set of standard monomials of some monomial ideal M.Suppose that M = 〈m1,..., m r〉. Since the monomials xα1,..., x αN are a basis for a quotient space, there is a unique representation of the mi as linear combinations of the xαj.This produces r polynomials f1,..., f r which belong to I(D). Question: are the only points which vanish on these polynomials the design points; that is, does V (f1,..., f r) = D?",
   "public_statement": "Question 47. [Pistone] Take a design of N distinct rational vectors D = {v1,..., v N } ⊆ Qd\n\nand xα1,..., x αN are monomials in Q[x1,..., x n] which are linearly independent modulo the design ideal I(D). Further suppose that these monomials are equal to the set of standard monomials of some monomial ideal M.Suppose that M = 〈m1,..., m r〉. Since the monomials xα1,..., x αN are a basis for a quotient space, there is a unique representation of the mi as linear combinations of the xαj.This produces r polynomials f1,..., f r which belong to I(D). Question: are the only points which vanish on these polynomials the design points; that is, does V (f1,..., f r) = D?",
   "evidence": "The official AIM page for the 2003 workshop *Computational algebraic statistics* records the following question (Question 47, attributed to Pistone). Let \\[ D=\\{v_1,\\ldots,v_N\\}\\subseteq\\mathbb Q^d \\] be a design of \\(N\\) distinct rational points. Let \\[ B=\\{x^{\\alpha_1},\\ldots,x^{\\alpha_N}\\} \\] be monomials that are linearly independent modulo the design ideal \\(I(D)\\), and suppose \\(B\\) is the set of standard monomials of a zero-dimensional monomial ideal \\[ M=\\langle m_1,\\ldots,m_r\\rangle. \\] Because \\(|B|=N=\\dim_{\\mathbb Q}\\mathbb Q[x]/I(D)\\), \\(B\\) is a quotient basis. Interpolate each \\(m_i\\) uniquely in the basis \\(B\\): \\[ m_i\\equiv h_i\\pmod {I(D)},\\qquad h_i\\in\\operatorname{span}_{\\mathbb Q}B, \\] and put \\(f_i=m_i-h_i\\in I(D)\\). Must \\[ V(f_1,\\ldots,f_r)=D? \\]",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-COMPUTATION-0062": {
   "statement_status": "exact",
   "original_statement": "Question 47 was answered in the negative by Bernd Sturmfels at the conference.",
   "clean_statement": "Question 47 was answered in the negative by Bernd Sturmfels at the conference.",
   "public_statement": "Question 47 was answered in the negative by Bernd Sturmfels at the conference.",
   "evidence": "The canonical record is not a new question. It says, in full:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-COMPUTATION-0063": {
   "statement_status": "exact",
   "original_statement": "Question 48 (Meek). Is there a polynomial time algorithm for describing the Markov basis of an arbitrary hierarchical model? Can you randomly choose elements from the Markov basis in polynomial time? Sullivant: Is the number of symmetry classes a polynomial in the size of the problem?",
   "clean_statement": "Question 48 (Meek). Is there a polynomial time algorithm for describing the Markov basis of an arbitrary hierarchical model? Can you randomly choose elements from the Markov basis in polynomial time? Sullivant: Is the number of symmetry classes a polynomial in the size of the problem?",
   "public_statement": "Question 48 (Meek). Is there a polynomial time algorithm for describing the Markov basis of an arbitrary hierarchical model? Can you randomly choose elements from the Markov basis in polynomial time? Sullivant: Is the number of symmetry classes a polynomial in the size of the problem?",
   "evidence": "The exact source is the AIM workshop *Computational algebraic statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-COMPUTATION-0064": {
   "statement_status": "exact",
   "original_statement": "Question 49 (Wynn). Why polynomials? What can be said about algebraic methods when applied to other rings, like rings of rational functions, or rings where Fourier analysis can be used? 11",
   "clean_statement": "Question 49 (Wynn). Why polynomials? What can be said about algebraic methods when applied to other rings, like rings of rational functions, or rings where Fourier analysis can be used? 11",
   "public_statement": "Question 49 (Wynn). Why polynomials? What can be said about algebraic methods when applied to other rings, like rings of rational functions, or rings where Fourier analysis can be used? 11",
   "evidence": "The corpus record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-COMPUTATION-0065": {
   "statement_status": "exact",
   "original_statement": "Question 50 (Kuhnt). What about adding continuous random variables into these prob-lems? Might an algebraic framework be useful? Dinwoodie: Can we try to apply algebraic techniques to microarray data? Meek: Gaussian (or similar) assumptions on the random variables might be possible to handle with algebraic techniques.",
   "clean_statement": "Question 50 (Kuhnt). What about adding continuous random variables into these prob-lems? Might an algebraic framework be useful? Dinwoodie: Can we try to apply algebraic techniques to microarray data? Meek: Gaussian (or similar) assumptions on the random variables might be possible to handle with algebraic techniques.",
   "public_statement": "Question 50 (Kuhnt). What about adding continuous random variables into these prob-lems? Might an algebraic framework be useful? Dinwoodie: Can we try to apply algebraic techniques to microarray data? Meek: Gaussian (or similar) assumptions on the random variables might be possible to handle with algebraic techniques.",
   "evidence": "The source is Question 50 from the AIM workshop list *Computational algebraic statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-COMPUTATION-0066": {
   "statement_status": "exact",
   "original_statement": "Question 51 (Meek). Is there a polynomial time algorithm for describing the constraints on the joint probability distribution on the observed variables in a graphical model with hidden variables?",
   "clean_statement": "Question 51 (Meek). Is there a polynomial time algorithm for describing the constraints on the joint probability distribution on the observed variables in a graphical model with hidden variables?",
   "public_statement": "Question 51 (Meek). Is there a polynomial time algorithm for describing the constraints on the joint probability distribution on the observed variables in a graphical model with hidden variables?",
   "evidence": "The exact AIM workshop question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-COMPUTATION-0067": {
   "statement_status": "exact",
   "original_statement": "Problem 52 (Pachter). Try applying the algebraic techniques to more restrictive families of models. For example, models with homogeneous transition matrices (that is, the same set of parameters is used for more than one edge/ clique in the graph), or genomic models.",
   "clean_statement": "Problem 52 (Pachter). Try applying the algebraic techniques to more restrictive families of models. For example, models with homogeneous transition matrices (that is, the same set of parameters is used for more than one edge/ clique in the graph), or genomic models.",
   "public_statement": "Problem 52 (Pachter). Try applying the algebraic techniques to more restrictive families of models. For example, models with homogeneous transition matrices (that is, the same set of parameters is used for more than one edge/ clique in the graph), or genomic models.",
   "evidence": "The canonical record is Problem 52 from the AIM workshop notes on computational algebraic statistics:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-COMPUTATION-0068": {
   "statement_status": "exact",
   "original_statement": "Question 53 (Fienberg). How can we make the algorithms and existing theoretical results apply to problems of sizes that are of practical interest? Scaling up to 5 10 tables is a hopeful goal.",
   "clean_statement": "Question 53 (Fienberg). How can we make the algorithms and existing theoretical results apply to problems of sizes that are of practical interest? Scaling up to 5 10 tables is a hopeful goal.",
   "public_statement": "Question 53 (Fienberg). How can we make the algorithms and existing theoretical results apply to problems of sizes that are of practical interest? Scaling up to 5 10 tables is a hopeful goal.",
   "evidence": "The canonical corpus record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-COMPUTATION-0069": {
   "statement_status": "exact",
   "original_statement": "Question 54 (Fienberg). How can we make it easier to identify \"new\" interesting details among long complicated computer output? Karr: can we take advantage of sparse representations of problems? \n\nSecond Open Problem Session",
   "clean_statement": "Question 54 (Fienberg). How can we make it easier to identify \"new\" interesting details among long complicated computer output? Karr: can we take advantage of sparse representations of problems?\n\nSecond Open Problem Session",
   "public_statement": "Question 54 (Fienberg). How can we make it easier to identify \"new\" interesting details among long complicated computer output? Karr: can we take advantage of sparse representations of problems?\n\nSecond Open Problem Session",
   "evidence": "The repository record contains:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-COMPUTATION-0070": {
   "statement_status": "exact",
   "original_statement": "Problem 55 (Karr). Find a set of strictly positive (or strictly > 5) margins for which the set of tables which has these margins has an arbitrarily large gap in a cell entry. Dinwoodie: Are there examples of this phenomenon in logistic regression?",
   "clean_statement": "Problem 55 (Karr). Find a set of strictly positive (or strictly > 5) margins for which the set of tables which has these margins has an arbitrarily large gap in a cell entry. Dinwoodie: Are there examples of this phenomenon in logistic regression?",
   "public_statement": "Problem 55 (Karr). Find a set of strictly positive (or strictly > 5) margins for which the set of tables which has these margins has an arbitrarily large gap in a cell entry. Dinwoodie: Are there examples of this phenomenon in logistic regression?",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 69,
   "attempt": 1
  },
  "AIM-COMPUTATION-0071": {
   "statement_status": "exact",
   "original_statement": "Problem 56 (Fienberg). Consider 3-way table missing data problem. This consists of a 3-way table with holes and 2-way margins which also have holes.. How can you estimate the likelihood and \"glue\" this information together into a good global picture of the whole table? Can this problem be formulated as an algebraic variety?",
   "clean_statement": "Problem 56 (Fienberg). Consider 3-way table missing data problem. This consists of a 3-way table with holes and 2-way margins which also have holes.. How can you estimate the likelihood and \"glue\" this information together into a good global picture of the whole table? Can this problem be formulated as an algebraic variety?",
   "public_statement": "Problem 56 (Fienberg). Consider 3-way table missing data problem. This consists of a 3-way table with holes and 2-way margins which also have holes.. How can you estimate the likelihood and \"glue\" this information together into a good global picture of the whole table? Can this problem be formulated as an algebraic variety?",
   "evidence": "The canonical record is Problem 56 from the AIM workshop *Computational algebraic statistics*. The repository record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 70,
   "attempt": 1
  },
  "AIM-COMPUTATION-0072": {
   "statement_status": "exact",
   "original_statement": "Problem 57. Gaussian missing data problem. Consider a continuous Gaussian population and a random sample of size n but only some of the characteristics of each individual are observed. Estimate the mean and covariance matrix. Fienberg: First approach the problem using a block structure. Steele: Look at a sequence of regressions as a sequence of polynomials. Can you solve the structure explicitly?",
   "clean_statement": "Problem 57. Gaussian missing data problem. Consider a continuous Gaussian population and a random sample of size n but only some of the characteristics of each individual are observed. Estimate the mean and covariance matrix. Fienberg: First approach the problem using a block structure. Steele: Look at a sequence of regressions as a sequence of polynomials. Can you solve the structure explicitly?",
   "public_statement": "Problem 57. Gaussian missing data problem. Consider a continuous Gaussian population and a random sample of size n but only some of the characteristics of each individual are observed. Estimate the mean and covariance matrix. Fienberg: First approach the problem using a block structure. Steele: Look at a sequence of regressions as a sequence of polynomials. Can you solve the structure explicitly?",
   "evidence": "The source record is Problem 57 in `aim-computation-notes.json`:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-COMPUTATION-0073": {
   "statement_status": "exact",
   "original_statement": "Problem 58 (Sturmfels). Compute a mixture of the S4 model from Diaconis' talk. What are the equations defining this secant variety? Is the likelihood multimodal? \n\nMLE Project \n\nOn Thursday afternoon, Bernd Sturmfels, Serkan Ho¸ sten, and Mathias Drton led a section on algebraic approaches to maximum likelihood estimations. The purpose was to introduce some algebraic techniques and possible small instances of statistical models to which one could apply these techniques. The eventual goal of this project is to try to come 12 \n\nto a better understanding of the maximum likelihood estimation problem using algebraic means. \n\nAlgebraic Methods for Optimization: \n\nA. Singular: Gr¨ obner bases and numerical eigenvalues for companion matrices or lexi-cographic solving B. Lagrange multipliers 1. Transformations out of the simplex? 2. Exponential representation for problems C. Resultants D. Homotopy Methods (e.g. PHCpack) E. Sum of Squares (e.g. SOStools) \n\nSmall Instances of Statistical Models \n\nA. σ3(P2 × P2 × P2), that is, the hidden variable naive Bayes model with three ob-served variables, where all the random variables have three states. This model has 20 parameters. B. Seemingly unrelated regression (SUR) model with three random variables. See ques-tions related to Mathias Drton's talk. This model has 3 parameters. C. Hidden Markov model with two hidden nodes. Hidden variables have 2 state and observed variables have 3 states. This model has 6 parameters. D. An unconditional independence model for 4 binary random variables. X1⊥X2 and \n\nX3⊥X4. This model has 13 parameters. E. Perfect tables on 3 random variables. X1⊥X2, X2⊥X3, and X1⊥X3.\n\nQuestions from the Working Sessions \n\nGraphical Models",
   "clean_statement": "Problem 58 (Sturmfels). Compute a mixture of the S4 model from Diaconis' talk. What are the equations defining this secant variety? Is the likelihood multimodal?\n\nMLE Project\n\nOn Thursday afternoon, Bernd Sturmfels, Serkan Ho¸ sten, and Mathias Drton led a section on algebraic approaches to maximum likelihood estimations. The purpose was to introduce some algebraic techniques and possible small instances of statistical models to which one could apply these techniques. The eventual goal of this project is to try to come 12\n\nto a better understanding of the maximum likelihood estimation problem using algebraic means.\n\nAlgebraic Methods for Optimization:\n\nA. Singular: Gr¨ obner bases and numerical eigenvalues for companion matrices or lexi-cographic solving B. Lagrange multipliers 1. Transformations out of the simplex? 2. Exponential representation for problems C. Resultants D. Homotopy Methods (e.g. PHCpack) E. Sum of Squares (e.g. SOStools)\n\nSmall Instances of Statistical Models\n\nA. σ3(P2 × P2 × P2), that is, the hidden variable naive Bayes model with three ob-served variables, where all the random variables have three states. This model has 20 parameters. B. Seemingly unrelated regression (SUR) model with three random variables. See ques-tions related to Mathias Drton's talk. This model has 3 parameters. C. Hidden Markov model with two hidden nodes. Hidden variables have 2 state and observed variables have 3 states. This model has 6 parameters. D. An unconditional independence model for 4 binary random variables. X1⊥X2 and\n\nX3⊥X4. This model has 13 parameters. E. Perfect tables on 3 random variables. X1⊥X2, X2⊥X3, and X1⊥X3.\n\nQuestions from the Working Sessions\n\nGraphical Models",
   "public_statement": "Problem 58 (Sturmfels). Compute a mixture of the S4 model from Diaconis' talk. What are the equations defining this secant variety? Is the likelihood multimodal?\n\nMLE Project\n\nOn Thursday afternoon, Bernd Sturmfels, Serkan Ho¸ sten, and Mathias Drton led a section on algebraic approaches to maximum likelihood estimations. The purpose was to introduce some algebraic techniques and possible small instances of statistical models to which one could apply these techniques. The eventual goal of this project is to try to come 12\n\nto a better understanding of the maximum likelihood estimation problem using algebraic means.\n\nAlgebraic Methods for Optimization:\n\nA. Singular: Gr¨ obner bases and numerical eigenvalues for companion matrices or lexi-cographic solving B. Lagrange multipliers 1. Transformations out of the simplex? 2. Exponential representation for problems C. Resultants D. Homotopy Methods (e.g. PHCpack) E. Sum of Squares (e.g. SOStools)\n\nSmall Instances of Statistical Models\n\nA. σ3(P2 × P2 × P2), that is, the hidden variable naive Bayes model with three ob-served variables, where all the random variables have three states. This model has 20 parameters. B. Seemingly unrelated regression (SUR) model with three random variables. See ques-tions related to Mathias Drton's talk. This model has 3 parameters. C. Hidden Markov model with two hidden nodes. Hidden variables have 2 state and observed variables have 3 states. This model has 6 parameters. D. An unconditional independence model for 4 binary random variables. X1⊥X2 and\n\nX3⊥X4. This model has 13 parameters. E. Perfect tables on 3 random variables. X1⊥X2, X2⊥X3, and X1⊥X3.\n\nQuestions from the Working Sessions\n\nGraphical Models",
   "evidence": "The source record begins:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 72,
   "attempt": 1
  },
  "AIM-COMPUTATION-0074": {
   "statement_status": "exact",
   "original_statement": "Question 59 (Ho¸ sten). What can algebra say about MLE's for models with hidden vari-ables?",
   "clean_statement": "Question 59 (Ho¸ sten). What can algebra say about MLE's for models with hidden vari-ables?",
   "public_statement": "Question 59 (Ho¸ sten). What can algebra say about MLE's for models with hidden vari-ables?",
   "evidence": "The exact extracted record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 73,
   "attempt": 1
  },
  "AIM-COMPUTATION-0075": {
   "statement_status": "exact",
   "original_statement": "Question 60 (Sullivant). For a log-linear model, what can be said about the structure of the set of b with Ax = b for some b and the solution space to the maximum likelihood equations is not zero-dimensional? Ho¸ sten: Look at the leading coefficients of polynomials in the elimination ideals (with parameters) and determine when these are zero.",
   "clean_statement": "Question 60 (Sullivant). For a log-linear model, what can be said about the structure of the set of b with Ax = b for some b and the solution space to the maximum likelihood equations is not zero-dimensional? Ho¸ sten: Look at the leading coefficients of polynomials in the elimination ideals (with parameters) and determine when these are zero.",
   "public_statement": "Question 60 (Sullivant). For a log-linear model, what can be said about the structure of the set of b with Ax = b for some b and the solution space to the maximum likelihood equations is not zero-dimensional? Ho¸ sten: Look at the leading coefficients of polynomials in the elimination ideals (with parameters) and determine when these are zero.",
   "evidence": "The canonical record is Question 60 from the AIM workshop *Computational algebraic statistics*. The exact record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 74,
   "attempt": 2
  },
  "AIM-COMPUTATION-0076": {
   "statement_status": "exact",
   "original_statement": "Question 61 (From graphical models notes). Is there any useful statistical measure that can be obtained from evaluating the polynomials that vanish on probabilities coming from a given model at an empirical distribution? What about \n\n∑ |f (̂p)|?\n\nLinear Polynomial Models \n\nA design D is a finite subset of Rd. The design ideal I(D) is the vanishing ideal of polynomial functions vanishing on D. The set of standard monomials modulo I(D) given a term order τ is denoted Est τ. A linear polynomial model L is a list of monomials. 13",
   "clean_statement": "Question 61 (From graphical models notes). Is there any useful statistical measure that can be obtained from evaluating the polynomials that vanish on probabilities coming from a given model at an empirical distribution? What about\n\n∑ |f (̂p)|?\n\nLinear Polynomial Models\n\nA design D is a finite subset of Rd. The design ideal I(D) is the vanishing ideal of polynomial functions vanishing on D. The set of standard monomials modulo I(D) given a term order τ is denoted Est τ. A linear polynomial model L is a list of monomials. 13",
   "public_statement": "Question 61 (From graphical models notes). Is there any useful statistical measure that can be obtained from evaluating the polynomials that vanish on probabilities coming from a given model at an empirical distribution? What about\n\n∑ |f (̂p)|?\n\nLinear Polynomial Models\n\nA design D is a finite subset of Rd. The design ideal I(D) is the vanishing ideal of polynomial functions vanishing on D. The set of standard monomials modulo I(D) given a term order τ is denoted Est τ. A linear polynomial model L is a list of monomials. 13",
   "evidence": "The recovered AIM question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-COMPUTATION-0077": {
   "statement_status": "exact",
   "original_statement": "Question 62 (Riccomagno). Is there an algebraic method to decide which Est τ factor is best when working with perturbed designs?",
   "clean_statement": "Question 62 (Riccomagno). Is there an algebraic method to decide which Est τ factor is best when working with perturbed designs?",
   "public_statement": "Question 62 (Riccomagno). Is there an algebraic method to decide which Est τ factor is best when working with perturbed designs?",
   "evidence": "The exact source record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-COMPUTATION-0078": {
   "statement_status": "exact",
   "original_statement": "Problem 63 (Riccomagno). Given a model L find a design D with special statistical prop-erties such that the monomials in L are linearly independent modulo I(D).",
   "clean_statement": "Problem 63 (Riccomagno). Given a model L find a design D with special statistical prop-erties such that the monomials in L are linearly independent modulo I(D).",
   "public_statement": "Problem 63 (Riccomagno). Given a model L find a design D with special statistical prop-erties such that the monomials in L are linearly independent modulo I(D).",
   "evidence": "The canonical record is Problem 63 from the AIM workshop list *Computational Algebraic Statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-COMPUTATION-0079": {
   "statement_status": "exact",
   "original_statement": "Question 64 (Laubenbacher). Choosing models modulo the design ideal depends on the term order chosen. How can we eliminate this choice of term order from the problem of finding a model that fits the data? Richards: Could you use random search or genetic algorithms to solve this problem?",
   "clean_statement": "Question 64 (Laubenbacher). Choosing models modulo the design ideal depends on the term order chosen. How can we eliminate this choice of term order from the problem of finding a model that fits the data? Richards: Could you use random search or genetic algorithms to solve this problem?",
   "public_statement": "Question 64 (Laubenbacher). Choosing models modulo the design ideal depends on the term order chosen. How can we eliminate this choice of term order from the problem of finding a model that fits the data? Richards: Could you use random search or genetic algorithms to solve this problem?",
   "evidence": "The canonical record is Question 64 from the AIM workshop problem list *Computational Algebraic Statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-COMPUTATION-0080": {
   "statement_status": "exact",
   "original_statement": "Question 65 (Laubenbacher). The model selection problem does not seem to be very effec-tive when only one time series is used but does seem to work well when multiple knockout time series are used in combination. Why?",
   "clean_statement": "Question 65 (Laubenbacher). The model selection problem does not seem to be very effec-tive when only one time series is used but does seem to work well when multiple knockout time series are used in combination. Why?",
   "public_statement": "Question 65 (Laubenbacher). The model selection problem does not seem to be very effec-tive when only one time series is used but does seem to work well when multiple knockout time series are used in combination. Why?",
   "evidence": "This is Question 65 in the AIM workshop list *Computational Algebraic Statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-COMPUTATION-0081": {
   "statement_status": "exact",
   "original_statement": "Question 66 (Laubenbacher). The model selection problem over finite fields is very sensitive to noise. Is there some sort of \"least squares\" over finite fields to deal with noisy data?",
   "clean_statement": "Question 66 (Laubenbacher). The model selection problem over finite fields is very sensitive to noise. Is there some sort of \"least squares\" over finite fields to deal with noisy data?",
   "public_statement": "Question 66 (Laubenbacher). The model selection problem over finite fields is very sensitive to noise. Is there some sort of \"least squares\" over finite fields to deal with noisy data?",
   "evidence": "The exact source record is Question 66 in the AIM workshop list *Computational Algebraic Statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-COMPUTATION-0082": {
   "statement_status": "exact",
   "original_statement": "Question 67 (Laubenbacher). What is the complexity of the algorithms used for time series modelling? How can we make the algorithms faster?",
   "clean_statement": "Question 67 (Laubenbacher). What is the complexity of the algorithms used for time series modelling? How can we make the algorithms faster?",
   "public_statement": "Question 67 (Laubenbacher). What is the complexity of the algorithms used for time series modelling? How can we make the algorithms faster?",
   "evidence": "The canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-COMPUTATION-0083": {
   "statement_status": "exact",
   "original_statement": "Question 68 (Sturmfels). In many of the problems for analyzing time series there are very few data points in a high dimensional space. How can techniques for reducing the dimensionality be used? \n\nSoftware for Algebraic Statistics",
   "clean_statement": "Question 68 (Sturmfels). In many of the problems for analyzing time series there are very few data points in a high dimensional space. How can techniques for reducing the dimensionality be used?\n\nSoftware for Algebraic Statistics",
   "public_statement": "Question 68 (Sturmfels). In many of the problems for analyzing time series there are very few data points in a high dimensional space. How can techniques for reducing the dimensionality be used?\n\nSoftware for Algebraic Statistics",
   "evidence": "The canonical record from `aim-computation-notes.json` is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 82,
   "attempt": 1
  },
  "AIM-COMPUTATION-0084": {
   "statement_status": "exact",
   "original_statement": "Question 69 (Roehrig). Why are there so many different systems for computational alge-bra? Couldn't they be add-ins to other systems? Stillman: Not likely to work since those other systems (Mathematica, Maple, etc.) do not give access to their kernels.",
   "clean_statement": "Question 69 (Roehrig). Why are there so many different systems for computational alge-bra? Couldn't they be add-ins to other systems? Stillman: Not likely to work since those other systems (Mathematica, Maple, etc.) do not give access to their kernels.",
   "public_statement": "Question 69 (Roehrig). Why are there so many different systems for computational alge-bra? Couldn't they be add-ins to other systems? Stillman: Not likely to work since those other systems (Mathematica, Maple, etc.) do not give access to their kernels.",
   "evidence": "The source is Question 69 in the AIM workshop list *Computational Algebraic Statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 83,
   "attempt": 1
  },
  "AIM-COMPUTATION-0085": {
   "statement_status": "exact",
   "original_statement": "Question 70 (Slavkovic). Are there scripts which convert between input formats for the different computational algebra systems?",
   "clean_statement": "Question 70 (Slavkovic). Are there scripts which convert between input formats for the different computational algebra systems?",
   "public_statement": "Question 70 (Slavkovic). Are there scripts which convert between input formats for the different computational algebra systems?",
   "evidence": "The canonical record is Question 70 from the AIM workshop list *Computational algebraic statistics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 84,
   "attempt": 1
  },
  "AIM-COMPUTATION-0086": {
   "statement_status": "exact",
   "original_statement": "Question 71 (Sturmfels). What should be the first algebraic functions available in R? Dinwoodie: Toric Markov bases Riccomagno: Ideal of points Pachter: A general R interface with algebra packages Riccomagno: Gr¨ obner bases over fraction fields, with algebraic numbers and parame-ters. Pachter: When do we find out when these things have been implemented? Allman: Improve the documentation!",
   "clean_statement": "Question 71 (Sturmfels). What should be the first algebraic functions available in R? Dinwoodie: Toric Markov bases Riccomagno: Ideal of points Pachter: A general R interface with algebra packages Riccomagno: Gr¨ obner bases over fraction fields, with algebraic numbers and parame-ters. Pachter: When do we find out when these things have been implemented? Allman: Improve the documentation!",
   "public_statement": "Question 71 (Sturmfels). What should be the first algebraic functions available in R? Dinwoodie: Toric Markov bases Riccomagno: Ideal of points Pachter: A general R interface with algebra packages Riccomagno: Gr¨ obner bases over fraction fields, with algebraic numbers and parame-ters. Pachter: When do we find out when these things have been implemented? Allman: Improve the documentation!",
   "evidence": "The canonical record, preserving its extraction defects, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 85,
   "attempt": 1
  },
  "AIM-COMPUTATION-0087": {
   "statement_status": "exact",
   "original_statement": "3. A test problem 4. Reduction of the dimension 5. References 3\n\nChapter A: Open problems \n\nA.1 High-dimensional Problems in Finance and extension \n\nA.1.a Some stochastic control problems in finance. Several examples were considered. \n\nOptimal Stopping and free boundary problems. \n\nLet's consider the following financial market containing a non-risky asset S0 \n\n> t\n\n= ert, and \n\nd risky assets (e.g. stocks) with prices modeled by a d-dimensional diffusion process S with dynamics \n\ndS t = rS tdt + diag( St)σ(t, S t)dW t,\n\nwhere W is a standard Brownian motion. An important problem in finance is the pricing of American options. Given a reward function g, an American option is a contract that provides to the owner the right to receive (from the seller) the amount g(St) at any time t if exercised before some fixed maturity T.This right can be exercised only once during the period [0, T ]. The price of this option at time t can be expressed as the value function associated to the optimal stopping problem \n\nv(t, S t) = sup \n\n> τ∈T [t,T ]\n\nE[e−r(τ −t)g(St) | St] (1.1.1) where T[t,T ] is the set of all stopping times with values in [ t, T ]. The associated value function v is the solution of the free boundary problem min {rv − vt − L v; v − g} = 0 on [0, T ) × [0, ∞)d\n\nv(T,. ) = g(.) on [0, ∞)d\n\nwhere L is the generator of the diffusion S.\n\nOptimal Investment and Hamilton-Jacobi-Bellman equations. \n\nAn other important issue in finance is that of optimal investment. Denoting by νt the number of stocks held by a given financial agent at time t, the associated wealth-process X ν\n\nhas dynamics given by \n\ndX t = νtdS t + ( Xt − ν∗ \n\n> t\n\nSt)dS 0\n\n> t\n\nwhere ∗ stands for transposition, and S may have a general dynamics of the form \n\ndS t = μ(t, S t, ν t)dt + σ(t, S t, ν t)dW t,\n\nin order to take into account a possible influence of the agent's financial strategy on the dynamics of the risky assets. Given a concave (utility) function V, the agent tries to maximize the expected utility of terminal wealth max E[V (XνT )] (1.1.2) over a set of admissible financial strategies ( νt) with values in some subset U of Rd. The associated value function v is the solution of Hamilton-Jacobi-Bellman equation \n\n−vt − sup \n\n> ν∈U\n\nLν v = 0 on [0, T ) × [0, ∞)d × Rv(T, s, x ) = V (x) for ( s, x ) ∈ [0, ∞)d × R\n\nwhere Lν is the generator of the diffusion ( S, X ν ). 4\n\nA.1.b Extensions and BSDE's. The value function of Problem (1.1.1) can be reformu-lated in terms of the Y component of the solution ( Y, Z, A ) of the Backward Stochastic Differential Equation \n\nYt = g(ST ) −\n\n∫ Tt\n\nZ∗ \n\n> t\n\ndW t + AT − At\n\nYt ≥ g(St)where A is a non-decreasing process satisfying \n\n∫ T\n\n> 0\n\n(Yt − g(St)) dA t = 0,\n\nsee e.g. [KKPPQ]. This leads us to consider the more general problem of approximating the solution of Reflected Forward - Backward Stochastic Differential Equations of the form \n\nXt = X0 +\n\n∫ Tt\n\nb(t, X t, Y t, Z t)dt +\n\n∫ Tt\n\na(t, X t, Y t, Z t)dW t\n\nYt = g(T, X T ) + \n\n∫ Tt\n\nf (t, X t, Y t, Z t)dt −\n\n∫ Tt\n\nZ∗ \n\n> t\n\ndW t + AT − At\n\nYt ≥ g(t, X t)where A is a non-decreasing process satisfying \n\n∫ T\n\n> 0\n\n(Yt − g(t, X t)) dA t = 0.\n\nThis framework partially includes control problems associated to HJB equations. It is related to non-linear PDE's of the form min {− vt(t, x ) − 1\n\n2Trace[ vxx (t, x )aa ∗(t, x, v (t, x ), θ (t, x ))] \n\n−b(t, x, v (t, x ), θ (t, x )) ∗vx(t, x ) − f (t, x, v (t, x ), θ (t, x )); v(t, x ) − g(t, x )} = 0 \n\nv(T,. ) = g(T,. )where θ(t, x ) solves \n\nvx(t, x )a(t, x, v (t, x ), θ (t, x )) = θ(t, x ),\n\nthrough the relations \n\nv(t, X t) = Yt and θ(t, X t)∗ = Zt.\n\nSee e.g. [MY]. \n\nA.2 Some New Methodologies \n\nA.2.a The case where b and a does not depend on Y and Z. Several methods were considered. \n\nPure Monte Carlo Methods. \n\nPure Monte-Carlo methods are based on a discrete time approximation of the forward-backward equation. Once discretized the forward process can be simulated. These sim-ulations are then used to compute the conditional expectations involved in the backward 5\n\ndiscrete time approximation of ( Y, Z ). Two different methods can be used to compute these conditional expectations. a. The Longstaff-Schwartz (Carriere) approach consists in approximating the conditional expectations by regressions on a given basis of functions. This provides a very powerful, and, easy to implement, algorithm for which convergence has been shown to hold in the case of the American option pricing problem. However, no rate of convergence is given and the choice of the basis is a quite difficult problem. It has been used successfully in up to 20 factors models. See [C], [LS], [CLP]. b. The Malliavin approach consists in re-writing the conditional expectations as the ratio of two unconditional expectations that can be estimated by standard Monte-Carlo methods. Upper bounds for the rate of convergence are proved. Contrary to the previous approach, this algorithm also provides good approximations for the greeks, i.e. the gradient of the associated value function (which is related to the Z, see above). So far, the existing algorithm has a too important complexity, which explains why it has only been tested in small dimensions (up to 5). Some numerical improvements have been proposed, reducing the complexity from \n\nN 2 to N ln( N )d, where N is the number of simulated paths. This work is still in progress, the aim being to develop an algorithm such that the major part of the work is done before a reward function is specified, so as to reduce as much as possible the effective computation time once a particular payoff is defined. See [BET], [BT]. \n\nGrid approximations. \n\nThe Quantization approach consists in approximating the original forward process by a discrete time process which evolves on some finite grid. The grid is constructed so has to provide the best Lp approximation. It was first applied to the pricing of American options in dimension up to 10. The construction of the optimal grid (and the computation of the associated transition probabilities) is very time consuming, but it can be done once for all. Once the grid, which is independent of the payoff function, is constructed, it provides a very quick algorithm for pricing American options on different payoffs, whenever they are written on the same assets. As in the Malliavin and Longstaff-Schwartz approach, the use of a good control variable is required. Extensions to non-linear filtering, optimal control and Asian-type options have also been studied. See [BP1], [BP2], [PP1], [PP2]. \n\nDual formulation. \n\nThis algorithm is based on a dual formulation for problem (1.1.1): \n\nv(0, S 0) = inf E\n\n[\n\nsup \n\n> t≤T\n\n(e−rt g(St) − Mt)\n\n]\n\nwhere the inf is taken over a well suited set of martingales M. This algorithm consists in providing a upper bound for v by choosing some martingale M ∗ and computing \n\nE\n\n[\n\nsup \n\n> t≤T\n\n(e−rt g(St) − M ∗ \n\n> t\n\n)\n\n]\n\nIn cases where a good martingale ˆM can be found, typically when \n\nE[e−r(T −t)g(ST )|St] =: ˆMt6\n\nis known as a function of S, this provides a quite sharp upper bound for the price of the American option. Numerical experiments, up to dimension 15, have been performed. See [R]. \n\nCubature on Wiener spaces. \n\nCubature formula on Wiener spaces have been developed in [VL] in order to construct probability measures with finite support which approximate the Wiener measure in the sense that the expectation of iterated Stratonovich integrals under the approximating measure and the Wiener measure are close. In a sense, this approach is similar to that of the quantization since it allows to reduce to a finite dimensional setting. So far, it has been used to develop high order numerical schemes for high dimensional SDE's and semi-elliptic PDE's. \n\nA.2.b The case where b or a depends on Y and Z. In that case, it is no more possible to approximate (or simulate) the forward process X since its dynamic depends on the solu-tion. In such situations, two different solutions have been proposed: a - Construct an a priori grid for X, possibly based on some a priori on the dynamics of Y\n\nand Z.b - Given an a priori solution Y0 and Z0, approximate (or simulate) the corresponding forward process X0 and use the above methodologies to construct the corresponding solution ( Y1, Z 1)of the BSDE. Then, use this solution ( Y1, Z 1) to approximate (or simulate) the corresponding forward process X1 and go on iterating this procedure. Under some mild assumptions, this algorithm should be convergent. However, it seems to be quite heavy to implement. Solution a. has already been applied in the quantization approach but, so far, does not provide very good results. See [PP1]. \n\nA.3 A test problem \n\nA test problem has been proposed to compare the performance of the different methods. It corresponds to the computation of a d dimensional at-the-money American min-put option in the Black-Scholes model. The assets are uncorrelated, have the same volatility and the same initial value: \n\nSit = 100 exp(( r − σ2/2) t − σW it ) for each i = 1,..., d \n\nwhere W = ( W 1,..., W d) is a standard Brownian motion. The payoff function is \n\ng(x) = [100 − min {x1,..., x d}]+.\n\nNumerical tests will be performed for different dimensions with σ = 20% and r = 5%. Results will be collected and compared by J. Cvitanic. \n\nA.4 Reduction of the dimension \n\nDifferent ways of reducing the dimension were proposed. aThe first consists in using Principal Component Analysis in order to \"aggregate\" a large number of random variables in a small number of principal directions. Also it is used in 7\n\npractice, it does not really solve the problem of the dimension, but only surrounds it, replac-ing the initial model by a more tractable, but different, one. bInstead of relying on PCA, dimension could be reduced by considering models where a few number of economic fundamentals could explain the behavior of many financial assets. cIn some cases, it is possible to reduce the dimension without changing the initial model. This is the case in P. Carr's Canadization approach for the computation of American options, where the randomization of the maturity allows to reduce to a time homogeneous problem. Extensions to Markovian control problems have been discussed. See [Carr] and [BKT]. \n\nA.5 References \n\n[BP1] Bally V. and G. Pages: A quantization algorithm for solving discrete time multidi-mensional optimal stopping problems, Bernoulli, (9) 6, 1-47, 2003. [BP2] Bally V. and G. Pages: Error analysis of the quantization algorithm for obstacle problems, Stochastic Processes and their Applications,106 (1), 1-40, 2003. [BT] Bouchard B. and N. Touzi: Discrete Time Approximation and Monte-Carlo Simulation of Backward Stochastic Differential Equations, Stochastic Processes and their Applications, to appear. [BET] Bouchard B., I. Ekeland and N. Touzi: On the Malliavin approach to Monte Carlo approximation of conditional expectations, Finance and Stochastics, 8 (1), 2004. [BKT] Bouchard B., N. El Karoui and N. Touzi: Maturity randomization for stochastic control problems, in preparation. [Carr] Carr P.: Randomization and the American Put, The Review of Financial Studies, 11, 597-626, 1997. [C] Carriere E.: Valuation of the Early-Exercise Price for Options using Simulations and Nonparametric Regression, Insurance: mathematics and Economics, 19, 19-30, 1996. [CLP] Cl´ ement E., D. Lamberton, and P. Protter: An analysis of a least squares regression method for American option pricing, Finance and Stochastics, 6, 449-472, 2002. [KKPPQ] El Karoui N., C. Kapoudjan, E. Pardoux, S. Peng and M.C. Quenez: Reflected solutions of backward stochastic differential equations and related obstacle problems for PDE's, Annals of Probability, 25, 702-737, 1997. [LS] Longstaff F.A. and R.S. Schwartz: Valuing American Options By Simulation: A simple Least-Square Approach, Review of Financial Studies, 14, 113-147, 2001. [MY] Ma J. and J. Yong: Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Math., 1702, Springer, 1999. [PP1] Pages G. and H. Pham: A quantization algorithm for multidimensional stochastic control problems, preprint LPMA n ◦697, 2001. [PP2] Pages G. and H. Pham: Optimal quantization methods for nonlinear filtering with discrete-time observations, preprint LPMA n ◦778, 2002. 8\n\n[R] Rogers L.C.G.: Monte Carlo valuation of American options, Mathematical Finance, 12, 271-286, 2002. [VL] Victoir N. and T. Lyons: Cubature on Wiener space, Proc. R. Soc. Lond. A, 460, 169-198, 2004.",
   "clean_statement": "3. A test problem 4. Reduction of the dimension 5. References 3\n\nChapter A: Open problems\n\nA.1 High-dimensional Problems in Finance and extension\n\nA.1.a Some stochastic control problems in finance. Several examples were considered.\n\nOptimal Stopping and free boundary problems.\n\nLet's consider the following financial market containing a non-risky asset S0\n\n> t\n\n= ert, and\n\nd risky assets (e.g. stocks) with prices modeled by a d-dimensional diffusion process S with dynamics\n\ndS t = rS tdt + diag( St)σ(t, S t)dW t,\n\nwhere W is a standard Brownian motion. An important problem in finance is the pricing of American options. Given a reward function g, an American option is a contract that provides to the owner the right to receive (from the seller) the amount g(St) at any time t if exercised before some fixed maturity T.This right can be exercised only once during the period [0, T ]. The price of this option at time t can be expressed as the value function associated to the optimal stopping problem\n\nv(t, S t) = sup\n\n> τ∈T [t,T ]\n\nE[e−r(τ −t)g(St) | St] (1.1.1) where T[t,T ] is the set of all stopping times with values in [ t, T ]. The associated value function v is the solution of the free boundary problem min {rv − vt − L v; v − g} = 0 on [0, T ) × [0, ∞)d\n\nv(T,. ) = g(.) on [0, ∞)d\n\nwhere L is the generator of the diffusion S.\n\nOptimal Investment and Hamilton-Jacobi-Bellman equations.\n\nAn other important issue in finance is that of optimal investment. Denoting by νt the number of stocks held by a given financial agent at time t, the associated wealth-process X ν\n\nhas dynamics given by\n\ndX t = νtdS t + ( Xt − ν∗\n\n> t\n\nSt)dS 0\n\n> t\n\nwhere ∗ stands for transposition, and S may have a general dynamics of the form\n\ndS t = μ(t, S t, ν t)dt + σ(t, S t, ν t)dW t,\n\nin order to take into account a possible influence of the agent's financial strategy on the dynamics of the risky assets. Given a concave (utility) function V, the agent tries to maximize the expected utility of terminal wealth max E[V (XνT )] (1.1.2) over a set of admissible financial strategies ( νt) with values in some subset U of Rd. The associated value function v is the solution of Hamilton-Jacobi-Bellman equation\n\n−vt − sup\n\n> ν∈U\n\nLν v = 0 on [0, T ) × [0, ∞)d × Rv(T, s, x ) = V (x) for ( s, x ) ∈ [0, ∞)d × R\n\nwhere Lν is the generator of the diffusion ( S, X ν ). 4\n\nA.1.b Extensions and BSDE's. The value function of Problem (1.1.1) can be reformu-lated in terms of the Y component of the solution ( Y, Z, A ) of the Backward Stochastic Differential Equation\n\nYt = g(ST ) −\n\n∫ Tt\n\nZ∗\n\n> t\n\ndW t + AT − At\n\nYt ≥ g(St)where A is a non-decreasing process satisfying\n\n∫ T\n\n> 0\n\n(Yt − g(St)) dA t = 0,\n\nsee e.g. [KKPPQ]. This leads us to consider the more general problem of approximating the solution of Reflected Forward - Backward Stochastic Differential Equations of the form\n\nXt = X0 +\n\n∫ Tt\n\nb(t, X t, Y t, Z t)dt +\n\n∫ Tt\n\na(t, X t, Y t, Z t)dW t\n\nYt = g(T, X T ) +\n\n∫ Tt\n\nf (t, X t, Y t, Z t)dt −\n\n∫ Tt\n\nZ∗\n\n> t\n\ndW t + AT − At\n\nYt ≥ g(t, X t)where A is a non-decreasing process satisfying\n\n∫ T\n\n> 0\n\n(Yt − g(t, X t)) dA t = 0.\n\nThis framework partially includes control problems associated to HJB equations. It is related to non-linear PDE's of the form min {− vt(t, x ) − 1\n\n2Trace[ vxx (t, x )aa ∗(t, x, v (t, x ), θ (t, x ))]\n\n−b(t, x, v (t, x ), θ (t, x )) ∗vx(t, x ) − f (t, x, v (t, x ), θ (t, x )); v(t, x ) − g(t, x )} = 0\n\nv(T,. ) = g(T,. )where θ(t, x ) solves\n\nvx(t, x )a(t, x, v (t, x ), θ (t, x )) = θ(t, x ),\n\nthrough the relations\n\nv(t, X t) = Yt and θ(t, X t)∗ = Zt.\n\nSee e.g. [MY].\n\nA.2 Some New Methodologies\n\nA.2.a The case where b and a does not depend on Y and Z. Several methods were considered.\n\nPure Monte Carlo Methods.\n\nPure Monte-Carlo methods are based on a discrete time approximation of the forward-backward equation. Once discretized the forward process can be simulated. These sim-ulations are then used to compute the conditional expectations involved in the backward 5\n\ndiscrete time approximation of ( Y, Z ). Two different methods can be used to compute these conditional expectations. a. The Longstaff-Schwartz (Carriere) approach consists in approximating the conditional expectations by regressions on a given basis of functions. This provides a very powerful, and, easy to implement, algorithm for which convergence has been shown to hold in the case of the American option pricing problem. However, no rate of convergence is given and the choice of the basis is a quite difficult problem. It has been used successfully in up to 20 factors models. See [C], [LS], [CLP]. b. The Malliavin approach consists in re-writing the conditional expectations as the ratio of two unconditional expectations that can be estimated by standard Monte-Carlo methods. Upper bounds for the rate of convergence are proved. Contrary to the previous approach, this algorithm also provides good approximations for the greeks, i.e. the gradient of the associated value function (which is related to the Z, see above). So far, the existing algorithm has a too important complexity, which explains why it has only been tested in small dimensions (up to 5). Some numerical improvements have been proposed, reducing the complexity from\n\nN 2 to N ln( N )d, where N is the number of simulated paths. This work is still in progress, the aim being to develop an algorithm such that the major part of the work is done before a reward function is specified, so as to reduce as much as possible the effective computation time once a particular payoff is defined. See [BET], [BT].\n\nGrid approximations.\n\nThe Quantization approach consists in approximating the original forward process by a discrete time process which evolves on some finite grid. The grid is constructed so has to provide the best Lp approximation. It was first applied to the pricing of American options in dimension up to 10. The construction of the optimal grid (and the computation of the associated transition probabilities) is very time consuming, but it can be done once for all. Once the grid, which is independent of the payoff function, is constructed, it provides a very quick algorithm for pricing American options on different payoffs, whenever they are written on the same assets. As in the Malliavin and Longstaff-Schwartz approach, the use of a good control variable is required. Extensions to non-linear filtering, optimal control and Asian-type options have also been studied. See [BP1], [BP2], [PP1], [PP2].\n\nDual formulation.\n\nThis algorithm is based on a dual formulation for problem (1.1.1):\n\nv(0, S 0) = inf E\n\n[\n\nsup\n\n> t≤T\n\n(e−rt g(St) − Mt)\n\n]\n\nwhere the inf is taken over a well suited set of martingales M. This algorithm consists in providing a upper bound for v by choosing some martingale M ∗ and computing\n\nE\n\n[\n\nsup\n\n> t≤T\n\n(e−rt g(St) − M ∗\n\n> t\n\n)\n\n]\n\nIn cases where a good martingale ˆM can be found, typically when\n\nE[e−r(T −t)g(ST )|St] =: ˆMt6\n\nis known as a function of S, this provides a quite sharp upper bound for the price of the American option. Numerical experiments, up to dimension 15, have been performed. See [R].\n\nCubature on Wiener spaces.\n\nCubature formula on Wiener spaces have been developed in [VL] in order to construct probability measures with finite support which approximate the Wiener measure in the sense that the expectation of iterated Stratonovich integrals under the approximating measure and the Wiener measure are close. In a sense, this approach is similar to that of the quantization since it allows to reduce to a finite dimensional setting. So far, it has been used to develop high order numerical schemes for high dimensional SDE's and semi-elliptic PDE's.\n\nA.2.b The case where b or a depends on Y and Z. In that case, it is no more possible to approximate (or simulate) the forward process X since its dynamic depends on the solu-tion. In such situations, two different solutions have been proposed: a - Construct an a priori grid for X, possibly based on some a priori on the dynamics of Y\n\nand Z.b - Given an a priori solution Y0 and Z0, approximate (or simulate) the corresponding forward process X0 and use the above methodologies to construct the corresponding solution ( Y1, Z 1)of the BSDE. Then, use this solution ( Y1, Z 1) to approximate (or simulate) the corresponding forward process X1 and go on iterating this procedure. Under some mild assumptions, this algorithm should be convergent. However, it seems to be quite heavy to implement. Solution a. has already been applied in the quantization approach but, so far, does not provide very good results. See [PP1].\n\nA.3 A test problem\n\nA test problem has been proposed to compare the performance of the different methods. It corresponds to the computation of a d dimensional at-the-money American min-put option in the Black-Scholes model. The assets are uncorrelated, have the same volatility and the same initial value:\n\nSit = 100 exp(( r − σ2/2) t − σW it ) for each i = 1,..., d\n\nwhere W = ( W 1,..., W d) is a standard Brownian motion. The payoff function is\n\ng(x) = [100 − min {x1,..., x d}]+.\n\nNumerical tests will be performed for different dimensions with σ = 20% and r = 5%. Results will be collected and compared by J. Cvitanic.\n\nA.4 Reduction of the dimension\n\nDifferent ways of reducing the dimension were proposed. aThe first consists in using Principal Component Analysis in order to \"aggregate\" a large number of random variables in a small number of principal directions. Also it is used in 7\n\npractice, it does not really solve the problem of the dimension, but only surrounds it, replac-ing the initial model by a more tractable, but different, one. bInstead of relying on PCA, dimension could be reduced by considering models where a few number of economic fundamentals could explain the behavior of many financial assets. cIn some cases, it is possible to reduce the dimension without changing the initial model. This is the case in P. Carr's Canadization approach for the computation of American options, where the randomization of the maturity allows to reduce to a time homogeneous problem. Extensions to Markovian control problems have been discussed. See [Carr] and [BKT].\n\nA.5 References\n\n[BP1] Bally V. and G. Pages: A quantization algorithm for solving discrete time multidi-mensional optimal stopping problems, Bernoulli, (9) 6, 1-47, 2003. [BP2] Bally V. and G. Pages: Error analysis of the quantization algorithm for obstacle problems, Stochastic Processes and their Applications,106 (1), 1-40, 2003. [BT] Bouchard B. and N. Touzi: Discrete Time Approximation and Monte-Carlo Simulation of Backward Stochastic Differential Equations, Stochastic Processes and their Applications, to appear. [BET] Bouchard B., I. Ekeland and N. Touzi: On the Malliavin approach to Monte Carlo approximation of conditional expectations, Finance and Stochastics, 8 (1), 2004. [BKT] Bouchard B., N. El Karoui and N. Touzi: Maturity randomization for stochastic control problems, in preparation. [Carr] Carr P.: Randomization and the American Put, The Review of Financial Studies, 11, 597-626, 1997. [C] Carriere E.: Valuation of the Early-Exercise Price for Options using Simulations and Nonparametric Regression, Insurance: mathematics and Economics, 19, 19-30, 1996. [CLP] Cl´ ement E., D. Lamberton, and P. Protter: An analysis of a least squares regression method for American option pricing, Finance and Stochastics, 6, 449-472, 2002. [KKPPQ] El Karoui N., C. Kapoudjan, E. Pardoux, S. Peng and M.C. Quenez: Reflected solutions of backward stochastic differential equations and related obstacle problems for PDE's, Annals of Probability, 25, 702-737, 1997. [LS] Longstaff F.A. and R.S. Schwartz: Valuing American Options By Simulation: A simple Least-Square Approach, Review of Financial Studies, 14, 113-147, 2001. [MY] Ma J. and J. Yong: Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Math., 1702, Springer, 1999. [PP1] Pages G. and H. Pham: A quantization algorithm for multidimensional stochastic control problems, preprint LPMA n ◦697, 2001. [PP2] Pages G. and H. Pham: Optimal quantization methods for nonlinear filtering with discrete-time observations, preprint LPMA n ◦778, 2002. 8\n\n[R] Rogers L.C.G.: Monte Carlo valuation of American options, Mathematical Finance, 12, 271-286, 2002. [VL] Victoir N. and T. Lyons: Cubature on Wiener space, Proc. R. Soc. Lond. A, 460, 169-198, 2004.",
   "public_statement": "3. A test problem 4. Reduction of the dimension 5. References 3\n\nChapter A: Open problems\n\nA.1 High-dimensional Problems in Finance and extension\n\nA.1.a Some stochastic control problems in finance. Several examples were considered.\n\nOptimal Stopping and free boundary problems.\n\nLet's consider the following financial market containing a non-risky asset S0\n\n> t\n\n= ert, and\n\nd risky assets (e.g. stocks) with prices modeled by a d-dimensional diffusion process S with dynamics\n\ndS t = rS tdt + diag( St)σ(t, S t)dW t,\n\nwhere W is a standard Brownian motion. An important problem in finance is the pricing of American options. Given a reward function g, an American option is a contract that provides to the owner the right to receive (from the seller) the amount g(St) at any time t if exercised before some fixed maturity T.This right can be exercised only once during the period [0, T ]. The price of this option at time t can be expressed as the value function associated to the optimal stopping problem\n\nv(t, S t) = sup\n\n> τ∈T [t,T ]\n\nE[e−r(τ −t)g(St) | St] (1.1.1) where T[t,T ] is the set of all stopping times with values in [ t, T ]. The associated value function v is the solution of the free boundary problem min {rv − vt − L v; v − g} = 0 on [0, T ) × [0, ∞)d\n\nv(T,. ) = g(.) on [0, ∞)d\n\nwhere L is the generator of the diffusion S.\n\nOptimal Investment and Hamilton-Jacobi-Bellman equations.\n\nAn other important issue in finance is that of optimal investment. Denoting by νt the number of stocks held by a given financial agent at time t, the associated wealth-process X ν\n\nhas dynamics given by\n\ndX t = νtdS t + ( Xt − ν∗\n\n> t\n\nSt)dS 0\n\n> t\n\nwhere ∗ stands for transposition, and S may have a general dynamics of the form\n\ndS t = μ(t, S t, ν t)dt + σ(t, S t, ν t)dW t,\n\nin order to take into account a possible influence of the agent's financial strategy on the dynamics of the risky assets. Given a concave (utility) function V, the agent tries to maximize the expected utility of terminal wealth max E[V (XνT )] (1.1.2) over a set of admissible financial strategies ( νt) with values in some subset U of Rd. The associated value function v is the solution of Hamilton-Jacobi-Bellman equation\n\n−vt − sup\n\n> ν∈U\n\nLν v = 0 on [0, T ) × [0, ∞)d × Rv(T, s, x ) = V (x) for ( s, x ) ∈ [0, ∞)d × R\n\nwhere Lν is the generator of the diffusion ( S, X ν ). 4\n\nA.1.b Extensions and BSDE's. The value function of Problem (1.1.1) can be reformu-lated in terms of the Y component of the solution ( Y, Z, A ) of the Backward Stochastic Differential Equation\n\nYt = g(ST ) −\n\n∫ Tt\n\nZ∗\n\n> t\n\ndW t + AT − At\n\nYt ≥ g(St)where A is a non-decreasing process satisfying\n\n∫ T\n\n> 0\n\n(Yt − g(St)) dA t = 0,\n\nsee e.g. [KKPPQ]. This leads us to consider the more general problem of approximating the solution of Reflected Forward - Backward Stochastic Differential Equations of the form\n\nXt = X0 +\n\n∫ Tt\n\nb(t, X t, Y t, Z t)dt +\n\n∫ Tt\n\na(t, X t, Y t, Z t)dW t\n\nYt = g(T, X T ) +\n\n∫ Tt\n\nf (t, X t, Y t, Z t)dt −\n\n∫ Tt\n\nZ∗\n\n> t\n\ndW t + AT − At\n\nYt ≥ g(t, X t)where A is a non-decreasing process satisfying\n\n∫ T\n\n> 0\n\n(Yt − g(t, X t)) dA t = 0.\n\nThis framework partially includes control problems associated to HJB equations. It is related to non-linear PDE's of the form min {− vt(t, x ) − 1\n\n2Trace[ vxx (t, x )aa ∗(t, x, v (t, x ), θ (t, x ))]\n\n−b(t, x, v (t, x ), θ (t, x )) ∗vx(t, x ) − f (t, x, v (t, x ), θ (t, x )); v(t, x ) − g(t, x )} = 0\n\nv(T,. ) = g(T,. )where θ(t, x ) solves\n\nvx(t, x )a(t, x, v (t, x ), θ (t, x )) = θ(t, x ),\n\nthrough the relations\n\nv(t, X t) = Yt and θ(t, X t)∗ = Zt.\n\nSee e.g. [MY].\n\nA.2 Some New Methodologies\n\nA.2.a The case where b and a does not depend on Y and Z. Several methods were considered.\n\nPure Monte Carlo Methods.\n\nPure Monte-Carlo methods are based on a discrete time approximation of the forward-backward equation. Once discretized the forward process can be simulated. These sim-ulations are then used to compute the conditional expectations involved in the backward 5\n\ndiscrete time approximation of ( Y, Z ). Two different methods can be used to compute these conditional expectations. a. The Longstaff-Schwartz (Carriere) approach consists in approximating the conditional expectations by regressions on a given basis of functions. This provides a very powerful, and, easy to implement, algorithm for which convergence has been shown to hold in the case of the American option pricing problem. However, no rate of convergence is given and the choice of the basis is a quite difficult problem. It has been used successfully in up to 20 factors models. See [C], [LS], [CLP]. b. The Malliavin approach consists in re-writing the conditional expectations as the ratio of two unconditional expectations that can be estimated by standard Monte-Carlo methods. Upper bounds for the rate of convergence are proved. Contrary to the previous approach, this algorithm also provides good approximations for the greeks, i.e. the gradient of the associated value function (which is related to the Z, see above). So far, the existing algorithm has a too important complexity, which explains why it has only been tested in small dimensions (up to 5). Some numerical improvements have been proposed, reducing the complexity from\n\nN 2 to N ln( N )d, where N is the number of simulated paths. This work is still in progress, the aim being to develop an algorithm such that the major part of the work is done before a reward function is specified, so as to reduce as much as possible the effective computation time once a particular payoff is defined. See [BET], [BT].\n\nGrid approximations.\n\nThe Quantization approach consists in approximating the original forward process by a discrete time process which evolves on some finite grid. The grid is constructed so has to provide the best Lp approximation. It was first applied to the pricing of American options in dimension up to 10. The construction of the optimal grid (and the computation of the associated transition probabilities) is very time consuming, but it can be done once for all. Once the grid, which is independent of the payoff function, is constructed, it provides a very quick algorithm for pricing American options on different payoffs, whenever they are written on the same assets. As in the Malliavin and Longstaff-Schwartz approach, the use of a good control variable is required. Extensions to non-linear filtering, optimal control and Asian-type options have also been studied. See [BP1], [BP2], [PP1], [PP2].\n\nDual formulation.\n\nThis algorithm is based on a dual formulation for problem (1.1.1):\n\nv(0, S 0) = inf E\n\n[\n\nsup\n\n> t≤T\n\n(e−rt g(St) − Mt)\n\n]\n\nwhere the inf is taken over a well suited set of martingales M. This algorithm consists in providing a upper bound for v by choosing some martingale M ∗ and computing\n\nE\n\n[\n\nsup\n\n> t≤T\n\n(e−rt g(St) − M ∗\n\n> t\n\n)\n\n]\n\nIn cases where a good martingale ˆM can be found, typically when\n\nE[e−r(T −t)g(ST )|St] =: ˆMt6\n\nis known as a function of S, this provides a quite sharp upper bound for the price of the American option. Numerical experiments, up to dimension 15, have been performed. See [R].\n\nCubature on Wiener spaces.\n\nCubature formula on Wiener spaces have been developed in [VL] in order to construct probability measures with finite support which approximate the Wiener measure in the sense that the expectation of iterated Stratonovich integrals under the approximating measure and the Wiener measure are close. In a sense, this approach is similar to that of the quantization since it allows to reduce to a finite dimensional setting. So far, it has been used to develop high order numerical schemes for high dimensional SDE's and semi-elliptic PDE's.\n\nA.2.b The case where b or a depends on Y and Z. In that case, it is no more possible to approximate (or simulate) the forward process X since its dynamic depends on the solu-tion. In such situations, two different solutions have been proposed: a - Construct an a priori grid for X, possibly based on some a priori on the dynamics of Y\n\nand Z.b - Given an a priori solution Y0 and Z0, approximate (or simulate) the corresponding forward process X0 and use the above methodologies to construct the corresponding solution ( Y1, Z 1)of the BSDE. Then, use this solution ( Y1, Z 1) to approximate (or simulate) the corresponding forward process X1 and go on iterating this procedure. Under some mild assumptions, this algorithm should be convergent. However, it seems to be quite heavy to implement. Solution a. has already been applied in the quantization approach but, so far, does not provide very good results. See [PP1].\n\nA.3 A test problem\n\nA test problem has been proposed to compare the performance of the different methods. It corresponds to the computation of a d dimensional at-the-money American min-put option in the Black-Scholes model. The assets are uncorrelated, have the same volatility and the same initial value:\n\nSit = 100 exp(( r − σ2/2) t − σW it ) for each i = 1,..., d\n\nwhere W = ( W 1,..., W d) is a standard Brownian motion. The payoff function is\n\ng(x) = [100 − min {x1,..., x d}]+.\n\nNumerical tests will be performed for different dimensions with σ = 20% and r = 5%. Results will be collected and compared by J. Cvitanic.\n\nA.4 Reduction of the dimension\n\nDifferent ways of reducing the dimension were proposed. aThe first consists in using Principal Component Analysis in order to \"aggregate\" a large number of random variables in a small number of principal directions. Also it is used in 7\n\npractice, it does not really solve the problem of the dimension, but only surrounds it, replac-ing the initial model by a more tractable, but different, one. bInstead of relying on PCA, dimension could be reduced by considering models where a few number of economic fundamentals could explain the behavior of many financial assets. cIn some cases, it is possible to reduce the dimension without changing the initial model. This is the case in P. Carr's Canadization approach for the computation of American options, where the randomization of the maturity allows to reduce to a time homogeneous problem. Extensions to Markovian control problems have been discussed. See [Carr] and [BKT].\n\nA.5 References\n\n[BP1] Bally V. and G. Pages: A quantization algorithm for solving discrete time multidi-mensional optimal stopping problems, Bernoulli, (9) 6, 1-47, 2003. [BP2] Bally V. and G. Pages: Error analysis of the quantization algorithm for obstacle problems, Stochastic Processes and their Applications,106 (1), 1-40, 2003. [BT] Bouchard B. and N. Touzi: Discrete Time Approximation and Monte-Carlo Simulation of Backward Stochastic Differential Equations, Stochastic Processes and their Applications, to appear. [BET] Bouchard B., I. Ekeland and N. Touzi: On the Malliavin approach to Monte Carlo approximation of conditional expectations, Finance and Stochastics, 8 (1), 2004. [BKT] Bouchard B., N. El Karoui and N. Touzi: Maturity randomization for stochastic control problems, in preparation. [Carr] Carr P.: Randomization and the American Put, The Review of Financial Studies, 11, 597-626, 1997. [C] Carriere E.: Valuation of the Early-Exercise Price for Options using Simulations and Nonparametric Regression, Insurance: mathematics and Economics, 19, 19-30, 1996. [CLP] Cl´ ement E., D. Lamberton, and P. Protter: An analysis of a least squares regression method for American option pricing, Finance and Stochastics, 6, 449-472, 2002. [KKPPQ] El Karoui N., C. Kapoudjan, E. Pardoux, S. Peng and M.C. Quenez: Reflected solutions of backward stochastic differential equations and related obstacle problems for PDE's, Annals of Probability, 25, 702-737, 1997. [LS] Longstaff F.A. and R.S. Schwartz: Valuing American Options By Simulation: A simple Least-Square Approach, Review of Financial Studies, 14, 113-147, 2001. [MY] Ma J. and J. Yong: Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Math., 1702, Springer, 1999. [PP1] Pages G. and H. Pham: A quantization algorithm for multidimensional stochastic control problems, preprint LPMA n ◦697, 2001. [PP2] Pages G. and H. Pham: Optimal quantization methods for nonlinear filtering with discrete-time observations, preprint LPMA n ◦778, 2002. 8\n\n[R] Rogers L.C.G.: Monte Carlo valuation of American options, Mathematical Finance, 12, 271-286, 2002. [VL] Victoir N. and T. Lyons: Cubature on Wiener space, Proc. R. Soc. Lond. A, 460, 169-198, 2004.",
   "evidence": "The canonical `problem` field is 12,517 characters long. It begins with a table-of-contents fragment and then reproduces almost all of “Chapter A: Open problems” from the AIM workshop list *Numerical probabilistic methods for high-dimensional problems in finance* [1]. It includes:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
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  "AIM-COMPUTATION-0088": {
   "statement_status": "exact",
   "original_statement": "Conjecture 3\n\nChapter A: Lecture Notes \n\nLecture notes were TeXed in real time by John Voight. \n\n#A.1 Agrawal: A Polynomial Time Algorithm for Testing Primality \n\nThe primality testing algorithm we present is based upon the following identity: n is prime if and only if (1 + X)n ≡ 1 + Xn (mod n),\n\nwhere we consider this congruence as an identity in the polynomial ring ( Z/n Z)[ X]. This easily gives a randomized algorithm which runs in polynomial time: rather than verifying it completely (which would be far too expensive), you verify it modulo a randomly chosen degree log n polynomial Q(X); repeating this sufficiently often, you expect to witness a failure of this congruence to hold fairly quickly if n is composite. \n\nConjecture. To prove that n is prime, it is enough to let Q(X) run over the set \n\n{X − 1, X 2 − 1,..., X log 2 n − 1}.\n\nWe were unable to prove this conjecture, but, instead the modified conjecture, which is then a proposition: \n\nProposition (Modified conjecture). To prove that n is prime, it is enough to let Q(X) run over the set \n\nR = {(X + a)r − 1: 1 ≤ r ≤ 16(log n)5, 1 ≤ a ≤ 8(log n)7/2},\n\nexcept when n has a 'small' prime factor ( ≤ (log n)5).",
   "clean_statement": "Conjecture 3\n\nChapter A: Lecture Notes\n\nLecture notes were TeXed in real time by John Voight.\n\n#A.1 Agrawal: A Polynomial Time Algorithm for Testing Primality\n\nThe primality testing algorithm we present is based upon the following identity: n is prime if and only if (1 + X)n ≡ 1 + Xn (mod n),\n\nwhere we consider this congruence as an identity in the polynomial ring ( Z/n Z)[ X]. This easily gives a randomized algorithm which runs in polynomial time: rather than verifying it completely (which would be far too expensive), you verify it modulo a randomly chosen degree log n polynomial Q(X); repeating this sufficiently often, you expect to witness a failure of this congruence to hold fairly quickly if n is composite.\n\nConjecture. To prove that n is prime, it is enough to let Q(X) run over the set\n\n{X − 1, X 2 − 1,..., X log 2 n − 1}.\n\nWe were unable to prove this conjecture, but, instead the modified conjecture, which is then a proposition:\n\nProposition (Modified conjecture). To prove that n is prime, it is enough to let Q(X) run over the set\n\nR = {(X + a)r − 1: 1 ≤ r ≤ 16(log n)5, 1 ≤ a ≤ 8(log n)7/2},\n\nexcept when n has a 'small' prime factor ( ≤ (log n)5).",
   "public_statement": "Conjecture 3\n\nChapter A: Lecture Notes\n\nLecture notes were TeXed in real time by John Voight.\n\n#A.1 Agrawal: A Polynomial Time Algorithm for Testing Primality\n\nThe primality testing algorithm we present is based upon the following identity: n is prime if and only if (1 + X)n ≡ 1 + Xn (mod n),\n\nwhere we consider this congruence as an identity in the polynomial ring ( Z/n Z)[ X]. This easily gives a randomized algorithm which runs in polynomial time: rather than verifying it completely (which would be far too expensive), you verify it modulo a randomly chosen degree log n polynomial Q(X); repeating this sufficiently often, you expect to witness a failure of this congruence to hold fairly quickly if n is composite.\n\nConjecture. To prove that n is prime, it is enough to let Q(X) run over the set\n\n{X − 1, X 2 − 1,..., X log 2 n − 1}.\n\nWe were unable to prove this conjecture, but, instead the modified conjecture, which is then a proposition:\n\nProposition (Modified conjecture). To prove that n is prime, it is enough to let Q(X) run over the set\n\nR = {(X + a)r − 1: 1 ≤ r ≤ 16(log n)5, 1 ≤ a ≤ 8(log n)7/2},\n\nexcept when n has a 'small' prime factor ( ≤ (log n)5).",
   "evidence": "The canonical record is an oversized parser aggregate from the AIM workshop *Future directions in algorithmic number theory* (24--28 March 2003). Its `problem` begins with `Conjecture 3`, includes the first Agrawal lecture and a proved modified proposition, while its `remarks` continue that proof and then absorb lectures A.2 through A.13 and the beginning of Chapter B. Those later lectures and problems are not parts of one conjecture.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
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  "AIM-COMPUTATION-0089": {
   "statement_status": "exact",
   "original_statement": "Question 1. If (X − 1) n ≡ Xn − 1 (mod n, X r − 1) and gcd( r, n ) = 1 does this imply that \n\nn2 ≡ 1 (mod r)when n is composite? (AKS)",
   "clean_statement": "Question 1. If (X − 1) n ≡ Xn − 1 (mod n, X r − 1) and gcd( r, n ) = 1 does this imply that\n\nn2 ≡ 1 (mod r)when n is composite? (AKS)",
   "public_statement": "Question 1. If (X − 1) n ≡ Xn − 1 (mod n, X r − 1) and gcd( r, n ) = 1 does this imply that\n\nn2 ≡ 1 (mod r)when n is composite? (AKS)",
   "evidence": "The canonical extraction reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 88,
   "attempt": 1
  },
  "AIM-COMPUTATION-0090": {
   "statement_status": "exact",
   "original_statement": "Question 2. Consider Fp and d ∈ Z>0, and S ⊂ Fp with S = d. Let h(X) ∈\n\nFp[X] with h(s) 6 = 0 for all s ∈ S. Let G be the group generated by X − s for s ∈ S. In the original AKS paper, we have G ≥ 2d. There are techniques for getting G ≥ (5.82) d\n\n(using lattice point counting); there seems to be much more room for improvement (perhaps using ABC). (Bernstein, Voloch) Perhaps (log G)/d → ∞ \n\nas d → ∞ uniformly over h.",
   "clean_statement": "Question 2. Consider Fp and d ∈ Z>0, and S ⊂ Fp with S = d. Let h(X) ∈\n\nFp[X] with h(s) 6 = 0 for all s ∈ S. Let G be the group generated by X − s for s ∈ S. In the original AKS paper, we have G ≥ 2d. There are techniques for getting G ≥ (5.82) d\n\n(using lattice point counting); there seems to be much more room for improvement (perhaps using ABC). (Bernstein, Voloch) Perhaps (log G)/d → ∞\n\nas d → ∞ uniformly over h.",
   "public_statement": "Question 2. Consider Fp and d ∈ Z>0, and S ⊂ Fp with S = d. Let h(X) ∈\n\nFp[X] with h(s) 6 = 0 for all s ∈ S. Let G be the group generated by X − s for s ∈ S. In the original AKS paper, we have G ≥ 2d. There are techniques for getting G ≥ (5.82) d\n\n(using lattice point counting); there seems to be much more room for improvement (perhaps using ABC). (Bernstein, Voloch) Perhaps (log G)/d → ∞\n\nas d → ∞ uniformly over h.",
   "evidence": "The canonical record comes from the AIM workshop list *Future directions in algorithmic number theory*, Question 2. The mathematical notation lost in the JSON extraction is recoverable from the official PDF and the official HTML transcription. The printed question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 89,
   "attempt": 1
  },
  "AIM-COMPUTATION-0091": {
   "statement_status": "exact",
   "original_statement": "Question 3. Find a deterministic polynomial time algorithm for recognizing perfect numbers. (Lenstra) Given two monic polynomials f, g ∈ Z[X] with no common irreducible factor, and two integers n, m ∈ Z>0, find in polynomial time all x ∈ Z such that f (x) | m and g(x) | n, with \n\n|x| < H (f, g, m, n ) for some function H.",
   "clean_statement": "Question 3. Find a deterministic polynomial time algorithm for recognizing perfect numbers. (Lenstra) Given two monic polynomials f, g ∈ Z[X] with no common irreducible factor, and two integers n, m ∈ Z>0, find in polynomial time all x ∈ Z such that f (x) | m and g(x) | n, with\n\n|x| < H (f, g, m, n ) for some function H.",
   "public_statement": "Question 3. Find a deterministic polynomial time algorithm for recognizing perfect numbers. (Lenstra) Given two monic polynomials f, g ∈ Z[X] with no common irreducible factor, and two integers n, m ∈ Z>0, find in polynomial time all x ∈ Z such that f (x) | m and g(x) | n, with\n\n|x| < H (f, g, m, n ) for some function H.",
   "evidence": "The canonical record is Question 3 from the AIM workshop *Future directions in algorithmic number theory*. The official AIM HTML and the PDF (printed page 25) agree that one numbered question contains two linked tasks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 90,
   "attempt": 1
  },
  "AIM-COMPUTATION-0092": {
   "statement_status": "corrected_verified",
   "original_statement": "Question 4. These questions are motivated by the questions posed by AKS concerning finding quadratic nonresidues modulo a prime p.(a) It is known that finding a single quadratic nonresidue for a given prime is polynomially equivalent to solving all quadratic equations. How far can one do this for finding a single bit of data (or few bits of data) for higher degrees? (Elkies) 26 \n\n(b) We do not know yet that there is a deterministic polynomial time algorithm for finding quadratic nonresidues. Is there a subexponential algorithm?",
   "clean_statement": "These questions are motivated by the questions posed by AKS concerning\nfinding quadratic nonresidues modulo a prime \\(p\\).\n\n(a) It is known that finding a single quadratic nonresidue for a given prime\nis polynomially equivalent to solving all quadratic equations. How far can\none do this for finding a single bit of data (or few bits of data) for higher\ndegrees?\n\n(b) We do not know yet that there is a deterministic polynomial time\nalgorithm for finding quadratic nonresidues. Is there a subexponential\nalgorithm?",
   "public_statement": "These questions are motivated by the questions posed by AKS concerning\nfinding quadratic nonresidues modulo a prime \\(p\\).\n\n(a) It is known that finding a single quadratic nonresidue for a given prime\nis polynomially equivalent to solving all quadratic equations. How far can\none do this for finding a single bit of data (or few bits of data) for higher\ndegrees?\n\n(b) We do not know yet that there is a deterministic polynomial time\nalgorithm for finding quadratic nonresidues. Is there a subexponential\nalgorithm?",
   "evidence": "The source is the AIM workshop list *Future directions in algorithmic number theory*, Question 4. The recovered statement is: This wording and the accompanying remarks were checked against the AIM PDF. The OCR fragment \\(p1/(4\\sqrt e)\\) in the JSON means \\(p^{1/(4\\sqrt e)}\\), and the fragment \\(X p-X\\) means \\(X^p-X\\). The trailing `Problem/` is an extraction artifact rather than part of the question. The source does not specify the computational model for “subexponential”; that ambiguity matters below.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-computation-notes.json",
   "source_index": 91,
   "attempt": 1
  },
  "AIM-COMPUTATION-0093": {
   "statement_status": "exact",
   "original_statement": "Question 5. Suppose that you have a nonconstant family of elliptic curves Eλ\n\nover Fp (e.g. the Frey curves), λ ∈ P1(Fp). Can you find deterministically λ such that Eλ\n\nand Eλ+1 are not isogenous (i.e. Eλ(Fp) 6 = Eλ+1 (Fp)). (Elkies)",
   "clean_statement": "Question 5. Suppose that you have a nonconstant family of elliptic curves Eλ\n\nover Fp (e.g. the Frey curves), λ ∈ P1(Fp). Can you find deterministically λ such that Eλ\n\nand Eλ+1 are not isogenous (i.e. Eλ(Fp) 6 = Eλ+1 (Fp)). (Elkies)",
   "public_statement": "Question 5. Suppose that you have a nonconstant family of elliptic curves Eλ\n\nover Fp (e.g. the Frey curves), λ ∈ P1(Fp). Can you find deterministically λ such that Eλ\n\nand Eλ+1 are not isogenous (i.e. Eλ(Fp) 6 = Eλ+1 (Fp)). (Elkies)",
   "evidence": "The canonical record is Question 5 in the AIM workshop list *Future directions in algorithmic number theory*. The official PDF and HTML transcription restore the notation lost by extraction:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-COMPUTATION-0094": {
   "statement_status": "exact",
   "original_statement": "Question 6. Let f (X, Y ) ∈ Fq[X, Y ] be irreducible. Let g(Y ) = f (aY + b, Y ). Count (or estimate) the number of pairs a, b over Fq such that g is irreducible over Fq. (Gao)",
   "clean_statement": "Question 6. Let f (X, Y ) ∈ Fq[X, Y ] be irreducible. Let g(Y ) = f (aY + b, Y ). Count (or estimate) the number of pairs a, b over Fq such that g is irreducible over Fq. (Gao)",
   "public_statement": "Question 6. Let f (X, Y ) ∈ Fq[X, Y ] be irreducible. Let g(Y ) = f (aY + b, Y ). Count (or estimate) the number of pairs a, b over Fq such that g is irreducible over Fq. (Gao)",
   "evidence": "The canonical record is Question 6 from the AIM workshop *Future directions in algorithmic number theory*. The official AIM HTML and workshop PDF give the following statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-COMPUTATION-0095": {
   "statement_status": "exact",
   "original_statement": "Question 7. Let m ≥ n ∈ Z. Prove there is a polynomial g ∈ Fq[x] of degree \n\n≤ 2 log n so that xm + g(x) has an irreducible factor of degree n. (Gao)",
   "clean_statement": "Question 7. Let m ≥ n ∈ Z. Prove there is a polynomial g ∈ Fq[x] of degree\n\n≤ 2 log n so that xm + g(x) has an irreducible factor of degree n. (Gao)",
   "public_statement": "Question 7. Let m ≥ n ∈ Z. Prove there is a polynomial g ∈ Fq[x] of degree\n\n≤ 2 log n so that xm + g(x) has an irreducible factor of degree n. (Gao)",
   "evidence": "The record is Question 7 in the AIM workshop list *Future directions in algorithmic number theory*. The official PDF and its HTML version give the statement",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 94,
   "attempt": 1
  },
  "AIM-COMPUTATION-0096": {
   "statement_status": "exact",
   "original_statement": "Question 8. Look at ∑ nρ over the zeros ρ of the Riemann zeta function with the imaginary part of ρ the interval [ T, 2T ]. This is ≈ T / (2 π)Λ( n). Can you make a primality test out of this? (Conrey) The sum ∑ nρζ′(ρ)/ζ ′′ (ρ) ≈ T / (2 π) log p log q if n = pq. Can you make a factoring algorithm out of this? \n\nProblem/",
   "clean_statement": "Question 8. Look at ∑ nρ over the zeros ρ of the Riemann zeta function with the imaginary part of ρ the interval [ T, 2T ]. This is ≈ T / (2 π)Λ( n). Can you make a primality test out of this? (Conrey) The sum ∑ nρζ′(ρ)/ζ ′′ (ρ) ≈ T / (2 π) log p log q if n = pq. Can you make a factoring algorithm out of this?\n\nProblem/",
   "public_statement": "Question 8. Look at ∑ nρ over the zeros ρ of the Riemann zeta function with the imaginary part of ρ the interval [ T, 2T ]. This is ≈ T / (2 π)Λ( n). Can you make a primality test out of this? (Conrey) The sum ∑ nρζ′(ρ)/ζ ′′ (ρ) ≈ T / (2 π) log p log q if n = pq. Can you make a factoring algorithm out of this?\n\nProblem/",
   "evidence": "The canonical record is Question 8 in the AIM workshop list *Future directions in algorithmic number theory*. The official PDF and HTML restore the extracted superscripts and derivatives as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-COMPUTATION-0097": {
   "statement_status": "exact",
   "original_statement": "Question 9. Compute the zeta function of Spec Z/n Z, namely \n\nζ(s) = ∏\n\n> p|n\n\n1\n\n1 − p−s,\n\nwithout knowing the prime factorization of n. Computing the special value s = 1 gives you \n\nφ(n) which is enough to factor n. (Wan) 28 \n\nReplace n by a polynomial f (x) ∈ Fp[x], so given \n\nζ(s) = ∏ \n\n> g|fgmonic,irreducible\n\n1\n\n1 − g\n\ncan you recover the factorization of f in Fq[x] in deterministic polynomial time? (Wan)",
   "clean_statement": "Question 9. Compute the zeta function of Spec Z/n Z, namely\n\nζ(s) = ∏\n\n> p|n\n\n1\n\n1 − p−s,\n\nwithout knowing the prime factorization of n. Computing the special value s = 1 gives you\n\nφ(n) which is enough to factor n. (Wan) 28\n\nReplace n by a polynomial f (x) ∈ Fp[x], so given\n\nζ(s) = ∏\n\n> g|fgmonic,irreducible\n\n1\n\n1 − g\n\ncan you recover the factorization of f in Fq[x] in deterministic polynomial time? (Wan)",
   "public_statement": "Question 9. Compute the zeta function of Spec Z/n Z, namely\n\nζ(s) = ∏\n\n> p|n\n\n1\n\n1 − p−s,\n\nwithout knowing the prime factorization of n. Computing the special value s = 1 gives you\n\nφ(n) which is enough to factor n. (Wan) 28\n\nReplace n by a polynomial f (x) ∈ Fp[x], so given\n\nζ(s) = ∏\n\n> g|fgmonic,irreducible\n\n1\n\n1 − g\n\ncan you recover the factorization of f in Fq[x] in deterministic polynomial time? (Wan)",
   "evidence": "This is Question 9 from the AIM workshop *Future directions in algorithmic number theory*. The official AIM HTML and the workshop PDF agree on the following formulas.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 96,
   "attempt": 1
  },
  "AIM-COMPUTATION-0098": {
   "statement_status": "exact",
   "original_statement": "Question 10. To speed up the elliptic curve factorization algorithm, pick E/ Q\n\nwith large torsion group so that its reduction modulo n is more likely to be smooth. Mazur's results show that this torsion subgroup can be no larger than 16. There is a result of Kamienny-Mazur-Merel: there is a bound on E(K)tors in terms of [ K: Q]. So try to find a number field K where p splits completely and such that E(K)tors large. (Pomerance)",
   "clean_statement": "Question 10. To speed up the elliptic curve factorization algorithm, pick E/ Q\n\nwith large torsion group so that its reduction modulo n is more likely to be smooth. Mazur's results show that this torsion subgroup can be no larger than 16. There is a result of Kamienny-Mazur-Merel: there is a bound on E(K)tors in terms of [ K: Q]. So try to find a number field K where p splits completely and such that E(K)tors large. (Pomerance)",
   "public_statement": "Question 10. To speed up the elliptic curve factorization algorithm, pick E/ Q\n\nwith large torsion group so that its reduction modulo n is more likely to be smooth. Mazur's results show that this torsion subgroup can be no larger than 16. There is a result of Kamienny-Mazur-Merel: there is a bound on E(K)tors in terms of [ K: Q]. So try to find a number field K where p splits completely and such that E(K)tors large. (Pomerance)",
   "evidence": "The source is Question 10 in the AIM workshop list *Future directions in algorithmic number theory*. The extracted record is faithful to the workshop PDF apart from lost typography. With notation restored, the question reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 97,
   "attempt": 1
  },
  "AIM-COMPUTATION-0099": {
   "statement_status": "exact",
   "original_statement": "Question 11. Given a (random) number n of 10000 digits, it may be impossible to find 5000 digit primes p, q such that n = pq. Much easier: find some 10000 digit primes \n\np, q such that n is the first 10000 digits of pq. Instead, find 7500 digit primes p, q such that \n\nn is the first 10000 digits of pq. (Coppersmith) Therefore, do this for 6000 digit primes. (Bernstein)",
   "clean_statement": "Question 11. Given a (random) number n of 10000 digits, it may be impossible to find 5000 digit primes p, q such that n = pq. Much easier: find some 10000 digit primes\n\np, q such that n is the first 10000 digits of pq. Instead, find 7500 digit primes p, q such that\n\nn is the first 10000 digits of pq. (Coppersmith) Therefore, do this for 6000 digit primes. (Bernstein)",
   "public_statement": "Question 11. Given a (random) number n of 10000 digits, it may be impossible to find 5000 digit primes p, q such that n = pq. Much easier: find some 10000 digit primes\n\np, q such that n is the first 10000 digits of pq. Instead, find 7500 digit primes p, q such that\n\nn is the first 10000 digits of pq. (Coppersmith) Therefore, do this for 6000 digit primes. (Bernstein)",
   "evidence": "This is Question 11 in the AIM problem list produced by the March 2003 workshop *Future directions in algorithmic number theory*. The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 98,
   "attempt": 1
  },
  "AIM-COMPUTATION-0100": {
   "statement_status": "exact",
   "original_statement": "Question 12. Let p be prime, write p − 1 = 2 `m where m is odd, and assume \n\n` ≥ 3. Find in deterministic polynomial time x ∈ Fp such that x2`−1\n\n+ 1 is not a 2 `−1th power. (Kedlaya)",
   "clean_statement": "Question 12. Let p be prime, write p − 1 = 2 `m where m is odd, and assume\n\n` ≥ 3. Find in deterministic polynomial time x ∈ Fp such that x2`−1\n\n+ 1 is not a 2 `−1th power. (Kedlaya)",
   "public_statement": "Question 12. Let p be prime, write p − 1 = 2 `m where m is odd, and assume\n\n` ≥ 3. Find in deterministic polynomial time x ∈ Fp such that x2`−1\n\n+ 1 is not a 2 `−1th power. (Kedlaya)",
   "evidence": "The OCR in the canonical record lost the superscripts and the letter \\(\\ell\\). The official AIM HTML transcription and workshop PDF agree on the following statement:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 99,
   "attempt": 1
  },
  "AIM-COMPUTATION-0101": {
   "statement_status": "exact",
   "original_statement": "Question 13. \n\n(a) Do the p-adic point counting methods of Lauder which allow you to go from one curve from another in a family apply to p-adic methods without the linear factor? (Edixhoven) (b) Do Fesenko's methods for proving good properties of Hasse-Weil L-functions of curves over number fields provide anything useful for computations? (Edixhoven)",
   "clean_statement": "Question 13.\n\n(a) Do the p-adic point counting methods of Lauder which allow you to go from one curve from another in a family apply to p-adic methods without the linear factor? (Edixhoven) (b) Do Fesenko's methods for proving good properties of Hasse-Weil L-functions of curves over number fields provide anything useful for computations? (Edixhoven)",
   "public_statement": "Question 13.\n\n(a) Do the p-adic point counting methods of Lauder which allow you to go from one curve from another in a family apply to p-adic methods without the linear factor? (Edixhoven) (b) Do Fesenko's methods for proving good properties of Hasse-Weil L-functions of curves over number fields provide anything useful for computations? (Edixhoven)",
   "evidence": "This record is Question 13 in the problem section of the American Institute of Mathematics workshop notes *Future directions in algorithmic number theory*. The workshop was held at AIM on 24--28 March 2003, and the PDF identifies itself as the version of 30 April 2003.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 100,
   "attempt": 1
  },
  "AIM-COMPUTATION-0102": {
   "statement_status": "exact",
   "original_statement": "Question 14. Pila's method for computing roots of unity is exponential in g. Is there any reason it can't be made linear? (Elkies)",
   "clean_statement": "Question 14. Pila's method for computing roots of unity is exponential in g. Is there any reason it can't be made linear? (Elkies)",
   "public_statement": "Question 14. Pila's method for computing roots of unity is exponential in g. Is there any reason it can't be made linear? (Elkies)",
   "evidence": "The official AIM workshop PDF and HTML both give the question literally as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 101,
   "attempt": 1
  },
  "AIM-COMPUTATION-0103": {
   "statement_status": "exact",
   "original_statement": "Question 15. Given a variety over a finite field, can you verify that a given function is its zeta function (i.e., is the question in NP)? (Lenstra, Edixhoven)",
   "clean_statement": "Question 15. Given a variety over a finite field, can you verify that a given function is its zeta function (i.e., is the question in NP)? (Lenstra, Edixhoven)",
   "public_statement": "Question 15. Given a variety over a finite field, can you verify that a given function is its zeta function (i.e., is the question in NP)? (Lenstra, Edixhoven)",
   "evidence": "The exact canonical record is Question 15 in *Future directions in algorithmic number theory*, source file `aim-computation-notes.json`, record index 102:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 102,
   "attempt": 1
  },
  "AIM-COMPUTATION-0104": {
   "statement_status": "exact",
   "original_statement": "Question 16. How can you compute with points as in Edixhoven's talk? In that application, you can avoid writing down explicit `-torsion points (only the fields of definition), but can you write them down for other transformation? (Edixhoven)",
   "clean_statement": "Question 16. How can you compute with points as in Edixhoven's talk? In that application, you can avoid writing down explicit `-torsion points (only the fields of definition), but can you write them down for other transformation? (Edixhoven)",
   "public_statement": "Question 16. How can you compute with points as in Edixhoven's talk? In that application, you can avoid writing down explicit `-torsion points (only the fields of definition), but can you write them down for other transformation? (Edixhoven)",
   "evidence": "The canonical record has lost several mathematical glyphs. The official AIM HTML transcription and the workshop PDF agree on the following restoration.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 103,
   "attempt": 1
  },
  "AIM-COMPUTATION-0105": {
   "statement_status": "exact",
   "original_statement": "Question 17. \n\n(a) Can deformation theory be applied `-adically? (Various) (b) Under what circumstances do Betti numbers stay the same under reduction modulo p?(Various) (c) Is finding the genus of a plane curve polynomial time (in the degree)? (Wan) \n\nProblem/",
   "clean_statement": "Question 17.\n\n(a) Can deformation theory be applied `-adically? (Various) (b) Under what circumstances do Betti numbers stay the same under reduction modulo p?(Various) (c) Is finding the genus of a plane curve polynomial time (in the degree)? (Wan)\n\nProblem/",
   "public_statement": "Question 17.\n\n(a) Can deformation theory be applied `-adically? (Various) (b) Under what circumstances do Betti numbers stay the same under reduction modulo p?(Various) (c) Is finding the genus of a plane curve polynomial time (in the degree)? (Wan)\n\nProblem/",
   "evidence": "This is Question 17 in the AIM workshop list *Future directions in algorithmic number theory* (workshop held March 24--28, 2003; source PDF version dated April 30, 2003). The source record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 104,
   "attempt": 1
  },
  "AIM-COMPUTATION-0106": {
   "statement_status": "exact",
   "original_statement": "Question 18. Can you compute roots modulo p of a fixed polynomial (` a la Schoof-Pila) in polynomial time, like x3 − 2? (Schoof) 30",
   "clean_statement": "Question 18. Can you compute roots modulo p of a fixed polynomial (` a la Schoof-Pila) in polynomial time, like x3 − 2? (Schoof) 30",
   "public_statement": "Question 18. Can you compute roots modulo p of a fixed polynomial (` a la Schoof-Pila) in polynomial time, like x3 − 2? (Schoof) 30",
   "evidence": "The canonical record comes from Question 18 of the AIM workshop notes *Future directions in algorithmic number theory*. The web version and the typeset PDF give the intended text as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-computation-notes.json",
   "source_index": 105,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0001": {
   "statement_status": "exact",
   "original_statement": "1. Find the central cross section u⊥:= {x ∈ Rn: x · u = 0 } of Bn \n\n> ∞\n\n=\n\n{x ∈ Rn: |xi| ≤ 1} with the largest perimeter. 2. What is the central cross section of Bn \n\n> ∞\n\nwith smallest perimeter? 3. What if instead of perimeter we take the mean width or intrinsic volume? 4. What about other codimensions? 5. What about other measures such as gaussian measures? 6. What about perimeter of projections? \n\n#2 Bounded Projection Inequality \n\nProposed by Mathieu Meyer \n\nLet Knos be the collection of all origin symmetric convex bodies in Rn.Find an asymptotic bound for the quantity \n\nmax \n\n> K∈Knos\n\nmin \n\n> u,v ∈Sn−1\n\nPv,u ⊥ (K)\n\nvol n−1(K ∩ u⊥), (1) where Pv,u ⊥ (K) is the projection of K on u⊥ in the direction v ∈ Sn−1.One should note that 2n is a rough upper bound for (1) because for every hyperplane H in Rn, there is a projection PH so that ‖PH ‖ ≤ 2.This would imply PH K ⊆ 2( K ∩ H) and hence ‖PH K‖ ≤ 2n−1.One should also note that \n\nmin \n\n> u,v ∈Sn−1\n\nPv,u ⊥ (K)\n\nvol n−1(K ∩ u⊥) (2) is affine invariant. If K is an ellipsoid then the quantity in (2) is 1 so (1) is always greater than or equal to 1.\n\n#3 Preservation of Convexity Under the Intersec-tion Body Operator \n\nProposed by Maria Alfonseca \n\n2Suppose K is an origin-symmetric starbody in Rn. let ρK (u):= max {a:\n\nau ∈ K} and define the intersection body IK of K as the body defined by \n\nρIK (u) = Vol n−1(K ∩ u⊥) ∀u ∈ Sn−1. (3) Define the convex kernel of K as the intersection of all convex bodies M\n\nsuch that m ⊆ K. What conditions on the convex kernel of K would imply \n\nIK is convex? \n\n#4 Cotype and Volume Ration of `np ⊗π `nq ⊗π `nr\n\nProposed by Carsten Schütt \n\nFind the cotype and volume ration of `np ⊗π `nq ⊗π `nr.\n\n#5 Banach-Mazur Distance Problem \n\nProposed by Mark Rudelson \n\nLet Knos denote the space of origin-symmetric, convex bodies in Rn and let dBM is the Banach-Mazur distance. Let α ∈ (0, 1) and n ∈ N. Suppose K, L ∈ Knos are such that there exists b > 0 so \n\ndBM (K ∩ E, L ∩ E) ≤ b ∀E ∈ Kαn os. (4) Does there exists a function f so that \n\ndBM (K, L ) ≤ f (b, α )? (5) It's worth noting that M. Rudelson showed during one of the problem sessions that this question is trivially true if dBM is replaced with the geometric distance dg.\n\n#6 Injective Operators which Correspond with the Intersection Body Operator \n\nProposed by Richard Gardner \n\nLet Kno denote the collection of convex bodies (i.e., compact convex sets with nonempty interior) in Rn and let Knos denote the origin-symmetric members of Kno. Is there an F: Kno → Knos such that F is continuous, affine invariant, injective (modulo translations and reflections in the origin), and \n\nF |Kn = I, where I is the intersection body operator. 37 Discrete Tomography \n\nProposed by Richard Gardner \n\n(Discrete Aleksandrov projection theorem.) If two n-dimensional cen-trally symmetric convex lattice sets in Zn are such that the cardinality of their projections on any lattice hyperplane is the same, are they equal up to translation? This question appears with the restriction n ≥ 3 in [GGZ]. For n = 2, the answer is negative, but only an isolated counterexample is known, consisting of three non-congruent origin-symmetric convex lattice sets, each with 11 points. See Figure 4 in the above paper. It is possible that the answer is affirmative when n = 2 for sets containing sufficiently many points. \n\n#8 Geometric Problems on Sections of Convex Bod-ies \n\nProposed by Richard Gardner \n\nLet G (n, k ) denote the set of k-dimensional subspaces of Rn, where \n\n2 ≤ k ≤ n−1 is fixed in the following problems. If K is a set let int K denote the interior of K. Also let o denote the origin in Rn and if S ∈ G (n, k ), let \n\nK|S denote the projection of K on S.",
   "clean_statement": "1. Find the central cross section u⊥:= {x ∈ Rn: x · u = 0 } of Bn\n\n> ∞\n\n=\n\n{x ∈ Rn: |xi| ≤ 1} with the largest perimeter. 2. What is the central cross section of Bn\n\n> ∞\n\nwith smallest perimeter? 3. What if instead of perimeter we take the mean width or intrinsic volume? 4. What about other codimensions? 5. What about other measures such as gaussian measures? 6. What about perimeter of projections?\n\n#2 Bounded Projection Inequality\n\nProposed by Mathieu Meyer\n\nLet Knos be the collection of all origin symmetric convex bodies in Rn.Find an asymptotic bound for the quantity\n\nmax\n\n> K∈Knos\n\nmin\n\n> u,v ∈Sn−1\n\nPv,u ⊥ (K)\n\nvol n−1(K ∩ u⊥), (1) where Pv,u ⊥ (K) is the projection of K on u⊥ in the direction v ∈ Sn−1.One should note that 2n is a rough upper bound for (1) because for every hyperplane H in Rn, there is a projection PH so that ‖PH ‖ ≤ 2.This would imply PH K ⊆ 2( K ∩ H) and hence ‖PH K‖ ≤ 2n−1.One should also note that\n\nmin\n\n> u,v ∈Sn−1\n\nPv,u ⊥ (K)\n\nvol n−1(K ∩ u⊥) (2) is affine invariant. If K is an ellipsoid then the quantity in (2) is 1 so (1) is always greater than or equal to 1.\n\n#3 Preservation of Convexity Under the Intersec-tion Body Operator\n\nProposed by Maria Alfonseca\n\n2Suppose K is an origin-symmetric starbody in Rn. let ρK (u):= max {a:\n\nau ∈ K} and define the intersection body IK of K as the body defined by\n\nρIK (u) = Vol n−1(K ∩ u⊥) ∀u ∈ Sn−1. (3) Define the convex kernel of K as the intersection of all convex bodies M\n\nsuch that m ⊆ K. What conditions on the convex kernel of K would imply\n\nIK is convex?\n\n#4 Cotype and Volume Ration of `np ⊗π `nq ⊗π `nr\n\nProposed by Carsten Schütt\n\nFind the cotype and volume ration of `np ⊗π `nq ⊗π `nr.\n\n#5 Banach-Mazur Distance Problem\n\nProposed by Mark Rudelson\n\nLet Knos denote the space of origin-symmetric, convex bodies in Rn and let dBM is the Banach-Mazur distance. Let α ∈ (0, 1) and n ∈ N. Suppose K, L ∈ Knos are such that there exists b > 0 so\n\ndBM (K ∩ E, L ∩ E) ≤ b ∀E ∈ Kαn os. (4) Does there exists a function f so that\n\ndBM (K, L ) ≤ f (b, α )? (5) It's worth noting that M. Rudelson showed during one of the problem sessions that this question is trivially true if dBM is replaced with the geometric distance dg.\n\n#6 Injective Operators which Correspond with the Intersection Body Operator\n\nProposed by Richard Gardner\n\nLet Kno denote the collection of convex bodies (i.e., compact convex sets with nonempty interior) in Rn and let Knos denote the origin-symmetric members of Kno. Is there an F: Kno → Knos such that F is continuous, affine invariant, injective (modulo translations and reflections in the origin), and\n\nF |Kn = I, where I is the intersection body operator. 37 Discrete Tomography\n\nProposed by Richard Gardner\n\n(Discrete Aleksandrov projection theorem.) If two n-dimensional cen-trally symmetric convex lattice sets in Zn are such that the cardinality of their projections on any lattice hyperplane is the same, are they equal up to translation? This question appears with the restriction n ≥ 3 in [GGZ]. For n = 2, the answer is negative, but only an isolated counterexample is known, consisting of three non-congruent origin-symmetric convex lattice sets, each with 11 points. See Figure 4 in the above paper. It is possible that the answer is affirmative when n = 2 for sets containing sufficiently many points.\n\n#8 Geometric Problems on Sections of Convex Bod-ies\n\nProposed by Richard Gardner\n\nLet G (n, k ) denote the set of k-dimensional subspaces of Rn, where\n\n2 ≤ k ≤ n−1 is fixed in the following problems. If K is a set let int K denote the interior of K. Also let o denote the origin in Rn and if S ∈ G (n, k ), let\n\nK|S denote the projection of K on S.",
   "public_statement": "1. Find the central cross section u⊥:= {x ∈ Rn: x · u = 0 } of Bn\n\n> ∞\n\n=\n\n{x ∈ Rn: |xi| ≤ 1} with the largest perimeter. 2. What is the central cross section of Bn\n\n> ∞\n\nwith smallest perimeter? 3. What if instead of perimeter we take the mean width or intrinsic volume? 4. What about other codimensions? 5. What about other measures such as gaussian measures? 6. What about perimeter of projections?\n\n#2 Bounded Projection Inequality\n\nProposed by Mathieu Meyer\n\nLet Knos be the collection of all origin symmetric convex bodies in Rn.Find an asymptotic bound for the quantity\n\nmax\n\n> K∈Knos\n\nmin\n\n> u,v ∈Sn−1\n\nPv,u ⊥ (K)\n\nvol n−1(K ∩ u⊥), (1) where Pv,u ⊥ (K) is the projection of K on u⊥ in the direction v ∈ Sn−1.One should note that 2n is a rough upper bound for (1) because for every hyperplane H in Rn, there is a projection PH so that ‖PH ‖ ≤ 2.This would imply PH K ⊆ 2( K ∩ H) and hence ‖PH K‖ ≤ 2n−1.One should also note that\n\nmin\n\n> u,v ∈Sn−1\n\nPv,u ⊥ (K)\n\nvol n−1(K ∩ u⊥) (2) is affine invariant. If K is an ellipsoid then the quantity in (2) is 1 so (1) is always greater than or equal to 1.\n\n#3 Preservation of Convexity Under the Intersec-tion Body Operator\n\nProposed by Maria Alfonseca\n\n2Suppose K is an origin-symmetric starbody in Rn. let ρK (u):= max {a:\n\nau ∈ K} and define the intersection body IK of K as the body defined by\n\nρIK (u) = Vol n−1(K ∩ u⊥) ∀u ∈ Sn−1. (3) Define the convex kernel of K as the intersection of all convex bodies M\n\nsuch that m ⊆ K. What conditions on the convex kernel of K would imply\n\nIK is convex?\n\n#4 Cotype and Volume Ration of `np ⊗π `nq ⊗π `nr\n\nProposed by Carsten Schütt\n\nFind the cotype and volume ration of `np ⊗π `nq ⊗π `nr.\n\n#5 Banach-Mazur Distance Problem\n\nProposed by Mark Rudelson\n\nLet Knos denote the space of origin-symmetric, convex bodies in Rn and let dBM is the Banach-Mazur distance. Let α ∈ (0, 1) and n ∈ N. Suppose K, L ∈ Knos are such that there exists b > 0 so\n\ndBM (K ∩ E, L ∩ E) ≤ b ∀E ∈ Kαn os. (4) Does there exists a function f so that\n\ndBM (K, L ) ≤ f (b, α )? (5) It's worth noting that M. Rudelson showed during one of the problem sessions that this question is trivially true if dBM is replaced with the geometric distance dg.\n\n#6 Injective Operators which Correspond with the Intersection Body Operator\n\nProposed by Richard Gardner\n\nLet Kno denote the collection of convex bodies (i.e., compact convex sets with nonempty interior) in Rn and let Knos denote the origin-symmetric members of Kno. Is there an F: Kno → Knos such that F is continuous, affine invariant, injective (modulo translations and reflections in the origin), and\n\nF |Kn = I, where I is the intersection body operator. 37 Discrete Tomography\n\nProposed by Richard Gardner\n\n(Discrete Aleksandrov projection theorem.) If two n-dimensional cen-trally symmetric convex lattice sets in Zn are such that the cardinality of their projections on any lattice hyperplane is the same, are they equal up to translation? This question appears with the restriction n ≥ 3 in [GGZ]. For n = 2, the answer is negative, but only an isolated counterexample is known, consisting of three non-congruent origin-symmetric convex lattice sets, each with 11 points. See Figure 4 in the above paper. It is possible that the answer is affirmative when n = 2 for sets containing sufficiently many points.\n\n#8 Geometric Problems on Sections of Convex Bod-ies\n\nProposed by Richard Gardner\n\nLet G (n, k ) denote the set of k-dimensional subspaces of Rn, where\n\n2 ≤ k ≤ n−1 is fixed in the following problems. If K is a set let int K denote the interior of K. Also let o denote the origin in Rn and if S ∈ G (n, k ), let\n\nK|S denote the projection of K on S.",
   "evidence": "The canonical JSON extraction runs past the end of Problem 1 and includes later problems from the same PDF. The official AIM source was therefore checked directly. On page 2 it gives the title **“Max/min Perimeter of central cross-sections,”** proposed by Hermann König, and asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0002": {
   "statement_status": "exact",
   "original_statement": "1. (See [G, \nProblem 7.4 and Note 7.2].) Suppose o ∈ int K and all sections K ∩ S, S ∈ G (n, k ), are affinely equivalent. Does it follow that K is an ellipsoid? Here K is a star body but the problem is also open when K is convex.",
   "clean_statement": "1. (See [G,\nProblem 7.4 and Note 7.2].) Suppose o ∈ int K and all sections K ∩ S, S ∈ G (n, k ), are affinely equivalent. Does it follow that K is an ellipsoid? Here K is a star body but the problem is also open when K is convex.",
   "public_statement": "1. (See [G,\nProblem 7.4 and Note 7.2].) Suppose o ∈ int K and all sections K ∩ S, S ∈ G (n, k ), are affinely equivalent. Does it follow that K is an ellipsoid? Here K is a star body but the problem is also open when K is convex.",
   "evidence": "The problem is Problem 1 in Section 8, “Geometric Problems on Sections of Convex Bodies,” of the AIM list *Sections of Convex Bodies* (August 2013). The section's introductory paragraph is part of the statement:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0003": {
   "statement_status": "exact",
   "original_statement": "2. (See [G, \nProblem 3.3 and Note 3.2].) Same problem as in (1) but for projections, i.e. if all projections K|S, S ∈ G (n, k ), are affinely equivalent, is K is an ellipsoid? Here K is convex.",
   "clean_statement": "2. (See [G,\nProblem 3.3 and Note 3.2].) Same problem as in (1) but for projections, i.e. if all projections K|S, S ∈ G (n, k ), are affinely equivalent, is K is an ellipsoid? Here K is convex.",
   "public_statement": "2. (See [G,\nProblem 3.3 and Note 3.2].) Same problem as in (1) but for projections, i.e. if all projections K|S, S ∈ G (n, k ), are affinely equivalent, is K is an ellipsoid? Here K is convex.",
   "evidence": "The official 2013 AIM problem list first fixes the conventions",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0004": {
   "statement_status": "exact",
   "original_statement": "3. (See [G, \nProblem 7.1 and Note 7.1].) Suppose K and L are star bodies and that K ∩ S is homethetic to L ∩ S for every G (n, k ). Does it follow that K is homothetic to L? The answer is affirmative if K\n\nand L are convex and o ∈ int K ∩ int L; see [G, Theorem 7.1.1].",
   "clean_statement": "3. (See [G,\nProblem 7.1 and Note 7.1].) Suppose K and L are star bodies and that K ∩ S is homethetic to L ∩ S for every G (n, k ). Does it follow that K is homothetic to L? The answer is affirmative if K\n\nand L are convex and o ∈ int K ∩ int L; see [G, Theorem 7.1.1].",
   "public_statement": "3. (See [G,\nProblem 7.1 and Note 7.1].) Suppose K and L are star bodies and that K ∩ S is homethetic to L ∩ S for every G (n, k ). Does it follow that K is homothetic to L? The answer is affirmative if K\n\nand L are convex and o ∈ int K ∩ int L; see [G, Theorem 7.1.1].",
   "evidence": "The source is Richard Gardner's problem list for the 2013 AIM workshop *Sections of Convex Bodies*. The introductory paragraph fixes an integer \\[ 2\\leq k\\leq n-1 \\] and writes \\(G(n,k)\\) for the Grassmannian of \\(k\\)-dimensional linear subspaces of \\(\\mathbb R^n\\). The extracted record has a grammatical omission: “for every \\(G(n,k)\\)” must read “for every \\(S\\in G(n,k)\\).” The official PDF confirms the surrounding notation and this reconstruction.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0005": {
   "statement_status": "corrected_verified",
   "original_statement": "4. (See [G, \nProblem 7.3 and Note 7.1].) Suppose K and L are star bodies and that K ∩ S is congruent to L ∩ S for every G (n, k ). Does it follow that K = ±L? It has been shown that the answer is affirmative for \n\nk = 2 when \"congruent to\" is replaced by \"a rotation of\". For more on this see [R]. \n4",
   "clean_statement": "Let \\(G(n,k)\\) be the Grassmannian of \\(k\\)-dimensional linear subspaces of\n\\(\\mathbb R^n\\), where \\(2\\leq k\\leq n-1\\) is fixed.  Suppose \\(K\\) and\n\\(L\\) are star bodies and, for every \\(S\\in G(n,k)\\), the sections\n\\(K\\cap S\\) and \\(L\\cap S\\) are congruent.  Must \\(K=\\pm L\\)?\n\nThe source adds that the answer is affirmative for \\(k=2\\) when\n“congruent” is replaced by “a rotation of,” citing [R].",
   "public_statement": "Let \\(G(n,k)\\) be the Grassmannian of \\(k\\)-dimensional linear subspaces of\n\\(\\mathbb R^n\\), where \\(2\\leq k\\leq n-1\\) is fixed.  Suppose \\(K\\) and\n\\(L\\) are star bodies and, for every \\(S\\in G(n,k)\\), the sections\n\\(K\\cap S\\) and \\(L\\cap S\\) are congruent.  Must \\(K=\\pm L\\)?\n\nThe source adds that the answer is affirmative for \\(k=2\\) when\n“congruent” is replaced by “a rotation of,” citing [R].",
   "evidence": "The exact source record in `input.json` is preserved, including two extraction defects. The notation and dimension range are inherited from the beginning of Section 8 of the official AIM PDF, and the final isolated `4` is the printed page number captured by OCR. The recovered statement is:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0006": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "5. (See [G, \nProblem 3.2 and Note 3.1].) Suppose K and L are convex bodies and that K|S is congruent to L|S for every G (n, k ). Does it follow that K is a translate of ±L? A corresponding affirmative answer for k = 2 and rotations is given by Ryabogin in [R]. More-over, significant progress on the question for congruent projections has been made recently by F. Nazarov; this information was commu-nicated by Dimtry Ryabogin. \n\n#9 Vertex Index Problems \n\nProposed by Alexander Litvak",
   "clean_statement": null,
   "public_statement": "5. (See [G,\nProblem 3.2 and Note 3.1].) Suppose K and L are convex bodies and that K|S is congruent to L|S for every G (n, k ). Does it follow that K is a translate of ±L? A corresponding affirmative answer for k = 2 and rotations is given by Ryabogin in [R]. More-over, significant progress on the question for congruent projections has been made recently by F. Nazarov; this information was commu-nicated by Dimtry Ryabogin.\n\n#9 Vertex Index Problems\n\nProposed by Alexander Litvak",
   "evidence": "The canonical record contains OCR hyphenation and then spills into the heading of the next section. The beginning of Section 8 of the official AIM PDF fixes an integer \\(k\\) with \\(2\\leq k\\leq n-1\\), writes \\(G(n,k)\\) for the Grassmannian of \\(k\\)-dimensional linear subspaces of \\(\\mathbb R^n\\), and writes \\(K\\mid S\\) for the orthogonal projection of \\(K\\) onto \\(S\\). With the missing variable restored, the problem is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0007": {
   "statement_status": "exact",
   "original_statement": "1. Prove that the vertex index of the n-dimensional Euclidean ball Bn \n\n> 2\n\nis \n\n2n3/2, where the vertex index of a centrally-symmetric convex body \n\nK = −K ⊂ Rn is defined as \n\nvein( K) = inf \n\n> {∑\n> i\n\n‖pi‖K | K ⊂ conv {pi}\n\n> }.",
   "clean_statement": "1. Prove that the vertex index of the n-dimensional Euclidean ball Bn\n\n> 2\n\nis\n\n2n3/2, where the vertex index of a centrally-symmetric convex body\n\nK = −K ⊂ Rn is defined as\n\nvein( K) = inf\n\n> {∑\n> i\n\n‖pi‖K | K ⊂ conv {pi}\n\n> }.",
   "public_statement": "1. Prove that the vertex index of the n-dimensional Euclidean ball Bn\n\n> 2\n\nis\n\n2n3/2, where the vertex index of a centrally-symmetric convex body\n\nK = −K ⊂ Rn is defined as\n\nvein( K) = inf\n\n> {∑\n> i\n\n‖pi‖K | K ⊂ conv {pi}\n\n> }.",
   "evidence": "The canonical JSON is damaged by PDF extraction. The official AIM problem PDF, Section 9 (“Vertex Index Problems”), Problem 1, has the following notation and exponent:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0008": {
   "statement_status": "exact",
   "original_statement": "2. Find the best possible upper bound for the vertex index of 3-dimensional centrally-symmetric convex body. 3. Provide an upper bound for the vertex index of a convex body K ⊂\n\nRn\n\nvein( K) = inf \n\n> {∑\n> i\n\n‖pi‖K−a | a ∈ K, K − a ⊂ conv {pi}\n\n> }.\n\n#10 Reconstruction of Polytopes with Few Facets (or Few Vertices) \n\nThese questions are motivated by the problem of efficiently estimating poly-topes with few facets or few vertices from random points.",
   "clean_statement": "2. Find the best possible upper bound for the vertex index of 3-dimensional centrally-symmetric convex body. 3. Provide an upper bound for the vertex index of a convex body K ⊂\n\nRn\n\nvein( K) = inf\n\n> {∑\n> i\n\n‖pi‖K−a | a ∈ K, K − a ⊂ conv {pi}\n\n> }.\n\n#10 Reconstruction of Polytopes with Few Facets (or Few Vertices)\n\nThese questions are motivated by the problem of efficiently estimating poly-topes with few facets or few vertices from random points.",
   "public_statement": "2. Find the best possible upper bound for the vertex index of 3-dimensional centrally-symmetric convex body. 3. Provide an upper bound for the vertex index of a convex body K ⊂\n\nRn\n\nvein( K) = inf\n\n> {∑\n> i\n\n‖pi‖K−a | a ∈ K, K − a ⊂ conv {pi}\n\n> }.\n\n#10 Reconstruction of Polytopes with Few Facets (or Few Vertices)\n\nThese questions are motivated by the problem of efficiently estimating poly-topes with few facets or few vertices from random points.",
   "evidence": "The canonical extraction merges two consecutive problems from Section 9 of the official 2013 AIM list and then spills into Section 10. The official PDF gives the following partition.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0009": {
   "statement_status": "exact",
   "original_statement": "1. (Identifiability from moments) Let the k-th moment tensor of a con-vex body be the k-th moment tensor of the uniform distribution on it. For any positive integer t, is there k = k(t) (independent of d) so that the following map is injective: The map that takes a d-dimensional polytope with dt facets and maps it to its first k moment tensors. Known: 5(a) [GLPR] Can recover a d-dimensional polytope with n vertices from O(dn ) moments along d random directions. (b) [FJK] Can estimate a parallelepiped efficiently from first 4 mo-ment tensors. (c) [AGR] Can estimate any simplex efficiently from first 3 moment tensors. In the affirmative case, is there an efficient reconstruction algorithm? Namely, is there an algorithm running in time polynomial in n whose input are the first k moment tensors of any given polytope P as before, and that outputs a polytope within 1/10 Hausdorff distance of P? What if the tensors are not known exactly but come from a sample approximation from a sample of size polynomial in n for any fixed t?",
   "clean_statement": "1. (Identifiability from moments) Let the k-th moment tensor of a con-vex body be the k-th moment tensor of the uniform distribution on it. For any positive integer t, is there k = k(t) (independent of d) so that the following map is injective: The map that takes a d-dimensional polytope with dt facets and maps it to its first k moment tensors. Known: 5(a) [GLPR] Can recover a d-dimensional polytope with n vertices from O(dn ) moments along d random directions. (b) [FJK] Can estimate a parallelepiped efficiently from first 4 mo-ment tensors. (c) [AGR] Can estimate any simplex efficiently from first 3 moment tensors. In the affirmative case, is there an efficient reconstruction algorithm? Namely, is there an algorithm running in time polynomial in n whose input are the first k moment tensors of any given polytope P as before, and that outputs a polytope within 1/10 Hausdorff distance of P? What if the tensors are not known exactly but come from a sample approximation from a sample of size polynomial in n for any fixed t?",
   "public_statement": "1. (Identifiability from moments) Let the k-th moment tensor of a con-vex body be the k-th moment tensor of the uniform distribution on it. For any positive integer t, is there k = k(t) (independent of d) so that the following map is injective: The map that takes a d-dimensional polytope with dt facets and maps it to its first k moment tensors. Known: 5(a) [GLPR] Can recover a d-dimensional polytope with n vertices from O(dn ) moments along d random directions. (b) [FJK] Can estimate a parallelepiped efficiently from first 4 mo-ment tensors. (c) [AGR] Can estimate any simplex efficiently from first 3 moment tensors. In the affirmative case, is there an efficient reconstruction algorithm? Namely, is there an algorithm running in time polynomial in n whose input are the first k moment tensors of any given polytope P as before, and that outputs a polytope within 1/10 Hausdorff distance of P? What if the tensors are not known exactly but come from a sample approximation from a sample of size polynomial in n for any fixed t?",
   "evidence": "The canonical record has lost superscript formatting and contains a variable inconsistency. The official AIM PDF, Section 10 (“Reconstruction of Polytopes with Few Facets (or Few Vertices)”), Problem 1, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0010": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "2. (Stability of reconstruction from relative central sectional areas) Is it true that for any positive integer t there is a positive integer t′ = t′(t)\n\nsuch that for any two isotropic centrally symmetric d-dimensional polytopes P, Q with at most dt facets we have: If for all θ ∈ Sd−1\n\n∣∣∣∣∣\n\nVol n−1(P ∩ θ⊥)\n\nVol (P ) − Vol n−1(Q ∩ θ⊥)\n\nVol (Q)\n\n∣∣∣∣∣ ≤ 1\n\ndt′,\n\nthen the Hausdorff distance between P and Q is at most 1/10?In the affirmative case, is there an efficient reconstruction algorithm? Namely, is there an algorithm running in time polynomial in n that when given access to the value of relative central sectional areas of any given polytope P as before, to within additive error 1/d t′, it outputs a polytope within 1/10 Hausdorff distance of P?\n\n#References \n\n> [AGR] Joseph Anderson, Navin Goyal, and Luis Rademacher, Efficient learning of simplices (2012). [FJK] Alan Frieze, Mark Jerrum, and Ravi Kannan, Learning linear transformations,Foundations of computer science, 1996. proceedings., 37th annual symposium on, 1996, pp. 359-368. [G] Richard J Gardner, Geometric tomography, Vol. 58, Cambridge University Press Cambridge, 1995. [GGZ] Richard J Gardner, Paolo Gronchi, and Chuanming Zong, Sums, projections, and sections of lattice sets, and the discrete covariogram, Discrete & Com-putational Geometry 34 (2005), no. 3, 391-409.\n\n6[GLPR] Nick Gravin, Jean Lasserre, Dmitrii V Pasechnik, and Sinai Robins, The inverse moment problem for convex polytopes, Discrete & Computational Geome-try 48 (2012), no. 3, 596-621. [R] Dmitry Ryabogin, On the continual rubik's cube, Advances in Mathematics 231 \n\n(2012), no. 6, 3429-3444. \n\n7",
   "clean_statement": null,
   "public_statement": "2. (Stability of reconstruction from relative central sectional areas) Is it true that for any positive integer t there is a positive integer t′ = t′(t)\n\nsuch that for any two isotropic centrally symmetric d-dimensional polytopes P, Q with at most dt facets we have: If for all θ ∈ Sd−1\n\n∣∣∣∣∣\n\nVol n−1(P ∩ θ⊥)\n\nVol (P ) − Vol n−1(Q ∩ θ⊥)\n\nVol (Q)\n\n∣∣∣∣∣ ≤ 1\n\ndt′,\n\nthen the Hausdorff distance between P and Q is at most 1/10?In the affirmative case, is there an efficient reconstruction algorithm? Namely, is there an algorithm running in time polynomial in n that when given access to the value of relative central sectional areas of any given polytope P as before, to within additive error 1/d t′, it outputs a polytope within 1/10 Hausdorff distance of P?\n\n#References\n\n> [AGR] Joseph Anderson, Navin Goyal, and Luis Rademacher, Efficient learning of simplices (2012). [FJK] Alan Frieze, Mark Jerrum, and Ravi Kannan, Learning linear transformations,Foundations of computer science, 1996. proceedings., 37th annual symposium on, 1996, pp. 359-368. [G] Richard J Gardner, Geometric tomography, Vol. 58, Cambridge University Press Cambridge, 1995. [GGZ] Richard J Gardner, Paolo Gronchi, and Chuanming Zong, Sums, projections, and sections of lattice sets, and the discrete covariogram, Discrete & Com-putational Geometry 34 (2005), no. 3, 391-409.\n\n6[GLPR] Nick Gravin, Jean Lasserre, Dmitrii V Pasechnik, and Sinai Robins, The inverse moment problem for convex polytopes, Discrete & Computational Geome-try 48 (2012), no. 3, 596-621. [R] Dmitry Ryabogin, On the continual rubik's cube, Advances in Mathematics 231\n\n(2012), no. 6, 3429-3444.\n\n7",
   "evidence": "The source literally says “running in time polynomial in \\(n\\)” in the last paragraph, although Problem 2 defines only \\(d\\). This is almost certainly a notation carryover: Problem 1 immediately before it uses \\(d\\) for dimension and \\(n\\) for the number of vertices. For Problem 2, “polynomial in \\(d\\)” is the natural reading, but this reconstruction is explicitly marked as an interpretation rather than a correction to the source.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0011": {
   "statement_status": "exact",
   "original_statement": "1. Is it true that if IK ⊂ IL and ΠK ⊂ ΠL then K ⊂ L?2. Is it true that if IK ⊂ IL and ΠK ⊂ ΠL then vol( K) 6 vol( L)? (True if n 6 4) 3. Is it true that for any convex (symmetric) body K there exists a direction u ∈ Sn−1\n\nsuch that vol(St u(K)∗) 6 Cvol( K∗)? 4. What about an average version of the previous question, i.e.: \n\n∫\n\n> Sn−1\n\nvol(St u(K)∗)du 6 Cvol( K∗)? 5. Is the average of P(K) for symmetric polytopes K greater than or equal to P(Bn\n\n> ∞\n\n)? More precisely, let K be a random symmetric polytope of a fixed number of vertices, where each vertex is independent Gaussian random vector is it true that EP(K) >\n\nP(Bn\n\n> ∞\n\n)? 6. Given even log-concave probability measure μ, estimate μ(K)μ(K∗) from above. (It is known that μ(K)μ(K∗) 6 μ(Bn \n\n> 2\n\n) if μ is unconditional) 7. In the above question, characterize the equality case and find lower bound for the un-conditional case. 18. Consider μ∗ in the sense of Artstein-Milman and answer to the same questions about \n\nμ(K)μ∗(K). 9. Let K and L be symmetric convex bodies in Rn and denote by N (K, L ) the covering number of K by L, the minimum number of the translates of L needed to cover K. Is it true that \n\nN (K, L ) 6 [N (L∗, cK ∗]δ,\n\nwhere c, δ are absolute constants? 10. Given random vectors X and Y uniformly distributed in K and K∗ respectively, look at 〈X, Y 〉, up to a \"100\"-moment. Are the first \"100\" moments the same (1 + ε) as those for Gaussian random vectors? 11. Prove Mahler's conjecture for finite-dimensional normed spaces that embed in Lp,\n\np < 1 and, in particular, for convex intersection bodies (which correspond to the case \n\np = −1). 12. What if 1 < p < 2? Is it true that P(BX ) > P(Bnp ) if X is embedded into Lp, 1 < p < 2and dimX = n?\n1",
   "clean_statement": "1. Is it true that if IK ⊂ IL and ΠK ⊂ ΠL then K ⊂ L?2. Is it true that if IK ⊂ IL and ΠK ⊂ ΠL then vol( K) 6 vol( L)? (True if n 6 4) 3. Is it true that for any convex (symmetric) body K there exists a direction u ∈ Sn−1\n\nsuch that vol(St u(K)∗) 6 Cvol( K∗)? 4. What about an average version of the previous question, i.e.:\n\n∫\n\n> Sn−1\n\nvol(St u(K)∗)du 6 Cvol( K∗)? 5. Is the average of P(K) for symmetric polytopes K greater than or equal to P(Bn\n\n> ∞\n\n)? More precisely, let K be a random symmetric polytope of a fixed number of vertices, where each vertex is independent Gaussian random vector is it true that EP(K) >\n\nP(Bn\n\n> ∞\n\n)? 6. Given even log-concave probability measure μ, estimate μ(K)μ(K∗) from above. (It is known that μ(K)μ(K∗) 6 μ(Bn\n\n> 2\n\n) if μ is unconditional) 7. In the above question, characterize the equality case and find lower bound for the un-conditional case. 18. Consider μ∗ in the sense of Artstein-Milman and answer to the same questions about\n\nμ(K)μ∗(K). 9. Let K and L be symmetric convex bodies in Rn and denote by N (K, L ) the covering number of K by L, the minimum number of the translates of L needed to cover K. Is it true that\n\nN (K, L ) 6 [N (L∗, cK ∗]δ,\n\nwhere c, δ are absolute constants? 10. Given random vectors X and Y uniformly distributed in K and K∗ respectively, look at 〈X, Y 〉, up to a \"100\"-moment. Are the first \"100\" moments the same (1 + ε) as those for Gaussian random vectors? 11. Prove Mahler's conjecture for finite-dimensional normed spaces that embed in Lp,\n\np < 1 and, in particular, for convex intersection bodies (which correspond to the case\n\np = −1). 12. What if 1 < p < 2? Is it true that P(BX ) > P(Bnp ) if X is embedded into Lp, 1 < p < 2and dimX = n?\n1",
   "public_statement": "1. Is it true that if IK ⊂ IL and ΠK ⊂ ΠL then K ⊂ L?2. Is it true that if IK ⊂ IL and ΠK ⊂ ΠL then vol( K) 6 vol( L)? (True if n 6 4) 3. Is it true that for any convex (symmetric) body K there exists a direction u ∈ Sn−1\n\nsuch that vol(St u(K)∗) 6 Cvol( K∗)? 4. What about an average version of the previous question, i.e.:\n\n∫\n\n> Sn−1\n\nvol(St u(K)∗)du 6 Cvol( K∗)? 5. Is the average of P(K) for symmetric polytopes K greater than or equal to P(Bn\n\n> ∞\n\n)? More precisely, let K be a random symmetric polytope of a fixed number of vertices, where each vertex is independent Gaussian random vector is it true that EP(K) >\n\nP(Bn\n\n> ∞\n\n)? 6. Given even log-concave probability measure μ, estimate μ(K)μ(K∗) from above. (It is known that μ(K)μ(K∗) 6 μ(Bn\n\n> 2\n\n) if μ is unconditional) 7. In the above question, characterize the equality case and find lower bound for the un-conditional case. 18. Consider μ∗ in the sense of Artstein-Milman and answer to the same questions about\n\nμ(K)μ∗(K). 9. Let K and L be symmetric convex bodies in Rn and denote by N (K, L ) the covering number of K by L, the minimum number of the translates of L needed to cover K. Is it true that\n\nN (K, L ) 6 [N (L∗, cK ∗]δ,\n\nwhere c, δ are absolute constants? 10. Given random vectors X and Y uniformly distributed in K and K∗ respectively, look at 〈X, Y 〉, up to a \"100\"-moment. Are the first \"100\" moments the same (1 + ε) as those for Gaussian random vectors? 11. Prove Mahler's conjecture for finite-dimensional normed spaces that embed in Lp,\n\np < 1 and, in particular, for convex intersection bodies (which correspond to the case\n\np = −1). 12. What if 1 < p < 2? Is it true that P(BX ) > P(Bnp ) if X is embedded into Lp, 1 < p < 2and dimX = n?\n1",
   "evidence": "The canonical record is the first extracted record from the AIM workshop *Mahler's conjecture and duality in convex geometry* (August 9--13, 2010), notes by Jaegil Kim. The extraction merged Questions 1--12 from the first two pages of the official four-page PDF. The unmodified OCR record is preserved in input.json.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 10,
   "attempt": 2
  },
  "AIM-CONVEX_GEOMETRY-0012": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "3. (T. Tao) Let K be a symmetric convex body in Rn. Consider the covering radius of \n\nZn for K∗, the smallest r > 0 such that \n\n⋃\n\n> v∈Zn\\K\n\nrK ∗ + v = Rn.\n\nIs it true that K∗ has the largest covering radius when K is a cube, among all sym-metric convex bodies in Rn?14. Does IK = K imply K = cB n \n\n> 2\n\nwhen n > 3? Notice that if ΠK is considered instead of IK, it is not true because the projection body of a cube is a dilate of a cube. It is a special case of \nProblem 8.7 in [G]. There are a few comments about the more general problem in [G, Note 8.6]. Moreover the analogous more general question for projection bodies is [G, \nProblem 4.5], and in [G, Note 4.6] it is stated that Weil solved this for polytopes. 15. For n > 3, is it true that ImK → Bn \n\n> 2\n\nin the Banach-Mazur distance as m → ∞?216. Let n > 5. Construct an example of a polytope K which is an intersection body, not a polar body of a zonotope. \n1",
   "clean_statement": null,
   "public_statement": "3. (T. Tao) Let K be a symmetric convex body in Rn. Consider the covering radius of\n\nZn for K∗, the smallest r > 0 such that\n\n⋃\n\n> v∈Zn\\K\n\nrK ∗ + v = Rn.\n\nIs it true that K∗ has the largest covering radius when K is a cube, among all sym-metric convex bodies in Rn?14. Does IK = K imply K = cB n\n\n> 2\n\nwhen n > 3? Notice that if ΠK is considered instead of IK, it is not true because the projection body of a cube is a dilate of a cube. It is a special case of\nProblem 8.7 in [G]. There are a few comments about the more general problem in [G, Note 8.6]. Moreover the analogous more general question for projection bodies is [G,\nProblem 4.5], and in [G, Note 4.6] it is stated that Weil solved this for polytopes. 15. For n > 3, is it true that ImK → Bn\n\n> 2\n\nin the Banach-Mazur distance as m → ∞?216. Let n > 5. Construct an example of a polytope K which is an intersection body, not a polar body of a zonotope.\n1",
   "evidence": "The canonical record is not one problem numbered 3. It is an extraction accident that concatenates Problems 13--16 from page 2 of the official AIM workshop PDF, *Mahler's conjecture and duality in convex geometry* (2010). The preceding canonical record contains Problems 1--12 and the following record starts with Problem 17. The four recovered statements are:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0013": {
   "statement_status": "exact",
   "original_statement": "7. (G. Kuperberg) Let K be a symmetric convex body in Rn. For random vectors X and \n\nY distributed in K and K∗ respectively, consider the random variable 〈X, Y 〉. Then the conjecture is that 〈X, Y 〉 has the largest variance when K is an ellipsoid. One may consider a weaker conjecture: the integral of |〈 X, Y 〉| 2 over K × K∗ is maximized when \n\nK is an ellipsoid. More generally, G. Paouris asked the same questions for the quantity \n\n|〈 X, Y 〉| p, (p > 0), instead of |〈 X, Y 〉| 2.Furthermore two other variations can be considered: Given 0 < c < 1, consider either the probability that 〈X, Y 〉 = c, or the volume of the region in which it is so. One may again conjecture that either the probability or at least the volume is maximized when \n\nK is an ellipsoid. \n1",
   "clean_statement": "7. (G. Kuperberg) Let K be a symmetric convex body in Rn. For random vectors X and\n\nY distributed in K and K∗ respectively, consider the random variable 〈X, Y 〉. Then the conjecture is that 〈X, Y 〉 has the largest variance when K is an ellipsoid. One may consider a weaker conjecture: the integral of |〈 X, Y 〉| 2 over K × K∗ is maximized when\n\nK is an ellipsoid. More generally, G. Paouris asked the same questions for the quantity\n\n|〈 X, Y 〉| p, (p > 0), instead of |〈 X, Y 〉| 2.Furthermore two other variations can be considered: Given 0 < c < 1, consider either the probability that 〈X, Y 〉 = c, or the volume of the region in which it is so. One may again conjecture that either the probability or at least the volume is maximized when\n\nK is an ellipsoid.\n1",
   "public_statement": "7. (G. Kuperberg) Let K be a symmetric convex body in Rn. For random vectors X and\n\nY distributed in K and K∗ respectively, consider the random variable 〈X, Y 〉. Then the conjecture is that 〈X, Y 〉 has the largest variance when K is an ellipsoid. One may consider a weaker conjecture: the integral of |〈 X, Y 〉| 2 over K × K∗ is maximized when\n\nK is an ellipsoid. More generally, G. Paouris asked the same questions for the quantity\n\n|〈 X, Y 〉| p, (p > 0), instead of |〈 X, Y 〉| 2.Furthermore two other variations can be considered: Given 0 < c < 1, consider either the probability that 〈X, Y 〉 = c, or the volume of the region in which it is so. One may again conjecture that either the probability or at least the volume is maximized when\n\nK is an ellipsoid.\n1",
   "evidence": "The canonical input preserves an OCR extraction beginning with “7.” Direct inspection of the official AIM PDF shows that this is **Problem 17** in the 2010 workshop list *Mahler's conjecture and duality in convex geometry*. The leading 1 was lost in extraction. The PDF uses \\(K^*\\) for the polar body in \\(\\mathbb R^n\\), and the exponents are \\(2\\) and \\(p>0\\).",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0014": {
   "statement_status": "exact",
   "original_statement": "8. (G. Kuperberg) In R3, consider a class A of centrally symmetric polytopes of a given combinatorial type, excluding the type of the cube and the octahedron. Then the con-jecture is that the volume product does not have a local minimum in A. Furthermore, one may ask if each such A has a unique critical point up to affine transformations, and it is a local maximum. \n1",
   "clean_statement": "8. (G. Kuperberg) In R3, consider a class A of centrally symmetric polytopes of a given combinatorial type, excluding the type of the cube and the octahedron. Then the con-jecture is that the volume product does not have a local minimum in A. Furthermore, one may ask if each such A has a unique critical point up to affine transformations, and it is a local maximum.\n1",
   "public_statement": "8. (G. Kuperberg) In R3, consider a class A of centrally symmetric polytopes of a given combinatorial type, excluding the type of the cube and the octahedron. Then the con-jecture is that the volume product does not have a local minimum in A. Furthermore, one may ask if each such A has a unique critical point up to affine transformations, and it is a local maximum.\n1",
   "evidence": "The canonical JSON extraction reads “8.” and ends with a stray “1”. Inspection of the official AIM workshop PDF shows that the leading digit was lost and that the terminal digit is a page-number spill. The official statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0015": {
   "statement_status": "exact",
   "original_statement": "9. (R.J. Gardner) Does the following theorem hold for star bodies, or perhaps more generally still? It comes from [G, \nProblem 7.1] and there are some relevant comments in [G, Note 7.1]. \n\nTheorem (Rogers, 1965.) Suppose that 2 6 k 6 n − 1 and that K and L are compact convex sets in Rn containing the origin in their relative interiors. If all k-sections K ∩S\n\nand L ∩ S of K and L are homothetic (or translates), then K and L are homothetic (or translates, respectively). \n\n2",
   "clean_statement": "9. (R.J. Gardner) Does the following theorem hold for star bodies, or perhaps more generally still? It comes from [G,\nProblem 7.1] and there are some relevant comments in [G, Note 7.1].\n\nTheorem (Rogers, 1965.) Suppose that 2 6 k 6 n − 1 and that K and L are compact convex sets in Rn containing the origin in their relative interiors. If all k-sections K ∩S\n\nand L ∩ S of K and L are homothetic (or translates), then K and L are homothetic (or translates, respectively).\n\n2",
   "public_statement": "9. (R.J. Gardner) Does the following theorem hold for star bodies, or perhaps more generally still? It comes from [G,\nProblem 7.1] and there are some relevant comments in [G, Note 7.1].\n\nTheorem (Rogers, 1965.) Suppose that 2 6 k 6 n − 1 and that K and L are compact convex sets in Rn containing the origin in their relative interiors. If all k-sections K ∩S\n\nand L ∩ S of K and L are homothetic (or translates), then K and L are homothetic (or translates, respectively).\n\n2",
   "evidence": "The canonical record is an OCR extraction from the official problem list for the 2010 AIM workshop *Mahler's conjecture and duality in convex geometry*. Three features of the extraction require correction:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0016": {
   "statement_status": "exact",
   "original_statement": "0. (R.J. Gardner)[G, \nProblem 7.6 ] If K and L are origin-symmetric star bodies in R3\n\nwhose sections by every plane through the origin have equal perimeters, is K = L? In particular, This was proved by R. Howard, F. Nazarov, D. Ryabogin and A. Zvavitch for bodies of revolution and by V. Yaskin for polytopes. \n2",
   "clean_statement": "0. (R.J. Gardner)[G,\nProblem 7.6 ] If K and L are origin-symmetric star bodies in R3\n\nwhose sections by every plane through the origin have equal perimeters, is K = L? In particular, This was proved by R. Howard, F. Nazarov, D. Ryabogin and A. Zvavitch for bodies of revolution and by V. Yaskin for polytopes.\n2",
   "public_statement": "0. (R.J. Gardner)[G,\nProblem 7.6 ] If K and L are origin-symmetric star bodies in R3\n\nwhose sections by every plane through the origin have equal perimeters, is K = L? In particular, This was proved by R. Howard, F. Nazarov, D. Ryabogin and A. Zvavitch for bodies of revolution and by V. Yaskin for polytopes.\n2",
   "evidence": "The canonical record is source index 15 of aim-convex-geometry-notes.json. Its number field is \"0\" and its text has stray page-number fragments. The official AIM PDF resolves the extraction error: this is **Problem 20**, not Problem 0, in *Problems from the workshop \"Mahler's conjecture and duality in convex geometry\"*, AIM, August 9--13, 2010, notes by Jaegil Kim.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0017": {
   "statement_status": "exact",
   "original_statement": "1. (S. Reisner) Improve the following theorem by finding the number of vertices N (n) >n + 3. \n\nTheorem (M. Meyer, S. Reisner)[MR] Let K be a convex polytope in Rn, with at most \n\nn + 3 vertices (or facets) and non-empty interior. Then \n\nP(K) > (n + 1) n+1 \n\n(n!) 2\n\n3with equality if and only if K is an n-dimensional simplex. \n\n2",
   "clean_statement": "1. (S. Reisner) Improve the following theorem by finding the number of vertices N (n) >n + 3.\n\nTheorem (M. Meyer, S. Reisner)[MR] Let K be a convex polytope in Rn, with at most\n\nn + 3 vertices (or facets) and non-empty interior. Then\n\nP(K) > (n + 1) n+1\n\n(n!) 2\n\n3with equality if and only if K is an n-dimensional simplex.\n\n2",
   "public_statement": "1. (S. Reisner) Improve the following theorem by finding the number of vertices N (n) >n + 3.\n\nTheorem (M. Meyer, S. Reisner)[MR] Let K be a convex polytope in Rn, with at most\n\nn + 3 vertices (or facets) and non-empty interior. Then\n\nP(K) > (n + 1) n+1\n\n(n!) 2\n\n3with equality if and only if K is an n-dimensional simplex.\n\n2",
   "evidence": "This record is Problem 21 in the AIM list from the workshop “Mahler's conjecture and duality in convex geometry.” The PDF extraction has lost superscripts and changed a non-strict inequality into a strict one. With \\[ K^z=\\{y\\in{\\mathbb R}^n:\\langle y,x-z\\rangle\\leq 1 \\text{ for every }x\\in K\\} \\] and \\[ {\\cal P}(K)=\\min_{z\\in\\operatorname{int}K}|K|\\,|K^z|, \\] the recovered problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0018": {
   "statement_status": "exact",
   "original_statement": "2. (S. Reisner) Quantify the shadow movement theorem (see [MR]). \n2",
   "clean_statement": "2. (S. Reisner) Quantify the shadow movement theorem (see [MR]).\n2",
   "public_statement": "2. (S. Reisner) Quantify the shadow movement theorem (see [MR]).\n2",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0019": {
   "statement_status": "exact",
   "original_statement": "3. (H. Koenig) For Q ⊂ Sn−1, define its dual Q◦ on the sphere Sn−1 by \n\nQ◦ = {x ∈ Sn−1: 〈x, y 〉 > 0 ∀y ∈ Q}\n\nand define its volume product by P(Q) = vol( Q)vol( Q◦). For any ∆ n ⊂ Sn−1 which is determined by distinct n hyperplanes passing through the origin, is it true P(∆ n) 6\n\nP(Rn \n\n> +\n\n∩ Sn−1)? In particular, if n = 3, it is true.",
   "clean_statement": "3. (H. Koenig) For Q ⊂ Sn−1, define its dual Q◦ on the sphere Sn−1 by\n\nQ◦ = {x ∈ Sn−1: 〈x, y 〉 > 0 ∀y ∈ Q}\n\nand define its volume product by P(Q) = vol( Q)vol( Q◦). For any ∆ n ⊂ Sn−1 which is determined by distinct n hyperplanes passing through the origin, is it true P(∆ n) 6\n\nP(Rn\n\n> +\n\n∩ Sn−1)? In particular, if n = 3, it is true.",
   "public_statement": "3. (H. Koenig) For Q ⊂ Sn−1, define its dual Q◦ on the sphere Sn−1 by\n\nQ◦ = {x ∈ Sn−1: 〈x, y 〉 > 0 ∀y ∈ Q}\n\nand define its volume product by P(Q) = vol( Q)vol( Q◦). For any ∆ n ⊂ Sn−1 which is determined by distinct n hyperplanes passing through the origin, is it true P(∆ n) 6\n\nP(Rn\n\n> +\n\n∩ Sn−1)? In particular, if n = 3, it is true.",
   "evidence": "The corpus record is an OCR-damaged extraction from the AIM workshop list *Mahler's conjecture and duality in convex geometry*. Direct inspection of the source PDF recovers the entry as Problem 23, attributed to H. Koenig:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0020": {
   "statement_status": "exact",
   "original_statement": "Problem 1 (Santosh Vempala) Let K ⊂ Rn be a convex body. How many uniformly randomly chosen points from K are needed to estimate the volume of K within a factor of \n\n2.",
   "clean_statement": "Problem 1 (Santosh Vempala) Let K ⊂ Rn be a convex body. How many uniformly randomly chosen points from K are needed to estimate the volume of K within a factor of\n\n2.",
   "public_statement": "Problem 1 (Santosh Vempala) Let K ⊂ Rn be a convex body. How many uniformly randomly chosen points from K are needed to estimate the volume of K within a factor of\n\n2.",
   "evidence": "The canonical record contains the line break",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0021": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 2 (Santosh Vempala) Let S ∈ Rn be compact, and let C ∈ argmin C∈K vol(∆( S, C )) \n\nbe a convex body closest to S (K stands for the set of compact convex sets with nonempty interior, and the empty set, and ∆( ·, ·) stands for the symmetric difference). We say that \n\nS is \u000f-convex if vol(∆( S, C )) ≤ \u000fvol( S). Assume that the center of gravity of S is at the origin. For a pair of points x, y 6 = 0 ∈ Rn, let the subspace spanned by them be H(x, y ) and define P (x, y ):= S ∩ H(x, y ).Let μ be the distribution on 2-dimensional sections P (x, y ) obtained by picking x and y\n\nuniformly at random from S. If \n\nPr \n\n> μ\n\n(P (x, y )is convex) ≥ 1 − \u000f, \n\nthen S is O(n\u000f )-convex.",
   "clean_statement": null,
   "public_statement": "Problem 2 (Santosh Vempala) Let S ∈ Rn be compact, and let C ∈ argmin C∈K vol(∆( S, C ))\n\nbe a convex body closest to S (K stands for the set of compact convex sets with nonempty interior, and the empty set, and ∆( ·, ·) stands for the symmetric difference). We say that\n\nS is [U+000F]-convex if vol(∆( S, C )) ≤ [U+000F]vol( S). Assume that the center of gravity of S is at the origin. For a pair of points x, y 6 = 0 ∈ Rn, let the subspace spanned by them be H(x, y ) and define P (x, y ):= S ∩ H(x, y ).Let μ be the distribution on 2-dimensional sections P (x, y ) obtained by picking x and y\n\nuniformly at random from S. If\n\nPr\n\n> μ\n\n(P (x, y )is convex) ≥ 1 − [U+000F],\n\nthen S is O(n[U+000F] )-convex.",
   "evidence": "The canonical record is Problem 2 from the AIM workshop “Algorithmic convex geometry.” Its extracted text contains the control character U+000F in place of a Greek letter and several damaged mathematical relations. Inspection of the official PDF and of its embedded TeX font encoding gives the following repairs.",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0022": {
   "statement_status": "exact",
   "original_statement": "Problem 3 (Van Vu) Let K ⊂ Rd (d fixed dimension) be a convex body. Let x1,..., x n\n\nbe uniformly random points from K, and let X:= vol(conv( x1,..., x n)). Does X satisfy the central limit theorem?",
   "clean_statement": "Problem 3 (Van Vu) Let K ⊂ Rd (d fixed dimension) be a convex body. Let x1,..., x n\n\nbe uniformly random points from K, and let X:= vol(conv( x1,..., x n)). Does X satisfy the central limit theorem?",
   "public_statement": "Problem 3 (Van Vu) Let K ⊂ Rd (d fixed dimension) be a convex body. Let x1,..., x n\n\nbe uniformly random points from K, and let X:= vol(conv( x1,..., x n)). Does X satisfy the central limit theorem?",
   "evidence": "The official AIM PDF, from the November 5--9, 2007 workshop *Algorithmic Convex Geometry*, gives the following as Problem 3, attributed to Van Vu:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0023": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 4 (Ryan O'Donnnell) Let K ⊂ Rn be a convex set, and let X be Gaussian random variable conditioned to lie in K. Does Var( Xθ) = 1 imply that K is a cylinder in direction θ? If Var( Xθ) = 1 − \u000f then does it mean that K has a small symmetric difference with a cylinder in direction θ?\n\n1Remarks. K is not necessarily symmetric; when it is symmetric, the solution is given by Sidak's lemma. It is known that Var( Xθ) ≤ 1 in every direction θ, where Xθ is the projection of X in direction θ. This can be proved in a number of ways, e.g., using Brascamp-Lieb inequality.",
   "clean_statement": null,
   "public_statement": "Problem 4 (Ryan O'Donnnell) Let K ⊂ Rn be a convex set, and let X be Gaussian random variable conditioned to lie in K. Does Var( Xθ) = 1 imply that K is a cylinder in direction θ? If Var( Xθ) = 1 − [U+000F] then does it mean that K has a small symmetric difference with a cylinder in direction θ?\n\n1Remarks. K is not necessarily symmetric; when it is symmetric, the solution is given by Sidak's lemma. It is known that Var( Xθ) ≤ 1 in every direction θ, where Xθ is the projection of X in direction θ. This can be proved in a number of ways, e.g., using Brascamp-Lieb inequality.",
   "evidence": "The canonical record is Problem 4 in the AIM workshop list *Algorithmic Convex Geometry* (workshop held November 5--9, 2007). The PDF asks, with notation restored,",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0024": {
   "statement_status": "exact",
   "original_statement": "Problem 5 (David Jerison) For a symmetric convex body K, let S be a least area surface that divides the volume of K into two halves (or any other fixed proportion). Is the surface a graph? (The surface in question may not be a hyperplane in general.)",
   "clean_statement": "Problem 5 (David Jerison) For a symmetric convex body K, let S be a least area surface that divides the volume of K into two halves (or any other fixed proportion). Is the surface a graph? (The surface in question may not be a hyperplane in general.)",
   "public_statement": "Problem 5 (David Jerison) For a symmetric convex body K, let S be a least area surface that divides the volume of K into two halves (or any other fixed proportion). Is the surface a graph? (The surface in question may not be a hyperplane in general.)",
   "evidence": "The canonical record is Problem 5, attributed to David Jerison, in the AIM workshop list *Problems from the Workshop on Algorithmic Convex Geometry* (version dated 31 October 2007). Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0025": {
   "statement_status": "exact",
   "original_statement": "Problem 6 (Mokshay Madiman) Let X1, X 2, X 3 be real-valued independent random vari-ables with densities. Is the following inequality true? \n\ne2H( X1+X2+X3) + e2H( X2)?\n\n≥ e2H( X1+X2) + e2H( X2+X3). (1)",
   "clean_statement": "Problem 6 (Mokshay Madiman) Let X1, X 2, X 3 be real-valued independent random vari-ables with densities. Is the following inequality true?\n\ne2H( X1+X2+X3) + e2H( X2)?\n\n≥ e2H( X1+X2) + e2H( X2+X3). (1)",
   "public_statement": "Problem 6 (Mokshay Madiman) Let X1, X 2, X 3 be real-valued independent random vari-ables with densities. Is the following inequality true?\n\ne2H( X1+X2+X3) + e2H( X2)?\n\n≥ e2H( X1+X2) + e2H( X2+X3). (1)",
   "evidence": "The AIM list records Problem 6, attributed to Mokshay Madiman. In the source's notation, \\(H\\) is differential entropy and the variables are independent, real-valued random variables with densities. The question is whether",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0026": {
   "statement_status": "exact",
   "original_statement": "Problem 7 (Grigoris Paouris) Which bodies are ψ2-bodies? Are zonoids ψ2-bodies? Are projections of ψ2-bodies ψ2-bodies? \n\n2Remark. It is known that the Bpn for p ≥ 2 is ψ2.",
   "clean_statement": "Problem 7 (Grigoris Paouris) Which bodies are ψ2-bodies? Are zonoids ψ2-bodies? Are projections of ψ2-bodies ψ2-bodies?\n\n2Remark. It is known that the Bpn for p ≥ 2 is ψ2.",
   "public_statement": "Problem 7 (Grigoris Paouris) Which bodies are ψ2-bodies? Are zonoids ψ2-bodies? Are projections of ψ2-bodies ψ2-bodies?\n\n2Remark. It is known that the Bpn for p ≥ 2 is ψ2.",
   "evidence": "The canonical record is Problem 7 in the AIM workshop list *Algorithmic Convex Geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0027": {
   "statement_status": "exact",
   "original_statement": "Problem 8 (Grigoris Paouris) Is it true that every compact set is anti-ψ2 in some di-rection?",
   "clean_statement": "Problem 8 (Grigoris Paouris) Is it true that every compact set is anti-ψ2 in some di-rection?",
   "public_statement": "Problem 8 (Grigoris Paouris) Is it true that every compact set is anti-ψ2 in some di-rection?",
   "evidence": "The AIM source states, verbatim apart from joining a line-broken word:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0028": {
   "statement_status": "exact",
   "original_statement": "Problem 9 (Yuval Peres) Consider lazy random walks on graphs or reversible Markov chains. Define the mixing time τTV (1 /2) as minimum t such that the total variation distance between pt(x, ·) and μ (stationary distribution) is at most 1/2.Does there exist c > 0 such that \n\nτTV (1 /2) ≤ c max \n\n> x,A:μ(A)≥1/2\n\nE TA,x,\n\nwhere TA,x is the hitting time for hitting A starting from x.",
   "clean_statement": "Problem 9 (Yuval Peres) Consider lazy random walks on graphs or reversible Markov chains. Define the mixing time τTV (1 /2) as minimum t such that the total variation distance between pt(x, ·) and μ (stationary distribution) is at most 1/2.Does there exist c > 0 such that\n\nτTV (1 /2) ≤ c max\n\n> x,A:μ(A)≥1/2\n\nE TA,x,\n\nwhere TA,x is the hitting time for hitting A starting from x.",
   "public_statement": "Problem 9 (Yuval Peres) Consider lazy random walks on graphs or reversible Markov chains. Define the mixing time τTV (1 /2) as minimum t such that the total variation distance between pt(x, ·) and μ (stationary distribution) is at most 1/2.Does there exist c > 0 such that\n\nτTV (1 /2) ≤ c max\n\n> x,A:μ(A)≥1/2\n\nE TA,x,\n\nwhere TA,x is the hitting time for hitting A starting from x.",
   "evidence": "The source is Problem 9, attributed to Yuval Peres, in *Problems from the AIM Workshop on Algorithmic Convex Geometry* (2007). The PDF asks about lazy random walks on graphs or reversible Markov chains. Its intended worst-case total-variation mixing time is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0029": {
   "statement_status": "exact",
   "original_statement": "Problem 10 (Yuval Peres) Consider random walks on undirected graphs where each node \n\nv is given a positive weight Wv (the probability of going from node u to node v is given by \n\n> WvP\n> w∈N(u)Ww, where N (u) is the set of neighbors of u in the graph). If we change the weights by a bounded factor, does the mixing time also change by a bounded factor?",
   "clean_statement": "Problem 10 (Yuval Peres) Consider random walks on undirected graphs where each node\n\nv is given a positive weight Wv (the probability of going from node u to node v is given by\n\n> WvP\n> w∈N(u)Ww, where N (u) is the set of neighbors of u in the graph). If we change the weights by a bounded factor, does the mixing time also change by a bounded factor?",
   "public_statement": "Problem 10 (Yuval Peres) Consider random walks on undirected graphs where each node\n\nv is given a positive weight Wv (the probability of going from node u to node v is given by\n\n> WvP\n> w∈N(u)Ww, where N (u) is the set of neighbors of u in the graph). If we change the weights by a bounded factor, does the mixing time also change by a bounded factor?",
   "evidence": "The source is Problem 10, attributed to Yuval Peres, in the AIM workshop list *Problems from the AIM Workshop on Algorithmic Convex Geometry*. The corpus transcription has lost the fraction bar. The PDF gives the following transition rule: on a finite undirected graph \\(G=(V,E)\\), give each vertex \\(v\\) a positive weight \\(W_v\\), and set",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0030": {
   "statement_status": "exact",
   "original_statement": "Problem 11 (Ryan O'Donnell) Is it true that every ellipsoid has Gaussian surface area bounded by a universal constant?",
   "clean_statement": "Problem 11 (Ryan O'Donnell) Is it true that every ellipsoid has Gaussian surface area bounded by a universal constant?",
   "public_statement": "Problem 11 (Ryan O'Donnell) Is it true that every ellipsoid has Gaussian surface area bounded by a universal constant?",
   "evidence": "The official AIM PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0031": {
   "statement_status": "exact",
   "original_statement": "1. (R. Schneider) Let Zn ⊂ Rn be a zonoid whose polar Zon is also a zonoid. Let d\n\ndenote the Banach-Mazur distance. Does d(Zn, B n \n\n> 2\n\n) converge to 1 as the dimension \n\nn tends to infinity?",
   "clean_statement": "1. (R. Schneider) Let Zn ⊂ Rn be a zonoid whose polar Zon is also a zonoid. Let d\n\ndenote the Banach-Mazur distance. Does d(Zn, B n\n\n> 2\n\n) converge to 1 as the dimension\n\nn tends to infinity?",
   "public_statement": "1. (R. Schneider) Let Zn ⊂ Rn be a zonoid whose polar Zon is also a zonoid. Let d\n\ndenote the Banach-Mazur distance. Does d(Zn, B n\n\n> 2\n\n) converge to 1 as the dimension\n\nn tends to infinity?",
   "evidence": "The canonical JSON record is visibly damaged by PDF extraction: `Zn`, `Zon`, and `B n > 2` have lost their subscript/superscript placement. The official AIM PDF gives the following statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0032": {
   "statement_status": "exact",
   "original_statement": "2. (R. Schneider) For an origin symmetric convex body K ∈ Rn, let λ(K) be the smallest number λ ≥ 1 such that there exists a zonoid Z with K ⊂ Z ⊂ λK. For example, for the cross-polytope Cn we have \n\nλ(Cn) = n\n\n2n−1\n\n((n−1) [( n−1) /2] \n\n)\n\n≈\n\n√2nπ\n\nas n → ∞. Is λ(K) maximal for the cross-polytope? \n\n> 12",
   "clean_statement": "2. (R. Schneider) For an origin symmetric convex body K ∈ Rn, let λ(K) be the smallest number λ ≥ 1 such that there exists a zonoid Z with K ⊂ Z ⊂ λK. For example, for the cross-polytope Cn we have\n\nλ(Cn) = n\n\n2n−1\n\n((n−1) [( n−1) /2]\n\n)\n\n≈\n\n√2nπ\n\nas n → ∞. Is λ(K) maximal for the cross-polytope?\n\n> 12",
   "public_statement": "2. (R. Schneider) For an origin symmetric convex body K ∈ Rn, let λ(K) be the smallest number λ ≥ 1 such that there exists a zonoid Z with K ⊂ Z ⊂ λK. For example, for the cross-polytope Cn we have\n\nλ(Cn) = n\n\n2n−1\n\n((n−1) [( n−1) /2]\n\n)\n\n≈\n\n√2nπ\n\nas n → ∞. Is λ(K) maximal for the cross-polytope?\n\n> 12",
   "evidence": "The canonical dataset record is item 2, attributed to R. Schneider, in the AIM problem list *Fourier analytic methods in convex geometry*. Its OCR text splits the displayed fraction across lines, renders the binomial coefficient as `((n-1) [(n-1)/2])`, collapses the square root to `sqrt(2n pi)`, and appends `12` after the problem.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 31,
   "attempt": 2
  },
  "AIM-CONVEX_GEOMETRY-0033": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "3. Let Kn denote the class of convex bodies in Rn. A Minkowski class M is a subset of \n\nKn that is closed in the Hausdorff metric, under Minkowski linear combinations and translations. A body K is a generalized M-body if there are two bodies M1, M 2 ∈ M \n\nsuch that K + M1 = M2.Let G be a subgroup of GL (n). We say that M is G-invariant if whenever K ∈ M\n\nand g ∈ G, we have gK ∈ M. Given B ∈ K n, G ⊂ GL (n), we define MB,G as the smallest Minkowski class containing B that is G-invariant. \n\nTheorem (Schneider, F. Schuster) Let B ∈ K n be non-symmetric. Every neighborhood of B contains an affine image B′ of B such that the generalized \n\nMB′,SO (n) bodies are dense in Kn.\n\nProblem: (R. Gardner) Dualize the theorem. If we replace generalized zonoids by \"generalized intersection bodies\", Minkowski sums by radial sums, etc, does a similar theorem hold? 4. In Rn, how many segments do we need to approximate a zonoid by zonotopes? Let \n\nK be a zonoid and Z a zonotope that is the sum of M segments. If d(Z, K ) ≤ 1 + \u000f,then M is of the order C(\u000f)n log n (Talagrand). If K is the Euclidean ball, this can be improved to C(\u000f)n. Is the extra log n in Talagrand's result necessary?",
   "clean_statement": null,
   "public_statement": "3. Let Kn denote the class of convex bodies in Rn. A Minkowski class M is a subset of\n\nKn that is closed in the Hausdorff metric, under Minkowski linear combinations and translations. A body K is a generalized M-body if there are two bodies M1, M 2 ∈ M\n\nsuch that K + M1 = M2.Let G be a subgroup of GL (n). We say that M is G-invariant if whenever K ∈ M\n\nand g ∈ G, we have gK ∈ M. Given B ∈ K n, G ⊂ GL (n), we define MB,G as the smallest Minkowski class containing B that is G-invariant.\n\nTheorem (Schneider, F. Schuster) Let B ∈ K n be non-symmetric. Every neighborhood of B contains an affine image B′ of B such that the generalized\n\nMB′,SO (n) bodies are dense in Kn.\n\nProblem: (R. Gardner) Dualize the theorem. If we replace generalized zonoids by \"generalized intersection bodies\", Minkowski sums by radial sums, etc, does a similar theorem hold? 4. In Rn, how many segments do we need to approximate a zonoid by zonotopes? Let\n\nK be a zonoid and Z a zonotope that is the sum of M segments. If d(Z, K ) ≤ 1 + [U+000F],then M is of the order C([U+000F])n log n (Talagrand). If K is the Euclidean ball, this can be improved to C([U+000F])n. Is the extra log n in Talagrand's result necessary?",
   "evidence": "The canonical JSON record has accidentally fused two consecutive problems from the AIM workshop list *Fourier analytic methods in convex geometry*. Inspection of the linked PDF separates them as follows.",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0034": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "5. (G. Schechtman) A theorem of Spencer states that if {Xi}ni=1 are in Bn \n\n> ∞\n\nand \u000fi = ±1then min \n\n> \u000fi\n\n‖\n\n> n\n\n∑\n\n> i=1\n\n\u000fiXi‖∞ ≤ C√n. \n\nDoes the same hold for any n-dimensional zonoid? i.e., does there exist a universal constant C such that for all n-dimensional centered zonoid Z, if {Xi}ni=1 ∈ Z then there are signs {\u000fi}ni=1 with ∑ni=1 \u000fiXi ∈ C√nZ?(If one can remove the log factor in problem 4 then the answer here is positive. As is, the best known substitute for C√n is C√n log log n.) 3",
   "clean_statement": null,
   "public_statement": "5. (G. Schechtman) A theorem of Spencer states that if {Xi}ni=1 are in Bn\n\n> ∞\n\nand [U+000F]i = ±1then min\n\n> [U+000F]i\n\n‖\n\n> n\n\n∑\n\n> i=1\n\n[U+000F]iXi‖∞ ≤ C√n.\n\nDoes the same hold for any n-dimensional zonoid? i.e., does there exist a universal constant C such that for all n-dimensional centered zonoid Z, if {Xi}ni=1 ∈ Z then there are signs {[U+000F]i}ni=1 with ∑ni=1 [U+000F]iXi ∈ C√nZ?(If one can remove the log factor in problem 4 then the answer here is positive. As is, the best known substitute for C√n is C√n log log n.) 3",
   "evidence": "The canonical JSON record contains flattened mathematical layout, U+000F control characters, and a trailing page number. The official AIM PDF verifies the following statement.",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0035": {
   "statement_status": "exact",
   "original_statement": "6. (M. Rudelson) Let K be a symmetric convex body. There exists a linear operator \n\nT such that \n\n∫\n\n> Sn−1\n\nhT K (θ) dσ (θ) ·\n\n∫\n\n> Sn−1\n\nh(T K )o (θ) dσ (θ) ≤ C log n. \n\nWhat is the upper bound for the above quantity for non-symmetric convex bodies \n\nK? (in this case we need to take T affine).",
   "clean_statement": "6. (M. Rudelson) Let K be a symmetric convex body. There exists a linear operator\n\nT such that\n\n∫\n\n> Sn−1\n\nhT K (θ) dσ (θ) ·\n\n∫\n\n> Sn−1\n\nh(T K )o (θ) dσ (θ) ≤ C log n.\n\nWhat is the upper bound for the above quantity for non-symmetric convex bodies\n\nK? (in this case we need to take T affine).",
   "public_statement": "6. (M. Rudelson) Let K be a symmetric convex body. There exists a linear operator\n\nT such that\n\n∫\n\n> Sn−1\n\nhT K (θ) dσ (θ) ·\n\n∫\n\n> Sn−1\n\nh(T K )o (θ) dσ (θ) ≤ C log n.\n\nWhat is the upper bound for the above quantity for non-symmetric convex bodies\n\nK? (in this case we need to take T affine).",
   "evidence": "The canonical record is problem 6, attributed to M. Rudelson, in the AIM list *Fourier analytic methods in convex geometry*. Inspection of the official PDF repairs the OCR and gives:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0036": {
   "statement_status": "exact",
   "original_statement": "7. (S. Robins) Given N, can we find explicitly N points on Sn−1 that are \"uniformly\" arranged? Are they the same as the vertices of the polytope with N vertices and maximum volume that is contained in Sn−1?8. Let f be a function on Sn−1. Can we decide from the Fourier transform of f if f is the support function of a convex body? Or if f is the radial function of a convex body?",
   "clean_statement": "7. (S. Robins) Given N, can we find explicitly N points on Sn−1 that are \"uniformly\" arranged? Are they the same as the vertices of the polytope with N vertices and maximum volume that is contained in Sn−1?8. Let f be a function on Sn−1. Can we decide from the Fourier transform of f if f is the support function of a convex body? Or if f is the radial function of a convex body?",
   "public_statement": "7. (S. Robins) Given N, can we find explicitly N points on Sn−1 that are \"uniformly\" arranged? Are they the same as the vertices of the polytope with N vertices and maximum volume that is contained in Sn−1?8. Let f be a function on Sn−1. Can we decide from the Fourier transform of f if f is the support function of a convex body? Or if f is the radial function of a convex body?",
   "evidence": "The canonical record is not one problem. It concatenates Problems 7 and 8 of the official 2007 AIM workshop list *Fourier analytic methods in convex geometry*. The PDF (printed page 4) reads, with its numbering restored:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0037": {
   "statement_status": "exact",
   "original_statement": "9. (R. Schneider) Classify all continuous T: Kn → K n with the following properties: \n\n• hT (K+L) = hT K + hT L \n\n• T (θK ) = θT K for every rotation θ.\n1",
   "clean_statement": "9. (R. Schneider) Classify all continuous T: Kn → K n with the following properties:\n\n• hT (K+L) = hT K + hT L\n\n• T (θK ) = θT K for every rotation θ.\n1",
   "public_statement": "9. (R. Schneider) Classify all continuous T: Kn → K n with the following properties:\n\n• hT (K+L) = hT K + hT L\n\n• T (θK ) = θT K for every rotation θ.\n1",
   "evidence": "The canonical record is Problem 9, attributed to Rolf Schneider, in the AIM workshop notes *Fourier Analytic Methods in Convexity*. The official PDF gives the following statement:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0038": {
   "statement_status": "exact",
   "original_statement": "0. (R. Schneider) Show that for most origin-symmetric convex K, the intersection body of K is not the polar of a zonoid. (\"Most\" means in the Baire category sense). \n1",
   "clean_statement": "0. (R. Schneider) Show that for most origin-symmetric convex K, the intersection body of K is not the polar of a zonoid. (\"Most\" means in the Baire category sense).\n1",
   "public_statement": "0. (R. Schneider) Show that for most origin-symmetric convex K, the intersection body of K is not the polar of a zonoid. (\"Most\" means in the Baire category sense).\n1",
   "evidence": "The canonical record reads",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0039": {
   "statement_status": "exact",
   "original_statement": "1. (S. Robins) (a) Let K be a polytope and consider the expansion of hK in spherical harmonics. If all of the coefficients have small absolute value, are all the facets of K simplices or close to simplices? (b) If, for each integer frequency, the Fourier transform of the indicator function 1 K\n\nis small, does the same conclusion hold? (This conjecture is made by analogy with pseudorandom number sequences which exhibit this behavior in dimension 1, and the intuition here is that thegiven hypothesis on the coefficients should exhibit a \"random\" polytope in some sense.) 4\n\n1",
   "clean_statement": "1. (S. Robins) (a) Let K be a polytope and consider the expansion of hK in spherical harmonics. If all of the coefficients have small absolute value, are all the facets of K simplices or close to simplices? (b) If, for each integer frequency, the Fourier transform of the indicator function 1 K\n\nis small, does the same conclusion hold? (This conjecture is made by analogy with pseudorandom number sequences which exhibit this behavior in dimension 1, and the intuition here is that thegiven hypothesis on the coefficients should exhibit a \"random\" polytope in some sense.) 4\n\n1",
   "public_statement": "1. (S. Robins) (a) Let K be a polytope and consider the expansion of hK in spherical harmonics. If all of the coefficients have small absolute value, are all the facets of K simplices or close to simplices? (b) If, for each integer frequency, the Fourier transform of the indicator function 1 K\n\nis small, does the same conclusion hold? (This conjecture is made by analogy with pseudorandom number sequences which exhibit this behavior in dimension 1, and the intuition here is that thegiven hypothesis on the coefficients should exhibit a \"random\" polytope in some sense.) 4\n\n1",
   "evidence": "The canonical JSON record is an OCR extraction from the AIM workshop compilation *Fourier analytic methods in convex geometry*. Comparison with the original PDF shows that this is Problem **11**, proposed by S. Robins, rather than Problem 1. The initial `1` in the JSON is the second digit of `11`; the trailing `4` is the printed page number, and the final `1` is extraction debris. The source also prints `thegiven`, which should be read as “the given.”",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0040": {
   "statement_status": "exact",
   "original_statement": "2. (Y. Gordon) Let K be a symmetric convex body. We define the zonoid ratio as \n\nzr (K) = min \n\n> Z∈Z,Z ⊂K\n\n|K|1/n \n\n|Z|1/n.\n\nThe volume ratio is defined as \n\nvr (K) = min \n\n> E ellipsoid, E ⊂K\n\n|K|1/n \n\n|E|1/n.\n\nClearly, vr (K) ≥ zr (K). Y. Gordon and M. Junge proved that there is a constant \n\nC0 > 0 such that every centrally symmetric convex body K in Rn has a section L\n\nof dimension [ n/ 2] such that zr (L) ≥ C0 vr (K). Since by K. Ball the local unconditional constant χu(L) is ≥ zr (L) for every convex L, and these parameters of the local theory are always hard to compute, it follows that C0 vr (K) yields an easy computational lower bound for some [ n/ 2] dimensional section L of K.Question: Suppose 0 < λ < 1 is given, what is the the greatest value f (λ) > 0such that for every n ≥ 3, every n-dimensional convex body K will have a section \n\nL of dimension [ λn ] for which zr (L) ≥ f (λ)vr (K)? \n1",
   "clean_statement": "2. (Y. Gordon) Let K be a symmetric convex body. We define the zonoid ratio as\n\nzr (K) = min\n\n> Z∈Z,Z ⊂K\n\n|K|1/n\n\n|Z|1/n.\n\nThe volume ratio is defined as\n\nvr (K) = min\n\n> E ellipsoid, E ⊂K\n\n|K|1/n\n\n|E|1/n.\n\nClearly, vr (K) ≥ zr (K). Y. Gordon and M. Junge proved that there is a constant\n\nC0 > 0 such that every centrally symmetric convex body K in Rn has a section L\n\nof dimension [ n/ 2] such that zr (L) ≥ C0 vr (K). Since by K. Ball the local unconditional constant χu(L) is ≥ zr (L) for every convex L, and these parameters of the local theory are always hard to compute, it follows that C0 vr (K) yields an easy computational lower bound for some [ n/ 2] dimensional section L of K.Question: Suppose 0 < λ < 1 is given, what is the the greatest value f (λ) > 0such that for every n ≥ 3, every n-dimensional convex body K will have a section\n\nL of dimension [ λn ] for which zr (L) ≥ f (λ)vr (K)?\n1",
   "public_statement": "2. (Y. Gordon) Let K be a symmetric convex body. We define the zonoid ratio as\n\nzr (K) = min\n\n> Z∈Z,Z ⊂K\n\n|K|1/n\n\n|Z|1/n.\n\nThe volume ratio is defined as\n\nvr (K) = min\n\n> E ellipsoid, E ⊂K\n\n|K|1/n\n\n|E|1/n.\n\nClearly, vr (K) ≥ zr (K). Y. Gordon and M. Junge proved that there is a constant\n\nC0 > 0 such that every centrally symmetric convex body K in Rn has a section L\n\nof dimension [ n/ 2] such that zr (L) ≥ C0 vr (K). Since by K. Ball the local unconditional constant χu(L) is ≥ zr (L) for every convex L, and these parameters of the local theory are always hard to compute, it follows that C0 vr (K) yields an easy computational lower bound for some [ n/ 2] dimensional section L of K.Question: Suppose 0 < λ < 1 is given, what is the the greatest value f (λ) > 0such that for every n ≥ 3, every n-dimensional convex body K will have a section\n\nL of dimension [ λn ] for which zr (L) ≥ f (λ)vr (K)?\n1",
   "evidence": "The canonical JSON labels this record as Problem 2, but the official AIM PDF places it on PDF page 4 as **Problem 12**, attributed to Y. Gordon. The printed statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0041": {
   "statement_status": "exact",
   "original_statement": "3. (S. Robins) Minkowski's Theorem in Zd. Given a finite number of vectors { ~ni} ⊂ \n\nZd and integers αi ∈ Z, is there a polytope such that { ~ni} are normal to the facets and the number of integer points on the ith-facet is αi? Do we need some extra constraints? 14. Which projection or/and section data are needed to determine non-symmetric con-vex bodies? Several known cases: A non-symmetric convex body is uniquely determined if we know: \n\n• The volume of sections passing through a given interior point and the centers of gravity of those sections. \n\n• The mean width and the Steiner points. \n\n• The brightness function and the illumination function. 5\n\nQuestions: \n\n(a) (Schneider) Do the areas of projections plus the centers of gravity determine a body? (b) What is the dual version of the third known case (brightness + illumination)? \n1",
   "clean_statement": "3. (S. Robins) Minkowski's Theorem in Zd. Given a finite number of vectors { ~ni} ⊂\n\nZd and integers αi ∈ Z, is there a polytope such that { ~ni} are normal to the facets and the number of integer points on the ith-facet is αi? Do we need some extra constraints? 14. Which projection or/and section data are needed to determine non-symmetric con-vex bodies? Several known cases: A non-symmetric convex body is uniquely determined if we know:\n\n• The volume of sections passing through a given interior point and the centers of gravity of those sections.\n\n• The mean width and the Steiner points.\n\n• The brightness function and the illumination function. 5\n\nQuestions:\n\n(a) (Schneider) Do the areas of projections plus the centers of gravity determine a body? (b) What is the dual version of the third known case (brightness + illumination)?\n1",
   "public_statement": "3. (S. Robins) Minkowski's Theorem in Zd. Given a finite number of vectors { ~ni} ⊂\n\nZd and integers αi ∈ Z, is there a polytope such that { ~ni} are normal to the facets and the number of integer points on the ith-facet is αi? Do we need some extra constraints? 14. Which projection or/and section data are needed to determine non-symmetric con-vex bodies? Several known cases: A non-symmetric convex body is uniquely determined if we know:\n\n• The volume of sections passing through a given interior point and the centers of gravity of those sections.\n\n• The mean width and the Steiner points.\n\n• The brightness function and the illumination function. 5\n\nQuestions:\n\n(a) (Schneider) Do the areas of projections plus the centers of gravity determine a body? (b) What is the dual version of the third known case (brightness + illumination)?\n1",
   "evidence": "The canonical record begins with `3. (S. Robins) Minkowski's Theorem in Zd` and then runs directly into `14. Which projection or/and section data...`. Inspection of the official AIM PDF shows that one extracted record has fused two independent consecutive problems. The official boundaries and repaired notation are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0042": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "5. (A. Zvavitch) Consider the Gaussian measure of sections of the cube Bn\n\n> ∞. It is known that if n ≥ 3, \n\nγn−1\n\n(Bn \n\n> ∞\n\n∩ θ⊥) ≤ γn−1\n\n(√ nn − 1Bn−1\n\n> ∞\n\n).\n\nIs this the best upper bound? If we introduce a dilation factor r > 0, for which \n\nθ = θ(r) is γn−1\n\n(rB n \n\n> ∞\n\n∩ θ⊥) maximal? 16. Minimum of slabs of the cube Bn \n\n> ∞\n\nGiven t ≤ 2√2 − 2, the minimal slab is in the direction of (1, 0,..., 0). Conjecture: there are numbers t1 and t2 such that for 2√2 − 2 < t < t 1, the minimum is in the direction of (1, 1, 0,..., 0); for t1 < t < t 2,the minimum is in the direccion of (1, 1, 1, 0,..., 0) and for t2 < t, the minimum is in the direction of (1, 1,..., 1). 17. Given n + k points x1,..., x n+k on Sn−1, with k ≤ n, we want to cover the sphere with caps of radius r = r(n, k ) centered at those points. What is the best possible position for the points that will minimize r? How does this minimal r behave as function of n and k? 18. Given n + 1 points x1,..., x n+1 on Sn−1, find the configuration such that the con-vex hull of x1,..., x n+1 has the largest mean width. Is it maximal for the regular simplex? 19. Let K ⊂ Rn be a non symmetric set with centroid 0. \n\nvol (K ∩ (−K)) ≥ 2−nvol (K).\n\nIs the simplex the extremal case? If not, what is it? 20. Find the smallest radius R = R(n) such that if a convex body K contains a ball of radius R, then the number of integer points in K is equivalent to the volume of K,up to a multiplicative factor (a polynomial of n). 6\n\n2",
   "clean_statement": null,
   "public_statement": "5. (A. Zvavitch) Consider the Gaussian measure of sections of the cube Bn\n\n> ∞. It is known that if n ≥ 3,\n\nγn−1\n\n(Bn\n\n> ∞\n\n∩ θ⊥) ≤ γn−1\n\n(√ nn − 1Bn−1\n\n> ∞\n\n).\n\nIs this the best upper bound? If we introduce a dilation factor r > 0, for which\n\nθ = θ(r) is γn−1\n\n(rB n\n\n> ∞\n\n∩ θ⊥) maximal? 16. Minimum of slabs of the cube Bn\n\n> ∞\n\nGiven t ≤ 2√2 − 2, the minimal slab is in the direction of (1, 0,..., 0). Conjecture: there are numbers t1 and t2 such that for 2√2 − 2 < t < t 1, the minimum is in the direction of (1, 1, 0,..., 0); for t1 < t < t 2,the minimum is in the direccion of (1, 1, 1, 0,..., 0) and for t2 < t, the minimum is in the direction of (1, 1,..., 1). 17. Given n + k points x1,..., x n+k on Sn−1, with k ≤ n, we want to cover the sphere with caps of radius r = r(n, k ) centered at those points. What is the best possible position for the points that will minimize r? How does this minimal r behave as function of n and k? 18. Given n + 1 points x1,..., x n+1 on Sn−1, find the configuration such that the con-vex hull of x1,..., x n+1 has the largest mean width. Is it maximal for the regular simplex? 19. Let K ⊂ Rn be a non symmetric set with centroid 0.\n\nvol (K ∩ (−K)) ≥ 2−nvol (K).\n\nIs the simplex the extremal case? If not, what is it? 20. Find the smallest radius R = R(n) such that if a convex body K contains a ball of radius R, then the number of integer points in K is equivalent to the volume of K,up to a multiplicative factor (a polynomial of n). 6\n\n2",
   "evidence": "The canonical record is an extraction accident: its field numbered 5 starts in the middle of official Problem 15 and then contains official Problems 16--20. The source is the AIM workshop list, Fourier analytic methods in convex geometry. Inspection of the PDF repairs the missing initial digit, display fractions, superscripts, and page debris, but does not otherwise rewrite the questions.",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0043": {
   "statement_status": "exact",
   "original_statement": "1. (W. Weil) Let Π K be the projection body of K. It is known that if K is a polytope and ΠΠ K is homothetic to K, then K is a direct sum of planar polygons (in even dimensions) or a direct sum of planar polygons and a segment (in odd dimensions). If \n\nK is an arbitrary convex body and ΠΠ K is homothetic to K, what can we conclude about K?\n2",
   "clean_statement": "1. (W. Weil) Let Π K be the projection body of K. It is known that if K is a polytope and ΠΠ K is homothetic to K, then K is a direct sum of planar polygons (in even dimensions) or a direct sum of planar polygons and a segment (in odd dimensions). If\n\nK is an arbitrary convex body and ΠΠ K is homothetic to K, what can we conclude about K?\n2",
   "public_statement": "1. (W. Weil) Let Π K be the projection body of K. It is known that if K is a polytope and ΠΠ K is homothetic to K, then K is a direct sum of planar polygons (in even dimensions) or a direct sum of planar polygons and a segment (in odd dimensions). If\n\nK is an arbitrary convex body and ΠΠ K is homothetic to K, what can we conclude about K?\n2",
   "evidence": "The JSON record has two extraction defects: its number is stored as `1` rather than `21`, and a terminal page marker `2` was appended to the problem text. The official AIM workshop PDF confirms the following statement as **Problem 21**:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0044": {
   "statement_status": "exact",
   "original_statement": "2. (A. Daurat and R. Gardner) We say that a measurable function f: R2 → R is a ridge function if it is constant in one direction, i.e., f (x) = g(〈x, v 〉) for some \n\ng: R → R, and v ∈ S1. Given n ridge functions f1,..., f n with directions v1,..., v n\n\nrespectively, the set E = {x ∈ R2: ( f1 + · · ·, f n)( x) ≥ 0} is uniquely determined among all measurable sets by the X-rays in the directions of v1,..., v n.It is known that there are sets of four directions such that every planar convex body can be determined by X-rays in those directions (for example, the directions of v1 = (1, 0), v2 = (0, 1), v3 = (2, 1) v4 = ( −1, 2) work). However, no set of three directions is enough. Is it possible to represent any convex body K ⊂ R2 in the form K = {x ∈ R2: ( f1 + f2 + f3 + f4)( x) ≥ 0} where f1,..., f 4 are ridge functions in the above four directions? 23. For t > 0, a ∈ Sn−1, consider the slab \n\nSl (a, t ) = {x ∈ Rn: ‖x‖∞ ≤ 1, |〈 x, a 〉| ≤ t}\n\nand the section \n\nS(a, t ) = {x ∈ Rn: ‖x‖∞ ≤ 1, 〈x, a 〉 = t}.\n\nDenote V (a, t ) = vol n(Sl (a, t )) and A(a, t ) = vol n−1(S(a, t )). Let fm = (1,..., 1, 0,..., 0) /√m ∈ Sn−1, where 1 ≤ m ≤ n and there are exactly \n\nm ones in fm. It is known (Hensley, Ball;Oleskiewicz-Pelcynski) that 1 ≤ A(a, 0) ≤√2, with equality in the left attained for a = f1, and on the right for a = f",
   "clean_statement": "2. (A. Daurat and R. Gardner) We say that a measurable function f: R2 → R is a ridge function if it is constant in one direction, i.e., f (x) = g(〈x, v 〉) for some\n\ng: R → R, and v ∈ S1. Given n ridge functions f1,..., f n with directions v1,..., v n\n\nrespectively, the set E = {x ∈ R2: ( f1 + · · ·, f n)( x) ≥ 0} is uniquely determined among all measurable sets by the X-rays in the directions of v1,..., v n.It is known that there are sets of four directions such that every planar convex body can be determined by X-rays in those directions (for example, the directions of v1 = (1, 0), v2 = (0, 1), v3 = (2, 1) v4 = ( −1, 2) work). However, no set of three directions is enough. Is it possible to represent any convex body K ⊂ R2 in the form K = {x ∈ R2: ( f1 + f2 + f3 + f4)( x) ≥ 0} where f1,..., f 4 are ridge functions in the above four directions? 23. For t > 0, a ∈ Sn−1, consider the slab\n\nSl (a, t ) = {x ∈ Rn: ‖x‖∞ ≤ 1, |〈 x, a 〉| ≤ t}\n\nand the section\n\nS(a, t ) = {x ∈ Rn: ‖x‖∞ ≤ 1, 〈x, a 〉 = t}.\n\nDenote V (a, t ) = vol n(Sl (a, t )) and A(a, t ) = vol n−1(S(a, t )). Let fm = (1,..., 1, 0,..., 0) /√m ∈ Sn−1, where 1 ≤ m ≤ n and there are exactly\n\nm ones in fm. It is known (Hensley, Ball;Oleskiewicz-Pelcynski) that 1 ≤ A(a, 0) ≤√2, with equality in the left attained for a = f1, and on the right for a = f",
   "public_statement": "2. (A. Daurat and R. Gardner) We say that a measurable function f: R2 → R is a ridge function if it is constant in one direction, i.e., f (x) = g(〈x, v 〉) for some\n\ng: R → R, and v ∈ S1. Given n ridge functions f1,..., f n with directions v1,..., v n\n\nrespectively, the set E = {x ∈ R2: ( f1 + · · ·, f n)( x) ≥ 0} is uniquely determined among all measurable sets by the X-rays in the directions of v1,..., v n.It is known that there are sets of four directions such that every planar convex body can be determined by X-rays in those directions (for example, the directions of v1 = (1, 0), v2 = (0, 1), v3 = (2, 1) v4 = ( −1, 2) work). However, no set of three directions is enough. Is it possible to represent any convex body K ⊂ R2 in the form K = {x ∈ R2: ( f1 + f2 + f3 + f4)( x) ≥ 0} where f1,..., f 4 are ridge functions in the above four directions? 23. For t > 0, a ∈ Sn−1, consider the slab\n\nSl (a, t ) = {x ∈ Rn: ‖x‖∞ ≤ 1, |〈 x, a 〉| ≤ t}\n\nand the section\n\nS(a, t ) = {x ∈ Rn: ‖x‖∞ ≤ 1, 〈x, a 〉 = t}.\n\nDenote V (a, t ) = vol n(Sl (a, t )) and A(a, t ) = vol n−1(S(a, t )). Let fm = (1,..., 1, 0,..., 0) /√m ∈ Sn−1, where 1 ≤ m ≤ n and there are exactly\n\nm ones in fm. It is known (Hensley, Ball;Oleskiewicz-Pelcynski) that 1 ≤ A(a, 0) ≤√2, with equality in the left attained for a = f1, and on the right for a = f",
   "evidence": "The canonical record accidentally concatenates all of Problem 22 with the beginning of Problem 23. The official AIM PDF shows that the part assigned here is Problem 22 and ends after the ridge-function question. In the PDF, a comma is also missing between the displayed vectors $v_3$ and $v_4$, and OCR has inserted a comma into the sum $f_1+\\cdots+f_n$. With those typographical repairs, the recovered problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-CONVEX_GEOMETRY-0045": {
   "statement_status": "exact",
   "original_statement": "2. (In the complex case, the result is 1 ≤ A(a, 0) ≤ 2). (a) (V. Milman) Is the minimum of V (a, t ) for fixed t > 0 and a ∈ Sn−1 (variable) always attained for a standard vector a = fi?7\n\n(b) If t ≤ 2√2 − 2, is min \n\n> a∈Sn−1\n\nV (a, t ) = V (f1, t )? This is true if t ≤ 3/4 (F. Barthe, A. Koldobsky). In the complex case, it is known to be true for t ≤ 4/5 (H. Koenig). Is it true for t ≤ t0 ≈ 0.867? Here t0\n\nis the solution of \n\nπt 2 = t(t2 − 1) √2 − t2 + (2 t2 − 1) arcsin( t√2 − t2).\n\n(c) (H. Koenig) Determine the best constants in the Khintchine-type inequality \n\nc−1 \n\n> p\n\n|a| ≤ ‖ \n\n> n\n\n∑\n\n> j=1\n\naj Xj ‖Lp\n\nwhere Xj are i. unif. dist. on Sm−1, −m + 1 < p < 0. (d) (H. Koenig) Find an integral formula for the slab-volume in Bn \n\n> 1\n\nand for the non-central sections of Bn \n\n> 1. Find the extremal directions for the slabs in Bn \n\n> 1.",
   "clean_statement": "2. (In the complex case, the result is 1 ≤ A(a, 0) ≤ 2). (a) (V. Milman) Is the minimum of V (a, t ) for fixed t > 0 and a ∈ Sn−1 (variable) always attained for a standard vector a = fi?7\n\n(b) If t ≤ 2√2 − 2, is min\n\n> a∈Sn−1\n\nV (a, t ) = V (f1, t )? This is true if t ≤ 3/4 (F. Barthe, A. Koldobsky). In the complex case, it is known to be true for t ≤ 4/5 (H. Koenig). Is it true for t ≤ t0 ≈ 0.867? Here t0\n\nis the solution of\n\nπt 2 = t(t2 − 1) √2 − t2 + (2 t2 − 1) arcsin( t√2 − t2).\n\n(c) (H. Koenig) Determine the best constants in the Khintchine-type inequality\n\nc−1\n\n> p\n\n|a| ≤ ‖\n\n> n\n\n∑\n\n> j=1\n\naj Xj ‖Lp\n\nwhere Xj are i. unif. dist. on Sm−1, −m + 1 < p < 0. (d) (H. Koenig) Find an integral formula for the slab-volume in Bn\n\n> 1\n\nand for the non-central sections of Bn\n\n> 1. Find the extremal directions for the slabs in Bn\n\n> 1.",
   "public_statement": "2. (In the complex case, the result is 1 ≤ A(a, 0) ≤ 2). (a) (V. Milman) Is the minimum of V (a, t ) for fixed t > 0 and a ∈ Sn−1 (variable) always attained for a standard vector a = fi?7\n\n(b) If t ≤ 2√2 − 2, is min\n\n> a∈Sn−1\n\nV (a, t ) = V (f1, t )? This is true if t ≤ 3/4 (F. Barthe, A. Koldobsky). In the complex case, it is known to be true for t ≤ 4/5 (H. Koenig). Is it true for t ≤ t0 ≈ 0.867? Here t0\n\nis the solution of\n\nπt 2 = t(t2 − 1) √2 − t2 + (2 t2 − 1) arcsin( t√2 − t2).\n\n(c) (H. Koenig) Determine the best constants in the Khintchine-type inequality\n\nc−1\n\n> p\n\n|a| ≤ ‖\n\n> n\n\n∑\n\n> j=1\n\naj Xj ‖Lp\n\nwhere Xj are i. unif. dist. on Sm−1, −m + 1 < p < 0. (d) (H. Koenig) Find an integral formula for the slab-volume in Bn\n\n> 1\n\nand for the non-central sections of Bn\n\n> 1. Find the extremal directions for the slabs in Bn\n\n> 1.",
   "evidence": "This record is the continuation of Problem 23 in the AIM workshop list *Fourier analytic methods in convex geometry*. The extraction starts in the middle of the preceding sentence and labels the record “2”; the official PDF shows that the number is 23 and that the isolated “7” after part (a) is a page-number artifact.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-convex-geometry-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0001": {
   "statement_status": "exact",
   "original_statement": "Parameter determination for post-quantum hash functions\n\nRamón Flores presented a Tillich-Zémor-style hash function using the Cayley graph of special linear groups in arbitrary dimension. Questions remain, however, about the optimal choice of parameters for use with this protocol.\n\nIn the design of the hash function one has control over two parameters; the dimension of the special linear group considered, and the characteristic of the underlying field. We would like to determine appropriate choices for both via two branches of cryptanalysis: first, by analysis of the multivariate system implied by long matrix products; and second by direct analysis of group-theoretic properties. We should also study the impact of the partial homomorphicity exhibited by the scheme.",
   "clean_statement": "Parameter determination for post-quantum hash functions\n\nRamón Flores presented a Tillich-Zémor-style hash function using the Cayley graph of special linear groups in arbitrary dimension. Questions remain, however, about the optimal choice of parameters for use with this protocol.\n\nIn the design of the hash function one has control over two parameters; the dimension of the special linear group considered, and the characteristic of the underlying field. We would like to determine appropriate choices for both via two branches of cryptanalysis: first, by analysis of the multivariate system implied by long matrix products; and second by direct analysis of group-theoretic properties. We should also study the impact of the partial homomorphicity exhibited by the scheme.",
   "public_statement": "Parameter determination for post-quantum hash functions\n\nRamón Flores presented a Tillich-Zémor-style hash function using the Cayley graph of special linear groups in arbitrary dimension. Questions remain, however, about the optimal choice of parameters for use with this protocol.\n\nIn the design of the hash function one has control over two parameters; the dimension of the special linear group considered, and the characteristic of the underlying field. We would like to determine appropriate choices for both via two branches of cryptanalysis: first, by analysis of the multivariate system implied by long matrix products; and second by direct analysis of group-theoretic properties. We should also study the impact of the partial homomorphicity exhibited by the scheme.",
   "evidence": "The exact canonical problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0002": {
   "statement_status": "exact",
   "original_statement": "Cryptanalysis of SDLP\n\nVarious problems relating to the cryptanalysis, both quantum and classical, of SDLP remain open.\n\nOn the quantum side the question of quantum equivalence of SDLP and the CDH-type problem underpinning semidirect product key exchange (SCDH) remains unproven, though we have strong cause to suspect their equivalence given the quantum equivalence of the heavily related vectorisation and parallelisation problems. The decomposition attack of Imran and Ivanyos, providing in some cases a reduction of SDLP to the abelian hidden subgroup problem, relies on the quantum ability to quickly compute a composition series of a solvable group. Whether similar quantum speedups in the computation series exist for non-solvable groups remains open.\n\nOn the classical side, we would like to develop a kind of Sylow theory for the projected sets implied by the semidirect product structure. It would also be useful to develop a classical method of computing the size of the acting group when one considers the semigroup variant of SDLP (currently we have only quantum methods).",
   "clean_statement": "Cryptanalysis of SDLP\n\nVarious problems relating to the cryptanalysis, both quantum and classical, of SDLP remain open.\n\nOn the quantum side the question of quantum equivalence of SDLP and the CDH-type problem underpinning semidirect product key exchange (SCDH) remains unproven, though we have strong cause to suspect their equivalence given the quantum equivalence of the heavily related vectorisation and parallelisation problems. The decomposition attack of Imran and Ivanyos, providing in some cases a reduction of SDLP to the abelian hidden subgroup problem, relies on the quantum ability to quickly compute a composition series of a solvable group. Whether similar quantum speedups in the computation series exist for non-solvable groups remains open.\n\nOn the classical side, we would like to develop a kind of Sylow theory for the projected sets implied by the semidirect product structure. It would also be useful to develop a classical method of computing the size of the acting group when one considers the semigroup variant of SDLP (currently we have only quantum methods).",
   "public_statement": "Cryptanalysis of SDLP\n\nVarious problems relating to the cryptanalysis, both quantum and classical, of SDLP remain open.\n\nOn the quantum side the question of quantum equivalence of SDLP and the CDH-type problem underpinning semidirect product key exchange (SCDH) remains unproven, though we have strong cause to suspect their equivalence given the quantum equivalence of the heavily related vectorisation and parallelisation problems. The decomposition attack of Imran and Ivanyos, providing in some cases a reduction of SDLP to the abelian hidden subgroup problem, relies on the quantum ability to quickly compute a composition series of a solvable group. Whether similar quantum speedups in the computation series exist for non-solvable groups remains open.\n\nOn the classical side, we would like to develop a kind of Sylow theory for the projected sets implied by the semidirect product structure. It would also be useful to develop a classical method of computing the size of the acting group when one considers the semigroup variant of SDLP (currently we have only quantum methods).",
   "evidence": "The canonical AIM record is problem 1.2, “Cryptanalysis of SDLP,” from the workshop *Post-quantum group-based cryptography*. It asks for progress on four related questions:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0003": {
   "statement_status": "exact",
   "original_statement": "Quantum lower bounds for group-theoretic problems in the generic group model\n\nWe would like to solidify the credentials of group-theoretic cryptography as post-quantum by developing lower bounds for the quantum complexity of problems arising in group theory. It would be especially nice to develop such a bound for the vectorisation problem, since this underpins many hardness problems in the area.",
   "clean_statement": "Quantum lower bounds for group-theoretic problems in the generic group model\n\nWe would like to solidify the credentials of group-theoretic cryptography as post-quantum by developing lower bounds for the quantum complexity of problems arising in group theory. It would be especially nice to develop such a bound for the vectorisation problem, since this underpins many hardness problems in the area.",
   "public_statement": "Quantum lower bounds for group-theoretic problems in the generic group model\n\nWe would like to solidify the credentials of group-theoretic cryptography as post-quantum by developing lower bounds for the quantum complexity of problems arising in group theory. It would be especially nice to develop such a bound for the vectorisation problem, since this underpins many hardness problems in the area.",
   "evidence": "The canonical record is AIM-CRYPTOGRAPHY-0003, item 1.3 in the Cryptanalysis section of the AIM workshop list *Post-quantum group-based cryptography*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0004": {
   "statement_status": "exact",
   "original_statement": "Machine learning-based cryptanalysis of group-theoretic protocols\n\nThere have been some results in the application of machine learning to the cryptanalysis of lattice-based schemes; whether these methods translate to the group-theoretic realm remains open.",
   "clean_statement": "Machine learning-based cryptanalysis of group-theoretic protocols\n\nThere have been some results in the application of machine learning to the cryptanalysis of lattice-based schemes; whether these methods translate to the group-theoretic realm remains open.",
   "public_statement": "Machine learning-based cryptanalysis of group-theoretic protocols\n\nThere have been some results in the application of machine learning to the cryptanalysis of lattice-based schemes; whether these methods translate to the group-theoretic realm remains open.",
   "evidence": "The exact AIM record, from the April 29--May 3, 2024 workshop *Post-quantum group-based cryptography*, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0005": {
   "statement_status": "exact",
   "original_statement": "Advanced digital signature functionality from group theory\n\nThe connection between SDLP and group actions suggests the availability of generic group-action tools to build more efficient signatures, or signatures with more specialised properties.\n\nWe would like to use generic group-action tools to build ring signatures, blind signatures, threshold signatures and more. We would also like to apply the \"twist\" technique present in similar isogeny schemes to improve the memory performance of group-based digital signatures.",
   "clean_statement": "Advanced digital signature functionality from group theory\n\nThe connection between SDLP and group actions suggests the availability of generic group-action tools to build more efficient signatures, or signatures with more specialised properties.\n\nWe would like to use generic group-action tools to build ring signatures, blind signatures, threshold signatures and more. We would also like to apply the \"twist\" technique present in similar isogeny schemes to improve the memory performance of group-based digital signatures.",
   "public_statement": "Advanced digital signature functionality from group theory\n\nThe connection between SDLP and group actions suggests the availability of generic group-action tools to build more efficient signatures, or signatures with more specialised properties.\n\nWe would like to use generic group-action tools to build ring signatures, blind signatures, threshold signatures and more. We would also like to apply the \"twist\" technique present in similar isogeny schemes to improve the memory performance of group-based digital signatures.",
   "evidence": "The exact canonical AIM record, problem 2.1 from the April--May 2024 workshop *Post-quantum group-based cryptography*, asks for generic group-action constructions of ring, blind, threshold, and other specialized signatures, and for application of a “twist” technique from isogeny schemes to improve the memory performance of group-based signatures.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0006": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Creation of challenge instances\n\nAcross the various group-based protocols there is a general lack of precision on which parameter specification. As the field matures we should make available challenge instances of each protocol.",
   "clean_statement": "Across the various group-based protocols there is a general lack of precision in parameter specification (or in which parameter specification to use). As the field matures, challenge instances should be made available for each protocol.",
   "public_statement": "Creation of challenge instances\n\nAcross the various group-based protocols there is a general lack of precision on which parameter specification. As the field matures we should make available challenge instances of each protocol.",
   "evidence": "The first sentence is grammatically incomplete. The official `source_url` returned an HTTP 502 error during this run, and exact-phrase searches did not locate an independently rendered copy. Thus the following is a **reconstruction, not verified source text**: Nearby canonical records concern advanced signatures, implementations, and key establishment but do not repair the sentence. The developed contribution below applies to the plausible reconstruction while retaining the original wording above.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0007": {
   "statement_status": "exact",
   "original_statement": "Implementation of group-based cryptography\n\nIn general very little work exists on efficient implementation of group-based protocols; over time we would like to build up a software library of such implementations.",
   "clean_statement": "Implementation of group-based cryptography\n\nIn general very little work exists on efficient implementation of group-based protocols; over time we would like to build up a software library of such implementations.",
   "public_statement": "Implementation of group-based cryptography\n\nIn general very little work exists on efficient implementation of group-based protocols; over time we would like to build up a software library of such implementations.",
   "evidence": "The canonical record is problem 2.3, “Implementation of group-based cryptography,” in the Design section of the AIM workshop *Post-quantum group-based cryptography*. Its exact mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0008": {
   "statement_status": "exact",
   "original_statement": "Methods of key establishment from group theory\n\nGenerally speaking the field does not seem to offer methods of key establishment. It remains open to construct standard key exchange protocols as well as more complicated multi-party variants.",
   "clean_statement": "Methods of key establishment from group theory\n\nGenerally speaking the field does not seem to offer methods of key establishment. It remains open to construct standard key exchange protocols as well as more complicated multi-party variants.",
   "public_statement": "Methods of key establishment from group theory\n\nGenerally speaking the field does not seem to offer methods of key establishment. It remains open to construct standard key exchange protocols as well as more complicated multi-party variants.",
   "evidence": "The exact AIM problem 2.4, recorded at the April--May 2024 workshop *Post-quantum group-based cryptography*, states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0009": {
   "statement_status": "exact",
   "original_statement": "Applications of infinite non-abelian groups to cryptography\n\nSeveral cryptanalytic methods against group-based cryptography suggest possible use cases for infinite non-abelian groups; for example, non-Hopfian groups. We would like to develop tools to understand how these infinite objects can fit into a cryptographic context, and whether their use can be used to address some of the cryptanalytic methods deployed against group-based cryptography.",
   "clean_statement": "Applications of infinite non-abelian groups to cryptography\n\nSeveral cryptanalytic methods against group-based cryptography suggest possible use cases for infinite non-abelian groups; for example, non-Hopfian groups. We would like to develop tools to understand how these infinite objects can fit into a cryptographic context, and whether their use can be used to address some of the cryptanalytic methods deployed against group-based cryptography.",
   "public_statement": "Applications of infinite non-abelian groups to cryptography\n\nSeveral cryptanalytic methods against group-based cryptography suggest possible use cases for infinite non-abelian groups; for example, non-Hopfian groups. We would like to develop tools to understand how these infinite objects can fit into a cryptographic context, and whether their use can be used to address some of the cryptanalytic methods deployed against group-based cryptography.",
   "evidence": "The canonical record is AIM Problem Lists, workshop “Post-quantum group-based cryptography,” section “Foundations,” Problem 3.1. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0010": {
   "statement_status": "exact",
   "original_statement": "Cryptographic building blocks from problems in group theory\n\nWe would like to use computational problems in group theory to derive fundamental cryptographic objects including one-way functions and multilinear maps.",
   "clean_statement": "Cryptographic building blocks from problems in group theory\n\nWe would like to use computational problems in group theory to derive fundamental cryptographic objects including one-way functions and multilinear maps.",
   "public_statement": "Cryptographic building blocks from problems in group theory\n\nWe would like to use computational problems in group theory to derive fundamental cryptographic objects including one-way functions and multilinear maps.",
   "evidence": "The canonical AIM record is problem 3.2 in the “Foundations” section of the workshop *Post-quantum group-based cryptography*. Its complete problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0011": {
   "statement_status": "exact",
   "original_statement": "Computational problems from graph groups\n\nIt is known that it is possible to establish correspondence between certain NP-hard graph problems and problems in group theory; we would like to expand our portfolio of such problems.",
   "clean_statement": "Computational problems from graph groups\n\nIt is known that it is possible to establish correspondence between certain NP-hard graph problems and problems in group theory; we would like to expand our portfolio of such problems.",
   "public_statement": "Computational problems from graph groups\n\nIt is known that it is possible to establish correspondence between certain NP-hard graph problems and problems in group theory; we would like to expand our portfolio of such problems.",
   "evidence": "The canonical source record is problem 3.3 in the Foundations section of the AIM workshop *Post-quantum group-based cryptography*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0012": {
   "statement_status": "exact",
   "original_statement": "Parameters: $n \\geq 1$, $r\\geq 1$, $\\mathbb F_{q^m}$, $t \\geq 1$. How quickly can we find $v\\in \\mathbb F_{q^m}^n$ with $Hv = s$ and $\\text{wt}(v) = t$, given $H\\in \\mathbb F_{q^m}^{r\\times n}$ and $s\\in \\mathbb F_{q^m}^r$?\n\nHere, $\\text{wt}(v)$ is defined to be the rank of the $v$, viewed as a matrix with $n$ columns.",
   "clean_statement": "Parameters: $n \\geq 1$, $r\\geq 1$, $\\mathbb F_{q^m}$, $t \\geq 1$. How quickly can we find $v\\in \\mathbb F_{q^m}^n$ with $Hv = s$ and $\\text{wt}(v) = t$, given $H\\in \\mathbb F_{q^m}^{r\\times n}$ and $s\\in \\mathbb F_{q^m}^r$?\n\nHere, $\\text{wt}(v)$ is defined to be the rank of the $v$, viewed as a matrix with $n$ columns.",
   "public_statement": "Parameters: $n \\geq 1$, $r\\geq 1$, $\\mathbb F_{q^m}$, $t \\geq 1$. How quickly can we find $v\\in \\mathbb F_{q^m}^n$ with $Hv = s$ and $\\text{wt}(v) = t$, given $H\\in \\mathbb F_{q^m}^{r\\times n}$ and $s\\in \\mathbb F_{q^m}^r$?\n\nHere, $\\text{wt}(v)$ is defined to be the rank of the $v$, viewed as a matrix with $n$ columns.",
   "evidence": "The canonical AIM record, from the workshop “Quantum algorithms for analysis of public-key crypto,” section “Codes,” Problem 1.1, says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0013": {
   "statement_status": "exact",
   "original_statement": "Given a parity-check matrix $H$, find the hidden Goppa code in $H$. How quickly can we decode $H$, assuming Goppa decoder for $H$ exists?",
   "clean_statement": "Given a parity-check matrix $H$, find the hidden Goppa code in $H$. How quickly can we decode $H$, assuming Goppa decoder for $H$ exists?",
   "public_statement": "Given a parity-check matrix $H$, find the hidden Goppa code in $H$. How quickly can we decode $H$, assuming Goppa decoder for $H$ exists?",
   "evidence": "The canonical record is problem 1.2 in the “Codes” section of the 2019 AIM workshop *Quantum algorithms for analysis of public-key crypto*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 12,
   "attempt": 2
  },
  "AIM-CRYPTOGRAPHY-0014": {
   "statement_status": "exact",
   "original_statement": "What witnesses are there of Goppa decodability or non-decodability?",
   "clean_statement": "What witnesses are there of Goppa decodability or non-decodability?",
   "public_statement": "What witnesses are there of Goppa decodability or non-decodability?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0015": {
   "statement_status": "exact",
   "original_statement": "Given a function $f:\\{0,1\\}^n \\to \\{0,1,2\\}$, find $x\\in \\{0,1\\}^n$ to maximize $f(x)$. Measure average resulting $f(x)$ if, e.g., $f$ has one 1, one 2, and all other values 0. Or consider the function $f:\\{0,1\\}^n \\to \\{0,1,2, \\dots, 1000\\}$. Or consider $f$ which is i.i.d.\n\nIs it possible to do better than Grover search for $x$ such that $f(x) \\geq T$ for threshold $T$?",
   "clean_statement": "Given a function $f:\\{0,1\\}^n \\to \\{0,1,2\\}$, find $x\\in \\{0,1\\}^n$ to maximize $f(x)$. Measure average resulting $f(x)$ if, e.g., $f$ has one 1, one 2, and all other values 0. Or consider the function $f:\\{0,1\\}^n \\to \\{0,1,2, \\dots, 1000\\}$. Or consider $f$ which is i.i.d.\n\nIs it possible to do better than Grover search for $x$ such that $f(x) \\geq T$ for threshold $T$?",
   "public_statement": "Given a function $f:\\{0,1\\}^n \\to \\{0,1,2\\}$, find $x\\in \\{0,1\\}^n$ to maximize $f(x)$. Measure average resulting $f(x)$ if, e.g., $f$ has one 1, one 2, and all other values 0. Or consider the function $f:\\{0,1\\}^n \\to \\{0,1,2, \\dots, 1000\\}$. Or consider $f$ which is i.i.d.\n\nIs it possible to do better than Grover search for $x$ such that $f(x) \\geq T$ for threshold $T$?",
   "evidence": "The exact canonical AIM text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0016": {
   "statement_status": "exact",
   "original_statement": "Find $x,y$ distinct with $f(x) = f(y) = 1$. Consider the case where there are many solutions at low density. How quickly can this be done? Faster than Grover?\n\nMore generally, find distinct $x_1,\\dots, x_m$ such that $f(x_1) = \\dots = f(x_m) = 1$. Can you do better than $m$ preimage searches?",
   "clean_statement": "Find $x,y$ distinct with $f(x) = f(y) = 1$. Consider the case where there are many solutions at low density. How quickly can this be done? Faster than Grover?\n\nMore generally, find distinct $x_1,\\dots, x_m$ such that $f(x_1) = \\dots = f(x_m) = 1$. Can you do better than $m$ preimage searches?",
   "public_statement": "Find $x,y$ distinct with $f(x) = f(y) = 1$. Consider the case where there are many solutions at low density. How quickly can this be done? Faster than Grover?\n\nMore generally, find distinct $x_1,\\dots, x_m$ such that $f(x_1) = \\dots = f(x_m) = 1$. Can you do better than $m$ preimage searches?",
   "evidence": "The source is the 2019 AIM workshop *Quantum algorithms for analysis of public-key crypto*, section “Competing with Grover's algorithm,” problem 2.3, attributed in the workshop report to Mike Hamburg. The extracted statement agrees with the report:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0017": {
   "statement_status": "exact",
   "original_statement": "What is the cost of hidden shift on $\\mathbb Z$ where the shift $s\\in [a,b]$, under binary cost of oracle?",
   "clean_statement": "What is the cost of hidden shift on $\\mathbb Z$ where the shift $s\\in [a,b]$, under binary cost of oracle?",
   "public_statement": "What is the cost of hidden shift on $\\mathbb Z$ where the shift $s\\in [a,b]$, under binary cost of oracle?",
   "evidence": "The exact canonical statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0018": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "How fast are approximate SVP attacks via hidden shift algorithms?",
   "clean_statement": null,
   "public_statement": "How fast are approximate SVP attacks via hidden shift algorithms?",
   "evidence": "The most conservative reconstruction is therefore:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0019": {
   "statement_status": "exact",
   "original_statement": "Hidden shift on $\\mathbb Z^d$ where $f_0$ and $f_1$ are periodic under unary oracle cost.",
   "clean_statement": "Hidden shift on $\\mathbb Z^d$ where $f_0$ and $f_1$ are periodic under unary oracle cost.",
   "public_statement": "Hidden shift on $\\mathbb Z^d$ where $f_0$ and $f_1$ are periodic under unary oracle cost.",
   "evidence": "The exact canonical statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0020": {
   "statement_status": "exact",
   "original_statement": "How much noise is tolerated in hidden shift algorithms? Consider too many solutions vs. too few.",
   "clean_statement": "How much noise is tolerated in hidden shift algorithms? Consider too many solutions vs. too few.",
   "public_statement": "How much noise is tolerated in hidden shift algorithms? Consider too many solutions vs. too few.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0021": {
   "statement_status": "exact",
   "original_statement": "How does Kuperberg's algorithm behave under multiple hidden shifts (with known relations between shifts, e.g. arithmetic progression) with Gaussians, using the work of Ivanyos, Prakash, and Santha?,",
   "clean_statement": "How does Kuperberg's algorithm behave under multiple hidden shifts (with known relations between shifts, e.g. arithmetic progression) with Gaussians, using the work of Ivanyos, Prakash, and Santha?,",
   "public_statement": "How does Kuperberg's algorithm behave under multiple hidden shifts (with known relations between shifts, e.g. arithmetic progression) with Gaussians, using the work of Ivanyos, Prakash, and Santha?,",
   "evidence": "The canonical JSON record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0022": {
   "statement_status": "exact",
   "original_statement": "Is there a fast hidden shift algorithm for Heisenberg group over $\\mathbb F_p$?",
   "clean_statement": "Is there a fast hidden shift algorithm for Heisenberg group over $\\mathbb F_p$?",
   "public_statement": "Is there a fast hidden shift algorithm for Heisenberg group over $\\mathbb F_p$?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0023": {
   "statement_status": "exact",
   "original_statement": "Can quantum algorithms be used to attack SIDH by using the extra points? Are there SIDH attacks better than claw-finding?",
   "clean_statement": "Can quantum algorithms be used to attack SIDH by using the extra points? Are there SIDH attacks better than claw-finding?",
   "public_statement": "Can quantum algorithms be used to attack SIDH by using the extra points? Are there SIDH attacks better than claw-finding?",
   "evidence": "The canonical record is problem 5.1 in the section “Isogeny-based cryptosystems” of the AIM workshop *Quantum algorithms for analysis of public-key crypto* (4--8 February 2019). The exact question, attributed there to Kirsten Eisentraeger, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0024": {
   "statement_status": "exact",
   "original_statement": "Are there CSIDH attacks better than hidden shift? What is the cost of the hidden shift attack?",
   "clean_statement": "Are there CSIDH attacks better than hidden shift? What is the cost of the hidden shift attack?",
   "public_statement": "Are there CSIDH attacks better than hidden shift? What is the cost of the hidden shift attack?",
   "evidence": "The canonical record is problem 5.2 in the AIM workshop list *Quantum algorithms for analysis of public-key crypto*. It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0025": {
   "statement_status": "exact",
   "original_statement": "Solve Decisional Diffie-Hellman (DDH) in the context of CSIDH: Let $E_0$ be a supersingular elliptic curve defined over $\\mathbb F_p$ with endomorphism ring $\\mathbb Z[\\pi]$. Distinguish between triples $(\\mathfrak a * E_0, \\mathfrak b* E_0,\\mathfrak c* E_0)$ and triples $(\\mathfrak a * E_0, \\mathfrak b* E_0,\\mathfrak a\\mathfrak b * E_0)$ where $\\mathfrak a, \\mathfrak b, \\mathfrak c$ are ideals in $\\mathbb Z[\\pi]$ of odd norm.",
   "clean_statement": "Solve Decisional Diffie-Hellman (DDH) in the context of CSIDH: Let $E_0$ be a supersingular elliptic curve defined over $\\mathbb F_p$ with endomorphism ring $\\mathbb Z[\\pi]$. Distinguish between triples $(\\mathfrak a * E_0, \\mathfrak b* E_0,\\mathfrak c* E_0)$ and triples $(\\mathfrak a * E_0, \\mathfrak b* E_0,\\mathfrak a\\mathfrak b * E_0)$ where $\\mathfrak a, \\mathfrak b, \\mathfrak c$ are ideals in $\\mathbb Z[\\pi]$ of odd norm.",
   "public_statement": "Solve Decisional Diffie-Hellman (DDH) in the context of CSIDH: Let $E_0$ be a supersingular elliptic curve defined over $\\mathbb F_p$ with endomorphism ring $\\mathbb Z[\\pi]$. Distinguish between triples $(\\mathfrak a * E_0, \\mathfrak b* E_0,\\mathfrak c* E_0)$ and triples $(\\mathfrak a * E_0, \\mathfrak b* E_0,\\mathfrak a\\mathfrak b * E_0)$ where $\\mathfrak a, \\mathfrak b, \\mathfrak c$ are ideals in $\\mathbb Z[\\pi]$ of odd norm.",
   "evidence": "The AIM record (Quantum algorithms for analysis of public-key crypto, Problem 5.3) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0026": {
   "statement_status": "exact",
   "original_statement": "Parameters: $n \\geq 2$ and $\\alpha > 0$. Given independent uniformly random $x_1, x_2, \\dots\\in S^{n-1} = \\{v \\in \\mathbb R^n : \\|v\\|_2 = 1\\}$, how quickly can we find a subsequence that covers $S^{n-1}$? (Here, ``covers\" means that every point of $S^{n-1}$ is within angle $\\alpha$ of a point in the subsequence.) For example, consider the case when $n = 1000$ and $\\alpha = 75^\\circ$.",
   "clean_statement": "Parameters: $n \\geq 2$ and $\\alpha > 0$. Given independent uniformly random $x_1, x_2, \\dots\\in S^{n-1} = \\{v \\in \\mathbb R^n : \\|v\\|_2 = 1\\}$, how quickly can we find a subsequence that covers $S^{n-1}$? (Here, ``covers\" means that every point of $S^{n-1}$ is within angle $\\alpha$ of a point in the subsequence.) For example, consider the case when $n = 1000$ and $\\alpha = 75^\\circ$.",
   "public_statement": "Parameters: $n \\geq 2$ and $\\alpha > 0$. Given independent uniformly random $x_1, x_2, \\dots\\in S^{n-1} = \\{v \\in \\mathbb R^n : \\|v\\|_2 = 1\\}$, how quickly can we find a subsequence that covers $S^{n-1}$? (Here, ``covers\" means that every point of $S^{n-1}$ is within angle $\\alpha$ of a point in the subsequence.) For example, consider the case when $n = 1000$ and $\\alpha = 75^\\circ$.",
   "evidence": "The canonical record is problem 6.1 in the “Miscellaneous” section of the AIM workshop *Quantum algorithms for analysis of public-key crypto* (4--8 February 2019). The official workshop summary attributes it to John Schanck and gives the same formulation [AIM19, p. 3]. The exact canonical wording is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0027": {
   "statement_status": "exact",
   "original_statement": "Does HHL break crypto? (See recent paper.)\n\nUnderstand condition number over $\\mathbb C$ of matrix of coefficients of $f_1, f_2, \\dots$ (original equations), $xf_1, xyf_2, \\dots$ (only monomial terms).",
   "clean_statement": "Does HHL break crypto? (See recent paper.)\n\nUnderstand condition number over $\\mathbb C$ of matrix of coefficients of $f_1, f_2, \\dots$ (original equations), $xf_1, xyf_2, \\dots$ (only monomial terms).",
   "public_statement": "Does HHL break crypto? (See recent paper.)\n\nUnderstand condition number over $\\mathbb C$ of matrix of coefficients of $f_1, f_2, \\dots$ (original equations), $xf_1, xyf_2, \\dots$ (only monomial terms).",
   "evidence": "The canonical repository record says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0028": {
   "statement_status": "exact",
   "original_statement": "Compute short units in the ring of integers of $\\mathbb Q[x]/(x^{312} - x^{156} -1)$. How short can they be, and how fast can you find them?",
   "clean_statement": "Compute short units in the ring of integers of $\\mathbb Q[x]/(x^{312} - x^{156} -1)$. How short can they be, and how fast can you find them?",
   "public_statement": "Compute short units in the ring of integers of $\\mathbb Q[x]/(x^{312} - x^{156} -1)$. How short can they be, and how fast can you find them?",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-CRYPTOGRAPHY-0029": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Is there a crypto problem that is solved by Simon's algorithm without superposition attackers? Which hidden subgroup problems have crypto applications? Or non-crypto instances?",
   "clean_statement": "1. **Simon/Q1 question.** Is there a cryptanalytic problem for which Simon's algorithm is useful when the attacker may run a quantum computer but may make only classical queries to the secret primitive, rather than quantum superposition queries?\n2. **Cryptographic-HSP question.** Which hidden-subgroup or closely related hidden-shift problems yield concrete cryptanalytic algorithms?\n3. **Explicit-instance question.** Which HSP algorithms have explicit, noncryptographic input functions, in the sense that Shor's period function \\(x\\mapsto a^x\\bmod N\\) is an efficiently implementable circuit rather than a formal black-box oracle?",
   "public_statement": "Is there a crypto problem that is solved by Simon's algorithm without superposition attackers? Which hidden subgroup problems have crypto applications? Or non-crypto instances?",
   "evidence": "The phrase is not standard English terminology. The most conservative reconstruction is:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-cryptography-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0001": {
   "statement_status": "exact",
   "original_statement": "Fixed point proportion of dynamical exceptional polynomials\n\nLet $G$ be a group acting on a infinite rooted tree, and denote $G_n$ the action of $G$ on the first $n$ levels of the tree. We define the \\textbf{fixed-point proportion} of $G$ as $$FPP(G) = \\lim_{n \\rightarrow +\\infty} \\frac{\\# \\left\\{g \\in G_n: \\text{ $g$ fixes at least one element on level $n$}\\right\\}}{\\# G_n}.$$\n\nLet $k$ be a field, $f \\in k(z)$ of degree at least $2$, and $t \\in k$. Define $k_n = k(f^{-n}(t))$, $G_n = Gal(k_n/k(t))$ and $G_\\infty = \\varprojlim G_n$. The group $G_n$ acts on $f^{-n}(t)$ via the natural action of the Galois group, so $G_\\infty$ acts on the infinite rooted tree of the $n$-th preimages of $t$ via $f$. \\\\\n\nIn particular, if $k = \\mathbb{C}(t)$ with $t$ transcendental over $\\mathbb{C}$, $G_\\infty$ is isomorphic to the closure of the iterated monodromy group of $f$, denoted $IMG(f)$ (see Proposition 6.4.2 in \\cite{MR2162164}). Results in this direction can be found in \\cite{arXiv:1204.2843}, where it is proved that for non dynamically exceptional rational functions, $FPP(IMG(f)) = 0$.\n\nLet $k$ be a field, $f \\in k(z)$ of degree at least $2$, and $t \\in k$. Define $k_n = k(f^{-n}(t))$, $G_n = Gal(k_n/k(t))$ and $G_\\infty = \\varprojlim G_n$. The group $G_n$ acts on $f^{-n}(t)$ via the natural action of the Galois group, so $G_\\infty$ acts on the infinite rooted tree of the $n$-th preimages of $t$ via $f$.\n\n1. What can we say about the fixed-point proportion of groups acting on rooted trees?\n\n2. Fixing the tree $T$, can we find a family of groups such that its fixed-point proportion is nonzero and its Hausdorff dimension converges to $1$?\n\n3. What can we say about the fixed-point proportion of Galois groups?\n\n4. What can we say about the fixed-point proportion of $IMG(f)$ of complex rational functions?\n\n5. What can we say about the fixed-point proportion of $IMG(f)$ of complex polynomials that are not dynamically exceptional?",
   "clean_statement": "Fixed point proportion of dynamical exceptional polynomials\n\nLet $G$ be a group acting on a infinite rooted tree, and denote $G_n$ the action of $G$ on the first $n$ levels of the tree. We define the \\textbf{fixed-point proportion} of $G$ as $$FPP(G) = \\lim_{n \\rightarrow +\\infty} \\frac{\\# \\left\\{g \\in G_n: \\text{ $g$ fixes at least one element on level $n$}\\right\\}}{\\# G_n}.$$\n\nLet $k$ be a field, $f \\in k(z)$ of degree at least $2$, and $t \\in k$. Define $k_n = k(f^{-n}(t))$, $G_n = Gal(k_n/k(t))$ and $G_\\infty = \\varprojlim G_n$. The group $G_n$ acts on $f^{-n}(t)$ via the natural action of the Galois group, so $G_\\infty$ acts on the infinite rooted tree of the $n$-th preimages of $t$ via $f$. \\\\\n\nIn particular, if $k = \\mathbb{C}(t)$ with $t$ transcendental over $\\mathbb{C}$, $G_\\infty$ is isomorphic to the closure of the iterated monodromy group of $f$, denoted $IMG(f)$ (see Proposition 6.4.2 in \\cite{MR2162164}). Results in this direction can be found in \\cite{arXiv:1204.2843}, where it is proved that for non dynamically exceptional rational functions, $FPP(IMG(f)) = 0$.\n\nLet $k$ be a field, $f \\in k(z)$ of degree at least $2$, and $t \\in k$. Define $k_n = k(f^{-n}(t))$, $G_n = Gal(k_n/k(t))$ and $G_\\infty = \\varprojlim G_n$. The group $G_n$ acts on $f^{-n}(t)$ via the natural action of the Galois group, so $G_\\infty$ acts on the infinite rooted tree of the $n$-th preimages of $t$ via $f$.\n\n1. What can we say about the fixed-point proportion of groups acting on rooted trees?\n\n2. Fixing the tree $T$, can we find a family of groups such that its fixed-point proportion is nonzero and its Hausdorff dimension converges to $1$?\n\n3. What can we say about the fixed-point proportion of Galois groups?\n\n4. What can we say about the fixed-point proportion of $IMG(f)$ of complex rational functions?\n\n5. What can we say about the fixed-point proportion of $IMG(f)$ of complex polynomials that are not dynamically exceptional?",
   "public_statement": "Fixed point proportion of dynamical exceptional polynomials\n\nLet $G$ be a group acting on a infinite rooted tree, and denote $G_n$ the action of $G$ on the first $n$ levels of the tree. We define the \\textbf{fixed-point proportion} of $G$ as $$FPP(G) = \\lim_{n \\rightarrow +\\infty} \\frac{\\# \\left\\{g \\in G_n: \\text{ $g$ fixes at least one element on level $n$}\\right\\}}{\\# G_n}.$$\n\nLet $k$ be a field, $f \\in k(z)$ of degree at least $2$, and $t \\in k$. Define $k_n = k(f^{-n}(t))$, $G_n = Gal(k_n/k(t))$ and $G_\\infty = \\varprojlim G_n$. The group $G_n$ acts on $f^{-n}(t)$ via the natural action of the Galois group, so $G_\\infty$ acts on the infinite rooted tree of the $n$-th preimages of $t$ via $f$. \\\\\n\nIn particular, if $k = \\mathbb{C}(t)$ with $t$ transcendental over $\\mathbb{C}$, $G_\\infty$ is isomorphic to the closure of the iterated monodromy group of $f$, denoted $IMG(f)$ (see Proposition 6.4.2 in \\cite{MR2162164}). Results in this direction can be found in \\cite{arXiv:1204.2843}, where it is proved that for non dynamically exceptional rational functions, $FPP(IMG(f)) = 0$.\n\nLet $k$ be a field, $f \\in k(z)$ of degree at least $2$, and $t \\in k$. Define $k_n = k(f^{-n}(t))$, $G_n = Gal(k_n/k(t))$ and $G_\\infty = \\varprojlim G_n$. The group $G_n$ acts on $f^{-n}(t)$ via the natural action of the Galois group, so $G_\\infty$ acts on the infinite rooted tree of the $n$-th preimages of $t$ via $f$.\n\n1. What can we say about the fixed-point proportion of groups acting on rooted trees?\n\n2. Fixing the tree $T$, can we find a family of groups such that its fixed-point proportion is nonzero and its Hausdorff dimension converges to $1$?\n\n3. What can we say about the fixed-point proportion of Galois groups?\n\n4. What can we say about the fixed-point proportion of $IMG(f)$ of complex rational functions?\n\n5. What can we say about the fixed-point proportion of $IMG(f)$ of complex polynomials that are not dynamically exceptional?",
   "evidence": "The canonical record comes from the AIM workshop report *Groups of dynamical origin*, section “The Fixed Point Proportion of Dynamically Exceptional Polynomials” (moderator Santiago Radi). The exact canonical JSON is preserved in `input.json`. It asks the following five broad questions after defining",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0002": {
   "statement_status": "exact",
   "original_statement": "Analogous of paths on iterated monodromy groups for iterated Galois groups of dynamical maps\n\nLet $k$ be a field, $f \\in k(z)$ of degree at least $2$, and $t$ be a transcendental number. Define $k_\\infty = k \\left( \\cup_{n \\geq 0} f^{-n}(t) \\right)$, $L = k_\\infty \\cap \\overline{k}$, $G_\\infty = Gal(k_\\infty/k(t))$ and $G_\\infty^{geom} = Gal(k_\\infty/L(t))$. \\\\\n\nIt was proved by Richard Pink (see \\cite[Proposition 6.4.2]{Self_similar_groups}) that when $k = \\mathbb{C}$, then $G_\\infty^{geom} = \\overline{IMG(f)}$ and for the iterated monodromy group of a continuous covering map, there is an explicit way to construct it by choosing paths between the chosen base point $z_0$ and $f^{-1}(z_0)$. \\\\\n\nFollowing the analogous of $IMG(f)$, we know the iterated monodromy group of $f$ can reconstruct the Julia set of $f$.\n\n1. Can we algebraically construct \"nice\" paths to describe $G_\\infty^{geom}$ by more explicit wreath recursions? More precisely, given $z\\in f^{-1}(t)$, can we find explicit endomorphisms of $K_\\infty$, perhaps recursively described, sending $t$ to $z$ with nice properties?",
   "clean_statement": "Analogous of paths on iterated monodromy groups for iterated Galois groups of dynamical maps\n\nLet $k$ be a field, $f \\in k(z)$ of degree at least $2$, and $t$ be a transcendental number. Define $k_\\infty = k \\left( \\cup_{n \\geq 0} f^{-n}(t) \\right)$, $L = k_\\infty \\cap \\overline{k}$, $G_\\infty = Gal(k_\\infty/k(t))$ and $G_\\infty^{geom} = Gal(k_\\infty/L(t))$. \\\\\n\nIt was proved by Richard Pink (see \\cite[Proposition 6.4.2]{Self_similar_groups}) that when $k = \\mathbb{C}$, then $G_\\infty^{geom} = \\overline{IMG(f)}$ and for the iterated monodromy group of a continuous covering map, there is an explicit way to construct it by choosing paths between the chosen base point $z_0$ and $f^{-1}(z_0)$. \\\\\n\nFollowing the analogous of $IMG(f)$, we know the iterated monodromy group of $f$ can reconstruct the Julia set of $f$.\n\n1. Can we algebraically construct \"nice\" paths to describe $G_\\infty^{geom}$ by more explicit wreath recursions? More precisely, given $z\\in f^{-1}(t)$, can we find explicit endomorphisms of $K_\\infty$, perhaps recursively described, sending $t$ to $z$ with nice properties?",
   "public_statement": "Analogous of paths on iterated monodromy groups for iterated Galois groups of dynamical maps\n\nLet $k$ be a field, $f \\in k(z)$ of degree at least $2$, and $t$ be a transcendental number. Define $k_\\infty = k \\left( \\cup_{n \\geq 0} f^{-n}(t) \\right)$, $L = k_\\infty \\cap \\overline{k}$, $G_\\infty = Gal(k_\\infty/k(t))$ and $G_\\infty^{geom} = Gal(k_\\infty/L(t))$. \\\\\n\nIt was proved by Richard Pink (see \\cite[Proposition 6.4.2]{Self_similar_groups}) that when $k = \\mathbb{C}$, then $G_\\infty^{geom} = \\overline{IMG(f)}$ and for the iterated monodromy group of a continuous covering map, there is an explicit way to construct it by choosing paths between the chosen base point $z_0$ and $f^{-1}(z_0)$. \\\\\n\nFollowing the analogous of $IMG(f)$, we know the iterated monodromy group of $f$ can reconstruct the Julia set of $f$.\n\n1. Can we algebraically construct \"nice\" paths to describe $G_\\infty^{geom}$ by more explicit wreath recursions? More precisely, given $z\\in f^{-1}(t)$, can we find explicit endomorphisms of $K_\\infty$, perhaps recursively described, sending $t$ to $z$ with nice properties?",
   "evidence": "The canonical record is Problem 1.2, “Analogous of paths on iterated monodromy groups for iterated Galois groups of dynamical maps,” from the AIM workshop *Groups of dynamical origin*. Its question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0003": {
   "statement_status": "exact",
   "original_statement": "Orbit equivalence for IMG\n\nLet $G,H$ two groups acting on the same set $X$. We say that $G$ and $H$ are \\textbf{orbit equivalent} if they have the same orbits. \\\\\n\nLet $f$ be a continuous map and $IMG(f)$ its iterated monodromy group. Assume $IMG(f)$ is countable and consider its action on $\\partial T$, the boundary of the tree. \\\\\n\n1. Can we find $\\varphi \\in Homeo(\\partial T)$ such that the orbit of the group generated by $\\varphi$ and $IMG(f)$ are orbit equivalent?\n\n2. Can we make $\\varphi$ explicit?",
   "clean_statement": "Orbit equivalence for IMG\n\nLet $G,H$ two groups acting on the same set $X$. We say that $G$ and $H$ are \\textbf{orbit equivalent} if they have the same orbits. \\\\\n\nLet $f$ be a continuous map and $IMG(f)$ its iterated monodromy group. Assume $IMG(f)$ is countable and consider its action on $\\partial T$, the boundary of the tree. \\\\\n\n1. Can we find $\\varphi \\in Homeo(\\partial T)$ such that the orbit of the group generated by $\\varphi$ and $IMG(f)$ are orbit equivalent?\n\n2. Can we make $\\varphi$ explicit?",
   "public_statement": "Orbit equivalence for IMG\n\nLet $G,H$ two groups acting on the same set $X$. We say that $G$ and $H$ are \\textbf{orbit equivalent} if they have the same orbits. \\\\\n\nLet $f$ be a continuous map and $IMG(f)$ its iterated monodromy group. Assume $IMG(f)$ is countable and consider its action on $\\partial T$, the boundary of the tree. \\\\\n\n1. Can we find $\\varphi \\in Homeo(\\partial T)$ such that the orbit of the group generated by $\\varphi$ and $IMG(f)$ are orbit equivalent?\n\n2. Can we make $\\varphi$ explicit?",
   "evidence": "### Canonical record, preserved verbatim",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0004": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Wreath recursion for Galois groups\n\n1. Contruct explicit wreath recursions for Galois groups\n\n2. Under what conditions is a group coming from a Galois arboreal representation a branch group?",
   "clean_statement": "Wreath recursion for Galois groups\n\n1. Construct. explicit wreath recursions for Galois groups\n\n2. Under what conditions is a group coming from a Galois arboreal representation a branch group?",
   "public_statement": "Wreath recursion for Galois groups\n\n1. Contruct explicit wreath recursions for Galois groups\n\n2. Under what conditions is a group coming from a Galois arboreal representation a branch group?",
   "evidence": "The exact canonical record is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 3,
   "attempt": 2
  },
  "AIM-DYNAMICAL_SYSTEMS-0005": {
   "statement_status": "exact",
   "original_statement": "Reconstructing the Julia set of a function from the iterated Galois group\n\nLet $k$ be a finite field extension of $\\mathbb{Q}_p$, $f \\in k(z)$ of degree at least $2$, and $t$ be a transcendental number. Define $k_\\infty = k \\left( \\cup_{n \\geq 0} f^{-n}(t) \\right)$, $L = k_\\infty \\cap \\overline{k}$, $G_\\infty = Gal(k_\\infty/k(t))$ and $G_\\infty^{geom} = Gal(k_\\infty/L(t))$. \\\\\n\nCan we reconstruct the $p$-adic Julia set of $f$ from $G_\\infty$ or $G_\\infty^{geom}$",
   "clean_statement": "Reconstructing the Julia set of a function from the iterated Galois group\n\nLet $k$ be a finite field extension of $\\mathbb{Q}_p$, $f \\in k(z)$ of degree at least $2$, and $t$ be a transcendental number. Define $k_\\infty = k \\left( \\cup_{n \\geq 0} f^{-n}(t) \\right)$, $L = k_\\infty \\cap \\overline{k}$, $G_\\infty = Gal(k_\\infty/k(t))$ and $G_\\infty^{geom} = Gal(k_\\infty/L(t))$. \\\\\n\nCan we reconstruct the $p$-adic Julia set of $f$ from $G_\\infty$ or $G_\\infty^{geom}$",
   "public_statement": "Reconstructing the Julia set of a function from the iterated Galois group\n\nLet $k$ be a finite field extension of $\\mathbb{Q}_p$, $f \\in k(z)$ of degree at least $2$, and $t$ be a transcendental number. Define $k_\\infty = k \\left( \\cup_{n \\geq 0} f^{-n}(t) \\right)$, $L = k_\\infty \\cap \\overline{k}$, $G_\\infty = Gal(k_\\infty/k(t))$ and $G_\\infty^{geom} = Gal(k_\\infty/L(t))$. \\\\\n\nCan we reconstruct the $p$-adic Julia set of $f$ from $G_\\infty$ or $G_\\infty^{geom}$",
   "evidence": "### Canonical record, preserved verbatim",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0006": {
   "statement_status": "unrecoverable",
   "original_statement": "Groups of subshifts not containing consecutive letters\n\nLet $A$ be a finite alphabet, $A^*$ the set of finite words in $A$ and $\\mathcal{F} \\subseteq A^*$ a set of forbidden patterns that contains the set $\\left\\{aa: a \\in A\\right\\}$. Let $$S_\\mathcal{F} = \\left\\{x \\in A^\\mathbb{Z}: \\text{ $x$ contains no subword in $\\mathcal{F}$}\\right\\}.$$\n\nDefine $\\varphi_a \\in Homeo(S_\\mathcal{F})$ as follows:\n\n$$\\varphi_a: \\left \\{ \\begin{matrix}\n\\text{ shift $x$ to the left} & \\text{ if $x(1) = a$} \\\\\n\\text{ shift $x$ to the right} & \\text{ if $x(0) = a$} \\\\\nx & \\text{ otherwise}\n\\end{matrix}\\right.$$\n\nand $G_\\mathcal{F} = \\left\\langle\\varphi_a: a \\in A \\right\\rangle$. \\\\\n\nRecall that the \\textbf{topological entropy} is defined as $$\\mathcal{H}_\\mathcal{F} := \\lim_{n \\rightarrow +\\infty} \\frac{\\log \\left( \\text{\\# of words of length $n$ that appear in some $x \\in S_\\mathcal{F}$ } \\right)}{n}.$$ and $F(G_\\mathcal{F})$ is the \\textbf{full topological group}. A group $G$ is called \\textbf{residually finite} if $$\\bigcap_{H \\leq G: [G:H] < \\infty} H = 1$$\n\n\\begin{enumerate}\n \\item Describe relations in $G_\\mathcal{F}$.\n \\item Can $\\mathcal{H}_\\mathcal{F}$ be algebraically interpreted as an invariant inside $G_\\mathcal{F}$?\n \\item What is \\textcolor{red}{in between (need explanation here)} minimal $S_\\mathcal{F}$ and shifts of finite type?\n \\item If $\\mathcal{H}_\\mathcal{F} = 0$, does either $G_\\mathcal{F}$ of $F(G_\\mathcal{F})$ not contain free subgroups?\n \\item When is $G_\\mathcal{F}$ residually finite?\n \\item When does $G_\\mathcal{F}$ contains a finitely generated subgroup of intermediate growth?\n\\end{enumerate}",
   "clean_statement": null,
   "public_statement": "Groups of subshifts not containing consecutive letters\n\nLet $A$ be a finite alphabet, $A^*$ the set of finite words in $A$ and $\\mathcal{F} \\subseteq A^*$ a set of forbidden patterns that contains the set $\\left\\{aa: a \\in A\\right\\}$. Let $$S_\\mathcal{F} = \\left\\{x \\in A^\\mathbb{Z}: \\text{ $x$ contains no subword in $\\mathcal{F}$}\\right\\}.$$\n\nDefine $\\varphi_a \\in Homeo(S_\\mathcal{F})$ as follows:\n\n$$\\varphi_a: \\left \\{ \\begin{matrix}\n\\text{ shift $x$ to the left} & \\text{ if $x(1) = a$} \\\\\n\\text{ shift $x$ to the right} & \\text{ if $x(0) = a$} \\\\\nx & \\text{ otherwise}\n\\end{matrix}\\right.$$\n\nand $G_\\mathcal{F} = \\left\\langle\\varphi_a: a \\in A \\right\\rangle$. \\\\\n\nRecall that the \\textbf{topological entropy} is defined as $$\\mathcal{H}_\\mathcal{F} := \\lim_{n \\rightarrow +\\infty} \\frac{\\log \\left( \\text{\\# of words of length $n$ that appear in some $x \\in S_\\mathcal{F}$ } \\right)}{n}.$$ and $F(G_\\mathcal{F})$ is the \\textbf{full topological group}. A group $G$ is called \\textbf{residually finite} if $$\\bigcap_{H \\leq G: [G:H] < \\infty} H = 1$$\n\n\\begin{enumerate}\n \\item Describe relations in $G_\\mathcal{F}$.\n \\item Can $\\mathcal{H}_\\mathcal{F}$ be algebraically interpreted as an invariant inside $G_\\mathcal{F}$?\n \\item What is \\textcolor{red}{in between (need explanation here)} minimal $S_\\mathcal{F}$ and shifts of finite type?\n \\item If $\\mathcal{H}_\\mathcal{F} = 0$, does either $G_\\mathcal{F}$ of $F(G_\\mathcal{F})$ not contain free subgroups?\n \\item When is $G_\\mathcal{F}$ residually finite?\n \\item When does $G_\\mathcal{F}$ contains a finitely generated subgroup of intermediate growth?\n\\end{enumerate}",
   "evidence": "The official 2024 AIM workshop report contains the same setup as Question 0.3 but lists only five questions: it omits canonical item 3. It retains “of” in item 4. Thus item 3 is an unrecovered editorial note, not a mathematical question. Item 4 is most naturally read with “or” in place of “of,” but both the grammar and the notation \\(F(G_{\\mathcal F})\\) remain ambiguous. The report calls this a “full topological group” without defining whether it means the full group of the \\(G_{\\mathcal F}\\)-action, its groupoid of germs, or the full group of the shift. Proposition 2 below shows that the natural groupoids have the same clopen pseudogroup, so their topological full groups agree; this is the interpretation used here.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0007": {
   "statement_status": "exact",
   "original_statement": "Defining relations for subgroups of full group of a shift\n\nLet $A$ be a finite alphabet and $\\Omega$ be a 2-sided subshift, i.e., a closed subset of $A^\\mathbb{Z}$ that is invariant under the shift map $\\sigma$. An element $g \\in Homeo(\\Omega)$ belong to the full group of the shift ($F(\\Omega)$) if each point in $\\Omega$ has an open neighborhood $U$ such that $g|_U = (\\sigma^n)|_U$ for some $n \\in \\mathbb{Z}$ depending on $U$ is equal to resembles a power of the shift map.\n\nIs it true that for all finitely generated subgroup $H \\leq F(\\Omega)$, $H$ is either elementary amenable or for any $K$ finitely presented subgroup such that $K \\twoheadrightarrow H$, $K$ contains a free subgroup?",
   "clean_statement": "Defining relations for subgroups of full group of a shift\n\nLet $A$ be a finite alphabet and $\\Omega$ be a 2-sided subshift, i.e., a closed subset of $A^\\mathbb{Z}$ that is invariant under the shift map $\\sigma$. An element $g \\in Homeo(\\Omega)$ belong to the full group of the shift ($F(\\Omega)$) if each point in $\\Omega$ has an open neighborhood $U$ such that $g|_U = (\\sigma^n)|_U$ for some $n \\in \\mathbb{Z}$ depending on $U$ is equal to resembles a power of the shift map.\n\nIs it true that for all finitely generated subgroup $H \\leq F(\\Omega)$, $H$ is either elementary amenable or for any $K$ finitely presented subgroup such that $K \\twoheadrightarrow H$, $K$ contains a free subgroup?",
   "public_statement": "Defining relations for subgroups of full group of a shift\n\nLet $A$ be a finite alphabet and $\\Omega$ be a 2-sided subshift, i.e., a closed subset of $A^\\mathbb{Z}$ that is invariant under the shift map $\\sigma$. An element $g \\in Homeo(\\Omega)$ belong to the full group of the shift ($F(\\Omega)$) if each point in $\\Omega$ has an open neighborhood $U$ such that $g|_U = (\\sigma^n)|_U$ for some $n \\in \\mathbb{Z}$ depending on $U$ is equal to resembles a power of the shift map.\n\nIs it true that for all finitely generated subgroup $H \\leq F(\\Omega)$, $H$ is either elementary amenable or for any $K$ finitely presented subgroup such that $K \\twoheadrightarrow H$, $K$ contains a free subgroup?",
   "evidence": "The canonical record, preserved verbatim despite its grammatical corruption, asks:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 6,
   "attempt": 2
  },
  "AIM-DYNAMICAL_SYSTEMS-0008": {
   "statement_status": "exact",
   "original_statement": "Entropy of subshifts of finite type\n\nLet $G$ be a finitely generated infinite ameanable group, $A$ be a finite alphabet and define $$\\mathcal{F} = \\left\\{f:B \\rightarrow A: B \\subseteq G \\text{ finite}\\right\\}$$ a finite set of forbidden patterns and $$S_{G,\\mathcal{F}, A} = \\left\\{x \\in A^G: x \\text{ contains no forbidden subpatterns}\\right\\}.$$\n\n$G$ acts on $S_{G,\\mathcal{F}, A}$ on both sides as the regular representation, namely, permuting the coordinates of the sequences by right or left translation. Choose $\\left\\{F_n\\right\\}_{n \\in \\mathbb{N}}$ a Følner sequence for $G$ and define $$\\mathcal{H}_{G, \\mathcal{F}, A} = \\lim_{n \\rightarrow +\\infty} \\frac{\\log \\left( \\# \\left\\{f: F_n \\rightarrow A: \\text{ $f$ appears in some $x \\in S_{G,\\mathcal{F}, A}$}\\right\\} \\right)}{n}.$$ \\\\\n\nIt is a fact that $\\mathcal{H}_{G, \\mathcal{F}, A}$ does not depend on the selected Følner sequence. Define also $$E_G = \\left\\{\\mathcal{H}_{G, \\mathcal{F}, A}: \\mathcal{F} \\text{ is a finite set of forbidden subwords}\\right\\}.$$\n\nIt is known (see \\cite{MR2680402}) that $$E_{\\mathbb{Z}} = \\left\\{q\\log(\\lambda): q \\in \\mathbb{Q}^+, \\text{ and $\\lambda$ is a Perron eigenvalue}\\right\\}$$ and $$E_{\\mathbb{Z}^2} = \\left\\{r \\in \\mathbb{R}^+: \\text{ $r$ is a right-recursively enumerable number}\\right\\},$$ where a Perron eigenvalue is the eigenvalue given by the Perron-Frobenius theorem of a square matrix with non-positive entries such that for some power of the matrix, the entries are all positive, and a right recursively enumerable number is the infimum of the image of a computable function $f: \\mathbb{N} \\rightarrow \\mathbb{Q}$.\n\n\\begin{enumerate}\n \\item Given a recursively presented and finitely generated infinite amenable group $G$, what is $E_G$?\n \\item Does there exist a recursively presented and finitely generated infinite amenable group $G$ such that $E_\\mathbb{Z} \\subsetneq E_G \\subsetneq E_{\\mathbb{Z}^2}$?\n\\end{enumerate}",
   "clean_statement": "Entropy of subshifts of finite type\n\nLet $G$ be a finitely generated infinite ameanable group, $A$ be a finite alphabet and define $$\\mathcal{F} = \\left\\{f:B \\rightarrow A: B \\subseteq G \\text{ finite}\\right\\}$$ a finite set of forbidden patterns and $$S_{G,\\mathcal{F}, A} = \\left\\{x \\in A^G: x \\text{ contains no forbidden subpatterns}\\right\\}.$$\n\n$G$ acts on $S_{G,\\mathcal{F}, A}$ on both sides as the regular representation, namely, permuting the coordinates of the sequences by right or left translation. Choose $\\left\\{F_n\\right\\}_{n \\in \\mathbb{N}}$ a Følner sequence for $G$ and define $$\\mathcal{H}_{G, \\mathcal{F}, A} = \\lim_{n \\rightarrow +\\infty} \\frac{\\log \\left( \\# \\left\\{f: F_n \\rightarrow A: \\text{ $f$ appears in some $x \\in S_{G,\\mathcal{F}, A}$}\\right\\} \\right)}{n}.$$ \\\\\n\nIt is a fact that $\\mathcal{H}_{G, \\mathcal{F}, A}$ does not depend on the selected Følner sequence. Define also $$E_G = \\left\\{\\mathcal{H}_{G, \\mathcal{F}, A}: \\mathcal{F} \\text{ is a finite set of forbidden subwords}\\right\\}.$$\n\nIt is known (see \\cite{MR2680402}) that $$E_{\\mathbb{Z}} = \\left\\{q\\log(\\lambda): q \\in \\mathbb{Q}^+, \\text{ and $\\lambda$ is a Perron eigenvalue}\\right\\}$$ and $$E_{\\mathbb{Z}^2} = \\left\\{r \\in \\mathbb{R}^+: \\text{ $r$ is a right-recursively enumerable number}\\right\\},$$ where a Perron eigenvalue is the eigenvalue given by the Perron-Frobenius theorem of a square matrix with non-positive entries such that for some power of the matrix, the entries are all positive, and a right recursively enumerable number is the infimum of the image of a computable function $f: \\mathbb{N} \\rightarrow \\mathbb{Q}$.\n\n\\begin{enumerate}\n \\item Given a recursively presented and finitely generated infinite amenable group $G$, what is $E_G$?\n \\item Does there exist a recursively presented and finitely generated infinite amenable group $G$ such that $E_\\mathbb{Z} \\subsetneq E_G \\subsetneq E_{\\mathbb{Z}^2}$?\n\\end{enumerate}",
   "public_statement": "Entropy of subshifts of finite type\n\nLet $G$ be a finitely generated infinite ameanable group, $A$ be a finite alphabet and define $$\\mathcal{F} = \\left\\{f:B \\rightarrow A: B \\subseteq G \\text{ finite}\\right\\}$$ a finite set of forbidden patterns and $$S_{G,\\mathcal{F}, A} = \\left\\{x \\in A^G: x \\text{ contains no forbidden subpatterns}\\right\\}.$$\n\n$G$ acts on $S_{G,\\mathcal{F}, A}$ on both sides as the regular representation, namely, permuting the coordinates of the sequences by right or left translation. Choose $\\left\\{F_n\\right\\}_{n \\in \\mathbb{N}}$ a Følner sequence for $G$ and define $$\\mathcal{H}_{G, \\mathcal{F}, A} = \\lim_{n \\rightarrow +\\infty} \\frac{\\log \\left( \\# \\left\\{f: F_n \\rightarrow A: \\text{ $f$ appears in some $x \\in S_{G,\\mathcal{F}, A}$}\\right\\} \\right)}{n}.$$ \\\\\n\nIt is a fact that $\\mathcal{H}_{G, \\mathcal{F}, A}$ does not depend on the selected Følner sequence. Define also $$E_G = \\left\\{\\mathcal{H}_{G, \\mathcal{F}, A}: \\mathcal{F} \\text{ is a finite set of forbidden subwords}\\right\\}.$$\n\nIt is known (see \\cite{MR2680402}) that $$E_{\\mathbb{Z}} = \\left\\{q\\log(\\lambda): q \\in \\mathbb{Q}^+, \\text{ and $\\lambda$ is a Perron eigenvalue}\\right\\}$$ and $$E_{\\mathbb{Z}^2} = \\left\\{r \\in \\mathbb{R}^+: \\text{ $r$ is a right-recursively enumerable number}\\right\\},$$ where a Perron eigenvalue is the eigenvalue given by the Perron-Frobenius theorem of a square matrix with non-positive entries such that for some power of the matrix, the entries are all positive, and a right recursively enumerable number is the infimum of the image of a computable function $f: \\mathbb{N} \\rightarrow \\mathbb{Q}$.\n\n\\begin{enumerate}\n \\item Given a recursively presented and finitely generated infinite amenable group $G$, what is $E_G$?\n \\item Does there exist a recursively presented and finitely generated infinite amenable group $G$ such that $E_\\mathbb{Z} \\subsetneq E_G \\subsetneq E_{\\mathbb{Z}^2}$?\n\\end{enumerate}",
   "evidence": "The canonical record is titled **“Entropy of subshifts of finite type.”** Its mathematical text reads, with the source’s wording preserved:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0009": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Properties of subgroups of topological full groups\n\nTry to understand relations, amenability and Liouville property of subgroups of topological full groups. In particular of $\\left\\langle \\delta_a: a \\in A \\right\\rangle$",
   "clean_statement": null,
   "public_statement": "Properties of subgroups of topological full groups\n\nTry to understand relations, amenability and Liouville property of subgroups of topological full groups. In particular of $\\left\\langle \\delta_a: a \\in A \\right\\rangle$",
   "evidence": "The literal prompt has no determinate truth value. The conservative reconstruction analyzed below is explicitly conditional:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0010": {
   "statement_status": "exact",
   "original_statement": "Commutator width of Thompson groups\n\nLet $G$ be $F,T$ or $V$ or a topological full group. Let $\\rho_1(g)$ be the commutator width of $g$ and $\\rho_2(g)$ the minimum number of involutions in a factorization of $g$.\n\n\\begin{enumerate}\n \\item Prove that $\\rho_1$ and $\\rho_2$ are bounded\n \\item Prove that $\\rho_1 = 1$ and $\\rho_2 = 3$\n \\item Is there a finitely presented simple group with commutator width greater or equal to $2$?\n\\end{enumerate}",
   "clean_statement": "Commutator width of Thompson groups\n\nLet $G$ be $F,T$ or $V$ or a topological full group. Let $\\rho_1(g)$ be the commutator width of $g$ and $\\rho_2(g)$ the minimum number of involutions in a factorization of $g$.\n\n\\begin{enumerate}\n \\item Prove that $\\rho_1$ and $\\rho_2$ are bounded\n \\item Prove that $\\rho_1 = 1$ and $\\rho_2 = 3$\n \\item Is there a finitely presented simple group with commutator width greater or equal to $2$?\n\\end{enumerate}",
   "public_statement": "Commutator width of Thompson groups\n\nLet $G$ be $F,T$ or $V$ or a topological full group. Let $\\rho_1(g)$ be the commutator width of $g$ and $\\rho_2(g)$ the minimum number of involutions in a factorization of $g$.\n\n\\begin{enumerate}\n \\item Prove that $\\rho_1$ and $\\rho_2$ are bounded\n \\item Prove that $\\rho_1 = 1$ and $\\rho_2 = 3$\n \\item Is there a finitely presented simple group with commutator width greater or equal to $2$?\n\\end{enumerate}",
   "evidence": "The canonical record is item 3.1, “Commutator width of Thompson groups,” from the AIM workshop *Groups of dynamical origin* (June 3--7, 2024). The official workshop report confirms the following wording; the issue discussed below is therefore mathematical rather than an OCR error:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0011": {
   "statement_status": "exact",
   "original_statement": "Maximal subgroups in Thompson groups\n\nFind new maximal subgroups of infinite index in Thompson groups",
   "clean_statement": "Maximal subgroups in Thompson groups\n\nFind new maximal subgroups of infinite index in Thompson groups",
   "public_statement": "Maximal subgroups in Thompson groups\n\nFind new maximal subgroups of infinite index in Thompson groups",
   "evidence": "The canonical AIM record (source file `aim-dynamical-systems-notes.json`, zero-based index 10) reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0012": {
   "statement_status": "exact",
   "original_statement": "Growths functions of repetitive graphs\n\nA graph $\\Gamma$ is called \\textbf{repetitive} if for all $r > 0$ there exists $d > 0$ such that for all $x,y \\in \\Gamma$, there exists an isomorphic copy of $B_r(x)$ (the ball of radius $r$ centered at $x$) at distance less or equal to $d$ from $y$. \\\\\n\nConsider two kind of growths functions for $\\Gamma$: pick $x_0 \\in \\Gamma$ and define $f_{x_0}(n) = \\# B_n(x_0)$ and $f(x) = \\max \\left\\{f_x(n): x \\in \\Gamma\\right\\}$\n\nWhat functions can be realized as growths functions of repetitive graphs?",
   "clean_statement": "Growths functions of repetitive graphs\n\nA graph $\\Gamma$ is called \\textbf{repetitive} if for all $r > 0$ there exists $d > 0$ such that for all $x,y \\in \\Gamma$, there exists an isomorphic copy of $B_r(x)$ (the ball of radius $r$ centered at $x$) at distance less or equal to $d$ from $y$. \\\\\n\nConsider two kind of growths functions for $\\Gamma$: pick $x_0 \\in \\Gamma$ and define $f_{x_0}(n) = \\# B_n(x_0)$ and $f(x) = \\max \\left\\{f_x(n): x \\in \\Gamma\\right\\}$\n\nWhat functions can be realized as growths functions of repetitive graphs?",
   "public_statement": "Growths functions of repetitive graphs\n\nA graph $\\Gamma$ is called \\textbf{repetitive} if for all $r > 0$ there exists $d > 0$ such that for all $x,y \\in \\Gamma$, there exists an isomorphic copy of $B_r(x)$ (the ball of radius $r$ centered at $x$) at distance less or equal to $d$ from $y$. \\\\\n\nConsider two kind of growths functions for $\\Gamma$: pick $x_0 \\in \\Gamma$ and define $f_{x_0}(n) = \\# B_n(x_0)$ and $f(x) = \\max \\left\\{f_x(n): x \\in \\Gamma\\right\\}$\n\nWhat functions can be realized as growths functions of repetitive graphs?",
   "evidence": "The canonical record, from the AIM workshop *Groups of dynamical origin*, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0013": {
   "statement_status": "exact",
   "original_statement": "Nekrashevych C*-algebras\n\nWhen is the Nekrashevych $C^*$-algebra of a contracting group simple?",
   "clean_statement": "Nekrashevych C*-algebras\n\nWhen is the Nekrashevych $C^*$-algebra of a contracting group simple?",
   "public_statement": "Nekrashevych C*-algebras\n\nWhen is the Nekrashevych $C^*$-algebra of a contracting group simple?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 12,
   "attempt": 2
  },
  "AIM-DYNAMICAL_SYSTEMS-0014": {
   "statement_status": "exact",
   "original_statement": "Describe\n\\[\nN(x)=\\#\\{p\\leq x : p|f^n(\\alpha) \\text { for some } n\\}\n\\]\nfor $p$ prime as $x\\rightarrow \\infty.$",
   "clean_statement": "Describe\n\\[\nN(x)=\\#\\{p\\leq x : p|f^n(\\alpha) \\text { for some } n\\}\n\\]\nfor $p$ prime as $x\\rightarrow \\infty.$",
   "public_statement": "Describe\n\\[\nN(x)=\\#\\{p\\leq x : p|f^n(\\alpha) \\text { for some } n\\}\n\\]\nfor $p$ prime as $x\\rightarrow \\infty.$",
   "evidence": "The canonical AIM record, in the workshop section **“Density results for PCF polynomials,”** asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0015": {
   "statement_status": "exact",
   "original_statement": "What is the growth rate of $N(x)$ for PCF and non PCF maps?",
   "clean_statement": "What is the growth rate of $N(x)$ for PCF and non PCF maps?",
   "public_statement": "What is the growth rate of $N(x)$ for PCF and non PCF maps?",
   "evidence": "The canonical record asks, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 14,
   "attempt": 2
  },
  "AIM-DYNAMICAL_SYSTEMS-0016": {
   "statement_status": "exact",
   "original_statement": "If $f(z)\\in\\mathbb{F}_q(T)$, what happens then? What happens with other fields?",
   "clean_statement": "If $f(z)\\in\\mathbb{F}_q(T)$, what happens then? What happens with other fields?",
   "public_statement": "If $f(z)\\in\\mathbb{F}_q(T)$, what happens then? What happens with other fields?",
   "evidence": "The archived AIM record, from the workshop *The Galois theory of orbits in arithmetic dynamics*, section “Density results for PCF polynomials,” asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0017": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Is Pink's generating function $\\Phi_w$ always rational for every $\\rho(\\text{Frob}_p)=w?$",
   "clean_statement": null,
   "public_statement": "Is Pink's generating function $\\Phi_w$ always rational for every $\\rho(\\text{Frob}_p)=w?$",
   "evidence": "The canonical record asks, verbatim:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0018": {
   "statement_status": "exact",
   "original_statement": "Cook up other conjugacy invariant things.",
   "clean_statement": "Cook up other conjugacy invariant things.",
   "public_statement": "Cook up other conjugacy invariant things.",
   "evidence": "The exact AIM record is Problem 2.2 in the workshop section “Conjugacy Invariants”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0019": {
   "statement_status": "exact",
   "original_statement": "Is there any example where $G$ does not have finite index in $G'?$",
   "clean_statement": "Is there any example where $G$ does not have finite index in $G'?$",
   "public_statement": "Is there any example where $G$ does not have finite index in $G'?$",
   "evidence": "The canonical record preserves only the sentence",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0020": {
   "statement_status": "exact",
   "original_statement": "Understand the relation with PCFness. Note this is related to Problem 1.1. Perhaps start with $f(z)=z^2-1.$",
   "clean_statement": "Understand the relation with PCFness. Note this is related to Problem 1.1. Perhaps start with $f(z)=z^2-1.$",
   "public_statement": "Understand the relation with PCFness. Note this is related to Problem 1.1. Perhaps start with $f(z)=z^2-1.$",
   "evidence": "The canonical record says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0021": {
   "statement_status": "exact",
   "original_statement": "\\begin{enumerate}\n\\item Analogy between dynatomic and modular curves (See Silverman, ADS Section 4.1 and 4.2 for dynatomic polynomials).\n\\item What are other proxies for level structure, e.g. marked critical points, for which bad primes are more natural?\n\\end{enumerate}",
   "clean_statement": "\\begin{enumerate}\n\\item Analogy between dynatomic and modular curves (See Silverman, ADS Section 4.1 and 4.2 for dynatomic polynomials).\n\\item What are other proxies for level structure, e.g. marked critical points, for which bad primes are more natural?\n\\end{enumerate}",
   "public_statement": "\\begin{enumerate}\n\\item Analogy between dynatomic and modular curves (See Silverman, ADS Section 4.1 and 4.2 for dynatomic polynomials).\n\\item What are other proxies for level structure, e.g. marked critical points, for which bad primes are more natural?\n\\end{enumerate}",
   "evidence": "The canonical AIM record is Question 4.1 in the section “Dynatomic Modular Curves.” Its two items are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0022": {
   "statement_status": "exact",
   "original_statement": "\\begin{enumerate}\n\\item For $p|N,$ $X^{\\text{dyn}}(N)$ has bad reduction at $p$ for the standard model, but evidence suggests there exists a different model with good reduction. Is this true? If not, why not? Is there a ``reason\" for this to be true?\n\\item(Silverman) In general, what are the primes of bad reduction and why?\n\\end{enumerate}",
   "clean_statement": "\\begin{enumerate}\n\\item For $p|N,$ $X^{\\text{dyn}}(N)$ has bad reduction at $p$ for the standard model, but evidence suggests there exists a different model with good reduction. Is this true? If not, why not? Is there a ``reason\" for this to be true?\n\\item(Silverman) In general, what are the primes of bad reduction and why?\n\\end{enumerate}",
   "public_statement": "\\begin{enumerate}\n\\item For $p|N,$ $X^{\\text{dyn}}(N)$ has bad reduction at $p$ for the standard model, but evidence suggests there exists a different model with good reduction. Is this true? If not, why not? Is there a ``reason\" for this to be true?\n\\item(Silverman) In general, what are the primes of bad reduction and why?\n\\end{enumerate}",
   "evidence": "The archived section itself has no introduction and really does omit a subscript on $X^{\\mathrm{dyn}}(N)$; this is not an extraction error. The 2016 AIM workshop summary removes the ambiguity: it writes $X^{\\mathrm{dyn}}_1(N)$, calls it the smooth completion of the vanishing locus of the $N$-th dynatomic polynomial $\\Phi_N(x,c)$, and displays the map \\[ X^{\\mathrm{dyn}}_1(N)\\longrightarrow \\mathbf P^1_c, \\qquad (x,c)\\longmapsto c. \\] The adjacent archived Problem 4.1 points to Sections 4.1--4.2 of Silverman's *The Arithmetic of Dynamical Systems*. Those sections use the quadratic family \\[ f_c(x)=x^2+c. \\] The workshop summary, Silverman's notation, and the subsequent paper [DKOPRSW19] therefore support the following reconstruction: the intended curve is the smooth projective quadratic dynatomic curve marking a point of formal period $N$. The generalization to $f_c(x)=x^m+c$ is discus...",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0023": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Give an algebraic description of the boundary and maps in the boundary in characteristic $p$ not necessarily $0.$ (Reference DeMarco). Try $d=3.$ Motivation: results for moduli curves use understanding of the boundary.",
   "clean_statement": "Describe the boundary $\\overline M_d\\setminus M_d$ of the GIT compactification algebraically over fields of arbitrary characteristic, identify the degenerate maps represented there, and describe the behavior of iteration on those boundary maps; begin with $d=3$ and account for inseparability in small characteristic.",
   "public_statement": "Give an algebraic description of the boundary and maps in the boundary in characteristic $p$ not necessarily $0.$ (Reference DeMarco). Try $d=3.$ Motivation: results for moduli curves use understanding of the boundary.",
   "evidence": "The prompt says $M_d$, asks to try $d=3$, and cites applications to moduli curves. The most plausible reconstruction is therefore:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0024": {
   "statement_status": "exact",
   "original_statement": "\\begin{enumerate}\n\\item (Want to look at a curve in $M_2$). For $f(z)=\\frac{z^2-1}{z^2-c},$ there is a collection of PCF maps which approach $c=1$ on the curve. Compare the arboreal representation for $f$ to the representation of the limit map.\n\\item More generally, how do these representations behave as you deform to the boundary?\n\\end{enumerate}",
   "clean_statement": "\\begin{enumerate}\n\\item (Want to look at a curve in $M_2$). For $f(z)=\\frac{z^2-1}{z^2-c},$ there is a collection of PCF maps which approach $c=1$ on the curve. Compare the arboreal representation for $f$ to the representation of the limit map.\n\\item More generally, how do these representations behave as you deform to the boundary?\n\\end{enumerate}",
   "public_statement": "\\begin{enumerate}\n\\item (Want to look at a curve in $M_2$). For $f(z)=\\frac{z^2-1}{z^2-c},$ there is a collection of PCF maps which approach $c=1$ on the curve. Compare the arboreal representation for $f$ to the representation of the limit map.\n\\item More generally, how do these representations behave as you deform to the boundary?\n\\end{enumerate}",
   "evidence": "The canonical record is Question 5.2, in the section “Boundary of \\(M_d\\),” from the May 2016 AIM workshop *The Galois theory of orbits in arithmetic dynamics*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0025": {
   "statement_status": "exact",
   "original_statement": "What is the influence of critical relations on the image of Galois over a number field (Pink condition)? More specifically which sorts of critical orbit relations force infinite index for the image of Galois?",
   "clean_statement": "What is the influence of critical relations on the image of Galois over a number field (Pink condition)? More specifically which sorts of critical orbit relations force infinite index for the image of Galois?",
   "public_statement": "What is the influence of critical relations on the image of Galois over a number field (Pink condition)? More specifically which sorts of critical orbit relations force infinite index for the image of Galois?",
   "evidence": "The canonical record is Problem 6.1, “Critical Relations,” from the AIM workshop *The Galois theory of orbits in arithmetic dynamics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0026": {
   "statement_status": "exact",
   "original_statement": "Bad reduction for models of critical orbit relation curves. e.g. $M_2.$",
   "clean_statement": "Bad reduction for models of critical orbit relation curves. e.g. $M_2.$",
   "public_statement": "Bad reduction for models of critical orbit relation curves. e.g. $M_2.$",
   "evidence": "The canonical AIM record says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0027": {
   "statement_status": "exact",
   "original_statement": "If $K_fK_g$ is finite over both $K_f$ and $K_g$, are $f$ and $g$ conjugate, an iterate, or conjugate to something that commutes?",
   "clean_statement": "If $K_fK_g$ is finite over both $K_f$ and $K_g$, are $f$ and $g$ conjugate, an iterate, or conjugate to something that commutes?",
   "public_statement": "If $K_fK_g$ is finite over both $K_f$ and $K_g$, are $f$ and $g$ conjugate, an iterate, or conjugate to something that commutes?",
   "evidence": "The canonical record is AIM Problem List question 7.1 from the 2016 workshop *The Galois theory of orbits in arithmetic dynamics*, stored at zero-based index 26 of aim-dynamical-systems-notes.json:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0028": {
   "statement_status": "exact",
   "original_statement": "What do you need to know about $K_f$ to figure out what $f$ was (or how close to $f$ can you get)?",
   "clean_statement": "What do you need to know about $K_f$ to figure out what $f$ was (or how close to $f$ can you get)?",
   "public_statement": "What do you need to know about $K_f$ to figure out what $f$ was (or how close to $f$ can you get)?",
   "evidence": "The canonical record is Problem 7.2 in the “Dynatomic side” section of the AIM workshop *The Galois theory of orbits in arithmetic dynamics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0029": {
   "statement_status": "exact",
   "original_statement": "Begin to classify those $f$ for which $\\text{Gal}(K_f/K)$ does not have finite index in the thing it's supposed to.\nAsk this question again for\n\\[\nK_f'=K(\\text{all periodic points of }f)\n\\]",
   "clean_statement": "Begin to classify those $f$ for which $\\text{Gal}(K_f/K)$ does not have finite index in the thing it's supposed to.\nAsk this question again for\n\\[\nK_f'=K(\\text{all periodic points of }f)\n\\]",
   "public_statement": "Begin to classify those $f$ for which $\\text{Gal}(K_f/K)$ does not have finite index in the thing it's supposed to.\nAsk this question again for\n\\[\nK_f'=K(\\text{all periodic points of }f)\n\\]",
   "evidence": "The canonical record is AIM Problem List question 7.3 from the 2016 workshop *The Galois theory of orbits in arithmetic dynamics*, stored at zero-based index 28 of aim-dynamical-systems-notes.json:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0030": {
   "statement_status": "exact",
   "original_statement": "The previous section's problem appropriately phrased for preimage extensions.",
   "clean_statement": "The previous section's problem appropriately phrased for preimage extensions.",
   "public_statement": "The previous section's problem appropriately phrased for preimage extensions.",
   "evidence": "The canonical AIM record is the following fragment, preserved verbatim:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0031": {
   "statement_status": "exact",
   "original_statement": "Is there a deformation theory for arboreal Galois representations?",
   "clean_statement": "Is there a deformation theory for arboreal Galois representations?",
   "public_statement": "Is there a deformation theory for arboreal Galois representations?",
   "evidence": "The canonical record is Problem 9.1 in the section “At what level do we see deformation?” of the AIM workshop *The Galois theory of orbits in arithmetic dynamics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0032": {
   "statement_status": "exact",
   "original_statement": "To what level can the representation associated to two quadratic polynomials agree?",
   "clean_statement": "To what level can the representation associated to two quadratic polynomials agree?",
   "public_statement": "To what level can the representation associated to two quadratic polynomials agree?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0033": {
   "statement_status": "exact",
   "original_statement": "Find classes of examples of polynomials or rational maps $d\\geq 3$ for which we can compute explicitly the arboreal Galois representation up to finite index over various fields.",
   "clean_statement": "Find classes of examples of polynomials or rational maps $d\\geq 3$ for which we can compute explicitly the arboreal Galois representation up to finite index over various fields.",
   "public_statement": "Find classes of examples of polynomials or rational maps $d\\geq 3$ for which we can compute explicitly the arboreal Galois representation up to finite index over various fields.",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0034": {
   "statement_status": "exact",
   "original_statement": "Generalizing: Can you find an example in higher dimensions?",
   "clean_statement": "Generalizing: Can you find an example in higher dimensions?",
   "public_statement": "Generalizing: Can you find an example in higher dimensions?",
   "evidence": "The canonical record is Problem 10.2 in the section “Find more arboreal Galois representations” of the AIM workshop *The Galois theory of orbits in arithmetic dynamics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0035": {
   "statement_status": "exact",
   "original_statement": "When is the image maximal?",
   "clean_statement": "When is the image maximal?",
   "public_statement": "When is the image maximal?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0036": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Describe $K_n\\cap K_{n'}.$ Prove that this is small.",
   "clean_statement": null,
   "public_statement": "Describe $K_n\\cap K_{n'}.$ Prove that this is small.",
   "evidence": "We use the following explicit reconstruction. Let $K$ be a characteristic-zero field and $f\\in K[x]$ have degree at least two. Define the dynatomic polynomial \\[ \\Phi^*_{f,n}(x) =\\prod_{r\\mid n}\\bigl(f^r(x)-x\\bigr)^{\\mu(n/r)}, \\tag{1.1} \\] which is a polynomial despite its quotient presentation. Let \\[ K_n(f/K)=\\text{the splitting field over $K$ of }\\Phi^*_{f,n}(x). \\tag{1.2} \\] When repeated roots occur, “splitting field” means the field generated by the distinct roots. Roots can have formal period $n$ but smaller exact period in parabolic cases [MP94, MS95]; this distinction is kept explicit. The question is meaningful for distinct $n,n'$. If $n=n'$, the intersection is tautologically $K_n$.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0037": {
   "statement_status": "exact",
   "original_statement": "Say something interesting about wild ramification for arboreal representations. e.g. study Galois representations over local fields of residue characteristic dividing degree of $f$ (or even less than or equal to the degree). e.g. An arboreal version of Sen's theorem.",
   "clean_statement": "Say something interesting about wild ramification for arboreal representations. e.g. study Galois representations over local fields of residue characteristic dividing degree of $f$ (or even less than or equal to the degree). e.g. An arboreal version of Sen's theorem.",
   "public_statement": "Say something interesting about wild ramification for arboreal representations. e.g. study Galois representations over local fields of residue characteristic dividing degree of $f$ (or even less than or equal to the degree). e.g. An arboreal version of Sen's theorem.",
   "evidence": "The exact AIM problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0038": {
   "statement_status": "exact",
   "original_statement": "Let $f(z)=z^2+1$ over $\\mathbb{Q}_2.$ What is the size, etc, of the arboreal representation?",
   "clean_statement": "Let $f(z)=z^2+1$ over $\\mathbb{Q}_2.$ What is the size, etc, of the arboreal representation?",
   "public_statement": "Let $f(z)=z^2+1$ over $\\mathbb{Q}_2.$ What is the size, etc, of the arboreal representation?",
   "evidence": "The canonical record is Problem 12.2 of the AIM workshop list *The Galois theory of orbits in arithmetic dynamics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0039": {
   "statement_status": "exact",
   "original_statement": "What operations are there on these ``representations''? And do some of them correspond to operations on $f?$",
   "clean_statement": "What operations are there on these ``representations''? And do some of them correspond to operations on $f?$",
   "public_statement": "What operations are there on these ``representations''? And do some of them correspond to operations on $f?$",
   "evidence": "The canonical AIM record is Problem 13.1, under the heading “Category of arboreal representations”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0040": {
   "statement_status": "exact",
   "original_statement": "Work over $\\mathbb{C}(x)$ for a start. Given a subgroup $G\\subseteq \\text{Aut}$(regularly rooted $d$-ary tree). Does there exist a polynomial $f(z)\\in K[z]$ with $G$ as its image of Galois representation? Interpret and solve.",
   "clean_statement": "Work over $\\mathbb{C}(x)$ for a start. Given a subgroup $G\\subseteq \\text{Aut}$(regularly rooted $d$-ary tree). Does there exist a polynomial $f(z)\\in K[z]$ with $G$ as its image of Galois representation? Interpret and solve.",
   "public_statement": "Work over $\\mathbb{C}(x)$ for a start. Given a subgroup $G\\subseteq \\text{Aut}$(regularly rooted $d$-ary tree). Does there exist a polynomial $f(z)\\in K[z]$ with $G$ as its image of Galois representation? Interpret and solve.",
   "evidence": "The AIM record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0041": {
   "statement_status": "exact",
   "original_statement": "Variation of problem 14.1: ask again only in a dynatomic setting. Formulate and solve.",
   "clean_statement": "Variation of problem 14.1: ask again only in a dynatomic setting. Formulate and solve.",
   "public_statement": "Variation of problem 14.1: ask again only in a dynatomic setting. Formulate and solve.",
   "evidence": "The canonical AIM record, Problem 14.2 of the workshop list *The Galois theory of orbits in arithmetic dynamics*, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0042": {
   "statement_status": "exact",
   "original_statement": "To what extent and when are the actions of Galois on preimages of $\\alpha$ independent from the action of the preimages of $\\beta?$.",
   "clean_statement": "To what extent and when are the actions of Galois on preimages of $\\alpha$ independent from the action of the preimages of $\\beta?$.",
   "public_statement": "To what extent and when are the actions of Galois on preimages of $\\alpha$ independent from the action of the preimages of $\\beta?$.",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0043": {
   "statement_status": "exact",
   "original_statement": "Given an arboreal Galois representations over $\\mathbf Q_p$ (for all or some $p$) what are some nonobvious, sufficient, or necessary conditions under which all of these come from some global representation attached to some $f?$",
   "clean_statement": "Given an arboreal Galois representations over $\\mathbf Q_p$ (for all or some $p$) what are some nonobvious, sufficient, or necessary conditions under which all of these come from some global representation attached to some $f?$",
   "public_statement": "Given an arboreal Galois representations over $\\mathbf Q_p$ (for all or some $p$) what are some nonobvious, sufficient, or necessary conditions under which all of these come from some global representation attached to some $f?$",
   "evidence": "The AIM record (workshop *The Galois theory of orbits in arithmetic dynamics*, section “Local Global Principle for arboreal Galois representations,” Problem 16.1) reads verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0044": {
   "statement_status": "exact",
   "original_statement": "Problem 1. Suppose the deterministic system of coupled ODEs \n\n˙x = f (x, y )˙y = g(x, y )\n\nhas a stable fixed point, at which the associated linear system has two complex eigenvalues. Then the linearized version of this system about this fixed point with a stochastic term [ dx dy \n\n]\n\n= A\n\n[ xy\n\n]\n\ndt +\n\n[ σ1dW 1\n\nσ2dW 2\n\n]\n\ncan be approximated by a constant times a rotation times pair of standard indepen-dent Ornstein-Uhlenbeck processes. What is the analogue of this result in the context of a discrete time reaction diffusion equation?",
   "clean_statement": "Problem 1. Suppose the deterministic system of coupled ODEs\n\n˙x = f (x, y )˙y = g(x, y )\n\nhas a stable fixed point, at which the associated linear system has two complex eigenvalues. Then the linearized version of this system about this fixed point with a stochastic term [ dx dy\n\n]\n\n= A\n\n[ xy\n\n]\n\ndt +\n\n[ σ1dW 1\n\nσ2dW 2\n\n]\n\ncan be approximated by a constant times a rotation times pair of standard indepen-dent Ornstein-Uhlenbeck processes. What is the analogue of this result in the context of a discrete time reaction diffusion equation?",
   "public_statement": "Problem 1. Suppose the deterministic system of coupled ODEs\n\n˙x = f (x, y )˙y = g(x, y )\n\nhas a stable fixed point, at which the associated linear system has two complex eigenvalues. Then the linearized version of this system about this fixed point with a stochastic term [ dx dy\n\n]\n\n= A\n\n[ xy\n\n]\n\ndt +\n\n[ σ1dW 1\n\nσ2dW 2\n\n]\n\ncan be approximated by a constant times a rotation times pair of standard indepen-dent Ornstein-Uhlenbeck processes. What is the analogue of this result in the context of a discrete time reaction diffusion equation?",
   "evidence": "The canonical record is Problem 1 in the AIM workshop notes *Stochastic methods for non-equilibrium dynamical systems*. The original PDF was inspected directly. Its damaged displays recover as",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0045": {
   "statement_status": "exact",
   "original_statement": "Problem 2. Given a Saini billard on a torus with a finite horizon and two masses \n\nm1 and m2, is there a way to find the system's viscosity, i.e. transfer of momentum. Additionally, is there a way to define heat in this model and can one check the Einstein relation? Lastly, if m1 6 = m2 then, in the limit, do the masses have the same kinetic energy?",
   "clean_statement": "Problem 2. Given a Saini billard on a torus with a finite horizon and two masses\n\nm1 and m2, is there a way to find the system's viscosity, i.e. transfer of momentum. Additionally, is there a way to define heat in this model and can one check the Einstein relation? Lastly, if m1 6 = m2 then, in the limit, do the masses have the same kinetic energy?",
   "public_statement": "Problem 2. Given a Saini billard on a torus with a finite horizon and two masses\n\nm1 and m2, is there a way to find the system's viscosity, i.e. transfer of momentum. Additionally, is there a way to define heat in this model and can one check the Einstein relation? Lastly, if m1 6 = m2 then, in the limit, do the masses have the same kinetic energy?",
   "evidence": "The AIM source is *Open Problems and Questions: Stochastic Methods for Non-Equilibrium Dynamical Systems*, notes by Ben Webb. Its Problem 2 reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0046": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 3. Suppose we are given two expanding maps Tβi: X → X for i = 1, 2 on \n\nX = [0, 1]. If these are randomly composed then, in this setting, there is a central limit theorem. Given an observable φ: X → R, can one show in the annealed dynamics that, for almost every P ∈ X × Ω, the quenched system has a central limit theorem?",
   "clean_statement": null,
   "public_statement": "Problem 3. Suppose we are given two expanding maps Tβi: X → X for i = 1, 2 on\n\nX = [0, 1]. If these are randomly composed then, in this setting, there is a central limit theorem. Given an observable φ: X → R, can one show in the annealed dynamics that, for almost every P ∈ X × Ω, the quenched system has a central limit theorem?",
   "evidence": "Nearby context matters. Page 2 contains Problem 14: “Quenched central limit theorem (CTL): Are the normalizing constants and variance almost surely the same?” This confirms that equality of centering and variance between annealed and quenched laws was an intended issue.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0047": {
   "statement_status": "exact",
   "original_statement": "Problem 4. For a discrete-time Lorentz gas with a finite horizon, is there always a measure of maximal entropy?",
   "clean_statement": "Problem 4. For a discrete-time Lorentz gas with a finite horizon, is there always a measure of maximal entropy?",
   "public_statement": "Problem 4. For a discrete-time Lorentz gas with a finite horizon, is there always a measure of maximal entropy?",
   "evidence": "The AIM PDF *Open Problems and Questions: Stochastic Methods for Non-Equilibrium Dynamical Systems*, notes by Ben Webb, says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0048": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 5. For a time dependent non-stationary process, is it possible to use coupling to study the statistical properties of this process?",
   "clean_statement": null,
   "public_statement": "Problem 5. For a time dependent non-stationary process, is it possible to use coupling to study the statistical properties of this process?",
   "evidence": "This wording is exact, including “time dependent” without a hyphen. The prompt does not specify a model, a coupling, or a statistical property, so it has no universal yes/no mathematical interpretation. Plausible readings include:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0049": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 6. Can we find Lasota-York type inequalities for a composition of oper-ators?",
   "clean_statement": null,
   "public_statement": "Problem 6. Can we find Lasota-York type inequalities for a composition of oper-ators?",
   "evidence": "The canonical record, preserved exactly, is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0050": {
   "statement_status": "exact",
   "original_statement": "Problem 7. Do stable laws exist for a non-stationary process? Also, is it possible to find a diffusion rate if there are is no limiting law?",
   "clean_statement": "Problem 7. Do stable laws exist for a non-stationary process? Also, is it possible to find a diffusion rate if there are is no limiting law?",
   "public_statement": "Problem 7. Do stable laws exist for a non-stationary process? Also, is it possible to find a diffusion rate if there are is no limiting law?",
   "evidence": "The primary three-page AIM PDF was inspected. The phrase “there are is” occurs in the PDF itself; it is not an extraction error. The minimally edited reading used below is:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0051": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 8. Is there an annealed almost sure invariant principle (asip) with rate \n\nn1/4 log n instead of n1/4+ δ?Suppose μ is an invariant measure of the system yn+1 = T y n. For \u000f > 0, how does the dynamics of the system \n\nx\u000f,y n+1 = x\u000f,y n + \u000ff (x\u000f,y n, y n)\n\n> 12\n\ncompare with the dynamics of the system \n\n¯xn+1 = ¯ xn + \u000f\n\n∫\n\nf (¯ xn, y n)dμ (y)? \n\nMoreover, what can be said about the quantity sup n< 1/\u000f |¯xn − x\u000f,y n |, specifically with respect to limit theorems?",
   "clean_statement": null,
   "public_statement": "Problem 8. Is there an annealed almost sure invariant principle (asip) with rate\n\nn1/4 log n instead of n1/4+ δ?Suppose μ is an invariant measure of the system yn+1 = T y n. For [U+000F] > 0, how does the dynamics of the system\n\nx[U+000F],y n+1 = x[U+000F],y n + [U+000F]f (x[U+000F],y n, y n)\n\n> 12\n\ncompare with the dynamics of the system\n\n¯xn+1 = ¯ xn + [U+000F]\n\n∫\n\nf (¯ xn, y n)dμ (y)?\n\nMoreover, what can be said about the quantity sup n< 1/[U+000F] |¯xn − x[U+000F],y n |, specifically with respect to limit theorems?",
   "evidence": "This is Problem 8 from the AIM workshop list *Stochastic methods for non-equilibrium dynamical systems*. The exact corpus field is preserved in `input.json`. It contains OCR damage: `n1/4`, `n1/4+ δ`, and the control character `\\u000f` denote mathematical superscripts and \\(\\epsilon\\), while the isolated text `> 12` is the page-number transition from page 1 to page 2, not a mathematical inequality.",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0052": {
   "statement_status": "exact",
   "original_statement": "Problem 9. Can one either determine or improve any estimates on the rate of convergence to the Poisson limit law in slowly mixing systems? For instance, from say a decay rate of 1/n 4 to 1/n.",
   "clean_statement": "Problem 9. Can one either determine or improve any estimates on the rate of convergence to the Poisson limit law in slowly mixing systems? For instance, from say a decay rate of 1/n 4 to 1/n.",
   "public_statement": "Problem 9. Can one either determine or improve any estimates on the rate of convergence to the Poisson limit law in slowly mixing systems? For instance, from say a decay rate of 1/n 4 to 1/n.",
   "evidence": "The canonical record is preserved exactly as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0053": {
   "statement_status": "exact",
   "original_statement": "Problem 10. What can be said about return times for coupled map lattices in the infinite case? Similarly, what can be said about almost sure invariant principles for such maps? That is, how fast does the signal go to infinity in such systems? Also, what the case of finite verses infinite coupling?",
   "clean_statement": "Problem 10. What can be said about return times for coupled map lattices in the infinite case? Similarly, what can be said about almost sure invariant principles for such maps? That is, how fast does the signal go to infinity in such systems? Also, what the case of finite verses infinite coupling?",
   "public_statement": "Problem 10. What can be said about return times for coupled map lattices in the infinite case? Similarly, what can be said about almost sure invariant principles for such maps? That is, how fast does the signal go to infinity in such systems? Also, what the case of finite verses infinite coupling?",
   "evidence": "The canonical record is Problem 10 of the AIM workshop notes *Stochastic Methods for Non-Equilibrium Dynamical Systems*. The exact source record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0054": {
   "statement_status": "exact",
   "original_statement": "Problem 11. Is there some limiting distribution and/or escape rate in the case that can be found in the case of eclipsing scatterers? \n\nOpen Problems and Questions: Tuesday",
   "clean_statement": "Problem 11. Is there some limiting distribution and/or escape rate in the case that can be found in the case of eclipsing scatterers?\n\nOpen Problems and Questions: Tuesday",
   "public_statement": "Problem 11. Is there some limiting distribution and/or escape rate in the case that can be found in the case of eclipsing scatterers?\n\nOpen Problems and Questions: Tuesday",
   "evidence": "The exact corpus field, preserved in input.json, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0055": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 12. Calculate transport coefficients or related physical quantities for a variety of billard systems.",
   "clean_statement": null,
   "public_statement": "Problem 12. Calculate transport coefficients or related physical quantities for a variety of billard systems.",
   "evidence": "The official AIM PDF itself prints “billard”; the spelling is therefore part of the raw source rather than an error introduced by corpus extraction. It is almost certainly a source misspelling of the standard English mathematical term “billiard.” The raw wording is changed nowhere above; only the reconstructed analysis below uses “billiard.”",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 54,
   "attempt": 2
  },
  "AIM-DYNAMICAL_SYSTEMS-0056": {
   "statement_status": "exact",
   "original_statement": "Problem 13. Suppose the deterministic system of coupled ODEs \n\n˙x = f (x, y )˙y = g(x, y )\n\nhas a stable fixed point, at which the associated linear system has two complex eigenvalues. Then the linearized version of this system about this fixed point with a stochastic term [ dx dy \n\n]\n\n= A\n\n[ xy\n\n]\n\ndt +\n\n[ σ1dW 1\n\nσ2dW 2\n\n]\n\ncan be approximated by a constant times a rotation times pair of standard indepen-dent Ornstein-Uhlenbeck processes. What is the analogue of this result in the context of a discrete time reaction diffusion equation? Also, what is the connection of this system to more general fast/slow systems?",
   "clean_statement": "Problem 13. Suppose the deterministic system of coupled ODEs\n\n˙x = f (x, y )˙y = g(x, y )\n\nhas a stable fixed point, at which the associated linear system has two complex eigenvalues. Then the linearized version of this system about this fixed point with a stochastic term [ dx dy\n\n]\n\n= A\n\n[ xy\n\n]\n\ndt +\n\n[ σ1dW 1\n\nσ2dW 2\n\n]\n\ncan be approximated by a constant times a rotation times pair of standard indepen-dent Ornstein-Uhlenbeck processes. What is the analogue of this result in the context of a discrete time reaction diffusion equation? Also, what is the connection of this system to more general fast/slow systems?",
   "public_statement": "Problem 13. Suppose the deterministic system of coupled ODEs\n\n˙x = f (x, y )˙y = g(x, y )\n\nhas a stable fixed point, at which the associated linear system has two complex eigenvalues. Then the linearized version of this system about this fixed point with a stochastic term [ dx dy\n\n]\n\n= A\n\n[ xy\n\n]\n\ndt +\n\n[ σ1dW 1\n\nσ2dW 2\n\n]\n\ncan be approximated by a constant times a rotation times pair of standard indepen-dent Ornstein-Uhlenbeck processes. What is the analogue of this result in the context of a discrete time reaction diffusion equation? Also, what is the connection of this system to more general fast/slow systems?",
   "evidence": "The canonical record is Problem 13 of the AIM workshop notes *Stochastic Methods for Non-Equilibrium Dynamical Systems*. Its PDF source was inspected directly, including the mathematical content stream on page 2. The damaged extraction recovers as",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0057": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 14. Quenched central limit theorem (CTL): Are the normalizing con-stants and variance almost surely the same?",
   "clean_statement": null,
   "public_statement": "Problem 14. Quenched central limit theorem (CTL): Are the normalizing con-stants and variance almost surely the same?",
   "evidence": "The canonical record is Problem 14 in the AIM list *Stochastic methods for non-equilibrium dynamical systems*:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0058": {
   "statement_status": "exact",
   "original_statement": "Problem 15. Can one use generalized function spaces to get martingales for con-catenations of random hyperbolic maps?",
   "clean_statement": "Problem 15. Can one use generalized function spaces to get martingales for con-catenations of random hyperbolic maps?",
   "public_statement": "Problem 15. Can one use generalized function spaces to get martingales for con-catenations of random hyperbolic maps?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0059": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 16. What kind of limit theorems can be found for random billards with moving/deforming scatterers?",
   "clean_statement": "**Problem 16.** What kind of limit theorems can be found for random billiards with moving/deforming scatterers?",
   "public_statement": "Problem 16. What kind of limit theorems can be found for random billards with moving/deforming scatterers?",
   "evidence": "The word `billards` is not an OCR invention. Inspection of the original PDF content stream shows the typeset glyph groups corresponding to `bil` + `lar` + `ds` on a single line; the same spelling occurs repeatedly elsewhere in the document. Since the standard mathematical term is *billiards*, and both the paper titles and the AIM workshop report use that terminology, the conservative reconstruction is:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0060": {
   "statement_status": "exact",
   "original_statement": "Problem 17. What new statistical properties can be found for slowly mixing sys-tems.",
   "clean_statement": "Problem 17. What new statistical properties can be found for slowly mixing sys-tems.",
   "public_statement": "Problem 17. What new statistical properties can be found for slowly mixing sys-tems.",
   "evidence": "The source is Problem 17 in the AIM workshop list *Stochastic methods for non-equilibrium dynamical systems*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0061": {
   "statement_status": "exact",
   "original_statement": "Problem 18. What can be said about spectral properties/Lyapunov spectrum in sequential or nonstationary settings? 3",
   "clean_statement": "Problem 18. What can be said about spectral properties/Lyapunov spectrum in sequential or nonstationary settings? 3",
   "public_statement": "Problem 18. What can be said about spectral properties/Lyapunov spectrum in sequential or nonstationary settings? 3",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0062": {
   "statement_status": "exact",
   "original_statement": "Problem 19. What can be said about return times in coupled map lattices (CML) in the infinite case? What about related problems regarding almost sure invariance principles? \n\nOpen Problems and Questions: Wednesday",
   "clean_statement": "Problem 19. What can be said about return times in coupled map lattices (CML) in the infinite case? What about related problems regarding almost sure invariance principles?\n\nOpen Problems and Questions: Wednesday",
   "public_statement": "Problem 19. What can be said about return times in coupled map lattices (CML) in the infinite case? What about related problems regarding almost sure invariance principles?\n\nOpen Problems and Questions: Wednesday",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0063": {
   "statement_status": "exact",
   "original_statement": "Problem 20. Calculate transport coefficients or related physical quantities for a variety of billard systems.",
   "clean_statement": "Problem 20. Calculate transport coefficients or related physical quantities for a variety of billard systems.",
   "public_statement": "Problem 20. Calculate transport coefficients or related physical quantities for a variety of billard systems.",
   "evidence": "The source is the AIM list *Stochastic methods for non-equilibrium dynamical systems*. The PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0064": {
   "statement_status": "exact",
   "original_statement": "Problem 21. Can one use function spaces to get martingale decompositions and concatenations for some class of hyperbolic maps?",
   "clean_statement": "Problem 21. Can one use function spaces to get martingale decompositions and concatenations for some class of hyperbolic maps?",
   "public_statement": "Problem 21. Can one use function spaces to get martingale decompositions and concatenations for some class of hyperbolic maps?",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0065": {
   "statement_status": "exact",
   "original_statement": "Problem 22. What kind of limit theorems can be found for random billards with moving/deforming scatterers?",
   "clean_statement": "Problem 22. What kind of limit theorems can be found for random billards with moving/deforming scatterers?",
   "public_statement": "Problem 22. What kind of limit theorems can be found for random billards with moving/deforming scatterers?",
   "evidence": "The canonical record is source index 64 of `aim-dynamical-systems-notes.json`, from the AIM workshop *Stochastic methods for non-equilibrium dynamical systems*. Its exact extracted statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0066": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 23. What type of limiting theorems/statistical properties can be found for slowly mixing systems.",
   "clean_statement": null,
   "public_statement": "Problem 23. What type of limiting theorems/statistical properties can be found for slowly mixing systems.",
   "evidence": "The source PDF for the AIM workshop *Stochastic methods for non-equilibrium dynamical systems* reads:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0067": {
   "statement_status": "exact",
   "original_statement": "Problem 24. What can be said about spectral properties/Lyapunov spectrum in sequential or nonstationary settings?",
   "clean_statement": "Problem 24. What can be said about spectral properties/Lyapunov spectrum in sequential or nonstationary settings?",
   "public_statement": "Problem 24. What can be said about spectral properties/Lyapunov spectrum in sequential or nonstationary settings?",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0068": {
   "statement_status": "exact",
   "original_statement": "Problem 25. What can be said about hitting times/EVT to shrinking targets in billards where the target is not a ball but a strip induced by a set in configuration space? \n\nOpen Problems and Questions: Thursday",
   "clean_statement": "Problem 25. What can be said about hitting times/EVT to shrinking targets in billards where the target is not a ball but a strip induced by a set in configuration space?\n\nOpen Problems and Questions: Thursday",
   "public_statement": "Problem 25. What can be said about hitting times/EVT to shrinking targets in billards where the target is not a ball but a strip induced by a set in configuration space?\n\nOpen Problems and Questions: Thursday",
   "evidence": "The canonical record is source index 67 of `aim-dynamical-systems-notes.json`, from the AIM workshop *Stochastic methods for non-equilibrium dynamical systems*. The exact stored `problem` field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0069": {
   "statement_status": "exact",
   "original_statement": "Problem 26. Calculate transport coefficients or related physical quantities for a variety of billard systems.",
   "clean_statement": "Problem 26. Calculate transport coefficients or related physical quantities for a variety of billard systems.",
   "public_statement": "Problem 26. Calculate transport coefficients or related physical quantities for a variety of billard systems.",
   "evidence": "The spelling **“billard” occurs in the PDF** and is not an extraction error. The standard English spelling “billiard” is used below except in the exact quotation. Problem 26 is the first problem following the heading “Open Problems and Questions: Thursday”; Problem 25 concerns shrinking strip targets in billiards, and Problem 27 concerns martingale decompositions.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0070": {
   "statement_status": "exact",
   "original_statement": "Problem 27. Can one use function spaces to get martingale decompositions and concatenations for some class of hyperbolic maps?",
   "clean_statement": "Problem 27. Can one use function spaces to get martingale decompositions and concatenations for some class of hyperbolic maps?",
   "public_statement": "Problem 27. Can one use function spaces to get martingale decompositions and concatenations for some class of hyperbolic maps?",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 69,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0071": {
   "statement_status": "exact",
   "original_statement": "Problem 28. Can one prove existence and/or physical properties of a measure of maximal entropy for dispersing billards? Connections to periodic orbits?",
   "clean_statement": "Problem 28. Can one prove existence and/or physical properties of a measure of maximal entropy for dispersing billards? Connections to periodic orbits?",
   "public_statement": "Problem 28. Can one prove existence and/or physical properties of a measure of maximal entropy for dispersing billards? Connections to periodic orbits?",
   "evidence": "The official AIM workshop problem list states, with its original spelling:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 70,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0072": {
   "statement_status": "exact",
   "original_statement": "Problem 29. How can one bound rates of convergence, using the Wasserstein distance, for Markov generators of the type, \n\nLA (x1,..., x n) = ∑\n\n> i\n\n∫\n\nωi(x, dy )[ A(y) − A(x)],\n\nwhich involve only a finite number of x′s and y′s near i.",
   "clean_statement": "Problem 29. How can one bound rates of convergence, using the Wasserstein distance, for Markov generators of the type,\n\nLA (x1,..., x n) = ∑\n\n> i\n\n∫\n\nωi(x, dy )[ A(y) − A(x)],\n\nwhich involve only a finite number of x′s and y′s near i.",
   "public_statement": "Problem 29. How can one bound rates of convergence, using the Wasserstein distance, for Markov generators of the type,\n\nLA (x1,..., x n) = ∑\n\n> i\n\n∫\n\nωi(x, dy )[ A(y) − A(x)],\n\nwhich involve only a finite number of x′s and y′s near i.",
   "evidence": "The canonical JSON record must be preserved as extracted:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0073": {
   "statement_status": "exact",
   "original_statement": "Problem 30. What type of limiting theorems/statistical properties can be found for slowly mixing systems?",
   "clean_statement": "Problem 30. What type of limiting theorems/statistical properties can be found for slowly mixing systems?",
   "public_statement": "Problem 30. What type of limiting theorems/statistical properties can be found for slowly mixing systems?",
   "evidence": "The canonical record is Problem 30 from the AIM workshop *Stochastic methods for non-equilibrium dynamical systems*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 72,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0074": {
   "statement_status": "exact",
   "original_statement": "Problem 31. What can be said about spectral properties/Lyapunov spectrum in sequential or nonstationary settings?",
   "clean_statement": "Problem 31. What can be said about spectral properties/Lyapunov spectrum in sequential or nonstationary settings?",
   "public_statement": "Problem 31. What can be said about spectral properties/Lyapunov spectrum in sequential or nonstationary settings?",
   "evidence": "The official AIM problem list states exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 73,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0075": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 32. Consider the following system of equations with constraint \n\n˙x = a(x, y ) + 1\n\n\u000f b(x)v(y)˙y = 1\n\n\u000f2 g(y),\n\n∫\n\nv(y)du (y) = 0.\n\nAssuming multiple correlations, at some given rate for μ, can one show \n\nx\u000f →ω X where dX = a(x)dt + b(x) ∗ dω?",
   "clean_statement": null,
   "public_statement": "Problem 32. Consider the following system of equations with constraint\n\n˙x = a(x, y ) + 1\n\n[U+000F] b(x)v(y)˙y = 1\n\n[U+000F]2 g(y),\n\n∫\n\nv(y)du (y) = 0.\n\nAssuming multiple correlations, at some given rate for μ, can one show\n\nx[U+000F] →ω X where dX = a(x)dt + b(x) ∗ dω?",
   "evidence": "The canonical JSON is corrupted by a control character at each occurrence of the scale parameter. Text extraction from the official three-page AIM PDF recovers Problem 32 as",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0076": {
   "statement_status": "exact",
   "original_statement": "Moduli height\n\nLet $X$ be a quasi-projective variety and $G$ a group action on $X$. Let $Y=X//G$ be a geometric quotient of $X$ by $G$. Is $h_Y(x) \\asymp \\min_{g \\in G} h_X(gx)$? (ample heights)\n\n In particular, fix an embedding $M_d \\subset \\mathbb{P}^M$. Then define a height $h_M:M_d \\to \\R$ using the embedding. Is this height function comparable to the minimum height of the maps in the class in $\\mathop{Rat}_d$?",
   "clean_statement": "Moduli height\n\nLet $X$ be a quasi-projective variety and $G$ a group action on $X$. Let $Y=X//G$ be a geometric quotient of $X$ by $G$. Is $h_Y(x) \\asymp \\min_{g \\in G} h_X(gx)$? (ample heights)\n\n In particular, fix an embedding $M_d \\subset \\mathbb{P}^M$. Then define a height $h_M:M_d \\to \\R$ using the embedding. Is this height function comparable to the minimum height of the maps in the class in $\\mathop{Rat}_d$?",
   "public_statement": "Moduli height\n\nLet $X$ be a quasi-projective variety and $G$ a group action on $X$. Let $Y=X//G$ be a geometric quotient of $X$ by $G$. Is $h_Y(x) \\asymp \\min_{g \\in G} h_X(gx)$? (ample heights)\n\n In particular, fix an embedding $M_d \\subset \\mathbb{P}^M$. Then define a height $h_M:M_d \\to \\R$ using the embedding. Is this height function comparable to the minimum height of the maps in the class in $\\mathop{Rat}_d$?",
   "evidence": "The canonical AIM record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0077": {
   "statement_status": "exact",
   "original_statement": "Critical height conjecture\n\nLet $f:\\mathbb{P}^1 \\to \\mathbb{P}^1$, not Latt\\`es. Define the critical height\n \\begin{equation*}\n \\hat{h}_{crit}(f) = \\sum_{c \\in \\text{crit}(f)} \\hat{h}(c).\n \\end{equation*}\n Does there exist $c_1,c_2 > 0$ depending on $d,h_M$ such that for all $f \\in M_d(\\bar{\\mathbb{Q}}) - \\{\\text{Latt\\`es}\\}$ such that\n \\begin{equation*}\n c_1h_M(f) - c_2 \\leq \\hat{h}_{crit}(f)?\n \\end{equation*}",
   "clean_statement": "Critical height conjecture\n\nLet $f:\\mathbb{P}^1 \\to \\mathbb{P}^1$, not Latt\\`es. Define the critical height\n \\begin{equation*}\n \\hat{h}_{crit}(f) = \\sum_{c \\in \\text{crit}(f)} \\hat{h}(c).\n \\end{equation*}\n Does there exist $c_1,c_2 > 0$ depending on $d,h_M$ such that for all $f \\in M_d(\\bar{\\mathbb{Q}}) - \\{\\text{Latt\\`es}\\}$ such that\n \\begin{equation*}\n c_1h_M(f) - c_2 \\leq \\hat{h}_{crit}(f)?\n \\end{equation*}",
   "public_statement": "Critical height conjecture\n\nLet $f:\\mathbb{P}^1 \\to \\mathbb{P}^1$, not Latt\\`es. Define the critical height\n \\begin{equation*}\n \\hat{h}_{crit}(f) = \\sum_{c \\in \\text{crit}(f)} \\hat{h}(c).\n \\end{equation*}\n Does there exist $c_1,c_2 > 0$ depending on $d,h_M$ such that for all $f \\in M_d(\\bar{\\mathbb{Q}}) - \\{\\text{Latt\\`es}\\}$ such that\n \\begin{equation*}\n c_1h_M(f) - c_2 \\leq \\hat{h}_{crit}(f)?\n \\end{equation*}",
   "evidence": "The canonical AIM record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0078": {
   "statement_status": "exact",
   "original_statement": "Equidistribution Problems\n\n\\begin{enumerate}\n \\item $f:\\mathbb{P}^1 \\to \\mathbb{P}^1$. Is the set PCF equidistributed on $M_d$ with respect to the bifurcation measure.\n \\item Same question for $f:\\mathbb{P}^N \\to \\mathbb{P}^N$ with respect to \\emph{what} measure?\n \\end{enumerate}",
   "clean_statement": "Equidistribution Problems\n\n\\begin{enumerate}\n \\item $f:\\mathbb{P}^1 \\to \\mathbb{P}^1$. Is the set PCF equidistributed on $M_d$ with respect to the bifurcation measure.\n \\item Same question for $f:\\mathbb{P}^N \\to \\mathbb{P}^N$ with respect to \\emph{what} measure?\n \\end{enumerate}",
   "public_statement": "Equidistribution Problems\n\n\\begin{enumerate}\n \\item $f:\\mathbb{P}^1 \\to \\mathbb{P}^1$. Is the set PCF equidistributed on $M_d$ with respect to the bifurcation measure.\n \\item Same question for $f:\\mathbb{P}^N \\to \\mathbb{P}^N$ with respect to \\emph{what} measure?\n \\end{enumerate}",
   "evidence": "The canonical record is AIM Problem List item 2.1, in the section “Moduli Problems” from the workshop *Postcritically finite maps in complex and arithmetic dynamics*. Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0079": {
   "statement_status": "exact",
   "original_statement": "Critical orbit relations\n\nAn algebraic subvariety in $M_d$ has a Zariski dense subset of PCF maps if and only if that subvariety is defined by critical orbit relations.\n\n \\begin{enumerate}\n \\item Let $X \\subset \\mathbb{A}^2$ be a curve. Is it true that you have infinitely many points $(c_1,c_2) \\in X$ such that $z^2+c_1$ and $z^2+c_2$ are both PCF if and only if it is vertical line, a horizontal line, or the diagonal?\n \\end{enumerate}",
   "clean_statement": "Critical orbit relations\n\nAn algebraic subvariety in $M_d$ has a Zariski dense subset of PCF maps if and only if that subvariety is defined by critical orbit relations.\n\n \\begin{enumerate}\n \\item Let $X \\subset \\mathbb{A}^2$ be a curve. Is it true that you have infinitely many points $(c_1,c_2) \\in X$ such that $z^2+c_1$ and $z^2+c_2$ are both PCF if and only if it is vertical line, a horizontal line, or the diagonal?\n \\end{enumerate}",
   "public_statement": "Critical orbit relations\n\nAn algebraic subvariety in $M_d$ has a Zariski dense subset of PCF maps if and only if that subvariety is defined by critical orbit relations.\n\n \\begin{enumerate}\n \\item Let $X \\subset \\mathbb{A}^2$ be a curve. Is it true that you have infinitely many points $(c_1,c_2) \\in X$ such that $z^2+c_1$ and $z^2+c_2$ are both PCF if and only if it is vertical line, a horizontal line, or the diagonal?\n \\end{enumerate}",
   "evidence": "The canonical record is item 2.2, “Critical orbit relations,” from the AIM problem list for the workshop *Postcritically finite maps in complex and arithmetic dynamics*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0080": {
   "statement_status": "exact",
   "original_statement": "PCF locus\n\n\\begin{enumerate}\n \\item How big is the PCF locus in $M_d^N$?\n \\item Given a ``critical portrait'' when is the set of PCF maps with that portrait zero dimensional? (what are the obvious counter examples?)\n \\end{enumerate}",
   "clean_statement": "PCF locus\n\n\\begin{enumerate}\n \\item How big is the PCF locus in $M_d^N$?\n \\item Given a ``critical portrait'' when is the set of PCF maps with that portrait zero dimensional? (what are the obvious counter examples?)\n \\end{enumerate}",
   "public_statement": "PCF locus\n\n\\begin{enumerate}\n \\item How big is the PCF locus in $M_d^N$?\n \\item Given a ``critical portrait'' when is the set of PCF maps with that portrait zero dimensional? (what are the obvious counter examples?)\n \\end{enumerate}",
   "evidence": "The exact record in aim-dynamical-systems-notes.json, index 79, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0081": {
   "statement_status": "exact",
   "original_statement": "$p$-adic PCF locus\n\nIs $M_d(\\Q_p) \\cap PCF$ (or polynomials $P_d(\\Q_p) \\cap PCF$) infinite for all $p$? (maybe no for $p > d$, but what about $p < d$?) What are the accumulation points?",
   "clean_statement": "$p$-adic PCF locus\n\nIs $M_d(\\Q_p) \\cap PCF$ (or polynomials $P_d(\\Q_p) \\cap PCF$) infinite for all $p$? (maybe no for $p > d$, but what about $p < d$?) What are the accumulation points?",
   "public_statement": "$p$-adic PCF locus\n\nIs $M_d(\\Q_p) \\cap PCF$ (or polynomials $P_d(\\Q_p) \\cap PCF$) infinite for all $p$? (maybe no for $p > d$, but what about $p < d$?) What are the accumulation points?",
   "evidence": "The canonical record is item 2.4, “$p$-adic PCF locus,” in the Moduli Problems section of the AIM problem list for *Postcritically finite maps in complex and arithmetic dynamics*. The exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0082": {
   "statement_status": "exact",
   "original_statement": "field of definition vs field of moduli\n\nAre all PCF maps defined over their field of moduli?",
   "clean_statement": "field of definition vs field of moduli\n\nAre all PCF maps defined over their field of moduli?",
   "public_statement": "field of definition vs field of moduli\n\nAre all PCF maps defined over their field of moduli?",
   "evidence": "The canonical record is **aim-dynamical-systems-notes.json**, zero-based index 81, from the AIM workshop *Postcritically finite maps in complex and arithmetic dynamics*, section “Moduli Problems,” problem 2.6. Its exact mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0083": {
   "statement_status": "exact",
   "original_statement": "Classification of moduli space\n\nFix $d$. If $n >>_d 1$, then is $M_d(n)$ of general type, where $M_d(n)$ marks periodic points of formal period $n$.",
   "clean_statement": "Classification of moduli space\n\nFix $d$. If $n >>_d 1$, then is $M_d(n)$ of general type, where $M_d(n)$ marks periodic points of formal period $n$.",
   "public_statement": "Classification of moduli space\n\nFix $d$. If $n >>_d 1$, then is $M_d(n)$ of general type, where $M_d(n)$ marks periodic points of formal period $n$.",
   "evidence": "The canonical record is item 2.7, “Classification of moduli space,” in the Moduli Problems section of the AIM list *Postcritically finite maps in complex and arithmetic dynamics*. Its exact wording is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 82,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0084": {
   "statement_status": "exact",
   "original_statement": "regular polynomial endomorphisms\n\nFix $d,N$. $f:\\mathbb{P}^N \\to \\mathbb{P}^N$ is a regular polynomial endomorphism if it leaves a hyperplane invariant, call the space of them $R_d^N$. If we restrict each $f$ to its hyperplane it gives a projection $\\pi: R_d^N \\to M_d^{N-1}$. In any given fiber of $\\pi$ is there a curve of PCF maps? (Known: Ingram-There are horizontal families, i.e. intersect every fiber in finitely many places. There are no PCF maps above the power map).\n\n Alternate formulation: If you have a map from a curve $C \\to R_d^N \\to M_d^{N-1}$ where the map $C \\to M_d^{N-1}$ is constant and $C$ lands in the PCF locus of $R_d^N$. Is the map $C \\to R_d^N$ already constant?",
   "clean_statement": "regular polynomial endomorphisms\n\nFix $d,N$. $f:\\mathbb{P}^N \\to \\mathbb{P}^N$ is a regular polynomial endomorphism if it leaves a hyperplane invariant, call the space of them $R_d^N$. If we restrict each $f$ to its hyperplane it gives a projection $\\pi: R_d^N \\to M_d^{N-1}$. In any given fiber of $\\pi$ is there a curve of PCF maps? (Known: Ingram-There are horizontal families, i.e. intersect every fiber in finitely many places. There are no PCF maps above the power map).\n\n Alternate formulation: If you have a map from a curve $C \\to R_d^N \\to M_d^{N-1}$ where the map $C \\to M_d^{N-1}$ is constant and $C$ lands in the PCF locus of $R_d^N$. Is the map $C \\to R_d^N$ already constant?",
   "public_statement": "regular polynomial endomorphisms\n\nFix $d,N$. $f:\\mathbb{P}^N \\to \\mathbb{P}^N$ is a regular polynomial endomorphism if it leaves a hyperplane invariant, call the space of them $R_d^N$. If we restrict each $f$ to its hyperplane it gives a projection $\\pi: R_d^N \\to M_d^{N-1}$. In any given fiber of $\\pi$ is there a curve of PCF maps? (Known: Ingram-There are horizontal families, i.e. intersect every fiber in finitely many places. There are no PCF maps above the power map).\n\n Alternate formulation: If you have a map from a curve $C \\to R_d^N \\to M_d^{N-1}$ where the map $C \\to M_d^{N-1}$ is constant and $C$ lands in the PCF locus of $R_d^N$. Is the map $C \\to R_d^N$ already constant?",
   "evidence": "The canonical record is **aim-dynamical-systems-notes.json**, zero-based index 83, from the AIM workshop *Postcritically finite maps in complex and arithmetic dynamics*, section “Moduli Problems,” problem 2.5. The live AIM page was retrieved and agrees exactly with the record:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 83,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0085": {
   "statement_status": "exact",
   "original_statement": "Irreducibility of Gleason polynomials\n\nDefine the Gleason polynomials as the polynomials in $c$ such that $f_c(x) = x^2+c$ is post-critically finite.\n\nAre the Gleason polynomials irreducible?",
   "clean_statement": "Irreducibility of Gleason polynomials\n\nDefine the Gleason polynomials as the polynomials in $c$ such that $f_c(x) = x^2+c$ is post-critically finite.\n\nAre the Gleason polynomials irreducible?",
   "public_statement": "Irreducibility of Gleason polynomials\n\nDefine the Gleason polynomials as the polynomials in $c$ such that $f_c(x) = x^2+c$ is post-critically finite.\n\nAre the Gleason polynomials irreducible?",
   "evidence": "The canonical record in aim-dynamical-systems-notes.json, zero-based index 84, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 84,
   "attempt": 2
  },
  "AIM-DYNAMICAL_SYSTEMS-0086": {
   "statement_status": "exact",
   "original_statement": "Algebraic proof of Thurston rigidity\n\nAlgebraic proof of infinitesimal Thurston Rigidity, i.e. if $f$ is PCF non-Latt\\`es and degree at least 2, then $1$ is not an eigenvalue of $f^{\\ast}:H^1(\\mathbb{P}^1,\\Theta_{P_f}) \\to H^1(\\mathbb{P}^1,\\Theta_{P_f})$ where $P_f$ is the post-critical set and $\\Theta$ is the sheaf of holomorphic vector fields with zeros along the post-critical set.",
   "clean_statement": "Algebraic proof of Thurston rigidity\n\nAlgebraic proof of infinitesimal Thurston Rigidity, i.e. if $f$ is PCF non-Latt\\`es and degree at least 2, then $1$ is not an eigenvalue of $f^{\\ast}:H^1(\\mathbb{P}^1,\\Theta_{P_f}) \\to H^1(\\mathbb{P}^1,\\Theta_{P_f})$ where $P_f$ is the post-critical set and $\\Theta$ is the sheaf of holomorphic vector fields with zeros along the post-critical set.",
   "public_statement": "Algebraic proof of Thurston rigidity\n\nAlgebraic proof of infinitesimal Thurston Rigidity, i.e. if $f$ is PCF non-Latt\\`es and degree at least 2, then $1$ is not an eigenvalue of $f^{\\ast}:H^1(\\mathbb{P}^1,\\Theta_{P_f}) \\to H^1(\\mathbb{P}^1,\\Theta_{P_f})$ where $P_f$ is the post-critical set and $\\Theta$ is the sheaf of holomorphic vector fields with zeros along the post-critical set.",
   "evidence": "Here \\[ P_f=\\bigcup_{n\\geq 1} f^n(C_f) \\] is understood as a reduced finite marked set. The live URL in the record, <http://aimpl.org/finitedynamics/3/>, timed out during this run, so the wording above is preserved verbatim from the canonical JSON. No reconstruction of its mathematical content is needed, but the notation $f^*$ requires an important clarification made in Section 3.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 85,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0087": {
   "statement_status": "corrected_verified",
   "original_statement": "Ramfication in pre-image towers\n\nLet $X$ be a projective variety, $f:X/K \\to X/K$ PCF. Let $\\alpha \\in X(K)$. Look at the tower of fields from adding the pull-backs of $\\alpha$\n \\begin{equation*}\n K_n = K(f^{-n}(\\alpha)).\n \\end{equation*}\n\nThere exists a finite set of primes $S$ such that all $K_n$ are unramified outside of $S$.\n\n (Known for $\\mathbb{P}^1$).\n \\begin{enumerate}\n \\item If not PCF and $X=\\mathbb{P}^1$ let $S_n = \\{\\text{primes of $K$ where $K_n/K$ is ramified}\\}$. Is the growth rate of $\\#S_n$ related to some height of $f$, specifically $\\hat{h}_{crit}$?\n \\item Assume conjecture ($X= \\mathbb{P}^1$). Can you arrange it so that all of the ramification in the tower is tame? (even for one example).\n \\end{enumerate}",
   "clean_statement": "Ramification in preimage towers, in pre-image towers\n\nLet $X$ be a projective variety, $f:X/K \\to X/K$ PCF. Let $\\alpha \\in X(K)$. Look at the tower of fields from adding the pull-backs of $\\alpha$\n \\begin{equation*}\n K_n = K(f^{-n}(\\alpha)).\n \\end{equation*}\n\nThere exists a finite set of primes $S$ such that all $K_n$ are unramified outside of $S$.\n\n (Known for $\\mathbb{P}^1$).\n \\begin{enumerate}\n \\item If not PCF and $X=\\mathbb{P}^1$ let $S_n = \\{\\text{primes of $K$ where $K_n/K$ is ramified}\\}$. Is the growth rate of $\\#S_n$ related to some height of $f$, specifically $\\hat{h}_{crit}$?\n \\item Assume conjecture ($X= \\mathbb{P}^1$). Can you arrange it so that all of the ramification in the tower is tame? (even for one example).\n \\end{enumerate}",
   "public_statement": "Ramification in preimage towers, in pre-image towers\n\nLet $X$ be a projective variety, $f:X/K \\to X/K$ PCF. Let $\\alpha \\in X(K)$. Look at the tower of fields from adding the pull-backs of $\\alpha$\n \\begin{equation*}\n K_n = K(f^{-n}(\\alpha)).\n \\end{equation*}\n\nThere exists a finite set of primes $S$ such that all $K_n$ are unramified outside of $S$.\n\n (Known for $\\mathbb{P}^1$).\n \\begin{enumerate}\n \\item If not PCF and $X=\\mathbb{P}^1$ let $S_n = \\{\\text{primes of $K$ where $K_n/K$ is ramified}\\}$. Is the growth rate of $\\#S_n$ related to some height of $f$, specifically $\\hat{h}_{crit}$?\n \\item Assume conjecture ($X= \\mathbb{P}^1$). Can you arrange it so that all of the ramification in the tower is tame? (even for one example).\n \\end{enumerate}",
   "evidence": "The canonical record is aim-dynamical-systems-notes.json, zero-based index 86, from the AIM workshop *Postcritically finite maps in complex and arithmetic dynamics*, section “Galois Problems,” Conjecture 4.1. The live AIM HTML was checked on 2 August 2026 and has exactly the same heading and body. In particular, the misspelling “Ramfication” is present on the live page and is not an extraction error. The canonical text is: The corrected English title is “Ramification in preimage towers,” but the source data is not altered.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 86,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0088": {
   "statement_status": "exact",
   "original_statement": "Detecting PCF maps\n\nLet $f$ be a PCF rational function defined over a number field. Is it ``algebraically detectable'' from $K_n = K(\\mathop{Per}_n(f))$ or the tower of such extensions that $f$ is PCF. Galois groups? Ramification?",
   "clean_statement": "Detecting PCF maps\n\nLet $f$ be a PCF rational function defined over a number field. Is it ``algebraically detectable'' from $K_n = K(\\mathop{Per}_n(f))$ or the tower of such extensions that $f$ is PCF. Galois groups? Ramification?",
   "public_statement": "Detecting PCF maps\n\nLet $f$ be a PCF rational function defined over a number field. Is it ``algebraically detectable'' from $K_n = K(\\mathop{Per}_n(f))$ or the tower of such extensions that $f$ is PCF. Galois groups? Ramification?",
   "evidence": "The canonical record in aim-dynamical-systems-notes.json, zero-based index 87, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 87,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0089": {
   "statement_status": "exact",
   "original_statement": "inertia groups for preimage towers\n\n\\begin{enumerate}\n \\item What can you say about inertia subgroups of Galois groups in towers of fields generated by $f^{-n}(\\alpha)$ for $f$ PCF?\n \\item What happens to decomposition groups?\n \\end{enumerate}",
   "clean_statement": "inertia groups for preimage towers\n\n\\begin{enumerate}\n \\item What can you say about inertia subgroups of Galois groups in towers of fields generated by $f^{-n}(\\alpha)$ for $f$ PCF?\n \\item What happens to decomposition groups?\n \\end{enumerate}",
   "public_statement": "inertia groups for preimage towers\n\n\\begin{enumerate}\n \\item What can you say about inertia subgroups of Galois groups in towers of fields generated by $f^{-n}(\\alpha)$ for $f$ PCF?\n \\item What happens to decomposition groups?\n \\end{enumerate}",
   "evidence": "The canonical AIM record, item 4.3 in the section “Galois Problems” of the workshop *Postcritically finite maps in complex and arithmetic dynamics*, reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 88,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0090": {
   "statement_status": "exact",
   "original_statement": "Notions of PCF\n\n$f:\\mathbb{P}^N \\to \\mathbb{P}^N$ morphism.\n \\begin{enumerate}\n \\item Defn: PCF if the orbit of the ramification locus $C$, $V=\\bigcup_{n \\geq 0} f^n(C)$ is a proper algebraic subvariety of $\\mathbb{P}^N$.\n \\item Defn: ``PCF all the way down'': If the restriction of $f$ to every periodic component of the post-critical component $V$ is PCF.\n \\end{enumerate}\n\n\\begin{enumerate}\n \\item Does (1) imply (2)?\n \\item Are PCF (1) or (2) Zariski dense in $M_d^N$?\n \\item Is there are more general definition than (\\ref{it1}) such that:\n\na) ``there are more of them''? (i.e. less restrictive)\n\nb) they are dense in any reasonable topology\n\nc) Andre-Oort property\n\n\\end{enumerate}",
   "clean_statement": "Notions of PCF\n\n$f:\\mathbb{P}^N \\to \\mathbb{P}^N$ morphism.\n \\begin{enumerate}\n \\item Defn: PCF if the orbit of the ramification locus $C$, $V=\\bigcup_{n \\geq 0} f^n(C)$ is a proper algebraic subvariety of $\\mathbb{P}^N$.\n \\item Defn: ``PCF all the way down'': If the restriction of $f$ to every periodic component of the post-critical component $V$ is PCF.\n \\end{enumerate}\n\n\\begin{enumerate}\n \\item Does (1) imply (2)?\n \\item Are PCF (1) or (2) Zariski dense in $M_d^N$?\n \\item Is there are more general definition than (\\ref{it1}) such that:\n\na) ``there are more of them''? (i.e. less restrictive)\n\nb) they are dense in any reasonable topology\n\nc) Andre-Oort property\n\n\\end{enumerate}",
   "public_statement": "Notions of PCF\n\n$f:\\mathbb{P}^N \\to \\mathbb{P}^N$ morphism.\n \\begin{enumerate}\n \\item Defn: PCF if the orbit of the ramification locus $C$, $V=\\bigcup_{n \\geq 0} f^n(C)$ is a proper algebraic subvariety of $\\mathbb{P}^N$.\n \\item Defn: ``PCF all the way down'': If the restriction of $f$ to every periodic component of the post-critical component $V$ is PCF.\n \\end{enumerate}\n\n\\begin{enumerate}\n \\item Does (1) imply (2)?\n \\item Are PCF (1) or (2) Zariski dense in $M_d^N$?\n \\item Is there are more general definition than (\\ref{it1}) such that:\n\na) ``there are more of them''? (i.e. less restrictive)\n\nb) they are dense in any reasonable topology\n\nc) Andre-Oort property\n\n\\end{enumerate}",
   "evidence": "The exact canonical record is item 5.1, “Notions of PCF,” in the “Higher dimensions” section of the AIM workshop *Postcritically finite maps in complex and arithmetic dynamics*. It asks, for a morphism \\[ f:\\mathbb P^N\\longrightarrow\\mathbb P^N, \\] whether ordinary PCF implies “PCF all the way down,” whether either class is Zariski dense in \\(M_d^N\\), and whether there is a less restrictive notion that is abundant, dense in a reasonable topology, and has an André–Oort property. The two definitions in the record are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 89,
   "attempt": 2
  },
  "AIM-DYNAMICAL_SYSTEMS-0091": {
   "statement_status": "exact",
   "original_statement": "Thurston Rigidity Exceptions\n\nWhat are the ``obvious'' exception to Thurston Rigidity in $M_d^N$?",
   "clean_statement": "Thurston Rigidity Exceptions\n\nWhat are the ``obvious'' exception to Thurston Rigidity in $M_d^N$?",
   "public_statement": "Thurston Rigidity Exceptions\n\nWhat are the ``obvious'' exception to Thurston Rigidity in $M_d^N$?",
   "evidence": "The canonical AIM Problem List record is item 5.2 in “Higher dimensions” from the workshop *Postcritically finite maps in complex and arithmetic dynamics*. Its exact text, including the singular/plural mismatch, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 90,
   "attempt": 3
  },
  "AIM-DYNAMICAL_SYSTEMS-0092": {
   "statement_status": "exact",
   "original_statement": "Flexible Family of PCF maps\n\nIs there a flexible family of PCF maps that does not come from $\\mathbb{P}^1$.",
   "clean_statement": "Flexible Family of PCF maps\n\nIs there a flexible family of PCF maps that does not come from $\\mathbb{P}^1$.",
   "public_statement": "Flexible Family of PCF maps\n\nIs there a flexible family of PCF maps that does not come from $\\mathbb{P}^1$.",
   "evidence": "The canonical AIM record (workshop *Postcritically finite maps in complex and arithmetic dynamics*, section 5.3, “Higher dimensions”) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 91,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0093": {
   "statement_status": "exact",
   "original_statement": "Entropy\n\n\\begin{enumerate}\n \\item Let $f(z) = z^2 + c$, $c \\in \\R$ and PCF with invariant $[a,b]$ with fixed point at one end. Which algebraic numbers arise as $e^s$ where $s$ is the entropy of $f$?\n \\item More generally, for $z^2+c$ on its Hubbard tree (or higher degree).\n \\item Is there an algebraic analogue of entropy (like dynamical degree vs. arithmetic degree in higher dimensions)?\n \\end{enumerate}",
   "clean_statement": "Entropy\n\n\\begin{enumerate}\n \\item Let $f(z) = z^2 + c$, $c \\in \\R$ and PCF with invariant $[a,b]$ with fixed point at one end. Which algebraic numbers arise as $e^s$ where $s$ is the entropy of $f$?\n \\item More generally, for $z^2+c$ on its Hubbard tree (or higher degree).\n \\item Is there an algebraic analogue of entropy (like dynamical degree vs. arithmetic degree in higher dimensions)?\n \\end{enumerate}",
   "public_statement": "Entropy\n\n\\begin{enumerate}\n \\item Let $f(z) = z^2 + c$, $c \\in \\R$ and PCF with invariant $[a,b]$ with fixed point at one end. Which algebraic numbers arise as $e^s$ where $s$ is the entropy of $f$?\n \\item More generally, for $z^2+c$ on its Hubbard tree (or higher degree).\n \\item Is there an algebraic analogue of entropy (like dynamical degree vs. arithmetic degree in higher dimensions)?\n \\end{enumerate}",
   "evidence": "The canonical AIM record, in the section “Other Questions” of the workshop *Postcritically finite maps in complex and arithmetic dynamics*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0094": {
   "statement_status": "exact",
   "original_statement": "Milnor's Characterisation of Lattes\n\nMilnor's characterization of Latt\\`es with 4 post-critical points\n\n \\begin{itemize}\n \\item no critical point is post-critical\n \\item all critical points are simple\n \\end{itemize}\n\n\\begin{enumerate}\n \\item Does this hold over positive characteristic?\n \\item What do you get from the dual of Frobenius when inseparable?\n \\item What about characteristic 2 and 3 and $\\mathop{Aut}(E)$ nonabelian?\n \\end{enumerate}",
   "clean_statement": "Milnor's Characterisation of Lattes\n\nMilnor's characterization of Latt\\`es with 4 post-critical points\n\n \\begin{itemize}\n \\item no critical point is post-critical\n \\item all critical points are simple\n \\end{itemize}\n\n\\begin{enumerate}\n \\item Does this hold over positive characteristic?\n \\item What do you get from the dual of Frobenius when inseparable?\n \\item What about characteristic 2 and 3 and $\\mathop{Aut}(E)$ nonabelian?\n \\end{enumerate}",
   "public_statement": "Milnor's Characterisation of Lattes\n\nMilnor's characterization of Latt\\`es with 4 post-critical points\n\n \\begin{itemize}\n \\item no critical point is post-critical\n \\item all critical points are simple\n \\end{itemize}\n\n\\begin{enumerate}\n \\item Does this hold over positive characteristic?\n \\item What do you get from the dual of Frobenius when inseparable?\n \\item What about characteristic 2 and 3 and $\\mathop{Aut}(E)$ nonabelian?\n \\end{enumerate}",
   "evidence": "The canonical AIM record is problem 6.2 in the workshop *Postcritically finite maps in complex and arithmetic dynamics*. Its mathematical content is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0095": {
   "statement_status": "exact",
   "original_statement": "portraits of rigid Latt\\`es\n\nWhat portraits occur for the rigid Latt\\`es maps? In particular, are any of them the flexible Latt\\`es portraits?",
   "clean_statement": "portraits of rigid Latt\\`es\n\nWhat portraits occur for the rigid Latt\\`es maps? In particular, are any of them the flexible Latt\\`es portraits?",
   "public_statement": "portraits of rigid Latt\\`es\n\nWhat portraits occur for the rigid Latt\\`es maps? In particular, are any of them the flexible Latt\\`es portraits?",
   "evidence": "The exact canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 94,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0096": {
   "statement_status": "exact",
   "original_statement": "Are there any more conspiracies?\n\nLet $\\mathop{Per}_n(\\lambda)$ be the locus in the moduli space of formal $n$-cycles with multiplier $\\lambda$. $\\mathop{Per}_n(\\lambda)$ is in general an irreducible cubic, $\\\\mathop{Per}_3(1)$ factors. Let $\\mathop{Per}^{\\ast}_n(\\lambda)$ be the same but with actual $n$-cycles instead of formal $n$-cycles. $\\mathop{Per}^{\\ast}_3(1) = \\mathop{Per}_2(-3)$ are the same line in $\\mathbb{A}^2$ (Milnor). $\\mathop{Per}_n(\\lambda) \\cap \\mathop{Per}_m(\\lambda')$ should be zero dimensional (need at least one multiplier outside the unit circle).\n\n Are there any other examples $n,m,\\lambda,\\lambda'$ where the intersection is not zero dimensional?",
   "clean_statement": "Are there any more conspiracies?\n\nLet $\\mathop{Per}_n(\\lambda)$ be the locus in the moduli space of formal $n$-cycles with multiplier $\\lambda$. $\\mathop{Per}_n(\\lambda)$ is in general an irreducible cubic, $\\\\mathop{Per}_3(1)$ factors. Let $\\mathop{Per}^{\\ast}_n(\\lambda)$ be the same but with actual $n$-cycles instead of formal $n$-cycles. $\\mathop{Per}^{\\ast}_3(1) = \\mathop{Per}_2(-3)$ are the same line in $\\mathbb{A}^2$ (Milnor). $\\mathop{Per}_n(\\lambda) \\cap \\mathop{Per}_m(\\lambda')$ should be zero dimensional (need at least one multiplier outside the unit circle).\n\n Are there any other examples $n,m,\\lambda,\\lambda'$ where the intersection is not zero dimensional?",
   "public_statement": "Are there any more conspiracies?\n\nLet $\\mathop{Per}_n(\\lambda)$ be the locus in the moduli space of formal $n$-cycles with multiplier $\\lambda$. $\\mathop{Per}_n(\\lambda)$ is in general an irreducible cubic, $\\\\mathop{Per}_3(1)$ factors. Let $\\mathop{Per}^{\\ast}_n(\\lambda)$ be the same but with actual $n$-cycles instead of formal $n$-cycles. $\\mathop{Per}^{\\ast}_3(1) = \\mathop{Per}_2(-3)$ are the same line in $\\mathbb{A}^2$ (Milnor). $\\mathop{Per}_n(\\lambda) \\cap \\mathop{Per}_m(\\lambda')$ should be zero dimensional (need at least one multiplier outside the unit circle).\n\n Are there any other examples $n,m,\\lambda,\\lambda'$ where the intersection is not zero dimensional?",
   "evidence": "The exact canonical AIM record is titled “Are there any more conspiracies?” and says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0097": {
   "statement_status": "exact",
   "original_statement": "(1) Existence and smoothness of local invariant manifolds at equilibria for neutral differential equations with state-dependent delay \n\nA particular class of equations is given by \n\nx′(t) = A (x′ (t − r(x(t)))) + f (x(t)) with a linear operator A, a delay function r and a smooth nonlinearity f.(2) Extension of \"lap-number\"-techniques to widest possible class of equations with delay \n\nInteresting classes are for instance the following: \n\n• equations with non-monotonic delay \n\n• equations with implicitly defined delay \n\n• equations with multiple delay Additionally, in this context numerical case studies could be useful for analytical considerations. (3) How does a state-dependent delay change the dynamics of an equation in contrast to a constant delay? \n\nHere, analytical as well as numerical studies are missing. Interesting situations are for instance \n\n• scalar equation with one delay such as Wright's equation or Mackey-Glass equa-tion \n\n• scalar equation with two or more delays \n\n• higher order equations with one or more delays (4) Approximation of delay dependence ( x(t + r(x(t))) ) by distributed/integral terms to improve smoothness properties",
   "clean_statement": "(1) Existence and smoothness of local invariant manifolds at equilibria for neutral differential equations with state-dependent delay\n\nA particular class of equations is given by\n\nx′(t) = A (x′ (t − r(x(t)))) + f (x(t)) with a linear operator A, a delay function r and a smooth nonlinearity f.(2) Extension of \"lap-number\"-techniques to widest possible class of equations with delay\n\nInteresting classes are for instance the following:\n\n• equations with non-monotonic delay\n\n• equations with implicitly defined delay\n\n• equations with multiple delay Additionally, in this context numerical case studies could be useful for analytical considerations. (3) How does a state-dependent delay change the dynamics of an equation in contrast to a constant delay?\n\nHere, analytical as well as numerical studies are missing. Interesting situations are for instance\n\n• scalar equation with one delay such as Wright's equation or Mackey-Glass equa-tion\n\n• scalar equation with two or more delays\n\n• higher order equations with one or more delays (4) Approximation of delay dependence ( x(t + r(x(t))) ) by distributed/integral terms to improve smoothness properties",
   "public_statement": "(1) Existence and smoothness of local invariant manifolds at equilibria for neutral differential equations with state-dependent delay\n\nA particular class of equations is given by\n\nx′(t) = A (x′ (t − r(x(t)))) + f (x(t)) with a linear operator A, a delay function r and a smooth nonlinearity f.(2) Extension of \"lap-number\"-techniques to widest possible class of equations with delay\n\nInteresting classes are for instance the following:\n\n• equations with non-monotonic delay\n\n• equations with implicitly defined delay\n\n• equations with multiple delay Additionally, in this context numerical case studies could be useful for analytical considerations. (3) How does a state-dependent delay change the dynamics of an equation in contrast to a constant delay?\n\nHere, analytical as well as numerical studies are missing. Interesting situations are for instance\n\n• scalar equation with one delay such as Wright's equation or Mackey-Glass equa-tion\n\n• scalar equation with two or more delays\n\n• higher order equations with one or more delays (4) Approximation of delay dependence ( x(t + r(x(t))) ) by distributed/integral terms to improve smoothness properties",
   "evidence": "This is record 1 from the AIM workshop *Low dimensional structures in dynamical systems with variable time lags* (June 2010). The linked PDF was checked directly. Its four subquestions, with line wrapping repaired but mathematical signs preserved, are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 96,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0098": {
   "statement_status": "exact",
   "original_statement": "(5) Global bifurcations in differential equations with state-dependent delay \n\nSome related issues are \n\n• continuation for homoclinic or heteroclinic solutions \n\n• continuation of (rapidly oscillating) periodic solutions for all relevant parameters (6) Analyticity of solutions of analytic delay differential equation (especially, with respect to the delay) \n\nIn particular, the \"Whiskey Problem\" suggested by Roger (7) Study of differential equations with unbounded finite delays as for instance arising in echo control 12 LIST OF PROBLEMS \n\nSome particular aspects are \n\n• equations with unbounded state-dependent delay \n\n• equation with delay given by an infinite integral (8) How can a given ODE dynamics be realized in a class of differential equa-tions with delay?",
   "clean_statement": "(5) Global bifurcations in differential equations with state-dependent delay\n\nSome related issues are\n\n• continuation for homoclinic or heteroclinic solutions\n\n• continuation of (rapidly oscillating) periodic solutions for all relevant parameters (6) Analyticity of solutions of analytic delay differential equation (especially, with respect to the delay)\n\nIn particular, the \"Whiskey Problem\" suggested by Roger (7) Study of differential equations with unbounded finite delays as for instance arising in echo control 12 LIST OF PROBLEMS\n\nSome particular aspects are\n\n• equations with unbounded state-dependent delay\n\n• equation with delay given by an infinite integral (8) How can a given ODE dynamics be realized in a class of differential equa-tions with delay?",
   "public_statement": "(5) Global bifurcations in differential equations with state-dependent delay\n\nSome related issues are\n\n• continuation for homoclinic or heteroclinic solutions\n\n• continuation of (rapidly oscillating) periodic solutions for all relevant parameters (6) Analyticity of solutions of analytic delay differential equation (especially, with respect to the delay)\n\nIn particular, the \"Whiskey Problem\" suggested by Roger (7) Study of differential equations with unbounded finite delays as for instance arising in echo control 12 LIST OF PROBLEMS\n\nSome particular aspects are\n\n• equations with unbounded state-dependent delay\n\n• equation with delay given by an infinite integral (8) How can a given ODE dynamics be realized in a class of differential equa-tions with delay?",
   "evidence": "The canonical record is item 97 (zero based) of `aim-dynamical-systems-notes.json`, extracted from the problem list of the June 7--11, 2010 AIM workshop *Low dimensional structures in dynamical systems with variable time lags*. Its literal `problem` field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 97,
   "attempt": 1
  },
  "AIM-DYNAMICAL_SYSTEMS-0099": {
   "statement_status": "exact",
   "original_statement": "(9) Alternative proofs of the existence of infinite dimensional local invariant manifolds at equilibria for differential equations with state-dependent delay \n\nFor instance, there is no result using the graph transform or Lyapunov-Perron technique to prove the existence of local stable manifolds at equilibria for differential equations with state-dependent delay.",
   "clean_statement": "(9) Alternative proofs of the existence of infinite dimensional local invariant manifolds at equilibria for differential equations with state-dependent delay\n\nFor instance, there is no result using the graph transform or Lyapunov-Perron technique to prove the existence of local stable manifolds at equilibria for differential equations with state-dependent delay.",
   "public_statement": "(9) Alternative proofs of the existence of infinite dimensional local invariant manifolds at equilibria for differential equations with state-dependent delay\n\nFor instance, there is no result using the graph transform or Lyapunov-Perron technique to prove the existence of local stable manifolds at equilibria for differential equations with state-dependent delay.",
   "evidence": "The official two-page AIM PDF has exactly this wording on page 2, modulo ordinary line wrapping. There is no substantive OCR error, missing formula, or spillover from a neighboring record. The mathematically standard typography would hyphenate “infinite-dimensional” and “state-dependent,” but those are editorial changes, not corrections to the source.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-dynamical-systems-notes.json",
   "source_index": 98,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0001": {
   "statement_status": "exact",
   "original_statement": "Log-Brunn-Minkowski Inequality\n\nFor convex bodies $K$ and $L$ in $\\R^n$ with support functions $h_K$ and $h_L$ the logarithmic sum $tK +_\\circ (1-t)L$ is defined as\n$$\ntK +_\\circ (1-t)L = \\underset{u \\in S^{n-1}}{\\cap} \\left\\{x \\in \\R^n:\\ \\langle x, u\\rangle \\leq h_K(u)^th_L(u)^{1-t} \\right\\}\n$$\nwhere $h_K(u) = \\max\\limits_{x \\in K} \\langle x, u\\rangle .$\n\nShow that if $K$ and $L$ symmetric convex bodies in $\\R^n$, then for all $0 \\leq t \\leq 1$ the following hods\n$$\n|tK +_\\circ (1-t)L| \\geq |K|^t |L|^{1-t}\n$$\nwith equality if and only if $K = L$ or $K$ and $L$ are parallelograms with parallel sides.",
   "clean_statement": "Log-Brunn-Minkowski Inequality\n\nFor convex bodies $K$ and $L$ in $\\R^n$ with support functions $h_K$ and $h_L$ the logarithmic sum $tK +_\\circ (1-t)L$ is defined as\n$$\ntK +_\\circ (1-t)L = \\underset{u \\in S^{n-1}}{\\cap} \\left\\{x \\in \\R^n:\\ \\langle x, u\\rangle \\leq h_K(u)^th_L(u)^{1-t} \\right\\}\n$$\nwhere $h_K(u) = \\max\\limits_{x \\in K} \\langle x, u\\rangle .$\n\nShow that if $K$ and $L$ symmetric convex bodies in $\\R^n$, then for all $0 \\leq t \\leq 1$ the following hods\n$$\n|tK +_\\circ (1-t)L| \\geq |K|^t |L|^{1-t}\n$$\nwith equality if and only if $K = L$ or $K$ and $L$ are parallelograms with parallel sides.",
   "public_statement": "Log-Brunn-Minkowski Inequality\n\nFor convex bodies $K$ and $L$ in $\\R^n$ with support functions $h_K$ and $h_L$ the logarithmic sum $tK +_\\circ (1-t)L$ is defined as\n$$\ntK +_\\circ (1-t)L = \\underset{u \\in S^{n-1}}{\\cap} \\left\\{x \\in \\R^n:\\ \\langle x, u\\rangle \\leq h_K(u)^th_L(u)^{1-t} \\right\\}\n$$\nwhere $h_K(u) = \\max\\limits_{x \\in K} \\langle x, u\\rangle .$\n\nShow that if $K$ and $L$ symmetric convex bodies in $\\R^n$, then for all $0 \\leq t \\leq 1$ the following hods\n$$\n|tK +_\\circ (1-t)L| \\geq |K|^t |L|^{1-t}\n$$\nwith equality if and only if $K = L$ or $K$ and $L$ are parallelograms with parallel sides.",
   "evidence": "The exact canonical record, Conjecture 1.05 in the AIM list *Symmetry and convexity in geometric inequalities*, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0002": {
   "statement_status": "exact",
   "original_statement": "Log-Minkowski inequality\n\nShow that if $K$ and $L$ symmetric convex bodies in $\\R^n$, then\n$$\n\\int\\limits_{S^{n-1}} \\log{\\frac{h_K(u)}{h_L(u)}}\\,d\\bar{V}_L(u) \\geq \\frac1n \\log{\\frac{|K|}{|L|}},\n$$\nwhere $\\bar{V}_L$ is the cone-volume probability measure of $L$.",
   "clean_statement": "Log-Minkowski inequality\n\nShow that if $K$ and $L$ symmetric convex bodies in $\\R^n$, then\n$$\n\\int\\limits_{S^{n-1}} \\log{\\frac{h_K(u)}{h_L(u)}}\\,d\\bar{V}_L(u) \\geq \\frac1n \\log{\\frac{|K|}{|L|}},\n$$\nwhere $\\bar{V}_L$ is the cone-volume probability measure of $L$.",
   "public_statement": "Log-Minkowski inequality\n\nShow that if $K$ and $L$ symmetric convex bodies in $\\R^n$, then\n$$\n\\int\\limits_{S^{n-1}} \\log{\\frac{h_K(u)}{h_L(u)}}\\,d\\bar{V}_L(u) \\geq \\frac1n \\log{\\frac{|K|}{|L|}},\n$$\nwhere $\\bar{V}_L$ is the cone-volume probability measure of $L$.",
   "evidence": "There is one terminological point that must not be left implicit. Here “symmetric” means **origin-symmetric**, \\(K=-K\\) and \\(L=-L\\), as in the originating paper. Merely being centrally symmetric about unspecified centers is not enough: support functions and cone-volume measures in the displayed formula are tied to the origin. No mathematical sign or normalization correction is needed.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0003": {
   "statement_status": "exact",
   "original_statement": "Cone-volume measure\n\nIf $K$ and $L$ symmetric convex bodies in $\\R^n$ such that $\\bar{V}_K=\\bar{V}_L$, then $K=L$ or $K$ and $L$ are parallelograms with parallel sides.",
   "clean_statement": "Cone-volume measure\n\nIf $K$ and $L$ symmetric convex bodies in $\\R^n$ such that $\\bar{V}_K=\\bar{V}_L$, then $K=L$ or $K$ and $L$ are parallelograms with parallel sides.",
   "public_statement": "Cone-volume measure\n\nIf $K$ and $L$ symmetric convex bodies in $\\R^n$ such that $\\bar{V}_K=\\bar{V}_L$, then $K=L$ or $K$ and $L$ are parallelograms with parallel sides.",
   "evidence": "The canonical record, AIM workshop *Symmetry and convexity in geometric inequalities*, Section 1 (Inequalities), Problem 1.2, says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0004": {
   "statement_status": "exact",
   "original_statement": "$L_p$-Brunn Minkowski inequality\n\nIf $K$ and $L$ are convex bodies that contain the origin in their interiors then the $L_p$-combination $tK +_p (1-t)L$ is defined by\n$$\ntK +_p (1-t)L = \\underset{u \\in S^{n-1}}{\\cap} \\left\\{ x \\in \\R^n:\\ \\left \\leq \\left(t h_K(u)^p + (1-t) h_L(u)^p \\right)^{\\frac1p} \\right\\}.\n$$\n\nShow that if $K$ and $L$ symmetric convex bodies in $\\R^n$, then for all $0 \\leq t \\leq 1$ and $p\\geq 0$\n$$\n|tK +_p (1-t)L| \\geq |K|^t |L|^{1-t}.\n$$",
   "clean_statement": "$L_p$-Brunn Minkowski inequality\n\nIf $K$ and $L$ are convex bodies that contain the origin in their interiors then the $L_p$-combination $tK +_p (1-t)L$ is defined by\n$$\ntK +_p (1-t)L = \\underset{u \\in S^{n-1}}{\\cap} \\left\\{ x \\in \\R^n:\\ \\left \\leq \\left(t h_K(u)^p + (1-t) h_L(u)^p \\right)^{\\frac1p} \\right\\}.\n$$\n\nShow that if $K$ and $L$ symmetric convex bodies in $\\R^n$, then for all $0 \\leq t \\leq 1$ and $p\\geq 0$\n$$\n|tK +_p (1-t)L| \\geq |K|^t |L|^{1-t}.\n$$",
   "public_statement": "$L_p$-Brunn Minkowski inequality\n\nIf $K$ and $L$ are convex bodies that contain the origin in their interiors then the $L_p$-combination $tK +_p (1-t)L$ is defined by\n$$\ntK +_p (1-t)L = \\underset{u \\in S^{n-1}}{\\cap} \\left\\{ x \\in \\R^n:\\ \\left \\leq \\left(t h_K(u)^p + (1-t) h_L(u)^p \\right)^{\\frac1p} \\right\\}.\n$$\n\nShow that if $K$ and $L$ symmetric convex bodies in $\\R^n$, then for all $0 \\leq t \\leq 1$ and $p\\geq 0$\n$$\n|tK +_p (1-t)L| \\geq |K|^t |L|^{1-t}.\n$$",
   "evidence": "The canonical `problem` field is preserved below exactly, including its malformed TeX:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0005": {
   "statement_status": "exact",
   "original_statement": "Conjecture due to D. Cordero-Erausquin\n\nIf $K$ and $L$ symmetric convex bodies in $\\R^n$, then\n$$\n|K\\cap L||K^\\circ \\cap L| \\leq |B^n_2\\cap L||B^n_2\\cap L|,\n$$\nwhere $K^\\circ =\\{y \\in \\R^n:\\ \\forall x \\in K \\, \\langle x ,y \\rangle \\leq 1 \\}$ is a polar body of $K$.",
   "clean_statement": "Conjecture due to D. Cordero-Erausquin\n\nIf $K$ and $L$ symmetric convex bodies in $\\R^n$, then\n$$\n|K\\cap L||K^\\circ \\cap L| \\leq |B^n_2\\cap L||B^n_2\\cap L|,\n$$\nwhere $K^\\circ =\\{y \\in \\R^n:\\ \\forall x \\in K \\, \\langle x ,y \\rangle \\leq 1 \\}$ is a polar body of $K$.",
   "public_statement": "Conjecture due to D. Cordero-Erausquin\n\nIf $K$ and $L$ symmetric convex bodies in $\\R^n$, then\n$$\n|K\\cap L||K^\\circ \\cap L| \\leq |B^n_2\\cap L||B^n_2\\cap L|,\n$$\nwhere $K^\\circ =\\{y \\in \\R^n:\\ \\forall x \\in K \\, \\langle x ,y \\rangle \\leq 1 \\}$ is a polar body of $K$.",
   "evidence": "The exact canonical record is Conjecture 1.35 in the AIM list *Symmetry and convexity in geometric inequalities*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0006": {
   "statement_status": "exact",
   "original_statement": "Dar's conjecture\n\nLet $\\mu(K, L)$ is defined by\n$\\mu(K, L) = \\max\\limits_{x \\in \\R^n} |K \\cap (x + L)|.$\nThen for convex bodies $K$ and $L$ in $\\R^n$\n$$\n|K + L|^{\\frac1n} \\geq \\mu(K, L) + \\frac{|K|^{\\frac1n} |L|^{\\frac1n}}{\\mu(K, L)}.\n$$",
   "clean_statement": "Dar's conjecture\n\nLet $\\mu(K, L)$ is defined by\n$\\mu(K, L) = \\max\\limits_{x \\in \\R^n} |K \\cap (x + L)|.$\nThen for convex bodies $K$ and $L$ in $\\R^n$\n$$\n|K + L|^{\\frac1n} \\geq \\mu(K, L) + \\frac{|K|^{\\frac1n} |L|^{\\frac1n}}{\\mu(K, L)}.\n$$",
   "public_statement": "Dar's conjecture\n\nLet $\\mu(K, L)$ is defined by\n$\\mu(K, L) = \\max\\limits_{x \\in \\R^n} |K \\cap (x + L)|.$\nThen for convex bodies $K$ and $L$ in $\\R^n$\n$$\n|K + L|^{\\frac1n} \\geq \\mu(K, L) + \\frac{|K|^{\\frac1n} |L|^{\\frac1n}}{\\mu(K, L)}.\n$$",
   "evidence": "This text is preserved exactly from aim-functional-analysis-notes.json, zero-based record index 5. A March 10, 2026 Internet Archive capture of the official AIM page contains exactly the same formula; hence the missing exponents are already present in the AIM source and are not an extraction error in this corpus.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0007": {
   "statement_status": "exact",
   "original_statement": "Brunn-Minkowski inequality for dual quermassintegrals\n\nShow that for convex bodies $K$ and $L$\n$$\n\\widetilde{W}_{n-i}(K + L)^{\\frac1i} \\geq \\widetilde{W}_{n-i}(K)^{\\frac1i} + \\widetilde{W}_{n-i}(L)^{\\frac1i},\n$$\nwhere $\\tilde{W}_{n-i}(K) = \\int\\limits_{S^{n-1}} \\rho_K(u)^i\\,du$.",
   "clean_statement": "Brunn-Minkowski inequality for dual quermassintegrals\n\nShow that for convex bodies $K$ and $L$\n$$\n\\widetilde{W}_{n-i}(K + L)^{\\frac1i} \\geq \\widetilde{W}_{n-i}(K)^{\\frac1i} + \\widetilde{W}_{n-i}(L)^{\\frac1i},\n$$\nwhere $\\tilde{W}_{n-i}(K) = \\int\\limits_{S^{n-1}} \\rho_K(u)^i\\,du$.",
   "public_statement": "Brunn-Minkowski inequality for dual quermassintegrals\n\nShow that for convex bodies $K$ and $L$\n$$\n\\widetilde{W}_{n-i}(K + L)^{\\frac1i} \\geq \\widetilde{W}_{n-i}(K)^{\\frac1i} + \\widetilde{W}_{n-i}(L)^{\\frac1i},\n$$\nwhere $\\tilde{W}_{n-i}(K) = \\int\\limits_{S^{n-1}} \\rho_K(u)^i\\,du$.",
   "evidence": "The canonical problem field is preserved exactly below:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0008": {
   "statement_status": "exact",
   "original_statement": "Brunn-Minkowski inequality for $L_p$ surface area\n\nShow that for convex bodies $K$ and $L$ in $\\R^n$ and for $0 \\leq p \\leq 1$\n$$\nS_p(K+L)^{\\frac1{n-p}} \\geq S_p(K)^{\\frac1{n-p}} + S_p(L)^{\\frac1{n-p}},\n$$\nwhere $S_p(K) = \\int\\limits_{S^{n-1}} h_K(u)^{1-p} \\, dS_K(u)$ is the $L_p$ surface area.",
   "clean_statement": "Brunn-Minkowski inequality for $L_p$ surface area\n\nShow that for convex bodies $K$ and $L$ in $\\R^n$ and for $0 \\leq p \\leq 1$\n$$\nS_p(K+L)^{\\frac1{n-p}} \\geq S_p(K)^{\\frac1{n-p}} + S_p(L)^{\\frac1{n-p}},\n$$\nwhere $S_p(K) = \\int\\limits_{S^{n-1}} h_K(u)^{1-p} \\, dS_K(u)$ is the $L_p$ surface area.",
   "public_statement": "Brunn-Minkowski inequality for $L_p$ surface area\n\nShow that for convex bodies $K$ and $L$ in $\\R^n$ and for $0 \\leq p \\leq 1$\n$$\nS_p(K+L)^{\\frac1{n-p}} \\geq S_p(K)^{\\frac1{n-p}} + S_p(L)^{\\frac1{n-p}},\n$$\nwhere $S_p(K) = \\int\\limits_{S^{n-1}} h_K(u)^{1-p} \\, dS_K(u)$ is the $L_p$ surface area.",
   "evidence": "The canonical record and the live AIM Problem Lists page both state Problem 1.3 as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0009": {
   "statement_status": "exact",
   "original_statement": "Convex bodies in $\\mathbb{C}^n$\n\nConsider a convex body in $\\mathbb{C}^n$ and fix a real subspace $E$ of $\\mathbb{C}^n$. Define $V_E(K) = \\int\\limits_{U(n)} |K|\\phi(E)| \\, d\\phi$, where $U(n)$ is a unitary group. Does Brunn-Minkowski type inequality hold for such quantities:\n$$\nV_E(K + L)^{\\frac1i} \\geq V_E(K)^{\\frac1i} +V_E(L)^{\\frac1i}?\n$$\nAlso, for what type of subspaces $E$ this will work?",
   "clean_statement": "Convex bodies in $\\mathbb{C}^n$\n\nConsider a convex body in $\\mathbb{C}^n$ and fix a real subspace $E$ of $\\mathbb{C}^n$. Define $V_E(K) = \\int\\limits_{U(n)} |K|\\phi(E)| \\, d\\phi$, where $U(n)$ is a unitary group. Does Brunn-Minkowski type inequality hold for such quantities:\n$$\nV_E(K + L)^{\\frac1i} \\geq V_E(K)^{\\frac1i} +V_E(L)^{\\frac1i}?\n$$\nAlso, for what type of subspaces $E$ this will work?",
   "public_statement": "Convex bodies in $\\mathbb{C}^n$\n\nConsider a convex body in $\\mathbb{C}^n$ and fix a real subspace $E$ of $\\mathbb{C}^n$. Define $V_E(K) = \\int\\limits_{U(n)} |K|\\phi(E)| \\, d\\phi$, where $U(n)$ is a unitary group. Does Brunn-Minkowski type inequality hold for such quantities:\n$$\nV_E(K + L)^{\\frac1i} \\geq V_E(K)^{\\frac1i} +V_E(L)^{\\frac1i}?\n$$\nAlso, for what type of subspaces $E$ this will work?",
   "evidence": "The archived AIM page has exactly the same malformed-looking notation and does not define $i$, so this is not an OCR error introduced by the corpus. There is nevertheless a high-confidence standard reconstruction. Put \\[ i=\\dim_{\\mathbb R}E, \\qquad V_E(K)=\\int_{U(n)}\\operatorname{vol}_i(P_{\\phi E}K)\\,d\\phi, \\tag{1} \\] where $P_F$ is orthogonal projection onto $F$ and $d\\phi$ is Haar probability measure. In convex geometry, $K\\mid F$ denotes $P_FK$, and $|K\\mid F|$ denotes its $i$-dimensional volume. Abardia--Wannerer use precisely this notation and precisely these unitary orbit averages in their treatment of the problem.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0010": {
   "statement_status": "exact",
   "original_statement": "Isoperimetric type problems for quermassintegrals in hyperbolic or spherical space.",
   "clean_statement": "Isoperimetric type problems for quermassintegrals in hyperbolic or spherical space.",
   "public_statement": "Isoperimetric type problems for quermassintegrals in hyperbolic or spherical space.",
   "evidence": "The exact canonical AIM text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0011": {
   "statement_status": "exact",
   "original_statement": "Polynomially integrable bodies\n\nLet $A_{K,u}(t) = |K \\cap \\{ tu + u^\\perp \\}|_{n-1}$ be a parallel section function of $K$, then an infinitely smooth convex body $K$ in $\\R^n$ is called polynomially integrable if its parallel section functions $A_{K,u}(t)$ are polynomials of $t$ (on its support) for all $u \\in S^{n-1}$.\n\nIt is known that the only smooth bodies in odd dimensions are ellipsoids, and in even dimensions, such bodies do not exist.\n\nReplacing volume of the sections by the surface area (or any intrinsic volume), is it true that the only bodies in even dimensions with such property are ellipsoids?",
   "clean_statement": "Polynomially integrable bodies\n\nLet $A_{K,u}(t) = |K \\cap \\{ tu + u^\\perp \\}|_{n-1}$ be a parallel section function of $K$, then an infinitely smooth convex body $K$ in $\\R^n$ is called polynomially integrable if its parallel section functions $A_{K,u}(t)$ are polynomials of $t$ (on its support) for all $u \\in S^{n-1}$.\n\nIt is known that the only smooth bodies in odd dimensions are ellipsoids, and in even dimensions, such bodies do not exist.\n\nReplacing volume of the sections by the surface area (or any intrinsic volume), is it true that the only bodies in even dimensions with such property are ellipsoids?",
   "public_statement": "Polynomially integrable bodies\n\nLet $A_{K,u}(t) = |K \\cap \\{ tu + u^\\perp \\}|_{n-1}$ be a parallel section function of $K$, then an infinitely smooth convex body $K$ in $\\R^n$ is called polynomially integrable if its parallel section functions $A_{K,u}(t)$ are polynomials of $t$ (on its support) for all $u \\in S^{n-1}$.\n\nIt is known that the only smooth bodies in odd dimensions are ellipsoids, and in even dimensions, such bodies do not exist.\n\nReplacing volume of the sections by the surface area (or any intrinsic volume), is it true that the only bodies in even dimensions with such property are ellipsoids?",
   "evidence": "The exact AIM record, item 2.1 “Polynomially integrable bodies,” defines \\[ A_{K,u}(t)=\\left|K\\cap\\{tu+u^\\perp\\}\\right|_{n-1} \\] and calls an infinitely smooth convex body $K\\subset\\mathbb R^n$ polynomially integrable when every $A_{K,u}$ is a polynomial in $t$ on its support. It then records the volume classification (ellipsoids in odd dimension and nonexistence in even dimension) and asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0012": {
   "statement_status": "exact",
   "original_statement": "Question by Makai and Martini\n\nIt is known that if for any direction $u \\in S^{n-1}$, a parallel section function $A_{K,u}(t)$ has maximum at $t = 0$, then body $K$ is symmetric.\n\nWhat if we pose the same question but replacing volumes by surface areas, that is $\\widetilde{A}_{K,u}(t) = |\\partial(K \\cap \\{tu + u^\\perp\\})|_{n-1}$, is body $K$ still symmetric?",
   "clean_statement": "Question by Makai and Martini\n\nIt is known that if for any direction $u \\in S^{n-1}$, a parallel section function $A_{K,u}(t)$ has maximum at $t = 0$, then body $K$ is symmetric.\n\nWhat if we pose the same question but replacing volumes by surface areas, that is $\\widetilde{A}_{K,u}(t) = |\\partial(K \\cap \\{tu + u^\\perp\\})|_{n-1}$, is body $K$ still symmetric?",
   "public_statement": "Question by Makai and Martini\n\nIt is known that if for any direction $u \\in S^{n-1}$, a parallel section function $A_{K,u}(t)$ has maximum at $t = 0$, then body $K$ is symmetric.\n\nWhat if we pose the same question but replacing volumes by surface areas, that is $\\widetilde{A}_{K,u}(t) = |\\partial(K \\cap \\{tu + u^\\perp\\})|_{n-1}$, is body $K$ still symmetric?",
   "evidence": "The canonical record and the live AIM Problem Lists page agree exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0013": {
   "statement_status": "exact",
   "original_statement": "Let $p > 1$ and $H \\subset \\R^n$ of dimension $k$. Consider a map\n$$\nf(t_1, \\dots, t_n) = \\left| \\begin{pmatrix}\ne^{t_1} & 0 & \\cdots & 0 \\\\\n0 & e^{t_2} & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & e^{t_n}\n\\end{pmatrix} B_p^n \\cap H \\right|_k,\n$$\nwhere $B_p^n = \\{ x\\in \\R^n: \\ \\sqrt[p]{|x_1|^p + \\cdots + |x_n|^p} \\leq 1\\}$.\n\nIs $f$ log-concave jointly in $t_1, \\dots, t_n$?",
   "clean_statement": "Let $p > 1$ and $H \\subset \\R^n$ of dimension $k$. Consider a map\n$$\nf(t_1, \\dots, t_n) = \\left| \\begin{pmatrix}\ne^{t_1} & 0 & \\cdots & 0 \\\\\n0 & e^{t_2} & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & e^{t_n}\n\\end{pmatrix} B_p^n \\cap H \\right|_k,\n$$\nwhere $B_p^n = \\{ x\\in \\R^n: \\ \\sqrt[p]{|x_1|^p + \\cdots + |x_n|^p} \\leq 1\\}$.\n\nIs $f$ log-concave jointly in $t_1, \\dots, t_n$?",
   "public_statement": "Let $p > 1$ and $H \\subset \\R^n$ of dimension $k$. Consider a map\n$$\nf(t_1, \\dots, t_n) = \\left| \\begin{pmatrix}\ne^{t_1} & 0 & \\cdots & 0 \\\\\n0 & e^{t_2} & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & e^{t_n}\n\\end{pmatrix} B_p^n \\cap H \\right|_k,\n$$\nwhere $B_p^n = \\{ x\\in \\R^n: \\ \\sqrt[p]{|x_1|^p + \\cdots + |x_n|^p} \\leq 1\\}$.\n\nIs $f$ log-concave jointly in $t_1, \\dots, t_n$?",
   "evidence": "The canonical AIM record asks the following.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0014": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Take $a_1, \\dots, a_n \\in \\R$ and $v_1, \\dots, v_n \\in \\R^n$ and define\n$$\nf(t) = \\int\\limits_{\\R^k} e^{-\\max \\limits_{ 1 \\leq i \\leq n } e^{ a_i t }|\\left |} \\, dx.\n$$\nThen $f(t)$ is log-concave is equivalent to log-Brunn-Minkowski. What is possible $f$ that we can use for Brunn-Minkowski?",
   "clean_statement": null,
   "public_statement": "Take $a_1, \\dots, a_n \\in \\R$ and $v_1, \\dots, v_n \\in \\R^n$ and define\n$$\nf(t) = \\int\\limits_{\\R^k} e^{-\\max \\limits_{ 1 \\leq i \\leq n } e^{ a_i t }|\\left |} \\, dx.\n$$\nThen $f(t)$ is log-concave is equivalent to log-Brunn-Minkowski. What is possible $f$ that we can use for Brunn-Minkowski?",
   "evidence": "The canonical JSON record literally says",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0015": {
   "statement_status": "exact",
   "original_statement": "\\begin{theorem}[Caffarelli]\n Let $T = \\nabla \\Phi: \\mu \\to \\nu$ be a Brenier map, where $\\mu$ and $\\nu$ are probability measures with densities $e^{-V(x)}$ and $e^{-W(x)}$. Suppose that\n $$\n \\nabla^2 W(x) \\geq K \\, Id,\n $$\nfor some constant $K$. Then for any $e \\in S^{n-1}$\n $$\n \\sup\\limits_{ x \\in \\R^n } \\Phi^2_{ e e } \\leq \\frac1K \\sup\\limits_{x \\in \\R^n} V_{ee},\n $$\n where $\\Phi_{e}$ denotes directional derivative.\n \\end{theorem}\n\nIf we replace condition in the theorem with $\\nabla^2 W(x) \\geq F(x) \\, Id$, can we obtain better results?",
   "clean_statement": "\\begin{theorem}[Caffarelli]\n Let $T = \\nabla \\Phi: \\mu \\to \\nu$ be a Brenier map, where $\\mu$ and $\\nu$ are probability measures with densities $e^{-V(x)}$ and $e^{-W(x)}$. Suppose that\n $$\n \\nabla^2 W(x) \\geq K \\, Id,\n $$\nfor some constant $K$. Then for any $e \\in S^{n-1}$\n $$\n \\sup\\limits_{ x \\in \\R^n } \\Phi^2_{ e e } \\leq \\frac1K \\sup\\limits_{x \\in \\R^n} V_{ee},\n $$\n where $\\Phi_{e}$ denotes directional derivative.\n \\end{theorem}\n\nIf we replace condition in the theorem with $\\nabla^2 W(x) \\geq F(x) \\, Id$, can we obtain better results?",
   "public_statement": "\\begin{theorem}[Caffarelli]\n Let $T = \\nabla \\Phi: \\mu \\to \\nu$ be a Brenier map, where $\\mu$ and $\\nu$ are probability measures with densities $e^{-V(x)}$ and $e^{-W(x)}$. Suppose that\n $$\n \\nabla^2 W(x) \\geq K \\, Id,\n $$\nfor some constant $K$. Then for any $e \\in S^{n-1}$\n $$\n \\sup\\limits_{ x \\in \\R^n } \\Phi^2_{ e e } \\leq \\frac1K \\sup\\limits_{x \\in \\R^n} V_{ee},\n $$\n where $\\Phi_{e}$ denotes directional derivative.\n \\end{theorem}\n\nIf we replace condition in the theorem with $\\nabla^2 W(x) \\geq F(x) \\, Id$, can we obtain better results?",
   "evidence": "The live AIM page for item 2.5 was checked on August 2, 2026. It agrees with the canonical record and has no status update or remark. It states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0016": {
   "statement_status": "exact",
   "original_statement": "For a smooth convex domain $K$ in $\\R^n$, Laplace equation with Dirichlet boundary conditions is\n$$\n\\begin{cases}\n\t\\Delta u = \\lambda u \\ &\\text{ on } K \\\\\n\tu \\equiv 0 &\\text{ on } \\partial K\n\\end{cases}\n$$\nChoose $u_1 > 0$ in $\\text{int}\\, K$, then\n$$\n\\text{Hess} \\, \\log{u_1} \\leq 0?\n$$\n\nTrue for $K = B_2$. Question is whether it is true for complex projective space $\\mathbb{C}\\mathbb{P}^n$ or for other symmetric K's.",
   "clean_statement": "For a smooth convex domain $K$ in $\\R^n$, Laplace equation with Dirichlet boundary conditions is\n$$\n\\begin{cases}\n\t\\Delta u = \\lambda u \\ &\\text{ on } K \\\\\n\tu \\equiv 0 &\\text{ on } \\partial K\n\\end{cases}\n$$\nChoose $u_1 > 0$ in $\\text{int}\\, K$, then\n$$\n\\text{Hess} \\, \\log{u_1} \\leq 0?\n$$\n\nTrue for $K = B_2$. Question is whether it is true for complex projective space $\\mathbb{C}\\mathbb{P}^n$ or for other symmetric K's.",
   "public_statement": "For a smooth convex domain $K$ in $\\R^n$, Laplace equation with Dirichlet boundary conditions is\n$$\n\\begin{cases}\n\t\\Delta u = \\lambda u \\ &\\text{ on } K \\\\\n\tu \\equiv 0 &\\text{ on } \\partial K\n\\end{cases}\n$$\nChoose $u_1 > 0$ in $\\text{int}\\, K$, then\n$$\n\\text{Hess} \\, \\log{u_1} \\leq 0?\n$$\n\nTrue for $K = B_2$. Question is whether it is true for complex projective space $\\mathbb{C}\\mathbb{P}^n$ or for other symmetric K's.",
   "evidence": "The canonical AIM record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0017": {
   "statement_status": "exact",
   "original_statement": "Does every separable C*-algebra embed into an ultrapower of\n$\\mathcal O_2$ with respect to an ultrafilter on $\\mathbb N$?",
   "clean_statement": "Does every separable C*-algebra embed into an ultrapower of\n$\\mathcal O_2$ with respect to an ultrafilter on $\\mathbb N$?",
   "public_statement": "Does every separable C*-algebra embed into an ultrapower of\n$\\mathcal O_2$ with respect to an ultrafilter on $\\mathbb N$?",
   "evidence": "The canonical AIM record (workshop *Set theory and C*-algebras*, section “Ultrapowers,” Problem 1.1) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0018": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "If $\\phi_j\\colon A\\to \\prod_{\\mathcal U} \\mathcal O_2$ are *-homomorphisms, write $\\phi_1\\leq\n\\phi_2$ if there is a partial isometry $u$ in $A$ such that $u^*\\phi_2\nu=\\phi_1$.\n\nAssume $A$ is separable and $A$ embeds into\n$\\prod_{\\mathcal U}\\mathcal O_2$. Is there a $\\leq$-maximal embedding $\\phi$ of $A$ into\n$\\prod_{\\mathcal U}\\mathcal O_2$?",
   "clean_statement": null,
   "public_statement": "If $\\phi_j\\colon A\\to \\prod_{\\mathcal U} \\mathcal O_2$ are *-homomorphisms, write $\\phi_1\\leq\n\\phi_2$ if there is a partial isometry $u$ in $A$ such that $u^*\\phi_2\nu=\\phi_1$.\n\nAssume $A$ is separable and $A$ embeds into\n$\\prod_{\\mathcal U}\\mathcal O_2$. Is there a $\\leq$-maximal embedding $\\phi$ of $A$ into\n$\\prod_{\\mathcal U}\\mathcal O_2$?",
   "evidence": "The canonical record is Problem 1.2 in the “Ultrapowers” section of the AIM workshop list *Set theory and C*-algebras*. Its exact mathematical text is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 17,
   "attempt": 2
  },
  "AIM-FUNCTIONAL_ANALYSIS-0019": {
   "statement_status": "exact",
   "original_statement": "Can one prove in ZFC that for some free ultrafilter $\\mathcal U$ on $\\mathbb N$ we have $\\mathcal{B}\\left( H\\right) '\\cap \\prod_{\\mathcal U}\\mathcal{B}\\left( H\\right) =\n\\mathbb C I$?",
   "clean_statement": "Can one prove in ZFC that for some free ultrafilter $\\mathcal U$ on $\\mathbb N$ we have $\\mathcal{B}\\left( H\\right) '\\cap \\prod_{\\mathcal U}\\mathcal{B}\\left( H\\right) =\n\\mathbb C I$?",
   "public_statement": "Can one prove in ZFC that for some free ultrafilter $\\mathcal U$ on $\\mathbb N$ we have $\\mathcal{B}\\left( H\\right) '\\cap \\prod_{\\mathcal U}\\mathcal{B}\\left( H\\right) =\n\\mathbb C I$?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0020": {
   "statement_status": "exact",
   "original_statement": "An ultrafilter $\\mathcal U$ on $\\mathbb N$ is \\emph{flat} if there are $h_n\\colon\n\\mathbb N\\searrow [0,1]$ such that\n\\begin{enumerate}\n\\item $h_n(0)=1$,\n\\item $\\lim_j h_n(j)=0$,\n\\item $(\\forall f\\colon \\mathbb N\\nearrow \\mathbb N)\\lim_{n\\to \\mathcal U}\n\\sup_{j\\in \\mathbb N} |h_n(j)-h_n(f(j))|=0$.\n\\end{enumerate}\n\nIs a nonprincipal ultrafilter such that $\\mathcal{B}\\left( H\\right) '\\cap \\mathcal{B}\\left( H\\right) ^{\\mathcal U}\\neq \\mathbb C I$ flat?",
   "clean_statement": "An ultrafilter $\\mathcal U$ on $\\mathbb N$ is \\emph{flat} if there are $h_n\\colon\n\\mathbb N\\searrow [0,1]$ such that\n\\begin{enumerate}\n\\item $h_n(0)=1$,\n\\item $\\lim_j h_n(j)=0$,\n\\item $(\\forall f\\colon \\mathbb N\\nearrow \\mathbb N)\\lim_{n\\to \\mathcal U}\n\\sup_{j\\in \\mathbb N} |h_n(j)-h_n(f(j))|=0$.\n\\end{enumerate}\n\nIs a nonprincipal ultrafilter such that $\\mathcal{B}\\left( H\\right) '\\cap \\mathcal{B}\\left( H\\right) ^{\\mathcal U}\\neq \\mathbb C I$ flat?",
   "public_statement": "An ultrafilter $\\mathcal U$ on $\\mathbb N$ is \\emph{flat} if there are $h_n\\colon\n\\mathbb N\\searrow [0,1]$ such that\n\\begin{enumerate}\n\\item $h_n(0)=1$,\n\\item $\\lim_j h_n(j)=0$,\n\\item $(\\forall f\\colon \\mathbb N\\nearrow \\mathbb N)\\lim_{n\\to \\mathcal U}\n\\sup_{j\\in \\mathbb N} |h_n(j)-h_n(f(j))|=0$.\n\\end{enumerate}\n\nIs a nonprincipal ultrafilter such that $\\mathcal{B}\\left( H\\right) '\\cap \\mathcal{B}\\left( H\\right) ^{\\mathcal U}\\neq \\mathbb C I$ flat?",
   "evidence": "The canonical AIM record, from the workshop *Set theory and C*-algebras*, section “Ultrapowers,” Problem 1.4, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0021": {
   "statement_status": "exact",
   "original_statement": "Is there a model of set theory in which the Calkin algebra has an\nautomorphism sending the unilateral shift $S$ to $S^{\\ast }$? Is the\nanalogous fact for $\\ell ^{\\infty }\\left/ c_{0}\\right. $ true?",
   "clean_statement": "Is there a model of set theory in which the Calkin algebra has an\nautomorphism sending the unilateral shift $S$ to $S^{\\ast }$? Is the\nanalogous fact for $\\ell ^{\\infty }\\left/ c_{0}\\right. $ true?",
   "public_statement": "Is there a model of set theory in which the Calkin algebra has an\nautomorphism sending the unilateral shift $S$ to $S^{\\ast }$? Is the\nanalogous fact for $\\ell ^{\\infty }\\left/ c_{0}\\right. $ true?",
   "evidence": "The canonical record, and also the live AIM page (Problem 2.1 in the \"Calkin algebra\" section), literally read:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0022": {
   "statement_status": "exact",
   "original_statement": "Is there a model of set theory in which the Calkin algebra has a not\napproximately inner automorphism preserving K-theory?",
   "clean_statement": "Is there a model of set theory in which the Calkin algebra has a not\napproximately inner automorphism preserving K-theory?",
   "public_statement": "Is there a model of set theory in which the Calkin algebra has a not\napproximately inner automorphism preserving K-theory?",
   "evidence": "The canonical AIM record reads verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0023": {
   "statement_status": "exact",
   "original_statement": "Is it consistent that, for a nonseparable Hilbert space $\\mathcal{H}$,\n$\\mathcal{B}\\left( \\mathcal{H}\\right) \\left/ \\mathcal{K}\\left( \\mathcal{H}%\n\\right) \\right. $ has an outer automorphism?",
   "clean_statement": "Is it consistent that, for a nonseparable Hilbert space $\\mathcal{H}$,\n$\\mathcal{B}\\left( \\mathcal{H}\\right) \\left/ \\mathcal{K}\\left( \\mathcal{H}%\n\\right) \\right. $ has an outer automorphism?",
   "public_statement": "Is it consistent that, for a nonseparable Hilbert space $\\mathcal{H}$,\n$\\mathcal{B}\\left( \\mathcal{H}\\right) \\left/ \\mathcal{K}\\left( \\mathcal{H}%\n\\right) \\right. $ has an outer automorphism?",
   "evidence": "The exact source is preserved below; the later display \\(\\mathcal B(H)/\\mathcal K(H)\\) is only a readability normalization and does not correct or alter the source notation.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0024": {
   "statement_status": "exact",
   "original_statement": "Does every masa in the Calkin algebra that is generated by its projections lift to a masa in $\\mathcal{B}\\left( H\\right) $?",
   "clean_statement": "Does every masa in the Calkin algebra that is generated by its projections lift to a masa in $\\mathcal{B}\\left( H\\right) $?",
   "public_statement": "Does every masa in the Calkin algebra that is generated by its projections lift to a masa in $\\mathcal{B}\\left( H\\right) $?",
   "evidence": "The canonical AIM record asks, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0025": {
   "statement_status": "exact",
   "original_statement": "Does the number of C*-norms on $A\\otimes _{alg}B$ for\n\n\\begin{itemize}\n\\item $A=B=\\mathcal{B}\\left( H\\right) $\n\n\\item $A=B=C^{\\ast }\\left( \\mathbb{F}_{\\infty }\\right) $, where $\\mathbb{F}%\n_{\\infty }$ is the free group on countably many generators\n\n\\item $A=B\\left( H\\right) $ and $B=\\mathcal{Q}\\left( H\\right) $ where $%\n\\mathcal{Q}\\left( H\\right) $ is the Calkin algebra\n\\end{itemize}\n\ndepend on the model of set theory?",
   "clean_statement": "Does the number of C*-norms on $A\\otimes _{alg}B$ for\n\n\\begin{itemize}\n\\item $A=B=\\mathcal{B}\\left( H\\right) $\n\n\\item $A=B=C^{\\ast }\\left( \\mathbb{F}_{\\infty }\\right) $, where $\\mathbb{F}%\n_{\\infty }$ is the free group on countably many generators\n\n\\item $A=B\\left( H\\right) $ and $B=\\mathcal{Q}\\left( H\\right) $ where $%\n\\mathcal{Q}\\left( H\\right) $ is the Calkin algebra\n\\end{itemize}\n\ndepend on the model of set theory?",
   "public_statement": "Does the number of C*-norms on $A\\otimes _{alg}B$ for\n\n\\begin{itemize}\n\\item $A=B=\\mathcal{B}\\left( H\\right) $\n\n\\item $A=B=C^{\\ast }\\left( \\mathbb{F}_{\\infty }\\right) $, where $\\mathbb{F}%\n_{\\infty }$ is the free group on countably many generators\n\n\\item $A=B\\left( H\\right) $ and $B=\\mathcal{Q}\\left( H\\right) $ where $%\n\\mathcal{Q}\\left( H\\right) $ is the Calkin algebra\n\\end{itemize}\n\ndepend on the model of set theory?",
   "evidence": "The canonical record is Problem 3.1 in the AIM workshop list *Set theory and C\\*-algebras*, section “Tensor products.” The exact extracted problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0026": {
   "statement_status": "exact",
   "original_statement": "What is the ideal structure of $\\mathcal{B}\\left( H\\right) \\otimes\n_{\\min }\\mathcal{Q}\\left( H\\right) $? Does it depend on the model of set\ntheory?",
   "clean_statement": "What is the ideal structure of $\\mathcal{B}\\left( H\\right) \\otimes\n_{\\min }\\mathcal{Q}\\left( H\\right) $? Does it depend on the model of set\ntheory?",
   "public_statement": "What is the ideal structure of $\\mathcal{B}\\left( H\\right) \\otimes\n_{\\min }\\mathcal{Q}\\left( H\\right) $? Does it depend on the model of set\ntheory?",
   "evidence": "The canonical AIM record is from the workshop *Set theory and C*-algebras*, section “Tensor products,” Problem 3.2:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0027": {
   "statement_status": "exact",
   "original_statement": "Does $\\otimes _{\\min }$ or $\\otimes _{\\max }$ commute with forcing?",
   "clean_statement": "Does $\\otimes _{\\min }$ or $\\otimes _{\\max }$ commute with forcing?",
   "public_statement": "Does $\\otimes _{\\min }$ or $\\otimes _{\\max }$ commute with forcing?",
   "evidence": "The exact canonical AIM record (workshop *Set theory and C\\(^*\\)-algebras*, section “Tensor products,” Problem 3.3) is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0028": {
   "statement_status": "exact",
   "original_statement": "Is every stably finite nuclear separable C*-algebra locally\napproximable by type I C*-algebras?",
   "clean_statement": "Is every stably finite nuclear separable C*-algebra locally\napproximable by type I C*-algebras?",
   "public_statement": "Is every stably finite nuclear separable C*-algebra locally\napproximable by type I C*-algebras?",
   "evidence": "The canonical AIM record (workshop *Set theory and C*-algebras*, section “Nuclearity,” Problem 4.1) asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0029": {
   "statement_status": "exact",
   "original_statement": "All presently known nuclear C*-algebras are obtained from $\\mathbb C$ by\ntaking closure under simple operations such as tensoring with\nfinite-dimensional matrix algebras and taking inductive limits. They\nbelong to the so-called bootstrap class.\n\nDoes every nuclear C*-algebra belong to the\nbootstrap class?",
   "clean_statement": "All presently known nuclear C*-algebras are obtained from $\\mathbb C$ by\ntaking closure under simple operations such as tensoring with\nfinite-dimensional matrix algebras and taking inductive limits. They\nbelong to the so-called bootstrap class.\n\nDoes every nuclear C*-algebra belong to the\nbootstrap class?",
   "public_statement": "All presently known nuclear C*-algebras are obtained from $\\mathbb C$ by\ntaking closure under simple operations such as tensoring with\nfinite-dimensional matrix algebras and taking inductive limits. They\nbelong to the so-called bootstrap class.\n\nDoes every nuclear C*-algebra belong to the\nbootstrap class?",
   "evidence": "The canonical record, AIM Problem 4.2 in the section “Nuclearity,” states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 28,
   "attempt": 2
  },
  "AIM-FUNCTIONAL_ANALYSIS-0030": {
   "statement_status": "exact",
   "original_statement": "Do all nuclear C*-algebras satisfy the UCT (Universal Coefficient\nTheorem)?",
   "clean_statement": "Do all nuclear C*-algebras satisfy the UCT (Universal Coefficient\nTheorem)?",
   "public_statement": "Do all nuclear C*-algebras satisfy the UCT (Universal Coefficient\nTheorem)?",
   "evidence": "The exact canonical AIM record, from the workshop *Set theory and C\\(^*\\)-algebras*, section “Nuclearity,” Problem 4.3, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0031": {
   "statement_status": "exact",
   "original_statement": "Does every pure state of the atomic masa in $\\mathcal{B}\\left( H\\right) $ extend uniquely to a pure\nstate of $\\mathcal{B}\\left( H\\right) $?",
   "clean_statement": "Does every pure state of the atomic masa in $\\mathcal{B}\\left( H\\right) $ extend uniquely to a pure\nstate of $\\mathcal{B}\\left( H\\right) $?",
   "public_statement": "Does every pure state of the atomic masa in $\\mathcal{B}\\left( H\\right) $ extend uniquely to a pure\nstate of $\\mathcal{B}\\left( H\\right) $?",
   "evidence": "The canonical AIM record, from the workshop *Set theory and C*-algebras*, section “Pure states,” Problem 5.1, asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0032": {
   "statement_status": "exact",
   "original_statement": "Consider the following statement: for every pure state $\\phi$ of $\\mathcal{B}\\left( H\\right) $ there is a masa $\\mathcal{A}$ such that\n$\\phi\\restriction \\mathcal{A}$ is multiplicative (i.e., pure). Is it consistent with ZFC?",
   "clean_statement": "Consider the following statement: for every pure state $\\phi$ of $\\mathcal{B}\\left( H\\right) $ there is a masa $\\mathcal{A}$ such that\n$\\phi\\restriction \\mathcal{A}$ is multiplicative (i.e., pure). Is it consistent with ZFC?",
   "public_statement": "Consider the following statement: for every pure state $\\phi$ of $\\mathcal{B}\\left( H\\right) $ there is a masa $\\mathcal{A}$ such that\n$\\phi\\restriction \\mathcal{A}$ is multiplicative (i.e., pure). Is it consistent with ZFC?",
   "evidence": "The exact canonical AIM record, from the workshop *Set theory and C*-algebras*, section “Pure states,” Problem 5.2, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0033": {
   "statement_status": "exact",
   "original_statement": "Consider the following statement: every pure state on $\\mathcal{B}\\left( H\\right) $ is diagonalizable. Is it consistent with with ZFC?",
   "clean_statement": "Consider the following statement: every pure state on $\\mathcal{B}\\left( H\\right) $ is diagonalizable. Is it consistent with with ZFC?",
   "public_statement": "Consider the following statement: every pure state on $\\mathcal{B}\\left( H\\right) $ is diagonalizable. Is it consistent with with ZFC?",
   "evidence": "The canonical record says, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0034": {
   "statement_status": "exact",
   "original_statement": "Is there a nonseparable AF C*-algebra or W*-algebra not isomorphic to\nits opposite algebra?",
   "clean_statement": "Is there a nonseparable AF C*-algebra or W*-algebra not isomorphic to\nits opposite algebra?",
   "public_statement": "Is there a nonseparable AF C*-algebra or W*-algebra not isomorphic to\nits opposite algebra?",
   "evidence": "The canonical AIM record, Problem 6.1 in the section “Nonseparable C\\*-algebras,” asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0035": {
   "statement_status": "exact",
   "original_statement": "Is the following relatively consistent with ZFC?\n\nEvery C*-algebra $A$ that has a unique irreducible representation up to the\nunitary equivalence is isomorphic to the algebra of compact operators on some\nHilbert space.",
   "clean_statement": "Is the following relatively consistent with ZFC?\n\nEvery C*-algebra $A$ that has a unique irreducible representation up to the\nunitary equivalence is isomorphic to the algebra of compact operators on some\nHilbert space.",
   "public_statement": "Is the following relatively consistent with ZFC?\n\nEvery C*-algebra $A$ that has a unique irreducible representation up to the\nunitary equivalence is isomorphic to the algebra of compact operators on some\nHilbert space.",
   "evidence": "The canonical record asks, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0036": {
   "statement_status": "exact",
   "original_statement": "Assume $A$ is a tensor product of algebras of the form $\\mathbb{M}_n(\\mathbb C)$, for $n\\in \\mathbb N$,\n $\\kappa<\\kappa'$ are cardinals\n and $\\bigotimes_{\\kappa'}\\mathbb{M}_2(\\mathbb C)$ unitally embeds into\n$A\\otimes \\bigotimes_{\\kappa}\\mathbb{M}_2(\\mathbb C)$. Can we conclude that there is a unital\nembedding of $\\bigotimes_\\kappa \\mathbb{M}_2(\\mathbb C)$ into $A$?",
   "clean_statement": "Assume $A$ is a tensor product of algebras of the form $\\mathbb{M}_n(\\mathbb C)$, for $n\\in \\mathbb N$,\n $\\kappa<\\kappa'$ are cardinals\n and $\\bigotimes_{\\kappa'}\\mathbb{M}_2(\\mathbb C)$ unitally embeds into\n$A\\otimes \\bigotimes_{\\kappa}\\mathbb{M}_2(\\mathbb C)$. Can we conclude that there is a unital\nembedding of $\\bigotimes_\\kappa \\mathbb{M}_2(\\mathbb C)$ into $A$?",
   "public_statement": "Assume $A$ is a tensor product of algebras of the form $\\mathbb{M}_n(\\mathbb C)$, for $n\\in \\mathbb N$,\n $\\kappa<\\kappa'$ are cardinals\n and $\\bigotimes_{\\kappa'}\\mathbb{M}_2(\\mathbb C)$ unitally embeds into\n$A\\otimes \\bigotimes_{\\kappa}\\mathbb{M}_2(\\mathbb C)$. Can we conclude that there is a unital\nembedding of $\\bigotimes_\\kappa \\mathbb{M}_2(\\mathbb C)$ into $A$?",
   "evidence": "Write \\[ D_p(\\lambda):=\\bigotimes_{\\lambda}M_p(\\mathbb C) \\] for the spatial tensor product, with the units used as reference vectors. The canonical AIM record asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0037": {
   "statement_status": "exact",
   "original_statement": "Is every exact C*-algebra a subalgebra of a nuclear C*-algebra?",
   "clean_statement": "Is every exact C*-algebra a subalgebra of a nuclear C*-algebra?",
   "public_statement": "Is every exact C*-algebra a subalgebra of a nuclear C*-algebra?",
   "evidence": "There is no apparent OCR corruption. Throughout, “subalgebra” means C*-subalgebra, equivalently the range of an injective *-homomorphism. The question is understood in ZFC and without a separability assumption.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0038": {
   "statement_status": "exact",
   "original_statement": "Is there a universal nuclear C*-algebra of character density $\\aleph_1$?\nMore generally, for which cardinals $\\kappa$ is there a universal\nnuclear C*-algebra of character density $\\kappa$? Similar question can be asked for exact algebras.",
   "clean_statement": "Is there a universal nuclear C*-algebra of character density $\\aleph_1$?\nMore generally, for which cardinals $\\kappa$ is there a universal\nnuclear C*-algebra of character density $\\kappa$? Similar question can be asked for exact algebras.",
   "public_statement": "Is there a universal nuclear C*-algebra of character density $\\aleph_1$?\nMore generally, for which cardinals $\\kappa$ is there a universal\nnuclear C*-algebra of character density $\\kappa$? Similar question can be asked for exact algebras.",
   "evidence": "The phrase **character density** is not an OCR error. It occurs in the circulated source problem list and means the least cardinality of a dense subset, now more commonly called the **density character**. I write \\[ \\operatorname{dens}(A)=\\min\\{|D|:D\\subseteq A\\text{ is norm-dense}\\}. \\]",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0039": {
   "statement_status": "exact",
   "original_statement": "Suppose a C*-algebra\nhas an approximate identity consisting of projections. Does it have an\nincreasing approximate identity (on some index set, possibly different)\nconsisting of projections?",
   "clean_statement": "Suppose a C*-algebra\nhas an approximate identity consisting of projections. Does it have an\nincreasing approximate identity (on some index set, possibly different)\nconsisting of projections?",
   "public_statement": "Suppose a C*-algebra\nhas an approximate identity consisting of projections. Does it have an\nincreasing approximate identity (on some index set, possibly different)\nconsisting of projections?",
   "evidence": "The record is Problem 6.6 on the AIM *Set theory and C*-algebras* page. The same wording occurs as Question 8.3 in Ilijas Farah's circulated 2008 list *Some problems about operator algebras with set-theoretic flavor*. The source adds that the answer is positive in the separable case and warns that even for a real-rank-zero algebra the full set of projections need not be directed. There is no visible corruption in the extracted statement.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0040": {
   "statement_status": "exact",
   "original_statement": "Is conjugacy by automorphism of masas of the hyperfinite II$_{1}$\nfactor classifiable by countable structures?",
   "clean_statement": "Is conjugacy by automorphism of masas of the hyperfinite II$_{1}$\nfactor classifiable by countable structures?",
   "public_statement": "Is conjugacy by automorphism of masas of the hyperfinite II$_{1}$\nfactor classifiable by countable structures?",
   "evidence": "There is no apparent corruption in the record. Throughout, \\(R\\) is the separable hyperfinite II\\(_1\\) factor with normalized trace \\(\\tau\\), and a masa is a maximal abelian von Neumann subalgebra of \\(R\\).",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0041": {
   "statement_status": "exact",
   "original_statement": "What is the right set-theoretic framework to deal with functorial\nclassification?",
   "clean_statement": "What is the right set-theoretic framework to deal with functorial\nclassification?",
   "public_statement": "What is the right set-theoretic framework to deal with functorial\nclassification?",
   "evidence": "The canonical record is Problem 7.1 in the “Borel complexity” section of the 2012 AIM workshop *Set theory and C*-algebras*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0042": {
   "statement_status": "exact",
   "original_statement": "Is there a constructive Borel proof of the $\\mathcal{O}_{2}$ embedding theorem?",
   "clean_statement": "Is there a constructive Borel proof of the $\\mathcal{O}_{2}$ embedding theorem?",
   "public_statement": "Is there a constructive Borel proof of the $\\mathcal{O}_{2}$ embedding theorem?",
   "evidence": "The exact canonical problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 41,
   "attempt": 2
  },
  "AIM-FUNCTIONAL_ANALYSIS-0043": {
   "statement_status": "exact",
   "original_statement": "Is the isomorphism relation in the following classes of separable\nC*-algebras:\n\n\\begin{itemize}\n\\item all\n\n\\item nuclear\n\n\\item exact\n\n\\item simple\n\n\\item simple exact\n\n\\item nuclear $\\mathcal{Z}$-stable\n\n\\item simple nuclear\n\\end{itemize}\n\ncomplete analytic? below a group actions? above all group actions?",
   "clean_statement": "Is the isomorphism relation in the following classes of separable\nC*-algebras:\n\n\\begin{itemize}\n\\item all\n\n\\item nuclear\n\n\\item exact\n\n\\item simple\n\n\\item simple exact\n\n\\item nuclear $\\mathcal{Z}$-stable\n\n\\item simple nuclear\n\\end{itemize}\n\ncomplete analytic? below a group actions? above all group actions?",
   "public_statement": "Is the isomorphism relation in the following classes of separable\nC*-algebras:\n\n\\begin{itemize}\n\\item all\n\n\\item nuclear\n\n\\item exact\n\n\\item simple\n\n\\item simple exact\n\n\\item nuclear $\\mathcal{Z}$-stable\n\n\\item simple nuclear\n\\end{itemize}\n\ncomplete analytic? below a group actions? above all group actions?",
   "evidence": "The canonical AIM record, Problem 7.2 in the 2012 workshop *Set theory and C\\(^*\\)-algebras*, asks verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0044": {
   "statement_status": "exact",
   "original_statement": "What is the complexity of isometric isomorphism of direct limits of\nnot necessarily self-adjoint subalgebras of finite dimensional C*-algebras?\nHow does it compare to the complexity of isomorphism of AF algebras?",
   "clean_statement": "What is the complexity of isometric isomorphism of direct limits of\nnot necessarily self-adjoint subalgebras of finite dimensional C*-algebras?\nHow does it compare to the complexity of isomorphism of AF algebras?",
   "public_statement": "What is the complexity of isometric isomorphism of direct limits of\nnot necessarily self-adjoint subalgebras of finite dimensional C*-algebras?\nHow does it compare to the complexity of isomorphism of AF algebras?",
   "evidence": "The exact AIM record (Set theory and C*-algebras, Problem 7.25) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0045": {
   "statement_status": "exact",
   "original_statement": "Is there a Borel inverse of the classification functor?",
   "clean_statement": "Is there a Borel inverse of the classification functor?",
   "public_statement": "Is there a Borel inverse of the classification functor?",
   "evidence": "The exact canonical sentence is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0046": {
   "statement_status": "exact",
   "original_statement": "For a separable C*-algebra $A$, what is the complexity of orbit\nequivalence relations associated to the action $\\mathrm{Aut}\\left( A\\right) $\non itself by conjugacy and to the action of $\\mathrm{Inn}\\left( A\\right) $\non $\\mathrm{\\mathrm{Au}t}\\left( A\\right) $ by left translation? Are these\naction turbulent? How are they related to structural properties of $A$?",
   "clean_statement": "For a separable C*-algebra $A$, what is the complexity of orbit\nequivalence relations associated to the action $\\mathrm{Aut}\\left( A\\right) $\non itself by conjugacy and to the action of $\\mathrm{Inn}\\left( A\\right) $\non $\\mathrm{\\mathrm{Au}t}\\left( A\\right) $ by left translation? Are these\naction turbulent? How are they related to structural properties of $A$?",
   "public_statement": "For a separable C*-algebra $A$, what is the complexity of orbit\nequivalence relations associated to the action $\\mathrm{Aut}\\left( A\\right) $\non itself by conjugacy and to the action of $\\mathrm{Inn}\\left( A\\right) $\non $\\mathrm{\\mathrm{Au}t}\\left( A\\right) $ by left translation? Are these\naction turbulent? How are they related to structural properties of $A$?",
   "evidence": "The canonical record is Problem 7.35 in the AIM list *Set theory and C*-algebras*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0047": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Is the Mackey Borel\nstructure on the spectrum of a simple separable C*-algebra always the same when\nit is not standard?",
   "clean_statement": null,
   "public_statement": "Is the Mackey Borel\nstructure on the spectrum of a simple separable C*-algebra always the same when\nit is not standard?",
   "evidence": "The exact canonical record asks:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0048": {
   "statement_status": "exact",
   "original_statement": "Does the complexity of the Mackey Borel structure of a simple separable\nC*-algebra increase as one goes from nuclear C*-algebras to exact ones to ones\nthat are not even exact?",
   "clean_statement": "Does the complexity of the Mackey Borel structure of a simple separable\nC*-algebra increase as one goes from nuclear C*-algebras to exact ones to ones\nthat are not even exact?",
   "public_statement": "Does the complexity of the Mackey Borel structure of a simple separable\nC*-algebra increase as one goes from nuclear C*-algebras to exact ones to ones\nthat are not even exact?",
   "evidence": "The AIM record (Set theory and C*-algebras, Borel complexity, Problem 7.45) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 47,
   "attempt": 2
  },
  "AIM-FUNCTIONAL_ANALYSIS-0049": {
   "statement_status": "exact",
   "original_statement": "If A is a C*-algebra, denote by $E_A$ the relation of unitary equivalence of pure states of A.\n\nAssume $A$ and $B$ are C*-algebras and $E_A$ is Borel-reducible to $E_B$. What does this fact imply about the\nrelation between $A$ and $B$?",
   "clean_statement": "If A is a C*-algebra, denote by $E_A$ the relation of unitary equivalence of pure states of A.\n\nAssume $A$ and $B$ are C*-algebras and $E_A$ is Borel-reducible to $E_B$. What does this fact imply about the\nrelation between $A$ and $B$?",
   "public_statement": "If A is a C*-algebra, denote by $E_A$ the relation of unitary equivalence of pure states of A.\n\nAssume $A$ and $B$ are C*-algebras and $E_A$ is Borel-reducible to $E_B$. What does this fact imply about the\nrelation between $A$ and $B$?",
   "evidence": "The exact source record is AIM Problem Lists, workshop *Set theory and C*-algebras*, section “Borel complexity,” problem 7.5:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-FUNCTIONAL_ANALYSIS-0050": {
   "statement_status": "exact",
   "original_statement": "A C*-algebra is called singly generated if it contains an element that is not contained in any proper sub-C*-algebra.\nA C*-algebra is called $\\mathcal{Z}$-stable if it absorbs the Jiang-Su algebra tensorially.\n\nWhich separable, unital, simple C*-algebras are singly generated? Is there a simple, nuclear C*-algebra that is singly generated, yet is not $\\mathcal{Z}$-stable? Is single generation connected to the regularity properties coming up in the classification of nuclear, simple C*-algebras?",
   "clean_statement": "A C*-algebra is called singly generated if it contains an element that is not contained in any proper sub-C*-algebra.\nA C*-algebra is called $\\mathcal{Z}$-stable if it absorbs the Jiang-Su algebra tensorially.\n\nWhich separable, unital, simple C*-algebras are singly generated? Is there a simple, nuclear C*-algebra that is singly generated, yet is not $\\mathcal{Z}$-stable? Is single generation connected to the regularity properties coming up in the classification of nuclear, simple C*-algebras?",
   "public_statement": "A C*-algebra is called singly generated if it contains an element that is not contained in any proper sub-C*-algebra.\nA C*-algebra is called $\\mathcal{Z}$-stable if it absorbs the Jiang-Su algebra tensorially.\n\nWhich separable, unital, simple C*-algebras are singly generated? Is there a simple, nuclear C*-algebra that is singly generated, yet is not $\\mathcal{Z}$-stable? Is single generation connected to the regularity properties coming up in the classification of nuclear, simple C*-algebras?",
   "evidence": "The canonical record is number 8.1, “Generators,” from the AIM workshop *Set theory and C\\(^*\\)-algebras*. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-functional-analysis-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0001": {
   "statement_status": "exact",
   "original_statement": "Quasi-convex subgroups\n\nA subgroup $F$ of a group $G$ is a \\emph{semi-fibre} if there exists an injective non-surjective homomorphism $\\varphi \\colon F \\to F$ such that $G$ is the HNN extension\n\\[G = F \\ast_{\\varphi}.\\]\nA subgroup $F$ of $G$ is a \\emph{virtual semi-fibre} if there exists a finite index subgroup $G'$ of $G$ such that $F$ is a semi-fibre of $G'$.\n\nLet $G$ be a hyperbolic free-by-cyclic group and $H$ a non-quasiconvex subgroup of $G$. Then $H$ contains a subgroup $F$ which is a virtual fibre or a virtual semi-fibre of a free-by-cyclic subgroup of $G$.",
   "clean_statement": "Quasi-convex subgroups\n\nA subgroup $F$ of a group $G$ is a \\emph{semi-fibre} if there exists an injective non-surjective homomorphism $\\varphi \\colon F \\to F$ such that $G$ is the HNN extension\n\\[G = F \\ast_{\\varphi}.\\]\nA subgroup $F$ of $G$ is a \\emph{virtual semi-fibre} if there exists a finite index subgroup $G'$ of $G$ such that $F$ is a semi-fibre of $G'$.\n\nLet $G$ be a hyperbolic free-by-cyclic group and $H$ a non-quasiconvex subgroup of $G$. Then $H$ contains a subgroup $F$ which is a virtual fibre or a virtual semi-fibre of a free-by-cyclic subgroup of $G$.",
   "public_statement": "Quasi-convex subgroups\n\nA subgroup $F$ of a group $G$ is a \\emph{semi-fibre} if there exists an injective non-surjective homomorphism $\\varphi \\colon F \\to F$ such that $G$ is the HNN extension\n\\[G = F \\ast_{\\varphi}.\\]\nA subgroup $F$ of $G$ is a \\emph{virtual semi-fibre} if there exists a finite index subgroup $G'$ of $G$ such that $F$ is a semi-fibre of $G'$.\n\nLet $G$ be a hyperbolic free-by-cyclic group and $H$ a non-quasiconvex subgroup of $G$. Then $H$ contains a subgroup $F$ which is a virtual fibre or a virtual semi-fibre of a free-by-cyclic subgroup of $G$.",
   "evidence": "The canonical AIM record is problem 1.1, “Quasi-convex subgroups,” from the 2023 workshop *Rigidity properties of free-by-cyclic groups*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 0,
   "attempt": 2
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0002": {
   "statement_status": "exact",
   "original_statement": "What are the possible distortion functions for subgroups of free-by-cyclic groups?",
   "clean_statement": "What are the possible distortion functions for subgroups of free-by-cyclic groups?",
   "public_statement": "What are the possible distortion functions for subgroups of free-by-cyclic groups?",
   "evidence": "The canonical AIM record asks, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0003": {
   "statement_status": "exact",
   "original_statement": "If $G$ is a hyperbolic free-by-cyclic group then does $G$ contain a closed surface subgroup?",
   "clean_statement": "If $G$ is a hyperbolic free-by-cyclic group then does $G$ contain a closed surface subgroup?",
   "public_statement": "If $G$ is a hyperbolic free-by-cyclic group then does $G$ contain a closed surface subgroup?",
   "evidence": "The canonical AIM record is problem 1.3 in the section “Subgroups of free-by-cyclic groups” from the workshop *Rigidity properties of free-by-cyclic groups*. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 2,
   "attempt": 2
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0004": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Effective coherence\n\nGiven a finite set $S$ of elements of a free-by-cyclic group $G$, algorithmically find a presentation of the sugroup of $G$ generated by $S$.",
   "clean_statement": null,
   "public_statement": "Effective coherence\n\nGiven a finite set $S$ of elements of a free-by-cyclic group $G$, algorithmically find a presentation of the sugroup of $G$ generated by $S$.",
   "evidence": "The canonical AIM record is Problem 1.4, “Effective coherence,” in the section “Subgroups of free-by-cyclic groups.” The exact source text is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0005": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Let $G$ be a hyperbolic free-by-cyclic groups. Which subgrups of $G$ admit Cannon--Thurston maps?",
   "clean_statement": null,
   "public_statement": "Let $G$ be a hyperbolic free-by-cyclic groups. Which subgrups of $G$ admit Cannon--Thurston maps?",
   "evidence": "The canonical record (AIM workshop *Rigidity properties of free-by-cyclic groups*, section “Subgroups of free-by-cyclic groups,” Problem 1.5) reads exactly:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0006": {
   "statement_status": "exact",
   "original_statement": "Which subgroups of free-by-cyclic groups are Morse?",
   "clean_statement": "Which subgroups of free-by-cyclic groups are Morse?",
   "public_statement": "Which subgroups of free-by-cyclic groups are Morse?",
   "evidence": "The canonical AIM record is Problem 1.6 in the section “Subgroups of free-by-cyclic groups” of the 2023 AIM list *Rigidity properties of free-by-cyclic groups*. Its complete problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0007": {
   "statement_status": "exact",
   "original_statement": "Relatively hyperbolic case\n\nIf $G$ is a LERF relatively hyperbolic free-by-cyclic group which is not hyperbolic then $G$ is the fundamental group of a hyperbolic 3-manifold.",
   "clean_statement": "Relatively hyperbolic case\n\nIf $G$ is a LERF relatively hyperbolic free-by-cyclic group which is not hyperbolic then $G$ is the fundamental group of a hyperbolic 3-manifold.",
   "public_statement": "Relatively hyperbolic case\n\nIf $G$ is a LERF relatively hyperbolic free-by-cyclic group which is not hyperbolic then $G$ is the fundamental group of a hyperbolic 3-manifold.",
   "evidence": "The canonical AIM record, in the section “Subgroup separability and finite quotients” of the workshop *Rigidity properties of free-by-cyclic groups*, states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0008": {
   "statement_status": "exact",
   "original_statement": "Hyperbolic case\n\nIf $G$ is a hyperbolic free-by-cyclic group with irreducible monodromy then $G$ is LERF if and only if the BNS invariant of every finite index subgroup of $G$ is symmetric.",
   "clean_statement": "Hyperbolic case\n\nIf $G$ is a hyperbolic free-by-cyclic group with irreducible monodromy then $G$ is LERF if and only if the BNS invariant of every finite index subgroup of $G$ is symmetric.",
   "public_statement": "Hyperbolic case\n\nIf $G$ is a hyperbolic free-by-cyclic group with irreducible monodromy then $G$ is LERF if and only if the BNS invariant of every finite index subgroup of $G$ is symmetric.",
   "evidence": "The canonical record is AIM problem 2.2 in the workshop list *Rigidity properties of free-by-cyclic groups*, section “Subgroup separability and finite quotients.” Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0009": {
   "statement_status": "exact",
   "original_statement": "Construct a hyperbolic free-by-cyclic group which is LERF.",
   "clean_statement": "Construct a hyperbolic free-by-cyclic group which is LERF.",
   "public_statement": "Construct a hyperbolic free-by-cyclic group which is LERF.",
   "evidence": "The canonical AIM record is Problem 2.3 in the section “Subgroup separability and finite quotients” of the 2023 list *Rigidity properties of free-by-cyclic groups*. Its exact problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0010": {
   "statement_status": "exact",
   "original_statement": "Congruence subgroup property\n\nLet $Q$ be a finite quotient of $\\mathrm{Out}(F_n)$ for $n > 2$. Does there exist a finite index characteristic subgroup $K$ of $F_n$ such that the quotient $\\mathrm{Out}(F_n) \\twoheadrightarrow Q$ factors the quotient $\\mathrm{Out}(F_n) \\twoheadrightarrow \\mathrm{Out}(F_n /K)$?",
   "clean_statement": "Congruence subgroup property\n\nLet $Q$ be a finite quotient of $\\mathrm{Out}(F_n)$ for $n > 2$. Does there exist a finite index characteristic subgroup $K$ of $F_n$ such that the quotient $\\mathrm{Out}(F_n) \\twoheadrightarrow Q$ factors the quotient $\\mathrm{Out}(F_n) \\twoheadrightarrow \\mathrm{Out}(F_n /K)$?",
   "public_statement": "Congruence subgroup property\n\nLet $Q$ be a finite quotient of $\\mathrm{Out}(F_n)$ for $n > 2$. Does there exist a finite index characteristic subgroup $K$ of $F_n$ such that the quotient $\\mathrm{Out}(F_n) \\twoheadrightarrow Q$ factors the quotient $\\mathrm{Out}(F_n) \\twoheadrightarrow \\mathrm{Out}(F_n /K)$?",
   "evidence": "The canonical record is Problem 2.4, “Congruence subgroup property,” in the section “Subgroup separability and finite quotients” of the AIM list *Rigidity properties of free-by-cyclic groups*. Its problem field is exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 9,
   "attempt": 2
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0011": {
   "statement_status": "exact",
   "original_statement": "Profinite invariants of free-by-cyclic groups\n\nWhich properties of free-by-cyclic groups are invariants of the profinite completion? How does the BNS invarian behave under profinite isomorphism of free-by-cyclic groups?",
   "clean_statement": "Profinite invariants of free-by-cyclic groups\n\nWhich properties of free-by-cyclic groups are invariants of the profinite completion? How does the BNS invarian behave under profinite isomorphism of free-by-cyclic groups?",
   "public_statement": "Profinite invariants of free-by-cyclic groups\n\nWhich properties of free-by-cyclic groups are invariants of the profinite completion? How does the BNS invarian behave under profinite isomorphism of free-by-cyclic groups?",
   "evidence": "The canonical AIM record and the live AIM page agree verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0012": {
   "statement_status": "exact",
   "original_statement": "Let $G$ and $H$ be free-by-cyclic groups which are quasi-isometric. Then $G$ admits irreducible and atoroidal monodromy if and only if $H$ admits irreducible and atoroidal monodromy.",
   "clean_statement": "Let $G$ and $H$ be free-by-cyclic groups which are quasi-isometric. Then $G$ admits irreducible and atoroidal monodromy if and only if $H$ admits irreducible and atoroidal monodromy.",
   "public_statement": "Let $G$ and $H$ be free-by-cyclic groups which are quasi-isometric. Then $G$ admits irreducible and atoroidal monodromy if and only if $H$ admits irreducible and atoroidal monodromy.",
   "evidence": "The live AIM page <http://aimpl.org/freebycyclic/3/> was checked on 2026-08-02. It reproduces this statement exactly, attributes it to Jean Pierre Mutanguha, and contains no status note. No textual correction is needed.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0013": {
   "statement_status": "exact",
   "original_statement": "If a group $G$ is quasi-isometric to a free-by-cyclic group then is it true that $G$ is virtually free-by-cyclic?",
   "clean_statement": "If a group $G$ is quasi-isometric to a free-by-cyclic group then is it true that $G$ is virtually free-by-cyclic?",
   "public_statement": "If a group $G$ is quasi-isometric to a free-by-cyclic group then is it true that $G$ is virtually free-by-cyclic?",
   "evidence": "The canonical AIM record (workshop *Rigidity properties of free-by-cyclic groups*, section “Quasi-isometric and measure equivalence rigidity,” Problem 3.2) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 12,
   "attempt": 2
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0014": {
   "statement_status": "exact",
   "original_statement": "Construct two hyperbolic irreducible free-by-cyclic groups which are not quasi-isometric.",
   "clean_statement": "Construct two hyperbolic irreducible free-by-cyclic groups which are not quasi-isometric.",
   "public_statement": "Construct two hyperbolic irreducible free-by-cyclic groups which are not quasi-isometric.",
   "evidence": "and irreducibility is a property of the outer automorphism of the finitely generated free group. This matters because Kielak--Linton use a broader convention in which the free kernel need not be finitely generated. Their 2024 theorem gives a striking near-solution, but not a solution to the statement in the workshop sense. No correction of the canonical text is needed.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0015": {
   "statement_status": "exact",
   "original_statement": "Let $F = F(a,b,c,d,e)$ be a free group of rank five. Define an automorphism $\\phi \\in \\mathrm{Aut}(F)$ such that\n\\[\\begin{split} \\phi \\colon F(a,b,c,d,e) &\\to F(a,b,c,d,e) \\\\\na &\\mapsto b\\\\\nb &\\mapsto c\\\\\nc &\\mapsto ab\\\\\nd &\\mapsto ea \\\\\ne &\\mapsto ed\n\\end{split}\\]\nLet $F'$ be the free factor of $F$ generated by $\\{a,b,c\\}$. Let $G = F \\rtimes_{\\phi} \\mathbb{Z}$ be the mapping torus of $\\phi$, and let $\\Gamma$ be the mapping torus of the restriction of $\\phi$ to $F'$. Then $\\Gamma$ and $G$ are not quasi-isometric.",
   "clean_statement": "Let $F = F(a,b,c,d,e)$ be a free group of rank five. Define an automorphism $\\phi \\in \\mathrm{Aut}(F)$ such that\n\\[\\begin{split} \\phi \\colon F(a,b,c,d,e) &\\to F(a,b,c,d,e) \\\\\na &\\mapsto b\\\\\nb &\\mapsto c\\\\\nc &\\mapsto ab\\\\\nd &\\mapsto ea \\\\\ne &\\mapsto ed\n\\end{split}\\]\nLet $F'$ be the free factor of $F$ generated by $\\{a,b,c\\}$. Let $G = F \\rtimes_{\\phi} \\mathbb{Z}$ be the mapping torus of $\\phi$, and let $\\Gamma$ be the mapping torus of the restriction of $\\phi$ to $F'$. Then $\\Gamma$ and $G$ are not quasi-isometric.",
   "public_statement": "Let $F = F(a,b,c,d,e)$ be a free group of rank five. Define an automorphism $\\phi \\in \\mathrm{Aut}(F)$ such that\n\\[\\begin{split} \\phi \\colon F(a,b,c,d,e) &\\to F(a,b,c,d,e) \\\\\na &\\mapsto b\\\\\nb &\\mapsto c\\\\\nc &\\mapsto ab\\\\\nd &\\mapsto ea \\\\\ne &\\mapsto ed\n\\end{split}\\]\nLet $F'$ be the free factor of $F$ generated by $\\{a,b,c\\}$. Let $G = F \\rtimes_{\\phi} \\mathbb{Z}$ be the mapping torus of $\\phi$, and let $\\Gamma$ be the mapping torus of the restriction of $\\phi$ to $F'$. Then $\\Gamma$ and $G$ are not quasi-isometric.",
   "evidence": "The canonical record is Conjecture 3.4 in the section “Quasi-isometric and measure equivalence rigidity” of the AIM list *Rigidity properties of free-by-cyclic groups*. The live AIM page attributes it to Jean Pierre Mutanguha and, as of 2026-08-02, displays exactly the same mathematical text as the corpus record:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0016": {
   "statement_status": "exact",
   "original_statement": "What are the measure equivalence classes of free-by-cyclic groups?",
   "clean_statement": "What are the measure equivalence classes of free-by-cyclic groups?",
   "public_statement": "What are the measure equivalence classes of free-by-cyclic groups?",
   "evidence": "The canonical record is AIM Problem List item 3.5 in the workshop *Rigidity properties of free-by-cyclic groups*, section “Quasi-isometric and measure equivalence rigidity.” Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0017": {
   "statement_status": "exact",
   "original_statement": "Let $G$ and $H$ be relatively hyperbolic non-hyperbolic free-by-cyclic groups with a trivial JSJ decomposition. If $G$ and $H$ are quasi-isometric then they are commensurable.",
   "clean_statement": "Let $G$ and $H$ be relatively hyperbolic non-hyperbolic free-by-cyclic groups with a trivial JSJ decomposition. If $G$ and $H$ are quasi-isometric then they are commensurable.",
   "public_statement": "Let $G$ and $H$ be relatively hyperbolic non-hyperbolic free-by-cyclic groups with a trivial JSJ decomposition. If $G$ and $H$ are quasi-isometric then they are commensurable.",
   "evidence": "The canonical record is AIM Problem List 3.6 from the workshop *Rigidity properties of free-by-cyclic groups*. The live AIM page was accessed on 2026-08-02. It gives Chris Leininger as proposer and states, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0018": {
   "statement_status": "exact",
   "original_statement": "$\\ell^2$-torsion\n\nClay shows that $\\ell^2$-torsion vanishes for free-by-cyclic groups with polynomially-growing monodromies \\cite{MR3667215}.\n\nIf $G$ is a free-by-cyclic group with exponentially growing monodromy then it has non-vanishing $\\ell^2$-torsion.",
   "clean_statement": "$\\ell^2$-torsion\n\nClay shows that $\\ell^2$-torsion vanishes for free-by-cyclic groups with polynomially-growing monodromies \\cite{MR3667215}.\n\nIf $G$ is a free-by-cyclic group with exponentially growing monodromy then it has non-vanishing $\\ell^2$-torsion.",
   "public_statement": "$\\ell^2$-torsion\n\nClay shows that $\\ell^2$-torsion vanishes for free-by-cyclic groups with polynomially-growing monodromies \\cite{MR3667215}.\n\nIf $G$ is a free-by-cyclic group with exponentially growing monodromy then it has non-vanishing $\\ell^2$-torsion.",
   "evidence": "The exact repository record is unambiguous and contains no apparent OCR corruption. The live AIM section URL did not render in the available browser, so the displayed wording above was checked against the canonical JSON record rather than silently reconstructed from a different version.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0019": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "If $G$ is a free-by-cyclic group with exponentially growing monodromy then does it have non-vashing homology torsion growth?",
   "clean_statement": null,
   "public_statement": "If $G$ is a free-by-cyclic group with exponentially growing monodromy then does it have non-vashing homology torsion growth?",
   "evidence": "Two plausible readings remain:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 18,
   "attempt": 2
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0020": {
   "statement_status": "exact",
   "original_statement": "Let $G$ be a non-geometric free-by-cyclic group and $A$ a finitely generated abelian group. Does there exist a finite index subgroup $G'$ of $G$ such that $A$ is isomorphic to a direct summand of the abelianisation of $G'$?",
   "clean_statement": "Let $G$ be a non-geometric free-by-cyclic group and $A$ a finitely generated abelian group. Does there exist a finite index subgroup $G'$ of $G$ such that $A$ is isomorphic to a direct summand of the abelianisation of $G'$?",
   "public_statement": "Let $G$ be a non-geometric free-by-cyclic group and $A$ a finitely generated abelian group. Does there exist a finite index subgroup $G'$ of $G$ such that $A$ is isomorphic to a direct summand of the abelianisation of $G'$?",
   "evidence": "The canonical record is Problem 4.3 in the “Homological properties” section of the AIM list *Rigidity properties of free-by-cyclic groups*. The live AIM page was accessed on 2026-08-02. It attributes the problem to Tam Cheetham-West and states, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0021": {
   "statement_status": "exact",
   "original_statement": "RFRS\n\nA group $G$ is \\emph{residually finite rationally solvable (RFRS)} if there exists a non-ascending residual chain $(H_i)_{i \\in \\mathbb{N}}$ of finite index normal subgroups $N_i \\trianglelefteq G$ with $H_0 = G$, such that $\\mathrm{ker}\\, \\alpha_i \\leq H_{i+1}$ for every $i$, where $\\alpha_i \\colon H_i \\to H_i^{\\mathrm{fab}}$ is the free abelianisation map.\n\nCharacterise the non-hyperbolic free-by-cyclic groups which are virtually RFRS.",
   "clean_statement": "RFRS\n\nA group $G$ is \\emph{residually finite rationally solvable (RFRS)} if there exists a non-ascending residual chain $(H_i)_{i \\in \\mathbb{N}}$ of finite index normal subgroups $N_i \\trianglelefteq G$ with $H_0 = G$, such that $\\mathrm{ker}\\, \\alpha_i \\leq H_{i+1}$ for every $i$, where $\\alpha_i \\colon H_i \\to H_i^{\\mathrm{fab}}$ is the free abelianisation map.\n\nCharacterise the non-hyperbolic free-by-cyclic groups which are virtually RFRS.",
   "public_statement": "RFRS\n\nA group $G$ is \\emph{residually finite rationally solvable (RFRS)} if there exists a non-ascending residual chain $(H_i)_{i \\in \\mathbb{N}}$ of finite index normal subgroups $N_i \\trianglelefteq G$ with $H_0 = G$, such that $\\mathrm{ker}\\, \\alpha_i \\leq H_{i+1}$ for every $i$, where $\\alpha_i \\colon H_i \\to H_i^{\\mathrm{fab}}$ is the free abelianisation map.\n\nCharacterise the non-hyperbolic free-by-cyclic groups which are virtually RFRS.",
   "evidence": "There is a genuine notational error on the source page: the chain is named \\((H_i)\\), but its terms are then called \\(N_i\\). This is not an OCR error in the repository. We do not silently repair the quotation.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0022": {
   "statement_status": "exact",
   "original_statement": "Are all free-by-cyclic groups virtually (residually torsion-free nilpotent)?",
   "clean_statement": "Are all free-by-cyclic groups virtually (residually torsion-free nilpotent)?",
   "public_statement": "Are all free-by-cyclic groups virtually (residually torsion-free nilpotent)?",
   "evidence": "The canonical AIM record asks, exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0023": {
   "statement_status": "exact",
   "original_statement": "Characterise the free-by-cyclic groups which act geometrically on CAT(0) spaces.",
   "clean_statement": "Characterise the free-by-cyclic groups which act geometrically on CAT(0) spaces.",
   "public_statement": "Characterise the free-by-cyclic groups which act geometrically on CAT(0) spaces.",
   "evidence": "There is no OCR corruption or missing formula in this record.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0024": {
   "statement_status": "exact",
   "original_statement": "Let $F = F(a,b,c)$ be a free group of rank 3. Define an automorphism $\\phi \\in \\mathrm{Aut}(F)$ such that\n\\[ \\begin{split} \\phi \\colon F(a,b,c) &\\to F(a,b,c) \\\\\na &\\mapsto a \\\\\nb &\\mapsto ab \\\\\nc &\\mapsto bcb.\n\\end{split}\n\\]\nDoes $G = F \\rtimes_{\\phi} \\mathbb{Z}$ act geometrically on a CAT(0) cube complex?",
   "clean_statement": "Let $F = F(a,b,c)$ be a free group of rank 3. Define an automorphism $\\phi \\in \\mathrm{Aut}(F)$ such that\n\\[ \\begin{split} \\phi \\colon F(a,b,c) &\\to F(a,b,c) \\\\\na &\\mapsto a \\\\\nb &\\mapsto ab \\\\\nc &\\mapsto bcb.\n\\end{split}\n\\]\nDoes $G = F \\rtimes_{\\phi} \\mathbb{Z}$ act geometrically on a CAT(0) cube complex?",
   "public_statement": "Let $F = F(a,b,c)$ be a free group of rank 3. Define an automorphism $\\phi \\in \\mathrm{Aut}(F)$ such that\n\\[ \\begin{split} \\phi \\colon F(a,b,c) &\\to F(a,b,c) \\\\\na &\\mapsto a \\\\\nb &\\mapsto ab \\\\\nc &\\mapsto bcb.\n\\end{split}\n\\]\nDoes $G = F \\rtimes_{\\phi} \\mathbb{Z}$ act geometrically on a CAT(0) cube complex?",
   "evidence": "The canonical record is Problem 5.2 in the “Geometry of free-by-cyclic groups” section of the AIM list *Rigidity properties of free-by-cyclic groups*. The live page was checked on 2026-08-02. It attributes the problem to Rylee Lyman and states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0025": {
   "statement_status": "exact",
   "original_statement": "What's the relationship between being CAT(0) and RFRS for free-by-cyclic groups?",
   "clean_statement": "What's the relationship between being CAT(0) and RFRS for free-by-cyclic groups?",
   "public_statement": "What's the relationship between being CAT(0) and RFRS for free-by-cyclic groups?",
   "evidence": "There is no OCR corruption in this record. There are, however, two essential scope ambiguities.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0026": {
   "statement_status": "exact",
   "original_statement": "Let $G$ be a free-by-cyclic group and $F$ a component of the BNS invariant of $G$. When is it the case that the stretch factor functions $\\lambda_{+}, \\lambda_{-} \\colon F \\to \\mathbb{R}$ agree on a convex subset of $F$?",
   "clean_statement": "Let $G$ be a free-by-cyclic group and $F$ a component of the BNS invariant of $G$. When is it the case that the stretch factor functions $\\lambda_{+}, \\lambda_{-} \\colon F \\to \\mathbb{R}$ agree on a convex subset of $F$?",
   "public_statement": "Let $G$ be a free-by-cyclic group and $F$ a component of the BNS invariant of $G$. When is it the case that the stretch factor functions $\\lambda_{+}, \\lambda_{-} \\colon F \\to \\mathbb{R}$ agree on a convex subset of $F$?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0027": {
   "statement_status": "exact",
   "original_statement": "Find families of examples of free-by-cyclic groups $G$ such that $b_1(G) > 2$ and the BNS invariant of $G$ has many components (up to the action of $\\mathrm{Out}(G)$).",
   "clean_statement": "Find families of examples of free-by-cyclic groups $G$ such that $b_1(G) > 2$ and the BNS invariant of $G$ has many components (up to the action of $\\mathrm{Out}(G)$).",
   "public_statement": "Find families of examples of free-by-cyclic groups $G$ such that $b_1(G) > 2$ and the BNS invariant of $G$ has many components (up to the action of $\\mathrm{Out}(G)$).",
   "evidence": "The canonical record is Problem 6.2 in the “BNS invariants” section of the AIM list *Rigidity properties of free-by-cyclic groups*. The live page was checked on 2026-08-02 and agrees exactly with the record:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0028": {
   "statement_status": "exact",
   "original_statement": "If $P$ is a polytope in $\\mathbb{R}^n$ with $n \\geq 2$ with a marking on its top dimensional faces, determine whether the cone over the marked faces of $P$ corresponds to a BNS invariant of a (free-by-cyclic) group.",
   "clean_statement": "If $P$ is a polytope in $\\mathbb{R}^n$ with $n \\geq 2$ with a marking on its top dimensional faces, determine whether the cone over the marked faces of $P$ corresponds to a BNS invariant of a (free-by-cyclic) group.",
   "public_statement": "If $P$ is a polytope in $\\mathbb{R}^n$ with $n \\geq 2$ with a marking on its top dimensional faces, determine whether the cone over the marked faces of $P$ corresponds to a BNS invariant of a (free-by-cyclic) group.",
   "evidence": "The AIM source (section 6, “BNS invariants,” Problem 6.3, attributed on the live page to Rylee Lyman) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0029": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A group $G$ is said to be of type $VF$ if there exists a finite index subgroup $G'$ of $G$ which admits a finite classifying space.\n\nIf $G$ is a free-by-cyclic group then $\\mathrm{Out}(G)$ is of type $VF$.",
   "clean_statement": null,
   "public_statement": "A group $G$ is said to be of type $VF$ if there exists a finite index subgroup $G'$ of $G$ which admits a finite classifying space.\n\nIf $G$ is a free-by-cyclic group then $\\mathrm{Out}(G)$ is of type $VF$.",
   "evidence": "The canonical record (AIM workshop *Rigidity properties of free-by-cyclic groups*, Miscellaneous 7.1) says:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0030": {
   "statement_status": "exact",
   "original_statement": "Solve the subgroup membership problem for free-by-cyclic groups.",
   "clean_statement": "Solve the subgroup membership problem for free-by-cyclic groups.",
   "public_statement": "Solve the subgroup membership problem for free-by-cyclic groups.",
   "evidence": "The canonical AIM record, Problem 7.2 in the “Miscellaneous” section of *Rigidity properties of free-by-cyclic groups*, says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0031": {
   "statement_status": "exact",
   "original_statement": "Are free-by-cyclic groups linear? When do they embed in $\\mathrm{SL}(2, \\mathbb{C})$?",
   "clean_statement": "Are free-by-cyclic groups linear? When do they embed in $\\mathrm{SL}(2, \\mathbb{C})$?",
   "public_statement": "Are free-by-cyclic groups linear? When do they embed in $\\mathrm{SL}(2, \\mathbb{C})$?",
   "evidence": "The canonical AIM record (workshop *Rigidity properties of free-by-cyclic groups*, section “Miscellaneous,” Problem 7.3) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0032": {
   "statement_status": "exact",
   "original_statement": "Every outer automorphism $\\phi \\in \\mathrm{Out}(F_n)$ admits a poset $P$ of attracting laminations. What is the length of the longest chain in $P$? Is the length an invariant of $G$?",
   "clean_statement": "Every outer automorphism $\\phi \\in \\mathrm{Out}(F_n)$ admits a poset $P$ of attracting laminations. What is the length of the longest chain in $P$? Is the length an invariant of $G$?",
   "public_statement": "Every outer automorphism $\\phi \\in \\mathrm{Out}(F_n)$ admits a poset $P$ of attracting laminations. What is the length of the longest chain in $P$? Is the length an invariant of $G$?",
   "evidence": "The canonical record is Problem 7.4 in the “Miscellaneous” section of the AIM list *Rigidity properties of free-by-cyclic groups*. The live AIM page was checked on 2026-08-02. It has no status remark and gives the following wording, attributed to Jean Pierre Mutanguha:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0033": {
   "statement_status": "exact",
   "original_statement": "Develop a fibred face theory for four-manifolds which fibre over the circle with fibers handlebodies or connect sums of $\\mathbb{S}^2 \\times \\mathbb{S}^1$.",
   "clean_statement": "Develop a fibred face theory for four-manifolds which fibre over the circle with fibers handlebodies or connect sums of $\\mathbb{S}^2 \\times \\mathbb{S}^1$.",
   "public_statement": "Develop a fibred face theory for four-manifolds which fibre over the circle with fibers handlebodies or connect sums of $\\mathbb{S}^2 \\times \\mathbb{S}^1$.",
   "evidence": "The canonical AIM record is Problem 7.5 in the “Miscellaneous” section of the 2023 workshop *Rigidity properties of free-by-cyclic groups*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0034": {
   "statement_status": "exact",
   "original_statement": "Does there exist an analogue of pseudo-Anosov maps for homeomorphisms of handlebodies or connect sums of $\\mathbb{S}^2 \\times \\mathbb{S}^1$?",
   "clean_statement": "Does there exist an analogue of pseudo-Anosov maps for homeomorphisms of handlebodies or connect sums of $\\mathbb{S}^2 \\times \\mathbb{S}^1$?",
   "public_statement": "Does there exist an analogue of pseudo-Anosov maps for homeomorphisms of handlebodies or connect sums of $\\mathbb{S}^2 \\times \\mathbb{S}^1$?",
   "evidence": "The canonical AIM record, from the workshop *Rigidity properties of free-by-cyclic groups*, section “Miscellaneous,” Problem 7.6, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 33,
   "attempt": 2
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0035": {
   "statement_status": "exact",
   "original_statement": "Does the $K(\\pi,1)$ conjecture hold for all Artin groups?",
   "clean_statement": "Does the $K(\\pi,1)$ conjecture hold for all Artin groups?",
   "public_statement": "Does the $K(\\pi,1)$ conjecture hold for all Artin groups?",
   "evidence": "The canonical record is from the AIM workshop *Geometry and topology of Artin groups*, section “The big questions,” Problem 1.1:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0036": {
   "statement_status": "exact",
   "original_statement": "Are all Artin groups torsion-free?",
   "clean_statement": "Are all Artin groups torsion-free?",
   "public_statement": "Are all Artin groups torsion-free?",
   "evidence": "The canonical record is Problem 1.2 in the section “The big questions” of the AIM list *Geometry and topology of Artin groups*. The live AIM page was checked on 2026-08-02. It contains no attribution, qualification, status note, or remark, and its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0037": {
   "statement_status": "exact",
   "original_statement": "Are the irreducible non-spherical Artin groups centerless? If not, describe the centers.",
   "clean_statement": "Are the irreducible non-spherical Artin groups centerless? If not, describe the centers.",
   "public_statement": "Are the irreducible non-spherical Artin groups centerless? If not, describe the centers.",
   "evidence": "The AIM record (workshop *Geometry and topology of Artin groups*, section “The big questions,” Problem 1.3) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0038": {
   "statement_status": "exact",
   "original_statement": "Is each Artin group isomorphic to its dual?",
   "clean_statement": "Is each Artin group isomorphic to its dual?",
   "public_statement": "Is each Artin group isomorphic to its dual?",
   "evidence": "The question is whether $\\Psi_c$ is an isomorphism for **every** $(W,S)$ and **every** choice of $c$. This is stronger and more precise than asking for some unspecified abstract isomorphism. The source record has no OCR corruption; its only defect is suppression of the essential parameter $c$ and of the word “canonical.”",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0039": {
   "statement_status": "exact",
   "original_statement": "Is the word problem solvable for all Artin groups?",
   "clean_statement": "Is the word problem solvable for all Artin groups?",
   "public_statement": "Is the word problem solvable for all Artin groups?",
   "evidence": "There is no OCR corruption or missing mathematical notation. The original AIM URL timed out when checked on 2 August 2026, so the wording above is verified from the canonical repository record, not from a fresh rendering of the webpage.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0040": {
   "statement_status": "exact",
   "original_statement": "Is there an explicit algorithm to solve the word problem for all Artin groups?",
   "clean_statement": "Is there an explicit algorithm to solve the word problem for all Artin groups?",
   "public_statement": "Is there an explicit algorithm to solve the word problem for all Artin groups?",
   "evidence": "The canonical record is AIM Problem 2.2 from the workshop *Geometry and topology of Artin groups*. The exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0041": {
   "statement_status": "exact",
   "original_statement": "(See Question 1.4 of \\cite{MR2983847} for the algorithm.)\n\nDoes Dehornoy-Godelle's algorithm solve the word problem for all Artin groups?",
   "clean_statement": "(See Question 1.4 of \\cite{MR2983847} for the algorithm.)\n\nDoes Dehornoy-Godelle's algorithm solve the word problem for all Artin groups?",
   "public_statement": "(See Question 1.4 of \\cite{MR2983847} for the algorithm.)\n\nDoes Dehornoy-Godelle's algorithm solve the word problem for all Artin groups?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 40,
   "attempt": 2
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0042": {
   "statement_status": "exact",
   "original_statement": "We call an Artin group \\emph{hyperbolic} if the quadratic form of the associated Coxeter group is rank $(n, 1)$.\n\nFind more examples of $\\geq 3$-dimensional hyperbolic Artin groups (whose diagrams have no edges labeled $\\infty$) where the word problem is solvable.",
   "clean_statement": "We call an Artin group \\emph{hyperbolic} if the quadratic form of the associated Coxeter group is rank $(n, 1)$.\n\nFind more examples of $\\geq 3$-dimensional hyperbolic Artin groups (whose diagrams have no edges labeled $\\infty$) where the word problem is solvable.",
   "public_statement": "We call an Artin group \\emph{hyperbolic} if the quadratic form of the associated Coxeter group is rank $(n, 1)$.\n\nFind more examples of $\\geq 3$-dimensional hyperbolic Artin groups (whose diagrams have no edges labeled $\\infty$) where the word problem is solvable.",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0043": {
   "statement_status": "exact",
   "original_statement": "If $g = a\\overline{b}$ for $a,b$ words in the positive monoid with no cancellation in the middle, are $a$ and $b$ unique?",
   "clean_statement": "If $g = a\\overline{b}$ for $a,b$ words in the positive monoid with no cancellation in the middle, are $a$ and $b$ unique?",
   "public_statement": "If $g = a\\overline{b}$ for $a,b$ words in the positive monoid with no cancellation in the middle, are $a$ and $b$ unique?",
   "evidence": "The canonical AIM record is Problem 2.5 in the section “The word problem” of the 2023 workshop *Geometry and topology of Artin groups*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0044": {
   "statement_status": "exact",
   "original_statement": "Is the class of spherical Artin groups closed under isomorphism? In other words, if an Artin group $A$ is isomorphic to a spherical Artin group, must $A$ be spherical?",
   "clean_statement": "Is the class of spherical Artin groups closed under isomorphism? In other words, if an Artin group $A$ is isomorphic to a spherical Artin group, must $A$ be spherical?",
   "public_statement": "Is the class of spherical Artin groups closed under isomorphism? In other words, if an Artin group $A$ is isomorphic to a spherical Artin group, must $A$ be spherical?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0045": {
   "statement_status": "exact",
   "original_statement": "Solve the isomorphism problem for 2-dimensional Artin groups.",
   "clean_statement": "Solve the isomorphism problem for 2-dimensional Artin groups.",
   "public_statement": "Solve the isomorphism problem for 2-dimensional Artin groups.",
   "evidence": "The canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0046": {
   "statement_status": "exact",
   "original_statement": "Solve the isomorphism problem for $\\infty$-free 2-dimensional Artin groups.",
   "clean_statement": "Solve the isomorphism problem for $\\infty$-free 2-dimensional Artin groups.",
   "public_statement": "Solve the isomorphism problem for $\\infty$-free 2-dimensional Artin groups.",
   "evidence": "The canonical AIM record (Geometry and topology of Artin groups, problem 3.3) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0047": {
   "statement_status": "exact",
   "original_statement": "For an Artin group $A$, when is $\\mathrm{Out}(A)$ finite/finitely generated? Find explicit generators.",
   "clean_statement": "For an Artin group $A$, when is $\\mathrm{Out}(A)$ finite/finitely generated? Find explicit generators.",
   "public_statement": "For an Artin group $A$, when is $\\mathrm{Out}(A)$ finite/finitely generated? Find explicit generators.",
   "evidence": "The record itself contains no qualifications on the Coxeter matrix. Thus it is a programmatic question about all finite-rank Artin groups, not a single yes/no conjecture. No corruption or ambiguity is visible in the source record. Throughout, an Artin group is assumed to have a finite standard generating set.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0048": {
   "statement_status": "exact",
   "original_statement": "For fixed $A$, which spherical and/or dihedral Artin groups embed in $A$?",
   "clean_statement": "For fixed $A$, which spherical and/or dihedral Artin groups embed in $A$?",
   "public_statement": "For fixed $A$, which spherical and/or dihedral Artin groups embed in $A$?",
   "evidence": "The canonical AIM record is problem 3.7 in the section “The isomorphism problem” from the workshop *Geometry and topology of Artin groups*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0049": {
   "statement_status": "exact",
   "original_statement": "For an Artin group $A$, which Artin groups $A'$ embed in $A$ such that $A'$ is not a parabolic subgroup?",
   "clean_statement": "For an Artin group $A$, which Artin groups $A'$ embed in $A$ such that $A'$ is not a parabolic subgroup?",
   "public_statement": "For an Artin group $A$, which Artin groups $A'$ embed in $A$ such that $A'$ is not a parabolic subgroup?",
   "evidence": "The exact canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0050": {
   "statement_status": "exact",
   "original_statement": "Classify $\\mathrm{End}(A)$.",
   "clean_statement": "Classify $\\mathrm{End}(A)$.",
   "public_statement": "Classify $\\mathrm{End}(A)$.",
   "evidence": "There is no OCR corruption. The notation \\(\\operatorname{End}(A)\\) is interpreted as the monoid of group endomorphisms under composition. The word “classify” is necessarily programmatic: \\(A\\) is not restricted to one Artin type, and in the literature a classification may be literal, up to conjugacy, or by a finite list of normal forms. The original AIM webpage returned a 502 error during this run, so the recovered wording is verified from the canonical repository record rather than a fresh copy of the webpage.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0051": {
   "statement_status": "exact",
   "original_statement": "Find the commensurability and Q.I. classification for $\\infty$-free Artin groups.\n\nA specific case: are $F_4$ and $H_4$ commensurable?",
   "clean_statement": "Find the commensurability and Q.I. classification for $\\infty$-free Artin groups.\n\nA specific case: are $F_4$ and $H_4$ commensurable?",
   "public_statement": "Find the commensurability and Q.I. classification for $\\infty$-free Artin groups.\n\nA specific case: are $F_4$ and $H_4$ commensurable?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0052": {
   "statement_status": "exact",
   "original_statement": "Are the braid groups $\\mathrm{CAT}(0)$?",
   "clean_statement": "Are the braid groups $\\mathrm{CAT}(0)$?",
   "public_statement": "Are the braid groups $\\mathrm{CAT}(0)$?",
   "evidence": "The canonical AIM record is problem 4.05 in the “Non-positive curvature” section of the workshop *Geometry and topology of Artin groups*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0053": {
   "statement_status": "exact",
   "original_statement": "Prove $Br_4$ does not act properly on a $\\mathrm{CAT}(0)$ cube complex.",
   "clean_statement": "Prove $Br_4$ does not act properly on a $\\mathrm{CAT}(0)$ cube complex.",
   "public_statement": "Prove $Br_4$ does not act properly on a $\\mathrm{CAT}(0)$ cube complex.",
   "evidence": "The canonical AIM record (Geometry and topology of Artin groups, section ``Non-positive curvature'', Problem 4.1) is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0054": {
   "statement_status": "exact",
   "original_statement": "Prove that hyperbolic triangle Artin groups do not act properly on a $\\mathrm{CAT}(0)$ cube complex.",
   "clean_statement": "Prove that hyperbolic triangle Artin groups do not act properly on a $\\mathrm{CAT}(0)$ cube complex.",
   "public_statement": "Prove that hyperbolic triangle Artin groups do not act properly on a $\\mathrm{CAT}(0)$ cube complex.",
   "evidence": "The exact canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0055": {
   "statement_status": "exact",
   "original_statement": "Does the $\\tilde{A}_2$ Artin group act properly and cocompactly on a Helly graph?",
   "clean_statement": "Does the $\\tilde{A}_2$ Artin group act properly and cocompactly on a Helly graph?",
   "public_statement": "Does the $\\tilde{A}_2$ Artin group act properly and cocompactly on a Helly graph?",
   "evidence": "The canonical record is AIM Problem 4.2 from the workshop list *Geometry and topology of Artin groups*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0056": {
   "statement_status": "exact",
   "original_statement": "Further classify the systolic Artin groups.",
   "clean_statement": "Further classify the systolic Artin groups.",
   "public_statement": "Further classify the systolic Artin groups.",
   "evidence": "The exact canonical record is AIM Problem 4.25 in the workshop list *Geometry and topology of Artin groups*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0057": {
   "statement_status": "exact",
   "original_statement": "Acylindrical hyperbolicity\n\nAll non-spherical irreducible Artin groups are acylindrically hyperbolic.",
   "clean_statement": "Acylindrical hyperbolicity\n\nAll non-spherical irreducible Artin groups are acylindrically hyperbolic.",
   "public_statement": "Acylindrical hyperbolicity\n\nAll non-spherical irreducible Artin groups are acylindrically hyperbolic.",
   "evidence": "The exact source record has no additional remarks. The linked AIM page timed out under both HTTP and HTTPS during this run, so no extra wording from the original page was available. There is no apparent OCR corruption in the canonical text.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0058": {
   "statement_status": "exact",
   "original_statement": "Describe the hyperbolic space on which the acylindrically hyperbolic Artin groups act.",
   "clean_statement": "Describe the hyperbolic space on which the acylindrically hyperbolic Artin groups act.",
   "public_statement": "Describe the hyperbolic space on which the acylindrically hyperbolic Artin groups act.",
   "evidence": "The canonical AIM record is problem 4.45 in the workshop *Geometry and topology of Artin groups*, section “Non-positive curvature”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0059": {
   "statement_status": "exact",
   "original_statement": "Is there a ``largest'' acylindrically hyperbolic space on which these Artin groups act that sees all loxodromic elements?",
   "clean_statement": "Is there a ``largest'' acylindrically hyperbolic space on which these Artin groups act that sees all loxodromic elements?",
   "public_statement": "Is there a ``largest'' acylindrically hyperbolic space on which these Artin groups act that sees all loxodromic elements?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0060": {
   "statement_status": "exact",
   "original_statement": "Which Artin groups are HHGs?",
   "clean_statement": "Which Artin groups are HHGs?",
   "public_statement": "Which Artin groups are HHGs?",
   "evidence": "The canonical AIM record (workshop *Geometry and topology of Artin groups*, section “Non-positive curvature,” Problem 4.3) asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0061": {
   "statement_status": "exact",
   "original_statement": "For $a,b \\in A$, say $a \\leq b$ if there is a positive word $c$ with $a = bc$. Is $(A,\\leq)$ a join-semilattice?",
   "clean_statement": "For $a,b \\in A$, say $a \\leq b$ if there is a positive word $c$ with $a = bc$. Is $(A,\\leq)$ a join-semilattice?",
   "public_statement": "For $a,b \\in A$, say $a \\leq b$ if there is a positive word $c$ with $a = bc$. Is $(A,\\leq)$ a join-semilattice?",
   "evidence": "No OCR corruption was found. The issue is under-specification, not damaged text.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0062": {
   "statement_status": "exact",
   "original_statement": "What kind of non-positive curvature might these complexes have? (systolic, Helly, CUB, $\\mathrm{CAT}(0)$?)",
   "clean_statement": "What kind of non-positive curvature might these complexes have? (systolic, Helly, CUB, $\\mathrm{CAT}(0)$?)",
   "public_statement": "What kind of non-positive curvature might these complexes have? (systolic, Helly, CUB, $\\mathrm{CAT}(0)$?)",
   "evidence": "The extracted record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0063": {
   "statement_status": "exact",
   "original_statement": "Find a direct combinatorial proof (i.e., without reference to mapping class groups) that $X$ is $\\delta$-hyperbolic for the braid groups.",
   "clean_statement": "Find a direct combinatorial proof (i.e., without reference to mapping class groups) that $X$ is $\\delta$-hyperbolic for the braid groups.",
   "public_statement": "Find a direct combinatorial proof (i.e., without reference to mapping class groups) that $X$ is $\\delta$-hyperbolic for the braid groups.",
   "evidence": "Thus \\(X\\) is the **complex of irreducible parabolic subgroups**. There is no OCR corruption in the displayed problem; the extraction merely omitted the section-level sentence defining the notation.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0064": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "If one of the complexes has non-positive curvature and all cell stabilizers have solvable word problem, does this mean the whole group has solvable word problem? Can this be generalized to arbitrary cocompact actions on NPC complexes? $\\mathrm{CAT}(0)$ cube complexes?",
   "clean_statement": null,
   "public_statement": "If one of the complexes has non-positive curvature and all cell stabilizers have solvable word problem, does this mean the whole group has solvable word problem? Can this be generalized to arbitrary cocompact actions on NPC complexes? $\\mathrm{CAT}(0)$ cube complexes?",
   "evidence": "The canonical record asks:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0065": {
   "statement_status": "exact",
   "original_statement": "Let $\\Sigma$ be the Davis complex of the associated Artin group. Take intersections of halfspaces in $\\Sigma$, then pull back via the natural surjection from $P$. What can we say about these subspaces of $P$? Specifically, when are they $\\pi_1$-injective?",
   "clean_statement": "Let $\\Sigma$ be the Davis complex of the associated Artin group. Take intersections of halfspaces in $\\Sigma$, then pull back via the natural surjection from $P$. What can we say about these subspaces of $P$? Specifically, when are they $\\pi_1$-injective?",
   "public_statement": "Let $\\Sigma$ be the Davis complex of the associated Artin group. Take intersections of halfspaces in $\\Sigma$, then pull back via the natural surjection from $P$. What can we say about these subspaces of $P$? Specifically, when are they $\\pi_1$-injective?",
   "evidence": "The AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0066": {
   "statement_status": "exact",
   "original_statement": "$K(\\pi,1)$ conjecturette: is $\\pi_2$ of Deligne complex/Salvetti complex/hyperplane complement always trivial? Find a proof using geometric/combinatorial techniques.",
   "clean_statement": "$K(\\pi,1)$ conjecturette: is $\\pi_2$ of Deligne complex/Salvetti complex/hyperplane complement always trivial? Find a proof using geometric/combinatorial techniques.",
   "public_statement": "$K(\\pi,1)$ conjecturette: is $\\pi_2$ of Deligne complex/Salvetti complex/hyperplane complement always trivial? Find a proof using geometric/combinatorial techniques.",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0067": {
   "statement_status": "exact",
   "original_statement": "Is the Moussong metric on the spherical Deligne complex for the type $B_3$ Artin group $\\mathrm{CAT}(1)$?",
   "clean_statement": "Is the Moussong metric on the spherical Deligne complex for the type $B_3$ Artin group $\\mathrm{CAT}(1)$?",
   "public_statement": "Is the Moussong metric on the spherical Deligne complex for the type $B_3$ Artin group $\\mathrm{CAT}(1)$?",
   "evidence": "The record is number 5.6 in the section “Complexes for Artin groups” of the AIM list *Geometry and topology of Artin groups*. The archived AIM page agrees with the record and adds, in March 2025, that the question was answered positively in arXiv:2503.15820. There is no apparent corruption or ambiguity in the extracted question.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0068": {
   "statement_status": "exact",
   "original_statement": "Does the $Br_4$ triangle complex embed in a (locally finite?) $\\tilde{A}_2$ building?",
   "clean_statement": "Does the $Br_4$ triangle complex embed in a (locally finite?) $\\tilde{A}_2$ building?",
   "public_statement": "Does the $Br_4$ triangle complex embed in a (locally finite?) $\\tilde{A}_2$ building?",
   "evidence": "This wording is reproduced by the archived AIM problem-list page, in the section “Complexes for Artin groups,” Problem 5.7. No OCR correction is needed. The page does not define “the $Br_4$ triangle complex.” From the terminology and the literature immediately relevant to the question, I interpret it as the two-dimensional equilateral-triangle complex $X$ obtained by projecting Brady's three-dimensional CAT(0) complex for the four-strand braid group along the central direction. Barré and Pichot call $X$ the **Brady complex**. Here $Br_4$ means the four-strand braid group, more usually written $B_4$.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0069": {
   "statement_status": "exact",
   "original_statement": "Is there a ``natural'' (cellular?) compactification of the hyperplane complement associated to an Artin group?",
   "clean_statement": "Is there a ``natural'' (cellular?) compactification of the hyperplane complement associated to an Artin group?",
   "public_statement": "Is there a ``natural'' (cellular?) compactification of the hyperplane complement associated to an Artin group?",
   "evidence": "The exact AIM problem (Geometry and topology of Artin groups, Section 5, Problem 5.8) is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0070": {
   "statement_status": "exact",
   "original_statement": "Which Artin groups embed in the mapping class group of a surface?",
   "clean_statement": "Which Artin groups embed in the mapping class group of a surface?",
   "public_statement": "Which Artin groups embed in the mapping class group of a surface?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 69,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0071": {
   "statement_status": "exact",
   "original_statement": "In \\cite{MR1911508}, several Artin groups are shown to be mapping class groups of certain orbifolds.\n\nFor an Artin group $A$ and orbifold $\\mathcal O$ with $A \\cong MCG(\\mathcal O)$, is there a (reasonably defined) curve complex for $\\mathcal O$ which agrees with the complex of irreducible parabolics for $A$?",
   "clean_statement": "In \\cite{MR1911508}, several Artin groups are shown to be mapping class groups of certain orbifolds.\n\nFor an Artin group $A$ and orbifold $\\mathcal O$ with $A \\cong MCG(\\mathcal O)$, is there a (reasonably defined) curve complex for $\\mathcal O$ which agrees with the complex of irreducible parabolics for $A$?",
   "public_statement": "In \\cite{MR1911508}, several Artin groups are shown to be mapping class groups of certain orbifolds.\n\nFor an Artin group $A$ and orbifold $\\mathcal O$ with $A \\cong MCG(\\mathcal O)$, is there a (reasonably defined) curve complex for $\\mathcal O$ which agrees with the complex of irreducible parabolics for $A$?",
   "evidence": "There is no OCR error in the AIM text, but there is a substantive **citation/premise mismatch**. Allcock defines an $n$-strand orbifold braid group",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 70,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0072": {
   "statement_status": "exact",
   "original_statement": "Are intersections of parabolic subgroups themselves parabolic? More specifically: is this true in the Euclidean case?",
   "clean_statement": "Are intersections of parabolic subgroups themselves parabolic? More specifically: is this true in the Euclidean case?",
   "public_statement": "Are intersections of parabolic subgroups themselves parabolic? More specifically: is this true in the Euclidean case?",
   "evidence": "The exact AIM problem, from the workshop *Geometry and topology of Artin groups*, Section 7, Problem 7.1, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0073": {
   "statement_status": "exact",
   "original_statement": "Are intersections of parabolic again parabolic when the Deligne complex has some notion of non-positive curvature? (i.e., systolic, injective, CUB, $\\mathrm{CAT}(0)$)",
   "clean_statement": "Are intersections of parabolic again parabolic when the Deligne complex has some notion of non-positive curvature? (i.e., systolic, injective, CUB, $\\mathrm{CAT}(0)$)",
   "public_statement": "Are intersections of parabolic again parabolic when the Deligne complex has some notion of non-positive curvature? (i.e., systolic, injective, CUB, $\\mathrm{CAT}(0)$)",
   "evidence": "The canonical AIM record (Geometry and topology of Artin groups, section \"Parabolic subgroups,\" Problem 7.2) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 72,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0074": {
   "statement_status": "exact",
   "original_statement": "Let $A$ be an Artin group with subgroup $H$. If for all $h \\in H$ there is a (spherical) parabolic containing $h$, does there exist a (spherical) parabolic containing $H$? Geometric version: If a group $G$ acts on $X$ and $H < G$ with each $h \\in H$ fixing a point of $X$, then does $H$ fix a point? What about when $X$ has non-positive curvature?",
   "clean_statement": "Let $A$ be an Artin group with subgroup $H$. If for all $h \\in H$ there is a (spherical) parabolic containing $h$, does there exist a (spherical) parabolic containing $H$? Geometric version: If a group $G$ acts on $X$ and $H < G$ with each $h \\in H$ fixing a point of $X$, then does $H$ fix a point? What about when $X$ has non-positive curvature?",
   "public_statement": "Let $A$ be an Artin group with subgroup $H$. If for all $h \\in H$ there is a (spherical) parabolic containing $h$, does there exist a (spherical) parabolic containing $H$? Geometric version: If a group $G$ acts on $X$ and $H < G$ with each $h \\in H$ fixing a point of $X$, then does $H$ fix a point? What about when $X$ has non-positive curvature?",
   "evidence": "No correction of the extracted text is needed. The parentheses leave two readings, with arbitrary parabolics or with spherical parabolics. The mathematically stronger geometric analogy is with **spherical** parabolics, because these are exactly the vertex stabilizers in the Deligne cube complex used below. All positive Artin statements in this report concern that reading.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 73,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0075": {
   "statement_status": "exact",
   "original_statement": "Are all Artin groups virtually residually finite?",
   "clean_statement": "Are all Artin groups virtually residually finite?",
   "public_statement": "Are all Artin groups virtually residually finite?",
   "evidence": "The canonical AIM record is problem 8.05 in the workshop section *Algebraic properties*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0076": {
   "statement_status": "exact",
   "original_statement": "Which Artin groups are profinitely rigid?",
   "clean_statement": "Which Artin groups are profinitely rigid?",
   "public_statement": "Which Artin groups are profinitely rigid?",
   "evidence": "The canonical record is AIM Problem List 8.15 from the workshop *Geometry and topology of Artin groups*, section “Algebraic properties”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0077": {
   "statement_status": "exact",
   "original_statement": "We say a group is \\emph{good in the sense of Serre} if it has the same cohomology as its profinite completion.\n\nAre all Artin groups good in the sense of Serre?",
   "clean_statement": "We say a group is \\emph{good in the sense of Serre} if it has the same cohomology as its profinite completion.\n\nAre all Artin groups good in the sense of Serre?",
   "public_statement": "We say a group is \\emph{good in the sense of Serre} if it has the same cohomology as its profinite completion.\n\nAre all Artin groups good in the sense of Serre?",
   "evidence": "The record has no apparent OCR corruption. The live AIM page was not retrievable during this run, so the quotation is verified against the exact repository record.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0078": {
   "statement_status": "exact",
   "original_statement": "Which Artin groups are Hopfian?",
   "clean_statement": "Which Artin groups are Hopfian?",
   "public_statement": "Which Artin groups are Hopfian?",
   "evidence": "The canonical AIM record is problem 8.2 in the workshop section *Algebraic properties*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0079": {
   "statement_status": "exact",
   "original_statement": "Does the group ring of an Artin group have zero divisors?",
   "clean_statement": "Does the group ring of an Artin group have zero divisors?",
   "public_statement": "Does the group ring of an Artin group have zero divisors?",
   "evidence": "The canonical AIM record is problem 8.25 in the workshop *Geometry and topology of Artin groups*, section “Algebraic properties”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0080": {
   "statement_status": "exact",
   "original_statement": "The generalized Tits conjecture",
   "clean_statement": "The generalized Tits conjecture",
   "public_statement": "The generalized Tits conjecture",
   "evidence": "The exact canonical AIM record says only:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0081": {
   "statement_status": "exact",
   "original_statement": "Does the Tits alternative hold for all Artin groups?",
   "clean_statement": "Does the Tits alternative hold for all Artin groups?",
   "public_statement": "Does the Tits alternative hold for all Artin groups?",
   "evidence": "The canonical AIM record is Problem 8.35 in the workshop list *Geometry and topology of Artin groups*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0082": {
   "statement_status": "exact",
   "original_statement": "Given $a,b \\in A$, is there an $n = n(a,b)$ such that $\\langle a^n, b^n \\rangle$ is free or free abelian?",
   "clean_statement": "Given $a,b \\in A$, is there an $n = n(a,b)$ such that $\\langle a^n, b^n \\rangle$ is free or free abelian?",
   "public_statement": "Given $a,b \\in A$, is there an $n = n(a,b)$ such that $\\langle a^n, b^n \\rangle$ is free or free abelian?",
   "evidence": "The canonical record and the live AIM page agree verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0083": {
   "statement_status": "exact",
   "original_statement": "Compute the centralizer of an arbitrary element.\n\nA more specific question: in a Euclidean Artin group, if two elements are contained in centers of spherical parabolic subgroups, when do the two elements commute?",
   "clean_statement": "Compute the centralizer of an arbitrary element.\n\nA more specific question: in a Euclidean Artin group, if two elements are contained in centers of spherical parabolic subgroups, when do the two elements commute?",
   "public_statement": "Compute the centralizer of an arbitrary element.\n\nA more specific question: in a Euclidean Artin group, if two elements are contained in centers of spherical parabolic subgroups, when do the two elements commute?",
   "evidence": "The canonical AIM record, numbered 8.45 in the repository, asks two questions:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 82,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0084": {
   "statement_status": "exact",
   "original_statement": "Find a ``good''/``natural'' (e.g., finite dimensional) linear representation for Artin groups.",
   "clean_statement": "Find a ``good''/``natural'' (e.g., finite dimensional) linear representation for Artin groups.",
   "public_statement": "Find a ``good''/``natural'' (e.g., finite dimensional) linear representation for Artin groups.",
   "evidence": "The record contains no remarks or attached literature. There is no visible OCR corruption. The words “good” and “natural” are intentionally not mathematical predicates, so this is not a single yes/no conjecture. I separate three questions:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 83,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0085": {
   "statement_status": "exact",
   "original_statement": "Prove Coxeter groups are residually finite without appealing to linearity.",
   "clean_statement": "Prove Coxeter groups are residually finite without appealing to linearity.",
   "public_statement": "Prove Coxeter groups are residually finite without appealing to linearity.",
   "evidence": "The canonical AIM record is Problem 8.55 in the workshop list *Geometry and topology of Artin groups*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 84,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0086": {
   "statement_status": "exact",
   "original_statement": "Which Artin groups have property $R_\\infty$?",
   "clean_statement": "Which Artin groups have property $R_\\infty$?",
   "public_statement": "Which Artin groups have property $R_\\infty$?",
   "evidence": "There is no visible OCR corruption. In the canonical array the nearby item numbers are \\(8.45,8.5,8.55,8.6\\). Thus the placement of \\(8.6\\) after \\(8.55\\) is consistent with decimal insertion/order and is not silently changed here. Both HTTP and HTTPS requests to the AIM page timed out during this run, so the wording and order were verified from the canonical repository record and its neighbors, not independently from the live page.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 85,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0087": {
   "statement_status": "exact",
   "original_statement": "Is the Thompson group $F$ amenable?",
   "clean_statement": "Is the Thompson group $F$ amenable?",
   "public_statement": "Is the Thompson group $F$ amenable?",
   "evidence": "The canonical AIM record is workshop *Amenability of discrete groups*, section *Thompson group F and groups of homeomorphisms of the interval and the circle*, Problem 1.1. Its statement is exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 86,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0088": {
   "statement_status": "exact",
   "original_statement": "What can be said about the structure of subgroups of $F$?",
   "clean_statement": "What can be said about the structure of subgroups of $F$?",
   "public_statement": "What can be said about the structure of subgroups of $F$?",
   "evidence": "The canonical record is workshop *Amenability of discrete groups*, section *Thompson group F and groups of homeomorphisms of the interval and the circle*, Problem 1.3. The exact problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 87,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0089": {
   "statement_status": "exact",
   "original_statement": "We say that a collection of groups $\\mathcal C$ is \\emph{quasi-ordered} if for any sequence $\\{G_i\\}_{i\\in\\mathbb{N}}$ of groups in $\\mathcal C$ there is $i< j$ such that $G_i$ embeds into $G_j$.\n\nIs it true that a subgroup of $F$ is elementary amenable if and only if it does not contain a copy of $F$?",
   "clean_statement": "We say that a collection of groups $\\mathcal C$ is \\emph{quasi-ordered} if for any sequence $\\{G_i\\}_{i\\in\\mathbb{N}}$ of groups in $\\mathcal C$ there is $i< j$ such that $G_i$ embeds into $G_j$.\n\nIs it true that a subgroup of $F$ is elementary amenable if and only if it does not contain a copy of $F$?",
   "public_statement": "We say that a collection of groups $\\mathcal C$ is \\emph{quasi-ordered} if for any sequence $\\{G_i\\}_{i\\in\\mathbb{N}}$ of groups in $\\mathcal C$ there is $i< j$ such that $G_i$ embeds into $G_j$.\n\nIs it true that a subgroup of $F$ is elementary amenable if and only if it does not contain a copy of $F$?",
   "evidence": "The canonical record is Problem 1.4 in the AIM list *Amenability of discrete groups*, in the section “Thompson group $F$ and groups of homeomorphisms of the interval and the circle.” Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 88,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0090": {
   "statement_status": "exact",
   "original_statement": "The \\emph{derivative} of a subset $S$ of a topological space $X$, denoted $S'$, is the set of limit points of $S$. By transfinite induction, one defines the $\\alpha$-derivative of $S$ for every ordinal $\\alpha$. The \\emph{Cantor-Bendixon} rank of $S$ is the smallest ordinal $\\beta$ such that the $\\beta$-derivative of $S$ and the\n$(\\beta+1)$-derivative of $S$ coincide.\n\nWhat is the Cantor-Bendixon rank of the space of Subgroups of $F$?",
   "clean_statement": "The \\emph{derivative} of a subset $S$ of a topological space $X$, denoted $S'$, is the set of limit points of $S$. By transfinite induction, one defines the $\\alpha$-derivative of $S$ for every ordinal $\\alpha$. The \\emph{Cantor-Bendixon} rank of $S$ is the smallest ordinal $\\beta$ such that the $\\beta$-derivative of $S$ and the\n$(\\beta+1)$-derivative of $S$ coincide.\n\nWhat is the Cantor-Bendixon rank of the space of Subgroups of $F$?",
   "public_statement": "The \\emph{derivative} of a subset $S$ of a topological space $X$, denoted $S'$, is the set of limit points of $S$. By transfinite induction, one defines the $\\alpha$-derivative of $S$ for every ordinal $\\alpha$. The \\emph{Cantor-Bendixon} rank of $S$ is the smallest ordinal $\\beta$ such that the $\\beta$-derivative of $S$ and the\n$(\\beta+1)$-derivative of $S$ coincide.\n\nWhat is the Cantor-Bendixon rank of the space of Subgroups of $F$?",
   "evidence": "The canonical AIM record, from the 2016 workshop *Amenability of discrete groups*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 89,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0091": {
   "statement_status": "exact",
   "original_statement": "Is there an invariant \\emph{continuous} probability measure for the action of $F$ acting on its perfect kernel (by conjugation)?",
   "clean_statement": "Is there an invariant \\emph{continuous} probability measure for the action of $F$ acting on its perfect kernel (by conjugation)?",
   "public_statement": "Is there an invariant \\emph{continuous} probability measure for the action of $F$ acting on its perfect kernel (by conjugation)?",
   "evidence": "The exact canonical AIM problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 90,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0092": {
   "statement_status": "exact",
   "original_statement": "What are the maximal subgroups of $F$, and specifically, are there elementary amenable maximal ones?",
   "clean_statement": "What are the maximal subgroups of $F$, and specifically, are there elementary amenable maximal ones?",
   "public_statement": "What are the maximal subgroups of $F$, and specifically, are there elementary amenable maximal ones?",
   "evidence": "The archived AIM page agrees verbatim with the JSON record. No mathematical symbols or qualifications had to be reconstructed. Throughout, “maximal subgroup” means a **maximal proper subgroup of \\(F\\)**, not a largest or maximum subgroup. The question concerns subgroups of Thompson's group \\(F\\) itself, not maximal subgroups of the ambient group \\(\\mathrm{PL}_o([0,1])\\).",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 91,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0093": {
   "statement_status": "exact",
   "original_statement": "Is $F$ sofic?",
   "clean_statement": "Is $F$ sofic?",
   "public_statement": "Is $F$ sofic?",
   "evidence": "The canonical record is AIM Problem Lists, workshop *Amenability of discrete groups*, section “Thompson group F and groups of homeomorphisms of the interval and the circle,” item 1.2. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0094": {
   "statement_status": "exact",
   "original_statement": "We say that a group $G$ is \\emph{invariably generated} by $\\{x_1,\\dots,x_n\\}$ if for any $g_1, \\ldots, g_n \\in F$, $\\{g_1^{-1}x_1g_1, \\ldots, g_n^{-1}x_ng_n\\}$ generates $G$.\n\nIs $F$ is invariably generated by a finite set?",
   "clean_statement": "We say that a group $G$ is \\emph{invariably generated} by $\\{x_1,\\dots,x_n\\}$ if for any $g_1, \\ldots, g_n \\in F$, $\\{g_1^{-1}x_1g_1, \\ldots, g_n^{-1}x_ng_n\\}$ generates $G$.\n\nIs $F$ is invariably generated by a finite set?",
   "public_statement": "We say that a group $G$ is \\emph{invariably generated} by $\\{x_1,\\dots,x_n\\}$ if for any $g_1, \\ldots, g_n \\in F$, $\\{g_1^{-1}x_1g_1, \\ldots, g_n^{-1}x_ng_n\\}$ generates $G$.\n\nIs $F$ is invariably generated by a finite set?",
   "evidence": "The canonical input preserves AIM Problem List item 1.9 exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0095": {
   "statement_status": "exact",
   "original_statement": "Let $H$ be the group of all piecewise-projective homeomorphisms of $S^1$ that fix $\\infty$.\n\nWhat is the subgroup structure of $H$?\n\nIs there an infinite anti-chain of subgroups of $H$?\n\nAre there only finitely many obstructions to amenability? (i.e., is there a finite collection of subgroups of $H$ such that every non amenable subgroup of $H$ contains a copy of one of them?) This question is also interesting for subgroups of Thompson's group $F$ or more generally, for subgroups of $\\mathrm{PL}_o(I)$ (the group of piecewise linear orientation preserving homeomorphisms of the interval $[0,1]$ with finitely many breakpoints).",
   "clean_statement": "Let $H$ be the group of all piecewise-projective homeomorphisms of $S^1$ that fix $\\infty$.\n\nWhat is the subgroup structure of $H$?\n\nIs there an infinite anti-chain of subgroups of $H$?\n\nAre there only finitely many obstructions to amenability? (i.e., is there a finite collection of subgroups of $H$ such that every non amenable subgroup of $H$ contains a copy of one of them?) This question is also interesting for subgroups of Thompson's group $F$ or more generally, for subgroups of $\\mathrm{PL}_o(I)$ (the group of piecewise linear orientation preserving homeomorphisms of the interval $[0,1]$ with finitely many breakpoints).",
   "public_statement": "Let $H$ be the group of all piecewise-projective homeomorphisms of $S^1$ that fix $\\infty$.\n\nWhat is the subgroup structure of $H$?\n\nIs there an infinite anti-chain of subgroups of $H$?\n\nAre there only finitely many obstructions to amenability? (i.e., is there a finite collection of subgroups of $H$ such that every non amenable subgroup of $H$ contains a copy of one of them?) This question is also interesting for subgroups of Thompson's group $F$ or more generally, for subgroups of $\\mathrm{PL}_o(I)$ (the group of piecewise linear orientation preserving homeomorphisms of the interval $[0,1]$ with finitely many breakpoints).",
   "evidence": "The canonical record is problem 1.8 in the section “Thompson group F and groups of homeomorphisms of the interval and the circle” from the 2016 AIM workshop *Amenability of discrete groups*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 94,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0096": {
   "statement_status": "exact",
   "original_statement": "Is non-amenability of a topological full group preserved under taking restrictions?\n\nTo be more precise: let $G$ act on a Cantor set $C$ minimally. Assume $G$ is equal to its full group and is non-amenable.\n\nTake a clopen subset $U$ of the Cantor set $C$, let $G_U$ be the restriction of $G$ to $U$. Is $G_U$ also non-amenable?",
   "clean_statement": "Is non-amenability of a topological full group preserved under taking restrictions?\n\nTo be more precise: let $G$ act on a Cantor set $C$ minimally. Assume $G$ is equal to its full group and is non-amenable.\n\nTake a clopen subset $U$ of the Cantor set $C$, let $G_U$ be the restriction of $G$ to $U$. Is $G_U$ also non-amenable?",
   "public_statement": "Is non-amenability of a topological full group preserved under taking restrictions?\n\nTo be more precise: let $G$ act on a Cantor set $C$ minimally. Assume $G$ is equal to its full group and is non-amenable.\n\nTake a clopen subset $U$ of the Cantor set $C$, let $G_U$ be the restriction of $G$ to $U$. Is $G_U$ also non-amenable?",
   "evidence": "The canonical AIM record is item 2.1 in the section “Topological Full group, IET and PRG” of the 2016 workshop *Amenability of discrete groups*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0097": {
   "statement_status": "exact",
   "original_statement": "Is the group of $IET$ (interval exchange transformations) amenable?",
   "clean_statement": "Is the group of $IET$ (interval exchange transformations) amenable?",
   "public_statement": "Is the group of $IET$ (interval exchange transformations) amenable?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 96,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0098": {
   "statement_status": "exact",
   "original_statement": "Is the group of polygon rearrangements (PRG) amenable?\nSpecifically, is the full group of the Penrose tiling amenable?",
   "clean_statement": "Is the group of polygon rearrangements (PRG) amenable?\nSpecifically, is the full group of the Penrose tiling amenable?",
   "public_statement": "Is the group of polygon rearrangements (PRG) amenable?\nSpecifically, is the full group of the Penrose tiling amenable?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 97,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0099": {
   "statement_status": "exact",
   "original_statement": "Does the topological full group of the (Morse, Chacon, Rudin-Shapiro) substitution subshift contain a subgroup of intermediate growth?",
   "clean_statement": "Does the topological full group of the (Morse, Chacon, Rudin-Shapiro) substitution subshift contain a subgroup of intermediate growth?",
   "public_statement": "Does the topological full group of the (Morse, Chacon, Rudin-Shapiro) substitution subshift contain a subgroup of intermediate growth?",
   "evidence": "The canonical AIM record, item 2.4 in “Topological Full group, IET and PRG” from the 2016 workshop *Amenability of discrete groups*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 98,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0100": {
   "statement_status": "exact",
   "original_statement": "Find an example of a $\\mathbf{Z}^2$-action on a Cantor set such that the commutator subgroup of the topological full group is f.p.",
   "clean_statement": "Find an example of a $\\mathbf{Z}^2$-action on a Cantor set such that the commutator subgroup of the topological full group is f.p.",
   "public_statement": "Find an example of a $\\mathbf{Z}^2$-action on a Cantor set such that the commutator subgroup of the topological full group is f.p.",
   "evidence": "The canonical AIM record is Problem 2.5 from the workshop list *Amenability of discrete groups*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 99,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0101": {
   "statement_status": "exact",
   "original_statement": "Are full groups of labeled graphs of \"low\" complexity (and \"low\" growth) amenable? (E.g. \"low\" = polynomial.)\n\nWhat about different interpretations of complexity?",
   "clean_statement": "Are full groups of labeled graphs of \"low\" complexity (and \"low\" growth) amenable? (E.g. \"low\" = polynomial.)\n\nWhat about different interpretations of complexity?",
   "public_statement": "Are full groups of labeled graphs of \"low\" complexity (and \"low\" growth) amenable? (E.g. \"low\" = polynomial.)\n\nWhat about different interpretations of complexity?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 100,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0102": {
   "statement_status": "exact",
   "original_statement": "Take a hyperfinite (ergodic) equivalence relation on $[0,1]$. Let $G$ be the full group of all measure-preserving transformations that respect the equivalence relation (i.e., whose graph is contained in the equivalence relation).\n\n\\textbf{Q:} Does $G$ contain any \\textit{discrete} non-amenable subgroups?\n\nHere, discrete means w.r.t. the distance\n\\[\nd(a,b) \\mathrel{\\mathop:}= \\mu \\{x\\ :\\ ax \\neq bx \\}\n\\]\n\n\\textbf{Juschenko:} If the answer is \\textit{no}, then IET is amenable.",
   "clean_statement": "Take a hyperfinite (ergodic) equivalence relation on $[0,1]$. Let $G$ be the full group of all measure-preserving transformations that respect the equivalence relation (i.e., whose graph is contained in the equivalence relation).\n\n\\textbf{Q:} Does $G$ contain any \\textit{discrete} non-amenable subgroups?\n\nHere, discrete means w.r.t. the distance\n\\[\nd(a,b) \\mathrel{\\mathop:}= \\mu \\{x\\ :\\ ax \\neq bx \\}\n\\]\n\n\\textbf{Juschenko:} If the answer is \\textit{no}, then IET is amenable.",
   "public_statement": "Take a hyperfinite (ergodic) equivalence relation on $[0,1]$. Let $G$ be the full group of all measure-preserving transformations that respect the equivalence relation (i.e., whose graph is contained in the equivalence relation).\n\n\\textbf{Q:} Does $G$ contain any \\textit{discrete} non-amenable subgroups?\n\nHere, discrete means w.r.t. the distance\n\\[\nd(a,b) \\mathrel{\\mathop:}= \\mu \\{x\\ :\\ ax \\neq bx \\}\n\\]\n\n\\textbf{Juschenko:} If the answer is \\textit{no}, then IET is amenable.",
   "evidence": "The AIM record asks the following.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 101,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0103": {
   "statement_status": "exact",
   "original_statement": "Is there a f.p. branch group?\nIs there a f.p. amenable branch group?",
   "clean_statement": "Is there a f.p. branch group?\nIs there a f.p. amenable branch group?",
   "public_statement": "Is there a f.p. branch group?\nIs there a f.p. amenable branch group?",
   "evidence": "The canonical AIM record (Problem 3.1 in the section “Grigorchuk's group, branch groups and groups of intermediate growth” from the workshop *Amenability of discrete groups*) asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 102,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0104": {
   "statement_status": "exact",
   "original_statement": "Is the universal Grigorchuk group amenable?",
   "clean_statement": "Is the universal Grigorchuk group amenable?",
   "public_statement": "Is the universal Grigorchuk group amenable?",
   "evidence": "The exact canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 103,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0105": {
   "statement_status": "exact",
   "original_statement": "Find bounds on Folner functions for the Grigorchuk group $G$.",
   "clean_statement": "Find bounds on Folner functions for the Grigorchuk group $G$.",
   "public_statement": "Find bounds on Folner functions for the Grigorchuk group $G$.",
   "evidence": "The exact canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 104,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0106": {
   "statement_status": "exact",
   "original_statement": "Is the sequence of balls $B_n(e)$ in Grigorchuk's group $G$ a Folner sequence?",
   "clean_statement": "Is the sequence of balls $B_n(e)$ in Grigorchuk's group $G$ a Folner sequence?",
   "public_statement": "Is the sequence of balls $B_n(e)$ in Grigorchuk's group $G$ a Folner sequence?",
   "evidence": "The canonical AIM record (workshop *Amenability of discrete groups*, section “Grigorchuk's group, branch groups and groups of intermediate growth,” Problem 3.5) reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 105,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0107": {
   "statement_status": "corrected_verified",
   "original_statement": "Is $\\rho_c<1$ for all groups of intermediate growth?",
   "clean_statement": "For every Cayley graph of every finitely generated group of\nintermediate growth, is the critical occupation probability $p_c$ for Bernoulli\npercolation strictly less than one?",
   "public_statement": "For every Cayley graph of every finitely generated group of\nintermediate growth, is the critical occupation probability $p_c$ for Bernoulli\npercolation strictly less than one?",
   "evidence": "This transcription is faithful to the source. The archived AIM page from 15 January 2017 itself displays `\\rho_c`, and supplies neither a definition nor a remark explaining the symbol. Thus `\\rho_c` is not an OCR error introduced by the corpus. The mathematical reconstruction used here is:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 106,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0108": {
   "statement_status": "exact",
   "original_statement": "Does the limit\n\\[\n\\lim_{n\\rightarrow\\infty} \\frac{\\log \\log (B_G(n))}{\\log n}\n\\]\nexist for the Grigorchuk group?",
   "clean_statement": "Does the limit\n\\[\n\\lim_{n\\rightarrow\\infty} \\frac{\\log \\log (B_G(n))}{\\log n}\n\\]\nexist for the Grigorchuk group?",
   "public_statement": "Does the limit\n\\[\n\\lim_{n\\rightarrow\\infty} \\frac{\\log \\log (B_G(n))}{\\log n}\n\\]\nexist for the Grigorchuk group?",
   "evidence": "The canonical AIM record (workshop *Amenability of discrete groups*, section “Grigorchuk's group, branch groups and groups of intermediate growth,” Problem 3.6) reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 107,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0109": {
   "statement_status": "exact",
   "original_statement": "Is there a finitely presented \\textit{amenable} group that surjects onto a group of intermediate growth?",
   "clean_statement": "Is there a finitely presented \\textit{amenable} group that surjects onto a group of intermediate growth?",
   "public_statement": "Is there a finitely presented \\textit{amenable} group that surjects onto a group of intermediate growth?",
   "evidence": "The canonical record is Problem 3.8 in the AIM workshop list *Amenability of discrete groups*, in the section “Grigorchuk's group, branch groups and groups of intermediate growth.” Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 108,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0110": {
   "statement_status": "exact",
   "original_statement": "Are polynomial activity automata groups amenable?",
   "clean_statement": "Are polynomial activity automata groups amenable?",
   "public_statement": "Are polynomial activity automata groups amenable?",
   "evidence": "The archived AIM page from 15 January 2017 contains precisely the same sentence as Problem 3.9 in the workshop list *Amenability of discrete groups*. There is no OCR error, missing formula, status note, or source-level qualification to reconstruct.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 109,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0111": {
   "statement_status": "exact",
   "original_statement": "\\textbf{Def:} A subgroup $H$ of a group $G$ is \\textit{commensurated} if $\\forall g \\in G$, $g^{-1}Hg \\cap H$ has finite index in $H$.\n\n\\textbf{Thm}(Wesolek): A finitely generated branch group is just infinite if and only if every commensurated subgroup is either finite or of finite index.\n\nFind a combinatorial proof of the following: every commensurated subgroup of a finitely generated branch group is either finite or of finite index.",
   "clean_statement": "\\textbf{Def:} A subgroup $H$ of a group $G$ is \\textit{commensurated} if $\\forall g \\in G$, $g^{-1}Hg \\cap H$ has finite index in $H$.\n\n\\textbf{Thm}(Wesolek): A finitely generated branch group is just infinite if and only if every commensurated subgroup is either finite or of finite index.\n\nFind a combinatorial proof of the following: every commensurated subgroup of a finitely generated branch group is either finite or of finite index.",
   "public_statement": "\\textbf{Def:} A subgroup $H$ of a group $G$ is \\textit{commensurated} if $\\forall g \\in G$, $g^{-1}Hg \\cap H$ has finite index in $H$.\n\n\\textbf{Thm}(Wesolek): A finitely generated branch group is just infinite if and only if every commensurated subgroup is either finite or of finite index.\n\nFind a combinatorial proof of the following: every commensurated subgroup of a finitely generated branch group is either finite or of finite index.",
   "evidence": "The exact corpus record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 110,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0112": {
   "statement_status": "exact",
   "original_statement": "\\textbf{Definition: } A group $G$ is \\textit{supramenable} if for every triplet $(G,X,E)$ (where $E$ is a nonempty subset of $X$ and $G \\def\\actson{\\curvearrowright} X$), there is an invariant measure normalized on $E$.\n\n\\textbf{Fact:} Subexponential growth implies supramenability.\n\n\\begin{enumerate}\n\\item \\textbf{Q:} (Rosenblatt 74') Is there a supramenable group of exponential growth?\n\\item Is the direct product of $2$ supramenable groups also supramenable?\n\\end{enumerate}",
   "clean_statement": "\\textbf{Definition: } A group $G$ is \\textit{supramenable} if for every triplet $(G,X,E)$ (where $E$ is a nonempty subset of $X$ and $G \\def\\actson{\\curvearrowright} X$), there is an invariant measure normalized on $E$.\n\n\\textbf{Fact:} Subexponential growth implies supramenability.\n\n\\begin{enumerate}\n\\item \\textbf{Q:} (Rosenblatt 74') Is there a supramenable group of exponential growth?\n\\item Is the direct product of $2$ supramenable groups also supramenable?\n\\end{enumerate}",
   "public_statement": "\\textbf{Definition: } A group $G$ is \\textit{supramenable} if for every triplet $(G,X,E)$ (where $E$ is a nonempty subset of $X$ and $G \\def\\actson{\\curvearrowright} X$), there is an invariant measure normalized on $E$.\n\n\\textbf{Fact:} Subexponential growth implies supramenability.\n\n\\begin{enumerate}\n\\item \\textbf{Q:} (Rosenblatt 74') Is there a supramenable group of exponential growth?\n\\item Is the direct product of $2$ supramenable groups also supramenable?\n\\end{enumerate}",
   "evidence": "The canonical record is Problem 4.05 in the “Amenability+” section of the AIM workshop list *Amenability of discrete groups*. Its exact extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 111,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0113": {
   "statement_status": "exact",
   "original_statement": "\\textbf{Definition:} A \\emph{hereditary just-infinite} group $G$ is a residually finite group such that every finite index subgroup of it is just-infinite.\n\nIs there an amenable hereditary just-infinite group that is not elementary amenable? Is there a finitely presented one?",
   "clean_statement": "\\textbf{Definition:} A \\emph{hereditary just-infinite} group $G$ is a residually finite group such that every finite index subgroup of it is just-infinite.\n\nIs there an amenable hereditary just-infinite group that is not elementary amenable? Is there a finitely presented one?",
   "public_statement": "\\textbf{Definition:} A \\emph{hereditary just-infinite} group $G$ is a residually finite group such that every finite index subgroup of it is just-infinite.\n\nIs there an amenable hereditary just-infinite group that is not elementary amenable? Is there a finitely presented one?",
   "evidence": "The exact source record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 112,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0114": {
   "statement_status": "exact",
   "original_statement": "Is there a f.p. amenable infinite simple group?\nA construction of Juschenko and Monod (2012) gives a finitely generated infinite simple amenable group.\n\\textbf{Conjecture:} The derived subgroup of the full group of the Penrose tiling is a candidate.",
   "clean_statement": "Is there a f.p. amenable infinite simple group?\nA construction of Juschenko and Monod (2012) gives a finitely generated infinite simple amenable group.\n\\textbf{Conjecture:} The derived subgroup of the full group of the Penrose tiling is a candidate.",
   "public_statement": "Is there a f.p. amenable infinite simple group?\nA construction of Juschenko and Monod (2012) gives a finitely generated infinite simple amenable group.\n\\textbf{Conjecture:} The derived subgroup of the full group of the Penrose tiling is a candidate.",
   "evidence": "The exact canonical record is AIM Problem 4.15 in the “Amenability+” section:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 113,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0115": {
   "statement_status": "exact",
   "original_statement": "Is there an infinite finitely generated amenable simple non-Liouville group (with respect to symmetric, finitely supported, generating measures)?",
   "clean_statement": "Is there an infinite finitely generated amenable simple non-Liouville group (with respect to symmetric, finitely supported, generating measures)?",
   "public_statement": "Is there an infinite finitely generated amenable simple non-Liouville group (with respect to symmetric, finitely supported, generating measures)?",
   "evidence": "The canonical AIM record (workshop *Amenability of discrete groups*, section “Amenability+”, Problem 4.2) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 114,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0116": {
   "statement_status": "exact",
   "original_statement": "\\textbf{Definition: } SQ-closure = closure under subgroups and quotients.\n\n\\begin{enumerate}\n\\item What are \"good\" SQ-closed families of amenable groups?\n\\item What is the SQ-closure of the set of bounded automata groups?\n\\end{enumerate}",
   "clean_statement": "\\textbf{Definition: } SQ-closure = closure under subgroups and quotients.\n\n\\begin{enumerate}\n\\item What are \"good\" SQ-closed families of amenable groups?\n\\item What is the SQ-closure of the set of bounded automata groups?\n\\end{enumerate}",
   "public_statement": "\\textbf{Definition: } SQ-closure = closure under subgroups and quotients.\n\n\\begin{enumerate}\n\\item What are \"good\" SQ-closed families of amenable groups?\n\\item What is the SQ-closure of the set of bounded automata groups?\n\\end{enumerate}",
   "evidence": "The canonical record is Problem 4.25 in the “Amenability+” section of the AIM list from the September 2016 workshop *Amenability of discrete groups*. The archived AIM HTML was inspected directly. It has no status line and no remarks, and reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 115,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0117": {
   "statement_status": "exact",
   "original_statement": "Are group actions with polynomial-growth Schreier graphs extensively amenable?\n\nExtensive amenability is known for recurrent actions.",
   "clean_statement": "Are group actions with polynomial-growth Schreier graphs extensively amenable?\n\nExtensive amenability is known for recurrent actions.",
   "public_statement": "Are group actions with polynomial-growth Schreier graphs extensively amenable?\n\nExtensive amenability is known for recurrent actions.",
   "evidence": "The canonical record is AIM Problem List 4.3 from the workshop *Amenability of discrete groups*. The archived AIM page was checked directly (Internet Archive capture dated 2017-01-15). Its wording is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 116,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0118": {
   "statement_status": "exact",
   "original_statement": "\\textbf{Definition: } let AE be the smallest class of locally compact groups s..t. the following holds:\n\\begin{enumerate}\n\\item AE contains all compact groups and AE contains all discrete amenable groups;\n\\item closed under the (top) elementray ops.\n\\end{enumerate}\n\n\\begin{enumerate}\n\\item is every ever locally compact amenable group in AE?\n\\item Find f.g. amenable groups with infinite commensurated subgroups with trivial normal core.\n\\item Is every commensurated subroup of the Basilica group commensurate with a normal subgroup?\n\\end{enumerate}",
   "clean_statement": "\\textbf{Definition: } let AE be the smallest class of locally compact groups s..t. the following holds:\n\\begin{enumerate}\n\\item AE contains all compact groups and AE contains all discrete amenable groups;\n\\item closed under the (top) elementray ops.\n\\end{enumerate}\n\n\\begin{enumerate}\n\\item is every ever locally compact amenable group in AE?\n\\item Find f.g. amenable groups with infinite commensurated subgroups with trivial normal core.\n\\item Is every commensurated subroup of the Basilica group commensurate with a normal subgroup?\n\\end{enumerate}",
   "public_statement": "\\textbf{Definition: } let AE be the smallest class of locally compact groups s..t. the following holds:\n\\begin{enumerate}\n\\item AE contains all compact groups and AE contains all discrete amenable groups;\n\\item closed under the (top) elementray ops.\n\\end{enumerate}\n\n\\begin{enumerate}\n\\item is every ever locally compact amenable group in AE?\n\\item Find f.g. amenable groups with infinite commensurated subgroups with trivial normal core.\n\\item Is every commensurated subroup of the Basilica group commensurate with a normal subgroup?\n\\end{enumerate}",
   "evidence": "The canonical record is AIM problem 4.35 from the 2016 workshop *Amenability of discrete groups*. The archived AIM page literally reads",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 117,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0119": {
   "statement_status": "exact",
   "original_statement": "Is there a non-amenable group that has a Liouville action with amenable stabilizers?",
   "clean_statement": "Is there a non-amenable group that has a Liouville action with amenable stabilizers?",
   "public_statement": "Is there a non-amenable group that has a Liouville action with amenable stabilizers?",
   "evidence": "The canonical AIM record, Section *Amenability+*, Problem 4.4, asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 118,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0120": {
   "statement_status": "exact",
   "original_statement": "Is there a transitive action $G \\def\\actson{\\curvearrowright} X$ that is $\\mu$-Liouville, but the action on some orbit of $G\\def\\actson{\\curvearrowright} \\subset X \\times X$ is not $\\mu^2$-Liouville?",
   "clean_statement": "Is there a transitive action $G \\def\\actson{\\curvearrowright} X$ that is $\\mu$-Liouville, but the action on some orbit of $G\\def\\actson{\\curvearrowright} \\subset X \\times X$ is not $\\mu^2$-Liouville?",
   "public_statement": "Is there a transitive action $G \\def\\actson{\\curvearrowright} X$ that is $\\mu$-Liouville, but the action on some orbit of $G\\def\\actson{\\curvearrowright} \\subset X \\times X$ is not $\\mu^2$-Liouville?",
   "evidence": "The canonical record (AIM, *Amenability of discrete groups*, item 4.45) literally reads",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 119,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0121": {
   "statement_status": "exact",
   "original_statement": "Is there a 2-generated infinite simple amenable group?",
   "clean_statement": "Is there a 2-generated infinite simple amenable group?",
   "public_statement": "Is there a 2-generated infinite simple amenable group?",
   "evidence": "The canonical AIM record is in the *Amenability+* section of the 2016 workshop *Amenability of discrete groups*. It asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 120,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0122": {
   "statement_status": "exact",
   "original_statement": "Are contracting groups amenable?",
   "clean_statement": "Are contracting groups amenable?",
   "public_statement": "Are contracting groups amenable?",
   "evidence": "The canonical AIM record is Problem 4.55 in the workshop list *Amenability of discrete groups*, section “Amenability+”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 121,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0123": {
   "statement_status": "exact",
   "original_statement": "Is the Liouville property stable under the choice of of symmetric, finitely supported, generating measure?",
   "clean_statement": "Is the Liouville property stable under the choice of of symmetric, finitely supported, generating measure?",
   "public_statement": "Is the Liouville property stable under the choice of of symmetric, finitely supported, generating measure?",
   "evidence": "The canonical AIM record is Problem 4.6 in the section “Amenability+” of the AIM workshop list *Amenability of discrete groups*. It reads verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 122,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0124": {
   "statement_status": "exact",
   "original_statement": "Is there an amenable group whose spectrum is a Cantor set?\n\n\\textbf{Remark} (Dudko): For Grigorchuk's group $G$, the spectrum is a union of two intervals: $[-1,0]\\cup[1/2,1]$; the question is whether there can be something a tad crazier.",
   "clean_statement": "Is there an amenable group whose spectrum is a Cantor set?\n\n\\textbf{Remark} (Dudko): For Grigorchuk's group $G$, the spectrum is a union of two intervals: $[-1,0]\\cup[1/2,1]$; the question is whether there can be something a tad crazier.",
   "public_statement": "Is there an amenable group whose spectrum is a Cantor set?\n\n\\textbf{Remark} (Dudko): For Grigorchuk's group $G$, the spectrum is a union of two intervals: $[-1,0]\\cup[1/2,1]$; the question is whether there can be something a tad crazier.",
   "evidence": "The exact AIM record is Problem 4.75 from the workshop *Amenability of discrete groups*, section “Amenability+”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 123,
   "attempt": 2
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0125": {
   "statement_status": "exact",
   "original_statement": "\\textbf{Conjecture:}A finitely presented group is either virtually nilpotent, or contains a free subsemigroup of order $2$.",
   "clean_statement": "\\textbf{Conjecture:}A finitely presented group is either virtually nilpotent, or contains a free subsemigroup of order $2$.",
   "public_statement": "\\textbf{Conjecture:}A finitely presented group is either virtually nilpotent, or contains a free subsemigroup of order $2$.",
   "evidence": "The live AIM archive gives the following exact wording in the workshop list *Amenability of discrete groups*, section “Other problems”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 124,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0126": {
   "statement_status": "exact",
   "original_statement": "Is there a finitely presented simple group that is not 2-generated?",
   "clean_statement": "Is there a finitely presented simple group that is not 2-generated?",
   "public_statement": "Is there a finitely presented simple group that is not 2-generated?",
   "evidence": "The canonical record is Problem 5.2 in the section “Other problems” of the AIM workshop list *Amenability of discrete groups*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 125,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0127": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Is true that every infinite amenable group contains an infinite abelian subgroup?",
   "clean_statement": "**Question.** Is it true that every infinite amenable group contains an infinite abelian subgroup?",
   "public_statement": "Is true that every infinite amenable group contains an infinite abelian subgroup?",
   "evidence": "The canonical AIM record is problem 5.3 in the section “Other problems” of the workshop “Amenability of discrete groups.” Its exact extracted text is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 126,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0128": {
   "statement_status": "exact",
   "original_statement": "$G \\curvearrowright X$, $p, x, y \\in X$.\nWhat can be said on $P(x \\in O_\\Lambda(p) \\operatorname{ and } y \\in O_{\\Lambda}(p))$? Are these events positively/negatively correlated?\n(Here $O_\\Lambda(p)$ is the inverted orbit of $p$.)",
   "clean_statement": "$G \\curvearrowright X$, $p, x, y \\in X$.\nWhat can be said on $P(x \\in O_\\Lambda(p) \\operatorname{ and } y \\in O_{\\Lambda}(p))$? Are these events positively/negatively correlated?\n(Here $O_\\Lambda(p)$ is the inverted orbit of $p$.)",
   "public_statement": "$G \\curvearrowright X$, $p, x, y \\in X$.\nWhat can be said on $P(x \\in O_\\Lambda(p) \\operatorname{ and } y \\in O_{\\Lambda}(p))$? Are these events positively/negatively correlated?\n(Here $O_\\Lambda(p)$ is the inverted orbit of $p$.)",
   "evidence": "The canonical record and the live AIM page give exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 127,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0129": {
   "statement_status": "exact",
   "original_statement": "Is there a f.p. infinite torsion group? \\textbf{Conjecture:}(Grigorchuk) no.",
   "clean_statement": "Is there a f.p. infinite torsion group? \\textbf{Conjecture:}(Grigorchuk) no.",
   "public_statement": "Is there a f.p. infinite torsion group? \\textbf{Conjecture:}(Grigorchuk) no.",
   "evidence": "The canonical record is problem 5.5 in the AIM workshop list *Amenability of discrete groups*, section “Other problems.” The archived AIM page and the canonical JSON agree on the wording:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 128,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0130": {
   "statement_status": "exact",
   "original_statement": "Is the Tarski number of $G \\times G$ equal to the Tarski number of $G$?",
   "clean_statement": "Is the Tarski number of $G \\times G$ equal to the Tarski number of $G$?",
   "public_statement": "Is the Tarski number of $G \\times G$ equal to the Tarski number of $G$?",
   "evidence": "The canonical AIM record (source file `aim-geometric-group-theory-notes.json`, zero-based index 129, workshop *Amenability of discrete groups*, Problem 5.6) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 129,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0131": {
   "statement_status": "exact",
   "original_statement": "Is there a group with Tarski number 7?\n\nIf this group is amenable, the Folner function of this group is universal bound.",
   "clean_statement": "Is there a group with Tarski number 7?\n\nIf this group is amenable, the Folner function of this group is universal bound.",
   "public_statement": "Is there a group with Tarski number 7?\n\nIf this group is amenable, the Folner function of this group is universal bound.",
   "evidence": "The canonical record is AIM problem 5.7 from the 2016 workshop *Amenability of discrete groups*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 130,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0132": {
   "statement_status": "exact",
   "original_statement": "\\textbf{Conjecture:}If the Folner function of a group is sub-exponential, then the group is virtually nilpotent.",
   "clean_statement": "\\textbf{Conjecture:}If the Folner function of a group is sub-exponential, then the group is virtually nilpotent.",
   "public_statement": "\\textbf{Conjecture:}If the Folner function of a group is sub-exponential, then the group is virtually nilpotent.",
   "evidence": "The exact canonical record (source file `aim-geometric-group-theory-notes.json`, zero-based index 131, workshop *Amenability of discrete groups*, Problem 5.8) is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 131,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0133": {
   "statement_status": "exact",
   "original_statement": "Are the Hanoi tower groups $H_n$, $n \\geq 4$ amenable?",
   "clean_statement": "Are the Hanoi tower groups $H_n$, $n \\geq 4$ amenable?",
   "public_statement": "Are the Hanoi tower groups $H_n$, $n \\geq 4$ amenable?",
   "evidence": "The canonical AIM record is problem 5.9 in the workshop list *Amenability of discrete groups*, section “Other problems.” The archived AIM page and the canonical JSON agree exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 132,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0134": {
   "statement_status": "exact",
   "original_statement": "Let $M$ be a closed Riemannian manifold and let $$\\Gamma=\\pi_1(M)>\\Gamma_1>\\Gamma_2>\\ldots$$\nbe a decreasing sequence of finite index normal subgroups of $\\Gamma$ with $\\bigcap\\Gamma_n=1$.\nLet $\\tilde{M}$ be the universal cover of $M$ and let $M_n=\\tilde{M}/\\Gamma_n$.\n\nIn addition, suppose that $M$ is aspherical with $\\dim M=2k+1$ and $\\beta_j^{(2)}=0$ for every $j$.\n\nDo we have\n$$\\rho^{(2)}(\\tilde{M})=(-1)^k\\lim \\frac{\\log(|tors (H_k(M_n))|)}{|\\Gamma:\\Gamma_n|}?$$",
   "clean_statement": "Let $M$ be a closed Riemannian manifold and let $$\\Gamma=\\pi_1(M)>\\Gamma_1>\\Gamma_2>\\ldots$$\nbe a decreasing sequence of finite index normal subgroups of $\\Gamma$ with $\\bigcap\\Gamma_n=1$.\nLet $\\tilde{M}$ be the universal cover of $M$ and let $M_n=\\tilde{M}/\\Gamma_n$.\n\nIn addition, suppose that $M$ is aspherical with $\\dim M=2k+1$ and $\\beta_j^{(2)}=0$ for every $j$.\n\nDo we have\n$$\\rho^{(2)}(\\tilde{M})=(-1)^k\\lim \\frac{\\log(|tors (H_k(M_n))|)}{|\\Gamma:\\Gamma_n|}?$$",
   "public_statement": "Let $M$ be a closed Riemannian manifold and let $$\\Gamma=\\pi_1(M)>\\Gamma_1>\\Gamma_2>\\ldots$$\nbe a decreasing sequence of finite index normal subgroups of $\\Gamma$ with $\\bigcap\\Gamma_n=1$.\nLet $\\tilde{M}$ be the universal cover of $M$ and let $M_n=\\tilde{M}/\\Gamma_n$.\n\nIn addition, suppose that $M$ is aspherical with $\\dim M=2k+1$ and $\\beta_j^{(2)}=0$ for every $j$.\n\nDo we have\n$$\\rho^{(2)}(\\tilde{M})=(-1)^k\\lim \\frac{\\log(|tors (H_k(M_n))|)}{|\\Gamma:\\Gamma_n|}?$$",
   "evidence": "The canonical record is AIM Problem 11.1 from the workshop *$L^2$ invariants and their relatives for finitely generated groups*. It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 133,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0135": {
   "statement_status": "exact",
   "original_statement": "Let $M$ be a closed Riemannian manifold and let $$\\Gamma=\\pi_1(M)>\\Gamma_1>\\Gamma_2>\\ldots$$\nbe a decreasing sequence of finite index normal subgroups of $\\Gamma$ with $\\bigcap\\Gamma_n=1$.\nLet $\\tilde{M}$ be the universal cover of $M$ and let $M_n=\\tilde{M}/\\Gamma_n$.\n\nIs $$\\rho^{(2)}(\\tilde{M})=\\lim\\limits_n \\frac{\\rho(M_n)}{|\\Gamma:\\Gamma_n|}?$$",
   "clean_statement": "Let $M$ be a closed Riemannian manifold and let $$\\Gamma=\\pi_1(M)>\\Gamma_1>\\Gamma_2>\\ldots$$\nbe a decreasing sequence of finite index normal subgroups of $\\Gamma$ with $\\bigcap\\Gamma_n=1$.\nLet $\\tilde{M}$ be the universal cover of $M$ and let $M_n=\\tilde{M}/\\Gamma_n$.\n\nIs $$\\rho^{(2)}(\\tilde{M})=\\lim\\limits_n \\frac{\\rho(M_n)}{|\\Gamma:\\Gamma_n|}?$$",
   "public_statement": "Let $M$ be a closed Riemannian manifold and let $$\\Gamma=\\pi_1(M)>\\Gamma_1>\\Gamma_2>\\ldots$$\nbe a decreasing sequence of finite index normal subgroups of $\\Gamma$ with $\\bigcap\\Gamma_n=1$.\nLet $\\tilde{M}$ be the universal cover of $M$ and let $M_n=\\tilde{M}/\\Gamma_n$.\n\nIs $$\\rho^{(2)}(\\tilde{M})=\\lim\\limits_n \\frac{\\rho(M_n)}{|\\Gamma:\\Gamma_n|}?$$",
   "evidence": "The canonical record is Problem 11.2 in the AIM workshop list *\\(L^2\\) invariants and their relatives for finitely generated groups*, in the section “Approximation of \\(L_2\\)-torsion.” Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 134,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0136": {
   "statement_status": "exact",
   "original_statement": "The following is equivalent to the previous problem.\n\nLet $A\\in M_l(\\mathbb{Z} \\Gamma)$, denote $A_k=A/\\Gamma_k$. Is it true that\n$$tr_{L\\Gamma}(\\log A^*A)=\\lim \\frac{tr(\\log A_k^*A_k)}{|\\Gamma:\\Gamma_k|}?$$",
   "clean_statement": "The following is equivalent to the previous problem.\n\nLet $A\\in M_l(\\mathbb{Z} \\Gamma)$, denote $A_k=A/\\Gamma_k$. Is it true that\n$$tr_{L\\Gamma}(\\log A^*A)=\\lim \\frac{tr(\\log A_k^*A_k)}{|\\Gamma:\\Gamma_k|}?$$",
   "public_statement": "The following is equivalent to the previous problem.\n\nLet $A\\in M_l(\\mathbb{Z} \\Gamma)$, denote $A_k=A/\\Gamma_k$. Is it true that\n$$tr_{L\\Gamma}(\\log A^*A)=\\lim \\frac{tr(\\log A_k^*A_k)}{|\\Gamma:\\Gamma_k|}?$$",
   "evidence": "The canonical record is Problem 11.3 in the section “Approximation of $L_2$-torsion” of the AIM list *$L^2$ invariants and their relatives for finitely generated groups*. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 135,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0137": {
   "statement_status": "exact",
   "original_statement": "Is $$\\lim \\frac{b_{F_p}(\\Gamma_n)}{|\\Gamma : \\Gamma_n|}=\\lim \\frac{rk(\\Gamma_n)}{|\\Gamma:\\Gamma_n|}$$ for any (not necessarily finitely generated) group?",
   "clean_statement": "Is $$\\lim \\frac{b_{F_p}(\\Gamma_n)}{|\\Gamma : \\Gamma_n|}=\\lim \\frac{rk(\\Gamma_n)}{|\\Gamma:\\Gamma_n|}$$ for any (not necessarily finitely generated) group?",
   "public_statement": "Is $$\\lim \\frac{b_{F_p}(\\Gamma_n)}{|\\Gamma : \\Gamma_n|}=\\lim \\frac{rk(\\Gamma_n)}{|\\Gamma:\\Gamma_n|}$$ for any (not necessarily finitely generated) group?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 136,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0138": {
   "statement_status": "exact",
   "original_statement": "Let $\\Gamma$ be a finitely presented residually $p$-group. Let $\\Gamma_n$ be a normal $p$-chain with $\\bigcap \\Gamma_n=1$, then\n$$\\lim \\frac{b_{\\mathbb{Q}}(\\Gamma_n)}{|\\Gamma:\\Gamma_n|}\\leq \\lim \\frac{b_{F_p}}{|\\Gamma: \\Gamma_n|}\\leq \\lim \\frac{rk(\\Gamma_n)}{|\\Gamma:\\Gamma_n|}$$\n\nCan these inequalities be strict?",
   "clean_statement": "Let $\\Gamma$ be a finitely presented residually $p$-group. Let $\\Gamma_n$ be a normal $p$-chain with $\\bigcap \\Gamma_n=1$, then\n$$\\lim \\frac{b_{\\mathbb{Q}}(\\Gamma_n)}{|\\Gamma:\\Gamma_n|}\\leq \\lim \\frac{b_{F_p}}{|\\Gamma: \\Gamma_n|}\\leq \\lim \\frac{rk(\\Gamma_n)}{|\\Gamma:\\Gamma_n|}$$\n\nCan these inequalities be strict?",
   "public_statement": "Let $\\Gamma$ be a finitely presented residually $p$-group. Let $\\Gamma_n$ be a normal $p$-chain with $\\bigcap \\Gamma_n=1$, then\n$$\\lim \\frac{b_{\\mathbb{Q}}(\\Gamma_n)}{|\\Gamma:\\Gamma_n|}\\leq \\lim \\frac{b_{F_p}}{|\\Gamma: \\Gamma_n|}\\leq \\lim \\frac{rk(\\Gamma_n)}{|\\Gamma:\\Gamma_n|}$$\n\nCan these inequalities be strict?",
   "evidence": "The canonical AIM record (workshop “$L^2$ invariants and their relatives for finitely generated groups,” problem 11.5) literally reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 137,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0139": {
   "statement_status": "exact",
   "original_statement": "Let $\\Gamma_1$ and $\\Gamma_2$ be infinite countable groups. Does $\\Gamma_1\\times\\Gamma_2$ have fixed price $1$?",
   "clean_statement": "Let $\\Gamma_1$ and $\\Gamma_2$ be infinite countable groups. Does $\\Gamma_1\\times\\Gamma_2$ have fixed price $1$?",
   "public_statement": "Let $\\Gamma_1$ and $\\Gamma_2$ be infinite countable groups. Does $\\Gamma_1\\times\\Gamma_2$ have fixed price $1$?",
   "evidence": "The canonical record is Problem 22.1 in the AIM section “Orbit Equivalence of Measure Preserving Actions.” Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 138,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0140": {
   "statement_status": "exact",
   "original_statement": "{\\bf Definition:} $\\Gamma$ is almost treable if it admits a free action on probability space such that the equivalence relation associated to this action is almost treeable, i.e., it is an increasing union of treeable subequivalence relations.\n\nThe property is stable under measure equivalence and taking subgroups. From the property it follows that $\\Gamma$ has the Haagerup property and is sofic. Moreover, $\\lambda_{cb}(\\Gamma)=1$ if $\\Gamma$ is almost treeable.\n\nExamples include: $\\mathbb{F}_2\\times H$, where $H$ is amenable.\n\nAre the fundamental groups of hyperbolic $3$-manifolds almost treeable?",
   "clean_statement": "{\\bf Definition:} $\\Gamma$ is almost treable if it admits a free action on probability space such that the equivalence relation associated to this action is almost treeable, i.e., it is an increasing union of treeable subequivalence relations.\n\nThe property is stable under measure equivalence and taking subgroups. From the property it follows that $\\Gamma$ has the Haagerup property and is sofic. Moreover, $\\lambda_{cb}(\\Gamma)=1$ if $\\Gamma$ is almost treeable.\n\nExamples include: $\\mathbb{F}_2\\times H$, where $H$ is amenable.\n\nAre the fundamental groups of hyperbolic $3$-manifolds almost treeable?",
   "public_statement": "{\\bf Definition:} $\\Gamma$ is almost treable if it admits a free action on probability space such that the equivalence relation associated to this action is almost treeable, i.e., it is an increasing union of treeable subequivalence relations.\n\nThe property is stable under measure equivalence and taking subgroups. From the property it follows that $\\Gamma$ has the Haagerup property and is sofic. Moreover, $\\lambda_{cb}(\\Gamma)=1$ if $\\Gamma$ is almost treeable.\n\nExamples include: $\\mathbb{F}_2\\times H$, where $H$ is amenable.\n\nAre the fundamental groups of hyperbolic $3$-manifolds almost treeable?",
   "evidence": "The canonical record reproduces the following 2011 AIM item:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 139,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0141": {
   "statement_status": "exact",
   "original_statement": "Are $2$- and $3$-generated groups topologically orbit equivalent?",
   "clean_statement": "Are $2$- and $3$-generated groups topologically orbit equivalent?",
   "public_statement": "Are $2$- and $3$-generated groups topologically orbit equivalent?",
   "evidence": "The canonical record says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 140,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0142": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Let $\\Gamma$ be a non-amenable group with a probability measure preserving action of $\\Gamma$ on $(X,\\mu)$.\n\nIs it true that for any $N$ there exist measurable subsets $A_g\\subseteq X$ $(g\\in\\Gamma)$ such that\n$\\prod\\limits_{x\\in A_g}(x,xg)$ is a forest with $\\sum\\limits_{g\\in \\Gamma} \\mu^2(A_g)>N$?",
   "clean_statement": null,
   "public_statement": "Let $\\Gamma$ be a non-amenable group with a probability measure preserving action of $\\Gamma$ on $(X,\\mu)$.\n\nIs it true that for any $N$ there exist measurable subsets $A_g\\subseteq X$ $(g\\in\\Gamma)$ such that\n$\\prod\\limits_{x\\in A_g}(x,xg)$ is a forest with $\\sum\\limits_{g\\in \\Gamma} \\mu^2(A_g)>N$?",
   "evidence": "The canonical record is numbered 22.4 in the corpus and reads literally:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 141,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0143": {
   "statement_status": "exact",
   "original_statement": "Do closed, aspherical manifolds with OE fundamental groups have the same simplicial volume?\n\nIf, in addition, the simplicial volumes are positive, are the dimensions of the manifolds equal?",
   "clean_statement": "Do closed, aspherical manifolds with OE fundamental groups have the same simplicial volume?\n\nIf, in addition, the simplicial volumes are positive, are the dimensions of the manifolds equal?",
   "public_statement": "Do closed, aspherical manifolds with OE fundamental groups have the same simplicial volume?\n\nIf, in addition, the simplicial volumes are positive, are the dimensions of the manifolds equal?",
   "evidence": "The canonical AIM record is problem 33.1 in the section “More problems” from the 2011 workshop *\\(L^2\\) invariants and their relatives for finitely generated groups*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 142,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0144": {
   "statement_status": "exact",
   "original_statement": "Let $\\Gamma/\\mathbb{Z}_2$ be a sofic group. Is $\\Gamma$ sofic?",
   "clean_statement": "Let $\\Gamma/\\mathbb{Z}_2$ be a sofic group. Is $\\Gamma$ sofic?",
   "public_statement": "Let $\\Gamma/\\mathbb{Z}_2$ be a sofic group. Is $\\Gamma$ sofic?",
   "evidence": "The canonical record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 143,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0145": {
   "statement_status": "exact",
   "original_statement": "It is known that for every finitely generated group we have $$\\beta_1^{(2)}(\\Gamma)\\leq d(\\Gamma)-1.$$\nwhere $d$ stands for the minimal number of generators of $\\Gamma$.\n\nA group $\\Gamma$ is normally generated by $S\\subseteq \\Gamma$ if the only normal subgroup of $\\Gamma$ containing $S$ is $\\Gamma$ itself. Let $nrk(\\Gamma)$ be the normal rank of $\\Gamma$, i.e., $nrk(\\Gamma)$ is the minimal number of normal generators.\n\nDo we have $\\beta_1^{(2)}(\\Gamma)\\leq nrk(\\Gamma)-1$ for a torsion free group $\\Gamma$?",
   "clean_statement": "It is known that for every finitely generated group we have $$\\beta_1^{(2)}(\\Gamma)\\leq d(\\Gamma)-1.$$\nwhere $d$ stands for the minimal number of generators of $\\Gamma$.\n\nA group $\\Gamma$ is normally generated by $S\\subseteq \\Gamma$ if the only normal subgroup of $\\Gamma$ containing $S$ is $\\Gamma$ itself. Let $nrk(\\Gamma)$ be the normal rank of $\\Gamma$, i.e., $nrk(\\Gamma)$ is the minimal number of normal generators.\n\nDo we have $\\beta_1^{(2)}(\\Gamma)\\leq nrk(\\Gamma)-1$ for a torsion free group $\\Gamma$?",
   "public_statement": "It is known that for every finitely generated group we have $$\\beta_1^{(2)}(\\Gamma)\\leq d(\\Gamma)-1.$$\nwhere $d$ stands for the minimal number of generators of $\\Gamma$.\n\nA group $\\Gamma$ is normally generated by $S\\subseteq \\Gamma$ if the only normal subgroup of $\\Gamma$ containing $S$ is $\\Gamma$ itself. Let $nrk(\\Gamma)$ be the normal rank of $\\Gamma$, i.e., $nrk(\\Gamma)$ is the minimal number of normal generators.\n\nDo we have $\\beta_1^{(2)}(\\Gamma)\\leq nrk(\\Gamma)-1$ for a torsion free group $\\Gamma$?",
   "evidence": "The canonical record comes from the AIM workshop *$L^2$ invariants and their relatives for finitely generated groups*, section “More problems.” The archived AIM page identifies Andreas Thom as the proposer and numbers the item **Problem 3.3**; the corpus value `33.3` is therefore an extraction artifact.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 144,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0146": {
   "statement_status": "exact",
   "original_statement": "The deep-fall property and the Atiyah Conjecture\n\nThis notion was used in the paper ``Sumultiplicativity and the Hanna Neumann Conjecture\".\nLet $\\hat{Y}$ be a complex with a free action by a left-orderable group~$\\Gamma$ and $i\\ge 0$.\nThis induces a $\\Gamma$-invariant total order on the set of $i$-cells in $\\hat{Y}$, $\\Sigma_i^{\\hat{Y}}$.\nFor $\\sigma\\in \\Sigma_i^{\\hat{Y}}$ and $E\\subseteq \\Sigma_i^{\\hat{Y}}\\setminus \\{\\sigma\\}$,\nlet\n$$\n[E<\\sigma] := \\{ \\tau\\in E |\\tau<\\sigma \\} .\n$$\nWe say that $\\sigma$ {\\em falls into} $E$ if\n$\\partial\\sigma\\in \\overline{\\partial(\\ell^2(E))}$.\nA cell $\\sigma\\in \\Sigma_i^{\\hat{Y}}$ is called {\\bf order-essential} if it falls into\n$[\\Sigma^{\\hat{Y}}_i< \\sigma]$, i.e.\n$$\\partial\\sigma\\in\n\\overline{\\partial(\\ell^2[\\Sigma^{\\hat{Y}}_i< \\sigma])}.$$ Call it {\\bf order-inessential} otherwise.\nLet $\\mathbb{E}^{\\hat{Y}}_i$ and $\\mathbb{I}^{\\hat{Y}}_i$ denote the sets of order-essential and order-inessential edges in $\\hat{Y}$, respectively.\nWe say that the $\\Gamma$-action on $\\hat{Y}$ has the {\\bf deep-fall property}, or more precisely $i$-deep-fall property, if for any\n$\\sigma\\in \\mathbb{E}^{\\hat{Y}}_i$ we have $\\partial\\sigma\\in\n\\overline{\\partial(\\ell^2[\\mathbb{I}^{\\hat{Y}}_i< \\sigma])}$.\n\nThe argument in ``Sumbmultiplicativity ...\" implies that each free action of a left-orderable group $\\Gamma$ on a complex with the deep-fall property for some $i$ provides an instance when the (integral) Atiyah Conjecture holds. (Represent the $i$th boundary map by a matrix with entries in $\\mathbb{Z}\\Gamma$ and see that the kernel has integral dimension.) So for future investigation we propose\n\nFind many examples of left-orderable groups and their free actions on complexes that have the deep-fall property in some dimension $i$. Find free actions by left orderable groups that do not have the deep-fall property. See what can be said about non-free actions on ordered complexes. The same will formally apply to matrices with entries in the group ring over $\\mathbb{C}$.",
   "clean_statement": "The deep-fall property and the Atiyah Conjecture\n\nThis notion was used in the paper ``Sumultiplicativity and the Hanna Neumann Conjecture\".\nLet $\\hat{Y}$ be a complex with a free action by a left-orderable group~$\\Gamma$ and $i\\ge 0$.\nThis induces a $\\Gamma$-invariant total order on the set of $i$-cells in $\\hat{Y}$, $\\Sigma_i^{\\hat{Y}}$.\nFor $\\sigma\\in \\Sigma_i^{\\hat{Y}}$ and $E\\subseteq \\Sigma_i^{\\hat{Y}}\\setminus \\{\\sigma\\}$,\nlet\n$$\n[E<\\sigma] := \\{ \\tau\\in E |\\tau<\\sigma \\} .\n$$\nWe say that $\\sigma$ {\\em falls into} $E$ if\n$\\partial\\sigma\\in \\overline{\\partial(\\ell^2(E))}$.\nA cell $\\sigma\\in \\Sigma_i^{\\hat{Y}}$ is called {\\bf order-essential} if it falls into\n$[\\Sigma^{\\hat{Y}}_i< \\sigma]$, i.e.\n$$\\partial\\sigma\\in\n\\overline{\\partial(\\ell^2[\\Sigma^{\\hat{Y}}_i< \\sigma])}.$$ Call it {\\bf order-inessential} otherwise.\nLet $\\mathbb{E}^{\\hat{Y}}_i$ and $\\mathbb{I}^{\\hat{Y}}_i$ denote the sets of order-essential and order-inessential edges in $\\hat{Y}$, respectively.\nWe say that the $\\Gamma$-action on $\\hat{Y}$ has the {\\bf deep-fall property}, or more precisely $i$-deep-fall property, if for any\n$\\sigma\\in \\mathbb{E}^{\\hat{Y}}_i$ we have $\\partial\\sigma\\in\n\\overline{\\partial(\\ell^2[\\mathbb{I}^{\\hat{Y}}_i< \\sigma])}$.\n\nThe argument in ``Sumbmultiplicativity ...\" implies that each free action of a left-orderable group $\\Gamma$ on a complex with the deep-fall property for some $i$ provides an instance when the (integral) Atiyah Conjecture holds. (Represent the $i$th boundary map by a matrix with entries in $\\mathbb{Z}\\Gamma$ and see that the kernel has integral dimension.) So for future investigation we propose\n\nFind many examples of left-orderable groups and their free actions on complexes that have the deep-fall property in some dimension $i$. Find free actions by left orderable groups that do not have the deep-fall property. See what can be said about non-free actions on ordered complexes. The same will formally apply to matrices with entries in the group ring over $\\mathbb{C}$.",
   "public_statement": "The deep-fall property and the Atiyah Conjecture\n\nThis notion was used in the paper ``Sumultiplicativity and the Hanna Neumann Conjecture\".\nLet $\\hat{Y}$ be a complex with a free action by a left-orderable group~$\\Gamma$ and $i\\ge 0$.\nThis induces a $\\Gamma$-invariant total order on the set of $i$-cells in $\\hat{Y}$, $\\Sigma_i^{\\hat{Y}}$.\nFor $\\sigma\\in \\Sigma_i^{\\hat{Y}}$ and $E\\subseteq \\Sigma_i^{\\hat{Y}}\\setminus \\{\\sigma\\}$,\nlet\n$$\n[E<\\sigma] := \\{ \\tau\\in E |\\tau<\\sigma \\} .\n$$\nWe say that $\\sigma$ {\\em falls into} $E$ if\n$\\partial\\sigma\\in \\overline{\\partial(\\ell^2(E))}$.\nA cell $\\sigma\\in \\Sigma_i^{\\hat{Y}}$ is called {\\bf order-essential} if it falls into\n$[\\Sigma^{\\hat{Y}}_i< \\sigma]$, i.e.\n$$\\partial\\sigma\\in\n\\overline{\\partial(\\ell^2[\\Sigma^{\\hat{Y}}_i< \\sigma])}.$$ Call it {\\bf order-inessential} otherwise.\nLet $\\mathbb{E}^{\\hat{Y}}_i$ and $\\mathbb{I}^{\\hat{Y}}_i$ denote the sets of order-essential and order-inessential edges in $\\hat{Y}$, respectively.\nWe say that the $\\Gamma$-action on $\\hat{Y}$ has the {\\bf deep-fall property}, or more precisely $i$-deep-fall property, if for any\n$\\sigma\\in \\mathbb{E}^{\\hat{Y}}_i$ we have $\\partial\\sigma\\in\n\\overline{\\partial(\\ell^2[\\mathbb{I}^{\\hat{Y}}_i< \\sigma])}$.\n\nThe argument in ``Sumbmultiplicativity ...\" implies that each free action of a left-orderable group $\\Gamma$ on a complex with the deep-fall property for some $i$ provides an instance when the (integral) Atiyah Conjecture holds. (Represent the $i$th boundary map by a matrix with entries in $\\mathbb{Z}\\Gamma$ and see that the kernel has integral dimension.) So for future investigation we propose\n\nFind many examples of left-orderable groups and their free actions on complexes that have the deep-fall property in some dimension $i$. Find free actions by left orderable groups that do not have the deep-fall property. See what can be said about non-free actions on ordered complexes. The same will formally apply to matrices with entries in the group ring over $\\mathbb{C}$.",
   "evidence": "The source record is AIM problem 44.1 from the 2011 workshop *\\(L^2\\) invariants and their relatives for finitely generated groups*. It asks for examples and nonexamples of the deep-fall property, its relation to the integral Atiyah conjecture, and analogues for nonfree ordered actions and matrices over \\(\\mathbb C\\Gamma\\).",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 145,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0147": {
   "statement_status": "unrecoverable",
   "original_statement": "Problem 1.1 intermingled with discussion on its potential applications. We try to capture some of this conversation in the remark. Finally, we listed several current candidate complexes, to ask which ones fulfill which of the sought properties.",
   "clean_statement": null,
   "public_statement": "Problem 1.1 intermingled with discussion on its potential applications. We try to capture some of this conversation in the remark. Finally, we listed several current candidate complexes, to ask which ones fulfill which of the sought properties.",
   "evidence": "This is not a mathematical problem. It is a damaged extraction of the introductory paragraph in Section 1, “Curve complex analogues,” of the 2010 AIM workshop problem list. Inspection of the original PDF shows that the paragraph says that the *development* of Problem 1.1 was intermingled with discussion. The actual Problem 1.1 begins in the next canonical record, `AIM-GEOMETRIC_GROUP_THEORY-0148`. It asks for a $\\delta$-hyperbolic graph with an $\\operatorname{Out}(F_n)$-action such that:",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 146,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0148": {
   "statement_status": "exact",
   "original_statement": "Problem 1.1. Construct a δ-hyperbolic graph on which Out (Fn) acts so that: (1) Every iwip γ has positive translation distance ( |γ| > 0)(2) Polynomially growing γ are elliptic ( |γ| = 0)(3) The construction is elegant/beautiful/useful/truthful (4) Local structure allows for inductive arguments (5) The graph is highly connected (with highly connected links)",
   "clean_statement": "Problem 1.1. Construct a δ-hyperbolic graph on which Out (Fn) acts so that: (1) Every iwip γ has positive translation distance ( |γ| > 0)(2) Polynomially growing γ are elliptic ( |γ| = 0)(3) The construction is elegant/beautiful/useful/truthful (4) Local structure allows for inductive arguments (5) The graph is highly connected (with highly connected links)",
   "public_statement": "Problem 1.1. Construct a δ-hyperbolic graph on which Out (Fn) acts so that: (1) Every iwip γ has positive translation distance ( |γ| > 0)(2) Polynomially growing γ are elliptic ( |γ| = 0)(3) The construction is elegant/beautiful/useful/truthful (4) Local structure allows for inductive arguments (5) The graph is highly connected (with highly connected links)",
   "evidence": "This is Problem 1.1 in the AIM workshop notes *The geometry of the outer automorphism group of a free group* (workshop of 25--29 October 2010, edited by Johanna Mangahas). The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 147,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0149": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 1.2. Do various candidate complexes satisfy the conditions of",
   "clean_statement": null,
   "public_statement": "Problem 1.2. Do various candidate complexes satisfy the conditions of",
   "evidence": "The exact canonical record is truncated:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 148,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0150": {
   "statement_status": "exact",
   "original_statement": "Problem 1.1? \n\n2. G EODESICS IN OUTER SPACE \n\nJuan Souto prompted another major discussion by asking to what extent Teichm¨ uller and WP geodesic analogies are comparable to geodesics in Outer Space with the Lipschitz metric. On the board, we compiled the multi-part question give below; the text that follows is meant to give a sense of the surrounding conversation.",
   "clean_statement": "Problem 1.1?\n\n2. G EODESICS IN OUTER SPACE\n\nJuan Souto prompted another major discussion by asking to what extent Teichm¨ uller and WP geodesic analogies are comparable to geodesics in Outer Space with the Lipschitz metric. On the board, we compiled the multi-part question give below; the text that follows is meant to give a sense of the surrounding conversation.",
   "public_statement": "Problem 1.1?\n\n2. G EODESICS IN OUTER SPACE\n\nJuan Souto prompted another major discussion by asking to what extent Teichm¨ uller and WP geodesic analogies are comparable to geodesics in Outer Space with the Lipschitz metric. On the board, we compiled the multi-part question give below; the text that follows is meant to give a sense of the surrounding conversation.",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 149,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0151": {
   "statement_status": "exact",
   "original_statement": "Problem 2.1. Understand analogies of Teichm¨ uller and WP geodesic behavior with behavior of geodesics of Out (Fn) in the Lipschitz metric. Specifically, \n\n(1) What are the \"geodesics\"/\"lines\" having strongly contracting projection functions? \n\n(2) Is there a \"good\" thick part? \n\n(3) Describe the relationship between behavior of geodesics and boundary theory. \n\n(4) Given an iwip φ, what is Min (φ): = {x|d(x, φ (x)) = inf y d(y, φ (y)) }? Is it, or is it in, a CAT(0) space, or an axis bundle? Is it quasi-isometric to a line? \n\n(5) How close are Min (φ) and Min (φ−1)? \n\nParts (1) and (2) were borne out of Juan's inquiry: is there a \"thick part\" which is hyperbolic? It is not enough for the definition to be a lower bound for injectivity radius. What about a thick geodesic? Every iwip axis should be thick with quantifiers. Can one find geodesics that go from thick to thin part? In general, to what extent are analogies truthful or not? THE GEOMETRY OF THE OUTER AUTOMORPHISM GROUP OF A FREE GROUP 3\n\nYael Algom-Kfir, to clarify Juan's question, said an axis is hyperbolic with some quantifiers (projection is exponentially contractible). Yael: So Juan wants, if you fix the quantifiers, find the geodesics for which the projection map is strongly contractible. Juan again: notion of \"thick ge-odesic\" is better than \"thick part.\" Thick part is too easy an analogy: you can run along strata, two rays parallel in thick part; this should not happen. As for motivation for the problem? Hap-piness, says Juan. Yael: he wants to understand the phenomena, for two iwips, when is one more contracting then the other? Meanwhile, Yair Minsky suggested (3). Kasra Rafi: if subsurface projections (meaning local structure) are bounded, heuristically, if you have an iwip spending not too much time in any one subsurface, then you are in the thick part. Juan disagreed, saying one should find which geodesics are contracting to figure out what subsurface projections should be. Regarding (4): this was originally stated as \"what is the set of axes of an iwip, what is its topology and how thick is it.\" The set of axes were clarified to mean the min set named in the problem. Handel's result gives that there are axes in outer space, but he was not sure if it is in Lipschitz metric homotopic to a line, or continuous vs. discrete. Yair asked, if you take powers, do the min sets change? Handel: there are graphs here that don't stretch by λ? Then there was some talk of train tracks for powers; axis bundles are associated to endpoints; having to take powers to make them train tracks. Handel: elements of axis bundle are train tracks with powers included. For irreducibles, are there graphs that are stretched by λ? Mladen Bestvina: no actual examples, but you're not done because of illegal turns. Handel: axis bundle diameter varies. Karen Vogtmann: understand geometry of the set. Handel: it is known what makes diameter bigger, quasi-isometric to a line. He thinks there's not a uniform bound to the diameter. Thierry Coulbois asked if (5) was too specific. Martin Lustig, perhaps in reference to (5)?, brought up the possibility of an analogue to the Masur criterion for non-uniquely ergodic points. 3. G ROUP PROPERTIES",
   "clean_statement": "Problem 2.1. Understand analogies of Teichm¨ uller and WP geodesic behavior with behavior of geodesics of Out (Fn) in the Lipschitz metric. Specifically,\n\n(1) What are the \"geodesics\"/\"lines\" having strongly contracting projection functions?\n\n(2) Is there a \"good\" thick part?\n\n(3) Describe the relationship between behavior of geodesics and boundary theory.\n\n(4) Given an iwip φ, what is Min (φ): = {x|d(x, φ (x)) = inf y d(y, φ (y)) }? Is it, or is it in, a CAT(0) space, or an axis bundle? Is it quasi-isometric to a line?\n\n(5) How close are Min (φ) and Min (φ−1)?\n\nParts (1) and (2) were borne out of Juan's inquiry: is there a \"thick part\" which is hyperbolic? It is not enough for the definition to be a lower bound for injectivity radius. What about a thick geodesic? Every iwip axis should be thick with quantifiers. Can one find geodesics that go from thick to thin part? In general, to what extent are analogies truthful or not? THE GEOMETRY OF THE OUTER AUTOMORPHISM GROUP OF A FREE GROUP 3\n\nYael Algom-Kfir, to clarify Juan's question, said an axis is hyperbolic with some quantifiers (projection is exponentially contractible). Yael: So Juan wants, if you fix the quantifiers, find the geodesics for which the projection map is strongly contractible. Juan again: notion of \"thick ge-odesic\" is better than \"thick part.\" Thick part is too easy an analogy: you can run along strata, two rays parallel in thick part; this should not happen. As for motivation for the problem? Hap-piness, says Juan. Yael: he wants to understand the phenomena, for two iwips, when is one more contracting then the other? Meanwhile, Yair Minsky suggested (3). Kasra Rafi: if subsurface projections (meaning local structure) are bounded, heuristically, if you have an iwip spending not too much time in any one subsurface, then you are in the thick part. Juan disagreed, saying one should find which geodesics are contracting to figure out what subsurface projections should be. Regarding (4): this was originally stated as \"what is the set of axes of an iwip, what is its topology and how thick is it.\" The set of axes were clarified to mean the min set named in the problem. Handel's result gives that there are axes in outer space, but he was not sure if it is in Lipschitz metric homotopic to a line, or continuous vs. discrete. Yair asked, if you take powers, do the min sets change? Handel: there are graphs here that don't stretch by λ? Then there was some talk of train tracks for powers; axis bundles are associated to endpoints; having to take powers to make them train tracks. Handel: elements of axis bundle are train tracks with powers included. For irreducibles, are there graphs that are stretched by λ? Mladen Bestvina: no actual examples, but you're not done because of illegal turns. Handel: axis bundle diameter varies. Karen Vogtmann: understand geometry of the set. Handel: it is known what makes diameter bigger, quasi-isometric to a line. He thinks there's not a uniform bound to the diameter. Thierry Coulbois asked if (5) was too specific. Martin Lustig, perhaps in reference to (5)?, brought up the possibility of an analogue to the Masur criterion for non-uniquely ergodic points. 3. G ROUP PROPERTIES",
   "public_statement": "Problem 2.1. Understand analogies of Teichm¨ uller and WP geodesic behavior with behavior of geodesics of Out (Fn) in the Lipschitz metric. Specifically,\n\n(1) What are the \"geodesics\"/\"lines\" having strongly contracting projection functions?\n\n(2) Is there a \"good\" thick part?\n\n(3) Describe the relationship between behavior of geodesics and boundary theory.\n\n(4) Given an iwip φ, what is Min (φ): = {x|d(x, φ (x)) = inf y d(y, φ (y)) }? Is it, or is it in, a CAT(0) space, or an axis bundle? Is it quasi-isometric to a line?\n\n(5) How close are Min (φ) and Min (φ−1)?\n\nParts (1) and (2) were borne out of Juan's inquiry: is there a \"thick part\" which is hyperbolic? It is not enough for the definition to be a lower bound for injectivity radius. What about a thick geodesic? Every iwip axis should be thick with quantifiers. Can one find geodesics that go from thick to thin part? In general, to what extent are analogies truthful or not? THE GEOMETRY OF THE OUTER AUTOMORPHISM GROUP OF A FREE GROUP 3\n\nYael Algom-Kfir, to clarify Juan's question, said an axis is hyperbolic with some quantifiers (projection is exponentially contractible). Yael: So Juan wants, if you fix the quantifiers, find the geodesics for which the projection map is strongly contractible. Juan again: notion of \"thick ge-odesic\" is better than \"thick part.\" Thick part is too easy an analogy: you can run along strata, two rays parallel in thick part; this should not happen. As for motivation for the problem? Hap-piness, says Juan. Yael: he wants to understand the phenomena, for two iwips, when is one more contracting then the other? Meanwhile, Yair Minsky suggested (3). Kasra Rafi: if subsurface projections (meaning local structure) are bounded, heuristically, if you have an iwip spending not too much time in any one subsurface, then you are in the thick part. Juan disagreed, saying one should find which geodesics are contracting to figure out what subsurface projections should be. Regarding (4): this was originally stated as \"what is the set of axes of an iwip, what is its topology and how thick is it.\" The set of axes were clarified to mean the min set named in the problem. Handel's result gives that there are axes in outer space, but he was not sure if it is in Lipschitz metric homotopic to a line, or continuous vs. discrete. Yair asked, if you take powers, do the min sets change? Handel: there are graphs here that don't stretch by λ? Then there was some talk of train tracks for powers; axis bundles are associated to endpoints; having to take powers to make them train tracks. Handel: elements of axis bundle are train tracks with powers included. For irreducibles, are there graphs that are stretched by λ? Mladen Bestvina: no actual examples, but you're not done because of illegal turns. Handel: axis bundle diameter varies. Karen Vogtmann: understand geometry of the set. Handel: it is known what makes diameter bigger, quasi-isometric to a line. He thinks there's not a uniform bound to the diameter. Thierry Coulbois asked if (5) was too specific. Martin Lustig, perhaps in reference to (5)?, brought up the possibility of an analogue to the Masur criterion for non-uniquely ergodic points. 3. G ROUP PROPERTIES",
   "evidence": "This is Problem 2.1 in the AIM workshop notes *The geometry of the outer automorphism group of a free group* (25--29 October 2010, edited by Johanna Mangahas). In normalized notation, it asks for analogies between Teichmüller/Weil--Petersson geodesics and geodesics in Culler--Vogtmann Outer space with the Lipschitz metric:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 150,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0152": {
   "statement_status": "exact",
   "original_statement": "Problem 3.1. Does Out (Fn) have Property (T) or (τ) if n ≥ 4?\n\nOut (F3) does not have Property (T) because it has a finite index subgroup Γ with |H1(Γ, Z)| = ∞.This motivates the question,",
   "clean_statement": "Problem 3.1. Does Out (Fn) have Property (T) or (τ) if n ≥ 4?\n\nOut (F3) does not have Property (T) because it has a finite index subgroup Γ with |H1(Γ, Z)| = ∞.This motivates the question,",
   "public_statement": "Problem 3.1. Does Out (Fn) have Property (T) or (τ) if n ≥ 4?\n\nOut (F3) does not have Property (T) because it has a finite index subgroup Γ with |H1(Γ, Z)| = ∞.This motivates the question,",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 151,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0153": {
   "statement_status": "exact",
   "original_statement": "Problem 3.2. Does there exist a finite index subgroup Γ < Out (Fn) with infinite abelianization? \n\nMore generally,",
   "clean_statement": "Problem 3.2. Does there exist a finite index subgroup Γ < Out (Fn) with infinite abelianization?\n\nMore generally,",
   "public_statement": "Problem 3.2. Does there exist a finite index subgroup Γ < Out (Fn) with infinite abelianization?\n\nMore generally,",
   "evidence": "The assigned corpus record reproduces:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 152,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0154": {
   "statement_status": "exact",
   "original_statement": "Problem 3.3. What are and what is known about the finite index subgroups of Out (Fn)?\n\nOne fact about finite-index subgroups concerns principal congruence subgroups. Given H acharacteristic finite-index subgroup of Fn, one has Γ = ker( Aut (Fn) → Aut (Fn/H)).",
   "clean_statement": "Problem 3.3. What are and what is known about the finite index subgroups of Out (Fn)?\n\nOne fact about finite-index subgroups concerns principal congruence subgroups. Given H acharacteristic finite-index subgroup of Fn, one has Γ = ker( Aut (Fn) → Aut (Fn/H)).",
   "public_statement": "Problem 3.3. What are and what is known about the finite index subgroups of Out (Fn)?\n\nOne fact about finite-index subgroups concerns principal congruence subgroups. Given H acharacteristic finite-index subgroup of Fn, one has Γ = ker( Aut (Fn) → Aut (Fn/H)).",
   "evidence": "The canonical record is Problem 3.3 from the AIM workshop *The geometry of the outer automorphism group of a free group*. The PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 153,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0155": {
   "statement_status": "exact",
   "original_statement": "Problem 3.4. Does every finite index subgroup contain one of these Γ?\n\nMladen Bestvina asked if there is an effective way to construct these subgroups. Verbal sub-groups, or pass to Burnside group, suggested Martin Bridson. Karen Vogtmann commented that you would classify characteristic subgroups of Out (Fn).4 EDITED BY JOHANNA MANGAHAS \n\nBridson offered a baby question in that direction: if n ≥ 4, is Out (Fn) large? Meaning, does there exist a finite-index subgroup which acts on a tree, thus maps onto a free group? On the board, he writes:",
   "clean_statement": "Problem 3.4. Does every finite index subgroup contain one of these Γ?\n\nMladen Bestvina asked if there is an effective way to construct these subgroups. Verbal sub-groups, or pass to Burnside group, suggested Martin Bridson. Karen Vogtmann commented that you would classify characteristic subgroups of Out (Fn).4 EDITED BY JOHANNA MANGAHAS\n\nBridson offered a baby question in that direction: if n ≥ 4, is Out (Fn) large? Meaning, does there exist a finite-index subgroup which acts on a tree, thus maps onto a free group? On the board, he writes:",
   "public_statement": "Problem 3.4. Does every finite index subgroup contain one of these Γ?\n\nMladen Bestvina asked if there is an effective way to construct these subgroups. Verbal sub-groups, or pass to Burnside group, suggested Martin Bridson. Karen Vogtmann commented that you would classify characteristic subgroups of Out (Fn).4 EDITED BY JOHANNA MANGAHAS\n\nBridson offered a baby question in that direction: if n ≥ 4, is Out (Fn) large? Meaning, does there exist a finite-index subgroup which acts on a tree, thus maps onto a free group? On the board, he writes:",
   "evidence": "The record comes from Johanna Mangahas's summary of the AIM workshop *The geometry of the outer automorphism group of a free group* (October 25--29, 2010), Problem 3.4. The exact extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 154,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0156": {
   "statement_status": "exact",
   "original_statement": "Problem 3.5. Does there exist a finite-index subgroup Γ < Out (Fn) such that Γ → F2?\n\nThierry Coulbois asked, what techniques would apply to these questions, and how does this relate to our machinery? Bridson offered a general point: the questions regard subgroups, whereas train tracks allow normal forms for understanding group elements. Juan Souto asked if one can find an infinite torsion quotient. This would imply non-bounded generation, he said, and don't we know this? Because we know bounded cohomology. The ques-tion is one way to ask how hyperbolic is Out (Fn):",
   "clean_statement": "Problem 3.5. Does there exist a finite-index subgroup Γ < Out (Fn) such that Γ → F2?\n\nThierry Coulbois asked, what techniques would apply to these questions, and how does this relate to our machinery? Bridson offered a general point: the questions regard subgroups, whereas train tracks allow normal forms for understanding group elements. Juan Souto asked if one can find an infinite torsion quotient. This would imply non-bounded generation, he said, and don't we know this? Because we know bounded cohomology. The ques-tion is one way to ask how hyperbolic is Out (Fn):",
   "public_statement": "Problem 3.5. Does there exist a finite-index subgroup Γ < Out (Fn) such that Γ → F2?\n\nThierry Coulbois asked, what techniques would apply to these questions, and how does this relate to our machinery? Bridson offered a general point: the questions regard subgroups, whereas train tracks allow normal forms for understanding group elements. Juan Souto asked if one can find an infinite torsion quotient. This would imply non-bounded generation, he said, and don't we know this? Because we know bounded cohomology. The ques-tion is one way to ask how hyperbolic is Out (Fn):",
   "evidence": "The unchanged canonical record begins:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 155,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0157": {
   "statement_status": "exact",
   "original_statement": "Problem 3.6. Does there exist p → 0 such that Out (Fn)/〈〈 γp〉〉, all γ ∈ Out (Fn), is infinite?",
   "clean_statement": "Problem 3.6. Does there exist p → 0 such that Out (Fn)/〈〈 γp〉〉, all γ ∈ Out (Fn), is infinite?",
   "public_statement": "Problem 3.6. Does there exist p → 0 such that Out (Fn)/〈〈 γp〉〉, all γ ∈ Out (Fn), is infinite?",
   "evidence": "The canonical record is Problem 3.6 from the AIM workshop report *The geometry of the outer automorphism group of a free group*. The PDF itself, not merely the JSON extraction, prints \\[ \\text{“Does there exist }p\\to0\\text{ such that } \\operatorname{Out}(F_n)/\\langle\\!\\langle\\gamma^p\\rangle\\!\\rangle, \\text{ all }\\gamma\\in\\operatorname{Out}(F_n),\\text{ is infinite?”} \\] The superscript \\(p\\), the double normal-closure brackets, and the scope “all \\(\\gamma\\in\\operatorname{Out}(F_n)\\)” are visible in the PDF. Thus the intended denominator is \\[ \\left\\langle\\!\\left\\langle \\gamma^p\\mid\\gamma\\in\\operatorname{Out}(F_n) \\right\\rangle\\!\\right\\rangle. \\] The symbol \\(p\\to0\\) is also genuinely present in the PDF: it is not an OCR substitution. It is nevertheless not meaningful after the existential quantifier if \\(p\\) is an integer exponent. Taking \\(p=0\\) would make every relator \\(\\gamma^0=1\\)...",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 156,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0158": {
   "statement_status": "exact",
   "original_statement": "Problem 3.7. Prove (uniform) uniform exponential growth for two-generator subgroups of Out (Fn),or in particular two Dehn twists. \n\nMatt Clay in his talk had proposed a \"baby question\" towards proving the Dehn twist case: bounding a certain ratio (which ought to be recalled here) appearing in his joint work with Alexandra Pettet. Martin Bridson suggested using linear quotients, as it is potentially easier to prove uniform exponential growth by mapping to well-understood subgroups. 4. H OMOMORPHISMS WITH MAPPING CLASS GROUPS \n\nJuan Souto asked if there are any non-obvious ways for the mapping class group to sit in Out (Fn).",
   "clean_statement": "Problem 3.7. Prove (uniform) uniform exponential growth for two-generator subgroups of Out (Fn),or in particular two Dehn twists.\n\nMatt Clay in his talk had proposed a \"baby question\" towards proving the Dehn twist case: bounding a certain ratio (which ought to be recalled here) appearing in his joint work with Alexandra Pettet. Martin Bridson suggested using linear quotients, as it is potentially easier to prove uniform exponential growth by mapping to well-understood subgroups. 4. H OMOMORPHISMS WITH MAPPING CLASS GROUPS\n\nJuan Souto asked if there are any non-obvious ways for the mapping class group to sit in Out (Fn).",
   "public_statement": "Problem 3.7. Prove (uniform) uniform exponential growth for two-generator subgroups of Out (Fn),or in particular two Dehn twists.\n\nMatt Clay in his talk had proposed a \"baby question\" towards proving the Dehn twist case: bounding a certain ratio (which ought to be recalled here) appearing in his joint work with Alexandra Pettet. Martin Bridson suggested using linear quotients, as it is potentially easier to prove uniform exponential growth by mapping to well-understood subgroups. 4. H OMOMORPHISMS WITH MAPPING CLASS GROUPS\n\nJuan Souto asked if there are any non-obvious ways for the mapping class group to sit in Out (Fn).",
   "evidence": "The source is the AIM workshop list *The geometry of the outer automorphism group of a free group*, Problem 3.7. The mathematical text in the original PDF is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 157,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0159": {
   "statement_status": "exact",
   "original_statement": "Problem 4.1. If S is closed, what is Hom (MCG (S ), Out (Fn))? What about injective homomor-phisms? \n\nIn the other direction, there are no injective homomorphisms from Out (Fn) to MCG (S ), because \n\nOut (Fn) has the Poison subgroups that preclude linearity, and these never appear in MCG (S ) by the result of Brendle and Hamidi-Tehrani. Nonetheless one can ask about non-injective homomor-phisms. An induction argument produces many homomorphisms with infinite image from Out (F3)\n\nto MCG (S ), but what about for n ≥ 4?",
   "clean_statement": "Problem 4.1. If S is closed, what is Hom (MCG (S ), Out (Fn))? What about injective homomor-phisms?\n\nIn the other direction, there are no injective homomorphisms from Out (Fn) to MCG (S ), because\n\nOut (Fn) has the Poison subgroups that preclude linearity, and these never appear in MCG (S ) by the result of Brendle and Hamidi-Tehrani. Nonetheless one can ask about non-injective homomor-phisms. An induction argument produces many homomorphisms with infinite image from Out (F3)\n\nto MCG (S ), but what about for n ≥ 4?",
   "public_statement": "Problem 4.1. If S is closed, what is Hom (MCG (S ), Out (Fn))? What about injective homomor-phisms?\n\nIn the other direction, there are no injective homomorphisms from Out (Fn) to MCG (S ), because\n\nOut (Fn) has the Poison subgroups that preclude linearity, and these never appear in MCG (S ) by the result of Brendle and Hamidi-Tehrani. Nonetheless one can ask about non-injective homomor-phisms. An induction argument produces many homomorphisms with infinite image from Out (F3)\n\nto MCG (S ), but what about for n ≥ 4?",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 158,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0160": {
   "statement_status": "exact",
   "original_statement": "Problem 4.2. What is Hom (Out (Fn), MCG (S ))? Can homomorphisms have infinite image if \n\nn ≥ 4?",
   "clean_statement": "Problem 4.2. What is Hom (Out (Fn), MCG (S ))? Can homomorphisms have infinite image if\n\nn ≥ 4?",
   "public_statement": "Problem 4.2. What is Hom (Out (Fn), MCG (S ))? Can homomorphisms have infinite image if\n\nn ≥ 4?",
   "evidence": "The canonical record comes from Section 4 of the AIM workshop list *The geometry of the outer automorphism group of a free group* (October 2010). The original PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 159,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0161": {
   "statement_status": "exact",
   "original_statement": "Problem 4.3. Does every homomorphism φ: Out (Fn) → MCG (S ) have virtually abelian image? \n\nMartin Bridson, I think, explained that if you fix n, the genus of S must be greater than or equal to n!. This is because Out (Fn) has the Weyl group Z2 o S n (where S n is the symmetries of the rose). \n\nZn · n! = Z2 o S n.It is known, by Martin and Juan's results, that all the Poison groups in Out (Fn) would give virtually abelian image under a homomorphism to MCG (S ). However, subgroups of finite index change the story.",
   "clean_statement": "Problem 4.3. Does every homomorphism φ: Out (Fn) → MCG (S ) have virtually abelian image?\n\nMartin Bridson, I think, explained that if you fix n, the genus of S must be greater than or equal to n!. This is because Out (Fn) has the Weyl group Z2 o S n (where S n is the symmetries of the rose).\n\nZn · n! = Z2 o S n.It is known, by Martin and Juan's results, that all the Poison groups in Out (Fn) would give virtually abelian image under a homomorphism to MCG (S ). However, subgroups of finite index change the story.",
   "public_statement": "Problem 4.3. Does every homomorphism φ: Out (Fn) → MCG (S ) have virtually abelian image?\n\nMartin Bridson, I think, explained that if you fix n, the genus of S must be greater than or equal to n!. This is because Out (Fn) has the Weyl group Z2 o S n (where S n is the symmetries of the rose).\n\nZn · n! = Z2 o S n.It is known, by Martin and Juan's results, that all the Poison groups in Out (Fn) would give virtually abelian image under a homomorphism to MCG (S ). However, subgroups of finite index change the story.",
   "evidence": "Thus `Z2 o S n` is the wreath product and `Zn · n!` is \\(2^n\\cdot n!\\), not a product involving \\(\\mathbb Z^n\\). The PDF really does say that the genus should be at least \\(n!\\); that part is not an OCR error.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 160,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0162": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 4.4. Same questions above for subgroups of finite index. \n\n5. C URRENTS, L AMINATIONS, AND HOROFUNCTION BOUNDARY \n\nArnaud Hilion was seeking a certain inequality. Take T a tree in outer space, and associate to it a Patterson-Sullivan current μT, with normalization choice i(T, μ T ) = 1.",
   "clean_statement": "Let \\(\\Gamma<\\operatorname{Out}(F_n)\\) have finite index. Determine \\(\\operatorname{Hom}(\\Gamma,\\operatorname{Mod}(S))\\). Can such a map have infinite image, especially for \\(n\\geq4\\)? Must its image be virtually abelian? What can be said about injectivity?",
   "public_statement": "Let \\(\\Gamma<\\operatorname{Out}(F_n)\\) have finite index. Determine \\(\\operatorname{Hom}(\\Gamma,\\operatorname{Mod}(S))\\). Can such a map have infinite image, especially for \\(n\\geq4\\)? Must its image be virtually abelian? What can be said about injectivity?",
   "evidence": "The official AIM PDF makes two boundaries clear. First, the Section 5 heading and the Patterson--Sullivan-current sentence begin a new section. They are extraction contamination and have no mathematical ownership in Problem 4.4.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 161,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0163": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 5.1. For all T, T ′ ∈ Xn as above, is it true i(T, μ T ′ ) · i(T ′, μ T ) ≥ 1? At least, where is the inequality true? \n\nThe hope is to use this to define a WP-metric on Xn.Kasra Rafi: Teichm¨ uller space embeds in currents, and Bonahon defines a metric which turns out to be the WP metric. If you try this with quadratic differentials and the Lipschitz metric, then inequality does not hold. The metric you get is not positive-Bonahon's construction doesn't work! That's a downer. But hope is that there is perhaps a subspace of outer space, perhaps a manifold, in which the inequality does hold. The subspace should be Out (Fn)-invariant. The next question comes from Martin Lustig, with the second part added by Arnaud. Lustig recalled that Huber[?] and Besson decompose an R-tree into components, and also we know an R-tree has a dual lamination characterizing its topological side in some sense.",
   "clean_statement": null,
   "public_statement": "Problem 5.1. For all T, T ′ ∈ Xn as above, is it true i(T, μ T ′ ) · i(T ′, μ T ) ≥ 1? At least, where is the inequality true?\n\nThe hope is to use this to define a WP-metric on Xn.Kasra Rafi: Teichm¨ uller space embeds in currents, and Bonahon defines a metric which turns out to be the WP metric. If you try this with quadratic differentials and the Lipschitz metric, then inequality does not hold. The metric you get is not positive-Bonahon's construction doesn't work! That's a downer. But hope is that there is perhaps a subspace of outer space, perhaps a manifold, in which the inequality does hold. The subspace should be Out (Fn)-invariant. The next question comes from Martin Lustig, with the second part added by Arnaud. Lustig recalled that Huber[?] and Besson decompose an R-tree into components, and also we know an R-tree has a dual lamination characterizing its topological side in some sense.",
   "evidence": "The canonical record is Problem 5.1 from the 2010 AIM workshop *The geometry of the outer automorphism group of a free group*. Its extracted text asks:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 162,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0164": {
   "statement_status": "exact",
   "original_statement": "Problem 5.2. Can one understand the GL-theory of FINE DECOMPOSITION in terms of dual laminations and vice versa? And in terms of systems of partial isometries? \n\nInterested in the horofunction boundary of outer space Xn in the Lipschitz metric, Cormac Walsh explained that it is coarser than the usual boundary. Mapping x 7 → d(−, x), outer space embeds as a subset X of C0(Xn)/∼, where ∼ is homotopy to a constant, and the topology is uniform convergence on compact subsets.",
   "clean_statement": "Problem 5.2. Can one understand the GL-theory of FINE DECOMPOSITION in terms of dual laminations and vice versa? And in terms of systems of partial isometries?\n\nInterested in the horofunction boundary of outer space Xn in the Lipschitz metric, Cormac Walsh explained that it is coarser than the usual boundary. Mapping x 7 → d(−, x), outer space embeds as a subset X of C0(Xn)/∼, where ∼ is homotopy to a constant, and the topology is uniform convergence on compact subsets.",
   "public_statement": "Problem 5.2. Can one understand the GL-theory of FINE DECOMPOSITION in terms of dual laminations and vice versa? And in terms of systems of partial isometries?\n\nInterested in the horofunction boundary of outer space Xn in the Lipschitz metric, Cormac Walsh explained that it is coarser than the usual boundary. Mapping x 7 → d(−, x), outer space embeds as a subset X of C0(Xn)/∼, where ∼ is homotopy to a constant, and the topology is uniform convergence on compact subsets.",
   "evidence": "The canonical record reproduces the following text, which is preserved here without silently correcting it:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 163,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0165": {
   "statement_status": "exact",
   "original_statement": "Problem 5.3. What is the closure X? What is X\\X, the horofunction boundary of outer space in the Lipschitz metric? \n\nThierry Coulbois asked if we know the horofunction boundary in the simplicial metric, and also if it is reasonable Busemann points could be the whole thing. For the latter, Cormac said yes, as it is true for Teichm¨ uller space. Specifically, a Busemann point Bσ is determined by a 'good' ray \n\nσ: [0, ∞) → Xn by Bσ(x) = lim n→∞ (d(x, σ (n)) − n). If X were CAT(0) or Teichm¨ uller space 6 EDITED BY JOHANNA MANGAHAS \n\nwith the Lipschitz metric, then X\\X would consist entirely of Busemann points. However, warned Cormac, even for normed spaces you can get points that are not Busemann points.",
   "clean_statement": "Problem 5.3. What is the closure X? What is X\\X, the horofunction boundary of outer space in the Lipschitz metric?\n\nThierry Coulbois asked if we know the horofunction boundary in the simplicial metric, and also if it is reasonable Busemann points could be the whole thing. For the latter, Cormac said yes, as it is true for Teichm¨ uller space. Specifically, a Busemann point Bσ is determined by a 'good' ray\n\nσ: [0, ∞) → Xn by Bσ(x) = lim n→∞ (d(x, σ (n)) − n). If X were CAT(0) or Teichm¨ uller space 6 EDITED BY JOHANNA MANGAHAS\n\nwith the Lipschitz metric, then X\\X would consist entirely of Busemann points. However, warned Cormac, even for normed spaces you can get points that are not Busemann points.",
   "public_statement": "Problem 5.3. What is the closure X? What is X\\X, the horofunction boundary of outer space in the Lipschitz metric?\n\nThierry Coulbois asked if we know the horofunction boundary in the simplicial metric, and also if it is reasonable Busemann points could be the whole thing. For the latter, Cormac said yes, as it is true for Teichm¨ uller space. Specifically, a Busemann point Bσ is determined by a 'good' ray\n\nσ: [0, ∞) → Xn by Bσ(x) = lim n→∞ (d(x, σ (n)) − n). If X were CAT(0) or Teichm¨ uller space 6 EDITED BY JOHANNA MANGAHAS\n\nwith the Lipschitz metric, then X\\X would consist entirely of Busemann points. However, warned Cormac, even for normed spaces you can get points that are not Busemann points.",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 164,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0166": {
   "statement_status": "exact",
   "original_statement": "1. Generalized Valette's conjecture: Let G be a locally compact compactly generated group and Γ be a cocompact lattice in G. If G has Property RD, then Γ has Property RD. The original Valette's conjecture concerns the case when G is a semi-simple Lie group. The conjecture has been proved [Ch1] for cocompact lattices in finite product of rank 1 simple Lie groups and [La1] for G = SL (3, K) where K = R, C, H and O. It is open in general for simple groups of higher rank and in particular for S0(2, 3) = Sp (4, R).\n\nAlso note that the converse to this conjecture is an easy result [Jo1]. 2. A weaker conjecture: Assume that Γ 1 and Γ 2 are two cocompact lattices in \n\nG. Then if Γ 1 has property RD, then so does Γ 2. This conjecture can be generalized to groups admitting a measurable topological coupling. Recall that two finitely generated groups Γ 1 and Γ 2 admit a measurable topological coupling if there exists a locally compact topological Γ 1 ×Γ2-space equipped with an invariant σ-compact Borel measure μ and such that the actions of Γ1 and Γ 2 are proper and cocompact. 3. Note that non-cocompact lattices often do not have RD since they contain solvable subgroups with exponential growth. 4. Problem: find a variation of RD that is true for every lattice in connected semisimple Lie groups and that still have interesting consequences. Possible answer: find a notion of relative RD. For instance, SL (n, Z) for n ≥ 3 does not have Property RD because of solvable subgroups of exponential growth, 1but maybe it would have RD \"relatively\" to a maximal solvable subgroup. Maybe solvable groups would have RD relatively to its exponentially dis-torted cyclic subgroups. 5. Let H be a cocompact normal subgroup of a locally compact, compactly generated group G with Property RD. Does H have RD? Note that the converse is easy to prove using induction of unitary representations. Indeed, if H has RD, then it is straightforward to see that the G-representation obtained by inducing to G the regular representation of H has RD and then to show that this implies that G has RD. 6. Another stability question: let 1 → H → Γ → P → 1be a short exact sequence of finitely generated groups such that P has RD. Is it true that Γ has RD if and only if H has RD for the induced length? 7. Is every co-compact lattice in a semi-simple Lie group (**)-relatively hy-perbolic in the sense of [DS2] with respect to quasi-flats? \n\n2 RD for unitary representations \n\nDuring the workshop, we emphasized the fact that RD can be formulated for any unitary representation ( π, H). \n\nDefinition 2.1. Let G be a locally compact group equipped with a length func-tion L. We say that a unitary representation ( π, H) of G has Property RD for the length function L if there exists s > 0 and C < ∞ such that \n\n∫\n\n> G\n\n〈π(g)v, w 〉\n\n1 + L(g)s dμ (g) ≤ C\n\nfor all unitary vectors v, w ∈ H and every g ∈ G.1. Stability properties: first, note that λG has RD if and only if the group G\n\nhas RD. If σ is weakly contained in π and if π has RD, then σ has RD. Conversely, if ( πi) is a decomposition of π into irreducible representations and if every πi have RD with uniform constants s and C, then π has RD with the same constants. 22.",
   "clean_statement": "1. Generalized Valette's conjecture: Let G be a locally compact compactly generated group and Γ be a cocompact lattice in G. If G has Property RD, then Γ has Property RD. The original Valette's conjecture concerns the case when G is a semi-simple Lie group. The conjecture has been proved [Ch1] for cocompact lattices in finite product of rank 1 simple Lie groups and [La1] for G = SL (3, K) where K = R, C, H and O. It is open in general for simple groups of higher rank and in particular for S0(2, 3) = Sp (4, R).\n\nAlso note that the converse to this conjecture is an easy result [Jo1]. 2. A weaker conjecture: Assume that Γ 1 and Γ 2 are two cocompact lattices in\n\nG. Then if Γ 1 has property RD, then so does Γ 2. This conjecture can be generalized to groups admitting a measurable topological coupling. Recall that two finitely generated groups Γ 1 and Γ 2 admit a measurable topological coupling if there exists a locally compact topological Γ 1 ×Γ2-space equipped with an invariant σ-compact Borel measure μ and such that the actions of Γ1 and Γ 2 are proper and cocompact. 3. Note that non-cocompact lattices often do not have RD since they contain solvable subgroups with exponential growth. 4. Problem: find a variation of RD that is true for every lattice in connected semisimple Lie groups and that still have interesting consequences. Possible answer: find a notion of relative RD. For instance, SL (n, Z) for n ≥ 3 does not have Property RD because of solvable subgroups of exponential growth, 1but maybe it would have RD \"relatively\" to a maximal solvable subgroup. Maybe solvable groups would have RD relatively to its exponentially dis-torted cyclic subgroups. 5. Let H be a cocompact normal subgroup of a locally compact, compactly generated group G with Property RD. Does H have RD? Note that the converse is easy to prove using induction of unitary representations. Indeed, if H has RD, then it is straightforward to see that the G-representation obtained by inducing to G the regular representation of H has RD and then to show that this implies that G has RD. 6. Another stability question: let 1 → H → Γ → P → 1be a short exact sequence of finitely generated groups such that P has RD. Is it true that Γ has RD if and only if H has RD for the induced length? 7. Is every co-compact lattice in a semi-simple Lie group (**)-relatively hy-perbolic in the sense of [DS2] with respect to quasi-flats?\n\n2 RD for unitary representations\n\nDuring the workshop, we emphasized the fact that RD can be formulated for any unitary representation ( π, H).\n\nDefinition 2.1. Let G be a locally compact group equipped with a length func-tion L. We say that a unitary representation ( π, H) of G has Property RD for the length function L if there exists s > 0 and C < ∞ such that\n\n∫\n\n> G\n\n〈π(g)v, w 〉\n\n1 + L(g)s dμ (g) ≤ C\n\nfor all unitary vectors v, w ∈ H and every g ∈ G.1. Stability properties: first, note that λG has RD if and only if the group G\n\nhas RD. If σ is weakly contained in π and if π has RD, then σ has RD. Conversely, if ( πi) is a decomposition of π into irreducible representations and if every πi have RD with uniform constants s and C, then π has RD with the same constants. 22.",
   "public_statement": "1. Generalized Valette's conjecture: Let G be a locally compact compactly generated group and Γ be a cocompact lattice in G. If G has Property RD, then Γ has Property RD. The original Valette's conjecture concerns the case when G is a semi-simple Lie group. The conjecture has been proved [Ch1] for cocompact lattices in finite product of rank 1 simple Lie groups and [La1] for G = SL (3, K) where K = R, C, H and O. It is open in general for simple groups of higher rank and in particular for S0(2, 3) = Sp (4, R).\n\nAlso note that the converse to this conjecture is an easy result [Jo1]. 2. A weaker conjecture: Assume that Γ 1 and Γ 2 are two cocompact lattices in\n\nG. Then if Γ 1 has property RD, then so does Γ 2. This conjecture can be generalized to groups admitting a measurable topological coupling. Recall that two finitely generated groups Γ 1 and Γ 2 admit a measurable topological coupling if there exists a locally compact topological Γ 1 ×Γ2-space equipped with an invariant σ-compact Borel measure μ and such that the actions of Γ1 and Γ 2 are proper and cocompact. 3. Note that non-cocompact lattices often do not have RD since they contain solvable subgroups with exponential growth. 4. Problem: find a variation of RD that is true for every lattice in connected semisimple Lie groups and that still have interesting consequences. Possible answer: find a notion of relative RD. For instance, SL (n, Z) for n ≥ 3 does not have Property RD because of solvable subgroups of exponential growth, 1but maybe it would have RD \"relatively\" to a maximal solvable subgroup. Maybe solvable groups would have RD relatively to its exponentially dis-torted cyclic subgroups. 5. Let H be a cocompact normal subgroup of a locally compact, compactly generated group G with Property RD. Does H have RD? Note that the converse is easy to prove using induction of unitary representations. Indeed, if H has RD, then it is straightforward to see that the G-representation obtained by inducing to G the regular representation of H has RD and then to show that this implies that G has RD. 6. Another stability question: let 1 → H → Γ → P → 1be a short exact sequence of finitely generated groups such that P has RD. Is it true that Γ has RD if and only if H has RD for the induced length? 7. Is every co-compact lattice in a semi-simple Lie group (**)-relatively hy-perbolic in the sense of [DS2] with respect to quasi-flats?\n\n2 RD for unitary representations\n\nDuring the workshop, we emphasized the fact that RD can be formulated for any unitary representation ( π, H).\n\nDefinition 2.1. Let G be a locally compact group equipped with a length func-tion L. We say that a unitary representation ( π, H) of G has Property RD for the length function L if there exists s > 0 and C < ∞ such that\n\n∫\n\n> G\n\n〈π(g)v, w 〉\n\n1 + L(g)s dμ (g) ≤ C\n\nfor all unitary vectors v, w ∈ H and every g ∈ G.1. Stability properties: first, note that λG has RD if and only if the group G\n\nhas RD. If σ is weakly contained in π and if π has RD, then σ has RD. Conversely, if ( πi) is a decomposition of π into irreducible representations and if every πi have RD with uniform constants s and C, then π has RD with the same constants. 22.",
   "evidence": "The source is the eight-page AIM workshop problem list *Property of rapid decay* (21 March 2006). The database record has accidentally concatenated all of Sections 1 and 2, page numbers, and the heading of Section 3. Comparing the record with pages 1--3 of the source gives the following boundary:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 165,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0167": {
   "statement_status": "exact",
   "original_statement": "1. First, observe that if G has RD for any length function L, then it has RD. Moreover, if Γ is finitely generated and has RD with respect to some length, then every subgroup has RD with respect to the induced length. In particular any cyclic subgroup of Γ is at most polynomially distorted. 2. An example: there exists a finitely group Γ which has RD for a length function that is exponentially distorted with respect to the word length. Namely, this is the case of the free group F2 equipped with the length induced from its inclusion in the group F2 o Z defined by the presentation \n\n〈a, b, c; ac = a2b, b c = ab 〉.\n\n4 Completely bounded property RD \n\nLet Γ be a discrete group equipped with a length function L. Denote by \n\nHsL(Γ) = {f: Γ → C, ∑\n\n> Γ\n\n|f (γ)|2\n\n1 + L(γ)s < ∞}.\n\nRecall that Γ has Property RD with respect to a length function l if and only if there exists s > 0 such that HsL(Γ) ↪→ C∗ \n\n> r\n\n(Γ) is a bounded operator. \n\nDefinition 4.1. Let X be a Banach space and let ( ‖·‖ n)n be a sequence of norms on Mn(X) such that \n\n‖Diag (V, W )‖n+m = max( ‖V ‖n, ‖W ‖m), ∀(V, W ) ∈ Mn(X) × Mm(X)3and \n\n‖αV β ‖m ≤ ‖ α‖‖ V ‖n‖β‖, ∀(α, V, β ) ∈ Mm,n (X) × Mn(X) × Mn,m (X)where ‖ · ‖ denotes the usual operator norm. A completely bounded map T:\n\nX → Y between two Banach spaces X and Y equipped with such sequences of norms is completely bounded if there exists K < ∞ such that \n\n‖T V ‖n ≤ K‖V ‖n, ∀V ∈ Mn(X)where T [·] = [ T (·)].",
   "clean_statement": "1. First, observe that if G has RD for any length function L, then it has RD. Moreover, if Γ is finitely generated and has RD with respect to some length, then every subgroup has RD with respect to the induced length. In particular any cyclic subgroup of Γ is at most polynomially distorted. 2. An example: there exists a finitely group Γ which has RD for a length function that is exponentially distorted with respect to the word length. Namely, this is the case of the free group F2 equipped with the length induced from its inclusion in the group F2 o Z defined by the presentation\n\n〈a, b, c; ac = a2b, b c = ab 〉.\n\n4 Completely bounded property RD\n\nLet Γ be a discrete group equipped with a length function L. Denote by\n\nHsL(Γ) = {f: Γ → C, ∑\n\n> Γ\n\n|f (γ)|2\n\n1 + L(γ)s < ∞}.\n\nRecall that Γ has Property RD with respect to a length function l if and only if there exists s > 0 such that HsL(Γ) ↪→ C∗\n\n> r\n\n(Γ) is a bounded operator.\n\nDefinition 4.1. Let X be a Banach space and let ( ‖·‖ n)n be a sequence of norms on Mn(X) such that\n\n‖Diag (V, W )‖n+m = max( ‖V ‖n, ‖W ‖m), ∀(V, W ) ∈ Mn(X) × Mm(X)3and\n\n‖αV β ‖m ≤ ‖ α‖‖ V ‖n‖β‖, ∀(α, V, β ) ∈ Mm,n (X) × Mn(X) × Mn,m (X)where ‖ · ‖ denotes the usual operator norm. A completely bounded map T:\n\nX → Y between two Banach spaces X and Y equipped with such sequences of norms is completely bounded if there exists K < ∞ such that\n\n‖T V ‖n ≤ K‖V ‖n, ∀V ∈ Mn(X)where T [·] = [ T (·)].",
   "public_statement": "1. First, observe that if G has RD for any length function L, then it has RD. Moreover, if Γ is finitely generated and has RD with respect to some length, then every subgroup has RD with respect to the induced length. In particular any cyclic subgroup of Γ is at most polynomially distorted. 2. An example: there exists a finitely group Γ which has RD for a length function that is exponentially distorted with respect to the word length. Namely, this is the case of the free group F2 equipped with the length induced from its inclusion in the group F2 o Z defined by the presentation\n\n〈a, b, c; ac = a2b, b c = ab 〉.\n\n4 Completely bounded property RD\n\nLet Γ be a discrete group equipped with a length function L. Denote by\n\nHsL(Γ) = {f: Γ → C, ∑\n\n> Γ\n\n|f (γ)|2\n\n1 + L(γ)s < ∞}.\n\nRecall that Γ has Property RD with respect to a length function l if and only if there exists s > 0 such that HsL(Γ) ↪→ C∗\n\n> r\n\n(Γ) is a bounded operator.\n\nDefinition 4.1. Let X be a Banach space and let ( ‖·‖ n)n be a sequence of norms on Mn(X) such that\n\n‖Diag (V, W )‖n+m = max( ‖V ‖n, ‖W ‖m), ∀(V, W ) ∈ Mn(X) × Mm(X)3and\n\n‖αV β ‖m ≤ ‖ α‖‖ V ‖n‖β‖, ∀(α, V, β ) ∈ Mm,n (X) × Mn(X) × Mn,m (X)where ‖ · ‖ denotes the usual operator norm. A completely bounded map T:\n\nX → Y between two Banach spaces X and Y equipped with such sequences of norms is completely bounded if there exists K < ∞ such that\n\n‖T V ‖n ≤ K‖V ‖n, ∀V ∈ Mn(X)where T [·] = [ T (·)].",
   "evidence": "The record merges the end of Section 3, all of Section 4, and the heading of Section 5 of the AIM workshop note *Property of rapid decay* (21 March 2006). The actual problem is the last paragraph of Section 4:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 166,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0168": {
   "statement_status": "exact",
   "original_statement": "1. Consider a building of type ˜A2 (e.g. Bruhat Tits building of SL (3, Qp)) Is the square root appearing in the expression 1 \n\n> 2\n\n(m+1)( n+1)( m+n+2) √(·)) (see [RRS]) necessary when bounding ‖f? g ‖2, g supported on words of shape ( m, n )? 2. Does 〈a, b, s, t; as = ( ab )2, b t = ( ab )2〉 (D. Wise's non-Hopfian group) have Property RD? This is a CAT(0) group, so in particular has no distorted subgroup. However, this group does not act on any cube complex and is not relatively hyperbolic. 3. Assume that Γ acts on a simplicial tree. What are the conditions on the vertex (and edge) stabilizers for Γ to have RD? (Maybe solved, but un-published by Steger and Talbi. The answer would be if and only if the stabilizers have RD for the induced length). For example, this would han-dle HNN extensions. 4. Can simplicial trees be replaced by δ-hyperbolic graphs with bounded de-gree? 46 RD and geometry",
   "clean_statement": "1. Consider a building of type ˜A2 (e.g. Bruhat Tits building of SL (3, Qp)) Is the square root appearing in the expression 1\n\n> 2\n\n(m+1)( n+1)( m+n+2) √(·)) (see [RRS]) necessary when bounding ‖f? g ‖2, g supported on words of shape ( m, n )? 2. Does 〈a, b, s, t; as = ( ab )2, b t = ( ab )2〉 (D. Wise's non-Hopfian group) have Property RD? This is a CAT(0) group, so in particular has no distorted subgroup. However, this group does not act on any cube complex and is not relatively hyperbolic. 3. Assume that Γ acts on a simplicial tree. What are the conditions on the vertex (and edge) stabilizers for Γ to have RD? (Maybe solved, but un-published by Steger and Talbi. The answer would be if and only if the stabilizers have RD for the induced length). For example, this would han-dle HNN extensions. 4. Can simplicial trees be replaced by δ-hyperbolic graphs with bounded de-gree? 46 RD and geometry",
   "public_statement": "1. Consider a building of type ˜A2 (e.g. Bruhat Tits building of SL (3, Qp)) Is the square root appearing in the expression 1\n\n> 2\n\n(m+1)( n+1)( m+n+2) √(·)) (see [RRS]) necessary when bounding ‖f? g ‖2, g supported on words of shape ( m, n )? 2. Does 〈a, b, s, t; as = ( ab )2, b t = ( ab )2〉 (D. Wise's non-Hopfian group) have Property RD? This is a CAT(0) group, so in particular has no distorted subgroup. However, this group does not act on any cube complex and is not relatively hyperbolic. 3. Assume that Γ acts on a simplicial tree. What are the conditions on the vertex (and edge) stabilizers for Γ to have RD? (Maybe solved, but un-published by Steger and Talbi. The answer would be if and only if the stabilizers have RD for the induced length). For example, this would han-dle HNN extensions. 4. Can simplicial trees be replaced by δ-hyperbolic graphs with bounded de-gree? 46 RD and geometry",
   "evidence": "This record is the four-question list in Section 5, “RD for groups acting on special metric spaces,” of the AIM workshop notes *The property of rapid decay*. The extracted record has several OCR defects. Comparison with the workshop PDF and the cited paper of Ramagge--Robertson--Steger (RRS) recovers the questions as follows.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 167,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0169": {
   "statement_status": "exact",
   "original_statement": "1. Is RD stable under quasi-isometry between finitely generated groups? If Γ and Λ have isometric Cayley graphs and if Γ has RD, does Λ have RD? 2. P. de la Harpe [Ha] has shown that hyperbolic groups have RD. This was extended to (**)-relatively hyperbolic groups with respect to RD subgroups by C. Drutu and M. Sapir [DS2]. Conversely, does RD have geometric consequences (such as for a space it acts on cocompactly)? 3. Define RD for a reasonable class of metric spaces such as graphs with bounded degree, or more generally, uniformly locally finite metric spaces. 4. Question (Lafforgue): Is RD equivalent to the existence of a polynomial P\n\nsuch that for all finite subspaces A, B, C of Γ such that A ⊂ B(r), \n\n|{ (a, b, c ) ∈ A × B × C, abc = 1 }| ≤ P (r)√|A|| B|| C|?Note that RD implies this property (apply RD to indicator functions). 5. Vague: is there a characterization of RD in terms of geometric properties such as some action on a boundary? Some action on a compact space? 6.",
   "clean_statement": "1. Is RD stable under quasi-isometry between finitely generated groups? If Γ and Λ have isometric Cayley graphs and if Γ has RD, does Λ have RD? 2. P. de la Harpe [Ha] has shown that hyperbolic groups have RD. This was extended to (**)-relatively hyperbolic groups with respect to RD subgroups by C. Drutu and M. Sapir [DS2]. Conversely, does RD have geometric consequences (such as for a space it acts on cocompactly)? 3. Define RD for a reasonable class of metric spaces such as graphs with bounded degree, or more generally, uniformly locally finite metric spaces. 4. Question (Lafforgue): Is RD equivalent to the existence of a polynomial P\n\nsuch that for all finite subspaces A, B, C of Γ such that A ⊂ B(r),\n\n|{ (a, b, c ) ∈ A × B × C, abc = 1 }| ≤ P (r)√|A|| B|| C|?Note that RD implies this property (apply RD to indicator functions). 5. Vague: is there a characterization of RD in terms of geometric properties such as some action on a boundary? Some action on a compact space? 6.",
   "public_statement": "1. Is RD stable under quasi-isometry between finitely generated groups? If Γ and Λ have isometric Cayley graphs and if Γ has RD, does Λ have RD? 2. P. de la Harpe [Ha] has shown that hyperbolic groups have RD. This was extended to (**)-relatively hyperbolic groups with respect to RD subgroups by C. Drutu and M. Sapir [DS2]. Conversely, does RD have geometric consequences (such as for a space it acts on cocompactly)? 3. Define RD for a reasonable class of metric spaces such as graphs with bounded degree, or more generally, uniformly locally finite metric spaces. 4. Question (Lafforgue): Is RD equivalent to the existence of a polynomial P\n\nsuch that for all finite subspaces A, B, C of Γ such that A ⊂ B(r),\n\n|{ (a, b, c ) ∈ A × B × C, abc = 1 }| ≤ P (r)√|A|| B|| C|?Note that RD implies this property (apply RD to indicator functions). 5. Vague: is there a characterization of RD in terms of geometric properties such as some action on a boundary? Some action on a compact space? 6.",
   "evidence": "The canonical record comes from the AIM workshop document *Property of rapid decay*, dated March 21, 2006. I inspected the official PDF, in particular PDF page 5. The extraction has joined a section heading and a following remark to the problem. The source actually reads as follows (typographical prose errors are retained here, while mathematical symbols are rendered in LaTeX).",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 168,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0170": {
   "statement_status": "exact",
   "original_statement": "1. The Rieffel problem: let Γ be a discrete group equipped with a length function L. Let D: l2(Γ) → l2(Γ) be the unbounded operator defined by \n\nDδ γ = L(γ)δγ.\n\nConnes showed that ( l2(Γ), D ) induces a (possibly unbounded) metric on the state space of C∗ \n\n> r\n\n(Γ). Rieffel noticed that this construction for a metric generalizes the Monge-Kantorovic metric on probability measures. Moti-vated by Kantorovic's result Rieffel found natural to ask when the the met-ric contructed by Connes will give the weak*-topology on the state space. Rieffel proved that this will happen exactly when the set \n\nL:= {a ∈ C∗ \n\n> r\n\n(Γ): tr( a) = 0, ‖[D, a ]‖ ≤ 1}\n\n5is precompact in norm topology. With this characterization at hand an interesting question is then to ask for what discrete groups we have fulfilled the foregoing precompactness. It seems natural to investigate groups with rapid decay, since the precompactness of L has already been established for Γ = Zd ([Ri]) and for Γ being a hyperbolic group ([OR]) (by completely different methods). Also it has been proved in [AC1] that RD implies that there exists a k0 ∈ N such that for any natural number k ≥ k0 the set \n\nLk:= {a ∈ C∗ \n\n> r\n\n(Γ): tr( a) = 0, ‖[D, [D,..., [D, a ]... ]] \n\n︸ ︷︷ ︸\n\n> k\n\n‖ ≤ 1}\n\nis precompact in the norm topology. 2. From Roe's book [Ro]: for what discrete groups do we have \n\nC∗ \n\n> r\n\nΓ = C∗\n\n> u\n\n(|Γ|) ∩ L(Γ) where C∗\n\n> u\n\n(|Γ|) is the uniform Roe's algebra of Γ and L(Γ) is the Von Neu-mann Algebra of Γ? Imitating Haagerup's proof for free groups yield this for any Γ that has RD for a conditionally negative length function. Recall that L is a conditionally negative length fonction if (Γ, √L) isometrically embeds into a Hilbert space. 3. Conjecture of Kaplansky: let Γ be a torsion free finitely generated group, then CΓ has no 0-divisor. Problem: prove it when Γ has RD. This problem is motivated by the fact proved by Lafforgue [La2] that RD + some (very general) geometric properties imply that the idempotents on CΓ are trivial. 4. Random walks: let Γ be a finitely generated group with Property RD and let ν be a finitely supported symmetric probability on Γ. Does there exist some constants d = d(ν) and c = c(ν) such that \n\nν(2 n)(e) ∼ cn −dρ2n\n\nwhere ρ is the spectral radius of the convolution operator associated to ν\n\non `2(Γ)? 5. Vague: is there a relation between RD and the entropy of ergodic actions of Γ on measure spaces? 68 Which one of these groups have RD?",
   "clean_statement": "1. The Rieffel problem: let Γ be a discrete group equipped with a length function L. Let D: l2(Γ) → l2(Γ) be the unbounded operator defined by\n\nDδ γ = L(γ)δγ.\n\nConnes showed that ( l2(Γ), D ) induces a (possibly unbounded) metric on the state space of C∗\n\n> r\n\n(Γ). Rieffel noticed that this construction for a metric generalizes the Monge-Kantorovic metric on probability measures. Moti-vated by Kantorovic's result Rieffel found natural to ask when the the met-ric contructed by Connes will give the weak*-topology on the state space. Rieffel proved that this will happen exactly when the set\n\nL:= {a ∈ C∗\n\n> r\n\n(Γ): tr( a) = 0, ‖[D, a ]‖ ≤ 1}\n\n5is precompact in norm topology. With this characterization at hand an interesting question is then to ask for what discrete groups we have fulfilled the foregoing precompactness. It seems natural to investigate groups with rapid decay, since the precompactness of L has already been established for Γ = Zd ([Ri]) and for Γ being a hyperbolic group ([OR]) (by completely different methods). Also it has been proved in [AC1] that RD implies that there exists a k0 ∈ N such that for any natural number k ≥ k0 the set\n\nLk:= {a ∈ C∗\n\n> r\n\n(Γ): tr( a) = 0, ‖[D, [D,..., [D, a ]... ]]\n\n︸ ︷︷ ︸\n\n> k\n\n‖ ≤ 1}\n\nis precompact in the norm topology. 2. From Roe's book [Ro]: for what discrete groups do we have\n\nC∗\n\n> r\n\nΓ = C∗\n\n> u\n\n(|Γ|) ∩ L(Γ) where C∗\n\n> u\n\n(|Γ|) is the uniform Roe's algebra of Γ and L(Γ) is the Von Neu-mann Algebra of Γ? Imitating Haagerup's proof for free groups yield this for any Γ that has RD for a conditionally negative length function. Recall that L is a conditionally negative length fonction if (Γ, √L) isometrically embeds into a Hilbert space. 3. Conjecture of Kaplansky: let Γ be a torsion free finitely generated group, then CΓ has no 0-divisor. Problem: prove it when Γ has RD. This problem is motivated by the fact proved by Lafforgue [La2] that RD + some (very general) geometric properties imply that the idempotents on CΓ are trivial. 4. Random walks: let Γ be a finitely generated group with Property RD and let ν be a finitely supported symmetric probability on Γ. Does there exist some constants d = d(ν) and c = c(ν) such that\n\nν(2 n)(e) ∼ cn −dρ2n\n\nwhere ρ is the spectral radius of the convolution operator associated to ν\n\non `2(Γ)? 5. Vague: is there a relation between RD and the entropy of ergodic actions of Γ on measure spaces? 68 Which one of these groups have RD?",
   "public_statement": "1. The Rieffel problem: let Γ be a discrete group equipped with a length function L. Let D: l2(Γ) → l2(Γ) be the unbounded operator defined by\n\nDδ γ = L(γ)δγ.\n\nConnes showed that ( l2(Γ), D ) induces a (possibly unbounded) metric on the state space of C∗\n\n> r\n\n(Γ). Rieffel noticed that this construction for a metric generalizes the Monge-Kantorovic metric on probability measures. Moti-vated by Kantorovic's result Rieffel found natural to ask when the the met-ric contructed by Connes will give the weak*-topology on the state space. Rieffel proved that this will happen exactly when the set\n\nL:= {a ∈ C∗\n\n> r\n\n(Γ): tr( a) = 0, ‖[D, a ]‖ ≤ 1}\n\n5is precompact in norm topology. With this characterization at hand an interesting question is then to ask for what discrete groups we have fulfilled the foregoing precompactness. It seems natural to investigate groups with rapid decay, since the precompactness of L has already been established for Γ = Zd ([Ri]) and for Γ being a hyperbolic group ([OR]) (by completely different methods). Also it has been proved in [AC1] that RD implies that there exists a k0 ∈ N such that for any natural number k ≥ k0 the set\n\nLk:= {a ∈ C∗\n\n> r\n\n(Γ): tr( a) = 0, ‖[D, [D,..., [D, a ]... ]]\n\n︸ ︷︷ ︸\n\n> k\n\n‖ ≤ 1}\n\nis precompact in the norm topology. 2. From Roe's book [Ro]: for what discrete groups do we have\n\nC∗\n\n> r\n\nΓ = C∗\n\n> u\n\n(|Γ|) ∩ L(Γ) where C∗\n\n> u\n\n(|Γ|) is the uniform Roe's algebra of Γ and L(Γ) is the Von Neu-mann Algebra of Γ? Imitating Haagerup's proof for free groups yield this for any Γ that has RD for a conditionally negative length function. Recall that L is a conditionally negative length fonction if (Γ, √L) isometrically embeds into a Hilbert space. 3. Conjecture of Kaplansky: let Γ be a torsion free finitely generated group, then CΓ has no 0-divisor. Problem: prove it when Γ has RD. This problem is motivated by the fact proved by Lafforgue [La2] that RD + some (very general) geometric properties imply that the idempotents on CΓ are trivial. 4. Random walks: let Γ be a finitely generated group with Property RD and let ν be a finitely supported symmetric probability on Γ. Does there exist some constants d = d(ν) and c = c(ν) such that\n\nν(2 n)(e) ∼ cn −dρ2n\n\nwhere ρ is the spectral radius of the convolution operator associated to ν\n\non `2(Γ)? 5. Vague: is there a relation between RD and the entropy of ergodic actions of Γ on measure spaces? 68 Which one of these groups have RD?",
   "evidence": "The canonical record is item 1 at zero-based index 169 of `aim-geometric-group-theory-notes.json`, from the AIM workshop *The property of rapid decay*. The extracted record has line-break/OCR damage (`l2`, split subscripts, and `nu(2 n)(e)`) and appends the heading of the next section. I checked the AIM source and recovered the following five questions.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 169,
   "attempt": 1
  },
  "AIM-GEOMETRIC_GROUP_THEORY-0171": {
   "statement_status": "exact",
   "original_statement": "1. Out (free group), 2. Aut (free group), 3. Braid groups Bn (B3 has RD), 4. Mapping class group, 5. Artin groups (right angled groups have RD since they act freely on cube complexes). Note that Coxeter groups have RD.",
   "clean_statement": "1. Out (free group), 2. Aut (free group), 3. Braid groups Bn (B3 has RD), 4. Mapping class group, 5. Artin groups (right angled groups have RD since they act freely on cube complexes). Note that Coxeter groups have RD.",
   "public_statement": "1. Out (free group), 2. Aut (free group), 3. Braid groups Bn (B3 has RD), 4. Mapping class group, 5. Artin groups (right angled groups have RD since they act freely on cube complexes). Note that Coxeter groups have RD.",
   "evidence": "The record is Section 8 of the AIM workshop notes *The property of rapid decay* (21 March 2006). The section heading occurs at the bottom of the preceding PDF page and was absorbed into corpus record 0170 as the string “68 Which one of these groups have RD?”: `6` is the printed page number and `8` is the section number. Reading the next page recovers the exact question:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometric-group-theory-notes.json",
   "source_index": 170,
   "attempt": 1
  },
  "AIM-GEOMETRY-0001": {
   "statement_status": "exact",
   "original_statement": "$L^1$ norm of first eigenfunction.\n\nLet $\\Omega\\subset \\mathbb{R}^n$ be an open set with finite measure, $\\lambda_1(\\Omega)$ be the first eigenvalue of the Laplacian with Dirichlet boundary conditions, and let\n\\[T(\\Omega)=\\int_{\\Omega}u\\]\nwhere $u:\\Omega\\to \\mathbb{R}$ is the torsion function given by the equation\n\\[\\begin{cases}\n-\\Delta u=1 & \\text{ in }\\Omega\\\\\nu=0 & \\text{ in }\\partial\\Omega\n\\end{cases}\\]\nLet $F(\\Omega) = \\lambda_1(\\Omega)T(\\Omega)/|\\Omega|$. Then it is known that $F(\\Omega)$ is bounded between $0$ and $1$, and these bounds are sharp. However when one restricts to convex subsets of $\\mathbb{R}^2$, there is a better bound\n\\[\\frac{\\pi^2}{32}\\leq F(\\Omega)\\leq 0.9967\\]\nThe left-hand side is due to [Brasco, Mazzoleni], and the right-hand side to [Ftouhi]. The conjecture raised by Phanuel Mariano is as follows: for any planar convex set $\\Omega$, does one has\n\\[\\frac{\\pi^2}{24}\\leq F(\\Omega)\\leq \\frac{\\pi^2}{12}\\]\nIf these bounds were verified, they would be sharp in the following sense: the right one is asymptotically reached for thin rectangles, while the left one is asymptotically reached for any sequence of triangles that collapses on the interval. More recently in [Banuelos, Mariano], these bounds were proven in the case where $\\Omega$ is either a triangle or a rectangle.",
   "clean_statement": "$L^1$ norm of first eigenfunction.\n\nLet $\\Omega\\subset \\mathbb{R}^n$ be an open set with finite measure, $\\lambda_1(\\Omega)$ be the first eigenvalue of the Laplacian with Dirichlet boundary conditions, and let\n\\[T(\\Omega)=\\int_{\\Omega}u\\]\nwhere $u:\\Omega\\to \\mathbb{R}$ is the torsion function given by the equation\n\\[\\begin{cases}\n-\\Delta u=1 & \\text{ in }\\Omega\\\\\nu=0 & \\text{ in }\\partial\\Omega\n\\end{cases}\\]\nLet $F(\\Omega) = \\lambda_1(\\Omega)T(\\Omega)/|\\Omega|$. Then it is known that $F(\\Omega)$ is bounded between $0$ and $1$, and these bounds are sharp. However when one restricts to convex subsets of $\\mathbb{R}^2$, there is a better bound\n\\[\\frac{\\pi^2}{32}\\leq F(\\Omega)\\leq 0.9967\\]\nThe left-hand side is due to [Brasco, Mazzoleni], and the right-hand side to [Ftouhi]. The conjecture raised by Phanuel Mariano is as follows: for any planar convex set $\\Omega$, does one has\n\\[\\frac{\\pi^2}{24}\\leq F(\\Omega)\\leq \\frac{\\pi^2}{12}\\]\nIf these bounds were verified, they would be sharp in the following sense: the right one is asymptotically reached for thin rectangles, while the left one is asymptotically reached for any sequence of triangles that collapses on the interval. More recently in [Banuelos, Mariano], these bounds were proven in the case where $\\Omega$ is either a triangle or a rectangle.",
   "public_statement": "$L^1$ norm of first eigenfunction.\n\nLet $\\Omega\\subset \\mathbb{R}^n$ be an open set with finite measure, $\\lambda_1(\\Omega)$ be the first eigenvalue of the Laplacian with Dirichlet boundary conditions, and let\n\\[T(\\Omega)=\\int_{\\Omega}u\\]\nwhere $u:\\Omega\\to \\mathbb{R}$ is the torsion function given by the equation\n\\[\\begin{cases}\n-\\Delta u=1 & \\text{ in }\\Omega\\\\\nu=0 & \\text{ in }\\partial\\Omega\n\\end{cases}\\]\nLet $F(\\Omega) = \\lambda_1(\\Omega)T(\\Omega)/|\\Omega|$. Then it is known that $F(\\Omega)$ is bounded between $0$ and $1$, and these bounds are sharp. However when one restricts to convex subsets of $\\mathbb{R}^2$, there is a better bound\n\\[\\frac{\\pi^2}{32}\\leq F(\\Omega)\\leq 0.9967\\]\nThe left-hand side is due to [Brasco, Mazzoleni], and the right-hand side to [Ftouhi]. The conjecture raised by Phanuel Mariano is as follows: for any planar convex set $\\Omega$, does one has\n\\[\\frac{\\pi^2}{24}\\leq F(\\Omega)\\leq \\frac{\\pi^2}{12}\\]\nIf these bounds were verified, they would be sharp in the following sense: the right one is asymptotically reached for thin rectangles, while the left one is asymptotically reached for any sequence of triangles that collapses on the interval. More recently in [Banuelos, Mariano], these bounds were proven in the case where $\\Omega$ is either a triangle or a rectangle.",
   "evidence": "The canonical record is AIM workshop problem 1.1 from *Symmetry-breaking of optimal shapes*. It asks about the scale-invariant Pólya functional \\[ F(\\Omega)=\\frac{\\lambda _1(\\Omega)T(\\Omega)}{|\\Omega|}, \\qquad T(\\Omega)=\\int_\\Omega u_\\Omega, \\] where \\(\\lambda _1(\\Omega)\\) is the first Dirichlet eigenvalue and the torsion function is the weak solution of \\[ -\\Delta u_\\Omega=1\\quad\\hbox{in }\\Omega, \\qquad u_\\Omega=0\\quad\\hbox{on }\\partial\\Omega. \\] For bounded convex planar domains the conjectured sharp bounds are \\[ \\boxed{\\frac{\\pi^2}{24}\\leq F(\\Omega)\\leq\\frac{\\pi^2}{12}}. \\tag{1} \\] The lower endpoint is approached by triangles collapsing to an interval, and the upper endpoint by elongating rectangles.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-GEOMETRY-0002": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Maximal gradient of the torsion\n\nThis is a report of a question that was raised in (Hoskins, Steinerberger - 2021), and the following statement is based on this paper.\n\nLet $\\Omega \\subset \\mathbb{R}^2$ be a convex set, and $u_\\Omega:\\Omega\\to \\mathbb{R}$ be the torsion function defined by\n%\n\\begin{align*}\n- \\Delta u_\\Omega = 1 , & \\qquad \\text{in $\\Omega$,} \\\\\nu_\\Omega=0 , & \\qquad \\text{on $\\partial \\Omega$.}\n\\end{align*}\n%\nIt is known that $\\lVert \\nabla u \\rVert_{L^\\infty(\\Omega)} \\leq c |\\Omega|^{1/2}$ for some constant $c<\\frac{1}{\\sqrt{2\\pi}}\\approx 0.398$, that cannot be taken smaller than $0.358$ (which is the best constant that is obtained numerically).\n\nOne could also obtain a slightly worse, explicit bound by looking at ellipses which is one of the few example with explicit torsion function $u_\\Omega$).\n\nThe question raised in (Hoskins, Steinerberger -2021) is the following: for a convex set of $\\mathbb{R}^2$ of given measure, how large can the gradient of the torsion function get ? In other words, what is\n\\[\\sup\\left\\{\\Vert \\nabla u_\\Omega\\Vert_{L^\\infty(\\Omega)},\\ \\Omega\\subset \\mathbb{R}^2\\text{ convex s.t. }|\\Omega|=1\\right\\}\\ ?\\]\n\nNumerically, it is known that the disk is not optimal (even among ellipses), and the optimal set obtained numerically seem to have some flat portion on the boundary.\n\nA particular point of interest is the location of the point of an optimal set where the maximal gradient is reached: is it possible to prove that such a point is necessarily on a flat part of the boundary in a well-quantified way ?",
   "clean_statement": null,
   "public_statement": "Maximal gradient of the torsion\n\nThis is a report of a question that was raised in (Hoskins, Steinerberger - 2021), and the following statement is based on this paper.\n\nLet $\\Omega \\subset \\mathbb{R}^2$ be a convex set, and $u_\\Omega:\\Omega\\to \\mathbb{R}$ be the torsion function defined by\n%\n\\begin{align*}\n- \\Delta u_\\Omega = 1 , & \\qquad \\text{in $\\Omega$,} \\\\\nu_\\Omega=0 , & \\qquad \\text{on $\\partial \\Omega$.}\n\\end{align*}\n%\nIt is known that $\\lVert \\nabla u \\rVert_{L^\\infty(\\Omega)} \\leq c |\\Omega|^{1/2}$ for some constant $c<\\frac{1}{\\sqrt{2\\pi}}\\approx 0.398$, that cannot be taken smaller than $0.358$ (which is the best constant that is obtained numerically).\n\nOne could also obtain a slightly worse, explicit bound by looking at ellipses which is one of the few example with explicit torsion function $u_\\Omega$).\n\nThe question raised in (Hoskins, Steinerberger -2021) is the following: for a convex set of $\\mathbb{R}^2$ of given measure, how large can the gradient of the torsion function get ? In other words, what is\n\\[\\sup\\left\\{\\Vert \\nabla u_\\Omega\\Vert_{L^\\infty(\\Omega)},\\ \\Omega\\subset \\mathbb{R}^2\\text{ convex s.t. }|\\Omega|=1\\right\\}\\ ?\\]\n\nNumerically, it is known that the disk is not optimal (even among ellipses), and the optimal set obtained numerically seem to have some flat portion on the boundary.\n\nA particular point of interest is the location of the point of an optimal set where the maximal gradient is reached: is it possible to prove that such a point is necessarily on a flat part of the boundary in a well-quantified way ?",
   "evidence": "The source record is AIM Problem Lists, workshop *Symmetry-breaking of optimal shapes*, section “Eigenfunctions,” Problem 1.2. The literal extracted boundary condition is",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-GEOMETRY-0003": {
   "statement_status": "exact",
   "original_statement": "Shape of the ground state\n\nLet $\\Omega\\subset\\mathbb{R}^n$ be a bounded convex set, and let $u$ be the lowest eigenfunction (ground state) of the Laplacian with Dirichet boundary conditions, with unit $L^2$ norm. A general question is to relate the domain $\\Omega$ and the ground state $u$ with a rectangular domain $R$ associated to a ground state with separated variables.\n\nFor this we introduce the following notations: let $\\xi \\in \\mathbb{R}^n$ be a unit vector ($|\\xi|=1$). Let\n\\[\nP(\\xi)=\\int_{\\partial \\Omega} |\\xi \\cdot \\nabla u|^2 \\, dS , \\qquad Q(\\xi)=\\int_\\Omega |\\xi \\cdot \\nabla u|^2 \\, dx .\n\\]\n%\\color{blue}It is known that $Q$ is the projection body function for volume.\n\nIt is known that\n\\[\nQ(\\Omega) \\leq \\frac{c}{\\text{inradius}(\\Omega)^2} .\n\\]\n\nA first question is the relation between $P$ and $Q$: the first is linked to the shape derivative of $\\lambda_1(\\Omega)$ in the direction of $\\xi$, while the second is related to the distribution of $\\nabla u $ in $\\Omega$.\n\nA second question is whether $\\log(u)$ is ``essentially quadratic'' on the interior of $\\Omega$ in the following sense: it is known that $\\log(u)$ is concave and reaches a unique maximum at some point $p^*$, the conjecture raised by David Jerison is whether there exists some constant $C_n>0$ such that for any $p\\in \\{u>\\frac{1}{2}u(p^*)\\}$, we have\n\\[\\frac{1}{C_n}\\left(-\\nabla^2 \\log\\ u(p)\\right)\\leq \\left(-\\nabla^2\\log\\ u(p^*)\\right)\\leq C_n\\left(-\\nabla^2\\log\\ u(p)\\right).\\]",
   "clean_statement": "Shape of the ground state\n\nLet $\\Omega\\subset\\mathbb{R}^n$ be a bounded convex set, and let $u$ be the lowest eigenfunction (ground state) of the Laplacian with Dirichet boundary conditions, with unit $L^2$ norm. A general question is to relate the domain $\\Omega$ and the ground state $u$ with a rectangular domain $R$ associated to a ground state with separated variables.\n\nFor this we introduce the following notations: let $\\xi \\in \\mathbb{R}^n$ be a unit vector ($|\\xi|=1$). Let\n\\[\nP(\\xi)=\\int_{\\partial \\Omega} |\\xi \\cdot \\nabla u|^2 \\, dS , \\qquad Q(\\xi)=\\int_\\Omega |\\xi \\cdot \\nabla u|^2 \\, dx .\n\\]\n%\\color{blue}It is known that $Q$ is the projection body function for volume.\n\nIt is known that\n\\[\nQ(\\Omega) \\leq \\frac{c}{\\text{inradius}(\\Omega)^2} .\n\\]\n\nA first question is the relation between $P$ and $Q$: the first is linked to the shape derivative of $\\lambda_1(\\Omega)$ in the direction of $\\xi$, while the second is related to the distribution of $\\nabla u $ in $\\Omega$.\n\nA second question is whether $\\log(u)$ is ``essentially quadratic'' on the interior of $\\Omega$ in the following sense: it is known that $\\log(u)$ is concave and reaches a unique maximum at some point $p^*$, the conjecture raised by David Jerison is whether there exists some constant $C_n>0$ such that for any $p\\in \\{u>\\frac{1}{2}u(p^*)\\}$, we have\n\\[\\frac{1}{C_n}\\left(-\\nabla^2 \\log\\ u(p)\\right)\\leq \\left(-\\nabla^2\\log\\ u(p^*)\\right)\\leq C_n\\left(-\\nabla^2\\log\\ u(p)\\right).\\]",
   "public_statement": "Shape of the ground state\n\nLet $\\Omega\\subset\\mathbb{R}^n$ be a bounded convex set, and let $u$ be the lowest eigenfunction (ground state) of the Laplacian with Dirichet boundary conditions, with unit $L^2$ norm. A general question is to relate the domain $\\Omega$ and the ground state $u$ with a rectangular domain $R$ associated to a ground state with separated variables.\n\nFor this we introduce the following notations: let $\\xi \\in \\mathbb{R}^n$ be a unit vector ($|\\xi|=1$). Let\n\\[\nP(\\xi)=\\int_{\\partial \\Omega} |\\xi \\cdot \\nabla u|^2 \\, dS , \\qquad Q(\\xi)=\\int_\\Omega |\\xi \\cdot \\nabla u|^2 \\, dx .\n\\]\n%\\color{blue}It is known that $Q$ is the projection body function for volume.\n\nIt is known that\n\\[\nQ(\\Omega) \\leq \\frac{c}{\\text{inradius}(\\Omega)^2} .\n\\]\n\nA first question is the relation between $P$ and $Q$: the first is linked to the shape derivative of $\\lambda_1(\\Omega)$ in the direction of $\\xi$, while the second is related to the distribution of $\\nabla u $ in $\\Omega$.\n\nA second question is whether $\\log(u)$ is ``essentially quadratic'' on the interior of $\\Omega$ in the following sense: it is known that $\\log(u)$ is concave and reaches a unique maximum at some point $p^*$, the conjecture raised by David Jerison is whether there exists some constant $C_n>0$ such that for any $p\\in \\{u>\\frac{1}{2}u(p^*)\\}$, we have\n\\[\\frac{1}{C_n}\\left(-\\nabla^2 \\log\\ u(p)\\right)\\leq \\left(-\\nabla^2\\log\\ u(p^*)\\right)\\leq C_n\\left(-\\nabla^2\\log\\ u(p)\\right).\\]",
   "evidence": "The canonical record, number 1.3 (“Shape of the ground state”) in the “Eigenfunctions” section of the AIM list *Symmetry-breaking of optimal shapes*, asks two related questions. For a bounded convex \\(\\Omega\\subset\\mathbb R^n\\), let \\[ -\\Delta u=\\lambda_1(\\Omega)u,\\qquad u|_{\\partial\\Omega}=0, \\qquad u>0,\\qquad \\|u\\|_{L^2(\\Omega)}=1, \\] and, for \\(|\\xi|=1\\), define \\[ P(\\xi)=\\int_{\\partial\\Omega}|\\xi\\cdot\\nabla u|^2\\,dS, \\qquad Q(\\xi)=\\int_\\Omega|\\xi\\cdot\\nabla u|^2\\,dx. \\tag{1} \\] The first question asks for a relation between these boundary and interior directional energies. The second is David Jerison's dimension-uniform Hessian conjecture: if \\(p^*\\) is the maximum point of \\(u\\), is there \\(C_n\\) such that, for every \\(p\\) with \\(u(p)>u(p^*)/2\\), \\[ \\frac1{C_n}\\bigl(-\\nabla^2\\log u(p)\\bigr) \\preceq -\\nabla^2\\log u(p^*) \\preceq C_n\\bigl(-\\nabla^2\\log u(p)\\bigr)? \\tag{2} \\] Here \\(...",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-GEOMETRY-0004": {
   "statement_status": "exact",
   "original_statement": "Curl-curl eigenvalue problem\n\nLet $\\Omega$ be a smooth compact domain of $\\mathbb{R}^3$, we define $(\\vec{E_k},\\lambda_k)_{k\\geq 1}$ to be the eigenpair of the Maxwell equation in $\\Omega$, as given by the equation\n\n\\begin{align*}\n\\nabla \\times \\nabla \\times \\vec{E}_k & = \\lambda_k \\vec{E}_k \\quad \\text{in $\\Omega$} \\\\\n\\nabla \\cdot \\vec{E}_k & = 0 \\qquad \\text{in $\\Omega$} \\\\\n\\vec{E}_k \\times \\vec{\\nu} & = 0 \\qquad \\text{on $\\partial \\Omega$}\n\\end{align*}\nwhere $\\vec{\\nu}$ is the outward normal vector of $\\Omega$. These eigenvalues and eigenvector may be defined variationally through the Rayleigh quotient\n\\[\\frac{\\int_\\Omega |\\nabla \\times \\vec{E}|^2 \\, dx}{\\int_\\Omega |\\vec{E}|^2 \\, dx}\\]\ntaken among vector fields $\\vec{E}\\in\\mathcal{C}^1(\\Omega,\\mathbb{R}^3)$ verifying the constraints\n\\[\\nabla \\cdot \\vec{E}=0\\text{ in }\\Omega,\\ \\vec{E}\\times \\vec{\\nu}=0\\text{ in }\\partial\\Omega\\]\nFew things seem to be known about the range of the first few eigenvalues of this operator depending on the geometric constraints on $\\Omega$: Nilima Nigam raises the questions of the optimal shape for $\\lambda_k(\\Omega)$ for $k=1,2,3$, under either a simple volume constraint, or a volume constraint among convex sets, and whether the ball may be expected to be optimal or not. Note also that this problem is a simplified version of Maxwell's equation, where permittivity (for instance) was not included.",
   "clean_statement": "Curl-curl eigenvalue problem\n\nLet $\\Omega$ be a smooth compact domain of $\\mathbb{R}^3$, we define $(\\vec{E_k},\\lambda_k)_{k\\geq 1}$ to be the eigenpair of the Maxwell equation in $\\Omega$, as given by the equation\n\n\\begin{align*}\n\\nabla \\times \\nabla \\times \\vec{E}_k & = \\lambda_k \\vec{E}_k \\quad \\text{in $\\Omega$} \\\\\n\\nabla \\cdot \\vec{E}_k & = 0 \\qquad \\text{in $\\Omega$} \\\\\n\\vec{E}_k \\times \\vec{\\nu} & = 0 \\qquad \\text{on $\\partial \\Omega$}\n\\end{align*}\nwhere $\\vec{\\nu}$ is the outward normal vector of $\\Omega$. These eigenvalues and eigenvector may be defined variationally through the Rayleigh quotient\n\\[\\frac{\\int_\\Omega |\\nabla \\times \\vec{E}|^2 \\, dx}{\\int_\\Omega |\\vec{E}|^2 \\, dx}\\]\ntaken among vector fields $\\vec{E}\\in\\mathcal{C}^1(\\Omega,\\mathbb{R}^3)$ verifying the constraints\n\\[\\nabla \\cdot \\vec{E}=0\\text{ in }\\Omega,\\ \\vec{E}\\times \\vec{\\nu}=0\\text{ in }\\partial\\Omega\\]\nFew things seem to be known about the range of the first few eigenvalues of this operator depending on the geometric constraints on $\\Omega$: Nilima Nigam raises the questions of the optimal shape for $\\lambda_k(\\Omega)$ for $k=1,2,3$, under either a simple volume constraint, or a volume constraint among convex sets, and whether the ball may be expected to be optimal or not. Note also that this problem is a simplified version of Maxwell's equation, where permittivity (for instance) was not included.",
   "public_statement": "Curl-curl eigenvalue problem\n\nLet $\\Omega$ be a smooth compact domain of $\\mathbb{R}^3$, we define $(\\vec{E_k},\\lambda_k)_{k\\geq 1}$ to be the eigenpair of the Maxwell equation in $\\Omega$, as given by the equation\n\n\\begin{align*}\n\\nabla \\times \\nabla \\times \\vec{E}_k & = \\lambda_k \\vec{E}_k \\quad \\text{in $\\Omega$} \\\\\n\\nabla \\cdot \\vec{E}_k & = 0 \\qquad \\text{in $\\Omega$} \\\\\n\\vec{E}_k \\times \\vec{\\nu} & = 0 \\qquad \\text{on $\\partial \\Omega$}\n\\end{align*}\nwhere $\\vec{\\nu}$ is the outward normal vector of $\\Omega$. These eigenvalues and eigenvector may be defined variationally through the Rayleigh quotient\n\\[\\frac{\\int_\\Omega |\\nabla \\times \\vec{E}|^2 \\, dx}{\\int_\\Omega |\\vec{E}|^2 \\, dx}\\]\ntaken among vector fields $\\vec{E}\\in\\mathcal{C}^1(\\Omega,\\mathbb{R}^3)$ verifying the constraints\n\\[\\nabla \\cdot \\vec{E}=0\\text{ in }\\Omega,\\ \\vec{E}\\times \\vec{\\nu}=0\\text{ in }\\partial\\Omega\\]\nFew things seem to be known about the range of the first few eigenvalues of this operator depending on the geometric constraints on $\\Omega$: Nilima Nigam raises the questions of the optimal shape for $\\lambda_k(\\Omega)$ for $k=1,2,3$, under either a simple volume constraint, or a volume constraint among convex sets, and whether the ball may be expected to be optimal or not. Note also that this problem is a simplified version of Maxwell's equation, where permittivity (for instance) was not included.",
   "evidence": "The AIM problem asks about the first three eigenvalues of the perfectly conducting electric cavity problem \\[ \\operatorname{curl}\\operatorname{curl}E=\\lambda E, \\qquad \\operatorname{div}E=0\\quad\\hbox{in }\\Omega, \\qquad E\\times\\nu=0\\quad\\hbox{on }\\partial\\Omega, \\] and asks which shapes optimize \\(\\lambda_k(\\Omega)\\), for \\(k=1,2,3\\), under a volume constraint, either without or with convexity. It asks in particular whether the ball should be expected to be optimal. This statement was checked against the live AIM page [AIM].",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-GEOMETRY-0005": {
   "statement_status": "exact",
   "original_statement": "Curl eigenvalue\n\nLet $\\Omega \\subset \\mathbb{R}^3$ be a bounded Lipschitz set with volume $1$, consider the curl eigenvalue problem\n%\n\\begin{align*}\n\\nabla \\times \\vec{u} & = \\mu \\vec{u} \\ \\text{ on $\\Omega$,} \\\\\n\\vec{\\nu} \\cdot \\vec{u} & = 0 \\ \\text{ on $\\partial \\Omega$}.\\\\\n\\int_{\\Omega}\\vec u\\cdot \\vec w&=0 \\ \\text{ for any }\\vec w\\in L^2(\\Omega)\\text{ with }\\nabla\\times \\vec w=0\\\\\n\\end{align*}\nMore details on the definition may be found in [Enciso, Gerner, Peralta--Salas]. There exists a sequence of eigenpairs $(\\mu_k,u_k)_{k\\in\\mathbb{Z}}$ such that\n\\[...\\leq \\mu_{-2}(\\Omega)\\leq \\mu_{-1}(\\Omega)<0<\\mu_1(\\Omega)\\leq \\mu_2(\\Omega)\\leq ...\\to\\infty\\]\n%\n\nWe remind the variational definition\n\\[\\min(\\mu_1(\\Omega)^2,\\mu_{-1}(\\Omega)^2)=\\inf\\left\\{\\frac{\\int_{\\Omega}|\\nabla \\times u|^2}{\\int_{\\Omega}|u|^2}, \\right\\}\\]\nwhere $u$ is taken in $\\mathcal{C}^1(\\Omega,\\mathbb{R}^3)$ with the constraints $\\vec{u}\\cdot\\vec{\\nu}=0$ on the boundary, $\\int_{\\Omega}\\vec{u}\\cdot\\vec{w}=0$ for any $\\vec{w}$ with $\\nabla\\times\\vec{w}=0$.\n\nThe goal is to minimize the first positive eigenvalue $\\mu_1(\\Omega)$ under volume constraint on $\\Omega$. In the case where $\\Omega$ is supposed to be convex, then it is known from [Enciso, Gerner, Peralta--Salas] that the ball is not optimal, and that there is an optimal domain that is not analytic. Without convexity constraint, the existence of optimal domain is only known (from [Enciso, Gerner, Peralta--Salas]) among uniformly Hölder sets, and remains to be investigated in general.\n\nMoreover, it is known (from the works of the same authors) that if $\\Omega$ is a $\\mathcal{C}^{2,\\alpha}$ optimal set, then $\\Omega$ is \\textbf{not} axisymmetric, and every connected component of $\\partial\\Omega$ is diffeomorphic to a torus.",
   "clean_statement": "Curl eigenvalue\n\nLet $\\Omega \\subset \\mathbb{R}^3$ be a bounded Lipschitz set with volume $1$, consider the curl eigenvalue problem\n%\n\\begin{align*}\n\\nabla \\times \\vec{u} & = \\mu \\vec{u} \\ \\text{ on $\\Omega$,} \\\\\n\\vec{\\nu} \\cdot \\vec{u} & = 0 \\ \\text{ on $\\partial \\Omega$}.\\\\\n\\int_{\\Omega}\\vec u\\cdot \\vec w&=0 \\ \\text{ for any }\\vec w\\in L^2(\\Omega)\\text{ with }\\nabla\\times \\vec w=0\\\\\n\\end{align*}\nMore details on the definition may be found in [Enciso, Gerner, Peralta--Salas]. There exists a sequence of eigenpairs $(\\mu_k,u_k)_{k\\in\\mathbb{Z}}$ such that\n\\[...\\leq \\mu_{-2}(\\Omega)\\leq \\mu_{-1}(\\Omega)<0<\\mu_1(\\Omega)\\leq \\mu_2(\\Omega)\\leq ...\\to\\infty\\]\n%\n\nWe remind the variational definition\n\\[\\min(\\mu_1(\\Omega)^2,\\mu_{-1}(\\Omega)^2)=\\inf\\left\\{\\frac{\\int_{\\Omega}|\\nabla \\times u|^2}{\\int_{\\Omega}|u|^2}, \\right\\}\\]\nwhere $u$ is taken in $\\mathcal{C}^1(\\Omega,\\mathbb{R}^3)$ with the constraints $\\vec{u}\\cdot\\vec{\\nu}=0$ on the boundary, $\\int_{\\Omega}\\vec{u}\\cdot\\vec{w}=0$ for any $\\vec{w}$ with $\\nabla\\times\\vec{w}=0$.\n\nThe goal is to minimize the first positive eigenvalue $\\mu_1(\\Omega)$ under volume constraint on $\\Omega$. In the case where $\\Omega$ is supposed to be convex, then it is known from [Enciso, Gerner, Peralta--Salas] that the ball is not optimal, and that there is an optimal domain that is not analytic. Without convexity constraint, the existence of optimal domain is only known (from [Enciso, Gerner, Peralta--Salas]) among uniformly Hölder sets, and remains to be investigated in general.\n\nMoreover, it is known (from the works of the same authors) that if $\\Omega$ is a $\\mathcal{C}^{2,\\alpha}$ optimal set, then $\\Omega$ is \\textbf{not} axisymmetric, and every connected component of $\\partial\\Omega$ is diffeomorphic to a torus.",
   "public_statement": "Curl eigenvalue\n\nLet $\\Omega \\subset \\mathbb{R}^3$ be a bounded Lipschitz set with volume $1$, consider the curl eigenvalue problem\n%\n\\begin{align*}\n\\nabla \\times \\vec{u} & = \\mu \\vec{u} \\ \\text{ on $\\Omega$,} \\\\\n\\vec{\\nu} \\cdot \\vec{u} & = 0 \\ \\text{ on $\\partial \\Omega$}.\\\\\n\\int_{\\Omega}\\vec u\\cdot \\vec w&=0 \\ \\text{ for any }\\vec w\\in L^2(\\Omega)\\text{ with }\\nabla\\times \\vec w=0\\\\\n\\end{align*}\nMore details on the definition may be found in [Enciso, Gerner, Peralta--Salas]. There exists a sequence of eigenpairs $(\\mu_k,u_k)_{k\\in\\mathbb{Z}}$ such that\n\\[...\\leq \\mu_{-2}(\\Omega)\\leq \\mu_{-1}(\\Omega)<0<\\mu_1(\\Omega)\\leq \\mu_2(\\Omega)\\leq ...\\to\\infty\\]\n%\n\nWe remind the variational definition\n\\[\\min(\\mu_1(\\Omega)^2,\\mu_{-1}(\\Omega)^2)=\\inf\\left\\{\\frac{\\int_{\\Omega}|\\nabla \\times u|^2}{\\int_{\\Omega}|u|^2}, \\right\\}\\]\nwhere $u$ is taken in $\\mathcal{C}^1(\\Omega,\\mathbb{R}^3)$ with the constraints $\\vec{u}\\cdot\\vec{\\nu}=0$ on the boundary, $\\int_{\\Omega}\\vec{u}\\cdot\\vec{w}=0$ for any $\\vec{w}$ with $\\nabla\\times\\vec{w}=0$.\n\nThe goal is to minimize the first positive eigenvalue $\\mu_1(\\Omega)$ under volume constraint on $\\Omega$. In the case where $\\Omega$ is supposed to be convex, then it is known from [Enciso, Gerner, Peralta--Salas] that the ball is not optimal, and that there is an optimal domain that is not analytic. Without convexity constraint, the existence of optimal domain is only known (from [Enciso, Gerner, Peralta--Salas]) among uniformly Hölder sets, and remains to be investigated in general.\n\nMoreover, it is known (from the works of the same authors) that if $\\Omega$ is a $\\mathcal{C}^{2,\\alpha}$ optimal set, then $\\Omega$ is \\textbf{not} axisymmetric, and every connected component of $\\partial\\Omega$ is diffeomorphic to a torus.",
   "evidence": "The canonical AIM record 2.2, “Curl eigenvalue,” considers a bounded Lipschitz set \\(\\Omega\\subset\\mathbb R^3\\) of volume \\(1\\) and formally writes \\[ \\operatorname{curl}u=\\mu u,\\qquad u\\cdot\\nu=0\\text{ on }\\partial\\Omega, \\qquad \\int_\\Omega u\\cdot w=0 \\quad\\text{for every }w\\in L^2(\\Omega)\\text{ with }\\operatorname{curl}w=0. \\tag{1} \\] It asks to minimize the first positive eigenvalue \\(\\mu _1(\\Omega)\\) under the volume constraint. It also gives the squared variational formula \\[ \\min\\{\\mu _1(\\Omega)^2,\\mu _{-1}(\\Omega)^2\\} =\\inf_{u\\ne0} \\frac{\\int_\\Omega|\\operatorname{curl}u|^2} {\\int_\\Omega|u|^2}, \\tag{2} \\] with the stated constraints.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-GEOMETRY-0006": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Exterior Robin problem\n\nLet $n \\geq 3$ and let $\\Omega\\subset\\mathbb{R}^3$ be a smooth compact domain. For a given $\\alpha \\in \\mathbb{R}$, we consider the Robin eigenvalue problem on the complement of $\\Omega$, of unknown $(\\lambda,u)$:\n%\n\\begin{align*}\n- \\Delta u & = \\lambda u \\qquad \\text{in $\\Omega^\\text{ext} = \\mathbb{R}^n \\setminus \\overline{\\Omega}$,} \\\\\n\\frac{\\partial u}{\\partial \\nu} & = \\alpha u \\qquad \\text{on $\\partial \\Omega$.}\n\\end{align*}\n%\nThe problem has essential spectrum $[0,\\infty)$, but does have some eigenvalues too, provided $\\alpha<\\alpha_*(n)<0$ where $\\alpha_*(n)$ is some dimensionnal constant. The lowest eigenvalue is given by the Rayleigh quotient\n\\[\n\\lambda_1^\\alpha(\\Omega^\\text{ext}) = \\min_{u \\in W^{1,2}(\\Omega^\\text{ext})} \\frac{\\int_{\\Omega^\\text{ext}} |\\nabla u|^2 \\, dx + \\alpha \\int_{\\partial \\Omega} u^2 \\, dS}{\\int_{\\Omega^\\text{ext}} u^2 \\, dx} .\n\\]\nFor $\\alpha<0$, D. Krejcirik and V. Lotoreichik have shown that the ball maximizes $\\lambda_1^\\alpha(\\Omega^\\text{ext})$ in dimension $n=2$ among all smooth, bounded, simply connected open sets of given measure and among all smooth, bounded, simply connected open sets of given perimeter.\n\nThis is no longer true in higher dimensions for sufficiently negative $\\alpha$: a counterexample is given by an ellipsoid, exploiting the asymptotic formula of $\\lambda_1^\\alpha(\\Omega^{\\text{ext}})$ that involves the maximum of the curvature of $\\Omega$.\n\nHowever the ball is still a local maximizer of $\\lambda_1^\\alpha(\\Omega^\\text{ext})$ among all nearly spherical domains of given measure. This situation raises two questions:\n%\n\\begin{itemize}\n\\item Does a global maximizer of $\\lambda_1^\\alpha(\\Omega^\\text{ext})$ (under volume constraint) exist in dimensions $n \\geq 3$ ? If so, what shape does it have ?\n\\item Since the ball is a local but not a global maximizer, does another critical domain exists as a consequence of Mountain Pass Theorem ? Must such a domain also be smooth ?\n\\end{itemize}",
   "clean_statement": "to maximize\n\\[\n\\lambda_1^\\alpha(D):=\\inf_{0\\ne u\\in H^1(D)}\n\\frac{q_{\\alpha,\\Omega}[u]}{\\|u\\|_{L^2(D)}^2}\n\\tag{1}\n\\]\nover a specified class of smooth bounded obstacles of prescribed volume, while distinguishing optimization of the *spectral bottom* from optimization restricted to obstacles for which that bottom is a negative eigenvalue.",
   "public_statement": "Exterior Robin problem\n\nLet $n \\geq 3$ and let $\\Omega\\subset\\mathbb{R}^3$ be a smooth compact domain. For a given $\\alpha \\in \\mathbb{R}$, we consider the Robin eigenvalue problem on the complement of $\\Omega$, of unknown $(\\lambda,u)$:\n%\n\\begin{align*}\n- \\Delta u & = \\lambda u \\qquad \\text{in $\\Omega^\\text{ext} = \\mathbb{R}^n \\setminus \\overline{\\Omega}$,} \\\\\n\\frac{\\partial u}{\\partial \\nu} & = \\alpha u \\qquad \\text{on $\\partial \\Omega$.}\n\\end{align*}\n%\nThe problem has essential spectrum $[0,\\infty)$, but does have some eigenvalues too, provided $\\alpha<\\alpha_*(n)<0$ where $\\alpha_*(n)$ is some dimensionnal constant. The lowest eigenvalue is given by the Rayleigh quotient\n\\[\n\\lambda_1^\\alpha(\\Omega^\\text{ext}) = \\min_{u \\in W^{1,2}(\\Omega^\\text{ext})} \\frac{\\int_{\\Omega^\\text{ext}} |\\nabla u|^2 \\, dx + \\alpha \\int_{\\partial \\Omega} u^2 \\, dS}{\\int_{\\Omega^\\text{ext}} u^2 \\, dx} .\n\\]\nFor $\\alpha<0$, D. Krejcirik and V. Lotoreichik have shown that the ball maximizes $\\lambda_1^\\alpha(\\Omega^\\text{ext})$ in dimension $n=2$ among all smooth, bounded, simply connected open sets of given measure and among all smooth, bounded, simply connected open sets of given perimeter.\n\nThis is no longer true in higher dimensions for sufficiently negative $\\alpha$: a counterexample is given by an ellipsoid, exploiting the asymptotic formula of $\\lambda_1^\\alpha(\\Omega^{\\text{ext}})$ that involves the maximum of the curvature of $\\Omega$.\n\nHowever the ball is still a local maximizer of $\\lambda_1^\\alpha(\\Omega^\\text{ext})$ among all nearly spherical domains of given measure. This situation raises two questions:\n%\n\\begin{itemize}\n\\item Does a global maximizer of $\\lambda_1^\\alpha(\\Omega^\\text{ext})$ (under volume constraint) exist in dimensions $n \\geq 3$ ? If so, what shape does it have ?\n\\item Since the ball is a local but not a global maximizer, does another critical domain exists as a consequence of Mountain Pass Theorem ? Must such a domain also be smooth ?\n\\end{itemize}",
   "evidence": "The AIM source URL was unavailable during this run (HTTP 502). The reconstruction above was checked against the primary papers cited below.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-GEOMETRY-0007": {
   "statement_status": "exact",
   "original_statement": "Maximizing Steklov $\\sigma_1$ on $N$-gons\n\nOne of the ``well-known'' conjecture of shape optimization is the optimization of $\\lambda_1$ (first eigenvalue of the Laplacian with Dirichlet boundary condition) among polygonal sets of fixed measure. A question raised by Iosif Polterovich on which even less is known is for the Steklov eigenvalue: given some integer $n\\geq 3$, what is the maximal value of the first Steklov eigenvalue $\\sigma_1$ among $n$-sided polygonal set of either fixed area or fixed perimeter ?\n\nEven the cases $n=3$ or $n=4$ are open, where one should be mindful of the behaviour of the corner in the shape derivative formula.",
   "clean_statement": "Maximizing Steklov $\\sigma_1$ on $N$-gons\n\nOne of the ``well-known'' conjecture of shape optimization is the optimization of $\\lambda_1$ (first eigenvalue of the Laplacian with Dirichlet boundary condition) among polygonal sets of fixed measure. A question raised by Iosif Polterovich on which even less is known is for the Steklov eigenvalue: given some integer $n\\geq 3$, what is the maximal value of the first Steklov eigenvalue $\\sigma_1$ among $n$-sided polygonal set of either fixed area or fixed perimeter ?\n\nEven the cases $n=3$ or $n=4$ are open, where one should be mindful of the behaviour of the corner in the shape derivative formula.",
   "public_statement": "Maximizing Steklov $\\sigma_1$ on $N$-gons\n\nOne of the ``well-known'' conjecture of shape optimization is the optimization of $\\lambda_1$ (first eigenvalue of the Laplacian with Dirichlet boundary condition) among polygonal sets of fixed measure. A question raised by Iosif Polterovich on which even less is known is for the Steklov eigenvalue: given some integer $n\\geq 3$, what is the maximal value of the first Steklov eigenvalue $\\sigma_1$ among $n$-sided polygonal set of either fixed area or fixed perimeter ?\n\nEven the cases $n=3$ or $n=4$ are open, where one should be mindful of the behaviour of the corner in the shape derivative formula.",
   "evidence": "The exact AIM record, attributed there to Iosif Polterovich, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-GEOMETRY-0008": {
   "statement_status": "exact",
   "original_statement": "Minimization of $p$-capacity\n\nLet $p\\in (1,2)$, and $\\Omega\\subset \\mathbb{R}^2$ a bounded convex set, we define its $p$-capacity\n\n\\[\\text{cap}_p(\\Omega)=\\inf\\left\\{\\int_{\\R^2}|\\nabla u|^p,\\ u\\in\\mathcal{C}^1_c(\\mathbb{R}^2),\\text{ such that }u\\geq 1\\text{ in a neighbourhood of }\\Omega\\right\\}\\]\n\nNote that this notion of capacity is not well-defined for $p=2$ (by the scale-invariance of $\\int_{\\R^2}|\\nabla u|^2$ and the fact that points have zero $2$-capacity), and is usually replaced with the logarithmic capacity.\n\nThe question is the following: among convex sets $\\Omega$ of fixed perimeter, is it true that $\\text{cap}_p(\\Omega)$ is minimal for segments ?\n\nNote that the perimeter should be counted twice when $\\Omega$ is a segment (or has unidimensional parts).\n\nIt is known from (Colesanti, Salani - 2003) that $\\Omega\\mapsto \\text{cap}_p(\\Omega)^{\\frac{1}{2-p}}$ satisfy the Brunn-Minkowsky inequality, meaning that the search for optimal sets may be restricted to the extremal points for Brunn-Minkowsky sum, meaning triangles (includes segments).\n\nIt is however unclear how to conclude even in the case of triangles.",
   "clean_statement": "Minimization of $p$-capacity\n\nLet $p\\in (1,2)$, and $\\Omega\\subset \\mathbb{R}^2$ a bounded convex set, we define its $p$-capacity\n\n\\[\\text{cap}_p(\\Omega)=\\inf\\left\\{\\int_{\\R^2}|\\nabla u|^p,\\ u\\in\\mathcal{C}^1_c(\\mathbb{R}^2),\\text{ such that }u\\geq 1\\text{ in a neighbourhood of }\\Omega\\right\\}\\]\n\nNote that this notion of capacity is not well-defined for $p=2$ (by the scale-invariance of $\\int_{\\R^2}|\\nabla u|^2$ and the fact that points have zero $2$-capacity), and is usually replaced with the logarithmic capacity.\n\nThe question is the following: among convex sets $\\Omega$ of fixed perimeter, is it true that $\\text{cap}_p(\\Omega)$ is minimal for segments ?\n\nNote that the perimeter should be counted twice when $\\Omega$ is a segment (or has unidimensional parts).\n\nIt is known from (Colesanti, Salani - 2003) that $\\Omega\\mapsto \\text{cap}_p(\\Omega)^{\\frac{1}{2-p}}$ satisfy the Brunn-Minkowsky inequality, meaning that the search for optimal sets may be restricted to the extremal points for Brunn-Minkowsky sum, meaning triangles (includes segments).\n\nIt is however unclear how to conclude even in the case of triangles.",
   "public_statement": "Minimization of $p$-capacity\n\nLet $p\\in (1,2)$, and $\\Omega\\subset \\mathbb{R}^2$ a bounded convex set, we define its $p$-capacity\n\n\\[\\text{cap}_p(\\Omega)=\\inf\\left\\{\\int_{\\R^2}|\\nabla u|^p,\\ u\\in\\mathcal{C}^1_c(\\mathbb{R}^2),\\text{ such that }u\\geq 1\\text{ in a neighbourhood of }\\Omega\\right\\}\\]\n\nNote that this notion of capacity is not well-defined for $p=2$ (by the scale-invariance of $\\int_{\\R^2}|\\nabla u|^2$ and the fact that points have zero $2$-capacity), and is usually replaced with the logarithmic capacity.\n\nThe question is the following: among convex sets $\\Omega$ of fixed perimeter, is it true that $\\text{cap}_p(\\Omega)$ is minimal for segments ?\n\nNote that the perimeter should be counted twice when $\\Omega$ is a segment (or has unidimensional parts).\n\nIt is known from (Colesanti, Salani - 2003) that $\\Omega\\mapsto \\text{cap}_p(\\Omega)^{\\frac{1}{2-p}}$ satisfy the Brunn-Minkowsky inequality, meaning that the search for optimal sets may be restricted to the extremal points for Brunn-Minkowsky sum, meaning triangles (includes segments).\n\nIt is however unclear how to conclude even in the case of triangles.",
   "evidence": "The source asks the following. For \\(1<p<2\\) and a bounded convex set \\(K\\subset\\mathbb R^2\\), define the homogeneous variational capacity \\[ \\operatorname{Cap}_p(K)=\\inf\\left\\{ \\int_{\\mathbb R^2}|\\nabla u|^p\\,dx: u\\in C_c^1(\\mathbb R^2),\\quad u\\geq1 \\text{ on a neighbourhood of }K\\right\\}. \\] Among convex sets of prescribed perimeter, is this capacity minimized by a segment, with a segment's perimeter counted twice?",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-GEOMETRY-0009": {
   "statement_status": "exact",
   "original_statement": "Cheeger's inequality on convex sets\n\nFor an open set of finite measure $\\Omega\\subset\\mathbb{R}^n$, denote $\\lambda_1(\\Omega)$ the first eigenvalue of the Laplacian with Dirichlet boundary conditions, and $h(\\Omega)$ the Cheeger constant which may be defined as\n\\[h(\\Omega)=\\inf_{A\\subset \\Omega}\\frac{|\\partial A|}{|A|}=\\inf_{u\\in\\mathcal{C}^\\infty_c(\\Omega)}\\frac{\\int_{\\Omega}|\\nabla u|}{\\int_{\\Omega}u}\\]\nCheeger's inequality is\n\\[\\lambda_1(\\Omega)\\geq \\frac{1}{4}h(\\Omega)^2\\]\nand the constant $\\frac{1}{4}$ is sharp in the sense that the ratio $\\frac{\\lambda_1(\\Omega)}{h(\\Omega)^2}$ tends to $\\frac{1}{4}$ when $\\Omega$ is taken to be the unit ball of $\\mathbb{R}^n$ for $n\\to +\\infty$. It is however not sharp when $\\Omega$ is restricted to be a planar convex sets, as is shown for instance by the result of I. Ftouhi:\n\\[\\frac{\\lambda_1(\\Omega)}{h(\\Omega)^2}\\geq 0.902\\]\nIlias Ftouhi's conjecture is that the optimal bound is $\\frac{2\\pi^2}{(2+\\sqrt{\\pi})^2}\\approx 1.387...$, which is reached exactly when $\\Omega$ is a square.\n\nAn optimal set is known to exist, by standard compactness arguments, but it is unknown whether this optimal set is a polygon. A first stepping stone would be to examine the case of polygonal sets, where the Cheeger set (the optimal set ``$A$'' in the definition) should coincide with the boundary of $\\Omega$ on some part, or leave the boundary tangentially following circle arcs (as described in [kawohl, lachand-robert]).",
   "clean_statement": "Cheeger's inequality on convex sets\n\nFor an open set of finite measure $\\Omega\\subset\\mathbb{R}^n$, denote $\\lambda_1(\\Omega)$ the first eigenvalue of the Laplacian with Dirichlet boundary conditions, and $h(\\Omega)$ the Cheeger constant which may be defined as\n\\[h(\\Omega)=\\inf_{A\\subset \\Omega}\\frac{|\\partial A|}{|A|}=\\inf_{u\\in\\mathcal{C}^\\infty_c(\\Omega)}\\frac{\\int_{\\Omega}|\\nabla u|}{\\int_{\\Omega}u}\\]\nCheeger's inequality is\n\\[\\lambda_1(\\Omega)\\geq \\frac{1}{4}h(\\Omega)^2\\]\nand the constant $\\frac{1}{4}$ is sharp in the sense that the ratio $\\frac{\\lambda_1(\\Omega)}{h(\\Omega)^2}$ tends to $\\frac{1}{4}$ when $\\Omega$ is taken to be the unit ball of $\\mathbb{R}^n$ for $n\\to +\\infty$. It is however not sharp when $\\Omega$ is restricted to be a planar convex sets, as is shown for instance by the result of I. Ftouhi:\n\\[\\frac{\\lambda_1(\\Omega)}{h(\\Omega)^2}\\geq 0.902\\]\nIlias Ftouhi's conjecture is that the optimal bound is $\\frac{2\\pi^2}{(2+\\sqrt{\\pi})^2}\\approx 1.387...$, which is reached exactly when $\\Omega$ is a square.\n\nAn optimal set is known to exist, by standard compactness arguments, but it is unknown whether this optimal set is a polygon. A first stepping stone would be to examine the case of polygonal sets, where the Cheeger set (the optimal set ``$A$'' in the definition) should coincide with the boundary of $\\Omega$ on some part, or leave the boundary tangentially following circle arcs (as described in [kawohl, lachand-robert]).",
   "public_statement": "Cheeger's inequality on convex sets\n\nFor an open set of finite measure $\\Omega\\subset\\mathbb{R}^n$, denote $\\lambda_1(\\Omega)$ the first eigenvalue of the Laplacian with Dirichlet boundary conditions, and $h(\\Omega)$ the Cheeger constant which may be defined as\n\\[h(\\Omega)=\\inf_{A\\subset \\Omega}\\frac{|\\partial A|}{|A|}=\\inf_{u\\in\\mathcal{C}^\\infty_c(\\Omega)}\\frac{\\int_{\\Omega}|\\nabla u|}{\\int_{\\Omega}u}\\]\nCheeger's inequality is\n\\[\\lambda_1(\\Omega)\\geq \\frac{1}{4}h(\\Omega)^2\\]\nand the constant $\\frac{1}{4}$ is sharp in the sense that the ratio $\\frac{\\lambda_1(\\Omega)}{h(\\Omega)^2}$ tends to $\\frac{1}{4}$ when $\\Omega$ is taken to be the unit ball of $\\mathbb{R}^n$ for $n\\to +\\infty$. It is however not sharp when $\\Omega$ is restricted to be a planar convex sets, as is shown for instance by the result of I. Ftouhi:\n\\[\\frac{\\lambda_1(\\Omega)}{h(\\Omega)^2}\\geq 0.902\\]\nIlias Ftouhi's conjecture is that the optimal bound is $\\frac{2\\pi^2}{(2+\\sqrt{\\pi})^2}\\approx 1.387...$, which is reached exactly when $\\Omega$ is a square.\n\nAn optimal set is known to exist, by standard compactness arguments, but it is unknown whether this optimal set is a polygon. A first stepping stone would be to examine the case of polygonal sets, where the Cheeger set (the optimal set ``$A$'' in the definition) should coincide with the boundary of $\\Omega$ on some part, or leave the boundary tangentially following circle arcs (as described in [kawohl, lachand-robert]).",
   "evidence": "The source record asks for the best lower bound on \\[ J(\\Omega):=\\frac{\\lambda _1(\\Omega)}{h(\\Omega)^2} \\] among planar convex open sets, conjecturing \\[ J(\\Omega)\\geq J(Q):=\\frac{2\\pi^2}{(2+\\sqrt\\pi)^2} =1.3870172735\\ldots, \\] with equality, up to similarities, only for a square. It also asks whether a minimizer is polygonal and suggests first studying polygons through the geometry of their Cheeger sets.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-GEOMETRY-0010": {
   "statement_status": "exact",
   "original_statement": "Slicing problem in 3 dimensions\n\nSuppose $\\Omega \\subset \\mathbb{R}^3$ is convex, normalized to have volume $1$, and with its center of mass at the origin. Let\n\\[\nA_{ij} = \\int_\\Omega x_i x_j \\, dx .\n\\]\nThe ball minimizes $\\det A$. The problem is to find the shape that maximizes $\\det A$.\n\nThis functional is affine invariant under linear maps with determinant $1$ (volume preserving). The scale invariant form of the functional is $\\det A/|\\Omega|^{n+2}$, where here $n=3$.\n\nIn two dimensions, the maximizer is the triangle. In dimensions $n \\geq 3$, the problem has been open since 1986: one possibility raised by Luis Rademacher is that it is reached by a simplex in dimension $n=3$.",
   "clean_statement": "Slicing problem in 3 dimensions\n\nSuppose $\\Omega \\subset \\mathbb{R}^3$ is convex, normalized to have volume $1$, and with its center of mass at the origin. Let\n\\[\nA_{ij} = \\int_\\Omega x_i x_j \\, dx .\n\\]\nThe ball minimizes $\\det A$. The problem is to find the shape that maximizes $\\det A$.\n\nThis functional is affine invariant under linear maps with determinant $1$ (volume preserving). The scale invariant form of the functional is $\\det A/|\\Omega|^{n+2}$, where here $n=3$.\n\nIn two dimensions, the maximizer is the triangle. In dimensions $n \\geq 3$, the problem has been open since 1986: one possibility raised by Luis Rademacher is that it is reached by a simplex in dimension $n=3$.",
   "public_statement": "Slicing problem in 3 dimensions\n\nSuppose $\\Omega \\subset \\mathbb{R}^3$ is convex, normalized to have volume $1$, and with its center of mass at the origin. Let\n\\[\nA_{ij} = \\int_\\Omega x_i x_j \\, dx .\n\\]\nThe ball minimizes $\\det A$. The problem is to find the shape that maximizes $\\det A$.\n\nThis functional is affine invariant under linear maps with determinant $1$ (volume preserving). The scale invariant form of the functional is $\\det A/|\\Omega|^{n+2}$, where here $n=3$.\n\nIn two dimensions, the maximizer is the triangle. In dimensions $n \\geq 3$, the problem has been open since 1986: one possibility raised by Luis Rademacher is that it is reached by a simplex in dimension $n=3$.",
   "evidence": "The canonical AIM record (Geometry, problem 3.2, source index 9) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-GEOMETRY-0011": {
   "statement_status": "exact",
   "original_statement": "Rearrangement problem\n\nLet $F:\\mathbb{R}\\to\\mathbb{R}$ be an even convex function, then it is known (from Xi, Zhao (2022)) that for any function $f,g\\in L^1(\\mathbb{R}^n)$ (for some $n\\geq 1$), denoting $f^*,g^*$ the symmetrically decreasing rearrangement of $f,g$, we have\n\\[\\iint_{R^n\\times\\mathbb{R}^n}F(x\\cdot y)f(x)g(y)dxdy\\geq \\iint_{R^n\\times\\mathbb{R}^n}F(x\\cdot y)f^*(x)g^*(y)dxdy\\]\nHowever it is unclear whether the convexity constraint on $F$ is sharp. The two questions raised by Paul Simanjuntak are as follows:\n\n1) Can the hypothesis on $F$ be weakened ?\n\n2) Does this inequality extend to other settings, for instance Gaussian measures ?",
   "clean_statement": "Rearrangement problem\n\nLet $F:\\mathbb{R}\\to\\mathbb{R}$ be an even convex function, then it is known (from Xi, Zhao (2022)) that for any function $f,g\\in L^1(\\mathbb{R}^n)$ (for some $n\\geq 1$), denoting $f^*,g^*$ the symmetrically decreasing rearrangement of $f,g$, we have\n\\[\\iint_{R^n\\times\\mathbb{R}^n}F(x\\cdot y)f(x)g(y)dxdy\\geq \\iint_{R^n\\times\\mathbb{R}^n}F(x\\cdot y)f^*(x)g^*(y)dxdy\\]\nHowever it is unclear whether the convexity constraint on $F$ is sharp. The two questions raised by Paul Simanjuntak are as follows:\n\n1) Can the hypothesis on $F$ be weakened ?\n\n2) Does this inequality extend to other settings, for instance Gaussian measures ?",
   "public_statement": "Rearrangement problem\n\nLet $F:\\mathbb{R}\\to\\mathbb{R}$ be an even convex function, then it is known (from Xi, Zhao (2022)) that for any function $f,g\\in L^1(\\mathbb{R}^n)$ (for some $n\\geq 1$), denoting $f^*,g^*$ the symmetrically decreasing rearrangement of $f,g$, we have\n\\[\\iint_{R^n\\times\\mathbb{R}^n}F(x\\cdot y)f(x)g(y)dxdy\\geq \\iint_{R^n\\times\\mathbb{R}^n}F(x\\cdot y)f^*(x)g^*(y)dxdy\\]\nHowever it is unclear whether the convexity constraint on $F$ is sharp. The two questions raised by Paul Simanjuntak are as follows:\n\n1) Can the hypothesis on $F$ be weakened ?\n\n2) Does this inequality extend to other settings, for instance Gaussian measures ?",
   "evidence": "The corpus record asks about the assertion",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-GEOMETRY-0012": {
   "statement_status": "exact",
   "original_statement": "Is there an association between the mutation equivalence of plabic graphs or the mutation equivalence of quivers to the invariants of isotopy classes of their associated links? Can these be used to produce invariants of quivers up to mutation equivalence?",
   "clean_statement": "Is there an association between the mutation equivalence of plabic graphs or the mutation equivalence of quivers to the invariants of isotopy classes of their associated links? Can these be used to produce invariants of quivers up to mutation equivalence?",
   "public_statement": "Is there an association between the mutation equivalence of plabic graphs or the mutation equivalence of quivers to the invariants of isotopy classes of their associated links? Can these be used to produce invariants of quivers up to mutation equivalence?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-GEOMETRY-0013": {
   "statement_status": "exact",
   "original_statement": "To a quiver, we can associate a point count polynomial defined recursively (Galashin-Lam). This is sometimes equal to the HOMFLY-PT polynomial of the related link. For any simple plabic graphs, are these polynomials always equivalent?",
   "clean_statement": "To a quiver, we can associate a point count polynomial defined recursively (Galashin-Lam). This is sometimes equal to the HOMFLY-PT polynomial of the related link. For any simple plabic graphs, are these polynomials always equivalent?",
   "public_statement": "To a quiver, we can associate a point count polynomial defined recursively (Galashin-Lam). This is sometimes equal to the HOMFLY-PT polynomial of the related link. For any simple plabic graphs, are these polynomials always equivalent?",
   "evidence": "The canonical AIM record asks, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-GEOMETRY-0014": {
   "statement_status": "exact",
   "original_statement": "In high levels of generality, varieties pick up a mixed Hodge structure (Lam-Speyer). When the cluster variety is adequately nice, the mixed Hodge structure has only two dimensions, and the point count polynomial corresponds to a certain polynomial, but not the HOMFLY-PT.\n\nWhat is the relationship between the mixed Hodge structure on the cohomology of a cluster variety coming from a plabic graph and knot invariants (eg. the Khovanov-Rozansky homology)?",
   "clean_statement": "In high levels of generality, varieties pick up a mixed Hodge structure (Lam-Speyer). When the cluster variety is adequately nice, the mixed Hodge structure has only two dimensions, and the point count polynomial corresponds to a certain polynomial, but not the HOMFLY-PT.\n\nWhat is the relationship between the mixed Hodge structure on the cohomology of a cluster variety coming from a plabic graph and knot invariants (eg. the Khovanov-Rozansky homology)?",
   "public_statement": "In high levels of generality, varieties pick up a mixed Hodge structure (Lam-Speyer). When the cluster variety is adequately nice, the mixed Hodge structure has only two dimensions, and the point count polynomial corresponds to a certain polynomial, but not the HOMFLY-PT.\n\nWhat is the relationship between the mixed Hodge structure on the cohomology of a cluster variety coming from a plabic graph and knot invariants (eg. the Khovanov-Rozansky homology)?",
   "evidence": "The AIM workshop *Cluster algebras and braid varieties* (23--27 January 2023), section 1.3, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-GEOMETRY-0015": {
   "statement_status": "exact",
   "original_statement": "When is the plabic cluster algebra locally acyclic? If it is not locally acyclic, is it still a polynomial count and can we say anything about a mixed Hodge structure?",
   "clean_statement": "When is the plabic cluster algebra locally acyclic? If it is not locally acyclic, is it still a polynomial count and can we say anything about a mixed Hodge structure?",
   "public_statement": "When is the plabic cluster algebra locally acyclic? If it is not locally acyclic, is it still a polynomial count and can we say anything about a mixed Hodge structure?",
   "evidence": "The canonical AIM record asks, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-GEOMETRY-0016": {
   "statement_status": "exact",
   "original_statement": "If we consider instead a braid variety, ie. a Legendrian associated to a $(-1)$-closure, does this still hold?",
   "clean_statement": "If we consider instead a braid variety, ie. a Legendrian associated to a $(-1)$-closure, does this still hold?",
   "public_statement": "If we consider instead a braid variety, ie. a Legendrian associated to a $(-1)$-closure, does this still hold?",
   "evidence": "The canonical record, AIM-GEOMETRY-0016, contains only the sentence",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-GEOMETRY-0017": {
   "statement_status": "exact",
   "original_statement": "For more general Legendrians, is the augmentation variety cluster?",
   "clean_statement": "For more general Legendrians, is the augmentation variety cluster?",
   "public_statement": "For more general Legendrians, is the augmentation variety cluster?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-GEOMETRY-0018": {
   "statement_status": "exact",
   "original_statement": "If $X$ is a general cluster variety, and $T\\subset X$ is a cluster torus. What can we say about $X - T$ (Eg. what are the irreducible components, intersections, etc.)?",
   "clean_statement": "If $X$ is a general cluster variety, and $T\\subset X$ is a cluster torus. What can we say about $X - T$ (Eg. what are the irreducible components, intersections, etc.)?",
   "public_statement": "If $X$ is a general cluster variety, and $T\\subset X$ is a cluster torus. What can we say about $X - T$ (Eg. what are the irreducible components, intersections, etc.)?",
   "evidence": "The canonical AIM record, from *Cluster algebras and braid varieties*, Section 2.3 (“Stratifications of braid varieties”), reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-GEOMETRY-0019": {
   "statement_status": "exact",
   "original_statement": "What can we say if $X$ is the cluster variety of a $(-1)$-closure, but $T$ comes from some weave other than the obvious/natural one?",
   "clean_statement": "What can we say if $X$ is the cluster variety of a $(-1)$-closure, but $T$ comes from some weave other than the obvious/natural one?",
   "public_statement": "What can we say if $X$ is the cluster variety of a $(-1)$-closure, but $T$ comes from some weave other than the obvious/natural one?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-GEOMETRY-0020": {
   "statement_status": "exact",
   "original_statement": "When is $P$ empty (in $G(2, 2k+1)$)?\n\nWhat about the big open positroid cell in $G(k,n)$?",
   "clean_statement": "When is $P$ empty (in $G(2, 2k+1)$)?\n\nWhat about the big open positroid cell in $G(k,n)$?",
   "public_statement": "When is $P$ empty (in $G(2, 2k+1)$)?\n\nWhat about the big open positroid cell in $G(k,n)$?",
   "evidence": "The canonical record, from the AIM problem list for the January 2023 workshop *Cluster algebras and braid varieties*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-GEOMETRY-0021": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "When $P$ is nonempty, when do we expect the complement of $P$ to be codimension 2?",
   "clean_statement": "When \\(P\\ne\\varnothing\\), when should the peripheral locus \\(P=X\\setminus M\\) itself have codimension two?",
   "public_statement": "When $P$ is nonempty, when do we expect the complement of $P$ to be codimension 2?",
   "evidence": "The canonical record, from the AIM list *Cluster algebras and braid varieties*, section 3.2 (“The peripheral locus”), reads:",
   "classification_method": "invalid_literal_with_repaired_reading",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-GEOMETRY-0022": {
   "statement_status": "exact",
   "original_statement": "Let $X$ be really full rank. This implies there is a torus of dimension the number of frozen variables acting on $X$ preserving cluster structure. $S$ acts freely on each cluster torus so any point with nontrivial $S$-stabilizers is in $P$.\n\nIn $G(k,n)$, nontrivial stabilizers correspond to disconnected matroids. So if $[v_1,\\dots, v_n]$ is a $k\\times n$ matrix where the matroid of $v_1,\\dots, v_n$ is disconnected, this point of $G(k,n)$ is in $P$.\n\nIs the converse also true (Is $P$ for the big positroid cell exactly the points with disconnected matroid)?",
   "clean_statement": "Let $X$ be really full rank. This implies there is a torus of dimension the number of frozen variables acting on $X$ preserving cluster structure. $S$ acts freely on each cluster torus so any point with nontrivial $S$-stabilizers is in $P$.\n\nIn $G(k,n)$, nontrivial stabilizers correspond to disconnected matroids. So if $[v_1,\\dots, v_n]$ is a $k\\times n$ matrix where the matroid of $v_1,\\dots, v_n$ is disconnected, this point of $G(k,n)$ is in $P$.\n\nIs the converse also true (Is $P$ for the big positroid cell exactly the points with disconnected matroid)?",
   "public_statement": "Let $X$ be really full rank. This implies there is a torus of dimension the number of frozen variables acting on $X$ preserving cluster structure. $S$ acts freely on each cluster torus so any point with nontrivial $S$-stabilizers is in $P$.\n\nIn $G(k,n)$, nontrivial stabilizers correspond to disconnected matroids. So if $[v_1,\\dots, v_n]$ is a $k\\times n$ matrix where the matroid of $v_1,\\dots, v_n$ is disconnected, this point of $G(k,n)$ is in $P$.\n\nIs the converse also true (Is $P$ for the big positroid cell exactly the points with disconnected matroid)?",
   "evidence": "The exact AIM record from the 2023 workshop *Cluster algebras and braid varieties* is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-GEOMETRY-0023": {
   "statement_status": "exact",
   "original_statement": "Is there a natural way to compactify a braid variety so that the smaller strata also have cluster structures?",
   "clean_statement": "Is there a natural way to compactify a braid variety so that the smaller strata also have cluster structures?",
   "public_statement": "Is there a natural way to compactify a braid variety so that the smaller strata also have cluster structures?",
   "evidence": "The canonical record is problem 4.1 from the AIM workshop *Cluster algebras and braid varieties*, in the section “Compactifications, tropical points, and \\(\\Theta\\)-functions”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-GEOMETRY-0024": {
   "statement_status": "exact",
   "original_statement": "Can we obtain brick manifolds by cluster-theoretic means?",
   "clean_statement": "Can we obtain brick manifolds by cluster-theoretic means?",
   "public_statement": "Can we obtain brick manifolds by cluster-theoretic means?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-GEOMETRY-0025": {
   "statement_status": "exact",
   "original_statement": "How does this relate to the combinatorics of the subword complex?",
   "clean_statement": "How does this relate to the combinatorics of the subword complex?",
   "public_statement": "How does this relate to the combinatorics of the subword complex?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-GEOMETRY-0026": {
   "statement_status": "exact",
   "original_statement": "How does this relate to compactifications coming from tropical geometry?",
   "clean_statement": "How does this relate to compactifications coming from tropical geometry?",
   "public_statement": "How does this relate to compactifications coming from tropical geometry?",
   "evidence": "The exact canonical problem is AIM workshop problem 4.4:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-GEOMETRY-0027": {
   "statement_status": "exact",
   "original_statement": "For each cluster torus, take a co-character lattice and glue them by piecewise linear maps coming from exchange relations. These tropical points label the Gross-Hacking-Keel-Kontsevich $\\Theta$-basis.\n\nWhat are the tropical points of the braid variety? What are the $\\Theta$-functions on the braid variety?",
   "clean_statement": "For each cluster torus, take a co-character lattice and glue them by piecewise linear maps coming from exchange relations. These tropical points label the Gross-Hacking-Keel-Kontsevich $\\Theta$-basis.\n\nWhat are the tropical points of the braid variety? What are the $\\Theta$-functions on the braid variety?",
   "public_statement": "For each cluster torus, take a co-character lattice and glue them by piecewise linear maps coming from exchange relations. These tropical points label the Gross-Hacking-Keel-Kontsevich $\\Theta$-basis.\n\nWhat are the tropical points of the braid variety? What are the $\\Theta$-functions on the braid variety?",
   "evidence": "The live AIM page did not render in the available browser, so the wording was checked against the exact repository record and its neighboring entries rather than silently reconstructed. The record contains no apparent OCR corruption. The symbol $\\Theta$ is read as the usual GHKK theta basis. One clarification is essential: the cocharacter lattices of the braid variety's cluster tori describe the tropicalization of that cluster variety, while theta functions on a cluster $\\mathcal A$-variety are naturally indexed by integral tropical points of the **Fock--Goncharov dual cluster $\\mathcal X$-variety**. These lattices may be identified after choosing a seed or, for braid varieties, through the cluster-ensemble isomorphisms, but they should not be silently conflated.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-GEOMETRY-0028": {
   "statement_status": "exact",
   "original_statement": "Let $U$ be a unipotent cell. Then $U$ can be thought of as a braid variety. Also $\\mathbb{C}[U]$ has a Mirkovi\\'c-Vilonen-basis.\n\nDoes the MV-basis equal the $\\Theta$-basis?",
   "clean_statement": "Let $U$ be a unipotent cell. Then $U$ can be thought of as a braid variety. Also $\\mathbb{C}[U]$ has a Mirkovi\\'c-Vilonen-basis.\n\nDoes the MV-basis equal the $\\Theta$-basis?",
   "public_statement": "Let $U$ be a unipotent cell. Then $U$ can be thought of as a braid variety. Also $\\mathbb{C}[U]$ has a Mirkovi\\'c-Vilonen-basis.\n\nDoes the MV-basis equal the $\\Theta$-basis?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-GEOMETRY-0029": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "What is the relationship between the piecewise linear combinatorics in the Casals-Gorsky-Gorsky-Le-Shen-Simental cluster algebra construction and the PL-combinatorics of MV-polytopes?",
   "clean_statement": "How much of the CGGLSS tropical Lusztig propagation is precisely the reduced-word PL atlas of MV polytopes, and what additional structure is introduced by the nonreduced \\(0\\)-Hecke vertices of a Demazure weave?",
   "public_statement": "What is the relationship between the piecewise linear combinatorics in the Casals-Gorsky-Gorsky-Le-Shen-Simental cluster algebra construction and the PL-combinatorics of MV-polytopes?",
   "evidence": "The exact canonical record is AIM workshop problem 4.7:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-GEOMETRY-0030": {
   "statement_status": "exact",
   "original_statement": "Plabic graphs and weaves give seeds in cluster structures on braid varieties.\n\nInvestigate:\n\\begin{enumerate}\n\\item cluster variables,\n\\item quivers,\n\\item positive parametrizations,\n\\item twist maps and Donaldson-Thomas transformations,\n\\item and the relationship between 3d-plabic graphs and weaves.\n\\end{enumerate}",
   "clean_statement": "Plabic graphs and weaves give seeds in cluster structures on braid varieties.\n\nInvestigate:\n\\begin{enumerate}\n\\item cluster variables,\n\\item quivers,\n\\item positive parametrizations,\n\\item twist maps and Donaldson-Thomas transformations,\n\\item and the relationship between 3d-plabic graphs and weaves.\n\\end{enumerate}",
   "public_statement": "Plabic graphs and weaves give seeds in cluster structures on braid varieties.\n\nInvestigate:\n\\begin{enumerate}\n\\item cluster variables,\n\\item quivers,\n\\item positive parametrizations,\n\\item twist maps and Donaldson-Thomas transformations,\n\\item and the relationship between 3d-plabic graphs and weaves.\n\\end{enumerate}",
   "evidence": "The canonical source record is aim-geometry-notes.json, zero-based index 29, from the AIM workshop *Cluster algebras and braid varieties*, Section 6.1, \"Combinatorics of cluster structures on braid varieties.\" The record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-GEOMETRY-0031": {
   "statement_status": "exact",
   "original_statement": "For each positroid stratum of a Grassmanian, there is a Legendrian link whose augmentation variety is that stratum. Positroid strata are also ordered with respect to closure.\n\nIs this partial order recoverable on the Legendrian links in terms of topology, for instance by (decomposable) Lagrangian cobordisms? Could the difference in dimension be related to the minimal genus of the cobordisms?",
   "clean_statement": "For each positroid stratum of a Grassmanian, there is a Legendrian link whose augmentation variety is that stratum. Positroid strata are also ordered with respect to closure.\n\nIs this partial order recoverable on the Legendrian links in terms of topology, for instance by (decomposable) Lagrangian cobordisms? Could the difference in dimension be related to the minimal genus of the cobordisms?",
   "public_statement": "For each positroid stratum of a Grassmanian, there is a Legendrian link whose augmentation variety is that stratum. Positroid strata are also ordered with respect to closure.\n\nIs this partial order recoverable on the Legendrian links in terms of topology, for instance by (decomposable) Lagrangian cobordisms? Could the difference in dimension be related to the minimal genus of the cobordisms?",
   "evidence": "The AIM record (Cluster algebras and braid varieties, Problem 7.1) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-GEOMETRY-0032": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Is there a local operation of plabic graphs which induces Lagrangian cobordisms between the associated Legendrian knots? In general, what can be done on the combinatorial side to guarantee a Lagrangian cobordism on the other side?",
   "clean_statement": "Which local edits of a reduced plabic graph can be certified combinatorially to give a directed orientable exact Lagrangian cobordism between the endpoint positroid Legendrians, and which familiar graph moves instead describe Legendrian isotopy or surgery between fillings of the same Legendrian?",
   "public_statement": "Is there a local operation of plabic graphs which induces Lagrangian cobordisms between the associated Legendrian knots? In general, what can be done on the combinatorial side to guarantee a Lagrangian cobordism on the other side?",
   "evidence": "The exact canonical AIM record (workshop *Cluster algebras and braid varieties*, Section 7, Problem 7.2) asks:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-GEOMETRY-0033": {
   "statement_status": "exact",
   "original_statement": "Is there a symplectic geometric application of 3d-plabic graphs? Do 3d-plabic graphs give some filling of a Legendrian link?",
   "clean_statement": "Is there a symplectic geometric application of 3d-plabic graphs? Do 3d-plabic graphs give some filling of a Legendrian link?",
   "public_statement": "Is there a symplectic geometric application of 3d-plabic graphs? Do 3d-plabic graphs give some filling of a Legendrian link?",
   "evidence": "The canonical record is AIM-GEOMETRY-0033, item 7.3 in the AIM workshop list *Cluster algebras and braid varieties*, section “Lagrangian cobordisms from plabic graphs.” Its exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-GEOMETRY-0034": {
   "statement_status": "exact",
   "original_statement": "For ADE types, all seeds in the cluster algebra can be realized by Lagrangian fillings. Can this be generalized?",
   "clean_statement": "For ADE types, all seeds in the cluster algebra can be realized by Lagrangian fillings. Can this be generalized?",
   "public_statement": "For ADE types, all seeds in the cluster algebra can be realized by Lagrangian fillings. Can this be generalized?",
   "evidence": "The exact AIM record (Cluster algebras and braid varieties, Problem 7.4) is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-GEOMETRY-0035": {
   "statement_status": "exact",
   "original_statement": "Can mutations of quivers with potentials be realized as a symplectic operation? Could we use this to write down potentials for quivers from braid varieties?",
   "clean_statement": "Can mutations of quivers with potentials be realized as a symplectic operation? Could we use this to write down potentials for quivers from braid varieties?",
   "public_statement": "Can mutations of quivers with potentials be realized as a symplectic operation? Could we use this to write down potentials for quivers from braid varieties?",
   "evidence": "The canonical record is Problem 7.5 in the AIM workshop list *Cluster algebras and braid varieties*, Section “Lagrangian cobordisms from plabic graphs”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-GEOMETRY-0036": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Does the cluster modular group act properly discontinuously on the positive real part of the braid variety? Does this group contain the centralizer of our braid?",
   "clean_statement": null,
   "public_statement": "Does the cluster modular group act properly discontinuously on the positive real part of the braid variety? Does this group contain the centralizer of our braid?",
   "evidence": "I use the following conservative reconstruction.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-GEOMETRY-0037": {
   "statement_status": "exact",
   "original_statement": "Given some automorphism of the braid variety, does symplectic geometry help inform if it's a quasi cluster automorphism?",
   "clean_statement": "Given some automorphism of the braid variety, does symplectic geometry help inform if it's a quasi cluster automorphism?",
   "public_statement": "Given some automorphism of the braid variety, does symplectic geometry help inform if it's a quasi cluster automorphism?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-GEOMETRY-0038": {
   "statement_status": "exact",
   "original_statement": "Describe the positive real/totally non negative part of the braid variety. Can this be done using motivation from the perspective of algebraic geometry? symplectic topology? linear algebra?",
   "clean_statement": "Describe the positive real/totally non negative part of the braid variety. Can this be done using motivation from the perspective of algebraic geometry? symplectic topology? linear algebra?",
   "public_statement": "Describe the positive real/totally non negative part of the braid variety. Can this be done using motivation from the perspective of algebraic geometry? symplectic topology? linear algebra?",
   "evidence": "The canonical record is AIM Problem List 9.1 from the 2023 workshop *Cluster algebras and braid varieties*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-GEOMETRY-0039": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "How can we define totally non negative braid varieties in a way which generalizes the totally non negative part of $G(k,n)/\\Pi_f$?",
   "clean_statement": null,
   "public_statement": "How can we define totally non negative braid varieties in a way which generalizes the totally non negative part of $G(k,n)/\\Pi_f$?",
   "evidence": "There are three plausible readings. 1. **Literal quotient.** Read \\(G(k,n)/\\Pi_f\\) as a quotient. This is not a defined object from the supplied data: \\(\\Pi_f\\) is a subvariety, not a group acting on \\(G(k,n)\\). This reading is rejected unless additional quotient data are supplied. 2. **Paired examples.** Read the slash informally as “\\(G(k,n)\\) / \\(\\Pi_f\\),” meaning the Grassmannian and its positroid varieties. This is the most plausible reading in context. 3. **Missing relation symbol.** The intended expression may have been \\(\\Pi_f\\subset G(k,n)\\), or may have meant the open positroid stratum \\(\\mathring{\\Pi}_f\\). The original page does not determine which.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-GEOMETRY-0040": {
   "statement_status": "exact",
   "original_statement": "Can we use different cluster structures or parametrizations to cover braid varieties or other interesting spaces by their different positive parts?",
   "clean_statement": "Can we use different cluster structures or parametrizations to cover braid varieties or other interesting spaces by their different positive parts?",
   "public_statement": "Can we use different cluster structures or parametrizations to cover braid varieties or other interesting spaces by their different positive parts?",
   "evidence": "The canonical record is question 9.3 in the AIM workshop list *Cluster algebras and braid varieties*, section “Totally non-negative parts of braid varieties”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-GEOMETRY-0041": {
   "statement_status": "exact",
   "original_statement": "How can we understand the connected components, topology, etc. of the real points in braid varieties?",
   "clean_statement": "How can we understand the connected components, topology, etc. of the real points in braid varieties?",
   "public_statement": "How can we understand the connected components, topology, etc. of the real points in braid varieties?",
   "evidence": "The original AIM page contains exactly the same sentence and no attached remarks, status claim, or additional notation. There is no visible OCR corruption. The neighboring questions ask for positive/totally-nonnegative loci and whether different positive parametrizations cover more of a braid variety. Thus “real points” naturally means the fixed locus \\(X(\\beta)(\\mathbb R)\\) of the standard real structure on a braid variety defined by real braid-matrix equations.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-GEOMETRY-0042": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Braid varieties are compactified by Brick manifolds. The strata here relate to positive braids.\n\nIs there a compactification of the cotangent bundle of Bott-Samelson varieties or the cotangent bundle of Brick manifolds related to all braids?",
   "clean_statement": null,
   "public_statement": "Braid varieties are compactified by Brick manifolds. The strata here relate to positive braids.\n\nIs there a compactification of the cotangent bundle of Bott-Samelson varieties or the cotangent bundle of Brick manifolds related to all braids?",
   "evidence": "The capitalization of “Brick” and the paragraph break are preserved. The nearby records do not supply a definition of “related to all braids,” and the workshop summary discusses positive braid words throughout. Thus the second sentence has at least three plausible readings:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-GEOMETRY-0043": {
   "statement_status": "exact",
   "original_statement": "What can we say about Soergel bimodules and braid varieties?",
   "clean_statement": "What can we say about Soergel bimodules and braid varieties?",
   "public_statement": "What can we say about Soergel bimodules and braid varieties?",
   "evidence": "The record has no remarks or supplied literature. Its source URL is <http://aimpl.org/clusterbraid/10/>; that page returned an HTTP 502 during this run, so the exact text above was verified from the canonical repository record and nearby records rather than from a live copy of the page. There is no visible OCR corruption.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-GEOMETRY-0044": {
   "statement_status": "exact",
   "original_statement": "Can we study parabolic braid varieties, ie. in $G/P$?\n\nDo these have cluster structures? Do we need to change the definition of a seed?",
   "clean_statement": "Can we study parabolic braid varieties, ie. in $G/P$?\n\nDo these have cluster structures? Do we need to change the definition of a seed?",
   "public_statement": "Can we study parabolic braid varieties, ie. in $G/P$?\n\nDo these have cluster structures? Do we need to change the definition of a seed?",
   "evidence": "The original AIM HTML contains exactly this wording, including “ie.” and the notation \\(G/P\\), and supplies no remarks or status update. There is no OCR corruption to repair. The neighboring questions concern compactifications and Soergel bimodules, so “parabolic” means replacing complete flags in \\(G/B\\) by partial flags in \\(G/P\\), not a parabolic subgroup of an Artin group.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-GEOMETRY-0045": {
   "statement_status": "exact",
   "original_statement": "What can be said about equilibrium states for the geometric potential associated to geodesic flows on finite volume, complete surfaces? More generally, manifolds?",
   "clean_statement": "What can be said about equilibrium states for the geometric potential associated to geodesic flows on finite volume, complete surfaces? More generally, manifolds?",
   "public_statement": "What can be said about equilibrium states for the geometric potential associated to geodesic flows on finite volume, complete surfaces? More generally, manifolds?",
   "evidence": "The canonical record is AIM-GEOMETRY-0045, item 1.1 in the AIM workshop section “Noncompact Phase Spaces”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-GEOMETRY-0046": {
   "statement_status": "exact",
   "original_statement": "Is the harmonic measure from Brownian motion in the finite-volume noncompact setting an equilibrium state?",
   "clean_statement": "Is the harmonic measure from Brownian motion in the finite-volume noncompact setting an equilibrium state?",
   "public_statement": "Is the harmonic measure from Brownian motion in the finite-volume noncompact setting an equilibrium state?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-GEOMETRY-0047": {
   "statement_status": "exact",
   "original_statement": "Let $M$ be a finite volume, complete, negatively curved manifold. Is the Liouville measure an equilibrium state? The case when the curvature is uniformly bounded below is known, so the case of curvature is unbounded is the remaining case.",
   "clean_statement": "Let $M$ be a finite volume, complete, negatively curved manifold. Is the Liouville measure an equilibrium state? The case when the curvature is uniformly bounded below is known, so the case of curvature is unbounded is the remaining case.",
   "public_statement": "Let $M$ be a finite volume, complete, negatively curved manifold. Is the Liouville measure an equilibrium state? The case when the curvature is uniformly bounded below is known, so the case of curvature is unbounded is the remaining case.",
   "evidence": "The exact canonical record is AIM-GEOMETRY-0047, item 1.3 in the AIM workshop list *Equilibrium states for dynamical systems arising from geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-GEOMETRY-0048": {
   "statement_status": "exact",
   "original_statement": "What can one say about extreme value theory for (non-uniformly) hyperbolic flows on noncompact spaces?",
   "clean_statement": "What can one say about extreme value theory for (non-uniformly) hyperbolic flows on noncompact spaces?",
   "public_statement": "What can one say about extreme value theory for (non-uniformly) hyperbolic flows on noncompact spaces?",
   "evidence": "The canonical AIM record (Geometry, workshop *Equilibrium states for dynamical systems arising from geometry*, section *Noncompact Phase Spaces*, Problem 1.4) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-GEOMETRY-0049": {
   "statement_status": "exact",
   "original_statement": "For geodesic flows on non-compact metric spaces, can the variational principle have an extremal measure and metric? That is, if $\\mathcal M_f$ is the set of $f$-invariant probability measures on $X$, $\\mathcal D_X$ is the space of distances on $X$ inducing the same topology, $h_d(f) = \\displaystyle \\sup_{K \\subset X} h_d(f,K)$ where the supremum is taken over the topological entropy of compact subsets $K$ with respect to the metric $d$, and:\n\n\\begin{eqnarray*}\nh_{\\operatorname{Borel}}(f) & = & \\sup_{\\mu \\in \\mathcal M_f} h_\\mu(f) \\\\\nh_{\\operatorname{top}}(f) & = & \\inf_{d \\in \\mathcal D_X} h_{d}(f)\n\\end{eqnarray*}\n\nThe Borel and topological entropies were shown to coincide in \\cite{MR1348316} for locally compact spaces. Under what conditions can one find a metric $d$ and measure $\\mu$ which realize the common value?",
   "clean_statement": "For geodesic flows on non-compact metric spaces, can the variational principle have an extremal measure and metric? That is, if $\\mathcal M_f$ is the set of $f$-invariant probability measures on $X$, $\\mathcal D_X$ is the space of distances on $X$ inducing the same topology, $h_d(f) = \\displaystyle \\sup_{K \\subset X} h_d(f,K)$ where the supremum is taken over the topological entropy of compact subsets $K$ with respect to the metric $d$, and:\n\n\\begin{eqnarray*}\nh_{\\operatorname{Borel}}(f) & = & \\sup_{\\mu \\in \\mathcal M_f} h_\\mu(f) \\\\\nh_{\\operatorname{top}}(f) & = & \\inf_{d \\in \\mathcal D_X} h_{d}(f)\n\\end{eqnarray*}\n\nThe Borel and topological entropies were shown to coincide in \\cite{MR1348316} for locally compact spaces. Under what conditions can one find a metric $d$ and measure $\\mu$ which realize the common value?",
   "public_statement": "For geodesic flows on non-compact metric spaces, can the variational principle have an extremal measure and metric? That is, if $\\mathcal M_f$ is the set of $f$-invariant probability measures on $X$, $\\mathcal D_X$ is the space of distances on $X$ inducing the same topology, $h_d(f) = \\displaystyle \\sup_{K \\subset X} h_d(f,K)$ where the supremum is taken over the topological entropy of compact subsets $K$ with respect to the metric $d$, and:\n\n\\begin{eqnarray*}\nh_{\\operatorname{Borel}}(f) & = & \\sup_{\\mu \\in \\mathcal M_f} h_\\mu(f) \\\\\nh_{\\operatorname{top}}(f) & = & \\inf_{d \\in \\mathcal D_X} h_{d}(f)\n\\end{eqnarray*}\n\nThe Borel and topological entropies were shown to coincide in \\cite{MR1348316} for locally compact spaces. Under what conditions can one find a metric $d$ and measure $\\mu$ which realize the common value?",
   "evidence": "The canonical AIM record asks whether the two extrema \\[ h_{\\mathrm{Borel}}(f) =\\sup_{\\mu\\in\\mathcal M_f}h_\\mu(f), \\qquad h_{\\mathrm{top}}(f) =\\inf_{d\\in\\mathcal D_X}h_d(f) \\] can be attained simultaneously for a geodesic flow on a noncompact metric space. Here \\[ h_d(f)=\\sup_{K\\subset X\\ \\mathrm{compact}}h_d(f,K), \\] \\(\\mathcal M_f\\) is the set of invariant Borel probability measures, and \\(\\mathcal D_X\\) is the set of distances inducing the topology of \\(X\\). The record says that the Borel and topological entropies coincide for locally compact spaces by the unresolved key \\(\\mathrm{MR1348316}\\).",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-GEOMETRY-0050": {
   "statement_status": "exact",
   "original_statement": "Give a complete picture of existence and uniqueness for H\\\"older potentials on the space of geodesics for the geodesic flow on noncompact $\\operatorname{CAT}(-1)$ spaces. In particular, consider potentials which depend not only the basepoint of a geodesic, but the geodesic itself.",
   "clean_statement": "Give a complete picture of existence and uniqueness for H\\\"older potentials on the space of geodesics for the geodesic flow on noncompact $\\operatorname{CAT}(-1)$ spaces. In particular, consider potentials which depend not only the basepoint of a geodesic, but the geodesic itself.",
   "public_statement": "Give a complete picture of existence and uniqueness for H\\\"older potentials on the space of geodesics for the geodesic flow on noncompact $\\operatorname{CAT}(-1)$ spaces. In particular, consider potentials which depend not only the basepoint of a geodesic, but the geodesic itself.",
   "evidence": "The exact AIM record (Geometry, workshop *Equilibrium states for dynamical systems arising from geometry*, section “Noncompact Phase Spaces,” Problem 1.6) is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-GEOMETRY-0051": {
   "statement_status": "exact",
   "original_statement": "Extend the results of \\cite{MR3124716} to noncompact spaces.",
   "clean_statement": "Extend the results of \\cite{MR3124716} to noncompact spaces.",
   "public_statement": "Extend the results of \\cite{MR3124716} to noncompact spaces.",
   "evidence": "The exact canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-GEOMETRY-0052": {
   "statement_status": "exact",
   "original_statement": "Let $\\tilde{M}$ be contractible negatively curved Riemannian manifold, and $\\Gamma \\subset \\operatorname{Isom}(\\tilde{M})$ be a discrete group of isometries which acts cocompactly.\n\n\\begin{enumerate}\n\\item Consider a nearest-neighbor random walk on $\\Gamma$, which produces an exit measure on the boundary $\\partial\\Gamma$. Is this measure a Gibbs state?\n\\item If $\\tilde{M}$ is nonpositively curved, what can be said about the measures induced on the boundary by Brownian motion? Is it a Gibbs state? What else can be said about it? (In the negative curvature case, this is known by //////)\n\\end{enumerate}",
   "clean_statement": "Let $\\tilde{M}$ be contractible negatively curved Riemannian manifold, and $\\Gamma \\subset \\operatorname{Isom}(\\tilde{M})$ be a discrete group of isometries which acts cocompactly.\n\n\\begin{enumerate}\n\\item Consider a nearest-neighbor random walk on $\\Gamma$, which produces an exit measure on the boundary $\\partial\\Gamma$. Is this measure a Gibbs state?\n\\item If $\\tilde{M}$ is nonpositively curved, what can be said about the measures induced on the boundary by Brownian motion? Is it a Gibbs state? What else can be said about it? (In the negative curvature case, this is known by //////)\n\\end{enumerate}",
   "public_statement": "Let $\\tilde{M}$ be contractible negatively curved Riemannian manifold, and $\\Gamma \\subset \\operatorname{Isom}(\\tilde{M})$ be a discrete group of isometries which acts cocompactly.\n\n\\begin{enumerate}\n\\item Consider a nearest-neighbor random walk on $\\Gamma$, which produces an exit measure on the boundary $\\partial\\Gamma$. Is this measure a Gibbs state?\n\\item If $\\tilde{M}$ is nonpositively curved, what can be said about the measures induced on the boundary by Brownian motion? Is it a Gibbs state? What else can be said about it? (In the negative curvature case, this is known by //////)\n\\end{enumerate}",
   "evidence": "The canonical record is item 2.1, “Geodesic Flows on Compact Spaces,” from the AIM workshop *Equilibrium states for dynamical systems arising from geometry*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-GEOMETRY-0053": {
   "statement_status": "exact",
   "original_statement": "Let $\\varphi_t$ be geodesic flow on the unit tangent bundle of a compact surface of constant negative curvature. Let $S_n$ denote the number of simple closed geodesics with length $\\le n$. Find precise growth rates for $S_n$. More generally, what can be said about variable negative curvature?",
   "clean_statement": "Let $\\varphi_t$ be geodesic flow on the unit tangent bundle of a compact surface of constant negative curvature. Let $S_n$ denote the number of simple closed geodesics with length $\\le n$. Find precise growth rates for $S_n$. More generally, what can be said about variable negative curvature?",
   "public_statement": "Let $\\varphi_t$ be geodesic flow on the unit tangent bundle of a compact surface of constant negative curvature. Let $S_n$ denote the number of simple closed geodesics with length $\\le n$. Find precise growth rates for $S_n$. More generally, what can be said about variable negative curvature?",
   "evidence": "The repository record and adjacent Problems 2.1--2.4 were checked. The original AIM URL was unavailable during this run, but there is no visible OCR corruption in this statement.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-GEOMETRY-0054": {
   "statement_status": "exact",
   "original_statement": "Glue together two hyperbolic manifolds along some geodesic segment, and consider the corresponding $\\operatorname{CAT}(-1)$ geodesic flow. Does the Bowen-Margulis measure satisfy exponential decay of correlations?",
   "clean_statement": "Glue together two hyperbolic manifolds along some geodesic segment, and consider the corresponding $\\operatorname{CAT}(-1)$ geodesic flow. Does the Bowen-Margulis measure satisfy exponential decay of correlations?",
   "public_statement": "Glue together two hyperbolic manifolds along some geodesic segment, and consider the corresponding $\\operatorname{CAT}(-1)$ geodesic flow. Does the Bowen-Margulis measure satisfy exponential decay of correlations?",
   "evidence": "The exact canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-GEOMETRY-0055": {
   "statement_status": "exact",
   "original_statement": "Consider the geodesic flow on a translation surface. The Lebesgue measure is a 0-entropy measure, and the measure of maximal entropy sits on saddle connections. Are there other interesting invariant measures with intermediate entropy?",
   "clean_statement": "Consider the geodesic flow on a translation surface. The Lebesgue measure is a 0-entropy measure, and the measure of maximal entropy sits on saddle connections. Are there other interesting invariant measures with intermediate entropy?",
   "public_statement": "Consider the geodesic flow on a translation surface. The Lebesgue measure is a 0-entropy measure, and the measure of maximal entropy sits on saddle connections. Are there other interesting invariant measures with intermediate entropy?",
   "evidence": "The exact canonical source record (AIM workshop section “Geodesic Flows on Compact Spaces,” problem 2.4) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-GEOMETRY-0056": {
   "statement_status": "exact",
   "original_statement": "Let $\\varphi_1,\\varphi_2 : X \\to \\R$ be two potentials. What interesting analysis can be said about the 2-parameter family of equilibrium states $\\mu_{q_1\\varphi_1 + q_2\\varphi_2}$? Can we find useful applications for these results?",
   "clean_statement": "Let $\\varphi_1,\\varphi_2 : X \\to \\R$ be two potentials. What interesting analysis can be said about the 2-parameter family of equilibrium states $\\mu_{q_1\\varphi_1 + q_2\\varphi_2}$? Can we find useful applications for these results?",
   "public_statement": "Let $\\varphi_1,\\varphi_2 : X \\to \\R$ be two potentials. What interesting analysis can be said about the 2-parameter family of equilibrium states $\\mu_{q_1\\varphi_1 + q_2\\varphi_2}$? Can we find useful applications for these results?",
   "evidence": "The repository record and adjacent “Specific Examples” problems were checked. The original AIM page was not recoverable during this run, but the statement has no visible OCR corruption. It does omit the map or flow on $X$, compactness, regularity of the potentials, and hypotheses giving existence or uniqueness. Therefore no universal family $\\mu_{q_1\\varphi_1+q_2\\varphi_2}$ is defined by the source alone.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-GEOMETRY-0057": {
   "statement_status": "exact",
   "original_statement": "Generalize the results of Bufetov and Gurevich in \\cite{MR2857792} to the space of quadratic differentials.",
   "clean_statement": "Generalize the results of Bufetov and Gurevich in \\cite{MR2857792} to the space of quadratic differentials.",
   "public_statement": "Generalize the results of Bufetov and Gurevich in \\cite{MR2857792} to the space of quadratic differentials.",
   "evidence": "The exact canonical AIM statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-GEOMETRY-0058": {
   "statement_status": "exact",
   "original_statement": "What can be said about interesting phase transitions for potentials $\\{q\\varphi\\}_{q\\in\\R}$? In particular, let $M$ be a compact surface of nonpositive curvature, and $\\varphi$ be the geometric potential, so that $q = 1$ is a phase transition.",
   "clean_statement": "What can be said about interesting phase transitions for potentials $\\{q\\varphi\\}_{q\\in\\R}$? In particular, let $M$ be a compact surface of nonpositive curvature, and $\\varphi$ be the geometric potential, so that $q = 1$ is a phase transition.",
   "public_statement": "What can be said about interesting phase transitions for potentials $\\{q\\varphi\\}_{q\\in\\R}$? In particular, let $M$ be a compact surface of nonpositive curvature, and $\\varphi$ be the geometric potential, so that $q = 1$ is a phase transition.",
   "evidence": "The exact canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-GEOMETRY-0059": {
   "statement_status": "exact",
   "original_statement": "Consider a dynamical billiard without cusps. Do (non-periodic) invariant probability measures for the billiard map satisfy the integrability condition for Oseledet's theorem?",
   "clean_statement": "Consider a dynamical billiard without cusps. Do (non-periodic) invariant probability measures for the billiard map satisfy the integrability condition for Oseledet's theorem?",
   "public_statement": "Consider a dynamical billiard without cusps. Do (non-periodic) invariant probability measures for the billiard map satisfy the integrability condition for Oseledet's theorem?",
   "evidence": "The canonical AIM record (workshop *Equilibrium states for dynamical systems arising from geometry*, section 3.4) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-GEOMETRY-0060": {
   "statement_status": "exact",
   "original_statement": "Are there examples of non-uniformly hyperbolic magnetic flows for which we can apply the tools of thermodynamical formalism?",
   "clean_statement": "Are there examples of non-uniformly hyperbolic magnetic flows for which we can apply the tools of thermodynamical formalism?",
   "public_statement": "Are there examples of non-uniformly hyperbolic magnetic flows for which we can apply the tools of thermodynamical formalism?",
   "evidence": "The exact canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-GEOMETRY-0061": {
   "statement_status": "exact",
   "original_statement": "Study the measure of maximal entropy for the Handel-Thurston Anosov flow. Does it coincide with Lebesgue measure? If not, what is its dimension?",
   "clean_statement": "Study the measure of maximal entropy for the Handel-Thurston Anosov flow. Does it coincide with Lebesgue measure? If not, what is its dimension?",
   "public_statement": "Study the measure of maximal entropy for the Handel-Thurston Anosov flow. Does it coincide with Lebesgue measure? If not, what is its dimension?",
   "evidence": "The exact canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-GEOMETRY-0062": {
   "statement_status": "exact",
   "original_statement": "Can one identify a geometric example of a system for which the set of phase transitions has positive Lebesgue measure?",
   "clean_statement": "Can one identify a geometric example of a system for which the set of phase transitions has positive Lebesgue measure?",
   "public_statement": "Can one identify a geometric example of a system for which the set of phase transitions has positive Lebesgue measure?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-GEOMETRY-0063": {
   "statement_status": "exact",
   "original_statement": "Let $S$ be a surface which has a flat cylinder of parallel periodic orbits. In \\cite{MR3856792}, a criterion for the existence and uniqueness of equilibrium states includes a pressure gap, $P_{\\operatorname{sing}}(\\varphi) < P(\\varphi)$. Can this be made explicit for the surface $S$?",
   "clean_statement": "Let $S$ be a surface which has a flat cylinder of parallel periodic orbits. In \\cite{MR3856792}, a criterion for the existence and uniqueness of equilibrium states includes a pressure gap, $P_{\\operatorname{sing}}(\\varphi) < P(\\varphi)$. Can this be made explicit for the surface $S$?",
   "public_statement": "Let $S$ be a surface which has a flat cylinder of parallel periodic orbits. In \\cite{MR3856792}, a criterion for the existence and uniqueness of equilibrium states includes a pressure gap, $P_{\\operatorname{sing}}(\\varphi) < P(\\varphi)$. Can this be made explicit for the surface $S$?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-GEOMETRY-0064": {
   "statement_status": "exact",
   "original_statement": "Define a decomposition and pressure gap for $\\mbox{CAT}(0)$ geodesic flows, and extend \\cite{MR3856792} for these flows.",
   "clean_statement": "Define a decomposition and pressure gap for $\\mbox{CAT}(0)$ geodesic flows, and extend \\cite{MR3856792} for these flows.",
   "public_statement": "Define a decomposition and pressure gap for $\\mbox{CAT}(0)$ geodesic flows, and extend \\cite{MR3856792} for these flows.",
   "evidence": "The exact canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-GEOMETRY-0065": {
   "statement_status": "exact",
   "original_statement": "Can one simplify the assumptions and/or arguments of \\cite{MR3046278} to the setting of \\cite{MR3856792}?",
   "clean_statement": "Can one simplify the assumptions and/or arguments of \\cite{MR3046278} to the setting of \\cite{MR3856792}?",
   "public_statement": "Can one simplify the assumptions and/or arguments of \\cite{MR3046278} to the setting of \\cite{MR3856792}?",
   "evidence": "There is no apparent OCR corruption, but “to the setting” is mathematically ambiguous. It can mean either (a) translate the 2013 symbolic uniqueness criterion to the continuous rank-one geodesic-flow setting, or (b) exploit the special geometric decomposition of the 2018 paper to shorten or weaken the already available flow argument. Reading (a) was substantially addressed before the 2018 paper by Climenhaga--Thompson’s 2016 flow theorem, which the 2018 paper invokes as its Theorem 2.6. The contribution below addresses the remaining concrete part of reading (b).",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-GEOMETRY-0066": {
   "statement_status": "exact",
   "original_statement": "Extend \\cite{MR3856792} to the case of no focal points. In particular, can one apply this to the case of the geometric potential? In particular, the Donnay-type sphere examples of surfaces with ``spherical'' caps.",
   "clean_statement": "Extend \\cite{MR3856792} to the case of no focal points. In particular, can one apply this to the case of the geometric potential? In particular, the Donnay-type sphere examples of surfaces with ``spherical'' caps.",
   "public_statement": "Extend \\cite{MR3856792} to the case of no focal points. In particular, can one apply this to the case of the geometric potential? In particular, the Donnay-type sphere examples of surfaces with ``spherical'' caps.",
   "evidence": "The last sentence is not an OCR error, but it needs a scope correction. Donnay’s examples are metrics on \\(S^2\\) built using focusing caps [Don88]. They are not examples without focal points. Donnay explicitly proves that a focusing cap has conjugate points (Proposition 6.1 and Remark 6.2), and every metric on \\(S^2\\) has conjugate points. Consequently, the no-focal extension and the Donnay-cap application are two different branches of the question.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-GEOMETRY-0067": {
   "statement_status": "exact",
   "original_statement": "Can one find a simplified proof of Burns-Gelfert in \\cite{MR3124716} using \\cite{MR3856792}, possibly with stronger conclusions? Similarly for the results of Paulin-Policott-Schapira in \\cite{MR3444431}.",
   "clean_statement": "Can one find a simplified proof of Burns-Gelfert in \\cite{MR3124716} using \\cite{MR3856792}, possibly with stronger conclusions? Similarly for the results of Paulin-Policott-Schapira in \\cite{MR3444431}.",
   "public_statement": "Can one find a simplified proof of Burns-Gelfert in \\cite{MR3124716} using \\cite{MR3856792}, possibly with stronger conclusions? Similarly for the results of Paulin-Policott-Schapira in \\cite{MR3444431}.",
   "evidence": "- **MR3124716:** Keith Burns and Katrin Gelfert, *Lyapunov spectrum for geodesic flows of rank 1 surfaces*, Discrete and Continuous Dynamical Systems 34 (2014), 1841--1872. The arXiv preprint 1106.0053 has the earlier title *Thermodynamics for geodesic flows of rank 1 surfaces*; this title difference is genuine, not an extraction error. - **MR3856792:** Keith Burns, Vaughn Climenhaga, Todd Fisher, and Daniel J. Thompson (BCFT), *Unique equilibrium states for geodesic flows in nonpositive curvature*, Geometric and Functional Analysis 28 (2018), 1209--1259. - **MR3444431:** Frédéric Paulin, Mark Pollicott, and Barbara Schapira (PPS), *Equilibrium states in negative curvature*, Astérisque 373 (2015), viii+281.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-GEOMETRY-0068": {
   "statement_status": "exact",
   "original_statement": "Let $f_1,f_2$ be commuting hyperbolic diffeomorphisms of a compact manifold $M$. One may construct coarse Lyapunov foliations $\\mathcal W$. Given a potential $\\varphi$, can one construct leaf-wise conditional measures $\\mu^{\\mathcal W}_\\varphi$ which satisfy Margulis-like cocycle properties?",
   "clean_statement": "Let $f_1,f_2$ be commuting hyperbolic diffeomorphisms of a compact manifold $M$. One may construct coarse Lyapunov foliations $\\mathcal W$. Given a potential $\\varphi$, can one construct leaf-wise conditional measures $\\mu^{\\mathcal W}_\\varphi$ which satisfy Margulis-like cocycle properties?",
   "public_statement": "Let $f_1,f_2$ be commuting hyperbolic diffeomorphisms of a compact manifold $M$. One may construct coarse Lyapunov foliations $\\mathcal W$. Given a potential $\\varphi$, can one construct leaf-wise conditional measures $\\mu^{\\mathcal W}_\\varphi$ which satisfy Margulis-like cocycle properties?",
   "evidence": "The source is Section 5.1, “Other Directions,” of the AIM problem list from the 2019 workshop *Equilibrium states for dynamical systems arising from geometry*. The text has no apparent OCR corruption, but it leaves four mathematical choices unstated.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-GEOMETRY-0069": {
   "statement_status": "exact",
   "original_statement": "For metric Anosov flows (Smale flows), under what conditions does orbit equivalence imply H\\\"older orbit equivalence? Can one find a Smale flow which is not H\\\"older covered by the suspension of a symbolic dynamical system.",
   "clean_statement": "For metric Anosov flows (Smale flows), under what conditions does orbit equivalence imply H\\\"older orbit equivalence? Can one find a Smale flow which is not H\\\"older covered by the suspension of a symbolic dynamical system.",
   "public_statement": "For metric Anosov flows (Smale flows), under what conditions does orbit equivalence imply H\\\"older orbit equivalence? Can one find a Smale flow which is not H\\\"older covered by the suspension of a symbolic dynamical system.",
   "evidence": "The exact problem field in the canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-GEOMETRY-0070": {
   "statement_status": "exact",
   "original_statement": "Let $f : \\mathbb{T}^2 \\to \\mathbb{T}^2$ be a nonuniformly hyperbolic diffeomorphism. Let $\\mu$ be the unstable SRB measure, $\\nu$ be the measure of maximal entropy and $m$ be the measure of maximal dimension (make assumptions on $f$ so that these exist and are unique). If $\\nu = \\mu$ or $m$, is $f$ smoothly conjugate to a linear automorphism?",
   "clean_statement": "Let $f : \\mathbb{T}^2 \\to \\mathbb{T}^2$ be a nonuniformly hyperbolic diffeomorphism. Let $\\mu$ be the unstable SRB measure, $\\nu$ be the measure of maximal entropy and $m$ be the measure of maximal dimension (make assumptions on $f$ so that these exist and are unique). If $\\nu = \\mu$ or $m$, is $f$ smoothly conjugate to a linear automorphism?",
   "public_statement": "Let $f : \\mathbb{T}^2 \\to \\mathbb{T}^2$ be a nonuniformly hyperbolic diffeomorphism. Let $\\mu$ be the unstable SRB measure, $\\nu$ be the measure of maximal entropy and $m$ be the measure of maximal dimension (make assumptions on $f$ so that these exist and are unique). If $\\nu = \\mu$ or $m$, is $f$ smoothly conjugate to a linear automorphism?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 69,
   "attempt": 1
  },
  "AIM-GEOMETRY-0071": {
   "statement_status": "exact",
   "original_statement": "Can one prove exponential decay of correlations for equilibrium states in settings where it it is known for the measure of maximal entropy? (eg, Weil-Petersson geodesic flows, noncompact spaces, non-positive curvature, etc)",
   "clean_statement": "Can one prove exponential decay of correlations for equilibrium states in settings where it it is known for the measure of maximal entropy? (eg, Weil-Petersson geodesic flows, noncompact spaces, non-positive curvature, etc)",
   "public_statement": "Can one prove exponential decay of correlations for equilibrium states in settings where it it is known for the measure of maximal entropy? (eg, Weil-Petersson geodesic flows, noncompact spaces, non-positive curvature, etc)",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 70,
   "attempt": 1
  },
  "AIM-GEOMETRY-0072": {
   "statement_status": "exact",
   "original_statement": "Can one find a dynamical interpretation of the pressure gap? For instance, can one relate the pressure gap to an essential spectral radius for a dynamical operator? Or some dynamically defined $\\zeta$-function?",
   "clean_statement": "Can one find a dynamical interpretation of the pressure gap? For instance, can one relate the pressure gap to an essential spectral radius for a dynamical operator? Or some dynamically defined $\\zeta$-function?",
   "public_statement": "Can one find a dynamical interpretation of the pressure gap? For instance, can one relate the pressure gap to an essential spectral radius for a dynamical operator? Or some dynamically defined $\\zeta$-function?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-GEOMETRY-0073": {
   "statement_status": "exact",
   "original_statement": "Describe the set of matrix cocycles which have multiple eqilibrium states. For simplicity, consider a subshift of finite type, and a locally constant, fiber bunched cocycle.",
   "clean_statement": "Describe the set of matrix cocycles which have multiple eqilibrium states. For simplicity, consider a subshift of finite type, and a locally constant, fiber bunched cocycle.",
   "public_statement": "Describe the set of matrix cocycles which have multiple eqilibrium states. For simplicity, consider a subshift of finite type, and a locally constant, fiber bunched cocycle.",
   "evidence": "The exact canonical AIM text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 72,
   "attempt": 1
  },
  "AIM-GEOMETRY-0074": {
   "statement_status": "exact",
   "original_statement": "build a moduli space for $C^{1,\\theta}$-perturbations of geodesic flows using potential functions (following McMullen's approach of expanding maps of the circle). In this case, can one identify the space of geodesic flows by a condition on their corresponding potential function?",
   "clean_statement": "build a moduli space for $C^{1,\\theta}$-perturbations of geodesic flows using potential functions (following McMullen's approach of expanding maps of the circle). In this case, can one identify the space of geodesic flows by a condition on their corresponding potential function?",
   "public_statement": "build a moduli space for $C^{1,\\theta}$-perturbations of geodesic flows using potential functions (following McMullen's approach of expanding maps of the circle). In this case, can one identify the space of geodesic flows by a condition on their corresponding potential function?",
   "evidence": "The exact problem field in the canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 73,
   "attempt": 1
  },
  "AIM-GEOMETRY-0075": {
   "statement_status": "exact",
   "original_statement": "Consider a finite-horizon billiards, and let $\\varphi$ be the geometric potential. Is there a jump in pressure $P(q\\varphi)$ and $q \\to 0^-$? Can one find perioidic orbits with arbitrarily large (but finite) Lyapunov exponent?",
   "clean_statement": "Consider a finite-horizon billiards, and let $\\varphi$ be the geometric potential. Is there a jump in pressure $P(q\\varphi)$ and $q \\to 0^-$? Can one find perioidic orbits with arbitrarily large (but finite) Lyapunov exponent?",
   "public_statement": "Consider a finite-horizon billiards, and let $\\varphi$ be the geometric potential. Is there a jump in pressure $P(q\\varphi)$ and $q \\to 0^-$? Can one find perioidic orbits with arbitrarily large (but finite) Lyapunov exponent?",
   "evidence": "The canonical record says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-GEOMETRY-0076": {
   "statement_status": "exact",
   "original_statement": "Develop the theory for weighting orbits in a subadditive (rather than additive) way. {\\rm Ask Kiho for references}",
   "clean_statement": "Develop the theory for weighting orbits in a subadditive (rather than additive) way. {\\rm Ask Kiho for references}",
   "public_statement": "Develop the theory for weighting orbits in a subadditive (rather than additive) way. {\\rm Ask Kiho for references}",
   "evidence": "The exact AIM record (workshop *Equilibrium states for dynamical systems arising from geometry*, section “Other Directions,” item 5.9) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-GEOMETRY-0077": {
   "statement_status": "exact",
   "original_statement": "Smale's problem: Rapidly generate $N$ points on $S^2$ whose logarithmic energy differs from the optimal energy by $O(log N)$.",
   "clean_statement": "Smale's problem: Rapidly generate $N$ points on $S^2$ whose logarithmic energy differs from the optimal energy by $O(log N)$.",
   "public_statement": "Smale's problem: Rapidly generate $N$ points on $S^2$ whose logarithmic energy differs from the optimal energy by $O(log N)$.",
   "evidence": "The exact AIM record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-GEOMETRY-0078": {
   "statement_status": "exact",
   "original_statement": "Can we extend the definition of universal optimality to potentials with slow decay (or functions that are not completely monotonic)?",
   "clean_statement": "Can we extend the definition of universal optimality to potentials with slow decay (or functions that are not completely monotonic)?",
   "public_statement": "Can we extend the definition of universal optimality to potentials with slow decay (or functions that are not completely monotonic)?",
   "evidence": "The exact canonical record is AIM Problem Lists, workshop **Discrete geometry and automorphic forms**, item 1.04:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-GEOMETRY-0079": {
   "statement_status": "exact",
   "original_statement": "Which lattices $\\Lambda$ in $\\mathbb R^d$ of determinant $1$ maximize\n\\[\n\\min_{y \\in \\mathbb R^d \\setminus \\Lambda} \\sum_{x \\in \\Lambda} \\frac{1}{|x-y|^s}?\n\\]\nFor $s \\to \\infty$ this is the covering problem. On $S^{n-1}$ this is called polarization.",
   "clean_statement": "Which lattices $\\Lambda$ in $\\mathbb R^d$ of determinant $1$ maximize\n\\[\n\\min_{y \\in \\mathbb R^d \\setminus \\Lambda} \\sum_{x \\in \\Lambda} \\frac{1}{|x-y|^s}?\n\\]\nFor $s \\to \\infty$ this is the covering problem. On $S^{n-1}$ this is called polarization.",
   "public_statement": "Which lattices $\\Lambda$ in $\\mathbb R^d$ of determinant $1$ maximize\n\\[\n\\min_{y \\in \\mathbb R^d \\setminus \\Lambda} \\sum_{x \\in \\Lambda} \\frac{1}{|x-y|^s}?\n\\]\nFor $s \\to \\infty$ this is the covering problem. On $S^{n-1}$ this is called polarization.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-GEOMETRY-0080": {
   "statement_status": "exact",
   "original_statement": "Are the Voronoi cells of the optimal $8$ or $24$ dimensional sphere packings as small as possible by volume?",
   "clean_statement": "Are the Voronoi cells of the optimal $8$ or $24$ dimensional sphere packings as small as possible by volume?",
   "public_statement": "Are the Voronoi cells of the optimal $8$ or $24$ dimensional sphere packings as small as possible by volume?",
   "evidence": "The exact source record, AIM workshop *Discrete geometry and automorphic forms*, item 1.08, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-GEOMETRY-0081": {
   "statement_status": "exact",
   "original_statement": "Prove $A_2$ is universally optimal. Or find magic function for the sphere packing poblem in $\\mathbb R^2$.",
   "clean_statement": "Prove $A_2$ is universally optimal. Or find magic function for the sphere packing poblem in $\\mathbb R^2$.",
   "public_statement": "Prove $A_2$ is universally optimal. Or find magic function for the sphere packing poblem in $\\mathbb R^2$.",
   "evidence": "The exact canonical AIM Problem Lists record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-GEOMETRY-0082": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Choose $\\Lambda \\subseteq \\mathbb R^d$ to minimize the Wasserstein (optimal transport) distance form $\\sum_{x \\in \\Lambda} \\delta_x$ to the Lebesque measure.",
   "clean_statement": null,
   "public_statement": "Choose $\\Lambda \\subseteq \\mathbb R^d$ to minimize the Wasserstein (optimal transport) distance form $\\sum_{x \\in \\Lambda} \\delta_x$ to the Lebesque measure.",
   "evidence": "The exact AIM record asks:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-GEOMETRY-0083": {
   "statement_status": "exact",
   "original_statement": "Use Siegel modular forms to improve bounds for the maximal dimension of an extremal even lattice.",
   "clean_statement": "Use Siegel modular forms to improve bounds for the maximal dimension of an extremal even lattice.",
   "public_statement": "Use Siegel modular forms to improve bounds for the maximal dimension of an extremal even lattice.",
   "evidence": "The exact AIM record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 82,
   "attempt": 1
  },
  "AIM-GEOMETRY-0084": {
   "statement_status": "exact",
   "original_statement": "To obtain LP bound in $\\mathbb R^n$ numerically, use linear combinations of different Gaussian (with complex parameters).",
   "clean_statement": "To obtain LP bound in $\\mathbb R^n$ numerically, use linear combinations of different Gaussian (with complex parameters).",
   "public_statement": "To obtain LP bound in $\\mathbb R^n$ numerically, use linear combinations of different Gaussian (with complex parameters).",
   "evidence": "The exact canonical AIM Problem Lists record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 83,
   "attempt": 1
  },
  "AIM-GEOMETRY-0085": {
   "statement_status": "exact",
   "original_statement": "Numerical LP bounds in high dimensions.",
   "clean_statement": "Numerical LP bounds in high dimensions.",
   "public_statement": "Numerical LP bounds in high dimensions.",
   "evidence": "The exact canonical record says only:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 84,
   "attempt": 1
  },
  "AIM-GEOMETRY-0086": {
   "statement_status": "exact",
   "original_statement": "Let $A_-(d)$ be the minimum or $r$ for which there exists a Schwartz function $f \\colon \\mathbb R^d \\to \\mathbb R$ such that $f(0) = \\hat f(0) = 1$, $f(x) \\leq 0$ for $|x| \\geq r$, and $\\hat f(y) \\geq 0$ for all $y$. For $d$ a non-integral positive number this bound is still defined by using radial functions and the Bessel transform. We know $A_-(1) = 1$, $A_-(8) = \\sqrt{2}$, and $A_-(24) = 2$. Numerically it seems to be the case that $A_-(2) = (4/3)^{1/4}$, and that $A_-'(8) = \\sqrt{2}/30 = (\\sqrt(2))^7/240$ and $A_-'(24) = 368/12285 = 23 * 2^8/196560$.",
   "clean_statement": "Let $A_-(d)$ be the minimum or $r$ for which there exists a Schwartz function $f \\colon \\mathbb R^d \\to \\mathbb R$ such that $f(0) = \\hat f(0) = 1$, $f(x) \\leq 0$ for $|x| \\geq r$, and $\\hat f(y) \\geq 0$ for all $y$. For $d$ a non-integral positive number this bound is still defined by using radial functions and the Bessel transform. We know $A_-(1) = 1$, $A_-(8) = \\sqrt{2}$, and $A_-(24) = 2$. Numerically it seems to be the case that $A_-(2) = (4/3)^{1/4}$, and that $A_-'(8) = \\sqrt{2}/30 = (\\sqrt(2))^7/240$ and $A_-'(24) = 368/12285 = 23 * 2^8/196560$.",
   "public_statement": "Let $A_-(d)$ be the minimum or $r$ for which there exists a Schwartz function $f \\colon \\mathbb R^d \\to \\mathbb R$ such that $f(0) = \\hat f(0) = 1$, $f(x) \\leq 0$ for $|x| \\geq r$, and $\\hat f(y) \\geq 0$ for all $y$. For $d$ a non-integral positive number this bound is still defined by using radial functions and the Bessel transform. We know $A_-(1) = 1$, $A_-(8) = \\sqrt{2}$, and $A_-(24) = 2$. Numerically it seems to be the case that $A_-(2) = (4/3)^{1/4}$, and that $A_-'(8) = \\sqrt{2}/30 = (\\sqrt(2))^7/240$ and $A_-'(24) = 368/12285 = 23 * 2^8/196560$.",
   "evidence": "The exact canonical AIM record is preserved in `input.json`. Its problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 85,
   "attempt": 1
  },
  "AIM-GEOMETRY-0087": {
   "statement_status": "exact",
   "original_statement": "It is know that $A_+(12) = \\sqrt{2}$ from the paper by Kumar and Gon\\c{c}alves. Numerically it seems $A_+'(12) = \\sqrt{8}/63 = (\\sqrt{2})^9/504$. Can we prove this?",
   "clean_statement": "It is know that $A_+(12) = \\sqrt{2}$ from the paper by Kumar and Gon\\c{c}alves. Numerically it seems $A_+'(12) = \\sqrt{8}/63 = (\\sqrt{2})^9/504$. Can we prove this?",
   "public_statement": "It is know that $A_+(12) = \\sqrt{2}$ from the paper by Kumar and Gon\\c{c}alves. Numerically it seems $A_+'(12) = \\sqrt{8}/63 = (\\sqrt{2})^9/504$. Can we prove this?",
   "evidence": "The exact canonical AIM Problem Lists record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 86,
   "attempt": 1
  },
  "AIM-GEOMETRY-0088": {
   "statement_status": "exact",
   "original_statement": "Is it true that if you remove one point of the $24$-cell, it is still optimal? Same question for removing up to $7$ points of $120$ cell. What about $E_8$ and $\\Lambda_{24}$?",
   "clean_statement": "Is it true that if you remove one point of the $24$-cell, it is still optimal? Same question for removing up to $7$ points of $120$ cell. What about $E_8$ and $\\Lambda_{24}$?",
   "public_statement": "Is it true that if you remove one point of the $24$-cell, it is still optimal? Same question for removing up to $7$ points of $120$ cell. What about $E_8$ and $\\Lambda_{24}$?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 87,
   "attempt": 1
  },
  "AIM-GEOMETRY-0089": {
   "statement_status": "exact",
   "original_statement": "Uniqueness of $4$-dimensional kissing configuration.",
   "clean_statement": "Uniqueness of $4$-dimensional kissing configuration.",
   "public_statement": "Uniqueness of $4$-dimensional kissing configuration.",
   "evidence": "The exact canonical AIM problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 88,
   "attempt": 1
  },
  "AIM-GEOMETRY-0090": {
   "statement_status": "exact",
   "original_statement": "Let $f$ be the magic function in $8$-dimension and $M_f(s) = \\int_0^\\infty f(x) x^{s-1} dx$. Conjecture of Cohn and Miller: $M_f(4) = \\frac{1}{15} = M_{\\hat{f}}(4)$. Can we prove this? The same question for the magic function in $24$-dimension, where $M_f(12) = M_{\\hat{f}}(12) = 0.17786094729650\\ldots$?",
   "clean_statement": "Let $f$ be the magic function in $8$-dimension and $M_f(s) = \\int_0^\\infty f(x) x^{s-1} dx$. Conjecture of Cohn and Miller: $M_f(4) = \\frac{1}{15} = M_{\\hat{f}}(4)$. Can we prove this? The same question for the magic function in $24$-dimension, where $M_f(12) = M_{\\hat{f}}(12) = 0.17786094729650\\ldots$?",
   "public_statement": "Let $f$ be the magic function in $8$-dimension and $M_f(s) = \\int_0^\\infty f(x) x^{s-1} dx$. Conjecture of Cohn and Miller: $M_f(4) = \\frac{1}{15} = M_{\\hat{f}}(4)$. Can we prove this? The same question for the magic function in $24$-dimension, where $M_f(12) = M_{\\hat{f}}(12) = 0.17786094729650\\ldots$?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 89,
   "attempt": 1
  },
  "AIM-GEOMETRY-0091": {
   "statement_status": "exact",
   "original_statement": "Does every lattice have a $\\mathbb{Z}$-basis of Voronoi vectors? (I.e., vectors defining facets of Voronoi cells)",
   "clean_statement": "Does every lattice have a $\\mathbb{Z}$-basis of Voronoi vectors? (I.e., vectors defining facets of Voronoi cells)",
   "public_statement": "Does every lattice have a $\\mathbb{Z}$-basis of Voronoi vectors? (I.e., vectors defining facets of Voronoi cells)",
   "evidence": "The canonical record in `aim-geometry-notes.json`, index 90, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 90,
   "attempt": 1
  },
  "AIM-GEOMETRY-0092": {
   "statement_status": "exact",
   "original_statement": "Consider the following $3$-point bound generalization of the Cohn-Elkies bound:\n\\[\n\\inf \\Big\\{f(0,0) : f \\in S(\\mathbb R^{2n}), \\, \\hat f(0, 0) = 1, \\, \\hat f \\geq 0, \\,f \\leq 0 \\text{ on } C_2 \\Big\\}^{1/2},\n\\]\nwhere\n\\[\nC_2 = \\big\\{(x,y) \\in \\mathbb R^{2n} : \\|x\\|, \\|y\\|, \\|x-y\\| \\in \\{0\\} \\cup [1, \\infty)\\big\\} \\setminus \\big\\{(0,0)\\big\\}.\n\\]\nThis gives an upper bound on the optimal Lattice sphere packing center density. Computational results suggest this is equal to the Cohn-Elkies bound, and the computed solutions are invariant under the action of the group $O(n) \\times O(n)$ on $\\mathbb R^{2n}$. Why are the solutions invariant under $O(n) \\times O(n)$. Can we use a solution of the Cohn-Elkies bound to formulate a solution for the above bound and vice versa?",
   "clean_statement": "Consider the following $3$-point bound generalization of the Cohn-Elkies bound:\n\\[\n\\inf \\Big\\{f(0,0) : f \\in S(\\mathbb R^{2n}), \\, \\hat f(0, 0) = 1, \\, \\hat f \\geq 0, \\,f \\leq 0 \\text{ on } C_2 \\Big\\}^{1/2},\n\\]\nwhere\n\\[\nC_2 = \\big\\{(x,y) \\in \\mathbb R^{2n} : \\|x\\|, \\|y\\|, \\|x-y\\| \\in \\{0\\} \\cup [1, \\infty)\\big\\} \\setminus \\big\\{(0,0)\\big\\}.\n\\]\nThis gives an upper bound on the optimal Lattice sphere packing center density. Computational results suggest this is equal to the Cohn-Elkies bound, and the computed solutions are invariant under the action of the group $O(n) \\times O(n)$ on $\\mathbb R^{2n}$. Why are the solutions invariant under $O(n) \\times O(n)$. Can we use a solution of the Cohn-Elkies bound to formulate a solution for the above bound and vice versa?",
   "public_statement": "Consider the following $3$-point bound generalization of the Cohn-Elkies bound:\n\\[\n\\inf \\Big\\{f(0,0) : f \\in S(\\mathbb R^{2n}), \\, \\hat f(0, 0) = 1, \\, \\hat f \\geq 0, \\,f \\leq 0 \\text{ on } C_2 \\Big\\}^{1/2},\n\\]\nwhere\n\\[\nC_2 = \\big\\{(x,y) \\in \\mathbb R^{2n} : \\|x\\|, \\|y\\|, \\|x-y\\| \\in \\{0\\} \\cup [1, \\infty)\\big\\} \\setminus \\big\\{(0,0)\\big\\}.\n\\]\nThis gives an upper bound on the optimal Lattice sphere packing center density. Computational results suggest this is equal to the Cohn-Elkies bound, and the computed solutions are invariant under the action of the group $O(n) \\times O(n)$ on $\\mathbb R^{2n}$. Why are the solutions invariant under $O(n) \\times O(n)$. Can we use a solution of the Cohn-Elkies bound to formulate a solution for the above bound and vice versa?",
   "evidence": "The canonical AIM record asks us to consider",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 91,
   "attempt": 1
  },
  "AIM-GEOMETRY-0093": {
   "statement_status": "exact",
   "original_statement": "Minimize energy for $f(r) = e^{-\\alpha r^2}$ in $\\mathbb{R}^n$ as $\\alpha \\rightarrow 0$",
   "clean_statement": "Minimize energy for $f(r) = e^{-\\alpha r^2}$ in $\\mathbb{R}^n$ as $\\alpha \\rightarrow 0$",
   "public_statement": "Minimize energy for $f(r) = e^{-\\alpha r^2}$ in $\\mathbb{R}^n$ as $\\alpha \\rightarrow 0$",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-GEOMETRY-0094": {
   "statement_status": "exact",
   "original_statement": "Behavior of energy as $n \\rightarrow \\infty$ with $\\alpha$ fixed. (Gaussian core model). Known for small $\\alpha$ by the paper of Cohn and de Courcy-Ireland.",
   "clean_statement": "Behavior of energy as $n \\rightarrow \\infty$ with $\\alpha$ fixed. (Gaussian core model). Known for small $\\alpha$ by the paper of Cohn and de Courcy-Ireland.",
   "public_statement": "Behavior of energy as $n \\rightarrow \\infty$ with $\\alpha$ fixed. (Gaussian core model). Known for small $\\alpha$ by the paper of Cohn and de Courcy-Ireland.",
   "evidence": "The exact canonical record in `aim-geometry-notes.json`, index 93, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-GEOMETRY-0095": {
   "statement_status": "exact",
   "original_statement": "Minimize energy for $5$ points on $S^2$ under $f(r) = r^{-s}$, for the values of $s$ where this problem is still open.",
   "clean_statement": "Minimize energy for $5$ points on $S^2$ under $f(r) = r^{-s}$, for the values of $s$ where this problem is still open.",
   "public_statement": "Minimize energy for $5$ points on $S^2$ under $f(r) = r^{-s}$, for the values of $s$ where this problem is still open.",
   "evidence": "The exact canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 94,
   "attempt": 1
  },
  "AIM-GEOMETRY-0096": {
   "statement_status": "exact",
   "original_statement": "Numerical computation of LP bound in $\\mathbb{H}^n$.",
   "clean_statement": "Numerical computation of LP bound in $\\mathbb{H}^n$.",
   "public_statement": "Numerical computation of LP bound in $\\mathbb{H}^n$.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-GEOMETRY-0097": {
   "statement_status": "exact",
   "original_statement": "Is it true that for $1 \\leq k \\leq d$, the only critical points for $d+k$ particles in $S^{d-1} \\subset \\mathbb{R}^d$ under some potential functions (logarthmic + others) are orthogonal unions of $k$ regular simplices?",
   "clean_statement": "Is it true that for $1 \\leq k \\leq d$, the only critical points for $d+k$ particles in $S^{d-1} \\subset \\mathbb{R}^d$ under some potential functions (logarthmic + others) are orthogonal unions of $k$ regular simplices?",
   "public_statement": "Is it true that for $1 \\leq k \\leq d$, the only critical points for $d+k$ particles in $S^{d-1} \\subset \\mathbb{R}^d$ under some potential functions (logarthmic + others) are orthogonal unions of $k$ regular simplices?",
   "evidence": "The exact canonical AIM text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 96,
   "attempt": 1
  },
  "AIM-GEOMETRY-0098": {
   "statement_status": "exact",
   "original_statement": "Prove there is no lattice whose shells are spherical $12$-designs",
   "clean_statement": "Prove there is no lattice whose shells are spherical $12$-designs",
   "public_statement": "Prove there is no lattice whose shells are spherical $12$-designs",
   "evidence": "The exact AIM record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 97,
   "attempt": 1
  },
  "AIM-GEOMETRY-0099": {
   "statement_status": "exact",
   "original_statement": "Prove there exist spherical $t$-designs on $S^2$ with $(\\frac{1}{2} + o(1))t^2$ points.",
   "clean_statement": "Prove there exist spherical $t$-designs on $S^2$ with $(\\frac{1}{2} + o(1))t^2$ points.",
   "public_statement": "Prove there exist spherical $t$-designs on $S^2$ with $(\\frac{1}{2} + o(1))t^2$ points.",
   "evidence": "The canonical AIM record is problem 1.46 from the list *Discrete geometry and automorphic forms*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 98,
   "attempt": 1
  },
  "AIM-GEOMETRY-0100": {
   "statement_status": "exact",
   "original_statement": "Is it true that no shell of $E_8$ is a $8$-design? (This is equivalent to Lehman's conjecture)",
   "clean_statement": "Is it true that no shell of $E_8$ is a $8$-design? (This is equivalent to Lehman's conjecture)",
   "public_statement": "Is it true that no shell of $E_8$ is a $8$-design? (This is equivalent to Lehman's conjecture)",
   "evidence": "The canonical AIM record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 99,
   "attempt": 1
  },
  "AIM-GEOMETRY-0101": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Suppose we have a radial function $f$ on $\\mathbb{R}^n$ ($0 1$, and $\\hat f$ has double roots at $\\sqrt{2k}$ for $k \\geq 1$. Then,\n$$ \\frac{f(0)}{\\hat f(0)} = - \\frac{(n^4 -56n^3 + 1184n^2 - 11200 n+ 40320}{16(n-10)(n-14)(n-18)}$$",
   "clean_statement": null,
   "public_statement": "Suppose we have a radial function $f$ on $\\mathbb{R}^n$ ($0 1$, and $\\hat f$ has double roots at $\\sqrt{2k}$ for $k \\geq 1$. Then,\n$$ \\frac{f(0)}{\\hat f(0)} = - \\frac{(n^4 -56n^3 + 1184n^2 - 11200 n+ 40320}{16(n-10)(n-14)(n-18)}$$",
   "evidence": "### 1.1 The corrupted canonical record",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 100,
   "attempt": 1
  },
  "AIM-GEOMETRY-0102": {
   "statement_status": "exact",
   "original_statement": "Prove/Disprove that for optimal $N$-point codes on $S^2$ non-hexagonal Voronoi cells are dense in $S^2$ as $N \\rightarrow \\infty$.",
   "clean_statement": "Prove/Disprove that for optimal $N$-point codes on $S^2$ non-hexagonal Voronoi cells are dense in $S^2$ as $N \\rightarrow \\infty$.",
   "public_statement": "Prove/Disprove that for optimal $N$-point codes on $S^2$ non-hexagonal Voronoi cells are dense in $S^2$ as $N \\rightarrow \\infty$.",
   "evidence": "The repository record is AIM Problem Lists, workshop and section *Discrete geometry and automorphic forms*, Problem 1.52. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 101,
   "attempt": 1
  },
  "AIM-GEOMETRY-0103": {
   "statement_status": "exact",
   "original_statement": "Pick $n$ points in $\\Lambda$ to maximize the number of adjacent pairs. What is the limit shape as $n \\rightarrow \\infty$?",
   "clean_statement": "Pick $n$ points in $\\Lambda$ to maximize the number of adjacent pairs. What is the limit shape as $n \\rightarrow \\infty$?",
   "public_statement": "Pick $n$ points in $\\Lambda$ to maximize the number of adjacent pairs. What is the limit shape as $n \\rightarrow \\infty$?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 102,
   "attempt": 1
  },
  "AIM-GEOMETRY-0104": {
   "statement_status": "exact",
   "original_statement": "(G. Kuperberg - Schramm) How large can the average kissing number be for packing with spheres of varying radii? It is known that in $\\mathbb{R}^3$ it is larger than $12$ and at most $24$. Is it true that it is strictly larger than the kissing number for dimension $4, 8,$ and $24$?",
   "clean_statement": "(G. Kuperberg - Schramm) How large can the average kissing number be for packing with spheres of varying radii? It is known that in $\\mathbb{R}^3$ it is larger than $12$ and at most $24$. Is it true that it is strictly larger than the kissing number for dimension $4, 8,$ and $24$?",
   "public_statement": "(G. Kuperberg - Schramm) How large can the average kissing number be for packing with spheres of varying radii? It is known that in $\\mathbb{R}^3$ it is larger than $12$ and at most $24$. Is it true that it is strictly larger than the kissing number for dimension $4, 8,$ and $24$?",
   "evidence": "The exact repository record is AIM Problem Lists, workshop and section *Discrete geometry and automorphic forms*, Problem 1.56:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 103,
   "attempt": 1
  },
  "AIM-GEOMETRY-0105": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "The soft dodecahedral conjecture\n\nThe dodecahedral conjecture, now a theorem of Hales and McLaughlin \\cite{MR2601036}, states that the minimal volume of a Voronoi cell in a sphere packing is at least as great as the volume of a regular circumscribing dodecahedron.\n\nProve the soft dodecahedral conjecture: The density of a Voronoi cell in a soft packing is maximized when the Voronoi cell is a regular circumscribing dodecahedron.",
   "clean_statement": null,
   "public_statement": "The soft dodecahedral conjecture\n\nThe dodecahedral conjecture, now a theorem of Hales and McLaughlin \\cite{MR2601036}, states that the minimal volume of a Voronoi cell in a sphere packing is at least as great as the volume of a regular circumscribing dodecahedron.\n\nProve the soft dodecahedral conjecture: The density of a Voronoi cell in a soft packing is maximized when the Voronoi cell is a regular circumscribing dodecahedron.",
   "evidence": "The canonical record is from the AIM problem list *Soft packings, nested clusters, and condensed matter*, Section 1, Problem 1.1, attributed on the webpage to Bezdek. Its exact `problem` field is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 104,
   "attempt": 1
  },
  "AIM-GEOMETRY-0106": {
   "statement_status": "exact",
   "original_statement": "Phase transitions\n\nThe best packing of hard spheres in $\\mathbb{R}^3$ is known to be achieved by the FCC lattice, while the thinnest covering is conjectured to be given by the BCC lattice. This implies that as $\\lambda$ is varied, for $0< \\lambda < \\sqrt{5/3} -1$ we expect there to be an FCC - BCC transition.\n\nDescribe the behavior of optimal arrangements of soft balls as $\\lambda$ varies.",
   "clean_statement": "Phase transitions\n\nThe best packing of hard spheres in $\\mathbb{R}^3$ is known to be achieved by the FCC lattice, while the thinnest covering is conjectured to be given by the BCC lattice. This implies that as $\\lambda$ is varied, for $0< \\lambda < \\sqrt{5/3} -1$ we expect there to be an FCC - BCC transition.\n\nDescribe the behavior of optimal arrangements of soft balls as $\\lambda$ varies.",
   "public_statement": "Phase transitions\n\nThe best packing of hard spheres in $\\mathbb{R}^3$ is known to be achieved by the FCC lattice, while the thinnest covering is conjectured to be given by the BCC lattice. This implies that as $\\lambda$ is varied, for $0< \\lambda < \\sqrt{5/3} -1$ we expect there to be an FCC - BCC transition.\n\nDescribe the behavior of optimal arrangements of soft balls as $\\lambda$ varies.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 105,
   "attempt": 1
  },
  "AIM-GEOMETRY-0107": {
   "statement_status": "corrected_verified",
   "original_statement": "Translative Packings\n\nWhat is the lowers dimension for which the densest translative packing of a convex body is denser than the densest lattice packing?",
   "clean_statement": "What is the **lowest** dimension $d$ for which there is a $d$-dimensional convex body $K$ whose translative packing density is strictly greater than its lattice packing density, i.e.\n\n\\[\n\\delta_T(K)>\\delta_L(K)?\n\\]",
   "public_statement": "What is the **lowest** dimension $d$ for which there is a $d$-dimensional convex body $K$ whose translative packing density is strictly greater than its lattice packing density, i.e.\n\n\\[\n\\delta_T(K)>\\delta_L(K)?\n\\]",
   "evidence": "The word “lowers” is visibly a typographical error. I use the following recovered statement, with that correction made explicitly and no other change in meaning: > **Recovered problem.** What is the **lowest** dimension $d$ for which there is a $d$-dimensional convex body $K$ whose translative packing density is strictly greater than its lattice packing density, i.e. > > \\[ > \\delta_T(K)>\\delta_L(K)? > \\]",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 106,
   "attempt": 1
  },
  "AIM-GEOMETRY-0108": {
   "statement_status": "exact",
   "original_statement": "Lattice Packings\n\nIt is know that the problem of finding the densest lattice packing of spheres in a fixed dimension is a solvable by a finite algorithm (Voronoi, Minkowski).\n\nCan the lattice soft ball packing problem be solved by a finite algorithm?\nAmongst lattice packings of unit balls, which maximizes the density with respect to soft balls?",
   "clean_statement": "Lattice Packings\n\nIt is know that the problem of finding the densest lattice packing of spheres in a fixed dimension is a solvable by a finite algorithm (Voronoi, Minkowski).\n\nCan the lattice soft ball packing problem be solved by a finite algorithm?\nAmongst lattice packings of unit balls, which maximizes the density with respect to soft balls?",
   "public_statement": "Lattice Packings\n\nIt is know that the problem of finding the densest lattice packing of spheres in a fixed dimension is a solvable by a finite algorithm (Voronoi, Minkowski).\n\nCan the lattice soft ball packing problem be solved by a finite algorithm?\nAmongst lattice packings of unit balls, which maximizes the density with respect to soft balls?",
   "evidence": "The exact canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 107,
   "attempt": 1
  },
  "AIM-GEOMETRY-0109": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Random packings\n\nIt is know from simulations and experiments that a \"random\" packing of spheres that cannot be \"locally improved\" achieves a density of 64%, well short of the maximal density of 74...%.\n\nWhat is a reasonable definition of a random jammed sphere packing? Can the experimental density of 64% in $\\mathbb{R}^3$ be justified? How is density related to contact number(is it)?",
   "clean_statement": null,
   "public_statement": "Random packings\n\nIt is know from simulations and experiments that a \"random\" packing of spheres that cannot be \"locally improved\" achieves a density of 64%, well short of the maximal density of 74...%.\n\nWhat is a reasonable definition of a random jammed sphere packing? Can the experimental density of 64% in $\\mathbb{R}^3$ be justified? How is density related to contact number(is it)?",
   "evidence": "The exact canonical record is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 108,
   "attempt": 1
  },
  "AIM-GEOMETRY-0110": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Aperiodic Jammed Packings\n\nFind aperiodic jammed packings in $\\mathbb{R}^2$. In $\\mathbb{R}^3,$ find a saturated jammed maximal packing that is aperiodic in all directions.",
   "clean_statement": null,
   "public_statement": "Aperiodic Jammed Packings\n\nFind aperiodic jammed packings in $\\mathbb{R}^2$. In $\\mathbb{R}^3,$ find a saturated jammed maximal packing that is aperiodic in all directions.",
   "evidence": "The AIM webpage timed out during this run, so this wording was verified from the exact repository record and nearby workshop records. The neighboring problem concerns jammed sphere packings, so the natural recovered object here is a packing of **congruent unit disks/balls**. That is an explicit reconstruction; the record itself does not name the packed body or define “jammed.”",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 109,
   "attempt": 1
  },
  "AIM-GEOMETRY-0111": {
   "statement_status": "exact",
   "original_statement": "Percolation\n\nMaximal density packings have connected interstices in dimensions greater than 2. Is there a critical $\\lambda$ such that optimal soft packings have path connected interstices? What about the existence of an infinite component?",
   "clean_statement": "Percolation\n\nMaximal density packings have connected interstices in dimensions greater than 2. Is there a critical $\\lambda$ such that optimal soft packings have path connected interstices? What about the existence of an infinite component?",
   "public_statement": "Percolation\n\nMaximal density packings have connected interstices in dimensions greater than 2. Is there a critical $\\lambda$ such that optimal soft packings have path connected interstices? What about the existence of an infinite component?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 110,
   "attempt": 1
  },
  "AIM-GEOMETRY-0112": {
   "statement_status": "exact",
   "original_statement": "Stability\n\nDo results for soft packings stabilize to results for hard packings as $\\lambda \\rightarrow 0$?",
   "clean_statement": "Stability\n\nDo results for soft packings stabilize to results for hard packings as $\\lambda \\rightarrow 0$?",
   "public_statement": "Stability\n\nDo results for soft packings stabilize to results for hard packings as $\\lambda \\rightarrow 0$?",
   "evidence": "The exact canonical record is problem 1.05, “Stability,” from the AIM workshop *Soft Packings, Nested Clusters, and Condensed Matter*. Its repository provenance is **aim-geometry-notes.json**, zero-based source index 111:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 111,
   "attempt": 1
  },
  "AIM-GEOMETRY-0113": {
   "statement_status": "exact",
   "original_statement": "Spherical and Hyperbolic Soft Packings\n\nThe methods of Rogers for hard spheres can be extended to curved space. Can the soft Rogers bound also be extended?",
   "clean_statement": "Spherical and Hyperbolic Soft Packings\n\nThe methods of Rogers for hard spheres can be extended to curved space. Can the soft Rogers bound also be extended?",
   "public_statement": "Spherical and Hyperbolic Soft Packings\n\nThe methods of Rogers for hard spheres can be extended to curved space. Can the soft Rogers bound also be extended?",
   "evidence": "The canonical AIM record reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 112,
   "attempt": 1
  },
  "AIM-GEOMETRY-0114": {
   "statement_status": "exact",
   "original_statement": "Potentials\n\nThere are other potentials that could reasonably describe a \"soft packing.\" Can we fit them into this framework?",
   "clean_statement": "Potentials\n\nThere are other potentials that could reasonably describe a \"soft packing.\" Can we fit them into this framework?",
   "public_statement": "Potentials\n\nThere are other potentials that could reasonably describe a \"soft packing.\" Can we fit them into this framework?",
   "evidence": "The exact canonical problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 113,
   "attempt": 1
  },
  "AIM-GEOMETRY-0115": {
   "statement_status": "exact",
   "original_statement": "Prime Clusters\n\nWhich clusters in complex crystals, periodic or aperiodic, are useful for understanding their growth? Which are useful for describing structure?",
   "clean_statement": "Prime Clusters\n\nWhich clusters in complex crystals, periodic or aperiodic, are useful for understanding their growth? Which are useful for describing structure?",
   "public_statement": "Prime Clusters\n\nWhich clusters in complex crystals, periodic or aperiodic, are useful for understanding their growth? Which are useful for describing structure?",
   "evidence": "The canonical record is problem 2.1, “Prime Clusters,” from the AIM workshop list *Soft packings, nested clusters, and condensed matter*. Its exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 114,
   "attempt": 1
  },
  "AIM-GEOMETRY-0116": {
   "statement_status": "exact",
   "original_statement": "Crystal Growth From Clusters\n\nCan connections be drawn between cluster structures that appear during complex crystal growth, and finite clusters that are optimal under other conditions? Are these finite clusters energy minimizers? Do they instead (or in addition) represent especially dense configurations under some packing constraints? Drawing these connections would help us build physical intuition for cluster growth mechanisms.",
   "clean_statement": "Crystal Growth From Clusters\n\nCan connections be drawn between cluster structures that appear during complex crystal growth, and finite clusters that are optimal under other conditions? Are these finite clusters energy minimizers? Do they instead (or in addition) represent especially dense configurations under some packing constraints? Drawing these connections would help us build physical intuition for cluster growth mechanisms.",
   "public_statement": "Crystal Growth From Clusters\n\nCan connections be drawn between cluster structures that appear during complex crystal growth, and finite clusters that are optimal under other conditions? Are these finite clusters energy minimizers? Do they instead (or in addition) represent especially dense configurations under some packing constraints? Drawing these connections would help us build physical intuition for cluster growth mechanisms.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 115,
   "attempt": 1
  },
  "AIM-GEOMETRY-0117": {
   "statement_status": "exact",
   "original_statement": "Frustration\n\nWhat, precisely, is geometric frustration? More specifically, how does it manifest itself in the context of nested cluster growth models of complex crystals? While the concept makes intuitive sense, it has evaded attempts at formalization.",
   "clean_statement": "Frustration\n\nWhat, precisely, is geometric frustration? More specifically, how does it manifest itself in the context of nested cluster growth models of complex crystals? While the concept makes intuitive sense, it has evaded attempts at formalization.",
   "public_statement": "Frustration\n\nWhat, precisely, is geometric frustration? More specifically, how does it manifest itself in the context of nested cluster growth models of complex crystals? While the concept makes intuitive sense, it has evaded attempts at formalization.",
   "evidence": "The canonical AIM record reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 116,
   "attempt": 1
  },
  "AIM-GEOMETRY-0118": {
   "statement_status": "exact",
   "original_statement": "Exotic Order\n\nEngineering structures by carefully designing the interactions of the particles making up the system, is seldom straightforward and often requires trial and error, as unintended crystal structures might still be favored. It is not clear which clusters frustrate the formation of simple crystals and favor exotic order. This mechanism for stabilization and growth of exotic order has recently been explored with some success in a lattice gas model, and we will work to explore the possibility of this scenario in off-lattice models of real materials.",
   "clean_statement": "Exotic Order\n\nEngineering structures by carefully designing the interactions of the particles making up the system, is seldom straightforward and often requires trial and error, as unintended crystal structures might still be favored. It is not clear which clusters frustrate the formation of simple crystals and favor exotic order. This mechanism for stabilization and growth of exotic order has recently been explored with some success in a lattice gas model, and we will work to explore the possibility of this scenario in off-lattice models of real materials.",
   "public_statement": "Exotic Order\n\nEngineering structures by carefully designing the interactions of the particles making up the system, is seldom straightforward and often requires trial and error, as unintended crystal structures might still be favored. It is not clear which clusters frustrate the formation of simple crystals and favor exotic order. This mechanism for stabilization and growth of exotic order has recently been explored with some success in a lattice gas model, and we will work to explore the possibility of this scenario in off-lattice models of real materials.",
   "evidence": "The canonical record is problem 2.4, “Exotic Order,” in the Clusters section of the September 2016 AIM problem list *Soft packings, nested clusters, and condensed matter*. The exact statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 117,
   "attempt": 1
  },
  "AIM-GEOMETRY-0119": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Building skeletal complexes\n\nThe notion of a skeletal complex can be naturally extended to Delone sets.\n\nStudy skeletal complexes based on Delone sets.",
   "clean_statement": null,
   "public_statement": "Building skeletal complexes\n\nThe notion of a skeletal complex can be naturally extended to Delone sets.\n\nStudy skeletal complexes based on Delone sets.",
   "evidence": "What the source does **not** specify is how edges and polygonal faces are to be selected from a bare Delone set. The assertion that the notion “can be naturally extended” is therefore a program, not a unique construction. I analyze two explicit reconstructions:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 118,
   "attempt": 1
  },
  "AIM-GEOMETRY-0120": {
   "statement_status": "exact",
   "original_statement": "For $d \\ge 4$, given a Delone set $(R,r)$ for which all $2R$- clusters are pairwise congruent, find upper bound on the order of the symmetry group of the clusters which does not depend on $R/r$.",
   "clean_statement": "For $d \\ge 4$, given a Delone set $(R,r)$ for which all $2R$- clusters are pairwise congruent, find upper bound on the order of the symmetry group of the clusters which does not depend on $R/r$.",
   "public_statement": "For $d \\ge 4$, given a Delone set $(R,r)$ for which all $2R$- clusters are pairwise congruent, find upper bound on the order of the symmetry group of the clusters which does not depend on $R/r$.",
   "evidence": "The canonical AIM record (Geometry, source index 119) says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 119,
   "attempt": 1
  },
  "AIM-GEOMETRY-0121": {
   "statement_status": "exact",
   "original_statement": "In 3 dimensions, it is known that if all $10R$ clusters are equivalent, the Delone set is regular.\n\nCan this result be improved for $\\rho < 10R$, e.g. $\\rho = 4R$? Is it possible to show similar results for full dimensional $r$-sets with conditions weaker than the $R$ condition?",
   "clean_statement": "In 3 dimensions, it is known that if all $10R$ clusters are equivalent, the Delone set is regular.\n\nCan this result be improved for $\\rho < 10R$, e.g. $\\rho = 4R$? Is it possible to show similar results for full dimensional $r$-sets with conditions weaker than the $R$ condition?",
   "public_statement": "In 3 dimensions, it is known that if all $10R$ clusters are equivalent, the Delone set is regular.\n\nCan this result be improved for $\\rho < 10R$, e.g. $\\rho = 4R$? Is it possible to show similar results for full dimensional $r$-sets with conditions weaker than the $R$ condition?",
   "evidence": "The canonical record is problem 3.3 in the AIM list *Soft packings, nested clusters, and condensed matter*, section “Delone Sets,” attributed on the live source page to Nikolay Dolbilin. The exact canonical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 120,
   "attempt": 1
  },
  "AIM-GEOMETRY-0122": {
   "statement_status": "exact",
   "original_statement": "Cycles in skeletal complexes\n\nWhat are the applications and physical significance of the cycles in skeletal polygonal complexes?",
   "clean_statement": "Cycles in skeletal complexes\n\nWhat are the applications and physical significance of the cycles in skeletal polygonal complexes?",
   "public_statement": "Cycles in skeletal complexes\n\nWhat are the applications and physical significance of the cycles in skeletal polygonal complexes?",
   "evidence": "The exact canonical record is AIM-GEOMETRY-0122, source file *aim-geometry-notes.json*, zero-based index 121, from the AIM workshop “Soft packings, nested clusters, and condensed matter,” section “Delone Sets,” problem 3.4:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 121,
   "attempt": 1
  },
  "AIM-GEOMETRY-0123": {
   "statement_status": "exact",
   "original_statement": "What is the precise relation between the complete hyperkahler metrics on the moduli spaces $\\mathcal{M^{\\ast }}$ and those on the full (wild) Hitchin moduli space $\\mathcal{M}$?\nCan one view the simpler metric on $\\mathcal{M^{\\ast }}$ as an approximation?",
   "clean_statement": "What is the precise relation between the complete hyperkahler metrics on the moduli spaces $\\mathcal{M^{\\ast }}$ and those on the full (wild) Hitchin moduli space $\\mathcal{M}$?\nCan one view the simpler metric on $\\mathcal{M^{\\ast }}$ as an approximation?",
   "public_statement": "What is the precise relation between the complete hyperkahler metrics on the moduli spaces $\\mathcal{M^{\\ast }}$ and those on the full (wild) Hitchin moduli space $\\mathcal{M}$?\nCan one view the simpler metric on $\\mathcal{M^{\\ast }}$ as an approximation?",
   "evidence": "The canonical AIM record (workshop *Spectral data for Higgs bundles*, problem 1.02) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 122,
   "attempt": 1
  },
  "AIM-GEOMETRY-0124": {
   "statement_status": "exact",
   "original_statement": "Prove: [Hitchin's projectively flat connection over $\\mathcal{M_{g}}$] $\\cong$ [Faltings' projectively flat connection over $\\mathcal{M_{g}}$]",
   "clean_statement": "Prove: [Hitchin's projectively flat connection over $\\mathcal{M_{g}}$] $\\cong$ [Faltings' projectively flat connection over $\\mathcal{M_{g}}$]",
   "public_statement": "Prove: [Hitchin's projectively flat connection over $\\mathcal{M_{g}}$] $\\cong$ [Faltings' projectively flat connection over $\\mathcal{M_{g}}$]",
   "evidence": "The canonical corpus record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 123,
   "attempt": 1
  },
  "AIM-GEOMETRY-0125": {
   "statement_status": "exact",
   "original_statement": "Extend Laszlo's proof of the relation between Hitchin's connection and TUY connection, to the case with punctures. Laszlo looks at the case with no marked points and group $SU_n$, genus at least 3.",
   "clean_statement": "Extend Laszlo's proof of the relation between Hitchin's connection and TUY connection, to the case with punctures. Laszlo looks at the case with no marked points and group $SU_n$, genus at least 3.",
   "public_statement": "Extend Laszlo's proof of the relation between Hitchin's connection and TUY connection, to the case with punctures. Laszlo looks at the case with no marked points and group $SU_n$, genus at least 3.",
   "evidence": "The exact canonical record, preserved verbatim, is:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 124,
   "attempt": 1
  },
  "AIM-GEOMETRY-0126": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Let $X$ be a surface and fix topological data. Then:\n\\[X\\rightarrow Bun_{G}\\left(X\\right)\\rightarrow H^{0}\\left( Bun_{G},\\alpha\\right)\\]\n\n\\[\\underline{X} \\rightarrow Bun_{G}\\left(X\\right)\\rightarrow U \\,\\,\\,\\,\\,\\,\\,\\, (\\star)\\]\n\nIs $(\\star)$ naturally projectively flat?",
   "clean_statement": "Let $X$ be a invalid_statement and fix topological data. Then:\n\\[X\\rightarrow Bun_{G}\\left(X\\right)\\rightarrow H^{0}\\left( Bun_{G},\\alpha\\right)\\]\n\n\\[\\underline{X} \\rightarrow Bun_{G}\\left(X\\right)\\rightarrow U \\,\\,\\,\\,\\,\\,\\,\\, (\\star)\\]\n\nIs $(\\star)$ naturally projectively flat?",
   "public_statement": "Let $X$ be a surface and fix topological data. Then:\n\\[X\\rightarrow Bun_{G}\\left(X\\right)\\rightarrow H^{0}\\left( Bun_{G},\\alpha\\right)\\]\n\n\\[\\underline{X} \\rightarrow Bun_{G}\\left(X\\right)\\rightarrow U \\,\\,\\,\\,\\,\\,\\,\\, (\\star)\\]\n\nIs $(\\star)$ naturally projectively flat?",
   "evidence": "There are three plausible readings, none source-verified. Accordingly, this attempt assigns the record status **`invalid_statement`**. It does not silently choose among these readings. The remainder records the present answer to the most plausible reading and proves a recovery-independent projective-flatness obstruction.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-geometry-notes.json",
   "source_index": 125,
   "attempt": 1
  },
  "AIM-GEOMETRY-0127": {
   "statement_status": "exact",
   "original_statement": "Let $X$ be a complex symplectic algebraic manifold and $G$ a complex reductive group with $G\\curvearrowright X$ in a Hamiltonian fashion, $\\mu:X\\rightarrow \\mathfrak{g}^{\\ast}$ a moment map and $Y=X//_{\\lambda}G=\\mu^{-1}(\\lambda)/G$.\n\nFind examples where:\n\ni) $Y$ is an algebraic completely integrable Hamiltonian system.\n\nii) $Y\\ncong \\mathcal{M^{\\ast}}(\\Sigma)$ (some moduli of tame/wild Higgs on decorated surface)",
   "clean_statement": "Let $X$ be a complex symplectic algebraic manifold and $G$ a complex reductive group with $G\\curvearrowright X$ in a Hamiltonian fashion, $\\mu:X\\rightarrow \\mathfrak{g}^{\\ast}$ a moment map and $Y=X//_{\\lambda}G=\\mu^{-1}(\\lambda)/G$.\n\nFind examples where:\n\ni) $Y$ is an algebraic completely integrable Hamiltonian system.\n\nii) $Y\\ncong \\mathcal{M^{\\ast}}(\\Sigma)$ (some moduli of tame/wild Higgs on decorated surface)",
   "public_statement": "Let $X$ be a complex symplectic algebraic manifold and $G$ a complex reductive group with $G\\curvearrowright X$ in a Hamiltonian fashion, $\\mu:X\\rightarrow \\mathfrak{g}^{\\ast}$ a moment map and $Y=X//_{\\lambda}G=\\mu^{-1}(\\lambda)/G$.\n\nFind examples where:\n\ni) $Y$ is an algebraic completely integrable Hamiltonian system.\n\nii) $Y\\ncong \\mathcal{M^{\\ast}}(\\Sigma)$ (some moduli of tame/wild Higgs on decorated surface)",
   "evidence": "The canonical AIM record is problem 1.1 in the workshop *Spectral data for Higgs bundles*. Its exact mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 126,
   "attempt": 1
  },
  "AIM-GEOMETRY-0128": {
   "statement_status": "exact",
   "original_statement": "Extend Hitchin section $B\\rightarrow\\mathcal{M}_{H}$ to a map $T^{\\ast}B\\rightarrow\\mathcal{M}_{H}$.\n\nDoes there exist a formal version?",
   "clean_statement": "Extend Hitchin section $B\\rightarrow\\mathcal{M}_{H}$ to a map $T^{\\ast}B\\rightarrow\\mathcal{M}_{H}$.\n\nDoes there exist a formal version?",
   "public_statement": "Extend Hitchin section $B\\rightarrow\\mathcal{M}_{H}$ to a map $T^{\\ast}B\\rightarrow\\mathcal{M}_{H}$.\n\nDoes there exist a formal version?",
   "evidence": "The AIM source page was refetched twice on 8 August 2026, but the live endpoint timed out. The exact canonical corpus record was therefore preserved verbatim; nearby records do not define the missing category or group. No emendation of the source wording is being made here.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 127,
   "attempt": 1
  },
  "AIM-GEOMETRY-0129": {
   "statement_status": "exact",
   "original_statement": "i) What is the Krull dimension of $\\mathcal{O}(T^{\\ast}\\overline{\\mathcal{M}}_{g,n})$?\n\nii) Is the affinization map an algebraic completely integrable Hamiltonian system?",
   "clean_statement": "i) What is the Krull dimension of $\\mathcal{O}(T^{\\ast}\\overline{\\mathcal{M}}_{g,n})$?\n\nii) Is the affinization map an algebraic completely integrable Hamiltonian system?",
   "public_statement": "i) What is the Krull dimension of $\\mathcal{O}(T^{\\ast}\\overline{\\mathcal{M}}_{g,n})$?\n\nii) Is the affinization map an algebraic completely integrable Hamiltonian system?",
   "evidence": "The canonical AIM record (source file `aim-geometry-notes.json`, zero-based index 128) reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 128,
   "attempt": 1
  },
  "AIM-GEOMETRY-0130": {
   "statement_status": "exact",
   "original_statement": "What happens on singular curves?\ne.g.\n\ni)$\\mathcal{O}(T^{\\ast}Bun_{G}(X))$\n\nii) relate to meromorphic Higgs on $\\tilde{X}$\n\niii) spectral data\n\niv) extending moduli space $/\\mathcal{M}_{g,n}$ to $/\\mathcal{\\overline{M}}_{g,n}$",
   "clean_statement": "What happens on singular curves?\ne.g.\n\ni)$\\mathcal{O}(T^{\\ast}Bun_{G}(X))$\n\nii) relate to meromorphic Higgs on $\\tilde{X}$\n\niii) spectral data\n\niv) extending moduli space $/\\mathcal{M}_{g,n}$ to $/\\mathcal{\\overline{M}}_{g,n}$",
   "public_statement": "What happens on singular curves?\ne.g.\n\ni)$\\mathcal{O}(T^{\\ast}Bun_{G}(X))$\n\nii) relate to meromorphic Higgs on $\\tilde{X}$\n\niii) spectral data\n\niv) extending moduli space $/\\mathcal{M}_{g,n}$ to $/\\mathcal{\\overline{M}}_{g,n}$",
   "evidence": "The canonical AIM record is problem 1.16 in the workshop *Spectral data for Higgs bundles*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 129,
   "attempt": 1
  },
  "AIM-GEOMETRY-0131": {
   "statement_status": "exact",
   "original_statement": "Let $\\Sigma$ a surface and $H$ a handlebody. Then:\n\\[\\mathcal{L}\\subset \\mathcal{M}_{B} (holomorphic) \\Rightarrow\\mathcal{L}\\subset \\mathcal{M}_{Dol} (real \\, Lagrangian)\\]\nfor $X^{3},\\partial(X)=\\Sigma$.\n\nQuestions:\n\ni) $H(\\mathcal{L}_X)$ is a subvariety of $B$ of half dimension. What is the intersection of $\\mathcal{L}$ and fibres of the Hitchin map, i.e. the connected components of $H^{-1}(b)\\cap \\mathcal{L}_X$? [do genus 1 at least]\n\nii) $RFM(\\mathcal{L})=?$",
   "clean_statement": "Let $\\Sigma$ a surface and $H$ a handlebody. Then:\n\\[\\mathcal{L}\\subset \\mathcal{M}_{B} (holomorphic) \\Rightarrow\\mathcal{L}\\subset \\mathcal{M}_{Dol} (real \\, Lagrangian)\\]\nfor $X^{3},\\partial(X)=\\Sigma$.\n\nQuestions:\n\ni) $H(\\mathcal{L}_X)$ is a subvariety of $B$ of half dimension. What is the intersection of $\\mathcal{L}$ and fibres of the Hitchin map, i.e. the connected components of $H^{-1}(b)\\cap \\mathcal{L}_X$? [do genus 1 at least]\n\nii) $RFM(\\mathcal{L})=?$",
   "public_statement": "Let $\\Sigma$ a surface and $H$ a handlebody. Then:\n\\[\\mathcal{L}\\subset \\mathcal{M}_{B} (holomorphic) \\Rightarrow\\mathcal{L}\\subset \\mathcal{M}_{Dol} (real \\, Lagrangian)\\]\nfor $X^{3},\\partial(X)=\\Sigma$.\n\nQuestions:\n\ni) $H(\\mathcal{L}_X)$ is a subvariety of $B$ of half dimension. What is the intersection of $\\mathcal{L}$ and fibres of the Hitchin map, i.e. the connected components of $H^{-1}(b)\\cap \\mathcal{L}_X$? [do genus 1 at least]\n\nii) $RFM(\\mathcal{L})=?$",
   "evidence": "The canonical AIM record (source file `aim-geometry-notes.json`, zero-based index 130) says verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 130,
   "attempt": 1
  },
  "AIM-GEOMETRY-0132": {
   "statement_status": "exact",
   "original_statement": "Let $\\Sigma, H$ as above.\n\ni) $\\mathcal{O}(\\mathcal{M}_{B})\\xrightarrow{def. quant.}\\mathcal{A}_{q}$\n\nii) $\\mathcal{L}\\subset\\mathcal{M}_{B}\\Rightarrow \\mathcal{I_{L}}\\subset\\mathcal{O}\n(\\mathcal{M}_{B})\\xrightarrow{def. quant.}\\mathcal{I_{L}^{q}}<\\mathcal{A}_{q} $ (left ideal)\n\niii) $\\mathcal{I_{L}^{q}}$ acts on $H^{0}(\\mathbb{L}^{k},Bun_{G}(\\Sigma))$ and this algebra annihilates $v_{H}\\in H^{0}(\\mathbb{L}^{k},Bun_{G}(\\Sigma))$",
   "clean_statement": "Let $\\Sigma, H$ as above.\n\ni) $\\mathcal{O}(\\mathcal{M}_{B})\\xrightarrow{def. quant.}\\mathcal{A}_{q}$\n\nii) $\\mathcal{L}\\subset\\mathcal{M}_{B}\\Rightarrow \\mathcal{I_{L}}\\subset\\mathcal{O}\n(\\mathcal{M}_{B})\\xrightarrow{def. quant.}\\mathcal{I_{L}^{q}}<\\mathcal{A}_{q} $ (left ideal)\n\niii) $\\mathcal{I_{L}^{q}}$ acts on $H^{0}(\\mathbb{L}^{k},Bun_{G}(\\Sigma))$ and this algebra annihilates $v_{H}\\in H^{0}(\\mathbb{L}^{k},Bun_{G}(\\Sigma))$",
   "public_statement": "Let $\\Sigma, H$ as above.\n\ni) $\\mathcal{O}(\\mathcal{M}_{B})\\xrightarrow{def. quant.}\\mathcal{A}_{q}$\n\nii) $\\mathcal{L}\\subset\\mathcal{M}_{B}\\Rightarrow \\mathcal{I_{L}}\\subset\\mathcal{O}\n(\\mathcal{M}_{B})\\xrightarrow{def. quant.}\\mathcal{I_{L}^{q}}<\\mathcal{A}_{q} $ (left ideal)\n\niii) $\\mathcal{I_{L}^{q}}$ acts on $H^{0}(\\mathbb{L}^{k},Bun_{G}(\\Sigma))$ and this algebra annihilates $v_{H}\\in H^{0}(\\mathbb{L}^{k},Bun_{G}(\\Sigma))$",
   "evidence": "The exact extracted problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 131,
   "attempt": 1
  },
  "AIM-GEOMETRY-0133": {
   "statement_status": "exact",
   "original_statement": "Study confluence of simple poles, e.g. hyperkahler metric on limit as limit of hyperkahler metrics.",
   "clean_statement": "Study confluence of simple poles, e.g. hyperkahler metric on limit as limit of hyperkahler metrics.",
   "public_statement": "Study confluence of simple poles, e.g. hyperkahler metric on limit as limit of hyperkahler metrics.",
   "evidence": "The canonical source record says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 132,
   "attempt": 1
  },
  "AIM-GEOMETRY-0134": {
   "statement_status": "exact",
   "original_statement": "Describe the Hitchin-Witten connection on the space of holomorphic sections on the moduli space of Higgs bundles.",
   "clean_statement": "Describe the Hitchin-Witten connection on the space of holomorphic sections on the moduli space of Higgs bundles.",
   "public_statement": "Describe the Hitchin-Witten connection on the space of holomorphic sections on the moduli space of Higgs bundles.",
   "evidence": "The exact canonical record (source file aim-geometry-notes.json, zero-based index 133) is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 133,
   "attempt": 1
  },
  "AIM-GEOMETRY-0135": {
   "statement_status": "exact",
   "original_statement": "Find the vector $v_{H}\\in Z(\\Sigma)$ associated to a handlebody $H$ bounding $\\Sigma$.",
   "clean_statement": "Find the vector $v_{H}\\in Z(\\Sigma)$ associated to a handlebody $H$ bounding $\\Sigma$.",
   "public_statement": "Find the vector $v_{H}\\in Z(\\Sigma)$ associated to a handlebody $H$ bounding $\\Sigma$.",
   "evidence": "The raw canonical JSON serialization is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 134,
   "attempt": 1
  },
  "AIM-GEOMETRY-0136": {
   "statement_status": "exact",
   "original_statement": "Describe the relation between the WKB, Eynard-Orantin, Dumitrescu-Mulase topological recursion and the (wild) non-abelian Hodge correspondence, at least in one example.",
   "clean_statement": "Describe the relation between the WKB, Eynard-Orantin, Dumitrescu-Mulase topological recursion and the (wild) non-abelian Hodge correspondence, at least in one example.",
   "public_statement": "Describe the relation between the WKB, Eynard-Orantin, Dumitrescu-Mulase topological recursion and the (wild) non-abelian Hodge correspondence, at least in one example.",
   "evidence": "The record contains no OCR corruption, but it compresses several genuinely different correspondences into one sentence. The original AIM page, <http://aimpl.org/spectralhiggs/1/>, did not return readable content during this run. I therefore preserve the sentence and make the following necessary distinctions.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 135,
   "attempt": 1
  },
  "AIM-GEOMETRY-0137": {
   "statement_status": "exact",
   "original_statement": "Determine the deformation families for branes in the Hitchin system (complex Lagrangian or hyper-holomorphic?).",
   "clean_statement": "Determine the deformation families for branes in the Hitchin system (complex Lagrangian or hyper-holomorphic?).",
   "public_statement": "Determine the deformation families for branes in the Hitchin system (complex Lagrangian or hyper-holomorphic?).",
   "evidence": "The exact canonical AIM record (source file aim-geometry-notes.json, zero-based index 136) is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 136,
   "attempt": 1
  },
  "AIM-GEOMETRY-0138": {
   "statement_status": "exact",
   "original_statement": "Langlands duality for meromorphic Higgs bundles\n\nThe existence of the hyperkahler metrics on the wild Hitchin spaces means the moduli spaces of meromorphic connection on curves (and the corresponding wild character varieties) have natural special Lagrangian torus fibrations, by hyperkahler rotation of the Hitchin fibration (shown to be proper in this generality by Nitsure).\n\nStudy the SYZ mirror symmetry proposal in this context. For example suppose we take the dual holomorphic Lagrangian fibration of the smooth fibres of the meromorphic Higgs bundle moduli space. Is the result another Hitchin fibration? What is the involution on the set of data (group, irregular curve, residue data) that corresponds to this T-duality?\n\nFor example for holomorphic Higgs bundles it is known one should just replace the group by the Langland dual group, so we are asking for an involution generalizing this.",
   "clean_statement": "Langlands duality for meromorphic Higgs bundles\n\nThe existence of the hyperkahler metrics on the wild Hitchin spaces means the moduli spaces of meromorphic connection on curves (and the corresponding wild character varieties) have natural special Lagrangian torus fibrations, by hyperkahler rotation of the Hitchin fibration (shown to be proper in this generality by Nitsure).\n\nStudy the SYZ mirror symmetry proposal in this context. For example suppose we take the dual holomorphic Lagrangian fibration of the smooth fibres of the meromorphic Higgs bundle moduli space. Is the result another Hitchin fibration? What is the involution on the set of data (group, irregular curve, residue data) that corresponds to this T-duality?\n\nFor example for holomorphic Higgs bundles it is known one should just replace the group by the Langland dual group, so we are asking for an involution generalizing this.",
   "public_statement": "Langlands duality for meromorphic Higgs bundles\n\nThe existence of the hyperkahler metrics on the wild Hitchin spaces means the moduli spaces of meromorphic connection on curves (and the corresponding wild character varieties) have natural special Lagrangian torus fibrations, by hyperkahler rotation of the Hitchin fibration (shown to be proper in this generality by Nitsure).\n\nStudy the SYZ mirror symmetry proposal in this context. For example suppose we take the dual holomorphic Lagrangian fibration of the smooth fibres of the meromorphic Higgs bundle moduli space. Is the result another Hitchin fibration? What is the involution on the set of data (group, irregular curve, residue data) that corresponds to this T-duality?\n\nFor example for holomorphic Higgs bundles it is known one should just replace the group by the Langland dual group, so we are asking for an involution generalizing this.",
   "evidence": "The canonical record is AIM Problem List 1.32, from the workshop “Spectral data for Higgs bundles.” Its problem field is reproduced verbatim, including “hyperkahler”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 137,
   "attempt": 1
  },
  "AIM-GEOMETRY-0139": {
   "statement_status": "exact",
   "original_statement": "Ways to construct hyper-holomorphic sheaves on branes, their mirrors and spectral data.",
   "clean_statement": "Ways to construct hyper-holomorphic sheaves on branes, their mirrors and spectral data.",
   "public_statement": "Ways to construct hyper-holomorphic sheaves on branes, their mirrors and spectral data.",
   "evidence": "The record has no remarks or literature field. The archived source URL recorded in the corpus, `http://aimpl.org/spectralhiggs/1/`, timed out when checked on 2026-08-08. The official AIM workshop page and workshop report confirm that the meeting concerned spectral data, Langlands duality, and dualities between branes, but they do not sharpen this one-sentence prompt. There is no visible OCR corruption to repair. The sentence is a research direction rather than a quantified conjecture, so it must not be reported as globally “solved.”",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 138,
   "attempt": 1
  },
  "AIM-GEOMETRY-0140": {
   "statement_status": "exact",
   "original_statement": "Construct and quantize Lagrangian in the moduli space of tame/wild parabolic Higgs bundles/quiver varieties.",
   "clean_statement": "Construct and quantize Lagrangian in the moduli space of tame/wild parabolic Higgs bundles/quiver varieties.",
   "public_statement": "Construct and quantize Lagrangian in the moduli space of tame/wild parabolic Higgs bundles/quiver varieties.",
   "evidence": "The canonical record is AIM-GEOMETRY-0140, item 1.36 in the AIM workshop list “Spectral data for Higgs bundles” (2015). Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 139,
   "attempt": 1
  },
  "AIM-GEOMETRY-0141": {
   "statement_status": "exact",
   "original_statement": "Understand the intersection of Lagrangians coming from 3-manifolds bounding the surface with fibres of the Hitchin fibration.",
   "clean_statement": "Understand the intersection of Lagrangians coming from 3-manifolds bounding the surface with fibres of the Hitchin fibration.",
   "public_statement": "Understand the intersection of Lagrangians coming from 3-manifolds bounding the surface with fibres of the Hitchin fibration.",
   "evidence": "The canonical AIM record (AIM Problem Lists, workshop *Spectral data for Higgs bundles*, Open problems 1.38) states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 140,
   "attempt": 1
  },
  "AIM-GEOMETRY-0142": {
   "statement_status": "exact",
   "original_statement": "Hilbert schemes of points on $K2$-surfaces.\n\nThere are $12$ known deformation classes of complete hyperkahler manifolds of real dimension four that arise as moduli spaces of meromorphic Higgs bundles.\nLets call them $K2$ surfaces.\nFor each of these $K2$ surfaces there is a sequence of Higgs bundle moduli space of dimension $4n$ for each $n$.\n\nIs each of these $4n$ dimensional hyperkahler manifolds diffeomorphic to (or a deformation of) the Hilbert scheme of $n$-points on the underlying $K2$ surface?",
   "clean_statement": "Hilbert schemes of points on $K2$-surfaces.\n\nThere are $12$ known deformation classes of complete hyperkahler manifolds of real dimension four that arise as moduli spaces of meromorphic Higgs bundles.\nLets call them $K2$ surfaces.\nFor each of these $K2$ surfaces there is a sequence of Higgs bundle moduli space of dimension $4n$ for each $n$.\n\nIs each of these $4n$ dimensional hyperkahler manifolds diffeomorphic to (or a deformation of) the Hilbert scheme of $n$-points on the underlying $K2$ surface?",
   "public_statement": "Hilbert schemes of points on $K2$-surfaces.\n\nThere are $12$ known deformation classes of complete hyperkahler manifolds of real dimension four that arise as moduli spaces of meromorphic Higgs bundles.\nLets call them $K2$ surfaces.\nFor each of these $K2$ surfaces there is a sequence of Higgs bundle moduli space of dimension $4n$ for each $n$.\n\nIs each of these $4n$ dimensional hyperkahler manifolds diffeomorphic to (or a deformation of) the Hilbert scheme of $n$-points on the underlying $K2$ surface?",
   "evidence": "Here “\\(K2\\) surface” is an ad hoc nickname introduced in the AIM question, not a standard surface class and not an OCR error for “K3 surface.” Neither of the two Boalch papers cited by the record uses “K2.” The proposed list contains noncompact surfaces, including \\(T^*E\\) for an elliptic curve \\(E\\); such a surface cannot be a K3 surface. Thus silently replacing “K2” by “K3” would corrupt the mathematics. Below, \\(S\\) denotes one of the two-complex-dimensional meromorphic Hitchin moduli spaces and \\(\\mathcal H_n\\) its proposed \\(2n\\)-complex-dimensional higher-rank counterpart.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 141,
   "attempt": 1
  },
  "AIM-GEOMETRY-0143": {
   "statement_status": "exact",
   "original_statement": "For $G$ other than $GL(n)$, do we have a smoothing of $\\overline{G-Hitchin}$?",
   "clean_statement": "For $G$ other than $GL(n)$, do we have a smoothing of $\\overline{G-Hitchin}$?",
   "public_statement": "For $G$ other than $GL(n)$, do we have a smoothing of $\\overline{G-Hitchin}$?",
   "evidence": "The canonical record, preserved verbatim, is:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 142,
   "attempt": 1
  },
  "AIM-GEOMETRY-0144": {
   "statement_status": "exact",
   "original_statement": "What are the algebraic/analytic automorphisms, $\\mathcal{M}_{n,d}^{dR}, \\mathcal{M}_{n,d}^{Betti}=?$",
   "clean_statement": "What are the algebraic/analytic automorphisms, $\\mathcal{M}_{n,d}^{dR}, \\mathcal{M}_{n,d}^{Betti}=?$",
   "public_statement": "What are the algebraic/analytic automorphisms, $\\mathcal{M}_{n,d}^{dR}, \\mathcal{M}_{n,d}^{Betti}=?$",
   "evidence": "The canonical AIM record, from the 2015 workshop *Spectral data for Higgs bundles*, reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 143,
   "attempt": 1
  },
  "AIM-GEOMETRY-0145": {
   "statement_status": "exact",
   "original_statement": "Does $\\mathcal{M}_{Higgs}^{ss}(SL(n,\\mathbb{C}))$ have a symplectic resolution?",
   "clean_statement": "Does $\\mathcal{M}_{Higgs}^{ss}(SL(n,\\mathbb{C}))$ have a symplectic resolution?",
   "public_statement": "Does $\\mathcal{M}_{Higgs}^{ss}(SL(n,\\mathbb{C}))$ have a symplectic resolution?",
   "evidence": "The canonical AIM record, Open Problem 1.48 from the 2015 workshop *Spectral data for Higgs bundles*, is exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 144,
   "attempt": 1
  },
  "AIM-GEOMETRY-0146": {
   "statement_status": "exact",
   "original_statement": "Compute $R=K_{\\mathbb{C}^{\\ast}}^{quant}(V^{\\otimes k}\\oplus Adj//_{U(N)})$ and describe what this ring is.",
   "clean_statement": "Compute $R=K_{\\mathbb{C}^{\\ast}}^{quant}(V^{\\otimes k}\\oplus Adj//_{U(N)})$ and describe what this ring is.",
   "public_statement": "Compute $R=K_{\\mathbb{C}^{\\ast}}^{quant}(V^{\\otimes k}\\oplus Adj//_{U(N)})$ and describe what this ring is.",
   "evidence": "The canonical AIM record says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 145,
   "attempt": 1
  },
  "AIM-GEOMETRY-0147": {
   "statement_status": "exact",
   "original_statement": "Is there a topological recursion formula for the $SL(2, \\mathbb{C})$-Verlinde's formula?",
   "clean_statement": "Is there a topological recursion formula for the $SL(2, \\mathbb{C})$-Verlinde's formula?",
   "public_statement": "Is there a topological recursion formula for the $SL(2, \\mathbb{C})$-Verlinde's formula?",
   "evidence": "This is problem 1.5 in the “Open problems” section of the 2015 AIM workshop *Spectral data for Higgs bundles*. The canonical record has no remarks or literature field. Its source URL is <http://aimpl.org/spectralhiggs/1/>. That problem-list URL timed out during this run; the exact repository record, nearby records, the official workshop page, and the workshop participant document were inspected. There is no visible OCR error in the formula. Nearby questions, however, concern Higgs moduli, quantum \\(K\\)-theory, and TQFT, so the group notation creates a substantive ambiguity.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 146,
   "attempt": 1
  },
  "AIM-GEOMETRY-0148": {
   "statement_status": "exact",
   "original_statement": "Given a measured foliation can we construct the Hitchin system?",
   "clean_statement": "Given a measured foliation can we construct the Hitchin system?",
   "public_statement": "Given a measured foliation can we construct the Hitchin system?",
   "evidence": "The canonical AIM record (workshop *Spectral data for Higgs bundles*, open problem 1.52) reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 147,
   "attempt": 1
  },
  "AIM-GEOMETRY-0149": {
   "statement_status": "exact",
   "original_statement": "Torelli theorems on $\\mathcal{M}_{dR}$ with parabolic structure.",
   "clean_statement": "Torelli theorems on $\\mathcal{M}_{dR}$ with parabolic structure.",
   "public_statement": "Torelli theorems on $\\mathcal{M}_{dR}$ with parabolic structure.",
   "evidence": "The archived AIM page from 9 September 2016 contains exactly the same one-line text, with no attribution, status, definitions, or hypotheses. This is not an OCR error. It is a research program rather than a proposition with a truth value: it does not specify the curve, group, residues, weights, determinant, stability condition, or the structure that an isomorphism must preserve.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 148,
   "attempt": 1
  },
  "AIM-GEOMETRY-0150": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "What is the mirror of the action $\\mathbb{C}^{\\ast}\\curvearrowright \\mathcal{M}_{G_{\\mathbb{C}}}$ in the Langlands dual side $\\mathcal{M}_{^{L}G_{\\mathbb{C}}}$?",
   "clean_statement": null,
   "public_statement": "What is the mirror of the action $\\mathbb{C}^{\\ast}\\curvearrowright \\mathcal{M}_{G_{\\mathbb{C}}}$ in the Langlands dual side $\\mathcal{M}_{^{L}G_{\\mathbb{C}}}$?",
   "evidence": "There is no visible OCR corruption. There is, however, genuine mathematical ambiguity. I use the following conservative reconstruction.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 149,
   "attempt": 2
  },
  "AIM-GEOMETRY-0151": {
   "statement_status": "exact",
   "original_statement": "The Lax project\n\nFor each classical integrable system there is often a \"Lax pair\", which is basically a meromorphic Higgs bundle on the trivial bundle on the Riemann sphere. Thus we can consider the corresponding wild Hitchin moduli space which is a complete hyperkahler manifold. Thus we have a map from integrable systems to complete hyperkahler manifolds.\n\nThe project is to go though the integrable systems literature and compile a list Lax pairs and the corresponding irregular curve and the resulting hyperkahler manifold.\nFurther in the cases where integrable systems have alternative Lax pairs, show that the corresponding hyperkahler manifolds are isomorphic.",
   "clean_statement": "The Lax project\n\nFor each classical integrable system there is often a \"Lax pair\", which is basically a meromorphic Higgs bundle on the trivial bundle on the Riemann sphere. Thus we can consider the corresponding wild Hitchin moduli space which is a complete hyperkahler manifold. Thus we have a map from integrable systems to complete hyperkahler manifolds.\n\nThe project is to go though the integrable systems literature and compile a list Lax pairs and the corresponding irregular curve and the resulting hyperkahler manifold.\nFurther in the cases where integrable systems have alternative Lax pairs, show that the corresponding hyperkahler manifolds are isomorphic.",
   "public_statement": "The Lax project\n\nFor each classical integrable system there is often a \"Lax pair\", which is basically a meromorphic Higgs bundle on the trivial bundle on the Riemann sphere. Thus we can consider the corresponding wild Hitchin moduli space which is a complete hyperkahler manifold. Thus we have a map from integrable systems to complete hyperkahler manifolds.\n\nThe project is to go though the integrable systems literature and compile a list Lax pairs and the corresponding irregular curve and the resulting hyperkahler manifold.\nFurther in the cases where integrable systems have alternative Lax pairs, show that the corresponding hyperkahler manifolds are isomorphic.",
   "evidence": "The canonical record is problem 1.58, “The Lax project,” from the AIM workshop *Spectral data for Higgs bundles*. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 150,
   "attempt": 2
  },
  "AIM-GEOMETRY-0152": {
   "statement_status": "exact",
   "original_statement": "Modularity conjecture\n\nAmongst the real 4d complete hyperkahler manifolds (gravitational instantons) all the known examples of type ALG and ALH are modular, for example the ALG examples are all moduli spaces of meromorphic Higgs bundles (although this has not been proved yet---the examples match precisely).\nProve that this holds in higher dimensions, i.e. define classes of complete hyperkahler manifolds (for example by their asymptotics), higherdiemsnional analogues of ALG and ALH, and show that all examples are modular (spaces of solutions of Hitchin's equations, or the Bogomolnyi equations or other reduction of the ASDYM equations)",
   "clean_statement": "Modularity conjecture\n\nAmongst the real 4d complete hyperkahler manifolds (gravitational instantons) all the known examples of type ALG and ALH are modular, for example the ALG examples are all moduli spaces of meromorphic Higgs bundles (although this has not been proved yet---the examples match precisely).\nProve that this holds in higher dimensions, i.e. define classes of complete hyperkahler manifolds (for example by their asymptotics), higherdiemsnional analogues of ALG and ALH, and show that all examples are modular (spaces of solutions of Hitchin's equations, or the Bogomolnyi equations or other reduction of the ASDYM equations)",
   "public_statement": "Modularity conjecture\n\nAmongst the real 4d complete hyperkahler manifolds (gravitational instantons) all the known examples of type ALG and ALH are modular, for example the ALG examples are all moduli spaces of meromorphic Higgs bundles (although this has not been proved yet---the examples match precisely).\nProve that this holds in higher dimensions, i.e. define classes of complete hyperkahler manifolds (for example by their asymptotics), higherdiemsnional analogues of ALG and ALH, and show that all examples are modular (spaces of solutions of Hitchin's equations, or the Bogomolnyi equations or other reduction of the ASDYM equations)",
   "evidence": "The exact canonical record is AIM-GEOMETRY-0152, problem 1.6 in the AIM workshop “Spectral data for Higgs bundles.” Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 151,
   "attempt": 1
  },
  "AIM-GEOMETRY-0153": {
   "statement_status": "exact",
   "original_statement": "Nonlinear representation theory\n\nSuppose we define an abstract notion of \"wild nonabelian Hodge structure\", consisting of a hyperkahler manifold with its various algebraic structures, such that it is a wild Hitchin moduli space for some choice of a group, an irregular curve and residue data. The choice of such a realisation of the abstract space can be viewed as a \"representation\" or \"realisation\" of the space. Study this representation theory.\n\nHere we are viewing these moduli spaces as global/nonlinear analogues of algebraic groups (as the Riemann--Hilbert problem suggests).\n\nFor example, take one simple example (such as the $D_4$ Hitchin space, the Painleve VI space, $GL_2$ with four simple poles on $P^1$), of complex dimension $2$ and find all the possible representations of this space.",
   "clean_statement": "Nonlinear representation theory\n\nSuppose we define an abstract notion of \"wild nonabelian Hodge structure\", consisting of a hyperkahler manifold with its various algebraic structures, such that it is a wild Hitchin moduli space for some choice of a group, an irregular curve and residue data. The choice of such a realisation of the abstract space can be viewed as a \"representation\" or \"realisation\" of the space. Study this representation theory.\n\nHere we are viewing these moduli spaces as global/nonlinear analogues of algebraic groups (as the Riemann--Hilbert problem suggests).\n\nFor example, take one simple example (such as the $D_4$ Hitchin space, the Painleve VI space, $GL_2$ with four simple poles on $P^1$), of complex dimension $2$ and find all the possible representations of this space.",
   "public_statement": "Nonlinear representation theory\n\nSuppose we define an abstract notion of \"wild nonabelian Hodge structure\", consisting of a hyperkahler manifold with its various algebraic structures, such that it is a wild Hitchin moduli space for some choice of a group, an irregular curve and residue data. The choice of such a realisation of the abstract space can be viewed as a \"representation\" or \"realisation\" of the space. Study this representation theory.\n\nHere we are viewing these moduli spaces as global/nonlinear analogues of algebraic groups (as the Riemann--Hilbert problem suggests).\n\nFor example, take one simple example (such as the $D_4$ Hitchin space, the Painleve VI space, $GL_2$ with four simple poles on $P^1$), of complex dimension $2$ and find all the possible representations of this space.",
   "evidence": "The canonical record is problem 1.62, “Nonlinear representation theory,” from the AIM workshop *Spectral data for Higgs bundles*. It asks for a theory of the different wild-Hitchin realizations of one abstract hyperkähler manifold and proposes, as a first case, the complex two-dimensional \\(D_4\\)/Painlevé VI space: rank-two logarithmic connections on \\(\\mathbb P^1\\) with four simple poles. The record asks in particular to “find all the possible representations of this space.” Its remarks propose rank-changing realizations via Fourier–Laplace transform or Katz middle convolution and cite the \\(G_2\\) realization in arXiv:1305.6594.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 152,
   "attempt": 1
  },
  "AIM-GEOMETRY-0154": {
   "statement_status": "exact",
   "original_statement": "Consider $(S^3,g)$ with scalar curvature $R\\geq 6$. The Ricci Flow exists on the maximal interval $(0,T)$. The scalar curvature evolves by\n$$\\partial_t R = \\Delta R + 2 |Rc|^2.$$\nOne can estimate that\n$$\\partial_t R = \\Delta R + 2 |Rc|^2 = \\Delta R + \\frac 2 3 R^2 + |\\mathring{Rc}|^2 \\geq \\Delta R + \\frac 2 3 R^2 .$$\nIt follows from the maximum principle that\n$$R\\geq \\frac{6}{1-4t} \\mbox{ and } T \\leq \\frac 1 4.$$\n\nIf $T$ is close to $\\frac 1 4$, is $g(0)$ close to the round metric in the Gromov-Hausdorff sense and in the $C^0$ sense? One can also ask the same question for other flows such as mean curvature flow. What would be a suitable topology?",
   "clean_statement": "Consider $(S^3,g)$ with scalar curvature $R\\geq 6$. The Ricci Flow exists on the maximal interval $(0,T)$. The scalar curvature evolves by\n$$\\partial_t R = \\Delta R + 2 |Rc|^2.$$\nOne can estimate that\n$$\\partial_t R = \\Delta R + 2 |Rc|^2 = \\Delta R + \\frac 2 3 R^2 + |\\mathring{Rc}|^2 \\geq \\Delta R + \\frac 2 3 R^2 .$$\nIt follows from the maximum principle that\n$$R\\geq \\frac{6}{1-4t} \\mbox{ and } T \\leq \\frac 1 4.$$\n\nIf $T$ is close to $\\frac 1 4$, is $g(0)$ close to the round metric in the Gromov-Hausdorff sense and in the $C^0$ sense? One can also ask the same question for other flows such as mean curvature flow. What would be a suitable topology?",
   "public_statement": "Consider $(S^3,g)$ with scalar curvature $R\\geq 6$. The Ricci Flow exists on the maximal interval $(0,T)$. The scalar curvature evolves by\n$$\\partial_t R = \\Delta R + 2 |Rc|^2.$$\nOne can estimate that\n$$\\partial_t R = \\Delta R + 2 |Rc|^2 = \\Delta R + \\frac 2 3 R^2 + |\\mathring{Rc}|^2 \\geq \\Delta R + \\frac 2 3 R^2 .$$\nIt follows from the maximum principle that\n$$R\\geq \\frac{6}{1-4t} \\mbox{ and } T \\leq \\frac 1 4.$$\n\nIf $T$ is close to $\\frac 1 4$, is $g(0)$ close to the round metric in the Gromov-Hausdorff sense and in the $C^0$ sense? One can also ask the same question for other flows such as mean curvature flow. What would be a suitable topology?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 153,
   "attempt": 1
  },
  "AIM-GEOMETRY-0155": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Let $K$ be a simplicial complex with metric $g$ such that each simplex has a metric with positive (or flat) curvature and $K$ has positive (or nonnegative) curvature in the Alexander sense. Does a Ricci flow $(K, g(t))$ exist such that for each $t>0$, $g(t)$ is a smooth orbifold metric and\n$$\\lim_{t\\to 0} g(t) =g $$\nin the Gromov-Hausdorff topology. Can one get rigidity in the nonnegative case?",
   "clean_statement": null,
   "public_statement": "Let $K$ be a simplicial complex with metric $g$ such that each simplex has a metric with positive (or flat) curvature and $K$ has positive (or nonnegative) curvature in the Alexander sense. Does a Ricci flow $(K, g(t))$ exist such that for each $t>0$, $g(t)$ is a smooth orbifold metric and\n$$\\lim_{t\\to 0} g(t) =g $$\nin the Gromov-Hausdorff topology. Can one get rigidity in the nonnegative case?",
   "evidence": "The canonical record says, verbatim:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 154,
   "attempt": 1
  },
  "AIM-GEOMETRY-0156": {
   "statement_status": "exact",
   "original_statement": "Does there exist a complete Ricci Flow that starts with uniformly bounded curvature that instantly has unbounded curvature?",
   "clean_statement": "Does there exist a complete Ricci Flow that starts with uniformly bounded curvature that instantly has unbounded curvature?",
   "public_statement": "Does there exist a complete Ricci Flow that starts with uniformly bounded curvature that instantly has unbounded curvature?",
   "evidence": "The canonical AIM record (workshop *Geometric flows and Riemannian geometry*, section *Ricci Flow*, problem 1.15) asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 155,
   "attempt": 1
  },
  "AIM-GEOMETRY-0157": {
   "statement_status": "exact",
   "original_statement": "Given a hypersurface $M\\subset \\mathbb{R}^{n+1}$, define\n$$F(M,x_0,t_0)=(4\\pi t_0)^{-\\frac n 2} \\int_M e^{-\\frac{|x-x_0|^2}{4t_0}} dH^{n},$$\n$$\\lambda(M) = \\sup_{(x_0,t_0)} F(M,x_0,t_0).$$\nQuestion: Is there a useful notion of $\\mu$-stability for Ricci Flow self-shrinkers analogous to Colding-Minicozzi for the $\\lambda$ entropy in mean curvature flow? What are the stable solitons?",
   "clean_statement": "Given a hypersurface $M\\subset \\mathbb{R}^{n+1}$, define\n$$F(M,x_0,t_0)=(4\\pi t_0)^{-\\frac n 2} \\int_M e^{-\\frac{|x-x_0|^2}{4t_0}} dH^{n},$$\n$$\\lambda(M) = \\sup_{(x_0,t_0)} F(M,x_0,t_0).$$\nQuestion: Is there a useful notion of $\\mu$-stability for Ricci Flow self-shrinkers analogous to Colding-Minicozzi for the $\\lambda$ entropy in mean curvature flow? What are the stable solitons?",
   "public_statement": "Given a hypersurface $M\\subset \\mathbb{R}^{n+1}$, define\n$$F(M,x_0,t_0)=(4\\pi t_0)^{-\\frac n 2} \\int_M e^{-\\frac{|x-x_0|^2}{4t_0}} dH^{n},$$\n$$\\lambda(M) = \\sup_{(x_0,t_0)} F(M,x_0,t_0).$$\nQuestion: Is there a useful notion of $\\mu$-stability for Ricci Flow self-shrinkers analogous to Colding-Minicozzi for the $\\lambda$ entropy in mean curvature flow? What are the stable solitons?",
   "evidence": "The canonical JSON has no visible OCR corruption. The linked AIM page could not be fetched on 2026-08-08, so the wording above was verified against the repository record, not against a live copy of the page.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 156,
   "attempt": 1
  },
  "AIM-GEOMETRY-0158": {
   "statement_status": "exact",
   "original_statement": "Understand singularities in Ricci Flow in dimension $n\\geq 4$.\nExamples are $S^3\\times \\mathbb{R}$, $S^2\\times \\mathbb{R}^2$, $4d$ Bryant soliton, $3d \\mbox{Bryant soliton} \\times \\mathbb{R}$ and FIK solitons.",
   "clean_statement": "Understand singularities in Ricci Flow in dimension $n\\geq 4$.\nExamples are $S^3\\times \\mathbb{R}$, $S^2\\times \\mathbb{R}^2$, $4d$ Bryant soliton, $3d \\mbox{Bryant soliton} \\times \\mathbb{R}$ and FIK solitons.",
   "public_statement": "Understand singularities in Ricci Flow in dimension $n\\geq 4$.\nExamples are $S^3\\times \\mathbb{R}$, $S^2\\times \\mathbb{R}^2$, $4d$ Bryant soliton, $3d \\mbox{Bryant soliton} \\times \\mathbb{R}$ and FIK solitons.",
   "evidence": "The canonical AIM record (Geometry, *Geometric flows and Riemannian geometry*, Ricci Flow, Problem 1.25) says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 157,
   "attempt": 1
  },
  "AIM-GEOMETRY-0159": {
   "statement_status": "exact",
   "original_statement": "In dimension $4$, can there be an ALE space as singularity for Ricci Flow?",
   "clean_statement": "In dimension $4$, can there be an ALE space as singularity for Ricci Flow?",
   "public_statement": "In dimension $4$, can there be an ALE space as singularity for Ricci Flow?",
   "evidence": "The record has no remarks or literature field. Its recorded source is <http://aimpl.org/flowriemannian/1/>. That page was unavailable during this run, so the wording above was verified against the repository record but not against a live copy of the original page. There is no visible OCR error. The ambiguity is mathematical: “ALE space as singularity” can mean at least four different things.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 158,
   "attempt": 1
  },
  "AIM-GEOMETRY-0160": {
   "statement_status": "exact",
   "original_statement": "Does $\\max_M R(\\cdot,t)$ blow up as $t\\to T$ the singular time?",
   "clean_statement": "Does $\\max_M R(\\cdot,t)$ blow up as $t\\to T$ the singular time?",
   "public_statement": "Does $\\max_M R(\\cdot,t)$ blow up as $t\\to T$ the singular time?",
   "evidence": "The canonical AIM record (Geometry, workshop *Geometric flows and Riemannian geometry*, Ricci Flow, Problem 1.35) reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 159,
   "attempt": 1
  },
  "AIM-GEOMETRY-0161": {
   "statement_status": "exact",
   "original_statement": "Let $(M^n,g_t)_{t\\in[0,T)}$ be a Ricci flow with $T<\\infty$. Assume $R <1$ on $M\\times [0,T)$. Characterize $g_t$ as $t\\to T$.",
   "clean_statement": "Let $(M^n,g_t)_{t\\in[0,T)}$ be a Ricci flow with $T<\\infty$. Assume $R <1$ on $M\\times [0,T)$. Characterize $g_t$ as $t\\to T$.",
   "public_statement": "Let $(M^n,g_t)_{t\\in[0,T)}$ be a Ricci flow with $T<\\infty$. Assume $R <1$ on $M\\times [0,T)$. Characterize $g_t$ as $t\\to T$.",
   "evidence": "The canonical AIM record says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 160,
   "attempt": 2
  },
  "AIM-GEOMETRY-0162": {
   "statement_status": "exact",
   "original_statement": "Let $(M^n,g_t)_{t\\in[0,T)}$ be a Ricci flow with $T<\\infty$. Assume $R< \\frac{C}{T-t}$. Characterize $(T-t)^{-1} g_t$ as $t\\to T$.\nThis is related to the Hamilton-Tian conjecture. Note that the Kahler Ricci flow on Fano manifolds satisfies $R< \\frac{C}{T-t}$.",
   "clean_statement": "Let $(M^n,g_t)_{t\\in[0,T)}$ be a Ricci flow with $T<\\infty$. Assume $R< \\frac{C}{T-t}$. Characterize $(T-t)^{-1} g_t$ as $t\\to T$.\nThis is related to the Hamilton-Tian conjecture. Note that the Kahler Ricci flow on Fano manifolds satisfies $R< \\frac{C}{T-t}$.",
   "public_statement": "Let $(M^n,g_t)_{t\\in[0,T)}$ be a Ricci flow with $T<\\infty$. Assume $R< \\frac{C}{T-t}$. Characterize $(T-t)^{-1} g_t$ as $t\\to T$.\nThis is related to the Hamilton-Tian conjecture. Note that the Kahler Ricci flow on Fano manifolds satisfies $R< \\frac{C}{T-t}$.",
   "evidence": "The canonical record is problem 1.45 in the Ricci-flow section of the AIM workshop *Geometric flows and Riemannian geometry*. Its exact problem text is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 161,
   "attempt": 1
  },
  "AIM-GEOMETRY-0163": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Classify singularities modulo singularities that in bounded scalar curvature setting.",
   "clean_statement": null,
   "public_statement": "Classify singularities modulo singularities that in bounded scalar curvature setting.",
   "evidence": "There are at least three plausible readings:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 162,
   "attempt": 1
  },
  "AIM-GEOMETRY-0164": {
   "statement_status": "exact",
   "original_statement": "R. Bamler and Q. Zhang proved a backwards pseudolocality theorem: Assume $R<1$. For $r \\in (0,1)$, if $|Rm| < r^{-2} $ on $B(x,t,r)$,\nthen $$|Rm| < (\\epsilon r)^{-2}$$ on $P(x,t,\\epsilon r,-(\\epsilon r)^2 ) =B(x,t,\\epsilon r) \\times [t-(\\epsilon r)^2, t)$.\nCan we remove $R<1$ assumption in the backwards pseudolocality theorem?",
   "clean_statement": "R. Bamler and Q. Zhang proved a backwards pseudolocality theorem: Assume $R<1$. For $r \\in (0,1)$, if $|Rm| < r^{-2} $ on $B(x,t,r)$,\nthen $$|Rm| < (\\epsilon r)^{-2}$$ on $P(x,t,\\epsilon r,-(\\epsilon r)^2 ) =B(x,t,\\epsilon r) \\times [t-(\\epsilon r)^2, t)$.\nCan we remove $R<1$ assumption in the backwards pseudolocality theorem?",
   "public_statement": "R. Bamler and Q. Zhang proved a backwards pseudolocality theorem: Assume $R<1$. For $r \\in (0,1)$, if $|Rm| < r^{-2} $ on $B(x,t,r)$,\nthen $$|Rm| < (\\epsilon r)^{-2}$$ on $P(x,t,\\epsilon r,-(\\epsilon r)^2 ) =B(x,t,\\epsilon r) \\times [t-(\\epsilon r)^2, t)$.\nCan we remove $R<1$ assumption in the backwards pseudolocality theorem?",
   "evidence": "The canonical AIM record (AIM Problem Lists, workshop *Geometric flows and Riemannian geometry*, Ricci Flow problem 1.55) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 163,
   "attempt": 1
  },
  "AIM-GEOMETRY-0165": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Can we remove $R<1$ assumption in R. Bamler and Q. Zhang's heat kernel bound and distance bound? For more information, see their papers on arxiv: HEAT KERNEL AND CURVATURE BOUNDS IN RICCI FLOWS WITH BOUNDED SCALAR CURVATURE, and HEAT KERNEL AND CURVATURE BOUNDS IN RICCI FLOWS WITH BOUNDED SCALAR CURVATURE-PART II.",
   "clean_statement": null,
   "public_statement": "Can we remove $R<1$ assumption in R. Bamler and Q. Zhang's heat kernel bound and distance bound? For more information, see their papers on arxiv: HEAT KERNEL AND CURVATURE BOUNDS IN RICCI FLOWS WITH BOUNDED SCALAR CURVATURE, and HEAT KERNEL AND CURVATURE BOUNDS IN RICCI FLOWS WITH BOUNDED SCALAR CURVATURE-PART II.",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 164,
   "attempt": 1
  },
  "AIM-GEOMETRY-0166": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Let $C\\subset \\mathbb{R}^2$ be the topologist's sine curve. More precisely,\n$$C=\\{(x,\\sin \\frac 1 x )| x\\in (0,\\frac{1}{2 \\pi})\\} \\cup \\Gamma,$$\nwhere $\\Gamma$ is a smooth curve connecting the origin and the point $(\\frac{1}{2 \\pi}, 0)$ that doesn't intersect $C\\setminus \\Gamma$.\nWhat happens to this when you apply level set flow? Does it become instantly smooth? More generally, what happens to a compact set $\\Omega \\subset \\mathbb{R}^2$ under level set flow?\n\nIt is known that Jordan curves of zero Lebesgue measure become instantly smooth. It is also known that any compact connected set $\\Omega$ with Lebesgue measure $H^2(\\Omega)=0$ is nonfattening, provided it separates $\\R^2$ into precisely two connected components. The level set flow of an arbitrary compact locally connected set is pretty well understood. See Joseph Lauer's paper on the arxiv for these facts and other background.",
   "clean_statement": "Let $C\\subset\\mathbb{R}^2$ be the topologist's sine curve. More precisely,\n\\[\nC=\\left\\{\\left(x,\\sin\\frac1x\\right):x\\in\\left(0,\\frac1{2\\pi}\\right)\\right\\}\\cup\\Gamma,\n\\]\nwhere $\\Gamma$ is a smooth curve connecting the origin and $(1/(2\\pi),0)$ that does not intersect $C\\setminus\\Gamma$. What happens under level-set flow? Does it become instantly smooth? More generally, what happens to a compact set in $\\mathbb R^2$ under level-set flow?",
   "public_statement": "Let $C\\subset \\mathbb{R}^2$ be the topologist's sine curve. More precisely,\n$$C=\\{(x,\\sin \\frac 1 x )| x\\in (0,\\frac{1}{2 \\pi})\\} \\cup \\Gamma,$$\nwhere $\\Gamma$ is a smooth curve connecting the origin and the point $(\\frac{1}{2 \\pi}, 0)$ that doesn't intersect $C\\setminus \\Gamma$.\nWhat happens to this when you apply level set flow? Does it become instantly smooth? More generally, what happens to a compact set $\\Omega \\subset \\mathbb{R}^2$ under level set flow?\n\nIt is known that Jordan curves of zero Lebesgue measure become instantly smooth. It is also known that any compact connected set $\\Omega$ with Lebesgue measure $H^2(\\Omega)=0$ is nonfattening, provided it separates $\\R^2$ into precisely two connected components. The level set flow of an arbitrary compact locally connected set is pretty well understood. See Joseph Lauer's paper on the arxiv for these facts and other background.",
   "evidence": "The canonical AIM record states:",
   "classification_method": "invalid_literal_with_repaired_reading",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 165,
   "attempt": 1
  },
  "AIM-GEOMETRY-0167": {
   "statement_status": "exact",
   "original_statement": "For $d=1$, if we have a bounded $C^{2,\\alpha}$ graph, let $c(t)$ be the curve shortening flow and let $u(t)$ be the solutions to be heat equation. Nara and Taniguchi proved that\n$$|c(t)-u(t)| \\to 0 \\mbox{ as } t \\to \\infty.$$\nIs there a higher dimension analog? The rotationally symmetric case is known.",
   "clean_statement": "For $d=1$, if we have a bounded $C^{2,\\alpha}$ graph, let $c(t)$ be the curve shortening flow and let $u(t)$ be the solutions to be heat equation. Nara and Taniguchi proved that\n$$|c(t)-u(t)| \\to 0 \\mbox{ as } t \\to \\infty.$$\nIs there a higher dimension analog? The rotationally symmetric case is known.",
   "public_statement": "For $d=1$, if we have a bounded $C^{2,\\alpha}$ graph, let $c(t)$ be the curve shortening flow and let $u(t)$ be the solutions to be heat equation. Nara and Taniguchi proved that\n$$|c(t)-u(t)| \\to 0 \\mbox{ as } t \\to \\infty.$$\nIs there a higher dimension analog? The rotationally symmetric case is known.",
   "evidence": "The canonical AIM record (workshop *Geometric flows and Riemannian geometry*, Mean Curvature Flow problem 2.2) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 166,
   "attempt": 1
  },
  "AIM-GEOMETRY-0168": {
   "statement_status": "exact",
   "original_statement": "Pick an integer $N$, and onsider a smooth embedded curve $\\gamma\\subset\\R^2$\nconstructed as follows. Let $\\alpha_\\pm:[0,2\\pi N]\\rightarrow\\R^2$ be a pair\nof disjoint embeddings that admit polar coordinate parametrizations\n$t\\mapsto (r_\\pm(t),\\theta_\\pm(t)) $ where $r_\\pm$ is increasing and has\n$C^2$ norm $<\\frac{1}{N}$, and $\\theta_\\pm(t)=t$. Thus $\\alpha_\\pm$ wraps around\nthe unit circle $N$ times, spiralling slowly outward. Now concatenate\n$\\alpha_+,\\alpha_-$\nwith a pair of curves $\\beta_+,\\beta_-$ of curvature and diameter $<10$ to form the\ncurve $\\gamma$.\n\nWhat happens to $\\gamma$ under curve shortening flow? It is known that it will shrink to a point,\nand become asymptotically round --- the problem is\nto provide a quantitative narrative for how this occurs. This should\ninclude estimates on how the length and total curvauture behave as a function of\ntime, and a geometric description of the structure of the curve.",
   "clean_statement": "Pick an integer $N$, and onsider a smooth embedded curve $\\gamma\\subset\\R^2$\nconstructed as follows. Let $\\alpha_\\pm:[0,2\\pi N]\\rightarrow\\R^2$ be a pair\nof disjoint embeddings that admit polar coordinate parametrizations\n$t\\mapsto (r_\\pm(t),\\theta_\\pm(t)) $ where $r_\\pm$ is increasing and has\n$C^2$ norm $<\\frac{1}{N}$, and $\\theta_\\pm(t)=t$. Thus $\\alpha_\\pm$ wraps around\nthe unit circle $N$ times, spiralling slowly outward. Now concatenate\n$\\alpha_+,\\alpha_-$\nwith a pair of curves $\\beta_+,\\beta_-$ of curvature and diameter $<10$ to form the\ncurve $\\gamma$.\n\nWhat happens to $\\gamma$ under curve shortening flow? It is known that it will shrink to a point,\nand become asymptotically round --- the problem is\nto provide a quantitative narrative for how this occurs. This should\ninclude estimates on how the length and total curvauture behave as a function of\ntime, and a geometric description of the structure of the curve.",
   "public_statement": "Pick an integer $N$, and onsider a smooth embedded curve $\\gamma\\subset\\R^2$\nconstructed as follows. Let $\\alpha_\\pm:[0,2\\pi N]\\rightarrow\\R^2$ be a pair\nof disjoint embeddings that admit polar coordinate parametrizations\n$t\\mapsto (r_\\pm(t),\\theta_\\pm(t)) $ where $r_\\pm$ is increasing and has\n$C^2$ norm $<\\frac{1}{N}$, and $\\theta_\\pm(t)=t$. Thus $\\alpha_\\pm$ wraps around\nthe unit circle $N$ times, spiralling slowly outward. Now concatenate\n$\\alpha_+,\\alpha_-$\nwith a pair of curves $\\beta_+,\\beta_-$ of curvature and diameter $<10$ to form the\ncurve $\\gamma$.\n\nWhat happens to $\\gamma$ under curve shortening flow? It is known that it will shrink to a point,\nand become asymptotically round --- the problem is\nto provide a quantitative narrative for how this occurs. This should\ninclude estimates on how the length and total curvauture behave as a function of\ntime, and a geometric description of the structure of the curve.",
   "evidence": "The exact canonical text is preserved in `input.json`. Its opening and closing sentences read:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 167,
   "attempt": 1
  },
  "AIM-GEOMETRY-0169": {
   "statement_status": "exact",
   "original_statement": "Recall that the mean curvature flow of a ``thin'' torus of revolution in $\\R^3$ goes singular everywhere simultaneously as it converges to a round circle. Under what conditions on a smooth embedded closed curve $\\gamma\\subset\\R^3$ does there exist a toroidal mean convex mean curvature flow $M_t$ that Hausdorff converges to $\\gamma$ at the blow-up time?",
   "clean_statement": "Recall that the mean curvature flow of a ``thin'' torus of revolution in $\\R^3$ goes singular everywhere simultaneously as it converges to a round circle. Under what conditions on a smooth embedded closed curve $\\gamma\\subset\\R^3$ does there exist a toroidal mean convex mean curvature flow $M_t$ that Hausdorff converges to $\\gamma$ at the blow-up time?",
   "public_statement": "Recall that the mean curvature flow of a ``thin'' torus of revolution in $\\R^3$ goes singular everywhere simultaneously as it converges to a round circle. Under what conditions on a smooth embedded closed curve $\\gamma\\subset\\R^3$ does there exist a toroidal mean convex mean curvature flow $M_t$ that Hausdorff converges to $\\gamma$ at the blow-up time?",
   "evidence": "The repository text is coherent and agrees with the “marriage ring” formulation used in recent literature; no OCR correction is needed. I interpret “blow-up time” as the first singular time $T$, “toroidal” as a smooth embedded genus-one surface for $t<T$, and Hausdorff convergence as",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 168,
   "attempt": 1
  },
  "AIM-GEOMETRY-0170": {
   "statement_status": "exact",
   "original_statement": "Must two Yang-Mills self-shrinkers that are asymptotic to the conical connection be actually identical?",
   "clean_statement": "Must two Yang-Mills self-shrinkers that are asymptotic to the conical connection be actually identical?",
   "public_statement": "Must two Yang-Mills self-shrinkers that are asymptotic to the conical connection be actually identical?",
   "evidence": "The canonical AIM record (Geometry, source index 169, workshop *Geometric flows and Riemannian geometry*, section “Other Geometric Flows,” Problem 3.1) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 169,
   "attempt": 1
  },
  "AIM-GEOMETRY-0171": {
   "statement_status": "exact",
   "original_statement": "Let $S^2$ be the 2 dimension unit sphere and $g$ be a metric on $S^2$ with positive Gauss curvature. Take $g=e^{2u} ds^2$.\nConsider the flow\n$$\\partial_t u=-1+\\frac{\\overline{K_u}}{K_u}$$\nwhere $\\overline{K_u}$ is the average of $K_u$.\nKnown: long time existence; some weak convergence (a priori estimate degenerate at $t=\\infty$).\n\nQuestion: Does the flow converge to round? (If the genus is greater than 1, the analogous flow is understood). What about higher dimensions? More information for $n=2$ can be found on the paper on arxiv.",
   "clean_statement": "Let $S^2$ be the 2 dimension unit sphere and $g$ be a metric on $S^2$ with positive Gauss curvature. Take $g=e^{2u} ds^2$.\nConsider the flow\n$$\\partial_t u=-1+\\frac{\\overline{K_u}}{K_u}$$\nwhere $\\overline{K_u}$ is the average of $K_u$.\nKnown: long time existence; some weak convergence (a priori estimate degenerate at $t=\\infty$).\n\nQuestion: Does the flow converge to round? (If the genus is greater than 1, the analogous flow is understood). What about higher dimensions? More information for $n=2$ can be found on the paper on arxiv.",
   "public_statement": "Let $S^2$ be the 2 dimension unit sphere and $g$ be a metric on $S^2$ with positive Gauss curvature. Take $g=e^{2u} ds^2$.\nConsider the flow\n$$\\partial_t u=-1+\\frac{\\overline{K_u}}{K_u}$$\nwhere $\\overline{K_u}$ is the average of $K_u$.\nKnown: long time existence; some weak convergence (a priori estimate degenerate at $t=\\infty$).\n\nQuestion: Does the flow converge to round? (If the genus is greater than 1, the analogous flow is understood). What about higher dimensions? More information for $n=2$ can be found on the paper on arxiv.",
   "evidence": "The canonical AIM record says, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 170,
   "attempt": 1
  },
  "AIM-GEOMETRY-0172": {
   "statement_status": "exact",
   "original_statement": "What are closed manifolds with almost nonnegative curvature operator?\n$$ \\mbox{Rm} \\cdot \\mbox{diam}^2 \\geq -\\epsilon.$$",
   "clean_statement": "What are closed manifolds with almost nonnegative curvature operator?\n$$ \\mbox{Rm} \\cdot \\mbox{diam}^2 \\geq -\\epsilon.$$",
   "public_statement": "What are closed manifolds with almost nonnegative curvature operator?\n$$ \\mbox{Rm} \\cdot \\mbox{diam}^2 \\geq -\\epsilon.$$",
   "evidence": "There is no apparent OCR error, but the displayed line is an abbreviated property rather than a complete definition: it omits the quantifier and suppresses that the inequality is an operator inequality. Following Definition 2.1 of Herrmann--Sebastian--Tuschmann, the standard reconstruction is:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 171,
   "attempt": 1
  },
  "AIM-GEOMETRY-0173": {
   "statement_status": "exact",
   "original_statement": "Let $M^3$ be a closed manifold. Outside flat points, it splits locally as $\\Sigma \\times \\mathbb{R}$. Does there exist a collection of totally geodesic flat surfaces $S_1, \\cdots, S_k, \\cdots $ such that\n$$M\\setminus (S_1 \\cup \\cdots \\cup S_k \\cup \\cdots ) \\mbox{ is } (\\Sigma \\times \\mathbb{R} )/\\mbox{group}?$$ where $\\Sigma$ is a surface (with possibly finitely or infinitely many components) with boundary a union of closed geodesics.\nMoreover, if $M$ has nonnegative curvature, then is it diffeomorphic to a lens space?",
   "clean_statement": "Let $M^3$ be a closed manifold. Outside flat points, it splits locally as $\\Sigma \\times \\mathbb{R}$. Does there exist a collection of totally geodesic flat surfaces $S_1, \\cdots, S_k, \\cdots $ such that\n$$M\\setminus (S_1 \\cup \\cdots \\cup S_k \\cup \\cdots ) \\mbox{ is } (\\Sigma \\times \\mathbb{R} )/\\mbox{group}?$$ where $\\Sigma$ is a surface (with possibly finitely or infinitely many components) with boundary a union of closed geodesics.\nMoreover, if $M$ has nonnegative curvature, then is it diffeomorphic to a lens space?",
   "public_statement": "Let $M^3$ be a closed manifold. Outside flat points, it splits locally as $\\Sigma \\times \\mathbb{R}$. Does there exist a collection of totally geodesic flat surfaces $S_1, \\cdots, S_k, \\cdots $ such that\n$$M\\setminus (S_1 \\cup \\cdots \\cup S_k \\cup \\cdots ) \\mbox{ is } (\\Sigma \\times \\mathbb{R} )/\\mbox{group}?$$ where $\\Sigma$ is a surface (with possibly finitely or infinitely many components) with boundary a union of closed geodesics.\nMoreover, if $M$ has nonnegative curvature, then is it diffeomorphic to a lens space?",
   "evidence": "The canonical AIM record (Geometry, source index 172, workshop *Geometric flows and Riemannian geometry*, section “Riemannian Geometry,” Problem 4.2) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 172,
   "attempt": 1
  },
  "AIM-GEOMETRY-0174": {
   "statement_status": "exact",
   "original_statement": "Let $M_\\epsilon =M\\times_{S^1} (\\epsilon S^1)$. As $\\epsilon \\to 0$, is Riem(Curvature Operator) bounded below?",
   "clean_statement": "Let $M_\\epsilon =M\\times_{S^1} (\\epsilon S^1)$. As $\\epsilon \\to 0$, is Riem(Curvature Operator) bounded below?",
   "public_statement": "Let $M_\\epsilon =M\\times_{S^1} (\\epsilon S^1)$. As $\\epsilon \\to 0$, is Riem(Curvature Operator) bounded below?",
   "evidence": "The exact canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 173,
   "attempt": 1
  },
  "AIM-GEOMETRY-0175": {
   "statement_status": "exact",
   "original_statement": "For the above problem, what about complex sectional curvature?",
   "clean_statement": "For the above problem, what about complex sectional curvature?",
   "public_statement": "For the above problem, what about complex sectional curvature?",
   "evidence": "The exact canonical record is aim-geometry-notes.json, zero-based index 174:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 174,
   "attempt": 1
  },
  "AIM-GEOMETRY-0176": {
   "statement_status": "exact",
   "original_statement": "What are ANCO(almost nonnegative curvature operator, i. e., there exists metric with\n$\\mbox{Riem} \\cdot \\mbox{diam}^2 \\geq -\\epsilon$) manifolds?\nConstructions: \\\\\n1. compact symmetric spaces\\\\\n2.almost flat manifolds\\\\\n3. products\\\\\n4. finite covers, finite quotients\\\\\n5. total space of principal $S^1$-bundle over an ANCO manifold",
   "clean_statement": "What are ANCO(almost nonnegative curvature operator, i. e., there exists metric with\n$\\mbox{Riem} \\cdot \\mbox{diam}^2 \\geq -\\epsilon$) manifolds?\nConstructions: \\\\\n1. compact symmetric spaces\\\\\n2.almost flat manifolds\\\\\n3. products\\\\\n4. finite covers, finite quotients\\\\\n5. total space of principal $S^1$-bundle over an ANCO manifold",
   "public_statement": "What are ANCO(almost nonnegative curvature operator, i. e., there exists metric with\n$\\mbox{Riem} \\cdot \\mbox{diam}^2 \\geq -\\epsilon$) manifolds?\nConstructions: \\\\\n1. compact symmetric spaces\\\\\n2.almost flat manifolds\\\\\n3. products\\\\\n4. finite covers, finite quotients\\\\\n5. total space of principal $S^1$-bundle over an ANCO manifold",
   "evidence": "The exact canonical text, including its awkward line breaks and extraction backslashes, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 175,
   "attempt": 1
  },
  "AIM-GEOMETRY-0177": {
   "statement_status": "exact",
   "original_statement": "Can we smooth metrics with $\\mbox{Riem} \\geq -\\mbox{Id}$ on volume scale,\ni.e., want nearby metrics with $|\\nabla^k Rm| \\leq C_k C_p^{-k-2}$?",
   "clean_statement": "Can we smooth metrics with $\\mbox{Riem} \\geq -\\mbox{Id}$ on volume scale,\ni.e., want nearby metrics with $|\\nabla^k Rm| \\leq C_k C_p^{-k-2}$?",
   "public_statement": "Can we smooth metrics with $\\mbox{Riem} \\geq -\\mbox{Id}$ on volume scale,\ni.e., want nearby metrics with $|\\nabla^k Rm| \\leq C_k C_p^{-k-2}$?",
   "evidence": "The exact canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 176,
   "attempt": 2
  },
  "AIM-GEOMETRY-0178": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Is there a flow related to Skeklov eigenvalue maximization? For closed surfaces in $S^3$, does Mean Curvature Flow improve $\\lambda_1 \\cdot Area$? What about balanced flow that keeps the center of mass?",
   "clean_statement": "Is there a flow related to Skeklov eigenvalue maximization? For closed surfaces in $S^3$, does Mean Curvature Flow improve $\\lambda_1 \\cdot Area$? What about improve flow that keeps the center of mass?",
   "public_statement": "Is there a flow related to Skeklov eigenvalue maximization? For closed surfaces in $S^3$, does Mean Curvature Flow improve $\\lambda_1 \\cdot Area$? What about balanced flow that keeps the center of mass?",
   "evidence": "The word **“Skeklov” is preserved here exactly as recorded**. It is almost certainly a typographical or OCR error for **Steklov**: Steklov eigenvalues are the boundary spectral invariant connected to free-boundary minimal surfaces and eigenvalue maximization. The supplied source URL, <http://aimpl.org/flowriemannian/5/>, returned a gateway error during this run, so that reconstruction could not be checked against the original rendered workshop page.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-geometry-notes.json",
   "source_index": 177,
   "attempt": 2
  },
  "AIM-GEOMETRY-0179": {
   "statement_status": "exact",
   "original_statement": "Existence of high genus free boundary minimal surface in $B^3$.",
   "clean_statement": "Existence of high genus free boundary minimal surface in $B^3$.",
   "public_statement": "Existence of high genus free boundary minimal surface in $B^3$.",
   "evidence": "There is no apparent OCR corruption, but the sentence suppresses important quantifiers and regularity conventions. I use the standard strong reading:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 178,
   "attempt": 3
  },
  "AIM-GEOMETRY-0180": {
   "statement_status": "exact",
   "original_statement": "Is the catenoid the only embedded free boundary minimal annulus in $B^3$ ?",
   "clean_statement": "Is the catenoid the only embedded free boundary minimal annulus in $B^3$ ?",
   "public_statement": "Is the catenoid the only embedded free boundary minimal annulus in $B^3$ ?",
   "evidence": "The canonical AIM record (Geometry, “Geometric flows and Riemannian geometry,” section “Eigenvalue Estimates,” Problem 5.3) asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 179,
   "attempt": 1
  },
  "AIM-GEOMETRY-0181": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Do $4$-dimension shrinkers have bounded scalar curvature?",
   "clean_statement": "If $(M^4,g,f)$ is a smooth, connected, complete four-real-dimensional gradient shrinking Ricci soliton\n\\[\n\\operatorname{Ric}+\\nabla^2f=\\frac12g,\n\\]\nmust its scalar curvature satisfy $\\sup_M R<\\infty$?",
   "public_statement": "Do $4$-dimension shrinkers have bounded scalar curvature?",
   "evidence": "This wording is preserved, including “$4$-dimension.” In the surrounding source records this appears under **Ricci Solitons**, between questions about examples and asymptotic splitting. The source record supplies no hypotheses or notation. The source URL was not retrievable during this run, so the following is an explicit reconstruction rather than a claim about missing words on the original page.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 180,
   "attempt": 1
  },
  "AIM-GEOMETRY-0182": {
   "statement_status": "exact",
   "original_statement": "Are there examples of shrinkers other than the one on $\\hat{\\mathbb{C}}$ in $4$ dimension?",
   "clean_statement": "Are there examples of shrinkers other than the one on $\\hat{\\mathbb{C}}$ in $4$ dimension?",
   "public_statement": "Are there examples of shrinkers other than the one on $\\hat{\\mathbb{C}}$ in $4$ dimension?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 181,
   "attempt": 2
  },
  "AIM-GEOMETRY-0183": {
   "statement_status": "exact",
   "original_statement": "For any dimension, is it true that a shrinker $(M,g)$ either is asymptotically conical, or it must split off a line?",
   "clean_statement": "For any dimension, is it true that a shrinker $(M,g)$ either is asymptotically conical, or it must split off a line?",
   "public_statement": "For any dimension, is it true that a shrinker $(M,g)$ either is asymptotically conical, or it must split off a line?",
   "evidence": "The canonical record is problem 6.3 in the “Ricci Solitons” section of the AIM workshop list *Geometric flows and Riemannian geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 182,
   "attempt": 1
  },
  "AIM-GEOMETRY-0184": {
   "statement_status": "exact",
   "original_statement": "Are there any (genuine) examples of shrinkers which are not Kahler?",
   "clean_statement": "Are there any (genuine) examples of shrinkers which are not Kahler?",
   "public_statement": "Are there any (genuine) examples of shrinkers which are not Kahler?",
   "evidence": "The canonical AIM record (section “Ricci Solitons,” item 6.4) asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 183,
   "attempt": 1
  },
  "AIM-GEOMETRY-0185": {
   "statement_status": "exact",
   "original_statement": "Do complete shrinking solitons split off a line?",
   "clean_statement": "Do complete shrinking solitons split off a line?",
   "public_statement": "Do complete shrinking solitons split off a line?",
   "evidence": "The canonical AIM record (section “Ricci Solitons,” item 6.5) asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 184,
   "attempt": 1
  },
  "AIM-GEOMETRY-0186": {
   "statement_status": "exact",
   "original_statement": "Are compact shrinkers with positive sectional curvature Einstein?",
   "clean_statement": "Are compact shrinkers with positive sectional curvature Einstein?",
   "public_statement": "Are compact shrinkers with positive sectional curvature Einstein?",
   "evidence": "The canonical record is problem 6.6 in the “Ricci Solitons” section of the AIM workshop list *Geometric flows and Riemannian geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 185,
   "attempt": 1
  },
  "AIM-GEOMETRY-0187": {
   "statement_status": "exact",
   "original_statement": "Is $\\mathbb{R} \\times \\Sigma$, for $\\Sigma$ the cigar soliton, the only collapsed $3$-dimension steady soliton?",
   "clean_statement": "Is $\\mathbb{R} \\times \\Sigma$, for $\\Sigma$ the cigar soliton, the only collapsed $3$-dimension steady soliton?",
   "public_statement": "Is $\\mathbb{R} \\times \\Sigma$, for $\\Sigma$ the cigar soliton, the only collapsed $3$-dimension steady soliton?",
   "evidence": "The canonical record is Problem 6.7 in the AIM list *Geometric flows and Riemannian geometry*, section “Ricci Solitons” (workshop dated September 2015). Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 186,
   "attempt": 1
  },
  "AIM-GEOMETRY-0188": {
   "statement_status": "exact",
   "original_statement": "1. (Farber) Configuration spaces of closed polygons in R3. Let ` = ( `1,..., ` n) be the length vector defining a closed polygon. The configuration space is M`\\Σ = tUi, where Ui are the connected components, and Σ is the set of configurations with self-intersections. Questions: Are the Ui contractible? What can we say about their topology?",
   "clean_statement": "1. (Farber) Configuration spaces of closed polygons in R3. Let ` = ( `1,..., ` n) be the length vector defining a closed polygon. The configuration space is M`\\Σ = tUi, where Ui are the connected components, and Σ is the set of configurations with self-intersections. Questions: Are the Ui contractible? What can we say about their topology?",
   "public_statement": "1. (Farber) Configuration spaces of closed polygons in R3. Let ` = ( `1,..., ` n) be the length vector defining a closed polygon. The configuration space is M`\\Σ = tUi, where Ui are the connected components, and Σ is the set of configurations with self-intersections. Questions: Are the Ui contractible? What can we say about their topology?",
   "evidence": "The canonical JSON record is visibly damaged by PDF font extraction: it replaces \\(\\ell\\) by a backtick and \\(\\bigsqcup\\) by `t`. Page 1 of the original AIM PDF was therefore checked directly. It reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 187,
   "attempt": 2
  },
  "AIM-GEOMETRY-0189": {
   "statement_status": "exact",
   "original_statement": "2. (Streinu) Same as the previous question, but with an open arm in R3, and `1 = `2 =\n\n· · · = `n.Question: Is the space of non-intersecting configurations connected?",
   "clean_statement": "2. (Streinu) Same as the previous question, but with an open arm in R3, and `1 = `2 =\n\n· · · = `n.Question: Is the space of non-intersecting configurations connected?",
   "public_statement": "2. (Streinu) Same as the previous question, but with an open arm in R3, and `1 = `2 =\n\n· · · = `n.Question: Is the space of non-intersecting configurations connected?",
   "evidence": "The canonical JSON has lost the letter `ell` in the length condition. The original AIM problem-list PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 188,
   "attempt": 1
  },
  "AIM-GEOMETRY-0190": {
   "statement_status": "exact",
   "original_statement": "3. (Farber/Panina) Consider a closed polygon in R3. According to Klyatchko, M` has a symplectic form for ω, and therefore for volume. \n\nV ol \n\n(\n\nt Ui\n\n)/ \n\nV ol (M`),\n\nwhere i corresponds to an unknot. (Note if `1 +`2 +· · · +`n−1 = `n, this is always an unknot and therefore the proportion above is exactly 1.(see below)) \n\nQuestion: try to understand this proportion.",
   "clean_statement": "3. (Farber/Panina) Consider a closed polygon in R3. According to Klyatchko, M` has a symplectic form for ω, and therefore for volume.\n\nV ol\n\n(\n\nt Ui\n\n)/\n\nV ol (M`),\n\nwhere i corresponds to an unknot. (Note if `1 +`2 +· · · +`n−1 = `n, this is always an unknot and therefore the proportion above is exactly 1.(see below))\n\nQuestion: try to understand this proportion.",
   "public_statement": "3. (Farber/Panina) Consider a closed polygon in R3. According to Klyatchko, M` has a symplectic form for ω, and therefore for volume.\n\nV ol\n\n(\n\nt Ui\n\n)/\n\nV ol (M`),\n\nwhere i corresponds to an unknot. (Note if `1 +`2 +· · · +`n−1 = `n, this is always an unknot and therefore the proportion above is exactly 1.(see below))\n\nQuestion: try to understand this proportion.",
   "evidence": "The canonical record is index 189 of `aim-geometry-notes.json`. Its extracted text is visibly damaged:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 189,
   "attempt": 1
  },
  "AIM-GEOMETRY-0191": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "4. (Holmes-Cerfon) Consider the configuration space of planar n-gons such that \n\nLi − \u000f ≤ `i ≤ Li + \u000f. ML,\u000f is a manifold with boundary. Question: Understand its topology and volume. \n1",
   "clean_statement": "**4. (Holmes-Cerfon)** Consider the configuration space of planar \\(n\\)-gons such\nthat\n\\[\nL_i-\\varepsilon\\leq \\ell_i\\leq L_i+\\varepsilon.\n\\]\n\\(M_{L,\\varepsilon}\\) is a manifold with boundary.  Question: Understand its\ntopology and volume.",
   "public_statement": "4. (Holmes-Cerfon) Consider the configuration space of planar n-gons such that\n\nLi − [U+000F] ≤ `i ≤ Li + [U+000F]. ML,[U+000F] is a manifold with boundary. Question: Understand its topology and volume.\n1",
   "evidence": "The exact canonical record is retained in `input.json`. It contains a form-feed control character in place of \\(\\varepsilon\\), backticks in place of \\(\\ell\\), and a trailing page number. Page 1 of the original AIM PDF was inspected directly. The source-verified reconstruction is:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 190,
   "attempt": 1
  },
  "AIM-GEOMETRY-0192": {
   "statement_status": "exact",
   "original_statement": "5. (Sitharam) Consider two polygons that share a \"chain\" (see below), or graphs of tree-width 2. \n\nQuestion: Apply Morse Theory ` a la Farber to derive the Betti numbers: a) Chambers b) Homologies",
   "clean_statement": "5. (Sitharam) Consider two polygons that share a \"chain\" (see below), or graphs of tree-width 2.\n\nQuestion: Apply Morse Theory ` a la Farber to derive the Betti numbers: a) Chambers b) Homologies",
   "public_statement": "5. (Sitharam) Consider two polygons that share a \"chain\" (see below), or graphs of tree-width 2.\n\nQuestion: Apply Morse Theory ` a la Farber to derive the Betti numbers: a) Chambers b) Homologies",
   "evidence": "The exact canonical JSON text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 191,
   "attempt": 1
  },
  "AIM-GEOMETRY-0193": {
   "statement_status": "exact",
   "original_statement": "6. (Farber) Question: Find asymptotic behaviour of Cn (the number of orbits of cham-bers in the case of polygonal linkages).",
   "clean_statement": "6. (Farber) Question: Find asymptotic behaviour of Cn (the number of orbits of cham-bers in the case of polygonal linkages).",
   "public_statement": "6. (Farber) Question: Find asymptotic behaviour of Cn (the number of orbits of cham-bers in the case of polygonal linkages).",
   "evidence": "The canonical record, including its line-break OCR artifact, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 192,
   "attempt": 1
  },
  "AIM-GEOMETRY-0194": {
   "statement_status": "exact",
   "original_statement": "7. (Thorpe) Consider a network of corner-sharing triangles in the plane, with holes of size 5, 6, 7, 8, 9 (and an average hole size of 6). If this is an infinite network, it is isostatic. \n\n2Now take a large finite piece of the framework. Experimentally, if we pin every other triangle boundary vertex (vertex of degree two) and run the pebble game, we get an isostatic network. Question: prove that this approach works in general and find other distributions of pins that also work. a) generic b) equilateral triangles Question (generalization): What about the class of graphs that have no proper rigid subgraphs, but that have more than two bodies at a pin? The underlying body-pin graph is no longer 3-regular.",
   "clean_statement": "7. (Thorpe) Consider a network of corner-sharing triangles in the plane, with holes of size 5, 6, 7, 8, 9 (and an average hole size of 6). If this is an infinite network, it is isostatic.\n\n2Now take a large finite piece of the framework. Experimentally, if we pin every other triangle boundary vertex (vertex of degree two) and run the pebble game, we get an isostatic network. Question: prove that this approach works in general and find other distributions of pins that also work. a) generic b) equilateral triangles Question (generalization): What about the class of graphs that have no proper rigid subgraphs, but that have more than two bodies at a pin? The underlying body-pin graph is no longer 3-regular.",
   "public_statement": "7. (Thorpe) Consider a network of corner-sharing triangles in the plane, with holes of size 5, 6, 7, 8, 9 (and an average hole size of 6). If this is an infinite network, it is isostatic.\n\n2Now take a large finite piece of the framework. Experimentally, if we pin every other triangle boundary vertex (vertex of degree two) and run the pebble game, we get an isostatic network. Question: prove that this approach works in general and find other distributions of pins that also work. a) generic b) equilateral triangles Question (generalization): What about the class of graphs that have no proper rigid subgraphs, but that have more than two bodies at a pin? The underlying body-pin graph is no longer 3-regular.",
   "evidence": "The exact canonical record is preserved in `input.json`. The original four-page AIM PDF was inspected directly. The source-verified statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 193,
   "attempt": 1
  },
  "AIM-GEOMETRY-0195": {
   "statement_status": "exact",
   "original_statement": "8. (Hempel) Consider a simplicial polyhedra with fixed combinatorics in R3.Question: characterize those collections of dihedral angles and face angles that can be realized by a polyhedron with these combinatorics. Conjecture: dimension = E − 1, where E is the number of edges. Needs more clarification.",
   "clean_statement": "8. (Hempel) Consider a simplicial polyhedra with fixed combinatorics in R3.Question: characterize those collections of dihedral angles and face angles that can be realized by a polyhedron with these combinatorics. Conjecture: dimension = E − 1, where E is the number of edges. Needs more clarification.",
   "public_statement": "8. (Hempel) Consider a simplicial polyhedra with fixed combinatorics in R3.Question: characterize those collections of dihedral angles and face angles that can be realized by a polyhedron with these combinatorics. Conjecture: dimension = E − 1, where E is the number of edges. Needs more clarification.",
   "evidence": "The canonical record is Problem 8 from the 2014 AIM workshop *Configuration spaces of linkages*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 194,
   "attempt": 1
  },
  "AIM-GEOMETRY-0196": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "9. (St. John) Consider a multi-robot formation that is a generically minimally rigid framework G, with diameter D = max (pi,p j ) || pi − pj ||. We remove an edge and obtain ¯G that is flexible. Let d = min diameter(configuration space of ¯G). Question: Understand the relationship between D and d, and find algorithms to detect what edge to delete for maximal change in diameter. More precisely, find find bar e and \"positioning\" q such that the diameter of ( G\\e, q ) is minimum under the constraints that bar lengths in G\\e are maintained. \n1",
   "clean_statement": null,
   "public_statement": "9. (St. John) Consider a multi-robot formation that is a generically minimally rigid framework G, with diameter D = max (pi,p j ) || pi − pj ||. We remove an edge and obtain ¯G that is flexible. Let d = min diameter(configuration space of ¯G). Question: Understand the relationship between D and d, and find algorithms to detect what edge to delete for maximal change in diameter. More precisely, find find bar e and \"positioning\" q such that the diameter of ( G\\e, q ) is minimum under the constraints that bar lengths in G\\e are maintained.\n1",
   "evidence": "The canonical JSON is an OCR extraction of Problem 9 in the AIM workshop open-problems PDF. The official PDF gives the following statement (notation normalized only by restoring subscripts, an overbar, and a set-minus sign):",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 195,
   "attempt": 2
  },
  "AIM-GEOMETRY-0197": {
   "statement_status": "exact",
   "original_statement": "0. (Theran) Consider a Delaunay triangulation (no vertex is inside the circumcircle of any triangle). Fix a combinatorial type of a triangulation. Question: What is the configuration space? \n\nd = 2 it is a ball \n\nd = 3 is it universal? \n1",
   "clean_statement": "0. (Theran) Consider a Delaunay triangulation (no vertex is inside the circumcircle of any triangle). Fix a combinatorial type of a triangulation. Question: What is the configuration space?\n\nd = 2 it is a ball\n\nd = 3 is it universal?\n1",
   "public_statement": "0. (Theran) Consider a Delaunay triangulation (no vertex is inside the circumcircle of any triangle). Fix a combinatorial type of a triangulation. Question: What is the configuration space?\n\nd = 2 it is a ball\n\nd = 3 is it universal?\n1",
   "evidence": "The canonical record is OCR-damaged. The original AIM problem list, *Configuration spaces of linkages*, contains the following as Problem 10 (not Problem 0):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 196,
   "attempt": 1
  },
  "AIM-GEOMETRY-0198": {
   "statement_status": "exact",
   "original_statement": "1. (Owen) Specialization: When is the Galois group of a particular rigid framework a subgroup of the Galois group of the graph? When is G(G, p ) ⊆ G (G) for any p with ( G, p ) isostatic? (Isostatic is sufficient but not necessary). Here p generic means p = x1,..., x n are algebraically independent over Q.\n1",
   "clean_statement": "1. (Owen) Specialization: When is the Galois group of a particular rigid framework a subgroup of the Galois group of the graph? When is G(G, p ) ⊆ G (G) for any p with ( G, p ) isostatic? (Isostatic is sufficient but not necessary). Here p generic means p = x1,..., x n are algebraically independent over Q.\n1",
   "public_statement": "1. (Owen) Specialization: When is the Galois group of a particular rigid framework a subgroup of the Galois group of the graph? When is G(G, p ) ⊆ G (G) for any p with ( G, p ) isostatic? (Isostatic is sufficient but not necessary). Here p generic means p = x1,..., x n are algebraically independent over Q.\n1",
   "evidence": "The canonical JSON record has lost a digit in the problem number and has an extraneous final `1`. The source PDF is the AIM workshop list *Configuration Spaces of Linkages: Open Problems*, notes by Elissa Ross, 25--31 October 2014. The source-verified item is problem 11 (not problem 1):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 197,
   "attempt": 1
  },
  "AIM-GEOMETRY-0199": {
   "statement_status": "exact",
   "original_statement": "2. (Whiteley) Conjecture: Given a symmetric bar-joint framework ( G, p ), the configu-ration space of ( G, p ) (with appropriate parts of the frame of reference fixed), has the symmetry of the most symmetric individual realization in the configuration space. 3\n1",
   "clean_statement": "2. (Whiteley) Conjecture: Given a symmetric bar-joint framework ( G, p ), the configu-ration space of ( G, p ) (with appropriate parts of the frame of reference fixed), has the symmetry of the most symmetric individual realization in the configuration space. 3\n1",
   "public_statement": "2. (Whiteley) Conjecture: Given a symmetric bar-joint framework ( G, p ), the configu-ration space of ( G, p ) (with appropriate parts of the frame of reference fixed), has the symmetry of the most symmetric individual realization in the configuration space. 3\n1",
   "evidence": "The canonical record is damaged at both ends. Inspection of the original AIM PDF shows that this is item **12**, not item 2:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 198,
   "attempt": 1
  },
  "AIM-GEOMETRY-0200": {
   "statement_status": "exact",
   "original_statement": "3. (Schulze) Understand the following question: does the pseudo triangulation algorithm for the Carpenter's Rule give some unfolding that preserves symmetries? (Note that Connelly, Demaine, Rote have non-algorithmic positive solution). \n1",
   "clean_statement": "3. (Schulze) Understand the following question: does the pseudo triangulation algorithm for the Carpenter's Rule give some unfolding that preserves symmetries? (Note that Connelly, Demaine, Rote have non-algorithmic positive solution).\n1",
   "public_statement": "3. (Schulze) Understand the following question: does the pseudo triangulation algorithm for the Carpenter's Rule give some unfolding that preserves symmetries? (Note that Connelly, Demaine, Rote have non-algorithmic positive solution).\n1",
   "evidence": "The canonical JSON record says “3.” and ends in a stray `1`. Inspection of the original four-page AIM problem list resolves both extraction defects. At the page break the printed page number was concatenated with the item number, and the intended item is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 199,
   "attempt": 1
  },
  "AIM-GEOMETRY-0201": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "4. (Schulze) Suppose a symmetric framework (linkage) has a 1DOF expansive mecha-nism. Does the mechanism preserve the symmetry? \n1",
   "clean_statement": null,
   "public_statement": "4. (Schulze) Suppose a symmetric framework (linkage) has a 1DOF expansive mecha-nism. Does the mechanism preserve the symmetry?\n1",
   "evidence": "The canonical record reads:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 200,
   "attempt": 1
  },
  "AIM-GEOMETRY-0202": {
   "statement_status": "exact",
   "original_statement": "5. (Schulze) Under what conditions is a linkage guaranteed to preserve the original symmetry throughout the configuration space? \n1",
   "clean_statement": "5. (Schulze) Under what conditions is a linkage guaranteed to preserve the original symmetry throughout the configuration space?\n1",
   "public_statement": "5. (Schulze) Under what conditions is a linkage guaranteed to preserve the original symmetry throughout the configuration space?\n1",
   "evidence": "The canonical record has lost the first digit of its item number. The original AIM PDF gives the source-verified problem as",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 201,
   "attempt": 1
  },
  "AIM-GEOMETRY-0203": {
   "statement_status": "exact",
   "original_statement": "6. (St. John/Schulze) Persistence theory: Group of connected agents, every agent has out-degree 2 (except for 2 agents, the leader and the co-leader). Question: Can symmetry of the configuration be exploited to reduce the computa-tion? What about a body-CAD version? 4",
   "clean_statement": "6. (St. John/Schulze) Persistence theory: Group of connected agents, every agent has out-degree 2 (except for 2 agents, the leader and the co-leader). Question: Can symmetry of the configuration be exploited to reduce the computa-tion? What about a body-CAD version? 4",
   "public_statement": "6. (St. John/Schulze) Persistence theory: Group of connected agents, every agent has out-degree 2 (except for 2 agents, the leader and the co-leader). Question: Can symmetry of the configuration be exploited to reduce the computa-tion? What about a body-CAD version? 4",
   "evidence": "The canonical JSON has three extraction defects. Inspection of the original AIM PDF shows that the item is **16**, not 6; “computa-tion” is a line-wrap hyphen and should read “computation”; and the terminal “4” is the printed page number. The verified statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 202,
   "attempt": 1
  },
  "AIM-GEOMETRY-0204": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "1. Zauner's Conjecture - Complex ETF's consisting of N = M 2 vectors exist for all M ∈ N.2. The Hadamard Conjecture - A Hadamard matrix of order 4k exists for every k ∈ N.3. The Paulsen Problem - If a frame is \u000f-close to being tight and \u000f-close to being equal-norm, then how far is it from being both equal-norm and tight? 4. Dustin Mixon: Does N < 4M − 4 imply that A: CM /T → RN is not injective? 5. Dan Edidin: At the transition around N = 4 M − 4, are there examples where the set of frames satisfying injectivity for A: CM /T → RN is open, but such that the complement has positive measure? 16. Dan Edidin: If B: PM −1/T →, [x] 7 → (|〈 x, f k〉| 2)Nk=1 is injective, what can can be con-cluded, if anything, about phase retrieval? 7. Bernhard Bodmann: Find N so that we have a quantitative measure for stability. 8. Dustin Mixon: Find N so that we have a quantitative measure for computational efficiency. 9. Ferenc Szollosi: Determine the maximum number of equiangular lines possible in R14. It is known that this number is 28, 29, or 30.10. Ferenc Szollosi: Give a full algebraic classification of all complex ETF's of small parameters \n\nM and N.11. Matt Fickus: Do real equi-modular ETF's exist (ie \" ±1 matrices\") whose redundancy is not approximately 2?12. Ferenc Szollosi: Can we construct ETF's from nonabelian difference sets? 13. Matt Fickus: Are there integrality conditions on the dimensions for the existence of complex equiangular frames? 14. Matt Fickus: Can we find an elementary proof showing that, for any 2( M − 2) 2-D planes in RM, there exists another 2-D plane that has principle angle of π/ 2 with each of the original planes? 15. Dustin Mixon: If we fix N and increase M, does the worst case coherence strictly decrease? 16. Matt Fickus: What is the threshold to know we are in the well of a global minimizer when we optimize over the UNTF's with cost function U (F ) = max m6 =n |〈 ϕn, ϕ m〉| 2?17. Zhiqiang Xu: Suppose x0 ∈ CM and that F = {fj }Nj=1 is a frame in CM. Furthermore, suppose that X0 is k-sparse (ie ‖x0‖0 ≤ k). Now set bj:= |〈 fj, x 0〉|, j = 1, 2,..., N. Then consider the following recovery problems (up to global phase factor): (a) What is the minimum N for which one can recover x0 uniquely for bj, j = 1, 2,..., N.(b) What is the minimum N such that one can recover x0 by solving the l1 minimization: \n\nmin ‖x‖1, s.t. |〈 fj, x 〉| = bj, j = 1, 2,...N.",
   "clean_statement": null,
   "public_statement": "1. Zauner's Conjecture - Complex ETF's consisting of N = M 2 vectors exist for all M ∈ N.2. The Hadamard Conjecture - A Hadamard matrix of order 4k exists for every k ∈ N.3. The Paulsen Problem - If a frame is [U+000F]-close to being tight and [U+000F]-close to being equal-norm, then how far is it from being both equal-norm and tight? 4. Dustin Mixon: Does N < 4M − 4 imply that A: CM /T → RN is not injective? 5. Dan Edidin: At the transition around N = 4 M − 4, are there examples where the set of frames satisfying injectivity for A: CM /T → RN is open, but such that the complement has positive measure? 16. Dan Edidin: If B: PM −1/T →, [x] 7 → (|〈 x, f k〉| 2)Nk=1 is injective, what can can be con-cluded, if anything, about phase retrieval? 7. Bernhard Bodmann: Find N so that we have a quantitative measure for stability. 8. Dustin Mixon: Find N so that we have a quantitative measure for computational efficiency. 9. Ferenc Szollosi: Determine the maximum number of equiangular lines possible in R14. It is known that this number is 28, 29, or 30.10. Ferenc Szollosi: Give a full algebraic classification of all complex ETF's of small parameters\n\nM and N.11. Matt Fickus: Do real equi-modular ETF's exist (ie \" ±1 matrices\") whose redundancy is not approximately 2?12. Ferenc Szollosi: Can we construct ETF's from nonabelian difference sets? 13. Matt Fickus: Are there integrality conditions on the dimensions for the existence of complex equiangular frames? 14. Matt Fickus: Can we find an elementary proof showing that, for any 2( M − 2) 2-D planes in RM, there exists another 2-D plane that has principle angle of π/ 2 with each of the original planes? 15. Dustin Mixon: If we fix N and increase M, does the worst case coherence strictly decrease? 16. Matt Fickus: What is the threshold to know we are in the well of a global minimizer when we optimize over the UNTF's with cost function U (F ) = max m6 =n |〈 ϕn, ϕ m〉| 2?17. Zhiqiang Xu: Suppose x0 ∈ CM and that F = {fj }Nj=1 is a frame in CM. Furthermore, suppose that X0 is k-sparse (ie ‖x0‖0 ≤ k). Now set bj:= |〈 fj, x 0〉|, j = 1, 2,..., N. Then consider the following recovery problems (up to global phase factor): (a) What is the minimum N for which one can recover x0 uniquely for bj, j = 1, 2,..., N.(b) What is the minimum N such that one can recover x0 by solving the l1 minimization:\n\nmin ‖x‖1, s.t. |〈 fj, x 〉| = bj, j = 1, 2,...N.",
   "evidence": "The canonical object is unusual: it combines problems 1--17 from a three-page workshop handout into one database record. The source is John Haas, *Open Problems in Frame Theory/Phase Retrieval*, compiled at the AIM workshop “Frame Theory Intersects Geometry” in August 2013 [AIM13]. The PDF continues with problems 18--21, but those are not part of this canonical record and are not claimed here.",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 203,
   "attempt": 1
  },
  "AIM-GEOMETRY-0205": {
   "statement_status": "exact",
   "original_statement": "18. Boumediene Et-Taoui: Let v(n, r, K) denote the maximum possible number of equi-isoclinic n-planes that can be imbedded in Kr, where K = R, C, or H.(a) For any r ≥ 4, find v(2, 2r, R).(b) Find v(2, 7, R).(c) For any r ≥ 3, find v(3, 3r, R).(d) Find the list of all regular v-tuples in G(4, 4v) whose symmetry groups are isomorphic to the symmetry group Sv.(e) Find v(3, 3r, K), where K = C or H.2(f) Find all (2 M, M ) complex equiangular tight frames. 19. Emily King: Let U denote the space of UNTF's with dimensions (N, M ). If it is known that complex ETF's do not exist in (N, M ), then what is the best one can do? In particular, optimize: \n\nmin \n\n> F∈U\n\nmax \n\n> i6=j\n\n|〈 fi, f j 〉|.",
   "clean_statement": "18. Boumediene Et-Taoui: Let v(n, r, K) denote the maximum possible number of equi-isoclinic n-planes that can be imbedded in Kr, where K = R, C, or H.(a) For any r ≥ 4, find v(2, 2r, R).(b) Find v(2, 7, R).(c) For any r ≥ 3, find v(3, 3r, R).(d) Find the list of all regular v-tuples in G(4, 4v) whose symmetry groups are isomorphic to the symmetry group Sv.(e) Find v(3, 3r, K), where K = C or H.2(f) Find all (2 M, M ) complex equiangular tight frames. 19. Emily King: Let U denote the space of UNTF's with dimensions (N, M ). If it is known that complex ETF's do not exist in (N, M ), then what is the best one can do? In particular, optimize:\n\nmin\n\n> F∈U\n\nmax\n\n> i6=j\n\n|〈 fi, f j 〉|.",
   "public_statement": "18. Boumediene Et-Taoui: Let v(n, r, K) denote the maximum possible number of equi-isoclinic n-planes that can be imbedded in Kr, where K = R, C, or H.(a) For any r ≥ 4, find v(2, 2r, R).(b) Find v(2, 7, R).(c) For any r ≥ 3, find v(3, 3r, R).(d) Find the list of all regular v-tuples in G(4, 4v) whose symmetry groups are isomorphic to the symmetry group Sv.(e) Find v(3, 3r, K), where K = C or H.2(f) Find all (2 M, M ) complex equiangular tight frames. 19. Emily King: Let U denote the space of UNTF's with dimensions (N, M ). If it is known that complex ETF's do not exist in (N, M ), then what is the best one can do? In particular, optimize:\n\nmin\n\n> F∈U\n\nmax\n\n> i6=j\n\n|〈 fi, f j 〉|.",
   "evidence": "The canonical record is an OCR concatenation of Problems 18 and 19 in the three-page AIM list *Open Problems in Frame Theory/Phase Retrieval*, prepared after the August 2013 AIM workshop “Frame theory intersects geometry” [1]. The following repairs were checked against the PDF itself:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 204,
   "attempt": 1
  },
  "AIM-GEOMETRY-0206": {
   "statement_status": "exact",
   "original_statement": "20. Do complex ETF's exist for the parameters M = 3, N = 8?21. Construct a table/catalogue which has entries corresponding to the different pairs (N, M )\n\nfor small N and M, indicating whether complex equiangular frames are known to exist, not to exist, or if the problem remains open. If the problem is settled, then the table should also describe by what methods the information is ascertained. For example, the entry corresponding to (7, 3) would indicate that they exist due to construction via difference sets. On the other hand, the entry for (8, 3) would indicate the question remains open. 3",
   "clean_statement": "20. Do complex ETF's exist for the parameters M = 3, N = 8?21. Construct a table/catalogue which has entries corresponding to the different pairs (N, M )\n\nfor small N and M, indicating whether complex equiangular frames are known to exist, not to exist, or if the problem remains open. If the problem is settled, then the table should also describe by what methods the information is ascertained. For example, the entry corresponding to (7, 3) would indicate that they exist due to construction via difference sets. On the other hand, the entry for (8, 3) would indicate the question remains open. 3",
   "public_statement": "20. Do complex ETF's exist for the parameters M = 3, N = 8?21. Construct a table/catalogue which has entries corresponding to the different pairs (N, M )\n\nfor small N and M, indicating whether complex equiangular frames are known to exist, not to exist, or if the problem remains open. If the problem is settled, then the table should also describe by what methods the information is ascertained. For example, the entry corresponding to (7, 3) would indicate that they exist due to construction via difference sets. On the other hand, the entry for (8, 3) would indicate the question remains open. 3",
   "evidence": "The canonical record is an OCR concatenation of Problems 20 and 21 on page 3 of the AIM list *Open Problems in Frame Theory/Phase Retrieval*, produced after the July--August 2013 workshop “Frame theory intersects geometry” [1]. Inspection of the PDF verifies the following source text, apart from normalized spacing:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 205,
   "attempt": 1
  },
  "AIM-GEOMETRY-0207": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 2.2.1 (Long, Bangert,",
   "clean_statement": null,
   "public_statement": "Conjecture 2.2.1 (Long, Bangert,",
   "evidence": "The canonical record is visibly truncated in the middle of its author attribution, immediately after Bangert's name.",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 206,
   "attempt": 1
  },
  "AIM-GEOMETRY-0208": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 15 from [ ´Alvarez2006]). Every irre-versible Finsler metric on S2 has either exactly 2 or infinitely many distinct prime closed geodesics. \n\nThere exist results supporting this conjecture. In particular, H. Hofer, K. Wysocki and E. Zehnder in [Ho-Wy-Ze2003] studied Reeb orbits on contact S3. Their result can be projected down to S2, and implies that the total number of distinct prime closed geodesics for a bumpy Finsler metric on S2 is either 2 or infinite, provided the stable and unstable manifolds of every hyperbolic closed geodesics intersect transversally. See also A. Harris and G. Paternain in [Har-Pat2008]. The closed geodesics in Katok's example are elliptic. 4 KEITH BURNS AND VLADIMIR S. MATVEEV",
   "clean_statement": "Problem 15 from [ ´Alvarez2006]). Every S2 Finsler metric on S2 has either exactly 2 or infinitely many distinct prime closed geodesics.\n\nThere exist results supporting this conjecture. In particular, H. Hofer, K. Wysocki and E. Zehnder in [Ho-Wy-Ze2003] studied Reeb orbits on contact S3. Their result can be projected down to S2, and implies that the total number of distinct prime closed geodesics for a bumpy Finsler metric on S2 is either 2 or infinite, provided the stable and unstable manifolds of every hyperbolic closed geodesics intersect transversally. See also A. Harris and G. Paternain in [Har-Pat2008]. The closed geodesics in Katok's example are elliptic. 4 KEITH BURNS AND VLADIMIR S. MATVEEV",
   "public_statement": "Problem 15 from [ ´Alvarez2006]). Every irre-versible Finsler metric on S2 has either exactly 2 or infinitely many distinct prime closed geodesics.\n\nThere exist results supporting this conjecture. In particular, H. Hofer, K. Wysocki and E. Zehnder in [Ho-Wy-Ze2003] studied Reeb orbits on contact S3. Their result can be projected down to S2, and implies that the total number of distinct prime closed geodesics for a bumpy Finsler metric on S2 is either 2 or infinite, provided the stable and unstable manifolds of every hyperbolic closed geodesics intersect transversally. See also A. Harris and G. Paternain in [Har-Pat2008]. The closed geodesics in Katok's example are elliptic. 4 KEITH BURNS AND VLADIMIR S. MATVEEV",
   "evidence": "The canonical record comes from Keith Burns and Vladimir S. Matveev, *Open problems and questions about geodesics*, an extended report from the August 2010 International Workshop on Geodesics [BM21]. The paper later appeared in *Ergodic Theory and Dynamical Systems* **41** (2021), 641--684. The relevant passage is on printed pages 3--4 and is numbered Conjecture 2.2.1, although the corpus uses the number 15 inherited from an earlier problem list.",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-geometry-notes.json",
   "source_index": 207,
   "attempt": 1
  },
  "AIM-GEOMETRY-0209": {
   "statement_status": "exact",
   "original_statement": "Conjecture 2.2.2 (Long). The existence of one hyperbolic prime closed geodesic on a Finsler S2 implies the existence of infinitely many distinct prime closed geodesics.",
   "clean_statement": "Conjecture 2.2.2 (Long). The existence of one hyperbolic prime closed geodesic on a Finsler S2 implies the existence of infinitely many distinct prime closed geodesics.",
   "public_statement": "Conjecture 2.2.2 (Long). The existence of one hyperbolic prime closed geodesic on a Finsler S2 implies the existence of infinitely many distinct prime closed geodesics.",
   "evidence": "The record is Conjecture 2.2.2 in the AIM workshop list *Geodesics* (source record 208 of `aim-geometry-notes.json`). The original PDF was checked to restore the superscript that is flattened in the JSON. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 208,
   "attempt": 1
  },
  "AIM-GEOMETRY-0210": {
   "statement_status": "exact",
   "original_statement": "Conjecture 2.2.3 (Long). Every Finsler S2 has at least one elliptic prime closed geodesic. \n\nThe conjecture agrees with a result of Y. Long and W. Wang who proved that there are always at least 2 elliptic prime closed geodesics on every irreversible Finsler S2, if the total number of prime closed geodesics is finite [Lon-Wan2008]. The conjecture does not contradict [Grjuntal1979] where an example of a metric such that all closed simple geodesic are hyperbolic is constructed. Indeed, for certain metrics on the sphere (and even for the metric of certain ellipsoids) most prime closed geodesics are not simple. 2.3. Of complete Riemannian metrics with finite volume.",
   "clean_statement": "Conjecture 2.2.3 (Long). Every Finsler S2 has at least one elliptic prime closed geodesic.\n\nThe conjecture agrees with a result of Y. Long and W. Wang who proved that there are always at least 2 elliptic prime closed geodesics on every irreversible Finsler S2, if the total number of prime closed geodesics is finite [Lon-Wan2008]. The conjecture does not contradict [Grjuntal1979] where an example of a metric such that all closed simple geodesic are hyperbolic is constructed. Indeed, for certain metrics on the sphere (and even for the metric of certain ellipsoids) most prime closed geodesics are not simple. 2.3. Of complete Riemannian metrics with finite volume.",
   "public_statement": "Conjecture 2.2.3 (Long). Every Finsler S2 has at least one elliptic prime closed geodesic.\n\nThe conjecture agrees with a result of Y. Long and W. Wang who proved that there are always at least 2 elliptic prime closed geodesics on every irreversible Finsler S2, if the total number of prime closed geodesics is finite [Lon-Wan2008]. The conjecture does not contradict [Grjuntal1979] where an example of a metric such that all closed simple geodesic are hyperbolic is constructed. Indeed, for certain metrics on the sphere (and even for the metric of certain ellipsoids) most prime closed geodesics are not simple. 2.3. Of complete Riemannian metrics with finite volume.",
   "evidence": "The exact mathematical statement on page 4 of the AIM source is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 209,
   "attempt": 1
  },
  "AIM-GEOMETRY-0211": {
   "statement_status": "exact",
   "original_statement": "Question 2.3.1 (Bangert). Does every complete Riemannian manifold with finite volume have at least one closed geodesic? \n\nThe question was answered affirmatively for dimension 2. Moreover, in dimen-sion 2 a complete Riemannian manifold of finite volume even has infinitely many geometrically different geodesics [Bangert1980]. The argument that was used in the proof is based on the Birkhoff minimax procedure we recalled in the beginning of §2, and does not work in dimensions ≥ 3. One can even hope to construct coun-terexamples in the class of Liouville-integrable geodesic flows. In this case, most orbits of the geodesic flow are rational or irrational windings on the Liouville tori; they are closed, if all of the corresponding frequencies are rational. Since there are essentially ( n − 1) frequencies in dimension n, one can hope that if n > 2 it would be possible to ensure that there is always at least one irrational frequency. Initial attempts to find a counterexample on T 2 × R with the metric of the form \n\na(r)dφ 2 + b(r)dψ 2 + dr 2 were, however, unsuccessful. A more difficult problem would be to prove that on any complete Riemannian manifold of finite volume there exist infinitely many closed geodesics.Questions of this nature are also interesting in the realm of Finsler geometry. One would expect the results for reversible Finsler metrics to be very similar to those for Riemannian metrics. For irreversible Finsler metrics, Katok's example shows that there are compact Finsler manifolds with only finitely many closed geodesics, but it is still possible that all non compact Finsler manifolds with finite volume might have infinitely many closed geodesics. On the other hand, the following question is completely open:",
   "clean_statement": "Question 2.3.1 (Bangert). Does every complete Riemannian manifold with finite volume have at least one closed geodesic?\n\nThe question was answered affirmatively for dimension 2. Moreover, in dimen-sion 2 a complete Riemannian manifold of finite volume even has infinitely many geometrically different geodesics [Bangert1980]. The argument that was used in the proof is based on the Birkhoff minimax procedure we recalled in the beginning of §2, and does not work in dimensions ≥ 3. One can even hope to construct coun-terexamples in the class of Liouville-integrable geodesic flows. In this case, most orbits of the geodesic flow are rational or irrational windings on the Liouville tori; they are closed, if all of the corresponding frequencies are rational. Since there are essentially ( n − 1) frequencies in dimension n, one can hope that if n > 2 it would be possible to ensure that there is always at least one irrational frequency. Initial attempts to find a counterexample on T 2 × R with the metric of the form\n\na(r)dφ 2 + b(r)dψ 2 + dr 2 were, however, unsuccessful. A more difficult problem would be to prove that on any complete Riemannian manifold of finite volume there exist infinitely many closed geodesics.Questions of this nature are also interesting in the realm of Finsler geometry. One would expect the results for reversible Finsler metrics to be very similar to those for Riemannian metrics. For irreversible Finsler metrics, Katok's example shows that there are compact Finsler manifolds with only finitely many closed geodesics, but it is still possible that all non compact Finsler manifolds with finite volume might have infinitely many closed geodesics. On the other hand, the following question is completely open:",
   "public_statement": "Question 2.3.1 (Bangert). Does every complete Riemannian manifold with finite volume have at least one closed geodesic?\n\nThe question was answered affirmatively for dimension 2. Moreover, in dimen-sion 2 a complete Riemannian manifold of finite volume even has infinitely many geometrically different geodesics [Bangert1980]. The argument that was used in the proof is based on the Birkhoff minimax procedure we recalled in the beginning of §2, and does not work in dimensions ≥ 3. One can even hope to construct coun-terexamples in the class of Liouville-integrable geodesic flows. In this case, most orbits of the geodesic flow are rational or irrational windings on the Liouville tori; they are closed, if all of the corresponding frequencies are rational. Since there are essentially ( n − 1) frequencies in dimension n, one can hope that if n > 2 it would be possible to ensure that there is always at least one irrational frequency. Initial attempts to find a counterexample on T 2 × R with the metric of the form\n\na(r)dφ 2 + b(r)dψ 2 + dr 2 were, however, unsuccessful. A more difficult problem would be to prove that on any complete Riemannian manifold of finite volume there exist infinitely many closed geodesics.Questions of this nature are also interesting in the realm of Finsler geometry. One would expect the results for reversible Finsler metrics to be very similar to those for Riemannian metrics. For irreversible Finsler metrics, Katok's example shows that there are compact Finsler manifolds with only finitely many closed geodesics, but it is still possible that all non compact Finsler manifolds with finite volume might have infinitely many closed geodesics. On the other hand, the following question is completely open:",
   "evidence": "This is Question 2.3.1, attributed to Victor Bangert, in the AIM *Geodesics* problem list. The exact `problem` field supplied to this attempt is reproduced verbatim below, including extraction artifacts:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 210,
   "attempt": 1
  },
  "AIM-GEOMETRY-0212": {
   "statement_status": "exact",
   "original_statement": "Question 2.3.2 (Bangert). Does there exist an irreversible Finsler metric of finite volume on R × S1 with no closed geodesics? \n\n2.4. Of magnetic flows on closed surfaces. It is known that the trajectory of a charged particle in the presence of magnetic forces (=\"magnetic geodesic\") is described by a Hamiltonian system with the \"kinetic\" Hamiltonian of the form ∑ \n\n> i,j\n\npipj gij on T ∗M with the symplectic form dp ∧dx +π∗ω, where ω is a closed (but not necessarily exact) form on M and π: T ∗M → M is the canonical projection. We assume that our surface M 2 is closed.",
   "clean_statement": "Question 2.3.2 (Bangert). Does there exist an irreversible Finsler metric of finite volume on R × S1 with no closed geodesics?\n\n2.4. Of magnetic flows on closed surfaces. It is known that the trajectory of a charged particle in the presence of magnetic forces (=\"magnetic geodesic\") is described by a Hamiltonian system with the \"kinetic\" Hamiltonian of the form ∑\n\n> i,j\n\npipj gij on T ∗M with the symplectic form dp ∧dx +π∗ω, where ω is a closed (but not necessarily exact) form on M and π: T ∗M → M is the canonical projection. We assume that our surface M 2 is closed.",
   "public_statement": "Question 2.3.2 (Bangert). Does there exist an irreversible Finsler metric of finite volume on R × S1 with no closed geodesics?\n\n2.4. Of magnetic flows on closed surfaces. It is known that the trajectory of a charged particle in the presence of magnetic forces (=\"magnetic geodesic\") is described by a Hamiltonian system with the \"kinetic\" Hamiltonian of the form ∑\n\n> i,j\n\npipj gij on T ∗M with the symplectic form dp ∧dx +π∗ω, where ω is a closed (but not necessarily exact) form on M and π: T ∗M → M is the canonical projection. We assume that our surface M 2 is closed.",
   "evidence": "The canonical record comes from Keith Burns and Vladimir S. Matveev, *Open problems and questions about geodesics*, an extended report from the August 2010 International Workshop on Geodesics [BM21]. The paper appeared in *Ergodic Theory and Dynamical Systems* **41** (2021), 641--684. The source PDF places the item in Section 2.3, headed “Of complete Riemannian metrics with finite volume.” Its exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 211,
   "attempt": 1
  },
  "AIM-GEOMETRY-0213": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 2.4.1 (Paternain). Is there at least one closed magnetic geodesic in every energy level? OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 5\n\nIf the form is exact, the affirmative answer was obtained by Contreras, Macarini and Paternain in [Co-Ma-Pa2004], which is partially based on [Taimanov1992]. It seems that the standard variational method to solve this problem does not work in this setting, because the form is not exact and therefore corresponds to no Lagrangian. One can try to solve this problem by applying a result of Hofer, Wysocki and Zehnder [Ho-Wy-Ze1993]. In view of this paper, it is sufficient to show that the flow is of contact type. We also refer the reader to a survey [Ginzburg1996]. 3. Path and loop spaces \n\nAs noted in the previous section, one of the main approaches to proving the existence of closed geodesics is to use topological complexity of the loop space Λ M\n\nto force the existence of critical points of the energy functional. Loops with length \n\n≤ T correspond to critical points in Λ T M. Similarly geodesics joining two points \n\np and q can be studied by investigating the path spaces Ω( p, q ) or Ω T (p, q ). The homology of these spaces have been much studied. 3.1. Sums of the Betti numbers. Let p, q be points in a Riemannian manifold. The space Ω T (p, q ) of paths from p to q with length ≤ T has the homotopy type of a finite complex (see eg. [Milnor1963]), and hence the sum of its Betti numbers is finite for each T. The same is true for the space Λ T M of loops with length at most T.",
   "clean_statement": null,
   "public_statement": "Question 2.4.1 (Paternain). Is there at least one closed magnetic geodesic in every energy level? OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 5\n\nIf the form is exact, the affirmative answer was obtained by Contreras, Macarini and Paternain in [Co-Ma-Pa2004], which is partially based on [Taimanov1992]. It seems that the standard variational method to solve this problem does not work in this setting, because the form is not exact and therefore corresponds to no Lagrangian. One can try to solve this problem by applying a result of Hofer, Wysocki and Zehnder [Ho-Wy-Ze1993]. In view of this paper, it is sufficient to show that the flow is of contact type. We also refer the reader to a survey [Ginzburg1996]. 3. Path and loop spaces\n\nAs noted in the previous section, one of the main approaches to proving the existence of closed geodesics is to use topological complexity of the loop space Λ M\n\nto force the existence of critical points of the energy functional. Loops with length\n\n≤ T correspond to critical points in Λ T M. Similarly geodesics joining two points\n\np and q can be studied by investigating the path spaces Ω( p, q ) or Ω T (p, q ). The homology of these spaces have been much studied. 3.1. Sums of the Betti numbers. Let p, q be points in a Riemannian manifold. The space Ω T (p, q ) of paths from p to q with length ≤ T has the homotopy type of a finite complex (see eg. [Milnor1963]), and hence the sum of its Betti numbers is finite for each T. The same is true for the space Λ T M of loops with length at most T.",
   "evidence": "The official AIM PDF places the item in Section 2.4, “Of magnetic flows on closed surfaces.” It first fixes a closed surface \\(M^2\\), the kinetic Hamiltonian on \\(T^*M\\), and the twisted symplectic form obtained from a closed, not necessarily exact, magnetic two-form. The exact question is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 212,
   "attempt": 1
  },
  "AIM-GEOMETRY-0214": {
   "statement_status": "exact",
   "original_statement": "Question 3.1.1 (Paternain). How do the sums of the Betti numbers for ΛT M and \n\nΩ( T (p, q ) grow as T → ∞? Does the growth depend on the metric? \n\nIt was shown by Gromov in [Gromov2001] that if M is simply connected there is a constant C such that sum of the Betti numbers of Λ CN is at least ∑Ni=0 bi(Λ).",
   "clean_statement": "Question 3.1.1 (Paternain). How do the sums of the Betti numbers for ΛT M and\n\nΩ( T (p, q ) grow as T → ∞? Does the growth depend on the metric?\n\nIt was shown by Gromov in [Gromov2001] that if M is simply connected there is a constant C such that sum of the Betti numbers of Λ CN is at least ∑Ni=0 bi(Λ).",
   "public_statement": "Question 3.1.1 (Paternain). How do the sums of the Betti numbers for ΛT M and\n\nΩ( T (p, q ) grow as T → ∞? Does the growth depend on the metric?\n\nIt was shown by Gromov in [Gromov2001] that if M is simply connected there is a constant C such that sum of the Betti numbers of Λ CN is at least ∑Ni=0 bi(Λ).",
   "evidence": "The exact `problem` field supplied for this attempt is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 213,
   "attempt": 1
  },
  "AIM-GEOMETRY-0215": {
   "statement_status": "exact",
   "original_statement": "Question 3.1.2 (Gromov). Does the number of closed geodesics with length ≤ T\n\ngrow exponentially as T → ∞ if the Betti numbers of the loop space grow exponen-tially? \n\nFrom [Gromov1978] it follows that the answer is positive for generic metrics. Here is a potentially interesting example. Consider f: S3 × S3 → S3 × S3 such that the induced action on H3(S3 × S3) is hyperbolic. Let M be the mapping torus for f, i.e. S3 × S3 × [0, 1] with ( x, 1) identified with ( f (x), 0) for each x ∈ S3 × S3.Then π1(M ) = Z and the universal cover ˜M is homotopy equivalent to S3 × S3.Hence the Betti numbers of the loop space grow polynomially. On the other hand, the hyperbolic action of f on H3 gives hope for exponential growth of the sum of the Betti numbers of Ω T (p, q ). 3.2. Stability of minimax levels (communicated by Nancy Hingston). \n\nLet M be a compact manifold with a Riemannian (or Finsler) metric g. Let \n\nG be a finitely generated abelian group. Given a nontrivial homology class \n\nX ∈ H∗(Λ M; G), the critical (minimax) level of X is \n\ncr X = inf {a: X ∈ Image H∗(Λ aM; G)}\n\n= inf \n\n> xǫX\n\nsup \n\n> γ∈Image x\n\nLength (γ)Here Λ aM is the subset of the free loop space Λ M consisting of loops whose length is at most a. The second definition is the minimax definition: the singular chain x6 KEITH BURNS AND VLADIMIR S. MATVEEV \n\nranges over all representatives of the homology class X, and γ over all the points in the image of x, which are loops in M.",
   "clean_statement": "Question 3.1.2 (Gromov). Does the number of closed geodesics with length ≤ T\n\ngrow exponentially as T → ∞ if the Betti numbers of the loop space grow exponen-tially?\n\nFrom [Gromov1978] it follows that the answer is positive for generic metrics. Here is a potentially interesting example. Consider f: S3 × S3 → S3 × S3 such that the induced action on H3(S3 × S3) is hyperbolic. Let M be the mapping torus for f, i.e. S3 × S3 × [0, 1] with ( x, 1) identified with ( f (x), 0) for each x ∈ S3 × S3.Then π1(M ) = Z and the universal cover ˜M is homotopy equivalent to S3 × S3.Hence the Betti numbers of the loop space grow polynomially. On the other hand, the hyperbolic action of f on H3 gives hope for exponential growth of the sum of the Betti numbers of Ω T (p, q ). 3.2. Stability of minimax levels (communicated by Nancy Hingston).\n\nLet M be a compact manifold with a Riemannian (or Finsler) metric g. Let\n\nG be a finitely generated abelian group. Given a nontrivial homology class\n\nX ∈ H∗(Λ M; G), the critical (minimax) level of X is\n\ncr X = inf {a: X ∈ Image H∗(Λ aM; G)}\n\n= inf\n\n> xǫX\n\nsup\n\n> γ∈Image x\n\nLength (γ)Here Λ aM is the subset of the free loop space Λ M consisting of loops whose length is at most a. The second definition is the minimax definition: the singular chain x6 KEITH BURNS AND VLADIMIR S. MATVEEV\n\nranges over all representatives of the homology class X, and γ over all the points in the image of x, which are loops in M.",
   "public_statement": "Question 3.1.2 (Gromov). Does the number of closed geodesics with length ≤ T\n\ngrow exponentially as T → ∞ if the Betti numbers of the loop space grow exponen-tially?\n\nFrom [Gromov1978] it follows that the answer is positive for generic metrics. Here is a potentially interesting example. Consider f: S3 × S3 → S3 × S3 such that the induced action on H3(S3 × S3) is hyperbolic. Let M be the mapping torus for f, i.e. S3 × S3 × [0, 1] with ( x, 1) identified with ( f (x), 0) for each x ∈ S3 × S3.Then π1(M ) = Z and the universal cover ˜M is homotopy equivalent to S3 × S3.Hence the Betti numbers of the loop space grow polynomially. On the other hand, the hyperbolic action of f on H3 gives hope for exponential growth of the sum of the Betti numbers of Ω T (p, q ). 3.2. Stability of minimax levels (communicated by Nancy Hingston).\n\nLet M be a compact manifold with a Riemannian (or Finsler) metric g. Let\n\nG be a finitely generated abelian group. Given a nontrivial homology class\n\nX ∈ H∗(Λ M; G), the critical (minimax) level of X is\n\ncr X = inf {a: X ∈ Image H∗(Λ aM; G)}\n\n= inf\n\n> xǫX\n\nsup\n\n> γ∈Image x\n\nLength (γ)Here Λ aM is the subset of the free loop space Λ M consisting of loops whose length is at most a. The second definition is the minimax definition: the singular chain x6 KEITH BURNS AND VLADIMIR S. MATVEEV\n\nranges over all representatives of the homology class X, and γ over all the points in the image of x, which are loops in M.",
   "evidence": "The record comes from Section 3.1, “Sums of the Betti numbers,” of Keith Burns and Vladimir S. Matveev, *Open problems and questions about geodesics*. The source defines \\(\\Lambda^T M\\) to be the free loops of length at most \\(T\\), notes that this space has finite total Betti number, and asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 214,
   "attempt": 1
  },
  "AIM-GEOMETRY-0216": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 3.2.1 (Hingston). Do there exist a metric on Sn and a homology class \n\nX ∈ H∗(Λ M; Z) with \n\n0 < cr (mX ) < cr (X)\n\nfor some m ∈ N? The simplest case is already interesting: Can we find a metric on \n\nS2 and m ∈ N so that cr (mX ) < cr (X), where X is a generator of H1(Λ S2; Z)?\n\nLet us explain how this question is related to closed geodesics. Given a metric g on M and a finitely generated abelian group G, the global mean frequency is defined as (3.1) αg,G = lim \n\n> deg X→∞\n\ndeg X\n\ncrX,\n\nwhere the limit is taken over all nontrivial homology classes X ∈ H∗(Λ Sn; G). The Resonance Theorem from [Hin-Rad2013] says that if M is a sphere and G\n\nis a field, the limit (3.1) exists. It is clear in this case that αg,G depends on the metric g. But does it really depend on the field G? The degree of X does not depend on anything but X. But what about the critical level cr X? Does cr X \n\ndepend on the coeficients? Let us note that for the spheres the nontrivial homology groups of the free loop space (with integer coefficients) are all Z or Z2 =: Z/2Z. (For odd spheres they are all Z.) Let us look at the case where X ∈ Hk(Λ; Z) = Z. For each m ∈ N\n\nthere is a critical level \n\ncr (mX ) = inf {a: mX ∈ Image H∗(Λ aM )}.\n\nIf j, m ∈ N, then clearly (since Image H∗(Λ a) is an additive subgroup of H∗(Λ)) \n\ncr (jmX ) ≤ cr (mX ). But can there be strict inequality? Here is a little \"example\" to show how this would affect the global mean frequency: Suppose it were the case that there were real numbers a < b < c < d with \n\ncr (mX ) = \n\n\n\nd if gcd( m, 3) = gcd( m, 7) = 1 \n\nc if 3 |m but gcd( m, 7) = 1 \n\nb if 7 |m but gcd( m, 3) = 1 \n\na if 21 |m\n\n",
   "clean_statement": null,
   "public_statement": "Question 3.2.1 (Hingston). Do there exist a metric on Sn and a homology class\n\nX ∈ H∗(Λ M; Z) with\n\n0 < cr (mX ) < cr (X)\n\nfor some m ∈ N? The simplest case is already interesting: Can we find a metric on\n\nS2 and m ∈ N so that cr (mX ) < cr (X), where X is a generator of H1(Λ S2; Z)?\n\nLet us explain how this question is related to closed geodesics. Given a metric g on M and a finitely generated abelian group G, the global mean frequency is defined as (3.1) αg,G = lim\n\n> deg X→∞\n\ndeg X\n\ncrX,\n\nwhere the limit is taken over all nontrivial homology classes X ∈ H∗(Λ Sn; G). The Resonance Theorem from [Hin-Rad2013] says that if M is a sphere and G\n\nis a field, the limit (3.1) exists. It is clear in this case that αg,G depends on the metric g. But does it really depend on the field G? The degree of X does not depend on anything but X. But what about the critical level cr X? Does cr X\n\ndepend on the coeficients? Let us note that for the spheres the nontrivial homology groups of the free loop space (with integer coefficients) are all Z or Z2 =: Z/2Z. (For odd spheres they are all Z.) Let us look at the case where X ∈ Hk(Λ; Z) = Z. For each m ∈ N\n\nthere is a critical level\n\ncr (mX ) = inf {a: mX ∈ Image H∗(Λ aM )}.\n\nIf j, m ∈ N, then clearly (since Image H∗(Λ a) is an additive subgroup of H∗(Λ))\n\ncr (jmX ) ≤ cr (mX ). But can there be strict inequality? Here is a little \"example\" to show how this would affect the global mean frequency: Suppose it were the case that there were real numbers a < b < c < d with\n\ncr (mX ) =\n\n\n\nd if gcd( m, 3) = gcd( m, 7) = 1\n\nc if 3 |m but gcd( m, 7) = 1\n\nb if 7 |m but gcd( m, 3) = 1\n\na if 21 |m\n\n",
   "evidence": "The canonical input is the record at zero-based index 215 of aim-geometry-notes.json. It comes from Section 3.2, “Stability of minimax levels,” of Keith Burns and Vladimir S. Matveev, *Open problems and questions about geodesics*. The source is the AIM PDF",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 215,
   "attempt": 1
  },
  "AIM-GEOMETRY-0217": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 3.2.2 (Hingston).\n\nαg,G = a if G = Q\n\nαg,G = b if G = Z3\n\nαg,G = c if G = Z7\n\nαg,G = a if G = Zp, p 6 = 3, 7.OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 7\n\n4. Curvature conditions and hyperbolicity of the geodesic flow \n\nMany results about geodesics and the geodesic flow assume that all sectional curvatures are negative or one of the following increasingly weaker properties: (1) all sectional curvatures are non positive, (2) no focal points, (3) no conjugate points. These properties can be characterized by the behaviour of Jacobi fields: \n(0) Negative curvature: the length of any (non trivial) Jacobi field orthogonal to a geodesic is a strictly convex function. \n(1) Non positive curvature: the length of any Jacobi field is a convex function. \n(2) No focal points: the length of an initially vanishing Jacobi field is a non decreasing function along a geodesic ray. \n(3) No conjugate points: a (non trivial) Jacobi field can vanish at most once. The no focal point property is equivalent to convexity of spheres in the universal cover. Most interesting results about manifolds with non positive curvature extend readily to manifolds with no focal points. For a compact manifold, negative curvature implies that the geodesic flow is uniformly hyperbolic, in other words an Anosov flow. This means that there is a \n\nDφ t-invariant splitting of the tangent bundle of the unit tangent bundle SM,\n\nT SM = Es ⊕ E0 ⊕ Eu,\n\nin which E0 is the one dimensional subbundle tangent to the orbits of the geodesic flow, and there are constants C ≥ 1 and λ > 0 such that for any t ≥ 0 and any vectors ξ ∈ Es and η ∈ Eu we have (4.1) ‖Dφ t(ξ)‖ ≤ Ce −λt ‖ξ‖ and ‖Dφ −t(η)‖ ≤ Ce −λt ‖η‖.\n\n(Here we have in mind the usual Sasaki metric; the same property would also hold for any equivalent metric with different constants C and λ.) This splitting is H¨ older-continuous, but usually not smooth. The bundles Es and Eu for an Anosov geodesic flow are integrable; their integral foliations are usually denoted by W s and W u. The lifts to the universal cover of the leaves of W s and W u are closely related to horospheres. If ˜v is the lift to S ˜M of \n\nv ∈ SM, the lifts to S ˜M of W s(v) and W u(v) are formed by the unit vectors that are normal to the appropriate horospheres orthogonal to ˜v and are on the same side of the horosphere as ˜v; see the picture below. 8 KEITH BURNS AND VLADIMIR S. MATVEEV \n\n4.1. Relations between these concepts. The notions introduced above are re-lated as follows: negative curvature + 3\n\n\u0013 non positive curvature + 3 no focal points \n\n\u0013\n\nAnosov geodesic flow + 3 no conjugate points That Anosov geodesic flow implies no conjugate points is proved in part B of [Ma˜ n´ e1987]. The class of compact manifolds that support metrics with variable negative curvature is much larger than the class that support hyperbolic metrics (of con-stant negative curvature). The earliest examples of manifolds that support variable but not constant negative curvature were given by Mostow-Siu [Mos-Siu1980] and Gromov-Thurston [Gro-Thu1987]. Recent work of Ontaneda has vastly increased the supply of examples [Ontaneda2011, Ontaneda2014].",
   "clean_statement": null,
   "public_statement": "Conjecture 3.2.2 (Hingston).\n\nαg,G = a if G = Q\n\nαg,G = b if G = Z3\n\nαg,G = c if G = Z7\n\nαg,G = a if G = Zp, p 6 = 3, 7.OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 7\n\n4. Curvature conditions and hyperbolicity of the geodesic flow\n\nMany results about geodesics and the geodesic flow assume that all sectional curvatures are negative or one of the following increasingly weaker properties: (1) all sectional curvatures are non positive, (2) no focal points, (3) no conjugate points. These properties can be characterized by the behaviour of Jacobi fields:\n(0) Negative curvature: the length of any (non trivial) Jacobi field orthogonal to a geodesic is a strictly convex function.\n(1) Non positive curvature: the length of any Jacobi field is a convex function.\n(2) No focal points: the length of an initially vanishing Jacobi field is a non decreasing function along a geodesic ray.\n(3) No conjugate points: a (non trivial) Jacobi field can vanish at most once. The no focal point property is equivalent to convexity of spheres in the universal cover. Most interesting results about manifolds with non positive curvature extend readily to manifolds with no focal points. For a compact manifold, negative curvature implies that the geodesic flow is uniformly hyperbolic, in other words an Anosov flow. This means that there is a\n\nDφ t-invariant splitting of the tangent bundle of the unit tangent bundle SM,\n\nT SM = Es ⊕ E0 ⊕ Eu,\n\nin which E0 is the one dimensional subbundle tangent to the orbits of the geodesic flow, and there are constants C ≥ 1 and λ > 0 such that for any t ≥ 0 and any vectors ξ ∈ Es and η ∈ Eu we have (4.1) ‖Dφ t(ξ)‖ ≤ Ce −λt ‖ξ‖ and ‖Dφ −t(η)‖ ≤ Ce −λt ‖η‖.\n\n(Here we have in mind the usual Sasaki metric; the same property would also hold for any equivalent metric with different constants C and λ.) This splitting is H¨ older-continuous, but usually not smooth. The bundles Es and Eu for an Anosov geodesic flow are integrable; their integral foliations are usually denoted by W s and W u. The lifts to the universal cover of the leaves of W s and W u are closely related to horospheres. If ˜v is the lift to S ˜M of\n\nv ∈ SM, the lifts to S ˜M of W s(v) and W u(v) are formed by the unit vectors that are normal to the appropriate horospheres orthogonal to ˜v and are on the same side of the horosphere as ˜v; see the picture below. 8 KEITH BURNS AND VLADIMIR S. MATVEEV\n\n4.1. Relations between these concepts. The notions introduced above are re-lated as follows: negative curvature + 3\n\n[U+0013] non positive curvature + 3 no focal points\n\n[U+0013]\n\nAnosov geodesic flow + 3 no conjugate points That Anosov geodesic flow implies no conjugate points is proved in part B of [Ma˜ n´ e1987]. The class of compact manifolds that support metrics with variable negative curvature is much larger than the class that support hyperbolic metrics (of con-stant negative curvature). The earliest examples of manifolds that support variable but not constant negative curvature were given by Mostow-Siu [Mos-Siu1980] and Gromov-Thurston [Gro-Thu1987]. Recent work of Ontaneda has vastly increased the supply of examples [Ontaneda2011, Ontaneda2014].",
   "evidence": "The canonical `problem` field begins as follows (line breaks and OCR are retained):",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 216,
   "attempt": 1
  },
  "AIM-GEOMETRY-0218": {
   "statement_status": "exact",
   "original_statement": "Conjecture 4.1.1 (Klingenberg). If a closed manifold admits a metric with Anosov geodesic flow, then it admits a metric with negative sectional curvature. \n\nAll known examples of metrics with Anosov geodesic flow are pertubations of metrics with negative curvature. It is not difficult to show using the uniformisa-tion theorem and the Thurston geometerisation theorem that conjecture is true in dimensions 2 and 3. Klingenberg [Klingenberg1974] showed that seven proper-ties of Riemannian manifolds with negative curvature extend to those with Anosov geodesic flow. One of these is Preissman's theorem [Preissman1943] that the fun-damental group of a manifold with negative sectional curvatures cannot contain a copy of Z × Z.Several examples of manifolds that admit metrics of non positive curvature but cannot support a metric of negative curvature (or with Anosov geodesic flow) can be found in the introduction to [Ba-Br-Eb1985]. These examples have a copy of \n\nZ × Z in their fundamental group. The simplest of them is due to Heintze who took two cusped hyperbolic 3-manifolds, cut off the cusps, glued the two pieces together OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 9\n\nalong their boundary tori and then smoothed out the metric to obtain a manifold with non positive curvature.",
   "clean_statement": "Conjecture 4.1.1 (Klingenberg). If a closed manifold admits a metric with Anosov geodesic flow, then it admits a metric with negative sectional curvature.\n\nAll known examples of metrics with Anosov geodesic flow are pertubations of metrics with negative curvature. It is not difficult to show using the uniformisa-tion theorem and the Thurston geometerisation theorem that conjecture is true in dimensions 2 and 3. Klingenberg [Klingenberg1974] showed that seven proper-ties of Riemannian manifolds with negative curvature extend to those with Anosov geodesic flow. One of these is Preissman's theorem [Preissman1943] that the fun-damental group of a manifold with negative sectional curvatures cannot contain a copy of Z × Z.Several examples of manifolds that admit metrics of non positive curvature but cannot support a metric of negative curvature (or with Anosov geodesic flow) can be found in the introduction to [Ba-Br-Eb1985]. These examples have a copy of\n\nZ × Z in their fundamental group. The simplest of them is due to Heintze who took two cusped hyperbolic 3-manifolds, cut off the cusps, glued the two pieces together OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 9\n\nalong their boundary tori and then smoothed out the metric to obtain a manifold with non positive curvature.",
   "public_statement": "Conjecture 4.1.1 (Klingenberg). If a closed manifold admits a metric with Anosov geodesic flow, then it admits a metric with negative sectional curvature.\n\nAll known examples of metrics with Anosov geodesic flow are pertubations of metrics with negative curvature. It is not difficult to show using the uniformisa-tion theorem and the Thurston geometerisation theorem that conjecture is true in dimensions 2 and 3. Klingenberg [Klingenberg1974] showed that seven proper-ties of Riemannian manifolds with negative curvature extend to those with Anosov geodesic flow. One of these is Preissman's theorem [Preissman1943] that the fun-damental group of a manifold with negative sectional curvatures cannot contain a copy of Z × Z.Several examples of manifolds that admit metrics of non positive curvature but cannot support a metric of negative curvature (or with Anosov geodesic flow) can be found in the introduction to [Ba-Br-Eb1985]. These examples have a copy of\n\nZ × Z in their fundamental group. The simplest of them is due to Heintze who took two cusped hyperbolic 3-manifolds, cut off the cusps, glued the two pieces together OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 9\n\nalong their boundary tori and then smoothed out the metric to obtain a manifold with non positive curvature.",
   "evidence": "The exact conjecture in the canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 217,
   "attempt": 1
  },
  "AIM-GEOMETRY-0219": {
   "statement_status": "exact",
   "original_statement": "Question 4.1.2. If (M, g ) has no conjugate points, does M carry a metric with nonpositive sectional curvature? \n\nAgain it is not difficult to use the uniformisation theorem to show that the answer is affirmative in dimension 2. But the problem is still open even in dimension 3. See [Cro-Sch1986], [Lebedeva2002] and [Iva-Kap2014] for results of the nature that the fundamental groups of manifolds with no conjugate points share properties with those of non positive curvature. Section 8 of [Iva-Kap2014] contains a number of open problems of which we mention:",
   "clean_statement": "Question 4.1.2. If (M, g ) has no conjugate points, does M carry a metric with nonpositive sectional curvature?\n\nAgain it is not difficult to use the uniformisation theorem to show that the answer is affirmative in dimension 2. But the problem is still open even in dimension 3. See [Cro-Sch1986], [Lebedeva2002] and [Iva-Kap2014] for results of the nature that the fundamental groups of manifolds with no conjugate points share properties with those of non positive curvature. Section 8 of [Iva-Kap2014] contains a number of open problems of which we mention:",
   "public_statement": "Question 4.1.2. If (M, g ) has no conjugate points, does M carry a metric with nonpositive sectional curvature?\n\nAgain it is not difficult to use the uniformisation theorem to show that the answer is affirmative in dimension 2. But the problem is still open even in dimension 3. See [Cro-Sch1986], [Lebedeva2002] and [Iva-Kap2014] for results of the nature that the fundamental groups of manifolds with no conjugate points share properties with those of non positive curvature. Section 8 of [Iva-Kap2014] contains a number of open problems of which we mention:",
   "evidence": "The canonical record is Question 4.1.2 in Keith Burns and Vladimir S. Matveev's problem list *Open problems and questions about geodesics*. Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 218,
   "attempt": 1
  },
  "AIM-GEOMETRY-0220": {
   "statement_status": "exact",
   "original_statement": "Question 4.1.3. Is the fundamental group of a closed manifold without conjugate points semihyperbolic? \n\nSemihyperbolicity is a condition introduced by Alonso and Bridson [Alo-Bri1995] to describe non-positive curvature in the large for an arbitrary metric space.",
   "clean_statement": "Question 4.1.3. Is the fundamental group of a closed manifold without conjugate points semihyperbolic?\n\nSemihyperbolicity is a condition introduced by Alonso and Bridson [Alo-Bri1995] to describe non-positive curvature in the large for an arbitrary metric space.",
   "public_statement": "Question 4.1.3. Is the fundamental group of a closed manifold without conjugate points semihyperbolic?\n\nSemihyperbolicity is a condition introduced by Alonso and Bridson [Alo-Bri1995] to describe non-positive curvature in the large for an arbitrary metric space.",
   "evidence": "The source is the AIM *Geodesics* problem list, Question 4.1.3. The record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 219,
   "attempt": 1
  },
  "AIM-GEOMETRY-0221": {
   "statement_status": "exact",
   "original_statement": "Question 4.1.4 (Hermann). Is there an example of a geodesic flow that is partially hyperbolic? \n\nThe geodesic flow is partially hyperbolic if there are a Dφ t-invariant splitting \n\nT SM = Es ⊕ Ec ⊕ Eu\n\nand constants C ≥ 1 and λ > μ > 0 such that (4.1) holds and in addition for all t\n\nand all ζ ∈ Ec we have \n\nC−1e−μ|t|‖ζ‖ ≤ ‖ Dφ t(ζ)‖ ≤ Ce μ|t|‖ζ‖.\n\nAnosov geodesic flows give a degenerate positive answer to the question. Gen-uine examples have been constructed by Carneiro and Pujals; see [Car-Puj2011] and [Car-Puj2013]. They deformed a higher rank locally symmetric space of non compact type. 5. Negative curvature and hyperbolicity of the geodesic flow \n\n5.1. Does the marked length spectrum determine the metric? Suppose (M, g ) is a closed Riemannian manifold. The marked length spectrum of M is the function that assigns to each free homotopy class of loops the infimum of the length of loops in the class (i.e. the length of the shortest closed geodesic lying in this class).",
   "clean_statement": "Question 4.1.4 (Hermann). Is there an example of a geodesic flow that is partially hyperbolic?\n\nThe geodesic flow is partially hyperbolic if there are a Dφ t-invariant splitting\n\nT SM = Es ⊕ Ec ⊕ Eu\n\nand constants C ≥ 1 and λ > μ > 0 such that (4.1) holds and in addition for all t\n\nand all ζ ∈ Ec we have\n\nC−1e−μ|t|‖ζ‖ ≤ ‖ Dφ t(ζ)‖ ≤ Ce μ|t|‖ζ‖.\n\nAnosov geodesic flows give a degenerate positive answer to the question. Gen-uine examples have been constructed by Carneiro and Pujals; see [Car-Puj2011] and [Car-Puj2013]. They deformed a higher rank locally symmetric space of non compact type. 5. Negative curvature and hyperbolicity of the geodesic flow\n\n5.1. Does the marked length spectrum determine the metric? Suppose (M, g ) is a closed Riemannian manifold. The marked length spectrum of M is the function that assigns to each free homotopy class of loops the infimum of the length of loops in the class (i.e. the length of the shortest closed geodesic lying in this class).",
   "public_statement": "Question 4.1.4 (Hermann). Is there an example of a geodesic flow that is partially hyperbolic?\n\nThe geodesic flow is partially hyperbolic if there are a Dφ t-invariant splitting\n\nT SM = Es ⊕ Ec ⊕ Eu\n\nand constants C ≥ 1 and λ > μ > 0 such that (4.1) holds and in addition for all t\n\nand all ζ ∈ Ec we have\n\nC−1e−μ|t|‖ζ‖ ≤ ‖ Dφ t(ζ)‖ ≤ Ce μ|t|‖ζ‖.\n\nAnosov geodesic flows give a degenerate positive answer to the question. Gen-uine examples have been constructed by Carneiro and Pujals; see [Car-Puj2011] and [Car-Puj2013]. They deformed a higher rank locally symmetric space of non compact type. 5. Negative curvature and hyperbolicity of the geodesic flow\n\n5.1. Does the marked length spectrum determine the metric? Suppose (M, g ) is a closed Riemannian manifold. The marked length spectrum of M is the function that assigns to each free homotopy class of loops the infimum of the length of loops in the class (i.e. the length of the shortest closed geodesic lying in this class).",
   "evidence": "The source is the AIM workshop list *Open problems and questions about geodesics*, Question 4.1.4 (attributed to Hermann). The PDF gives the following definition and question. If \\(\\phi^t:SM\\to SM\\) is the geodesic flow, it asks whether there is a \\(D\\phi^t\\)-invariant splitting \\[ T(SM)=E^s\\oplus E^c\\oplus E^u \\] and constants \\(C\\geq 1\\) and \\(\\lambda>\\mu>0\\) such that, for \\(t\\geq0\\), \\[ \\|D\\phi^t\\xi\\|\\leq Ce^{-\\lambda t}\\|\\xi\\|\\quad(\\xi\\in E^s), \\qquad \\|D\\phi^{-t}\\eta\\|\\leq Ce^{-\\lambda t}\\|\\eta\\|\\quad(\\eta\\in E^u), \\tag{4.1} \\] and, for every \\(t\\in\\mathbb R\\) and \\(\\zeta\\in E^c\\), \\[ C^{-1}e^{-\\mu |t|}\\|\\zeta\\| \\leq \\|D\\phi^t\\zeta\\| \\leq Ce^{\\mu |t|}\\|\\zeta\\|. \\tag{4.2} \\] The question is: **is there an example of a geodesic flow satisfying this condition?** The source then observes that an Anosov geodesic flow is a degenerate positive answer and says that genuine examples were...",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 220,
   "attempt": 1
  },
  "AIM-GEOMETRY-0222": {
   "statement_status": "exact",
   "original_statement": "Conjecture 5.1.1 ([Bur-Kat1985]). Two metrics with negative curvature on a compact manifold must be isometric if they have the same marked length spectrum. \n\nCroke and Otal showed that this conjecture is true for metrics on surfaces [Otal1990, Croke1990]. Indeed it is enough to assume that the metrics have non-positive curvature. The problem is open in higher dimensions. Hamenst¨ adt showed that the geodesic flows of the two metrics must be C0-conjugate, thereby reducing the problem to",
   "clean_statement": "Conjecture 5.1.1 ([Bur-Kat1985]). Two metrics with negative curvature on a compact manifold must be isometric if they have the same marked length spectrum.\n\nCroke and Otal showed that this conjecture is true for metrics on surfaces [Otal1990, Croke1990]. Indeed it is enough to assume that the metrics have non-positive curvature. The problem is open in higher dimensions. Hamenst¨ adt showed that the geodesic flows of the two metrics must be C0-conjugate, thereby reducing the problem to",
   "public_statement": "Conjecture 5.1.1 ([Bur-Kat1985]). Two metrics with negative curvature on a compact manifold must be isometric if they have the same marked length spectrum.\n\nCroke and Otal showed that this conjecture is true for metrics on surfaces [Otal1990, Croke1990]. Indeed it is enough to assume that the metrics have non-positive curvature. The problem is open in higher dimensions. Hamenst¨ adt showed that the geodesic flows of the two metrics must be C0-conjugate, thereby reducing the problem to",
   "evidence": "The canonical record is Conjecture 5.1.1 in Keith Burns and Vladimir S. Matveev's problem list *Open problems and questions about geodesics*. The exact canonical input is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 221,
   "attempt": 1
  },
  "AIM-GEOMETRY-0223": {
   "statement_status": "exact",
   "original_statement": "Conjecture 5.2.1 in the next subsection [Hamenst¨ adt1991]. One obtains a natural interesting modification of this conjecture by replacing negative curvature by nonexistence of conjugate points. On the other hand some restriction on the metrics is necessary as the examples of Croke and Kleiner which we describe in the next subsection provide non-isometric metrics with the same 10 KEITH BURNS AND VLADIMIR S. MATVEEV \n\nmarked length spectrum. It is also necessary to consider the marked length spec-trum rather than the length spectrum. Vign´ eras gave examples of non-isometric hyperbolic surfaces that have the same set of lengths for their closed geodesics [Vign´ eras1980]. One can ask a similar question for non-manifolds. Consider a 2-dimensional (metrical) simplicial complex such that every simplex is hyperbolic with geodesic edges and such that CAT (−1) condition holds on every vertex. Assume in addition that every edge is contained in at least two simplices.",
   "clean_statement": "Conjecture 5.2.1 in the next subsection [Hamenst¨ adt1991]. One obtains a natural interesting modification of this conjecture by replacing negative curvature by nonexistence of conjugate points. On the other hand some restriction on the metrics is necessary as the examples of Croke and Kleiner which we describe in the next subsection provide non-isometric metrics with the same 10 KEITH BURNS AND VLADIMIR S. MATVEEV\n\nmarked length spectrum. It is also necessary to consider the marked length spec-trum rather than the length spectrum. Vign´ eras gave examples of non-isometric hyperbolic surfaces that have the same set of lengths for their closed geodesics [Vign´ eras1980]. One can ask a similar question for non-manifolds. Consider a 2-dimensional (metrical) simplicial complex such that every simplex is hyperbolic with geodesic edges and such that CAT (−1) condition holds on every vertex. Assume in addition that every edge is contained in at least two simplices.",
   "public_statement": "Conjecture 5.2.1 in the next subsection [Hamenst¨ adt1991]. One obtains a natural interesting modification of this conjecture by replacing negative curvature by nonexistence of conjugate points. On the other hand some restriction on the metrics is necessary as the examples of Croke and Kleiner which we describe in the next subsection provide non-isometric metrics with the same 10 KEITH BURNS AND VLADIMIR S. MATVEEV\n\nmarked length spectrum. It is also necessary to consider the marked length spec-trum rather than the length spectrum. Vign´ eras gave examples of non-isometric hyperbolic surfaces that have the same set of lengths for their closed geodesics [Vign´ eras1980]. One can ask a similar question for non-manifolds. Consider a 2-dimensional (metrical) simplicial complex such that every simplex is hyperbolic with geodesic edges and such that CAT (−1) condition holds on every vertex. Assume in addition that every edge is contained in at least two simplices.",
   "evidence": "The canonical record is preserved verbatim in `input.json`. It begins",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 222,
   "attempt": 1
  },
  "AIM-GEOMETRY-0224": {
   "statement_status": "exact",
   "original_statement": "Question 5.1.2 (Schmidt). Does the marked length spectrum determine such a metric (in the class of all locally CAT (−1) metrics on this space, or in the class of all 2-dimensional (metrical) simplicial complexes with the above properties homeo-morphic to the given complex)? \n\nIf the simplicial complex is (topologically) a manifold, the answer is positive and is due to [Her-Pau1997]. One can also also ask the question in higher dimensions. An easier, but still interesting version of the question is when we assume that every edge is contained in at least three simplices. The questions above are closely related to the boundary rigidity problem, which we now recall. Given a compact manifold M with smooth boundary N, a Riemann-ian metric g on M induces a nonnegative real valued function d on N × N where \n\nd(p, q ) is the distance in ( M, g ) between p and q. We call ( M, g ) boundary rigid if a Riemannian manifold g′ on M that induces the same function on N × N must be isometric to g′.",
   "clean_statement": "Question 5.1.2 (Schmidt). Does the marked length spectrum determine such a metric (in the class of all locally CAT (−1) metrics on this space, or in the class of all 2-dimensional (metrical) simplicial complexes with the above properties homeo-morphic to the given complex)?\n\nIf the simplicial complex is (topologically) a manifold, the answer is positive and is due to [Her-Pau1997]. One can also also ask the question in higher dimensions. An easier, but still interesting version of the question is when we assume that every edge is contained in at least three simplices. The questions above are closely related to the boundary rigidity problem, which we now recall. Given a compact manifold M with smooth boundary N, a Riemann-ian metric g on M induces a nonnegative real valued function d on N × N where\n\nd(p, q ) is the distance in ( M, g ) between p and q. We call ( M, g ) boundary rigid if a Riemannian manifold g′ on M that induces the same function on N × N must be isometric to g′.",
   "public_statement": "Question 5.1.2 (Schmidt). Does the marked length spectrum determine such a metric (in the class of all locally CAT (−1) metrics on this space, or in the class of all 2-dimensional (metrical) simplicial complexes with the above properties homeo-morphic to the given complex)?\n\nIf the simplicial complex is (topologically) a manifold, the answer is positive and is due to [Her-Pau1997]. One can also also ask the question in higher dimensions. An easier, but still interesting version of the question is when we assume that every edge is contained in at least three simplices. The questions above are closely related to the boundary rigidity problem, which we now recall. Given a compact manifold M with smooth boundary N, a Riemann-ian metric g on M induces a nonnegative real valued function d on N × N where\n\nd(p, q ) is the distance in ( M, g ) between p and q. We call ( M, g ) boundary rigid if a Riemannian manifold g′ on M that induces the same function on N × N must be isometric to g′.",
   "evidence": "The exact record begins:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 223,
   "attempt": 1
  },
  "AIM-GEOMETRY-0225": {
   "statement_status": "exact",
   "original_statement": "Question 5.1.3 ([Michel1981]). What conditions on M, N and g imply boundary rigidity? \n\nOne can modify this question by requiring that the other Riemannian metric \n\ng′ in the definition of boundary rigid manifolds above also satisfies some addi-tional assumption. Special cases of the last question were answered in [Croke1990, Cr-Da-Sh2000, Sha-Uhl2000, Bur-Iva2013]. One can also ask the boundary rigidity question for Finsler metrics; recent ref-erences with nontrivial results include [Coo-Del2010, Bur-Iva2010]. 5.2. The conjugacy problem. Two Riemannian manifolds ( M1, g 1) and ( M2, g 2)have Ck-conjugate geodesic flows if there is an invertible map h: SM \n\n> 1\n\n→ SM \n\n> 2\n\nsuch that h and h−1 are Ck and h ◦ φ1 \n\n> t\n\n= φ2 \n\n> t\n\n◦ h for all t. Here φit is the geodesic flow for the metric gi. A long standing conjecture is",
   "clean_statement": "Question 5.1.3 ([Michel1981]). What conditions on M, N and g imply boundary rigidity?\n\nOne can modify this question by requiring that the other Riemannian metric\n\ng′ in the definition of boundary rigid manifolds above also satisfies some addi-tional assumption. Special cases of the last question were answered in [Croke1990, Cr-Da-Sh2000, Sha-Uhl2000, Bur-Iva2013]. One can also ask the boundary rigidity question for Finsler metrics; recent ref-erences with nontrivial results include [Coo-Del2010, Bur-Iva2010]. 5.2. The conjugacy problem. Two Riemannian manifolds ( M1, g 1) and ( M2, g 2)have Ck-conjugate geodesic flows if there is an invertible map h: SM\n\n> 1\n\n→ SM\n\n> 2\n\nsuch that h and h−1 are Ck and h ◦ φ1\n\n> t\n\n= φ2\n\n> t\n\n◦ h for all t. Here φit is the geodesic flow for the metric gi. A long standing conjecture is",
   "public_statement": "Question 5.1.3 ([Michel1981]). What conditions on M, N and g imply boundary rigidity?\n\nOne can modify this question by requiring that the other Riemannian metric\n\ng′ in the definition of boundary rigid manifolds above also satisfies some addi-tional assumption. Special cases of the last question were answered in [Croke1990, Cr-Da-Sh2000, Sha-Uhl2000, Bur-Iva2013]. One can also ask the boundary rigidity question for Finsler metrics; recent ref-erences with nontrivial results include [Coo-Del2010, Bur-Iva2010]. 5.2. The conjugacy problem. Two Riemannian manifolds ( M1, g 1) and ( M2, g 2)have Ck-conjugate geodesic flows if there is an invertible map h: SM\n\n> 1\n\n→ SM\n\n> 2\n\nsuch that h and h−1 are Ck and h ◦ φ1\n\n> t\n\n= φ2\n\n> t\n\n◦ h for all t. Here φit is the geodesic flow for the metric gi. A long standing conjecture is",
   "evidence": "The canonical record comes from Question 5.1.3 of the AIM problem list *Geometry of geodesics and related topics*. In the official PDF, the preceding paragraph defines the data. If (M) is a compact manifold with smooth boundary (N=\\partial M), a Riemannian metric (g) determines \\[ d_g^\\partial(p,q)=d_g(p,q),\\qquad (p,q)\\in N\\times N. \\] The question itself is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 224,
   "attempt": 1
  },
  "AIM-GEOMETRY-0226": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 5.2.1. Compact Riemannian manifolds with negative curvature must be isometric if they have C0-conjugate geodesic flows. As with",
   "clean_statement": null,
   "public_statement": "Conjecture 5.2.1. Compact Riemannian manifolds with negative curvature must be isometric if they have C0-conjugate geodesic flows. As with",
   "evidence": "The exact canonical record is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 225,
   "attempt": 1
  },
  "AIM-GEOMETRY-0227": {
   "statement_status": "exact",
   "original_statement": "Conjecture 5.1.1 it may be possible to relax the hypothesis of nega-tive curvature to no conjugate points, but some restriction on the metrics is re-quired. Weinstein pointed out that all of the Zoll metrics on S2 have conjugate geodesic flows. Using the explicit rotationally-symmetric examples of Zoll metrics (Tannery metrics in the terminology of [Besse1978, Chapter 4]) Croke and Kleiner constructed examples of different metrics on an arbitrary closed manifold such that their geodesic flows are conjugate and their marked length spectra coincide, see [Cro-Klei1994]. The conjecture is true in dimension two; indeed it is enough to assume that one of the manifolds has nonpositive curvature [Cr-Fa-Fe1992]. It follows from OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 11 \n\nthe theorem of Besson-Courtois-Gallot discussed in the next subsection that the conjecture holds if one of the metrics is locally symmetric. Croke and Kleiner showed that C1 conjugacy implies that the volumes of the manifolds are the same with no restrictions on the metrics and implies isometry of the metrics if one of them has a global parallel vector (as is the case in a Riemannian product) [Cro-Klei1994]. 5.3. Entropy and locally symmetric spaces with negative curvature. Let \n\nhLiou denote the entropy of the geodesic flow with respect to the Liouville measure, \n\nhtop its topological entropy and hvol the volume entropy \n\nhvol = lim \n\n> r→∞\n\n1\n\nr log Vol B˜M (p, r ),\n\nwhere B˜M (p, r ) is the ball of radius r around a point p in the universal cover ˜M.It is always true that hvol ≤ htop and they are equal if there are no conjugate points, in particular if the curvature is negative [Manning1979, Fre-Ma˜ n1982]. By the variational principle for entropy hLiou ≤ htop for any metric; these entropies are equal for a locally symmetric metric. Katok [Katok1982] showed that if g is an arbitrary metric and g0 a metric of constant negative curvature on a surface of genus ≥ 2 such that area( g) = area( g0), then \n\nhLiou (g) ≤ hLiou (g0) = htop (g0) ≤ htop (g)and both inequalities are strict unless g also has constant negative curvature. He explicitly formulated the following influential and still open conjecture:",
   "clean_statement": "Conjecture 5.1.1 it may be possible to relax the hypothesis of nega-tive curvature to no conjugate points, but some restriction on the metrics is re-quired. Weinstein pointed out that all of the Zoll metrics on S2 have conjugate geodesic flows. Using the explicit rotationally-symmetric examples of Zoll metrics (Tannery metrics in the terminology of [Besse1978, Chapter 4]) Croke and Kleiner constructed examples of different metrics on an arbitrary closed manifold such that their geodesic flows are conjugate and their marked length spectra coincide, see [Cro-Klei1994]. The conjecture is true in dimension two; indeed it is enough to assume that one of the manifolds has nonpositive curvature [Cr-Fa-Fe1992]. It follows from OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 11\n\nthe theorem of Besson-Courtois-Gallot discussed in the next subsection that the conjecture holds if one of the metrics is locally symmetric. Croke and Kleiner showed that C1 conjugacy implies that the volumes of the manifolds are the same with no restrictions on the metrics and implies isometry of the metrics if one of them has a global parallel vector (as is the case in a Riemannian product) [Cro-Klei1994]. 5.3. Entropy and locally symmetric spaces with negative curvature. Let\n\nhLiou denote the entropy of the geodesic flow with respect to the Liouville measure,\n\nhtop its topological entropy and hvol the volume entropy\n\nhvol = lim\n\n> r→∞\n\n1\n\nr log Vol B˜M (p, r ),\n\nwhere B˜M (p, r ) is the ball of radius r around a point p in the universal cover ˜M.It is always true that hvol ≤ htop and they are equal if there are no conjugate points, in particular if the curvature is negative [Manning1979, Fre-Ma˜ n1982]. By the variational principle for entropy hLiou ≤ htop for any metric; these entropies are equal for a locally symmetric metric. Katok [Katok1982] showed that if g is an arbitrary metric and g0 a metric of constant negative curvature on a surface of genus ≥ 2 such that area( g) = area( g0), then\n\nhLiou (g) ≤ hLiou (g0) = htop (g0) ≤ htop (g)and both inequalities are strict unless g also has constant negative curvature. He explicitly formulated the following influential and still open conjecture:",
   "public_statement": "Conjecture 5.1.1 it may be possible to relax the hypothesis of nega-tive curvature to no conjugate points, but some restriction on the metrics is re-quired. Weinstein pointed out that all of the Zoll metrics on S2 have conjugate geodesic flows. Using the explicit rotationally-symmetric examples of Zoll metrics (Tannery metrics in the terminology of [Besse1978, Chapter 4]) Croke and Kleiner constructed examples of different metrics on an arbitrary closed manifold such that their geodesic flows are conjugate and their marked length spectra coincide, see [Cro-Klei1994]. The conjecture is true in dimension two; indeed it is enough to assume that one of the manifolds has nonpositive curvature [Cr-Fa-Fe1992]. It follows from OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 11\n\nthe theorem of Besson-Courtois-Gallot discussed in the next subsection that the conjecture holds if one of the metrics is locally symmetric. Croke and Kleiner showed that C1 conjugacy implies that the volumes of the manifolds are the same with no restrictions on the metrics and implies isometry of the metrics if one of them has a global parallel vector (as is the case in a Riemannian product) [Cro-Klei1994]. 5.3. Entropy and locally symmetric spaces with negative curvature. Let\n\nhLiou denote the entropy of the geodesic flow with respect to the Liouville measure,\n\nhtop its topological entropy and hvol the volume entropy\n\nhvol = lim\n\n> r→∞\n\n1\n\nr log Vol B˜M (p, r ),\n\nwhere B˜M (p, r ) is the ball of radius r around a point p in the universal cover ˜M.It is always true that hvol ≤ htop and they are equal if there are no conjugate points, in particular if the curvature is negative [Manning1979, Fre-Ma˜ n1982]. By the variational principle for entropy hLiou ≤ htop for any metric; these entropies are equal for a locally symmetric metric. Katok [Katok1982] showed that if g is an arbitrary metric and g0 a metric of constant negative curvature on a surface of genus ≥ 2 such that area( g) = area( g0), then\n\nhLiou (g) ≤ hLiou (g0) = htop (g0) ≤ htop (g)and both inequalities are strict unless g also has constant negative curvature. He explicitly formulated the following influential and still open conjecture:",
   "evidence": "The exact extracted object is preserved in `input.json`. It is not an independent conjecture. Comparison with the adjacent canonical records and with the original AIM PDF gives the following boundary audit.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 226,
   "attempt": 1
  },
  "AIM-GEOMETRY-0228": {
   "statement_status": "exact",
   "original_statement": "Conjecture 5.3.1 ([Katok1982]). htop = hLiou for a metric of negative curvature on a compact manifold if and only if it is locally symmetric. Katok's arguments extend to higher dimensions provided the two metrics are conformally equivalent. Flaminio showed that",
   "clean_statement": "Conjecture 5.3.1 ([Katok1982]). htop = hLiou for a metric of negative curvature on a compact manifold if and only if it is locally symmetric. Katok's arguments extend to higher dimensions provided the two metrics are conformally equivalent. Flaminio showed that",
   "public_statement": "Conjecture 5.3.1 ([Katok1982]). htop = hLiou for a metric of negative curvature on a compact manifold if and only if it is locally symmetric. Katok's arguments extend to higher dimensions provided the two metrics are conformally equivalent. Flaminio showed that",
   "evidence": "The canonical JSON record stops in the middle of a sentence:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 227,
   "attempt": 1
  },
  "AIM-GEOMETRY-0229": {
   "statement_status": "exact",
   "original_statement": "Conjecture 5.3.1 holds for deforma-tions of constant curvature metrics [Flaminio1995]. Katok's paper implicity raised the questions of whether in higher dimensions \n\nhtop is minimized and hLiou maximized (among metrics of negative curvature of fixed volume on a given manifold) by the locally symmetric metrics. It is now known that the locally symmetric metrics are critical points for both entropies [Ka-Kn-We1991], but the locally symmetric spaces do not maximize hLiou except in the two dimensional case [Flaminio1995]. The topological entropy, however, is minimized. Gromov [Gromov1983] conjectured that if f: ( M, g ) → (M0, g 0) is a continuous map of degree d 6 = 0 between compact connected oriented n-dimensional Riemann-ian manifolds and M0 has constant negative curvature, then \n\nhvol (g)nVol( M, g ) ≥ | d|hvol (g0)nVol( M0, g 0)with equality if and only if g also has constant negative curvature and f is homotopic to a d-sheeted covering. The quantity hnVol has the advantage of being invariant under homothetic rescaling of the manifold; this obviates the assumption that the two manifolds have the same volume. \n\nTheorem 5.3.2 ([Be-Co-Ga1995]). The above conjecture of Gromov is true even when (M0, g 0) is allowed to be a locally symmetric space of negative curvature. \n\nWe refer the reader to Section 9 of [Be-Co-Ga1995] and the survey [Eberlein2001] for a list of the many corollaries of this remarkable theorem. 12 KEITH BURNS AND VLADIMIR S. MATVEEV \n\n5.4. Regularity of the Anosov structure. An extensive list of results about the regularity of the Anosov splitting for the geodesic flow in negative curvature (and for other Anosov systems) can be found in the introduction to [Hasselblatt1994]. In dimension two, or if the curvature is 1 /4-pinched, the splitting is always at least \n\nC1. In higher dimensions it is always H¨ older continuous but typically not C1. It is natural to ask:",
   "clean_statement": "Conjecture 5.3.1 holds for deforma-tions of constant curvature metrics [Flaminio1995]. Katok's paper implicity raised the questions of whether in higher dimensions\n\nhtop is minimized and hLiou maximized (among metrics of negative curvature of fixed volume on a given manifold) by the locally symmetric metrics. It is now known that the locally symmetric metrics are critical points for both entropies [Ka-Kn-We1991], but the locally symmetric spaces do not maximize hLiou except in the two dimensional case [Flaminio1995]. The topological entropy, however, is minimized. Gromov [Gromov1983] conjectured that if f: ( M, g ) → (M0, g 0) is a continuous map of degree d 6 = 0 between compact connected oriented n-dimensional Riemann-ian manifolds and M0 has constant negative curvature, then\n\nhvol (g)nVol( M, g ) ≥ | d|hvol (g0)nVol( M0, g 0)with equality if and only if g also has constant negative curvature and f is homotopic to a d-sheeted covering. The quantity hnVol has the advantage of being invariant under homothetic rescaling of the manifold; this obviates the assumption that the two manifolds have the same volume.\n\nTheorem 5.3.2 ([Be-Co-Ga1995]). The above conjecture of Gromov is true even when (M0, g 0) is allowed to be a locally symmetric space of negative curvature.\n\nWe refer the reader to Section 9 of [Be-Co-Ga1995] and the survey [Eberlein2001] for a list of the many corollaries of this remarkable theorem. 12 KEITH BURNS AND VLADIMIR S. MATVEEV\n\n5.4. Regularity of the Anosov structure. An extensive list of results about the regularity of the Anosov splitting for the geodesic flow in negative curvature (and for other Anosov systems) can be found in the introduction to [Hasselblatt1994]. In dimension two, or if the curvature is 1 /4-pinched, the splitting is always at least\n\nC1. In higher dimensions it is always H¨ older continuous but typically not C1. It is natural to ask:",
   "public_statement": "Conjecture 5.3.1 holds for deforma-tions of constant curvature metrics [Flaminio1995]. Katok's paper implicity raised the questions of whether in higher dimensions\n\nhtop is minimized and hLiou maximized (among metrics of negative curvature of fixed volume on a given manifold) by the locally symmetric metrics. It is now known that the locally symmetric metrics are critical points for both entropies [Ka-Kn-We1991], but the locally symmetric spaces do not maximize hLiou except in the two dimensional case [Flaminio1995]. The topological entropy, however, is minimized. Gromov [Gromov1983] conjectured that if f: ( M, g ) → (M0, g 0) is a continuous map of degree d 6 = 0 between compact connected oriented n-dimensional Riemann-ian manifolds and M0 has constant negative curvature, then\n\nhvol (g)nVol( M, g ) ≥ | d|hvol (g0)nVol( M0, g 0)with equality if and only if g also has constant negative curvature and f is homotopic to a d-sheeted covering. The quantity hnVol has the advantage of being invariant under homothetic rescaling of the manifold; this obviates the assumption that the two manifolds have the same volume.\n\nTheorem 5.3.2 ([Be-Co-Ga1995]). The above conjecture of Gromov is true even when (M0, g 0) is allowed to be a locally symmetric space of negative curvature.\n\nWe refer the reader to Section 9 of [Be-Co-Ga1995] and the survey [Eberlein2001] for a list of the many corollaries of this remarkable theorem. 12 KEITH BURNS AND VLADIMIR S. MATVEEV\n\n5.4. Regularity of the Anosov structure. An extensive list of results about the regularity of the Anosov splitting for the geodesic flow in negative curvature (and for other Anosov systems) can be found in the introduction to [Hasselblatt1994]. In dimension two, or if the curvature is 1 /4-pinched, the splitting is always at least\n\nC1. In higher dimensions it is always H¨ older continuous but typically not C1. It is natural to ask:",
   "evidence": "The canonical record is index 228 (zero based) in `aim-geometry-notes.json`, extracted from Keith Burns and Vladimir S. Matveev, *Open Problems and Questions About Geodesics*, Section 5.3 and the first paragraph of Section 5.4. Its exact extracted `problem` field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 228,
   "attempt": 1
  },
  "AIM-GEOMETRY-0230": {
   "statement_status": "exact",
   "original_statement": "Question 5.4.1. Does there exist a closed Riemannian manifold of negative curva-ture such that Es and Eu are C1 smooth, but the dimension is > 2 and the metric is not 1/4-pinched?Another motivation for this question is the question of Pollicott in the next subsection. A program to construct open sets of metrics for whose geodesic flows have Anosov splitting that have low regularity of the Anosove splitting on large subsets of the unit tangent bundle is explained in section 4 of [Has-Wil1999]; see in particular Proposition 12. The idea is to perturb a suitable base example. This base example must have directions in its unstable manifolds with widely different expansion rates. Unfortunately there are no known examples of geodesic flows that are suitable bases for the perturbation argument; see the discussion at the end of section 4 in [Has-Wil1999]. Along all geodesics in complex hyperbolic space one has Lyapunov exponents of 1 and 2 corresponding to parallel families of planes with curvature −1and −4 respectively.",
   "clean_statement": "Question 5.4.1. Does there exist a closed Riemannian manifold of negative curva-ture such that Es and Eu are C1 smooth, but the dimension is > 2 and the metric is not 1/4-pinched?Another motivation for this question is the question of Pollicott in the next subsection. A program to construct open sets of metrics for whose geodesic flows have Anosov splitting that have low regularity of the Anosove splitting on large subsets of the unit tangent bundle is explained in section 4 of [Has-Wil1999]; see in particular Proposition 12. The idea is to perturb a suitable base example. This base example must have directions in its unstable manifolds with widely different expansion rates. Unfortunately there are no known examples of geodesic flows that are suitable bases for the perturbation argument; see the discussion at the end of section 4 in [Has-Wil1999]. Along all geodesics in complex hyperbolic space one has Lyapunov exponents of 1 and 2 corresponding to parallel families of planes with curvature −1and −4 respectively.",
   "public_statement": "Question 5.4.1. Does there exist a closed Riemannian manifold of negative curva-ture such that Es and Eu are C1 smooth, but the dimension is > 2 and the metric is not 1/4-pinched?Another motivation for this question is the question of Pollicott in the next subsection. A program to construct open sets of metrics for whose geodesic flows have Anosov splitting that have low regularity of the Anosove splitting on large subsets of the unit tangent bundle is explained in section 4 of [Has-Wil1999]; see in particular Proposition 12. The idea is to perturb a suitable base example. This base example must have directions in its unstable manifolds with widely different expansion rates. Unfortunately there are no known examples of geodesic flows that are suitable bases for the perturbation argument; see the discussion at the end of section 4 in [Has-Wil1999]. Along all geodesics in complex hyperbolic space one has Lyapunov exponents of 1 and 2 corresponding to parallel families of planes with curvature −1and −4 respectively.",
   "evidence": "The record is Question 5.4.1 in Keith Burns and Vladimir Matveev, *Open problems and questions about geodesics*. The AIM PDF was inspected at the start of Section 5.4 and through the beginning of Question 5.4.2 so that the record boundary was clear. The mathematical question, with only OCR repairs, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 229,
   "attempt": 1
  },
  "AIM-GEOMETRY-0231": {
   "statement_status": "exact",
   "original_statement": "Question 5.4.2. Is there a metric of negative curvature for which the ratio of largest positive Lyapunov exponent to smallest positive Lyapunov exponent is greater than 2 for typical geodesics (not just exceptional closed geodesics)? The Anosov splitting is C∞ only if and only the metric is locally symmetric. This follows from [Be-Fo-La1992] and [Be-Co-Ga1995]. For surfaces Hurder and Katok [Hur-Kat1990] showed that the splitting must be C∞ if one of the stable or unstable foliations is C1+ o(x log |x|). In higher dimensions it is expected that the splitting must be C∞ if it is C2, but this question still seems to be open. 5.5. How many closed geodesics of length ≤ T exist? (Communicated by Pollicott). Let ( M, g ) be a closed Riemannian manifold with negative sectional curvatures. It is known [Margulis2004] that the number N (T ) of closed geodesics of length ≤ T grows asymptotically as ehT /hT.",
   "clean_statement": "Question 5.4.2. Is there a metric of negative curvature for which the ratio of largest positive Lyapunov exponent to smallest positive Lyapunov exponent is greater than 2 for typical geodesics (not just exceptional closed geodesics)? The Anosov splitting is C∞ only if and only the metric is locally symmetric. This follows from [Be-Fo-La1992] and [Be-Co-Ga1995]. For surfaces Hurder and Katok [Hur-Kat1990] showed that the splitting must be C∞ if one of the stable or unstable foliations is C1+ o(x log |x|). In higher dimensions it is expected that the splitting must be C∞ if it is C2, but this question still seems to be open. 5.5. How many closed geodesics of length ≤ T exist? (Communicated by Pollicott). Let ( M, g ) be a closed Riemannian manifold with negative sectional curvatures. It is known [Margulis2004] that the number N (T ) of closed geodesics of length ≤ T grows asymptotically as ehT /hT.",
   "public_statement": "Question 5.4.2. Is there a metric of negative curvature for which the ratio of largest positive Lyapunov exponent to smallest positive Lyapunov exponent is greater than 2 for typical geodesics (not just exceptional closed geodesics)? The Anosov splitting is C∞ only if and only the metric is locally symmetric. This follows from [Be-Fo-La1992] and [Be-Co-Ga1995]. For surfaces Hurder and Katok [Hur-Kat1990] showed that the splitting must be C∞ if one of the stable or unstable foliations is C1+ o(x log |x|). In higher dimensions it is expected that the splitting must be C∞ if it is C2, but this question still seems to be open. 5.5. How many closed geodesics of length ≤ T exist? (Communicated by Pollicott). Let ( M, g ) be a closed Riemannian manifold with negative sectional curvatures. It is known [Margulis2004] that the number N (T ) of closed geodesics of length ≤ T grows asymptotically as ehT /hT.",
   "evidence": "The canonical record is Question 5.4.2 in Keith Burns and Vladimir Matveev's *Open problems and questions about geodesics*, source file `aim-geometry-notes.json`, zero-based index 230. The exact question in the source PDF is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 230,
   "attempt": 1
  },
  "AIM-GEOMETRY-0232": {
   "statement_status": "exact",
   "original_statement": "Conjecture 5.5.1. N (T ) = (1 + O(e−εT )) \n\n∫ ehT \n\n> 2\n\ndu \n\nlog u.\n\nThe conjecture is true for all metrics such that the stable and unstable bundles \n\nEs and Eu are C1 smooth, which is the case for surfaces and for 1/4 pinched metrics.",
   "clean_statement": "Conjecture 5.5.1. N (T ) = (1 + O(e−εT ))\n\n∫ ehT\n\n> 2\n\ndu\n\nlog u.\n\nThe conjecture is true for all metrics such that the stable and unstable bundles\n\nEs and Eu are C1 smooth, which is the case for surfaces and for 1/4 pinched metrics.",
   "public_statement": "Conjecture 5.5.1. N (T ) = (1 + O(e−εT ))\n\n∫ ehT\n\n> 2\n\ndu\n\nlog u.\n\nThe conjecture is true for all metrics such that the stable and unstable bundles\n\nEs and Eu are C1 smooth, which is the case for surfaces and for 1/4 pinched metrics.",
   "evidence": "The canonical record is zero-based index 231 of aim-geometry-notes.json. Its exact extracted problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 231,
   "attempt": 1
  },
  "AIM-GEOMETRY-0233": {
   "statement_status": "exact",
   "original_statement": "Question 5.4.1 above asks whether there are other examples. 5.6. Infinitely many simple closed geodesics.",
   "clean_statement": "Question 5.4.1 above asks whether there are other examples. 5.6. Infinitely many simple closed geodesics.",
   "public_statement": "Question 5.4.1 above asks whether there are other examples. 5.6. Infinitely many simple closed geodesics.",
   "evidence": "The source text contains no unresolved assertion, interrogative sentence, or quantifier of its own. The correct classification is therefore `context_only`, with `problem_status_at_run: not_a_problem`. The extracted text is preserved verbatim in `input.json`; the reconstruction above separates rather than rewrites its two fragments.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 232,
   "attempt": 1
  },
  "AIM-GEOMETRY-0234": {
   "statement_status": "exact",
   "original_statement": "Question 5.6.1 (Miller [Miller2001], Reid). Are there infinitely many simple closed geodesics in every hyperbolic three-manifold of finite volume? \n\nOne can ask the same question for manifolds of variable negative curvature in any dimension. The answer is positive for surfaces, since in this case a free homotopy OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 13 \n\nclass contains a simple closed geodesic if and only if it contains a simple closed curve. The answer is also positive for generic metrics of negative curvature. Reid [Reid1993] has examples of (arithmetic) hyperbolic manifolds of finite vol-ume in which every closed geodesic is simple. 6. Nonpositive curvature and non uniform hyperbolicity of the geodesic flow \n\nRecall that the rank of a vector v in such a manifold is the dimension of the space of Jacobi fields along the geodesic γv that are covariantly constant. It is easily seen that rank is upper semi continuous. All vectors have rank ≥ 1, since the velocity vector field is a covariantly constant Jacobi field. The rank of the manifold is the minimum rank of a vector. The set of vectors of minimum rank is obviously open and is known to be dense [Ballmann1982]. The definitions generalize the classical notion of rank for locally symmetric spaces of non compact type. 6.1. Ergodicity of geodesic flows.",
   "clean_statement": "Question 5.6.1 (Miller [Miller2001], Reid). Are there infinitely many simple closed geodesics in every hyperbolic three-manifold of finite volume?\n\nOne can ask the same question for manifolds of variable negative curvature in any dimension. The answer is positive for surfaces, since in this case a free homotopy OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 13\n\nclass contains a simple closed geodesic if and only if it contains a simple closed curve. The answer is also positive for generic metrics of negative curvature. Reid [Reid1993] has examples of (arithmetic) hyperbolic manifolds of finite vol-ume in which every closed geodesic is simple. 6. Nonpositive curvature and non uniform hyperbolicity of the geodesic flow\n\nRecall that the rank of a vector v in such a manifold is the dimension of the space of Jacobi fields along the geodesic γv that are covariantly constant. It is easily seen that rank is upper semi continuous. All vectors have rank ≥ 1, since the velocity vector field is a covariantly constant Jacobi field. The rank of the manifold is the minimum rank of a vector. The set of vectors of minimum rank is obviously open and is known to be dense [Ballmann1982]. The definitions generalize the classical notion of rank for locally symmetric spaces of non compact type. 6.1. Ergodicity of geodesic flows.",
   "public_statement": "Question 5.6.1 (Miller [Miller2001], Reid). Are there infinitely many simple closed geodesics in every hyperbolic three-manifold of finite volume?\n\nOne can ask the same question for manifolds of variable negative curvature in any dimension. The answer is positive for surfaces, since in this case a free homotopy OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 13\n\nclass contains a simple closed geodesic if and only if it contains a simple closed curve. The answer is also positive for generic metrics of negative curvature. Reid [Reid1993] has examples of (arithmetic) hyperbolic manifolds of finite vol-ume in which every closed geodesic is simple. 6. Nonpositive curvature and non uniform hyperbolicity of the geodesic flow\n\nRecall that the rank of a vector v in such a manifold is the dimension of the space of Jacobi fields along the geodesic γv that are covariantly constant. It is easily seen that rank is upper semi continuous. All vectors have rank ≥ 1, since the velocity vector field is a covariantly constant Jacobi field. The rank of the manifold is the minimum rank of a vector. The set of vectors of minimum rank is obviously open and is known to be dense [Ballmann1982]. The definitions generalize the classical notion of rank for locally symmetric spaces of non compact type. 6.1. Ergodicity of geodesic flows.",
   "evidence": "The canonical record is Question 5.6.1 in Keith Burns and Vladimir S. Matveev's problem list. After checking the original PDF, the recoverable statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 233,
   "attempt": 1
  },
  "AIM-GEOMETRY-0235": {
   "statement_status": "exact",
   "original_statement": "Question 6.1.1. Is the geodesic flow of a metric of non positive curvature on a closed surface of genus ≥ 2 ergodic with respect to the Liouville measure? \n\nIt is known that the flow is ergodic on the set of geodesics passing through points where the curvature is negative [Pesin1977]. The complement of this set consists of vectors tangent to zero curvature geodesics, i.e. geodesics along which the Gaussian curvature is always zero. It is not known whether this set must have measure zero. There is an analogous question in higher dimensions: is the geodesic flow of a closed rank one manifold of non positive curvature ergodic? Here it is known that the flow is ergodic on the set of rank one vectors [Bal-Bri1982, Burns1983], but it is not known if the complementary set (of higher rank vectors) must have measure zero. Several papers published in the 1980s (notably [Burns1983], [Bal-Bri1982], and [Ba-Br-Eb1985]) stated that this problem had been solved. These claims were based on an incorrect proof that the set of higher rank vectors must have measure zero. The measure considered above is the Liouville measure. There is a (unique) measure of maximal entropy for the geodesic flow of a rank one manifold. It was constructed by Knieper, who proved that it is ergodic [Knieper1998]. 6.2. Zero curvature geodesics and flat strips. Consider a closed surface of genus ≥ 2.",
   "clean_statement": "Question 6.1.1. Is the geodesic flow of a metric of non positive curvature on a closed surface of genus ≥ 2 ergodic with respect to the Liouville measure?\n\nIt is known that the flow is ergodic on the set of geodesics passing through points where the curvature is negative [Pesin1977]. The complement of this set consists of vectors tangent to zero curvature geodesics, i.e. geodesics along which the Gaussian curvature is always zero. It is not known whether this set must have measure zero. There is an analogous question in higher dimensions: is the geodesic flow of a closed rank one manifold of non positive curvature ergodic? Here it is known that the flow is ergodic on the set of rank one vectors [Bal-Bri1982, Burns1983], but it is not known if the complementary set (of higher rank vectors) must have measure zero. Several papers published in the 1980s (notably [Burns1983], [Bal-Bri1982], and [Ba-Br-Eb1985]) stated that this problem had been solved. These claims were based on an incorrect proof that the set of higher rank vectors must have measure zero. The measure considered above is the Liouville measure. There is a (unique) measure of maximal entropy for the geodesic flow of a rank one manifold. It was constructed by Knieper, who proved that it is ergodic [Knieper1998]. 6.2. Zero curvature geodesics and flat strips. Consider a closed surface of genus ≥ 2.",
   "public_statement": "Question 6.1.1. Is the geodesic flow of a metric of non positive curvature on a closed surface of genus ≥ 2 ergodic with respect to the Liouville measure?\n\nIt is known that the flow is ergodic on the set of geodesics passing through points where the curvature is negative [Pesin1977]. The complement of this set consists of vectors tangent to zero curvature geodesics, i.e. geodesics along which the Gaussian curvature is always zero. It is not known whether this set must have measure zero. There is an analogous question in higher dimensions: is the geodesic flow of a closed rank one manifold of non positive curvature ergodic? Here it is known that the flow is ergodic on the set of rank one vectors [Bal-Bri1982, Burns1983], but it is not known if the complementary set (of higher rank vectors) must have measure zero. Several papers published in the 1980s (notably [Burns1983], [Bal-Bri1982], and [Ba-Br-Eb1985]) stated that this problem had been solved. These claims were based on an incorrect proof that the set of higher rank vectors must have measure zero. The measure considered above is the Liouville measure. There is a (unique) measure of maximal entropy for the geodesic flow of a rank one manifold. It was constructed by Knieper, who proved that it is ergodic [Knieper1998]. 6.2. Zero curvature geodesics and flat strips. Consider a closed surface of genus ≥ 2.",
   "evidence": "The canonical record comes from Keith Burns and Vladimir S. Matveev, *Open problems and questions about geodesics*, §6.1, Question 6.1.1. Its mathematical question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 234,
   "attempt": 1
  },
  "AIM-GEOMETRY-0236": {
   "statement_status": "exact",
   "original_statement": "Question 6.2.1. [Burns] Is there a C∞ (or at least Ck, k ≥ 3) metric for which there exists a non closed geodesic along which the Gaussian curvature is everywhere zero? \n\nA negative answer to this question would give a positive answer to",
   "clean_statement": "Question 6.2.1. [Burns] Is there a C∞ (or at least Ck, k ≥ 3) metric for which there exists a non closed geodesic along which the Gaussian curvature is everywhere zero?\n\nA negative answer to this question would give a positive answer to",
   "public_statement": "Question 6.2.1. [Burns] Is there a C∞ (or at least Ck, k ≥ 3) metric for which there exists a non closed geodesic along which the Gaussian curvature is everywhere zero?\n\nA negative answer to this question would give a positive answer to",
   "evidence": "The canonical JSON record is visibly truncated. Its exact `problem` field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 235,
   "attempt": 1
  },
  "AIM-GEOMETRY-0237": {
   "statement_status": "exact",
   "original_statement": "Question 6.1.1. If we assume that the curvature is only C0, then a metric with such a geodesic can be constructed. Each end of the geodesic is asymptotic to a closed geodesic. But a geodesic along which the curvature is zero cannot spiral into a closed geodesic if the curvature is C2; see [Ruggiero1998]. Wu [Wu2013] has recently given a negative answer to the question under the assumption that the subset of the surface where the curvature is negative has finitely many components; he also shows under this hypothesis that only finitely many free homotopy classes can contain zero curvature closed geodesics. 14 KEITH BURNS AND VLADIMIR S. MATVEEV \n\nA flat strip is a totally geodesic isometric immersion of the Riemannian product of R with an interval. Zero curvature geodesics can be viewed as flat strips with infinitesimal width. It is known that all flat strips (with positive width) must close up in the R-direction; in other words they are really immersions of the product of S1 with an interval. Furthermore for each δ > 0 there are only finitely many flat strips with width ≥ δ. Proofs can be found in a preprint of Cao and Xavier [Cao-Xav]. The main idea is that if two flat strips of width δ cross each other at a very shallow angle, then their intersection contains a long rectangle with width close to 2 δ.It is possible that there are only finitely many flat strips. This is true when the set where the curvature is negative has finitely many components [Wu2013], but the only effort to prove it in general [Rodriguez Hertz2003] was unsuccessful.",
   "clean_statement": "Question 6.1.1. If we assume that the curvature is only C0, then a metric with such a geodesic can be constructed. Each end of the geodesic is asymptotic to a closed geodesic. But a geodesic along which the curvature is zero cannot spiral into a closed geodesic if the curvature is C2; see [Ruggiero1998]. Wu [Wu2013] has recently given a negative answer to the question under the assumption that the subset of the surface where the curvature is negative has finitely many components; he also shows under this hypothesis that only finitely many free homotopy classes can contain zero curvature closed geodesics. 14 KEITH BURNS AND VLADIMIR S. MATVEEV\n\nA flat strip is a totally geodesic isometric immersion of the Riemannian product of R with an interval. Zero curvature geodesics can be viewed as flat strips with infinitesimal width. It is known that all flat strips (with positive width) must close up in the R-direction; in other words they are really immersions of the product of S1 with an interval. Furthermore for each δ > 0 there are only finitely many flat strips with width ≥ δ. Proofs can be found in a preprint of Cao and Xavier [Cao-Xav]. The main idea is that if two flat strips of width δ cross each other at a very shallow angle, then their intersection contains a long rectangle with width close to 2 δ.It is possible that there are only finitely many flat strips. This is true when the set where the curvature is negative has finitely many components [Wu2013], but the only effort to prove it in general [Rodriguez Hertz2003] was unsuccessful.",
   "public_statement": "Question 6.1.1. If we assume that the curvature is only C0, then a metric with such a geodesic can be constructed. Each end of the geodesic is asymptotic to a closed geodesic. But a geodesic along which the curvature is zero cannot spiral into a closed geodesic if the curvature is C2; see [Ruggiero1998]. Wu [Wu2013] has recently given a negative answer to the question under the assumption that the subset of the surface where the curvature is negative has finitely many components; he also shows under this hypothesis that only finitely many free homotopy classes can contain zero curvature closed geodesics. 14 KEITH BURNS AND VLADIMIR S. MATVEEV\n\nA flat strip is a totally geodesic isometric immersion of the Riemannian product of R with an interval. Zero curvature geodesics can be viewed as flat strips with infinitesimal width. It is known that all flat strips (with positive width) must close up in the R-direction; in other words they are really immersions of the product of S1 with an interval. Furthermore for each δ > 0 there are only finitely many flat strips with width ≥ δ. Proofs can be found in a preprint of Cao and Xavier [Cao-Xav]. The main idea is that if two flat strips of width δ cross each other at a very shallow angle, then their intersection contains a long rectangle with width close to 2 δ.It is possible that there are only finitely many flat strips. This is true when the set where the curvature is negative has finitely many components [Wu2013], but the only effort to prove it in general [Rodriguez Hertz2003] was unsuccessful.",
   "evidence": "This record is not an independent question. It is the continuation of the discussion following Question 6.2.1 in Keith Burns and Vladimir Matveev's AIM problem list *Open Problems and Questions About Geodesics*. The PDF places the material in Section 6.2, “Zero curvature geodesics and flat strips.” Question 6.2.1, which belongs to the preceding canonical record, asks whether a closed surface of genus at least two admits a smooth (or at least \\(C^k\\), \\(k\\geq 3\\)) nonpositively curved metric with a nonclosed geodesic on which the Gaussian curvature vanishes identically.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 236,
   "attempt": 1
  },
  "AIM-GEOMETRY-0238": {
   "statement_status": "exact",
   "original_statement": "Question 6.2.1 still makes sense even if the curvature of the manifold is not restricted to be non positive. It might generalize also to higher dimensions. What can be said about a geodesic along which all sectional curvatures are 0? Or less stringently the sectional curvatures of planes containing the tangent vector? 6.3. Flats in rank one manifolds with nonpositive curvature. A flat is a flat strip of infinite width, in other words a totally geodesic isometric immersion of the Euclidean plane. Eberlein asked whether a compact rank one manifold that contains a flat must contain a closed flat, i.e. a totally geodesic isometric immersion of a flat torus. Bangert and Schroeder gave an affirmative answer for real analytic metrics [Ban-Sch1991]. The problem is open for C∞ metrics. 6.4. Besson-Courtois-Gallot. Does their rigidity theorem from [Be-Co-Ga1995], Theorem 5.3.2 in this paper, generalize to higher rank symmetric spaces of non compact type? Connell and Farb [Con-Far2003a, Con-Far2003b] extended the barycenter method, which plays a vital role in [Be-Co-Ga1995], and proved that the theorem holds for a product in which each factor is a symmetric space with negative curvature and dimension ≥ 3. We also refer the reader to their extensive survey in [Con-Far2003c]. A recent preprint of Merlin [Merlin2014] uses a calibration argument to show that h4\n\n> vol\n\nVol is minimized by the locally symmetric metric on a compact quotient of the product of two hyperbolic planes. 6.5. Closed geodesics. Consider a closed surface of genus ≥ 2 with a metric of nonpositive curvature. Let N (T ) be the number of free homotopy classes containing closed geodesics along which the Gaussian curvature is everywhere zero.",
   "clean_statement": "Question 6.2.1 still makes sense even if the curvature of the manifold is not restricted to be non positive. It might generalize also to higher dimensions. What can be said about a geodesic along which all sectional curvatures are 0? Or less stringently the sectional curvatures of planes containing the tangent vector? 6.3. Flats in rank one manifolds with nonpositive curvature. A flat is a flat strip of infinite width, in other words a totally geodesic isometric immersion of the Euclidean plane. Eberlein asked whether a compact rank one manifold that contains a flat must contain a closed flat, i.e. a totally geodesic isometric immersion of a flat torus. Bangert and Schroeder gave an affirmative answer for real analytic metrics [Ban-Sch1991]. The problem is open for C∞ metrics. 6.4. Besson-Courtois-Gallot. Does their rigidity theorem from [Be-Co-Ga1995], Theorem 5.3.2 in this paper, generalize to higher rank symmetric spaces of non compact type? Connell and Farb [Con-Far2003a, Con-Far2003b] extended the barycenter method, which plays a vital role in [Be-Co-Ga1995], and proved that the theorem holds for a product in which each factor is a symmetric space with negative curvature and dimension ≥ 3. We also refer the reader to their extensive survey in [Con-Far2003c]. A recent preprint of Merlin [Merlin2014] uses a calibration argument to show that h4\n\n> vol\n\nVol is minimized by the locally symmetric metric on a compact quotient of the product of two hyperbolic planes. 6.5. Closed geodesics. Consider a closed surface of genus ≥ 2 with a metric of nonpositive curvature. Let N (T ) be the number of free homotopy classes containing closed geodesics along which the Gaussian curvature is everywhere zero.",
   "public_statement": "Question 6.2.1 still makes sense even if the curvature of the manifold is not restricted to be non positive. It might generalize also to higher dimensions. What can be said about a geodesic along which all sectional curvatures are 0? Or less stringently the sectional curvatures of planes containing the tangent vector? 6.3. Flats in rank one manifolds with nonpositive curvature. A flat is a flat strip of infinite width, in other words a totally geodesic isometric immersion of the Euclidean plane. Eberlein asked whether a compact rank one manifold that contains a flat must contain a closed flat, i.e. a totally geodesic isometric immersion of a flat torus. Bangert and Schroeder gave an affirmative answer for real analytic metrics [Ban-Sch1991]. The problem is open for C∞ metrics. 6.4. Besson-Courtois-Gallot. Does their rigidity theorem from [Be-Co-Ga1995], Theorem 5.3.2 in this paper, generalize to higher rank symmetric spaces of non compact type? Connell and Farb [Con-Far2003a, Con-Far2003b] extended the barycenter method, which plays a vital role in [Be-Co-Ga1995], and proved that the theorem holds for a product in which each factor is a symmetric space with negative curvature and dimension ≥ 3. We also refer the reader to their extensive survey in [Con-Far2003c]. A recent preprint of Merlin [Merlin2014] uses a calibration argument to show that h4\n\n> vol\n\nVol is minimized by the locally symmetric metric on a compact quotient of the product of two hyperbolic planes. 6.5. Closed geodesics. Consider a closed surface of genus ≥ 2 with a metric of nonpositive curvature. Let N (T ) be the number of free homotopy classes containing closed geodesics along which the Gaussian curvature is everywhere zero.",
   "evidence": "The canonical record is a merge of several consecutive passages on page 14 of the AIM problem list (PDF page index 13). Its boundaries matter.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 237,
   "attempt": 1
  },
  "AIM-GEOMETRY-0239": {
   "statement_status": "exact",
   "original_statement": "Question 6.5.1 (Knieper). Suppose N (T ) grows subexponentially, i.e. \n\nlim \n\n> T→∞\n\nlog N (T )\n\nT = 0.\n\nIs there a quadratic upper bound on N (T ), i.e., does there exist C > 0 such that \n\nN (T ) ≤ C · T 2?\n\nThe answer is positive for the torus. One can also study this problem in the Finsler category. OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 15 \n\n7. Manifolds without conjugate points (rigidity conjectures) \n\nThere are many rigidity results for manifolds with non positive curvature that might extend to manifolds with no conjugate points. One still has the basic setting of a universal cover homeomorphic to Rn in which any two points are joined by a unique geodesic. However the convexity of the length of Jacobi fields, which is the basis of many arguments used in non positive curvature, is no longer available. Two major results of this nature are: \n\nTheorem 7.0.2. A Riemannian metric with no conjugate points on a torus is flat. \n\nTheorem 7.0.3. Let g be a complete Riemannian metric without conjugate points on the plane R2. Then for every point p\n\nlim inf \n\n> r→∞\n\narea B(p, r )\n\nπr 2 ≥ 1,\n\nwith equality if and only if g is flat. \n\nThe two dimensional case of Theorem 7.0.2 was proved by E. Hopf [Hopf1948] and the general case by Burago and Ivanov [Bur-Iva1994]. The Lorentzian ana-logue of Theorem 7.0.2 is false; two dimensional counterexamples are constructed in [Bav-Mou2013]. Theorem 7.0.3 is a recent result of Bangert and Emmerich [Ban-Emm2013], which greatly improved on earlier results in [Bur-Kni1991] and [Ban-Emm2011]. Both results are easy for manifolds with non positive curvature but require subtle arguments in the context of no conjugate points. Bangert and Emmerich's work was motivated by the following conjecture, which they prove using their theorem:",
   "clean_statement": "Question 6.5.1 (Knieper). Suppose N (T ) grows subexponentially, i.e.\n\nlim\n\n> T→∞\n\nlog N (T )\n\nT = 0.\n\nIs there a quadratic upper bound on N (T ), i.e., does there exist C > 0 such that\n\nN (T ) ≤ C · T 2?\n\nThe answer is positive for the torus. One can also study this problem in the Finsler category. OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 15\n\n7. Manifolds without conjugate points (rigidity conjectures)\n\nThere are many rigidity results for manifolds with non positive curvature that might extend to manifolds with no conjugate points. One still has the basic setting of a universal cover homeomorphic to Rn in which any two points are joined by a unique geodesic. However the convexity of the length of Jacobi fields, which is the basis of many arguments used in non positive curvature, is no longer available. Two major results of this nature are:\n\nTheorem 7.0.2. A Riemannian metric with no conjugate points on a torus is flat.\n\nTheorem 7.0.3. Let g be a complete Riemannian metric without conjugate points on the plane R2. Then for every point p\n\nlim inf\n\n> r→∞\n\narea B(p, r )\n\nπr 2 ≥ 1,\n\nwith equality if and only if g is flat.\n\nThe two dimensional case of Theorem 7.0.2 was proved by E. Hopf [Hopf1948] and the general case by Burago and Ivanov [Bur-Iva1994]. The Lorentzian ana-logue of Theorem 7.0.2 is false; two dimensional counterexamples are constructed in [Bav-Mou2013]. Theorem 7.0.3 is a recent result of Bangert and Emmerich [Ban-Emm2013], which greatly improved on earlier results in [Bur-Kni1991] and [Ban-Emm2011]. Both results are easy for manifolds with non positive curvature but require subtle arguments in the context of no conjugate points. Bangert and Emmerich's work was motivated by the following conjecture, which they prove using their theorem:",
   "public_statement": "Question 6.5.1 (Knieper). Suppose N (T ) grows subexponentially, i.e.\n\nlim\n\n> T→∞\n\nlog N (T )\n\nT = 0.\n\nIs there a quadratic upper bound on N (T ), i.e., does there exist C > 0 such that\n\nN (T ) ≤ C · T 2?\n\nThe answer is positive for the torus. One can also study this problem in the Finsler category. OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 15\n\n7. Manifolds without conjugate points (rigidity conjectures)\n\nThere are many rigidity results for manifolds with non positive curvature that might extend to manifolds with no conjugate points. One still has the basic setting of a universal cover homeomorphic to Rn in which any two points are joined by a unique geodesic. However the convexity of the length of Jacobi fields, which is the basis of many arguments used in non positive curvature, is no longer available. Two major results of this nature are:\n\nTheorem 7.0.2. A Riemannian metric with no conjugate points on a torus is flat.\n\nTheorem 7.0.3. Let g be a complete Riemannian metric without conjugate points on the plane R2. Then for every point p\n\nlim inf\n\n> r→∞\n\narea B(p, r )\n\nπr 2 ≥ 1,\n\nwith equality if and only if g is flat.\n\nThe two dimensional case of Theorem 7.0.2 was proved by E. Hopf [Hopf1948] and the general case by Burago and Ivanov [Bur-Iva1994]. The Lorentzian ana-logue of Theorem 7.0.2 is false; two dimensional counterexamples are constructed in [Bav-Mou2013]. Theorem 7.0.3 is a recent result of Bangert and Emmerich [Ban-Emm2013], which greatly improved on earlier results in [Bur-Kni1991] and [Ban-Emm2011]. Both results are easy for manifolds with non positive curvature but require subtle arguments in the context of no conjugate points. Bangert and Emmerich's work was motivated by the following conjecture, which they prove using their theorem:",
   "evidence": "The original AIM PDF places this record at the end of Section 6.5, “Closed geodesics.” The preceding two sentences, which the canonical extraction assigned to AIM-GEOMETRY-0238, are indispensable:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 238,
   "attempt": 1
  },
  "AIM-GEOMETRY-0240": {
   "statement_status": "exact",
   "original_statement": "Conjecture 7.0.4 (Bangert, Burns and Knieper). Consider a complete Riemann-ian metric without conjugate points on the cylinder R × S1. Assume that the ends spread sublinearly, i.e. \n\nlim \n\n> d(p,p 0)→∞\n\nl(p)\n\ndist (p, p 0) = 0,\n\nwhere l(p) is the length of the shortest geodesic loop with the based at p. Then the metric is flat. \n\n7.1. Divergence of geodesics. Let α and β be two geodesics in a complete simply connected Riemannian manifold without conjugate points.",
   "clean_statement": "Conjecture 7.0.4 (Bangert, Burns and Knieper). Consider a complete Riemann-ian metric without conjugate points on the cylinder R × S1. Assume that the ends spread sublinearly, i.e.\n\nlim\n\n> d(p,p 0)→∞\n\nl(p)\n\ndist (p, p 0) = 0,\n\nwhere l(p) is the length of the shortest geodesic loop with the based at p. Then the metric is flat.\n\n7.1. Divergence of geodesics. Let α and β be two geodesics in a complete simply connected Riemannian manifold without conjugate points.",
   "public_statement": "Conjecture 7.0.4 (Bangert, Burns and Knieper). Consider a complete Riemann-ian metric without conjugate points on the cylinder R × S1. Assume that the ends spread sublinearly, i.e.\n\nlim\n\n> d(p,p 0)→∞\n\nl(p)\n\ndist (p, p 0) = 0,\n\nwhere l(p) is the length of the shortest geodesic loop with the based at p. Then the metric is flat.\n\n7.1. Divergence of geodesics. Let α and β be two geodesics in a complete simply connected Riemannian manifold without conjugate points.",
   "evidence": "The canonical record contains OCR and extraction defects:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 239,
   "attempt": 1
  },
  "AIM-GEOMETRY-0241": {
   "statement_status": "exact",
   "original_statement": "Question 7.1.1. Suppose α(0) = β(0). Does dist (α(t), β (t)) → ∞ as t → ∞?\n\nThe answer is positive in dimension 2 in [Green1954]. The question is open in higher dimensions; the proof in [Green1956] is incorrect. A closely related question is:",
   "clean_statement": "Question 7.1.1. Suppose α(0) = β(0). Does dist (α(t), β (t)) → ∞ as t → ∞?\n\nThe answer is positive in dimension 2 in [Green1954]. The question is open in higher dimensions; the proof in [Green1956] is incorrect. A closely related question is:",
   "public_statement": "Question 7.1.1. Suppose α(0) = β(0). Does dist (α(t), β (t)) → ∞ as t → ∞?\n\nThe answer is positive in dimension 2 in [Green1954]. The question is open in higher dimensions; the proof in [Green1956] is incorrect. A closely related question is:",
   "evidence": "The original AIM PDF places the record in §7, “Manifolds without conjugate points (rigidity conjectures).” Immediately before Question 7.1.1 it says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 240,
   "attempt": 1
  },
  "AIM-GEOMETRY-0242": {
   "statement_status": "exact",
   "original_statement": "Question 7.1.2. Suppose dist (α(t), β (t)) → 0 as t → −∞. Does dist (α(t), β (t)) →∞ as t → ∞?\n\nWithout positive anwers to these questions there would seem to be little hope for a satisfactory analogue of the sphere at infinity, which plays a prominent role in the case of non positive curvature. 16 KEITH BURNS AND VLADIMIR S. MATVEEV \n\n7.2. Parallel postulate questions. \n\nThe flat strip theorem is another basic tool in the study of manifolds with non positive curvature. It states that \"parallel geodesics\" must bound a flat strip. More precisely if if α and β are two geodesics in a simply connected manifold with non positive curvature such that dist (α(t), β (t)) is bounded for all t ∈ R, then the two geodesics are the edges of a totally geodesic isometric immersion of the Riemannian product of R with an interval. The flat strip theorem fails for manifolds with no conjugate points (although it does extend fairly easily to manifolds without focal points). Counter examples have been given in [Burns1992] and by Kleiner (unpublished). Kleiner's example has a copy of Z × Z in its fundamental group, but does not contain a corresponding flat torus. He perturbs the Heintze example (Example 4 in [Ba-Br-Eb1985]), which we described in subsection 4.1. The following basic question appears to be open in general (although Proposition 4 in [Eschenburg1977] suggests an affirmative answer under some extra hypotheses).",
   "clean_statement": "Question 7.1.2. Suppose dist (α(t), β (t)) → 0 as t → −∞. Does dist (α(t), β (t)) →∞ as t → ∞?\n\nWithout positive anwers to these questions there would seem to be little hope for a satisfactory analogue of the sphere at infinity, which plays a prominent role in the case of non positive curvature. 16 KEITH BURNS AND VLADIMIR S. MATVEEV\n\n7.2. Parallel postulate questions.\n\nThe flat strip theorem is another basic tool in the study of manifolds with non positive curvature. It states that \"parallel geodesics\" must bound a flat strip. More precisely if if α and β are two geodesics in a simply connected manifold with non positive curvature such that dist (α(t), β (t)) is bounded for all t ∈ R, then the two geodesics are the edges of a totally geodesic isometric immersion of the Riemannian product of R with an interval. The flat strip theorem fails for manifolds with no conjugate points (although it does extend fairly easily to manifolds without focal points). Counter examples have been given in [Burns1992] and by Kleiner (unpublished). Kleiner's example has a copy of Z × Z in its fundamental group, but does not contain a corresponding flat torus. He perturbs the Heintze example (Example 4 in [Ba-Br-Eb1985]), which we described in subsection 4.1. The following basic question appears to be open in general (although Proposition 4 in [Eschenburg1977] suggests an affirmative answer under some extra hypotheses).",
   "public_statement": "Question 7.1.2. Suppose dist (α(t), β (t)) → 0 as t → −∞. Does dist (α(t), β (t)) →∞ as t → ∞?\n\nWithout positive anwers to these questions there would seem to be little hope for a satisfactory analogue of the sphere at infinity, which plays a prominent role in the case of non positive curvature. 16 KEITH BURNS AND VLADIMIR S. MATVEEV\n\n7.2. Parallel postulate questions.\n\nThe flat strip theorem is another basic tool in the study of manifolds with non positive curvature. It states that \"parallel geodesics\" must bound a flat strip. More precisely if if α and β are two geodesics in a simply connected manifold with non positive curvature such that dist (α(t), β (t)) is bounded for all t ∈ R, then the two geodesics are the edges of a totally geodesic isometric immersion of the Riemannian product of R with an interval. The flat strip theorem fails for manifolds with no conjugate points (although it does extend fairly easily to manifolds without focal points). Counter examples have been given in [Burns1992] and by Kleiner (unpublished). Kleiner's example has a copy of Z × Z in its fundamental group, but does not contain a corresponding flat torus. He perturbs the Heintze example (Example 4 in [Ba-Br-Eb1985]), which we described in subsection 4.1. The following basic question appears to be open in general (although Proposition 4 in [Eschenburg1977] suggests an affirmative answer under some extra hypotheses).",
   "evidence": "The canonical record contains the following question:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 241,
   "attempt": 1
  },
  "AIM-GEOMETRY-0243": {
   "statement_status": "exact",
   "original_statement": "Question 7.2.1. Must homotopic closed geodesics in a manifold with no conjugate point have the same length? \n\nOne can still hope for a version of the theorem in manifolds with no conjugate points in which there is a large enough family of \"parallel\" geodesics. One would then hope to find a totally geodesic isometric immersion of the Euclidean plane. Rigidity might hold in the large, even though the examples above show that it breaks down locally. The simplest question of this type asks if a Riemannian plane satisfying Euclid's 5th postulate must be flat. Consider the plane R2 with a complete Riemannian metric. Assume that for every geodesic and for every point not on the geodesic there exists precisely one nontrivial geodesic that passes through the point and does not intersect the geodesic. This assumption implies that the metric has no conjugate points.",
   "clean_statement": "Question 7.2.1. Must homotopic closed geodesics in a manifold with no conjugate point have the same length?\n\nOne can still hope for a version of the theorem in manifolds with no conjugate points in which there is a large enough family of \"parallel\" geodesics. One would then hope to find a totally geodesic isometric immersion of the Euclidean plane. Rigidity might hold in the large, even though the examples above show that it breaks down locally. The simplest question of this type asks if a Riemannian plane satisfying Euclid's 5th postulate must be flat. Consider the plane R2 with a complete Riemannian metric. Assume that for every geodesic and for every point not on the geodesic there exists precisely one nontrivial geodesic that passes through the point and does not intersect the geodesic. This assumption implies that the metric has no conjugate points.",
   "public_statement": "Question 7.2.1. Must homotopic closed geodesics in a manifold with no conjugate point have the same length?\n\nOne can still hope for a version of the theorem in manifolds with no conjugate points in which there is a large enough family of \"parallel\" geodesics. One would then hope to find a totally geodesic isometric immersion of the Euclidean plane. Rigidity might hold in the large, even though the examples above show that it breaks down locally. The simplest question of this type asks if a Riemannian plane satisfying Euclid's 5th postulate must be flat. Consider the plane R2 with a complete Riemannian metric. Assume that for every geodesic and for every point not on the geodesic there exists precisely one nontrivial geodesic that passes through the point and does not intersect the geodesic. This assumption implies that the metric has no conjugate points.",
   "evidence": "The canonical record, from Keith Burns and Vladimir S. Matveev's problem list, contains the following question and then a transition paragraph:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 242,
   "attempt": 1
  },
  "AIM-GEOMETRY-0244": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 7.2.2 ([Bur-Kni1991]). Must a metric satisfying this version of the par-allel postulate be flat? \n\nThe question looks like a question in the synthetic geometry, but it is not, since we do not require a priori that the other axioms of the Euclidean geometry are fulfilled (for example the congruence axioms). 7.2.1. Higher rank rigidity.",
   "clean_statement": "Question (Playfair rigidity).** Let \\((P,g)\\) be a complete Riemannian surface diffeomorphic to \\(\\mathbb R^2\\). Suppose that for every complete geodesic image \\(\\ell\\subset P\\) and every \\(p\\notin\\ell\\), there is exactly one nonconstant complete geodesic image through \\(p\\) disjoint from \\(\\ell\\). Must \\((P,g)\\) be isometric to the Euclidean plane?",
   "public_statement": "Question 7.2.2 ([Bur-Kni1991]). Must a metric satisfying this version of the par-allel postulate be flat?\n\nThe question looks like a question in the synthetic geometry, but it is not, since we do not require a priori that the other axioms of the Euclidean geometry are fulfilled (for example the congruence axioms). 7.2.1. Higher rank rigidity.",
   "evidence": "The exact canonical record is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 243,
   "attempt": 1
  },
  "AIM-GEOMETRY-0245": {
   "statement_status": "exact",
   "original_statement": "Conjecture 7.2.3 (Spatzier). Let (M, g ) be a closed symmetric space of noncom-pact type and of higher rank. Then the only metrics on M with no conjugate points are homothetic rescalings of g.\n\n7.3. Is M × S with no conjugate points a direct product? Consider the product of a closed surface M 2 of genus ≥ 2 with the circle S. Let g be a Riemannian metric on M × S with no conjugate points.",
   "clean_statement": "Conjecture 7.2.3 (Spatzier). Let (M, g ) be a closed symmetric space of noncom-pact type and of higher rank. Then the only metrics on M with no conjugate points are homothetic rescalings of g.\n\n7.3. Is M × S with no conjugate points a direct product? Consider the product of a closed surface M 2 of genus ≥ 2 with the circle S. Let g be a Riemannian metric on M × S with no conjugate points.",
   "public_statement": "Conjecture 7.2.3 (Spatzier). Let (M, g ) be a closed symmetric space of noncom-pact type and of higher rank. Then the only metrics on M with no conjugate points are homothetic rescalings of g.\n\n7.3. Is M × S with no conjugate points a direct product? Consider the product of a closed surface M 2 of genus ≥ 2 with the circle S. Let g be a Riemannian metric on M × S with no conjugate points.",
   "evidence": "The canonical record begins:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 244,
   "attempt": 1
  },
  "AIM-GEOMETRY-0246": {
   "statement_status": "exact",
   "original_statement": "Conjecture 7.3.1 (Burago-Kleiner). The metric on the Z-cover corresponding to \n\nS1 factor is a direct product: (M × R, g ) = ( M, g 1) × (R, dt 2).\n\nOne can of course make more general conjectures, e.g. about metrics without conjugate points on products, or on manifolds which admit nonpositively curved OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 17 \n\nmetrics with higher rank, but the above conjecture with M × S1 seems to be the easiest one (and is probably still hard to prove). 7.4. Magnetic geodesics without conjugate points. One can generalize the notion \"conjugate points\" for magnetic geodesics (even for arbitrary natural Hamil-tonian systems.)",
   "clean_statement": "Conjecture 7.3.1 (Burago-Kleiner). The metric on the Z-cover corresponding to\n\nS1 factor is a direct product: (M × R, g ) = ( M, g 1) × (R, dt 2).\n\nOne can of course make more general conjectures, e.g. about metrics without conjugate points on products, or on manifolds which admit nonpositively curved OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 17\n\nmetrics with higher rank, but the above conjecture with M × S1 seems to be the easiest one (and is probably still hard to prove). 7.4. Magnetic geodesics without conjugate points. One can generalize the notion \"conjugate points\" for magnetic geodesics (even for arbitrary natural Hamil-tonian systems.)",
   "public_statement": "Conjecture 7.3.1 (Burago-Kleiner). The metric on the Z-cover corresponding to\n\nS1 factor is a direct product: (M × R, g ) = ( M, g 1) × (R, dt 2).\n\nOne can of course make more general conjectures, e.g. about metrics without conjugate points on products, or on manifolds which admit nonpositively curved OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 17\n\nmetrics with higher rank, but the above conjecture with M × S1 seems to be the easiest one (and is probably still hard to prove). 7.4. Magnetic geodesics without conjugate points. One can generalize the notion \"conjugate points\" for magnetic geodesics (even for arbitrary natural Hamil-tonian systems.)",
   "evidence": "The canonical record is an OCR extraction from §7.3 of the AIM list *Open Problems and Questions about Geodesics*. The preceding record supplies the hypotheses that were lost at the record boundary:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 245,
   "attempt": 1
  },
  "AIM-GEOMETRY-0247": {
   "statement_status": "exact",
   "original_statement": "Question 7.4.1 (Paternain). Consider a magnetic flow on a closed surface of genus \n\n≥ 2 and an energy level such that the topological entropy vanishes. Suppose that the magnetic geodesics on this level do not have conjugate points. Is the system locally symmetric in the sense that the metric has constant curvature and the magnetic form is a constant multiple of the volume form? \n\nMany of the earlier questions in this section can also be asked about magnetic geodesics lying on a certain (possibly sufficiently high) energy level. Let us men-tion [Bialy2000], where a natural analog of Theorem 7.0.2 was proved under the assumption that the metric is conformally flat, and it was conjectured that this assumption is not essential. 8. Unrestricted curvature \n\n8.1. Ergodic geodesic flows. Does every closed manifold (with dimension ≥ 2)admit a metric (Riemannian or Finsler) with ergodic geodesic flow? \n\nThis is known for surfaces [Donnay1988II], for 3-manifolds [Katok1994], for prod-uct manifolds in which the factors have dimenson ≤ 3 [Bur-Ger1994], and for spheres [Bur-Ged]. Donnay and Pugh have shown that any embedded surface can be perturbed, in the C0 topology to an embedded surface whose geodesic flow is ergodic [Don-Pugh2004]. These constructions all make essential use of the focusing caps introduced by Donnay in [Donnay1988I]. The general problem is still wide open despite some reports of its solution (p. 87 of [Berger2000] and Section 10.9 of [Berger2003]). 8.2. Positive curvature.",
   "clean_statement": "Question 7.4.1 (Paternain). Consider a magnetic flow on a closed surface of genus\n\n≥ 2 and an energy level such that the topological entropy vanishes. Suppose that the magnetic geodesics on this level do not have conjugate points. Is the system locally symmetric in the sense that the metric has constant curvature and the magnetic form is a constant multiple of the volume form?\n\nMany of the earlier questions in this section can also be asked about magnetic geodesics lying on a certain (possibly sufficiently high) energy level. Let us men-tion [Bialy2000], where a natural analog of Theorem 7.0.2 was proved under the assumption that the metric is conformally flat, and it was conjectured that this assumption is not essential. 8. Unrestricted curvature\n\n8.1. Ergodic geodesic flows. Does every closed manifold (with dimension ≥ 2)admit a metric (Riemannian or Finsler) with ergodic geodesic flow?\n\nThis is known for surfaces [Donnay1988II], for 3-manifolds [Katok1994], for prod-uct manifolds in which the factors have dimenson ≤ 3 [Bur-Ger1994], and for spheres [Bur-Ged]. Donnay and Pugh have shown that any embedded surface can be perturbed, in the C0 topology to an embedded surface whose geodesic flow is ergodic [Don-Pugh2004]. These constructions all make essential use of the focusing caps introduced by Donnay in [Donnay1988I]. The general problem is still wide open despite some reports of its solution (p. 87 of [Berger2000] and Section 10.9 of [Berger2003]). 8.2. Positive curvature.",
   "public_statement": "Question 7.4.1 (Paternain). Consider a magnetic flow on a closed surface of genus\n\n≥ 2 and an energy level such that the topological entropy vanishes. Suppose that the magnetic geodesics on this level do not have conjugate points. Is the system locally symmetric in the sense that the metric has constant curvature and the magnetic form is a constant multiple of the volume form?\n\nMany of the earlier questions in this section can also be asked about magnetic geodesics lying on a certain (possibly sufficiently high) energy level. Let us men-tion [Bialy2000], where a natural analog of Theorem 7.0.2 was proved under the assumption that the metric is conformally flat, and it was conjectured that this assumption is not essential. 8. Unrestricted curvature\n\n8.1. Ergodic geodesic flows. Does every closed manifold (with dimension ≥ 2)admit a metric (Riemannian or Finsler) with ergodic geodesic flow?\n\nThis is known for surfaces [Donnay1988II], for 3-manifolds [Katok1994], for prod-uct manifolds in which the factors have dimenson ≤ 3 [Bur-Ger1994], and for spheres [Bur-Ged]. Donnay and Pugh have shown that any embedded surface can be perturbed, in the C0 topology to an embedded surface whose geodesic flow is ergodic [Don-Pugh2004]. These constructions all make essential use of the focusing caps introduced by Donnay in [Donnay1988I]. The general problem is still wide open despite some reports of its solution (p. 87 of [Berger2000] and Section 10.9 of [Berger2003]). 8.2. Positive curvature.",
   "evidence": "The canonical corpus record is source index 246 of `aim-geometry-notes.json`. Its `problem` field is preserved here exactly, including OCR line breaks, hyphenation, and text accidentally captured from the next section:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 246,
   "attempt": 1
  },
  "AIM-GEOMETRY-0248": {
   "statement_status": "exact",
   "original_statement": "Question 8.2.1. Is there a Riemannian metric on a closed manifold with every-where positive sectional curvatures and ergodic geodesic flow? \n\nMetrics close to the standard metric on S2 would be especially interesting. 8.3. The measure of transitive and recurrent sets. Let ( M 2, g ) be a closed surface with ergodic geodesic flow. Then every tangent vector lies in one of the following sets: \n\nTb:= {v ∈ SM | any lift of γv stays in a bounded subset of ˜M },Tp:= {v ∈ SM | any lift of γv is unbounded, and approaches infinity properly },Ti:= T M \\ (Tb ∪ Tp).\n\nAll three sets are measurable and invariant under the geodesic flow. By ergodicity one of them has full measure.",
   "clean_statement": "Question 8.2.1. Is there a Riemannian metric on a closed manifold with every-where positive sectional curvatures and ergodic geodesic flow?\n\nMetrics close to the standard metric on S2 would be especially interesting. 8.3. The measure of transitive and recurrent sets. Let ( M 2, g ) be a closed surface with ergodic geodesic flow. Then every tangent vector lies in one of the following sets:\n\nTb:= {v ∈ SM | any lift of γv stays in a bounded subset of ˜M },Tp:= {v ∈ SM | any lift of γv is unbounded, and approaches infinity properly },Ti:= T M \\ (Tb ∪ Tp).\n\nAll three sets are measurable and invariant under the geodesic flow. By ergodicity one of them has full measure.",
   "public_statement": "Question 8.2.1. Is there a Riemannian metric on a closed manifold with every-where positive sectional curvatures and ergodic geodesic flow?\n\nMetrics close to the standard metric on S2 would be especially interesting. 8.3. The measure of transitive and recurrent sets. Let ( M 2, g ) be a closed surface with ergodic geodesic flow. Then every tangent vector lies in one of the following sets:\n\nTb:= {v ∈ SM | any lift of γv stays in a bounded subset of ˜M },Tp:= {v ∈ SM | any lift of γv is unbounded, and approaches infinity properly },Ti:= T M \\ (Tb ∪ Tp).\n\nAll three sets are measurable and invariant under the geodesic flow. By ergodicity one of them has full measure.",
   "evidence": "The ensuing definitions of \\(T_b,T_p,T_i\\), the statement that these sets are invariant, and the full-measure observation are context for Question 8.3.1. The PDF places Question 8.3.1 immediately after them. They are therefore adjacent-section contamination in this record, not part of Question 8.2.1. They are preserved verbatim in `input.json` but are not used as hypotheses here.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 247,
   "attempt": 2
  },
  "AIM-GEOMETRY-0249": {
   "statement_status": "exact",
   "original_statement": "Question 8.3.1 (Schmidt). Which of these sets has full measure? 18 KEITH BURNS AND VLADIMIR S. MATVEEV \n\nIt is expected that the answer depends on the genus of the surface: on the torus the set Ti has full measure, and on the surfaces of higher genus Tp has full measure. In this context it is natural to consider only minimizing geodesics: if one shows that the set of minimizing geodesics has nonzero measure, then it has full measure by ergodicity of the geodesic flow. However, there exist examples of metrics on the torus such that the set of minimising geodesics has small Hausdorff dimension. The question seems to make sense in any dimension. 8.4. Generic metrics. Consider a smooth closed manifold M and denote by G\n\nthe space of all smooth Riemannian metrics on M. One can equip this space with a natural Ck-topology: locally, in a coordinate chart it is induced by a Ck-norm of the metric g viewed as an n × n matrix in this coordinate chart. More precisely, let us consider a finite number U1,..., U m of charts that cover the manifold. We say that two metrics g and g′ are ǫ-close if for any coordinate chart Us and for all i, j ≤ n the functions gij and g′ \n\n> ij\n\n(the ( i, j )-components of the metrics in the coordinate chart Us) are ǫ-closed in the standard Ck-norm, i.e., the difference of the values of these functions and all their partial derivatives up to the order k is less then ǫ). The notion of ǫ-closeness induces a notion of a ball in the space of the metrics and also a topology in the space of the metrics: a ball with center g is the set of g′ that are ǫ-close to g. As usual, we call a subset of G open if for any point of the subset a ball around the point is contained in the subset. It is an easy exercise to show that the topology does not depend on the choice of the charts U1,..., U m used to define it.",
   "clean_statement": "Question 8.3.1 (Schmidt). Which of these sets has full measure? 18 KEITH BURNS AND VLADIMIR S. MATVEEV\n\nIt is expected that the answer depends on the genus of the surface: on the torus the set Ti has full measure, and on the surfaces of higher genus Tp has full measure. In this context it is natural to consider only minimizing geodesics: if one shows that the set of minimizing geodesics has nonzero measure, then it has full measure by ergodicity of the geodesic flow. However, there exist examples of metrics on the torus such that the set of minimising geodesics has small Hausdorff dimension. The question seems to make sense in any dimension. 8.4. Generic metrics. Consider a smooth closed manifold M and denote by G\n\nthe space of all smooth Riemannian metrics on M. One can equip this space with a natural Ck-topology: locally, in a coordinate chart it is induced by a Ck-norm of the metric g viewed as an n × n matrix in this coordinate chart. More precisely, let us consider a finite number U1,..., U m of charts that cover the manifold. We say that two metrics g and g′ are ǫ-close if for any coordinate chart Us and for all i, j ≤ n the functions gij and g′\n\n> ij\n\n(the ( i, j )-components of the metrics in the coordinate chart Us) are ǫ-closed in the standard Ck-norm, i.e., the difference of the values of these functions and all their partial derivatives up to the order k is less then ǫ). The notion of ǫ-closeness induces a notion of a ball in the space of the metrics and also a topology in the space of the metrics: a ball with center g is the set of g′ that are ǫ-close to g. As usual, we call a subset of G open if for any point of the subset a ball around the point is contained in the subset. It is an easy exercise to show that the topology does not depend on the choice of the charts U1,..., U m used to define it.",
   "public_statement": "Question 8.3.1 (Schmidt). Which of these sets has full measure? 18 KEITH BURNS AND VLADIMIR S. MATVEEV\n\nIt is expected that the answer depends on the genus of the surface: on the torus the set Ti has full measure, and on the surfaces of higher genus Tp has full measure. In this context it is natural to consider only minimizing geodesics: if one shows that the set of minimizing geodesics has nonzero measure, then it has full measure by ergodicity of the geodesic flow. However, there exist examples of metrics on the torus such that the set of minimising geodesics has small Hausdorff dimension. The question seems to make sense in any dimension. 8.4. Generic metrics. Consider a smooth closed manifold M and denote by G\n\nthe space of all smooth Riemannian metrics on M. One can equip this space with a natural Ck-topology: locally, in a coordinate chart it is induced by a Ck-norm of the metric g viewed as an n × n matrix in this coordinate chart. More precisely, let us consider a finite number U1,..., U m of charts that cover the manifold. We say that two metrics g and g′ are ǫ-close if for any coordinate chart Us and for all i, j ≤ n the functions gij and g′\n\n> ij\n\n(the ( i, j )-components of the metrics in the coordinate chart Us) are ǫ-closed in the standard Ck-norm, i.e., the difference of the values of these functions and all their partial derivatives up to the order k is less then ǫ). The notion of ǫ-closeness induces a notion of a ball in the space of the metrics and also a topology in the space of the metrics: a ball with center g is the set of g′ that are ǫ-close to g. As usual, we call a subset of G open if for any point of the subset a ball around the point is contained in the subset. It is an easy exercise to show that the topology does not depend on the choice of the charts U1,..., U m used to define it.",
   "evidence": "The canonical record begins in the middle of §8.3 of the AIM list *Open Problems and Questions about Geodesics*. The preceding canonical record and the official PDF supply the definitions.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 248,
   "attempt": 1
  },
  "AIM-GEOMETRY-0250": {
   "statement_status": "exact",
   "original_statement": "Question 8.4.1. Do the metrics with positive entropy form a dense subset of G?This question can be asked for any k. For k = 2 it was recently positively answered in all dimensions in [Contreras2010]; see also [Con-Pat2002]. For k = ∞\n\nand dimension 2, it was answered positively in [Kni-Wei2002].",
   "clean_statement": "Question 8.4.1. Do the metrics with positive entropy form a dense subset of G?This question can be asked for any k. For k = 2 it was recently positively answered in all dimensions in [Contreras2010]; see also [Con-Pat2002]. For k = ∞\n\nand dimension 2, it was answered positively in [Kni-Wei2002].",
   "public_statement": "Question 8.4.1. Do the metrics with positive entropy form a dense subset of G?This question can be asked for any k. For k = 2 it was recently positively answered in all dimensions in [Contreras2010]; see also [Con-Pat2002]. For k = ∞\n\nand dimension 2, it was answered positively in [Kni-Wei2002].",
   "evidence": "The exact canonical OCR field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 249,
   "attempt": 1
  },
  "AIM-GEOMETRY-0251": {
   "statement_status": "exact",
   "original_statement": "Question 8.4.2. Assuming n = dim( M ) ≥ 3, do the metrics for which the geodesic flow is transitive form a dense subset of G?Recall that a geodesic flow is transitive, if there exists a geodesic γ with | ˙γ| = 1 such that the set of its tangent vectors is dense on SM.If n = dim( M ) = 2, the answer is negative: if we take a metric with integrable geodesic flow such that certain Liouville tori are irrational, any small perturbation of the metric has nontransitive geodesic flow by the KAM theory. This question is also closely related to the famous Arnold diffusion conjecture.",
   "clean_statement": "Question 8.4.2. Assuming n = dim( M ) ≥ 3, do the metrics for which the geodesic flow is transitive form a dense subset of G?Recall that a geodesic flow is transitive, if there exists a geodesic γ with | ˙γ| = 1 such that the set of its tangent vectors is dense on SM.If n = dim( M ) = 2, the answer is negative: if we take a metric with integrable geodesic flow such that certain Liouville tori are irrational, any small perturbation of the metric has nontransitive geodesic flow by the KAM theory. This question is also closely related to the famous Arnold diffusion conjecture.",
   "public_statement": "Question 8.4.2. Assuming n = dim( M ) ≥ 3, do the metrics for which the geodesic flow is transitive form a dense subset of G?Recall that a geodesic flow is transitive, if there exists a geodesic γ with | ˙γ| = 1 such that the set of its tangent vectors is dense on SM.If n = dim( M ) = 2, the answer is negative: if we take a metric with integrable geodesic flow such that certain Liouville tori are irrational, any small perturbation of the metric has nontransitive geodesic flow by the KAM theory. This question is also closely related to the famous Arnold diffusion conjecture.",
   "evidence": "The source is Keith Burns and Vladimir S. Matveev, *Open problems and questions about geodesics*, Question 8.4.2. The surrounding conventions are important. In the paper, \\(M\\) is connected; Section 8.4 assumes that \\(M\\) is smooth and closed and writes",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 250,
   "attempt": 1
  },
  "AIM-GEOMETRY-0252": {
   "statement_status": "exact",
   "original_statement": "Question 8.4.3. Is there a dense subset of metrics in G for which the tangent vectors to the closed geodesics are dense in SM?\n\nNote that is relatively easy to construct, see e.g. [Weinstein1970], a metric on any manifold such that a certain open subset of SM contains no vectors tangent to a closed geodesic. 8.5. Density in the manifold. In the previous section we discussed the unit tangent bundle SM. In this section we ask similar questions about M itself, and we will not assume that the metrics are generic.",
   "clean_statement": "Question 8.4.3. Is there a dense subset of metrics in G for which the tangent vectors to the closed geodesics are dense in SM?\n\nNote that is relatively easy to construct, see e.g. [Weinstein1970], a metric on any manifold such that a certain open subset of SM contains no vectors tangent to a closed geodesic. 8.5. Density in the manifold. In the previous section we discussed the unit tangent bundle SM. In this section we ask similar questions about M itself, and we will not assume that the metrics are generic.",
   "public_statement": "Question 8.4.3. Is there a dense subset of metrics in G for which the tangent vectors to the closed geodesics are dense in SM?\n\nNote that is relatively easy to construct, see e.g. [Weinstein1970], a metric on any manifold such that a certain open subset of SM contains no vectors tangent to a closed geodesic. 8.5. Density in the manifold. In the previous section we discussed the unit tangent bundle SM. In this section we ask similar questions about M itself, and we will not assume that the metrics are generic.",
   "evidence": "The canonical record is Question 8.4.3 from the AIM workshop list *Open Problems and Questions about Geodesics*. The source section fixes a smooth closed manifold $M$, lets $\\mathcal G$ be the space of smooth Riemannian metrics on $M$, and equips $\\mathcal G$ with a finite $C^k$ topology (with the usual $C^\\infty$ interpretation when that topology is discussed). For $g\\in\\mathcal G$, $S_gM$ is its unit tangent bundle.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 251,
   "attempt": 1
  },
  "AIM-GEOMETRY-0253": {
   "statement_status": "exact",
   "original_statement": "Question 8.5.1. Is the union of the closed geodesics always dense in the manifold?",
   "clean_statement": "Question 8.5.1. Is the union of the closed geodesics always dense in the manifold?",
   "public_statement": "Question 8.5.1. Is the union of the closed geodesics always dense in the manifold?",
   "evidence": "The source record is Question 8.5.1 in Keith Burns and Vladimir S. Matveev's AIM problem list:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 252,
   "attempt": 1
  },
  "AIM-GEOMETRY-0254": {
   "statement_status": "exact",
   "original_statement": "Question 8.5.2. Does every metric on a compact surface of positive genus have a geodesic that is dense in the surface? OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 19 \n\n8.6. Gaidukov in higher dimensions. We consider a Riemannian metric on an oriented closed surface M of genus ≥ 1. Let Γ be a nontrivial free homotopy class and p a point in M. A theorem of Gaidukov [Gaidukov1966] says that there exist a closed geodesic γ: R → M in Γ and a ray β: [0, ∞) → M with β(0) = p such that \n\ndist (β(t), γ (t)) → 0 as t → ∞. As explained in [Bia-Pol1986], Gajdukov's results follow easily from the classical results of [Morse1924] and [Hedlund1932].",
   "clean_statement": "Question 8.5.2. Does every metric on a compact surface of positive genus have a geodesic that is dense in the surface? OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 19\n\n8.6. Gaidukov in higher dimensions. We consider a Riemannian metric on an oriented closed surface M of genus ≥ 1. Let Γ be a nontrivial free homotopy class and p a point in M. A theorem of Gaidukov [Gaidukov1966] says that there exist a closed geodesic γ: R → M in Γ and a ray β: [0, ∞) → M with β(0) = p such that\n\ndist (β(t), γ (t)) → 0 as t → ∞. As explained in [Bia-Pol1986], Gajdukov's results follow easily from the classical results of [Morse1924] and [Hedlund1932].",
   "public_statement": "Question 8.5.2. Does every metric on a compact surface of positive genus have a geodesic that is dense in the surface? OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 19\n\n8.6. Gaidukov in higher dimensions. We consider a Riemannian metric on an oriented closed surface M of genus ≥ 1. Let Γ be a nontrivial free homotopy class and p a point in M. A theorem of Gaidukov [Gaidukov1966] says that there exist a closed geodesic γ: R → M in Γ and a ray β: [0, ∞) → M with β(0) = p such that\n\ndist (β(t), γ (t)) → 0 as t → ∞. As explained in [Bia-Pol1986], Gajdukov's results follow easily from the classical results of [Morse1924] and [Hedlund1932].",
   "evidence": "The canonical record comes from Keith Burns and Vladimir Matveev, *Open problems and questions about geodesics*, Question 8.5.2. Inspection of the source PDF and the adjacent records recovers the question as:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 253,
   "attempt": 1
  },
  "AIM-GEOMETRY-0255": {
   "statement_status": "exact",
   "original_statement": "Problem 8.6.1 (Schmidt). Generalize this statement to higher dimensions. \n\nGaidukov's proof is profoundly two-dimensional and cannot be generalized. But Mather's work on minimizing orbits of Lagrangian systems (see eg. [Mather1991] or [Con-Itu1999]) might offer an approach. Instead of a single closed geodesic one should consider a minimizing set, namely the support of a minimal measure. 8.7. Systolic and diastolic inequalities for surfaces (communicated by Guth, Rotman and Sabourau). Let ( M, g ) be a compact Riemannian surface. The inequalities in question compare the length of certain short closed geodesics with the square root of the area of ( M, g ), Recall that the systole sys (M, g ) is the least length of a non trivial closed geodesic. Two other geometrically meaningful constants can be defined by the following minimax procedure: \n\nL(M, g ) = inf \n\n> f\n\nmax \n\n> t∈R\n\nF [f −1(t)],\n\nin which the infimum is taken over all (Morse) functions f: M → R and the functional F is either (a) the total length of f −1(t) or (b) the length of its longest component. In both cases L(M, g ) is realized as the length of a certain union of closed geodesics. In case (a), L(M, g ) is one definition of the diastole dias( M, g ) of the Riemannian surface (at least two different notions of diastole have been studied; see [Bal-Sab2010]). By a result originally due to [Croke1988] and later improved in [Nab-Rot2002], [Sabourau2004] and [Rotman2006], for every Riemannian metric g on the sphere \n\nS2 one has sys( S2, g ) ≤ √32 √area( S2, g ).\n\nActually, in [Croke1988] it was suggested that the constant √32 in the above in-equality can be replaced by √2√3. The following example due to [Croke1988] shows that one can not go below √2√3: take two congruent equilateral triangles and glue them along their boundaries. The example is not smooth and suggests the study of systolic inequalities on the space of smooth metrics with conical singularities. On the space of such metrics, let us consider the function \n\nσ(g) = area( S2, g )\n\nsys( S2, g )2.\n\nThis function has many nice properties; for example it is Lipschitz with respect to the appropriate distance on the space of metrics. By [Croke1988], the function has a positive minimum, and the natural conjecture is that the minimum is attained on the sphere constructed from two equilateral triangles as described above. But other critical points of this functions are also interesting: for example the round metric of the sphere is a critical point because of the huge set of Zoll metrics having the same value of σ as the round metric, see [Balacheff2006]. 20 KEITH BURNS AND VLADIMIR S. MATVEEV",
   "clean_statement": "Problem 8.6.1 (Schmidt). Generalize this statement to higher dimensions.\n\nGaidukov's proof is profoundly two-dimensional and cannot be generalized. But Mather's work on minimizing orbits of Lagrangian systems (see eg. [Mather1991] or [Con-Itu1999]) might offer an approach. Instead of a single closed geodesic one should consider a minimizing set, namely the support of a minimal measure. 8.7. Systolic and diastolic inequalities for surfaces (communicated by Guth, Rotman and Sabourau). Let ( M, g ) be a compact Riemannian surface. The inequalities in question compare the length of certain short closed geodesics with the square root of the area of ( M, g ), Recall that the systole sys (M, g ) is the least length of a non trivial closed geodesic. Two other geometrically meaningful constants can be defined by the following minimax procedure:\n\nL(M, g ) = inf\n\n> f\n\nmax\n\n> t∈R\n\nF [f −1(t)],\n\nin which the infimum is taken over all (Morse) functions f: M → R and the functional F is either (a) the total length of f −1(t) or (b) the length of its longest component. In both cases L(M, g ) is realized as the length of a certain union of closed geodesics. In case (a), L(M, g ) is one definition of the diastole dias( M, g ) of the Riemannian surface (at least two different notions of diastole have been studied; see [Bal-Sab2010]). By a result originally due to [Croke1988] and later improved in [Nab-Rot2002], [Sabourau2004] and [Rotman2006], for every Riemannian metric g on the sphere\n\nS2 one has sys( S2, g ) ≤ √32 √area( S2, g ).\n\nActually, in [Croke1988] it was suggested that the constant √32 in the above in-equality can be replaced by √2√3. The following example due to [Croke1988] shows that one can not go below √2√3: take two congruent equilateral triangles and glue them along their boundaries. The example is not smooth and suggests the study of systolic inequalities on the space of smooth metrics with conical singularities. On the space of such metrics, let us consider the function\n\nσ(g) = area( S2, g )\n\nsys( S2, g )2.\n\nThis function has many nice properties; for example it is Lipschitz with respect to the appropriate distance on the space of metrics. By [Croke1988], the function has a positive minimum, and the natural conjecture is that the minimum is attained on the sphere constructed from two equilateral triangles as described above. But other critical points of this functions are also interesting: for example the round metric of the sphere is a critical point because of the huge set of Zoll metrics having the same value of σ as the round metric, see [Balacheff2006]. 20 KEITH BURNS AND VLADIMIR S. MATVEEV",
   "public_statement": "Problem 8.6.1 (Schmidt). Generalize this statement to higher dimensions.\n\nGaidukov's proof is profoundly two-dimensional and cannot be generalized. But Mather's work on minimizing orbits of Lagrangian systems (see eg. [Mather1991] or [Con-Itu1999]) might offer an approach. Instead of a single closed geodesic one should consider a minimizing set, namely the support of a minimal measure. 8.7. Systolic and diastolic inequalities for surfaces (communicated by Guth, Rotman and Sabourau). Let ( M, g ) be a compact Riemannian surface. The inequalities in question compare the length of certain short closed geodesics with the square root of the area of ( M, g ), Recall that the systole sys (M, g ) is the least length of a non trivial closed geodesic. Two other geometrically meaningful constants can be defined by the following minimax procedure:\n\nL(M, g ) = inf\n\n> f\n\nmax\n\n> t∈R\n\nF [f −1(t)],\n\nin which the infimum is taken over all (Morse) functions f: M → R and the functional F is either (a) the total length of f −1(t) or (b) the length of its longest component. In both cases L(M, g ) is realized as the length of a certain union of closed geodesics. In case (a), L(M, g ) is one definition of the diastole dias( M, g ) of the Riemannian surface (at least two different notions of diastole have been studied; see [Bal-Sab2010]). By a result originally due to [Croke1988] and later improved in [Nab-Rot2002], [Sabourau2004] and [Rotman2006], for every Riemannian metric g on the sphere\n\nS2 one has sys( S2, g ) ≤ √32 √area( S2, g ).\n\nActually, in [Croke1988] it was suggested that the constant √32 in the above in-equality can be replaced by √2√3. The following example due to [Croke1988] shows that one can not go below √2√3: take two congruent equilateral triangles and glue them along their boundaries. The example is not smooth and suggests the study of systolic inequalities on the space of smooth metrics with conical singularities. On the space of such metrics, let us consider the function\n\nσ(g) = area( S2, g )\n\nsys( S2, g )2.\n\nThis function has many nice properties; for example it is Lipschitz with respect to the appropriate distance on the space of metrics. By [Croke1988], the function has a positive minimum, and the natural conjecture is that the minimum is attained on the sphere constructed from two equilateral triangles as described above. But other critical points of this functions are also interesting: for example the round metric of the sphere is a critical point because of the huge set of Zoll metrics having the same value of σ as the round metric, see [Balacheff2006]. 20 KEITH BURNS AND VLADIMIR S. MATVEEV",
   "evidence": "The canonical record is extracted from Section 8.6 of Keith Burns and Vladimir Matveev's AIM list *Open problems and questions about geodesics*. Its literal problem text begins:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 254,
   "attempt": 1
  },
  "AIM-GEOMETRY-0256": {
   "statement_status": "exact",
   "original_statement": "Question 8.7.1 (Babenko). Does there exist, in the space of metrics with conical singularities, a (continuous or smooth) family of metrics gt such that g0 is the round metric of the sphere, g1 is the metric from the example above, and σ(gt) is a decreasing function of t.\n\nA simpler version of this question is",
   "clean_statement": "Question 8.7.1 (Babenko). Does there exist, in the space of metrics with conical singularities, a (continuous or smooth) family of metrics gt such that g0 is the round metric of the sphere, g1 is the metric from the example above, and σ(gt) is a decreasing function of t.\n\nA simpler version of this question is",
   "public_statement": "Question 8.7.1 (Babenko). Does there exist, in the space of metrics with conical singularities, a (continuous or smooth) family of metrics gt such that g0 is the round metric of the sphere, g1 is the metric from the example above, and σ(gt) is a decreasing function of t.\n\nA simpler version of this question is",
   "evidence": "The exact canonical OCR record is preserved in `input.json`. It ends with the words “A simpler version of this question is,” so it cannot be read without the neighboring source context. Inspection of the original AIM PDF, §8.7, pages 18–19, and of the adjacent canonical records gives the following source-verified reconstruction.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 255,
   "attempt": 1
  },
  "AIM-GEOMETRY-0257": {
   "statement_status": "exact",
   "original_statement": "Question 8.7.2 (Babenko). Does every Riemannian metric on S2 with area 4π\n\nthat is close enough to the round metric have a closed geodesic with length ≤ 2π?\n\nFor surfaces of higher genus Gromov proved the existence of a constant C such that sys( M, g ) ≤ C log(genus( M )) √area( M, g ); the dependence on the genus of M in this inequality is sharp. For the diastole defined by (a), Balacheff and Sabourau showed in [Bal-Sab2010] that there is a constant C such that dias( M, g ) ≤ C√genus( M )) √area( M, g ); the dependence of this inequality on the genus is again optimal.",
   "clean_statement": "Question 8.7.2 (Babenko). Does every Riemannian metric on S2 with area 4π\n\nthat is close enough to the round metric have a closed geodesic with length ≤ 2π?\n\nFor surfaces of higher genus Gromov proved the existence of a constant C such that sys( M, g ) ≤ C log(genus( M )) √area( M, g ); the dependence on the genus of M in this inequality is sharp. For the diastole defined by (a), Balacheff and Sabourau showed in [Bal-Sab2010] that there is a constant C such that dias( M, g ) ≤ C√genus( M )) √area( M, g ); the dependence of this inequality on the genus is again optimal.",
   "public_statement": "Question 8.7.2 (Babenko). Does every Riemannian metric on S2 with area 4π\n\nthat is close enough to the round metric have a closed geodesic with length ≤ 2π?\n\nFor surfaces of higher genus Gromov proved the existence of a constant C such that sys( M, g ) ≤ C log(genus( M )) √area( M, g ); the dependence on the genus of M in this inequality is sharp. For the diastole defined by (a), Balacheff and Sabourau showed in [Bal-Sab2010] that there is a constant C such that dias( M, g ) ≤ C√genus( M )) √area( M, g ); the dependence of this inequality on the genus is again optimal.",
   "evidence": "The canonical record is Question 8.7.2 in the AIM problem list *Open Problems and Questions about Geodesics*. The source first defines $\\operatorname{sys}(S^2,g)$ as the least length of a nontrivial (equivalently here, nonconstant) closed geodesic. The exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 256,
   "attempt": 1
  },
  "AIM-GEOMETRY-0258": {
   "statement_status": "exact",
   "original_statement": "Question 8.7.3 (Guth). Is there a constant C such that the invariant defined by (b) above is bounded from above by C√area( M, g )?\n\nAn affirmative answer to Guth's question would mean that the three quantities under consideration all depend on the genus in different ways, and are therefore measuring different features of the surface. A positive answer would also show that one can always find a pants decomposition of a closed Riemannian surface of genus \n\ng into 3 g − 3 disjoint closed geodesics of length at most C√area( M ). This would precisely give the optimal Bers' constant for a genus g surface. Even for hyperbolic surfaces, this question is still open (see [Buser1992] for partial results). The question above also makes sense in higher dimensions, but the relation with closed geodesics is not clear in this context. In this case f −1(t) would be an ( n − 1)-complex, where n is the dimension of the manifold, and F would measure its total volume or the volume of its largest component. 8.8. Questions related to the systole in higher dimensions.",
   "clean_statement": "Question 8.7.3 (Guth). Is there a constant C such that the invariant defined by (b) above is bounded from above by C√area( M, g )?\n\nAn affirmative answer to Guth's question would mean that the three quantities under consideration all depend on the genus in different ways, and are therefore measuring different features of the surface. A positive answer would also show that one can always find a pants decomposition of a closed Riemannian surface of genus\n\ng into 3 g − 3 disjoint closed geodesics of length at most C√area( M ). This would precisely give the optimal Bers' constant for a genus g surface. Even for hyperbolic surfaces, this question is still open (see [Buser1992] for partial results). The question above also makes sense in higher dimensions, but the relation with closed geodesics is not clear in this context. In this case f −1(t) would be an ( n − 1)-complex, where n is the dimension of the manifold, and F would measure its total volume or the volume of its largest component. 8.8. Questions related to the systole in higher dimensions.",
   "public_statement": "Question 8.7.3 (Guth). Is there a constant C such that the invariant defined by (b) above is bounded from above by C√area( M, g )?\n\nAn affirmative answer to Guth's question would mean that the three quantities under consideration all depend on the genus in different ways, and are therefore measuring different features of the surface. A positive answer would also show that one can always find a pants decomposition of a closed Riemannian surface of genus\n\ng into 3 g − 3 disjoint closed geodesics of length at most C√area( M ). This would precisely give the optimal Bers' constant for a genus g surface. Even for hyperbolic surfaces, this question is still open (see [Buser1992] for partial results). The question above also makes sense in higher dimensions, but the relation with closed geodesics is not clear in this context. In this case f −1(t) would be an ( n − 1)-complex, where n is the dimension of the manifold, and F would measure its total volume or the volume of its largest component. 8.8. Questions related to the systole in higher dimensions.",
   "evidence": "The source is Question 8.7.3, attributed to Larry Guth, in Burns--Matveev's list *Open problems and questions about geodesics*. Section 8.7 first defines, for a compact Riemannian surface \\((M,g)\\), \\[ L_F(M,g)=\\inf_{f:M\\to\\mathbb R\\ \\mathrm{Morse}}\\ \\max_t F(f^{-1}(t)). \\] In case (b), \\(F(f^{-1}(t))\\) is **the length of the longest connected component** of the level set. Thus the invariant asked about is \\[ L_b(M,g)=\\inf_f W_b(f),\\qquad W_b(f)=\\sup_{t\\in\\mathbb R}\\max_{C\\in\\pi_0(f^{-1}(t))} \\mathcal H^1_g(C). \\tag{1} \\] At a critical value a component may be a finite graph; its length in (1) is its one-dimensional Hausdorff measure. Writing a supremum rather than a maximum avoids an irrelevant attainment issue.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 257,
   "attempt": 1
  },
  "AIM-GEOMETRY-0259": {
   "statement_status": "exact",
   "original_statement": "Question 4.11 of [Gromov2001]). Does a Riemannian metric on a real projective space with the same volume as the canonical metric have a closed geodesic with length ≤ π?\n\nIn dimension two, an affirmative answer is in [Gromov2001, Proposition 4.10].",
   "clean_statement": "Question 4.11 of [Gromov2001]). Does a Riemannian metric on a real projective space with the same volume as the canonical metric have a closed geodesic with length ≤ π?\n\nIn dimension two, an affirmative answer is in [Gromov2001, Proposition 4.10].",
   "public_statement": "Question 4.11 of [Gromov2001]). Does a Riemannian metric on a real projective space with the same volume as the canonical metric have a closed geodesic with length ≤ π?\n\nIn dimension two, an affirmative answer is in [Gromov2001, Proposition 4.10].",
   "evidence": "The canonical JSON record has lost the beginning of the label and consequently starts with an unmatched parenthesis. Inspection of page 20 of the original AIM PDF recovers the exact displayed item as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 258,
   "attempt": 1
  },
  "AIM-GEOMETRY-0260": {
   "statement_status": "exact",
   "original_statement": "Question 8.8.2 (Alvarez Paiva). Can there be Riemannian metrics on S3 or S2 ×\n\nS1 all of whose closed geodesics are long? More specifically, does every metric on these spaces have a closed geodesic with length ≤ 10 24 if the volume is 1?\n\n8.9. Metrics such that all geodesics are closed. This is a classical topic - the first examples go back at least to [Zoll1903]; see [Besse1978] for details. The book [Besse1978] is still up to date, and many problems/questions listed in it (in particular in Chapter 0 §D) are still open. As the most interesting question from their list we suggest: OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 21",
   "clean_statement": "Question 8.8.2 (Alvarez Paiva). Can there be Riemannian metrics on S3 or S2 ×\n\nS1 all of whose closed geodesics are long? More specifically, does every metric on these spaces have a closed geodesic with length ≤ 10 24 if the volume is 1?\n\n8.9. Metrics such that all geodesics are closed. This is a classical topic - the first examples go back at least to [Zoll1903]; see [Besse1978] for details. The book [Besse1978] is still up to date, and many problems/questions listed in it (in particular in Chapter 0 §D) are still open. As the most interesting question from their list we suggest: OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 21",
   "public_statement": "Question 8.8.2 (Alvarez Paiva). Can there be Riemannian metrics on S3 or S2 ×\n\nS1 all of whose closed geodesics are long? More specifically, does every metric on these spaces have a closed geodesic with length ≤ 10 24 if the volume is 1?\n\n8.9. Metrics such that all geodesics are closed. This is a classical topic - the first examples go back at least to [Zoll1903]; see [Besse1978] for details. The book [Besse1978] is still up to date, and many problems/questions listed in it (in particular in Chapter 0 §D) are still open. As the most interesting question from their list we suggest: OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 21",
   "evidence": "The record is Question 8.8.2 from the AIM list *Open Problems and Questions about Geodesics*. The raw extraction has two defects: superscripts were flattened in the numerical bound, and the beginning of Section 8.9 was appended to the record. Inspection of the official PDF recovers the mathematical question as follows.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 259,
   "attempt": 1
  },
  "AIM-GEOMETRY-0261": {
   "statement_status": "exact",
   "original_statement": "Question 8.9.1. Let M be a closed manifold not covered by a sphere. Let g be a metric on M for which all geodesics are closed. Is (M, g ) a CROSS (=compact rank one symmetric) manifold (with the standard metric)? \n\nOf course, variants of this question can be asked about Finsler and Lorentzian metrics. In the Finsler setting, one may ask for example to describe all Finsler met-rics on CP (n) such that their geodesics are geodesics of the standard (Fubini-Studi) metric on CP (n). In the Lorenzian setting, one can ask to construct all manifolds such that all light-like geodesics are closed, see e.g. [Mou-Suh2013, Suhr2013b]. 9. Lorentzian metrics and metrics of arbitrary signature. \n\n9.1. Closed geodesics. Most of the questions we asked about the Riemannian and Finsler metric can be modified such that they are also interesting in the semi-Riemannian metrics (i.e. of arbitrary signature) and in particular when the signa-ture is Lorentzian. It appears though that the answers in the Lorentzian case are sometimes very different from those in the Riemannian case. Many methods that were effectively used in the Riemannian case, for example the variational methods, do not work in the case of general signature. Let us consider as an example the question analogous to the one we considered in Section 2: how many geometrically different closed geodesics must there be for a Lorenzian metric on a closed manifold. First let us note that there are two possible natural notions of closed geodesic in the Lorentzian setting: one may define a closed geodesic as an embedding γ:\n\nS1 → M such that ∇ ˙γ ˙γ = 0, or as a curve γ: S1 → M that can be locally reparameterized in such a way that it satisfies the equation ∇ ˙γ ˙γ = 0. In the Riemannian case, these definitions are essentially equivalent, since this reparameterization can always be made \"global\". If the signature is indefinite, it is easy to construct examples of an embedding γ: S1 → M such that locally γ\n\ncan be reparameterized so that it becomes an affinely-parameterized geodesic, but globally such reparameterisation is impossible: the velocity vector of the geodesic after returning to the same point is proportional but not equal to the initial velocity vector. Of course, this is possible only if the velocity vector is light-like. We will follow most publications and define a closed geodesic as an embedding \n\nγ: S1 → M such that ∇ ˙γ ˙γ = 0. To the best of our knowledge, the existence of a closed geodesic on a Lorentzian manifold is a quite complicated problem and nothing is known in dimensions ≥ 3. In dimension 2, a closed orientable manifold with a metric of signature (+, −) is homeomorphic to the torus. By [Suhr2013a], every Lorentzian 2-dimensional torus has at least two simple closed geodesics one of which is definite, i.e. timelike or spacelike. Moreover, explicit examples show the optimality of this claim. We therefore ask the following",
   "clean_statement": "Question 8.9.1. Let M be a closed manifold not covered by a sphere. Let g be a metric on M for which all geodesics are closed. Is (M, g ) a CROSS (=compact rank one symmetric) manifold (with the standard metric)?\n\nOf course, variants of this question can be asked about Finsler and Lorentzian metrics. In the Finsler setting, one may ask for example to describe all Finsler met-rics on CP (n) such that their geodesics are geodesics of the standard (Fubini-Studi) metric on CP (n). In the Lorenzian setting, one can ask to construct all manifolds such that all light-like geodesics are closed, see e.g. [Mou-Suh2013, Suhr2013b]. 9. Lorentzian metrics and metrics of arbitrary signature.\n\n9.1. Closed geodesics. Most of the questions we asked about the Riemannian and Finsler metric can be modified such that they are also interesting in the semi-Riemannian metrics (i.e. of arbitrary signature) and in particular when the signa-ture is Lorentzian. It appears though that the answers in the Lorentzian case are sometimes very different from those in the Riemannian case. Many methods that were effectively used in the Riemannian case, for example the variational methods, do not work in the case of general signature. Let us consider as an example the question analogous to the one we considered in Section 2: how many geometrically different closed geodesics must there be for a Lorenzian metric on a closed manifold. First let us note that there are two possible natural notions of closed geodesic in the Lorentzian setting: one may define a closed geodesic as an embedding γ:\n\nS1 → M such that ∇ ˙γ ˙γ = 0, or as a curve γ: S1 → M that can be locally reparameterized in such a way that it satisfies the equation ∇ ˙γ ˙γ = 0. In the Riemannian case, these definitions are essentially equivalent, since this reparameterization can always be made \"global\". If the signature is indefinite, it is easy to construct examples of an embedding γ: S1 → M such that locally γ\n\ncan be reparameterized so that it becomes an affinely-parameterized geodesic, but globally such reparameterisation is impossible: the velocity vector of the geodesic after returning to the same point is proportional but not equal to the initial velocity vector. Of course, this is possible only if the velocity vector is light-like. We will follow most publications and define a closed geodesic as an embedding\n\nγ: S1 → M such that ∇ ˙γ ˙γ = 0. To the best of our knowledge, the existence of a closed geodesic on a Lorentzian manifold is a quite complicated problem and nothing is known in dimensions ≥ 3. In dimension 2, a closed orientable manifold with a metric of signature (+, −) is homeomorphic to the torus. By [Suhr2013a], every Lorentzian 2-dimensional torus has at least two simple closed geodesics one of which is definite, i.e. timelike or spacelike. Moreover, explicit examples show the optimality of this claim. We therefore ask the following",
   "public_statement": "Question 8.9.1. Let M be a closed manifold not covered by a sphere. Let g be a metric on M for which all geodesics are closed. Is (M, g ) a CROSS (=compact rank one symmetric) manifold (with the standard metric)?\n\nOf course, variants of this question can be asked about Finsler and Lorentzian metrics. In the Finsler setting, one may ask for example to describe all Finsler met-rics on CP (n) such that their geodesics are geodesics of the standard (Fubini-Studi) metric on CP (n). In the Lorenzian setting, one can ask to construct all manifolds such that all light-like geodesics are closed, see e.g. [Mou-Suh2013, Suhr2013b]. 9. Lorentzian metrics and metrics of arbitrary signature.\n\n9.1. Closed geodesics. Most of the questions we asked about the Riemannian and Finsler metric can be modified such that they are also interesting in the semi-Riemannian metrics (i.e. of arbitrary signature) and in particular when the signa-ture is Lorentzian. It appears though that the answers in the Lorentzian case are sometimes very different from those in the Riemannian case. Many methods that were effectively used in the Riemannian case, for example the variational methods, do not work in the case of general signature. Let us consider as an example the question analogous to the one we considered in Section 2: how many geometrically different closed geodesics must there be for a Lorenzian metric on a closed manifold. First let us note that there are two possible natural notions of closed geodesic in the Lorentzian setting: one may define a closed geodesic as an embedding γ:\n\nS1 → M such that ∇ ˙γ ˙γ = 0, or as a curve γ: S1 → M that can be locally reparameterized in such a way that it satisfies the equation ∇ ˙γ ˙γ = 0. In the Riemannian case, these definitions are essentially equivalent, since this reparameterization can always be made \"global\". If the signature is indefinite, it is easy to construct examples of an embedding γ: S1 → M such that locally γ\n\ncan be reparameterized so that it becomes an affinely-parameterized geodesic, but globally such reparameterisation is impossible: the velocity vector of the geodesic after returning to the same point is proportional but not equal to the initial velocity vector. Of course, this is possible only if the velocity vector is light-like. We will follow most publications and define a closed geodesic as an embedding\n\nγ: S1 → M such that ∇ ˙γ ˙γ = 0. To the best of our knowledge, the existence of a closed geodesic on a Lorentzian manifold is a quite complicated problem and nothing is known in dimensions ≥ 3. In dimension 2, a closed orientable manifold with a metric of signature (+, −) is homeomorphic to the torus. By [Suhr2013a], every Lorentzian 2-dimensional torus has at least two simple closed geodesics one of which is definite, i.e. timelike or spacelike. Moreover, explicit examples show the optimality of this claim. We therefore ask the following",
   "evidence": "The canonical JSON record contains Question 8.9.1, the short Finsler and Lorentzian variants that follow it, and then text accidentally spilled from the next section of the source. Page 21 of the AIM problem list gives the actual Riemannian question as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 260,
   "attempt": 1
  },
  "AIM-GEOMETRY-0262": {
   "statement_status": "exact",
   "original_statement": "Question 9.1.1. Does every closed Lorentzian manifold have at least one closed geodesic? \n\nOne of course can ask this question about manifolds of arbitrary signature and also about complete (note that in the Lorentzian setting there are many different nonequivalent notions of completeness) manifolds of finite volume. See the introduction to the paper [Fl-Ja-Pi2011] for a list of known results about the existence of closed time-like geodesics under additional assumptions. 22 KEITH BURNS AND VLADIMIR S. MATVEEV \n\n9.2. Light-like geodesics of semi-Riemannian metrics. Let M be a closed manifold with semi-riemannian metric g of indefinite signature.",
   "clean_statement": "Question 9.1.1. Does every closed Lorentzian manifold have at least one closed geodesic?\n\nOne of course can ask this question about manifolds of arbitrary signature and also about complete (note that in the Lorentzian setting there are many different nonequivalent notions of completeness) manifolds of finite volume. See the introduction to the paper [Fl-Ja-Pi2011] for a list of known results about the existence of closed time-like geodesics under additional assumptions. 22 KEITH BURNS AND VLADIMIR S. MATVEEV\n\n9.2. Light-like geodesics of semi-Riemannian metrics. Let M be a closed manifold with semi-riemannian metric g of indefinite signature.",
   "public_statement": "Question 9.1.1. Does every closed Lorentzian manifold have at least one closed geodesic?\n\nOne of course can ask this question about manifolds of arbitrary signature and also about complete (note that in the Lorentzian setting there are many different nonequivalent notions of completeness) manifolds of finite volume. See the introduction to the paper [Fl-Ja-Pi2011] for a list of known results about the existence of closed time-like geodesics under additional assumptions. 22 KEITH BURNS AND VLADIMIR S. MATVEEV\n\n9.2. Light-like geodesics of semi-Riemannian metrics. Let M be a closed manifold with semi-riemannian metric g of indefinite signature.",
   "evidence": "The record is Question 9.1.1 in the AIM list *Open Problems and Questions about Geodesics*. The official PDF first distinguishes two notions that coincide in the Riemannian case but not in indefinite signature. The convention adopted there is an affinely parametrized, simple closed geodesic: an embedding $\\gamma:S^1\\to M$ satisfying $\\nabla_{\\dot\\gamma}\\dot\\gamma=0$. With that convention, the recovered question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 261,
   "attempt": 1
  },
  "AIM-GEOMETRY-0263": {
   "statement_status": "exact",
   "original_statement": "Question 9.2.1. Can there exist a complete semi-Riemannian metric g and a nontrivial 1-form η on a closed manifold such that for every light-like geodesic γ(t)\n\nthe function η( ˙ γ(t)) grows linearly in both directions: i.e. for every geodesic there exist C1 6 = 0, C 2 such that η( ˙ γ(t)) = C1 · t + C2?\n\nA negative answer to this question would give an easy proof of the semi-Riemannian version of the projective Lichnerowicz-Obata conjecture: \n\nLet a connected Lie group G act on a complete manifold (M n≥2, g ) by projective transformations (diffeomorphisms take geodesics to geodesics without necessarily preserving the parametrization). Then G acts by isometries, or g has constant sectional curvature. \n\nIf g is Riemannian, the conjecture was proved in [Matveev2005, Matveev2007]. The proof is complicated, and does not generalize to the semi-Riemannian setting in dimensions ≥ 3 (in dimension 2, under the assumption that the manifold is closed, the conjecture was proved in [Matveev2012b]). Fortunately, in the semi-Riemannian case the following argument gives a proof for closed manifolds provided that the answer to the question above is positive. It is known (see, for example, [Matveev2007]) that a 1-form ηi generates a one-parameter group of projective transformations, if and only if \n\nηi,jk + ηj,ik − 2\n\n(n + 1) ηℓ,ℓk gij = λigjk + λj gik \n\n(for a certain 1-form λi). We take a light line geodesic γ(t), multiply the above equation by ˙ γi ˙γj ˙γk (and sum with respect to repeating indexes) at every point γ(t)of the geodesic. The terms with the metric g disappear since gij ˙γi ˙γj = 0, so we obtain the equation d2 \n\n> dt 2\n\nη( ˙ γ(t)) = 0 implying that η( ˙ γ(t)) = t · C1 + C2. A negative answer to the question above implies that C1 = 0. Hence η( ˙ γ(t)) is constant, which in turn implies that ηi,j + ηj,i is proportional to gij. Then the covector field ηi\n\ngenerates a one-parameter group of conformal transformations. Finally, the proof of the conjecture follows from a classical observation of H. Weyl [Weyl1921] that every transformation that is projective and conformal is a homothety. A version of the question above is whether, for a complete semi-Riemannian metric on a closed manifold M, the tangent vector of almost every geodesic remains in a bounded set of T M. A positive answer on this question immediately implies that the answer to the initial question is negative, thereby proving the projective Lichnerowicz-Obata conjecture on closed manifolds. 9.3. Completeness of closed manifolds of arbitrary siganature (commu-nicated by H. Baum). F It is well-known that a closed Riemannian manifold is geodesically complete, in the sense that every geodesic γ: ( a, b ) → M can be extended to a geodesic ˜ γ: R → M such that ˜ γ|[a,b ] = γ. It is also well known that for any indefinite signature there exist metrics on closed manifolds that are not geodesically complete.",
   "clean_statement": "Question 9.2.1. Can there exist a complete semi-Riemannian metric g and a nontrivial 1-form η on a closed manifold such that for every light-like geodesic γ(t)\n\nthe function η( ˙ γ(t)) grows linearly in both directions: i.e. for every geodesic there exist C1 6 = 0, C 2 such that η( ˙ γ(t)) = C1 · t + C2?\n\nA negative answer to this question would give an easy proof of the semi-Riemannian version of the projective Lichnerowicz-Obata conjecture:\n\nLet a connected Lie group G act on a complete manifold (M n≥2, g ) by projective transformations (diffeomorphisms take geodesics to geodesics without necessarily preserving the parametrization). Then G acts by isometries, or g has constant sectional curvature.\n\nIf g is Riemannian, the conjecture was proved in [Matveev2005, Matveev2007]. The proof is complicated, and does not generalize to the semi-Riemannian setting in dimensions ≥ 3 (in dimension 2, under the assumption that the manifold is closed, the conjecture was proved in [Matveev2012b]). Fortunately, in the semi-Riemannian case the following argument gives a proof for closed manifolds provided that the answer to the question above is positive. It is known (see, for example, [Matveev2007]) that a 1-form ηi generates a one-parameter group of projective transformations, if and only if\n\nηi,jk + ηj,ik − 2\n\n(n + 1) ηℓ,ℓk gij = λigjk + λj gik\n\n(for a certain 1-form λi). We take a light line geodesic γ(t), multiply the above equation by ˙ γi ˙γj ˙γk (and sum with respect to repeating indexes) at every point γ(t)of the geodesic. The terms with the metric g disappear since gij ˙γi ˙γj = 0, so we obtain the equation d2\n\n> dt 2\n\nη( ˙ γ(t)) = 0 implying that η( ˙ γ(t)) = t · C1 + C2. A negative answer to the question above implies that C1 = 0. Hence η( ˙ γ(t)) is constant, which in turn implies that ηi,j + ηj,i is proportional to gij. Then the covector field ηi\n\ngenerates a one-parameter group of conformal transformations. Finally, the proof of the conjecture follows from a classical observation of H. Weyl [Weyl1921] that every transformation that is projective and conformal is a homothety. A version of the question above is whether, for a complete semi-Riemannian metric on a closed manifold M, the tangent vector of almost every geodesic remains in a bounded set of T M. A positive answer on this question immediately implies that the answer to the initial question is negative, thereby proving the projective Lichnerowicz-Obata conjecture on closed manifolds. 9.3. Completeness of closed manifolds of arbitrary siganature (commu-nicated by H. Baum). F It is well-known that a closed Riemannian manifold is geodesically complete, in the sense that every geodesic γ: ( a, b ) → M can be extended to a geodesic ˜ γ: R → M such that ˜ γ|[a,b ] = γ. It is also well known that for any indefinite signature there exist metrics on closed manifolds that are not geodesically complete.",
   "public_statement": "Question 9.2.1. Can there exist a complete semi-Riemannian metric g and a nontrivial 1-form η on a closed manifold such that for every light-like geodesic γ(t)\n\nthe function η( ˙ γ(t)) grows linearly in both directions: i.e. for every geodesic there exist C1 6 = 0, C 2 such that η( ˙ γ(t)) = C1 · t + C2?\n\nA negative answer to this question would give an easy proof of the semi-Riemannian version of the projective Lichnerowicz-Obata conjecture:\n\nLet a connected Lie group G act on a complete manifold (M n≥2, g ) by projective transformations (diffeomorphisms take geodesics to geodesics without necessarily preserving the parametrization). Then G acts by isometries, or g has constant sectional curvature.\n\nIf g is Riemannian, the conjecture was proved in [Matveev2005, Matveev2007]. The proof is complicated, and does not generalize to the semi-Riemannian setting in dimensions ≥ 3 (in dimension 2, under the assumption that the manifold is closed, the conjecture was proved in [Matveev2012b]). Fortunately, in the semi-Riemannian case the following argument gives a proof for closed manifolds provided that the answer to the question above is positive. It is known (see, for example, [Matveev2007]) that a 1-form ηi generates a one-parameter group of projective transformations, if and only if\n\nηi,jk + ηj,ik − 2\n\n(n + 1) ηℓ,ℓk gij = λigjk + λj gik\n\n(for a certain 1-form λi). We take a light line geodesic γ(t), multiply the above equation by ˙ γi ˙γj ˙γk (and sum with respect to repeating indexes) at every point γ(t)of the geodesic. The terms with the metric g disappear since gij ˙γi ˙γj = 0, so we obtain the equation d2\n\n> dt 2\n\nη( ˙ γ(t)) = 0 implying that η( ˙ γ(t)) = t · C1 + C2. A negative answer to the question above implies that C1 = 0. Hence η( ˙ γ(t)) is constant, which in turn implies that ηi,j + ηj,i is proportional to gij. Then the covector field ηi\n\ngenerates a one-parameter group of conformal transformations. Finally, the proof of the conjecture follows from a classical observation of H. Weyl [Weyl1921] that every transformation that is projective and conformal is a homothety. A version of the question above is whether, for a complete semi-Riemannian metric on a closed manifold M, the tangent vector of almost every geodesic remains in a bounded set of T M. A positive answer on this question immediately implies that the answer to the initial question is negative, thereby proving the projective Lichnerowicz-Obata conjecture on closed manifolds. 9.3. Completeness of closed manifolds of arbitrary siganature (commu-nicated by H. Baum). F It is well-known that a closed Riemannian manifold is geodesically complete, in the sense that every geodesic γ: ( a, b ) → M can be extended to a geodesic ˜ γ: R → M such that ˜ γ|[a,b ] = γ. It is also well known that for any indefinite signature there exist metrics on closed manifolds that are not geodesically complete.",
   "evidence": "The canonical record is Question 9.2.1 in Keith Burns and Vladimir Matveev, *Open problems and questions about geodesics*. The official AIM PDF, p. 22 of the printed article (PDF page 21), confirms that the OCR string `C1 6 = 0` is \\(C_1\\ne0\\). The literal question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 262,
   "attempt": 1
  },
  "AIM-GEOMETRY-0264": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 9.3.1. What geometric assumptions imply that a metric (possibly, of a fixed signature) on a closed manifold is geodesically complete? \n\nWe of course are interested in geometric assumptions that are easy to check or which are fulfilled for many interesting metrics. OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 23 \n\nLet us mention a few classical results. For compact homogeneous manifolds, ge-odesic completeness was established in [Marsden1972]. For compact Lorentz mani-folds of constant curvature geodesic completeness was proved in [Carri` ere1989] (flat case) and [Klingler1996] (general case). We refer to the paper [Sanchez2013] for a list of interesting results on this topic and for a list of open questions from which we repeat here only:",
   "clean_statement": null,
   "public_statement": "Question 9.3.1. What geometric assumptions imply that a metric (possibly, of a fixed signature) on a closed manifold is geodesically complete?\n\nWe of course are interested in geometric assumptions that are easy to check or which are fulfilled for many interesting metrics. OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 23\n\nLet us mention a few classical results. For compact homogeneous manifolds, ge-odesic completeness was established in [Marsden1972]. For compact Lorentz mani-folds of constant curvature geodesic completeness was proved in [Carri` ere1989] (flat case) and [Klingler1996] (general case). We refer to the paper [Sanchez2013] for a list of interesting results on this topic and for a list of open questions from which we repeat here only:",
   "evidence": "The source PDF has the section heading “9.3. Completeness of closed manifolds of arbitrary signature (communicated by H. Baum).” Its introductory paragraph, which the JSON extraction attached to the preceding record, says that closed Riemannian manifolds are geodesically complete, whereas every indefinite signature admits incomplete metrics on closed manifolds. After removing a page header and line-break hyphenation, the exact question belonging to this record is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 263,
   "attempt": 1
  },
  "AIM-GEOMETRY-0265": {
   "statement_status": "corrected_verified",
   "original_statement": "Question 9.3.2 ([Sanchez2013]). Assume that a compact Lorentzian manifold is globally conformal to a manifold of constant curvature. Must it be geodesically complete? \n\nNote also that in the noncompact case homogeneous manifolds of indefinite signa-ture are not necessary geodesically complete; see for example [Sanchez2013, Exam-ple 2 in §4]. It is interesting to understand whether completeness of a homogeneous manifold can follow from algebraic properties of the isometry group. 10. Integrability and ergodicity of geodesic flows on surfaces of higher genus \n\n10.1. Metrics with integrable geodesic flow on surfaces of genus ≥ 2.",
   "clean_statement": "**Question 9.3.2 ([Sanchez2013]).** Assume that a compact Lorentzian manifold is globally conformal to a manifold of constant curvature. Must it be geodesically complete?",
   "public_statement": "**Question 9.3.2 ([Sanchez2013]).** Assume that a compact Lorentzian manifold is globally conformal to a manifold of constant curvature. Must it be geodesically complete?",
   "evidence": "The official AIM PDF, page 23 (PDF page index 22), gives the following question: The next paragraph in the PDF concerns noncompact homogeneous indefinite metrics and is contextual commentary, not part of Question 9.3.2. The extracted text beginning “10. Integrability and ergodicity…” is the next section and is not part of this record. The PDF's line-break hyphenation in “signature” and “Example” has also been removed in this recovery.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 264,
   "attempt": 1
  },
  "AIM-GEOMETRY-0266": {
   "statement_status": "exact",
   "original_statement": "Question 10.1.1 (Bangert). Does there exist a Riemannian metric on a closed surface of genus ≥ 2 whose geodesic flow is completely integrable? \n\nThe answer may depend on what we understand by \"completely integrable\": whether the integral is functionally independent of the Hamiltonian on an open everywhere dense subset, or we additionally assume that the subset has full measure. Ma˜ n´ e showed that a Hamiltonian flow on surfaces is generically Anosov or has zero Liouville exponents [Ma˜ n´ e1996]. This suggests that an easier version of the above question would be the following",
   "clean_statement": "Question 10.1.1 (Bangert). Does there exist a Riemannian metric on a closed surface of genus ≥ 2 whose geodesic flow is completely integrable?\n\nThe answer may depend on what we understand by \"completely integrable\": whether the integral is functionally independent of the Hamiltonian on an open everywhere dense subset, or we additionally assume that the subset has full measure. Ma˜ n´ e showed that a Hamiltonian flow on surfaces is generically Anosov or has zero Liouville exponents [Ma˜ n´ e1996]. This suggests that an easier version of the above question would be the following",
   "public_statement": "Question 10.1.1 (Bangert). Does there exist a Riemannian metric on a closed surface of genus ≥ 2 whose geodesic flow is completely integrable?\n\nThe answer may depend on what we understand by \"completely integrable\": whether the integral is functionally independent of the Hamiltonian on an open everywhere dense subset, or we additionally assume that the subset has full measure. Ma˜ n´ e showed that a Hamiltonian flow on surfaces is generically Anosov or has zero Liouville exponents [Ma˜ n´ e1996]. This suggests that an easier version of the above question would be the following",
   "evidence": "The canonical record is Question 10.1.1 from Keith Burns and Vladimir S. Matveev, *Open problems and questions about geodesics*, originating in the AIM workshop on Geodesics. The journal version is Ergodic Theory and Dynamical Systems **41** (2021), 641--684, DOI 10.1017/etds.2019.73. The official PDF gives the following text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 265,
   "attempt": 1
  },
  "AIM-GEOMETRY-0267": {
   "statement_status": "exact",
   "original_statement": "Question 10.1.2 (Paternain). Is there a Riemannian metric on a closed surface of genus ≥ 2 whose geodesic flow has zero Liouville entropy? \n\nBangert and Paternain have outlined a nonconstructive proof that there are Finsler metrics with this property; finding a Riemannian metric is certainly harder. A special case of the Finsler version of the question above would be:",
   "clean_statement": "Question 10.1.2 (Paternain). Is there a Riemannian metric on a closed surface of genus ≥ 2 whose geodesic flow has zero Liouville entropy?\n\nBangert and Paternain have outlined a nonconstructive proof that there are Finsler metrics with this property; finding a Riemannian metric is certainly harder. A special case of the Finsler version of the question above would be:",
   "public_statement": "Question 10.1.2 (Paternain). Is there a Riemannian metric on a closed surface of genus ≥ 2 whose geodesic flow has zero Liouville entropy?\n\nBangert and Paternain have outlined a nonconstructive proof that there are Finsler metrics with this property; finding a Riemannian metric is certainly harder. A special case of the Finsler version of the question above would be:",
   "evidence": "The canonical record is Question 10.1.2 in Keith Burns and Vladimir S. Matveev's problem list, from the workshop section “Metrics with integrable geodesic flow on surfaces of genus \\(\\ge 2\\).” The source PDF gives the following question:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 266,
   "attempt": 1
  },
  "AIM-GEOMETRY-0268": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 10.1.3. Let F1, F2 be Finsler metrics on a closed surface of genus ≥ 2.Assume every (unparameterized) F1-geodesic is an F2-geodesic. Must F1 be obtained from F2 by multiplication by a constant and adding a closed form? \n\nOne can also ask this question for arbitrary closed manifolds that can carry a hy-perbolic metric (if F1, F2 are Riemannian, the answer is affirmative [Matveev2003]). The previous question is related to the other questions in this section because of the following observation from [Mat-Top1998]: one can use the second metric to construct an integral of the geodesic flow of the first one. If both metrics are Riemannian, the integral is quadratic in velocities and the affirmative answer follows from [Kolokoltsov1983]; see [Mat-Top2000].",
   "clean_statement": null,
   "public_statement": "Question 10.1.3. Let F1, F2 be Finsler metrics on a closed surface of genus ≥ 2.Assume every (unparameterized) F1-geodesic is an F2-geodesic. Must F1 be obtained from F2 by multiplication by a constant and adding a closed form?\n\nOne can also ask this question for arbitrary closed manifolds that can carry a hy-perbolic metric (if F1, F2 are Riemannian, the answer is affirmative [Matveev2003]). The previous question is related to the other questions in this section because of the following observation from [Mat-Top1998]: one can use the second metric to construct an integral of the geodesic flow of the first one. If both metrics are Riemannian, the integral is quadratic in velocities and the affirmative answer follows from [Kolokoltsov1983]; see [Mat-Top2000].",
   "evidence": "The official AIM workshop PDF gives the following statement (spacing normalized, but mathematical wording unchanged):",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 267,
   "attempt": 1
  },
  "AIM-GEOMETRY-0269": {
   "statement_status": "exact",
   "original_statement": "Question 10.1.4. Does there exist a nonriemannian Finsler metric satisfying the Landsberg condition on a surface of genus ≥ 2?24 KEITH BURNS AND VLADIMIR S. MATVEEV \n\nThe Landsberg condition is defined in [Bao2007]. It implies the existence of an integral for the geodesic flow of the metric, which is the relation of this question with the present section. See [Gom-Rug2013] for a negative answer to the question assuming nonexistence of conjugate points. 10.2. Integrable geodesic flows with integrals of higher degree.",
   "clean_statement": "Question 10.1.4. Does there exist a nonriemannian Finsler metric satisfying the Landsberg condition on a surface of genus ≥ 2?24 KEITH BURNS AND VLADIMIR S. MATVEEV\n\nThe Landsberg condition is defined in [Bao2007]. It implies the existence of an integral for the geodesic flow of the metric, which is the relation of this question with the present section. See [Gom-Rug2013] for a negative answer to the question assuming nonexistence of conjugate points. 10.2. Integrable geodesic flows with integrals of higher degree.",
   "public_statement": "Question 10.1.4. Does there exist a nonriemannian Finsler metric satisfying the Landsberg condition on a surface of genus ≥ 2?24 KEITH BURNS AND VLADIMIR S. MATVEEV\n\nThe Landsberg condition is defined in [Bao2007]. It implies the existence of an integral for the geodesic flow of the metric, which is the relation of this question with the present section. See [Gom-Rug2013] for a negative answer to the question assuming nonexistence of conjugate points. 10.2. Integrable geodesic flows with integrals of higher degree.",
   "evidence": "The canonical record is extracted from Keith Burns and Vladimir S. Matveev, *Open problems and questions about geodesics*, Question 10.1.4. Comparison with the source PDF gives the intended text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 268,
   "attempt": 1
  },
  "AIM-GEOMETRY-0270": {
   "statement_status": "exact",
   "original_statement": "Conjecture 10.2.1 ([Bo-Ko-Fo1995]). If the geodesic flow of a Riemannian metric on the torus T 2 admits an integral that is polynomial of degree 3 in the velocities, then the metric admits a Killing vector field. \n\nA Killing vector field V allows us to construct an integral \n\nI: T M → R, I (ξ):= g(V, ξ )that is evidently linear in velocities; its third power is then an integral cubic in velocities. The motivation to study metrics admitting integrals polynomial in velocities comes from the following observation (which dates back at least to Darboux and Whittaker): if the geodesic flow admits an integral that is analytic in velocities, then each component of this integral that is homogeneous in velocities is also an integral. The natural idea is then to study the integrals that are polynomial in velocities of low degree. By the result of Kolokoltsov [Kolokoltsov1983], no metric on a surface of genus ≥\n\n2 admits an integral that is polynomial in velocities and is functionally independent of the energy integral. The state of the art if the surface is the sphere or the torus can be explained by the following table: \n\nSphere S2 Torus T 2\n\nDegree 1 All is known All is known \n\nDegree 2 All is known All is known \n\nDegree 3 Series of examples Partial negative results \n\nDegree 4 Series of examples Partial negative results \n\nDegree ≥ 5 Nothing is known Nothing is known \n\nIn the table, \"Degree\" means the smallest degree of a nontrivial integral polyno-mial in velocities. \"All is known\" means that there exists an effective description and classification (which can be found in [Bo-Ma-Fo1998]). A simpler version of the question is when we replace the geodesic flow in the question above by a Lagrangian system with the Lagrangian of the form L(x, ξ ):= \n\nK + U = ∑ gij ξiξj + U (x). In this case we assume that the integral is a sum of polynomials of degrees 3 and 1 in velocities. The \"partial negative results\" in the table above correspond to this case; moreover, in most cases it is assumed that the metric gij is flat, see for example [Bialy1987, Mironov2010, De-Ko-Tr2012] (and [Bialy2010] for results that do not require this assumption). Similar conjectures could be posed for integrals of every degree. If the degree d\n\nis odd, the conjecture is that the existence of an integral that is polynomial of the degree d in velocities implies the existence of a Killing vector field. If the degree \n\nd is even, the conjecture is that the existence of an integral that is polynomial of the degree d in velocities implies the existence of an integral quadratic in velocities and not proportional to the energy integral. OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 25 \n\n10.3. Metrics such that one can explicitly find all geodesics. Geodesics of a metric are solutions of a nonlinear ordinary differential equation ∇ ˙γ ˙γ = 0 which can not be explicitly solved in most cases. There are only a few examples of 2-dimensional metrics for which one can explicitly find all geodesics using elementary functions: they are metrics of constant curvature and Darboux-superintegrable metrics (see [Br-Ma-Ma2008] for definition). An interesting problem is to construct other examples of metrics such that all geodesics are explicitly known. A related problem (asked recently in [Tao2010]) is to construct metrics with an explicitly given distance function. 11. Stationary nets (Communicated by Rotman). \n\nA graph G in a Riemannian surface ( S, g ) is called a stationary net, if every edge is a geodesic and if at every vertex the sum of diverging unit vectors is zero. Vertices must have valence at least 3. \n\n11.1. Stationary Θ-nets. A Θ-graph is a graph that looks like the Greek letter \"Theta\": two vertices connected by three edges, see the picture below.",
   "clean_statement": "Conjecture 10.2.1 ([Bo-Ko-Fo1995]). If the geodesic flow of a Riemannian metric on the torus T 2 admits an integral that is polynomial of degree 3 in the velocities, then the metric admits a Killing vector field.\n\nA Killing vector field V allows us to construct an integral\n\nI: T M → R, I (ξ):= g(V, ξ )that is evidently linear in velocities; its third power is then an integral cubic in velocities. The motivation to study metrics admitting integrals polynomial in velocities comes from the following observation (which dates back at least to Darboux and Whittaker): if the geodesic flow admits an integral that is analytic in velocities, then each component of this integral that is homogeneous in velocities is also an integral. The natural idea is then to study the integrals that are polynomial in velocities of low degree. By the result of Kolokoltsov [Kolokoltsov1983], no metric on a surface of genus ≥\n\n2 admits an integral that is polynomial in velocities and is functionally independent of the energy integral. The state of the art if the surface is the sphere or the torus can be explained by the following table:\n\nSphere S2 Torus T 2\n\nDegree 1 All is known All is known\n\nDegree 2 All is known All is known\n\nDegree 3 Series of examples Partial negative results\n\nDegree 4 Series of examples Partial negative results\n\nDegree ≥ 5 Nothing is known Nothing is known\n\nIn the table, \"Degree\" means the smallest degree of a nontrivial integral polyno-mial in velocities. \"All is known\" means that there exists an effective description and classification (which can be found in [Bo-Ma-Fo1998]). A simpler version of the question is when we replace the geodesic flow in the question above by a Lagrangian system with the Lagrangian of the form L(x, ξ ):=\n\nK + U = ∑ gij ξiξj + U (x). In this case we assume that the integral is a sum of polynomials of degrees 3 and 1 in velocities. The \"partial negative results\" in the table above correspond to this case; moreover, in most cases it is assumed that the metric gij is flat, see for example [Bialy1987, Mironov2010, De-Ko-Tr2012] (and [Bialy2010] for results that do not require this assumption). Similar conjectures could be posed for integrals of every degree. If the degree d\n\nis odd, the conjecture is that the existence of an integral that is polynomial of the degree d in velocities implies the existence of a Killing vector field. If the degree\n\nd is even, the conjecture is that the existence of an integral that is polynomial of the degree d in velocities implies the existence of an integral quadratic in velocities and not proportional to the energy integral. OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 25\n\n10.3. Metrics such that one can explicitly find all geodesics. Geodesics of a metric are solutions of a nonlinear ordinary differential equation ∇ ˙γ ˙γ = 0 which can not be explicitly solved in most cases. There are only a few examples of 2-dimensional metrics for which one can explicitly find all geodesics using elementary functions: they are metrics of constant curvature and Darboux-superintegrable metrics (see [Br-Ma-Ma2008] for definition). An interesting problem is to construct other examples of metrics such that all geodesics are explicitly known. A related problem (asked recently in [Tao2010]) is to construct metrics with an explicitly given distance function. 11. Stationary nets (Communicated by Rotman).\n\nA graph G in a Riemannian surface ( S, g ) is called a stationary net, if every edge is a geodesic and if at every vertex the sum of diverging unit vectors is zero. Vertices must have valence at least 3.\n\n11.1. Stationary Θ-nets. A Θ-graph is a graph that looks like the Greek letter \"Theta\": two vertices connected by three edges, see the picture below.",
   "public_statement": "Conjecture 10.2.1 ([Bo-Ko-Fo1995]). If the geodesic flow of a Riemannian metric on the torus T 2 admits an integral that is polynomial of degree 3 in the velocities, then the metric admits a Killing vector field.\n\nA Killing vector field V allows us to construct an integral\n\nI: T M → R, I (ξ):= g(V, ξ )that is evidently linear in velocities; its third power is then an integral cubic in velocities. The motivation to study metrics admitting integrals polynomial in velocities comes from the following observation (which dates back at least to Darboux and Whittaker): if the geodesic flow admits an integral that is analytic in velocities, then each component of this integral that is homogeneous in velocities is also an integral. The natural idea is then to study the integrals that are polynomial in velocities of low degree. By the result of Kolokoltsov [Kolokoltsov1983], no metric on a surface of genus ≥\n\n2 admits an integral that is polynomial in velocities and is functionally independent of the energy integral. The state of the art if the surface is the sphere or the torus can be explained by the following table:\n\nSphere S2 Torus T 2\n\nDegree 1 All is known All is known\n\nDegree 2 All is known All is known\n\nDegree 3 Series of examples Partial negative results\n\nDegree 4 Series of examples Partial negative results\n\nDegree ≥ 5 Nothing is known Nothing is known\n\nIn the table, \"Degree\" means the smallest degree of a nontrivial integral polyno-mial in velocities. \"All is known\" means that there exists an effective description and classification (which can be found in [Bo-Ma-Fo1998]). A simpler version of the question is when we replace the geodesic flow in the question above by a Lagrangian system with the Lagrangian of the form L(x, ξ ):=\n\nK + U = ∑ gij ξiξj + U (x). In this case we assume that the integral is a sum of polynomials of degrees 3 and 1 in velocities. The \"partial negative results\" in the table above correspond to this case; moreover, in most cases it is assumed that the metric gij is flat, see for example [Bialy1987, Mironov2010, De-Ko-Tr2012] (and [Bialy2010] for results that do not require this assumption). Similar conjectures could be posed for integrals of every degree. If the degree d\n\nis odd, the conjecture is that the existence of an integral that is polynomial of the degree d in velocities implies the existence of a Killing vector field. If the degree\n\nd is even, the conjecture is that the existence of an integral that is polynomial of the degree d in velocities implies the existence of an integral quadratic in velocities and not proportional to the energy integral. OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 25\n\n10.3. Metrics such that one can explicitly find all geodesics. Geodesics of a metric are solutions of a nonlinear ordinary differential equation ∇ ˙γ ˙γ = 0 which can not be explicitly solved in most cases. There are only a few examples of 2-dimensional metrics for which one can explicitly find all geodesics using elementary functions: they are metrics of constant curvature and Darboux-superintegrable metrics (see [Br-Ma-Ma2008] for definition). An interesting problem is to construct other examples of metrics such that all geodesics are explicitly known. A related problem (asked recently in [Tao2010]) is to construct metrics with an explicitly given distance function. 11. Stationary nets (Communicated by Rotman).\n\nA graph G in a Riemannian surface ( S, g ) is called a stationary net, if every edge is a geodesic and if at every vertex the sum of diverging unit vectors is zero. Vertices must have valence at least 3.\n\n11.1. Stationary Θ-nets. A Θ-graph is a graph that looks like the Greek letter \"Theta\": two vertices connected by three edges, see the picture below.",
   "evidence": "The mathematical record begins with the following statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 269,
   "attempt": 1
  },
  "AIM-GEOMETRY-0271": {
   "statement_status": "exact",
   "original_statement": "Question 11.1.1 ([Has-Mor1996]). Does every metric on S2 admit a stationary \n\nΘ-graph? \n\nPartial results in this direction were obtained in [Has-Mor1996]: they showed that on every sphere of positive curvature there exists a stationary Θ-net, or a stationary eight curve net, or a stationary eyeglasses net. \n\n11.2. Density of stationary nets and of closed geodesics. We call a stationary net nontrivial if it has at least one vertex of valency ≥ 3, in other words if it is not a disjoint union of simple closed geodesics.",
   "clean_statement": "Question 11.1.1 ([Has-Mor1996]). Does every metric on S2 admit a stationary\n\nΘ-graph?\n\nPartial results in this direction were obtained in [Has-Mor1996]: they showed that on every sphere of positive curvature there exists a stationary Θ-net, or a stationary eight curve net, or a stationary eyeglasses net.\n\n11.2. Density of stationary nets and of closed geodesics. We call a stationary net nontrivial if it has at least one vertex of valency ≥ 3, in other words if it is not a disjoint union of simple closed geodesics.",
   "public_statement": "Question 11.1.1 ([Has-Mor1996]). Does every metric on S2 admit a stationary\n\nΘ-graph?\n\nPartial results in this direction were obtained in [Has-Mor1996]: they showed that on every sphere of positive curvature there exists a stationary Θ-net, or a stationary eight curve net, or a stationary eyeglasses net.\n\n11.2. Density of stationary nets and of closed geodesics. We call a stationary net nontrivial if it has at least one vertex of valency ≥ 3, in other words if it is not a disjoint union of simple closed geodesics.",
   "evidence": "The official AIM PDF first defines a stationary net and then asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 270,
   "attempt": 1
  },
  "AIM-GEOMETRY-0272": {
   "statement_status": "exact",
   "original_statement": "Question 11.2.1 (Gromov). Are nontrivial stationary nets dense on any closed Riemannian surface? In other words, for each non empty open subset U, is there a stationary net intersecting U?26 KEITH BURNS AND VLADIMIR S. MATVEEV \n\nThe answer is evidently positive for every surface of constant curvature. Indeed, in this case tangent vectors to closed geodesics are dense in the unit tangent bundle and we can take the union of closed intersecting geodesics as a stationary net. On other manifolds, the answer is less trivial, for example because in higher dimensions periodic geodesics may not intersect. It is also not known whether the union of the images of the closed geodesics is always a dense subset of the manifold. Weinstein showed that any manifold carries a metric (even a bumpy metric) for which the tangent vectors to the closed geodesics are not dense in the unit tangent bundle [Weinstein1970], but his argument says nothing about the projection of the union of the closed geodesics to the manifold. The following question is also still open:",
   "clean_statement": "Question 11.2.1 (Gromov). Are nontrivial stationary nets dense on any closed Riemannian surface? In other words, for each non empty open subset U, is there a stationary net intersecting U?26 KEITH BURNS AND VLADIMIR S. MATVEEV\n\nThe answer is evidently positive for every surface of constant curvature. Indeed, in this case tangent vectors to closed geodesics are dense in the unit tangent bundle and we can take the union of closed intersecting geodesics as a stationary net. On other manifolds, the answer is less trivial, for example because in higher dimensions periodic geodesics may not intersect. It is also not known whether the union of the images of the closed geodesics is always a dense subset of the manifold. Weinstein showed that any manifold carries a metric (even a bumpy metric) for which the tangent vectors to the closed geodesics are not dense in the unit tangent bundle [Weinstein1970], but his argument says nothing about the projection of the union of the closed geodesics to the manifold. The following question is also still open:",
   "public_statement": "Question 11.2.1 (Gromov). Are nontrivial stationary nets dense on any closed Riemannian surface? In other words, for each non empty open subset U, is there a stationary net intersecting U?26 KEITH BURNS AND VLADIMIR S. MATVEEV\n\nThe answer is evidently positive for every surface of constant curvature. Indeed, in this case tangent vectors to closed geodesics are dense in the unit tangent bundle and we can take the union of closed intersecting geodesics as a stationary net. On other manifolds, the answer is less trivial, for example because in higher dimensions periodic geodesics may not intersect. It is also not known whether the union of the images of the closed geodesics is always a dense subset of the manifold. Weinstein showed that any manifold carries a metric (even a bumpy metric) for which the tangent vectors to the closed geodesics are not dense in the unit tangent bundle [Weinstein1970], but his argument says nothing about the projection of the union of the closed geodesics to the manifold. The following question is also still open:",
   "evidence": "The primary PDF gives the following definition immediately before the question. A graph \\(G\\) in a Riemannian surface \\((S,g)\\) is a **stationary net** if every edge is a geodesic and, at every vertex, the sum of the unit tangent vectors directed outward along all incident half-edges is zero. Vertices have valence at least three. It then says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 271,
   "attempt": 1
  },
  "AIM-GEOMETRY-0273": {
   "statement_status": "exact",
   "original_statement": "Question 11.2.2. Are the tangent vectors to closed geodesics dense in the tangent bundle for a generic Riemannian metric on a closed manifold? \n\nIt is known that this is the case for metrics with nonuniformly hyperbolic geodesic flows. 12. How to reconstruct a metric from its geodesics. \n\nA (Riemannian or semi-Riemannian) metric allows one to construct geodesics. Every nonzero vector at every point is tangent to a unique curve from the family of geodesics. This section is dedicated to open problems related to the following questions: given a family of curves with this property, are they the geodesics for a metric; and, if there is such a metric, is it unique and how can it be reconstructed? These questions can also be posed for Finsler metrics, but in that context most natural problems are either solved or seem to be out of reach. Let us introduce some notation. By a path structure we will understand (follow-ing [Thomas1925]) a family of curves γα(t) such that for any point x and for any vector v ∈ TxM, v 6 = 0, there exists a unique curve γ from this family such that the for a certain t we have γ(t) = x and ˙ γ(t) ∈ span( v). We think that the curves are smooth and smoothly depend on the parameters α = ( α1,..., α 2n−2). We may insist that the parametrization of the curves stays fixed, or we may be willing to reparametrize them. The first step is to find an affine connection for which the curves are the geodesics. The problem reduces to a system of linear equations (whether one can actually write down and solve them depends on the form in which the curves are given.) We outline the main ideas. First consider the case in which the parametrization of the curves is fixed. If all the curves γ from this path structure are affinely parameterised geodesics of a connection ∇ =\n\n(\n\nΓijk \n\n), then at any point p ∈ M the Christoffel symbols satisfy the system of equations (12.1) ¨γ(t)i + Γ ijk ˙γ(t)j ˙γ(t)k = 0 for all curves γ from the path structure such that γ(t) = p. This is a system of \n\nn linear equations in the n2(n+1) \n\n> 2\n\nunknowns Γ( p)ijk. Consequently n(n+1) \n\n> 2\n\ncurves \n\nγα from the path structure give us a system of n2 (n+1) \n\n> 2\n\nlinear equations in n2 (n+1) \n\n> 2\n\nunknowns Γ( p)ijk. It is an easy exercise to see that if the velocity vectors of the curves at the point p are in general position, then this system is uniquely solvable; by solving it we obtain the Christoffel symbols at this point. Clearly, these Christoffel OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 27 \n\nsymbols should satisfy the equation (12.1) for all curves γ from the path structure (so generic path structures do not come from a connection). It depends on how the curves γ are given whether it is possibly to check this. For example, if all the curves are given by explicit formulas that depend algebraically on t and on α, then this is an algorithmically doable but computationally complicated task. Note that the above considerations show that reconstruction of a symmetric affine connection from affinely parameterized geodesics is unique. Let us now deal with the reconstruction of a connection from unparameterized curves. Our goal is to find a connection ∇ =\n\n(\n\nΓijk \n\n)\n\nfor which each curve γα from our path structure, after an appropriate reparameterization, is a geodesic. In this case, a similar idea works. The analog of (12.1) is (12.2) ¨γ(t)i + Γ ijk ˙γ(t)j ˙γ(t)k = c ˙γi,\n\nwhere the unknowns are Γ( p)ijk and c (though we are interested only in Γ( p)ijk ). For one curve γ containing p ∈ M we have therefore n equations in n2(n+1) \n\n> 2\n\n+ 1 unknowns Γ( p)ijk and c. For two curves γ1, γ 2 we obtain then 2 n equations in \n\n> n2(n+1)\n> 2\n\n+ 2 unknowns Γ( p)ijk, c 1, c 2 and so on. We see that for k > n2 (n+1) \n\n> 2( n−1)\n\ncurves \n\nγα from the path structure (passing through the point p) we have more equations than unknowns; by solving this system (if it is solvable) we obtain a connection. See [Matveev2012a, §2.1] for more details. Note that (as was already known to [Levi-Civita1896] and [Weyl1921]) the so-lution Γ ijk of (12.2), if it exists, is not unique: two connections ∇ =\n\n(\n\nΓijk \n\n)\n\nand ¯∇ =\n\n(¯Γijk \n\n)\n\nhave the same unparameterized geodesics, if and only if there exists a (0, 1)-tensorfield φi such that (12.3) ¯Γijk = Γ ijk + δikφj + δij φk.\n\nThus, the freedom in reconstructing of a connection is an arbitrary choice of a 1-form φi.Connections ∇ =\n\n(\n\nΓijk \n\n)\n\nand ¯∇ =\n\n(¯Γijk \n\n)\n\nrelated by (12.3) are called projectively equivalent; projective equivalence of two connections means that they have the same geodesics considered as unparameterized curves. Let us now touch on the question of whether/how one can reconstruct a met-ric (Riemannian or of arbitrary signature) from a path structure. As we explained above, it is relatively easy to reconstruct an affine connection (resp. a class of projec-tively equivalent affine connections) from affinely (resp. arbitrary) parameterized geodesics, so the actual question is how to reconstruct a metric from its affine connection (resp. a class of projectively equivalent affine connections), when it is possible, and what is the freedom. Let us first discuss how/whether it is possible to reconstruct a metric parallel with respect to a given symmetric affine connection. This question is well-studied; see for example the answers of Bryant and Thurston in [Bry-Thu2011]. A theoretical answer is as follows: the affine connection deter-mines the holonomy group. The existence of a metric with a given affine connection is equivalent to the existence of a nondegenerate bilinear form preserved by the ho-lonomy group, see e.g. [Schmidt1973]. A practical test for the existence of the metric is as follows: the connection allows to construct the curvature tensor Rijkℓ,and if the connection is the Levi-Civita connection of a metric, then this metric 28 KEITH BURNS AND VLADIMIR S. MATVEEV \n\nsatisfies the following equations: (12.4) Rsjkℓ gsi = −Rsikℓ gsj, R sjkℓ gsi = Rsℓij gsk.\n\nThese equations are essentially the algebraic symmetries of the Riemann curvature tensor: the first one corresponds to Rijkm = −Rjikm, and the second corresponds to Rijkm = Rkmij. One should view these equations as linear equations in the unknowns gij; the number of equations is bigger than the number of unknowns so it is expected that the system has no nonzero or nondegenerate solution (and in this case there exists no metric compatible with this connection). In many cases the solution is unique (up to multiplication by a conformal coefficient) and in this case we already have the conformal class of the metrics. Now, having the conformal class of the metric we have the conformal class of the volume form and it is easy to reconstruct the metric using the condition that the volume form is parallel, which immediately reduces to the condition that a certain 1-form is closed; see also [Mat-Trau2014]. Note that if the equations (12.4) do not give enough information one could consider their \"derivatives\" (12.5) Rsjkℓ,m gsi = −Rsikℓ,m gsj, R sjkℓ,m gsi = Rsℓij,m gsk \n\nwhich gives us again a huge system of equations in the same unknowns gij. If necessary one then considers higher order derivatives until there is enough infor-mation. The general theory says that in the analytic category the existence of a nondegenerate solution of the resulting system of equations implies the existence of a metric whose Levi-Civita connection is the given one. In the nonanalytic setting, however, there exist C∞ counterexamples. Let us now discuss how unique is the reconstruction of a metric from affinely parameterized geodesics. Locally, the answer is known: for Riemannian metrics, if was understood already by Cartan and Eisenhart [Eisenhart1923]; for metrics of arbitrary signature, the answer is in the recent papers [Boubel2012, Boubel2014]. Both results are an explicit local description of all metrics having the same Levi-Civita connection. In the Riemannian case, a global analog of the result of Cartan is due to [DeRham1952]. As the only interesting unsolved problem in this topic we suggest",
   "clean_statement": "Question 11.2.2. Are the tangent vectors to closed geodesics dense in the tangent bundle for a generic Riemannian metric on a closed manifold?\n\nIt is known that this is the case for metrics with nonuniformly hyperbolic geodesic flows. 12. How to reconstruct a metric from its geodesics.\n\nA (Riemannian or semi-Riemannian) metric allows one to construct geodesics. Every nonzero vector at every point is tangent to a unique curve from the family of geodesics. This section is dedicated to open problems related to the following questions: given a family of curves with this property, are they the geodesics for a metric; and, if there is such a metric, is it unique and how can it be reconstructed? These questions can also be posed for Finsler metrics, but in that context most natural problems are either solved or seem to be out of reach. Let us introduce some notation. By a path structure we will understand (follow-ing [Thomas1925]) a family of curves γα(t) such that for any point x and for any vector v ∈ TxM, v 6 = 0, there exists a unique curve γ from this family such that the for a certain t we have γ(t) = x and ˙ γ(t) ∈ span( v). We think that the curves are smooth and smoothly depend on the parameters α = ( α1,..., α 2n−2). We may insist that the parametrization of the curves stays fixed, or we may be willing to reparametrize them. The first step is to find an affine connection for which the curves are the geodesics. The problem reduces to a system of linear equations (whether one can actually write down and solve them depends on the form in which the curves are given.) We outline the main ideas. First consider the case in which the parametrization of the curves is fixed. If all the curves γ from this path structure are affinely parameterised geodesics of a connection ∇ =\n\n(\n\nΓijk\n\n), then at any point p ∈ M the Christoffel symbols satisfy the system of equations (12.1) ¨γ(t)i + Γ ijk ˙γ(t)j ˙γ(t)k = 0 for all curves γ from the path structure such that γ(t) = p. This is a system of\n\nn linear equations in the n2(n+1)\n\n> 2\n\nunknowns Γ( p)ijk. Consequently n(n+1)\n\n> 2\n\ncurves\n\nγα from the path structure give us a system of n2 (n+1)\n\n> 2\n\nlinear equations in n2 (n+1)\n\n> 2\n\nunknowns Γ( p)ijk. It is an easy exercise to see that if the velocity vectors of the curves at the point p are in general position, then this system is uniquely solvable; by solving it we obtain the Christoffel symbols at this point. Clearly, these Christoffel OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 27\n\nsymbols should satisfy the equation (12.1) for all curves γ from the path structure (so generic path structures do not come from a connection). It depends on how the curves γ are given whether it is possibly to check this. For example, if all the curves are given by explicit formulas that depend algebraically on t and on α, then this is an algorithmically doable but computationally complicated task. Note that the above considerations show that reconstruction of a symmetric affine connection from affinely parameterized geodesics is unique. Let us now deal with the reconstruction of a connection from unparameterized curves. Our goal is to find a connection ∇ =\n\n(\n\nΓijk\n\n)\n\nfor which each curve γα from our path structure, after an appropriate reparameterization, is a geodesic. In this case, a similar idea works. The analog of (12.1) is (12.2) ¨γ(t)i + Γ ijk ˙γ(t)j ˙γ(t)k = c ˙γi,\n\nwhere the unknowns are Γ( p)ijk and c (though we are interested only in Γ( p)ijk ). For one curve γ containing p ∈ M we have therefore n equations in n2(n+1)\n\n> 2\n\n+ 1 unknowns Γ( p)ijk and c. For two curves γ1, γ 2 we obtain then 2 n equations in\n\n> n2(n+1)\n> 2\n\n+ 2 unknowns Γ( p)ijk, c 1, c 2 and so on. We see that for k > n2 (n+1)\n\n> 2( n−1)\n\ncurves\n\nγα from the path structure (passing through the point p) we have more equations than unknowns; by solving this system (if it is solvable) we obtain a connection. See [Matveev2012a, §2.1] for more details. Note that (as was already known to [Levi-Civita1896] and [Weyl1921]) the so-lution Γ ijk of (12.2), if it exists, is not unique: two connections ∇ =\n\n(\n\nΓijk\n\n)\n\nand ¯∇ =\n\n(¯Γijk\n\n)\n\nhave the same unparameterized geodesics, if and only if there exists a (0, 1)-tensorfield φi such that (12.3) ¯Γijk = Γ ijk + δikφj + δij φk.\n\nThus, the freedom in reconstructing of a connection is an arbitrary choice of a 1-form φi.Connections ∇ =\n\n(\n\nΓijk\n\n)\n\nand ¯∇ =\n\n(¯Γijk\n\n)\n\nrelated by (12.3) are called projectively equivalent; projective equivalence of two connections means that they have the same geodesics considered as unparameterized curves. Let us now touch on the question of whether/how one can reconstruct a met-ric (Riemannian or of arbitrary signature) from a path structure. As we explained above, it is relatively easy to reconstruct an affine connection (resp. a class of projec-tively equivalent affine connections) from affinely (resp. arbitrary) parameterized geodesics, so the actual question is how to reconstruct a metric from its affine connection (resp. a class of projectively equivalent affine connections), when it is possible, and what is the freedom. Let us first discuss how/whether it is possible to reconstruct a metric parallel with respect to a given symmetric affine connection. This question is well-studied; see for example the answers of Bryant and Thurston in [Bry-Thu2011]. A theoretical answer is as follows: the affine connection deter-mines the holonomy group. The existence of a metric with a given affine connection is equivalent to the existence of a nondegenerate bilinear form preserved by the ho-lonomy group, see e.g. [Schmidt1973]. A practical test for the existence of the metric is as follows: the connection allows to construct the curvature tensor Rijkℓ,and if the connection is the Levi-Civita connection of a metric, then this metric 28 KEITH BURNS AND VLADIMIR S. MATVEEV\n\nsatisfies the following equations: (12.4) Rsjkℓ gsi = −Rsikℓ gsj, R sjkℓ gsi = Rsℓij gsk.\n\nThese equations are essentially the algebraic symmetries of the Riemann curvature tensor: the first one corresponds to Rijkm = −Rjikm, and the second corresponds to Rijkm = Rkmij. One should view these equations as linear equations in the unknowns gij; the number of equations is bigger than the number of unknowns so it is expected that the system has no nonzero or nondegenerate solution (and in this case there exists no metric compatible with this connection). In many cases the solution is unique (up to multiplication by a conformal coefficient) and in this case we already have the conformal class of the metrics. Now, having the conformal class of the metric we have the conformal class of the volume form and it is easy to reconstruct the metric using the condition that the volume form is parallel, which immediately reduces to the condition that a certain 1-form is closed; see also [Mat-Trau2014]. Note that if the equations (12.4) do not give enough information one could consider their \"derivatives\" (12.5) Rsjkℓ,m gsi = −Rsikℓ,m gsj, R sjkℓ,m gsi = Rsℓij,m gsk\n\nwhich gives us again a huge system of equations in the same unknowns gij. If necessary one then considers higher order derivatives until there is enough infor-mation. The general theory says that in the analytic category the existence of a nondegenerate solution of the resulting system of equations implies the existence of a metric whose Levi-Civita connection is the given one. In the nonanalytic setting, however, there exist C∞ counterexamples. Let us now discuss how unique is the reconstruction of a metric from affinely parameterized geodesics. Locally, the answer is known: for Riemannian metrics, if was understood already by Cartan and Eisenhart [Eisenhart1923]; for metrics of arbitrary signature, the answer is in the recent papers [Boubel2012, Boubel2014]. Both results are an explicit local description of all metrics having the same Levi-Civita connection. In the Riemannian case, a global analog of the result of Cartan is due to [DeRham1952]. As the only interesting unsolved problem in this topic we suggest",
   "public_statement": "Question 11.2.2. Are the tangent vectors to closed geodesics dense in the tangent bundle for a generic Riemannian metric on a closed manifold?\n\nIt is known that this is the case for metrics with nonuniformly hyperbolic geodesic flows. 12. How to reconstruct a metric from its geodesics.\n\nA (Riemannian or semi-Riemannian) metric allows one to construct geodesics. Every nonzero vector at every point is tangent to a unique curve from the family of geodesics. This section is dedicated to open problems related to the following questions: given a family of curves with this property, are they the geodesics for a metric; and, if there is such a metric, is it unique and how can it be reconstructed? These questions can also be posed for Finsler metrics, but in that context most natural problems are either solved or seem to be out of reach. Let us introduce some notation. By a path structure we will understand (follow-ing [Thomas1925]) a family of curves γα(t) such that for any point x and for any vector v ∈ TxM, v 6 = 0, there exists a unique curve γ from this family such that the for a certain t we have γ(t) = x and ˙ γ(t) ∈ span( v). We think that the curves are smooth and smoothly depend on the parameters α = ( α1,..., α 2n−2). We may insist that the parametrization of the curves stays fixed, or we may be willing to reparametrize them. The first step is to find an affine connection for which the curves are the geodesics. The problem reduces to a system of linear equations (whether one can actually write down and solve them depends on the form in which the curves are given.) We outline the main ideas. First consider the case in which the parametrization of the curves is fixed. If all the curves γ from this path structure are affinely parameterised geodesics of a connection ∇ =\n\n(\n\nΓijk\n\n), then at any point p ∈ M the Christoffel symbols satisfy the system of equations (12.1) ¨γ(t)i + Γ ijk ˙γ(t)j ˙γ(t)k = 0 for all curves γ from the path structure such that γ(t) = p. This is a system of\n\nn linear equations in the n2(n+1)\n\n> 2\n\nunknowns Γ( p)ijk. Consequently n(n+1)\n\n> 2\n\ncurves\n\nγα from the path structure give us a system of n2 (n+1)\n\n> 2\n\nlinear equations in n2 (n+1)\n\n> 2\n\nunknowns Γ( p)ijk. It is an easy exercise to see that if the velocity vectors of the curves at the point p are in general position, then this system is uniquely solvable; by solving it we obtain the Christoffel symbols at this point. Clearly, these Christoffel OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 27\n\nsymbols should satisfy the equation (12.1) for all curves γ from the path structure (so generic path structures do not come from a connection). It depends on how the curves γ are given whether it is possibly to check this. For example, if all the curves are given by explicit formulas that depend algebraically on t and on α, then this is an algorithmically doable but computationally complicated task. Note that the above considerations show that reconstruction of a symmetric affine connection from affinely parameterized geodesics is unique. Let us now deal with the reconstruction of a connection from unparameterized curves. Our goal is to find a connection ∇ =\n\n(\n\nΓijk\n\n)\n\nfor which each curve γα from our path structure, after an appropriate reparameterization, is a geodesic. In this case, a similar idea works. The analog of (12.1) is (12.2) ¨γ(t)i + Γ ijk ˙γ(t)j ˙γ(t)k = c ˙γi,\n\nwhere the unknowns are Γ( p)ijk and c (though we are interested only in Γ( p)ijk ). For one curve γ containing p ∈ M we have therefore n equations in n2(n+1)\n\n> 2\n\n+ 1 unknowns Γ( p)ijk and c. For two curves γ1, γ 2 we obtain then 2 n equations in\n\n> n2(n+1)\n> 2\n\n+ 2 unknowns Γ( p)ijk, c 1, c 2 and so on. We see that for k > n2 (n+1)\n\n> 2( n−1)\n\ncurves\n\nγα from the path structure (passing through the point p) we have more equations than unknowns; by solving this system (if it is solvable) we obtain a connection. See [Matveev2012a, §2.1] for more details. Note that (as was already known to [Levi-Civita1896] and [Weyl1921]) the so-lution Γ ijk of (12.2), if it exists, is not unique: two connections ∇ =\n\n(\n\nΓijk\n\n)\n\nand ¯∇ =\n\n(¯Γijk\n\n)\n\nhave the same unparameterized geodesics, if and only if there exists a (0, 1)-tensorfield φi such that (12.3) ¯Γijk = Γ ijk + δikφj + δij φk.\n\nThus, the freedom in reconstructing of a connection is an arbitrary choice of a 1-form φi.Connections ∇ =\n\n(\n\nΓijk\n\n)\n\nand ¯∇ =\n\n(¯Γijk\n\n)\n\nrelated by (12.3) are called projectively equivalent; projective equivalence of two connections means that they have the same geodesics considered as unparameterized curves. Let us now touch on the question of whether/how one can reconstruct a met-ric (Riemannian or of arbitrary signature) from a path structure. As we explained above, it is relatively easy to reconstruct an affine connection (resp. a class of projec-tively equivalent affine connections) from affinely (resp. arbitrary) parameterized geodesics, so the actual question is how to reconstruct a metric from its affine connection (resp. a class of projectively equivalent affine connections), when it is possible, and what is the freedom. Let us first discuss how/whether it is possible to reconstruct a metric parallel with respect to a given symmetric affine connection. This question is well-studied; see for example the answers of Bryant and Thurston in [Bry-Thu2011]. A theoretical answer is as follows: the affine connection deter-mines the holonomy group. The existence of a metric with a given affine connection is equivalent to the existence of a nondegenerate bilinear form preserved by the ho-lonomy group, see e.g. [Schmidt1973]. A practical test for the existence of the metric is as follows: the connection allows to construct the curvature tensor Rijkℓ,and if the connection is the Levi-Civita connection of a metric, then this metric 28 KEITH BURNS AND VLADIMIR S. MATVEEV\n\nsatisfies the following equations: (12.4) Rsjkℓ gsi = −Rsikℓ gsj, R sjkℓ gsi = Rsℓij gsk.\n\nThese equations are essentially the algebraic symmetries of the Riemann curvature tensor: the first one corresponds to Rijkm = −Rjikm, and the second corresponds to Rijkm = Rkmij. One should view these equations as linear equations in the unknowns gij; the number of equations is bigger than the number of unknowns so it is expected that the system has no nonzero or nondegenerate solution (and in this case there exists no metric compatible with this connection). In many cases the solution is unique (up to multiplication by a conformal coefficient) and in this case we already have the conformal class of the metrics. Now, having the conformal class of the metric we have the conformal class of the volume form and it is easy to reconstruct the metric using the condition that the volume form is parallel, which immediately reduces to the condition that a certain 1-form is closed; see also [Mat-Trau2014]. Note that if the equations (12.4) do not give enough information one could consider their \"derivatives\" (12.5) Rsjkℓ,m gsi = −Rsikℓ,m gsj, R sjkℓ,m gsi = Rsℓij,m gsk\n\nwhich gives us again a huge system of equations in the same unknowns gij. If necessary one then considers higher order derivatives until there is enough infor-mation. The general theory says that in the analytic category the existence of a nondegenerate solution of the resulting system of equations implies the existence of a metric whose Levi-Civita connection is the given one. In the nonanalytic setting, however, there exist C∞ counterexamples. Let us now discuss how unique is the reconstruction of a metric from affinely parameterized geodesics. Locally, the answer is known: for Riemannian metrics, if was understood already by Cartan and Eisenhart [Eisenhart1923]; for metrics of arbitrary signature, the answer is in the recent papers [Boubel2012, Boubel2014]. Both results are an explicit local description of all metrics having the same Levi-Civita connection. In the Riemannian case, a global analog of the result of Cartan is due to [DeRham1952]. As the only interesting unsolved problem in this topic we suggest",
   "evidence": "The canonical input is Question 11.2.2 from Keith Burns and Vladimir Matveev:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 272,
   "attempt": 1
  },
  "AIM-GEOMETRY-0274": {
   "statement_status": "exact",
   "original_statement": "Question 12.0.3. Suppose a closed manifold (M, g ) admits a nonzero (1, 1) -tensor field that is self-adjoint with respect to g, parallel and nilpotent. Does this manifold or its double cover admit a nonzero light-like parallel vector field? \n\nAlso in the Finsler case parameterised geodesics determine the connection; we will call two Finsler metrics affinely equivalent if any geodesic of the first metric (considered as a curved parametrized such that the length of the velocity vector is a constant) is a geodesic of the second metric.",
   "clean_statement": "Question 12.0.3. Suppose a closed manifold (M, g ) admits a nonzero (1, 1) -tensor field that is self-adjoint with respect to g, parallel and nilpotent. Does this manifold or its double cover admit a nonzero light-like parallel vector field?\n\nAlso in the Finsler case parameterised geodesics determine the connection; we will call two Finsler metrics affinely equivalent if any geodesic of the first metric (considered as a curved parametrized such that the length of the velocity vector is a constant) is a geodesic of the second metric.",
   "public_statement": "Question 12.0.3. Suppose a closed manifold (M, g ) admits a nonzero (1, 1) -tensor field that is self-adjoint with respect to g, parallel and nilpotent. Does this manifold or its double cover admit a nonzero light-like parallel vector field?\n\nAlso in the Finsler case parameterised geodesics determine the connection; we will call two Finsler metrics affinely equivalent if any geodesic of the first metric (considered as a curved parametrized such that the length of the velocity vector is a constant) is a geodesic of the second metric.",
   "evidence": "The canonical record is Question 12.0.3 in Keith Burns and Vladimir Matveev, *Open problems and questions about geodesics*. Inspection of the source PDF shows that the question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 273,
   "attempt": 1
  },
  "AIM-GEOMETRY-0275": {
   "statement_status": "exact",
   "original_statement": "Problem 12.0.4. Describe all affinely equivalent Finsler metrics. \n\nThe \"unparameterized\" versions of these problems for Riemannian metrics are the subject of the recent survey [Matveev2012a]. Roughly speaking, the situation is similar to the one in the \"parameterized\" case: locally, a general strategy for reconstructing a metric is understood: the existence of a metric compatible with a path structure is equivalent to the existence of (nondegenerate) parallel sections of a certain tensor bundle [Eas-Mat2008]. A connection on this tensor bundle is OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 29 \n\nconstructed by the projective structure constructed by the path structure. Thus, a theoretic answer is to see whether the holonomy group of this connection preserves a certain nondegenerate element of the fiber, and a practical method is to construct the curvatures of the connection and look for elements of the fiber compatible with the curvature. As an interesting open problem we suggest:",
   "clean_statement": "Problem 12.0.4. Describe all affinely equivalent Finsler metrics.\n\nThe \"unparameterized\" versions of these problems for Riemannian metrics are the subject of the recent survey [Matveev2012a]. Roughly speaking, the situation is similar to the one in the \"parameterized\" case: locally, a general strategy for reconstructing a metric is understood: the existence of a metric compatible with a path structure is equivalent to the existence of (nondegenerate) parallel sections of a certain tensor bundle [Eas-Mat2008]. A connection on this tensor bundle is OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 29\n\nconstructed by the projective structure constructed by the path structure. Thus, a theoretic answer is to see whether the holonomy group of this connection preserves a certain nondegenerate element of the fiber, and a practical method is to construct the curvatures of the connection and look for elements of the fiber compatible with the curvature. As an interesting open problem we suggest:",
   "public_statement": "Problem 12.0.4. Describe all affinely equivalent Finsler metrics.\n\nThe \"unparameterized\" versions of these problems for Riemannian metrics are the subject of the recent survey [Matveev2012a]. Roughly speaking, the situation is similar to the one in the \"parameterized\" case: locally, a general strategy for reconstructing a metric is understood: the existence of a metric compatible with a path structure is equivalent to the existence of (nondegenerate) parallel sections of a certain tensor bundle [Eas-Mat2008]. A connection on this tensor bundle is OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS 29\n\nconstructed by the projective structure constructed by the path structure. Thus, a theoretic answer is to see whether the holonomy group of this connection preserves a certain nondegenerate element of the fiber, and a practical method is to construct the curvatures of the connection and look for elements of the fiber compatible with the curvature. As an interesting open problem we suggest:",
   "evidence": "The primary Burns--Matveev PDF first defines affine equivalence in the preceding lines: two Finsler metrics are affinely equivalent when a geodesic of the first, with its constant-speed affine parametrization, is a geodesic of the second. It then states the entire target:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 274,
   "attempt": 1
  },
  "AIM-GEOMETRY-0276": {
   "statement_status": "exact",
   "original_statement": "Problem 12.0.5. Construct a system of scalar invariants of a projective structure that vanish if and only if there exists (locally, in a neighborhood of almost every point) a metric compatible with a given projective structure. \n\nIn dimension two, the problem was solved in [Br-Du-Eas2009], and the system of invariants is quite complicated - the simplest invariant has degree 5 in derivatives. It is possible that in higher dimensions the system of invariants could be easier in some ways, since in this case the PDE-system corresponding to the existence of a metric for a projective structure has a higher degree of overdeterminacy. In particular fewer differentiations are needed to construct the first obstruction to the existence of a metric class. See the recent paper [Dun-Eas2014]. Let us now discuss the freedom in reconstructing the metric by its geodesics; an equivalent question is how many different metrics can have the same geodesics considered as unparameterized curves. Of course, the metrics g and const · g have the same geodesics; in [Matveev2012a] it was shown that for a generic metric any projectively equivalent metric is proportional to it with a constant coefficient. There exist local (the first examples are due already to Lagrange, Beltrami and Dini) and global examples of nonproportional projectively equivalent metrics. Locally, in the Riemannian case, a complete description of projectively equivalent metrics is due to Levi-Civita [Levi-Civita1896] and in arbitrary signature is due to [Bol-Mat2013]. Globally, in the Riemannian case, the situation is also pretty clear, but if the metrics have arbitrary signature, virtually nothing is known. A general problem is to understand topology of closed manifolds admitting nonproptional projectively equivalent metrics of indefinite signature and as the simplest version of this problem we suggest",
   "clean_statement": "Problem 12.0.5. Construct a system of scalar invariants of a projective structure that vanish if and only if there exists (locally, in a neighborhood of almost every point) a metric compatible with a given projective structure.\n\nIn dimension two, the problem was solved in [Br-Du-Eas2009], and the system of invariants is quite complicated - the simplest invariant has degree 5 in derivatives. It is possible that in higher dimensions the system of invariants could be easier in some ways, since in this case the PDE-system corresponding to the existence of a metric for a projective structure has a higher degree of overdeterminacy. In particular fewer differentiations are needed to construct the first obstruction to the existence of a metric class. See the recent paper [Dun-Eas2014]. Let us now discuss the freedom in reconstructing the metric by its geodesics; an equivalent question is how many different metrics can have the same geodesics considered as unparameterized curves. Of course, the metrics g and const · g have the same geodesics; in [Matveev2012a] it was shown that for a generic metric any projectively equivalent metric is proportional to it with a constant coefficient. There exist local (the first examples are due already to Lagrange, Beltrami and Dini) and global examples of nonproportional projectively equivalent metrics. Locally, in the Riemannian case, a complete description of projectively equivalent metrics is due to Levi-Civita [Levi-Civita1896] and in arbitrary signature is due to [Bol-Mat2013]. Globally, in the Riemannian case, the situation is also pretty clear, but if the metrics have arbitrary signature, virtually nothing is known. A general problem is to understand topology of closed manifolds admitting nonproptional projectively equivalent metrics of indefinite signature and as the simplest version of this problem we suggest",
   "public_statement": "Problem 12.0.5. Construct a system of scalar invariants of a projective structure that vanish if and only if there exists (locally, in a neighborhood of almost every point) a metric compatible with a given projective structure.\n\nIn dimension two, the problem was solved in [Br-Du-Eas2009], and the system of invariants is quite complicated - the simplest invariant has degree 5 in derivatives. It is possible that in higher dimensions the system of invariants could be easier in some ways, since in this case the PDE-system corresponding to the existence of a metric for a projective structure has a higher degree of overdeterminacy. In particular fewer differentiations are needed to construct the first obstruction to the existence of a metric class. See the recent paper [Dun-Eas2014]. Let us now discuss the freedom in reconstructing the metric by its geodesics; an equivalent question is how many different metrics can have the same geodesics considered as unparameterized curves. Of course, the metrics g and const · g have the same geodesics; in [Matveev2012a] it was shown that for a generic metric any projectively equivalent metric is proportional to it with a constant coefficient. There exist local (the first examples are due already to Lagrange, Beltrami and Dini) and global examples of nonproportional projectively equivalent metrics. Locally, in the Riemannian case, a complete description of projectively equivalent metrics is due to Levi-Civita [Levi-Civita1896] and in arbitrary signature is due to [Bol-Mat2013]. Globally, in the Riemannian case, the situation is also pretty clear, but if the metrics have arbitrary signature, virtually nothing is known. A general problem is to understand topology of closed manifolds admitting nonproptional projectively equivalent metrics of indefinite signature and as the simplest version of this problem we suggest",
   "evidence": "The canonical input is Problem 12.0.5 from the AIM problem list *Open Problems and Questions about Geodesics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 275,
   "attempt": 1
  },
  "AIM-GEOMETRY-0277": {
   "statement_status": "exact",
   "original_statement": "Question 12.0.6. Can a 3-dimensional sphere admit two nonproportional projec-tively equivalent metrics of indefinite signature? \n\nAcknowledgements. We thank all the participants, especially Victor Bangert, Charles Boubel, Nancy Hingston, Gerhard Knieper, Yiming Long, Gabriel Pater-nain, Mark Pollicott, Regina Rotman, St´ ephane Sabourau, Benjamin Schmidt, for active participation in the problem sessions. We also thank non participants who have helped: Juan-Carlos Alvarez Paiva, Ivan Babenko, Florent Balacheff, Helga Baum, Michael Bialy, Dima Burago, Maciej Dunajski, Misha Gromov, Larry Guth, Sergei Ivanov, Bruce Kleiner, Alan Reid, Stefan Suhr. Most of the writing of the paper was done during three visits by KB to the Friedrich-Schiller-Universit¨ at Jena, supported by DFG (SPP 1154 and GK 1523).",
   "clean_statement": "Question 12.0.6. Can a 3-dimensional sphere admit two nonproportional projec-tively equivalent metrics of indefinite signature?\n\nAcknowledgements. We thank all the participants, especially Victor Bangert, Charles Boubel, Nancy Hingston, Gerhard Knieper, Yiming Long, Gabriel Pater-nain, Mark Pollicott, Regina Rotman, St´ ephane Sabourau, Benjamin Schmidt, for active participation in the problem sessions. We also thank non participants who have helped: Juan-Carlos Alvarez Paiva, Ivan Babenko, Florent Balacheff, Helga Baum, Michael Bialy, Dima Burago, Maciej Dunajski, Misha Gromov, Larry Guth, Sergei Ivanov, Bruce Kleiner, Alan Reid, Stefan Suhr. Most of the writing of the paper was done during three visits by KB to the Friedrich-Schiller-Universit¨ at Jena, supported by DFG (SPP 1154 and GK 1523).",
   "public_statement": "Question 12.0.6. Can a 3-dimensional sphere admit two nonproportional projec-tively equivalent metrics of indefinite signature?\n\nAcknowledgements. We thank all the participants, especially Victor Bangert, Charles Boubel, Nancy Hingston, Gerhard Knieper, Yiming Long, Gabriel Pater-nain, Mark Pollicott, Regina Rotman, St´ ephane Sabourau, Benjamin Schmidt, for active participation in the problem sessions. We also thank non participants who have helped: Juan-Carlos Alvarez Paiva, Ivan Babenko, Florent Balacheff, Helga Baum, Michael Bialy, Dima Burago, Maciej Dunajski, Misha Gromov, Larry Guth, Sergei Ivanov, Bruce Kleiner, Alan Reid, Stefan Suhr. Most of the writing of the paper was done during three visits by KB to the Friedrich-Schiller-Universit¨ at Jena, supported by DFG (SPP 1154 and GK 1523).",
   "evidence": "The canonical record comes from the last question in Keith Burns and Vladimir Matveev, *Open problems and questions about geodesics*. The primary PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 276,
   "attempt": 1
  },
  "AIM-GEOMETRY-0278": {
   "statement_status": "exact",
   "original_statement": "1. Elliptically fibred CY manifolds and their SLAG submanifolds \n\n• What can be said about SLAGs in a CY 3-fold that happens to be elliptically fibred or K3 fibred? \n\n• Are there nice characterizations of such situations? \n\n• Do we know anything special about the SLAGs in these types of CY manifolds? For example, are they fibred? Even if they are not fibred in general, maybe we can make a nice theory out of the ones that are fibred. Can we make a nice derived category using the Lagrangians that respect the elliptic structure? \n\n• More specifically, suppose M 6 is a CY 3-fold fibreing over CP 1 with K3 fibres. Imag-ine that L3 is a SLAG in M that fibres over a curve C in CP 1, with fibres which are real T 2 torii in the K3's. We understand torii in K3, so perhaps we can assemble L\n\nas 1-parameter families? \n\n• The question probably makes the most sense in a family of CY manifolds or in a limit of such a family. Here is a related question. Consider a family Mt of such fibred CY's, fibreing over the same base B with K3 fibres, which collapse in some \"adiabatic limit\" as t → 0. That is, M0 = B. Suppose we have a family Lt of SLAGs in Mt, fibreing over a curve C in B. What is the curve B in the limit as t → 0? \n\n• Can we find a way to make the limiting geometry yield information about the SLAGs in the family of K3-fibred CY 3-folds? 2. Yau-Zaslow number for K3 manifolds \n\n• Let X be a K3 surface. Then a SLAG L in X is a holomorphic curve with respect to some other complex structure J on X.\n\n• Can we use this fact to compute the Yau-Zaslow number? (The number of holomor-phic curves representing a given homology class in X.) 13. Backlund transformations and integrable systems in calibrated Geometry \n\n• Are there Backlund type transformations that relate different calibrated geometries? In other words, are there equivalences or relations among PDE systems for calibrated objects? \n\n• For example, there is a relationship between pseudo-holomorphic curves in S6 and SLAGS fibred by great circles. Then the cone on them is coassociative in R7.\n\n• More generally, what is the role of integrable systems in calibrations? \n\n• In the case of SLAGs in C3, it might be important. Can integrable systems be used to investigate the index problems for SLAGs? \n\n• A related question: Given a calibration, what is the best way to understand what it calibrates? How can we express the calibrated condition in a useful way? For example, as an exterior differential system, or as completing inequalities to equalities. 4. Analogue of Floer homology in G2 geometry \n\n• Consider two coassociative submanifolds N and N ′ in a G2 manifold M 7 which are sufficiently close. How many associative submanifolds A of M with boundary in both \n\nN and N ′ are there? This problem might be equivalent to Seiberg-Witten data on the coassociatives. \n\n• Another way to phrase this is to consider two coassociatives N and N ′ with associa-tive \"necks\" joining them. In general, N and N ′ will intersect in circles. We expect the associative necks joining them to visit the circles. \n\n• It might be more appropriate to generalize this question to the context of closed G2\n\nstructures, not necessarily co-closed. This might be the right area for transversality arguments to work. \n\n• Assume that N ′ is essentially the graph of a small closed self-dual 2-form on N\n\nand consider only associative necks of small volume. Do these compute the Seiberg-Witten invariants of N? This is clear by Taubes \"GW = SW\" if N and N ′ are non-intersecting. (If the self-dual 2-form has no zeroes.) \n\n• It would also be interesting to investigate the quantum field theory analogue of this situation in M-theory. 5. Globally defined calibrated submanifolds in Euclidean spaces \n\n• Describe the graphs of h: R3 → R4 which are associative, and globally defined. They do not have to be linear. For example, the Cartesian product of the graph of a holomorphic function with a line. \n\n• These are Bernstein-type questions. Perhaps we can consider graphs with growth conditions, such as polynomial volume growth? \n\n• For example, three-dimensional special Lagrangian graphs (submanifolds) in R6 ⊂\n\nIm( O) are always associative, thus one has plenty of associative global graphs if the potential function of three variables for the special Lagrangian graphs are taken as harmonic functions of the first two variables. • We can formulate the Liouville-Bernstein question: Are all associative global graphs with linear growth affine? The answer is yes, at least if one has the gradient bound. One may also formulate the growth condition in terms of the volume. 6. Geometric Measure Theory and Calibrations \n\n• Do singularities of calibrated currents simplify if we impose some genericity condi-tions on the calibrated manifold? \n\n• Can we close the gap between general calibrated cycles and more well-behaved ones? \n\n• For example, is the generic calibrated cycle in an infinite family of cycles for a family of calibrations? \n\n• Another question: Consider a finite-dimensional family of calibrated currents in an infinite dimensional family of calibrations. For a generic calibration, is the generic current nicely structured? (Well behaved in some sense?) \n\n• A generic calibration here could mean, for example, Joyce's notion of almost Calabi-Yau, or a closed but not necessarily co-closed G2 structure. \n\n• Suppose that for a family of calibrations, we get a family of calibrated submanifolds existing. What does this mean? \n\n• What about uniqueness of tangent cones for calibrated currents? Rivi` ere proved the epiperimetric inequality in complex dimension n ≤ 2 (for almost complex subman-ifolds.) There are some other results known (Tian-Rivi` ere). Uniqueness of tangent cones is true in the complex case. 7. Special Lagrangian cones \n\n• An interesting question is the analyticity of special Lagrangian cones. There exist examples of SLAG cones over torii which are not analytic. Are these cones going to be semi-analytic? (This means that if they were reflection invariant, they would be analytic. So they are \"half\" of an analytic cone.) \n\n• We can consider semi-analyticity of cones in calibrated geometry in general. For example, let Σ be an almost complex curve in S6. The cone C(Σ) is associative in \n\nR7. Is C(Σ) a component of an algebraic variety? \n\n• One question is to classify all the SLAG T 2 cones in C3 which are semi-analytic. The fact that they are cones over T 2 allows you to use integrable systems. The con-struction involves some algebraic map from a generalized Jacobi variety to projective space. \n\n• It would be useful to prove that the index of a cone was proportional to area. We would need a more effective proof of a 2-sided bound on area in terms of the index. \n\n• Are there any higher genus special Legendrian submanifolds in S5 with λ1 = 2, where \n\nλ1 is the first eigenvalue of the Laplacian? 8. Regularity of calibrated cycles satisfying extra conditions \n\n• Do we have better regularity results for calibrated cycles satisfying some transver-sality condition? For example, calibrated submanifolds whose tangent planes lie in some geometrically defined open subset of the full calibrated Grassmanian.",
   "clean_statement": "1. Elliptically fibred CY manifolds and their SLAG submanifolds\n\n• What can be said about SLAGs in a CY 3-fold that happens to be elliptically fibred or K3 fibred?\n\n• Are there nice characterizations of such situations?\n\n• Do we know anything special about the SLAGs in these types of CY manifolds? For example, are they fibred? Even if they are not fibred in general, maybe we can make a nice theory out of the ones that are fibred. Can we make a nice derived category using the Lagrangians that respect the elliptic structure?\n\n• More specifically, suppose M 6 is a CY 3-fold fibreing over CP 1 with K3 fibres. Imag-ine that L3 is a SLAG in M that fibres over a curve C in CP 1, with fibres which are real T 2 torii in the K3's. We understand torii in K3, so perhaps we can assemble L\n\nas 1-parameter families?\n\n• The question probably makes the most sense in a family of CY manifolds or in a limit of such a family. Here is a related question. Consider a family Mt of such fibred CY's, fibreing over the same base B with K3 fibres, which collapse in some \"adiabatic limit\" as t → 0. That is, M0 = B. Suppose we have a family Lt of SLAGs in Mt, fibreing over a curve C in B. What is the curve B in the limit as t → 0?\n\n• Can we find a way to make the limiting geometry yield information about the SLAGs in the family of K3-fibred CY 3-folds? 2. Yau-Zaslow number for K3 manifolds\n\n• Let X be a K3 surface. Then a SLAG L in X is a holomorphic curve with respect to some other complex structure J on X.\n\n• Can we use this fact to compute the Yau-Zaslow number? (The number of holomor-phic curves representing a given homology class in X.) 13. Backlund transformations and integrable systems in calibrated Geometry\n\n• Are there Backlund type transformations that relate different calibrated geometries? In other words, are there equivalences or relations among PDE systems for calibrated objects?\n\n• For example, there is a relationship between pseudo-holomorphic curves in S6 and SLAGS fibred by great circles. Then the cone on them is coassociative in R7.\n\n• More generally, what is the role of integrable systems in calibrations?\n\n• In the case of SLAGs in C3, it might be important. Can integrable systems be used to investigate the index problems for SLAGs?\n\n• A related question: Given a calibration, what is the best way to understand what it calibrates? How can we express the calibrated condition in a useful way? For example, as an exterior differential system, or as completing inequalities to equalities. 4. Analogue of Floer homology in G2 geometry\n\n• Consider two coassociative submanifolds N and N ′ in a G2 manifold M 7 which are sufficiently close. How many associative submanifolds A of M with boundary in both\n\nN and N ′ are there? This problem might be equivalent to Seiberg-Witten data on the coassociatives.\n\n• Another way to phrase this is to consider two coassociatives N and N ′ with associa-tive \"necks\" joining them. In general, N and N ′ will intersect in circles. We expect the associative necks joining them to visit the circles.\n\n• It might be more appropriate to generalize this question to the context of closed G2\n\nstructures, not necessarily co-closed. This might be the right area for transversality arguments to work.\n\n• Assume that N ′ is essentially the graph of a small closed self-dual 2-form on N\n\nand consider only associative necks of small volume. Do these compute the Seiberg-Witten invariants of N? This is clear by Taubes \"GW = SW\" if N and N ′ are non-intersecting. (If the self-dual 2-form has no zeroes.)\n\n• It would also be interesting to investigate the quantum field theory analogue of this situation in M-theory. 5. Globally defined calibrated submanifolds in Euclidean spaces\n\n• Describe the graphs of h: R3 → R4 which are associative, and globally defined. They do not have to be linear. For example, the Cartesian product of the graph of a holomorphic function with a line.\n\n• These are Bernstein-type questions. Perhaps we can consider graphs with growth conditions, such as polynomial volume growth?\n\n• For example, three-dimensional special Lagrangian graphs (submanifolds) in R6 ⊂\n\nIm( O) are always associative, thus one has plenty of associative global graphs if the potential function of three variables for the special Lagrangian graphs are taken as harmonic functions of the first two variables. • We can formulate the Liouville-Bernstein question: Are all associative global graphs with linear growth affine? The answer is yes, at least if one has the gradient bound. One may also formulate the growth condition in terms of the volume. 6. Geometric Measure Theory and Calibrations\n\n• Do singularities of calibrated currents simplify if we impose some genericity condi-tions on the calibrated manifold?\n\n• Can we close the gap between general calibrated cycles and more well-behaved ones?\n\n• For example, is the generic calibrated cycle in an infinite family of cycles for a family of calibrations?\n\n• Another question: Consider a finite-dimensional family of calibrated currents in an infinite dimensional family of calibrations. For a generic calibration, is the generic current nicely structured? (Well behaved in some sense?)\n\n• A generic calibration here could mean, for example, Joyce's notion of almost Calabi-Yau, or a closed but not necessarily co-closed G2 structure.\n\n• Suppose that for a family of calibrations, we get a family of calibrated submanifolds existing. What does this mean?\n\n• What about uniqueness of tangent cones for calibrated currents? Rivi` ere proved the epiperimetric inequality in complex dimension n ≤ 2 (for almost complex subman-ifolds.) There are some other results known (Tian-Rivi` ere). Uniqueness of tangent cones is true in the complex case. 7. Special Lagrangian cones\n\n• An interesting question is the analyticity of special Lagrangian cones. There exist examples of SLAG cones over torii which are not analytic. Are these cones going to be semi-analytic? (This means that if they were reflection invariant, they would be analytic. So they are \"half\" of an analytic cone.)\n\n• We can consider semi-analyticity of cones in calibrated geometry in general. For example, let Σ be an almost complex curve in S6. The cone C(Σ) is associative in\n\nR7. Is C(Σ) a component of an algebraic variety?\n\n• One question is to classify all the SLAG T 2 cones in C3 which are semi-analytic. The fact that they are cones over T 2 allows you to use integrable systems. The con-struction involves some algebraic map from a generalized Jacobi variety to projective space.\n\n• It would be useful to prove that the index of a cone was proportional to area. We would need a more effective proof of a 2-sided bound on area in terms of the index.\n\n• Are there any higher genus special Legendrian submanifolds in S5 with λ1 = 2, where\n\nλ1 is the first eigenvalue of the Laplacian? 8. Regularity of calibrated cycles satisfying extra conditions\n\n• Do we have better regularity results for calibrated cycles satisfying some transver-sality condition? For example, calibrated submanifolds whose tangent planes lie in some geometrically defined open subset of the full calibrated Grassmanian.",
   "public_statement": "1. Elliptically fibred CY manifolds and their SLAG submanifolds\n\n• What can be said about SLAGs in a CY 3-fold that happens to be elliptically fibred or K3 fibred?\n\n• Are there nice characterizations of such situations?\n\n• Do we know anything special about the SLAGs in these types of CY manifolds? For example, are they fibred? Even if they are not fibred in general, maybe we can make a nice theory out of the ones that are fibred. Can we make a nice derived category using the Lagrangians that respect the elliptic structure?\n\n• More specifically, suppose M 6 is a CY 3-fold fibreing over CP 1 with K3 fibres. Imag-ine that L3 is a SLAG in M that fibres over a curve C in CP 1, with fibres which are real T 2 torii in the K3's. We understand torii in K3, so perhaps we can assemble L\n\nas 1-parameter families?\n\n• The question probably makes the most sense in a family of CY manifolds or in a limit of such a family. Here is a related question. Consider a family Mt of such fibred CY's, fibreing over the same base B with K3 fibres, which collapse in some \"adiabatic limit\" as t → 0. That is, M0 = B. Suppose we have a family Lt of SLAGs in Mt, fibreing over a curve C in B. What is the curve B in the limit as t → 0?\n\n• Can we find a way to make the limiting geometry yield information about the SLAGs in the family of K3-fibred CY 3-folds? 2. Yau-Zaslow number for K3 manifolds\n\n• Let X be a K3 surface. Then a SLAG L in X is a holomorphic curve with respect to some other complex structure J on X.\n\n• Can we use this fact to compute the Yau-Zaslow number? (The number of holomor-phic curves representing a given homology class in X.) 13. Backlund transformations and integrable systems in calibrated Geometry\n\n• Are there Backlund type transformations that relate different calibrated geometries? In other words, are there equivalences or relations among PDE systems for calibrated objects?\n\n• For example, there is a relationship between pseudo-holomorphic curves in S6 and SLAGS fibred by great circles. Then the cone on them is coassociative in R7.\n\n• More generally, what is the role of integrable systems in calibrations?\n\n• In the case of SLAGs in C3, it might be important. Can integrable systems be used to investigate the index problems for SLAGs?\n\n• A related question: Given a calibration, what is the best way to understand what it calibrates? How can we express the calibrated condition in a useful way? For example, as an exterior differential system, or as completing inequalities to equalities. 4. Analogue of Floer homology in G2 geometry\n\n• Consider two coassociative submanifolds N and N ′ in a G2 manifold M 7 which are sufficiently close. How many associative submanifolds A of M with boundary in both\n\nN and N ′ are there? This problem might be equivalent to Seiberg-Witten data on the coassociatives.\n\n• Another way to phrase this is to consider two coassociatives N and N ′ with associa-tive \"necks\" joining them. In general, N and N ′ will intersect in circles. We expect the associative necks joining them to visit the circles.\n\n• It might be more appropriate to generalize this question to the context of closed G2\n\nstructures, not necessarily co-closed. This might be the right area for transversality arguments to work.\n\n• Assume that N ′ is essentially the graph of a small closed self-dual 2-form on N\n\nand consider only associative necks of small volume. Do these compute the Seiberg-Witten invariants of N? This is clear by Taubes \"GW = SW\" if N and N ′ are non-intersecting. (If the self-dual 2-form has no zeroes.)\n\n• It would also be interesting to investigate the quantum field theory analogue of this situation in M-theory. 5. Globally defined calibrated submanifolds in Euclidean spaces\n\n• Describe the graphs of h: R3 → R4 which are associative, and globally defined. They do not have to be linear. For example, the Cartesian product of the graph of a holomorphic function with a line.\n\n• These are Bernstein-type questions. Perhaps we can consider graphs with growth conditions, such as polynomial volume growth?\n\n• For example, three-dimensional special Lagrangian graphs (submanifolds) in R6 ⊂\n\nIm( O) are always associative, thus one has plenty of associative global graphs if the potential function of three variables for the special Lagrangian graphs are taken as harmonic functions of the first two variables. • We can formulate the Liouville-Bernstein question: Are all associative global graphs with linear growth affine? The answer is yes, at least if one has the gradient bound. One may also formulate the growth condition in terms of the volume. 6. Geometric Measure Theory and Calibrations\n\n• Do singularities of calibrated currents simplify if we impose some genericity condi-tions on the calibrated manifold?\n\n• Can we close the gap between general calibrated cycles and more well-behaved ones?\n\n• For example, is the generic calibrated cycle in an infinite family of cycles for a family of calibrations?\n\n• Another question: Consider a finite-dimensional family of calibrated currents in an infinite dimensional family of calibrations. For a generic calibration, is the generic current nicely structured? (Well behaved in some sense?)\n\n• A generic calibration here could mean, for example, Joyce's notion of almost Calabi-Yau, or a closed but not necessarily co-closed G2 structure.\n\n• Suppose that for a family of calibrations, we get a family of calibrated submanifolds existing. What does this mean?\n\n• What about uniqueness of tangent cones for calibrated currents? Rivi` ere proved the epiperimetric inequality in complex dimension n ≤ 2 (for almost complex subman-ifolds.) There are some other results known (Tian-Rivi` ere). Uniqueness of tangent cones is true in the complex case. 7. Special Lagrangian cones\n\n• An interesting question is the analyticity of special Lagrangian cones. There exist examples of SLAG cones over torii which are not analytic. Are these cones going to be semi-analytic? (This means that if they were reflection invariant, they would be analytic. So they are \"half\" of an analytic cone.)\n\n• We can consider semi-analyticity of cones in calibrated geometry in general. For example, let Σ be an almost complex curve in S6. The cone C(Σ) is associative in\n\nR7. Is C(Σ) a component of an algebraic variety?\n\n• One question is to classify all the SLAG T 2 cones in C3 which are semi-analytic. The fact that they are cones over T 2 allows you to use integrable systems. The con-struction involves some algebraic map from a generalized Jacobi variety to projective space.\n\n• It would be useful to prove that the index of a cone was proportional to area. We would need a more effective proof of a 2-sided bound on area in terms of the index.\n\n• Are there any higher genus special Legendrian submanifolds in S5 with λ1 = 2, where\n\nλ1 is the first eigenvalue of the Laplacian? 8. Regularity of calibrated cycles satisfying extra conditions\n\n• Do we have better regularity results for calibrated cycles satisfying some transver-sality condition? For example, calibrated submanifolds whose tangent planes lie in some geometrically defined open subset of the full calibrated Grassmanian.",
   "evidence": "The primary three-page PDF is an AIM Workshop on Calibrations problem list compiled by Spiro Karigiannis in 2006. Lines 9--30 of page 1 form one numbered item:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 277,
   "attempt": 1
  },
  "AIM-GEOMETRY-0279": {
   "statement_status": "exact",
   "original_statement": "Question 1 implies the complete solution to the sphere packing problem in dimensions 8 and 24.",
   "clean_statement": "Question 1 implies the complete solution to the sphere packing problem in dimensions 8 and 24.",
   "public_statement": "Question 1 implies the complete solution to the sphere packing problem in dimensions 8 and 24.",
   "evidence": "The exact canonical corpus record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 278,
   "attempt": 1
  },
  "AIM-GEOMETRY-0280": {
   "statement_status": "exact",
   "original_statement": "Question 1. Can the technique of Cohn-Elkies prove that the E8 and Leech lattices have the highest sphere packing densities in dimensions 8 and 24, respectively?",
   "clean_statement": "Question 1. Can the technique of Cohn-Elkies prove that the E8 and Leech lattices have the highest sphere packing densities in dimensions 8 and 24, respectively?",
   "public_statement": "Question 1. Can the technique of Cohn-Elkies prove that the E8 and Leech lattices have the highest sphere packing densities in dimensions 8 and 24, respectively?",
   "evidence": "The official AIM PDF states, exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 279,
   "attempt": 1
  },
  "AIM-GEOMETRY-0281": {
   "statement_status": "exact",
   "original_statement": "Question 2. Settle the Cohn-Elkies conjecture: that there exist 'optimal functions' for use in their theorem in these dimensions. (If yes, the answer to",
   "clean_statement": "Question 2. Settle the Cohn-Elkies conjecture: that there exist 'optimal functions' for use in their theorem in these dimensions. (If yes, the answer to",
   "public_statement": "Question 2. Settle the Cohn-Elkies conjecture: that there exist 'optimal functions' for use in their theorem in these dimensions. (If yes, the answer to",
   "evidence": "The exact canonical record is truncated:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 280,
   "attempt": 1
  },
  "AIM-GEOMETRY-0282": {
   "statement_status": "exact",
   "original_statement": "Question 3. Further conjectures of Cohn-Miller: A: Can their determinants be rescaled according to their conjecture 3.2? B: Do their rescaled determinants converge uniformly on compact sets to a limiting function? C: Does this limiting function have any unexpected sign changes (i.e. sign changes other than the forced ones). If 'yes', 'yes', and 'no', respectively, then the answer to",
   "clean_statement": "Question 3. Further conjectures of Cohn-Miller: A: Can their determinants be rescaled according to their conjecture 3.2? B: Do their rescaled determinants converge uniformly on compact sets to a limiting function? C: Does this limiting function have any unexpected sign changes (i.e. sign changes other than the forced ones). If 'yes', 'yes', and 'no', respectively, then the answer to",
   "public_statement": "Question 3. Further conjectures of Cohn-Miller: A: Can their determinants be rescaled according to their conjecture 3.2? B: Do their rescaled determinants converge uniformly on compact sets to a limiting function? C: Does this limiting function have any unexpected sign changes (i.e. sign changes other than the forced ones). If 'yes', 'yes', and 'no', respectively, then the answer to",
   "evidence": "The exact canonical record is truncated:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 281,
   "attempt": 1
  },
  "AIM-GEOMETRY-0283": {
   "statement_status": "exact",
   "original_statement": "Question 4. What are the purported limits of the Cohn-Miller sequences? Can the observed rationalities in their Taylor coefficients be explained? Are there others outside from the quadratic terms? Contributed by Peter Sarnak and Andreas Str¨ ombergsson \n\nThe Epstein-zeta function of a lattice is a Mellin transform of the theta function of the lattice. These functions depend only on the length spectrum of the vectors in the lattice. Let E(L, s ) be the Epstein Zeta function for a lattice L of covolume 1 in Rn. Here s is positive and E(L, s ) = ∑ \n\n> m∈L−{ 0}\n\n|m|−2s. This converges for s > n/ 2 and makes sense by analytic continuation for all s (this is due to Epstein).",
   "clean_statement": "Question 4. What are the purported limits of the Cohn-Miller sequences? Can the observed rationalities in their Taylor coefficients be explained? Are there others outside from the quadratic terms? Contributed by Peter Sarnak and Andreas Str¨ ombergsson\n\nThe Epstein-zeta function of a lattice is a Mellin transform of the theta function of the lattice. These functions depend only on the length spectrum of the vectors in the lattice. Let E(L, s ) be the Epstein Zeta function for a lattice L of covolume 1 in Rn. Here s is positive and E(L, s ) = ∑\n\n> m∈L−{ 0}\n\n|m|−2s. This converges for s > n/ 2 and makes sense by analytic continuation for all s (this is due to Epstein).",
   "public_statement": "Question 4. What are the purported limits of the Cohn-Miller sequences? Can the observed rationalities in their Taylor coefficients be explained? Are there others outside from the quadratic terms? Contributed by Peter Sarnak and Andreas Str¨ ombergsson\n\nThe Epstein-zeta function of a lattice is a Mellin transform of the theta function of the lattice. These functions depend only on the length spectrum of the vectors in the lattice. Let E(L, s ) be the Epstein Zeta function for a lattice L of covolume 1 in Rn. Here s is positive and E(L, s ) = ∑\n\n> m∈L−{ 0}\n\n|m|−2s. This converges for s > n/ 2 and makes sense by analytic continuation for all s (this is due to Epstein).",
   "evidence": "The canonical record literally reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 282,
   "attempt": 1
  },
  "AIM-GEOMETRY-0284": {
   "statement_status": "exact",
   "original_statement": "Question 0: For a fixed s, which L yields the minimum value of E(L, s )? For s → ∞ this becomes the familiar densest lattice packing problem. For n = 2 Rankin and Cassels showed that the hexagonal lattice is the unique minimizer for all s > 0. What we can show is that D4 (when n = 4), L = E8 (when n = 8) and the Leech lattice (when \n\nn = 24) are local optima for E(L, s ), for every s > 0.",
   "clean_statement": "Question 0: For a fixed s, which L yields the minimum value of E(L, s )? For s → ∞ this becomes the familiar densest lattice packing problem. For n = 2 Rankin and Cassels showed that the hexagonal lattice is the unique minimizer for all s > 0. What we can show is that D4 (when n = 4), L = E8 (when n = 8) and the Leech lattice (when\n\nn = 24) are local optima for E(L, s ), for every s > 0.",
   "public_statement": "Question 0: For a fixed s, which L yields the minimum value of E(L, s )? For s → ∞ this becomes the familiar densest lattice packing problem. For n = 2 Rankin and Cassels showed that the hexagonal lattice is the unique minimizer for all s > 0. What we can show is that D4 (when n = 4), L = E8 (when n = 8) and the Leech lattice (when\n\nn = 24) are local optima for E(L, s ), for every s > 0.",
   "evidence": "The exact canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 283,
   "attempt": 1
  },
  "AIM-GEOMETRY-0285": {
   "statement_status": "exact",
   "original_statement": "Conjecture 1: These lattices in dimensions 4, 8 and 24 are global minimizers of E(L, s ), for any fixed s > \n0. (The corresponding statement is false in dimension n = 3; the fcc lattice is not the global minimizer of E(L, s ) for any 0 < s < 3/4.)",
   "clean_statement": "Conjecture 1: These lattices in dimensions 4, 8 and 24 are global minimizers of E(L, s ), for any fixed s >\n0. (The corresponding statement is false in dimension n = 3; the fcc lattice is not the global minimizer of E(L, s ) for any 0 < s < 3/4.)",
   "public_statement": "Conjecture 1: These lattices in dimensions 4, 8 and 24 are global minimizers of E(L, s ), for any fixed s >\n0. (The corresponding statement is false in dimension n = 3; the fcc lattice is not the global minimizer of E(L, s ) for any 0 < s < 3/4.)",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 284,
   "attempt": 1
  },
  "AIM-GEOMETRY-0286": {
   "statement_status": "exact",
   "original_statement": "Problem 2: Does there exist in each dimension a lattice L for which E(L, s ) has no zeroes in (0, ∞)? (Note: One can show that if a lattice L′ is a local minimizer of E(L, s ) for all positive \n\ns, then at least E(L′, s ) has no odd order zero in s on (0, ∞).) \n\nContributed by Y.-Z. Huang, Jim Lepowsky and Arne Meurman \n\nCan the completely-extendable conformal intertwining algebras introduced by Y.-Z. Huang be used to prove the conjecture of Frenkel, Lepowsky and Meurman that the Moonshine module vertex operator algebra has a uniqueness property analogous to the uniqueness of the Golay code and of the Leech lattice? (For instance, one already has analogs of 'duality' and 'self-duality' (as in lattices) for vertex operator algebras). \n\nContributed by John Conway \n\nWhat is the analog of 'genus' for vertex operator algebras? What is the analog of Kneser's 'neighbour theorem' in lattices for vertex operator algebras? What is the analo-gous notion to Voronoi cells and Voronoi vectors for vertex operator algebras? \n\nContributed by Geoff Mason \n\nConjecture: Any 'good' vertex operator algebra V has finite automorphism group if and only if V1 = 0. Here 'good' means 'has semi-simple representation theory'. \n\nContributed by John Conway, Noam Elkies, and Simon Norton \n\nLet Γ be the Thompson-Smith lattice, an even unimodular lattice. The construction of Γ is 'local' in nature, and there is little global information known. For example, the rank and determinant of Γ are known for every value of p. However, the minimal vectors of Γ are not known and the theta function is not known. It was recently shown by G. Nebe that Γ contains a norm 12 vector. Is there a norm 14 vector in Γ? Are the norm 12 and norm 16 orbits unique? Does the vertex operator algebra for Γ have any interesting structure? \n\nContributed by John Conway and Noam Elkies \n\nLet M be the Monster simple group. Then M has a subgroup 2 · M24 × S2 where M24 \n\nis the automorphism group of the Golay code. The group Co 0 has a subgroup 2 12 · M24 \n\nwhere 2 12 ∼= C, the Golay code, and F i 24 also has a subgroup 2 12 · M24 where 2 12 ∼= C∗,the Golay cocode: 2 · M24 × S2 ⊆ M¡ ¬\n\n212 · M24 ⊆ Co 0 212 · M24 ⊆ F i 24 \n\n¬ ¡\n\nM24 \n\nThese are automorphisms of \n\nV OA \n\n¡ ¬\n\nLattice X\n\n¬ ¡\n\nCode \n\nWhat is X?",
   "clean_statement": "Problem 2: Does there exist in each dimension a lattice L for which E(L, s ) has no zeroes in (0, ∞)? (Note: One can show that if a lattice L′ is a local minimizer of E(L, s ) for all positive\n\ns, then at least E(L′, s ) has no odd order zero in s on (0, ∞).)\n\nContributed by Y.-Z. Huang, Jim Lepowsky and Arne Meurman\n\nCan the completely-extendable conformal intertwining algebras introduced by Y.-Z. Huang be used to prove the conjecture of Frenkel, Lepowsky and Meurman that the Moonshine module vertex operator algebra has a uniqueness property analogous to the uniqueness of the Golay code and of the Leech lattice? (For instance, one already has analogs of 'duality' and 'self-duality' (as in lattices) for vertex operator algebras).\n\nContributed by John Conway\n\nWhat is the analog of 'genus' for vertex operator algebras? What is the analog of Kneser's 'neighbour theorem' in lattices for vertex operator algebras? What is the analo-gous notion to Voronoi cells and Voronoi vectors for vertex operator algebras?\n\nContributed by Geoff Mason\n\nConjecture: Any 'good' vertex operator algebra V has finite automorphism group if and only if V1 = 0. Here 'good' means 'has semi-simple representation theory'.\n\nContributed by John Conway, Noam Elkies, and Simon Norton\n\nLet Γ be the Thompson-Smith lattice, an even unimodular lattice. The construction of Γ is 'local' in nature, and there is little global information known. For example, the rank and determinant of Γ are known for every value of p. However, the minimal vectors of Γ are not known and the theta function is not known. It was recently shown by G. Nebe that Γ contains a norm 12 vector. Is there a norm 14 vector in Γ? Are the norm 12 and norm 16 orbits unique? Does the vertex operator algebra for Γ have any interesting structure?\n\nContributed by John Conway and Noam Elkies\n\nLet M be the Monster simple group. Then M has a subgroup 2 · M24 × S2 where M24\n\nis the automorphism group of the Golay code. The group Co 0 has a subgroup 2 12 · M24\n\nwhere 2 12 ∼= C, the Golay code, and F i 24 also has a subgroup 2 12 · M24 where 2 12 ∼= C∗,the Golay cocode: 2 · M24 × S2 ⊆ M¡ ¬\n\n212 · M24 ⊆ Co 0 212 · M24 ⊆ F i 24\n\n¬ ¡\n\nM24\n\nThese are automorphisms of\n\nV OA\n\n¡ ¬\n\nLattice X\n\n¬ ¡\n\nCode\n\nWhat is X?",
   "public_statement": "Problem 2: Does there exist in each dimension a lattice L for which E(L, s ) has no zeroes in (0, ∞)? (Note: One can show that if a lattice L′ is a local minimizer of E(L, s ) for all positive\n\ns, then at least E(L′, s ) has no odd order zero in s on (0, ∞).)\n\nContributed by Y.-Z. Huang, Jim Lepowsky and Arne Meurman\n\nCan the completely-extendable conformal intertwining algebras introduced by Y.-Z. Huang be used to prove the conjecture of Frenkel, Lepowsky and Meurman that the Moonshine module vertex operator algebra has a uniqueness property analogous to the uniqueness of the Golay code and of the Leech lattice? (For instance, one already has analogs of 'duality' and 'self-duality' (as in lattices) for vertex operator algebras).\n\nContributed by John Conway\n\nWhat is the analog of 'genus' for vertex operator algebras? What is the analog of Kneser's 'neighbour theorem' in lattices for vertex operator algebras? What is the analo-gous notion to Voronoi cells and Voronoi vectors for vertex operator algebras?\n\nContributed by Geoff Mason\n\nConjecture: Any 'good' vertex operator algebra V has finite automorphism group if and only if V1 = 0. Here 'good' means 'has semi-simple representation theory'.\n\nContributed by John Conway, Noam Elkies, and Simon Norton\n\nLet Γ be the Thompson-Smith lattice, an even unimodular lattice. The construction of Γ is 'local' in nature, and there is little global information known. For example, the rank and determinant of Γ are known for every value of p. However, the minimal vectors of Γ are not known and the theta function is not known. It was recently shown by G. Nebe that Γ contains a norm 12 vector. Is there a norm 14 vector in Γ? Are the norm 12 and norm 16 orbits unique? Does the vertex operator algebra for Γ have any interesting structure?\n\nContributed by John Conway and Noam Elkies\n\nLet M be the Monster simple group. Then M has a subgroup 2 · M24 × S2 where M24\n\nis the automorphism group of the Golay code. The group Co 0 has a subgroup 2 12 · M24\n\nwhere 2 12 ∼= C, the Golay code, and F i 24 also has a subgroup 2 12 · M24 where 2 12 ∼= C∗,the Golay cocode: 2 · M24 × S2 ⊆ M¡ ¬\n\n212 · M24 ⊆ Co 0 212 · M24 ⊆ F i 24\n\n¬ ¡\n\nM24\n\nThese are automorphisms of\n\nV OA\n\n¡ ¬\n\nLattice X\n\n¬ ¡\n\nCode\n\nWhat is X?",
   "evidence": "The canonical record contains several accidentally concatenated problems. Its genuine first item is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 285,
   "attempt": 1
  },
  "AIM-GEOMETRY-0287": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 1.1 Let X be a hyperK¨ ahler manifold with a hyperHamiltonian action of a compact Lie group G. Let f denote the norm-square of the hyperK¨ ahler moment map. Sup-pose that for every x ∈ X the forward trajectory of x under the negative gradient flow of f\n\nis contained in a compact subset of X. Then there is a surjection \n\nH∗ \n\n> K\n\n(X) → H∗ \n\n> K\n\n(μ−1 \n\n> HK\n\n(0)) \n\nin K-equivariant cohomology. \n\nComment 1.2 Kirwan, in stating her theorems, actually requires that one of the following hold: 1. The original hyperK¨ ahler manifold is compact. 2. The norm-square of the hyperK¨ ahler moment map is proper. 3. For every x ∈ X, the forward trajectory of x under the negative gradient flow of the norm-square of the hyperK¨ ahler moment map is contained in a compact subset of X.(Note that 1 implies 2 implies 3.) The first and second hypotheses are almost never fulfilled, so the only relevant hypothesis is the last one. We have therefore stated",
   "clean_statement": null,
   "public_statement": "Conjecture 1.1 Let X be a hyperK¨ ahler manifold with a hyperHamiltonian action of a compact Lie group G. Let f denote the norm-square of the hyperK¨ ahler moment map. Sup-pose that for every x ∈ X the forward trajectory of x under the negative gradient flow of f\n\nis contained in a compact subset of X. Then there is a surjection\n\nH∗\n\n> K\n\n(X) → H∗\n\n> K\n\n(μ−1\n\n> HK\n\n(0))\n\nin K-equivariant cohomology.\n\nComment 1.2 Kirwan, in stating her theorems, actually requires that one of the following hold: 1. The original hyperK¨ ahler manifold is compact. 2. The norm-square of the hyperK¨ ahler moment map is proper. 3. For every x ∈ X, the forward trajectory of x under the negative gradient flow of the norm-square of the hyperK¨ ahler moment map is contained in a compact subset of X.(Note that 1 implies 2 implies 3.) The first and second hypotheses are almost never fulfilled, so the only relevant hypothesis is the last one. We have therefore stated",
   "evidence": "There is a genuine notation defect in the source: the acting group is called $G$, while the cohomology group is indexed by an undefined $K$. In this report, **the conjecture is reconstructed with $K=G$**. This is the only natural reading, but it is an explicit reconstruction rather than a silent correction.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 286,
   "attempt": 1
  },
  "AIM-GEOMETRY-0288": {
   "statement_status": "exact",
   "original_statement": "Conjecture 1.1 using this last hypothesis. 2Comment 1.3 Section 3 of Kirwan's manuscript deals with the case where the group is abelian. The participants at the AIM workshop tentatively agreed that in this case, under the hypotheses of the conjecture above, the argument seems to be identical to the one given by Kirwan for the original surjectivity (for the symplectic case). The only difference is that we replace the moment map by the hyperK¨ ahler moment map. However, we (the AIM participants) did not check the details in this case. \n\nComment 1.4 Kirwan's third hypothesis (that the gradient flows are contained in compact sets) holds for linear G-actions on T ∗Cn, by an argument similar to the one given by Sjamaar [Sja98] for linear G-actions on Cn. Here, G can be nonabelian. Thus, if the conjecture is correct, we have Kirwan surjectivity for quiver varieties. \n\nComment 1.5 Section 5 of Kirwan's manuscript, which gives her second proof of hy-perK¨ ahler Kirwan surjectivity, uses the \"plus construction\" of Carrell-Goresky [CG83] when \n\nμ−1 \n\n> C\n\n(0) is singular. (In almost all examples of interest, μ−1 \n\n> C\n\n(0) is indeed singular.) The work-shop participants were confused on the following points: first, the results in Carrell-Goresky are stated for homology, not cohomology. Second, their results are not stated equivariantly. Regarding the first point, S. Wu believes that having the result for homology is not a prob-lem; as long as there is a filtration, there are spectral sequences for both homology and cohomology. Regarding the second, S. Wu and J. Weitsman had a discussion during the workshop in which they seemed to conclude that the flow of the square of the moment map seems to define an R2 action rather than a C∗ action, so we did not understand why the theorem in [CG83] can be applied. The workshop participants hope to come to a better understanding on these points. \n\n2 3-Sasakian surjectivity",
   "clean_statement": "Conjecture 1.1 using this last hypothesis. 2Comment 1.3 Section 3 of Kirwan's manuscript deals with the case where the group is abelian. The participants at the AIM workshop tentatively agreed that in this case, under the hypotheses of the conjecture above, the argument seems to be identical to the one given by Kirwan for the original surjectivity (for the symplectic case). The only difference is that we replace the moment map by the hyperK¨ ahler moment map. However, we (the AIM participants) did not check the details in this case.\n\nComment 1.4 Kirwan's third hypothesis (that the gradient flows are contained in compact sets) holds for linear G-actions on T ∗Cn, by an argument similar to the one given by Sjamaar [Sja98] for linear G-actions on Cn. Here, G can be nonabelian. Thus, if the conjecture is correct, we have Kirwan surjectivity for quiver varieties.\n\nComment 1.5 Section 5 of Kirwan's manuscript, which gives her second proof of hy-perK¨ ahler Kirwan surjectivity, uses the \"plus construction\" of Carrell-Goresky [CG83] when\n\nμ−1\n\n> C\n\n(0) is singular. (In almost all examples of interest, μ−1\n\n> C\n\n(0) is indeed singular.) The work-shop participants were confused on the following points: first, the results in Carrell-Goresky are stated for homology, not cohomology. Second, their results are not stated equivariantly. Regarding the first point, S. Wu believes that having the result for homology is not a prob-lem; as long as there is a filtration, there are spectral sequences for both homology and cohomology. Regarding the second, S. Wu and J. Weitsman had a discussion during the workshop in which they seemed to conclude that the flow of the square of the moment map seems to define an R2 action rather than a C∗ action, so we did not understand why the theorem in [CG83] can be applied. The workshop participants hope to come to a better understanding on these points.\n\n2 3-Sasakian surjectivity",
   "public_statement": "Conjecture 1.1 using this last hypothesis. 2Comment 1.3 Section 3 of Kirwan's manuscript deals with the case where the group is abelian. The participants at the AIM workshop tentatively agreed that in this case, under the hypotheses of the conjecture above, the argument seems to be identical to the one given by Kirwan for the original surjectivity (for the symplectic case). The only difference is that we replace the moment map by the hyperK¨ ahler moment map. However, we (the AIM participants) did not check the details in this case.\n\nComment 1.4 Kirwan's third hypothesis (that the gradient flows are contained in compact sets) holds for linear G-actions on T ∗Cn, by an argument similar to the one given by Sjamaar [Sja98] for linear G-actions on Cn. Here, G can be nonabelian. Thus, if the conjecture is correct, we have Kirwan surjectivity for quiver varieties.\n\nComment 1.5 Section 5 of Kirwan's manuscript, which gives her second proof of hy-perK¨ ahler Kirwan surjectivity, uses the \"plus construction\" of Carrell-Goresky [CG83] when\n\nμ−1\n\n> C\n\n(0) is singular. (In almost all examples of interest, μ−1\n\n> C\n\n(0) is indeed singular.) The work-shop participants were confused on the following points: first, the results in Carrell-Goresky are stated for homology, not cohomology. Second, their results are not stated equivariantly. Regarding the first point, S. Wu believes that having the result for homology is not a prob-lem; as long as there is a filtration, there are spectral sequences for both homology and cohomology. Regarding the second, S. Wu and J. Weitsman had a discussion during the workshop in which they seemed to conclude that the flow of the square of the moment map seems to define an R2 action rather than a C∗ action, so we did not understand why the theorem in [CG83] can be applied. The workshop participants hope to come to a better understanding on these points.\n\n2 3-Sasakian surjectivity",
   "evidence": "This canonical record is not a new standalone conjecture. It is an extraction of Comments 1.3--1.5 following Conjecture 1.1 in the AIM workshop notes *Moment maps and surjectivity in various geometries* (workshop held August 9--13, 2004). The exact `input.json` begins with the final sentence of Comment 1.2, which belongs to `AIM-GEOMETRY-0287`, and ends with the heading “2 3-Sasakian surjectivity,” which begins the next section. Inspection of the official PDF fixes the owned text as Comments 1.3, 1.4, and 1.5 only.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 287,
   "attempt": 1
  },
  "AIM-GEOMETRY-0289": {
   "statement_status": "exact",
   "original_statement": "Conjecture 2.1 (C. Boyer) Let S be a complete compact 3-Sasakian manifold with a con-nected compact Lie group G ⊂ Aut (S) acting on S. Consider the 3-Sasakian reduction Sred \n\n(reduction at 0). Then the 3-Sasakian Kirwan map \n\nH∗\n\n> G\n\n(S) → H∗(Sred )\n\nis surjective up to the middle dimension. Here we take either Q or R coefficients. \n\nComment 2.2 (C. Boyer) We know the conjecture is true for S1-reductions of spheres. Note also that in odd dimension, \"up to the middle dimension\" is all we can hope for. Boyer, Galicki, and Piccinni [BGP02] have constructed some examples of 3-Sasakian quotients for which they don't know how to compute cohomology, and Kirwan surjectivity would be a nice way to get at some cohomology. \n\nComment 2.3 (C. Boyer and T. Hausel) We might need some additional hypotheses for the nonabelian case, but T. Hausel has some ideas about how to prove something like this for abelian reductions. Namely, the surjectivity in the 3-Sasakian case should be related to surjectivity for hyperK¨ ahler quotients. 3We consider a more specific example. Suppose there is an action of Sp (n + 1) acting on S4n+3. Restrict to an action of a compact subgroup G of this Sp (n + 1). Consider the 3-Sasakian reduction S4n+3 ///G, and additionally assume that this is orbifold. We conjecture that, in this case, the 3-Sasakian Kirwan map is surjective to the middle dimension of the quotient. Here is a tentative idea of the proof. Look at the correponding hyperK¨ ahler quotient. Consider the same group G acting now on Hn+1, which containes S4n+3 as its sphere. Let X\n\ndenote now the hyperK¨ ahler quotient. The space X might be singular since G might not have a center. But now suppose there exists some ξ in the center of G so that the hyperK¨ ahler quotient is orbifold. Assume also that the quotient is hypercompact (see Definition 6.9), so the core is middle-dimensional. Then, in this case, the surjectivity of the hyperK¨ ahler Kirwan map should imply the surjectivity (up to middle dimension) of the 3-Sasakian Kirwan map. T. Hausel thinks this should be an elementary argument. C. Boyer and T. Hausel will be pursuing this line of thought after the AIM Workshop ends.",
   "clean_statement": "Conjecture 2.1 (C. Boyer) Let S be a complete compact 3-Sasakian manifold with a con-nected compact Lie group G ⊂ Aut (S) acting on S. Consider the 3-Sasakian reduction Sred\n\n(reduction at 0). Then the 3-Sasakian Kirwan map\n\nH∗\n\n> G\n\n(S) → H∗(Sred )\n\nis surjective up to the middle dimension. Here we take either Q or R coefficients.\n\nComment 2.2 (C. Boyer) We know the conjecture is true for S1-reductions of spheres. Note also that in odd dimension, \"up to the middle dimension\" is all we can hope for. Boyer, Galicki, and Piccinni [BGP02] have constructed some examples of 3-Sasakian quotients for which they don't know how to compute cohomology, and Kirwan surjectivity would be a nice way to get at some cohomology.\n\nComment 2.3 (C. Boyer and T. Hausel) We might need some additional hypotheses for the nonabelian case, but T. Hausel has some ideas about how to prove something like this for abelian reductions. Namely, the surjectivity in the 3-Sasakian case should be related to surjectivity for hyperK¨ ahler quotients. 3We consider a more specific example. Suppose there is an action of Sp (n + 1) acting on S4n+3. Restrict to an action of a compact subgroup G of this Sp (n + 1). Consider the 3-Sasakian reduction S4n+3 ///G, and additionally assume that this is orbifold. We conjecture that, in this case, the 3-Sasakian Kirwan map is surjective to the middle dimension of the quotient. Here is a tentative idea of the proof. Look at the correponding hyperK¨ ahler quotient. Consider the same group G acting now on Hn+1, which containes S4n+3 as its sphere. Let X\n\ndenote now the hyperK¨ ahler quotient. The space X might be singular since G might not have a center. But now suppose there exists some ξ in the center of G so that the hyperK¨ ahler quotient is orbifold. Assume also that the quotient is hypercompact (see Definition 6.9), so the core is middle-dimensional. Then, in this case, the surjectivity of the hyperK¨ ahler Kirwan map should imply the surjectivity (up to middle dimension) of the 3-Sasakian Kirwan map. T. Hausel thinks this should be an elementary argument. C. Boyer and T. Hausel will be pursuing this line of thought after the AIM Workshop ends.",
   "public_statement": "Conjecture 2.1 (C. Boyer) Let S be a complete compact 3-Sasakian manifold with a con-nected compact Lie group G ⊂ Aut (S) acting on S. Consider the 3-Sasakian reduction Sred\n\n(reduction at 0). Then the 3-Sasakian Kirwan map\n\nH∗\n\n> G\n\n(S) → H∗(Sred )\n\nis surjective up to the middle dimension. Here we take either Q or R coefficients.\n\nComment 2.2 (C. Boyer) We know the conjecture is true for S1-reductions of spheres. Note also that in odd dimension, \"up to the middle dimension\" is all we can hope for. Boyer, Galicki, and Piccinni [BGP02] have constructed some examples of 3-Sasakian quotients for which they don't know how to compute cohomology, and Kirwan surjectivity would be a nice way to get at some cohomology.\n\nComment 2.3 (C. Boyer and T. Hausel) We might need some additional hypotheses for the nonabelian case, but T. Hausel has some ideas about how to prove something like this for abelian reductions. Namely, the surjectivity in the 3-Sasakian case should be related to surjectivity for hyperK¨ ahler quotients. 3We consider a more specific example. Suppose there is an action of Sp (n + 1) acting on S4n+3. Restrict to an action of a compact subgroup G of this Sp (n + 1). Consider the 3-Sasakian reduction S4n+3 ///G, and additionally assume that this is orbifold. We conjecture that, in this case, the 3-Sasakian Kirwan map is surjective to the middle dimension of the quotient. Here is a tentative idea of the proof. Look at the correponding hyperK¨ ahler quotient. Consider the same group G acting now on Hn+1, which containes S4n+3 as its sphere. Let X\n\ndenote now the hyperK¨ ahler quotient. The space X might be singular since G might not have a center. But now suppose there exists some ξ in the center of G so that the hyperK¨ ahler quotient is orbifold. Assume also that the quotient is hypercompact (see Definition 6.9), so the core is middle-dimensional. Then, in this case, the surjectivity of the hyperK¨ ahler Kirwan map should imply the surjectivity (up to middle dimension) of the 3-Sasakian Kirwan map. T. Hausel thinks this should be an elementary argument. C. Boyer and T. Hausel will be pursuing this line of thought after the AIM Workshop ends.",
   "evidence": "The canonical record comes from the AIM workshop *Moment maps and surjectivity in various geometries*, Conjecture 2.1 and Comments 2.2--2.3. The source is the official AIM problem-list PDF: <https://aimath.org/WWN/momentmaps/momentmaps.pdf>.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 288,
   "attempt": 1
  },
  "AIM-GEOMETRY-0290": {
   "statement_status": "exact",
   "original_statement": "Question 2.4 (T. Hausel) Is there a Martin theorem for 3-Sasakian quotients? What about for Sasakian quotients? Getting a Martin-type formula requires Kirwan surjectivity, but perhaps surjectivity \"up to middle dimension\" would be enough to get a Martin-type theorem for the 3-Sasakian case. \n\nComment 2.5 (J. Munn) Martin's theorem probably should be extendable analytically in the polysymplectic case. Since 3-Sasakian manifolds can arise as boundaries of hyperK¨ ahler manifolds, we can look at compact cohomology based on the 3-Sasakian manifold (regarded as a boundary). Here we can ignore cone points and do integration theory.",
   "clean_statement": "Question 2.4 (T. Hausel) Is there a Martin theorem for 3-Sasakian quotients? What about for Sasakian quotients? Getting a Martin-type formula requires Kirwan surjectivity, but perhaps surjectivity \"up to middle dimension\" would be enough to get a Martin-type theorem for the 3-Sasakian case.\n\nComment 2.5 (J. Munn) Martin's theorem probably should be extendable analytically in the polysymplectic case. Since 3-Sasakian manifolds can arise as boundaries of hyperK¨ ahler manifolds, we can look at compact cohomology based on the 3-Sasakian manifold (regarded as a boundary). Here we can ignore cone points and do integration theory.",
   "public_statement": "Question 2.4 (T. Hausel) Is there a Martin theorem for 3-Sasakian quotients? What about for Sasakian quotients? Getting a Martin-type formula requires Kirwan surjectivity, but perhaps surjectivity \"up to middle dimension\" would be enough to get a Martin-type theorem for the 3-Sasakian case.\n\nComment 2.5 (J. Munn) Martin's theorem probably should be extendable analytically in the polysymplectic case. Since 3-Sasakian manifolds can arise as boundaries of hyperK¨ ahler manifolds, we can look at compact cohomology based on the 3-Sasakian manifold (regarded as a boundary). Here we can ignore cone points and do integration theory.",
   "evidence": "The exact OCR-extracted record, including its line breaks and encoding defect, is preserved in **input.json**. A verified mathematical transcription is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 289,
   "attempt": 1
  },
  "AIM-GEOMETRY-0291": {
   "statement_status": "exact",
   "original_statement": "Question 2.6 (C. Boyer) In the case of 3-Sasakian reduction by abelian groups, would it be possible to extract information about the cohomology of the reductions by using fixed points of smaller tori? The question is motivated by the fact that some cases of 3-Sasakian reductions of spheres by S1 is well-understood. Suppose there is a circle action on S4n+3, where the action is locally free on the whole sphere, and free on the zero level set. Then one can see explicitly that H∗ \n\n> S1\n\n(S4n+3; Q) surjects onto H∗(S4n+3 ///S 1; Q), up to the middle dimension. The idea would be to look at the fixed point set M H for subtori H ⊂ T, where we start with the abelian group T acting on the 3-Sasakian manifold M. Note that the fixed point sets M H are contact, and for the Sasakian case, known examples of M H are spheres. \n\nComment 2.7 (C. Boyer) Main references for 3-Sasakian toral reductions are [BG99, BGM94, BGMR98]. \n\n3 Kirwan surjectivity for contact quotients",
   "clean_statement": "Question 2.6 (C. Boyer) In the case of 3-Sasakian reduction by abelian groups, would it be possible to extract information about the cohomology of the reductions by using fixed points of smaller tori? The question is motivated by the fact that some cases of 3-Sasakian reductions of spheres by S1 is well-understood. Suppose there is a circle action on S4n+3, where the action is locally free on the whole sphere, and free on the zero level set. Then one can see explicitly that H∗\n\n> S1\n\n(S4n+3; Q) surjects onto H∗(S4n+3 ///S 1; Q), up to the middle dimension. The idea would be to look at the fixed point set M H for subtori H ⊂ T, where we start with the abelian group T acting on the 3-Sasakian manifold M. Note that the fixed point sets M H are contact, and for the Sasakian case, known examples of M H are spheres.\n\nComment 2.7 (C. Boyer) Main references for 3-Sasakian toral reductions are [BG99, BGM94, BGMR98].\n\n3 Kirwan surjectivity for contact quotients",
   "public_statement": "Question 2.6 (C. Boyer) In the case of 3-Sasakian reduction by abelian groups, would it be possible to extract information about the cohomology of the reductions by using fixed points of smaller tori? The question is motivated by the fact that some cases of 3-Sasakian reductions of spheres by S1 is well-understood. Suppose there is a circle action on S4n+3, where the action is locally free on the whole sphere, and free on the zero level set. Then one can see explicitly that H∗\n\n> S1\n\n(S4n+3; Q) surjects onto H∗(S4n+3 ///S 1; Q), up to the middle dimension. The idea would be to look at the fixed point set M H for subtori H ⊂ T, where we start with the abelian group T acting on the 3-Sasakian manifold M. Note that the fixed point sets M H are contact, and for the Sasakian case, known examples of M H are spheres.\n\nComment 2.7 (C. Boyer) Main references for 3-Sasakian toral reductions are [BG99, BGM94, BGMR98].\n\n3 Kirwan surjectivity for contact quotients",
   "evidence": "The record is Question 2.6 and Comment 2.7 of the official AIM workshop problem list *Moment maps and surjectivity in various geometries* (August 2004): <https://aimath.org/WWN/momentmaps/momentmaps.pdf>.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 290,
   "attempt": 1
  },
  "AIM-GEOMETRY-0292": {
   "statement_status": "exact",
   "original_statement": "Question 3.1 (E. Lerman) Kirwan surjecitivty cannot work for contact quotients. Here is a counterexample: consider S3 ⊂ C2 with the action of S1 given by λ · (z1, z 2) = ( λz 1, λ −1z2).\n\nThe S1-equivariant cohomology of this S3 is the ordinary cohomology of P1. On the other 4hand, the contact moment map is μ: ( z1, z 2) 7 → | z1|2 − | z2|2. Hence the contact quotient is \n\nμ−1(0) /S 1 = S1. There is no surjective map from H∗(P1) to H∗(S1).\n\nNevertheless, there are still interesting questions to ask: what is the kernel? What is the cokernel? \n\nComment 3.2 (C. Boyer) It would also be interesting to ask under what conditions a contact Kirwan surjectivity would hold. There may be a class of spaces for which surjectivity would hold in the 3-Sasakian case (although perhaps with some dimension condition, as in the \"up to middle dimension\" clause in the 3-Sasakian surjectivity conjecture above). \n\n4 Orbifold cohomology and surjectivity \n\nThe starting point of this discussion is the theorem of Goldin, Holm, and Knutson. \n\nTheorem 4.1 (Goldin-Holm-Knutson) Let (M, ω ) be a compact Hamiltonian T -space with moment map μ. Let α be a regular value of μ. Then the direct sum \n\n⊕g∈T H∗ \n\n> T\n\n(M g; Q)\n\nhas a ring structure such that there exists a natural ring map \n\n⊕g∈T H∗ \n\n> T\n\n(M g; Q) → H∗\n\n> orb\n\n(M / /αT; Q)\n\nis a surjection. Here M g denotes the points in M fixed by the element g ∈ T, and orbifold cohomology H∗ \n\n> orb\n\nis in the sense of Chen and Ruan. \n\nComment 4.2 (E. Lerman) A more standard definition of Horb ∗ is due to Haefliger; it precedes Chen-Ruan's definition by at least a decade or two.",
   "clean_statement": "Question 3.1 (E. Lerman) Kirwan surjecitivty cannot work for contact quotients. Here is a counterexample: consider S3 ⊂ C2 with the action of S1 given by λ · (z1, z 2) = ( λz 1, λ −1z2).\n\nThe S1-equivariant cohomology of this S3 is the ordinary cohomology of P1. On the other 4hand, the contact moment map is μ: ( z1, z 2) 7 → | z1|2 − | z2|2. Hence the contact quotient is\n\nμ−1(0) /S 1 = S1. There is no surjective map from H∗(P1) to H∗(S1).\n\nNevertheless, there are still interesting questions to ask: what is the kernel? What is the cokernel?\n\nComment 3.2 (C. Boyer) It would also be interesting to ask under what conditions a contact Kirwan surjectivity would hold. There may be a class of spaces for which surjectivity would hold in the 3-Sasakian case (although perhaps with some dimension condition, as in the \"up to middle dimension\" clause in the 3-Sasakian surjectivity conjecture above).\n\n4 Orbifold cohomology and surjectivity\n\nThe starting point of this discussion is the theorem of Goldin, Holm, and Knutson.\n\nTheorem 4.1 (Goldin-Holm-Knutson) Let (M, ω ) be a compact Hamiltonian T -space with moment map μ. Let α be a regular value of μ. Then the direct sum\n\n⊕g∈T H∗\n\n> T\n\n(M g; Q)\n\nhas a ring structure such that there exists a natural ring map\n\n⊕g∈T H∗\n\n> T\n\n(M g; Q) → H∗\n\n> orb\n\n(M / /αT; Q)\n\nis a surjection. Here M g denotes the points in M fixed by the element g ∈ T, and orbifold cohomology H∗\n\n> orb\n\nis in the sense of Chen and Ruan.\n\nComment 4.2 (E. Lerman) A more standard definition of Horb ∗ is due to Haefliger; it precedes Chen-Ruan's definition by at least a decade or two.",
   "public_statement": "Question 3.1 (E. Lerman) Kirwan surjecitivty cannot work for contact quotients. Here is a counterexample: consider S3 ⊂ C2 with the action of S1 given by λ · (z1, z 2) = ( λz 1, λ −1z2).\n\nThe S1-equivariant cohomology of this S3 is the ordinary cohomology of P1. On the other 4hand, the contact moment map is μ: ( z1, z 2) 7 → | z1|2 − | z2|2. Hence the contact quotient is\n\nμ−1(0) /S 1 = S1. There is no surjective map from H∗(P1) to H∗(S1).\n\nNevertheless, there are still interesting questions to ask: what is the kernel? What is the cokernel?\n\nComment 3.2 (C. Boyer) It would also be interesting to ask under what conditions a contact Kirwan surjectivity would hold. There may be a class of spaces for which surjectivity would hold in the 3-Sasakian case (although perhaps with some dimension condition, as in the \"up to middle dimension\" clause in the 3-Sasakian surjectivity conjecture above).\n\n4 Orbifold cohomology and surjectivity\n\nThe starting point of this discussion is the theorem of Goldin, Holm, and Knutson.\n\nTheorem 4.1 (Goldin-Holm-Knutson) Let (M, ω ) be a compact Hamiltonian T -space with moment map μ. Let α be a regular value of μ. Then the direct sum\n\n⊕g∈T H∗\n\n> T\n\n(M g; Q)\n\nhas a ring structure such that there exists a natural ring map\n\n⊕g∈T H∗\n\n> T\n\n(M g; Q) → H∗\n\n> orb\n\n(M / /αT; Q)\n\nis a surjection. Here M g denotes the points in M fixed by the element g ∈ T, and orbifold cohomology H∗\n\n> orb\n\nis in the sense of Chen and Ruan.\n\nComment 4.2 (E. Lerman) A more standard definition of Horb ∗ is due to Haefliger; it precedes Chen-Ruan's definition by at least a decade or two.",
   "evidence": "The canonical JSON record overcaptures the beginning of Section 4 of the source PDF. The owned record is exactly Question 3.1 and Comment 3.2 on page 4 of the AIM workshop document. The heading “4 Orbifold cohomology and surjectivity,” Theorem 4.1 of Goldin--Holm--Knutson, and Comment 4.2 belong to later records and are excluded here.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 291,
   "attempt": 1
  },
  "AIM-GEOMETRY-0293": {
   "statement_status": "exact",
   "original_statement": "Question 4.3 (M. Pflaum) Is there an analogous theorem for compact non-abelian Lie group G?\n\nComment 4.4 (M. Pflaum) The generalization to non-abelian G would be interesting from the point of view of quantization of singular reduced spaces and cross product algebras, as they appear in symplectic orbifold theory. \n\nComment 4.5 (R. Goldin, T. Holm) The main problem would be in defining the product structure on the ring in the LHS of the statement of the theorem. In the theorem of Goldin-Holm-Knutson, they use in a fundamental way the commutativity of the group structure when defining this product structure. \n\nComment 4.6 (R. Goldin, T. Holm, M. Pflaum) We expect a theorem of this nature to be true if we take only the additive structure. Note also that it is clear that we can't simply replace T by G naively. One idea for how to adjust the theorem would be to take the product in the LHS over conjugacy classes 5of G instead of all elements of G. Another idea would be to take the product on the LHS to be ∏\n\n> g∈G\n\nH∗\n\n> G\n\n(∪h∈GM hgh −1\n\n)instead. This question on a non-abelian version of the theorem of Goldin-Holm-Knutson was a topic during the small group discussion sessions at the AIM workshop. We now present some of the conclusions and comments arising from this discussion (R. Goldin, T. Holm, M. Pflaum). One possible approach to this problem would be to use the groupoid language for orbifold cohomology. In other words, given an orbifold which is a symplectic reduction by a torus, we may translate the theorem of Goldin-Holm-Knutson into the language of proper ´etale Lie groupoids. We may then try to use the approach of Moerdijk on descriptions of orbifolds by proper ´ etale Lie groupoids [MM03]. This approach could give a new description of the ring structure of the orbifold cohomology - at least, of orbifolds appearing as global quotients by a torus. It is possible that this new description would also suggest a product structure for the non-abelian case. \n\nComment 4.7 (C. Boyer) There may be some subtleties regarding Morita equivalent groupoids representing the same orbifold. Another approach would be to use methods from Hochschild and cyclic homology theory. One can use \n\nHC ∗(C∞ o G)to obtain the orbifold cohomology H∗\n\n> orb\n\n(X, C).\n\nStill another approach would be to use crepant resolutions (i.e. the crepant resolution conjecture for orbifold cohomology). \n\n5 Topological aspects of moment map theory",
   "clean_statement": "Question 4.3 (M. Pflaum) Is there an analogous theorem for compact non-abelian Lie group G?\n\nComment 4.4 (M. Pflaum) The generalization to non-abelian G would be interesting from the point of view of quantization of singular reduced spaces and cross product algebras, as they appear in symplectic orbifold theory.\n\nComment 4.5 (R. Goldin, T. Holm) The main problem would be in defining the product structure on the ring in the LHS of the statement of the theorem. In the theorem of Goldin-Holm-Knutson, they use in a fundamental way the commutativity of the group structure when defining this product structure.\n\nComment 4.6 (R. Goldin, T. Holm, M. Pflaum) We expect a theorem of this nature to be true if we take only the additive structure. Note also that it is clear that we can't simply replace T by G naively. One idea for how to adjust the theorem would be to take the product in the LHS over conjugacy classes 5of G instead of all elements of G. Another idea would be to take the product on the LHS to be ∏\n\n> g∈G\n\nH∗\n\n> G\n\n(∪h∈GM hgh −1\n\n)instead. This question on a non-abelian version of the theorem of Goldin-Holm-Knutson was a topic during the small group discussion sessions at the AIM workshop. We now present some of the conclusions and comments arising from this discussion (R. Goldin, T. Holm, M. Pflaum). One possible approach to this problem would be to use the groupoid language for orbifold cohomology. In other words, given an orbifold which is a symplectic reduction by a torus, we may translate the theorem of Goldin-Holm-Knutson into the language of proper ´etale Lie groupoids. We may then try to use the approach of Moerdijk on descriptions of orbifolds by proper ´ etale Lie groupoids [MM03]. This approach could give a new description of the ring structure of the orbifold cohomology - at least, of orbifolds appearing as global quotients by a torus. It is possible that this new description would also suggest a product structure for the non-abelian case.\n\nComment 4.7 (C. Boyer) There may be some subtleties regarding Morita equivalent groupoids representing the same orbifold. Another approach would be to use methods from Hochschild and cyclic homology theory. One can use\n\nHC ∗(C∞ o G)to obtain the orbifold cohomology H∗\n\n> orb\n\n(X, C).\n\nStill another approach would be to use crepant resolutions (i.e. the crepant resolution conjecture for orbifold cohomology).\n\n5 Topological aspects of moment map theory",
   "public_statement": "Question 4.3 (M. Pflaum) Is there an analogous theorem for compact non-abelian Lie group G?\n\nComment 4.4 (M. Pflaum) The generalization to non-abelian G would be interesting from the point of view of quantization of singular reduced spaces and cross product algebras, as they appear in symplectic orbifold theory.\n\nComment 4.5 (R. Goldin, T. Holm) The main problem would be in defining the product structure on the ring in the LHS of the statement of the theorem. In the theorem of Goldin-Holm-Knutson, they use in a fundamental way the commutativity of the group structure when defining this product structure.\n\nComment 4.6 (R. Goldin, T. Holm, M. Pflaum) We expect a theorem of this nature to be true if we take only the additive structure. Note also that it is clear that we can't simply replace T by G naively. One idea for how to adjust the theorem would be to take the product in the LHS over conjugacy classes 5of G instead of all elements of G. Another idea would be to take the product on the LHS to be ∏\n\n> g∈G\n\nH∗\n\n> G\n\n(∪h∈GM hgh −1\n\n)instead. This question on a non-abelian version of the theorem of Goldin-Holm-Knutson was a topic during the small group discussion sessions at the AIM workshop. We now present some of the conclusions and comments arising from this discussion (R. Goldin, T. Holm, M. Pflaum). One possible approach to this problem would be to use the groupoid language for orbifold cohomology. In other words, given an orbifold which is a symplectic reduction by a torus, we may translate the theorem of Goldin-Holm-Knutson into the language of proper ´etale Lie groupoids. We may then try to use the approach of Moerdijk on descriptions of orbifolds by proper ´ etale Lie groupoids [MM03]. This approach could give a new description of the ring structure of the orbifold cohomology - at least, of orbifolds appearing as global quotients by a torus. It is possible that this new description would also suggest a product structure for the non-abelian case.\n\nComment 4.7 (C. Boyer) There may be some subtleties regarding Morita equivalent groupoids representing the same orbifold. Another approach would be to use methods from Hochschild and cyclic homology theory. One can use\n\nHC ∗(C∞ o G)to obtain the orbifold cohomology H∗\n\n> orb\n\n(X, C).\n\nStill another approach would be to use crepant resolutions (i.e. the crepant resolution conjecture for orbifold cohomology).\n\n5 Topological aspects of moment map theory",
   "evidence": "The source is the AIM workshop list *Moment maps and surjectivity in various geometries*, Question 4.3 and Comments 4.4--4.7. The question follows Theorem 4.1, which records the Goldin--Holm--Knutson theorem: for a compact Hamiltonian torus space and a regular value, inertial equivariant cohomology maps surjectively **as a ring** to the Chen--Ruan cohomology of the quotient.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 292,
   "attempt": 1
  },
  "AIM-GEOMETRY-0294": {
   "statement_status": "exact",
   "original_statement": "Question 5.1 (E. Lerman) Kirwan surjectivity is not really a \"symplectic\" result - it's more a topological result which relies on certain topological properties of the moment map. So a similar result can be obtained for maps other than a moment map. See [LT97] for an example, where the topology of a small resolution of singular symplectic reduced space was computed using a map which was not a moment map. Can one prove similar results in the hyperK¨ ahler case? There is a theory of \"abstract moment maps\" developed in [GGK02] which isolates those properties of moment maps that make the familiar theorems hold. The setting is roughly as follows; see [GGK02] for details. Suppose given an action of G on a manifold M.Note that M does not necessarily have a symplectic structure. An abstract moment map is a map μ: M → g∗ satisfying G-equivariance as well as some conditions involving subgroups \n\nH of G and its fixed points M H. However, a valid example of an abstract moment map is 6the constant function on R2, where the group S1 acts by rotation. Thus, to get a function suitable for Morse-Bott theory, we must add some condition of non-degeneracy. We define, following [GGK02], an abstract moment map to be non-degenerate if its components are Morse-Bott, and for each component, the critical sets for that component correspond to the fixed point set M G.",
   "clean_statement": "Question 5.1 (E. Lerman) Kirwan surjectivity is not really a \"symplectic\" result - it's more a topological result which relies on certain topological properties of the moment map. So a similar result can be obtained for maps other than a moment map. See [LT97] for an example, where the topology of a small resolution of singular symplectic reduced space was computed using a map which was not a moment map. Can one prove similar results in the hyperK¨ ahler case? There is a theory of \"abstract moment maps\" developed in [GGK02] which isolates those properties of moment maps that make the familiar theorems hold. The setting is roughly as follows; see [GGK02] for details. Suppose given an action of G on a manifold M.Note that M does not necessarily have a symplectic structure. An abstract moment map is a map μ: M → g∗ satisfying G-equivariance as well as some conditions involving subgroups\n\nH of G and its fixed points M H. However, a valid example of an abstract moment map is 6the constant function on R2, where the group S1 acts by rotation. Thus, to get a function suitable for Morse-Bott theory, we must add some condition of non-degeneracy. We define, following [GGK02], an abstract moment map to be non-degenerate if its components are Morse-Bott, and for each component, the critical sets for that component correspond to the fixed point set M G.",
   "public_statement": "Question 5.1 (E. Lerman) Kirwan surjectivity is not really a \"symplectic\" result - it's more a topological result which relies on certain topological properties of the moment map. So a similar result can be obtained for maps other than a moment map. See [LT97] for an example, where the topology of a small resolution of singular symplectic reduced space was computed using a map which was not a moment map. Can one prove similar results in the hyperK¨ ahler case? There is a theory of \"abstract moment maps\" developed in [GGK02] which isolates those properties of moment maps that make the familiar theorems hold. The setting is roughly as follows; see [GGK02] for details. Suppose given an action of G on a manifold M.Note that M does not necessarily have a symplectic structure. An abstract moment map is a map μ: M → g∗ satisfying G-equivariance as well as some conditions involving subgroups\n\nH of G and its fixed points M H. However, a valid example of an abstract moment map is 6the constant function on R2, where the group S1 acts by rotation. Thus, to get a function suitable for Morse-Bott theory, we must add some condition of non-degeneracy. We define, following [GGK02], an abstract moment map to be non-degenerate if its components are Morse-Bott, and for each component, the critical sets for that component correspond to the fixed point set M G.",
   "evidence": "This record is Question 5.1, attributed to E. Lerman, in the AIM workshop list *Moment maps and surjectivity in various geometries*. The exact extracted `problem` field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 293,
   "attempt": 1
  },
  "AIM-GEOMETRY-0295": {
   "statement_status": "exact",
   "original_statement": "Conjecture 5.2 (G. Landweber) Kirwan surjectivity holds under these conditions. \n\nComment 5.3 (M. Harada, T. Holm, L. Mare) For S1-actions, this is shown (under the additional assumption that M is compact) in [GGK02]. The proof for T -actions in [GGK02] contains an error. \n\nComment 5.4 (G. Landweber) Given an action of G on M with abstract moment map, can one find symplectic forms at least locally? If so, can we do the necessary geometry just using these local symplectic forms? \n\nComment 5.5 (M. Harada) The first question in the comment above is addressed in The-orem G.22 of [GGK02]. Roughly, it says that if the abstract moment map is non-degenerate and symplectic slices admit an invariant complex structure, then there is a symplectic form in a neighborhood of an orbit associated to the abstract moment map.",
   "clean_statement": "Conjecture 5.2 (G. Landweber) Kirwan surjectivity holds under these conditions.\n\nComment 5.3 (M. Harada, T. Holm, L. Mare) For S1-actions, this is shown (under the additional assumption that M is compact) in [GGK02]. The proof for T -actions in [GGK02] contains an error.\n\nComment 5.4 (G. Landweber) Given an action of G on M with abstract moment map, can one find symplectic forms at least locally? If so, can we do the necessary geometry just using these local symplectic forms?\n\nComment 5.5 (M. Harada) The first question in the comment above is addressed in The-orem G.22 of [GGK02]. Roughly, it says that if the abstract moment map is non-degenerate and symplectic slices admit an invariant complex structure, then there is a symplectic form in a neighborhood of an orbit associated to the abstract moment map.",
   "public_statement": "Conjecture 5.2 (G. Landweber) Kirwan surjectivity holds under these conditions.\n\nComment 5.3 (M. Harada, T. Holm, L. Mare) For S1-actions, this is shown (under the additional assumption that M is compact) in [GGK02]. The proof for T -actions in [GGK02] contains an error.\n\nComment 5.4 (G. Landweber) Given an action of G on M with abstract moment map, can one find symplectic forms at least locally? If so, can we do the necessary geometry just using these local symplectic forms?\n\nComment 5.5 (M. Harada) The first question in the comment above is addressed in The-orem G.22 of [GGK02]. Roughly, it says that if the abstract moment map is non-degenerate and symplectic slices admit an invariant complex structure, then there is a symplectic form in a neighborhood of an orbit associated to the abstract moment map.",
   "evidence": "The exact canonical record in **input.json** contains Conjecture 5.2 and Comments 5.3--5.5 from the 2004 AIM workshop document. Its phrase “these conditions” refers to the immediately preceding Question 5.1. That adjacent record is used only to recover the antecedent; no result for it is claimed here.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 294,
   "attempt": 1
  },
  "AIM-GEOMETRY-0296": {
   "statement_status": "exact",
   "original_statement": "Question 5.6 (G. Landweber) In standard Morse theory, every function is arbitrarily close to a Morse function. In the space of abstract moment maps, is every abstract moment map arbitrarily close to a non-degenerate abstract moment map?",
   "clean_statement": "Question 5.6 (G. Landweber) In standard Morse theory, every function is arbitrarily close to a Morse function. In the space of abstract moment maps, is every abstract moment map arbitrarily close to a non-degenerate abstract moment map?",
   "public_statement": "Question 5.6 (G. Landweber) In standard Morse theory, every function is arbitrarily close to a Morse function. In the space of abstract moment maps, is every abstract moment map arbitrarily close to a non-degenerate abstract moment map?",
   "evidence": "The source is the AIM workshop list *Moment maps and surjectivity in various geometries*, Section 5, “Topological aspects of moment map theory.” The literal statement in the official PDF is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 295,
   "attempt": 1
  },
  "AIM-GEOMETRY-0297": {
   "statement_status": "corrected_verified",
   "original_statement": "Question 5.7 (G. Landweber) We can model the local structure on M using abstract mo-ment maps. Is there an analogous statement for contact moment maps? Is there a theory of abstract contact moment maps?",
   "clean_statement": "We can model the local structure on \\(M\\) using abstract moment maps. Is there an analogous statement for contact moment maps? Is there a theory of abstract contact moment maps?",
   "public_statement": "We can model the local structure on \\(M\\) using abstract moment maps. Is there an analogous statement for contact moment maps? Is there a theory of abstract contact moment maps?",
   "evidence": "The source PDF confirms this text. The only repair needed is the line-break hyphenation “mo-ment,” which I reconstruct as “moment.” Thus the recovered question is:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 296,
   "attempt": 1
  },
  "AIM-GEOMETRY-0298": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 5.8 (E. Lerman) Can we prove Kirwan surjectivity without Morse theory? The motivation for this question comes from the fact that two fundamental results in the theory of symplectic moment maps - connectedness and convexity, which were originally proved using Morse theory have an alternative proof [CDM88]. This alternative approach works well in the contact setting (equivalently in the setting of symplectic cones) where Morse theory fails. The reasons for the failure in the contact setting are due to the fact are that the contact moment maps are not Morse and that there is no relationship between critical points and isotropy groups. In the equivalent setting of symplectic cones the moment maps are not proper. \n\nComment 5.9 (E. Lerman) As mentioned in",
   "clean_statement": null,
   "public_statement": "Question 5.8 (E. Lerman) Can we prove Kirwan surjectivity without Morse theory? The motivation for this question comes from the fact that two fundamental results in the theory of symplectic moment maps - connectedness and convexity, which were originally proved using Morse theory have an alternative proof [CDM88]. This alternative approach works well in the contact setting (equivalently in the setting of symplectic cones) where Morse theory fails. The reasons for the failure in the contact setting are due to the fact are that the contact moment maps are not Morse and that there is no relationship between critical points and isotropy groups. In the equivalent setting of symplectic cones the moment maps are not proper.\n\nComment 5.9 (E. Lerman) As mentioned in",
   "evidence": "The exact canonical **problem** field in **input.json** is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 297,
   "attempt": 1
  },
  "AIM-GEOMETRY-0299": {
   "statement_status": "exact",
   "original_statement": "Question 5.8 above, for contact moment maps, one can show the convexity of the contact moment map image [Ler02] for tori of high enough dimension using methods of [CDM88]. A quick note on what \"high enough\" means: convexity fails for 2-tori and connectedness fails for circles. However, both are true for tori of dimension 3 and higher, as long as zero level set of the moment map is empty (which is true in the toric case). \n\nComment 5.10 (C. Boyer). There exists a version of Morse theory \"through a range,\" as in the instanton moduli space. Perhaps this is applicable in the contact case. 76 The cohomology of hyperK¨ ahler quotients \n\nThis section is an edited version of T. Hausel's talk given at AIM during the workshop, in which he listed many conjectures and open problems. The main examples of hyperK¨ ahler manifolds considered in his talk were the moduli spaces of Higgs bundles on a Riemann surface, Nakajima's quiver varieties, and hypertoric manifolds (introduced by Bielawski-Dancer). Note that all examples hyperK¨ ahler quotients that T. Hausel considers here take the form T ∗A/ ///G, i.e. they are hyperK¨ ahler reductions of cotangent bundles to an affine space. \n\n#6.1 Generators for the cohomology ring of the quotient \n\nA version of this conjecture was already presented as",
   "clean_statement": "Question 5.8 above, for contact moment maps, one can show the convexity of the contact moment map image [Ler02] for tori of high enough dimension using methods of [CDM88]. A quick note on what \"high enough\" means: convexity fails for 2-tori and connectedness fails for circles. However, both are true for tori of dimension 3 and higher, as long as zero level set of the moment map is empty (which is true in the toric case).\n\nComment 5.10 (C. Boyer). There exists a version of Morse theory \"through a range,\" as in the instanton moduli space. Perhaps this is applicable in the contact case. 76 The cohomology of hyperK¨ ahler quotients\n\nThis section is an edited version of T. Hausel's talk given at AIM during the workshop, in which he listed many conjectures and open problems. The main examples of hyperK¨ ahler manifolds considered in his talk were the moduli spaces of Higgs bundles on a Riemann surface, Nakajima's quiver varieties, and hypertoric manifolds (introduced by Bielawski-Dancer). Note that all examples hyperK¨ ahler quotients that T. Hausel considers here take the form T ∗A/ ///G, i.e. they are hyperK¨ ahler reductions of cotangent bundles to an affine space.\n\n#6.1 Generators for the cohomology ring of the quotient\n\nA version of this conjecture was already presented as",
   "public_statement": "Question 5.8 above, for contact moment maps, one can show the convexity of the contact moment map image [Ler02] for tori of high enough dimension using methods of [CDM88]. A quick note on what \"high enough\" means: convexity fails for 2-tori and connectedness fails for circles. However, both are true for tori of dimension 3 and higher, as long as zero level set of the moment map is empty (which is true in the toric case).\n\nComment 5.10 (C. Boyer). There exists a version of Morse theory \"through a range,\" as in the instanton moduli space. Perhaps this is applicable in the contact case. 76 The cohomology of hyperK¨ ahler quotients\n\nThis section is an edited version of T. Hausel's talk given at AIM during the workshop, in which he listed many conjectures and open problems. The main examples of hyperK¨ ahler manifolds considered in his talk were the moduli spaces of Higgs bundles on a Riemann surface, Nakajima's quiver varieties, and hypertoric manifolds (introduced by Bielawski-Dancer). Note that all examples hyperK¨ ahler quotients that T. Hausel considers here take the form T ∗A/ ///G, i.e. they are hyperK¨ ahler reductions of cotangent bundles to an affine space.\n\n#6.1 Generators for the cohomology ring of the quotient\n\nA version of this conjecture was already presented as",
   "evidence": "The canonical corpus field `problem` is reproduced verbatim below. It is a mixed extraction rather than a single mathematical question:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
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   "source_file": "aim-geometry-notes.json",
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  },
  "AIM-GEOMETRY-0300": {
   "statement_status": "exact",
   "original_statement": "Conjecture 1.1, but we state it again here in T. Hausel's formulation, for good measure:",
   "clean_statement": "Conjecture 1.1, but we state it again here in T. Hausel's formulation, for good measure:",
   "public_statement": "Conjecture 1.1, but we state it again here in T. Hausel's formulation, for good measure:",
   "evidence": "The canonical record contains exactly the following text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
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   "source_file": "aim-geometry-notes.json",
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  "AIM-GEOMETRY-0301": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.1 (T. Hausel) The hyperk¨ ahler Kirwan map κ: H ∗\n\n> G\n\n(T ∗A) ∼= H∗(BG ) →\n\nH∗(T ∗A//// ξ G) is surjective. \n\nComment 6.2 Here is what is known about the three main examples (moduli spaces, quiver varieties, and hypertoric varieties). \n\n• It is known for M1\n\n> Dol\n\n(GL (2, C)) by (Hausel-Thaddeus 2000), for MdDol (GL (n, C)) by (Markman 2001) \n\n• For quiver varieties it is conjectured by (Nakajima 2002), for hyperpolygon spaces it is proven in (Konno 2000, Hausel-Proudfoot 2003) \n\n• For hypertoric manifolds M(A, ξ ) it is known by (Konno 2000, Hausel-Sturmfels 2002) \n\n#6.2 Integration theory on hyperK¨ ahler manifolds \n\nTo state the conjectures here, we must first make a few definitions. \n\nDefinition 6.3 A smooth oriented manifold M with a circle action U (1) on M is circle-compact if the set of fixed points M U (1) is compact. Note that the natural circle action on the fiber directions of T ∗A induces a natural circle action on the hyperk¨ ahler quotients. With this circle action all of our examples (moduli spaces, quiver varieties, hypertoric varieties) are circle-compact. \n\nDefinition 6.4 (T. Hausel, N. Proudfoot) Let M be an oriented manifold with a U (1) action such that M is circle-compact. Then the rationalized U (1) equivariant cohomology is defined as the vector space ˆH∗ \n\n> U(1)\n\n(M ):= H∗ \n\n> U(1)\n\n(M ) ⊗Q[u] Q(u),\n\n8over the field Q(u) of rational functions. For α ∈ ˆH∗ \n\n> U(1)\n\n(M ) we define \n\n∫\n\n> M\n\nα:= ∑\n\n> F\n\n∫\n\n> F\n\ni∗ \n\n> F\n\n(α)\n\nE(NF ) ∈ Q(u)\n\nComment 6.5 (S. Wu) There is related work of E. Prato and S. Wu [PW94] which takes the approach of interpreting the right-hand side of the definition above as a tempered dis-tribution. \n\nComment 6.6 (T. Hausel) It would be interesting to make sense of this definition in terms of equivariant differential forms.",
   "clean_statement": "Conjecture 6.1 (T. Hausel) The hyperk¨ ahler Kirwan map κ: H ∗\n\n> G\n\n(T ∗A) ∼= H∗(BG ) →\n\nH∗(T ∗A//// ξ G) is surjective.\n\nComment 6.2 Here is what is known about the three main examples (moduli spaces, quiver varieties, and hypertoric varieties).\n\n• It is known for M1\n\n> Dol\n\n(GL (2, C)) by (Hausel-Thaddeus 2000), for MdDol (GL (n, C)) by (Markman 2001)\n\n• For quiver varieties it is conjectured by (Nakajima 2002), for hyperpolygon spaces it is proven in (Konno 2000, Hausel-Proudfoot 2003)\n\n• For hypertoric manifolds M(A, ξ ) it is known by (Konno 2000, Hausel-Sturmfels 2002)\n\n#6.2 Integration theory on hyperK¨ ahler manifolds\n\nTo state the conjectures here, we must first make a few definitions.\n\nDefinition 6.3 A smooth oriented manifold M with a circle action U (1) on M is circle-compact if the set of fixed points M U (1) is compact. Note that the natural circle action on the fiber directions of T ∗A induces a natural circle action on the hyperk¨ ahler quotients. With this circle action all of our examples (moduli spaces, quiver varieties, hypertoric varieties) are circle-compact.\n\nDefinition 6.4 (T. Hausel, N. Proudfoot) Let M be an oriented manifold with a U (1) action such that M is circle-compact. Then the rationalized U (1) equivariant cohomology is defined as the vector space ˆH∗\n\n> U(1)\n\n(M ):= H∗\n\n> U(1)\n\n(M ) ⊗Q[u] Q(u),\n\n8over the field Q(u) of rational functions. For α ∈ ˆH∗\n\n> U(1)\n\n(M ) we define\n\n∫\n\n> M\n\nα:= ∑\n\n> F\n\n∫\n\n> F\n\ni∗\n\n> F\n\n(α)\n\nE(NF ) ∈ Q(u)\n\nComment 6.5 (S. Wu) There is related work of E. Prato and S. Wu [PW94] which takes the approach of interpreting the right-hand side of the definition above as a tempered dis-tribution.\n\nComment 6.6 (T. Hausel) It would be interesting to make sense of this definition in terms of equivariant differential forms.",
   "public_statement": "Conjecture 6.1 (T. Hausel) The hyperk¨ ahler Kirwan map κ: H ∗\n\n> G\n\n(T ∗A) ∼= H∗(BG ) →\n\nH∗(T ∗A//// ξ G) is surjective.\n\nComment 6.2 Here is what is known about the three main examples (moduli spaces, quiver varieties, and hypertoric varieties).\n\n• It is known for M1\n\n> Dol\n\n(GL (2, C)) by (Hausel-Thaddeus 2000), for MdDol (GL (n, C)) by (Markman 2001)\n\n• For quiver varieties it is conjectured by (Nakajima 2002), for hyperpolygon spaces it is proven in (Konno 2000, Hausel-Proudfoot 2003)\n\n• For hypertoric manifolds M(A, ξ ) it is known by (Konno 2000, Hausel-Sturmfels 2002)\n\n#6.2 Integration theory on hyperK¨ ahler manifolds\n\nTo state the conjectures here, we must first make a few definitions.\n\nDefinition 6.3 A smooth oriented manifold M with a circle action U (1) on M is circle-compact if the set of fixed points M U (1) is compact. Note that the natural circle action on the fiber directions of T ∗A induces a natural circle action on the hyperk¨ ahler quotients. With this circle action all of our examples (moduli spaces, quiver varieties, hypertoric varieties) are circle-compact.\n\nDefinition 6.4 (T. Hausel, N. Proudfoot) Let M be an oriented manifold with a U (1) action such that M is circle-compact. Then the rationalized U (1) equivariant cohomology is defined as the vector space ˆH∗\n\n> U(1)\n\n(M ):= H∗\n\n> U(1)\n\n(M ) ⊗Q[u] Q(u),\n\n8over the field Q(u) of rational functions. For α ∈ ˆH∗\n\n> U(1)\n\n(M ) we define\n\n∫\n\n> M\n\nα:= ∑\n\n> F\n\n∫\n\n> F\n\ni∗\n\n> F\n\n(α)\n\nE(NF ) ∈ Q(u)\n\nComment 6.5 (S. Wu) There is related work of E. Prato and S. Wu [PW94] which takes the approach of interpreting the right-hand side of the definition above as a tempered dis-tribution.\n\nComment 6.6 (T. Hausel) It would be interesting to make sense of this definition in terms of equivariant differential forms.",
   "evidence": "The assigned record is Conjecture 6.1, attributed to T. Hausel, in the AIM workshop report *Moment maps and surjectivity in various geometries*. The exact mathematical sentence in the extracted record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 300,
   "attempt": 1
  },
  "AIM-GEOMETRY-0302": {
   "statement_status": "exact",
   "original_statement": "Question 6.7 (M. Libine) Here is another way to view integration theory. Let M be a noncompact symplectic manifold which is real algebraic. Let T be a compact torus, acting on M Hamiltonianly with a proper moment map μ. Assume that μ is semi-algebraic. Suppose \n\nM T is compact. Let α be an equivariant form on M which is semi-algebraic as a map from g\n\nto the space of differential forms on M (smooth, not necessarily polynomial). Then it should be possible to define the integral ∫\n\n> M\n\nα\n\ngeometrically, so that the localization theorem holds. The integration takes values in smooth functions on the Lie algebra. The hypotheses given here should not be very restrictive. The first result is \n\nTheorem 6.8 (Hausel-Proudfoot 2003) The pairing on ˆH∗ \n\n> U(1)\n\n(M ) given by \n\n∫\n\n> M\n\nα ∧ β\n\nis non-degenarate. \n\nThis gives us a \"Poincar´ e duality\" for this pairing and allows us to do kernel compu-tations. \n\nDefinition 6.9 A hyperk¨ ahler manifold M is hypercompact for the complex structure I, if there is a ωI -Hamiltonian circle action on M, with proper moment map with finitely many critical points and a minimum, such that the holomorphic symplectic form ωC:= ωJ + iω K,for λ ∈ C× satisfies λ∗ωC = λω C.\n\nUsing this definition, we make the following",
   "clean_statement": "Question 6.7 (M. Libine) Here is another way to view integration theory. Let M be a noncompact symplectic manifold which is real algebraic. Let T be a compact torus, acting on M Hamiltonianly with a proper moment map μ. Assume that μ is semi-algebraic. Suppose\n\nM T is compact. Let α be an equivariant form on M which is semi-algebraic as a map from g\n\nto the space of differential forms on M (smooth, not necessarily polynomial). Then it should be possible to define the integral ∫\n\n> M\n\nα\n\ngeometrically, so that the localization theorem holds. The integration takes values in smooth functions on the Lie algebra. The hypotheses given here should not be very restrictive. The first result is\n\nTheorem 6.8 (Hausel-Proudfoot 2003) The pairing on ˆH∗\n\n> U(1)\n\n(M ) given by\n\n∫\n\n> M\n\nα ∧ β\n\nis non-degenarate.\n\nThis gives us a \"Poincar´ e duality\" for this pairing and allows us to do kernel compu-tations.\n\nDefinition 6.9 A hyperk¨ ahler manifold M is hypercompact for the complex structure I, if there is a ωI -Hamiltonian circle action on M, with proper moment map with finitely many critical points and a minimum, such that the holomorphic symplectic form ωC:= ωJ + iω K,for λ ∈ C× satisfies λ∗ωC = λω C.\n\nUsing this definition, we make the following",
   "public_statement": "Question 6.7 (M. Libine) Here is another way to view integration theory. Let M be a noncompact symplectic manifold which is real algebraic. Let T be a compact torus, acting on M Hamiltonianly with a proper moment map μ. Assume that μ is semi-algebraic. Suppose\n\nM T is compact. Let α be an equivariant form on M which is semi-algebraic as a map from g\n\nto the space of differential forms on M (smooth, not necessarily polynomial). Then it should be possible to define the integral ∫\n\n> M\n\nα\n\ngeometrically, so that the localization theorem holds. The integration takes values in smooth functions on the Lie algebra. The hypotheses given here should not be very restrictive. The first result is\n\nTheorem 6.8 (Hausel-Proudfoot 2003) The pairing on ˆH∗\n\n> U(1)\n\n(M ) given by\n\n∫\n\n> M\n\nα ∧ β\n\nis non-degenarate.\n\nThis gives us a \"Poincar´ e duality\" for this pairing and allows us to do kernel compu-tations.\n\nDefinition 6.9 A hyperk¨ ahler manifold M is hypercompact for the complex structure I, if there is a ωI -Hamiltonian circle action on M, with proper moment map with finitely many critical points and a minimum, such that the holomorphic symplectic form ωC:= ωJ + iω K,for λ ∈ C× satisfies λ∗ωC = λω C.\n\nUsing this definition, we make the following",
   "evidence": "The canonical `problem` field is preserved verbatim below, including OCR damage and the spillover into later contextual material:",
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   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
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  },
  "AIM-GEOMETRY-0303": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.10 (T. Hausel 2003) Let M 4n be a hyper-compact hyperk¨ ahler manifold and \n\nσ(M ) denote the signature (corresponding to the ordering, given by the sign of the leading term) of the pairing on ˆH∗ \n\n> U(1)\n\n(M ). Then \n\n(−1) nσ(M ) ≥ 0.\n\nComment 6.11 (T. Hausel) This conjecture is known in the hypertoric case, where it is in fact the Brown-Colbourn inequality for the the h-numbers of a matroid, which was first proved in the context of reliability of computer networks. 96.3 Abelianization \n\nWe now state some theorems and conjectures related to the abelianization procedure, given in the symplectic case by S. Martin. \n\nTheorem 6.12 (Hausel-Proudfoot 2003) In the construction of hyperk¨ ahler quotients let A\n\nbe finite dimensional and G compact. Let T ⊂ G be a maximal torus of G. Suppose that \n\nT ∗A////G and T ∗A////T are both circle compact. If α ∈̂ H∗ \n\n> U(1) ×G\n\n(T ∗A), then \n\n∫ \n\n> T∗A////G\n\nˆκG(α) = 1\n\n|W |\n\n∫ \n\n> T∗A////T\n\nˆκT (α) ∧ e, \n\nwhere \n\ne = ∏\n\n> a∈∆\n\na(u − a) ∈ (Sym t∗)W ⊗ Q[u] ∼= HU (1) ×G(pt ).\n\nTheorem 6.13 (Hausel-Proudfoot 2003) Suppose that T ∗A////G and T ∗A////T are equiv-ariantly formal, circle compact, and that the Kirwan map κG: H∗(T ∗A) → H∗(T ∗A////G )\n\nis surjective. Then \n\nH∗ \n\n> U(1)\n\n(T ∗A////G ) ∼= H∗ \n\n> U(1)\n\n(T ∗A////T )W\n\nAnn (e).\n\nThe ring H∗ \n\n> U(1)\n\n(T ∗A////T ) has been calculated in (Harada-Proudfoot 2002). Thus surjectivity of the hyperk¨ ahler Kirwan map ⇒ description of the cohomology ring of the hyperk¨ ahler quotient. This program has been completed only in the case of the hyperpolygon spaces of Konno by (Hausel-Proudfoot 2003), obtaining the circle equivariant cohomology ring of the hyperpolygon space of (Harada-Proudfoot 2003).",
   "clean_statement": "Conjecture 6.10 (T. Hausel 2003) Let M 4n be a hyper-compact hyperk¨ ahler manifold and\n\nσ(M ) denote the signature (corresponding to the ordering, given by the sign of the leading term) of the pairing on ˆH∗\n\n> U(1)\n\n(M ). Then\n\n(−1) nσ(M ) ≥ 0.\n\nComment 6.11 (T. Hausel) This conjecture is known in the hypertoric case, where it is in fact the Brown-Colbourn inequality for the the h-numbers of a matroid, which was first proved in the context of reliability of computer networks. 96.3 Abelianization\n\nWe now state some theorems and conjectures related to the abelianization procedure, given in the symplectic case by S. Martin.\n\nTheorem 6.12 (Hausel-Proudfoot 2003) In the construction of hyperk¨ ahler quotients let A\n\nbe finite dimensional and G compact. Let T ⊂ G be a maximal torus of G. Suppose that\n\nT ∗A////G and T ∗A////T are both circle compact. If α ∈̂ H∗\n\n> U(1) ×G\n\n(T ∗A), then\n\n∫\n\n> T∗A////G\n\nˆκG(α) = 1\n\n|W |\n\n∫\n\n> T∗A////T\n\nˆκT (α) ∧ e,\n\nwhere\n\ne = ∏\n\n> a∈∆\n\na(u − a) ∈ (Sym t∗)W ⊗ Q[u] ∼= HU (1) ×G(pt ).\n\nTheorem 6.13 (Hausel-Proudfoot 2003) Suppose that T ∗A////G and T ∗A////T are equiv-ariantly formal, circle compact, and that the Kirwan map κG: H∗(T ∗A) → H∗(T ∗A////G )\n\nis surjective. Then\n\nH∗\n\n> U(1)\n\n(T ∗A////G ) ∼= H∗\n\n> U(1)\n\n(T ∗A////T )W\n\nAnn (e).\n\nThe ring H∗\n\n> U(1)\n\n(T ∗A////T ) has been calculated in (Harada-Proudfoot 2002). Thus surjectivity of the hyperk¨ ahler Kirwan map ⇒ description of the cohomology ring of the hyperk¨ ahler quotient. This program has been completed only in the case of the hyperpolygon spaces of Konno by (Hausel-Proudfoot 2003), obtaining the circle equivariant cohomology ring of the hyperpolygon space of (Harada-Proudfoot 2003).",
   "public_statement": "Conjecture 6.10 (T. Hausel 2003) Let M 4n be a hyper-compact hyperk¨ ahler manifold and\n\nσ(M ) denote the signature (corresponding to the ordering, given by the sign of the leading term) of the pairing on ˆH∗\n\n> U(1)\n\n(M ). Then\n\n(−1) nσ(M ) ≥ 0.\n\nComment 6.11 (T. Hausel) This conjecture is known in the hypertoric case, where it is in fact the Brown-Colbourn inequality for the the h-numbers of a matroid, which was first proved in the context of reliability of computer networks. 96.3 Abelianization\n\nWe now state some theorems and conjectures related to the abelianization procedure, given in the symplectic case by S. Martin.\n\nTheorem 6.12 (Hausel-Proudfoot 2003) In the construction of hyperk¨ ahler quotients let A\n\nbe finite dimensional and G compact. Let T ⊂ G be a maximal torus of G. Suppose that\n\nT ∗A////G and T ∗A////T are both circle compact. If α ∈̂ H∗\n\n> U(1) ×G\n\n(T ∗A), then\n\n∫\n\n> T∗A////G\n\nˆκG(α) = 1\n\n|W |\n\n∫\n\n> T∗A////T\n\nˆκT (α) ∧ e,\n\nwhere\n\ne = ∏\n\n> a∈∆\n\na(u − a) ∈ (Sym t∗)W ⊗ Q[u] ∼= HU (1) ×G(pt ).\n\nTheorem 6.13 (Hausel-Proudfoot 2003) Suppose that T ∗A////G and T ∗A////T are equiv-ariantly formal, circle compact, and that the Kirwan map κG: H∗(T ∗A) → H∗(T ∗A////G )\n\nis surjective. Then\n\nH∗\n\n> U(1)\n\n(T ∗A////G ) ∼= H∗\n\n> U(1)\n\n(T ∗A////T )W\n\nAnn (e).\n\nThe ring H∗\n\n> U(1)\n\n(T ∗A////T ) has been calculated in (Harada-Proudfoot 2002). Thus surjectivity of the hyperk¨ ahler Kirwan map ⇒ description of the cohomology ring of the hyperk¨ ahler quotient. This program has been completed only in the case of the hyperpolygon spaces of Konno by (Hausel-Proudfoot 2003), obtaining the circle equivariant cohomology ring of the hyperpolygon space of (Harada-Proudfoot 2003).",
   "evidence": "The canonical corpus field is preserved below verbatim (including OCR artifacts and the material from the next subsection that was attached to this record):",
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   "source_file": "aim-geometry-notes.json",
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  },
  "AIM-GEOMETRY-0304": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.14 (Hausel 2000) Suppose that both T ∗A////G and T ∗A////T are hyper-compact. Then \n\nH∗(T ∗A////G ) ∼= H∗(T ∗A////T )W\n\nAnn (˜ e),\n\nwhere \n\n˜e = ∏\n\n> a∈∆\n\na ∈ (Sym t∗)W ∼= HT (pt )W.\n\n#6.4 Equivariant intersection numbers on MdDol (SL (n, C)) \n\nThe following theorems and conjectures are motivated by similar work by Witten on other moduli spaces. \n\nTheorem 6.15 (Hausel-Szenes 2003) Let α ∈ H2 \n\n> U(1)\n\n(M1\n\n> Dol\n\n(SL (2, C)) ∼= Z be the positive integral generator, M:= M1\n\n> Dol\n\n(SL (2, C)) \n\nBA (y) = \n\n( 2\n\nu\n\n)g−1\n\n( 2 \n\n> (1 −y/u )2\n\n+ u\n\n)g\n\n(\n\ney u +y \n\n> u−y\n\n− e−y u −y\n\n> u+y\n\n)\n\ny2g−2(u2 − y2)g−1,\n\n10 we have \n\n∫\n\n> M\n\neα = Res \n\n> y=0\n\nBA (y) + Res \n\n> y=−u\n\nBA (y) + Res \n\n> y=u\n\nBA (y)= − ∑\n\n> b\n\nRes \n\n> y=b\n\nBA (y),\n\nwhere the last sum is taken over the solutions of the Bethe-Ansatz equations: \n\neb u + b\n\nu − b = e−b u − b\n\nu + b",
   "clean_statement": "Conjecture 6.14 (Hausel 2000) Suppose that both T ∗A////G and T ∗A////T are hyper-compact. Then\n\nH∗(T ∗A////G ) ∼= H∗(T ∗A////T )W\n\nAnn (˜ e),\n\nwhere\n\n˜e = ∏\n\n> a∈∆\n\na ∈ (Sym t∗)W ∼= HT (pt )W.\n\n#6.4 Equivariant intersection numbers on MdDol (SL (n, C))\n\nThe following theorems and conjectures are motivated by similar work by Witten on other moduli spaces.\n\nTheorem 6.15 (Hausel-Szenes 2003) Let α ∈ H2\n\n> U(1)\n\n(M1\n\n> Dol\n\n(SL (2, C)) ∼= Z be the positive integral generator, M:= M1\n\n> Dol\n\n(SL (2, C))\n\nBA (y) =\n\n( 2\n\nu\n\n)g−1\n\n( 2\n\n> (1 −y/u )2\n\n+ u\n\n)g\n\n(\n\ney u +y\n\n> u−y\n\n− e−y u −y\n\n> u+y\n\n)\n\ny2g−2(u2 − y2)g−1,\n\n10 we have\n\n∫\n\n> M\n\neα = Res\n\n> y=0\n\nBA (y) + Res\n\n> y=−u\n\nBA (y) + Res\n\n> y=u\n\nBA (y)= − ∑\n\n> b\n\nRes\n\n> y=b\n\nBA (y),\n\nwhere the last sum is taken over the solutions of the Bethe-Ansatz equations:\n\neb u + b\n\nu − b = e−b u − b\n\nu + b",
   "public_statement": "Conjecture 6.14 (Hausel 2000) Suppose that both T ∗A////G and T ∗A////T are hyper-compact. Then\n\nH∗(T ∗A////G ) ∼= H∗(T ∗A////T )W\n\nAnn (˜ e),\n\nwhere\n\n˜e = ∏\n\n> a∈∆\n\na ∈ (Sym t∗)W ∼= HT (pt )W.\n\n#6.4 Equivariant intersection numbers on MdDol (SL (n, C))\n\nThe following theorems and conjectures are motivated by similar work by Witten on other moduli spaces.\n\nTheorem 6.15 (Hausel-Szenes 2003) Let α ∈ H2\n\n> U(1)\n\n(M1\n\n> Dol\n\n(SL (2, C)) ∼= Z be the positive integral generator, M:= M1\n\n> Dol\n\n(SL (2, C))\n\nBA (y) =\n\n( 2\n\nu\n\n)g−1\n\n( 2\n\n> (1 −y/u )2\n\n+ u\n\n)g\n\n(\n\ney u +y\n\n> u−y\n\n− e−y u −y\n\n> u+y\n\n)\n\ny2g−2(u2 − y2)g−1,\n\n10 we have\n\n∫\n\n> M\n\neα = Res\n\n> y=0\n\nBA (y) + Res\n\n> y=−u\n\nBA (y) + Res\n\n> y=u\n\nBA (y)= − ∑\n\n> b\n\nRes\n\n> y=b\n\nBA (y),\n\nwhere the last sum is taken over the solutions of the Bethe-Ansatz equations:\n\neb u + b\n\nu − b = e−b u − b\n\nu + b",
   "evidence": "The canonical record is Conjecture 6.14 in the AIM workshop list *Moment Maps and Surjectivity in Various Geometries*. The exact mathematical part of the PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 303,
   "attempt": 1
  },
  "AIM-GEOMETRY-0305": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.16 (Nekrasov-Shatasvili-Moore 1998, Hausel-Szenes 2004) The equivariant volume of MdDol (SL (n, C)): ∫\n\n> M\n\neα = ∑\n\n> F\n\n∫\n\n> F\n\neα\n\nE(NF )\n\n(and indeed all equivariant intersection numbers of MdDol (SL (n, C)) ), can be described as an iterated residue of a certain expression \n\nBA (y1, y 2,..., y n).\n\nSome of the poles of the expression BA are in one-to-one correspondence with ordered par-titions of n, and the rest are the solutions of certain Bethe Ansatz equations. The iterated residue taken at a pole corresponding to the ordered partition n = λ1 + · · · + λk, agrees with the contribution to the equivariant volume by the components of type (λ1, λ 2,..., λ k). More-over minus of the sum of the residues at the Bethe poles give the equivariant volume of the Higgs moduli space. \n\n#6.5 Arithmetic approach \n\nFix a compact Riemann surface Σ and a pair of relatively prime integers n and d. Carlos Simpson's nonabelian Hodge theory provides a diffeomorphism between MdDol (GL (n, C)), the moduli space of rank n degree d Higgs bundles, and MdB (GL (n, C)), the moduli space of twisted n-dimensional representations of π1(Σ): \n\nMdB (GL (n, C)):= {A1, B 1,..., A g, B g ∈ GL (n, C)|\n\nA−11 B−11 A1B1... A −1 \n\n> g\n\nB−1 \n\n> g\n\nAgBg = ξnId }/GL (n, C)The strategy of (Hausel-Rodriguez-Villegas 2003) for getting the Betti numbers of \n\nMdB (GL (n, C)) is to count the rational points of the variety over a finite field Fq. Thus we have to count points of \n\nMB (GL (n, Fq)):= {A1, B 1,..., A g, B g ∈ GL (n, Fq)|\n\nA−11 B−11 A1B1... A −1 \n\n> g\n\nB−1 \n\n> g\n\nAgBg = ξnId }/GL (n, Fq),\n\n11 for which (Frobenius-Schur 1907) gives: #{M B (GL (n, Fq)) } = ∑ \n\n> χ∈Irr (GL (n, Fq))\n\n|GL (n, Fq)|2g−2\n\nχ(1) 2g−1 χ(ξn)Deligne's mixed Hodge structure for \n\nM:= MdB (GL (n, C)) gives two filtrations on the cohomology Hk(M, C) whose associated graded is \n\n⊕\n\n> p,q\n\nHp,q;k(M ),\n\nwe denote by hp,q;k the dimension of Hp,q;k(M ). \n\nHn(x, y, t ):= ∑\n\n> p,q,k\n\nhp,q;k(M )xpyqtk,\n\nis the mixed Hodge polynomial. \n\nTheorem 6.17 (Hausel-Rodriguez-Villegas 2003) \n\nHn(√q, √q, −1) = #{M B (GL (n, Fq)) }\n\n= ∑ \n\n> χ∈Irr (GL (n, Fq))\n\n|GL (n, Fq)|2g−2\n\nχ(1) 2g−1 χ(ξn)We now define a function Hn(q, t ) which will conjecturally be related to the mixed Hodge numbers. We define \n\nVn(q, t ) = Hn(q, t ) (qt 2)(1 −g)n(n−1) \n\n(qt 2 − 1)( q − 1),\n\nand \n\nZn(q, t, T ) = exp \n\n(∑\n\n> r≥1\n\nVn(qr, −(−t)r)T r\n\nr\n\n).\n\nThen let \n\nHλg (q, t ) = ∏\n\n> z∈d(λ)\n\n(qt 2)(2 −2g)`(z)(1 + qh(z)t2`(z)+1 )2g\n\n(1 − qh(z)t2`(z)+2 )(1 − qh(z)t2`(z)).\n\nHere `(z) denotes the leg length and h(z) is the hook length, and h(z) = a(z)+ `(z)−1,\n\nwhere a(z) is the arm length, as in the figure below. \n\n• • • • •\n\n• z • • • • a(z)\n\n• • • •• • •\n\n• `(z)\n\n12 We finally define Hn(q, t ), in generating function form: \n\n> ∞\n\n∏\n\n> n=1\n\nZn(q, t, T n) = ∑\n\n> λ∈P\n\nHλg (q, t )T |λ|.\n\nUsing this notation, we have the following conjectures.",
   "clean_statement": "Conjecture 6.16 (Nekrasov-Shatasvili-Moore 1998, Hausel-Szenes 2004) The equivariant volume of MdDol (SL (n, C)): ∫\n\n> M\n\neα = ∑\n\n> F\n\n∫\n\n> F\n\neα\n\nE(NF )\n\n(and indeed all equivariant intersection numbers of MdDol (SL (n, C)) ), can be described as an iterated residue of a certain expression\n\nBA (y1, y 2,..., y n).\n\nSome of the poles of the expression BA are in one-to-one correspondence with ordered par-titions of n, and the rest are the solutions of certain Bethe Ansatz equations. The iterated residue taken at a pole corresponding to the ordered partition n = λ1 + · · · + λk, agrees with the contribution to the equivariant volume by the components of type (λ1, λ 2,..., λ k). More-over minus of the sum of the residues at the Bethe poles give the equivariant volume of the Higgs moduli space.\n\n#6.5 Arithmetic approach\n\nFix a compact Riemann surface Σ and a pair of relatively prime integers n and d. Carlos Simpson's nonabelian Hodge theory provides a diffeomorphism between MdDol (GL (n, C)), the moduli space of rank n degree d Higgs bundles, and MdB (GL (n, C)), the moduli space of twisted n-dimensional representations of π1(Σ):\n\nMdB (GL (n, C)):= {A1, B 1,..., A g, B g ∈ GL (n, C)|\n\nA−11 B−11 A1B1... A −1\n\n> g\n\nB−1\n\n> g\n\nAgBg = ξnId }/GL (n, C)The strategy of (Hausel-Rodriguez-Villegas 2003) for getting the Betti numbers of\n\nMdB (GL (n, C)) is to count the rational points of the variety over a finite field Fq. Thus we have to count points of\n\nMB (GL (n, Fq)):= {A1, B 1,..., A g, B g ∈ GL (n, Fq)|\n\nA−11 B−11 A1B1... A −1\n\n> g\n\nB−1\n\n> g\n\nAgBg = ξnId }/GL (n, Fq),\n\n11 for which (Frobenius-Schur 1907) gives: #{M B (GL (n, Fq)) } = ∑\n\n> χ∈Irr (GL (n, Fq))\n\n|GL (n, Fq)|2g−2\n\nχ(1) 2g−1 χ(ξn)Deligne's mixed Hodge structure for\n\nM:= MdB (GL (n, C)) gives two filtrations on the cohomology Hk(M, C) whose associated graded is\n\n⊕\n\n> p,q\n\nHp,q;k(M ),\n\nwe denote by hp,q;k the dimension of Hp,q;k(M ).\n\nHn(x, y, t ):= ∑\n\n> p,q,k\n\nhp,q;k(M )xpyqtk,\n\nis the mixed Hodge polynomial.\n\nTheorem 6.17 (Hausel-Rodriguez-Villegas 2003)\n\nHn(√q, √q, −1) = #{M B (GL (n, Fq)) }\n\n= ∑\n\n> χ∈Irr (GL (n, Fq))\n\n|GL (n, Fq)|2g−2\n\nχ(1) 2g−1 χ(ξn)We now define a function Hn(q, t ) which will conjecturally be related to the mixed Hodge numbers. We define\n\nVn(q, t ) = Hn(q, t ) (qt 2)(1 −g)n(n−1)\n\n(qt 2 − 1)( q − 1),\n\nand\n\nZn(q, t, T ) = exp\n\n(∑\n\n> r≥1\n\nVn(qr, −(−t)r)T r\n\nr\n\n).\n\nThen let\n\nHλg (q, t ) = ∏\n\n> z∈d(λ)\n\n(qt 2)(2 −2g)`(z)(1 + qh(z)t2`(z)+1 )2g\n\n(1 − qh(z)t2`(z)+2 )(1 − qh(z)t2`(z)).\n\nHere `(z) denotes the leg length and h(z) is the hook length, and h(z) = a(z)+ `(z)−1,\n\nwhere a(z) is the arm length, as in the figure below.\n\n• • • • •\n\n• z • • • • a(z)\n\n• • • •• • •\n\n• `(z)\n\n12 We finally define Hn(q, t ), in generating function form:\n\n> ∞\n\n∏\n\n> n=1\n\nZn(q, t, T n) = ∑\n\n> λ∈P\n\nHλg (q, t )T |λ|.\n\nUsing this notation, we have the following conjectures.",
   "public_statement": "Conjecture 6.16 (Nekrasov-Shatasvili-Moore 1998, Hausel-Szenes 2004) The equivariant volume of MdDol (SL (n, C)): ∫\n\n> M\n\neα = ∑\n\n> F\n\n∫\n\n> F\n\neα\n\nE(NF )\n\n(and indeed all equivariant intersection numbers of MdDol (SL (n, C)) ), can be described as an iterated residue of a certain expression\n\nBA (y1, y 2,..., y n).\n\nSome of the poles of the expression BA are in one-to-one correspondence with ordered par-titions of n, and the rest are the solutions of certain Bethe Ansatz equations. The iterated residue taken at a pole corresponding to the ordered partition n = λ1 + · · · + λk, agrees with the contribution to the equivariant volume by the components of type (λ1, λ 2,..., λ k). More-over minus of the sum of the residues at the Bethe poles give the equivariant volume of the Higgs moduli space.\n\n#6.5 Arithmetic approach\n\nFix a compact Riemann surface Σ and a pair of relatively prime integers n and d. Carlos Simpson's nonabelian Hodge theory provides a diffeomorphism between MdDol (GL (n, C)), the moduli space of rank n degree d Higgs bundles, and MdB (GL (n, C)), the moduli space of twisted n-dimensional representations of π1(Σ):\n\nMdB (GL (n, C)):= {A1, B 1,..., A g, B g ∈ GL (n, C)|\n\nA−11 B−11 A1B1... A −1\n\n> g\n\nB−1\n\n> g\n\nAgBg = ξnId }/GL (n, C)The strategy of (Hausel-Rodriguez-Villegas 2003) for getting the Betti numbers of\n\nMdB (GL (n, C)) is to count the rational points of the variety over a finite field Fq. Thus we have to count points of\n\nMB (GL (n, Fq)):= {A1, B 1,..., A g, B g ∈ GL (n, Fq)|\n\nA−11 B−11 A1B1... A −1\n\n> g\n\nB−1\n\n> g\n\nAgBg = ξnId }/GL (n, Fq),\n\n11 for which (Frobenius-Schur 1907) gives: #{M B (GL (n, Fq)) } = ∑\n\n> χ∈Irr (GL (n, Fq))\n\n|GL (n, Fq)|2g−2\n\nχ(1) 2g−1 χ(ξn)Deligne's mixed Hodge structure for\n\nM:= MdB (GL (n, C)) gives two filtrations on the cohomology Hk(M, C) whose associated graded is\n\n⊕\n\n> p,q\n\nHp,q;k(M ),\n\nwe denote by hp,q;k the dimension of Hp,q;k(M ).\n\nHn(x, y, t ):= ∑\n\n> p,q,k\n\nhp,q;k(M )xpyqtk,\n\nis the mixed Hodge polynomial.\n\nTheorem 6.17 (Hausel-Rodriguez-Villegas 2003)\n\nHn(√q, √q, −1) = #{M B (GL (n, Fq)) }\n\n= ∑\n\n> χ∈Irr (GL (n, Fq))\n\n|GL (n, Fq)|2g−2\n\nχ(1) 2g−1 χ(ξn)We now define a function Hn(q, t ) which will conjecturally be related to the mixed Hodge numbers. We define\n\nVn(q, t ) = Hn(q, t ) (qt 2)(1 −g)n(n−1)\n\n(qt 2 − 1)( q − 1),\n\nand\n\nZn(q, t, T ) = exp\n\n(∑\n\n> r≥1\n\nVn(qr, −(−t)r)T r\n\nr\n\n).\n\nThen let\n\nHλg (q, t ) = ∏\n\n> z∈d(λ)\n\n(qt 2)(2 −2g)`(z)(1 + qh(z)t2`(z)+1 )2g\n\n(1 − qh(z)t2`(z)+2 )(1 − qh(z)t2`(z)).\n\nHere `(z) denotes the leg length and h(z) is the hook length, and h(z) = a(z)+ `(z)−1,\n\nwhere a(z) is the arm length, as in the figure below.\n\n• • • • •\n\n• z • • • • a(z)\n\n• • • •• • •\n\n• `(z)\n\n12 We finally define Hn(q, t ), in generating function form:\n\n> ∞\n\n∏\n\n> n=1\n\nZn(q, t, T n) = ∑\n\n> λ∈P\n\nHλg (q, t )T |λ|.\n\nUsing this notation, we have the following conjectures.",
   "evidence": "The source is Conjecture 6.16 in the AIM workshop list *Moment maps and surjectivity in various geometries*. The mathematical content of the record, with typography repaired but without changing its scope, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 304,
   "attempt": 1
  },
  "AIM-GEOMETRY-0306": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 6.18 (Hausel-Rodriguez-Villegas 2004) The mixed Hodge polynomial of MdB (GL (n, C)),\n\nis given by \n\nHn(√q, √q, t ) = Hn(q, t )\n\nExample 6.19 The case n = 2 follows from (Hausel-Thaddeus 2000): \n\nH2(√q, √q, t )/(qt + 1) 2g = (q2t3 + 1) 2g\n\n(q2t2 − 1)( q2t4 − 1) + q2g−2t4g−4(q2t + 1) 2g\n\n(q2 − 1)( q2t2 − 1) \n\n−1\n\n2\n\nq2g−2t4g−4(qt + 1) 2g\n\n(qt 2 − 1)( q − 1) − 1\n\n2\n\nq2g−2t4g−4(qt − 1) 2g\n\n(q + 1)( qt 2 + 1),\n\nand when g = 3: \n\nH2(√q, √q, t )/(qt + 1) 6 = t12 q12 + t12 q10 + 6 t11 q10 + t12 q8 + t10 q10 \n\n+6 t11 q8 + 16 t10 q8 + 6 t9q8 + t10 q6 + t8q8 + 26 t9q6\n\n+16 t8q6 + 6 t7q6 + t8q4 + t6q6 + 6 t7q4 + 16 t6q4 ++6 t5q4 + t4q4 + t4q2 + 6 t3q2 + t2q2 + 1.",
   "clean_statement": null,
   "public_statement": "Conjecture 6.18 (Hausel-Rodriguez-Villegas 2004) The mixed Hodge polynomial of MdB (GL (n, C)),\n\nis given by\n\nHn(√q, √q, t ) = Hn(q, t )\n\nExample 6.19 The case n = 2 follows from (Hausel-Thaddeus 2000):\n\nH2(√q, √q, t )/(qt + 1) 2g = (q2t3 + 1) 2g\n\n(q2t2 − 1)( q2t4 − 1) + q2g−2t4g−4(q2t + 1) 2g\n\n(q2 − 1)( q2t2 − 1)\n\n−1\n\n2\n\nq2g−2t4g−4(qt + 1) 2g\n\n(qt 2 − 1)( q − 1) − 1\n\n2\n\nq2g−2t4g−4(qt − 1) 2g\n\n(q + 1)( qt 2 + 1),\n\nand when g = 3:\n\nH2(√q, √q, t )/(qt + 1) 6 = t12 q12 + t12 q10 + 6 t11 q10 + t12 q8 + t10 q10\n\n+6 t11 q8 + 16 t10 q8 + 6 t9q8 + t10 q6 + t8q8 + 26 t9q6\n\n+16 t8q6 + 6 t7q6 + t8q4 + t6q6 + 6 t7q4 + 16 t6q4 ++6 t5q4 + t4q4 + t4q2 + 6 t3q2 + t2q2 + 1.",
   "evidence": "The canonical AIM record is Conjecture 6.18 from the 2004 workshop *Moment maps and surjectivity in various geometries*. Its exact extracted problem field is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 305,
   "attempt": 1
  },
  "AIM-GEOMETRY-0307": {
   "statement_status": "exact",
   "original_statement": "Conjecture 6.20 (T. Hausel) The Pure rings of MdDol (GL (n, C)) and N d(GL (n, C)) (the moduli space of rank n stable bundles of degree d), i.e. the subrings of the cohomology rings generated by the classes a2,..., a n are isomorphic. In particular, unlike the whole cohomology ring of N d(GL (n, C)), it does not depend on d. Moreover the Poincar´ e polynomial P P n(t)\n\nof the pure ring is given by: \n\nP V n(t) = P P n(t) t2(1 −g)n(n−1) \n\n(t2 − 1),P Z n(t, T ) = exp \n\n(∑\n\n> r≥1\n\nP V n(tr)T r\n\nr\n\n).\n\nPHλg (t) = t4(1 −g)n(λ′) ∏\n\n> x∈d(λ); a(x)=0\n\n1\n\n(1 − t2h(x)),n(λ′):= ∑\n\n> z∈d(λ)\n\n`(z).\n\n> ∞\n\n∏\n\n> n=1\n\nP Z n(t, T n) = ∑\n\n> λ∈P\n\nPHλg (t)T |λ|.\n\n13 Example 6.21 For the case n = 2, the pure ring is generated by β = a2, and βg = 0, this is the famous Newstead conjecture for N 1(GL (2, C)), was first proved by (Kirwan 1992, Thaddeus 1992), while for M1\n\n> Dol\n\n(GL (2, C)) it was proved in (Hausel-Thaddeus 2000) \n\nExample 6.22 For the case n > 2, similar vanishings for the pure ring of N d(GL (n, C)) was proved by (Earl-Kirwan 1999) using (Jeffrey-Kirwan 1998), which also follow from the conjecture. \n\nP P 3(t) = 1\n\n(t6 − 1) ( t4 − 1) + t12 g−12 \n\n− t8 g−8\n\nt2 − 1 + 1\n\n3\n\nt12 g−12 \n\n(t2 − 1) 2 − 1\n\n3\n\nt12 g−12 \n\nt4 + t2 + 1 \n\n− t8 g−8\n\n(t4 − 1) ( t2 − 1) + t12 g−12 \n\nt2 − 1\n\n7 Kernel computations for Kirwan maps \n\nSuppose a compact Lie group G acts linearly on an affine space Cn Hamiltonianly. Consider the symplectic quotient Cn//αG, for α central and regular. We can also view this space as a GIT quotient (Cn \\ { the α-unstable locus }) /G C.\n\nEither way, we get a Kirwan map κα\n\nH∗\n\n> G\n\n(Cn) ∼= H∗\n\n> G\n\n(pt ) → H∗(Cn//αG).\n\nThis gives us a different maps κα and different ideals Iα:= ker( κα) depending on our choice of α.Note that, from the viewpoint of the GIT quotient, the α-unstable locus for any central regular α contains points in Cn on which G fails to act locally freely. So instead, let's look at \n\nX:= ( Cn \\ { points where G fails to act locally freely }) /G C.\n\nUsing this space, we also get a map \n\nf: H∗(Cn) ∼= H∗\n\n> G\n\n(pt ) → H∗(X/G ),\n\nand from the description of the space X it is clear that ker( f ) ⊂ ker( κα), ∀α.",
   "clean_statement": "Conjecture 6.20 (T. Hausel) The Pure rings of MdDol (GL (n, C)) and N d(GL (n, C)) (the moduli space of rank n stable bundles of degree d), i.e. the subrings of the cohomology rings generated by the classes a2,..., a n are isomorphic. In particular, unlike the whole cohomology ring of N d(GL (n, C)), it does not depend on d. Moreover the Poincar´ e polynomial P P n(t)\n\nof the pure ring is given by:\n\nP V n(t) = P P n(t) t2(1 −g)n(n−1)\n\n(t2 − 1),P Z n(t, T ) = exp\n\n(∑\n\n> r≥1\n\nP V n(tr)T r\n\nr\n\n).\n\nPHλg (t) = t4(1 −g)n(λ′) ∏\n\n> x∈d(λ); a(x)=0\n\n1\n\n(1 − t2h(x)),n(λ′):= ∑\n\n> z∈d(λ)\n\n`(z).\n\n> ∞\n\n∏\n\n> n=1\n\nP Z n(t, T n) = ∑\n\n> λ∈P\n\nPHλg (t)T |λ|.\n\n13 Example 6.21 For the case n = 2, the pure ring is generated by β = a2, and βg = 0, this is the famous Newstead conjecture for N 1(GL (2, C)), was first proved by (Kirwan 1992, Thaddeus 1992), while for M1\n\n> Dol\n\n(GL (2, C)) it was proved in (Hausel-Thaddeus 2000)\n\nExample 6.22 For the case n > 2, similar vanishings for the pure ring of N d(GL (n, C)) was proved by (Earl-Kirwan 1999) using (Jeffrey-Kirwan 1998), which also follow from the conjecture.\n\nP P 3(t) = 1\n\n(t6 − 1) ( t4 − 1) + t12 g−12\n\n− t8 g−8\n\nt2 − 1 + 1\n\n3\n\nt12 g−12\n\n(t2 − 1) 2 − 1\n\n3\n\nt12 g−12\n\nt4 + t2 + 1\n\n− t8 g−8\n\n(t4 − 1) ( t2 − 1) + t12 g−12\n\nt2 − 1\n\n7 Kernel computations for Kirwan maps\n\nSuppose a compact Lie group G acts linearly on an affine space Cn Hamiltonianly. Consider the symplectic quotient Cn//αG, for α central and regular. We can also view this space as a GIT quotient (Cn \\ { the α-unstable locus }) /G C.\n\nEither way, we get a Kirwan map κα\n\nH∗\n\n> G\n\n(Cn) ∼= H∗\n\n> G\n\n(pt ) → H∗(Cn//αG).\n\nThis gives us a different maps κα and different ideals Iα:= ker( κα) depending on our choice of α.Note that, from the viewpoint of the GIT quotient, the α-unstable locus for any central regular α contains points in Cn on which G fails to act locally freely. So instead, let's look at\n\nX:= ( Cn \\ { points where G fails to act locally freely }) /G C.\n\nUsing this space, we also get a map\n\nf: H∗(Cn) ∼= H∗\n\n> G\n\n(pt ) → H∗(X/G ),\n\nand from the description of the space X it is clear that ker( f ) ⊂ ker( κα), ∀α.",
   "public_statement": "Conjecture 6.20 (T. Hausel) The Pure rings of MdDol (GL (n, C)) and N d(GL (n, C)) (the moduli space of rank n stable bundles of degree d), i.e. the subrings of the cohomology rings generated by the classes a2,..., a n are isomorphic. In particular, unlike the whole cohomology ring of N d(GL (n, C)), it does not depend on d. Moreover the Poincar´ e polynomial P P n(t)\n\nof the pure ring is given by:\n\nP V n(t) = P P n(t) t2(1 −g)n(n−1)\n\n(t2 − 1),P Z n(t, T ) = exp\n\n(∑\n\n> r≥1\n\nP V n(tr)T r\n\nr\n\n).\n\nPHλg (t) = t4(1 −g)n(λ′) ∏\n\n> x∈d(λ); a(x)=0\n\n1\n\n(1 − t2h(x)),n(λ′):= ∑\n\n> z∈d(λ)\n\n`(z).\n\n> ∞\n\n∏\n\n> n=1\n\nP Z n(t, T n) = ∑\n\n> λ∈P\n\nPHλg (t)T |λ|.\n\n13 Example 6.21 For the case n = 2, the pure ring is generated by β = a2, and βg = 0, this is the famous Newstead conjecture for N 1(GL (2, C)), was first proved by (Kirwan 1992, Thaddeus 1992), while for M1\n\n> Dol\n\n(GL (2, C)) it was proved in (Hausel-Thaddeus 2000)\n\nExample 6.22 For the case n > 2, similar vanishings for the pure ring of N d(GL (n, C)) was proved by (Earl-Kirwan 1999) using (Jeffrey-Kirwan 1998), which also follow from the conjecture.\n\nP P 3(t) = 1\n\n(t6 − 1) ( t4 − 1) + t12 g−12\n\n− t8 g−8\n\nt2 − 1 + 1\n\n3\n\nt12 g−12\n\n(t2 − 1) 2 − 1\n\n3\n\nt12 g−12\n\nt4 + t2 + 1\n\n− t8 g−8\n\n(t4 − 1) ( t2 − 1) + t12 g−12\n\nt2 − 1\n\n7 Kernel computations for Kirwan maps\n\nSuppose a compact Lie group G acts linearly on an affine space Cn Hamiltonianly. Consider the symplectic quotient Cn//αG, for α central and regular. We can also view this space as a GIT quotient (Cn \\ { the α-unstable locus }) /G C.\n\nEither way, we get a Kirwan map κα\n\nH∗\n\n> G\n\n(Cn) ∼= H∗\n\n> G\n\n(pt ) → H∗(Cn//αG).\n\nThis gives us a different maps κα and different ideals Iα:= ker( κα) depending on our choice of α.Note that, from the viewpoint of the GIT quotient, the α-unstable locus for any central regular α contains points in Cn on which G fails to act locally freely. So instead, let's look at\n\nX:= ( Cn \\ { points where G fails to act locally freely }) /G C.\n\nUsing this space, we also get a map\n\nf: H∗(Cn) ∼= H∗\n\n> G\n\n(pt ) → H∗(X/G ),\n\nand from the description of the space X it is clear that ker( f ) ⊂ ker( κα), ∀α.",
   "evidence": "The canonical record is Conjecture 6.20 in the AIM workshop notes *Moment maps and surjectivity in various geometries*. The JSON extraction has lost superscripts, accents, line breaks, and much of the layout of the displayed generating series. It also appends the beginning of Section 7, “Kernel computations for Kirwan maps,” which starts immediately after Example 6.22 in the source PDF and is not part of Conjecture 6.20.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 306,
   "attempt": 1
  },
  "AIM-GEOMETRY-0308": {
   "statement_status": "exact",
   "original_statement": "Question 7.1 (N. Proudfoot) In this setting, is it true that we have the equality ker( f ) = ∩αker( κα).\n\nwhere the intersection is over central regular values α?14 Comment 7.2 (N. Proudfoot) When G = T is abelian, the conjecture is true. The proof requires hyperK¨ ahler geometry, despite the fact that the statement doesn't involve anything hyperK¨ ahler. Is it possible to prove it directly?",
   "clean_statement": "Question 7.1 (N. Proudfoot) In this setting, is it true that we have the equality ker( f ) = ∩αker( κα).\n\nwhere the intersection is over central regular values α?14 Comment 7.2 (N. Proudfoot) When G = T is abelian, the conjecture is true. The proof requires hyperK¨ ahler geometry, despite the fact that the statement doesn't involve anything hyperK¨ ahler. Is it possible to prove it directly?",
   "public_statement": "Question 7.1 (N. Proudfoot) In this setting, is it true that we have the equality ker( f ) = ∩αker( κα).\n\nwhere the intersection is over central regular values α?14 Comment 7.2 (N. Proudfoot) When G = T is abelian, the conjecture is true. The proof requires hyperK¨ ahler geometry, despite the fact that the statement doesn't involve anything hyperK¨ ahler. Is it possible to prove it directly?",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 307,
   "attempt": 1
  },
  "AIM-GEOMETRY-0309": {
   "statement_status": "exact",
   "original_statement": "Question 7.3 (R Goldin) Let G act on Cn as above, and X be as defined above. Then is the map \n\nH∗\n\n> G\n\n(Cn) → H∗(X/G ) = H∗\n\n> G\n\n(X)surjective? What is the kernel? \n\nComment 7.4 (M. Libine) We should probably begin by looking at irreducible representa-tions. Note also that it is possible that X is empty without some additional assumptions on the action of G.\n\n8 Higgs bundles and relations to gauge theory",
   "clean_statement": "Question 7.3 (R Goldin) Let G act on Cn as above, and X be as defined above. Then is the map\n\nH∗\n\n> G\n\n(Cn) → H∗(X/G ) = H∗\n\n> G\n\n(X)surjective? What is the kernel?\n\nComment 7.4 (M. Libine) We should probably begin by looking at irreducible representa-tions. Note also that it is possible that X is empty without some additional assumptions on the action of G.\n\n8 Higgs bundles and relations to gauge theory",
   "public_statement": "Question 7.3 (R Goldin) Let G act on Cn as above, and X be as defined above. Then is the map\n\nH∗\n\n> G\n\n(Cn) → H∗(X/G ) = H∗\n\n> G\n\n(X)surjective? What is the kernel?\n\nComment 7.4 (M. Libine) We should probably begin by looking at irreducible representa-tions. Note also that it is possible that X is empty without some additional assumptions on the action of G.\n\n8 Higgs bundles and relations to gauge theory",
   "evidence": "The canonical record is Question 7.3 in the AIM 2004 workshop list *Moment maps and surjectivity in various geometries*. Its exact extracted problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 308,
   "attempt": 1
  },
  "AIM-GEOMETRY-0310": {
   "statement_status": "exact",
   "original_statement": "Question 8.1 (G. Daskalopolous) For a 3-manifold, the theorem of Corlette gives a corre-spondence between representations of the fundamental group in SL (2, C) with the space of Higgs bundles. The question is if under the existence of a contact structure, there is any additional structure on the space of Higgs bundles as in the surface case. For example, is there a contact interpretation of the ( L2) norm of the Higgs field? Is there a description of the critical points of the norm of the Higgs field? Can this be used in any way to show existence of representations of the fundamental group into SU (2) or P SL (2, R)? \n\n9 Intersection Cohomology",
   "clean_statement": "Question 8.1 (G. Daskalopolous) For a 3-manifold, the theorem of Corlette gives a corre-spondence between representations of the fundamental group in SL (2, C) with the space of Higgs bundles. The question is if under the existence of a contact structure, there is any additional structure on the space of Higgs bundles as in the surface case. For example, is there a contact interpretation of the ( L2) norm of the Higgs field? Is there a description of the critical points of the norm of the Higgs field? Can this be used in any way to show existence of representations of the fundamental group into SU (2) or P SL (2, R)?\n\n9 Intersection Cohomology",
   "public_statement": "Question 8.1 (G. Daskalopolous) For a 3-manifold, the theorem of Corlette gives a corre-spondence between representations of the fundamental group in SL (2, C) with the space of Higgs bundles. The question is if under the existence of a contact structure, there is any additional structure on the space of Higgs bundles as in the surface case. For example, is there a contact interpretation of the ( L2) norm of the Higgs field? Is there a description of the critical points of the norm of the Higgs field? Can this be used in any way to show existence of representations of the fundamental group into SU (2) or P SL (2, R)?\n\n9 Intersection Cohomology",
   "evidence": "The canonical record is Question 8.1 in the AIM workshop notes *Moment maps and surjectivity in various geometries* (August 2004). The typeset PDF prints the attribution as “G. Daskalopolous”; the mathematician's official Brown University profile verifies the spelling **Georgios Daskalopoulos**. The PDF also verifies the intended notation \\(L^2\\), \\(SL(2,\\mathbb C)\\), \\(SU(2)\\), and \\(PSL(2,\\mathbb R)\\). The heading “9 Intersection Cohomology” begins the next section and is not part of Question 8.1.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 309,
   "attempt": 1
  },
  "AIM-GEOMETRY-0311": {
   "statement_status": "exact",
   "original_statement": "Question 9.1 (N. Proudfoot) The intersection cohomology of a singular hypertoric vari-eties has a \"seemingly natural\" ring structure (as a quotient of the equivariant cohomology \n\nH∗ \n\n> T\n\n(T ∗Cn) of the original space). Is there, in general, a natural ring structure on the inter-section cohomology group of a singular hyperK¨ ahler quotient? \n\nComment 9.2 (N. Proudfoot) One setting in which one gets a ring structure on intersection cohomology is when one has a small resolution, but in the hypertoric examples, Nick is not aware of any such small resolution.",
   "clean_statement": "Question 9.1 (N. Proudfoot) The intersection cohomology of a singular hypertoric vari-eties has a \"seemingly natural\" ring structure (as a quotient of the equivariant cohomology\n\nH∗\n\n> T\n\n(T ∗Cn) of the original space). Is there, in general, a natural ring structure on the inter-section cohomology group of a singular hyperK¨ ahler quotient?\n\nComment 9.2 (N. Proudfoot) One setting in which one gets a ring structure on intersection cohomology is when one has a small resolution, but in the hypertoric examples, Nick is not aware of any such small resolution.",
   "public_statement": "Question 9.1 (N. Proudfoot) The intersection cohomology of a singular hypertoric vari-eties has a \"seemingly natural\" ring structure (as a quotient of the equivariant cohomology\n\nH∗\n\n> T\n\n(T ∗Cn) of the original space). Is there, in general, a natural ring structure on the inter-section cohomology group of a singular hyperK¨ ahler quotient?\n\nComment 9.2 (N. Proudfoot) One setting in which one gets a ring structure on intersection cohomology is when one has a small resolution, but in the hypertoric examples, Nick is not aware of any such small resolution.",
   "evidence": "The canonical record is Question 9.1 from the AIM workshop list *Moment maps and surjectivity in various geometries*. The PDF reads (with line-break hyphenation silently repaired):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 310,
   "attempt": 1
  },
  "AIM-GEOMETRY-0312": {
   "statement_status": "exact",
   "original_statement": "Question 9.3 (T. Holm) Is there a natural ring structure for the intersection cohomology of singular K¨ ahler quotients? \n\nComment 9.4 (E. Lerman) In [LT00], E. Lerman and S. Tolman construct a small resolu-tion for S1-reduced spaces and thus obtain a ring structure on the intersection cohomology of the S1-reduction, but their results are special to the S1 case. \n\nComment 9.5 (R. Sjamaar) In their paper \"ntersection cohomology of symplectic quotients by circle actions\" (to be published in J. London Math. Soc. ), Kiem and Woolf produce an example of a singular symplectic (in fact K¨ ahler) quotient which has two small resolutions 15 with distinct cohomology rings. So there appears to be no natural ring structure on the intersection homology of a singular quotient. \n\n10 Computations over Z",
   "clean_statement": "Question 9.3 (T. Holm) Is there a natural ring structure for the intersection cohomology of singular K¨ ahler quotients?\n\nComment 9.4 (E. Lerman) In [LT00], E. Lerman and S. Tolman construct a small resolu-tion for S1-reduced spaces and thus obtain a ring structure on the intersection cohomology of the S1-reduction, but their results are special to the S1 case.\n\nComment 9.5 (R. Sjamaar) In their paper \"ntersection cohomology of symplectic quotients by circle actions\" (to be published in J. London Math. Soc. ), Kiem and Woolf produce an example of a singular symplectic (in fact K¨ ahler) quotient which has two small resolutions 15 with distinct cohomology rings. So there appears to be no natural ring structure on the intersection homology of a singular quotient.\n\n10 Computations over Z",
   "public_statement": "Question 9.3 (T. Holm) Is there a natural ring structure for the intersection cohomology of singular K¨ ahler quotients?\n\nComment 9.4 (E. Lerman) In [LT00], E. Lerman and S. Tolman construct a small resolu-tion for S1-reduced spaces and thus obtain a ring structure on the intersection cohomology of the S1-reduction, but their results are special to the S1 case.\n\nComment 9.5 (R. Sjamaar) In their paper \"ntersection cohomology of symplectic quotients by circle actions\" (to be published in J. London Math. Soc. ), Kiem and Woolf produce an example of a singular symplectic (in fact K¨ ahler) quotient which has two small resolutions 15 with distinct cohomology rings. So there appears to be no natural ring structure on the intersection homology of a singular quotient.\n\n10 Computations over Z",
   "evidence": "The canonical JSON record preserves the following OCR text, including its errors and a spillover heading:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 311,
   "attempt": 1
  },
  "AIM-GEOMETRY-0313": {
   "statement_status": "exact",
   "original_statement": "Problem 10.1 (R. Goldin, S. Tolman, J. Weitsman) Describe all the conditions under which Kirwan surjectivity holds for symplectic reductions over the integers. What are the most general conditions under which it holds?",
   "clean_statement": "Problem 10.1 (R. Goldin, S. Tolman, J. Weitsman) Describe all the conditions under which Kirwan surjectivity holds for symplectic reductions over the integers. What are the most general conditions under which it holds?",
   "public_statement": "Problem 10.1 (R. Goldin, S. Tolman, J. Weitsman) Describe all the conditions under which Kirwan surjectivity holds for symplectic reductions over the integers. What are the most general conditions under which it holds?",
   "evidence": "The canonical record is Problem 10.1 from the AIM workshop *Moment maps and surjectivity in various geometries*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 312,
   "attempt": 1
  },
  "AIM-GEOMETRY-0314": {
   "statement_status": "exact",
   "original_statement": "Question 10.2 (R. Goldin) Consider the case of smooth toric varieties Cn//T. Is it always true for all toric varieties?",
   "clean_statement": "Question 10.2 (R. Goldin) Consider the case of smooth toric varieties Cn//T. Is it always true for all toric varieties?",
   "public_statement": "Question 10.2 (R. Goldin) Consider the case of smooth toric varieties Cn//T. Is it always true for all toric varieties?",
   "evidence": "The exact canonical OCR record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 313,
   "attempt": 1
  },
  "AIM-GEOMETRY-0315": {
   "statement_status": "exact",
   "original_statement": "Question 10.3 (R. Goldin) For which symplectic toric orbifolds [LT97] does the surjectiv-ity hold over Z? Actually, perhaps the more appropriate question involves looking at the analogous statement which uses the orbifold cohomology instead of ordinary cohomology. Also, what happens if we take Z/p Z coefficients? \n\nComment 10.4 (E. Lerman) Note that in",
   "clean_statement": "Question 10.3 (R. Goldin) For which symplectic toric orbifolds [LT97] does the surjectiv-ity hold over Z? Actually, perhaps the more appropriate question involves looking at the analogous statement which uses the orbifold cohomology instead of ordinary cohomology. Also, what happens if we take Z/p Z coefficients?\n\nComment 10.4 (E. Lerman) Note that in",
   "public_statement": "Question 10.3 (R. Goldin) For which symplectic toric orbifolds [LT97] does the surjectiv-ity hold over Z? Actually, perhaps the more appropriate question involves looking at the analogous statement which uses the orbifold cohomology instead of ordinary cohomology. Also, what happens if we take Z/p Z coefficients?\n\nComment 10.4 (E. Lerman) Note that in",
   "evidence": "The canonical record is Question 10.3 from the AIM workshop list *Moment maps and surjectivity in various geometries*. Its exact extracted problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 314,
   "attempt": 1
  },
  "AIM-GEOMETRY-0316": {
   "statement_status": "exact",
   "original_statement": "Question 10.3, the issue is not just one of sur-jectivity but also of the choice of the cohomology theory for the reduced space. I.e. should one take Chen-Ruan orbifold cohomology? Or that of Haefliger? One approach to addressing the question would be to look for a good (= optimal) condition on critical sets of the norm-square of the moment map. \n\n11 Localization formulas for non-compact groups",
   "clean_statement": "Question 10.3, the issue is not just one of sur-jectivity but also of the choice of the cohomology theory for the reduced space. I.e. should one take Chen-Ruan orbifold cohomology? Or that of Haefliger? One approach to addressing the question would be to look for a good (= optimal) condition on critical sets of the norm-square of the moment map.\n\n11 Localization formulas for non-compact groups",
   "public_statement": "Question 10.3, the issue is not just one of sur-jectivity but also of the choice of the cohomology theory for the reduced space. I.e. should one take Chen-Ruan orbifold cohomology? Or that of Haefliger? One approach to addressing the question would be to look for a good (= optimal) condition on critical sets of the norm-square of the moment map.\n\n11 Localization formulas for non-compact groups",
   "evidence": "The JSON record contains this OCR text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 315,
   "attempt": 1
  },
  "AIM-GEOMETRY-0317": {
   "statement_status": "exact",
   "original_statement": "Question 11.1 (M. Libine) Let M be a compact manifold, and consider T ∗M. Let σ be the canonical symplectic form on T ∗M. For any other exact symplectic form ω = dα on T ∗M,with α|M exact, is it possible to find a diffeomorphism on T ∗M preserving M sending ω to \n\n±σ?",
   "clean_statement": "Question 11.1 (M. Libine) Let M be a compact manifold, and consider T ∗M. Let σ be the canonical symplectic form on T ∗M. For any other exact symplectic form ω = dα on T ∗M,with α|M exact, is it possible to find a diffeomorphism on T ∗M preserving M sending ω to\n\n±σ?",
   "public_statement": "Question 11.1 (M. Libine) Let M be a compact manifold, and consider T ∗M. Let σ be the canonical symplectic form on T ∗M. For any other exact symplectic form ω = dα on T ∗M,with α|M exact, is it possible to find a diffeomorphism on T ∗M preserving M sending ω to\n\n±σ?",
   "evidence": "The record is Question 11.1, attributed to M. Libine, in the American Institute of Mathematics problem list *Moment maps and surjectivity in various geometries*. The original PDF was checked directly. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 316,
   "attempt": 1
  },
  "AIM-GEOMETRY-0318": {
   "statement_status": "exact",
   "original_statement": "Question 11.2 (M. Libine) If there is no such diffeomorphism, how can we parametrize symplectic forms on T ∗M up to this equivalence? \n\nComment 11.3 (M. Libine) Answering this question would provide further extensions to the Berligne-Vergne localization formula extended to non-compact (reductive) group actions. \n\n12 Volume growth of hyperK¨ ahler manifolds",
   "clean_statement": "Question 11.2 (M. Libine) If there is no such diffeomorphism, how can we parametrize symplectic forms on T ∗M up to this equivalence?\n\nComment 11.3 (M. Libine) Answering this question would provide further extensions to the Berligne-Vergne localization formula extended to non-compact (reductive) group actions.\n\n12 Volume growth of hyperK¨ ahler manifolds",
   "public_statement": "Question 11.2 (M. Libine) If there is no such diffeomorphism, how can we parametrize symplectic forms on T ∗M up to this equivalence?\n\nComment 11.3 (M. Libine) Answering this question would provide further extensions to the Berligne-Vergne localization formula extended to non-compact (reductive) group actions.\n\n12 Volume growth of hyperK¨ ahler manifolds",
   "evidence": "The canonical record is Question 11.2 from the AIM workshop list *Moment maps and surjectivity in various geometries*. Its exact extracted problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 317,
   "attempt": 1
  },
  "AIM-GEOMETRY-0319": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 12.1 (H. Konno) Let M be a connected noncompact hyperK¨ ahler manifold. Fix a point p ∈ M. Consider the open ball B(p, r ) of radius r around p in M. Describe the asymptotic behavior of the volume of the ball V ol (B(p, r )) as r → ∞. (Fact: this is indepen-dent of the choice of p ∈ M. ) It would be interesting to search for examples of hyperK¨ ahler manifolds with different volume growth. 16 13 Hodge theory",
   "clean_statement": "Let \\(M\\) be a connected noncompact hyperkähler manifold. Fix a point \\(p\\in M\\). Consider the open ball \\(B(p,r)\\) of radius \\(r\\) around \\(p\\) in \\(M\\). Describe the asymptotic behavior of \\(\\operatorname{Vol}(B(p,r))\\) as \\(r\\to\\infty\\). (Fact: this is independent of the choice of \\(p\\in M\\).) It would be interesting to search for examples of hyperkähler manifolds with different volume growth.",
   "public_statement": "Question 12.1 (H. Konno) Let M be a connected noncompact hyperK¨ ahler manifold. Fix a point p ∈ M. Consider the open ball B(p, r ) of radius r around p in M. Describe the asymptotic behavior of the volume of the ball V ol (B(p, r )) as r → ∞. (Fact: this is indepen-dent of the choice of p ∈ M. ) It would be interesting to search for examples of hyperK¨ ahler manifolds with different volume growth. 16 13 Hodge theory",
   "evidence": "The record is Question 12.1, attributed to H. Konno, in the American Institute of Mathematics problem list *Moment maps and surjectivity in various geometries*. The original PDF was checked directly. The recovered statement is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 318,
   "attempt": 1
  },
  "AIM-GEOMETRY-0320": {
   "statement_status": "exact",
   "original_statement": "Question 13.1 (J. Weitsman) Suppose M is a compact K¨ ahler manifold and a Hamiltonian \n\nG-space. Suppose all the ordinary cohomology is of type ( p, p ). Can we say anything about the cohomology of the quotient? \n\nComment 13.2 (R. Sjamaar) Usual quantization-commutes-with-reduction states that the cohomology H∗(M; L) with coefficients in the sheaf of the prequantum line bundle L is a \n\nG-module, and further, that the G-invariant part is the cohomology of the GIT quotient with coefficients in the sheaf of the induced prequantum line bundle Lred. Teleman [Tel00] says that something similar should work not just for the prequantum line bundle but also for a sheaf such as L ⊗ Ωq, where L is still the sheaf of the prequantum line bundle, and Ω q\n\nis the sheaf of holomorphic q-forms. But then since Hp,q (M ) = Hp(M; Ω q), perhaps some quantization-commutes-with-reduction argument, using just the sheaf Ω q instead of a sheaf tensored with L, could also be used to show that the appropriate cohomology also vanishes downstairs. \n\nComment 13.3 (D. Burns, J. Weitsman) Burns and Weitsman have found one method for doing this, though it may already be implicit, as suggested by Sjamaar in Comment 13.2, in earlier work.",
   "clean_statement": "Question 13.1 (J. Weitsman) Suppose M is a compact K¨ ahler manifold and a Hamiltonian\n\nG-space. Suppose all the ordinary cohomology is of type ( p, p ). Can we say anything about the cohomology of the quotient?\n\nComment 13.2 (R. Sjamaar) Usual quantization-commutes-with-reduction states that the cohomology H∗(M; L) with coefficients in the sheaf of the prequantum line bundle L is a\n\nG-module, and further, that the G-invariant part is the cohomology of the GIT quotient with coefficients in the sheaf of the induced prequantum line bundle Lred. Teleman [Tel00] says that something similar should work not just for the prequantum line bundle but also for a sheaf such as L ⊗ Ωq, where L is still the sheaf of the prequantum line bundle, and Ω q\n\nis the sheaf of holomorphic q-forms. But then since Hp,q (M ) = Hp(M; Ω q), perhaps some quantization-commutes-with-reduction argument, using just the sheaf Ω q instead of a sheaf tensored with L, could also be used to show that the appropriate cohomology also vanishes downstairs.\n\nComment 13.3 (D. Burns, J. Weitsman) Burns and Weitsman have found one method for doing this, though it may already be implicit, as suggested by Sjamaar in Comment 13.2, in earlier work.",
   "public_statement": "Question 13.1 (J. Weitsman) Suppose M is a compact K¨ ahler manifold and a Hamiltonian\n\nG-space. Suppose all the ordinary cohomology is of type ( p, p ). Can we say anything about the cohomology of the quotient?\n\nComment 13.2 (R. Sjamaar) Usual quantization-commutes-with-reduction states that the cohomology H∗(M; L) with coefficients in the sheaf of the prequantum line bundle L is a\n\nG-module, and further, that the G-invariant part is the cohomology of the GIT quotient with coefficients in the sheaf of the induced prequantum line bundle Lred. Teleman [Tel00] says that something similar should work not just for the prequantum line bundle but also for a sheaf such as L ⊗ Ωq, where L is still the sheaf of the prequantum line bundle, and Ω q\n\nis the sheaf of holomorphic q-forms. But then since Hp,q (M ) = Hp(M; Ω q), perhaps some quantization-commutes-with-reduction argument, using just the sheaf Ω q instead of a sheaf tensored with L, could also be used to show that the appropriate cohomology also vanishes downstairs.\n\nComment 13.3 (D. Burns, J. Weitsman) Burns and Weitsman have found one method for doing this, though it may already be implicit, as suggested by Sjamaar in Comment 13.2, in earlier work.",
   "evidence": "The canonical record is Question 13.1 from the AIM workshop *Moment maps and surjectivity in various geometries*. The exact OCR extraction is preserved in `input.json`. Comparison with page 16 of the [AIM source PDF](https://aimath.org/WWN/momentmaps/momentmaps.pdf) gives the following recovered question:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 319,
   "attempt": 1
  },
  "AIM-GEOMETRY-0321": {
   "statement_status": "exact",
   "original_statement": "A.1 Combinatorics of linear tropical varieties \n\nWe know that the Bergman complex of a variety (as defined in Chapter 9 of Sturmfels' book on \"Solving systems of polynomial equations\"), i.e. the tropical variety, is a polyhedral complex. For linear subspaces, the paper of Ardila and Klivans shows that this Bergman complex has a very nice combinatorial structure: a subdivision of it is the order complex of the lattice of flats of the associated matroid, a very well understood combinatorial object. (As a corollary we get the topology, etc.) Question: Can we find a similar combinatorial description for other classes of Bergman complexes? (contributed by Federico Ardila)",
   "clean_statement": "A.1 Combinatorics of linear tropical varieties\n\nWe know that the Bergman complex of a variety (as defined in Chapter 9 of Sturmfels' book on \"Solving systems of polynomial equations\"), i.e. the tropical variety, is a polyhedral complex. For linear subspaces, the paper of Ardila and Klivans shows that this Bergman complex has a very nice combinatorial structure: a subdivision of it is the order complex of the lattice of flats of the associated matroid, a very well understood combinatorial object. (As a corollary we get the topology, etc.) Question: Can we find a similar combinatorial description for other classes of Bergman complexes? (contributed by Federico Ardila)",
   "public_statement": "A.1 Combinatorics of linear tropical varieties\n\nWe know that the Bergman complex of a variety (as defined in Chapter 9 of Sturmfels' book on \"Solving systems of polynomial equations\"), i.e. the tropical variety, is a polyhedral complex. For linear subspaces, the paper of Ardila and Klivans shows that this Bergman complex has a very nice combinatorial structure: a subdivision of it is the order complex of the lattice of flats of the associated matroid, a very well understood combinatorial object. (As a corollary we get the topology, etc.) Question: Can we find a similar combinatorial description for other classes of Bergman complexes? (contributed by Federico Ardila)",
   "evidence": "The exact AIM PDF, page 3, and the AIM HTML transcription agree. The record is headed “A.1 Combinatorics of linear tropical varieties” and asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 320,
   "attempt": 1
  },
  "AIM-GEOMETRY-0322": {
   "statement_status": "exact",
   "original_statement": "A.2 Monge-Amp` ere measure and mixed cells \n\nBackground: There exists a Bernstein theorem for tropical varieties (see Sturmfels' book on \"Solving systems of polynomial equations\"), there also exists a mixed Monge-Amp` ere measure whose value at any connected compact component K of the intersection of the considered amoebas is the number of solutions of the corresponding polynomial system in the pre-image (in the complex torus) by Log of K (see the paper \"Amoebas, Monge-Amp` ere measures and triangulations of the Newton polytope\" from M. Passare and H. Rullg˚ ard). Questions: Is it true that this value coincides with the volume of the mixed cell corre-sponding to K (this volume participates in the Bernstein theorem for tropical varieties)? Is there a one-to-one correspondence with the solutions of our system in Log −1(K) and the so-lutions of a binomial system corresponding to the mixed cell, and which sends real solutions to real solutions? (contributed by Frederic Bihan)",
   "clean_statement": "A.2 Monge-Amp` ere measure and mixed cells\n\nBackground: There exists a Bernstein theorem for tropical varieties (see Sturmfels' book on \"Solving systems of polynomial equations\"), there also exists a mixed Monge-Amp` ere measure whose value at any connected compact component K of the intersection of the considered amoebas is the number of solutions of the corresponding polynomial system in the pre-image (in the complex torus) by Log of K (see the paper \"Amoebas, Monge-Amp` ere measures and triangulations of the Newton polytope\" from M. Passare and H. Rullg˚ ard). Questions: Is it true that this value coincides with the volume of the mixed cell corre-sponding to K (this volume participates in the Bernstein theorem for tropical varieties)? Is there a one-to-one correspondence with the solutions of our system in Log −1(K) and the so-lutions of a binomial system corresponding to the mixed cell, and which sends real solutions to real solutions? (contributed by Frederic Bihan)",
   "public_statement": "A.2 Monge-Amp` ere measure and mixed cells\n\nBackground: There exists a Bernstein theorem for tropical varieties (see Sturmfels' book on \"Solving systems of polynomial equations\"), there also exists a mixed Monge-Amp` ere measure whose value at any connected compact component K of the intersection of the considered amoebas is the number of solutions of the corresponding polynomial system in the pre-image (in the complex torus) by Log of K (see the paper \"Amoebas, Monge-Amp` ere measures and triangulations of the Newton polytope\" from M. Passare and H. Rullg˚ ard). Questions: Is it true that this value coincides with the volume of the mixed cell corre-sponding to K (this volume participates in the Bernstein theorem for tropical varieties)? Is there a one-to-one correspondence with the solutions of our system in Log −1(K) and the so-lutions of a binomial system corresponding to the mixed cell, and which sends real solutions to real solutions? (contributed by Frederic Bihan)",
   "evidence": "This is Question A.2 from the AIM workshop list *Amoebas and tropical geometry* (January 2004), contributed by F. Bihan. The source record is tagged `section`, but it contains two genuine research questions.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 321,
   "attempt": 1
  },
  "AIM-GEOMETRY-0323": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A.3 Membership problems \n\nBackground: For every ideal a in Rd = Z[x±11,... x ±1 \n\n> d\n\n] there is a related dynamical system generated by d commuting automorphisms of a compact abelian group via Pontryagin duality. For dynamics it is very important to determine when such systems have a finiteness condition called expansiveness. A theorem of Klaus Schmidt states in effect that when a\n\ncontains no nonzero integers, then the system is expansive if and only if the complex amoeba of a does not contain the origin. 4\n\nQuestion: Is there an algorithm to determine whether the complex amoeba of an ideal \n\na in Rd contains the origin? (contributed by Manfred Einsiedler and Doug Lind)",
   "clean_statement": "**A.3 Membership problems.** For every ideal $\\mathfrak a$ in\n\\[\nR_d=\\mathbb Z[x_1^{\\pm1},\\ldots,x_d^{\\pm1}]\n\\]\nthere is a related dynamical system generated by $d$ commuting automorphisms of a compact abelian group via Pontryagin duality. For dynamics it is very important to determine when such systems have a finiteness condition called expansiveness. A theorem of Klaus Schmidt states in effect that when $\\mathfrak a$ contains no nonzero integers, then the system is expansive if and only if the complex amoeba of $\\mathfrak a$ does not contain the origin.\n\n**Question.** Is there an algorithm to determine whether the complex amoeba of an ideal $\\mathfrak a$ in $R_d$ contains the origin? (Contributed by Manfred Einsiedler and Doug Lind.)",
   "public_statement": "A.3 Membership problems\n\nBackground: For every ideal a in Rd = Z[x±11,... x ±1\n\n> d\n\n] there is a related dynamical system generated by d commuting automorphisms of a compact abelian group via Pontryagin duality. For dynamics it is very important to determine when such systems have a finiteness condition called expansiveness. A theorem of Klaus Schmidt states in effect that when a\n\ncontains no nonzero integers, then the system is expansive if and only if the complex amoeba of a does not contain the origin. 4\n\nQuestion: Is there an algorithm to determine whether the complex amoeba of an ideal\n\na in Rd contains the origin? (contributed by Manfred Einsiedler and Doug Lind)",
   "evidence": "The canonical OCR record is preserved verbatim in input.json. Comparison with the original [AIM problem list](https://aimath.org/WWN/amoebas/amoebas.pdf), including its surrounding page layout, gives this recovered statement:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 322,
   "attempt": 1
  },
  "AIM-GEOMETRY-0324": {
   "statement_status": "exact",
   "original_statement": "A.4 Recognition problems \n\nBackground: Let k be an algebraically closed non-archimedean field, and p be a prime ideal in k[x±11,... x ±1 \n\n> d\n\n]. We know that the non-archimedean amoeba of p coincides with the Bieri-Groves set of the algebra A = k[x±11,... x ±1 \n\n> d\n\n]/p, and is thus a homogeneous polyhedral complex whose dimension is the Krull dimension of A, and which is rationally defined over the value group of k. It also has the geometric property of total concavity, a sort of harmonic condition of spreading for the complex. Question: Given a homogeneous polyhedral complex that is rationally defined over a dense subgroup of the reals and is also totally concave, what further conditions are necessary in order for it to be the amoeba of a prime ideal in the ring of Laurent polynomials over an algebraically closed non-archimedean field? (contributed by Manfred Einsiedler and Doug Lind)",
   "clean_statement": "A.4 Recognition problems\n\nBackground: Let k be an algebraically closed non-archimedean field, and p be a prime ideal in k[x±11,... x ±1\n\n> d\n\n]. We know that the non-archimedean amoeba of p coincides with the Bieri-Groves set of the algebra A = k[x±11,... x ±1\n\n> d\n\n]/p, and is thus a homogeneous polyhedral complex whose dimension is the Krull dimension of A, and which is rationally defined over the value group of k. It also has the geometric property of total concavity, a sort of harmonic condition of spreading for the complex. Question: Given a homogeneous polyhedral complex that is rationally defined over a dense subgroup of the reals and is also totally concave, what further conditions are necessary in order for it to be the amoeba of a prime ideal in the ring of Laurent polynomials over an algebraically closed non-archimedean field? (contributed by Manfred Einsiedler and Doug Lind)",
   "public_statement": "A.4 Recognition problems\n\nBackground: Let k be an algebraically closed non-archimedean field, and p be a prime ideal in k[x±11,... x ±1\n\n> d\n\n]. We know that the non-archimedean amoeba of p coincides with the Bieri-Groves set of the algebra A = k[x±11,... x ±1\n\n> d\n\n]/p, and is thus a homogeneous polyhedral complex whose dimension is the Krull dimension of A, and which is rationally defined over the value group of k. It also has the geometric property of total concavity, a sort of harmonic condition of spreading for the complex. Question: Given a homogeneous polyhedral complex that is rationally defined over a dense subgroup of the reals and is also totally concave, what further conditions are necessary in order for it to be the amoeba of a prime ideal in the ring of Laurent polynomials over an algebraically closed non-archimedean field? (contributed by Manfred Einsiedler and Doug Lind)",
   "evidence": "The canonical corpus record is AIM-GEOMETRY-0324, item A.4 of the AIM workshop “Amoebas and tropical geometry.” The raw JSON faithfully retains a damaged extraction, including",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 323,
   "attempt": 1
  },
  "AIM-GEOMETRY-0325": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A.5 Half-space behavior of amoebas \n\nBackground: Let Rd = Z[x±11,... x ±1 \n\n> d\n\n] and a be an ideal in Rd with a ∩ Z = {0}.The adelic amoeba of a is the union of its complex amoeba and its p-adic amoebas over all rational primes p. If a = 〈f 〉 is principal, an argument from dynamics shows that every 1-dimensional ray from the origin must intersect the adelic amoeba of f. There should be a version of this for general ideals, and it is enough to state this for prime ideals. Question: Let p be a prime ideal in Rd, and r denote the Krull dimension of Rd/p.Then for every subspace of Rd with dimension d−r +1, does every half-space of the subspace intersect the adelic amoeba of p?(contributed by Manfred Einsiedler and Doug Lind)",
   "clean_statement": null,
   "public_statement": "A.5 Half-space behavior of amoebas\n\nBackground: Let Rd = Z[x±11,... x ±1\n\n> d\n\n] and a be an ideal in Rd with a ∩ Z = {0}.The adelic amoeba of a is the union of its complex amoeba and its p-adic amoebas over all rational primes p. If a = 〈f 〉 is principal, an argument from dynamics shows that every 1-dimensional ray from the origin must intersect the adelic amoeba of f. There should be a version of this for general ideals, and it is enough to state this for prime ideals. Question: Let p be a prime ideal in Rd, and r denote the Krull dimension of Rd/p.Then for every subspace of Rd with dimension d−r +1, does every half-space of the subspace intersect the adelic amoeba of p?(contributed by Manfred Einsiedler and Doug Lind)",
   "evidence": "This is Question A.5, “Half-space behavior of amoebas,” from the AIM workshop list *Amoebas and tropical geometry* (version dated January 14, 2004), contributed by Manfred Einsiedler and Doug Lind. The exact database record is preserved in input.json; it is tagged section but contains a genuine mathematical question.",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 324,
   "attempt": 1
  },
  "AIM-GEOMETRY-0326": {
   "statement_status": "exact",
   "original_statement": "A.6 Higher order connectedness of amoebas \n\nBackground: Let k be an algebraically closed non-archimedean field, and p be a prime ideal in R = k[x±11,... x ±1 \n\n> d\n\n]. An argument in a forthcoming paper by Einsiedler, Lind, and Kapranov shows that the non-archimedean amoeba of p is a connected set in Rd. We also know that it is a homogeneous polyhedral complex of dimension r, where r is the Krull dimension of R/ p. But examples show that the amoeba may always have a higher type of connectivity as well. Question: Let p be a prime ideal in R, and r be the Krull dimension of R/ p. Form the finite graph whose vertices are the r-dimensional faces of the amoeba of p, and for which two vertices are joined if they share an ( r − 1)-face of the amoeba. Then is this graph always connected? (contributed by Manfred Einsiedler, Doug Lind, and Rekha Thomas) 5",
   "clean_statement": "A.6 Higher order connectedness of amoebas\n\nBackground: Let k be an algebraically closed non-archimedean field, and p be a prime ideal in R = k[x±11,... x ±1\n\n> d\n\n]. An argument in a forthcoming paper by Einsiedler, Lind, and Kapranov shows that the non-archimedean amoeba of p is a connected set in Rd. We also know that it is a homogeneous polyhedral complex of dimension r, where r is the Krull dimension of R/ p. But examples show that the amoeba may always have a higher type of connectivity as well. Question: Let p be a prime ideal in R, and r be the Krull dimension of R/ p. Form the finite graph whose vertices are the r-dimensional faces of the amoeba of p, and for which two vertices are joined if they share an ( r − 1)-face of the amoeba. Then is this graph always connected? (contributed by Manfred Einsiedler, Doug Lind, and Rekha Thomas) 5",
   "public_statement": "A.6 Higher order connectedness of amoebas\n\nBackground: Let k be an algebraically closed non-archimedean field, and p be a prime ideal in R = k[x±11,... x ±1\n\n> d\n\n]. An argument in a forthcoming paper by Einsiedler, Lind, and Kapranov shows that the non-archimedean amoeba of p is a connected set in Rd. We also know that it is a homogeneous polyhedral complex of dimension r, where r is the Krull dimension of R/ p. But examples show that the amoeba may always have a higher type of connectivity as well. Question: Let p be a prime ideal in R, and r be the Krull dimension of R/ p. Form the finite graph whose vertices are the r-dimensional faces of the amoeba of p, and for which two vertices are joined if they share an ( r − 1)-face of the amoeba. Then is this graph always connected? (contributed by Manfred Einsiedler, Doug Lind, and Rekha Thomas) 5",
   "evidence": "The source record is item A.6, “Higher order connectedness of amoebas,” in the AIM workshop list on amoebas. The exact AIM HTML resolves the OCR damage in the extracted JSON. In modern notation the question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 325,
   "attempt": 1
  },
  "AIM-GEOMETRY-0327": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A.7 What does the Riemann-Roch theorem say in the tropical world? \n\nBackground: One way to approach this would be to build up the machinery of line bundles (or maybe coherent sheaves?). A different, more immediately geometric approach might be called the \"Brill-Noether\" approach. This requires only two ingredients: A. \"plane curve with ordinary nodes\" and B. If A, B and C are curves with a common point of intersection, what is \" A∩B−A∩C\", the \"residual intersection of C in A ∩ B\". If Ox is the local ring of x on A, and fB,\n\nfC are the images in Ox of the equations of B and C, then in the classical case the parts of the intersections A ∩ B and A ∩ C supported at x are represented by the ideals ( fB ) and ( fC ) in Ox, and the residual is represented by the ideal (fB: fC ):= {g ∈ O x | gf C ∈ (fB )}.\n\nThe early work on Riemann-Roch treated only the case where A is smooth at x. Then \n\nA ∩ B at x is represented just by a multiplicity, and residuation is just subtraction. When A is arbitrary, things still work because Ox is a Gorenstein ring for any smooth curve, no matter how singular. Question: How do these notions play out for tropical plane curves? (contributed by David Eisenbud)",
   "clean_statement": "“for any **plane** curve, no matter how singular”: locally a plane curve ring is a hypersurface quotient of a regular local ring, hence is Gorenstein. This replacement is an inference, not a verified correction, so the analysis below separates the unambiguous divisor question from the more delicate local colon-ideal question.",
   "public_statement": "A.7 What does the Riemann-Roch theorem say in the tropical world?\n\nBackground: One way to approach this would be to build up the machinery of line bundles (or maybe coherent sheaves?). A different, more immediately geometric approach might be called the \"Brill-Noether\" approach. This requires only two ingredients: A. \"plane curve with ordinary nodes\" and B. If A, B and C are curves with a common point of intersection, what is \" A∩B−A∩C\", the \"residual intersection of C in A ∩ B\". If Ox is the local ring of x on A, and fB,\n\nfC are the images in Ox of the equations of B and C, then in the classical case the parts of the intersections A ∩ B and A ∩ C supported at x are represented by the ideals ( fB ) and ( fC ) in Ox, and the residual is represented by the ideal (fB: fC ):= {g ∈ O x | gf C ∈ (fB )}.\n\nThe early work on Riemann-Roch treated only the case where A is smooth at x. Then\n\nA ∩ B at x is represented just by a multiplicity, and residuation is just subtraction. When A is arbitrary, things still work because Ox is a Gorenstein ring for any smooth curve, no matter how singular. Question: How do these notions play out for tropical plane curves? (contributed by David Eisenbud)",
   "evidence": "There is an internal contradiction in the source itself, not merely in the extracted JSON: “a Gorenstein ring for any smooth curve, no matter how singular.” A point cannot simultaneously be smooth and singular. A plausible reconstruction is “for any **plane** curve, no matter how singular”: locally a plane curve ring is a hypersurface quotient of a regular local ring, hence is Gorenstein. This replacement is an inference, not a verified correction, so the analysis below separates the unambiguous divisor question from the more delicate local colon-ideal question.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 326,
   "attempt": 1
  },
  "AIM-GEOMETRY-0328": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A.8 Tropical Calabi-Yau manifolds and tropical line bundles \n\nAn affine manifold is a real manifold with coordinate charts whose transition maps are in Aff( Rn). We will call a tropical Calabi-Yau manifold a real manifold B with a dense open subset \n\nB0 ⊆ B which has an affine structure with transition maps in Rn o GL n(Z), and such that \n\nB \\ B0 =: ∆ is a locally finite union of locally closed submanifolds of B.It makes sense to call B0 a tropical variety. Certainly B0 locally looks like tropical affine space, and maps in Rn o GL n(Z) look like maps defined by tropical monomials, so this seems natural. One can additionally talk about the sheaf of piecewise linear functions on \n\nB0 with integral slope, or the sheaf of continuous functions on B which restrict to piecewise linear functions on B0 with integral slope. This should play the role of the structure sheaf. Question (Sturmfels): Is it natural to call B a tropical Calabi-Yau variety? In other words, do these singularities make sense in the tropical context? This is related to Zharkov's question of cutting tentacles. Let Aff( B, R) denote the sheaf of functions on B which are continuous and restrict to affine linear functions with integral slope on B0. We define a tropical line bundle to be an element of H1(B, Aff( B, R)). Representing an element by a ˇCech 1-cocycle (αij ) for an open cover {Ui}, a section of this tropical line bundle is a collection of tropical functions si on Ui such that si − sj = αij. (Here this is ordinary subtraction). We saw how sections of tropical line bundles over tori are tropical theta functions. Question (Eisenbud, see also the question on Riemann-Roch 33 ): What is tropical Riemann-Roch? \n\n> 33 page 4, What does the Riemann-Roch theorem say in the tropical world? 6\n\nThe above discussion should go over to tropical varieties in general, if we have the right definitions. The same question applies. Questions: What is the notion of an ample line bundle? Is it interesting to study embeddings into tropical projective space? Exercise: Consider a tropical plane cubic, say \n\n−6x3 − 4x2y − 3xy 2 − 6y3 − 4y2z − 3yz 2 − 0xyz − 3x2z − 1xz 2 − 3z3.\n\nDraw a picture of this curve. Cut off the infinite rays, to get a polygon. The affine length of each edge is defined as follows. For the vertices of an edge, v and w, write v − w = ld, where \n\nd is a primitive integral vector and l is a real number. Then the affine length is |l|. Check that the sum of the affine lengths of the edges is 13. Show this polygon can be obtained as an embedding R/13 Z → T P2, using three tropical sections of a tropical line bundle of degree \n\nfive.Question: This seems a bit strange, doesn't it? Observation: If one uses a line bundle of degree 3 to try to map to T P2, certain line segments in the circle will be contracted! Does this mean that the line bundle of degree 3 isn't very ample? Given such a B, we can form two manifolds of twice the dimension, both torus bundles over B0. Let Λ ⊆ T B0 be a family of lattices in the tangent bundle generated locally by \n\n∂/∂y 1,..., ∂/∂y n where y1,..., y n are local affine coordinates on B0. Because of the GL n(Z)restriction on transition functions, this is well-defined. Let X(B0) = TB0 /Λ. This carries a complex structure which interchanges horizontal and vertical directions in the tangent bundle. Similarly, let ˇΛ ⊆ T ∗ \n\n> B0\n\nbe the dual family of lattices generated by dy 1,..., dy n. Then we set ˇX(B0) = T ∗ \n\n> B0\n\n/ˇΛ. This is canonically a symplectic manifold. One particularly important question relevant for the Strominger-Yau-Zaslow conjecture is the following. We would like to find classical sections of tropical line bundles (i.e. smooth functions ( Ui, s i) with si − sj = αij ) satisfying the Monge-Amp` ere equation det( ∂2si/∂y j ∂y k) = constant. \n\nIf one does this, then pulling back the functions si to X(B0) will give K¨ ahler potentials for Ricci-flat metrics. Question (Gross, Siebert, Zharkov): Is there a tropical Monge-Amp` ere equation? Fix-ing the line bundle L, can one find a sequence of sections si ∈ Γ( B, Ln) (hopefully satisfying this tropical form) such that ℏsn converges to a classical solution of the equation. (Here \n\nℏ = 1 /n ). Write a computer program to produce numerical solutions in this way, and draw a picture of a genuine Ricci-flat metric!",
   "clean_statement": null,
   "public_statement": "A.8 Tropical Calabi-Yau manifolds and tropical line bundles\n\nAn affine manifold is a real manifold with coordinate charts whose transition maps are in Aff( Rn). We will call a tropical Calabi-Yau manifold a real manifold B with a dense open subset\n\nB0 ⊆ B which has an affine structure with transition maps in Rn o GL n(Z), and such that\n\nB \\ B0 =: ∆ is a locally finite union of locally closed submanifolds of B.It makes sense to call B0 a tropical variety. Certainly B0 locally looks like tropical affine space, and maps in Rn o GL n(Z) look like maps defined by tropical monomials, so this seems natural. One can additionally talk about the sheaf of piecewise linear functions on\n\nB0 with integral slope, or the sheaf of continuous functions on B which restrict to piecewise linear functions on B0 with integral slope. This should play the role of the structure sheaf. Question (Sturmfels): Is it natural to call B a tropical Calabi-Yau variety? In other words, do these singularities make sense in the tropical context? This is related to Zharkov's question of cutting tentacles. Let Aff( B, R) denote the sheaf of functions on B which are continuous and restrict to affine linear functions with integral slope on B0. We define a tropical line bundle to be an element of H1(B, Aff( B, R)). Representing an element by a ˇCech 1-cocycle (αij ) for an open cover {Ui}, a section of this tropical line bundle is a collection of tropical functions si on Ui such that si − sj = αij. (Here this is ordinary subtraction). We saw how sections of tropical line bundles over tori are tropical theta functions. Question (Eisenbud, see also the question on Riemann-Roch 33 ): What is tropical Riemann-Roch?\n\n> 33 page 4, What does the Riemann-Roch theorem say in the tropical world? 6\n\nThe above discussion should go over to tropical varieties in general, if we have the right definitions. The same question applies. Questions: What is the notion of an ample line bundle? Is it interesting to study embeddings into tropical projective space? Exercise: Consider a tropical plane cubic, say\n\n−6x3 − 4x2y − 3xy 2 − 6y3 − 4y2z − 3yz 2 − 0xyz − 3x2z − 1xz 2 − 3z3.\n\nDraw a picture of this curve. Cut off the infinite rays, to get a polygon. The affine length of each edge is defined as follows. For the vertices of an edge, v and w, write v − w = ld, where\n\nd is a primitive integral vector and l is a real number. Then the affine length is |l|. Check that the sum of the affine lengths of the edges is 13. Show this polygon can be obtained as an embedding R/13 Z → T P2, using three tropical sections of a tropical line bundle of degree\n\nfive.Question: This seems a bit strange, doesn't it? Observation: If one uses a line bundle of degree 3 to try to map to T P2, certain line segments in the circle will be contracted! Does this mean that the line bundle of degree 3 isn't very ample? Given such a B, we can form two manifolds of twice the dimension, both torus bundles over B0. Let Λ ⊆ T B0 be a family of lattices in the tangent bundle generated locally by\n\n∂/∂y 1,..., ∂/∂y n where y1,..., y n are local affine coordinates on B0. Because of the GL n(Z)restriction on transition functions, this is well-defined. Let X(B0) = TB0 /Λ. This carries a complex structure which interchanges horizontal and vertical directions in the tangent bundle. Similarly, let ˇΛ ⊆ T ∗\n\n> B0\n\nbe the dual family of lattices generated by dy 1,..., dy n. Then we set ˇX(B0) = T ∗\n\n> B0\n\n/ˇΛ. This is canonically a symplectic manifold. One particularly important question relevant for the Strominger-Yau-Zaslow conjecture is the following. We would like to find classical sections of tropical line bundles (i.e. smooth functions ( Ui, s i) with si − sj = αij ) satisfying the Monge-Amp` ere equation det( ∂2si/∂y j ∂y k) = constant.\n\nIf one does this, then pulling back the functions si to X(B0) will give K¨ ahler potentials for Ricci-flat metrics. Question (Gross, Siebert, Zharkov): Is there a tropical Monge-Amp` ere equation? Fix-ing the line bundle L, can one find a sequence of sections si ∈ Γ( B, Ln) (hopefully satisfying this tropical form) such that ℏsn converges to a classical solution of the equation. (Here\n\nℏ = 1 /n ). Write a computer program to produce numerical solutions in this way, and draw a picture of a genuine Ricci-flat metric!",
   "evidence": "The PDF extraction has page numbers $6$ and $7$ inside the prose; they are not mathematical data. The HTML displays an index mismatch near the last question, writing $s_i\\in\\Gamma(B,\\mathcal L^n)$ and then $s_n$. The sequence notation $s_n\\in\\Gamma(B,\\mathcal L^{\\otimes n})$ used above is an explicit reconstruction from the stated scaling $\\hbar=1/n$.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 327,
   "attempt": 1
  },
  "AIM-GEOMETRY-0329": {
   "statement_status": "exact",
   "original_statement": "A.9 Real tropical varieties \n\nReal tropical hypersurfaces are directly related to T-hypersurfaces (piecewise-linear hypersurfaces arising in the combinatorial patchworking). Many restrictions on the topology of T-hypersurfaces are known. It would be interesting to look at these restrictions from the point of view of tropical geometry and to study the topology of real tropical varieties. For example, the following question arises. Question: What can be said about Betti numbers of a real tropical variety? (contributed by Ilia Itenberg)",
   "clean_statement": "A.9 Real tropical varieties\n\nReal tropical hypersurfaces are directly related to T-hypersurfaces (piecewise-linear hypersurfaces arising in the combinatorial patchworking). Many restrictions on the topology of T-hypersurfaces are known. It would be interesting to look at these restrictions from the point of view of tropical geometry and to study the topology of real tropical varieties. For example, the following question arises. Question: What can be said about Betti numbers of a real tropical variety? (contributed by Ilia Itenberg)",
   "public_statement": "A.9 Real tropical varieties\n\nReal tropical hypersurfaces are directly related to T-hypersurfaces (piecewise-linear hypersurfaces arising in the combinatorial patchworking). Many restrictions on the topology of T-hypersurfaces are known. It would be interesting to look at these restrictions from the point of view of tropical geometry and to study the topology of real tropical varieties. For example, the following question arises. Question: What can be said about Betti numbers of a real tropical variety? (contributed by Ilia Itenberg)",
   "evidence": "This record is item A.9 of the AIM workshop list *Amoebas and tropical geometry*. The AIM HTML page and the supplied PDF agree. There is no material OCR ambiguity. The exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 328,
   "attempt": 1
  },
  "AIM-GEOMETRY-0330": {
   "statement_status": "exact",
   "original_statement": "A.10 The tropical Grassmannian \n\nThe tropical Grassmannian, studied by Speyer and Sturmfels, turns out, at least in the cases they study, (2, n ) and (3, 6) to have strong combinatorial connections with the Kapranov's Chow quotient G(r, n )// (C∗)n.Question (Eugene Tevelev and Sean Keel): Try to understand the precise relationship. (contributed by Sean Keel)",
   "clean_statement": "A.10 The tropical Grassmannian\n\nThe tropical Grassmannian, studied by Speyer and Sturmfels, turns out, at least in the cases they study, (2, n ) and (3, 6) to have strong combinatorial connections with the Kapranov's Chow quotient G(r, n )// (C∗)n.Question (Eugene Tevelev and Sean Keel): Try to understand the precise relationship. (contributed by Sean Keel)",
   "public_statement": "A.10 The tropical Grassmannian\n\nThe tropical Grassmannian, studied by Speyer and Sturmfels, turns out, at least in the cases they study, (2, n ) and (3, 6) to have strong combinatorial connections with the Kapranov's Chow quotient G(r, n )// (C∗)n.Question (Eugene Tevelev and Sean Keel): Try to understand the precise relationship. (contributed by Sean Keel)",
   "evidence": "This is Problem A.10 in the AIM workshop list *Amoebas and tropical geometry*, PDF version dated January 14, 2004. The PDF was checked directly. It reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 329,
   "attempt": 1
  },
  "AIM-GEOMETRY-0331": {
   "statement_status": "exact",
   "original_statement": "A.11 Real Gromov-Witten invariants and tropical geometry \n\nBackground: G. Tian and S. Kwon recently defined a real Gromov-Witten invariant on each chamber in the real Chow cycles' parameter space when the target space is CP 2.That is a real enumerative invariant, counting the number of intersection points of pull back of real Chow cycles in the real part of the Kontsevich's moduli space of stable maps from genus 0 curves. To use Mikhalkin's work on counting plane rational nodal curves, we showed that the classical nodal Severi variety is embedded as a Zariski open dense subset in the Kontsevich's moduli space. Question: It will be interesting to develop techniques to calculate real Gromov-Witten invariants by using tropical geometry. (contributed by Seongchun Kwon)",
   "clean_statement": "A.11 Real Gromov-Witten invariants and tropical geometry\n\nBackground: G. Tian and S. Kwon recently defined a real Gromov-Witten invariant on each chamber in the real Chow cycles' parameter space when the target space is CP 2.That is a real enumerative invariant, counting the number of intersection points of pull back of real Chow cycles in the real part of the Kontsevich's moduli space of stable maps from genus 0 curves. To use Mikhalkin's work on counting plane rational nodal curves, we showed that the classical nodal Severi variety is embedded as a Zariski open dense subset in the Kontsevich's moduli space. Question: It will be interesting to develop techniques to calculate real Gromov-Witten invariants by using tropical geometry. (contributed by Seongchun Kwon)",
   "public_statement": "A.11 Real Gromov-Witten invariants and tropical geometry\n\nBackground: G. Tian and S. Kwon recently defined a real Gromov-Witten invariant on each chamber in the real Chow cycles' parameter space when the target space is CP 2.That is a real enumerative invariant, counting the number of intersection points of pull back of real Chow cycles in the real part of the Kontsevich's moduli space of stable maps from genus 0 curves. To use Mikhalkin's work on counting plane rational nodal curves, we showed that the classical nodal Severi variety is embedded as a Zariski open dense subset in the Kontsevich's moduli space. Question: It will be interesting to develop techniques to calculate real Gromov-Witten invariants by using tropical geometry. (contributed by Seongchun Kwon)",
   "evidence": "This record is item A.11 of the AIM workshop list *Amoebas and tropical geometry*. The source record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 330,
   "attempt": 1
  },
  "AIM-GEOMETRY-0332": {
   "statement_status": "exact",
   "original_statement": "A.12 Idempotent geometry \n\nQuestions: 1. Is it possible to construct a version of algebraic geometry over a class of algebraically closed idempotent semifields (not only tropical semifields)?",
   "clean_statement": "A.12 Idempotent geometry\n\nQuestions: 1. Is it possible to construct a version of algebraic geometry over a class of algebraically closed idempotent semifields (not only tropical semifields)?",
   "public_statement": "A.12 Idempotent geometry\n\nQuestions: 1. Is it possible to construct a version of algebraic geometry over a class of algebraically closed idempotent semifields (not only tropical semifields)?",
   "evidence": "The record is item A.12, “Idempotent geometry,” contributed by G. L. Litvinov in cooperation with G. B. Shpiz to the AIM workshop list *Amoebas and tropical geometry*. The source asks a six-part foundational program:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 331,
   "attempt": 1
  },
  "AIM-GEOMETRY-0333": {
   "statement_status": "exact",
   "original_statement": "A.13 Moduli space of holomorphic polygons \n\nBackground: In the paper with Fukaya \"Zero loop open strings in the cotangent bundle and Morse homotopy\", Asian J. Math. 1 (1997), 96 - 180, we proved that \"The moduli space of holomorphic polygons with boundary lying on k-tuples of La-grangian graphs of k-Morse functions is diffeomorphic to that of graph flows of the Morse functions in the adiabatic limit or (in the large complex structure limit). The projections, near the limit, of the holomorphic polygons on the base of the cotangent bundle resembles amoeba-type shapes and it shrinks to the graphs of Morse flows in the limit.\" In the paper, we dealt with the case of discs, i.e., open Riemann surfaces of genus zero. Problem: Study the similar degeneration problem for the higher genus case. (contributed by Yong-Geun Oh)",
   "clean_statement": "A.13 Moduli space of holomorphic polygons\n\nBackground: In the paper with Fukaya \"Zero loop open strings in the cotangent bundle and Morse homotopy\", Asian J. Math. 1 (1997), 96 - 180, we proved that \"The moduli space of holomorphic polygons with boundary lying on k-tuples of La-grangian graphs of k-Morse functions is diffeomorphic to that of graph flows of the Morse functions in the adiabatic limit or (in the large complex structure limit). The projections, near the limit, of the holomorphic polygons on the base of the cotangent bundle resembles amoeba-type shapes and it shrinks to the graphs of Morse flows in the limit.\" In the paper, we dealt with the case of discs, i.e., open Riemann surfaces of genus zero. Problem: Study the similar degeneration problem for the higher genus case. (contributed by Yong-Geun Oh)",
   "public_statement": "A.13 Moduli space of holomorphic polygons\n\nBackground: In the paper with Fukaya \"Zero loop open strings in the cotangent bundle and Morse homotopy\", Asian J. Math. 1 (1997), 96 - 180, we proved that \"The moduli space of holomorphic polygons with boundary lying on k-tuples of La-grangian graphs of k-Morse functions is diffeomorphic to that of graph flows of the Morse functions in the adiabatic limit or (in the large complex structure limit). The projections, near the limit, of the holomorphic polygons on the base of the cotangent bundle resembles amoeba-type shapes and it shrinks to the graphs of Morse flows in the limit.\" In the paper, we dealt with the case of discs, i.e., open Riemann surfaces of genus zero. Problem: Study the similar degeneration problem for the higher genus case. (contributed by Yong-Geun Oh)",
   "evidence": "The canonical record is item A.13, contributed by Yong-Geun Oh, in the AIM workshop list *Amoebas and tropical geometry*. The stored record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 332,
   "attempt": 1
  },
  "AIM-GEOMETRY-0334": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A.14 Solidness of amoebas of maximally sparse polynomials \n\nLet f (z) = ∑ \n\n> α∈A\n\naαzα, with A a finite subset of the integer lattice Zn, be a complex Laurent polynomial. Its amoeba is the subset of Rn obtained as the image of {f (z) = 0 }\n\nunder the mapping ( z1,..., z n) 7 → (log |z1|,..., log |zn|). The amoeba is said to be solid \n\nif the number of connected components of its complement is minimal, that is, equal to the number of vertices of the Newton polytope ∆ f of f. Solid amoebas are particularly well adapted to tropical geometry. The polynomial f is said to be maximally sparse if the support of summation A is minimal, that is, equal to the set of vertices of ∆ f. When n = 1 a maximally sparse polynomial is a binomial. 9\n\nQuestion: Does every maximally sparse polynomial have a solid amoeba? The conjecture is mainly based on empirical data (=computer pictures). I did prove with Hans Rullg˚ ard that if the number of vertices is less than or equal to n + 2, then the tropical spine is contained in the amoeba. (So it would seem very plausible that the number of complement components is minimal for maximally sparse polynomials with at most n + 2 terms.) (contributed by Mikael Passare)",
   "clean_statement": null,
   "public_statement": "A.14 Solidness of amoebas of maximally sparse polynomials\n\nLet f (z) = ∑\n\n> α∈A\n\naαzα, with A a finite subset of the integer lattice Zn, be a complex Laurent polynomial. Its amoeba is the subset of Rn obtained as the image of {f (z) = 0 }\n\nunder the mapping ( z1,..., z n) 7 → (log |z1|,..., log |zn|). The amoeba is said to be solid\n\nif the number of connected components of its complement is minimal, that is, equal to the number of vertices of the Newton polytope ∆ f of f. Solid amoebas are particularly well adapted to tropical geometry. The polynomial f is said to be maximally sparse if the support of summation A is minimal, that is, equal to the set of vertices of ∆ f. When n = 1 a maximally sparse polynomial is a binomial. 9\n\nQuestion: Does every maximally sparse polynomial have a solid amoeba? The conjecture is mainly based on empirical data (=computer pictures). I did prove with Hans Rullg˚ ard that if the number of vertices is less than or equal to n + 2, then the tropical spine is contained in the amoeba. (So it would seem very plausible that the number of complement components is minimal for maximally sparse polynomials with at most n + 2 terms.) (contributed by Mikael Passare)",
   "evidence": "The canonical record is item A.14 of the AIM workshop list *Amoebas and tropical geometry*, contributed by Mikael Passare. The source PDF is <https://aimath.org/WWN/amoebas/amoebas.pdf>.",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 333,
   "attempt": 1
  },
  "AIM-GEOMETRY-0335": {
   "statement_status": "exact",
   "original_statement": "A.15 Topology of amoebas of linear spaces \n\nConsider a d-dimensional linear subspace V of Cn, and let M be the intersection of \n\nV with ( C∗)n. Then M is the complement of a collection H of n hyperplanes in V, and (virtually) any arrangement of hyperplanes arises in this way. It is a classical problem to study the topology of M in terms of the combinatorics (for example the matroid) of H.Questions: A. What are the fibers of the map Log: M → A, where A is the amoeba of V?B. What conditions on H will guarantee that this map is a homeomorphism? C. What can we say in general about the topology of the amoeba of a linear space? D. How does this relate to Federico Ardila's characterization of the tropicalization of V\n\nin terms of the matroid of H?When d = 1, the answer to (1) is easy. In this case, H is a collection of points on a complex line. If there exist three points that do not lie on a common real line, then Log is injective. If all n points lie on a real line, then the fibers of Log are the orbits of the Z2\n\naction given by reflection over this line. Higher dimensional examples of hyperplane arrangements such that Log is injective can be constructed by taking a product of d copies of three generic points on a complex line, and then adding arbitrarily many more hyperplanes to this collection of 3d-hyperplanes in \n\nV = Cd. But there should be many examples that are simpler than these. (contributed by Nicholas Proudfoot)",
   "clean_statement": "A.15 Topology of amoebas of linear spaces\n\nConsider a d-dimensional linear subspace V of Cn, and let M be the intersection of\n\nV with ( C∗)n. Then M is the complement of a collection H of n hyperplanes in V, and (virtually) any arrangement of hyperplanes arises in this way. It is a classical problem to study the topology of M in terms of the combinatorics (for example the matroid) of H.Questions: A. What are the fibers of the map Log: M → A, where A is the amoeba of V?B. What conditions on H will guarantee that this map is a homeomorphism? C. What can we say in general about the topology of the amoeba of a linear space? D. How does this relate to Federico Ardila's characterization of the tropicalization of V\n\nin terms of the matroid of H?When d = 1, the answer to (1) is easy. In this case, H is a collection of points on a complex line. If there exist three points that do not lie on a common real line, then Log is injective. If all n points lie on a real line, then the fibers of Log are the orbits of the Z2\n\naction given by reflection over this line. Higher dimensional examples of hyperplane arrangements such that Log is injective can be constructed by taking a product of d copies of three generic points on a complex line, and then adding arbitrarily many more hyperplanes to this collection of 3d-hyperplanes in\n\nV = Cd. But there should be many examples that are simpler than these. (contributed by Nicholas Proudfoot)",
   "public_statement": "A.15 Topology of amoebas of linear spaces\n\nConsider a d-dimensional linear subspace V of Cn, and let M be the intersection of\n\nV with ( C∗)n. Then M is the complement of a collection H of n hyperplanes in V, and (virtually) any arrangement of hyperplanes arises in this way. It is a classical problem to study the topology of M in terms of the combinatorics (for example the matroid) of H.Questions: A. What are the fibers of the map Log: M → A, where A is the amoeba of V?B. What conditions on H will guarantee that this map is a homeomorphism? C. What can we say in general about the topology of the amoeba of a linear space? D. How does this relate to Federico Ardila's characterization of the tropicalization of V\n\nin terms of the matroid of H?When d = 1, the answer to (1) is easy. In this case, H is a collection of points on a complex line. If there exist three points that do not lie on a common real line, then Log is injective. If all n points lie on a real line, then the fibers of Log are the orbits of the Z2\n\naction given by reflection over this line. Higher dimensional examples of hyperplane arrangements such that Log is injective can be constructed by taking a product of d copies of three generic points on a complex line, and then adding arbitrarily many more hyperplanes to this collection of 3d-hyperplanes in\n\nV = Cd. But there should be many examples that are simpler than these. (contributed by Nicholas Proudfoot)",
   "evidence": "The canonical record is item A.15, contributed by Nicholas Proudfoot, in the AIM workshop list *Amoebas and tropical geometry*. It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 334,
   "attempt": 1
  },
  "AIM-GEOMETRY-0336": {
   "statement_status": "exact",
   "original_statement": "A.16 Nullstellensatz for amoebas \n\nLet f (x1,..., x n) be a Laurent polynomial, and write f (x1,..., x n) = ∑ln=1 mi(x), where mi(x) are the monomial terms of x. Given a point a ∈ Rn, let f {a} denote the list of positive reals [ |m1(Log −1(a)) |,..., |ml(Log −1(a)) |]. Note this is well defined, even though Log is not injective. We say that a list of positive numbers satisfies the polygon condition if it is possible to make a polygon with those side lengths, i.e. no number is greater than the sum of all the others. \n\nTheorem 1. Let I be an ideal, and A(I) its amoeba. Then a ∈ A(I) if and only if \n\nf {a} satisfies the polygon condition for all f ∈ I.Let P (f ) = {a ∈ Rn: f {a} satisfies the polygon condition }. Think of this as an approximation to the amoeba of a hypersurface. \n\nTheorem 2. Let A(f ) be the amoeba of a hypersurface. Let \n\nfm(x1,..., x n) = the product of f (u1x1,..., u nxn)10 \n\nover all ui such that umi = 1. The family P (fm) converges uniformly (in the Euclidean norm) to A(f ). Questions: A. Is there a version of theorem 2 (an explicit family approximating the amoeba) in the higher codimension case? B. An analogous statement to theorem 1 is known for non-archimedean Amoebas. Is theorem 1 true in an even more general context? C. The convergence of the family in theorem 2 is of order O(log m/m ), at least in worst case situations. How fast does this family converge for a randomly chosen f? If the approximation is within ( a log m + b)/m of the actual amoeba, what are a and b, in typical examples? D. What open problems can this be used to solve? (contributed by Kevin Purbhoo)",
   "clean_statement": "A.16 Nullstellensatz for amoebas\n\nLet f (x1,..., x n) be a Laurent polynomial, and write f (x1,..., x n) = ∑ln=1 mi(x), where mi(x) are the monomial terms of x. Given a point a ∈ Rn, let f {a} denote the list of positive reals [ |m1(Log −1(a)) |,..., |ml(Log −1(a)) |]. Note this is well defined, even though Log is not injective. We say that a list of positive numbers satisfies the polygon condition if it is possible to make a polygon with those side lengths, i.e. no number is greater than the sum of all the others.\n\nTheorem 1. Let I be an ideal, and A(I) its amoeba. Then a ∈ A(I) if and only if\n\nf {a} satisfies the polygon condition for all f ∈ I.Let P (f ) = {a ∈ Rn: f {a} satisfies the polygon condition }. Think of this as an approximation to the amoeba of a hypersurface.\n\nTheorem 2. Let A(f ) be the amoeba of a hypersurface. Let\n\nfm(x1,..., x n) = the product of f (u1x1,..., u nxn)10\n\nover all ui such that umi = 1. The family P (fm) converges uniformly (in the Euclidean norm) to A(f ). Questions: A. Is there a version of theorem 2 (an explicit family approximating the amoeba) in the higher codimension case? B. An analogous statement to theorem 1 is known for non-archimedean Amoebas. Is theorem 1 true in an even more general context? C. The convergence of the family in theorem 2 is of order O(log m/m ), at least in worst case situations. How fast does this family converge for a randomly chosen f? If the approximation is within ( a log m + b)/m of the actual amoeba, what are a and b, in typical examples? D. What open problems can this be used to solve? (contributed by Kevin Purbhoo)",
   "public_statement": "A.16 Nullstellensatz for amoebas\n\nLet f (x1,..., x n) be a Laurent polynomial, and write f (x1,..., x n) = ∑ln=1 mi(x), where mi(x) are the monomial terms of x. Given a point a ∈ Rn, let f {a} denote the list of positive reals [ |m1(Log −1(a)) |,..., |ml(Log −1(a)) |]. Note this is well defined, even though Log is not injective. We say that a list of positive numbers satisfies the polygon condition if it is possible to make a polygon with those side lengths, i.e. no number is greater than the sum of all the others.\n\nTheorem 1. Let I be an ideal, and A(I) its amoeba. Then a ∈ A(I) if and only if\n\nf {a} satisfies the polygon condition for all f ∈ I.Let P (f ) = {a ∈ Rn: f {a} satisfies the polygon condition }. Think of this as an approximation to the amoeba of a hypersurface.\n\nTheorem 2. Let A(f ) be the amoeba of a hypersurface. Let\n\nfm(x1,..., x n) = the product of f (u1x1,..., u nxn)10\n\nover all ui such that umi = 1. The family P (fm) converges uniformly (in the Euclidean norm) to A(f ). Questions: A. Is there a version of theorem 2 (an explicit family approximating the amoeba) in the higher codimension case? B. An analogous statement to theorem 1 is known for non-archimedean Amoebas. Is theorem 1 true in an even more general context? C. The convergence of the family in theorem 2 is of order O(log m/m ), at least in worst case situations. How fast does this family converge for a randomly chosen f? If the approximation is within ( a log m + b)/m of the actual amoeba, what are a and b, in typical examples? D. What open problems can this be used to solve? (contributed by Kevin Purbhoo)",
   "evidence": "Item A.16, “Nullstellensatz for amoebas,” was contributed by Kevin Purbhoo to the AIM list *Amoebas and tropical geometry*. The repository text has several extraction errors. Comparing the AIM PDF and HTML entry with Purbhoo’s primary paper gives the following recovered statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 335,
   "attempt": 1
  },
  "AIM-GEOMETRY-0337": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A.17 Tropical Calabi-Yau structures \n\nAn example of a tropical Calabi-Yau is the base of a Lagrangian fibered K3 surface. This is a sphere with, generically, an affine structure A on the complement of 24 points where the singularity at each point has a structure specified by two features: A. The monodromy in the affine structure A along a simple loop around a singular point is conjugate to ( 1 10 1\n\n)\n\nand B. there is an injective map Φ: ( U − R, A) → (R2, A0) where U is a neighborhood of the singularity and R is a ray based at the singular point. (Here the map Φ is assumed to be a local isomorphism of the affine structures A and A0.) The injectivity follows from an argument involving three-dimensional contact geometry. A natural question is what closed surfaces admit such a singular affine structure, and how many singular points there can be on such a surface. In fact, the possibilities are: a torus or Klein bottle with no singular points, a sphere with 24 singular points, or an RP 2\n\nwith 12 singular points. Each one can be realized as the base of a (singular) Lagrangian fibration. The singular fibers in each are diffeomorphic to the singular fibers in a genus one Lefschetz fibration, i.e. they are spheres with one positive self-intersection. Question: What can one say about the geometry or topology of the set of tropical Calabi-Yau structures on S2?",
   "clean_statement": null,
   "public_statement": "A.17 Tropical Calabi-Yau structures\n\nAn example of a tropical Calabi-Yau is the base of a Lagrangian fibered K3 surface. This is a sphere with, generically, an affine structure A on the complement of 24 points where the singularity at each point has a structure specified by two features: A. The monodromy in the affine structure A along a simple loop around a singular point is conjugate to ( 1 10 1\n\n)\n\nand B. there is an injective map Φ: ( U − R, A) → (R2, A0) where U is a neighborhood of the singularity and R is a ray based at the singular point. (Here the map Φ is assumed to be a local isomorphism of the affine structures A and A0.) The injectivity follows from an argument involving three-dimensional contact geometry. A natural question is what closed surfaces admit such a singular affine structure, and how many singular points there can be on such a surface. In fact, the possibilities are: a torus or Klein bottle with no singular points, a sphere with 24 singular points, or an RP 2\n\nwith 12 singular points. Each one can be realized as the base of a (singular) Lagrangian fibration. The singular fibers in each are diffeomorphic to the singular fibers in a genus one Lefschetz fibration, i.e. they are spheres with one positive self-intersection. Question: What can one say about the geometry or topology of the set of tropical Calabi-Yau structures on S2?",
   "evidence": "The canonical record is item A.17, “Tropical Calabi--Yau structures,” in the AIM workshop problem list *Amoebas and tropical geometry*, contributed by Margaret Symington. I checked both the source PDF <https://aimath.org/WWN/amoebas/amoebas.pdf> and AIM's HTML rendering <https://www.aimath.org/WWN/amoebas/articles/html/43a/>.",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 336,
   "attempt": 1
  },
  "AIM-GEOMETRY-0338": {
   "statement_status": "exact",
   "original_statement": "A.18 Contour of an amoeba \n\nFor f ∈ C[x1, x 2], let Cf ⊂ R2 denote the contour of the amoeba of f, i.e., the locus of the critical points of the Gauss map. The singular points V on Cf naturally divides Cf into several arcs E, and thus ( V, E ) defines a planar graph. Question: What combinatorial properties does the graph ( V, E ) have? Which graphs can be realized by some function f?Background: Some examples of the contour can be found e.g., in T. Theobald, Com-puting amoebas, Exp. Math. 11:513-526, 2002, or in M. Passare and A. Tsikh, Amoebas: their spines and their contours, Preprint, 2003. Since tracing the contour can be used to (nu-merically) compute the boundary of the amoeba,understanding the combinatorial properties of the contour helps to compute the boundary of the amoeba. Question: How does this generalize to higher dimension? (contributed by Thorsten Theobald)",
   "clean_statement": "A.18 Contour of an amoeba\n\nFor f ∈ C[x1, x 2], let Cf ⊂ R2 denote the contour of the amoeba of f, i.e., the locus of the critical points of the Gauss map. The singular points V on Cf naturally divides Cf into several arcs E, and thus ( V, E ) defines a planar graph. Question: What combinatorial properties does the graph ( V, E ) have? Which graphs can be realized by some function f?Background: Some examples of the contour can be found e.g., in T. Theobald, Com-puting amoebas, Exp. Math. 11:513-526, 2002, or in M. Passare and A. Tsikh, Amoebas: their spines and their contours, Preprint, 2003. Since tracing the contour can be used to (nu-merically) compute the boundary of the amoeba,understanding the combinatorial properties of the contour helps to compute the boundary of the amoeba. Question: How does this generalize to higher dimension? (contributed by Thorsten Theobald)",
   "public_statement": "A.18 Contour of an amoeba\n\nFor f ∈ C[x1, x 2], let Cf ⊂ R2 denote the contour of the amoeba of f, i.e., the locus of the critical points of the Gauss map. The singular points V on Cf naturally divides Cf into several arcs E, and thus ( V, E ) defines a planar graph. Question: What combinatorial properties does the graph ( V, E ) have? Which graphs can be realized by some function f?Background: Some examples of the contour can be found e.g., in T. Theobald, Com-puting amoebas, Exp. Math. 11:513-526, 2002, or in M. Passare and A. Tsikh, Amoebas: their spines and their contours, Preprint, 2003. Since tracing the contour can be used to (nu-merically) compute the boundary of the amoeba,understanding the combinatorial properties of the contour helps to compute the boundary of the amoeba. Question: How does this generalize to higher dimension? (contributed by Thorsten Theobald)",
   "evidence": "The AIM source (problem A.18 from the workshop *Amoebas and tropical geometry*) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 337,
   "attempt": 1
  },
  "AIM-GEOMETRY-0339": {
   "statement_status": "exact",
   "original_statement": "A.19 Tropical bases \n\nLet I ⊂ C[x1,..., x n] be an ideal. The problem is to characterize/compute subsets J\n\nof I which suffice to define the tropical variety T (I), i.e. T (I) = ∩j∈J T (j). \n\nTheorem. The 3 × 3-minors of an n × n-matrix of indeterminates (which are not a not a universal Gr¨ obner basis) suffice to define the tropical variety of that ideal. Question (Sturmfels): Do the 4 × 4-minors of a 5 × 5-matrix of indeterminates (which are far from a universal Gr¨ obner basis) suffice to define the tropical variety? Question: Find a characterization of a (smaller) sufficient set (which should be eas-ier/better to compute). (contributed by Rekha Thomas)",
   "clean_statement": "A.19 Tropical bases\n\nLet I ⊂ C[x1,..., x n] be an ideal. The problem is to characterize/compute subsets J\n\nof I which suffice to define the tropical variety T (I), i.e. T (I) = ∩j∈J T (j).\n\nTheorem. The 3 × 3-minors of an n × n-matrix of indeterminates (which are not a not a universal Gr¨ obner basis) suffice to define the tropical variety of that ideal. Question (Sturmfels): Do the 4 × 4-minors of a 5 × 5-matrix of indeterminates (which are far from a universal Gr¨ obner basis) suffice to define the tropical variety? Question: Find a characterization of a (smaller) sufficient set (which should be eas-ier/better to compute). (contributed by Rekha Thomas)",
   "public_statement": "A.19 Tropical bases\n\nLet I ⊂ C[x1,..., x n] be an ideal. The problem is to characterize/compute subsets J\n\nof I which suffice to define the tropical variety T (I), i.e. T (I) = ∩j∈J T (j).\n\nTheorem. The 3 × 3-minors of an n × n-matrix of indeterminates (which are not a not a universal Gr¨ obner basis) suffice to define the tropical variety of that ideal. Question (Sturmfels): Do the 4 × 4-minors of a 5 × 5-matrix of indeterminates (which are far from a universal Gr¨ obner basis) suffice to define the tropical variety? Question: Find a characterization of a (smaller) sufficient set (which should be eas-ier/better to compute). (contributed by Rekha Thomas)",
   "evidence": "The stored AIM record is a section-level record headed “A.19 Tropical bases.” Its mathematical content is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 338,
   "attempt": 1
  },
  "AIM-GEOMETRY-0340": {
   "statement_status": "exact",
   "original_statement": "A.20 Real enumerative invariants \n\nIn the lecture I gave at the AIM workshop on Amoebas and tropical geometry, I defined some enumerative invariants of real algebraic convex 3-manifolds. For example, through a generic configuration of 2 d real points in the complex projective space, there passes only finitely many irreducible real rational curves of degree d. Their real parts provide a collection of embedded knots in RP 3. Equip this real projective space with a spin structure. Then it is possible to define a spinor orientation on these knots. Indeed, considering a real subholomorphic line bundle of maximal degree in the normal bundle of the curves, one first defines a framing on these knots. From this framing, one can then build a loop in the \n\nSO 3(R)-principal bundle of orthonormal frames of RP 3. Then, the spinor orientation of the real curve is the obstruction to lift this loop as a loop of the Spin 3-principal bundle given by 12 \n\nthe spin structure. Now the algebraic number of real curves, counted with respect to their spinor orientation, turns out to be independent of the choice of the configuration of points and this is my invariant. Question: Is it possible to compute this invariant with the help of tropical algebraic geometry? (contributed by Jean-Yves Welschinger)",
   "clean_statement": "A.20 Real enumerative invariants\n\nIn the lecture I gave at the AIM workshop on Amoebas and tropical geometry, I defined some enumerative invariants of real algebraic convex 3-manifolds. For example, through a generic configuration of 2 d real points in the complex projective space, there passes only finitely many irreducible real rational curves of degree d. Their real parts provide a collection of embedded knots in RP 3. Equip this real projective space with a spin structure. Then it is possible to define a spinor orientation on these knots. Indeed, considering a real subholomorphic line bundle of maximal degree in the normal bundle of the curves, one first defines a framing on these knots. From this framing, one can then build a loop in the\n\nSO 3(R)-principal bundle of orthonormal frames of RP 3. Then, the spinor orientation of the real curve is the obstruction to lift this loop as a loop of the Spin 3-principal bundle given by 12\n\nthe spin structure. Now the algebraic number of real curves, counted with respect to their spinor orientation, turns out to be independent of the choice of the configuration of points and this is my invariant. Question: Is it possible to compute this invariant with the help of tropical algebraic geometry? (contributed by Jean-Yves Welschinger)",
   "public_statement": "A.20 Real enumerative invariants\n\nIn the lecture I gave at the AIM workshop on Amoebas and tropical geometry, I defined some enumerative invariants of real algebraic convex 3-manifolds. For example, through a generic configuration of 2 d real points in the complex projective space, there passes only finitely many irreducible real rational curves of degree d. Their real parts provide a collection of embedded knots in RP 3. Equip this real projective space with a spin structure. Then it is possible to define a spinor orientation on these knots. Indeed, considering a real subholomorphic line bundle of maximal degree in the normal bundle of the curves, one first defines a framing on these knots. From this framing, one can then build a loop in the\n\nSO 3(R)-principal bundle of orthonormal frames of RP 3. Then, the spinor orientation of the real curve is the obstruction to lift this loop as a loop of the Spin 3-principal bundle given by 12\n\nthe spin structure. Now the algebraic number of real curves, counted with respect to their spinor orientation, turns out to be independent of the choice of the configuration of points and this is my invariant. Question: Is it possible to compute this invariant with the help of tropical algebraic geometry? (contributed by Jean-Yves Welschinger)",
   "evidence": "This is item A.20, “Real enumerative invariants,” contributed by Jean-Yves Welschinger to the 2003 AIM workshop *Amoebas and tropical geometry*. The exact canonical extraction is preserved in `input.json`. Comparison with the AIM PDF and HTML entry and with Welschinger’s primary paper gives the following recovered statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 339,
   "attempt": 1
  },
  "AIM-GEOMETRY-0341": {
   "statement_status": "exact",
   "original_statement": "A.21 Positive tropical varieties and cluster algebras \n\nQuestion: What is the connection between the tropicalization of the totally positive part of a variety, and the cluster algebra structure of the variety? Background: In joint work with David Speyer, we have described the tropicalization of the totally positive part of the Grassmannian G(k, n ). When k = 2, we get a fan which is closely related to the type A associahedron. For G(3, 6) and G(3, 7), we get fans which are related to the type D4 and type E6 associahedra. Our results seem to be related to the results of Joshua Scott, who showed that the cluster algebra structure of the Grassmannians \n\nG(2, n ), G(3, 6), and G(3, 7) are of types A, D4, and E6, respectively. (contributed by Lauren Williams)",
   "clean_statement": "A.21 Positive tropical varieties and cluster algebras\n\nQuestion: What is the connection between the tropicalization of the totally positive part of a variety, and the cluster algebra structure of the variety? Background: In joint work with David Speyer, we have described the tropicalization of the totally positive part of the Grassmannian G(k, n ). When k = 2, we get a fan which is closely related to the type A associahedron. For G(3, 6) and G(3, 7), we get fans which are related to the type D4 and type E6 associahedra. Our results seem to be related to the results of Joshua Scott, who showed that the cluster algebra structure of the Grassmannians\n\nG(2, n ), G(3, 6), and G(3, 7) are of types A, D4, and E6, respectively. (contributed by Lauren Williams)",
   "public_statement": "A.21 Positive tropical varieties and cluster algebras\n\nQuestion: What is the connection between the tropicalization of the totally positive part of a variety, and the cluster algebra structure of the variety? Background: In joint work with David Speyer, we have described the tropicalization of the totally positive part of the Grassmannian G(k, n ). When k = 2, we get a fan which is closely related to the type A associahedron. For G(3, 6) and G(3, 7), we get fans which are related to the type D4 and type E6 associahedra. Our results seem to be related to the results of Joshua Scott, who showed that the cluster algebra structure of the Grassmannians\n\nG(2, n ), G(3, 6), and G(3, 7) are of types A, D4, and E6, respectively. (contributed by Lauren Williams)",
   "evidence": "The exact AIM HTML page, the workshop PDF, and the stored record in `input.json` were compared. The recovered question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 340,
   "attempt": 1
  },
  "AIM-GEOMETRY-0342": {
   "statement_status": "exact",
   "original_statement": "A.22 Statistical algebraic geometry \n\nBackground: In statistical algebraic geometry, we put a Gaussian probability measure on the space of polynomials of degree N in m real or complex variables. For simplicity, we think mainly of the U (m + 1)-invariant Gaussian measure in the complex case and the \n\nO(m + 1) invariant measure in the real case. We then consider probabilities and expected values for interesting random variables. The real and complex cases are quite different, since deterministic problems in the complex case can become random in the real case. Questions for real algebraic plane curves: \n\n• Consider the ensemble of plane algebraic curves of degree N. Let the random variable be: the number of components of the curve. What is the most probable number of connected components? What is the expected number? \n\n• Consider random spherical harmonics of degree N. Let the random variable be the number of nodal domains (i.e. components of the complement of the zero set of the harmonic). What is the most probable number of nodal domains? The expected number? \n\nRandom real fewnomials. We fix a number f. In dimension m, we select m real fewnomials of degree N at random, each with at most f monomials. We pick the spectrum of each fewnomial at random ( f lattice points in Zm \n\n> +\n\n∩ N Σ). We then pick the coefficients of these fewnomials at random from the O(m + 1) ensemble. The problem is: \n\nQuestion: What is the expected number of real zeros of a random fewnomial system of degree N with f monomials in each fewnomial? The current bound, due to Khovanski, is #real zeros ≤ 2m2f (f −1) /2(m + 1) f.\n\nIt is believed to be an enormous over-estimate. 13 \n\nZeros of random real fewnomials with fixed Newton polytope. We now pick m random fewnomials p1,..., p m with prescribed Newton polytopes ∆ 1,..., ∆m and fixed fewnomial number f. How does the number of simultaneous zeros behave as the polytopes are dilated, ∆j → N ∆j. I.e. we increase the degrees, but keep the fewnomial number f fixed and keep the spectra in the dilates of the polytopes. \n\nZeros of random real Kac fewnomials. We ask the same questions but define random real fewnomial as ∑ \n\n> α\n\ncαxα where cα are normal. That is, we do not use projective space to define norms of monomials. [The number of real zeros then goes way down.] Current result: Shiffman and I currently have an exact formula for the expected number of real zeros of random fewnomial ensembles, but we have not yet found its asymptotics. \n\nProposition. The density KNf (x) of real zeros of random f -fewnomial systems of degree N is given by: \n\nKNf (x) = 1 \n\n> πknm\n> Q\n> |S (N,f )|\n\n∑ \n\n> S∈S (N,f )\n\n√det ∇x∇y log Π N |S (x,y )|x=y\n\n> [\n\n√ΠN |S (x,x )] m,\n\nwhere Q:= ∫ \n\n> Rm\n\n|ξ| exp ( −〈 ξ, ξ 〉) dξ, and where Π N |S is the Szeg¨ o kernel for the spectrum S\n\nΠN |S (x, y ) = ∑\n\n> β∈S\n\n(Nβ\n\n)\n\nxβ yβ.\n\nHere, S(N, f ) is the set of possible spectra. \n\nCritical points of holomorphic sections. Critical points of holomorphic functions have a long history (Picard-Lefschetz, Milnor-Orlik, Arnold, etc.). Critical points of holomorphic sections have arisen recently in string theory, where they are 'supersymmetric vacua'. Mike Douglas has posed the problem of counting them, finding how they are distributed, and many other statistic problems relevant in string/M theory. Critical points of a holomorphic section s ∈ H0(M, L ) of a holomorphic line bundle depend on a choice of Hermitian metric h or connection ∇, which we usually pick to be the metric connection. The equation reads ∇s(z) = 0 and hence the number of critical points depends on the connection or metric. (contributed by Steve Zelditch)",
   "clean_statement": "A.22 Statistical algebraic geometry\n\nBackground: In statistical algebraic geometry, we put a Gaussian probability measure on the space of polynomials of degree N in m real or complex variables. For simplicity, we think mainly of the U (m + 1)-invariant Gaussian measure in the complex case and the\n\nO(m + 1) invariant measure in the real case. We then consider probabilities and expected values for interesting random variables. The real and complex cases are quite different, since deterministic problems in the complex case can become random in the real case. Questions for real algebraic plane curves:\n\n• Consider the ensemble of plane algebraic curves of degree N. Let the random variable be: the number of components of the curve. What is the most probable number of connected components? What is the expected number?\n\n• Consider random spherical harmonics of degree N. Let the random variable be the number of nodal domains (i.e. components of the complement of the zero set of the harmonic). What is the most probable number of nodal domains? The expected number?\n\nRandom real fewnomials. We fix a number f. In dimension m, we select m real fewnomials of degree N at random, each with at most f monomials. We pick the spectrum of each fewnomial at random ( f lattice points in Zm\n\n> +\n\n∩ N Σ). We then pick the coefficients of these fewnomials at random from the O(m + 1) ensemble. The problem is:\n\nQuestion: What is the expected number of real zeros of a random fewnomial system of degree N with f monomials in each fewnomial? The current bound, due to Khovanski, is #real zeros ≤ 2m2f (f −1) /2(m + 1) f.\n\nIt is believed to be an enormous over-estimate. 13\n\nZeros of random real fewnomials with fixed Newton polytope. We now pick m random fewnomials p1,..., p m with prescribed Newton polytopes ∆ 1,..., ∆m and fixed fewnomial number f. How does the number of simultaneous zeros behave as the polytopes are dilated, ∆j → N ∆j. I.e. we increase the degrees, but keep the fewnomial number f fixed and keep the spectra in the dilates of the polytopes.\n\nZeros of random real Kac fewnomials. We ask the same questions but define random real fewnomial as ∑\n\n> α\n\ncαxα where cα are normal. That is, we do not use projective space to define norms of monomials. [The number of real zeros then goes way down.] Current result: Shiffman and I currently have an exact formula for the expected number of real zeros of random fewnomial ensembles, but we have not yet found its asymptotics.\n\nProposition. The density KNf (x) of real zeros of random f -fewnomial systems of degree N is given by:\n\nKNf (x) = 1\n\n> πknm\n> Q\n> |S (N,f )|\n\n∑\n\n> S∈S (N,f )\n\n√det ∇x∇y log Π N |S (x,y )|x=y\n\n> [\n\n√ΠN |S (x,x )] m,\n\nwhere Q:= ∫\n\n> Rm\n\n|ξ| exp ( −〈 ξ, ξ 〉) dξ, and where Π N |S is the Szeg¨ o kernel for the spectrum S\n\nΠN |S (x, y ) = ∑\n\n> β∈S\n\n(Nβ\n\n)\n\nxβ yβ.\n\nHere, S(N, f ) is the set of possible spectra.\n\nCritical points of holomorphic sections. Critical points of holomorphic functions have a long history (Picard-Lefschetz, Milnor-Orlik, Arnold, etc.). Critical points of holomorphic sections have arisen recently in string theory, where they are 'supersymmetric vacua'. Mike Douglas has posed the problem of counting them, finding how they are distributed, and many other statistic problems relevant in string/M theory. Critical points of a holomorphic section s ∈ H0(M, L ) of a holomorphic line bundle depend on a choice of Hermitian metric h or connection ∇, which we usually pick to be the metric connection. The equation reads ∇s(z) = 0 and hence the number of critical points depends on the connection or metric. (contributed by Steve Zelditch)",
   "public_statement": "A.22 Statistical algebraic geometry\n\nBackground: In statistical algebraic geometry, we put a Gaussian probability measure on the space of polynomials of degree N in m real or complex variables. For simplicity, we think mainly of the U (m + 1)-invariant Gaussian measure in the complex case and the\n\nO(m + 1) invariant measure in the real case. We then consider probabilities and expected values for interesting random variables. The real and complex cases are quite different, since deterministic problems in the complex case can become random in the real case. Questions for real algebraic plane curves:\n\n• Consider the ensemble of plane algebraic curves of degree N. Let the random variable be: the number of components of the curve. What is the most probable number of connected components? What is the expected number?\n\n• Consider random spherical harmonics of degree N. Let the random variable be the number of nodal domains (i.e. components of the complement of the zero set of the harmonic). What is the most probable number of nodal domains? The expected number?\n\nRandom real fewnomials. We fix a number f. In dimension m, we select m real fewnomials of degree N at random, each with at most f monomials. We pick the spectrum of each fewnomial at random ( f lattice points in Zm\n\n> +\n\n∩ N Σ). We then pick the coefficients of these fewnomials at random from the O(m + 1) ensemble. The problem is:\n\nQuestion: What is the expected number of real zeros of a random fewnomial system of degree N with f monomials in each fewnomial? The current bound, due to Khovanski, is #real zeros ≤ 2m2f (f −1) /2(m + 1) f.\n\nIt is believed to be an enormous over-estimate. 13\n\nZeros of random real fewnomials with fixed Newton polytope. We now pick m random fewnomials p1,..., p m with prescribed Newton polytopes ∆ 1,..., ∆m and fixed fewnomial number f. How does the number of simultaneous zeros behave as the polytopes are dilated, ∆j → N ∆j. I.e. we increase the degrees, but keep the fewnomial number f fixed and keep the spectra in the dilates of the polytopes.\n\nZeros of random real Kac fewnomials. We ask the same questions but define random real fewnomial as ∑\n\n> α\n\ncαxα where cα are normal. That is, we do not use projective space to define norms of monomials. [The number of real zeros then goes way down.] Current result: Shiffman and I currently have an exact formula for the expected number of real zeros of random fewnomial ensembles, but we have not yet found its asymptotics.\n\nProposition. The density KNf (x) of real zeros of random f -fewnomial systems of degree N is given by:\n\nKNf (x) = 1\n\n> πknm\n> Q\n> |S (N,f )|\n\n∑\n\n> S∈S (N,f )\n\n√det ∇x∇y log Π N |S (x,y )|x=y\n\n> [\n\n√ΠN |S (x,x )] m,\n\nwhere Q:= ∫\n\n> Rm\n\n|ξ| exp ( −〈 ξ, ξ 〉) dξ, and where Π N |S is the Szeg¨ o kernel for the spectrum S\n\nΠN |S (x, y ) = ∑\n\n> β∈S\n\n(Nβ\n\n)\n\nxβ yβ.\n\nHere, S(N, f ) is the set of possible spectra.\n\nCritical points of holomorphic sections. Critical points of holomorphic functions have a long history (Picard-Lefschetz, Milnor-Orlik, Arnold, etc.). Critical points of holomorphic sections have arisen recently in string theory, where they are 'supersymmetric vacua'. Mike Douglas has posed the problem of counting them, finding how they are distributed, and many other statistic problems relevant in string/M theory. Critical points of a holomorphic section s ∈ H0(M, L ) of a holomorphic line bundle depend on a choice of Hermitian metric h or connection ∇, which we usually pick to be the metric connection. The equation reads ∇s(z) = 0 and hence the number of critical points depends on the connection or metric. (contributed by Steve Zelditch)",
   "evidence": "This record is Section A.22, “Statistical algebraic geometry,” contributed by Steve Zelditch to the AIM workshop list *Amoebas and Tropical Geometry*. It is a collection of four related research questions rather than a single assertion.",
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   "original_statement": "A.23 Compact tropical varieties, Monge-Amp` ere equation, Calabi conjecture and curve counting \n\nBackground: We adapt Gross's definition of tropical Calabi-Yau manifolds as well as notations (see his contribution on tropical Calabi-Yau manifolds and tropical line bundles). \n\nCompact tropical varieties. The natural question is how to make sense of compact tropical manifolds, not necessarily Calabi-Yau. There has to be a procedure of deleting pseudo-pods and leaving as much of affine structure as possible. My guess is that this will require a choice of polarization (tropical K¨ ahler class). But the affine structure should not depend on this choice and has to be of purely algebro-geometric nature. Question: How to modify naturally the valuation map for compact tropical varieties? \n\nTropical Monge-Amp` ere equation. Let me also add to Gross's question on Monge-Amp` ere equation. The beauty and importance of the real (and complex) MA equation is 14 \n\nthat given a boundary conditions it has a unique solution. This is crucial in proving various Calabi conjectures. In the differential-geometric picture a metric is determined locally in an affine chart \n\nU ⊂ B0 by the graph of the differential of a potential dK ⊂ Rn × (Rn)∗ (we can consistently identify tangent spaces at different points in U with Rn and cotangent spaces - with ( Rn)∗). In the tropical world the C∞ graphs should be replaced by piece-wise linear ones, which will define distribution-like metrics (or measures) on B. The Monge-Amp` ere condition - the equality of euclidean measures on Rn and ( Rn)∗ provided by the graph dK - has to be understood in this distributional sense as well. Question: (Tropical Calabi conjecture) Is there a way to define a tropical Monge-Amp` ere operator whose solution gives a (unique?) Monge-Amp` ere measure on B in a given polarization class in H1(B, Aff( B, R))? The uniqueness seems to be false in an obvious assumption that the bendings of the potentials are regulated by the integral lattice. On a torus this corresponds to several possible Voronoi cell decompositions in dimension higher than one. \n\nCurve counting. The symplectic area of a straight line interval is easily seen to be proportional to the scalar product of the primitive vector along the interval and the distance vector between the end points in the dual structure. The behavior of a curve near the singular locus is very restrictive. Namely, it can end on a singular point only coming from a unique (eigen) direction. Question: Given this can we perform a tropical Gromov-Witten calculation on a Calabi-Yau? As was shown by Mikhalkin the tropical invariants coincide with the genuine ones for curves in surfaces. In higher dimensions, however, there are tropical curves which are not limits of true holomorphic curves. Question: Is there a simple recipe deciding which tropical curves are the limits of classical ones? (contributed by Ilia Zharkov) \n\nChapter B: Snapshot of the pre-open problem session \n\nRelevant aspects: \n\nAmoebas \n\n• maximally sparse polynomials \n\n• amoebas for fewnomials \n\n• spine \n\n• for general varieties (non-hypersurfaces): what are spine, solid, maximally sparse? \n\n• topological structure of amoebas; convexity \n\n• for specific classes of varieties? \n\n• discriminants and amoebas 15 \n\nTropical geometry \n\n• tropical linear algebra \n\n• line bundles and vector bundles \n\n• variations of tropical varieties \n\nAmoebas vs. tropical geometry \n\n• What is gained or lost in the transition? \n\nChapter C: Snapshot of the open problem session \n\nThe workshop included a moderated, open-problem discussion session.",
   "clean_statement": "A.23 Compact tropical varieties, Monge-Amp` ere equation, Calabi conjecture and curve counting\n\nBackground: We adapt Gross's definition of tropical Calabi-Yau manifolds as well as notations (see his contribution on tropical Calabi-Yau manifolds and tropical line bundles).\n\nCompact tropical varieties. The natural question is how to make sense of compact tropical manifolds, not necessarily Calabi-Yau. There has to be a procedure of Monge-Amp and leaving as much of affine structure as possible. My guess is that this will require a choice of polarization (tropical K¨ ahler class). But the affine structure should not depend on this choice and has to be of purely algebro-geometric nature. Question: How to modify naturally the valuation map for compact tropical varieties?\n\nTropical Monge-Amp` ere equation. Let me also add to Gross's question on Monge-Amp` ere equation. The beauty and importance of the real (and complex) MA equation is 14\n\nthat given a boundary conditions it has a unique solution. This is crucial in proving various Calabi conjectures. In the differential-geometric picture a metric is determined locally in an affine chart\n\nU ⊂ B0 by the graph of the differential of a potential dK ⊂ Rn × (Rn)∗ (we can consistently identify tangent spaces at different points in U with Rn and cotangent spaces - with ( Rn)∗). In the tropical world the C∞ graphs should be replaced by piece-wise linear ones, which will define distribution-like metrics (or measures) on B. The Monge-Amp` ere condition - the equality of euclidean measures on Rn and ( Rn)∗ provided by the graph dK - has to be understood in this distributional sense as well. Question: (Tropical Calabi conjecture) Is there a way to define a tropical Monge-Amp` ere operator whose solution gives a (unique?) Monge-Amp` ere measure on B in a given polarization class in H1(B, Aff( B, R))? The uniqueness seems to be false in an obvious assumption that the bendings of the potentials are regulated by the integral lattice. On a torus this corresponds to several possible Voronoi cell decompositions in dimension higher than one.\n\nCurve counting. The symplectic area of a straight line interval is easily seen to be proportional to the scalar product of the primitive vector along the interval and the distance vector between the end points in the dual structure. The behavior of a curve near the singular locus is very restrictive. Namely, it can end on a singular point only coming from a unique (eigen) direction. Question: Given this can we perform a tropical Gromov-Witten calculation on a Calabi-Yau? As was shown by Mikhalkin the tropical invariants coincide with the genuine ones for curves in surfaces. In higher dimensions, however, there are tropical curves which are not limits of true holomorphic curves. Question: Is there a simple recipe deciding which tropical curves are the limits of classical ones? (contributed by Ilia Zharkov)\n\nChapter B: Snapshot of the pre-open problem session\n\nRelevant aspects:\n\nAmoebas\n\n• maximally sparse polynomials\n\n• amoebas for fewnomials\n\n• spine\n\n• for general varieties (non-hypersurfaces): what are spine, solid, maximally sparse?\n\n• topological structure of amoebas; convexity\n\n• for specific classes of varieties?\n\n• discriminants and amoebas 15\n\nTropical geometry\n\n• tropical linear algebra\n\n• line bundles and vector bundles\n\n• variations of tropical varieties\n\nAmoebas vs. tropical geometry\n\n• What is gained or lost in the transition?\n\nChapter C: Snapshot of the open problem session\n\nThe workshop included a moderated, open-problem discussion session.",
   "public_statement": "A.23 Compact tropical varieties, Monge-Amp` ere equation, Calabi conjecture and curve counting\n\nBackground: We adapt Gross's definition of tropical Calabi-Yau manifolds as well as notations (see his contribution on tropical Calabi-Yau manifolds and tropical line bundles).\n\nCompact tropical varieties. The natural question is how to make sense of compact tropical manifolds, not necessarily Calabi-Yau. There has to be a procedure of Monge-Amp and leaving as much of affine structure as possible. My guess is that this will require a choice of polarization (tropical K¨ ahler class). But the affine structure should not depend on this choice and has to be of purely algebro-geometric nature. Question: How to modify naturally the valuation map for compact tropical varieties?\n\nTropical Monge-Amp` ere equation. Let me also add to Gross's question on Monge-Amp` ere equation. The beauty and importance of the real (and complex) MA equation is 14\n\nthat given a boundary conditions it has a unique solution. This is crucial in proving various Calabi conjectures. In the differential-geometric picture a metric is determined locally in an affine chart\n\nU ⊂ B0 by the graph of the differential of a potential dK ⊂ Rn × (Rn)∗ (we can consistently identify tangent spaces at different points in U with Rn and cotangent spaces - with ( Rn)∗). In the tropical world the C∞ graphs should be replaced by piece-wise linear ones, which will define distribution-like metrics (or measures) on B. The Monge-Amp` ere condition - the equality of euclidean measures on Rn and ( Rn)∗ provided by the graph dK - has to be understood in this distributional sense as well. Question: (Tropical Calabi conjecture) Is there a way to define a tropical Monge-Amp` ere operator whose solution gives a (unique?) Monge-Amp` ere measure on B in a given polarization class in H1(B, Aff( B, R))? The uniqueness seems to be false in an obvious assumption that the bendings of the potentials are regulated by the integral lattice. On a torus this corresponds to several possible Voronoi cell decompositions in dimension higher than one.\n\nCurve counting. The symplectic area of a straight line interval is easily seen to be proportional to the scalar product of the primitive vector along the interval and the distance vector between the end points in the dual structure. The behavior of a curve near the singular locus is very restrictive. Namely, it can end on a singular point only coming from a unique (eigen) direction. Question: Given this can we perform a tropical Gromov-Witten calculation on a Calabi-Yau? As was shown by Mikhalkin the tropical invariants coincide with the genuine ones for curves in surfaces. In higher dimensions, however, there are tropical curves which are not limits of true holomorphic curves. Question: Is there a simple recipe deciding which tropical curves are the limits of classical ones? (contributed by Ilia Zharkov)\n\nChapter B: Snapshot of the pre-open problem session\n\nRelevant aspects:\n\nAmoebas\n\n• maximally sparse polynomials\n\n• amoebas for fewnomials\n\n• spine\n\n• for general varieties (non-hypersurfaces): what are spine, solid, maximally sparse?\n\n• topological structure of amoebas; convexity\n\n• for specific classes of varieties?\n\n• discriminants and amoebas 15\n\nTropical geometry\n\n• tropical linear algebra\n\n• line bundles and vector bundles\n\n• variations of tropical varieties\n\nAmoebas vs. tropical geometry\n\n• What is gained or lost in the transition?\n\nChapter C: Snapshot of the open problem session\n\nThe workshop included a moderated, open-problem discussion session.",
   "evidence": "This record is item A.23, contributed by Ilia Zharkov, in the AIM workshop notes *Amoebas and tropical geometry*. The canonical JSON has several encoding defects and also appends the beginning of Chapters B and C. Comparison with the source PDF shows that “Chapter B: Snapshot of the pre-open problem session” begins a new chapter and is not part of A.23. The present job therefore owns only A.23, not the trailing chapter material. With typography repaired but wording otherwise preserved, A.23 asks three groups of questions.",
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  "AIM-GEOMETRY-0344": {
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   "original_statement": "C.1 Relevant lines of research \n\n• Basic definitions \n\n• Computational issues \n\n• Amoebas of higher codimension \n\n• Families of examples \n\n• Applications of abstract data types in tropical and idempotent calculus \n\n• Recognition problems \n\n• Applications to complex algebraic geometry \n\n• Applications to real algebraic geometry \n\n• Applications to dynamical systems \n\n• Applications to differential equations \n\n• Applications to optimization and control theory \n\n• Applications to representation theory \n\n• Applications to number theory \n\n• Applications to statistical mechanics \n\n• Tropical representation theory",
   "clean_statement": "C.1 Relevant lines of research\n\n• Basic definitions\n\n• Computational issues\n\n• Amoebas of higher codimension\n\n• Families of examples\n\n• Applications of abstract data types in tropical and idempotent calculus\n\n• Recognition problems\n\n• Applications to complex algebraic geometry\n\n• Applications to real algebraic geometry\n\n• Applications to dynamical systems\n\n• Applications to differential equations\n\n• Applications to optimization and control theory\n\n• Applications to representation theory\n\n• Applications to number theory\n\n• Applications to statistical mechanics\n\n• Tropical representation theory",
   "public_statement": "C.1 Relevant lines of research\n\n• Basic definitions\n\n• Computational issues\n\n• Amoebas of higher codimension\n\n• Families of examples\n\n• Applications of abstract data types in tropical and idempotent calculus\n\n• Recognition problems\n\n• Applications to complex algebraic geometry\n\n• Applications to real algebraic geometry\n\n• Applications to dynamical systems\n\n• Applications to differential equations\n\n• Applications to optimization and control theory\n\n• Applications to representation theory\n\n• Applications to number theory\n\n• Applications to statistical mechanics\n\n• Tropical representation theory",
   "evidence": "The canonical record is C.1 of the American Institute of Mathematics problem list *Amoebas and tropical geometry*, produced from the AIM workshop of October 23--26, 2003. The source is <https://aimath.org/WWN/amoebas/amoebas.pdf>. The canonical metadata are `source_file = aim-geometry-notes.json`, zero-based `source_index = 343`, and `tag = section`.",
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  "AIM-GEOMETRY-0345": {
   "statement_status": "exact",
   "original_statement": "C.2 Basic definitions \n\nA. What is an abstract tropical variety (without embedding, as ringed spaces, rigid analytical space, non-archimedean field given by a covering of charts)? What is a good family of local models? Gluing maps. B. What is an abstract idempotent variety? \n\nDetailed discussion (moderated and contributed by Margaret Symington; see also the figure of the white board of that session): \n\nIn the subsequent discussion of the basic definition of tropical varieties, the workshop participants expressed a desire to lay out a couple of definitions, sorting out the names of different objects arising in the tropical realm. Here is what was proposed (in the lower right hand section of the white board): Given an ideal I in K∗ ⊂ K = C(t) (the Puiseux series), consider two maps, val: K∗ → R∗16 \n\nand (val, phase): K∗ → C∗\n\nwhere the valuation map \"val\" takes the value of the smallest exponent in a Puiseux series and \"phase\" takes the argument of the coefficient of the term with the smallest exponent. Then \n\n• a tropical variety is the image of val, \n\n• a complex tropical variety is the image of (val, phase), and \n\n• a real tropical variety is the subset of a complex tropical variety on which the phase is real-valued. In the p-adic setting (lower center of the white board) one has analogous objects of interest: the image of the valuation map in Q and the image of the valuation and the phase in Q × F ∗ \n\n> p\n\nwhere F ∗ \n\n> p\n\nis the closure of the set of multiplicative generators of the algebraic closure of Fp.More generally (in characteristic zero), let I be an ideal in the Laurent polynomial ring \n\nK[z±11,..., z ±1 \n\n> n\n\n] and let V (I) be its affine variety V (I) ⊂ (K∗)n. Then the corresponding tropical and complex tropical varieties are the images of val and (val, phase) in ( Q∗)n and (C∗)n. The fiber of the projection from such a complex tropical variety to the corresponding tropical variety is a torus, while the fiber of the projection from the real tropical variety is a subset of Zn \n\n> 2.Now view an ideal I in K[z±11,..., z ±1 \n\n> n\n\n] as a family of ideals It in C[z±11,..., z ±1 \n\n> n\n\n]. As \n\nt → 0 the complex algebraic varieties V (It) converge in the Hausdorff limit to the complex tropical variety for I. In general the complex tropical variety is not homeomorphic to V (It)for t 6 = 0. For curves, if the complex tropical variety is smooth, then it is homeomorphic to \n\nV (It) for small t.Question: When is the complex tropical variety homeomorphic or homotopic to the algebraic variety V (It) for small t?Question: Can one define a Hodge theory for complex tropical varieties that is consis-tent with the limit of the classical Hodge theory? Moreover, the following questions and needs were stated: A. Patchworking theorem: Is there a refinement of tropical geometry that will give information about a single (classical) hypersurface? B. Need to define maps, line bundles, vector bundles",
   "clean_statement": "C.2 Basic definitions\n\nA. What is an abstract tropical variety (without embedding, as ringed spaces, rigid analytical space, non-archimedean field given by a covering of charts)? What is a good family of local models? Gluing maps. B. What is an abstract idempotent variety?\n\nDetailed discussion (moderated and contributed by Margaret Symington; see also the figure of the white board of that session):\n\nIn the subsequent discussion of the basic definition of tropical varieties, the workshop participants expressed a desire to lay out a couple of definitions, sorting out the names of different objects arising in the tropical realm. Here is what was proposed (in the lower right hand section of the white board): Given an ideal I in K∗ ⊂ K = C(t) (the Puiseux series), consider two maps, val: K∗ → R∗16\n\nand (val, phase): K∗ → C∗\n\nwhere the valuation map \"val\" takes the value of the smallest exponent in a Puiseux series and \"phase\" takes the argument of the coefficient of the term with the smallest exponent. Then\n\n• a tropical variety is the image of val,\n\n• a complex tropical variety is the image of (val, phase), and\n\n• a real tropical variety is the subset of a complex tropical variety on which the phase is real-valued. In the p-adic setting (lower center of the white board) one has analogous objects of interest: the image of the valuation map in Q and the image of the valuation and the phase in Q × F ∗\n\n> p\n\nwhere F ∗\n\n> p\n\nis the closure of the set of multiplicative generators of the algebraic closure of Fp.More generally (in characteristic zero), let I be an ideal in the Laurent polynomial ring\n\nK[z±11,..., z ±1\n\n> n\n\n] and let V (I) be its affine variety V (I) ⊂ (K∗)n. Then the corresponding tropical and complex tropical varieties are the images of val and (val, phase) in ( Q∗)n and (C∗)n. The fiber of the projection from such a complex tropical variety to the corresponding tropical variety is a torus, while the fiber of the projection from the real tropical variety is a subset of Zn\n\n> 2.Now view an ideal I in K[z±11,..., z ±1\n\n> n\n\n] as a family of ideals It in C[z±11,..., z ±1\n\n> n\n\n]. As\n\nt → 0 the complex algebraic varieties V (It) converge in the Hausdorff limit to the complex tropical variety for I. In general the complex tropical variety is not homeomorphic to V (It)for t 6 = 0. For curves, if the complex tropical variety is smooth, then it is homeomorphic to\n\nV (It) for small t.Question: When is the complex tropical variety homeomorphic or homotopic to the algebraic variety V (It) for small t?Question: Can one define a Hodge theory for complex tropical varieties that is consis-tent with the limit of the classical Hodge theory? Moreover, the following questions and needs were stated: A. Patchworking theorem: Is there a refinement of tropical geometry that will give information about a single (classical) hypersurface? B. Need to define maps, line bundles, vector bundles",
   "public_statement": "C.2 Basic definitions\n\nA. What is an abstract tropical variety (without embedding, as ringed spaces, rigid analytical space, non-archimedean field given by a covering of charts)? What is a good family of local models? Gluing maps. B. What is an abstract idempotent variety?\n\nDetailed discussion (moderated and contributed by Margaret Symington; see also the figure of the white board of that session):\n\nIn the subsequent discussion of the basic definition of tropical varieties, the workshop participants expressed a desire to lay out a couple of definitions, sorting out the names of different objects arising in the tropical realm. Here is what was proposed (in the lower right hand section of the white board): Given an ideal I in K∗ ⊂ K = C(t) (the Puiseux series), consider two maps, val: K∗ → R∗16\n\nand (val, phase): K∗ → C∗\n\nwhere the valuation map \"val\" takes the value of the smallest exponent in a Puiseux series and \"phase\" takes the argument of the coefficient of the term with the smallest exponent. Then\n\n• a tropical variety is the image of val,\n\n• a complex tropical variety is the image of (val, phase), and\n\n• a real tropical variety is the subset of a complex tropical variety on which the phase is real-valued. In the p-adic setting (lower center of the white board) one has analogous objects of interest: the image of the valuation map in Q and the image of the valuation and the phase in Q × F ∗\n\n> p\n\nwhere F ∗\n\n> p\n\nis the closure of the set of multiplicative generators of the algebraic closure of Fp.More generally (in characteristic zero), let I be an ideal in the Laurent polynomial ring\n\nK[z±11,..., z ±1\n\n> n\n\n] and let V (I) be its affine variety V (I) ⊂ (K∗)n. Then the corresponding tropical and complex tropical varieties are the images of val and (val, phase) in ( Q∗)n and (C∗)n. The fiber of the projection from such a complex tropical variety to the corresponding tropical variety is a torus, while the fiber of the projection from the real tropical variety is a subset of Zn\n\n> 2.Now view an ideal I in K[z±11,..., z ±1\n\n> n\n\n] as a family of ideals It in C[z±11,..., z ±1\n\n> n\n\n]. As\n\nt → 0 the complex algebraic varieties V (It) converge in the Hausdorff limit to the complex tropical variety for I. In general the complex tropical variety is not homeomorphic to V (It)for t 6 = 0. For curves, if the complex tropical variety is smooth, then it is homeomorphic to\n\nV (It) for small t.Question: When is the complex tropical variety homeomorphic or homotopic to the algebraic variety V (It) for small t?Question: Can one define a Hodge theory for complex tropical varieties that is consis-tent with the limit of the classical Hodge theory? Moreover, the following questions and needs were stated: A. Patchworking theorem: Is there a refinement of tropical geometry that will give information about a single (classical) hypersurface? B. Need to define maps, line bundles, vector bundles",
   "evidence": "This record is Section C.2, “Basic definitions,” in the 14 January 2004 AIM workshop report *Amoebas and Tropical Geometry*. It records workshop proposals and questions, not a settled definition. The questions are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 344,
   "attempt": 1
  },
  "AIM-GEOMETRY-0346": {
   "statement_status": "exact",
   "original_statement": "C.3 Computational issues \n\nA. How to compute examples of amoebas and tropical varieties? B. Special features to compute: \n\n• homology groups of the complement; \n\n• is there a Hermitean form whose signature is equal to the number of comple-ment components? C. Which classical varieties are expansive (i.e., is 0 is in the amoeba?)? E.g., \n\n• Grassmannians, generalized flag manifolds, spherical varieties, determinantal varieties, Schubert varieties; \n\n• Calabi-Yaus. 17 \n\nD. Develop the theory of smooth curves. E. What is the complexity of deciding the vanishing of a resultant? F. When does Log( X) lie in a lower-dimensional subspace? G. How hard is it to compute (i.e., what is the complexity of computing) the distance of a point to the discriminantal variety?",
   "clean_statement": "C.3 Computational issues\n\nA. How to compute examples of amoebas and tropical varieties? B. Special features to compute:\n\n• homology groups of the complement;\n\n• is there a Hermitean form whose signature is equal to the number of comple-ment components? C. Which classical varieties are expansive (i.e., is 0 is in the amoeba?)? E.g.,\n\n• Grassmannians, generalized flag manifolds, spherical varieties, determinantal varieties, Schubert varieties;\n\n• Calabi-Yaus. 17\n\nD. Develop the theory of smooth curves. E. What is the complexity of deciding the vanishing of a resultant? F. When does Log( X) lie in a lower-dimensional subspace? G. How hard is it to compute (i.e., what is the complexity of computing) the distance of a point to the discriminantal variety?",
   "public_statement": "C.3 Computational issues\n\nA. How to compute examples of amoebas and tropical varieties? B. Special features to compute:\n\n• homology groups of the complement;\n\n• is there a Hermitean form whose signature is equal to the number of comple-ment components? C. Which classical varieties are expansive (i.e., is 0 is in the amoeba?)? E.g.,\n\n• Grassmannians, generalized flag manifolds, spherical varieties, determinantal varieties, Schubert varieties;\n\n• Calabi-Yaus. 17\n\nD. Develop the theory of smooth curves. E. What is the complexity of deciding the vanishing of a resultant? F. When does Log( X) lie in a lower-dimensional subspace? G. How hard is it to compute (i.e., what is the complexity of computing) the distance of a point to the discriminantal variety?",
   "evidence": "Section C.3 of the AIM list *Amoebas and tropical geometry* asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 345,
   "attempt": 1
  },
  "AIM-GEOMETRY-0347": {
   "statement_status": "exact",
   "original_statement": "C.4 Recognition problems \n\nA. How to recognize whether a a pure d-complex is a tropical variety? B. Characterize the image of linear subspaces w.r.t. under the phase map (\"phlat\"). i.e., characterize π(L) where L ⊂ C∗ is a linear subspace, and π is the phase map \n\nπ: C∗ → T n (T n: n-dimensional torus).",
   "clean_statement": "C.4 Recognition problems\n\nA. How to recognize whether a a pure d-complex is a tropical variety? B. Characterize the image of linear subspaces w.r.t. under the phase map (\"phlat\"). i.e., characterize π(L) where L ⊂ C∗ is a linear subspace, and π is the phase map\n\nπ: C∗ → T n (T n: n-dimensional torus).",
   "public_statement": "C.4 Recognition problems\n\nA. How to recognize whether a a pure d-complex is a tropical variety? B. Characterize the image of linear subspaces w.r.t. under the phase map (\"phlat\"). i.e., characterize π(L) where L ⊂ C∗ is a linear subspace, and π is the phase map\n\nπ: C∗ → T n (T n: n-dimensional torus).",
   "evidence": "The canonical record is C.4 of the American Institute of Mathematics problem list *Amoebas and tropical geometry*, from the AIM workshop of October 23--26, 2003. Its source is <https://aimath.org/WWN/amoebas/amoebas.pdf>. The canonical metadata are `source_file = aim-geometry-notes.json`, zero-based `source_index = 346`, and `attempt = 1`.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 346,
   "attempt": 1
  },
  "AIM-GEOMETRY-0348": {
   "statement_status": "exact",
   "original_statement": "C.5 Applications \n\nA. From number theory: Consider one polynomial in one variable with integer coeffi-cients, written as a straight line program of complexity τ. How many roots are there which are congruent to 1 modulo p, as a function of τ?There exist estimates using amoeba theory, by work of Maurice Rojas. B. Prove Calabi conjecture for tropical Calabi-Yaus: in each metric class there is a unique metric which solves the tropical Monge-Amp` ere measure. In the real case this corresponds to det(Hessian) = const.",
   "clean_statement": "C.5 Applications\n\nA. From number theory: Consider one polynomial in one variable with integer coeffi-cients, written as a straight line program of complexity τ. How many roots are there which are congruent to 1 modulo p, as a function of τ?There exist estimates using amoeba theory, by work of Maurice Rojas. B. Prove Calabi conjecture for tropical Calabi-Yaus: in each metric class there is a unique metric which solves the tropical Monge-Amp` ere measure. In the real case this corresponds to det(Hessian) = const.",
   "public_statement": "C.5 Applications\n\nA. From number theory: Consider one polynomial in one variable with integer coeffi-cients, written as a straight line program of complexity τ. How many roots are there which are congruent to 1 modulo p, as a function of τ?There exist estimates using amoeba theory, by work of Maurice Rojas. B. Prove Calabi conjecture for tropical Calabi-Yaus: in each metric class there is a unique metric which solves the tropical Monge-Amp` ere measure. In the real case this corresponds to det(Hessian) = const.",
   "evidence": "This record is item C.5, “Applications,” from the AIM workshop list *Amoebas and tropical geometry*. The canonical JSON reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 347,
   "attempt": 1
  },
  "AIM-GEOMETRY-0349": {
   "statement_status": "exact",
   "original_statement": "Conjecture 1. If S is a natural n-form and ∫ S is conformally invariant, then \n\nS = const · Q + L + G,\n\nwhere L is a local conformal invariant and G has a local conformal primitive. That is, there is a local invariant F for which the conformal variation of ∫ F is ∫ ωG.The point of separating these 3 kinds of terms is that ∫ ωL will have a very banal conformal primitive, namely itself, or (to write it in a way that makes the properties of a conformal primitive more apparent), 1\n\n2\n\n∫\n\nc(ˆ g, g )( Lˆg + Lg).18 \n\nQ has an interesting conformal primitive, as discussed above. Q is not uniquely defined, but the The Q in the statement of the conjecture could be anyone's favorite version of Q. In fact, Q is well-defined up to addition of an L.Then there are the following related conjectures:",
   "clean_statement": "Conjecture 1. If S is a natural n-form and ∫ S is conformally invariant, then\n\nS = const · Q + L + G,\n\nwhere L is a local conformal invariant and G has a local conformal primitive. That is, there is a local invariant F for which the conformal variation of ∫ F is ∫ ωG.The point of separating these 3 kinds of terms is that ∫ ωL will have a very banal conformal primitive, namely itself, or (to write it in a way that makes the properties of a conformal primitive more apparent), 1\n\n2\n\n∫\n\nc(ˆ g, g )( Lˆg + Lg).18\n\nQ has an interesting conformal primitive, as discussed above. Q is not uniquely defined, but the The Q in the statement of the conjecture could be anyone's favorite version of Q. In fact, Q is well-defined up to addition of an L.Then there are the following related conjectures:",
   "public_statement": "Conjecture 1. If S is a natural n-form and ∫ S is conformally invariant, then\n\nS = const · Q + L + G,\n\nwhere L is a local conformal invariant and G has a local conformal primitive. That is, there is a local invariant F for which the conformal variation of ∫ F is ∫ ωG.The point of separating these 3 kinds of terms is that ∫ ωL will have a very banal conformal primitive, namely itself, or (to write it in a way that makes the properties of a conformal primitive more apparent), 1\n\n2\n\n∫\n\nc(ˆ g, g )( Lˆg + Lg).18\n\nQ has an interesting conformal primitive, as discussed above. Q is not uniquely defined, but the The Q in the statement of the conjecture could be anyone's favorite version of Q. In fact, Q is well-defined up to addition of an L.Then there are the following related conjectures:",
   "evidence": "The canonical record reproduces Conjecture 1 from the 2003 AIM workshop document *Conformal Structure in Geometry, Analysis, and Physics*. The exact extracted text is preserved in `input.json`. Inspection of the linked PDF and its preceding definitions supports the following reconstruction.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 348,
   "attempt": 1
  },
  "AIM-GEOMETRY-0350": {
   "statement_status": "exact",
   "original_statement": "Conjecture 2. Any S as above may be written const · Q + L + V,\n\nwhere V is an exact divergence.",
   "clean_statement": "Conjecture 2. Any S as above may be written const · Q + L + V,\n\nwhere V is an exact divergence.",
   "public_statement": "Conjecture 2. Any S as above may be written const · Q + L + V,\n\nwhere V is an exact divergence.",
   "evidence": "The canonical corpus record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 349,
   "attempt": 1
  },
  "AIM-GEOMETRY-0351": {
   "statement_status": "exact",
   "original_statement": "Conjecture 3. Any S as above may be written const · Pff + L + V.\n\nThere are at least 2 filtrations of the local invariants of this type that should be relevant. First, any invariant can be written as a sum of monomial expressions in R and ∇ with \n\nk∇ + 2 kR = n, where k∇ (resp. kR) is the number of occurrences of ∇ (resp. R) in the monomial. If an invariant T can be written with k∇ ≤ p for each monomial term, let's say \n\nT ∈ P p. Then \n\nP0 ⊂ P 2 ⊂ · · · ⊂ P n−2, Podd = 0.\n\nPff is in the most elite space, P0. The exact divergences inject into P2/P0.\n\nExercise 6. Use the conformal change law for Q to show that the class of Q in \n\nPn−2/Pn−4 is nontrivial, and agrees with the class of (∆ (n−2) /2J)dv g. This establishes that \n\nPff and Q are \"at opposite ends\" of the P-filtration. The other filtration is by the degree of the conformal change law. If ˆT = T + T1(dω ) + · · · + Tn(dω ),\n\nwith Ti of homogeneity i with respect to scalar multiples of ω, then say T ∈ L p if Tq = 0 for q > p. Then local conformal invariants are in L0, and Q is in L1. Pff on the other hand does not look so great in this filtration. \n\nOther routes to Q and its variants \n\nThere is an alternative definition of Q which avoids dimensional continuation. We write \n\nE for the space of smooth functions, E 1 for space of smooth 1-forms and define the special section \n\nIg:= \n\n 2 − n\n\n0\n\nJ\n\n\n\nof the direct sum bundle E ⊕ E 1 ⊕ E. Let us first set the dimension to be 4, simply present the some results and then explain how this works. Then we get \n\n§Ig =\n\n 00\n\nQ4\n\n,\n\nwhere § is the coupled conformal Laplacian operator. More precisely § = −∇ a∇a + ( n −\n\n2) K/ (4 n − 4), which appears to be the usual formula for the conformal Laplacian (cf. Y\n\nabove), but now ∇ is a connection which couples the usual metric connection with the 19 \n\nconnection \n\n∇a\n\n σμτ\n\n =\n\n ∇σ − μ\n\n∇μ + gτ + Pσ\n\n∇τ − μ−| P\n\n.\n\non the sum bundle T:= E ⊕ E 1 ⊕ E. This bundle T is called the standard tractor bundle and this connection is usually termed the (normal conformal) tractor connection. It is equivalent to a principal bundle structure known as the (normal conformal) Cartan connection. For those who know about Cartan connections we can say that the tractor bundle and connection is an associated bundle and connection for the Cartan bundle. We have used a metric \n\ng to express these objects in terms of a Riemannian structure but in fact the bundle and connection are conformally invariant and so descend to well defined structures on a conformal manifold. In fact, to be more accurate, the decomposition of the standard tractor bundle T\n\nis really T = E[1] ⊕ E 1 ⊗ E [1] ⊕ E [−1] where E[w] indicates the space of conformal densities of weight w. The field Ig is a section of T ⊗ E [−1]. In this section and the next we are allowing tensors and tractor fields to be density valued, to simplify the notation, but partially suppressing the details of weights involved. This construction generalises. In each even dimension n there is a conformally invariant differential operator §n−2 so that for any metric g we have \n\n§n−2Ig =\n\n 00\n\nQn\n\n. (32) Here Ig is as above, while §n−2 has the form ∆ n/ 2−1 + LOT. The tractor field Ig is not conformally invariant, but it does have an interesting conformal transformation. If ˆ g is a metric related to g conformally according to ˆ g = e2ωg (ω a smooth function) then \n\nIˆg = Ig + Dω, (33) where D is a well known second order conformally invariant linear differential operator known as the tractor D operator. From this and (41) it follows that the Q-curvature ˆQn, for ˆ g,differs from Qn by a linear conformally invariant operator acting on ω. In fact it follows easily from the definition of §n−2 that \n\n§n−2Dω =\n\n 00\n\nPnω\n\n\n\nwhere Pn is the GJMS operator of order n. So we have recovered the now famous property ˆQn = Qn + Pnω (cf. (28)). As a final comment on the above story we should clarify the origins of the tractor field \n\nIg defined and used above. For those who are familiar with tractors a more enlightening alternative definition is \n\nIgA:= − 1\n\nnDAσ−1DAB σ. \n\nHere DA is the tractor D operator and DAB is the so-called fundamental D operator. 0 6 =\n\nσ ∈ ΓE[1] is the conformal scale corresponding to the metric g. The point is that these operators are both conformally invariant and under g 7 → ˆg = e2ωg we have σ 7 → eωσ.Since DAB satisfies a Leibniz rule and σ−1DAB σ is a \"logarithmic derivative\" the conformal transformation law of Ig is no surprise. 20 \n\nThe main results above are derived via the ambient metric construction of Fefferman and Graham. Explaining this construction would be a significant detour at this point. Suffice to say that this construction geometrically associates to an n-dimensional conformal manifold M an ( n + 2)-dimensional pseudo-Riemannian manifold ˜M. The GJMS operators \n\nP2` arise from powers of the Laplacian ˜∆`, of ˜M, acting on suitably homogeneous functions. The operators §2` arise in a similar way from ˜∆` on appropriately homogeneous sections of the tangent bundle T ˜M. Such homogeneous sections correspond to tractor fields on the conformal manifold M. The results above are given by an easy calculation on the ambient manifold. Thus we can take (41) as a definition of the Q-curvature; it is simply the natural scalar field that turns up on the right hand side. While this definition avoids dimensional continuation, there is still the issue of getting a formula for Qn. There is an effective algorithm for re-expressing the ambient results in terms of tractors which then expand easily into formulae in terms of the underlying Riemannian curvature and its covariant derivatives. This solves the problem for small n. For example \n\nQ6 = 8 Pij |kPij |k + 16 Pij Pij |kk − 32 Pij PikPj k − 16 Pij Pij J+8J3 − 8J|kkJ + J|j j kk + 16 Pij Pkl Cikj l.\n\nWhile such formulae shed some light on the nature of the Q-curvature it would clearly be ideal to give a general formula or simple inductive formula. From the angle discussed here the missing information is a general formula for the operators §n−2.",
   "clean_statement": "Conjecture 3. Any S as above may be written const · Pff + L + V.\n\nThere are at least 2 filtrations of the local invariants of this type that should be relevant. First, any invariant can be written as a sum of monomial expressions in R and ∇ with\n\nk∇ + 2 kR = n, where k∇ (resp. kR) is the number of occurrences of ∇ (resp. R) in the monomial. If an invariant T can be written with k∇ ≤ p for each monomial term, let's say\n\nT ∈ P p. Then\n\nP0 ⊂ P 2 ⊂ · · · ⊂ P n−2, Podd = 0.\n\nPff is in the most elite space, P0. The exact divergences inject into P2/P0.\n\nExercise 6. Use the conformal change law for Q to show that the class of Q in\n\nPn−2/Pn−4 is nontrivial, and agrees with the class of (∆ (n−2) /2J)dv g. This establishes that\n\nPff and Q are \"at opposite ends\" of the P-filtration. The other filtration is by the degree of the conformal change law. If ˆT = T + T1(dω ) + · · · + Tn(dω ),\n\nwith Ti of homogeneity i with respect to scalar multiples of ω, then say T ∈ L p if Tq = 0 for q > p. Then local conformal invariants are in L0, and Q is in L1. Pff on the other hand does not look so great in this filtration.\n\nOther routes to Q and its variants\n\nThere is an alternative definition of Q which avoids dimensional continuation. We write\n\nE for the space of smooth functions, E 1 for space of smooth 1-forms and define the special section\n\nIg:=\n\n 2 − n\n\n0\n\nJ\n\n\n\nof the direct sum bundle E ⊕ E 1 ⊕ E. Let us first set the dimension to be 4, simply present the some results and then explain how this works. Then we get\n\n§Ig =\n\n 00\n\nQ4\n\n,\n\nwhere § is the coupled conformal Laplacian operator. More precisely § = −∇ a∇a + ( n −\n\n2) K/ (4 n − 4), which appears to be the usual formula for the conformal Laplacian (cf. Y\n\nabove), but now ∇ is a connection which couples the usual metric connection with the 19\n\nconnection\n\n∇a\n\n σμτ\n\n =\n\n ∇σ − μ\n\n∇μ + gτ + Pσ\n\n∇τ − μ−| P\n\n.\n\non the sum bundle T:= E ⊕ E 1 ⊕ E. This bundle T is called the standard tractor bundle and this connection is usually termed the (normal conformal) tractor connection. It is equivalent to a principal bundle structure known as the (normal conformal) Cartan connection. For those who know about Cartan connections we can say that the tractor bundle and connection is an associated bundle and connection for the Cartan bundle. We have used a metric\n\ng to express these objects in terms of a Riemannian structure but in fact the bundle and connection are conformally invariant and so descend to well defined structures on a conformal manifold. In fact, to be more accurate, the decomposition of the standard tractor bundle T\n\nis really T = E[1] ⊕ E 1 ⊗ E [1] ⊕ E [−1] where E[w] indicates the space of conformal densities of weight w. The field Ig is a section of T ⊗ E [−1]. In this section and the next we are allowing tensors and tractor fields to be density valued, to simplify the notation, but partially suppressing the details of weights involved. This construction generalises. In each even dimension n there is a conformally invariant differential operator §n−2 so that for any metric g we have\n\n§n−2Ig =\n\n 00\n\nQn\n\n. (32) Here Ig is as above, while §n−2 has the form ∆ n/ 2−1 + LOT. The tractor field Ig is not conformally invariant, but it does have an interesting conformal transformation. If ˆ g is a metric related to g conformally according to ˆ g = e2ωg (ω a smooth function) then\n\nIˆg = Ig + Dω, (33) where D is a well known second order conformally invariant linear differential operator known as the tractor D operator. From this and (41) it follows that the Q-curvature ˆQn, for ˆ g,differs from Qn by a linear conformally invariant operator acting on ω. In fact it follows easily from the definition of §n−2 that\n\n§n−2Dω =\n\n 00\n\nPnω\n\n\n\nwhere Pn is the GJMS operator of order n. So we have recovered the now famous property ˆQn = Qn + Pnω (cf. (28)). As a final comment on the above story we should clarify the origins of the tractor field\n\nIg defined and used above. For those who are familiar with tractors a more enlightening alternative definition is\n\nIgA:= − 1\n\nnDAσ−1DAB σ.\n\nHere DA is the tractor D operator and DAB is the so-called fundamental D operator. 0 6 =\n\nσ ∈ ΓE[1] is the conformal scale corresponding to the metric g. The point is that these operators are both conformally invariant and under g 7 → ˆg = e2ωg we have σ 7 → eωσ.Since DAB satisfies a Leibniz rule and σ−1DAB σ is a \"logarithmic derivative\" the conformal transformation law of Ig is no surprise. 20\n\nThe main results above are derived via the ambient metric construction of Fefferman and Graham. Explaining this construction would be a significant detour at this point. Suffice to say that this construction geometrically associates to an n-dimensional conformal manifold M an ( n + 2)-dimensional pseudo-Riemannian manifold ˜M. The GJMS operators\n\nP2` arise from powers of the Laplacian ˜∆`, of ˜M, acting on suitably homogeneous functions. The operators §2` arise in a similar way from ˜∆` on appropriately homogeneous sections of the tangent bundle T ˜M. Such homogeneous sections correspond to tractor fields on the conformal manifold M. The results above are given by an easy calculation on the ambient manifold. Thus we can take (41) as a definition of the Q-curvature; it is simply the natural scalar field that turns up on the right hand side. While this definition avoids dimensional continuation, there is still the issue of getting a formula for Qn. There is an effective algorithm for re-expressing the ambient results in terms of tractors which then expand easily into formulae in terms of the underlying Riemannian curvature and its covariant derivatives. This solves the problem for small n. For example\n\nQ6 = 8 Pij |kPij |k + 16 Pij Pij |kk − 32 Pij PikPj k − 16 Pij Pij J+8J3 − 8J|kkJ + J|j j kk + 16 Pij Pkl Cikj l.\n\nWhile such formulae shed some light on the nature of the Q-curvature it would clearly be ideal to give a general formula or simple inductive formula. From the angle discussed here the missing information is a general formula for the operators §n−2.",
   "public_statement": "Conjecture 3. Any S as above may be written const · Pff + L + V.\n\nThere are at least 2 filtrations of the local invariants of this type that should be relevant. First, any invariant can be written as a sum of monomial expressions in R and ∇ with\n\nk∇ + 2 kR = n, where k∇ (resp. kR) is the number of occurrences of ∇ (resp. R) in the monomial. If an invariant T can be written with k∇ ≤ p for each monomial term, let's say\n\nT ∈ P p. Then\n\nP0 ⊂ P 2 ⊂ · · · ⊂ P n−2, Podd = 0.\n\nPff is in the most elite space, P0. The exact divergences inject into P2/P0.\n\nExercise 6. Use the conformal change law for Q to show that the class of Q in\n\nPn−2/Pn−4 is nontrivial, and agrees with the class of (∆ (n−2) /2J)dv g. This establishes that\n\nPff and Q are \"at opposite ends\" of the P-filtration. The other filtration is by the degree of the conformal change law. If ˆT = T + T1(dω ) + · · · + Tn(dω ),\n\nwith Ti of homogeneity i with respect to scalar multiples of ω, then say T ∈ L p if Tq = 0 for q > p. Then local conformal invariants are in L0, and Q is in L1. Pff on the other hand does not look so great in this filtration.\n\nOther routes to Q and its variants\n\nThere is an alternative definition of Q which avoids dimensional continuation. We write\n\nE for the space of smooth functions, E 1 for space of smooth 1-forms and define the special section\n\nIg:=\n\n 2 − n\n\n0\n\nJ\n\n\n\nof the direct sum bundle E ⊕ E 1 ⊕ E. Let us first set the dimension to be 4, simply present the some results and then explain how this works. Then we get\n\n§Ig =\n\n 00\n\nQ4\n\n,\n\nwhere § is the coupled conformal Laplacian operator. More precisely § = −∇ a∇a + ( n −\n\n2) K/ (4 n − 4), which appears to be the usual formula for the conformal Laplacian (cf. Y\n\nabove), but now ∇ is a connection which couples the usual metric connection with the 19\n\nconnection\n\n∇a\n\n σμτ\n\n =\n\n ∇σ − μ\n\n∇μ + gτ + Pσ\n\n∇τ − μ−| P\n\n.\n\non the sum bundle T:= E ⊕ E 1 ⊕ E. This bundle T is called the standard tractor bundle and this connection is usually termed the (normal conformal) tractor connection. It is equivalent to a principal bundle structure known as the (normal conformal) Cartan connection. For those who know about Cartan connections we can say that the tractor bundle and connection is an associated bundle and connection for the Cartan bundle. We have used a metric\n\ng to express these objects in terms of a Riemannian structure but in fact the bundle and connection are conformally invariant and so descend to well defined structures on a conformal manifold. In fact, to be more accurate, the decomposition of the standard tractor bundle T\n\nis really T = E[1] ⊕ E 1 ⊗ E [1] ⊕ E [−1] where E[w] indicates the space of conformal densities of weight w. The field Ig is a section of T ⊗ E [−1]. In this section and the next we are allowing tensors and tractor fields to be density valued, to simplify the notation, but partially suppressing the details of weights involved. This construction generalises. In each even dimension n there is a conformally invariant differential operator §n−2 so that for any metric g we have\n\n§n−2Ig =\n\n 00\n\nQn\n\n. (32) Here Ig is as above, while §n−2 has the form ∆ n/ 2−1 + LOT. The tractor field Ig is not conformally invariant, but it does have an interesting conformal transformation. If ˆ g is a metric related to g conformally according to ˆ g = e2ωg (ω a smooth function) then\n\nIˆg = Ig + Dω, (33) where D is a well known second order conformally invariant linear differential operator known as the tractor D operator. From this and (41) it follows that the Q-curvature ˆQn, for ˆ g,differs from Qn by a linear conformally invariant operator acting on ω. In fact it follows easily from the definition of §n−2 that\n\n§n−2Dω =\n\n 00\n\nPnω\n\n\n\nwhere Pn is the GJMS operator of order n. So we have recovered the now famous property ˆQn = Qn + Pnω (cf. (28)). As a final comment on the above story we should clarify the origins of the tractor field\n\nIg defined and used above. For those who are familiar with tractors a more enlightening alternative definition is\n\nIgA:= − 1\n\nnDAσ−1DAB σ.\n\nHere DA is the tractor D operator and DAB is the so-called fundamental D operator. 0 6 =\n\nσ ∈ ΓE[1] is the conformal scale corresponding to the metric g. The point is that these operators are both conformally invariant and under g 7 → ˆg = e2ωg we have σ 7 → eωσ.Since DAB satisfies a Leibniz rule and σ−1DAB σ is a \"logarithmic derivative\" the conformal transformation law of Ig is no surprise. 20\n\nThe main results above are derived via the ambient metric construction of Fefferman and Graham. Explaining this construction would be a significant detour at this point. Suffice to say that this construction geometrically associates to an n-dimensional conformal manifold M an ( n + 2)-dimensional pseudo-Riemannian manifold ˜M. The GJMS operators\n\nP2` arise from powers of the Laplacian ˜∆`, of ˜M, acting on suitably homogeneous functions. The operators §2` arise in a similar way from ˜∆` on appropriately homogeneous sections of the tangent bundle T ˜M. Such homogeneous sections correspond to tractor fields on the conformal manifold M. The results above are given by an easy calculation on the ambient manifold. Thus we can take (41) as a definition of the Q-curvature; it is simply the natural scalar field that turns up on the right hand side. While this definition avoids dimensional continuation, there is still the issue of getting a formula for Qn. There is an effective algorithm for re-expressing the ambient results in terms of tractors which then expand easily into formulae in terms of the underlying Riemannian curvature and its covariant derivatives. This solves the problem for small n. For example\n\nQ6 = 8 Pij |kPij |k + 16 Pij Pij |kk − 32 Pij PikPj k − 16 Pij Pij J+8J3 − 8J|kkJ + J|j j kk + 16 Pij Pkl Cikj l.\n\nWhile such formulae shed some light on the nature of the Q-curvature it would clearly be ideal to give a general formula or simple inductive formula. From the angle discussed here the missing information is a general formula for the operators §n−2.",
   "evidence": "The assigned record begins with:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 350,
   "attempt": 1
  },
  "AIM-GEOMETRY-0352": {
   "statement_status": "exact",
   "original_statement": "Problem 1: Give general formulae or inductive formulae for the operators §2`.This seems to be a difficult problem. In another direction there is another exercise to which we already have some answers. One of the features of the Q-curvature is that it \"transforms by a linear operator\" within a conformal class. More precisely, it is an example of a natural Riemannian tensor-density field with a transformation law \n\nN ˆg = N g + Lω, (34) \n\nL being some universal linear differential operator. (Here ω has the usual meaning; ˆ g = e2ωg.)",
   "clean_statement": "Problem 1: Give general formulae or inductive formulae for the operators §2`.This seems to be a difficult problem. In another direction there is another exercise to which we already have some answers. One of the features of the Q-curvature is that it \"transforms by a linear operator\" within a conformal class. More precisely, it is an example of a natural Riemannian tensor-density field with a transformation law\n\nN ˆg = N g + Lω, (34)\n\nL being some universal linear differential operator. (Here ω has the usual meaning; ˆ g = e2ωg.)",
   "public_statement": "Problem 1: Give general formulae or inductive formulae for the operators §2`.This seems to be a difficult problem. In another direction there is another exercise to which we already have some answers. One of the features of the Q-curvature is that it \"transforms by a linear operator\" within a conformal class. More precisely, it is an example of a natural Riemannian tensor-density field with a transformation law\n\nN ˆg = N g + Lω, (34)\n\nL being some universal linear differential operator. (Here ω has the usual meaning; ˆ g = e2ωg.)",
   "evidence": "The canonical record joins one actual problem to the beginning of the setup for the next problem. Inspection of the hard-copy AIM PDF and of the mathematical `ALT` text in AIM's contemporaneous HTML conversion recovers the first sentence as",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 351,
   "attempt": 1
  },
  "AIM-GEOMETRY-0353": {
   "statement_status": "exact",
   "original_statement": "Problem 2: Construct other natural tensor-densities which transform according to (43). (Note that any solution yields a conformally invariant natural operator L.) From the transformation law for Ig above, we can evidently manufacture solutions to this problem. We have observed already that Ig is a section of the bundle T [−1]:= T ⊗E [−1]. If P is any scalar (or rather density) valued natural conformally invariant differential operator which acts on T [−1] then P can act on Ig, and P I g has a conformal transformation of the the form (43). Using the calculus naturally associated to tractor bundles (or equally effectively, using the ambient metric) it is in fact a simple matter to write down examples, and the possibilities increase with dimension. This is most interesting when the resulting scalar field gives a possible modification to the original Q-curvature. For those familiar with densities this means that P should take values in densities of weight −n; this is the weight at which densities that can be integrated on a conformal manifold. For example, in any dimension we may take P to be ι(D)|C|2 where ι(D) indicates a contracted action of the tractor D\n\noperator and the square of the Weyl curvature is here viewed as a multiplication operator. In dimension 6 this takes values in E[−6] and we have \n\nι(D)|C|2Ig = 4∆ |C|221 \n\nand \n\nι(D)|C|2Iˆg = 4∆ |C|2 − 16 δ|C|2dω. \n\nNote that in this example the conformally invariant \" L-operator\" δ|C|2d is formally self-adjoint. So for any constant α, Q6 + αι (D)|C|2Ig is another scalar field with almost the same properties as Q6. It is not so closely related to the GJMS operator P6, but it is related instead to a modification of P6 by δ|C|2d. It is clear that solutions to",
   "clean_statement": "Problem 2: Construct other natural tensor-densities which transform according to (43). (Note that any solution yields a conformally invariant natural operator L.) From the transformation law for Ig above, we can evidently manufacture solutions to this problem. We have observed already that Ig is a section of the bundle T [−1]:= T ⊗E [−1]. If P is any scalar (or rather density) valued natural conformally invariant differential operator which acts on T [−1] then P can act on Ig, and P I g has a conformal transformation of the the form (43). Using the calculus naturally associated to tractor bundles (or equally effectively, using the ambient metric) it is in fact a simple matter to write down examples, and the possibilities increase with dimension. This is most interesting when the resulting scalar field gives a possible modification to the original Q-curvature. For those familiar with densities this means that P should take values in densities of weight −n; this is the weight at which densities that can be integrated on a conformal manifold. For example, in any dimension we may take P to be ι(D)|C|2 where ι(D) indicates a contracted action of the tractor D\n\noperator and the square of the Weyl curvature is here viewed as a multiplication operator. In dimension 6 this takes values in E[−6] and we have\n\nι(D)|C|2Ig = 4∆ |C|221\n\nand\n\nι(D)|C|2Iˆg = 4∆ |C|2 − 16 δ|C|2dω.\n\nNote that in this example the conformally invariant \" L-operator\" δ|C|2d is formally self-adjoint. So for any constant α, Q6 + αι (D)|C|2Ig is another scalar field with almost the same properties as Q6. It is not so closely related to the GJMS operator P6, but it is related instead to a modification of P6 by δ|C|2d. It is clear that solutions to",
   "public_statement": "Problem 2: Construct other natural tensor-densities which transform according to (43). (Note that any solution yields a conformally invariant natural operator L.) From the transformation law for Ig above, we can evidently manufacture solutions to this problem. We have observed already that Ig is a section of the bundle T [−1]:= T ⊗E [−1]. If P is any scalar (or rather density) valued natural conformally invariant differential operator which acts on T [−1] then P can act on Ig, and P I g has a conformal transformation of the the form (43). Using the calculus naturally associated to tractor bundles (or equally effectively, using the ambient metric) it is in fact a simple matter to write down examples, and the possibilities increase with dimension. This is most interesting when the resulting scalar field gives a possible modification to the original Q-curvature. For those familiar with densities this means that P should take values in densities of weight −n; this is the weight at which densities that can be integrated on a conformal manifold. For example, in any dimension we may take P to be ι(D)|C|2 where ι(D) indicates a contracted action of the tractor D\n\noperator and the square of the Weyl curvature is here viewed as a multiplication operator. In dimension 6 this takes values in E[−6] and we have\n\nι(D)|C|2Ig = 4∆ |C|221\n\nand\n\nι(D)|C|2Iˆg = 4∆ |C|2 − 16 δ|C|2dω.\n\nNote that in this example the conformally invariant \" L-operator\" δ|C|2d is formally self-adjoint. So for any constant α, Q6 + αι (D)|C|2Ig is another scalar field with almost the same properties as Q6. It is not so closely related to the GJMS operator P6, but it is related instead to a modification of P6 by δ|C|2d. It is clear that solutions to",
   "evidence": "The canonical record is extracted from the American Institute of Mathematics workshop notes *Conformal Structure in Geometry, Analysis, and Physics*, version 15 October 2003, in the section “Other routes to \\(Q\\) and its variants.” The extraction stops in the middle of the last sentence. Inspection of printed pages 20--21 gives the intended statement and resolves three defects in the record.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 352,
   "attempt": 1
  },
  "AIM-GEOMETRY-0354": {
   "statement_status": "exact",
   "original_statement": "Problem 2 have a role to play in the problem of characterising the Q-curvature and the GJMS operators. \n\nA generalisation: maps like Q\n\nSo far we have viewed the Q-curvature as a natural scalar field. It turns out that if instead we view it as an operator then it fits naturally into a bigger picture. To simplify matters suppose we are working with a compact, oriented, but not necessarily connected, manifold of even dimension n. We fix n and so omit n in the notation for Q. We can view Q\n\nas a multiplication operator from the closed 0-forms C0 (i.e. the locally constant functions) into the space of n-forms En (which we identify with E[−n] via the conformal Hodge?). With the observations above we have the following properties: A. Q: C0 → E n is not conformally invariant but ˆQ = Q + Pnω, where Pn is a formally self-adjoint operator from 0-forms to n-forms. Pn has the form dM d which implies the next properties. B. Q: C0 → Hn(M ) is conformally invariant and non-trivial in general. C. If c ∈ C 0 and u ∈ N (Pn) then ∫ uQc is conformally invariant. D. (See the discussion immediately below.) In each choice of metric Q: E 0 → E [−n] is formally self-adjoint. E. (See the discussion immediately below.) Q1 is the Q-curvature. The last properties are trivial since the operator is multiplication by a scalar field and by definition Q1 = Q. However we should note that we can add to Q any differential operator that annihilates constants, and properties 1-3 will be unaffected. Property 4 is suggesting that if we do that, then we should insist that the result is formally self-adjoint. The idea now is to look for analogous operators on other forms. We write Ck for the space of closed k-forms. Consider the operator \n\nM g = dδ + 2 J − 4P]: En/ 2−1 → E n/ 2−1[−2]. (35) (Note that by the conformal Hodge star E n/ 2−1[−2] ∼= En/ 2+1, so we can also view this as an operator into ( n/ 2 + 1)-forms.) Then: \n\nExercise 7. On Cn/ 2−1 we have \n\nM ˆg = M g + βδdω, \n\nwhere β is some nonzero constant, ˆ g = e2ωg, and in the display ω is viewed as a multiplication operator. Note that the conformal variation term δd is the Maxwell operator and is formally self-adjoint. So M g satisfies the analogue of property 1 above. The analogue of property 3 is an immediate consequence, i.e., ∫ 〈u, M c 〉 is conformally invariant where now c is a closed (n/ 2−1)-form and u ∈ N (δd ) (so in fact by compactness u, c are both closed). Next observe, by inspection, that M g is formally self-adjoint. So we have analogues for 1,3,4. There is also a bonus property, which is clear from the transformation law displayed: 22 \n\nδM gd: En/ 2−2 → E n/ 2−2[−4] is a non-trivial conformally invariant operator. In dimension 4 this is the Paneitz operator. So finally we need an analogue for property 2. It is clear that M g is conformally invariant as a map Cn/ 2−1 → E n/ 2−1[−2] /R(δ), so this is an analogue. But we can do more. There is no reason to suppose the image is co-closed. On the other hand note that δM g is conformally invariant on Cn/ 2−1 and so we have the following: \n\nM g: Hn/ 2−1 → Hn/ 2+1 (M ) with \n\nHn/ 2−1:= N (δM: Cn/ 2−1 → E n/ 2−2[−4]) (36) is conformally invariant. The space Hn/ 2−1 may be viewed as the space of \"conformal har-monics\". Evidently dim( Hn/ 2−1) is not always the same as the Betti number bn/ 2−1, but the elliptic coercivity of the pair ( d, δM ) gives it a good chance of returning the Betti number off a set of conformal structures that is somehow small. One should also check that the map (36) is non-trivial. \n\nFact: Let M = Sp × Sq, where p = n/ 2 − 1, q = n/ 2 + 1, with the standard Riemannian structure. Then φ ∈ H p if and only if φ is harmonic. Furthermore, the map (36) is non-trivial. In some recent work the authors have used the ambient metric, and its relationship to tractors, to show that the above construction generalises along the following lines: There are operators M gk: Ek → E n−k (k ≤ n/ 2 − 1), given by a uniform construction, with the following properties: A. M gk: Ck → E n−k has the conformal transformation law M ˆgk = M gk + Lkω, where Lk\n\nis a formally self-adjoint operator from k-forms to ( n − k)-forms, and is a constant multiple of dM gk+1 d.B. Hk:= N (dM gk: Ck → E n−k+1 ) is a conformally invariant subspace of Ck and Mk:\n\nHk → Hn−k(M ) is conformally invariant. There are conformal manifolds on which \n\nMk is non-trivial. C. If c ∈ C k and u ∈ N (Lk) then \n\n∫\n\n〈u, M gk c〉\n\nis conformally invariant. D. For each choice of metric g, M gk: Ek → E n−k is formally self-adjoint. E. M g \n\n> 0\n\n1 is the Q-curvature. From the uniqueness of the Maxwell operator at leading order (as a conformally invariant operator En/ 2−1 → E n/ 2−1[−2]), and the explicit formula (44) it is clear M gn/ 2−1 is not the difference between any conformally invariant differential operator and a divergence (even as an operator on closed forms). A similar argument applies to the Mk generally. Thus, from the point of view that the Q-curvature is a non-conformally invariant object that in a deep sense cannot be made conformally invariant, but one which nevertheless determines a global conformal invariant, the operators M gk give a genuine generalisation of the Q-curvature to an operator on closed forms. 23 \n\nProblems k: There are analogues for the operators M gk of most of the conundrums and problems for the Q-curvature. \n\nChapter C: Open problems \n\nConformal Structure in Geometry, Analysis, and Physics \n\nAugust 12 to 16, 2003 at the American Institute of Mathematics, Palo Alto, California \n\nI. Problems suggested by the participants Thomas Branson. Anti-conformal perturbations.",
   "clean_statement": "Problem 2 have a role to play in the problem of characterising the Q-curvature and the GJMS operators.\n\nA generalisation: maps like Q\n\nSo far we have viewed the Q-curvature as a natural scalar field. It turns out that if instead we view it as an operator then it fits naturally into a bigger picture. To simplify matters suppose we are working with a compact, oriented, but not necessarily connected, manifold of even dimension n. We fix n and so omit n in the notation for Q. We can view Q\n\nas a multiplication operator from the closed 0-forms C0 (i.e. the locally constant functions) into the space of n-forms En (which we identify with E[−n] via the conformal Hodge?). With the observations above we have the following properties: A. Q: C0 → E n is not conformally invariant but ˆQ = Q + Pnω, where Pn is a formally self-adjoint operator from 0-forms to n-forms. Pn has the form dM d which implies the next properties. B. Q: C0 → Hn(M ) is conformally invariant and non-trivial in general. C. If c ∈ C 0 and u ∈ N (Pn) then ∫ uQc is conformally invariant. D. (See the discussion immediately below.) In each choice of metric Q: E 0 → E [−n] is formally self-adjoint. E. (See the discussion immediately below.) Q1 is the Q-curvature. The last properties are trivial since the operator is multiplication by a scalar field and by definition Q1 = Q. However we should note that we can add to Q any differential operator that annihilates constants, and properties 1-3 will be unaffected. Property 4 is suggesting that if we do that, then we should insist that the result is formally self-adjoint. The idea now is to look for analogous operators on other forms. We write Ck for the space of closed k-forms. Consider the operator\n\nM g = dδ + 2 J − 4P]: En/ 2−1 → E n/ 2−1[−2]. (35) (Note that by the conformal Hodge star E n/ 2−1[−2] ∼= En/ 2+1, so we can also view this as an operator into ( n/ 2 + 1)-forms.) Then:\n\nExercise 7. On Cn/ 2−1 we have\n\nM ˆg = M g + βδdω,\n\nwhere β is some nonzero constant, ˆ g = e2ωg, and in the display ω is viewed as a multiplication operator. Note that the conformal variation term δd is the Maxwell operator and is formally self-adjoint. So M g satisfies the analogue of property 1 above. The analogue of property 3 is an immediate consequence, i.e., ∫ 〈u, M c 〉 is conformally invariant where now c is a closed (n/ 2−1)-form and u ∈ N (δd ) (so in fact by compactness u, c are both closed). Next observe, by inspection, that M g is formally self-adjoint. So we have analogues for 1,3,4. There is also a bonus property, which is clear from the transformation law displayed: 22\n\nδM gd: En/ 2−2 → E n/ 2−2[−4] is a non-trivial conformally invariant operator. In dimension 4 this is the Paneitz operator. So finally we need an analogue for property 2. It is clear that M g is conformally invariant as a map Cn/ 2−1 → E n/ 2−1[−2] /R(δ), so this is an analogue. But we can do more. There is no reason to suppose the image is co-closed. On the other hand note that δM g is conformally invariant on Cn/ 2−1 and so we have the following:\n\nM g: Hn/ 2−1 → Hn/ 2+1 (M ) with\n\nHn/ 2−1:= N (δM: Cn/ 2−1 → E n/ 2−2[−4]) (36) is conformally invariant. The space Hn/ 2−1 may be viewed as the space of \"conformal har-monics\". Evidently dim( Hn/ 2−1) is not always the same as the Betti number bn/ 2−1, but the elliptic coercivity of the pair ( d, δM ) gives it a good chance of returning the Betti number off a set of conformal structures that is somehow small. One should also check that the map (36) is non-trivial.\n\nFact: Let M = Sp × Sq, where p = n/ 2 − 1, q = n/ 2 + 1, with the standard Riemannian structure. Then φ ∈ H p if and only if φ is harmonic. Furthermore, the map (36) is non-trivial. In some recent work the authors have used the ambient metric, and its relationship to tractors, to show that the above construction generalises along the following lines: There are operators M gk: Ek → E n−k (k ≤ n/ 2 − 1), given by a uniform construction, with the following properties: A. M gk: Ck → E n−k has the conformal transformation law M ˆgk = M gk + Lkω, where Lk\n\nis a formally self-adjoint operator from k-forms to ( n − k)-forms, and is a constant multiple of dM gk+1 d.B. Hk:= N (dM gk: Ck → E n−k+1 ) is a conformally invariant subspace of Ck and Mk:\n\nHk → Hn−k(M ) is conformally invariant. There are conformal manifolds on which\n\nMk is non-trivial. C. If c ∈ C k and u ∈ N (Lk) then\n\n∫\n\n〈u, M gk c〉\n\nis conformally invariant. D. For each choice of metric g, M gk: Ek → E n−k is formally self-adjoint. E. M g\n\n> 0\n\n1 is the Q-curvature. From the uniqueness of the Maxwell operator at leading order (as a conformally invariant operator En/ 2−1 → E n/ 2−1[−2]), and the explicit formula (44) it is clear M gn/ 2−1 is not the difference between any conformally invariant differential operator and a divergence (even as an operator on closed forms). A similar argument applies to the Mk generally. Thus, from the point of view that the Q-curvature is a non-conformally invariant object that in a deep sense cannot be made conformally invariant, but one which nevertheless determines a global conformal invariant, the operators M gk give a genuine generalisation of the Q-curvature to an operator on closed forms. 23\n\nProblems k: There are analogues for the operators M gk of most of the conundrums and problems for the Q-curvature.\n\nChapter C: Open problems\n\nConformal Structure in Geometry, Analysis, and Physics\n\nAugust 12 to 16, 2003 at the American Institute of Mathematics, Palo Alto, California\n\nI. Problems suggested by the participants Thomas Branson. Anti-conformal perturbations.",
   "public_statement": "Problem 2 have a role to play in the problem of characterising the Q-curvature and the GJMS operators.\n\nA generalisation: maps like Q\n\nSo far we have viewed the Q-curvature as a natural scalar field. It turns out that if instead we view it as an operator then it fits naturally into a bigger picture. To simplify matters suppose we are working with a compact, oriented, but not necessarily connected, manifold of even dimension n. We fix n and so omit n in the notation for Q. We can view Q\n\nas a multiplication operator from the closed 0-forms C0 (i.e. the locally constant functions) into the space of n-forms En (which we identify with E[−n] via the conformal Hodge?). With the observations above we have the following properties: A. Q: C0 → E n is not conformally invariant but ˆQ = Q + Pnω, where Pn is a formally self-adjoint operator from 0-forms to n-forms. Pn has the form dM d which implies the next properties. B. Q: C0 → Hn(M ) is conformally invariant and non-trivial in general. C. If c ∈ C 0 and u ∈ N (Pn) then ∫ uQc is conformally invariant. D. (See the discussion immediately below.) In each choice of metric Q: E 0 → E [−n] is formally self-adjoint. E. (See the discussion immediately below.) Q1 is the Q-curvature. The last properties are trivial since the operator is multiplication by a scalar field and by definition Q1 = Q. However we should note that we can add to Q any differential operator that annihilates constants, and properties 1-3 will be unaffected. Property 4 is suggesting that if we do that, then we should insist that the result is formally self-adjoint. The idea now is to look for analogous operators on other forms. We write Ck for the space of closed k-forms. Consider the operator\n\nM g = dδ + 2 J − 4P]: En/ 2−1 → E n/ 2−1[−2]. (35) (Note that by the conformal Hodge star E n/ 2−1[−2] ∼= En/ 2+1, so we can also view this as an operator into ( n/ 2 + 1)-forms.) Then:\n\nExercise 7. On Cn/ 2−1 we have\n\nM ˆg = M g + βδdω,\n\nwhere β is some nonzero constant, ˆ g = e2ωg, and in the display ω is viewed as a multiplication operator. Note that the conformal variation term δd is the Maxwell operator and is formally self-adjoint. So M g satisfies the analogue of property 1 above. The analogue of property 3 is an immediate consequence, i.e., ∫ 〈u, M c 〉 is conformally invariant where now c is a closed (n/ 2−1)-form and u ∈ N (δd ) (so in fact by compactness u, c are both closed). Next observe, by inspection, that M g is formally self-adjoint. So we have analogues for 1,3,4. There is also a bonus property, which is clear from the transformation law displayed: 22\n\nδM gd: En/ 2−2 → E n/ 2−2[−4] is a non-trivial conformally invariant operator. In dimension 4 this is the Paneitz operator. So finally we need an analogue for property 2. It is clear that M g is conformally invariant as a map Cn/ 2−1 → E n/ 2−1[−2] /R(δ), so this is an analogue. But we can do more. There is no reason to suppose the image is co-closed. On the other hand note that δM g is conformally invariant on Cn/ 2−1 and so we have the following:\n\nM g: Hn/ 2−1 → Hn/ 2+1 (M ) with\n\nHn/ 2−1:= N (δM: Cn/ 2−1 → E n/ 2−2[−4]) (36) is conformally invariant. The space Hn/ 2−1 may be viewed as the space of \"conformal har-monics\". Evidently dim( Hn/ 2−1) is not always the same as the Betti number bn/ 2−1, but the elliptic coercivity of the pair ( d, δM ) gives it a good chance of returning the Betti number off a set of conformal structures that is somehow small. One should also check that the map (36) is non-trivial.\n\nFact: Let M = Sp × Sq, where p = n/ 2 − 1, q = n/ 2 + 1, with the standard Riemannian structure. Then φ ∈ H p if and only if φ is harmonic. Furthermore, the map (36) is non-trivial. In some recent work the authors have used the ambient metric, and its relationship to tractors, to show that the above construction generalises along the following lines: There are operators M gk: Ek → E n−k (k ≤ n/ 2 − 1), given by a uniform construction, with the following properties: A. M gk: Ck → E n−k has the conformal transformation law M ˆgk = M gk + Lkω, where Lk\n\nis a formally self-adjoint operator from k-forms to ( n − k)-forms, and is a constant multiple of dM gk+1 d.B. Hk:= N (dM gk: Ck → E n−k+1 ) is a conformally invariant subspace of Ck and Mk:\n\nHk → Hn−k(M ) is conformally invariant. There are conformal manifolds on which\n\nMk is non-trivial. C. If c ∈ C k and u ∈ N (Lk) then\n\n∫\n\n〈u, M gk c〉\n\nis conformally invariant. D. For each choice of metric g, M gk: Ek → E n−k is formally self-adjoint. E. M g\n\n> 0\n\n1 is the Q-curvature. From the uniqueness of the Maxwell operator at leading order (as a conformally invariant operator En/ 2−1 → E n/ 2−1[−2]), and the explicit formula (44) it is clear M gn/ 2−1 is not the difference between any conformally invariant differential operator and a divergence (even as an operator on closed forms). A similar argument applies to the Mk generally. Thus, from the point of view that the Q-curvature is a non-conformally invariant object that in a deep sense cannot be made conformally invariant, but one which nevertheless determines a global conformal invariant, the operators M gk give a genuine generalisation of the Q-curvature to an operator on closed forms. 23\n\nProblems k: There are analogues for the operators M gk of most of the conundrums and problems for the Q-curvature.\n\nChapter C: Open problems\n\nConformal Structure in Geometry, Analysis, and Physics\n\nAugust 12 to 16, 2003 at the American Institute of Mathematics, Palo Alto, California\n\nI. Problems suggested by the participants Thomas Branson. Anti-conformal perturbations.",
   "evidence": "This record is an OCR extraction of the final part of Chapter B of the 2003 AIM workshop list *Conformal Structure in Geometry, Analysis, and Physics*. The official HTML transcription and the source PDF resolve several important ambiguities.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 353,
   "attempt": 1
  },
  "AIM-GEOMETRY-0355": {
   "statement_status": "exact",
   "original_statement": "Problem 1a: Given any functional of the metric that is well understood conformally, is there information that can arise going across conformal classes? If the functional is the integral of a local invariant we can obtain information by computing its anti-conformal variation. If the functional is a nonlocal spectral invariant, like the functional determinant, then it is even a challenge to compute the anti-conformal deformation.",
   "clean_statement": "Problem 1a: Given any functional of the metric that is well understood conformally, is there information that can arise going across conformal classes? If the functional is the integral of a local invariant we can obtain information by computing its anti-conformal variation. If the functional is a nonlocal spectral invariant, like the functional determinant, then it is even a challenge to compute the anti-conformal deformation.",
   "public_statement": "Problem 1a: Given any functional of the metric that is well understood conformally, is there information that can arise going across conformal classes? If the functional is the integral of a local invariant we can obtain information by computing its anti-conformal variation. If the functional is a nonlocal spectral invariant, like the functional determinant, then it is even a challenge to compute the anti-conformal deformation.",
   "evidence": "The record comes from the AIM workshop *Conformal structure in geometry, analysis, and physics* (August 12--16, 2003), in the subsection headed “Thomas Branson. Anti-conformal perturbations.” The exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 354,
   "attempt": 1
  },
  "AIM-GEOMETRY-0356": {
   "statement_status": "exact",
   "original_statement": "Problem 1b: How to obtain the information that arises going across conformal classes?",
   "clean_statement": "Problem 1b: How to obtain the information that arises going across conformal classes?",
   "public_statement": "Problem 1b: How to obtain the information that arises going across conformal classes?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 355,
   "attempt": 1
  },
  "AIM-GEOMETRY-0357": {
   "statement_status": "exact",
   "original_statement": "Problem 1c: Study variational problems arising from conformally invariant problems. \n\nMichael Eastwood.",
   "clean_statement": "Problem 1c: Study variational problems arising from conformally invariant problems.\n\nMichael Eastwood.",
   "public_statement": "Problem 1c: Study variational problems arising from conformally invariant problems.\n\nMichael Eastwood.",
   "evidence": "The exact canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 356,
   "attempt": 1
  },
  "AIM-GEOMETRY-0358": {
   "statement_status": "exact",
   "original_statement": "Problem 2: Find an explicit relation between Q and Pff (R) in the conformally flat case.",
   "clean_statement": "Problem 2: Find an explicit relation between Q and Pff (R) in the conformally flat case.",
   "public_statement": "Problem 2: Find an explicit relation between Q and Pff (R) in the conformally flat case.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 357,
   "attempt": 1
  },
  "AIM-GEOMETRY-0359": {
   "statement_status": "exact",
   "original_statement": "Problem 3: Is there a global ambient metric construction?",
   "clean_statement": "Problem 3: Is there a global ambient metric construction?",
   "public_statement": "Problem 3: Is there a global ambient metric construction?",
   "evidence": "The canonical record comes from the AIM workshop *Conformal structure in geometry, analysis, and physics*. In the official workshop problem list it appears, under Michael Eastwood's contribution, exactly as",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 358,
   "attempt": 1
  },
  "AIM-GEOMETRY-0360": {
   "statement_status": "exact",
   "original_statement": "Problem 4: Can we explicitly write Q in dimension 6 uniquely as constant times Pff (R)plus local conformally invariant plus divergence? \n\nAnswer to problem 4: Robin Graham reports the answer to be YES. \n\nAlice Chang. General problems in conformal geometry:",
   "clean_statement": "Problem 4: Can we explicitly write Q in dimension 6 uniquely as constant times Pff (R)plus local conformally invariant plus divergence?\n\nAnswer to problem 4: Robin Graham reports the answer to be YES.\n\nAlice Chang. General problems in conformal geometry:",
   "public_statement": "Problem 4: Can we explicitly write Q in dimension 6 uniquely as constant times Pff (R)plus local conformally invariant plus divergence?\n\nAnswer to problem 4: Robin Graham reports the answer to be YES.\n\nAlice Chang. General problems in conformal geometry:",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 359,
   "attempt": 1
  },
  "AIM-GEOMETRY-0361": {
   "statement_status": "exact",
   "original_statement": "Problem 5a: How to decide which curvature invariants have a conformal primitive? For example on manifold M, we have ∆( J n/ 2−1) has 2 \n\n> n\n\nJn/ 2 as a conformal primitive, i.e. \n\n(∫ 2\n\nnJn/ 2\n\n)•\n\n(ω) = ∆( Jn/ 2−1)for all smooth function ω on M, see \"Origins, applications and generalizations of the Q-curvature\" by T. Branson and R. Gover. Available through http://www.aimath.org.",
   "clean_statement": "Problem 5a: How to decide which curvature invariants have a conformal primitive? For example on manifold M, we have ∆( J n/ 2−1) has 2\n\n> n\n\nJn/ 2 as a conformal primitive, i.e.\n\n(∫ 2\n\nnJn/ 2\n\n)•\n\n(ω) = ∆( Jn/ 2−1)for all smooth function ω on M, see \"Origins, applications and generalizations of the Q-curvature\" by T. Branson and R. Gover. Available through http://www.aimath.org.",
   "public_statement": "Problem 5a: How to decide which curvature invariants have a conformal primitive? For example on manifold M, we have ∆( J n/ 2−1) has 2\n\n> n\n\nJn/ 2 as a conformal primitive, i.e.\n\n(∫ 2\n\nnJn/ 2\n\n)•\n\n(ω) = ∆( Jn/ 2−1)for all smooth function ω on M, see \"Origins, applications and generalizations of the Q-curvature\" by T. Branson and R. Gover. Available through http://www.aimath.org.",
   "evidence": "The canonical record is visibly damaged. Its problem field is preserved verbatim here, including line breaks and the stray extraction marker:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 360,
   "attempt": 1
  },
  "AIM-GEOMETRY-0362": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 5b: What characterizes such curvature invariants? A related problem is posed by T. Branson: On M n, Q curvature is a local invariant (of density weight −n) which does not have a conformal primitive. The local invariants that have conformal primitives form a vector subspace, say L′, of the space of local invariants L. Thus the quotient space L/L′ is the 24 \n\nspace which measures \"how many things\" do not have a conformal primitive. There are also local conformal invariants, L′′ say.",
   "clean_statement": "**Problem 5b.** What characterizes such curvature invariants? A related problem is posed by T. Branson: On \\(M^n\\), \\(Q\\)-curvature is a local invariant (of density weight \\(-n\\)) which does not have a conformal primitive. The local invariants that have conformal primitives form a vector subspace, say \\(\\mathcal L'\\), of the space of local invariants \\(\\mathcal L\\). Thus the quotient space \\(\\mathcal L/\\mathcal L'\\) is the space which measures “how many things” do not have a conformal primitive. There are also local conformal invariants, \\(\\mathcal L''\\), say.",
   "public_statement": "**Problem 5b.** What characterizes such curvature invariants? A related problem is posed by T. Branson: On \\(M^n\\), \\(Q\\)-curvature is a local invariant (of density weight \\(-n\\)) which does not have a conformal primitive. The local invariants that have conformal primitives form a vector subspace, say \\(\\mathcal L'\\), of the space of local invariants \\(\\mathcal L\\). Thus the quotient space \\(\\mathcal L/\\mathcal L'\\) is the space which measures “how many things” do not have a conformal primitive. There are also local conformal invariants, \\(\\mathcal L''\\), say.",
   "evidence": "The canonical record is Problem 5b from the 2003 AIM workshop *Conformal structure in geometry, analysis, and physics*. The exact recovered statement is: This recovery was checked against both the official AIM HTML rendering and the official PDF. Three extraction defects in the canonical JSON are thereby resolved:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 361,
   "attempt": 1
  },
  "AIM-GEOMETRY-0363": {
   "statement_status": "exact",
   "original_statement": "Problem 6: Is L/(L′ + L′′ ) one-dimensional and generated by the class of Q?",
   "clean_statement": "Problem 6: Is L/(L′ + L′′ ) one-dimensional and generated by the class of Q?",
   "public_statement": "Problem 6: Is L/(L′ + L′′ ) one-dimensional and generated by the class of Q?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 362,
   "attempt": 1
  },
  "AIM-GEOMETRY-0364": {
   "statement_status": "exact",
   "original_statement": "Problem 7: On M 4, Gursky (\"The principal eigenvalue of a conformally invariant dif-ferential operator, with an application to semilinear elliptic PDE.\" Comm. Math. Phys., 207(1):131-143, 1999.) proved that if Sc > 0, and if ∫ Q > 0, then the Paneitz operator \n\nP4 is positive with its kernel consisting of constants. The original proof given by Gursky depends on estimates of solution of some non-linear PDE. Can one also see this fact from the construction method of the general GJMS operators? \n\nClaude LeBrun.",
   "clean_statement": "Problem 7: On M 4, Gursky (\"The principal eigenvalue of a conformally invariant dif-ferential operator, with an application to semilinear elliptic PDE.\" Comm. Math. Phys., 207(1):131-143, 1999.) proved that if Sc > 0, and if ∫ Q > 0, then the Paneitz operator\n\nP4 is positive with its kernel consisting of constants. The original proof given by Gursky depends on estimates of solution of some non-linear PDE. Can one also see this fact from the construction method of the general GJMS operators?\n\nClaude LeBrun.",
   "public_statement": "Problem 7: On M 4, Gursky (\"The principal eigenvalue of a conformally invariant dif-ferential operator, with an application to semilinear elliptic PDE.\" Comm. Math. Phys., 207(1):131-143, 1999.) proved that if Sc > 0, and if ∫ Q > 0, then the Paneitz operator\n\nP4 is positive with its kernel consisting of constants. The original proof given by Gursky depends on estimates of solution of some non-linear PDE. Can one also see this fact from the construction method of the general GJMS operators?\n\nClaude LeBrun.",
   "evidence": "The exact corpus record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 363,
   "attempt": 1
  },
  "AIM-GEOMETRY-0365": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 8: Explicitly expess the Gauss-Bonnet integrand as a sum of σn/ 2(P) plus terms in-volving the Weyl curvature, and then use this to explicitly understand relationships between \n\nQ and topology.",
   "clean_statement": "Problem 8: Explicitly in-volving, the Gauss-Bonnet integrand as a sum of σn/ 2(P) plus terms in-volving the Weyl curvature, and then use this to explicitly understand relationships between\n\nQ and topology.",
   "public_statement": "Problem 8: Explicitly in-volving, the Gauss-Bonnet integrand as a sum of σn/ 2(P) plus terms in-volving the Weyl curvature, and then use this to explicitly understand relationships between\n\nQ and topology.",
   "evidence": "The official AIM PDF and its HTML rendering were checked. The typo “expess” occurs in both official versions and presumably means “express”; this report does not silently treat it as an extraction error. The PDF line break accounts for “in-volving,” which the HTML renders as “involving.” The typography `σn/ 2(P)` means \\(\\sigma_{n/2}(P)\\). Thus the recovered mathematical request is: Here a complete answer is given for the first clause as a finite contraction formula in every even dimension, and a convention-complete answer to both clauses is proved in dimension four. A complete all-even-dimensional simplification of every Weyl correction is not claimed.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-geometry-notes.json",
   "source_index": 364,
   "attempt": 1
  },
  "AIM-GEOMETRY-0366": {
   "statement_status": "exact",
   "original_statement": "Problem 9: Given a compact manifold of even dimension > 2, show that there exists a sequence of metrics such that ∫ Q → +∞.\n\nRobin Graham",
   "clean_statement": "Problem 9: Given a compact manifold of even dimension > 2, show that there exists a sequence of metrics such that ∫ Q → +∞.\n\nRobin Graham",
   "public_statement": "Problem 9: Given a compact manifold of even dimension > 2, show that there exists a sequence of metrics such that ∫ Q → +∞.\n\nRobin Graham",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 365,
   "attempt": 1
  },
  "AIM-GEOMETRY-0367": {
   "statement_status": "exact",
   "original_statement": "Problem 10: If n ≥ 4 is even, is there a nonzero scalar conformal invariant of weight −n\n\nwhich is expressible as a linear combination of complete contractions of the tensors ∇lP, \n\nl ≥ 0? If the answer to this question is no, then the Q-curvature defined via the ambient metric construction is uniquely determined by its transformation law in terms of the GJMS operator \n\nPn and the fact that it can be written just in terms of P and its derivatives. The answer is no if n = 4. It is worth pointing out that there are scalar conformal invariants of more negative weight which can be so expressed: the norm squared of the Bach tensor is of this form if \n\nn = 4, as is the norm squared of the ambient obstruction tensor in higher even dimensions.",
   "clean_statement": "Problem 10: If n ≥ 4 is even, is there a nonzero scalar conformal invariant of weight −n\n\nwhich is expressible as a linear combination of complete contractions of the tensors ∇lP,\n\nl ≥ 0? If the answer to this question is no, then the Q-curvature defined via the ambient metric construction is uniquely determined by its transformation law in terms of the GJMS operator\n\nPn and the fact that it can be written just in terms of P and its derivatives. The answer is no if n = 4. It is worth pointing out that there are scalar conformal invariants of more negative weight which can be so expressed: the norm squared of the Bach tensor is of this form if\n\nn = 4, as is the norm squared of the ambient obstruction tensor in higher even dimensions.",
   "public_statement": "Problem 10: If n ≥ 4 is even, is there a nonzero scalar conformal invariant of weight −n\n\nwhich is expressible as a linear combination of complete contractions of the tensors ∇lP,\n\nl ≥ 0? If the answer to this question is no, then the Q-curvature defined via the ambient metric construction is uniquely determined by its transformation law in terms of the GJMS operator\n\nPn and the fact that it can be written just in terms of P and its derivatives. The answer is no if n = 4. It is worth pointing out that there are scalar conformal invariants of more negative weight which can be so expressed: the norm squared of the Bach tensor is of this form if\n\nn = 4, as is the norm squared of the ambient obstruction tensor in higher even dimensions.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 366,
   "attempt": 1
  },
  "AIM-GEOMETRY-0368": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 11: If n ≥ 4 is even, is the GJMS operator Pn the only natural differential operator with principal part ∆ n/ 2 whose coefficients can be expressed purely in terms of the tensors \n\n∇lP, l ≥ 0, and which is conformally invariant from E(0) to E(−n)? If the answer is yes, then this gives a characterization of the GJMS operator Pn. Combined with a negative answer to",
   "clean_statement": null,
   "public_statement": "Problem 11: If n ≥ 4 is even, is the GJMS operator Pn the only natural differential operator with principal part ∆ n/ 2 whose coefficients can be expressed purely in terms of the tensors\n\n∇lP, l ≥ 0, and which is conformally invariant from E(0) to E(−n)? If the answer is yes, then this gives a characterization of the GJMS operator Pn. Combined with a negative answer to",
   "evidence": "This reconstruction is source-verified, but the self-reference is almost certainly a typographical error: the preceding Problem 10 asks whether a weight-\\(-n\\) scalar conformal invariant can be made purely from \\(\\nabla^lP\\), and that is exactly the complementary condition needed to specify \\(Q\\). Replacing the printed second “Problem 11” by “Problem 10” is therefore a logical inference, not verified wording.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 367,
   "attempt": 1
  },
  "AIM-GEOMETRY-0369": {
   "statement_status": "exact",
   "original_statement": "Problem 11, this would provide a unique specification of Q.\n\nRod Gover Alice Chang and Jie Qing have an order 3 operator P3, on 3-manifolds (boundary of a 4-dimensional manifold, or embedded in a 4-dimensional manifold). There is a version of Q3 associated to this P3.",
   "clean_statement": "Problem 11, this would provide a unique specification of Q.\n\nRod Gover Alice Chang and Jie Qing have an order 3 operator P3, on 3-manifolds (boundary of a 4-dimensional manifold, or embedded in a 4-dimensional manifold). There is a version of Q3 associated to this P3.",
   "public_statement": "Problem 11, this would provide a unique specification of Q.\n\nRod Gover Alice Chang and Jie Qing have an order 3 operator P3, on 3-manifolds (boundary of a 4-dimensional manifold, or embedded in a 4-dimensional manifold). There is a version of Q3 associated to this P3.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 368,
   "attempt": 1
  },
  "AIM-GEOMETRY-0370": {
   "statement_status": "exact",
   "original_statement": "Problem 12: What sort of information is encoded by Q3 and/or ∫ Q3?25 \n\nHelga Baum On a spin manifold ( M, g ) with spin bundle S, we have two conformally covariant operators. The Dirac operator Dg and the twistor operator Pg. If ∇S represents the spin connection then, \n\n∇S: Γ( S) → Γ( T ∗M ⊗ S) ∼= Γ( S) ⊕ Γ( Tw)and we define Dg = pr 1∇S and Pg = pr 2∇S, with pr i the projection on the i-th factor. Let h(g) be the dimension of ker( Dg) (harmonic spinors) and let t(g) be the dimension of ker( Pg) (twistor spinors/conformal Killing spinors). Both numbers are conformal invariants. In case of Riemannian conformal structures these invariants are rather well studied. In the Lorentzian case much less is known.",
   "clean_statement": "Problem 12: What sort of information is encoded by Q3 and/or ∫ Q3?25\n\nHelga Baum On a spin manifold ( M, g ) with spin bundle S, we have two conformally covariant operators. The Dirac operator Dg and the twistor operator Pg. If ∇S represents the spin connection then,\n\n∇S: Γ( S) → Γ( T ∗M ⊗ S) ∼= Γ( S) ⊕ Γ( Tw)and we define Dg = pr 1∇S and Pg = pr 2∇S, with pr i the projection on the i-th factor. Let h(g) be the dimension of ker( Dg) (harmonic spinors) and let t(g) be the dimension of ker( Pg) (twistor spinors/conformal Killing spinors). Both numbers are conformal invariants. In case of Riemannian conformal structures these invariants are rather well studied. In the Lorentzian case much less is known.",
   "public_statement": "Problem 12: What sort of information is encoded by Q3 and/or ∫ Q3?25\n\nHelga Baum On a spin manifold ( M, g ) with spin bundle S, we have two conformally covariant operators. The Dirac operator Dg and the twistor operator Pg. If ∇S represents the spin connection then,\n\n∇S: Γ( S) → Γ( T ∗M ⊗ S) ∼= Γ( S) ⊕ Γ( Tw)and we define Dg = pr 1∇S and Pg = pr 2∇S, with pr i the projection on the i-th factor. Let h(g) be the dimension of ker( Dg) (harmonic spinors) and let t(g) be the dimension of ker( Pg) (twistor spinors/conformal Killing spinors). Both numbers are conformal invariants. In case of Riemannian conformal structures these invariants are rather well studied. In the Lorentzian case much less is known.",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 369,
   "attempt": 1
  },
  "AIM-GEOMETRY-0371": {
   "statement_status": "exact",
   "original_statement": "Problem 13: Find all Lorentzian conformal structures ( M, [g]) with t(g) > 0 or h(g) > 0.",
   "clean_statement": "Problem 13: Find all Lorentzian conformal structures ( M, [g]) with t(g) > 0 or h(g) > 0.",
   "public_statement": "Problem 13: Find all Lorentzian conformal structures ( M, [g]) with t(g) > 0 or h(g) > 0.",
   "evidence": "The AIM source states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 370,
   "attempt": 1
  },
  "AIM-GEOMETRY-0372": {
   "statement_status": "exact",
   "original_statement": "Problem 14: How t(g) and h(g) relate to other conformal invariants?",
   "clean_statement": "Problem 14: How t(g) and h(g) relate to other conformal invariants?",
   "public_statement": "Problem 14: How t(g) and h(g) relate to other conformal invariants?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 371,
   "attempt": 1
  },
  "AIM-GEOMETRY-0373": {
   "statement_status": "exact",
   "original_statement": "Problem 15: Relate t(g) to the holonomy of conformal Cartan connections.",
   "clean_statement": "Problem 15: Relate t(g) to the holonomy of conformal Cartan connections.",
   "public_statement": "Problem 15: Relate t(g) to the holonomy of conformal Cartan connections.",
   "evidence": "The official AIM workshop PDF, version 15 October 2003, states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 372,
   "attempt": 1
  },
  "AIM-GEOMETRY-0374": {
   "statement_status": "exact",
   "original_statement": "Problem 16: Relate h(g) to the dynamic of null geodesics.",
   "clean_statement": "Problem 16: Relate h(g) to the dynamic of null geodesics.",
   "public_statement": "Problem 16: Relate h(g) to the dynamic of null geodesics.",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 373,
   "attempt": 1
  },
  "AIM-GEOMETRY-0375": {
   "statement_status": "unrecoverable",
   "original_statement": "Problem 17: Describe conformally flat Lorentzian manifolds with h(g) > 0 or t(g) > 0. \n\nII. Problems extracted from the document \"A Primer on Q-curvature\" by M. Eastwood and J. Slov` ack. 1\n\nIn the conformally flat case, locally by setting gab = Ω 2ηab where ηab is flat, then \n\nQ = ∆ n/ 2 log Ω, (37) where ∆ is the ordinary Laplacian in Euclidean space with ηab as metric. For this construction of Q to be well-defined it is necessary that, if also gab =̂ Ω2̂ ηab, then ∆n/ 2 log Ω = ̂ ∆n/ 2 log ̂ Ω.\n\nThis reduces to two facts:- \n\nfact 1:: ∆n/ 2 is conformally invariant on flat space. \n\nfact 2:: if gab is itself flat, then ∆ n/ 2 log Ω = 0. The second of these is necessary in order that (37) be well-defined. There is a Lie alge-braic proof of fact 1. It corresponds to the existence of a homomorphism between certain generalized Verma modules for so (n + 1, 1).",
   "clean_statement": null,
   "public_statement": "Problem 17: Describe conformally flat Lorentzian manifolds with h(g) > 0 or t(g) > 0.\n\nII. Problems extracted from the document \"A Primer on Q-curvature\" by M. Eastwood and J. Slov` ack. 1\n\nIn the conformally flat case, locally by setting gab = Ω 2ηab where ηab is flat, then\n\nQ = ∆ n/ 2 log Ω, (37) where ∆ is the ordinary Laplacian in Euclidean space with ηab as metric. For this construction of Q to be well-defined it is necessary that, if also gab =̂ Ω2̂ ηab, then ∆n/ 2 log Ω = ̂ ∆n/ 2 log ̂ Ω.\n\nThis reduces to two facts:-\n\nfact 1:: ∆n/ 2 is conformally invariant on flat space.\n\nfact 2:: if gab is itself flat, then ∆ n/ 2 log Ω = 0. The second of these is necessary in order that (37) be well-defined. There is a Lie alge-braic proof of fact 1. It corresponds to the existence of a homomorphism between certain generalized Verma modules for so (n + 1, 1).",
   "evidence": "This merged extraction is not one mathematical problem. The official AIM HTML places Problem 17 on line 70 and begins a new section, “II. Problems extracted from the document ‘A Primer on Q-curvature’,” on line 71. The Q-curvature paragraphs therefore belong to the following section and are extraction spillover. They are preserved above without substantive correction, including OCR and accent errors, but are not used below.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-geometry-notes.json",
   "source_index": 374,
   "attempt": 1
  },
  "AIM-GEOMETRY-0376": {
   "statement_status": "exact",
   "original_statement": "Problem 18: Deduce fact 2 from fact 1 or vice versa. Alternatively, construct a Lie algebraic proof of fact 2.About a formula for Q, Eastwood and Slov´ ack have deduce: ∆2 log Ω = −̂ ∆̂ P − (n − 2) ̂ Pab ̂ Pab + 2 ̂ P2\n\n+ 2( n − 4)Υ â ∇â P + 2( n − 4)Υ aΥâ P\n\n− (n − 2)( n − 4)Υ aΥb̂ Pab + 1 \n\n> 4\n\n(n − 2)( n − 4)Υ aΥaΥbΥb.\n\n(38) \n\n> 1This section is the recompilation of the conundra in that document. Refer to the original for more details. 26\n\nThough it is only guaranteed that this formula is valid in the conformally flat case, in fact it agrees with the general expression in dimension 4, \n\nQ = 2P 2 − 2P ab Pab − ∆P. (39) It is possible, by further differentiating (38), to obtain a formula for ∆ k log Ω expressed in terms of complete contractions of ̂ Pab, its hatted derivatives, and Υ a. With increasing k,this gets rapidly out of hand. Moreover, it is only guaranteed to give Q in the conformally flat case. Indeed, when n = 6 this naive derivation of Q fails for a general metric.",
   "clean_statement": "Problem 18: Deduce fact 2 from fact 1 or vice versa. Alternatively, construct a Lie algebraic proof of fact 2.About a formula for Q, Eastwood and Slov´ ack have deduce: ∆2 log Ω = −̂ ∆̂ P − (n − 2) ̂ Pab ̂ Pab + 2 ̂ P2\n\n+ 2( n − 4)Υ â ∇â P + 2( n − 4)Υ aΥâ P\n\n− (n − 2)( n − 4)Υ aΥb̂ Pab + 1\n\n> 4\n\n(n − 2)( n − 4)Υ aΥaΥbΥb.\n\n(38)\n\n> 1This section is the recompilation of the conundra in that document. Refer to the original for more details. 26\n\nThough it is only guaranteed that this formula is valid in the conformally flat case, in fact it agrees with the general expression in dimension 4,\n\nQ = 2P 2 − 2P ab Pab − ∆P. (39) It is possible, by further differentiating (38), to obtain a formula for ∆ k log Ω expressed in terms of complete contractions of ̂ Pab, its hatted derivatives, and Υ a. With increasing k,this gets rapidly out of hand. Moreover, it is only guaranteed to give Q in the conformally flat case. Indeed, when n = 6 this naive derivation of Q fails for a general metric.",
   "public_statement": "Problem 18: Deduce fact 2 from fact 1 or vice versa. Alternatively, construct a Lie algebraic proof of fact 2.About a formula for Q, Eastwood and Slov´ ack have deduce: ∆2 log Ω = −̂ ∆̂ P − (n − 2) ̂ Pab ̂ Pab + 2 ̂ P2\n\n+ 2( n − 4)Υ â ∇â P + 2( n − 4)Υ aΥâ P\n\n− (n − 2)( n − 4)Υ aΥb̂ Pab + 1\n\n> 4\n\n(n − 2)( n − 4)Υ aΥaΥbΥb.\n\n(38)\n\n> 1This section is the recompilation of the conundra in that document. Refer to the original for more details. 26\n\nThough it is only guaranteed that this formula is valid in the conformally flat case, in fact it agrees with the general expression in dimension 4,\n\nQ = 2P 2 − 2P ab Pab − ∆P. (39) It is possible, by further differentiating (38), to obtain a formula for ∆ k log Ω expressed in terms of complete contractions of ̂ Pab, its hatted derivatives, and Υ a. With increasing k,this gets rapidly out of hand. Moreover, it is only guaranteed to give Q in the conformally flat case. Indeed, when n = 6 this naive derivation of Q fails for a general metric.",
   "evidence": "The canonical record is aim-geometry-notes.json, zero-based index 375. Its problem field is preserved verbatim in input.json. The beginning of that field is",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 375,
   "attempt": 1
  },
  "AIM-GEOMETRY-0377": {
   "statement_status": "exact",
   "original_statement": "Problem 19: Find a formula for Q in the conformally flat case. Show that the procedure outlined by Eastwood and Slov´ ack produces a formula for Q.In the conformally flat case, it follows from a theorem of Branson, Gilkey, and Pohjan-pelto that Q must be a multiple of the Pfaffian plus a divergence.",
   "clean_statement": "Problem 19: Find a formula for Q in the conformally flat case. Show that the procedure outlined by Eastwood and Slov´ ack produces a formula for Q.In the conformally flat case, it follows from a theorem of Branson, Gilkey, and Pohjan-pelto that Q must be a multiple of the Pfaffian plus a divergence.",
   "public_statement": "Problem 19: Find a formula for Q in the conformally flat case. Show that the procedure outlined by Eastwood and Slov´ ack produces a formula for Q.In the conformally flat case, it follows from a theorem of Branson, Gilkey, and Pohjan-pelto that Q must be a multiple of the Pfaffian plus a divergence.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 376,
   "attempt": 1
  },
  "AIM-GEOMETRY-0378": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 20: Find a direct link between Q and the Pfaffian in the conformally flat case. Prove directly that ∫ \n\n> M\n\nQ is a topological invariant in this case.",
   "clean_statement": "**Problem 20.** Find a direct link between \\(Q\\) and the Pfaffian in the conformally flat case. Prove directly that \\(\\int_M Q\\) is a topological invariant in this case.",
   "public_statement": "Problem 20: Find a direct link between Q and the Pfaffian in the conformally flat case. Prove directly that ∫\n\n> M\n\nQ is a topological invariant in this case.",
   "evidence": "The exact canonical record is preserved in input.json. Its problem field, with line breaks rendered and no correction of the extraction marker, is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 377,
   "attempt": 1
  },
  "AIM-GEOMETRY-0379": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 21: Is it true that, on a general Riemannian manifold, Q may be written as a multiple of the Pfaffian plus a local conformal invariant plus a divergence? See",
   "clean_statement": null,
   "public_statement": "Problem 21: Is it true that, on a general Riemannian manifold, Q may be written as a multiple of the Pfaffian plus a local conformal invariant plus a divergence? See",
   "evidence": "The exact canonical record in aim-geometry-notes.json, zero-based index 378, is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 378,
   "attempt": 1
  },
  "AIM-GEOMETRY-0380": {
   "statement_status": "exact",
   "original_statement": "Problem 4 for the 6 dimensional case. Also, T. Branson has appointed that if it is true that any local invariant L of density weight −n has the form constant LPff + divergence L + (local conformal invariant) L\n\nwhere L signals the dependence on L then, in this decomposition for Q, we have constant Q 6 =0. In fact we know constant Q exactly, since we know (the constant values of) Q and Pff on the sphere. \n\nHow is Q-curvature related to Weyl structures? Q may be defined for a Weyl structure as follows. Since Q is a Riemannian invariant, the differential operator P is necessarily of the form f 7 → Sa∇af for some Riemannian invariant linear differential operator from 1-forms to \n\nn-forms. Now, if [ gab, α a] is a Weyl structure, choose a representative metric gab and consider the n-form \n\nQ − Saαa,\n\nwhere Q is the Riemannian Q-curvature associated to gab and αa is the 1-form associated to gab. If ̂ gab = Ω 2gab, then ̂\n\nQ −̂ Sâ αa = Q −̂ Saαa.\n\nIn dimension 4, Eastwood and Slov´ ack have appointed that \n\nQ − Saαa + 4 ∇a(αb∇[aαb])is an invariant of the Weyl structure that agrees with Q when the Weyl structure arises from a Riemannian structure.",
   "clean_statement": "Problem 4 for the 6 dimensional case. Also, T. Branson has appointed that if it is true that any local invariant L of density weight −n has the form constant LPff + divergence L + (local conformal invariant) L\n\nwhere L signals the dependence on L then, in this decomposition for Q, we have constant Q 6 =0. In fact we know constant Q exactly, since we know (the constant values of) Q and Pff on the sphere.\n\nHow is Q-curvature related to Weyl structures? Q may be defined for a Weyl structure as follows. Since Q is a Riemannian invariant, the differential operator P is necessarily of the form f 7 → Sa∇af for some Riemannian invariant linear differential operator from 1-forms to\n\nn-forms. Now, if [ gab, α a] is a Weyl structure, choose a representative metric gab and consider the n-form\n\nQ − Saαa,\n\nwhere Q is the Riemannian Q-curvature associated to gab and αa is the 1-form associated to gab. If ̂ gab = Ω 2gab, then ̂\n\nQ −̂ Sâ αa = Q −̂ Saαa.\n\nIn dimension 4, Eastwood and Slov´ ack have appointed that\n\nQ − Saαa + 4 ∇a(αb∇[aαb])is an invariant of the Weyl structure that agrees with Q when the Weyl structure arises from a Riemannian structure.",
   "public_statement": "Problem 4 for the 6 dimensional case. Also, T. Branson has appointed that if it is true that any local invariant L of density weight −n has the form constant LPff + divergence L + (local conformal invariant) L\n\nwhere L signals the dependence on L then, in this decomposition for Q, we have constant Q 6 =0. In fact we know constant Q exactly, since we know (the constant values of) Q and Pff on the sphere.\n\nHow is Q-curvature related to Weyl structures? Q may be defined for a Weyl structure as follows. Since Q is a Riemannian invariant, the differential operator P is necessarily of the form f 7 → Sa∇af for some Riemannian invariant linear differential operator from 1-forms to\n\nn-forms. Now, if [ gab, α a] is a Weyl structure, choose a representative metric gab and consider the n-form\n\nQ − Saαa,\n\nwhere Q is the Riemannian Q-curvature associated to gab and αa is the 1-form associated to gab. If ̂ gab = Ω 2gab, then ̂\n\nQ −̂ Sâ αa = Q −̂ Saαa.\n\nIn dimension 4, Eastwood and Slov´ ack have appointed that\n\nQ − Saαa + 4 ∇a(αb∇[aαb])is an invariant of the Weyl structure that agrees with Q when the Weyl structure arises from a Riemannian structure.",
   "evidence": "The canonical record is a merged extraction from the 2003 AIM workshop document *Conformal Structure in Geometry, Analysis, and Physics*. It contains two logically separate pieces.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 379,
   "attempt": 1
  },
  "AIM-GEOMETRY-0381": {
   "statement_status": "exact",
   "original_statement": "Problem 22: Can we find such a Q in general even dimensions? Presumably, this would restrict the choice of Riemannian Q.27 \n\nThough Q is an invariant of the Weyl structure, it is not manifestly so. With a detailed calculation, Eastwood and Slov'ack have shown that in dimension 4: \n\nQ = 2 P2 − 2Pab Pba − DaDaP\n\na manifest invariant of the Weyl structure.",
   "clean_statement": "Problem 22: Can we find such a Q in general even dimensions? Presumably, this would restrict the choice of Riemannian Q.27\n\nThough Q is an invariant of the Weyl structure, it is not manifestly so. With a detailed calculation, Eastwood and Slov'ack have shown that in dimension 4:\n\nQ = 2 P2 − 2Pab Pba − DaDaP\n\na manifest invariant of the Weyl structure.",
   "public_statement": "Problem 22: Can we find such a Q in general even dimensions? Presumably, this would restrict the choice of Riemannian Q.27\n\nThough Q is an invariant of the Weyl structure, it is not manifestly so. With a detailed calculation, Eastwood and Slov'ack have shown that in dimension 4:\n\nQ = 2 P2 − 2Pab Pba − DaDaP\n\na manifest invariant of the Weyl structure.",
   "evidence": "The exact canonical problem field is preserved in input.json. With its line breaks displayed, it reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 380,
   "attempt": 1
  },
  "AIM-GEOMETRY-0382": {
   "statement_status": "exact",
   "original_statement": "Problem 23: Did we really need to go through that detailed calculation? What are the implications, if any, for the operator S: 1-forms → 4-forms?",
   "clean_statement": "Problem 23: Did we really need to go through that detailed calculation? What are the implications, if any, for the operator S: 1-forms → 4-forms?",
   "public_statement": "Problem 23: Did we really need to go through that detailed calculation? What are the implications, if any, for the operator S: 1-forms → 4-forms?",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 381,
   "attempt": 1
  },
  "AIM-GEOMETRY-0383": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 24 a: Can we characterise the Riemannian Q by sufficiently many properties?",
   "clean_statement": "Find a geometrically meaningful, noncircular package of properties that\nuniquely selects Branson's critical \\(Q\\)-curvature from other natural\nweight-\\(-n\\) Riemannian scalar densities. Determine which familiar\nproperties fail to give uniqueness and what additional normalization removes\nthe ambiguity.",
   "public_statement": "Find a geometrically meaningful, noncircular package of properties that\nuniquely selects Branson's critical \\(Q\\)-curvature from other natural\nweight-\\(-n\\) Riemannian scalar densities. Determine which familiar\nproperties fail to give uniqueness and what additional normalization removes\nthe ambiguity.",
   "evidence": "The official AIM HTML reproduces exactly this sentence, including the British spelling “characterise,” as Problem 24a. There is no evident OCR corruption in this record. The nearby official text determines what the short question is asking. Problems 22 and 23 discuss extending Branson's Riemannian \\(Q\\)-curvature form to Weyl structures. Problem 24b immediately asks whether Weyl structures help with the characterization and records Branson's cocycle",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 382,
   "attempt": 1
  },
  "AIM-GEOMETRY-0384": {
   "statement_status": "exact",
   "original_statement": "Problem 24 b: Do Weyl structures help in this regard? Tom Branson has suggested that, for two metrics g and ̂ g = Ω 2g in the same conformal class on a compact manifold M, one should consider the quantity \n\nH[̂g, g ] = \n\n∫\n\n> M\n\n(log Ω)( ̂ Q + Q).\n\nThat it is a cocycle, \n\nH[̂̂ g, ̂ g] + H[̂g, g ] = H[̂̂ g, g ],\n\nis easily seen to be equivalent to the GJMS operators P being self-adjoint.",
   "clean_statement": "Problem 24 b: Do Weyl structures help in this regard? Tom Branson has suggested that, for two metrics g and ̂ g = Ω 2g in the same conformal class on a compact manifold M, one should consider the quantity\n\nH[̂g, g ] =\n\n∫\n\n> M\n\n(log Ω)( ̂ Q + Q).\n\nThat it is a cocycle,\n\nH[̂̂ g, ̂ g] + H[̂g, g ] = H[̂̂ g, g ],\n\nis easily seen to be equivalent to the GJMS operators P being self-adjoint.",
   "public_statement": "Problem 24 b: Do Weyl structures help in this regard? Tom Branson has suggested that, for two metrics g and ̂ g = Ω 2g in the same conformal class on a compact manifold M, one should consider the quantity\n\nH[̂g, g ] =\n\n∫\n\n> M\n\n(log Ω)( ̂ Q + Q).\n\nThat it is a cocycle,\n\nH[̂̂ g, ̂ g] + H[̂g, g ] = H[̂̂ g, g ],\n\nis easily seen to be equivalent to the GJMS operators P being self-adjoint.",
   "evidence": "The canonical record is Problem 24(b) from the AIM workshop *Conformal structure in geometry, analysis, and physics*. With the damaged PDF extraction normalized, it asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 383,
   "attempt": 1
  },
  "AIM-GEOMETRY-0385": {
   "statement_status": "exact",
   "original_statement": "Problem 25: Are there any deeper properties of Branson's cocycle H[̂g, g ]? One possible rˆ ole for Q is in a curvature prescription problem:",
   "clean_statement": "Problem 25: Are there any deeper properties of Branson's cocycle H[̂g, g ]? One possible rˆ ole for Q is in a curvature prescription problem:",
   "public_statement": "Problem 25: Are there any deeper properties of Branson's cocycle H[̂g, g ]? One possible rˆ ole for Q is in a curvature prescription problem:",
   "evidence": "The exact canonical problem string is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 384,
   "attempt": 1
  },
  "AIM-GEOMETRY-0386": {
   "statement_status": "exact",
   "original_statement": "Problem 26: On a given manifold M, can one find a metric with specified Q?One can also ask this question within a given conformal class or within the realm of confor-mally flat metrics though, of course, if M is compact, then ∫ \n\n> M\n\nQ must be as specified by the conformal class and the topology of M.",
   "clean_statement": "Problem 26: On a given manifold M, can one find a metric with specified Q?One can also ask this question within a given conformal class or within the realm of confor-mally flat metrics though, of course, if M is compact, then ∫\n\n> M\n\nQ must be as specified by the conformal class and the topology of M.",
   "public_statement": "Problem 26: On a given manifold M, can one find a metric with specified Q?One can also ask this question within a given conformal class or within the realm of confor-mally flat metrics though, of course, if M is compact, then ∫\n\n> M\n\nQ must be as specified by the conformal class and the topology of M.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 385,
   "attempt": 1
  },
  "AIM-GEOMETRY-0387": {
   "statement_status": "exact",
   "original_statement": "Problem 27: When does Q determine the metric up to constant rescaling within a given conformal class? Since we know how Q changes under conformal rescaling: ̂\n\nQ = Q + P log Ω,\n\nwhere P is a linear differential operator from functions to n-forms whose symbol is ∆ n/ 2 this question is equivalent to",
   "clean_statement": "Problem 27: When does Q determine the metric up to constant rescaling within a given conformal class? Since we know how Q changes under conformal rescaling: ̂\n\nQ = Q + P log Ω,\n\nwhere P is a linear differential operator from functions to n-forms whose symbol is ∆ n/ 2 this question is equivalent to",
   "public_statement": "Problem 27: When does Q determine the metric up to constant rescaling within a given conformal class? Since we know how Q changes under conformal rescaling: ̂\n\nQ = Q + P log Ω,\n\nwhere P is a linear differential operator from functions to n-forms whose symbol is ∆ n/ 2 this question is equivalent to",
   "evidence": "The canonical record (source index 386 of aim-geometry-notes.json) reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 386,
   "attempt": 1
  },
  "AIM-GEOMETRY-0388": {
   "statement_status": "exact",
   "original_statement": "Problem 28: When does the equation P f = 0 have only constant solutions? On a compact manifold in two dimensions this is always true: harmonic functions are con-stant. In four dimensions, though there are conditions under which P f = 0 has only constant solutions, there are also counterexamples, even on conformally flat manifolds. 28 \n\nIII. Problems extracted from the document \"Origins, applications, and general-izations of the Q-curvature\" by T. Branson and R. Gover. 2\n\nLet A be a natural differential operator with positive definite leading symbol, and suppose \n\nA is a positive power of a conformally invariant operator. For example, A could be one of the GJMS operators, or it could be the square of the Dirac operator. Then in dimensions 2,4,6, \n\n− log det ˆA\n\ndet A = α\n\n{1\n\n2\n\n∫\n\nωP ω dv +\n\n∫\n\nωQ dv \n\n}\n\n+\n\n∫ (F dv − F dv ) + H, (40) where α is a constant, F is a local scalar invariant, and H is a term depending on the null space of A. In particular, if the conformally invariant condition N (A) = 0 is satisfied, then H = 0. The determinant involved is the zeta-regularized functional determinant of a positively elliptic operator.",
   "clean_statement": "Problem 28: When does the equation P f = 0 have only constant solutions? On a compact manifold in two dimensions this is always true: harmonic functions are con-stant. In four dimensions, though there are conditions under which P f = 0 has only constant solutions, there are also counterexamples, even on conformally flat manifolds. 28\n\nIII. Problems extracted from the document \"Origins, applications, and general-izations of the Q-curvature\" by T. Branson and R. Gover. 2\n\nLet A be a natural differential operator with positive definite leading symbol, and suppose\n\nA is a positive power of a conformally invariant operator. For example, A could be one of the GJMS operators, or it could be the square of the Dirac operator. Then in dimensions 2,4,6,\n\n− log det ˆA\n\ndet A = α\n\n{1\n\n2\n\n∫\n\nωP ω dv +\n\n∫\n\nωQ dv\n\n}\n\n+\n\n∫ (F dv − F dv ) + H, (40) where α is a constant, F is a local scalar invariant, and H is a term depending on the null space of A. In particular, if the conformally invariant condition N (A) = 0 is satisfied, then H = 0. The determinant involved is the zeta-regularized functional determinant of a positively elliptic operator.",
   "public_statement": "Problem 28: When does the equation P f = 0 have only constant solutions? On a compact manifold in two dimensions this is always true: harmonic functions are con-stant. In four dimensions, though there are conditions under which P f = 0 has only constant solutions, there are also counterexamples, even on conformally flat manifolds. 28\n\nIII. Problems extracted from the document \"Origins, applications, and general-izations of the Q-curvature\" by T. Branson and R. Gover. 2\n\nLet A be a natural differential operator with positive definite leading symbol, and suppose\n\nA is a positive power of a conformally invariant operator. For example, A could be one of the GJMS operators, or it could be the square of the Dirac operator. Then in dimensions 2,4,6,\n\n− log det ˆA\n\ndet A = α\n\n{1\n\n2\n\n∫\n\nωP ω dv +\n\n∫\n\nωQ dv\n\n}\n\n+\n\n∫ (F dv − F dv ) + H, (40) where α is a constant, F is a local scalar invariant, and H is a term depending on the null space of A. In particular, if the conformally invariant condition N (A) = 0 is satisfied, then H = 0. The determinant involved is the zeta-regularized functional determinant of a positively elliptic operator.",
   "evidence": "The canonical record is `aim-geometry-notes.json`, zero-based index 387. Its genuine question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 387,
   "attempt": 1
  },
  "AIM-GEOMETRY-0389": {
   "statement_status": "exact",
   "original_statement": "Problem 29: 3 Is (40) true in higher even dimensions? The following conjecture would be enough to answer the previous problem.",
   "clean_statement": "Problem 29: 3 Is (40) true in higher even dimensions? The following conjecture would be enough to answer the previous problem.",
   "public_statement": "Problem 29: 3 Is (40) true in higher even dimensions? The following conjecture would be enough to answer the previous problem.",
   "evidence": "The canonical extracted record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 388,
   "attempt": 1
  },
  "AIM-GEOMETRY-0390": {
   "statement_status": "exact",
   "original_statement": "Problem 30: If S is a natural n-form and ∫ S is conformally invariant, then \n\nS = const · Q + L + G,\n\nwhere L is a local conformal invariant and G has a local conformal primitive. That is, there is a local invariant F for which the conformal variation of ∫ F is ∫ ωG.",
   "clean_statement": "Problem 30: If S is a natural n-form and ∫ S is conformally invariant, then\n\nS = const · Q + L + G,\n\nwhere L is a local conformal invariant and G has a local conformal primitive. That is, there is a local invariant F for which the conformal variation of ∫ F is ∫ ωG.",
   "public_statement": "Problem 30: If S is a natural n-form and ∫ S is conformally invariant, then\n\nS = const · Q + L + G,\n\nwhere L is a local conformal invariant and G has a local conformal primitive. That is, there is a local invariant F for which the conformal variation of ∫ F is ∫ ωG.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 389,
   "attempt": 1
  },
  "AIM-GEOMETRY-0391": {
   "statement_status": "exact",
   "original_statement": "Problem 31: Is it possible to write any S, as in",
   "clean_statement": "Problem 31: Is it possible to write any S, as in",
   "public_statement": "Problem 31: Is it possible to write any S, as in",
   "evidence": "The exact canonical record is visibly incomplete:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 390,
   "attempt": 1
  },
  "AIM-GEOMETRY-0392": {
   "statement_status": "exact",
   "original_statement": "Problem 30, in the form const · Q + L + V,\n\nwhere V is an exact divergence?",
   "clean_statement": "Problem 30, in the form const · Q + L + V,\n\nwhere V is an exact divergence?",
   "public_statement": "Problem 30, in the form const · Q + L + V,\n\nwhere V is an exact divergence?",
   "evidence": "The canonical record is a page-break fragment:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 391,
   "attempt": 1
  },
  "AIM-GEOMETRY-0393": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 32: Is it possible to write any S, as in",
   "clean_statement": "If \\(\\mathbf S\\) is a natural critical \\(n\\)-density and \\(\\int_M\\mathbf S\\) is conformally invariant, can one write, pointwise and universally,\n\\[\n\\mathbf S=c\\,\\mathbf{Pf}_g+\\mathbf L_g+\\mathbf V_g,\n\\]\nwhere \\(\\mathbf L\\) is a local conformal invariant and \\(\\mathbf V\\) is the exact divergence of a natural vector field?",
   "public_statement": "If \\(\\mathbf S\\) is a natural critical \\(n\\)-density and \\(\\int_M\\mathbf S\\) is conformally invariant, can one write, pointwise and universally,\n\\[\n\\mathbf S=c\\,\\mathbf{Pf}_g+\\mathbf L_g+\\mathbf V_g,\n\\]\nwhere \\(\\mathbf L\\) is a local conformal invariant and \\(\\mathbf V\\) is the exact divergence of a natural vector field?",
   "evidence": "The exact canonical record assigned to this attempt is truncated: This fragment is preserved rather than silently repaired. The two preceding canonical records contain Problem 31 and its continuation, and the following canonical record begins with the continuation of Problem 32. The official AIM PDF gives the complete text on page 27:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 392,
   "attempt": 1
  },
  "AIM-GEOMETRY-0394": {
   "statement_status": "exact",
   "original_statement": "Problem 30, in the form const · Pff + L + V?\n\nOther routes to Q. There is an alternative definition of Q which avoids dimensional con-tinuation. Let E be the space of smooth functions, let E 1 be space of smooth 1-forms and define the special section \n\nIg:= \n\n 2 − n\n\n0\n\nJ\n\n\n\nof the direct sum bundle E ⊕ E 1 ⊕ E. In dimension 4: \n\n§Ig =\n\n 00\n\nQ4\n\n,\n\n> 2This section is the recompilation of the problems and conjectures in that document. We strongly suggest its lecture for a better understanding of the following problems.\n> 3Problems 29 - 32 are actually conjectures that T. Branson and R Gover address in their document. 29\n\nwhere \n\n§ = −∇ a∇a + ( n − 2) K/ (4 n − 4),\n\nwhich appears to be the usual formula for the conformal Laplacian, but now ∇ is a connection which couples the usual metric connection with the connection \n\n∇a\n\n σμτ\n\n =\n\n ∇σ − μ\n\n∇μ + gτ + V σ \n\n∇τ − μ−| P\n\n\n\non the sum bundle T:= E ⊕E 1 ⊕E. In any even dimension n, there is a conformally invariant differential operator §n−2 so that for any metric g:\n\n§n−2Ig =\n\n 00\n\nQn\n\n. (41) Here Ig is as above, while §n−2 has the form ∆ n/ 2−1 + lot (with lot = \"lower order terms\"). If ˆ g is a metric related to g conformally according to ˆ g = e2ωg (ω a smooth function) then \n\nIˆg = Ig + Dω, (42) where D is a well known second order conformally invariant linear differential operator (the tractor D operator). From this and (41) it follows that the Q-curvature ˆQn, for ˆ g, differs from Qn by a linear conformally invariant operator acting on ω. In fact \n\n§n−2Dω =\n\n 00\n\nPnω\n\n\n\nwhere Pn is the GJMS operator of order n, recovering the property ˆQn = Qn + Pnω. \n\nWhile this definition avoids dimensional continuation, there is still the issue of getting a formula for Qn. There is an effective algorithm for re-expressing the ambient results in terms of tractors which then expand easily into formulae in terms of the underlying Riemannian curvature and its covariant derivatives, solving the problem for small n.",
   "clean_statement": "Problem 30, in the form const · Pff + L + V?\n\nOther routes to Q. There is an alternative definition of Q which avoids dimensional con-tinuation. Let E be the space of smooth functions, let E 1 be space of smooth 1-forms and define the special section\n\nIg:=\n\n 2 − n\n\n0\n\nJ\n\n\n\nof the direct sum bundle E ⊕ E 1 ⊕ E. In dimension 4:\n\n§Ig =\n\n 00\n\nQ4\n\n,\n\n> 2This section is the recompilation of the problems and conjectures in that document. We strongly suggest its lecture for a better understanding of the following problems.\n> 3Problems 29 - 32 are actually conjectures that T. Branson and R Gover address in their document. 29\n\nwhere\n\n§ = −∇ a∇a + ( n − 2) K/ (4 n − 4),\n\nwhich appears to be the usual formula for the conformal Laplacian, but now ∇ is a connection which couples the usual metric connection with the connection\n\n∇a\n\n σμτ\n\n =\n\n ∇σ − μ\n\n∇μ + gτ + V σ\n\n∇τ − μ−| P\n\n\n\non the sum bundle T:= E ⊕E 1 ⊕E. In any even dimension n, there is a conformally invariant differential operator §n−2 so that for any metric g:\n\n§n−2Ig =\n\n 00\n\nQn\n\n. (41) Here Ig is as above, while §n−2 has the form ∆ n/ 2−1 + lot (with lot = \"lower order terms\"). If ˆ g is a metric related to g conformally according to ˆ g = e2ωg (ω a smooth function) then\n\nIˆg = Ig + Dω, (42) where D is a well known second order conformally invariant linear differential operator (the tractor D operator). From this and (41) it follows that the Q-curvature ˆQn, for ˆ g, differs from Qn by a linear conformally invariant operator acting on ω. In fact\n\n§n−2Dω =\n\n 00\n\nPnω\n\n\n\nwhere Pn is the GJMS operator of order n, recovering the property ˆQn = Qn + Pnω.\n\nWhile this definition avoids dimensional continuation, there is still the issue of getting a formula for Qn. There is an effective algorithm for re-expressing the ambient results in terms of tractors which then expand easily into formulae in terms of the underlying Riemannian curvature and its covariant derivatives, solving the problem for small n.",
   "public_statement": "Problem 30, in the form const · Pff + L + V?\n\nOther routes to Q. There is an alternative definition of Q which avoids dimensional con-tinuation. Let E be the space of smooth functions, let E 1 be space of smooth 1-forms and define the special section\n\nIg:=\n\n 2 − n\n\n0\n\nJ\n\n\n\nof the direct sum bundle E ⊕ E 1 ⊕ E. In dimension 4:\n\n§Ig =\n\n 00\n\nQ4\n\n,\n\n> 2This section is the recompilation of the problems and conjectures in that document. We strongly suggest its lecture for a better understanding of the following problems.\n> 3Problems 29 - 32 are actually conjectures that T. Branson and R Gover address in their document. 29\n\nwhere\n\n§ = −∇ a∇a + ( n − 2) K/ (4 n − 4),\n\nwhich appears to be the usual formula for the conformal Laplacian, but now ∇ is a connection which couples the usual metric connection with the connection\n\n∇a\n\n σμτ\n\n =\n\n ∇σ − μ\n\n∇μ + gτ + V σ\n\n∇τ − μ−| P\n\n\n\non the sum bundle T:= E ⊕E 1 ⊕E. In any even dimension n, there is a conformally invariant differential operator §n−2 so that for any metric g:\n\n§n−2Ig =\n\n 00\n\nQn\n\n. (41) Here Ig is as above, while §n−2 has the form ∆ n/ 2−1 + lot (with lot = \"lower order terms\"). If ˆ g is a metric related to g conformally according to ˆ g = e2ωg (ω a smooth function) then\n\nIˆg = Ig + Dω, (42) where D is a well known second order conformally invariant linear differential operator (the tractor D operator). From this and (41) it follows that the Q-curvature ˆQn, for ˆ g, differs from Qn by a linear conformally invariant operator acting on ω. In fact\n\n§n−2Dω =\n\n 00\n\nPnω\n\n\n\nwhere Pn is the GJMS operator of order n, recovering the property ˆQn = Qn + Pnω.\n\nWhile this definition avoids dimensional continuation, there is still the issue of getting a formula for Qn. There is an effective algorithm for re-expressing the ambient results in terms of tractors which then expand easily into formulae in terms of the underlying Riemannian curvature and its covariant derivatives, solving the problem for small n.",
   "evidence": "This canonical record is a page-split composite, not a single new problem. The official AIM source separates its contents as follows.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 393,
   "attempt": 1
  },
  "AIM-GEOMETRY-0395": {
   "statement_status": "exact",
   "original_statement": "Problem 33: Give general formulae or inductive formulae for the operators §2`.In another direction there is another exercise to which already are some answers. One of the features of the Q-curvature is that it \"transforms by a linear operator\" within a conformal class. More precisely, it is an example of a natural Riemannian tensor-density field with a transformation law \n\nN ˆg = N g + Lω, (43) \n\nL being some universal linear differential operator. (Here ω has the usual meaning; ˆ g = e2ωg.)",
   "clean_statement": "Problem 33: Give general formulae or inductive formulae for the operators §2`.In another direction there is another exercise to which already are some answers. One of the features of the Q-curvature is that it \"transforms by a linear operator\" within a conformal class. More precisely, it is an example of a natural Riemannian tensor-density field with a transformation law\n\nN ˆg = N g + Lω, (43)\n\nL being some universal linear differential operator. (Here ω has the usual meaning; ˆ g = e2ωg.)",
   "public_statement": "Problem 33: Give general formulae or inductive formulae for the operators §2`.In another direction there is another exercise to which already are some answers. One of the features of the Q-curvature is that it \"transforms by a linear operator\" within a conformal class. More precisely, it is an example of a natural Riemannian tensor-density field with a transformation law\n\nN ˆg = N g + Lω, (43)\n\nL being some universal linear differential operator. (Here ω has the usual meaning; ˆ g = e2ωg.)",
   "evidence": "The canonical JSON record is OCR-damaged and also contains the opening paragraph of the next problem. Its exact `problem` field begins",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 394,
   "attempt": 1
  },
  "AIM-GEOMETRY-0396": {
   "statement_status": "exact",
   "original_statement": "Problem 34: Construct other natural tensor-densities which transform according to (43). (Note that any solution yields a conformally invariant natural operator L.) Solutions to",
   "clean_statement": "Problem 34: Construct other natural tensor-densities which transform according to (43). (Note that any solution yields a conformally invariant natural operator L.) Solutions to",
   "public_statement": "Problem 34: Construct other natural tensor-densities which transform according to (43). (Note that any solution yields a conformally invariant natural operator L.) Solutions to",
   "evidence": "The exact canonical record is truncated and is preserved in input.json:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 395,
   "attempt": 1
  },
  "AIM-GEOMETRY-0397": {
   "statement_status": "exact",
   "original_statement": "Problem 34 have a role to play in the problem of characterizing the Q-curvature and the GJMS operators. 30 \n\nGeneralizations of Q. In a compact, oriented, but not necessarily connected, manifold of even dimension n, Q can be seen as a multiplication operator from the closed 0-forms C0\n\n(i.e. the locally constant functions) into the space of n-forms E n (identified with E[−n] via the conformal Hodge?). This operator has the following properties: A. Q: C0 → E n is not conformally invariant but ˆQ = Q + Pnω, where Pn is a formally self-adjoint operator from 0-forms to n-forms. Pn has the form dM d which implies the next properties. B. Q: C0 → Hn(M ) is conformally invariant and non-trivial in general. C. If c ∈ C 0 and u ∈ N (Pn) then ∫ uQc is conformally invariant. D. In each choice of metric Q: E0 → E [−n] is formally self-adjoint. E. Q1 is the Q-curvature. The idea now is to look for analogous operators on other forms. T. Branson and R. Gover (see math.DG/0309085) have used the ambient metric, and its relationship to tractors, to show that the previous generalizes along the following lines: There are operators M gk: Ek → E n−k\n\n(k ≤ n/ 2 − 1), given by a uniform construction, with the following properties: A. M gk: Ck → E n−k has the conformal transformation law M ˆgk = M gk + Lkω, where Lk\n\nis a formally self-adjoint operator from k-forms to ( n − k)-forms, and is a constant multiple of dM gk+1 d. Here Ck is the space of closed k-forms, B. Hk:= N (dM gk: Ck → E n−k+1 ) is a conformally invariant subspace of Ck and Mk:\n\nHk → Hn−k(M ) is conformally invariant. There are conformal manifolds on which \n\nMk is non-trivial. C. If c ∈ C k and u ∈ N (Lk) then ∫\n\n〈u, M gk c〉\n\nis conformally invariant. D. For each choice of metric g, M gk: Ek → E n−k is formally self-adjoint. E. M g \n\n> 0\n\n1 is the Q-curvature. From the uniqueness of the Maxwell operator at leading order (as a conformally invariant operator En/ 2−1 → E n/ 2−1[−2]), and the explicit formula \n\nM g = dδ + 2 J − 4P ]: En/ 2−1 → E n/ 2−1[−2], (44) it is clear M gn/ 2−1 is not the difference between any conformally invariant differential operator and a divergence (even as an operator on closed forms). A similar argument applies to the Mk\n\ngenerally. Thus, from the point of view that the Q-curvature is a non-conformally invariant object that in a deep sense cannot be made conformally invariant, but one which nevertheless determines a global conformal invariant, the operators M gk give a genuine generalization of the Q-curvature to an operator on closed forms.",
   "clean_statement": "Problem 34 have a role to play in the problem of characterizing the Q-curvature and the GJMS operators. 30\n\nGeneralizations of Q. In a compact, oriented, but not necessarily connected, manifold of even dimension n, Q can be seen as a multiplication operator from the closed 0-forms C0\n\n(i.e. the locally constant functions) into the space of n-forms E n (identified with E[−n] via the conformal Hodge?). This operator has the following properties: A. Q: C0 → E n is not conformally invariant but ˆQ = Q + Pnω, where Pn is a formally self-adjoint operator from 0-forms to n-forms. Pn has the form dM d which implies the next properties. B. Q: C0 → Hn(M ) is conformally invariant and non-trivial in general. C. If c ∈ C 0 and u ∈ N (Pn) then ∫ uQc is conformally invariant. D. In each choice of metric Q: E0 → E [−n] is formally self-adjoint. E. Q1 is the Q-curvature. The idea now is to look for analogous operators on other forms. T. Branson and R. Gover (see math.DG/0309085) have used the ambient metric, and its relationship to tractors, to show that the previous generalizes along the following lines: There are operators M gk: Ek → E n−k\n\n(k ≤ n/ 2 − 1), given by a uniform construction, with the following properties: A. M gk: Ck → E n−k has the conformal transformation law M ˆgk = M gk + Lkω, where Lk\n\nis a formally self-adjoint operator from k-forms to ( n − k)-forms, and is a constant multiple of dM gk+1 d. Here Ck is the space of closed k-forms, B. Hk:= N (dM gk: Ck → E n−k+1 ) is a conformally invariant subspace of Ck and Mk:\n\nHk → Hn−k(M ) is conformally invariant. There are conformal manifolds on which\n\nMk is non-trivial. C. If c ∈ C k and u ∈ N (Lk) then ∫\n\n〈u, M gk c〉\n\nis conformally invariant. D. For each choice of metric g, M gk: Ek → E n−k is formally self-adjoint. E. M g\n\n> 0\n\n1 is the Q-curvature. From the uniqueness of the Maxwell operator at leading order (as a conformally invariant operator En/ 2−1 → E n/ 2−1[−2]), and the explicit formula\n\nM g = dδ + 2 J − 4P ]: En/ 2−1 → E n/ 2−1[−2], (44) it is clear M gn/ 2−1 is not the difference between any conformally invariant differential operator and a divergence (even as an operator on closed forms). A similar argument applies to the Mk\n\ngenerally. Thus, from the point of view that the Q-curvature is a non-conformally invariant object that in a deep sense cannot be made conformally invariant, but one which nevertheless determines a global conformal invariant, the operators M gk give a genuine generalization of the Q-curvature to an operator on closed forms.",
   "public_statement": "Problem 34 have a role to play in the problem of characterizing the Q-curvature and the GJMS operators. 30\n\nGeneralizations of Q. In a compact, oriented, but not necessarily connected, manifold of even dimension n, Q can be seen as a multiplication operator from the closed 0-forms C0\n\n(i.e. the locally constant functions) into the space of n-forms E n (identified with E[−n] via the conformal Hodge?). This operator has the following properties: A. Q: C0 → E n is not conformally invariant but ˆQ = Q + Pnω, where Pn is a formally self-adjoint operator from 0-forms to n-forms. Pn has the form dM d which implies the next properties. B. Q: C0 → Hn(M ) is conformally invariant and non-trivial in general. C. If c ∈ C 0 and u ∈ N (Pn) then ∫ uQc is conformally invariant. D. In each choice of metric Q: E0 → E [−n] is formally self-adjoint. E. Q1 is the Q-curvature. The idea now is to look for analogous operators on other forms. T. Branson and R. Gover (see math.DG/0309085) have used the ambient metric, and its relationship to tractors, to show that the previous generalizes along the following lines: There are operators M gk: Ek → E n−k\n\n(k ≤ n/ 2 − 1), given by a uniform construction, with the following properties: A. M gk: Ck → E n−k has the conformal transformation law M ˆgk = M gk + Lkω, where Lk\n\nis a formally self-adjoint operator from k-forms to ( n − k)-forms, and is a constant multiple of dM gk+1 d. Here Ck is the space of closed k-forms, B. Hk:= N (dM gk: Ck → E n−k+1 ) is a conformally invariant subspace of Ck and Mk:\n\nHk → Hn−k(M ) is conformally invariant. There are conformal manifolds on which\n\nMk is non-trivial. C. If c ∈ C k and u ∈ N (Lk) then ∫\n\n〈u, M gk c〉\n\nis conformally invariant. D. For each choice of metric g, M gk: Ek → E n−k is formally self-adjoint. E. M g\n\n> 0\n\n1 is the Q-curvature. From the uniqueness of the Maxwell operator at leading order (as a conformally invariant operator En/ 2−1 → E n/ 2−1[−2]), and the explicit formula\n\nM g = dδ + 2 J − 4P ]: En/ 2−1 → E n/ 2−1[−2], (44) it is clear M gn/ 2−1 is not the difference between any conformally invariant differential operator and a divergence (even as an operator on closed forms). A similar argument applies to the Mk\n\ngenerally. Thus, from the point of view that the Q-curvature is a non-conformally invariant object that in a deep sense cannot be made conformally invariant, but one which nevertheless determines a global conformal invariant, the operators M gk give a genuine generalization of the Q-curvature to an operator on closed forms.",
   "evidence": "This canonical record is not itself an open problem. It is the explanatory paragraph between **Problem 34** and **Problem 35** in the 2003 AIM workshop notes *Conformal Structure in Geometry, Analysis, and Physics*. The first words lost in extraction are “Solutions to,” so the record begins:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 396,
   "attempt": 1
  },
  "AIM-GEOMETRY-0398": {
   "statement_status": "exact",
   "original_statement": "Problem 35: There are analogues for the operators M gk of most of the Problems in Sections II. and III. for the Q-curvature. \n\nChapter D: Reference list \n\nA list of references related to the topic of the workshop is available at http://www.und.nodak.edu/instruct/lapeters/bib",
   "clean_statement": "Problem 35: There are analogues for the operators M gk of most of the Problems in Sections II. and III. for the Q-curvature.\n\nChapter D: Reference list\n\nA list of references related to the topic of the workshop is available at http://www.und.nodak.edu/instruct/lapeters/bib",
   "public_statement": "Problem 35: There are analogues for the operators M gk of most of the Problems in Sections II. and III. for the Q-curvature.\n\nChapter D: Reference list\n\nA list of references related to the topic of the workshop is available at http://www.und.nodak.edu/instruct/lapeters/bib",
   "evidence": "The exact canonical record is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 397,
   "attempt": 1
  },
  "AIM-GEOMETRY-0399": {
   "statement_status": "exact",
   "original_statement": "Question 1.1. Tobias Ekholm asked [something approximating] the following: If L ⊂ J 1(M )\n\nis {0} cross the 0-section of T ∗M and L′ ⊂ J1(M ) is Legendrian isotopic to L, must the projections of L and L′ to T ∗M intersect? The discussion apparently concluded that the answer is yes (if M is compact?), so at least one question seems to have been answered at the workshop! [Eliashberg points out that the answer to the question as written above is obvious from the existence of a generating function, so I probably don't have the statement of the question exactly right, and there was some nontrivial proof of something which I missed.] \n\nIf E → M is a fiber bundle and f: E → R is a function, then the set of fiberwise critical values of f (or the Cerf diagram) defines a subset of M × R, which is the front projection of a Legendrian submanifold L ⊂ J 1(M ). We say that f is a generating function \n\nfor L.",
   "clean_statement": "Question 1.1. Tobias Ekholm asked [something approximating] the following: If L ⊂ J 1(M )\n\nis {0} cross the 0-section of T ∗M and L′ ⊂ J1(M ) is Legendrian isotopic to L, must the projections of L and L′ to T ∗M intersect? The discussion apparently concluded that the answer is yes (if M is compact?), so at least one question seems to have been answered at the workshop! [Eliashberg points out that the answer to the question as written above is obvious from the existence of a generating function, so I probably don't have the statement of the question exactly right, and there was some nontrivial proof of something which I missed.]\n\nIf E → M is a fiber bundle and f: E → R is a function, then the set of fiberwise critical values of f (or the Cerf diagram) defines a subset of M × R, which is the front projection of a Legendrian submanifold L ⊂ J 1(M ). We say that f is a generating function\n\nfor L.",
   "public_statement": "Question 1.1. Tobias Ekholm asked [something approximating] the following: If L ⊂ J 1(M )\n\nis {0} cross the 0-section of T ∗M and L′ ⊂ J1(M ) is Legendrian isotopic to L, must the projections of L and L′ to T ∗M intersect? The discussion apparently concluded that the answer is yes (if M is compact?), so at least one question seems to have been answered at the workshop! [Eliashberg points out that the answer to the question as written above is obvious from the existence of a generating function, so I probably don't have the statement of the question exactly right, and there was some nontrivial proof of something which I missed.]\n\nIf E → M is a fiber bundle and f: E → R is a function, then the set of fiberwise critical values of f (or the Cerf diagram) defines a subset of M × R, which is the front projection of a Legendrian submanifold L ⊂ J 1(M ). We say that f is a generating function\n\nfor L.",
   "evidence": "The official 2003 AIM workshop PDF says, deliberately uncertainly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 398,
   "attempt": 1
  },
  "AIM-GEOMETRY-0400": {
   "statement_status": "exact",
   "original_statement": "Question 1.2. Which L admit generating functions? One wants to impose some restrictions on the generating function. (Typically one wants the generating function to be quadratic at infinity, although it might be interesting to explore other conditions at infinity. Quadratic at infinity imposes some restrictions on the Legendrians that can be realized, for example stabi-lizations (zig zags) are somehow prohibited in dimension 3.) As Yasha Eliashberg explained, there is a theorem of Giroux which gives a necessary and sufficient condition for existence of a finite-dimensional generating function f if no condition on the behavior of f at infinity is imposed. If one allows an infinite-dimensional generating function, then one can write an explicit formula for it using the action functional. \n\nWhen the Legendrian L admits a [suitable?] generating function, one can use Morse theory of the generating function to define various invariants of L.\n\nTheme 1.3. Lisa Traynor discussed how there is a mysterious [or not?] connection be-tween her polynomial invariants of Legendrian submanifolds defined in terms of generating functions and other polynomial invariants defined out of Linearized Contact Homology [def-inition?]. In some sense the two invariants encode the same information. So maybe there is some more precise question to ask here about sorting this out.",
   "clean_statement": "Question 1.2. Which L admit generating functions? One wants to impose some restrictions on the generating function. (Typically one wants the generating function to be quadratic at infinity, although it might be interesting to explore other conditions at infinity. Quadratic at infinity imposes some restrictions on the Legendrians that can be realized, for example stabi-lizations (zig zags) are somehow prohibited in dimension 3.) As Yasha Eliashberg explained, there is a theorem of Giroux which gives a necessary and sufficient condition for existence of a finite-dimensional generating function f if no condition on the behavior of f at infinity is imposed. If one allows an infinite-dimensional generating function, then one can write an explicit formula for it using the action functional.\n\nWhen the Legendrian L admits a [suitable?] generating function, one can use Morse theory of the generating function to define various invariants of L.\n\nTheme 1.3. Lisa Traynor discussed how there is a mysterious [or not?] connection be-tween her polynomial invariants of Legendrian submanifolds defined in terms of generating functions and other polynomial invariants defined out of Linearized Contact Homology [def-inition?]. In some sense the two invariants encode the same information. So maybe there is some more precise question to ask here about sorting this out.",
   "public_statement": "Question 1.2. Which L admit generating functions? One wants to impose some restrictions on the generating function. (Typically one wants the generating function to be quadratic at infinity, although it might be interesting to explore other conditions at infinity. Quadratic at infinity imposes some restrictions on the Legendrians that can be realized, for example stabi-lizations (zig zags) are somehow prohibited in dimension 3.) As Yasha Eliashberg explained, there is a theorem of Giroux which gives a necessary and sufficient condition for existence of a finite-dimensional generating function f if no condition on the behavior of f at infinity is imposed. If one allows an infinite-dimensional generating function, then one can write an explicit formula for it using the action functional.\n\nWhen the Legendrian L admits a [suitable?] generating function, one can use Morse theory of the generating function to define various invariants of L.\n\nTheme 1.3. Lisa Traynor discussed how there is a mysterious [or not?] connection be-tween her polynomial invariants of Legendrian submanifolds defined in terms of generating functions and other polynomial invariants defined out of Linearized Contact Homology [def-inition?]. In some sense the two invariants encode the same information. So maybe there is some more precise question to ask here about sorting this out.",
   "evidence": "The source is Question 1.2 in the AIM workshop notes *Holomorphic curves in contact geometry*. The PDF was checked against the extracted record. Its question is deliberately tentative:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 399,
   "attempt": 1
  },
  "AIM-GEOMETRY-0401": {
   "statement_status": "corrected_verified",
   "original_statement": "Question 1.4. Ekholm asked: (a) Is Linearized Contact Homology always equal to Generating Function Homology [defini-tion?] for a (compact?) Legendrian L ⊂ J 1(Rn)? As evidence for this conjecture, it was proved in the case where L is the double of a compact n-manifold with boundary M ⊂ Rn.4\n\n(b) For a Legendrian L ⊂ Rn, does L admit a generating function [satisfying what restric-tions?] if / only if the Linearized Contact Homology of L is (uniquely) defined? \n\nTheme 1.5. A recurrent theme discussed by several people during the first day was the possibility of understanding holomorphic curves by taking some limit in which they degenerate to simpler objects. For example, Fukaya-Oh, inspired by a paper of Witten, studied how in a certain situation, holomorphic discs degenerate to \"gradient trees\". This idea is currently being applied to relative contact homology in the work of Lenny Ng and Zhu Ke. Also, there is Mikhalkin's work on amoebas, which similarly understands holomorphic curves in a Lagrangian fibration by shrinking the fibers. In a similar vein, there is also recent work relating the Chas-Sullivan product on the loop space to the cup product in Floer homology. Anyway, while we are not asking a question here, this seems to be an important theme for future work.",
   "clean_statement": "Question 1.Theme 1.5. Ekholm asked: (a) Is Linearized Contact Homology always equal to Generating Function Homology [defini-tion?] for a (compact?) Legendrian L ⊂ J 1(Rn)? As evidence for this conjecture, it was proved in the case where L is the double of a compact n-manifold with boundary M ⊂ Rn.Theme 1.5\n\n(b) For a Legendrian L ⊂ Rn, does L admit a generating function [satisfying what restric-tions?] if / only if the Linearized Contact Homology of L is (uniquely) defined?\n\nTheme 1.5. A recurrent theme discussed by several people during the first day was the possibility of understanding holomorphic curves by taking some limit in which they degenerate to simpler objects. For example, Fukaya-Oh, inspired by a paper of Witten, studied how in a certain situation, holomorphic discs degenerate to \"gradient trees\". This idea is currently being applied to relative contact homology in the work of Lenny Ng and Zhu Ke. Also, there is Mikhalkin's work on amoebas, which similarly understands holomorphic curves in a Lagrangian fibration by shrinking the fibers. In a similar vein, there is also recent work relating the Chas-Sullivan product on the loop space to the cup product in Floer homology. Anyway, while we are not asking a question here, this seems to be an important theme for future work.",
   "public_statement": "Question 1.Theme 1.5. Ekholm asked: (a) Is Linearized Contact Homology always equal to Generating Function Homology [defini-tion?] for a (compact?) Legendrian L ⊂ J 1(Rn)? As evidence for this conjecture, it was proved in the case where L is the double of a compact n-manifold with boundary M ⊂ Rn.Theme 1.5\n\n(b) For a Legendrian L ⊂ Rn, does L admit a generating function [satisfying what restric-tions?] if / only if the Linearized Contact Homology of L is (uniquely) defined?\n\nTheme 1.5. A recurrent theme discussed by several people during the first day was the possibility of understanding holomorphic curves by taking some limit in which they degenerate to simpler objects. For example, Fukaya-Oh, inspired by a paper of Witten, studied how in a certain situation, holomorphic discs degenerate to \"gradient trees\". This idea is currently being applied to relative contact homology in the work of Lenny Ng and Zhu Ke. Also, there is Mikhalkin's work on amoebas, which similarly understands holomorphic curves in a Lagrangian fibration by shrinking the fibers. In a similar vein, there is also recent work relating the Chas-Sullivan product on the loop space to the cup product in Floer homology. Anyway, while we are not asking a question here, this seems to be an important theme for future work.",
   "evidence": "The official AIM PDF uses the convention \\[ J^1(M)=\\mathbb R_u\\times T^*M, \\qquad \\alpha=du-p\\,dq, \\] which is the usual \\(T^*M\\times\\mathbb R_z\\) with contact form \\(dz-p\\,dq\\), up to the order and name of the coordinates. It prints Question 1.4 as follows (typographical line-break hyphens suppressed, but the tentative brackets preserved): The superscript-like “4” after part (a) in the extracted record is a footnote marker, not an exponent. The PDF itself really does say \\(L\\subset\\mathbb R^n\\) in part (b); this is not an extraction error. Literally, however, that phrase does not specify a contact structure or the appropriate Legendrian dimension. The surrounding definition and part (a) strongly suggest the intended ambient space was \\(J^1(\\mathbb R^n)\\). All mathematical conclusions below explicitly state their ambient space.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-geometry-notes.json",
   "source_index": 400,
   "attempt": 1
  },
  "AIM-GEOMETRY-0402": {
   "statement_status": "exact",
   "original_statement": "Question 1.6. Ko Honda spoke on results with Etnyre that certain knot types are not transversally simple. The proof is indirect and there is no invariant here. Eliashberg asked if an invariant can be constructed [to distinguish transversal knots]. \n\nSecond day",
   "clean_statement": "Question 1.6. Ko Honda spoke on results with Etnyre that certain knot types are not transversally simple. The proof is indirect and there is no invariant here. Eliashberg asked if an invariant can be constructed [to distinguish transversal knots].\n\nSecond day",
   "public_statement": "Question 1.6. Ko Honda spoke on results with Etnyre that certain knot types are not transversally simple. The proof is indirect and there is no invariant here. Eliashberg asked if an invariant can be constructed [to distinguish transversal knots].\n\nSecond day",
   "evidence": "The canonical record is Question 1.6 from the AIM workshop *Holomorphic curves in contact geometry*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 401,
   "attempt": 1
  },
  "AIM-GEOMETRY-0403": {
   "statement_status": "exact",
   "original_statement": "Question 1.7. Paul Biran asked, following up some results presented by Emmanuel Giroux, if one can give a definition of \"overtwisted\" in higher dimensions in terms of open books. [Giroux proposed some definition, see also what he said on the sixth day.]",
   "clean_statement": "Question 1.7. Paul Biran asked, following up some results presented by Emmanuel Giroux, if one can give a definition of \"overtwisted\" in higher dimensions in terms of open books. [Giroux proposed some definition, see also what he said on the sixth day.]",
   "public_statement": "Question 1.7. Paul Biran asked, following up some results presented by Emmanuel Giroux, if one can give a definition of \"overtwisted\" in higher dimensions in terms of open books. [Giroux proposed some definition, see also what he said on the sixth day.]",
   "evidence": "The canonical record is Question 1.7 in the American Institute of Mathematics problem list *Holomorphic curves in contact geometry*, version 15 October 2003:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 402,
   "attempt": 1
  },
  "AIM-GEOMETRY-0404": {
   "statement_status": "exact",
   "original_statement": "Question 1.8. There was a fair bit of discussion on how to possibly compute contact ho-mology in terms of (contact) open books. For example, Denis Auroux pointed out that if you have a Lagrangian L in a page, then this is a Legendrian in the open book, and the Reeb chords are the intersections of L with φk(L) where φ is the monodromy. This might be a starting point for computing the relative contact homology of L, in the complement of the binding, in terms of things related to symplectic Floer homology of Lagrangians. There is then the problem of understanding what happens when one puts back the binding, but this might correspond to an \" A∞ deformation\". \n\nNext, Eliashberg discussed the following questions:",
   "clean_statement": "Question 1.8. There was a fair bit of discussion on how to possibly compute contact ho-mology in terms of (contact) open books. For example, Denis Auroux pointed out that if you have a Lagrangian L in a page, then this is a Legendrian in the open book, and the Reeb chords are the intersections of L with φk(L) where φ is the monodromy. This might be a starting point for computing the relative contact homology of L, in the complement of the binding, in terms of things related to symplectic Floer homology of Lagrangians. There is then the problem of understanding what happens when one puts back the binding, but this might correspond to an \" A∞ deformation\".\n\nNext, Eliashberg discussed the following questions:",
   "public_statement": "Question 1.8. There was a fair bit of discussion on how to possibly compute contact ho-mology in terms of (contact) open books. For example, Denis Auroux pointed out that if you have a Lagrangian L in a page, then this is a Legendrian in the open book, and the Reeb chords are the intersections of L with φk(L) where φ is the monodromy. This might be a starting point for computing the relative contact homology of L, in the complement of the binding, in terms of things related to symplectic Floer homology of Lagrangians. There is then the problem of understanding what happens when one puts back the binding, but this might correspond to an \" A∞ deformation\".\n\nNext, Eliashberg discussed the following questions:",
   "evidence": "The canonical record is Question 1.8 from the 2003 AIM workshop *Holomorphic curves in contact geometry*. Its exact extracted text is preserved in `input.json`. The official PDF gives the following mathematical proposal:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 403,
   "attempt": 1
  },
  "AIM-GEOMETRY-0405": {
   "statement_status": "exact",
   "original_statement": "Question 1.9. Conjecture: for any Stein fillable contact manifold, the cylindrical contact homology is defined. (As some have used it in the past, CCH is defined if there exists a contact form with no contractible Reeb orbits of certain Maslov indices. If such a form exists, then any two such forms give the same CCH. For this conjecture, one might not be able to eliminate bad contractible orbits, but one could hope that one can make them somehow \"algebraically cancel\".) If so, then counting pairs of pants etc. gives operations on the CCH, as in Floer theory of symplectomorphisms.",
   "clean_statement": "Question 1.9. Conjecture: for any Stein fillable contact manifold, the cylindrical contact homology is defined. (As some have used it in the past, CCH is defined if there exists a contact form with no contractible Reeb orbits of certain Maslov indices. If such a form exists, then any two such forms give the same CCH. For this conjecture, one might not be able to eliminate bad contractible orbits, but one could hope that one can make them somehow \"algebraically cancel\".) If so, then counting pairs of pants etc. gives operations on the CCH, as in Floer theory of symplectomorphisms.",
   "public_statement": "Question 1.9. Conjecture: for any Stein fillable contact manifold, the cylindrical contact homology is defined. (As some have used it in the past, CCH is defined if there exists a contact form with no contractible Reeb orbits of certain Maslov indices. If such a form exists, then any two such forms give the same CCH. For this conjecture, one might not be able to eliminate bad contractible orbits, but one could hope that one can make them somehow \"algebraically cancel\".) If so, then counting pairs of pants etc. gives operations on the CCH, as in Floer theory of symplectomorphisms.",
   "evidence": "This is Question 1.9 from the AIM workshop list *Holomorphic curves in contact geometry*. The canonical record states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 404,
   "attempt": 1
  },
  "AIM-GEOMETRY-0406": {
   "statement_status": "exact",
   "original_statement": "Question 1.10. This leads to the problem of computing contact homology of Stein fillable contact manifolds. If W is a Stein filling, a pseudoconvex function ϕ: W → R gives a handle decomposition. One could try to understand what happens to the contact homology as one attaches handles. The subcritical case is more or less understood by the work of Mei-Lin Yau and Frederic Bourgeois. The point is that when you attach a subcritical handle, there is a contact sphere in the middle of the handle, so you basically get the homology of the manifold with coefficients in the contact homology of these spheres. 5\n\nThe interesting part is to understand what happens when you attach a handle of critical index along a Legendrian sphere L. In this case a Reeb trajectory can hit L, enter the handle, and leave the handle anywhere else along L. So the new closed Reeb orbits are unions of Reeb chords of L. This is highly suggestive that there is some surgery formula in terms of the relative contact homology of L. What is the surgery formula??? Someone made some analogy with adding exceptional fibers to a Lefschetz fibration...",
   "clean_statement": "Question 1.10. This leads to the problem of computing contact homology of Stein fillable contact manifolds. If W is a Stein filling, a pseudoconvex function ϕ: W → R gives a handle decomposition. One could try to understand what happens to the contact homology as one attaches handles. The subcritical case is more or less understood by the work of Mei-Lin Yau and Frederic Bourgeois. The point is that when you attach a subcritical handle, there is a contact sphere in the middle of the handle, so you basically get the homology of the manifold with coefficients in the contact homology of these spheres. 5\n\nThe interesting part is to understand what happens when you attach a handle of critical index along a Legendrian sphere L. In this case a Reeb trajectory can hit L, enter the handle, and leave the handle anywhere else along L. So the new closed Reeb orbits are unions of Reeb chords of L. This is highly suggestive that there is some surgery formula in terms of the relative contact homology of L. What is the surgery formula??? Someone made some analogy with adding exceptional fibers to a Lefschetz fibration...",
   "public_statement": "Question 1.10. This leads to the problem of computing contact homology of Stein fillable contact manifolds. If W is a Stein filling, a pseudoconvex function ϕ: W → R gives a handle decomposition. One could try to understand what happens to the contact homology as one attaches handles. The subcritical case is more or less understood by the work of Mei-Lin Yau and Frederic Bourgeois. The point is that when you attach a subcritical handle, there is a contact sphere in the middle of the handle, so you basically get the homology of the manifold with coefficients in the contact homology of these spheres. 5\n\nThe interesting part is to understand what happens when you attach a handle of critical index along a Legendrian sphere L. In this case a Reeb trajectory can hit L, enter the handle, and leave the handle anywhere else along L. So the new closed Reeb orbits are unions of Reeb chords of L. This is highly suggestive that there is some surgery formula in terms of the relative contact homology of L. What is the surgery formula??? Someone made some analogy with adding exceptional fibers to a Lefschetz fibration...",
   "evidence": "This is Question 1.10 in the AIM workshop list *Holomorphic curves in contact geometry*. It asks how contact homology changes under the handle decomposition of a Stein filling. The critical case is attachment of a middle-index Weinstein handle along a Legendrian sphere \\(L\\); the source observes that new closed Reeb orbits should be concatenations of Reeb chords of \\(L\\), and asks for a surgery formula in terms of the relative contact homology of \\(L\\).",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 405,
   "attempt": 1
  },
  "AIM-GEOMETRY-0407": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 1.11. Generalizing the theme of computing things, one could try to extend sym-plectic field theory to an \"extended field theory\". Recall that a TQFT assigns to an n-manifold (possibly with some extra structure) a number, and to an (n−1) -manifold a vector space, sat-isfying various axioms which allow one to compute the invariant of an n-manifold by cutting it up along (n − 1) -manifolds. But then one is the left with the problem of understanding the \n\n(n − 1) -dimensional invariant. In an extended TQFT, one can compute the latter by cutting along (n − 2) -dimensional manifolds, to each of which is assigned a category. (One can continue by assigning 2-categories to (n − 3) -manifolds and so forth, so that the manifolds get simpler while the theory gets more complicated...) Now the question is, how can one do this for SFT?",
   "clean_statement": null,
   "public_statement": "Question 1.11. Generalizing the theme of computing things, one could try to extend sym-plectic field theory to an \"extended field theory\". Recall that a TQFT assigns to an n-manifold (possibly with some extra structure) a number, and to an (n−1) -manifold a vector space, sat-isfying various axioms which allow one to compute the invariant of an n-manifold by cutting it up along (n − 1) -manifolds. But then one is the left with the problem of understanding the\n\n(n − 1) -dimensional invariant. In an extended TQFT, one can compute the latter by cutting along (n − 2) -dimensional manifolds, to each of which is assigned a category. (One can continue by assigning 2-categories to (n − 3) -manifolds and so forth, so that the manifolds get simpler while the theory gets more complicated...) Now the question is, how can one do this for SFT?",
   "evidence": "The record is Question 1.11 in the AIM problem list *Holomorphic curves in contact geometry* (version dated 15 October 2003). The following is a conservative reconstruction from the official PDF:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 406,
   "attempt": 1
  },
  "AIM-GEOMETRY-0408": {
   "statement_status": "exact",
   "original_statement": "Question 1.12. In setting up such an extended field theory picture, it is important to for-mulate holomorphic curve theory for manifolds with boundary, or open manifolds with some asymptotic conditions. The basic idea is that when you replace boundary conditions with asymptotic conditions, you get more information about and control over how you approach the boundary. For example, if you are looking at holomorphic curves with boundary in a Lagrangian, it is sometimes better to consider curves with asymptotics in the unit cotangent bundle of the Lagrangian. More generally, for the extended field theory picture, one may need to consider holomorphic curves with boundary along some Lagrangian cylinders, and then you have holomorphic curves with corners, two different types of asymptotic conditions...",
   "clean_statement": "Question 1.12. In setting up such an extended field theory picture, it is important to for-mulate holomorphic curve theory for manifolds with boundary, or open manifolds with some asymptotic conditions. The basic idea is that when you replace boundary conditions with asymptotic conditions, you get more information about and control over how you approach the boundary. For example, if you are looking at holomorphic curves with boundary in a Lagrangian, it is sometimes better to consider curves with asymptotics in the unit cotangent bundle of the Lagrangian. More generally, for the extended field theory picture, one may need to consider holomorphic curves with boundary along some Lagrangian cylinders, and then you have holomorphic curves with corners, two different types of asymptotic conditions...",
   "public_statement": "Question 1.12. In setting up such an extended field theory picture, it is important to for-mulate holomorphic curve theory for manifolds with boundary, or open manifolds with some asymptotic conditions. The basic idea is that when you replace boundary conditions with asymptotic conditions, you get more information about and control over how you approach the boundary. For example, if you are looking at holomorphic curves with boundary in a Lagrangian, it is sometimes better to consider curves with asymptotics in the unit cotangent bundle of the Lagrangian. More generally, for the extended field theory picture, one may need to consider holomorphic curves with boundary along some Lagrangian cylinders, and then you have holomorphic curves with corners, two different types of asymptotic conditions...",
   "evidence": "The source is Question 1.12 in the AIM workshop list *Holomorphic curves in contact geometry*. The official AIM PDF was checked against the extracted record. The PDF reads (with only the line-break artifact repaired):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 407,
   "attempt": 1
  },
  "AIM-GEOMETRY-0409": {
   "statement_status": "corrected_verified",
   "original_statement": "Question 1.13. How unique are the open books corresponding to contact manifolds? Giroux explained that the open books produced by Donaldson's construction, which depend on a pos-itive integer parameter k, are unique up to stabilization when k is sufficiently large.",
   "clean_statement": "**Question 1.13.** How unique are the open books corresponding to contact manifolds? Giroux explained that the open books produced by Donaldson's construction, which depend on a positive integer parameter \\(k\\), are unique up to stabilization when \\(k\\) is sufficiently large.",
   "public_statement": "**Question 1.13.** How unique are the open books corresponding to contact manifolds? Giroux explained that the open books produced by Donaldson's construction, which depend on a positive integer parameter \\(k\\), are unique up to stabilization when \\(k\\) is sufficiently large.",
   "evidence": "The record is Question 1.13 from the AIM workshop *Holomorphic curves in contact geometry*. The official AIM PDF and HTML version agree, apart from a line-break OCR error in the corpus. The recovered statement is: The corpus text has `pos-itive`; this has been repaired to `positive`. There is no further missing formula in the statement. In this context “stabilization” must be read as **positive stabilization**, together with the usual conjugations/isotopies (and, in the modern higher-dimensional formulation, Weinstein homotopies). An unqualified negative stabilization does not belong to the equivalence relation.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 408,
   "attempt": 1
  },
  "AIM-GEOMETRY-0410": {
   "statement_status": "unrecoverable",
   "original_statement": "Question 1.14. Stein is to Weinstein as plurisubharmonic is to what? [This is not a mathematical problem, just a question of what term to use; the definition we want is clear. Perhaps this question is not worthy of this problem list.] \n\nTakao Akahori asked the following:",
   "clean_statement": null,
   "public_statement": "Question 1.14. Stein is to Weinstein as plurisubharmonic is to what? [This is not a mathematical problem, just a question of what term to use; the definition we want is clear. Perhaps this question is not worthy of this problem list.]\n\nTakao Akahori asked the following:",
   "evidence": "> **Question 1.14.** Stein is to Weinstein as plurisubharmonic is to what? [This is not a mathematical problem, just a question of what term to use; the definition we want is clear. Perhaps this question is not worthy of this problem list.] > > Takao Akahori asked the following: > **Recovered Question 1.14.** Stein is to Weinstein as plurisubharmonic is to what? [This is not a mathematical problem, just a question of what term to use; the definition we want is clear. Perhaps this question is not worthy of this problem list.]",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-geometry-notes.json",
   "source_index": 409,
   "attempt": 1
  },
  "AIM-GEOMETRY-0411": {
   "statement_status": "exact",
   "original_statement": "Question 1.15. (a) Make a theory of \"Weinstein spaces\", as opposed to Weinstein mani-folds, by analogy with Stein manifolds and Stein spaces. (b) Is the space of psh functions on a Weinstein manifold connected? (No critical points at infinity.) What if we assume the same contact structure at infinity? There exists an example of a manifold diffeomorphic to Cn for which there is more than 1 critical point for any psh function, and this bears on uniqueness.",
   "clean_statement": "Question 1.15. (a) Make a theory of \"Weinstein spaces\", as opposed to Weinstein mani-folds, by analogy with Stein manifolds and Stein spaces. (b) Is the space of psh functions on a Weinstein manifold connected? (No critical points at infinity.) What if we assume the same contact structure at infinity? There exists an example of a manifold diffeomorphic to Cn for which there is more than 1 critical point for any psh function, and this bears on uniqueness.",
   "public_statement": "Question 1.15. (a) Make a theory of \"Weinstein spaces\", as opposed to Weinstein mani-folds, by analogy with Stein manifolds and Stein spaces. (b) Is the space of psh functions on a Weinstein manifold connected? (No critical points at infinity.) What if we assume the same contact structure at infinity? There exists an example of a manifold diffeomorphic to Cn for which there is more than 1 critical point for any psh function, and this bears on uniqueness.",
   "evidence": "The canonical record is Question 1.15 in Michael Hutchings, with help from Yasha Eliashberg and John Etnyre, *Holomorphic curves in contact geometry*, AIM workshop problem list, version 15 October 2003:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 410,
   "attempt": 1
  },
  "AIM-GEOMETRY-0412": {
   "statement_status": "exact",
   "original_statement": "Question 1.16. Auroux asked (and others participated in the discussion) if there is some category for Legendrians in open books, by analogy with the Fukaya category of Lagrangians in a sympletic manifold. It is difficult to separate out the two Legendrians (i.e. just look at Reeb chords from one to the other), so you would have to have coefficients in the contact homology of the individual Legendrians. One can make use of the extra function direction in the open 6\n\nbook, e.g. one can filter CH by the amount of rotation of the Reeb chords around the open book. If the two Legendrians are on the same page, the CH should not agree with the Lagrangian Floer homology, because one has to take into account the images of the Legendrians under the iterated monodromy. And again, putting in the binding might somehow correspond to a deformation of an A∞ category. Also note that in general, the relative contact homology should be a module over the absolute contact homology. Eliashberg discussed an approach to the geometry of gluing in the binding. For the sym-plectization of the complement of the binding, you have convex, flat, and concave boundary components. However you can round corners to absorb the flat component into the convex component. Then, when you glue in the binding, you \"partially glue\" onto part of the convex component. \n\nThird day \n\nAt the end of his talk, Paul Biran asked the following questions:",
   "clean_statement": "Question 1.16. Auroux asked (and others participated in the discussion) if there is some category for Legendrians in open books, by analogy with the Fukaya category of Lagrangians in a sympletic manifold. It is difficult to separate out the two Legendrians (i.e. just look at Reeb chords from one to the other), so you would have to have coefficients in the contact homology of the individual Legendrians. One can make use of the extra function direction in the open 6\n\nbook, e.g. one can filter CH by the amount of rotation of the Reeb chords around the open book. If the two Legendrians are on the same page, the CH should not agree with the Lagrangian Floer homology, because one has to take into account the images of the Legendrians under the iterated monodromy. And again, putting in the binding might somehow correspond to a deformation of an A∞ category. Also note that in general, the relative contact homology should be a module over the absolute contact homology. Eliashberg discussed an approach to the geometry of gluing in the binding. For the sym-plectization of the complement of the binding, you have convex, flat, and concave boundary components. However you can round corners to absorb the flat component into the convex component. Then, when you glue in the binding, you \"partially glue\" onto part of the convex component.\n\nThird day\n\nAt the end of his talk, Paul Biran asked the following questions:",
   "public_statement": "Question 1.16. Auroux asked (and others participated in the discussion) if there is some category for Legendrians in open books, by analogy with the Fukaya category of Lagrangians in a sympletic manifold. It is difficult to separate out the two Legendrians (i.e. just look at Reeb chords from one to the other), so you would have to have coefficients in the contact homology of the individual Legendrians. One can make use of the extra function direction in the open 6\n\nbook, e.g. one can filter CH by the amount of rotation of the Reeb chords around the open book. If the two Legendrians are on the same page, the CH should not agree with the Lagrangian Floer homology, because one has to take into account the images of the Legendrians under the iterated monodromy. And again, putting in the binding might somehow correspond to a deformation of an A∞ category. Also note that in general, the relative contact homology should be a module over the absolute contact homology. Eliashberg discussed an approach to the geometry of gluing in the binding. For the sym-plectization of the complement of the binding, you have convex, flat, and concave boundary components. However you can round corners to absorb the flat component into the convex component. Then, when you glue in the binding, you \"partially glue\" onto part of the convex component.\n\nThird day\n\nAt the end of his talk, Paul Biran asked the following questions:",
   "evidence": "This is Question 1.16 in the AIM workshop report *Holomorphic curves in contact geometry*. The JSON extraction contains a page-number artifact and text from the next workshop day. Comparison with pages 5--6 of the official PDF gives the following recovery.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 411,
   "attempt": 1
  },
  "AIM-GEOMETRY-0413": {
   "statement_status": "exact",
   "original_statement": "Question 1.17. Let P 5 be the contact S1-bundle over T 4 with c1 equal to the class of the symplectic form on T 4. Conjecture: P has no Stein filling.",
   "clean_statement": "Question 1.17. Let P 5 be the contact S1-bundle over T 4 with c1 equal to the class of the symplectic form on T 4. Conjecture: P has no Stein filling.",
   "public_statement": "Question 1.17. Let P 5 be the contact S1-bundle over T 4 with c1 equal to the class of the symplectic form on T 4. Conjecture: P has no Stein filling.",
   "evidence": "The canonical record is `aim-geometry-notes.json`, record 412 (zero-based). Its extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 412,
   "attempt": 1
  },
  "AIM-GEOMETRY-0414": {
   "statement_status": "exact",
   "original_statement": "Question 1.18. Let Qn = {z20 + · · · + z2 \n\n> n+1\n\n= 0 } ⊂ CP n. Conjecture: Qn does not contain two disjoint Lagrangian spheres. (This is trivial if n is even.) \n\nAt the end of his talk, Leonid Polterovich asked the following question:",
   "clean_statement": "Question 1.18. Let Qn = {z20 + · · · + z2\n\n> n+1\n\n= 0 } ⊂ CP n. Conjecture: Qn does not contain two disjoint Lagrangian spheres. (This is trivial if n is even.)\n\nAt the end of his talk, Leonid Polterovich asked the following question:",
   "public_statement": "Question 1.18. Let Qn = {z20 + · · · + z2\n\n> n+1\n\n= 0 } ⊂ CP n. Conjecture: Qn does not contain two disjoint Lagrangian spheres. (This is trivial if n is even.)\n\nAt the end of his talk, Leonid Polterovich asked the following question:",
   "evidence": "The canonical record is source index 413 of \\`aim-geometry-notes.json\\). It displays \\[ Q^n=\\{z_0^2+\\cdots+z_{n+1}^2=0\\}\\subset\\mathbb CP^n \\] and conjectures that \\(Q^n\\) does not contain two disjoint Lagrangian spheres.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 413,
   "attempt": 1
  },
  "AIM-GEOMETRY-0415": {
   "statement_status": "exact",
   "original_statement": "Question 1.19. (a) Let M be the unit ball in R4 and let L = {p21 + q21 = p22 + q22 = 1 /2} ⊂ R4\n\nbe the Clifford torus. Is L weakly boundary rigid? That is, if K is Hamiltonian isotopic to L\n\nin M, then must K ⊂ ∂M? (There are several proofs that one cannot have K ⊂ M \\ ∂M.) [See Eliashberg's discussion at the end of the day.] (b) Similarly investigate Lagrangian tori in other ellipsoids in R2n, and explore nonremovable intersections in this setting. \n\nAt the end of his talk, Alex Ivrii mentioned the following open questions:",
   "clean_statement": "Question 1.19. (a) Let M be the unit ball in R4 and let L = {p21 + q21 = p22 + q22 = 1 /2} ⊂ R4\n\nbe the Clifford torus. Is L weakly boundary rigid? That is, if K is Hamiltonian isotopic to L\n\nin M, then must K ⊂ ∂M? (There are several proofs that one cannot have K ⊂ M \\ ∂M.) [See Eliashberg's discussion at the end of the day.] (b) Similarly investigate Lagrangian tori in other ellipsoids in R2n, and explore nonremovable intersections in this setting.\n\nAt the end of his talk, Alex Ivrii mentioned the following open questions:",
   "public_statement": "Question 1.19. (a) Let M be the unit ball in R4 and let L = {p21 + q21 = p22 + q22 = 1 /2} ⊂ R4\n\nbe the Clifford torus. Is L weakly boundary rigid? That is, if K is Hamiltonian isotopic to L\n\nin M, then must K ⊂ ∂M? (There are several proofs that one cannot have K ⊂ M \\ ∂M.) [See Eliashberg's discussion at the end of the day.] (b) Similarly investigate Lagrangian tori in other ellipsoids in R2n, and explore nonremovable intersections in this setting.\n\nAt the end of his talk, Alex Ivrii mentioned the following open questions:",
   "evidence": "The canonical record is `aim-geometry-notes.json`, record 414 (zero-based). Its formula is visibly damaged by extraction. The official AIM PDF, *Holomorphic curves in contact geometry*, page 6, gives the following statement:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 414,
   "attempt": 1
  },
  "AIM-GEOMETRY-0416": {
   "statement_status": "exact",
   "original_statement": "Question 1.20. (a) If L ⊂ T ∗Σg is a Lagrangian that is homologous to the 0-section, and \n\ng > 1, must L be Lagrangian isotopic to the 0-section? (This is now known for g = 0 and \n\ng = 1 by the works of Richard Hind and Alex Ivrii respectively.) (b) Is there a Lagrangian T n ⊂ R2n which is not Lagrangian isotopic to the Clifford torus when n > 2? The answer is no for n = 2 by Ivrii. However current holomorphic curve techniques do not seem applicable to higher n. We do not even know: (c) Are there any local Lagrangian knots in R2n for n > 2, i.e. Lagrangian submanifolds that are asymptotic to a Lagrangian n-plane but not Lagrangian isotopic to one? (Eliashberg and Polterovich showed that the answer is no when n = 2.)",
   "clean_statement": "Question 1.20. (a) If L ⊂ T ∗Σg is a Lagrangian that is homologous to the 0-section, and\n\ng > 1, must L be Lagrangian isotopic to the 0-section? (This is now known for g = 0 and\n\ng = 1 by the works of Richard Hind and Alex Ivrii respectively.) (b) Is there a Lagrangian T n ⊂ R2n which is not Lagrangian isotopic to the Clifford torus when n > 2? The answer is no for n = 2 by Ivrii. However current holomorphic curve techniques do not seem applicable to higher n. We do not even know: (c) Are there any local Lagrangian knots in R2n for n > 2, i.e. Lagrangian submanifolds that are asymptotic to a Lagrangian n-plane but not Lagrangian isotopic to one? (Eliashberg and Polterovich showed that the answer is no when n = 2.)",
   "public_statement": "Question 1.20. (a) If L ⊂ T ∗Σg is a Lagrangian that is homologous to the 0-section, and\n\ng > 1, must L be Lagrangian isotopic to the 0-section? (This is now known for g = 0 and\n\ng = 1 by the works of Richard Hind and Alex Ivrii respectively.) (b) Is there a Lagrangian T n ⊂ R2n which is not Lagrangian isotopic to the Clifford torus when n > 2? The answer is no for n = 2 by Ivrii. However current holomorphic curve techniques do not seem applicable to higher n. We do not even know: (c) Are there any local Lagrangian knots in R2n for n > 2, i.e. Lagrangian submanifolds that are asymptotic to a Lagrangian n-plane but not Lagrangian isotopic to one? (Eliashberg and Polterovich showed that the answer is no when n = 2.)",
   "evidence": "The source is Michael Hutchings, with help from Yasha Eliashberg and John Etnyre, *Holomorphic curves in contact geometry*, AIM workshop problem list, version 15 October 2003, Question 1.20. The official HTML and PDF agree on the following mathematical statement (typography restored, wording unchanged):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 415,
   "attempt": 1
  },
  "AIM-GEOMETRY-0417": {
   "statement_status": "exact",
   "original_statement": "Question 1.21. Margaret Symington asked a series of questions centered around the question of whether a locally toric structure is helpful for counting holomorphic curves. This is inspired by the success of using amoebas in Mikhalkin's work; locally toric pictures have a lot of the same structure Mikhalkin used. [Unfortunately I do not have good notes on this.]",
   "clean_statement": "Question 1.21. Margaret Symington asked a series of questions centered around the question of whether a locally toric structure is helpful for counting holomorphic curves. This is inspired by the success of using amoebas in Mikhalkin's work; locally toric pictures have a lot of the same structure Mikhalkin used. [Unfortunately I do not have good notes on this.]",
   "public_statement": "Question 1.21. Margaret Symington asked a series of questions centered around the question of whether a locally toric structure is helpful for counting holomorphic curves. This is inspired by the success of using amoebas in Mikhalkin's work; locally toric pictures have a lot of the same structure Mikhalkin used. [Unfortunately I do not have good notes on this.]",
   "evidence": "The source is Michael Hutchings, with help from Yasha Eliashberg and John Etnyre, *Holomorphic curves in contact geometry*, AIM workshop problem list, version 15 October 2003, Question 1.21. The official AIM HTML and PDF contain the same text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 416,
   "attempt": 1
  },
  "AIM-GEOMETRY-0418": {
   "statement_status": "exact",
   "original_statement": "Question 1.22. Polterovich mentioned spectral invariants in Floer homology. Namely, for \n\nf ∈ Ham( M ) and α ∈ QH ∗(M ), define c(f, α ) to be the smallest c such that α appears in HF ∗(A ≤ c), where A is the action functional on a suitable cover of the loop space. Question: study the asymptotic properties of c(f n, α ).7",
   "clean_statement": "Question 1.22. Polterovich mentioned spectral invariants in Floer homology. Namely, for\n\nf ∈ Ham( M ) and α ∈ QH ∗(M ), define c(f, α ) to be the smallest c such that α appears in HF ∗(A ≤ c), where A is the action functional on a suitable cover of the loop space. Question: study the asymptotic properties of c(f n, α ).7",
   "public_statement": "Question 1.22. Polterovich mentioned spectral invariants in Floer homology. Namely, for\n\nf ∈ Ham( M ) and α ∈ QH ∗(M ), define c(f, α ) to be the smallest c such that α appears in HF ∗(A ≤ c), where A is the action functional on a suitable cover of the loop space. Question: study the asymptotic properties of c(f n, α ).7",
   "evidence": "This is Question 1.22 in the AIM problem list *Holomorphic curves in contact geometry* (attributed in the document to Michael Hutchings, with help from Yasha Eliashberg and John Etnyre). The PDF statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 417,
   "attempt": 1
  },
  "AIM-GEOMETRY-0419": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 1.23. Polterovich also suggested that it if Φ: M → B is a [what kind of?] fibration, then the set {Φ∗H | H: M → R} is a \"maximal torus\" in Ham( M ) and should provide a good source of examples for calculations in Floer homology. [There was then some further discussion of locally toric fibrations by various people of which I do not have good notes.]",
   "clean_statement": null,
   "public_statement": "Question 1.23. Polterovich also suggested that it if Φ: M → B is a [what kind of?] fibration, then the set {Φ∗H | H: M → R} is a \"maximal torus\" in Ham( M ) and should provide a good source of examples for calculations in Floer homology. [There was then some further discussion of locally toric fibrations by various people of which I do not have good notes.]",
   "evidence": "The source is Michael Hutchings, with help from Yasha Eliashberg and John Etnyre, *Holomorphic curves in contact geometry*, AIM workshop problem list, version 15 October 2003, Question 1.23. The official AIM PDF and HTML both literally print:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 418,
   "attempt": 1
  },
  "AIM-GEOMETRY-0420": {
   "statement_status": "exact",
   "original_statement": "Question 1.24. Polterovich then suggested that Hofer's geometry on the space of Lagrangian submanifolds (as opposed to Hamiltonian symplectomorphisms) has not really been studied properly. In particular, the following 2-dimensional problem is unsolved. Let L denote the space of simple closed curves in S2 that divide S2 into two regions of equal area. Define a metric d on L as follows. Suppose F (x, t ) is a Hamiltonian generating an isotopy {Lt} from \n\nL0 to L1. Define the length of the path {Lt} by \n\nlength {Lt}:= \n\n∫ 10\n\n(max Ft|Lt − min Ft|Lt ) dt. \n\nFinally, define \n\nd(L0, L 1):= inf γ length( γ)\n\nwhere the infimum is taken over any path γ from L0 to L1.Question: what is the diameter of the metric space (L, d )? It is known that Ham( S2)\n\nhas infinite diameter, but there is no natural candidate for a path going to ∞ in L. (The following possibility was proposed: take a vertical great circle and then stretch it by spinning a neighborhood of the equator a lot. However there was some skepticism that this would give a path of infinite length.) For a given pair of curves in L, one can apparently get an upper bound on the distance between them in terms of combinatorics (meanders). If this diameter is finite, then it might also be interesting to study the analogous in-variant for a general pair (M, L ) where M is a symplectic manifold and L is a Lagrangian submanifold of M.For example it is interesting to ask the same question for (CP 2, RP 2).Also note that there is a natural map \n\nHam( M, ω ) −→ L (M × M, ω ⊕ − ω).\n\nThis preserves the lengths of smooth paths, but is not an isometry; the work of Ostrover shows that this map is highly distorting.",
   "clean_statement": "Question 1.24. Polterovich then suggested that Hofer's geometry on the space of Lagrangian submanifolds (as opposed to Hamiltonian symplectomorphisms) has not really been studied properly. In particular, the following 2-dimensional problem is unsolved. Let L denote the space of simple closed curves in S2 that divide S2 into two regions of equal area. Define a metric d on L as follows. Suppose F (x, t ) is a Hamiltonian generating an isotopy {Lt} from\n\nL0 to L1. Define the length of the path {Lt} by\n\nlength {Lt}:=\n\n∫ 10\n\n(max Ft|Lt − min Ft|Lt ) dt.\n\nFinally, define\n\nd(L0, L 1):= inf γ length( γ)\n\nwhere the infimum is taken over any path γ from L0 to L1.Question: what is the diameter of the metric space (L, d )? It is known that Ham( S2)\n\nhas infinite diameter, but there is no natural candidate for a path going to ∞ in L. (The following possibility was proposed: take a vertical great circle and then stretch it by spinning a neighborhood of the equator a lot. However there was some skepticism that this would give a path of infinite length.) For a given pair of curves in L, one can apparently get an upper bound on the distance between them in terms of combinatorics (meanders). If this diameter is finite, then it might also be interesting to study the analogous in-variant for a general pair (M, L ) where M is a symplectic manifold and L is a Lagrangian submanifold of M.For example it is interesting to ask the same question for (CP 2, RP 2).Also note that there is a natural map\n\nHam( M, ω ) −→ L (M × M, ω ⊕ − ω).\n\nThis preserves the lengths of smooth paths, but is not an isometry; the work of Ostrover shows that this map is highly distorting.",
   "public_statement": "Question 1.24. Polterovich then suggested that Hofer's geometry on the space of Lagrangian submanifolds (as opposed to Hamiltonian symplectomorphisms) has not really been studied properly. In particular, the following 2-dimensional problem is unsolved. Let L denote the space of simple closed curves in S2 that divide S2 into two regions of equal area. Define a metric d on L as follows. Suppose F (x, t ) is a Hamiltonian generating an isotopy {Lt} from\n\nL0 to L1. Define the length of the path {Lt} by\n\nlength {Lt}:=\n\n∫ 10\n\n(max Ft|Lt − min Ft|Lt ) dt.\n\nFinally, define\n\nd(L0, L 1):= inf γ length( γ)\n\nwhere the infimum is taken over any path γ from L0 to L1.Question: what is the diameter of the metric space (L, d )? It is known that Ham( S2)\n\nhas infinite diameter, but there is no natural candidate for a path going to ∞ in L. (The following possibility was proposed: take a vertical great circle and then stretch it by spinning a neighborhood of the equator a lot. However there was some skepticism that this would give a path of infinite length.) For a given pair of curves in L, one can apparently get an upper bound on the distance between them in terms of combinatorics (meanders). If this diameter is finite, then it might also be interesting to study the analogous in-variant for a general pair (M, L ) where M is a symplectic manifold and L is a Lagrangian submanifold of M.For example it is interesting to ask the same question for (CP 2, RP 2).Also note that there is a natural map\n\nHam( M, ω ) −→ L (M × M, ω ⊕ − ω).\n\nThis preserves the lengths of smooth paths, but is not an isometry; the work of Ostrover shows that this map is highly distorting.",
   "evidence": "This is Question 1.24 in Michael Hutchings's AIM outline *Holomorphic curves in contact geometry*, written with help from Yasha Eliashberg and John Etnyre. The official PDF and HTML give the following formulas, repairing the corpus OCR:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 419,
   "attempt": 1
  },
  "AIM-GEOMETRY-0421": {
   "statement_status": "exact",
   "original_statement": "Question 1.25. Biran pointed out that the following conjecture of Audin has not yet been proved in full generality: if L is a Lagrangian torus in Cn then the minimal Maslov number \n\nNL = 2. In the monotone case, this follows by a simple argument using Oh's spectral sequence.",
   "clean_statement": "Question 1.25. Biran pointed out that the following conjecture of Audin has not yet been proved in full generality: if L is a Lagrangian torus in Cn then the minimal Maslov number\n\nNL = 2. In the monotone case, this follows by a simple argument using Oh's spectral sequence.",
   "public_statement": "Question 1.25. Biran pointed out that the following conjecture of Audin has not yet been proved in full generality: if L is a Lagrangian torus in Cn then the minimal Maslov number\n\nNL = 2. In the monotone case, this follows by a simple argument using Oh's spectral sequence.",
   "evidence": "The canonical record is Question 1.25 from the AIM workshop list *Holomorphic curves in contact geometry*. The extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 420,
   "attempt": 1
  },
  "AIM-GEOMETRY-0422": {
   "statement_status": "exact",
   "original_statement": "Question 1.26. Eliashberg discussed using SFT to prove the Audin conjecture and the weak boundary rigidity of the Clifford torus in S3. [How much of this requires the as-yet unfinished foundations of SFT, and how much of it is merely inspired by SFT and does not require that much machinery?] The basic idea is that instead of considering holomorphic curves with boundary in L one should consider holomorphic curves with asymptotic conditions in T ∗L.One then has to study holomorphic curves in T ∗(T n) just once. However even this remains \"stupidly open\". (It was also asked if one can do something with CR structures...) 8\n\nFourth day",
   "clean_statement": "Question 1.26. Eliashberg discussed using SFT to prove the Audin conjecture and the weak boundary rigidity of the Clifford torus in S3. [How much of this requires the as-yet unfinished foundations of SFT, and how much of it is merely inspired by SFT and does not require that much machinery?] The basic idea is that instead of considering holomorphic curves with boundary in L one should consider holomorphic curves with asymptotic conditions in T ∗L.One then has to study holomorphic curves in T ∗(T n) just once. However even this remains \"stupidly open\". (It was also asked if one can do something with CR structures...) 8\n\nFourth day",
   "public_statement": "Question 1.26. Eliashberg discussed using SFT to prove the Audin conjecture and the weak boundary rigidity of the Clifford torus in S3. [How much of this requires the as-yet unfinished foundations of SFT, and how much of it is merely inspired by SFT and does not require that much machinery?] The basic idea is that instead of considering holomorphic curves with boundary in L one should consider holomorphic curves with asymptotic conditions in T ∗L.One then has to study holomorphic curves in T ∗(T n) just once. However even this remains \"stupidly open\". (It was also asked if one can do something with CR structures...) 8\n\nFourth day",
   "evidence": "This is Question 1.26 in the AIM workshop notes *Holomorphic curves in contact geometry* (version dated 15 October 2003). The corpus transcription has three recoverable OCR/layout errors:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 421,
   "attempt": 1
  },
  "AIM-GEOMETRY-0423": {
   "statement_status": "exact",
   "original_statement": "Question 1.27. At the end of his talk, Hutchings mentioned that one could try to generalize the methods therein to compute the Embedded Contact Homology of S1 × S2, torus bundles over S1, or unit cotangent bundles of surfaces of genus g > 1. It might also be interesting to try to further understand the holomorphic curves in R × T 3 in terms of amoebas etc.",
   "clean_statement": "Question 1.27. At the end of his talk, Hutchings mentioned that one could try to generalize the methods therein to compute the Embedded Contact Homology of S1 × S2, torus bundles over S1, or unit cotangent bundles of surfaces of genus g > 1. It might also be interesting to try to further understand the holomorphic curves in R × T 3 in terms of amoebas etc.",
   "public_statement": "Question 1.27. At the end of his talk, Hutchings mentioned that one could try to generalize the methods therein to compute the Embedded Contact Homology of S1 × S2, torus bundles over S1, or unit cotangent bundles of surfaces of genus g > 1. It might also be interesting to try to further understand the holomorphic curves in R × T 3 in terms of amoebas etc.",
   "evidence": "The canonical record is Question 1.27 from the AIM workshop notes *Holomorphic curves in contact geometry*. The official AIM HTML and PDF put the question immediately after the heading “Fourth day.” Restoring the superscripts and blackboard-bold font lost in the corpus extraction, the statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 422,
   "attempt": 1
  },
  "AIM-GEOMETRY-0424": {
   "statement_status": "exact",
   "original_statement": "Question 1.28. Peter Ozsvath discussed the problem of computing Ozsvath and Szabo's invariants combinatorially or axiomatically. Their theory lies somewhere between combina-torial invariants, which are easy to compute but not very useful, and invariants defined in terms of PDE's, which are very useful but hard to compute. \"Right next door\" is Khovanov's categorification of the Jones polynomial, which looks like a Floer theory but is defined purely combinatorially. In fact there is the following bridge between them: given a link K in S3,there is a spectral sequence whose E2 term is the Khovanov homology of K (with Z/2 co-efficients) and which converges to ̂ HF (Σ 2(K)), where Σ2(K) denotes the double cover of \n\nS3 branched along K. Question: can one compute the differentials in the spectral sequence combinatorially? Also, ̂ HF (Σ 2(K)) has applications to slice genus bounds: can Khovanov homology say anything about the 4-ball genus?",
   "clean_statement": "Question 1.28. Peter Ozsvath discussed the problem of computing Ozsvath and Szabo's invariants combinatorially or axiomatically. Their theory lies somewhere between combina-torial invariants, which are easy to compute but not very useful, and invariants defined in terms of PDE's, which are very useful but hard to compute. \"Right next door\" is Khovanov's categorification of the Jones polynomial, which looks like a Floer theory but is defined purely combinatorially. In fact there is the following bridge between them: given a link K in S3,there is a spectral sequence whose E2 term is the Khovanov homology of K (with Z/2 co-efficients) and which converges to ̂ HF (Σ 2(K)), where Σ2(K) denotes the double cover of\n\nS3 branched along K. Question: can one compute the differentials in the spectral sequence combinatorially? Also, ̂ HF (Σ 2(K)) has applications to slice genus bounds: can Khovanov homology say anything about the 4-ball genus?",
   "public_statement": "Question 1.28. Peter Ozsvath discussed the problem of computing Ozsvath and Szabo's invariants combinatorially or axiomatically. Their theory lies somewhere between combina-torial invariants, which are easy to compute but not very useful, and invariants defined in terms of PDE's, which are very useful but hard to compute. \"Right next door\" is Khovanov's categorification of the Jones polynomial, which looks like a Floer theory but is defined purely combinatorially. In fact there is the following bridge between them: given a link K in S3,there is a spectral sequence whose E2 term is the Khovanov homology of K (with Z/2 co-efficients) and which converges to ̂ HF (Σ 2(K)), where Σ2(K) denotes the double cover of\n\nS3 branched along K. Question: can one compute the differentials in the spectral sequence combinatorially? Also, ̂ HF (Σ 2(K)) has applications to slice genus bounds: can Khovanov homology say anything about the 4-ball genus?",
   "evidence": "The canonical record is Question 1.28 from the AIM workshop list *Holomorphic curves in contact geometry*. The JSON preserves the source text but loses mathematical typography and contains line-break hyphenation:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 423,
   "attempt": 1
  },
  "AIM-GEOMETRY-0425": {
   "statement_status": "exact",
   "original_statement": "Question 1.29. A big open problem is to prove the conjectured equivalence between Ozsvath-Szabo theory and Seiberg-Witten theory. Yi-Jen Lee discussed a possible approach to this. \n\nFifth day",
   "clean_statement": "Question 1.29. A big open problem is to prove the conjectured equivalence between Ozsvath-Szabo theory and Seiberg-Witten theory. Yi-Jen Lee discussed a possible approach to this.\n\nFifth day",
   "public_statement": "Question 1.29. A big open problem is to prove the conjectured equivalence between Ozsvath-Szabo theory and Seiberg-Witten theory. Yi-Jen Lee discussed a possible approach to this.\n\nFifth day",
   "evidence": "The source is the AIM workshop list *Holomorphic curves in contact geometry*, version dated 15 October 2003. The database record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 424,
   "attempt": 1
  },
  "AIM-GEOMETRY-0426": {
   "statement_status": "exact",
   "original_statement": "Question 1.\n3\n0. (Eliashberg) Do symplectic and contact invariants of the cotangent bun-dle of a smooth manifold M remember the topology of M? For example, if M1 and M2\n\nare homeomorphic but not diffeomorphic (e.g. exotic spheres), can one detect this by show-ing that T ∗M1 and T ∗M2 are not symplectomorphic, or that ST ∗M1 and ST ∗M2 are not contactomorphic? Related question: is the natural map from knots in S3 to Legendrian tori in ST ∗S3 '\n\nS3 × S2 injective? Ng's work is the first result in this direction.",
   "clean_statement": "Question 1.\n3\n0. (Eliashberg) Do symplectic and contact invariants of the cotangent bun-dle of a smooth manifold M remember the topology of M? For example, if M1 and M2\n\nare homeomorphic but not diffeomorphic (e.g. exotic spheres), can one detect this by show-ing that T ∗M1 and T ∗M2 are not symplectomorphic, or that ST ∗M1 and ST ∗M2 are not contactomorphic? Related question: is the natural map from knots in S3 to Legendrian tori in ST ∗S3 '\n\nS3 × S2 injective? Ng's work is the first result in this direction.",
   "public_statement": "Question 1.\n3\n0. (Eliashberg) Do symplectic and contact invariants of the cotangent bun-dle of a smooth manifold M remember the topology of M? For example, if M1 and M2\n\nare homeomorphic but not diffeomorphic (e.g. exotic spheres), can one detect this by show-ing that T ∗M1 and T ∗M2 are not symplectomorphic, or that ST ∗M1 and ST ∗M2 are not contactomorphic? Related question: is the natural map from knots in S3 to Legendrian tori in ST ∗S3 '\n\nS3 × S2 injective? Ng's work is the first result in this direction.",
   "evidence": "The OCR record splits the number “1.30,” hyphenates words in the middle of lines, and turns the isomorphism sign into a stray apostrophe. The official AIM workshop page gives the following recovered question (notation normalized only by typesetting):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 425,
   "attempt": 1
  },
  "AIM-GEOMETRY-0427": {
   "statement_status": "exact",
   "original_statement": "Question 1.31. Eliashberg discussed trying to understand the Gopakumar-Vafa picture in terms of symplectic field theory. [I missed a lot here...] One question which is important for this is to try to define Relative Contact Homology using higher genus curves, not just rational curves. \n\nSixth day",
   "clean_statement": "Question 1.31. Eliashberg discussed trying to understand the Gopakumar-Vafa picture in terms of symplectic field theory. [I missed a lot here...] One question which is important for this is to try to define Relative Contact Homology using higher genus curves, not just rational curves.\n\nSixth day",
   "public_statement": "Question 1.31. Eliashberg discussed trying to understand the Gopakumar-Vafa picture in terms of symplectic field theory. [I missed a lot here...] One question which is important for this is to try to define Relative Contact Homology using higher genus curves, not just rational curves.\n\nSixth day",
   "evidence": "The canonical record is Question 1.31 from the AIM workshop list *Holomorphic curves in contact geometry*. The source record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 426,
   "attempt": 1
  },
  "AIM-GEOMETRY-0428": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 1.32. Akahori discussed some problems involving understanding the geometry of Kohn-Rossi cohomology.",
   "clean_statement": null,
   "public_statement": "Question 1.32. Akahori discussed some problems involving understanding the geometry of Kohn-Rossi cohomology.",
   "evidence": "The complete entry in the official AIM workshop report *Holomorphic curves in contact geometry* is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-geometry-notes.json",
   "source_index": 427,
   "attempt": 1
  },
  "AIM-GEOMETRY-0429": {
   "statement_status": "exact",
   "original_statement": "Question 1.\n3\n3. (Giroux) (a) A contact structure on a closed manifold is equivalent to a symplectomorphism of a Stein manifold, equal to the identity on the boundary, modulo some kind of stabilization. What are interesting examples of symplectomorphisms of Stein manifolds that are equal to the identity on the boundary? One example of such a symplectomorphism is the symplectic Dehn twist around a (parametrized) Lagrangian sphere. Do these generate the group of all such symplectomorphisms? 9\n\nOf course, this would imply that the group of all such symplectomorphisms of the 6-ball is trivial, which we do not know. (Eliashberg: is there some formulation of the above question modulo the seemingly hopeless problem of understanding symplectomorphisms of higher-dimensional balls?) For example, consider a closed symplectic manifold (W, ω ) with [ω] integral. Let Hk\n\ndenote the degree k Donaldson hyperplane section for k >> 0. Then Fk:= W \\ Hk is Stein. A neighborhood of ∂F k is a symplectic annulus bundle over Hk. Suppose you do a Dehn twist in each annulus fiber, then you get a symplectomorphism of Fk which is the identity on the boundary. (This is the monodromy of a certain canonical open book...) Question: is this symplectomorphism a product of symplectic Dehn twists? In some cases, Fk is a subcritical submanifold, so by Biran-Cieliebak, contains no Lagrangian sphere. In particular, in this case, is the above symplectomorphism isotopic (even topologically) to the identity? (b) Here is a possible procedure for constructing Ustilovsky's infinitely many contact struc-tures on S4n+1 in terms of open books. The standard contact structure on S2n+1 comes from the positive symplectic Dehn twist φ on T ∗Sn. Think that φ3, φ 5,... give Ustilovsky's con-tact structures. Can one distinguish φ from φ3 using Floer homology? Is there a stabilized version of Floer homology distinguishing the contact structures, and how does this compare to contact homology? More generally, this is a procedure for constructing many new contact structures out of an old one, by replacing each Dehn twist by an odd iterate of it. (c) Here is a way to possibly produce nonfillable contact structures in higher dimensions, analogous to overtwisted contact structures. Let W be a Stein manifold, φ a symplectomor-phism of W equal to the identity on the boundary, and Dn a Lagrangian ball with ∂D ⊂ ∂W.On ∂D, attach a handle outside, to get a new Stein manifold with a Lagrangian sphere. Compose φ with a left-handed Dehn twist. We get the same manifold, but are the con-tact structures different? Unfillable? In dimension 3, one can get all overtwisted contact structures by this construction. (Eliashberg: Try S5, one left-handed twist in U T ∗(S2).)",
   "clean_statement": "Question 1.\n3\n3. (Giroux) (a) A contact structure on a closed manifold is equivalent to a symplectomorphism of a Stein manifold, equal to the identity on the boundary, modulo some kind of stabilization. What are interesting examples of symplectomorphisms of Stein manifolds that are equal to the identity on the boundary? One example of such a symplectomorphism is the symplectic Dehn twist around a (parametrized) Lagrangian sphere. Do these generate the group of all such symplectomorphisms? 9\n\nOf course, this would imply that the group of all such symplectomorphisms of the 6-ball is trivial, which we do not know. (Eliashberg: is there some formulation of the above question modulo the seemingly hopeless problem of understanding symplectomorphisms of higher-dimensional balls?) For example, consider a closed symplectic manifold (W, ω ) with [ω] integral. Let Hk\n\ndenote the degree k Donaldson hyperplane section for k >> 0. Then Fk:= W \\ Hk is Stein. A neighborhood of ∂F k is a symplectic annulus bundle over Hk. Suppose you do a Dehn twist in each annulus fiber, then you get a symplectomorphism of Fk which is the identity on the boundary. (This is the monodromy of a certain canonical open book...) Question: is this symplectomorphism a product of symplectic Dehn twists? In some cases, Fk is a subcritical submanifold, so by Biran-Cieliebak, contains no Lagrangian sphere. In particular, in this case, is the above symplectomorphism isotopic (even topologically) to the identity? (b) Here is a possible procedure for constructing Ustilovsky's infinitely many contact struc-tures on S4n+1 in terms of open books. The standard contact structure on S2n+1 comes from the positive symplectic Dehn twist φ on T ∗Sn. Think that φ3, φ 5,... give Ustilovsky's con-tact structures. Can one distinguish φ from φ3 using Floer homology? Is there a stabilized version of Floer homology distinguishing the contact structures, and how does this compare to contact homology? More generally, this is a procedure for constructing many new contact structures out of an old one, by replacing each Dehn twist by an odd iterate of it. (c) Here is a way to possibly produce nonfillable contact structures in higher dimensions, analogous to overtwisted contact structures. Let W be a Stein manifold, φ a symplectomor-phism of W equal to the identity on the boundary, and Dn a Lagrangian ball with ∂D ⊂ ∂W.On ∂D, attach a handle outside, to get a new Stein manifold with a Lagrangian sphere. Compose φ with a left-handed Dehn twist. We get the same manifold, but are the con-tact structures different? Unfillable? In dimension 3, one can get all overtwisted contact structures by this construction. (Eliashberg: Try S5, one left-handed twist in U T ∗(S2).)",
   "public_statement": "Question 1.\n3\n3. (Giroux) (a) A contact structure on a closed manifold is equivalent to a symplectomorphism of a Stein manifold, equal to the identity on the boundary, modulo some kind of stabilization. What are interesting examples of symplectomorphisms of Stein manifolds that are equal to the identity on the boundary? One example of such a symplectomorphism is the symplectic Dehn twist around a (parametrized) Lagrangian sphere. Do these generate the group of all such symplectomorphisms? 9\n\nOf course, this would imply that the group of all such symplectomorphisms of the 6-ball is trivial, which we do not know. (Eliashberg: is there some formulation of the above question modulo the seemingly hopeless problem of understanding symplectomorphisms of higher-dimensional balls?) For example, consider a closed symplectic manifold (W, ω ) with [ω] integral. Let Hk\n\ndenote the degree k Donaldson hyperplane section for k >> 0. Then Fk:= W \\ Hk is Stein. A neighborhood of ∂F k is a symplectic annulus bundle over Hk. Suppose you do a Dehn twist in each annulus fiber, then you get a symplectomorphism of Fk which is the identity on the boundary. (This is the monodromy of a certain canonical open book...) Question: is this symplectomorphism a product of symplectic Dehn twists? In some cases, Fk is a subcritical submanifold, so by Biran-Cieliebak, contains no Lagrangian sphere. In particular, in this case, is the above symplectomorphism isotopic (even topologically) to the identity? (b) Here is a possible procedure for constructing Ustilovsky's infinitely many contact struc-tures on S4n+1 in terms of open books. The standard contact structure on S2n+1 comes from the positive symplectic Dehn twist φ on T ∗Sn. Think that φ3, φ 5,... give Ustilovsky's con-tact structures. Can one distinguish φ from φ3 using Floer homology? Is there a stabilized version of Floer homology distinguishing the contact structures, and how does this compare to contact homology? More generally, this is a procedure for constructing many new contact structures out of an old one, by replacing each Dehn twist by an odd iterate of it. (c) Here is a way to possibly produce nonfillable contact structures in higher dimensions, analogous to overtwisted contact structures. Let W be a Stein manifold, φ a symplectomor-phism of W equal to the identity on the boundary, and Dn a Lagrangian ball with ∂D ⊂ ∂W.On ∂D, attach a handle outside, to get a new Stein manifold with a Lagrangian sphere. Compose φ with a left-handed Dehn twist. We get the same manifold, but are the con-tact structures different? Unfillable? In dimension 3, one can get all overtwisted contact structures by this construction. (Eliashberg: Try S5, one left-handed twist in U T ∗(S2).)",
   "evidence": "This is Question 1.33 (Giroux), under “Sixth day,” in the AIM workshop notes *Holomorphic Curves in Contact Geometry*. The corpus extraction is badly broken: “1.33” is split across lines, a printed page number `9` appears in the middle of part (a), superscripts and subscripts are flattened, and several words are hyphenated across line breaks. Reading the source PDF gives the following unambiguous mathematical content.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 428,
   "attempt": 1
  },
  "AIM-GEOMETRY-0430": {
   "statement_status": "exact",
   "original_statement": "Question 1.34. Auroux discussed how, roughly, a certain subgroup of the braid group gives automorphisms of a symplectic Lefschetz pencil, asked whether we can do something like this with open books or contact pencils, and in connection with this explained how Giroux's question (a) above has a positive answer in a certain precise case. \n\n[The following is from the notes of John Etnyre and David Farris, since I wasn't there.]",
   "clean_statement": "Question 1.34. Auroux discussed how, roughly, a certain subgroup of the braid group gives automorphisms of a symplectic Lefschetz pencil, asked whether we can do something like this with open books or contact pencils, and in connection with this explained how Giroux's question (a) above has a positive answer in a certain precise case.\n\n[The following is from the notes of John Etnyre and David Farris, since I wasn't there.]",
   "public_statement": "Question 1.34. Auroux discussed how, roughly, a certain subgroup of the braid group gives automorphisms of a symplectic Lefschetz pencil, asked whether we can do something like this with open books or contact pencils, and in connection with this explained how Giroux's question (a) above has a positive answer in a certain precise case.\n\n[The following is from the notes of John Etnyre and David Farris, since I wasn't there.]",
   "evidence": "The official AIM report *Holomorphic curves in contact geometry* contains:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 429,
   "attempt": 1
  },
  "AIM-GEOMETRY-0431": {
   "statement_status": "exact",
   "original_statement": "Question 1.35. Eliashberg made some remarks on the above topics. \n\nBanyaga discussed locally conformal symplectic geometry (c.s.s.). A manifold has a local conformal symplectic structure if it supports a non degenerate 2-form Ω such that \n\ndΩ = −ω ∧ Ω for some closed 1-form ω. An example is constructed starting with a contact manifold ( N, α ). Let X = N × S1, θ be the pull back of α and set Ω = dθ + ω ∧ θ where ω\n\nis the pull back of the volume from on S1.\n\nDefine Lichnerowicz cohomology as follows. Fix a closed 1-form ω now define dω by \n\ndωθ = dθ + ω ∧ θ. This is a differential on forms and if Ω is a c.s.s. then dωΩ = 0. So a conformal symplectic form is closed with respect to some differential. Another example of a c.s.s is constructed by starting with a symplectic manifold ( X, ω )then let Ω = f ω for any positive function f on X. Such a c.s.s. is called a global c.s.s.",
   "clean_statement": "Question 1.35. Eliashberg made some remarks on the above topics.\n\nBanyaga discussed locally conformal symplectic geometry (c.s.s.). A manifold has a local conformal symplectic structure if it supports a non degenerate 2-form Ω such that\n\ndΩ = −ω ∧ Ω for some closed 1-form ω. An example is constructed starting with a contact manifold ( N, α ). Let X = N × S1, θ be the pull back of α and set Ω = dθ + ω ∧ θ where ω\n\nis the pull back of the volume from on S1.\n\nDefine Lichnerowicz cohomology as follows. Fix a closed 1-form ω now define dω by\n\ndωθ = dθ + ω ∧ θ. This is a differential on forms and if Ω is a c.s.s. then dωΩ = 0. So a conformal symplectic form is closed with respect to some differential. Another example of a c.s.s is constructed by starting with a symplectic manifold ( X, ω )then let Ω = f ω for any positive function f on X. Such a c.s.s. is called a global c.s.s.",
   "public_statement": "Question 1.35. Eliashberg made some remarks on the above topics.\n\nBanyaga discussed locally conformal symplectic geometry (c.s.s.). A manifold has a local conformal symplectic structure if it supports a non degenerate 2-form Ω such that\n\ndΩ = −ω ∧ Ω for some closed 1-form ω. An example is constructed starting with a contact manifold ( N, α ). Let X = N × S1, θ be the pull back of α and set Ω = dθ + ω ∧ θ where ω\n\nis the pull back of the volume from on S1.\n\nDefine Lichnerowicz cohomology as follows. Fix a closed 1-form ω now define dω by\n\ndωθ = dθ + ω ∧ θ. This is a differential on forms and if Ω is a c.s.s. then dωΩ = 0. So a conformal symplectic form is closed with respect to some differential. Another example of a c.s.s is constructed by starting with a symplectic manifold ( X, ω )then let Ω = f ω for any positive function f on X. Such a c.s.s. is called a global c.s.s.",
   "evidence": "The canonical record is source index 430 of `aim-geometry-notes.json`, extracted from the AIM workshop notes *Holomorphic curves in contact geometry*. The official PDF and HTML have the following layout:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 430,
   "attempt": 1
  },
  "AIM-GEOMETRY-0432": {
   "statement_status": "exact",
   "original_statement": "Question 1.36. Given a c.s.s. is it global? 10 \n\nThere is the following result: a c.s.s. Ω is global if and only if ω is exact. Are there other conditions? Banyaga has constructed c.s.s. for which Ω is non zero in the Lichnerowicz cohomology, but such examples seem few and far between.",
   "clean_statement": "Question 1.36. Given a c.s.s. is it global? 10\n\nThere is the following result: a c.s.s. Ω is global if and only if ω is exact. Are there other conditions? Banyaga has constructed c.s.s. for which Ω is non zero in the Lichnerowicz cohomology, but such examples seem few and far between.",
   "public_statement": "Question 1.36. Given a c.s.s. is it global? 10\n\nThere is the following result: a c.s.s. Ω is global if and only if ω is exact. Are there other conditions? Banyaga has constructed c.s.s. for which Ω is non zero in the Lichnerowicz cohomology, but such examples seem few and far between.",
   "evidence": "The official AIM HTML and PDF agree on the following text in the sixth-day notes:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 431,
   "attempt": 1
  },
  "AIM-GEOMETRY-0433": {
   "statement_status": "exact",
   "original_statement": "Question 1.37. Find other constructions of c.s.s. that represent a non zero class in Lich-nerowicz cohomology. \n\nGiven a c.s.s. one can consider compatible almost complex structures, just as one does for symplectic structures.",
   "clean_statement": "Question 1.37. Find other constructions of c.s.s. that represent a non zero class in Lich-nerowicz cohomology.\n\nGiven a c.s.s. one can consider compatible almost complex structures, just as one does for symplectic structures.",
   "public_statement": "Question 1.37. Find other constructions of c.s.s. that represent a non zero class in Lich-nerowicz cohomology.\n\nGiven a c.s.s. one can consider compatible almost complex structures, just as one does for symplectic structures.",
   "evidence": "1. `c.s.s.` is present in the official PDF; it is not an extraction error. The preceding Question 1.35 defines it as a “local conformal symplectic structure.” Modern usage is **locally conformally symplectic structure**, abbreviated **l.c.s.** I use l.c.s. below. 2. `Lich-nerowicz` in the extracted record is only end-of-line hyphenation. The word is *Lichnerowicz*.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 432,
   "attempt": 1
  },
  "AIM-GEOMETRY-0434": {
   "statement_status": "corrected_verified",
   "original_statement": "Question 1.38. What can be said about holomorphic curves for theis compatible almost complex structure? Are they useful tools for studying c.s.s.? Eliashberg thinks it is unlikely they will be able to say much.",
   "clean_statement": "What can be said about \\(J\\)-holomorphic curves for an almost complex\nstructure compatible with a locally conformal symplectic form? Can those\ncurves be used to study l.c.s. geometry?",
   "public_statement": "What can be said about \\(J\\)-holomorphic curves for an almost complex\nstructure compatible with a locally conformal symplectic form? Can those\ncurves be used to study l.c.s. geometry?",
   "evidence": "The canonical record is source index 433 of `aim-geometry-notes.json`, from the AIM workshop notes *Holomorphic curves in contact geometry*. The official PDF places it immediately after two pieces of context: The word “theis” occurs in both the official PDF and its HTML transcription and is plainly a typo for **“this.”** The abbreviation “c.s.s.” is not a one-off OCR error: the surrounding workshop notes use it consistently for conformal or locally conformal symplectic structures. The now-standard abbreviation is **l.c.s.** Thus the recovered question is:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-geometry-notes.json",
   "source_index": 433,
   "attempt": 1
  },
  "AIM-GEOMETRY-0435": {
   "statement_status": "exact",
   "original_statement": "Question 1.39. Polterovich made the following conjecture and gave some discussion of examples and related results. Conjecture: let (M, ξ ) be a closed contact manifold and ϕ ∈\n\nCont( M, ξ ) a contactomorphism. Suppose that the induced action on the contact homology \n\nCH (M, ξ ) is hyperbolic. (I think part of the problem is to formulate what \"hyperbolic\" should mean in this situation.) Then ϕ is strongly dissipative, i.e. there does not exist a smooth \n\nϕ-invariant volume form. This question is part of a larger theme. There is a kind of map from the smooth topological world to the contact world. For example given a manifold M one can consider its unit cotangent bundle X. X is a contact manifold. Given a submanifold K of M one can consider its unit conormal bundle LK in X. This is a Legendrian submanifold of X. \n\nGiven a diffeomorphism f of M one can lift it to a contactomorphism of X. These are just a few examples of the \"functor\" between the smooth world and the contact world. [Compare Eliashberg's first question on the fifth day.] The main question of Polterovich is: How much information is lost by this functor? The above question is just one possible way of showing that some subtle information is preserved. [How exactly does this relate to the previous paragraph?]",
   "clean_statement": "Question 1.39. Polterovich made the following conjecture and gave some discussion of examples and related results. Conjecture: let (M, ξ ) be a closed contact manifold and ϕ ∈\n\nCont( M, ξ ) a contactomorphism. Suppose that the induced action on the contact homology\n\nCH (M, ξ ) is hyperbolic. (I think part of the problem is to formulate what \"hyperbolic\" should mean in this situation.) Then ϕ is strongly dissipative, i.e. there does not exist a smooth\n\nϕ-invariant volume form. This question is part of a larger theme. There is a kind of map from the smooth topological world to the contact world. For example given a manifold M one can consider its unit cotangent bundle X. X is a contact manifold. Given a submanifold K of M one can consider its unit conormal bundle LK in X. This is a Legendrian submanifold of X.\n\nGiven a diffeomorphism f of M one can lift it to a contactomorphism of X. These are just a few examples of the \"functor\" between the smooth world and the contact world. [Compare Eliashberg's first question on the fifth day.] The main question of Polterovich is: How much information is lost by this functor? The above question is just one possible way of showing that some subtle information is preserved. [How exactly does this relate to the previous paragraph?]",
   "public_statement": "Question 1.39. Polterovich made the following conjecture and gave some discussion of examples and related results. Conjecture: let (M, ξ ) be a closed contact manifold and ϕ ∈\n\nCont( M, ξ ) a contactomorphism. Suppose that the induced action on the contact homology\n\nCH (M, ξ ) is hyperbolic. (I think part of the problem is to formulate what \"hyperbolic\" should mean in this situation.) Then ϕ is strongly dissipative, i.e. there does not exist a smooth\n\nϕ-invariant volume form. This question is part of a larger theme. There is a kind of map from the smooth topological world to the contact world. For example given a manifold M one can consider its unit cotangent bundle X. X is a contact manifold. Given a submanifold K of M one can consider its unit conormal bundle LK in X. This is a Legendrian submanifold of X.\n\nGiven a diffeomorphism f of M one can lift it to a contactomorphism of X. These are just a few examples of the \"functor\" between the smooth world and the contact world. [Compare Eliashberg's first question on the fifth day.] The main question of Polterovich is: How much information is lost by this functor? The above question is just one possible way of showing that some subtle information is preserved. [How exactly does this relate to the previous paragraph?]",
   "evidence": "The canonical record is Question 1.39 from the AIM workshop *Holomorphic curves in contact geometry*. The AIM HTML page and the workshop PDF agree. The line breaks in the corpus record inside `Cont(M,xi)`, `CH(M,xi)`, and “phi-invariant” are extraction artifacts; no mathematical symbol had to be guessed. The bracketed comments and, in particular, the warning that “hyperbolic” still has to be formulated are in the source itself.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 434,
   "attempt": 1
  },
  "AIM-GEOMETRY-0436": {
   "statement_status": "exact",
   "original_statement": "Question 1.40. Mitsumatsu discussed something about linking numbers of orbits of vector fields in connection with Reeb vector fields of contact structures and foliations.",
   "clean_statement": "Question 1.40. Mitsumatsu discussed something about linking numbers of orbits of vector fields in connection with Reeb vector fields of contact structures and foliations.",
   "public_statement": "Question 1.40. Mitsumatsu discussed something about linking numbers of orbits of vector fields in connection with Reeb vector fields of contact structures and foliations.",
   "evidence": "The complete canonical record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 435,
   "attempt": 1
  },
  "AIM-GEOMETRY-0437": {
   "statement_status": "exact",
   "original_statement": "Question 1.41. Yi-Jen Lee gave some further discussion of how to possibly relate Ozsvath-Szabo and Seiberg-Witten theory.",
   "clean_statement": "Question 1.41. Yi-Jen Lee gave some further discussion of how to possibly relate Ozsvath-Szabo and Seiberg-Witten theory.",
   "public_statement": "Question 1.41. Yi-Jen Lee gave some further discussion of how to possibly relate Ozsvath-Szabo and Seiberg-Witten theory.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 436,
   "attempt": 1
  },
  "AIM-GEOMETRY-0438": {
   "statement_status": "exact",
   "original_statement": "Question 1.42. Peter Ozsvath said that it would be nice to see a spectrum theory (a la C. Manolescu) for Lagrangian intersections.",
   "clean_statement": "Question 1.42. Peter Ozsvath said that it would be nice to see a spectrum theory (a la C. Manolescu) for Lagrangian intersections.",
   "public_statement": "Question 1.42. Peter Ozsvath said that it would be nice to see a spectrum theory (a la C. Manolescu) for Lagrangian intersections.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-geometry-notes.json",
   "source_index": 437,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0001": {
   "statement_status": "exact",
   "original_statement": "1. a) What are the \"other tools\"? \n\nb) Open source only? \n\n- Free to use only? \n\n- Total cost of ownership? \n\n- Open access vs open source \n\n- Risk of owner changes",
   "clean_statement": "1. a) What are the \"other tools\"?\n\nb) Open source only?\n\n- Free to use only?\n\n- Total cost of ownership?\n\n- Open access vs open source\n\n- Risk of owner changes",
   "public_statement": "1. a) What are the \"other tools\"?\n\nb) Open source only?\n\n- Free to use only?\n\n- Total cost of ownership?\n\n- Open access vs open source\n\n- Risk of owner changes",
   "evidence": "There is no substantive OCR corruption. The corpus preserves the wording; only list line breaks were normalized above.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0002": {
   "statement_status": "exact",
   "original_statement": "2. a) How can the tools change the systems: \n\n- When assessment occurs \n\n- How people limit \n\n- Are students ready for a change? \n\n- Is anyone ready? \n\nb) The tool vs how we use it: \n\n- Adoption vs implementation \n\n- Underlying theory vs the software vs the community \n\nc) Barriers to adoption? \n\nd) Can online assessment interpret unconventional phrasing?",
   "clean_statement": "2. a) How can the tools change the systems:\n\n- When assessment occurs\n\n- How people limit\n\n- Are students ready for a change?\n\n- Is anyone ready?\n\nb) The tool vs how we use it:\n\n- Adoption vs implementation\n\n- Underlying theory vs the software vs the community\n\nc) Barriers to adoption?\n\nd) Can online assessment interpret unconventional phrasing?",
   "public_statement": "2. a) How can the tools change the systems:\n\n- When assessment occurs\n\n- How people limit\n\n- Are students ready for a change?\n\n- Is anyone ready?\n\nb) The tool vs how we use it:\n\n- Adoption vs implementation\n\n- Underlying theory vs the software vs the community\n\nc) Barriers to adoption?\n\nd) Can online assessment interpret unconventional phrasing?",
   "evidence": "The canonical record is item 2 from the AIM workshop *Open source mathematics curriculum and assessment tools* (Maseno University, Kisumu, Kenya, 5--9 August 2024). The official problem-list PDF labels the page “Monday Afternoon Discussion (Open Problem Session)” and describes its contents as brainstorming “Discussion Items (list).” The exact item is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0003": {
   "statement_status": "exact",
   "original_statement": "3. Technology as it relates to how we motivate students",
   "clean_statement": "3. Technology as it relates to how we motivate students",
   "public_statement": "3. Technology as it relates to how we motivate students",
   "evidence": "The canonical record is item 3 in the official AIM problem-list PDF for the workshop *Open source mathematics curriculum and assessment tools* (Maseno University, Kisumu, Kenya, 5--9 August 2024). Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0004": {
   "statement_status": "unrecoverable",
   "original_statement": "4. How to gather and share data? \n\n- Why do we want the data? \n\n- What are the questions? \n\n- Is the error due to a lack of prior knowledge? \n\n- How can technology detect and adapt to such errors? \n\n- combining data",
   "clean_statement": null,
   "public_statement": "4. How to gather and share data?\n\n- Why do we want the data?\n\n- What are the questions?\n\n- Is the error due to a lack of prior knowledge?\n\n- How can technology detect and adapt to such errors?\n\n- combining data",
   "evidence": "The workshop took place at Maseno University in Kisumu, Kenya, 5--9 August 2024. Its public page identifies cross-institutional sharing of anonymized data, interoperability, learning research, and responsible AI as workshop themes. The later report describes data sharing and analysis as a major topic. This context confirms that item 4 is a research-and-governance agenda, not a single mathematical problem. The conservative classification is `context_only`, with present status `not_a_problem`.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0005": {
   "statement_status": "exact",
   "original_statement": "5. How to give input as the technology is developed? \n\n- What studies will help to answer that question?",
   "clean_statement": "5. How to give input as the technology is developed?\n\n- What studies will help to answer that question?",
   "public_statement": "5. How to give input as the technology is developed?\n\n- What studies will help to answer that question?",
   "evidence": "The corpus record has no substantive OCR corruption. The nearby items ask how to gather and combine data, how to structure feedback to improve learning, how to make work transferable, and how to connect similar courses. The official workshop page says the meeting brought developers, implementers, and mathematics-education researchers together to improve open educational technology and emphasized research questions, data sharing, interoperability, and implementation challenges. The workshop report also calls for structured collaboration, quality control, accessibility, and materials that do not constrain users to a single tool.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0006": {
   "statement_status": "exact",
   "original_statement": "6. How to structure feedback to improve learning? \n\n- What studies will help to answer that question?",
   "clean_statement": "6. How to structure feedback to improve learning?\n\n- What studies will help to answer that question?",
   "public_statement": "6. How to structure feedback to improve learning?\n\n- What studies will help to answer that question?",
   "evidence": "The canonical record comes from item 6 of the AIM workshop list *Open source mathematics curriculum and assessment tools*. The recovered text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0007": {
   "statement_status": "exact",
   "original_statement": "7. a) How can I structure my work so that it is transferable? \n\nb) How to do better than \"top down\" (i.e. just sharing a course packet) \n\nc) How to connect across apparently similar courses? \n\n- Interoperability, e.g. of questions",
   "clean_statement": "7. a) How can I structure my work so that it is transferable?\n\nb) How to do better than \"top down\" (i.e. just sharing a course packet)\n\nc) How to connect across apparently similar courses?\n\n- Interoperability, e.g. of questions",
   "public_statement": "7. a) How can I structure my work so that it is transferable?\n\nb) How to do better than \"top down\" (i.e. just sharing a course packet)\n\nc) How to connect across apparently similar courses?\n\n- Interoperability, e.g. of questions",
   "evidence": "The assigned record is item 7 in the official AIM document *Monday Afternoon Discussion (Open Problem Session): Discussion Items*. The exact record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0008": {
   "statement_status": "exact",
   "original_statement": "9. a) How to have students work together? \n\n- Technology enables group interaction - Group interaction is a strict subset of collaboration \n\nb) Collaboration among instructors \n\nc) Student involvement in content creation",
   "clean_statement": "9. a) How to have students work together?\n\n- Technology enables group interaction - Group interaction is a strict subset of collaboration\n\nb) Collaboration among instructors\n\nc) Student involvement in content creation",
   "public_statement": "9. a) How to have students work together?\n\n- Technology enables group interaction - Group interaction is a strict subset of collaboration\n\nb) Collaboration among instructors\n\nc) Student involvement in content creation",
   "evidence": "The canonical record comes from the official AIM PDF *Monday Afternoon Discussion (Open Problem Session)* for the workshop *Open source mathematics curriculum and assessment tools*, held at Maseno University in Kisumu, Kenya, 5--9 August 2024. The canonical record reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0009": {
   "statement_status": "exact",
   "original_statement": "10. How to ensure the student provided the answer? \"Authenticity\" \n\n- Some answers are self-assessing (i.e. \"does this work\")",
   "clean_statement": "10. How to ensure the student provided the answer? \"Authenticity\"\n\n- Some answers are self-assessing (i.e. \"does this work\")",
   "public_statement": "10. How to ensure the student provided the answer? \"Authenticity\"\n\n- Some answers are self-assessing (i.e. \"does this work\")",
   "evidence": "The canonical record is item 10 in the AIM workshop discussion list *Open source mathematics curriculum and assessment tools*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0010": {
   "statement_status": "exact",
   "original_statement": "11. Are we convinced that Stack actually works? \n\n- Another variable: how do students actually use the tool? (e.g. are they \n\nbeing goofy) \n\n- Stack plays different roles at different institutions",
   "clean_statement": "11. Are we convinced that Stack actually works?\n\n- Another variable: how do students actually use the tool? (e.g. are they\n\nbeing goofy)\n\n- Stack plays different roles at different institutions",
   "public_statement": "11. Are we convinced that Stack actually works?\n\n- Another variable: how do students actually use the tool? (e.g. are they\n\nbeing goofy)\n\n- Stack plays different roles at different institutions",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0011": {
   "statement_status": "exact",
   "original_statement": "12. Longitudinal study (individuals, cohorts) \n\n- Goal: find best practices",
   "clean_statement": "12. Longitudinal study (individuals, cohorts)\n\n- Goal: find best practices",
   "public_statement": "12. Longitudinal study (individuals, cohorts)\n\n- Goal: find best practices",
   "evidence": "The canonical record is item 12 of the AIM workshop list *Open source mathematics curriculum and assessment tools*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0012": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "13. Exams (weights vary) \n\n- Alignment of final exam with mid-course assessment (in content and \n\nformat)",
   "clean_statement": null,
   "public_statement": "13. Exams (weights vary)\n\n- Alignment of final exam with mid-course assessment (in content and\n\nformat)",
   "evidence": "The canonical record is item 13 in the AIM workshop discussion list *Open source mathematics curriculum and assessment tools*. Its exact text is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0013": {
   "statement_status": "exact",
   "original_statement": "14. Do short online assessments impair later long-form work? \n\n- Connect to questions about collaborative work (communication \n\nchanges how you think)",
   "clean_statement": "14. Do short online assessments impair later long-form work?\n\n- Connect to questions about collaborative work (communication\n\nchanges how you think)",
   "public_statement": "14. Do short online assessments impair later long-form work?\n\n- Connect to questions about collaborative work (communication\n\nchanges how you think)",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0014": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "15. Relate (any!) questions to the students' long term goals",
   "clean_statement": null,
   "public_statement": "15. Relate (any!) questions to the students' long term goals",
   "evidence": "The canonical record is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0015": {
   "statement_status": "exact",
   "original_statement": "16. Does the technology put distance between the student and instructor?",
   "clean_statement": "16. Does the technology put distance between the student and instructor?",
   "public_statement": "16. Does the technology put distance between the student and instructor?",
   "evidence": "The canonical record is item 16 of the AIM workshop *Open source mathematics curriculum and assessment tools*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0016": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "17. Think in terms of suites of tools",
   "clean_statement": null,
   "public_statement": "17. Think in terms of suites of tools",
   "evidence": "Because the source does not state a yes/no proposition, this report uses the following **explicit reconstruction**, which is an inference from that context and not a quotation:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0017": {
   "statement_status": "exact",
   "original_statement": "18. a) Community of Stack users \n\nb) Do instructors without stats knowledge understand what Stack is doing? \n\n- Make a more accessible dashboard \n\n- Better: make a good report",
   "clean_statement": "18. a) Community of Stack users\n\nb) Do instructors without stats knowledge understand what Stack is doing?\n\n- Make a more accessible dashboard\n\n- Better: make a good report",
   "public_statement": "18. a) Community of Stack users\n\nb) Do instructors without stats knowledge understand what Stack is doing?\n\n- Make a more accessible dashboard\n\n- Better: make a good report",
   "evidence": "The official AIM PDF has the same wording and line structure, except that it prints the item number without a following space. There is no apparent OCR loss. This report preserves the source styling “Stack.” Current official project materials style the name in capitals as **STACK**; that convention is used below when referring to the present software and community, without silently rewriting the source.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0018": {
   "statement_status": "exact",
   "original_statement": "19. How does Stack change instruction? How does Stack reinforce instruction?",
   "clean_statement": "19. How does Stack change instruction? How does Stack reinforce instruction?",
   "public_statement": "19. How does Stack change instruction? How does Stack reinforce instruction?",
   "evidence": "The canonical record is item 19 of the AIM workshop *Open source mathematics curriculum and assessment tools*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0019": {
   "statement_status": "exact",
   "original_statement": "20. Understand what type of engagement the technology enhances/supports",
   "clean_statement": "20. Understand what type of engagement the technology enhances/supports",
   "public_statement": "20. Understand what type of engagement the technology enhances/supports",
   "evidence": "The canonical record is item 20 from the AIM workshop *Open source mathematics curriculum and assessment tools*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0020": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "22. a) PD for instructors, on the use of online tools \n\nb) What other support is needed?",
   "clean_statement": null,
   "public_statement": "22. a) PD for instructors, on the use of online tools\n\nb) What other support is needed?",
   "evidence": "Because this is a design question rather than a formal conjecture, the following is an **explicit reconstruction**, not a quotation:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0021": {
   "statement_status": "exact",
   "original_statement": "23. What stops other lecturers from using online assessment tools?",
   "clean_statement": "23. What stops other lecturers from using online assessment tools?",
   "public_statement": "23. What stops other lecturers from using online assessment tools?",
   "evidence": "The canonical AIM record is problem 23 from the workshop *Open source mathematics curriculum and assessment tools*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0022": {
   "statement_status": "exact",
   "original_statement": "24. How to encourage collaboration when it is a competition?",
   "clean_statement": "24. How to encourage collaboration when it is a competition?",
   "public_statement": "24. How to encourage collaboration when it is a competition?",
   "evidence": "The canonical record is item 24 from the AIM workshop *Open source mathematics curriculum and assessment tools*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0023": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "25. The mistaken impression that active learning does not cover enough material \n\n(counteract the \"curriculum focus\")",
   "clean_statement": null,
   "public_statement": "25. The mistaken impression that active learning does not cover enough material\n\n(counteract the \"curriculum focus\")",
   "evidence": "That operational question is an explicit reconstruction, not a replacement for the exact source statement.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0024": {
   "statement_status": "exact",
   "original_statement": "26. Accessibility of online tools for special needs",
   "clean_statement": "26. Accessibility of online tools for special needs",
   "public_statement": "26. Accessibility of online tools for special needs",
   "evidence": "The canonical AIM record is problem 26 from the August 5–9, 2024 workshop *Open source mathematics curriculum and assessment tools*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0025": {
   "statement_status": "exact",
   "original_statement": "27. Certification of student knowledge (outside of formal courses)",
   "clean_statement": "27. Certification of student knowledge (outside of formal courses)",
   "public_statement": "27. Certification of student knowledge (outside of formal courses)",
   "evidence": "The canonical record is item 27 from the AIM workshop *Open source mathematics curriculum and assessment tools*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0026": {
   "statement_status": "exact",
   "original_statement": "28. Structured pedagogy (using material prepared by somebody else) at the \n\nelementary level \n\n- Can it be adapted to higher level? \n\n- When is it appropriate?",
   "clean_statement": "28. Structured pedagogy (using material prepared by somebody else) at the\n\nelementary level\n\n- Can it be adapted to higher level?\n\n- When is it appropriate?",
   "public_statement": "28. Structured pedagogy (using material prepared by somebody else) at the\n\nelementary level\n\n- Can it be adapted to higher level?\n\n- When is it appropriate?",
   "evidence": "The canonical record is problem 28 from the AIM workshop *Open source mathematics curriculum and assessment tools*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0027": {
   "statement_status": "exact",
   "original_statement": "29. Collaborative invention of mathematics (by the students)",
   "clean_statement": "29. Collaborative invention of mathematics (by the students)",
   "public_statement": "29. Collaborative invention of mathematics (by the students)",
   "evidence": "The canonical record is item 29 from the AIM workshop *Open source mathematics curriculum and assessment tools*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0028": {
   "statement_status": "exact",
   "original_statement": "30. Embedding research in curriculum innovation",
   "clean_statement": "30. Embedding research in curriculum innovation",
   "public_statement": "30. Embedding research in curriculum innovation",
   "evidence": "The canonical record is item 30 from the AIM workshop *Open source mathematics curriculum and assessment tools*, held August 5--9, 2024:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0029": {
   "statement_status": "exact",
   "original_statement": "31. Beyond maths (STEM? more?)",
   "clean_statement": "31. Beyond maths (STEM? more?)",
   "public_statement": "31. Beyond maths (STEM? more?)",
   "evidence": "The exact canonical statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0030": {
   "statement_status": "exact",
   "original_statement": "32. Expanding to other countries",
   "clean_statement": "32. Expanding to other countries",
   "public_statement": "32. Expanding to other countries",
   "evidence": "The canonical statement is preserved verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0031": {
   "statement_status": "exact",
   "original_statement": "33. Engage more people in maths education research 34. Evaluate and improve individual tasks \n\n- What more tasks are needed? \n\n- (Embedded research)",
   "clean_statement": "33. Engage more people in maths education research 34. Evaluate and improve individual tasks\n\n- What more tasks are needed?\n\n- (Embedded research)",
   "public_statement": "33. Engage more people in maths education research 34. Evaluate and improve individual tasks\n\n- What more tasks are needed?\n\n- (Embedded research)",
   "evidence": "The canonical record is source index 30 in `aim-infrastructure-notes.json`, from the AIM workshop *Open source mathematics curriculum and assessment tools* (August 5--9, 2024). Its exact problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0032": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "35. Problems which are not \"task oriented\" (e.g. not computational)",
   "clean_statement": null,
   "public_statement": "35. Problems which are not \"task oriented\" (e.g. not computational)",
   "evidence": "The source gives no further sentence under item 35, so its meaning is not uniquely determined. Nearby items ask how to evaluate and improve individual tasks, what tasks are needed, what general principles make a good task, whether a task improves learning, and how assessment can reinforce and motivate. The fuller transcript also discusses proof-like communication, peer explanation, collaboration, contextual nuance lost by online systems, and tasks going beyond simple calculations. I therefore distinguish two plausible readings rather than silently expanding the source:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0033": {
   "statement_status": "exact",
   "original_statement": "36. General principles to guide how to write a good task",
   "clean_statement": "36. General principles to guide how to write a good task",
   "public_statement": "36. General principles to guide how to write a good task",
   "evidence": "The canonical record is item 36 of the AIM workshop notes *Open source mathematics curriculum and assessment tools*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0034": {
   "statement_status": "exact",
   "original_statement": "37. How to tell if a particular task (positively) impacts learning",
   "clean_statement": "37. How to tell if a particular task (positively) impacts learning",
   "public_statement": "37. How to tell if a particular task (positively) impacts learning",
   "evidence": "The canonical source record is item 37 from the AIM workshop *Open source mathematics curriculum and assessment tools*, held August 5--9, 2024 at Maseno University in Kisumu, Kenya:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0035": {
   "statement_status": "exact",
   "original_statement": "38. All assessment should provide positive reinforcement, to encourage and \n\nmotivate",
   "clean_statement": "38. All assessment should provide positive reinforcement, to encourage and\n\nmotivate",
   "public_statement": "38. All assessment should provide positive reinforcement, to encourage and\n\nmotivate",
   "evidence": "The canonical record contains the following exact problem text, including two newline characters between the last two fragments:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0036": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "39. Inspire more students to like maths, and to be motivated to learn \n\n(\"engagement\")",
   "clean_statement": null,
   "public_statement": "39. Inspire more students to like maths, and to be motivated to learn\n\n(\"engagement\")",
   "evidence": "The exact canonical `problem` string is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0037": {
   "statement_status": "exact",
   "original_statement": "40. Engaging institutions \n\nSummarized Notes \n\nEducational Tools and Open Source vs. Commercial Solutions \n\nThe workshop began with a discussion on educational tools like STACK and \n\nWebWork, debating whether to focus exclusively on open-source tools. Participants \n\ndistinguished between open-source tools, which can be modified, and freely \n\navailable tools, which are not editable. They recognized that while open-source tools \n\noffer control over long-term costs, they still incur expenses related to servers and \n\nexpertise. The focus was on ensuring that tools are not only available but also \n\neffectively implemented with proper training and support. \n\nImplementation and Effectiveness \n\nThere was a consensus that the success of educational tools depends on their \n\nimplementation rather than the tools themselves. Effective use requires thoughtful \n\nintegration into the curriculum, considering usability and user community. The \n\ndiscussion highlighted that diverse assessment methods are needed, and merely \n\nproviding tools is not sufficient; critical thinking and training are essential. \n\nCollaborative Problem-Solving and Tool Adaptation \n\nParticipants explored the potential of tools designed for collaborative \n\nproblem-solving, suggesting that students should be able to pass problems to peers \n\nfor continued work. They emphasized the need for technologies that support group \n\ninteractions and improve collaborative learning. Additionally, the importance of \n\nadapting tools based on user feedback and ensuring they meet the needs of diverse \n\nlearners was highlighted. \n\nAssessment and Data Utilization \n\nThe workshop addressed the role of assessments in evaluating student learning, \n\ndiscussing both intrinsic assessments (where task completion itself verifies correctness) and traditional evaluations. The need for research on the alignment \n\nbetween digital and traditional assessments was noted, as well as the importance of \n\nintegrating meaningful research into teaching practices. Participants stressed the \n\nnecessity of understanding how digital tools affect student engagement and learning \n\noutcomes. \n\nCultural and Contextual Considerations \n\nParticipants discussed the cultural aspect of mathematics education, emphasizing \n\nthe need to make math relatable and engaging through real-world scenarios. The \n\nconversation also covered the importance of contextualizing online assessments to \n\naddress language and cultural differences, and how hybrid methods combining \n\ntraditional and technological tools could be beneficial. \n\nProfessional Development and Collaboration \n\nThe discussion included the need for professional development in using AI and other \n\ntechnological tools in education. Participants noted the challenges faced by \n\neducators in adopting new methods and stressed the importance of fostering a \n\ncollaborative culture in education. Building support networks and addressing \n\nattitudes towards new practices were identified as crucial for effective \n\nimplementation. \n\nFuture Directions and Research Needs \n\nThe workshop concluded with a call for further research into the effectiveness of \n\neducational tools and assessment methods. Participants discussed the need for \n\nlongitudinal studies, exploring the impact of technology on different educational \n\ncontexts and demographics. They also highlighted the importance of international \n\ncollaboration and the need for ongoing evaluation and refinement of educational \n\nstrategies. \n\nThe workshop aimed to refine ideas into actionable projects and proposals, with a \n\nfocus on future collaboration and continued development of educational practices \n\nand tools. \n\nFully Transcripted Notes \n\nDiscussion started with highlighting the two primary tools (STACK and Web work) \n\nbut questioned the potential for multiple tools to address challenges in education. The group considered whether to focus on open source tools exclusively and if open \n\nsource should be a qualifying criterion. \n\nThere was a discussion about the distinction between open source tools, which allow \n\nfor code modification, and freely available tools, which may not be editable. The \n\ngroup aimed to clarify whether \"open\" means free to use or also includes the ability \n\nto edit and customize the tool. \n\nThe total cost of ownership for open source tools was addressed, noting that despite \n\nbeing free to use, they require servers and expertise, which can be expensive. It was \n\nacknowledged that while open source tools offer control over long-term costs, they \n\nare not completely free, and these costs must be considered by policymakers. \n\nThere was a focus on the feasibility and impact of educational interventions, \n\nparticularly the use of online tools and assessments. A key point raised by Chris \n\nhighlighted the challenge of ensuring these tools are responsive to students' learning \n\nneeds and attainment levels. He suggested that the way we use these tools, rather \n\nthan the tools themselves, significantly affects their outcomes. \n\nIt was emphasized that the tool's effectiveness depends on its implementation and \n\nthe policies guiding its use. There was a consensus that simply providing the tools is \n\ninsufficient; critical thinking and training on their use are essential. \n\nTwo main themes emerged: adoption and implementation. Adoption refers to \n\nwhether the tool is used or not, while implementation concerns how the tool is used. \n\nEffective implementation requires considering the theoretical foundations, \n\nuser-friendliness, and the community of users. \n\nParticipants agreed that technology should be designed to adapt based on feedback \n\nand be supported with appropriate training and resources. The discussion \n\nunderscored the need for a comprehensive approach that considers curriculum \n\nviews, software usability, and the user community. \n\nIt was highlighted that it isn't solely about open source but rather the cost of use \n\nand the implications of maintaining and supporting these tools. The importance of \n\nconsidering both immediate and long-term costs was emphasized for effective \n\ndecision-making in educational contexts. \n\nFurthermore, there was a discussion on the integration of lectureship positions with \n\ncurriculum design, emphasizing the importance of creating transferable resources. \n\nThe idea is to develop open-source course packs, such as those for linear algebra, \n\nthat instructors can download and use. These resources should not only be \n\nwell-packaged for use but also designed for sharing and community collaboration, \n\nallowing for feedback and continuous improvement. A key point raised was the need for a two-way conversation in resource sharing. \n\nInstead of a top-down approach, where a package is distributed for everyone to use, \n\nthe focus should be on building a collaborative loop. This involves educators \n\ncontributing to and refining shared resources. \n\nAdditionally, the workshop highlighted the challenge of authoring questions, which is \n\ntime-consuming for educators. There was a discussion about the interoperability of \n\neducational content across different learning systems. This includes the potential for \n\nimporting and adapting course materials from one platform to another, ensuring that \n\ncontent is reusable and efficient. \n\nThere was a call for both technological and community-based solutions to facilitate \n\nthe sharing of educational resources. The goal is to improve the quality and volume \n\nof shared content, thereby saving time and enhancing the overall educational \n\nexperience. \n\nAnother discussion on accessing education data emerged and emphasized the \n\nimportance of tailoring educational tools and data collection to different contexts to \n\nmotivate learners effectively. This contextual approach ensures that data is \n\nrepresentative of diverse environments, aiding in comprehensive analysis. \n\nParticipants highlighted the need for large datasets to train effective models. \n\nCollaboration with institutions is essential to gather socio-demographic information, \n\nwhich can enhance the utility of data for various purposes. A key question raised was \n\nthe broader objectives of collecting combined data sets and the types of questions \n\nsuch data could help answer. \n\nOne significant barrier to technology adoption in education is the lack of adaptability \n\nto individual learner levels. Technologies often fail to identify specific areas where \n\nstudents struggle, unlike human teachers who can provide personalized guidance. \n\nAddressing this gap could involve using data to adapt educational technologies to \n\nmeet individual learning needs more effectively. \n\nOverall, the discussion underscored the necessity of actionable data to identify and \n\naddress learning gaps, enhancing the adaptability of educational tools to support \n\nstudent success. \n\nThe concept of intrinsic assessment was discussed, particularly in the context of \n\nproblem-solving and modeling activities. Intrinsic assessment occurs when the \n\ncompletion of a task inherently demonstrates its correctness. For instance, in coding, \n\nthe functionality of the code serves as its own assessment-if it works, it meets the \n\nrequired standards. This self-assessing nature is valuable but should be one of many \n\ntools in an educational portfolio. Discussion on authenticity and functionality highlighted that for an artifact to be \n\nconsidered authentic, it must function correctly. This idea challenges the traditional \n\n\"us versus them\" model of assessment, where an external party evaluates the work. \n\nInstead, the artifact's ability to perform its intended function serves as a measure of \n\nits authenticity and correctness. \n\nBroadening assessment tools discussions underscored the importance of having a \n\ndiverse set of assessment tools. Intrinsic assessments, while valuable, cannot stand \n\nalone. Educators should incorporate various methods to ensure comprehensive \n\nevaluation and support student learning. \n\nThere was also a discussion on the need to research the effectiveness of new \n\neducational tools, such as STACK, in enhancing student learning. Concerns were \n\nraised about potential unintended consequences of these innovations. It was \n\nsuggested that thorough testing and research are necessary to understand their \n\nimpact fully and to address any negative outcomes. \n\nThe interaction between students and educational tools was another key topic. The \n\nimportance of structured time and focused engagement was emphasized to prevent \n\nstudents from rushing through tasks without understanding. The debate about the \n\nquality of online math practice compared to traditional methods was also addressed, \n\nwith suggestions that research could help validate the effectiveness of online tools. \n\nThere was a consensus on the need for diverse assessment methods, careful \n\nimplementation of educational innovations, and thorough research to ensure these \n\ntools positively impact student learning. The discussions highlighted the complexities \n\nof modern education and the necessity of a multifaceted approach to teaching and \n\nassessment. \n\nIncorporating math education research at the development stage of technologies can \n\nprovide valuable feedback to improve teaching and learning. One key area needing \n\nresearch is the development of teachers' content knowledge. For instance, \n\nunderstanding how to effectively teach fractions and identifying common student \n\nmistakes can be challenging. Technology can help by collecting and analyzing data \n\non student performance, which can then be used to inform teacher training and \n\nimprove instructional methods. \n\nIn the Kenyan context, the shift from a summative to a formative assessment \n\napproach under the Competency-Based Curriculum (CBC) highlights the need for \n\nbetter utilization of assessment data. By analyzing data from formative assessments, \n\nthe government can provide feedback to teachers, helping them address specific \n\nareas of student weakness. This approach can enhance both individual and national \n\neducation outcomes. Research should also focus on the specific features of assessment tools that support \n\nstudent engagement with mathematical ideas. Understanding how feedback is \n\nstructured and presented can be crucial. Qualitative research, such as interviewing \n\nstudents about their experiences, can provide insights into what supports or hinders \n\ntheir learning. This information can guide the design of more effective feedback \n\nmechanisms, ultimately improving student learning outcomes. \n\nOnline assessments often fail to connect with students due to contextual differences. \n\nOne significant issue is the language used in these assessments, which is \n\npredominantly English. The expectations for how responses should be input can be a \n\nbarrier, especially if the student's way of expressing themselves isn't aligned with \n\nconventional standards. Educators who know their students well can often infer their \n\nintended meaning, but this nuance is lost in automated online assessments. \n\nThere is a need to explore ways to bridge this gap and make online assessments \n\nmore context-sensitive. One suggestion is to incorporate tools that allow for some \n\nlevel of interpretation or personalization based on the student's context. Additionally, \n\ninnovative approaches to teaching and assessment that consider the specific context \n\nand needs of students should be developed. Hybrid methods combining traditional \n\nand technological tools could be beneficial. \n\nMathematics should not just be viewed as a subject but as a cultural element that \n\ninfluences various professions. The discussion highlighted that individuals exposed \n\nto mathematical thinking from an early age tend to excel in their fields, even if those \n\nfields are not directly related to mathematics. This cultural aspect of mathematics \n\nhelps individuals develop better problem-solving skills and analytical thinking, which \n\nare valuable in any profession. \n\nThe example of using a golf ball to teach mathematics illustrates the importance of \n\nmaking math relatable and applicable to real-world scenarios. This approach can \n\nchange students' perceptions of mathematics and make it more engaging and \n\nrelevant to their lives and future careers. \n\nDiscussions here emphasized the importance of contextualizing online assessments, \n\nusing hybrid methods, and promoting the cultural aspects of mathematics to enhance \n\nlearning outcomes. These strategies can help bridge the gap between students' \n\nunderstanding and conventional assessment methods, ultimately fostering a deeper \n\nappreciation and proficiency in mathematics. \n\nParticipants discussed the potential of designing educational tools that facilitate \n\ncollaborative problem-solving. One idea presented was a tool allowing students to \n\nstart solving a problem individually and, if they get stuck, pass it on to a peer who \n\ncontinues the work. This approach would enable students to learn from each other's problem-solving methods. The discussion emphasized shifting from technology \n\ndesigned for individual use to technology that supports group interactions. This shift \n\ncould enhance collaboration, particularly in the context of competency-based \n\ncurricula and 21st-century skills. \n\nCollaboration in mathematics goes beyond group work; it involves students sharing \n\nand building on each other's ideas. Technologies that support this kind of interaction \n\ncan foster deeper collaboration and improve learning outcomes. Participants \n\nexplored the idea of involving students in content creation, not just as consumers. \n\nThis approach could address language barriers and content accessibility, making \n\nlearning materials more relevant and authentic. The discussions addressed the \n\nchallenge of ensuring that student feedback and answers in assessments are \n\nauthentic. Participants discussed the need for reliable electronic tools that accurately \n\nreflect students' understanding and performance. \n\nThe discussion highlights a key issue: the gap between current technology use and \n\nthe experience of educators who may not have been trained in modern tech-based \n\nteaching methods. The focus needs to be on a holistic approach that considers the \n\nentire educational system, including policymakers, educators, and students. There is \n\nan observed resistance or lack of familiarity with new methods among educators, not \n\nnecessarily due to opposition but because they have not been exposed to or trained \n\nin these modern approaches. This suggests the need for a shift in training programs \n\nfor future educators to better integrate contemporary practices. The conversation \n\nalso emphasized the importance of addressing attitudes and values in educational \n\nchange. Successful implementation of new practices, whether technology-based or \n\nnot, requires attention to the attitudes of those involved. This means incorporating \n\nthese aspects into the design and deployment of educational initiatives to ensure \n\neffective adoption and application. \n\nThere were discussions revolving around how to effectively build and sustain a \n\ncommunity around educational technologies like STACK, ensuring high adoption and \n\nongoing development. A major concern is how educators using STACK can interpret \n\nthe data analytics it provides, especially since not all users have a background in \n\nstatistics. The goal is to simplify this data so that educators, regardless of their \n\nstatistical expertise, can easily understand and apply the insights to address specific \n\nissues their students may face. \n\nThe question posed is how to make the analytics from tools like STACK more \n\naccessible and useful for educators. It is essential to explore ways to automate or \n\nsimplify the process of interpreting and sharing insights from these tools. Additionally, \n\nunderstanding how these tools impact different types of engagement-emotional, \n\ncognitive, and behavioral-is important. This includes examining whether these tools \n\naffect engagement levels differently and using this understanding to guide future \n\nimprovements. In summary, the discussion sought to address how to enhance the usability of \n\neducational tools and the analytics they provide, focusing on improving their \n\naccessibility for educators and understanding their impact on student engagement. \n\nAgain, the discussions highlighted several key issues around communication and \n\nstudent engagement in educational settings. One notable point was the impact of \n\ntransitioning to digital tools on student-instructor relationships. An example \n\nmentioned was about how a professor shared that switching to online homework \n\nsubmissions reduced their familiarity with student names, demonstrating how \n\ntechnology can affect personal interactions. \n\nDiscussions emphasized the importance of maintaining student interaction, even \n\nwhen integrating new technological tools. It was argued that while digital tools can \n\nenhance learning, they should not replace face-to-face engagement. This balance is \n\ncritical to ensuring students feel heard and supported. \n\nParticipants suggested that instead of relying on a single comprehensive tool, a suite \n\nof complementary tools might better address various educational needs. This \n\napproach allows instructors to choose the most appropriate tools for quizzes, group \n\nwork, assessments, and content delivery based on their specific class context. \n\nOne proposed strategy was using tools to foster student interaction and \n\ncollaboration, where students receive additional points for helping their peers. This \n\nmethod encourages active participation and peer support, contributing to a more \n\ndynamic learning environment. \n\nThe discussion concluded with a call for a stable, long-term platform for educators to \n\nshare and receive feedback on the effective use of technological tools. This platform \n\ncould help educators adapt and improve their teaching strategies, ensuring that \n\ntechnology enhances rather than detracts from the learning experience. \n\nDiscussions further highlighted the effectiveness of structured pedagogical activities \n\nfor teachers. By following well-designed activities step-by-step, even less \n\nexperienced teachers can see improvements in teaching and learning outcomes. \n\nHowever, this approach may limit opportunities for innovation and creativity, which \n\ncould be a drawback for confident teachers looking to enhance their sessions further. \n\nA key topic was the concept of \"collaboratively invented mathematics,\" where \n\nstudents use digital tools to collaboratively discover mathematical concepts, such as \n\nthe Taylor series. This method shifts from traditional pedagogy to a more engaging, \n\nexploratory approach, allowing students to invent mathematics that historically took \n\ncenturies to develop. There was a strong emphasis on embedding meaningful research into teaching \n\npractices. This includes designing assessments with digital tools to evaluate \n\nstudents' understanding effectively. The integration of research can inform \n\neducational innovations and improve teaching methodologies. \n\nParticipants also discussed the importance of international collaboration in \n\nmathematics education. Countries interested in adopting these innovative teaching \n\nmethods need support to integrate and implement them effectively. The potential for \n\nusing online tools to facilitate these collaborations was considered crucial for \n\nbroadening the impact of these educational innovations. \n\nIt was then concluded from this discourse that structured pedagogy can significantly \n\nimprove teaching outcomes, but there is a need to balance this with opportunities for \n\nteacher innovation. The collaborative invention of mathematics and embedding \n\nresearch into teaching practices were highlighted as promising approaches. Global \n\ncollaboration and effective implementation of these methods are essential for their \n\nsuccess. \n\nA participant had mentioned the need to discuss the positive uses of AI in teaching \n\nand its potential benefits. Participants highlighted the importance of professional \n\ndevelopment for lecturers and teachers to effectively use AI tools, including both \n\npre-service and in-service training. \n\nA concern was raised about why only a few lecturers consistently use new \n\ntechnological tools while others do not. The discussion also explored the support \n\navailable for African institutions wishing to adopt technology in teaching. Building \n\nsupport networks and fostering collaborative learning were identified as crucial \n\nelements. \n\nThe conversation noted that education systems often promote individualism over \n\ncollaboration. This mentality persists into higher education and research, making \n\ncollaboration challenging. Encouraging a culture of helping and sharing knowledge \n\nfrom a young age was seen as vital. \n\nAlso, it was mentioned that some teachers appreciated innovative teaching methods \n\nthat do not rely on technology but expressed concerns about time constraints. They \n\nfeared that creative teaching methods might reduce the amount of content covered \n\nduring class. The group questioned ways to balance innovative teaching with \n\ncurriculum requirements, aiming to inspire students to explore concepts \n\nindependently. \n\nOne issue discussed was the alignment between final exam results and outcomes \n\nfrom online assessments. The concern is whether traditional exams provide the \n\nsame results as digital formative assessments. This disparity could be an interesting research question, particularly in contexts like the Open University, where online \n\nassessments are prevalent. Understanding how long-term use of digital tools affects \n\nmathematical communication and writing could be another research avenue. \n\nA key point raised was the relationship between formative assessments and \n\ntraditional examinations. There is interest in researching how these different forms of \n\nassessment align with each other, especially if one is digitized and the other is not. \n\nThis could reveal important insights into the effectiveness and consistency of various \n\nassessment methods. \n\nAnother discussion topic was the role of digital tools in enhancing or hindering \n\nmathematical communication. The group considered how these tools impact \n\nstudents' abilities to communicate mathematical ideas effectively. There was a \n\nsuggestion to explore ways to leverage student collaboration to improve \n\ncommunication skills, potentially by rewarding students for explaining concepts to \n\npeers. \n\nThe conversation also touched on how improving collaborative learning can \n\nsimultaneously enhance mathematical communication skills. Encouraging students \n\nto work together and explain their reasoning can create a virtuous cycle of improved \n\ncommunication and understanding. This approach could be beneficial in fostering \n\nboth collaboration and competency in mathematics. \n\nFinally, participants highlighted the need to understand different levels of \n\nmathematical education. From high school students aiming for basic competency to \n\nthose pursuing careers in mathematics, it's important to consider how various tools \n\nand methods support different educational goals. This broader understanding can \n\nhelp tailor educational strategies to meet diverse student needs. \n\nThe math education researchers discussed the importance of building capacity \n\namong mathematicians who currently teach but may lack certain skills. This involves \n\nengaging more individuals in math education research beyond just the math \n\neducation researchers. \n\nParticipants considered conducting a research project aimed at improving the quality \n\nof math tasks. This includes identifying the best and worst tasks and determining \n\nwhere new tasks should be developed. Research on individual tasks can help \n\npinpoint those that are most effective. There are challenges in designing tasks for \n\ncertain areas of mathematics, such as abstract algebra and geometry. The \n\ndiscussion covered the need to create effective tasks that go beyond simple \n\ncalculations and consider the specific content of each course. An idea was proposed \n\nto identify a set of principles for designing good tasks that promote learning, \n\nregardless of the course context. For instance, out of 200 derivative problems, \n\nfinding the top 20 that best promote learning. The workshop emphasized the importance of maintaining the human element in \n\nassessments. Positive reinforcement can generate new ideas and support students \n\neffectively. One participant shared their teaching approach, which involves assigning \n\nseminar topics to groups of students. These groups work on their topics throughout \n\nthe semester and present their findings, fostering collaboration and deeper \n\nunderstanding. \n\nDuring the workshop, discussions focused on the integration of education technology \n\nand its impact on student learning. One key point raised was the need to assess \n\nwhether these technologies are genuinely beneficial or potentially harmful. Although \n\ninitial plans to collaborate with a math education researcher were not realized in \n\ntime, the intention is to pilot the technology with first-year students and conduct a \n\nfollow-up assessment in the second year. \n\nParticipants emphasized the importance of understanding the distinct roles education \n\ntechnology plays in different institutional contexts, such as CalTech versus other \n\nuniversities. This understanding is crucial for determining the effectiveness of such \n\ntechnologies in enhancing learning experiences. \n\nAnother significant idea was the implementation of longitudinal studies to track \n\nstudent progress over several years. These studies could help identify best practices \n\nand measure the long-term impact of education technologies on learning outcomes. \n\nFor example, tracking the same cohort of students through a four-year degree \n\nprogram could reveal valuable insights into their learning journeys. \n\nThe workshop also highlighted the potential to investigate specific issues, such as \n\ngender disparities in STEM fields. By comparing data from different universities and \n\ncontexts, researchers could analyze how online tools and other interventions \n\ninfluence retention rates and learning experiences for different student \n\ndemographics. \n\nIn conclusion, the discussions underscored the need for rigorous educational \n\nresearch to identify effective practices and understand how various factors influence \n\nstudent learning across different contexts. \n\nDiscussions highlighted the tendency to treat students as a homogenous group, \n\nwhich overlooks individual progress. Mary raised a point about the uniform speed of \n\nstudent progress enforced by current assessment systems. Unlike learning to drive \n\nin the UK, where individuals take their driving test when ready, school exams are \n\nscheduled uniformly for all students. This system's rigidity does not account for \n\nindividual readiness and progress. The conversation explored the potential of \n\nelectronic assessment tools to transform not only learning but also assessment \n\nsystems. The current system, rooted in historical practices, necessitates uniform \n\nexam schedules. Electronic tools, however, offer the flexibility to tailor assessments to individual progress, potentially leading to a more personalized and effective \n\neducation system. \n\nMary's quote about play sparked a discussion on the nature of compulsory \n\nparticipation. True play requires freedom-emotional, economic, and choice \n\nfreedom. Compulsory education systems often lack these freedoms, making \n\nparticipation feel forced. The group pondered whether new tools could introduce \n\nmore freedom and playfulness into learning, thereby enhancing engagement and \n\neffectiveness. \n\nThe discussion extended to the broader implications of changing educational \n\nsystems. It was suggested that learning could be more context-specific and playful, \n\nintegrating disciplines in meaningful ways. For instance, learning math through \n\nhistorical contexts could make it more relevant and engaging for students. \n\nA key concern was whether stakeholders-students, educators, institutions, and \n\nsocieties-are ready for such a transformation. The readiness in terms of attitude, \n\ncapacity, and resources was questioned, especially considering the challenges faced \n\nby educational systems in regions like Africa. The discussion concluded with a call to \n\nassess the readiness and willingness of all stakeholders to transition from traditional \n\nto technology-integrated assessments. \n\nThe workshop highlighted a critical distinction between commercial and open-source \n\nsolutions, particularly concerning future cost implications. Participants debated \n\nwhether the focus should be on Open Access or open-source terms, considering \n\ntheir impact on accessibility and contribution rights. \n\nA key point raised was the risk of relying on commercial software that might become \n\ncostly or inaccessible if terms change. In contrast, open-source software offers more \n\nstability and control, allowing modifications and reducing dependency on external \n\nvendors. \n\nThe discussion also touched on the need to understand how open-source principles \n\ncould inform educational software choices. The idea is to draw from the open \n\ncommunity's experiences to determine what makes software truly open and \n\nsustainable. \n\nParticipants expressed interest in identifying qualifying criteria for evaluating different \n\nsoftware options. The upcoming sessions will aim to clarify these criteria, explore \n\nalternative platforms, and discuss their functionalities to make informed decisions. \n\nOverall, there is a call to explore how open-source and open-access principles can \n\nbetter serve educational institutions and to determine the most suitable approach \n\nbased on their needs and trade-offs. Another discussion highlighted the tension between competitive academic systems \n\nand the need for collaboration. There is a concern about how to foster collaboration \n\nfrom a young age within a system that traditionally emphasizes competition. This \n\nraises questions about how competitive academic structures can adapt to support \n\ncollaborative learning. \n\nIt was noted that shifting teaching approaches might not require technology but a \n\nchange from a strict curriculum focus. Teachers accustomed to curriculum-based \n\ninstruction may struggle to integrate new methods. The question arises whether \n\nthese new approaches can fit within the existing curriculum or if they require a \n\ncomplete overhaul. \n\nThe conversation addressed the need to adapt digital tools for students with special \n\nneeds. Ensuring that digital educational resources are accessible to all learners is \n\ncrucial, and there is interest in how these tools can be modified to meet the needs of \n\nindividuals with disabilities. \n\nA question was raised about whether students could receive certification for \n\ncompleting modules from open educational resources outside traditional institutions. \n\nThis discussion explores the potential for recognizing and certifying informal or \n\nself-directed learning experiences. \n\nRecent evidence suggests that structured pedagogy, which includes detailed lesson \n\nplans and specific instructional strategies, improves teaching and learning at \n\nfoundational levels. The inquiry is whether similar structured approaches could \n\nenhance education at higher levels, combining structured methods with more \n\nadvanced pedagogical strategies. \n\nThe discussion highlighted the importance of exploring alternative assessment \n\nmethods beyond digital tools. Emphasis was placed on incorporating human \n\ninteractions and experiences, which are often difficult to quantify. These \n\nassessments should inspire and motivate rather than merely evaluate. \n\nThere is a need for online tools that not only assess but also engage and inspire \n\nusers. This involves considering how institutions can be involved in data sharing \n\nagreements and fostering better engagement at an institutional level, rather than \n\nfocusing solely on individuals. \n\nAs the session concluded, there was a brief discussion on record-keeping and the \n\nneed for capturing high-speed data. The final points stressed were the importance of \n\nintegrating motivational aspects into assessments and the need for institutional \n\ninvolvement in data sharing. \n\nThe workshop aims to explore and develop a variety of significant and \n\nwell-considered ideas. Participants will engage in open-ended discussions and collaborative activities to refine these ideas. The goal is to convert these ideas into \n\nactionable projects, such as grant proposals or collaborative efforts. \n\nThe moderators and organizers will review the collected ideas and organize them \n\ninto key topics for group discussions scheduled for tomorrow. This process will \n\ninvolve multiple cycles of group work to generate viable projects. \n\nBy the end of the workshop, participants are expected to develop detailed plans and \n\npotential projects. However, given the scope of the topics, the workshop will primarily \n\nserve as a starting point, with continued work beyond the event. \n\nParticipants should use the current session to propose and refine ideas they are \n\ninterested in. The workshop will provide a foundation for future collaboration, with the \n\nunderstanding that comprehensive solutions will evolve over time. \n\nA report summarizing the workshop outcomes will be available, detailing the \n\nproposed topics and next steps. Participants are encouraged to bring forward any \n\nnew ideas or questions they have.",
   "clean_statement": "40. Engaging institutions\n\nSummarized Notes\n\nEducational Tools and Open Source vs. Commercial Solutions\n\nThe workshop began with a discussion on educational tools like STACK and\n\nWebWork, debating whether to focus exclusively on open-source tools. Participants\n\ndistinguished between open-source tools, which can be modified, and freely\n\navailable tools, which are not editable. They recognized that while open-source tools\n\noffer control over long-term costs, they still incur expenses related to servers and\n\nexpertise. The focus was on ensuring that tools are not only available but also\n\neffectively implemented with proper training and support.\n\nImplementation and Effectiveness\n\nThere was a consensus that the success of educational tools depends on their\n\nimplementation rather than the tools themselves. Effective use requires thoughtful\n\nintegration into the curriculum, considering usability and user community. The\n\ndiscussion highlighted that diverse assessment methods are needed, and merely\n\nproviding tools is not sufficient; critical thinking and training are essential.\n\nCollaborative Problem-Solving and Tool Adaptation\n\nParticipants explored the potential of tools designed for collaborative\n\nproblem-solving, suggesting that students should be able to pass problems to peers\n\nfor continued work. They emphasized the need for technologies that support group\n\ninteractions and improve collaborative learning. Additionally, the importance of\n\nadapting tools based on user feedback and ensuring they meet the needs of diverse\n\nlearners was highlighted.\n\nAssessment and Data Utilization\n\nThe workshop addressed the role of assessments in evaluating student learning,\n\ndiscussing both intrinsic assessments (where task completion itself verifies correctness) and traditional evaluations. The need for research on the alignment\n\nbetween digital and traditional assessments was noted, as well as the importance of\n\nintegrating meaningful research into teaching practices. Participants stressed the\n\nnecessity of understanding how digital tools affect student engagement and learning\n\noutcomes.\n\nCultural and Contextual Considerations\n\nParticipants discussed the cultural aspect of mathematics education, emphasizing\n\nthe need to make math relatable and engaging through real-world scenarios. The\n\nconversation also covered the importance of contextualizing online assessments to\n\naddress language and cultural differences, and how hybrid methods combining\n\ntraditional and technological tools could be beneficial.\n\nProfessional Development and Collaboration\n\nThe discussion included the need for professional development in using AI and other\n\ntechnological tools in education. Participants noted the challenges faced by\n\neducators in adopting new methods and stressed the importance of fostering a\n\ncollaborative culture in education. Building support networks and addressing\n\nattitudes towards new practices were identified as crucial for effective\n\nimplementation.\n\nFuture Directions and Research Needs\n\nThe workshop concluded with a call for further research into the effectiveness of\n\neducational tools and assessment methods. Participants discussed the need for\n\nlongitudinal studies, exploring the impact of technology on different educational\n\ncontexts and demographics. They also highlighted the importance of international\n\ncollaboration and the need for ongoing evaluation and refinement of educational\n\nstrategies.\n\nThe workshop aimed to refine ideas into actionable projects and proposals, with a\n\nfocus on future collaboration and continued development of educational practices\n\nand tools.\n\nFully Transcripted Notes\n\nDiscussion started with highlighting the two primary tools (STACK and Web work)\n\nbut questioned the potential for multiple tools to address challenges in education. The group considered whether to focus on open source tools exclusively and if open\n\nsource should be a qualifying criterion.\n\nThere was a discussion about the distinction between open source tools, which allow\n\nfor code modification, and freely available tools, which may not be editable. The\n\ngroup aimed to clarify whether \"open\" means free to use or also includes the ability\n\nto edit and customize the tool.\n\nThe total cost of ownership for open source tools was addressed, noting that despite\n\nbeing free to use, they require servers and expertise, which can be expensive. It was\n\nacknowledged that while open source tools offer control over long-term costs, they\n\nare not completely free, and these costs must be considered by policymakers.\n\nThere was a focus on the feasibility and impact of educational interventions,\n\nparticularly the use of online tools and assessments. A key point raised by Chris\n\nhighlighted the challenge of ensuring these tools are responsive to students' learning\n\nneeds and attainment levels. He suggested that the way we use these tools, rather\n\nthan the tools themselves, significantly affects their outcomes.\n\nIt was emphasized that the tool's effectiveness depends on its implementation and\n\nthe policies guiding its use. There was a consensus that simply providing the tools is\n\ninsufficient; critical thinking and training on their use are essential.\n\nTwo main themes emerged: adoption and implementation. Adoption refers to\n\nwhether the tool is used or not, while implementation concerns how the tool is used.\n\nEffective implementation requires considering the theoretical foundations,\n\nuser-friendliness, and the community of users.\n\nParticipants agreed that technology should be designed to adapt based on feedback\n\nand be supported with appropriate training and resources. The discussion\n\nunderscored the need for a comprehensive approach that considers curriculum\n\nviews, software usability, and the user community.\n\nIt was highlighted that it isn't solely about open source but rather the cost of use\n\nand the implications of maintaining and supporting these tools. The importance of\n\nconsidering both immediate and long-term costs was emphasized for effective\n\ndecision-making in educational contexts.\n\nFurthermore, there was a discussion on the integration of lectureship positions with\n\ncurriculum design, emphasizing the importance of creating transferable resources.\n\nThe idea is to develop open-source course packs, such as those for linear algebra,\n\nthat instructors can download and use. These resources should not only be\n\nwell-packaged for use but also designed for sharing and community collaboration,\n\nallowing for feedback and continuous improvement. A key point raised was the need for a two-way conversation in resource sharing.\n\nInstead of a top-down approach, where a package is distributed for everyone to use,\n\nthe focus should be on building a collaborative loop. This involves educators\n\ncontributing to and refining shared resources.\n\nAdditionally, the workshop highlighted the challenge of authoring questions, which is\n\ntime-consuming for educators. There was a discussion about the interoperability of\n\neducational content across different learning systems. This includes the potential for\n\nimporting and adapting course materials from one platform to another, ensuring that\n\ncontent is reusable and efficient.\n\nThere was a call for both technological and community-based solutions to facilitate\n\nthe sharing of educational resources. The goal is to improve the quality and volume\n\nof shared content, thereby saving time and enhancing the overall educational\n\nexperience.\n\nAnother discussion on accessing education data emerged and emphasized the\n\nimportance of tailoring educational tools and data collection to different contexts to\n\nmotivate learners effectively. This contextual approach ensures that data is\n\nrepresentative of diverse environments, aiding in comprehensive analysis.\n\nParticipants highlighted the need for large datasets to train effective models.\n\nCollaboration with institutions is essential to gather socio-demographic information,\n\nwhich can enhance the utility of data for various purposes. A key question raised was\n\nthe broader objectives of collecting combined data sets and the types of questions\n\nsuch data could help answer.\n\nOne significant barrier to technology adoption in education is the lack of adaptability\n\nto individual learner levels. Technologies often fail to identify specific areas where\n\nstudents struggle, unlike human teachers who can provide personalized guidance.\n\nAddressing this gap could involve using data to adapt educational technologies to\n\nmeet individual learning needs more effectively.\n\nOverall, the discussion underscored the necessity of actionable data to identify and\n\naddress learning gaps, enhancing the adaptability of educational tools to support\n\nstudent success.\n\nThe concept of intrinsic assessment was discussed, particularly in the context of\n\nproblem-solving and modeling activities. Intrinsic assessment occurs when the\n\ncompletion of a task inherently demonstrates its correctness. For instance, in coding,\n\nthe functionality of the code serves as its own assessment-if it works, it meets the\n\nrequired standards. This self-assessing nature is valuable but should be one of many\n\ntools in an educational portfolio. Discussion on authenticity and functionality highlighted that for an artifact to be\n\nconsidered authentic, it must function correctly. This idea challenges the traditional\n\n\"us versus them\" model of assessment, where an external party evaluates the work.\n\nInstead, the artifact's ability to perform its intended function serves as a measure of\n\nits authenticity and correctness.\n\nBroadening assessment tools discussions underscored the importance of having a\n\ndiverse set of assessment tools. Intrinsic assessments, while valuable, cannot stand\n\nalone. Educators should incorporate various methods to ensure comprehensive\n\nevaluation and support student learning.\n\nThere was also a discussion on the need to research the effectiveness of new\n\neducational tools, such as STACK, in enhancing student learning. Concerns were\n\nraised about potential unintended consequences of these innovations. It was\n\nsuggested that thorough testing and research are necessary to understand their\n\nimpact fully and to address any negative outcomes.\n\nThe interaction between students and educational tools was another key topic. The\n\nimportance of structured time and focused engagement was emphasized to prevent\n\nstudents from rushing through tasks without understanding. The debate about the\n\nquality of online math practice compared to traditional methods was also addressed,\n\nwith suggestions that research could help validate the effectiveness of online tools.\n\nThere was a consensus on the need for diverse assessment methods, careful\n\nimplementation of educational innovations, and thorough research to ensure these\n\ntools positively impact student learning. The discussions highlighted the complexities\n\nof modern education and the necessity of a multifaceted approach to teaching and\n\nassessment.\n\nIncorporating math education research at the development stage of technologies can\n\nprovide valuable feedback to improve teaching and learning. One key area needing\n\nresearch is the development of teachers' content knowledge. For instance,\n\nunderstanding how to effectively teach fractions and identifying common student\n\nmistakes can be challenging. Technology can help by collecting and analyzing data\n\non student performance, which can then be used to inform teacher training and\n\nimprove instructional methods.\n\nIn the Kenyan context, the shift from a summative to a formative assessment\n\napproach under the Competency-Based Curriculum (CBC) highlights the need for\n\nbetter utilization of assessment data. By analyzing data from formative assessments,\n\nthe government can provide feedback to teachers, helping them address specific\n\nareas of student weakness. This approach can enhance both individual and national\n\neducation outcomes. Research should also focus on the specific features of assessment tools that support\n\nstudent engagement with mathematical ideas. Understanding how feedback is\n\nstructured and presented can be crucial. Qualitative research, such as interviewing\n\nstudents about their experiences, can provide insights into what supports or hinders\n\ntheir learning. This information can guide the design of more effective feedback\n\nmechanisms, ultimately improving student learning outcomes.\n\nOnline assessments often fail to connect with students due to contextual differences.\n\nOne significant issue is the language used in these assessments, which is\n\npredominantly English. The expectations for how responses should be input can be a\n\nbarrier, especially if the student's way of expressing themselves isn't aligned with\n\nconventional standards. Educators who know their students well can often infer their\n\nintended meaning, but this nuance is lost in automated online assessments.\n\nThere is a need to explore ways to bridge this gap and make online assessments\n\nmore context-sensitive. One suggestion is to incorporate tools that allow for some\n\nlevel of interpretation or personalization based on the student's context. Additionally,\n\ninnovative approaches to teaching and assessment that consider the specific context\n\nand needs of students should be developed. Hybrid methods combining traditional\n\nand technological tools could be beneficial.\n\nMathematics should not just be viewed as a subject but as a cultural element that\n\ninfluences various professions. The discussion highlighted that individuals exposed\n\nto mathematical thinking from an early age tend to excel in their fields, even if those\n\nfields are not directly related to mathematics. This cultural aspect of mathematics\n\nhelps individuals develop better problem-solving skills and analytical thinking, which\n\nare valuable in any profession.\n\nThe example of using a golf ball to teach mathematics illustrates the importance of\n\nmaking math relatable and applicable to real-world scenarios. This approach can\n\nchange students' perceptions of mathematics and make it more engaging and\n\nrelevant to their lives and future careers.\n\nDiscussions here emphasized the importance of contextualizing online assessments,\n\nusing hybrid methods, and promoting the cultural aspects of mathematics to enhance\n\nlearning outcomes. These strategies can help bridge the gap between students'\n\nunderstanding and conventional assessment methods, ultimately fostering a deeper\n\nappreciation and proficiency in mathematics.\n\nParticipants discussed the potential of designing educational tools that facilitate\n\ncollaborative problem-solving. One idea presented was a tool allowing students to\n\nstart solving a problem individually and, if they get stuck, pass it on to a peer who\n\ncontinues the work. This approach would enable students to learn from each other's problem-solving methods. The discussion emphasized shifting from technology\n\ndesigned for individual use to technology that supports group interactions. This shift\n\ncould enhance collaboration, particularly in the context of competency-based\n\ncurricula and 21st-century skills.\n\nCollaboration in mathematics goes beyond group work; it involves students sharing\n\nand building on each other's ideas. Technologies that support this kind of interaction\n\ncan foster deeper collaboration and improve learning outcomes. Participants\n\nexplored the idea of involving students in content creation, not just as consumers.\n\nThis approach could address language barriers and content accessibility, making\n\nlearning materials more relevant and authentic. The discussions addressed the\n\nchallenge of ensuring that student feedback and answers in assessments are\n\nauthentic. Participants discussed the need for reliable electronic tools that accurately\n\nreflect students' understanding and performance.\n\nThe discussion highlights a key issue: the gap between current technology use and\n\nthe experience of educators who may not have been trained in modern tech-based\n\nteaching methods. The focus needs to be on a holistic approach that considers the\n\nentire educational system, including policymakers, educators, and students. There is\n\nan observed resistance or lack of familiarity with new methods among educators, not\n\nnecessarily due to opposition but because they have not been exposed to or trained\n\nin these modern approaches. This suggests the need for a shift in training programs\n\nfor future educators to better integrate contemporary practices. The conversation\n\nalso emphasized the importance of addressing attitudes and values in educational\n\nchange. Successful implementation of new practices, whether technology-based or\n\nnot, requires attention to the attitudes of those involved. This means incorporating\n\nthese aspects into the design and deployment of educational initiatives to ensure\n\neffective adoption and application.\n\nThere were discussions revolving around how to effectively build and sustain a\n\ncommunity around educational technologies like STACK, ensuring high adoption and\n\nongoing development. A major concern is how educators using STACK can interpret\n\nthe data analytics it provides, especially since not all users have a background in\n\nstatistics. The goal is to simplify this data so that educators, regardless of their\n\nstatistical expertise, can easily understand and apply the insights to address specific\n\nissues their students may face.\n\nThe question posed is how to make the analytics from tools like STACK more\n\naccessible and useful for educators. It is essential to explore ways to automate or\n\nsimplify the process of interpreting and sharing insights from these tools. Additionally,\n\nunderstanding how these tools impact different types of engagement-emotional,\n\ncognitive, and behavioral-is important. This includes examining whether these tools\n\naffect engagement levels differently and using this understanding to guide future\n\nimprovements. In summary, the discussion sought to address how to enhance the usability of\n\neducational tools and the analytics they provide, focusing on improving their\n\naccessibility for educators and understanding their impact on student engagement.\n\nAgain, the discussions highlighted several key issues around communication and\n\nstudent engagement in educational settings. One notable point was the impact of\n\ntransitioning to digital tools on student-instructor relationships. An example\n\nmentioned was about how a professor shared that switching to online homework\n\nsubmissions reduced their familiarity with student names, demonstrating how\n\ntechnology can affect personal interactions.\n\nDiscussions emphasized the importance of maintaining student interaction, even\n\nwhen integrating new technological tools. It was argued that while digital tools can\n\nenhance learning, they should not replace face-to-face engagement. This balance is\n\ncritical to ensuring students feel heard and supported.\n\nParticipants suggested that instead of relying on a single comprehensive tool, a suite\n\nof complementary tools might better address various educational needs. This\n\napproach allows instructors to choose the most appropriate tools for quizzes, group\n\nwork, assessments, and content delivery based on their specific class context.\n\nOne proposed strategy was using tools to foster student interaction and\n\ncollaboration, where students receive additional points for helping their peers. This\n\nmethod encourages active participation and peer support, contributing to a more\n\ndynamic learning environment.\n\nThe discussion concluded with a call for a stable, long-term platform for educators to\n\nshare and receive feedback on the effective use of technological tools. This platform\n\ncould help educators adapt and improve their teaching strategies, ensuring that\n\ntechnology enhances rather than detracts from the learning experience.\n\nDiscussions further highlighted the effectiveness of structured pedagogical activities\n\nfor teachers. By following well-designed activities step-by-step, even less\n\nexperienced teachers can see improvements in teaching and learning outcomes.\n\nHowever, this approach may limit opportunities for innovation and creativity, which\n\ncould be a drawback for confident teachers looking to enhance their sessions further.\n\nA key topic was the concept of \"collaboratively invented mathematics,\" where\n\nstudents use digital tools to collaboratively discover mathematical concepts, such as\n\nthe Taylor series. This method shifts from traditional pedagogy to a more engaging,\n\nexploratory approach, allowing students to invent mathematics that historically took\n\ncenturies to develop. There was a strong emphasis on embedding meaningful research into teaching\n\npractices. This includes designing assessments with digital tools to evaluate\n\nstudents' understanding effectively. The integration of research can inform\n\neducational innovations and improve teaching methodologies.\n\nParticipants also discussed the importance of international collaboration in\n\nmathematics education. Countries interested in adopting these innovative teaching\n\nmethods need support to integrate and implement them effectively. The potential for\n\nusing online tools to facilitate these collaborations was considered crucial for\n\nbroadening the impact of these educational innovations.\n\nIt was then concluded from this discourse that structured pedagogy can significantly\n\nimprove teaching outcomes, but there is a need to balance this with opportunities for\n\nteacher innovation. The collaborative invention of mathematics and embedding\n\nresearch into teaching practices were highlighted as promising approaches. Global\n\ncollaboration and effective implementation of these methods are essential for their\n\nsuccess.\n\nA participant had mentioned the need to discuss the positive uses of AI in teaching\n\nand its potential benefits. Participants highlighted the importance of professional\n\ndevelopment for lecturers and teachers to effectively use AI tools, including both\n\npre-service and in-service training.\n\nA concern was raised about why only a few lecturers consistently use new\n\ntechnological tools while others do not. The discussion also explored the support\n\navailable for African institutions wishing to adopt technology in teaching. Building\n\nsupport networks and fostering collaborative learning were identified as crucial\n\nelements.\n\nThe conversation noted that education systems often promote individualism over\n\ncollaboration. This mentality persists into higher education and research, making\n\ncollaboration challenging. Encouraging a culture of helping and sharing knowledge\n\nfrom a young age was seen as vital.\n\nAlso, it was mentioned that some teachers appreciated innovative teaching methods\n\nthat do not rely on technology but expressed concerns about time constraints. They\n\nfeared that creative teaching methods might reduce the amount of content covered\n\nduring class. The group questioned ways to balance innovative teaching with\n\ncurriculum requirements, aiming to inspire students to explore concepts\n\nindependently.\n\nOne issue discussed was the alignment between final exam results and outcomes\n\nfrom online assessments. The concern is whether traditional exams provide the\n\nsame results as digital formative assessments. This disparity could be an interesting research question, particularly in contexts like the Open University, where online\n\nassessments are prevalent. Understanding how long-term use of digital tools affects\n\nmathematical communication and writing could be another research avenue.\n\nA key point raised was the relationship between formative assessments and\n\ntraditional examinations. There is interest in researching how these different forms of\n\nassessment align with each other, especially if one is digitized and the other is not.\n\nThis could reveal important insights into the effectiveness and consistency of various\n\nassessment methods.\n\nAnother discussion topic was the role of digital tools in enhancing or hindering\n\nmathematical communication. The group considered how these tools impact\n\nstudents' abilities to communicate mathematical ideas effectively. There was a\n\nsuggestion to explore ways to leverage student collaboration to improve\n\ncommunication skills, potentially by rewarding students for explaining concepts to\n\npeers.\n\nThe conversation also touched on how improving collaborative learning can\n\nsimultaneously enhance mathematical communication skills. Encouraging students\n\nto work together and explain their reasoning can create a virtuous cycle of improved\n\ncommunication and understanding. This approach could be beneficial in fostering\n\nboth collaboration and competency in mathematics.\n\nFinally, participants highlighted the need to understand different levels of\n\nmathematical education. From high school students aiming for basic competency to\n\nthose pursuing careers in mathematics, it's important to consider how various tools\n\nand methods support different educational goals. This broader understanding can\n\nhelp tailor educational strategies to meet diverse student needs.\n\nThe math education researchers discussed the importance of building capacity\n\namong mathematicians who currently teach but may lack certain skills. This involves\n\nengaging more individuals in math education research beyond just the math\n\neducation researchers.\n\nParticipants considered conducting a research project aimed at improving the quality\n\nof math tasks. This includes identifying the best and worst tasks and determining\n\nwhere new tasks should be developed. Research on individual tasks can help\n\npinpoint those that are most effective. There are challenges in designing tasks for\n\ncertain areas of mathematics, such as abstract algebra and geometry. The\n\ndiscussion covered the need to create effective tasks that go beyond simple\n\ncalculations and consider the specific content of each course. An idea was proposed\n\nto identify a set of principles for designing good tasks that promote learning,\n\nregardless of the course context. For instance, out of 200 derivative problems,\n\nfinding the top 20 that best promote learning. The workshop emphasized the importance of maintaining the human element in\n\nassessments. Positive reinforcement can generate new ideas and support students\n\neffectively. One participant shared their teaching approach, which involves assigning\n\nseminar topics to groups of students. These groups work on their topics throughout\n\nthe semester and present their findings, fostering collaboration and deeper\n\nunderstanding.\n\nDuring the workshop, discussions focused on the integration of education technology\n\nand its impact on student learning. One key point raised was the need to assess\n\nwhether these technologies are genuinely beneficial or potentially harmful. Although\n\ninitial plans to collaborate with a math education researcher were not realized in\n\ntime, the intention is to pilot the technology with first-year students and conduct a\n\nfollow-up assessment in the second year.\n\nParticipants emphasized the importance of understanding the distinct roles education\n\ntechnology plays in different institutional contexts, such as CalTech versus other\n\nuniversities. This understanding is crucial for determining the effectiveness of such\n\ntechnologies in enhancing learning experiences.\n\nAnother significant idea was the implementation of longitudinal studies to track\n\nstudent progress over several years. These studies could help identify best practices\n\nand measure the long-term impact of education technologies on learning outcomes.\n\nFor example, tracking the same cohort of students through a four-year degree\n\nprogram could reveal valuable insights into their learning journeys.\n\nThe workshop also highlighted the potential to investigate specific issues, such as\n\ngender disparities in STEM fields. By comparing data from different universities and\n\ncontexts, researchers could analyze how online tools and other interventions\n\ninfluence retention rates and learning experiences for different student\n\ndemographics.\n\nIn conclusion, the discussions underscored the need for rigorous educational\n\nresearch to identify effective practices and understand how various factors influence\n\nstudent learning across different contexts.\n\nDiscussions highlighted the tendency to treat students as a homogenous group,\n\nwhich overlooks individual progress. Mary raised a point about the uniform speed of\n\nstudent progress enforced by current assessment systems. Unlike learning to drive\n\nin the UK, where individuals take their driving test when ready, school exams are\n\nscheduled uniformly for all students. This system's rigidity does not account for\n\nindividual readiness and progress. The conversation explored the potential of\n\nelectronic assessment tools to transform not only learning but also assessment\n\nsystems. The current system, rooted in historical practices, necessitates uniform\n\nexam schedules. Electronic tools, however, offer the flexibility to tailor assessments to individual progress, potentially leading to a more personalized and effective\n\neducation system.\n\nMary's quote about play sparked a discussion on the nature of compulsory\n\nparticipation. True play requires freedom-emotional, economic, and choice\n\nfreedom. Compulsory education systems often lack these freedoms, making\n\nparticipation feel forced. The group pondered whether new tools could introduce\n\nmore freedom and playfulness into learning, thereby enhancing engagement and\n\neffectiveness.\n\nThe discussion extended to the broader implications of changing educational\n\nsystems. It was suggested that learning could be more context-specific and playful,\n\nintegrating disciplines in meaningful ways. For instance, learning math through\n\nhistorical contexts could make it more relevant and engaging for students.\n\nA key concern was whether stakeholders-students, educators, institutions, and\n\nsocieties-are ready for such a transformation. The readiness in terms of attitude,\n\ncapacity, and resources was questioned, especially considering the challenges faced\n\nby educational systems in regions like Africa. The discussion concluded with a call to\n\nassess the readiness and willingness of all stakeholders to transition from traditional\n\nto technology-integrated assessments.\n\nThe workshop highlighted a critical distinction between commercial and open-source\n\nsolutions, particularly concerning future cost implications. Participants debated\n\nwhether the focus should be on Open Access or open-source terms, considering\n\ntheir impact on accessibility and contribution rights.\n\nA key point raised was the risk of relying on commercial software that might become\n\ncostly or inaccessible if terms change. In contrast, open-source software offers more\n\nstability and control, allowing modifications and reducing dependency on external\n\nvendors.\n\nThe discussion also touched on the need to understand how open-source principles\n\ncould inform educational software choices. The idea is to draw from the open\n\ncommunity's experiences to determine what makes software truly open and\n\nsustainable.\n\nParticipants expressed interest in identifying qualifying criteria for evaluating different\n\nsoftware options. The upcoming sessions will aim to clarify these criteria, explore\n\nalternative platforms, and discuss their functionalities to make informed decisions.\n\nOverall, there is a call to explore how open-source and open-access principles can\n\nbetter serve educational institutions and to determine the most suitable approach\n\nbased on their needs and trade-offs. Another discussion highlighted the tension between competitive academic systems\n\nand the need for collaboration. There is a concern about how to foster collaboration\n\nfrom a young age within a system that traditionally emphasizes competition. This\n\nraises questions about how competitive academic structures can adapt to support\n\ncollaborative learning.\n\nIt was noted that shifting teaching approaches might not require technology but a\n\nchange from a strict curriculum focus. Teachers accustomed to curriculum-based\n\ninstruction may struggle to integrate new methods. The question arises whether\n\nthese new approaches can fit within the existing curriculum or if they require a\n\ncomplete overhaul.\n\nThe conversation addressed the need to adapt digital tools for students with special\n\nneeds. Ensuring that digital educational resources are accessible to all learners is\n\ncrucial, and there is interest in how these tools can be modified to meet the needs of\n\nindividuals with disabilities.\n\nA question was raised about whether students could receive certification for\n\ncompleting modules from open educational resources outside traditional institutions.\n\nThis discussion explores the potential for recognizing and certifying informal or\n\nself-directed learning experiences.\n\nRecent evidence suggests that structured pedagogy, which includes detailed lesson\n\nplans and specific instructional strategies, improves teaching and learning at\n\nfoundational levels. The inquiry is whether similar structured approaches could\n\nenhance education at higher levels, combining structured methods with more\n\nadvanced pedagogical strategies.\n\nThe discussion highlighted the importance of exploring alternative assessment\n\nmethods beyond digital tools. Emphasis was placed on incorporating human\n\ninteractions and experiences, which are often difficult to quantify. These\n\nassessments should inspire and motivate rather than merely evaluate.\n\nThere is a need for online tools that not only assess but also engage and inspire\n\nusers. This involves considering how institutions can be involved in data sharing\n\nagreements and fostering better engagement at an institutional level, rather than\n\nfocusing solely on individuals.\n\nAs the session concluded, there was a brief discussion on record-keeping and the\n\nneed for capturing high-speed data. The final points stressed were the importance of\n\nintegrating motivational aspects into assessments and the need for institutional\n\ninvolvement in data sharing.\n\nThe workshop aims to explore and develop a variety of significant and\n\nwell-considered ideas. Participants will engage in open-ended discussions and collaborative activities to refine these ideas. The goal is to convert these ideas into\n\nactionable projects, such as grant proposals or collaborative efforts.\n\nThe moderators and organizers will review the collected ideas and organize them\n\ninto key topics for group discussions scheduled for tomorrow. This process will\n\ninvolve multiple cycles of group work to generate viable projects.\n\nBy the end of the workshop, participants are expected to develop detailed plans and\n\npotential projects. However, given the scope of the topics, the workshop will primarily\n\nserve as a starting point, with continued work beyond the event.\n\nParticipants should use the current session to propose and refine ideas they are\n\ninterested in. The workshop will provide a foundation for future collaboration, with the\n\nunderstanding that comprehensive solutions will evolve over time.\n\nA report summarizing the workshop outcomes will be available, detailing the\n\nproposed topics and next steps. Participants are encouraged to bring forward any\n\nnew ideas or questions they have.",
   "public_statement": "40. Engaging institutions\n\nSummarized Notes\n\nEducational Tools and Open Source vs. Commercial Solutions\n\nThe workshop began with a discussion on educational tools like STACK and\n\nWebWork, debating whether to focus exclusively on open-source tools. Participants\n\ndistinguished between open-source tools, which can be modified, and freely\n\navailable tools, which are not editable. They recognized that while open-source tools\n\noffer control over long-term costs, they still incur expenses related to servers and\n\nexpertise. The focus was on ensuring that tools are not only available but also\n\neffectively implemented with proper training and support.\n\nImplementation and Effectiveness\n\nThere was a consensus that the success of educational tools depends on their\n\nimplementation rather than the tools themselves. Effective use requires thoughtful\n\nintegration into the curriculum, considering usability and user community. The\n\ndiscussion highlighted that diverse assessment methods are needed, and merely\n\nproviding tools is not sufficient; critical thinking and training are essential.\n\nCollaborative Problem-Solving and Tool Adaptation\n\nParticipants explored the potential of tools designed for collaborative\n\nproblem-solving, suggesting that students should be able to pass problems to peers\n\nfor continued work. They emphasized the need for technologies that support group\n\ninteractions and improve collaborative learning. Additionally, the importance of\n\nadapting tools based on user feedback and ensuring they meet the needs of diverse\n\nlearners was highlighted.\n\nAssessment and Data Utilization\n\nThe workshop addressed the role of assessments in evaluating student learning,\n\ndiscussing both intrinsic assessments (where task completion itself verifies correctness) and traditional evaluations. The need for research on the alignment\n\nbetween digital and traditional assessments was noted, as well as the importance of\n\nintegrating meaningful research into teaching practices. Participants stressed the\n\nnecessity of understanding how digital tools affect student engagement and learning\n\noutcomes.\n\nCultural and Contextual Considerations\n\nParticipants discussed the cultural aspect of mathematics education, emphasizing\n\nthe need to make math relatable and engaging through real-world scenarios. The\n\nconversation also covered the importance of contextualizing online assessments to\n\naddress language and cultural differences, and how hybrid methods combining\n\ntraditional and technological tools could be beneficial.\n\nProfessional Development and Collaboration\n\nThe discussion included the need for professional development in using AI and other\n\ntechnological tools in education. Participants noted the challenges faced by\n\neducators in adopting new methods and stressed the importance of fostering a\n\ncollaborative culture in education. Building support networks and addressing\n\nattitudes towards new practices were identified as crucial for effective\n\nimplementation.\n\nFuture Directions and Research Needs\n\nThe workshop concluded with a call for further research into the effectiveness of\n\neducational tools and assessment methods. Participants discussed the need for\n\nlongitudinal studies, exploring the impact of technology on different educational\n\ncontexts and demographics. They also highlighted the importance of international\n\ncollaboration and the need for ongoing evaluation and refinement of educational\n\nstrategies.\n\nThe workshop aimed to refine ideas into actionable projects and proposals, with a\n\nfocus on future collaboration and continued development of educational practices\n\nand tools.\n\nFully Transcripted Notes\n\nDiscussion started with highlighting the two primary tools (STACK and Web work)\n\nbut questioned the potential for multiple tools to address challenges in education. The group considered whether to focus on open source tools exclusively and if open\n\nsource should be a qualifying criterion.\n\nThere was a discussion about the distinction between open source tools, which allow\n\nfor code modification, and freely available tools, which may not be editable. The\n\ngroup aimed to clarify whether \"open\" means free to use or also includes the ability\n\nto edit and customize the tool.\n\nThe total cost of ownership for open source tools was addressed, noting that despite\n\nbeing free to use, they require servers and expertise, which can be expensive. It was\n\nacknowledged that while open source tools offer control over long-term costs, they\n\nare not completely free, and these costs must be considered by policymakers.\n\nThere was a focus on the feasibility and impact of educational interventions,\n\nparticularly the use of online tools and assessments. A key point raised by Chris\n\nhighlighted the challenge of ensuring these tools are responsive to students' learning\n\nneeds and attainment levels. He suggested that the way we use these tools, rather\n\nthan the tools themselves, significantly affects their outcomes.\n\nIt was emphasized that the tool's effectiveness depends on its implementation and\n\nthe policies guiding its use. There was a consensus that simply providing the tools is\n\ninsufficient; critical thinking and training on their use are essential.\n\nTwo main themes emerged: adoption and implementation. Adoption refers to\n\nwhether the tool is used or not, while implementation concerns how the tool is used.\n\nEffective implementation requires considering the theoretical foundations,\n\nuser-friendliness, and the community of users.\n\nParticipants agreed that technology should be designed to adapt based on feedback\n\nand be supported with appropriate training and resources. The discussion\n\nunderscored the need for a comprehensive approach that considers curriculum\n\nviews, software usability, and the user community.\n\nIt was highlighted that it isn't solely about open source but rather the cost of use\n\nand the implications of maintaining and supporting these tools. The importance of\n\nconsidering both immediate and long-term costs was emphasized for effective\n\ndecision-making in educational contexts.\n\nFurthermore, there was a discussion on the integration of lectureship positions with\n\ncurriculum design, emphasizing the importance of creating transferable resources.\n\nThe idea is to develop open-source course packs, such as those for linear algebra,\n\nthat instructors can download and use. These resources should not only be\n\nwell-packaged for use but also designed for sharing and community collaboration,\n\nallowing for feedback and continuous improvement. A key point raised was the need for a two-way conversation in resource sharing.\n\nInstead of a top-down approach, where a package is distributed for everyone to use,\n\nthe focus should be on building a collaborative loop. This involves educators\n\ncontributing to and refining shared resources.\n\nAdditionally, the workshop highlighted the challenge of authoring questions, which is\n\ntime-consuming for educators. There was a discussion about the interoperability of\n\neducational content across different learning systems. This includes the potential for\n\nimporting and adapting course materials from one platform to another, ensuring that\n\ncontent is reusable and efficient.\n\nThere was a call for both technological and community-based solutions to facilitate\n\nthe sharing of educational resources. The goal is to improve the quality and volume\n\nof shared content, thereby saving time and enhancing the overall educational\n\nexperience.\n\nAnother discussion on accessing education data emerged and emphasized the\n\nimportance of tailoring educational tools and data collection to different contexts to\n\nmotivate learners effectively. This contextual approach ensures that data is\n\nrepresentative of diverse environments, aiding in comprehensive analysis.\n\nParticipants highlighted the need for large datasets to train effective models.\n\nCollaboration with institutions is essential to gather socio-demographic information,\n\nwhich can enhance the utility of data for various purposes. A key question raised was\n\nthe broader objectives of collecting combined data sets and the types of questions\n\nsuch data could help answer.\n\nOne significant barrier to technology adoption in education is the lack of adaptability\n\nto individual learner levels. Technologies often fail to identify specific areas where\n\nstudents struggle, unlike human teachers who can provide personalized guidance.\n\nAddressing this gap could involve using data to adapt educational technologies to\n\nmeet individual learning needs more effectively.\n\nOverall, the discussion underscored the necessity of actionable data to identify and\n\naddress learning gaps, enhancing the adaptability of educational tools to support\n\nstudent success.\n\nThe concept of intrinsic assessment was discussed, particularly in the context of\n\nproblem-solving and modeling activities. Intrinsic assessment occurs when the\n\ncompletion of a task inherently demonstrates its correctness. For instance, in coding,\n\nthe functionality of the code serves as its own assessment-if it works, it meets the\n\nrequired standards. This self-assessing nature is valuable but should be one of many\n\ntools in an educational portfolio. Discussion on authenticity and functionality highlighted that for an artifact to be\n\nconsidered authentic, it must function correctly. This idea challenges the traditional\n\n\"us versus them\" model of assessment, where an external party evaluates the work.\n\nInstead, the artifact's ability to perform its intended function serves as a measure of\n\nits authenticity and correctness.\n\nBroadening assessment tools discussions underscored the importance of having a\n\ndiverse set of assessment tools. Intrinsic assessments, while valuable, cannot stand\n\nalone. Educators should incorporate various methods to ensure comprehensive\n\nevaluation and support student learning.\n\nThere was also a discussion on the need to research the effectiveness of new\n\neducational tools, such as STACK, in enhancing student learning. Concerns were\n\nraised about potential unintended consequences of these innovations. It was\n\nsuggested that thorough testing and research are necessary to understand their\n\nimpact fully and to address any negative outcomes.\n\nThe interaction between students and educational tools was another key topic. The\n\nimportance of structured time and focused engagement was emphasized to prevent\n\nstudents from rushing through tasks without understanding. The debate about the\n\nquality of online math practice compared to traditional methods was also addressed,\n\nwith suggestions that research could help validate the effectiveness of online tools.\n\nThere was a consensus on the need for diverse assessment methods, careful\n\nimplementation of educational innovations, and thorough research to ensure these\n\ntools positively impact student learning. The discussions highlighted the complexities\n\nof modern education and the necessity of a multifaceted approach to teaching and\n\nassessment.\n\nIncorporating math education research at the development stage of technologies can\n\nprovide valuable feedback to improve teaching and learning. One key area needing\n\nresearch is the development of teachers' content knowledge. For instance,\n\nunderstanding how to effectively teach fractions and identifying common student\n\nmistakes can be challenging. Technology can help by collecting and analyzing data\n\non student performance, which can then be used to inform teacher training and\n\nimprove instructional methods.\n\nIn the Kenyan context, the shift from a summative to a formative assessment\n\napproach under the Competency-Based Curriculum (CBC) highlights the need for\n\nbetter utilization of assessment data. By analyzing data from formative assessments,\n\nthe government can provide feedback to teachers, helping them address specific\n\nareas of student weakness. This approach can enhance both individual and national\n\neducation outcomes. Research should also focus on the specific features of assessment tools that support\n\nstudent engagement with mathematical ideas. Understanding how feedback is\n\nstructured and presented can be crucial. Qualitative research, such as interviewing\n\nstudents about their experiences, can provide insights into what supports or hinders\n\ntheir learning. This information can guide the design of more effective feedback\n\nmechanisms, ultimately improving student learning outcomes.\n\nOnline assessments often fail to connect with students due to contextual differences.\n\nOne significant issue is the language used in these assessments, which is\n\npredominantly English. The expectations for how responses should be input can be a\n\nbarrier, especially if the student's way of expressing themselves isn't aligned with\n\nconventional standards. Educators who know their students well can often infer their\n\nintended meaning, but this nuance is lost in automated online assessments.\n\nThere is a need to explore ways to bridge this gap and make online assessments\n\nmore context-sensitive. One suggestion is to incorporate tools that allow for some\n\nlevel of interpretation or personalization based on the student's context. Additionally,\n\ninnovative approaches to teaching and assessment that consider the specific context\n\nand needs of students should be developed. Hybrid methods combining traditional\n\nand technological tools could be beneficial.\n\nMathematics should not just be viewed as a subject but as a cultural element that\n\ninfluences various professions. The discussion highlighted that individuals exposed\n\nto mathematical thinking from an early age tend to excel in their fields, even if those\n\nfields are not directly related to mathematics. This cultural aspect of mathematics\n\nhelps individuals develop better problem-solving skills and analytical thinking, which\n\nare valuable in any profession.\n\nThe example of using a golf ball to teach mathematics illustrates the importance of\n\nmaking math relatable and applicable to real-world scenarios. This approach can\n\nchange students' perceptions of mathematics and make it more engaging and\n\nrelevant to their lives and future careers.\n\nDiscussions here emphasized the importance of contextualizing online assessments,\n\nusing hybrid methods, and promoting the cultural aspects of mathematics to enhance\n\nlearning outcomes. These strategies can help bridge the gap between students'\n\nunderstanding and conventional assessment methods, ultimately fostering a deeper\n\nappreciation and proficiency in mathematics.\n\nParticipants discussed the potential of designing educational tools that facilitate\n\ncollaborative problem-solving. One idea presented was a tool allowing students to\n\nstart solving a problem individually and, if they get stuck, pass it on to a peer who\n\ncontinues the work. This approach would enable students to learn from each other's problem-solving methods. The discussion emphasized shifting from technology\n\ndesigned for individual use to technology that supports group interactions. This shift\n\ncould enhance collaboration, particularly in the context of competency-based\n\ncurricula and 21st-century skills.\n\nCollaboration in mathematics goes beyond group work; it involves students sharing\n\nand building on each other's ideas. Technologies that support this kind of interaction\n\ncan foster deeper collaboration and improve learning outcomes. Participants\n\nexplored the idea of involving students in content creation, not just as consumers.\n\nThis approach could address language barriers and content accessibility, making\n\nlearning materials more relevant and authentic. The discussions addressed the\n\nchallenge of ensuring that student feedback and answers in assessments are\n\nauthentic. Participants discussed the need for reliable electronic tools that accurately\n\nreflect students' understanding and performance.\n\nThe discussion highlights a key issue: the gap between current technology use and\n\nthe experience of educators who may not have been trained in modern tech-based\n\nteaching methods. The focus needs to be on a holistic approach that considers the\n\nentire educational system, including policymakers, educators, and students. There is\n\nan observed resistance or lack of familiarity with new methods among educators, not\n\nnecessarily due to opposition but because they have not been exposed to or trained\n\nin these modern approaches. This suggests the need for a shift in training programs\n\nfor future educators to better integrate contemporary practices. The conversation\n\nalso emphasized the importance of addressing attitudes and values in educational\n\nchange. Successful implementation of new practices, whether technology-based or\n\nnot, requires attention to the attitudes of those involved. This means incorporating\n\nthese aspects into the design and deployment of educational initiatives to ensure\n\neffective adoption and application.\n\nThere were discussions revolving around how to effectively build and sustain a\n\ncommunity around educational technologies like STACK, ensuring high adoption and\n\nongoing development. A major concern is how educators using STACK can interpret\n\nthe data analytics it provides, especially since not all users have a background in\n\nstatistics. The goal is to simplify this data so that educators, regardless of their\n\nstatistical expertise, can easily understand and apply the insights to address specific\n\nissues their students may face.\n\nThe question posed is how to make the analytics from tools like STACK more\n\naccessible and useful for educators. It is essential to explore ways to automate or\n\nsimplify the process of interpreting and sharing insights from these tools. Additionally,\n\nunderstanding how these tools impact different types of engagement-emotional,\n\ncognitive, and behavioral-is important. This includes examining whether these tools\n\naffect engagement levels differently and using this understanding to guide future\n\nimprovements. In summary, the discussion sought to address how to enhance the usability of\n\neducational tools and the analytics they provide, focusing on improving their\n\naccessibility for educators and understanding their impact on student engagement.\n\nAgain, the discussions highlighted several key issues around communication and\n\nstudent engagement in educational settings. One notable point was the impact of\n\ntransitioning to digital tools on student-instructor relationships. An example\n\nmentioned was about how a professor shared that switching to online homework\n\nsubmissions reduced their familiarity with student names, demonstrating how\n\ntechnology can affect personal interactions.\n\nDiscussions emphasized the importance of maintaining student interaction, even\n\nwhen integrating new technological tools. It was argued that while digital tools can\n\nenhance learning, they should not replace face-to-face engagement. This balance is\n\ncritical to ensuring students feel heard and supported.\n\nParticipants suggested that instead of relying on a single comprehensive tool, a suite\n\nof complementary tools might better address various educational needs. This\n\napproach allows instructors to choose the most appropriate tools for quizzes, group\n\nwork, assessments, and content delivery based on their specific class context.\n\nOne proposed strategy was using tools to foster student interaction and\n\ncollaboration, where students receive additional points for helping their peers. This\n\nmethod encourages active participation and peer support, contributing to a more\n\ndynamic learning environment.\n\nThe discussion concluded with a call for a stable, long-term platform for educators to\n\nshare and receive feedback on the effective use of technological tools. This platform\n\ncould help educators adapt and improve their teaching strategies, ensuring that\n\ntechnology enhances rather than detracts from the learning experience.\n\nDiscussions further highlighted the effectiveness of structured pedagogical activities\n\nfor teachers. By following well-designed activities step-by-step, even less\n\nexperienced teachers can see improvements in teaching and learning outcomes.\n\nHowever, this approach may limit opportunities for innovation and creativity, which\n\ncould be a drawback for confident teachers looking to enhance their sessions further.\n\nA key topic was the concept of \"collaboratively invented mathematics,\" where\n\nstudents use digital tools to collaboratively discover mathematical concepts, such as\n\nthe Taylor series. This method shifts from traditional pedagogy to a more engaging,\n\nexploratory approach, allowing students to invent mathematics that historically took\n\ncenturies to develop. There was a strong emphasis on embedding meaningful research into teaching\n\npractices. This includes designing assessments with digital tools to evaluate\n\nstudents' understanding effectively. The integration of research can inform\n\neducational innovations and improve teaching methodologies.\n\nParticipants also discussed the importance of international collaboration in\n\nmathematics education. Countries interested in adopting these innovative teaching\n\nmethods need support to integrate and implement them effectively. The potential for\n\nusing online tools to facilitate these collaborations was considered crucial for\n\nbroadening the impact of these educational innovations.\n\nIt was then concluded from this discourse that structured pedagogy can significantly\n\nimprove teaching outcomes, but there is a need to balance this with opportunities for\n\nteacher innovation. The collaborative invention of mathematics and embedding\n\nresearch into teaching practices were highlighted as promising approaches. Global\n\ncollaboration and effective implementation of these methods are essential for their\n\nsuccess.\n\nA participant had mentioned the need to discuss the positive uses of AI in teaching\n\nand its potential benefits. Participants highlighted the importance of professional\n\ndevelopment for lecturers and teachers to effectively use AI tools, including both\n\npre-service and in-service training.\n\nA concern was raised about why only a few lecturers consistently use new\n\ntechnological tools while others do not. The discussion also explored the support\n\navailable for African institutions wishing to adopt technology in teaching. Building\n\nsupport networks and fostering collaborative learning were identified as crucial\n\nelements.\n\nThe conversation noted that education systems often promote individualism over\n\ncollaboration. This mentality persists into higher education and research, making\n\ncollaboration challenging. Encouraging a culture of helping and sharing knowledge\n\nfrom a young age was seen as vital.\n\nAlso, it was mentioned that some teachers appreciated innovative teaching methods\n\nthat do not rely on technology but expressed concerns about time constraints. They\n\nfeared that creative teaching methods might reduce the amount of content covered\n\nduring class. The group questioned ways to balance innovative teaching with\n\ncurriculum requirements, aiming to inspire students to explore concepts\n\nindependently.\n\nOne issue discussed was the alignment between final exam results and outcomes\n\nfrom online assessments. The concern is whether traditional exams provide the\n\nsame results as digital formative assessments. This disparity could be an interesting research question, particularly in contexts like the Open University, where online\n\nassessments are prevalent. Understanding how long-term use of digital tools affects\n\nmathematical communication and writing could be another research avenue.\n\nA key point raised was the relationship between formative assessments and\n\ntraditional examinations. There is interest in researching how these different forms of\n\nassessment align with each other, especially if one is digitized and the other is not.\n\nThis could reveal important insights into the effectiveness and consistency of various\n\nassessment methods.\n\nAnother discussion topic was the role of digital tools in enhancing or hindering\n\nmathematical communication. The group considered how these tools impact\n\nstudents' abilities to communicate mathematical ideas effectively. There was a\n\nsuggestion to explore ways to leverage student collaboration to improve\n\ncommunication skills, potentially by rewarding students for explaining concepts to\n\npeers.\n\nThe conversation also touched on how improving collaborative learning can\n\nsimultaneously enhance mathematical communication skills. Encouraging students\n\nto work together and explain their reasoning can create a virtuous cycle of improved\n\ncommunication and understanding. This approach could be beneficial in fostering\n\nboth collaboration and competency in mathematics.\n\nFinally, participants highlighted the need to understand different levels of\n\nmathematical education. From high school students aiming for basic competency to\n\nthose pursuing careers in mathematics, it's important to consider how various tools\n\nand methods support different educational goals. This broader understanding can\n\nhelp tailor educational strategies to meet diverse student needs.\n\nThe math education researchers discussed the importance of building capacity\n\namong mathematicians who currently teach but may lack certain skills. This involves\n\nengaging more individuals in math education research beyond just the math\n\neducation researchers.\n\nParticipants considered conducting a research project aimed at improving the quality\n\nof math tasks. This includes identifying the best and worst tasks and determining\n\nwhere new tasks should be developed. Research on individual tasks can help\n\npinpoint those that are most effective. There are challenges in designing tasks for\n\ncertain areas of mathematics, such as abstract algebra and geometry. The\n\ndiscussion covered the need to create effective tasks that go beyond simple\n\ncalculations and consider the specific content of each course. An idea was proposed\n\nto identify a set of principles for designing good tasks that promote learning,\n\nregardless of the course context. For instance, out of 200 derivative problems,\n\nfinding the top 20 that best promote learning. The workshop emphasized the importance of maintaining the human element in\n\nassessments. Positive reinforcement can generate new ideas and support students\n\neffectively. One participant shared their teaching approach, which involves assigning\n\nseminar topics to groups of students. These groups work on their topics throughout\n\nthe semester and present their findings, fostering collaboration and deeper\n\nunderstanding.\n\nDuring the workshop, discussions focused on the integration of education technology\n\nand its impact on student learning. One key point raised was the need to assess\n\nwhether these technologies are genuinely beneficial or potentially harmful. Although\n\ninitial plans to collaborate with a math education researcher were not realized in\n\ntime, the intention is to pilot the technology with first-year students and conduct a\n\nfollow-up assessment in the second year.\n\nParticipants emphasized the importance of understanding the distinct roles education\n\ntechnology plays in different institutional contexts, such as CalTech versus other\n\nuniversities. This understanding is crucial for determining the effectiveness of such\n\ntechnologies in enhancing learning experiences.\n\nAnother significant idea was the implementation of longitudinal studies to track\n\nstudent progress over several years. These studies could help identify best practices\n\nand measure the long-term impact of education technologies on learning outcomes.\n\nFor example, tracking the same cohort of students through a four-year degree\n\nprogram could reveal valuable insights into their learning journeys.\n\nThe workshop also highlighted the potential to investigate specific issues, such as\n\ngender disparities in STEM fields. By comparing data from different universities and\n\ncontexts, researchers could analyze how online tools and other interventions\n\ninfluence retention rates and learning experiences for different student\n\ndemographics.\n\nIn conclusion, the discussions underscored the need for rigorous educational\n\nresearch to identify effective practices and understand how various factors influence\n\nstudent learning across different contexts.\n\nDiscussions highlighted the tendency to treat students as a homogenous group,\n\nwhich overlooks individual progress. Mary raised a point about the uniform speed of\n\nstudent progress enforced by current assessment systems. Unlike learning to drive\n\nin the UK, where individuals take their driving test when ready, school exams are\n\nscheduled uniformly for all students. This system's rigidity does not account for\n\nindividual readiness and progress. The conversation explored the potential of\n\nelectronic assessment tools to transform not only learning but also assessment\n\nsystems. The current system, rooted in historical practices, necessitates uniform\n\nexam schedules. Electronic tools, however, offer the flexibility to tailor assessments to individual progress, potentially leading to a more personalized and effective\n\neducation system.\n\nMary's quote about play sparked a discussion on the nature of compulsory\n\nparticipation. True play requires freedom-emotional, economic, and choice\n\nfreedom. Compulsory education systems often lack these freedoms, making\n\nparticipation feel forced. The group pondered whether new tools could introduce\n\nmore freedom and playfulness into learning, thereby enhancing engagement and\n\neffectiveness.\n\nThe discussion extended to the broader implications of changing educational\n\nsystems. It was suggested that learning could be more context-specific and playful,\n\nintegrating disciplines in meaningful ways. For instance, learning math through\n\nhistorical contexts could make it more relevant and engaging for students.\n\nA key concern was whether stakeholders-students, educators, institutions, and\n\nsocieties-are ready for such a transformation. The readiness in terms of attitude,\n\ncapacity, and resources was questioned, especially considering the challenges faced\n\nby educational systems in regions like Africa. The discussion concluded with a call to\n\nassess the readiness and willingness of all stakeholders to transition from traditional\n\nto technology-integrated assessments.\n\nThe workshop highlighted a critical distinction between commercial and open-source\n\nsolutions, particularly concerning future cost implications. Participants debated\n\nwhether the focus should be on Open Access or open-source terms, considering\n\ntheir impact on accessibility and contribution rights.\n\nA key point raised was the risk of relying on commercial software that might become\n\ncostly or inaccessible if terms change. In contrast, open-source software offers more\n\nstability and control, allowing modifications and reducing dependency on external\n\nvendors.\n\nThe discussion also touched on the need to understand how open-source principles\n\ncould inform educational software choices. The idea is to draw from the open\n\ncommunity's experiences to determine what makes software truly open and\n\nsustainable.\n\nParticipants expressed interest in identifying qualifying criteria for evaluating different\n\nsoftware options. The upcoming sessions will aim to clarify these criteria, explore\n\nalternative platforms, and discuss their functionalities to make informed decisions.\n\nOverall, there is a call to explore how open-source and open-access principles can\n\nbetter serve educational institutions and to determine the most suitable approach\n\nbased on their needs and trade-offs. Another discussion highlighted the tension between competitive academic systems\n\nand the need for collaboration. There is a concern about how to foster collaboration\n\nfrom a young age within a system that traditionally emphasizes competition. This\n\nraises questions about how competitive academic structures can adapt to support\n\ncollaborative learning.\n\nIt was noted that shifting teaching approaches might not require technology but a\n\nchange from a strict curriculum focus. Teachers accustomed to curriculum-based\n\ninstruction may struggle to integrate new methods. The question arises whether\n\nthese new approaches can fit within the existing curriculum or if they require a\n\ncomplete overhaul.\n\nThe conversation addressed the need to adapt digital tools for students with special\n\nneeds. Ensuring that digital educational resources are accessible to all learners is\n\ncrucial, and there is interest in how these tools can be modified to meet the needs of\n\nindividuals with disabilities.\n\nA question was raised about whether students could receive certification for\n\ncompleting modules from open educational resources outside traditional institutions.\n\nThis discussion explores the potential for recognizing and certifying informal or\n\nself-directed learning experiences.\n\nRecent evidence suggests that structured pedagogy, which includes detailed lesson\n\nplans and specific instructional strategies, improves teaching and learning at\n\nfoundational levels. The inquiry is whether similar structured approaches could\n\nenhance education at higher levels, combining structured methods with more\n\nadvanced pedagogical strategies.\n\nThe discussion highlighted the importance of exploring alternative assessment\n\nmethods beyond digital tools. Emphasis was placed on incorporating human\n\ninteractions and experiences, which are often difficult to quantify. These\n\nassessments should inspire and motivate rather than merely evaluate.\n\nThere is a need for online tools that not only assess but also engage and inspire\n\nusers. This involves considering how institutions can be involved in data sharing\n\nagreements and fostering better engagement at an institutional level, rather than\n\nfocusing solely on individuals.\n\nAs the session concluded, there was a brief discussion on record-keeping and the\n\nneed for capturing high-speed data. The final points stressed were the importance of\n\nintegrating motivational aspects into assessments and the need for institutional\n\ninvolvement in data sharing.\n\nThe workshop aims to explore and develop a variety of significant and\n\nwell-considered ideas. Participants will engage in open-ended discussions and collaborative activities to refine these ideas. The goal is to convert these ideas into\n\nactionable projects, such as grant proposals or collaborative efforts.\n\nThe moderators and organizers will review the collected ideas and organize them\n\ninto key topics for group discussions scheduled for tomorrow. This process will\n\ninvolve multiple cycles of group work to generate viable projects.\n\nBy the end of the workshop, participants are expected to develop detailed plans and\n\npotential projects. However, given the scope of the topics, the workshop will primarily\n\nserve as a starting point, with continued work beyond the event.\n\nParticipants should use the current session to propose and refine ideas they are\n\ninterested in. The workshop will provide a foundation for future collaboration, with the\n\nunderstanding that comprehensive solutions will evolve over time.\n\nA report summarizing the workshop outcomes will be available, detailing the\n\nproposed topics and next steps. Participants are encouraged to bring forward any\n\nnew ideas or questions they have.",
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   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0044": {
   "statement_status": "exact",
   "original_statement": "The sole object of this group is to decide whether or not to finish the software SpaceMath.",
   "clean_statement": "The sole object of this group is to decide whether or not to finish the software SpaceMath.",
   "public_statement": "The sole object of this group is to decide whether or not to finish the software SpaceMath.",
   "evidence": "The source file is `aim-infrastructure-notes.json`, record index 43 (zero based). The linked AIM Problem Lists page was not retrievable during this run, so the repository record is reproduced without silently repairing it. No mathematical notation appears corrupted. The prompt does leave “finish,” the decision owner, time horizon, budget, and success criteria undefined; those omissions are decision variables, not OCR errors.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0045": {
   "statement_status": "exact",
   "original_statement": "The goal of this group is to create a survey of what work has been done on educational proof software and to describe the goals which spurred its creation. The group should aim to end with a concrete idea of what exists in this area now, what is being worked on, and should have ideas on how these systems may be brought together or used.",
   "clean_statement": "The goal of this group is to create a survey of what work has been done on educational proof software and to describe the goals which spurred its creation. The group should aim to end with a concrete idea of what exists in this area now, what is being worked on, and should have ideas on how these systems may be brought together or used.",
   "public_statement": "The goal of this group is to create a survey of what work has been done on educational proof software and to describe the goals which spurred its creation. The group should aim to end with a concrete idea of what exists in this area now, what is being worked on, and should have ideas on how these systems may be brought together or used.",
   "evidence": "The statement is intact and requires no reconstruction. It is an agenda rather than a mathematical conjecture. The source URL, `http://aimpl.org/cyberinfrastructure/8/`, returned a 502 error during this run. The official [AIM workshop page](https://aimath.org/pastworkshops/cyberinfrastructure.html) confirms the workshop and links its [four-page activity report](https://aimath.org/pastworkshops/cyberinfrastructurerep.pdf). That report does not contain a dedicated “Proof Software in Education” working-group summary. We therefore do not infer a workshop outcome that the available official report does not state.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0046": {
   "statement_status": "exact",
   "original_statement": "The goal of this group should be to investigate what tools exist for knowledge tracing and should outline what tools could be developed to continue this pursuit, now that interest in it has been renewed.",
   "clean_statement": "The goal of this group should be to investigate what tools exist for knowledge tracing and should outline what tools could be developed to continue this pursuit, now that interest in it has been renewed.",
   "public_statement": "The goal of this group should be to investigate what tools exist for knowledge tracing and should outline what tools could be developed to continue this pursuit, now that interest in it has been renewed.",
   "evidence": "There are no remarks or supplied literature. The sentence is intact and shows no visible OCR corruption, but “knowledge tracing” and “this pursuit” are not defined in the record. The supplied AimPL page, `http://aimpl.org/cyberinfrastructure/9/`, could not be retrieved on 2026-08-09. The official AIM workshop report resolves the intended domain: the group examined how to model a student’s understanding while the student interacts with new material, asked how automated assessment could improve understanding of mastery of mathematical concepts, discussed problems and successes of ALEKS, and brainstormed an ideal interface and process. The report points to a GitHub wiki page, but that page was not retrievable in this run; no details are attributed to it.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0047": {
   "statement_status": "exact",
   "original_statement": "This group should aim to understand to what extent LaTeX does and does not meet the needs of current mathematicians. After such an understanding is had, it should aim to create a list of resources that can supplement TeX, or replace it outright. This group may also venture into ideas on what tools can be developed to address points from the first objective and to create or plan to create said tools.",
   "clean_statement": "This group should aim to understand to what extent LaTeX does and does not meet the needs of current mathematicians. After such an understanding is had, it should aim to create a list of resources that can supplement TeX, or replace it outright. This group may also venture into ideas on what tools can be developed to address points from the first objective and to create or plan to create said tools.",
   "public_statement": "This group should aim to understand to what extent LaTeX does and does not meet the needs of current mathematicians. After such an understanding is had, it should aim to create a list of resources that can supplement TeX, or replace it outright. This group may also venture into ideas on what tools can be developed to address points from the first objective and to create or plan to create said tools.",
   "evidence": "The source is `aim-infrastructure-notes.json`, zero-based index 46. The linked AIM Problem Lists page returned a gateway error during this run, so the repository text is preserved verbatim rather than silently reconstructed. The official workshop page confirms the 4–8 December 2023 event and its open-source, collaboration, maintenance, and inclusion remit. The official four-page report names LaTeX and PreTeXt as authoring tools but does not record the outcome of this particular group. That silence is not evidence that the group reached no conclusion elsewhere. The text has no visible OCR corruption, but it has three substantive ambiguities:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0048": {
   "statement_status": "exact",
   "original_statement": "From a book with worksheets or activities, extract a \"workbook\". It might have alternate frontmatter, but it should also have links to the original source.",
   "clean_statement": "From a book with worksheets or activities, extract a \"workbook\". It might have alternate frontmatter, but it should also have links to the original source.",
   "public_statement": "From a book with worksheets or activities, extract a \"workbook\". It might have alternate frontmatter, but it should also have links to the original source.",
   "evidence": "The record is legible and contains no apparent OCR corruption. The linked community-wiki page could not be retrieved through the research browser, so the exact record above is the verified statement used here. Neighboring records concern activities, stand-alone worksheets, and print/online variants, which supports reading “workbook” as a new PreTeXt document collecting selected worksheet/activity divisions rather than as a spreadsheet.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0049": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Currently the activities element does not allow @workspace , maybe it should?",
   "clean_statement": "Currently the activities element does not allow `@workspace`, maybe it should?",
   "public_statement": "Currently the activities element does not allow @workspace , maybe it should?",
   "evidence": "The canonical wording is preserved above. Two small differences are extraction artifacts: the wiki uses code formatting around `@workspace` and has no space before the comma. More importantly, **`activities` is not the name of a schema element** in either the 8 July 2024 schema snapshot or the current schema inspected here. The actual PreTeXt element is singular `<activity>`. It shares the `ProjectLike` content pattern with `<investigation>`, `<exploration>`, and `<project>`. The most conservative reconstruction is therefore:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0050": {
   "statement_status": "exact",
   "original_statement": "There should also be \"stand alone\" worksheets.",
   "clean_statement": "There should also be \"stand alone\" worksheets.",
   "public_statement": "There should also be \"stand alone\" worksheets.",
   "evidence": "The record is short but legible, with no apparent OCR corruption. The linked historical community-wiki page could not be retrieved through the research browser, so no missing wording is silently reconstructed. Nearby records discuss extracting a workbook from a book, worksheet workspace, compiling a fragment while ignoring a larger preamble, and print versus interactive behavior. They make the intended object clear enough to distinguish from a spreadsheet, but they do not settle what “stand alone” meant technically.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0051": {
   "statement_status": "exact",
   "original_statement": "\"Standalone\" as a way to compile a fragment, and ignore the preamble when in a large document. (Can this be managed by \"versions\"?). Related, an Overlay journal: material wrapped around a paper that appears elsewhere.",
   "clean_statement": "\"Standalone\" as a way to compile a fragment, and ignore the preamble when in a large document. (Can this be managed by \"versions\"?). Related, an Overlay journal: material wrapped around a paper that appears elsewhere.",
   "public_statement": "\"Standalone\" as a way to compile a fragment, and ignore the preamble when in a large document. (Can this be managed by \"versions\"?). Related, an Overlay journal: material wrapped around a paper that appears elsewhere.",
   "evidence": "The canonical record is item 4 from the AIM workshop list “PreTeXt for small documents”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0052": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Worksheets as print with workspace or online interactive exercises. (Also good to print these, but online, they do not need blank space.)",
   "clean_statement": "Recovered design question (explicit reconstruction).** Can one semantic exercise source support (i) a static print realization that retains measured writing workspace and (ii) an ordinary online realization that retains the interactive response mechanism but suppresses paper-only blank space, while leaving a printable static realization of the interactive exercise available?",
   "public_statement": "Worksheets as print with workspace or online interactive exercises. (Also good to print these, but online, they do not need blank space.)",
   "evidence": "**Recovered design question (explicit reconstruction).** Can one semantic exercise source support (i) a static print realization that retains measured writing workspace and (ii) an ordinary online realization that retains the interactive response mechanism but suppresses paper-only blank space, while leaving a printable static realization of the interactive exercise available?",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0053": {
   "statement_status": "exact",
   "original_statement": "Syllabus: What is a syllabus? Need many more tags!",
   "clean_statement": "Syllabus: What is a syllabus? Need many more tags!",
   "public_statement": "Syllabus: What is a syllabus? Need many more tags!",
   "evidence": "The exact canonical record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0054": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Worksheet: what is a worksheet? e.g., Name, grade/marking",
   "clean_statement": "therefore treated as a content-model problem: which information belongs to a\nreusable worksheet source, and which belongs to a delivered copy, learner\nattempt, or evaluation?",
   "public_statement": "Worksheet: what is a worksheet? e.g., Name, grade/marking",
   "evidence": "The canonical record is item 7 from the AIM workshop list “PreTeXt for small documents”:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0055": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "User experience: single source file document that is easy to share.",
   "clean_statement": null,
   "public_statement": "User experience: single source file document that is easy to share.",
   "evidence": "**Recovered question (explicit reconstruction).** For a small PreTeXt document, which conditions make (a) one editable source pathname sufficient to hand off and rebuild, and (b) one rendered pathname sufficient to deliver and view with the promised behavior? How do those conditions change under offline, reproducibility, LMS-policy, and trust requirements?",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0056": {
   "statement_status": "exact",
   "original_statement": "Landing page (this sort of exists, but it could be better).",
   "clean_statement": "Landing page (this sort of exists, but it could be better).",
   "public_statement": "Landing page (this sort of exists, but it could be better).",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0057": {
   "statement_status": "exact",
   "original_statement": "Using a journal style on your latex (this could go in the publisher file). What is the \\documentclass{?} , a journal .sty file. How does \\begin{theorem} look (does it have a label?)",
   "clean_statement": "Using a journal style on your latex (this could go in the publisher file). What is the \\documentclass{?} , a journal .sty file. How does \\begin{theorem} look (does it have a label?)",
   "public_statement": "Using a journal style on your latex (this could go in the publisher file). What is the \\documentclass{?} , a journal .sty file. How does \\begin{theorem} look (does it have a label?)",
   "evidence": "The canonical corpus record (source index 56 in `aim-infrastructure-notes.json`) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0058": {
   "statement_status": "exact",
   "original_statement": "PreTeXt to JATS (Journal Article Tag Suite). It's an XML format.",
   "clean_statement": "PreTeXt to JATS (Journal Article Tag Suite). It's an XML format.",
   "public_statement": "PreTeXt to JATS (Journal Article Tag Suite). It's an XML format.",
   "evidence": "The canonical `problem` field says, exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0059": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Other conversions to/from PreTeXt. We should make a poster/diagram.",
   "clean_statement": null,
   "public_statement": "Other conversions to/from PreTeXt. We should make a poster/diagram.",
   "evidence": "**Recovered task (explicit reconstruction).** Specify a versioned, direction-sensitive diagram of the known routes into and out of PreTeXt, in which each arrow states who implements it, its maturity, its supported input profile, its verified semantic guarantees, its known losses, and the evidence date. Give a rule for what can truthfully be inferred about a multi-arrow path and for when a round trip is impossible.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0060": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Want to make a link to the full version.",
   "clean_statement": null,
   "public_statement": "Want to make a link to the full version.",
   "evidence": "**Recovered task (explicit reconstruction).** When a version/extraction retains an `<xref>` but omits its target, provide a sound output-appropriate reference to that exact target in a designated full publication. Preserve readable reference text, detect stale or ambiguous mappings, and never silently guess a destination. The full publication might be a book, workbook parent, or another designated component-version; the publisher must identify which. This reconstruction is verified from the wiki hierarchy, but the phrase itself does not specify a target edition, output format, deployment, numbering policy, or persistence promise.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0061": {
   "statement_status": "exact",
   "original_statement": "Numbering when extracting. Preserve or renumber? Levels of numbering (figure for example). Both need to be publisher options.",
   "clean_statement": "Numbering when extracting. Preserve or renumber? Levels of numbering (figure for example). Both need to be publisher options.",
   "public_statement": "Numbering when extracting. Preserve or renumber? Levels of numbering (figure for example). Both need to be publisher options.",
   "evidence": "The record belongs to the AIM workshop list “PreTeXt for small documents.” I checked it against the PreTeXt Community Wiki clone at commit `9093b9cb8b56a54e019ef1696a81f0a710d1102c` (2024-07-17). The wiki has the same sentence, with only capitalization and doubled-space differences, as a sub-bullet of item 13, “Cross references when extracting.” Thus the corpus field `number: \"14\"` is not the displayed number of this item on the checked wiki; it appears to be a consequence of flattening the workshop bullets. There is no apparent OCR corruption or missing symbol, and the canonical statement is not silently rewritten here.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0062": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "What should the reference look like? Name or number?",
   "clean_statement": null,
   "public_statement": "What should the reference look like? Name or number?",
   "evidence": "**Recovered task (explicit reconstruction).** Choose the reader-visible text of a cross-reference when making an extraction or small version. In particular, decide whether a reference should show a generic type name, an authored title/name, a number, or a combination; make that choice consistent with whether the actual destination is local to the extract or external in a designated full publication; and keep it usable in both linked and unlinked output. This reconstruction is strongly supported by the parent bullet, but the terse source does not specify output medium, candidate audience scope, number-preservation policy, or the meaning of “name.”",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0063": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Reference for intermediate users: what can go here?",
   "clean_statement": null,
   "public_statement": "Reference for intermediate users: what can go here?",
   "evidence": "The canonical record says, exactly:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0064": {
   "statement_status": "exact",
   "original_statement": "There are multiple sections of the guide that are still \"todo\"",
   "clean_statement": "There are multiple sections of the guide that are still \"todo\"",
   "public_statement": "There are multiple sections of the guide that are still \"todo\"",
   "evidence": "The record is source index 63 of `aim-infrastructure-notes.json`, with canonical number 17. There is no OCR corruption in the sentence. There is, however, an extraction-context issue. The live PreTeXt community wiki at commit `9093b9cb8b56a54e019ef1696a81f0a710d1102c` (2024-07-17) places this sentence as one bullet under item 14, **Documentation**, alongside quick starts, samples, snippets, autocomplete, and other documentation requests. Thus the source is best recovered as a historical workshop observation and maintenance request, not as a mathematical problem with quantified hypotheses. The corpus has promoted the bullet to its own numbered record; this report preserves the canonical wording while restoring that parent context.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0065": {
   "statement_status": "exact",
   "original_statement": "Quick start for specific document types.",
   "clean_statement": "Quick start for specific document types.",
   "public_statement": "Quick start for specific document types.",
   "evidence": "The live wiki was checked on 2026-08-09. The wording is exact: there is no apparent OCR corruption. On the live page it is a sub-bullet of item 14, “Documentation,” immediately after “Quick start (less than 5 minutes)” and before “Easy to find samples (e.g. annotated book).” Thus the canonical number 18 is an extraction ordinal, not the current top-level wiki number.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0066": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Easy to find samples (e.g. annotated book)",
   "clean_statement": null,
   "public_statement": "Easy to find samples (e.g. annotated book)",
   "evidence": "The canonical record says, exactly:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0067": {
   "statement_status": "exact",
   "original_statement": "More copy/paste snippets (in vscode)",
   "clean_statement": "More copy/paste snippets (in vscode)",
   "public_statement": "More copy/paste snippets (in vscode)",
   "evidence": "The sentence has no visible OCR corruption. Its capitalization and wording are preserved above. It does, however, lose a material hierarchy when flattened. The official community-wiki source inspected at commit `9093b9cb8b56a54e019ef1696a81f0a710d1102c` (2024-07-17) places it under top-level item 14:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0068": {
   "statement_status": "exact",
   "original_statement": "Sample small documents to copy/modify. Maybe as templates in the CLI or in a contributed repository",
   "clean_statement": "Sample small documents to copy/modify. Maybe as templates in the CLI or in a contributed repository",
   "public_statement": "Sample small documents to copy/modify. Maybe as templates in the CLI or in a contributed repository",
   "evidence": "This is zero-based record 67 of `aim-infrastructure-notes.json`, with canonical number `21` and source URL <https://github.com/PreTeXtBook/community/wiki/Moderated-Problem-Session>. There is no apparent OCR error. The punctuation differs slightly from the pinned wiki: the wiki has two spaces after the first period, a harmless Markdown detail.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0069": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Converting from latex/markdown easier than pandoc.",
   "clean_statement": null,
   "public_statement": "Converting from latex/markdown easier than pandoc.",
   "evidence": "The exact canonical record is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0070": {
   "statement_status": "exact",
   "original_statement": "Converters from \"lite\" languages: highlight part of a document and hit convert. Perhaps AI to go from lite documents to pretext?",
   "clean_statement": "Converters from \"lite\" languages: highlight part of a document and hit convert. Perhaps AI to go from lite documents to pretext?",
   "public_statement": "Converters from \"lite\" languages: highlight part of a document and hit convert. Perhaps AI to go from lite documents to pretext?",
   "evidence": "This is record `AIM-INFRASTRUCTURE-0070`, zero-based source index 69 in `aim-infrastructure-notes.json`. Its `remarks` list and `literature` field are empty. The source text is intelligible and shows no apparent OCR error. The official PreTeXt Community Wiki hierarchy at revision `9093b9cb8b56a54e019ef1696a81f0a710d1102c` places it next to, but distinguishes it from, whole-document LaTeX/Markdown conversion. The following wiki item mentions YAML and Markdown as possible “lite” formats. Thus “lite” is deliberately open-ended rather than a corrupted technical term.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 69,
   "attempt": 2
  },
  "AIM-INFRASTRUCTURE-0071": {
   "statement_status": "exact",
   "original_statement": "What subset of LaTeX converts to PreTeXt?",
   "clean_statement": "What subset of LaTeX converts to PreTeXt?",
   "public_statement": "What subset of LaTeX converts to PreTeXt?",
   "evidence": "The canonical record is `AIM-INFRASTRUCTURE-0071`, source file `aim-infrastructure-notes.json`, zero-based index 70, displayed as flattened item 24. No OCR correction is needed.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 70,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0072": {
   "statement_status": "exact",
   "original_statement": "Why write in PreTeXt? Interactivity, Accessibility.",
   "clean_statement": "Why write in PreTeXt? Interactivity, Accessibility.",
   "public_statement": "Why write in PreTeXt? Interactivity, Accessibility.",
   "evidence": "This is record `AIM-INFRASTRUCTURE-0072`, zero-based index 71 of `aim-infrastructure-notes.json`. Its `remarks` and `literature` fields are empty. There is no apparent OCR corruption.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0073": {
   "statement_status": "exact",
   "original_statement": "Create by CLI: syllabus, worksheets.",
   "clean_statement": "Create by CLI: syllabus, worksheets.",
   "public_statement": "Create by CLI: syllabus, worksheets.",
   "evidence": "The canonical record is preserved verbatim: It is record 72 (zero-based) of `aim-infrastructure-notes.json`, with canonical `number: \"26\"`, workshop “PreTeXt for small documents,” and source URL <https://github.com/PreTeXtBook/community/wiki/Moderated-Problem-Session>. There is no visible OCR corruption.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 72,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0074": {
   "statement_status": "exact",
   "original_statement": "How to share outside a \"Course\" context (like sharing a .sty file).",
   "clean_statement": "How to share outside a \"Course\" context (like sharing a .sty file).",
   "public_statement": "How to share outside a \"Course\" context (like sharing a .sty file).",
   "evidence": "The exact canonical problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 73,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0075": {
   "statement_status": "exact",
   "original_statement": "Start a document with <publication> .",
   "clean_statement": "Start a document with <publication> .",
   "public_statement": "Start a document with <publication> .",
   "evidence": "The canonical record is preserved verbatim, including its unusual space before the period:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0076": {
   "statement_status": "exact",
   "original_statement": "Need to pay more attention to the Instructor (\"Private Publishing\")",
   "clean_statement": "Need to pay more attention to the Instructor (\"Private Publishing\")",
   "public_statement": "Need to pay more attention to the Instructor (\"Private Publishing\")",
   "evidence": "The canonical record is preserved verbatim:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0077": {
   "statement_status": "exact",
   "original_statement": "Assembling legacy material (book of worksheets).",
   "clean_statement": "Assembling legacy material (book of worksheets).",
   "public_statement": "Assembling legacy material (book of worksheets).",
   "evidence": "The exact canonical problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0078": {
   "statement_status": "exact",
   "original_statement": "Programming Languages: Which are available, what are their capabilities?",
   "clean_statement": "Programming Languages: Which are available, what are their capabilities?",
   "public_statement": "Programming Languages: Which are available, what are their capabilities?",
   "evidence": "Thus there is no apparent OCR corruption in the sentence. The disagreement between canonical number 31 and wiki item 23 is a numbering/extraction artifact, not a mathematical change. The statement is genuinely ambiguous in a more important way: “available” can mean at least schema-admissible, statically displayed, syntax-highlighted, editable/executable, traceable, or testable. Those meanings are not equivalent.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0079": {
   "statement_status": "exact",
   "original_statement": "Other embeddables: Penrose diagrams, Lurch, Sonification for data (multimodal), large data sets (see Runestone data file object).",
   "clean_statement": "Other embeddables: Penrose diagrams, Lurch, Sonification for data (multimodal), large data sets (see Runestone data file object).",
   "public_statement": "Other embeddables: Penrose diagrams, Lurch, Sonification for data (multimodal), large data sets (see Runestone data file object).",
   "evidence": "The official wiki repository at commit `9093b9cb8b56a54e019ef1696a81f0a710d1102c` contains exactly the same sentence as numbered item 24. It follows “Programming Languages: Which are available, what are their capabilities?” and precedes the one-page-output item. Thus the sentence is a top-level agenda item about additional embedded media and tools, not a child of the programming-language item or the one-page-output item. The canonical number 32 is an extractor-assigned record number; its difference from wiki item 24 is not an OCR error. Capitalization, punctuation, and the parenthetical phrase are source-verified.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0080": {
   "statement_status": "exact",
   "original_statement": "Is ePub good enough? No, because of knowls.",
   "clean_statement": "Is ePub good enough? No, because of knowls.",
   "public_statement": "Is ePub good enough? No, because of knowls.",
   "evidence": "The canonical record is preserved verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0081": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Other use cases: slides on a thumb drive",
   "clean_statement": null,
   "public_statement": "Other use cases: slides on a thumb drive",
   "evidence": "The phrase is a use case, not a formal specification. “On a thumb drive” may mean only that the files are carried on removable storage. It does not literally say that the presentation computer has no network, that the deck is one file, or that it is opened with a `file:` URL rather than a loopback web server. Nevertheless, the one-page parent and its bundle/external-resource contrast support this explicit reconstruction: This reconstruction is used below but is not asserted to be the only intended meaning of the short source text.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0082": {
   "statement_status": "exact",
   "original_statement": "Put a slide in a book? Really gather all slides in one place so you can share all slides for a course in a single link, that is updated regularly. This is really like creating a \"course\".",
   "clean_statement": "Put a slide in a book? Really gather all slides in one place so you can share all slides for a course in a single link, that is updated regularly. This is really like creating a \"course\".",
   "public_statement": "Put a slide in a book? Really gather all slides in one place so you can share all slides for a course in a single link, that is updated regularly. This is really like creating a \"course\".",
   "evidence": "The canonical problem text is preserved verbatim: The live raw Markdown of the official PreTeXt community wiki was inspected on 2026-08-09. The wording above is exact; there is no apparent OCR corruption. It is top-level item 26, immediately after item 25, “One page output (HTML including CSS & JS),” whose last sub-bullet is “Other use cases: slides on a thumb drive.” The next item is “LTI or LMS integration.” The corpus number 35 is therefore a flattened extraction ordinal rather than the displayed wiki item number.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0083": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Parellelization\n\nIt would be nice to have parallel computation in the M2 core.\n\nHow to parellize Macaulay2? Do we need a new garbage collector to do this? Which algorithms are inherently parallelizable and which need more thought?",
   "clean_statement": null,
   "public_statement": "Parellelization\n\nIt would be nice to have parallel computation in the M2 core.\n\nHow to parellize Macaulay2? Do we need a new garbage collector to do this? Which algorithms are inherently parallelizable and which need more thought?",
   "evidence": "The analysis below uses the conservative reconstruction",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 82,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0084": {
   "statement_status": "exact",
   "original_statement": "In addition, the \"top level\" user interface would benefit from parellization.\n\nWhat features at the user level should be parallelized? For which computations do we want to run multiple algorithms simultaneously and take whichever finishes first? Can we do other calculations in the same instance while one calculation is running?",
   "clean_statement": "In addition, the \"top level\" user interface would benefit from parellization.\n\nWhat features at the user level should be parallelized? For which computations do we want to run multiple algorithms simultaneously and take whichever finishes first? Can we do other calculations in the same instance while one calculation is running?",
   "public_statement": "In addition, the \"top level\" user interface would benefit from parellization.\n\nWhat features at the user level should be parallelized? For which computations do we want to run multiple algorithms simultaneously and take whichever finishes first? Can we do other calculations in the same instance while one calculation is running?",
   "evidence": "The canonical record is item 1.2 in the section “Macaulay2 internals and benchmarks” of the AIM workshop list *Macaulay2: expanded functionality and improved efficiency*. The exact canonical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 83,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0085": {
   "statement_status": "exact",
   "original_statement": "Other internal improvements\n\nThe current \"basis\" method can be quite slow in situations where it should be nearly instantaneous (see Github issue #2899).\n\nHow can we implement a faster basis calculation without breaking anything?",
   "clean_statement": "Other internal improvements\n\nThe current \"basis\" method can be quite slow in situations where it should be nearly instantaneous (see Github issue #2899).\n\nHow can we implement a faster basis calculation without breaking anything?",
   "public_statement": "Other internal improvements\n\nThe current \"basis\" method can be quite slow in situations where it should be nearly instantaneous (see Github issue #2899).\n\nHow can we implement a faster basis calculation without breaking anything?",
   "evidence": "The record is plain text and shows no sign of OCR corruption. The linked AIM page could not be fetched during this run: HTTPS reported an expired certificate on one attempt and DNS lookup failed on another. Consequently the canonical text above is preserved, not silently “corrected.” The workshop and date were independently confirmed on the current AIM workshop page.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 84,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0086": {
   "statement_status": "exact",
   "original_statement": "Currently, \"kernel\", \"Hom\", and several other commands have no options for \"DegreeLimit\".\n\nWhat commands should we add \"DegreeLimit\" to, and how can we better document \"DegreeLimit\" and its expected behavior in these functions?",
   "clean_statement": "Currently, \"kernel\", \"Hom\", and several other commands have no options for \"DegreeLimit\".\n\nWhat commands should we add \"DegreeLimit\" to, and how can we better document \"DegreeLimit\" and its expected behavior in these functions?",
   "public_statement": "Currently, \"kernel\", \"Hom\", and several other commands have no options for \"DegreeLimit\".\n\nWhat commands should we add \"DegreeLimit\" to, and how can we better document \"DegreeLimit\" and its expected behavior in these functions?",
   "evidence": "The canonical statement is grammatical and contains no visible OCR corruption. Nearby records confirm its workshop and section context. The supplied AIMPL URL, `http://aimpl.org/macaulay2efie/1/`, redirected to HTTPS and returned a 502 error when checked on 2026-08-09, so the wording could not be compared with the live problem page. The official AIM workshop page still links to an open problem list. I therefore preserve the canonical wording rather than silently modernizing it.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 85,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0087": {
   "statement_status": "exact",
   "original_statement": "Benchmarks\n\nBy a \"benchmark\" we mean an interesting calculation (e.g., one occurring in someone's research) which pushes the capabilities of M2. These are especially useful if they come in a natural family, which we can compute up to some degree but not beyond at the moment.\n\nWhat \"benchmark\" calculations should be tracked in M2? What is the best way to keep track of these benchmarks (e.g., wiki, on the website, etc.)?",
   "clean_statement": "Benchmarks\n\nBy a \"benchmark\" we mean an interesting calculation (e.g., one occurring in someone's research) which pushes the capabilities of M2. These are especially useful if they come in a natural family, which we can compute up to some degree but not beyond at the moment.\n\nWhat \"benchmark\" calculations should be tracked in M2? What is the best way to keep track of these benchmarks (e.g., wiki, on the website, etc.)?",
   "public_statement": "Benchmarks\n\nBy a \"benchmark\" we mean an interesting calculation (e.g., one occurring in someone's research) which pushes the capabilities of M2. These are especially useful if they come in a natural family, which we can compute up to some degree but not beyond at the moment.\n\nWhat \"benchmark\" calculations should be tracked in M2? What is the best way to keep track of these benchmarks (e.g., wiki, on the website, etc.)?",
   "evidence": "The canonical record is internally coherent and has no apparent OCR corruption. Its listed AIMPL URL, `http://aimpl.org/macaulay2efie/1/`, did not return a usable page during this run, so the wording above is verified only against `input.json` and the canonical repository record. Nearby records confirm that this is an infrastructure question in the stated section; they do not change its meaning.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 86,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0088": {
   "statement_status": "exact",
   "original_statement": "Applying differential operators\n\nCurrently, the D-module package(s) doesn't have a built-in way to act on ring elements by differential operators.\n\nCan we add a convenient way to let the differential operators on a ring R act on elements of R? Can we have them act on elements of the fraction field?",
   "clean_statement": "Applying differential operators\n\nCurrently, the D-module package(s) doesn't have a built-in way to act on ring elements by differential operators.\n\nCan we add a convenient way to let the differential operators on a ring R act on elements of R? Can we have them act on elements of the fraction field?",
   "public_statement": "Applying differential operators\n\nCurrently, the D-module package(s) doesn't have a built-in way to act on ring elements by differential operators.\n\nCan we add a convenient way to let the differential operators on a ring R act on elements of R? Can we have them act on elements of the fraction field?",
   "evidence": "The live AIM HTML page was checked on 2026-08-09. It has the same wording (apart from a typographic apostrophe and trailing-space differences), contains no status update or remark, and confirms the numbering. There is no OCR corruption to repair. The word “ring” is broader than the software representations involved, however. The theorem below treats the polynomial-domain case actually modeled by a Weyl algebra, and Section 8 explains why an arbitrary quotient ring cannot silently be treated in the same way.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 87,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0089": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Computing local cohomology via D-modules\n\nThe current method for computing local cohomology requires the iterated computation of Bernstein--Sato polynomials at every step of the Cech complex.\n\nCan computing just the necessary steps of the Cech complex speed up the computation of local cohomology? What about computing just the Budur--Mustata--Saito b-function just once?",
   "clean_statement": null,
   "public_statement": "Computing local cohomology via D-modules\n\nThe current method for computing local cohomology requires the iterated computation of Bernstein--Sato polynomials at every step of the Cech complex.\n\nCan computing just the necessary steps of the Cech complex speed up the computation of local cohomology? What about computing just the Budur--Mustata--Saito b-function just once?",
   "evidence": "This record is Problem 2.1, “Computing local cohomology via D-modules,” from the AIM workshop *Macaulay2: expanded functionality and improved efficiency*, section “Local cohomology and differential operators.” The canonical record asks:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 88,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0090": {
   "statement_status": "exact",
   "original_statement": "Local cohomology in characteristic p\n\nIf R is a regular ring of characteristic p, then the local cohomology modules have many nice finiteness properties; one can describe many of these properties via the theory of unit $F$-modules. Work on implementing these calculations in M2 was begun at the 2023 Minneapolis workshop.\n\nGiven an F-finite F-module, how can one efficiently compute a unit F-module? For example, given the map $\\mathrm{Ext}^i(R/I,R)\\to \\mathrm{Ext}^i(R/I^{[p]},R)$, how can one find a root for the local cohomology module $H^i_I(R)$?",
   "clean_statement": "Local cohomology in characteristic p\n\nIf R is a regular ring of characteristic p, then the local cohomology modules have many nice finiteness properties; one can describe many of these properties via the theory of unit $F$-modules. Work on implementing these calculations in M2 was begun at the 2023 Minneapolis workshop.\n\nGiven an F-finite F-module, how can one efficiently compute a unit F-module? For example, given the map $\\mathrm{Ext}^i(R/I,R)\\to \\mathrm{Ext}^i(R/I^{[p]},R)$, how can one find a root for the local cohomology module $H^i_I(R)$?",
   "public_statement": "Local cohomology in characteristic p\n\nIf R is a regular ring of characteristic p, then the local cohomology modules have many nice finiteness properties; one can describe many of these properties via the theory of unit $F$-modules. Work on implementing these calculations in M2 was begun at the 2023 Minneapolis workshop.\n\nGiven an F-finite F-module, how can one efficiently compute a unit F-module? For example, given the map $\\mathrm{Ext}^i(R/I,R)\\to \\mathrm{Ext}^i(R/I^{[p]},R)$, how can one find a root for the local cohomology module $H^i_I(R)$?",
   "evidence": "The canonical record is problem 2.2, “Local cohomology in characteristic p,” in the AIM list for the workshop *Macaulay2: expanded functionality and improved efficiency*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 89,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0091": {
   "statement_status": "exact",
   "original_statement": "Can we compute the n-th step of a resolution quickly, without computing the previous steps?",
   "clean_statement": "Can we compute the n-th step of a resolution quickly, without computing the previous steps?",
   "public_statement": "Can we compute the n-th step of a resolution quickly, without computing the previous steps?",
   "evidence": "The sentence is syntactically intact; there is no sign of OCR corruption. The original AIMPL page at `http://aimpl.org/macaulay2efie/3/` was unavailable during this run. The official AIM workshop page and final report confirm that resolutions over nonregular rings and DG-algebra methods were central topics. Nearby canonical records ask about DG modules, semifree resolutions, and simplicial resolutions, which supports the section assignment but does not remove the main ambiguity in the terse question.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 90,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0092": {
   "statement_status": "exact",
   "original_statement": "What functionality should be added to the DGAlgebras package? Can we handle DG ideals and modules? What about semifree resolutions? Can it be better incorporated with the AInfinity and EagonResolution packages? Can we add higher-order Massey operations?",
   "clean_statement": "What functionality should be added to the DGAlgebras package? Can we handle DG ideals and modules? What about semifree resolutions? Can it be better incorporated with the AInfinity and EagonResolution packages? Can we add higher-order Massey operations?",
   "public_statement": "What functionality should be added to the DGAlgebras package? Can we handle DG ideals and modules? What about semifree resolutions? Can it be better incorporated with the AInfinity and EagonResolution packages? Can we add higher-order Massey operations?",
   "evidence": "The record is a broad, multi-part software-research agenda, not a single proposition. Its wording is coherent and has no apparent OCR corruption. The original AIMPL page was unavailable during this run, so the exact repository record is the recovered statement; no missing mathematical symbols were inferred.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 91,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0093": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Can we add simplicial resolutions to Macaulay2?",
   "clean_statement": null,
   "public_statement": "Can we add simplicial resolutions to Macaulay2?",
   "evidence": "The AIM page for the workshop *Macaulay2: expanded functionality and improved efficiency*, section “DG Algebras and Resolutions,” gives Problem 3.3 exactly as:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0094": {
   "statement_status": "exact",
   "original_statement": "M2 currently has sheaves on projective varieties, but no morphisms between them, and no way to compute the associated maps on cohomology.\n\nWhat is the best way to implement a \"morphism of sheaves\" data type? What's the best way to calculate the maps on sheaf cohomology induced by such a morphism on a projective variety?",
   "clean_statement": "M2 currently has sheaves on projective varieties, but no morphisms between them, and no way to compute the associated maps on cohomology.\n\nWhat is the best way to implement a \"morphism of sheaves\" data type? What's the best way to calculate the maps on sheaf cohomology induced by such a morphism on a projective variety?",
   "public_statement": "M2 currently has sheaves on projective varieties, but no morphisms between them, and no way to compute the associated maps on cohomology.\n\nWhat is the best way to implement a \"morphism of sheaves\" data type? What's the best way to calculate the maps on sheaf cohomology induced by such a morphism on a projective variety?",
   "evidence": "The canonical statement is coherent and contains no apparent OCR corruption. The original AIMPL page listed in the record was unavailable during this run, so the statement above was checked against the repository record and the later official AIM workshop summary rather than silently reconstructed.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0095": {
   "statement_status": "exact",
   "original_statement": "There is preliminary code to decompose a module into indecomposables, but it would benefit from being much faster. Right now the bottleneck is the computation of the endomorphisms of module M.\n\nCan we compute all the degree-0 (or degree-≤ e) homomorphisms of a graded module M without computing the entirety of End M? What are other ways to speed up the decomposition of modules into indecomposable summands? If M is the direct sum of $N_1$ and $N_2$, how can we extract $\\mathrm{End}(N_1)$ and $\\mathrm{End}(N_2)$ from $\\mathrm{End}(M)$ without recalculating?",
   "clean_statement": "There is preliminary code to decompose a module into indecomposables, but it would benefit from being much faster. Right now the bottleneck is the computation of the endomorphisms of module M.\n\nCan we compute all the degree-0 (or degree-≤ e) homomorphisms of a graded module M without computing the entirety of End M? What are other ways to speed up the decomposition of modules into indecomposable summands? If M is the direct sum of $N_1$ and $N_2$, how can we extract $\\mathrm{End}(N_1)$ and $\\mathrm{End}(N_2)$ from $\\mathrm{End}(M)$ without recalculating?",
   "public_statement": "There is preliminary code to decompose a module into indecomposables, but it would benefit from being much faster. Right now the bottleneck is the computation of the endomorphisms of module M.\n\nCan we compute all the degree-0 (or degree-≤ e) homomorphisms of a graded module M without computing the entirety of End M? What are other ways to speed up the decomposition of modules into indecomposable summands? If M is the direct sum of $N_1$ and $N_2$, how can we extract $\\mathrm{End}(N_1)$ and $\\mathrm{End}(N_2)$ from $\\mathrm{End}(M)$ without recalculating?",
   "evidence": "The text is coherent and has no apparent OCR error. The original AIMPL page was unavailable during this run, but the exact repository record and the later official AIM workshop report agree on the direct-summands project.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 94,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0096": {
   "statement_status": "unrecoverable",
   "original_statement": "Cases of Harassment in the Community\n\nHow can we support people who have been harassed? How do we as a community handle working with people who have been accused (including by way of warnings over the \"whisper network\")? Is there room for restorative justice?",
   "clean_statement": null,
   "public_statement": "Cases of Harassment in the Community\n\nHow can we support people who have been harassed? How do we as a community handle working with people who have been accused (including by way of warnings over the \"whisper network\")? Is there room for restorative justice?",
   "evidence": "This is a policy and community-safety question, not a mathematical problem. It has no universal answer independent of employment law, education law, collective agreements, professional-society authority, safeguarding duties, privacy rules, and the country or state involved. This report therefore offers a testable governance design, not legal advice. Every adopting body must have qualified local personnel map the design to its jurisdiction, insurance, funder, employer, venue, union, and institutional obligations before use.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0097": {
   "statement_status": "exact",
   "original_statement": "Taking Intent to Action\n\nHow can we go from ideas/intention to action with our DEI work? In particular, can we create a list of ways to take action and/or find literature on best practices for taking intent to action and then translate for mathematicians' use?",
   "clean_statement": "Taking Intent to Action\n\nHow can we go from ideas/intention to action with our DEI work? In particular, can we create a list of ways to take action and/or find literature on best practices for taking intent to action and then translate for mathematicians' use?",
   "public_statement": "Taking Intent to Action\n\nHow can we go from ideas/intention to action with our DEI work? In particular, can we create a list of ways to take action and/or find literature on best practices for taking intent to action and then translate for mathematicians' use?",
   "evidence": "The text is coherent and contains no apparent OCR error. The original AIMPL item did not load during this run. The official AIM workshop page confirms that the March 27–31, 2023 workshop joined combinatorics research with discussion of gender equity, intersectionality, and the experiences of trans and non-binary mathematicians. The exact wording above is preserved from the canonical repository record; nothing was silently reconstructed.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 96,
   "attempt": 2
  },
  "AIM-INFRASTRUCTURE-0098": {
   "statement_status": "exact",
   "original_statement": "The History of Women in Combinatorics\n\nWhat is the history of women in combinatorics? Answering this question might involve researchers from many different areas, including historians, sociologists, mathematicians, and data scientists. As other possible guiding questions: What departments are producing the most women PhDs in math? What are they doing right? What is going wrong at other departments? How can we ensure our gender data is accurate for this study? Will this study be US-centered, and if so, how can we perform similar studies outside the US?",
   "clean_statement": "The History of Women in Combinatorics\n\nWhat is the history of women in combinatorics? Answering this question might involve researchers from many different areas, including historians, sociologists, mathematicians, and data scientists. As other possible guiding questions: What departments are producing the most women PhDs in math? What are they doing right? What is going wrong at other departments? How can we ensure our gender data is accurate for this study? Will this study be US-centered, and if so, how can we perform similar studies outside the US?",
   "public_statement": "The History of Women in Combinatorics\n\nWhat is the history of women in combinatorics? Answering this question might involve researchers from many different areas, including historians, sociologists, mathematicians, and data scientists. As other possible guiding questions: What departments are producing the most women PhDs in math? What are they doing right? What is going wrong at other departments? How can we ensure our gender data is accurate for this study? Will this study be US-centered, and if so, how can we perform similar studies outside the US?",
   "evidence": "The repository record is internally legible and shows no apparent OCR corruption. The legacy URL `http://aimpl.org/gemscombin/7/` did not render during this run, so the exact wording above is verified against the canonical repository input rather than a live copy of that page. AIM’s surviving workshop page verifies the context: GEMS ran 27–31 March 2023 and aimed to address gender equity in combinatorics. Importantly, that page explicitly broadens the workshop’s scope beyond “women in mathematics” to people who self-identify as gender minorities, including trans and non-binary mathematicians [AIM2023]. That broader workshop aim does not authorize silently changing this record’s title. A history of women and a study of gender minorities are related but distinct projects; any combined infrastructure must keep their target constructs separately labelled.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 97,
   "attempt": 2
  },
  "AIM-INFRASTRUCTURE-0099": {
   "statement_status": "exact",
   "original_statement": "Starting DEI Conversations\n\nHow can we start DEI conversations in departments outside the US?",
   "clean_statement": "Starting DEI Conversations\n\nHow can we start DEI conversations in departments outside the US?",
   "public_statement": "Starting DEI Conversations\n\nHow can we start DEI conversations in departments outside the US?",
   "evidence": "The canonical JSON and the live AIM HTML agree verbatim. The live entry has no status note or remark. There is no visible OCR corruption, missing notation, or truncation. The acronym “DEI” is therefore preserved as part of the source question; this report does **not** assume that the English acronym, its usual U.S. expansion, U.S. demographic categories, or U.S. legal and institutional assumptions have a safe or meaningful counterpart elsewhere.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 98,
   "attempt": 2
  },
  "AIM-INFRASTRUCTURE-0100": {
   "statement_status": "unrecoverable",
   "original_statement": "Improving Conference Organization/Resources for Organizers\n\nWhat might it look like to make our events (conferences, workshops, etc.) accessible to all participants? How can we make it accessible to parent participants, trans participants, participants from the Global South, non-US participants, ? Can we make a \"best practices\" document for anyone interested in organizing an equitable event in math?",
   "clean_statement": null,
   "public_statement": "Improving Conference Organization/Resources for Organizers\n\nWhat might it look like to make our events (conferences, workshops, etc.) accessible to all participants? How can we make it accessible to parent participants, trans participants, participants from the Global South, non-US participants, ? Can we make a \"best practices\" document for anyone interested in organizing an equitable event in math?",
   "evidence": "No word is inserted after the comma: the missing category is unrecoverable from the sources checked. The verified named groups are parent participants, trans participants, participants from the Global South, and non-US participants. Disability and universal design, caregiving beyond parenting, language, time zones, religion and culture, cost, digital access, health, and safety are treated below as a present-day expansion needed to answer the clear general question, not as reconstruction of the missing text.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 99,
   "attempt": 2
  },
  "AIM-INFRASTRUCTURE-0101": {
   "statement_status": "exact",
   "original_statement": "Less Actionable But Nonetheless Important Issues\n\nHow do we deal with the structural problem of expecting mathematicians to move every 2 years for post-docs? How can we work to change the gender pay gap? How can we financially support mathematicians who would like to have kids, especially those who have kids in ways that are financially draining? How can we provide better access to healthcare for trans mathematicians?",
   "clean_statement": "Less Actionable But Nonetheless Important Issues\n\nHow do we deal with the structural problem of expecting mathematicians to move every 2 years for post-docs? How can we work to change the gender pay gap? How can we financially support mathematicians who would like to have kids, especially those who have kids in ways that are financially draining? How can we provide better access to healthcare for trans mathematicians?",
   "public_statement": "Less Actionable But Nonetheless Important Issues\n\nHow do we deal with the structural problem of expecting mathematicians to move every 2 years for post-docs? How can we work to change the gender pay gap? How can we financially support mathematicians who would like to have kids, especially those who have kids in ways that are financially draining? How can we provide better access to healthcare for trans mathematicians?",
   "evidence": "The canonical repository record and its neighboring Equity Questions records are legible; there is no apparent OCR corruption or truncation. The legacy source URL did not render during this run, so the exact text is verified from the repository. AIM’s current GEMS page verifies the March 2023 workshop context and explicitly includes self-identified gender minorities, including trans and non-binary mathematicians [AIM2023].",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 100,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0102": {
   "statement_status": "exact",
   "original_statement": "Gender Diversity Beyond the Gender Binary\n\nHow can we address and work towards gender diversity outside of the gender binary? Can \"Women in Math\" organizations clarify what they mean by \"women?\" And can we push for greater inclusivity in these organizations?",
   "clean_statement": "Gender Diversity Beyond the Gender Binary\n\nHow can we address and work towards gender diversity outside of the gender binary? Can \"Women in Math\" organizations clarify what they mean by \"women?\" And can we push for greater inclusivity in these organizations?",
   "public_statement": "Gender Diversity Beyond the Gender Binary\n\nHow can we address and work towards gender diversity outside of the gender binary? Can \"Women in Math\" organizations clarify what they mean by \"women?\" And can we push for greater inclusivity in these organizations?",
   "evidence": "The canonical record and live AIM HTML agree. The live entry has no status note or remark, and there is no visible OCR corruption or truncation. This report does not silently replace the question by a different one.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 101,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0103": {
   "statement_status": "exact",
   "original_statement": "How Does the Gender Binary Influence the Math we Create?\n\nHow does the gender binary (and more generally how we perceive the world) influence the math we create? (not just how we teach and share it)",
   "clean_statement": "How Does the Gender Binary Influence the Math we Create?\n\nHow does the gender binary (and more generally how we perceive the world) influence the math we create? (not just how we teach and share it)",
   "public_statement": "How Does the Gender Binary Influence the Math we Create?\n\nHow does the gender binary (and more generally how we perceive the world) influence the math we create? (not just how we teach and share it)",
   "evidence": "The canonical record is item 7.8 in the “Equity Questions” section of the 2023 AIM workshop *Gems of combinatorics*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 102,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0104": {
   "statement_status": "exact",
   "original_statement": "(1) What strategies do IBL instructors use to create and sustain an appropriate classroom culture in IBL real analysis courses? (2) What are the essential aspects of appropriate classroom culture that faculty make explicit moves to establish and maintain, regardless of classroom structure or strategy for doing so? \n\nAbstract: Because IBL instruction departs from the traditional (lecture-style) mathemat-ics classroom contract between teacher and students (and among peers), we anticipate that all IBL instructors use various strategies for creating and maintaining a classroom culture conducive to the expectations and goals of IBL pedagogy. While discussion and self-report of experienced IBL instructors suggests that these strategies are diverse, we hypothesize that the diverse strategies are used to accomplish a more universal set of goals. We pro-pose a study of these teaching moves, their purposes, and their efficacy through qualitative observation of IBL classrooms. We anticipate doing so by identifying key points in which faculty make explicit moves to create and sustain a desirable classroom culture before con-ducting faculty and student interviews around video clips of such events. Eliciting student interpretation of such events and how they arise to breaches of the intended classroom ex-pectations, we anticipate being able to learn about students understanding and adoption of these expectations for learning. \n\n• Video record early classes and then a sampling later in the semester. \n\n• Attend to initial solicitation for buy in to IBL and then to breaches of classroom contract that are negotiated. 3\n\n• Conduct member check interviews for instructor and student interpretations of class-room expectations and the justification for/sources of such expectations. It was proposed to measure student perceptions twice per semester, possibly after 4 and 12 weeks (based upon IBL instructors experiences of the recurrent progression of students affective responses in such courses). \n\n• Possible student questioning strategy: If you could choose how classroom time is spent in the mathematics course you are taking, what percentage of the time would you spend on: professor lecture, student presentations, group work on challenging problems, etc? \n\n• We assume many students will come into an IBL real analysis course with a baseline set of expectations informed by traditional mathematics instruction in which students attempt tasks after receiving relevant direct instruction, students are assessed on private work without publishing student work to other students, and the professor and textbook are the primary sources of new mathematics in the class. IBL is taken to entail the expectation that students attempt proof tasks for which they have not received direct instruction, students present their own work to the class for evaluation, and that students author new mathematical content into the classroom. \n\n• Relevant categories we anticipate may arise in the study: \n\n- Legitimizing failure (or linking success to hard work and repeated attempts) \n\n- Soliciting buy in to the alternative classroom expectations \n\n- Negotiating classroom expectations (of students and teacher) \n\n- Endorsing standards for acceptable proof \n\n- Curtailing undesirable mathematical practices \n\n• We anticipate observing IBL instructors of various levels of experience. Novice in-structors might provide more opportunity to identify breaches and difficulties that require intervention. Observing experienced IBL instructors will aide in identifying successful strategies, since documenting and publishing such strategies is an impor-tant research product. \n\nLearning through proof presentations. The overarching goal of this investigation is to under-stand what students learn in IBL courses when listening to peer proof presentations. Two relevant hypotheses were identified: \n\nHypothesis 1: Students learn more from proof presentations in an IBL course because they have already attempted the proof tasks before seeing the completed solution. \n\nHypothesis 2: Students learn more from proof presentations in an IBL course because they pay closer attention to find and identify mistakes, which may depend upon how student listening is guided by the instructor. We propose two possibly independent studies for these two hypotheses. \n\nResearch Questions:",
   "clean_statement": "(1) What strategies do IBL instructors use to create and sustain an appropriate classroom culture in IBL real analysis courses? (2) What are the essential aspects of appropriate classroom culture that faculty make explicit moves to establish and maintain, regardless of classroom structure or strategy for doing so?\n\nAbstract: Because IBL instruction departs from the traditional (lecture-style) mathemat-ics classroom contract between teacher and students (and among peers), we anticipate that all IBL instructors use various strategies for creating and maintaining a classroom culture conducive to the expectations and goals of IBL pedagogy. While discussion and self-report of experienced IBL instructors suggests that these strategies are diverse, we hypothesize that the diverse strategies are used to accomplish a more universal set of goals. We pro-pose a study of these teaching moves, their purposes, and their efficacy through qualitative observation of IBL classrooms. We anticipate doing so by identifying key points in which faculty make explicit moves to create and sustain a desirable classroom culture before con-ducting faculty and student interviews around video clips of such events. Eliciting student interpretation of such events and how they arise to breaches of the intended classroom ex-pectations, we anticipate being able to learn about students understanding and adoption of these expectations for learning.\n\n• Video record early classes and then a sampling later in the semester.\n\n• Attend to initial solicitation for buy in to IBL and then to breaches of classroom contract that are negotiated. 3\n\n• Conduct member check interviews for instructor and student interpretations of class-room expectations and the justification for/sources of such expectations. It was proposed to measure student perceptions twice per semester, possibly after 4 and 12 weeks (based upon IBL instructors experiences of the recurrent progression of students affective responses in such courses).\n\n• Possible student questioning strategy: If you could choose how classroom time is spent in the mathematics course you are taking, what percentage of the time would you spend on: professor lecture, student presentations, group work on challenging problems, etc?\n\n• We assume many students will come into an IBL real analysis course with a baseline set of expectations informed by traditional mathematics instruction in which students attempt tasks after receiving relevant direct instruction, students are assessed on private work without publishing student work to other students, and the professor and textbook are the primary sources of new mathematics in the class. IBL is taken to entail the expectation that students attempt proof tasks for which they have not received direct instruction, students present their own work to the class for evaluation, and that students author new mathematical content into the classroom.\n\n• Relevant categories we anticipate may arise in the study:\n\n- Legitimizing failure (or linking success to hard work and repeated attempts)\n\n- Soliciting buy in to the alternative classroom expectations\n\n- Negotiating classroom expectations (of students and teacher)\n\n- Endorsing standards for acceptable proof\n\n- Curtailing undesirable mathematical practices\n\n• We anticipate observing IBL instructors of various levels of experience. Novice in-structors might provide more opportunity to identify breaches and difficulties that require intervention. Observing experienced IBL instructors will aide in identifying successful strategies, since documenting and publishing such strategies is an impor-tant research product.\n\nLearning through proof presentations. The overarching goal of this investigation is to under-stand what students learn in IBL courses when listening to peer proof presentations. Two relevant hypotheses were identified:\n\nHypothesis 1: Students learn more from proof presentations in an IBL course because they have already attempted the proof tasks before seeing the completed solution.\n\nHypothesis 2: Students learn more from proof presentations in an IBL course because they pay closer attention to find and identify mistakes, which may depend upon how student listening is guided by the instructor. We propose two possibly independent studies for these two hypotheses.\n\nResearch Questions:",
   "public_statement": "(1) What strategies do IBL instructors use to create and sustain an appropriate classroom culture in IBL real analysis courses? (2) What are the essential aspects of appropriate classroom culture that faculty make explicit moves to establish and maintain, regardless of classroom structure or strategy for doing so?\n\nAbstract: Because IBL instruction departs from the traditional (lecture-style) mathemat-ics classroom contract between teacher and students (and among peers), we anticipate that all IBL instructors use various strategies for creating and maintaining a classroom culture conducive to the expectations and goals of IBL pedagogy. While discussion and self-report of experienced IBL instructors suggests that these strategies are diverse, we hypothesize that the diverse strategies are used to accomplish a more universal set of goals. We pro-pose a study of these teaching moves, their purposes, and their efficacy through qualitative observation of IBL classrooms. We anticipate doing so by identifying key points in which faculty make explicit moves to create and sustain a desirable classroom culture before con-ducting faculty and student interviews around video clips of such events. Eliciting student interpretation of such events and how they arise to breaches of the intended classroom ex-pectations, we anticipate being able to learn about students understanding and adoption of these expectations for learning.\n\n• Video record early classes and then a sampling later in the semester.\n\n• Attend to initial solicitation for buy in to IBL and then to breaches of classroom contract that are negotiated. 3\n\n• Conduct member check interviews for instructor and student interpretations of class-room expectations and the justification for/sources of such expectations. It was proposed to measure student perceptions twice per semester, possibly after 4 and 12 weeks (based upon IBL instructors experiences of the recurrent progression of students affective responses in such courses).\n\n• Possible student questioning strategy: If you could choose how classroom time is spent in the mathematics course you are taking, what percentage of the time would you spend on: professor lecture, student presentations, group work on challenging problems, etc?\n\n• We assume many students will come into an IBL real analysis course with a baseline set of expectations informed by traditional mathematics instruction in which students attempt tasks after receiving relevant direct instruction, students are assessed on private work without publishing student work to other students, and the professor and textbook are the primary sources of new mathematics in the class. IBL is taken to entail the expectation that students attempt proof tasks for which they have not received direct instruction, students present their own work to the class for evaluation, and that students author new mathematical content into the classroom.\n\n• Relevant categories we anticipate may arise in the study:\n\n- Legitimizing failure (or linking success to hard work and repeated attempts)\n\n- Soliciting buy in to the alternative classroom expectations\n\n- Negotiating classroom expectations (of students and teacher)\n\n- Endorsing standards for acceptable proof\n\n- Curtailing undesirable mathematical practices\n\n• We anticipate observing IBL instructors of various levels of experience. Novice in-structors might provide more opportunity to identify breaches and difficulties that require intervention. Observing experienced IBL instructors will aide in identifying successful strategies, since documenting and publishing such strategies is an impor-tant research product.\n\nLearning through proof presentations. The overarching goal of this investigation is to under-stand what students learn in IBL courses when listening to peer proof presentations. Two relevant hypotheses were identified:\n\nHypothesis 1: Students learn more from proof presentations in an IBL course because they have already attempted the proof tasks before seeing the completed solution.\n\nHypothesis 2: Students learn more from proof presentations in an IBL course because they pay closer attention to find and identify mistakes, which may depend upon how student listening is guided by the instructor. We propose two possibly independent studies for these two hypotheses.\n\nResearch Questions:",
   "evidence": "The canonical record is source index 103 of **aim-infrastructure-notes.json**, extracted from the American Institute of Mathematics workshop summary *Research on inquiry based learning in undergraduate real analysis* (7--11 December 2015). The exact canonical record is preserved without alteration in **input.json**. Its recoverable research questions are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 103,
   "attempt": 1
  },
  "AIM-INFRASTRUCTURE-0105": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "(1) How does students having attempted a proof production influence their learning from a proof presentation? (2) How do students ongoing classroom experiences (in lecture or IBL classrooms) influ-ence their learning from proof production and observation of a proof presentation? 4\n\nAbstract: IBL instructors anticipate that students learn from their peers proof presenta-tions in part because they have already attempted the task themselves. To investigate this, we propose that two groups of students attempt to (a) produce a proof of a claim and (b) view a presentation of a proof of that claim, in alternate orders. The condition of proof production before presentation mimics the IBL listening environment. After both learning experiences, Mejia-Ramos et al.s proof comprehension instrument will be administered to assess various dimensions of student learning about the proof. We further propose that stu-dents currently enrolled in both IBL and lecture-style courses participate in the study. This will make the study sensitive to the possibility that students ongoing learning practice may determine their optimal learning conditions (rather than the learning conditions themselves). \n\n• Alternative possible conditions: proof presentation by an experienced instructor or proof presentation by a student (with or without mistakes). \n\n• The study design may need to attend to how students time listening to their peers proof presentations is structured by the instructor (e.g. by assigning roles, by pro-viding rubrics, by inviting peer-peer feedback). \n\n• We anticipate that the choice of mathematical topic, proof task, and that tasks relative difficulty will be crucial to the outcome and viability of the study. \n\nResearch Question: How does student learning from a proof presentation differ between presentations made by expert mathematics faculty and presentations made by other students? \n\nAbstract: Students are the primary authors of mathematical proofs in IBL courses, meaning that students will more often see imperfect proofs rather than valid proofs that exhibit standard mathematical form. Observing peer proofs may invite more active listenership since students are expected to question and give feedback on peer proofs. Observing expert proofs may improve learning because they serve as models of appropriate proof writing. We propose a study that compares students proof comprehension, as measured by Mejia-Ramos et al.s instrument, after viewing the two types of proof presentations. We anticipate that conducting the study with students from both IBL and lecture-style courses will benefit the study. It will be important to distinguish the comparative benefits of presentation conditions from the possibility that students simply learn how to listen effectively within their native learning environment. \n\n• This study was conceptualized with reference to similar studies in physics education and elsewhere where listening to peers produced greater gains in comprehension over a short time interval relative to some measures of learning. \n\n• We anticipate that the 2x2 design coupled with the multi-dimensional measure of proof comprehension will provide a rich set of possible outcomes leading to different inferences about the nature of student learning from listening. \n\nPersistence & identity. Exploratory study of student development. \n\nResearch Questions: \n\n• What are the student developmental categories in a Moore Method course? \n\n• How can we refine and explain these categories? \n\n• How can we explain and describe student advancement between developmental cat-egories? 5\n\nAbstract: Many Moore Method 1 instructors report seeing dramatic transformations ex-perienced by students when they make significant steps forward in their levels of mathe-matical achievement. Through qualitative observation of several Moore Method classrooms, we anticipate identifying and refining definitions of student developmental categories. We anticipate this study will develop baselines for subsequent studies to document transforma-tions to higher developmental categories in Moore Method classes, record them in detail, and investigate the conditions that foster such transformations. We propose a preliminary list of developmental categories as follows: \n\nBeginners:: Students who have not been able do any presentations successfully. \n\nNovices:: Students who successfully present a proof that is a follow your nose type proof (they know what a definition is and definitions are and the logic and format of a proof are and can put them together). \n\nApprentices:: Students who successfully present a proof that requires some significant insight or ingenuity to answer. \n\nMasters:: Students who initiate interest in mathematical problems that they want to do themselves. Note that these levels somewhat parallel Lee May's levels of performance, but differ in a practical way. Assessing a student for May's levels require a carefully designed assessment procedure in addition to the classwork itself. In contrast, instructors can classify students according to the levels proposed here based only on what they see from the student's class performance. \n\nData Sources: Ted Mahavier has video we might use for exploratory studies of his and his father's courses. Anneliese Spaeth and Padraig McLoughlin have volunteered to use their Analysis courses and possibly record them in Spring 2016. Ted Mahavier is on sabbatical during Spring 2016, and has volunteered some time. Ted Mahavier is teaching Real Analysis in Fall 2016, and has volunteered his course for participation in this study. We invite others to participate by contributing observations of these transformations in their own Moore Method classes. \n\nCase study of the impact of IBL on student development. \n\nResearch Questions: How does taking a Moore Method course affect student: \n\n• Confidence: Willingness to engage and belief in eventual success \n\n• Independence: Prove theorems and solve problems (and verify on their own) \n\n• Identity: Participation in mathematical culture \n\n• Willingness to try and fail (and see its worth) \n\n• Resilience: Willingness to try after failure \n\n• Perception of self worth/worth of their work \n\n• Locus of control \n\n• Perception of the nature of mathematics \n\n> 1By Moore Method, we mean an Inquiry Based Learning (IBL) course centered around individual student presentations. 6\n\nAbstract: The Colorado Study (2014) 2 established that students from IBL courses attain greater success in subsequent math courses than students from more traditional, lecture-based courses. In an effort to explain in more depth why this might be the case, we propose a study tracking any of several student attributes identified by mathematicians as important for student success in mathematics. Above we have begun a preliminary description of these attributes. We propose a series of case studies of students making transitions from one category (Beginner, Novice, Apprentice, Master) to another, including cases of failure to transition. \n\nBenefits of MMM over Lecture for Strong Students. \n\nResearch Questions:",
   "clean_statement": null,
   "public_statement": "(1) How does students having attempted a proof production influence their learning from a proof presentation? (2) How do students ongoing classroom experiences (in lecture or IBL classrooms) influ-ence their learning from proof production and observation of a proof presentation? 4\n\nAbstract: IBL instructors anticipate that students learn from their peers proof presenta-tions in part because they have already attempted the task themselves. To investigate this, we propose that two groups of students attempt to (a) produce a proof of a claim and (b) view a presentation of a proof of that claim, in alternate orders. The condition of proof production before presentation mimics the IBL listening environment. After both learning experiences, Mejia-Ramos et al.s proof comprehension instrument will be administered to assess various dimensions of student learning about the proof. We further propose that stu-dents currently enrolled in both IBL and lecture-style courses participate in the study. This will make the study sensitive to the possibility that students ongoing learning practice may determine their optimal learning conditions (rather than the learning conditions themselves).\n\n• Alternative possible conditions: proof presentation by an experienced instructor or proof presentation by a student (with or without mistakes).\n\n• The study design may need to attend to how students time listening to their peers proof presentations is structured by the instructor (e.g. by assigning roles, by pro-viding rubrics, by inviting peer-peer feedback).\n\n• We anticipate that the choice of mathematical topic, proof task, and that tasks relative difficulty will be crucial to the outcome and viability of the study.\n\nResearch Question: How does student learning from a proof presentation differ between presentations made by expert mathematics faculty and presentations made by other students?\n\nAbstract: Students are the primary authors of mathematical proofs in IBL courses, meaning that students will more often see imperfect proofs rather than valid proofs that exhibit standard mathematical form. Observing peer proofs may invite more active listenership since students are expected to question and give feedback on peer proofs. Observing expert proofs may improve learning because they serve as models of appropriate proof writing. We propose a study that compares students proof comprehension, as measured by Mejia-Ramos et al.s instrument, after viewing the two types of proof presentations. We anticipate that conducting the study with students from both IBL and lecture-style courses will benefit the study. It will be important to distinguish the comparative benefits of presentation conditions from the possibility that students simply learn how to listen effectively within their native learning environment.\n\n• This study was conceptualized with reference to similar studies in physics education and elsewhere where listening to peers produced greater gains in comprehension over a short time interval relative to some measures of learning.\n\n• We anticipate that the 2x2 design coupled with the multi-dimensional measure of proof comprehension will provide a rich set of possible outcomes leading to different inferences about the nature of student learning from listening.\n\nPersistence & identity. Exploratory study of student development.\n\nResearch Questions:\n\n• What are the student developmental categories in a Moore Method course?\n\n• How can we refine and explain these categories?\n\n• How can we explain and describe student advancement between developmental cat-egories? 5\n\nAbstract: Many Moore Method 1 instructors report seeing dramatic transformations ex-perienced by students when they make significant steps forward in their levels of mathe-matical achievement. Through qualitative observation of several Moore Method classrooms, we anticipate identifying and refining definitions of student developmental categories. We anticipate this study will develop baselines for subsequent studies to document transforma-tions to higher developmental categories in Moore Method classes, record them in detail, and investigate the conditions that foster such transformations. We propose a preliminary list of developmental categories as follows:\n\nBeginners:: Students who have not been able do any presentations successfully.\n\nNovices:: Students who successfully present a proof that is a follow your nose type proof (they know what a definition is and definitions are and the logic and format of a proof are and can put them together).\n\nApprentices:: Students who successfully present a proof that requires some significant insight or ingenuity to answer.\n\nMasters:: Students who initiate interest in mathematical problems that they want to do themselves. Note that these levels somewhat parallel Lee May's levels of performance, but differ in a practical way. Assessing a student for May's levels require a carefully designed assessment procedure in addition to the classwork itself. In contrast, instructors can classify students according to the levels proposed here based only on what they see from the student's class performance.\n\nData Sources: Ted Mahavier has video we might use for exploratory studies of his and his father's courses. Anneliese Spaeth and Padraig McLoughlin have volunteered to use their Analysis courses and possibly record them in Spring 2016. Ted Mahavier is on sabbatical during Spring 2016, and has volunteered some time. Ted Mahavier is teaching Real Analysis in Fall 2016, and has volunteered his course for participation in this study. We invite others to participate by contributing observations of these transformations in their own Moore Method classes.\n\nCase study of the impact of IBL on student development.\n\nResearch Questions: How does taking a Moore Method course affect student:\n\n• Confidence: Willingness to engage and belief in eventual success\n\n• Independence: Prove theorems and solve problems (and verify on their own)\n\n• Identity: Participation in mathematical culture\n\n• Willingness to try and fail (and see its worth)\n\n• Resilience: Willingness to try after failure\n\n• Perception of self worth/worth of their work\n\n• Locus of control\n\n• Perception of the nature of mathematics\n\n> 1By Moore Method, we mean an Inquiry Based Learning (IBL) course centered around individual student presentations. 6\n\nAbstract: The Colorado Study (2014) 2 established that students from IBL courses attain greater success in subsequent math courses than students from more traditional, lecture-based courses. In an effort to explain in more depth why this might be the case, we propose a study tracking any of several student attributes identified by mathematicians as important for student success in mathematics. Above we have begun a preliminary description of these attributes. We propose a series of case studies of students making transitions from one category (Beginner, Novice, Apprentice, Master) to another, including cases of failure to transition.\n\nBenefits of MMM over Lecture for Strong Students.\n\nResearch Questions:",
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   "original_statement": "(1) What are long term effects of MMM instruction upon strong students? (2) How do these effects compare with those of more traditional, lecture-based instruction (and possibly more general IBL)? (3) How can we further describe student development within the strong/high-achieving category? (see Tranformation Study) (4) In what ways do MMM courses support this development, specifically for strong students? \n\nAbstract: The following statement appears in the final report of the Colorado Study, Chapter 9: Summary of Findings 3\n\nOverall, it appeared that non-IBL courses tended to reinforce prior achieve-ment patterns, helping the \"rich\" to get \"richer.\" In contrast, IBL courses seemed to offer an extra boost to lower achieving students, especially among pre-service teachers. Yet there was no evidence of harm done to the strongest students. Indeed, high-achieving students may be encouraged by an IBL expe-rience to take more mathematics courses, especially more IBL courses (6.6.2) again, consistent with instructor observations that strong students found the IBL approach stimulating (8.2.5). The study did recognize and record different forms of IBL, primarily individual pre-sentation and group work. However, it went on to aggregate them under the label \"IBL,\" in order to provide enough data to draw statistically significant conclusions. We applaud the findings concerning lower achieving students 4. However, we also rec-ognize that the strongest students of today will be the leaders of tomorrow, and that it is important to assist them in achieving their full potential. Doing them \"no harm\" does not seem likely to achieve that. We would like to investigate the possible benefits of IBL to these students mentioned in the above quote from the Colorado Study. The idea of this study is that these benefits might be more pronounced and easier to establish if it is focused on the benefits of the MMM, rather than IBL in general. \n\n> 2Laursen, S. L., Hassi, M.-L., Kogan, M., & Weston, T. J. (2014). Benefits for women and men of inquiry-based learning in college mathematics: A Multi-institution study. Journal for Research in Mathematics Education, 45(4), 406418.\n> 3S. Laursen, M. Hassi, M. Kogan, A. Hunter, Evaluation of the IBL Mathematics Project, Assessment & Evaluation Center for IBL in Mathematics, Ethnography and Evaluation Research and Tim Weston ATLAS Assessment and Research Center, University of Colorado at Boulder (2011), p169.\n> 4Laursen, et al., divided students into Low achieving (Math GPA 2.5 or below), High achieving (Math GPA 2.5 or above) categories. 7\n\nThe purpose of this study is to determine if there is validity to the commonly held belief by MMM instructors that, among students previously identified as strongest, MMM students tend subsequently to fulfill more of their potential than those taking the same courses in a traditional, lecture-based setting. To that end, we wish to more carefully examine the benefits of Modified Moore Method courses to strong students. By \"strong\" students we mean those students that are either hard-working and diligent, or talented and smart, or both. Below we propose some possible proxy measures for these traits. Thus far we have put together two possible approaches: (1) We wish to compare the effect of MMM vs. lecture-based instruction upon strong students. (2) We wish to examine high-achieving students (after becoming informed by our previous transformation study), and accomplish a similar goal of describing developmental categories at a higher resolution for this group of students. In order to investigate this pair of research questions, it will first be necessary to determine the following: (1) How are strong incoming freshmen best identified? (Perhaps by high school grades? SAT grades? Other?) (2) How is realization of potential by college graduates who declared a mathematics major best measured? (Perhaps by completion of a mathematics degree after declaring the major? By taking mathematics courses or doing individualized work in mathematics beyond the requirements for the degree? By acceptance to graduate school? By individual testimony? Other?) There are many ways achievement of potential could be manifest. The study will rely on identifying ways that are practical to measure. Once satisfactory metrics have been identified and described, the study may be carried out in the following steps: (1) Institutions will be identified which have mathematics courses that are taught some-times by MMM and sometimes in a traditional, lecture-based format. (2) For those institutions, records will be gathered of recent graduates who had at some point declared a mathematics major. (3) From those records, students who qualify as \"strong\" will be selected. (4) From the selected students, two groups will be formed. One group will consist of those who had taken the most MMM courses, and the other will consist of those who had taken the most lecture-based courses. (5) Data on realization of potential will then be analyzed and used to compare MMM-students with lecture-based-students. The investigation of research questions 3 and 4 above will be informed by our planned \"Exploratory Study of Student Development Under the Modified Moore Method\". That study will describe student developmental categories and describe how students advance through these categories. We wish to build on that study by examining the highest-achieving 8\n\ncategory in greater resolution. That is, how can we explain and describe student development in a MMM course within the highest-achieving category? \n\nProblem sequences & learning trajectories. Intellectual cross-training. \n\nResearch Question: When interweaving content areas within a problem sequence, what are the impacts on students success, conceptual development, relationships between related concepts, their images of the roles of definitions and theorems, and equivalence of approaches? \n\nAbstract: IBL Real Analysis instructors often interweave problems on different concepts rather than treating each concept independently. We propose to study whether doing so increases the odds of success for individual students and, if so, why. Perhaps having prob-lems spanning multiple concepts on which to work each evening allows students to make connections between these concepts thereby increasing the students' understanding of each concept as well as the interplay between them. What is an optimal amount of interweaving? We illustrate the phenomena we wish to study via two examples. (1) An instructor might alternate, within a problem sequence, problems related to limit points and problems related to sequences prior to stating a problem that ties the two concepts together, such as every infinite bounded sequence has a limit point. (2) An instructor might have problems intertwined between continuity and differentia-bility, with the concluding problem being to prove that every differentiable function is continuous. \n\nStrategic Walls. \n\nResearch Question: How does an instructor create an adequate number of conjec-tures that will effectively demonstrate students need for mathematical need for proof in order to verify or invalidate their intuition? What are the features and timing of these conjectures for optimal effectiveness? \n\nAbstract: Pedagogical conjecturing activities are intended to both develop students intu-itions about the content in real analysis and reinforce students understanding of the need for proof or counterexample. Real analysis is a particularly appropriate site for this question since students start with significant intuitions (from calculus) that is not necessarily consis-tent with the foundations of analysis (completeness, monsters, etc.). Specific types of these activities may include \n\nTraps:: contradicts standard obvious intuitions \n\nHooks:: build new (perhaps surprising) intuitions, e.g., the cantor set \n\nGeneralizing:: asking whether a recently proven statement holds for a broader class of tasks What does implementing such strategies communicate to students about (1) correct proof? (2) their own responsibility for their learning? (3) the instructor's responsibility for their learning? (4) mathematical authority? (5) what it means to \"do math?\" (6) potentially, conceptual understanding? 9\n\n(7) workload? (8) the difficulty of the concepts? (9) the amount of coverage accomplished by the course? In an extreme version of this question, we seek to understand the differences in the devel-opment of students habits of mind when approaching mathematical tasks if the course is structured as statements, with many (1/4? 1/2?) are false. \n\nBeyond proof. \n\nResearch Question: What are the impacts of incorporating mathematical activity such as defining or conjecturing at different points or with different frequency in an IBL course? \n\nAbstract: The classic Moore Method course involves the distribution of a sequence of def-initions and problems students are charged with solving. Other IBL strategies may ask students to generate the definitions and problems themselves. The choice of strategies on this continuum may depend on the instructors goal for the course. For example, requiring students to participate in the defining and conjecturing tasks may result in a different con-ception of the nature of mathematical inquiry and the culture of mathematics. On the other hand, it may be the case that requiring students to generate every definition in a course may discourage students or require an undesirable amount of time. We wish to study the optimal frequency, locations, and depth of these experiences, both defining and conjecturing. Many effects on student perception would be of interest, including perception of: student and in-structor responsibility for learning, mathematical authority, what it means to do math, their understanding, course difficulty, course coverage. \n\nDesigning in the Zone of Proximal Development. \n\nResearch Question: When teaching from a set of notes, how does an experienced IBL instructor identify when a problem sequence needs to be reconstructed to support the particular learning needs of their students, how do they accomplish it, and how do they assess its effectiveness? \n\nAbstract: Course notes are more static than actual problem sequences implemented by IBL instructors. Effective instructors often modify problems in response to the ways that particular classes develop. For example a particular problem may be too difficult as stated and need to be broken into sub-steps to allow student success. Alternately, an instructor may realize that some of their students may be better served by using an alternate definition to a particular term, thus changing the trajectory of problems (e.g., the old definition now becomes a theorem). We seek to understand what aspects of students understanding an instructor must attend to in order effectively identify the need for these modifications and to understand the design principles that allow them to successfully modify the sequence. We also seek to understand any aspects of students or topics that may affect this process such as differences in when working with content for which students have a rich concept image vs. when a concept image is relatively absent. In addition to studying how this process 10 \n\nunfolds during a course, we may also explore how the standard versions of course notes may be modified outside of a particular implementation. \n\nProof. How does an IBL class impact students' understanding of proof?. \n\nResearch Questions: In what ways does an IBL class 5:(1) impact students' proof schemes? (2) affect students' understanding of the contextual meanings in mathematics of the words \"axioms, definitions, conjecture, theorem\" and their role in mathematics? (3) help students develop competence at proof? What features of IBL create these impacts? \n\nAbstract: Harel & Sowder have proposed a framework to classify and examine students' proof schemes. They have looked mostly at students in traditional courses, or as part of teaching experiments. We conjecture that in those implementations of IBL that provide experiences to students in which they must prove propositions on their own, students' proof schemes will improve, as will their understanding of proof, and their competence at writing proofs. We propose a qualitative study (a case study possibly) of a Real Analysis IBL course. Data will include documentation of the IBL implementation (syllabus, problem sequence, classroom observations, instructor's interview), evidence of students' procedural and concep-tual understanding of proof (artifacts such as proofs worked in tests or homework assign-ments, clinical interviews, preand post-surveys). Things to think about: \n\n• If the study is done in two sites (e.g., Texas and India), how to ensure that classrooms observations are reliable and consistent? (Ideas: video, Skype interviews.) \n\n• Why Real Analysis? It could be done in other courses, but RA is a good one: students have some experience with proofs, they are more mature mathematically, course content is challenging enough. \n\n• Data should include students' demographic information, previous IBL experience, academic perfomance prior to the course. \n\n• Teasing out what features of IBL cause change may be more difficult than gathering evidence of such change. \n\n• We need to consider follow up evidence of the robustness of change or impact. \n\n• Data collection instruments (interview protocols, proof tasks, etc.) will be piloted both in IBL and non-IBL courses, as possible. \n\n• A more comprehensive literature review should be conducted to inform data collection and data analysis strategies, as well as useful frameworks to use or to adapt for this study.",
   "clean_statement": "(1) What are long term effects of MMM instruction upon strong students? (2) How do these effects compare with those of more traditional, lecture-based instruction (and possibly more general IBL)? (3) How can we further describe student development within the strong/high-achieving category? (see Tranformation Study) (4) In what ways do MMM courses support this development, specifically for strong students?\n\nAbstract: The following statement appears in the final report of the Colorado Study, Chapter 9: Summary of Findings 3\n\nOverall, it appeared that non-IBL courses tended to reinforce prior achieve-ment patterns, helping the \"rich\" to get \"richer.\" In contrast, IBL courses seemed to offer an extra boost to lower achieving students, especially among pre-service teachers. Yet there was no evidence of harm done to the strongest students. Indeed, high-achieving students may be encouraged by an IBL expe-rience to take more mathematics courses, especially more IBL courses (6.6.2) again, consistent with instructor observations that strong students found the IBL approach stimulating (8.2.5). The study did recognize and record different forms of IBL, primarily individual pre-sentation and group work. However, it went on to aggregate them under the label \"IBL,\" in order to provide enough data to draw statistically significant conclusions. We applaud the findings concerning lower achieving students 4. However, we also rec-ognize that the strongest students of today will be the leaders of tomorrow, and that it is important to assist them in achieving their full potential. Doing them \"no harm\" does not seem likely to achieve that. We would like to investigate the possible benefits of IBL to these students mentioned in the above quote from the Colorado Study. The idea of this study is that these benefits might be more pronounced and easier to establish if it is focused on the benefits of the MMM, rather than IBL in general.\n2Laursen, S. L., Hassi, M.-L., Kogan, M., & Weston, T. J. (2014). Benefits for women and men of inquiry-based learning in college mathematics: A Multi-institution study. Journal for Research in Mathematics Education, 45(4), 406418.\n3S. Laursen, M. Hassi, M. Kogan, A. Hunter, Evaluation of the IBL Mathematics Project, Assessment & Evaluation Center for IBL in Mathematics, Ethnography and Evaluation Research and Tim Weston ATLAS Assessment and Research Center, University of Colorado at Boulder (2011), p169.\n4Laursen, et al., divided students into Low achieving (Math GPA 2.5 or below), High achieving (Math GPA 2.5 or above) categories. 7\n\nThe purpose of this study is to determine if there is validity to the commonly held belief by MMM instructors that, among students previously identified as strongest, MMM students tend subsequently to fulfill more of their potential than those taking the same courses in a traditional, lecture-based setting. To that end, we wish to more carefully examine the benefits of Modified Moore Method courses to strong students. By \"strong\" students we mean those students that are either hard-working and diligent, or talented and smart, or both. Below we propose some possible proxy measures for these traits. Thus far we have put together two possible approaches: (1) We wish to compare the effect of MMM vs. lecture-based instruction upon strong students. (2) We wish to examine high-achieving students (after becoming informed by our previous transformation study), and accomplish a similar goal of describing developmental categories at a higher resolution for this group of students. In order to investigate this pair of research questions, it will first be necessary to determine the following: (1) How are strong incoming freshmen best identified? (Perhaps by high school grades? SAT grades? Other?) (2) How is realization of potential by college graduates who declared a mathematics major best measured? (Perhaps by completion of a mathematics degree after declaring the major? By taking mathematics courses or doing individualized work in mathematics beyond the requirements for the degree? By acceptance to graduate school? By individual testimony? Other?) There are many ways achievement of potential could be manifest. The study will rely on identifying ways that are practical to measure. Once satisfactory metrics have been identified and described, the study may be carried out in the following steps: (1) Institutions will be identified which have mathematics courses that are taught some-times by MMM and sometimes in a traditional, lecture-based format. (2) For those institutions, records will be gathered of recent graduates who had at some point declared a mathematics major. (3) From those records, students who qualify as \"strong\" will be selected. (4) From the selected students, two groups will be formed. One group will consist of those who had taken the most MMM courses, and the other will consist of those who had taken the most lecture-based courses. (5) Data on realization of potential will then be analyzed and used to compare MMM-students with lecture-based-students. The investigation of research questions 3 and 4 above will be informed by our planned \"Exploratory Study of Student Development Under the Modified Moore Method\". That study will describe student developmental categories and describe how students advance through these categories. We wish to build on that study by examining the highest-achieving 8\n\ncategory in greater resolution. That is, how can we explain and describe student development in a MMM course within the highest-achieving category?\n\nProblem sequences & learning trajectories. Intellectual cross-training.\n\nResearch Question: When interweaving content areas within a problem sequence, what are the impacts on students success, conceptual development, relationships between related concepts, their images of the roles of definitions and theorems, and equivalence of approaches?\n\nAbstract: IBL Real Analysis instructors often interweave problems on different concepts rather than treating each concept independently. We propose to study whether doing so increases the odds of success for individual students and, if so, why. Perhaps having prob-lems spanning multiple concepts on which to work each evening allows students to make connections between these concepts thereby increasing the students' understanding of each concept as well as the interplay between them. What is an optimal amount of interweaving? We illustrate the phenomena we wish to study via two examples. (1) An instructor might alternate, within a problem sequence, problems related to limit points and problems related to sequences prior to stating a problem that ties the two concepts together, such as every infinite bounded sequence has a limit point. (2) An instructor might have problems intertwined between continuity and differentia-bility, with the concluding problem being to prove that every differentiable function is continuous.\n\nStrategic Walls.\n\nResearch Question: How does an instructor create an adequate number of conjec-tures that will effectively demonstrate students need for mathematical need for proof in order to verify or invalidate their intuition? What are the features and timing of these conjectures for optimal effectiveness?\n\nAbstract: Pedagogical conjecturing activities are intended to both develop students intu-itions about the content in real analysis and reinforce students understanding of the need for proof or counterexample. Real analysis is a particularly appropriate site for this question since students start with significant intuitions (from calculus) that is not necessarily consis-tent with the foundations of analysis (completeness, monsters, etc.). Specific types of these activities may include\n\nTraps:: contradicts standard obvious intuitions\n\nHooks:: build new (perhaps surprising) intuitions, e.g., the cantor set\n\nGeneralizing:: asking whether a recently proven statement holds for a broader class of tasks What does implementing such strategies communicate to students about (1) correct proof? (2) their own responsibility for their learning? (3) the instructor's responsibility for their learning? (4) mathematical authority? (5) what it means to \"proof schemes? (2) affect students\" (6) potentially, conceptual understanding? 9\n\n(7) workload? (8) the difficulty of the concepts? (9) the amount of coverage accomplished by the course? In an extreme version of this question, we seek to understand the differences in the devel-opment of students habits of mind when approaching mathematical tasks if the course is structured as statements, with many (1/4? 1/2?) are false.\n\nBeyond proof.\n\nResearch Question: What are the impacts of incorporating mathematical activity such as defining or conjecturing at different points or with different frequency in an IBL course?\n\nAbstract: The classic Moore Method course involves the distribution of a sequence of def-initions and problems students are charged with solving. Other IBL strategies may ask students to generate the definitions and problems themselves. The choice of strategies on this continuum may depend on the instructors goal for the course. For example, requiring students to participate in the defining and conjecturing tasks may result in a different con-ception of the nature of mathematical inquiry and the culture of mathematics. On the other hand, it may be the case that requiring students to generate every definition in a course may discourage students or require an undesirable amount of time. We wish to study the optimal frequency, locations, and depth of these experiences, both defining and conjecturing. Many effects on student perception would be of interest, including perception of: student and in-structor responsibility for learning, mathematical authority, what it means to do math, their understanding, course difficulty, course coverage.\n\nDesigning in the Zone of Proximal Development.\n\nResearch Question: When teaching from a set of notes, how does an experienced IBL instructor identify when a problem sequence needs to be reconstructed to support the particular learning needs of their students, how do they accomplish it, and how do they assess its effectiveness?\n\nAbstract: Course notes are more static than actual problem sequences implemented by IBL instructors. Effective instructors often modify problems in response to the ways that particular classes develop. For example a particular problem may be too difficult as stated and need to be broken into sub-steps to allow student success. Alternately, an instructor may realize that some of their students may be better served by using an alternate definition to a particular term, thus changing the trajectory of problems (e.g., the old definition now becomes a theorem). We seek to understand what aspects of students understanding an instructor must attend to in order effectively identify the need for these modifications and to understand the design principles that allow them to successfully modify the sequence. We also seek to understand any aspects of students or topics that may affect this process such as differences in when working with content for which students have a rich concept image vs. when a concept image is relatively absent. In addition to studying how this process 10\n\nunfolds during a course, we may also explore how the standard versions of course notes may be modified outside of a particular implementation.\n\nProof. How does an IBL class impact students' understanding of proof?.\n\nResearch Questions: In what ways does an IBL class 5:(1) impact students' proof schemes? (2) affect students' understanding of the contextual meanings in mathematics of the words \"axioms, definitions, conjecture, theorem\" and their role in mathematics? (3) help students develop competence at proof? What features of IBL create these impacts?\n\nAbstract: Harel & Sowder have proposed a framework to classify and examine students' proof schemes. They have looked mostly at students in traditional courses, or as part of teaching experiments. We conjecture that in those implementations of IBL that provide experiences to students in which they must prove propositions on their own, students' proof schemes will improve, as will their understanding of proof, and their competence at writing proofs. We propose a qualitative study (a case study possibly) of a Real Analysis IBL course. Data will include documentation of the IBL implementation (syllabus, problem sequence, classroom observations, instructor's interview), evidence of students' procedural and concep-tual understanding of proof (artifacts such as proofs worked in tests or homework assign-ments, clinical interviews, preand post-surveys). Things to think about:\n\n• If the study is done in two sites (e.g., Texas and India), how to ensure that classrooms observations are reliable and consistent? (Ideas: video, Skype interviews.)\n\n• Why Real Analysis? It could be done in other courses, but RA is a good one: students have some experience with proofs, they are more mature mathematically, course content is challenging enough.\n\n• Data should include students' demographic information, previous IBL experience, academic perfomance prior to the course.\n\n• Teasing out what features of IBL cause change may be more difficult than gathering evidence of such change.\n\n• We need to consider follow up evidence of the robustness of change or impact.\n\n• Data collection instruments (interview protocols, proof tasks, etc.) will be piloted both in IBL and non-IBL courses, as possible.\n\n• A more comprehensive literature review should be conducted to inform data collection and data analysis strategies, as well as useful frameworks to use or to adapt for this study.",
   "public_statement": "(1) What are long term effects of MMM instruction upon strong students? (2) How do these effects compare with those of more traditional, lecture-based instruction (and possibly more general IBL)? (3) How can we further describe student development within the strong/high-achieving category? (see Tranformation Study) (4) In what ways do MMM courses support this development, specifically for strong students?\n\nAbstract: The following statement appears in the final report of the Colorado Study, Chapter 9: Summary of Findings 3\n\nOverall, it appeared that non-IBL courses tended to reinforce prior achieve-ment patterns, helping the \"rich\" to get \"richer.\" In contrast, IBL courses seemed to offer an extra boost to lower achieving students, especially among pre-service teachers. Yet there was no evidence of harm done to the strongest students. Indeed, high-achieving students may be encouraged by an IBL expe-rience to take more mathematics courses, especially more IBL courses (6.6.2) again, consistent with instructor observations that strong students found the IBL approach stimulating (8.2.5). The study did recognize and record different forms of IBL, primarily individual pre-sentation and group work. However, it went on to aggregate them under the label \"IBL,\" in order to provide enough data to draw statistically significant conclusions. We applaud the findings concerning lower achieving students 4. However, we also rec-ognize that the strongest students of today will be the leaders of tomorrow, and that it is important to assist them in achieving their full potential. Doing them \"no harm\" does not seem likely to achieve that. We would like to investigate the possible benefits of IBL to these students mentioned in the above quote from the Colorado Study. The idea of this study is that these benefits might be more pronounced and easier to establish if it is focused on the benefits of the MMM, rather than IBL in general.\n\n> 2Laursen, S. L., Hassi, M.-L., Kogan, M., & Weston, T. J. (2014). Benefits for women and men of inquiry-based learning in college mathematics: A Multi-institution study. Journal for Research in Mathematics Education, 45(4), 406418.\n> 3S. Laursen, M. Hassi, M. Kogan, A. Hunter, Evaluation of the IBL Mathematics Project, Assessment & Evaluation Center for IBL in Mathematics, Ethnography and Evaluation Research and Tim Weston ATLAS Assessment and Research Center, University of Colorado at Boulder (2011), p169.\n> 4Laursen, et al., divided students into Low achieving (Math GPA 2.5 or below), High achieving (Math GPA 2.5 or above) categories. 7\n\nThe purpose of this study is to determine if there is validity to the commonly held belief by MMM instructors that, among students previously identified as strongest, MMM students tend subsequently to fulfill more of their potential than those taking the same courses in a traditional, lecture-based setting. To that end, we wish to more carefully examine the benefits of Modified Moore Method courses to strong students. By \"strong\" students we mean those students that are either hard-working and diligent, or talented and smart, or both. Below we propose some possible proxy measures for these traits. Thus far we have put together two possible approaches: (1) We wish to compare the effect of MMM vs. lecture-based instruction upon strong students. (2) We wish to examine high-achieving students (after becoming informed by our previous transformation study), and accomplish a similar goal of describing developmental categories at a higher resolution for this group of students. In order to investigate this pair of research questions, it will first be necessary to determine the following: (1) How are strong incoming freshmen best identified? (Perhaps by high school grades? SAT grades? Other?) (2) How is realization of potential by college graduates who declared a mathematics major best measured? (Perhaps by completion of a mathematics degree after declaring the major? By taking mathematics courses or doing individualized work in mathematics beyond the requirements for the degree? By acceptance to graduate school? By individual testimony? Other?) There are many ways achievement of potential could be manifest. The study will rely on identifying ways that are practical to measure. Once satisfactory metrics have been identified and described, the study may be carried out in the following steps: (1) Institutions will be identified which have mathematics courses that are taught some-times by MMM and sometimes in a traditional, lecture-based format. (2) For those institutions, records will be gathered of recent graduates who had at some point declared a mathematics major. (3) From those records, students who qualify as \"strong\" will be selected. (4) From the selected students, two groups will be formed. One group will consist of those who had taken the most MMM courses, and the other will consist of those who had taken the most lecture-based courses. (5) Data on realization of potential will then be analyzed and used to compare MMM-students with lecture-based-students. The investigation of research questions 3 and 4 above will be informed by our planned \"Exploratory Study of Student Development Under the Modified Moore Method\". That study will describe student developmental categories and describe how students advance through these categories. We wish to build on that study by examining the highest-achieving 8\n\ncategory in greater resolution. That is, how can we explain and describe student development in a MMM course within the highest-achieving category?\n\nProblem sequences & learning trajectories. Intellectual cross-training.\n\nResearch Question: When interweaving content areas within a problem sequence, what are the impacts on students success, conceptual development, relationships between related concepts, their images of the roles of definitions and theorems, and equivalence of approaches?\n\nAbstract: IBL Real Analysis instructors often interweave problems on different concepts rather than treating each concept independently. We propose to study whether doing so increases the odds of success for individual students and, if so, why. Perhaps having prob-lems spanning multiple concepts on which to work each evening allows students to make connections between these concepts thereby increasing the students' understanding of each concept as well as the interplay between them. What is an optimal amount of interweaving? We illustrate the phenomena we wish to study via two examples. (1) An instructor might alternate, within a problem sequence, problems related to limit points and problems related to sequences prior to stating a problem that ties the two concepts together, such as every infinite bounded sequence has a limit point. (2) An instructor might have problems intertwined between continuity and differentia-bility, with the concluding problem being to prove that every differentiable function is continuous.\n\nStrategic Walls.\n\nResearch Question: How does an instructor create an adequate number of conjec-tures that will effectively demonstrate students need for mathematical need for proof in order to verify or invalidate their intuition? What are the features and timing of these conjectures for optimal effectiveness?\n\nAbstract: Pedagogical conjecturing activities are intended to both develop students intu-itions about the content in real analysis and reinforce students understanding of the need for proof or counterexample. Real analysis is a particularly appropriate site for this question since students start with significant intuitions (from calculus) that is not necessarily consis-tent with the foundations of analysis (completeness, monsters, etc.). Specific types of these activities may include\n\nTraps:: contradicts standard obvious intuitions\n\nHooks:: build new (perhaps surprising) intuitions, e.g., the cantor set\n\nGeneralizing:: asking whether a recently proven statement holds for a broader class of tasks What does implementing such strategies communicate to students about (1) correct proof? (2) their own responsibility for their learning? (3) the instructor's responsibility for their learning? (4) mathematical authority? (5) what it means to \"do math?\" (6) potentially, conceptual understanding? 9\n\n(7) workload? (8) the difficulty of the concepts? (9) the amount of coverage accomplished by the course? In an extreme version of this question, we seek to understand the differences in the devel-opment of students habits of mind when approaching mathematical tasks if the course is structured as statements, with many (1/4? 1/2?) are false.\n\nBeyond proof.\n\nResearch Question: What are the impacts of incorporating mathematical activity such as defining or conjecturing at different points or with different frequency in an IBL course?\n\nAbstract: The classic Moore Method course involves the distribution of a sequence of def-initions and problems students are charged with solving. Other IBL strategies may ask students to generate the definitions and problems themselves. The choice of strategies on this continuum may depend on the instructors goal for the course. For example, requiring students to participate in the defining and conjecturing tasks may result in a different con-ception of the nature of mathematical inquiry and the culture of mathematics. On the other hand, it may be the case that requiring students to generate every definition in a course may discourage students or require an undesirable amount of time. We wish to study the optimal frequency, locations, and depth of these experiences, both defining and conjecturing. Many effects on student perception would be of interest, including perception of: student and in-structor responsibility for learning, mathematical authority, what it means to do math, their understanding, course difficulty, course coverage.\n\nDesigning in the Zone of Proximal Development.\n\nResearch Question: When teaching from a set of notes, how does an experienced IBL instructor identify when a problem sequence needs to be reconstructed to support the particular learning needs of their students, how do they accomplish it, and how do they assess its effectiveness?\n\nAbstract: Course notes are more static than actual problem sequences implemented by IBL instructors. Effective instructors often modify problems in response to the ways that particular classes develop. For example a particular problem may be too difficult as stated and need to be broken into sub-steps to allow student success. Alternately, an instructor may realize that some of their students may be better served by using an alternate definition to a particular term, thus changing the trajectory of problems (e.g., the old definition now becomes a theorem). We seek to understand what aspects of students understanding an instructor must attend to in order effectively identify the need for these modifications and to understand the design principles that allow them to successfully modify the sequence. We also seek to understand any aspects of students or topics that may affect this process such as differences in when working with content for which students have a rich concept image vs. when a concept image is relatively absent. In addition to studying how this process 10\n\nunfolds during a course, we may also explore how the standard versions of course notes may be modified outside of a particular implementation.\n\nProof. How does an IBL class impact students' understanding of proof?.\n\nResearch Questions: In what ways does an IBL class 5:(1) impact students' proof schemes? (2) affect students' understanding of the contextual meanings in mathematics of the words \"axioms, definitions, conjecture, theorem\" and their role in mathematics? (3) help students develop competence at proof? What features of IBL create these impacts?\n\nAbstract: Harel & Sowder have proposed a framework to classify and examine students' proof schemes. They have looked mostly at students in traditional courses, or as part of teaching experiments. We conjecture that in those implementations of IBL that provide experiences to students in which they must prove propositions on their own, students' proof schemes will improve, as will their understanding of proof, and their competence at writing proofs. We propose a qualitative study (a case study possibly) of a Real Analysis IBL course. Data will include documentation of the IBL implementation (syllabus, problem sequence, classroom observations, instructor's interview), evidence of students' procedural and concep-tual understanding of proof (artifacts such as proofs worked in tests or homework assign-ments, clinical interviews, preand post-surveys). Things to think about:\n\n• If the study is done in two sites (e.g., Texas and India), how to ensure that classrooms observations are reliable and consistent? (Ideas: video, Skype interviews.)\n\n• Why Real Analysis? It could be done in other courses, but RA is a good one: students have some experience with proofs, they are more mature mathematically, course content is challenging enough.\n\n• Data should include students' demographic information, previous IBL experience, academic perfomance prior to the course.\n\n• Teasing out what features of IBL cause change may be more difficult than gathering evidence of such change.\n\n• We need to consider follow up evidence of the robustness of change or impact.\n\n• Data collection instruments (interview protocols, proof tasks, etc.) will be piloted both in IBL and non-IBL courses, as possible.\n\n• A more comprehensive literature review should be conducted to inform data collection and data analysis strategies, as well as useful frameworks to use or to adapt for this study.",
   "evidence": "The canonical record is source index 105 of `aim-infrastructure-notes.json`, extracted from the AIM workshop report *Research on inquiry based learning in undergraduate real analysis*. The exact canonical `problem` field is preserved below, including source spelling, line-break hyphenation, page numbers, footnote markers, and the material that was accidentally merged into the record.",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-infrastructure-notes.json",
   "source_index": 105,
   "attempt": 1
  },
  "AIM-LINEAR_ALGEBRA-0001": {
   "statement_status": "exact",
   "original_statement": "Second Eigenvalue Multiplicity\n\nLet $G$ be a graph with maximum degree bounded by $\\Delta$. For the following choices of Hermitian matrix $M_G$ associated with $G$, what is the maximum multiplicity of the second largest eigenvalue of $M_G$?\nFor the following choices of Hermitian matrix $M_G$ associated with $G$, what is the maximum multiplicity of the second largest eigenvalue of $M_G$?\n \\begin{enumerate}\n \\item The adjacency matrix: in this case a sublinear upper bound was proved by \\cite{JTYZZ21}, with an improvement in the case of regular graphs provided by \\cite{MRS21} and also for the normalized adjacency matrix of general bounded degree graphs. Constructions of graphs establishing lower bounds were given by \\cite{HSZZ21}.\n \\item The Laplacian matrix.\n \\item Schr\\\"{o}dinger operators: any matrix of the form $D+A_G$ where $A_G$ is the adjacency matrix and $D$ is an arbitrary diagonal matrix.\n \\item Weighted adjacency matrices: any Hermitian matrix $M_G$ such that $M_G[i,j] = 0$ if $\\{i,j\\}\\notin E(G)$.\n\\end{enumerate}",
   "clean_statement": "Second Eigenvalue Multiplicity\n\nLet $G$ be a graph with maximum degree bounded by $\\Delta$. For the following choices of Hermitian matrix $M_G$ associated with $G$, what is the maximum multiplicity of the second largest eigenvalue of $M_G$?\nFor the following choices of Hermitian matrix $M_G$ associated with $G$, what is the maximum multiplicity of the second largest eigenvalue of $M_G$?\n \\begin{enumerate}\n \\item The adjacency matrix: in this case a sublinear upper bound was proved by \\cite{JTYZZ21}, with an improvement in the case of regular graphs provided by \\cite{MRS21} and also for the normalized adjacency matrix of general bounded degree graphs. Constructions of graphs establishing lower bounds were given by \\cite{HSZZ21}.\n \\item The Laplacian matrix.\n \\item Schr\\\"{o}dinger operators: any matrix of the form $D+A_G$ where $A_G$ is the adjacency matrix and $D$ is an arbitrary diagonal matrix.\n \\item Weighted adjacency matrices: any Hermitian matrix $M_G$ such that $M_G[i,j] = 0$ if $\\{i,j\\}\\notin E(G)$.\n\\end{enumerate}",
   "public_statement": "Second Eigenvalue Multiplicity\n\nLet $G$ be a graph with maximum degree bounded by $\\Delta$. For the following choices of Hermitian matrix $M_G$ associated with $G$, what is the maximum multiplicity of the second largest eigenvalue of $M_G$?\nFor the following choices of Hermitian matrix $M_G$ associated with $G$, what is the maximum multiplicity of the second largest eigenvalue of $M_G$?\n \\begin{enumerate}\n \\item The adjacency matrix: in this case a sublinear upper bound was proved by \\cite{JTYZZ21}, with an improvement in the case of regular graphs provided by \\cite{MRS21} and also for the normalized adjacency matrix of general bounded degree graphs. Constructions of graphs establishing lower bounds were given by \\cite{HSZZ21}.\n \\item The Laplacian matrix.\n \\item Schr\\\"{o}dinger operators: any matrix of the form $D+A_G$ where $A_G$ is the adjacency matrix and $D$ is an arbitrary diagonal matrix.\n \\item Weighted adjacency matrices: any Hermitian matrix $M_G$ such that $M_G[i,j] = 0$ if $\\{i,j\\}\\notin E(G)$.\n\\end{enumerate}",
   "evidence": "There are two further ambiguities in the source itself, not OCR errors:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-LINEAR_ALGEBRA-0002": {
   "statement_status": "exact",
   "original_statement": "Nonbacktracking Matrix\n\nBasic questions about the eigenvalues and eigenvectors of the nonbacktracking matrix of irregular graphs remain open.\n\nLet $G$ be any graph on $n$ vertices and $m$ edges, and let $B_G$ denote the \\emph{nonbacktracking matrix} of $G$, which is a $2m\\times 2m$ matrix whose rows and columns are indexed by directed edges of $G$, defined as follows:\n\\[\n B_G[uv, xy] =\n \\begin{cases}\n 1 & \\text{if $v = x$, $u\\ne y$} \\\\\n 0 &\\text{otherwise}.\n \\end{cases}\n\\]\n\\begin{enumerate}\n \\item It was proved by \\cite{LP16} that if $G$ is a regular graph, then every Jordan block of $B_G$ has size at most $2$. Prove upper and lower bounds on the function:\n \\[\n f(n) := \\max_{\\substack{G~\\text{$n$-vertex graph} \\\\ \\text{with no leaves}}} \\max_{S~\\text{Jordan block of $B_G$}} |S|.\n \\]\n Is $f(n)$ bounded by a constant or is it growing with $n$? If $f(n)$ is indeed growing with $n$, as a weak step towards proving a bound, can we show that it is sublinear in $n$?\n \\item Prove an Alon--Boppana bound for $B_G$. Concretely, show that:\n \\[\n |\\lambda|_2(B_G) \\ge \\sqrt{\\rho(B_G)} - o_n(1)\n \\]\n where $\\rho(B_G)$ is the spectral radius of $B_G$.\n\\end{enumerate}",
   "clean_statement": "Nonbacktracking Matrix\n\nBasic questions about the eigenvalues and eigenvectors of the nonbacktracking matrix of irregular graphs remain open.\n\nLet $G$ be any graph on $n$ vertices and $m$ edges, and let $B_G$ denote the \\emph{nonbacktracking matrix} of $G$, which is a $2m\\times 2m$ matrix whose rows and columns are indexed by directed edges of $G$, defined as follows:\n\\[\n B_G[uv, xy] =\n \\begin{cases}\n 1 & \\text{if $v = x$, $u\\ne y$} \\\\\n 0 &\\text{otherwise}.\n \\end{cases}\n\\]\n\\begin{enumerate}\n \\item It was proved by \\cite{LP16} that if $G$ is a regular graph, then every Jordan block of $B_G$ has size at most $2$. Prove upper and lower bounds on the function:\n \\[\n f(n) := \\max_{\\substack{G~\\text{$n$-vertex graph} \\\\ \\text{with no leaves}}} \\max_{S~\\text{Jordan block of $B_G$}} |S|.\n \\]\n Is $f(n)$ bounded by a constant or is it growing with $n$? If $f(n)$ is indeed growing with $n$, as a weak step towards proving a bound, can we show that it is sublinear in $n$?\n \\item Prove an Alon--Boppana bound for $B_G$. Concretely, show that:\n \\[\n |\\lambda|_2(B_G) \\ge \\sqrt{\\rho(B_G)} - o_n(1)\n \\]\n where $\\rho(B_G)$ is the spectral radius of $B_G$.\n\\end{enumerate}",
   "public_statement": "Nonbacktracking Matrix\n\nBasic questions about the eigenvalues and eigenvectors of the nonbacktracking matrix of irregular graphs remain open.\n\nLet $G$ be any graph on $n$ vertices and $m$ edges, and let $B_G$ denote the \\emph{nonbacktracking matrix} of $G$, which is a $2m\\times 2m$ matrix whose rows and columns are indexed by directed edges of $G$, defined as follows:\n\\[\n B_G[uv, xy] =\n \\begin{cases}\n 1 & \\text{if $v = x$, $u\\ne y$} \\\\\n 0 &\\text{otherwise}.\n \\end{cases}\n\\]\n\\begin{enumerate}\n \\item It was proved by \\cite{LP16} that if $G$ is a regular graph, then every Jordan block of $B_G$ has size at most $2$. Prove upper and lower bounds on the function:\n \\[\n f(n) := \\max_{\\substack{G~\\text{$n$-vertex graph} \\\\ \\text{with no leaves}}} \\max_{S~\\text{Jordan block of $B_G$}} |S|.\n \\]\n Is $f(n)$ bounded by a constant or is it growing with $n$? If $f(n)$ is indeed growing with $n$, as a weak step towards proving a bound, can we show that it is sublinear in $n$?\n \\item Prove an Alon--Boppana bound for $B_G$. Concretely, show that:\n \\[\n |\\lambda|_2(B_G) \\ge \\sqrt{\\rho(B_G)} - o_n(1)\n \\]\n where $\\rho(B_G)$ is the spectral radius of $B_G$.\n\\end{enumerate}",
   "evidence": "The canonical record is aim-linear-algebra-notes.json, zero-based index 1. The live AIM page was inspected on August 10, 2026. Its problem body is identical to the canonical record and has no status text or approved remarks. The exact statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-LINEAR_ALGEBRA-0003": {
   "statement_status": "exact",
   "original_statement": "For a fixed choice of parameters $\\rho$ and $\\Delta$, what is the largest possible multiplicity of the second eigenvalue of the adjacency matrix $A_G$ with the constraint that $\\lambda_2(A_G) = \\rho$.",
   "clean_statement": "For a fixed choice of parameters $\\rho$ and $\\Delta$, what is the largest possible multiplicity of the second eigenvalue of the adjacency matrix $A_G$ with the constraint that $\\lambda_2(A_G) = \\rho$.",
   "public_statement": "For a fixed choice of parameters $\\rho$ and $\\Delta$, what is the largest possible multiplicity of the second eigenvalue of the adjacency matrix $A_G$ with the constraint that $\\lambda_2(A_G) = \\rho$.",
   "evidence": "The exact canonical problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-LINEAR_ALGEBRA-0004": {
   "statement_status": "exact",
   "original_statement": "Positive and negative $\\ell_2$-mass on spectrum\n\nLet $G$ be a connected graph and let $\\lambda_1\\ge\\dots\\ge\\lambda_s\\ge 0>\\lambda_{s+1}\\ge\\dots\\ge\\lambda_n$ be the eigenvalues of the adjacency matrix of $G$. Defining:\n\\begin{align*}\n S^+(G) &:=\\sum_{i=1}^s \\lambda_i^2\\\\\n S^-(G) &:= \\sum_{i=s+1}^n \\lambda_i^2.\n\\end{align*}\nSome conjectures and questions of interest are the following.\n\\begin{enumerate}\n \\item Prove that $\\min\\{S^+(G), S^-(G)\\} \\ge n-1$.\n \\item Denoting $E(\\overline{G}) = \\{e_1,\\dots,e_{\\ell}\\}$, and defining $G_i:= G\\cup\\{e_1,\\dots,e_i\\}$, prove that $S^+(G_i)$ is a monotonically increasing sequence, and that $S^-(G_i)$ is a unimodal sequence. Note that this would imply the first statement by choosing $G'$ as a tree, and the sequence $e_1,\\dots,e_{\\ell}$ so that some $G'_i=G$.\n \\item For any $r\\ge 1$, is $\\sum_{t=1}^r \\lambda_t(G_i)^2$ monotonically increasing?\n \\item Given some threshold $\\tau$, is $\\sum_{t:\\lambda_t\\ge\\tau} \\lambda_t(G_i)^2$ monotonically increasing?\n\\end{enumerate}",
   "clean_statement": "Positive and negative $\\ell_2$-mass on spectrum\n\nLet $G$ be a connected graph and let $\\lambda_1\\ge\\dots\\ge\\lambda_s\\ge 0>\\lambda_{s+1}\\ge\\dots\\ge\\lambda_n$ be the eigenvalues of the adjacency matrix of $G$. Defining:\n\\begin{align*}\n S^+(G) &:=\\sum_{i=1}^s \\lambda_i^2\\\\\n S^-(G) &:= \\sum_{i=s+1}^n \\lambda_i^2.\n\\end{align*}\nSome conjectures and questions of interest are the following.\n\\begin{enumerate}\n \\item Prove that $\\min\\{S^+(G), S^-(G)\\} \\ge n-1$.\n \\item Denoting $E(\\overline{G}) = \\{e_1,\\dots,e_{\\ell}\\}$, and defining $G_i:= G\\cup\\{e_1,\\dots,e_i\\}$, prove that $S^+(G_i)$ is a monotonically increasing sequence, and that $S^-(G_i)$ is a unimodal sequence. Note that this would imply the first statement by choosing $G'$ as a tree, and the sequence $e_1,\\dots,e_{\\ell}$ so that some $G'_i=G$.\n \\item For any $r\\ge 1$, is $\\sum_{t=1}^r \\lambda_t(G_i)^2$ monotonically increasing?\n \\item Given some threshold $\\tau$, is $\\sum_{t:\\lambda_t\\ge\\tau} \\lambda_t(G_i)^2$ monotonically increasing?\n\\end{enumerate}",
   "public_statement": "Positive and negative $\\ell_2$-mass on spectrum\n\nLet $G$ be a connected graph and let $\\lambda_1\\ge\\dots\\ge\\lambda_s\\ge 0>\\lambda_{s+1}\\ge\\dots\\ge\\lambda_n$ be the eigenvalues of the adjacency matrix of $G$. Defining:\n\\begin{align*}\n S^+(G) &:=\\sum_{i=1}^s \\lambda_i^2\\\\\n S^-(G) &:= \\sum_{i=s+1}^n \\lambda_i^2.\n\\end{align*}\nSome conjectures and questions of interest are the following.\n\\begin{enumerate}\n \\item Prove that $\\min\\{S^+(G), S^-(G)\\} \\ge n-1$.\n \\item Denoting $E(\\overline{G}) = \\{e_1,\\dots,e_{\\ell}\\}$, and defining $G_i:= G\\cup\\{e_1,\\dots,e_i\\}$, prove that $S^+(G_i)$ is a monotonically increasing sequence, and that $S^-(G_i)$ is a unimodal sequence. Note that this would imply the first statement by choosing $G'$ as a tree, and the sequence $e_1,\\dots,e_{\\ell}$ so that some $G'_i=G$.\n \\item For any $r\\ge 1$, is $\\sum_{t=1}^r \\lambda_t(G_i)^2$ monotonically increasing?\n \\item Given some threshold $\\tau$, is $\\sum_{t:\\lambda_t\\ge\\tau} \\lambda_t(G_i)^2$ monotonically increasing?\n\\end{enumerate}",
   "evidence": "The canonical record asks about a connected graph $G$ with adjacency eigenvalues $$ \\lambda_1\\geq\\cdots\\geq\\lambda_s\\geq0>\\lambda_{s+1}\\geq\\cdots\\geq\\lambda_n $$ and $$ S^+(G)=\\sum_{i=1}^s\\lambda_i^2, \\qquad S^-(G)=\\sum_{i=s+1}^n\\lambda_i^2. $$ It contains four items:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-LINEAR_ALGEBRA-0005": {
   "statement_status": "exact",
   "original_statement": "\\begin{itemize}\n \\item (i) Let $\\mathcal{P}_d := \\left\\{f(z)=\n \\sum\\limits_{j=0}^{d}a_j z^j \\ | \\ a_j \\geq 0~\n \\forall j, ~~f(z)=0 \\implies z \\in \\mathbb{R}\\right\\}$.\n Which functions $F: \\mathbb{R}_+ \\longrightarrow\n \\mathbb{R}_+$ such that $F(0)=0$ map $\\mathcal{P}_d$ into\n itself, where $F[f](z) :=\n \\sum\\limits_{j=0}^{d}F(a_j)z^j$?\n\n Equivalently, let \\[\n T_{\\bf a} :=\n {\\scriptsize \\begin{pmatrix}\n a_0 & 0 & 0 & \\cdots\\\\\n a_1 & a_0 & 0 & \\cdots\\\\\n a_2 & a_1 & a_0 & \\cdots\\\\\n \\vdots & \\vdots & \\vdots & \\ddots\n \\end{pmatrix}} \\in TN,\n \\]\n where ${\\bf a}= (a_0,\\ldots,a_d)\\in (0,\\infty)^{d+1}$. When is $F[T_{\\bf a}]\\in TN$ for all such $\\bf a$?\n \\item (ii) The same question for fixed $d$.\n \\item (iii) One can replace $\\mathcal{P}_d$ with $\\tilde{\\mathcal{P}}_d:=\\left\\{f(z)= \\sum\\limits_{j=0}^{d}\\frac{a_jz^j}{j !}~|~a_j\\geq 0~ \\forall j, ~~f(z)=0 \\implies z \\in \\mathbb{R}\\right\\}$ or replace $F[f](z)$ with $\\tilde{F}[f](z):=\\sum\\limits_{j=0}^{d}\\frac{F(a_j)z^j}{j !}$ and consider the above questions (four possible cases).\n \\item (iv) What about preservers of the form $F_g[f](z)=\\sum\\limits_{j=0}^{d}g(j)F(a_j)z^j$? Such $g$ are called multiplier sequences and have been classified (P\\'olya, Laguerre and Schur).\n \\item (v) Let $h: \\mathbb{R}^3 \\longrightarrow \\mathbb{R}$ and let $F_h[\\sum\\limits_{j=0}^{d}a_jz^j]=\\sum\\limits_{j=0}^{d}h(a_{j-1},a_j,a_{j+1})z^j$, where $a_{-1}=a_{d+1}=0$. E.g., if $h(x,y,z)=y^2-xz$, then $F_h[-]$ preserves real rooted polynomials.\n\nWhat other $h$ work?\n\nWhat about higher order determinants?\n\n \\end{itemize}",
   "clean_statement": "\\begin{itemize}\n \\item (i) Let $\\mathcal{P}_d := \\left\\{f(z)=\n \\sum\\limits_{j=0}^{d}a_j z^j \\ | \\ a_j \\geq 0~\n \\forall j, ~~f(z)=0 \\implies z \\in \\mathbb{R}\\right\\}$.\n Which functions $F: \\mathbb{R}_+ \\longrightarrow\n \\mathbb{R}_+$ such that $F(0)=0$ map $\\mathcal{P}_d$ into\n itself, where $F[f](z) :=\n \\sum\\limits_{j=0}^{d}F(a_j)z^j$?\n\n Equivalently, let \\[\n T_{\\bf a} :=\n {\\scriptsize \\begin{pmatrix}\n a_0 & 0 & 0 & \\cdots\\\\\n a_1 & a_0 & 0 & \\cdots\\\\\n a_2 & a_1 & a_0 & \\cdots\\\\\n \\vdots & \\vdots & \\vdots & \\ddots\n \\end{pmatrix}} \\in TN,\n \\]\n where ${\\bf a}= (a_0,\\ldots,a_d)\\in (0,\\infty)^{d+1}$. When is $F[T_{\\bf a}]\\in TN$ for all such $\\bf a$?\n \\item (ii) The same question for fixed $d$.\n \\item (iii) One can replace $\\mathcal{P}_d$ with $\\tilde{\\mathcal{P}}_d:=\\left\\{f(z)= \\sum\\limits_{j=0}^{d}\\frac{a_jz^j}{j !}~|~a_j\\geq 0~ \\forall j, ~~f(z)=0 \\implies z \\in \\mathbb{R}\\right\\}$ or replace $F[f](z)$ with $\\tilde{F}[f](z):=\\sum\\limits_{j=0}^{d}\\frac{F(a_j)z^j}{j !}$ and consider the above questions (four possible cases).\n \\item (iv) What about preservers of the form $F_g[f](z)=\\sum\\limits_{j=0}^{d}g(j)F(a_j)z^j$? Such $g$ are called multiplier sequences and have been classified (P\\'olya, Laguerre and Schur).\n \\item (v) Let $h: \\mathbb{R}^3 \\longrightarrow \\mathbb{R}$ and let $F_h[\\sum\\limits_{j=0}^{d}a_jz^j]=\\sum\\limits_{j=0}^{d}h(a_{j-1},a_j,a_{j+1})z^j$, where $a_{-1}=a_{d+1}=0$. E.g., if $h(x,y,z)=y^2-xz$, then $F_h[-]$ preserves real rooted polynomials.\n\nWhat other $h$ work?\n\nWhat about higher order determinants?\n\n \\end{itemize}",
   "public_statement": "\\begin{itemize}\n \\item (i) Let $\\mathcal{P}_d := \\left\\{f(z)=\n \\sum\\limits_{j=0}^{d}a_j z^j \\ | \\ a_j \\geq 0~\n \\forall j, ~~f(z)=0 \\implies z \\in \\mathbb{R}\\right\\}$.\n Which functions $F: \\mathbb{R}_+ \\longrightarrow\n \\mathbb{R}_+$ such that $F(0)=0$ map $\\mathcal{P}_d$ into\n itself, where $F[f](z) :=\n \\sum\\limits_{j=0}^{d}F(a_j)z^j$?\n\n Equivalently, let \\[\n T_{\\bf a} :=\n {\\scriptsize \\begin{pmatrix}\n a_0 & 0 & 0 & \\cdots\\\\\n a_1 & a_0 & 0 & \\cdots\\\\\n a_2 & a_1 & a_0 & \\cdots\\\\\n \\vdots & \\vdots & \\vdots & \\ddots\n \\end{pmatrix}} \\in TN,\n \\]\n where ${\\bf a}= (a_0,\\ldots,a_d)\\in (0,\\infty)^{d+1}$. When is $F[T_{\\bf a}]\\in TN$ for all such $\\bf a$?\n \\item (ii) The same question for fixed $d$.\n \\item (iii) One can replace $\\mathcal{P}_d$ with $\\tilde{\\mathcal{P}}_d:=\\left\\{f(z)= \\sum\\limits_{j=0}^{d}\\frac{a_jz^j}{j !}~|~a_j\\geq 0~ \\forall j, ~~f(z)=0 \\implies z \\in \\mathbb{R}\\right\\}$ or replace $F[f](z)$ with $\\tilde{F}[f](z):=\\sum\\limits_{j=0}^{d}\\frac{F(a_j)z^j}{j !}$ and consider the above questions (four possible cases).\n \\item (iv) What about preservers of the form $F_g[f](z)=\\sum\\limits_{j=0}^{d}g(j)F(a_j)z^j$? Such $g$ are called multiplier sequences and have been classified (P\\'olya, Laguerre and Schur).\n \\item (v) Let $h: \\mathbb{R}^3 \\longrightarrow \\mathbb{R}$ and let $F_h[\\sum\\limits_{j=0}^{d}a_jz^j]=\\sum\\limits_{j=0}^{d}h(a_{j-1},a_j,a_{j+1})z^j$, where $a_{-1}=a_{d+1}=0$. E.g., if $h(x,y,z)=y^2-xz$, then $F_h[-]$ preserves real rooted polynomials.\n\nWhat other $h$ work?\n\nWhat about higher order determinants?\n\n \\end{itemize}",
   "evidence": "The canonical record is `aim-linear-algebra-notes.json`, zero-based index 4. The live AIM page was inspected on August 10, 2026. Its current problem record (revision 96) agrees with the canonical input and has no status or remarks. No OCR correction is needed.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 4,
   "attempt": 2
  },
  "AIM-LINEAR_ALGEBRA-0006": {
   "statement_status": "exact",
   "original_statement": "The Aissen--Edrei--Schoenberg--Whitney (AESW) theorem states that\na real sequence $(a_0,a_1,a_2,\\ldots)$ with $a_0 \\neq 0$ is a one-sided\nPF sequence $\\Leftrightarrow F(z)=a_0 e^{\\delta z}\n\\prod\\limits_{j=0}^{\\infty}\\frac{1+\\alpha_j z}{1-\\beta_jz}$ for\n$\\alpha_j, \\beta_j, \\delta, a_0 \\geq 0$ and $\\sum_j (\\alpha_j + \\beta_j)\n< \\infty$. Find a proof that does not use Nevanlinna theory: in other\nwords, the fact that if $F(z)$ is entire of genus at most one, and has no\nzeros, then it is an exponential function $ae^{\\delta z}$ for some\n$\\delta \\geq 0$.",
   "clean_statement": "The Aissen--Edrei--Schoenberg--Whitney (AESW) theorem states that\na real sequence $(a_0,a_1,a_2,\\ldots)$ with $a_0 \\neq 0$ is a one-sided\nPF sequence $\\Leftrightarrow F(z)=a_0 e^{\\delta z}\n\\prod\\limits_{j=0}^{\\infty}\\frac{1+\\alpha_j z}{1-\\beta_jz}$ for\n$\\alpha_j, \\beta_j, \\delta, a_0 \\geq 0$ and $\\sum_j (\\alpha_j + \\beta_j)\n< \\infty$. Find a proof that does not use Nevanlinna theory: in other\nwords, the fact that if $F(z)$ is entire of genus at most one, and has no\nzeros, then it is an exponential function $ae^{\\delta z}$ for some\n$\\delta \\geq 0$.",
   "public_statement": "The Aissen--Edrei--Schoenberg--Whitney (AESW) theorem states that\na real sequence $(a_0,a_1,a_2,\\ldots)$ with $a_0 \\neq 0$ is a one-sided\nPF sequence $\\Leftrightarrow F(z)=a_0 e^{\\delta z}\n\\prod\\limits_{j=0}^{\\infty}\\frac{1+\\alpha_j z}{1-\\beta_jz}$ for\n$\\alpha_j, \\beta_j, \\delta, a_0 \\geq 0$ and $\\sum_j (\\alpha_j + \\beta_j)\n< \\infty$. Find a proof that does not use Nevanlinna theory: in other\nwords, the fact that if $F(z)$ is entire of genus at most one, and has no\nzeros, then it is an exponential function $ae^{\\delta z}$ for some\n$\\delta \\geq 0$.",
   "evidence": "The exact canonical problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-LINEAR_ALGEBRA-0007": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Let \\[\nM_{\\bf A} :=\n{\\scriptsize \\begin{pmatrix}\n A_0 & 0 & 0 & \\cdots\\\\\n A_1 & A_0 & 0 & \\cdots\\\\\n A_2 & A_1 & A_0 & \\cdots\\\\\n \\vdots & \\vdots & \\vdots & \\ddots\n\\end{pmatrix}} \\in TN,\n\\] and $F(t)=A_0+tA_1+t^2A_2+\\cdots \\in M_2[[t]],$ where $A_0$ is invertible.\nIs there an AESW-type factorization for $F(t)$?",
   "clean_statement": null,
   "public_statement": "Let \\[\nM_{\\bf A} :=\n{\\scriptsize \\begin{pmatrix}\n A_0 & 0 & 0 & \\cdots\\\\\n A_1 & A_0 & 0 & \\cdots\\\\\n A_2 & A_1 & A_0 & \\cdots\\\\\n \\vdots & \\vdots & \\vdots & \\ddots\n\\end{pmatrix}} \\in TN,\n\\] and $F(t)=A_0+tA_1+t^2A_2+\\cdots \\in M_2[[t]],$ where $A_0$ is invertible.\nIs there an AESW-type factorization for $F(t)$?",
   "evidence": "The AIMPL page returned HTTP 502 during this run. The official 2023 AIM workshop report confirms that the workshop generated thirteen problems and points to that page, but it does not reproduce this particular statement. Thus the exact extracted record is preserved, and the two conventions above are explicitly labeled reconstructions rather than additional source text.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-LINEAR_ALGEBRA-0008": {
   "statement_status": "exact",
   "original_statement": "An algorithmic process is known to identify inequalities that compare the products of two minors for all $TN$ matrices.\n\nClassify real linear combinations of products of minors that are\n\tnonnegative for all $TN$ matrices. More concretely, given a\n\tpolynomial $p( \\{ x_{I \\times J} \\ | \\ I,J \\subset [n], |I| = |J| \\}\n\t)$, when can one certify, via some constructive algorithmic process, that evaluating $p$ at $x_{I \\times J} =\n\t\\det A_{I \\times J}$ yields a non-negative output for all $TN$\n\tmatrices $A$.",
   "clean_statement": "An algorithmic process is known to identify inequalities that compare the products of two minors for all $TN$ matrices.\n\nClassify real linear combinations of products of minors that are\n\tnonnegative for all $TN$ matrices. More concretely, given a\n\tpolynomial $p( \\{ x_{I \\times J} \\ | \\ I,J \\subset [n], |I| = |J| \\}\n\t)$, when can one certify, via some constructive algorithmic process, that evaluating $p$ at $x_{I \\times J} =\n\t\\det A_{I \\times J}$ yields a non-negative output for all $TN$\n\tmatrices $A$.",
   "public_statement": "An algorithmic process is known to identify inequalities that compare the products of two minors for all $TN$ matrices.\n\nClassify real linear combinations of products of minors that are\n\tnonnegative for all $TN$ matrices. More concretely, given a\n\tpolynomial $p( \\{ x_{I \\times J} \\ | \\ I,J \\subset [n], |I| = |J| \\}\n\t)$, when can one certify, via some constructive algorithmic process, that evaluating $p$ at $x_{I \\times J} =\n\t\\det A_{I \\times J}$ yields a non-negative output for all $TN$\n\tmatrices $A$.",
   "evidence": "The canonical record is aim-linear-algebra-notes.json, zero-based index 7, Problem 1.25 from the AIM workshop *Theory and applications of total positivity*. The live AIM page was inspected on August 10, 2026. Its current record (revision 78, attributed there to Prateek Kumar Vishwakarma) agrees with the canonical text and has no status or remarks.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 7,
   "attempt": 2
  },
  "AIM-LINEAR_ALGEBRA-0009": {
   "statement_status": "exact",
   "original_statement": "Consider $TN$ Toeplitz matrices. What are the allowed patterns of\n\tzero and nonzero minors (i.e., which positroid cells contain\n\tToeplitz matrices)? How does this relate to P\\'olya frequency\n\tsequences? (Check work by Rietsch for the lower triangular\n\tToeplitz case.)\n\tIs it enough to check all the corner minors to identify $TN$\n\tToeplitz matrices?",
   "clean_statement": "Consider $TN$ Toeplitz matrices. What are the allowed patterns of\n\tzero and nonzero minors (i.e., which positroid cells contain\n\tToeplitz matrices)? How does this relate to P\\'olya frequency\n\tsequences? (Check work by Rietsch for the lower triangular\n\tToeplitz case.)\n\tIs it enough to check all the corner minors to identify $TN$\n\tToeplitz matrices?",
   "public_statement": "Consider $TN$ Toeplitz matrices. What are the allowed patterns of\n\tzero and nonzero minors (i.e., which positroid cells contain\n\tToeplitz matrices)? How does this relate to P\\'olya frequency\n\tsequences? (Check work by Rietsch for the lower triangular\n\tToeplitz case.)\n\tIs it enough to check all the corner minors to identify $TN$\n\tToeplitz matrices?",
   "evidence": "The canonical record, from `aim-linear-algebra-notes.json` at zero-based index 8, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 8,
   "attempt": 2
  },
  "AIM-LINEAR_ALGEBRA-0010": {
   "statement_status": "exact",
   "original_statement": "Consider semi-infinite lower triangular $TN$ Toeplitz matrices. Under what conditions can a non-trivial $k\\times k$ minor be zero?",
   "clean_statement": "Consider semi-infinite lower triangular $TN$ Toeplitz matrices. Under what conditions can a non-trivial $k\\times k$ minor be zero?",
   "public_statement": "Consider semi-infinite lower triangular $TN$ Toeplitz matrices. Under what conditions can a non-trivial $k\\times k$ minor be zero?",
   "evidence": "The record has no remarks or literature field. The listed AIMPL page, `http://aimpl.org/totalpos/1/`, was unavailable during the run (HTTP 502/timeout on 2026-08-10). The official 2023 AIM workshop report confirms the problem-list URL and the workshop context, but does not reproduce this particular question. There is no visible OCR error in the repository statement.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-LINEAR_ALGEBRA-0011": {
   "statement_status": "exact",
   "original_statement": "Given positive scalars $a_0, \\dots, a_d$ such that\n$\\frac{a_k^2}{a_{k-1}a_{k+1}}\\geq 4$ for $0 0$\nsuch that if $\\lambda \\geq \\lambda_d$, then the ``$\\lambda$-rhombus\nlog-concavity'' of the Maclaurin coefficients of a homogeneous polynomial $p(x,y,z)$ implies that $p$ is real-stable.\n\nCan one remove the dependence of $\\lambda_d$ on $d$? If yes, then what\nabout for $n > 3$ variables?",
   "clean_statement": "Given positive scalars $a_0, \\dots, a_d$ such that\n$\\frac{a_k^2}{a_{k-1}a_{k+1}}\\geq 4$ for $0 0$\nsuch that if $\\lambda \\geq \\lambda_d$, then the ``$\\lambda$-rhombus\nlog-concavity'' of the Maclaurin coefficients of a homogeneous polynomial $p(x,y,z)$ implies that $p$ is real-stable.\n\nCan one remove the dependence of $\\lambda_d$ on $d$? If yes, then what\nabout for $n > 3$ variables?",
   "public_statement": "Given positive scalars $a_0, \\dots, a_d$ such that\n$\\frac{a_k^2}{a_{k-1}a_{k+1}}\\geq 4$ for $0 0$\nsuch that if $\\lambda \\geq \\lambda_d$, then the ``$\\lambda$-rhombus\nlog-concavity'' of the Maclaurin coefficients of a homogeneous polynomial $p(x,y,z)$ implies that $p$ is real-stable.\n\nCan one remove the dependence of $\\lambda_d$ on $d$? If yes, then what\nabout for $n > 3$ variables?",
   "evidence": "The canonical record is visibly corrupted. It contains",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-LINEAR_ALGEBRA-0012": {
   "statement_status": "exact",
   "original_statement": "There are many classical theorems about operations that preserve\nnegative-real-rootedness of polynomials: for instance, $d/dx + a$\nor $x \\cdot d/dx + a$ with $a \\ge 0$; or Hadamard product; or Hadamard\nproduct with an extra $n!$ (sometimes known as Schur composition);\nor Br\\\"and\\'en's (2011) log-concavity operation $\\ldots$. These can\nbe reinterpreted as statements about \\emph{pointwise} total positivity\nfor certain Toeplitz matrices involving elementary symmetric functions\n(e.g. $(n+a) e_n$). Can these results be upgraded to total positivity\nin the \\emph{monomial} basis? Computational tests suggest that the\nanswer is yes. But we have been unable, thus far, to come up with\na plausible strategy for proving any of these conjectures.",
   "clean_statement": "There are many classical theorems about operations that preserve\nnegative-real-rootedness of polynomials: for instance, $d/dx + a$\nor $x \\cdot d/dx + a$ with $a \\ge 0$; or Hadamard product; or Hadamard\nproduct with an extra $n!$ (sometimes known as Schur composition);\nor Br\\\"and\\'en's (2011) log-concavity operation $\\ldots$. These can\nbe reinterpreted as statements about \\emph{pointwise} total positivity\nfor certain Toeplitz matrices involving elementary symmetric functions\n(e.g. $(n+a) e_n$). Can these results be upgraded to total positivity\nin the \\emph{monomial} basis? Computational tests suggest that the\nanswer is yes. But we have been unable, thus far, to come up with\na plausible strategy for proving any of these conjectures.",
   "public_statement": "There are many classical theorems about operations that preserve\nnegative-real-rootedness of polynomials: for instance, $d/dx + a$\nor $x \\cdot d/dx + a$ with $a \\ge 0$; or Hadamard product; or Hadamard\nproduct with an extra $n!$ (sometimes known as Schur composition);\nor Br\\\"and\\'en's (2011) log-concavity operation $\\ldots$. These can\nbe reinterpreted as statements about \\emph{pointwise} total positivity\nfor certain Toeplitz matrices involving elementary symmetric functions\n(e.g. $(n+a) e_n$). Can these results be upgraded to total positivity\nin the \\emph{monomial} basis? Computational tests suggest that the\nanswer is yes. But we have been unable, thus far, to come up with\na plausible strategy for proving any of these conjectures.",
   "evidence": "There is no substantive OCR corruption in this record. The source is a research-program question rather than one fully quantified conjecture: it lists several operators and asks for coefficientwise total-nonnegativity statements. Consequently, the result below addresses the displayed family \\((n+a)e_n\\) and does not purport to settle every conjecture in the record.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-LINEAR_ALGEBRA-0013": {
   "statement_status": "exact",
   "original_statement": "It was recently shown that if $\\lambda, \\mu$ are $N$-tuples of\n\tpositive integers, then the ratio of Schur polynomials $s_\\lambda\n\t/ s_\\mu$, when evaluated at $N$ real variables with values in\n\t$(0,1]^N$ (the ``log-negative orthant''), is minimized at\n\t$(1,\\dots,1)$ if and only if $-\\lambda$ weakly majorizes $-\\mu$.\n\n\tThe analogous statement, in which the minimum over the entire\n\tpositive orthant $(0,\\infty)^N$ is achieved at $(1,\\dots,1)$, is\n\tequivalent to $\\lambda$ majorizing $\\mu$.\n\tWhat can one say these domains are replaced by $I^N$ for other\n\tintervals $I \\subset (0,\\infty)$?",
   "clean_statement": "It was recently shown that if $\\lambda, \\mu$ are $N$-tuples of\n\tpositive integers, then the ratio of Schur polynomials $s_\\lambda\n\t/ s_\\mu$, when evaluated at $N$ real variables with values in\n\t$(0,1]^N$ (the ``log-negative orthant''), is minimized at\n\t$(1,\\dots,1)$ if and only if $-\\lambda$ weakly majorizes $-\\mu$.\n\n\tThe analogous statement, in which the minimum over the entire\n\tpositive orthant $(0,\\infty)^N$ is achieved at $(1,\\dots,1)$, is\n\tequivalent to $\\lambda$ majorizing $\\mu$.\n\tWhat can one say these domains are replaced by $I^N$ for other\n\tintervals $I \\subset (0,\\infty)$?",
   "public_statement": "It was recently shown that if $\\lambda, \\mu$ are $N$-tuples of\n\tpositive integers, then the ratio of Schur polynomials $s_\\lambda\n\t/ s_\\mu$, when evaluated at $N$ real variables with values in\n\t$(0,1]^N$ (the ``log-negative orthant''), is minimized at\n\t$(1,\\dots,1)$ if and only if $-\\lambda$ weakly majorizes $-\\mu$.\n\n\tThe analogous statement, in which the minimum over the entire\n\tpositive orthant $(0,\\infty)^N$ is achieved at $(1,\\dots,1)$, is\n\tequivalent to $\\lambda$ majorizing $\\mu$.\n\tWhat can one say these domains are replaced by $I^N$ for other\n\tintervals $I \\subset (0,\\infty)$?",
   "evidence": "The exact repository record (AIM Problem Lists, workshop *Theory and applications of total positivity*, problem 1.45) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-LINEAR_ALGEBRA-0014": {
   "statement_status": "exact",
   "original_statement": "Consider a bipartite graph $G$ on an annulus, with a fixed\n\t``base'' perfect matching $\\mu$. Given any other $\\mu'$,\n\tsuperimpose it on $\\mu$; this gives a bunch of cycles. Count the\n\tnumber of non-contractible cycles $=f(\\mu')$. Running over all\n\t$\\mu' \\neq \\mu$, we get\n\t$$ \\sum_{k \\geq 0} \\\n\\sum_{\\mu'\\neq \\mu \\ | \\ f(\\mu')=k} z^{k} = \\sum_{k\\geq 0} a_{k}z^{k}.\n\t$$\t(This is known to be independent of the base matching $\\mu$.)\n\tIs it true that $\\sum_{k\\geq 0} a_{k}z^{k}$ generates a P\\'olya\n\tfrequency sequence?\n\tIf one attaches weights to the edges, does one get a\n\tcoefficientwise-$TN$ infinite triangular $TN$ Toeplitz matrix? (It is\n\tknown that this Toeplitz matrix is coefficientwise ${TN}_2.$)",
   "clean_statement": "Consider a bipartite graph $G$ on an annulus, with a fixed\n\t``base'' perfect matching $\\mu$. Given any other $\\mu'$,\n\tsuperimpose it on $\\mu$; this gives a bunch of cycles. Count the\n\tnumber of non-contractible cycles $=f(\\mu')$. Running over all\n\t$\\mu' \\neq \\mu$, we get\n\t$$ \\sum_{k \\geq 0} \\\n\\sum_{\\mu'\\neq \\mu \\ | \\ f(\\mu')=k} z^{k} = \\sum_{k\\geq 0} a_{k}z^{k}.\n\t$$\t(This is known to be independent of the base matching $\\mu$.)\n\tIs it true that $\\sum_{k\\geq 0} a_{k}z^{k}$ generates a P\\'olya\n\tfrequency sequence?\n\tIf one attaches weights to the edges, does one get a\n\tcoefficientwise-$TN$ infinite triangular $TN$ Toeplitz matrix? (It is\n\tknown that this Toeplitz matrix is coefficientwise ${TN}_2.$)",
   "public_statement": "Consider a bipartite graph $G$ on an annulus, with a fixed\n\t``base'' perfect matching $\\mu$. Given any other $\\mu'$,\n\tsuperimpose it on $\\mu$; this gives a bunch of cycles. Count the\n\tnumber of non-contractible cycles $=f(\\mu')$. Running over all\n\t$\\mu' \\neq \\mu$, we get\n\t$$ \\sum_{k \\geq 0} \\\n\\sum_{\\mu'\\neq \\mu \\ | \\ f(\\mu')=k} z^{k} = \\sum_{k\\geq 0} a_{k}z^{k}.\n\t$$\t(This is known to be independent of the base matching $\\mu$.)\n\tIs it true that $\\sum_{k\\geq 0} a_{k}z^{k}$ generates a P\\'olya\n\tfrequency sequence?\n\tIf one attaches weights to the edges, does one get a\n\tcoefficientwise-$TN$ infinite triangular $TN$ Toeplitz matrix? (It is\n\tknown that this Toeplitz matrix is coefficientwise ${TN}_2.$)",
   "evidence": "The record is Problem 1.5 in the AIM workshop list *Theory and applications of total positivity* and is attributed there to Pavlo Pylyavskyy. I compared the corpus record with the [archived AIM page](https://web.archive.org/web/20240208020313id_/http://aimpl.org/totalpos/1/). The mathematical text, including the restriction \\(\\mu'\\ne\\mu\\), agrees; there is no consequential OCR corruption. The source display is a nested sum: for each \\(k\\), its inner sum contributes one copy of \\(z^k\\) for every matching with \\(f(\\mu')=k\\). Equivalently,",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 13,
   "attempt": 2
  },
  "AIM-LINEAR_ALGEBRA-0015": {
   "statement_status": "exact",
   "original_statement": "Suppose $n,r \\geq 1$ are integers, and $(\\lambda_1, \\dots,\n\t\\lambda_r)$ and $(\\mu_1, \\dots, \\mu_r)$ are (non-increasing)\n\tpartitions of $n$. If $\\mu$ majorizes $\\lambda$, then for\n\tall $TN$ matrices $A$ we have:\n\t\\[\n\t\\lambda_1! \\cdots \\lambda_r! \\sum_I \\prod_{k=1}^r \\det A_{I_k\n\t\\times I_k} \\geq\n\t\\mu_1! \\cdots \\mu_r! \\sum_J \\prod_{k=1}^r \\det A_{J_k\n\t\\times J_k},\n\t\\]\n\tfor all partitions $I = (I_1, \\dots, I_r)$ and $J = (J_1, \\dots,\n\tJ_r)$ of $[n]$ with $|I_k| = \\lambda_k, |J_k| = \\mu_k$ for all\n\t$k$.\n\n\tIs the converse true?",
   "clean_statement": "Suppose $n,r \\geq 1$ are integers, and $(\\lambda_1, \\dots,\n\t\\lambda_r)$ and $(\\mu_1, \\dots, \\mu_r)$ are (non-increasing)\n\tpartitions of $n$. If $\\mu$ majorizes $\\lambda$, then for\n\tall $TN$ matrices $A$ we have:\n\t\\[\n\t\\lambda_1! \\cdots \\lambda_r! \\sum_I \\prod_{k=1}^r \\det A_{I_k\n\t\\times I_k} \\geq\n\t\\mu_1! \\cdots \\mu_r! \\sum_J \\prod_{k=1}^r \\det A_{J_k\n\t\\times J_k},\n\t\\]\n\tfor all partitions $I = (I_1, \\dots, I_r)$ and $J = (J_1, \\dots,\n\tJ_r)$ of $[n]$ with $|I_k| = \\lambda_k, |J_k| = \\mu_k$ for all\n\t$k$.\n\n\tIs the converse true?",
   "public_statement": "Suppose $n,r \\geq 1$ are integers, and $(\\lambda_1, \\dots,\n\t\\lambda_r)$ and $(\\mu_1, \\dots, \\mu_r)$ are (non-increasing)\n\tpartitions of $n$. If $\\mu$ majorizes $\\lambda$, then for\n\tall $TN$ matrices $A$ we have:\n\t\\[\n\t\\lambda_1! \\cdots \\lambda_r! \\sum_I \\prod_{k=1}^r \\det A_{I_k\n\t\\times I_k} \\geq\n\t\\mu_1! \\cdots \\mu_r! \\sum_J \\prod_{k=1}^r \\det A_{J_k\n\t\\times J_k},\n\t\\]\n\tfor all partitions $I = (I_1, \\dots, I_r)$ and $J = (J_1, \\dots,\n\tJ_r)$ of $[n]$ with $|I_k| = \\lambda_k, |J_k| = \\mu_k$ for all\n\t$k$.\n\n\tIs the converse true?",
   "evidence": "This is Problem 1.55 in the AIM list from the workshop *Theory and applications of total positivity*. The canonical record is aim-linear-algebra-notes.json, zero-based index 14. Its literature field says, “This has now been answered affirmatively.”",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-LINEAR_ALGEBRA-0016": {
   "statement_status": "exact",
   "original_statement": "Is Fisk's conjecture true for $3 \\times 3$ minors / $n \\times n$\n\tminors? Namely, if $\\sum_{k=0}^d a_k z^k$ is a real-rooted\n\tpolynomial with all $a_k > 0$, then Br\\\"and\\'en showed the ``$2\n\t\\times 2$ analogue'':\n\t\\[\n\t\\sum_{k \\geq 0} \\left| \\begin{matrix} a_k & a_{k-1} \\\\ a_{k+1} &\n\ta_k \\end{matrix} \\right| z^k\n\t\\]\n\tis also real-rooted. The conjecture by Fisk asks if the $3 \\times\n\t3$ analogue (or $n \\times n$ for higher $n$) is true: is\n\t\\[\n\t\\sum_{k \\geq 0} \\left| \\begin{matrix} a_k & a_{k-1} & a_{k-2} \\\\\n\ta_{k+1} & a_k & a_{k-1} \\\\ a_{k+2} & a_{k+1} & a_k \\end{matrix}\n\t\\right| z^k\n\t\\]\n\talso real-rooted?",
   "clean_statement": "Is Fisk's conjecture true for $3 \\times 3$ minors / $n \\times n$\n\tminors? Namely, if $\\sum_{k=0}^d a_k z^k$ is a real-rooted\n\tpolynomial with all $a_k > 0$, then Br\\\"and\\'en showed the ``$2\n\t\\times 2$ analogue'':\n\t\\[\n\t\\sum_{k \\geq 0} \\left| \\begin{matrix} a_k & a_{k-1} \\\\ a_{k+1} &\n\ta_k \\end{matrix} \\right| z^k\n\t\\]\n\tis also real-rooted. The conjecture by Fisk asks if the $3 \\times\n\t3$ analogue (or $n \\times n$ for higher $n$) is true: is\n\t\\[\n\t\\sum_{k \\geq 0} \\left| \\begin{matrix} a_k & a_{k-1} & a_{k-2} \\\\\n\ta_{k+1} & a_k & a_{k-1} \\\\ a_{k+2} & a_{k+1} & a_k \\end{matrix}\n\t\\right| z^k\n\t\\]\n\talso real-rooted?",
   "public_statement": "Is Fisk's conjecture true for $3 \\times 3$ minors / $n \\times n$\n\tminors? Namely, if $\\sum_{k=0}^d a_k z^k$ is a real-rooted\n\tpolynomial with all $a_k > 0$, then Br\\\"and\\'en showed the ``$2\n\t\\times 2$ analogue'':\n\t\\[\n\t\\sum_{k \\geq 0} \\left| \\begin{matrix} a_k & a_{k-1} \\\\ a_{k+1} &\n\ta_k \\end{matrix} \\right| z^k\n\t\\]\n\tis also real-rooted. The conjecture by Fisk asks if the $3 \\times\n\t3$ analogue (or $n \\times n$ for higher $n$) is true: is\n\t\\[\n\t\\sum_{k \\geq 0} \\left| \\begin{matrix} a_k & a_{k-1} & a_{k-2} \\\\\n\ta_{k+1} & a_k & a_{k-1} \\\\ a_{k+2} & a_{k+1} & a_k \\end{matrix}\n\t\\right| z^k\n\t\\]\n\talso real-rooted?",
   "evidence": "The canonical AIM record (source file `aim-linear-algebra-notes.json`, zero-based index 15) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 15,
   "attempt": 2
  },
  "AIM-LINEAR_ALGEBRA-0017": {
   "statement_status": "exact",
   "original_statement": "Suppose $\\lambda \\geq 4$. Given a (strictly) ${TP}_1$ square matrix\n\t$A$, it is known that if $ad - \\lambda bc > 0$ for every $2 \\times 2$ submatrix\n\t$\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$ of $A$, then $A$\n\tis (strictly) TP.\n\n\tThe question is if this fact can be extended to higher orders of\n\ttotal positivity. Namely, can the above result be extended to\n\tfinding a quantitative (in $\\lambda > 0$) sufficient condition,\n\tideally involving $k \\times k$ or smaller minors of $A$, such\n\tthat every (strictly) ${TP}_{k-1}$ matrix satisfying this condition\n\tis automatically (strictly) $TP$? If yes, what is the smallest\n\tvalue of $\\lambda$ for each such $k$?",
   "clean_statement": "Suppose $\\lambda \\geq 4$. Given a (strictly) ${TP}_1$ square matrix\n\t$A$, it is known that if $ad - \\lambda bc > 0$ for every $2 \\times 2$ submatrix\n\t$\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$ of $A$, then $A$\n\tis (strictly) TP.\n\n\tThe question is if this fact can be extended to higher orders of\n\ttotal positivity. Namely, can the above result be extended to\n\tfinding a quantitative (in $\\lambda > 0$) sufficient condition,\n\tideally involving $k \\times k$ or smaller minors of $A$, such\n\tthat every (strictly) ${TP}_{k-1}$ matrix satisfying this condition\n\tis automatically (strictly) $TP$? If yes, what is the smallest\n\tvalue of $\\lambda$ for each such $k$?",
   "public_statement": "Suppose $\\lambda \\geq 4$. Given a (strictly) ${TP}_1$ square matrix\n\t$A$, it is known that if $ad - \\lambda bc > 0$ for every $2 \\times 2$ submatrix\n\t$\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$ of $A$, then $A$\n\tis (strictly) TP.\n\n\tThe question is if this fact can be extended to higher orders of\n\ttotal positivity. Namely, can the above result be extended to\n\tfinding a quantitative (in $\\lambda > 0$) sufficient condition,\n\tideally involving $k \\times k$ or smaller minors of $A$, such\n\tthat every (strictly) ${TP}_{k-1}$ matrix satisfying this condition\n\tis automatically (strictly) $TP$? If yes, what is the smallest\n\tvalue of $\\lambda$ for each such $k$?",
   "evidence": "This is Problem 1.65 of the AIM list *Theory and applications of total positivity*, attributed on the [archived AIM page](https://web.archive.org/web/20240208020313id_/http://aimpl.org/totalpos/1/) to Charles Johnson and Steven Karp. The corpus record agrees with that page; there is no OCR corruption. In the strict convention used below, \\(TP_r\\) means that every minor of order at most \\(r\\) is positive, and \\(TP\\) means that every minor is positive. There is an important formulation distinction. The AIM problem literally says “every \\(2\\times2\\) submatrix,” which ordinarily means every choice of two rows and two columns. The primary theorem of Katkova--Vishnyakova needs only the **local adjacent-entry** inequalities \\[ a_{ij}a_{i+1,j+1}>c\\,a_{i,j+1}a_{i+1,j}. \\tag{1.1} \\] Thus the literal AIM hypothesis is stronger. This is not an OCR error; the workshop report and the cited theorem...",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-LINEAR_ALGEBRA-0018": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "6. (Grone-Merris conjecture) Is λ majorized by d∗?\n\nIt was shown at the workshop that if the Grone-Merris conjecture is true, then \n\nEL(G) ≤\n\n> n\n\nX\n\n> i=1\n\n˛˛˛˛d∗ \n\n> i\n\n− 2m\n\nn\n\n˛˛˛˛.\n\n5",
   "clean_statement": "**Question 6 (Grone--Merris conjecture).** Is \\(\\lambda\\) majorized by\n\\(d^*\\)?\n\nIt was shown at the workshop that, if the Grone--Merris conjecture is true,\nthen\n\\[\nE_L(G)\\leq \\sum_{i=1}^n\n\\left|d_i^*-\\frac{2m}{n}\\right|.\n\\]",
   "public_statement": "6. (Grone-Merris conjecture) Is λ majorized by d∗?\n\nIt was shown at the workshop that if the Grone-Merris conjecture is true, then\n\nEL(G) ≤\n\n> n\n\nX\n\n> i=1\n\n˛˛˛˛d∗\n\n> i\n\n− 2m\n\nn\n\n˛˛˛˛.\n\n5",
   "evidence": "The exact canonical record is retained in `input.json`. Its extracted problem field reads:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-linear-algebra-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-LOGIC-0001": {
   "statement_status": "exact",
   "original_statement": "Let $\\kappa$ be a regular cardinal and $\\gamma$ an infinite ordinal. Consider the following two-player game (called the \\emph{Welch game}): Player $I$ plays $\\kappa$-algebras $\\mathscr{A}_i$, while Player II plays increasing $\\kappa$-complete filters $\\mathcal{F}_i$ on $\\mathscr{A}_i$. The game goes on for $\\gamma$ many steps. The first player to break one of the rules loses.\n\n\\begin{enumerate}\n\\item For what values of $\\kappa$ and $\\gamma$ are these games determined?\n\\item How can these games be generalized to, for example, extender sequences? What about strong compactness and supercompactness?\n\\end{enumerate}",
   "clean_statement": "Let $\\kappa$ be a regular cardinal and $\\gamma$ an infinite ordinal. Consider the following two-player game (called the \\emph{Welch game}): Player $I$ plays $\\kappa$-algebras $\\mathscr{A}_i$, while Player II plays increasing $\\kappa$-complete filters $\\mathcal{F}_i$ on $\\mathscr{A}_i$. The game goes on for $\\gamma$ many steps. The first player to break one of the rules loses.\n\n\\begin{enumerate}\n\\item For what values of $\\kappa$ and $\\gamma$ are these games determined?\n\\item How can these games be generalized to, for example, extender sequences? What about strong compactness and supercompactness?\n\\end{enumerate}",
   "public_statement": "Let $\\kappa$ be a regular cardinal and $\\gamma$ an infinite ordinal. Consider the following two-player game (called the \\emph{Welch game}): Player $I$ plays $\\kappa$-algebras $\\mathscr{A}_i$, while Player II plays increasing $\\kappa$-complete filters $\\mathcal{F}_i$ on $\\mathscr{A}_i$. The game goes on for $\\gamma$ many steps. The first player to break one of the rules loses.\n\n\\begin{enumerate}\n\\item For what values of $\\kappa$ and $\\gamma$ are these games determined?\n\\item How can these games be generalized to, for example, extender sequences? What about strong compactness and supercompactness?\n\\end{enumerate}",
   "evidence": "The canonical AIM record asks about a game of length an infinite ordinal \\(\\gamma\\) on a regular cardinal \\(\\kappa\\), saying only that Player I plays “\\(\\kappa\\)-algebras” and Player II plays increasing \\(\\kappa\\)-complete filters. That extraction is materially incomplete. The formal definition in Foreman--Magidor--Zeman (FMZ) is as follows.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-LOGIC-0002": {
   "statement_status": "exact",
   "original_statement": "Suppose $\\kappa$ is singular with cofinality $\\omega_1$. Let $\\lambda=\\kappa^+$. Must $\\square_{\\omega_1}^*$ hold in a forcing extension where $\\lambda$ is $\\aleph_2$?",
   "clean_statement": "Suppose $\\kappa$ is singular with cofinality $\\omega_1$. Let $\\lambda=\\kappa^+$. Must $\\square_{\\omega_1}^*$ hold in a forcing extension where $\\lambda$ is $\\aleph_2$?",
   "public_statement": "Suppose $\\kappa$ is singular with cofinality $\\omega_1$. Let $\\lambda=\\kappa^+$. Must $\\square_{\\omega_1}^*$ hold in a forcing extension where $\\lambda$ is $\\aleph_2$?",
   "evidence": "There is no visible OCR corruption. The original AIM problem page timed out during this run, but the official 2023 workshop report independently gives the underlying question in the following two-model form:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 1,
   "attempt": 2
  },
  "AIM-LOGIC-0003": {
   "statement_status": "exact",
   "original_statement": "Let $\\kappa$ be an infinite cardinal and put $\\lambda=\\kappa^+$. Define $u_2(\\kappa)=\\sup\\left\\{(\\lambda^+)^{L[A]}:A\\subseteq\\kappa\\right\\}$.\n\n\\begin{enumerate}\n\\item Is $u_2(\\omega_1)=\\omega_3$ consistent with (sufficiently) large cardinals?\n\\item How to increase $u_2(\\omega_1)$ without collapsing $\\omega_1$ or $\\omega_2$?\n\\end{enumerate}",
   "clean_statement": "Let $\\kappa$ be an infinite cardinal and put $\\lambda=\\kappa^+$. Define $u_2(\\kappa)=\\sup\\left\\{(\\lambda^+)^{L[A]}:A\\subseteq\\kappa\\right\\}$.\n\n\\begin{enumerate}\n\\item Is $u_2(\\omega_1)=\\omega_3$ consistent with (sufficiently) large cardinals?\n\\item How to increase $u_2(\\omega_1)$ without collapsing $\\omega_1$ or $\\omega_2$?\n\\end{enumerate}",
   "public_statement": "Let $\\kappa$ be an infinite cardinal and put $\\lambda=\\kappa^+$. Define $u_2(\\kappa)=\\sup\\left\\{(\\lambda^+)^{L[A]}:A\\subseteq\\kappa\\right\\}$.\n\n\\begin{enumerate}\n\\item Is $u_2(\\omega_1)=\\omega_3$ consistent with (sufficiently) large cardinals?\n\\item How to increase $u_2(\\omega_1)$ without collapsing $\\omega_1$ or $\\omega_2$?\n\\end{enumerate}",
   "evidence": "The statement is mathematically coherent and shows no substantive OCR corruption. One source remark ends “a generic embedding with critical point $\\omega_1$ which fixed $\\omega_3$.” In present-tense mathematical prose this should read “which **fixes** $\\omega_3$.” This report preserves the source claim but does not silently treat the grammatical correction as a mathematical change. The referent of “it has a sharp” in the preceding remark is not explicit in the extracted record, and the record supplies no bibliography for that assertion.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 2,
   "attempt": 2
  },
  "AIM-LOGIC-0004": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Let $\\phi(n)$ be the statement ``for every graph of size $\\aleph_{\\omega+1}$, if every subgraph of size $<\\aleph_{\\omega+1}$ has chromatic number $\\le\\aleph_n$, then the entire graph has chromatic number $\\aleph_n$.''\n\nIs $\\phi(0)$ consistent?",
   "clean_statement": null,
   "public_statement": "Let $\\phi(n)$ be the statement ``for every graph of size $\\aleph_{\\omega+1}$, if every subgraph of size $<\\aleph_{\\omega+1}$ has chromatic number $\\le\\aleph_n$, then the entire graph has chromatic number $\\aleph_n$.''\n\nIs $\\phi(0)$ consistent?",
   "evidence": "The canonical record is AIM Problem 5.55, attributed on the original page to Magidor. The archived AIM page gives the following text (including the final equality):",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-LOGIC-0005": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Assume $\\text{NS}_{\\omega_1}$ is precipitous. Is it possible to force $\\text{NS}_{\\omega_1}$ to be non-precipitous without adding subsets of $\\omega_1$?",
   "clean_statement": null,
   "public_statement": "Assume $\\text{NS}_{\\omega_1}$ is precipitous. Is it possible to force $\\text{NS}_{\\omega_1}$ to be non-precipitous without adding subsets of $\\omega_1$?",
   "evidence": "1. \\(\\mathrm{NS}_{\\omega_1}^{+}\\) means the stationary/positive cone, normally viewed modulo the nonstationary ideal. 2. “A subset of size \\(\\omega_1\\)” has no stated ambient set. It cannot mean a new subset of \\(\\omega_1\\), since that is expressly forbidden in the question. Plausible readings are a new set of ordinals of cardinality \\(\\omega_1\\), or a new \\(\\omega_1\\)-sequence of ordinals. Nothing below silently chooses between these readings.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-LOGIC-0006": {
   "statement_status": "exact",
   "original_statement": "\\begin{enumerate}\n\\item Does Con(ZFC) imply Con(MM for posets of size $\\aleph_1$)?\n\n\\item Is there a stationary set preserving poset of size $\\aleph_1$ which is not semiproper?\n\\end{enumerate}",
   "clean_statement": "\\begin{enumerate}\n\\item Does Con(ZFC) imply Con(MM for posets of size $\\aleph_1$)?\n\n\\item Is there a stationary set preserving poset of size $\\aleph_1$ which is not semiproper?\n\\end{enumerate}",
   "public_statement": "\\begin{enumerate}\n\\item Does Con(ZFC) imply Con(MM for posets of size $\\aleph_1$)?\n\n\\item Is there a stationary set preserving poset of size $\\aleph_1$ which is not semiproper?\n\\end{enumerate}",
   "evidence": "The mathematical statement itself shows no substantive OCR corruption. I preserve it rather than silently modernizing it. Following Dobrinen--Krueger--Marun--Mota--Zapletal, I write $\\mathsf{MM}(\\omega_1)$ for “MM for posets of cardinality $\\omega_1$.” The original AIMPL detail URL was unavailable during this check, but the canonical JSON, the official AIM workshop page and report, and the subsequent paper agree on the intended question.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 5,
   "attempt": 2
  },
  "AIM-LOGIC-0007": {
   "statement_status": "exact",
   "original_statement": "Given $f\\in \\mathbb Z[x_1,\\dots, x_n]$, define the following sets.\n$$\n\tA(f) = \\{\\text{subrings of }\\mathbb Q \\text{ where $f$ has a solution}\\}\n$$\n$$\n\t\\mathcal C(f) = \\text{The interior of the complement of $A(f)$}\n$$\n$$\n\tB(f) =\\text{The boundary of the complement of $A(f)$}\n\t%= \\text{``the boundary set\"}\n$$\n\\begin{enumerate}\n\t\\item Can the Lebesgue measure of $B(f)$ be positive?\n\t\\item If so, what is the maximal complexity of the measure of $B(f)$?\n\t\\item Is $\\mathcal C(f)$ always a \\emph{finite} union of basic open sets?\n\\end{enumerate}",
   "clean_statement": "Given $f\\in \\mathbb Z[x_1,\\dots, x_n]$, define the following sets.\n$$\n\tA(f) = \\{\\text{subrings of }\\mathbb Q \\text{ where $f$ has a solution}\\}\n$$\n$$\n\t\\mathcal C(f) = \\text{The interior of the complement of $A(f)$}\n$$\n$$\n\tB(f) =\\text{The boundary of the complement of $A(f)$}\n\t%= \\text{``the boundary set\"}\n$$\n\\begin{enumerate}\n\t\\item Can the Lebesgue measure of $B(f)$ be positive?\n\t\\item If so, what is the maximal complexity of the measure of $B(f)$?\n\t\\item Is $\\mathcal C(f)$ always a \\emph{finite} union of basic open sets?\n\\end{enumerate}",
   "public_statement": "Given $f\\in \\mathbb Z[x_1,\\dots, x_n]$, define the following sets.\n$$\n\tA(f) = \\{\\text{subrings of }\\mathbb Q \\text{ where $f$ has a solution}\\}\n$$\n$$\n\t\\mathcal C(f) = \\text{The interior of the complement of $A(f)$}\n$$\n$$\n\tB(f) =\\text{The boundary of the complement of $A(f)$}\n\t%= \\text{``the boundary set\"}\n$$\n\\begin{enumerate}\n\t\\item Can the Lebesgue measure of $B(f)$ be positive?\n\t\\item If so, what is the maximal complexity of the measure of $B(f)$?\n\t\\item Is $\\mathcal C(f)$ always a \\emph{finite} union of basic open sets?\n\\end{enumerate}",
   "evidence": "The canonical record is Problem 1.1 in the “Computability” section of the AIM list *Definability and decidability problems in number theory*. The current AIM page attributes the question to Russell Miller. The source record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-LOGIC-0008": {
   "statement_status": "exact",
   "original_statement": "Let $K$ be an algebraically closed field of characteristic 0, and let $G$ be an ordered divisible group. Consider the Hahn field, defined as follows:\n$$\n K((G)) = \\left\\{\\sum_{g\\in S}a_g t^g \\mid S\\subseteq G \\text{ well-ordered}, a_g\\in K\\right\\}\n$$\n\nIf $K$ and $G$ are computable and $f\\in K((G))[X]$, how complex is it to determine a root of $f$?\nAt each step, the procedure should produce the next term in a root of $f$, based on the $a_g$ in the input coefficients of $f$. For each step $\\alpha$, how many jumps over these coefficients are needed to do step $\\alpha$?",
   "clean_statement": "Let $K$ be an algebraically closed field of characteristic 0, and let $G$ be an ordered divisible group. Consider the Hahn field, defined as follows:\n$$\n K((G)) = \\left\\{\\sum_{g\\in S}a_g t^g \\mid S\\subseteq G \\text{ well-ordered}, a_g\\in K\\right\\}\n$$\n\nIf $K$ and $G$ are computable and $f\\in K((G))[X]$, how complex is it to determine a root of $f$?\nAt each step, the procedure should produce the next term in a root of $f$, based on the $a_g$ in the input coefficients of $f$. For each step $\\alpha$, how many jumps over these coefficients are needed to do step $\\alpha$?",
   "public_statement": "Let $K$ be an algebraically closed field of characteristic 0, and let $G$ be an ordered divisible group. Consider the Hahn field, defined as follows:\n$$\n K((G)) = \\left\\{\\sum_{g\\in S}a_g t^g \\mid S\\subseteq G \\text{ well-ordered}, a_g\\in K\\right\\}\n$$\n\nIf $K$ and $G$ are computable and $f\\in K((G))[X]$, how complex is it to determine a root of $f$?\nAt each step, the procedure should produce the next term in a root of $f$, based on the $a_g$ in the input coefficients of $f$. For each step $\\alpha$, how many jumps over these coefficients are needed to do step $\\alpha$?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-LOGIC-0009": {
   "statement_status": "exact",
   "original_statement": "Given an elliptic curve $E$ given by $y^2 = x^3 + Ax + B$:\n\\begin{enumerate}\n\t\\item Is there are algorithm that decides if $E(\\mathbb Z) \\setminus\\{\\infty\\}$ is empty?\n\t\\item Is there an algorithm for enumerating $E(\\mathbb Z)$?\n\t\\item Can these questions be answered if $E$ is replaced with an arbitrary curve?\n\\end{enumerate}",
   "clean_statement": "Given an elliptic curve $E$ given by $y^2 = x^3 + Ax + B$:\n\\begin{enumerate}\n\t\\item Is there are algorithm that decides if $E(\\mathbb Z) \\setminus\\{\\infty\\}$ is empty?\n\t\\item Is there an algorithm for enumerating $E(\\mathbb Z)$?\n\t\\item Can these questions be answered if $E$ is replaced with an arbitrary curve?\n\\end{enumerate}",
   "public_statement": "Given an elliptic curve $E$ given by $y^2 = x^3 + Ax + B$:\n\\begin{enumerate}\n\t\\item Is there are algorithm that decides if $E(\\mathbb Z) \\setminus\\{\\infty\\}$ is empty?\n\t\\item Is there an algorithm for enumerating $E(\\mathbb Z)$?\n\t\\item Can these questions be answered if $E$ is replaced with an arbitrary curve?\n\\end{enumerate}",
   "evidence": "The canonical record is AIM Problem List 3.1 from the 2019 workshop *Definability and decidability problems in number theory*. An archived copy of the AIM page attributes the problem to Chris Hall and Alexandra Shlapentokh and gives the following wording:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-LOGIC-0010": {
   "statement_status": "exact",
   "original_statement": "\\begin{enumerate}\n\\item Is the first order theory of $\\mathbb C(t)$ decidable in signature with $1$ and $t$?\n\\item Does it help if $\\frac{\\partial}{\\partial t_i}$ is added to the signature?\n\\end{enumerate}",
   "clean_statement": "\\begin{enumerate}\n\\item Is the first order theory of $\\mathbb C(t)$ decidable in signature with $1$ and $t$?\n\\item Does it help if $\\frac{\\partial}{\\partial t_i}$ is added to the signature?\n\\end{enumerate}",
   "public_statement": "\\begin{enumerate}\n\\item Is the first order theory of $\\mathbb C(t)$ decidable in signature with $1$ and $t$?\n\\item Does it help if $\\frac{\\partial}{\\partial t_i}$ is added to the signature?\n\\end{enumerate}",
   "evidence": "The live AIM page was unavailable during this attempt, but the archived page was recovered. Its earliest available capture, dated 10 December 2019, and a 7 December 2023 capture have the same archived content digest and contain exactly the displayed wording. Thus the subscript \\(i\\) is not an OCR error introduced into this repository; the ambiguity is in the source itself.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 9,
   "attempt": 2
  },
  "AIM-LOGIC-0011": {
   "statement_status": "exact",
   "original_statement": "Is the existential theory of $\\{\\text{complex holomorphic functions over $\\mathbb C$}\\}$ in signature $0,1,+,\\cdot, z$ decidable?",
   "clean_statement": "Is the existential theory of $\\{\\text{complex holomorphic functions over $\\mathbb C$}\\}$ in signature $0,1,+,\\cdot, z$ decidable?",
   "public_statement": "Is the existential theory of $\\{\\text{complex holomorphic functions over $\\mathbb C$}\\}$ in signature $0,1,+,\\cdot, z$ decidable?",
   "evidence": "The AIM source page was checked against the canonical record. No OCR correction is needed, and the page supplies no additional remarks or attribution.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 10,
   "attempt": 2
  },
  "AIM-LOGIC-0012": {
   "statement_status": "exact",
   "original_statement": "Is $\\exists \\text{Th}(\\mathbb F_p(t))$, in the language of rings without a name for $t$, decidable?",
   "clean_statement": "Is $\\exists \\text{Th}(\\mathbb F_p(t))$, in the language of rings without a name for $t$, decidable?",
   "public_statement": "Is $\\exists \\text{Th}(\\mathbb F_p(t))$, in the language of rings without a name for $t$, decidable?",
   "evidence": "The page attributes the problem to Arno Fehm. The archived wording matches the repository record; no OCR corruption was found.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-LOGIC-0013": {
   "statement_status": "exact",
   "original_statement": "Let $p$ be a fixed prime. Define\n$$\n\tx \\mid_p y \\Leftrightarrow \\exists s\\in \\mathbb N, y = p^sx\n$$\nDoes $(\\mathbb Z, +, \\mid_p, 0, 1)$ have a decidable existential theory?",
   "clean_statement": "Let $p$ be a fixed prime. Define\n$$\n\tx \\mid_p y \\Leftrightarrow \\exists s\\in \\mathbb N, y = p^sx\n$$\nDoes $(\\mathbb Z, +, \\mid_p, 0, 1)$ have a decidable existential theory?",
   "public_statement": "Let $p$ be a fixed prime. Define\n$$\n\tx \\mid_p y \\Leftrightarrow \\exists s\\in \\mathbb N, y = p^sx\n$$\nDoes $(\\mathbb Z, +, \\mid_p, 0, 1)$ have a decidable existential theory?",
   "evidence": "The earliest available archived AIM page, from 10 December 2019, contains this exact wording and attributes the problem to Thanasis Pheidas. The tab before \\(x\\) is only source formatting; no mathematical OCR correction is needed.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-LOGIC-0014": {
   "statement_status": "exact",
   "original_statement": "Which fragments of $\\text{Th}(\\mathbb Q)$ are undecidable? Consider the number of quantifiers, not just alternations.",
   "clean_statement": "Which fragments of $\\text{Th}(\\mathbb Q)$ are undecidable? Consider the number of quantifiers, not just alternations.",
   "public_statement": "Which fragments of $\\text{Th}(\\mathbb Q)$ are undecidable? Consider the number of quantifiers, not just alternations.",
   "evidence": "This wording was checked against the archived AIM Problem List page for section 3, “Decidability,” captured on 2019-12-10. It agrees verbatim, is numbered Problem 3.6, and is attributed there to Arno Fehm. There is no OCR corruption and no source remark.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-LOGIC-0015": {
   "statement_status": "exact",
   "original_statement": "\\begin{enumerate}\n\\item Is $\\mathbb C[t_1,\\dots, t_n]$ definable in $\\mathbb C(t_1,\\dots, t_n)$?\n\\item Is there a diophantine definition?\n\\end{enumerate}",
   "clean_statement": "\\begin{enumerate}\n\\item Is $\\mathbb C[t_1,\\dots, t_n]$ definable in $\\mathbb C(t_1,\\dots, t_n)$?\n\\item Is there a diophantine definition?\n\\end{enumerate}",
   "public_statement": "\\begin{enumerate}\n\\item Is $\\mathbb C[t_1,\\dots, t_n]$ definable in $\\mathbb C(t_1,\\dots, t_n)$?\n\\item Is there a diophantine definition?\n\\end{enumerate}",
   "evidence": "The live AIM page was checked against the record. It has exactly this text, no remarks, and no specification of language or permitted parameters. There is no OCR error.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-LOGIC-0016": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Define a non-trivial valuation on $\\mathbb C(t_1,\\dots, t_n)$.",
   "clean_statement": null,
   "public_statement": "Define a non-trivial valuation on $\\mathbb C(t_1,\\dots, t_n)$.",
   "evidence": "The canonical record, `aim-logic-notes.json`, index 15, gives exactly:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-LOGIC-0017": {
   "statement_status": "exact",
   "original_statement": "Is $(\\mathbb Z, +, \\cdot)$ definable as a subring by first-order formulas in $\\mathbb C(t_1,\\dots, t_k)$?",
   "clean_statement": "Is $(\\mathbb Z, +, \\cdot)$ definable as a subring by first-order formulas in $\\mathbb C(t_1,\\dots, t_k)$?",
   "public_statement": "Is $(\\mathbb Z, +, \\cdot)$ definable as a subring by first-order formulas in $\\mathbb C(t_1,\\dots, t_k)$?",
   "evidence": "The archived AIM page was checked against the JSON record. The wording agrees exactly; the page gives no attribution, status, remarks, or language convention. There is no OCR error. Nearby Problems 4.1 and 4.2 ask, respectively, about defining the polynomial ring and a nontrivial valuation. The official report of the May 2019 workshop describes work on the polynomial-ring problem but does not report progress on Problem 4.3 [AIM-2019].",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-LOGIC-0018": {
   "statement_status": "exact",
   "original_statement": "Is there a diophantine subset $X$ of $\\mathbb Q$ such that $\\sup\\{x\\mid x\\in X\\}$ is transcendental?",
   "clean_statement": "Is there a diophantine subset $X$ of $\\mathbb Q$ such that $\\sup\\{x\\mid x\\in X\\}$ is transcendental?",
   "public_statement": "Is there a diophantine subset $X$ of $\\mathbb Q$ such that $\\sup\\{x\\mid x\\in X\\}$ is transcendental?",
   "evidence": "The canonical record is AIM Problem 5.2 in the section “Hilbert’s Tenth Problem for Subrings of \\(\\mathbb Q\\)” of the workshop *Definability and decidability problems in number theory*. The archived AIM page attributes the question to Hector Pasten and reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-LOGIC-0019": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Let $X/\\mathbb Q$ be a variety, and let $\\{Y_a\\}_{a\\in A}$ be a set of uniformly definable subsets of $X$.\n\n\\begin{enumerate}\n\\item Is $\\overline{Y_a(\\mathbb Q)} = Y_a(\\mathbb R)$?\n\\item Does this imply that $\\mathbb Z$ is not diophantine in $\\mathbb Q$?\n\\end{enumerate}",
   "clean_statement": null,
   "public_statement": "Let $X/\\mathbb Q$ be a variety, and let $\\{Y_a\\}_{a\\in A}$ be a set of uniformly definable subsets of $X$.\n\n\\begin{enumerate}\n\\item Is $\\overline{Y_a(\\mathbb Q)} = Y_a(\\mathbb R)$?\n\\item Does this imply that $\\mathbb Z$ is not diophantine in $\\mathbb Q$?\n\\end{enumerate}",
   "evidence": "The canonical record, `aim-logic-notes.json`, zero-based index 18, gives exactly:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-LOGIC-0020": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "What is the structure of HTP for big rings in $\\mathbb Q$ under $\\leq_T$? (Here, ``big rings\" mean rings where infinitely many primes are inverted.)",
   "clean_statement": null,
   "public_statement": "What is the structure of HTP for big rings in $\\mathbb Q$ under $\\leq_T$? (Here, ``big rings\" mean rings where infinitely many primes are inverted.)",
   "evidence": "The **recovered statement** is AIM Problem List 5.6 from the workshop *Definability and decidability problems in number theory*:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-LOGIC-0021": {
   "statement_status": "exact",
   "original_statement": "Prove or disprove the following conjecture.\n\n Let $K$ be a global field, let $v$ be a place of $K$, and let $X$ and $Y$ be projective varieties which are not singletons. Suppose that $X(K)$ is Zariski dense in $X$. Given a dominant rational map $f: X \\dashrightarrow Y$ and a nonempty Zariski open set $U\\subset X$, there is a Cartier divisor $D > 0$ on $Y$, defined over $K$, such that some sequence of points in $(U \\setminus f^*D)(K)$ approaches $f^*D$ $v$-adically.",
   "clean_statement": "Prove or disprove the following conjecture.\n\n Let $K$ be a global field, let $v$ be a place of $K$, and let $X$ and $Y$ be projective varieties which are not singletons. Suppose that $X(K)$ is Zariski dense in $X$. Given a dominant rational map $f: X \\dashrightarrow Y$ and a nonempty Zariski open set $U\\subset X$, there is a Cartier divisor $D > 0$ on $Y$, defined over $K$, such that some sequence of points in $(U \\setminus f^*D)(K)$ approaches $f^*D$ $v$-adically.",
   "public_statement": "Prove or disprove the following conjecture.\n\n Let $K$ be a global field, let $v$ be a place of $K$, and let $X$ and $Y$ be projective varieties which are not singletons. Suppose that $X(K)$ is Zariski dense in $X$. Given a dominant rational map $f: X \\dashrightarrow Y$ and a nonempty Zariski open set $U\\subset X$, there is a Cartier divisor $D > 0$ on $Y$, defined over $K$, such that some sequence of points in $(U \\setminus f^*D)(K)$ approaches $f^*D$ $v$-adically.",
   "evidence": "The archived AIM page reproduces this wording exactly, so there is no OCR error in the repository record. The page attributes Problem 6.1 to Hector Pasten. The more precise published formulation in Pasten's 2022 paper is the intended **recovered statement**:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-LOGIC-0022": {
   "statement_status": "exact",
   "original_statement": "\\begin{enumerate}\n\t\\item What are the automorphisms of cohesive powers $\\prod_C F$ for finite algebraic extensions $F$ of $\\mathbb Q$?\n\t\\item Are there any that do not arise from automorphisms of $F$?\n\\end{enumerate}",
   "clean_statement": "\\begin{enumerate}\n\t\\item What are the automorphisms of cohesive powers $\\prod_C F$ for finite algebraic extensions $F$ of $\\mathbb Q$?\n\t\\item Are there any that do not arise from automorphisms of $F$?\n\\end{enumerate}",
   "public_statement": "\\begin{enumerate}\n\t\\item What are the automorphisms of cohesive powers $\\prod_C F$ for finite algebraic extensions $F$ of $\\mathbb Q$?\n\t\\item Are there any that do not arise from automorphisms of $F$?\n\\end{enumerate}",
   "evidence": "The workshop report confirms this reading and explains that the 2019 working group had obtained a conditional negative answer to the second question, assuming a Diophantine definition of $\\mathbb Z$ over the ring of integers of $F$ [AIM19]. There is no substantive OCR error in the canonical record. The broken line in the report merely reflects PDF text extraction.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-LOGIC-0023": {
   "statement_status": "exact",
   "original_statement": "Consider the system of equations\n$$\n\t\\left\\{\n \\begin{array}{ll}\n a_1s_1 + \\dots + a_4s_4 &= c\\\\\n b_1s_1 + \\dots + b_4s_4 &= d\n \\end{array}\n \\right.\n$$\nwhere $a_i, b_j, c,d\\in \\mathbb Z$ for all $1\\leq i,j\\leq 4$. Suppose that this system is solvable with the $s_j$ in $\\mathbb Z$ modulo every prime $p$. Fix $k \\geq 4$, and let $s_j$ be the $j$-th symmetric function of $x_1, \\dots, x_k$. (E.g., $s_1 = x_1 + \\dots + x_k$.) Must this system have integer solutions for $x_1, \\dots, x_k$? What happens if we add more rows or columns?",
   "clean_statement": "Consider the system of equations\n$$\n\t\\left\\{\n \\begin{array}{ll}\n a_1s_1 + \\dots + a_4s_4 &= c\\\\\n b_1s_1 + \\dots + b_4s_4 &= d\n \\end{array}\n \\right.\n$$\nwhere $a_i, b_j, c,d\\in \\mathbb Z$ for all $1\\leq i,j\\leq 4$. Suppose that this system is solvable with the $s_j$ in $\\mathbb Z$ modulo every prime $p$. Fix $k \\geq 4$, and let $s_j$ be the $j$-th symmetric function of $x_1, \\dots, x_k$. (E.g., $s_1 = x_1 + \\dots + x_k$.) Must this system have integer solutions for $x_1, \\dots, x_k$? What happens if we add more rows or columns?",
   "public_statement": "Consider the system of equations\n$$\n\t\\left\\{\n \\begin{array}{ll}\n a_1s_1 + \\dots + a_4s_4 &= c\\\\\n b_1s_1 + \\dots + b_4s_4 &= d\n \\end{array}\n \\right.\n$$\nwhere $a_i, b_j, c,d\\in \\mathbb Z$ for all $1\\leq i,j\\leq 4$. Suppose that this system is solvable with the $s_j$ in $\\mathbb Z$ modulo every prime $p$. Fix $k \\geq 4$, and let $s_j$ be the $j$-th symmetric function of $x_1, \\dots, x_k$. (E.g., $s_1 = x_1 + \\dots + x_k$.) Must this system have integer solutions for $x_1, \\dots, x_k$? What happens if we add more rows or columns?",
   "evidence": "The **recovered statement** is AIM Problem 6.5 from the workshop *Definability and decidability problems in number theory*, section “Miscellaneous”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-LOGIC-0024": {
   "statement_status": "exact",
   "original_statement": "5. Is it consistent that for every sequence 〈Sn|n ∈ ω〉 with each Sn ⊆ ℵ n+2 ∩\n\ncof( ω1), each Sn, the sequence is mutually stationary? Lower bounds are known: inner model with infinitely many cardinals κn\n\nsuch that for all m the class of measurables λ < κ n with Mitchell order at least κm is stationary in V for n > m. (Koepke-Welch) 6. Is M M (c+) consistent with Woodin's Axiom ( ∗)? Known: Assume M M ++ for arbitrary partial orders, weak UBH (a proper class of Woodins, extender sequences witnessing Woodinness; then UBH holds for those extender sequences.); let Γ ∞ be the universally Baire sets. Suppose θuB > ℵ1. Then ( ∗) holds. (Schindler-Woodin) 7. Does Th( L(Γ uB )) = Th( L(Γ uB )) V P\n\n(with constant symbols for each uB set) for all P, plus a proper class of Woodin cardinals, plus M M ++, imply cof( θuB ) > ℵ1?Known: M M ++ + weak UBH + proper class of Woodins =⇒ TFAE: (a) cof( θuB ) > ℵ1\n\n(b) ∃ semiproper P adding uB A such that A > w B for all uB B in V\n\n(conjecture: both are true)",
   "clean_statement": "5. Is it consistent that for every sequence 〈Sn|n ∈ ω〉 with each Sn ⊆ ℵ n+2 ∩\n\ncof( ω1), each Sn, the sequence is mutually stationary? Lower bounds are known: inner model with infinitely many cardinals κn\n\nsuch that for all m the class of measurables λ < κ n with Mitchell order at least κm is stationary in V for n > m. (Koepke-Welch) 6. Is M M (c+) consistent with Woodin's Axiom ( ∗)? Known: Assume M M ++ for arbitrary partial orders, weak UBH (a proper class of Woodins, extender sequences witnessing Woodinness; then UBH holds for those extender sequences.); let Γ ∞ be the universally Baire sets. Suppose θuB > ℵ1. Then ( ∗) holds. (Schindler-Woodin) 7. Does Th( L(Γ uB )) = Th( L(Γ uB )) V P\n\n(with constant symbols for each uB set) for all P, plus a proper class of Woodin cardinals, plus M M ++, imply cof( θuB ) > ℵ1?Known: M M ++ + weak UBH + proper class of Woodins =⇒ TFAE: (a) cof( θuB ) > ℵ1\n\n(b) ∃ semiproper P adding uB A such that A > w B for all uB B in V\n\n(conjecture: both are true)",
   "public_statement": "5. Is it consistent that for every sequence 〈Sn|n ∈ ω〉 with each Sn ⊆ ℵ n+2 ∩\n\ncof( ω1), each Sn, the sequence is mutually stationary? Lower bounds are known: inner model with infinitely many cardinals κn\n\nsuch that for all m the class of measurables λ < κ n with Mitchell order at least κm is stationary in V for n > m. (Koepke-Welch) 6. Is M M (c+) consistent with Woodin's Axiom ( ∗)? Known: Assume M M ++ for arbitrary partial orders, weak UBH (a proper class of Woodins, extender sequences witnessing Woodinness; then UBH holds for those extender sequences.); let Γ ∞ be the universally Baire sets. Suppose θuB > ℵ1. Then ( ∗) holds. (Schindler-Woodin) 7. Does Th( L(Γ uB )) = Th( L(Γ uB )) V P\n\n(with constant symbols for each uB set) for all P, plus a proper class of Woodin cardinals, plus M M ++, imply cof( θuB ) > ℵ1?Known: M M ++ + weak UBH + proper class of Woodins =⇒ TFAE: (a) cof( θuB ) > ℵ1\n\n(b) ∃ semiproper P adding uB A such that A > w B for all uB B in V\n\n(conjecture: both are true)",
   "evidence": "This canonical record is an extraction accident: it combines printed Problems 5, 6, and 7, followed by Remark 1, from the AIM workshop notes *Descriptive Inner Model Theory* (June 2--6, 2014). It remains one canonical job here. The exact repository `problem` field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-LOGIC-0025": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "8. What is the consistency strength of M M (c)? Upper bound: AD R + Θ is regular (Woodin: Pmax book) Lower bound: AD L(R) is safe (Steel-Zoble), more may be known. 9. What is the consistency strength of ¬\u0003ω2 + ¬\u0003(ω2) + 2 ω1 = ω2?Upper bound: weaker than AD R + Θ is Mahlo. \n\n{α|cof( θα) ≥ ℵ 2 + θα regular in HOD }\n\nLower bound: PD (maybe AD L(R)?) \n\n> 3Take an elementary substructure where the cofinalities alternate. It never projects in L;get an elementary embedding L→L.",
   "clean_statement": null,
   "public_statement": "8. What is the consistency strength of M M (c)? Upper bound: AD R + Θ is regular (Woodin: Pmax book) Lower bound: AD L(R) is safe (Steel-Zoble), more may be known. 9. What is the consistency strength of ¬[U+0003]ω2 + ¬[U+0003](ω2) + 2 ω1 = ω2?Upper bound: weaker than AD R + Θ is Mahlo.\n\n{α|cof( θα) ≥ ℵ 2 + θα regular in HOD }\n\nLower bound: PD (maybe AD L(R)?)\n\n> 3Take an elementary substructure where the cofinalities alternate. It never projects in L;get an elementary embedding L→L.",
   "evidence": "The canonical record is corrupted and merges two consecutive printed questions. To preserve it exactly, its `problem` field is reproduced here in JSON-escaped form (so the control character is visible):",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 24,
   "attempt": 2
  },
  "AIM-LOGIC-0026": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "210. What is the consistency strength of \" ℵ2 and ℵ3 both have the tree prop-erty\"? Upper bound: weakly compact above a supercompact. (Abraham) Lower bound: nowadays the argument in Foreman-Magidor-Schindler would give a Woodin cardinal. (2 is open) 11. Is there a unique model L(R, μ ) such that L(R, μ ) satisfies μ is a normal fine measure on Pω1 (R)? What is the consistency strength of such a pair? Lower bound: ω2 Woodins. Known: If L(R, μ ) and L(R, ν ) are two such models, then P(R)∩L(R, μ ) ⊆\n\nL(R, ν ) or vice versa. 12. Does BMM =⇒ 0¶ exists? Upper bound: BMM gives an inner model with a strong cardinal. (Schindler) Lower bound: BMM is consistent from ω + 1 Woodins plus a measurable. (Woodin) 13. \"Dual covering theorem\" for ( M, λ, δ ) is the statement: For every λ, there is f: λ<ω → λ such that ∀X ⊆ Ord closed under f, X is a union of \n\nδ-many sets in M.For reasonable inner models M, can you get the failure of dual covering for ( M, ℵ3, ℵ1) from some large cardinals? E.g.: (a) Assuming no proper class model with a Woodin cardinal, M is the one-Woodin K.(b) Assuming no proper class model with a strong cardinal, M is the one-Woodin K?14. The Axiom of Strong Condensation: ∀κ > ω there is a bijection h: κ →\n\nH(κ) such that for all X ≺ (H(κ), h ), π[X ∩ h] = h \u0016 ot (X ∩ κ), for π the uncollapse. Suppose N is an inner model satisfying strong condensation, and covering fails relative to N. Must N exist? 4",
   "clean_statement": null,
   "public_statement": "210. What is the consistency strength of \" ℵ2 and ℵ3 both have the tree prop-erty\"? Upper bound: weakly compact above a supercompact. (Abraham) Lower bound: nowadays the argument in Foreman-Magidor-Schindler would give a Woodin cardinal. (2 is open) 11. Is there a unique model L(R, μ ) such that L(R, μ ) satisfies μ is a normal fine measure on Pω1 (R)? What is the consistency strength of such a pair? Lower bound: ω2 Woodins. Known: If L(R, μ ) and L(R, ν ) are two such models, then P(R)∩L(R, μ ) ⊆\n\nL(R, ν ) or vice versa. 12. Does BMM =⇒ 0¶ exists? Upper bound: BMM gives an inner model with a strong cardinal. (Schindler) Lower bound: BMM is consistent from ω + 1 Woodins plus a measurable. (Woodin) 13. \"Dual covering theorem\" for ( M, λ, δ ) is the statement: For every λ, there is f: λ<ω → λ such that ∀X ⊆ Ord closed under f, X is a union of\n\nδ-many sets in M.For reasonable inner models M, can you get the failure of dual covering for ( M, ℵ3, ℵ1) from some large cardinals? E.g.: (a) Assuming no proper class model with a Woodin cardinal, M is the one-Woodin K.(b) Assuming no proper class model with a strong cardinal, M is the one-Woodin K?14. The Axiom of Strong Condensation: ∀κ > ω there is a bijection h: κ →\n\nH(κ) such that for all X ≺ (H(κ), h ), π[X ∩ h] = h [U+0016] ot (X ∩ κ), for π the uncollapse. Suppose N is an inner model satisfying strong condensation, and covering fails relative to N. Must N exist? 4",
   "evidence": "The canonical record is a damaged extraction from the official four-page PDF *Problem Session Notes*, AIM Workshop on Descriptive Inner Model Theory, June 2--6, 2014. It merges Questions 10--14. The exact OCR record remains unchanged in *input.json*; in particular, its raw Question 14 contains the byte U+0016, represented here safely as '<U+0016>' rather than copied into this artifact.",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 25,
   "attempt": 2
  },
  "AIM-LOGIC-0027": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "15. Suppose there is no inner model with a Woodin cardinal, and let κ be a singular cardinal in K. Suppose κ is a singular cardinal in V. Must κ be measurable in K?For K below 0 ¶ this is known (Cox). \n\n> 4If Nis a model of condensation there is a function which witnesses it uniformly for all κ\n> - so indiscernibles relative to that would do.",
   "clean_statement": null,
   "public_statement": "15. Suppose there is no inner model with a Woodin cardinal, and let κ be a singular cardinal in K. Suppose κ is a singular cardinal in V. Must κ be measurable in K?For K below 0 ¶ this is known (Cox).\n\n> 4If Nis a model of condensation there is a function which witnesses it uniformly for all κ\n> - so indiscernibles relative to that would do.",
   "evidence": "The canonical repository record is a corrupted extraction of printed Problem 15 in the AIM workshop notes *Descriptive Inner Model Theory* (June 2--6, 2014). Its exact `problem` field is preserved here:",
   "classification_method": "invalid_literal_with_repaired_reading",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-LOGIC-0028": {
   "statement_status": "exact",
   "original_statement": "316. Suppose there is no inner model with a Woodin cardinal, and κ is a singular strong limit of uncountable cofinality, with 2 κ = λ, some regular λ > κ +.Must o(κ)K ≥ λ?A negative answer may have applications in pcf theory. Known below 0 ¶ (Gitik-Mitchell). 4",
   "clean_statement": "316. Suppose there is no inner model with a Woodin cardinal, and κ is a singular strong limit of uncountable cofinality, with 2 κ = λ, some regular λ > κ +.Must o(κ)K ≥ λ?A negative answer may have applications in pcf theory. Known below 0 ¶ (Gitik-Mitchell). 4",
   "public_statement": "316. Suppose there is no inner model with a Woodin cardinal, and κ is a singular strong limit of uncountable cofinality, with 2 κ = λ, some regular λ > κ +.Must o(κ)K ≥ λ?A negative answer may have applications in pcf theory. Known below 0 ¶ (Gitik-Mitchell). 4",
   "evidence": "The exact corpus field is visibly corrupted:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-LOGIC-0029": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Let $\\mathbb{C} = (C,\\le)$ be a linear order of size $\\kappa$ and consider its Cartesian square under the ordering $(x_1,y_1) \\le (x_2,y_2)$ iff $x_1 \\le x_2$ and $y_1 \\le y_2$. $\\mathbb C$ is a \\emph{Countryman line} if this Cartesian square is the union of less than $\\kappa$-many chains (i.e. linearly ordered subsets).\n\nIs it consistent (with the continuum hypothesis) that there is a minimal $\\aleph_2$-Countryman line-that is, an $\\aleph_2$-Countryman line that order-embeds into all others?",
   "clean_statement": null,
   "public_statement": "Let $\\mathbb{C} = (C,\\le)$ be a linear order of size $\\kappa$ and consider its Cartesian square under the ordering $(x_1,y_1) \\le (x_2,y_2)$ iff $x_1 \\le x_2$ and $y_1 \\le y_2$. $\\mathbb C$ is a \\emph{Countryman line} if this Cartesian square is the union of less than $\\kappa$-many chains (i.e. linearly ordered subsets).\n\nIs it consistent (with the continuum hypothesis) that there is a minimal $\\aleph_2$-Countryman line-that is, an $\\aleph_2$-Countryman line that order-embeds into all others?",
   "evidence": "The canonical statement is well formed. Its exact repository `problem` field is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-LOGIC-0030": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Two linear orders $\\mathbb{C}_1$ and $\\mathbb{C}_2$ are \\emph{near} if there is another linear order $\\mathbb{C}_0$ that embeds into both of them. $\\mathbb{C}_1$ and $\\mathbb{C}_2$ are \\emph{co-near} if there is a linear order embedding into $\\mathbb{C}_1$ and $\\mathbb{C}_2^\\ast$, where $\\mathbb{C}_2^\\ast$ is the reverse of $\\mathbb{C}_2$.\n\nIs it consistent with the continuum hypothesis that any two $\\aleph_2$-Countryman lines are near or co-near?",
   "clean_statement": null,
   "public_statement": "Two linear orders $\\mathbb{C}_1$ and $\\mathbb{C}_2$ are \\emph{near} if there is another linear order $\\mathbb{C}_0$ that embeds into both of them. $\\mathbb{C}_1$ and $\\mathbb{C}_2$ are \\emph{co-near} if there is a linear order embedding into $\\mathbb{C}_1$ and $\\mathbb{C}_2^\\ast$, where $\\mathbb{C}_2^\\ast$ is the reverse of $\\mathbb{C}_2$.\n\nIs it consistent with the continuum hypothesis that any two $\\aleph_2$-Countryman lines are near or co-near?",
   "evidence": "The exact canonical statement is:",
   "classification_method": "invalid_literal_with_repaired_reading",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-LOGIC-0031": {
   "statement_status": "exact",
   "original_statement": "Strong homology is a homology theory satisfying the Eilenberg-Steenrod axioms and is invariant under strong shape.\n\nIs strong homology consistently additive for closed subspaces of Euclidean space, or more generally for locally compact metric spaces?",
   "clean_statement": "Strong homology is a homology theory satisfying the Eilenberg-Steenrod axioms and is invariant under strong shape.\n\nIs strong homology consistently additive for closed subspaces of Euclidean space, or more generally for locally compact metric spaces?",
   "public_statement": "Strong homology is a homology theory satisfying the Eilenberg-Steenrod axioms and is invariant under strong shape.\n\nIs strong homology consistently additive for closed subspaces of Euclidean space, or more generally for locally compact metric spaces?",
   "evidence": "The live AIM Problem Lists page was fetched and inspected. It contains the same wording and literature note, attributes Problem 1.15 to Justin Moore, and places it in “Problems in Low Forcing.” No OCR correction is needed. The old statement about \\(\\mathfrak d\\) is preserved exactly above, but Section 2 treats it as a historical heuristic rather than a theorem.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-LOGIC-0032": {
   "statement_status": "exact",
   "original_statement": "$\\mathbb A = \\langle A_f : f \\in \\omega^\\omega \\rangle$ is the following inverse system of abelian groups:\n\n$$ A_f = \\bigoplus_{n \\in \\omega} \\bigoplus_{i < f(n)} \\mathbb Z = \\bigoplus_{n \\in \\omega} {\\mathbb Z}^{f(n)}$$\n\nLet ${\\mathbb Z}^{\\omega \\times \\omega}/fin$ be ${\\mathbb Z}^{\\omega \\times \\omega}$ modulo finite equivalence, let $G_f = \\prod_{n \\in \\omega} {\\mathbb Z}^{f(n)}/fin$, and consider the chain complex\n\n$${\\mathbb Z}^{\\omega \\times \\omega}/fin \\xrightarrow{\\delta} \\prod_{f_0 \\in \\omega^\\omega} G_{f_0} \\xrightarrow{\\delta} \\prod_{f_0 \\le f_1} G_{f_0} \\xrightarrow{\\delta} \\prod_{f_0 \\le f_1 \\le f_2} G_{f_0} \\xrightarrow{\\delta} \\ldots$$\n\nwhere\n\n$$\\delta_s(f_0,f_1,\\ldots,f_n) = \\sum^n_{i = 0}(-1)^i s(f_0,\\ldots,\\hat{f}_i,\\ldots,f_n)$$\n\nfor $s \\in \\prod_{f_0 \\le \\ldots \\le f_{n-1}} G_{f_0}$.\n\nWe define $\\lim^p \\mathbb A \\cong \\ker(\\delta^p)/\\text{im}(\\delta^{p-1})$.\n\nIs it consistent to have $\\lim^p \\mathbb{A} = 0$ for all $p$?",
   "clean_statement": "$\\mathbb A = \\langle A_f : f \\in \\omega^\\omega \\rangle$ is the following inverse system of abelian groups:\n\n$$ A_f = \\bigoplus_{n \\in \\omega} \\bigoplus_{i < f(n)} \\mathbb Z = \\bigoplus_{n \\in \\omega} {\\mathbb Z}^{f(n)}$$\n\nLet ${\\mathbb Z}^{\\omega \\times \\omega}/fin$ be ${\\mathbb Z}^{\\omega \\times \\omega}$ modulo finite equivalence, let $G_f = \\prod_{n \\in \\omega} {\\mathbb Z}^{f(n)}/fin$, and consider the chain complex\n\n$${\\mathbb Z}^{\\omega \\times \\omega}/fin \\xrightarrow{\\delta} \\prod_{f_0 \\in \\omega^\\omega} G_{f_0} \\xrightarrow{\\delta} \\prod_{f_0 \\le f_1} G_{f_0} \\xrightarrow{\\delta} \\prod_{f_0 \\le f_1 \\le f_2} G_{f_0} \\xrightarrow{\\delta} \\ldots$$\n\nwhere\n\n$$\\delta_s(f_0,f_1,\\ldots,f_n) = \\sum^n_{i = 0}(-1)^i s(f_0,\\ldots,\\hat{f}_i,\\ldots,f_n)$$\n\nfor $s \\in \\prod_{f_0 \\le \\ldots \\le f_{n-1}} G_{f_0}$.\n\nWe define $\\lim^p \\mathbb A \\cong \\ker(\\delta^p)/\\text{im}(\\delta^{p-1})$.\n\nIs it consistent to have $\\lim^p \\mathbb{A} = 0$ for all $p$?",
   "public_statement": "$\\mathbb A = \\langle A_f : f \\in \\omega^\\omega \\rangle$ is the following inverse system of abelian groups:\n\n$$ A_f = \\bigoplus_{n \\in \\omega} \\bigoplus_{i < f(n)} \\mathbb Z = \\bigoplus_{n \\in \\omega} {\\mathbb Z}^{f(n)}$$\n\nLet ${\\mathbb Z}^{\\omega \\times \\omega}/fin$ be ${\\mathbb Z}^{\\omega \\times \\omega}$ modulo finite equivalence, let $G_f = \\prod_{n \\in \\omega} {\\mathbb Z}^{f(n)}/fin$, and consider the chain complex\n\n$${\\mathbb Z}^{\\omega \\times \\omega}/fin \\xrightarrow{\\delta} \\prod_{f_0 \\in \\omega^\\omega} G_{f_0} \\xrightarrow{\\delta} \\prod_{f_0 \\le f_1} G_{f_0} \\xrightarrow{\\delta} \\prod_{f_0 \\le f_1 \\le f_2} G_{f_0} \\xrightarrow{\\delta} \\ldots$$\n\nwhere\n\n$$\\delta_s(f_0,f_1,\\ldots,f_n) = \\sum^n_{i = 0}(-1)^i s(f_0,\\ldots,\\hat{f}_i,\\ldots,f_n)$$\n\nfor $s \\in \\prod_{f_0 \\le \\ldots \\le f_{n-1}} G_{f_0}$.\n\nWe define $\\lim^p \\mathbb A \\cong \\ker(\\delta^p)/\\text{im}(\\delta^{p-1})$.\n\nIs it consistent to have $\\lim^p \\mathbb{A} = 0$ for all $p$?",
   "evidence": "The exact corpus record is AIM Problem List question 1.2 from the workshop *High and low forcing*, source file `aim-logic-notes.json`, zero-based record index 31. The supplied source URL is <http://aimpl.org/highlowforcing/1/>. Both its HTTP and HTTPS forms returned a 502 error during this run, so the live page could not be compared with the corpus record. The record itself is preserved verbatim in `input.json`.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 31,
   "attempt": 2
  },
  "AIM-LOGIC-0033": {
   "statement_status": "exact",
   "original_statement": "Is it consistent that there is a poset which adds a club in $\\omega_3$ which contains no infinite ground model subset over a model of GCH?",
   "clean_statement": "Is it consistent that there is a poset which adds a club in $\\omega_3$ which contains no infinite ground model subset over a model of GCH?",
   "public_statement": "Is it consistent that there is a poset which adds a club in $\\omega_3$ which contains no infinite ground model subset over a model of GCH?",
   "evidence": "The current AIM page has exactly this wording, with no attribution, status note, or preservation hypothesis. There is no visible OCR corruption. There is, however, a mathematically decisive ambiguity: does \\(\\omega_3\\) mean the fixed ground-model ordinal \\((\\omega_3)^V\\), with collapse or singularization allowed, or must the forcing preserve cardinals through \\(\\omega_3\\), so that the same ordinal is \\(\\omega_3\\) and regular in the extension? The neighboring AIM questions concern Namba forcing and its iteration, which makes the second, preservation-sensitive reading plausible, but it does not supply a missing hypothesis.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-LOGIC-0034": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Is there a large cardinal hypothesis that proves the bounded forcing axiom for Namba forcing?",
   "clean_statement": null,
   "public_statement": "Is there a large cardinal hypothesis that proves the bounded forcing axiom for Namba forcing?",
   "evidence": "The canonical record is AIM-LOGIC-0034, source file `aim-logic-notes.json`, zero-based index 33. Its statement is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-LOGIC-0035": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Can Namba forcing be iterated with side conditions? Does this work if we replace Namba forcing with a forcing satisfying Shelah's $S$-condition?",
   "clean_statement": null,
   "public_statement": "Can Namba forcing be iterated with side conditions? Does this work if we replace Namba forcing with a forcing satisfying Shelah's $S$-condition?",
   "evidence": "This is Problem 1.35 in the AIM workshop *High and low forcing* (January 11--15, 2016), section “Problems in Low Forcing.” The exact corpus text is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 34,
   "attempt": 2
  },
  "AIM-LOGIC-0036": {
   "statement_status": "exact",
   "original_statement": "Assume CH. Is there a strongest forcing axiom of $\\sigma$-closed posets meeting $\\aleph_2$-many dense sets? What if we do not assume CH?",
   "clean_statement": "Assume CH. Is there a strongest forcing axiom of $\\sigma$-closed posets meeting $\\aleph_2$-many dense sets? What if we do not assume CH?",
   "public_statement": "Assume CH. Is there a strongest forcing axiom of $\\sigma$-closed posets meeting $\\aleph_2$-many dense sets? What if we do not assume CH?",
   "evidence": "The canonical repository record is AIM Problem List item 1.4 from the workshop *High and low forcing*, section “Problems in Low Forcing.” Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-LOGIC-0037": {
   "statement_status": "exact",
   "original_statement": "The following is inconsistent: ``For every $\\Sigma_1$ statement $\\varphi$ with $\\aleph_1$ and $\\aleph_2$ as parameters, if $\\varphi$ can be forced by an $\\aleph_1, \\aleph_2$-preserving forcing, then $\\varphi$ holds.''",
   "clean_statement": "The following is inconsistent: ``For every $\\Sigma_1$ statement $\\varphi$ with $\\aleph_1$ and $\\aleph_2$ as parameters, if $\\varphi$ can be forced by an $\\aleph_1, \\aleph_2$-preserving forcing, then $\\varphi$ holds.''",
   "public_statement": "The following is inconsistent: ``For every $\\Sigma_1$ statement $\\varphi$ with $\\aleph_1$ and $\\aleph_2$ as parameters, if $\\varphi$ can be forced by an $\\aleph_1, \\aleph_2$-preserving forcing, then $\\varphi$ holds.''",
   "evidence": "The current AIM page agrees verbatim with the repository record and supplies no attribution, definitions, remarks, or status update. There is no OCR error, but four conventions must be made explicit.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-LOGIC-0038": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "If $X$ is a topological space, let $s(X)=\\sup\\{|Y|:Y \\text{ is a discrete subspace of }X\\}$. It is a theorem that $|X| \\le 2^{2^{s(X)}}$.\n\nWhen can one obtain $|X|\\le 2^{s(X)}$?",
   "clean_statement": "For which infinite Hausdorff spaces $X$ can the classical bound\n$|X|\\leq 2^{2^{s(X)}}$ be improved to $|X|\\leq 2^{s(X)}$?  In\nparticular, is it consistent that this improvement holds for every\nHausdorff (or every regular) space?",
   "public_statement": "If $X$ is a topological space, let $s(X)=\\sup\\{|Y|:Y \\text{ is a discrete subspace of }X\\}$. It is a theorem that $|X| \\le 2^{2^{s(X)}}$.\n\nWhen can one obtain $|X|\\le 2^{s(X)}$?",
   "evidence": "The canonical record says, verbatim:",
   "classification_method": "invalid_literal_with_repaired_reading",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-logic-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-LOGIC-0039": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Do any of the analogs of PFA at $\\aleph_2$ give us $|X| \\le 2^{\\aleph_1}$ for every Hausdorff space with no discrete subspaces of size $\\aleph_2$?",
   "clean_statement": null,
   "public_statement": "Do any of the analogs of PFA at $\\aleph_2$ give us $|X| \\le 2^{\\aleph_1}$ for every Hausdorff space with no discrete subspaces of size $\\aleph_2$?",
   "evidence": "“Analogs of PFA at \\(\\aleph_2\\)” has at least three plausible readings that must not be conflated:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 38,
   "attempt": 2
  },
  "AIM-LOGIC-0040": {
   "statement_status": "exact",
   "original_statement": "It is a fact that a compact Hausdorff space $X$ is countably tight if and only if $X$ contains no $\\omega_1$-free sequence.\n\nSuppose $X$ is a compact Hausdorff space of tightness $\\aleph_1$ (i.e. there are no $\\omega_2$-free sequences) and density $\\aleph_1$. Is it true that $|X| \\le 2^{\\aleph_1}$?",
   "clean_statement": "It is a fact that a compact Hausdorff space $X$ is countably tight if and only if $X$ contains no $\\omega_1$-free sequence.\n\nSuppose $X$ is a compact Hausdorff space of tightness $\\aleph_1$ (i.e. there are no $\\omega_2$-free sequences) and density $\\aleph_1$. Is it true that $|X| \\le 2^{\\aleph_1}$?",
   "public_statement": "It is a fact that a compact Hausdorff space $X$ is countably tight if and only if $X$ contains no $\\omega_1$-free sequence.\n\nSuppose $X$ is a compact Hausdorff space of tightness $\\aleph_1$ (i.e. there are no $\\omega_2$-free sequences) and density $\\aleph_1$. Is it true that $|X| \\le 2^{\\aleph_1}$?",
   "evidence": "There is no apparent OCR corruption. There is, however, a convention issue. If \\[ F(X)=\\sup\\{|S|:S\\text{ is the range of a free sequence in }X\\}, \\] then the compact-space theorem \\(F(X)=t(X)\\) makes \\[ \\text{“no \\(\\omega_2\\)-free sequence”}\\quad\\Longleftrightarrow\\quad t(X)\\leq\\aleph_1. \\] Thus the parenthetical gloss expresses an upper bound, not the literal equality \\(t(X)=\\aleph_1\\). The density phrase could likewise be read as either equality or an upper bound. The exactification theorem below proves that these readings give equivalent cardinal-bound questions at \\(\\aleph_1\\).",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-LOGIC-0041": {
   "statement_status": "exact",
   "original_statement": "Under a given analog of PFA for $\\aleph_2$, how many cofinal types of directed sets of cardinality $\\aleph_2$ are there?",
   "clean_statement": "Under a given analog of PFA for $\\aleph_2$, how many cofinal types of directed sets of cardinality $\\aleph_2$ are there?",
   "public_statement": "Under a given analog of PFA for $\\aleph_2$, how many cofinal types of directed sets of cardinality $\\aleph_2$ are there?",
   "evidence": "The canonical record is `aim-logic-notes.json`, zero-based index 40, from the AIM workshop *High and low forcing*, section “Problems in Low Forcing,” Problem 1.65. Its exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-LOGIC-0042": {
   "statement_status": "exact",
   "original_statement": "$\\kappa \\rightarrow (\\alpha,\\beta)^n$ means that for every coloring $F:[\\kappa]^n \\rightarrow \\{0,1\\}$, then then there is either a homogeneous set $H_0$ of order-type $\\alpha$ such that $F\"H_0=\\{0\\}$, or else a homogeneous set $H_1$ such that $F\"H_1=\\{1\\}$.\n\nCan $\\omega_2 \\rightarrow (\\omega_2,\\alpha)^2$ be deduced from one of the forcing axioms at $\\aleph_2$.",
   "clean_statement": "$\\kappa \\rightarrow (\\alpha,\\beta)^n$ means that for every coloring $F:[\\kappa]^n \\rightarrow \\{0,1\\}$, then then there is either a homogeneous set $H_0$ of order-type $\\alpha$ such that $F\"H_0=\\{0\\}$, or else a homogeneous set $H_1$ such that $F\"H_1=\\{1\\}$.\n\nCan $\\omega_2 \\rightarrow (\\omega_2,\\alpha)^2$ be deduced from one of the forcing axioms at $\\aleph_2$.",
   "public_statement": "$\\kappa \\rightarrow (\\alpha,\\beta)^n$ means that for every coloring $F:[\\kappa]^n \\rightarrow \\{0,1\\}$, then then there is either a homogeneous set $H_0$ of order-type $\\alpha$ such that $F\"H_0=\\{0\\}$, or else a homogeneous set $H_1$ such that $F\"H_1=\\{1\\}$.\n\nCan $\\omega_2 \\rightarrow (\\omega_2,\\alpha)^2$ be deduced from one of the forcing axioms at $\\aleph_2$.",
   "evidence": "The exact canonical AIM record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-LOGIC-0043": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "It is known under PFA that the gaps-spectrum of $P(\\omega)/fin$ consists of $(\\omega_1,\\omega_1^\\ast),(\\omega_2,\\omega^\\ast)$, and $(\\omega,\\omega_2^\\ast)$.\n\nAssuming an appropriate generalization of PFA, determine the gaps spectrum of $P(\\omega)/fin$.",
   "clean_statement": null,
   "public_statement": "It is known under PFA that the gaps-spectrum of $P(\\omega)/fin$ consists of $(\\omega_1,\\omega_1^\\ast),(\\omega_2,\\omega^\\ast)$, and $(\\omega,\\omega_2^\\ast)$.\n\nAssuming an appropriate generalization of PFA, determine the gaps spectrum of $P(\\omega)/fin$.",
   "evidence": "The canonical record is `aim-logic-notes.json`, zero-based index 42, from the AIM workshop *High and low forcing*, section “Problems in Low Forcing,” Problem 1.75. Its exact text is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-LOGIC-0044": {
   "statement_status": "exact",
   "original_statement": "Is it consistent (relative to large cardinals) that $\\text{TP}_{\\aleph_{\\omega+1}}$ holds along with reflection for all stationary subsets of $\\aleph_{\\omega+1}$?",
   "clean_statement": "Is it consistent (relative to large cardinals) that $\\text{TP}_{\\aleph_{\\omega+1}}$ holds along with reflection for all stationary subsets of $\\aleph_{\\omega+1}$?",
   "public_statement": "Is it consistent (relative to large cardinals) that $\\text{TP}_{\\aleph_{\\omega+1}}$ holds along with reflection for all stationary subsets of $\\aleph_{\\omega+1}$?",
   "evidence": "There is no visible OCR corruption in this record. The intended reflection principle is the standard **individual** principle \\[ \\operatorname{Refl}(\\kappa): \\quad\\text{every stationary }S\\subseteq\\kappa\\text{ reflects at some }\\delta<\\kappa \\text{ of uncountable cofinality}. \\] Thus the reflection point may depend on $S$. This is not the stronger assertion that every finite or countable family of stationary sets has a common reflection point. This interpretation agrees with the definition used in the current Poveda--Sinapova manuscript and with the terminology in the cited primary literature.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 43,
   "attempt": 2
  },
  "AIM-LOGIC-0045": {
   "statement_status": "exact",
   "original_statement": "Is it consistent to have the failure of both $\\text{SCH}$ and $\\square_{\\aleph_\\omega}^\\ast$?",
   "clean_statement": "Is it consistent to have the failure of both $\\text{SCH}$ and $\\square_{\\aleph_\\omega}^\\ast$?",
   "public_statement": "Is it consistent to have the failure of both $\\text{SCH}$ and $\\square_{\\aleph_\\omega}^\\ast$?",
   "evidence": "The exact canonical problem record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-LOGIC-0046": {
   "statement_status": "exact",
   "original_statement": "Is $\\neg \\square_\\kappa^\\ast + \\neg SCH_\\kappa + GCH_{<\\kappa}$ consistent for $\\kappa$ singular?",
   "clean_statement": "Is $\\neg \\square_\\kappa^\\ast + \\neg SCH_\\kappa + GCH_{<\\kappa}$ consistent for $\\kappa$ singular?",
   "public_statement": "Is $\\neg \\square_\\kappa^\\ast + \\neg SCH_\\kappa + GCH_{<\\kappa}$ consistent for $\\kappa$ singular?",
   "evidence": "Here the plus signs denote conjunction. Inspection of the exact record and its nearby source records reveals no OCR corruption in the displayed question. I use the following standard reading:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-LOGIC-0047": {
   "statement_status": "exact",
   "original_statement": "Is it consistent that there is a poset turning $\\aleph_{\\omega+1}$ into $\\aleph_2$?",
   "clean_statement": "Is it consistent that there is a poset turning $\\aleph_{\\omega+1}$ into $\\aleph_2$?",
   "public_statement": "Is it consistent that there is a poset turning $\\aleph_{\\omega+1}$ into $\\aleph_2$?",
   "evidence": "The intended strong reading is verified by a primary source. Cummings formulated the motivating question in *Collapsing successors of singulars* as follows: \\[ \\text{Can }V\\subseteq W\\text{ be models of ZFC with } (\\aleph_{\\omega+1})^V=(\\aleph_2)^W? \\tag{1.1} \\] The official AIM workshop report repeats “can one turn \\(\\aleph_{\\omega+1}\\) into \\(\\aleph_2\\)?” among the ambitious questions on which no progress was made. Thus “turning” is forcing shorthand for the cardinal-preserving equality (1.1), not an OCR error and not the weak routine collapse.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-LOGIC-0048": {
   "statement_status": "exact",
   "original_statement": "Find a model of $\\text{TP}_\\kappa$ for all regular $\\kappa \\in [\\aleph_2,\\aleph_{\\omega^2+2}]$ where $\\aleph_{\\omega^2}$ is the first strong limit.",
   "clean_statement": "Find a model of $\\text{TP}_\\kappa$ for all regular $\\kappa \\in [\\aleph_2,\\aleph_{\\omega^2+2}]$ where $\\aleph_{\\omega^2}$ is the first strong limit.",
   "public_statement": "Find a model of $\\text{TP}_\\kappa$ for all regular $\\kappa \\in [\\aleph_2,\\aleph_{\\omega^2+2}]$ where $\\aleph_{\\omega^2}$ is the first strong limit.",
   "evidence": "The exact AIM record (workshop *High and low forcing*, section “Problems in High Forcing,” Problem 2.25) is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-LOGIC-0049": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Is it consistent that every poset either adds a real or collapses a cardinal?",
   "clean_statement": null,
   "public_statement": "Is it consistent that every poset either adds a real or collapses a cardinal?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-LOGIC-0050": {
   "statement_status": "exact",
   "original_statement": "A (thin) $(\\kappa,\\lambda)$-tree is a set $F \\subset \\{f: x \\to 2 | x \\in P_\\kappa(\\lambda)\\}$ such that:\n\n\\begin{enumerate}\n\\item $\\forall f \\in F, \\forall x \\subset \\text{dom}(f), f \\upharpoonleft x \\in F$;\n\\item $\\forall x \\in P_\\kappa(\\lambda), \\exists f \\in F$ such that $\\text{dom}(f)=x$;\n\\item $\\forall x \\in P_\\kappa(\\lambda), |\\text{Lev}_x(F)|<\\kappa$ where $\\text{Lev}_x(F) = \\{f \\in F: \\text{dom}(f)=x \\}$.\n\\end{enumerate}\n\nA \\emph{cofinal branch} of a $(\\kappa,\\lambda)$-tree $F$ is a function $b:\\lambda \\to 2$ such that $\\forall x \\in P_\\kappa(\\lambda)$, $b \\upharpoonright x \\in \\text{Lev}_x(F)$. Given an assignment $x \\mapsto f_x \\in \\text{Lev}_x(F)$, a cofinal branch $b$ is \\emph{ineffable} if $\\{x \\in P_\\kappa(\\lambda): b \\upharpoonright x = f_x\\}$ is stationary.\n\nIf $\\kappa$ is regular it has the \\emph{strong tree property} if $\\forall \\lambda \\ge \\kappa$, every $(\\kappa,\\lambda)$-tree has a cofinal branch. $\\kappa$ has the \\emph{super tree property} if $\\forall \\lambda \\ge \\kappa$ and every $(\\kappa,\\lambda)$-tree, there is a cofinal branch $b$ which is ineffable.\n\nCan we obtain the super tree property at $\\aleph_{\\omega+1}$?",
   "clean_statement": "A (thin) $(\\kappa,\\lambda)$-tree is a set $F \\subset \\{f: x \\to 2 | x \\in P_\\kappa(\\lambda)\\}$ such that:\n\n\\begin{enumerate}\n\\item $\\forall f \\in F, \\forall x \\subset \\text{dom}(f), f \\upharpoonleft x \\in F$;\n\\item $\\forall x \\in P_\\kappa(\\lambda), \\exists f \\in F$ such that $\\text{dom}(f)=x$;\n\\item $\\forall x \\in P_\\kappa(\\lambda), |\\text{Lev}_x(F)|<\\kappa$ where $\\text{Lev}_x(F) = \\{f \\in F: \\text{dom}(f)=x \\}$.\n\\end{enumerate}\n\nA \\emph{cofinal branch} of a $(\\kappa,\\lambda)$-tree $F$ is a function $b:\\lambda \\to 2$ such that $\\forall x \\in P_\\kappa(\\lambda)$, $b \\upharpoonright x \\in \\text{Lev}_x(F)$. Given an assignment $x \\mapsto f_x \\in \\text{Lev}_x(F)$, a cofinal branch $b$ is \\emph{ineffable} if $\\{x \\in P_\\kappa(\\lambda): b \\upharpoonright x = f_x\\}$ is stationary.\n\nIf $\\kappa$ is regular it has the \\emph{strong tree property} if $\\forall \\lambda \\ge \\kappa$, every $(\\kappa,\\lambda)$-tree has a cofinal branch. $\\kappa$ has the \\emph{super tree property} if $\\forall \\lambda \\ge \\kappa$ and every $(\\kappa,\\lambda)$-tree, there is a cofinal branch $b$ which is ineffable.\n\nCan we obtain the super tree property at $\\aleph_{\\omega+1}$?",
   "public_statement": "A (thin) $(\\kappa,\\lambda)$-tree is a set $F \\subset \\{f: x \\to 2 | x \\in P_\\kappa(\\lambda)\\}$ such that:\n\n\\begin{enumerate}\n\\item $\\forall f \\in F, \\forall x \\subset \\text{dom}(f), f \\upharpoonleft x \\in F$;\n\\item $\\forall x \\in P_\\kappa(\\lambda), \\exists f \\in F$ such that $\\text{dom}(f)=x$;\n\\item $\\forall x \\in P_\\kappa(\\lambda), |\\text{Lev}_x(F)|<\\kappa$ where $\\text{Lev}_x(F) = \\{f \\in F: \\text{dom}(f)=x \\}$.\n\\end{enumerate}\n\nA \\emph{cofinal branch} of a $(\\kappa,\\lambda)$-tree $F$ is a function $b:\\lambda \\to 2$ such that $\\forall x \\in P_\\kappa(\\lambda)$, $b \\upharpoonright x \\in \\text{Lev}_x(F)$. Given an assignment $x \\mapsto f_x \\in \\text{Lev}_x(F)$, a cofinal branch $b$ is \\emph{ineffable} if $\\{x \\in P_\\kappa(\\lambda): b \\upharpoonright x = f_x\\}$ is stationary.\n\nIf $\\kappa$ is regular it has the \\emph{strong tree property} if $\\forall \\lambda \\ge \\kappa$, every $(\\kappa,\\lambda)$-tree has a cofinal branch. $\\kappa$ has the \\emph{super tree property} if $\\forall \\lambda \\ge \\kappa$ and every $(\\kappa,\\lambda)$-tree, there is a cofinal branch $b$ which is ineffable.\n\nCan we obtain the super tree property at $\\aleph_{\\omega+1}$?",
   "evidence": "The AIM record (workshop *High and low forcing*, section “Problems in High Forcing,” Problem 2.35) defines a thin $(\\kappa,\\lambda)$-tree as a set",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-LOGIC-0051": {
   "statement_status": "exact",
   "original_statement": "There is a lemma of Magidor and Shelah that if $\\lambda = \\sup_{n<\\omega}\\kappa_n$ for supercompact cardinals $\\kappa_n$, then $\\lambda^+$ has the tree property. Fontanella proved that if the $\\kappa_n$'s are strongly compact then $\\lambda^+$ has the strong tree property.\n\nIs it possible to obtain the super tree property at a successor of a singular cardinal?",
   "clean_statement": "There is a lemma of Magidor and Shelah that if $\\lambda = \\sup_{n<\\omega}\\kappa_n$ for supercompact cardinals $\\kappa_n$, then $\\lambda^+$ has the tree property. Fontanella proved that if the $\\kappa_n$'s are strongly compact then $\\lambda^+$ has the strong tree property.\n\nIs it possible to obtain the super tree property at a successor of a singular cardinal?",
   "public_statement": "There is a lemma of Magidor and Shelah that if $\\lambda = \\sup_{n<\\omega}\\kappa_n$ for supercompact cardinals $\\kappa_n$, then $\\lambda^+$ has the tree property. Fontanella proved that if the $\\kappa_n$'s are strongly compact then $\\lambda^+$ has the strong tree property.\n\nIs it possible to obtain the super tree property at a successor of a singular cardinal?",
   "evidence": "The canonical record is AIM Problem Lists, workshop *High and low forcing*, section “Problems in High Forcing,” problem 2.4. Its exact mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-LOGIC-0052": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Suppose $\\forall n<\\omega$, $kappa_n$ has the super tree property property. If $\\lambda = \\sup_{n<\\omega}$, does $\\lambda^+$ have the tree property?",
   "clean_statement": "Suppose $\\langle\\kappa_n:n<\\omega\\rangle$ is an increasing sequence of regular cardinals, every $\\kappa_n$ has the super tree property, and\n\\[\n\\lambda=\\sup_{n<\\omega}\\kappa_n.\n\\]\nMust $\\lambda^+$ have the (ordinary) tree property?",
   "public_statement": "Suppose $\\forall n<\\omega$, $kappa_n$ has the super tree property property. If $\\lambda = \\sup_{n<\\omega}$, does $\\lambda^+$ have the tree property?",
   "evidence": "The canonical AIM record (High and Low Forcing, Problem 2.45) reads verbatim:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-logic-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-LOGIC-0053": {
   "statement_status": "exact",
   "original_statement": "Suppose $\\kappa$ has the strong tree property. Does SCH hold above $\\kappa$? (If yes, then the strong tree property at $\\aleph_2$ implies SCH.)",
   "clean_statement": "Suppose $\\kappa$ has the strong tree property. Does SCH hold above $\\kappa$? (If yes, then the strong tree property at $\\aleph_2$ implies SCH.)",
   "public_statement": "Suppose $\\kappa$ has the strong tree property. Does SCH hold above $\\kappa$? (If yes, then the strong tree property at $\\aleph_2$ implies SCH.)",
   "evidence": "The workshop report repeats the question as “Does the strong tree property at $\\kappa$ imply SCH above $\\kappa$?” and explains the intended comparison with Solovay’s theorem for strongly compact cardinals. There is no substantive OCR corruption in the canonical text.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-LOGIC-0054": {
   "statement_status": "exact",
   "original_statement": "Is it consistent to obtain $TP_{\\aleph_2}$ together with the existence of a saturated ideal on $\\aleph_2$?",
   "clean_statement": "Is it consistent to obtain $TP_{\\aleph_2}$ together with the existence of a saturated ideal on $\\aleph_2$?",
   "public_statement": "Is it consistent to obtain $TP_{\\aleph_2}$ together with the existence of a saturated ideal on $\\aleph_2$?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-LOGIC-0055": {
   "statement_status": "exact",
   "original_statement": "Is there a model for $\\text{TP}_{\\kappa^{++}}$, with $\\kappa$ a singular strong limit of countable cofinality, where we use side conditions in place of the Mitchell poset?",
   "clean_statement": "Is there a model for $\\text{TP}_{\\kappa^{++}}$, with $\\kappa$ a singular strong limit of countable cofinality, where we use side conditions in place of the Mitchell poset?",
   "public_statement": "Is there a model for $\\text{TP}_{\\kappa^{++}}$, with $\\kappa$ a singular strong limit of countable cofinality, where we use side conditions in place of the Mitchell poset?",
   "evidence": "Here \\(\\mathrm{TP}_{\\lambda}\\) means that every tree of height \\(\\lambda\\), whose levels have size less than \\(\\lambda\\), has a cofinal branch. The statement is legible and agrees with the surrounding section, “Problems in the Overlap of High and Low Forcing.” No OCR correction is needed.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-LOGIC-0056": {
   "statement_status": "exact",
   "original_statement": "Is there a model for $\\text{TP}_{\\aleph_2} + \\text{TP}_{\\aleph_3}$ using side conditions?",
   "clean_statement": "Is there a model for $\\text{TP}_{\\aleph_2} + \\text{TP}_{\\aleph_3}$ using side conditions?",
   "public_statement": "Is there a model for $\\text{TP}_{\\aleph_2} + \\text{TP}_{\\aleph_3}$ using side conditions?",
   "evidence": "The canonical record is Problem 3.2 in the AIM workshop list *High and low forcing*, section “Problems in the Overlap of High and Low Forcing.” Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-LOGIC-0057": {
   "statement_status": "exact",
   "original_statement": "Is there a model where both $I[\\aleph_2]$ and $I[\\aleph_3]$ are trivial? In other words, is $\\text{AP}_{\\aleph_2} + \\text{AP}_{\\aleph_3}$ consistent?",
   "clean_statement": "Is there a model where both $I[\\aleph_2]$ and $I[\\aleph_3]$ are trivial? In other words, is $\\text{AP}_{\\aleph_2} + \\text{AP}_{\\aleph_3}$ consistent?",
   "public_statement": "Is there a model where both $I[\\aleph_2]$ and $I[\\aleph_3]$ are trivial? In other words, is $\\text{AP}_{\\aleph_2} + \\text{AP}_{\\aleph_3}$ consistent?",
   "evidence": "The canonical AIM record says, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-LOGIC-0058": {
   "statement_status": "corrected_verified",
   "original_statement": "Is the forcing axiom for proper posets of size $\\aleph_1$ with $\\aleph_2$-many dense sets consistent? If so, what if we replace the size-$\\aleph_2$ requirement with the $\\aleph_2$-chain condition?",
   "clean_statement": "Is the forcing axiom for proper posets of size $\\aleph_1$ with $\\aleph_2$-many dense sets consistent? If so, what if we replace the size $\\aleph_1$ with size $\\aleph_2$.?",
   "public_statement": "Is the forcing axiom for proper posets of size $\\aleph_1$ with $\\aleph_2$-many dense sets consistent? If so, what if we replace the size $\\aleph_1$ with size $\\aleph_2$.?",
   "evidence": "This is not an OCR error introduced by the repository: the live AIM page contains exactly the same words. The second sentence is nevertheless internally defective, because the first sentence contains no “size-$\\aleph_2$ requirement” to replace. The strongest surviving source evidence for the intended correction is the status paragraph immediately following Problem 3.5 on the same AIM page. It refers explicitly to",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-logic-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-LOGIC-0059": {
   "statement_status": "unrecoverable",
   "original_statement": "There is a coloring $F:[\\aleph_2]^2 \\rightarrow {0,1}$ of pairs from $\\aleph_2$ in $2$ colors so that the poset of finite approximations to a $0$- or $1$-homogeneous set has the $\\aleph_2$-chain condition. (We mean either the poset $\\mathbb P$ of finite functions $f:\\omega \\rightarrow F^{-1}(0)$ or the poset of $f:\\omega \\rightarrow F^{-1}(0)$, ordered by inclusion)\n\nCan this forcing be made proper using side conditions? Does this preserve the $\\aleph_2$-chain condition?",
   "clean_statement": null,
   "public_statement": "There is a coloring $F:[\\aleph_2]^2 \\rightarrow {0,1}$ of pairs from $\\aleph_2$ in $2$ colors so that the poset of finite approximations to a $0$- or $1$-homogeneous set has the $\\aleph_2$-chain condition. (We mean either the poset $\\mathbb P$ of finite functions $f:\\omega \\rightarrow F^{-1}(0)$ or the poset of $f:\\omega \\rightarrow F^{-1}(0)$, ordered by inclusion)\n\nCan this forcing be made proper using side conditions? Does this preserve the $\\aleph_2$-chain condition?",
   "evidence": "No authoritative symbol-level correction was found. Accordingly the exact record is treated as an **invalid statement**, not silently replaced by a conjectural repair.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-logic-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-LOGIC-0060": {
   "statement_status": "exact",
   "original_statement": "Suppose $\\aleph_\\omega$ is a strong limit. Is there a poset ${\\mathbb P}$ of size $<\\aleph_\\omega$, which adds a club guessing sequence for $S^{\\omega_2}_{\\omega_1}$, preserving $\\omega_1$, $\\omega_2$, and $\\omega_3$.",
   "clean_statement": "Suppose $\\aleph_\\omega$ is a strong limit. Is there a poset ${\\mathbb P}$ of size $<\\aleph_\\omega$, which adds a club guessing sequence for $S^{\\omega_2}_{\\omega_1}$, preserving $\\omega_1$, $\\omega_2$, and $\\omega_3$.",
   "public_statement": "Suppose $\\aleph_\\omega$ is a strong limit. Is there a poset ${\\mathbb P}$ of size $<\\aleph_\\omega$, which adds a club guessing sequence for $S^{\\omega_2}_{\\omega_1}$, preserving $\\omega_1$, $\\omega_2$, and $\\omega_3$.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-LOGIC-0061": {
   "statement_status": "exact",
   "original_statement": "Question 1 (Shlapentokh, Eisentr¨ ager). Let K be a number field. (a) Does there exist a ( ∀x)( ∃~y)-formula θK (x) such that \n\nθK (a) ⇐⇒ a ∈ O K?(b) Same question with Z in place of OK.(c) Does there exist a ∀-formula ϕK (x) such that \n\nθK (a) ⇐⇒ a ∈ Z?",
   "clean_statement": "Question 1 (Shlapentokh, Eisentr¨ ager). Let K be a number field. (a) Does there exist a ( ∀x)( ∃~y)-formula θK (x) such that\n\nθK (a) ⇐⇒ a ∈ O K?(b) Same question with Z in place of OK.(c) Does there exist a ∀-formula ϕK (x) such that\n\nθK (a) ⇐⇒ a ∈ Z?",
   "public_statement": "Question 1 (Shlapentokh, Eisentr¨ ager). Let K be a number field. (a) Does there exist a ( ∀x)( ∃~y)-formula θK (x) such that\n\nθK (a) ⇐⇒ a ∈ O K?(b) Same question with Z in place of OK.(c) Does there exist a ∀-formula ϕK (x) such that\n\nθK (a) ⇐⇒ a ∈ Z?",
   "evidence": "The canonical record is an OCR extraction from the first page of the AIM workshop notes *Problems related to “Definability and Decidability Problems in Number Theory”* (workshop of September 9--13, 2013, moderated by T. Scanlon, notes by J. Demeyer). The PDF prints:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-LOGIC-0062": {
   "statement_status": "exact",
   "original_statement": "Question 2 (Pheidas). Let K be a number field. Does there exist an ∃-formula δ(x, y ) such that \n\nδ(a, b ) ⇐⇒ (for every p above a prime 1 mod 4)( vp(a) < 0 → vp(b) < 0).\n\nThis is known (more or less) for K = Q for primes 3 mod 4. A positive answer would imply existential definability of Z in Q, hence a negative answer to HTP over Q.",
   "clean_statement": "Question 2 (Pheidas). Let K be a number field. Does there exist an ∃-formula δ(x, y ) such that\n\nδ(a, b ) ⇐⇒ (for every p above a prime 1 mod 4)( vp(a) < 0 → vp(b) < 0).\n\nThis is known (more or less) for K = Q for primes 3 mod 4. A positive answer would imply existential definability of Z in Q, hence a negative answer to HTP over Q.",
   "public_statement": "Question 2 (Pheidas). Let K be a number field. Does there exist an ∃-formula δ(x, y ) such that\n\nδ(a, b ) ⇐⇒ (for every p above a prime 1 mod 4)( vp(a) < 0 → vp(b) < 0).\n\nThis is known (more or less) for K = Q for primes 3 mod 4. A positive answer would imply existential definability of Z in Q, hence a negative answer to HTP over Q.",
   "evidence": "The corpus transcription is faithful. There is no substantive OCR error. To remove notational ambiguity, below \\(p\\) is written \\(\\mathfrak p\\) for a nonzero prime ideal of \\(\\mathcal O_K\\), \\(q\\) denotes the rational prime below it, and \\(v_{\\mathfrak p}\\) is the normalized additive valuation. Put",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-LOGIC-0063": {
   "statement_status": "exact",
   "original_statement": "Question 3 (Demeyer). A uniform statement of the theorem that r.e. sets are diophantine for Fp[t]. How should we even state this precisely?",
   "clean_statement": "Question 3 (Demeyer). A uniform statement of the theorem that r.e. sets are diophantine for Fp[t]. How should we even state this precisely?",
   "public_statement": "Question 3 (Demeyer). A uniform statement of the theorem that r.e. sets are diophantine for Fp[t]. How should we even state this precisely?",
   "evidence": "The source is the three-page AIM problem list *Problems related to “Definability and Decidability Problems in Number Theory”*, moderated by T. Scanlon, notes by J. Demeyer, September 9--13, 2013. On page 1 the PDF reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-LOGIC-0064": {
   "statement_status": "exact",
   "original_statement": "Question 4 (Pheidas). Is Q existentially definable inside Q(t) (using the language of rings with t)? It is known to be definable (even without t in the language). Using elliptic curves, one can easily existentially define dense subsets of \n\nQ.1",
   "clean_statement": "Question 4 (Pheidas). Is Q existentially definable inside Q(t) (using the language of rings with t)? It is known to be definable (even without t in the language). Using elliptic curves, one can easily existentially define dense subsets of\n\nQ.1",
   "public_statement": "Question 4 (Pheidas). Is Q existentially definable inside Q(t) (using the language of rings with t)? It is known to be definable (even without t in the language). Using elliptic curves, one can easily existentially define dense subsets of\n\nQ.1",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-LOGIC-0065": {
   "statement_status": "exact",
   "original_statement": "Question 5 (Koenigsmann). Given Q∗ ⊆ Q∗∗, both elementary equivalent to Q (in the language of rings). Does it imply that ( Q∗)alg ∩ Q∗∗ = Q∗ (i.e. is \n\nQ∗ relatively algebraically closed inside Q∗∗ )? If the answer is \"NO\" in some case, then we know that Z is not ∃-definable in Q.We know that Q∗ is quadratically closed in Q∗∗.",
   "clean_statement": "Question 5 (Koenigsmann). Given Q∗ ⊆ Q∗∗, both elementary equivalent to Q (in the language of rings). Does it imply that ( Q∗)alg ∩ Q∗∗ = Q∗ (i.e. is\n\nQ∗ relatively algebraically closed inside Q∗∗ )? If the answer is \"NO\" in some case, then we know that Z is not ∃-definable in Q.We know that Q∗ is quadratically closed in Q∗∗.",
   "public_statement": "Question 5 (Koenigsmann). Given Q∗ ⊆ Q∗∗, both elementary equivalent to Q (in the language of rings). Does it imply that ( Q∗)alg ∩ Q∗∗ = Q∗ (i.e. is\n\nQ∗ relatively algebraically closed inside Q∗∗ )? If the answer is \"NO\" in some case, then we know that Z is not ∃-definable in Q.We know that Q∗ is quadratically closed in Q∗∗.",
   "evidence": "The source is the AIM workshop problem list *Definability and Decidability Problems in Number Theory* (September 9--13, 2013), moderated by T. Scanlon and with notes by J. Demeyer. The original PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-LOGIC-0066": {
   "statement_status": "exact",
   "original_statement": "Question 6 (Koenigsmann). Consider the language O2 = {0, 1, +, P 2} where \n\nP2 is the set of squares. Is every ∃-O2-definable set in Q already ∃+-O2-definable? Equivalently, are the following sets ∃+-O2-definable: \n\n• { x ∈ Q | x 6 = 0 },\n\n• { x ∈ Q | (∀y)( y2 6 = x)}.",
   "clean_statement": "Question 6 (Koenigsmann). Consider the language O2 = {0, 1, +, P 2} where\n\nP2 is the set of squares. Is every ∃-O2-definable set in Q already ∃+-O2-definable? Equivalently, are the following sets ∃+-O2-definable:\n\n• { x ∈ Q | x 6 = 0 },\n\n• { x ∈ Q | (∀y)( y2 6 = x)}.",
   "public_statement": "Question 6 (Koenigsmann). Consider the language O2 = {0, 1, +, P 2} where\n\nP2 is the set of squares. Is every ∃-O2-definable set in Q already ∃+-O2-definable? Equivalently, are the following sets ∃+-O2-definable:\n\n• { x ∈ Q | x 6 = 0 },\n\n• { x ∈ Q | (∀y)( y2 6 = x)}.",
   "evidence": "The canonical JSON record has several OCR substitutions: it prints `O2`, `P 2`, and `6 =` where the PDF has mathematical glyphs. Direct inspection of the PDF text stream and its embedded fonts gives the following statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-LOGIC-0067": {
   "statement_status": "exact",
   "original_statement": "Question 7 (Miller). Given a countable graph ( V, E ), find a field K (of characteristic 0), polynomials P ∈ Q[x, t ], Q, R ∈ Q[x, y, z ] and a bijection \n\nα: V → { x ∈ K | (∃t ∈ K)( P (x, t ) = 0) } such that for all v, w ∈ V:1. ( v, w ) ∈ E ↔ (∃z)( R(α(v), α (w), z ) = 0). 2. ( v, w ) 6 ∈ E ↔ (∃z)( Q(α(v), α (w), z ) = 0). Poonen: let P define a curve without non-trivial automorphism. For the empty graph: Find a polynomial P ∈ Q[x, t ] such that K is a field generated by \n\nx1, x 2,..., t 1, t 2,... such that all {x1, x 2,... } are algebraically independent and such that P (xi, t i) = 0 for all i and such that 1. For every permutation π of N there is a unique automorphism σπ of K\n\nsuch that σπ(xi) = xπ(i).2. {xi | i ∈ N} = {y ∈ K | (∃t)( P (y, t ) = 0) }.",
   "clean_statement": "Question 7 (Miller). Given a countable graph ( V, E ), find a field K (of characteristic 0), polynomials P ∈ Q[x, t ], Q, R ∈ Q[x, y, z ] and a bijection\n\nα: V → { x ∈ K | (∃t ∈ K)( P (x, t ) = 0) } such that for all v, w ∈ V:1. ( v, w ) ∈ E ↔ (∃z)( R(α(v), α (w), z ) = 0). 2. ( v, w ) 6 ∈ E ↔ (∃z)( Q(α(v), α (w), z ) = 0). Poonen: let P define a curve without non-trivial automorphism. For the empty graph: Find a polynomial P ∈ Q[x, t ] such that K is a field generated by\n\nx1, x 2,..., t 1, t 2,... such that all {x1, x 2,... } are algebraically independent and such that P (xi, t i) = 0 for all i and such that 1. For every permutation π of N there is a unique automorphism σπ of K\n\nsuch that σπ(xi) = xπ(i).2. {xi | i ∈ N} = {y ∈ K | (∃t)( P (y, t ) = 0) }.",
   "public_statement": "Question 7 (Miller). Given a countable graph ( V, E ), find a field K (of characteristic 0), polynomials P ∈ Q[x, t ], Q, R ∈ Q[x, y, z ] and a bijection\n\nα: V → { x ∈ K | (∃t ∈ K)( P (x, t ) = 0) } such that for all v, w ∈ V:1. ( v, w ) ∈ E ↔ (∃z)( R(α(v), α (w), z ) = 0). 2. ( v, w ) 6 ∈ E ↔ (∃z)( Q(α(v), α (w), z ) = 0). Poonen: let P define a curve without non-trivial automorphism. For the empty graph: Find a polynomial P ∈ Q[x, t ] such that K is a field generated by\n\nx1, x 2,..., t 1, t 2,... such that all {x1, x 2,... } are algebraically independent and such that P (xi, t i) = 0 for all i and such that 1. For every permutation π of N there is a unique automorphism σπ of K\n\nsuch that σπ(xi) = xπ(i).2. {xi | i ∈ N} = {y ∈ K | (∃t)( P (y, t ) = 0) }.",
   "evidence": "There is one genuine syntactic ambiguity, not an OCR error. In model-theoretic shorthand, \\(z\\) often denotes a finite tuple of witnesses. Read this way, the later construction answers the question. If the printed membership \\(Q,R\\in\\mathbb Q[x,y,z]\\) is instead required literally with one scalar witness \\(z\\), the published construction does not directly prove that arity-one strengthening: its marker predicate asks for the two coordinates of a point on a plane curve. This distinction is maintained throughout.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-LOGIC-0068": {
   "statement_status": "exact",
   "original_statement": "Question 8 (Pasten). Do we know an r.e. subset of Q which is not diophan-tine? There is some evidence that Z is not diophantine, but we have no proof yet. What is the \"thinnest\" subset of Q known to be diophantine? Does Bombieri-Lang imply that the diophantine subsets of Q are either finite or have \"fast\" growth rates? This would give a different proof, assuming Bombieri-Lang, that Z is not diophantine. 2",
   "clean_statement": "Question 8 (Pasten). Do we know an r.e. subset of Q which is not diophan-tine? There is some evidence that Z is not diophantine, but we have no proof yet. What is the \"thinnest\" subset of Q known to be diophantine? Does Bombieri-Lang imply that the diophantine subsets of Q are either finite or have \"fast\" growth rates? This would give a different proof, assuming Bombieri-Lang, that Z is not diophantine. 2",
   "public_statement": "Question 8 (Pasten). Do we know an r.e. subset of Q which is not diophan-tine? There is some evidence that Z is not diophantine, but we have no proof yet. What is the \"thinnest\" subset of Q known to be diophantine? Does Bombieri-Lang imply that the diophantine subsets of Q are either finite or have \"fast\" growth rates? This would give a different proof, assuming Bombieri-Lang, that Z is not diophantine. 2",
   "evidence": "The source is the AIM problem list *Problems related to “Definability and Decidability Problems in Number Theory”*, moderated by T. Scanlon with notes by J. Demeyer, September 9--13, 2013. The original PDF says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-LOGIC-0069": {
   "statement_status": "exact",
   "original_statement": "Question 9 (Flenner). Is there a sentence ϕ in the language of rings such that Q(t1,..., t n) satisfies ϕ if and only if n is even? Using ultrapowers, one might show this is not possible.",
   "clean_statement": "Question 9 (Flenner). Is there a sentence ϕ in the language of rings such that Q(t1,..., t n) satisfies ϕ if and only if n is even? Using ultrapowers, one might show this is not possible.",
   "public_statement": "Question 9 (Flenner). Is there a sentence ϕ in the language of rings such that Q(t1,..., t n) satisfies ϕ if and only if n is even? Using ultrapowers, one might show this is not possible.",
   "evidence": "The source PDF gives the following statement (subscripts and punctuation restored from the mathematical fonts rather than inferred from the JSON OCR):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-LOGIC-0070": {
   "statement_status": "exact",
   "original_statement": "Question 10 (Pop). For K and L finitely generated fields, does K ≡ L\n\nimply K ∼= L?",
   "clean_statement": "Question 10 (Pop). For K and L finitely generated fields, does K ≡ L\n\nimply K ∼= L?",
   "public_statement": "Question 10 (Pop). For K and L finitely generated fields, does K ≡ L\n\nimply K ∼= L?",
   "evidence": "The canonical AIM record is Question 10, attributed to Florian Pop, from the AIM workshop *Definability and decidability problems in number theory*. The PDF and the JSON record read:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 69,
   "attempt": 1
  },
  "AIM-LOGIC-0071": {
   "statement_status": "exact",
   "original_statement": "Question 11 (Flenner). Does there exist a formula ϕ(x, y ) such that for any finitely generated field K and every rank one divisorial valuation v on \n\nK, there exists a parameter a such that Ov = {b ∈ K | ϕ(b, a )}?",
   "clean_statement": "Question 11 (Flenner). Does there exist a formula ϕ(x, y ) such that for any finitely generated field K and every rank one divisorial valuation v on\n\nK, there exists a parameter a such that Ov = {b ∈ K | ϕ(b, a )}?",
   "public_statement": "Question 11 (Flenner). Does there exist a formula ϕ(x, y ) such that for any finitely generated field K and every rank one divisorial valuation v on\n\nK, there exists a parameter a such that Ov = {b ∈ K | ϕ(b, a )}?",
   "evidence": "The source is Question 11 in the AIM workshop list *Definability and decidability problems in number theory*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 70,
   "attempt": 1
  },
  "AIM-LOGIC-0072": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 12 (Videla). Is there an existential analogue of Robinson's Q?",
   "clean_statement": null,
   "public_statement": "Question 12 (Videla). Is there an existential analogue of Robinson's Q?",
   "evidence": "The exact text on page 3 of the AIM problem list *Definability and Decidability Problems in Number Theory* (September 9--13, 2013), moderated by Thomas Scanlon with notes by Jeroen Demeyer, is:",
   "classification_method": "invalid_literal_with_repaired_reading",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-LOGIC-0073": {
   "statement_status": "exact",
   "original_statement": "Question 13 (Vidaux). Applications of exceptional sets for B¨ uchi's problem for higher powers in characteristic p?",
   "clean_statement": "Question 13 (Vidaux). Applications of exceptional sets for B¨ uchi's problem for higher powers in characteristic p?",
   "public_statement": "Question 13 (Vidaux). Applications of exceptional sets for B¨ uchi's problem for higher powers in characteristic p?",
   "evidence": "The canonical record reproduces this one-line item from the 2013 AIM workshop *Definability and Decidability Problems in Number Theory*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 72,
   "attempt": 1
  },
  "AIM-LOGIC-0074": {
   "statement_status": "exact",
   "original_statement": "Question 14 (Pasten). Existential undecidability of K(t) in the language \n\n{0, 1, +, ·, T } where T (f ) ↔ f 6 ∈ K? This question is open for every field \n\nK.",
   "clean_statement": "Question 14 (Pasten). Existential undecidability of K(t) in the language\n\n{0, 1, +, ·, T } where T (f ) ↔ f 6 ∈ K? This question is open for every field\n\nK.",
   "public_statement": "Question 14 (Pasten). Existential undecidability of K(t) in the language\n\n{0, 1, +, ·, T } where T (f ) ↔ f 6 ∈ K? This question is open for every field\n\nK.",
   "evidence": "The canonical record is Question 14 (Pasten) in the AIM problem list *Problems related to “Definability and Decidability Problems in Number Theory”*, moderated by T. Scanlon and recorded by J. Demeyer (September 9--13, 2013). The extracted text is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 73,
   "attempt": 1
  },
  "AIM-LOGIC-0075": {
   "statement_status": "exact",
   "original_statement": "Question 15 (Demeyer). For various fields K, consider the relation on K[t]defined by \n\nR(f, g ) ⇐⇒ deg( f ) = deg( g).\n\n(a) Is this relation (existentially) definable for K = C?(b) For K = Fp, is there an existential definition, uniform in p?",
   "clean_statement": "Question 15 (Demeyer). For various fields K, consider the relation on K[t]defined by\n\nR(f, g ) ⇐⇒ deg( f ) = deg( g).\n\n(a) Is this relation (existentially) definable for K = C?(b) For K = Fp, is there an existential definition, uniform in p?",
   "public_statement": "Question 15 (Demeyer). For various fields K, consider the relation on K[t]defined by\n\nR(f, g ) ⇐⇒ deg( f ) = deg( g).\n\n(a) Is this relation (existentially) definable for K = C?(b) For K = Fp, is there an existential definition, uniform in p?",
   "evidence": "The statement on page 3 of the AIM list *Definability and Decidability Problems in Number Theory* (September 9--13, 2013), moderated by Thomas Scanlon with notes by Jeroen Demeyer, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-LOGIC-0076": {
   "statement_status": "exact",
   "original_statement": "Question 16 (Miller). Situate with respect to ≤T the following sets: \n\nHT P ∞(Q), HT P ∞,1(Q), HT P ∞(Z), HT P ∞,1(Z).\n\n(known: 0 ′ ≤T HT P ∞(Z) ≤T 0′′ )",
   "clean_statement": "Question 16 (Miller). Situate with respect to ≤T the following sets:\n\nHT P ∞(Q), HT P ∞,1(Q), HT P ∞(Z), HT P ∞,1(Z).\n\n(known: 0 ′ ≤T HT P ∞(Z) ≤T 0′′ )",
   "public_statement": "Question 16 (Miller). Situate with respect to ≤T the following sets:\n\nHT P ∞(Q), HT P ∞,1(Q), HT P ∞(Z), HT P ∞,1(Z).\n\n(known: 0 ′ ≤T HT P ∞(Z) ≤T 0′′ )",
   "evidence": "The canonical record is Question 16 from the 2013 AIM workshop problem list *Definability and Decidability Problems in Number Theory*. The original PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-LOGIC-0077": {
   "statement_status": "exact",
   "original_statement": "Question 17 (Koenigsmann). Decidability of Qsolv, Qab, Zsolv and Zab.",
   "clean_statement": "Question 17 (Koenigsmann). Decidability of Qsolv, Qab, Zsolv and Zab.",
   "public_statement": "Question 17 (Koenigsmann). Decidability of Qsolv, Qab, Zsolv and Zab.",
   "evidence": "The canonical record is Question 17 (Koenigsmann) in the AIM list *Problems related to “Definability and Decidability Problems in Number Theory”*, moderated by T. Scanlon and recorded by J. Demeyer (September 9--13, 2013). The database extraction reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-LOGIC-0078": {
   "statement_status": "exact",
   "original_statement": "Question 18 (Jarden). Various questions from Jarden's talk. 3",
   "clean_statement": "Question 18 (Jarden). Various questions from Jarden's talk. 3",
   "public_statement": "Question 18 (Jarden). Various questions from Jarden's talk. 3",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-LOGIC-0079": {
   "statement_status": "exact",
   "original_statement": "Question 1. Is is the case that every computable structure is com-putable approximable? \n\n> Date: August 12, 2013.\n> 1COMPUTABLE STABILITY THEORY PROBLEM SESSION 2",
   "clean_statement": "Question 1. Is is the case that every computable structure is com-putable approximable?\n\n> Date: August 12, 2013.\n> 1COMPUTABLE STABILITY THEORY PROBLEM SESSION 2",
   "public_statement": "Question 1. Is is the case that every computable structure is com-putable approximable?\n\n> Date: August 12, 2013.\n> 1COMPUTABLE STABILITY THEORY PROBLEM SESSION 2",
   "evidence": "The canonical JSON has merged Question 1 with a date line and the next-page header, and it preserves a line-break hyphen in “com-putable.” The [original AIM PDF](https://aimath.org/pastworkshops/computestabproblems.pdf), page 1, has the subsection heading **“1.1. Computable approximability (Calvert)”** followed by:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-LOGIC-0080": {
   "statement_status": "exact",
   "original_statement": "Question 2. Let ϕ ∈ L ω1,ω be a satisfiable sentence of quantifier rank \n\nα, and suppose that either ω1 > β > α or β > α + ω. Is there an \n\nM | = ϕ such that SR( M) < β?\n\n1.2. Strongly minimal nontrivial locally modular nonorthogo-nal groups (Medvedev). \n\nLet M be a strongly minimal nontrivial locally modular structure. There is a nonorthogonal interpretable strongly minimal group G.",
   "clean_statement": "Question 2. Let ϕ ∈ L ω1,ω be a satisfiable sentence of quantifier rank\n\nα, and suppose that either ω1 > β > α or β > α + ω. Is there an\n\nM | = ϕ such that SR( M) < β?\n\n1.2. Strongly minimal nontrivial locally modular nonorthogo-nal groups (Medvedev).\n\nLet M be a strongly minimal nontrivial locally modular structure. There is a nonorthogonal interpretable strongly minimal group G.",
   "public_statement": "Question 2. Let ϕ ∈ L ω1,ω be a satisfiable sentence of quantifier rank\n\nα, and suppose that either ω1 > β > α or β > α + ω. Is there an\n\nM | = ϕ such that SR( M) < β?\n\n1.2. Strongly minimal nontrivial locally modular nonorthogo-nal groups (Medvedev).\n\nLet M be a strongly minimal nontrivial locally modular structure. There is a nonorthogonal interpretable strongly minimal group G.",
   "evidence": "The canonical record contains two distinct pieces of the 2013 AIM problem list. It begins with Question 2 from Section 1.1, “Computable approximability (Calvert),” and then accidentally continues into the heading and introductory paragraph of Section 1.2, “Strongly minimal nontrivial locally modular nonorthogonal groups (Medvedev).” The exact canonical record is preserved in `input.json`; the reconstruction below removes only this demonstrable next-section contamination.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-LOGIC-0081": {
   "statement_status": "exact",
   "original_statement": "Question 3. How difficult is it to find a presentation of G in terms of \n\nM?",
   "clean_statement": "Question 3. How difficult is it to find a presentation of G in terms of\n\nM?",
   "public_statement": "Question 3. How difficult is it to find a presentation of G in terms of\n\nM?",
   "evidence": "The next two questions ask conversely for a presentation of \\(M\\) from \\(G\\), and then discuss a three-to-one map from a strongly minimal group \\((M,\\oplus)\\) to \\((\\mathbb Q,+)\\). This confirms that “presentation” is meant in the computable-model-theoretic sense and that finite covers/quotients are central. There is no OCR corruption in Question 3.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-LOGIC-0082": {
   "statement_status": "exact",
   "original_statement": "Question 4. How difficult is it to find a presentation of M in terms of G?\n\nConsider a three-to-one map ( Q, +) ← (M, ⊕) where ( M, ⊕) is strongly minimal.",
   "clean_statement": "Question 4. How difficult is it to find a presentation of M in terms of G?\n\nConsider a three-to-one map ( Q, +) ← (M, ⊕) where ( M, ⊕) is strongly minimal.",
   "public_statement": "Question 4. How difficult is it to find a presentation of M in terms of G?\n\nConsider a three-to-one map ( Q, +) ← (M, ⊕) where ( M, ⊕) is strongly minimal.",
   "evidence": "The canonical record is Question 4 from the AIM workshop *Computable Stability Theory* (problem session, dated August 12, 2013):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-LOGIC-0083": {
   "statement_status": "exact",
   "original_statement": "Question 5. Must ⊕ be definable in some Q × F, where F is a finite set? \n\n1.3. Continuous sections (Miller). \n\nConsider T stable. Then for all M ≺ N |= T, the map S1(N ) →\n\nS1(M) has a continuous section, which sends p to the unique non-forking extension.",
   "clean_statement": "Question 5. Must ⊕ be definable in some Q × F, where F is a finite set?\n\n1.3. Continuous sections (Miller).\n\nConsider T stable. Then for all M ≺ N |= T, the map S1(N ) →\n\nS1(M) has a continuous section, which sends p to the unique non-forking extension.",
   "public_statement": "Question 5. Must ⊕ be definable in some Q × F, where F is a finite set?\n\n1.3. Continuous sections (Miller).\n\nConsider T stable. Then for all M ≺ N |= T, the map S1(N ) →\n\nS1(M) has a continuous section, which sends p to the unique non-forking extension.",
   "evidence": "The canonical JSON is not a faithful boundary extraction. It gives Question 5 and then appends the heading and opening paragraph of Section 1.3, “Continuous sections (Miller).” Inspection of page 2 of the original AIM problem-session PDF shows that the relevant source passage is instead the end of Section 1.2:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 82,
   "attempt": 1
  },
  "AIM-LOGIC-0084": {
   "statement_status": "exact",
   "original_statement": "Question 6. Is there a computable section?",
   "clean_statement": "Question 6. Is there a computable section?",
   "public_statement": "Question 6. Is there a computable section?",
   "evidence": "The record is Question 6 in Section 1.3, “Continuous sections (Miller),” of the AIM problem list *Computable stability theory*. The surrounding text recovers the statement as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 83,
   "attempt": 1
  },
  "AIM-LOGIC-0085": {
   "statement_status": "exact",
   "original_statement": "Question 7. How complicated is the map ϕ 7 → dpϕ (possibly with uniformity in p)?",
   "clean_statement": "Question 7. How complicated is the map ϕ 7 → dpϕ (possibly with uniformity in p)?",
   "public_statement": "Question 7. How complicated is the map ϕ 7 → dpϕ (possibly with uniformity in p)?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 84,
   "attempt": 1
  },
  "AIM-LOGIC-0086": {
   "statement_status": "exact",
   "original_statement": "Question 8. Does the existence of a computable section give other computable information for other characterizations of stability? \n\n1.4. κ+-computable categoricity (Knight). Definition. Let κ be a cardinal. Then a set is κ+-recursively enu-merable when it is Σ1 on Lκ+.\n\nDefinition. K is relatively κ+-categorical when for any two κ+-computable N, M of cardinality κ+, they are isomorphic in \n\nLκ+ (M, N ).COMPUTABLE STABILITY THEORY PROBLEM SESSION 3\n\nRecall that when an AEC is quasiminimal excellent, it is κ-categorical for all uncountable κ.",
   "clean_statement": "Question 8. Does the existence of a computable section give other computable information for other characterizations of stability?\n\n1.4. κ+-computable categoricity (Knight). Definition. Let κ be a cardinal. Then a set is κ+-recursively enu-merable when it is Σ1 on Lκ+.\n\nDefinition. K is relatively κ+-categorical when for any two κ+-computable N, M of cardinality κ+, they are isomorphic in\n\nLκ+ (M, N ).COMPUTABLE STABILITY THEORY PROBLEM SESSION 3\n\nRecall that when an AEC is quasiminimal excellent, it is κ-categorical for all uncountable κ.",
   "public_statement": "Question 8. Does the existence of a computable section give other computable information for other characterizations of stability?\n\n1.4. κ+-computable categoricity (Knight). Definition. Let κ be a cardinal. Then a set is κ+-recursively enu-merable when it is Σ1 on Lκ+.\n\nDefinition. K is relatively κ+-categorical when for any two κ+-computable N, M of cardinality κ+, they are isomorphic in\n\nLκ+ (M, N ).COMPUTABLE STABILITY THEORY PROBLEM SESSION 3\n\nRecall that when an AEC is quasiminimal excellent, it is κ-categorical for all uncountable κ.",
   "evidence": "The canonical JSON record is visibly overlong. Inspection of page 2 of the original AIM problem-session PDF shows that the material beginning “1.4. \\(\\kappa^+\\)-computable categoricity (Knight)” belongs to the next section and is extraction contamination.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 85,
   "attempt": 1
  },
  "AIM-LOGIC-0087": {
   "statement_status": "exact",
   "original_statement": "Question 9. Let K be a quasiminimal excellent class, and λ < κ.Suppose that K is κ+-computably categorical. Must K be λ+-computably categorical?",
   "clean_statement": "Question 9. Let K be a quasiminimal excellent class, and λ < κ.Suppose that K is κ+-computably categorical. Must K be λ+-computably categorical?",
   "public_statement": "Question 9. Let K be a quasiminimal excellent class, and λ < κ.Suppose that K is κ+-computably categorical. Must K be λ+-computably categorical?",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 86,
   "attempt": 1
  },
  "AIM-LOGIC-0088": {
   "statement_status": "corrected_verified",
   "original_statement": "Question 10. Suppose that K is relatively κ+-computably categorical. Must K be relatively λ+-computably categorical? \n\n1.5. Σ -definable isomorphisms for copies of C (Goncharov).",
   "clean_statement": "**Question 10.** Suppose that $K$ is relatively $\\kappa^+$-computably categorical. Must $K$ be relatively $\\lambda^+$-computably categorical?",
   "public_statement": "**Question 10.** Suppose that $K$ is relatively $\\kappa^+$-computably categorical. Must $K$ be relatively $\\lambda^+$-computably categorical?",
   "evidence": "The canonical JSON record contains an OCR/extraction error: after Question 10 it appends the next section heading, “1.5. $\\Sigma$-definable isomorphisms for copies of $C$ (Goncharov).” Inspection of the original AIM workshop PDF shows that this heading is not part of Question 10. The recovered problem is therefore:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-logic-notes.json",
   "source_index": 87,
   "attempt": 1
  },
  "AIM-LOGIC-0089": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 11. Let A = HF( C), and suppose K ∼= C is Σ-definable in \n\nA. Is there a Σ-definable isomorphism? \n\nThe answer is yes if we replace C by R and both ( K, ⊕, ) ∼= R and \n\nK ⊆ R hold. It is open if merely K ⊆ R2.1.6. λ-many models of each cardinality λ ≥ ℵ 1 (Greenberg).",
   "clean_statement": null,
   "public_statement": "Question 11. Let A = HF( C), and suppose K ∼= C is Σ-definable in\n\nA. Is there a Σ-definable isomorphism?\n\nThe answer is yes if we replace C by R and both ( K, ⊕, ) ∼= R and\n\nK ⊆ R hold. It is open if merely K ⊆ R2.1.6. λ-many models of each cardinality λ ≥ ℵ 1 (Greenberg).",
   "evidence": "The repository record is visibly corrupted: it drops a binary-operation symbol, joins the exponent in \\(\\mathbb R^2\\) to the following heading, and then absorbs the beginning of Section 1.6. Inspection of page 3 (PDF page index 2) of the official AIM problem-session PDF, including the embedded math-font encoding, recovers the passage as follows:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 88,
   "attempt": 1
  },
  "AIM-LOGIC-0090": {
   "statement_status": "exact",
   "original_statement": "Question \n1\n2. (Assume V = L if it makes things easier.) Suppose that for all λ ≥ ℵ 1 a theory T has at most λ-many models of cardinality λ.Must each such model have a λ-computable presentation? \n\nSuch a T is necessarily ω-stable and non-multidimensional. The question is true if T is ℵ1-categorical.",
   "clean_statement": "Question\n1\n2. (Assume V = L if it makes things easier.) Suppose that for all λ ≥ ℵ 1 a theory T has at most λ-many models of cardinality λ.Must each such model have a λ-computable presentation?\n\nSuch a T is necessarily ω-stable and non-multidimensional. The question is true if T is ℵ1-categorical.",
   "public_statement": "Question\n1\n2. (Assume V = L if it makes things easier.) Suppose that for all λ ≥ ℵ 1 a theory T has at most λ-many models of cardinality λ.Must each such model have a λ-computable presentation?\n\nSuch a T is necessarily ω-stable and non-multidimensional. The question is true if T is ℵ1-categorical.",
   "evidence": "The canonical JSON is OCR-corrupted: the question number is split across lines as “1” and “2,” and the sentence boundary after the second occurrence of \\(\\lambda\\) has lost a space. The original AIM PDF verifies the following recovered statement in Section 1.6, “\\(\\lambda\\)-many models of each cardinality \\(\\lambda\\geq\\aleph_1\\) (Greenberg)”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 89,
   "attempt": 1
  },
  "AIM-LOGIC-0091": {
   "statement_status": "exact",
   "original_statement": "Question 13. What if T has finitely many models in ℵ1? (Maybe look at models in ℵn.) \n\n1.7. Non-abelian free groups (Knight). Consider n-generated groups with a single relator of length at most t. (For each t, n there are finitely many such groups.) For every sentence ϕ define \n\nhn,t,ϕ = |{ G ∈ Hn,t: G |= ϕ}| |Hn,t |.",
   "clean_statement": "Question 13. What if T has finitely many models in ℵ1? (Maybe look at models in ℵn.)\n\n1.7. Non-abelian free groups (Knight). Consider n-generated groups with a single relator of length at most t. (For each t, n there are finitely many such groups.) For every sentence ϕ define\n\nhn,t,ϕ = |{ G ∈ Hn,t: G |= ϕ}| |Hn,t |.",
   "public_statement": "Question 13. What if T has finitely many models in ℵ1? (Maybe look at models in ℵn.)\n\n1.7. Non-abelian free groups (Knight). Consider n-generated groups with a single relator of length at most t. (For each t, n there are finitely many such groups.) For every sentence ϕ define\n\nhn,t,ϕ = |{ G ∈ Hn,t: G |= ϕ}| |Hn,t |.",
   "evidence": "The repository record contains the beginning of the next, unrelated section on non-abelian free groups. The official AIM PDF separates the material as follows. Section 1.6 is titled “\\(\\lambda\\)-many models of each cardinality \\(\\lambda\\geq\\aleph_1\\) (Greenberg)” and reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 90,
   "attempt": 1
  },
  "AIM-LOGIC-0092": {
   "statement_status": "exact",
   "original_statement": "Conjecture 14. The limit lim t→∞ hn,t,ϕ exists, and always takes the value 0 or 1, moreover in a way that may depend on ϕ but not on n.",
   "clean_statement": "Conjecture 14. The limit lim t→∞ hn,t,ϕ exists, and always takes the value 0 or 1, moreover in a way that may depend on ϕ but not on n.",
   "public_statement": "Conjecture 14. The limit lim t→∞ hn,t,ϕ exists, and always takes the value 0 or 1, moreover in a way that may depend on ϕ but not on n.",
   "evidence": "The canonical record is a faithful plain-text rendering of Conjecture 14, but it omits notation introduced immediately before it. The original AIM PDF says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 91,
   "attempt": 1
  },
  "AIM-LOGIC-0093": {
   "statement_status": "exact",
   "original_statement": "Conjecture 15. Furthermore, the asymptotically almost sure (a.a.s.) theory determined by this 0 − 1 law is that of F2.COMPUTABLE STABILITY THEORY PROBLEM SESSION 4\n\n1.8. Standard systems of RCF (Marker).",
   "clean_statement": "Conjecture 15. Furthermore, the asymptotically almost sure (a.a.s.) theory determined by this 0 − 1 law is that of F2.COMPUTABLE STABILITY THEORY PROBLEM SESSION 4\n\n1.8. Standard systems of RCF (Marker).",
   "public_statement": "Conjecture 15. Furthermore, the asymptotically almost sure (a.a.s.) theory determined by this 0 − 1 law is that of F2.COMPUTABLE STABILITY THEORY PROBLEM SESSION 4\n\n1.8. Standard systems of RCF (Marker).",
   "evidence": "The JSON record is visibly contaminated by a page break. It reads",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-LOGIC-0094": {
   "statement_status": "exact",
   "original_statement": "Question 16. What are the possible standard systems of recursively saturated real closed fields? (This is essentially asking: What are the possible sets of cuts of Q that are realized in some models?) \n\nThe ideal answer might be \"all Scott sets\". 1.9. Models of ℵ1-categorical theories. (Andrews).",
   "clean_statement": "Question 16. What are the possible standard systems of recursively saturated real closed fields? (This is essentially asking: What are the possible sets of cuts of Q that are realized in some models?)\n\nThe ideal answer might be \"all Scott sets\". 1.9. Models of ℵ1-categorical theories. (Andrews).",
   "public_statement": "Question 16. What are the possible standard systems of recursively saturated real closed fields? (This is essentially asking: What are the possible sets of cuts of Q that are realized in some models?)\n\nThe ideal answer might be \"all Scott sets\". 1.9. Models of ℵ1-categorical theories. (Andrews).",
   "evidence": "The repository record has absorbed the heading of the next, unrelated section. The official AIM PDF separates the text as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-LOGIC-0095": {
   "statement_status": "exact",
   "original_statement": "Conjecture 17. For any ℵ1-categorical T there is an n such that if T\n\nhas a computable model then every countable model has a presentation computable in 0(n).\n\nNote that if T is strongly minimal then n = 4 works. 1.10. Computable prime models (Andrews).",
   "clean_statement": "Conjecture 17. For any ℵ1-categorical T there is an n such that if T\n\nhas a computable model then every countable model has a presentation computable in 0(n).\n\nNote that if T is strongly minimal then n = 4 works. 1.10. Computable prime models (Andrews).",
   "public_statement": "Conjecture 17. For any ℵ1-categorical T there is an n such that if T\n\nhas a computable model then every countable model has a presentation computable in 0(n).\n\nNote that if T is strongly minimal then n = 4 works. 1.10. Computable prime models (Andrews).",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 94,
   "attempt": 1
  },
  "AIM-LOGIC-0096": {
   "statement_status": "exact",
   "original_statement": "Question 18 (Millar). Let T be a decidable theory having countably many countable models. When must the prime model have a decidable presentation? (Note that ω-stability suffices.)",
   "clean_statement": "Question 18 (Millar). Let T be a decidable theory having countably many countable models. When must the prime model have a decidable presentation? (Note that ω-stability suffices.)",
   "public_statement": "Question 18 (Millar). Let T be a decidable theory having countably many countable models. When must the prime model have a decidable presentation? (Note that ω-stability suffices.)",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-LOGIC-0097": {
   "statement_status": "exact",
   "original_statement": "Question 19. Let T be a decidable theory having countably many countable models. What do we need to know to build a computable prime model of T? (Of course ω-stability again suffices.) \n\n1.11. Turing degrees of DCFs (Calvert).",
   "clean_statement": "Question 19. Let T be a decidable theory having countably many countable models. What do we need to know to build a computable prime model of T? (Of course ω-stability again suffices.)\n\n1.11. Turing degrees of DCFs (Calvert).",
   "public_statement": "Question 19. Let T be a decidable theory having countably many countable models. What do we need to know to build a computable prime model of T? (Of course ω-stability again suffices.)\n\n1.11. Turing degrees of DCFs (Calvert).",
   "evidence": "The repository record contains the beginning of the next, unrelated section. The official AIM PDF separates the text as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 96,
   "attempt": 1
  },
  "AIM-LOGIC-0098": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 20 (Harizanov). Let d be a Turing degree. Is there a differ-entially closed field with a copy that is computable in d and such that every copy computes d?\n\n1.12. Spectrum of totally categorical theories (Andrews). Definition. Spec( T ) = {d: there is a model of T computable in d}.",
   "clean_statement": "Question 20 (Harizanov). Let d be a Turing degree. Is there a 1.11. Turing degrees of DCFs (Calvert). closed field with a copy that is computable in d and such that every copy computes d?\n\n1.11. Turing degrees of DCFs (Calvert). (Andrews). Definition. Spec( T ) = {d: there is a model of T computable in d}.",
   "public_statement": "Question 20 (Harizanov). Let d be a Turing degree. Is there a differ-entially closed field with a copy that is computable in d and such that every copy computes d?\n\n1.12. Spectrum of totally categorical theories (Andrews). Definition. Spec( T ) = {d: there is a model of T computable in d}.",
   "evidence": "The canonical record preserves the following OCR extraction:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-logic-notes.json",
   "source_index": 97,
   "attempt": 1
  },
  "AIM-LOGIC-0099": {
   "statement_status": "exact",
   "original_statement": "Conjecture 21. If T is totally categorical then Spec( T ) is a cone. \n\nThis is true in a finite language. COMPUTABLE STABILITY THEORY PROBLEM SESSION 5\n\n1.13. Model theoretic consequences of Erd˝ os-Rado (Greenberg). Task 22 (Hirschfeldt). Find proofs in second-order arithmetic of model-theoretic consequences of Erd˝ os-Rado (e.g., forking = dividing in simple theories). \n\n1.14. Borel complexity of isomorphism (Marker).",
   "clean_statement": "Conjecture 21. If T is totally categorical then Spec( T ) is a cone.\n\nThis is true in a finite language. COMPUTABLE STABILITY THEORY PROBLEM SESSION 5\n\n1.13. Model theoretic consequences of Erd˝ os-Rado (Greenberg). Task 22 (Hirschfeldt). Find proofs in second-order arithmetic of model-theoretic consequences of Erd˝ os-Rado (e.g., forking = dividing in simple theories).\n\n1.14. Borel complexity of isomorphism (Marker).",
   "public_statement": "Conjecture 21. If T is totally categorical then Spec( T ) is a cone.\n\nThis is true in a finite language. COMPUTABLE STABILITY THEORY PROBLEM SESSION 5\n\n1.13. Model theoretic consequences of Erd˝ os-Rado (Greenberg). Task 22 (Hirschfeldt). Find proofs in second-order arithmetic of model-theoretic consequences of Erd˝ os-Rado (e.g., forking = dividing in simple theories).\n\n1.14. Borel complexity of isomorphism (Marker).",
   "evidence": "The canonical record contains the text of Conjecture 21 followed by material from the next PDF page. Inspection of the original AIM workshop PDF, *Computable stability theory*, page 4, gives the relevant section exactly as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 98,
   "attempt": 1
  },
  "AIM-LOGIC-0100": {
   "statement_status": "exact",
   "original_statement": "Question 23. Let T ′ be an expansion of T by one constant. Suppose \n\n∼=T ′ is Borel complete. Is ∼=T Borel complete?",
   "clean_statement": "Question 23. Let T ′ be an expansion of T by one constant. Suppose\n\n∼=T ′ is Borel complete. Is ∼=T Borel complete?",
   "public_statement": "Question 23. Let T ′ be an expansion of T by one constant. Suppose\n\n∼=T ′ is Borel complete. Is ∼=T Borel complete?",
   "evidence": "The official AIM PDF places the item in §1.14, “Borel complexity of isomorphism (Marker),” and reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 99,
   "attempt": 1
  },
  "AIM-LOGIC-0101": {
   "statement_status": "exact",
   "original_statement": "Question 24. Suppose ∼=T is Borel complete. Is ∼=T ′ Borel complete?",
   "clean_statement": "Question 24. Suppose ∼=T is Borel complete. Is ∼=T ′ Borel complete?",
   "public_statement": "Question 24. Suppose ∼=T is Borel complete. Is ∼=T ′ Borel complete?",
   "evidence": "The canonical OCR record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 100,
   "attempt": 1
  },
  "AIM-LOGIC-0102": {
   "statement_status": "exact",
   "original_statement": "1. Let T be a complete superstable metric theory, let σ be a new symbol for a unitary function and let Tσ be the theory \n\nT ∪ \"σ is an automorphism\". Assume that TA, the model companion of Tσ exists. Is TA supersimple up to perturba-tions of the automorphism? \n\nBackground: In the first order context, if T is superstable and TA exists, then TA is supersimple (Pillay-Chatzidakis). When T is the theory of probability algebras, T is supersta-ble and TA exists, but TA is NOT superstable (Ben-Yaacov). But TA is superstable up to perturbations of the automorp-hism (Ben-Yaacov, Berenstein). 2. The Ehrenfeucht constructions provide examples of non-countably categorical theories with finitely many countable models. \n\na) Is there an analogue of this construction in the metric setting? \n\nb) We may measure the distance between to models up to perturbation. This gives rise to a pseudometric in the collection of separable models. What can we say about the theory if the quotient of the space of sepa-rable structures modulo distance zero is a compact (or finite) space? \n\nBackground:",
   "clean_statement": "1. Let T be a complete superstable metric theory, let σ be a new symbol for a unitary function and let Tσ be the theory\n\nT ∪ \"σ is an automorphism\". Assume that TA, the model companion of Tσ exists. Is TA supersimple up to perturba-tions of the automorphism?\n\nBackground: In the first order context, if T is superstable and TA exists, then TA is supersimple (Pillay-Chatzidakis). When T is the theory of probability algebras, T is supersta-ble and TA exists, but TA is NOT superstable (Ben-Yaacov). But TA is superstable up to perturbations of the automorp-hism (Ben-Yaacov, Berenstein). 2. The Ehrenfeucht constructions provide examples of non-countably categorical theories with finitely many countable models.\n\na) Is there an analogue of this construction in the metric setting?\n\nb) We may measure the distance between to models up to perturbation. This gives rise to a pseudometric in the collection of separable models. What can we say about the theory if the quotient of the space of sepa-rable structures modulo distance zero is a compact (or finite) space?\n\nBackground:",
   "public_statement": "1. Let T be a complete superstable metric theory, let σ be a new symbol for a unitary function and let Tσ be the theory\n\nT ∪ \"σ is an automorphism\". Assume that TA, the model companion of Tσ exists. Is TA supersimple up to perturba-tions of the automorphism?\n\nBackground: In the first order context, if T is superstable and TA exists, then TA is supersimple (Pillay-Chatzidakis). When T is the theory of probability algebras, T is supersta-ble and TA exists, but TA is NOT superstable (Ben-Yaacov). But TA is superstable up to perturbations of the automorp-hism (Ben-Yaacov, Berenstein). 2. The Ehrenfeucht constructions provide examples of non-countably categorical theories with finitely many countable models.\n\na) Is there an analogue of this construction in the metric setting?\n\nb) We may measure the distance between to models up to perturbation. This gives rise to a pseudometric in the collection of separable models. What can we say about the theory if the quotient of the space of sepa-rable structures modulo distance zero is a compact (or finite) space?\n\nBackground:",
   "evidence": "The exact \"problem\" field in \"input.json\" is preserved here, including its OCR line breaks and the material from Problem 2 that was appended to this record:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 101,
   "attempt": 1
  },
  "AIM-LOGIC-0103": {
   "statement_status": "exact",
   "original_statement": "3. Find connections between different notions of stability: quan-tifier free stability, stability of structures, stability of theo-ries. What does each of these notions imply? \n\n> 1\n\nBackground: Assume M is a normed space. If the norm of M is a stable formula, then there is an isometric copy of lp inside an ultrapower of M for some p ≥ 1 (Krivine). The group working on \"Definable norms and stable Banach spaces\"worked on this subject. 4. Let B be a Banach space. \n\na) Find the implications of the stability of T h (B). \n\nb) Find the implications of the ℵ1-categoricity of T h (B). \n\nBackground: The stability of T h (B) implies the existen-ce of isometric copies of lp in ultrapowers of the models of T h (B) (see previous entry). ℵ1-categoricity of T h (B)implies uncountable categoricity of T h (B) (Ben-Yaacov). Henson conjectured that such uncountable categorical theo-ries are essentially the theory of a Hilbert space. 5. What can we say about theories whose models are isomorp-hic in the sense of Banach space theory (linearly homeo-morphic, not necessarily isometric). \n\nBackground: There is a version of Morley's Theorem for \n\nℵ1-categorical (isometric isomorphism) theories. What can be said when a theory T has a unique model of density chraracter ℵ1 up to linear homeomorphism? 6. Assume a theory is categorical in the sense that any two se-parable models are linearly homeomorphic. Is there a bound on the coefficients of the deformations? 7. Understand Tsirelson space. Is its theory categorical up to small perturbations? \n\n> 2\n\nBackground: There was a talk given by William Johnson in this subject. 8. Find examples of ℵ1-categorical structures which are essen-tially different from Hilbert space. \n\nBackground: See entry 4. 9. Do the randomization of structures due to Keisler generate new examples of categorical or stable structures? \n\nBackground: If the theory of a first order structure is countably categorical, then the theory of its randomization is also separably categorical (Keisler). If the theory is ω-stable then the theory of its randomization is also ω-stable (Keisler). Does it preserve λ-stability? Keisler randomiza-tion was studied by the group working on Weak First Order and Probability Structures. 10. Find examples of not essentially separably categorical theo-ries. (Are Nakano spaces an example?) \n\nBackground: The model theory of Nakano spaces is stu-died in Pedro Poitevin's thesis (UIUC 2006). Nakano spaces generate some theories that are not separably categorical. Are these theories separably categorical up to perturbations of the norm? There was a talk given by C. Ward Henson in this subject. 11. Find examples of stable theories which are not essentially \n\nω-stable theories. (Are Nakano spaces an example?) \n\nBackground: Nakano spaces (see previous entry) generate some theories that are 2 ℵ0 stable not ω-stable. Are these theories ω-stable up to perturbation of the norm? \n\n> 3",
   "clean_statement": "3. Find connections between different notions of stability: quan-tifier free stability, stability of structures, stability of theo-ries. What does each of these notions imply?\n\n> 1\n\nBackground: Assume M is a normed space. If the norm of M is a stable formula, then there is an isometric copy of lp inside an ultrapower of M for some p ≥ 1 (Krivine). The group working on \"Definable norms and stable Banach spaces\"worked on this subject. 4. Let B be a Banach space.\n\na) Find the implications of the stability of T h (B).\n\nb) Find the implications of the ℵ1-categoricity of T h (B).\n\nBackground: The stability of T h (B) implies the existen-ce of isometric copies of lp in ultrapowers of the models of T h (B) (see previous entry). ℵ1-categoricity of T h (B)implies uncountable categoricity of T h (B) (Ben-Yaacov). Henson conjectured that such uncountable categorical theo-ries are essentially the theory of a Hilbert space. 5. What can we say about theories whose models are isomorp-hic in the sense of Banach space theory (linearly homeo-morphic, not necessarily isometric).\n\nBackground: There is a version of Morley's Theorem for\n\nℵ1-categorical (isometric isomorphism) theories. What can be said when a theory T has a unique model of density chraracter ℵ1 up to linear homeomorphism? 6. Assume a theory is categorical in the sense that any two se-parable models are linearly homeomorphic. Is there a bound on the coefficients of the deformations? 7. Understand Tsirelson space. Is its theory categorical up to small perturbations?\n\n> 2\n\nBackground: There was a talk given by William Johnson in this subject. 8. Find examples of ℵ1-categorical structures which are essen-tially different from Hilbert space.\n\nBackground: See entry 4. 9. Do the randomization of structures due to Keisler generate new examples of categorical or stable structures?\n\nBackground: If the theory of a first order structure is countably categorical, then the theory of its randomization is also separably categorical (Keisler). If the theory is ω-stable then the theory of its randomization is also ω-stable (Keisler). Does it preserve λ-stability? Keisler randomiza-tion was studied by the group working on Weak First Order and Probability Structures. 10. Find examples of not essentially separably categorical theo-ries. (Are Nakano spaces an example?)\n\nBackground: The model theory of Nakano spaces is stu-died in Pedro Poitevin's thesis (UIUC 2006). Nakano spaces generate some theories that are not separably categorical. Are these theories separably categorical up to perturbations of the norm? There was a talk given by C. Ward Henson in this subject. 11. Find examples of stable theories which are not essentially\n\nω-stable theories. (Are Nakano spaces an example?)\n\nBackground: Nakano spaces (see previous entry) generate some theories that are 2 ℵ0 stable not ω-stable. Are these theories ω-stable up to perturbation of the norm?\n\n> 3",
   "public_statement": "3. Find connections between different notions of stability: quan-tifier free stability, stability of structures, stability of theo-ries. What does each of these notions imply?\n\n> 1\n\nBackground: Assume M is a normed space. If the norm of M is a stable formula, then there is an isometric copy of lp inside an ultrapower of M for some p ≥ 1 (Krivine). The group working on \"Definable norms and stable Banach spaces\"worked on this subject. 4. Let B be a Banach space.\n\na) Find the implications of the stability of T h (B).\n\nb) Find the implications of the ℵ1-categoricity of T h (B).\n\nBackground: The stability of T h (B) implies the existen-ce of isometric copies of lp in ultrapowers of the models of T h (B) (see previous entry). ℵ1-categoricity of T h (B)implies uncountable categoricity of T h (B) (Ben-Yaacov). Henson conjectured that such uncountable categorical theo-ries are essentially the theory of a Hilbert space. 5. What can we say about theories whose models are isomorp-hic in the sense of Banach space theory (linearly homeo-morphic, not necessarily isometric).\n\nBackground: There is a version of Morley's Theorem for\n\nℵ1-categorical (isometric isomorphism) theories. What can be said when a theory T has a unique model of density chraracter ℵ1 up to linear homeomorphism? 6. Assume a theory is categorical in the sense that any two se-parable models are linearly homeomorphic. Is there a bound on the coefficients of the deformations? 7. Understand Tsirelson space. Is its theory categorical up to small perturbations?\n\n> 2\n\nBackground: There was a talk given by William Johnson in this subject. 8. Find examples of ℵ1-categorical structures which are essen-tially different from Hilbert space.\n\nBackground: See entry 4. 9. Do the randomization of structures due to Keisler generate new examples of categorical or stable structures?\n\nBackground: If the theory of a first order structure is countably categorical, then the theory of its randomization is also separably categorical (Keisler). If the theory is ω-stable then the theory of its randomization is also ω-stable (Keisler). Does it preserve λ-stability? Keisler randomiza-tion was studied by the group working on Weak First Order and Probability Structures. 10. Find examples of not essentially separably categorical theo-ries. (Are Nakano spaces an example?)\n\nBackground: The model theory of Nakano spaces is stu-died in Pedro Poitevin's thesis (UIUC 2006). Nakano spaces generate some theories that are not separably categorical. Are these theories separably categorical up to perturbations of the norm? There was a talk given by C. Ward Henson in this subject. 11. Find examples of stable theories which are not essentially\n\nω-stable theories. (Are Nakano spaces an example?)\n\nBackground: Nakano spaces (see previous entry) generate some theories that are 2 ℵ0 stable not ω-stable. Are these theories ω-stable up to perturbation of the norm?\n\n> 3",
   "evidence": "The exact canonical record in input.json is contaminated after that background: it appends the independent Problems 4 through 11 and the page markers **> 2** and **> 3**. In the official PDF, Problem 4 begins immediately after the final background sentence of Problem 3. The appended text is therefore not part of AIM-LOGIC-0103, but it remains preserved verbatim in input.json.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 102,
   "attempt": 1
  },
  "AIM-LOGIC-0104": {
   "statement_status": "exact",
   "original_statement": "12. Find examples of \"Non-Modular\"stable theories. That is, theories whose beautiful pairs are not essentially separably categorical. \n\nBackground: Probability algebras and Hilbert spaces are \n\nω stable structures whose beatiful pairs are separably cate-gorical. Lp spaces have two non isomorphic beautiful sepa-rable pairs, but these two structures are isomorphic up to perturbations of the norm. 13. Build analogues of Hrushovski's constructions in the metric setting. \n\nBackground:",
   "clean_statement": "12. Find examples of \"Non-Modular\"stable theories. That is, theories whose beautiful pairs are not essentially separably categorical.\n\nBackground: Probability algebras and Hilbert spaces are\n\nω stable structures whose beatiful pairs are separably cate-gorical. Lp spaces have two non isomorphic beautiful sepa-rable pairs, but these two structures are isomorphic up to perturbations of the norm. 13. Build analogues of Hrushovski's constructions in the metric setting.\n\nBackground:",
   "public_statement": "12. Find examples of \"Non-Modular\"stable theories. That is, theories whose beautiful pairs are not essentially separably categorical.\n\nBackground: Probability algebras and Hilbert spaces are\n\nω stable structures whose beatiful pairs are separably cate-gorical. Lp spaces have two non isomorphic beautiful sepa-rable pairs, but these two structures are isomorphic up to perturbations of the norm. 13. Build analogues of Hrushovski's constructions in the metric setting.\n\nBackground:",
   "evidence": "The original AIM PDF, *Questions on Model theory for metric structures*, page 4, contains:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 103,
   "attempt": 1
  },
  "AIM-LOGIC-0105": {
   "statement_status": "corrected_verified",
   "original_statement": "14. Understand the enconding of graphs in metric structures. \n\nBackground:",
   "clean_statement": "Understand the **encoding** of graphs in metric structures.",
   "public_statement": "Understand the **encoding** of graphs in metric structures.",
   "evidence": "Inspection of page 4 of the source PDF confirms that **“enconding” occurs in the PDF itself**. It is not a repository-only OCR error. The evident correction is: > **Recovered statement.** Understand the **encoding** of graphs in metric structures.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-logic-notes.json",
   "source_index": 104,
   "attempt": 1
  },
  "AIM-LOGIC-0106": {
   "statement_status": "exact",
   "original_statement": "15. Study the group configuration theorem in the setting of continuous logic. \n\nBackground:",
   "clean_statement": "15. Study the group configuration theorem in the setting of continuous logic.\n\nBackground:",
   "public_statement": "15. Study the group configuration theorem in the setting of continuous logic.\n\nBackground:",
   "evidence": "The canonical record is Problem 15 from the AIM workshop *Model theory of metric structures*. Its exact extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 105,
   "attempt": 1
  },
  "AIM-LOGIC-0107": {
   "statement_status": "exact",
   "original_statement": "16. Study the existence of ample generics in the setting of ω-stable separably categorical continuous theory. \n\nBackground: Generalize the ideas of Hodkinson, Hodges, Lascar and Shelah presented in the paper \"The small in-dex property for ω-stable, ω-categorical structures and the random graph\"to the setting of metric structures. 17. Is there a model theoretic understanding of parametric fa-milies of probability measures. In particular, find ways to compare different experiments. \n\nBackground: \n\n> 4",
   "clean_statement": "16. Study the existence of ample generics in the setting of ω-stable separably categorical continuous theory.\n\nBackground: Generalize the ideas of Hodkinson, Hodges, Lascar and Shelah presented in the paper \"The small in-dex property for ω-stable, ω-categorical structures and the random graph\"to the setting of metric structures. 17. Is there a model theoretic understanding of parametric fa-milies of probability measures. In particular, find ways to compare different experiments.\n\nBackground:\n\n> 4",
   "public_statement": "16. Study the existence of ample generics in the setting of ω-stable separably categorical continuous theory.\n\nBackground: Generalize the ideas of Hodkinson, Hodges, Lascar and Shelah presented in the paper \"The small in-dex property for ω-stable, ω-categorical structures and the random graph\"to the setting of metric structures. 17. Is there a model theoretic understanding of parametric fa-milies of probability measures. In particular, find ways to compare different experiments.\n\nBackground:\n\n> 4",
   "evidence": "The official AIM PDF, *Questions on Model theory for metric structures*, contains:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 106,
   "attempt": 1
  },
  "AIM-LOGIC-0108": {
   "statement_status": "exact",
   "original_statement": "18. Find connections between different notions of genericity. In particular, the notion of genericity of Vershik and the notion of being existentially closed. Urysohn space is gene-ric in both senses. \n\nBackground:",
   "clean_statement": "18. Find connections between different notions of genericity. In particular, the notion of genericity of Vershik and the notion of being existentially closed. Urysohn space is gene-ric in both senses.\n\nBackground:",
   "public_statement": "18. Find connections between different notions of genericity. In particular, the notion of genericity of Vershik and the notion of being existentially closed. Urysohn space is gene-ric in both senses.\n\nBackground:",
   "evidence": "The canonical record is problem 18 in the AIM list *Questions on Model theory for metric structures*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 107,
   "attempt": 1
  },
  "AIM-LOGIC-0109": {
   "statement_status": "exact",
   "original_statement": "19. Study the definable sets in the expansion of the Urysohn space with a predicate for an involution map. \n\nBackground:",
   "clean_statement": "19. Study the definable sets in the expansion of the Urysohn space with a predicate for an involution map.\n\nBackground:",
   "public_statement": "19. Study the definable sets in the expansion of the Urysohn space with a predicate for an involution map.\n\nBackground:",
   "evidence": "The canonical record is Problem 19 from the AIM workshop *Model theory of metric structures*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 108,
   "attempt": 1
  },
  "AIM-LOGIC-0110": {
   "statement_status": "exact",
   "original_statement": "20. Study non commutative probability spaces. \n\nBackground: The working group on Non Commutative Probability Spaces worked in this subject. 21. Find examples of continuous theories with the NIP. \n\nBackground: If M is first order structure whose theory has the NIP, does the randomization of M have the NIP? 22. Understand basic results of stable/ ω-stable groups in the setting of continuous logic. \n\nBackground: The working group on Stable Groups worked in this subject. 23. Study definable sets. \n\nBackground: The working group on Definable Sets worked in this subject. 24. Let L be a metric language. A weak L-structure is a 2-sorted structure ( M, [0, 1],... ) where we have the usual in-terpretation for all the symbols in L and we add a symbol for every continuous function from [0, 1] n into [0, 1]. We work with non-standard models of the theory of such a \n\n> 5\n\nstructure and consider the metric structure obtained by ta-king the standard part map. Is there a justification of why one uses the quantifiers sup, ´ ınf? Are there clear proofs of the keisler-Shelah Theorem or the omitting types Theorem? \n\nBackground: The working group on Weak First Order and Probability Structures worked in this subject. 25. Understand integrals as oppose to types in continuous logic. \n\nBackground:",
   "clean_statement": "20. Study non commutative probability spaces.\n\nBackground: The working group on Non Commutative Probability Spaces worked in this subject. 21. Find examples of continuous theories with the NIP.\n\nBackground: If M is first order structure whose theory has the NIP, does the randomization of M have the NIP? 22. Understand basic results of stable/ ω-stable groups in the setting of continuous logic.\n\nBackground: The working group on Stable Groups worked in this subject. 23. Study definable sets.\n\nBackground: The working group on Definable Sets worked in this subject. 24. Let L be a metric language. A weak L-structure is a 2-sorted structure ( M, [0, 1],... ) where we have the usual in-terpretation for all the symbols in L and we add a symbol for every continuous function from [0, 1] n into [0, 1]. We work with non-standard models of the theory of such a\n\n> 5\n\nstructure and consider the metric structure obtained by ta-king the standard part map. Is there a justification of why one uses the quantifiers sup, ´ ınf? Are there clear proofs of the keisler-Shelah Theorem or the omitting types Theorem?\n\nBackground: The working group on Weak First Order and Probability Structures worked in this subject. 25. Understand integrals as oppose to types in continuous logic.\n\nBackground:",
   "public_statement": "20. Study non commutative probability spaces.\n\nBackground: The working group on Non Commutative Probability Spaces worked in this subject. 21. Find examples of continuous theories with the NIP.\n\nBackground: If M is first order structure whose theory has the NIP, does the randomization of M have the NIP? 22. Understand basic results of stable/ ω-stable groups in the setting of continuous logic.\n\nBackground: The working group on Stable Groups worked in this subject. 23. Study definable sets.\n\nBackground: The working group on Definable Sets worked in this subject. 24. Let L be a metric language. A weak L-structure is a 2-sorted structure ( M, [0, 1],... ) where we have the usual in-terpretation for all the symbols in L and we add a symbol for every continuous function from [0, 1] n into [0, 1]. We work with non-standard models of the theory of such a\n\n> 5\n\nstructure and consider the metric structure obtained by ta-king the standard part map. Is there a justification of why one uses the quantifiers sup, ´ ınf? Are there clear proofs of the keisler-Shelah Theorem or the omitting types Theorem?\n\nBackground: The working group on Weak First Order and Probability Structures worked in this subject. 25. Understand integrals as oppose to types in continuous logic.\n\nBackground:",
   "evidence": "The canonical JSON record accidentally concatenates several later numbered items. The official AIM workshop PDF, page 4, separates them. The recovered statement owned by this attempt is exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 109,
   "attempt": 1
  },
  "AIM-LOGIC-0111": {
   "statement_status": "exact",
   "original_statement": "26. Under which circumstances are there enough definable sets? (i.e. under which assumptions are types determined by the definable sets they belong to?). In particular, do we get a positive answer when T is ω-stable? \n\nBackground:",
   "clean_statement": "26. Under which circumstances are there enough definable sets? (i.e. under which assumptions are types determined by the definable sets they belong to?). In particular, do we get a positive answer when T is ω-stable?\n\nBackground:",
   "public_statement": "26. Under which circumstances are there enough definable sets? (i.e. under which assumptions are types determined by the definable sets they belong to?). In particular, do we get a positive answer when T is ω-stable?\n\nBackground:",
   "evidence": "The official AIM workshop PDF, page 5, states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 110,
   "attempt": 1
  },
  "AIM-LOGIC-0112": {
   "statement_status": "exact",
   "original_statement": "27. Find a mechanism for deducing an \"almost isometry theo-rem\"from an isometric theorem using compactness (an ul-traproduct). \n\nBackground:",
   "clean_statement": "27. Find a mechanism for deducing an \"almost isometry theo-rem\"from an isometric theorem using compactness (an ul-traproduct).\n\nBackground:",
   "public_statement": "27. Find a mechanism for deducing an \"almost isometry theo-rem\"from an isometric theorem using compactness (an ul-traproduct).\n\nBackground:",
   "evidence": "The canonical record is Problem 27 from the AIM workshop *Model theory of metric structures*. The OCR extraction reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 111,
   "attempt": 1
  },
  "AIM-LOGIC-0113": {
   "statement_status": "exact",
   "original_statement": "28. Is there a notion of \"explicitly defined norm\" so that any Banach space with an explictly defined norm hereditarily contains an lp space? \n\nBackground: The group working on Definable Norms and Stable Banach Spaces worked on this subject. 29. Study asymtotic cones. \n\nBackground: The group working on Asymptotic cones worked on this subject.",
   "clean_statement": "28. Is there a notion of \"explicitly defined norm\" so that any Banach space with an explictly defined norm hereditarily contains an lp space?\n\nBackground: The group working on Definable Norms and Stable Banach Spaces worked on this subject. 29. Study asymtotic cones.\n\nBackground: The group working on Asymptotic cones worked on this subject.",
   "public_statement": "28. Is there a notion of \"explicitly defined norm\" so that any Banach space with an explictly defined norm hereditarily contains an lp space?\n\nBackground: The group working on Definable Norms and Stable Banach Spaces worked on this subject. 29. Study asymtotic cones.\n\nBackground: The group working on Asymptotic cones worked on this subject.",
   "evidence": "The exact record in `input.json` joins two numbered items. Inspection of the official AIM questions PDF separates them. Problem 28 reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 112,
   "attempt": 1
  },
  "AIM-LOGIC-0114": {
   "statement_status": "exact",
   "original_statement": "Question 1 (D'Aquino). Fermat's little theorem states that \n\nxp ≡ x mod p\n\nProof 1: F∗ \n\n> p\n\nis cyclic using the fact that \n\n#{x | P (x) = 0 } ≤ deg( P )\n\nProof 2: List R = {1, 2,..., p − 1}. Show that for a ∈ R, multiplication by a is a permuation. Then p−1\n\n∏\n\n> i=1\n\ni ≡\n\n> p−1\n\n∏\n\n> i=1\n\n(ai ) mod p\n\nFrom this follows that \n\n(p − 1)! ≡ ap−1(p − 1)! mod p\n\nGive a \"simple\" definition of n! mod p (this is OK for exponentiation). Proof 3: Use \n\n(x + y)p = xp + yp mod p\n\nFind other proofs.",
   "clean_statement": "Question 1 (D'Aquino). Fermat's little theorem states that\n\nxp ≡ x mod p\n\nProof 1: F∗\n\n> p\n\nis cyclic using the fact that\n\n#{x | P (x) = 0 } ≤ deg( P )\n\nProof 2: List R = {1, 2,..., p − 1}. Show that for a ∈ R, multiplication by a is a permuation. Then p−1\n\n∏\n\n> i=1\n\ni ≡\n\n> p−1\n\n∏\n\n> i=1\n\n(ai ) mod p\n\nFrom this follows that\n\n(p − 1)! ≡ ap−1(p − 1)! mod p\n\nGive a \"simple\" definition of n! mod p (this is OK for exponentiation). Proof 3: Use\n\n(x + y)p = xp + yp mod p\n\nFind other proofs.",
   "public_statement": "Question 1 (D'Aquino). Fermat's little theorem states that\n\nxp ≡ x mod p\n\nProof 1: F∗\n\n> p\n\nis cyclic using the fact that\n\n#{x | P (x) = 0 } ≤ deg( P )\n\nProof 2: List R = {1, 2,..., p − 1}. Show that for a ∈ R, multiplication by a is a permuation. Then p−1\n\n∏\n\n> i=1\n\ni ≡\n\n> p−1\n\n∏\n\n> i=1\n\n(ai ) mod p\n\nFrom this follows that\n\n(p − 1)! ≡ ap−1(p − 1)! mod p\n\nGive a \"simple\" definition of n! mod p (this is OK for exponentiation). Proof 3: Use\n\n(x + y)p = xp + yp mod p\n\nFind other proofs.",
   "evidence": "The canonical record is Question 1 attributed to Paola D'Aquino in the problem list from the 21--25 March 2005 AIM workshop *Extensions of Hilbert's Tenth Problem*. The record's plain-text extraction is visibly damaged: \\(x^p\\) lost its superscript, \\(\\mathbb F_p^*\\) was split across lines, and product limits became quote blocks. I checked page 1 of the original PDF. It reads, with formulas restored,",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 113,
   "attempt": 1
  },
  "AIM-LOGIC-0115": {
   "statement_status": "exact",
   "original_statement": "Question 2 (D'Aquino). Is DPRM a theorem of I∆0? This is Peano arithmetic with the induction axiom for every first order formula ϕ(x) with bounded quantifiers \n\nI(ϕ): \n\n[\n\nϕ(0) ∧ (∀x)(ϕ(x) → ϕ(x + 1) )]\n\n→ (∀x)(ϕ(x))\n\nA positive answer would imply that NP is equal to co-NP. Given a Σ1 formula ψ(~x), does there exist a polynomial P (~x, ~ y) such that \n\nI∆0 ` (∀~x)\n\n(\n\nψ(~x) ↔ (∃~y)(P (~x, ~ y) = 0 ))\n\nConsider the language L = {+, ·, 0, 1, #, ≤}, where \n\n#( x, y ):= xblog( y)c",
   "clean_statement": "Question 2 (D'Aquino). Is DPRM a theorem of I∆0? This is Peano arithmetic with the induction axiom for every first order formula ϕ(x) with bounded quantifiers\n\nI(ϕ):\n\n[\n\nϕ(0) ∧ (∀x)(ϕ(x) → ϕ(x + 1) )]\n\n→ (∀x)(ϕ(x))\n\nA positive answer would imply that NP is equal to co-NP. Given a Σ1 formula ψ(~x), does there exist a polynomial P (~x, ~ y) such that\n\nI∆0 ` (∀~x)\n\n(\n\nψ(~x) ↔ (∃~y)(P (~x, ~ y) = 0 ))\n\nConsider the language L = {+, ·, 0, 1, #, ≤}, where\n\n#( x, y ):= xblog( y)c",
   "public_statement": "Question 2 (D'Aquino). Is DPRM a theorem of I∆0? This is Peano arithmetic with the induction axiom for every first order formula ϕ(x) with bounded quantifiers\n\nI(ϕ):\n\n[\n\nϕ(0) ∧ (∀x)(ϕ(x) → ϕ(x + 1) )]\n\n→ (∀x)(ϕ(x))\n\nA positive answer would imply that NP is equal to co-NP. Given a Σ1 formula ψ(~x), does there exist a polynomial P (~x, ~ y) such that\n\nI∆0 ` (∀~x)\n\n(\n\nψ(~x) ↔ (∃~y)(P (~x, ~ y) = 0 ))\n\nConsider the language L = {+, ·, 0, 1, #, ≤}, where\n\n#( x, y ):= xblog( y)c",
   "evidence": "The source is the AIM workshop note *Problems related to “Extensions of Hilbert's Tenth Problem”*, moderated by B. Poonen and T. Scanlon and recorded by J. Demeyer. On page 1, Question 2 is attributed to P. D'Aquino. The PDF gives the following question (notation expanded but mathematical content preserved):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 114,
   "attempt": 1
  },
  "AIM-LOGIC-0116": {
   "statement_status": "exact",
   "original_statement": "Question 3 (Demeyer). Consider the ring Fq[W, Z ]. Does there exist a Diophantine predicate \n\nα(f, ~ g) with f ∈ Fq[W, Z ] and ~g ∈ Fq[Z]n such that \n\n(1) For all f ∈ Fq[W, Z ], there exists a ~g ∈ Fq[Z]n such that α(f, ~ g) holds. \n\n(2) For all ~g ∈ Fq[Z]n, the set {f ∈ Fq[W, Z ] | α(f, ~ g) holds } is finite. This will imply that r.e. = Diophantine for Fq[W, Z ].It is possible to give such a Diophantine predicate if \" α(· · · ) holds\" is replaced with \"α(· · · ) does not hold\". \n\n> 12MODERATED BY B. POONEN AND T. SCANLON, NOTES BY J. DEMEYER",
   "clean_statement": "Question 3 (Demeyer). Consider the ring Fq[W, Z ]. Does there exist a Diophantine predicate\n\nα(f, ~ g) with f ∈ Fq[W, Z ] and ~g ∈ Fq[Z]n such that\n\n(1) For all f ∈ Fq[W, Z ], there exists a ~g ∈ Fq[Z]n such that α(f, ~ g) holds.\n\n(2) For all ~g ∈ Fq[Z]n, the set {f ∈ Fq[W, Z ] | α(f, ~ g) holds } is finite. This will imply that r.e. = Diophantine for Fq[W, Z ].It is possible to give such a Diophantine predicate if \" α(· · · ) holds\" is replaced with \"α(· · · ) does not hold\".\n\n> 12MODERATED BY B. POONEN AND T. SCANLON, NOTES BY J. DEMEYER",
   "public_statement": "Question 3 (Demeyer). Consider the ring Fq[W, Z ]. Does there exist a Diophantine predicate\n\nα(f, ~ g) with f ∈ Fq[W, Z ] and ~g ∈ Fq[Z]n such that\n\n(1) For all f ∈ Fq[W, Z ], there exists a ~g ∈ Fq[Z]n such that α(f, ~ g) holds.\n\n(2) For all ~g ∈ Fq[Z]n, the set {f ∈ Fq[W, Z ] | α(f, ~ g) holds } is finite. This will imply that r.e. = Diophantine for Fq[W, Z ].It is possible to give such a Diophantine predicate if \" α(· · · ) holds\" is replaced with \"α(· · · ) does not hold\".\n\n> 12MODERATED BY B. POONEN AND T. SCANLON, NOTES BY J. DEMEYER",
   "evidence": "The official AIM PDF, on its first page, gives the following question.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 115,
   "attempt": 1
  },
  "AIM-LOGIC-0117": {
   "statement_status": "exact",
   "original_statement": "Question 4 (Demeyer). Fix a prime p. Is there a Diophantine model of Fq[Z] over Fp[Z],when q is a power of p, uniformly in q?In other words, do there exist polynomials f (t, ~ x, ~ x′), g(t, ~ y, ~ y′) and h(t, ~ z, ~ z′) such that: \n\n• For every power q of p, Sq:= {~x | f (Zq, ~ x, ~ x′) = 0 } is in bijection with Fq[Z].\n\n• { ~y | g(Zq, ~ y, ~ y′) = 0 } ⊆ S3 \n\n> q\n\ncorresponds to the graph of addition on Fq[Z].\n\n• { ~z | h(Zq, ~ z, ~ z′) = 0 } ⊆ S3 \n\n> q\n\ncorresponds to the graph of multiplication on Fq[Z].Or with \" Zq\" replaced by some other reasonable function {powers of p} → Fp[Z].This might imply that r.e. = Diophantine for Fp[Z].",
   "clean_statement": "Question 4 (Demeyer). Fix a prime p. Is there a Diophantine model of Fq[Z] over Fp[Z],when q is a power of p, uniformly in q?In other words, do there exist polynomials f (t, ~ x, ~ x′), g(t, ~ y, ~ y′) and h(t, ~ z, ~ z′) such that:\n\n• For every power q of p, Sq:= {~x | f (Zq, ~ x, ~ x′) = 0 } is in bijection with Fq[Z].\n\n• { ~y | g(Zq, ~ y, ~ y′) = 0 } ⊆ S3\n\n> q\n\ncorresponds to the graph of addition on Fq[Z].\n\n• { ~z | h(Zq, ~ z, ~ z′) = 0 } ⊆ S3\n\n> q\n\ncorresponds to the graph of multiplication on Fq[Z].Or with \" Zq\" replaced by some other reasonable function {powers of p} → Fp[Z].This might imply that r.e. = Diophantine for Fp[Z].",
   "public_statement": "Question 4 (Demeyer). Fix a prime p. Is there a Diophantine model of Fq[Z] over Fp[Z],when q is a power of p, uniformly in q?In other words, do there exist polynomials f (t, ~ x, ~ x′), g(t, ~ y, ~ y′) and h(t, ~ z, ~ z′) such that:\n\n• For every power q of p, Sq:= {~x | f (Zq, ~ x, ~ x′) = 0 } is in bijection with Fq[Z].\n\n• { ~y | g(Zq, ~ y, ~ y′) = 0 } ⊆ S3\n\n> q\n\ncorresponds to the graph of addition on Fq[Z].\n\n• { ~z | h(Zq, ~ z, ~ z′) = 0 } ⊆ S3\n\n> q\n\ncorresponds to the graph of multiplication on Fq[Z].Or with \" Zq\" replaced by some other reasonable function {powers of p} → Fp[Z].This might imply that r.e. = Diophantine for Fp[Z].",
   "evidence": "The source is Question 4, attributed to J. Demeyer, in the AIM notes *Problems related to “Extensions of Hilbert's Tenth Problem”*. The official PDF gives the following statement after repairing OCR layout while preserving the mathematical text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 116,
   "attempt": 1
  },
  "AIM-LOGIC-0118": {
   "statement_status": "exact",
   "original_statement": "Question 5 (Pheidas). An additive polynomial in Fp[Z] is a polynomial of the form \n\nF (Z) = α0Z + α1Zp + α2Zp2\n\n+ · · · + αnZpn\n\n(αi ∈ Fp)\n\nThese are the polynomials that satisfy f (A + B) = f (A) + f (B) for all A, B ∈ Fp[Z]. Can we Diophantinely define the additive polynomials? (Demeyer) The following suggestion by Pheidas does not work: \n\n(∃A, B, C, L, M, N ∈ Fp[Z])( ∃α, β, γ, λ, μ, ν ∈ Fp)\n\nF = ( Ap − A) + αZ \n\n∧ F 2 = ( Bp − B) + βZ + ( Cp − C)Z + γZ 2\n\n∧ F 3 = ( Lp − L) + λZ + ( M p − M )Z + μZ 2 + ( N p − N )Z2 + νZ 3...Continue this up to some power F n. All additive polynomials satisfy this predicate, but also the following non-additive polynomial satisfies, no matter how many equations you add: \n\n> p−1\n\n∑\n\n> i=0\n\n(Z2p − Zp+1 )pi\n\nFact 6 (Cornelissen). Here is an example of a non-commutative undecidable theory. Let \n\nL be any field of characteristic p > 0. Let AL denote the ring of additive polynomials with coefficients from L (a ring for addition and composition). Then f ◦ Zp = Zp ◦ f is a Diophantine definition of AFp\n\n∼= Fp[Z] in AL.The same works in the quotient skew field QL of AL. Hence the Diophantine theory of \n\nAL and QL in a ring language augmented by a symbol for Z is undecidable (since the theories of Fq[Z] and Fq(Z) are by Denef and Pheidas). If one can therefore give a Diophantine definition of AL or QL in L[Z] or L(Z), the theory of the latter would be undecidable.",
   "clean_statement": "Question 5 (Pheidas). An additive polynomial in Fp[Z] is a polynomial of the form\n\nF (Z) = α0Z + α1Zp + α2Zp2\n\n+ · · · + αnZpn\n\n(αi ∈ Fp)\n\nThese are the polynomials that satisfy f (A + B) = f (A) + f (B) for all A, B ∈ Fp[Z]. Can we Diophantinely define the additive polynomials? (Demeyer) The following suggestion by Pheidas does not work:\n\n(∃A, B, C, L, M, N ∈ Fp[Z])( ∃α, β, γ, λ, μ, ν ∈ Fp)\n\nF = ( Ap − A) + αZ\n\n∧ F 2 = ( Bp − B) + βZ + ( Cp − C)Z + γZ 2\n\n∧ F 3 = ( Lp − L) + λZ + ( M p − M )Z + μZ 2 + ( N p − N )Z2 + νZ 3...Continue this up to some power F n. All additive polynomials satisfy this predicate, but also the following non-additive polynomial satisfies, no matter how many equations you add:\n\n> p−1\n\n∑\n\n> i=0\n\n(Z2p − Zp+1 )pi\n\nFact 6 (Cornelissen). Here is an example of a non-commutative undecidable theory. Let\n\nL be any field of characteristic p > 0. Let AL denote the ring of additive polynomials with coefficients from L (a ring for addition and composition). Then f ◦ Zp = Zp ◦ f is a Diophantine definition of AFp\n\n∼= Fp[Z] in AL.The same works in the quotient skew field QL of AL. Hence the Diophantine theory of\n\nAL and QL in a ring language augmented by a symbol for Z is undecidable (since the theories of Fq[Z] and Fq(Z) are by Denef and Pheidas). If one can therefore give a Diophantine definition of AL or QL in L[Z] or L(Z), the theory of the latter would be undecidable.",
   "public_statement": "Question 5 (Pheidas). An additive polynomial in Fp[Z] is a polynomial of the form\n\nF (Z) = α0Z + α1Zp + α2Zp2\n\n+ · · · + αnZpn\n\n(αi ∈ Fp)\n\nThese are the polynomials that satisfy f (A + B) = f (A) + f (B) for all A, B ∈ Fp[Z]. Can we Diophantinely define the additive polynomials? (Demeyer) The following suggestion by Pheidas does not work:\n\n(∃A, B, C, L, M, N ∈ Fp[Z])( ∃α, β, γ, λ, μ, ν ∈ Fp)\n\nF = ( Ap − A) + αZ\n\n∧ F 2 = ( Bp − B) + βZ + ( Cp − C)Z + γZ 2\n\n∧ F 3 = ( Lp − L) + λZ + ( M p − M )Z + μZ 2 + ( N p − N )Z2 + νZ 3...Continue this up to some power F n. All additive polynomials satisfy this predicate, but also the following non-additive polynomial satisfies, no matter how many equations you add:\n\n> p−1\n\n∑\n\n> i=0\n\n(Z2p − Zp+1 )pi\n\nFact 6 (Cornelissen). Here is an example of a non-commutative undecidable theory. Let\n\nL be any field of characteristic p > 0. Let AL denote the ring of additive polynomials with coefficients from L (a ring for addition and composition). Then f ◦ Zp = Zp ◦ f is a Diophantine definition of AFp\n\n∼= Fp[Z] in AL.The same works in the quotient skew field QL of AL. Hence the Diophantine theory of\n\nAL and QL in a ring language augmented by a symbol for Z is undecidable (since the theories of Fq[Z] and Fq(Z) are by Denef and Pheidas). If one can therefore give a Diophantine definition of AL or QL in L[Z] or L(Z), the theory of the latter would be undecidable.",
   "evidence": "The canonical record merges Question 5 with the following Fact 6 and loses many superscripts. I checked page 2 of the official 2005 AIM problem-list PDF. Question 5 is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 117,
   "attempt": 1
  },
  "AIM-LOGIC-0119": {
   "statement_status": "exact",
   "original_statement": "Question 5 of Pheidas tries to define the set AL. For cognescenti: this works more generally if \" f ◦ Zp = Zp ◦ f \" is replaced by f ◦ ρT = ρT ◦ f for ρ a Drinfeld Fq[T ]-module over L.",
   "clean_statement": "Question 5 of Pheidas tries to define the set AL. For cognescenti: this works more generally if \" f ◦ Zp = Zp ◦ f \" is replaced by f ◦ ρT = ρT ◦ f for ρ a Drinfeld Fq[T ]-module over L.",
   "public_statement": "Question 5 of Pheidas tries to define the set AL. For cognescenti: this works more generally if \" f ◦ Zp = Zp ◦ f \" is replaced by f ◦ ρT = ρT ◦ f for ρ a Drinfeld Fq[T ]-module over L.",
   "evidence": "The canonical record is an OCR extraction from the AIM workshop notes *Problems related to “Extensions of Hilbert's tenth problem”*, moderated by B. Poonen and T. Scanlon and recorded by J. Demeyer. The official PDF places the record at the end of Question 5, immediately after Fact 6 and immediately before Question 7. It is therefore explanatory context, not a separately posed open question.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 118,
   "attempt": 1
  },
  "AIM-LOGIC-0120": {
   "statement_status": "exact",
   "original_statement": "Question 7 (Davis). Let H be the quaternions over Q, and \n\nO = Z + iZ + jZ + kZ\n\n(1) Is there an algorithm to decide whether a noncommutative polynomial equation f (x1,..., x n) = 0 with coefficients in Q has a solution in H?\n\n(2) Is there an algorithm to decide whether a noncommutative polynomial equation f (x1,..., x n) = 0 with coefficients in Q has a solution in O?PROBLEMS RELATED TO \"EXTENSIONS OF HILBERT'S TENTH PROBLEM\" 3\n\n(3) Is there an algorithm to decide whether a noncommutative polynomial equation f (x1,..., x n) = 0 with coefficients in H has a solution in H?\n\n(4) Is there an algorithm to decide whether a noncommutative polynomial equation f (x1,..., x n) = 0 with coefficients in H has a solution in O?Is Z existentially definable in O? This probably works: \n\nx ∈ Z ⇐⇒ (∃I, J, K )( I2 = −1 ∧ J2 = −1 ∧ IJ = −JI ∧ xI = Ix ∧ xJ = Jx )\n\nVery likely done by D. Tunc. This solves the problems 2 and 4. In an analogous way, Q should be Diophantine in H. So, 1 and 3 are equivalent with Hilbert's Tenth Problem over Q.Same questions for the matrix rings Mn(Z) and Mn(Q).",
   "clean_statement": "Question 7 (Davis). Let H be the quaternions over Q, and\n\nO = Z + iZ + jZ + kZ\n\n(1) Is there an algorithm to decide whether a noncommutative polynomial equation f (x1,..., x n) = 0 with coefficients in Q has a solution in H?\n\n(2) Is there an algorithm to decide whether a noncommutative polynomial equation f (x1,..., x n) = 0 with coefficients in Q has a solution in O?PROBLEMS RELATED TO \"EXTENSIONS OF HILBERT'S TENTH PROBLEM\" 3\n\n(3) Is there an algorithm to decide whether a noncommutative polynomial equation f (x1,..., x n) = 0 with coefficients in H has a solution in H?\n\n(4) Is there an algorithm to decide whether a noncommutative polynomial equation f (x1,..., x n) = 0 with coefficients in H has a solution in O?Is Z existentially definable in O? This probably works:\n\nx ∈ Z ⇐⇒ (∃I, J, K )( I2 = −1 ∧ J2 = −1 ∧ IJ = −JI ∧ xI = Ix ∧ xJ = Jx )\n\nVery likely done by D. Tunc. This solves the problems 2 and 4. In an analogous way, Q should be Diophantine in H. So, 1 and 3 are equivalent with Hilbert's Tenth Problem over Q.Same questions for the matrix rings Mn(Z) and Mn(Q).",
   "public_statement": "Question 7 (Davis). Let H be the quaternions over Q, and\n\nO = Z + iZ + jZ + kZ\n\n(1) Is there an algorithm to decide whether a noncommutative polynomial equation f (x1,..., x n) = 0 with coefficients in Q has a solution in H?\n\n(2) Is there an algorithm to decide whether a noncommutative polynomial equation f (x1,..., x n) = 0 with coefficients in Q has a solution in O?PROBLEMS RELATED TO \"EXTENSIONS OF HILBERT'S TENTH PROBLEM\" 3\n\n(3) Is there an algorithm to decide whether a noncommutative polynomial equation f (x1,..., x n) = 0 with coefficients in H has a solution in H?\n\n(4) Is there an algorithm to decide whether a noncommutative polynomial equation f (x1,..., x n) = 0 with coefficients in H has a solution in O?Is Z existentially definable in O? This probably works:\n\nx ∈ Z ⇐⇒ (∃I, J, K )( I2 = −1 ∧ J2 = −1 ∧ IJ = −JI ∧ xI = Ix ∧ xJ = Jx )\n\nVery likely done by D. Tunc. This solves the problems 2 and 4. In an analogous way, Q should be Diophantine in H. So, 1 and 3 are equivalent with Hilbert's Tenth Problem over Q.Same questions for the matrix rings Mn(Z) and Mn(Q).",
   "evidence": "The official AIM PDF, *Problems related to “Extensions of Hilbert's Tenth Problem”*, Question 7 (Davis), asks the following. Let",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 119,
   "attempt": 1
  },
  "AIM-LOGIC-0121": {
   "statement_status": "exact",
   "original_statement": "Question 8 (Pheidas). Is the following problem decidable: Given P (~x) ∈ Z[~x], do there exist n1,..., n m ∈ N such that P (2 n1,..., 2nm ) = 0?The answer is YES: this is related to the Mordell-Lang conjecture for tori.",
   "clean_statement": "Question 8 (Pheidas). Is the following problem decidable: Given P (~x) ∈ Z[~x], do there exist n1,..., n m ∈ N such that P (2 n1,..., 2nm ) = 0?The answer is YES: this is related to the Mordell-Lang conjecture for tori.",
   "public_statement": "Question 8 (Pheidas). Is the following problem decidable: Given P (~x) ∈ Z[~x], do there exist n1,..., n m ∈ N such that P (2 n1,..., 2nm ) = 0?The answer is YES: this is related to the Mordell-Lang conjecture for tori.",
   "evidence": "The canonical record has lost exponent superscripts. I checked page 3 (PDF page index 2) of the official 2005 AIM problem list. The intended statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 120,
   "attempt": 1
  },
  "AIM-LOGIC-0122": {
   "statement_status": "exact",
   "original_statement": "Question 9 (Pheidas). Can we redo the proof of Hilbert's Tenth Problem over Z, using elliptic curves instead of Pell equations? Hopefully, this would lead to a lower number of variables and/or lower degree. Can this give a finite-fold Diophantine definition of all r.e. sets?",
   "clean_statement": "Question 9 (Pheidas). Can we redo the proof of Hilbert's Tenth Problem over Z, using elliptic curves instead of Pell equations? Hopefully, this would lead to a lower number of variables and/or lower degree. Can this give a finite-fold Diophantine definition of all r.e. sets?",
   "public_statement": "Question 9 (Pheidas). Can we redo the proof of Hilbert's Tenth Problem over Z, using elliptic curves instead of Pell equations? Hopefully, this would lead to a lower number of variables and/or lower degree. Can this give a finite-fold Diophantine definition of all r.e. sets?",
   "evidence": "The canonical record comes from the AIM workshop list *Problems related to “Extensions of Hilbert's tenth problem”*, moderated by B. Poonen and T. Scanlon and recorded by J. Demeyer. The official PDF gives the following Question 9:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 121,
   "attempt": 1
  },
  "AIM-LOGIC-0123": {
   "statement_status": "exact",
   "original_statement": "Question 10 (Davis). Find a native proof of DPRM in Z, instead of referring to N.Prove DPRM for some class of rings abstractly, with no reference to N.",
   "clean_statement": "Question 10 (Davis). Find a native proof of DPRM in Z, instead of referring to N.Prove DPRM for some class of rings abstractly, with no reference to N.",
   "public_statement": "Question 10 (Davis). Find a native proof of DPRM in Z, instead of referring to N.Prove DPRM for some class of rings abstractly, with no reference to N.",
   "evidence": "The repository record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 122,
   "attempt": 1
  },
  "AIM-LOGIC-0124": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 11 (Davis). A subset S ⊆ N is called simple if and only if: \n\n(1) S is r.e. \n\n(2) N \\ S is infinite. \n\n(3) If T ⊆ N \\ S is r.e., then T is finite. Take a simple set S ⊆ O K and an embedding f: OK ↪→ R, for some ring R. Let \n\nS = {x ∈ O K | (∃~y ∈ O nK )( P (x, ~ y) = 0) }\n\nand consider \n\nS′ = {x ∈ O K | (∃~y ∈ Rn)( P (x, ~ y) = 0) }\n\nClearly, f (S) ⊆ S′. Either S′ is simple (hence not recursive) or its complement is finite. In particular, if P (x, ~ y) ∈ Z[x, ~ y] is such that \n\n{x ∈ Z | (∃~y ∈ Zn)( P (x, ~ y) = 0) }\n\nis simple and \n\nZ \\ { x ∈ Z | (∃~y ∈ Qn)( P (x, ~ y) = 0) }\n\nis infinite, then Hilbert's Tenth Problem for Q has a negative answer. Reference: Davis, Putnam, \"Diophantine sets over polynomial rings\".",
   "clean_statement": null,
   "public_statement": "Question 11 (Davis). A subset S ⊆ N is called simple if and only if:\n\n(1) S is r.e.\n\n(2) N \\ S is infinite.\n\n(3) If T ⊆ N \\ S is r.e., then T is finite. Take a simple set S ⊆ O K and an embedding f: OK ↪→ R, for some ring R. Let\n\nS = {x ∈ O K | (∃~y ∈ O nK )( P (x, ~ y) = 0) }\n\nand consider\n\nS′ = {x ∈ O K | (∃~y ∈ Rn)( P (x, ~ y) = 0) }\n\nClearly, f (S) ⊆ S′. Either S′ is simple (hence not recursive) or its complement is finite. In particular, if P (x, ~ y) ∈ Z[x, ~ y] is such that\n\n{x ∈ Z | (∃~y ∈ Zn)( P (x, ~ y) = 0) }\n\nis simple and\n\nZ \\ { x ∈ Z | (∃~y ∈ Qn)( P (x, ~ y) = 0) }\n\nis infinite, then Hilbert's Tenth Problem for Q has a negative answer. Reference: Davis, Putnam, \"Diophantine sets over polynomial rings\".",
   "evidence": "The canonical source record has been preserved in `input.json`; the corrections above are explicit reconstructions, not silent changes to it.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 123,
   "attempt": 1
  },
  "AIM-LOGIC-0125": {
   "statement_status": "exact",
   "original_statement": "Question 12 (Cornelissen). If Z admits a Diophantine interpretation in Q (that is, using an equivalence relation), does it follow that Mazur's conjecture is wrong? See Cornelissen-Zahidi, Contemp. Math. 270 253-260. 4 MODERATED BY B. POONEN AND T. SCANLON, NOTES BY J. DEMEYER",
   "clean_statement": "Question 12 (Cornelissen). If Z admits a Diophantine interpretation in Q (that is, using an equivalence relation), does it follow that Mazur's conjecture is wrong? See Cornelissen-Zahidi, Contemp. Math. 270 253-260. 4 MODERATED BY B. POONEN AND T. SCANLON, NOTES BY J. DEMEYER",
   "public_statement": "Question 12 (Cornelissen). If Z admits a Diophantine interpretation in Q (that is, using an equivalence relation), does it follow that Mazur's conjecture is wrong? See Cornelissen-Zahidi, Contemp. Math. 270 253-260. 4 MODERATED BY B. POONEN AND T. SCANLON, NOTES BY J. DEMEYER",
   "evidence": "The official AIM PDF gives the following question on page 3 (PDF page index 2):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 124,
   "attempt": 1
  },
  "AIM-LOGIC-0126": {
   "statement_status": "exact",
   "original_statement": "Question 13 (Cornelissen). Solve in integers A, B, X, Y:\n\n(A2 + B2)( A2 + 11 B2) = 9 · 25 · (X2 − 5Y 2)2\n\nThis is related to defining the integers in the rational numbers by a Σ+3 -formula, see Cornelissen-Zahidi, ArXiv:math.NT/0412473.",
   "clean_statement": "Question 13 (Cornelissen). Solve in integers A, B, X, Y:\n\n(A2 + B2)( A2 + 11 B2) = 9 · 25 · (X2 − 5Y 2)2\n\nThis is related to defining the integers in the rational numbers by a Σ+3 -formula, see Cornelissen-Zahidi, ArXiv:math.NT/0412473.",
   "public_statement": "Question 13 (Cornelissen). Solve in integers A, B, X, Y:\n\n(A2 + B2)( A2 + 11 B2) = 9 · 25 · (X2 − 5Y 2)2\n\nThis is related to defining the integers in the rational numbers by a Σ+3 -formula, see Cornelissen-Zahidi, ArXiv:math.NT/0412473.",
   "evidence": "The repository OCR reads",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 125,
   "attempt": 1
  },
  "AIM-LOGIC-0127": {
   "statement_status": "exact",
   "original_statement": "Question 14 (Cornelissen). Jeroen Demeyer has observed that the existence of a polynomial bijection N2 → N implies that any first order formula over N in positive prenex form is equivalent to one in which every block of consecutive universal quantifiers is replaced by just one (and the number of existential quantifiers goes up). Such a polynomial bijection can be found in Davis, Math. Monthly 80, 236-237. Does something similar work for Q, in other words, can we find a Diophantine injection \n\nQ2 ↪→ Q? There are some observations related to this in C.R.A.S. Paris 328, 3-8 (1999); for example, this would follow from the generalized abc-conjecture.",
   "clean_statement": "Question 14 (Cornelissen). Jeroen Demeyer has observed that the existence of a polynomial bijection N2 → N implies that any first order formula over N in positive prenex form is equivalent to one in which every block of consecutive universal quantifiers is replaced by just one (and the number of existential quantifiers goes up). Such a polynomial bijection can be found in Davis, Math. Monthly 80, 236-237. Does something similar work for Q, in other words, can we find a Diophantine injection\n\nQ2 ↪→ Q? There are some observations related to this in C.R.A.S. Paris 328, 3-8 (1999); for example, this would follow from the generalized abc-conjecture.",
   "public_statement": "Question 14 (Cornelissen). Jeroen Demeyer has observed that the existence of a polynomial bijection N2 → N implies that any first order formula over N in positive prenex form is equivalent to one in which every block of consecutive universal quantifiers is replaced by just one (and the number of existential quantifiers goes up). Such a polynomial bijection can be found in Davis, Math. Monthly 80, 236-237. Does something similar work for Q, in other words, can we find a Diophantine injection\n\nQ2 ↪→ Q? There are some observations related to this in C.R.A.S. Paris 328, 3-8 (1999); for example, this would follow from the generalized abc-conjecture.",
   "evidence": "The official AIM PDF gives the following statement (Question 14, Cornelissen):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 126,
   "attempt": 1
  },
  "AIM-LOGIC-0128": {
   "statement_status": "exact",
   "original_statement": "Question 15 (Rojas). What is the smallest n such that Hilbert's Tenth Problem over Z\n\nrestricted to one polynomial in n variables is undecidable? Minimal n is known to be 2 ≤ n ≤ 22 by Matijaseviˇ c, and probably 2 ≤ n ≤ 11 by some Chinese. There is some evidence that n = 3.",
   "clean_statement": "Question 15 (Rojas). What is the smallest n such that Hilbert's Tenth Problem over Z\n\nrestricted to one polynomial in n variables is undecidable? Minimal n is known to be 2 ≤ n ≤ 22 by Matijaseviˇ c, and probably 2 ≤ n ≤ 11 by some Chinese. There is some evidence that n = 3.",
   "public_statement": "Question 15 (Rojas). What is the smallest n such that Hilbert's Tenth Problem over Z\n\nrestricted to one polynomial in n variables is undecidable? Minimal n is known to be 2 ≤ n ≤ 22 by Matijaseviˇ c, and probably 2 ≤ n ≤ 11 by some Chinese. There is some evidence that n = 3.",
   "evidence": "This record is Question 15 from the problem list associated with the American Institute of Mathematics workshop *Extensions of Hilbert's Tenth Problem*, held 21--25 March 2005. The list is titled *Problems related to \"Extensions of Hilbert's Tenth Problem\"*, moderated by B. Poonen and T. Scanlon, with notes by J. Demeyer.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 127,
   "attempt": 1
  },
  "AIM-LOGIC-0129": {
   "statement_status": "exact",
   "original_statement": "Question 16 (Rojas). Consider sequences in Z[x] of the form \n\n1, x, g 1, g 2,... \n\nwhere each gi is a sum, difference or product of 2 earlier terms in the sequence. Let \n\nτ (f ):= min {n | there exists such a sequence with gn = f }\n\nConjecture: there exists a constant c such that the number of integer zeros of f is at most (1 + τ (f ))c, where f is not identically zero.",
   "clean_statement": "Question 16 (Rojas). Consider sequences in Z[x] of the form\n\n1, x, g 1, g 2,...\n\nwhere each gi is a sum, difference or product of 2 earlier terms in the sequence. Let\n\nτ (f ):= min {n | there exists such a sequence with gn = f }\n\nConjecture: there exists a constant c such that the number of integer zeros of f is at most (1 + τ (f ))c, where f is not identically zero.",
   "public_statement": "Question 16 (Rojas). Consider sequences in Z[x] of the form\n\n1, x, g 1, g 2,...\n\nwhere each gi is a sum, difference or product of 2 earlier terms in the sequence. Let\n\nτ (f ):= min {n | there exists such a sequence with gn = f }\n\nConjecture: there exists a constant c such that the number of integer zeros of f is at most (1 + τ (f ))c, where f is not identically zero.",
   "evidence": "The canonical record is Question 16 (Rojas) in the AIM problem list *Problems related to “Extensions of Hilbert's Tenth Problem”*, moderated by B. Poonen and T. Scanlon and recorded by J. Demeyer. The text extraction lost subscripts on the \\(g_i\\)'s and, more importantly, printed the final exponent as an ordinary `c`. The official PDF, p. 4 (PDF page index 3), displays",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 128,
   "attempt": 1
  },
  "AIM-LOGIC-0130": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 17 (Rojas). Let cj ∈ Z and consider polynomials of the form \n\nP (x1,..., x n) = \n\n> n+1\n\n∏\n\n> j=1\n\ncj~x ~aj\n\nwhere ~a1,..., ~an+1 ∈ Nn are affinely independent. Can we decide in polynomial time (for fixed p) whether there exists a ~x ∈ Qnp such that \n\nP (~x) = 0?Answer: NO, because the 0/1 knapsack problem can be encoded as a subproblem of this (Poonen). Over R this is in NP, and probably in P (modulo some technicalities). Can we decide whether there exists a ~x ∈ Qn such that P (~x) = 0?This includes the unsolved problem of deciding whether a genus 1 curve of the form \n\nax 3 + by 3 = 1 has a rational point, so it is probably very hard.",
   "clean_statement": null,
   "public_statement": "Question 17 (Rojas). Let cj ∈ Z and consider polynomials of the form\n\nP (x1,..., x n) =\n\n> n+1\n\n∏\n\n> j=1\n\ncj~x ~aj\n\nwhere ~a1,..., ~an+1 ∈ Nn are affinely independent. Can we decide in polynomial time (for fixed p) whether there exists a ~x ∈ Qnp such that\n\nP (~x) = 0?Answer: NO, because the 0/1 knapsack problem can be encoded as a subproblem of this (Poonen). Over R this is in NP, and probably in P (modulo some technicalities). Can we decide whether there exists a ~x ∈ Qn such that P (~x) = 0?This includes the unsolved problem of deciding whether a genus 1 curve of the form\n\nax 3 + by 3 = 1 has a rational point, so it is probably very hard.",
   "evidence": "The canonical JSON is visibly corrupted by PDF extraction. In particular, the displayed sum was read as a product, vector notation was flattened, and \\(\\mathbb Q_p^n\\) was read as \\(\\mathbb Q^{np}\\). Inspection of Question 17 in the official AIM workshop PDF recovers the following statement.",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 129,
   "attempt": 1
  },
  "AIM-LOGIC-0131": {
   "statement_status": "exact",
   "original_statement": "Question 18 (Rojas). Is there a computable bound (in function of f ) on the size of the largest integer solution to f (x, y ) = 0, when there are finitely many solutions? This is already done for genus 1 curves. There exists an algorithm to decide finiteness of the set of solutions. For rational points, there are papers by Minhyong Kim from Arizona: \"Relating decision and search algorithms for rational points on curves of higher genus\", Arch. Math. Logic 42 (2003), no. 6, 563-568 PROBLEMS RELATED TO \"EXTENSIONS OF HILBERT'S TENTH PROBLEM\" 5\n\n\"On relative computability for curves\", ArXiv:math.NT/0502224",
   "clean_statement": "Question 18 (Rojas). Is there a computable bound (in function of f ) on the size of the largest integer solution to f (x, y ) = 0, when there are finitely many solutions? This is already done for genus 1 curves. There exists an algorithm to decide finiteness of the set of solutions. For rational points, there are papers by Minhyong Kim from Arizona: \"Relating decision and search algorithms for rational points on curves of higher genus\", Arch. Math. Logic 42 (2003), no. 6, 563-568 PROBLEMS RELATED TO \"EXTENSIONS OF HILBERT'S TENTH PROBLEM\" 5\n\n\"On relative computability for curves\", ArXiv:math.NT/0502224",
   "public_statement": "Question 18 (Rojas). Is there a computable bound (in function of f ) on the size of the largest integer solution to f (x, y ) = 0, when there are finitely many solutions? This is already done for genus 1 curves. There exists an algorithm to decide finiteness of the set of solutions. For rational points, there are papers by Minhyong Kim from Arizona: \"Relating decision and search algorithms for rational points on curves of higher genus\", Arch. Math. Logic 42 (2003), no. 6, 563-568 PROBLEMS RELATED TO \"EXTENSIONS OF HILBERT'S TENTH PROBLEM\" 5\n\n\"On relative computability for curves\", ArXiv:math.NT/0502224",
   "evidence": "The canonical record is Question 18 from the AIM workshop *Extensions of Hilbert's tenth problem*. The official AIM PDF gives the following question (typography normalized, but wording preserved):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 130,
   "attempt": 1
  },
  "AIM-LOGIC-0132": {
   "statement_status": "exact",
   "original_statement": "Question 19 (Jarden). Is there an algorithm to decide whether f (x, y ) = 0 has infinitely many Q-rational solutions? This seems to be very hard for genus 1 curves. It has been done in other cases. Possible if X is finite for all elliptic curves over Q.",
   "clean_statement": "Question 19 (Jarden). Is there an algorithm to decide whether f (x, y ) = 0 has infinitely many Q-rational solutions? This seems to be very hard for genus 1 curves. It has been done in other cases. Possible if X is finite for all elliptic curves over Q.",
   "public_statement": "Question 19 (Jarden). Is there an algorithm to decide whether f (x, y ) = 0 has infinitely many Q-rational solutions? This seems to be very hard for genus 1 curves. It has been done in other cases. Possible if X is finite for all elliptic curves over Q.",
   "evidence": "The canonical record is Question 19 (Jarden) from the 2005 AIM workshop *Extensions of Hilbert's tenth problem*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 131,
   "attempt": 1
  },
  "AIM-LOGIC-0133": {
   "statement_status": "exact",
   "original_statement": "Question 20 (Shlapentokh). Let E be an elliptic curve over Q of rank 2. Does there exist an existentially definable rank 1 subgroup?",
   "clean_statement": "Question 20 (Shlapentokh). Let E be an elliptic curve over Q of rank 2. Does there exist an existentially definable rank 1 subgroup?",
   "public_statement": "Question 20 (Shlapentokh). Let E be an elliptic curve over Q of rank 2. Does there exist an existentially definable rank 1 subgroup?",
   "evidence": "The official AIM workshop PDF states, without OCR damage:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 132,
   "attempt": 1
  },
  "AIM-LOGIC-0134": {
   "statement_status": "exact",
   "original_statement": "Question 21 (Shlapentokh). Let E be an elliptic curve over Q of rank 2. Can we find a subset S of (infinitely many) primes such that the subgroup generated by E(Z[S−1]) has rank one? If S is finite, the Siegel-Mahler theorem states that E(Z[S−1]) is finite. Suppose S is infinite, but of density 0. Is E(Z[S−1]) still \"small\"?",
   "clean_statement": "Question 21 (Shlapentokh). Let E be an elliptic curve over Q of rank 2. Can we find a subset S of (infinitely many) primes such that the subgroup generated by E(Z[S−1]) has rank one? If S is finite, the Siegel-Mahler theorem states that E(Z[S−1]) is finite. Suppose S is infinite, but of density 0. Is E(Z[S−1]) still \"small\"?",
   "public_statement": "Question 21 (Shlapentokh). Let E be an elliptic curve over Q of rank 2. Can we find a subset S of (infinitely many) primes such that the subgroup generated by E(Z[S−1]) has rank one? If S is finite, the Siegel-Mahler theorem states that E(Z[S−1]) is finite. Suppose S is infinite, but of density 0. Is E(Z[S−1]) still \"small\"?",
   "evidence": "Question 21 (Shlapentokh) in the 2005 AIM list *Extensions of Hilbert's tenth problem* reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 133,
   "attempt": 1
  },
  "AIM-LOGIC-0135": {
   "statement_status": "exact",
   "original_statement": "Question 22 (Zahidi). Look at the Denef curve \n\nE: f (t)Y 2 = f (X)\n\nwhere f is a cubic. If we choose the curve in a good way, then E(k(t)) has rank 1.Define \n\nEu: f (u)Y 2 = f (X)\n\nTry to give conditions on u ∈ k(t) such that Eu(k(t)) also has rank 1.",
   "clean_statement": "Question 22 (Zahidi). Look at the Denef curve\n\nE: f (t)Y 2 = f (X)\n\nwhere f is a cubic. If we choose the curve in a good way, then E(k(t)) has rank 1.Define\n\nEu: f (u)Y 2 = f (X)\n\nTry to give conditions on u ∈ k(t) such that Eu(k(t)) also has rank 1.",
   "public_statement": "Question 22 (Zahidi). Look at the Denef curve\n\nE: f (t)Y 2 = f (X)\n\nwhere f is a cubic. If we choose the curve in a good way, then E(k(t)) has rank 1.Define\n\nEu: f (u)Y 2 = f (X)\n\nTry to give conditions on u ∈ k(t) such that Eu(k(t)) also has rank 1.",
   "evidence": "The canonical JSON record has lost superscript and subscript formatting, but the official AIM workshop PDF gives the following unambiguous statement:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 134,
   "attempt": 1
  },
  "AIM-LOGIC-0136": {
   "statement_status": "exact",
   "original_statement": "Question 23 (Pheidas). Consider the elliptic curve \n\nE: Y 2 = X3 + aX + b\n\nThe following statement is Diophantine: \" End( E)/(2 End( E)) has more than 2 elements\". Because End( E) is a free finitely generated Z-module, this is equivalent with \" End( E) 6 = Z\". So, we can existentially define the following set in C(Z):\n\n{j ∈ C | j is the j-invariant of a CM elliptic curve }\n\nCan we do anything with this set?",
   "clean_statement": "Question 23 (Pheidas). Consider the elliptic curve\n\nE: Y 2 = X3 + aX + b\n\nThe following statement is Diophantine: \" End( E)/(2 End( E)) has more than 2 elements\". Because End( E) is a free finitely generated Z-module, this is equivalent with \" End( E) 6 = Z\". So, we can existentially define the following set in C(Z):\n\n{j ∈ C | j is the j-invariant of a CM elliptic curve }\n\nCan we do anything with this set?",
   "public_statement": "Question 23 (Pheidas). Consider the elliptic curve\n\nE: Y 2 = X3 + aX + b\n\nThe following statement is Diophantine: \" End( E)/(2 End( E)) has more than 2 elements\". Because End( E) is a free finitely generated Z-module, this is equivalent with \" End( E) 6 = Z\". So, we can existentially define the following set in C(Z):\n\n{j ∈ C | j is the j-invariant of a CM elliptic curve }\n\nCan we do anything with this set?",
   "evidence": "The canonical record is Question 23 in the AIM workshop list *Extensions of Hilbert's tenth problem*. Direct inspection of page 5 of the official PDF gives:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 135,
   "attempt": 1
  },
  "AIM-LOGIC-0137": {
   "statement_status": "exact",
   "original_statement": "Question 24 (Pheidas). If x ∈ C(Z), then \n\nord Z=0 \n\n( 1 + Zx 2\n\n1 − Zx 2\n\n)\n\n= ord Z=∞\n\n( 1 + Zx 2\n\n1 − Zx 2\n\n)\n\n= 0 \n\nCan every f ∈ C(Z) with ord Z=0 (f ) = ord Z=∞(f ) even be written as (obviously, the number 1000 can be changed to any other integer) \n\nf = u21000 ∏\n\n> i=1\n\n1 + Zx 2\n\n> i\n\n1 − Zx 2\n\n> i\n\nWeaker version: is this true at least for f ∈ Q(Z), with u, x i ∈ C(Z)?This would imply that the existential theory of C(Z) is undecidable.",
   "clean_statement": "Question 24 (Pheidas). If x ∈ C(Z), then\n\nord Z=0\n\n( 1 + Zx 2\n\n1 − Zx 2\n\n)\n\n= ord Z=∞\n\n( 1 + Zx 2\n\n1 − Zx 2\n\n)\n\n= 0\n\nCan every f ∈ C(Z) with ord Z=0 (f ) = ord Z=∞(f ) even be written as (obviously, the number 1000 can be changed to any other integer)\n\nf = u21000 ∏\n\n> i=1\n\n1 + Zx 2\n\n> i\n\n1 − Zx 2\n\n> i\n\nWeaker version: is this true at least for f ∈ Q(Z), with u, x i ∈ C(Z)?This would imply that the existential theory of C(Z) is undecidable.",
   "public_statement": "Question 24 (Pheidas). If x ∈ C(Z), then\n\nord Z=0\n\n( 1 + Zx 2\n\n1 − Zx 2\n\n)\n\n= ord Z=∞\n\n( 1 + Zx 2\n\n1 − Zx 2\n\n)\n\n= 0\n\nCan every f ∈ C(Z) with ord Z=0 (f ) = ord Z=∞(f ) even be written as (obviously, the number 1000 can be changed to any other integer)\n\nf = u21000 ∏\n\n> i=1\n\n1 + Zx 2\n\n> i\n\n1 − Zx 2\n\n> i\n\nWeaker version: is this true at least for f ∈ Q(Z), with u, x i ∈ C(Z)?This would imply that the existential theory of C(Z) is undecidable.",
   "evidence": "This is Question 24 in the problem list from the March 2005 AIM workshop *Extensions of Hilbert's Tenth Problem*. The canonical JSON record has a layout/OCR corruption: `u21000` is not an exponent, and the `1000` belongs above a product sign. Inspection of the official PDF gives the following normalized statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 136,
   "attempt": 1
  },
  "AIM-LOGIC-0138": {
   "statement_status": "exact",
   "original_statement": "Question 25 (Pheidas). Is {f ∈ C(Z) | ord Z=0 (f ) ≥ 0} (existentially) definable in C(Z),where there is a symbol for Z in the language? 6 MODERATED BY B. POONEN AND T. SCANLON, NOTES BY J. DEMEYER",
   "clean_statement": "Question 25 (Pheidas). Is {f ∈ C(Z) | ord Z=0 (f ) ≥ 0} (existentially) definable in C(Z),where there is a symbol for Z in the language? 6 MODERATED BY B. POONEN AND T. SCANLON, NOTES BY J. DEMEYER",
   "public_statement": "Question 25 (Pheidas). Is {f ∈ C(Z) | ord Z=0 (f ) ≥ 0} (existentially) definable in C(Z),where there is a symbol for Z in the language? 6 MODERATED BY B. POONEN AND T. SCANLON, NOTES BY J. DEMEYER",
   "evidence": "The official AIM workshop PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 137,
   "attempt": 1
  },
  "AIM-LOGIC-0139": {
   "statement_status": "exact",
   "original_statement": "Question 26 (Moret-Bailly). Is there a nontrivial valuation ring \n\nR ⊂ Frac R[x, y ]\n\n(x2 + y2 + 1) \n\nwhich is definable? Same question for \"semi-local ring\" (finite intersection of valuation rings) instead of \"valuation ring\"? This is equivalent with the problem for valuation rings.",
   "clean_statement": "Question 26 (Moret-Bailly). Is there a nontrivial valuation ring\n\nR ⊂ Frac R[x, y ]\n\n(x2 + y2 + 1)\n\nwhich is definable? Same question for \"semi-local ring\" (finite intersection of valuation rings) instead of \"valuation ring\"? This is equivalent with the problem for valuation rings.",
   "public_statement": "Question 26 (Moret-Bailly). Is there a nontrivial valuation ring\n\nR ⊂ Frac R[x, y ]\n\n(x2 + y2 + 1)\n\nwhich is definable? Same question for \"semi-local ring\" (finite intersection of valuation rings) instead of \"valuation ring\"? This is equivalent with the problem for valuation rings.",
   "evidence": "The canonical JSON record has lost a typographical distinction and displays both the sought valuation ring and the real constant field as `R`. The official AIM PDF was downloaded and its page-6 PDF content stream was inspected. The first symbol is set in the ordinary math-italic font CMMI12, whereas the numerator uses the blackboard-bold font MSBM10. Thus the recovered statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 138,
   "attempt": 1
  },
  "AIM-LOGIC-0140": {
   "statement_status": "exact",
   "original_statement": "Question 27 (Shlapentokh). Can one find an algebraically closed field K and a nontrivial valuation ring R ⊂ K(Z) (or a finite extension), which is definable in K(Z)?",
   "clean_statement": "Question 27 (Shlapentokh). Can one find an algebraically closed field K and a nontrivial valuation ring R ⊂ K(Z) (or a finite extension), which is definable in K(Z)?",
   "public_statement": "Question 27 (Shlapentokh). Can one find an algebraically closed field K and a nontrivial valuation ring R ⊂ K(Z) (or a finite extension), which is definable in K(Z)?",
   "evidence": "The exact source is the AIM problem list *Problems related to “Extensions of Hilbert's Tenth Problem”*, moderated by B. Poonen and T. Scanlon, with notes by J. Demeyer. The source PDF was checked directly. Question 27 reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 139,
   "attempt": 1
  },
  "AIM-LOGIC-0141": {
   "statement_status": "exact",
   "original_statement": "Question 28 (Shlapentokh). Is there an algebraic extension K of Q and a nontrivial valu-ation ring R ⊂ K, such that the residue field of R is algebraically closed and R is definable over K?Answer: YES. Inside Qalg \n\n> p\n\n= Q ∩ Qp ⊆ Qp, the ring Zalg \n\n> p\n\nis definable. \n\nFact 29 (Pheidas). C[[ Z]] is definable in C(( Z)):\n\nx ∈ C[[ Z]] ⇐⇒ (∃y)(1 + Zx 2 = y2)\n\nProven using Hensel's lemma.",
   "clean_statement": "Question 28 (Shlapentokh). Is there an algebraic extension K of Q and a nontrivial valu-ation ring R ⊂ K, such that the residue field of R is algebraically closed and R is definable over K?Answer: YES. Inside Qalg\n\n> p\n\n= Q ∩ Qp ⊆ Qp, the ring Zalg\n\n> p\n\nis definable.\n\nFact 29 (Pheidas). C[[ Z]] is definable in C(( Z)):\n\nx ∈ C[[ Z]] ⇐⇒ (∃y)(1 + Zx 2 = y2)\n\nProven using Hensel's lemma.",
   "public_statement": "Question 28 (Shlapentokh). Is there an algebraic extension K of Q and a nontrivial valu-ation ring R ⊂ K, such that the residue field of R is algebraically closed and R is definable over K?Answer: YES. Inside Qalg\n\n> p\n\n= Q ∩ Qp ⊆ Qp, the ring Zalg\n\n> p\n\nis definable.\n\nFact 29 (Pheidas). C[[ Z]] is definable in C(( Z)):\n\nx ∈ C[[ Z]] ⇐⇒ (∃y)(1 + Zx 2 = y2)\n\nProven using Hensel's lemma.",
   "evidence": "The canonical record combines two consecutive items from the 2005 AIM workshop list *Extensions of Hilbert's Tenth Problem*. Visual inspection of the official PDF shows the boundary clearly.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 140,
   "attempt": 1
  },
  "AIM-LOGIC-0142": {
   "statement_status": "exact",
   "original_statement": "Question 30 (Shlapentokh). Let K be a number field and OK its ring of integers. Fix an embedding K ↪ → C, with K 6 ⊆ R. Is {α ∈ O K | | α| ≤ 1} Diophantine in OK?If this is true for all K, then Hilbert's Tenth Problem is undecidable for all OK.",
   "clean_statement": "Question 30 (Shlapentokh). Let K be a number field and OK its ring of integers. Fix an embedding K ↪ → C, with K 6 ⊆ R. Is {α ∈ O K | | α| ≤ 1} Diophantine in OK?If this is true for all K, then Hilbert's Tenth Problem is undecidable for all OK.",
   "public_statement": "Question 30 (Shlapentokh). Let K be a number field and OK its ring of integers. Fix an embedding K ↪ → C, with K 6 ⊆ R. Is {α ∈ O K | | α| ≤ 1} Diophantine in OK?If this is true for all K, then Hilbert's Tenth Problem is undecidable for all OK.",
   "evidence": "The exact source is the AIM problem list *Problems related to “Extensions of Hilbert's Tenth Problem”*, moderated by B. Poonen and T. Scanlon, with notes by J. Demeyer. The source PDF was checked directly. Question 30 reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 141,
   "attempt": 1
  },
  "AIM-LOGIC-0143": {
   "statement_status": "exact",
   "original_statement": "Question 31 (Cornelissen). Let K be a number field and OK its ring of integers. A set \n\nA ⊆ O K is said to be division-ample if \n\n• It is Diophantine over OK.\n\n• Any x ∈ O K divides some a ∈ A.\n\n• There exists a positive integer l such that for any a ∈ A, there exists ˜a ∈ Z with ˜a|a\n\nand N (a) ≤ | ˜a|l.Observe that if A ⊆ Z, then one can dispose of the last condition by choosing ˜a = a\n\nand l = [ K: Q].Question: give an example of such A where for any finite S ⊆ O K, A is not a subset of O∗ \n\n> K\n\n· (Z ∪ S).Cornelissen-Pheidas-Zahidi have shown that HTP( OK ) has a negative answer if such \n\nA exists and there exists an elliptic curve of rank one over K.",
   "clean_statement": "Question 31 (Cornelissen). Let K be a number field and OK its ring of integers. A set\n\nA ⊆ O K is said to be division-ample if\n\n• It is Diophantine over OK.\n\n• Any x ∈ O K divides some a ∈ A.\n\n• There exists a positive integer l such that for any a ∈ A, there exists ˜a ∈ Z with ˜a|a\n\nand N (a) ≤ | ˜a|l.Observe that if A ⊆ Z, then one can dispose of the last condition by choosing ˜a = a\n\nand l = [ K: Q].Question: give an example of such A where for any finite S ⊆ O K, A is not a subset of O∗\n\n> K\n\n· (Z ∪ S).Cornelissen-Pheidas-Zahidi have shown that HTP( OK ) has a negative answer if such\n\nA exists and there exists an elliptic curve of rank one over K.",
   "public_statement": "Question 31 (Cornelissen). Let K be a number field and OK its ring of integers. A set\n\nA ⊆ O K is said to be division-ample if\n\n• It is Diophantine over OK.\n\n• Any x ∈ O K divides some a ∈ A.\n\n• There exists a positive integer l such that for any a ∈ A, there exists ˜a ∈ Z with ˜a|a\n\nand N (a) ≤ | ˜a|l.Observe that if A ⊆ Z, then one can dispose of the last condition by choosing ˜a = a\n\nand l = [ K: Q].Question: give an example of such A where for any finite S ⊆ O K, A is not a subset of O∗\n\n> K\n\n· (Z ∪ S).Cornelissen-Pheidas-Zahidi have shown that HTP( OK ) has a negative answer if such\n\nA exists and there exists an elliptic curve of rank one over K.",
   "evidence": "The official AIM PDF was checked at the level of its page-6 content stream. The symbols lost or displaced by OCR are \\(\\widetilde a\\), \\(\\mathbb Z\\), and the superscript star in \\(\\mathcal O_K^*\\). The recovered statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 142,
   "attempt": 1
  },
  "AIM-LOGIC-0144": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 32 (Poonen). Is is true that for all number fields K, there exists a variety X\n\n(scheme of finite type) over Z such that \n\n(1) X(Z) is infinite. \n\n(2) X(OK ) = X(Z).",
   "clean_statement": "**Question 32 (Poonen).** Is it true that for all number fields $K$, there exists a variety $X$ (scheme of finite type) over $\\mathbb Z$ such that\n\n1. $X(\\mathbb Z)$ is infinite.\n2. $X(\\mathcal O_K)=X(\\mathbb Z)$.",
   "public_statement": "Question 32 (Poonen). Is is true that for all number fields K, there exists a variety X\n\n(scheme of finite type) over Z such that\n\n(1) X(Z) is infinite.\n\n(2) X(OK ) = X(Z).",
   "evidence": "The canonical JSON extraction reads “Is is true” and writes the ring of integers as `OK`. Inspection of the AIM source identifies these as extraction/OCR defects. The recovered statement is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-logic-notes.json",
   "source_index": 143,
   "attempt": 1
  },
  "AIM-LOGIC-0145": {
   "statement_status": "exact",
   "original_statement": "Question 33 (Videla). Let K ⊆ Qtot. real ⊆ Q. Define \n\nAK:= {s ∈ R>0 | There exist infinitely many α ∈ O K\n\nsuch that α and its conjugates are all in [0, s ]}\n\nQuestion of Julia Robinson: Is the infimum of AK an element of AK? If so, the first order theory of OK is undecidable. For K = Qtot. real, inf( AK ) = 4 ∈ AK.PROBLEMS RELATED TO \"EXTENSIONS OF HILBERT'S TENTH PROBLEM\" 7",
   "clean_statement": "Question 33 (Videla). Let K ⊆ Qtot. real ⊆ Q. Define\n\nAK:= {s ∈ R>0 | There exist infinitely many α ∈ O K\n\nsuch that α and its conjugates are all in [0, s ]}\n\nQuestion of Julia Robinson: Is the infimum of AK an element of AK? If so, the first order theory of OK is undecidable. For K = Qtot. real, inf( AK ) = 4 ∈ AK.PROBLEMS RELATED TO \"EXTENSIONS OF HILBERT'S TENTH PROBLEM\" 7",
   "public_statement": "Question 33 (Videla). Let K ⊆ Qtot. real ⊆ Q. Define\n\nAK:= {s ∈ R>0 | There exist infinitely many α ∈ O K\n\nsuch that α and its conjugates are all in [0, s ]}\n\nQuestion of Julia Robinson: Is the infimum of AK an element of AK? If so, the first order theory of OK is undecidable. For K = Qtot. real, inf( AK ) = 4 ∈ AK.PROBLEMS RELATED TO \"EXTENSIONS OF HILBERT'S TENTH PROBLEM\" 7",
   "evidence": "The source is the AIM problem list *Problems related to “Extensions of Hilbert's Tenth Problem”*, moderated by B. Poonen and T. Scanlon, with notes by J. Demeyer. The typeset PDF gives the following statement:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 144,
   "attempt": 1
  },
  "AIM-LOGIC-0146": {
   "statement_status": "exact",
   "original_statement": "Question 34 (Zahidi). Let Ralg:= Q ∩ R. It is known that Ralg ≡ R (elementary equiv-alence), but that Ralg (t) 6 ≡ R(t). On the other hand, the existential theories of Ralg (t) and \n\nR(t) are the same. What is the minimal quantifier complexity for which Ralg (t) and R(t)\n\nhave different theories? Another question is the minimal number of variables one needs.",
   "clean_statement": "Question 34 (Zahidi). Let Ralg:= Q ∩ R. It is known that Ralg ≡ R (elementary equiv-alence), but that Ralg (t) 6 ≡ R(t). On the other hand, the existential theories of Ralg (t) and\n\nR(t) are the same. What is the minimal quantifier complexity for which Ralg (t) and R(t)\n\nhave different theories? Another question is the minimal number of variables one needs.",
   "public_statement": "Question 34 (Zahidi). Let Ralg:= Q ∩ R. It is known that Ralg ≡ R (elementary equiv-alence), but that Ralg (t) 6 ≡ R(t). On the other hand, the existential theories of Ralg (t) and\n\nR(t) are the same. What is the minimal quantifier complexity for which Ralg (t) and R(t)\n\nhave different theories? Another question is the minimal number of variables one needs.",
   "evidence": "The OCR record has lost a conjugation bar and has turned `\\(\\not\\equiv\\)` into `6 equiv`. Page 7 of the official AIM PDF gives the following recovered statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 145,
   "attempt": 1
  },
  "AIM-LOGIC-0147": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 35 (Pheidas). Let X be a variety over Q. Call X hyperbolic iff there is no nonconstant holomorphic map C → X(C). Is there an algorithm which can decide whether a variety X/ Q over hyperbolic?",
   "clean_statement": "**Is there an algorithm which can decide whether a variety\n\\(X/\\mathbb Q\\) is hyperbolic?**",
   "public_statement": "Question 35 (Pheidas). Let X be a variety over Q. Call X hyperbolic iff there is no nonconstant holomorphic map C → X(C). Is there an algorithm which can decide whether a variety X/ Q over hyperbolic?",
   "evidence": "The canonical record comes from Question 35 of the AIM problem list *Problems related to “Extensions of Hilbert's Tenth Problem”*, moderated by B. Poonen and T. Scanlon, with notes by J. Demeyer. The PDF text reads:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-logic-notes.json",
   "source_index": 146,
   "attempt": 1
  },
  "AIM-LOGIC-0148": {
   "statement_status": "corrected_verified",
   "original_statement": "Question 36 (Jarden). Given f1,..., f n ∈ C[x1,..., x m] which are homogeneous of degree \n\nd. Assume that the only common zero of the fi is (0,..., 0). Prove that \n\nV (f1(~x) = b1,..., f n(~x) = bn)\n\nis finite, for all b1,..., b n ∈ C.Solution: If it were infinite, then the variety in Pm defined by the homogenizations of the equations would be positive-dimensional, and then it would have to intersect the hyperplane at infinity, which would mean that the fi have a common zero.",
   "clean_statement": "**Question 36 (Jarden).** Given $f_1,\\ldots,f_n\\in\\mathbb C[x_1,\\ldots,x_m]$ which are homogeneous of degree $d$, assume that their only common zero is $(0,\\ldots,0)$. Prove that\n\\[\nV\\bigl(f_1(\\vec x)=b_1,\\ldots,f_n(\\vec x)=b_n\\bigr)\n\\]\nis finite for all $b_1,\\ldots,b_n\\in\\mathbb C$.",
   "public_statement": "**Question 36 (Jarden).** Given $f_1,\\ldots,f_n\\in\\mathbb C[x_1,\\ldots,x_m]$ which are homogeneous of degree $d$, assume that their only common zero is $(0,\\ldots,0)$. Prove that\n\\[\nV\\bigl(f_1(\\vec x)=b_1,\\ldots,f_n(\\vec x)=b_n\\bigr)\n\\]\nis finite for all $b_1,\\ldots,b_n\\in\\mathbb C$.",
   "evidence": "The canonical JSON preserves the source record but contains extraction artifacts: `f n`, `x m`, `C.Solution`, and `f_i(~x)`. The official AIM PDF confirms the intended subscripts, spacing, and vector notation. The recovered statement is:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-logic-notes.json",
   "source_index": 147,
   "attempt": 1
  },
  "AIM-LOGIC-0149": {
   "statement_status": "exact",
   "original_statement": "A.1 Aspero, David \n\nMost of Hugh Woodin's deep work on core model theory remains unpublished. I believe this workshop will be an excellent opportunity for me to learn about the Cofinal Branches Hypothesis (about which I don't know much) and about Woodin's refutation of it.",
   "clean_statement": "A.1 Aspero, David\n\nMost of Hugh Woodin's deep work on core model theory remains unpublished. I believe this workshop will be an excellent opportunity for me to learn about the Cofinal Branches Hypothesis (about which I don't know much) and about Woodin's refutation of it.",
   "public_statement": "A.1 Aspero, David\n\nMost of Hugh Woodin's deep work on core model theory remains unpublished. I believe this workshop will be an excellent opportunity for me to learn about the Cofinal Branches Hypothesis (about which I don't know much) and about Woodin's refutation of it.",
   "evidence": "The canonical record is source index 148 of **aim-logic-notes.json**. Its source is the official AIM PDF *Recent Advances in Core Model Theory*, version dated 29 November 2004. The PDF table of contents calls Chapter A “Participant Contributions”; page 3 begins with the same heading and then lists the participants by name. The entries immediately following A.1 are also first-person descriptions of what participants hoped to learn at the workshop.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 148,
   "attempt": 1
  },
  "AIM-LOGIC-0150": {
   "statement_status": "exact",
   "original_statement": "A.2 Brooke-Taylor, Andrew \n\nI find progress on the continuum hypothesis very interesting, and so learning what the ramifications of this work are for Woodin's Omega conjecture is particularly enticing. Also, my current area of study is at the level of I3, so it will be good to know more about the difficulties in trying to build models with superstrong cardinals, with an eye to continuing the ascent.",
   "clean_statement": "A.2 Brooke-Taylor, Andrew\n\nI find progress on the continuum hypothesis very interesting, and so learning what the ramifications of this work are for Woodin's Omega conjecture is particularly enticing. Also, my current area of study is at the level of I3, so it will be good to know more about the difficulties in trying to build models with superstrong cardinals, with an eye to continuing the ascent.",
   "public_statement": "A.2 Brooke-Taylor, Andrew\n\nI find progress on the continuum hypothesis very interesting, and so learning what the ramifications of this work are for Woodin's Omega conjecture is particularly enticing. Also, my current area of study is at the level of I3, so it will be good to know more about the difficulties in trying to build models with superstrong cardinals, with an eye to continuing the ascent.",
   "evidence": "The source is the four-page AIM document *Recent Advances in Core Model Theory*, version 29 November 2004. Its table of contents labels Chapter A “Participant Contributions.” On page 3, item A.2 occurs between the analogous personal statements A.1 and A.3. The exact recovered statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 149,
   "attempt": 1
  },
  "AIM-LOGIC-0151": {
   "statement_status": "exact",
   "original_statement": "A.3 Brown, Elizabeth \n\nCore Model Theory as such is largely a new topic to me; what I know about CMT specifically I have learnt in preparation for this workshop. I am particularly interested in CMT as part of the general study of large cardinals and their equivalencies, and in the implications of CMT for analysis.",
   "clean_statement": "A.3 Brown, Elizabeth\n\nCore Model Theory as such is largely a new topic to me; what I know about CMT specifically I have learnt in preparation for this workshop. I am particularly interested in CMT as part of the general study of large cardinals and their equivalencies, and in the implications of CMT for analysis.",
   "public_statement": "A.3 Brown, Elizabeth\n\nCore Model Theory as such is largely a new topic to me; what I know about CMT specifically I have learnt in preparation for this workshop. I am particularly interested in CMT as part of the general study of large cardinals and their equivalencies, and in the implications of CMT for analysis.",
   "evidence": "The canonical record is item A.3 in the AIM workshop document *Recent advances in core model theory*. The source PDF labels its appendix “Participant Contributions”; A.1, A.2, A.3, and A.4 are individual participants' statements. The text of A.3 is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 150,
   "attempt": 1
  },
  "AIM-LOGIC-0152": {
   "statement_status": "corrected_verified",
   "original_statement": "A.4 Cummings, James \n\nSome of my goals/questions for the workshop: A. I would like a better understanding of the relationship between models of the form HOD M and the classical L[ ~E] models. B. Very strong axioms are used in the refutations of UBH and CBH (at least in the versions of these results which I have heard about). How exactly are they used? Are they necessary? C. I would like to know more about Woodin's recent work on promising extender se-quences. D. I would like to get a picture of the status and significance of the Ω conjecture.",
   "clean_statement": "Some of my goals/questions for the workshop:\n\nA. I would like a better understanding of the relationship between models of the form \\(\\mathrm{HOD}^M\\) and the classical \\(L[\\vec E]\\) models.\n\nB. Very strong axioms are used in the refutations of UBH and CBH (at least in the versions of these results which I have heard about). How exactly are they used? Are they necessary?\n\nC. I would like to know more about Woodin's recent work on promising extender sequences.\n\nD. I would like to get a picture of the status and significance of the \\(\\Omega\\) conjecture.",
   "public_statement": "Some of my goals/questions for the workshop:\n\nA. I would like a better understanding of the relationship between models of the form \\(\\mathrm{HOD}^M\\) and the classical \\(L[\\vec E]\\) models.\n\nB. Very strong axioms are used in the refutations of UBH and CBH (at least in the versions of these results which I have heard about). How exactly are they used? Are they necessary?\n\nC. I would like to know more about Woodin's recent work on promising extender sequences.\n\nD. I would like to get a picture of the status and significance of the \\(\\Omega\\) conjecture.",
   "evidence": "The record is item A.4, James Cummings's contribution to the AIM workshop *Recent advances in core model theory*. The canonical extraction is substantially readable, but it loses two pieces of mathematical typography and introduces a line-break hyphen. Inspection of the official AIM PDF verifies the following recovered statement: The PDF content stream places the \\(M\\) as a superscript on HOD and places a vector accent over \\(E\\); thus the intended expressions are \\(\\mathrm{HOD}^M\\) and \\(L[\\vec E]\\), not “HOD M” and a literal \\(L[\\widetilde E]\\). “se-quences” is only a line-break artifact. These corrections are source-verified rather than silent rewrites.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-logic-notes.json",
   "source_index": 151,
   "attempt": 1
  },
  "AIM-LOGIC-0153": {
   "statement_status": "unrecoverable",
   "original_statement": "A.5 Dobrinen, Natasha \n\nI am delighted to participate in the ARCC Workshop. My general goal is, naturally, to gain a deeper and better understanding of the state of the art in core model research and find some open problems to work on. In particular, my research with games related to distributive laws in Boolean algebras is leading me to look at large cardinals. My hope is to learn techniques for working with large cardinals which will help in settling questions about relationships between games, distributive laws, and stationary subsets of Pκλ.",
   "clean_statement": null,
   "public_statement": "A.5 Dobrinen, Natasha\n\nI am delighted to participate in the ARCC Workshop. My general goal is, naturally, to gain a deeper and better understanding of the state of the art in core model research and find some open problems to work on. In particular, my research with games related to distributive laws in Boolean algebras is leading me to look at large cardinals. My hope is to learn techniques for working with large cardinals which will help in settling questions about relationships between games, distributive laws, and stationary subsets of Pκλ.",
   "evidence": "This is a research-interest statement, not a mathematical question with specified hypotheses or a requested conclusion. Thus the correct status is `context_only`, with `problem_status_at_run` equal to `not_a_problem`.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-logic-notes.json",
   "source_index": 152,
   "attempt": 1
  },
  "AIM-LOGIC-0154": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A.6 Dzamonja, Mirna \n\nThe primary interest of my research is combinatorial set theory. The fields of core model theory and combinatorial set theory may look rather distant at a first glance, but in fact the work in the last ten or so years have shown that there is a large overlap. I am very 4\n\npleased to participate in the workshop with the idea of learning more methods that have been invented within the core model theory and understanding their combinatorial nature.",
   "clean_statement": null,
   "public_statement": "A.6 Dzamonja, Mirna\n\nThe primary interest of my research is combinatorial set theory. The fields of core model theory and combinatorial set theory may look rather distant at a first glance, but in fact the work in the last ten or so years have shown that there is a large overlap. I am very 4\n\npleased to participate in the workshop with the idea of learning more methods that have been invented within the core model theory and understanding their combinatorial nature.",
   "evidence": "The canonical record is item A.6 of the AIM workshop document *Recent advances in core model theory*. The PDF places A.6 in Chapter A, “Participant Contributions.” After correcting only a page-layout artifact, the recovered statement is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 153,
   "attempt": 1
  },
  "AIM-LOGIC-0155": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A.7 Fuchs, Gunter \n\nOf the scope of the conference, two topics are of main importance to me. Firstly, getting familiar with the method used to refute the CBH is essential. I am looking forward to learning about this. Secondly, the theory of inner models constructed relative to a sequence of extenders together with partial iteration strategies seems to get more and more important. This is an intriguing area, and I hope to be able to do some research here, using a very widely applicable form of fine structure theory.",
   "clean_statement": null,
   "public_statement": "A.7 Fuchs, Gunter\n\nOf the scope of the conference, two topics are of main importance to me. Firstly, getting familiar with the method used to refute the CBH is essential. I am looking forward to learning about this. Secondly, the theory of inner models constructed relative to a sequence of extenders together with partial iteration strategies seems to get more and more important. This is an intriguing area, and I hope to be able to do some research here, using a very widely applicable form of fine structure theory.",
   "evidence": "The official AIM PDF confirms that A.7 is Gunter Fuchs's participant contribution to the 2004 workshop *Recent advances in core model theory*. The recovered statement is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-logic-notes.json",
   "source_index": 154,
   "attempt": 1
  },
  "AIM-LOGIC-0156": {
   "statement_status": "exact",
   "original_statement": "A.8 Greenberg, Noam \n\nMy mathematical training is mainly in recursion theory on admissible ordinals, in which fine-structural notions come into play. I am also in general interested in set theory; inner model theory in particular has some flavor of recursion theory, and so I hope to gain understanding of the field. As I am a beginner I'm afraid I don't have much to contribute in the way of suggesting problems or issues for consideration.",
   "clean_statement": "A.8 Greenberg, Noam\n\nMy mathematical training is mainly in recursion theory on admissible ordinals, in which fine-structural notions come into play. I am also in general interested in set theory; inner model theory in particular has some flavor of recursion theory, and so I hope to gain understanding of the field. As I am a beginner I'm afraid I don't have much to contribute in the way of suggesting problems or issues for consideration.",
   "public_statement": "A.8 Greenberg, Noam\n\nMy mathematical training is mainly in recursion theory on admissible ordinals, in which fine-structural notions come into play. I am also in general interested in set theory; inner model theory in particular has some flavor of recursion theory, and so I hope to gain understanding of the field. As I am a beginner I'm afraid I don't have much to contribute in the way of suggesting problems or issues for consideration.",
   "evidence": "The official AIM PDF *Recent Advances in Core Model Theory* was checked directly. Item A.8 occurs in Chapter A, “Participant Contributions,” and reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 155,
   "attempt": 1
  },
  "AIM-LOGIC-0157": {
   "statement_status": "exact",
   "original_statement": "A.9 Koellner, Peter \n\nI am interested in Woodin's HOD -analysis. In particular, I would like to come away from the workshop with an understanding of the analysis of HOD L(R) (under the assumption of AD L(R)) and the proof that HOD L(R) thinks V = HOD.",
   "clean_statement": "A.9 Koellner, Peter\n\nI am interested in Woodin's HOD -analysis. In particular, I would like to come away from the workshop with an understanding of the analysis of HOD L(R) (under the assumption of AD L(R)) and the proof that HOD L(R) thinks V = HOD.",
   "public_statement": "A.9 Koellner, Peter\n\nI am interested in Woodin's HOD -analysis. In particular, I would like to come away from the workshop with an understanding of the analysis of HOD L(R) (under the assumption of AD L(R)) and the proof that HOD L(R) thinks V = HOD.",
   "evidence": "This record is item A.9 in the appendix “Participant Contributions” of the AIM workshop report *Recent advances in core model theory*. The canonical JSON lost superscripts. Inspection of the official PDF typography gives the following recovered statement (the superscripts are the only reconstruction):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 156,
   "attempt": 1
  },
  "AIM-LOGIC-0158": {
   "statement_status": "exact",
   "original_statement": "A.10 Sargsyan, Grigor \n\nI am a second year graduate student at UC, Berkeley, and have taken inner model theory seminars with Professor Steel and Professor Woodin. Having learnt the basics of inner model theory, I hope to learn, during this workshop, more advanced topics in core model theory.",
   "clean_statement": "A.10 Sargsyan, Grigor\n\nI am a second year graduate student at UC, Berkeley, and have taken inner model theory seminars with Professor Steel and Professor Woodin. Having learnt the basics of inner model theory, I hope to learn, during this workshop, more advanced topics in core model theory.",
   "public_statement": "A.10 Sargsyan, Grigor\n\nI am a second year graduate student at UC, Berkeley, and have taken inner model theory seminars with Professor Steel and Professor Woodin. Having learnt the basics of inner model theory, I hope to learn, during this workshop, more advanced topics in core model theory.",
   "evidence": "The official AIM PDF verifies the canonical A.10 entry as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 157,
   "attempt": 1
  },
  "AIM-LOGIC-0159": {
   "statement_status": "exact",
   "original_statement": "A.11 Schimmerling, Ernest \n\nPlease see http://www.math.cmu.edu/ ∼eschimme/AIM/Index.html",
   "clean_statement": "A.11 Schimmerling, Ernest\n\nPlease see http://www.math.cmu.edu/ ∼eschimme/AIM/Index.html",
   "public_statement": "A.11 Schimmerling, Ernest\n\nPlease see http://www.math.cmu.edu/ ∼eschimme/AIM/Index.html",
   "evidence": "The canonical record is preserved verbatim in `input.json`:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 158,
   "attempt": 1
  },
  "AIM-LOGIC-0160": {
   "statement_status": "exact",
   "original_statement": "A.12 Yoshinobu, Yasuo \n\nI am working in core model theory. My interests are exactly those described by the organizers in their detailed conference description.",
   "clean_statement": "A.12 Yoshinobu, Yasuo\n\nI am working in core model theory. My interests are exactly those described by the organizers in their detailed conference description.",
   "public_statement": "A.12 Yoshinobu, Yasuo\n\nI am working in core model theory. My interests are exactly those described by the organizers in their detailed conference description.",
   "evidence": "The canonical record is item A.12 in Chapter A, “Participant Contributions,” of the official AIM PDF *Recent Advances in Core Model Theory*. The PDF was checked directly and reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-logic-notes.json",
   "source_index": 159,
   "attempt": 1
  },
  "AIM-OPTIMIZATION-0001": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 1.10.1. Characterize all d-dimensional, pointed, closed, convex cones in \n\nRd which admit a lifted-LMI representation. I believe that this set of cones strictly contain the hyperbolic cones. \n\nOpen",
   "clean_statement": "**Open Problem 1.10.1.** Characterize all \\(d\\)-dimensional, pointed, closed, convex cones in \\(\\mathbb R^d\\) which admit a lifted-LMI representation. I believe that this set of cones strictly contain the hyperbolic cones.",
   "public_statement": "**Open Problem 1.10.1.** Characterize all \\(d\\)-dimensional, pointed, closed, convex cones in \\(\\mathbb R^d\\) which admit a lifted-LMI representation. I believe that this set of cones strictly contain the hyperbolic cones.",
   "evidence": "The official AIM PDF, *Theory and Algorithms of Linear Matrix Inequalities*, pp. 10--11, places this contribution in Levent Tunçel's section 1.10, “Representation Theory for LMIs.” The PDF first gives the following definition, with notation normalized but mathematical content preserved: The JSON text “Rd” is OCR loss for \\(\\mathbb R^d\\); it also omits the preceding definition. The source's phrase “strictly contain” is reproduced rather than silently corrected. “\\(d\\)-dimensional in \\(\\mathbb R^d\\)” means full-dimensional. Importantly, the source defines the representation through the **strictly feasible interior**, not merely by a weak inequality on all boundary points.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-optimization-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-OPTIMIZATION-0002": {
   "statement_status": "exact",
   "original_statement": "Problem 1.10.2. Characterize all d-dimensional, pointed, closed, convex cones in \n\nRd which admit a poly-time, lifted-LMI representation. I believe that this set of cones strictly contain the hyperbolic cones. \n\nMost specifically: \n\nOpen",
   "clean_statement": "Problem 1.10.2. Characterize all d-dimensional, pointed, closed, convex cones in\n\nRd which admit a poly-time, lifted-LMI representation. I believe that this set of cones strictly contain the hyperbolic cones.\n\nMost specifically:\n\nOpen",
   "public_statement": "Problem 1.10.2. Characterize all d-dimensional, pointed, closed, convex cones in\n\nRd which admit a poly-time, lifted-LMI representation. I believe that this set of cones strictly contain the hyperbolic cones.\n\nMost specifically:\n\nOpen",
   "evidence": "The canonical JSON record has two extraction defects. The string `Rd` is \\(\\mathbb R^d\\), and the trailing words “Most specifically: Open” do not belong to Problem 1.10.2. In the official AIM PDF, “Most specifically:” is a transition to the separately numbered Problem 1.10.3. The recovered statement of the assigned problem is therefore:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-optimization-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-OPTIMIZATION-0003": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 1.10.3. Are all Hyperbolic Feasibility Problems polynomial-time equivalent to LMI problems? \n\nThis last question needs some definitions and clarifications. \n\nDefinition 1.10.2. Let p1, p 2,..., p m: Rd → R be given polynomials. Then the problem \"does there exist x ∈ Rd such that pi(x) ≥ 0, ∀i ∈ { 1, 2,..., m } is a Hyperbolic Feasibility Problem (HFP) if every pi is a hyperbolic polynomial. \n\nNext, we define the size( HF P ). The \"size\" should involve the basic complexity measures needed to bound the amount of computational effort required (in the Blum-Shub-Smale real computation model) to \"solve\" HFP to \u000f ∈ (0, 1) accuracy using some general class of well-established algorithms. For instance, we can define size( HF P ):= max {m, ln(1 /\u000f ), ln( R)},\n\nwhere R > 1 denotes the volume of a given ellipsoid E0 which determines the region in which we will decide the solvability of HFP. I.e., our problem is to find ¯ x ∈ E0 satisfying all the inequalities. We require that after poly(size( HLP )) operations the algorithm either outputs ¯ x ∈ Rd such that pi(¯ x) ≥ 0, ∀i ∈ { 1, 2,..., m } or it outputs \"there does not exist a ball of volume at least \u000f which is contained in \n\nE0 ∩ {x ∈ Rd: pi(x) ≥ 0, ∀i ∈ { 1, 2,..., m }}.′′ \n\nIn this context, when we say HFP is polynomial-time equivalent to LMI we mean that for every HFP (with m, R and a given \u000f ∈ (0, 1)), we can explicitly describe an LMI such that 11 • the formulated LMI can be solved to \u000f accuracy in time poly (size( HF P )), \n\n• solving the LMI within accuracy \u000f, solves the original HFP. This notion of poly-time equivalence is quite important in optimization theory. A problem analogous to",
   "clean_statement": null,
   "public_statement": "Problem 1.10.3. Are all Hyperbolic Feasibility Problems polynomial-time equivalent to LMI problems?\n\nThis last question needs some definitions and clarifications.\n\nDefinition 1.10.2. Let p1, p 2,..., p m: Rd → R be given polynomials. Then the problem \"does there exist x ∈ Rd such that pi(x) ≥ 0, ∀i ∈ { 1, 2,..., m } is a Hyperbolic Feasibility Problem (HFP) if every pi is a hyperbolic polynomial.\n\nNext, we define the size( HF P ). The \"size\" should involve the basic complexity measures needed to bound the amount of computational effort required (in the Blum-Shub-Smale real computation model) to \"solve\" HFP to [U+000F] ∈ (0, 1) accuracy using some general class of well-established algorithms. For instance, we can define size( HF P ):= max {m, ln(1 /[U+000F] ), ln( R)},\n\nwhere R > 1 denotes the volume of a given ellipsoid E0 which determines the region in which we will decide the solvability of HFP. I.e., our problem is to find ¯ x ∈ E0 satisfying all the inequalities. We require that after poly(size( HLP )) operations the algorithm either outputs ¯ x ∈ Rd such that pi(¯ x) ≥ 0, ∀i ∈ { 1, 2,..., m } or it outputs \"there does not exist a ball of volume at least [U+000F] which is contained in\n\nE0 ∩ {x ∈ Rd: pi(x) ≥ 0, ∀i ∈ { 1, 2,..., m }}.′′\n\nIn this context, when we say HFP is polynomial-time equivalent to LMI we mean that for every HFP (with m, R and a given [U+000F] ∈ (0, 1)), we can explicitly describe an LMI such that 11 • the formulated LMI can be solved to [U+000F] accuracy in time poly (size( HF P )),\n\n• solving the LMI within accuracy [U+000F], solves the original HFP. This notion of poly-time equivalence is quite important in optimization theory. A problem analogous to",
   "evidence": "The canonical record comes from the AIM workshop notes *Theory and Algorithms of Linear Matrix Inequalities*, Open Problem 1.10.3 (printed pp. 10--11; PDF pages 11--12), version dated March 12, 2006. The mathematical core is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-optimization-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-OPTIMIZATION-0004": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 1.10.3 was solved in [5] by showing that Second Order Cone Programming is poly.-time equivalent to Linear Programming. \n\n1.11 Hugo Woerdeman \n\nOne of the questions I am interested in is how to approximate numerically the Schur com-plement of a positive definite operator supported on a finite subspace. Of course there is the formula that for a block matrix [A B; C D] the Schur complement is A B inv(D) C, but this requires determining the inverse of the infinite operator D. The way this question arose is through attempts to develop multivariable analogs of the Gohberg-Semencul formula. One way to prove the Gohberg-Semencul formula is by determining the Schur complement of the inverse of a Toeplitz operator whose symbol is the reciprocal of a positive trigonometric poly-nomial. In several variables the analogous attempt runs into difficulties. In order to at least get a decent numerical approximation of the inverse of a doubly Toeplitz matrix, one may try to find reasonable numerical methods to determine finitely based Schur complements of infinite operators. 12 Chapter 2 Ideas for Teaching \n\nMihai: OPEN FOR ADDITIONS, CORRECTIONS, REARRANGEMENTS ToDo \n\nA sketch of a plan: 1. General convexity (Hahn Banach, Minkowski separation theorem, Caratheodory's the-orem on generators of convex hulls) 2. Weighted sums of squares in free *-algebras 3. The spectral theorem for commuting self-adjoint operators. Note the spectral measure in physical terms is just the power spectral density. 4. Multivariate moment problems and their dual: weighted SOS decompositions of poly-nomials 5. Applications (optimization, Lyapunov functions,...) 6. Real algebra, logic and the full Positivestellensatz 7. More optimization (see Tuncel, Henrion, Lasserre) 13 Chapter 3 Other",
   "clean_statement": "**1.11 Hugo Woerdeman.** One of the questions I am interested in is how to approximate numerically the Schur complement of a positive definite operator supported on a finite subspace. Of course there is the formula that for a block matrix\n\\[\n\\begin{pmatrix}A&B\\\\ C&D\\end{pmatrix}\n\\]\nthe Schur complement is \\(A-BD^{-1}C\\), but this requires determining the inverse of the infinite operator \\(D\\). The way this question arose is through attempts to develop multivariable analogs of the Gohberg--Semencul formula. One way to prove the Gohberg--Semencul formula is by determining the Schur complement of the inverse of a Toeplitz operator whose symbol is the reciprocal of a positive trigonometric polynomial. In several variables the analogous attempt runs into difficulties. In order to at least get a decent numerical approximation of the inverse of a doubly Toeplitz matrix, one may try to find reasonable numerical methods to determine finitely based Schur complements of infinite operators.",
   "public_statement": "**1.11 Hugo Woerdeman.** One of the questions I am interested in is how to approximate numerically the Schur complement of a positive definite operator supported on a finite subspace. Of course there is the formula that for a block matrix\n\\[\n\\begin{pmatrix}A&B\\\\ C&D\\end{pmatrix}\n\\]\nthe Schur complement is \\(A-BD^{-1}C\\), but this requires determining the inverse of the infinite operator \\(D\\). The way this question arose is through attempts to develop multivariable analogs of the Gohberg--Semencul formula. One way to prove the Gohberg--Semencul formula is by determining the Schur complement of the inverse of a Toeplitz operator whose symbol is the reciprocal of a positive trigonometric polynomial. In several variables the analogous attempt runs into difficulties. In order to at least get a decent numerical approximation of the inverse of a doubly Toeplitz matrix, one may try to find reasonable numerical methods to determine finitely based Schur complements of infinite operators.",
   "evidence": "This canonical record is a genuine extraction-boundary collision. The exact record in the assigned input contains all of the following, and none is silently discarded: The official AIM PDF fixes the boundaries exactly. On PDF p. 12, the sentence",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-optimization-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-PDES-0001": {
   "statement_status": "exact",
   "original_statement": "Nonlinear enhanced dissipation for the 2D Euler equation with horizontal viscosity",
   "clean_statement": "Nonlinear enhanced dissipation for the 2D Euler equation with horizontal viscosity",
   "public_statement": "Nonlinear enhanced dissipation for the 2D Euler equation with horizontal viscosity",
   "evidence": "The canonical record contains only the title. The original AimPL page is no longer available at its live URL, but the Internet Archive snapshot dated 2024-08-28 recovers the page. Its complete mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-PDES-0002": {
   "statement_status": "exact",
   "original_statement": "2D stationary Navier-Stokes with viscosity-independent forcing",
   "clean_statement": "2D stationary Navier-Stokes with viscosity-independent forcing",
   "public_statement": "2D stationary Navier-Stokes with viscosity-independent forcing",
   "evidence": "The canonical record contains only the title “2D stationary Navier--Stokes with viscosity-independent forcing.” The archived official AIM Problem Lists page, captured on 28 August 2024, gives the following statement:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-PDES-0003": {
   "statement_status": "exact",
   "original_statement": "Large data global wellposedness for 2D Burgers with hyper-dissipation",
   "clean_statement": "Large data global wellposedness for 2D Burgers with hyper-dissipation",
   "public_statement": "Large data global wellposedness for 2D Burgers with hyper-dissipation",
   "evidence": "The canonical record says only:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 2,
   "attempt": 2
  },
  "AIM-PDES-0004": {
   "statement_status": "exact",
   "original_statement": "Existence/nonexistence of self-similar solutions for 3D Navier-Stokes",
   "clean_statement": "Existence/nonexistence of self-similar solutions for 3D Navier-Stokes",
   "public_statement": "Existence/nonexistence of self-similar solutions for 3D Navier-Stokes",
   "evidence": "The canonical record has the title",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-PDES-0005": {
   "statement_status": "exact",
   "original_statement": "Extending illposedness for SQG to QG",
   "clean_statement": "Extending illposedness for SQG to QG",
   "public_statement": "Extending illposedness for SQG to QG",
   "evidence": "The canonical record gives only the title “Extending illposedness for SQG to QG.” The archived official AIM Problem Lists page, captured on 28 August 2024, contains exactly one mathematical sentence:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-PDES-0006": {
   "statement_status": "exact",
   "original_statement": "Exponential growth of the length of Lagrangian flow map for 2D Euler",
   "clean_statement": "Exponential growth of the length of Lagrangian flow map for 2D Euler",
   "public_statement": "Exponential growth of the length of Lagrangian flow map for 2D Euler",
   "evidence": "The canonical corpus record contains only the title",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-PDES-0007": {
   "statement_status": "exact",
   "original_statement": "The QG equation with Ekman layer with topography",
   "clean_statement": "The QG equation with Ekman layer with topography",
   "public_statement": "The QG equation with Ekman layer with topography",
   "evidence": "The canonical AIM record says, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 6,
   "attempt": 2
  },
  "AIM-PDES-0008": {
   "statement_status": "exact",
   "original_statement": "2D Euler norm inflation",
   "clean_statement": "2D Euler norm inflation",
   "public_statement": "2D Euler norm inflation",
   "evidence": "The AIM workshop report independently confirms the intended range by naming the working-group problem “Norm inflation for 2D Euler, for \\(u\\in H^s\\), \\(s\\in(0,1)\\).” It says that the difficulty is precisely the low velocity regularity between the two conserved endpoint norms and contrasts the question with the then-recent result for \\(s\\in(1,2)\\). Thus \\(0<s<1\\) is not an OCR error. The corpus extraction lost the body of a section-type record; it must not be silently replaced by the heading.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-PDES-0009": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Vanishing viscosity limit in a setting like Temam--Wang",
   "clean_statement": null,
   "public_statement": "Vanishing viscosity limit in a setting like Temam--Wang",
   "evidence": "The canonical record contains only the heading",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-pdes-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-PDES-0010": {
   "statement_status": "exact",
   "original_statement": "Prove existence and stability of pulled fronts for the complex Ginzburg-Landau equation.",
   "clean_statement": "Prove existence and stability of pulled fronts for the complex Ginzburg-Landau equation.",
   "public_statement": "Prove existence and stability of pulled fronts for the complex Ginzburg-Landau equation.",
   "evidence": "The canonical AIM record says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-PDES-0011": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A starting point for the above problem would be\n\nProve existence and stability for pushed or pulled fronts for a toy model using a computer assisted proof.",
   "clean_statement": null,
   "public_statement": "A starting point for the above problem would be\n\nProve existence and stability for pushed or pulled fronts for a toy model using a computer assisted proof.",
   "evidence": "The official section heading is exactly \"Existence and marginally stability of pushed and pulled fronts.\" The grammatical defect is present on the source page and is therefore not an OCR error. The intended phrase is almost surely \"existence and marginal stability,\" but this reconstruction is not silently substituted for the source.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-pdes-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-PDES-0012": {
   "statement_status": "exact",
   "original_statement": "Use computer assisted methods to extend spatial dynamics in infinite\ndimensions from the perturbative regime.",
   "clean_statement": "Use computer assisted methods to extend spatial dynamics in infinite\ndimensions from the perturbative regime.",
   "public_statement": "Use computer assisted methods to extend spatial dynamics in infinite\ndimensions from the perturbative regime.",
   "evidence": "The newline between “infinite” and “dimensions” in `input.json` is a source line wrap, not an OCR error. The archived page contains no model equation, hypotheses, boundary conditions, or indication of whether the desired object is a center manifold, periodic orbit, front, pulse, or other bounded spatial trajectory. Accordingly, this report does **not** silently choose a particular PDE or claim a model-specific existence result.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-PDES-0013": {
   "statement_status": "exact",
   "original_statement": "Develop rigorous computer assisted methods for computing the Maslov index in two dimensions.",
   "clean_statement": "Develop rigorous computer assisted methods for computing the Maslov index in two dimensions.",
   "public_statement": "Develop rigorous computer assisted methods for computing the Maslov index in two dimensions.",
   "evidence": "The canonical AIM record says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-PDES-0014": {
   "statement_status": "exact",
   "original_statement": "Develop rigorous computer assisted methods for computing the Evans function in two dimensions.",
   "clean_statement": "Develop rigorous computer assisted methods for computing the Evans function in two dimensions.",
   "public_statement": "Develop rigorous computer assisted methods for computing the Evans function in two dimensions.",
   "evidence": "The recovered statement is AIM Problem 4.2 in the section “Computing Maslov indices and Evans functions in 2 dimensions” of the 2023 workshop *Computer assisted proofs for stability analysis of nonlinear waves*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-PDES-0015": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Prove existence of a traveling wave for Gross-Pitaevskii equation in\n\\(\\mathbb{R}^2\\) for all wave speeds \\(c \\in [0, \\sqrt{2}]\\).",
   "clean_statement": null,
   "public_statement": "Prove existence of a traveling wave for Gross-Pitaevskii equation in\n\\(\\mathbb{R}^2\\) for all wave speeds \\(c \\in [0, \\sqrt{2}]\\).",
   "evidence": "The canonical record in `aim-pdes-notes.json` (zero-based index 14) states exactly:",
   "classification_method": "invalid_literal_with_repaired_reading",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-pdes-notes.json",
   "source_index": 14,
   "attempt": 2
  },
  "AIM-PDES-0016": {
   "statement_status": "exact",
   "original_statement": "Prove that the spectrum for spatially periodic solutions to\n generalised KdV lies entirely on the imaginary axis.",
   "clean_statement": "Prove that the spectrum for spatially periodic solutions to\n generalised KdV lies entirely on the imaginary axis.",
   "public_statement": "Prove that the spectrum for spatially periodic solutions to\n generalised KdV lies entirely on the imaginary axis.",
   "evidence": "The original AIM page was checked. It contains the same sentence, attributes the item to Jared Bronski, and supplies no equation, nonlinearity, wave family, period, or perturbation class. Thus “generalised” is the source's British spelling, not an OCR error, but the mathematical statement is under-specified. In particular, it does not say whether “the spectrum” means:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-PDES-0017": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Estimate size of the basin of attraction for each equilibria.",
   "clean_statement": null,
   "public_statement": "Estimate size of the basin of attraction for each equilibria.",
   "evidence": "The archived AIM page gives Problem 7.1 in the section “Finite-dimensional gradient flows with many stable equilibria,” attributed to Keith Promislow:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-pdes-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-PDES-0018": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A starting point would be the following problem.\n\nProve existence of quasiperiodic patters for the 1d Swift-Hogenberg.",
   "clean_statement": "A starting point would be the following problem.\n\nProve existence of pattern - for the 1d Swift-Hogenberg.",
   "public_statement": "A starting point would be the following problem.\n\nProve existence of quasiperiodic patters for the 1d Swift-Hogenberg.",
   "evidence": "The canonical record states exactly:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-pdes-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-PDES-0019": {
   "statement_status": "exact",
   "original_statement": "Develop a geometric approach to stability that generalizes to higher dimensions.",
   "clean_statement": "Develop a geometric approach to stability that generalizes to higher dimensions.",
   "public_statement": "Develop a geometric approach to stability that generalizes to higher dimensions.",
   "evidence": "The original AIM page was checked. It contains exactly this sentence, attributes it to Christopher Jones, and gives no introduction, remarks, equations, or status. This is not an OCR error, but it is not a mathematically closed proposition. The workshop list separately contains a section titled “Computing Maslov indices and Evans functions in 2 dimensions,” which makes an Evans/Maslov interpretation plausible but does not settle what “dimensions” means here.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-PDES-0020": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "The Maslow index allow you to prove stability results for symmetric\npulses in Hamiltonian systems. Can this be done also for non-symmetric\npulses?",
   "clean_statement": null,
   "public_statement": "The Maslow index allow you to prove stability results for symmetric\npulses in Hamiltonian systems. Can this be done also for non-symmetric\npulses?",
   "evidence": "The archived AIM page gives Problem 10.1 in the section “Maslov index for an asymmetric pulse,” attributed to Jonathan Jaquette:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-pdes-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-PDES-0021": {
   "statement_status": "exact",
   "original_statement": "Develop rigorous computer assisted methods for handling center\nmanifolds.",
   "clean_statement": "Develop rigorous computer assisted methods for handling center\nmanifolds.",
   "public_statement": "Develop rigorous computer assisted methods for handling center\nmanifolds.",
   "evidence": "The canonical record is problem 11.1 in the AIM workshop section “Center manifolds”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-PDES-0022": {
   "statement_status": "exact",
   "original_statement": "Landau equation",
   "clean_statement": "Landau equation",
   "public_statement": "Landau equation",
   "evidence": "The exact canonical record says only",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-PDES-0023": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Nonlocal aggregation-diffusion equations",
   "clean_statement": null,
   "public_statement": "Nonlocal aggregation-diffusion equations",
   "evidence": "The exact canonical record is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-pdes-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-PDES-0024": {
   "statement_status": "exact",
   "original_statement": "Applied Models",
   "clean_statement": "Applied Models",
   "public_statement": "Applied Models",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-PDES-0025": {
   "statement_status": "exact",
   "original_statement": "Let $u$ plurisubharmonic satisfy\n\\[\n\\mathrm{det}(u_{j \\overline k}) = f(u)\n\\] on $B_1(0)$ with Dirichlet boundary conditions, then $u$ is rotationally symmetric",
   "clean_statement": "Let $u$ plurisubharmonic satisfy\n\\[\n\\mathrm{det}(u_{j \\overline k}) = f(u)\n\\] on $B_1(0)$ with Dirichlet boundary conditions, then $u$ is rotationally symmetric",
   "public_statement": "Let $u$ plurisubharmonic satisfy\n\\[\n\\mathrm{det}(u_{j \\overline k}) = f(u)\n\\] on $B_1(0)$ with Dirichlet boundary conditions, then $u$ is rotationally symmetric",
   "evidence": "The exact canonical statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-PDES-0026": {
   "statement_status": "exact",
   "original_statement": "Let $u$ strictly convex with Dirichlet boundary conditions solve\n\\[\\mathrm{det}(u_{ij}) = 1 + \\delta_{x_0}, \\]\nthen $u=\\varphi+w$ for $\\varphi$ smooth uniformly convex cone, and $w$ smooth.",
   "clean_statement": "Let $u$ strictly convex with Dirichlet boundary conditions solve\n\\[\\mathrm{det}(u_{ij}) = 1 + \\delta_{x_0}, \\]\nthen $u=\\varphi+w$ for $\\varphi$ smooth uniformly convex cone, and $w$ smooth.",
   "public_statement": "Let $u$ strictly convex with Dirichlet boundary conditions solve\n\\[\\mathrm{det}(u_{ij}) = 1 + \\delta_{x_0}, \\]\nthen $u=\\varphi+w$ for $\\varphi$ smooth uniformly convex cone, and $w$ smooth.",
   "evidence": "The archived AIM page reproduces exactly this wording and attributes the problem to Xu-Jia Wang. Thus the omissions are not an OCR error: the page does not specify the domain, dimension, boundary values, solution concept, normalization of the Dirac mass, or the meanings of “smooth uniformly convex cone” and “smooth.”",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-PDES-0027": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "For $u$ strictly convex, determine the existence and regularity of solutions to \\[\\mathrm{det}(D^2 u) u^{ij}u_i u_j = 1.\\]",
   "clean_statement": null,
   "public_statement": "For $u$ strictly convex, determine the existence and regularity of solutions to \\[\\mathrm{det}(D^2 u) u^{ij}u_i u_j = 1.\\]",
   "evidence": "The canonical record is Problem 1.15 in the “Monge-Ampère equations” section of the AIM problem list *Nonlinear PDEs in real and complex geometry*. The current AIM page attributes it to Tristan Collins and displays:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-pdes-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-PDES-0028": {
   "statement_status": "exact",
   "original_statement": "A function $u:\\Omega \\to \\mathbb R$ satisfies the special Lagrangian equation if \\[\\sum_i \\mathrm{arctan}(\\lambda_i)= c\\] for a constant $c$, where $\\lambda_i$ are the eigenvalues of $D^2u$.\n\n(\\textit{Conjecture of Nadirashvili-Vl\\u{a}du\\c{t}}) Any viscosity solution of the special Lagrangian equation on the unit ball is $C^1(B)$ for sufficiently smooth boundary data.",
   "clean_statement": "A function $u:\\Omega \\to \\mathbb R$ satisfies the special Lagrangian equation if \\[\\sum_i \\mathrm{arctan}(\\lambda_i)= c\\] for a constant $c$, where $\\lambda_i$ are the eigenvalues of $D^2u$.\n\n(\\textit{Conjecture of Nadirashvili-Vl\\u{a}du\\c{t}}) Any viscosity solution of the special Lagrangian equation on the unit ball is $C^1(B)$ for sufficiently smooth boundary data.",
   "public_statement": "A function $u:\\Omega \\to \\mathbb R$ satisfies the special Lagrangian equation if \\[\\sum_i \\mathrm{arctan}(\\lambda_i)= c\\] for a constant $c$, where $\\lambda_i$ are the eigenvalues of $D^2u$.\n\n(\\textit{Conjecture of Nadirashvili-Vl\\u{a}du\\c{t}}) Any viscosity solution of the special Lagrangian equation on the unit ball is $C^1(B)$ for sufficiently smooth boundary data.",
   "evidence": "The canonical AIM record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-PDES-0029": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "(\\textit{Conjecture of Nadirashvili-Tkachev-Vl\\u{a}du\\c{t}}) A uniformly elliptic equation in divergence form admits a solution in $W^{1,p}$ for some $p>1$.",
   "clean_statement": "If $u\\in W^{1,2}_{\\mathrm{loc}}(\\Omega)$ solves $\\operatorname{div}(A(x)\\nabla u)=0$, where $A$ is bounded measurable and uniformly elliptic with fixed constants, does $\\nabla u$ belong locally to $L^{2+\\varepsilon}$ for some $\\varepsilon>0$ depending only on dimension and ellipticity?",
   "public_statement": "(\\textit{Conjecture of Nadirashvili-Tkachev-Vl\\u{a}du\\c{t}}) A uniformly elliptic equation in divergence form admits a solution in $W^{1,p}$ for some $p>1$.",
   "evidence": "The exact canonical record is:",
   "classification_method": "invalid_literal_with_repaired_reading",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-pdes-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-PDES-0030": {
   "statement_status": "exact",
   "original_statement": "Let $u$ be a solution to \\[u_{tt}+\\mathrm{det}(u_{ij}) = 0\\] on $T^n\\times[0,1]$ with convex boundary data. Prove a $C^2$ estimate for $u(\\cdot,t)$ for $t$ fixed.",
   "clean_statement": "Let $u$ be a solution to \\[u_{tt}+\\mathrm{det}(u_{ij}) = 0\\] on $T^n\\times[0,1]$ with convex boundary data. Prove a $C^2$ estimate for $u(\\cdot,t)$ for $t$ fixed.",
   "public_statement": "Let $u$ be a solution to \\[u_{tt}+\\mathrm{det}(u_{ij}) = 0\\] on $T^n\\times[0,1]$ with convex boundary data. Prove a $C^2$ estimate for $u(\\cdot,t)$ for $t$ fixed.",
   "evidence": "Thus the repository transcription is faithful: this is not an OCR error introduced by the corpus. The statement on AIM itself is missing definitions essential to a mathematical boundary-value problem:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-PDES-0031": {
   "statement_status": "exact",
   "original_statement": "Find conditions that allow for $C^2$ regularity for degenerate Monge-Ampère equation (both real and complex).",
   "clean_statement": "Find conditions that allow for $C^2$ regularity for degenerate Monge-Ampère equation (both real and complex).",
   "public_statement": "Find conditions that allow for $C^2$ regularity for degenerate Monge-Ampère equation (both real and complex).",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-PDES-0032": {
   "statement_status": "exact",
   "original_statement": "Let $u$ be plurisubharmonic solution to $\\mathrm{det}(u_{\\overline k j}) =1$ on $\\mathbb C^n$ satisfying\n\\[ C^{-1} (|z|^2+1) \\leq u \\leq C (|z|^2+1)\n\\] then $u$ is quadratic.",
   "clean_statement": "Let $u$ be plurisubharmonic solution to $\\mathrm{det}(u_{\\overline k j}) =1$ on $\\mathbb C^n$ satisfying\n\\[ C^{-1} (|z|^2+1) \\leq u \\leq C (|z|^2+1)\n\\] then $u$ is quadratic.",
   "public_statement": "Let $u$ be plurisubharmonic solution to $\\mathrm{det}(u_{\\overline k j}) =1$ on $\\mathbb C^n$ satisfying\n\\[ C^{-1} (|z|^2+1) \\leq u \\leq C (|z|^2+1)\n\\] then $u$ is quadratic.",
   "evidence": "This display agrees with the archived AIM formulation and with Question 1.1 quoted by Li--Sheng; there is no apparent OCR corruption. The reversal \\(u_{\\bar k j}\\) versus the more usual \\(u_{j\\bar k}\\) is harmless. There are, however, three source-level ambiguities or omissions:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-PDES-0033": {
   "statement_status": "exact",
   "original_statement": "Let $u:\\mathbb R^2 \\to \\mathbb R$ be a solution of\n\\[\nu_{11} u_{22} = 1.\n\\]\nIf $u(x) \\leq C(1 + |x|^2)$, then $u$ must be quadratic.\n\nCan the growth condition be removed?\n\nIf one considers instead solutions $u:\\Omega \\to \\mathbb R$ for $\\Omega \\subseteq \\mathbb R^2$ convex, and bounded, and $u$ with Dirichlet boundary conditions, can one obtain a $C^2$ boundary estimate?",
   "clean_statement": "Let $u:\\mathbb R^2 \\to \\mathbb R$ be a solution of\n\\[\nu_{11} u_{22} = 1.\n\\]\nIf $u(x) \\leq C(1 + |x|^2)$, then $u$ must be quadratic.\n\nCan the growth condition be removed?\n\nIf one considers instead solutions $u:\\Omega \\to \\mathbb R$ for $\\Omega \\subseteq \\mathbb R^2$ convex, and bounded, and $u$ with Dirichlet boundary conditions, can one obtain a $C^2$ boundary estimate?",
   "public_statement": "Let $u:\\mathbb R^2 \\to \\mathbb R$ be a solution of\n\\[\nu_{11} u_{22} = 1.\n\\]\nIf $u(x) \\leq C(1 + |x|^2)$, then $u$ must be quadratic.\n\nCan the growth condition be removed?\n\nIf one considers instead solutions $u:\\Omega \\to \\mathbb R$ for $\\Omega \\subseteq \\mathbb R^2$ convex, and bounded, and $u$ with Dirichlet boundary conditions, can one obtain a $C^2$ boundary estimate?",
   "evidence": "The canonical record is AIM Problem List 1.45 in the section “Monge–Ampère equations” of the workshop *Nonlinear PDEs in real and complex geometry*. The live AIM page was checked on 2026-08-10. It agrees with the corpus and attributes the problem to **Xiangwen Zhang**. The displayed line break before \\(u_{11}u_{22}=1\\) is formatting, not a mathematical symbol.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-PDES-0034": {
   "statement_status": "exact",
   "original_statement": "Let $\\mathrm{det}(u_{\\overline k j}) = \\frac {f(z)}{|z|^\\alpha}$ for $f(z)$ smooth. What is the optimal regularity of $u$?",
   "clean_statement": "Let $\\mathrm{det}(u_{\\overline k j}) = \\frac {f(z)}{|z|^\\alpha}$ for $f(z)$ smooth. What is the optimal regularity of $u$?",
   "public_statement": "Let $\\mathrm{det}(u_{\\overline k j}) = \\frac {f(z)}{|z|^\\alpha}$ for $f(z)$ smooth. What is the optimal regularity of $u$?",
   "evidence": "The exact canonical AIM record, from the workshop *Nonlinear PDEs in real and complex geometry*, Monge--Ampère equations, Problem 1.5, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-PDES-0035": {
   "statement_status": "exact",
   "original_statement": "Find a toric proof of the toric case of the orbifold compactness theorem proved by Chen-Weber.",
   "clean_statement": "Find a toric proof of the toric case of the orbifold compactness theorem proved by Chen-Weber.",
   "public_statement": "Find a toric proof of the toric case of the orbifold compactness theorem proved by Chen-Weber.",
   "evidence": "The exact canonical prompt is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-PDES-0036": {
   "statement_status": "exact",
   "original_statement": "A complex manifold is called \\textit{almost K\\\"ahler} if it admits a complex structure $J$, not necessarily integrable, and a Hermitian metric $g$ such that $g(\\cdot,J\\cdot)$ defines a closed 2-form. One says that $g$ and $J$ are \\textit{compatible}.\n\n(\\textit{Goldberg Conjecture}) Is a compact, almost K\\\"ahler, Einstein manifold necessarily K\\\"ahler?",
   "clean_statement": "A complex manifold is called \\textit{almost K\\\"ahler} if it admits a complex structure $J$, not necessarily integrable, and a Hermitian metric $g$ such that $g(\\cdot,J\\cdot)$ defines a closed 2-form. One says that $g$ and $J$ are \\textit{compatible}.\n\n(\\textit{Goldberg Conjecture}) Is a compact, almost K\\\"ahler, Einstein manifold necessarily K\\\"ahler?",
   "public_statement": "A complex manifold is called \\textit{almost K\\\"ahler} if it admits a complex structure $J$, not necessarily integrable, and a Hermitian metric $g$ such that $g(\\cdot,J\\cdot)$ defines a closed 2-form. One says that $g$ and $J$ are \\textit{compatible}.\n\n(\\textit{Goldberg Conjecture}) Is a compact, almost K\\\"ahler, Einstein manifold necessarily K\\\"ahler?",
   "evidence": "The canonical AIM record (PDEs, workshop *Nonlinear PDEs in real and complex geometry*, Section 2, Problem 2.2) says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-PDES-0037": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "(Question of Donaldson) Find smooth solutions to the equation\n\\[\\ddot{\\varphi}- \\frac 12 |\\nabla \\dot \\varphi|^2_{\\omega_\\varphi}=-\\lambda R_{\\omega_\\varphi}\\] for $\\lambda>0$ on $M \\times [0,1]$ with $\\varphi|_{M\\times\\{0\\}}=0$, $\\varphi|_{M\\times\\{0\\}}=\\varphi_0$.",
   "clean_statement": null,
   "public_statement": "(Question of Donaldson) Find smooth solutions to the equation\n\\[\\ddot{\\varphi}- \\frac 12 |\\nabla \\dot \\varphi|^2_{\\omega_\\varphi}=-\\lambda R_{\\omega_\\varphi}\\] for $\\lambda>0$ on $M \\times [0,1]$ with $\\varphi|_{M\\times\\{0\\}}=0$, $\\varphi|_{M\\times\\{0\\}}=\\varphi_0$.",
   "evidence": "No corrected primary or archived formulation was located in the searches described below. The following natural **reconstruction is therefore an inference, not verified source text**: Accordingly, the literal AIM problem has status `invalid_statement`. The rest of this report gives rigorous consequences and a linearized solution theory for the explicitly labeled reconstruction (P); it does not claim nonlinear existence for (P).",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-pdes-notes.json",
   "source_index": 36,
   "attempt": 2
  },
  "AIM-PDES-0038": {
   "statement_status": "exact",
   "original_statement": "$M$ K\\\"ahler, $\\theta$ big cohomology class. Can we define a family of distances generalizing the Finsler geometry of K\\\"ahler potentials and compatible with weak Mabuchi geodesics?",
   "clean_statement": "$M$ K\\\"ahler, $\\theta$ big cohomology class. Can we define a family of distances generalizing the Finsler geometry of K\\\"ahler potentials and compatible with weak Mabuchi geodesics?",
   "public_statement": "$M$ K\\\"ahler, $\\theta$ big cohomology class. Can we define a family of distances generalizing the Finsler geometry of K\\\"ahler potentials and compatible with weak Mabuchi geodesics?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-PDES-0039": {
   "statement_status": "exact",
   "original_statement": "Let $u$ be a plurisubharmonic function satisfying\n\\[\n(i \\partial \\overline \\partial u)^n = \\psi\n\\]\nwith $\\psi>0$ and smooth. Assume $u$ is $C^{1,\\alpha}$ for $\\alpha > 1 - \\frac 2 n$, then $u\\in C^\\infty$.",
   "clean_statement": "Let $u$ be a plurisubharmonic function satisfying\n\\[\n(i \\partial \\overline \\partial u)^n = \\psi\n\\]\nwith $\\psi>0$ and smooth. Assume $u$ is $C^{1,\\alpha}$ for $\\alpha > 1 - \\frac 2 n$, then $u\\in C^\\infty$.",
   "public_statement": "Let $u$ be a plurisubharmonic function satisfying\n\\[\n(i \\partial \\overline \\partial u)^n = \\psi\n\\]\nwith $\\psi>0$ and smooth. Assume $u$ is $C^{1,\\alpha}$ for $\\alpha > 1 - \\frac 2 n$, then $u\\in C^\\infty$.",
   "evidence": "The exact canonical text is declarative:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-PDES-0040": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Find a counterexample to the maximal rank conjecture.\n\n\\textbf{\\emph{Conjecture}} (Maximal Rank Conjecture)\n\nLet $A\\subseteq\\mathbb C$ be the annulus $A = \\{10$ independent of $\\delta$.",
   "clean_statement": null,
   "public_statement": "Find a counterexample to the maximal rank conjecture.\n\n\\textbf{\\emph{Conjecture}} (Maximal Rank Conjecture)\n\nLet $A\\subseteq\\mathbb C$ be the annulus $A = \\{10$ independent of $\\delta$.",
   "evidence": "The canonical record is visibly corrupted. It ends with",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-pdes-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-PDES-0041": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Let $(M,\\chi)$ compact K\\\"ahler, and $\\omega$ another K\\\"ahler metric on $M$. There are necessary and sufficient conditions for solving\n\\[\n\\mathrm{tr}_{\\omega_\\varphi} \\chi = c\n\\]\nwhere $c = n \\frac {[\\omega]^{n-1} \\cap [\\chi]} {[\\omega]^n}$ is a constant (Song-Weinkove) in terms of a positivity condition. However, this may be hard to check in general.\n\n(Conjecture of Lejmi-Sz\\'ekelyhidi) There exists a solution if and only if\n\\[\n\\int_V{(c \\omega^p - p\\omega^{p-1}\\wedge \\chi)}>0\n\\]\nfor all proper subvarieties $V \\subset M$, where $p = \\mathrm{dim}(V)$",
   "clean_statement": null,
   "public_statement": "Let $(M,\\chi)$ compact K\\\"ahler, and $\\omega$ another K\\\"ahler metric on $M$. There are necessary and sufficient conditions for solving\n\\[\n\\mathrm{tr}_{\\omega_\\varphi} \\chi = c\n\\]\nwhere $c = n \\frac {[\\omega]^{n-1} \\cap [\\chi]} {[\\omega]^n}$ is a constant (Song-Weinkove) in terms of a positivity condition. However, this may be hard to check in general.\n\n(Conjecture of Lejmi-Sz\\'ekelyhidi) There exists a solution if and only if\n\\[\n\\int_V{(c \\omega^p - p\\omega^{p-1}\\wedge \\chi)}>0\n\\]\nfor all proper subvarieties $V \\subset M$, where $p = \\mathrm{dim}(V)$",
   "evidence": "The canonical AIM record is problem 2.7 in the “Complex geometry” section of the workshop *Nonlinear PDEs in real and complex geometry*. Its wording has minor extraction defects: the phrase “Let \\((M,\\chi)\\) compact Kähler” is missing “be,” and the cap symbol denotes a cohomological intersection. The displayed mathematics agrees with Lejmi--Székelyhidi's original formulation.",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-pdes-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-PDES-0042": {
   "statement_status": "exact",
   "original_statement": "Can one establish regularity properties of geodesics in the space of K\\\"ahler potentials for singular K\\\"ahler varieties?",
   "clean_statement": "Can one establish regularity properties of geodesics in the space of K\\\"ahler potentials for singular K\\\"ahler varieties?",
   "public_statement": "Can one establish regularity properties of geodesics in the space of K\\\"ahler potentials for singular K\\\"ahler varieties?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-PDES-0043": {
   "statement_status": "exact",
   "original_statement": "Let $(M,\\omega)$ and $(M',\\omega')$ be compact K\\\"ahler-Einstein manifolds which are $\\varepsilon$-close in Gromov-Hausdorff distance. Is there a holomorphic family $M_t$ with $M_0 = M$ and $M_1=M'$?",
   "clean_statement": "Let $(M,\\omega)$ and $(M',\\omega')$ be compact K\\\"ahler-Einstein manifolds which are $\\varepsilon$-close in Gromov-Hausdorff distance. Is there a holomorphic family $M_t$ with $M_0 = M$ and $M_1=M'$?",
   "public_statement": "Let $(M,\\omega)$ and $(M',\\omega')$ be compact K\\\"ahler-Einstein manifolds which are $\\varepsilon$-close in Gromov-Hausdorff distance. Is there a holomorphic family $M_t$ with $M_0 = M$ and $M_1=M'$?",
   "evidence": "The canonical record, AIM Problem List 2.9, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-PDES-0044": {
   "statement_status": "exact",
   "original_statement": "(Problem originally proposed by Fang-Hua Lin) Let $(T^2,g_0) \\subseteq (\\mathbb R^3, g_{\\mathrm{eucl}})$ be an isometric embedding and $g$ be a metric on $T^2$ which is $C^0$ close to $g_0$. Is there an isometric embedding of $(T^2,g)$ in $\\mathbb R^3$?",
   "clean_statement": "(Problem originally proposed by Fang-Hua Lin) Let $(T^2,g_0) \\subseteq (\\mathbb R^3, g_{\\mathrm{eucl}})$ be an isometric embedding and $g$ be a metric on $T^2$ which is $C^0$ close to $g_0$. Is there an isometric embedding of $(T^2,g)$ in $\\mathbb R^3$?",
   "public_statement": "(Problem originally proposed by Fang-Hua Lin) Let $(T^2,g_0) \\subseteq (\\mathbb R^3, g_{\\mathrm{eucl}})$ be an isometric embedding and $g$ be a metric on $T^2$ which is $C^0$ close to $g_0$. Is there an isometric embedding of $(T^2,g)$ in $\\mathbb R^3$?",
   "evidence": "The canonical AIM record (PDEs, workshop *Nonlinear PDEs in real and complex geometry*, Geometry 3.1) says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-PDES-0045": {
   "statement_status": "exact",
   "original_statement": "The \\textit{Chern-Ricci curvature} of a Hermitian metric $g$ is defined by\n\\[\nR^{\\mathcal{Ch}}_{\\overline k j} = -\\partial_j \\partial_{\\overline k} \\mathrm{log det}(g).\n\\]\nA family of Hermitian metrics satisfies the Chern-Ricci flow if\n\\[\\frac \\partial {\\partial t} g_{\\overline k j} = -R^{\\mathcal{Ch}}_{\\overline k j}.\\]\n\nWhat is the behavior of the Chern-Ricci flow on the simplest Hopf surface, $\\mathbb (\\mathbb C^2 \\setminus \\{ 0\\})/((z_1,z_2) \\sim (2z_1,2z_2))$?",
   "clean_statement": "The \\textit{Chern-Ricci curvature} of a Hermitian metric $g$ is defined by\n\\[\nR^{\\mathcal{Ch}}_{\\overline k j} = -\\partial_j \\partial_{\\overline k} \\mathrm{log det}(g).\n\\]\nA family of Hermitian metrics satisfies the Chern-Ricci flow if\n\\[\\frac \\partial {\\partial t} g_{\\overline k j} = -R^{\\mathcal{Ch}}_{\\overline k j}.\\]\n\nWhat is the behavior of the Chern-Ricci flow on the simplest Hopf surface, $\\mathbb (\\mathbb C^2 \\setminus \\{ 0\\})/((z_1,z_2) \\sim (2z_1,2z_2))$?",
   "public_statement": "The \\textit{Chern-Ricci curvature} of a Hermitian metric $g$ is defined by\n\\[\nR^{\\mathcal{Ch}}_{\\overline k j} = -\\partial_j \\partial_{\\overline k} \\mathrm{log det}(g).\n\\]\nA family of Hermitian metrics satisfies the Chern-Ricci flow if\n\\[\\frac \\partial {\\partial t} g_{\\overline k j} = -R^{\\mathcal{Ch}}_{\\overline k j}.\\]\n\nWhat is the behavior of the Chern-Ricci flow on the simplest Hopf surface, $\\mathbb (\\mathbb C^2 \\setminus \\{ 0\\})/((z_1,z_2) \\sim (2z_1,2z_2))$?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-PDES-0046": {
   "statement_status": "exact",
   "original_statement": "What is the behavior of $g_{j \\overline k}(t)$ solving the Calabi flow,\n\\[\\frac \\partial {\\partial t} g_{j \\overline k} = \\partial_j \\partial_{\\overline k} R?\\]\nCan long time existence be established for toric surfaces?",
   "clean_statement": "What is the behavior of $g_{j \\overline k}(t)$ solving the Calabi flow,\n\\[\\frac \\partial {\\partial t} g_{j \\overline k} = \\partial_j \\partial_{\\overline k} R?\\]\nCan long time existence be established for toric surfaces?",
   "public_statement": "What is the behavior of $g_{j \\overline k}(t)$ solving the Calabi flow,\n\\[\\frac \\partial {\\partial t} g_{j \\overline k} = \\partial_j \\partial_{\\overline k} R?\\]\nCan long time existence be established for toric surfaces?",
   "evidence": "The repository text has no visible OCR corruption. The live source URL was unavailable during this run, so the wording was checked against the exact canonical record, not silently altered. The mathematical question is nevertheless under-specified in several important ways. The following is the recovered smooth invariant reading used below.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-PDES-0047": {
   "statement_status": "corrected_verified",
   "original_statement": "Consider the flow of inverse Hermitian metrics on a compact complex manifold:\n\\[\n\\frac \\partial {\\partial t} g^{j \\overline k} = g^{m \\overline n} \\partial_m \\partial_{\\overline n} g^{j \\overline k} - \\partial_m g^{j \\overline k} \\partial_{\\overline n} g^{m \\overline k}.\n\\]\nOne has short time existence, and it is known that at the maximal time of existence\n\\[\n| \\mathrm{Rm}^{\\mathcal{Ch}} | + |T| + |\\nabla T| \\to \\infty.\n\\]\nCan this be improved? Can one find analogues of Perelman's $\\mathcal F$ and $\\mathcal W$ functionals?",
   "clean_statement": "Can the Streets--Tian continuation criterion for \\(HCF_+\\) be reduced to\ncontrol of fewer geometric quantities, and does this exact flow possess\nPerelman-type \\(\\mathcal F\\) and \\(\\mathcal W\\) functionals?",
   "public_statement": "Can the Streets--Tian continuation criterion for \\(HCF_+\\) be reduced to\ncontrol of fewer geometric quantities, and does this exact flow possess\nPerelman-type \\(\\mathcal F\\) and \\(\\mathcal W\\) functionals?",
   "evidence": "This is source corruption, not merely corruption introduced into the JSON: the archived AIM page from 14 December 2019 contains exactly the same bad indices and attributes Problem 4.3 to Yuri Ustinovskiy. The intended equation is independently verified by Proposition 3.9 and Corollary 3.10 of Ustinovskiy's 2018 Princeton thesis. The recovered inverse-metric equation is \\[ \\boxed{\\quad \\frac{\\partial}{\\partial t}g^{i\\bar j} =g^{m\\bar n}\\partial_m\\partial_{\\bar n}g^{i\\bar j} -\\bigl(\\partial_m g^{i\\bar n}\\bigr) \\bigl(\\partial_{\\bar n}g^{m\\bar j}\\bigr). \\quad} \\tag{HCF\\(_+^{-1}\\)} \\] Thus the first \\(g^{j\\bar k}\\) in the bad quadratic term must be \\(g^{j\\bar n}\\). All contracted and free indices then occur correctly. Invariantly, this is Ustinovskiy's distinguished Hermitian curvature flow, now often called the **positive Hermitian curvature flow** \\(HCF_+\\): \\[ \\frac{\\partial}{\\partial t...",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-pdes-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-PDES-0048": {
   "statement_status": "exact",
   "original_statement": "Is the diameter bounded at finite time singularities of the K\\\"ahler-Ricci flow on compact K\\\"ahler manifolds?",
   "clean_statement": "Is the diameter bounded at finite time singularities of the K\\\"ahler-Ricci flow on compact K\\\"ahler manifolds?",
   "public_statement": "Is the diameter bounded at finite time singularities of the K\\\"ahler-Ricci flow on compact K\\\"ahler manifolds?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-PDES-0049": {
   "statement_status": "exact",
   "original_statement": "Open Problem:Do there exist stationary spherical solutions to the above problem (the idea being that stationary spherical solutions should be the simplest multi-D objects to study). It is not clear how the source terms s and f\n\nmust be chosen in order to have a viscous or inviscid shock. Could f or s be identically zero? What is the role of curvature in the stability of the front and existence of nearby perturbed spherical fronts? Note that in the short-time analysis of Majda, curvature does not play a role. A possible scenario is that as the radius of the spherical shock gets bigger, stability is more likely, and there is likely to be a transition from stability to instability as the radius decreases to a critical radius r∗. Can one prove or disprove this?",
   "clean_statement": "Open Problem:Do there exist stationary spherical solutions to the above problem (the idea being that stationary spherical solutions should be the simplest multi-D objects to study). It is not clear how the source terms s and f\n\nmust be chosen in order to have a viscous or inviscid shock. Could f or s be identically zero? What is the role of curvature in the stability of the front and existence of nearby perturbed spherical fronts? Note that in the short-time analysis of Majda, curvature does not play a role. A possible scenario is that as the radius of the spherical shock gets bigger, stability is more likely, and there is likely to be a transition from stability to instability as the radius decreases to a critical radius r∗. Can one prove or disprove this?",
   "public_statement": "Open Problem:Do there exist stationary spherical solutions to the above problem (the idea being that stationary spherical solutions should be the simplest multi-D objects to study). It is not clear how the source terms s and f\n\nmust be chosen in order to have a viscous or inviscid shock. Could f or s be identically zero? What is the role of curvature in the stability of the front and existence of nearby perturbed spherical fronts? Note that in the short-time analysis of Majda, curvature does not play a role. A possible scenario is that as the radius of the spherical shock gets bigger, stability is more likely, and there is likely to be a transition from stability to instability as the radius decreases to a critical radius r∗. Can one prove or disprove this?",
   "evidence": "The record comes from the problem list *Open Problems in Multidimensional Stability of Waves and Patterns*, compiled by N. Costanzino after the AIM workshop of May 16--20, 2005. The PDF supplies equations that were omitted from the extracted record. In space dimension \\(d=2\\) or \\(3\\), put \\(m=d-1\\), write the density as \\(\\rho(r,t)\\), and take the velocity vector to be \\(u(r,t)x/r\\). The displayed barotropic system is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-PDES-0050": {
   "statement_status": "exact",
   "original_statement": "Open Problem:Determine the relationship between the Lopatinski determinant for stability and recent work on stability of spherical waves using spherical harmonics (c.f. C.C. Wu & P. H. Roberts, Bubble shape instability and sonoluminescence, Phys. Lett. A 250, 131 (1998)).",
   "clean_statement": "Open Problem:Determine the relationship between the Lopatinski determinant for stability and recent work on stability of spherical waves using spherical harmonics (c.f. C.C. Wu & P. H. Roberts, Bubble shape instability and sonoluminescence, Phys. Lett. A 250, 131 (1998)).",
   "public_statement": "Open Problem:Determine the relationship between the Lopatinski determinant for stability and recent work on stability of spherical waves using spherical harmonics (c.f. C.C. Wu & P. H. Roberts, Bubble shape instability and sonoluminescence, Phys. Lett. A 250, 131 (1998)).",
   "evidence": "The spacing after “Problem:” is compressed, but comparison with the original seven-page AIM PDF shows no substantive OCR error in this sentence. The canonical record does, however, omit the setup immediately preceding it. The PDF places the problem under S. Benzoni-Gavage, H. K. Jenssen, and M. Williams, “Existence and Stability of Spherical Fronts.” They propose studying curved shock, reactive, or viscous fronts in the simplest curved geometry. In space dimension two or three they reduce barotropic gas dynamics to radial variables: \\[ \\rho_t+(\\rho u)_r+\\frac{(d-1)\\rho u}{r}=s(r,t), \\tag{1} \\] \\[ (\\rho u)_t+(\\rho u^2+P(\\rho))_r +\\frac{(d-1)\\rho u^2}{r} =\\nu\\left(u_r+\\frac{(d-1)u}{r}\\right)_r+F(r,t). \\tag{2} \\] The first open problem asks for stationary spherical fronts and for the role of curvature and the sources. The present record is the second open problem. The third asks about no...",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-PDES-0051": {
   "statement_status": "exact",
   "original_statement": "Open Problem:A classical and apparently very hard problem is the analysis of non stationary spherical waves, e.g. a focusing inviscid shock (c.f. R. Courant & K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, New York, (1948) or L.D. Landau & E.M. Lifshitz Fluid Mechanics, Pergamon Press, Oxford (1959)) Are such waves stable? 1ARCC Workshop on Multidimensional Stability of Waves and Patterns 2\n\nP. Szmolyan: (Kinetic and Boltzmann Equations) \n\nSetup:Consider the kinetic equation \n\nft + v · fxQ(f, f )where f = f (x, v, t ), x ∈ Rd. Profiles solve (v − c)f ′ = Q(f, f )",
   "clean_statement": "Open Problem:A classical and apparently very hard problem is the analysis of non stationary spherical waves, e.g. a focusing inviscid shock (c.f. R. Courant & K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, New York, (1948) or L.D. Landau & E.M. Lifshitz Fluid Mechanics, Pergamon Press, Oxford (1959)) Are such waves stable? 1ARCC Workshop on Multidimensional Stability of Waves and Patterns 2\n\nP. Szmolyan: (Kinetic and Boltzmann Equations)\n\nSetup:Consider the kinetic equation\n\nft + v · fxQ(f, f )where f = f (x, v, t ), x ∈ Rd. Profiles solve (v − c)f ′ = Q(f, f )",
   "public_statement": "Open Problem:A classical and apparently very hard problem is the analysis of non stationary spherical waves, e.g. a focusing inviscid shock (c.f. R. Courant & K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, New York, (1948) or L.D. Landau & E.M. Lifshitz Fluid Mechanics, Pergamon Press, Oxford (1959)) Are such waves stable? 1ARCC Workshop on Multidimensional Stability of Waves and Patterns 2\n\nP. Szmolyan: (Kinetic and Boltzmann Equations)\n\nSetup:Consider the kinetic equation\n\nft + v · fxQ(f, f )where f = f (x, v, t ), x ∈ Rd. Profiles solve (v − c)f ′ = Q(f, f )",
   "evidence": "The canonical OCR record must be preserved, but it joins three different pieces of the source PDF. The original first page contains the following complete problem under S. Benzoni-Gavage, H. K. Jenssen, and M. Williams:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-PDES-0052": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Open Problem:Under what conditions do there exist profiles for this equation? Are they stable to multidimensional pertur-bations? What does the Evans function look like. (As a first step, one might start with considering discrete velocity models.) \n\nY. Li: (Stability of travelling waves of the full water wave problem near the critical case) \n\nSetup:Consider the water wave equations \n\nηt = GΦΦt + Φ2 \n\n> x\n\n+ 2 ηxΦxGΦ − (GΦ) 2\n\n1 + η2\n\n> x\n\n+ gη = 0 where ̂ G(k) = k tanh( k). The equation admits travelling wave solutions ( ηc, Φc) = ( η(x − ct ), Φc(x − ct )) for a variety of wave speeds.",
   "clean_statement": "1. under what collision, end-state, and speed conditions does (1.3) have a\n   positive heteroclinic profile;\n2. when is that planar profile stable to perturbations depending on transverse\n   spatial variables; and\n3. how should an Evans function be constructed for the resulting kinetic\n   spectral problem?",
   "public_statement": "Open Problem:Under what conditions do there exist profiles for this equation? Are they stable to multidimensional pertur-bations? What does the Evans function look like. (As a first step, one might start with considering discrete velocity models.)\n\nY. Li: (Stability of travelling waves of the full water wave problem near the critical case)\n\nSetup:Consider the water wave equations\n\nηt = GΦΦt + Φ2\n\n> x\n\n+ 2 ηxΦxGΦ − (GΦ) 2\n\n1 + η2\n\n> x\n\n+ gη = 0 where ̂ G(k) = k tanh( k). The equation admits travelling wave solutions ( ηc, Φc) = ( η(x − ct ), Φc(x − ct )) for a variety of wave speeds.",
   "evidence": "The canonical JSON record is visibly spliced across a page/presenter boundary. It begins with",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-pdes-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-PDES-0053": {
   "statement_status": "exact",
   "original_statement": "Open Problem:How we determine the stability of such waves near the critical speed c∗.\n\nS. Benzoni-Gavage: (Shocks with Capillarity) \n\nSetup:Consider the equations \n\n∂t% + ∇ · (%u) = 0 \n\n∂tu + ( u · ∇ )u + ∇P = ∇\n\n(\n\nK(%)∇% + 12\n\n∂K (%)\n\n∂% |∇ %|2\n\n)\n\nwhere u ∈ R3 is the velocity, % > 0 is the density of the fluid, and P = P (%) is the pressure. The equations admits a planar profile ( %, u). It can be shown that the spectrum of the resulting linear operator obtained by linearization of the equations about the planar profile must lie on the imaginary axis.",
   "clean_statement": "Open Problem:How we determine the stability of such waves near the critical speed c∗.\n\nS. Benzoni-Gavage: (Shocks with Capillarity)\n\nSetup:Consider the equations\n\n∂t% + ∇ · (%u) = 0\n\n∂tu + ( u · ∇ )u + ∇P = ∇\n\n(\n\nK(%)∇% + 12\n\n∂K (%)\n\n∂% |∇ %|2\n\n)\n\nwhere u ∈ R3 is the velocity, % > 0 is the density of the fluid, and P = P (%) is the pressure. The equations admits a planar profile ( %, u). It can be shown that the spectrum of the resulting linear operator obtained by linearization of the equations about the planar profile must lie on the imaginary axis.",
   "public_statement": "Open Problem:How we determine the stability of such waves near the critical speed c∗.\n\nS. Benzoni-Gavage: (Shocks with Capillarity)\n\nSetup:Consider the equations\n\n∂t% + ∇ · (%u) = 0\n\n∂tu + ( u · ∇ )u + ∇P = ∇\n\n(\n\nK(%)∇% + 12\n\n∂K (%)\n\n∂% |∇ %|2\n\n)\n\nwhere u ∈ R3 is the velocity, % > 0 is the density of the fluid, and P = P (%) is the pressure. The equations admits a planar profile ( %, u). It can be shown that the spectrum of the resulting linear operator obtained by linearization of the equations about the planar profile must lie on the imaginary axis.",
   "evidence": "The exact canonical record is preserved in `input.json`, but it crosses a presenter boundary. The original AIM PDF puts the following material consecutively:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-PDES-0054": {
   "statement_status": "exact",
   "original_statement": "Open Problem:Are there any eigenvalues on the imaginary axis? Is the planar profile spectrally stable or unstable? Does linear information tell the whole story? That is, can one prove full nonlinear stability from the linear information. ARCC Workshop on Multidimensional Stability of Waves and Patterns 3\n\nR. Pego: (Stability of Toda Lattice Solitons)",
   "clean_statement": "Open Problem:Are there any eigenvalues on the imaginary axis? Is the planar profile spectrally stable or unstable? Does linear information tell the whole story? That is, can one prove full nonlinear stability from the linear information. ARCC Workshop on Multidimensional Stability of Waves and Patterns 3\n\nR. Pego: (Stability of Toda Lattice Solitons)",
   "public_statement": "Open Problem:Are there any eigenvalues on the imaginary axis? Is the planar profile spectrally stable or unstable? Does linear information tell the whole story? That is, can one prove full nonlinear stability from the linear information. ARCC Workshop on Multidimensional Stability of Waves and Patterns 3\n\nR. Pego: (Stability of Toda Lattice Solitons)",
   "evidence": "The canonical record is an OCR extraction from the AIM workshop list *Open Problems in Multidimensional Stability of Waves and Patterns* (May 16--20, 2005), under S. Benzoni-Gavage's heading “Shocks with Capillarity.” The PDF gives the mass equation and a velocity-form capillary equation for density \\(\\rho>0\\) and velocity \\(u\\in\\mathbb R^3\\). The dimensionally consistent recovery is \\[ \\rho_t+\\nabla\\!\\cdot(\\rho u)=0, \\qquad u_t+(u\\!\\cdot\\!\\nabla)u+\\nabla P(\\rho) =\\nabla\\!\\left(K(\\rho)\\Delta\\rho+ \\frac12K'(\\rho)|\\nabla\\rho|^2\\right). \\tag{EK} \\] The source's glyph before \\(\\rho\\) was extracted as \\(\\nabla\\), but that would add a vector to a scalar inside the outer gradient. The standard Euler--Korteweg formula in Benzoni-Gavage--Danchin--Descombes--Jamet (2005, equation (1.3)) confirms that the intended glyph is \\(\\Delta\\). The workshop calls \\(P\\) “pressure”; in this velocity formula...",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-PDES-0055": {
   "statement_status": "exact",
   "original_statement": "Open Problem:The Toda lattice ¨qk = exp( qk+1 − qk) − exp( qk − qk−1)has the well known solutions \n\nqk = log \n\n( cosh( β(k − ct + 1)) cosh ( β(k − ct )) \n\n), c = sinh ββ\n\nWhat are their spectral stability properties? \n\nM. Williams: (Two Interacting Shocks in 1D) \n\nSetup:Consider the inviscid conservation law \n\n∂tu + ∂xf (u) = 0 (A) and the viscous regularization \n\n∂tuε + ∂xf (uε) = ε∆uε (B) where for simplicity we may take u ∈ R2 and x ∈ R. Take a two-shock solution to the viscous conservation law of the form below: \n\nFigure 1: The Two Shock Setup",
   "clean_statement": "Open Problem:The Toda lattice ¨qk = exp( qk+1 − qk) − exp( qk − qk−1)has the well known solutions\n\nqk = log\n\n( cosh( β(k − ct + 1)) cosh ( β(k − ct ))\n\n), c = sinh ββ\n\nWhat are their spectral stability properties?\n\nM. Williams: (Two Interacting Shocks in 1D)\n\nSetup:Consider the inviscid conservation law\n\n∂tu + ∂xf (u) = 0 (A) and the viscous regularization\n\n∂tuε + ∂xf (uε) = ε∆uε (B) where for simplicity we may take u ∈ R2 and x ∈ R. Take a two-shock solution to the viscous conservation law of the form below:\n\nFigure 1: The Two Shock Setup",
   "public_statement": "Open Problem:The Toda lattice ¨qk = exp( qk+1 − qk) − exp( qk − qk−1)has the well known solutions\n\nqk = log\n\n( cosh( β(k − ct + 1)) cosh ( β(k − ct ))\n\n), c = sinh ββ\n\nWhat are their spectral stability properties?\n\nM. Williams: (Two Interacting Shocks in 1D)\n\nSetup:Consider the inviscid conservation law\n\n∂tu + ∂xf (u) = 0 (A) and the viscous regularization\n\n∂tuε + ∂xf (uε) = ε∆uε (B) where for simplicity we may take u ∈ R2 and x ∈ R. Take a two-shock solution to the viscous conservation law of the form below:\n\nFigure 1: The Two Shock Setup",
   "evidence": "The canonical JSON has lost a fraction bar and then runs into the next speaker's problem. Inspection of page 3 of the original AIM workshop PDF recovers R. Pego's item as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-PDES-0056": {
   "statement_status": "exact",
   "original_statement": "Open Problem:Can we construct explicit solutions to the viscous problem (B), say of the form uε(t, x, t/ε, x/ε ), which converge in a reasonable sense to solutions u(t, x ) of the inviscid problem (A) as ε → 0? Note: this question has been answered abstractly (c.f. the C.I.M.E. notes by A. Bressan ), but our goal is to explicitly resolve the dynamics at the corner where Ul and Ur meet in the xt -plane. ARCC Workshop on Multidimensional Stability of Waves and Patterns 4\n\nJ. Humphreys: (Stability for strong viscous shocks of the p-system) \n\nConsider the p-system \n\nvt − ux = 0 \n\nut + p(v)x = ( b(v)u)x\n\nwhere p′ < 0, p′′ > 0. Can we show that strong shocks for the p-system are stable or unstable? What role, if any, does symmetrizability play in the onset of instability? \n\nK. Promislow: (MultiD front dynamics in optical resonance) \n\nA model for pattern formation in an optical cavity near resonance is given by \n\niϕ t − 12 ∆lϕ + |ϕ|2ϕ + ( i − a)ϕ − γϕ ∗ = 0 where ∆ l = ∂2 \n\n> x\n\n+ l−1∂2 \n\n> y\n\nand a, γ are real constants. The equation admits a planar front solution. Consider a perturbation of the planar front. What is the Evans function for the perturbed front? What are the dynamics of the front? \n\nB. Sandstede: (Dynamical interpretation of the roots of the D(λ) embedded in the absolute spectra) \n\nSetup:In many circumstances, the Evans function can be extended into branch points of the linear dispersion relation. It is natural to ask what role, if any, roots of the Evans function at branch points play for the temporal dynamics of the linear or nonlinear evolution. It has been shown by Murata [Tohoku Math J 37 (1985) 151-195] that temporal decay rates of scalar linear heat equations depend very much on the presence of these roots.",
   "clean_statement": "Open Problem:Can we construct explicit solutions to the viscous problem (B), say of the form uε(t, x, t/ε, x/ε ), which converge in a reasonable sense to solutions u(t, x ) of the inviscid problem (A) as ε → 0? Note: this question has been answered abstractly (c.f. the C.I.M.E. notes by A. Bressan ), but our goal is to explicitly resolve the dynamics at the corner where Ul and Ur meet in the xt -plane. ARCC Workshop on Multidimensional Stability of Waves and Patterns 4\n\nJ. Humphreys: (Stability for strong viscous shocks of the p-system)\n\nConsider the p-system\n\nvt − ux = 0\n\nut + p(v)x = ( b(v)u)x\n\nwhere p′ < 0, p′′ > 0. Can we show that strong shocks for the p-system are stable or unstable? What role, if any, does symmetrizability play in the onset of instability?\n\nK. Promislow: (MultiD front dynamics in optical resonance)\n\nA model for pattern formation in an optical cavity near resonance is given by\n\niϕ t − 12 ∆lϕ + |ϕ|2ϕ + ( i − a)ϕ − γϕ ∗ = 0 where ∆ l = ∂2\n\n> x\n\n+ l−1∂2\n\n> y\n\nand a, γ are real constants. The equation admits a planar front solution. Consider a perturbation of the planar front. What is the Evans function for the perturbed front? What are the dynamics of the front?\n\nB. Sandstede: (Dynamical interpretation of the roots of the D(λ) embedded in the absolute spectra)\n\nSetup:In many circumstances, the Evans function can be extended into branch points of the linear dispersion relation. It is natural to ask what role, if any, roots of the Evans function at branch points play for the temporal dynamics of the linear or nonlinear evolution. It has been shown by Murata [Tohoku Math J 37 (1985) 151-195] that temporal decay rates of scalar linear heat equations depend very much on the presence of these roots.",
   "public_statement": "Open Problem:Can we construct explicit solutions to the viscous problem (B), say of the form uε(t, x, t/ε, x/ε ), which converge in a reasonable sense to solutions u(t, x ) of the inviscid problem (A) as ε → 0? Note: this question has been answered abstractly (c.f. the C.I.M.E. notes by A. Bressan ), but our goal is to explicitly resolve the dynamics at the corner where Ul and Ur meet in the xt -plane. ARCC Workshop on Multidimensional Stability of Waves and Patterns 4\n\nJ. Humphreys: (Stability for strong viscous shocks of the p-system)\n\nConsider the p-system\n\nvt − ux = 0\n\nut + p(v)x = ( b(v)u)x\n\nwhere p′ < 0, p′′ > 0. Can we show that strong shocks for the p-system are stable or unstable? What role, if any, does symmetrizability play in the onset of instability?\n\nK. Promislow: (MultiD front dynamics in optical resonance)\n\nA model for pattern formation in an optical cavity near resonance is given by\n\niϕ t − 12 ∆lϕ + |ϕ|2ϕ + ( i − a)ϕ − γϕ ∗ = 0 where ∆ l = ∂2\n\n> x\n\n+ l−1∂2\n\n> y\n\nand a, γ are real constants. The equation admits a planar front solution. Consider a perturbation of the planar front. What is the Evans function for the perturbed front? What are the dynamics of the front?\n\nB. Sandstede: (Dynamical interpretation of the roots of the D(λ) embedded in the absolute spectra)\n\nSetup:In many circumstances, the Evans function can be extended into branch points of the linear dispersion relation. It is natural to ask what role, if any, roots of the Evans function at branch points play for the temporal dynamics of the linear or nonlinear evolution. It has been shown by Murata [Tohoku Math J 37 (1985) 151-195] that temporal decay rates of scalar linear heat equations depend very much on the presence of these roots.",
   "evidence": "The source is M. Williams, “Two Interacting Shocks in 1D,” in the AIM workshop list *Open Problems in Multidimensional Stability of Waves and Patterns* (2005). The setup is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-PDES-0057": {
   "statement_status": "exact",
   "original_statement": "Open Problem: To what extent is this true for more general parabolic PDEs and for other nonlinear equation? \n\nS. Malham: (Biscale chaos) \n\nSetup:Consider the coupled reaction diffusion equations \n\nut = δ∆u − uv 2\n\nvt = ∆ v + uv 2\n\nwhich is a model of autocatalysis. Here the paramteter δ is a certain ratio of the speed of the autocatalyst molecules and the fuel molecules. When δ ∼ 8, small perturbations of a planar front evolve into a very complex front. It has been suggested (c.f. Biscale chaos in propagating fronts, Phys. Rev. E 52, (1995), pp. 4724 -4735) that the wrinkles that form from the perturbed planar interface exhibit spatial-temporal chaotic behaviour characterized by two length scales. This is called biscale chaos.",
   "clean_statement": "Open Problem: To what extent is this true for more general parabolic PDEs and for other nonlinear equation?\n\nS. Malham: (Biscale chaos)\n\nSetup:Consider the coupled reaction diffusion equations\n\nut = δ∆u − uv 2\n\nvt = ∆ v + uv 2\n\nwhich is a model of autocatalysis. Here the paramteter δ is a certain ratio of the speed of the autocatalyst molecules and the fuel molecules. When δ ∼ 8, small perturbations of a planar front evolve into a very complex front. It has been suggested (c.f. Biscale chaos in propagating fronts, Phys. Rev. E 52, (1995), pp. 4724 -4735) that the wrinkles that form from the perturbed planar interface exhibit spatial-temporal chaotic behaviour characterized by two length scales. This is called biscale chaos.",
   "public_statement": "Open Problem: To what extent is this true for more general parabolic PDEs and for other nonlinear equation?\n\nS. Malham: (Biscale chaos)\n\nSetup:Consider the coupled reaction diffusion equations\n\nut = δ∆u − uv 2\n\nvt = ∆ v + uv 2\n\nwhich is a model of autocatalysis. Here the paramteter δ is a certain ratio of the speed of the autocatalyst molecules and the fuel molecules. When δ ∼ 8, small perturbations of a planar front evolve into a very complex front. It has been suggested (c.f. Biscale chaos in propagating fronts, Phys. Rev. E 52, (1995), pp. 4724 -4735) that the wrinkles that form from the perturbed planar interface exhibit spatial-temporal chaotic behaviour characterized by two length scales. This is called biscale chaos.",
   "evidence": "The canonical JSON record is corrupted at a record boundary. In the original AIM PDF, the entry is headed by B. Sandstede and begins with the observation that an Evans function can often be extended to branch points of the linear dispersion relation. It then says that Murata showed that the temporal decay rates for scalar linear heat equations depend strongly on the presence of roots at those branch points. The open problem is, verbatim,",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-PDES-0058": {
   "statement_status": "exact",
   "original_statement": "Open Problem:What are the stability properties of the planar front? Can we understand the secondary instability by getting an Evans function for the front that arises from the secondary instability? ARCC Workshop on Multidimensional Stability of Waves and Patterns 5\n\nM. Haragus: (Stability for KP-I profiles with periodic structure) \n\nSetup:Consider the KP-I equation (ut − uxxx + cu x + uu x)x + uyy = 0 Clearly the function u(x, y, t ) = φ(z) where φ(z) is the usual KdV soliton solution is a solution to the KP-I equations with no variation in the y direction. This line soliton is known to be unstable to transverse per-turbations. A family of y-periodic waves bifurcates from it and connects to the well-known lump solution, which is presumably stable.",
   "clean_statement": "Open Problem:What are the stability properties of the planar front? Can we understand the secondary instability by getting an Evans function for the front that arises from the secondary instability? ARCC Workshop on Multidimensional Stability of Waves and Patterns 5\n\nM. Haragus: (Stability for KP-I profiles with periodic structure)\n\nSetup:Consider the KP-I equation (ut − uxxx + cu x + uu x)x + uyy = 0 Clearly the function u(x, y, t ) = φ(z) where φ(z) is the usual KdV soliton solution is a solution to the KP-I equations with no variation in the y direction. This line soliton is known to be unstable to transverse per-turbations. A family of y-periodic waves bifurcates from it and connects to the well-known lump solution, which is presumably stable.",
   "public_statement": "Open Problem:What are the stability properties of the planar front? Can we understand the secondary instability by getting an Evans function for the front that arises from the secondary instability? ARCC Workshop on Multidimensional Stability of Waves and Patterns 5\n\nM. Haragus: (Stability for KP-I profiles with periodic structure)\n\nSetup:Consider the KP-I equation (ut − uxxx + cu x + uu x)x + uyy = 0 Clearly the function u(x, y, t ) = φ(z) where φ(z) is the usual KdV soliton solution is a solution to the KP-I equations with no variation in the y direction. This line soliton is known to be unstable to transverse per-turbations. A family of y-periodic waves bifurcates from it and connects to the well-known lump solution, which is presumably stable.",
   "evidence": "The canonical record begins in the middle of S. Malham's presentation and then appends the next speaker's KP-I problem. Pages 4--5 of the original AIM workshop PDF recover the intended item as follows.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-PDES-0059": {
   "statement_status": "exact",
   "original_statement": "Open Problem:What is the stability of the KP-I soliton solution with periodic structure? \n\nJ. Albert: (Benjamin-Ono type equations) \n\nSetup:Consider the Benjamin-Ono equation \n\nut + uu x − K ux = 0 where ̂ Ku(k) = |k|ˆu(k). This equation supports travelling waves of the form φ(z) = 41+ z2. The resulting eigenvalue problem is (cv + Kv − φv )z = λv (A) where v is the perturbation v:= u − φ.",
   "clean_statement": "Open Problem:What is the stability of the KP-I soliton solution with periodic structure?\n\nJ. Albert: (Benjamin-Ono type equations)\n\nSetup:Consider the Benjamin-Ono equation\n\nut + uu x − K ux = 0 where ̂ Ku(k) = |k|ˆu(k). This equation supports travelling waves of the form φ(z) = 41+ z2. The resulting eigenvalue problem is (cv + Kv − φv )z = λv (A) where v is the perturbation v:= u − φ.",
   "public_statement": "Open Problem:What is the stability of the KP-I soliton solution with periodic structure?\n\nJ. Albert: (Benjamin-Ono type equations)\n\nSetup:Consider the Benjamin-Ono equation\n\nut + uu x − K ux = 0 where ̂ Ku(k) = |k|ˆu(k). This equation supports travelling waves of the form φ(z) = 41+ z2. The resulting eigenvalue problem is (cv + Kv − φv )z = λv (A) where v is the perturbation v:= u − φ.",
   "evidence": "The canonical JSON record is damaged by a record-boundary error. The AIM PDF puts the following material under M. Haragus, “Stability for KP-I profiles with periodic structure”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-PDES-0060": {
   "statement_status": "exact",
   "original_statement": "Open Problem:What is the spectrum of (A) and how does it relate to the spectrum of \n\ncv + Kv − φv = λv \n\nwhich is well known. A possible approach would be to treat the eigenvalue equation as a two dimensional problem. \n\nK. Zumbrun: (Strong Shocks in one or multiD of viscous conservation laws) \n\nThe Setup:Consider the system of viscous conservation laws in one or several space dimensions d,\n\nut +\n\n> d\n\n∑\n\n> j=1\n\nf j (u)xj =\n\n> d\n\n∑\n\n> j,k =1\n\n(Bjk (u)uxk )xj\n\nwhere u ∈ Rn, f is a smooth mapping from Rn to Rn and Bj,k is a smooth mapping from Rn to Rn×n.The stability of weak shocks in one spatial dimension has been investigated recently by H. Freisthler & P. Szmolyan (Spectral stability of small shock waves, I, Arch. Rat. Mech. Analysis, 164, (2002), 287-309) and R. Plaza & K. Zumbrun (An Evans function approach to spectral stability of small-amplitude shock profiles, J. Discrete and Continuous Dynamical Systems 10, 2004, no. 4, 885-924). The weak shock assumption induces a fast-slow structure (with the small parameter being the strength of the shock) that can be exploited in the calculations. It is not immediately apparent that there is fast-slow structure hidden somewhere for strong shock profiles that can exploited.",
   "clean_statement": "Open Problem:What is the spectrum of (A) and how does it relate to the spectrum of\n\ncv + Kv − φv = λv\n\nwhich is well known. A possible approach would be to treat the eigenvalue equation as a two dimensional problem.\n\nK. Zumbrun: (Strong Shocks in one or multiD of viscous conservation laws)\n\nThe Setup:Consider the system of viscous conservation laws in one or several space dimensions d,\n\nut +\n\n> d\n\n∑\n\n> j=1\n\nf j (u)xj =\n\n> d\n\n∑\n\n> j,k =1\n\n(Bjk (u)uxk )xj\n\nwhere u ∈ Rn, f is a smooth mapping from Rn to Rn and Bj,k is a smooth mapping from Rn to Rn×n.The stability of weak shocks in one spatial dimension has been investigated recently by H. Freisthler & P. Szmolyan (Spectral stability of small shock waves, I, Arch. Rat. Mech. Analysis, 164, (2002), 287-309) and R. Plaza & K. Zumbrun (An Evans function approach to spectral stability of small-amplitude shock profiles, J. Discrete and Continuous Dynamical Systems 10, 2004, no. 4, 885-924). The weak shock assumption induces a fast-slow structure (with the small parameter being the strength of the shock) that can be exploited in the calculations. It is not immediately apparent that there is fast-slow structure hidden somewhere for strong shock profiles that can exploited.",
   "public_statement": "Open Problem:What is the spectrum of (A) and how does it relate to the spectrum of\n\ncv + Kv − φv = λv\n\nwhich is well known. A possible approach would be to treat the eigenvalue equation as a two dimensional problem.\n\nK. Zumbrun: (Strong Shocks in one or multiD of viscous conservation laws)\n\nThe Setup:Consider the system of viscous conservation laws in one or several space dimensions d,\n\nut +\n\n> d\n\n∑\n\n> j=1\n\nf j (u)xj =\n\n> d\n\n∑\n\n> j,k =1\n\n(Bjk (u)uxk )xj\n\nwhere u ∈ Rn, f is a smooth mapping from Rn to Rn and Bj,k is a smooth mapping from Rn to Rn×n.The stability of weak shocks in one spatial dimension has been investigated recently by H. Freisthler & P. Szmolyan (Spectral stability of small shock waves, I, Arch. Rat. Mech. Analysis, 164, (2002), 287-309) and R. Plaza & K. Zumbrun (An Evans function approach to spectral stability of small-amplitude shock profiles, J. Discrete and Continuous Dynamical Systems 10, 2004, no. 4, 885-924). The weak shock assumption induces a fast-slow structure (with the small parameter being the strength of the shock) that can be exploited in the calculations. It is not immediately apparent that there is fast-slow structure hidden somewhere for strong shock profiles that can exploited.",
   "evidence": "The canonical JSON record crosses a presenter boundary. Inspection of the original AIM workshop PDF and the preceding canonical record AIM-PDES-0059 recovers the following J. Albert entry.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-PDES-0061": {
   "statement_status": "exact",
   "original_statement": "Open Problem:Determine the stability properties of strong shocks for viscous conservation laws in one or several space dimensions. ARCC Workshop on Multidimensional Stability of Waves and Patterns 6\n\nM. Wechselberger: (Loss of hyperbolicity, algebraic decay and the Evans function) \n\nSetup:For certain values of p, the generalized KdV equation \n\nut − uxxx + up x = 0 admits standing waves φ(x) which decay exponentially to zero as |x| → ∞. However, the eigenfunctions of the operator obtained by linearizing about φ need not decay exponentially and depending on the power p\n\n(say p=5) may only decay algebraically as |x| → ∞.",
   "clean_statement": "Open Problem:Determine the stability properties of strong shocks for viscous conservation laws in one or several space dimensions. ARCC Workshop on Multidimensional Stability of Waves and Patterns 6\n\nM. Wechselberger: (Loss of hyperbolicity, algebraic decay and the Evans function)\n\nSetup:For certain values of p, the generalized KdV equation\n\nut − uxxx + up x = 0 admits standing waves φ(x) which decay exponentially to zero as |x| → ∞. However, the eigenfunctions of the operator obtained by linearizing about φ need not decay exponentially and depending on the power p\n\n(say p=5) may only decay algebraically as |x| → ∞.",
   "public_statement": "Open Problem:Determine the stability properties of strong shocks for viscous conservation laws in one or several space dimensions. ARCC Workshop on Multidimensional Stability of Waves and Patterns 6\n\nM. Wechselberger: (Loss of hyperbolicity, algebraic decay and the Evans function)\n\nSetup:For certain values of p, the generalized KdV equation\n\nut − uxxx + up x = 0 admits standing waves φ(x) which decay exponentially to zero as |x| → ∞. However, the eigenfunctions of the operator obtained by linearizing about φ need not decay exponentially and depending on the power p\n\n(say p=5) may only decay algebraically as |x| → ∞.",
   "evidence": "The canonical record is preserved verbatim in input.json, but its problem field straddles two speakers. The original seven-page AIM document was checked directly. On PDF page 5 (printed workshop page 5) the entry is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-PDES-0062": {
   "statement_status": "exact",
   "original_statement": "Open Problem:Can one construct an Evans function for this problem? If so, how does it behave? Note that for this problem the absolute spectrum touches the essential spectrum so there is no exponential dichotomy that we can exploit. \n\nP. Howard: (Combination structures in viscous conservation laws)",
   "clean_statement": "Open Problem:Can one construct an Evans function for this problem? If so, how does it behave? Note that for this problem the absolute spectrum touches the essential spectrum so there is no exponential dichotomy that we can exploit.\n\nP. Howard: (Combination structures in viscous conservation laws)",
   "public_statement": "Open Problem:Can one construct an Evans function for this problem? If so, how does it behave? Note that for this problem the absolute spectrum touches the essential spectrum so there is no exponential dichotomy that we can exploit.\n\nP. Howard: (Combination structures in viscous conservation laws)",
   "evidence": "The canonical record is an OCR extraction from the AIM workshop list “Stability criteria for multi-dimensional waves and patterns.” As stored, it reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-PDES-0063": {
   "statement_status": "exact",
   "original_statement": "Open Problem:The thin film equation \n\nut + ( u2 − u3)x = −ε(u3uxxx )x\n\nsupports solutions that are comprised of a Lax shock moving to the left with speed s1 and an undercom-pressive shock moving to the right with speed s2 as in Figure 2. Can one prove the stability or instability of such a structure? \n\nFigure 2: The Combination Structure \n\nC. Jones: (Stability of energized states of NLS) \n\nConsider \n\niu t = urr = n − 1\n\nr ur + f (|u|)u + ωu \n\nIs there really no spectrum off of iR? What is the meaning of this for stability? Does the linear information give us full stability? Opposite Krein signature eigenvalues for NLS. ARCC Workshop on Multidimensional Stability of Waves and Patterns 7\n\nT. Kapitula: (Transient dynamics for vortex patterns) \n\nSetup: Start with the Gross-Pitaevskii equation \n\niq t + ∆ q ± | q|2q = V (x)q\n\nNumerical investigations show that if one starts with a \"nice\" vortex pattern with a lot of symmetry, it evolves to a \"ugly\" vortex pattern with little symmetry or structure, and then evolves further to to a \"nice\" pattern again.",
   "clean_statement": "Open Problem:The thin film equation\n\nut + ( u2 − u3)x = −ε(u3uxxx )x\n\nsupports solutions that are comprised of a Lax shock moving to the left with speed s1 and an undercom-pressive shock moving to the right with speed s2 as in Figure 2. Can one prove the stability or instability of such a structure?\n\nFigure 2: The Combination Structure\n\nC. Jones: (Stability of energized states of NLS)\n\nConsider\n\niu t = urr = n − 1\n\nr ur + f (|u|)u + ωu\n\nIs there really no spectrum off of iR? What is the meaning of this for stability? Does the linear information give us full stability? Opposite Krein signature eigenvalues for NLS. ARCC Workshop on Multidimensional Stability of Waves and Patterns 7\n\nT. Kapitula: (Transient dynamics for vortex patterns)\n\nSetup: Start with the Gross-Pitaevskii equation\n\niq t + ∆ q ± | q|2q = V (x)q\n\nNumerical investigations show that if one starts with a \"nice\" vortex pattern with a lot of symmetry, it evolves to a \"ugly\" vortex pattern with little symmetry or structure, and then evolves further to to a \"nice\" pattern again.",
   "public_statement": "Open Problem:The thin film equation\n\nut + ( u2 − u3)x = −ε(u3uxxx )x\n\nsupports solutions that are comprised of a Lax shock moving to the left with speed s1 and an undercom-pressive shock moving to the right with speed s2 as in Figure 2. Can one prove the stability or instability of such a structure?\n\nFigure 2: The Combination Structure\n\nC. Jones: (Stability of energized states of NLS)\n\nConsider\n\niu t = urr = n − 1\n\nr ur + f (|u|)u + ωu\n\nIs there really no spectrum off of iR? What is the meaning of this for stability? Does the linear information give us full stability? Opposite Krein signature eigenvalues for NLS. ARCC Workshop on Multidimensional Stability of Waves and Patterns 7\n\nT. Kapitula: (Transient dynamics for vortex patterns)\n\nSetup: Start with the Gross-Pitaevskii equation\n\niq t + ∆ q ± | q|2q = V (x)q\n\nNumerical investigations show that if one starts with a \"nice\" vortex pattern with a lot of symmetry, it evolves to a \"ugly\" vortex pattern with little symmetry or structure, and then evolves further to to a \"nice\" pattern again.",
   "evidence": "The canonical record is an OCR extraction from the AIM workshop report *Open Problems in Multidimensional Stability of Waves and Patterns*. Inspection of the original PDF confirms that P. Howard's problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-PDES-0064": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem: By what mechanism does this happen and how can one capture the general dynamics? \n\nH. Warchall: (Stability of Encapsulated-Vortex Solutions) \n\nSetup:Consider the equations \n\nJu t = ∆ u + g(u) (NLS) \n\nutt = ∆ u + g(u) (NLKG) where u ∈ RN +1 → RM, g: RM → RM continuous satisfying g(y) = h(|y|2)ˆ y, with h: [0, ∞) → R and ˆy ≡ y/ |y|. J is an invertible M × M skew symmetric matrix. Consider standing-wave solutions to (NLS) and (NLKG) of the form \n\nu(x, t ) = eμKt ˆψ(ˆ x)w(r)with μ ∈ R a constant, and K a real skew-symmetric M × M matrix with \n\nK = J−1 for (NLS) \n\nK2 = −I for (NLKG) where w: [0, ∞) → R and ˆψ: SN −1 → SM −1. Here ˆψ is a unit vector valued eigenfunction of the Laplacian on the sphere SN −1 ⊂ RN, with ∆S ˆψ = −l(l + N − 2) ˆψ\n\nWe remark that the possible values of l are limited by the dimension M of range space (see J. Iaia & H.A. Warchall, Encapsulated-vortex solutions to equivariant wave equations: existence, SIAM J. Math. Anal., \n\n30 (1999), 118-139). If u satisfies (NLS) or (NLKG) then the spatial profile w satisfies \n\nw′′ + N − 1\n\nr w′ − l(l + N − 2) \n\nr2 w + f (w) = 0 where f (y) = g(y) + ωy with \n\nω =\n\n{ μ for (NLS) \n\nμ2 for (NLKG) Note: Traveling waves are generated by Galilean or Lorentz boosts. The idea is to generalize, e.g. standing wave solutions u(x, t ) = eiωt eimθ w(r) of −iu t − ∆u = g(u) whose stability was analysed by R.L. Pego & H.A.Warchall (Spectrally stable encapsulated vortices for nonlinear Schr¨ odinger equations, J.Nonlinear Sci., 12 (2002), 347-394). Under appropriate conditions on f there exist smooth exponentially localized solutions w to the profile ODE. One essentially needs \n\nf ′(0) < 0 F (t) = \n\n∫ t\n\n> 0\n\nf (s) ds > 0 for some t > 0",
   "clean_statement": null,
   "public_statement": "Problem: By what mechanism does this happen and how can one capture the general dynamics?\n\nH. Warchall: (Stability of Encapsulated-Vortex Solutions)\n\nSetup:Consider the equations\n\nJu t = ∆ u + g(u) (NLS)\n\nutt = ∆ u + g(u) (NLKG) where u ∈ RN +1 → RM, g: RM → RM continuous satisfying g(y) = h(|y|2)ˆ y, with h: [0, ∞) → R and ˆy ≡ y/ |y|. J is an invertible M × M skew symmetric matrix. Consider standing-wave solutions to (NLS) and (NLKG) of the form\n\nu(x, t ) = eμKt ˆψ(ˆ x)w(r)with μ ∈ R a constant, and K a real skew-symmetric M × M matrix with\n\nK = J−1 for (NLS)\n\nK2 = −I for (NLKG) where w: [0, ∞) → R and ˆψ: SN −1 → SM −1. Here ˆψ is a unit vector valued eigenfunction of the Laplacian on the sphere SN −1 ⊂ RN, with ∆S ˆψ = −l(l + N − 2) ˆψ\n\nWe remark that the possible values of l are limited by the dimension M of range space (see J. Iaia & H.A. Warchall, Encapsulated-vortex solutions to equivariant wave equations: existence, SIAM J. Math. Anal.,\n\n30 (1999), 118-139). If u satisfies (NLS) or (NLKG) then the spatial profile w satisfies\n\nw′′ + N − 1\n\nr w′ − l(l + N − 2)\n\nr2 w + f (w) = 0 where f (y) = g(y) + ωy with\n\nω =\n\n{ μ for (NLS)\n\nμ2 for (NLKG) Note: Traveling waves are generated by Galilean or Lorentz boosts. The idea is to generalize, e.g. standing wave solutions u(x, t ) = eiωt eimθ w(r) of −iu t − ∆u = g(u) whose stability was analysed by R.L. Pego & H.A.Warchall (Spectrally stable encapsulated vortices for nonlinear Schr¨ odinger equations, J.Nonlinear Sci., 12 (2002), 347-394). Under appropriate conditions on f there exist smooth exponentially localized solutions w to the profile ODE. One essentially needs\n\nf ′(0) < 0 F (t) =\n\n∫ t\n\n> 0\n\nf (s) ds > 0 for some t > 0",
   "evidence": "The canonical record is split across an extraction boundary. The exact record in `input.json` starts with the final Kapitula question and then incorrectly includes the beginning of H. Warchall's next, unrelated entry. Comparing the preceding canonical record, AIM-PDES-0063, with page 7 of the original AIM workshop PDF recovers the Kapitula entry as follows (typographical spacing is normalized, but the sign is not changed):",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-pdes-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-PDES-0065": {
   "statement_status": "exact",
   "original_statement": "Open Problem:Under what conditions are the encapsulated-vortex solutions u(x, t ) = eμKt ˆψ(ˆ x)w(r) of (NLS) or (NLKG) with N ≥ 3 stable or unstable?",
   "clean_statement": "Open Problem:Under what conditions are the encapsulated-vortex solutions u(x, t ) = eμKt ˆψ(ˆ x)w(r) of (NLS) or (NLKG) with N ≥ 3 stable or unstable?",
   "public_statement": "Open Problem:Under what conditions are the encapsulated-vortex solutions u(x, t ) = eμKt ˆψ(ˆ x)w(r) of (NLS) or (NLKG) with N ≥ 3 stable or unstable?",
   "evidence": "The canonical record contains only H. Warchall's final open question. Its setup was lost across the record boundary and appears at the end of AIM-PDES-0064. Page 7 of the original AIM workshop PDF verifies the following reconstruction. Let \\[ u:\\mathbb R^{N+1}\\longrightarrow\\mathbb R^M \\] solve either \\[ J u_t=\\Delta u+g(u) \\quad\\text{(NLS)},\\qquad u_{tt}=\\Delta u+g(u) \\quad\\text{(NLKG)}, \\tag{1.1} \\] where \\(J\\) is an invertible skew-symmetric \\(M\\times M\\) matrix and \\[ g(y)=h(|y|^2)\\widehat y,\\qquad \\widehat y=y/|y|. \\tag{1.2} \\] The source assumes \\(g\\) continuous. It considers \\[ u(x,t)=e^{\\mu Kt}\\widehat\\psi(\\widehat x)w(r),\\qquad r=|x|,\\quad \\widehat x=x/r, \\tag{1.3} \\] where \\(K=J^{-1}\\) for NLS, \\(K^2=-I\\) for NLKG, and \\(K\\) is real skew-symmetric. The angular map \\[ \\widehat\\psi:S^{N-1}\\to S^{M-1},\\qquad \\Delta_S\\widehat\\psi=-l(l+N-2)\\widehat\\psi \\tag{1.4} \\] is a unit-vect...",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-PDES-0066": {
   "statement_status": "exact",
   "original_statement": "A.1 Amadori, Debora \n\nI would be interested in the numerical approximation of the scalar equation \n\nut + f (u)x = 1\n\nε h\n\n(x\n\nε\n\n)\n\n(1) with h continuous and 1-periodic, with zero average; f ∈ C1(R), u · f ′(u) > 0, f (u) → +∞\n\nas |u| → +∞, and with 1-periodic initial data. I can contribute some recent results on the pointwise behavior of the oscillation, as \n\nε → 0, of the solutions to (1); this analysis could be of help in the numerical approximation of (1). More generally, I am interested in the analysis of problems that exibit resonant phe-nomena.",
   "clean_statement": "A.1 Amadori, Debora\n\nI would be interested in the numerical approximation of the scalar equation\n\nut + f (u)x = 1\n\nε h\n\n(x\n\nε\n\n)\n\n(1) with h continuous and 1-periodic, with zero average; f ∈ C1(R), u · f ′(u) > 0, f (u) → +∞\n\nas |u| → +∞, and with 1-periodic initial data. I can contribute some recent results on the pointwise behavior of the oscillation, as\n\nε → 0, of the solutions to (1); this analysis could be of help in the numerical approximation of (1). More generally, I am interested in the analysis of problems that exibit resonant phe-nomena.",
   "public_statement": "A.1 Amadori, Debora\n\nI would be interested in the numerical approximation of the scalar equation\n\nut + f (u)x = 1\n\nε h\n\n(x\n\nε\n\n)\n\n(1) with h continuous and 1-periodic, with zero average; f ∈ C1(R), u · f ′(u) > 0, f (u) → +∞\n\nas |u| → +∞, and with 1-periodic initial data. I can contribute some recent results on the pointwise behavior of the oscillation, as\n\nε → 0, of the solutions to (1); this analysis could be of help in the numerical approximation of (1). More generally, I am interested in the analysis of problems that exibit resonant phe-nomena.",
   "evidence": "The original AIM PDF, *Stiff Sources and Numerical Methods for Conservation Laws*, version dated April 1, 2005, contains the following participant contribution by Debora Amadori: \\[ u_t+\\partial_x f(u)=\\frac1\\varepsilon h\\!\\left(\\frac{x}{\\varepsilon}\\right). \\tag{1.1} \\] It asks for numerical approximation when \\(h\\) is continuous, 1-periodic, and has zero average; \\(f\\in C^1(\\mathbb R)\\), \\(u f'(u)>0\\), and \\(f(u)\\to+\\infty\\) as \\(|u|\\to\\infty\\); and the initial data are 1-periodic. It mentions recent results on the pointwise behavior of the oscillations as \\(\\varepsilon\\to0\\), and a broader interest in resonance.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-PDES-0067": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A.2 Chertock, Alina \n\nOne of the projects I am working on is aimed at developing a hybrid finite-volume-particle method for systems of conservation or balance laws coupled with a nonlinear trans-port equation. Solutions of such systems are usually nonsmooth: they may contain shocks, rarefaction waves and contact discontinuities. The presence of a stiff source term adds another level of complexity to the model. Such problems arise, for instance, in modeling transport of a passive pollutant in shallow water (in which case the source of the pollutant may be even a point-source modeled by a delta-function) or compressible inviscid reacting gases (in which case the source is usually stiff, since the reaction is fast and the time scale as-sociated with the reaction is much smaller than that associated with the fluid advection). It is well known that numerical dissipation present in shock-capturing methods may not only seriously degrade the quality of the computed solution but may also lead to nonphysical states, which in turn, may completely destroy the numerical solution. The core idea of the new method is to use a finite-volume method to numerically integrate a system of conservation (balance) laws and a particle method to solve transport equations coupled with the system. This way the specific advantages of each scheme are utilized at the right place. Particle methods applied to transport equations, can ameliorate most of the problems posed by the presence of numerical viscosity since particles provide a non-dissipative approximation of the convection. In these methods, the solution is sought in the form of a linear combination of the delta-functions, whose positions and coefficients represent locations and weights of the particles, respectively. The locations and weights of the particles are then evolved in time according to a system of ODEs, obtained from the weak formulation of the transport equations. We have successfully implemented the finite-volume-particle method for the above as well as some other (inviscid) models and we plan to apply the method to more realistic advection-diffusion-reaction models. This extension is not straightforward since it will in-volve the treatment of diffusion and reaction terms that may appear in the equation. There was a number of attempts in the past to use particle methods for approximating solutions of convection-diffusion models, but each of them has its own drawback, associated primarily with the reconstruction of the point values of the computed solution from its particle distri-bution. Most of known recovering procedures, suitable for smooth functions, typically fail 4\n\nto produce reasonable results in the nonsmooth case. In the purely convective case, we were able to overcome this difficulty using the concept of the dual equation, but it is not clear whether this approach can be generalized for the viscous case. Another (theoretical) difficulty one may encounter while implementing the finite-volume-particle method is related to the lack of smoothness in the right-hand side of the ODE system that describes the evolution of particles and their weights. While the existence of a general-ized solution is guaranteed by the theory of Filippov, the uniqueness can only be obtained via a proper regularization. The presence of a point-source term makes the problem even more challenging and a theoretical justification of the particle method in this case is a wide open problem.",
   "clean_statement": null,
   "public_statement": "A.2 Chertock, Alina\n\nOne of the projects I am working on is aimed at developing a hybrid finite-volume-particle method for systems of conservation or balance laws coupled with a nonlinear trans-port equation. Solutions of such systems are usually nonsmooth: they may contain shocks, rarefaction waves and contact discontinuities. The presence of a stiff source term adds another level of complexity to the model. Such problems arise, for instance, in modeling transport of a passive pollutant in shallow water (in which case the source of the pollutant may be even a point-source modeled by a delta-function) or compressible inviscid reacting gases (in which case the source is usually stiff, since the reaction is fast and the time scale as-sociated with the reaction is much smaller than that associated with the fluid advection). It is well known that numerical dissipation present in shock-capturing methods may not only seriously degrade the quality of the computed solution but may also lead to nonphysical states, which in turn, may completely destroy the numerical solution. The core idea of the new method is to use a finite-volume method to numerically integrate a system of conservation (balance) laws and a particle method to solve transport equations coupled with the system. This way the specific advantages of each scheme are utilized at the right place. Particle methods applied to transport equations, can ameliorate most of the problems posed by the presence of numerical viscosity since particles provide a non-dissipative approximation of the convection. In these methods, the solution is sought in the form of a linear combination of the delta-functions, whose positions and coefficients represent locations and weights of the particles, respectively. The locations and weights of the particles are then evolved in time according to a system of ODEs, obtained from the weak formulation of the transport equations. We have successfully implemented the finite-volume-particle method for the above as well as some other (inviscid) models and we plan to apply the method to more realistic advection-diffusion-reaction models. This extension is not straightforward since it will in-volve the treatment of diffusion and reaction terms that may appear in the equation. There was a number of attempts in the past to use particle methods for approximating solutions of convection-diffusion models, but each of them has its own drawback, associated primarily with the reconstruction of the point values of the computed solution from its particle distri-bution. Most of known recovering procedures, suitable for smooth functions, typically fail 4\n\nto produce reasonable results in the nonsmooth case. In the purely convective case, we were able to overcome this difficulty using the concept of the dual equation, but it is not clear whether this approach can be generalized for the viscous case. Another (theoretical) difficulty one may encounter while implementing the finite-volume-particle method is related to the lack of smoothness in the right-hand side of the ODE system that describes the evolution of particles and their weights. While the existence of a general-ized solution is guaranteed by the theory of Filippov, the uniqueness can only be obtained via a proper regularization. The presence of a point-source term makes the problem even more challenging and a theoretical justification of the particle method in this case is a wide open problem.",
   "evidence": "The canonical record is a participant contribution in the AIM workshop report *Stiff Sources and Numerical Methods for Conservation Laws*, version dated April 1, 2005. Its exact stored problem field is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-pdes-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-PDES-0068": {
   "statement_status": "exact",
   "original_statement": "A.3 Christoforou, Cleopatra \n\nMy area of interest is the theory of hyperbolic conservation laws. My current research is an application of the method of vanishing viscosity: The aim is to construct solutions of hyperbolic systems of balance laws with dissipative source terms \n\nut + ( f (u)) x + g(u) = 0 as limits of solutions of parabolic systems \n\nu≤t + ( f (u≤)) x + g(u≤) = ≤u ≤xx \n\nwith viscosity ≤ tending to zero. The analysis of the vanishing viscosity method of Bianchini and Bressan [BiB] is extended to this class of systems. Because of the presence of the dissipative source terms, supplementary Lyapunov functionals are constructed and additional techniques are employed to those already devised in [BiB]. Moreover, an exponential decay of the total variation of vanishing viscosity approximations is established. I am interested in applying these techniques to other systems of conservation laws. A very challenging question would be the case of systems with physical viscosity. Finally, I am very interested in increasing my knowledge in mathematical biology. It is a challenging, fast-growing area that together with numerical methods will introduce me to new tools and assist me to improve my ability to work with physical problems.",
   "clean_statement": "A.3 Christoforou, Cleopatra\n\nMy area of interest is the theory of hyperbolic conservation laws. My current research is an application of the method of vanishing viscosity: The aim is to construct solutions of hyperbolic systems of balance laws with dissipative source terms\n\nut + ( f (u)) x + g(u) = 0 as limits of solutions of parabolic systems\n\nu≤t + ( f (u≤)) x + g(u≤) = ≤u ≤xx\n\nwith viscosity ≤ tending to zero. The analysis of the vanishing viscosity method of Bianchini and Bressan [BiB] is extended to this class of systems. Because of the presence of the dissipative source terms, supplementary Lyapunov functionals are constructed and additional techniques are employed to those already devised in [BiB]. Moreover, an exponential decay of the total variation of vanishing viscosity approximations is established. I am interested in applying these techniques to other systems of conservation laws. A very challenging question would be the case of systems with physical viscosity. Finally, I am very interested in increasing my knowledge in mathematical biology. It is a challenging, fast-growing area that together with numerical methods will introduce me to new tools and assist me to improve my ability to work with physical problems.",
   "public_statement": "A.3 Christoforou, Cleopatra\n\nMy area of interest is the theory of hyperbolic conservation laws. My current research is an application of the method of vanishing viscosity: The aim is to construct solutions of hyperbolic systems of balance laws with dissipative source terms\n\nut + ( f (u)) x + g(u) = 0 as limits of solutions of parabolic systems\n\nu≤t + ( f (u≤)) x + g(u≤) = ≤u ≤xx\n\nwith viscosity ≤ tending to zero. The analysis of the vanishing viscosity method of Bianchini and Bressan [BiB] is extended to this class of systems. Because of the presence of the dissipative source terms, supplementary Lyapunov functionals are constructed and additional techniques are employed to those already devised in [BiB]. Moreover, an exponential decay of the total variation of vanishing viscosity approximations is established. I am interested in applying these techniques to other systems of conservation laws. A very challenging question would be the case of systems with physical viscosity. Finally, I am very interested in increasing my knowledge in mathematical biology. It is a challenging, fast-growing area that together with numerical methods will introduce me to new tools and assist me to improve my ability to work with physical problems.",
   "evidence": "This record is item A.3 in the AIM workshop document *Stiff Sources and Numerical Methods for Conservation Laws*. The canonical JSON has an OCR substitution in which every occurrence of the Greek letter epsilon became the symbol `<=`. The original PDF was checked directly. Its equations are",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-PDES-0069": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A.4 Despres, Bruno \n\nMy interests go in two directions. The first one is not directly related to the subject of the Workshop. It is about the theory of convergence of Finite Volume schemes by means of the old consistency+stability-implies-convergence approach. It helps to get a linear approach of the convegrence of these methods and it is possible to prove quite accurate results even for a non linear scalar con-servation law. The other one is directly related to the subject of Workshop. With a colleague (Christophe Buet) we are currently working on the numerical approximation of the model problem { ut + 1 \n\n> ε\n\nvx = 0,vt + 1 \n\n> ε\n\nf (u, v ) = − σ \n\n> ε2\n\nv. \n\nOur idea is that we absolutely need an implicite solver for time step requirements of the diffusion limit. Various stability criteria are possible. If the model is the moment modeliza-tion of some kinetic equation, then |v| \n\n> u\n\n≤ 1 is natural. We have develop a 1D solver for this 5\n\nsystem: the solver is implicit (we only solve a linear system with ad-hoc frozen coefficients), stable ( |v| \n\n> u\n\n≤ 1), and has the correct diffusion limit. I will be happy to compare this approach with others.",
   "clean_statement": null,
   "public_statement": "A.4 Despres, Bruno\n\nMy interests go in two directions. The first one is not directly related to the subject of the Workshop. It is about the theory of convergence of Finite Volume schemes by means of the old consistency+stability-implies-convergence approach. It helps to get a linear approach of the convegrence of these methods and it is possible to prove quite accurate results even for a non linear scalar con-servation law. The other one is directly related to the subject of Workshop. With a colleague (Christophe Buet) we are currently working on the numerical approximation of the model problem { ut + 1\n\n> ε\n\nvx = 0,vt + 1\n\n> ε\n\nf (u, v ) = − σ\n\n> ε2\n\nv.\n\nOur idea is that we absolutely need an implicite solver for time step requirements of the diffusion limit. Various stability criteria are possible. If the model is the moment modeliza-tion of some kinetic equation, then |v|\n\n> u\n\n≤ 1 is natural. We have develop a 1D solver for this 5\n\nsystem: the solver is implicit (we only solve a linear system with ad-hoc frozen coefficients), stable ( |v|\n\n> u\n\n≤ 1), and has the correct diffusion limit. I will be happy to compare this approach with others.",
   "evidence": "The extracted second equation lacks a spatial derivative. That literal reading cannot yield the claimed diffusion limit because relaxation of $v$ supplies no spatial constitutive law. The balance-law context and the later two-moment radiation equations of Buet and Després both put a spatial derivative on the pressure/second-moment flux. We therefore analyze the explicitly labeled reconstruction",
   "classification_method": "explicit_uncertainty_evidence",
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   "source_file": "aim-pdes-notes.json",
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  "AIM-PDES-0070": {
   "statement_status": "exact",
   "original_statement": "A.5 Filbet, Francis \n\nApproximation of Hyperbolic Models for Chemosensitive Movement Numerical methods with different orders of accuracy are proposed to approximate hy-perbolic models for chemosensitive movements. On the one hand, first and second order well-balanced finite volume schemes are presented. This approach provides an exact con-servation of the steady state solutions. On the other hand, a high order finite difference weighted essentially non-oscillatory (WENO) scheme is constructed and the well-balanced reconstruction is adapted to this scheme in order to exactly preserve steady states and to retain high order accuracy. Numerical simulations are performed to verify accuracy and the well-balanced property of the proposed schemes and to observe the formation of networks in the hyperbolic models similar to those observed in the experiments. Keywords: chemotaxis, hyperbolic systems, finite volume methods, finite difference methods, WENO schemes, well-balanced schemes. This work is in collaboration with Chi-Wang Shu, Brown University",
   "clean_statement": "A.5 Filbet, Francis\n\nApproximation of Hyperbolic Models for Chemosensitive Movement Numerical methods with different orders of accuracy are proposed to approximate hy-perbolic models for chemosensitive movements. On the one hand, first and second order well-balanced finite volume schemes are presented. This approach provides an exact con-servation of the steady state solutions. On the other hand, a high order finite difference weighted essentially non-oscillatory (WENO) scheme is constructed and the well-balanced reconstruction is adapted to this scheme in order to exactly preserve steady states and to retain high order accuracy. Numerical simulations are performed to verify accuracy and the well-balanced property of the proposed schemes and to observe the formation of networks in the hyperbolic models similar to those observed in the experiments. Keywords: chemotaxis, hyperbolic systems, finite volume methods, finite difference methods, WENO schemes, well-balanced schemes. This work is in collaboration with Chi-Wang Shu, Brown University",
   "public_statement": "A.5 Filbet, Francis\n\nApproximation of Hyperbolic Models for Chemosensitive Movement Numerical methods with different orders of accuracy are proposed to approximate hy-perbolic models for chemosensitive movements. On the one hand, first and second order well-balanced finite volume schemes are presented. This approach provides an exact con-servation of the steady state solutions. On the other hand, a high order finite difference weighted essentially non-oscillatory (WENO) scheme is constructed and the well-balanced reconstruction is adapted to this scheme in order to exactly preserve steady states and to retain high order accuracy. Numerical simulations are performed to verify accuracy and the well-balanced property of the proposed schemes and to observe the formation of networks in the hyperbolic models similar to those observed in the experiments. Keywords: chemotaxis, hyperbolic systems, finite volume methods, finite difference methods, WENO schemes, well-balanced schemes. This work is in collaboration with Chi-Wang Shu, Brown University",
   "evidence": "The canonical record is tagged `section`, and inspection of the original AIM workshop PDF confirms that it is a contributed-talk abstract, not an open-problem question. The record begins “A.5 Filbet, Francis” and summarizes numerical methods developed with Chi-Wang Shu under the title *Approximation of Hyperbolic Models for Chemosensitive Movement*. The two strings `hy-perbolic` and `con-servation` in the canonical JSON are line-break OCR artifacts. With only those artifacts repaired, the mathematical content says that first- and second-order well-balanced finite-volume schemes and a high-order finite-difference WENO scheme are proposed for hyperbolic chemotaxis, with tests of accuracy, preservation of steady states, and network formation.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 69,
   "attempt": 1
  },
  "AIM-PDES-0071": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A.6 Gamba, Irene \n\nNon-equilibrium time dependent reactive kinetic-Poisson systems appear in the mod-eling of such diverse areas as electron transport in solids, biological transport, granular and energy dissipative flows. These non-conservative systems exhibit a common feature: their steady or self similar states are given by statistical Stationary Non-equilibrium States (SNS), meaning they are far deviated from Gaussian probability distributions. They entice approx-imating hydrodynamic models modeling \"source\" representing momentum and energy gain or dissipation due to strong friction or forcing scales. When these non-equilibrium regimes take over, classical hydrodynamic models (based on Gaussians/Maxwellian closures) do not apply and need to be corrected to account for the non-equilibrium statistics. I am interested in issues related to the mathematical properties of these models such as existence, uniqueness, self-similarity, stability and to analyze their higher order moment equations (hydro-dynamical corrections) and corresponding boundary value problems; as well as to investigate optimal numerical simulation methods for corresponding quantum, kinetic and macroscopic (hydrodynamic) models. I will present a very recent work related to item (1.i) below: \"Deterministic solvers to transient Boltzmann-Poisson equations\" Abstract: The Boltzmann-Poisson system is the most reliable model for the flow of charged particles in semiconductors devices. Real device models have not already been simulated by deterministic computations due to its high computational cost, although is very well known and general practice to solve these models by Monte-Carlo (DSMC) methods. We focus in a rather easy and fast deterministic solver for a channel flow: one and two-dimension and three-velocity dimension. The system of equations reduces to a linear kinetic (non-local) equation solved by WENO methods coupled with the Poisson equation for the force field acting on the particles accounting for long range interactions. We will 6\n\nfocus on the development of the method, simulation results for diodes and MESFET as well as comparisons to other classical models in the field. In particular we compute, determin-istically, the evolution probability density function with its first three moments. Boundary singularities for 2-space dimensions models are accurately computed. This work has been done in collaboration with J.A. Carrillo, A. Majorana and C.-W. Shu. Finally, I will present work in progress on computations by DG schemes of linear Boltzmann equations, work in collaboration with Jennifer Proft and Ross Heath. Some other issues I am and have been studying, and I am interested in learning more, are: 1)Self-consistent models of kinetic charged transport. Perturbations of Stationary Non-Equilibrium States (SNS). 1.i) Numerical implementation of deterministic kinetic-Poisson systems and compar-isons to DSMC simulations by WENO schemes, and more recently, developing Discontinuous Galerkin (DG) schemes. 1.ii)Boundary value problems, existence and hydrodynamics limits for strong force fields. Coupling of hyperbolic (SNS) and diffusion (SES) scaling regimes by kinetic layers. 1.iii)Biological transport of charged molecules and Chemotaxis kinetic transport. 2) Quantum Trajectory Models (QTM) for charged transport and Quantum hydrody-namics (QHD) from a semi-classical picture: thermalization and Bose-Einstein condensates models. Existence and non-existence to dispersion/diffusion models. Applications and com-putations. 2.i) Strong force field (Chapman-Enskog) expansion to the semi-classical Wigner trans-port equation 2.ii) Finite time flow up for the QHD equations under high velocity data 2.iii)Numerical calculations for quantum states. 3) The Boltzmann equation for energy dissipative flows, such as inelastic collisions in the modeling of rapid granular flows or elastic collisions in mixtures. 3.i) Energy dissipative Maxwell model type-solutions with power like tails-Levy distri-butions. Trends to equilibrium for energy dissipative Pseudo Maxwell models. 3.ii) Point-wise upper bounds for variable hard spheres, both in the elastic and inelastic case. Boundary value problems. Space inhomogeneous equation. 3.iii) Numerical implementations by spectral methods References can be found at www.ma.utexas.edu/users/gamba/research.html",
   "clean_statement": null,
   "public_statement": "A.6 Gamba, Irene\n\nNon-equilibrium time dependent reactive kinetic-Poisson systems appear in the mod-eling of such diverse areas as electron transport in solids, biological transport, granular and energy dissipative flows. These non-conservative systems exhibit a common feature: their steady or self similar states are given by statistical Stationary Non-equilibrium States (SNS), meaning they are far deviated from Gaussian probability distributions. They entice approx-imating hydrodynamic models modeling \"source\" representing momentum and energy gain or dissipation due to strong friction or forcing scales. When these non-equilibrium regimes take over, classical hydrodynamic models (based on Gaussians/Maxwellian closures) do not apply and need to be corrected to account for the non-equilibrium statistics. I am interested in issues related to the mathematical properties of these models such as existence, uniqueness, self-similarity, stability and to analyze their higher order moment equations (hydro-dynamical corrections) and corresponding boundary value problems; as well as to investigate optimal numerical simulation methods for corresponding quantum, kinetic and macroscopic (hydrodynamic) models. I will present a very recent work related to item (1.i) below: \"Deterministic solvers to transient Boltzmann-Poisson equations\" Abstract: The Boltzmann-Poisson system is the most reliable model for the flow of charged particles in semiconductors devices. Real device models have not already been simulated by deterministic computations due to its high computational cost, although is very well known and general practice to solve these models by Monte-Carlo (DSMC) methods. We focus in a rather easy and fast deterministic solver for a channel flow: one and two-dimension and three-velocity dimension. The system of equations reduces to a linear kinetic (non-local) equation solved by WENO methods coupled with the Poisson equation for the force field acting on the particles accounting for long range interactions. We will 6\n\nfocus on the development of the method, simulation results for diodes and MESFET as well as comparisons to other classical models in the field. In particular we compute, determin-istically, the evolution probability density function with its first three moments. Boundary singularities for 2-space dimensions models are accurately computed. This work has been done in collaboration with J.A. Carrillo, A. Majorana and C.-W. Shu. Finally, I will present work in progress on computations by DG schemes of linear Boltzmann equations, work in collaboration with Jennifer Proft and Ross Heath. Some other issues I am and have been studying, and I am interested in learning more, are: 1)Self-consistent models of kinetic charged transport. Perturbations of Stationary Non-Equilibrium States (SNS). 1.i) Numerical implementation of deterministic kinetic-Poisson systems and compar-isons to DSMC simulations by WENO schemes, and more recently, developing Discontinuous Galerkin (DG) schemes. 1.ii)Boundary value problems, existence and hydrodynamics limits for strong force fields. Coupling of hyperbolic (SNS) and diffusion (SES) scaling regimes by kinetic layers. 1.iii)Biological transport of charged molecules and Chemotaxis kinetic transport. 2) Quantum Trajectory Models (QTM) for charged transport and Quantum hydrody-namics (QHD) from a semi-classical picture: thermalization and Bose-Einstein condensates models. Existence and non-existence to dispersion/diffusion models. Applications and com-putations. 2.i) Strong force field (Chapman-Enskog) expansion to the semi-classical Wigner trans-port equation 2.ii) Finite time flow up for the QHD equations under high velocity data 2.iii)Numerical calculations for quantum states. 3) The Boltzmann equation for energy dissipative flows, such as inelastic collisions in the modeling of rapid granular flows or elastic collisions in mixtures. 3.i) Energy dissipative Maxwell model type-solutions with power like tails-Levy distri-butions. Trends to equilibrium for energy dissipative Pseudo Maxwell models. 3.ii) Point-wise upper bounds for variable hard spheres, both in the elastic and inelastic case. Boundary value problems. Space inhomogeneous equation. 3.iii) Numerical implementations by spectral methods References can be found at www.ma.utexas.edu/users/gamba/research.html",
   "evidence": "The canonical record is item A.6, “Gamba, Irene,” in the 1 April 2005 AIM workshop document *Stiff Sources and Numerical Methods for Conservation Laws*. The original PDF was inspected on pages 5--6 of the PDF (printed pages 6--7).",
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   "source_index": 70,
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  "AIM-PDES-0072": {
   "statement_status": "exact",
   "original_statement": "A.7 Gelb, Anne \n\nI am most interested in shallow water equations as they pertain to environmental fluid dynamics. Specifically I am interested in global and local methods as they can be applied to spheres. I am also interested in problems of long term simulations.",
   "clean_statement": "A.7 Gelb, Anne\n\nI am most interested in shallow water equations as they pertain to environmental fluid dynamics. Specifically I am interested in global and local methods as they can be applied to spheres. I am also interested in problems of long term simulations.",
   "public_statement": "A.7 Gelb, Anne\n\nI am most interested in shallow water equations as they pertain to environmental fluid dynamics. Specifically I am interested in global and local methods as they can be applied to spheres. I am also interested in problems of long term simulations.",
   "evidence": "The exact canonical text is:",
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   "source_file": "aim-pdes-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-PDES-0073": {
   "statement_status": "exact",
   "original_statement": "A.8 Gerritsen, Margot \n\nJoint contribution with Rami Younis. Our general area of interest is the efficient numerical solution of flow and transport equations in reservoirs. At the moment, we focus primarily on the design of accurate meth-ods for simulation of miscible gas injection and in-situ combustion (or fire-flooding), which 7\n\nare important Enhanced Oil Recovery processes. These processes are inherently multi-scale. They are generally modeled by two sets of equations. One describing the general flow and energy (if non isothermal) in the reservoir, which either parabolic in character. The sec-ond set models transport of components present in the oil, gas and water in the reservoir. The resulting equations are (weakly) hyperbolic, and very strongly nonlinear. In in-situ combustion processes additional reaction terms render the system also very stiff. We are ex-ploring Euler-Lagrangian type methods for gas injection processes, and splitting techniques for treatment of the equations governing in-situ combustion.",
   "clean_statement": "A.8 Gerritsen, Margot\n\nJoint contribution with Rami Younis. Our general area of interest is the efficient numerical solution of flow and transport equations in reservoirs. At the moment, we focus primarily on the design of accurate meth-ods for simulation of miscible gas injection and in-situ combustion (or fire-flooding), which 7\n\nare important Enhanced Oil Recovery processes. These processes are inherently multi-scale. They are generally modeled by two sets of equations. One describing the general flow and energy (if non isothermal) in the reservoir, which either parabolic in character. The sec-ond set models transport of components present in the oil, gas and water in the reservoir. The resulting equations are (weakly) hyperbolic, and very strongly nonlinear. In in-situ combustion processes additional reaction terms render the system also very stiff. We are ex-ploring Euler-Lagrangian type methods for gas injection processes, and splitting techniques for treatment of the equations governing in-situ combustion.",
   "public_statement": "A.8 Gerritsen, Margot\n\nJoint contribution with Rami Younis. Our general area of interest is the efficient numerical solution of flow and transport equations in reservoirs. At the moment, we focus primarily on the design of accurate meth-ods for simulation of miscible gas injection and in-situ combustion (or fire-flooding), which 7\n\nare important Enhanced Oil Recovery processes. These processes are inherently multi-scale. They are generally modeled by two sets of equations. One describing the general flow and energy (if non isothermal) in the reservoir, which either parabolic in character. The sec-ond set models transport of components present in the oil, gas and water in the reservoir. The resulting equations are (weakly) hyperbolic, and very strongly nonlinear. In in-situ combustion processes additional reaction terms render the system also very stiff. We are ex-ploring Euler-Lagrangian type methods for gas injection processes, and splitting techniques for treatment of the equations governing in-situ combustion.",
   "evidence": "This record is participant statement A.8 by Margot Gerritsen, jointly with Rami Younis, in the AIM workshop document *Stiff Sources and Numerical Methods for Conservation Laws*, version dated 1 April 2005. It is a research-interest statement, not an explicit open problem.",
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   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 72,
   "attempt": 1
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  "AIM-PDES-0074": {
   "statement_status": "exact",
   "original_statement": "A.9 Hauck, Cory \n\nMy interest in this workshop stems from work on hydrodynamic models of electron transport in semiconductors. These models are balance laws that approximate the evolu-tion of a kinetic distribution of electrons by tracking a given set of velocity and/or energy moments. Hydrodynamic equations contain conservative terms, relaxation terms that arise from collisions, and drift terms derived from a electrical potential that satifies a Poisson equation. Hydrodynamic equations suffer from several difficulties. First of all, most electronic devices contain some type of abrupt material interface which make it difficult to devise a numerical scheme that properly captures the balance of forces found at the differential level. For high-field, transition-regime devices, numerical results are characterized by current oscillations that pollute the solution and can even cause breakdown via negative temperatures and densities. It would very helpful to know if this issue can be overcome with a well-balanced numerical scheme or whether it is a model defect. Second, in the low-field, high-density limit, the hydrodynamic equations recover the well-known drift-diffusion model. Hydrodynamic equations become stiff in this limit, and this includes stiff flux terms. Thus, with the restrictions given by a CFL condition, it is not straightforward how to implement an implicit scheme in an efficient way. Finally, I am interested in finding reasonable approximations to relaxation terms and, in particular, relaxation times. In gas dynamics there is an issue of obtaining the correct transport coefficients predicted by the kinetic model in the fluid limit. More specifically, one would like to recover an appropriate ratio of thermal conductivity to viscosity. In electron transport, the drift-diffusion equations contain only one transport coefficient - the mobility - and therefore this is not an issue. However, the behavior of a numerical solution, especially the current and the temperature, is still drastically affecting by the choice of transport coefficients. Although this is more of a physical modeling issue, I think it is important to understand the effects of relaxation times on numerical solutions.",
   "clean_statement": "A.9 Hauck, Cory\n\nMy interest in this workshop stems from work on hydrodynamic models of electron transport in semiconductors. These models are balance laws that approximate the evolu-tion of a kinetic distribution of electrons by tracking a given set of velocity and/or energy moments. Hydrodynamic equations contain conservative terms, relaxation terms that arise from collisions, and drift terms derived from a electrical potential that satifies a Poisson equation. Hydrodynamic equations suffer from several difficulties. First of all, most electronic devices contain some type of abrupt material interface which make it difficult to devise a numerical scheme that properly captures the balance of forces found at the differential level. For high-field, transition-regime devices, numerical results are characterized by current oscillations that pollute the solution and can even cause breakdown via negative temperatures and densities. It would very helpful to know if this issue can be overcome with a well-balanced numerical scheme or whether it is a model defect. Second, in the low-field, high-density limit, the hydrodynamic equations recover the well-known drift-diffusion model. Hydrodynamic equations become stiff in this limit, and this includes stiff flux terms. Thus, with the restrictions given by a CFL condition, it is not straightforward how to implement an implicit scheme in an efficient way. Finally, I am interested in finding reasonable approximations to relaxation terms and, in particular, relaxation times. In gas dynamics there is an issue of obtaining the correct transport coefficients predicted by the kinetic model in the fluid limit. More specifically, one would like to recover an appropriate ratio of thermal conductivity to viscosity. In electron transport, the drift-diffusion equations contain only one transport coefficient - the mobility - and therefore this is not an issue. However, the behavior of a numerical solution, especially the current and the temperature, is still drastically affecting by the choice of transport coefficients. Although this is more of a physical modeling issue, I think it is important to understand the effects of relaxation times on numerical solutions.",
   "public_statement": "A.9 Hauck, Cory\n\nMy interest in this workshop stems from work on hydrodynamic models of electron transport in semiconductors. These models are balance laws that approximate the evolu-tion of a kinetic distribution of electrons by tracking a given set of velocity and/or energy moments. Hydrodynamic equations contain conservative terms, relaxation terms that arise from collisions, and drift terms derived from a electrical potential that satifies a Poisson equation. Hydrodynamic equations suffer from several difficulties. First of all, most electronic devices contain some type of abrupt material interface which make it difficult to devise a numerical scheme that properly captures the balance of forces found at the differential level. For high-field, transition-regime devices, numerical results are characterized by current oscillations that pollute the solution and can even cause breakdown via negative temperatures and densities. It would very helpful to know if this issue can be overcome with a well-balanced numerical scheme or whether it is a model defect. Second, in the low-field, high-density limit, the hydrodynamic equations recover the well-known drift-diffusion model. Hydrodynamic equations become stiff in this limit, and this includes stiff flux terms. Thus, with the restrictions given by a CFL condition, it is not straightforward how to implement an implicit scheme in an efficient way. Finally, I am interested in finding reasonable approximations to relaxation terms and, in particular, relaxation times. In gas dynamics there is an issue of obtaining the correct transport coefficients predicted by the kinetic model in the fluid limit. More specifically, one would like to recover an appropriate ratio of thermal conductivity to viscosity. In electron transport, the drift-diffusion equations contain only one transport coefficient - the mobility - and therefore this is not an issue. However, the behavior of a numerical solution, especially the current and the temperature, is still drastically affecting by the choice of transport coefficients. Although this is more of a physical modeling issue, I think it is important to understand the effects of relaxation times on numerical solutions.",
   "evidence": "This record is participant statement A.9 by Cory Hauck in the AIM workshop document *Stiff Sources and Numerical Methods for Conservation Laws* (version dated April 1, 2005). It is a research agenda, not a single formally quantified problem. It asks about three related issues for hydrodynamic electron-transport models:",
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  "AIM-PDES-0075": {
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   "original_statement": "A.10 Jin, Shi \n\nMy recent interests include numerical methods for physical problems involving multiple scales. In particular I am interested in the transition from quantum to classical mechanics, from kinetic theory to hydrodynamics, and its numerical relavance. In the workshop I will present recent results on numerical methods for Liouville equations with singular Hamiltoni-ans (which arise either from a discontinuous potential or a discontinuous local wave speed). 8",
   "clean_statement": "A.10 Jin, Shi\n\nMy recent interests include numerical methods for physical problems involving multiple scales. In particular I am interested in the transition from quantum to classical mechanics, from kinetic theory to hydrodynamics, and its numerical relavance. In the workshop I will present recent results on numerical methods for Liouville equations with singular Hamiltoni-ans (which arise either from a discontinuous potential or a discontinuous local wave speed). 8",
   "public_statement": "A.10 Jin, Shi\n\nMy recent interests include numerical methods for physical problems involving multiple scales. In particular I am interested in the transition from quantum to classical mechanics, from kinetic theory to hydrodynamics, and its numerical relavance. In the workshop I will present recent results on numerical methods for Liouville equations with singular Hamiltoni-ans (which arise either from a discontinuous potential or a discontinuous local wave speed). 8",
   "evidence": "The canonical extraction is preserved here verbatim:",
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   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-PDES-0076": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "A.11 Katsaounis, Theodoros \n\nBalance laws appear as mathematical models in a great number of applications areas such as gas dynamics, mechanics, geophysics, biology. In recent years there has been enor-mous activity on developing numerical methods for capturing correctly the properties and features of the analytical solutions. I am particularly interested in developing numerical schemes for balance laws using relaxation approximation. The starting point of our approach is the class of relaxation schemes, introduced in [JX], which are based on the relaxation approximation to the nonlinear conservation law, that has a linear convection term and needs neither a Riemann solver nor the characteristic decomposition and thus enjoys great simplicity in the expense of increasing the number of unknowns. The stabilization mechanisms are the regularization by wave operators. The idea is to use a local relaxation approximation to construct linear hyperbolic system with a stiff lower order term that approximates the original nonlinear system with a small dissipative correction. Relaxation is a flux approximation and relaxation linearizes the Riemann prob-lem. This simplicity can be of great significance when one has to solve large-scale engineering problems. The numerical schemes are based on finite volume and finite element discretizations of the relaxation models. In [DK1], [DK2] the finite volume(difference) method is used to descritize the relaxation approximation of the shallow water equations in one and two space dimensions respectively. The source term is treated in two different ways. The numerical schemes are of first or second order in space and time, do not need Riemann solvers, they are able to treat the dry bed case(vacuum case) with no extra effort, and satisfy the steady states, an important feature of the analytical solution, within the accuracy of the relaxation parameter ≤.In [DK3] the numerical schemes presented in [DK1], [DK2] are used to compute the transport and diffusion of a passive pollutant by a water flow. The flow is modeled by the well-known shallow water equations and the pollutant propagation is described by a transport equation. It's worth mentioning that that no special treatment is needed for the transport equation in order to obtain accurate results. The relaxation approximation of conservation laws provide a natural setting for apply-ing the finite element method. We apply the standard finite element method combined with appropriate Runge-Kutta methods for the time discretization. Adaptive mesh refinement strategies based on a-posteriori indicators and inverse inequalities are employed for resolv-ing accurately regions with shocks. The resulting schemes have a regularization mechanism with finite speed of propagation, do not need the solution of approximate local Riemann problems, can be formulated as low order or high order schemes, or even a combination of them (h-p methods), and can be extended in multi-dimensions by using the finite element framework, [AKM], [KM], [GM]. Some properties of these schemes, concerning stability and convergence are presented in [AMT]. Simulating a shear band(a narrow layer of intense shearing in a material, not a crack though) is another topic of interest. The mathematical model consists of a system of con-servation laws, close related to that of elastodynamics. The highly nonlinear model has a internal diffusion mechanism which collapses on the shear band. The temperature and the strain rate grow (blow up?) while the velocity develops a δ-function behavior. It is an 9\n\nopen question whether the temperature and the strain rate blow up in finite or infinite time. Numerical simulation of this singular behavior is a challenge, [BKT].",
   "clean_statement": null,
   "public_statement": "A.11 Katsaounis, Theodoros\n\nBalance laws appear as mathematical models in a great number of applications areas such as gas dynamics, mechanics, geophysics, biology. In recent years there has been enor-mous activity on developing numerical methods for capturing correctly the properties and features of the analytical solutions. I am particularly interested in developing numerical schemes for balance laws using relaxation approximation. The starting point of our approach is the class of relaxation schemes, introduced in [JX], which are based on the relaxation approximation to the nonlinear conservation law, that has a linear convection term and needs neither a Riemann solver nor the characteristic decomposition and thus enjoys great simplicity in the expense of increasing the number of unknowns. The stabilization mechanisms are the regularization by wave operators. The idea is to use a local relaxation approximation to construct linear hyperbolic system with a stiff lower order term that approximates the original nonlinear system with a small dissipative correction. Relaxation is a flux approximation and relaxation linearizes the Riemann prob-lem. This simplicity can be of great significance when one has to solve large-scale engineering problems. The numerical schemes are based on finite volume and finite element discretizations of the relaxation models. In [DK1], [DK2] the finite volume(difference) method is used to descritize the relaxation approximation of the shallow water equations in one and two space dimensions respectively. The source term is treated in two different ways. The numerical schemes are of first or second order in space and time, do not need Riemann solvers, they are able to treat the dry bed case(vacuum case) with no extra effort, and satisfy the steady states, an important feature of the analytical solution, within the accuracy of the relaxation parameter ≤.In [DK3] the numerical schemes presented in [DK1], [DK2] are used to compute the transport and diffusion of a passive pollutant by a water flow. The flow is modeled by the well-known shallow water equations and the pollutant propagation is described by a transport equation. It's worth mentioning that that no special treatment is needed for the transport equation in order to obtain accurate results. The relaxation approximation of conservation laws provide a natural setting for apply-ing the finite element method. We apply the standard finite element method combined with appropriate Runge-Kutta methods for the time discretization. Adaptive mesh refinement strategies based on a-posteriori indicators and inverse inequalities are employed for resolv-ing accurately regions with shocks. The resulting schemes have a regularization mechanism with finite speed of propagation, do not need the solution of approximate local Riemann problems, can be formulated as low order or high order schemes, or even a combination of them (h-p methods), and can be extended in multi-dimensions by using the finite element framework, [AKM], [KM], [GM]. Some properties of these schemes, concerning stability and convergence are presented in [AMT]. Simulating a shear band(a narrow layer of intense shearing in a material, not a crack though) is another topic of interest. The mathematical model consists of a system of con-servation laws, close related to that of elastodynamics. The highly nonlinear model has a internal diffusion mechanism which collapses on the shear band. The temperature and the strain rate grow (blow up?) while the velocity develops a δ-function behavior. It is an 9\n\nopen question whether the temperature and the strain rate blow up in finite or infinite time. Numerical simulation of this singular behavior is a challenge, [BKT].",
   "evidence": "The canonical record is participant contribution A.11 by Theodoros Katsaounis in the AIM workshop document *Stiff Sources and Numerical Methods for Conservation Laws* (version dated 1 April 2005). Most of the record surveys relaxation schemes. Its final paragraph contains a genuine research question:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-pdes-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-PDES-0077": {
   "statement_status": "exact",
   "original_statement": "1. (Albert Fathi) Let f be a smooth vector field on a compact manifold M and let \n\nφtf: M → M be its flow. There is a standard way to include the flow φtf in the flow of a positive definite Lagrangian system. Take a Riemannian metric on M and let \n\n‖ ‖ be the corresponding norm. Define a Lagrangian L: T M → R by \n\nL(x, v ) = 1\n\n2‖v − f (x)‖2.\n\nThen the zero section M ⊂ T M of the tangent bundle is an invariant manifold for the Lagrangian flow. The Aubry set A0 ⊂ T M corresponding to the zero cohomology class 0 ∈ H1(M, R) is contained in M.Problem: give a characterization of A0 in terms of the dynamics of the flow φtf. In particular, does A0 contain the chain recurrent set of the flow φtf?2. Let H: T ∗M → R,\n\nH(x, p ) = 1\n\n2‖p‖2 − f (x) · p\n\nbe the Hamiltonian corresponding to L. Consider the Hamilton-Jacobi equation \n\nH(x, Du (x)) = 1\n\n2‖Du (x)‖2 − f (x) · Du (x) = 0.\n\nQuestion: Under what conditions is u = 0 the unique viscosity solution?",
   "clean_statement": "1. (Albert Fathi) Let f be a smooth vector field on a compact manifold M and let\n\nφtf: M → M be its flow. There is a standard way to include the flow φtf in the flow of a positive definite Lagrangian system. Take a Riemannian metric on M and let\n\n‖ ‖ be the corresponding norm. Define a Lagrangian L: T M → R by\n\nL(x, v ) = 1\n\n2‖v − f (x)‖2.\n\nThen the zero section M ⊂ T M of the tangent bundle is an invariant manifold for the Lagrangian flow. The Aubry set A0 ⊂ T M corresponding to the zero cohomology class 0 ∈ H1(M, R) is contained in M.Problem: give a characterization of A0 in terms of the dynamics of the flow φtf. In particular, does A0 contain the chain recurrent set of the flow φtf?2. Let H: T ∗M → R,\n\nH(x, p ) = 1\n\n2‖p‖2 − f (x) · p\n\nbe the Hamiltonian corresponding to L. Consider the Hamilton-Jacobi equation\n\nH(x, Du (x)) = 1\n\n2‖Du (x)‖2 − f (x) · Du (x) = 0.\n\nQuestion: Under what conditions is u = 0 the unique viscosity solution?",
   "public_statement": "1. (Albert Fathi) Let f be a smooth vector field on a compact manifold M and let\n\nφtf: M → M be its flow. There is a standard way to include the flow φtf in the flow of a positive definite Lagrangian system. Take a Riemannian metric on M and let\n\n‖ ‖ be the corresponding norm. Define a Lagrangian L: T M → R by\n\nL(x, v ) = 1\n\n2‖v − f (x)‖2.\n\nThen the zero section M ⊂ T M of the tangent bundle is an invariant manifold for the Lagrangian flow. The Aubry set A0 ⊂ T M corresponding to the zero cohomology class 0 ∈ H1(M, R) is contained in M.Problem: give a characterization of A0 in terms of the dynamics of the flow φtf. In particular, does A0 contain the chain recurrent set of the flow φtf?2. Let H: T ∗M → R,\n\nH(x, p ) = 1\n\n2‖p‖2 − f (x) · p\n\nbe the Hamiltonian corresponding to L. Consider the Hamilton-Jacobi equation\n\nH(x, Du (x)) = 1\n\n2‖Du (x)‖2 − f (x) · Du (x) = 0.\n\nQuestion: Under what conditions is u = 0 the unique viscosity solution?",
   "evidence": "The primary AIM PDF was checked directly. With typographical layout normalized, it asks about \\[ L_f(x,v)=\\frac12\\lVert v-f(x)\\rVert_x^2 \\] for a smooth vector field \\(f\\) on a compact Riemannian manifold \\(M\\), the relation between its zero-class Aubry set and recurrence of the flow \\(\\phi^t\\) of \\(f\\), and uniqueness of the zero solution of a stationary Hamilton--Jacobi equation. The PDF literally says that the zero section in \\(TM\\) is invariant and literally prints \\[ H_-(x,p)=\\frac12\\lVert p\\rVert_x^2-p(f(x)). \\]",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-PDES-0078": {
   "statement_status": "exact",
   "original_statement": "3. (John Mather) Let M be a compact manifold and L: T M ×T → R a C∞ Lagrangian satisfying the usual hypotheses of the Mather theory - convexity, superlinearity and completeness. Let Ac ⊂ M be the Aubry set corresponding to the cohomology class \n\nc ∈ H1(M, R). Define a pseudo metric ρc on Ac as follows. Modify the Lagrangian by subtracting a closed 1-form from the cohomology class c and adding a constant \n\nα(c) so that for the new Lc we have inf \n\n{∫ \n\nLc dμ | μ is an invariant probablility measure on T M × T\n\n}\n\n= 0.\n\nThen set \n\nhnc (x, y ) = inf \n\n{∫ n\n\n> 0\n\nLc(γ(t), ˙γ(t), t ) dt | γ: [0, n ] → M connects x and y\n\n}\n\nand \n\nρc(x, y ) = lim inf \n\n> n→∞\n\nhnc (x, y ) + lim inf \n\n> n→∞\n\nhnc (y, x ).\n\nDefine an equivalence relation on Ac by x ∼ y iff ρc(x, y ) = 0. Let ¯Ac be the corresponding quotient space. Question: Is it true that ¯Ac is totally disconnected? Does ¯Ac have zero Hausdorff dimension?",
   "clean_statement": "3. (John Mather) Let M be a compact manifold and L: T M ×T → R a C∞ Lagrangian satisfying the usual hypotheses of the Mather theory - convexity, superlinearity and completeness. Let Ac ⊂ M be the Aubry set corresponding to the cohomology class\n\nc ∈ H1(M, R). Define a pseudo metric ρc on Ac as follows. Modify the Lagrangian by subtracting a closed 1-form from the cohomology class c and adding a constant\n\nα(c) so that for the new Lc we have inf\n\n{∫\n\nLc dμ | μ is an invariant probablility measure on T M × T\n\n}\n\n= 0.\n\nThen set\n\nhnc (x, y ) = inf\n\n{∫ n\n\n> 0\n\nLc(γ(t), ˙γ(t), t ) dt | γ: [0, n ] → M connects x and y\n\n}\n\nand\n\nρc(x, y ) = lim inf\n\n> n→∞\n\nhnc (x, y ) + lim inf\n\n> n→∞\n\nhnc (y, x ).\n\nDefine an equivalence relation on Ac by x ∼ y iff ρc(x, y ) = 0. Let ¯Ac be the corresponding quotient space. Question: Is it true that ¯Ac is totally disconnected? Does ¯Ac have zero Hausdorff dimension?",
   "public_statement": "3. (John Mather) Let M be a compact manifold and L: T M ×T → R a C∞ Lagrangian satisfying the usual hypotheses of the Mather theory - convexity, superlinearity and completeness. Let Ac ⊂ M be the Aubry set corresponding to the cohomology class\n\nc ∈ H1(M, R). Define a pseudo metric ρc on Ac as follows. Modify the Lagrangian by subtracting a closed 1-form from the cohomology class c and adding a constant\n\nα(c) so that for the new Lc we have inf\n\n{∫\n\nLc dμ | μ is an invariant probablility measure on T M × T\n\n}\n\n= 0.\n\nThen set\n\nhnc (x, y ) = inf\n\n{∫ n\n\n> 0\n\nLc(γ(t), ˙γ(t), t ) dt | γ: [0, n ] → M connects x and y\n\n}\n\nand\n\nρc(x, y ) = lim inf\n\n> n→∞\n\nhnc (x, y ) + lim inf\n\n> n→∞\n\nhnc (y, x ).\n\nDefine an equivalence relation on Ac by x ∼ y iff ρc(x, y ) = 0. Let ¯Ac be the corresponding quotient space. Question: Is it true that ¯Ac is totally disconnected? Does ¯Ac have zero Hausdorff dimension?",
   "evidence": "The canonical extraction, preserved without silent correction, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-PDES-0079": {
   "statement_status": "exact",
   "original_statement": "4. (Albert Fathi) Is it true that for generic L ∈ C∞ there exists only a one-parameter family of viscosity solutions of the corresponding Hamilton-Jacobi equation? 4",
   "clean_statement": "4. (Albert Fathi) Is it true that for generic L ∈ C∞ there exists only a one-parameter family of viscosity solutions of the corresponding Hamilton-Jacobi equation? 4",
   "public_statement": "4. (Albert Fathi) Is it true that for generic L ∈ C∞ there exists only a one-parameter family of viscosity solutions of the corresponding Hamilton-Jacobi equation? 4",
   "evidence": "The AIM list *New connections between dynamical systems and PDE's* (version dated 15 August 2003) prints:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-PDES-0080": {
   "statement_status": "exact",
   "original_statement": "5. (Patrick Bernard) Consider an analytic positive definite Lagrangian L: Tn × Rn ×\n\nT → R, L = L(x, v, t ), satisfying Mather's conditions. Let \n\nαL(c) = − inf \n\n> μ\n\n∫\n\n(L(x, v, t ) − c · v) dμ \n\nbe Mather's function αL: Rn → R.Question: Is it true that if αL is analytic with positive definite second derivative, then the Lagrangian system is completely integrable?",
   "clean_statement": "5. (Patrick Bernard) Consider an analytic positive definite Lagrangian L: Tn × Rn ×\n\nT → R, L = L(x, v, t ), satisfying Mather's conditions. Let\n\nαL(c) = − inf\n\n> μ\n\n∫\n\n(L(x, v, t ) − c · v) dμ\n\nbe Mather's function αL: Rn → R.Question: Is it true that if αL is analytic with positive definite second derivative, then the Lagrangian system is completely integrable?",
   "public_statement": "5. (Patrick Bernard) Consider an analytic positive definite Lagrangian L: Tn × Rn ×\n\nT → R, L = L(x, v, t ), satisfying Mather's conditions. Let\n\nαL(c) = − inf\n\n> μ\n\n∫\n\n(L(x, v, t ) − c · v) dμ\n\nbe Mather's function αL: Rn → R.Question: Is it true that if αL is analytic with positive definite second derivative, then the Lagrangian system is completely integrable?",
   "evidence": "The AIM problem list *New connections between dynamical systems and PDE's* (version dated 15 August 2003) states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-PDES-0081": {
   "statement_status": "exact",
   "original_statement": "6. (Gonzalo Contreras) Is it true that under the same condition on αL, each Aubry set \n\nAc covers the whole configuration space Tn?",
   "clean_statement": "6. (Gonzalo Contreras) Is it true that under the same condition on αL, each Aubry set\n\nAc covers the whole configuration space Tn?",
   "public_statement": "6. (Gonzalo Contreras) Is it true that under the same condition on αL, each Aubry set\n\nAc covers the whole configuration space Tn?",
   "evidence": "The canonical extraction is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-PDES-0082": {
   "statement_status": "exact",
   "original_statement": "7. (John Mather) Let f: T × R → T × R be an analytic twist map. Suppose that \n\nT × R is foliated by invariant curves Γ c (which are Aubry-Mather sets). Are these curves Γ c necessarily analytic?",
   "clean_statement": "7. (John Mather) Let f: T × R → T × R be an analytic twist map. Suppose that\n\nT × R is foliated by invariant curves Γ c (which are Aubry-Mather sets). Are these curves Γ c necessarily analytic?",
   "public_statement": "7. (John Mather) Let f: T × R → T × R be an analytic twist map. Suppose that\n\nT × R is foliated by invariant curves Γ c (which are Aubry-Mather sets). Are these curves Γ c necessarily analytic?",
   "evidence": "The AIM workshop list, Problem 7 attributed to John Mather, states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-PDES-0083": {
   "statement_status": "exact",
   "original_statement": "8. (Vadim Kaloshin) Consider a billiard inside a region bounded by a closed convex curve Γ ⊂ R2. Suppose that a neighborhood of Γ is foliated by caustics for the billiard. Is it true that then Γ is an ellipse and hence the caustics are analytic curves? The billiard in Γ is described by a twist map f, and caustics correspond to invariant curves of f.Hence this questions is a particular case of the previous one.",
   "clean_statement": "8. (Vadim Kaloshin) Consider a billiard inside a region bounded by a closed convex curve Γ ⊂ R2. Suppose that a neighborhood of Γ is foliated by caustics for the billiard. Is it true that then Γ is an ellipse and hence the caustics are analytic curves? The billiard in Γ is described by a twist map f, and caustics correspond to invariant curves of f.Hence this questions is a particular case of the previous one.",
   "public_statement": "8. (Vadim Kaloshin) Consider a billiard inside a region bounded by a closed convex curve Γ ⊂ R2. Suppose that a neighborhood of Γ is foliated by caustics for the billiard. Is it true that then Γ is an ellipse and hence the caustics are analytic curves? The billiard in Γ is described by a twist map f, and caustics correspond to invariant curves of f.Hence this questions is a particular case of the previous one.",
   "evidence": "The AIM list *New connections between dynamical systems and PDE's*, Problem 8 (attributed to Vadim Kaloshin), asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 82,
   "attempt": 1
  },
  "AIM-PDES-0084": {
   "statement_status": "exact",
   "original_statement": "9. (John Mather) Does there exist a C3 twist map f: T × R → T × R which has an invariant curve Γ with irrational rotation number and such that f |Γ is not conjugate to a rotation? By the Denjoy example there exist C1 maps of a circle with irrational rotation number not conjugate to a rotation. Can this happen for invariant curves of twist maps? \n1",
   "clean_statement": "9. (John Mather) Does there exist a C3 twist map f: T × R → T × R which has an invariant curve Γ with irrational rotation number and such that f |Γ is not conjugate to a rotation? By the Denjoy example there exist C1 maps of a circle with irrational rotation number not conjugate to a rotation. Can this happen for invariant curves of twist maps?\n1",
   "public_statement": "9. (John Mather) Does there exist a C3 twist map f: T × R → T × R which has an invariant curve Γ with irrational rotation number and such that f |Γ is not conjugate to a rotation? By the Denjoy example there exist C1 maps of a circle with irrational rotation number not conjugate to a rotation. Can this happen for invariant curves of twist maps?\n1",
   "evidence": "The canonical record is item 9, attributed to John Mather, in the AIM workshop list *New connections between dynamical systems and PDE's*. Its exact extracted problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 83,
   "attempt": 2
  },
  "AIM-PDES-0085": {
   "statement_status": "exact",
   "original_statement": "0. (Patrick Bernard) The existence of KAM invariant tori of a Hamiltonian system on \n\nTn × Rn is equivalent to the existence of regular solutions of the Hamilton-Jacobi equation H(x, c + Du (x)) = α(c). Hence KAM theory can be regarded as regularity result for viscosity solutions. Question: Is it possible to obtain proofs of KAM results using this connection? \n1",
   "clean_statement": "0. (Patrick Bernard) The existence of KAM invariant tori of a Hamiltonian system on\n\nTn × Rn is equivalent to the existence of regular solutions of the Hamilton-Jacobi equation H(x, c + Du (x)) = α(c). Hence KAM theory can be regarded as regularity result for viscosity solutions. Question: Is it possible to obtain proofs of KAM results using this connection?\n1",
   "public_statement": "0. (Patrick Bernard) The existence of KAM invariant tori of a Hamiltonian system on\n\nTn × Rn is equivalent to the existence of regular solutions of the Hamilton-Jacobi equation H(x, c + Du (x)) = α(c). Hence KAM theory can be regarded as regularity result for viscosity solutions. Question: Is it possible to obtain proofs of KAM results using this connection?\n1",
   "evidence": "The original AIM PDF, version dated 15 August 2003, contains the following as Problem 10 (attributed to Patrick Bernard):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 84,
   "attempt": 1
  },
  "AIM-PDES-0086": {
   "statement_status": "exact",
   "original_statement": "1. (Albert Fathi) For a Hamiltonian H(x, p ) on Tn×Rn, which is superlinear and strictly convex in p, we have a fairly good description both in dynamical terms (Mather theory) and in PDE terms (viscosity solutions) of the Aubry set. It would be nice to understand the situation where H is still superlinear but not necessarily convex: \n\n• Does there exist an Aubry set from the PDE point of view (for example as a uniqueness set for viscosity solutions)? \n\n• Does there exist an Aubry set from the point of view of dynamics, i.e. acanonical graph invariant under the Hamiltonian flow? \n\n• Is there a relationship between the two sets if they exist? 12. Let c0 be the infimum of the c's such that the Hamilton-Jacobi equation H(x, Du (x)) = \n\nc with superlinear H admits a global viscosity subsolution u: Tn → R. Let SS be the set of global viscosity subsolutions of H(x, Du (x)) = c0. Define a function S on \n\nTn × Tn by \n\nS(x, y ) = sup {u(x) − u(y) | u ∈ SS }.5\n\nThen for fixed x, the function S(x, ·) is a viscosity subsolution of H(x, Du (x)) = c0\n\non Tn, and a viscosity solution on Tn \\ { 0}. When H is convex in p, the Aubry set \n\nA0 is the set of x ∈ Tn such that S(x, ·) is a global viscosity solution on the whole \n\nTn.Question: If H is not necessarily convex, does there exist x ∈ Tn such that S(x, ·) is a viscosity solution on the whole Tn?\n1",
   "clean_statement": "1. (Albert Fathi) For a Hamiltonian H(x, p ) on Tn×Rn, which is superlinear and strictly convex in p, we have a fairly good description both in dynamical terms (Mather theory) and in PDE terms (viscosity solutions) of the Aubry set. It would be nice to understand the situation where H is still superlinear but not necessarily convex:\n\n• Does there exist an Aubry set from the PDE point of view (for example as a uniqueness set for viscosity solutions)?\n\n• Does there exist an Aubry set from the point of view of dynamics, i.e. acanonical graph invariant under the Hamiltonian flow?\n\n• Is there a relationship between the two sets if they exist? 12. Let c0 be the infimum of the c's such that the Hamilton-Jacobi equation H(x, Du (x)) =\n\nc with superlinear H admits a global viscosity subsolution u: Tn → R. Let SS be the set of global viscosity subsolutions of H(x, Du (x)) = c0. Define a function S on\n\nTn × Tn by\n\nS(x, y ) = sup {u(x) − u(y) | u ∈ SS }.5\n\nThen for fixed x, the function S(x, ·) is a viscosity subsolution of H(x, Du (x)) = c0\n\non Tn, and a viscosity solution on Tn \\ { 0}. When H is convex in p, the Aubry set\n\nA0 is the set of x ∈ Tn such that S(x, ·) is a global viscosity solution on the whole\n\nTn.Question: If H is not necessarily convex, does there exist x ∈ Tn such that S(x, ·) is a viscosity solution on the whole Tn?\n1",
   "public_statement": "1. (Albert Fathi) For a Hamiltonian H(x, p ) on Tn×Rn, which is superlinear and strictly convex in p, we have a fairly good description both in dynamical terms (Mather theory) and in PDE terms (viscosity solutions) of the Aubry set. It would be nice to understand the situation where H is still superlinear but not necessarily convex:\n\n• Does there exist an Aubry set from the PDE point of view (for example as a uniqueness set for viscosity solutions)?\n\n• Does there exist an Aubry set from the point of view of dynamics, i.e. acanonical graph invariant under the Hamiltonian flow?\n\n• Is there a relationship between the two sets if they exist? 12. Let c0 be the infimum of the c's such that the Hamilton-Jacobi equation H(x, Du (x)) =\n\nc with superlinear H admits a global viscosity subsolution u: Tn → R. Let SS be the set of global viscosity subsolutions of H(x, Du (x)) = c0. Define a function S on\n\nTn × Tn by\n\nS(x, y ) = sup {u(x) − u(y) | u ∈ SS }.5\n\nThen for fixed x, the function S(x, ·) is a viscosity subsolution of H(x, Du (x)) = c0\n\non Tn, and a viscosity solution on Tn \\ { 0}. When H is convex in p, the Aubry set\n\nA0 is the set of x ∈ Tn such that S(x, ·) is a global viscosity solution on the whole\n\nTn.Question: If H is not necessarily convex, does there exist x ∈ Tn such that S(x, ·) is a viscosity solution on the whole Tn?\n1",
   "evidence": "The canonical record is an extraction of two consecutive questions from the 2003 AIM list. It must remain one corpus job, but mathematically it separates as follows.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 85,
   "attempt": 1
  },
  "AIM-PDES-0087": {
   "statement_status": "exact",
   "original_statement": "3. (Walter Craig) Consider a positive definite Lagrangian L: T M × T → R. Let \n\nα: H1(M, R) → R be Mather's α-function. Problem: Relate the regularity of the Mather set Mc and the arithmetic properties of the frequency set ω = Dα (c) ⊂ H1(M, R). \n1",
   "clean_statement": "3. (Walter Craig) Consider a positive definite Lagrangian L: T M × T → R. Let\n\nα: H1(M, R) → R be Mather's α-function. Problem: Relate the regularity of the Mather set Mc and the arithmetic properties of the frequency set ω = Dα (c) ⊂ H1(M, R).\n1",
   "public_statement": "3. (Walter Craig) Consider a positive definite Lagrangian L: T M × T → R. Let\n\nα: H1(M, R) → R be Mather's α-function. Problem: Relate the regularity of the Mather set Mc and the arithmetic properties of the frequency set ω = Dα (c) ⊂ H1(M, R).\n1",
   "evidence": "The canonical record is an OCR extraction from Walter Craig's Problem 13 in the AIM workshop list *New connections between dynamical systems and PDE's* (version dated 15 August 2003). The PDF reads, modulo lost mathematical typography:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 86,
   "attempt": 1
  },
  "AIM-PDES-0088": {
   "statement_status": "exact",
   "original_statement": "4. (Walter Craig) This question concerns the minimax Birkhoff orbits in an area pre-serving twist map of an annulus. It is well known that \n\n• There exist at least two periodic orbits of a twist map with given rational rotation number α = p/q, namely a minimal orbit and a minimax orbit. \n\n• Taking a limit αj → ω ∈ R \\ Q of rational rotation numbers, the minimal Birkhoff orbits converge to the Mather set Π ω.\n\n• If Π ω is an invariant circle, then the minimax orbits also converge to Π ω.Question: What is the fate of the minimax orbits for the case when the Mather set is a Cantor set? \n1",
   "clean_statement": "4. (Walter Craig) This question concerns the minimax Birkhoff orbits in an area pre-serving twist map of an annulus. It is well known that\n\n• There exist at least two periodic orbits of a twist map with given rational rotation number α = p/q, namely a minimal orbit and a minimax orbit.\n\n• Taking a limit αj → ω ∈ R \\ Q of rational rotation numbers, the minimal Birkhoff orbits converge to the Mather set Π ω.\n\n• If Π ω is an invariant circle, then the minimax orbits also converge to Π ω.Question: What is the fate of the minimax orbits for the case when the Mather set is a Cantor set?\n1",
   "public_statement": "4. (Walter Craig) This question concerns the minimax Birkhoff orbits in an area pre-serving twist map of an annulus. It is well known that\n\n• There exist at least two periodic orbits of a twist map with given rational rotation number α = p/q, namely a minimal orbit and a minimax orbit.\n\n• Taking a limit αj → ω ∈ R \\ Q of rational rotation numbers, the minimal Birkhoff orbits converge to the Mather set Π ω.\n\n• If Π ω is an invariant circle, then the minimax orbits also converge to Π ω.Question: What is the fate of the minimax orbits for the case when the Mather set is a Cantor set?\n1",
   "evidence": "The canonical JSON record is a damaged extraction of Problem 14 in the AIM workshop notes *New connections between dynamical systems and PDE's*. The source page and PDF give the following statement (with only typographical normalization):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 87,
   "attempt": 1
  },
  "AIM-PDES-0089": {
   "statement_status": "exact",
   "original_statement": "5. (Gonzalo Contreras) Let M be a compact manifold, L: T M → R a convex superlin-ear Lagrangian, and ˜L: T ˜M → R - lift of L to the universal cover ˜M of M. Let cu\n\nbe the Mane critical value for ˜L.Question: Is it true that \n\ncu = inf {h ∈ R | ∃ x ∈ ˜M ∀y ∈ M ∃trajectory with energy h joining x to y}.",
   "clean_statement": "5. (Gonzalo Contreras) Let M be a compact manifold, L: T M → R a convex superlin-ear Lagrangian, and ˜L: T ˜M → R - lift of L to the universal cover ˜M of M. Let cu\n\nbe the Mane critical value for ˜L.Question: Is it true that\n\ncu = inf {h ∈ R | ∃ x ∈ ˜M ∀y ∈ M ∃trajectory with energy h joining x to y}.",
   "public_statement": "5. (Gonzalo Contreras) Let M be a compact manifold, L: T M → R a convex superlin-ear Lagrangian, and ˜L: T ˜M → R - lift of L to the universal cover ˜M of M. Let cu\n\nbe the Mane critical value for ˜L.Question: Is it true that\n\ncu = inf {h ∈ R | ∃ x ∈ ˜M ∀y ∈ M ∃trajectory with energy h joining x to y}.",
   "evidence": "The record has two recoverable OCR defects: the original item number 15 became 5, and the symbols \\(\\widetilde L\\), \\(\\widetilde M\\), and \\(c_u\\) were partly flattened. More importantly, the mixed quantifiers \\(x\\in\\widetilde M\\) and \\(y\\in M\\) occur in both the official PDF and HTML. They are therefore not an OCR error, but (1) is not literally well typed: a curve in \\(\\widetilde M\\) cannot have an endpoint \\(y\\in M\\) without specifying a lift or asking only that its projected endpoint be \\(y\\).",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 88,
   "attempt": 1
  },
  "AIM-PDES-0090": {
   "statement_status": "exact",
   "original_statement": "16. Let M be a compact manifold, L: T M → R a convex superlinear Lagrangian. The only example known of an energy level without periodic orbits or singularities is \n\n{H = cu} with cu a Mane critical value for the lift of the Lagrangian to ˜M.Question: Is there an example with other energy levels without periodic orbits or singularities? 17. Let M be a compact manifold, H: T M → R a convex superlinear Hamiltonian. The energy level Σ h = {H = h} is of contact type if there exists a 1-form λ on Σ h\n\nsuch that dλ is the symplectic 2-form on T ∗M and λ(XH ) > 0, where XH is the Hamiltonian vector field. Let cu be the critical value on ˜M and let \n\ne0 = min {h ∈ R | π(Σ h) = M }.\n\nQuestion: Is it true that if M 6 = T2, then for all e0 < h < c u the energy level Σ h is not of contact type? 18. Let M be a non-compact manifold and L: T M → R a convex superlinear Lagrangian satisfying appropriate completeness conditions at infinity. There is a compactification 6\n\nof M by adding an \"extended Aubry set\" whose points correspond to \"Busemann viscosity solutions\" of the Hamilton-Jacobi equation (Calc. Var. 13 (2001), 427-458). Problem: Understand the geometry of this compactification. 19. Let L: Tn × Rn → R be an autonomous convex superlinear Lagrangian and H the corresponding Hamiltonian. Let c0 = α(0) be the Mane critical value. Question: Is it true that for any h < c 0 the set \n\n{(x, p ) ∈ R2n | H(x, p ) < h }\n\nhas finite symplectic capacity? 20. Let H: Tn × Rn → R be an autonomous convex superlinear Hamiltonian. Suppose that the Aubry set A0 satisfies π(A0) = Tn and there exists a unique (up to a constant) viscosity solution of the Hamilton-Jacobi equation H(x, Du (x)) = α(0). For c ∈ Rn, let Ac be the Aubry set corresponding to the Hamiltonian Hc(x, p ) = \n\nH(x, p − c). Let vc be a viscosity solution of Hc(x, Dv c(x)) = α(c). Question: is it true that lim \n\n> |c|→ 0\n\n‖Dv c − Du ‖Lip ( π(Ac)) = 0.",
   "clean_statement": "16. Let M be a compact manifold, L: T M → R a convex superlinear Lagrangian. The only example known of an energy level without periodic orbits or singularities is\n\n{H = cu} with cu a Mane critical value for the lift of the Lagrangian to ˜M.Question: Is there an example with other energy levels without periodic orbits or singularities? 17. Let M be a compact manifold, H: T M → R a convex superlinear Hamiltonian. The energy level Σ h = {H = h} is of contact type if there exists a 1-form λ on Σ h\n\nsuch that dλ is the symplectic 2-form on T ∗M and λ(XH ) > 0, where XH is the Hamiltonian vector field. Let cu be the critical value on ˜M and let\n\ne0 = min {h ∈ R | π(Σ h) = M }.\n\nQuestion: Is it true that if M 6 = T2, then for all e0 < h < c u the energy level Σ h is not of contact type? 18. Let M be a non-compact manifold and L: T M → R a convex superlinear Lagrangian satisfying appropriate completeness conditions at infinity. There is a compactification 6\n\nof M by adding an \"extended Aubry set\" whose points correspond to \"Busemann viscosity solutions\" of the Hamilton-Jacobi equation (Calc. Var. 13 (2001), 427-458). Problem: Understand the geometry of this compactification. 19. Let L: Tn × Rn → R be an autonomous convex superlinear Lagrangian and H the corresponding Hamiltonian. Let c0 = α(0) be the Mane critical value. Question: Is it true that for any h < c 0 the set\n\n{(x, p ) ∈ R2n | H(x, p ) < h }\n\nhas finite symplectic capacity? 20. Let H: Tn × Rn → R be an autonomous convex superlinear Hamiltonian. Suppose that the Aubry set A0 satisfies π(A0) = Tn and there exists a unique (up to a constant) viscosity solution of the Hamilton-Jacobi equation H(x, Du (x)) = α(0). For c ∈ Rn, let Ac be the Aubry set corresponding to the Hamiltonian Hc(x, p ) =\n\nH(x, p − c). Let vc be a viscosity solution of Hc(x, Dv c(x)) = α(c). Question: is it true that lim\n\n> |c|→ 0\n\n‖Dv c − Du ‖Lip ( π(Ac)) = 0.",
   "public_statement": "16. Let M be a compact manifold, L: T M → R a convex superlinear Lagrangian. The only example known of an energy level without periodic orbits or singularities is\n\n{H = cu} with cu a Mane critical value for the lift of the Lagrangian to ˜M.Question: Is there an example with other energy levels without periodic orbits or singularities? 17. Let M be a compact manifold, H: T M → R a convex superlinear Hamiltonian. The energy level Σ h = {H = h} is of contact type if there exists a 1-form λ on Σ h\n\nsuch that dλ is the symplectic 2-form on T ∗M and λ(XH ) > 0, where XH is the Hamiltonian vector field. Let cu be the critical value on ˜M and let\n\ne0 = min {h ∈ R | π(Σ h) = M }.\n\nQuestion: Is it true that if M 6 = T2, then for all e0 < h < c u the energy level Σ h is not of contact type? 18. Let M be a non-compact manifold and L: T M → R a convex superlinear Lagrangian satisfying appropriate completeness conditions at infinity. There is a compactification 6\n\nof M by adding an \"extended Aubry set\" whose points correspond to \"Busemann viscosity solutions\" of the Hamilton-Jacobi equation (Calc. Var. 13 (2001), 427-458). Problem: Understand the geometry of this compactification. 19. Let L: Tn × Rn → R be an autonomous convex superlinear Lagrangian and H the corresponding Hamiltonian. Let c0 = α(0) be the Mane critical value. Question: Is it true that for any h < c 0 the set\n\n{(x, p ) ∈ R2n | H(x, p ) < h }\n\nhas finite symplectic capacity? 20. Let H: Tn × Rn → R be an autonomous convex superlinear Hamiltonian. Suppose that the Aubry set A0 satisfies π(A0) = Tn and there exists a unique (up to a constant) viscosity solution of the Hamilton-Jacobi equation H(x, Du (x)) = α(0). For c ∈ Rn, let Ac be the Aubry set corresponding to the Hamiltonian Hc(x, p ) =\n\nH(x, p − c). Let vc be a viscosity solution of Hc(x, Dv c(x)) = α(c). Question: is it true that lim\n\n> |c|→ 0\n\n‖Dv c − Du ‖Lip ( π(Ac)) = 0.",
   "evidence": "The canonical JSON record merges five consecutive questions, Problems 16--20 in the 2003 AIM workshop list *New connections between dynamical systems and PDE's*. The complete record is preserved as one job, but its components are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 89,
   "attempt": 1
  },
  "AIM-PDES-0091": {
   "statement_status": "exact",
   "original_statement": "21. Mane proved that for a generic positive definite Lagrangian L: T M × T → R and generic c ∈ H1(M, R), the minimizing measure in the Mather set Mc is unique. Problem: For generic L and c is the unique minimizing measure in Mc supported in a periodic orbit? Conjecture (Jeff Xia): For a generic L and all c ∈ H1(M, R), the number of ergodic invariant measures in Mc is finite. \n2",
   "clean_statement": "21. Mane proved that for a generic positive definite Lagrangian L: T M × T → R and generic c ∈ H1(M, R), the minimizing measure in the Mather set Mc is unique. Problem: For generic L and c is the unique minimizing measure in Mc supported in a periodic orbit? Conjecture (Jeff Xia): For a generic L and all c ∈ H1(M, R), the number of ergodic invariant measures in Mc is finite.\n2",
   "public_statement": "21. Mane proved that for a generic positive definite Lagrangian L: T M × T → R and generic c ∈ H1(M, R), the minimizing measure in the Mather set Mc is unique. Problem: For generic L and c is the unique minimizing measure in Mc supported in a periodic orbit? Conjecture (Jeff Xia): For a generic L and all c ∈ H1(M, R), the number of ergodic invariant measures in Mc is finite.\n2",
   "evidence": "The canonical JSON record is an OCR extraction of Problem 21 from the AIM workshop list *New connections between dynamical systems and PDE's*. The original AIM HTML version confirms the following statement (notation modernized only typographically):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 90,
   "attempt": 1
  },
  "AIM-PDES-0092": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "2. (Kostia Khanin) Consider a convex superlinear positive definite Lagrangian on Tn ×\n\nRn with white noise perturbation: \n\nL(x, v, t ) = L0(x, v ) + \n\n> N\n\n∑\n\n> i=1\n\nFi(x) ˙ wi(t),\n\nwhere wi(t) are independent Brownian motions. Suppose that the map F = ( F1,..., F n): \n\nTn → RN is an embedding. Then with probability 1 there exists a unique global minimizer γ: R → M.Conjecture: With probability 1, the minimizer γ is a hyperbolic trajectory of the Lagrangian flow. This is proved (E-Khanin-Mazel-Sinai) for n = 1. \n2",
   "clean_statement": null,
   "public_statement": "2. (Kostia Khanin) Consider a convex superlinear positive definite Lagrangian on Tn ×\n\nRn with white noise perturbation:\n\nL(x, v, t ) = L0(x, v ) +\n\n> N\n\n∑\n\n> i=1\n\nFi(x) ˙ wi(t),\n\nwhere wi(t) are independent Brownian motions. Suppose that the map F = ( F1,..., F n):\n\nTn → RN is an embedding. Then with probability 1 there exists a unique global minimizer γ: R → M.Conjecture: With probability 1, the minimizer γ is a hyperbolic trajectory of the Lagrangian flow. This is proved (E-Khanin-Mazel-Sinai) for n = 1.\n2",
   "evidence": "The official AIM PDF, version dated 15 August 2003, gives this as Problem 22, proposed by Kostia Khanin:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-pdes-notes.json",
   "source_index": 91,
   "attempt": 1
  },
  "AIM-PDES-0093": {
   "statement_status": "exact",
   "original_statement": "3. (Arnold diffusion) Consider a Cr, 3 ≤ r ≤ ω, Hamiltonian Hε: Tn × Rn × T → R\n\nof the form \n\nHε(x, p, t ) = H0(p) + εH 1(x, p, t ).\n\nSuppose that H0 is positive definite and superlinear. 7\n\nConjecture: Suppose that n ≥ 2. Then for given open sets U0, U 1 ⊂ Rn and typical in the Cr topology H1, there exists ε0 > 0 such that for any ε ∈ (0, ε 0), there exists a trajectory ( x(t), p (t)) of the Hamiltonian system such that p(t0) ∈ U0 and p(t1) ∈ U1.This conjecture seems very hard in the Cω category, when the change of the action is exponentially slow in ε by the Nekhoroshev theorem. The case of finite r should be easier. A precise definition of \"typical\" needs to be established. Mather proved this conjec-ture for n = 2 and for a cusp residual set of perturbations εH 1 in the C∞ topology. This doesn't prove the above conjecture: for given H1 the set of admissible ε does not cover an interval (0, ε 0). Question: Does Mather's theorem holds for an εH 1 in a cusp residual set of trigono-metric polynomials of high order N? Exponentially small perturbations of coefficients are allowed, so the transcendental problem of exponentially small separatrice splitting is avoided. 24. Consider an apriori unstable Hamiltonian of the form \n\nH(θ, I, x, y ) = H0(I) + F (x, y ) + εH 1(θ, I, x, y ),\n\non Tn × Rn × R2, where H0 is convex and superlinear, the Hamiltonian F: R2 → R\n\nhas a separatrix loop, and H1 is a generic perturbation. The large gap problem of Arnold diffusion was overpassed recently for such systems by variational methods of Mather (Xia), by geometrical methods using secondary KAM-tori (de la Llave, Delshams, Seara), and by the method of separatrix map (Treschev). Problem: Understand the relation between the variational method and the method of separatrix map. They seem similar in spirit. \n2",
   "clean_statement": "3. (Arnold diffusion) Consider a Cr, 3 ≤ r ≤ ω, Hamiltonian Hε: Tn × Rn × T → R\n\nof the form\n\nHε(x, p, t ) = H0(p) + εH 1(x, p, t ).\n\nSuppose that H0 is positive definite and superlinear. 7\n\nConjecture: Suppose that n ≥ 2. Then for given open sets U0, U 1 ⊂ Rn and typical in the Cr topology H1, there exists ε0 > 0 such that for any ε ∈ (0, ε 0), there exists a trajectory ( x(t), p (t)) of the Hamiltonian system such that p(t0) ∈ U0 and p(t1) ∈ U1.This conjecture seems very hard in the Cω category, when the change of the action is exponentially slow in ε by the Nekhoroshev theorem. The case of finite r should be easier. A precise definition of \"typical\" needs to be established. Mather proved this conjec-ture for n = 2 and for a cusp residual set of perturbations εH 1 in the C∞ topology. This doesn't prove the above conjecture: for given H1 the set of admissible ε does not cover an interval (0, ε 0). Question: Does Mather's theorem holds for an εH 1 in a cusp residual set of trigono-metric polynomials of high order N? Exponentially small perturbations of coefficients are allowed, so the transcendental problem of exponentially small separatrice splitting is avoided. 24. Consider an apriori unstable Hamiltonian of the form\n\nH(θ, I, x, y ) = H0(I) + F (x, y ) + εH 1(θ, I, x, y ),\n\non Tn × Rn × R2, where H0 is convex and superlinear, the Hamiltonian F: R2 → R\n\nhas a separatrix loop, and H1 is a generic perturbation. The large gap problem of Arnold diffusion was overpassed recently for such systems by variational methods of Mather (Xia), by geometrical methods using secondary KAM-tori (de la Llave, Delshams, Seara), and by the method of separatrix map (Treschev). Problem: Understand the relation between the variational method and the method of separatrix map. They seem similar in spirit.\n2",
   "public_statement": "3. (Arnold diffusion) Consider a Cr, 3 ≤ r ≤ ω, Hamiltonian Hε: Tn × Rn × T → R\n\nof the form\n\nHε(x, p, t ) = H0(p) + εH 1(x, p, t ).\n\nSuppose that H0 is positive definite and superlinear. 7\n\nConjecture: Suppose that n ≥ 2. Then for given open sets U0, U 1 ⊂ Rn and typical in the Cr topology H1, there exists ε0 > 0 such that for any ε ∈ (0, ε 0), there exists a trajectory ( x(t), p (t)) of the Hamiltonian system such that p(t0) ∈ U0 and p(t1) ∈ U1.This conjecture seems very hard in the Cω category, when the change of the action is exponentially slow in ε by the Nekhoroshev theorem. The case of finite r should be easier. A precise definition of \"typical\" needs to be established. Mather proved this conjec-ture for n = 2 and for a cusp residual set of perturbations εH 1 in the C∞ topology. This doesn't prove the above conjecture: for given H1 the set of admissible ε does not cover an interval (0, ε 0). Question: Does Mather's theorem holds for an εH 1 in a cusp residual set of trigono-metric polynomials of high order N? Exponentially small perturbations of coefficients are allowed, so the transcendental problem of exponentially small separatrice splitting is avoided. 24. Consider an apriori unstable Hamiltonian of the form\n\nH(θ, I, x, y ) = H0(I) + F (x, y ) + εH 1(θ, I, x, y ),\n\non Tn × Rn × R2, where H0 is convex and superlinear, the Hamiltonian F: R2 → R\n\nhas a separatrix loop, and H1 is a generic perturbation. The large gap problem of Arnold diffusion was overpassed recently for such systems by variational methods of Mather (Xia), by geometrical methods using secondary KAM-tori (de la Llave, Delshams, Seara), and by the method of separatrix map (Treschev). Problem: Understand the relation between the variational method and the method of separatrix map. They seem similar in spirit.\n2",
   "evidence": "The canonical record is not one problem. It is an OCR merge of two consecutive entries in the AIM workshop list *New connections between dynamical systems and PDE's* (notes by S. Bolotin). The original AIM HTML page presents them as separate list items. In the PDF numbering they are Problems 23 and 24: the extraction lost the leading “2” in “23.” but retained “24.” inside the same JSON string.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-PDES-0094": {
   "statement_status": "exact",
   "original_statement": "5. (Paul Rabinowitz) Is there a reasonable PDE analogue of Arnold's diffusion? It seems that a version of finite dimensional transition chains is not the right mech-anism for PDE. There are results of Kuksin which prove \"diffusion\" for PDE with respect to high order Sobolev norms. The same problem for infinite lattices like Fermi-Pasta-Ulam system. \n2",
   "clean_statement": "5. (Paul Rabinowitz) Is there a reasonable PDE analogue of Arnold's diffusion? It seems that a version of finite dimensional transition chains is not the right mech-anism for PDE. There are results of Kuksin which prove \"diffusion\" for PDE with respect to high order Sobolev norms. The same problem for infinite lattices like Fermi-Pasta-Ulam system.\n2",
   "public_statement": "5. (Paul Rabinowitz) Is there a reasonable PDE analogue of Arnold's diffusion? It seems that a version of finite dimensional transition chains is not the right mech-anism for PDE. There are results of Kuksin which prove \"diffusion\" for PDE with respect to high order Sobolev norms. The same problem for infinite lattices like Fermi-Pasta-Ulam system.\n2",
   "evidence": "The canonical JSON record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-PDES-0095": {
   "statement_status": "exact",
   "original_statement": "6. (Victor Bangert) Let L: Tn × T × Rn → R, L = L(x, u, p ) be a smooth multi-dimensional Lagrangian satisfying the usual convexity assumptions in p ∈ Rn. For example, \n\nL(x, u, p ) = 1\n\n2|p|2 + F (x, u ),\n\nwhere the potential F: Tn+1 → R is periodic in all variables. A function u: Rn → R is called minimal if it minimizes the action functional \n\n∫\n\nL(x, u (x), Du (x)) dx \n\nfor all variations with compact support. The set of minimizers u \"without self-intersections\" is very well understood. Here u: Rn → R is said to be \"without self-intersections\" if the projection to Tn+1 of graph( u) ⊂ Rn+1 is a hypersurface without 8\n\nself-intersections. In particular, for every minimizer u without self-intersections there exists a \"rotation vector\" c ∈ (Rn)∗ such that \n\n|u(x) − u(0) − c · x|\n\nis bounded by a constant that only depends on L. Conversely, for every c ∈ (Rn)∗\n\nthere exists a minimizer u without self-intersections such that |u(x)−c·x| is bounded. Question: Is there an analytical condition implying that the minimizers satisfying this condition have no self-intersections. More concrete question: Suppose u: Rn → R is minimal and |Du (x)| is bounded. Is it true that the graph of u in Tn+1 has no self-intersections. For partial results see V. Bangert, Ann. Inst. Henri Poincar´ e - Analyse non lin´ eaire 6(1989), 95-138, in particular Sect. 8. \n2",
   "clean_statement": "6. (Victor Bangert) Let L: Tn × T × Rn → R, L = L(x, u, p ) be a smooth multi-dimensional Lagrangian satisfying the usual convexity assumptions in p ∈ Rn. For example,\n\nL(x, u, p ) = 1\n\n2|p|2 + F (x, u ),\n\nwhere the potential F: Tn+1 → R is periodic in all variables. A function u: Rn → R is called minimal if it minimizes the action functional\n\n∫\n\nL(x, u (x), Du (x)) dx\n\nfor all variations with compact support. The set of minimizers u \"without self-intersections\" is very well understood. Here u: Rn → R is said to be \"without self-intersections\" if the projection to Tn+1 of graph( u) ⊂ Rn+1 is a hypersurface without 8\n\nself-intersections. In particular, for every minimizer u without self-intersections there exists a \"rotation vector\" c ∈ (Rn)∗ such that\n\n|u(x) − u(0) − c · x|\n\nis bounded by a constant that only depends on L. Conversely, for every c ∈ (Rn)∗\n\nthere exists a minimizer u without self-intersections such that |u(x)−c·x| is bounded. Question: Is there an analytical condition implying that the minimizers satisfying this condition have no self-intersections. More concrete question: Suppose u: Rn → R is minimal and |Du (x)| is bounded. Is it true that the graph of u in Tn+1 has no self-intersections. For partial results see V. Bangert, Ann. Inst. Henri Poincar´ e - Analyse non lin´ eaire 6(1989), 95-138, in particular Sect. 8.\n2",
   "public_statement": "6. (Victor Bangert) Let L: Tn × T × Rn → R, L = L(x, u, p ) be a smooth multi-dimensional Lagrangian satisfying the usual convexity assumptions in p ∈ Rn. For example,\n\nL(x, u, p ) = 1\n\n2|p|2 + F (x, u ),\n\nwhere the potential F: Tn+1 → R is periodic in all variables. A function u: Rn → R is called minimal if it minimizes the action functional\n\n∫\n\nL(x, u (x), Du (x)) dx\n\nfor all variations with compact support. The set of minimizers u \"without self-intersections\" is very well understood. Here u: Rn → R is said to be \"without self-intersections\" if the projection to Tn+1 of graph( u) ⊂ Rn+1 is a hypersurface without 8\n\nself-intersections. In particular, for every minimizer u without self-intersections there exists a \"rotation vector\" c ∈ (Rn)∗ such that\n\n|u(x) − u(0) − c · x|\n\nis bounded by a constant that only depends on L. Conversely, for every c ∈ (Rn)∗\n\nthere exists a minimizer u without self-intersections such that |u(x)−c·x| is bounded. Question: Is there an analytical condition implying that the minimizers satisfying this condition have no self-intersections. More concrete question: Suppose u: Rn → R is minimal and |Du (x)| is bounded. Is it true that the graph of u in Tn+1 has no self-intersections. For partial results see V. Bangert, Ann. Inst. Henri Poincar´ e - Analyse non lin´ eaire 6(1989), 95-138, in particular Sect. 8.\n2",
   "evidence": "The canonical JSON is an OCR extraction of Problem 26 (Victor Bangert) in the AIM list *New connections between dynamical systems and PDE's*. The JSON number “6” has lost its leading “2”; the isolated “8” and final “2” in the prose are page-number artifacts. The official AIM PDF gives the following problem.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 94,
   "attempt": 1
  },
  "AIM-PDES-0096": {
   "statement_status": "exact",
   "original_statement": "7. (Victor Bangert) Let M n be a compact Riemannian manifold and let CqM be the set of closed q-currents on M:\n\nCqM = {T ∈ (Ω q(M )) ∗: T (dα ) = 0 for all α ∈ Ωq−1(M )}.\n\nEvery q-current T defines a homology class [ T ] ∈ Hq(M, R) and a mass \n\nM (T ) = sup {T (ω): ω ∈ Ωq(M ), ‖ω‖∞ = 1 }.\n\nEvery homology class h ∈ Hq(M, R) has a representative with minimal mass. For q = 1 the supports of minimal currents consist of minimal geodesics. For q = n−1any minimizer T is given by a measured lamination by minimizing hypersurfaces (possibly with singularities). Problem: What can one say about minimal currents for 1 < q < n − 1. \n2",
   "clean_statement": "7. (Victor Bangert) Let M n be a compact Riemannian manifold and let CqM be the set of closed q-currents on M:\n\nCqM = {T ∈ (Ω q(M )) ∗: T (dα ) = 0 for all α ∈ Ωq−1(M )}.\n\nEvery q-current T defines a homology class [ T ] ∈ Hq(M, R) and a mass\n\nM (T ) = sup {T (ω): ω ∈ Ωq(M ), ‖ω‖∞ = 1 }.\n\nEvery homology class h ∈ Hq(M, R) has a representative with minimal mass. For q = 1 the supports of minimal currents consist of minimal geodesics. For q = n−1any minimizer T is given by a measured lamination by minimizing hypersurfaces (possibly with singularities). Problem: What can one say about minimal currents for 1 < q < n − 1.\n2",
   "public_statement": "7. (Victor Bangert) Let M n be a compact Riemannian manifold and let CqM be the set of closed q-currents on M:\n\nCqM = {T ∈ (Ω q(M )) ∗: T (dα ) = 0 for all α ∈ Ωq−1(M )}.\n\nEvery q-current T defines a homology class [ T ] ∈ Hq(M, R) and a mass\n\nM (T ) = sup {T (ω): ω ∈ Ωq(M ), ‖ω‖∞ = 1 }.\n\nEvery homology class h ∈ Hq(M, R) has a representative with minimal mass. For q = 1 the supports of minimal currents consist of minimal geodesics. For q = n−1any minimizer T is given by a measured lamination by minimizing hypersurfaces (possibly with singularities). Problem: What can one say about minimal currents for 1 < q < n − 1.\n2",
   "evidence": "The canonical record is OCR from Problem 27 of the AIM workshop list *New connections between dynamical systems and PDE's* (notes by S. Bolotin). The extraction lost the leading “2” in “27.” The original AIM HTML confirms the following notation and question:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-PDES-0097": {
   "statement_status": "exact",
   "original_statement": "8. (Franz Auer) Conjecture: If T ∈ Cn−1M is a minimizing closed current, then the Hausdorff dimension of the corresponding singularity set is at most n − 7. 29. Define a stable norm on Hq(M, R) by \n\n‖h‖ = inf {M (T ): [ T ] = h}.\n\nThis norm is an analog of Mather's β-function). Problem: Study convexity and differentiability properties of the unit ball \n\nB = {h ∈ Hq(M, R): ‖h‖ ≤ 1}.\n\n3",
   "clean_statement": "8. (Franz Auer) Conjecture: If T ∈ Cn−1M is a minimizing closed current, then the Hausdorff dimension of the corresponding singularity set is at most n − 7. 29. Define a stable norm on Hq(M, R) by\n\n‖h‖ = inf {M (T ): [ T ] = h}.\n\nThis norm is an analog of Mather's β-function). Problem: Study convexity and differentiability properties of the unit ball\n\nB = {h ∈ Hq(M, R): ‖h‖ ≤ 1}.\n\n3",
   "public_statement": "8. (Franz Auer) Conjecture: If T ∈ Cn−1M is a minimizing closed current, then the Hausdorff dimension of the corresponding singularity set is at most n − 7. 29. Define a stable norm on Hq(M, R) by\n\n‖h‖ = inf {M (T ): [ T ] = h}.\n\nThis norm is an analog of Mather's β-function). Problem: Study convexity and differentiability properties of the unit ball\n\nB = {h ∈ Hq(M, R): ‖h‖ ≤ 1}.\n\n3",
   "evidence": "The canonical JSON record is visibly concatenated. It begins with",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 96,
   "attempt": 1
  },
  "AIM-PDES-0098": {
   "statement_status": "exact",
   "original_statement": "0. (Victor Bangert) Let M 2n be a manifold with an almost complex structure J. Apseudo-holomorphic line is a map f: R2 → M satisfying the PDE: fy = Jf x. Moser proved that for an almost complex structure J on T2n which is close to a standard complex structure J0, and any sufficiently irrational vector v ∈ R2n, there exists a foliation of T2n by pseudo-holomorphic curves which is conjugate to a linear foliation by real 2-planes containing the vector v.Problem: Develop a global (non-perturbative) theory of such foliations or lamina-tions. The theory is expected to be particularly rich for the case T4 due to the positivity of the intersection number of pseudoholomorphic curves. This is an analog of the 9\n\nintersection number for geodesics on T2 - the basis for Hedlund's results on minimal geodesics. \n3",
   "clean_statement": "0. (Victor Bangert) Let M 2n be a manifold with an almost complex structure J. Apseudo-holomorphic line is a map f: R2 → M satisfying the PDE: fy = Jf x. Moser proved that for an almost complex structure J on T2n which is close to a standard complex structure J0, and any sufficiently irrational vector v ∈ R2n, there exists a foliation of T2n by pseudo-holomorphic curves which is conjugate to a linear foliation by real 2-planes containing the vector v.Problem: Develop a global (non-perturbative) theory of such foliations or lamina-tions. The theory is expected to be particularly rich for the case T4 due to the positivity of the intersection number of pseudoholomorphic curves. This is an analog of the 9\n\nintersection number for geodesics on T2 - the basis for Hedlund's results on minimal geodesics.\n3",
   "public_statement": "0. (Victor Bangert) Let M 2n be a manifold with an almost complex structure J. Apseudo-holomorphic line is a map f: R2 → M satisfying the PDE: fy = Jf x. Moser proved that for an almost complex structure J on T2n which is close to a standard complex structure J0, and any sufficiently irrational vector v ∈ R2n, there exists a foliation of T2n by pseudo-holomorphic curves which is conjugate to a linear foliation by real 2-planes containing the vector v.Problem: Develop a global (non-perturbative) theory of such foliations or lamina-tions. The theory is expected to be particularly rich for the case T4 due to the positivity of the intersection number of pseudoholomorphic curves. This is an analog of the 9\n\nintersection number for geodesics on T2 - the basis for Hedlund's results on minimal geodesics.\n3",
   "evidence": "The canonical JSON record is an OCR extraction from Problem 30 in the AIM workshop list *New connections between dynamical systems and PDE's*. It reads “0.” because the leading digit 3 was lost. It also contains the page artifacts “9” and “3,” splits “laminations,” and suppresses superscripts and subscripts.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 97,
   "attempt": 1
  },
  "AIM-PDES-0099": {
   "statement_status": "exact",
   "original_statement": "1. (Craig Evans) Consider the Hamiltonian H(x, p ) = 1 \n\n> 2\n\n|p|2 + V (x), where the potential \n\nV is Tn periodic. The corresponding Hamiltonian operator in quantum mechanics is then − h2 \n\n> 2\n\n∆ + V (x). Much exciting research in \"semiclassical analysis\" concerns studying the limit as \n\nh → 0 of solutions u(h) of the eigenvalue problem \n\nh2\n\n2 ∆u(h) + V (x)u(h) = E(h)u(h) (∗)and finding connections with classical Hamiltonian dynamics. Problem: Do Mather sets play any role here? Or, conversely, can we somehow \"quantize\" Mather sets? This would presumably mean to build quasimodes (ie ap-proximate solutions of (*)) corresponding to Mather's sets and to prove good error bounds. Would some sort of Diophantine condition be useful here? \n3",
   "clean_statement": "1. (Craig Evans) Consider the Hamiltonian H(x, p ) = 1\n\n> 2\n\n|p|2 + V (x), where the potential\n\nV is Tn periodic. The corresponding Hamiltonian operator in quantum mechanics is then − h2\n\n> 2\n\n∆ + V (x). Much exciting research in \"semiclassical analysis\" concerns studying the limit as\n\nh → 0 of solutions u(h) of the eigenvalue problem\n\nh2\n\n2 ∆u(h) + V (x)u(h) = E(h)u(h) (∗)and finding connections with classical Hamiltonian dynamics. Problem: Do Mather sets play any role here? Or, conversely, can we somehow \"quantize\" Mather sets? This would presumably mean to build quasimodes (ie ap-proximate solutions of (*)) corresponding to Mather's sets and to prove good error bounds. Would some sort of Diophantine condition be useful here?\n3",
   "public_statement": "1. (Craig Evans) Consider the Hamiltonian H(x, p ) = 1\n\n> 2\n\n|p|2 + V (x), where the potential\n\nV is Tn periodic. The corresponding Hamiltonian operator in quantum mechanics is then − h2\n\n> 2\n\n∆ + V (x). Much exciting research in \"semiclassical analysis\" concerns studying the limit as\n\nh → 0 of solutions u(h) of the eigenvalue problem\n\nh2\n\n2 ∆u(h) + V (x)u(h) = E(h)u(h) (∗)and finding connections with classical Hamiltonian dynamics. Problem: Do Mather sets play any role here? Or, conversely, can we somehow \"quantize\" Mather sets? This would presumably mean to build quasimodes (ie ap-proximate solutions of (*)) corresponding to Mather's sets and to prove good error bounds. Would some sort of Diophantine condition be useful here?\n3",
   "evidence": "The canonical record is source index 98 of `aim-pdes-notes.json`. It is Craig Evans's problem from the AIM workshop *New connections between dynamical systems and PDE's*. The source PDF/HTML labels it as Problem 31, although the extracted record has lost the leading digit and says “1.”",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 98,
   "attempt": 1
  },
  "AIM-PDES-0100": {
   "statement_status": "exact",
   "original_statement": "2. (Takis Souganidis) Homogenization problem for random Hamilton-Jacobi equation \n\nF (D2u, Du, u, x/ε, x, ω ) = 0 with non-convex stationary ergodic in x/ε Hamiltonian F.\n3",
   "clean_statement": "2. (Takis Souganidis) Homogenization problem for random Hamilton-Jacobi equation\n\nF (D2u, Du, u, x/ε, x, ω ) = 0 with non-convex stationary ergodic in x/ε Hamiltonian F.\n3",
   "public_statement": "2. (Takis Souganidis) Homogenization problem for random Hamilton-Jacobi equation\n\nF (D2u, Du, u, x/ε, x, ω ) = 0 with non-convex stationary ergodic in x/ε Hamiltonian F.\n3",
   "evidence": "The canonical record is an OCR extraction from the AIM workshop list *New connections between dynamical systems and PDE's*. The official AIM HTML and PDF agree on the following text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 99,
   "attempt": 1
  },
  "AIM-PDES-0101": {
   "statement_status": "exact",
   "original_statement": "3. (Elena Kosygina) Let Ω be a probability space with probability measure P ergodic under a shift transformation group τx: Ω → Ω, x ∈ Rn. Consider the stochastic Hamilton-Jacobi equation \n\nut + H(Dxu, τ xω) = 0, (x, t ) ∈ Rn × (0, ∞),\n\nwhere H(p, ω ) is convex in p and satisfies some regularity assumptions, for example \n\nH(p, ω ) = 1\n\n2|p|2 − V (ω).\n\nA homogenization result for this problem was obtained by Lions and Souganidis. For the Hamiltonian above homogenization is equivalent to a large deviation result for Brownian motion among random obstacles (Snitzman). Problem: Derive a large deviation result for general stationary ergodic setting without any independence or mixing assumptions on P under the shifts. \n3",
   "clean_statement": "3. (Elena Kosygina) Let Ω be a probability space with probability measure P ergodic under a shift transformation group τx: Ω → Ω, x ∈ Rn. Consider the stochastic Hamilton-Jacobi equation\n\nut + H(Dxu, τ xω) = 0, (x, t ) ∈ Rn × (0, ∞),\n\nwhere H(p, ω ) is convex in p and satisfies some regularity assumptions, for example\n\nH(p, ω ) = 1\n\n2|p|2 − V (ω).\n\nA homogenization result for this problem was obtained by Lions and Souganidis. For the Hamiltonian above homogenization is equivalent to a large deviation result for Brownian motion among random obstacles (Snitzman). Problem: Derive a large deviation result for general stationary ergodic setting without any independence or mixing assumptions on P under the shifts.\n3",
   "public_statement": "3. (Elena Kosygina) Let Ω be a probability space with probability measure P ergodic under a shift transformation group τx: Ω → Ω, x ∈ Rn. Consider the stochastic Hamilton-Jacobi equation\n\nut + H(Dxu, τ xω) = 0, (x, t ) ∈ Rn × (0, ∞),\n\nwhere H(p, ω ) is convex in p and satisfies some regularity assumptions, for example\n\nH(p, ω ) = 1\n\n2|p|2 − V (ω).\n\nA homogenization result for this problem was obtained by Lions and Souganidis. For the Hamiltonian above homogenization is equivalent to a large deviation result for Brownian motion among random obstacles (Snitzman). Problem: Derive a large deviation result for general stationary ergodic setting without any independence or mixing assumptions on P under the shifts.\n3",
   "evidence": "The canonical JSON has lost the tens digit in the problem number and contains a terminal OCR artifact. The official AIM HTML and the workshop PDF show that this record is **Problem 33**, not Problem 3. The mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 100,
   "attempt": 1
  },
  "AIM-PDES-0102": {
   "statement_status": "exact",
   "original_statement": "4. (Diogo Gomes) Investigate possible extensions of the techniques used in the Aubry- Mather theory and Monge-Kantorovich problems to study linear programming prob-lems in infinite dimensions. An example of such extensions are the stochastic Mather measures which can be used to analyze second order Hamilton-Jacobi equations. \n3",
   "clean_statement": "4. (Diogo Gomes) Investigate possible extensions of the techniques used in the Aubry- Mather theory and Monge-Kantorovich problems to study linear programming prob-lems in infinite dimensions. An example of such extensions are the stochastic Mather measures which can be used to analyze second order Hamilton-Jacobi equations.\n3",
   "public_statement": "4. (Diogo Gomes) Investigate possible extensions of the techniques used in the Aubry- Mather theory and Monge-Kantorovich problems to study linear programming prob-lems in infinite dimensions. An example of such extensions are the stochastic Mather measures which can be used to analyze second order Hamilton-Jacobi equations.\n3",
   "evidence": "The canonical record is source index 101 of aim-pdes-notes.json. The original AIM PDF, *New connections between dynamical systems and PDE's* (version of August 15, 2003), verifies that this is **Problem 34**, proposed by Diogo Gomes:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 101,
   "attempt": 1
  },
  "AIM-PDES-0103": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "5. (Diogo Gomes) For a small perturbation of a completely integrable Hamiltonian system a viscosity solution corresponding to a non-resonant unperturbed torus can be uniformly approximated by formal expansions. Similar expansions can be constructed for the density of the Mather measures. However, it is not known how well Mather measures themselves are approximated by these formal expansions. \n3",
   "clean_statement": "**Problem 35 (Diogo Gomes).** For a small perturbation of a completely integrable Hamiltonian system a viscosity solution corresponding to a non-resonant unperturbed torus can be uniformly approximated by formal expansions. Similar expansions can be constructed for the density of the Mather measures. However, it is not known how well Mather measures themselves are approximated by these formal expansions.",
   "public_statement": "5. (Diogo Gomes) For a small perturbation of a completely integrable Hamiltonian system a viscosity solution corresponding to a non-resonant unperturbed torus can be uniformly approximated by formal expansions. Similar expansions can be constructed for the density of the Mather measures. However, it is not known how well Mather measures themselves are approximated by these formal expansions.\n3",
   "evidence": "The canonical JSON record is affected by a small OCR/page-boundary error: its displayed number is `5` and it ends with an isolated `3`. The official AIM workshop page identifies it as Problem 35 and gives the following statement:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-pdes-notes.json",
   "source_index": 102,
   "attempt": 1
  },
  "AIM-PDES-0104": {
   "statement_status": "exact",
   "original_statement": "6. (Massimiliano Berti) For a nonlinear wave equation \n\nutt − uxx = f (u), f (0) = f ′(0) = 0 10 \n\nwith Dirichlet boundary conditions and general nonlinearity there exist a large num-ber of small amplitude periodic orbit with fixed period. Could one find small ampli-tude periodic orbits of large period and quasi-periodic solutions for the wave equa-tion?",
   "clean_statement": "6. (Massimiliano Berti) For a nonlinear wave equation\n\nutt − uxx = f (u), f (0) = f ′(0) = 0 10\n\nwith Dirichlet boundary conditions and general nonlinearity there exist a large num-ber of small amplitude periodic orbit with fixed period. Could one find small ampli-tude periodic orbits of large period and quasi-periodic solutions for the wave equa-tion?",
   "public_statement": "6. (Massimiliano Berti) For a nonlinear wave equation\n\nutt − uxx = f (u), f (0) = f ′(0) = 0 10\n\nwith Dirichlet boundary conditions and general nonlinearity there exist a large num-ber of small amplitude periodic orbit with fixed period. Could one find small ampli-tude periodic orbits of large period and quasi-periodic solutions for the wave equa-tion?",
   "evidence": "The canonical record is affected by OCR. The official AIM web rendering and the linked workshop PDF identify this as **Problem 36**, not Problem 6. The isolated `10` after the displayed formula is a page number, and the broken words “num-ber” and “ampli-tude” are line-end hyphenation. The recovered statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-pdes-notes.json",
   "source_index": 103,
   "attempt": 1
  },
  "AIM-PHYSICS-0001": {
   "statement_status": "exact",
   "original_statement": "Eigenvalues and Curves\n\nDo the eigenvalues of $a^w(x,hD)$ lie on or cluster around curves in the complex plane $\\mathbb{C}$?",
   "clean_statement": "Eigenvalues and Curves\n\nDo the eigenvalues of $a^w(x,hD)$ lie on or cluster around curves in the complex plane $\\mathbb{C}$?",
   "public_statement": "Eigenvalues and Curves\n\nDo the eigenvalues of $a^w(x,hD)$ lie on or cluster around curves in the complex plane $\\mathbb{C}$?",
   "evidence": "The canonical AIM record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-PHYSICS-0002": {
   "statement_status": "exact",
   "original_statement": "Bohr-Sommerfeld\n\nIs there a Bohr-Sommerfeld quantization rule for $a^w(x,hD)$? Is there a link to the usual Bohr-Sommerfeld rule for Hermitian operators?",
   "clean_statement": "Bohr-Sommerfeld\n\nIs there a Bohr-Sommerfeld quantization rule for $a^w(x,hD)$? Is there a link to the usual Bohr-Sommerfeld rule for Hermitian operators?",
   "public_statement": "Bohr-Sommerfeld\n\nIs there a Bohr-Sommerfeld quantization rule for $a^w(x,hD)$? Is there a link to the usual Bohr-Sommerfeld rule for Hermitian operators?",
   "evidence": "The exact corpus record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-PHYSICS-0003": {
   "statement_status": "exact",
   "original_statement": "A non-Hermitian Operator from Physics\n\nWhat is the distribution of eigenvalues of the non-Hermitian Hamiltonian\n\\[\nH = \\omega a^\\dagger a + \\chi (a^\\dagger a)^2 - i \\gamma a a^\\dagger + \\beta (a^\\dagger + a)?\n\\]\nHere $a$ and $a^\\dagger$ are the raising and lowering operators respectively and $\\omega$, $\\chi$, $\\gamma$, and $\\beta$ are real parameters.",
   "clean_statement": "A non-Hermitian Operator from Physics\n\nWhat is the distribution of eigenvalues of the non-Hermitian Hamiltonian\n\\[\nH = \\omega a^\\dagger a + \\chi (a^\\dagger a)^2 - i \\gamma a a^\\dagger + \\beta (a^\\dagger + a)?\n\\]\nHere $a$ and $a^\\dagger$ are the raising and lowering operators respectively and $\\omega$, $\\chi$, $\\gamma$, and $\\beta$ are real parameters.",
   "public_statement": "A non-Hermitian Operator from Physics\n\nWhat is the distribution of eigenvalues of the non-Hermitian Hamiltonian\n\\[\nH = \\omega a^\\dagger a + \\chi (a^\\dagger a)^2 - i \\gamma a a^\\dagger + \\beta (a^\\dagger + a)?\n\\]\nHere $a$ and $a^\\dagger$ are the raising and lowering operators respectively and $\\omega$, $\\chi$, $\\gamma$, and $\\beta$ are real parameters.",
   "evidence": "The canonical AIMPL record asks for the eigenvalue distribution of \\[ H=\\omega a^\\dagger a+\\chi(a^\\dagger a)^2-i\\gamma aa^\\dagger +\\beta(a^\\dagger+a), \\qquad \\omega,\\chi,\\gamma,\\beta\\in\\mathbb R. \\] With the standard bosonic commutation relation \\([a,a^\\dagger]=I\\) and number operator \\(N=a^\\dagger a\\), one has \\[ aa^\\dagger=N+1. \\] Therefore the exact canonical operator is \\[ \\boxed{H=\\chi N^2+(\\omega-i\\gamma)N-i\\gamma I+\\beta(a+a^\\dagger).} \\]",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-PHYSICS-0004": {
   "statement_status": "exact",
   "original_statement": "Hyper-K\\\"{a}hler Structure on $T^* M$\n\nLet $M$ be a K\\\"{a}hler manifold. Must $T^*M$ admit a hyper-K\\\"{a}hler structure?",
   "clean_statement": "Hyper-K\\\"{a}hler Structure on $T^* M$\n\nLet $M$ be a K\\\"{a}hler manifold. Must $T^*M$ admit a hyper-K\\\"{a}hler structure?",
   "public_statement": "Hyper-K\\\"{a}hler Structure on $T^* M$\n\nLet $M$ be a K\\\"{a}hler manifold. Must $T^*M$ admit a hyper-K\\\"{a}hler structure?",
   "evidence": "The canonical record is source index 3 of `aim-physics-notes.json`, from the AIM workshop *Non-Hermitian quantum mechanics and symplectic geometry*, section “Hyper-Kähler Structures,” problem 2.1. Its exact mathematical text is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-PHYSICS-0005": {
   "statement_status": "exact",
   "original_statement": "Evolution of Coherent States on SU(2)\n\nDetermine the quantum evolution in the semiclassical limit of initial coherent states on SU(2) with a Hamiltonian that is either non-Hermitian or nonlinear in the Lie algebra generators.",
   "clean_statement": "Evolution of Coherent States on SU(2)\n\nDetermine the quantum evolution in the semiclassical limit of initial coherent states on SU(2) with a Hamiltonian that is either non-Hermitian or nonlinear in the Lie algebra generators.",
   "public_statement": "Evolution of Coherent States on SU(2)\n\nDetermine the quantum evolution in the semiclassical limit of initial coherent states on SU(2) with a Hamiltonian that is either non-Hermitian or nonlinear in the Lie algebra generators.",
   "evidence": "The canonical AIM record states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-PHYSICS-0006": {
   "statement_status": "unrecoverable",
   "original_statement": "1. Yaniv Almog \n\n1.1. Completeness of eigenfunctions for Schr¨ odinger operators with complex potentials. For \n\nα > 0 consider Aα:= −d2/dx2 + i |x|α in R (or R+ with Dirichlet boundary condition at 0). If α > 2/3, then it is known that the eigenfunctions form a complete system. \n\nOpen problem: Is the same true for 0 < α ≤ 2/3? 1.2. Magnetic Schr¨ odinger operator. Consider \n\nA:= − ∂2\n\n∂x 2 −\n\n( ∂∂y − ix2\n\n2\n\n)2\n\n+ i cy, D(A):= H10 (R2+) ∩ { u: Au ∈ L2(R2+)}\n\nwhere R2+ = {(x, y ) ∈ R2: y > 0}.\n\nOpen problem: Is σ(A) 6 = ∅?It is known that σ(A) 6 = ∅ if |c| << 1 or |c| >> 1. 2. Lyonell Boulton \n\n2.1. Schauder bases of periodic functions and multipliers. Let en(x):= √2 sin( nπx ). Then {en}\n\nis a Schauder basis of Lp(0, 1) for all p > 1. Let f ∈ C(R, C) satisfy f (x + 2) = f (x), f (−x) = −f (x), \n\nf (1 /2 + x) = f (1 /2 − x) and define fn(x):= f (nx ). Let A: Lp(0, 1) → Lp(0, 1) be the linear extension of the map Ae n = fn. Then {fn} is a Schauder basis of Lp(0, 1) if and only if A: Lp(0, 1) −→ Lp(0, 1) is a bounded operator with a bounded inverse. Let {ck} be the Fourier coefficients of f. Then A can be written as A = ∑ \n\n> k\n\nckMk where Mk are the linear extensions of the map Mken = ekn.\n\nOpen problem: Find necessary and sufficient conditions on {ck} for 0 /∈ σ(A) whenever p 6 = 2. 3. Amin Boumenir \n\n3.1. Non-self-adjoint inverse problems. We are interested in identifying a non-self-adjoint operator associated with an evolution equation (parabolic or hyperbolic) through \"observations\" of the solution as time evolves. Thus for example in a certain Hilbert space we have \n\nu′(t) = Au (t) and u(0) = f (1) where, for simplicity, we assume that \n\nA = L + B\n\nwith L is a given (known) self-adjoint operator with \"nice properties\" while B is an unknown non-self-adjoint perturbation. For example Ay (x) = y′′ (x) − q(x)y(x) or Au = ∆ u − q(x)u with Im q(x) 6 = 0. We assume that we can observe the solution through a functional 〈·, g 〉 say \n\nω(t) = 〈u(t), g 〉.\n\nFor example if u(x, t ) is the solution of a heat equation, where x ∈ Ω ⊂ Rn, and p ∈ ∂Ω, then ω(t) = u(p, t )(temperature) or ω(t) = ∂nu (p, t ) (heat transfer) are usual observations/readings of the solution on the boundary. Thus we want to recover A or at least its spectrum σA = {λn} ⊂ C from the observation mapping \n\nu(0) → ω(t).\n\n> Date: June 8 - 12, 2015, American Institute of Mathematics, San Jose, California.\n> 1\n\nTo do so, although we do NOT know A, we assume that it has a discrete spectrum {λn} ⊂ C, and in general Im λn → 0 as n → ∞, while Re λn → −∞. If we denote its eigenfunctions by ϕn, 0 and its associated eigenfunctions (roots) by ϕn,ν for ν = 1,..., m n − 1, where mn is the multiplicity of the eigenvalue λn, then we can write a formal solution to the evolution equation \n\nu (t) = ∑\n\n> n≥1\n\neλntmn−1∑\n\n> ν=0\n\ncnν (f ) pnν (t)ϕnν (2) where the Fourier coefficients are cnν (f ) = 〈f, ψ nν 〉 and {ψnν } is the biorthogonal system to {ϕn,ν }. Here \n\npnν are polynomials generated by the multiplicity of the eigenvalue λn. The observation then is given by \n\nω(t) = ∑\n\n> n≥1\n\neλntmn−1∑\n\n> ν=0\n\ncnν (f ) pnν (t)〈ϕnν, g 〉. (3) In the best case, when all cnν (f ) 6 = 0 and 〈ϕnν, g 〉 6 = 0 then it is possible to evaluate/extract all the λn from the observation (2). \n\nOpen problems: i) How do you choose the initial condition f, so we can observe all eλnt, that is all cnν (f ) 6 = 0? We need to know something about the biorthogonal system {ψnν }.ii) How do you choose the observation g so all 〈ϕnν, g 〉 6 = 0? We need to know something about the root functions {ϕn,ν }.iii) How smooth is the sum (2), so we can choose g? We need some information on the type of convergence in (2) so (3) holds. iv) How do we extract the λn and their multiplicity from a given signal given by (3) in finite time? When \n\nλn are complex values and the sum contains polynomials in t, it is much harder than the real case. v) Find the best f and g that allow the identification of A by using the smallest number of observations. Evolution equations are often found in control theory, and for that purpose, we need finite number of observations done in finite time. 4. Marina Chugunova \n\n4.1. Computations of the instability index for a non-self-adjoint operators. The stability of steady states is a basic question about the dynamics of any partial differential equation that models the evolution of a physical system. In order to numerically evaluate the instability index of a given differential operator A, its computation should be reduced to a problem of linear algebra. Particularly for problems with periodic boundary condi-tions, it seems natural to restrict the operator A to a finite-dimensional space of trigonometric polynomials. \n\nOpen problem: Under what conditions the instability index (the total number of unstable eigenvalues) can be computed from the resulting finite dimensional matrix? One difficulty is that the entries of the infinite matrix corresponding to the differential operator A grow with the row and column index, so that any truncation is not a small perturbation. If A is a self-adjoint semi-bounded differential operator of even order, then the instability index can be estimated by variational methods, or computed directly from the zeros of the corresponding Evans function. Understanding the spectrum of a non-self-adjoint operator is a much harder problem. It is not at all obvious how to restrict the computation of its instability index to a finite-dimensional subspace, or how to even estimate its dimension. Furthermore, the numerical calculation of eigenvalues can be extremely ill-conditioned even in finite dimensions. 5. Michael Demuth \n\n5.1. Spectral radius and operator norm. Let A be a bounded linear operator on a Banach space X. Its spectral radius is defined by spr( A):= max {| z|: z ∈ σ(A)}.\n\nGelfand proved the classical formula spr( A) = lim \n\n> n→∞\n\n‖An‖ 1 \n\n> n.\n\n> 2\n\nObviously, 0 ≤ spr( A) ≤ ‖ A‖. The question arises: What is the gap between ‖A‖ and spr( A)? Introduce the denotation gap( A):= ‖A‖ − spr( A).\n\nOpen problems: i) For which class of operators holds gap( A) > 0 or gap( A) = 0,\n\nrespectively. ii) What is the smallest m ∈ (0, 1], such that spr( A) ≤ m‖A‖\n\nor gap( A) ≥ (1 − m)‖A‖?\n\nExample 1. Let X = `1(N) and A be the weighted shift-operator defined according to the canonical standard basis by the infinite matrix \n\n\n\n0\n\nb1 0\n\nb2 0\n\nb1 0\n\nb2 0......\n\n\n\nwhere b1, b 2 > 0 and b1b2 = 1. In this case ‖A‖ = max {b1, b 2} and σ(A) = {z ∈ C: |z| ≤ 1} and therefore spr( A) = 1. Thus \n\n• gap( A) = 0: If b1 = b2 = 1 then ‖A‖ = spr( A). \n\n• gap( A) > 0: If b1 6 = b2 then ‖A‖ > spr( A). This kind of estimates are useful in the following situation. Let K be a compact perturbation of A. Study the discrete spectrum of B:= A + K. We are able to analyze the moments and the number of eigenvalues of B outside a ball of radius ‖A‖. It is more interesting and also natural to enlarge this region up to the complement of a ball with radius spr( A). 6. Mark Embree \n\n6.1. Davies' conjecture about approximate diagonalization. Consider a non-normal matrix A ∈\n\nCn×n. Define \n\ns(A, ε ):= inf \n\n> ∆,V V−1(A+∆) Vdiagonal\n\n‖V ‖ ‖ V −1‖ε + ‖∆‖.\n\nOpen problem: Prove Davies' conjecture (2007): There exists a constant Cn > 0, independent of A ∈ Cn×n,such that s(A, ε ) ≤ Cn\n\n√ε.It is known that the conjecture holds for Jordan blocks (then Cn = 2 suffices) and for 3 × 3 matrices with \n\n‖A‖ ≤ 1 (then Cn = 4 suffices). 6.2. Crouzeix' conjecture about the norm of matrix functions. Let A be a bounded linear operator. It is known that ‖Ak‖ ≤ 2 max z∈W (A) |zk|; W (A) denotes the numerical range of A.\n\nOpen problem: Prove Crouzeix' conjecture: There exists a constant C ≥ 2 such that for all analytic functions \n\nf: W (A) → C holds ‖f (A)‖ ≤ C max z∈W (A) |f (z)|.Crouzeix conjectured further that C ≤ 11.08. \n\n> 3",
   "clean_statement": null,
   "public_statement": "1. Yaniv Almog\n\n1.1. Completeness of eigenfunctions for Schr¨ odinger operators with complex potentials. For\n\nα > 0 consider Aα:= −d2/dx2 + i |x|α in R (or R+ with Dirichlet boundary condition at 0). If α > 2/3, then it is known that the eigenfunctions form a complete system.\n\nOpen problem: Is the same true for 0 < α ≤ 2/3? 1.2. Magnetic Schr¨ odinger operator. Consider\n\nA:= − ∂2\n\n∂x 2 −\n\n( ∂∂y − ix2\n\n2\n\n)2\n\n+ i cy, D(A):= H10 (R2+) ∩ { u: Au ∈ L2(R2+)}\n\nwhere R2+ = {(x, y ) ∈ R2: y > 0}.\n\nOpen problem: Is σ(A) 6 = ∅?It is known that σ(A) 6 = ∅ if |c| << 1 or |c| >> 1. 2. Lyonell Boulton\n\n2.1. Schauder bases of periodic functions and multipliers. Let en(x):= √2 sin( nπx ). Then {en}\n\nis a Schauder basis of Lp(0, 1) for all p > 1. Let f ∈ C(R, C) satisfy f (x + 2) = f (x), f (−x) = −f (x),\n\nf (1 /2 + x) = f (1 /2 − x) and define fn(x):= f (nx ). Let A: Lp(0, 1) → Lp(0, 1) be the linear extension of the map Ae n = fn. Then {fn} is a Schauder basis of Lp(0, 1) if and only if A: Lp(0, 1) −→ Lp(0, 1) is a bounded operator with a bounded inverse. Let {ck} be the Fourier coefficients of f. Then A can be written as A = ∑\n\n> k\n\nckMk where Mk are the linear extensions of the map Mken = ekn.\n\nOpen problem: Find necessary and sufficient conditions on {ck} for 0 /∈ σ(A) whenever p 6 = 2. 3. Amin Boumenir\n\n3.1. Non-self-adjoint inverse problems. We are interested in identifying a non-self-adjoint operator associated with an evolution equation (parabolic or hyperbolic) through \"observations\" of the solution as time evolves. Thus for example in a certain Hilbert space we have\n\nu′(t) = Au (t) and u(0) = f (1) where, for simplicity, we assume that\n\nA = L + B\n\nwith L is a given (known) self-adjoint operator with \"nice properties\" while B is an unknown non-self-adjoint perturbation. For example Ay (x) = y′′ (x) − q(x)y(x) or Au = ∆ u − q(x)u with Im q(x) 6 = 0. We assume that we can observe the solution through a functional 〈·, g 〉 say\n\nω(t) = 〈u(t), g 〉.\n\nFor example if u(x, t ) is the solution of a heat equation, where x ∈ Ω ⊂ Rn, and p ∈ ∂Ω, then ω(t) = u(p, t )(temperature) or ω(t) = ∂nu (p, t ) (heat transfer) are usual observations/readings of the solution on the boundary. Thus we want to recover A or at least its spectrum σA = {λn} ⊂ C from the observation mapping\n\nu(0) → ω(t).\n\n> Date: June 8 - 12, 2015, American Institute of Mathematics, San Jose, California.\n> 1\n\nTo do so, although we do NOT know A, we assume that it has a discrete spectrum {λn} ⊂ C, and in general Im λn → 0 as n → ∞, while Re λn → −∞. If we denote its eigenfunctions by ϕn, 0 and its associated eigenfunctions (roots) by ϕn,ν for ν = 1,..., m n − 1, where mn is the multiplicity of the eigenvalue λn, then we can write a formal solution to the evolution equation\n\nu (t) = ∑\n\n> n≥1\n\neλntmn−1∑\n\n> ν=0\n\ncnν (f ) pnν (t)ϕnν (2) where the Fourier coefficients are cnν (f ) = 〈f, ψ nν 〉 and {ψnν } is the biorthogonal system to {ϕn,ν }. Here\n\npnν are polynomials generated by the multiplicity of the eigenvalue λn. The observation then is given by\n\nω(t) = ∑\n\n> n≥1\n\neλntmn−1∑\n\n> ν=0\n\ncnν (f ) pnν (t)〈ϕnν, g 〉. (3) In the best case, when all cnν (f ) 6 = 0 and 〈ϕnν, g 〉 6 = 0 then it is possible to evaluate/extract all the λn from the observation (2).\n\nOpen problems: i) How do you choose the initial condition f, so we can observe all eλnt, that is all cnν (f ) 6 = 0? We need to know something about the biorthogonal system {ψnν }.ii) How do you choose the observation g so all 〈ϕnν, g 〉 6 = 0? We need to know something about the root functions {ϕn,ν }.iii) How smooth is the sum (2), so we can choose g? We need some information on the type of convergence in (2) so (3) holds. iv) How do we extract the λn and their multiplicity from a given signal given by (3) in finite time? When\n\nλn are complex values and the sum contains polynomials in t, it is much harder than the real case. v) Find the best f and g that allow the identification of A by using the smallest number of observations. Evolution equations are often found in control theory, and for that purpose, we need finite number of observations done in finite time. 4. Marina Chugunova\n\n4.1. Computations of the instability index for a non-self-adjoint operators. The stability of steady states is a basic question about the dynamics of any partial differential equation that models the evolution of a physical system. In order to numerically evaluate the instability index of a given differential operator A, its computation should be reduced to a problem of linear algebra. Particularly for problems with periodic boundary condi-tions, it seems natural to restrict the operator A to a finite-dimensional space of trigonometric polynomials.\n\nOpen problem: Under what conditions the instability index (the total number of unstable eigenvalues) can be computed from the resulting finite dimensional matrix? One difficulty is that the entries of the infinite matrix corresponding to the differential operator A grow with the row and column index, so that any truncation is not a small perturbation. If A is a self-adjoint semi-bounded differential operator of even order, then the instability index can be estimated by variational methods, or computed directly from the zeros of the corresponding Evans function. Understanding the spectrum of a non-self-adjoint operator is a much harder problem. It is not at all obvious how to restrict the computation of its instability index to a finite-dimensional subspace, or how to even estimate its dimension. Furthermore, the numerical calculation of eigenvalues can be extremely ill-conditioned even in finite dimensions. 5. Michael Demuth\n\n5.1. Spectral radius and operator norm. Let A be a bounded linear operator on a Banach space X. Its spectral radius is defined by spr( A):= max {| z|: z ∈ σ(A)}.\n\nGelfand proved the classical formula spr( A) = lim\n\n> n→∞\n\n‖An‖ 1\n\n> n.\n\n> 2\n\nObviously, 0 ≤ spr( A) ≤ ‖ A‖. The question arises: What is the gap between ‖A‖ and spr( A)? Introduce the denotation gap( A):= ‖A‖ − spr( A).\n\nOpen problems: i) For which class of operators holds gap( A) > 0 or gap( A) = 0,\n\nrespectively. ii) What is the smallest m ∈ (0, 1], such that spr( A) ≤ m‖A‖\n\nor gap( A) ≥ (1 − m)‖A‖?\n\nExample 1. Let X = `1(N) and A be the weighted shift-operator defined according to the canonical standard basis by the infinite matrix\n\n\n\n0\n\nb1 0\n\nb2 0\n\nb1 0\n\nb2 0......\n\n\n\nwhere b1, b 2 > 0 and b1b2 = 1. In this case ‖A‖ = max {b1, b 2} and σ(A) = {z ∈ C: |z| ≤ 1} and therefore spr( A) = 1. Thus\n\n• gap( A) = 0: If b1 = b2 = 1 then ‖A‖ = spr( A).\n\n• gap( A) > 0: If b1 6 = b2 then ‖A‖ > spr( A). This kind of estimates are useful in the following situation. Let K be a compact perturbation of A. Study the discrete spectrum of B:= A + K. We are able to analyze the moments and the number of eigenvalues of B outside a ball of radius ‖A‖. It is more interesting and also natural to enlarge this region up to the complement of a ball with radius spr( A). 6. Mark Embree\n\n6.1. Davies' conjecture about approximate diagonalization. Consider a non-normal matrix A ∈\n\nCn×n. Define\n\ns(A, ε ):= inf\n\n> ∆,V V−1(A+∆) Vdiagonal\n\n‖V ‖ ‖ V −1‖ε + ‖∆‖.\n\nOpen problem: Prove Davies' conjecture (2007): There exists a constant Cn > 0, independent of A ∈ Cn×n,such that s(A, ε ) ≤ Cn\n\n√ε.It is known that the conjecture holds for Jordan blocks (then Cn = 2 suffices) and for 3 × 3 matrices with\n\n‖A‖ ≤ 1 (then Cn = 4 suffices). 6.2. Crouzeix' conjecture about the norm of matrix functions. Let A be a bounded linear operator. It is known that ‖Ak‖ ≤ 2 max z∈W (A) |zk|; W (A) denotes the numerical range of A.\n\nOpen problem: Prove Crouzeix' conjecture: There exists a constant C ≥ 2 such that for all analytic functions\n\nf: W (A) → C holds ‖f (A)‖ ≤ C max z∈W (A) |f (z)|.Crouzeix conjectured further that C ≤ 11.08.\n\n> 3",
   "evidence": "The canonical JSON record is not one mathematical problem. It is an OCR extraction of the first three pages of the 2015 AIM workshop list *Mathematical aspects of physics with non-self-adjoint operators*. It starts with Yaniv Almog's item 1, but then runs through items contributed by five other participants. The exact OCR text is preserved in `input.json`; it is not silently rewritten here.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-physics-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-PHYSICS-0007": {
   "statement_status": "exact",
   "original_statement": "7. Rupert L. Frank \n\n7.1. Laptev-Safronov conjecture. Consider a Schr¨ odinger operator −∆ + V in Rd with a complex po-tential V.\n\nOpen problems: i) What is the largest p such that all non-real eigenvalues lie in a disk around 0 of radius \n\nD(∫ \n\n> Rd\n\n|V |p dx)(p−d/ 2) −1\n\n(the constant D > 0 shall not depend on V )? ii) What happens to embedded eigenvalues of self-adjoint Schr¨ odinger operators under non-self-adjoint perturbations? Both problems are related to the Laptev-Safronov conjecture which states that for every d ∈ N and 0 < γ ≤ d/ 2 there exists a constant Dγ,d > 0 such that for every potential V every non-real eigenvalue λ\n\nsatisfies \n\n|λ|γ ≤ Dγ,d \n\n∫\n\n> Rd\n\n|V |γ+ d \n\n> 2\n\ndx;here γ = p − d/ 2 with p from problem i). The Laptev-Safronov conjecture is known to be true in dimension \n\nd = 1 if γ = 1 /2 and in dimension d ≥ 2 if 0 < γ ≤ 1/2. 8. Marcel Hansmann \n\n8.1. Tensor trick for perturbed operators. Let A be a bounded self-adjoint operator in a Hilbert space and let K be a perturbation which is of trace class. Assume that for any ε > 0 there exists a constant \n\nC(ε) > 0 such that ∑\n\n> λ∈σd(A+K)\n\ndist( λ, σ (A)) 1+ ε ≤ C(ε)‖K‖1+ ε\n\n> 1+ ε.\n\nOpen problem: Does it follow that there exists C > 0 such that \n\n∑\n\n> λ∈σd(A+K)\n\ndist( λ, σ (A)) ≤ C‖K‖1?9. Michael Hitrik \n\n9.1. Upper bounds on the norm of the resolvent. It is well known that the spectrum of a non-self-adjoint operator does not control its resolvent and that the latter may become very large far from the spectrum. Some general upper bounds on resolvents are provided by the abstract operator theory, and restricting the attention to the setting of semiclassical operators on Rn, let us give a rough statement of such bounds. Assume that P = pw(x, hD x) is the semiclassical Weyl quantization on Rn of a nice symbol p\n\nwith Re p ≥ 0, say. Then the norm of the resolvent of P is bounded from above by a quantity of the form \n\nO(1) exp ( O(1) h−n), provided that z ∈ neigh(0, C) is not too close to the spectrum of P. On the other hand, the available lower bounds on the resolvent of P, in the interior of the range of the symbol, coming from the pseudospectral considerations, are typically of the form C−1 \n\n> N\n\nh−N, N ∈ N, or (1 /C )e1/(Ch ), provided that p enjoys some analyticity properties, [DSZ04]. There appears to be therefore a substantial gap between the available upper and lower bounds on the resolvent, especially when n ≥ 2, which, to the best of my knowledge, has so far only been bridged in the very special case of elliptic quadratic differential operators, see [HSV13]. \n\nOpen problem: Is the upper bound sharp (especially for dimension n ≥ 2)? 10. David Krejˇ ciˇ r´ ık \n\n10.1. Semiclassical pseudomodes of Schr¨ odinger operators with discontinuous potentials. For smooth potentials there exists a quite general theory on the construction of semiclassical pseudomodes, see [DSZ04]. \n\nOpen problem: Can the technique be adapted to discontinuous potentials? In my joint paper with Henry [HK15] we have a non-trivial pseudospectrum in a toy model (complex Heaviside-type potential). However, our technique is restricted to the particular situation and the non-trivial pseudospectrum is rather generated by the behavior of the potential at infinity. \n\n> 4",
   "clean_statement": "7. Rupert L. Frank\n\n7.1. Laptev-Safronov conjecture. Consider a Schr¨ odinger operator −∆ + V in Rd with a complex po-tential V.\n\nOpen problems: i) What is the largest p such that all non-real eigenvalues lie in a disk around 0 of radius\n\nD(∫\n\n> Rd\n\n|V |p dx)(p−d/ 2) −1\n\n(the constant D > 0 shall not depend on V )? ii) What happens to embedded eigenvalues of self-adjoint Schr¨ odinger operators under non-self-adjoint perturbations? Both problems are related to the Laptev-Safronov conjecture which states that for every d ∈ N and 0 < γ ≤ d/ 2 there exists a constant Dγ,d > 0 such that for every potential V every non-real eigenvalue λ\n\nsatisfies\n\n|λ|γ ≤ Dγ,d\n\n∫\n\n> Rd\n\n|V |γ+ d\n\n> 2\n\ndx;here γ = p − d/ 2 with p from problem i). The Laptev-Safronov conjecture is known to be true in dimension\n\nd = 1 if γ = 1 /2 and in dimension d ≥ 2 if 0 < γ ≤ 1/2. 8. Marcel Hansmann\n\n8.1. Tensor trick for perturbed operators. Let A be a bounded self-adjoint operator in a Hilbert space and let K be a perturbation which is of trace class. Assume that for any ε > 0 there exists a constant\n\nC(ε) > 0 such that ∑\n\n> λ∈σd(A+K)\n\ndist( λ, σ (A)) 1+ ε ≤ C(ε)‖K‖1+ ε\n\n> 1+ ε.\n\nOpen problem: Does it follow that there exists C > 0 such that\n\n∑\n\n> λ∈σd(A+K)\n\ndist( λ, σ (A)) ≤ C‖K‖1?9. Michael Hitrik\n\n9.1. Upper bounds on the norm of the resolvent. It is well known that the spectrum of a non-self-adjoint operator does not control its resolvent and that the latter may become very large far from the spectrum. Some general upper bounds on resolvents are provided by the abstract operator theory, and restricting the attention to the setting of semiclassical operators on Rn, let us give a rough statement of such bounds. Assume that P = pw(x, hD x) is the semiclassical Weyl quantization on Rn of a nice symbol p\n\nwith Re p ≥ 0, say. Then the norm of the resolvent of P is bounded from above by a quantity of the form\n\nO(1) exp ( O(1) h−n), provided that z ∈ neigh(0, C) is not too close to the spectrum of P. On the other hand, the available lower bounds on the resolvent of P, in the interior of the range of the symbol, coming from the pseudospectral considerations, are typically of the form C−1\n\n> N\n\nh−N, N ∈ N, or (1 /C )e1/(Ch ), provided that p enjoys some analyticity properties, [DSZ04]. There appears to be therefore a substantial gap between the available upper and lower bounds on the resolvent, especially when n ≥ 2, which, to the best of my knowledge, has so far only been bridged in the very special case of elliptic quadratic differential operators, see [HSV13].\n\nOpen problem: Is the upper bound sharp (especially for dimension n ≥ 2)? 10. David Krejˇ ciˇ r´ ık\n\n10.1. Semiclassical pseudomodes of Schr¨ odinger operators with discontinuous potentials. For smooth potentials there exists a quite general theory on the construction of semiclassical pseudomodes, see [DSZ04].\n\nOpen problem: Can the technique be adapted to discontinuous potentials? In my joint paper with Henry [HK15] we have a non-trivial pseudospectrum in a toy model (complex Heaviside-type potential). However, our technique is restricted to the particular situation and the non-trivial pseudospectrum is rather generated by the behavior of the potential at infinity.\n\n> 4",
   "public_statement": "7. Rupert L. Frank\n\n7.1. Laptev-Safronov conjecture. Consider a Schr¨ odinger operator −∆ + V in Rd with a complex po-tential V.\n\nOpen problems: i) What is the largest p such that all non-real eigenvalues lie in a disk around 0 of radius\n\nD(∫\n\n> Rd\n\n|V |p dx)(p−d/ 2) −1\n\n(the constant D > 0 shall not depend on V )? ii) What happens to embedded eigenvalues of self-adjoint Schr¨ odinger operators under non-self-adjoint perturbations? Both problems are related to the Laptev-Safronov conjecture which states that for every d ∈ N and 0 < γ ≤ d/ 2 there exists a constant Dγ,d > 0 such that for every potential V every non-real eigenvalue λ\n\nsatisfies\n\n|λ|γ ≤ Dγ,d\n\n∫\n\n> Rd\n\n|V |γ+ d\n\n> 2\n\ndx;here γ = p − d/ 2 with p from problem i). The Laptev-Safronov conjecture is known to be true in dimension\n\nd = 1 if γ = 1 /2 and in dimension d ≥ 2 if 0 < γ ≤ 1/2. 8. Marcel Hansmann\n\n8.1. Tensor trick for perturbed operators. Let A be a bounded self-adjoint operator in a Hilbert space and let K be a perturbation which is of trace class. Assume that for any ε > 0 there exists a constant\n\nC(ε) > 0 such that ∑\n\n> λ∈σd(A+K)\n\ndist( λ, σ (A)) 1+ ε ≤ C(ε)‖K‖1+ ε\n\n> 1+ ε.\n\nOpen problem: Does it follow that there exists C > 0 such that\n\n∑\n\n> λ∈σd(A+K)\n\ndist( λ, σ (A)) ≤ C‖K‖1?9. Michael Hitrik\n\n9.1. Upper bounds on the norm of the resolvent. It is well known that the spectrum of a non-self-adjoint operator does not control its resolvent and that the latter may become very large far from the spectrum. Some general upper bounds on resolvents are provided by the abstract operator theory, and restricting the attention to the setting of semiclassical operators on Rn, let us give a rough statement of such bounds. Assume that P = pw(x, hD x) is the semiclassical Weyl quantization on Rn of a nice symbol p\n\nwith Re p ≥ 0, say. Then the norm of the resolvent of P is bounded from above by a quantity of the form\n\nO(1) exp ( O(1) h−n), provided that z ∈ neigh(0, C) is not too close to the spectrum of P. On the other hand, the available lower bounds on the resolvent of P, in the interior of the range of the symbol, coming from the pseudospectral considerations, are typically of the form C−1\n\n> N\n\nh−N, N ∈ N, or (1 /C )e1/(Ch ), provided that p enjoys some analyticity properties, [DSZ04]. There appears to be therefore a substantial gap between the available upper and lower bounds on the resolvent, especially when n ≥ 2, which, to the best of my knowledge, has so far only been bridged in the very special case of elliptic quadratic differential operators, see [HSV13].\n\nOpen problem: Is the upper bound sharp (especially for dimension n ≥ 2)? 10. David Krejˇ ciˇ r´ ık\n\n10.1. Semiclassical pseudomodes of Schr¨ odinger operators with discontinuous potentials. For smooth potentials there exists a quite general theory on the construction of semiclassical pseudomodes, see [DSZ04].\n\nOpen problem: Can the technique be adapted to discontinuous potentials? In my joint paper with Henry [HK15] we have a non-trivial pseudospectrum in a toy model (complex Heaviside-type potential). However, our technique is restricted to the particular situation and the non-trivial pseudospectrum is rather generated by the behavior of the potential at infinity.\n\n> 4",
   "evidence": "The canonical record is visibly corrupted by page-level OCR: it begins with Rupert L. Frank's item 7.1 and then appends the complete items 8.1 (Marcel Hansmann), 9.1 (Michael Hitrik), and 10.1 (David Krejčiřík). The record key is `number: 7`, so this attempt treats only item 7.1. The later items are extraction spillover, not additional assignments.",
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  "AIM-PHYSICS-0008": {
   "statement_status": "exact",
   "original_statement": "11. Michael Levitin \n\n11.1. Complex eigenvalues of indefinite pencil. For 0 < c < 2 define the 2 n × 2n matrices \n\nA:= \n\n\n\nc 11 c 11 c......... 11 c\n\n, B:= \n\n\n\n1...1\n\n−1...\n\n−1\n\n\n\nwhere the numbers of 1 and −1 in B coincide (equal to n). The eigenvalues of the pencil λ 7 → A − λB are the eigenvalues of B−1A = BA. The spectrum is symmetric with respect to both R and i R. The non-real eigenvalues are contained in the union of the two closed disks {λ ∈ C: |λ ± c| ≤ 2} whereas numerical examples suggest that they lie in their intersection. \n\nOpen problem: Prove that |λ − c| ≤ 2 and |λ + c| ≤ 2 holds for all λ ∈ σ(BA )\\R.11.2. Indefinite Sturm-Liouville. Recently, we have proved some conjectures related to the generalized eigenvalue problem ( − d2 \n\n> dx 2\n\n+ c \n\n> 1+ |x|\n\n)ψ = λsgn( x)ψ.\n\nOpen problem: Replace the potential term by a more general function. 12. Marco Marletta \n\n12.1. Zeros and poles of Nevanlinna functions. Let m1, m 2,... be an infinite sequence of meromorphic functions with Im mj (λ) > 0 if Im λ > 0 and Im mj (λ) < 0 if Im λ < 0 (Nevanlinna functions). Suppose that there exists a non-empty interval I ⊆ R such that for every non-empty subinterval J ⊆ I holds lim \n\n> j→∞\n\n#{pole of mj in J} = ∞.\n\nLet g be a function which is analytic in a complex neighborhood of I.\n\nOpen problems: i) Show that for every open complex neighborhood U of J,lim \n\n> j→∞\n\n#{zero of ( mj − g) in U } = ∞.\n\nii) Show that there exists a constant C > 0, independent of j, such that for every open complex neighborhood \n\nU of J, ∣∣#{zero of ( mj − g) in U } − #{pole of mj in U }∣∣ ≤ C. \n\nThe result is known to hold if μ(J):= lim j→∞ j−1 #{pole of mj in J} exists. 13. Boris Mityagin \n\n13.1. Schauder basis for Schr¨ odinger operators with periodic boundary conditions. Consider the Schr¨ odinger operator −d2/dx2 + V on (0, π ) with periodic boundary conditions, where the potential V is a trigonometric polynomial, i.e., \n\nV (x) = \n\n> m\n\n∑\n\n> k=−m\n\nvke2ikx; vk ∈ C, |k| ≤ m. \n\nOpen Problem: For which sets of coefficients {vk}mk=−m do the eigenfunctions form a Schauder basis for \n\nL2(0, π )? Known Case: Let V (x):= e−2ix + be 2ix. Then the answer is \"yes\" if and only if |b| = 1. \n\n> 5",
   "clean_statement": "11. Michael Levitin\n\n11.1. Complex eigenvalues of indefinite pencil. For 0 < c < 2 define the 2 n × 2n matrices\n\nA:=\n\n\n\nc 11 c 11 c......... 11 c\n\n, B:=\n\n\n\n1...1\n\n−1...\n\n−1\n\n\n\nwhere the numbers of 1 and −1 in B coincide (equal to n). The eigenvalues of the pencil λ 7 → A − λB are the eigenvalues of B−1A = BA. The spectrum is symmetric with respect to both R and i R. The non-real eigenvalues are contained in the union of the two closed disks {λ ∈ C: |λ ± c| ≤ 2} whereas numerical examples suggest that they lie in their intersection.\n\nOpen problem: Prove that |λ − c| ≤ 2 and |λ + c| ≤ 2 holds for all λ ∈ σ(BA )\\R.11.2. Indefinite Sturm-Liouville. Recently, we have proved some conjectures related to the generalized eigenvalue problem ( − d2\n\n> dx 2\n\n+ c\n\n> 1+ |x|\n\n)ψ = λsgn( x)ψ.\n\nOpen problem: Replace the potential term by a more general function. 12. Marco Marletta\n\n12.1. Zeros and poles of Nevanlinna functions. Let m1, m 2,... be an infinite sequence of meromorphic functions with Im mj (λ) > 0 if Im λ > 0 and Im mj (λ) < 0 if Im λ < 0 (Nevanlinna functions). Suppose that there exists a non-empty interval I ⊆ R such that for every non-empty subinterval J ⊆ I holds lim\n\n> j→∞\n\n#{pole of mj in J} = ∞.\n\nLet g be a function which is analytic in a complex neighborhood of I.\n\nOpen problems: i) Show that for every open complex neighborhood U of J,lim\n\n> j→∞\n\n#{zero of ( mj − g) in U } = ∞.\n\nii) Show that there exists a constant C > 0, independent of j, such that for every open complex neighborhood\n\nU of J, ∣∣#{zero of ( mj − g) in U } − #{pole of mj in U }∣∣ ≤ C.\n\nThe result is known to hold if μ(J):= lim j→∞ j−1 #{pole of mj in J} exists. 13. Boris Mityagin\n\n13.1. Schauder basis for Schr¨ odinger operators with periodic boundary conditions. Consider the Schr¨ odinger operator −d2/dx2 + V on (0, π ) with periodic boundary conditions, where the potential V is a trigonometric polynomial, i.e.,\n\nV (x) =\n\n> m\n\n∑\n\n> k=−m\n\nvke2ikx; vk ∈ C, |k| ≤ m.\n\nOpen Problem: For which sets of coefficients {vk}mk=−m do the eigenfunctions form a Schauder basis for\n\nL2(0, π )? Known Case: Let V (x):= e−2ix + be 2ix. Then the answer is \"yes\" if and only if |b| = 1.\n\n> 5",
   "public_statement": "11. Michael Levitin\n\n11.1. Complex eigenvalues of indefinite pencil. For 0 < c < 2 define the 2 n × 2n matrices\n\nA:=\n\n\n\nc 11 c 11 c......... 11 c\n\n, B:=\n\n\n\n1...1\n\n−1...\n\n−1\n\n\n\nwhere the numbers of 1 and −1 in B coincide (equal to n). The eigenvalues of the pencil λ 7 → A − λB are the eigenvalues of B−1A = BA. The spectrum is symmetric with respect to both R and i R. The non-real eigenvalues are contained in the union of the two closed disks {λ ∈ C: |λ ± c| ≤ 2} whereas numerical examples suggest that they lie in their intersection.\n\nOpen problem: Prove that |λ − c| ≤ 2 and |λ + c| ≤ 2 holds for all λ ∈ σ(BA )\\R.11.2. Indefinite Sturm-Liouville. Recently, we have proved some conjectures related to the generalized eigenvalue problem ( − d2\n\n> dx 2\n\n+ c\n\n> 1+ |x|\n\n)ψ = λsgn( x)ψ.\n\nOpen problem: Replace the potential term by a more general function. 12. Marco Marletta\n\n12.1. Zeros and poles of Nevanlinna functions. Let m1, m 2,... be an infinite sequence of meromorphic functions with Im mj (λ) > 0 if Im λ > 0 and Im mj (λ) < 0 if Im λ < 0 (Nevanlinna functions). Suppose that there exists a non-empty interval I ⊆ R such that for every non-empty subinterval J ⊆ I holds lim\n\n> j→∞\n\n#{pole of mj in J} = ∞.\n\nLet g be a function which is analytic in a complex neighborhood of I.\n\nOpen problems: i) Show that for every open complex neighborhood U of J,lim\n\n> j→∞\n\n#{zero of ( mj − g) in U } = ∞.\n\nii) Show that there exists a constant C > 0, independent of j, such that for every open complex neighborhood\n\nU of J, ∣∣#{zero of ( mj − g) in U } − #{pole of mj in U }∣∣ ≤ C.\n\nThe result is known to hold if μ(J):= lim j→∞ j−1 #{pole of mj in J} exists. 13. Boris Mityagin\n\n13.1. Schauder basis for Schr¨ odinger operators with periodic boundary conditions. Consider the Schr¨ odinger operator −d2/dx2 + V on (0, π ) with periodic boundary conditions, where the potential V is a trigonometric polynomial, i.e.,\n\nV (x) =\n\n> m\n\n∑\n\n> k=−m\n\nvke2ikx; vk ∈ C, |k| ≤ m.\n\nOpen Problem: For which sets of coefficients {vk}mk=−m do the eigenfunctions form a Schauder basis for\n\nL2(0, π )? Known Case: Let V (x):= e−2ix + be 2ix. Then the answer is \"yes\" if and only if |b| = 1.\n\n> 5",
   "evidence": "The canonical record is zero-based source index 7 of `aim-physics-notes.json`. It is a composite OCR extraction from page 5 of the AIM PDF *List of Open Problems: Mathematical Aspects of Physics with Non-Self-Adjoint Operators*. The raw record begins with Michael Levitin's item 11 but continues through items 12 and 13 because several page entries were merged into one JSON object.",
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  "AIM-PHYSICS-0009": {
   "statement_status": "exact",
   "original_statement": "14. Kwang Shin \n\n14.1. Non-polynomial complex potentials. Consider \n\nH = − d2\n\ndx 2 + xm + a1xm−1 + · · · + am\n\nin L2(R+) with y(0) cos θ + y′(0) sin θ = 0, where aj ∈ C and θ ∈ C. It is known that H has infinitely many eigenvalues. Moreover, all eigenvalues are real if and only if aj ∈ R for all j and θ ∈ R.\n\nOpen Problem: Is there any non-self-adjoint non-polynomial potential case H, either in L2(R) or in L2(R+),\n\nthat generates infinitely many real eigenvalues and at most finitely many non-real eigenvalues? 15. Petr Siegl \n\n15.1. Riesz basis for Schr¨ odinger operator with complex potential. Consider the self-adjoint har-monic oscillator A0:= −d2/dx2 + x2 in L2(R). Define A:= A0 + V with a complex-valued potential \n\nV ∈ L∞(R). \n\nOpen problem: Is the eigensystem of A a Riesz basis? It is known that the answer is 'yes' if V ∈ Lp(R) for some 1 ≤ p < ∞.16. David A. Smith \n\n16.1. Spectral representation of two-point differential operators. Augmented eigenfunctions are a class of spectral functionals which have been shown to be useful in expressing solutions of initial-boundary value problems [FS15, PSss, Smi14]. This is particularly important in the case where the spatial differential operator is degenerate irregular in the sense of Locker [Loc08], as no other effective solution representation is known. They also provide a spectral theorem where the inverse of the operator is diagonalized. \n\nOpen problems: i) Are there other applications for augmented eigenfunctions? ii) Can a spectral theory be developed using augmented eigenfunctions? \n\n> 6\n\n*Other Open Problems",
   "clean_statement": "14. Kwang Shin\n\n14.1. Non-polynomial complex potentials. Consider\n\nH = − d2\n\ndx 2 + xm + a1xm−1 + · · · + am\n\nin L2(R+) with y(0) cos θ + y′(0) sin θ = 0, where aj ∈ C and θ ∈ C. It is known that H has infinitely many eigenvalues. Moreover, all eigenvalues are real if and only if aj ∈ R for all j and θ ∈ R.\n\nOpen Problem: Is there any non-self-adjoint non-polynomial potential case H, either in L2(R) or in L2(R+),\n\nthat generates infinitely many real eigenvalues and at most finitely many non-real eigenvalues? 15. Petr Siegl\n\n15.1. Riesz basis for Schr¨ odinger operator with complex potential. Consider the self-adjoint har-monic oscillator A0:= −d2/dx2 + x2 in L2(R). Define A:= A0 + V with a complex-valued potential\n\nV ∈ L∞(R).\n\nOpen problem: Is the eigensystem of A a Riesz basis? It is known that the answer is 'yes' if V ∈ Lp(R) for some 1 ≤ p < ∞.16. David A. Smith\n\n16.1. Spectral representation of two-point differential operators. Augmented eigenfunctions are a class of spectral functionals which have been shown to be useful in expressing solutions of initial-boundary value problems [FS15, PSss, Smi14]. This is particularly important in the case where the spatial differential operator is degenerate irregular in the sense of Locker [Loc08], as no other effective solution representation is known. They also provide a spectral theorem where the inverse of the operator is diagonalized.\n\nOpen problems: i) Are there other applications for augmented eigenfunctions? ii) Can a spectral theory be developed using augmented eigenfunctions?\n\n> 6\n\n*Other Open Problems",
   "public_statement": "14. Kwang Shin\n\n14.1. Non-polynomial complex potentials. Consider\n\nH = − d2\n\ndx 2 + xm + a1xm−1 + · · · + am\n\nin L2(R+) with y(0) cos θ + y′(0) sin θ = 0, where aj ∈ C and θ ∈ C. It is known that H has infinitely many eigenvalues. Moreover, all eigenvalues are real if and only if aj ∈ R for all j and θ ∈ R.\n\nOpen Problem: Is there any non-self-adjoint non-polynomial potential case H, either in L2(R) or in L2(R+),\n\nthat generates infinitely many real eigenvalues and at most finitely many non-real eigenvalues? 15. Petr Siegl\n\n15.1. Riesz basis for Schr¨ odinger operator with complex potential. Consider the self-adjoint har-monic oscillator A0:= −d2/dx2 + x2 in L2(R). Define A:= A0 + V with a complex-valued potential\n\nV ∈ L∞(R).\n\nOpen problem: Is the eigensystem of A a Riesz basis? It is known that the answer is 'yes' if V ∈ Lp(R) for some 1 ≤ p < ∞.16. David A. Smith\n\n16.1. Spectral representation of two-point differential operators. Augmented eigenfunctions are a class of spectral functionals which have been shown to be useful in expressing solutions of initial-boundary value problems [FS15, PSss, Smi14]. This is particularly important in the case where the spatial differential operator is degenerate irregular in the sense of Locker [Loc08], as no other effective solution representation is known. They also provide a spectral theorem where the inverse of the operator is diagonalized.\n\nOpen problems: i) Are there other applications for augmented eigenfunctions? ii) Can a spectral theory be developed using augmented eigenfunctions?\n\n> 6\n\n*Other Open Problems",
   "evidence": "The canonical record is item 14.1, attributed to Kwang Shin, in the AIM workshop list *Mathematical aspects of physics with non-self-adjoint operators*. The PDF asks first about the polynomial half-line operator",
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  "AIM-PHYSICS-0010": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "17. Lyonell Boulton \n\n17.1. Numerical approximation of rigorous enclosures for the spectrum of J-self-adjoint opera-tors. The aim of this theme is to device strategies for computing rigorous (hopefully sharp) bounds/enclosures for the spectrum of J-self-adjoint operators by means of projected space methods. The theory of computa-tion for spectra of self-adjoint operators is classical and well developed. J-self-adjoint operators share many properties with their self-adjoint counterpart. I am interested in discussing to what extent general strategies for numerically estimating spectra of J-self-adjoint operators can be devised. \n\nOpen problems: i) Computation of rigorous rough enclosures for J-self-adjoint operators, taking into account the structure of the conjugation J into the projection scheme. ii) Computation of sharp (numerically relevant) enclosures in specific or generic cases. iii) Impact in the study of evolution problems for J-self-adjoint operators. 18. Tanya J. Christiansen \n\n18.1. Isoresonant potentials. Consider the Schr¨ odinger operator −∆ + V on Rd, where the potential \n\nV ∈ L∞ \n\n> 0\n\n(Rd). If V ∈ C∞ \n\n> c\n\n(Rd; R), then if V is non-trivial the Schr¨ odinger operator has infinitely many resonances. However, if d ≥ 2 there are non-trivial complex-valued potentials V ∈ C∞ \n\n> c\n\n(Rd) for which the corresponding Schr¨ odinger operator has no resonances. More generally, one can explicitly construct families of isoresonant, compactly supported complex-valued potentials in dimensions at least 2. \n\nOpen problems: Is there some other data related in some way to spectral or pseudo spectral properties of the operators that distinguish elements (potentials) in these sets? There are related families of isospectral Schr¨ odinger operators in other settings- on the unit circle, for example. One can ask the same question there. 19. Michael Demuth \n\n19.1. Estimates for the resolvent near the spectrum. Let A be a linear operator on a Banach space. Let K be a compact perturbation of A. The approximation numbers of K are defined by \n\nαN (K):= inf {‖ K − F ‖, rank( F ) < N }.\n\nWe consider only compact operators K with lim N →∞ αN (K) = 0. The objective is to estimate the numbers of eigenvalues of the perturbed operator B:= A + K in cer-tain regions of the complex plane. Let Ω t = {λ ∈ C, |λ| > t }. Denote spr( A):= max {| λ|, λ ∈ σ(A)} and assume spr( A) < t < s. De-note by nB (s) the number of eigenvalues of B in Ω s. In [DHHK15] we obtained \n\nnB (s) ≤ (2 e) p\n\n> 2\n\nlog st\n\nsup λ∈Ωt ‖(λ − A)−1‖p\n\n(1 − αN +1 (K) sup λ∈Ωt ‖(λ − A)−1‖)pN∑\n\n> j=1\n\n(αN +1 (K) + αj (K)) p. (4) Here N has to be so large that \n\nαN +1 (K) sup \n\n> λ∈Ωt\n\n‖(λ − A)−1‖ < 1.\n\nThe optimal result depends on the behavior of ‖(λ−A)−1‖ near the spectrum of A, i.e. on Ω s. This is typical for many spectral considerations. It is also related to the pseudospectrum of A. For instance if |λ| > ‖A‖\n\nthen \n\n‖(λ − A)−1‖ ≤ 1\n\n|λ| − ‖ A‖\n\n> 7\n\nand therefore (4) becomes \n\nnB (s) ≤ (2 e) p\n\n> 2\n\nlog st (t − (‖A‖ + αN +1 (K))) pN∑\n\n> j=1\n\n(αN +1 (K) + αj (K)) p.\n\nOpen problems: i) Classify the operators for which the resolvent is polynomially bounded if λ → σ(A)? ii) Classify the operators for which one can find an M ≥ 1 such that \n\n‖(λ − A)−1‖ ≤ M\n\ndist ( λ, σ (A)) for all λ ∈ res( A) or \n\n‖(λ − A)−1‖ ≤ M\n\n|λ| − spr( A)for |λ| > spr( A).",
   "clean_statement": "17. Lyonell Boulton\n\n17.1. Numerical approximation of rigorous enclosures for the spectrum of J-self-adjoint opera-tors. The aim of this theme is to devise strategies for computing rigorous (hopefully sharp) bounds/enclosures for the spectrum of J-self-adjoint operators by means of projected space methods. The theory of computa-tion for spectra of self-adjoint operators is classical and well developed. J-self-adjoint operators share many properties with their self-adjoint counterpart. I am interested in discussing to what extent general strategies for numerically estimating spectra of J-self-adjoint operators can be devised.\n\nOpen problems: i) Computation of rigorous rough enclosures for J-self-adjoint operators, taking into account the structure of the conjugation J into the projection scheme. ii) Computation of sharp (numerically relevant) enclosures in specific or generic cases. iii) Impact in the study of evolution problems for J-self-adjoint operators. 18. Tanya J. Christiansen\n\n18.1. Isoresonant potentials. Consider the Schr¨ odinger operator −∆ + V on Rd, where the potential\n\nV ∈ L∞\n0\n\n(Rd). If V ∈ C∞\nc\n\n(Rd; R), then if V is non-trivial the Schr¨ odinger operator has infinitely many resonances. However, if d ≥ 2 there are non-trivial complex-valued potentials V ∈ C∞\nc\n\n(Rd) for which the corresponding Schr¨ odinger operator has no resonances. More generally, one can explicitly construct families of isoresonant, compactly supported complex-valued potentials in dimensions at least 2.\n\nOpen problems: Is there some other data related in some way to spectral or pseudo spectral properties of the operators that distinguish elements (potentials) in these sets? There are related families of isospectral Schr¨ odinger operators in other settings- on the unit circle, for example. One can ask the same question there. 19. Michael Demuth\n\n19.1. Estimates for the resolvent near the spectrum. Let A be a linear operator on a Banach space. Let K be a compact perturbation of A. The approximation numbers of K are defined by\n\nαN (K):= inf {‖ K − F ‖, rank( F ) < N }.\n\nWe consider only compact operators K with lim N →∞ αN (K) = 0. The objective is to estimate the numbers of eigenvalues of the perturbed operator B:= A + K in cer-tain regions of the complex plane. Let Ω t = {λ ∈ C, |λ| > t }. Denote spr( A):= max {| λ|, λ ∈ σ(A)} and assume spr( A) < t < s. De-note by nB (s) the number of eigenvalues of B in Ω s. In [DHHK15] we obtained\n\nnB (s) ≤ (2 e) p\n2\n\nlog st\n\nsup λ∈Ωt ‖(λ − A)−1‖p\n\n(1 − αN +1 (K) sup λ∈Ωt ‖(λ − A)−1‖)pN∑\nj=1\n\n(αN +1 (K) + αj (K)) p. (4) Here N has to be so large that\n\nαN +1 (K) sup\nλ∈Ωt\n\n‖(λ − A)−1‖ < 1.\n\nThe optimal result depends on the behavior of ‖(λ−A)−1‖ near the spectrum of A, i.e. on Ω s. This is typical for many spectral considerations. It is also related to the pseudospectrum of A. For instance if |λ| > ‖A‖\n\nthen\n\n‖(λ − A)−1‖ ≤ 1\n\n|λ| − ‖ A‖\n7\n\nand therefore (4) becomes\n\nnB (s) ≤ (2 e) p\n2\n\nlog st (t − (‖A‖ + αN +1 (K))) pN∑\nj=1\n\n(αN +1 (K) + αj (K)) p.\n\nOpen problems: i) Classify the operators for which the resolvent is polynomially bounded if λ → σ(A)? ii) Classify the operators for which one can find an M ≥ 1 such that\n\n‖(λ − A)−1‖ ≤ M\n\ndist ( λ, σ (A)) for all λ ∈ res( A) or\n\n‖(λ − A)−1‖ ≤ M\n\n|λ| − spr( A)for |λ| > spr( A).",
   "public_statement": "17. Lyonell Boulton\n\n17.1. Numerical approximation of rigorous enclosures for the spectrum of J-self-adjoint opera-tors. The aim of this theme is to device strategies for computing rigorous (hopefully sharp) bounds/enclosures for the spectrum of J-self-adjoint operators by means of projected space methods. The theory of computa-tion for spectra of self-adjoint operators is classical and well developed. J-self-adjoint operators share many properties with their self-adjoint counterpart. I am interested in discussing to what extent general strategies for numerically estimating spectra of J-self-adjoint operators can be devised.\n\nOpen problems: i) Computation of rigorous rough enclosures for J-self-adjoint operators, taking into account the structure of the conjugation J into the projection scheme. ii) Computation of sharp (numerically relevant) enclosures in specific or generic cases. iii) Impact in the study of evolution problems for J-self-adjoint operators. 18. Tanya J. Christiansen\n\n18.1. Isoresonant potentials. Consider the Schr¨ odinger operator −∆ + V on Rd, where the potential\n\nV ∈ L∞\n\n> 0\n\n(Rd). If V ∈ C∞\n\n> c\n\n(Rd; R), then if V is non-trivial the Schr¨ odinger operator has infinitely many resonances. However, if d ≥ 2 there are non-trivial complex-valued potentials V ∈ C∞\n\n> c\n\n(Rd) for which the corresponding Schr¨ odinger operator has no resonances. More generally, one can explicitly construct families of isoresonant, compactly supported complex-valued potentials in dimensions at least 2.\n\nOpen problems: Is there some other data related in some way to spectral or pseudo spectral properties of the operators that distinguish elements (potentials) in these sets? There are related families of isospectral Schr¨ odinger operators in other settings- on the unit circle, for example. One can ask the same question there. 19. Michael Demuth\n\n19.1. Estimates for the resolvent near the spectrum. Let A be a linear operator on a Banach space. Let K be a compact perturbation of A. The approximation numbers of K are defined by\n\nαN (K):= inf {‖ K − F ‖, rank( F ) < N }.\n\nWe consider only compact operators K with lim N →∞ αN (K) = 0. The objective is to estimate the numbers of eigenvalues of the perturbed operator B:= A + K in cer-tain regions of the complex plane. Let Ω t = {λ ∈ C, |λ| > t }. Denote spr( A):= max {| λ|, λ ∈ σ(A)} and assume spr( A) < t < s. De-note by nB (s) the number of eigenvalues of B in Ω s. In [DHHK15] we obtained\n\nnB (s) ≤ (2 e) p\n\n> 2\n\nlog st\n\nsup λ∈Ωt ‖(λ − A)−1‖p\n\n(1 − αN +1 (K) sup λ∈Ωt ‖(λ − A)−1‖)pN∑\n\n> j=1\n\n(αN +1 (K) + αj (K)) p. (4) Here N has to be so large that\n\nαN +1 (K) sup\n\n> λ∈Ωt\n\n‖(λ − A)−1‖ < 1.\n\nThe optimal result depends on the behavior of ‖(λ−A)−1‖ near the spectrum of A, i.e. on Ω s. This is typical for many spectral considerations. It is also related to the pseudospectrum of A. For instance if |λ| > ‖A‖\n\nthen\n\n‖(λ − A)−1‖ ≤ 1\n\n|λ| − ‖ A‖\n\n> 7\n\nand therefore (4) becomes\n\nnB (s) ≤ (2 e) p\n\n> 2\n\nlog st (t − (‖A‖ + αN +1 (K))) pN∑\n\n> j=1\n\n(αN +1 (K) + αj (K)) p.\n\nOpen problems: i) Classify the operators for which the resolvent is polynomially bounded if λ → σ(A)? ii) Classify the operators for which one can find an M ≥ 1 such that\n\n‖(λ − A)−1‖ ≤ M\n\ndist ( λ, σ (A)) for all λ ∈ res( A) or\n\n‖(λ − A)−1‖ ≤ M\n\n|λ| − spr( A)for |λ| > spr( A).",
   "evidence": "The canonical record is an OCR concatenation. Its primary item is Lyonell Boulton's item 17.1 from the 2015 AIM workshop *Mathematical aspects of physics with non-self-adjoint operators*:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-physics-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-PHYSICS-0011": {
   "statement_status": "exact",
   "original_statement": "Does the holomorphic anomaly equation \"integrate\" over elliptic fibrations?",
   "clean_statement": "Does the holomorphic anomaly equation \"integrate\" over elliptic fibrations?",
   "public_statement": "Does the holomorphic anomaly equation \"integrate\" over elliptic fibrations?",
   "evidence": "The original AimPL URL is currently unavailable through the web interface. The local canonical extraction is internally consistent and contains no visible OCR corruption. The next records are essential context:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-PHYSICS-0012": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Eta-products and root systems and holomorphic anomaly equation\n\nAs it turns out mock modular forms (such as eta-products) do not satisfy the holomorphic anaomaly equation.\n\nDo they satisfy other differential equations?",
   "clean_statement": null,
   "public_statement": "Eta-products and root systems and holomorphic anomaly equation\n\nAs it turns out mock modular forms (such as eta-products) do not satisfy the holomorphic anaomaly equation.\n\nDo they satisfy other differential equations?",
   "evidence": "This is a plausible reconstruction, not a claim that the inaccessible AimPL page explicitly named Saito. Macdonald's affine-root-system eta identities and Saito's elliptic-root-system eta-products show that the title has a standard mathematical referent.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-physics-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-PHYSICS-0013": {
   "statement_status": "exact",
   "original_statement": "Investigate the Holomorphic anomaly equation (HAE) in fibered Calabi-Yau 3-folds and BPS counting. \\label{hae-cy}",
   "clean_statement": "Investigate the Holomorphic anomaly equation (HAE) in fibered Calabi-Yau 3-folds and BPS counting. \\label{hae-cy}",
   "public_statement": "Investigate the Holomorphic anomaly equation (HAE) in fibered Calabi-Yau 3-folds and BPS counting. \\label{hae-cy}",
   "evidence": "The exact canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-PHYSICS-0014": {
   "statement_status": "exact",
   "original_statement": "When can the elliptic genus be defined for noncompact Calabi-Yau manifolds?",
   "clean_statement": "When can the elliptic genus be defined for noncompact Calabi-Yau manifolds?",
   "public_statement": "When can the elliptic genus be defined for noncompact Calabi-Yau manifolds?",
   "evidence": "The exact canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-PHYSICS-0015": {
   "statement_status": "exact",
   "original_statement": "When can modular forms on Calabi Yau 3-folds moduli spaces be reduced to lower dimensional forms?",
   "clean_statement": "When can modular forms on Calabi Yau 3-folds moduli spaces be reduced to lower dimensional forms?",
   "public_statement": "When can modular forms on Calabi Yau 3-folds moduli spaces be reduced to lower dimensional forms?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-PHYSICS-0016": {
   "statement_status": "exact",
   "original_statement": "Is there an index theoretic interpretation of elliptic genus? When it can, how do we compute it and what is the geometric interpretation of the coefficients? Check that this matches CFT calculations.",
   "clean_statement": "Is there an index theoretic interpretation of elliptic genus? When it can, how do we compute it and what is the geometric interpretation of the coefficients? Check that this matches CFT calculations.",
   "public_statement": "Is there an index theoretic interpretation of elliptic genus? When it can, how do we compute it and what is the geometric interpretation of the coefficients? Check that this matches CFT calculations.",
   "evidence": "The exact canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-PHYSICS-0017": {
   "statement_status": "exact",
   "original_statement": "Investigate the role of paramodular groups in counting problems in Calabi Yau 3-folds",
   "clean_statement": "Investigate the role of paramodular groups in counting problems in Calabi Yau 3-folds",
   "public_statement": "Investigate the role of paramodular groups in counting problems in Calabi Yau 3-folds",
   "evidence": "The canonical record is problem 3.4 in the section “Calabi--Yau manifolds”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-PHYSICS-0018": {
   "statement_status": "exact",
   "original_statement": "Can we calculate the leading terms of $F_g^B(q)$ at the orbifold point $z=0$ of the quintic\n\\[\n(\\sum x_i^5 + z \\prod x_i =0 )/ \\mathbb{Z}_5^3?\n\\]",
   "clean_statement": "Can we calculate the leading terms of $F_g^B(q)$ at the orbifold point $z=0$ of the quintic\n\\[\n(\\sum x_i^5 + z \\prod x_i =0 )/ \\mathbb{Z}_5^3?\n\\]",
   "public_statement": "Can we calculate the leading terms of $F_g^B(q)$ at the orbifold point $z=0$ of the quintic\n\\[\n(\\sum x_i^5 + z \\prod x_i =0 )/ \\mathbb{Z}_5^3?\n\\]",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-PHYSICS-0019": {
   "statement_status": "exact",
   "original_statement": "Here $h_m$ is the $m$-th Hurwitz class number, i.e. the number of equivalence classes of positive definite binary quadratic forms of discriminant $-m$ with the class containing $x^2+y^2$ weighted by 1/2 and the class containing $x^2 + xy + y^2$ weighted by 1/3. Moreover by convention $h_0=-1/12.$\n\nFor more information in regards to the connection between physics and number theory, see the following work of\n\n Kathrin Bringmann and Ben Kane\nhttp://arxiv.org/pdf/1305.0112v1.pdf\n\nKatrin Bringmann and Sameer Murthy\nhttp://arxiv.org/pdf/1208.3476v2.pdf\n\nKatrin Bringmann and Jan Manschot\nhttp://arxiv.org/pdf/1304.7208v1.pdf\n\nDo the coefficients of $h_m$ have enumerative significance? Can we find $h_m$ for $m \\geq 9$?",
   "clean_statement": "Here $h_m$ is the $m$-th Hurwitz class number, i.e. the number of equivalence classes of positive definite binary quadratic forms of discriminant $-m$ with the class containing $x^2+y^2$ weighted by 1/2 and the class containing $x^2 + xy + y^2$ weighted by 1/3. Moreover by convention $h_0=-1/12.$\n\nFor more information in regards to the connection between physics and number theory, see the following work of\n\n Kathrin Bringmann and Ben Kane\nhttp://arxiv.org/pdf/1305.0112v1.pdf\n\nKatrin Bringmann and Sameer Murthy\nhttp://arxiv.org/pdf/1208.3476v2.pdf\n\nKatrin Bringmann and Jan Manschot\nhttp://arxiv.org/pdf/1304.7208v1.pdf\n\nDo the coefficients of $h_m$ have enumerative significance? Can we find $h_m$ for $m \\geq 9$?",
   "public_statement": "Here $h_m$ is the $m$-th Hurwitz class number, i.e. the number of equivalence classes of positive definite binary quadratic forms of discriminant $-m$ with the class containing $x^2+y^2$ weighted by 1/2 and the class containing $x^2 + xy + y^2$ weighted by 1/3. Moreover by convention $h_0=-1/12.$\n\nFor more information in regards to the connection between physics and number theory, see the following work of\n\n Kathrin Bringmann and Ben Kane\nhttp://arxiv.org/pdf/1305.0112v1.pdf\n\nKatrin Bringmann and Sameer Murthy\nhttp://arxiv.org/pdf/1208.3476v2.pdf\n\nKatrin Bringmann and Jan Manschot\nhttp://arxiv.org/pdf/1304.7208v1.pdf\n\nDo the coefficients of $h_m$ have enumerative significance? Can we find $h_m$ for $m \\geq 9$?",
   "evidence": "The exact canonical record (item 4.2, “Specific functions”) defines \\(h_m\\) as the weighted number of positive-definite integral binary quadratic forms of discriminant \\(-m\\), gives the exceptional weights \\(1/2\\) for \\(x^2+y^2\\) and \\(1/3\\) for \\(x^2+xy+y^2\\), sets \\(h_0=-1/12\\), cites three papers, and asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-PHYSICS-0020": {
   "statement_status": "exact",
   "original_statement": "Does $F= \\sum f_g(\\tau)\\lambda^{2g-2}$ have transformation properties with respect to $\\lambda$?",
   "clean_statement": "Does $F= \\sum f_g(\\tau)\\lambda^{2g-2}$ have transformation properties with respect to $\\lambda$?",
   "public_statement": "Does $F= \\sum f_g(\\tau)\\lambda^{2g-2}$ have transformation properties with respect to $\\lambda$?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-PHYSICS-0021": {
   "statement_status": "exact",
   "original_statement": "Is $\\sum \\frac{(5n)!}{(n!)^5}z^n$ related to (non-holomorphic) forms? Is it related to automorphic objects?\nDo other solutions to the Picard-Fuchs equation have modular properties?",
   "clean_statement": "Is $\\sum \\frac{(5n)!}{(n!)^5}z^n$ related to (non-holomorphic) forms? Is it related to automorphic objects?\nDo other solutions to the Picard-Fuchs equation have modular properties?",
   "public_statement": "Is $\\sum \\frac{(5n)!}{(n!)^5}z^n$ related to (non-holomorphic) forms? Is it related to automorphic objects?\nDo other solutions to the Picard-Fuchs equation have modular properties?",
   "evidence": "The exact canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-PHYSICS-0022": {
   "statement_status": "exact",
   "original_statement": "Do the L-series of mixed mock modular forms have interesting properties?",
   "clean_statement": "Do the L-series of mixed mock modular forms have interesting properties?",
   "public_statement": "Do the L-series of mixed mock modular forms have interesting properties?",
   "evidence": "The sentence has no apparent OCR corruption, but it is mathematically underspecified. It does not define “mixed mock modular form,” choose a cusp or multiplier, or say which of several inequivalent objects is the “L-series.” The nearby records ask whether mixed mock modular forms satisfy differential equations and how mock modular forms relate to geometric invariants. They add motivation but no definitions.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-PHYSICS-0023": {
   "statement_status": "exact",
   "original_statement": "Do mixed mock modular forms satisfy differential equations with respect to a modular function?",
   "clean_statement": "Do mixed mock modular forms satisfy differential equations with respect to a modular function?",
   "public_statement": "Do mixed mock modular forms satisfy differential equations with respect to a modular function?",
   "evidence": "The exact canonical record (`aim-physics-notes.json`, zero-based index 22) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-PHYSICS-0024": {
   "statement_status": "exact",
   "original_statement": "Determine the connection between Mock modular forms and geometric invariants like Gromov-Witten, Donaldson, and OSV conjecture.",
   "clean_statement": "Determine the connection between Mock modular forms and geometric invariants like Gromov-Witten, Donaldson, and OSV conjecture.",
   "public_statement": "Determine the connection between Mock modular forms and geometric invariants like Gromov-Witten, Donaldson, and OSV conjecture.",
   "evidence": "The exact canonical record is item 5.3 in the section “Other problems”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-PHYSICS-0025": {
   "statement_status": "exact",
   "original_statement": "(5) Let G be a finite graph with maximum degree ∆. This problem is about choosing a random coloring from the uniform distribution. Work with Glauber dynamics: pick a site at random, choose a color at random, and try to change the color at the chosen site. If this results in a proper coloring, make the change. If not, keep the current coloring. For q ≥ ∆ + 2 it is known that this Markov chain is ergodic, i.e., \n\n> 12\n\nthat it connects the set of proper colorings. Let t = t(q) be the mixing time: the smallest time such that the total variation distance from the uniform distribution is at most 1 /e from any starting configuration. The question is: does t(q) decrease as \n\nq increases? Even on the square lattice with n sites there are curious gaps in our knowledge: we know that t(3) is polynomial in n (Luby, Randall and Sinclair: Proc. FOCS 1995) and we know that t(q) = O(n log n) for q ≥ 6. But, for q = 4 and q = 5 we do not even have a polynomial upper bound, even though there is overwhelming numerical evidence that there is a finite correlation length, and therefore that the mixing time is O(n log n). (6) Is there a variant of k-SAT that describes the scaling properties of jamming? In other words, is k-SAT a good \"Boolean idealization\" of the jamming process? Should this involve k-SAT on a lattice? (See Schwartz and Middleton, Phys. Rev. E 70 (2004), 035103(R).) (7) Find a model with constraints and a phase transition, which we can perturb by \"soft-ening\" the constraints. Study the sensitivity of this model's behavior with respect to this perturbation. (8) In constraint satisfaction problems such as k-SAT, what are the useful response/correlation functions? Do they show singular, or power-law, behavior at the transition, analo-gous to spatial correlations and response functions in lattice systems such as the Ising model? (9) The second moment method in k-SAT gives a highly accurate lower bound for the critical density, when the approach of (Achlioptas and Moore: Proc. FOCS 2002) is refined by (Achlioptas and Peres: Proc. STOC 2003). This refinement can be thought of as adding a finite (not infinitesimal) external field which discourages lit-erals from being true. Is there any physical meaning to this value of the external field? For instance, does it minimize fluctuations in, say, the free energy at zero temperature? (10) We believe (and, in some cases, have proved) that below the satisfiability/unsatisfiability transition in k-SAT there is a \"clustering\" transition, in which the solutions clump together in isolated groups. It has been conjectured that this clustering phenomenon makes it hard for search algorithms to find solutions. Can we find algorithms which, in simulation, find solutions in, say, linear time, for values of k and densities at which clustering has been rigorously established? (11) (Following up on the previous question) Can we prove that clustering occurs at some density below the satisfiability transition for k = 3, 4, 5? So far we only have proofs of clustering for larger values of k.(12) Prove an order/disorder phase transition for the equilibrium statistical mechanics model of hard spheres (or hard disks). Simulation evidence of such a transition is 3\n\ngenerally believed to be convincing in both cases, though the nature of the transition is controversial for disks. (13) How should we add friction to jamming? (14) Consider the Ising model on the d-dimensional lattice. Fix β (and perhaps the ex-ternal field strength h) and consider Glauber dynamics. The cutoff phenomenon in Markov chains refers to the existence of a time at which the total variation distance from the stationary (Gibbs) distribution jumps from 1 to 0 in the limit where the size of the lattice goes to infinity. We ask whether this sharp cutoff exists if and only if there is a unique Gibbs state in the infinite system. (15) Give a good definition of the thermodynamic limit of k-SAT. That is, clearly define the mathematical object that corresponds to random formulas Fk(n, m = rn ) with constant density r in the limit n → ∞.(16) Consider the Ising model on a finite box in Zd. Set β > β c and no external field. With fixed boundary conditions where all the boundary sites are set to +1, it is known that Glauber dynamics can take a long time to mix if the initial state consists of a large droplet of −1's. But with a \"warm start,\" where the initial condition consists of all +1's or is random, is the mixing time optimal, i.e., O(n log n) where n is the number of sites in the box?",
   "clean_statement": "(5) Let G be a finite graph with maximum degree ∆. This problem is about choosing a random coloring from the uniform distribution. Work with Glauber dynamics: pick a site at random, choose a color at random, and try to change the color at the chosen site. If this results in a proper coloring, make the change. If not, keep the current coloring. For q ≥ ∆ + 2 it is known that this Markov chain is ergodic, i.e.,\n\n> 12\n\nthat it connects the set of proper colorings. Let t = t(q) be the mixing time: the smallest time such that the total variation distance from the uniform distribution is at most 1 /e from any starting configuration. The question is: does t(q) decrease as\n\nq increases? Even on the square lattice with n sites there are curious gaps in our knowledge: we know that t(3) is polynomial in n (Luby, Randall and Sinclair: Proc. FOCS 1995) and we know that t(q) = O(n log n) for q ≥ 6. But, for q = 4 and q = 5 we do not even have a polynomial upper bound, even though there is overwhelming numerical evidence that there is a finite correlation length, and therefore that the mixing time is O(n log n). (6) Is there a variant of k-SAT that describes the scaling properties of jamming? In other words, is k-SAT a good \"Boolean idealization\" of the jamming process? Should this involve k-SAT on a lattice? (See Schwartz and Middleton, Phys. Rev. E 70 (2004), 035103(R).) (7) Find a model with constraints and a phase transition, which we can perturb by \"soft-ening\" the constraints. Study the sensitivity of this model's behavior with respect to this perturbation. (8) In constraint satisfaction problems such as k-SAT, what are the useful response/correlation functions? Do they show singular, or power-law, behavior at the transition, analo-gous to spatial correlations and response functions in lattice systems such as the Ising model? (9) The second moment method in k-SAT gives a highly accurate lower bound for the critical density, when the approach of (Achlioptas and Moore: Proc. FOCS 2002) is refined by (Achlioptas and Peres: Proc. STOC 2003). This refinement can be thought of as adding a finite (not infinitesimal) external field which discourages lit-erals from being true. Is there any physical meaning to this value of the external field? For instance, does it minimize fluctuations in, say, the free energy at zero temperature? (10) We believe (and, in some cases, have proved) that below the satisfiability/unsatisfiability transition in k-SAT there is a \"clustering\" transition, in which the solutions clump together in isolated groups. It has been conjectured that this clustering phenomenon makes it hard for search algorithms to find solutions. Can we find algorithms which, in simulation, find solutions in, say, linear time, for values of k and densities at which clustering has been rigorously established? (11) (Following up on the previous question) Can we prove that clustering occurs at some density below the satisfiability transition for k = 3, 4, 5? So far we only have proofs of clustering for larger values of k.(12) Prove an order/disorder phase transition for the equilibrium statistical mechanics model of hard spheres (or hard disks). Simulation evidence of such a transition is 3\n\ngenerally believed to be convincing in both cases, though the nature of the transition is controversial for disks. (13) How should we add friction to jamming? (14) Consider the Ising model on the d-dimensional lattice. Fix β (and perhaps the ex-ternal field strength h) and consider Glauber dynamics. The cutoff phenomenon in Markov chains refers to the existence of a time at which the total variation distance from the stationary (Gibbs) distribution jumps from 1 to 0 in the limit where the size of the lattice goes to infinity. We ask whether this sharp cutoff exists if and only if there is a unique Gibbs state in the infinite system. (15) Give a good definition of the thermodynamic limit of k-SAT. That is, clearly define the mathematical object that corresponds to random formulas Fk(n, m = rn ) with constant density r in the limit n → ∞.(16) Consider the Ising model on a finite box in Zd. Set β > β c and no external field. With fixed boundary conditions where all the boundary sites are set to +1, it is known that Glauber dynamics can take a long time to mix if the initial state consists of a large droplet of −1's. But with a \"warm start,\" where the initial condition consists of all +1's or is random, is the mixing time optimal, i.e., O(n log n) where n is the number of sites in the box?",
   "public_statement": "(5) Let G be a finite graph with maximum degree ∆. This problem is about choosing a random coloring from the uniform distribution. Work with Glauber dynamics: pick a site at random, choose a color at random, and try to change the color at the chosen site. If this results in a proper coloring, make the change. If not, keep the current coloring. For q ≥ ∆ + 2 it is known that this Markov chain is ergodic, i.e.,\n\n> 12\n\nthat it connects the set of proper colorings. Let t = t(q) be the mixing time: the smallest time such that the total variation distance from the uniform distribution is at most 1 /e from any starting configuration. The question is: does t(q) decrease as\n\nq increases? Even on the square lattice with n sites there are curious gaps in our knowledge: we know that t(3) is polynomial in n (Luby, Randall and Sinclair: Proc. FOCS 1995) and we know that t(q) = O(n log n) for q ≥ 6. But, for q = 4 and q = 5 we do not even have a polynomial upper bound, even though there is overwhelming numerical evidence that there is a finite correlation length, and therefore that the mixing time is O(n log n). (6) Is there a variant of k-SAT that describes the scaling properties of jamming? In other words, is k-SAT a good \"Boolean idealization\" of the jamming process? Should this involve k-SAT on a lattice? (See Schwartz and Middleton, Phys. Rev. E 70 (2004), 035103(R).) (7) Find a model with constraints and a phase transition, which we can perturb by \"soft-ening\" the constraints. Study the sensitivity of this model's behavior with respect to this perturbation. (8) In constraint satisfaction problems such as k-SAT, what are the useful response/correlation functions? Do they show singular, or power-law, behavior at the transition, analo-gous to spatial correlations and response functions in lattice systems such as the Ising model? (9) The second moment method in k-SAT gives a highly accurate lower bound for the critical density, when the approach of (Achlioptas and Moore: Proc. FOCS 2002) is refined by (Achlioptas and Peres: Proc. STOC 2003). This refinement can be thought of as adding a finite (not infinitesimal) external field which discourages lit-erals from being true. Is there any physical meaning to this value of the external field? For instance, does it minimize fluctuations in, say, the free energy at zero temperature? (10) We believe (and, in some cases, have proved) that below the satisfiability/unsatisfiability transition in k-SAT there is a \"clustering\" transition, in which the solutions clump together in isolated groups. It has been conjectured that this clustering phenomenon makes it hard for search algorithms to find solutions. Can we find algorithms which, in simulation, find solutions in, say, linear time, for values of k and densities at which clustering has been rigorously established? (11) (Following up on the previous question) Can we prove that clustering occurs at some density below the satisfiability transition for k = 3, 4, 5? So far we only have proofs of clustering for larger values of k.(12) Prove an order/disorder phase transition for the equilibrium statistical mechanics model of hard spheres (or hard disks). Simulation evidence of such a transition is 3\n\ngenerally believed to be convincing in both cases, though the nature of the transition is controversial for disks. (13) How should we add friction to jamming? (14) Consider the Ising model on the d-dimensional lattice. Fix β (and perhaps the ex-ternal field strength h) and consider Glauber dynamics. The cutoff phenomenon in Markov chains refers to the existence of a time at which the total variation distance from the stationary (Gibbs) distribution jumps from 1 to 0 in the limit where the size of the lattice goes to infinity. We ask whether this sharp cutoff exists if and only if there is a unique Gibbs state in the infinite system. (15) Give a good definition of the thermodynamic limit of k-SAT. That is, clearly define the mathematical object that corresponds to random formulas Fk(n, m = rn ) with constant density r in the limit n → ∞.(16) Consider the Ising model on a finite box in Zd. Set β > β c and no external field. With fixed boundary conditions where all the boundary sites are set to +1, it is known that Glauber dynamics can take a long time to mix if the initial state consists of a large droplet of −1's. But with a \"warm start,\" where the initial condition consists of all +1's or is random, is the mixing time optimal, i.e., O(n log n) where n is the number of sites in the box?",
   "evidence": "The canonical JSON record is a contaminated extraction from the three-page AIM workshop PDF *Phase transitions*. Two page numbers were fused into the text as the block `> 12`, and printed problems (6)--(16) were appended to problem (5). Inspection of the original PDF shows that problem (5) ends with the sentence ending “the mixing time is \\(O(n\\log n)\\).” The next paragraph, beginning “(6) Is there a variant of \\(k\\)-SAT…”, is a separate problem and is not analyzed here.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-PHYSICS-0026": {
   "statement_status": "exact",
   "original_statement": "(1) No analytic description of relativistic caustics (except for the intersections on the equatorial plane. (2) No analytic description of gravitational lensing out of the equatorial plane. (3) No idea (analytical or numerical) for what happens to higher order caustics. (4) Is strong field gravitational lensing observable? (5) Analytical shape of the caustics in Strong Deflection Limit? (6) Gravitational lensing far from the equatorial plane in the Strong Deflection Limit? (7) Description of additional images in the Strong Deflection Limit? (8) What coordinates are best suited for Lensing in Kerr spacetime? (9) Projects for MAXIM? (10) Non-disk lensing events for MAXIM? (11) Gas trajectories in the accretion disk - fully GR code or MHD/Kerr? (12) Are the time delays the same for Kerr and \"Shifted Schwarzschild\"? To what order? (13) Are caustics/critical curves in first order strong field limit equivalent to \"shifted Schwarzschild\"?",
   "clean_statement": "(1) No analytic description of relativistic caustics (except for the intersections on the equatorial plane. (2) No analytic description of gravitational lensing out of the equatorial plane. (3) No idea (analytical or numerical) for what happens to higher order caustics. (4) Is strong field gravitational lensing observable? (5) Analytical shape of the caustics in Strong Deflection Limit? (6) Gravitational lensing far from the equatorial plane in the Strong Deflection Limit? (7) Description of additional images in the Strong Deflection Limit? (8) What coordinates are best suited for Lensing in Kerr spacetime? (9) Projects for MAXIM? (10) Non-disk lensing events for MAXIM? (11) Gas trajectories in the accretion disk - fully GR code or MHD/Kerr? (12) Are the time delays the same for Kerr and \"Shifted Schwarzschild\"? To what order? (13) Are caustics/critical curves in first order strong field limit equivalent to \"shifted Schwarzschild\"?",
   "public_statement": "(1) No analytic description of relativistic caustics (except for the intersections on the equatorial plane. (2) No analytic description of gravitational lensing out of the equatorial plane. (3) No idea (analytical or numerical) for what happens to higher order caustics. (4) Is strong field gravitational lensing observable? (5) Analytical shape of the caustics in Strong Deflection Limit? (6) Gravitational lensing far from the equatorial plane in the Strong Deflection Limit? (7) Description of additional images in the Strong Deflection Limit? (8) What coordinates are best suited for Lensing in Kerr spacetime? (9) Projects for MAXIM? (10) Non-disk lensing events for MAXIM? (11) Gas trajectories in the accretion disk - fully GR code or MHD/Kerr? (12) Are the time delays the same for Kerr and \"Shifted Schwarzschild\"? To what order? (13) Are caustics/critical curves in first order strong field limit equivalent to \"shifted Schwarzschild\"?",
   "evidence": "The canonical problem field concatenates thirteen numbered entries from a one-page AIM list. The original PDF was inspected directly. Its heading is “Gravitational Lensing in the Kerr Spacetime Geometry — Short list of problems/issues,” and its first four lines are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-PHYSICS-0027": {
   "statement_status": "exact",
   "original_statement": "(14) Plotting the caustics from conjugate points: Which photons (light rays) form the caustics? (15) What does the lightcone look like? (16) Plot higher order caustic sheets numerically. (17) Plot primary caustic sheet for observer close to Black hole. (18) How do caustics look for observers outside the equatorial plane? (19) How are the astroid caustics formed? (20) Is the birth process of the astroid caustic sheet for Kerr stable? (21) Clarify GR definitions of twist/shear/convergence in terms of lensing definition. (22) Stability of caustics with respect to metric perturbations? (23) Are all stable caustic sheets lightlike?",
   "clean_statement": "(14) Plotting the caustics from conjugate points: Which photons (light rays) form the caustics? (15) What does the lightcone look like? (16) Plot higher order caustic sheets numerically. (17) Plot primary caustic sheet for observer close to Black hole. (18) How do caustics look for observers outside the equatorial plane? (19) How are the astroid caustics formed? (20) Is the birth process of the astroid caustic sheet for Kerr stable? (21) Clarify GR definitions of twist/shear/convergence in terms of lensing definition. (22) Stability of caustics with respect to metric perturbations? (23) Are all stable caustic sheets lightlike?",
   "public_statement": "(14) Plotting the caustics from conjugate points: Which photons (light rays) form the caustics? (15) What does the lightcone look like? (16) Plot higher order caustic sheets numerically. (17) Plot primary caustic sheet for observer close to Black hole. (18) How do caustics look for observers outside the equatorial plane? (19) How are the astroid caustics formed? (20) Is the birth process of the astroid caustic sheet for Kerr stable? (21) Clarify GR definitions of twist/shear/convergence in terms of lensing definition. (22) Stability of caustics with respect to metric perturbations? (23) Are all stable caustic sheets lightlike?",
   "evidence": "The canonical JSON record concatenates ten separately numbered questions from the one-page AIM problem list *Gravitational Lensing in the Kerr Spacetime Geometry*. Inspection of the original PDF shows the exact source boundary:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-PHYSICS-0028": {
   "statement_status": "exact",
   "original_statement": "1. (Carron) One needs geometrical assumptions on ( M, g ) to make this work. It turns out that bounded Ricci-curvature is not a sufficient condition. Donnelly [3] had shown that if a complete non-compact Riemannian manifold of dimension n has Ric ≥ − (n − 1) g then the essential spectrum of the Laplace operator (on functions) has a non-empty intersection with [0, (n − 1) 2/4]. Moreover, Donnelly and Li [4] had shown that if a Cartan-Hadamard manifold (i.e. a complete simply-connected Riemannian manifold with non-positive curvature) has its curvature going to −∞ at infinity then the essential spectrum of the Laplace operator is empty. 2. It can also be generalized introducing weighted Sobolev spaces - with special conditions on the weights depending on the geometry at infinity (see Joyce's book [7] for the case of ALE manifolds).",
   "clean_statement": "1. (Carron) One needs geometrical assumptions on ( M, g ) to make this work. It turns out that bounded Ricci-curvature is not a sufficient condition. Donnelly [3] had shown that if a complete non-compact Riemannian manifold of dimension n has Ric ≥ − (n − 1) g then the essential spectrum of the Laplace operator (on functions) has a non-empty intersection with [0, (n − 1) 2/4]. Moreover, Donnelly and Li [4] had shown that if a Cartan-Hadamard manifold (i.e. a complete simply-connected Riemannian manifold with non-positive curvature) has its curvature going to −∞ at infinity then the essential spectrum of the Laplace operator is empty. 2. It can also be generalized introducing weighted Sobolev spaces - with special conditions on the weights depending on the geometry at infinity (see Joyce's book [7] for the case of ALE manifolds).",
   "public_statement": "1. (Carron) One needs geometrical assumptions on ( M, g ) to make this work. It turns out that bounded Ricci-curvature is not a sufficient condition. Donnelly [3] had shown that if a complete non-compact Riemannian manifold of dimension n has Ric ≥ − (n − 1) g then the essential spectrum of the Laplace operator (on functions) has a non-empty intersection with [0, (n − 1) 2/4]. Moreover, Donnelly and Li [4] had shown that if a Cartan-Hadamard manifold (i.e. a complete simply-connected Riemannian manifold with non-positive curvature) has its curvature going to −∞ at infinity then the essential spectrum of the Laplace operator is empty. 2. It can also be generalized introducing weighted Sobolev spaces - with special conditions on the weights depending on the geometry at infinity (see Joyce's book [7] for the case of ALE manifolds).",
   "evidence": "The canonical record is not a self-contained open problem. It is the first two answers to a question on page 1 of the AIM workshop notes *Questions arising in open problem sessions in AIM workshop on $L^2$-harmonic forms in geometry and string theory* (notes by Anda Degeratu and Mark Haskins, 17 March 2004). The missing antecedent is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-PHYSICS-0029": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "3. (Melrose) There is no good general answer. \n\nQuestion [ L2-cohomology]: \n\nIs there a proper way to define L2-homology as opposed to L2-cohomology? \n\nMotivation: There is a way to define it for triangulated manifolds ( M, T ), by going up to the universal cover ( ˜M, ˜T ) and taking the L2-chains to be \n\nCi\n\n> (2)\n\n( ˜M ) = {∑ aiσi| ∑ |ai|2 < ∞}.\n\n(and possibly some boundary terms condition coming in also?) \n\nQuestion to ask: How does this depend on the metric? And when do you get something dual to the \n\nL2-cohomology. \n\nQuestion [ L2-cohomology]: \n\nIs there any relation between L2-cohomology and group-cohomology. \n\nAnswer: see L¨ uck in his book [8]. \n\nQuestions [Non-parabolicity and exactness of the excision sequence for reduced L2-cohomology - G. Carron's lecture]: \n\n1Let ( M, g ) Riemannian manifold so that d+d∗ is non-parabolic at infinity with respect to K. For a compact \n\n˜K so that K ⊂ ˜K we define the norm \n\nN ˜K (α):= || α|| L2 ( ˜K) + || (d + d∗)α|| L2 (M \\ ˜K).\n\nThen on C∞ \n\n> 0\n\n(Λ( M )) all these norms are equivalent. Let W be the completion of C∞ \n\n> 0\n\n(Λ( M )) with respect one of them. One of the main points was that \n\nd + d∗: W → L2\n\nis Fredholm. Several related questions arose: 1. Is there any other description for W?2. Is the reduced L2-cohomology ˙Hk\n\n> (2)\n\n(M, g ) the cohomology of a complex/ of many complexes? 3. Can the space W be put into a complex? \n\nQuestion [reduced L2-cohomology - G. Carron's lecture]: \n\nWhen is there a Mayer-Vietoris sequence for ˙H(2) (M, g )? \n\nAnswer: (Carron) The condition of non-parabolicity at infinity is not sufficient for this. One needs a better control over noncompact overlaps. Perhaps if also Range( d) is closed on U ∩ V then it is sufficient?? \n\nQuestions [Self-dual Gravitational Instantons - S. Cherkis's lecture]:",
   "clean_statement": null,
   "public_statement": "3. (Melrose) There is no good general answer.\n\nQuestion [ L2-cohomology]:\n\nIs there a proper way to define L2-homology as opposed to L2-cohomology?\n\nMotivation: There is a way to define it for triangulated manifolds ( M, T ), by going up to the universal cover ( ˜M, ˜T ) and taking the L2-chains to be\n\nCi\n\n> (2)\n\n( ˜M ) = {∑ aiσi| ∑ |ai|2 < ∞}.\n\n(and possibly some boundary terms condition coming in also?)\n\nQuestion to ask: How does this depend on the metric? And when do you get something dual to the\n\nL2-cohomology.\n\nQuestion [ L2-cohomology]:\n\nIs there any relation between L2-cohomology and group-cohomology.\n\nAnswer: see L¨ uck in his book [8].\n\nQuestions [Non-parabolicity and exactness of the excision sequence for reduced L2-cohomology - G. Carron's lecture]:\n\n1Let ( M, g ) Riemannian manifold so that d+d∗ is non-parabolic at infinity with respect to K. For a compact\n\n˜K so that K ⊂ ˜K we define the norm\n\nN ˜K (α):= || α|| L2 ( ˜K) + || (d + d∗)α|| L2 (M \\ ˜K).\n\nThen on C∞\n\n> 0\n\n(Λ( M )) all these norms are equivalent. Let W be the completion of C∞\n\n> 0\n\n(Λ( M )) with respect one of them. One of the main points was that\n\nd + d∗: W → L2\n\nis Fredholm. Several related questions arose: 1. Is there any other description for W?2. Is the reduced L2-cohomology ˙Hk\n\n> (2)\n\n(M, g ) the cohomology of a complex/ of many complexes? 3. Can the space W be put into a complex?\n\nQuestion [reduced L2-cohomology - G. Carron's lecture]:\n\nWhen is there a Mayer-Vietoris sequence for ˙H(2) (M, g )?\n\nAnswer: (Carron) The condition of non-parabolicity at infinity is not sufficient for this. One needs a better control over noncompact overlaps. Perhaps if also Range( d) is closed on U ∩ V then it is sufficient??\n\nQuestions [Self-dual Gravitational Instantons - S. Cherkis's lecture]:",
   "evidence": "The canonical JSON field is corrupted by a record-boundary error. It begins",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-physics-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-PHYSICS-0030": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "1. Why do you expect them to arise as monopole moduli spaces? \n\nAnswer: (Cherkis) string theory argument. 2. Are all 8-dimensional hyperk¨ ahler manifolds (non-compact, with some sort of well-behaved asymptotic behaviour) moduli spaces? \n\nAnswer: (Cherkis) yes, from string theory intuition. 3. What about dimension > 8? 4. One way to get gravitational instantons is by solving the vacuum Einstein equation in Lorentz space, and then performing a wick rotation (meaning t → it ). This worked in the case of Lorentzian Taub-NUT to get the Riemannian Taub-NUT. Which other instantons arrise this way? \n\nAnswer: In order to get a Riemannian metric, one needs t → t + c (time translation) be an isometry. Therefore it does work for Ak-cases but not for the others ( Dk, E6, E 7, E 8). 5. Can they arrive as \"non-trivial\" hyperk¨ ahler reductions of finite dimensional linear spaces? \n\nAnswer: Yes, for ALE spaces (Kronheimer's construction). Yes, for ALF if certain (Dancer) spaces are included with the linear spaces. 6. Is it true that every gravitational instanton has an asymptotic local triholomorphic isometry? \n\nReason: If you have a local R-action at infinity, then the metric is given in terms of a harmonic function, which arises from the hyperk¨ ahler moment map associated to this action (see the work of Gibbons). 7. When is it true that {Solutions of the Einstein equation } > {Self-Dual metrics }?\n\nReason: In general ≥.\n\nAnswers: \n\n(1) In the case of ALF the answer is >, since the Euclidean Schwarzschild metric is not self-dual. (2) Nakajima conjectured that in dimension 4, ALE and Ricci-flat implies self-dual. (3) In dimension > 4 and even, the conjecture is that ALE and Ricci-flat implies K¨ ahler. (4) (Carron) For odd dimensional manifolds an ALE Ricci-flat metric has to be flat since (a) by the Cheeger-Gromoll theorem this manifold has only one end (if not it splits isometrically as \n\nR × N with N compact); (b) The topology at infinity must be of the type ( R, ∞) × S2n/G where G is a finite subgroup of \n\nO(2 n + 1) acting freely on S2n. If g ∈ SO (2 n + 1) it must have 1 as an eigenvalue hence the only such G is \n\n{I, −I}; but the quotient is the real projective space, and there is no odd dimensional compact manifold whose boundary is the real projective space. Therefore G is trivial and the Bishop-Gromov inequality 2implies that M is the Euclidean space. Recall that the Bishop-Gromov inequality states that in a manifold \n\nM n with non-negative Ricci curvature r 7 → vol B(x, r )/w nrn is decreasing with equality everywhere if and only if M n is the Euclidean space. This ratio goes to 1 when r → 0, and in the ALE case we have that this ratio goes to 1 /card G when r → ∞.\n\nQuestions [of S. Cherkis]:",
   "clean_statement": "**Question 1.** Why do you expect self-dual gravitational instantons to arise as monopole moduli spaces?\n\n**Answer (Cherkis).** String theory argument.",
   "public_statement": "1. Why do you expect them to arise as monopole moduli spaces?\n\nAnswer: (Cherkis) string theory argument. 2. Are all 8-dimensional hyperk¨ ahler manifolds (non-compact, with some sort of well-behaved asymptotic behaviour) moduli spaces?\n\nAnswer: (Cherkis) yes, from string theory intuition. 3. What about dimension > 8? 4. One way to get gravitational instantons is by solving the vacuum Einstein equation in Lorentz space, and then performing a wick rotation (meaning t → it ). This worked in the case of Lorentzian Taub-NUT to get the Riemannian Taub-NUT. Which other instantons arrise this way?\n\nAnswer: In order to get a Riemannian metric, one needs t → t + c (time translation) be an isometry. Therefore it does work for Ak-cases but not for the others ( Dk, E6, E 7, E 8). 5. Can they arrive as \"non-trivial\" hyperk¨ ahler reductions of finite dimensional linear spaces?\n\nAnswer: Yes, for ALE spaces (Kronheimer's construction). Yes, for ALF if certain (Dancer) spaces are included with the linear spaces. 6. Is it true that every gravitational instanton has an asymptotic local triholomorphic isometry?\n\nReason: If you have a local R-action at infinity, then the metric is given in terms of a harmonic function, which arises from the hyperk¨ ahler moment map associated to this action (see the work of Gibbons). 7. When is it true that {Solutions of the Einstein equation } > {Self-Dual metrics }?\n\nReason: In general ≥.\n\nAnswers:\n\n(1) In the case of ALF the answer is >, since the Euclidean Schwarzschild metric is not self-dual. (2) Nakajima conjectured that in dimension 4, ALE and Ricci-flat implies self-dual. (3) In dimension > 4 and even, the conjecture is that ALE and Ricci-flat implies K¨ ahler. (4) (Carron) For odd dimensional manifolds an ALE Ricci-flat metric has to be flat since (a) by the Cheeger-Gromoll theorem this manifold has only one end (if not it splits isometrically as\n\nR × N with N compact); (b) The topology at infinity must be of the type ( R, ∞) × S2n/G where G is a finite subgroup of\n\nO(2 n + 1) acting freely on S2n. If g ∈ SO (2 n + 1) it must have 1 as an eigenvalue hence the only such G is\n\n{I, −I}; but the quotient is the real projective space, and there is no odd dimensional compact manifold whose boundary is the real projective space. Therefore G is trivial and the Bishop-Gromov inequality 2implies that M is the Euclidean space. Recall that the Bishop-Gromov inequality states that in a manifold\n\nM n with non-negative Ricci curvature r 7 → vol B(x, r )/w nrn is decreasing with equality everywhere if and only if M n is the Euclidean space. This ratio goes to 1 when r → 0, and in the ALE case we have that this ratio goes to 1 /card G when r → ∞.\n\nQuestions [of S. Cherkis]:",
   "evidence": "The canonical extraction begins:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-physics-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-PHYSICS-0031": {
   "statement_status": "exact",
   "original_statement": "1. There exists an explicit construction of Atiyah [1] and Page [10] for the Green's function on certain ALE \n\nand ALF spaces. Page's construction comes from physics and constructs by hand the Green's functions for both AkALE and AkALF. Atiyah has a formalism which in theory allows one to construct the Green's function on any self-dual gravitational instanton - this formalism involves Serre classes on the corresponding twistor spaces. In the paper he constructs only the Green's function in the AkALE case. (According to Etesi the use of Atiyah's Serre-class formalism is significantly more difficult for the ALF case and has not yet been carried out. The main problem he says is that one cannot find a nice compacification for the twistor space of an ALF self-dual space. Carrying out this construction would allow us to construct L2\n\nharmonic forms over the new DkALF gravitational instantons constructed by Cherkis and Hitchin.) \n\nQuestion: Is there a way to generalize this formalism to 8 dimensional hyperk¨ ahler manifolds? \n\nReason: In the non-compact 4-dimensional case or K3 case, the coefficient of the leading order term in the Green's function gave information about the values of the parameters (e.g. sizes of the cycles, metric). 2. The same story for quaternionic manifolds. How to modify Atiyah's formalism? \n\nReason: There is a reasonable twistor theory for these manifolds. 3. Is it true that any 4-dimensional hyperk¨ ahler manifold (non-compact, with reasonable decay at infinity) can be deformed to a quaternionic one? \n\nFor example: HK has Ric = 0, while quaternionic manifolds satisfy Ric = λg with λ a constant (non-zero cosmological constant). One would like to obtain any HK as a limit of quaternionic spaces with λ → 0. \n\nExample: R4 is the limit of HP 1 (positive λ's) and HH 1 (negative λ's). \n\n2 March 18, 2004 - S. Cherkis as moderator. \n\nQuestion (R. Mazzeo): \n\nWhat are the asymptotics at infinity of the multi-monopole moduli spaces? \n\nAnswer: (Cherkis) They should behave like a N-body problem. \n\nQuestions (K. Lee) \n\nFor these questions, R4 \n\n> B\n\ndenotes the non-commutative R4. Here B = Bμν dx μdx ν is a 2-form which gives the non-commutative structure. Then, [ xμ, x ν ] = Bμν. We define non-commutative instantons using the Nahm transform of instanton data \n\nμ1 = [ T1, T 2] − [T3, T 4] − j∗j = B12,\n\nplus two other similar equations (see the work Nekrasov-Schwarz on non-commutative instantons [9]). Let \n\nMk,n B be the moduli space of centered k-instantons on R4 \n\n> B\n\nwith gauge group G = U (n). (The fact that the instantons are centered means we consider only instantons with center of mass at the origin) 1. What is the instanton number in the non-commutative set-up? \n\nMotivation: The space Mk,n B is singular for B = 0 but non-singular for B 6 = 0. One would like to understand the limit as B → 0. 2. Does there exists a unique harmonic form in the middle dimension of Mk,n B?\n\nNote: The answer to this question is Yes for the following cases: (i) 1-instantons and any G;(ii) 2-instantons and G = U (1) (see the work of Lee et al). 33. Conjecture: The moduli space of centered 1-instantons on R3 × S1 \n\n> B\n\nwith \"given holonomy around S1\n\n> ∞\n\n\"(holonomy is not in the center of G = U (n)) has exactly n L 2-harmonic forms in middle dimension.",
   "clean_statement": "1. There exists an explicit construction of Atiyah [1] and Page [10] for the Green's function on certain ALE\n\nand ALF spaces. Page's construction comes from physics and constructs by hand the Green's functions for both AkALE and AkALF. Atiyah has a formalism which in theory allows one to construct the Green's function on any self-dual gravitational instanton - this formalism involves Serre classes on the corresponding twistor spaces. In the paper he constructs only the Green's function in the AkALE case. (According to Etesi the use of Atiyah's Serre-class formalism is significantly more difficult for the ALF case and has not yet been carried out. The main problem he says is that one cannot find a nice compacification for the twistor space of an ALF self-dual space. Carrying out this construction would allow us to construct L2\n\nharmonic forms over the new DkALF gravitational instantons constructed by Cherkis and Hitchin.)\n\nQuestion: Is there a way to generalize this formalism to 8 dimensional hyperk¨ ahler manifolds?\n\nReason: In the non-compact 4-dimensional case or K3 case, the coefficient of the leading order term in the Green's function gave information about the values of the parameters (e.g. sizes of the cycles, metric). 2. The same story for quaternionic manifolds. How to modify Atiyah's formalism?\n\nReason: There is a reasonable twistor theory for these manifolds. 3. Is it true that any 4-dimensional hyperk¨ ahler manifold (non-compact, with reasonable decay at infinity) can be deformed to a quaternionic one?\n\nFor example: HK has Ric = 0, while quaternionic manifolds satisfy Ric = λg with λ a constant (non-zero cosmological constant). One would like to obtain any HK as a limit of quaternionic spaces with λ → 0.\n\nExample: R4 is the limit of HP 1 (positive λ's) and HH 1 (negative λ's).\n\n2 March 18, 2004 - S. Cherkis as moderator.\n\nQuestion (R. Mazzeo):\n\nWhat are the asymptotics at infinity of the multi-monopole moduli spaces?\n\nAnswer: (Cherkis) They should behave like a N-body problem.\n\nQuestions (K. Lee)\n\nFor these questions, R4\n\n> B\n\ndenotes the non-commutative R4. Here B = Bμν dx μdx ν is a 2-form which gives the non-commutative structure. Then, [ xμ, x ν ] = Bμν. We define non-commutative instantons using the Nahm transform of instanton data\n\nμ1 = [ T1, T 2] − [T3, T 4] − j∗j = B12,\n\nplus two other similar equations (see the work Nekrasov-Schwarz on non-commutative instantons [9]). Let\n\nMk,n B be the moduli space of centered k-instantons on R4\n\n> B\n\nwith gauge group G = U (n). (The fact that the instantons are centered means we consider only instantons with center of mass at the origin) 1. What is the instanton number in the non-commutative set-up?\n\nMotivation: The space Mk,n B is singular for B = 0 but non-singular for B 6 = 0. One would like to understand the limit as B → 0. 2. Does there exists a unique harmonic form in the middle dimension of Mk,n B?\n\nNote: The answer to this question is Yes for the following cases: (i) 1-instantons and any G;(ii) 2-instantons and G = U (1) (see the work of Lee et al). 33. Conjecture: The moduli space of centered 1-instantons on R3 × S1\n\n> B\n\nwith \"given holonomy around S1\n\n> ∞\n\n\"(holonomy is not in the center of G = U (n)) has exactly n L 2-harmonic forms in middle dimension.",
   "public_statement": "1. There exists an explicit construction of Atiyah [1] and Page [10] for the Green's function on certain ALE\n\nand ALF spaces. Page's construction comes from physics and constructs by hand the Green's functions for both AkALE and AkALF. Atiyah has a formalism which in theory allows one to construct the Green's function on any self-dual gravitational instanton - this formalism involves Serre classes on the corresponding twistor spaces. In the paper he constructs only the Green's function in the AkALE case. (According to Etesi the use of Atiyah's Serre-class formalism is significantly more difficult for the ALF case and has not yet been carried out. The main problem he says is that one cannot find a nice compacification for the twistor space of an ALF self-dual space. Carrying out this construction would allow us to construct L2\n\nharmonic forms over the new DkALF gravitational instantons constructed by Cherkis and Hitchin.)\n\nQuestion: Is there a way to generalize this formalism to 8 dimensional hyperk¨ ahler manifolds?\n\nReason: In the non-compact 4-dimensional case or K3 case, the coefficient of the leading order term in the Green's function gave information about the values of the parameters (e.g. sizes of the cycles, metric). 2. The same story for quaternionic manifolds. How to modify Atiyah's formalism?\n\nReason: There is a reasonable twistor theory for these manifolds. 3. Is it true that any 4-dimensional hyperk¨ ahler manifold (non-compact, with reasonable decay at infinity) can be deformed to a quaternionic one?\n\nFor example: HK has Ric = 0, while quaternionic manifolds satisfy Ric = λg with λ a constant (non-zero cosmological constant). One would like to obtain any HK as a limit of quaternionic spaces with λ → 0.\n\nExample: R4 is the limit of HP 1 (positive λ's) and HH 1 (negative λ's).\n\n2 March 18, 2004 - S. Cherkis as moderator.\n\nQuestion (R. Mazzeo):\n\nWhat are the asymptotics at infinity of the multi-monopole moduli spaces?\n\nAnswer: (Cherkis) They should behave like a N-body problem.\n\nQuestions (K. Lee)\n\nFor these questions, R4\n\n> B\n\ndenotes the non-commutative R4. Here B = Bμν dx μdx ν is a 2-form which gives the non-commutative structure. Then, [ xμ, x ν ] = Bμν. We define non-commutative instantons using the Nahm transform of instanton data\n\nμ1 = [ T1, T 2] − [T3, T 4] − j∗j = B12,\n\nplus two other similar equations (see the work Nekrasov-Schwarz on non-commutative instantons [9]). Let\n\nMk,n B be the moduli space of centered k-instantons on R4\n\n> B\n\nwith gauge group G = U (n). (The fact that the instantons are centered means we consider only instantons with center of mass at the origin) 1. What is the instanton number in the non-commutative set-up?\n\nMotivation: The space Mk,n B is singular for B = 0 but non-singular for B 6 = 0. One would like to understand the limit as B → 0. 2. Does there exists a unique harmonic form in the middle dimension of Mk,n B?\n\nNote: The answer to this question is Yes for the following cases: (i) 1-instantons and any G;(ii) 2-instantons and G = U (1) (see the work of Lee et al). 33. Conjecture: The moduli space of centered 1-instantons on R3 × S1\n\n> B\n\nwith \"given holonomy around S1\n\n> ∞\n\n\"(holonomy is not in the center of G = U (n)) has exactly n L 2-harmonic forms in middle dimension.",
   "evidence": "The canonical JSON record is contaminated: after the genuine first item in S. Cherkis's list it contains separately numbered items 2 and 3, material from a March 18 session, and questions attributed to K. Lee and M. Singer. The original six-page AIM workshop PDF was checked directly. On page 2, under “Questions [of S. Cherkis],” item 1 discusses Page's explicit Green functions on certain ALE/ALF gravitational instantons and Atiyah's twistor/Serre-class construction on self-dual four-manifolds. The question and its immediately attached reason are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-PHYSICS-0032": {
   "statement_status": "exact",
   "original_statement": "1. When M is a 3-manifold are there some 2-dimensional objects that are related to Spec(∆) in a similar way? 2. Does there exist some analogue for the \"area spectrum\" of minimal surfaces in M 3 (with some suitable geometric assumptions imposed on M 3)? 3. Compute the number of minimal surfaces N (n1, n 2, g ) in T 4 = T 2 × T 2 endowed with the flat metric which represent the homology class n1[T1] + n2[T2], where Ti are the two 2-tori factors of T 4.",
   "clean_statement": "1. When M is a 3-manifold are there some 2-dimensional objects that are related to Spec(∆) in a similar way? 2. Does there exist some analogue for the \"area spectrum\" of minimal surfaces in M 3 (with some suitable geometric assumptions imposed on M 3)? 3. Compute the number of minimal surfaces N (n1, n 2, g ) in T 4 = T 2 × T 2 endowed with the flat metric which represent the homology class n1[T1] + n2[T2], where Ti are the two 2-tori factors of T 4.",
   "public_statement": "1. When M is a 3-manifold are there some 2-dimensional objects that are related to Spec(∆) in a similar way? 2. Does there exist some analogue for the \"area spectrum\" of minimal surfaces in M 3 (with some suitable geometric assumptions imposed on M 3)? 3. Compute the number of minimal surfaces N (n1, n 2, g ) in T 4 = T 2 × T 2 endowed with the flat metric which represent the homology class n1[T1] + n2[T2], where Ti are the two 2-tori factors of T 4.",
   "evidence": "The exact canonical `problem` field contains three numbered questions. This attempt owns only the first:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-PHYSICS-0033": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "1. Understand the asymptotics of these spaces (for a start see work of Bielawksi-Dancer [2] and Gibbons-Rychnenkova [6]).",
   "clean_statement": "develop systematic asymptotic descriptions of toric hyperkähler (hypertoric) quotients, in service of their \\(L^2\\) Hodge theory. The problem ranges over dimensions and quotient data. This report proves a quantitative result only for the classical four-dimensional Gibbons--Hawking finite-centre subfamily.",
   "public_statement": "1. Understand the asymptotics of these spaces (for a start see work of Bielawksi-Dancer [2] and Gibbons-Rychnenkova [6]).",
   "evidence": "The canonical record reads:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-physics-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-PHYSICS-0034": {
   "statement_status": "exact",
   "original_statement": "2. (Mazzeo) Combine asymptotics with N -body techniques.",
   "clean_statement": "2. (Mazzeo) Combine asymptotics with N -body techniques.",
   "public_statement": "2. (Mazzeo) Combine asymptotics with N -body techniques.",
   "evidence": "The raw canonical record is preserved in `input.json` and reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-PHYSICS-0035": {
   "statement_status": "exact",
   "original_statement": "3. (Hausel) We know that all the L2 Hodge cohomology of these spaces must be in the middle dimension and that any L2-harmonic form in middle dimension is ± self-dual. So it is enough to understand the \n\nL2-signature. This will have consequences for the sign of σL2 (X) and is related to combinatorics and matroids. \n\nMore general problem: Prove a suitable index theorem on such manifolds. Will there be contributions from subsytems?? \n\nQuestions (G. Etesi):",
   "clean_statement": "3. (Hausel) We know that all the L2 Hodge cohomology of these spaces must be in the middle dimension and that any L2-harmonic form in middle dimension is ± self-dual. So it is enough to understand the\n\nL2-signature. This will have consequences for the sign of σL2 (X) and is related to combinatorics and matroids.\n\nMore general problem: Prove a suitable index theorem on such manifolds. Will there be contributions from subsytems??\n\nQuestions (G. Etesi):",
   "public_statement": "3. (Hausel) We know that all the L2 Hodge cohomology of these spaces must be in the middle dimension and that any L2-harmonic form in middle dimension is ± self-dual. So it is enough to understand the\n\nL2-signature. This will have consequences for the sign of σL2 (X) and is related to combinatorics and matroids.\n\nMore general problem: Prove a suitable index theorem on such manifolds. Will there be contributions from subsytems??\n\nQuestions (G. Etesi):",
   "evidence": "The canonical record is item 3 in T. Hausel's question in the wrap-up session of the 2004 AIM workshop *\\(L^2\\) harmonic forms in geometry and string theory*. The preceding lines are necessary to resolve the phrase “these spaces.” They say that “toric” hyperkähler quotients are among the simplest complete hyperkähler metrics and ask for a systematic understanding of their Hodge cohomology. The recovered item is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-PHYSICS-0036": {
   "statement_status": "exact",
   "original_statement": "1. It is known that there exists a good compactification of ALF spaces for L2-cohomology. What about its twistor space Q?(a) Does it compactify? (b) Does the complex structure extend over the compactification? 2. Study the Yang-Mills functional on open manifolds. For example, for manifolds which are conformally equivalent to a manifold with one cylindrical end, the energy of Yang-Mills instantons (if finite) is con-gruent mod Z to one of the Chern-Simons invariants of the boundary. What about different asymptotic geometries (ALF spaces, etc)?",
   "clean_statement": "1. It is known that there exists a good compactification of ALF spaces for L2-cohomology. What about its twistor space Q?(a) Does it compactify? (b) Does the complex structure extend over the compactification? 2. Study the Yang-Mills functional on open manifolds. For example, for manifolds which are conformally equivalent to a manifold with one cylindrical end, the energy of Yang-Mills instantons (if finite) is con-gruent mod Z to one of the Chern-Simons invariants of the boundary. What about different asymptotic geometries (ALF spaces, etc)?",
   "public_statement": "1. It is known that there exists a good compactification of ALF spaces for L2-cohomology. What about its twistor space Q?(a) Does it compactify? (b) Does the complex structure extend over the compactification? 2. Study the Yang-Mills functional on open manifolds. For example, for manifolds which are conformally equivalent to a manifold with one cylindrical end, the energy of Yang-Mills instantons (if finite) is con-gruent mod Z to one of the Chern-Simons invariants of the boundary. What about different asymptotic geometries (ALF spaces, etc)?",
   "evidence": "The canonical JSON record is contaminated by the next numbered item. Inspection of page 4 of the original AIM workshop PDF gives the complete relevant text as follows (line wrapping and the typography of \\(L^2\\) are normalized, but the symbol \\(Q\\) is preserved):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-PHYSICS-0037": {
   "statement_status": "exact",
   "original_statement": "1. Is there a topological interpretation for the L2-cohomology of higher dimensional monopole modulii spaces? (i) What are the right tools to use here? (ii) Can we use Saper's L-module techniques? (see [11, Section 12]) 2. These questions involve complete manifolds with special holonomy. (i) Understand asymptotic geometry of these manifolds. 5(ii) Understand their L2-cohomology. (iii) Find general vanishing theorems (possibly even if we do not understand part 1 completely).",
   "clean_statement": "1. Is there a topological interpretation for the L2-cohomology of higher dimensional monopole modulii spaces? (i) What are the right tools to use here? (ii) Can we use Saper's L-module techniques? (see [11, Section 12]) 2. These questions involve complete manifolds with special holonomy. (i) Understand asymptotic geometry of these manifolds. 5(ii) Understand their L2-cohomology. (iii) Find general vanishing theorems (possibly even if we do not understand part 1 completely).",
   "public_statement": "1. Is there a topological interpretation for the L2-cohomology of higher dimensional monopole modulii spaces? (i) What are the right tools to use here? (ii) Can we use Saper's L-module techniques? (see [11, Section 12]) 2. These questions involve complete manifolds with special holonomy. (i) Understand asymptotic geometry of these manifolds. 5(ii) Understand their L2-cohomology. (iii) Find general vanishing theorems (possibly even if we do not understand part 1 completely).",
   "evidence": "The original AIM PDF places this record under the heading **“Questions (E. Hunsicker)”**. The genuine item 1 is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-PHYSICS-0038": {
   "statement_status": "exact",
   "original_statement": "Problem 1. Consider the three-body problem. Let E be the energy, C be the angular momentum, and L = C2E for circular Euler motion. An exchange orbit is an unbounded solution with the property that the distance between one mass m3 and the binary {m1, m 2}\n\ngoes to infinity as t → −∞, m3 exchange with m2 and the distance between m2 and the the binary {m1, m 3} goes to infinity as t → +∞. A solution has Hill type stability if the three bodies cannot undergo an exchange of binary. For exchange orbits, how close to 1 can C2E\n\n> L\n\nbe? (Christian Marchal)",
   "clean_statement": "Problem 1. Consider the three-body problem. Let E be the energy, C be the angular momentum, and L = C2E for circular Euler motion. An exchange orbit is an unbounded solution with the property that the distance between one mass m3 and the binary {m1, m 2}\n\ngoes to infinity as t → −∞, m3 exchange with m2 and the distance between m2 and the the binary {m1, m 3} goes to infinity as t → +∞. A solution has Hill type stability if the three bodies cannot undergo an exchange of binary. For exchange orbits, how close to 1 can C2E\n\n> L\n\nbe? (Christian Marchal)",
   "public_statement": "Problem 1. Consider the three-body problem. Let E be the energy, C be the angular momentum, and L = C2E for circular Euler motion. An exchange orbit is an unbounded solution with the property that the distance between one mass m3 and the binary {m1, m 2}\n\ngoes to infinity as t → −∞, m3 exchange with m2 and the distance between m2 and the the binary {m1, m 3} goes to infinity as t → +∞. A solution has Hill type stability if the three bodies cannot undergo an exchange of binary. For exchange orbits, how close to 1 can C2E\n\n> L\n\nbe? (Christian Marchal)",
   "evidence": "The canonical record is visibly flattened. I checked the compressed text stream of the official AIM PDF itself, not only the extracted JSON. The mathematical content on page 3 is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-PHYSICS-0039": {
   "statement_status": "exact",
   "original_statement": "Problem 2. Consider the planar restricted three-body problem. For Jacobi constants such that the Hill's region looks like a barbell, do transit orbits always exist? (Rick Moeckel)",
   "clean_statement": "Problem 2. Consider the planar restricted three-body problem. For Jacobi constants such that the Hill's region looks like a barbell, do transit orbits always exist? (Rick Moeckel)",
   "public_statement": "Problem 2. Consider the planar restricted three-body problem. For Jacobi constants such that the Hill's region looks like a barbell, do transit orbits always exist? (Rick Moeckel)",
   "evidence": "There is no substantive OCR corruption in the canonical record. The phrase “and moving” is grammatical in the PDF exactly as extracted. Figure 1 is a schematic tunnel with an invariant set and a crossing orbit; the text extraction retains only the label “invariant set” and caption. The use of a conserved Jacobi constant identifies the intended model as the **planar circular restricted three-body problem** (PCR3BP), not the elliptic restricted problem or the full three-body problem. This reading is confirmed by Moeckel's paper written in direct response to the question.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-PHYSICS-0040": {
   "statement_status": "exact",
   "original_statement": "Problem 3. Prove the existence of super-eight solution for the 4-body problem with equal masses. (Joseph Gerver)",
   "clean_statement": "Problem 3. Prove the existence of super-eight solution for the 4-body problem with equal masses. (Joseph Gerver)",
   "public_statement": "Problem 3. Prove the existence of super-eight solution for the 4-body problem with equal masses. (Joseph Gerver)",
   "evidence": "The source-verified AIM item is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-PHYSICS-0041": {
   "statement_status": "exact",
   "original_statement": "Problem \n4. (Due to Poincar´ e, 1896) Can we obtain periodic solutions by minimizing while fixing the kij 's? (Alain Chenciner)",
   "clean_statement": "Problem\n4. (Due to Poincar´ e, 1896) Can we obtain periodic solutions by minimizing while fixing the kij 's? (Alain Chenciner)",
   "public_statement": "Problem\n4. (Due to Poincar´ e, 1896) Can we obtain periodic solutions by minimizing while fixing the kij 's? (Alain Chenciner)",
   "evidence": "The canonical JSON record is visibly missing the sentence that defines its notation. I checked the immediately preceding paragraph on page 4 of the official AIM workshop PDF. The printed text reads “Poincare characterize homology class ...”; in normalized English, its mathematical content is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-PHYSICS-0042": {
   "statement_status": "exact",
   "original_statement": "Problem 5. Is the figure-8 orbit a minimizer with homology class (0, 0, 0)? (Alain Chenciner)",
   "clean_statement": "Problem 5. Is the figure-8 orbit a minimizer with homology class (0, 0, 0)? (Alain Chenciner)",
   "public_statement": "Problem 5. Is the figure-8 orbit a minimizer with homology class (0, 0, 0)? (Alain Chenciner)",
   "evidence": "The source-verified AIM item is exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-PHYSICS-0043": {
   "statement_status": "exact",
   "original_statement": "Problem 6. Prove the existence of Broucke-H´ enon orbit for the planar 3-body problem. Do Broucke-H´ enon orbits exist for small + big masses? Is the Broucke-H´ enon solution a minimizer in the class (1, 0, 1)? (Andrea Venturelli)",
   "clean_statement": "Problem 6. Prove the existence of Broucke-H´ enon orbit for the planar 3-body problem. Do Broucke-H´ enon orbits exist for small + big masses? Is the Broucke-H´ enon solution a minimizer in the class (1, 0, 1)? (Andrea Venturelli)",
   "public_statement": "Problem 6. Prove the existence of Broucke-H´ enon orbit for the planar 3-body problem. Do Broucke-H´ enon orbits exist for small + big masses? Is the Broucke-H´ enon solution a minimizer in the class (1, 0, 1)? (Andrea Venturelli)",
   "evidence": "The official AIM workshop PDF, *Variational Methods in Celestial Mechanics*, gives the following problem (Problem 6):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-PHYSICS-0044": {
   "statement_status": "exact",
   "original_statement": "Problem 7. Is (1, 0, −1) realized by a collision-free periodic orbit (see figure 3)? If exists, is it minimizing or not? Is trivial braid realized by a collision-free periodic solution? (Andrea Venturelli)",
   "clean_statement": "Problem 7. Is (1, 0, −1) realized by a collision-free periodic orbit (see figure 3)? If exists, is it minimizing or not? Is trivial braid realized by a collision-free periodic solution? (Andrea Venturelli)",
   "public_statement": "Problem 7. Is (1, 0, −1) realized by a collision-free periodic orbit (see figure 3)? If exists, is it minimizing or not? Is trivial braid realized by a collision-free periodic solution? (Andrea Venturelli)",
   "evidence": "The official AIM workshop PDF gives the needed definition immediately before Problems 4--7:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-PHYSICS-0045": {
   "statement_status": "exact",
   "original_statement": "Problem 8. Can any symbol sequence be realized in the planar three-body problem? Are they minimizing or not? Do they have zero angular momentum? (Richard Montgomery)",
   "clean_statement": "Problem 8. Can any symbol sequence be realized in the planar three-body problem? Are they minimizing or not? Do they have zero angular momentum? (Richard Montgomery)",
   "public_statement": "Problem 8. Can any symbol sequence be realized in the planar three-body problem? Are they minimizing or not? Do they have zero angular momentum? (Richard Montgomery)",
   "evidence": "The source-verified AIM item reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-PHYSICS-0046": {
   "statement_status": "exact",
   "original_statement": "Problem 9. Often the minimizers in a symmetry class has more symmetry that the class asks for. Is that a coincidence? (Alain Chenciner)",
   "clean_statement": "Problem 9. Often the minimizers in a symmetry class has more symmetry that the class asks for. Is that a coincidence? (Alain Chenciner)",
   "public_statement": "Problem 9. Often the minimizers in a symmetry class has more symmetry that the class asks for. Is that a coincidence? (Alain Chenciner)",
   "evidence": "The canonical record is Problem 9 in the AIM workshop notes *Variational Methods in Celestial Mechanics*, available at <https://aimath.org/WWN/varcelest/varcelest.pdf>. The printed source reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-PHYSICS-0047": {
   "statement_status": "exact",
   "original_statement": "Problem 10. Is the central configuration for equal masses with minimal potential on the ellipsoid of moment of inertia 1 necessarily symmetric? (Rick Moeckel)",
   "clean_statement": "Problem 10. Is the central configuration for equal masses with minimal potential on the ellipsoid of moment of inertia 1 necessarily symmetric? (Rick Moeckel)",
   "public_statement": "Problem 10. Is the central configuration for equal masses with minimal potential on the ellipsoid of moment of inertia 1 necessarily symmetric? (Rick Moeckel)",
   "evidence": "Thus the number \\(47\\) is present in the source and is not an OCR error. The PDF is dated 2003 and attributes the notes to Kuo-Chang Chen. It gives no coordinates, proof, citation, ambient dimension, or definition of “symmetric,” “orbit,” or “normalized potential.” In particular, the \\(n=47\\) sentence is evidence of an informal computation or example known at the workshop, not a verifiable counterexample by itself.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-PHYSICS-0048": {
   "statement_status": "exact",
   "original_statement": "Problem 11. Do choreographies imply equal masses? (Alain Chenciner)",
   "clean_statement": "Problem 11. Do choreographies imply equal masses? (Alain Chenciner)",
   "public_statement": "Problem 11. Do choreographies imply equal masses? (Alain Chenciner)",
   "evidence": "The wording and all three remarks were checked against the AIM PDF; there is no OCR corruption in this record.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-PHYSICS-0049": {
   "statement_status": "exact",
   "original_statement": "Problem 12. Existence of choreographies with distinct time shifts.",
   "clean_statement": "Problem 12. Existence of choreographies with distinct time shifts.",
   "public_statement": "Problem 12. Existence of choreographies with distinct time shifts.",
   "evidence": "The canonical record is Problem 12 in the AIM workshop notes *Variational Methods in Celestial Mechanics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-PHYSICS-0050": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 13. What subgroups of O(2) ×O(d)×Sn can be realized as symmetries of solutions? (Davide Ferrario) Can symmetry group arise differently? (Alain Chenciner)",
   "clean_statement": null,
   "public_statement": "Problem 13. What subgroups of O(2) ×O(d)×Sn can be realized as symmetries of solutions? (Davide Ferrario) Can symmetry group arise differently? (Alain Chenciner)",
   "evidence": "The official AIM PDF *Variational Methods in Celestial Mechanics*, version dated June 22, 2003, gives:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-physics-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-PHYSICS-0051": {
   "statement_status": "exact",
   "original_statement": "Problem 14. Is there a conceptual proof for Saari's conjecture? Why not fix the moment of inertia tensor and ask the same question (maybe in higher dimensions)? (Alain Chenciner)",
   "clean_statement": "Problem 14. Is there a conceptual proof for Saari's conjecture? Why not fix the moment of inertia tensor and ask the same question (maybe in higher dimensions)? (Alain Chenciner)",
   "public_statement": "Problem 14. Is there a conceptual proof for Saari's conjecture? Why not fix the moment of inertia tensor and ask the same question (maybe in higher dimensions)? (Alain Chenciner)",
   "evidence": "The wording was checked against the AIM workshop PDF. No OCR correction is needed. The mathematical conventions are not stated in the one-line prompt, so the following standard reading is used.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-PHYSICS-0052": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 15. Are there solutions to the n-body problem that fall in a certain affine class that are not relative equilibrium or some symmetric solutions? (c.f. Gerver's super-8) (Alain Chenciner) 6",
   "clean_statement": null,
   "public_statement": "Problem 15. Are there solutions to the n-body problem that fall in a certain affine class that are not relative equilibrium or some symmetric solutions? (c.f. Gerver's super-8) (Alain Chenciner) 6",
   "evidence": "This reconstruction is consistent with the super-eight comparison: the equal-mass four-body super-eight belongs to the centrally symmetric/parallelogram class. It is not verified as Chenciner's uniquely intended definition, so all conclusions below are conditional on this reading.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-physics-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-PHYSICS-0053": {
   "statement_status": "exact",
   "original_statement": "Problem 16. Is the only solution satisfying (i), (ii) and having no syzygies (i.e. eclipses) the Lagrange homothety solutions? (i) Energy < 0. (ii) Angular momentum = \n0. (Richard Montgomery)",
   "clean_statement": "Problem 16. Is the only solution satisfying (i), (ii) and having no syzygies (i.e. eclipses) the Lagrange homothety solutions? (i) Energy < 0. (ii) Angular momentum =\n0. (Richard Montgomery)",
   "public_statement": "Problem 16. Is the only solution satisfying (i), (ii) and having no syzygies (i.e. eclipses) the Lagrange homothety solutions? (i) Energy < 0. (ii) Angular momentum =\n0. (Richard Montgomery)",
   "evidence": "The canonical record is Problem 16 from the AIM workshop list *Variational Methods in Celestial Mechanics*. It reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-PHYSICS-0054": {
   "statement_status": "exact",
   "original_statement": "Problem 17. Problems on the shape space for the three-body problem: (Wu-Yi Hsiang) - Are there solutions whose α-(or ω-)limit set is a limit cycle? - The area of the unit shape sphere is π. Conjecture: for any 0 < A 0 ≤ π, there is a solution whose shape curve has its closure with area A0.- Can two closed shape curves with the same homotopy class be deformed to one another by closed shape curves? Conjecture: No. But then determine connected components.",
   "clean_statement": "Problem 17. Problems on the shape space for the three-body problem: (Wu-Yi Hsiang) - Are there solutions whose α-(or ω-)limit set is a limit cycle? - The area of the unit shape sphere is π. Conjecture: for any 0 < A 0 ≤ π, there is a solution whose shape curve has its closure with area A0.- Can two closed shape curves with the same homotopy class be deformed to one another by closed shape curves? Conjecture: No. But then determine connected components.",
   "public_statement": "Problem 17. Problems on the shape space for the three-body problem: (Wu-Yi Hsiang) - Are there solutions whose α-(or ω-)limit set is a limit cycle? - The area of the unit shape sphere is π. Conjecture: for any 0 < A 0 ≤ π, there is a solution whose shape curve has its closure with area A0.- Can two closed shape curves with the same homotopy class be deformed to one another by closed shape curves? Conjecture: No. But then determine connected components.",
   "evidence": "The canonical record is a flattened extraction of Problem 17 in the AIM workshop list *Variational Methods in Celestial Mechanics*. The official PDF, including its page layout, was inspected on PDF page 6. It contains three separate en-dash bullets:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-PHYSICS-0055": {
   "statement_status": "exact",
   "original_statement": "Problem 18. Let ∆ be the area of the triangle, I be the moment of inertia. Find periodic solutions to the (planar or spatial) three-body problem such that ∆ \n\n> I\n\nis a constant. (Wu-Yi Hsiang)",
   "clean_statement": "Problem 18. Let ∆ be the area of the triangle, I be the moment of inertia. Find periodic solutions to the (planar or spatial) three-body problem such that ∆\n\n> I\n\nis a constant. (Wu-Yi Hsiang)",
   "public_statement": "Problem 18. Let ∆ be the area of the triangle, I be the moment of inertia. Find periodic solutions to the (planar or spatial) three-body problem such that ∆\n\n> I\n\nis a constant. (Wu-Yi Hsiang)",
   "evidence": "The exact canonical extraction is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-PHYSICS-0056": {
   "statement_status": "exact",
   "original_statement": "Problem 19. What can be said about the volume of the tetrahedron in the four-body problem? Can it be nonzero forever and stay bounded away from 0 and ∞? (Joseph Gerver)",
   "clean_statement": "Problem 19. What can be said about the volume of the tetrahedron in the four-body problem? Can it be nonzero forever and stay bounded away from 0 and ∞? (Joseph Gerver)",
   "public_statement": "Problem 19. What can be said about the volume of the tetrahedron in the four-body problem? Can it be nonzero forever and stay bounded away from 0 and ∞? (Joseph Gerver)",
   "evidence": "The canonical record is Problem 19 from the AIM workshop *Variational Methods in Celestial Mechanics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-PHYSICS-0057": {
   "statement_status": "exact",
   "original_statement": "Problem 20. For circular restricted three-body problem, there is a Jacobi integral. For a complete integrability, need two more integrals. Extra integral exists when m1 + m2 = 0. Is there one more integral? (Christian Marchal)",
   "clean_statement": "Problem 20. For circular restricted three-body problem, there is a Jacobi integral. For a complete integrability, need two more integrals. Extra integral exists when m1 + m2 = 0. Is there one more integral? (Christian Marchal)",
   "public_statement": "Problem 20. For circular restricted three-body problem, there is a Jacobi integral. For a complete integrability, need two more integrals. Extra integral exists when m1 + m2 = 0. Is there one more integral? (Christian Marchal)",
   "evidence": "The official AIM PDF, *Variational Methods in Celestial Mechanics*, version of 22 June 2003, p. 6 of the PDF (printed page 6), was inspected directly. It says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-PHYSICS-0058": {
   "statement_status": "exact",
   "original_statement": "Problem 21. Does there exist potential U (x) in the plane such that if ˙ x(0) is tangential to the level curve of U, then U remains constant? (Mark Levi)",
   "clean_statement": "Problem 21. Does there exist potential U (x) in the plane such that if ˙ x(0) is tangential to the level curve of U, then U remains constant? (Mark Levi)",
   "public_statement": "Problem 21. Does there exist potential U (x) in the plane such that if ˙ x(0) is tangential to the level curve of U, then U remains constant? (Mark Levi)",
   "evidence": "The corpus text has placed the dot before the letter, `˙ x(0)`; inspection of the original PDF verifies that the intended symbol is \\(\\dot x(0)\\). No other substantive OCR correction is needed.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-PHYSICS-0059": {
   "statement_status": "exact",
   "original_statement": "Problem 22. Assume E = 0 and angular momentum = 0. Can Jacobi's metric give new insights beyond what McGehee's coordinates give? (Richard Montgomery)",
   "clean_statement": "Problem 22. Assume E = 0 and angular momentum = 0. Can Jacobi's metric give new insights beyond what McGehee's coordinates give? (Richard Montgomery)",
   "public_statement": "Problem 22. Assume E = 0 and angular momentum = 0. Can Jacobi's metric give new insights beyond what McGehee's coordinates give? (Richard Montgomery)",
   "evidence": "The canonical record agrees with the PDF; no OCR correction is needed.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-PHYSICS-0060": {
   "statement_status": "exact",
   "original_statement": "Problem 23. Among the many interesting open questions is the understanding of the nat-ural limits of the minimization method: (Alain Chenciner) - To what extent is it connected to symmetry constraints? - May topological constraints be imposed without forcing the minimizers to have collisions? - To what extent interesting results may be obtained for arbitrary masses? - What can be said of the hyperbolic and elliptic dimensions of a minimizer? - Is it interesting to look at minimizers with fixed energy?",
   "clean_statement": "Problem 23. Among the many interesting open questions is the understanding of the nat-ural limits of the minimization method: (Alain Chenciner) - To what extent is it connected to symmetry constraints? - May topological constraints be imposed without forcing the minimizers to have collisions? - To what extent interesting results may be obtained for arbitrary masses? - What can be said of the hyperbolic and elliptic dimensions of a minimizer? - Is it interesting to look at minimizers with fixed energy?",
   "public_statement": "Problem 23. Among the many interesting open questions is the understanding of the nat-ural limits of the minimization method: (Alain Chenciner) - To what extent is it connected to symmetry constraints? - May topological constraints be imposed without forcing the minimizers to have collisions? - To what extent interesting results may be obtained for arbitrary masses? - What can be said of the hyperbolic and elliptic dimensions of a minimizer? - Is it interesting to look at minimizers with fixed energy?",
   "evidence": "The canonical record is Problem 23 in the 2003 AIM workshop list *Variational Methods in Celestial Mechanics*. The exact record is preserved in `input.json`. Inspection of page 6 of the official PDF shows that the corpus string `nat-ural` is only line-break hyphenation. With that typographical repair, the source reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-PHYSICS-0061": {
   "statement_status": "exact",
   "original_statement": "Problem 24. For the n-body problem, n ≥ 4, show that the number of central configurations is finite for all choices of masses mi > 0 (or find a counterexample). (Rick Moeckel, Marshall Hampton)",
   "clean_statement": "Problem 24. For the n-body problem, n ≥ 4, show that the number of central configurations is finite for all choices of masses mi > 0 (or find a counterexample). (Rick Moeckel, Marshall Hampton)",
   "public_statement": "Problem 24. For the n-body problem, n ≥ 4, show that the number of central configurations is finite for all choices of masses mi > 0 (or find a counterexample). (Rick Moeckel, Marshall Hampton)",
   "evidence": "The source adds that for \\(n>4\\) even generic finiteness was then open, and that the question can be posed algebraically. Comparison with the original PDF found no OCR corruption or missing symbol.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-PHYSICS-0062": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 25. Give a sharp upper bound for the Morse index of a nonplanar central config-uration (as a critical point of the potential on the normalized configuration space). Give a sharp lower bound for the Morse index of a planar central configuration when it is viewed 7\n\nas part of the nonplanar configuration space. Is the Morse index related to the stability of the rigidly rotating periodic orbits? For example, does a linearly stable relative equilibrium necessarily arise from a minimum of the Newtonian potential? (Rick Moeckel)",
   "clean_statement": null,
   "public_statement": "Problem 25. Give a sharp upper bound for the Morse index of a nonplanar central config-uration (as a critical point of the potential on the normalized configuration space). Give a sharp lower bound for the Morse index of a planar central configuration when it is viewed 7\n\nas part of the nonplanar configuration space. Is the Morse index related to the stability of the rigidly rotating periodic orbits? For example, does a linearly stable relative equilibrium necessarily arise from a minimum of the Newtonian potential? (Rick Moeckel)",
   "evidence": "The canonical record is Problem 25 from the AIM workshop list *Variational Methods in Celestial Mechanics*. The official PDF, pages 5--6, gives the following statement after repairing only extraction artifacts:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-physics-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-PHYSICS-0063": {
   "statement_status": "exact",
   "original_statement": "Problem 26. Existence proof for homoclinic and heteroclinic orbits between unstable rela-tive equilibrium solutions. (Rick Moeckel)",
   "clean_statement": "Problem 26. Existence proof for homoclinic and heteroclinic orbits between unstable rela-tive equilibrium solutions. (Rick Moeckel)",
   "public_statement": "Problem 26. Existence proof for homoclinic and heteroclinic orbits between unstable rela-tive equilibrium solutions. (Rick Moeckel)",
   "evidence": "The canonical record is Problem 26 from the AIM workshop list *Variational Methods in Celestial Mechanics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-PHYSICS-0064": {
   "statement_status": "exact",
   "original_statement": "Problem 27. Existence proof for the \"halo orbits\" of the three-dimensional restricted three-body problem. (Rick Moeckel)",
   "clean_statement": "Problem 27. Existence proof for the \"halo orbits\" of the three-dimensional restricted three-body problem. (Rick Moeckel)",
   "public_statement": "Problem 27. Existence proof for the \"halo orbits\" of the three-dimensional restricted three-body problem. (Rick Moeckel)",
   "evidence": "The canonical record is Problem 27 from the AIM workshop *Variational Methods in Celestial Mechanics* (PDF version dated June 22, 2003):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-PHYSICS-0065": {
   "statement_status": "exact",
   "original_statement": "Problem 28. Consider negative energy three-body orbits which are unbound in both direc-tions of time. The two Jacobi vectors of such an orbit then asymptote to Keplerian orbits in both the distant past and the distant future, and so associated to such an orbit we have pair of Kepler elements in the past (one elliptic the other hyperbolic) and another pair in the future. The \"direct scattering\" problem is: which pair of Kepler elements can be connected to each other in this way? (Richard Montgomery)",
   "clean_statement": "Problem 28. Consider negative energy three-body orbits which are unbound in both direc-tions of time. The two Jacobi vectors of such an orbit then asymptote to Keplerian orbits in both the distant past and the distant future, and so associated to such an orbit we have pair of Kepler elements in the past (one elliptic the other hyperbolic) and another pair in the future. The \"direct scattering\" problem is: which pair of Kepler elements can be connected to each other in this way? (Richard Montgomery)",
   "public_statement": "Problem 28. Consider negative energy three-body orbits which are unbound in both direc-tions of time. The two Jacobi vectors of such an orbit then asymptote to Keplerian orbits in both the distant past and the distant future, and so associated to such an orbit we have pair of Kepler elements in the past (one elliptic the other hyperbolic) and another pair in the future. The \"direct scattering\" problem is: which pair of Kepler elements can be connected to each other in this way? (Richard Montgomery)",
   "evidence": "The canonical record is Problem 28 in the AIM workshop list *Variational Methods in Celestial Mechanics*. Inspection of page 7 of the official PDF confirms that the only textual defect is line-break hyphenation: “direc-tions” means “directions.” The exact unmodified record remains in input.json. With only that repair, the problem reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-PHYSICS-0066": {
   "statement_status": "exact",
   "original_statement": "Problem 29. Given three different masses that are comparable in size, prove the existence of prograde orbits without assuming one ratio of mutual distances is nearly zero. (Kuo-Chang Chen)",
   "clean_statement": "Problem 29. Given three different masses that are comparable in size, prove the existence of prograde orbits without assuming one ratio of mutual distances is nearly zero. (Kuo-Chang Chen)",
   "public_statement": "Problem 29. Given three different masses that are comparable in size, prove the existence of prograde orbits without assuming one ratio of mutual distances is nearly zero. (Kuo-Chang Chen)",
   "evidence": "The canonical record is Problem 29 in the AIM workshop list *Variational Methods in Celestial Mechanics* (version dated June 22, 2003):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-physics-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-PROBABILITY-0001": {
   "statement_status": "exact",
   "original_statement": "Is $RBM_{4,3}$ a universal approximator?\n\nDoes the closure of $RBM_{4,3}$ fill the simplex $\\Delta_{15},$ making it a \\textit{universal approximator}?",
   "clean_statement": "Is $RBM_{4,3}$ a universal approximator?\n\nDoes the closure of $RBM_{4,3}$ fill the simplex $\\Delta_{15},$ making it a \\textit{universal approximator}?",
   "public_statement": "Is $RBM_{4,3}$ a universal approximator?\n\nDoes the closure of $RBM_{4,3}$ fill the simplex $\\Delta_{15},$ making it a \\textit{universal approximator}?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-PROBABILITY-0002": {
   "statement_status": "exact",
   "original_statement": "Determine the maximum divergence of $RBM_{3,1}.$\n\nDetermine the maximum divergence of $RBM_{3,1}.$",
   "clean_statement": "Determine the maximum divergence of $RBM_{3,1}.$\n\nDetermine the maximum divergence of $RBM_{3,1}.$",
   "public_statement": "Determine the maximum divergence of $RBM_{3,1}.$\n\nDetermine the maximum divergence of $RBM_{3,1}.$",
   "evidence": "The canonical record contains the sentence",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-PROBABILITY-0003": {
   "statement_status": "exact",
   "original_statement": "What kind of distributions can be represented by an RBM as opposed to directed models?\n\nWhat kind of distributions can be represented by an RBM as opposed to directed models?",
   "clean_statement": "What kind of distributions can be represented by an RBM as opposed to directed models?\n\nWhat kind of distributions can be represented by an RBM as opposed to directed models?",
   "public_statement": "What kind of distributions can be represented by an RBM as opposed to directed models?\n\nWhat kind of distributions can be represented by an RBM as opposed to directed models?",
   "evidence": "The canonical record is item 1.3, in the section “Representational Capacity” of the AIM workshop list *Boltzmann Machines*. Its `problem` field is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-PROBABILITY-0004": {
   "statement_status": "exact",
   "original_statement": "Characterize the limiting set of distributions for $\\lim_{n\\to\\infty} RBM_{n,n}.$\n\nConsider the limit as $n \\to \\infty$ for $RBM_{n,n}.$ It was suggested that this limiting set of distributions can be seen as a set of measures on the unit interval. Do all possible measures belong to this set? If not, which ones do?",
   "clean_statement": "Characterize the limiting set of distributions for $\\lim_{n\\to\\infty} RBM_{n,n}.$\n\nConsider the limit as $n \\to \\infty$ for $RBM_{n,n}.$ It was suggested that this limiting set of distributions can be seen as a set of measures on the unit interval. Do all possible measures belong to this set? If not, which ones do?",
   "public_statement": "Characterize the limiting set of distributions for $\\lim_{n\\to\\infty} RBM_{n,n}.$\n\nConsider the limit as $n \\to \\infty$ for $RBM_{n,n}.$ It was suggested that this limiting set of distributions can be seen as a set of measures on the unit interval. Do all possible measures belong to this set? If not, which ones do?",
   "evidence": "The exact canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-PROBABILITY-0005": {
   "statement_status": "exact",
   "original_statement": "Can we characterize tensors of non-negative rank at most 3 using the description of $RBM_{3,2}?$\n\nIs it possible to use the implicit description of $RBM_{3,2}$ to determine the characterizing properties of non-negative rank $\\leq 3$ tensors?",
   "clean_statement": "Can we characterize tensors of non-negative rank at most 3 using the description of $RBM_{3,2}?$\n\nIs it possible to use the implicit description of $RBM_{3,2}$ to determine the characterizing properties of non-negative rank $\\leq 3$ tensors?",
   "public_statement": "Can we characterize tensors of non-negative rank at most 3 using the description of $RBM_{3,2}?$\n\nIs it possible to use the implicit description of $RBM_{3,2}$ to determine the characterizing properties of non-negative rank $\\leq 3$ tensors?",
   "evidence": "The canonical AIM record contains two equivalent questions:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-PROBABILITY-0006": {
   "statement_status": "exact",
   "original_statement": "What are the algebraic structures relevant for studying learning algorithms for RBMs?\n\nWhat are the algebraic structures relevant for studying learning algorithms for RBMs?",
   "clean_statement": "What are the algebraic structures relevant for studying learning algorithms for RBMs?\n\nWhat are the algebraic structures relevant for studying learning algorithms for RBMs?",
   "public_statement": "What are the algebraic structures relevant for studying learning algorithms for RBMs?\n\nWhat are the algebraic structures relevant for studying learning algorithms for RBMs?",
   "evidence": "The canonical record is item 2.2 in the AIM *Boltzmann Machines* section “Algebraic Statistics and Tensor Characterizations.” Its problem field is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-PROBABILITY-0007": {
   "statement_status": "exact",
   "original_statement": "Efficient methods to train RBMs with fixed, sparse connectivity\n\nWhat are the most efficient learning algorithms for training sparsely connected RBMs?",
   "clean_statement": "Efficient methods to train RBMs with fixed, sparse connectivity\n\nWhat are the most efficient learning algorithms for training sparsely connected RBMs?",
   "public_statement": "Efficient methods to train RBMs with fixed, sparse connectivity\n\nWhat are the most efficient learning algorithms for training sparsely connected RBMs?",
   "evidence": "The canonical record is AIM Problem Lists, *Boltzmann Machines*, section 3.1, “Effects of Network Connectivity”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-PROBABILITY-0008": {
   "statement_status": "exact",
   "original_statement": "How do RBMs with quantum effects differ from classical RBMs?\n\nHow are the following models different from classical RBMs:\n\n1) RBMs with sparse connectivity.\n\n2) RBMs with sparse connectivity, and with quantum effects.\n\n3) RBMs with sparse connectivity, and with quantum effects and quantum training algorithm.",
   "clean_statement": "How do RBMs with quantum effects differ from classical RBMs?\n\nHow are the following models different from classical RBMs:\n\n1) RBMs with sparse connectivity.\n\n2) RBMs with sparse connectivity, and with quantum effects.\n\n3) RBMs with sparse connectivity, and with quantum effects and quantum training algorithm.",
   "public_statement": "How do RBMs with quantum effects differ from classical RBMs?\n\nHow are the following models different from classical RBMs:\n\n1) RBMs with sparse connectivity.\n\n2) RBMs with sparse connectivity, and with quantum effects.\n\n3) RBMs with sparse connectivity, and with quantum effects and quantum training algorithm.",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-PROBABILITY-0009": {
   "statement_status": "exact",
   "original_statement": "Given a fixed connectivity structure, how many inference functions can an RBM model compute?\n\nHow many inference functions can an RBM model compute, constrained by a fixed connectivity structure?",
   "clean_statement": "Given a fixed connectivity structure, how many inference functions can an RBM model compute?\n\nHow many inference functions can an RBM model compute, constrained by a fixed connectivity structure?",
   "public_statement": "Given a fixed connectivity structure, how many inference functions can an RBM model compute?\n\nHow many inference functions can an RBM model compute, constrained by a fixed connectivity structure?",
   "evidence": "The canonical AIM record (Boltzmann Machines workshop, section “Effects of Network Connectivity,” problem 3.3) contains:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-PROBABILITY-0010": {
   "statement_status": "exact",
   "original_statement": "Given two RBMs with the same number of units but different connectivities, how much do these statistical models overlap?\n\nHow to quantify function approximation given a network topology? In other words, how to measure the proximity between two functions when each are computed by separate networks? Then, can one use this measure to quantify the effects of altering a network's connectivity structure?",
   "clean_statement": "Given two RBMs with the same number of units but different connectivities, how much do these statistical models overlap?\n\nHow to quantify function approximation given a network topology? In other words, how to measure the proximity between two functions when each are computed by separate networks? Then, can one use this measure to quantify the effects of altering a network's connectivity structure?",
   "public_statement": "Given two RBMs with the same number of units but different connectivities, how much do these statistical models overlap?\n\nHow to quantify function approximation given a network topology? In other words, how to measure the proximity between two functions when each are computed by separate networks? Then, can one use this measure to quantify the effects of altering a network's connectivity structure?",
   "evidence": "The accompanying note says that the question was motivated by biological brains having fine-scale topological differences but broadly similar computational properties. The source record is internally coherent and contains no apparent OCR corruption. Nearby records concern the number of inference functions available under fixed connectivity and related representational questions, which supports reading “function” here as the visible probability function or its log-weight/free-energy representative.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-PROBABILITY-0011": {
   "statement_status": "exact",
   "original_statement": "Is it true that tropical RBMs always have expected dimension?\n\nIs it true that tropical RBMs always have expected dimension, meaning equal to the number of parameters within the model or the dimension of the ambient probability simplex?",
   "clean_statement": "Is it true that tropical RBMs always have expected dimension?\n\nIs it true that tropical RBMs always have expected dimension, meaning equal to the number of parameters within the model or the dimension of the ambient probability simplex?",
   "public_statement": "Is it true that tropical RBMs always have expected dimension?\n\nIs it true that tropical RBMs always have expected dimension, meaning equal to the number of parameters within the model or the dimension of the ambient probability simplex?",
   "evidence": "The two sentences are a short question followed by its clarification; there is no apparent OCR corruption. The original URL, <http://aimpl.org/boltzmann/4/>, returned a 502 gateway error when checked on 11 August 2026. The terminology and parameter count agree exactly with the conjecture of Cueto, Morton, and Sturmfels [CMS10].",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-PROBABILITY-0012": {
   "statement_status": "exact",
   "original_statement": "What tropical objects characterize all tropical RBMs?\n\nWhat tropical objects (varieties) characterize all tropical RBMs?",
   "clean_statement": "What tropical objects characterize all tropical RBMs?\n\nWhat tropical objects (varieties) characterize all tropical RBMs?",
   "public_statement": "What tropical objects characterize all tropical RBMs?\n\nWhat tropical objects (varieties) characterize all tropical RBMs?",
   "evidence": "The canonical AIM record is from the 2018 Boltzmann Machines workshop, section “Tropical RBMs,” problem 4.2. It reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-PROBABILITY-0013": {
   "statement_status": "exact",
   "original_statement": "Characterizing the optimization landscape for RBMs\n\nHow do we characterize local optima on the optimization landscape for RBMs as a function of the architecture and number of empirical samples?",
   "clean_statement": "Characterizing the optimization landscape for RBMs\n\nHow do we characterize local optima on the optimization landscape for RBMs as a function of the architecture and number of empirical samples?",
   "public_statement": "Characterizing the optimization landscape for RBMs\n\nHow do we characterize local optima on the optimization landscape for RBMs as a function of the architecture and number of empirical samples?",
   "evidence": "The canonical AIM record, from the 2018 workshop *Boltzmann Machines*, section “RBM Optimization,” problem 5.2, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-PROBABILITY-0014": {
   "statement_status": "exact",
   "original_statement": "What are the critical points of the EM algorithm for $RBM_{3,2}$?\n\nWhat are the critical points of the EM algorithm for $RBM_{3,2}$? Can the MLE degree for the strata of $RBM_{3,2}$ be determined?",
   "clean_statement": "What are the critical points of the EM algorithm for $RBM_{3,2}$?\n\nWhat are the critical points of the EM algorithm for $RBM_{3,2}$? Can the MLE degree for the strata of $RBM_{3,2}$ be determined?",
   "public_statement": "What are the critical points of the EM algorithm for $RBM_{3,2}$?\n\nWhat are the critical points of the EM algorithm for $RBM_{3,2}$? Can the MLE degree for the strata of $RBM_{3,2}$ be determined?",
   "evidence": "The canonical AIM record is from the 2018 Boltzmann Machines workshop, section “RBM Optimization,” problem 5.1. It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-PROBABILITY-0015": {
   "statement_status": "exact",
   "original_statement": "Is using dropout during RBM training equivalent to adding a regularizing penalty term on to the training objective function?\n\nIs using dropout during RBM training equivalent to adding a regularizing penalty term on to the training objective function?",
   "clean_statement": "Is using dropout during RBM training equivalent to adding a regularizing penalty term on to the training objective function?\n\nIs using dropout during RBM training equivalent to adding a regularizing penalty term on to the training objective function?",
   "public_statement": "Is using dropout during RBM training equivalent to adding a regularizing penalty term on to the training objective function?\n\nIs using dropout during RBM training equivalent to adding a regularizing penalty term on to the training objective function?",
   "evidence": "The canonical AIM record (Boltzmann Machines workshop, section “RBM Optimization,” Problem 5.3) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-PROBABILITY-0016": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "How do we use the Wasserstein geometry on RMBs to approximation PDE solutions?\n\nHow do we use the Wasserstein geometry on RMBs to approximation PDE solutions?",
   "clean_statement": null,
   "public_statement": "How do we use the Wasserstein geometry on RMBs to approximation PDE solutions?\n\nHow do we use the Wasserstein geometry on RMBs to approximation PDE solutions?",
   "evidence": "This is visibly corrupt: “RMBs” is almost certainly “RBMs,” and “to approximation” is almost certainly “to approximate.” The original AIM problem-list page was unavailable during this run, so those corrections could not be verified against the original wording. The official 2018 AIM workshop report does verify the surrounding context: a working group on Wasserstein distance and optimal transport studied Wasserstein natural gradients for Boltzmann machines [AIM18]. Accordingly, the **plausible but not source-verified reconstruction** used here is:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-PROBABILITY-0017": {
   "statement_status": "exact",
   "original_statement": "Can we design Boltzmann machines that are interpretable?\n\nCan we design Boltzmann machines that are interpretable? In other words, can we construct useful Boltzmann machines that admit a simple explanation in human terms, and behaves in a predictable manner with certainty?",
   "clean_statement": "Can we design Boltzmann machines that are interpretable?\n\nCan we design Boltzmann machines that are interpretable? In other words, can we construct useful Boltzmann machines that admit a simple explanation in human terms, and behaves in a predictable manner with certainty?",
   "public_statement": "Can we design Boltzmann machines that are interpretable?\n\nCan we design Boltzmann machines that are interpretable? In other words, can we construct useful Boltzmann machines that admit a simple explanation in human terms, and behaves in a predictable manner with certainty?",
   "evidence": "The canonical AIM record is problem 7.1 in the “More philosophical questions” section of the Boltzmann Machines workshop list. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-PROBABILITY-0018": {
   "statement_status": "exact",
   "original_statement": "Are there biologically plausible extensions of Boltzmann Machines for neural networks?\n\nAre there biologically plausible extensions of Boltzmann Machines for neural networks?",
   "clean_statement": "Are there biologically plausible extensions of Boltzmann Machines for neural networks?\n\nAre there biologically plausible extensions of Boltzmann Machines for neural networks?",
   "public_statement": "Are there biologically plausible extensions of Boltzmann Machines for neural networks?\n\nAre there biologically plausible extensions of Boltzmann Machines for neural networks?",
   "evidence": "The question is duplicated verbatim in the extracted `problem` field. This is harmless source duplication; there is no visible OCR corruption. The wording is deliberately broad. “Biologically plausible” is not a mathematical predicate until one specifies which biological constraints are required, and “extension” could refer to neuron dynamics, learning, architecture, or all three. This report therefore gives a conditional existence answer under explicit criteria, not a claim that a particular mechanism is used by an actual brain.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-PROBABILITY-0019": {
   "statement_status": "exact",
   "original_statement": "How to use RBMs to do statistical inference for a physical system, or properties of a network?\n\nHow to use RBMs to do statistical inference for a physical system, or properties of a network?",
   "clean_statement": "How to use RBMs to do statistical inference for a physical system, or properties of a network?\n\nHow to use RBMs to do statistical inference for a physical system, or properties of a network?",
   "public_statement": "How to use RBMs to do statistical inference for a physical system, or properties of a network?\n\nHow to use RBMs to do statistical inference for a physical system, or properties of a network?",
   "evidence": "The canonical AIM record is problem 7.3 in the “More philosophical questions” section of the Boltzmann Machines workshop list. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-PROBABILITY-0020": {
   "statement_status": "exact",
   "original_statement": "The differences between RBM variants\n\nWhat are the differences between classical RBMs, Quantum RBMs, and Complex valued RBMs with quantum states?",
   "clean_statement": "The differences between RBM variants\n\nWhat are the differences between classical RBMs, Quantum RBMs, and Complex valued RBMs with quantum states?",
   "public_statement": "The differences between RBM variants\n\nWhat are the differences between classical RBMs, Quantum RBMs, and Complex valued RBMs with quantum states?",
   "evidence": "The canonical record is item 7.4, \"The differences between RBM variants,\" from the AIM workshop *Boltzmann Machines*, under \"More philosophical questions.\" Its exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-PROBABILITY-0021": {
   "statement_status": "exact",
   "original_statement": "Let $\\mu$ be a probability density on $\\mathbb{R}$, with positive density near 0. Let $X_1, X_2, \\ldots, \\overset{i.i.d}\\sim \\mu$, and define the harmonic mean\n\\[H_n = \\frac{n}{\\frac{1}{X_1} + \\cdots + \\frac{1}{X_n}}.\\]\nProve, using Stein's method, that $H_n$ converges to a Cauchy distribution as $n \\rightarrow \\infty$, and find its rate.",
   "clean_statement": "Let $\\mu$ be a probability density on $\\mathbb{R}$, with positive density near 0. Let $X_1, X_2, \\ldots, \\overset{i.i.d}\\sim \\mu$, and define the harmonic mean\n\\[H_n = \\frac{n}{\\frac{1}{X_1} + \\cdots + \\frac{1}{X_n}}.\\]\nProve, using Stein's method, that $H_n$ converges to a Cauchy distribution as $n \\rightarrow \\infty$, and find its rate.",
   "public_statement": "Let $\\mu$ be a probability density on $\\mathbb{R}$, with positive density near 0. Let $X_1, X_2, \\ldots, \\overset{i.i.d}\\sim \\mu$, and define the harmonic mean\n\\[H_n = \\frac{n}{\\frac{1}{X_1} + \\cdots + \\frac{1}{X_n}}.\\]\nProve, using Stein's method, that $H_n$ converges to a Cauchy distribution as $n \\rightarrow \\infty$, and find its rate.",
   "evidence": "The canonical AIM record is Problem 1.05 from the 2018 workshop *Stein's method and applications in high-dimensional statistics*. It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-PROBABILITY-0022": {
   "statement_status": "exact",
   "original_statement": "Let $\\pi$ be a random permutation in $S_n$. For each $1 \\le i \\le n - 1$, define the random variables\n\\begin{align*}\nX_i &=\n\\begin{cases}\n1 &\\mbox{if } \\pi(i + 1) 0\\}}{n} \\rightarrow \\text{Beta}(1/2, 1/2),\\]\nand compute the rate of convergence.",
   "clean_statement": "Let $\\pi$ be a random permutation in $S_n$. For each $1 \\le i \\le n - 1$, define the random variables\n\\begin{align*}\nX_i &=\n\\begin{cases}\n1 &\\mbox{if } \\pi(i + 1) 0\\}}{n} \\rightarrow \\text{Beta}(1/2, 1/2),\\]\nand compute the rate of convergence.",
   "public_statement": "Let $\\pi$ be a random permutation in $S_n$. For each $1 \\le i \\le n - 1$, define the random variables\n\\begin{align*}\nX_i &=\n\\begin{cases}\n1 &\\mbox{if } \\pi(i + 1) 0\\}}{n} \\rightarrow \\text{Beta}(1/2, 1/2),\\]\nand compute the rate of convergence.",
   "evidence": "The canonical record is source index 21 of `aim-probability-notes.json`, from the AIM workshop *Stein's method and applications in high-dimensional statistics*. Its `problem` field is preserved verbatim below. It is visibly truncated and is not a syntactically complete mathematical statement.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-PROBABILITY-0023": {
   "statement_status": "exact",
   "original_statement": "Let $X_1, X_2, \\ldots$ be i.i.d. symmetric Cauchy random variables, and let\n\\begin{align*}\nS_t &= \\sum_{i = 1}^t X_i \\\\\nV_t &= \\sum_{i = 1}^t X_i^2\n\\end{align*}\nUsing Stein's method, prove that for all $m, x \\ge 0$,\n\\[\\mathbb{P}\\left( \\exists t \\in \\mathbb{N}: \\frac{S_t}{V_t + m} > x \\right) \\le e^{-2mx^2}.\\]",
   "clean_statement": "Let $X_1, X_2, \\ldots$ be i.i.d. symmetric Cauchy random variables, and let\n\\begin{align*}\nS_t &= \\sum_{i = 1}^t X_i \\\\\nV_t &= \\sum_{i = 1}^t X_i^2\n\\end{align*}\nUsing Stein's method, prove that for all $m, x \\ge 0$,\n\\[\\mathbb{P}\\left( \\exists t \\in \\mathbb{N}: \\frac{S_t}{V_t + m} > x \\right) \\le e^{-2mx^2}.\\]",
   "public_statement": "Let $X_1, X_2, \\ldots$ be i.i.d. symmetric Cauchy random variables, and let\n\\begin{align*}\nS_t &= \\sum_{i = 1}^t X_i \\\\\nV_t &= \\sum_{i = 1}^t X_i^2\n\\end{align*}\nUsing Stein's method, prove that for all $m, x \\ge 0$,\n\\[\\mathbb{P}\\left( \\exists t \\in \\mathbb{N}: \\frac{S_t}{V_t + m} > x \\right) \\le e^{-2mx^2}.\\]",
   "evidence": "The canonical record is Problem 1.15 from the 2018 AIM workshop *Stein's method and applications in high-dimensional statistics*. It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-PROBABILITY-0024": {
   "statement_status": "exact",
   "original_statement": "Let $X \\sim MVN(\\theta, I_d)$, where $I_d$ denotes the $d$-dimensional identity matrix. Consider the problem of estimating $\\theta$ using estimators of the form $X + h(X)$, where $h$ is in some class $\\mathcal{H}$ of functions. By SURE,\n\\[\\hat{r}_h = d + 2\\nabla \\cdot h(X) + \\|h(X)\\|^2\\]\nis an unbiased estimator of the risk associated with $h$. Let\n\\begin{align*}\nh^* &= \\text{arg min} (h \\in \\mathcal{H}: \\hat{r}_h) \\\\\n\\hat{\\theta}^* &= X + h^*(X).\n\\end{align*}\nFind an unbiased estimator $r^{**}$ of the risk $\\mathbb{E}(\\|\\hat{\\theta}^* - \\theta\\|^2)$ of $\\hat{\\theta}^*$.",
   "clean_statement": "Let $X \\sim MVN(\\theta, I_d)$, where $I_d$ denotes the $d$-dimensional identity matrix. Consider the problem of estimating $\\theta$ using estimators of the form $X + h(X)$, where $h$ is in some class $\\mathcal{H}$ of functions. By SURE,\n\\[\\hat{r}_h = d + 2\\nabla \\cdot h(X) + \\|h(X)\\|^2\\]\nis an unbiased estimator of the risk associated with $h$. Let\n\\begin{align*}\nh^* &= \\text{arg min} (h \\in \\mathcal{H}: \\hat{r}_h) \\\\\n\\hat{\\theta}^* &= X + h^*(X).\n\\end{align*}\nFind an unbiased estimator $r^{**}$ of the risk $\\mathbb{E}(\\|\\hat{\\theta}^* - \\theta\\|^2)$ of $\\hat{\\theta}^*$.",
   "public_statement": "Let $X \\sim MVN(\\theta, I_d)$, where $I_d$ denotes the $d$-dimensional identity matrix. Consider the problem of estimating $\\theta$ using estimators of the form $X + h(X)$, where $h$ is in some class $\\mathcal{H}$ of functions. By SURE,\n\\[\\hat{r}_h = d + 2\\nabla \\cdot h(X) + \\|h(X)\\|^2\\]\nis an unbiased estimator of the risk associated with $h$. Let\n\\begin{align*}\nh^* &= \\text{arg min} (h \\in \\mathcal{H}: \\hat{r}_h) \\\\\n\\hat{\\theta}^* &= X + h^*(X).\n\\end{align*}\nFind an unbiased estimator $r^{**}$ of the risk $\\mathbb{E}(\\|\\hat{\\theta}^* - \\theta\\|^2)$ of $\\hat{\\theta}^*$.",
   "evidence": "The canonical source is Problem 1.2 from the AIM workshop *Stein's method and applications in high-dimensional statistics*. The exact mathematical request is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-PROBABILITY-0025": {
   "statement_status": "exact",
   "original_statement": "Let $t_r$ denote the $t$ distribution with $r$ degrees of freedom. As $r \\rightarrow \\infty$, we know that $t_r \\overset{d}\\rightarrow Z$, where $Z$ is distributed as $N(0, 1)$.\n\\begin{itemize}\n\\item[(a)]\nShow that there exists a constant $c > 0$ for which\n\\[d_{TV}(t_r, Z) \\le c/r.\\]\n\\item[(b)]\nFind the best such constant.\n\\item[(c)]\nDoes there exist a constant $c'$ for which\n\\[d_{TV}(t_r, Z) \\ge c'/r?\\]\n\\end{itemize}",
   "clean_statement": "Let $t_r$ denote the $t$ distribution with $r$ degrees of freedom. As $r \\rightarrow \\infty$, we know that $t_r \\overset{d}\\rightarrow Z$, where $Z$ is distributed as $N(0, 1)$.\n\\begin{itemize}\n\\item[(a)]\nShow that there exists a constant $c > 0$ for which\n\\[d_{TV}(t_r, Z) \\le c/r.\\]\n\\item[(b)]\nFind the best such constant.\n\\item[(c)]\nDoes there exist a constant $c'$ for which\n\\[d_{TV}(t_r, Z) \\ge c'/r?\\]\n\\end{itemize}",
   "public_statement": "Let $t_r$ denote the $t$ distribution with $r$ degrees of freedom. As $r \\rightarrow \\infty$, we know that $t_r \\overset{d}\\rightarrow Z$, where $Z$ is distributed as $N(0, 1)$.\n\\begin{itemize}\n\\item[(a)]\nShow that there exists a constant $c > 0$ for which\n\\[d_{TV}(t_r, Z) \\le c/r.\\]\n\\item[(b)]\nFind the best such constant.\n\\item[(c)]\nDoes there exist a constant $c'$ for which\n\\[d_{TV}(t_r, Z) \\ge c'/r?\\]\n\\end{itemize}",
   "evidence": "The canonical AIM record is Problem 1.25 from the 2018 workshop *Stein's method and applications in high-dimensional statistics*. It asks, for Student \\(t_r\\) with \\(r\\) degrees of freedom and \\(Z\\sim N(0,1)\\):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-PROBABILITY-0026": {
   "statement_status": "exact",
   "original_statement": "A random variable $X$ with distribution $f$ is called $s$-concave if, for every $\\lambda \\in [0, 1]$ and all $x$ and $y$ such that $f(x), f(y) > 0$,\n\\[f((1 - \\lambda)x + \\lambda y) > ((1 - \\lambda) f(x)^s + \\lambda f(y)^s)^{1/s}.\\]\nSuppose $Y$ is a random variable with characteristic function\n\\[\\mathbb{E}(e^{itY}) = e^{-|t|^{\\alpha}}.\\]\nShow that $Y$ is $s$-concave for some $s = s(\\alpha)$. Further, if $Y$ has density $f$, show that the curvature of $f$ at its mode is negative.",
   "clean_statement": "A random variable $X$ with distribution $f$ is called $s$-concave if, for every $\\lambda \\in [0, 1]$ and all $x$ and $y$ such that $f(x), f(y) > 0$,\n\\[f((1 - \\lambda)x + \\lambda y) > ((1 - \\lambda) f(x)^s + \\lambda f(y)^s)^{1/s}.\\]\nSuppose $Y$ is a random variable with characteristic function\n\\[\\mathbb{E}(e^{itY}) = e^{-|t|^{\\alpha}}.\\]\nShow that $Y$ is $s$-concave for some $s = s(\\alpha)$. Further, if $Y$ has density $f$, show that the curvature of $f$ at its mode is negative.",
   "public_statement": "A random variable $X$ with distribution $f$ is called $s$-concave if, for every $\\lambda \\in [0, 1]$ and all $x$ and $y$ such that $f(x), f(y) > 0$,\n\\[f((1 - \\lambda)x + \\lambda y) > ((1 - \\lambda) f(x)^s + \\lambda f(y)^s)^{1/s}.\\]\nSuppose $Y$ is a random variable with characteristic function\n\\[\\mathbb{E}(e^{itY}) = e^{-|t|^{\\alpha}}.\\]\nShow that $Y$ is $s$-concave for some $s = s(\\alpha)$. Further, if $Y$ has density $f$, show that the curvature of $f$ at its mode is negative.",
   "evidence": "The canonical record is source index 25 of `aim-probability-notes.json`. Its problem field is preserved exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-PROBABILITY-0027": {
   "statement_status": "exact",
   "original_statement": "Consider a Markov queueing system where there are $N$ servers who serve at rate 1, and new customers arrive into the system at a constant rate of $\\lambda$. When a new customer arrives, they choose to join the queue behind the server with the shortest existing length. If there is more than one queue with the shortest length, simply choose one uniformly at random. Let us assume that $\\lambda < N$ so we have a stationary distribution for this chain. Denote $Q = (q_1, q_2, \\ldots)$ as the stationary distribution of this process, where $q_i$ is the number of queues with \\emph{at least} $i$ people in the queue. This model is known as the Join the shortest queue (JSQ) model.\n\nAs a simplification to this process, suppose that when a new customer arrives, and \\emph{all} of the server queues have 2 people, then this customer leaves straight away. Let $Q_2$ denote the corresponding stationary distribution for this modified process. This is a commonly used approximation for the full JSQ model. Can we use Stein's method to explicitly bound the difference $|\\mathbb{E} f(Q) - \\mathbb{E} f(Q_2)|$ for suitable $f$?\n\nConsider another simplification of the full JSQ process, where an new customer joins the shortest of $k$ randomly chosen queues. Let $Q_3$ denote the stationary distribution of this process. What can we say about $|\\mathbb{E} f(Q) - \\mathbb{E}f(Q_3)|$?",
   "clean_statement": "Consider a Markov queueing system where there are $N$ servers who serve at rate 1, and new customers arrive into the system at a constant rate of $\\lambda$. When a new customer arrives, they choose to join the queue behind the server with the shortest existing length. If there is more than one queue with the shortest length, simply choose one uniformly at random. Let us assume that $\\lambda < N$ so we have a stationary distribution for this chain. Denote $Q = (q_1, q_2, \\ldots)$ as the stationary distribution of this process, where $q_i$ is the number of queues with \\emph{at least} $i$ people in the queue. This model is known as the Join the shortest queue (JSQ) model.\n\nAs a simplification to this process, suppose that when a new customer arrives, and \\emph{all} of the server queues have 2 people, then this customer leaves straight away. Let $Q_2$ denote the corresponding stationary distribution for this modified process. This is a commonly used approximation for the full JSQ model. Can we use Stein's method to explicitly bound the difference $|\\mathbb{E} f(Q) - \\mathbb{E} f(Q_2)|$ for suitable $f$?\n\nConsider another simplification of the full JSQ process, where an new customer joins the shortest of $k$ randomly chosen queues. Let $Q_3$ denote the stationary distribution of this process. What can we say about $|\\mathbb{E} f(Q) - \\mathbb{E}f(Q_3)|$?",
   "public_statement": "Consider a Markov queueing system where there are $N$ servers who serve at rate 1, and new customers arrive into the system at a constant rate of $\\lambda$. When a new customer arrives, they choose to join the queue behind the server with the shortest existing length. If there is more than one queue with the shortest length, simply choose one uniformly at random. Let us assume that $\\lambda < N$ so we have a stationary distribution for this chain. Denote $Q = (q_1, q_2, \\ldots)$ as the stationary distribution of this process, where $q_i$ is the number of queues with \\emph{at least} $i$ people in the queue. This model is known as the Join the shortest queue (JSQ) model.\n\nAs a simplification to this process, suppose that when a new customer arrives, and \\emph{all} of the server queues have 2 people, then this customer leaves straight away. Let $Q_2$ denote the corresponding stationary distribution for this modified process. This is a commonly used approximation for the full JSQ model. Can we use Stein's method to explicitly bound the difference $|\\mathbb{E} f(Q) - \\mathbb{E} f(Q_2)|$ for suitable $f$?\n\nConsider another simplification of the full JSQ process, where an new customer joins the shortest of $k$ randomly chosen queues. Let $Q_3$ denote the stationary distribution of this process. What can we say about $|\\mathbb{E} f(Q) - \\mathbb{E}f(Q_3)|$?",
   "evidence": "The source is Problem 1.35 from the AIM workshop list *Stein's method and applications in high-dimensional statistics*. It asks about \\(N\\) unit-rate exponential servers, a Poisson arrival stream, and the occupancy vector",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 26,
   "attempt": 2
  },
  "AIM-PROBABILITY-0028": {
   "statement_status": "exact",
   "original_statement": "Two approaches to generalize Stein's method to non-Gaussian distributions are the following:\n\\begin{itemize}\n\\item[(i)]\nStein coefficients: for a random variable $Y$, the \\emph{Stein coefficient} $T(Y)$ satisfies\n\\[\\mathbb{E}(Y f(Y)) = \\mathbb{E}(T f'(Y)).\\]\n\\item[(ii)]\nZero-biasing: for $Y$ of mean 0 and variance $\\sigma^2$, compute the $Y^*$ that satisfies\n\\[\\mathbb{E}(Yf(Y)) = \\sigma^2 \\mathbb{E} f'(Y^*).\\]\n\nLet $Y_1, \\ldots, Y_d \\overset{i.i.d.}\\sim Y$, and let $\\overline{Y} = (Y_1, \\ldots, Y_d)$. Suppose that $\\mathbb{E} Y_i = 0$, and let $X = Y + \\theta$. Here we will assume $Y$ is a known distribution while $\\theta$ is unknown.\n\nConsider estimators $\\hat{\\theta}$ of $\\Theta$ of the form $\\hat{\\theta} = X + h(X)$. Recall that SURE for Gaussian random variables gives the unbiased estimate of risk\n\\[\\hat{r}_h = d + 2\\nabla \\cdot h(X) + \\|h(X)\\|^2.\\]\nCalculate the bias\n\\[e_d := \\mathbb{E}(\\hat{r}_h - \\mathbb{E} \\|\\hat{\\theta} - \\theta\\|^2)\\]\nusing Stein coefficients or zero-biasing. Does $\\frac{e_d}{d}$ converge to 0 as $d \\rightarrow \\infty$? Can we apply this to wavelet shrinkage?\n\\end{itemize}",
   "clean_statement": "Two approaches to generalize Stein's method to non-Gaussian distributions are the following:\n\\begin{itemize}\n\\item[(i)]\nStein coefficients: for a random variable $Y$, the \\emph{Stein coefficient} $T(Y)$ satisfies\n\\[\\mathbb{E}(Y f(Y)) = \\mathbb{E}(T f'(Y)).\\]\n\\item[(ii)]\nZero-biasing: for $Y$ of mean 0 and variance $\\sigma^2$, compute the $Y^*$ that satisfies\n\\[\\mathbb{E}(Yf(Y)) = \\sigma^2 \\mathbb{E} f'(Y^*).\\]\n\nLet $Y_1, \\ldots, Y_d \\overset{i.i.d.}\\sim Y$, and let $\\overline{Y} = (Y_1, \\ldots, Y_d)$. Suppose that $\\mathbb{E} Y_i = 0$, and let $X = Y + \\theta$. Here we will assume $Y$ is a known distribution while $\\theta$ is unknown.\n\nConsider estimators $\\hat{\\theta}$ of $\\Theta$ of the form $\\hat{\\theta} = X + h(X)$. Recall that SURE for Gaussian random variables gives the unbiased estimate of risk\n\\[\\hat{r}_h = d + 2\\nabla \\cdot h(X) + \\|h(X)\\|^2.\\]\nCalculate the bias\n\\[e_d := \\mathbb{E}(\\hat{r}_h - \\mathbb{E} \\|\\hat{\\theta} - \\theta\\|^2)\\]\nusing Stein coefficients or zero-biasing. Does $\\frac{e_d}{d}$ converge to 0 as $d \\rightarrow \\infty$? Can we apply this to wavelet shrinkage?\n\\end{itemize}",
   "public_statement": "Two approaches to generalize Stein's method to non-Gaussian distributions are the following:\n\\begin{itemize}\n\\item[(i)]\nStein coefficients: for a random variable $Y$, the \\emph{Stein coefficient} $T(Y)$ satisfies\n\\[\\mathbb{E}(Y f(Y)) = \\mathbb{E}(T f'(Y)).\\]\n\\item[(ii)]\nZero-biasing: for $Y$ of mean 0 and variance $\\sigma^2$, compute the $Y^*$ that satisfies\n\\[\\mathbb{E}(Yf(Y)) = \\sigma^2 \\mathbb{E} f'(Y^*).\\]\n\nLet $Y_1, \\ldots, Y_d \\overset{i.i.d.}\\sim Y$, and let $\\overline{Y} = (Y_1, \\ldots, Y_d)$. Suppose that $\\mathbb{E} Y_i = 0$, and let $X = Y + \\theta$. Here we will assume $Y$ is a known distribution while $\\theta$ is unknown.\n\nConsider estimators $\\hat{\\theta}$ of $\\Theta$ of the form $\\hat{\\theta} = X + h(X)$. Recall that SURE for Gaussian random variables gives the unbiased estimate of risk\n\\[\\hat{r}_h = d + 2\\nabla \\cdot h(X) + \\|h(X)\\|^2.\\]\nCalculate the bias\n\\[e_d := \\mathbb{E}(\\hat{r}_h - \\mathbb{E} \\|\\hat{\\theta} - \\theta\\|^2)\\]\nusing Stein coefficients or zero-biasing. Does $\\frac{e_d}{d}$ converge to 0 as $d \\rightarrow \\infty$? Can we apply this to wavelet shrinkage?\n\\end{itemize}",
   "evidence": "The canonical AIM record is Problem 1.4 from the workshop *Stein's method and applications in high-dimensional statistics*. It proposes two non-Gaussian integration-by-parts devices,",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-PROBABILITY-0029": {
   "statement_status": "exact",
   "original_statement": "Let $X_1, X_2, \\ldots$ be i.i.d. random variables with mean 0 and variance 1, and let $S_n = \\sum_{i = 1}^n X_i$.\n\nFor $b > 0$, let $T_b$ be a sequence of random times. The following theorem is due to Renyi (1961):\n\\begin{theorem}\nIf $\\frac{T_b}{n_b} \\overset{p}\\rightarrow 1$ as $b \\rightarrow \\infty$, where $n_b \\rightarrow \\infty$ is deterministic, then\n\\[\\frac{S_{T_b}}{\\sqrt{n_b}} \\rightarrow N(0, 1).\\]\nCan we get a bound on $d\\left( \\frac{S_{T_b}}{n_b}, Z\\right)$?\n\\end{theorem}\n\nSome remarks: if $T_b$ is independent of $X_1, X_2, \\ldots$, then Doebler has some results. Also, see ``Randomly Stopped Sums'' by Allan Gut",
   "clean_statement": "Let $X_1, X_2, \\ldots$ be i.i.d. random variables with mean 0 and variance 1, and let $S_n = \\sum_{i = 1}^n X_i$.\n\nFor $b > 0$, let $T_b$ be a sequence of random times. The following theorem is due to Renyi (1961):\n\\begin{theorem}\nIf $\\frac{T_b}{n_b} \\overset{p}\\rightarrow 1$ as $b \\rightarrow \\infty$, where $n_b \\rightarrow \\infty$ is deterministic, then\n\\[\\frac{S_{T_b}}{\\sqrt{n_b}} \\rightarrow N(0, 1).\\]\nCan we get a bound on $d\\left( \\frac{S_{T_b}}{n_b}, Z\\right)$?\n\\end{theorem}\n\nSome remarks: if $T_b$ is independent of $X_1, X_2, \\ldots$, then Doebler has some results. Also, see ``Randomly Stopped Sums'' by Allan Gut",
   "public_statement": "Let $X_1, X_2, \\ldots$ be i.i.d. random variables with mean 0 and variance 1, and let $S_n = \\sum_{i = 1}^n X_i$.\n\nFor $b > 0$, let $T_b$ be a sequence of random times. The following theorem is due to Renyi (1961):\n\\begin{theorem}\nIf $\\frac{T_b}{n_b} \\overset{p}\\rightarrow 1$ as $b \\rightarrow \\infty$, where $n_b \\rightarrow \\infty$ is deterministic, then\n\\[\\frac{S_{T_b}}{\\sqrt{n_b}} \\rightarrow N(0, 1).\\]\nCan we get a bound on $d\\left( \\frac{S_{T_b}}{n_b}, Z\\right)$?\n\\end{theorem}\n\nSome remarks: if $T_b$ is independent of $X_1, X_2, \\ldots$, then Doebler has some results. Also, see ``Randomly Stopped Sums'' by Allan Gut",
   "evidence": "The canonical record is source index 28 of aim-probability-notes.json. Its problem field says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-PROBABILITY-0030": {
   "statement_status": "exact",
   "original_statement": "Let $(X_i, Y_i)_{i = 1}^n$ be i.i.d. pairs of random variables, where $X_i \\in \\mathbb{R}^p$ and $Y_i \\in \\mathbb{R}^q$.\n\\begin{itemize}\n\\item[1.]\nFind $a \\in \\mathbb{R}^p$ and $b \\in \\mathbb{R}^1$ such that\n\\[\\text{corr}\\left( a^TX, b^T Y \\right)\\]\nis maximized.\n\\item[2.]\nPick $\\hat{a}, \\hat{b}$ such that $\\sum \\hat{a}_i = \\sum \\hat{b}_i = 1$ so that\n\\[\\hat{c} := \\text{corr}\\left( \\{\\hat{a}^TX_i, \\hat{b}^TY_i\\}_{i = 1}^n \\right)\\]\nis maximized.\n\nIt is shown in Bartlett(1939) and Hotelling(1936) that if $X_i$ is independent of $Y_i$, and both have finite fourth moments, then\n\\[A_n := -\\log(1 - \\hat{c})\\left( n - \\frac{1}{2}(p + q + 3) \\right) \\overset{d}\\rightarrow \\chi^2_{p + q - 2}\\]\nas $n \\rightarrow \\infty$. Can we get bounds on $d(A_n, \\chi^2_{p + q - 2})$?\n\\end{itemize}",
   "clean_statement": "Let $(X_i, Y_i)_{i = 1}^n$ be i.i.d. pairs of random variables, where $X_i \\in \\mathbb{R}^p$ and $Y_i \\in \\mathbb{R}^q$.\n\\begin{itemize}\n\\item[1.]\nFind $a \\in \\mathbb{R}^p$ and $b \\in \\mathbb{R}^1$ such that\n\\[\\text{corr}\\left( a^TX, b^T Y \\right)\\]\nis maximized.\n\\item[2.]\nPick $\\hat{a}, \\hat{b}$ such that $\\sum \\hat{a}_i = \\sum \\hat{b}_i = 1$ so that\n\\[\\hat{c} := \\text{corr}\\left( \\{\\hat{a}^TX_i, \\hat{b}^TY_i\\}_{i = 1}^n \\right)\\]\nis maximized.\n\nIt is shown in Bartlett(1939) and Hotelling(1936) that if $X_i$ is independent of $Y_i$, and both have finite fourth moments, then\n\\[A_n := -\\log(1 - \\hat{c})\\left( n - \\frac{1}{2}(p + q + 3) \\right) \\overset{d}\\rightarrow \\chi^2_{p + q - 2}\\]\nas $n \\rightarrow \\infty$. Can we get bounds on $d(A_n, \\chi^2_{p + q - 2})$?\n\\end{itemize}",
   "public_statement": "Let $(X_i, Y_i)_{i = 1}^n$ be i.i.d. pairs of random variables, where $X_i \\in \\mathbb{R}^p$ and $Y_i \\in \\mathbb{R}^q$.\n\\begin{itemize}\n\\item[1.]\nFind $a \\in \\mathbb{R}^p$ and $b \\in \\mathbb{R}^1$ such that\n\\[\\text{corr}\\left( a^TX, b^T Y \\right)\\]\nis maximized.\n\\item[2.]\nPick $\\hat{a}, \\hat{b}$ such that $\\sum \\hat{a}_i = \\sum \\hat{b}_i = 1$ so that\n\\[\\hat{c} := \\text{corr}\\left( \\{\\hat{a}^TX_i, \\hat{b}^TY_i\\}_{i = 1}^n \\right)\\]\nis maximized.\n\nIt is shown in Bartlett(1939) and Hotelling(1936) that if $X_i$ is independent of $Y_i$, and both have finite fourth moments, then\n\\[A_n := -\\log(1 - \\hat{c})\\left( n - \\frac{1}{2}(p + q + 3) \\right) \\overset{d}\\rightarrow \\chi^2_{p + q - 2}\\]\nas $n \\rightarrow \\infty$. Can we get bounds on $d(A_n, \\chi^2_{p + q - 2})$?\n\\end{itemize}",
   "evidence": "The canonical record asks the following (notation preserved, including apparent errors). For i.i.d. pairs $(X_i,Y_i)$ with $X_i\\in\\mathbb R^p$ and $Y_i\\in\\mathbb R^q$:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-PROBABILITY-0031": {
   "statement_status": "exact",
   "original_statement": "Universal lower bound for Potts\n\nConsider the $q$-Potts model or uniform $q$ proper colorings on a general graph $G$. Is there a universal $\\asymp n\\log n$ lower bound on the mixing time uniform in the temperature? Is there always an $\\frac {n\\log n}2$ lower bound?",
   "clean_statement": "Universal lower bound for Potts\n\nConsider the $q$-Potts model or uniform $q$ proper colorings on a general graph $G$. Is there a universal $\\asymp n\\log n$ lower bound on the mixing time uniform in the temperature? Is there always an $\\frac {n\\log n}2$ lower bound?",
   "public_statement": "Universal lower bound for Potts\n\nConsider the $q$-Potts model or uniform $q$ proper colorings on a general graph $G$. Is there a universal $\\asymp n\\log n$ lower bound on the mixing time uniform in the temperature? Is there always an $\\frac {n\\log n}2$ lower bound?",
   "evidence": "The canonical AIM record (workshop *Markov chain mixing times*, section *Spin systems*, problem 1.1) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 30,
   "attempt": 2
  },
  "AIM-PROBABILITY-0032": {
   "statement_status": "exact",
   "original_statement": "Fast mixing for anti-ferromagnetic Ising at high temperature\n\nTake a (random) $d$-regular graph, and consider the anti-ferromagnetic Ising model, \\textit{i.e.} $J_{ij}=-1$ for all $i,j$. Can we show show fast mixing $O(n\\log n)$ up to\n$$(d-1)\\tanh(\\beta)<1 \\, ?$$",
   "clean_statement": "Fast mixing for anti-ferromagnetic Ising at high temperature\n\nTake a (random) $d$-regular graph, and consider the anti-ferromagnetic Ising model, \\textit{i.e.} $J_{ij}=-1$ for all $i,j$. Can we show show fast mixing $O(n\\log n)$ up to\n$$(d-1)\\tanh(\\beta)<1 \\, ?$$",
   "public_statement": "Fast mixing for anti-ferromagnetic Ising at high temperature\n\nTake a (random) $d$-regular graph, and consider the anti-ferromagnetic Ising model, \\textit{i.e.} $J_{ij}=-1$ for all $i,j$. Can we show show fast mixing $O(n\\log n)$ up to\n$$(d-1)\\tanh(\\beta)<1 \\, ?$$",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 31,
   "attempt": 2
  },
  "AIM-PROBABILITY-0033": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Spin glass with i.i.d. couplings\n\nConsider the spin glass model on $\\mathbb{Z}_n^d$ with i.i.d. couplings, \\textit{i.e.} with Hamiltonian given by\n$$\nH_n(\\sigma)= \\sum_{(i,j)\\in E_n} J_{ij}\\sigma_i\\sigma_j\\, ,\n$$\nwhere $J=(J_{ij})$ is a random symmetric matrix with i.i.d. $\\pm 1$ entries. The Gibbs distribution is given by\n$$\n\\mu_n(\\sigma)= \\frac{1}{Z_n}\\exp(\\beta H_n)\\, .\n$$\nFix a realization of $J_{ij}$ and consider the Glauber dynamics. Does there exist $\\beta_0$ and $\\varepsilon>0$ such that, for all $\\beta\\geq\\beta_0$, the mixing time of this chain is at least $\\exp(n^\\varepsilon)$?",
   "clean_statement": null,
   "public_statement": "Spin glass with i.i.d. couplings\n\nConsider the spin glass model on $\\mathbb{Z}_n^d$ with i.i.d. couplings, \\textit{i.e.} with Hamiltonian given by\n$$\nH_n(\\sigma)= \\sum_{(i,j)\\in E_n} J_{ij}\\sigma_i\\sigma_j\\, ,\n$$\nwhere $J=(J_{ij})$ is a random symmetric matrix with i.i.d. $\\pm 1$ entries. The Gibbs distribution is given by\n$$\n\\mu_n(\\sigma)= \\frac{1}{Z_n}\\exp(\\beta H_n)\\, .\n$$\nFix a realization of $J_{ij}$ and consider the Glauber dynamics. Does there exist $\\beta_0$ and $\\varepsilon>0$ such that, for all $\\beta\\geq\\beta_0$, the mixing time of this chain is at least $\\exp(n^\\varepsilon)$?",
   "evidence": "The canonical record, titled **“Spin glass with i.i.d. couplings,”** considers nearest-neighbor spins on $\\mathbb Z_n^d$ with",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-PROBABILITY-0034": {
   "statement_status": "exact",
   "original_statement": "Mixing time for ferromagnetic Ising at critical temperature\n\nWhat is the mixing time for the ferromagnetic Ising model at critical temperature $\\beta=\\beta_c$ on a random $d$-regular graph? Is it $n^c$?",
   "clean_statement": "Mixing time for ferromagnetic Ising at critical temperature\n\nWhat is the mixing time for the ferromagnetic Ising model at critical temperature $\\beta=\\beta_c$ on a random $d$-regular graph? Is it $n^c$?",
   "public_statement": "Mixing time for ferromagnetic Ising at critical temperature\n\nWhat is the mixing time for the ferromagnetic Ising model at critical temperature $\\beta=\\beta_c$ on a random $d$-regular graph? Is it $n^c$?",
   "evidence": "The repository text is internally coherent; no OCR correction is needed. The original AIM page (`http://aimpl.org/markovmixing/1/`) returned HTTP 502 when checked on 2026-08-11, so the wording above is verified against the canonical repository record rather than the live page.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-PROBABILITY-0035": {
   "statement_status": "exact",
   "original_statement": "Censoring for the Potts model\n\nConsider the $q$-state Potts model (say with $q=3$ for concreteness) and start from all green. Is it the case that deterministically censoring a sequence of spin flips can only increase the total variation distance to stationarity?",
   "clean_statement": "Censoring for the Potts model\n\nConsider the $q$-state Potts model (say with $q=3$ for concreteness) and start from all green. Is it the case that deterministically censoring a sequence of spin flips can only increase the total variation distance to stationarity?",
   "public_statement": "Censoring for the Potts model\n\nConsider the $q$-state Potts model (say with $q=3$ for concreteness) and start from all green. Is it the case that deterministically censoring a sequence of spin flips can only increase the total variation distance to stationarity?",
   "evidence": "The canonical AIM record (workshop *Markov chain mixing times*, section *Spin systems*, problem 1.5) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-PROBABILITY-0036": {
   "statement_status": "exact",
   "original_statement": "Diagnostics\n\nWe seek diagnostic tests for knowing that a general spin system is mixed on a general graph $G$.\n\n1. Consider the ferromagnetic Ising model on a general graph $G$. Which two $x,y\\in\\Omega$ maximize $\\|P^t(x,\\cdot)-P^t(y,\\cdot)\\|_{TV}$? More generally, is it true that in any monotone reversible chain, the maximal and minimal initial configurations maximize the total variation distance between two chains?\n\n2. Can one use a diagnostic to estimate $t_{\\mbox{mix}}$ up to $O(1)$? Of course, answering question (1) would answer this as well.\n\n3. Consider the $3$-Potts model for a general graph $G$. Can one devise a diagnostic to differentiate between $t_{\\mbox{mix}}=O(n\\log n)$ and $\\exp(\\Omega(n))$?",
   "clean_statement": "Diagnostics\n\nWe seek diagnostic tests for knowing that a general spin system is mixed on a general graph $G$.\n\n1. Consider the ferromagnetic Ising model on a general graph $G$. Which two $x,y\\in\\Omega$ maximize $\\|P^t(x,\\cdot)-P^t(y,\\cdot)\\|_{TV}$? More generally, is it true that in any monotone reversible chain, the maximal and minimal initial configurations maximize the total variation distance between two chains?\n\n2. Can one use a diagnostic to estimate $t_{\\mbox{mix}}$ up to $O(1)$? Of course, answering question (1) would answer this as well.\n\n3. Consider the $3$-Potts model for a general graph $G$. Can one devise a diagnostic to differentiate between $t_{\\mbox{mix}}=O(n\\log n)$ and $\\exp(\\Omega(n))$?",
   "public_statement": "Diagnostics\n\nWe seek diagnostic tests for knowing that a general spin system is mixed on a general graph $G$.\n\n1. Consider the ferromagnetic Ising model on a general graph $G$. Which two $x,y\\in\\Omega$ maximize $\\|P^t(x,\\cdot)-P^t(y,\\cdot)\\|_{TV}$? More generally, is it true that in any monotone reversible chain, the maximal and minimal initial configurations maximize the total variation distance between two chains?\n\n2. Can one use a diagnostic to estimate $t_{\\mbox{mix}}$ up to $O(1)$? Of course, answering question (1) would answer this as well.\n\n3. Consider the $3$-Potts model for a general graph $G$. Can one devise a diagnostic to differentiate between $t_{\\mbox{mix}}=O(n\\log n)$ and $\\exp(\\Omega(n))$?",
   "evidence": "The canonical record is item 1.6, “Diagnostics,” in the “Spin systems” section of the AIM workshop *Markov chain mixing times*. Its three questions are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-PROBABILITY-0037": {
   "statement_status": "exact",
   "original_statement": "Noisy majority model\n\nThe noisy majority model with parameter $\\epsilon\\in (0,1)$ is a spin system on a graph $G$ that is the stationary distribution of the following Markov chain on $\\Omega=\\{\\pm 1\\}^{V(G)}$: assign each vertex a rate-$1$ Poisson clock. When the clock at a site rings, the spin at that vertex randomizes according to a $\\mbox{Ber}(\\frac 12)$ with probability $\\epsilon$ and chooses the majority of the spins of its neighbors and itself with probability $1-\\epsilon$ (flipping a coin in the event of a tie).\n\nConsider the noisy majority model on $\\mathbb Z_n^d$ for $d>1$. Show there exists a fixed $\\epsilon>0$ such that $t_{\\mbox{mix}}\\gtrsim \\exp(n^{\\epsilon})$ (should be true for $\\exp(cn)$).",
   "clean_statement": "Noisy majority model\n\nThe noisy majority model with parameter $\\epsilon\\in (0,1)$ is a spin system on a graph $G$ that is the stationary distribution of the following Markov chain on $\\Omega=\\{\\pm 1\\}^{V(G)}$: assign each vertex a rate-$1$ Poisson clock. When the clock at a site rings, the spin at that vertex randomizes according to a $\\mbox{Ber}(\\frac 12)$ with probability $\\epsilon$ and chooses the majority of the spins of its neighbors and itself with probability $1-\\epsilon$ (flipping a coin in the event of a tie).\n\nConsider the noisy majority model on $\\mathbb Z_n^d$ for $d>1$. Show there exists a fixed $\\epsilon>0$ such that $t_{\\mbox{mix}}\\gtrsim \\exp(n^{\\epsilon})$ (should be true for $\\exp(cn)$).",
   "public_statement": "Noisy majority model\n\nThe noisy majority model with parameter $\\epsilon\\in (0,1)$ is a spin system on a graph $G$ that is the stationary distribution of the following Markov chain on $\\Omega=\\{\\pm 1\\}^{V(G)}$: assign each vertex a rate-$1$ Poisson clock. When the clock at a site rings, the spin at that vertex randomizes according to a $\\mbox{Ber}(\\frac 12)$ with probability $\\epsilon$ and chooses the majority of the spins of its neighbors and itself with probability $1-\\epsilon$ (flipping a coin in the event of a tie).\n\nConsider the noisy majority model on $\\mathbb Z_n^d$ for $d>1$. Show there exists a fixed $\\epsilon>0$ such that $t_{\\mbox{mix}}\\gtrsim \\exp(n^{\\epsilon})$ (should be true for $\\exp(cn)$).",
   "evidence": "The canonical source record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-PROBABILITY-0038": {
   "statement_status": "exact",
   "original_statement": "Edge-flip chain on dyadic tilings and rectangular dissections (Sarah Cannon)\n\nConsider rectangular dissections of an $n\\times n$ lattice into $n$ rectangles of area $n$, where $n=2^k$. The edge-flip chain proceeds by picking an edge bordering two rectangles and replacing it by its bisector. In the \\emph{dyadic} case, the edge is flipped provided the resulting tiling remains dyadic. In the weighted setting, a parameter $\\lambda>0$ is fixed, and the weight of a dissection is given by\n$$\n\\pi(\\sigma)=\\frac{\\lambda^{|\\sigma |}}{Z} ,\n$$\nwhere $|\\sigma|$ is the total edge-length. The chains (in the dyadic and general cases) then correspond to the Glauber dynamics.\n\nFor $\\lambda=1$, can we establish a polynomial upper-bound (fast-mixing) for the mixing time of the edge-flip chain on dyadic tilings or on rectangular dissections?",
   "clean_statement": "Edge-flip chain on dyadic tilings and rectangular dissections (Sarah Cannon)\n\nConsider rectangular dissections of an $n\\times n$ lattice into $n$ rectangles of area $n$, where $n=2^k$. The edge-flip chain proceeds by picking an edge bordering two rectangles and replacing it by its bisector. In the \\emph{dyadic} case, the edge is flipped provided the resulting tiling remains dyadic. In the weighted setting, a parameter $\\lambda>0$ is fixed, and the weight of a dissection is given by\n$$\n\\pi(\\sigma)=\\frac{\\lambda^{|\\sigma |}}{Z} ,\n$$\nwhere $|\\sigma|$ is the total edge-length. The chains (in the dyadic and general cases) then correspond to the Glauber dynamics.\n\nFor $\\lambda=1$, can we establish a polynomial upper-bound (fast-mixing) for the mixing time of the edge-flip chain on dyadic tilings or on rectangular dissections?",
   "public_statement": "Edge-flip chain on dyadic tilings and rectangular dissections (Sarah Cannon)\n\nConsider rectangular dissections of an $n\\times n$ lattice into $n$ rectangles of area $n$, where $n=2^k$. The edge-flip chain proceeds by picking an edge bordering two rectangles and replacing it by its bisector. In the \\emph{dyadic} case, the edge is flipped provided the resulting tiling remains dyadic. In the weighted setting, a parameter $\\lambda>0$ is fixed, and the weight of a dissection is given by\n$$\n\\pi(\\sigma)=\\frac{\\lambda^{|\\sigma |}}{Z} ,\n$$\nwhere $|\\sigma|$ is the total edge-length. The chains (in the dyadic and general cases) then correspond to the Glauber dynamics.\n\nFor $\\lambda=1$, can we establish a polynomial upper-bound (fast-mixing) for the mixing time of the edge-flip chain on dyadic tilings or on rectangular dissections?",
   "evidence": "The preserved input asks about equitable rectangular dissections of an \\(n\\times n\\) lattice square into \\(n\\) rectangles of area \\(n\\), with \\(n=2^k\\), and the edge-flip Glauber chain having stationary weight \\(\\pi(\\sigma)\\propto\\lambda^{|\\sigma|}\\). At \\(\\lambda=1\\), it asks for a polynomial mixing-time upper bound in either of two state spaces:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-PROBABILITY-0039": {
   "statement_status": "exact",
   "original_statement": "Random walks on Dyck's paths\n\nConsider the Markov chain on the set of Dyck's paths, which moves by choosing uniformly at random two coordinates and exchanging their value ($+$ or $-1$) provided the resulting path remains a Dyck's path. What is the mixing time?",
   "clean_statement": "Random walks on Dyck's paths\n\nConsider the Markov chain on the set of Dyck's paths, which moves by choosing uniformly at random two coordinates and exchanging their value ($+$ or $-1$) provided the resulting path remains a Dyck's path. What is the mixing time?",
   "public_statement": "Random walks on Dyck's paths\n\nConsider the Markov chain on the set of Dyck's paths, which moves by choosing uniformly at random two coordinates and exchanging their value ($+$ or $-1$) provided the resulting path remains a Dyck's path. What is the mixing time?",
   "evidence": "The canonical record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-PROBABILITY-0040": {
   "statement_status": "exact",
   "original_statement": "Edge-flip on triangulations of a convex polygon\n\nConsider the random walk on triangulations of a convex polygon, driven by flips of a randomly chosen diagonal. What is the mixing time?",
   "clean_statement": "Edge-flip on triangulations of a convex polygon\n\nConsider the random walk on triangulations of a convex polygon, driven by flips of a randomly chosen diagonal. What is the mixing time?",
   "public_statement": "Edge-flip on triangulations of a convex polygon\n\nConsider the random walk on triangulations of a convex polygon, driven by flips of a randomly chosen diagonal. What is the mixing time?",
   "evidence": "The exact source record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-PROBABILITY-0041": {
   "statement_status": "exact",
   "original_statement": "Edge-flip chain on triangulations\n\nConsider the chain over triangulations of $[0,n]^2$, in which, at each step, an edge is randomly chosen and, if it is the diagonal of convex quadrilateral, flipped to the opposite diagonal. Does this chain have polynomial mixing time?",
   "clean_statement": "Edge-flip chain on triangulations\n\nConsider the chain over triangulations of $[0,n]^2$, in which, at each step, an edge is randomly chosen and, if it is the diagonal of convex quadrilateral, flipped to the opposite diagonal. Does this chain have polynomial mixing time?",
   "public_statement": "Edge-flip chain on triangulations\n\nConsider the chain over triangulations of $[0,n]^2$, in which, at each step, an edge is randomly chosen and, if it is the diagonal of convex quadrilateral, flipped to the opposite diagonal. Does this chain have polynomial mixing time?",
   "evidence": "The exact canonical problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-PROBABILITY-0042": {
   "statement_status": "exact",
   "original_statement": "Hitting time of trajectory\n\nConsider a nonreversible (say $\\frac 23$--$\\frac 13$) random walk on an Eulerian digraph $(V,E)$. Prove that the first time, $\\tau_{(X^{(y)}_t)}$ the random walk hits a moving target $X^{(y)}_t$ with $X_0=y$ satisfies the following: $\\max_{x,y} \\mathbb E_x [\\tau_{(X_t^{(y)})}] \\leq |E|\\,|V|$.",
   "clean_statement": "Hitting time of trajectory\n\nConsider a nonreversible (say $\\frac 23$--$\\frac 13$) random walk on an Eulerian digraph $(V,E)$. Prove that the first time, $\\tau_{(X^{(y)}_t)}$ the random walk hits a moving target $X^{(y)}_t$ with $X_0=y$ satisfies the following: $\\max_{x,y} \\mathbb E_x [\\tau_{(X_t^{(y)})}] \\leq |E|\\,|V|$.",
   "public_statement": "Hitting time of trajectory\n\nConsider a nonreversible (say $\\frac 23$--$\\frac 13$) random walk on an Eulerian digraph $(V,E)$. Prove that the first time, $\\tau_{(X^{(y)}_t)}$ the random walk hits a moving target $X^{(y)}_t$ with $X_0=y$ satisfies the following: $\\max_{x,y} \\mathbb E_x [\\tau_{(X_t^{(y)})}] \\leq |E|\\,|V|$.",
   "evidence": "The source record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-PROBABILITY-0043": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Cycle + Erdos-Renyi\n\nConsider the graph $G=\\mathbb Z_n \\cup ER(n,p)$ and make the simple random walk on $G$ nonreversible by adding a counterclockwise drift to the cycle part.\n\nProve that for $p=\\frac \\epsilon n$, $\\epsilon>0$ fixed, the nonreversible chain has $t_{\\mbox{mix}}\\asymp \\log n$.\n\nProve that when $p=\\epsilon n^{-\\frac 32}$ the nonreversible chain has $t_{\\mbox{mix}}=\\tilde O(\\sqrt n)$.",
   "clean_statement": "**Cycle + Erdos-Renyi.** Consider the graph \\(G=\\mathbb Z_n\\cup ER(n,p)\\) and make the simple random walk on \\(G\\) nonreversible by adding a counterclockwise drift to the cycle part.\n\nProve that for \\(p=\\epsilon/n\\), \\(\\epsilon>0\\) fixed, the nonreversible chain has \\(t_{\\mathrm{mix}}\\asymp\\log n\\).\n\nProve that when \\(p=\\epsilon n^{-3/2}\\) the nonreversible chain has \\(t_{\\mathrm{mix}}=\\widetilde O(\\sqrt n)\\).",
   "public_statement": "Cycle + Erdos-Renyi\n\nConsider the graph $G=\\mathbb Z_n \\cup ER(n,p)$ and make the simple random walk on $G$ nonreversible by adding a counterclockwise drift to the cycle part.\n\nProve that for $p=\\frac \\epsilon n$, $\\epsilon>0$ fixed, the nonreversible chain has $t_{\\mbox{mix}}\\asymp \\log n$.\n\nProve that when $p=\\epsilon n^{-\\frac 32}$ the nonreversible chain has $t_{\\mbox{mix}}=\\tilde O(\\sqrt n)$.",
   "evidence": "The mathematical formulas are readable; the important defect is not OCR but under-specification. “Adding a counterclockwise drift” does not determine transition probabilities, a stationary measure, whether time is discrete or continuous, or whether the claim is quenched or annealed. The original AIM page was unavailable (HTTP 502) when checked on 2026-08-11. The workshop report mentions a working group on “cutoff on the small world,” but supplies no missing kernel definition. Thus the results below use a precise, explicitly labeled reconstruction.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-probability-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-PROBABILITY-0044": {
   "statement_status": "exact",
   "original_statement": "Nonreversible vs. reversible chains\n\nLet $P$ be the transition matrix of a general nonreversible chain with, say, bounded degree vertices and transition probabilities uniformly bounded from below. Denote its mixing time by $t_{\\mbox{mix}}^P$ and let $\\tilde t_{\\mbox{mix}}$ be the mixing time of its symmetrization.\n\nIs it always true that $t_{\\mbox{mix}}^P\\lesssim \\tilde t_{\\mbox{mix}}$? Is the same also true of the cover time?",
   "clean_statement": "Nonreversible vs. reversible chains\n\nLet $P$ be the transition matrix of a general nonreversible chain with, say, bounded degree vertices and transition probabilities uniformly bounded from below. Denote its mixing time by $t_{\\mbox{mix}}^P$ and let $\\tilde t_{\\mbox{mix}}$ be the mixing time of its symmetrization.\n\nIs it always true that $t_{\\mbox{mix}}^P\\lesssim \\tilde t_{\\mbox{mix}}$? Is the same also true of the cover time?",
   "public_statement": "Nonreversible vs. reversible chains\n\nLet $P$ be the transition matrix of a general nonreversible chain with, say, bounded degree vertices and transition probabilities uniformly bounded from below. Denote its mixing time by $t_{\\mbox{mix}}^P$ and let $\\tilde t_{\\mbox{mix}}$ be the mixing time of its symmetrization.\n\nIs it always true that $t_{\\mbox{mix}}^P\\lesssim \\tilde t_{\\mbox{mix}}$? Is the same also true of the cover time?",
   "evidence": "The canonical record is AIM-PROBABILITY-0044, item 3.3 in the AIM workshop list *Markov chain mixing times*, section “Non-reversible chains.” Its problem field reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-PROBABILITY-0045": {
   "statement_status": "exact",
   "original_statement": "Non-backtracking vs. simple random walk\n\nIs it true that the lazy non-backtracking random walk (NBRW) mixes faster than the lazy simple random walk (SRW)?",
   "clean_statement": "Non-backtracking vs. simple random walk\n\nIs it true that the lazy non-backtracking random walk (NBRW) mixes faster than the lazy simple random walk (SRW)?",
   "public_statement": "Non-backtracking vs. simple random walk\n\nIs it true that the lazy non-backtracking random walk (NBRW) mixes faster than the lazy simple random walk (SRW)?",
   "evidence": "The exact source question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-PROBABILITY-0046": {
   "statement_status": "exact",
   "original_statement": "Comparison of interchange to SRW and Exclusion\n\nConsider the interchange process with $n$ particles on $n$ vertices. Can you say the following?\n\n\\begin{align*}\nt_{\\mbox{mix}}^{IP}\\lesssim & t_{\\mbox{mix}}^{SRW} \\log n\\,, \\\\\nt_{\\mbox{mix}}^{IP} \\lesssim & t_{\\mbox{mix}}^{SRW,n}\\,, \\\\\nt_{\\mbox{mix}}^{IP} \\asymp & t_{\\mbox{mix}}^{SSEP,n/2}\\,.\n\\end{align*}",
   "clean_statement": "Comparison of interchange to SRW and Exclusion\n\nConsider the interchange process with $n$ particles on $n$ vertices. Can you say the following?\n\n\\begin{align*}\nt_{\\mbox{mix}}^{IP}\\lesssim & t_{\\mbox{mix}}^{SRW} \\log n\\,, \\\\\nt_{\\mbox{mix}}^{IP} \\lesssim & t_{\\mbox{mix}}^{SRW,n}\\,, \\\\\nt_{\\mbox{mix}}^{IP} \\asymp & t_{\\mbox{mix}}^{SSEP,n/2}\\,.\n\\end{align*}",
   "public_statement": "Comparison of interchange to SRW and Exclusion\n\nConsider the interchange process with $n$ particles on $n$ vertices. Can you say the following?\n\n\\begin{align*}\nt_{\\mbox{mix}}^{IP}\\lesssim & t_{\\mbox{mix}}^{SRW} \\log n\\,, \\\\\nt_{\\mbox{mix}}^{IP} \\lesssim & t_{\\mbox{mix}}^{SRW,n}\\,, \\\\\nt_{\\mbox{mix}}^{IP} \\asymp & t_{\\mbox{mix}}^{SSEP,n/2}\\,.\n\\end{align*}",
   "evidence": "There is no visible OCR corruption, but several mathematical conventions are omitted:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-PROBABILITY-0047": {
   "statement_status": "exact",
   "original_statement": "Interchange process on the hypercube\n\nObtain good bounds on $t_{\\mbox{mix}}^{IP}$ on the hypercube $\\{0,1\\}^n$.",
   "clean_statement": "Interchange process on the hypercube\n\nObtain good bounds on $t_{\\mbox{mix}}^{IP}$ on the hypercube $\\{0,1\\}^n$.",
   "public_statement": "Interchange process on the hypercube\n\nObtain good bounds on $t_{\\mbox{mix}}^{IP}$ on the hypercube $\\{0,1\\}^n$.",
   "evidence": "The canonical record is AIM-PROBABILITY-0047, item 4.2 in the AIM workshop list *Markov chain mixing times*, section “Exclusion and Interchange processes.” Its complete mathematical prompt is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-PROBABILITY-0048": {
   "statement_status": "exact",
   "original_statement": "Hitting time comparison\n\nCan you compare the hitting times of the interchange process to hitting times for the SRW on a graph $G$.",
   "clean_statement": "Hitting time comparison\n\nCan you compare the hitting times of the interchange process to hitting times for the SRW on a graph $G$.",
   "public_statement": "Hitting time comparison\n\nCan you compare the hitting times of the interchange process to hitting times for the SRW on a graph $G$.",
   "evidence": "The canonical record is item 4.3 in the section “Exclusion and Interchange processes” of the AIM workshop *Markov chain mixing times*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-PROBABILITY-0049": {
   "statement_status": "exact",
   "original_statement": "Cutoff for random-to-random shuffle\n\nThe random-to-random card shuffle on $S_n$ has cutoff at $\\frac {3n}4 \\log n$.",
   "clean_statement": "Cutoff for random-to-random shuffle\n\nThe random-to-random card shuffle on $S_n$ has cutoff at $\\frac {3n}4 \\log n$.",
   "public_statement": "Cutoff for random-to-random shuffle\n\nThe random-to-random card shuffle on $S_n$ has cutoff at $\\frac {3n}4 \\log n$.",
   "evidence": "The canonical record is item 5.1 in the AIM workshop section “Chains on \\(S_n\\)”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-PROBABILITY-0050": {
   "statement_status": "exact",
   "original_statement": "Random biased transpositions\n\nThe random uniform transposition walk on $S_n$ is the discrete time Markov chain that at each time step applies a transposition $(i,j)$ uniformly at random.\n\nThe random nearly-uniform (biased) transposition walk is the corresponding chain that at each time step applies a transposition $(i,j)$ with probability $p_{ij}$ for $p_{ij}$ comparable to the uniform rate up to uniformly bounded constants.\n\nDoes the random nearly-uniform transposition walk on $S_n$ exhibit cutoff?\n\nWhat about in the simpler case where $p_{ij}=p_{i}p_{j}$ for each $1\\leq i,j\\leq n$?",
   "clean_statement": "Random biased transpositions\n\nThe random uniform transposition walk on $S_n$ is the discrete time Markov chain that at each time step applies a transposition $(i,j)$ uniformly at random.\n\nThe random nearly-uniform (biased) transposition walk is the corresponding chain that at each time step applies a transposition $(i,j)$ with probability $p_{ij}$ for $p_{ij}$ comparable to the uniform rate up to uniformly bounded constants.\n\nDoes the random nearly-uniform transposition walk on $S_n$ exhibit cutoff?\n\nWhat about in the simpler case where $p_{ij}=p_{i}p_{j}$ for each $1\\leq i,j\\leq n$?",
   "public_statement": "Random biased transpositions\n\nThe random uniform transposition walk on $S_n$ is the discrete time Markov chain that at each time step applies a transposition $(i,j)$ uniformly at random.\n\nThe random nearly-uniform (biased) transposition walk is the corresponding chain that at each time step applies a transposition $(i,j)$ with probability $p_{ij}$ for $p_{ij}$ comparable to the uniform rate up to uniformly bounded constants.\n\nDoes the random nearly-uniform transposition walk on $S_n$ exhibit cutoff?\n\nWhat about in the simpler case where $p_{ij}=p_{i}p_{j}$ for each $1\\leq i,j\\leq n$?",
   "evidence": "The local JSON record is internally legible and shows no OCR corruption. The linked AIM page returned an HTTP 502 during this run, so I could not compare its current rendering with the preserved record.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-PROBABILITY-0051": {
   "statement_status": "exact",
   "original_statement": "Cutoff profile of random transpositions\n\nCan we obtain the cutoff profile for the uniform random transposition walk on $S_n$?",
   "clean_statement": "Cutoff profile of random transpositions\n\nCan we obtain the cutoff profile for the uniform random transposition walk on $S_n$?",
   "public_statement": "Cutoff profile of random transpositions\n\nCan we obtain the cutoff profile for the uniform random transposition walk on $S_n$?",
   "evidence": "The canonical record is Problem 5.3 in the AIM workshop list “Markov chain mixing times,” section “Chains on \\(S_n\\)”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-PROBABILITY-0052": {
   "statement_status": "exact",
   "original_statement": "Kac's random walk\n\nConsider the Kac's random walk on the $n$-sphere defined as follows: for a vector $v\\in \\mathcal S^n$, pick two coordinates uniformly at random, and an angle uniformly at random on $[0,2\\pi)$. Then rotate in the circle generated by the two coordinates by that angle.\n\nDoes Kac's random walk on $\\mathcal S^n$ exhibit cutoff?\n\nDoes the analogously defined random walk on $SO(n)$ exhibit cutoff?",
   "clean_statement": "Kac's random walk\n\nConsider the Kac's random walk on the $n$-sphere defined as follows: for a vector $v\\in \\mathcal S^n$, pick two coordinates uniformly at random, and an angle uniformly at random on $[0,2\\pi)$. Then rotate in the circle generated by the two coordinates by that angle.\n\nDoes Kac's random walk on $\\mathcal S^n$ exhibit cutoff?\n\nDoes the analogously defined random walk on $SO(n)$ exhibit cutoff?",
   "public_statement": "Kac's random walk\n\nConsider the Kac's random walk on the $n$-sphere defined as follows: for a vector $v\\in \\mathcal S^n$, pick two coordinates uniformly at random, and an angle uniformly at random on $[0,2\\pi)$. Then rotate in the circle generated by the two coordinates by that angle.\n\nDoes Kac's random walk on $\\mathcal S^n$ exhibit cutoff?\n\nDoes the analogously defined random walk on $SO(n)$ exhibit cutoff?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-PROBABILITY-0053": {
   "statement_status": "exact",
   "original_statement": "Metropolis on $[0,1]^2$\n\nConsider the stationary distribution on $[0,1]^2$ given by\n\\[\\pi_A(p_1,p_2)=\\mathcal Z^{-1} \\exp(-A|p_1-p_2|^2)\n\\]\nparametrized by $A\\in [10,1000]$.\n\nObtain a good bound on the mixing time of the Metropolis algorithm for this stationary distribution in terms of $A$. Generalize this to $[0,1]^d$.",
   "clean_statement": "Metropolis on $[0,1]^2$\n\nConsider the stationary distribution on $[0,1]^2$ given by\n\\[\\pi_A(p_1,p_2)=\\mathcal Z^{-1} \\exp(-A|p_1-p_2|^2)\n\\]\nparametrized by $A\\in [10,1000]$.\n\nObtain a good bound on the mixing time of the Metropolis algorithm for this stationary distribution in terms of $A$. Generalize this to $[0,1]^d$.",
   "public_statement": "Metropolis on $[0,1]^2$\n\nConsider the stationary distribution on $[0,1]^2$ given by\n\\[\\pi_A(p_1,p_2)=\\mathcal Z^{-1} \\exp(-A|p_1-p_2|^2)\n\\]\nparametrized by $A\\in [10,1000]$.\n\nObtain a good bound on the mixing time of the Metropolis algorithm for this stationary distribution in terms of $A$. Generalize this to $[0,1]^d$.",
   "evidence": "The canonical record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-PROBABILITY-0054": {
   "statement_status": "exact",
   "original_statement": "Function-specific mixing time and concentration\n\nLet $(X_n)$ be a stationary Markov chain on a finite state space $\\Omega$ and $f\\colon\\Omega\\to [-1,1]$. Consider\n$$\nS_n=\\frac{1}{n} \\sum_{i=1}^n f(X_i) .\n$$\n\nDoes there exist an absolute constant $c>0$ such that, for all $\\varepsilon>0$ and $n\\geq 1$,\n$$\n\\mathbb{P}\\left(\\Big| S_n-\\mathbb{E} S_n\\Big| \\geq\\varepsilon\\right)\\leq 2\\exp\\left(-\\frac{cn\\varepsilon^2}{t_f(\\delta)}\\right)\\, ,\n$$\nwhere $t_f(\\delta)=\\sup_{x\\in\\Omega}\\left\\{n\\geq 0,\\, \\Big|\\mathbb{E}_x f(X_n) -\\mathbb{E}_\\pi f\\Big|\\leq\\delta\\right\\}$ ? Does it hold for $\\delta=1/4$?",
   "clean_statement": "Function-specific mixing time and concentration\n\nLet $(X_n)$ be a stationary Markov chain on a finite state space $\\Omega$ and $f\\colon\\Omega\\to [-1,1]$. Consider\n$$\nS_n=\\frac{1}{n} \\sum_{i=1}^n f(X_i) .\n$$\n\nDoes there exist an absolute constant $c>0$ such that, for all $\\varepsilon>0$ and $n\\geq 1$,\n$$\n\\mathbb{P}\\left(\\Big| S_n-\\mathbb{E} S_n\\Big| \\geq\\varepsilon\\right)\\leq 2\\exp\\left(-\\frac{cn\\varepsilon^2}{t_f(\\delta)}\\right)\\, ,\n$$\nwhere $t_f(\\delta)=\\sup_{x\\in\\Omega}\\left\\{n\\geq 0,\\, \\Big|\\mathbb{E}_x f(X_n) -\\mathbb{E}_\\pi f\\Big|\\leq\\delta\\right\\}$ ? Does it hold for $\\delta=1/4$?",
   "public_statement": "Function-specific mixing time and concentration\n\nLet $(X_n)$ be a stationary Markov chain on a finite state space $\\Omega$ and $f\\colon\\Omega\\to [-1,1]$. Consider\n$$\nS_n=\\frac{1}{n} \\sum_{i=1}^n f(X_i) .\n$$\n\nDoes there exist an absolute constant $c>0$ such that, for all $\\varepsilon>0$ and $n\\geq 1$,\n$$\n\\mathbb{P}\\left(\\Big| S_n-\\mathbb{E} S_n\\Big| \\geq\\varepsilon\\right)\\leq 2\\exp\\left(-\\frac{cn\\varepsilon^2}{t_f(\\delta)}\\right)\\, ,\n$$\nwhere $t_f(\\delta)=\\sup_{x\\in\\Omega}\\left\\{n\\geq 0,\\, \\Big|\\mathbb{E}_x f(X_n) -\\mathbb{E}_\\pi f\\Big|\\leq\\delta\\right\\}$ ? Does it hold for $\\delta=1/4$?",
   "evidence": "The canonical record asks the following. Let \\((X_i)\\) be a stationary Markov chain on a finite state space \\(\\Omega\\), let \\(f:\\Omega\\to[-1,1]\\), and put \\[ S_N=\\frac1N\\sum_{i=1}^N f(X_i). \\] Is there an absolute constant \\(c>0\\) such that \\[ \\Pr\\bigl(|S_N-\\mathbb E S_N|\\geq\\varepsilon\\bigr) \\leq 2\\exp\\!\\left(-\\frac{cN\\varepsilon^2}{t_f(\\delta)}\\right) \\tag{1} \\] for every \\(\\varepsilon>0\\) and \\(N\\geq1\\)? In particular, can one take the fixed tolerance \\(\\delta=1/4\\)?",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-PROBABILITY-0055": {
   "statement_status": "exact",
   "original_statement": "Random walk on monotone subsets of $\\{0,1\\}^n$\n\nLet $A\\subset \\{0,1\\}^n$ be a monotone subset of the hypercube: for all $x\\in A$, if $y$ satisfies $y_i\\geq x_i$ for all $1\\leq i\\leq n$, then $y\\in A$. The random walk picks a coordinate at random, flip it, and accepts the move only if it remains in $A$.\n\nCan we establish an upper bound in $O(n\\log n)$ for the mixing time ?",
   "clean_statement": "Random walk on monotone subsets of $\\{0,1\\}^n$\n\nLet $A\\subset \\{0,1\\}^n$ be a monotone subset of the hypercube: for all $x\\in A$, if $y$ satisfies $y_i\\geq x_i$ for all $1\\leq i\\leq n$, then $y\\in A$. The random walk picks a coordinate at random, flip it, and accepts the move only if it remains in $A$.\n\nCan we establish an upper bound in $O(n\\log n)$ for the mixing time ?",
   "public_statement": "Random walk on monotone subsets of $\\{0,1\\}^n$\n\nLet $A\\subset \\{0,1\\}^n$ be a monotone subset of the hypercube: for all $x\\in A$, if $y$ satisfies $y_i\\geq x_i$ for all $1\\leq i\\leq n$, then $y\\in A$. The random walk picks a coordinate at random, flip it, and accepts the move only if it remains in $A$.\n\nCan we establish an upper bound in $O(n\\log n)$ for the mixing time ?",
   "evidence": "The canonical record states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-PROBABILITY-0056": {
   "statement_status": "exact",
   "original_statement": "Two matrix model with non symmetric potentials\n\nStudy eigenvalue properties (in particular, the limiting density of eigenvalues) of $M_1$ in a two matrix model given by the probability density\n$$\n\\frac{1}{Z_N}e^{-N\\textrm{tr}[V(M_1)+W(M_2)+tM_1M_2]}dM_1 dM_2.\n$$\nIn this context, $V(x)=x^2/2$ and $W(y)$ is a general polynomial of even degree (no symmetry). More in general, consider this model when $V(x)$ is an even quartic polynomial and $W(y)$ is a general polynomial with no symmetries. \\label{two-matrix-x2}",
   "clean_statement": "Two matrix model with non symmetric potentials\n\nStudy eigenvalue properties (in particular, the limiting density of eigenvalues) of $M_1$ in a two matrix model given by the probability density\n$$\n\\frac{1}{Z_N}e^{-N\\textrm{tr}[V(M_1)+W(M_2)+tM_1M_2]}dM_1 dM_2.\n$$\nIn this context, $V(x)=x^2/2$ and $W(y)$ is a general polynomial of even degree (no symmetry). More in general, consider this model when $V(x)$ is an even quartic polynomial and $W(y)$ is a general polynomial with no symmetries. \\label{two-matrix-x2}",
   "public_statement": "Two matrix model with non symmetric potentials\n\nStudy eigenvalue properties (in particular, the limiting density of eigenvalues) of $M_1$ in a two matrix model given by the probability density\n$$\n\\frac{1}{Z_N}e^{-N\\textrm{tr}[V(M_1)+W(M_2)+tM_1M_2]}dM_1 dM_2.\n$$\nIn this context, $V(x)=x^2/2$ and $W(y)$ is a general polynomial of even degree (no symmetry). More in general, consider this model when $V(x)$ is an even quartic polynomial and $W(y)$ is a general polynomial with no symmetries. \\label{two-matrix-x2}",
   "evidence": "The AIM record asks for eigenvalue information, especially the limiting eigenvalue density of \\(M_1\\), in the Hermitian two-matrix model",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-PROBABILITY-0057": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Coupled random matrix model with negative potentials\n\nAnalysis of the coupled matrix model similar to problem \\ref{two-matrix-x2}, but with potentials\n$$\nV(x)=-(x^4-ax^2), \\qquad W(y)=V(y).\n$$\nIs there any new critical behavior for some value of $a$?",
   "clean_statement": null,
   "public_statement": "Coupled random matrix model with negative potentials\n\nAnalysis of the coupled matrix model similar to problem \\ref{two-matrix-x2}, but with potentials\n$$\nV(x)=-(x^4-ax^2), \\qquad W(y)=V(y).\n$$\nIs there any new critical behavior for some value of $a$?",
   "evidence": "The canonical record is:",
   "classification_method": "invalid_literal_with_repaired_reading",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-PROBABILITY-0058": {
   "statement_status": "exact",
   "original_statement": "$2+1/2$ random matrix models\n\nA general question about Riemann-Hilbert techniques to study the two-matrix plus external source model.",
   "clean_statement": "$2+1/2$ random matrix models\n\nA general question about Riemann-Hilbert techniques to study the two-matrix plus external source model.",
   "public_statement": "$2+1/2$ random matrix models\n\nA general question about Riemann-Hilbert techniques to study the two-matrix plus external source model.",
   "evidence": "The canonical record, in section “Two-matrix models” of the AIM workshop *Vector equilibrium problems and their applications to random matrix models*, reads in full:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-PROBABILITY-0059": {
   "statement_status": "exact",
   "original_statement": "Random matrices with external source. More general cases\n\nStudy the random matrix model with probability measure\n$$\n\\frac{1}{Z_N}e^{-N\\textrm{tr}(V(M)-AM)},\n$$\nwhere $A$ is an $N\\times N$ diagonal matrix with eigenvalues $\\pm a$ (each with equal multiplicity $N/2$, assuming $N$ is even), and $V(M)$ is a general polynomial.",
   "clean_statement": "Random matrices with external source. More general cases\n\nStudy the random matrix model with probability measure\n$$\n\\frac{1}{Z_N}e^{-N\\textrm{tr}(V(M)-AM)},\n$$\nwhere $A$ is an $N\\times N$ diagonal matrix with eigenvalues $\\pm a$ (each with equal multiplicity $N/2$, assuming $N$ is even), and $V(M)$ is a general polynomial.",
   "public_statement": "Random matrices with external source. More general cases\n\nStudy the random matrix model with probability measure\n$$\n\\frac{1}{Z_N}e^{-N\\textrm{tr}(V(M)-AM)},\n$$\nwhere $A$ is an $N\\times N$ diagonal matrix with eigenvalues $\\pm a$ (each with equal multiplicity $N/2$, assuming $N$ is even), and $V(M)$ is a general polynomial.",
   "evidence": "The exact AIM record asks to study the Hermitian external-source ensemble",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-PROBABILITY-0060": {
   "statement_status": "exact",
   "original_statement": "Normal matrix model and d-bar problems\n\nIs it possible to use the theory of $\\overline{\\partial}$-problems to analyze the normal matrix models?",
   "clean_statement": "Normal matrix model and d-bar problems\n\nIs it possible to use the theory of $\\overline{\\partial}$-problems to analyze the normal matrix models?",
   "public_statement": "Normal matrix model and d-bar problems\n\nIs it possible to use the theory of $\\overline{\\partial}$-problems to analyze the normal matrix models?",
   "evidence": "The canonical AIM record is problem 3.1 in the section “Normal matrix model” of the workshop *Vector equilibrium problems and their applications to random matrix models*. Its question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-PROBABILITY-0061": {
   "statement_status": "exact",
   "original_statement": "$S$-curves for complex potentials in the scalar case\n\nAnalysis of existence and properties of $S$-curves in the scalar case when the potential is $\\phi(z)=\\textrm{Re}\\, z^p$, for $p\\geq 3$, and we take $\\tau$ as the set of continua that join neighboring valleys of $\\phi(z)$ (where $\\phi(z)\\to-\\infty$) in the complex plane. In particular, prove that the support of the equilibrium measure on a curve with the $S$-property is one-cut case, or give bounds on the number of cuts depending on $p$.",
   "clean_statement": "$S$-curves for complex potentials in the scalar case\n\nAnalysis of existence and properties of $S$-curves in the scalar case when the potential is $\\phi(z)=\\textrm{Re}\\, z^p$, for $p\\geq 3$, and we take $\\tau$ as the set of continua that join neighboring valleys of $\\phi(z)$ (where $\\phi(z)\\to-\\infty$) in the complex plane. In particular, prove that the support of the equilibrium measure on a curve with the $S$-property is one-cut case, or give bounds on the number of cuts depending on $p$.",
   "public_statement": "$S$-curves for complex potentials in the scalar case\n\nAnalysis of existence and properties of $S$-curves in the scalar case when the potential is $\\phi(z)=\\textrm{Re}\\, z^p$, for $p\\geq 3$, and we take $\\tau$ as the set of continua that join neighboring valleys of $\\phi(z)$ (where $\\phi(z)\\to-\\infty$) in the complex plane. In particular, prove that the support of the equilibrium measure on a curve with the $S$-property is one-cut case, or give bounds on the number of cuts depending on $p$.",
   "evidence": "The canonical AIM record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-PROBABILITY-0062": {
   "statement_status": "exact",
   "original_statement": "Numerical computation of $S$-curves\n\nA general question about numerical methods for vector equilibrium problems and computation of corresponding $S$-curves. More in particular, how to produce critical trajectories in a reliable way, given the spectral curve of the problem?",
   "clean_statement": "Numerical computation of $S$-curves\n\nA general question about numerical methods for vector equilibrium problems and computation of corresponding $S$-curves. More in particular, how to produce critical trajectories in a reliable way, given the spectral curve of the problem?",
   "public_statement": "Numerical computation of $S$-curves\n\nA general question about numerical methods for vector equilibrium problems and computation of corresponding $S$-curves. More in particular, how to produce critical trajectories in a reliable way, given the spectral curve of the problem?",
   "evidence": "The canonical AIM record, problem 4.2 in the section “\\(S\\)-curves,” asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-PROBABILITY-0063": {
   "statement_status": "exact",
   "original_statement": "$S$-curves with piecewise harmonic external field\n\nConsider the external potential\n$$\n\\phi(z)=\\begin{cases} -k_1\\, \\textrm{Re}\\, z, & \\textrm{Re}\\, z>0,\\\\ k_2\\, \\textrm{Re}\\, z, & \\textrm{Re}\\, z0$. Is there a continuum with the $S$-property connecting the points $z_1=-\\varepsilon_1+i$ and $z_2=\\varepsilon_2-i$ in the complex plane, for small positive $\\varepsilon_1$ and $\\varepsilon_2$? Also, consider the max-min problem in this setting. Does the curve obtained have a portion along the imaginary axis?",
   "clean_statement": "$S$-curves with piecewise harmonic external field\n\nConsider the external potential\n$$\n\\phi(z)=\\begin{cases} -k_1\\, \\textrm{Re}\\, z, & \\textrm{Re}\\, z>0,\\\\ k_2\\, \\textrm{Re}\\, z, & \\textrm{Re}\\, z0$. Is there a continuum with the $S$-property connecting the points $z_1=-\\varepsilon_1+i$ and $z_2=\\varepsilon_2-i$ in the complex plane, for small positive $\\varepsilon_1$ and $\\varepsilon_2$? Also, consider the max-min problem in this setting. Does the curve obtained have a portion along the imaginary axis?",
   "public_statement": "$S$-curves with piecewise harmonic external field\n\nConsider the external potential\n$$\n\\phi(z)=\\begin{cases} -k_1\\, \\textrm{Re}\\, z, & \\textrm{Re}\\, z>0,\\\\ k_2\\, \\textrm{Re}\\, z, & \\textrm{Re}\\, z0$. Is there a continuum with the $S$-property connecting the points $z_1=-\\varepsilon_1+i$ and $z_2=\\varepsilon_2-i$ in the complex plane, for small positive $\\varepsilon_1$ and $\\varepsilon_2$? Also, consider the max-min problem in this setting. Does the curve obtained have a portion along the imaginary axis?",
   "evidence": "The canonical record is visibly truncated. Its formula ends with the literal text",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-PROBABILITY-0064": {
   "statement_status": "exact",
   "original_statement": "Green's potential\n\nExistence theory for $S$-curves in a general sense, if the logarithmic potential is replaced by Green's potential.",
   "clean_statement": "Green's potential\n\nExistence theory for $S$-curves in a general sense, if the logarithmic potential is replaced by Green's potential.",
   "public_statement": "Green's potential\n\nExistence theory for $S$-curves in a general sense, if the logarithmic potential is replaced by Green's potential.",
   "evidence": "There is no apparent OCR error, but the statement omits essential data: the Greenian domain \\(D\\), the normalization of its kernel, the external field, whether admissible continua may meet \\(\\partial D\\), the topology of the admissible class, and whether the request concerns equilibrium on a fixed continuum or a max--min free boundary.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-PROBABILITY-0065": {
   "statement_status": "exact",
   "original_statement": "In a general sense, is it possible to find some explicit connection between the notions of string equations, spectral curve, Lax pair, WKB approximation, etc, and the vector equilibrium problems that appear in the two-matrix or external source models? For example, would it be possible to use string equations and asymptotic information together with the spectral curve, to find heuristically the equilibrium measure?",
   "clean_statement": "In a general sense, is it possible to find some explicit connection between the notions of string equations, spectral curve, Lax pair, WKB approximation, etc, and the vector equilibrium problems that appear in the two-matrix or external source models? For example, would it be possible to use string equations and asymptotic information together with the spectral curve, to find heuristically the equilibrium measure?",
   "public_statement": "In a general sense, is it possible to find some explicit connection between the notions of string equations, spectral curve, Lax pair, WKB approximation, etc, and the vector equilibrium problems that appear in the two-matrix or external source models? For example, would it be possible to use string equations and asymptotic information together with the spectral curve, to find heuristically the equilibrium measure?",
   "evidence": "The AIM record asks whether string equations, spectral curves, Lax pairs, and WKB asymptotics can be connected explicitly to vector equilibrium problems in two-matrix or external-source models, and whether string equations plus asymptotic spectral-curve data can heuristically recover the equilibrium measure.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-PROBABILITY-0066": {
   "statement_status": "exact",
   "original_statement": "Is it possible to relate multiple orthogonal polynomials to standard orthogonal polynomials, via the so-called ``convolution product\"?",
   "clean_statement": "Is it possible to relate multiple orthogonal polynomials to standard orthogonal polynomials, via the so-called ``convolution product\"?",
   "public_statement": "Is it possible to relate multiple orthogonal polynomials to standard orthogonal polynomials, via the so-called ``convolution product\"?",
   "evidence": "The canonical AIM record asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-PROBABILITY-0067": {
   "statement_status": "exact",
   "original_statement": "Statistics of KPZ equation in $1+1$ dimension:\n\nCompute statistics for different initial data including $\\mathcal{Z}_0(x)=1$ (flat) and $\\mathcal{Z}_0(x)=e^{B(x)}$ for $B(x)$ a two-sided Brownian motion (equilibrium).",
   "clean_statement": "Statistics of KPZ equation in $1+1$ dimension:\n\nCompute statistics for different initial data including $\\mathcal{Z}_0(x)=1$ (flat) and $\\mathcal{Z}_0(x)=e^{B(x)}$ for $B(x)$ a two-sided Brownian motion (equilibrium).",
   "public_statement": "Statistics of KPZ equation in $1+1$ dimension:\n\nCompute statistics for different initial data including $\\mathcal{Z}_0(x)=1$ (flat) and $\\mathcal{Z}_0(x)=e^{B(x)}$ for $B(x)$ a two-sided Brownian motion (equilibrium).",
   "evidence": "This is Problem 11.05 in the AIM workshop list *The Kardar--Parisi--Zhang equation and universality class* (canonical source record `aim-probability-notes.json`, index 66). The exact recovered mathematical request is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-PROBABILITY-0068": {
   "statement_status": "exact",
   "original_statement": "Compute multi-point (spatial) distribution for KPZ with various initial data. For example, with $\\mathcal{Z}_0(x)=\\delta_{x=0}$, let\n$$\\mathcal{Z}(t,x)=e^{-\\frac{x^2}{2t}+t^{1/3}A_t(t^{-2/3}x)-\\frac{t}{24}}.$$\nWhat is the distribution $F(\\xi_1, \\xi_2)$ such that\n$$\\mathbb{P}\\left(A_t(x_1)\\leq\\xi_1, A_t(x_2)\\leq\\xi_2\\right)=F(\\xi_1,\\xi_2)?$$\nPerhaps easier, show that as $t$ goes to infinity and space is scaled like $t^{2/3}x$, the process $A_{t}(t^{2/3}x)$ converges to the Airy$_2$ process in $x$.",
   "clean_statement": "Compute multi-point (spatial) distribution for KPZ with various initial data. For example, with $\\mathcal{Z}_0(x)=\\delta_{x=0}$, let\n$$\\mathcal{Z}(t,x)=e^{-\\frac{x^2}{2t}+t^{1/3}A_t(t^{-2/3}x)-\\frac{t}{24}}.$$\nWhat is the distribution $F(\\xi_1, \\xi_2)$ such that\n$$\\mathbb{P}\\left(A_t(x_1)\\leq\\xi_1, A_t(x_2)\\leq\\xi_2\\right)=F(\\xi_1,\\xi_2)?$$\nPerhaps easier, show that as $t$ goes to infinity and space is scaled like $t^{2/3}x$, the process $A_{t}(t^{2/3}x)$ converges to the Airy$_2$ process in $x$.",
   "public_statement": "Compute multi-point (spatial) distribution for KPZ with various initial data. For example, with $\\mathcal{Z}_0(x)=\\delta_{x=0}$, let\n$$\\mathcal{Z}(t,x)=e^{-\\frac{x^2}{2t}+t^{1/3}A_t(t^{-2/3}x)-\\frac{t}{24}}.$$\nWhat is the distribution $F(\\xi_1, \\xi_2)$ such that\n$$\\mathbb{P}\\left(A_t(x_1)\\leq\\xi_1, A_t(x_2)\\leq\\xi_2\\right)=F(\\xi_1,\\xi_2)?$$\nPerhaps easier, show that as $t$ goes to infinity and space is scaled like $t^{2/3}x$, the process $A_{t}(t^{2/3}x)$ converges to the Airy$_2$ process in $x$.",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-PROBABILITY-0069": {
   "statement_status": "exact",
   "original_statement": "Compute multi-time distribution for KPZ. What is the distribution $F'_{t_1, t_2}(\\xi_1, \\xi_2)$ such that\n$$\\mathbb{P}\\left(A_{t_1}(x)\\leq\\xi_1, A_{t_2}(x)\\leq\\xi_2\\right)=F'_{t_1, t_2}(\\xi_1,\\xi_2)?$$",
   "clean_statement": "Compute multi-time distribution for KPZ. What is the distribution $F'_{t_1, t_2}(\\xi_1, \\xi_2)$ such that\n$$\\mathbb{P}\\left(A_{t_1}(x)\\leq\\xi_1, A_{t_2}(x)\\leq\\xi_2\\right)=F'_{t_1, t_2}(\\xi_1,\\xi_2)?$$",
   "public_statement": "Compute multi-time distribution for KPZ. What is the distribution $F'_{t_1, t_2}(\\xi_1, \\xi_2)$ such that\n$$\\mathbb{P}\\left(A_{t_1}(x)\\leq\\xi_1, A_{t_2}(x)\\leq\\xi_2\\right)=F'_{t_1, t_2}(\\xi_1,\\xi_2)?$$",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-PROBABILITY-0070": {
   "statement_status": "exact",
   "original_statement": "Higher dimensions\n\nMake rigorous sense of the anisotropic KPZ equation\n$$\\partial_t h=(\\partial_x h)^2-(\\partial_y h)^2+\\Delta h+\\xi$$\nand show that the Gaussian free field is invariant.\nIs there a way of getting this equation out of the 2d Schur process dynamics of Borodin-Ferrari? Or perhaps out of the 2d q-Whittaker process dynamics of Borodin-Corwin? Is this related to $2d$ quantum Toda chain?",
   "clean_statement": "Higher dimensions\n\nMake rigorous sense of the anisotropic KPZ equation\n$$\\partial_t h=(\\partial_x h)^2-(\\partial_y h)^2+\\Delta h+\\xi$$\nand show that the Gaussian free field is invariant.\nIs there a way of getting this equation out of the 2d Schur process dynamics of Borodin-Ferrari? Or perhaps out of the 2d q-Whittaker process dynamics of Borodin-Corwin? Is this related to $2d$ quantum Toda chain?",
   "public_statement": "Higher dimensions\n\nMake rigorous sense of the anisotropic KPZ equation\n$$\\partial_t h=(\\partial_x h)^2-(\\partial_y h)^2+\\Delta h+\\xi$$\nand show that the Gaussian free field is invariant.\nIs there a way of getting this equation out of the 2d Schur process dynamics of Borodin-Ferrari? Or perhaps out of the 2d q-Whittaker process dynamics of Borodin-Corwin? Is this related to $2d$ quantum Toda chain?",
   "evidence": "This is Problem 11.2 in the AIM workshop list *The Kardar--Parisi--Zhang equation and universality class*. The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 69,
   "attempt": 1
  },
  "AIM-PROBABILITY-0071": {
   "statement_status": "exact",
   "original_statement": "Prove that above 2 spatial dimensions, the KPZ equation is trivial (i.e., discretizations limit to the linearized equation, perhaps with a larger variance in the noise).",
   "clean_statement": "Prove that above 2 spatial dimensions, the KPZ equation is trivial (i.e., discretizations limit to the linearized equation, perhaps with a larger variance in the noise).",
   "public_statement": "Prove that above 2 spatial dimensions, the KPZ equation is trivial (i.e., discretizations limit to the linearized equation, perhaps with a larger variance in the noise).",
   "evidence": "The AIM record, from the workshop *The Kardar--Parisi--Zhang equation and universality class* (problem 11.25), asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 70,
   "attempt": 1
  },
  "AIM-PROBABILITY-0072": {
   "statement_status": "exact",
   "original_statement": "In 2 spatial dimensions is the measure valued solution of the stochastic heat equation related to the ``exponential'' of the dynamic GFF and hence quantum Louiville gravity?",
   "clean_statement": "In 2 spatial dimensions is the measure valued solution of the stochastic heat equation related to the ``exponential'' of the dynamic GFF and hence quantum Louiville gravity?",
   "public_statement": "In 2 spatial dimensions is the measure valued solution of the stochastic heat equation related to the ``exponential'' of the dynamic GFF and hence quantum Louiville gravity?",
   "evidence": "The canonical AIM record (workshop *The Kardar--Parisi--Zhang equation and universality class*, “Big picture questions,” item 11.3) asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-PROBABILITY-0073": {
   "statement_status": "exact",
   "original_statement": "Study polymer free energy and growth model fluctuation exponents in higher dimension? Compute limit shapes and fluctuation exponents.",
   "clean_statement": "Study polymer free energy and growth model fluctuation exponents in higher dimension? Compute limit shapes and fluctuation exponents.",
   "public_statement": "Study polymer free energy and growth model fluctuation exponents in higher dimension? Compute limit shapes and fluctuation exponents.",
   "evidence": "This is Problem 11.35 from the AIM workshop list *The Kardar--Parisi--Zhang equation and universality class*. The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 72,
   "attempt": 1
  },
  "AIM-PROBABILITY-0074": {
   "statement_status": "exact",
   "original_statement": "Structure at positive temperature / asymmetry:\n\nPositive temperature polymers have triangular arrays associated with them (which are limits of the Macdonald processes). The marginal measure on a given level is a tropical analog of random matrix eigenvalue ensembles (such as GUE or LUE) which are determinantal point processes. These tropical point processes are no longer determinantal yet, there are still Fredholm determinants for Laplace type transforms of the analog of the largest or smallest eigenvalues. Information about all of the tropical eigenvalues should be accessible via Fredholm determinants (tropical analogs of gap probabilities for example).\n\nWhat is the structure which replaces determinantal point processes and correlation functions and which allows for such computations?",
   "clean_statement": "Structure at positive temperature / asymmetry:\n\nPositive temperature polymers have triangular arrays associated with them (which are limits of the Macdonald processes). The marginal measure on a given level is a tropical analog of random matrix eigenvalue ensembles (such as GUE or LUE) which are determinantal point processes. These tropical point processes are no longer determinantal yet, there are still Fredholm determinants for Laplace type transforms of the analog of the largest or smallest eigenvalues. Information about all of the tropical eigenvalues should be accessible via Fredholm determinants (tropical analogs of gap probabilities for example).\n\nWhat is the structure which replaces determinantal point processes and correlation functions and which allows for such computations?",
   "public_statement": "Structure at positive temperature / asymmetry:\n\nPositive temperature polymers have triangular arrays associated with them (which are limits of the Macdonald processes). The marginal measure on a given level is a tropical analog of random matrix eigenvalue ensembles (such as GUE or LUE) which are determinantal point processes. These tropical point processes are no longer determinantal yet, there are still Fredholm determinants for Laplace type transforms of the analog of the largest or smallest eigenvalues. Information about all of the tropical eigenvalues should be accessible via Fredholm determinants (tropical analogs of gap probabilities for example).\n\nWhat is the structure which replaces determinantal point processes and correlation functions and which allows for such computations?",
   "evidence": "The AIM record asks what replaces determinantal point processes and correlation functions for triangular arrays associated with positive-temperature polymers. These arrays arise as limits of Macdonald processes. Their fixed-level laws resemble GUE/LUE eigenvalue ensembles, but are not determinantal; nevertheless, Laplace transforms of an extremal coordinate can have Fredholm determinant formulas. The question asks for a structure making all analogous eigenvalues, especially joint gap probabilities, accessible.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 73,
   "attempt": 1
  },
  "AIM-PROBABILITY-0075": {
   "statement_status": "exact",
   "original_statement": "Associated to q-TASEP there is a triangular array (the q-Whittaker process). This whole array is helpful in computing things about q-TASEP.\n\nIs there such an array for ASEP or for some sort of transformed version of ASEP?",
   "clean_statement": "Associated to q-TASEP there is a triangular array (the q-Whittaker process). This whole array is helpful in computing things about q-TASEP.\n\nIs there such an array for ASEP or for some sort of transformed version of ASEP?",
   "public_statement": "Associated to q-TASEP there is a triangular array (the q-Whittaker process). This whole array is helpful in computing things about q-TASEP.\n\nIs there such an array for ASEP or for some sort of transformed version of ASEP?",
   "evidence": "The record is Problem 11.45 in the AIM workshop list *The Kardar-Parisi-Zhang equation and universality class*. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-PROBABILITY-0076": {
   "statement_status": "exact",
   "original_statement": "The limit of the triangular array is the diffusion of O'Connell based on the quantum Toda lattice, or (after another limit) the KPZ$_T$ line ensemble. Both have Brownian Gibbs properties allowing paths to cross but at exponential cost (a soft analog of the non-intersecting Brownian Gibbs property for the Airy line ensemble.)The Karlin-McGregor formula underlies the solvability of zero temperature system since it writes non-intersecting line ensembles in terms of determinants.\n\nIs there an analog of the Karlin-McGregor formula when the non-intersecting conditioning is replaced by a softer form of conditioning? Does this explain the solvability and occurrence of Fredholm determinants?",
   "clean_statement": "The limit of the triangular array is the diffusion of O'Connell based on the quantum Toda lattice, or (after another limit) the KPZ$_T$ line ensemble. Both have Brownian Gibbs properties allowing paths to cross but at exponential cost (a soft analog of the non-intersecting Brownian Gibbs property for the Airy line ensemble.)The Karlin-McGregor formula underlies the solvability of zero temperature system since it writes non-intersecting line ensembles in terms of determinants.\n\nIs there an analog of the Karlin-McGregor formula when the non-intersecting conditioning is replaced by a softer form of conditioning? Does this explain the solvability and occurrence of Fredholm determinants?",
   "public_statement": "The limit of the triangular array is the diffusion of O'Connell based on the quantum Toda lattice, or (after another limit) the KPZ$_T$ line ensemble. Both have Brownian Gibbs properties allowing paths to cross but at exponential cost (a soft analog of the non-intersecting Brownian Gibbs property for the Airy line ensemble.)The Karlin-McGregor formula underlies the solvability of zero temperature system since it writes non-intersecting line ensembles in terms of determinants.\n\nIs there an analog of the Karlin-McGregor formula when the non-intersecting conditioning is replaced by a softer form of conditioning? Does this explain the solvability and occurrence of Fredholm determinants?",
   "evidence": "This record is problem 11.5 in the AIM workshop list *The Kardar–Parisi–Zhang equation and universality class*, section “Big picture questions.” The exact extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-PROBABILITY-0077": {
   "statement_status": "exact",
   "original_statement": "Expand the universality of the KPZ equation to:\n\nGrowth model such as ballistic deposition.",
   "clean_statement": "Expand the universality of the KPZ equation to:\n\nGrowth model such as ballistic deposition.",
   "public_statement": "Expand the universality of the KPZ equation to:\n\nGrowth model such as ballistic deposition.",
   "evidence": "The exact AIM record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-PROBABILITY-0078": {
   "statement_status": "exact",
   "original_statement": "Height functions associated to exclusion processes with longer range jumps, or environment dependent speed-changes.",
   "clean_statement": "Height functions associated to exclusion processes with longer range jumps, or environment dependent speed-changes.",
   "public_statement": "Height functions associated to exclusion processes with longer range jumps, or environment dependent speed-changes.",
   "evidence": "The exact AIM record is the one-line prompt:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-PROBABILITY-0079": {
   "statement_status": "exact",
   "original_statement": "Stochastic Hamilton-Jacobi equations $\\partial_th=F(\\nabla h)+\\Delta h+\\xi_\\epsilon$ with $F$ scaled appropriately.",
   "clean_statement": "Stochastic Hamilton-Jacobi equations $\\partial_th=F(\\nabla h)+\\Delta h+\\xi_\\epsilon$ with $F$ scaled appropriately.",
   "public_statement": "Stochastic Hamilton-Jacobi equations $\\partial_th=F(\\nabla h)+\\Delta h+\\xi_\\epsilon$ with $F$ scaled appropriately.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-PROBABILITY-0080": {
   "statement_status": "exact",
   "original_statement": "Eden model.",
   "clean_statement": "Eden model.",
   "public_statement": "Eden model.",
   "evidence": "The exact canonical record says only:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-PROBABILITY-0081": {
   "statement_status": "exact",
   "original_statement": "Universality of the KPZ universality class:\n\nProve the existence and uniqueness of the KPZ universality class fixed point introduced by Corwin-Quastel in 1 spatial dimension. That is to say, consider any growth process $h(t,x)$ and show that for an appropriate centering function $\\overline{h}_{\\epsilon}$,\n$$\\lim_{\\epsilon\\rightarrow\\infty}\\epsilon^{1/2}h(\\epsilon^{-3/2}t, \\epsilon^{-1}x)-\\overline{h}_\\epsilon$$\nhas a limit as $\\epsilon\\to 0$ as a space-time process? Then show that the properties of this process identify it uniquely (see the conjectured properties in Corwin-Quastel).",
   "clean_statement": "Universality of the KPZ universality class:\n\nProve the existence and uniqueness of the KPZ universality class fixed point introduced by Corwin-Quastel in 1 spatial dimension. That is to say, consider any growth process $h(t,x)$ and show that for an appropriate centering function $\\overline{h}_{\\epsilon}$,\n$$\\lim_{\\epsilon\\rightarrow\\infty}\\epsilon^{1/2}h(\\epsilon^{-3/2}t, \\epsilon^{-1}x)-\\overline{h}_\\epsilon$$\nhas a limit as $\\epsilon\\to 0$ as a space-time process? Then show that the properties of this process identify it uniquely (see the conjectured properties in Corwin-Quastel).",
   "public_statement": "Universality of the KPZ universality class:\n\nProve the existence and uniqueness of the KPZ universality class fixed point introduced by Corwin-Quastel in 1 spatial dimension. That is to say, consider any growth process $h(t,x)$ and show that for an appropriate centering function $\\overline{h}_{\\epsilon}$,\n$$\\lim_{\\epsilon\\rightarrow\\infty}\\epsilon^{1/2}h(\\epsilon^{-3/2}t, \\epsilon^{-1}x)-\\overline{h}_\\epsilon$$\nhas a limit as $\\epsilon\\to 0$ as a space-time process? Then show that the properties of this process identify it uniquely (see the conjectured properties in Corwin-Quastel).",
   "evidence": "The canonical record (AIM workshop *The Kardar--Parisi--Zhang equation and universality class*, problem 11.75) says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-PROBABILITY-0082": {
   "statement_status": "exact",
   "original_statement": "Prove that the fixed point is attractive (i.e., universal in some class of models).",
   "clean_statement": "Prove that the fixed point is attractive (i.e., universal in some class of models).",
   "public_statement": "Prove that the fixed point is attractive (i.e., universal in some class of models).",
   "evidence": "The canonical AIM record states verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-PROBABILITY-0083": {
   "statement_status": "exact",
   "original_statement": "For example, prove universality of LPP and polymer (with respect to weight distributions).",
   "clean_statement": "For example, prove universality of LPP and polymer (with respect to weight distributions).",
   "public_statement": "For example, prove universality of LPP and polymer (with respect to weight distributions).",
   "evidence": "The exact canonical statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 82,
   "attempt": 1
  },
  "AIM-PROBABILITY-0084": {
   "statement_status": "exact",
   "original_statement": "Provide a variational explanation for the actual form of the GUE Tracy-Widom distribution in terms of the properties of the fixed point. Reduce the exact solvability to purely probabilistic terms.",
   "clean_statement": "Provide a variational explanation for the actual form of the GUE Tracy-Widom distribution in terms of the properties of the fixed point. Reduce the exact solvability to purely probabilistic terms.",
   "public_statement": "Provide a variational explanation for the actual form of the GUE Tracy-Widom distribution in terms of the properties of the fixed point. Reduce the exact solvability to purely probabilistic terms.",
   "evidence": "The canonical AIM record (workshop *The Kardar--Parisi--Zhang equation and universality class*, problem 11.9) is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 83,
   "attempt": 1
  },
  "AIM-PROBABILITY-0085": {
   "statement_status": "exact",
   "original_statement": "Exact solvability\n\nUse the Macdonald processes formulas to prove the Baik-Ben Arous-P\\'{e}ch\\'{e} transition for the semi-discrete polymer with a finite number of non-zero drifts for the underlying Brownian motions (tuned critically). In the intermediate disorder scaling, derive the formula for the statistics of the stochastic heat equation started with $\\mathcal{Z}_0(X) = {\\bf 1}_{X\\geq 0} Z^r(X)$ where $Z^N(X)$ is the partition function for the semi-discrete polymer with $r$ levels at time $X$ (the $r$ here represents the number of spikes in BBP and the drifts here of the $r$ Brownian motions can be generally chosen).",
   "clean_statement": "Exact solvability\n\nUse the Macdonald processes formulas to prove the Baik-Ben Arous-P\\'{e}ch\\'{e} transition for the semi-discrete polymer with a finite number of non-zero drifts for the underlying Brownian motions (tuned critically). In the intermediate disorder scaling, derive the formula for the statistics of the stochastic heat equation started with $\\mathcal{Z}_0(X) = {\\bf 1}_{X\\geq 0} Z^r(X)$ where $Z^N(X)$ is the partition function for the semi-discrete polymer with $r$ levels at time $X$ (the $r$ here represents the number of spikes in BBP and the drifts here of the $r$ Brownian motions can be generally chosen).",
   "public_statement": "Exact solvability\n\nUse the Macdonald processes formulas to prove the Baik-Ben Arous-P\\'{e}ch\\'{e} transition for the semi-discrete polymer with a finite number of non-zero drifts for the underlying Brownian motions (tuned critically). In the intermediate disorder scaling, derive the formula for the statistics of the stochastic heat equation started with $\\mathcal{Z}_0(X) = {\\bf 1}_{X\\geq 0} Z^r(X)$ where $Z^N(X)$ is the partition function for the semi-discrete polymer with $r$ levels at time $X$ (the $r$ here represents the number of spikes in BBP and the drifts here of the $r$ Brownian motions can be generally chosen).",
   "evidence": "The canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 84,
   "attempt": 1
  },
  "AIM-PROBABILITY-0086": {
   "statement_status": "exact",
   "original_statement": "Stochastic analysis\n\nConsider the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren. It is defined via chaos series -- show that it can also be defined by spatially smoothing white-noise and then making sense of the Wick exponential in its formulation as a partition function for non-intersecting Brownian bridges in a space time white-noise environment.",
   "clean_statement": "Stochastic analysis\n\nConsider the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren. It is defined via chaos series -- show that it can also be defined by spatially smoothing white-noise and then making sense of the Wick exponential in its formulation as a partition function for non-intersecting Brownian bridges in a space time white-noise environment.",
   "public_statement": "Stochastic analysis\n\nConsider the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren. It is defined via chaos series -- show that it can also be defined by spatially smoothing white-noise and then making sense of the Wick exponential in its formulation as a partition function for non-intersecting Brownian bridges in a space time white-noise environment.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 85,
   "attempt": 1
  },
  "AIM-PROBABILITY-0087": {
   "statement_status": "exact",
   "original_statement": "Exact solvability\n\nMotivated by success in solvable positive temperature polymers, try to study the polynuclear growth model with an underlying Brownian path measure -- is there a Burkes theorem in this setting? Specifically, study the Poissonian last passage model but at positive temperature and with the Poisson points having random weights.",
   "clean_statement": "Exact solvability\n\nMotivated by success in solvable positive temperature polymers, try to study the polynuclear growth model with an underlying Brownian path measure -- is there a Burkes theorem in this setting? Specifically, study the Poissonian last passage model but at positive temperature and with the Poisson points having random weights.",
   "public_statement": "Exact solvability\n\nMotivated by success in solvable positive temperature polymers, try to study the polynuclear growth model with an underlying Brownian path measure -- is there a Burkes theorem in this setting? Specifically, study the Poissonian last passage model but at positive temperature and with the Poisson points having random weights.",
   "evidence": "The canonical AIM record, from the workshop *The Kardar--Parisi--Zhang equation and universality class*, problem 22.06, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 86,
   "attempt": 1
  },
  "AIM-PROBABILITY-0088": {
   "statement_status": "exact",
   "original_statement": "Exact solvability\n\nStudy the overlap for replicated paths of a polymer with respect to the quenched Gibbs measure. What is the distribution of the intersection local time for the continuum polymer? Is there replica symmetry or replica breaking in the strong disorder regime or intermediate disorder regime? Try to use the Macdonald processes contour integral formulas to compute more than just the expected overlap.",
   "clean_statement": "Exact solvability\n\nStudy the overlap for replicated paths of a polymer with respect to the quenched Gibbs measure. What is the distribution of the intersection local time for the continuum polymer? Is there replica symmetry or replica breaking in the strong disorder regime or intermediate disorder regime? Try to use the Macdonald processes contour integral formulas to compute more than just the expected overlap.",
   "public_statement": "Exact solvability\n\nStudy the overlap for replicated paths of a polymer with respect to the quenched Gibbs measure. What is the distribution of the intersection local time for the continuum polymer? Is there replica symmetry or replica breaking in the strong disorder regime or intermediate disorder regime? Try to use the Macdonald processes contour integral formulas to compute more than just the expected overlap.",
   "evidence": "The canonical record (AIM workshop *The Kardar--Parisi--Zhang equation and universality class*, Open Problems 22.08) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 87,
   "attempt": 1
  },
  "AIM-PROBABILITY-0089": {
   "statement_status": "exact",
   "original_statement": "Stochastic analysis\n\nFind a microscopic G\\\"{a}rtner transform (discrete Hopf-Cole transform) which turns the dynamics of $q$-TASEP into a discrete stochastic heat equation or better yet a polymer. Then prove this converges to the continuum stochastic heat equation.",
   "clean_statement": "Stochastic analysis\n\nFind a microscopic G\\\"{a}rtner transform (discrete Hopf-Cole transform) which turns the dynamics of $q$-TASEP into a discrete stochastic heat equation or better yet a polymer. Then prove this converges to the continuum stochastic heat equation.",
   "public_statement": "Stochastic analysis\n\nFind a microscopic G\\\"{a}rtner transform (discrete Hopf-Cole transform) which turns the dynamics of $q$-TASEP into a discrete stochastic heat equation or better yet a polymer. Then prove this converges to the continuum stochastic heat equation.",
   "evidence": "This is Problem 22.1, “Stochastic analysis,” from the AIM workshop *The Kardar–Parisi–Zhang equation and universality class*. The source record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 88,
   "attempt": 1
  },
  "AIM-PROBABILITY-0090": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Interacting particle systems\n\nBulk of $q$-Whittaker $2d$ dynamics. Invariant measure.",
   "clean_statement": null,
   "public_statement": "Interacting particle systems\n\nBulk of $q$-Whittaker $2d$ dynamics. Invariant measure.",
   "evidence": "The AIM page was unavailable (HTTP 502 on 2026-08-12), and nearby records only confirm that this item belongs to the q-TASEP/q-Whittaker interacting-particle-system cluster of the workshop. The fragment omits the state space, update rule, boundary conditions, and meaning of “bulk.” There are at least three plausible readings:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 89,
   "attempt": 1
  },
  "AIM-PROBABILITY-0091": {
   "statement_status": "exact",
   "original_statement": "Interacting particle systems\n\nStudy $q$-TASEP hydrodynamics and other interacting particle systems properties such as second class particles, invariant distributions, and Burkes type theorem.",
   "clean_statement": "Interacting particle systems\n\nStudy $q$-TASEP hydrodynamics and other interacting particle systems properties such as second class particles, invariant distributions, and Burkes type theorem.",
   "public_statement": "Interacting particle systems\n\nStudy $q$-TASEP hydrodynamics and other interacting particle systems properties such as second class particles, invariant distributions, and Burkes type theorem.",
   "evidence": "The exact canonical record (AIM Problem Lists, workshop *The Kardar--Parisi--Zhang equation and universality class*, Open Problem 22.14) says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 90,
   "attempt": 1
  },
  "AIM-PROBABILITY-0092": {
   "statement_status": "exact",
   "original_statement": "Interacting particle systems\n\nWhat about $q$-ASEP, where particle can jump to right and left and feel a repulsion from their right or left neighbors: Is there Burkes Theorem? Do second class particle methods work?",
   "clean_statement": "Interacting particle systems\n\nWhat about $q$-ASEP, where particle can jump to right and left and feel a repulsion from their right or left neighbors: Is there Burkes Theorem? Do second class particle methods work?",
   "public_statement": "Interacting particle systems\n\nWhat about $q$-ASEP, where particle can jump to right and left and feel a repulsion from their right or left neighbors: Is there Burkes Theorem? Do second class particle methods work?",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 91,
   "attempt": 1
  },
  "AIM-PROBABILITY-0093": {
   "statement_status": "exact",
   "original_statement": "Gibbs line ensembles\n\nConsider a Brownian excursion on time interval $[-N,N]$ condition on fixed area $N$. Take $1/3, 2/3$ scalings around the edge. Do we get an ``Airy-like'' limiting fluctuation?",
   "clean_statement": "Gibbs line ensembles\n\nConsider a Brownian excursion on time interval $[-N,N]$ condition on fixed area $N$. Take $1/3, 2/3$ scalings around the edge. Do we get an ``Airy-like'' limiting fluctuation?",
   "public_statement": "Gibbs line ensembles\n\nConsider a Brownian excursion on time interval $[-N,N]$ condition on fixed area $N$. Take $1/3, 2/3$ scalings around the edge. Do we get an ``Airy-like'' limiting fluctuation?",
   "evidence": "This is Problem 22.18, “Gibbs line ensembles,” from the AIM workshop *The Kardar–Parisi–Zhang equation and universality class*. The exact source record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-PROBABILITY-0094": {
   "statement_status": "exact",
   "original_statement": "Stochastic analysis\n\nUse the well-posedness theory for the KPZ equation to redo Bertini-Giacomin or to prove convergence of simpler discretizations of the KPZ equation such as the one previously studied by Sasamoto and Spohn.",
   "clean_statement": "Stochastic analysis\n\nUse the well-posedness theory for the KPZ equation to redo Bertini-Giacomin or to prove convergence of simpler discretizations of the KPZ equation such as the one previously studied by Sasamoto and Spohn.",
   "public_statement": "Stochastic analysis\n\nUse the well-posedness theory for the KPZ equation to redo Bertini-Giacomin or to prove convergence of simpler discretizations of the KPZ equation such as the one previously studied by Sasamoto and Spohn.",
   "evidence": "The exact canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-PROBABILITY-0095": {
   "statement_status": "exact",
   "original_statement": "Polymer universality\n\nConsider intermediate scaling for polymer with $\\beta=n^{-1/4}c(n)$ where $c(n)$ is any function which tends to infinity with $n$. Show universality of the $F_{GUE}$ fluctuations with a fluctuation exponent which may depend on $c(n)$.",
   "clean_statement": "Polymer universality\n\nConsider intermediate scaling for polymer with $\\beta=n^{-1/4}c(n)$ where $c(n)$ is any function which tends to infinity with $n$. Show universality of the $F_{GUE}$ fluctuations with a fluctuation exponent which may depend on $c(n)$.",
   "public_statement": "Polymer universality\n\nConsider intermediate scaling for polymer with $\\beta=n^{-1/4}c(n)$ where $c(n)$ is any function which tends to infinity with $n$. Show universality of the $F_{GUE}$ fluctuations with a fluctuation exponent which may depend on $c(n)$.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 94,
   "attempt": 1
  },
  "AIM-PROBABILITY-0096": {
   "statement_status": "exact",
   "original_statement": "Exact solvability\n\nInvestigate the relationship between the tropical RSK correspondence and mirror symmetry: The entrance law for the Markov process which arises from adding columns under the tropical RSK correspondence arises via a critical point calculation.",
   "clean_statement": "Exact solvability\n\nInvestigate the relationship between the tropical RSK correspondence and mirror symmetry: The entrance law for the Markov process which arises from adding columns under the tropical RSK correspondence arises via a critical point calculation.",
   "public_statement": "Exact solvability\n\nInvestigate the relationship between the tropical RSK correspondence and mirror symmetry: The entrance law for the Markov process which arises from adding columns under the tropical RSK correspondence arises via a critical point calculation.",
   "evidence": "The exact canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-PROBABILITY-0097": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Universality of initial data for (T)ASEP.\n\nConsider two initial conditions for (T)ASEP corresponding to height functions $h_1(x;t=0)$ and $h_2(x;t=0)$ which can be random, but are independent of each other. Hydrodynamic theory says that if for $i=1,2$, $\\epsilon^{-1} h_i(\\epsilon^{-1} x;t=0)\\to \\bar{h}(x;t=0)$ as $\\epsilon\\to 0$, then so does $\\epsilon^{-1} h_i(\\epsilon^{-1} x;t)\\to \\bar{h}(x;t)$ where $\\bar{h}$ solves a Hamilton-Jacobi conservation law with quadratic flux. We would like a similar result, but for fluctuations. This result would show that if we assume for $i=1,2$, $$\\epsilon^{1/2}\\left[h_i(\\epsilon^{-1}x;t=0)-\\epsilon^{-3/2}\\bar{h}(\\epsilon^{1/2} x;0)\\right]\\to \\tilde{h}(x;t=0)$$ as $\\epsilon\\to 0$, then so does $$\\epsilon^{1/2}\\left[h_i(\\epsilon^{-1}x;\\epsilon^{-3/2}t)-\\epsilon^{-3/2}\\bar{h}(\\epsilon^{1/2}x;t)\\right]\\to \\tilde{h}(x;t).$$",
   "clean_statement": null,
   "public_statement": "Universality of initial data for (T)ASEP.\n\nConsider two initial conditions for (T)ASEP corresponding to height functions $h_1(x;t=0)$ and $h_2(x;t=0)$ which can be random, but are independent of each other. Hydrodynamic theory says that if for $i=1,2$, $\\epsilon^{-1} h_i(\\epsilon^{-1} x;t=0)\\to \\bar{h}(x;t=0)$ as $\\epsilon\\to 0$, then so does $\\epsilon^{-1} h_i(\\epsilon^{-1} x;t)\\to \\bar{h}(x;t)$ where $\\bar{h}$ solves a Hamilton-Jacobi conservation law with quadratic flux. We would like a similar result, but for fluctuations. This result would show that if we assume for $i=1,2$, $$\\epsilon^{1/2}\\left[h_i(\\epsilon^{-1}x;t=0)-\\epsilon^{-3/2}\\bar{h}(\\epsilon^{1/2} x;0)\\right]\\to \\tilde{h}(x;t=0)$$ as $\\epsilon\\to 0$, then so does $$\\epsilon^{1/2}\\left[h_i(\\epsilon^{-1}x;\\epsilon^{-3/2}t)-\\epsilon^{-3/2}\\bar{h}(\\epsilon^{1/2}x;t)\\right]\\to \\tilde{h}(x;t).$$",
   "evidence": "The first display is almost certainly malformed as written. A microscopic exclusion height has order \\(\\epsilon^{-1}\\) on Euler distance \\(\\epsilon^{-1}\\), so the standard Euler rescaling multiplies it by \\(\\epsilon\\), not \\(\\epsilon^{-1}\\); Euler time is also normally rescaled. This possible typo is not silently repaired here.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 96,
   "attempt": 2
  },
  "AIM-PROBABILITY-0098": {
   "statement_status": "exact",
   "original_statement": "Universality of initial data for (T)ASEP.\n\nShow $T^{1/3}$ fluctuations for KPZ equation slightly out of equilibrium by using a variant of the second class particle method for WASEP near equilibrium.",
   "clean_statement": "Universality of initial data for (T)ASEP.\n\nShow $T^{1/3}$ fluctuations for KPZ equation slightly out of equilibrium by using a variant of the second class particle method for WASEP near equilibrium.",
   "public_statement": "Universality of initial data for (T)ASEP.\n\nShow $T^{1/3}$ fluctuations for KPZ equation slightly out of equilibrium by using a variant of the second class particle method for WASEP near equilibrium.",
   "evidence": "This is AIM Problem 22.32 from the workshop *The Kardar--Parisi--Zhang equation and universality class*. The exact source record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 97,
   "attempt": 1
  },
  "AIM-PROBABILITY-0099": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Gibbs line ensembles\n\nShow tightness and limiting Gibbs property for line ensembles besides the Airy line ensemble (e.g. Bessel process, Peary process, Sine process).",
   "clean_statement": null,
   "public_statement": "Gibbs line ensembles\n\nShow tightness and limiting Gibbs property for line ensembles besides the Airy line ensemble (e.g. Bessel process, Peary process, Sine process).",
   "evidence": "“Peary process” is not a recognized process in the relevant probability literature and is almost certainly an OCR/transcription error for the **Pearcey process**, the cusp scaling limit of nonintersecting Brownian paths. The input is preserved unchanged; only the analysis uses this explicit reconstruction. “Bessel process” is interpreted as the hard-edge extended Bessel process, or more precisely a line ensemble whose one-time section is the Bessel point process. “Sine process” is interpreted as the bulk extended-sine determinantal diffusion, not merely a single fixed-time sine point process.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 98,
   "attempt": 1
  },
  "AIM-PROBABILITY-0100": {
   "statement_status": "exact",
   "original_statement": "Exact solvability\n\nExplain the occurrence of a boundary value problem (BVP) in the kernel for the continuum statistics for the Airy$_2$ process: $\\mathbb{P}\\left(A_2(\\cdot)\\leq g(\\cdot)\\right)=\\det(I-M),$ where $M$ has a kernel given in terms of a BVP involving $g$. Is this due to the Karlin-McGregor formula for this infinite ensemble of lines? Study the analogous continuum statistics for the top curve of $N$ non-intersecting Brownian bridges or Brownian motions. This might explain where this BVP comes from.",
   "clean_statement": "Exact solvability\n\nExplain the occurrence of a boundary value problem (BVP) in the kernel for the continuum statistics for the Airy$_2$ process: $\\mathbb{P}\\left(A_2(\\cdot)\\leq g(\\cdot)\\right)=\\det(I-M),$ where $M$ has a kernel given in terms of a BVP involving $g$. Is this due to the Karlin-McGregor formula for this infinite ensemble of lines? Study the analogous continuum statistics for the top curve of $N$ non-intersecting Brownian bridges or Brownian motions. This might explain where this BVP comes from.",
   "public_statement": "Exact solvability\n\nExplain the occurrence of a boundary value problem (BVP) in the kernel for the continuum statistics for the Airy$_2$ process: $\\mathbb{P}\\left(A_2(\\cdot)\\leq g(\\cdot)\\right)=\\det(I-M),$ where $M$ has a kernel given in terms of a BVP involving $g$. Is this due to the Karlin-McGregor formula for this infinite ensemble of lines? Study the analogous continuum statistics for the top curve of $N$ non-intersecting Brownian bridges or Brownian motions. This might explain where this BVP comes from.",
   "evidence": "The exact canonical AIM record is Problem 22.36 from the workshop *The Kardar--Parisi--Zhang equation and universality class*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 99,
   "attempt": 1
  },
  "AIM-PROBABILITY-0101": {
   "statement_status": "exact",
   "original_statement": "Exact solvability\n\nStudy solvable variants of the log-Gamma polymer with different types of polymer paths.",
   "clean_statement": "Exact solvability\n\nStudy solvable variants of the log-Gamma polymer with different types of polymer paths.",
   "public_statement": "Exact solvability\n\nStudy solvable variants of the log-Gamma polymer with different types of polymer paths.",
   "evidence": "The canonical record is AIM Probability problem 22.38 from the workshop *The Kardar-Parisi-Zhang equation and universality class*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 100,
   "attempt": 1
  },
  "AIM-PROBABILITY-0102": {
   "statement_status": "exact",
   "original_statement": "Polymer universality\n\nDetermine limiting behavior of polymer without a sixth moment, and with $\\beta\\rightarrow0$ at the correct rate to have a limit.",
   "clean_statement": "Polymer universality\n\nDetermine limiting behavior of polymer without a sixth moment, and with $\\beta\\rightarrow0$ at the correct rate to have a limit.",
   "public_statement": "Polymer universality\n\nDetermine limiting behavior of polymer without a sixth moment, and with $\\beta\\rightarrow0$ at the correct rate to have a limit.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 101,
   "attempt": 1
  },
  "AIM-PROBABILITY-0103": {
   "statement_status": "exact",
   "original_statement": "Q: In a von Neumann algebra M with a faithful normal trace-state τ let X =\n\nX∗ ∈ M and let 1 ∈ B ⊂ M be an infinite-dimensional von Neumann subalgebra so that B and X are free in the algebraic sense and M = W ∗(X, B ). Assume that ∂X:B is closable in L2(M, τ ) (this is the case for instantce if \n\nX is a free semicircular perturbation X = X0 + εS, with S a semicircular free from X0 and B). Under what conditions are the L2 solutions of \n\n∂X:B u = 0 in L2(B, τ )? A related question about a stronger condition: when does the free Poincare inequality \n\nC‖∂X:B ξ‖2 ≥ ‖ ξ − EB ξ‖2\n\nhold for ξ ∈ B〈X〉?\n\n#0.2 \"Large Deviations\", Guionnet, Hiai, Cabanal-Duvillard.",
   "clean_statement": "Q: In a von Neumann algebra M with a faithful normal trace-state τ let X =\n\nX∗ ∈ M and let 1 ∈ B ⊂ M be an infinite-dimensional von Neumann subalgebra so that B and X are free in the algebraic sense and M = W ∗(X, B ). Assume that ∂X:B is closable in L2(M, τ ) (this is the case for instantce if\n\nX is a free semicircular perturbation X = X0 + εS, with S a semicircular free from X0 and B). Under what conditions are the L2 solutions of\n\n∂X:B u = 0 in L2(B, τ )? A related question about a stronger condition: when does the free Poincare inequality\n\nC‖∂X:B ξ‖2 ≥ ‖ ξ − EB ξ‖2\n\nhold for ξ ∈ B〈X〉?\n\n#0.2 \"Large Deviations\", Guionnet, Hiai, Cabanal-Duvillard.",
   "public_statement": "Q: In a von Neumann algebra M with a faithful normal trace-state τ let X =\n\nX∗ ∈ M and let 1 ∈ B ⊂ M be an infinite-dimensional von Neumann subalgebra so that B and X are free in the algebraic sense and M = W ∗(X, B ). Assume that ∂X:B is closable in L2(M, τ ) (this is the case for instantce if\n\nX is a free semicircular perturbation X = X0 + εS, with S a semicircular free from X0 and B). Under what conditions are the L2 solutions of\n\n∂X:B u = 0 in L2(B, τ )? A related question about a stronger condition: when does the free Poincare inequality\n\nC‖∂X:B ξ‖2 ≥ ‖ ξ − EB ξ‖2\n\nhold for ξ ∈ B〈X〉?\n\n#0.2 \"Large Deviations\", Guionnet, Hiai, Cabanal-Duvillard.",
   "evidence": "The canonical input is the first item of the AIM workshop list *Free Analysis* (24 August 2006), under the heading “X-constants and free Poincare inequality” (Voiculescu). The stored extraction says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 102,
   "attempt": 1
  },
  "AIM-PROBABILITY-0104": {
   "statement_status": "exact",
   "original_statement": "Q: Given a tracial state τ corresponding to a free stochastic process, does there exist a sequence of tracial states τn → τ with χ∗\n\n> p\n\n(τn) → χ∗\n\n> p\n\n(τ ) where τn\n\ncorresponds to the process dA i(t) = dS i(t) + kt(A1(s),..., A m(s)) s≤tdt with kt\n\nstepwise constant in s, and χ∗ \n\n> p\n\ndenotes the quantity χ∗ defined for processes in the paper of Guionnet and Cabanal-Duvillard.",
   "clean_statement": "Q: Given a tracial state τ corresponding to a free stochastic process, does there exist a sequence of tracial states τn → τ with χ∗\n\n> p\n\n(τn) → χ∗\n\n> p\n\n(τ ) where τn\n\ncorresponds to the process dA i(t) = dS i(t) + kt(A1(s),..., A m(s)) s≤tdt with kt\n\nstepwise constant in s, and χ∗\n\n> p\n\ndenotes the quantity χ∗ defined for processes in the paper of Guionnet and Cabanal-Duvillard.",
   "public_statement": "Q: Given a tracial state τ corresponding to a free stochastic process, does there exist a sequence of tracial states τn → τ with χ∗\n\n> p\n\n(τn) → χ∗\n\n> p\n\n(τ ) where τn\n\ncorresponds to the process dA i(t) = dS i(t) + kt(A1(s),..., A m(s)) s≤tdt with kt\n\nstepwise constant in s, and χ∗\n\n> p\n\ndenotes the quantity χ∗ defined for processes in the paper of Guionnet and Cabanal-Duvillard.",
   "evidence": "The canonical JSON record has severe line-break/OCR damage around the entropy symbol. The original AIM workshop PDF, *Problems* (August 24, 2006), gives the question as follows:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 103,
   "attempt": 1
  },
  "AIM-PROBABILITY-0105": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Q: In the one variable case, if A(t) follows a process dA (t) = dS (t) + \n\nkt(A(s)) s≤t then replacing A(t) with A(t) + C\u000f (with C having Cauchy distri-bution and free from A(t)) then kt is replaced by k\u000ft = τ (kt|A(t) + C\u000f). Thus, \n\nk\u000ft is smooth. Is there an analog of this smoothing in the several-variable case?",
   "clean_statement": null,
   "public_statement": "Q: In the one variable case, if A(t) follows a process dA (t) = dS (t) +\n\nkt(A(s)) s≤t then replacing A(t) with A(t) + C[U+000F] (with C having Cauchy distri-bution and free from A(t)) then kt is replaced by k[U+000F]t = τ (kt|A(t) + C[U+000F]). Thus,\n\nk[U+000F]t is smooth. Is there an analog of this smoothing in the several-variable case?",
   "evidence": "The original AIM PDF, dated August 24, 2006, was inspected directly. Its embedded font also defeats text extraction at precisely the epsilon glyph, but the page layout and the immediately following question use the same notation in the usual Poisson kernel with denominator $(y-x)^2+\\varepsilon^2$. The conservative reconstruction is",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 104,
   "attempt": 1
  },
  "AIM-PROBABILITY-0106": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Q: We know that if f: R → R and A is an n × n Hermitian random matrix, then there exists a random matrix C\u000f with Cauchy distribution such that Ef (A + C\u000f) = P\u000ff (A) with P\u000ff (x) = ∫ f (y) \n\n> (y−x)2+i\u000f 2\n\ndy the usual Cauchy (Poisson) kernel. Can this be done for several variables? 1Q: Given x1,..., x m ∈ (A, τ ) a tracial unital vN algebra, do the conjugate variables belong to the L2 closure of cyclic gradient space? i.e. do there exist \n\nHk ∈ C 〈α1,..., α m〉 such that J (xi) = lim k DiHk where ∂xi: L2(A, τ ) →\n\nL2(A, τ ) ⊗ L2(A, τ ) by xj 7 → δij 1 ⊗ 1 as a densely defined operator, J (xi) = \n\n∂∗ \n\n> xi\n\n(1 ⊗ 1), and Di = m ◦ ∂xi (m is the flip-multiplication x ⊗ y 7 → yx ).",
   "clean_statement": null,
   "public_statement": "Q: We know that if f: R → R and A is an n × n Hermitian random matrix, then there exists a random matrix C[U+000F] with Cauchy distribution such that Ef (A + C[U+000F]) = P[U+000F]f (A) with P[U+000F]f (x) = ∫ f (y)\n\n> (y−x)2+i[U+000F] 2\n\ndy the usual Cauchy (Poisson) kernel. Can this be done for several variables? 1Q: Given x1,..., x m ∈ (A, τ ) a tracial unital vN algebra, do the conjugate variables belong to the L2 closure of cyclic gradient space? i.e. do there exist\n\nHk ∈ C 〈α1,..., α m〉 such that J (xi) = lim k DiHk where ∂xi: L2(A, τ ) →\n\nL2(A, τ ) ⊗ L2(A, τ ) by xj 7 → δij 1 ⊗ 1 as a densely defined operator, J (xi) =\n\n∂∗\n\n> xi\n\n(1 ⊗ 1), and Di = m ◦ ∂xi (m is the flip-multiplication x ⊗ y 7 → yx ).",
   "evidence": "The canonical JSON record is not one coherent problem. It contains two consecutive questions from the 2006 AIM workshop list *Free Analysis*. In the PDF, the first question ends at the bottom of page 1 (PDF index 0), and the second starts on page 2. During extraction, the printed page number `1` was attached to the next `Q:`, producing `1Q:`. The control character U+000F in the JSON is a failed extraction of the parameter \\(\\varepsilon\\), and `7 \\to` is a failed `\\(\\mapsto\\)`.",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 105,
   "attempt": 1
  },
  "AIM-PROBABILITY-0107": {
   "statement_status": "exact",
   "original_statement": "Q: Does the change of variables formula for χ also hold for χ∗?",
   "clean_statement": "Q: Does the change of variables formula for χ also hold for χ∗?",
   "public_statement": "Q: Does the change of variables formula for χ also hold for χ∗?",
   "evidence": "The original AIM Free Analysis workshop PDF, dated August 24, 2006, contains the exact question:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 106,
   "attempt": 1
  },
  "AIM-PROBABILITY-0108": {
   "statement_status": "exact",
   "original_statement": "Q: Is there a change of variables formula for processes? i.e. suppose that we start with random variables x1,..., x m ∈ (A, τ ) which can be reached by a pro-cess dA i(t) = dS i(t) + kt(A1(s),..., A m(s)) s≤t, μA1(1),...,A m(1) = μx1,...,x m. We define new random variables via functional calculus y1 = f1(x1,..., x m),..., y m =\n\nfm(x1,..., x m). Can we apply a function P to kt to get dB i(t) = dS i(t) + \n\nP (kt(B1(s),..., B m(s)) s≤t) such that μB1(1),...,B m(1) = μy1,...,y m.\n\nOpen Problem: Can we replace lim sup with lim inf in the microstates definition of the free entropy χ?\n\nQ Hiai introduced the free pressure πR(h) for a self-adjoint element (re-garded as a free hamiltonian) h of the universal free product C∗-algebra A(n) =\n\nFni=1 C([ −R, R ]), and defined a free entropy-like quantity ηR(τ ) of a tracial state \n\nτ ∈ T S (A(n)). The inequality ηR(τ ) ≥ χ(τ ) holds. τ is called an equillibrium tracial state with respect to h if the variational equality ηR(τ ) = τ (h) + πR(h)holds. Such a τ always exists for each h. For which h there is a unique equilib-rium tracial state? A way to prove this is the free transportation inequality.",
   "clean_statement": "Q: Is there a change of variables formula for processes? i.e. suppose that we start with random variables x1,..., x m ∈ (A, τ ) which can be reached by a pro-cess dA i(t) = dS i(t) + kt(A1(s),..., A m(s)) s≤t, μA1(1),...,A m(1) = μx1,...,x m. We define new random variables via functional calculus y1 = f1(x1,..., x m),..., y m =\n\nfm(x1,..., x m). Can we apply a function P to kt to get dB i(t) = dS i(t) +\n\nP (kt(B1(s),..., B m(s)) s≤t) such that μB1(1),...,B m(1) = μy1,...,y m.\n\nOpen Problem: Can we replace lim sup with lim inf in the microstates definition of the free entropy χ?\n\nQ Hiai introduced the free pressure πR(h) for a self-adjoint element (re-garded as a free hamiltonian) h of the universal free product C∗-algebra A(n) =\n\nFni=1 C([ −R, R ]), and defined a free entropy-like quantity ηR(τ ) of a tracial state\n\nτ ∈ T S (A(n)). The inequality ηR(τ ) ≥ χ(τ ) holds. τ is called an equillibrium tracial state with respect to h if the variational equality ηR(τ ) = τ (h) + πR(h)holds. Such a τ always exists for each h. For which h there is a unique equilib-rium tracial state? A way to prove this is the free transportation inequality.",
   "public_statement": "Q: Is there a change of variables formula for processes? i.e. suppose that we start with random variables x1,..., x m ∈ (A, τ ) which can be reached by a pro-cess dA i(t) = dS i(t) + kt(A1(s),..., A m(s)) s≤t, μA1(1),...,A m(1) = μx1,...,x m. We define new random variables via functional calculus y1 = f1(x1,..., x m),..., y m =\n\nfm(x1,..., x m). Can we apply a function P to kt to get dB i(t) = dS i(t) +\n\nP (kt(B1(s),..., B m(s)) s≤t) such that μB1(1),...,B m(1) = μy1,...,y m.\n\nOpen Problem: Can we replace lim sup with lim inf in the microstates definition of the free entropy χ?\n\nQ Hiai introduced the free pressure πR(h) for a self-adjoint element (re-garded as a free hamiltonian) h of the universal free product C∗-algebra A(n) =\n\nFni=1 C([ −R, R ]), and defined a free entropy-like quantity ηR(τ ) of a tracial state\n\nτ ∈ T S (A(n)). The inequality ηR(τ ) ≥ χ(τ ) holds. τ is called an equillibrium tracial state with respect to h if the variational equality ηR(τ ) = τ (h) + πR(h)holds. Such a τ always exists for each h. For which h there is a unique equilib-rium tracial state? A way to prove this is the free transportation inequality.",
   "evidence": "The canonical record is an extraction fusion. The AIM workshop PDF, *Free Analysis* (section 0.2, “Large Deviations”), places three separate unnumbered items consecutively; the next item begins immediately after them. The canonical problem field has joined all three into one record. The source PDF also drops a visible \\(dt\\) after each drift and prints a single \\(k_t\\) where a vector drift is apparently intended. The following separates the questions and records the minimal reconstruction used below.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 107,
   "attempt": 1
  },
  "AIM-PROBABILITY-0109": {
   "statement_status": "exact",
   "original_statement": "Q: It was recently shown by Guionnet and Maurel-Segala that for the vN algebra ( A, τ ) generated by m free semicirculars, sup \n\n> τ∈T S (A)\n\n{\n\nχ(τ ) − τ (∑ tiqi)\n\n}\n\n= ∑\n\n> p1,...,p m\n\n∏\n\n> k1,...,k m\n\n(ti)pi\n\nki! C(q, k 1,..., k m)where C(q, k 1,..., k m) enumerated planar maps with colored edges and vertices of types q, k 1,..., k m. Is there a similar interpretation for the non-microstates analog sup \n\n> τ∈T S (A)\n\n{\n\nχ∗(τ ) − τ (∑ tiqi)\n\n}?\n\n#0.3 \"Free von Neumann Algebras\", Dykema, Ricard.",
   "clean_statement": "Q: It was recently shown by Guionnet and Maurel-Segala that for the vN algebra ( A, τ ) generated by m free semicirculars, sup\n\n> τ∈T S (A)\n\n{\n\nχ(τ ) − τ (∑ tiqi)\n\n}\n\n= ∑\n\n> p1,...,p m\n\n∏\n\n> k1,...,k m\n\n(ti)pi\n\nki! C(q, k 1,..., k m)where C(q, k 1,..., k m) enumerated planar maps with colored edges and vertices of types q, k 1,..., k m. Is there a similar interpretation for the non-microstates analog sup\n\n> τ∈T S (A)\n\n{\n\nχ∗(τ ) − τ (∑ tiqi)\n\n}?\n\n#0.3 \"Free von Neumann Algebras\", Dykema, Ricard.",
   "public_statement": "Q: It was recently shown by Guionnet and Maurel-Segala that for the vN algebra ( A, τ ) generated by m free semicirculars, sup\n\n> τ∈T S (A)\n\n{\n\nχ(τ ) − τ (∑ tiqi)\n\n}\n\n= ∑\n\n> p1,...,p m\n\n∏\n\n> k1,...,k m\n\n(ti)pi\n\nki! C(q, k 1,..., k m)where C(q, k 1,..., k m) enumerated planar maps with colored edges and vertices of types q, k 1,..., k m. Is there a similar interpretation for the non-microstates analog sup\n\n> τ∈T S (A)\n\n{\n\nχ∗(τ ) − τ (∑ tiqi)\n\n}?\n\n#0.3 \"Free von Neumann Algebras\", Dykema, Ricard.",
   "evidence": "The canonical record is preserved verbatim in `input.json`. It is an OCR extraction from page 2 (PDF index 1) of the 24 August 2006 AIM list *Free Analysis*. The terminal text `#0.3 \"Free von Neumann Algebras\", Dykema, Ricard` is the next section heading and is not part of the problem.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 108,
   "attempt": 1
  },
  "AIM-PROBABILITY-0110": {
   "statement_status": "exact",
   "original_statement": "Q: Given A, B free group factors with a common diffuse subalgebra D ⊂ A, B,what conditions on A, B, D guarantee that A? D B is a free group factor? \n\n2Q: for a regular weakly-rigid (in the sense of Popa) subalgebra of a von Neumann algebra, is the free entropy dimension ≤ 1? \n\nOpen Problem: for generators γ1,..., γ n ∈ Γ with the first L2 -Betti number β1(Γ) large, is the microstates free entropy dimension of this family of generators large? (This is known for the non-microstates free entropy dimension [work of Mineyev-Shlyakhtenko]).",
   "clean_statement": "Q: Given A, B free group factors with a common diffuse subalgebra D ⊂ A, B,what conditions on A, B, D guarantee that A? D B is a free group factor?\n\n2Q: for a regular weakly-rigid (in the sense of Popa) subalgebra of a von Neumann algebra, is the free entropy dimension ≤ 1?\n\nOpen Problem: for generators γ1,..., γ n ∈ Γ with the first L2 -Betti number β1(Γ) large, is the microstates free entropy dimension of this family of generators large? (This is known for the non-microstates free entropy dimension [work of Mineyev-Shlyakhtenko]).",
   "public_statement": "Q: Given A, B free group factors with a common diffuse subalgebra D ⊂ A, B,what conditions on A, B, D guarantee that A? D B is a free group factor?\n\n2Q: for a regular weakly-rigid (in the sense of Popa) subalgebra of a von Neumann algebra, is the free entropy dimension ≤ 1?\n\nOpen Problem: for generators γ1,..., γ n ∈ Γ with the first L2 -Betti number β1(Γ) large, is the microstates free entropy dimension of this family of generators large? (This is known for the non-microstates free entropy dimension [work of Mineyev-Shlyakhtenko]).",
   "evidence": "The canonical JSON record is an OCR extraction of three consecutive questions in Section 0.3, “Free von Neumann Algebras” (Dykema–Ricard), of the AIM workshop notes *Problems* (24 August 2006). Inspection of pages 1–2 of the original PDF recovers the questions as follows.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 109,
   "attempt": 1
  },
  "AIM-PROBABILITY-0111": {
   "statement_status": "exact",
   "original_statement": "Q: Consider ∆ = ∑mi=1 ∂∗ \n\n> xi\n\n∂xi and the corresponding completely posi-tive map ϕt = exp( −t∆), where ( x1,..., x m) have finite Free Fisher Infor-mation. Can ϕt converge uniformly to the identity map on the unit ball of \n\nW ∗(x1,..., x m)? If no, it follows that the von Neumann algebra generated by (x1,..., x m) is not weakly rigid if it is non-hyperfinite.",
   "clean_statement": "Q: Consider ∆ = ∑mi=1 ∂∗\n\n> xi\n\n∂xi and the corresponding completely posi-tive map ϕt = exp( −t∆), where ( x1,..., x m) have finite Free Fisher Infor-mation. Can ϕt converge uniformly to the identity map on the unit ball of\n\nW ∗(x1,..., x m)? If no, it follows that the von Neumann algebra generated by (x1,..., x m) is not weakly rigid if it is non-hyperfinite.",
   "public_statement": "Q: Consider ∆ = ∑mi=1 ∂∗\n\n> xi\n\n∂xi and the corresponding completely posi-tive map ϕt = exp( −t∆), where ( x1,..., x m) have finite Free Fisher Infor-mation. Can ϕt converge uniformly to the identity map on the unit ball of\n\nW ∗(x1,..., x m)? If no, it follows that the von Neumann algebra generated by (x1,..., x m) is not weakly rigid if it is non-hyperfinite.",
   "evidence": "The canonical record comes from page 3 of the 2006 AIM workshop list *Free Analysis*, in section 0.3, “Free von Neumann Algebras.” The PDF asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 110,
   "attempt": 1
  },
  "AIM-PROBABILITY-0112": {
   "statement_status": "exact",
   "original_statement": "Q: Let Γ q,n = W ∗(sq (g) = l(g) + l(g)∗|g ∈ H R) with n = dim HR, −1 <q < 1 be the von Neumann algebra generated by fields operators acting on a q-deformed Fock space. Does Γ q,n depend on q? A way to approach this question could come from the following observation. In the free case, q = 0, the natural orthormal basis of the Fock space consists of vectors ei = e⊗α1 \n\n> i1\n\n⊗... ⊗ e⊗αk\n\n> ik\n\nwith i1 6 =... 6 = ik and α1 > 0. This basis can be recoved from the algebra as ei = Tα1 (s0(e1))...T αk (s0(eq ))Ω, where Tk are Chebytchev polynomials. It would be interesting to find an analogue for these formulas in the general case and to unterstand the underlying combinatorics. The q-deformation leads to the commutation relations l(e)∗l(f ) = ql (f )l(e)∗+\n\n〈f, e 〉Id. Instead consider themore general relations l(ei)∗l(ej ) = ∑ \n\n> s,t\n\nts,t i,j l(es)l(et)∗+\n\nδi,j Id. When does the C∗-algebra generated by these operators is an extension of a Cuntz algebra by compacts? When does the fields operators associated to them produce a type II 1 factor? Consider the projection Pk from Γ q,n to its subspace consisting of x such that \n\nx. Ω has length at most k in the Fock space. Is ‖Pk‖cb polynomially bounded in \n\nk? This would prove the CBAP for the associated Lp spaces (1 < p < ∞) and the exactness of the C∗-algebra generated by q-gaussians.",
   "clean_statement": "Q: Let Γ q,n = W ∗(sq (g) = l(g) + l(g)∗|g ∈ H R) with n = dim HR, −1 <q < 1 be the von Neumann algebra generated by fields operators acting on a q-deformed Fock space. Does Γ q,n depend on q? A way to approach this question could come from the following observation. In the free case, q = 0, the natural orthormal basis of the Fock space consists of vectors ei = e⊗α1\n\n> i1\n\n⊗... ⊗ e⊗αk\n\n> ik\n\nwith i1 6 =... 6 = ik and α1 > 0. This basis can be recoved from the algebra as ei = Tα1 (s0(e1))...T αk (s0(eq ))Ω, where Tk are Chebytchev polynomials. It would be interesting to find an analogue for these formulas in the general case and to unterstand the underlying combinatorics. The q-deformation leads to the commutation relations l(e)∗l(f ) = ql (f )l(e)∗+\n\n〈f, e 〉Id. Instead consider themore general relations l(ei)∗l(ej ) = ∑\n\n> s,t\n\nts,t i,j l(es)l(et)∗+\n\nδi,j Id. When does the C∗-algebra generated by these operators is an extension of a Cuntz algebra by compacts? When does the fields operators associated to them produce a type II 1 factor? Consider the projection Pk from Γ q,n to its subspace consisting of x such that\n\nx. Ω has length at most k in the Fock space. Is ‖Pk‖cb polynomially bounded in\n\nk? This would prove the CBAP for the associated Lp spaces (1 < p < ∞) and the exactness of the C∗-algebra generated by q-gaussians.",
   "public_statement": "Q: Let Γ q,n = W ∗(sq (g) = l(g) + l(g)∗|g ∈ H R) with n = dim HR, −1 <q < 1 be the von Neumann algebra generated by fields operators acting on a q-deformed Fock space. Does Γ q,n depend on q? A way to approach this question could come from the following observation. In the free case, q = 0, the natural orthormal basis of the Fock space consists of vectors ei = e⊗α1\n\n> i1\n\n⊗... ⊗ e⊗αk\n\n> ik\n\nwith i1 6 =... 6 = ik and α1 > 0. This basis can be recoved from the algebra as ei = Tα1 (s0(e1))...T αk (s0(eq ))Ω, where Tk are Chebytchev polynomials. It would be interesting to find an analogue for these formulas in the general case and to unterstand the underlying combinatorics. The q-deformation leads to the commutation relations l(e)∗l(f ) = ql (f )l(e)∗+\n\n〈f, e 〉Id. Instead consider themore general relations l(ei)∗l(ej ) = ∑\n\n> s,t\n\nts,t i,j l(es)l(et)∗+\n\nδi,j Id. When does the C∗-algebra generated by these operators is an extension of a Cuntz algebra by compacts? When does the fields operators associated to them produce a type II 1 factor? Consider the projection Pk from Γ q,n to its subspace consisting of x such that\n\nx. Ω has length at most k in the Fock space. Is ‖Pk‖cb polynomially bounded in\n\nk? This would prove the CBAP for the associated Lp spaces (1 < p < ∞) and the exactness of the C∗-algebra generated by q-gaussians.",
   "evidence": "The canonical JSON record is a damaged extraction of Question 10 in the AIM workshop list *Free Analysis: Problems* (24 August 2006). The original PDF, page 3 (zero-based PDF page 2), was checked directly through its indexed text. The question has four related parts.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 111,
   "attempt": 1
  },
  "AIM-PROBABILITY-0113": {
   "statement_status": "exact",
   "original_statement": "Q: To prove the existence of an embedding Γ q,n → R ω, one uses Speicher's central limit theorem. In this procedure, is it possible to find explictely uni-formly bounded matrix whose mixed moments approach those of q-gaussians? More precisely, let ci,j be unitary generators of the CAR-algebra (or −1-gaussians), are the matrices 1√n [ci,j ]i,j ≤n uniformly bounded?",
   "clean_statement": "Q: To prove the existence of an embedding Γ q,n → R ω, one uses Speicher's central limit theorem. In this procedure, is it possible to find explictely uni-formly bounded matrix whose mixed moments approach those of q-gaussians? More precisely, let ci,j be unitary generators of the CAR-algebra (or −1-gaussians), are the matrices 1√n [ci,j ]i,j ≤n uniformly bounded?",
   "public_statement": "Q: To prove the existence of an embedding Γ q,n → R ω, one uses Speicher's central limit theorem. In this procedure, is it possible to find explictely uni-formly bounded matrix whose mixed moments approach those of q-gaussians? More precisely, let ci,j be unitary generators of the CAR-algebra (or −1-gaussians), are the matrices 1√n [ci,j ]i,j ≤n uniformly bounded?",
   "evidence": "The record comes from the AIM workshop *Free Analysis* (June 19--23, 2006), question 11. The PDF text reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 112,
   "attempt": 1
  },
  "AIM-PROBABILITY-0114": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Q: For the random matrix model exp( −nT r (p(A1, A ∗\n\n> 1,..., A m, A ∗\n\n> m\n\n)) we know that the conjugate variables satisfy Ji = DiP. Is the operator exp( −t ∑ ∂∗ \n\n> j\n\n∂j )compact in the limit n → ∞ (where ∂j is Voiculescu's partial difference quo-tient on the limit algebra with respect to the limit of Aj )? As a starting point, consider P = ∑ A2 \n\n> i\n\n+ ∑ tiqi(A1,..., A m) where Guionnet and Maurel-Segala have shown convergence of the model. 30.4 Focus Group on Free Entropy (day 3) \n\nOpen Problem: Is δ∗ = δ?? Here \n\nδ∗ = n − lim sup \n\n> t↓0\n\nχ∗(x1 + √ts 1,..., x n + √ts m)\n\nlog t1/2\n\nand \n\nδ? = n − lim sup \n\n> t→0\n> n\n\n∑\n\n> i=1\n\ntΦ∗(x1 + √ts 1,... x m + √ts m).",
   "clean_statement": null,
   "public_statement": "Q: For the random matrix model exp( −nT r (p(A1, A ∗\n\n> 1,..., A m, A ∗\n\n> m\n\n)) we know that the conjugate variables satisfy Ji = DiP. Is the operator exp( −t ∑ ∂∗\n\n> j\n\n∂j )compact in the limit n → ∞ (where ∂j is Voiculescu's partial difference quo-tient on the limit algebra with respect to the limit of Aj )? As a starting point, consider P = ∑ A2\n\n> i\n\n+ ∑ tiqi(A1,..., A m) where Guionnet and Maurel-Segala have shown convergence of the model. 30.4 Focus Group on Free Entropy (day 3)\n\nOpen Problem: Is δ∗ = δ?? Here\n\nδ∗ = n − lim sup\n\n> t↓0\n\nχ∗(x1 + √ts 1,..., x n + √ts m)\n\nlog t1/2\n\nand\n\nδ? = n − lim sup\n\n> t→0\n> n\n\n∑\n\n> i=1\n\ntΦ∗(x1 + √ts 1,... x m + √ts m).",
   "evidence": "The standard definitions in Voiculescu's theory, for an \\(n\\)-tuple \\(X=(x_1,\\ldots,x_n)\\) and a variance-one semicircular tuple \\(S=(s_1,\\ldots,s_n)\\) free from \\(X\\), are \\[ \\delta^*(X) =n-\\liminf_{t\\downarrow0} \\frac{\\chi^*(X+\\sqrt t\\,S)}{\\log\\sqrt t}, \\tag{1.6} \\] and \\[ \\delta^\\star(X) =n-\\liminf_{t\\downarrow0}t\\Phi^*(X+\\sqrt t\\,S). \\tag{1.7} \\] Equations (1.6)--(1.7), not the malformed displays (1.4)--(1.5), are used below. This is an explicit reconstruction, not a silent alteration of `input.json`, which preserves the exact canonical record.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 113,
   "attempt": 1
  },
  "AIM-PROBABILITY-0115": {
   "statement_status": "exact",
   "original_statement": "Q: What is the non-microstates analogue of free entropy in the presence, \n\nχ(x1,..., x n: y1,..., y n)? \n\n#0.5 Focus Group on Operator Theory (day 3)",
   "clean_statement": "Q: What is the non-microstates analogue of free entropy in the presence,\n\nχ(x1,..., x n: y1,..., y n)?\n\n#0.5 Focus Group on Operator Theory (day 3)",
   "public_statement": "Q: What is the non-microstates analogue of free entropy in the presence,\n\nχ(x1,..., x n: y1,..., y n)?\n\n#0.5 Focus Group on Operator Theory (day 3)",
   "evidence": "The canonical JSON record ends with a line break followed by “Focus Group on Operator Theory (day 3),” and its plain-text layout makes the number of conditioning variables slightly uncertain. Inspection of page 3 of the original AIM workshop PDF, *Problems* (24 August 2006), recovers the complete question as",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 114,
   "attempt": 1
  },
  "AIM-PROBABILITY-0116": {
   "statement_status": "exact",
   "original_statement": "Q: What is the boundary behavior of the subordination functions which appear in free convolution of operator-valued random variables?",
   "clean_statement": "Q: What is the boundary behavior of the subordination functions which appear in free convolution of operator-valued random variables?",
   "public_statement": "Q: What is the boundary behavior of the subordination functions which appear in free convolution of operator-valued random variables?",
   "evidence": "The exact source is question 14 in the AIM workshop problem list *Free Analysis*, dated August 24, 2006:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 115,
   "attempt": 1
  },
  "AIM-PROBABILITY-0117": {
   "statement_status": "exact",
   "original_statement": "Q: What are examples/conditions for freely strongly unimodal variables, i.e. unimodal random variables that when freely convolved with a unimodal vari-ables remain unimodal? (Unimodal means that the law of the random variable has a smooth density with a unique maximum; example: Gaussian law or the semicircle law).",
   "clean_statement": "Q: What are examples/conditions for freely strongly unimodal variables, i.e. unimodal random variables that when freely convolved with a unimodal vari-ables remain unimodal? (Unimodal means that the law of the random variable has a smooth density with a unique maximum; example: Gaussian law or the semicircle law).",
   "public_statement": "Q: What are examples/conditions for freely strongly unimodal variables, i.e. unimodal random variables that when freely convolved with a unimodal vari-ables remain unimodal? (Unimodal means that the law of the random variable has a smooth density with a unique maximum; example: Gaussian law or the semicircle law).",
   "evidence": "The canonical record is question 15 from the AIM workshop list *Free Analysis* (24 August 2006). The exact extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 116,
   "attempt": 1
  },
  "AIM-PROBABILITY-0118": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Q: More specifically, if μ, ν are symmetric unimodal distribution, is μ \u0001 ν\n\nunimodal? \n\n#0.6 \"Invariant Subspaces for an Operator\", Haagerup",
   "clean_statement": "secure from the local context and is independently confirmed by Hasebe--Ueda, who state the identical question as Conjecture 3.5.",
   "public_statement": "Q: More specifically, if μ, ν are symmetric unimodal distribution, is μ [U+0001] ν\n\nunimodal?\n\n#0.6 \"Invariant Subspaces for an Operator\", Haagerup",
   "evidence": "The canonical JSON record is corrupted at the binary operation and has absorbed the next section heading. It reads, in relevant part,",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-probability-notes.json",
   "source_index": 117,
   "attempt": 1
  },
  "AIM-PROBABILITY-0119": {
   "statement_status": "exact",
   "original_statement": "Q: Let x, y be two free circular elements, and let S, T be two operators in a II 1\n\nfactor, which is free from x, y. In the Haagerup-Schultz estimate (?? ) ∥∥(S + xy −1)−1 − (T + xy −1)−1∥∥p ≤ c(p) ‖S − T ‖p < ∞\n\nwith 0 < p < 2 \n\n> 3, can one use x instead of xy −1?",
   "clean_statement": "Q: Let x, y be two free circular elements, and let S, T be two operators in a II 1\n\nfactor, which is free from x, y. In the Haagerup-Schultz estimate (?? ) ∥∥(S + xy −1)−1 − (T + xy −1)−1∥∥p ≤ c(p) ‖S − T ‖p < ∞\n\nwith 0 < p < 2\n\n> 3, can one use x instead of xy −1?",
   "public_statement": "Q: Let x, y be two free circular elements, and let S, T be two operators in a II 1\n\nfactor, which is free from x, y. In the Haagerup-Schultz estimate (?? ) ∥∥(S + xy −1)−1 − (T + xy −1)−1∥∥p ≤ c(p) ‖S − T ‖p < ∞\n\nwith 0 < p < 2\n\n> 3, can one use x instead of xy −1?",
   "evidence": "The canonical JSON record is visibly damaged by PDF extraction: it turns the displayed label into “(??),” separates the fraction $2/3$, and obscures the placement of inverse signs and norm subscripts. I therefore checked page 4 of the original seven-page AIM PDF visually. The source is the problem list dated 24 August 2006, section 0.6, “Invariant Subspaces for an Operator,” attributed to Haagerup. Its displayed question is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 118,
   "attempt": 1
  },
  "AIM-PROBABILITY-0120": {
   "statement_status": "exact",
   "original_statement": "Q: (Brown measure of unbounded operators): As defined by (Haagerup and Schultz), ∆( T ) makes sense for T ∈ M ∆ where M ∆ =\n\n{\n\nT ∈ ˜M | ∫ ∞ \n\n> 0\n\nlog t dμ T (t) < ∞\n\n}.Then ∆( T ) = exp( ∫ ∞ \n\n> 0\n\nlog t dμ T (t)) ∈ [0, ∞]. Can one make sense of μT for such unbounded T?",
   "clean_statement": "Q: (Brown measure of unbounded operators): As defined by (Haagerup and Schultz), ∆( T ) makes sense for T ∈ M ∆ where M ∆ =\n\n{\n\nT ∈ ˜M | ∫ ∞\n\n> 0\n\nlog t dμ T (t) < ∞\n\n}.Then ∆( T ) = exp( ∫ ∞\n\n> 0\n\nlog t dμ T (t)) ∈ [0, ∞]. Can one make sense of μT for such unbounded T?",
   "public_statement": "Q: (Brown measure of unbounded operators): As defined by (Haagerup and Schultz), ∆( T ) makes sense for T ∈ M ∆ where M ∆ =\n\n{\n\nT ∈ ˜M | ∫ ∞\n\n> 0\n\nlog t dμ T (t) < ∞\n\n}.Then ∆( T ) = exp( ∫ ∞\n\n> 0\n\nlog t dμ T (t)) ∈ [0, ∞]. Can one make sense of μT for such unbounded T?",
   "evidence": "This is Question 18 from the AIM workshop list *Free analysis*. The supplied record is visibly damaged by mathematical-text extraction. In particular, it prints the defining integral as fragments such as",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 119,
   "attempt": 1
  },
  "AIM-PROBABILITY-0121": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Q: Does the main result of (Haagerup and Schultz) hold for T ∈ LpM (some or all p)? T ∈ M ∆? T ∈ ˜M?40.7 \"Free Group Factors\", Ozawa \n\nConj: if H an M -M bimodule M = LFn, and M HM \u0016 L2M ⊗ L2M, (weak containment) then Hom( M H ⊗ \n\n> M\n\nH ⊗ \n\n> M\n\nHM, L 2M ⊗ L2M ) 6 = 0. Note that the assumption of weak containment is equivalent that the map \n\nx ⊗ y 7 → (λ(x)ρ(y): HM 3 h 7 → xhy ) ∈ B(M HM )is continuous for the min-tensor product on M ⊗ M. Examples of bimodules with this property come from the basic construction \n\n> M\n\nHM = M ⊗A M\n\nover a hyperfinite subalgebra A ⊂ M.\n\n#0.8 Focus Group on Combinatorics of Random Matrix Models (day 4) \n\nGiven random matrices An and Bn with corresponding measures μAn and μBn\n\non Mn(C), we define their Itzykson-Zuber integral as \n\nIZ (An, B n) = \n\n∫\n\nexp( −nT r (AU ∗BU )) dμ An (A)dμ Bn (B). Thm (Guionnet and Zeitouni): if ‖An‖ < c, ‖Bn‖ < c then IZ (An, B n) ∼\n\nexp( −nψ ).",
   "clean_statement": null,
   "public_statement": "Q: Does the main result of (Haagerup and Schultz) hold for T ∈ LpM (some or all p)? T ∈ M ∆? T ∈ ˜M?40.7 \"Free Group Factors\", Ozawa\n\nConj: if H an M -M bimodule M = LFn, and M HM [U+0016] L2M ⊗ L2M, (weak containment) then Hom( M H ⊗\n\n> M\n\nH ⊗\n\n> M\n\nHM, L 2M ⊗ L2M ) 6 = 0. Note that the assumption of weak containment is equivalent that the map\n\nx ⊗ y 7 → (λ(x)ρ(y): HM 3 h 7 → xhy ) ∈ B(M HM )is continuous for the min-tensor product on M ⊗ M. Examples of bimodules with this property come from the basic construction\n\n> M\n\nHM = M ⊗A M\n\nover a hyperfinite subalgebra A ⊂ M.\n\n#0.8 Focus Group on Combinatorics of Random Matrix Models (day 4)\n\nGiven random matrices An and Bn with corresponding measures μAn and μBn\n\non Mn(C), we define their Itzykson-Zuber integral as\n\nIZ (An, B n) =\n\n∫\n\nexp( −nT r (AU ∗BU )) dμ An (A)dμ Bn (B). Thm (Guionnet and Zeitouni): if ‖An‖ < c, ‖Bn‖ < c then IZ (An, B n) ∼\n\nexp( −nψ ).",
   "evidence": "The canonical JSON record fuses the end of Section 0.6 with Section 0.7 and the opening paragraph of Section 0.8 of the AIM workshop PDF. The string 40.7 is a page number 4 followed by the new section number 0.7. The assigned question is only Question 19:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 120,
   "attempt": 1
  },
  "AIM-PROBABILITY-0122": {
   "statement_status": "exact",
   "original_statement": "Q: There is another result that states that \n\n∂n\n\n∂t n log IZ (tA n, B n)|t=0 converges. Does this expression match ψ above? Can we extend Guionnet and Zeitouni's result to complex parameters?",
   "clean_statement": "Q: There is another result that states that\n\n∂n\n\n∂t n log IZ (tA n, B n)|t=0 converges. Does this expression match ψ above? Can we extend Guionnet and Zeitouni's result to complex parameters?",
   "public_statement": "Q: There is another result that states that\n\n∂n\n\n∂t n log IZ (tA n, B n)|t=0 converges. Does this expression match ψ above? Can we extend Guionnet and Zeitouni's result to complex parameters?",
   "evidence": "This is Question 20 in the 2006 AIM workshop list *Free analysis*, immediately after the definition of an Itzykson--Zuber integral and a summary of the Guionnet--Zeitouni theorem. The supplied record says",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 121,
   "attempt": 1
  },
  "AIM-PROBABILITY-0123": {
   "statement_status": "exact",
   "original_statement": "Q: Extend the model exp( −nT r (P (A1,..., A m) + 1\n\n> 2\n\n∑mi=1 A2 \n\n> i\n\n)) dA 1... dA m\n\nof Guionnet and Maurel-Segala to non-selfadjoint P (i.e.polynomials with com-plex coefficients).",
   "clean_statement": "Q: Extend the model exp( −nT r (P (A1,..., A m) + 1\n\n> 2\n\n∑mi=1 A2\n\n> i\n\n)) dA 1... dA m\n\nof Guionnet and Maurel-Segala to non-selfadjoint P (i.e.polynomials with com-plex coefficients).",
   "public_statement": "Q: Extend the model exp( −nT r (P (A1,..., A m) + 1\n\n> 2\n\n∑mi=1 A2\n\n> i\n\n)) dA 1... dA m\n\nof Guionnet and Maurel-Segala to non-selfadjoint P (i.e.polynomials with com-plex coefficients).",
   "evidence": "The exact text on physical page 5 of the AIM *Free Analysis* problem list, in subsection 0.8, “Focus Group on Combinatorics of Random Matrix Models (day 4),” is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 122,
   "attempt": 1
  },
  "AIM-PROBABILITY-0124": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Q: Is there a combinatorial interpretation of free cumulants in terms of enumeration of maps and operations on maps? Consider the spherical integrals \n\nIn(z, E n):= \n\n∫\n\nexp {ntr( U D nU ∗En)}dmn (U ),\n\n5where Dn = diag(z, 0, 0,..., 0), z ∈ C, and En is a sequence of n × n selfadjoint (diagonal) matrices, with spectrum uniformly bounded in n, and converging in distribution to μE\n\nThe sequence of functions of zfn(z) = ∂z\n\n1\n\nn log In(z, E n),\n\nhas been shown by Guionnet and Maida to converge to RμE (z) for |z| small enough. Questions: What is the largest domain in the complex plane on which this convergence takes place? If μE is \u0001-infinitely divisible, is the convergence hap-pening on all the upper half-plane? Is there any possible generalization to mea-sures with noncompact support? (one could probably approach this problem by trying to study the normality of the family/sequence fn)\n\n#0.9 Focus Group on Invariant Subspaces (day 4) \n\nIf M is a II 1 factor, T1,..., T n ∈ M, [ Ti, T j ] = 0, then we have the \"Brown Measure\" defined as the unique measure on Cn such that (?) log ∆(1 − ∑ αiTi) = \n\n∫\n\nlog(1 − ∑ αiζi)dμ T!,...,T n (ζ1,..., ζ n).",
   "clean_statement": "Is there a combinatorial interpretation of free cumulants in terms of enumeration of maps and operations on maps? Consider\n\\[\nI_n(z,E_n)=\\int_{\\mathcal U(n)}\n  \\exp\\{n\\operatorname{Tr}(UD_nU^*E_n)\\}\\,dm_n(U),\n\\qquad D_n=\\operatorname{diag}(z,0,\\ldots,0),\\quad z\\in\\mathbb C,\n\\]\nwhere \\(E_n\\) is self-adjoint (and may be taken diagonal), its spectrum is uniformly bounded, and its empirical spectral distribution converges to \\(\\mu_E\\). Guionnet and Ma\\u00efda showed that\n\\[\nf_n(z)=\\partial_z\\left(\\frac1n\\log I_n(z,E_n)\\right)\n\\]\nconverges to \\(R_{\\mu_E}(z)\\) for \\(|z|\\) small enough. What is the largest complex domain of convergence? If \\(\\mu_E\\) is \\(\\boxplus\\)-infinitely divisible, does convergence hold on the whole upper half-plane? Can one generalize to noncompactly supported measures? Perhaps this can be approached through normality of \\((f_n)\\).",
   "public_statement": "Q: Is there a combinatorial interpretation of free cumulants in terms of enumeration of maps and operations on maps? Consider the spherical integrals\n\nIn(z, E n):=\n\n∫\n\nexp {ntr( U D nU ∗En)}dmn (U ),\n\n5where Dn = diag(z, 0, 0,..., 0), z ∈ C, and En is a sequence of n × n selfadjoint (diagonal) matrices, with spectrum uniformly bounded in n, and converging in distribution to μE\n\nThe sequence of functions of zfn(z) = ∂z\n\n1\n\nn log In(z, E n),\n\nhas been shown by Guionnet and Maida to converge to RμE (z) for |z| small enough. Questions: What is the largest domain in the complex plane on which this convergence takes place? If μE is [U+0001]-infinitely divisible, is the convergence hap-pening on all the upper half-plane? Is there any possible generalization to mea-sures with noncompact support? (one could probably approach this problem by trying to study the normality of the family/sequence fn)\n\n#0.9 Focus Group on Invariant Subspaces (day 4)\n\nIf M is a II 1 factor, T1,..., T n ∈ M, [ Ti, T j ] = 0, then we have the \"Brown Measure\" defined as the unique measure on Cn such that (?) log ∆(1 − ∑ αiTi) =\n\n∫\n\nlog(1 − ∑ αiζi)dμ T!,...,T n (ζ1,..., ζ n).",
   "evidence": "This record is Question 22 in the AIM workshop notes *Free analysis*. Inspection of the original PDF shows that the first sentence and the spherical-integral paragraph form one uninterrupted question block; there is no intervening question number or section heading. With display structure and OCR repaired, the question is:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-probability-notes.json",
   "source_index": 123,
   "attempt": 1
  },
  "AIM-PROBABILITY-0125": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Q: Is supp μT1,...,T n ⊂ σ(T1,..., T n), the Taylor spectrum of T1,..., T n?",
   "clean_statement": "For a commuting tuple \\(T=(T_1,\\ldots,T_n)\\) in a \\(\\mathrm{II}_1\\) factor, is\n\\[\n   \\operatorname{supp}\\nu_T\\subseteq \\operatorname{Sp}(T),\n\\]\nwhere \\(\\operatorname{Sp}(T)\\) is the Taylor joint spectrum?",
   "public_statement": "Q: Is supp μT1,...,T n ⊂ σ(T1,..., T n), the Taylor spectrum of T1,..., T n?",
   "evidence": "There is a material typesetting/OCR issue in the source: Question 22 prints a complex logarithm without absolute-value signs. Formula (1), with \\(\\log|\\cdot|\\) on both sides and extended-real values allowed, is the formulation proved by Schultz and repeated in Charlesworth--Dykema--Sukochev--Zanin. A branch of complex logarithm cannot in general make the printed formula meaningful on all of \\(\\mathbb C^n\\). No change has been made to `input.json`; this is an explicit reconstruction from the neighboring question and the primary literature.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-probability-notes.json",
   "source_index": 124,
   "attempt": 1
  },
  "AIM-PROBABILITY-0126": {
   "statement_status": "exact",
   "original_statement": "Q: Which functions on Cn have an integral representation as in (?)?",
   "clean_statement": "Q: Which functions on Cn have an integral representation as in (?)?",
   "public_statement": "Q: Which functions on Cn have an integral representation as in (?)?",
   "evidence": "The exact source is physical page 6 of the AIM *Free Analysis* problem list, subsection 0.9, “Focus Group on Invariant Subspaces (day 4).” It says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 125,
   "attempt": 1
  },
  "AIM-PROBABILITY-0127": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Q: M a II 1 factor and T ∈ M. Define \n\nK(T, r ) = \n\n{\n\nξ ∈ H|∃ ξn ∈ H s.t. ‖ξn − ξ‖2 → 0 and lim sup ‖T nξn‖1/n → 0\n\n},and E(T, r ) = \n\n{\n\nξ ∈ H| lim sup ‖T nξn‖1/n → 0\n\n}.Does K(T, r ) = E(T, r )? The DT quasinilpotent operator may be a counterex-ample.",
   "clean_statement": null,
   "public_statement": "Q: M a II 1 factor and T ∈ M. Define\n\nK(T, r ) =\n\n{\n\nξ ∈ H|∃ ξn ∈ H s.t. ‖ξn − ξ‖2 → 0 and lim sup ‖T nξn‖1/n → 0\n\n},and E(T, r ) =\n\n{\n\nξ ∈ H| lim sup ‖T nξn‖1/n → 0\n\n}.Does K(T, r ) = E(T, r )? The DT quasinilpotent operator may be a counterex-ample.",
   "evidence": "The exact extracted record is visibly corrupted:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 126,
   "attempt": 1
  },
  "AIM-PROBABILITY-0128": {
   "statement_status": "exact",
   "original_statement": "Q: Let c be a circular element ( σ(c) = ¯D), and let f ∈ C∞(C). Can we make sense of f (c) as an (unbounded) operator affiliated with {c}′′?",
   "clean_statement": "Q: Let c be a circular element ( σ(c) = ¯D), and let f ∈ C∞(C). Can we make sense of f (c) as an (unbounded) operator affiliated with {c}′′?",
   "public_statement": "Q: Let c be a circular element ( σ(c) = ¯D), and let f ∈ C∞(C). Can we make sense of f (c) as an (unbounded) operator affiliated with {c}′′?",
   "evidence": "The original AIM *Free analysis* PDF places this as a standalone question in Section 0.9, “Focus Group on Invariant Subspaces (day 4).” The statement, with only typographical notation restored, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 127,
   "attempt": 1
  },
  "AIM-PROBABILITY-0129": {
   "statement_status": "exact",
   "original_statement": "Q: Let (Γ, τ ) be a II 1 factor, T ∈ Γ, μT = δ0. Does T have a non-trivial invariant subspace affiliated with Γ?",
   "clean_statement": "Q: Let (Γ, τ ) be a II 1 factor, T ∈ Γ, μT = δ0. Does T have a non-trivial invariant subspace affiliated with Γ?",
   "public_statement": "Q: Let (Γ, τ ) be a II 1 factor, T ∈ Γ, μT = δ0. Does T have a non-trivial invariant subspace affiliated with Γ?",
   "evidence": "The original AIM *Free Analysis* PDF, physical page 6, subsection 0.9 “Focus Group on Invariant Subspaces (day 4),” states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 128,
   "attempt": 1
  },
  "AIM-PROBABILITY-0130": {
   "statement_status": "exact",
   "original_statement": "Q: Let Bcbe a band limited operator obtained from c a circular element, and let D be the band limited operator obtained from the identity. Then D is uniformly distributed on [0, 1] and?-free from {Bc, B ∗ \n\n> c\n\n}. Is D ∈ W ∗(Bc)? Or is W ∗(Bc) = LFt with t = 1 + 2 c(1 − c \n\n> 2\n\n)? 60.10 \"Infinite Divisibility\", Nica.",
   "clean_statement": "Q: Let Bcbe a band limited operator obtained from c a circular element, and let D be the band limited operator obtained from the identity. Then D is uniformly distributed on [0, 1] and?-free from {Bc, B ∗\n\n> c\n\n}. Is D ∈ W ∗(Bc)? Or is W ∗(Bc) = LFt with t = 1 + 2 c(1 − c\n\n> 2\n\n)? 60.10 \"Infinite Divisibility\", Nica.",
   "public_statement": "Q: Let Bcbe a band limited operator obtained from c a circular element, and let D be the band limited operator obtained from the identity. Then D is uniformly distributed on [0, 1] and?-free from {Bc, B ∗\n\n> c\n\n}. Is D ∈ W ∗(Bc)? Or is W ∗(Bc) = LFt with t = 1 + 2 c(1 − c\n\n> 2\n\n)? 60.10 \"Infinite Divisibility\", Nica.",
   "evidence": "The canonical JSON extraction is badly damaged at precisely the important symbols. I checked the original AIM PDF and its embedded Computer Modern font encoding. The problem on page 6 of the PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 129,
   "attempt": 1
  },
  "AIM-PROBABILITY-0131": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Q: Given x1,..., x k and y1,..., y k in a vNa such that {x1,..., x k} is tensor-independent of {y1,..., y k} and such that μx1,...,x k, ν y1,...,y k are freely infinitely divisible, we can apply the Fourier transform to get the power-series of the classical convolution of μx1,...,x k and νy1,...,y k. How do such power-series relate to the noncommutative power series obtained from free convolution? (In other words how does the set of classically obtainable power-series relate to the set of freely obtainable power-series?)",
   "clean_statement": "compare the all-partition cumulant series which linearizes tensor convolution with the noncrossing-partition $R$-series which linearizes free convolution, and identify what the scalar Fourier series forgets.",
   "public_statement": "Q: Given x1,..., x k and y1,..., y k in a vNa such that {x1,..., x k} is tensor-independent of {y1,..., y k} and such that μx1,...,x k, ν y1,...,y k are freely infinitely divisible, we can apply the Fourier transform to get the power-series of the classical convolution of μx1,...,x k and νy1,...,y k. How do such power-series relate to the noncommutative power series obtained from free convolution? (In other words how does the set of classically obtainable power-series relate to the set of freely obtainable power-series?)",
   "evidence": "The canonical record comes from the AIM workshop list *Free analysis*, subsection 0.10, “Infinite Divisibility,” attributed to Nica. The original PDF was inspected visually at printed page 7. It says (with only typographical spacing normalized):",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-probability-notes.json",
   "source_index": 130,
   "attempt": 1
  },
  "AIM-PROBABILITY-0132": {
   "statement_status": "exact",
   "original_statement": "Q: Can we make sense of the R-transform for x1, x 2 unbounded (power-series are insufficient to encode all the information)? Easier question is for infinitely divisible unbounded operators.",
   "clean_statement": "Q: Can we make sense of the R-transform for x1, x 2 unbounded (power-series are insufficient to encode all the information)? Easier question is for infinitely divisible unbounded operators.",
   "public_statement": "Q: Can we make sense of the R-transform for x1, x 2 unbounded (power-series are insufficient to encode all the information)? Easier question is for infinitely divisible unbounded operators.",
   "evidence": "The canonical record is the second question in subsection 0.10, “Infinite Divisibility,” of the AIM workshop list *Free analysis*. The original PDF was inspected visually at printed page 7. It says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 131,
   "attempt": 1
  },
  "AIM-PROBABILITY-0133": {
   "statement_status": "exact",
   "original_statement": "Q: If c is unbounded R-diagonal, what is the R-transform of c, c ∗?\n\n#0.11 Focus Group on Dirichlet Forms, from Classical to Quantum (day 5)",
   "clean_statement": "Q: If c is unbounded R-diagonal, what is the R-transform of c, c ∗?\n\n#0.11 Focus Group on Dirichlet Forms, from Classical to Quantum (day 5)",
   "public_statement": "Q: If c is unbounded R-diagonal, what is the R-transform of c, c ∗?\n\n#0.11 Focus Group on Dirichlet Forms, from Classical to Quantum (day 5)",
   "evidence": "The original AIM *Free Analysis* problem list was inspected visually at physical/PDF page 7. The relevant consecutive text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 132,
   "attempt": 1
  },
  "AIM-PROBABILITY-0134": {
   "statement_status": "exact",
   "original_statement": "Q: For the q-deformed semicircular, the analogue of ∂∗∂ exists (it is the number operator). Describe explicitly the associated ∂ (which exists by the work of Sauvageot).",
   "clean_statement": "Q: For the q-deformed semicircular, the analogue of ∂∗∂ exists (it is the number operator). Describe explicitly the associated ∂ (which exists by the work of Sauvageot).",
   "public_statement": "Q: For the q-deformed semicircular, the analogue of ∂∗∂ exists (it is the number operator). Describe explicitly the associated ∂ (which exists by the work of Sauvageot).",
   "evidence": "The source is the problem list from the AIM workshop *Free Analysis*, held June 19--23, 2006. Inspection of the original PDF, rather than only the extracted JSON, gives the following text in Section 0.11, “Focus Group on Dirichlet Forms, from Classical to Quantum (day 5)”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 133,
   "attempt": 1
  },
  "AIM-PROBABILITY-0135": {
   "statement_status": "exact",
   "original_statement": "Q: More generally, given a negative definite function on a group Γ (i.e. aDirichlet form), we know it gives a representation by affine actions on L2Γ. When is it a multiple of the left regular representation? What conditions on the negative definite function guarantee this?",
   "clean_statement": "Q: More generally, given a negative definite function on a group Γ (i.e. aDirichlet form), we know it gives a representation by affine actions on L2Γ. When is it a multiple of the left regular representation? What conditions on the negative definite function guarantee this?",
   "public_statement": "Q: More generally, given a negative definite function on a group Γ (i.e. aDirichlet form), we know it gives a representation by affine actions on L2Γ. When is it a multiple of the left regular representation? What conditions on the negative definite function guarantee this?",
   "evidence": "The canonical record is question 33 in the AIM workshop list *Free analysis*, section 0.11, “Focus Group on Dirichlet Forms, from Classical to Quantum (day 5).” The original PDF was inspected directly. It reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 134,
   "attempt": 1
  },
  "AIM-PROBABILITY-0136": {
   "statement_status": "exact",
   "original_statement": "Q: What conditions on a Dirichlet form δ∗δ guarantee that the bimodule associated to δ embeds into ⊕ L2N ⊗ L2N?",
   "clean_statement": "Q: What conditions on a Dirichlet form δ∗δ guarantee that the bimodule associated to δ embeds into ⊕ L2N ⊗ L2N?",
   "public_statement": "Q: What conditions on a Dirichlet form δ∗δ guarantee that the bimodule associated to δ embeds into ⊕ L2N ⊗ L2N?",
   "evidence": "The record comes from the AIM workshop *Free analysis* (August 2006), Section 0.11, “Focus Group on Dirichlet Forms, from Classical to Quantum (day 5).” The source asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 135,
   "attempt": 1
  },
  "AIM-PROBABILITY-0137": {
   "statement_status": "exact",
   "original_statement": "Q: What is the analogue of the Bakry-Emery criterion in the noncommuta-tive case? i.e. what is Γ 2 for noncommutative Dirichlet forms?",
   "clean_statement": "Q: What is the analogue of the Bakry-Emery criterion in the noncommuta-tive case? i.e. what is Γ 2 for noncommutative Dirichlet forms?",
   "public_statement": "Q: What is the analogue of the Bakry-Emery criterion in the noncommuta-tive case? i.e. what is Γ 2 for noncommutative Dirichlet forms?",
   "evidence": "The original AIM *Free Analysis* problem list was inspected visually at physical/PDF page 7. Question 35, in Section 0.11 “Focus Group on Dirichlet Forms, from Classical to Quantum (day 5),” reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 136,
   "attempt": 1
  },
  "AIM-PROBABILITY-0138": {
   "statement_status": "exact",
   "original_statement": "Q: Let ∂: M → L2(M ) ¯ ⊗L2(M o) be a closable derivation, and let ∆ = ∂∗∂,\n\nSt = exp( −t∆). If the semigroup St converges uniformly to the identity in \n\n‖·‖ 2 on the unit ball, is the derivation inner when considered with values in the algebra of unbounded operators affiliated to M ¯⊗M o?7",
   "clean_statement": "Q: Let ∂: M → L2(M ) ¯ ⊗L2(M o) be a closable derivation, and let ∆ = ∂∗∂,\n\nSt = exp( −t∆). If the semigroup St converges uniformly to the identity in\n\n‖·‖ 2 on the unit ball, is the derivation inner when considered with values in the algebra of unbounded operators affiliated to M ¯⊗M o?7",
   "public_statement": "Q: Let ∂: M → L2(M ) ¯ ⊗L2(M o) be a closable derivation, and let ∆ = ∂∗∂,\n\nSt = exp( −t∆). If the semigroup St converges uniformly to the identity in\n\n‖·‖ 2 on the unit ball, is the derivation inner when considered with values in the algebra of unbounded operators affiliated to M ¯⊗M o?7",
   "evidence": "The source is Question 36 in the AIM workshop list *Free analysis*, in the focus-group section “Dirichlet Forms, from Classical to Quantum.” Inspection of the original PDF gives the following statement (notation modernized only typographically):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 137,
   "attempt": 1
  },
  "AIM-PROBABILITY-0139": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 1.1. Explain the numerology. The cardinality of N C W, the cardinality of N N W\n\nand the number of facets of ∆ W are all equal to the Catalan number Cat( W ). The rank numbers of N C W, the height numbers of N N W (in general, N N W is not graded), and the h-vector of ∆ W are all the same, given by the Narayana numbers (for which there is no known closed formula, in general). The enumerative coincidences are quite extensive, and quite mysterious, as there is still no theoreretical connection between these objects. In fact, only for N N W and its relatives is there any proof whatsoever of the enumerative formulas that is not case-by-case, using the finite type classification. Find bijections between these objects which preserve the numerology. Is there some theoretical algebraic framework behind the scenes, as yet undiscovered? David Bessis has suggested a notion of \"dual\" Coxeter systems [6]. Is there a way to formalize this notion? The exponents of W are one below the corresponding degrees of the fundamental polynomial invariants of W (see [31]). Does the number Cat( W ) have any significance in an invariant theory context?",
   "clean_statement": null,
   "public_statement": "Problem 1.1. Explain the numerology. The cardinality of N C W, the cardinality of N N W\n\nand the number of facets of ∆ W are all equal to the Catalan number Cat( W ). The rank numbers of N C W, the height numbers of N N W (in general, N N W is not graded), and the h-vector of ∆ W are all the same, given by the Narayana numbers (for which there is no known closed formula, in general). The enumerative coincidences are quite extensive, and quite mysterious, as there is still no theoreretical connection between these objects. In fact, only for N N W and its relatives is there any proof whatsoever of the enumerative formulas that is not case-by-case, using the finite type classification. Find bijections between these objects which preserve the numerology. Is there some theoretical algebraic framework behind the scenes, as yet undiscovered? David Bessis has suggested a notion of \"dual\" Coxeter systems [6]. Is there a way to formalize this notion? The exponents of W are one below the corresponding degrees of the fundamental polynomial invariants of W (see [31]). Does the number Cat( W ) have any significance in an invariant theory context?",
   "evidence": "This record is Problem 1.1, “Explain the numerology,” in the AIM workshop report *Braid Groups, Clusters, and Free Probability*. The canonical JSON has OCR spacing such as “\\(N C W\\),” “\\(N N W\\),” and “\\(\\Delta W\\).” Inspection of the official PDF recovers these as \\(NC_W\\), \\(NN_W\\), and \\(\\Delta_W\\), and recovers the intersection-flat map as \\[ g(A)=\\bigcap_{\\alpha\\in A}\\alpha^\\perp . \\]",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 138,
   "attempt": 1
  },
  "AIM-PROBABILITY-0140": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 3.2). Also, Tom Brady and Colum Watt have given a new definition of ∆ W in terms of noncrossing partitions [14]. This may provide some connection between the structure of N C W and ∆ W.",
   "clean_statement": null,
   "public_statement": "Problem 3.2). Also, Tom Brady and Colum Watt have given a new definition of ∆ W in terms of noncrossing partitions [14]. This may provide some connection between the structure of N C W and ∆ W.",
   "evidence": "The exact canonical input reads:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 139,
   "attempt": 1
  },
  "AIM-PROBABILITY-0141": {
   "statement_status": "exact",
   "original_statement": "Problem 1.2. What are the largest natural domains of definition for the families N C W,\n\nN N W and ∆ W, and for their corresponding applications? In a sense, the broadest setting possible for the numerology is finite groups generated by pseudoreflections. (A pseudoreflec-tion is a unitary operator on an n-dimensional complex vector space whose eigenvalues are 0 with multiplicity n − 1, and −1 with multiplicity 1.) It is a classical result of Shephard and Todd that the ring of invariants of a group W is a polynomial ring precisely when the group is of this type. And in this case the sequence of degrees d1, d 2,..., d n of fundamental invariants is unique [43]. In the general (complex) case, David Bessis suggests that the Catalan number should be Cat( W ):= \n\n> n\n\n∏\n\n> i=1\n\nh + di\n\ndi,\n\nwhere we set h equal to the highest degree dn. This agrees with our earlier definition in the real types. However, this may apply only when W is a duality group (or a well-generated group ), since otherwise Cat( W ) may fail to be an integer. See the paper [7] by David Bessis for more information. \n\n• The noncrossing partitions are currently the most general of the Catalan families. The poset N C W is defined for all finite Coxeter groups, and the definition makes sense in principle for any finitely generated Coxeter group (although the definition may not be unique when W is infinite [11]). David Bessis and Ruth Corran gave a combinatorial realization of N C W for an infinite class of complex reflection groups in [8], and Bessis has suggested a uniform definition for N C W whenever W is a well-generated complex reflection group [7]. \n\n• Can one generalize free probability beyond types A and B? The combinatorics of free probability is naturally expressed in terms of the type A noncrossing partitions [47], and some work has been done on a type B free probability [10]. Does it make sense to generalize further? One would presumably need to express Roland Speicher's work on multiplicative functions [48] in the completely general case. See",
   "clean_statement": "Problem 1.2. What are the largest natural domains of definition for the families N C W,\n\nN N W and ∆ W, and for their corresponding applications? In a sense, the broadest setting possible for the numerology is finite groups generated by pseudoreflections. (A pseudoreflec-tion is a unitary operator on an n-dimensional complex vector space whose eigenvalues are 0 with multiplicity n − 1, and −1 with multiplicity 1.) It is a classical result of Shephard and Todd that the ring of invariants of a group W is a polynomial ring precisely when the group is of this type. And in this case the sequence of degrees d1, d 2,..., d n of fundamental invariants is unique [43]. In the general (complex) case, David Bessis suggests that the Catalan number should be Cat( W ):=\n\n> n\n\n∏\n\n> i=1\n\nh + di\n\ndi,\n\nwhere we set h equal to the highest degree dn. This agrees with our earlier definition in the real types. However, this may apply only when W is a duality group (or a well-generated group ), since otherwise Cat( W ) may fail to be an integer. See the paper [7] by David Bessis for more information.\n\n• The noncrossing partitions are currently the most general of the Catalan families. The poset N C W is defined for all finite Coxeter groups, and the definition makes sense in principle for any finitely generated Coxeter group (although the definition may not be unique when W is infinite [11]). David Bessis and Ruth Corran gave a combinatorial realization of N C W for an infinite class of complex reflection groups in [8], and Bessis has suggested a uniform definition for N C W whenever W is a well-generated complex reflection group [7].\n\n• Can one generalize free probability beyond types A and B? The combinatorics of free probability is naturally expressed in terms of the type A noncrossing partitions [47], and some work has been done on a type B free probability [10]. Does it make sense to generalize further? One would presumably need to express Roland Speicher's work on multiplicative functions [48] in the completely general case. See",
   "public_statement": "Problem 1.2. What are the largest natural domains of definition for the families N C W,\n\nN N W and ∆ W, and for their corresponding applications? In a sense, the broadest setting possible for the numerology is finite groups generated by pseudoreflections. (A pseudoreflec-tion is a unitary operator on an n-dimensional complex vector space whose eigenvalues are 0 with multiplicity n − 1, and −1 with multiplicity 1.) It is a classical result of Shephard and Todd that the ring of invariants of a group W is a polynomial ring precisely when the group is of this type. And in this case the sequence of degrees d1, d 2,..., d n of fundamental invariants is unique [43]. In the general (complex) case, David Bessis suggests that the Catalan number should be Cat( W ):=\n\n> n\n\n∏\n\n> i=1\n\nh + di\n\ndi,\n\nwhere we set h equal to the highest degree dn. This agrees with our earlier definition in the real types. However, this may apply only when W is a duality group (or a well-generated group ), since otherwise Cat( W ) may fail to be an integer. See the paper [7] by David Bessis for more information.\n\n• The noncrossing partitions are currently the most general of the Catalan families. The poset N C W is defined for all finite Coxeter groups, and the definition makes sense in principle for any finitely generated Coxeter group (although the definition may not be unique when W is infinite [11]). David Bessis and Ruth Corran gave a combinatorial realization of N C W for an infinite class of complex reflection groups in [8], and Bessis has suggested a uniform definition for N C W whenever W is a well-generated complex reflection group [7].\n\n• Can one generalize free probability beyond types A and B? The combinatorics of free probability is naturally expressed in terms of the type A noncrossing partitions [47], and some work has been done on a type B free probability [10]. Does it make sense to generalize further? One would presumably need to express Roland Speicher's work on multiplicative functions [48] in the completely general case. See",
   "evidence": "The source is Problem 1.2 in the 2005 AIM workshop report *Braid Groups, Clusters, and Free Probability*. The canonical JSON ends in the middle of a bullet, and it contains a mathematically impossible definition. Direct inspection of pages 4--5 of the original PDF gives the following verified reconstruction.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 140,
   "attempt": 1
  },
  "AIM-PROBABILITY-0142": {
   "statement_status": "exact",
   "original_statement": "Problem 5.1. \n\n• Explain the theory of cluster algebras in infinite types. See",
   "clean_statement": "Problem 5.1.\n\n• Explain the theory of cluster algebras in infinite types. See",
   "public_statement": "Problem 5.1.\n\n• Explain the theory of cluster algebras in infinite types. See",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 141,
   "attempt": 1
  },
  "AIM-PROBABILITY-0143": {
   "statement_status": "exact",
   "original_statement": "Problem 6.5. \n\n• The most glaring case of this problem is the seeming dependence of N N W and its relatives on the crystallographic structure of W. When W is a Weyl group, there are amazing enumerative correspondences with the other Catalan objects (see the 5\n\nremarks following",
   "clean_statement": "Problem 6.5.\n\n• The most glaring case of this problem is the seeming dependence of N N W and its relatives on the crystallographic structure of W. When W is a Weyl group, there are amazing enumerative correspondences with the other Catalan objects (see the 5\n\nremarks following",
   "public_statement": "Problem 6.5.\n\n• The most glaring case of this problem is the seeming dependence of N N W and its relatives on the crystallographic structure of W. When W is a Weyl group, there are amazing enumerative correspondences with the other Catalan objects (see the 5\n\nremarks following",
   "evidence": "**Source.** *Braid groups, clusters, and free probability*, AIM workshop problem list (2005), printed pages 4--5, Problem 1.2. The canonical record is from `aim-probability-notes.json`, zero-based index 142.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 142,
   "attempt": 1
  },
  "AIM-PROBABILITY-0144": {
   "statement_status": "exact",
   "original_statement": "Problem 1.1). But there is currently no idea how to generalize these objects to the noncrystallographic types. To what extent can the nonnesting partitions and root order be generalized to noncrystallographic types? Presumably, there are objects which can not be general-ized in their current form, such as Lie algebras and affine hyperplane arrangements. Generalize where possible, and explain where there are essential barriers to this gen-eralization. Cathy Kriloff and Arun Ram have dealt with some of these issues in studying the representation theory of noncrystallographic types [33]. Fr´ ed´ eric Chapoton's conjecture gives a way to define the H-triangle for all finite Coxeter groups [18]. What object is it counting in the noncrystallographic types?",
   "clean_statement": "Problem 1.1). But there is currently no idea how to generalize these objects to the noncrystallographic types. To what extent can the nonnesting partitions and root order be generalized to noncrystallographic types? Presumably, there are objects which can not be general-ized in their current form, such as Lie algebras and affine hyperplane arrangements. Generalize where possible, and explain where there are essential barriers to this gen-eralization. Cathy Kriloff and Arun Ram have dealt with some of these issues in studying the representation theory of noncrystallographic types [33]. Fr´ ed´ eric Chapoton's conjecture gives a way to define the H-triangle for all finite Coxeter groups [18]. What object is it counting in the noncrystallographic types?",
   "public_statement": "Problem 1.1). But there is currently no idea how to generalize these objects to the noncrystallographic types. To what extent can the nonnesting partitions and root order be generalized to noncrystallographic types? Presumably, there are objects which can not be general-ized in their current form, such as Lie algebras and affine hyperplane arrangements. Generalize where possible, and explain where there are essential barriers to this gen-eralization. Cathy Kriloff and Arun Ram have dealt with some of these issues in studying the representation theory of noncrystallographic types [33]. Fr´ ed´ eric Chapoton's conjecture gives a way to define the H-triangle for all finite Coxeter groups [18]. What object is it counting in the noncrystallographic types?",
   "evidence": "The canonical record is not an independent problem. Its exact extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 143,
   "attempt": 1
  },
  "AIM-PROBABILITY-0145": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 1.3. What are the most natural generalizations of the families N C W, N N W, and ∆W? Classical combinatorics is full of enumerative generalizations of the Catalan numbers. Which of these is relevant in the reflection group setting? Define the Fuss-Catalan numbers \n\nCat (k)(W ):= \n\n> n\n\n∏\n\n> i=1\n\nkh + ei + 1 \n\nei + 1,\n\nwhere k is a positive integer. In type A, these generalize the classical Fuss numbers and the Catalan numbers [21, 30]. As seen from the formula, Cat (k)(W ) is a very natural generaliza-tion of the Catalan numbers in the reflection group context. Recently these numbers have shown up in all three of the Catalan families. (1) Drew Armstrong has defined a generalization of the noncrossing partitions N C (k) \n\n> W,called the k-divisible noncrossing partitions [1]. This is a graded join-semilattice which is counted by Cat (k)(W ). Call the rank numbers the Fuss-Narayana numbers. In types \n\nA and B, N C (k) \n\n> W\n\nis isomorphic to the poset of k-divisible noncrossing set partitions (partitions in which each block has size divisible by k). (2) Sergey Fomin and Nathan Reading have defined a simplicial complex ∆ (k) \n\n> W\n\nwhich is a generalization of the simplicial associahedron [21]. The facets of ∆ (k) \n\n> W\n\nare counted by the Fuss-Catalan numbers, and the entries of the h-vector are given by the Fuss-Narayana numbers. In types A and B, this complex is defined in terms of ( k +2)-angulations of a regular polygon, and has been studied independently by Eleni Tzanaki [51]. (3) The Fuss-Catalan numbers appear in many places in the N N W family of objects. Let W be a finite Weyl group. Christos Athanasiadis suggested the definition of the Fuss-Narayana numbers in this context, and proved that these numbers count several objects, including positive regions in a certain affine deformation of the Coxeter hyperplane arrangement, as well as co-filtered multichains of ideals in the root order [2, 3]. Mark Haiman has shown that the Fuss-Catalan numbers count orbits in the quotient ˇQ/ (kh + 1) ˇQ of the coroot lattice ˇQ [28], and Eric Sommers has encountered these numbers in the study of Lie algebras [46]. Repeat Problems 1.1 and 1.2 in this more general setting. Any theoretical relation-ships found between N C W, N N W, and ∆ W, must generalize to explain the Fuss-Catalan combinatorics. Given that Cat (k)(W ) is naturally defined in terms of the exponents of W,is there an underlying algebraic framework that explains these numbers? 6",
   "clean_statement": null,
   "public_statement": "Problem 1.3. What are the most natural generalizations of the families N C W, N N W, and ∆W? Classical combinatorics is full of enumerative generalizations of the Catalan numbers. Which of these is relevant in the reflection group setting? Define the Fuss-Catalan numbers\n\nCat (k)(W ):=\n\n> n\n\n∏\n\n> i=1\n\nkh + ei + 1\n\nei + 1,\n\nwhere k is a positive integer. In type A, these generalize the classical Fuss numbers and the Catalan numbers [21, 30]. As seen from the formula, Cat (k)(W ) is a very natural generaliza-tion of the Catalan numbers in the reflection group context. Recently these numbers have shown up in all three of the Catalan families. (1) Drew Armstrong has defined a generalization of the noncrossing partitions N C (k)\n\n> W,called the k-divisible noncrossing partitions [1]. This is a graded join-semilattice which is counted by Cat (k)(W ). Call the rank numbers the Fuss-Narayana numbers. In types\n\nA and B, N C (k)\n\n> W\n\nis isomorphic to the poset of k-divisible noncrossing set partitions (partitions in which each block has size divisible by k). (2) Sergey Fomin and Nathan Reading have defined a simplicial complex ∆ (k)\n\n> W\n\nwhich is a generalization of the simplicial associahedron [21]. The facets of ∆ (k)\n\n> W\n\nare counted by the Fuss-Catalan numbers, and the entries of the h-vector are given by the Fuss-Narayana numbers. In types A and B, this complex is defined in terms of ( k +2)-angulations of a regular polygon, and has been studied independently by Eleni Tzanaki [51]. (3) The Fuss-Catalan numbers appear in many places in the N N W family of objects. Let W be a finite Weyl group. Christos Athanasiadis suggested the definition of the Fuss-Narayana numbers in this context, and proved that these numbers count several objects, including positive regions in a certain affine deformation of the Coxeter hyperplane arrangement, as well as co-filtered multichains of ideals in the root order [2, 3]. Mark Haiman has shown that the Fuss-Catalan numbers count orbits in the quotient ˇQ/ (kh + 1) ˇQ of the coroot lattice ˇQ [28], and Eric Sommers has encountered these numbers in the study of Lie algebras [46]. Repeat Problems 1.1 and 1.2 in this more general setting. Any theoretical relation-ships found between N C W, N N W, and ∆ W, must generalize to explain the Fuss-Catalan combinatorics. Given that Cat (k)(W ) is naturally defined in terms of the exponents of W,is there an underlying algebraic framework that explains these numbers? 6",
   "evidence": "This record is Problem 1.3 in the AIM workshop report *Braid Groups, Clusters, and Free Probability*. The canonical JSON has OCR artifacts such as spaced symbols \\(NC_W\\), \\(NN_W\\), and \\(\\Delta_W\\), a stray printed page number ``6,'' and a truncated remarks field. Inspection of the official PDF recovers the notation as \\[ NC_W,\\qquad NN_W,\\qquad \\Delta_W, \\] and the displayed number as \\[ \\operatorname{Cat}^{(k)}(W) =\\prod_{i=1}^{n}\\frac{kh+e_i+1}{e_i+1}, \\qquad k\\in\\mathbb Z_{>0}. \\] For an irreducible finite real reflection group, the invariant degrees satisfy \\(d_i=e_i+1\\), so this is equivalently \\(\\prod_i(kh+d_i)/d_i\\).",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 144,
   "attempt": 1
  },
  "AIM-PROBABILITY-0146": {
   "statement_status": "exact",
   "original_statement": "Problem 5.3, in free probability. \n\n• Is there a natural generalization of the poset of nonnesting partitions N N (k) \n\n> W? In type \n\nA, one may take k-divisible nonnesting set partitions under refinement (mimicking \n\nN C (k) \n\n> An−1\n\n). In the general case, perhaps this is isomorphic to a partial order on co-filtered multichains of ideals in the root order. \n\n• Christos Athanasiadis and Stavros Garoufallidis have suggested a q-version of the Catalan combinatorics. See",
   "clean_statement": "Problem 5.3, in free probability.\n\n• Is there a natural generalization of the poset of nonnesting partitions N N (k)\n\n> W? In type\n\nA, one may take k-divisible nonnesting set partitions under refinement (mimicking\n\nN C (k)\n\n> An−1\n\n). In the general case, perhaps this is isomorphic to a partial order on co-filtered multichains of ideals in the root order.\n\n• Christos Athanasiadis and Stavros Garoufallidis have suggested a q-version of the Catalan combinatorics. See",
   "public_statement": "Problem 5.3, in free probability.\n\n• Is there a natural generalization of the poset of nonnesting partitions N N (k)\n\n> W? In type\n\nA, one may take k-divisible nonnesting set partitions under refinement (mimicking\n\nN C (k)\n\n> An−1\n\n). In the general case, perhaps this is isomorphic to a partial order on co-filtered multichains of ideals in the root order.\n\n• Christos Athanasiadis and Stavros Garoufallidis have suggested a q-version of the Catalan combinatorics. See",
   "evidence": "The exact canonical field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 145,
   "attempt": 1
  },
  "AIM-PROBABILITY-0147": {
   "statement_status": "exact",
   "original_statement": "Problem 2.1 below. 2. Enumerative Combinatorics",
   "clean_statement": "Problem 2.1 below. 2. Enumerative Combinatorics",
   "public_statement": "Problem 2.1 below. 2. Enumerative Combinatorics",
   "evidence": "The canonical record contains exactly",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 146,
   "attempt": 1
  },
  "AIM-PROBABILITY-0148": {
   "statement_status": "exact",
   "original_statement": "Problem 2.\n1. (C. Athanasiadis) Define the q-Fuss-Catalan numbers \n\nq-Cat (k)(W ):= \n\n> n\n\n∏\n\n> i=1\n\n[kh + ei + 1] q\n\n[ei + 1] q, (1) where [ n]q = q + q2 + · · · + qn is the usual q-analogue of the positive integer n. Show that \n\nq-Cat (k)(W ) is a polynomial in q with nonnegative integer coefficients. This is known in the classical A, B, and D cases. In type A with k = 1, this coincides (up to a power of q) with the q, t -Catalan number of Adriano Garsia and Mark Haiman [25], with the specialization \n\nt = 1 /q.",
   "clean_statement": "Problem 2.\n1. (C. Athanasiadis) Define the q-Fuss-Catalan numbers\n\nq-Cat (k)(W ):=\n\n> n\n\n∏\n\n> i=1\n\n[kh + ei + 1] q\n\n[ei + 1] q, (1) where [ n]q = q + q2 + · · · + qn is the usual q-analogue of the positive integer n. Show that\n\nq-Cat (k)(W ) is a polynomial in q with nonnegative integer coefficients. This is known in the classical A, B, and D cases. In type A with k = 1, this coincides (up to a power of q) with the q, t -Catalan number of Adriano Garsia and Mark Haiman [25], with the specialization\n\nt = 1 /q.",
   "public_statement": "Problem 2.\n1. (C. Athanasiadis) Define the q-Fuss-Catalan numbers\n\nq-Cat (k)(W ):=\n\n> n\n\n∏\n\n> i=1\n\n[kh + ei + 1] q\n\n[ei + 1] q, (1) where [ n]q = q + q2 + · · · + qn is the usual q-analogue of the positive integer n. Show that\n\nq-Cat (k)(W ) is a polynomial in q with nonnegative integer coefficients. This is known in the classical A, B, and D cases. In type A with k = 1, this coincides (up to a power of q) with the q, t -Catalan number of Adriano Garsia and Mark Haiman [25], with the specialization\n\nt = 1 /q.",
   "evidence": "The record comes from Problem 2.1 of Drew Armstrong's outline of the January 2005 AIM workshop *Braid Groups, Clusters, and Free Probability*. I checked the original PDF, including the displayed signs and indices. The canonical JSON has an OCR line break in the number (`Problem 2.\\n1`), but the source reads **Problem 2.1**. It asks, for a finite Coxeter group \\(W\\) of rank \\(n\\), Coxeter number \\(h\\), exponents \\(e_1,\\ldots,e_n\\), and a positive integer \\(k\\), to define",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 147,
   "attempt": 1
  },
  "AIM-PROBABILITY-0149": {
   "statement_status": "exact",
   "original_statement": "Problem 2.\n2. (C. Kriloff, V. Reiner) This is a possible systematic approach to",
   "clean_statement": "Problem 2.\n2. (C. Kriloff, V. Reiner) This is a possible systematic approach to",
   "public_statement": "Problem 2.\n2. (C. Kriloff, V. Reiner) This is a possible systematic approach to",
   "evidence": "The exact canonical field is truncated:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 148,
   "attempt": 1
  },
  "AIM-PROBABILITY-0150": {
   "statement_status": "exact",
   "original_statement": "Problem 2.1. As mentioned, in type A with k = 1, the numbers (1) correspond (up to a power of \n\nq, and specialized at t = 1 /q ) with the q, t -Catalan numbers of Garsia and Haiman, which are given by the q, t -bigraded Hilbert series for the sign-isotypic component of the ring of 7\n\ndiagonal harmonics C[V ⊕ V ]/(C[V ⊕ V ]W \n\n> +\n\n) [29]. Can this situation be generalized to other \n\nW?When W is a symmetric group An−1, it is known that the action of W on the (ungraded) diagonal harmonics has the same irreducible decomposition as the action of W on the \"finite torus\" Q/ (h + 1) Q, where Q is the root lattice. Mark Haiman noted that this does not hold in type B [29]. However, Iain Gordon has shown that the problem may be feasible for general \n\nW, since it is possible to take a further quotient which does give the right combinatorics [26].",
   "clean_statement": "Problem 2.1. As mentioned, in type A with k = 1, the numbers (1) correspond (up to a power of\n\nq, and specialized at t = 1 /q ) with the q, t -Catalan numbers of Garsia and Haiman, which are given by the q, t -bigraded Hilbert series for the sign-isotypic component of the ring of 7\n\ndiagonal harmonics C[V ⊕ V ]/(C[V ⊕ V ]W\n\n> +\n\n) [29]. Can this situation be generalized to other\n\nW?When W is a symmetric group An−1, it is known that the action of W on the (ungraded) diagonal harmonics has the same irreducible decomposition as the action of W on the \"finite torus\" Q/ (h + 1) Q, where Q is the root lattice. Mark Haiman noted that this does not hold in type B [29]. However, Iain Gordon has shown that the problem may be feasible for general\n\nW, since it is possible to take a further quotient which does give the right combinatorics [26].",
   "public_statement": "Problem 2.1. As mentioned, in type A with k = 1, the numbers (1) correspond (up to a power of\n\nq, and specialized at t = 1 /q ) with the q, t -Catalan numbers of Garsia and Haiman, which are given by the q, t -bigraded Hilbert series for the sign-isotypic component of the ring of 7\n\ndiagonal harmonics C[V ⊕ V ]/(C[V ⊕ V ]W\n\n> +\n\n) [29]. Can this situation be generalized to other\n\nW?When W is a symmetric group An−1, it is known that the action of W on the (ungraded) diagonal harmonics has the same irreducible decomposition as the action of W on the \"finite torus\" Q/ (h + 1) Q, where Q is the root lattice. Mark Haiman noted that this does not hold in type B [29]. However, Iain Gordon has shown that the problem may be feasible for general\n\nW, since it is possible to take a further quotient which does give the right combinatorics [26].",
   "evidence": "The canonical record is an OCR-damaged continuation of a question in the AIM workshop notes *Braid groups, clusters and free probability*. The header occurs in the preceding corpus fragment: this is Problem 2.2, not Problem 2.1.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 149,
   "attempt": 1
  },
  "AIM-PROBABILITY-0151": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 2.3. The following are two elementary combinatorial facts, for which it would be nice to have elementary explanations. Both problems are unique to type B, and concern centrally symmetric structures on polygons (structures that are invariant under the antipodal map). (1) (S. Fomin) Among the centrally symmetric partial ( k + 2)-angulations of a regular (2 kn + 2)-gon containing i orbits (under the antipodal map) of k-admissible chords [21, 51], the proportion that contain a diameter is i/n. (A k-admissible chord is one that may be present in a full ( k + 2)-angulation.) Give an elementary proof. (2) (D. Armstrong) Among the centrally symmetric k-divisible noncrossing partitions of a 2 kn -gon with i orbits (under the antipodal map) of nonzero blocks [1, 42], the proportion that contain a zero block is i/n. (A zero block is a block that contains a diameter.) Give an elementary proof.",
   "clean_statement": null,
   "public_statement": "Problem 2.3. The following are two elementary combinatorial facts, for which it would be nice to have elementary explanations. Both problems are unique to type B, and concern centrally symmetric structures on polygons (structures that are invariant under the antipodal map). (1) (S. Fomin) Among the centrally symmetric partial ( k + 2)-angulations of a regular (2 kn + 2)-gon containing i orbits (under the antipodal map) of k-admissible chords [21, 51], the proportion that contain a diameter is i/n. (A k-admissible chord is one that may be present in a full ( k + 2)-angulation.) Give an elementary proof. (2) (D. Armstrong) Among the centrally symmetric k-divisible noncrossing partitions of a 2 kn -gon with i orbits (under the antipodal map) of nonzero blocks [1, 42], the proportion that contain a zero block is i/n. (A zero block is a block that contains a diameter.) Give an elementary proof.",
   "evidence": "The canonical record is Problem 2.3 from the AIM workshop list *Braid groups, clusters and free probability*. With the notation normalized but the words unchanged, it asks for elementary explanations of the following two claims.",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 150,
   "attempt": 1
  },
  "AIM-PROBABILITY-0152": {
   "statement_status": "exact",
   "original_statement": "Problem 2.\n4. (H.T. Hall) Suppose that a stream has 2 n bridges across it. A classical \n\nmeander is (the homotopy class of) a closed path which crosses each bridge once without intersecting itself. On each side of the stream, the meander is given by a noncrossing pairing of the set [2 n]:= {1, 2,..., 2n}. Noncrossing pairings are naturally in bijection with type A\n\nnoncrossing partitions of the set [ n]. Every ordered pair of noncrossing partitions defines a path (with possibly multiple components) which crosses each bridge exactly once. There is a bijection which says that the meanders (the paths with only one connected component) correspond exactly to pairs of noncrossing partitions that are maximally separated in the Hasse diagram of N C An−1 (they are diameters in the graph theoretical sense). Does this bijection suggest a new way to count meanders? One may also use this bijection to define meanders of type W (they are the ordered diameters of the Hasse diagram of N C W ). Is there some combinatorial object that this corresponds to? Is there a type B meander?",
   "clean_statement": "Problem 2.\n4. (H.T. Hall) Suppose that a stream has 2 n bridges across it. A classical\n\nmeander is (the homotopy class of) a closed path which crosses each bridge once without intersecting itself. On each side of the stream, the meander is given by a noncrossing pairing of the set [2 n]:= {1, 2,..., 2n}. Noncrossing pairings are naturally in bijection with type A\n\nnoncrossing partitions of the set [ n]. Every ordered pair of noncrossing partitions defines a path (with possibly multiple components) which crosses each bridge exactly once. There is a bijection which says that the meanders (the paths with only one connected component) correspond exactly to pairs of noncrossing partitions that are maximally separated in the Hasse diagram of N C An−1 (they are diameters in the graph theoretical sense). Does this bijection suggest a new way to count meanders? One may also use this bijection to define meanders of type W (they are the ordered diameters of the Hasse diagram of N C W ). Is there some combinatorial object that this corresponds to? Is there a type B meander?",
   "public_statement": "Problem 2.\n4. (H.T. Hall) Suppose that a stream has 2 n bridges across it. A classical\n\nmeander is (the homotopy class of) a closed path which crosses each bridge once without intersecting itself. On each side of the stream, the meander is given by a noncrossing pairing of the set [2 n]:= {1, 2,..., 2n}. Noncrossing pairings are naturally in bijection with type A\n\nnoncrossing partitions of the set [ n]. Every ordered pair of noncrossing partitions defines a path (with possibly multiple components) which crosses each bridge exactly once. There is a bijection which says that the meanders (the paths with only one connected component) correspond exactly to pairs of noncrossing partitions that are maximally separated in the Hasse diagram of N C An−1 (they are diameters in the graph theoretical sense). Does this bijection suggest a new way to count meanders? One may also use this bijection to define meanders of type W (they are the ordered diameters of the Hasse diagram of N C W ). Is there some combinatorial object that this corresponds to? Is there a type B meander?",
   "evidence": "The canonical record is an OCR-damaged extraction of page 6 and the top of page 7 of the AIM workshop problem list *Braid Groups, Clusters, and Free Probability* (January 2005). The original PDF gives:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 151,
   "attempt": 1
  },
  "AIM-PROBABILITY-0153": {
   "statement_status": "exact",
   "original_statement": "Problem 3.\n1. (V. Reiner) Let W be a finite Coxeter group, and let T be the generating set of all reflections, as in Section 1. Again, let ` denote the word length on W with respect to T. This is often called the absolute length on W. In general, for all u, v in W, we have the triangle inequality `(uv ) ≤ `(u) + `(v). Define the absolute length poset, as before, by setting a ≤ b whenever `(b) = `(a) + \n\n`(a−1b). This is a partial order on W whose Hasse diagram is the Cayley graph of W with respect to T. The poset is graded with rank function given by `.What is the topology of this poset? In types A and B is there an EL -labelling which exhibits a shelling of the order complex? It is known that the absolute length poset is not shellable in type D. Perhaps this can be fixed in a uniform way by considering only the subposet which is the order ideal of parabolic Coxeter elements (elements of W which are a Coxeter element in some parabolic subgroup).",
   "clean_statement": "Problem 3.\n1. (V. Reiner) Let W be a finite Coxeter group, and let T be the generating set of all reflections, as in Section 1. Again, let ` denote the word length on W with respect to T. This is often called the absolute length on W. In general, for all u, v in W, we have the triangle inequality `(uv ) ≤ `(u) + `(v). Define the absolute length poset, as before, by setting a ≤ b whenever `(b) = `(a) +\n\n`(a−1b). This is a partial order on W whose Hasse diagram is the Cayley graph of W with respect to T. The poset is graded with rank function given by `.What is the topology of this poset? In types A and B is there an EL -labelling which exhibits a shelling of the order complex? It is known that the absolute length poset is not shellable in type D. Perhaps this can be fixed in a uniform way by considering only the subposet which is the order ideal of parabolic Coxeter elements (elements of W which are a Coxeter element in some parabolic subgroup).",
   "public_statement": "Problem 3.\n1. (V. Reiner) Let W be a finite Coxeter group, and let T be the generating set of all reflections, as in Section 1. Again, let ` denote the word length on W with respect to T. This is often called the absolute length on W. In general, for all u, v in W, we have the triangle inequality `(uv ) ≤ `(u) + `(v). Define the absolute length poset, as before, by setting a ≤ b whenever `(b) = `(a) +\n\n`(a−1b). This is a partial order on W whose Hasse diagram is the Cayley graph of W with respect to T. The poset is graded with rank function given by `.What is the topology of this poset? In types A and B is there an EL -labelling which exhibits a shelling of the order complex? It is known that the absolute length poset is not shellable in type D. Perhaps this can be fixed in a uniform way by considering only the subposet which is the order ideal of parabolic Coxeter elements (elements of W which are a Coxeter element in some parabolic subgroup).",
   "evidence": "The canonical extraction merges a section heading and damages several symbols. Inspection of the official AIM PDF gives the following reconstruction.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 152,
   "attempt": 1
  },
  "AIM-PROBABILITY-0154": {
   "statement_status": "exact",
   "original_statement": "Problem 3.\n2. (N. Reading) Let W be a finite Coxeter group. In [41], Nathan Reading defines the notion of Coxeter-sortability for elements of W, relative to some Coxeter element \n\nc.There are natural maps nc and cl from the Coxeter-sortable elements of W to the noncrossing partitions N C W, and to the set of clusters of type W, respectively. In [41], these maps are concretely defined, but the proof that they are bijections is case-by-case, using the fact that both objects are known to be counted by the Catalan number Cat( W ). (1) Give a uniform proof that the Coxeter-sorted elements are counted by Cat( W ). (2) Give a uniform proof that the map nc is well-defined. (3) Give a uniform proof that the maps nc and cl are bijections. (4) The notion of Coxeter-sortable elements, and the maps nc and cl can be defined for infinite type Coxeter groups. What happens in this case?",
   "clean_statement": "Problem 3.\n2. (N. Reading) Let W be a finite Coxeter group. In [41], Nathan Reading defines the notion of Coxeter-sortability for elements of W, relative to some Coxeter element\n\nc.There are natural maps nc and cl from the Coxeter-sortable elements of W to the noncrossing partitions N C W, and to the set of clusters of type W, respectively. In [41], these maps are concretely defined, but the proof that they are bijections is case-by-case, using the fact that both objects are known to be counted by the Catalan number Cat( W ). (1) Give a uniform proof that the Coxeter-sorted elements are counted by Cat( W ). (2) Give a uniform proof that the map nc is well-defined. (3) Give a uniform proof that the maps nc and cl are bijections. (4) The notion of Coxeter-sortable elements, and the maps nc and cl can be defined for infinite type Coxeter groups. What happens in this case?",
   "public_statement": "Problem 3.\n2. (N. Reading) Let W be a finite Coxeter group. In [41], Nathan Reading defines the notion of Coxeter-sortability for elements of W, relative to some Coxeter element\n\nc.There are natural maps nc and cl from the Coxeter-sortable elements of W to the noncrossing partitions N C W, and to the set of clusters of type W, respectively. In [41], these maps are concretely defined, but the proof that they are bijections is case-by-case, using the fact that both objects are known to be counted by the Catalan number Cat( W ). (1) Give a uniform proof that the Coxeter-sorted elements are counted by Cat( W ). (2) Give a uniform proof that the map nc is well-defined. (3) Give a uniform proof that the maps nc and cl are bijections. (4) The notion of Coxeter-sortable elements, and the maps nc and cl can be defined for infinite type Coxeter groups. What happens in this case?",
   "evidence": "The canonical extraction splits the number and spacing. Inspection of the official AIM PDF recovers the header as **Problem 3.2 (N. Reading)** under “3. Reflection Groups.” In modern notation the problem fixes a finite Coxeter system \\((W,S)\\) and a Coxeter element \\(c\\), and asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 153,
   "attempt": 1
  },
  "AIM-PROBABILITY-0155": {
   "statement_status": "exact",
   "original_statement": "Problem 3.\n3. (D. Bessis, F. Chapoton) The Lyashko-Looijenga mapping associates to any complex-valued function on a manifold the polynomial in one variable whose roots are the critical values of the function. The main theorem in [34] states that this mapping is a ramified covering for some families of functions. There is a known relationship between the Lyashko-Looijenga covering of the complex sphere, and the combinatorial cacti of Ian Goulden and David Jackson [27]. Interpret the combinatorics of the type A noncrossing partitions in terms of the Lyashko-Looijenga covering of the sphere. The degree of the covering is nn−2, which is also the number 9\n\nof maximal chains in N C An−1. This number is known to count many things, including la-belled trees, and cacti. Do these combinatorics generalize to other types?",
   "clean_statement": "Problem 3.\n3. (D. Bessis, F. Chapoton) The Lyashko-Looijenga mapping associates to any complex-valued function on a manifold the polynomial in one variable whose roots are the critical values of the function. The main theorem in [34] states that this mapping is a ramified covering for some families of functions. There is a known relationship between the Lyashko-Looijenga covering of the complex sphere, and the combinatorial cacti of Ian Goulden and David Jackson [27]. Interpret the combinatorics of the type A noncrossing partitions in terms of the Lyashko-Looijenga covering of the sphere. The degree of the covering is nn−2, which is also the number 9\n\nof maximal chains in N C An−1. This number is known to count many things, including la-belled trees, and cacti. Do these combinatorics generalize to other types?",
   "public_statement": "Problem 3.\n3. (D. Bessis, F. Chapoton) The Lyashko-Looijenga mapping associates to any complex-valued function on a manifold the polynomial in one variable whose roots are the critical values of the function. The main theorem in [34] states that this mapping is a ramified covering for some families of functions. There is a known relationship between the Lyashko-Looijenga covering of the complex sphere, and the combinatorial cacti of Ian Goulden and David Jackson [27]. Interpret the combinatorics of the type A noncrossing partitions in terms of the Lyashko-Looijenga covering of the sphere. The degree of the covering is nn−2, which is also the number 9\n\nof maximal chains in N C An−1. This number is known to count many things, including la-belled trees, and cacti. Do these combinatorics generalize to other types?",
   "evidence": "The canonical JSON is an OCR extraction of **Problem 3.3**, attributed to D. Bessis and F. Chapoton, in the AIM workshop problem list *Braid groups, clusters and free probability*. The PDF gives the following question (typography normalized, wording retained):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 154,
   "attempt": 1
  },
  "AIM-PROBABILITY-0156": {
   "statement_status": "exact",
   "original_statement": "Problem 3.\n4. (D. Bessis) Is there a structure theory of Lie groups and algebraic groups that is analogous to the dual braid monoid [6]? Is there some dual notion of BN -pairs? 4. Garside Structures \n\nAs mentioned, the lattice of noncrossing partitions N C W in its full generality was defined by David Bessis [6] and Tom Brady [12] in order to study the Artin group A(W )corresponding to the Coxeter group W. It turns out that the properties of the poset N C W\n\nhave many consequences for the group theory, including a nice algorithmic solution to the word and conjugacy problems. In general, every poset P together with a labelling of the edges in its Hasse diagram generates a monoid M (P ) and a group G(P ). When this labelling has certain properties, P\n\nis called a combinatorial Garside structure. Having such a Garside structure gives a powerful tool for studying the monoid M (P ) and the group G(P ). This is an emerging subject with interest to combinatorics and group theory. The survey article [35] by Jon McCammond gives a good introduction to these topics.",
   "clean_statement": "Problem 3.\n4. (D. Bessis) Is there a structure theory of Lie groups and algebraic groups that is analogous to the dual braid monoid [6]? Is there some dual notion of BN -pairs? 4. Garside Structures\n\nAs mentioned, the lattice of noncrossing partitions N C W in its full generality was defined by David Bessis [6] and Tom Brady [12] in order to study the Artin group A(W )corresponding to the Coxeter group W. It turns out that the properties of the poset N C W\n\nhave many consequences for the group theory, including a nice algorithmic solution to the word and conjugacy problems. In general, every poset P together with a labelling of the edges in its Hasse diagram generates a monoid M (P ) and a group G(P ). When this labelling has certain properties, P\n\nis called a combinatorial Garside structure. Having such a Garside structure gives a powerful tool for studying the monoid M (P ) and the group G(P ). This is an emerging subject with interest to combinatorics and group theory. The survey article [35] by Jon McCammond gives a good introduction to these topics.",
   "public_statement": "Problem 3.\n4. (D. Bessis) Is there a structure theory of Lie groups and algebraic groups that is analogous to the dual braid monoid [6]? Is there some dual notion of BN -pairs? 4. Garside Structures\n\nAs mentioned, the lattice of noncrossing partitions N C W in its full generality was defined by David Bessis [6] and Tom Brady [12] in order to study the Artin group A(W )corresponding to the Coxeter group W. It turns out that the properties of the poset N C W\n\nhave many consequences for the group theory, including a nice algorithmic solution to the word and conjugacy problems. In general, every poset P together with a labelling of the edges in its Hasse diagram generates a monoid M (P ) and a group G(P ). When this labelling has certain properties, P\n\nis called a combinatorial Garside structure. Having such a Garside structure gives a powerful tool for studying the monoid M (P ) and the group G(P ). This is an emerging subject with interest to combinatorics and group theory. The survey article [35] by Jon McCammond gives a good introduction to these topics.",
   "evidence": "The canonical record combines one genuine question with prose from the next section. Inspection of page 8 of the official AIM PDF gives the exact problem:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 155,
   "attempt": 1
  },
  "AIM-PROBABILITY-0157": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 4.\n1. (R. Charney) Questions about classification. (1) Given an arbitrary poset P, when can it be given a Garside labelling? When such a labelling exists, say that P is a Garside poset.(2) Given a Garside poset P, what are the relationships between its inequivalent Garside labellings? When does P have a unique Garside labelling? (3) Given a poset with an edge labelling, when can this be embedded in a Garside struc-ture? When do the corresponding monoids/groups embed? What are the minimal obstructions to doing this?",
   "clean_statement": "**Problem 4.1 (R. Charney), Questions about classification.**\n(1) Given an arbitrary poset \\(P\\), when can it be given a Garside labelling? When such a labelling exists, say that \\(P\\) is a Garside poset.\n(2) Given a Garside poset \\(P\\), what are the relationships between its inequivalent Garside labellings? When does \\(P\\) have a unique Garside labelling?\n(3) Given a poset with an edge labelling, when can this be embedded in a Garside structure? When do the corresponding monoids/groups embed? What are the minimal obstructions to doing this?",
   "public_statement": "**Problem 4.1 (R. Charney), Questions about classification.**\n(1) Given an arbitrary poset \\(P\\), when can it be given a Garside labelling? When such a labelling exists, say that \\(P\\) is a Garside poset.\n(2) Given a Garside poset \\(P\\), what are the relationships between its inequivalent Garside labellings? When does \\(P\\) have a unique Garside labelling?\n(3) Given a poset with an edge labelling, when can this be embedded in a Garside structure? When do the corresponding monoids/groups embed? What are the minimal obstructions to doing this?",
   "evidence": "The input extraction has three presentational OCR/layout artifacts: `Problem 4.\\n1.` is `Problem 4.1.`, spaces were lost before (2) and (3), and `struc-ture` is a line-break hyphenation of `structure`. These repairs were checked against page 8 of the source PDF. No mathematical symbol or quantifier needed reconstruction.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-probability-notes.json",
   "source_index": 156,
   "attempt": 1
  },
  "AIM-PROBABILITY-0158": {
   "statement_status": "exact",
   "original_statement": "Problem 4.\n2. (P. Dehornoy) \n\n(1) Given a cancellative, finitely-generated monoid M in which lcm's exist, is M neces-sarily a Garside monoid? That is, does there exist a Garside element ∆ in M?(2) In the case of Artin groups, the nicest Garside structures come from the Cayley graph of the corresponding Coxeter group. Is there a way to systematize this? Is there some notion of a \"Coxeter group\" corresponding to each Garside group? (3) Let M be a Garside monoid with Garside element ∆, and let ` be a length on M (M\n\nis atomic). Is it always true that `(∆ k) ≤ Ck |∆| for some constant C?",
   "clean_statement": "Problem 4.\n2. (P. Dehornoy)\n\n(1) Given a cancellative, finitely-generated monoid M in which lcm's exist, is M neces-sarily a Garside monoid? That is, does there exist a Garside element ∆ in M?(2) In the case of Artin groups, the nicest Garside structures come from the Cayley graph of the corresponding Coxeter group. Is there a way to systematize this? Is there some notion of a \"Coxeter group\" corresponding to each Garside group? (3) Let M be a Garside monoid with Garside element ∆, and let ` be a length on M (M\n\nis atomic). Is it always true that `(∆ k) ≤ Ck |∆| for some constant C?",
   "public_statement": "Problem 4.\n2. (P. Dehornoy)\n\n(1) Given a cancellative, finitely-generated monoid M in which lcm's exist, is M neces-sarily a Garside monoid? That is, does there exist a Garside element ∆ in M?(2) In the case of Artin groups, the nicest Garside structures come from the Cayley graph of the corresponding Coxeter group. Is there a way to systematize this? Is there some notion of a \"Coxeter group\" corresponding to each Garside group? (3) Let M be a Garside monoid with Garside element ∆, and let ` be a length on M (M\n\nis atomic). Is it always true that `(∆ k) ≤ Ck |∆| for some constant C?",
   "evidence": "The canonical JSON is an OCR extraction from the AIM workshop proceedings *Braid groups, clusters and free probability*. Inspection of the original PDF shows that the item is **Problem 4.2**, not two separate headings “Problem 4.” and “2.” The line-break hyphen in “neces-sarily” is typographical. The source reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 157,
   "attempt": 1
  },
  "AIM-PROBABILITY-0159": {
   "statement_status": "exact",
   "original_statement": "Problem 4.\n3. (J. McCammond) In general, the most difficult property of a Garside structure to establish is the lattice property. Call an edge-labelled poset a quasi-Garside structure if it satisfies all properties except the lattice property. There is a large natural source of quasi-Garside structures. Let G be a group, generated by a finite, conjugate-closed generating set T. Then any interval in the Cayley graph of G\n\nwith respect to T is a quasi-Garside structure. Many of these have the lattice property, and many do not. Are there natural conditions on G and T that imply the lattice property? Find a natural class of these posets in which the presence or absence of the lattice property can be explained. 10",
   "clean_statement": "Problem 4.\n3. (J. McCammond) In general, the most difficult property of a Garside structure to establish is the lattice property. Call an edge-labelled poset a quasi-Garside structure if it satisfies all properties except the lattice property. There is a large natural source of quasi-Garside structures. Let G be a group, generated by a finite, conjugate-closed generating set T. Then any interval in the Cayley graph of G\n\nwith respect to T is a quasi-Garside structure. Many of these have the lattice property, and many do not. Are there natural conditions on G and T that imply the lattice property? Find a natural class of these posets in which the presence or absence of the lattice property can be explained. 10",
   "public_statement": "Problem 4.\n3. (J. McCammond) In general, the most difficult property of a Garside structure to establish is the lattice property. Call an edge-labelled poset a quasi-Garside structure if it satisfies all properties except the lattice property. There is a large natural source of quasi-Garside structures. Let G be a group, generated by a finite, conjugate-closed generating set T. Then any interval in the Cayley graph of G\n\nwith respect to T is a quasi-Garside structure. Many of these have the lattice property, and many do not. Are there natural conditions on G and T that imply the lattice property? Find a natural class of these posets in which the presence or absence of the lattice property can be explained. 10",
   "evidence": "The canonical input is record 158 (zero-based) of `aim-probability-notes.json`, extracted from the AIM workshop *Braid groups, clusters and free probability*. The PDF itself shows that the split OCR heading “Problem 4.\\n3” is **Problem 4.3**, attributed to J. McCammond. The terminal “10” in the extracted problem is the printed page number, not part of the problem or a footnote. The notation in the remark is normalized as \\(NC_W\\), the noncrossing-partition interval for \\(W\\).",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 158,
   "attempt": 1
  },
  "AIM-PROBABILITY-0160": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 4.\n4. (D. Armstrong) As above, let G be a group generated by T, where T is finite and closed under conjugation. Then every interval in the Cayley graph of ( G, T ) is a locally self-dual poset (every interval in the poset is self-dual). In particular, to each element g of G, associate the poset Pg which is the interval [1, g ]in the Cayley graph of ( G, T ). Note that Pg and Ph are isomorphic whenever g and h are conjugate. Now, associate to each Pg its Ehrenborg quasisymmetric function \n\nF (Pg):= ∑\n\n> k\n\n∑ \n\n> 1≤g0≤g1≤···≤ gk≤g\n\nx`(g−10 g1)1 x`(g−11 g2)2 · · · x`(g−1 \n\n> k−1gk)\n> k.\n\nIt is known that the Ehrenborg function of a self-dual poset must, in fact, be a symmetric function (see [49]). So F is a map from conjugacy classes of G to the ring of symmetric functions. What is the structure of this map? Does it preserve some Hopf algebra structure? 5. Free Probability \n\nFree probability, initiated by Dan Voiculescu, is a subject in functional analysis which has been used successfully to study von Neumann algebras. It is a noncommutative analogue of probability in which the role of random variables is played by operators in some ∗-algebra (typically a C∗-algebra). The theory naturally describes the asymptotics of large random matrices, as well as the asymptotics of representations of large symmetric groups. Roland Speicher showed that the combinatorics of free probability is governed by the lattice of type A noncrossing partitions, in a role which is analogous to the role played by the lattice of unrestricted set partitions in classical probability. Many of the natural transforms on free algebras of random variables can be understood in terms of M¨ obius inversion in the incidence algebra of N C An−1. See the survey [47] for more information.",
   "clean_statement": null,
   "public_statement": "Problem 4.\n4. (D. Armstrong) As above, let G be a group generated by T, where T is finite and closed under conjugation. Then every interval in the Cayley graph of ( G, T ) is a locally self-dual poset (every interval in the poset is self-dual). In particular, to each element g of G, associate the poset Pg which is the interval [1, g ]in the Cayley graph of ( G, T ). Note that Pg and Ph are isomorphic whenever g and h are conjugate. Now, associate to each Pg its Ehrenborg quasisymmetric function\n\nF (Pg):= ∑\n\n> k\n\n∑\n\n> 1≤g0≤g1≤···≤ gk≤g\n\nx`(g−10 g1)1 x`(g−11 g2)2 · · · x`(g−1\n\n> k−1gk)\n> k.\n\nIt is known that the Ehrenborg function of a self-dual poset must, in fact, be a symmetric function (see [49]). So F is a map from conjugacy classes of G to the ring of symmetric functions. What is the structure of this map? Does it preserve some Hopf algebra structure? 5. Free Probability\n\nFree probability, initiated by Dan Voiculescu, is a subject in functional analysis which has been used successfully to study von Neumann algebras. It is a noncommutative analogue of probability in which the role of random variables is played by operators in some ∗-algebra (typically a C∗-algebra). The theory naturally describes the asymptotics of large random matrices, as well as the asymptotics of representations of large symmetric groups. Roland Speicher showed that the combinatorics of free probability is governed by the lattice of type A noncrossing partitions, in a role which is analogous to the role played by the lattice of unrestricted set partitions in classical probability. Many of the natural transforms on free algebras of random variables can be understood in terms of M¨ obius inversion in the incidence algebra of N C An−1. See the survey [47] for more information.",
   "evidence": "These repairs are explicit reconstructions; the exact source record in `input.json` has not been altered.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 159,
   "attempt": 1
  },
  "AIM-PROBABILITY-0161": {
   "statement_status": "exact",
   "original_statement": "Problem 5.\n1. (F. Goodman, P. Sniady) Philippe Biane, Fred Goodman, and Alexan-dru Nica have defined a type B analogue of free probability [10]. The definition has been motivated by the combinatorics, and there is currently no model of this theory (as the large random matrices are a model for type A free probability). Find a natural model for type B free probability, which motivates the combinatorics. Is there a corresponding notion of free probability in other types?",
   "clean_statement": "Problem 5.\n1. (F. Goodman, P. Sniady) Philippe Biane, Fred Goodman, and Alexan-dru Nica have defined a type B analogue of free probability [10]. The definition has been motivated by the combinatorics, and there is currently no model of this theory (as the large random matrices are a model for type A free probability). Find a natural model for type B free probability, which motivates the combinatorics. Is there a corresponding notion of free probability in other types?",
   "public_statement": "Problem 5.\n1. (F. Goodman, P. Sniady) Philippe Biane, Fred Goodman, and Alexan-dru Nica have defined a type B analogue of free probability [10]. The definition has been motivated by the combinatorics, and there is currently no model of this theory (as the large random matrices are a model for type A free probability). Find a natural model for type B free probability, which motivates the combinatorics. Is there a corresponding notion of free probability in other types?",
   "evidence": "The corpus record is Problem 5.1 from the AIM workshop list *Braid groups, clusters and free probability*. Inspection of page 10 of the original PDF resolves the line break in “5.\\n1” and the OCR split in Alexandru Nica's name. The recovered statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 160,
   "attempt": 1
  },
  "AIM-PROBABILITY-0162": {
   "statement_status": "exact",
   "original_statement": "Problem 5.\n2. (A. Nica) Let ( A, ϕ ) be a ∗-probability space. That is, A is some ∗-algebra, and ϕ is a linear functional on A which plays the role of \"expectation\". Let C0〈〈 z1,..., z s〉〉 11 \n\ndenote the set of power series in s noncommuting variables which have zero constant term. For each s-tuple of elements a1,,..., a s in A, there is a function Ra1,...,a s called the R-transform, which is an element of C0〈〈 z1,..., z s〉〉. See [39] for details. There is a unique binary operation?s defined on C0〈〈 z1,..., z s〉〉 with the property that for any two families {a1,..., a s} and {b1,..., b s} of freely independent random variables, we have \n\nRa1,...,a s?s Rb1,...,b s = Ra1b1,...,a sbs.\n\nThe operation?s is associative, and has a unit ∆ s(z1,..., z s):= z1 + · · · + zs. In [39], Alexandru Nica and Roland Speicher show that, in general, the coefficients of f?s g can be described combinatorially, using a summation over noncrossing partitions of type A.Describe the structure of the group of invertible elements in the semigroup ( C0〈〈 z1,..., z s〉〉,?s).",
   "clean_statement": "Problem 5.\n2. (A. Nica) Let ( A, ϕ ) be a ∗-probability space. That is, A is some ∗-algebra, and ϕ is a linear functional on A which plays the role of \"expectation\". Let C0〈〈 z1,..., z s〉〉 11\n\ndenote the set of power series in s noncommuting variables which have zero constant term. For each s-tuple of elements a1,,..., a s in A, there is a function Ra1,...,a s called the R-transform, which is an element of C0〈〈 z1,..., z s〉〉. See [39] for details. There is a unique binary operation?s defined on C0〈〈 z1,..., z s〉〉 with the property that for any two families {a1,..., a s} and {b1,..., b s} of freely independent random variables, we have\n\nRa1,...,a s?s Rb1,...,b s = Ra1b1,...,a sbs.\n\nThe operation?s is associative, and has a unit ∆ s(z1,..., z s):= z1 + · · · + zs. In [39], Alexandru Nica and Roland Speicher show that, in general, the coefficients of f?s g can be described combinatorially, using a summation over noncrossing partitions of type A.Describe the structure of the group of invertible elements in the semigroup ( C0〈〈 z1,..., z s〉〉,?s).",
   "public_statement": "Problem 5.\n2. (A. Nica) Let ( A, ϕ ) be a ∗-probability space. That is, A is some ∗-algebra, and ϕ is a linear functional on A which plays the role of \"expectation\". Let C0〈〈 z1,..., z s〉〉 11\n\ndenote the set of power series in s noncommuting variables which have zero constant term. For each s-tuple of elements a1,,..., a s in A, there is a function Ra1,...,a s called the R-transform, which is an element of C0〈〈 z1,..., z s〉〉. See [39] for details. There is a unique binary operation?s defined on C0〈〈 z1,..., z s〉〉 with the property that for any two families {a1,..., a s} and {b1,..., b s} of freely independent random variables, we have\n\nRa1,...,a s?s Rb1,...,b s = Ra1b1,...,a sbs.\n\nThe operation?s is associative, and has a unit ∆ s(z1,..., z s):= z1 + · · · + zs. In [39], Alexandru Nica and Roland Speicher show that, in general, the coefficients of f?s g can be described combinatorially, using a summation over noncrossing partitions of type A.Describe the structure of the group of invertible elements in the semigroup ( C0〈〈 z1,..., z s〉〉,?s).",
   "evidence": "The canonical input is record 161 (zero-based) of `aim-probability-notes.json`, from the AIM workshop *Braid groups, clusters and free probability*. The official PDF identifies it as **Problem 5.2**, attributed to A. Nica.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 161,
   "attempt": 1
  },
  "AIM-PROBABILITY-0163": {
   "statement_status": "exact",
   "original_statement": "Problem 5.\n3. (A. Nica) Let u be a unitary element of a ∗-probability space ( A, ϕ ). Sup-pose u has order k and that ϕ(ui) = 0 for 1 ≤ i < k. Let κn denote the multilinear cumulant functionals of ( A, ϕ ). Give a combinatorial way to compute the cumulants in u and u∗. Equivalently, give a formula for the R-transform of ( u, u ∗).",
   "clean_statement": "Problem 5.\n3. (A. Nica) Let u be a unitary element of a ∗-probability space ( A, ϕ ). Sup-pose u has order k and that ϕ(ui) = 0 for 1 ≤ i < k. Let κn denote the multilinear cumulant functionals of ( A, ϕ ). Give a combinatorial way to compute the cumulants in u and u∗. Equivalently, give a formula for the R-transform of ( u, u ∗).",
   "public_statement": "Problem 5.\n3. (A. Nica) Let u be a unitary element of a ∗-probability space ( A, ϕ ). Sup-pose u has order k and that ϕ(ui) = 0 for 1 ≤ i < k. Let κn denote the multilinear cumulant functionals of ( A, ϕ ). Give a combinatorial way to compute the cumulants in u and u∗. Equivalently, give a formula for the R-transform of ( u, u ∗).",
   "evidence": "The AIM source contains the following problem (Problem 5.3, attributed to A. Nica), after repairing line-break OCR:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 162,
   "attempt": 1
  },
  "AIM-PROBABILITY-0164": {
   "statement_status": "exact",
   "original_statement": "Problem 6.\n1. (H. Thomas, A. Zelevinsky) When W is a noncrystallographic finite Coxeter group, give a geometric construction that realizes ∆ W as a convex polytope. There is a realization of ∆ W in all types as a complete simplicial fan, but it is not clear whether this fan is polytopal in the noncrystallographic types.",
   "clean_statement": "Problem 6.\n1. (H. Thomas, A. Zelevinsky) When W is a noncrystallographic finite Coxeter group, give a geometric construction that realizes ∆ W as a convex polytope. There is a realization of ∆ W in all types as a complete simplicial fan, but it is not clear whether this fan is polytopal in the noncrystallographic types.",
   "public_statement": "Problem 6.\n1. (H. Thomas, A. Zelevinsky) When W is a noncrystallographic finite Coxeter group, give a geometric construction that realizes ∆ W as a convex polytope. There is a realization of ∆ W in all types as a complete simplicial fan, but it is not clear whether this fan is polytopal in the noncrystallographic types.",
   "evidence": "There is no substantive OCR corruption, but `∆ W` means the subscripted cluster complex \\(\\Delta_W\\), and the split lines `Problem 6.` and `1.` form Problem 6.1. The preceding source paragraph defines \\(\\Delta_W\\) as the flag complex of compatible subsets of the almost-positive roots \\[ \\Phi_{\\ge-1}=\\Phi^+\\cup(-\\Pi). \\] It also says that in types \\(A,B\\) these complexes generalize the **duals** of the classical associahedron and cyclohedron. Thus the precise reading is: construct a simple convex polytope whose polar boundary is \\(\\Delta_W\\), and, more strongly, make the displayed complete fan its normal fan. Confusing the simplicial complex with the face lattice of the simple polytope rather than its dual reverses incidences.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 163,
   "attempt": 1
  },
  "AIM-PROBABILITY-0165": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 6.\n2. (A. Zelevinsky) In the classical types ( A, B, C, and D), the associahedron ∆W has a visually transparent realization in terms of regular plane polygons and their triangulations. Find a similar interpratation in the exceptional types.",
   "clean_statement": "**Problem 6.2 (A. Zelevinsky).** In the classical types (A, B, C, and D),\nthe associahedron \\(\\Delta_W\\) has a visually transparent realization in\nterms of regular plane polygons and their triangulations. Find a similar\ninterpratation in the exceptional types.",
   "public_statement": "Problem 6.\n2. (A. Zelevinsky) In the classical types ( A, B, C, and D), the associahedron ∆W has a visually transparent realization in terms of regular plane polygons and their triangulations. Find a similar interpratation in the exceptional types.",
   "evidence": "The canonical record is the second item in Section 6 of the January 2005 AIM workshop report *Braid Groups, Clusters, and Free Probability*. Page 12 of the original PDF reads:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-probability-notes.json",
   "source_index": 164,
   "attempt": 1
  },
  "AIM-PROBABILITY-0166": {
   "statement_status": "exact",
   "original_statement": "Problem 6.\n3. (S. Fomin) Conjecture: The Fomin-Reading generalization of the associa-hedron ∆ (k) \n\n> W\n\n(see",
   "clean_statement": "Problem 6.\n3. (S. Fomin) Conjecture: The Fomin-Reading generalization of the associa-hedron ∆ (k)\n\n> W\n\n(see",
   "public_statement": "Problem 6.\n3. (S. Fomin) Conjecture: The Fomin-Reading generalization of the associa-hedron ∆ (k)\n\n> W\n\n(see",
   "evidence": "The canonical input is record 165 (zero-based) of `aim-probability-notes.json`. It ends after the word “see” and is not a complete mathematical statement. Inspection of the official AIM PDF shows that one printed problem was split across canonical records 165--167. The source reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 165,
   "attempt": 1
  },
  "AIM-PROBABILITY-0167": {
   "statement_status": "exact",
   "original_statement": "Problem 1.3) is Cohen-Macaulay, and is homotopy equivalent to a wedge of Cat (k−1) (W ) = \n\n> n\n\n∏\n\n> i=1\n\n(k − 1) h + ei + 1 \n\nei + 1 spheres. This has been proved by Eleni Tzanaki in types A and B using shelling methods [51]. Moreover, ∆ (k) \n\n> W\n\nseems to be the skeleton of a polytopal manifold. Can this be realized geometrically? (This generalizes",
   "clean_statement": "Problem 1.3) is Cohen-Macaulay, and is homotopy equivalent to a wedge of Cat (k−1) (W ) =\n\n> n\n\n∏\n\n> i=1\n\n(k − 1) h + ei + 1\n\nei + 1 spheres. This has been proved by Eleni Tzanaki in types A and B using shelling methods [51]. Moreover, ∆ (k)\n\n> W\n\nseems to be the skeleton of a polytopal manifold. Can this be realized geometrically? (This generalizes",
   "public_statement": "Problem 1.3) is Cohen-Macaulay, and is homotopy equivalent to a wedge of Cat (k−1) (W ) =\n\n> n\n\n∏\n\n> i=1\n\n(k − 1) h + ei + 1\n\nei + 1 spheres. This has been proved by Eleni Tzanaki in types A and B using shelling methods [51]. Moreover, ∆ (k)\n\n> W\n\nseems to be the skeleton of a polytopal manifold. Can this be realized geometrically? (This generalizes",
   "evidence": "The canonical input is not a self-contained problem. Its problem field, with OCR layout normalized but wording preserved, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 166,
   "attempt": 1
  },
  "AIM-PROBABILITY-0168": {
   "statement_status": "exact",
   "original_statement": "Problem 6.1 above.)",
   "clean_statement": "Problem 6.1 above.)",
   "public_statement": "Problem 6.1 above.)",
   "evidence": "The canonical record is visibly fragmented. Its exact `problem` field is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 167,
   "attempt": 1
  },
  "AIM-PROBABILITY-0169": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 6.\n4. (D. Bessis, C. Kriloff ) There is no construction of a cluster algebra in the noncrystallographic finite types. What happens when one applies matrix mutations to the Cartan matrix of a noncrystallographic finite Coxeter group? Are there recurrences? 13",
   "clean_statement": null,
   "public_statement": "Problem 6.\n4. (D. Bessis, C. Kriloff ) There is no construction of a cluster algebra in the noncrystallographic finite types. What happens when one applies matrix mutations to the Cartan matrix of a noncrystallographic finite Coxeter group? Are there recurrences? 13",
   "evidence": "The canonical record reads exactly:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-probability-notes.json",
   "source_index": 168,
   "attempt": 1
  },
  "AIM-PROBABILITY-0170": {
   "statement_status": "exact",
   "original_statement": "Problem 6.\n5. (A. Zelevinsky) Describe a classification of infinite type cluster algebras as tame or wild. This should generalize the notions of tame/wild Artin groups, tame/wild quivers, etc.",
   "clean_statement": "Problem 6.\n5. (A. Zelevinsky) Describe a classification of infinite type cluster algebras as tame or wild. This should generalize the notions of tame/wild Artin groups, tame/wild quivers, etc.",
   "public_statement": "Problem 6.\n5. (A. Zelevinsky) Describe a classification of infinite type cluster algebras as tame or wild. This should generalize the notions of tame/wild Artin groups, tame/wild quivers, etc.",
   "evidence": "The record comes from the AIM workshop *Braid groups, clusters and free probability*. The official workshop problem list gives the following statement (the line break between “6.” and “5.” in the JSON record is only an extraction artifact):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-probability-notes.json",
   "source_index": 169,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0001": {
   "statement_status": "exact",
   "original_statement": "Central derivative of Rankin--Selberg $L$-functions for weight-$2$ modular forms and congruent forms\n\nThe hope is to show the non-vanishing of the central-value derivative of the Rankin--Selberg $L$-function of a weight $2$ modular form tensored with a theta series coming from a class group character via the Rankin--Selberg $L$-function with a congruent modular form on a definite quaternion algebra.\n\nLet $f$ be a modular form of weight $2$ and let $\\chi$ be a character of the class group of an imaginary quadratic number field $K$. Formulate a direct connection\n\\[\n L'(f \\otimes \\theta_\\chi, \\frac{1}{2}) \\longleftrightarrow L(g \\otimes \\theta_\\chi, \\frac{1}{2}),\n\\]\nfor a modular form $g$ on a definite quaternion algebra with some congruence to $f$.",
   "clean_statement": "Central derivative of Rankin--Selberg $L$-functions for weight-$2$ modular forms and congruent forms\n\nThe hope is to show the non-vanishing of the central-value derivative of the Rankin--Selberg $L$-function of a weight $2$ modular form tensored with a theta series coming from a class group character via the Rankin--Selberg $L$-function with a congruent modular form on a definite quaternion algebra.\n\nLet $f$ be a modular form of weight $2$ and let $\\chi$ be a character of the class group of an imaginary quadratic number field $K$. Formulate a direct connection\n\\[\n L'(f \\otimes \\theta_\\chi, \\frac{1}{2}) \\longleftrightarrow L(g \\otimes \\theta_\\chi, \\frac{1}{2}),\n\\]\nfor a modular form $g$ on a definite quaternion algebra with some congruence to $f$.",
   "public_statement": "Central derivative of Rankin--Selberg $L$-functions for weight-$2$ modular forms and congruent forms\n\nThe hope is to show the non-vanishing of the central-value derivative of the Rankin--Selberg $L$-function of a weight $2$ modular form tensored with a theta series coming from a class group character via the Rankin--Selberg $L$-function with a congruent modular form on a definite quaternion algebra.\n\nLet $f$ be a modular form of weight $2$ and let $\\chi$ be a character of the class group of an imaginary quadratic number field $K$. Formulate a direct connection\n\\[\n L'(f \\otimes \\theta_\\chi, \\frac{1}{2}) \\longleftrightarrow L(g \\otimes \\theta_\\chi, \\frac{1}{2}),\n\\]\nfor a modular form $g$ on a definite quaternion algebra with some congruence to $f$.",
   "evidence": "The AIM record (workshop *Analytic, arithmetic, and geometric aspects of automorphic forms*, Group Problems 1.1) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0002": {
   "statement_status": "exact",
   "original_statement": "Relative conductor of $p$-adic groups\n\nLet $H \\leq G$ be groups over $\\mathbb{Q}_p$, $\\pi$ be an admissible representation of $G$, and $K_G$ be a maximal compact of $G$. Define\n\\[\n K_H(p^n) := \\{g \\in K_G \\mid (g \\mod p^n) \\in H\\}.\n\\]\n\nDoes $\\pi$ have a non-zero vector fixed by $K_H(p^n)$ for some $n$?\n\nIf so, what's the smallest such $n$?",
   "clean_statement": "Relative conductor of $p$-adic groups\n\nLet $H \\leq G$ be groups over $\\mathbb{Q}_p$, $\\pi$ be an admissible representation of $G$, and $K_G$ be a maximal compact of $G$. Define\n\\[\n K_H(p^n) := \\{g \\in K_G \\mid (g \\mod p^n) \\in H\\}.\n\\]\n\nDoes $\\pi$ have a non-zero vector fixed by $K_H(p^n)$ for some $n$?\n\nIf so, what's the smallest such $n$?",
   "public_statement": "Relative conductor of $p$-adic groups\n\nLet $H \\leq G$ be groups over $\\mathbb{Q}_p$, $\\pi$ be an admissible representation of $G$, and $K_G$ be a maximal compact of $G$. Define\n\\[\n K_H(p^n) := \\{g \\in K_G \\mid (g \\mod p^n) \\in H\\}.\n\\]\n\nDoes $\\pi$ have a non-zero vector fixed by $K_H(p^n)$ for some $n$?\n\nIf so, what's the smallest such $n$?",
   "evidence": "The AIM record, in the section “Group problems” of *Analytic, arithmetic, and geometric aspects of automorphic forms*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0003": {
   "statement_status": "exact",
   "original_statement": "Non-vanishing of the central $L$-derivative for $\\mathrm{U}(n) \\times \\mathrm{U}(n-1)$\n\nGiven an automorphic representation $\\pi$ of $\\mathrm{U}(n)$, show that there exists an automorphic representation $\\sigma$ of $\\mathrm{U}(n-1)$ such that\n\\[\n L'(\\pi \\otimes \\sigma, \\frac{1}{2}) \\neq 0.\n\\]",
   "clean_statement": "Non-vanishing of the central $L$-derivative for $\\mathrm{U}(n) \\times \\mathrm{U}(n-1)$\n\nGiven an automorphic representation $\\pi$ of $\\mathrm{U}(n)$, show that there exists an automorphic representation $\\sigma$ of $\\mathrm{U}(n-1)$ such that\n\\[\n L'(\\pi \\otimes \\sigma, \\frac{1}{2}) \\neq 0.\n\\]",
   "public_statement": "Non-vanishing of the central $L$-derivative for $\\mathrm{U}(n) \\times \\mathrm{U}(n-1)$\n\nGiven an automorphic representation $\\pi$ of $\\mathrm{U}(n)$, show that there exists an automorphic representation $\\sigma$ of $\\mathrm{U}(n-1)$ such that\n\\[\n L'(\\pi \\otimes \\sigma, \\frac{1}{2}) \\neq 0.\n\\]",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0004": {
   "statement_status": "exact",
   "original_statement": "$p$-indivisibility of $L(\\chi, 0)$\n\nFix a prime $p$. Let $q$ be a prime and let $\\chi$ be an odd character of $(\\Z/q\\Z)^\\times$.\n\nHow often is $L(\\chi, 0)$ indivisible by $p$?",
   "clean_statement": "$p$-indivisibility of $L(\\chi, 0)$\n\nFix a prime $p$. Let $q$ be a prime and let $\\chi$ be an odd character of $(\\Z/q\\Z)^\\times$.\n\nHow often is $L(\\chi, 0)$ indivisible by $p$?",
   "public_statement": "$p$-indivisibility of $L(\\chi, 0)$\n\nFix a prime $p$. Let $q$ be a prime and let $\\chi$ be an odd character of $(\\Z/q\\Z)^\\times$.\n\nHow often is $L(\\chi, 0)$ indivisible by $p$?",
   "evidence": "The accompanying note says that a lower bound $q^{1/2-\\epsilon}$ is known and that a positive proportion is expected. The repository text is coherent and shows no apparent OCR error. The original HTTP page was unavailable during this run (HTTP 502), so the wording above is the exact repository record rather than a new transcription from the page.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0005": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Beilinson conjecture for Hecke characters of quartic CM fields\n\nProve the Beilinson conjecture for Hecke characters of quartic CM fields.",
   "clean_statement": null,
   "public_statement": "Beilinson conjecture for Hecke characters of quartic CM fields\n\nProve the Beilinson conjecture for Hecke characters of quartic CM fields.",
   "evidence": "The literature note about generalizing the Eisenstein symbol strongly suggests the following plausible reading. For an algebraic \\(\\psi\\) of weight \\(w\\) and an integer \\(n>w/2+1\\), construct motivic classes in the \\(\\psi\\)-part of the cohomology of a CM abelian surface, and prove that their Deligne-regulator determinant gives \\(L_K(\\psi,n)\\), modulo the coefficient field. At a Deligne-critical \\(n\\), this becomes a period-algebraicity statement; at a noncritical \\(n\\), it is the Deninger-style weak Beilinson regulator statement. The latter is the reading used below. It is a reconstruction, not text verified on the unavailable source page.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0006": {
   "statement_status": "exact",
   "original_statement": "Construction of abelian surfaces for paramodular newforms\n\nGiven a paramodular newform $F$ of weight $2$ with rational coefficients, find a construction of the abelian surface to which it corresponds.",
   "clean_statement": "Construction of abelian surfaces for paramodular newforms\n\nGiven a paramodular newform $F$ of weight $2$ with rational coefficients, find a construction of the abelian surface to which it corresponds.",
   "public_statement": "Construction of abelian surfaces for paramodular newforms\n\nGiven a paramodular newform $F$ of weight $2$ with rational coefficients, find a construction of the abelian surface to which it corresponds.",
   "evidence": "The AIM record, in the section “Fixed vectors in representations of \\(p\\)-adic groups” of *Analytic, arithmetic, and geometric aspects of automorphic forms*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0007": {
   "statement_status": "exact",
   "original_statement": "Fixed vectors for $\\mathrm{GSp}_{2n}$\n\nLet $\\pi$ be a irreducible admissible infinite-dimensional representation of $\\mathrm{GSp}_{2n}(\\mathbb{Q}_p)$. Consider the subgroup\n\\[\n R(n) := \\{g \\equiv (\\text{matrix with $1$ on the diagonal for first $n$ entries, $a$ on the diagonal for the last $n$ entries}) \\pmod{p^n} \\text{ for some } a \\in \\mathbb{Z}_p^\\times \\}.\n\\]\n\nDoes $\\pi$ have a $R(n)$-fixed vector for some $n$?",
   "clean_statement": "Fixed vectors for $\\mathrm{GSp}_{2n}$\n\nLet $\\pi$ be a irreducible admissible infinite-dimensional representation of $\\mathrm{GSp}_{2n}(\\mathbb{Q}_p)$. Consider the subgroup\n\\[\n R(n) := \\{g \\equiv (\\text{matrix with $1$ on the diagonal for first $n$ entries, $a$ on the diagonal for the last $n$ entries}) \\pmod{p^n} \\text{ for some } a \\in \\mathbb{Z}_p^\\times \\}.\n\\]\n\nDoes $\\pi$ have a $R(n)$-fixed vector for some $n$?",
   "public_statement": "Fixed vectors for $\\mathrm{GSp}_{2n}$\n\nLet $\\pi$ be a irreducible admissible infinite-dimensional representation of $\\mathrm{GSp}_{2n}(\\mathbb{Q}_p)$. Consider the subgroup\n\\[\n R(n) := \\{g \\equiv (\\text{matrix with $1$ on the diagonal for first $n$ entries, $a$ on the diagonal for the last $n$ entries}) \\pmod{p^n} \\text{ for some } a \\in \\mathbb{Z}_p^\\times \\}.\n\\]\n\nDoes $\\pi$ have a $R(n)$-fixed vector for some $n$?",
   "evidence": "The exact repository record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0008": {
   "statement_status": "exact",
   "original_statement": "Arithmetic applications of horizontal equidistribution results\n\nLet $K$ be an imaginary quadratic number fields. There are equidistribution results for CM points with square-free order of conductor $C$ as $C \\rightarrow \\infty$.\n\nIs there an interesting arithmetic application for this?",
   "clean_statement": "Arithmetic applications of horizontal equidistribution results\n\nLet $K$ be an imaginary quadratic number fields. There are equidistribution results for CM points with square-free order of conductor $C$ as $C \\rightarrow \\infty$.\n\nIs there an interesting arithmetic application for this?",
   "public_statement": "Arithmetic applications of horizontal equidistribution results\n\nLet $K$ be an imaginary quadratic number fields. There are equidistribution results for CM points with square-free order of conductor $C$ as $C \\rightarrow \\infty$.\n\nIs there an interesting arithmetic application for this?",
   "evidence": "The AIM record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0009": {
   "statement_status": "exact",
   "original_statement": "Joint equidistribution of CM points\n\nDetermine the joint equidistribution of CM points over $K$ of products of modular curves over $F$ as $\\mathrm{disc}(K) \\rightarrow \\infty$.",
   "clean_statement": "Joint equidistribution of CM points\n\nDetermine the joint equidistribution of CM points over $K$ of products of modular curves over $F$ as $\\mathrm{disc}(K) \\rightarrow \\infty$.",
   "public_statement": "Joint equidistribution of CM points\n\nDetermine the joint equidistribution of CM points over $K$ of products of modular curves over $F$ as $\\mathrm{disc}(K) \\rightarrow \\infty$.",
   "evidence": "The canonical AIM record (source file `aim-representation-theory-notes.json`, zero-based index 8) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0010": {
   "statement_status": "exact",
   "original_statement": "Computing Fourier coefficients from $L$-values\n\nLet $d$ be a positive integer and let $F$ be a Siegel modular form with Fourier coefficients $a(F)$. In many cases, we know the (generalized) B\\\"{o}cherer conjecture:\n\\[\n \\left \\lvert \\sum_{S \\in \\mathrm{Cl}_d} a(F, S) \\Lambda(S) \\right \\rvert^2 = L\\big(\\pi_F \\times \\theta_\\Lambda, \\frac{1}{2}\\big) = L\\big(\\mathrm{BC}_{K/\\mathbb{Q}} (\\pi_F) \\times \\Lambda, \\frac{1}{2}\\big).\n\\]\n\nKnowing this equality and values of $L(\\pi_F \\times \\theta_\\Lambda, \\frac{1}{2})$ for many $\\Lambda$, can we compute the Fourier coefficients $a(F, S)$?",
   "clean_statement": "Computing Fourier coefficients from $L$-values\n\nLet $d$ be a positive integer and let $F$ be a Siegel modular form with Fourier coefficients $a(F)$. In many cases, we know the (generalized) B\\\"{o}cherer conjecture:\n\\[\n \\left \\lvert \\sum_{S \\in \\mathrm{Cl}_d} a(F, S) \\Lambda(S) \\right \\rvert^2 = L\\big(\\pi_F \\times \\theta_\\Lambda, \\frac{1}{2}\\big) = L\\big(\\mathrm{BC}_{K/\\mathbb{Q}} (\\pi_F) \\times \\Lambda, \\frac{1}{2}\\big).\n\\]\n\nKnowing this equality and values of $L(\\pi_F \\times \\theta_\\Lambda, \\frac{1}{2})$ for many $\\Lambda$, can we compute the Fourier coefficients $a(F, S)$?",
   "public_statement": "Computing Fourier coefficients from $L$-values\n\nLet $d$ be a positive integer and let $F$ be a Siegel modular form with Fourier coefficients $a(F)$. In many cases, we know the (generalized) B\\\"{o}cherer conjecture:\n\\[\n \\left \\lvert \\sum_{S \\in \\mathrm{Cl}_d} a(F, S) \\Lambda(S) \\right \\rvert^2 = L\\big(\\pi_F \\times \\theta_\\Lambda, \\frac{1}{2}\\big) = L\\big(\\mathrm{BC}_{K/\\mathbb{Q}} (\\pi_F) \\times \\Lambda, \\frac{1}{2}\\big).\n\\]\n\nKnowing this equality and values of $L(\\pi_F \\times \\theta_\\Lambda, \\frac{1}{2})$ for many $\\Lambda$, can we compute the Fourier coefficients $a(F, S)$?",
   "evidence": "The canonical record is Problem 5.1, “Computing Fourier coefficients from \\(L\\)-values,” from the AIM workshop list *Analytic, arithmetic, and geometric aspects of automorphic forms*. It asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0011": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Siegel modular forms\n\nLet $F$ be a Siegel cusp form of full level and weight $k$ that is a Hecke eigenform. Assume any standard conjecture on $L$-values (e.g. GRH) and all conjectural period formulae.\n\nProve that for some $\\delta > 0$,\n\\[\n \\lvert a(F, S) \\rvert \\ll_{F} \\det(S)^{\\frac{K}{2}-\\frac{1}{2}-\\delta}.\n\\]",
   "clean_statement": null,
   "public_statement": "Siegel modular forms\n\nLet $F$ be a Siegel cusp form of full level and weight $k$ that is a Hecke eigenform. Assume any standard conjecture on $L$-values (e.g. GRH) and all conjectural period formulae.\n\nProve that for some $\\delta > 0$,\n\\[\n \\lvert a(F, S) \\rvert \\ll_{F} \\det(S)^{\\frac{K}{2}-\\frac{1}{2}-\\delta}.\n\\]",
   "evidence": "The uppercase \\(K\\) in the exponent has no definition and is almost certainly a typographical/OCR error for the weight \\(k\\). The neighboring Problem 5.1 uses ideal-class characters and generalized Böcherer formulae, which identifies the intended setting as scalar-valued Siegel modular forms of **degree 2**. The natural reconstruction is therefore \\[ F(Z)=\\sum_{S\\in\\Lambda_2^+}a(F,S)e^{2\\pi i\\operatorname{tr}(SZ)}, \\qquad F\\in S_k(\\operatorname{Sp}_4(\\mathbf Z)), \\tag{1.1} \\] where \\[ S=\\begin{pmatrix}a&b/2\\\\b/2&c\\end{pmatrix}>0, \\qquad a,b,c\\in\\mathbf Z, \\] and the requested estimate is \\[ |a(F,S)|\\ll_F\\det(S)^{k/2-1/2-\\delta}. \\tag{1.2} \\]",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0012": {
   "statement_status": "exact",
   "original_statement": "Applications of the Hecke orbit conjecture\n\nAre there applications of the recent progress on the Hecke orbit conjecture to vertical mod-$p$ non-vanishing (as in Iwasawa theory)?",
   "clean_statement": "Applications of the Hecke orbit conjecture\n\nAre there applications of the recent progress on the Hecke orbit conjecture to vertical mod-$p$ non-vanishing (as in Iwasawa theory)?",
   "public_statement": "Applications of the Hecke orbit conjecture\n\nAre there applications of the recent progress on the Hecke orbit conjecture to vertical mod-$p$ non-vanishing (as in Iwasawa theory)?",
   "evidence": "The canonical AIM record (source file `aim-representation-theory-notes.json`, zero-based index 11) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0013": {
   "statement_status": "exact",
   "original_statement": "Normalization of Bianchi modular forms\n\nConsider a Bianchi modular form, is there a correct way to normalize its coefficients in its Whittaker expansion (in $H^2$)?",
   "clean_statement": "Normalization of Bianchi modular forms\n\nConsider a Bianchi modular form, is there a correct way to normalize its coefficients in its Whittaker expansion (in $H^2$)?",
   "public_statement": "Normalization of Bianchi modular forms\n\nConsider a Bianchi modular form, is there a correct way to normalize its coefficients in its Whittaker expansion (in $H^2$)?",
   "evidence": "The canonical AIM record, Problem 6.1 in “More problems” from the workshop *Analytic, arithmetic, and geometric aspects of automorphic forms*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0014": {
   "statement_status": "exact",
   "original_statement": "$p$-adic differential operators for higher-rank groups\n\nFor a higher-rank group (e.g. $\\mathrm{GSp}_4$), there are some standard $p$-adic differential operators. Do the images of these differential operators contain special automorphic forms coming from lower-rank groups (e.g. Yoshida lifts, Saito--Kurokawa lifts)?",
   "clean_statement": "$p$-adic differential operators for higher-rank groups\n\nFor a higher-rank group (e.g. $\\mathrm{GSp}_4$), there are some standard $p$-adic differential operators. Do the images of these differential operators contain special automorphic forms coming from lower-rank groups (e.g. Yoshida lifts, Saito--Kurokawa lifts)?",
   "public_statement": "$p$-adic differential operators for higher-rank groups\n\nFor a higher-rank group (e.g. $\\mathrm{GSp}_4$), there are some standard $p$-adic differential operators. Do the images of these differential operators contain special automorphic forms coming from lower-rank groups (e.g. Yoshida lifts, Saito--Kurokawa lifts)?",
   "evidence": "The canonical record is problem 6.3 in the “More problems” section of the AIM workshop *Analytic, arithmetic, and geometric aspects of automorphic forms*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0015": {
   "statement_status": "exact",
   "original_statement": "Local-global description of Stark units\n\nLet $f$ be a weight-$1$ modular form with associated Galois representation $\\rho_f$. The Harris--Venkatesh conjecture gives a description of Stark units for $\\mathrm{Ad}^0(\\rho_f)$ modulo $p$ for almost all $p$. In explicit exotic cases, can this be used to describe the Stark units for $\\mathrm{Ad}^0(\\rho_f)$ themselves?",
   "clean_statement": "Local-global description of Stark units\n\nLet $f$ be a weight-$1$ modular form with associated Galois representation $\\rho_f$. The Harris--Venkatesh conjecture gives a description of Stark units for $\\mathrm{Ad}^0(\\rho_f)$ modulo $p$ for almost all $p$. In explicit exotic cases, can this be used to describe the Stark units for $\\mathrm{Ad}^0(\\rho_f)$ themselves?",
   "public_statement": "Local-global description of Stark units\n\nLet $f$ be a weight-$1$ modular form with associated Galois representation $\\rho_f$. The Harris--Venkatesh conjecture gives a description of Stark units for $\\mathrm{Ad}^0(\\rho_f)$ modulo $p$ for almost all $p$. In explicit exotic cases, can this be used to describe the Stark units for $\\mathrm{Ad}^0(\\rho_f)$ themselves?",
   "evidence": "The canonical AIM record (Representation theory, workshop *Analytic, arithmetic, and geometric aspects of automorphic forms*, section “More problems,” problem 6.5) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0016": {
   "statement_status": "exact",
   "original_statement": "Nate Harman's question on lifting $\\mathrm{FI}$-modules from $\\mathbb{C}$ to $\\mathbb{Z}$\n\nGiven a map $\\phi_{\\mathbb{Q}} \\colon V_{\\mathbb{Q}} \\to W_{\\mathbb{Q}}$ of finitely generated $\\mathrm{FI}$-modules over $\\mathbb{Q}$ (or may be over an algebraically closed field), and choices $V, W$ of lattices in $V_{\\mathbb{Q}}$ and $W_{\\mathbb{Q}}$. If $p$ is sufficiently large, is $ V \\otimes_{\\mathbb{Z}} \\mathbb{F}_p \\to W \\otimes_{\\mathbb{Z}} \\mathbb{F}_p$ independent of the choice of $V$ and $W$ up to isomorphism?",
   "clean_statement": "Nate Harman's question on lifting $\\mathrm{FI}$-modules from $\\mathbb{C}$ to $\\mathbb{Z}$\n\nGiven a map $\\phi_{\\mathbb{Q}} \\colon V_{\\mathbb{Q}} \\to W_{\\mathbb{Q}}$ of finitely generated $\\mathrm{FI}$-modules over $\\mathbb{Q}$ (or may be over an algebraically closed field), and choices $V, W$ of lattices in $V_{\\mathbb{Q}}$ and $W_{\\mathbb{Q}}$. If $p$ is sufficiently large, is $ V \\otimes_{\\mathbb{Z}} \\mathbb{F}_p \\to W \\otimes_{\\mathbb{Z}} \\mathbb{F}_p$ independent of the choice of $V$ and $W$ up to isomorphism?",
   "public_statement": "Nate Harman's question on lifting $\\mathrm{FI}$-modules from $\\mathbb{C}$ to $\\mathbb{Z}$\n\nGiven a map $\\phi_{\\mathbb{Q}} \\colon V_{\\mathbb{Q}} \\to W_{\\mathbb{Q}}$ of finitely generated $\\mathrm{FI}$-modules over $\\mathbb{Q}$ (or may be over an algebraically closed field), and choices $V, W$ of lattices in $V_{\\mathbb{Q}}$ and $W_{\\mathbb{Q}}$. If $p$ is sufficiently large, is $ V \\otimes_{\\mathbb{Z}} \\mathbb{F}_p \\to W \\otimes_{\\mathbb{Z}} \\mathbb{F}_p$ independent of the choice of $V$ and $W$ up to isomorphism?",
   "evidence": "The AIM record (Representation stability workshop, Section 1.1) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0017": {
   "statement_status": "exact",
   "original_statement": "Noetherianity of $\\mathrm{FI}$-algebras\n\nLet $V_n = \\mathbf{k}[x_1 , x_2, \\ldots x_n]$. Find a concrete example of a sub-$\\mathrm{FI}$-algebra which is not noetherian. Is there a version for finitely presented modules? Does it depend on characteristic?",
   "clean_statement": "Noetherianity of $\\mathrm{FI}$-algebras\n\nLet $V_n = \\mathbf{k}[x_1 , x_2, \\ldots x_n]$. Find a concrete example of a sub-$\\mathrm{FI}$-algebra which is not noetherian. Is there a version for finitely presented modules? Does it depend on characteristic?",
   "public_statement": "Noetherianity of $\\mathrm{FI}$-algebras\n\nLet $V_n = \\mathbf{k}[x_1 , x_2, \\ldots x_n]$. Find a concrete example of a sub-$\\mathrm{FI}$-algebra which is not noetherian. Is there a version for finitely presented modules? Does it depend on characteristic?",
   "evidence": "The canonical AIM record (workshop *Representation stability*, section “\\(\\mathrm{FI}\\)-modules and tca's,” Problem 1.2) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0018": {
   "statement_status": "exact",
   "original_statement": "Classify torsion free injectives\n\nWhat's the injective hull of the trivial $\\mathrm{FI}$-modules ($V_n = \\mathrm{triv}_n$ for each $n$)? In general, classify all indecomposable injectives. Is it true that these indecomposable injectives are degree-wise finitely generated (same question over a more general combinatorial category).",
   "clean_statement": "Classify torsion free injectives\n\nWhat's the injective hull of the trivial $\\mathrm{FI}$-modules ($V_n = \\mathrm{triv}_n$ for each $n$)? In general, classify all indecomposable injectives. Is it true that these indecomposable injectives are degree-wise finitely generated (same question over a more general combinatorial category).",
   "public_statement": "Classify torsion free injectives\n\nWhat's the injective hull of the trivial $\\mathrm{FI}$-modules ($V_n = \\mathrm{triv}_n$ for each $n$)? In general, classify all indecomposable injectives. Is it true that these indecomposable injectives are degree-wise finitely generated (same question over a more general combinatorial category).",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0019": {
   "statement_status": "exact",
   "original_statement": "Highest weight category\n\nWorking over a field. Is the Serre quotient $\\mathrm{Mod}_{\\mathrm{FI}}/\\mathrm{Mod}_{\\mathrm{FI}}^{\\mathrm{tors}}$ a highest weight category?",
   "clean_statement": "Highest weight category\n\nWorking over a field. Is the Serre quotient $\\mathrm{Mod}_{\\mathrm{FI}}/\\mathrm{Mod}_{\\mathrm{FI}}^{\\mathrm{tors}}$ a highest weight category?",
   "public_statement": "Highest weight category\n\nWorking over a field. Is the Serre quotient $\\mathrm{Mod}_{\\mathrm{FI}}/\\mathrm{Mod}_{\\mathrm{FI}}^{\\mathrm{tors}}$ a highest weight category?",
   "evidence": "The exact AIM record is Problem 1.4 in the workshop section “FI-modules and tca's”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0020": {
   "statement_status": "exact",
   "original_statement": "Noetherianity of degree two tca's\n\n\\begin{enumerate}\n\\item[{T. Church}] Is $\\mathrm{Sym}(\\mathrm{Sym}^2)$ noetherian over $\\mathbb{Z}$ $($or over a general noetherian ring$)$?\n\\item[{J. Wilson}] Same question for $\\bigwedge(\\mathrm{Sym}^2)$.\n\\end{enumerate}",
   "clean_statement": "Noetherianity of degree two tca's\n\n\\begin{enumerate}\n\\item[{T. Church}] Is $\\mathrm{Sym}(\\mathrm{Sym}^2)$ noetherian over $\\mathbb{Z}$ $($or over a general noetherian ring$)$?\n\\item[{J. Wilson}] Same question for $\\bigwedge(\\mathrm{Sym}^2)$.\n\\end{enumerate}",
   "public_statement": "Noetherianity of degree two tca's\n\n\\begin{enumerate}\n\\item[{T. Church}] Is $\\mathrm{Sym}(\\mathrm{Sym}^2)$ noetherian over $\\mathbb{Z}$ $($or over a general noetherian ring$)$?\n\\item[{J. Wilson}] Same question for $\\bigwedge(\\mathrm{Sym}^2)$.\n\\end{enumerate}",
   "evidence": "The repository record has no apparent OCR corruption. The original AIM URL returned an HTTP error during this run, so the wording above was not independently re-extracted from the page. No mathematical reconstruction of the stored wording was needed.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0021": {
   "statement_status": "exact",
   "original_statement": "Bounds on syzygies of modules over degree two tca's\n\nLet $A = \\mathrm{Sym}(\\mathrm{Sym}^2)$ or $\\bigwedge(\\mathrm{Sym}^2)$, and let $M$ be an $A$-module generated in degree $d$ and related in degree $r$. Are there any good bounds on the syzygies? In other words, bound generators of the terms of the free resolutions. Can start with characteristic $0$.",
   "clean_statement": "Bounds on syzygies of modules over degree two tca's\n\nLet $A = \\mathrm{Sym}(\\mathrm{Sym}^2)$ or $\\bigwedge(\\mathrm{Sym}^2)$, and let $M$ be an $A$-module generated in degree $d$ and related in degree $r$. Are there any good bounds on the syzygies? In other words, bound generators of the terms of the free resolutions. Can start with characteristic $0$.",
   "public_statement": "Bounds on syzygies of modules over degree two tca's\n\nLet $A = \\mathrm{Sym}(\\mathrm{Sym}^2)$ or $\\bigwedge(\\mathrm{Sym}^2)$, and let $M$ be an $A$-module generated in degree $d$ and related in degree $r$. Are there any good bounds on the syzygies? In other words, bound generators of the terms of the free resolutions. Can start with characteristic $0$.",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0022": {
   "statement_status": "exact",
   "original_statement": "Hilbert series of modules over degree two tca's\n\nWhat can we say about Hilbert series of modules over $A = \\mathrm{Sym}(\\mathrm{Sym}^2)$. More precisely, take $M$ to be $A$ modulo a determinantal ideal. Is it true that its $($non-exponential$)$ Hilbert series is algebraic? Assume characteristic $0$.",
   "clean_statement": "Hilbert series of modules over degree two tca's\n\nWhat can we say about Hilbert series of modules over $A = \\mathrm{Sym}(\\mathrm{Sym}^2)$. More precisely, take $M$ to be $A$ modulo a determinantal ideal. Is it true that its $($non-exponential$)$ Hilbert series is algebraic? Assume characteristic $0$.",
   "public_statement": "Hilbert series of modules over degree two tca's\n\nWhat can we say about Hilbert series of modules over $A = \\mathrm{Sym}(\\mathrm{Sym}^2)$. More precisely, take $M$ to be $A$ modulo a determinantal ideal. Is it true that its $($non-exponential$)$ Hilbert series is algebraic? Assume characteristic $0$.",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0023": {
   "statement_status": "exact",
   "original_statement": "$q$-tca's\n\nBerenstein--Zwicknagl define quantum analog of $\\mathrm{Sym}(\\mathbf{S}_{\\lambda})$ in [BZ]. For small $\\lambda$, this behaves like the classical case, but for large $\\lambda$ (and $q$ generic) its isotypic decomposition has smaller multiplicities than the classical case.\n\nAre modules over this algebra $($as $q$-tca's$)$ equivalent to modules in the classical case $($as tca's$)$?",
   "clean_statement": "$q$-tca's\n\nBerenstein--Zwicknagl define quantum analog of $\\mathrm{Sym}(\\mathbf{S}_{\\lambda})$ in [BZ]. For small $\\lambda$, this behaves like the classical case, but for large $\\lambda$ (and $q$ generic) its isotypic decomposition has smaller multiplicities than the classical case.\n\nAre modules over this algebra $($as $q$-tca's$)$ equivalent to modules in the classical case $($as tca's$)$?",
   "public_statement": "$q$-tca's\n\nBerenstein--Zwicknagl define quantum analog of $\\mathrm{Sym}(\\mathbf{S}_{\\lambda})$ in [BZ]. For small $\\lambda$, this behaves like the classical case, but for large $\\lambda$ (and $q$ generic) its isotypic decomposition has smaller multiplicities than the classical case.\n\nAre modules over this algebra $($as $q$-tca's$)$ equivalent to modules in the classical case $($as tca's$)$?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0024": {
   "statement_status": "exact",
   "original_statement": "Quasi-polynomial behavior\n\nLet $V_n = \\wedge^2 \\mathbb{Q}^{n-1} = M_{n-2,1,1}$. Then $a_n := \\langle \\mathrm{H}^i(\\mathrm{PConf}^n(\\mathbb{C}), V_n\\rangle_{S_n}$ is independent of $n$ eventually. More precisely,\n\n\\begin{align*}\na_n = \\begin{cases}\n2 i -2 &\\mbox{if } i \\equiv 0 \\pmod{4} \\\\\n2 i -3 &\\mbox{if } i \\equiv 1 \\pmod{4} \\\\\n2 i -2 &\\mbox{if } i \\equiv 2 \\pmod{4} \\\\\n2 i -1 &\\mbox{if } i \\equiv 3 \\pmod{4}\n\\end{cases}.\n\\end{align*}\n\nIn fact, for any $\\lambda$, $\\lim_{n \\to \\infty }\\langle \\mathrm{H}^i(\\mathrm{PConf}^n(\\mathbb{C}), V(\\lambda)\\rangle_{S_n}$ is a quasi-polynomial in $i$ if $i \\ge 2$.\n\n\\begin{enumerate}\n\\item Does the same hold if we replace $\\mathbb{C}$ by a manifold $X$ $($for some class of manifolds$)$?\n\\item For a general finitely generated $\\mathrm{FI}$-algebra $A$, it is not true that $\\lim_{n \\to \\infty } \\langle A_n, V(\\lambda)\\rangle_{S_n}$ is a quasi-polynomial.\nIs there some hypothesis on $A$ that makes this true?\n\\end{enumerate}",
   "clean_statement": "Quasi-polynomial behavior\n\nLet $V_n = \\wedge^2 \\mathbb{Q}^{n-1} = M_{n-2,1,1}$. Then $a_n := \\langle \\mathrm{H}^i(\\mathrm{PConf}^n(\\mathbb{C}), V_n\\rangle_{S_n}$ is independent of $n$ eventually. More precisely,\n\n\\begin{align*}\na_n = \\begin{cases}\n2 i -2 &\\mbox{if } i \\equiv 0 \\pmod{4} \\\\\n2 i -3 &\\mbox{if } i \\equiv 1 \\pmod{4} \\\\\n2 i -2 &\\mbox{if } i \\equiv 2 \\pmod{4} \\\\\n2 i -1 &\\mbox{if } i \\equiv 3 \\pmod{4}\n\\end{cases}.\n\\end{align*}\n\nIn fact, for any $\\lambda$, $\\lim_{n \\to \\infty }\\langle \\mathrm{H}^i(\\mathrm{PConf}^n(\\mathbb{C}), V(\\lambda)\\rangle_{S_n}$ is a quasi-polynomial in $i$ if $i \\ge 2$.\n\n\\begin{enumerate}\n\\item Does the same hold if we replace $\\mathbb{C}$ by a manifold $X$ $($for some class of manifolds$)$?\n\\item For a general finitely generated $\\mathrm{FI}$-algebra $A$, it is not true that $\\lim_{n \\to \\infty } \\langle A_n, V(\\lambda)\\rangle_{S_n}$ is a quasi-polynomial.\nIs there some hypothesis on $A$ that makes this true?\n\\end{enumerate}",
   "public_statement": "Quasi-polynomial behavior\n\nLet $V_n = \\wedge^2 \\mathbb{Q}^{n-1} = M_{n-2,1,1}$. Then $a_n := \\langle \\mathrm{H}^i(\\mathrm{PConf}^n(\\mathbb{C}), V_n\\rangle_{S_n}$ is independent of $n$ eventually. More precisely,\n\n\\begin{align*}\na_n = \\begin{cases}\n2 i -2 &\\mbox{if } i \\equiv 0 \\pmod{4} \\\\\n2 i -3 &\\mbox{if } i \\equiv 1 \\pmod{4} \\\\\n2 i -2 &\\mbox{if } i \\equiv 2 \\pmod{4} \\\\\n2 i -1 &\\mbox{if } i \\equiv 3 \\pmod{4}\n\\end{cases}.\n\\end{align*}\n\nIn fact, for any $\\lambda$, $\\lim_{n \\to \\infty }\\langle \\mathrm{H}^i(\\mathrm{PConf}^n(\\mathbb{C}), V(\\lambda)\\rangle_{S_n}$ is a quasi-polynomial in $i$ if $i \\ge 2$.\n\n\\begin{enumerate}\n\\item Does the same hold if we replace $\\mathbb{C}$ by a manifold $X$ $($for some class of manifolds$)$?\n\\item For a general finitely generated $\\mathrm{FI}$-algebra $A$, it is not true that $\\lim_{n \\to \\infty } \\langle A_n, V(\\lambda)\\rangle_{S_n}$ is a quasi-polynomial.\nIs there some hypothesis on $A$ that makes this true?\n\\end{enumerate}",
   "evidence": "The AIM record is Problem 2.1, “Quasi-polynomial behavior,” in the Topology section of the Representation Stability workshop list. Its first example concerns",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0025": {
   "statement_status": "exact",
   "original_statement": "Naturally occurring non-trivial $\\mathrm{FI}_2$ modules\n\nLet $X$ be a manifold with two specified boundary components. Claim: $V_n = \\mathrm{H}_i(\\mathrm{PConf}^n(X))$ has an $\\mathrm{FI}_2$-module structure. In particular, if $X = S^1 \\times I$ is the cylinder and $i=1$. Then $V$ has the following properties:\n\\begin{enumerate}\n\\item There is an exact sequence $0 \\to \\tilde{M}(1) \\to V \\to \\tilde{M}(2) \\to 0$ where $\\tilde{M}(1)$ and $\\tilde{M}(2)$ are the pull-backs of $M(1)$ and $M(2)$ along the natural forgetful map $\\mathrm{FI}_2 \\to \\mathrm{FI}$. (See [CEF] for $M$ notation.)\n\\item $V$ has an $\\mathrm{FI}$-module structure and $V$ is not in the image of the natural map $\\mathrm{Mod}_{\\mathrm{FI}} \\to \\mathrm{Mod}_{\\mathrm{FI}_2}$ induced by the forgetful map $\\mathrm{FI}_2 \\to \\mathrm{FI}$.\n\\end{enumerate}\n\nIn other words, $V$ is filtered by things that are pull-backs from $\\mathrm{FI}$ but it is not itself a pull-back from $\\mathrm{FI}$.\n\n\\begin{enumerate}[\\rm (a)]\n\\item What can we say about $\\mathrm{H}_i(\\mathrm{PConf}(X))$ generally along these lines?\n\\item Can we calculate $\\mathrm{Ext}(M,N)$ for $\\mathrm{FI}_2$-modules?\n\\end{enumerate}",
   "clean_statement": "Naturally occurring non-trivial $\\mathrm{FI}_2$ modules\n\nLet $X$ be a manifold with two specified boundary components. Claim: $V_n = \\mathrm{H}_i(\\mathrm{PConf}^n(X))$ has an $\\mathrm{FI}_2$-module structure. In particular, if $X = S^1 \\times I$ is the cylinder and $i=1$. Then $V$ has the following properties:\n\\begin{enumerate}\n\\item There is an exact sequence $0 \\to \\tilde{M}(1) \\to V \\to \\tilde{M}(2) \\to 0$ where $\\tilde{M}(1)$ and $\\tilde{M}(2)$ are the pull-backs of $M(1)$ and $M(2)$ along the natural forgetful map $\\mathrm{FI}_2 \\to \\mathrm{FI}$. (See [CEF] for $M$ notation.)\n\\item $V$ has an $\\mathrm{FI}$-module structure and $V$ is not in the image of the natural map $\\mathrm{Mod}_{\\mathrm{FI}} \\to \\mathrm{Mod}_{\\mathrm{FI}_2}$ induced by the forgetful map $\\mathrm{FI}_2 \\to \\mathrm{FI}$.\n\\end{enumerate}\n\nIn other words, $V$ is filtered by things that are pull-backs from $\\mathrm{FI}$ but it is not itself a pull-back from $\\mathrm{FI}$.\n\n\\begin{enumerate}[\\rm (a)]\n\\item What can we say about $\\mathrm{H}_i(\\mathrm{PConf}(X))$ generally along these lines?\n\\item Can we calculate $\\mathrm{Ext}(M,N)$ for $\\mathrm{FI}_2$-modules?\n\\end{enumerate}",
   "public_statement": "Naturally occurring non-trivial $\\mathrm{FI}_2$ modules\n\nLet $X$ be a manifold with two specified boundary components. Claim: $V_n = \\mathrm{H}_i(\\mathrm{PConf}^n(X))$ has an $\\mathrm{FI}_2$-module structure. In particular, if $X = S^1 \\times I$ is the cylinder and $i=1$. Then $V$ has the following properties:\n\\begin{enumerate}\n\\item There is an exact sequence $0 \\to \\tilde{M}(1) \\to V \\to \\tilde{M}(2) \\to 0$ where $\\tilde{M}(1)$ and $\\tilde{M}(2)$ are the pull-backs of $M(1)$ and $M(2)$ along the natural forgetful map $\\mathrm{FI}_2 \\to \\mathrm{FI}$. (See [CEF] for $M$ notation.)\n\\item $V$ has an $\\mathrm{FI}$-module structure and $V$ is not in the image of the natural map $\\mathrm{Mod}_{\\mathrm{FI}} \\to \\mathrm{Mod}_{\\mathrm{FI}_2}$ induced by the forgetful map $\\mathrm{FI}_2 \\to \\mathrm{FI}$.\n\\end{enumerate}\n\nIn other words, $V$ is filtered by things that are pull-backs from $\\mathrm{FI}$ but it is not itself a pull-back from $\\mathrm{FI}$.\n\n\\begin{enumerate}[\\rm (a)]\n\\item What can we say about $\\mathrm{H}_i(\\mathrm{PConf}(X))$ generally along these lines?\n\\item Can we calculate $\\mathrm{Ext}(M,N)$ for $\\mathrm{FI}_2$-modules?\n\\end{enumerate}",
   "evidence": "The canonical record is problem 2.2 in the “Topology” section of the AIM Representation Stability list (`aim-representation-theory-notes.json`, zero-based index 24). It asks for naturally occurring nontrivial \\(\\mathrm{FI}_2\\)-modules. For a manifold \\(X\\) with two specified boundary components it claims that",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0026": {
   "statement_status": "exact",
   "original_statement": "Cohomology of $\\mathrm{PConf}^n$\n\nFind the Specht module decomposition for $\\mathrm{H}^i(\\mathrm{PConf}^n \\mathbb{C})$, and its $\\mathrm{FI}$-module generators. It might involve a derangement recurrence. Recurrence is going to involve both $i$ and $n$. $($This question might relate to W. Chen's question.$)$",
   "clean_statement": "Cohomology of $\\mathrm{PConf}^n$\n\nFind the Specht module decomposition for $\\mathrm{H}^i(\\mathrm{PConf}^n \\mathbb{C})$, and its $\\mathrm{FI}$-module generators. It might involve a derangement recurrence. Recurrence is going to involve both $i$ and $n$. $($This question might relate to W. Chen's question.$)$",
   "public_statement": "Cohomology of $\\mathrm{PConf}^n$\n\nFind the Specht module decomposition for $\\mathrm{H}^i(\\mathrm{PConf}^n \\mathbb{C})$, and its $\\mathrm{FI}$-module generators. It might involve a derangement recurrence. Recurrence is going to involve both $i$ and $n$. $($This question might relate to W. Chen's question.$)$",
   "evidence": "The canonical AIM record (Representation stability workshop, Topology, Problem 2.3) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0027": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "$S_{\\infty}$ structure and configuration spaces\n\nCompute $S_{\\infty}$ representations of configuration spaces directly.",
   "clean_statement": "an inference, not a correction of OCR. The source page supplied in the record did not return usable content during this run, and the record has no remarks or literature field from which to recover a more specific convention.",
   "public_statement": "$S_{\\infty}$ structure and configuration spaces\n\nCompute $S_{\\infty}$ representations of configuration spaces directly.",
   "evidence": "The nearby AIM records concern \\(H^i(\\operatorname{PConf}_n(\\mathbb C);\\mathbb Q)\\), its Specht decomposition, and its FI-module structure. I therefore adopt the following explicit reconstruction:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0028": {
   "statement_status": "exact",
   "original_statement": "Highly acyclic complexes\n\nWorking over a field $\\mathbf{k}$, define $\\mathrm{PBC}_{\\bullet}^n$ with\n\\[\n\\mathrm{PBC}_{p}^n = \\{(v_1, \\ldots, v_p, C) \\colon v_i \\mbox{ are } \\mathbf{k}\\mbox{-linearly independent, } C \\mbox{ a complement of span of } v_i \\} .\n\\]\nThen $\\mathrm{PBC}^n_{\\bullet}$ is a semi-simplicial $\\mathbf{VIC}(\\mathbf{k})$-set. It is highly connected ($(n-3)/2$-connected). The corresponding augmented chain complex\n\\[\n\\mathbb{Z}[\\mathrm{PBC}_{\\bullet}] \\xrightarrow{\\epsilon} \\mathbb{Z} \\to 0\n\\]\nis acyclic (exact). The boundary maps are the signed sums of all ways to take a vector and put it in the complement. This all holds if we replace the field $\\mathbf{k}$ by a general ring $R$, and in this case we have $\\mathbb{Z}[\\mathrm{PBC}_{p}] = \\mathbb{Z}\\mathrm{GL}_n(R) \\otimes_{\\mathbb{Z}\\mathrm{GL}_{n - (p+1)}(R)} \\mathbb{Z} $. Generalizing this (and changing the coefficient ring from $\\mathbb{Z}$ to $R$) we get the following question:\n\nIs $R\\mathrm{GL}_n(R) \\otimes_{R\\mathrm{GL}_{n - (\\bullet+1)}(R)} R^{n - (\\bullet+1)} \\to R^n \\to 0$ highly acyclic?",
   "clean_statement": "Highly acyclic complexes\n\nWorking over a field $\\mathbf{k}$, define $\\mathrm{PBC}_{\\bullet}^n$ with\n\\[\n\\mathrm{PBC}_{p}^n = \\{(v_1, \\ldots, v_p, C) \\colon v_i \\mbox{ are } \\mathbf{k}\\mbox{-linearly independent, } C \\mbox{ a complement of span of } v_i \\} .\n\\]\nThen $\\mathrm{PBC}^n_{\\bullet}$ is a semi-simplicial $\\mathbf{VIC}(\\mathbf{k})$-set. It is highly connected ($(n-3)/2$-connected). The corresponding augmented chain complex\n\\[\n\\mathbb{Z}[\\mathrm{PBC}_{\\bullet}] \\xrightarrow{\\epsilon} \\mathbb{Z} \\to 0\n\\]\nis acyclic (exact). The boundary maps are the signed sums of all ways to take a vector and put it in the complement. This all holds if we replace the field $\\mathbf{k}$ by a general ring $R$, and in this case we have $\\mathbb{Z}[\\mathrm{PBC}_{p}] = \\mathbb{Z}\\mathrm{GL}_n(R) \\otimes_{\\mathbb{Z}\\mathrm{GL}_{n - (p+1)}(R)} \\mathbb{Z} $. Generalizing this (and changing the coefficient ring from $\\mathbb{Z}$ to $R$) we get the following question:\n\nIs $R\\mathrm{GL}_n(R) \\otimes_{R\\mathrm{GL}_{n - (\\bullet+1)}(R)} R^{n - (\\bullet+1)} \\to R^n \\to 0$ highly acyclic?",
   "public_statement": "Highly acyclic complexes\n\nWorking over a field $\\mathbf{k}$, define $\\mathrm{PBC}_{\\bullet}^n$ with\n\\[\n\\mathrm{PBC}_{p}^n = \\{(v_1, \\ldots, v_p, C) \\colon v_i \\mbox{ are } \\mathbf{k}\\mbox{-linearly independent, } C \\mbox{ a complement of span of } v_i \\} .\n\\]\nThen $\\mathrm{PBC}^n_{\\bullet}$ is a semi-simplicial $\\mathbf{VIC}(\\mathbf{k})$-set. It is highly connected ($(n-3)/2$-connected). The corresponding augmented chain complex\n\\[\n\\mathbb{Z}[\\mathrm{PBC}_{\\bullet}] \\xrightarrow{\\epsilon} \\mathbb{Z} \\to 0\n\\]\nis acyclic (exact). The boundary maps are the signed sums of all ways to take a vector and put it in the complement. This all holds if we replace the field $\\mathbf{k}$ by a general ring $R$, and in this case we have $\\mathbb{Z}[\\mathrm{PBC}_{p}] = \\mathbb{Z}\\mathrm{GL}_n(R) \\otimes_{\\mathbb{Z}\\mathrm{GL}_{n - (p+1)}(R)} \\mathbb{Z} $. Generalizing this (and changing the coefficient ring from $\\mathbb{Z}$ to $R$) we get the following question:\n\nIs $R\\mathrm{GL}_n(R) \\otimes_{R\\mathrm{GL}_{n - (\\bullet+1)}(R)} R^{n - (\\bullet+1)} \\to R^n \\to 0$ highly acyclic?",
   "evidence": "The canonical record is problem 2.5, “Highly acyclic complexes,” in the Topology section of the AIM Representation Stability list (`aim-representation-theory-notes.json`, zero-based index 27). It starts over a field \\(\\mathbf k\\) with",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0029": {
   "statement_status": "exact",
   "original_statement": "Schur weyl duals\n\nAre there direct constructions for the Schur--Weyl dual of $\\mathrm{FI}$-modules like $\\mathrm{H}^{\\bullet}(\\mathrm{PConf}^n{X}, \\mathbb{Q})$? E.g., can one construct a $\\mathbf{GL}_{\\infty}$-space and a $\\mathbf{GL}_{\\infty}$-equivariant sheaf whose cohomology is the Schur--Weyl dual?",
   "clean_statement": "Schur weyl duals\n\nAre there direct constructions for the Schur--Weyl dual of $\\mathrm{FI}$-modules like $\\mathrm{H}^{\\bullet}(\\mathrm{PConf}^n{X}, \\mathbb{Q})$? E.g., can one construct a $\\mathbf{GL}_{\\infty}$-space and a $\\mathbf{GL}_{\\infty}$-equivariant sheaf whose cohomology is the Schur--Weyl dual?",
   "public_statement": "Schur weyl duals\n\nAre there direct constructions for the Schur--Weyl dual of $\\mathrm{FI}$-modules like $\\mathrm{H}^{\\bullet}(\\mathrm{PConf}^n{X}, \\mathbb{Q})$? E.g., can one construct a $\\mathbf{GL}_{\\infty}$-space and a $\\mathbf{GL}_{\\infty}$-equivariant sheaf whose cohomology is the Schur--Weyl dual?",
   "evidence": "The source is the AIM workshop list *Representation stability*, section \"Topology,\" problem 2.6. Its extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0030": {
   "statement_status": "exact",
   "original_statement": "Coherence\n\nBy [Theorem B, Ra], the category of $\\mathrm{FI}$-modules presented in finite degree is abelian.\n\nIs this true for other categories?",
   "clean_statement": "Coherence\n\nBy [Theorem B, Ra], the category of $\\mathrm{FI}$-modules presented in finite degree is abelian.\n\nIs this true for other categories?",
   "public_statement": "Coherence\n\nBy [Theorem B, Ra], the category of $\\mathrm{FI}$-modules presented in finite degree is abelian.\n\nIs this true for other categories?",
   "evidence": "There is no OCR corruption apparent in this record. The question is intentionally broad: “other categories” does not specify a class of categories or a coefficient ring. I therefore separate two precise tasks:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0031": {
   "statement_status": "exact",
   "original_statement": "Enrichments of $\\mathrm{VIC}(\\mathbb{F})$\n\nLook at enrichments of $\\mathrm{VIC}(\\mathbb{F})$ $(\\mathbb{F}$ a field$)$ over schemes $($i.e., use algebraic representations of $\\mathbf{GL})$. What can you say about their representations?",
   "clean_statement": "Enrichments of $\\mathrm{VIC}(\\mathbb{F})$\n\nLook at enrichments of $\\mathrm{VIC}(\\mathbb{F})$ $(\\mathbb{F}$ a field$)$ over schemes $($i.e., use algebraic representations of $\\mathbf{GL})$. What can you say about their representations?",
   "public_statement": "Enrichments of $\\mathrm{VIC}(\\mathbb{F})$\n\nLook at enrichments of $\\mathrm{VIC}(\\mathbb{F})$ $(\\mathbb{F}$ a field$)$ over schemes $($i.e., use algebraic representations of $\\mathbf{GL})$. What can you say about their representations?",
   "evidence": "The exact extracted AIM problem, from the workshop *Representation stability*, section \"Representation theory,\" problem 3.2, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0032": {
   "statement_status": "exact",
   "original_statement": "$\\mathrm{VI}$ like categories\n\n$\\mathrm{VI}$ is obtained from $\\mathrm{FI}$ by replacing finite sets with finite vector spaces. In combinatorics, often better to replace finite sets by finite projective spaces.\n\nDoes this lead to an interesting analogue of $\\mathrm{VI}$-modules?",
   "clean_statement": "$\\mathrm{VI}$ like categories\n\n$\\mathrm{VI}$ is obtained from $\\mathrm{FI}$ by replacing finite sets with finite vector spaces. In combinatorics, often better to replace finite sets by finite projective spaces.\n\nDoes this lead to an interesting analogue of $\\mathrm{VI}$-modules?",
   "public_statement": "$\\mathrm{VI}$ like categories\n\n$\\mathrm{VI}$ is obtained from $\\mathrm{FI}$ by replacing finite sets with finite vector spaces. In combinatorics, often better to replace finite sets by finite projective spaces.\n\nDoes this lead to an interesting analogue of $\\mathrm{VI}$-modules?",
   "evidence": "The source record is `aim-representation-theory-notes.json`, record 31 (zero-based):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0033": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "$\\mathrm{FI}$ like categories\n\nModify $\\mathrm{FI}$ to use double cover of symmetric groups.",
   "clean_statement": null,
   "public_statement": "$\\mathrm{FI}$ like categories\n\nModify $\\mathrm{FI}$ to use double cover of symmetric groups.",
   "evidence": "The original AIM URL was unavailable during this run, and the record gives no definitions or literature. We therefore distinguish three plausible readings.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0034": {
   "statement_status": "corrected_verified",
   "original_statement": "Whitehouse modules\n\nV.~Reiner described Whitehouse modules as virtual representations. Is it clear why the negative part is a submodule of the positive part, i.e., can we fine explicit embedding? Is there a way to think about this in terms of $\\mathrm{FI}$-modules?",
   "clean_statement": "V. Reiner described Whitehouse modules as virtual representations. Is it clear why the negative part is a submodule of the positive part, i.e., can we fine explicit embedding? Is there a way to think about this in terms of \\(\\mathrm{FI}\\)-modules?",
   "public_statement": "V. Reiner described Whitehouse modules as virtual representations. Is it clear why the negative part is a submodule of the positive part, i.e., can we fine explicit embedding? Is there a way to think about this in terms of \\(\\mathrm{FI}\\)-modules?",
   "evidence": "The word “fine” occurs on the live source page and is therefore not an OCR error. It is an evident typographical error; below it is emended to **find**. No mathematical symbol needs reconstruction.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0035": {
   "statement_status": "exact",
   "original_statement": "Stable decomposition into irreducibles\n\nHard problem in modular $S_n$ representation theory: determining simple factors of Specht modules. Are stable values (for example if the representations are coming from a finitely generated $\\mathrm{FI}$-module) easier?",
   "clean_statement": "Stable decomposition into irreducibles\n\nHard problem in modular $S_n$ representation theory: determining simple factors of Specht modules. Are stable values (for example if the representations are coming from a finitely generated $\\mathrm{FI}$-module) easier?",
   "public_statement": "Stable decomposition into irreducibles\n\nHard problem in modular $S_n$ representation theory: determining simple factors of Specht modules. Are stable values (for example if the representations are coming from a finitely generated $\\mathrm{FI}$-module) easier?",
   "evidence": "The repository record agrees with the AIM source and has no visible OCR corruption. The phrase “stable values” is intentionally informal. In fixed characteristic \\(p\\), at least four distinct interpretations must be separated:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0036": {
   "statement_status": "exact",
   "original_statement": "What does the induction theorem of Achar--Riche/Hodge--Karuppuchamy--Scott imply about the relationship between representations of $G(\\F_q)$ and $B(\\F_q)$ in defining characteristic?",
   "clean_statement": "What does the induction theorem of Achar--Riche/Hodge--Karuppuchamy--Scott imply about the relationship between representations of $G(\\F_q)$ and $B(\\F_q)$ in defining characteristic?",
   "public_statement": "What does the induction theorem of Achar--Riche/Hodge--Karuppuchamy--Scott imply about the relationship between representations of $G(\\F_q)$ and $B(\\F_q)$ in defining characteristic?",
   "evidence": "“Hodge” is not an extraction error. It is Terrell L. Hodge, coauthor with Paramasamy Karuppuchamy and Leonard L. Scott of *Remarks on the ABG Induction Theorem*. The slash in “Achar--Riche/Hodge--Karuppuchamy--Scott” is best read as referring to two proofs/formulations of the modular algebraic-group induction theorem. Achar--Riche explicitly cite the Hodge--Karuppuchamy--Scott proof in the Borel case.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0037": {
   "statement_status": "exact",
   "original_statement": "Let $P$ be a parabolic subgroup of $G$. Is it true in characteristic $p$ that $R^i\\operatorname{Ind}_P^G\\operatorname{Sym}(\\mathfrak{n}_P^\\bullet) = 0$ for all $i > 0$? (Known for $P = B$ and for certain other parabolics. Possible connection with normality of nilpotent orbit closures.)",
   "clean_statement": "Let $P$ be a parabolic subgroup of $G$. Is it true in characteristic $p$ that $R^i\\operatorname{Ind}_P^G\\operatorname{Sym}(\\mathfrak{n}_P^\\bullet) = 0$ for all $i > 0$? (Known for $P = B$ and for certain other parabolics. Possible connection with normality of nilpotent orbit closures.)",
   "public_statement": "Let $P$ be a parabolic subgroup of $G$. Is it true in characteristic $p$ that $R^i\\operatorname{Ind}_P^G\\operatorname{Sym}(\\mathfrak{n}_P^\\bullet) = 0$ for all $i > 0$? (Known for $P = B$ and for certain other parabolics. Possible connection with normality of nilpotent orbit closures.)",
   "evidence": "The exact repository record, from the AIM workshop *Sheaves and modular representations of reductive groups*, section \"Rational Representations: Induction, Cohomology Vanishing,\" problem 1.5, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0038": {
   "statement_status": "exact",
   "original_statement": "What representations of $G(\\F_q)$ arise by restricting a tilting module of $G$?",
   "clean_statement": "What representations of $G(\\F_q)$ arise by restricting a tilting module of $G$?",
   "public_statement": "What representations of $G(\\F_q)$ arise by restricting a tilting module of $G$?",
   "evidence": "The canonical record is AIM workshop problem 1.2 from *Sheaves and modular representations of reductive groups*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0039": {
   "statement_status": "exact",
   "original_statement": "What rational representations of $B$ arise by restricting a tilting module of $G$?",
   "clean_statement": "What rational representations of $B$ arise by restricting a tilting module of $G$?",
   "public_statement": "What rational representations of $B$ arise by restricting a tilting module of $G$?",
   "evidence": "The AIM problem record is from the 2016 workshop *Sheaves and modular representations of reductive groups*, section “Rational Representations: Induction, Cohomology Vanishing,” Problem 1.3. Its complete question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0040": {
   "statement_status": "exact",
   "original_statement": "What objects of the derived category of representations of $B$ correspond to tilting modules of $G$?",
   "clean_statement": "What objects of the derived category of representations of $B$ correspond to tilting modules of $G$?",
   "public_statement": "What objects of the derived category of representations of $B$ correspond to tilting modules of $G$?",
   "evidence": "The AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0041": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Is $\\mathrm{Rep}_0(G_1T)$ equivalent to a category of perverse sheaves on a space of quasi-maps from $\\mathbb{P}^1$ to $G/B$?",
   "clean_statement": null,
   "public_statement": "Is $\\mathrm{Rep}_0(G_1T)$ equivalent to a category of perverse sheaves on a space of quasi-maps from $\\mathbb{P}^1$ to $G/B$?",
   "evidence": "The canonical record asks:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0042": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "What geometric category is equivalent to $\\mathrm{Rep}_0(G_rT)$ or $\\mathrm{Rep}_{\\mathrm{res}}(\\mathfrak{g})$?",
   "clean_statement": "What geometric categgeometric category.y is equivalent to $\\mathrm{Rep}_0(G_rT)$ geometric category. $\\mathrm{Rep}_{\\mathrm{res}}(\\mathfrak{g})$?",
   "public_statement": "What geometric category is equivalent to $\\mathrm{Rep}_0(G_rT)$ or $\\mathrm{Rep}_{\\mathrm{res}}(\\mathfrak{g})$?",
   "evidence": "The canonical record (AIM problem-list item 2.2 from the workshop *Sheaves and modular representations of reductive groups*) asks:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0043": {
   "statement_status": "exact",
   "original_statement": "Is there a homological interpretation of the $p$-Kazhdan--Lusztig polynomials?",
   "clean_statement": "Is there a homological interpretation of the $p$-Kazhdan--Lusztig polynomials?",
   "public_statement": "Is there a homological interpretation of the $p$-Kazhdan--Lusztig polynomials?",
   "evidence": "The canonical AIM record is Problem 4.1 from the workshop *Sheaves and modular representations of reductive groups*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0044": {
   "statement_status": "exact",
   "original_statement": "\\begin{enumerate}\n \\item For which primes $p$ is it true that all indecomposable $B$-constructible parity $\\F_p$ sheaves on $G/B$ are perverse?\n \\item Same question for $I$-constructible parity $\\F_p$-sheaves on $Gr$.\n\\end{enumerate}\n\n(Expect this to be true for large $p$.)",
   "clean_statement": "\\begin{enumerate}\n \\item For which primes $p$ is it true that all indecomposable $B$-constructible parity $\\F_p$ sheaves on $G/B$ are perverse?\n \\item Same question for $I$-constructible parity $\\F_p$-sheaves on $Gr$.\n\\end{enumerate}\n\n(Expect this to be true for large $p$.)",
   "public_statement": "\\begin{enumerate}\n \\item For which primes $p$ is it true that all indecomposable $B$-constructible parity $\\F_p$ sheaves on $G/B$ are perverse?\n \\item Same question for $I$-constructible parity $\\F_p$-sheaves on $Gr$.\n\\end{enumerate}\n\n(Expect this to be true for large $p$.)",
   "evidence": "The source record is Problem 5.1 from the AIM workshop *Sheaves and modular representations of reductive groups*, section “Parity Sheaves and Torsion in IC Sheaves”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0045": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "For which primes $p$ do the IC sheaves on $\\operatorname{Perv}_I(Gr, \\Z_p)$ have torsion-free stalks?\n\n(Related to Problem 10.1 part 2, and to Ext-vanishing between reduced standard and costandard modules.)",
   "clean_statement": "Fix the affine Grassmannian \\(Gr\\) and its stratification by orbits of an\nIwahori subgroup \\(I\\). For which primes \\(p\\) do all integral intersection\ncohomology objects\n\\[\n\\mathrm{IC}_y(\\mathcal O)\\in\\operatorname{Perv}_I(Gr,\\mathcal O),\n\\]\none for each \\(I\\)-orbit closure \\(\\overline{Gr_y}\\), have torsion-free\nstalk cohomology?",
   "public_statement": "For which primes $p$ do the IC sheaves on $\\operatorname{Perv}_I(Gr, \\Z_p)$ have torsion-free stalks?\n\n(Related to Problem 10.1 part 2, and to Ext-vanishing between reduced standard and costandard modules.)",
   "evidence": "Literally, a sheaf is not “on” a category. The neighboring problems separately discuss \\(I\\)-constructible parity sheaves on \\(Gr\\), so the conservative reconstruction is:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0046": {
   "statement_status": "exact",
   "original_statement": "Give nontrivial sufficient conditions for indecomposable parity sheaves on $G/B$ or $Gr$ or $Fl$ to be simple perverse sheaves.",
   "clean_statement": "Give nontrivial sufficient conditions for indecomposable parity sheaves on $G/B$ or $Gr$ or $Fl$ to be simple perverse sheaves.",
   "public_statement": "Give nontrivial sufficient conditions for indecomposable parity sheaves on $G/B$ or $Gr$ or $Fl$ to be simple perverse sheaves.",
   "evidence": "The source record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0047": {
   "statement_status": "exact",
   "original_statement": "Is there a faster algorithm to compute indecomposable parity sheaves on $\\operatorname{Perv}_{\\mathrm{sph}}(Gr, \\F_p)$ than by computing the $p$-canonical basis?\n\n(Motivated by A. Broer's algorithm in characteristic 0.)",
   "clean_statement": "Is there a faster algorithm to compute indecomposable parity sheaves on $\\operatorname{Perv}_{\\mathrm{sph}}(Gr, \\F_p)$ than by computing the $p$-canonical basis?\n\n(Motivated by A. Broer's algorithm in characteristic 0.)",
   "public_statement": "Is there a faster algorithm to compute indecomposable parity sheaves on $\\operatorname{Perv}_{\\mathrm{sph}}(Gr, \\F_p)$ than by computing the $p$-canonical basis?\n\n(Motivated by A. Broer's algorithm in characteristic 0.)",
   "evidence": "The exact canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0048": {
   "statement_status": "exact",
   "original_statement": "Is there a geometric description of the center of $\\mathrm{Rep}_0G$? Possibly related to affine Springer fibers?",
   "clean_statement": "Is there a geometric description of the center of $\\mathrm{Rep}_0G$? Possibly related to affine Springer fibers?",
   "public_statement": "Is there a geometric description of the center of $\\mathrm{Rep}_0G$? Possibly related to affine Springer fibers?",
   "evidence": "There is no visible OCR corruption in this record. Its real ambiguity is mathematical: which “center” is intended? We take",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0049": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Does this help to describe the center of $\\operatorname{Dist}(G)$?",
   "clean_statement": null,
   "public_statement": "Does this help to describe the center of $\\operatorname{Dist}(G)$?",
   "evidence": "Thus the conservative reconstruction of 6.2 is:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0050": {
   "statement_status": "exact",
   "original_statement": "Under geometric Satake, what operation on perverse sheaves corresponds to Frobenius twist?",
   "clean_statement": "Under geometric Satake, what operation on perverse sheaves corresponds to Frobenius twist?",
   "public_statement": "Under geometric Satake, what operation on perverse sheaves corresponds to Frobenius twist?",
   "evidence": "The canonical AIM record is problem 7.1 in the workshop list *Sheaves and modular representations of reductive groups*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0051": {
   "statement_status": "exact",
   "original_statement": "\\begin{enumerate}\n \\item Is there a version of the derived equivalences of Arkhipov--Bezrukavnikov--Ginzburg that applies when $\\ell$ (the order of the root of unity) is less than or equal to $h$ (the Coxeter number)?\n \\item Same question for singular blocks rather than the principal block.\n\\end{enumerate}",
   "clean_statement": "\\begin{enumerate}\n \\item Is there a version of the derived equivalences of Arkhipov--Bezrukavnikov--Ginzburg that applies when $\\ell$ (the order of the root of unity) is less than or equal to $h$ (the Coxeter number)?\n \\item Same question for singular blocks rather than the principal block.\n\\end{enumerate}",
   "public_statement": "\\begin{enumerate}\n \\item Is there a version of the derived equivalences of Arkhipov--Bezrukavnikov--Ginzburg that applies when $\\ell$ (the order of the root of unity) is less than or equal to $h$ (the Coxeter number)?\n \\item Same question for singular blocks rather than the principal block.\n\\end{enumerate}",
   "evidence": "There is no apparent OCR corruption in this record. The important ambiguity is mathematical rather than textual: “the ABG equivalences” can mean the big Lusztig quantum group/coherent-sheaf equivalence, its constructible counterpart, or later small-quantum-group descendants. These must not be conflated.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0052": {
   "statement_status": "exact",
   "original_statement": "Is there a geometric proof (i.e.~avoiding Geometric Satake) that $\\operatorname{Perv}_{\\mathrm{sph}}(Gr, k)$ is highest-weight?",
   "clean_statement": "Is there a geometric proof (i.e.~avoiding Geometric Satake) that $\\operatorname{Perv}_{\\mathrm{sph}}(Gr, k)$ is highest-weight?",
   "public_statement": "Is there a geometric proof (i.e.~avoiding Geometric Satake) that $\\operatorname{Perv}_{\\mathrm{sph}}(Gr, k)$ is highest-weight?",
   "evidence": "The exact canonical AIM record, Problem 9.1 from *Sheaves and modular representations of reductive groups*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0053": {
   "statement_status": "exact",
   "original_statement": "Given a symplectic resolution $\\pi\\colon Y \\to X$, with $G = \\operatorname{Aut}(X, \\omega)$, consider the subcategory of $\\operatorname{Perv}_G(X, k)$ generated by subquotients of $\\pi_*\\underline{k}_Y$. When is this highest-weight?",
   "clean_statement": "Given a symplectic resolution $\\pi\\colon Y \\to X$, with $G = \\operatorname{Aut}(X, \\omega)$, consider the subcategory of $\\operatorname{Perv}_G(X, k)$ generated by subquotients of $\\pi_*\\underline{k}_Y$. When is this highest-weight?",
   "public_statement": "Given a symplectic resolution $\\pi\\colon Y \\to X$, with $G = \\operatorname{Aut}(X, \\omega)$, consider the subcategory of $\\operatorname{Perv}_G(X, k)$ generated by subquotients of $\\pi_*\\underline{k}_Y$. When is this highest-weight?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0054": {
   "statement_status": "exact",
   "original_statement": "More generally, when is a category of perverse sheaves constructible with respect to a fixed stratification highest-weight?",
   "clean_statement": "More generally, when is a category of perverse sheaves constructible with respect to a fixed stratification highest-weight?",
   "public_statement": "More generally, when is a category of perverse sheaves constructible with respect to a fixed stratification highest-weight?",
   "evidence": "There is no apparent OCR error. The original AIM page was unavailable during this run, so the intended scope was reconstructed from the canonical record and its two neighboring questions in the same section. Problem 9.1 asks whether the spherical perverse category on the affine Grassmannian is highest-weight without using geometric Satake; problem 9.3 asks the same for a subcategory associated with a symplectic resolution. This strongly suggests middle-perversity, coefficients in a field \\(k\\) (including modular coefficients), and the closure order on strata or supports. That reconstruction is an inference, not text recovered from the unavailable web page.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0055": {
   "statement_status": "exact",
   "original_statement": "Given an algebraic group $G$ acting on a variety $X$ with finitely many orbits and a $G$-equivariant morphism $\\pi\\colon \\widetilde{X} \\to X$ where $\\widetilde{X}$ is nonsingular (not necessarily connected), consider $A = \\operatorname{End}^\\bullet_{D^\\mathrm{b}_G(X)}(\\pi_* \\underline{k}_{\\widetilde{X}})$, an algebra over $H^G(pt, k)$. What geometric condition on $\\pi$ is equivalent to $A$ being quasi-hereditary over $H^G(pt, k)$?",
   "clean_statement": "Given an algebraic group $G$ acting on a variety $X$ with finitely many orbits and a $G$-equivariant morphism $\\pi\\colon \\widetilde{X} \\to X$ where $\\widetilde{X}$ is nonsingular (not necessarily connected), consider $A = \\operatorname{End}^\\bullet_{D^\\mathrm{b}_G(X)}(\\pi_* \\underline{k}_{\\widetilde{X}})$, an algebra over $H^G(pt, k)$. What geometric condition on $\\pi$ is equivalent to $A$ being quasi-hereditary over $H^G(pt, k)$?",
   "public_statement": "Given an algebraic group $G$ acting on a variety $X$ with finitely many orbits and a $G$-equivariant morphism $\\pi\\colon \\widetilde{X} \\to X$ where $\\widetilde{X}$ is nonsingular (not necessarily connected), consider $A = \\operatorname{End}^\\bullet_{D^\\mathrm{b}_G(X)}(\\pi_* \\underline{k}_{\\widetilde{X}})$, an algebra over $H^G(pt, k)$. What geometric condition on $\\pi$ is equivalent to $A$ being quasi-hereditary over $H^G(pt, k)$?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0056": {
   "statement_status": "exact",
   "original_statement": "Let $X$ be the moduli stack of torsion coherent sheaves on $\\mathbb{P}^1$ over $\\C$. Is it true that $H^{\\mathrm{odd}}(X, k) = 0$ for all fields $k$?\n\n(Motivated by KLR algebras for affine $\\mathfrak{sl}_2$.)",
   "clean_statement": "Let $X$ be the moduli stack of torsion coherent sheaves on $\\mathbb{P}^1$ over $\\C$. Is it true that $H^{\\mathrm{odd}}(X, k) = 0$ for all fields $k$?\n\n(Motivated by KLR algebras for affine $\\mathfrak{sl}_2$.)",
   "public_statement": "Let $X$ be the moduli stack of torsion coherent sheaves on $\\mathbb{P}^1$ over $\\C$. Is it true that $H^{\\mathrm{odd}}(X, k) = 0$ for all fields $k$?\n\n(Motivated by KLR algebras for affine $\\mathfrak{sl}_2$.)",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0057": {
   "statement_status": "exact",
   "original_statement": "Can one develop a theory of modular character sheaves on $G$ that relates to the representation theory of $G(\\F_p)$ in non-defining characteristic?",
   "clean_statement": "Can one develop a theory of modular character sheaves on $G$ that relates to the representation theory of $G(\\F_p)$ in non-defining characteristic?",
   "public_statement": "Can one develop a theory of modular character sheaves on $G$ that relates to the representation theory of $G(\\F_p)$ in non-defining characteristic?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0058": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "\\begin{enumerate}\n \\item What is the representation-theoretic meaning of the Steenrod algebra action on mod-p cohomology? For example, on the cohomology of $Gr$?\n \\item Does the Steenrod algebra act on the cohomology of any parity sheaf? (Note: examples of Goresky show that this fails for mod-p IC sheaves.)\n\\end{enumerate}",
   "clean_statement": null,
   "public_statement": "\\begin{enumerate}\n \\item What is the representation-theoretic meaning of the Steenrod algebra action on mod-p cohomology? For example, on the cohomology of $Gr$?\n \\item Does the Steenrod algebra act on the cohomology of any parity sheaf? (Note: examples of Goresky show that this fails for mod-p IC sheaves.)\n\\end{enumerate}",
   "evidence": "There is no visible OCR corruption. The source page was not retrievable during this run, so four pieces of scope must be reconstructed.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0059": {
   "statement_status": "exact",
   "original_statement": "(1) What does one need to know about \"basic functions\" to put them in the trace formula? What are asymptotics of the \"basic functions\"? (2) Define and explain the Fourier transform for Vinberg monoids. Find their relations to the Langlands-Shahidi method and the Rankin-Selberg method. \n\nEndoscopy and Beyond:",
   "clean_statement": "(1) What does one need to know about \"basic functions\" to put them in the trace formula? What are asymptotics of the \"basic functions\"? (2) Define and explain the Fourier transform for Vinberg monoids. Find their relations to the Langlands-Shahidi method and the Rankin-Selberg method.\n\nEndoscopy and Beyond:",
   "public_statement": "(1) What does one need to know about \"basic functions\" to put them in the trace formula? What are asymptotics of the \"basic functions\"? (2) Define and explain the Fourier transform for Vinberg monoids. Find their relations to the Langlands-Shahidi method and the Rankin-Selberg method.\n\nEndoscopy and Beyond:",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0060": {
   "statement_status": "exact",
   "original_statement": "(1) Can beyond endoscopy be used to prove that the R(θ)-bounds in the work of Jiang-Liu are sharp?",
   "clean_statement": "(1) Can beyond endoscopy be used to prove that the R(θ)-bounds in the work of Jiang-Liu are sharp?",
   "public_statement": "(1) Can beyond endoscopy be used to prove that the R(θ)-bounds in the work of Jiang-Liu are sharp?",
   "evidence": "The canonical AIM record is the following one-line question from the December 2015 workshop *Automorphic kernel functions*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0061": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "(2) Relative endoscopy: Develop the theory of relative endoscopy in conjunc-tion with the relative trace formula. (3) Referring to MR3117742: there are transfer factors from an elliptic torus in GL 2 to GL 2. What about transfer factors from GL 2 to an elliptic torus? (4) What are the relations between the intertwining operators and character relations? See Labesse-Langlands. \n\nBeyond Endoscopy:",
   "clean_statement": "**(2)** Develop relative endoscopy in conjunction with the relative trace\nformula.  **(3)** Referring to MR3117742, understand transfer in the\ndirection from \\(\\mathrm{GL}_2\\) to an elliptic torus, as opposed to the\ntorus-to-\\(\\mathrm{GL}_2\\) direction.  **(4)** Explain the relation between\nintertwining operators and character relations, with Labesse--Langlands as\nthe rank-one model.",
   "public_statement": "(2) Relative endoscopy: Develop the theory of relative endoscopy in conjunc-tion with the relative trace formula. (3) Referring to MR3117742: there are transfer factors from an elliptic torus in GL 2 to GL 2. What about transfer factors from GL 2 to an elliptic torus? (4) What are the relations between the intertwining operators and character relations? See Labesse-Langlands.\n\nBeyond Endoscopy:",
   "evidence": "The canonical record is a composite of three questions from the AIM workshop *Automorphic Kernel Functions*. Its literal text is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0062": {
   "statement_status": "exact",
   "original_statement": "(1) Isolate contributions of the continuous \"special\" representations on the geo-metric side of the trace formula. Here, special refers to larger poles of rela-tive L-functions. Some special cases may be related to the Rankin-Selberg integral method, for instance, the doubling integral method of Piatetski-Shapiro and Rallis on GL n or GL 2.(2) Take the current literature on beyond endoscopy, and make a list of the precise local and global statements required. For example: local matching and global matching. (3) Arthur has introduced the so-called r-trace formula. Describe (and name!) the r-\"beyond endoscopy\" groups and discuss the meaning of geometric matching. (4) How does the functional equation give an expression for the r-trace formula? In other words, how does the r-Fourier transform play roles here? (5) Isolate the contribution of hyperkloosterman sums to the Kuznetsov trace formula and explain it spectrally.",
   "clean_statement": "(1) Isolate contributions of the continuous \"special\" representations on the geo-metric side of the trace formula. Here, special refers to larger poles of rela-tive L-functions. Some special cases may be related to the Rankin-Selberg integral method, for instance, the doubling integral method of Piatetski-Shapiro and Rallis on GL n or GL 2.(2) Take the current literature on beyond endoscopy, and make a list of the precise local and global statements required. For example: local matching and global matching. (3) Arthur has introduced the so-called r-trace formula. Describe (and name!) the r-\"beyond endoscopy\" groups and discuss the meaning of geometric matching. (4) How does the functional equation give an expression for the r-trace formula? In other words, how does the r-Fourier transform play roles here? (5) Isolate the contribution of hyperkloosterman sums to the Kuznetsov trace formula and explain it spectrally.",
   "public_statement": "(1) Isolate contributions of the continuous \"special\" representations on the geo-metric side of the trace formula. Here, special refers to larger poles of rela-tive L-functions. Some special cases may be related to the Rankin-Selberg integral method, for instance, the doubling integral method of Piatetski-Shapiro and Rallis on GL n or GL 2.(2) Take the current literature on beyond endoscopy, and make a list of the precise local and global statements required. For example: local matching and global matching. (3) Arthur has introduced the so-called r-trace formula. Describe (and name!) the r-\"beyond endoscopy\" groups and discuss the meaning of geometric matching. (4) How does the functional equation give an expression for the r-trace formula? In other words, how does the r-Fourier transform play roles here? (5) Isolate the contribution of hyperkloosterman sums to the Kuznetsov trace formula and explain it spectrally.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0063": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Iwahori-Matsumoto presentation\n\nWhat is the ``Iwahori-Matsumoto Presentation'' of Iwahori-Hecke algebra for $\\widetilde{G}_F$ for split $G_F$?",
   "clean_statement": "For a split connected reductive group $G/F$ and a finite Brylinski--Deligne central cover $\\widetilde G$, describe the Hecke algebra attached to a split Iwahori and a fixed genuine central character, first in the tame/unramified case.",
   "public_statement": "Iwahori-Matsumoto presentation\n\nWhat is the ``Iwahori-Matsumoto Presentation'' of Iwahori-Hecke algebra for $\\widetilde{G}_F$ for split $G_F$?",
   "evidence": "The record itself does **not** specify the degree or construction of the cover, the residual characteristic, a splitting of an Iwahori subgroup, a genuine central character, or a type. Consequently there is no single Hecke algebra determined by the literal question. The nearby workshop problems use $\\widetilde G_F$ for nonlinear central covers of reductive groups; Problem 3.1 later explicitly imposes degree prime to the residual characteristic. The official workshop summary also identifies Brylinski--Deligne central extensions as the natural framework. Thus the most conservative reconstruction is:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0064": {
   "statement_status": "exact",
   "original_statement": "Construct supercuspidal representations of $\\widetilde{G}_{F}$.",
   "clean_statement": "Construct supercuspidal representations of $\\widetilde{G}_{F}$.",
   "public_statement": "Construct supercuspidal representations of $\\widetilde{G}_{F}$.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0065": {
   "statement_status": "exact",
   "original_statement": "What is the category of depth-zero unipotent representations for $\\widetilde{G}_{F}$ for a $p$-adic field $F$?",
   "clean_statement": "What is the category of depth-zero unipotent representations for $\\widetilde{G}_{F}$ for a $p$-adic field $F$?",
   "public_statement": "What is the category of depth-zero unipotent representations for $\\widetilde{G}_{F}$ for a $p$-adic field $F$?",
   "evidence": "The canonical record is preserved verbatim:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0066": {
   "statement_status": "exact",
   "original_statement": "Steinberg representation\n\nDefine Steinberg representation of $\\widetilde{G}_{F}$. What is the formal degree of Steinberg representation of $\\widetilde{G}_{F}$ when $G$ is split over $F$?\n\nWhat is analogue of the trivial representation?",
   "clean_statement": "Steinberg representation\n\nDefine Steinberg representation of $\\widetilde{G}_{F}$. What is the formal degree of Steinberg representation of $\\widetilde{G}_{F}$ when $G$ is split over $F$?\n\nWhat is analogue of the trivial representation?",
   "public_statement": "Steinberg representation\n\nDefine Steinberg representation of $\\widetilde{G}_{F}$. What is the formal degree of Steinberg representation of $\\widetilde{G}_{F}$ when $G$ is split over $F$?\n\nWhat is analogue of the trivial representation?",
   "evidence": "The canonical AIM record (workshop *Automorphic forms and harmonic analysis on covering groups*, section \"Representations\", Problem 1.4) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0067": {
   "statement_status": "exact",
   "original_statement": "What is theta correspondence for $\\widetilde{G}$?\n\ne.g.) $SL_2 \\times \\widetilde{SL}^{(3)}_2 \\hookrightarrow \\widetilde{G_2}^{(3)}$ (G-R-S)\n\n$SL_3 \\times \\widetilde{SL}^{(2)}_3 \\hookrightarrow \\widetilde{F_4}^{(2)}$\n\n$\\widetilde{Spin}^{(2)}_{2a+1} \\times \\widetilde{Spin}^{(2)}_{2b} \\hookrightarrow \\widetilde{Spin}^{(2)}_{2(a+b)+1}$ (Loke-Savin)\n\n$\\widetilde{Spin}^{(2)}_{5} \\times \\widetilde{Spin}^{(2)}_{4} \\hookrightarrow \\widetilde{Spin}^{(2)}_{9}$",
   "clean_statement": "What is theta correspondence for $\\widetilde{G}$?\n\ne.g.) $SL_2 \\times \\widetilde{SL}^{(3)}_2 \\hookrightarrow \\widetilde{G_2}^{(3)}$ (G-R-S)\n\n$SL_3 \\times \\widetilde{SL}^{(2)}_3 \\hookrightarrow \\widetilde{F_4}^{(2)}$\n\n$\\widetilde{Spin}^{(2)}_{2a+1} \\times \\widetilde{Spin}^{(2)}_{2b} \\hookrightarrow \\widetilde{Spin}^{(2)}_{2(a+b)+1}$ (Loke-Savin)\n\n$\\widetilde{Spin}^{(2)}_{5} \\times \\widetilde{Spin}^{(2)}_{4} \\hookrightarrow \\widetilde{Spin}^{(2)}_{9}$",
   "public_statement": "What is theta correspondence for $\\widetilde{G}$?\n\ne.g.) $SL_2 \\times \\widetilde{SL}^{(3)}_2 \\hookrightarrow \\widetilde{G_2}^{(3)}$ (G-R-S)\n\n$SL_3 \\times \\widetilde{SL}^{(2)}_3 \\hookrightarrow \\widetilde{F_4}^{(2)}$\n\n$\\widetilde{Spin}^{(2)}_{2a+1} \\times \\widetilde{Spin}^{(2)}_{2b} \\hookrightarrow \\widetilde{Spin}^{(2)}_{2(a+b)+1}$ (Loke-Savin)\n\n$\\widetilde{Spin}^{(2)}_{5} \\times \\widetilde{Spin}^{(2)}_{4} \\hookrightarrow \\widetilde{Spin}^{(2)}_{9}$",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0068": {
   "statement_status": "exact",
   "original_statement": "Determine notion of stable conjugacy when stable conjugacy classes in $\\widetilde{G}_F$ and $\\widetilde{H}_L$ naturally are related.",
   "clean_statement": "Determine notion of stable conjugacy when stable conjugacy classes in $\\widetilde{G}_F$ and $\\widetilde{H}_L$ naturally are related.",
   "public_statement": "Determine notion of stable conjugacy when stable conjugacy classes in $\\widetilde{G}_F$ and $\\widetilde{H}_L$ naturally are related.",
   "evidence": "The exact AIM record (workshop *Automorphic forms and harmonic analysis on covering groups*, section “Fundamental properties,” Problem 2.1) says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0069": {
   "statement_status": "exact",
   "original_statement": "In the real case, given a cover of $G_{\\mathbb{R}}$, describe Brylinski-Deligne data.",
   "clean_statement": "In the real case, given a cover of $G_{\\mathbb{R}}$, describe Brylinski-Deligne data.",
   "public_statement": "In the real case, given a cover of $G_{\\mathbb{R}}$, describe Brylinski-Deligne data.",
   "evidence": "The canonical AIM record (workshop *Automorphic forms and harmonic analysis on covering groups*, section \"Fundamental properties\", Problem 2.2) asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0070": {
   "statement_status": "exact",
   "original_statement": "ABV for covering groups\n\nThere is a canonical isomorphism as $K$-groups\n\\[\nKRep \\widetilde{G}_{\\mathbb R} \\cong \\Big( KPer_{H} (X) \\Big)^*,\n\\]\nwhere $X$ is a $\\mathbb C$-algebraic variety and $H$ is a $\\mathbb C$-algebraic group (for the trivial cover, $H={^{\\vee}}G$).",
   "clean_statement": "ABV for covering groups\n\nThere is a canonical isomorphism as $K$-groups\n\\[\nKRep \\widetilde{G}_{\\mathbb R} \\cong \\Big( KPer_{H} (X) \\Big)^*,\n\\]\nwhere $X$ is a $\\mathbb C$-algebraic variety and $H$ is a $\\mathbb C$-algebraic group (for the trivial cover, $H={^{\\vee}}G$).",
   "public_statement": "ABV for covering groups\n\nThere is a canonical isomorphism as $K$-groups\n\\[\nKRep \\widetilde{G}_{\\mathbb R} \\cong \\Big( KPer_{H} (X) \\Big)^*,\n\\]\nwhere $X$ is a $\\mathbb C$-algebraic variety and $H$ is a $\\mathbb C$-algebraic group (for the trivial cover, $H={^{\\vee}}G$).",
   "evidence": "The original AIMPL URL returned a 502 error when checked on 12 August 2026. The local corpus record and nearby records were therefore preserved without alteration. There is no visible OCR error. The formula is a compressed version of the Adams--Barbasch--Vogan (ABV) perfect pairing: $KRep$ is a Grothendieck group of finite-length representations, $KPer$ is a Grothendieck group of equivariant perverse sheaves on a geometric parameter space, and ${}^*$ is an algebraic dual.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 69,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0071": {
   "statement_status": "unrecoverable",
   "original_statement": "Lurie Conjecture\n\nTwisted Whittaker models by D. Gaitsgory",
   "clean_statement": null,
   "public_statement": "Lurie Conjecture\n\nTwisted Whittaker models by D. Gaitsgory",
   "evidence": "It is item 2.4, under **Fundamental properties**, in the problem list from the 2013 AIM workshop *Automorphic forms and harmonic analysis on covering groups*. This is not a mathematical statement as extracted: it is a title followed by a bibliographic pointer. The supplied `source_url` currently returns an error, and nearby records do not add notation. The source record has therefore been preserved exactly rather than silently expanded.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 70,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0072": {
   "statement_status": "exact",
   "original_statement": "Covering group of tori\n\nLet $T$ be a (not necessary split) torus over a $p$--adic field. Construct $\\widetilde{T}$\n\\[\n1 \\longrightarrow \\mu_F \\longrightarrow \\widetilde{T}\n\\longrightarrow T \\longrightarrow 1\n\\]\nexplicitly within Brylinski-Deligne framework.",
   "clean_statement": "Covering group of tori\n\nLet $T$ be a (not necessary split) torus over a $p$--adic field. Construct $\\widetilde{T}$\n\\[\n1 \\longrightarrow \\mu_F \\longrightarrow \\widetilde{T}\n\\longrightarrow T \\longrightarrow 1\n\\]\nexplicitly within Brylinski-Deligne framework.",
   "public_statement": "Covering group of tori\n\nLet $T$ be a (not necessary split) torus over a $p$--adic field. Construct $\\widetilde{T}$\n\\[\n1 \\longrightarrow \\mu_F \\longrightarrow \\widetilde{T}\n\\longrightarrow T \\longrightarrow 1\n\\]\nexplicitly within Brylinski-Deligne framework.",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0073": {
   "statement_status": "exact",
   "original_statement": "What is special about $2$-fold covers?",
   "clean_statement": "What is special about $2$-fold covers?",
   "public_statement": "What is special about $2$-fold covers?",
   "evidence": "The canonical record is problem 2.6 in the “Fundamental properties” section of the 2013 AIM workshop *Automorphic forms and harmonic analysis on covering groups*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 72,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0074": {
   "statement_status": "exact",
   "original_statement": "Base change\n\n\\begin{itemize}\n\\item $F$ : a $p$-adic field of characteristic $0.$\n\\item $G_F$ : an unramified linear group (quasi-split over $F$ and split over a finite unramified extension $L$ of $F.$)\n\\item $\\widetilde{G}_F$ : a covering group of degree coprime to $p.$\n\\end{itemize}\n\nGiven a genuine irreducible representation $\\widetilde{\\pi}$ of $\\widetilde{G}_F,$ describe $Lift_{L/F}(\\widetilde{\\pi})$ the base change lift of $\\widetilde{\\pi}.$ Note that $Lift_{L/F}(\\widetilde{\\pi})$ is supposed to be the virtual representation of $\\widetilde{G}_L.$\\label{basechange}",
   "clean_statement": "Base change\n\n\\begin{itemize}\n\\item $F$ : a $p$-adic field of characteristic $0.$\n\\item $G_F$ : an unramified linear group (quasi-split over $F$ and split over a finite unramified extension $L$ of $F.$)\n\\item $\\widetilde{G}_F$ : a covering group of degree coprime to $p.$\n\\end{itemize}\n\nGiven a genuine irreducible representation $\\widetilde{\\pi}$ of $\\widetilde{G}_F,$ describe $Lift_{L/F}(\\widetilde{\\pi})$ the base change lift of $\\widetilde{\\pi}.$ Note that $Lift_{L/F}(\\widetilde{\\pi})$ is supposed to be the virtual representation of $\\widetilde{G}_L.$\\label{basechange}",
   "public_statement": "Base change\n\n\\begin{itemize}\n\\item $F$ : a $p$-adic field of characteristic $0.$\n\\item $G_F$ : an unramified linear group (quasi-split over $F$ and split over a finite unramified extension $L$ of $F.$)\n\\item $\\widetilde{G}_F$ : a covering group of degree coprime to $p.$\n\\end{itemize}\n\nGiven a genuine irreducible representation $\\widetilde{\\pi}$ of $\\widetilde{G}_F,$ describe $Lift_{L/F}(\\widetilde{\\pi})$ the base change lift of $\\widetilde{\\pi}.$ Note that $Lift_{L/F}(\\widetilde{\\pi})$ is supposed to be the virtual representation of $\\widetilde{G}_L.$\\label{basechange}",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 73,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0075": {
   "statement_status": "exact",
   "original_statement": "This is a preliminary problem to Problem \\ref{basechange}\n\nDoes $Gal(L/F)$ act on $\\widetilde{G}_L$?",
   "clean_statement": "This is a preliminary problem to Problem \\ref{basechange}\n\nDoes $Gal(L/F)$ act on $\\widetilde{G}_L$?",
   "public_statement": "This is a preliminary problem to Problem \\ref{basechange}\n\nDoes $Gal(L/F)$ act on $\\widetilde{G}_L$?",
   "evidence": "The exact canonical AIM record is problem 3.2 in the “Lifts” section of the 2013 workshop *Automorphic forms and harmonic analysis on covering groups*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0076": {
   "statement_status": "exact",
   "original_statement": "Make sense of transferring representation from $\\widetilde{G}_F$ to $\\widetilde{G'}_F,$\nwhere two linear groups $G_F$ and $G'_F$ are inner forms each other.",
   "clean_statement": "Make sense of transferring representation from $\\widetilde{G}_F$ to $\\widetilde{G'}_F,$\nwhere two linear groups $G_F$ and $G'_F$ are inner forms each other.",
   "public_statement": "Make sense of transferring representation from $\\widetilde{G}_F$ to $\\widetilde{G'}_F,$\nwhere two linear groups $G_F$ and $G'_F$ are inner forms each other.",
   "evidence": "There is no apparent OCR corruption. The wording is deliberately underspecified: an inner twist relates the **linear algebraic groups**, but it does not by itself specify a relation between two topological central extensions. The old AIM problem-list URL was unavailable during this run, so the exact record in input.json and its neighboring “Lifts” questions are the verified source.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0077": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Is there a natural lifting from representations of $G_F$ to $\\widetilde{G'}_F$?",
   "clean_statement": null,
   "public_statement": "Is there a natural lifting from representations of $G_F$ to $\\widetilde{G'}_F$?",
   "evidence": "The record itself does not define $G'$, the cover, the class of representations, or the sense of “natural.” The immediately preceding AIM problem (3.3) asks for transfer from $\\widetilde G_F$ to $\\widetilde{G'}_F$ “where two linear groups $G_F$ and $G'_F$ are inner forms each other,” and warns that the centers of the covers may differ. I therefore use the following conservative reconstruction:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0078": {
   "statement_status": "exact",
   "original_statement": "What is the local character identity for $Mp(2n)$?",
   "clean_statement": "What is the local character identity for $Mp(2n)$?",
   "public_statement": "What is the local character identity for $Mp(2n)$?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0079": {
   "statement_status": "exact",
   "original_statement": "Are there instances of Rankin-Selberg methods for higher covers?",
   "clean_statement": "Are there instances of Rankin-Selberg methods for higher covers?",
   "public_statement": "Are there instances of Rankin-Selberg methods for higher covers?",
   "evidence": "It is item 4.1 in the “Applications” section of the AIM list *Automorphic forms and harmonic analysis on covering groups*. The neighboring items ask for arithmetic and trace-formula applications of global covering groups, so “Rankin--Selberg methods” means global integral representations that unfold and produce local or global \\(L\\)-functions. The phrase “higher covers” is not defined in the record. I use the standard interpretation: finite central covers of degree \\(m>2\\), especially Matsumoto or Brylinski--Deligne covers. No OCR correction is needed. The original AIM problem-list URL returned an HTTP 502 error during this run, so this reconstruction uses the exact canonical record, its neighbors, and the workshop summary.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0080": {
   "statement_status": "exact",
   "original_statement": "What global covering groups have arithmetic applications?",
   "clean_statement": "What global covering groups have arithmetic applications?",
   "public_statement": "What global covering groups have arithmetic applications?",
   "evidence": "The source record has no remarks or literature field. There is no OCR error. The neighboring AIM questions ask about higher-cover Rankin--Selberg methods, trace formulas, formal degrees, Whittaker models, Langlands--Shahidi methods, and covers of split tori. Thus “arithmetic applications” should be read broadly but mathematically: special values or derivatives of $L$-functions, arithmetic Fourier coefficients, representation numbers, arithmetic cycles, automorphic products, and multiple Dirichlet series all qualify.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0081": {
   "statement_status": "exact",
   "original_statement": "Are there applications of the invariant trace formula for $\\widetilde{G}_{F}$?",
   "clean_statement": "Are there applications of the invariant trace formula for $\\widetilde{G}_{F}$?",
   "public_statement": "Are there applications of the invariant trace formula for $\\widetilde{G}_{F}$?",
   "evidence": "There is no visible OCR corruption. There is, however, a notation ambiguity. An Arthur--Selberg trace formula is global: for a number field $F$ it is attached to an adelic cover",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0082": {
   "statement_status": "exact",
   "original_statement": "Interpret formal degrees of representations of $\\widetilde{G}_{F}$ on level of Hecke algebra.\n(\\`a la Opdam, Reeder, et al)",
   "clean_statement": "Interpret formal degrees of representations of $\\widetilde{G}_{F}$ on level of Hecke algebra.\n(\\`a la Opdam, Reeder, et al)",
   "public_statement": "Interpret formal degrees of representations of $\\widetilde{G}_{F}$ on level of Hecke algebra.\n(\\`a la Opdam, Reeder, et al)",
   "evidence": "Here $\\widetilde G_F$ is read as the group of $F$-points of a covering group. The source sequence backslash--backtick--`a` is the usual TeX accent in “à la,” not an OCR error, and its newline is only formatting. The broader workshop summary makes the motivating example more precise: the Bernstein blocks containing the even and odd Weil representations of the two-fold metaplectic group should be compared with Iwahori-spherical blocks of equal-rank odd orthogonal groups. The summary singles out preservation of the natural $L^2$ norm under the Hecke-algebra isomorphism, because that is exactly what transports Plancherel measure and hence formal degrees.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0083": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Are there global applications of the trace formula, automorphic forms, Whittaker models, the Langlands-Shahidi method for $\\widetilde{G}$ ?\n\n-- Classify automorphic representations of covers of split tori.",
   "clean_statement": null,
   "public_statement": "Are there global applications of the trace formula, automorphic forms, Whittaker models, the Langlands-Shahidi method for $\\widetilde{G}$ ?\n\n-- Classify automorphic representations of covers of split tori.",
   "evidence": "The canonical record is item 4.5 in the ``Applications'' section of the AIM workshop *Automorphic forms and harmonic analysis on covering groups*. Its exact problem text is",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 82,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0084": {
   "statement_status": "exact",
   "original_statement": "Study the residual spectrum for $\\widetilde{GL}(n)$.\n\n-- Can the lifting of Levi subgroups be done over $p$-adic? If so, is it one of the Brylinski-Deligne extensions?\n\n-- Describe covers of $GL(n)$ not arising from Kazhdan-Petterson.",
   "clean_statement": "Study the residual spectrum for $\\widetilde{GL}(n)$.\n\n-- Can the lifting of Levi subgroups be done over $p$-adic? If so, is it one of the Brylinski-Deligne extensions?\n\n-- Describe covers of $GL(n)$ not arising from Kazhdan-Petterson.",
   "public_statement": "Study the residual spectrum for $\\widetilde{GL}(n)$.\n\n-- Can the lifting of Levi subgroups be done over $p$-adic? If so, is it one of the Brylinski-Deligne extensions?\n\n-- Describe covers of $GL(n)$ not arising from Kazhdan-Petterson.",
   "evidence": "The exact canonical record (AIM workshop *Automorphic forms and harmonic analysis on covering groups*, Applications 4.6) says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 83,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0085": {
   "statement_status": "exact",
   "original_statement": "Problem 1.1. Let G be a finite group. Let X be a partition of Irr (G) (such that it forms a supercharacter theory). Let σX = ∑χ∈X χ(1) χ. When is it true that the product of two of these σ's is a positive integer combination of σ's?",
   "clean_statement": "Problem 1.1. Let G be a finite group. Let X be a partition of Irr (G) (such that it forms a supercharacter theory). Let σX = ∑χ∈X χ(1) χ. When is it true that the product of two of these σ's is a positive integer combination of σ's?",
   "public_statement": "Problem 1.1. Let G be a finite group. Let X be a partition of Irr (G) (such that it forms a supercharacter theory). Let σX = ∑χ∈X χ(1) χ. When is it true that the product of two of these σ's is a positive integer combination of σ's?",
   "evidence": "The AIM workshop list asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 84,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0086": {
   "statement_status": "exact",
   "original_statement": "Problem 1.2. Can one construct a nested supercharacter theory that would realize the Hopf algebra NCQSym (indexed by ordered partitions)?",
   "clean_statement": "Problem 1.2. Can one construct a nested supercharacter theory that would realize the Hopf algebra NCQSym (indexed by ordered partitions)?",
   "public_statement": "Problem 1.2. Can one construct a nested supercharacter theory that would realize the Hopf algebra NCQSym (indexed by ordered partitions)?",
   "evidence": "The canonical record is Problem 1.2 from the AIM workshop report *Supercharacters and combinatorial Hopf algebras* (May 17--21, 2010):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 85,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0087": {
   "statement_status": "exact",
   "original_statement": "Problem 1.3. \n\nIn the original supercharacter theory of Andre, the supercharacters/superclasses associated to single boxes above the identity are irreducible. Is Andre's supercharacter theory the coarsest containing this?. If not, what is the coarsest supercharacter theory containing the irreducibles?",
   "clean_statement": "Problem 1.3.\n\nIn the original supercharacter theory of Andre, the supercharacters/superclasses associated to single boxes above the identity are irreducible. Is Andre's supercharacter theory the coarsest containing this?. If not, what is the coarsest supercharacter theory containing the irreducibles?",
   "public_statement": "Problem 1.3.\n\nIn the original supercharacter theory of Andre, the supercharacters/superclasses associated to single boxes above the identity are irreducible. Is Andre's supercharacter theory the coarsest containing this?. If not, what is the coarsest supercharacter theory containing the irreducibles?",
   "evidence": "The canonical record says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 86,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0088": {
   "statement_status": "exact",
   "original_statement": "Problem 1.4. \n\nCharacterize the Andre/Yan supercharacter theory -along the lines of the previous problem-.",
   "clean_statement": "Problem 1.4.\n\nCharacterize the Andre/Yan supercharacter theory -along the lines of the previous problem-.",
   "public_statement": "Problem 1.4.\n\nCharacterize the Andre/Yan supercharacter theory -along the lines of the previous problem-.",
   "evidence": "The canonical record, transcribed from the AIM workshop list, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 87,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0089": {
   "statement_status": "exact",
   "original_statement": "Problem 1.5. Consider all the automorphisms of the unitriangular group. Do these automor-phisms give additional nice cumpling (coarsification) to the theory?",
   "clean_statement": "Problem 1.5. Consider all the automorphisms of the unitriangular group. Do these automor-phisms give additional nice cumpling (coarsification) to the theory?",
   "public_statement": "Problem 1.5. Consider all the automorphisms of the unitriangular group. Do these automor-phisms give additional nice cumpling (coarsification) to the theory?",
   "evidence": "The exact repository record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 88,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0090": {
   "statement_status": "exact",
   "original_statement": "Problem 1.6. \n\nFor Un every supercharacter is a product of supercharacters that happen to be irreducible. This is not true for every pattern groups. When is it true (even for algebra groups)?",
   "clean_statement": "Problem 1.6.\n\nFor Un every supercharacter is a product of supercharacters that happen to be irreducible. This is not true for every pattern groups. When is it true (even for algebra groups)?",
   "public_statement": "Problem 1.6.\n\nFor Un every supercharacter is a product of supercharacters that happen to be irreducible. This is not true for every pattern groups. When is it true (even for algebra groups)?",
   "evidence": "The AIM source is Problem 1.6 from the workshop list *Supercharacters and combinatorial Hopf algebras*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 89,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0091": {
   "statement_status": "exact",
   "original_statement": "Problem 1.7. \n\nMake sense out of superinduction and restriction more generally than for algebra groups.",
   "clean_statement": "Problem 1.7.\n\nMake sense out of superinduction and restriction more generally than for algebra groups.",
   "public_statement": "Problem 1.7.\n\nMake sense out of superinduction and restriction more generally than for algebra groups.",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 90,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0092": {
   "statement_status": "exact",
   "original_statement": "Problem 1.8. Find natural supercharacter theory for Sylow p-subgroups of S pn. Are they nested? Does the restricition of a supercharacter for S pn to S pn−1 split into supercharacters?",
   "clean_statement": "Problem 1.8. Find natural supercharacter theory for Sylow p-subgroups of S pn. Are they nested? Does the restricition of a supercharacter for S pn to S pn−1 split into supercharacters?",
   "public_statement": "Problem 1.8. Find natural supercharacter theory for Sylow p-subgroups of S pn. Are they nested? Does the restricition of a supercharacter for S pn to S pn−1 split into supercharacters?",
   "evidence": "The extracted record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 91,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0093": {
   "statement_status": "exact",
   "original_statement": "Problem 1.9. Describe irreducible characters of maximal degree of pattern groups (as super-characters) using dimension vectors and polynomial equations.",
   "clean_statement": "Problem 1.9. Describe irreducible characters of maximal degree of pattern groups (as super-characters) using dimension vectors and polynomial equations.",
   "public_statement": "Problem 1.9. Describe irreducible characters of maximal degree of pattern groups (as super-characters) using dimension vectors and polynomial equations.",
   "evidence": "The AIM problem list for the workshop *Supercharacters and combinatorial Hopf algebras* gives:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0094": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 1.10. Try to carry over all of the Un work to the Borel subgroup Bn of GL n. Is there a similar (module to a) Hopf algebra?",
   "clean_statement": null,
   "public_statement": "Problem 1.10. Try to carry over all of the Un work to the Borel subgroup Bn of GL n. Is there a similar (module to a) Hopf algebra?",
   "evidence": "The odd parenthetical “(module to a)” is present in the original AIM PDF, not introduced by the JSON extraction. It is therefore an editorial ambiguity rather than a correctable OCR error. Two plausible readings are “a similar Hopf algebra” and “a similar module over/attached to a Hopf algebra.” The literature below answers the first, stronger reading. The split Hopf projection proved in this report also gives a precise module/comodule interpretation of the second reading.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0095": {
   "statement_status": "exact",
   "original_statement": "Problem 1.11. In what sense is describing the conjugacy classes of Un difficult? Is the number of conjugacy classes difficult?",
   "clean_statement": "Problem 1.11. In what sense is describing the conjugacy classes of Un difficult? Is the number of conjugacy classes difficult?",
   "public_statement": "Problem 1.11. In what sense is describing the conjugacy classes of Un difficult? Is the number of conjugacy classes difficult?",
   "evidence": "The exact extracted record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 94,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0096": {
   "statement_status": "exact",
   "original_statement": "Problem 1.12. Is superclass theory for Un tame?. That is, is there a tame algebra whose category of modules is equivalent to the category of supercharacters of Un?.",
   "clean_statement": "Problem 1.12. Is superclass theory for Un tame?. That is, is there a tame algebra whose category of modules is equivalent to the category of supercharacters of Un?.",
   "public_statement": "Problem 1.12. Is superclass theory for Un tame?. That is, is there a tame algebra whose category of modules is equivalent to the category of supercharacters of Un?.",
   "evidence": "The AIM workshop PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0097": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 1.13. Consider the two presentations for NCSym given below: \n\nMμ =\n\n∑\n\n> ∇ω=μ\n\nω (1.1) \n\nwhere w ∈ A? and A = {a1, a2, · · · } non commuting. \n\nUμ =\n\n∑ \n\n> σ∈Sn, λ (σ)=μ\n\nx1σ(1) x2σ2 · · · (1.2) \n\nwhere {xi j } are commutative variables such that xi j xl j = 0 if i, j or xi j xik = 0 if j, k.Is it possible to describe the Hopf isomorphism S C (2)? −→ Π? with Hopf and internal comul-tiplication? SUPERCHARACTERS AND COMBINATORIAL HOPF ALGEBRAS 3",
   "clean_statement": "Describe explicitly the Hopf isomorphism\n\\[\nSC^{(2)*}\\longrightarrow \\Pi^*\\cong\\Pi QSym\n\\]\nin these two presentations, and make it compatible with an internal comultiplication.",
   "public_statement": "Describe explicitly the Hopf isomorphism\n\\[\nSC^{(2)*}\\longrightarrow \\Pi^*\\cong\\Pi QSym\n\\]\nin these two presentations, and make it compatible with an internal comultiplication.",
   "evidence": "Inspection of the original AIM PDF confirms that several question marks and missing conditions are OCR errors. Comparing the display with the later polynomial realization in Aguiar et al. [AAB] gives the following unambiguous reconstruction. Thus the recovered question is:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 96,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0098": {
   "statement_status": "exact",
   "original_statement": "Problem 1.14. To get a better understanding of the supercharacter basis, give a basis in NC-Sym that is nicely related to supercharacters. For example, see remark about p-basis pp.20 in \n\nSupercharacters, symmetric functions in non commuting variables, and related Hopf algebras.",
   "clean_statement": "Problem 1.14. To get a better understanding of the supercharacter basis, give a basis in NC-Sym that is nicely related to supercharacters. For example, see remark about p-basis pp.20 in\n\nSupercharacters, symmetric functions in non commuting variables, and related Hopf algebras.",
   "public_statement": "Problem 1.14. To get a better understanding of the supercharacter basis, give a basis in NC-Sym that is nicely related to supercharacters. For example, see remark about p-basis pp.20 in\n\nSupercharacters, symmetric functions in non commuting variables, and related Hopf algebras.",
   "evidence": "The canonical AIM record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 97,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0099": {
   "statement_status": "exact",
   "original_statement": "Problem 1.15. Is there any relation between supercharacter theory of Un and properties of \n\nGL n?. Is there an analogous problem where this has been worked out?.",
   "clean_statement": "Problem 1.15. Is there any relation between supercharacter theory of Un and properties of\n\nGL n?. Is there an analogous problem where this has been worked out?.",
   "public_statement": "Problem 1.15. Is there any relation between supercharacter theory of Un and properties of\n\nGL n?. Is there an analogous problem where this has been worked out?.",
   "evidence": "The AIM workshop PDF says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 98,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0100": {
   "statement_status": "exact",
   "original_statement": "Problem 1.16. Tung Le and Kay Magaard have a way of refining supercharacters for Un. Is there an analog for algebra groups?",
   "clean_statement": "Problem 1.16. Tung Le and Kay Magaard have a way of refining supercharacters for Un. Is there an analog for algebra groups?",
   "public_statement": "Problem 1.16. Tung Le and Kay Magaard have a way of refining supercharacters for Un. Is there an analog for algebra groups?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 99,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0101": {
   "statement_status": "exact",
   "original_statement": "Problem 1.17. Is it true that there are four infinite families of supercharacter theories for symmetric groups and 6 or 7 exceptions? Are those four families nested?. See Supercharacter theories of cyclic p-groups by A. Hendrickson.",
   "clean_statement": "Problem 1.17. Is it true that there are four infinite families of supercharacter theories for symmetric groups and 6 or 7 exceptions? Are those four families nested?. See Supercharacter theories of cyclic p-groups by A. Hendrickson.",
   "public_statement": "Problem 1.17. Is it true that there are four infinite families of supercharacter theories for symmetric groups and 6 or 7 exceptions? Are those four families nested?. See Supercharacter theories of cyclic p-groups by A. Hendrickson.",
   "evidence": "This wording was checked against the original three-page workshop PDF; “symmetric groups” is not an extraction error. The adjacent Problem 1.18 again refers to “the four infinite families of supercharacter theories of \\(S_n\\),” and Problem 1.2 says that “nested” means a projective system of groups. Thus the cyclic-\\(p\\)-group citation supplies construction/lattice background rather than changing the intended family of groups.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 100,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0102": {
   "statement_status": "exact",
   "original_statement": "Problem 1.18. Can the four infinite families of supercharacter theories of S n be realized as isomorphic to a Hopf quotient of S ym?. Can we define multiplication and comultiplication in one of the families to obtain Hopf algebra structure?.",
   "clean_statement": "Problem 1.18. Can the four infinite families of supercharacter theories of S n be realized as isomorphic to a Hopf quotient of S ym?. Can we define multiplication and comultiplication in one of the families to obtain Hopf algebra structure?.",
   "public_statement": "Problem 1.18. Can the four infinite families of supercharacter theories of S n be realized as isomorphic to a Hopf quotient of S ym?. Can we define multiplication and comultiplication in one of the families to obtain Hopf algebra structure?.",
   "evidence": "The canonical JSON record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 101,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0103": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 1 (Breuil-M´ ezard).\n\nμGal = μAut.\n\nGenerally, one can apply global arguments to prove that μGal ≥ μAut, the reverse inequality is considerably more difficult, and is in essence equivalent to proving a modularity lifting theorem. Suppose that τ: IQp → GL 2(E) is of Galois type. W let R\u0003,ψ (k, τ, ρ) be a certain (uniquely defined) quotient of R\u0003(ρ) ⊗W (F) O - where R\u0003(ρ) is the universal framed deformation ring, i.e. the ring representing the functor which associates to a local Artin ring A with residue field F the set of isomorphism classes of deformations VA of ρ to A, together with a lifting to VA of a some fixed choice of basis for VF.The following conjecture generalizes the Breuil-M´ ezard conjecture to the situation where ρ has nontrivial endomorphisms and is central in this approach to the Fontaine-Mazur conjecture:",
   "clean_statement": null,
   "public_statement": "Conjecture 1 (Breuil-M´ ezard).\n\nμGal = μAut.\n\nGenerally, one can apply global arguments to prove that μGal ≥ μAut, the reverse inequality is considerably more difficult, and is in essence equivalent to proving a modularity lifting theorem. Suppose that τ: IQp → GL 2(E) is of Galois type. W let R[U+0003],ψ (k, τ, ρ) be a certain (uniquely defined) quotient of R[U+0003](ρ) ⊗W (F) O - where R[U+0003](ρ) is the universal framed deformation ring, i.e. the ring representing the functor which associates to a local Artin ring A with residue field F the set of isomorphism classes of deformations VA of ρ to A, together with a lifting to VA of a some fixed choice of basis for VF.The following conjecture generalizes the Breuil-M´ ezard conjecture to the situation where ρ has nontrivial endomorphisms and is central in this approach to the Fontaine-Mazur conjecture:",
   "evidence": "The canonical record is number 1 in the AIM workshop notes *$p$-adic representations, modularity, and beyond*. Its core text is",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 102,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0104": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 2 (Kisin). The Hilbert-Samuel multiplicity of R\u0003,ψ (k, τ, ρ)/(π) is equal to μAut.\n\nMost cases of this conjecture are proved in Kisin's preprint. Indeed, by the same reasoning as above, the difficulty lies in proving the single inequality: e(R\u0003,ψ (k, τ, ρ)/(π)) ≤ μAut - where e\n\ndenotes 'Hilbert-Samuel multiplicity'. 1.2. Colmez's functor and an expectation. One of the main inputs into Kisin's proof (without the assumption that the representation becomes semi-stable over an abelian extension) of the above inequality is the following construction of Colmez. Let G = GL 2(Qp), K = GL 2(Zp) and let Z be the center of G. If σ is a representation of KZ on a finite dimensional vector space Vσ over F, then write I(σ) = Ind GKZ σ for the compact induction of σ.Put σ = Sym rF, and let χ: Q× \n\n> p\n\n→ F× be a character, let λ ∈ F. For x ∈ F we put μx: Q× \n\n> p\n\n→ F×\n\n- the unramified character sending p ∈ Q× \n\n> p\n\nto x. Now set π(r, λ, χ ) = I(σ)/(T − λ)I(σ) ⊗ χ ◦ det. Let Π be a representation of GL 2(Qp) on a W (F)-module. The representation Π is admissible if Π has finite length and each of its Jordan-H¨ older factors has a central character. Equivalently, Π is admissible when it is of finite length and the Jordan-H¨ older factors of Π are either one-dimensional or an infinite dimensional subquotient of some π(r, λ, χ ). \n\nTheorem 2 (Colmez). There exists an exact contravariant functor V ∗ from the category of fi-nite length, admissible GL 2(Qp)-representations to the category of finite length representations of \n\nW (F)[ GQp ]. Moreover, we have \n\n(1) V ∗(Π) = 0 if Π is one-dimensional, \n\n(2) V ∗(π(r, λ, χ )) = χμ λ−1 if λ 6 = 0,\n\n(3) V ∗(π(r, 0, χ )) = Ind GQp\n\n> GQp2\n\nωr+1 2 ⊗ χ.\n\nOne can reinterpret Colmez's functor as a covariant functor as follows: Fix a character ψ: GQp →O× (regarded as a character of Q× \n\n> p\n\nvia local class field theory), suppose that Π is a finite length \n\nO[GL 2(Qp)]-module, which is admissible as a W (F)[1 /p ]-module. Define Vψ(Π) = ( V ∗(Π)) ∗(χcyc ψ)where V ∗(Π) ∗ is the Pontryagin dual of the finite length O-module V ∗(Π). Suppose that Π is now a representation of GL 2(Qp) on a W (F)-module, put Π n = Π ⊗Z Z/p nZ.Assume that Π is p-adically complete and separated, in particular Π = proj lim n Πn, and Π n is admissible (and of finite length) for each n. We write Vψ(Π) = proj lim Vψ(Π n). Since admissible p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 3\n\nrepresentations have finite length, projective limits are exact, thus Vψ(Π) /pV ψ(Π) = Vψ(Π 1), in particular Vψ(Π) is a finite generated W (F)-module, as it is p-adically separated. Such a repre-sentation Π will be called an admissible lattice. If in addition Π is an O-module, we call it an \n\nadmissible O-lattice.The following result (in its full generality) is still pending: \n\nTheorem 3 (Colmez(?)). Let E′/E be a finite extension and let V be a two-dimensional E′-vector space with a continuous GQp -action. Suppose that V is potentially semi-stable of type τ with Hodge-Tate weights 0 and k − 1 (k ≥ 2) and that det V = ψχ.Then there exists an admissible OE′ -lattice Π with central character ψ such that Vψ(Π) ⊗Zp\n\nQp ˜→ V. If Π′ is another such lattice, then there exists a continuous isomorphism of E′[GL 2(Qp)] -modules Π′ ⊗Zp Qp ˜→ Π ⊗Zp Qp.\n\nMoreover, there exists a GL 2(Zp)-equivariant inclusion σ(k, τ ) ↪→ Π ⊗Zp Qp.\n\nThis result is known for triganuline representations. 2. Emerton: Part one. \n\nCaveat: - This session started with Matthew Emerton fielding questions from the audience, thus this section consisted largely of open discussion, and consequently the narrative suffered. RIBET: Where does m live? Let N be an integer. Define T(N ) to be the Hecke algebra of level N. We have the following diagram of maps: \n\nH(N ) = ⊗`-N H(GL 2(Q`)// GL 2(Z`)) \n\n> \u000f\n> \u000f\n\nT / / T(N )mN\n\nRΣN\n\n> O\n> O\n> O\n> O\n\nThis is compact with N enlarging: \n\nH(N ′) / / / / \n\n> \u000f\n> \u000f\n\nT(N ′)m \n\n> \u000f\n> \u000f\n\nRΣN ′\n\n> oooo\n> \u000f\n> \u000f\n\nH(N ) / / / / T(N )m RΣN\n\n> o\n> o\n> o\n> o\n\nBUZZARD: Why let all primes ramify? This is the whole picture, but in practice, only finitely many primes are used. TAYLOR: Explain Colmez. 2.1. Definition of Colmez' functor. GL 2(Qp)-representations over A (where A is some artinian ring lifting F)Def( π) ˜ →Def( ρ)FALSE START Let MF = V ∗ be the functor from Kisin's talk, consider an admissible finite length representation \n\nπ(r, λ, χ ). There is a diagram of functors: 4 NOTES BY MICHAEL VOLPATO \n\n{fin. lgth, smth, cntrlly cofin. /w J-H factors in list }{fin. lgth, adms. W (F)[GL 2(Qp)]-reps } \n\n> 2\n> 2\n> dddddddddddddddddddddddddddddd\n> V∗\n>,\n>,\n> ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ\n\n{finite length W (F)[ GQp ]-mods }\n\n> O\n> O\n\nThen (1) {admissible J-H factors } ⊆ { J-H factors of π(r, λ, χ )}.\n\nRecall that irreducible admissible is the same as irreducible, smooth with central character. \n\nSmooth: every vector fixed by an open sub-group. \n\nAdmissible: above with finite length. Finite length admissible is equivalent to finite length, smooth and centrally cofinite. These are, in turn, the same as finite length with Jordan-Holder factors in (1). The finite dimensional clause doesn't matter on the Galois side - this is (maybe?) a local analogue of Ihara's lemma. 2.1.1. Deformation theory. Let A be an artin ring. Let π/A be finite free over A. Apply MF, gives \n\nρ/A, deforming: \n\nπ ↔ ρ\n\nConsider Hom( π, A ), if A was killed by A/p n, then consider Hom( π, Z/p nZ). We have Hom(lim \n\n> →\n\nAn, A ) = lim \n\n> ←\n\nHom( An, A ) ∼= An\n\nDefinition of MF: Define MF( π) = V ∗(π ⊗A Hom( A, Qp/Zp)).\n\nTake \n\nπ = Hom( π∗, A ) \n\n> \u000f\n> \u000f\n\nP//oo\n\n> \u000f\n> \u000f\n\nπF / / π∗ MF ′ \n\n> //\n\nρ\n\nHom( π, F)\n\nwe have GL 2(Qp) on both sides, where P is category of pro-free A-modules. In fact, action is integral, thus we actually have an action of F[[GL 2(Zp)]], the latter functor being covariant. Need to be careful about changing scalars - analogous to defining Hom's of sheaves. 2.2. The (mod p) correspondence. Let G = GL 2(Qp), B =\n\n( ∗ ∗\n\n0 ∗\n\n)\n\nand B =\n\n( ∗ 0\n\n∗ ∗\n\n). We want \n\nρ =\n\n( χ ∗\n\n0 ψ\n\n)?? \n\n→ π\n\nIf χψ −1 6 = ω 6 = 1 then 0 / / Ind G \n\n> B\n\nχ ⊗ ψω / / π / / Ind G \n\n> B\n\nψ ⊗ χω / / 0p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 5\n\n0 ⊆ St ⊆ · \n\n︸ ︷︷ ︸\n\n> 1\n\n⊆ π\n\n︸︷︷︸ \n\n> Ind G\n> Bω−1⊗ω.\n\n2.3. Jacquet Modules. Let T =\n\n( ∗ 00 ∗\n\n). Consider Ind BG χ ⊗ ψω \n\nThen Hom G(V, Ind G \n\n> B\n\n(χ ⊗ ψω )) ∼= Hom B (V, ψω ⊗ χ)and (Ind G \n\n> B\n\nχ ⊗ ψω )N = ψω ⊗ χ where N =\n\n( 1 ∗\n\n0 1\n\n)\n\nThere is a action of T on the left-hand side. We define the ordinary Jacquet functor as \n\nJord (V ) = \n\nV\n\n1 Zp\n\n0 1\n\n>!\n\n\n\n> ord\n\nwith an action of Up.\n\nJord (Ind G \n\n> B\n\n(χ ⊗ χω )) = χ ⊗ ψω \n\nHom(Ind G \n\n> B\n\nU, V ) = Hom T (U, J ord (V )) One can compute: \n\nR1Jord (V ) = ( VN )( ω−1 ⊗ ω).\n\nWhere R1 is the first derived functor of the ordinary Jacquet functor. N.B. the ordinary Jacquet functor has cohomological dimension 2. Does \n\nH2(GL 2(Qp), F p) = 0? Is the 'bar' irrelevant? Computing cohomology difficult because complicated interactions with the topology and the representation theory. 3. Some open problems \n\n3.1. Conjecture: Emerton.",
   "clean_statement": null,
   "public_statement": "Conjecture 2 (Kisin). The Hilbert-Samuel multiplicity of R[U+0003],ψ (k, τ, ρ)/(π) is equal to μAut.\n\nMost cases of this conjecture are proved in Kisin's preprint. Indeed, by the same reasoning as above, the difficulty lies in proving the single inequality: e(R[U+0003],ψ (k, τ, ρ)/(π)) ≤ μAut - where e\n\ndenotes 'Hilbert-Samuel multiplicity'. 1.2. Colmez's functor and an expectation. One of the main inputs into Kisin's proof (without the assumption that the representation becomes semi-stable over an abelian extension) of the above inequality is the following construction of Colmez. Let G = GL 2(Qp), K = GL 2(Zp) and let Z be the center of G. If σ is a representation of KZ on a finite dimensional vector space Vσ over F, then write I(σ) = Ind GKZ σ for the compact induction of σ.Put σ = Sym rF, and let χ: Q×\n\n> p\n\n→ F× be a character, let λ ∈ F. For x ∈ F we put μx: Q×\n\n> p\n\n→ F×\n\n- the unramified character sending p ∈ Q×\n\n> p\n\nto x. Now set π(r, λ, χ ) = I(σ)/(T − λ)I(σ) ⊗ χ ◦ det. Let Π be a representation of GL 2(Qp) on a W (F)-module. The representation Π is admissible if Π has finite length and each of its Jordan-H¨ older factors has a central character. Equivalently, Π is admissible when it is of finite length and the Jordan-H¨ older factors of Π are either one-dimensional or an infinite dimensional subquotient of some π(r, λ, χ ).\n\nTheorem 2 (Colmez). There exists an exact contravariant functor V ∗ from the category of fi-nite length, admissible GL 2(Qp)-representations to the category of finite length representations of\n\nW (F)[ GQp ]. Moreover, we have\n\n(1) V ∗(Π) = 0 if Π is one-dimensional,\n\n(2) V ∗(π(r, λ, χ )) = χμ λ−1 if λ 6 = 0,\n\n(3) V ∗(π(r, 0, χ )) = Ind GQp\n\n> GQp2\n\nωr+1 2 ⊗ χ.\n\nOne can reinterpret Colmez's functor as a covariant functor as follows: Fix a character ψ: GQp →O× (regarded as a character of Q×\n\n> p\n\nvia local class field theory), suppose that Π is a finite length\n\nO[GL 2(Qp)]-module, which is admissible as a W (F)[1 /p ]-module. Define Vψ(Π) = ( V ∗(Π)) ∗(χcyc ψ)where V ∗(Π) ∗ is the Pontryagin dual of the finite length O-module V ∗(Π). Suppose that Π is now a representation of GL 2(Qp) on a W (F)-module, put Π n = Π ⊗Z Z/p nZ.Assume that Π is p-adically complete and separated, in particular Π = proj lim n Πn, and Π n is admissible (and of finite length) for each n. We write Vψ(Π) = proj lim Vψ(Π n). Since admissible p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 3\n\nrepresentations have finite length, projective limits are exact, thus Vψ(Π) /pV ψ(Π) = Vψ(Π 1), in particular Vψ(Π) is a finite generated W (F)-module, as it is p-adically separated. Such a repre-sentation Π will be called an admissible lattice. If in addition Π is an O-module, we call it an\n\nadmissible O-lattice.The following result (in its full generality) is still pending:\n\nTheorem 3 (Colmez(?)). Let E′/E be a finite extension and let V be a two-dimensional E′-vector space with a continuous GQp -action. Suppose that V is potentially semi-stable of type τ with Hodge-Tate weights 0 and k − 1 (k ≥ 2) and that det V = ψχ.Then there exists an admissible OE′ -lattice Π with central character ψ such that Vψ(Π) ⊗Zp\n\nQp ˜→ V. If Π′ is another such lattice, then there exists a continuous isomorphism of E′[GL 2(Qp)] -modules Π′ ⊗Zp Qp ˜→ Π ⊗Zp Qp.\n\nMoreover, there exists a GL 2(Zp)-equivariant inclusion σ(k, τ ) ↪→ Π ⊗Zp Qp.\n\nThis result is known for triganuline representations. 2. Emerton: Part one.\n\nCaveat: - This session started with Matthew Emerton fielding questions from the audience, thus this section consisted largely of open discussion, and consequently the narrative suffered. RIBET: Where does m live? Let N be an integer. Define T(N ) to be the Hecke algebra of level N. We have the following diagram of maps:\n\nH(N ) = ⊗`-N H(GL 2(Q`)// GL 2(Z`))\n\n> [U+000F]\n> [U+000F]\n\nT / / T(N )mN\n\nRΣN\n\n> O\n> O\n> O\n> O\n\nThis is compact with N enlarging:\n\nH(N ′) / / / /\n\n> [U+000F]\n> [U+000F]\n\nT(N ′)m\n\n> [U+000F]\n> [U+000F]\n\nRΣN ′\n\n> oooo\n> [U+000F]\n> [U+000F]\n\nH(N ) / / / / T(N )m RΣN\n\n> o\n> o\n> o\n> o\n\nBUZZARD: Why let all primes ramify? This is the whole picture, but in practice, only finitely many primes are used. TAYLOR: Explain Colmez. 2.1. Definition of Colmez' functor. GL 2(Qp)-representations over A (where A is some artinian ring lifting F)Def( π) ˜ →Def( ρ)FALSE START Let MF = V ∗ be the functor from Kisin's talk, consider an admissible finite length representation\n\nπ(r, λ, χ ). There is a diagram of functors: 4 NOTES BY MICHAEL VOLPATO\n\n{fin. lgth, smth, cntrlly cofin. /w J-H factors in list }{fin. lgth, adms. W (F)[GL 2(Qp)]-reps }\n\n> 2\n> 2\n> dddddddddddddddddddddddddddddd\n> V∗\n>,\n>,\n> ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ\n\n{finite length W (F)[ GQp ]-mods }\n\n> O\n> O\n\nThen (1) {admissible J-H factors } ⊆ { J-H factors of π(r, λ, χ )}.\n\nRecall that irreducible admissible is the same as irreducible, smooth with central character.\n\nSmooth: every vector fixed by an open sub-group.\n\nAdmissible: above with finite length. Finite length admissible is equivalent to finite length, smooth and centrally cofinite. These are, in turn, the same as finite length with Jordan-Holder factors in (1). The finite dimensional clause doesn't matter on the Galois side - this is (maybe?) a local analogue of Ihara's lemma. 2.1.1. Deformation theory. Let A be an artin ring. Let π/A be finite free over A. Apply MF, gives\n\nρ/A, deforming:\n\nπ ↔ ρ\n\nConsider Hom( π, A ), if A was killed by A/p n, then consider Hom( π, Z/p nZ). We have Hom(lim\n\n> →\n\nAn, A ) = lim\n\n> ←\n\nHom( An, A ) ∼= An\n\nDefinition of MF: Define MF( π) = V ∗(π ⊗A Hom( A, Qp/Zp)).\n\nTake\n\nπ = Hom( π∗, A )\n\n> [U+000F]\n> [U+000F]\n\nP//oo\n\n> [U+000F]\n> [U+000F]\n\nπF / / π∗ MF ′\n\n> //\n\nρ\n\nHom( π, F)\n\nwe have GL 2(Qp) on both sides, where P is category of pro-free A-modules. In fact, action is integral, thus we actually have an action of F[[GL 2(Zp)]], the latter functor being covariant. Need to be careful about changing scalars - analogous to defining Hom's of sheaves. 2.2. The (mod p) correspondence. Let G = GL 2(Qp), B =\n\n( ∗ ∗\n\n0 ∗\n\n)\n\nand B =\n\n( ∗ 0\n\n∗ ∗\n\n). We want\n\nρ =\n\n( χ ∗\n\n0 ψ\n\n)??\n\n→ π\n\nIf χψ −1 6 = ω 6 = 1 then 0 / / Ind G\n\n> B\n\nχ ⊗ ψω / / π / / Ind G\n\n> B\n\nψ ⊗ χω / / 0p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 5\n\n0 ⊆ St ⊆ ·\n\n︸ ︷︷ ︸\n\n> 1\n\n⊆ π\n\n︸︷︷︸\n\n> Ind G\n> Bω−1⊗ω.\n\n2.3. Jacquet Modules. Let T =\n\n( ∗ 00 ∗\n\n). Consider Ind BG χ ⊗ ψω\n\nThen Hom G(V, Ind G\n\n> B\n\n(χ ⊗ ψω )) ∼= Hom B (V, ψω ⊗ χ)and (Ind G\n\n> B\n\nχ ⊗ ψω )N = ψω ⊗ χ where N =\n\n( 1 ∗\n\n0 1\n\n)\n\nThere is a action of T on the left-hand side. We define the ordinary Jacquet functor as\n\nJord (V ) =\n\nV\n\n1 Zp\n\n0 1\n\n>!\n\n\n\n> ord\n\nwith an action of Up.\n\nJord (Ind G\n\n> B\n\n(χ ⊗ χω )) = χ ⊗ ψω\n\nHom(Ind G\n\n> B\n\nU, V ) = Hom T (U, J ord (V )) One can compute:\n\nR1Jord (V ) = ( VN )( ω−1 ⊗ ω).\n\nWhere R1 is the first derived functor of the ordinary Jacquet functor. N.B. the ordinary Jacquet functor has cohomological dimension 2. Does\n\nH2(GL 2(Qp), F p) = 0? Is the 'bar' irrelevant? Computing cohomology difficult because complicated interactions with the topology and the representation theory. 3. Some open problems\n\n3.1. Conjecture: Emerton.",
   "evidence": "The canonical input is an overlong extraction from Michael Volpato's notes for the AIM workshop *p-adic Representations, Modularity, and Beyond* (20--24 February 2006). The original PDF identifies the relevant item on page 2 (PDF page index 1), immediately before the heading `1.2. Colmez's functor and an expectation.' The actual record ends there. Everything in the input beginning with that heading---including Colmez's functor, the later Emerton discussion, and the later open problems---belongs to other sections and is extraction spillover. It is not part of Conjecture 2 and is not treated as an assigned problem here.",
   "classification_method": "damaged_source_without_verified_clean_repair",
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  "AIM-REPRESENTATION_THEORY-0105": {
   "statement_status": "exact",
   "original_statement": "Conjecture 3 (Emerton). Let ρ be absolutely irreducible odd residual two-dimensional represen-tation of GQ. Let mN ⊆ TN ⊗ Fp correspond to ρ (mN generated by (T` − Tr ρ(Frob `)) `-N )inj lim \n\n> N\n\n[\n\nH1(X(N )Q, Fp)[ m]\n\n] ∼=\n\n(⊗ ′\n\nπmodified \n\n> `\n\n(\n\nρ|GQ`\n\n)) \n\n⊗ ρ\n\nwhere πmodified \n\n> `\n\n(\n\nρ|GQ`\n\n)\n\nis a finite length admissible smooth representation of GL 2(Q`). (Emerton gives a specific πmodified \n\n> `\n\n)\n\n3.1.1. Calegari. - From the explicit list of these πmod \n\n> `\n\ncan one calculate explicitly the multiplicities of the m-eigenspaces in H1(X(N )Q, Fp) at finite level? (Emerton: away from p \"easy\", however at \n\np is a thesis question.) 6 NOTES BY MICHAEL VOLPATO \n\n3.1.2. Diamond. - What about reducible representations? Further: what about a version for Shimura curves? (For quaternion algebras unramified over p)Would this help understand what happens at p?3.2. Berger. In the notation of Breuil-Berger:",
   "clean_statement": "Conjecture 3 (Emerton). Let ρ be absolutely irreducible odd residual two-dimensional represen-tation of GQ. Let mN ⊆ TN ⊗ Fp correspond to ρ (mN generated by (T` − Tr ρ(Frob `)) `-N )inj lim\n\n> N\n\n[\n\nH1(X(N )Q, Fp)[ m]\n\n] ∼=\n\n(⊗ ′\n\nπmodified\n\n> `\n\n(\n\nρ|GQ`\n\n))\n\n⊗ ρ\n\nwhere πmodified\n\n> `\n\n(\n\nρ|GQ`\n\n)\n\nis a finite length admissible smooth representation of GL 2(Q`). (Emerton gives a specific πmodified\n\n> `\n\n)\n\n3.1.1. Calegari. - From the explicit list of these πmod\n\n> `\n\ncan one calculate explicitly the multiplicities of the m-eigenspaces in H1(X(N )Q, Fp) at finite level? (Emerton: away from p \"easy\", however at\n\np is a thesis question.) 6 NOTES BY MICHAEL VOLPATO\n\n3.1.2. Diamond. - What about reducible representations? Further: what about a version for Shimura curves? (For quaternion algebras unramified over p)Would this help understand what happens at p?3.2. Berger. In the notation of Breuil-Berger:",
   "public_statement": "Conjecture 3 (Emerton). Let ρ be absolutely irreducible odd residual two-dimensional represen-tation of GQ. Let mN ⊆ TN ⊗ Fp correspond to ρ (mN generated by (T` − Tr ρ(Frob `)) `-N )inj lim\n\n> N\n\n[\n\nH1(X(N )Q, Fp)[ m]\n\n] ∼=\n\n(⊗ ′\n\nπmodified\n\n> `\n\n(\n\nρ|GQ`\n\n))\n\n⊗ ρ\n\nwhere πmodified\n\n> `\n\n(\n\nρ|GQ`\n\n)\n\nis a finite length admissible smooth representation of GL 2(Q`). (Emerton gives a specific πmodified\n\n> `\n\n)\n\n3.1.1. Calegari. - From the explicit list of these πmod\n\n> `\n\ncan one calculate explicitly the multiplicities of the m-eigenspaces in H1(X(N )Q, Fp) at finite level? (Emerton: away from p \"easy\", however at\n\np is a thesis question.) 6 NOTES BY MICHAEL VOLPATO\n\n3.1.2. Diamond. - What about reducible representations? Further: what about a version for Shimura curves? (For quaternion algebras unramified over p)Would this help understand what happens at p?3.2. Berger. In the notation of Breuil-Berger:",
   "evidence": "The record is Conjecture 3 in Michael Volpato's notes from the 2006 AIM workshop *(p)-adic representations, modularity, and beyond*, followed by questions of Calegari and Diamond. The PDF warns that the notes may contain transcription errors. Its text extraction also drops overlines and some subscripts. With those losses restored in the conventional way, the conjecture reads as follows.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
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  "AIM-REPRESENTATION_THEORY-0106": {
   "statement_status": "exact",
   "original_statement": "Question 1 (Berger). Let V/Fp be an irreducible Galois representation of GQp of arbitrary dimen-sion. Consider Ω( V ) = \n\n(\n\nproj lim \n\n> ψ\n\nD(V )\n\n)b\n\n> ∗\n\n= Hom Cont \n\n(\n\nproj lim \n\n> ψ\n\nD(V )\n\n)b, Fp\n\n\n\nLet P =\n\n( ∗ ∗\n\n0 1\n\n), and Ω( V ) is a smooth irreducible admissible representation of P. Which such P -representations occur as Ω( V ) for some V?3.2.1. Ribet. - Does the P -action extend to a larger group in any natural way? For a oneor two-dimensional V, the answer is YES. 3.3. Buzzard. Why ( ϕ, Γ)-modules?! Is there a useful generalization? In particular, can one replace Γ by higher dimensional p-adic Lie groups, and get a \"( ϕ, Γ)-module\". This has to classify \n\np-adic Glaois representations, not just modulo p.Kisin recalls for us that Fontaine gives a procedure using a norm field that produces a theory that does this for mod p representations for any p-adic Lie group Γ. 3.4. Breuil.",
   "clean_statement": "Question 1 (Berger). Let V/Fp be an irreducible Galois representation of GQp of arbitrary dimen-sion. Consider Ω( V ) =\n\n(\n\nproj lim\n\n> ψ\n\nD(V )\n\n)b\n\n> ∗\n\n= Hom Cont\n\n(\n\nproj lim\n\n> ψ\n\nD(V )\n\n)b, Fp\n\n\n\nLet P =\n\n( ∗ ∗\n\n0 1\n\n), and Ω( V ) is a smooth irreducible admissible representation of P. Which such P -representations occur as Ω( V ) for some V?3.2.1. Ribet. - Does the P -action extend to a larger group in any natural way? For a oneor two-dimensional V, the answer is YES. 3.3. Buzzard. Why ( ϕ, Γ)-modules?! Is there a useful generalization? In particular, can one replace Γ by higher dimensional p-adic Lie groups, and get a \"( ϕ, Γ)-module\". This has to classify\n\np-adic Glaois representations, not just modulo p.Kisin recalls for us that Fontaine gives a procedure using a norm field that produces a theory that does this for mod p representations for any p-adic Lie group Γ. 3.4. Breuil.",
   "public_statement": "Question 1 (Berger). Let V/Fp be an irreducible Galois representation of GQp of arbitrary dimen-sion. Consider Ω( V ) =\n\n(\n\nproj lim\n\n> ψ\n\nD(V )\n\n)b\n\n> ∗\n\n= Hom Cont\n\n(\n\nproj lim\n\n> ψ\n\nD(V )\n\n)b, Fp\n\n\n\nLet P =\n\n( ∗ ∗\n\n0 1\n\n), and Ω( V ) is a smooth irreducible admissible representation of P. Which such P -representations occur as Ω( V ) for some V?3.2.1. Ribet. - Does the P -action extend to a larger group in any natural way? For a oneor two-dimensional V, the answer is YES. 3.3. Buzzard. Why ( ϕ, Γ)-modules?! Is there a useful generalization? In particular, can one replace Γ by higher dimensional p-adic Lie groups, and get a \"( ϕ, Γ)-module\". This has to classify\n\np-adic Glaois representations, not just modulo p.Kisin recalls for us that Fontaine gives a procedure using a norm field that produces a theory that does this for mod p representations for any p-adic Lie group Γ. 3.4. Breuil.",
   "evidence": "The canonical record is extracted from §3.2 of Michael Volpato's notes for the 2006 AIM workshop *$p$-adic representations, modularity, and beyond*. The primary PDF gives the following question before starting the separately labelled §3.2.1 and §3.3 discussions.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
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  },
  "AIM-REPRESENTATION_THEORY-0107": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 2. Let V be a two-dimensional irreducible potentially crystalline representation of GQp,assume that it is of supercuspidal type (i.e. the associated WD-group representation is irreducible) Let B(V ) be the associated (conjecturally) irreducible admissible Banach space representation. Can one prove that the locally analytic vectors in B(V ) determine the Hodge filtration on Dpcris (V )? 3.4.1. Emerton. More generally: relate B(V )an to Drig (V ). Can one relate ( B(V )an )′ to the de Rham cohomology of the coverings of the p-adic upper half-plane? (Where ' ′' is the dual.) 4. Serre's conjecture: Ribet \n\nLet p be a prime, for instance, let p = 5. Then suppose we have a Galois representation \n\nρ: GQ → GL 2(Fp)which is irreducible, odd, the question is, is it modular? Khare induction on the prime. Tate proved for p = 2, 3 Tate and Serre proved that this is vacuously the case. Start at ρ lift to ˜ ρ: GQ → GL 2(Ep) - want ˜ ρ to be minimal, i.e. with prescribed Serre level and weight k for 2 ≤ k ≤ p + 1 - it should be E-rational, compatible. This representation should lift to a Galois representation which is geometric etc... ˜ ρ = ˜ ρp we have a family (˜ ρp). Should look as if it comes from a modular form. Need to interweave Taylor's potential modularity theorem with deformation theory. Then use an analogue of Wiles' 3-5 trick. One technical obstacle, is you may get a reducible representation, then one has to apply Skinner-Wiles - which means you must check the hypothesis! Khare inducts simultaneously on the weight and the prime characteristic. Ideally, one wants to move to a lower prime, and simultaneously control the weight, in particular, reduce it. Then induct. p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 7\n\n4.1. Interplay between Taylor's theorem and deformation theory. Consider level N = 1 and residue characteristic p = 3 - then it is a theorem of Serre that Serre's conjecture is true in both the strong and weak formulations. In particular, every residual Galois representation is modular (because there are none!). Consider ρp, take a minimal lift ρp with level one and weight 2. Then include ρp in a strictly compatible family {ρp}. In general one must keep track of ramification. In this case, however S = ∅.Taylor's potential modularity states that given a Barsotti-Tate representation ρp there exists a totally real field F such that ρp|GF is modular. In particular, there is a Hilbert modular form over \n\nF of parallel weight two such that ρp|GF\n\n∼= ρf, p for some p | p. One can do this process to insure that the images of ρp and ρp|GF have the same image. 5. Buzzard: Serre's conjecture over Q\n\nFix an algebraic closure Qp of Qp and let F denote an unramified extension of Qp. Let Gal( Qp/Qp) ⊇ I ⊆ Gal( F /F ) = GF. Local class field theory gives us a canonical isomorphism \n\nGab \n\n> F\n\n∼= F ×, the image of I in Gab \n\n> F\n\nis identified with O× \n\n> F. Therefore, there exists a canonical quotient \n\nIn of I identified with k× where k denotes the residue field of F (where k = pn). We say that a character χ: I → F× \n\n> p\n\nhas level n if it factors as \n\nI → In → F× \n\n> p.\n\nThere are pn − 1 characters of level n. We have \n\nI / / / / In = k×.\n\nA character of level n is fundamental if the induced group homomorphism k× → F× \n\n> p\n\nextends to an injection of fields k ↪ → F p.Let F/ Qp be an unramified extension. \n\nLemma 4. If ρ: Gal( F /F ) → GL 2(Fp) is continuous, then either \n\nρ ∼=\n\n( χ1 ∗\n\n0 χ2\n\n)\n\nwhere χ1|I and χ|I have level n1; or ρ is irreducible and \n\nρ|I ∼=\n\n( χ 00 χpn\n\n)\n\nwhere χ of level 2n.\n\nIf f = ∑ \n\n> n≥1\n\nanqn ∈ Fp[[ q]] is a mod p modular cusp form of level N, p - N and a1 = 1 and f is an eigenform. Then there exists a Galois representation ρF = ρ associated to FρF: GQ → GL 2(Fp)continuous, odd, semisimple. If ` is a prime and ` - N p, then Tr( ρF (Frob arith \n\n> `\n\n)) is a` and det( ρF ) = \n\nχk−1cyc a dirichlet character of level n.What can we say about ρ|Dp? Good results 2 ≤ k ≤ p + 1. Answer: in this case if ap is nonzero, then ρ|Dp is reducible ( χ1 ∗\n\n0 χ2\n\n)\n\nand χ|I = ωk−1 and χ2|I = trivial, where ω is the mod p cyclotomic character. If ap = 0, then ρ|Dp\n\nis irreducible and \n\nρ|Ip ∼=\n\n( ψk−1 00 ψp(k−1) \n\n)8 NOTES BY MICHAEL VOLPATO \n\nwhere ψ is fundamental of level 2. \n\nSome facts: - If f = ∑ anqn is a mod p weight k cusp form, then Af = ∑ anqn is a mod p\n\nweight k + ( p − 1) cusp form. And Θ f = ∑ na nqn is a mod p weight k + ( p + 1) cusp form. \n\nρAf ∼= ρf and ρΘf ∼= ρf ⊗ ω\n\nSo if ρ ∼= ρf for some f of weight k, then ρ ⊗ ω modular weight k + ( p + 1) and ρ is modular of weight k + ( p − 1). These are the ingredients of Serre's precise conjecture. If ρ: GQ → GL 2(Fp) is continuous, odd and irreducible, then Serre predicts ρ is modular and furthermore predicts the precise weight k(ρ)for which there exists an f of weight k such that ρ ∼= ρF.\n\nIdea for k(ρ): - say \n\nρ|Ip ∼=\n\n( ωa ∗\n\n0 ψp(k−1) \n\n)\n\nand (ω−b ⊗ ρ)|Ip ∼\n\n( ω(a−b) ∗\n\n0 1\n\n)\n\nlooks modular of weight a − b + 1, therefore ρ looks modular of weight ( a − b + 1) + b(p + 1), if furthermore ∗ = 0 then \n\nρ|Ip ∼\n\n( ωb ∗\n\n0 ωa\n\n)\n\nand same trick gives another k.\n\nHow do you generalize to totally real fields? Annoying fact: - if f is a characteristic zero Hilbert modular form of weight ( k1,..., k α) and all ki congruent modulo 2, of level prime to p. Then for w ∈ Z (which is congruent to k mod 2) there exists an automorphic form πf,α associated to f and ρπf,α is crystalline at all places of F\n\nabove p thne det ρπf,w = ωinteger × char conductor prime to p.\n\nProblem: - typically there are mod p totally odd representations of Gal( F /F ) whose deter-minant is not the reduction of ωint × (prime to p). Therefore, the naive generalization of Serre's conjecture should NOT say that for all ρ: GF → GL 2(Fp) continuous totally odd irreducible are modular coming from a Hilbert modular form of level prime to p.\n\nFred's fix: - totally rethink the notion of weight. Say f is a weight k classical mod p modular form, where 2 < k < p + 1, of level N prime to \n\np. One can lift f to some characteristic zero form F of weight 2 and level Γ 1(N p ), of character ω\n\nat p.. Can find ρf in Jac( Xn(N p ))[ p]. Assume from now on that everything has an implicit level \n\nN. One can find f is a certain subspace of Pic ◦(X(p))[ p] where X(p)/Q is the non-geometrically connected modular curve of level Γ 1(N ) ×\n\n( 1 00 1\n\n)\n\n(mod p) over Q.The group Pic ◦(X(p))[ p]( Q) has an action of Gal( Q/Q) and a commuting action of GL 2(Fp), and our modular ρF lives in the subspace of this Pic ◦ where \n\n( ∗ ∗\n\n0 ∗\n\n)\n\n⊆ GL 2(Fp) is acting in a certain explicit way. One can now write down an explicit irreducible mod p representation of GL 2(Fp), say Vk, such that ρ ⊆ Hom G(V, Pic ◦(X(p))[ p]( Q)). This latter space is the one which generalizes to the totally real setting. \n\nEmerton: - ρ modular at weight V - where V is any irreducible mod p representation of GL 2(Fp), if ρ ⊆ Hom G(V, H 1et (X(p), Fp)). \n\nDiamond: - ρ modular of weight V if \n\nρ ⊆ (VFp ⊗ Pic ◦(X(p))[ p]( Q)) G = Hom G(V ∗, Pic ◦(X(p))[ p]( Q)) p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 9\n\nOne gets a reformulation of Serre's conjecture: If ρ is continuous, odd, irreducible GQ → GL 2(Fp), then ρ is modular, and furthermore it is modular for weight V - for which we have a recipe. The recipe now looks \"nicer\", because the irreducible mod p representations of GL 2(Fp) are all of the form det a ⊗Sym b−1(F2\n\n> p\n\n) where 0 ≤ a ≤ p − 2 and 1 ≤ b ≤ p.Get a simpler picture, e.g. in the irreducible case: \n\nρ|I = ωa\n\n( ψb 00 ψpb \n\n)\n\nwhere ω is fundamental of level 1 and ψ fundamental level 2. Fred predicts weight V = det a ⊗Sym b−1.6. Gee: Proof of Buzzard-Diamond-Jarvis \n\nLet F be totally real, and let p > 2 be unramified in F. Let \n\nρ: GF → GL 2(Fp),\n\nbe modular of weights {V } where V are irreducible characteristic p representations of GL 2(OF /p )and V = ⊗ \n\n> V|p\n\nVav,bv where av, bv are [ kv: Fp]-tuples indexed by σ: kv ↪→ Fp, where 0 ≤ av ≤ p−1, not all av = p − 1 and 1 ≤ bv ≤ p.\n\nVav,bv = ⊗ \n\n> σ:kv↪→Fp\n\n(\n\ndet av Sym bv −1k2\n\n> v\n\n)\n\n⊗σ Fp.\n\nThere exists explicit recipe for ρ|Gv to {Vvp respresentations of GL 2(kv)} and ρ to {V = ⊗Vv}.Assume p inert: If ρ|Gp is irreducible then there are 2 f weights, f = [ F: Q]. If ρ|Gp is reducible, there are ≤ 2f weights (generically). \n\n( ψ1 ∗\n\n0 ψ2\n\n)\n\nif ∗ = 0 you get all the weights, if ∗ is generic, get 1 weights. Say a weight ⊗v|pVav,bv is regular if 2 ≤ bv ≤ p − 2 for all v.\n\nTheorem 5 (Gee). Assume further that ρ(GF ) is non-solvable. (Can be removed). Assume also that for each v: ρ|Gv is not scalar. If V is a regular weight, ρ is irreducible, ρ is modular of weight \n\nV if and only if B-D-J predicted that it is. \n\nTheorem 6 (Gee). For p > 2 and F a totally real field, with p unramified in F. Let E/ Qp be a finite extension, and let O denote the ring of integers of E. Let \n\nρ: GF → GL 2(O),\n\nbe continuous, unramified outside of a finite set of primes, and det ρ = (cyc)( finite order ). Suppose that \n\n(1) ρ|Gv is potentially Barsotti-Tate for all v | p.\n\n(2) ρ is modular \n\n(3) ρ|GF (ζp ) is absolutely irreducible. then ρ is modular. \n\nAssume 2 ≤ k ≤ p, p > 2 if a modular newform of level Γ 1(N ), p - N and weight k.\n\nρf: GQ → GL 2(Fp),10 NOTES BY MICHAEL VOLPATO \n\nassume ρf also irreducible. Assume \n\nρf |Gp ∼=\n\n( ψ1ωk−1 00 ψ2\n\n)\n\nwhere ψ1 and ψ2 are unramified an ωk−1ψ1 6 = ψ2.then ( ρf ⊗ ωk′−1)|Gp ∼=\n\n( ψ2ωk′−1 00 ψ1\n\n)\n\nwhere \n\nk′ =\n\n{\n\np + 1 − k if k 6 = pp if k = p\n\nSerre predicts that there exists an eigenform of weight k′, of level Γ 1(N ), such that ρg ∼= ρf ⊗\n\nωk′−1. If k = p, the Up-eigenvalue of g is congruent to ψ2(Frob p) modulo p.By using Hida theory it suffices to find g′ of level Γ 1(N p ), and weight 2 with \n\nρg′ ∼= ρf ⊗ ωk′−1.\n\nthen \n\nρg′ ∼=\n\n( ˜ψ2 ˜ωk′−2χcyc ∗\n\n0 ˜ψ1\n\n)\n\nwhere˜stands for Teichm¨ uller lifts. Assume that ρ(GQ) is non-solvable. Now: (1) find ρg′, then (2) prove ρg′ is modular. For (2) we simply check the hypothesis of the earlier theorem. (1) follows from a theorem of Ramakrishna (and Taylor). In essence on has to check that the local deformation ring at p is large enough - a dimension calculation. For B-D-J one has to consider many lifts. In fact, the lifts we want to consider are potentially Barsotti-Tate of a specified type. (These types are always tame). Starting the a residual rep-resentation considers all lifts of this type. Then using combinatorial arguments you control the weights. 7. Buzzard: p-adic Local Langlands \n\nFor GL 2(K), where K/ Qp finite: it bijects supercuspidal (infinite dimensional) representations of GL 2(K) with irreducible 2-dimensional C-representations of the Weil group WK. This first set is contained in the set of smooth irreducible admissible representations of GL 2(K). The latter set is contained in the set of F -semisimple 2-dimensional Weil-Delgine representations. Vigneras: situation is also good when we replace C by F` for ` 6 = p.What about Fp? What about a \"mod p local Langlands?\" Objects on the right-hand side: {continuous ρ: Gal( K/K ) → GL 2(Fp)}, this set contains the irreducible representations. At least for K/ Qp unramified, then ρ gives rise to a finite set of irreducible representations of GL 2(k) where k denotes the residue field of K.e.g. When K = Qp and ρ irreducible \n\nρ|I =\n\n( ψb 00 ψpb \n\n)\n\nyou get Sym b−1 and also twist of Sym p+b.Take F/ Qp unramified, with ring of integers O. Let K = GL 2(O), and Z = F × ↪→ G = GL 2(F )and k be the residue field of F. If V is a finite dimensional representation of GL 2(k) over Fp.Then make V a representation of K by letting K act via GL 2(k) and then a representation of KZ \n\nby letting O× act via K and letting \n\n( p 00 p\n\n)\n\nact trivially. Define c − Ind GKZ V to be the set of p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 11 \n\nfunctions f: G → V such that f (kg ) = k ∗ f (g) for all k ∈ KZ, where ∗ is the action of KZ on V;and such that the support of f is a finite union of costs of KZ. We define a G-action on c − Ind by putting ( gf )( g′) = f (g′g). Note that c − Ind is an infinite-dimensional representation of G.Consider all the embeddings k σ\n\n↪→ Fp. Now assume that V = ⊗ \n\n> σ\n\nσ ◦ Sym rσ k2 where 0 ≤ rσ ≤\n\np − 1. [Up to twists this is all the irreducible representations of GL 2(k). Explicitly: homogeneous polynomials of degree rσ in two-variables x and y such that \n\n(( a bc d\n\n)\n\nf\n\n)\n\n(x, y ) = f (ax + cy, bx + dy )Define an Fp-linear map U = ⊗Uσ: V → V, by \n\nUσ(xiyrσ −i) = 0 if i > 0 and \n\nUσ(yrσ ) = yrσ.\n\nNow define a map \n\nϕ: G → End Fp (V )by \n\nϕ\n\n(( 1 00 p−1\n\n)) \n\n= U\n\nExtend to KZ \n\n( 1 00 p−1\n\n)\n\nKZ by ϕ(ka −1k′) = k ∗ ϕ(a−1 ∗ k′. Then extend to G by 0. So ϕ gives rise to a G-endomorphism of c − Ind GKZ V in a natural way - call this T.Wonderful observations of Barthel-Livn´ e: Let Wr:= c − Ind GKZ ⊗ Sym rσ.\n\nIf λ ∈ Fp, λ 6 = 0, then Wr/(T − λ) is almost always an irreducible smooth admissible represen-tation of G (call these principal series ), except occasionally it has length 2, 1 − d subquotient and Steinberg subquotient. An irreducible representation of G is supersingular if its a quotient of Wr/(T ). The principal series are never isomorphic to 1 − d which are never isomorphic to the Steinberg which is never isomorphic to principal series representations and none are ever isomorphic to supersingular. Within the principal series, 1 − d, Steinberg understood. Supersingular case: mysterious Wr/(T ) does have infinite length if K 6 = Qp.Breuil: restricts to K = Qp and r = r ∈ { 0,..., p −1}, Breuil finishes the story: he shows Wr/(T )is irreducible and Wr/(T ) is isomorphic to twist of Ws/(T ) if and only if r = s or r + s = p − 1 and know exactly the twist. B-L observe that: any smooth irreducible admissible Fp-representation of G = GL 2(F ), with a central character is 1 − d, principal series, Steinberg, or supersingular - up to twist. For F = Qp we can write down all smooth admissible irreducible representations of GL 2(Qp)and all two-dimensional representations of GQp.Restrict to the semi-simple case: Choose a lift F of Frobenius in Gal( Qp/Qp) and restrict to representations ρ of GQp such that det ρ(F ) = 1. If ρ|I =\n\n( ψr+1 00 ψp(r+1) \n\n), where ψ is fundamental of level 2, then match with Wr/(T ). If ρ =\n\n( ωr+1 × unr( λ) 00 unr( λ−1)\n\n)\n\nthen match with (Wr/(T − λ)) ss ⊕ (Wp−3−r/(T − λ−1) ⊗ ωr+1 )ss.\n\nThis all gives us a semi-simple local Langlands conjecture (theorem for GL 2(Qp)). 12 NOTES BY MICHAEL VOLPATO \n\nMatthew Emerton's picture: \n\nHΓ( p) H\n\nV \u001f  / / H1(X(p), Fp)ρ \u001f  / / inj lim H1(X(pr), Fp)ρ\n\nThese groups all have a GL 2(F) action on them. Finally, consider GL 2(F ) where F = Qp2 and r = ( r0, r 1), and let Vr be a representation of GL 2(Fp2 ). It is known that Wr/(T ) has infinite length. This is too big. Barthel-Livn´ e suggest that we should consider quotients of this object. Paskunas: writes downa particular irreducible quotient of Wr/(T ) call it Pr. Paskunas showed that Pr is isomorphic to a twist of Ps if and only if r = s or r = p − 1 − s where r ∈ { 0,..., p − 1}2.Up to unramified twist we get exactly q(q − 1) /2 irreducible non-isomorphic supersingular rep-resentations of GL 2(F ). Now we guess: irreducible representations of GF ←→ Pr -- WRONG. \n\nVa,b = det a0 ×σ ◦ det a1 ×Sym b0−1 ⊗ σ ◦ Sym b1−1. Fp2 ↪→ Fp\n\nρ irreducible and (2) ρ|I =\n\n( ψr0+1+ p(r1+1) 4 00 ψ4p2(this )\n\n)\n\nIf ρ → Pr for some r, then because Pr is isomorphic to a twist of Pp−1−r we must see 2 lines such that sum of b's is ( p − 1, p − 1) Fred predicts Va,b:\n\na0 a1 b0 − 1 b1 − 1\n\n0 0 r0 r1\n\nr0 + 1 −1 p − 2 − r0 r1 + 1 \n\n0 r1 + 1 r0 − 1 p − 2 − r1\n\nr0 r1 + 1 p − 1 − r0 p − 3 − r1\n\nIf ρ is reducible, then \n\nρ|I ∼=\n\n( ψr0+1+ p(r1+1) 2 00 1\n\n)\n\n←→ P S ⊕ P S ⊕ Pr\n\nFred predicts: \n\na0 a1 b0 − 1 b1 − 1\n\n0 0 r0 r1\n\nr0 + 1 r1 + 1 p − 3 − r0 p − 3 − r1\n\np − 1 r1 r0 + 1 p − 2 − r\n\nr0 p − 1 p − 2 − r0 r1 + 1 \n\nThus (2) corresponds to a new quotient of Wr/(T ). Matt had an insighful diagram with (too!) much commentary: \n\n8. Kisin: Pseudo-representations \n\nPseudo-representations: - G a group and A a ring, or more generally an A-algebra R. Consider functions T: R → A, satisfying two conditions: (1) T (xy ) = T (yx ), (2) depends on d, (3) \n\nT (1) = d ∈ N. Furthermore we require d! is invertible in A. For d = 2, thus, we suppose that 2 is invertible in A. If you have an element σ ∈ R then we can define T (σ). p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 13 \n\nDefine S(σ):= 1 \n\n> 2\n\n(T (σ)2 − T (σ2)), then S is a character, i.e. S(σ1)S(σ2) = S(σ1σ2). We have \n\nX2 − T (σ)X + S(σ).\n\nDefine Ker( T ) = {x ∈ R: T (xy ) = 0 ∀y ∈ R}. Then we have a map \n\nR = R/ Ker( T ) → A, \n\nfor σ ∈ R: Pσ(x) - the characteristic polynomial of σ. Ask the following question: is Pσ(σ) = 0. \n\nTheorem 7 (Taylor). If A is an algebraically closed field, then there is a correspondence between: the set of pseudo-representations and the set of semi-simple representations. \n\nAlways genuine representations give pseudo-representations. 8.1.",
   "clean_statement": null,
   "public_statement": "Question 2. Let V be a two-dimensional irreducible potentially crystalline representation of GQp,assume that it is of supercuspidal type (i.e. the associated WD-group representation is irreducible) Let B(V ) be the associated (conjecturally) irreducible admissible Banach space representation. Can one prove that the locally analytic vectors in B(V ) determine the Hodge filtration on Dpcris (V )? 3.4.1. Emerton. More generally: relate B(V )an to Drig (V ). Can one relate ( B(V )an )′ to the de Rham cohomology of the coverings of the p-adic upper half-plane? (Where ' ′' is the dual.) 4. Serre's conjecture: Ribet\n\nLet p be a prime, for instance, let p = 5. Then suppose we have a Galois representation\n\nρ: GQ → GL 2(Fp)which is irreducible, odd, the question is, is it modular? Khare induction on the prime. Tate proved for p = 2, 3 Tate and Serre proved that this is vacuously the case. Start at ρ lift to ˜ ρ: GQ → GL 2(Ep) - want ˜ ρ to be minimal, i.e. with prescribed Serre level and weight k for 2 ≤ k ≤ p + 1 - it should be E-rational, compatible. This representation should lift to a Galois representation which is geometric etc... ˜ ρ = ˜ ρp we have a family (˜ ρp). Should look as if it comes from a modular form. Need to interweave Taylor's potential modularity theorem with deformation theory. Then use an analogue of Wiles' 3-5 trick. One technical obstacle, is you may get a reducible representation, then one has to apply Skinner-Wiles - which means you must check the hypothesis! Khare inducts simultaneously on the weight and the prime characteristic. Ideally, one wants to move to a lower prime, and simultaneously control the weight, in particular, reduce it. Then induct. p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 7\n\n4.1. Interplay between Taylor's theorem and deformation theory. Consider level N = 1 and residue characteristic p = 3 - then it is a theorem of Serre that Serre's conjecture is true in both the strong and weak formulations. In particular, every residual Galois representation is modular (because there are none!). Consider ρp, take a minimal lift ρp with level one and weight 2. Then include ρp in a strictly compatible family {ρp}. In general one must keep track of ramification. In this case, however S = ∅.Taylor's potential modularity states that given a Barsotti-Tate representation ρp there exists a totally real field F such that ρp|GF is modular. In particular, there is a Hilbert modular form over\n\nF of parallel weight two such that ρp|GF\n\n∼= ρf, p for some p | p. One can do this process to insure that the images of ρp and ρp|GF have the same image. 5. Buzzard: Serre's conjecture over Q\n\nFix an algebraic closure Qp of Qp and let F denote an unramified extension of Qp. Let Gal( Qp/Qp) ⊇ I ⊆ Gal( F /F ) = GF. Local class field theory gives us a canonical isomorphism\n\nGab\n\n> F\n\n∼= F ×, the image of I in Gab\n\n> F\n\nis identified with O×\n\n> F. Therefore, there exists a canonical quotient\n\nIn of I identified with k× where k denotes the residue field of F (where k = pn). We say that a character χ: I → F×\n\n> p\n\nhas level n if it factors as\n\nI → In → F×\n\n> p.\n\nThere are pn − 1 characters of level n. We have\n\nI / / / / In = k×.\n\nA character of level n is fundamental if the induced group homomorphism k× → F×\n\n> p\n\nextends to an injection of fields k ↪ → F p.Let F/ Qp be an unramified extension.\n\nLemma 4. If ρ: Gal( F /F ) → GL 2(Fp) is continuous, then either\n\nρ ∼=\n\n( χ1 ∗\n\n0 χ2\n\n)\n\nwhere χ1|I and χ|I have level n1; or ρ is irreducible and\n\nρ|I ∼=\n\n( χ 00 χpn\n\n)\n\nwhere χ of level 2n.\n\nIf f = ∑\n\n> n≥1\n\nanqn ∈ Fp[[ q]] is a mod p modular cusp form of level N, p - N and a1 = 1 and f is an eigenform. Then there exists a Galois representation ρF = ρ associated to FρF: GQ → GL 2(Fp)continuous, odd, semisimple. If ` is a prime and ` - N p, then Tr( ρF (Frob arith\n\n> `\n\n)) is a` and det( ρF ) =\n\nχk−1cyc a dirichlet character of level n.What can we say about ρ|Dp? Good results 2 ≤ k ≤ p + 1. Answer: in this case if ap is nonzero, then ρ|Dp is reducible ( χ1 ∗\n\n0 χ2\n\n)\n\nand χ|I = ωk−1 and χ2|I = trivial, where ω is the mod p cyclotomic character. If ap = 0, then ρ|Dp\n\nis irreducible and\n\nρ|Ip ∼=\n\n( ψk−1 00 ψp(k−1)\n\n)8 NOTES BY MICHAEL VOLPATO\n\nwhere ψ is fundamental of level 2.\n\nSome facts: - If f = ∑ anqn is a mod p weight k cusp form, then Af = ∑ anqn is a mod p\n\nweight k + ( p − 1) cusp form. And Θ f = ∑ na nqn is a mod p weight k + ( p + 1) cusp form.\n\nρAf ∼= ρf and ρΘf ∼= ρf ⊗ ω\n\nSo if ρ ∼= ρf for some f of weight k, then ρ ⊗ ω modular weight k + ( p + 1) and ρ is modular of weight k + ( p − 1). These are the ingredients of Serre's precise conjecture. If ρ: GQ → GL 2(Fp) is continuous, odd and irreducible, then Serre predicts ρ is modular and furthermore predicts the precise weight k(ρ)for which there exists an f of weight k such that ρ ∼= ρF.\n\nIdea for k(ρ): - say\n\nρ|Ip ∼=\n\n( ωa ∗\n\n0 ψp(k−1)\n\n)\n\nand (ω−b ⊗ ρ)|Ip ∼\n\n( ω(a−b) ∗\n\n0 1\n\n)\n\nlooks modular of weight a − b + 1, therefore ρ looks modular of weight ( a − b + 1) + b(p + 1), if furthermore ∗ = 0 then\n\nρ|Ip ∼\n\n( ωb ∗\n\n0 ωa\n\n)\n\nand same trick gives another k.\n\nHow do you generalize to totally real fields? Annoying fact: - if f is a characteristic zero Hilbert modular form of weight ( k1,..., k α) and all ki congruent modulo 2, of level prime to p. Then for w ∈ Z (which is congruent to k mod 2) there exists an automorphic form πf,α associated to f and ρπf,α is crystalline at all places of F\n\nabove p thne det ρπf,w = ωinteger × char conductor prime to p.\n\nProblem: - typically there are mod p totally odd representations of Gal( F /F ) whose deter-minant is not the reduction of ωint × (prime to p). Therefore, the naive generalization of Serre's conjecture should NOT say that for all ρ: GF → GL 2(Fp) continuous totally odd irreducible are modular coming from a Hilbert modular form of level prime to p.\n\nFred's fix: - totally rethink the notion of weight. Say f is a weight k classical mod p modular form, where 2 < k < p + 1, of level N prime to\n\np. One can lift f to some characteristic zero form F of weight 2 and level Γ 1(N p ), of character ω\n\nat p.. Can find ρf in Jac( Xn(N p ))[ p]. Assume from now on that everything has an implicit level\n\nN. One can find f is a certain subspace of Pic ◦(X(p))[ p] where X(p)/Q is the non-geometrically connected modular curve of level Γ 1(N ) ×\n\n( 1 00 1\n\n)\n\n(mod p) over Q.The group Pic ◦(X(p))[ p]( Q) has an action of Gal( Q/Q) and a commuting action of GL 2(Fp), and our modular ρF lives in the subspace of this Pic ◦ where\n\n( ∗ ∗\n\n0 ∗\n\n)\n\n⊆ GL 2(Fp) is acting in a certain explicit way. One can now write down an explicit irreducible mod p representation of GL 2(Fp), say Vk, such that ρ ⊆ Hom G(V, Pic ◦(X(p))[ p]( Q)). This latter space is the one which generalizes to the totally real setting.\n\nEmerton: - ρ modular at weight V - where V is any irreducible mod p representation of GL 2(Fp), if ρ ⊆ Hom G(V, H 1et (X(p), Fp)).\n\nDiamond: - ρ modular of weight V if\n\nρ ⊆ (VFp ⊗ Pic ◦(X(p))[ p]( Q)) G = Hom G(V ∗, Pic ◦(X(p))[ p]( Q)) p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 9\n\nOne gets a reformulation of Serre's conjecture: If ρ is continuous, odd, irreducible GQ → GL 2(Fp), then ρ is modular, and furthermore it is modular for weight V - for which we have a recipe. The recipe now looks \"nicer\", because the irreducible mod p representations of GL 2(Fp) are all of the form det a ⊗Sym b−1(F2\n\n> p\n\n) where 0 ≤ a ≤ p − 2 and 1 ≤ b ≤ p.Get a simpler picture, e.g. in the irreducible case:\n\nρ|I = ωa\n\n( ψb 00 ψpb\n\n)\n\nwhere ω is fundamental of level 1 and ψ fundamental level 2. Fred predicts weight V = det a ⊗Sym b−1.6. Gee: Proof of Buzzard-Diamond-Jarvis\n\nLet F be totally real, and let p > 2 be unramified in F. Let\n\nρ: GF → GL 2(Fp),\n\nbe modular of weights {V } where V are irreducible characteristic p representations of GL 2(OF /p )and V = ⊗\n\n> V|p\n\nVav,bv where av, bv are [ kv: Fp]-tuples indexed by σ: kv ↪→ Fp, where 0 ≤ av ≤ p−1, not all av = p − 1 and 1 ≤ bv ≤ p.\n\nVav,bv = ⊗\n\n> σ:kv↪→Fp\n\n(\n\ndet av Sym bv −1k2\n\n> v\n\n)\n\n⊗σ Fp.\n\nThere exists explicit recipe for ρ|Gv to {Vvp respresentations of GL 2(kv)} and ρ to {V = ⊗Vv}.Assume p inert: If ρ|Gp is irreducible then there are 2 f weights, f = [ F: Q]. If ρ|Gp is reducible, there are ≤ 2f weights (generically).\n\n( ψ1 ∗\n\n0 ψ2\n\n)\n\nif ∗ = 0 you get all the weights, if ∗ is generic, get 1 weights. Say a weight ⊗v|pVav,bv is regular if 2 ≤ bv ≤ p − 2 for all v.\n\nTheorem 5 (Gee). Assume further that ρ(GF ) is non-solvable. (Can be removed). Assume also that for each v: ρ|Gv is not scalar. If V is a regular weight, ρ is irreducible, ρ is modular of weight\n\nV if and only if B-D-J predicted that it is.\n\nTheorem 6 (Gee). For p > 2 and F a totally real field, with p unramified in F. Let E/ Qp be a finite extension, and let O denote the ring of integers of E. Let\n\nρ: GF → GL 2(O),\n\nbe continuous, unramified outside of a finite set of primes, and det ρ = (cyc)( finite order ). Suppose that\n\n(1) ρ|Gv is potentially Barsotti-Tate for all v | p.\n\n(2) ρ is modular\n\n(3) ρ|GF (ζp ) is absolutely irreducible. then ρ is modular.\n\nAssume 2 ≤ k ≤ p, p > 2 if a modular newform of level Γ 1(N ), p - N and weight k.\n\nρf: GQ → GL 2(Fp),10 NOTES BY MICHAEL VOLPATO\n\nassume ρf also irreducible. Assume\n\nρf |Gp ∼=\n\n( ψ1ωk−1 00 ψ2\n\n)\n\nwhere ψ1 and ψ2 are unramified an ωk−1ψ1 6 = ψ2.then ( ρf ⊗ ωk′−1)|Gp ∼=\n\n( ψ2ωk′−1 00 ψ1\n\n)\n\nwhere\n\nk′ =\n\n{\n\np + 1 − k if k 6 = pp if k = p\n\nSerre predicts that there exists an eigenform of weight k′, of level Γ 1(N ), such that ρg ∼= ρf ⊗\n\nωk′−1. If k = p, the Up-eigenvalue of g is congruent to ψ2(Frob p) modulo p.By using Hida theory it suffices to find g′ of level Γ 1(N p ), and weight 2 with\n\nρg′ ∼= ρf ⊗ ωk′−1.\n\nthen\n\nρg′ ∼=\n\n( ˜ψ2 ˜ωk′−2χcyc ∗\n\n0 ˜ψ1\n\n)\n\nwhere˜stands for Teichm¨ uller lifts. Assume that ρ(GQ) is non-solvable. Now: (1) find ρg′, then (2) prove ρg′ is modular. For (2) we simply check the hypothesis of the earlier theorem. (1) follows from a theorem of Ramakrishna (and Taylor). In essence on has to check that the local deformation ring at p is large enough - a dimension calculation. For B-D-J one has to consider many lifts. In fact, the lifts we want to consider are potentially Barsotti-Tate of a specified type. (These types are always tame). Starting the a residual rep-resentation considers all lifts of this type. Then using combinatorial arguments you control the weights. 7. Buzzard: p-adic Local Langlands\n\nFor GL 2(K), where K/ Qp finite: it bijects supercuspidal (infinite dimensional) representations of GL 2(K) with irreducible 2-dimensional C-representations of the Weil group WK. This first set is contained in the set of smooth irreducible admissible representations of GL 2(K). The latter set is contained in the set of F -semisimple 2-dimensional Weil-Delgine representations. Vigneras: situation is also good when we replace C by F` for ` 6 = p.What about Fp? What about a \"mod p local Langlands?\" Objects on the right-hand side: {continuous ρ: Gal( K/K ) → GL 2(Fp)}, this set contains the irreducible representations. At least for K/ Qp unramified, then ρ gives rise to a finite set of irreducible representations of GL 2(k) where k denotes the residue field of K.e.g. When K = Qp and ρ irreducible\n\nρ|I =\n\n( ψb 00 ψpb\n\n)\n\nyou get Sym b−1 and also twist of Sym p+b.Take F/ Qp unramified, with ring of integers O. Let K = GL 2(O), and Z = F × ↪→ G = GL 2(F )and k be the residue field of F. If V is a finite dimensional representation of GL 2(k) over Fp.Then make V a representation of K by letting K act via GL 2(k) and then a representation of KZ\n\nby letting O× act via K and letting\n\n( p 00 p\n\n)\n\nact trivially. Define c − Ind GKZ V to be the set of p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 11\n\nfunctions f: G → V such that f (kg ) = k ∗ f (g) for all k ∈ KZ, where ∗ is the action of KZ on V;and such that the support of f is a finite union of costs of KZ. We define a G-action on c − Ind by putting ( gf )( g′) = f (g′g). Note that c − Ind is an infinite-dimensional representation of G.Consider all the embeddings k σ\n\n↪→ Fp. Now assume that V = ⊗\n\n> σ\n\nσ ◦ Sym rσ k2 where 0 ≤ rσ ≤\n\np − 1. [Up to twists this is all the irreducible representations of GL 2(k). Explicitly: homogeneous polynomials of degree rσ in two-variables x and y such that\n\n(( a bc d\n\n)\n\nf\n\n)\n\n(x, y ) = f (ax + cy, bx + dy )Define an Fp-linear map U = ⊗Uσ: V → V, by\n\nUσ(xiyrσ −i) = 0 if i > 0 and\n\nUσ(yrσ ) = yrσ.\n\nNow define a map\n\nϕ: G → End Fp (V )by\n\nϕ\n\n(( 1 00 p−1\n\n))\n\n= U\n\nExtend to KZ\n\n( 1 00 p−1\n\n)\n\nKZ by ϕ(ka −1k′) = k ∗ ϕ(a−1 ∗ k′. Then extend to G by 0. So ϕ gives rise to a G-endomorphism of c − Ind GKZ V in a natural way - call this T.Wonderful observations of Barthel-Livn´ e: Let Wr:= c − Ind GKZ ⊗ Sym rσ.\n\nIf λ ∈ Fp, λ 6 = 0, then Wr/(T − λ) is almost always an irreducible smooth admissible represen-tation of G (call these principal series ), except occasionally it has length 2, 1 − d subquotient and Steinberg subquotient. An irreducible representation of G is supersingular if its a quotient of Wr/(T ). The principal series are never isomorphic to 1 − d which are never isomorphic to the Steinberg which is never isomorphic to principal series representations and none are ever isomorphic to supersingular. Within the principal series, 1 − d, Steinberg understood. Supersingular case: mysterious Wr/(T ) does have infinite length if K 6 = Qp.Breuil: restricts to K = Qp and r = r ∈ { 0,..., p −1}, Breuil finishes the story: he shows Wr/(T )is irreducible and Wr/(T ) is isomorphic to twist of Ws/(T ) if and only if r = s or r + s = p − 1 and know exactly the twist. B-L observe that: any smooth irreducible admissible Fp-representation of G = GL 2(F ), with a central character is 1 − d, principal series, Steinberg, or supersingular - up to twist. For F = Qp we can write down all smooth admissible irreducible representations of GL 2(Qp)and all two-dimensional representations of GQp.Restrict to the semi-simple case: Choose a lift F of Frobenius in Gal( Qp/Qp) and restrict to representations ρ of GQp such that det ρ(F ) = 1. If ρ|I =\n\n( ψr+1 00 ψp(r+1)\n\n), where ψ is fundamental of level 2, then match with Wr/(T ). If ρ =\n\n( ωr+1 × unr( λ) 00 unr( λ−1)\n\n)\n\nthen match with (Wr/(T − λ)) ss ⊕ (Wp−3−r/(T − λ−1) ⊗ ωr+1 )ss.\n\nThis all gives us a semi-simple local Langlands conjecture (theorem for GL 2(Qp)). 12 NOTES BY MICHAEL VOLPATO\n\nMatthew Emerton's picture:\n\nHΓ( p) H\n\nV [U+001F]  / / H1(X(p), Fp)ρ [U+001F]  / / inj lim H1(X(pr), Fp)ρ\n\nThese groups all have a GL 2(F) action on them. Finally, consider GL 2(F ) where F = Qp2 and r = ( r0, r 1), and let Vr be a representation of GL 2(Fp2 ). It is known that Wr/(T ) has infinite length. This is too big. Barthel-Livn´ e suggest that we should consider quotients of this object. Paskunas: writes downa particular irreducible quotient of Wr/(T ) call it Pr. Paskunas showed that Pr is isomorphic to a twist of Ps if and only if r = s or r = p − 1 − s where r ∈ { 0,..., p − 1}2.Up to unramified twist we get exactly q(q − 1) /2 irreducible non-isomorphic supersingular rep-resentations of GL 2(F ). Now we guess: irreducible representations of GF ←→ Pr -- WRONG.\n\nVa,b = det a0 ×σ ◦ det a1 ×Sym b0−1 ⊗ σ ◦ Sym b1−1. Fp2 ↪→ Fp\n\nρ irreducible and (2) ρ|I =\n\n( ψr0+1+ p(r1+1) 4 00 ψ4p2(this )\n\n)\n\nIf ρ → Pr for some r, then because Pr is isomorphic to a twist of Pp−1−r we must see 2 lines such that sum of b's is ( p − 1, p − 1) Fred predicts Va,b:\n\na0 a1 b0 − 1 b1 − 1\n\n0 0 r0 r1\n\nr0 + 1 −1 p − 2 − r0 r1 + 1\n\n0 r1 + 1 r0 − 1 p − 2 − r1\n\nr0 r1 + 1 p − 1 − r0 p − 3 − r1\n\nIf ρ is reducible, then\n\nρ|I ∼=\n\n( ψr0+1+ p(r1+1) 2 00 1\n\n)\n\n←→ P S ⊕ P S ⊕ Pr\n\nFred predicts:\n\na0 a1 b0 − 1 b1 − 1\n\n0 0 r0 r1\n\nr0 + 1 r1 + 1 p − 3 − r0 p − 3 − r1\n\np − 1 r1 r0 + 1 p − 2 − r\n\nr0 p − 1 p − 2 − r0 r1 + 1\n\nThus (2) corresponds to a new quotient of Wr/(T ). Matt had an insighful diagram with (too!) much commentary:\n\n8. Kisin: Pseudo-representations\n\nPseudo-representations: - G a group and A a ring, or more generally an A-algebra R. Consider functions T: R → A, satisfying two conditions: (1) T (xy ) = T (yx ), (2) depends on d, (3)\n\nT (1) = d ∈ N. Furthermore we require d! is invertible in A. For d = 2, thus, we suppose that 2 is invertible in A. If you have an element σ ∈ R then we can define T (σ). p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 13\n\nDefine S(σ):= 1\n\n> 2\n\n(T (σ)2 − T (σ2)), then S is a character, i.e. S(σ1)S(σ2) = S(σ1σ2). We have\n\nX2 − T (σ)X + S(σ).\n\nDefine Ker( T ) = {x ∈ R: T (xy ) = 0 ∀y ∈ R}. Then we have a map\n\nR = R/ Ker( T ) → A,\n\nfor σ ∈ R: Pσ(x) - the characteristic polynomial of σ. Ask the following question: is Pσ(σ) = 0.\n\nTheorem 7 (Taylor). If A is an algebraically closed field, then there is a correspondence between: the set of pseudo-representations and the set of semi-simple representations.\n\nAlways genuine representations give pseudo-representations. 8.1.",
   "evidence": "The canonical JSON record is severely overlong: after the intended question it absorbs all of Sections 4--7 and the beginning of Section 8 of the workshop notes. The primary source is Michael Volpato's notes from the 2006 AIM workshop *\\(p\\)-adic representations, modularity, and beyond*. On printed page 6 (PDF page 5), Section 3.4 contains exactly the following question and follow-up:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
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  },
  "AIM-REPRESENTATION_THEORY-0108": {
   "statement_status": "exact",
   "original_statement": "Problem 1: Recast the theory so that d!−1 is not necessarily in A. [[This is not simply a problem about divided powers]]. 8.2.",
   "clean_statement": "Problem 1: Recast the theory so that d!−1 is not necessarily in A. [[This is not simply a problem about divided powers]]. 8.2.",
   "public_statement": "Problem 1: Recast the theory so that d!−1 is not necessarily in A. [[This is not simply a problem about divided powers]]. 8.2.",
   "evidence": "The record comes from Section 8, “Kisin: Pseudo-representations,” of the AIM workshop notes *\\(p\\)-adic representations, modularity, and beyond*. The preceding discussion fixes a group \\(G\\), a commutative coefficient ring \\(A\\), and more generally an \\(A\\)-algebra \\(R\\). It describes a dimension-\\(d\\) trace-like function and explicitly assumes that \\(d!\\) is invertible in \\(A\\).",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 107,
   "attempt": 1
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  "AIM-REPRESENTATION_THEORY-0109": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 2: The relationship between moduli of pseudo-representations and representations. We have the following theorem: \n\nTheorem 8 (Nyssen, Rouquire). If the representation is absolutely irreducible, then these moduli are equivalent. \n\nLet F be a (finite) field, for example \n\nVF = ω1 ⊕ ω2 7 → TF\n\nsuch that ω1 and ω2 are distinct characters. Look at representations whose reduction is a nontrivial extension of ω1 by ω2. Let A be an W (F)-algebra. Consider the following diagram: \n\nXω2\n\n> \u000f\n> \u000f\n\nSpec( R(TF)) Note that Ext 1(ω1, ω 2) \\ { 0}/F×. We have the following: [[Mark drew a picture of a cone (the special fiber) projecting on to a disc.]] Taking a point x on the special fiber. Then x gives a representation Vx (= extensions of ω1 by \n\nω2). Completing we have ̂\n\nRx = universal deformation ring of Vx.Now Xω2 carries a universal rank 2 vector bundle Vω2. Taking the direct image: π∗Vω2, we get a bundle on Spec( R(TF)). Mark suggested the following diagram: \n\nXω2 \n\n> &\n> &\n> LLLLLLLLLL\n\nXω1\n\n> x\n> x\n> rrrrrrrrrr\n\nSpec( R(TF)) and that maybe one could glue along the trivial extension and get some geometric object - perhaps an algebraic stack. Richard ask if Xω2 was even proper, Mark insisted it should be. In fact, by the valuative criterion of properness, this is indeed the case. 8.3.",
   "clean_statement": null,
   "public_statement": "Problem 2: The relationship between moduli of pseudo-representations and representations. We have the following theorem:\n\nTheorem 8 (Nyssen, Rouquire). If the representation is absolutely irreducible, then these moduli are equivalent.\n\nLet F be a (finite) field, for example\n\nVF = ω1 ⊕ ω2 7 → TF\n\nsuch that ω1 and ω2 are distinct characters. Look at representations whose reduction is a nontrivial extension of ω1 by ω2. Let A be an W (F)-algebra. Consider the following diagram:\n\nXω2\n\n> [U+000F]\n> [U+000F]\n\nSpec( R(TF)) Note that Ext 1(ω1, ω 2) \\ { 0}/F×. We have the following: [[Mark drew a picture of a cone (the special fiber) projecting on to a disc.]] Taking a point x on the special fiber. Then x gives a representation Vx (= extensions of ω1 by\n\nω2). Completing we have ̂\n\nRx = universal deformation ring of Vx.Now Xω2 carries a universal rank 2 vector bundle Vω2. Taking the direct image: π∗Vω2, we get a bundle on Spec( R(TF)). Mark suggested the following diagram:\n\nXω2\n\n> &\n> &\n> LLLLLLLLLL\n\nXω1\n\n> x\n> x\n> rrrrrrrrrr\n\nSpec( R(TF)) and that maybe one could glue along the trivial extension and get some geometric object - perhaps an algebraic stack. Richard ask if Xω2 was even proper, Mark insisted it should be. In fact, by the valuative criterion of properness, this is indeed the case. 8.3.",
   "evidence": "The source is §8.2, Problem 2, of the AIM workshop notes *p-adic representations, modularity, and beyond*. The corpus transcription has several OCR defects: “Rouquire” is **Rouquier**, the displayed map is",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 108,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0110": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 3: Classically, if one is looking at deformation rings of Galois representations one has various cohomological tools. One would like analogous of these in the situation of pseudo-representations. 14 NOTES BY MICHAEL VOLPATO \n\n9. Kedlaya: (ϕ, Γ) -modules \n\n9.1. Motivation. \n\n9.1.1. Dieudonn´ e-Manin classification. - Let k be an algebraically closed field of characteristic \n\np > 0, and let K be a finite extension of the field of fractions of the Witt vectors W (k), fix a uniformizer π of W (k). Let ϕ denote a lifting of the Frobenius endomorphism. \n\nDefinition 1. A ϕ-module over K is a finite free K-module M equipped with a semilinear ϕ-action, i.e. M → M such that ϕ∗(M ) = M ⊗ϕ K ˜→M.\n\nTheorem 9 (Dieudonn´ e-Manin classification). For r = a \n\n> b\n\n∈ Q (b > 0 and (a, b ) = 0 ), let Mr be the ϕ-module defined by: \n\n0 πa\n\n1.........\n\n1 0\n\n\n\nThen every ϕ-module over K is isomorphic to a direct sum of Mr's. In particular Ext 1(Mr, M s) = 0. Moreover Mr ∼= Ms if and only if r = s.\n\nDefine the degree of a ϕ-module as follows: if M has rank one, say M = ( x), i.e. pick v ∈ M,then ϕ(v) = xv. Define deg( M ) = vp(x). Where vp is the p-adic valuation. Define deg( M ) = deg( ∧rank( M ) M ). If Mr ∼= Ms implies that their ranks are equal, in particular their degrees are equal. We define the slope of a ϕ-module M as the quotient deg( M )/rank( M ). Note that the decomposition of a ϕ-module is not unique. For example: \n\n( 1 00 1\n\n)\n\nhas fixed vectors \n\nKϕe1 + Kϕe2.\n\nNow assume that k is only assumed to be perfect, and not algebraically closed. Apply the D-M classification over ̂ Kunr get isotypical decomposition of M ⊗K̂ Kunr, which descends to a decomposition \"pure slope decomposition.\" Alternative characterization of \"pure.\" Say M has rank r and degree d, then M is pure of slope \n\nr/d if there exists an OK -lattice L of M such that π−rϕd acts on L and \n\n(\n\nπ−rϕd)∗\n\nL → L\n\ni.e. in some basis π−rϕd acts via an invertible matrix over OK.Exercise: M is pure of slope r/d if and only if M ⊗K̂ Kunr ∼= ( Mr/d )⊕i for some i. \"Pure of slope zero\" = \"´ etale\" = \"unit-root.\" Also (pure) ⊗(pure)=(pure). Now let k be an arbitrary field of characteristic p > 0, and let K be a finite extension of the field of fractions of a Cohen ring. For instance, for F a finite extension of Qp we can define \n\nE =̂ OF [[ t]][ t−1][ 1\n\np ].\n\nApply D-M classification over ̂ Lunr where L =̂ inj lim ϕ K. We get an isotypical decomposition of \n\nM ⊗ L but not of M itself. \n\n( 1 y\n\n0 1\n\n)−1 ( 1 x\n\n0 p\n\n) ( 1 y\n\n0 1\n\n)ϕ\n\n=\n\n( 1 x + ϕ(y) − py \n\n0 p\n\n)p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 15 \n\nwhich does not split, but \n\n( p x\n\n0 1\n\n)\n\nsplits. Get on M a slope filtration: 0 = M0 ⊂ M1 ⊂ M2 ⊂ · · · ⊂ M` = M\n\nwhere each Mi/M i−1 is pure of slope si, with s1 < s 2 < · · · < s `.Recall that to a p-adic Galois representation ρ: GQp → GL 2(Qp) one can associate an ´ etale (ϕ, Γ)-module over the ring E, where Γ = Gal( Qp(μp∞ )/Qp). Define E† to be those power series in \n\nt which converge and are bounded in some annulus ∗ ≤ | t| < 1. This ring is not complete for the \n\np-adic topology, but it is henselian. If you complete it for the p-adic topology, then you get E. Also define the Robba ring R to be those power series in t which converge in some annulus ∗ ≤ | t| < 1. Note that R is not a field, however its units are bounded, i.e. belong to E †. In particular the concept of degree still makes sense over the Robba ring R as then does the concept of slope. When you define, however, \"pure of some slope\" over R, the lattice should be over OE†.\n\nTheorem 10. There is a functor: \n\n{´etale ϕ-modules }/E† → { ´etale ϕ-modules }/R\n\ngiven by \"tensor with R\", which is an equivalence of categories. The same is then true for (ϕ, Γ) -modules. \n\nNote that there is still a functor from the category of ´ etale ϕ-modules over E † to the category of ´etale ϕ-modules over E, however, this is only fully faithful, but not essentially surjective. Restricting this latter functor to ( ϕ, Γ)-modules gives the functor of Colmez. In fact, this restricted functor is \n\nan equivalence of categories (Theorem of Chernonnier-Colmez). \n\nR \n\n> >>>>>>>>\n\n(E†)unr \n\n> yyyyyyyyy\n\n˜R\n\nWe can get a DM-classification over ˜R, then we descend \n\nTheorem 11. Let M be a ϕ-module over R, then there exists a unique filtration \n\n0 = M0 ⊂ M1 ⊂ M2 ⊂ · · · ⊂ M` = M\n\nwhere each Mi/M i−1 is pure of slope si and s1 < s 2 < · · · < s `.\n\nWe can also have filtrations going the \"wrong way.\" 0 / / M1 / / M / / M2 / / 0where M1 is pure of rank 1 slope 1, M is pure of rank 2 and slope 0, and M2 pure of rank 1 and slope 2. For example, ( ϕ, Γ)-modules, modular form of weight 3 and ap ≡ 0 (mod p). If, on the other hand, ap 6 = 0 modulo p then the representation well be ordinary, and you get an exact sequence: 0 / / M1 / / M / / M2 / / 0each ϕ-module pure of slope zero. The Newton polygon looks like: \n\n• 0 •• \n\n> −1\n> @@@@@@@+1\n> ~~~~~~~16 NOTES BY MICHAEL VOLPATO\n\nwhere as the Hodge polygon looks like: \n\n• 1 ••\n\n> 0\n\n@@@@@@@ 2\n\n~~~~~~~\n\nIn principle: The Newton polygon is not equal to a Hodge polygon. 9.2. ( ϕ, Γ) -module from Galois representation. Fix embeddings: Qp ⊂ Qp(μp∞ ) ⊂ Qp ⊂ Cp.Let ρ: GQp → GL( V ) be a finite dimensional p-adic Galois representation. Consider OCp /p OCp -this has a ϕ-p-power Frobenius action on it. Fix a compatible system of units: ε = (..., ε 1, ε 0), and put t = [ ε] − 1, then construct: \n\nW\n\n(\n\nFrac \n\n(\n\nproj lim \n\n> ϕ\n\nOCp /p OCp\n\n)) [ 1\n\np\n\n]\n\n⊃ E:= Zp[[ t]] [t−1] [ 1\n\np\n\n]\n\n⊃̂ Eunr \n\nthis has a GQp action on it. \n\nTheorem 12 (Fontaine). Then we define the following ´ etale (ϕ, Γ) -module: \n\nD(V ):= \n\n(\n\nV ⊗Qp̂ Eunr \n\n)H\n\n= finite free E-module of rank dim Qp (V )\n\nand this is equivalent to (\n\nD(V ) ⊗Ê Eunr \n\n)ϕ=1 ∼= V, \n\nwhere Γ acts on the first factor and GQp acts on the second. \n\n10. Emerton part II \n\n10.1. Set up. Let F be a finite field extension of Fp, fix O = OK ⊂ K, where F is the residue field of K and K is a finite extension of Qp. Let ρ: GQ → GL 2(F) be an absolutely irreducible, modular Galois representation. Denote by R the universal deformation ring of ρ unramified outside some set Σ. Define ˆH1 = ˆH1Σ,ρ = inj lim n1,...n s H1(X(qn1 \n\n> 1\n\n· · · qns \n\n> s\n\n), O)mρ where mρ is the maximal ideal in \n\nT(qn1 \n\n> 1\n\n· · · qns \n\n> s\n\n) corresponding to ρ - where T(qn1 \n\n> 1\n\n· · · qns \n\n> s\n\n) the algebra generated by T` for all ` 6 = Σ. \n\nR / / / /\n\n\u000f\n\n\u000f TΣ,ρ = proj lim T(qn1 \n\n> 1\n\n· · · qns \n\n> s\n\n)mρ\n\nRmod \n\n5\n\n5\n\njjjjjjjjjjjjjjjj\n\nGeometrically, we have Spec( Rmod ) ↪→ Spec R.\n\nThis is the Zariski closure of all x ∈ Spec R corresponding to classical modular forms lifting ρ\n\nunramified outside Σ. \n\nTheorem 13 (Boeckle). If p > 2 and ρ|GQp is flat or ordinary not \n\n( ω−1 ∗\n\n0 1\n\n), and if ρ|GQp(√p∗)\n\nis irreducible, then we have an isomorphism R ˜→R mod.Argument: Taylor-Wiles gets lots of points in Rmod, infinite fern lets you fill this out in families. Let Rmod /I be the ring of a component of Spec Rmod, then we have the following conjecture: p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 17",
   "clean_statement": null,
   "public_statement": "Problem 3: Classically, if one is looking at deformation rings of Galois representations one has various cohomological tools. One would like analogous of these in the situation of pseudo-representations. 14 NOTES BY MICHAEL VOLPATO\n\n9. Kedlaya: (ϕ, Γ) -modules\n\n9.1. Motivation.\n\n9.1.1. Dieudonn´ e-Manin classification. - Let k be an algebraically closed field of characteristic\n\np > 0, and let K be a finite extension of the field of fractions of the Witt vectors W (k), fix a uniformizer π of W (k). Let ϕ denote a lifting of the Frobenius endomorphism.\n\nDefinition 1. A ϕ-module over K is a finite free K-module M equipped with a semilinear ϕ-action, i.e. M → M such that ϕ∗(M ) = M ⊗ϕ K ˜→M.\n\nTheorem 9 (Dieudonn´ e-Manin classification). For r = a\n\n> b\n\n∈ Q (b > 0 and (a, b ) = 0 ), let Mr be the ϕ-module defined by: \n\n0 πa\n\n1.........\n\n1 0\n\n\n\nThen every ϕ-module over K is isomorphic to a direct sum of Mr's. In particular Ext 1(Mr, M s) = 0. Moreover Mr ∼= Ms if and only if r = s.\n\nDefine the degree of a ϕ-module as follows: if M has rank one, say M = ( x), i.e. pick v ∈ M,then ϕ(v) = xv. Define deg( M ) = vp(x). Where vp is the p-adic valuation. Define deg( M ) = deg( ∧rank( M ) M ). If Mr ∼= Ms implies that their ranks are equal, in particular their degrees are equal. We define the slope of a ϕ-module M as the quotient deg( M )/rank( M ). Note that the decomposition of a ϕ-module is not unique. For example:\n\n( 1 00 1\n\n)\n\nhas fixed vectors\n\nKϕe1 + Kϕe2.\n\nNow assume that k is only assumed to be perfect, and not algebraically closed. Apply the D-M classification over ̂ Kunr get isotypical decomposition of M ⊗K̂ Kunr, which descends to a decomposition \"pure slope decomposition.\" Alternative characterization of \"pure.\" Say M has rank r and degree d, then M is pure of slope\n\nr/d if there exists an OK -lattice L of M such that π−rϕd acts on L and\n\n(\n\nπ−rϕd)∗\n\nL → L\n\ni.e. in some basis π−rϕd acts via an invertible matrix over OK.Exercise: M is pure of slope r/d if and only if M ⊗K̂ Kunr ∼= ( Mr/d )⊕i for some i. \"Pure of slope zero\" = \"´ etale\" = \"unit-root.\" Also (pure) ⊗(pure)=(pure). Now let k be an arbitrary field of characteristic p > 0, and let K be a finite extension of the field of fractions of a Cohen ring. For instance, for F a finite extension of Qp we can define\n\nE =̂ OF [[ t]][ t−1][ 1\n\np ].\n\nApply D-M classification over ̂ Lunr where L =̂ inj lim ϕ K. We get an isotypical decomposition of\n\nM ⊗ L but not of M itself.\n\n( 1 y\n\n0 1\n\n)−1 ( 1 x\n\n0 p\n\n) ( 1 y\n\n0 1\n\n)ϕ\n\n=\n\n( 1 x + ϕ(y) − py\n\n0 p\n\n)p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 15\n\nwhich does not split, but\n\n( p x\n\n0 1\n\n)\n\nsplits. Get on M a slope filtration: 0 = M0 ⊂ M1 ⊂ M2 ⊂ · · · ⊂ M` = M\n\nwhere each Mi/M i−1 is pure of slope si, with s1 < s 2 < · · · < s `.Recall that to a p-adic Galois representation ρ: GQp → GL 2(Qp) one can associate an ´ etale (ϕ, Γ)-module over the ring E, where Γ = Gal( Qp(μp∞ )/Qp). Define E† to be those power series in\n\nt which converge and are bounded in some annulus ∗ ≤ | t| < 1. This ring is not complete for the\n\np-adic topology, but it is henselian. If you complete it for the p-adic topology, then you get E. Also define the Robba ring R to be those power series in t which converge in some annulus ∗ ≤ | t| < 1. Note that R is not a field, however its units are bounded, i.e. belong to E †. In particular the concept of degree still makes sense over the Robba ring R as then does the concept of slope. When you define, however, \"pure of some slope\" over R, the lattice should be over OE†.\n\nTheorem 10. There is a functor:\n\n{´etale ϕ-modules }/E† → { ´etale ϕ-modules }/R\n\ngiven by \"tensor with R\", which is an equivalence of categories. The same is then true for (ϕ, Γ) -modules.\n\nNote that there is still a functor from the category of ´ etale ϕ-modules over E † to the category of ´etale ϕ-modules over E, however, this is only fully faithful, but not essentially surjective. Restricting this latter functor to ( ϕ, Γ)-modules gives the functor of Colmez. In fact, this restricted functor is\n\nan equivalence of categories (Theorem of Chernonnier-Colmez).\n\nR\n\n> >>>>>>>>\n\n(E†)unr\n\n> yyyyyyyyy\n\n˜R\n\nWe can get a DM-classification over ˜R, then we descend\n\nTheorem 11. Let M be a ϕ-module over R, then there exists a unique filtration\n\n0 = M0 ⊂ M1 ⊂ M2 ⊂ · · · ⊂ M` = M\n\nwhere each Mi/M i−1 is pure of slope si and s1 < s 2 < · · · < s `.\n\nWe can also have filtrations going the \"wrong way.\" 0 / / M1 / / M / / M2 / / 0where M1 is pure of rank 1 slope 1, M is pure of rank 2 and slope 0, and M2 pure of rank 1 and slope 2. For example, ( ϕ, Γ)-modules, modular form of weight 3 and ap ≡ 0 (mod p). If, on the other hand, ap 6 = 0 modulo p then the representation well be ordinary, and you get an exact sequence: 0 / / M1 / / M / / M2 / / 0each ϕ-module pure of slope zero. The Newton polygon looks like:\n\n• 0 ••\n\n> −1\n> @@@@@@@+1\n> ~~~~~~~16 NOTES BY MICHAEL VOLPATO\n\nwhere as the Hodge polygon looks like:\n\n• 1 ••\n\n> 0\n\n@@@@@@@ 2\n\n~~~~~~~\n\nIn principle: The Newton polygon is not equal to a Hodge polygon. 9.2. ( ϕ, Γ) -module from Galois representation. Fix embeddings: Qp ⊂ Qp(μp∞ ) ⊂ Qp ⊂ Cp.Let ρ: GQp → GL( V ) be a finite dimensional p-adic Galois representation. Consider OCp /p OCp -this has a ϕ-p-power Frobenius action on it. Fix a compatible system of units: ε = (..., ε 1, ε 0), and put t = [ ε] − 1, then construct:\n\nW\n\n(\n\nFrac\n\n(\n\nproj lim\n\n> ϕ\n\nOCp /p OCp\n\n)) [ 1\n\np\n\n]\n\n⊃ E:= Zp[[ t]] [t−1] [ 1\n\np\n\n]\n\n⊃̂ Eunr\n\nthis has a GQp action on it.\n\nTheorem 12 (Fontaine). Then we define the following ´ etale (ϕ, Γ) -module:\n\nD(V ):=\n\n(\n\nV ⊗Qp̂ Eunr\n\n)H\n\n= finite free E-module of rank dim Qp (V )\n\nand this is equivalent to (\n\nD(V ) ⊗Ê Eunr\n\n)ϕ=1 ∼= V,\n\nwhere Γ acts on the first factor and GQp acts on the second.\n\n10. Emerton part II\n\n10.1. Set up. Let F be a finite field extension of Fp, fix O = OK ⊂ K, where F is the residue field of K and K is a finite extension of Qp. Let ρ: GQ → GL 2(F) be an absolutely irreducible, modular Galois representation. Denote by R the universal deformation ring of ρ unramified outside some set Σ. Define ˆH1 = ˆH1Σ,ρ = inj lim n1,...n s H1(X(qn1\n\n> 1\n\n· · · qns\n\n> s\n\n), O)mρ where mρ is the maximal ideal in\n\nT(qn1\n\n> 1\n\n· · · qns\n\n> s\n\n) corresponding to ρ - where T(qn1\n\n> 1\n\n· · · qns\n\n> s\n\n) the algebra generated by T` for all ` 6 = Σ.\n\nR / / / /\n\n[U+000F]\n\n[U+000F] TΣ,ρ = proj lim T(qn1\n\n> 1\n\n· · · qns\n\n> s\n\n)mρ\n\nRmod\n\n5\n\n5\n\njjjjjjjjjjjjjjjj\n\nGeometrically, we have Spec( Rmod ) ↪→ Spec R.\n\nThis is the Zariski closure of all x ∈ Spec R corresponding to classical modular forms lifting ρ\n\nunramified outside Σ.\n\nTheorem 13 (Boeckle). If p > 2 and ρ|GQp is flat or ordinary not\n\n( ω−1 ∗\n\n0 1\n\n), and if ρ|GQp(√p∗)\n\nis irreducible, then we have an isomorphism R ˜→R mod.Argument: Taylor-Wiles gets lots of points in Rmod, infinite fern lets you fill this out in families. Let Rmod /I be the ring of a component of Spec Rmod, then we have the following conjecture: p-ADIC REPRESENTATIONS, MODULARITY AND BEYOND 17",
   "evidence": "The assigned record comes from Section 8, “Kisin: Pseudo-representations,” of the AIM workshop notes *p-adic representations, modularity, and beyond*. The exact problem, checked against the [official AIM PDF](https://aimath.org/WWN/padicmodularity/padicmodularity.pdf), is:",
   "classification_method": "damaged_source_without_verified_clean_repair",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 109,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0111": {
   "statement_status": "exact",
   "original_statement": "Conjecture 4. There is an equivariant isomorphism: \n\n(3) ˆH1[I]?? \n\n∼= ρRmod /I ˆ⊗Rmod /I ˆπp\n\n(\n\nρRmod /I |GQp\n\n) ˆ⊗\n\n(̂ ⊗ \n\n> `6=p,` ∈Σ\n\nˆπmod \n\n> `\n\n(\n\nρRmod /I |GQ`\n\n))\n\nwhere ρRmod /I is the universal deformation over Spec( Rmod /I ), and ˆπp is an orthonomalizable \n\nRmod /I -Banach module, attached to ρRmod /I |GQp via p-adic local Langlands. Also, πmod \n\n> `\n\nis the m-adic completion of modified local Langlands at ` applied to \"generic fiber of ρRmod /I, descended to \n\nRmod /I.\" Note that the term on the left hand-side has an action of GQ × ∏ \n\n> q∈Σ\n\nGL 2(Qq), and the terms on the right hand-side have an action of GQ, GL 2(Qp), GL 2(Qq) respectively. Hypotheses:(1) Assume that ρ|GQp has scalar endomorphisms - irreducible \n\n( χ ∗\n\n0 ψ\n\n)\n\nwhere χ 6 = ψ and \n\n∗ 6 = 0 and it is not a twist of \n\n( ω−1 ∗\n\n0 1\n\n).(2) Suppose there exists ˆ πp(Rmod /I )-orthonormalizable m-adically complete Rmod /I -module. Such that for all classical modular forms f unramified outside Σ − { p} of weight k ≥ 2, deforming ρ giving rise to \n\nϕf: Rmod → Kf\n\nwhere φf factors through Rmod /I, we have ˆ πp(Rmod /I ) ˆ ⊗ϕf Kf ∼= B(ρf |GQp ) - Berger-Breuil-Emerton. \n\nTheorem 14. Assume the two above hypotheses. Then (3) holds.",
   "clean_statement": "Conjecture 4. There is an equivariant isomorphism:\n\n(3) ˆH1[I]??\n\n∼= ρRmod /I ˆ⊗Rmod /I ˆπp\n\n(\n\nρRmod /I |GQp\n\n) ˆ⊗\n\n(̂ ⊗\n\n> `6=p,` ∈Σ\n\nˆπmod\n\n> `\n\n(\n\nρRmod /I |GQ`\n\n))\n\nwhere ρRmod /I is the universal deformation over Spec( Rmod /I ), and ˆπp is an orthonomalizable\n\nRmod /I -Banach module, attached to ρRmod /I |GQp via p-adic local Langlands. Also, πmod\n\n> `\n\nis the m-adic completion of modified local Langlands at ` applied to \"generic fiber of ρRmod /I, descended to\n\nRmod /I.\" Note that the term on the left hand-side has an action of GQ × ∏\n\n> q∈Σ\n\nGL 2(Qq), and the terms on the right hand-side have an action of GQ, GL 2(Qp), GL 2(Qq) respectively. Hypotheses:(1) Assume that ρ|GQp has scalar endomorphisms - irreducible\n\n( χ ∗\n\n0 ψ\n\n)\n\nwhere χ 6 = ψ and\n\n∗ 6 = 0 and it is not a twist of\n\n( ω−1 ∗\n\n0 1\n\n).(2) Suppose there exists ˆ πp(Rmod /I )-orthonormalizable m-adically complete Rmod /I -module. Such that for all classical modular forms f unramified outside Σ − { p} of weight k ≥ 2, deforming ρ giving rise to\n\nϕf: Rmod → Kf\n\nwhere φf factors through Rmod /I, we have ˆ πp(Rmod /I ) ˆ ⊗ϕf Kf ∼= B(ρf |GQp ) - Berger-Breuil-Emerton.\n\nTheorem 14. Assume the two above hypotheses. Then (3) holds.",
   "public_statement": "Conjecture 4. There is an equivariant isomorphism:\n\n(3) ˆH1[I]??\n\n∼= ρRmod /I ˆ⊗Rmod /I ˆπp\n\n(\n\nρRmod /I |GQp\n\n) ˆ⊗\n\n(̂ ⊗\n\n> `6=p,` ∈Σ\n\nˆπmod\n\n> `\n\n(\n\nρRmod /I |GQ`\n\n))\n\nwhere ρRmod /I is the universal deformation over Spec( Rmod /I ), and ˆπp is an orthonomalizable\n\nRmod /I -Banach module, attached to ρRmod /I |GQp via p-adic local Langlands. Also, πmod\n\n> `\n\nis the m-adic completion of modified local Langlands at ` applied to \"generic fiber of ρRmod /I, descended to\n\nRmod /I.\" Note that the term on the left hand-side has an action of GQ × ∏\n\n> q∈Σ\n\nGL 2(Qq), and the terms on the right hand-side have an action of GQ, GL 2(Qp), GL 2(Qq) respectively. Hypotheses:(1) Assume that ρ|GQp has scalar endomorphisms - irreducible\n\n( χ ∗\n\n0 ψ\n\n)\n\nwhere χ 6 = ψ and\n\n∗ 6 = 0 and it is not a twist of\n\n( ω−1 ∗\n\n0 1\n\n).(2) Suppose there exists ˆ πp(Rmod /I )-orthonormalizable m-adically complete Rmod /I -module. Such that for all classical modular forms f unramified outside Σ − { p} of weight k ≥ 2, deforming ρ giving rise to\n\nϕf: Rmod → Kf\n\nwhere φf factors through Rmod /I, we have ˆ πp(Rmod /I ) ˆ ⊗ϕf Kf ∼= B(ρf |GQp ) - Berger-Breuil-Emerton.\n\nTheorem 14. Assume the two above hypotheses. Then (3) holds.",
   "evidence": "The source is the AIM workshop notes *\\(p\\)-adic representations, modularity, and beyond* (February 20--24, 2006), notes by Michael Volpato. The document itself warns that it was typeset during the talks and may contain transcription errors.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 110,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0112": {
   "statement_status": "exact",
   "original_statement": "Conjecture 5. Let p be a prime not dividing N. If σ is a modular representation of GL 2(Fp) then: \n\nH1(Γ 1(N ), σ ) contains a Hecke eigenclass with eigenvalue of T` equal to Tr( ρ(Frob `)) for all ` - N p \n\nif and only if N (ρ) | N and JH (σ) ∩ Wt( ρ|Ip ) 6 = ∅.\n\nThink about Wt( ρ|Ip ) ⊂ Wt( ρ|ss \n\n> Ip\n\n); Matt gave another way to think about this earlier. How to generalize this? ASH et. al.: \n\nρ: GQ → GL n(Fp)irreducible. We require that \n\n| dim ρc=1 − dim ρc=−1 |≤ 1Then N (ρ) ∈ Z>0 correpsonds with {ρ|I`: ` 6 = p}, and Wt( ρ|Ip ) ⊂ Wt(GL n(Fp)the latter being an irreducible mod p representation of GL n(Fp). Set Γ1(N ) = \n\n{\n\ng ∈ SL n(Z) | g ≡\n\n( ∗ ∗\n\n0 1\n\n)\n\n(mod N )\n\n}.\n\nFor ` - N p we write \n\nT` =\n\nΓ1(N )\n\n\n\n` 0 · · · 00 1............... 00 · · · 0 1\n\n Γ1(N )\n\n.",
   "clean_statement": "Conjecture 5. Let p be a prime not dividing N. If σ is a modular representation of GL 2(Fp) then:\n\nH1(Γ 1(N ), σ ) contains a Hecke eigenclass with eigenvalue of T` equal to Tr( ρ(Frob `)) for all ` - N p\n\nif and only if N (ρ) | N and JH (σ) ∩ Wt( ρ|Ip ) 6 = ∅.\n\nThink about Wt( ρ|Ip ) ⊂ Wt( ρ|ss\n\n> Ip\n\n); Matt gave another way to think about this earlier. How to generalize this? ASH et. al.:\n\nρ: GQ → GL n(Fp)irreducible. We require that\n\n| dim ρc=1 − dim ρc=−1 |≤ 1Then N (ρ) ∈ Z>0 correpsonds with {ρ|I`: ` 6 = p}, and Wt( ρ|Ip ) ⊂ Wt(GL n(Fp)the latter being an irreducible mod p representation of GL n(Fp). Set Γ1(N ) =\n\n{\n\ng ∈ SL n(Z) | g ≡\n\n( ∗ ∗\n\n0 1\n\n)\n\n(mod N )\n\n}.\n\nFor ` - N p we write\n\nT` =\n\nΓ1(N )\n\n\n\n` 0 · · · 00 1............... 00 · · · 0 1\n\n Γ1(N )\n\n.",
   "public_statement": "Conjecture 5. Let p be a prime not dividing N. If σ is a modular representation of GL 2(Fp) then:\n\nH1(Γ 1(N ), σ ) contains a Hecke eigenclass with eigenvalue of T` equal to Tr( ρ(Frob `)) for all ` - N p\n\nif and only if N (ρ) | N and JH (σ) ∩ Wt( ρ|Ip ) 6 = ∅.\n\nThink about Wt( ρ|Ip ) ⊂ Wt( ρ|ss\n\n> Ip\n\n); Matt gave another way to think about this earlier. How to generalize this? ASH et. al.:\n\nρ: GQ → GL n(Fp)irreducible. We require that\n\n| dim ρc=1 − dim ρc=−1 |≤ 1Then N (ρ) ∈ Z>0 correpsonds with {ρ|I`: ` 6 = p}, and Wt( ρ|Ip ) ⊂ Wt(GL n(Fp)the latter being an irreducible mod p representation of GL n(Fp). Set Γ1(N ) =\n\n{\n\ng ∈ SL n(Z) | g ≡\n\n( ∗ ∗\n\n0 1\n\n)\n\n(mod N )\n\n}.\n\nFor ` - N p we write\n\nT` =\n\nΓ1(N )\n\n\n\n` 0 · · · 00 1............... 00 · · · 0 1\n\n Γ1(N )\n\n.",
   "evidence": "The source is Section 12, “Taylor: Florian Herzig's Thesis,” of the AIM workshop notes *\\(p\\)-adic representations, modularity, and beyond* (February 2006). The section declares \\(p>2\\) and uses arithmetic Frobenius. Its goal is to generalize the weight part of Serre's conjecture from \\(\\mathrm{GL}_2/\\mathbf Q\\) to \\(\\mathrm{GL}_n/\\mathbf Q\\).",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 111,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0113": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 6 (approx.). If σ is any mod p representation of GL n(Fp), then: there is a Hecke eigenclass x in Hd(Γ 1(N ), σ ) for some d with T`x = Tr( ρ(Frob `)) x for all ` - N p, and p - N if and only if N (ρ) | N and JH (σ) ∩ Wt( ρ|Ip ) 6 = ∅.",
   "clean_statement": null,
   "public_statement": "Conjecture 6 (approx.). If σ is any mod p representation of GL n(Fp), then: there is a Hecke eigenclass x in Hd(Γ 1(N ), σ ) for some d with T`x = Tr( ρ(Frob `)) x for all ` - N p, and p - N if and only if N (ρ) | N and JH (σ) ∩ Wt( ρ|Ip ) 6 = ∅.",
   "evidence": "The source is the AIM workshop notes *$p$-adic representations, modularity, and beyond*, pp. 21--24. The extracted record contains substantial OCR corruption. Reading the PDF and its immediately preceding setup gives the following statement.",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 112,
   "attempt": 1
  },
  "AIM-REPRESENTATION_THEORY-0114": {
   "statement_status": "exact",
   "original_statement": "Conjecture 7 (Herzig(?)).\n\nWt( ρ|ss \n\n> Ip\n\n)reg = R\n\n(\n\nJH \n\n(\n\nv\n\n(\n\nρ|ss \n\n> Ip\n\n)\n\n(mod p)\n\n)).\n\nEx:\n\n• n = 3, q = p: Can extend R, it is a one-to-many map, and you get all weights predicted. \n\n• n = 2, q = pr: Consistent with Fred's conjecture, can extend R as above, get all weights, same as Fred's prediction. (Currently, there are multiple extensions, and it's unclear which is right.) (",
   "clean_statement": "Conjecture 7 (Herzig(?)).\n\nWt( ρ|ss\n\n> Ip\n\n)reg = R\n\n(\n\nJH\n\n(\n\nv\n\n(\n\nρ|ss\n\n> Ip\n\n)\n\n(mod p)\n\n)).\n\nEx:\n\n• n = 3, q = p: Can extend R, it is a one-to-many map, and you get all weights predicted.\n\n• n = 2, q = pr: Consistent with Fred's conjecture, can extend R as above, get all weights, same as Fred's prediction. (Currently, there are multiple extensions, and it's unclear which is right.) (",
   "public_statement": "Conjecture 7 (Herzig(?)).\n\nWt( ρ|ss\n\n> Ip\n\n)reg = R\n\n(\n\nJH\n\n(\n\nv\n\n(\n\nρ|ss\n\n> Ip\n\n)\n\n(mod p)\n\n)).\n\nEx:\n\n• n = 3, q = p: Can extend R, it is a one-to-many map, and you get all weights predicted.\n\n• n = 2, q = pr: Consistent with Fred's conjecture, can extend R as above, get all weights, same as Fred's prediction. (Currently, there are multiple extensions, and it's unclear which is right.) (",
   "evidence": "The source is the AIM workshop document *\\(p\\)-adic representations, modularity, and beyond*, notes by Michael Volpato from February 20--24, 2006. The document warns that it was typeset during the talks and may contain transcription errors.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-representation-theory-notes.json",
   "source_index": 113,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0001": {
   "statement_status": "exact",
   "original_statement": "Let $\\Omega$ be a pseudoconvex domain in $\\mathbb{C}^n$ that contains $0$ and let $\\phi\\in L^1_{loc}(\\Omega)$ be a weight function on $\\Omega$. Let $H$ be a hyperplane through $0$ and $f$ be a holomorphic function on $H\\cap\\Omega$ such that $$\\int_{H\\cap\\Omega}|f|^2e^{-\\phi}dV<\\infty.$$\nSuppose that for some constant $C$, $$i\\partial\\overline{\\partial}\\phi\\geq -Ci\\partial\\overline{\\partial}\\log B_{\\Omega}(z,z)$$\nwhere $B_{\\Omega}(z,z)$ denotes the Bergman kernel of $\\Omega$ on the diagonal.\nHow big can $C$ be so that there exists a holomorphic function $F$ on $\\Omega$ such that $$F=f \\text{ on }H\\cap\\Omega \\text{ and }\\int_{\\Omega}|F|^2e^{-\\phi}dV\\leq \\int_{H\\cap\\Omega}|f|^2e^{-\\phi}dV?$$\nIs it possible to find a universal constant $C$?\n\n$\\bullet$When $C=0$, this is the Ohsawa-Takegoshi extension theorem (see \\cite{MR2743817}).\n\n$\\bullet$ If $\\Omega$ is strictly pseudoconvex then $C\\geq\\frac{1}{n+1}$ works (see \\cite{MR2743817}); however, it is not known whether this is sharp.\n\n$\\bullet$ It is known that such a constant $C$ exists when the domain is convex and finite type in $\\mathbb{C}^n$ or just finite type in $\\mathbb{C}^2$.",
   "clean_statement": "Let $\\Omega$ be a pseudoconvex domain in $\\mathbb{C}^n$ that contains $0$ and let $\\phi\\in L^1_{loc}(\\Omega)$ be a weight function on $\\Omega$. Let $H$ be a hyperplane through $0$ and $f$ be a holomorphic function on $H\\cap\\Omega$ such that $$\\int_{H\\cap\\Omega}|f|^2e^{-\\phi}dV<\\infty.$$\nSuppose that for some constant $C$, $$i\\partial\\overline{\\partial}\\phi\\geq -Ci\\partial\\overline{\\partial}\\log B_{\\Omega}(z,z)$$\nwhere $B_{\\Omega}(z,z)$ denotes the Bergman kernel of $\\Omega$ on the diagonal.\nHow big can $C$ be so that there exists a holomorphic function $F$ on $\\Omega$ such that $$F=f \\text{ on }H\\cap\\Omega \\text{ and }\\int_{\\Omega}|F|^2e^{-\\phi}dV\\leq \\int_{H\\cap\\Omega}|f|^2e^{-\\phi}dV?$$\nIs it possible to find a universal constant $C$?\n\n$\\bullet$When $C=0$, this is the Ohsawa-Takegoshi extension theorem (see \\cite{MR2743817}).\n\n$\\bullet$ If $\\Omega$ is strictly pseudoconvex then $C\\geq\\frac{1}{n+1}$ works (see \\cite{MR2743817}); however, it is not known whether this is sharp.\n\n$\\bullet$ It is known that such a constant $C$ exists when the domain is convex and finite type in $\\mathbb{C}^n$ or just finite type in $\\mathbb{C}^2$.",
   "public_statement": "Let $\\Omega$ be a pseudoconvex domain in $\\mathbb{C}^n$ that contains $0$ and let $\\phi\\in L^1_{loc}(\\Omega)$ be a weight function on $\\Omega$. Let $H$ be a hyperplane through $0$ and $f$ be a holomorphic function on $H\\cap\\Omega$ such that $$\\int_{H\\cap\\Omega}|f|^2e^{-\\phi}dV<\\infty.$$\nSuppose that for some constant $C$, $$i\\partial\\overline{\\partial}\\phi\\geq -Ci\\partial\\overline{\\partial}\\log B_{\\Omega}(z,z)$$\nwhere $B_{\\Omega}(z,z)$ denotes the Bergman kernel of $\\Omega$ on the diagonal.\nHow big can $C$ be so that there exists a holomorphic function $F$ on $\\Omega$ such that $$F=f \\text{ on }H\\cap\\Omega \\text{ and }\\int_{\\Omega}|F|^2e^{-\\phi}dV\\leq \\int_{H\\cap\\Omega}|f|^2e^{-\\phi}dV?$$\nIs it possible to find a universal constant $C$?\n\n$\\bullet$When $C=0$, this is the Ohsawa-Takegoshi extension theorem (see \\cite{MR2743817}).\n\n$\\bullet$ If $\\Omega$ is strictly pseudoconvex then $C\\geq\\frac{1}{n+1}$ works (see \\cite{MR2743817}); however, it is not known whether this is sharp.\n\n$\\bullet$ It is known that such a constant $C$ exists when the domain is convex and finite type in $\\mathbb{C}^n$ or just finite type in $\\mathbb{C}^2$.",
   "evidence": "The canonical record asks the following. Let \\(\\Omega\\subset\\mathbb C^n\\) be pseudoconvex and contain \\(0\\), let \\(H\\) be a hyperplane through \\(0\\), and let \\(\\phi\\in L^1_{\\mathrm{loc}}(\\Omega)\\). Assuming \\[ i\\partial\\bar\\partial\\phi\\geq -C\\,i\\partial\\bar\\partial\\log B_\\Omega(z,z), \\] how large may \\(C\\) be while every holomorphic \\(f\\) on \\(H\\cap\\Omega\\) of finite weighted \\(L^2\\)-norm has an extension \\(F\\in\\mathcal O(\\Omega)\\) satisfying the coefficient-one estimate \\[ \\int_\\Omega |F|^2e^{-\\phi}\\,dV_n \\leq \\int_{H\\cap\\Omega}|f|^2e^{-\\phi}\\,dV_{n-1}? \\tag{1} \\] It also asks whether \\(C\\) can be universal. The source says that \\(C=0\\) is Ohsawa--Takegoshi and that, for strictly pseudoconvex \\(\\Omega\\), ``\\(C\\geq 1/(n+1)\\) works.''",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0002": {
   "statement_status": "exact",
   "original_statement": "Let $\\mathcal{W}$ be the smooth Diederich--Fornaess worm domain in $\\mathbb{C}^2$ (see \\cite{MR0430315}) and $\\rho$ be a defining function for $\\mathcal{W}$. Let $g_{\\rho}$ be the K\\\"ahler metric on $\\mathcal{W}$ whose K\\\"ahler 2-form is $i\\partial\\overline{\\partial}\\log(-\\rho).$\nStudy the $\\mathcal{C}^{\\infty}$-regularity up the boundary for the Dirichlet problem\n\\begin{align*}\n\\Delta_{\\rho}u&=0 \\text{ in } \\mathcal{W},\\\\\nu&=f \\text{ on } b\\mathcal{W},\n\\end{align*}\nwhere $f\\in \\mathcal{C}^{\\infty}(b\\mathcal{W})$ and $\\Delta_{\\rho}$ is the Laplace-Beltrami operator corresponding to $g_{\\rho}$. Start by analyzing the pseudohermitian invariants of the leaves of the foliation given by the level sets of $\\rho$ and in particular understand the geometry of the weakly pseudoconvex locus of this foliation.",
   "clean_statement": "Let $\\mathcal{W}$ be the smooth Diederich--Fornaess worm domain in $\\mathbb{C}^2$ (see \\cite{MR0430315}) and $\\rho$ be a defining function for $\\mathcal{W}$. Let $g_{\\rho}$ be the K\\\"ahler metric on $\\mathcal{W}$ whose K\\\"ahler 2-form is $i\\partial\\overline{\\partial}\\log(-\\rho).$\nStudy the $\\mathcal{C}^{\\infty}$-regularity up the boundary for the Dirichlet problem\n\\begin{align*}\n\\Delta_{\\rho}u&=0 \\text{ in } \\mathcal{W},\\\\\nu&=f \\text{ on } b\\mathcal{W},\n\\end{align*}\nwhere $f\\in \\mathcal{C}^{\\infty}(b\\mathcal{W})$ and $\\Delta_{\\rho}$ is the Laplace-Beltrami operator corresponding to $g_{\\rho}$. Start by analyzing the pseudohermitian invariants of the leaves of the foliation given by the level sets of $\\rho$ and in particular understand the geometry of the weakly pseudoconvex locus of this foliation.",
   "public_statement": "Let $\\mathcal{W}$ be the smooth Diederich--Fornaess worm domain in $\\mathbb{C}^2$ (see \\cite{MR0430315}) and $\\rho$ be a defining function for $\\mathcal{W}$. Let $g_{\\rho}$ be the K\\\"ahler metric on $\\mathcal{W}$ whose K\\\"ahler 2-form is $i\\partial\\overline{\\partial}\\log(-\\rho).$\nStudy the $\\mathcal{C}^{\\infty}$-regularity up the boundary for the Dirichlet problem\n\\begin{align*}\n\\Delta_{\\rho}u&=0 \\text{ in } \\mathcal{W},\\\\\nu&=f \\text{ on } b\\mathcal{W},\n\\end{align*}\nwhere $f\\in \\mathcal{C}^{\\infty}(b\\mathcal{W})$ and $\\Delta_{\\rho}$ is the Laplace-Beltrami operator corresponding to $g_{\\rho}$. Start by analyzing the pseudohermitian invariants of the leaves of the foliation given by the level sets of $\\rho$ and in particular understand the geometry of the weakly pseudoconvex locus of this foliation.",
   "evidence": "The supplied AIM record asks for boundary regularity on the smooth Diederich--Fornæss worm \\(\\mathcal W\\subset\\mathbb C^2\\) for the Laplace--Beltrami operator associated with the displayed form \\[ i\\partial\\bar\\partial\\log(-\\rho), \\] where \\(\\rho<0\\) in \\(\\mathcal W\\), and prints the boundary condition as `\\(\\nu=f\\)`.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0003": {
   "statement_status": "exact",
   "original_statement": "Let $\\Omega$ be a smooth bounded domain in $\\mathbb{C}^n$ such that the automorphism group of $\\Omega$ is non-compact. What can be said about $\\Omega$?\nSee \\cite{MR1706680}.",
   "clean_statement": "Let $\\Omega$ be a smooth bounded domain in $\\mathbb{C}^n$ such that the automorphism group of $\\Omega$ is non-compact. What can be said about $\\Omega$?\nSee \\cite{MR1706680}.",
   "public_statement": "Let $\\Omega$ be a smooth bounded domain in $\\mathbb{C}^n$ such that the automorphism group of $\\Omega$ is non-compact. What can be said about $\\Omega$?\nSee \\cite{MR1706680}.",
   "evidence": "The canonical record is problem 1.3 in the section “Estimates for \\(\\overline{\\partial}\\)” of the AIM workshop list *The Cauchy--Riemann equations in several variables*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0004": {
   "statement_status": "exact",
   "original_statement": "Let $\\mathcal{W}$ be the smooth Diederich--Fornaess worm domain in $\\mathbb{C}^2$ and $\\mathbf{B}_{\\mathcal{W}}$ denote the Bergman projection operator on $\\mathcal{W}$. Prove or disprove that $\\mathbf{B}_{\\mathcal{W}}(\\log \\overline{z_1})$ is in $\\mathcal{C}^{\\infty}(\\overline{\\mathcal{W}})$. Note that if it fails to be smooth up to the boundary then an alternative proof is obtained for the fact that the Condition R fails on $\\mathcal{W}$. See \\cite{MR3130312} and \\cite{MR1370592}.",
   "clean_statement": "Let $\\mathcal{W}$ be the smooth Diederich--Fornaess worm domain in $\\mathbb{C}^2$ and $\\mathbf{B}_{\\mathcal{W}}$ denote the Bergman projection operator on $\\mathcal{W}$. Prove or disprove that $\\mathbf{B}_{\\mathcal{W}}(\\log \\overline{z_1})$ is in $\\mathcal{C}^{\\infty}(\\overline{\\mathcal{W}})$. Note that if it fails to be smooth up to the boundary then an alternative proof is obtained for the fact that the Condition R fails on $\\mathcal{W}$. See \\cite{MR3130312} and \\cite{MR1370592}.",
   "public_statement": "Let $\\mathcal{W}$ be the smooth Diederich--Fornaess worm domain in $\\mathbb{C}^2$ and $\\mathbf{B}_{\\mathcal{W}}$ denote the Bergman projection operator on $\\mathcal{W}$. Prove or disprove that $\\mathbf{B}_{\\mathcal{W}}(\\log \\overline{z_1})$ is in $\\mathcal{C}^{\\infty}(\\overline{\\mathcal{W}})$. Note that if it fails to be smooth up to the boundary then an alternative proof is obtained for the fact that the Condition R fails on $\\mathcal{W}$. See \\cite{MR3130312} and \\cite{MR1370592}.",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0005": {
   "statement_status": "exact",
   "original_statement": "Let $\\Omega$ be a smooth bounded pseudoconvex domain such that for any $0<\\eta<1$, there exists a defining function $\\rho_{\\eta}$ such that $-(-\\rho_{\\eta})^{\\eta}$ is plurisubharmonic on $\\Omega$. Note that such an $\\eta$ always exists \\cite{MR0430315}; however, it could be small. Determine minimal assumptions on $\\Omega$ to show that $\\overline{\\partial}$-Neumann operator $N$ (or the Bergman projection operator $\\mathbf{B}$) is globally regular. Determine minimal assumptions on $\\Omega$ to show existence of a Stein neighborhood basis for the closure. Are there any domains such that $\\mathbf{B}$ is globally regular but no such a family of defining functions exists?",
   "clean_statement": "Let $\\Omega$ be a smooth bounded pseudoconvex domain such that for any $0<\\eta<1$, there exists a defining function $\\rho_{\\eta}$ such that $-(-\\rho_{\\eta})^{\\eta}$ is plurisubharmonic on $\\Omega$. Note that such an $\\eta$ always exists \\cite{MR0430315}; however, it could be small. Determine minimal assumptions on $\\Omega$ to show that $\\overline{\\partial}$-Neumann operator $N$ (or the Bergman projection operator $\\mathbf{B}$) is globally regular. Determine minimal assumptions on $\\Omega$ to show existence of a Stein neighborhood basis for the closure. Are there any domains such that $\\mathbf{B}$ is globally regular but no such a family of defining functions exists?",
   "public_statement": "Let $\\Omega$ be a smooth bounded pseudoconvex domain such that for any $0<\\eta<1$, there exists a defining function $\\rho_{\\eta}$ such that $-(-\\rho_{\\eta})^{\\eta}$ is plurisubharmonic on $\\Omega$. Note that such an $\\eta$ always exists \\cite{MR0430315}; however, it could be small. Determine minimal assumptions on $\\Omega$ to show that $\\overline{\\partial}$-Neumann operator $N$ (or the Bergman projection operator $\\mathbf{B}$) is globally regular. Determine minimal assumptions on $\\Omega$ to show existence of a Stein neighborhood basis for the closure. Are there any domains such that $\\mathbf{B}$ is globally regular but no such a family of defining functions exists?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0006": {
   "statement_status": "exact",
   "original_statement": "Let $\\Omega$ be a smooth bounded pseudoconvex domain, let $\\mathbf{B}_{\\Omega}$ denote the Bergman projection operator and $M_{\\psi}$ denote the multiplication operator by $\\psi$. Suppose that $[\\mathbf{B}_{\\Omega},M_{\\psi}]$ is compact on $L^2(\\Omega)$ for all $\\psi\\in C(\\overline{\\Omega})$. Is $N_1$ compact on $L^2_{(0,1)}(\\Omega)$?",
   "clean_statement": "Let $\\Omega$ be a smooth bounded pseudoconvex domain, let $\\mathbf{B}_{\\Omega}$ denote the Bergman projection operator and $M_{\\psi}$ denote the multiplication operator by $\\psi$. Suppose that $[\\mathbf{B}_{\\Omega},M_{\\psi}]$ is compact on $L^2(\\Omega)$ for all $\\psi\\in C(\\overline{\\Omega})$. Is $N_1$ compact on $L^2_{(0,1)}(\\Omega)$?",
   "public_statement": "Let $\\Omega$ be a smooth bounded pseudoconvex domain, let $\\mathbf{B}_{\\Omega}$ denote the Bergman projection operator and $M_{\\psi}$ denote the multiplication operator by $\\psi$. Suppose that $[\\mathbf{B}_{\\Omega},M_{\\psi}]$ is compact on $L^2(\\Omega)$ for all $\\psi\\in C(\\overline{\\Omega})$. Is $N_1$ compact on $L^2_{(0,1)}(\\Omega)$?",
   "evidence": "The canonical record is problem 2.1, in the section “Obstruction to Compactness” of the AIM list *The Cauchy--Riemann equations in several variables*. Its exact mathematical question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0007": {
   "statement_status": "exact",
   "original_statement": "Let $\\Omega_2\\subset \\Omega_1$ be two smooth bounded pseudoconvex domains that share a boundary point $p$. If the restriction map from $A^2(\\Omega_1)$ into $A^2(\\Omega_2)$ is not compact, is the D'Angelo type of $p$ the same with respect to $\\Omega_1$ and $\\Omega_2$?\n\n$\\bullet$ When the inside domain $\\Omega_2$ is a ball, is $p$ strictly pseudoconvex with respect to $\\Omega_1$?",
   "clean_statement": "Let $\\Omega_2\\subset \\Omega_1$ be two smooth bounded pseudoconvex domains that share a boundary point $p$. If the restriction map from $A^2(\\Omega_1)$ into $A^2(\\Omega_2)$ is not compact, is the D'Angelo type of $p$ the same with respect to $\\Omega_1$ and $\\Omega_2$?\n\n$\\bullet$ When the inside domain $\\Omega_2$ is a ball, is $p$ strictly pseudoconvex with respect to $\\Omega_1$?",
   "public_statement": "Let $\\Omega_2\\subset \\Omega_1$ be two smooth bounded pseudoconvex domains that share a boundary point $p$. If the restriction map from $A^2(\\Omega_1)$ into $A^2(\\Omega_2)$ is not compact, is the D'Angelo type of $p$ the same with respect to $\\Omega_1$ and $\\Omega_2$?\n\n$\\bullet$ When the inside domain $\\Omega_2$ is a ball, is $p$ strictly pseudoconvex with respect to $\\Omega_1$?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0008": {
   "statement_status": "exact",
   "original_statement": "Describe the relation between compactness of the $\\overline{\\partial}$-Neumann operator and existence of a Stein neighborhood basis. In particular, does the geometric sufficient condition for compactness in \\cite{MR2097419} imply existence of a Stein neighborhood basis?",
   "clean_statement": "Describe the relation between compactness of the $\\overline{\\partial}$-Neumann operator and existence of a Stein neighborhood basis. In particular, does the geometric sufficient condition for compactness in \\cite{MR2097419} imply existence of a Stein neighborhood basis?",
   "public_statement": "Describe the relation between compactness of the $\\overline{\\partial}$-Neumann operator and existence of a Stein neighborhood basis. In particular, does the geometric sufficient condition for compactness in \\cite{MR2097419} imply existence of a Stein neighborhood basis?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0009": {
   "statement_status": "exact",
   "original_statement": "Is there a smooth bounded pseudoconvex domain $\\Omega$ in $\\mathbb{C}^n~ (n\\geq 3$) that contains a non-trivial analytic disc in $b\\Omega$ and yet the $\\overline{\\partial}$-Neumann operator $N_1$ is compact? When $n=2$, this is not possible \\cite{MR2603659}.",
   "clean_statement": "Is there a smooth bounded pseudoconvex domain $\\Omega$ in $\\mathbb{C}^n~ (n\\geq 3$) that contains a non-trivial analytic disc in $b\\Omega$ and yet the $\\overline{\\partial}$-Neumann operator $N_1$ is compact? When $n=2$, this is not possible \\cite{MR2603659}.",
   "public_statement": "Is there a smooth bounded pseudoconvex domain $\\Omega$ in $\\mathbb{C}^n~ (n\\geq 3$) that contains a non-trivial analytic disc in $b\\Omega$ and yet the $\\overline{\\partial}$-Neumann operator $N_1$ is compact? When $n=2$, this is not possible \\cite{MR2603659}.",
   "evidence": "The canonical record is problem 2.4 in the AIM workshop list *The Cauchy--Riemann equations in several variables*, section “Obstruction to Compactness.” Its exact extracted problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0010": {
   "statement_status": "exact",
   "original_statement": "Given proper holomorphic maps $f,g: \\mathbb{B}^n\\to \\mathbb{B}^N$, how does one show that $f$ and $g$ are not homotopic? Find homotopy invariants for proper holomorphic maps between balls.\n\n$\\bullet$ Special case (Lebl): Is the Faran map from $\\mathbb{B}^2$ to $\\mathbb{B}^4$ given by $$(z,w) \\to (z^3,\\sqrt{3}zw, w^3,0)$$ homotopic to the map $(z,w)\\to (z,w, 0, 0)$?",
   "clean_statement": "Given proper holomorphic maps $f,g: \\mathbb{B}^n\\to \\mathbb{B}^N$, how does one show that $f$ and $g$ are not homotopic? Find homotopy invariants for proper holomorphic maps between balls.\n\n$\\bullet$ Special case (Lebl): Is the Faran map from $\\mathbb{B}^2$ to $\\mathbb{B}^4$ given by $$(z,w) \\to (z^3,\\sqrt{3}zw, w^3,0)$$ homotopic to the map $(z,w)\\to (z,w, 0, 0)$?",
   "public_statement": "Given proper holomorphic maps $f,g: \\mathbb{B}^n\\to \\mathbb{B}^N$, how does one show that $f$ and $g$ are not homotopic? Find homotopy invariants for proper holomorphic maps between balls.\n\n$\\bullet$ Special case (Lebl): Is the Faran map from $\\mathbb{B}^2$ to $\\mathbb{B}^4$ given by $$(z,w) \\to (z^3,\\sqrt{3}zw, w^3,0)$$ homotopic to the map $(z,w)\\to (z,w, 0, 0)$?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0011": {
   "statement_status": "exact",
   "original_statement": "Let $R(z,\\overline{z})$ be a bihomogeneous polynomial. Find necessary and sufficient conditions such that there exists an integer $N$ with\n$$(R(z,\\overline{z}))^N=\\sum_{j=1}^{m}|p_j(z)|^2.$$\nHere $p_j(z)$'s are linearly independent holomorphic polynomials. See \\cite{MR1682713} and \\cite{MR2770459}.",
   "clean_statement": "Let $R(z,\\overline{z})$ be a bihomogeneous polynomial. Find necessary and sufficient conditions such that there exists an integer $N$ with\n$$(R(z,\\overline{z}))^N=\\sum_{j=1}^{m}|p_j(z)|^2.$$\nHere $p_j(z)$'s are linearly independent holomorphic polynomials. See \\cite{MR1682713} and \\cite{MR2770459}.",
   "public_statement": "Let $R(z,\\overline{z})$ be a bihomogeneous polynomial. Find necessary and sufficient conditions such that there exists an integer $N$ with\n$$(R(z,\\overline{z}))^N=\\sum_{j=1}^{m}|p_j(z)|^2.$$\nHere $p_j(z)$'s are linearly independent holomorphic polynomials. See \\cite{MR1682713} and \\cite{MR2770459}.",
   "evidence": "The exact record is problem 3.2, “Mappings,” from the AIM workshop *The Cauchy--Riemann equations in several variables*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0012": {
   "statement_status": "exact",
   "original_statement": "Let $R(z,\\overline{z})$ be a Hermitian polynomial that is a sum of squares (SOS)\n$$R(z,\\overline{z})=\\sum_{j=1}^m|p_j(z)|^2,$$\nwhere $p_j(z)$'s are linearly independent holomorphic polynomials. Suppose that $R(z,\\overline{z})$ is divisible by $||z||^2$ (i.e. $R(z,\\overline{z})=||z||^2A(z,\\overline{z})$). What are the possible values of $m$?\n\n$\\bullet$ Huang's lemma \\cite{MR1703603} implies that $m$ is either 0 or at least $n$.\n\n$\\bullet$ See \\cite{MR2869101} for a generalization of the Huang's lemma when $||z||^2$ is replaced by $||z||^{2d}$. Consider the same question in this case.",
   "clean_statement": "Let $R(z,\\overline{z})$ be a Hermitian polynomial that is a sum of squares (SOS)\n$$R(z,\\overline{z})=\\sum_{j=1}^m|p_j(z)|^2,$$\nwhere $p_j(z)$'s are linearly independent holomorphic polynomials. Suppose that $R(z,\\overline{z})$ is divisible by $||z||^2$ (i.e. $R(z,\\overline{z})=||z||^2A(z,\\overline{z})$). What are the possible values of $m$?\n\n$\\bullet$ Huang's lemma \\cite{MR1703603} implies that $m$ is either 0 or at least $n$.\n\n$\\bullet$ See \\cite{MR2869101} for a generalization of the Huang's lemma when $||z||^2$ is replaced by $||z||^{2d}$. Consider the same question in this case.",
   "public_statement": "Let $R(z,\\overline{z})$ be a Hermitian polynomial that is a sum of squares (SOS)\n$$R(z,\\overline{z})=\\sum_{j=1}^m|p_j(z)|^2,$$\nwhere $p_j(z)$'s are linearly independent holomorphic polynomials. Suppose that $R(z,\\overline{z})$ is divisible by $||z||^2$ (i.e. $R(z,\\overline{z})=||z||^2A(z,\\overline{z})$). What are the possible values of $m$?\n\n$\\bullet$ Huang's lemma \\cite{MR1703603} implies that $m$ is either 0 or at least $n$.\n\n$\\bullet$ See \\cite{MR2869101} for a generalization of the Huang's lemma when $||z||^2$ is replaced by $||z||^{2d}$. Consider the same question in this case.",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0013": {
   "statement_status": "exact",
   "original_statement": "Let $D_1$ and $D_2$ be two bounded symmetric domains such that neither is a ball. Let $F:D_1\\to D_2$ be a proper holomorphic map. Is $F$ a \\textit{trivial map}, that is $F(z)=(z,g(z))$, in suitable coordinates, where $g(z)$ is a vector valued holomorphic function? By a result of Tsai, the answer is yes when the rank (as a bounded symmetric domain) of $D_2$ is not greater than the rank of $D_1$.",
   "clean_statement": "Let $D_1$ and $D_2$ be two bounded symmetric domains such that neither is a ball. Let $F:D_1\\to D_2$ be a proper holomorphic map. Is $F$ a \\textit{trivial map}, that is $F(z)=(z,g(z))$, in suitable coordinates, where $g(z)$ is a vector valued holomorphic function? By a result of Tsai, the answer is yes when the rank (as a bounded symmetric domain) of $D_2$ is not greater than the rank of $D_1$.",
   "public_statement": "Let $D_1$ and $D_2$ be two bounded symmetric domains such that neither is a ball. Let $F:D_1\\to D_2$ be a proper holomorphic map. Is $F$ a \\textit{trivial map}, that is $F(z)=(z,g(z))$, in suitable coordinates, where $g(z)$ is a vector valued holomorphic function? By a result of Tsai, the answer is yes when the rank (as a bounded symmetric domain) of $D_2$ is not greater than the rank of $D_1$.",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0014": {
   "statement_status": "exact",
   "original_statement": "It was proven by Eliashberg that any embeddable CR-structure on $\\mathbb{S}^3$ bounds a Stein manifold $X\\simeq \\mathbb{B}^4$ with a strictly plurisubharmonic exhaustion function with a single critical point. Can $X$ always be embedded into $\\mathbb{C}^2$?",
   "clean_statement": "It was proven by Eliashberg that any embeddable CR-structure on $\\mathbb{S}^3$ bounds a Stein manifold $X\\simeq \\mathbb{B}^4$ with a strictly plurisubharmonic exhaustion function with a single critical point. Can $X$ always be embedded into $\\mathbb{C}^2$?",
   "public_statement": "It was proven by Eliashberg that any embeddable CR-structure on $\\mathbb{S}^3$ bounds a Stein manifold $X\\simeq \\mathbb{B}^4$ with a strictly plurisubharmonic exhaustion function with a single critical point. Can $X$ always be embedded into $\\mathbb{C}^2$?",
   "evidence": "The local JSON record and its neighboring records were inspected. There is no apparent OCR corruption. The old page `http://aimpl.org/crscv/3/` did not load during this run, so the displayed record could not be compared with a currently served original page.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0015": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Let $C$ be a class of hypersurfaces defined by a finite order condition. A normal form is a subclass $C_0$ where normal representatives are determined up to a finite dimensional group. For example, for Levi non-degenerate hypersurfaces there is the Chern-Moser normal form and for finite type hypersurfaces in $\\mathbb{C}^2$ Kollar presented a normal form. Can you find a class where it can be proved that there is no convergent normal form?",
   "clean_statement": null,
   "public_statement": "Let $C$ be a class of hypersurfaces defined by a finite order condition. A normal form is a subclass $C_0$ where normal representatives are determined up to a finite dimensional group. For example, for Levi non-degenerate hypersurfaces there is the Chern-Moser normal form and for finite type hypersurfaces in $\\mathbb{C}^2$ Kollar presented a normal form. Can you find a class where it can be proved that there is no convergent normal form?",
   "evidence": "The exact AIM record, problem 5.1 in the section “Normal Forms,” asks:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0016": {
   "statement_status": "exact",
   "original_statement": "Compute the spectrum of the $\\overline{\\partial}$-Neumann Laplacian on the Hartogs triangle and on the Diederich-Fornaess worm domain. In particular, determine whether or not the corresponding spectra are discrete.",
   "clean_statement": "Compute the spectrum of the $\\overline{\\partial}$-Neumann Laplacian on the Hartogs triangle and on the Diederich-Fornaess worm domain. In particular, determine whether or not the corresponding spectra are discrete.",
   "public_statement": "Compute the spectrum of the $\\overline{\\partial}$-Neumann Laplacian on the Hartogs triangle and on the Diederich-Fornaess worm domain. In particular, determine whether or not the corresponding spectra are discrete.",
   "evidence": "The canonical AIM record (workshop *The Cauchy--Riemann equations in several variables*, section “Spectrum of the $\\overline\\partial$-Neumann Laplacian,” problem 6.1) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0017": {
   "statement_status": "exact",
   "original_statement": "Relate the infimum of the essential spectrum of the $\\overline{\\partial}$-Neumann Laplacian on the worm domain to the winding of the domain.",
   "clean_statement": "Relate the infimum of the essential spectrum of the $\\overline{\\partial}$-Neumann Laplacian on the worm domain to the winding of the domain.",
   "public_statement": "Relate the infimum of the essential spectrum of the $\\overline{\\partial}$-Neumann Laplacian on the worm domain to the winding of the domain.",
   "evidence": "It is problem 6.2 in the section “Spectrum of $\\overline\\partial$-Neumann Laplacian” of the AIM list *The Cauchy--Riemann equations in several variables*. The neighboring problems ask for the spectrum on the Hartogs triangle and worm domain (6.1) and whether the spectrum is always discrete on a smooth bounded pseudoconvex domain (6.3). The text has no visible OCR error, but it is mathematically under-specified in four ways:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0018": {
   "statement_status": "exact",
   "original_statement": "Is the spectrum of the $\\overline{\\partial}$-Neumann Laplacian on a bounded smooth pseudoconvex domain in $\\mathbb{C}^n$ always discrete? Is the infimum of the spectrum always a point eigenvalue (ground state energy)?",
   "clean_statement": "Is the spectrum of the $\\overline{\\partial}$-Neumann Laplacian on a bounded smooth pseudoconvex domain in $\\mathbb{C}^n$ always discrete? Is the infimum of the spectrum always a point eigenvalue (ground state energy)?",
   "public_statement": "Is the spectrum of the $\\overline{\\partial}$-Neumann Laplacian on a bounded smooth pseudoconvex domain in $\\mathbb{C}^n$ always discrete? Is the infimum of the spectrum always a point eigenvalue (ground state energy)?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0019": {
   "statement_status": "exact",
   "original_statement": "Let $\\phi:\\mathbb{C}^n\\to\\mathbb{R}$ be a $C^2$ plurisubharmonic function. Determine the relation between the eigenvalues of the complex Hesssian of $\\phi$ and the compactness of the $\\overline{\\partial}$-Neumann operator on $L^2_{(0,q)}(\\mathbb{C}^n,e^{-\\phi})$. See \\cite{MR3029187} for $n=1$. Determine the relation between the spectrum of the complex Laplacian $\\Box_{\\phi}$ and the weight function $\\phi$. In particular, understand the bottom (infimum) of the essential spectrum.",
   "clean_statement": "Let $\\phi:\\mathbb{C}^n\\to\\mathbb{R}$ be a $C^2$ plurisubharmonic function. Determine the relation between the eigenvalues of the complex Hesssian of $\\phi$ and the compactness of the $\\overline{\\partial}$-Neumann operator on $L^2_{(0,q)}(\\mathbb{C}^n,e^{-\\phi})$. See \\cite{MR3029187} for $n=1$. Determine the relation between the spectrum of the complex Laplacian $\\Box_{\\phi}$ and the weight function $\\phi$. In particular, understand the bottom (infimum) of the essential spectrum.",
   "public_statement": "Let $\\phi:\\mathbb{C}^n\\to\\mathbb{R}$ be a $C^2$ plurisubharmonic function. Determine the relation between the eigenvalues of the complex Hesssian of $\\phi$ and the compactness of the $\\overline{\\partial}$-Neumann operator on $L^2_{(0,q)}(\\mathbb{C}^n,e^{-\\phi})$. See \\cite{MR3029187} for $n=1$. Determine the relation between the spectrum of the complex Laplacian $\\Box_{\\phi}$ and the weight function $\\phi$. In particular, understand the bottom (infimum) of the essential spectrum.",
   "evidence": "The source is Problem 6.4, “Spectrum of \\(\\overline\\partial\\)-Neumann Laplacian,” from the AIM workshop *The Cauchy–Riemann equations in several variables*. Apart from correcting the typographical error “Hesssian” to “Hessian,” the problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0020": {
   "statement_status": "exact",
   "original_statement": "1. (B. Lamel) Let n ≥ 2. \n\n• Given a proper rational map f: Bn → BN, find the smallest number k(n, N ) such that f is determined by its k-jet at the origin. \n\n• Let Ω, Ω′ in Cn and CN respectively be strongly pseudoconvex domains. If f: Ω → Ω′\n\nis a proper holomorphic map which extends smoothly to ∂Ω, does there exist a k\n\nsuch that f is determined by its k-jet at a point?",
   "clean_statement": "1. (B. Lamel) Let n ≥ 2.\n\n• Given a proper rational map f: Bn → BN, find the smallest number k(n, N ) such that f is determined by its k-jet at the origin.\n\n• Let Ω, Ω′ in Cn and CN respectively be strongly pseudoconvex domains. If f: Ω → Ω′\n\nis a proper holomorphic map which extends smoothly to ∂Ω, does there exist a k\n\nsuch that f is determined by its k-jet at a point?",
   "public_statement": "1. (B. Lamel) Let n ≥ 2.\n\n• Given a proper rational map f: Bn → BN, find the smallest number k(n, N ) such that f is determined by its k-jet at the origin.\n\n• Let Ω, Ω′ in Cn and CN respectively be strongly pseudoconvex domains. If f: Ω → Ω′\n\nis a proper holomorphic map which extends smoothly to ∂Ω, does there exist a k\n\nsuch that f is determined by its k-jet at a point?",
   "evidence": "This is Problem 1 proposed by Bernhard Lamel in the 2010 AIM workshop *Emerging Applications of Complexity for CR Mappings*. The original three-page AIM PDF says that \\(\\mathbb B^n\\) is the unit ball in \\(\\mathbb C^n\\), and asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 19,
   "attempt": 2
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0021": {
   "statement_status": "exact",
   "original_statement": "2. (F. Meylan) Assume M ⊆ Cn is a real-analytic generic submanifold, f = pq: Cn → CN\n\nis the germ of a meromorphic map at z ∈ M, and f is holomorphic in a one-sided wedge \n\nW attached to M near z. Assume that ‖f (w)‖2 increases to 1 as w → M in W. Does f\n\nextend to a full neighborhood of z in Cn?\n\n•",
   "clean_statement": "2. (F. Meylan) Assume M ⊆ Cn is a real-analytic generic submanifold, f = pq: Cn → CN\n\nis the germ of a meromorphic map at z ∈ M, and f is holomorphic in a one-sided wedge\n\nW attached to M near z. Assume that ‖f (w)‖2 increases to 1 as w → M in W. Does f\n\nextend to a full neighborhood of z in Cn?\n\n•",
   "public_statement": "2. (F. Meylan) Assume M ⊆ Cn is a real-analytic generic submanifold, f = pq: Cn → CN\n\nis the germ of a meromorphic map at z ∈ M, and f is holomorphic in a one-sided wedge\n\nW attached to M near z. Assume that ‖f (w)‖2 increases to 1 as w → M in W. Does f\n\nextend to a full neighborhood of z in Cn?\n\n•",
   "evidence": "The official AIM workshop PDF is *Emerging applications of complexity for CR mappings* (Palo Alto, August 9--13, 2010). Its Section \"Mappings between balls\" defines \\(\\mathbb B^n\\) to be the unit ball and gives the following Problem 2, attributed to F. Meylan:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0022": {
   "statement_status": "exact",
   "original_statement": "3. (X. Huang) Does there exist a universal constant t such that a proper holomorphic map between Bn and BN (with 1 < n < N ) which is Ct up to the boundary is actually a rational map?",
   "clean_statement": "3. (X. Huang) Does there exist a universal constant t such that a proper holomorphic map between Bn and BN (with 1 < n < N ) which is Ct up to the boundary is actually a rational map?",
   "public_statement": "3. (X. Huang) Does there exist a universal constant t such that a proper holomorphic map between Bn and BN (with 1 < n < N ) which is Ct up to the boundary is actually a rational map?",
   "evidence": "There is no substantive OCR corruption. The notation \\(C^t\\) in this literature normally means \\(t\\) continuous derivatives on the closed ball, with \\(t\\) a nonnegative integer. If \\(t=k+\\alpha\\) is allowed to be nonintegral, the appropriate interpretation is the Hölder class \\(C^{k,\\alpha}\\); that is a distinct, stronger quantitative version of the question.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0023": {
   "statement_status": "exact",
   "original_statement": "4. (X. Huang) Assume Γ ⊆ Aut( BN ) is a discrete subgroup and f: Bn → BN is a proper holomorphic embedding such that Γ( f (Bn)) ⊆ f (Bn) and f (Bn)/Γ is compact. Is f linear?",
   "clean_statement": "4. (X. Huang) Assume Γ ⊆ Aut( BN ) is a discrete subgroup and f: Bn → BN is a proper holomorphic embedding such that Γ( f (Bn)) ⊆ f (Bn) and f (Bn)/Γ is compact. Is f linear?",
   "public_statement": "4. (X. Huang) Assume Γ ⊆ Aut( BN ) is a discrete subgroup and f: Bn → BN is a proper holomorphic embedding such that Γ( f (Bn)) ⊆ f (Bn) and f (Bn)/Γ is compact. Is f linear?",
   "evidence": "The canonical record was extracted from the AIM list for the 2010 workshop “Emerging Applications of Complexity for CR Mappings.” The source PDF defines \\(\\mathbb B^k\\) to be the unit ball in \\(\\mathbb C^k\\), and Problem 4 reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 22,
   "attempt": 2
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0024": {
   "statement_status": "exact",
   "original_statement": "5. (J. D'Angelo) Let f: Bn → BN be a proper rational map. Is f homotopic to a polynomial map p of the same degree? That is, ∀t ∈ [0, 1], each ft: Bn → BN is proper with f0 = f and f1 = p.\n\nSquared norms",
   "clean_statement": "5. (J. D'Angelo) Let f: Bn → BN be a proper rational map. Is f homotopic to a polynomial map p of the same degree? That is, ∀t ∈ [0, 1], each ft: Bn → BN is proper with f0 = f and f1 = p.\n\nSquared norms",
   "public_statement": "5. (J. D'Angelo) Let f: Bn → BN be a proper rational map. Is f homotopic to a polynomial map p of the same degree? That is, ∀t ∈ [0, 1], each ft: Bn → BN is proper with f0 = f and f1 = p.\n\nSquared norms",
   "evidence": "The official AIM PDF *Emerging applications of complexity for CR mappings* (2010) begins by declaring that \\(\\mathbb B^n\\) denotes the unit ball in \\(\\mathbb C^n\\). In its section \"Mappings between balls,\" Problem 5, attributed to J. D'Angelo, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0025": {
   "statement_status": "exact",
   "original_statement": "1. (D. Grundmeier and J. Lebl) For n ≥ 2, define the rank of a real polynomial p(z, ¯z)on Cn to be the smallest integer N such that p(z, ¯z) = ∑Nj=1 ±| pj (z)|2. Given p: Cn → R\n\nreal analytic, suppose that for all complex hyperplanes H, p|H has rank ≤ k. Does there exist c(n, k ) such that rank( p) ≤ c(n, k )? \n\n•",
   "clean_statement": "1. (D. Grundmeier and J. Lebl) For n ≥ 2, define the rank of a real polynomial p(z, ¯z)on Cn to be the smallest integer N such that p(z, ¯z) = ∑Nj=1 ±| pj (z)|2. Given p: Cn → R\n\nreal analytic, suppose that for all complex hyperplanes H, p|H has rank ≤ k. Does there exist c(n, k ) such that rank( p) ≤ c(n, k )?\n\n•",
   "public_statement": "1. (D. Grundmeier and J. Lebl) For n ≥ 2, define the rank of a real polynomial p(z, ¯z)on Cn to be the smallest integer N such that p(z, ¯z) = ∑Nj=1 ±| pj (z)|2. Given p: Cn → R\n\nreal analytic, suppose that for all complex hyperplanes H, p|H has rank ≤ k. Does there exist c(n, k ) such that rank( p) ≤ c(n, k )?\n\n•",
   "evidence": "The AIM workshop list asks the following question (with OCR spacing and notation normalized, but no mathematical change):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0026": {
   "statement_status": "exact",
   "original_statement": "2. (J. D'Angelo) Find necessary and sufficient conditions on a real-valued polynomial \n\nR(z, ¯z) such that there exists N with RN = ∑kj=1 |pj (z)|2.",
   "clean_statement": "2. (J. D'Angelo) Find necessary and sufficient conditions on a real-valued polynomial\n\nR(z, ¯z) such that there exists N with RN = ∑kj=1 |pj (z)|2.",
   "public_statement": "2. (J. D'Angelo) Find necessary and sufficient conditions on a real-valued polynomial\n\nR(z, ¯z) such that there exists N with RN = ∑kj=1 |pj (z)|2.",
   "evidence": "The canonical JSON record is an OCR extraction from the AIM workshop list *Emerging applications of complexity for CR mappings*. The extraction lost superscripts and the lower and upper placement of the sum indices. The original PDF gives Problem 2, attributed to J. D'Angelo:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0027": {
   "statement_status": "exact",
   "original_statement": "3. (P. Ebenfelt) Let S(z, ¯z) be a harmonic hermitian polynomial of degree ( d, d ) on Cn.For any given k, how \"many\" polynomials A(z, ¯z) of type ( d − 1, d − 1) exist such that \n\nS + A‖z‖2 = ∑kj=1 ‖fj ‖2? Which properties of S determine this? 1•",
   "clean_statement": "3. (P. Ebenfelt) Let S(z, ¯z) be a harmonic hermitian polynomial of degree ( d, d ) on Cn.For any given k, how \"many\" polynomials A(z, ¯z) of type ( d − 1, d − 1) exist such that\n\nS + A‖z‖2 = ∑kj=1 ‖fj ‖2? Which properties of S determine this? 1•",
   "public_statement": "3. (P. Ebenfelt) Let S(z, ¯z) be a harmonic hermitian polynomial of degree ( d, d ) on Cn.For any given k, how \"many\" polynomials A(z, ¯z) of type ( d − 1, d − 1) exist such that\n\nS + A‖z‖2 = ∑kj=1 ‖fj ‖2? Which properties of S determine this? 1•",
   "evidence": "The AIM list asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0028": {
   "statement_status": "exact",
   "original_statement": "1. (B. Stensones) Let p(z, ¯z, w, ¯w) be a plurisubharmonic, homogeneous, degree 2 k poly-nomial on C2. Let Σ be an algebraic curve not containing the origin. If p|Σ ≡ 0, then is Σ the level set of a homogeneous holomorphic polynomial?",
   "clean_statement": "1. (B. Stensones) Let p(z, ¯z, w, ¯w) be a plurisubharmonic, homogeneous, degree 2 k poly-nomial on C2. Let Σ be an algebraic curve not containing the origin. If p|Σ ≡ 0, then is Σ the level set of a homogeneous holomorphic polynomial?",
   "public_statement": "1. (B. Stensones) Let p(z, ¯z, w, ¯w) be a plurisubharmonic, homogeneous, degree 2 k poly-nomial on C2. Let Σ be an algebraic curve not containing the origin. If p|Σ ≡ 0, then is Σ the level set of a homogeneous holomorphic polynomial?",
   "evidence": "The AIM workshop PDF has a section headed “Plurisubharmonic polynomials” and asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0029": {
   "statement_status": "exact",
   "original_statement": "2. (B. Stensones) Let p(z, ¯z, w, ¯w) = pαβγδ zα ¯zβ wγ ¯wδ be a plurisubharmonic real-valued polynomial on C2. Write N (p) = {(a, b ) ∈ N2: a = α + β, b = γ + δ, p αβγδ 6 = 0 }. Let Γ 1, Γ2\n\nbe extreme edges of N (p) such that Γ 1 ∩ Γ2 6 = ∅, where an extreme edge is a line in N (p)with no points in N (p) below. Can q = ∑ \n\n> (α+β,γ +δ)∈Γ1∪Γ2\n\npαβγδ zα ¯zβ wγ ¯wδ be written as the sum of two plurisubharmonic weighted homogeneous polynomials? \n\n•",
   "clean_statement": "2. (B. Stensones) Let p(z, ¯z, w, ¯w) = pαβγδ zα ¯zβ wγ ¯wδ be a plurisubharmonic real-valued polynomial on C2. Write N (p) = {(a, b ) ∈ N2: a = α + β, b = γ + δ, p αβγδ 6 = 0 }. Let Γ 1, Γ2\n\nbe extreme edges of N (p) such that Γ 1 ∩ Γ2 6 = ∅, where an extreme edge is a line in N (p)with no points in N (p) below. Can q = ∑\n\n> (α+β,γ +δ)∈Γ1∪Γ2\n\npαβγδ zα ¯zβ wγ ¯wδ be written as the sum of two plurisubharmonic weighted homogeneous polynomials?\n\n•",
   "public_statement": "2. (B. Stensones) Let p(z, ¯z, w, ¯w) = pαβγδ zα ¯zβ wγ ¯wδ be a plurisubharmonic real-valued polynomial on C2. Write N (p) = {(a, b ) ∈ N2: a = α + β, b = γ + δ, p αβγδ 6 = 0 }. Let Γ 1, Γ2\n\nbe extreme edges of N (p) such that Γ 1 ∩ Γ2 6 = ∅, where an extreme edge is a line in N (p)with no points in N (p) below. Can q = ∑\n\n> (α+β,γ +δ)∈Γ1∪Γ2\n\npαβγδ zα ¯zβ wγ ¯wδ be written as the sum of two plurisubharmonic weighted homogeneous polynomials?\n\n•",
   "evidence": "The canonical JSON is a visibly corrupted OCR extraction. In particular, it contains `6 =` twice, a stray `>` before the summation condition, and the phrase “a line in \\(N(p)\\) with no points ... below.” Inspection of the original AIM PDF resolves the mathematical symbols. Most importantly, the source says \\[ \\Gamma_1\\cap\\Gamma_2\\ne\\varnothing, \\] not that the edges are disjoint. The source-verified statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0030": {
   "statement_status": "exact",
   "original_statement": "1. (S. Berhanu) Let M 2n−1 ⊆ Cn be a real hypersurface satisfying the maximum principle for CR functions in the strong sense (continuous CR functions attaining a (weak) local maximum must be constant). Is it necessary that any local continuous CR function is open? \n\n•",
   "clean_statement": "1. (S. Berhanu) Let M 2n−1 ⊆ Cn be a real hypersurface satisfying the maximum principle for CR functions in the strong sense (continuous CR functions attaining a (weak) local maximum must be constant). Is it necessary that any local continuous CR function is open?\n\n•",
   "public_statement": "1. (S. Berhanu) Let M 2n−1 ⊆ Cn be a real hypersurface satisfying the maximum principle for CR functions in the strong sense (continuous CR functions attaining a (weak) local maximum must be constant). Is it necessary that any local continuous CR function is open?\n\n•",
   "evidence": "The source is the 2010 AIM workshop list *Emerging Applications of Complexity for CR Mappings*. Its preamble says that manifolds are smooth unless otherwise stated. In the section “CR manifolds and mappings,” the PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0031": {
   "statement_status": "exact",
   "original_statement": "2. (D. Zaitsev) Let n ≥ 2 and M ⊆ Cn be a compact CR manifold which is maximally complex (so the CR codimension is 1). Suppose in the class of maximally complex CR manifolds, M is homotopic to a sphere in the complex m-plane, where m − 1 is the CR dimension. Must M be the boundary of a smooth analytic submanifold?",
   "clean_statement": "2. (D. Zaitsev) Let n ≥ 2 and M ⊆ Cn be a compact CR manifold which is maximally complex (so the CR codimension is 1). Suppose in the class of maximally complex CR manifolds, M is homotopic to a sphere in the complex m-plane, where m − 1 is the CR dimension. Must M be the boundary of a smooth analytic submanifold?",
   "public_statement": "2. (D. Zaitsev) Let n ≥ 2 and M ⊆ Cn be a compact CR manifold which is maximally complex (so the CR codimension is 1). Suppose in the class of maximally complex CR manifolds, M is homotopic to a sphere in the complex m-plane, where m − 1 is the CR dimension. Must M be the boundary of a smooth analytic submanifold?",
   "evidence": "The AIM PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0032": {
   "statement_status": "exact",
   "original_statement": "3. (S. Ji) Let M ⊆ Cn be a compact CR manifold defined by polynomials, with n ≥ 2 and CR codimension greater or equal to 1. The Harvey-Lawson variety may have isolated singularities. What is the relationship between the degree of the defining polynomials and the number of singularities?",
   "clean_statement": "3. (S. Ji) Let M ⊆ Cn be a compact CR manifold defined by polynomials, with n ≥ 2 and CR codimension greater or equal to 1. The Harvey-Lawson variety may have isolated singularities. What is the relationship between the degree of the defining polynomials and the number of singularities?",
   "public_statement": "3. (S. Ji) Let M ⊆ Cn be a compact CR manifold defined by polynomials, with n ≥ 2 and CR codimension greater or equal to 1. The Harvey-Lawson variety may have isolated singularities. What is the relationship between the degree of the defining polynomials and the number of singularities?",
   "evidence": "This agrees with the repository record, apart from the repository's plain-text loss of superscripting in \\(\\mathbb C^n\\). No substantive OCR error was found.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0033": {
   "statement_status": "exact",
   "original_statement": "4. (D. Zaitsev) Let M be a locally homogeneous CR manifold. That is, suppose for any two points p, q ∈ M, there exists a CR diffeomorphism h: ( U, p ) → (V, q ) for some neighborhoods U of p and V of q. Does it follow that the Lie algebra of infinitesimal automorphisms spans TpM for one (and hence all) p ∈ M.\n\n•",
   "clean_statement": "4. (D. Zaitsev) Let M be a locally homogeneous CR manifold. That is, suppose for any two points p, q ∈ M, there exists a CR diffeomorphism h: ( U, p ) → (V, q ) for some neighborhoods U of p and V of q. Does it follow that the Lie algebra of infinitesimal automorphisms spans TpM for one (and hence all) p ∈ M.\n\n•",
   "public_statement": "4. (D. Zaitsev) Let M be a locally homogeneous CR manifold. That is, suppose for any two points p, q ∈ M, there exists a CR diffeomorphism h: ( U, p ) → (V, q ) for some neighborhoods U of p and V of q. Does it follow that the Lie algebra of infinitesimal automorphisms spans TpM for one (and hence all) p ∈ M.\n\n•",
   "evidence": "The source is the AIM workshop list *Emerging applications of complexity for CR mappings*. Its preamble says that manifolds are smooth unless otherwise stated. On page 2 (PDF page index 1), Problem 4 reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0034": {
   "statement_status": "exact",
   "original_statement": "5. (D. Zaitsev) Given M a CR manifold and p ∈ M, let O(p) be the set of points q ∈ M\n\nfor which there exists a CR automorphism h: ( U, p ) → (V, q ) for some neighborhoods U\n\nof p and V of q. Is O(p) always a manifold? In that case, is the span of the Lie algebra of infinitesimal automorphisms of M equal to TqO(p) if q ∈ O(p)? \n\n•",
   "clean_statement": "5. (D. Zaitsev) Given M a CR manifold and p ∈ M, let O(p) be the set of points q ∈ M\n\nfor which there exists a CR automorphism h: ( U, p ) → (V, q ) for some neighborhoods U\n\nof p and V of q. Is O(p) always a manifold? In that case, is the span of the Lie algebra of infinitesimal automorphisms of M equal to TqO(p) if q ∈ O(p)?\n\n•",
   "public_statement": "5. (D. Zaitsev) Given M a CR manifold and p ∈ M, let O(p) be the set of points q ∈ M\n\nfor which there exists a CR automorphism h: ( U, p ) → (V, q ) for some neighborhoods U\n\nof p and V of q. Is O(p) always a manifold? In that case, is the span of the Lie algebra of infinitesimal automorphisms of M equal to TqO(p) if q ∈ O(p)?\n\n•",
   "evidence": "The original AIM workshop PDF was inspected directly. It states globally that manifolds are smooth unless otherwise specified. Problem 5 in “CR manifolds and mappings” reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0035": {
   "statement_status": "exact",
   "original_statement": "6. (D. Zaitsev) Let ( M, p ), (M ′, p ′) be germs of CR submanifolds in Cn which are formally equivalent. Are they biholomorphically equivalent? \n\n•",
   "clean_statement": "6. (D. Zaitsev) Let ( M, p ), (M ′, p ′) be germs of CR submanifolds in Cn which are formally equivalent. Are they biholomorphically equivalent?\n\n•",
   "public_statement": "6. (D. Zaitsev) Let ( M, p ), (M ′, p ′) be germs of CR submanifolds in Cn which are formally equivalent. Are they biholomorphically equivalent?\n\n•",
   "evidence": "The AIM PDF says in its preamble that manifolds are smooth unless otherwise stated. On printed page 3, under “CR manifolds and mappings,” the source reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0036": {
   "statement_status": "exact",
   "original_statement": "7. (D. Zaitsev) Let M be a real submanifold of Cn which is generically minimal and holo-morphically nondegenerate. What can you say about Aut( M, p ) at nonminimal points? \n\nCR embeddings",
   "clean_statement": "7. (D. Zaitsev) Let M be a real submanifold of Cn which is generically minimal and holo-morphically nondegenerate. What can you say about Aut( M, p ) at nonminimal points?\n\nCR embeddings",
   "public_statement": "7. (D. Zaitsev) Let M be a real submanifold of Cn which is generically minimal and holo-morphically nondegenerate. What can you say about Aut( M, p ) at nonminimal points?\n\nCR embeddings",
   "evidence": "The assigned AIM record is Problem 7 of the workshop list *Emerging applications of complexity for CR mappings*. The PDF was inspected directly. Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0037": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "1. (M. Kolar) Assume M ⊆ C2 is a real analytic hypersurface that admits a nonlinearizable automorphism H. Does there exist a hyperquadric Q ⊆ Cn such that M ↪ → Q nontrivially and H is induced by an automorphism of Q? Can a bound on n be found?",
   "clean_statement": null,
   "public_statement": "1. (M. Kolar) Assume M ⊆ C2 is a real analytic hypersurface that admits a nonlinearizable automorphism H. Does there exist a hyperquadric Q ⊆ Cn such that M ↪ → Q nontrivially and H is induced by an automorphism of Q? Can a bound on n be found?",
   "evidence": "The record comes from the AIM workshop list *Emerging applications of complexity for CR mappings*, in the section “CR embeddings.” Inspection of the original PDF recovers the statement as",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0038": {
   "statement_status": "exact",
   "original_statement": "2. (D. Zaitsev) Assume M ⊆ C2 admits a nontrivial map tangent to the identity of order 2. Does M have an embedding into a hyperquadric Q as in the preceding problem?",
   "clean_statement": "2. (D. Zaitsev) Assume M ⊆ C2 admits a nontrivial map tangent to the identity of order 2. Does M have an embedding into a hyperquadric Q as in the preceding problem?",
   "public_statement": "2. (D. Zaitsev) Assume M ⊆ C2 admits a nontrivial map tangent to the identity of order 2. Does M have an embedding into a hyperquadric Q as in the preceding problem?",
   "evidence": "The AIM PDF has a section headed **CR embeddings**. Its two consecutive questions are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0039": {
   "statement_status": "corrected_verified",
   "original_statement": "3. (S. Dragomir) Which Pontrjagin forms of the Fefferman metric of a strictly pseudocon-vex (abstract) hypersurface M are obstructions to (global) embeddability of M?\n\nApproximation",
   "clean_statement": "**Problem (S. Dragomir).** Which Pontryagin forms of the Fefferman metric of a strictly pseudoconvex abstract CR hypersurface \\(M\\) are obstructions to global embeddability of \\(M\\)?",
   "public_statement": "**Problem (S. Dragomir).** Which Pontryagin forms of the Fefferman metric of a strictly pseudoconvex abstract CR hypersurface \\(M\\) are obstructions to global embeddability of \\(M\\)?",
   "evidence": "Inspection of the AIM workshop PDF shows that “pseudocon-vex” is only a line-break hyphenation and that **Approximation** is the heading of the next section, not part of Problem 3. The recovered statement is therefore:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0040": {
   "statement_status": "exact",
   "original_statement": "1. (J.E. Fornaess) Let ω be a holomorphic 1-form on a polydisc in C3 with ω ∧dω = 0. Can \n\nω be approximated on compact subsets by polynomial 1-forms h satisfying h ∧ dh = 0? \n\n•",
   "clean_statement": "1. (J.E. Fornaess) Let ω be a holomorphic 1-form on a polydisc in C3 with ω ∧dω = 0. Can\n\nω be approximated on compact subsets by polynomial 1-forms h satisfying h ∧ dh = 0?\n\n•",
   "public_statement": "1. (J.E. Fornaess) Let ω be a holomorphic 1-form on a polydisc in C3 with ω ∧dω = 0. Can\n\nω be approximated on compact subsets by polynomial 1-forms h satisfying h ∧ dh = 0?\n\n•",
   "evidence": "The record is from the “Approximation” section of the AIM problem list for the 2010 workshop *Emerging Applications of Complexity for CR Mappings*. Inspection of page 3 of the original PDF gives:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0041": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "1. Sums of squares of polynomials. (Proposed by John D'Angelo) Let Ω ⊂ Cn be a strongly pseudoconvex domain with compact algebraic boundary. Let \n\nR(z, z ) be a real polynomial which is positive on the boundary of Ω. Does there exist a positive integer k and polynomials p1(z), p 2(z),..., p k(z) such that \n\nR(z, z ) = \n\n> k\n\n∑\n\n> j=1\n\n|pj (z)|2\n\non Ω? The answer is known to be yes if Ω is the unit ball in Cn. (See [CD].)",
   "clean_statement": null,
   "public_statement": "1. Sums of squares of polynomials. (Proposed by John D'Angelo) Let Ω ⊂ Cn be a strongly pseudoconvex domain with compact algebraic boundary. Let\n\nR(z, z ) be a real polynomial which is positive on the boundary of Ω. Does there exist a positive integer k and polynomials p1(z), p 2(z),..., p k(z) such that\n\nR(z, z ) =\n\n> k\n\n∑\n\n> j=1\n\n|pj (z)|2\n\non Ω? The answer is known to be yes if Ω is the unit ball in Cn. (See [CD].)",
   "evidence": "The exact extracted record is preserved in input.json. Inspection of page 1 and the bibliography of the original AIM PDF recovers the mathematical typography as follows:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0042": {
   "statement_status": "exact",
   "original_statement": "2. The radical of the set of squared norms of polynomial mappings.(Proposed by John D'Angelo and Dror Varolin) Let P denote the set of Hermitian symmetric polynomials R(z, w ), and let S ⊂ P denote the set of squared norms of holomorphic polynomial mappings. For R ∈ P, give a necessary and sufficient condition for the existence of an integer N such that RN ∈ S.In other words, when do there exist an N and holomorphic polynomials pj such that: \n\nR(z, z )N =\n\n> k\n\n∑\n\n> j=1\n\n|pj (z)|2.\n\nIt is known that, if RN ∈ S, then R satisfies the inequality \n\n|R(z, w )|2 ≤ R(z, z )R(w, w ).\n\nThis inequality can hold without R being a squared norm. The following is a necessary and sufficient condition for R ∈ S: For all choices of points {zj } ∈ Cn, and for all k, the k by k matrix with ( i, j ) entry equal to \n\nR(zi, z j ) is nonnegative definite. See [CD] and [DV] for related additional information. Another necessary and sufficient condition in the case N = 1: write [ A] = (A1, A 2,..., A n) where the Ai are r × r commuting matrices. Then R can be written as a sum of square norms of polynomials if and only if R([ A], [A∗]) ≥ 0holds for all possible choices of such commuting A′ \n\n> j\n\ns. (Convention: write adjoints first in the calculation of R([ A], [A∗]).) 1OPEN PROBLEMS 2",
   "clean_statement": "2. The radical of the set of squared norms of polynomial mappings.(Proposed by John D'Angelo and Dror Varolin) Let P denote the set of Hermitian symmetric polynomials R(z, w ), and let S ⊂ P denote the set of squared norms of holomorphic polynomial mappings. For R ∈ P, give a necessary and sufficient condition for the existence of an integer N such that RN ∈ S.In other words, when do there exist an N and holomorphic polynomials pj such that:\n\nR(z, z )N =\n\n> k\n\n∑\n\n> j=1\n\n|pj (z)|2.\n\nIt is known that, if RN ∈ S, then R satisfies the inequality\n\n|R(z, w )|2 ≤ R(z, z )R(w, w ).\n\nThis inequality can hold without R being a squared norm. The following is a necessary and sufficient condition for R ∈ S: For all choices of points {zj } ∈ Cn, and for all k, the k by k matrix with ( i, j ) entry equal to\n\nR(zi, z j ) is nonnegative definite. See [CD] and [DV] for related additional information. Another necessary and sufficient condition in the case N = 1: write [ A] = (A1, A 2,..., A n) where the Ai are r × r commuting matrices. Then R can be written as a sum of square norms of polynomials if and only if R([ A], [A∗]) ≥ 0holds for all possible choices of such commuting A′\n\n> j\n\ns. (Convention: write adjoints first in the calculation of R([ A], [A∗]).) 1OPEN PROBLEMS 2",
   "public_statement": "2. The radical of the set of squared norms of polynomial mappings.(Proposed by John D'Angelo and Dror Varolin) Let P denote the set of Hermitian symmetric polynomials R(z, w ), and let S ⊂ P denote the set of squared norms of holomorphic polynomial mappings. For R ∈ P, give a necessary and sufficient condition for the existence of an integer N such that RN ∈ S.In other words, when do there exist an N and holomorphic polynomials pj such that:\n\nR(z, z )N =\n\n> k\n\n∑\n\n> j=1\n\n|pj (z)|2.\n\nIt is known that, if RN ∈ S, then R satisfies the inequality\n\n|R(z, w )|2 ≤ R(z, z )R(w, w ).\n\nThis inequality can hold without R being a squared norm. The following is a necessary and sufficient condition for R ∈ S: For all choices of points {zj } ∈ Cn, and for all k, the k by k matrix with ( i, j ) entry equal to\n\nR(zi, z j ) is nonnegative definite. See [CD] and [DV] for related additional information. Another necessary and sufficient condition in the case N = 1: write [ A] = (A1, A 2,..., A n) where the Ai are r × r commuting matrices. Then R can be written as a sum of square norms of polynomials if and only if R([ A], [A∗]) ≥ 0holds for all possible choices of such commuting A′\n\n> j\n\ns. (Convention: write adjoints first in the calculation of R([ A], [A∗]).) 1OPEN PROBLEMS 2",
   "evidence": "The exact canonical input is preserved in `input.json`. It has lost conjugation bars, superscripts, and some line layout during PDF extraction. Inspection of page 1 of the original AIM PDF and its references on pages 5--6 gives the following reconstruction.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0043": {
   "statement_status": "exact",
   "original_statement": "3. Finite mappings. (Proposed by Linda Rothschild) Suppose f: ( Cn, 0) →\n\n(Cn, 0) is a finite mapping, that V is a complex variety. Suppose that f −1(V ) is a smooth manifold. Does this imply that V is a manifold? The following counterexample in characteristic 2 shows that an \"algebra-only\" argument will not do. Let k = 3, f (x) = ( z21, z 22, z 3 + z1z2) and V = {(w1, w 2, w 3) ∈\n\nC3: w23 = w1w2}. Then f −1(V ) = {(z1, z 2, z 3) ∈ C3: z3 = 0 } ∪ { (z1, z 2, z 3) ∈ C3:\n\nz3 + 2 z1z2 = 0 } which is the complex hyperplane where z3 = 0 in the event that 2 = 0. If the dimension of V is 1 then the answer is yes. Some motivation for this: Given a (germ of a) real analytic submanifold M ⊂ Cn\n\nnear 0, and f as above. A natural question to ask is: when is f (M ) a manifold? See [ER].",
   "clean_statement": "3. Finite mappings. (Proposed by Linda Rothschild) Suppose f: ( Cn, 0) →\n\n(Cn, 0) is a finite mapping, that V is a complex variety. Suppose that f −1(V ) is a smooth manifold. Does this imply that V is a manifold? The following counterexample in characteristic 2 shows that an \"algebra-only\" argument will not do. Let k = 3, f (x) = ( z21, z 22, z 3 + z1z2) and V = {(w1, w 2, w 3) ∈\n\nC3: w23 = w1w2}. Then f −1(V ) = {(z1, z 2, z 3) ∈ C3: z3 = 0 } ∪ { (z1, z 2, z 3) ∈ C3:\n\nz3 + 2 z1z2 = 0 } which is the complex hyperplane where z3 = 0 in the event that 2 = 0. If the dimension of V is 1 then the answer is yes. Some motivation for this: Given a (germ of a) real analytic submanifold M ⊂ Cn\n\nnear 0, and f as above. A natural question to ask is: when is f (M ) a manifold? See [ER].",
   "public_statement": "3. Finite mappings. (Proposed by Linda Rothschild) Suppose f: ( Cn, 0) →\n\n(Cn, 0) is a finite mapping, that V is a complex variety. Suppose that f −1(V ) is a smooth manifold. Does this imply that V is a manifold? The following counterexample in characteristic 2 shows that an \"algebra-only\" argument will not do. Let k = 3, f (x) = ( z21, z 22, z 3 + z1z2) and V = {(w1, w 2, w 3) ∈\n\nC3: w23 = w1w2}. Then f −1(V ) = {(z1, z 2, z 3) ∈ C3: z3 = 0 } ∪ { (z1, z 2, z 3) ∈ C3:\n\nz3 + 2 z1z2 = 0 } which is the complex hyperplane where z3 = 0 in the event that 2 = 0. If the dimension of V is 1 then the answer is yes. Some motivation for this: Given a (germ of a) real analytic submanifold M ⊂ Cn\n\nnear 0, and f as above. A natural question to ask is: when is f (M ) a manifold? See [ER].",
   "evidence": "The AIM PDF asks the following question, proposed by Linda Rothschild.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0044": {
   "statement_status": "exact",
   "original_statement": "4. Mappings between Riemann surfaces. (Proposed by Dror Varolin) Let \n\nX, Y be compact Riemann surfaces and {aj } ∈ X points considered to have finite multiplicity mj > 0. Does there exist f: X → Y so that near aj, f (z) = zmj +1,and f has no other critical points? Next, assume φ: X → R is C∞, ∆ φ ≥ 0, ∆ φ > 0 away from the aj and ∆φ\n\n|z − aj |2mj\n\nnear the aj. Then does there exist such an f?",
   "clean_statement": "4. Mappings between Riemann surfaces. (Proposed by Dror Varolin) Let\n\nX, Y be compact Riemann surfaces and {aj } ∈ X points considered to have finite multiplicity mj > 0. Does there exist f: X → Y so that near aj, f (z) = zmj +1,and f has no other critical points? Next, assume φ: X → R is C∞, ∆ φ ≥ 0, ∆ φ > 0 away from the aj and ∆φ\n\n|z − aj |2mj\n\nnear the aj. Then does there exist such an f?",
   "public_statement": "4. Mappings between Riemann surfaces. (Proposed by Dror Varolin) Let\n\nX, Y be compact Riemann surfaces and {aj } ∈ X points considered to have finite multiplicity mj > 0. Does there exist f: X → Y so that near aj, f (z) = zmj +1,and f has no other critical points? Next, assume φ: X → R is C∞, ∆ φ ≥ 0, ∆ φ > 0 away from the aj and ∆φ\n\n|z − aj |2mj\n\nnear the aj. Then does there exist such an f?",
   "evidence": "The source is Problem 4, proposed by Dror Varolin, in the AIM workshop list *Complexity of mappings in CR geometry*. The JSON extraction corrupts superscripts and loses a comparison symbol. The first part can nevertheless be recovered unambiguously as follows.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0045": {
   "statement_status": "exact",
   "original_statement": "5. Holomorphic continuation of CR maps. (Proposed by Francine Meylan) Let ( M, p ) ⊂ Cn be a generic real analytic submanifold, and suppose f: M → ∂B N,\n\nn ≤ N, is a CR and continuous mapping that extends as a meromorphic function in a neighborhood of p. Then when does f extend as a holomorphic function near \n\np? In particular, does f extend if M is minimal? Comment. When M is of codimension 1, this is always true. (See [Ch].)",
   "clean_statement": "5. Holomorphic continuation of CR maps. (Proposed by Francine Meylan) Let ( M, p ) ⊂ Cn be a generic real analytic submanifold, and suppose f: M → ∂B N,\n\nn ≤ N, is a CR and continuous mapping that extends as a meromorphic function in a neighborhood of p. Then when does f extend as a holomorphic function near\n\np? In particular, does f extend if M is minimal? Comment. When M is of codimension 1, this is always true. (See [Ch].)",
   "public_statement": "5. Holomorphic continuation of CR maps. (Proposed by Francine Meylan) Let ( M, p ) ⊂ Cn be a generic real analytic submanifold, and suppose f: M → ∂B N,\n\nn ≤ N, is a CR and continuous mapping that extends as a meromorphic function in a neighborhood of p. Then when does f extend as a holomorphic function near\n\np? In particular, does f extend if M is minimal? Comment. When M is of codimension 1, this is always true. (See [Ch].)",
   "evidence": "The source is Problem 5 in the AIM workshop list *Complexity of mappings in CR geometry* (proposed by Francine Meylan). With the typography restored, the statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0046": {
   "statement_status": "exact",
   "original_statement": "6. Determination by finite order jets. (Proposed by Bernhard Lamel) Let \n\nM be a generic real analytic manifold, holomorphically nondegenerate, of finite AIM WORKSHOP ON COMPLEXITY OF MAPPINGS IN CR GEOMETRY 3\n\ntype and connected. Then for all p ∈ M there exists k(p) such that \n\nAut M (p) → Gk(Cn)\n\nf 7 → jkp f\n\nwhere k(p) is the least k such that the above mapping is injective. Does there exist an M in C2 with a sequence {pj } in M such that k(pj ) → ∞ as j → ∞?Comment. Such an M does exist in higher dimensions. Next, if M is only generically of finite type, does there exist an M as above with the pj converging to a point of M?",
   "clean_statement": "6. Determination by finite order jets. (Proposed by Bernhard Lamel) Let\n\nM be a generic real analytic manifold, holomorphically nondegenerate, of finite AIM WORKSHOP ON COMPLEXITY OF MAPPINGS IN CR GEOMETRY 3\n\ntype and connected. Then for all p ∈ M there exists k(p) such that\n\nAut M (p) → Gk(Cn)\n\nf 7 → jkp f\n\nwhere k(p) is the least k such that the above mapping is injective. Does there exist an M in C2 with a sequence {pj } in M such that k(pj ) → ∞ as j → ∞?Comment. Such an M does exist in higher dimensions. Next, if M is only generically of finite type, does there exist an M as above with the pj converging to a point of M?",
   "public_statement": "6. Determination by finite order jets. (Proposed by Bernhard Lamel) Let\n\nM be a generic real analytic manifold, holomorphically nondegenerate, of finite AIM WORKSHOP ON COMPLEXITY OF MAPPINGS IN CR GEOMETRY 3\n\ntype and connected. Then for all p ∈ M there exists k(p) such that\n\nAut M (p) → Gk(Cn)\n\nf 7 → jkp f\n\nwhere k(p) is the least k such that the above mapping is injective. Does there exist an M in C2 with a sequence {pj } in M such that k(pj ) → ∞ as j → ∞?Comment. Such an M does exist in higher dimensions. Next, if M is only generically of finite type, does there exist an M as above with the pj converging to a point of M?",
   "evidence": "The raw JSON record is preserved in input.json. It contains the running header “AIM WORKSHOP ON COMPLEXITY OF MAPPINGS IN CR GEOMETRY 3” inside the phrase “of finite type” and flattens subscripts and superscripts in the jet map. Reading pages 2--3 of the original PDF gives the following reconstruction.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0047": {
   "statement_status": "exact",
   "original_statement": "7. Approximation of formal mappings with convergent mappings. \n\n(Proposed by Nordine Mir) Let 0 ∈ M ⊂ Cn, 0 ∈ M ′ ⊂ CN be real ana-lytic manifolds with real analytic defining functions ρ, ρ ′. Suppose also that both are of (Bloom-Graham-Kohn) finite type, f: ( Cn, 0) → (CN, 0) a formal holo-morphic mapping such that f (M ) ⊂ (M ′) formally. (This means that f (z) = (f1[z], f 2[z],..., f N ([ z])) is an N -tuple of formal power series where for some formal power series a(z, z ), ρ′(f (z), f (z)) = a(z, z )ρ(z, z ) as formal power series.) Then the question is: does there exist an Artin-type approximation theorem, i.e., for each \n\n` > 0 can we find a (convergent) holomorphic mapping f `: ( Cn, 0) → (Cn, 0) such that f (M ) ⊂ M ′ and j`\n\n> 0\n\nf ` = j`\n\n> 0\n\nf?If M ′ is real algebraic, the answer is yes. (See [MMZ].) Suppose that 0 ∈ A0 ⊂ Cn × CN, the graph of f lies in A0 (formally?) and we let c(f ):= dim (A0) − N (the \"complexity\" of f ). Then c(f ) = 0 if and only if f is convergent.",
   "clean_statement": "7. Approximation of formal mappings with convergent mappings.\n\n(Proposed by Nordine Mir) Let 0 ∈ M ⊂ Cn, 0 ∈ M ′ ⊂ CN be real ana-lytic manifolds with real analytic defining functions ρ, ρ ′. Suppose also that both are of (Bloom-Graham-Kohn) finite type, f: ( Cn, 0) → (CN, 0) a formal holo-morphic mapping such that f (M ) ⊂ (M ′) formally. (This means that f (z) = (f1[z], f 2[z],..., f N ([ z])) is an N -tuple of formal power series where for some formal power series a(z, z ), ρ′(f (z), f (z)) = a(z, z )ρ(z, z ) as formal power series.) Then the question is: does there exist an Artin-type approximation theorem, i.e., for each\n\n` > 0 can we find a (convergent) holomorphic mapping f `: ( Cn, 0) → (Cn, 0) such that f (M ) ⊂ M ′ and j`\n\n> 0\n\nf ` = j`\n\n> 0\n\nf?If M ′ is real algebraic, the answer is yes. (See [MMZ].) Suppose that 0 ∈ A0 ⊂ Cn × CN, the graph of f lies in A0 (formally?) and we let c(f ):= dim (A0) − N (the \"complexity\" of f ). Then c(f ) = 0 if and only if f is convergent.",
   "public_statement": "7. Approximation of formal mappings with convergent mappings.\n\n(Proposed by Nordine Mir) Let 0 ∈ M ⊂ Cn, 0 ∈ M ′ ⊂ CN be real ana-lytic manifolds with real analytic defining functions ρ, ρ ′. Suppose also that both are of (Bloom-Graham-Kohn) finite type, f: ( Cn, 0) → (CN, 0) a formal holo-morphic mapping such that f (M ) ⊂ (M ′) formally. (This means that f (z) = (f1[z], f 2[z],..., f N ([ z])) is an N -tuple of formal power series where for some formal power series a(z, z ), ρ′(f (z), f (z)) = a(z, z )ρ(z, z ) as formal power series.) Then the question is: does there exist an Artin-type approximation theorem, i.e., for each\n\n` > 0 can we find a (convergent) holomorphic mapping f `: ( Cn, 0) → (Cn, 0) such that f (M ) ⊂ M ′ and j`\n\n> 0\n\nf ` = j`\n\n> 0\n\nf?If M ′ is real algebraic, the answer is yes. (See [MMZ].) Suppose that 0 ∈ A0 ⊂ Cn × CN, the graph of f lies in A0 (formally?) and we let c(f ):= dim (A0) − N (the \"complexity\" of f ). Then c(f ) = 0 if and only if f is convergent.",
   "evidence": "The source is Problem 7, proposed by Nordine Mir, in the AIM workshop list *Complexity of mappings in CR geometry*. The supplied JSON has lost conjugation bars, superscripts, and jet subscripts. Inspection of the original PDF gives the following intended question.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0048": {
   "statement_status": "exact",
   "original_statement": "8. Rational mappings of balls. (Proposed by Han Peters) Let R: Bn → BN,\n\nn ≥ 3 be a rational proper mapping: R(z) = p(z)/q (z). Let d = d0(R) be the degree of R. \nQuestion 8a: is d ≤ N −1 \n\n> n−1? \nQuestion 8b: Is d ≤ N −1 \n\n> n−1\n\nif we also assume that \n\nn ≥ 2d2 + 2 d? Characterize all R(z) such that this inequality is sharp. For monomials the answer to question 8b is yes. (See [DLP].) For n = 2 it is known for monomial maps that d ≤ 2N − 3 and that this is sharp. (See [DKR].) OPEN PROBLEMS 4\n\nA further question: let f: Bn → BN be proper, and suppose f is rational, polynomial or monomial with d:= d0(f ) > 1. Does there exist a proper g: Bn →\n\nBN which is rational, polynomial or monomial such that d0(g) = d0(f ) − 1?",
   "clean_statement": "8. Rational mappings of balls. (Proposed by Han Peters) Let R: Bn → BN,\n\nn ≥ 3 be a rational proper mapping: R(z) = p(z)/q (z). Let d = d0(R) be the degree of R.\nQuestion 8a: is d ≤ N −1\n\n> n−1?\nQuestion 8b: Is d ≤ N −1\n\n> n−1\n\nif we also assume that\n\nn ≥ 2d2 + 2 d? Characterize all R(z) such that this inequality is sharp. For monomials the answer to question 8b is yes. (See [DLP].) For n = 2 it is known for monomial maps that d ≤ 2N − 3 and that this is sharp. (See [DKR].) OPEN PROBLEMS 4\n\nA further question: let f: Bn → BN be proper, and suppose f is rational, polynomial or monomial with d:= d0(f ) > 1. Does there exist a proper g: Bn →\n\nBN which is rational, polynomial or monomial such that d0(g) = d0(f ) − 1?",
   "public_statement": "8. Rational mappings of balls. (Proposed by Han Peters) Let R: Bn → BN,\n\nn ≥ 3 be a rational proper mapping: R(z) = p(z)/q (z). Let d = d0(R) be the degree of R.\nQuestion 8a: is d ≤ N −1\n\n> n−1?\nQuestion 8b: Is d ≤ N −1\n\n> n−1\n\nif we also assume that\n\nn ≥ 2d2 + 2 d? Characterize all R(z) such that this inequality is sharp. For monomials the answer to question 8b is yes. (See [DLP].) For n = 2 it is known for monomial maps that d ≤ 2N − 3 and that this is sharp. (See [DKR].) OPEN PROBLEMS 4\n\nA further question: let f: Bn → BN be proper, and suppose f is rational, polynomial or monomial with d:= d0(f ) > 1. Does there exist a proper g: Bn →\n\nBN which is rational, polynomial or monomial such that d0(g) = d0(f ) − 1?",
   "evidence": "The raw record is OCR from Problem 8 of the AIM workshop list *Complexity of mappings in CR geometry* (proposed by Han Peters). Inspection of the original PDF repairs the ball superscripts, two displayed fractions, the subscript on the degree, and one running page header. The recovered statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0049": {
   "statement_status": "exact",
   "original_statement": "9. Finite jet determination. (Proposed by Bernhard Lamel) Suppose that \n\nR, ˜R: Bn → BN are proper rational functions. Find the smallest integer r such that if jr \n\n> 0\n\nR = jr \n\n> 0\n\n˜R then R = ˜R. The same question could be asked if R, ˜R are merely proper (and not rational.)",
   "clean_statement": "9. Finite jet determination. (Proposed by Bernhard Lamel) Suppose that\n\nR, ˜R: Bn → BN are proper rational functions. Find the smallest integer r such that if jr\n\n> 0\n\nR = jr\n\n> 0\n\n˜R then R = ˜R. The same question could be asked if R, ˜R are merely proper (and not rational.)",
   "public_statement": "9. Finite jet determination. (Proposed by Bernhard Lamel) Suppose that\n\nR, ˜R: Bn → BN are proper rational functions. Find the smallest integer r such that if jr\n\n> 0\n\nR = jr\n\n> 0\n\n˜R then R = ˜R. The same question could be asked if R, ˜R are merely proper (and not rational.)",
   "evidence": "The canonical JSON has line-break and character-extraction damage. Inspection of the original AIM PDF recovers the statement as follows (with \\(\\mathbb B^n\\) the unit ball in \\(\\mathbb C^n\\)):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0050": {
   "statement_status": "exact",
   "original_statement": "10. Deformation of varieties. (Proposed by Dmitri Zaitsev) Let n ≥ 3 and suppose that V ⊂ Cn is a complex analytic subvariety of codimension 1. Assume that 0 ∈ V and that 0 is an isolated singularity of V. Suppose that ∂V is compact and smooth. Can V be 'deformed' into Bn−1 ⊂ Cn−1? A smooth deformation (Vt, ∂V t) of such varieties with t in an interval I is defined to satisfy: (1) In a neighborhood of every ( p0, t 0) with t0 ∈ I, p0 ∈ Vt0, there exists a smooth complex function F (p, t ), holomorphic in p, such that Vt = {p: F (p, t ) = 0 }.(2) In a neighborhood of every ( p0, t 0) with t0 ∈ I, p0 ∈ ∂V t0, there exists a smooth complex function F (p, t ) as above together with a smooth real function \n\nr(p, t ), such that Vt = {p: F (p, t ) = 0, r (p, t ) < 0} and such that the partial derivative vectors ∂F/∂p and ∂r/∂p are linearly independent.",
   "clean_statement": "10. Deformation of varieties. (Proposed by Dmitri Zaitsev) Let n ≥ 3 and suppose that V ⊂ Cn is a complex analytic subvariety of codimension 1. Assume that 0 ∈ V and that 0 is an isolated singularity of V. Suppose that ∂V is compact and smooth. Can V be 'deformed' into Bn−1 ⊂ Cn−1? A smooth deformation (Vt, ∂V t) of such varieties with t in an interval I is defined to satisfy: (1) In a neighborhood of every ( p0, t 0) with t0 ∈ I, p0 ∈ Vt0, there exists a smooth complex function F (p, t ), holomorphic in p, such that Vt = {p: F (p, t ) = 0 }.(2) In a neighborhood of every ( p0, t 0) with t0 ∈ I, p0 ∈ ∂V t0, there exists a smooth complex function F (p, t ) as above together with a smooth real function\n\nr(p, t ), such that Vt = {p: F (p, t ) = 0, r (p, t ) < 0} and such that the partial derivative vectors ∂F/∂p and ∂r/∂p are linearly independent.",
   "public_statement": "10. Deformation of varieties. (Proposed by Dmitri Zaitsev) Let n ≥ 3 and suppose that V ⊂ Cn is a complex analytic subvariety of codimension 1. Assume that 0 ∈ V and that 0 is an isolated singularity of V. Suppose that ∂V is compact and smooth. Can V be 'deformed' into Bn−1 ⊂ Cn−1? A smooth deformation (Vt, ∂V t) of such varieties with t in an interval I is defined to satisfy: (1) In a neighborhood of every ( p0, t 0) with t0 ∈ I, p0 ∈ Vt0, there exists a smooth complex function F (p, t ), holomorphic in p, such that Vt = {p: F (p, t ) = 0 }.(2) In a neighborhood of every ( p0, t 0) with t0 ∈ I, p0 ∈ ∂V t0, there exists a smooth complex function F (p, t ) as above together with a smooth real function\n\nr(p, t ), such that Vt = {p: F (p, t ) = 0, r (p, t ) < 0} and such that the partial derivative vectors ∂F/∂p and ∂r/∂p are linearly independent.",
   "evidence": "The record is Problem 10, proposed by Dmitri Zaitsev, in the AIM workshop list *Complexity of mappings in CR geometry*. With notation restored but no mathematical hypotheses added, it asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0051": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "11. Solutions to ∂. (Proposed by Mei-Chi Shaw) Let Ω ⊂ CP n be pseudo-convex with C∞ boundary. Let us solve ∂u = f in Ω. If f is a ( p, q ) form such that \n\n∂f = 0, f ∈ C∞(Ω), then does there exist u ∈ W 1(Ω) such that ∂u = f?It is known that there exists such a u in L2(Ω), and also that there exists such a u in W \u000f(Ω) for some \u000f > 0, where \u000f depends on Ω. See [CSW]. The answer is not even known if ∂Ω is real analytic.",
   "clean_statement": null,
   "public_statement": "11. Solutions to ∂. (Proposed by Mei-Chi Shaw) Let Ω ⊂ CP n be pseudo-convex with C∞ boundary. Let us solve ∂u = f in Ω. If f is a ( p, q ) form such that\n\n∂f = 0, f ∈ C∞(Ω), then does there exist u ∈ W 1(Ω) such that ∂u = f?It is known that there exists such a u in L2(Ω), and also that there exists such a u in W [U+000F](Ω) for some [U+000F] > 0, where [U+000F] depends on Ω. See [CSW]. The answer is not even known if ∂Ω is real analytic.",
   "evidence": "The canonical JSON record is preserved in `input.json`. It contains OCR damage: the bar over the Cauchy--Riemann operator is lost, superscripts and closure bars are flattened, and the Greek letter epsilon appears as the control character U+000F.",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0052": {
   "statement_status": "exact",
   "original_statement": "12. Transversality. (Proposed by Peter Ebenfelt) Suppose that 0 ∈ M ⊂\n\nCn+1, 0 ∈ M ′ ⊂ CN +1, where M, M ′ are hypersurfaces ( C∞ or real analytic), and n ≤ N. Let H: ( Cn+1, 0) → (CN +1, 0) be holomorphic, and H(M ) ⊂ M ′.Suppose that H does not map a full neighborhood of 0 in Cn+1 into M ′. Then give AIM WORKSHOP ON COMPLEXITY OF MAPPINGS IN CR GEOMETRY 5\n\nconditions on M, M ′ such that H is transversal at 0 to M ′ for all such H.The statement ' H is transversal to M ′ at p' means that \n\ndpH[TpCn+1 ] + TH(p)M ′ = TH(p)CN +1.\n\nIn the case that N = n and M or M ′ is of finite type at 0, then transversality holds under a finite map H.If M and M ′ are strictly pseudoconvex, H is transversal. Example. Consider the mapping H: C2 → C3 given by \n\nH(z, w ) = ( z + z2 + i\n\n2 w, z − z2 − i\n\n2 w, −2zw ).\n\nLet M = {(z, w ) ∈ C2: Im( w) = |z|2} and \n\nM ′ = {(z1, z 2, w ) ∈ C3: Im( w) = −| z1|2 + |z2|2}.\n\nThe H is transversal on M \\ { (z, w ): Re( z) = 0 }.",
   "clean_statement": "12. Transversality. (Proposed by Peter Ebenfelt) Suppose that 0 ∈ M ⊂\n\nCn+1, 0 ∈ M ′ ⊂ CN +1, where M, M ′ are hypersurfaces ( C∞ or real analytic), and n ≤ N. Let H: ( Cn+1, 0) → (CN +1, 0) be holomorphic, and H(M ) ⊂ M ′.Suppose that H does not map a full neighborhood of 0 in Cn+1 into M ′. Then give AIM WORKSHOP ON COMPLEXITY OF MAPPINGS IN CR GEOMETRY 5\n\nconditions on M, M ′ such that H is transversal at 0 to M ′ for all such H.The statement ' H is transversal to M ′ at p' means that\n\ndpH[TpCn+1 ] + TH(p)M ′ = TH(p)CN +1.\n\nIn the case that N = n and M or M ′ is of finite type at 0, then transversality holds under a finite map H.If M and M ′ are strictly pseudoconvex, H is transversal. Example. Consider the mapping H: C2 → C3 given by\n\nH(z, w ) = ( z + z2 + i\n\n2 w, z − z2 − i\n\n2 w, −2zw ).\n\nLet M = {(z, w ) ∈ C2: Im( w) = |z|2} and\n\nM ′ = {(z1, z 2, w ) ∈ C3: Im( w) = −| z1|2 + |z2|2}.\n\nThe H is transversal on M \\ { (z, w ): Re( z) = 0 }.",
   "public_statement": "12. Transversality. (Proposed by Peter Ebenfelt) Suppose that 0 ∈ M ⊂\n\nCn+1, 0 ∈ M ′ ⊂ CN +1, where M, M ′ are hypersurfaces ( C∞ or real analytic), and n ≤ N. Let H: ( Cn+1, 0) → (CN +1, 0) be holomorphic, and H(M ) ⊂ M ′.Suppose that H does not map a full neighborhood of 0 in Cn+1 into M ′. Then give AIM WORKSHOP ON COMPLEXITY OF MAPPINGS IN CR GEOMETRY 5\n\nconditions on M, M ′ such that H is transversal at 0 to M ′ for all such H.The statement ' H is transversal to M ′ at p' means that\n\ndpH[TpCn+1 ] + TH(p)M ′ = TH(p)CN +1.\n\nIn the case that N = n and M or M ′ is of finite type at 0, then transversality holds under a finite map H.If M and M ′ are strictly pseudoconvex, H is transversal. Example. Consider the mapping H: C2 → C3 given by\n\nH(z, w ) = ( z + z2 + i\n\n2 w, z − z2 − i\n\n2 w, −2zw ).\n\nLet M = {(z, w ) ∈ C2: Im( w) = |z|2} and\n\nM ′ = {(z1, z 2, w ) ∈ C3: Im( w) = −| z1|2 + |z2|2}.\n\nThe H is transversal on M \\ { (z, w ): Re( z) = 0 }.",
   "evidence": "The canonical record contains damaged superscripts, primes, fractions, tangent-space subscripts, and an inserted page header. Inspection of the original AIM PDF, cross-checked against the subsequently published Example 2.4 of Baouendi--Ebenfelt--Rothschild, gives the following statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-SEVERAL_COMPLEX_VARIABLES-0053": {
   "statement_status": "exact",
   "original_statement": "13. Singularities of varieties. (Proposed by Xiaojun Huang) Let M ⊂ CN\n\nbe a compact real analytic spherical CR manifold of hypersurface type and suppose that M is the boundary of an analytic variety V. Then what kind of singularities can V have? It is known that if M is algebraic then V has at most one isolated singular point. See [HJ]. See also a survey paper [H]. Example. Consider the mapping f: B2 → B3 given by f (z1, z 2) = ( z21, √2z1z2, z 22 ). Then the image of f is {(w1, w 2, w 3) ∈ C3: Q(w1, w 2, w 3) ≡ w22 − 2w1w3 = 0 }.Let M = f (∂B 2) and V = f (B2). Then V and M satisfy the conditions above. Furthermore, M has exactly one singularity: ( ∂Q ∂w 1, ∂Q ∂w 2, ∂Q ∂w 3 ) = ( −2w3, 2w2, −2w1), and this is zero only at the point (0, 0, 0). This example shows that the result in [HJ] is sharp.",
   "clean_statement": "13. Singularities of varieties. (Proposed by Xiaojun Huang) Let M ⊂ CN\n\nbe a compact real analytic spherical CR manifold of hypersurface type and suppose that M is the boundary of an analytic variety V. Then what kind of singularities can V have? It is known that if M is algebraic then V has at most one isolated singular point. See [HJ]. See also a survey paper [H]. Example. Consider the mapping f: B2 → B3 given by f (z1, z 2) = ( z21, √2z1z2, z 22 ). Then the image of f is {(w1, w 2, w 3) ∈ C3: Q(w1, w 2, w 3) ≡ w22 − 2w1w3 = 0 }.Let M = f (∂B 2) and V = f (B2). Then V and M satisfy the conditions above. Furthermore, M has exactly one singularity: ( ∂Q ∂w 1, ∂Q ∂w 2, ∂Q ∂w 3 ) = ( −2w3, 2w2, −2w1), and this is zero only at the point (0, 0, 0). This example shows that the result in [HJ] is sharp.",
   "public_statement": "13. Singularities of varieties. (Proposed by Xiaojun Huang) Let M ⊂ CN\n\nbe a compact real analytic spherical CR manifold of hypersurface type and suppose that M is the boundary of an analytic variety V. Then what kind of singularities can V have? It is known that if M is algebraic then V has at most one isolated singular point. See [HJ]. See also a survey paper [H]. Example. Consider the mapping f: B2 → B3 given by f (z1, z 2) = ( z21, √2z1z2, z 22 ). Then the image of f is {(w1, w 2, w 3) ∈ C3: Q(w1, w 2, w 3) ≡ w22 − 2w1w3 = 0 }.Let M = f (∂B 2) and V = f (B2). Then V and M satisfy the conditions above. Furthermore, M has exactly one singularity: ( ∂Q ∂w 1, ∂Q ∂w 2, ∂Q ∂w 3 ) = ( −2w3, 2w2, −2w1), and this is zero only at the point (0, 0, 0). This example shows that the result in [HJ] is sharp.",
   "evidence": "Problem 13 in the AIM workshop list *Complexity of mappings in CR geometry*, proposed by Xiaojun Huang, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-several-complex-variables-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0001": {
   "statement_status": "exact",
   "original_statement": "(1) Find a crystal structure on ribbon tableaux compatible with the spin statistic. This has been solved for type $A$ domino tableaux.",
   "clean_statement": "(1) Find a crystal structure on ribbon tableaux compatible with the spin statistic. This has been solved for type $A$ domino tableaux.",
   "public_statement": "(1) Find a crystal structure on ribbon tableaux compatible with the spin statistic. This has been solved for type $A$ domino tableaux.",
   "evidence": "The canonical AIM record reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0002": {
   "statement_status": "exact",
   "original_statement": "(2) Prove that the four definitions of $k$-Schur functions (see Morse's lecture notes) are indeed equivalent.",
   "clean_statement": "(2) Prove that the four definitions of $k$-Schur functions (see Morse's lecture notes) are indeed equivalent.",
   "public_statement": "(2) Prove that the four definitions of $k$-Schur functions (see Morse's lecture notes) are indeed equivalent.",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0003": {
   "statement_status": "exact",
   "original_statement": "(3) Prove that the $k$-atoms (see the first definition of $k$-Schur functions in Morse's lecture notes) form a basis for the linear span of $\\{J_\\mu:\\mu_1\\le k\\}$. Prove also that the expansion coefficients of the $J_\\lambda$'s in terms of the $k$-atoms refine the $q,t$-Kostka numbers.",
   "clean_statement": "(3) Prove that the $k$-atoms (see the first definition of $k$-Schur functions in Morse's lecture notes) form a basis for the linear span of $\\{J_\\mu:\\mu_1\\le k\\}$. Prove also that the expansion coefficients of the $J_\\lambda$'s in terms of the $k$-atoms refine the $q,t$-Kostka numbers.",
   "public_statement": "(3) Prove that the $k$-atoms (see the first definition of $k$-Schur functions in Morse's lecture notes) form a basis for the linear span of $\\{J_\\mu:\\mu_1\\le k\\}$. Prove also that the expansion coefficients of the $J_\\lambda$'s in terms of the $k$-atoms refine the $q,t$-Kostka numbers.",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0004": {
   "statement_status": "exact",
   "original_statement": "(4) Prove that the $k$-$q,t$-Kostka polynomials lie in $\\mathbb{N}[q,t]$.",
   "clean_statement": "(4) Prove that the $k$-$q,t$-Kostka polynomials lie in $\\mathbb{N}[q,t]$.",
   "public_statement": "(4) Prove that the $k$-$q,t$-Kostka polynomials lie in $\\mathbb{N}[q,t]$.",
   "evidence": "The canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0005": {
   "statement_status": "exact",
   "original_statement": "(5) Prove that the dual $k$-Schur functions are the symmetric component of affine Schubert polynomials. Prove also that the $k$-Schur functions are a Schubert basis of the homology of the affine Grassmannian, and that the dual $k$-Schur functions give a Schubert basis of the cohomology of the affine Grassmannian.",
   "clean_statement": "(5) Prove that the dual $k$-Schur functions are the symmetric component of affine Schubert polynomials. Prove also that the $k$-Schur functions are a Schubert basis of the homology of the affine Grassmannian, and that the dual $k$-Schur functions give a Schubert basis of the cohomology of the affine Grassmannian.",
   "public_statement": "(5) Prove that the dual $k$-Schur functions are the symmetric component of affine Schubert polynomials. Prove also that the $k$-Schur functions are a Schubert basis of the homology of the affine Grassmannian, and that the dual $k$-Schur functions give a Schubert basis of the cohomology of the affine Grassmannian.",
   "evidence": "The AIM *Generalized Kostka polynomials* problem list asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0006": {
   "statement_status": "exact",
   "original_statement": "(6) Generalize $k$-Schur functions to root systems other than type $A$.",
   "clean_statement": "(6) Generalize $k$-Schur functions to root systems other than type $A$.",
   "public_statement": "(6) Generalize $k$-Schur functions to root systems other than type $A$.",
   "evidence": "The canonical AIM record is problem 6 from the workshop list *Generalized Kostka polynomials*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0007": {
   "statement_status": "exact",
   "original_statement": "(7) Elucidate the representation-theoretical significance of the $k$-Schur functions. The answer may involve either representation theory of symmetric groups or representation theory of Lie algebras.",
   "clean_statement": "(7) Elucidate the representation-theoretical significance of the $k$-Schur functions. The answer may involve either representation theory of symmetric groups or representation theory of Lie algebras.",
   "public_statement": "(7) Elucidate the representation-theoretical significance of the $k$-Schur functions. The answer may involve either representation theory of symmetric groups or representation theory of Lie algebras.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0008": {
   "statement_status": "exact",
   "original_statement": "(8) Describe the expansions of LLT polynomials in terms of $k$-Schur functions. This problem may serve as a stepping-stone toward the problem of computing Schur expansions of LLT polynomials. For instance, LLT polynomials are $k$-Schur functions in certain special cases.",
   "clean_statement": "(8) Describe the expansions of LLT polynomials in terms of $k$-Schur functions. This problem may serve as a stepping-stone toward the problem of computing Schur expansions of LLT polynomials. For instance, LLT polynomials are $k$-Schur functions in certain special cases.",
   "public_statement": "(8) Describe the expansions of LLT polynomials in terms of $k$-Schur functions. This problem may serve as a stepping-stone toward the problem of computing Schur expansions of LLT polynomials. For instance, LLT polynomials are $k$-Schur functions in certain special cases.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0009": {
   "statement_status": "exact",
   "original_statement": "(9) The Schur basis of $\\Lambda$ is contained in the spanning set consisting of all LLT polynomials. Are there other interesting bases that have some simple relation to LLT polynomials? Lam remarked that all Hall-Littlewood polynomials are LLT polynomials, as are all skew Schur functions. Morse remarked that not every $k$-Schur function is an LLT polynomial, for example $s^{(3)}_{(2,1,1)}$. Modified Macdonald polynomials and, conjecturally, $\\nabla(e_n)$ can be expressed as weighted sums of LLT polynomials indexed by tuples of ribbons and tuples of shifted columns, respectively.",
   "clean_statement": "(9) The Schur basis of $\\Lambda$ is contained in the spanning set consisting of all LLT polynomials. Are there other interesting bases that have some simple relation to LLT polynomials? Lam remarked that all Hall-Littlewood polynomials are LLT polynomials, as are all skew Schur functions. Morse remarked that not every $k$-Schur function is an LLT polynomial, for example $s^{(3)}_{(2,1,1)}$. Modified Macdonald polynomials and, conjecturally, $\\nabla(e_n)$ can be expressed as weighted sums of LLT polynomials indexed by tuples of ribbons and tuples of shifted columns, respectively.",
   "public_statement": "(9) The Schur basis of $\\Lambda$ is contained in the spanning set consisting of all LLT polynomials. Are there other interesting bases that have some simple relation to LLT polynomials? Lam remarked that all Hall-Littlewood polynomials are LLT polynomials, as are all skew Schur functions. Morse remarked that not every $k$-Schur function is an LLT polynomial, for example $s^{(3)}_{(2,1,1)}$. Modified Macdonald polynomials and, conjecturally, $\\nabla(e_n)$ can be expressed as weighted sums of LLT polynomials indexed by tuples of ribbons and tuples of shifted columns, respectively.",
   "evidence": "The repository transcription agrees with the original AIM text; there is no visible OCR error. The wording is exploratory rather than a proposition with a unique yes/no resolution. It also suppresses several conventions that matter:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0010": {
   "statement_status": "exact",
   "original_statement": "(10) What is the relationship between $k$-Schur functions and $k$-level-restricted Schur functions? These may be essentially the same: it was asserted that some of the structure constants for $k$-Schur functions, the $k$-Littlewood-Richardson coefficients, give all of the structure constants for level-restricted Schur functions. A paper by Goodman and Wenzl on Iwahori-Hecke algebras of type $A$ at roots of unity (J. Algebra 215 (1999), 694-734) may be relevant in this context.",
   "clean_statement": "(10) What is the relationship between $k$-Schur functions and $k$-level-restricted Schur functions? These may be essentially the same: it was asserted that some of the structure constants for $k$-Schur functions, the $k$-Littlewood-Richardson coefficients, give all of the structure constants for level-restricted Schur functions. A paper by Goodman and Wenzl on Iwahori-Hecke algebras of type $A$ at roots of unity (J. Algebra 215 (1999), 694-734) may be relevant in this context.",
   "public_statement": "(10) What is the relationship between $k$-Schur functions and $k$-level-restricted Schur functions? These may be essentially the same: it was asserted that some of the structure constants for $k$-Schur functions, the $k$-Littlewood-Richardson coefficients, give all of the structure constants for level-restricted Schur functions. A paper by Goodman and Wenzl on Iwahori-Hecke algebras of type $A$ at roots of unity (J. Algebra 215 (1999), 694-734) may be relevant in this context.",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0011": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "(11) Prove Buch's weakened version of Knutson's false conjecture for the Schubert structure constants that arise in Schubert calculus on flag manifolds. The original conjecture turns out to be false for full flags, but seems to be OK for two-step flags $V^a\\subseteq V^b\\subseteq \\mathbb{C}^n$. A paper by Buch, Kresch, Tamvakis, and Yong (Duke Math. J. 122 (2004), 125-143) reduces $q$-Schubert calculus on Grassmannians to this two-step case.",
   "clean_statement": null,
   "public_statement": "(11) Prove Buch's weakened version of Knutson's false conjecture for the Schubert structure constants that arise in Schubert calculus on flag manifolds. The original conjecture turns out to be false for full flags, but seems to be OK for two-step flags $V^a\\subseteq V^b\\subseteq \\mathbb{C}^n$. A paper by Buch, Kresch, Tamvakis, and Yong (Duke Math. J. 122 (2004), 125-143) reduces $q$-Schubert calculus on Grassmannians to this two-step case.",
   "evidence": "The exact AIM record is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0012": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "(12) Consider the product $B=B^{r_\\ell s_\\ell}\\otimes_k\\cdots\\otimes_k B^{r_1 s_1}$. In this case, where the factors are products of rectangles in type $A$, there are known combinatorial interpretations and fermionic formulas for the fusion coefficients via rigged configurations (Schilling et al.). There exist conjectural formulas in other types (Hatayama et al.). The problem of generalizing the grading of the tensor product to the two-variable $(q,t)$ case is open.",
   "clean_statement": "construct a meaningful two-variable refinement of the one-variable grading on a level-$k$ fusion product of rectangular type-$A$ factors, preferably with a combinatorial or representation-theoretic interpretation.",
   "public_statement": "(12) Consider the product $B=B^{r_\\ell s_\\ell}\\otimes_k\\cdots\\otimes_k B^{r_1 s_1}$. In this case, where the factors are products of rectangles in type $A$, there are known combinatorial interpretations and fermionic formulas for the fusion coefficients via rigged configurations (Schilling et al.). There exist conjectural formulas in other types (Hatayama et al.). The problem of generalizing the grading of the tensor product to the two-variable $(q,t)$ case is open.",
   "evidence": "The canonical record reads:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0013": {
   "statement_status": "exact",
   "original_statement": "(13) Suspicions were voiced at the workshop that the two gradings of products of Schur functions - one from LLT polynomials and one from Morse and Lapointe's $t$-statistic - may be related. Can this relationship be stated precisely? What role does the affine Hecke algebra play? What is the connection between $q$-Littlewood-Richardson coefficients, Macdonald polynomials, and the definition of $k$-Schur functions via generalized Kostka polynomials?",
   "clean_statement": "(13) Suspicions were voiced at the workshop that the two gradings of products of Schur functions - one from LLT polynomials and one from Morse and Lapointe's $t$-statistic - may be related. Can this relationship be stated precisely? What role does the affine Hecke algebra play? What is the connection between $q$-Littlewood-Richardson coefficients, Macdonald polynomials, and the definition of $k$-Schur functions via generalized Kostka polynomials?",
   "public_statement": "(13) Suspicions were voiced at the workshop that the two gradings of products of Schur functions - one from LLT polynomials and one from Morse and Lapointe's $t$-statistic - may be related. Can this relationship be stated precisely? What role does the affine Hecke algebra play? What is the connection between $q$-Littlewood-Richardson coefficients, Macdonald polynomials, and the definition of $k$-Schur functions via generalized Kostka polynomials?",
   "evidence": "The canonical record is number 13 in the AIM workshop list *Generalized Kostka polynomials*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0014": {
   "statement_status": "exact",
   "original_statement": "(14) Are there any nice relationships between $k$-Schur functions and diagonal harmonics modules or Garsia-Haiman modules, whose Frobenius series are given by $\\nabla(e_n)$ and $\\widetilde{H}_\\mu$, respectively?",
   "clean_statement": "(14) Are there any nice relationships between $k$-Schur functions and diagonal harmonics modules or Garsia-Haiman modules, whose Frobenius series are given by $\\nabla(e_n)$ and $\\widetilde{H}_\\mu$, respectively?",
   "public_statement": "(14) Are there any nice relationships between $k$-Schur functions and diagonal harmonics modules or Garsia-Haiman modules, whose Frobenius series are given by $\\nabla(e_n)$ and $\\widetilde{H}_\\mu$, respectively?",
   "evidence": "The canonical record in `aim-special-functions-notes.json`, index 13, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0015": {
   "statement_status": "exact",
   "original_statement": "(15) Define a charge statistic on galleries. Recall that the tableaux form a subset of the galleries under a natural injection, so the new statistic should restrict to the usual charge statistic on tableaux. Galleries, in turn, can be viewed as Littelmann paths, so one could even ask for a charge statistic on the latter objects.",
   "clean_statement": "(15) Define a charge statistic on galleries. Recall that the tableaux form a subset of the galleries under a natural injection, so the new statistic should restrict to the usual charge statistic on tableaux. Galleries, in turn, can be viewed as Littelmann paths, so one could even ask for a charge statistic on the latter objects.",
   "public_statement": "(15) Define a charge statistic on galleries. Recall that the tableaux form a subset of the galleries under a natural injection, so the new statistic should restrict to the usual charge statistic on tableaux. Galleries, in turn, can be viewed as Littelmann paths, so one could even ask for a charge statistic on the latter objects.",
   "evidence": "Comparison with the AIM workshop page, the contemporaneous workshop report, and the neighboring records reveals no OCR error. The report confirms that this was posed after a workshop presentation on galleries and that a participant proposed pursuing the Littelmann-path direction.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0016": {
   "statement_status": "exact",
   "original_statement": "(16) Give a non-miraculous representation-theoretical explanation of charge. There exists a well-defined parametrizing set for a basis of the representation for each weight space and filtration. Yet the canonical Lusztig-Kashiwara bases are not compatible. How can these be related?",
   "clean_statement": "(16) Give a non-miraculous representation-theoretical explanation of charge. There exists a well-defined parametrizing set for a basis of the representation for each weight space and filtration. Yet the canonical Lusztig-Kashiwara bases are not compatible. How can these be related?",
   "public_statement": "(16) Give a non-miraculous representation-theoretical explanation of charge. There exists a well-defined parametrizing set for a basis of the representation for each weight space and filtration. Yet the canonical Lusztig-Kashiwara bases are not compatible. How can these be related?",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-SPECIAL_FUNCTIONS-0017": {
   "statement_status": "exact",
   "original_statement": "(17) Find a good definition of level-restricted $q,t$-Kostka numbers. Does this make sense for root systems other than type $A$? Is there any connection to $k$-$q,t$-Kostka numbers?",
   "clean_statement": "(17) Find a good definition of level-restricted $q,t$-Kostka numbers. Does this make sense for root systems other than type $A$? Is there any connection to $k$-$q,t$-Kostka numbers?",
   "public_statement": "(17) Find a good definition of level-restricted $q,t$-Kostka numbers. Does this make sense for root systems other than type $A$? Is there any connection to $k$-$q,t$-Kostka numbers?",
   "evidence": "The canonical record, from the AIM workshop *Generalized Kostka Polynomials*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-special-functions-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0001": {
   "statement_status": "exact",
   "original_statement": "Taxonomize h-principles for contact submanifolds: Write down a full h-principle for twisted contact submanifolds for some (or any) definition of twisted.",
   "clean_statement": "Taxonomize h-principles for contact submanifolds: Write down a full h-principle for twisted contact submanifolds for some (or any) definition of twisted.",
   "public_statement": "Taxonomize h-principles for contact submanifolds: Write down a full h-principle for twisted contact submanifolds for some (or any) definition of twisted.",
   "evidence": "The canonical AIM record is Problem 1.1 in the section “Contact submanifolds” of the 2024 AIM workshop *Higher-dimensional contact topology*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0002": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Can hypersurfaces be $c^\\infty$-approximated by Weinstein convex hypersurfaces?",
   "clean_statement": null,
   "public_statement": "Can hypersurfaces be $c^\\infty$-approximated by Weinstein convex hypersurfaces?",
   "evidence": "The exact canonical AIM record asks:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0003": {
   "statement_status": "exact",
   "original_statement": "For what $k$ can we $c^k$-approximate the mapping tori of ergodic time-1 flows of contact vector fields by a contactomorphism whose mapping torus is convex?",
   "clean_statement": "For what $k$ can we $c^k$-approximate the mapping tori of ergodic time-1 flows of contact vector fields by a contactomorphism whose mapping torus is convex?",
   "public_statement": "For what $k$ can we $c^k$-approximate the mapping tori of ergodic time-1 flows of contact vector fields by a contactomorphism whose mapping torus is convex?",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0004": {
   "statement_status": "exact",
   "original_statement": "Explicitly describe Weinstein convex perturbations of Liouville convex hypersurfaces. For instance, doubles of Liouville manifolds.",
   "clean_statement": "Explicitly describe Weinstein convex perturbations of Liouville convex hypersurfaces. For instance, doubles of Liouville manifolds.",
   "public_statement": "Explicitly describe Weinstein convex perturbations of Liouville convex hypersurfaces. For instance, doubles of Liouville manifolds.",
   "evidence": "The exact canonical AIM record is Problem 3.1 in the “Liouville vs Weinstein” section of the April 2024 workshop *Higher-dimensional contact topology*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0005": {
   "statement_status": "exact",
   "original_statement": "Are all Liouville manifolds with half-dimensional homotopy type Liouville homotopic to Weinstein manifolds? 1-parametric version?",
   "clean_statement": "Are all Liouville manifolds with half-dimensional homotopy type Liouville homotopic to Weinstein manifolds? 1-parametric version?",
   "public_statement": "Are all Liouville manifolds with half-dimensional homotopy type Liouville homotopic to Weinstein manifolds? 1-parametric version?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0006": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "What about in the special case where the skeleton is stratified by manifolds of dimension at most half?",
   "clean_statement": "Let \\((X^{2n},\\lambda)\\) be a Liouville manifold of the half-dimensional Morse/homotopy type contemplated in problem 3.2. Suppose, for the given Liouville form \\(\\lambda\\), that its skeleton (core) is stratified by smooth manifolds of dimension at most \\(n\\). Is \\(\\lambda\\) Liouville homotopic to a Weinstein structure? Is there a one-parameter version?",
   "public_statement": "What about in the special case where the skeleton is stratified by manifolds of dimension at most half?",
   "evidence": "The most conservative reconstruction is:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-topology-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0007": {
   "statement_status": "exact",
   "original_statement": "Can we characterize convex hypersurfaces with tight neighbourhoods?",
   "clean_statement": "Can we characterize convex hypersurfaces with tight neighbourhoods?",
   "public_statement": "Can we characterize convex hypersurfaces with tight neighbourhoods?",
   "evidence": "The canonical AIM record is number 4.1 in the section “Tightness criteria via convex hypersurface theory” of the workshop list *Higher-dimensional contact topology*. Its exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0008": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Given the existence of one tight structure in this almost contact class, must there be infinitely many?",
   "clean_statement": "therefore the following universal question. Fix a closed cooriented manifold \\(M^{2n+1}\\) and a homotopy class \\(J\\) of almost contact structures. If \\(J\\) contains one tight contact structure, must it contain infinitely many distinct tight contact structures?",
   "public_statement": "Given the existence of one tight structure in this almost contact class, must there be infinitely many?",
   "evidence": "The most conservative reconstruction is therefore the following universal question. Fix a closed cooriented manifold \\(M^{2n+1}\\) and a homotopy class \\(J\\) of almost contact structures. If \\(J\\) contains one tight contact structure, must it contain infinitely many distinct tight contact structures?",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-topology-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0009": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Must every almost contact class admit a tight structure?",
   "clean_statement": null,
   "public_statement": "Must every almost contact class admit a tight structure?",
   "evidence": "The old AIM problem-list page could not be inspected in the available interface. The official 2024 workshop page and report were checked, and the dimensional convention above is therefore an explicit reconstruction, not a silent change to the canonical text. This report treats both readings:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0010": {
   "statement_status": "exact",
   "original_statement": "Do there exist tight but not fillable contact structures whose fillability is not obstructed by SFT?",
   "clean_statement": "Do there exist tight but not fillable contact structures whose fillability is not obstructed by SFT?",
   "public_statement": "Do there exist tight but not fillable contact structures whose fillability is not obstructed by SFT?",
   "evidence": "The canonical AIM record is problem 5.1 in the section *Tightness and symplectic field theory* of the workshop *Higher-dimensional contact topology*. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0011": {
   "statement_status": "exact",
   "original_statement": "Can Donaldson's theorem be proved on a closed symplectic manifolds via CHT techniques?",
   "clean_statement": "Can Donaldson's theorem be proved on a closed symplectic manifolds via CHT techniques?",
   "public_statement": "Can Donaldson's theorem be proved on a closed symplectic manifolds via CHT techniques?",
   "evidence": "The canonical record is preserved verbatim:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0012": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Is it true that given a Liouville domain, there is a codimension two Liouville submanifold whose complement is Weinstein?",
   "clean_statement": null,
   "public_statement": "Is it true that given a Liouville domain, there is a codimension two Liouville submanifold whose complement is Weinstein?",
   "evidence": "The adjacent problem asks whether Donaldson's theorem for closed symplectic manifolds can be proved using convex hypersurface theory. This makes the intended analogy with a Donaldson divisor and its Weinstein complement very likely, but it does not fix the boundary conventions. The linked AIM problem page timed out during this run. The official workshop announcement and summary were inspected, but they do not restate problem 6.2. Thus the following choices are reconstructions, not verified additions to the source.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0013": {
   "statement_status": "exact",
   "original_statement": "Give a bypass description of convex surfaces foliating the region in $(S^5,\\xi_k):=\\text{OBD}(T^*S^2, \\tau^{2k+1})$ between two Darboux balls.",
   "clean_statement": "Give a bypass description of convex surfaces foliating the region in $(S^5,\\xi_k):=\\text{OBD}(T^*S^2, \\tau^{2k+1})$ between two Darboux balls.",
   "public_statement": "Give a bypass description of convex surfaces foliating the region in $(S^5,\\xi_k):=\\text{OBD}(T^*S^2, \\tau^{2k+1})$ between two Darboux balls.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0014": {
   "statement_status": "exact",
   "original_statement": "Find other interesting explicit bypass decompositions of contact manifolds.",
   "clean_statement": "Find other interesting explicit bypass decompositions of contact manifolds.",
   "public_statement": "Find other interesting explicit bypass decompositions of contact manifolds.",
   "evidence": "The record has no remarks or literature field. There is no visible OCR corruption. The legacy `source_url` could not be opened with the available web tooling, so the canonical record was checked against the official workshop report instead. The prompt is deliberately open-ended: “other” refers to the preceding Problem 7.1, which asks for a bypass description of the layers between two Darboux balls in",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0015": {
   "statement_status": "exact",
   "original_statement": "Can we characterize trivial bypasses in terms of the belt sphere of $n$-handle and the attaching sphere of the $n+1$ handle?",
   "clean_statement": "Can we characterize trivial bypasses in terms of the belt sphere of $n$-handle and the attaching sphere of the $n+1$ handle?",
   "public_statement": "Can we characterize trivial bypasses in terms of the belt sphere of $n$-handle and the attaching sphere of the $n+1$ handle?",
   "evidence": "The exact canonical AIM question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0016": {
   "statement_status": "exact",
   "original_statement": "Can triviality be detected via capacities on balls around the intersection point?",
   "clean_statement": "Can triviality be detected via capacities on balls around the intersection point?",
   "public_statement": "Can triviality be detected via capacities on balls around the intersection point?",
   "evidence": "The canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0017": {
   "statement_status": "exact",
   "original_statement": "Understand contact connected sum in high dimensions: can non-standard structures on $S^5$ connect sum to give $\\xi_{std}$?",
   "clean_statement": "Understand contact connected sum in high dimensions: can non-standard structures on $S^5$ connect sum to give $\\xi_{std}$?",
   "public_statement": "Understand contact connected sum in high dimensions: can non-standard structures on $S^5$ connect sum to give $\\xi_{std}$?",
   "evidence": "The canonical record is AIM Problem Lists, workshop *Higher-dimensional contact topology*, section *Explicit contact handlebodies*, Problem 7.5:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0018": {
   "statement_status": "exact",
   "original_statement": "Does connected sum preserve tightness?",
   "clean_statement": "Does connected sum preserve tightness?",
   "public_statement": "Does connected sum preserve tightness?",
   "evidence": "The exact canonical AIM question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0019": {
   "statement_status": "exact",
   "original_statement": "Can non-destabilizable OBDs connect sum to a destabilizable (or otherwise non-minimal) OBD?",
   "clean_statement": "Can non-destabilizable OBDs connect sum to a destabilizable (or otherwise non-minimal) OBD?",
   "public_statement": "Can non-destabilizable OBDs connect sum to a destabilizable (or otherwise non-minimal) OBD?",
   "evidence": "The canonical AIM record is Problem 7.7 from the workshop *Higher-dimensional contact topology*, section “Explicit contact handlebodies”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0020": {
   "statement_status": "exact",
   "original_statement": "Can we define an appropriate notion for overtwistedness for conformally symplectic manifolds? Two potential options: the existence of an embedded model, having a contactization which is overtwisted.",
   "clean_statement": "Can we define an appropriate notion for overtwistedness for conformally symplectic manifolds? Two potential options: the existence of an embedded model, having a contactization which is overtwisted.",
   "public_statement": "Can we define an appropriate notion for overtwistedness for conformally symplectic manifolds? Two potential options: the existence of an embedded model, having a contactization which is overtwisted.",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0021": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Given a Weinstein domain, can we construct a distinct Weinstein filling of its countact boundary and/or a contactomorphism of its boundary inducing the same augmentation.",
   "clean_statement": null,
   "public_statement": "Given a Weinstein domain, can we construct a distinct Weinstein filling of its countact boundary and/or a contactomorphism of its boundary inducing the same augmentation.",
   "evidence": "The exact canonical record is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0022": {
   "statement_status": "exact",
   "original_statement": "Classify symplectic fillings of contact submanifolds of $(S^5,\\xi_{std})$ in its standard $D^6$ filling. E.g. Show that the standard $S^3\\subset S^5$ has a unique filling up to Hamiltonian isotopy.",
   "clean_statement": "Classify symplectic fillings of contact submanifolds of $(S^5,\\xi_{std})$ in its standard $D^6$ filling. E.g. Show that the standard $S^3\\subset S^5$ has a unique filling up to Hamiltonian isotopy.",
   "public_statement": "Classify symplectic fillings of contact submanifolds of $(S^5,\\xi_{std})$ in its standard $D^6$ filling. E.g. Show that the standard $S^3\\subset S^5$ has a unique filling up to Hamiltonian isotopy.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0023": {
   "statement_status": "exact",
   "original_statement": "Can we restrict the topology of symplectic fillings using convex surfaces in their boundaries? (c.f. filling by holomorphic curves)",
   "clean_statement": "Can we restrict the topology of symplectic fillings using convex surfaces in their boundaries? (c.f. filling by holomorphic curves)",
   "public_statement": "Can we restrict the topology of symplectic fillings using convex surfaces in their boundaries? (c.f. filling by holomorphic curves)",
   "evidence": "This is Problem 9.3 in the AIM list *Higher-dimensional contact topology*, section “Symplectic fillings of contact submanifolds.” The original AIM URL is currently unavailable, but the 2024-08-01 Wayback snapshot verifies the wording (apart from trailing whitespace) and attributes the problem to Gironella [AIM24a]. The canonical input file has been preserved verbatim. There is no apparent OCR error. There is, however, genuine scope ambiguity:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0024": {
   "statement_status": "exact",
   "original_statement": "In higher dimensions, do intersections of convex hypersurfaces admit a standard model? Can we describe edge rounding?",
   "clean_statement": "In higher dimensions, do intersections of convex hypersurfaces admit a standard model? Can we describe edge rounding?",
   "public_statement": "In higher dimensions, do intersections of convex hypersurfaces admit a standard model? Can we describe edge rounding?",
   "evidence": "This is Problem 10.1 in the “Miscellaneous” section of the 2024 AIM workshop *Higher-dimensional contact topology*. An archived August 2024 copy of the original AIM page agrees verbatim and attributes the question to Sheel Ganatra; it supplies no further hypotheses or remarks. The current original URL was unavailable during this run.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0025": {
   "statement_status": "exact",
   "original_statement": "Flexibility\n\nDo properly embedded codimension 2 Weinstein domains satisfy an existence h-principle?",
   "clean_statement": "Flexibility\n\nDo properly embedded codimension 2 Weinstein domains satisfy an existence h-principle?",
   "public_statement": "Flexibility\n\nDo properly embedded codimension 2 Weinstein domains satisfy an existence h-principle?",
   "evidence": "The canonical record is problem 10.2 in the “Miscellaneous” section of the AIM list from the workshop *Higher-dimensional contact topology*. Its complete mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0026": {
   "statement_status": "exact",
   "original_statement": "Do Liouville domains of dimension at least six satisfy an existence h-principle?",
   "clean_statement": "Do Liouville domains of dimension at least six satisfy an existence h-principle?",
   "public_statement": "Do Liouville domains of dimension at least six satisfy an existence h-principle?",
   "evidence": "The archived AIM page confirms this wording. There is no apparent OCR corruption. The ambiguity is mathematical rather than textual: the page does not define the formal objects, and “existence h-principle” could mean either absolute existence up to formal homotopy or the stronger relative/parametric statement customary for an open differential relation. These readings have different answers below.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 25,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0027": {
   "statement_status": "exact",
   "original_statement": "What do CHT techniques tell us about the map $$\\text{Cont}(M,\\xi)\\to \\text{Diff}(M)?$$ Is $\\pi_1(\\text{Cont}(S^5, \\xi))\\to \\pi_1(\\text{Diff}(S^5))$ a surjection?",
   "clean_statement": "What do CHT techniques tell us about the map $$\\text{Cont}(M,\\xi)\\to \\text{Diff}(M)?$$ Is $\\pi_1(\\text{Cont}(S^5, \\xi))\\to \\pi_1(\\text{Diff}(S^5))$ a surjection?",
   "public_statement": "What do CHT techniques tell us about the map $$\\text{Cont}(M,\\xi)\\to \\text{Diff}(M)?$$ Is $\\pi_1(\\text{Cont}(S^5, \\xi))\\to \\pi_1(\\text{Diff}(S^5))$ a surjection?",
   "evidence": "The canonical record, AIM workshop *Higher-dimensional contact topology*, Miscellaneous 10.4, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0028": {
   "statement_status": "exact",
   "original_statement": "The Colored Jones Polynomial Volume Conjecture for even root of unity\n\nLet $L \\in S^3$ be a hyperbolic link. For n even, let $J_n(L) \\in \\mathbb{Z}[t^{\\pm}]$ be the n-th normalized Colored Jones Polynomial.\nThen, as $n \\rightarrow \\infty$,\n$$ J_n(L, t = e^{\\frac{2\\pi i}{n}}) \\sim \\text{exp} \\left[\\frac{n}{2\\pi} (\\text{vol}(S^3 - L) + i CS(S^3 - L) )\\right] $$",
   "clean_statement": "The Colored Jones Polynomial Volume Conjecture for even root of unity\n\nLet $L \\in S^3$ be a hyperbolic link. For n even, let $J_n(L) \\in \\mathbb{Z}[t^{\\pm}]$ be the n-th normalized Colored Jones Polynomial.\nThen, as $n \\rightarrow \\infty$,\n$$ J_n(L, t = e^{\\frac{2\\pi i}{n}}) \\sim \\text{exp} \\left[\\frac{n}{2\\pi} (\\text{vol}(S^3 - L) + i CS(S^3 - L) )\\right] $$",
   "public_statement": "The Colored Jones Polynomial Volume Conjecture for even root of unity\n\nLet $L \\in S^3$ be a hyperbolic link. For n even, let $J_n(L) \\in \\mathbb{Z}[t^{\\pm}]$ be the n-th normalized Colored Jones Polynomial.\nThen, as $n \\rightarrow \\infty$,\n$$ J_n(L, t = e^{\\frac{2\\pi i}{n}}) \\sim \\text{exp} \\left[\\frac{n}{2\\pi} (\\text{vol}(S^3 - L) + i CS(S^3 - L) )\\right] $$",
   "evidence": "The canonical AIM record is Conjecture 1.1 in the “Volume Conjecture” section of the 2023 workshop *Quantum invariants and low-dimensional topology*. Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0029": {
   "statement_status": "exact",
   "original_statement": "The Colored Jones Polynomial Volume Conjecture for odd root of unity\n\nLet $L \\in S^3$ be a hyperbolic link. For n odd, let $J_n(L) \\in \\mathbb{Z}[t^{\\pm}]$ be the n-th normalized Colored Jones Polynomial.\nThen, as $n \\rightarrow \\infty$,\n$$ J_n(L, t = e^{\\frac{2\\pi i}{n}}) \\sim \\text{exp} \\left[\\frac{n}{2\\pi} (\\text{vol}(S^3 - L) + i CS(S^3 - L) )\\right] $$",
   "clean_statement": "The Colored Jones Polynomial Volume Conjecture for odd root of unity\n\nLet $L \\in S^3$ be a hyperbolic link. For n odd, let $J_n(L) \\in \\mathbb{Z}[t^{\\pm}]$ be the n-th normalized Colored Jones Polynomial.\nThen, as $n \\rightarrow \\infty$,\n$$ J_n(L, t = e^{\\frac{2\\pi i}{n}}) \\sim \\text{exp} \\left[\\frac{n}{2\\pi} (\\text{vol}(S^3 - L) + i CS(S^3 - L) )\\right] $$",
   "public_statement": "The Colored Jones Polynomial Volume Conjecture for odd root of unity\n\nLet $L \\in S^3$ be a hyperbolic link. For n odd, let $J_n(L) \\in \\mathbb{Z}[t^{\\pm}]$ be the n-th normalized Colored Jones Polynomial.\nThen, as $n \\rightarrow \\infty$,\n$$ J_n(L, t = e^{\\frac{2\\pi i}{n}}) \\sim \\text{exp} \\left[\\frac{n}{2\\pi} (\\text{vol}(S^3 - L) + i CS(S^3 - L) )\\right] $$",
   "evidence": "The archived 1 August 2024 AIM page matches this wording, so the issue below is not an OCR error. The notation is nevertheless underspecified:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0030": {
   "statement_status": "exact",
   "original_statement": "Asymptotics of Reshetikhin-Turaev and Turaev-Viro invariants for hyperbolic 3-manifolds\n\nLet M be a hyperbolic 3-manifold. What are the asymptotics of the Turaev-Viro and Reshetikhin-Turaev invariants for this manifold?",
   "clean_statement": "Asymptotics of Reshetikhin-Turaev and Turaev-Viro invariants for hyperbolic 3-manifolds\n\nLet M be a hyperbolic 3-manifold. What are the asymptotics of the Turaev-Viro and Reshetikhin-Turaev invariants for this manifold?",
   "public_statement": "Asymptotics of Reshetikhin-Turaev and Turaev-Viro invariants for hyperbolic 3-manifolds\n\nLet M be a hyperbolic 3-manifold. What are the asymptotics of the Turaev-Viro and Reshetikhin-Turaev invariants for this manifold?",
   "evidence": "The canonical record is AIM-TOPOLOGY-0030, problem 1.15 in the AIM workshop list *Quantum invariants and low-dimensional topology*, section “Volume Conjecture”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0031": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Teichm\\\"uller TQFT volume conjecture for Fundamental Shadow Link complements\n\nFormulate and prove the volume conjecture coming from the Teichm\\\"uller TQFT for Fundamental Shadow Link complements",
   "clean_statement": "3-manifold bound efficiently\"uller TQFT volume conjecture for Fundamental Shadow Link complements\n\nFormulate and prove the volume conjecture coming from the 3-manifold bound efficiently\"uller TQFT for Fundamental Shadow Link complements",
   "public_statement": "Teichm\\\"uller TQFT volume conjecture for Fundamental Shadow Link complements\n\nFormulate and prove the volume conjecture coming from the Teichm\\\"uller TQFT for Fundamental Shadow Link complements",
   "evidence": "The canonical AIM record (Volume Conjecture section, item 1.35) is preserved verbatim in `input.json`:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-topology-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0032": {
   "statement_status": "exact",
   "original_statement": "Understanding connections between Baseilhac-Benedetti invariants and other quantum invariants\n\nWhat are the connections between Baseilhac-Benedetti invariants and\n1. ADO invariant?\n2. BWY invariant?\n3. modified TV invariant?",
   "clean_statement": "Understanding connections between Baseilhac-Benedetti invariants and other quantum invariants\n\nWhat are the connections between Baseilhac-Benedetti invariants and\n1. ADO invariant?\n2. BWY invariant?\n3. modified TV invariant?",
   "public_statement": "Understanding connections between Baseilhac-Benedetti invariants and other quantum invariants\n\nWhat are the connections between Baseilhac-Benedetti invariants and\n1. ADO invariant?\n2. BWY invariant?\n3. modified TV invariant?",
   "evidence": "The record identifies ADO as Akutsu--Deguchi--Ohtsuki and identifies modified TV with Geer--Patureau (2010). The source text is intelligible and shows no apparent OCR corruption. There is, however, a mathematical ambiguity: “Baseilhac--Benedetti invariants” can refer both to quantum-hyperbolic link/3-manifold invariants and to the mapping-class or fibred-cusped-manifold invariants later compared with quantum Teichmüller invariants. The three questions do not all concern exactly the same category of decorated objects. This report keeps those variants distinct.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 31,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0033": {
   "statement_status": "exact",
   "original_statement": "For links with diffeomorphic complements, how are their colored Jones polynomial (asymptotics) related?\n\nIf two links have diffeomorphic complements in $S^3$, relate the asymptotics of their colored Jones Polynomial.",
   "clean_statement": "For links with diffeomorphic complements, how are their colored Jones polynomial (asymptotics) related?\n\nIf two links have diffeomorphic complements in $S^3$, relate the asymptotics of their colored Jones Polynomial.",
   "public_statement": "For links with diffeomorphic complements, how are their colored Jones polynomial (asymptotics) related?\n\nIf two links have diffeomorphic complements in $S^3$, relate the asymptotics of their colored Jones Polynomial.",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0034": {
   "statement_status": "exact",
   "original_statement": "Pairs of 3-manifold having the same volume but different TV invariants\n\nFor two manifolds with same volume and different Turaev-Viro invariants, how are they related?",
   "clean_statement": "Pairs of 3-manifold having the same volume but different TV invariants\n\nFor two manifolds with same volume and different Turaev-Viro invariants, how are they related?",
   "public_statement": "Pairs of 3-manifold having the same volume but different TV invariants\n\nFor two manifolds with same volume and different Turaev-Viro invariants, how are they related?",
   "evidence": "The canonical AIM record is intact. Its exact mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 33,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0035": {
   "statement_status": "exact",
   "original_statement": "Find a shadow formula for Teichm\\\"uller TQFT",
   "clean_statement": "Find a shadow formula for Teichm\\\"uller TQFT",
   "public_statement": "Find a shadow formula for Teichm\\\"uller TQFT",
   "evidence": "The exact canonical record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0036": {
   "statement_status": "exact",
   "original_statement": "Two variable invariants and their connections\n\nAre the Habiro 2-variable invariants for knots and Gukov-Manolescu invariant (also known as the F-invariant) the same?",
   "clean_statement": "Two variable invariants and their connections\n\nAre the Habiro 2-variable invariants for knots and Gukov-Manolescu invariant (also known as the F-invariant) the same?",
   "public_statement": "Two variable invariants and their connections\n\nAre the Habiro 2-variable invariants for knots and Gukov-Manolescu invariant (also known as the F-invariant) the same?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0037": {
   "statement_status": "exact",
   "original_statement": "State and prove the volume conjecture for the Gukov-Manolescu invariant.",
   "clean_statement": "State and prove the volume conjecture for the Gukov-Manolescu invariant.",
   "public_statement": "State and prove the volume conjecture for the Gukov-Manolescu invariant.",
   "evidence": "The canonical AIM record, item 1.55 in the “Volume Conjecture” section of the workshop list *Quantum invariants and low-dimensional topology*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0038": {
   "statement_status": "exact",
   "original_statement": "Generalize Teichm\\\"uller TQFT to wider families e.g. cone manifolds or fundamental shadow link complements",
   "clean_statement": "Generalize Teichm\\\"uller TQFT to wider families e.g. cone manifolds or fundamental shadow link complements",
   "public_statement": "Generalize Teichm\\\"uller TQFT to wider families e.g. cone manifolds or fundamental shadow link complements",
   "evidence": "The exact canonical record in input.json says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0039": {
   "statement_status": "exact",
   "original_statement": "Peripheral ideal in Kauffman bracket skein algebra\n\nFind non-trivial explicit element in the peripheral ideal in the Kauffman bracket skein algebra of the boundary torus for two bridge knots.",
   "clean_statement": "Peripheral ideal in Kauffman bracket skein algebra\n\nFind non-trivial explicit element in the peripheral ideal in the Kauffman bracket skein algebra of the boundary torus for two bridge knots.",
   "public_statement": "Peripheral ideal in Kauffman bracket skein algebra\n\nFind non-trivial explicit element in the peripheral ideal in the Kauffman bracket skein algebra of the boundary torus for two bridge knots.",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0040": {
   "statement_status": "exact",
   "original_statement": "Find other classes of knots for which the peripheral ideal is non trivial.\nFor example, trying to find a knot which is not a 2-bridge knot",
   "clean_statement": "Find other classes of knots for which the peripheral ideal is non trivial.\nFor example, trying to find a knot which is not a 2-bridge knot",
   "public_statement": "Find other classes of knots for which the peripheral ideal is non trivial.\nFor example, trying to find a knot which is not a 2-bridge knot",
   "evidence": "The canonical AIM record, in the section “Skein modules and algebra” of the workshop *Quantum invariants and low-dimensional topology*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0041": {
   "statement_status": "exact",
   "original_statement": "Indecomposable $SL_3$ webs\n\nFor a triangulated punctured surface, find the indecomposable $SL_3$ webs for the Knutsen-Tao cone",
   "clean_statement": "Indecomposable $SL_3$ webs\n\nFor a triangulated punctured surface, find the indecomposable $SL_3$ webs for the Knutsen-Tao cone",
   "public_statement": "Indecomposable $SL_3$ webs\n\nFor a triangulated punctured surface, find the indecomposable $SL_3$ webs for the Knutsen-Tao cone",
   "evidence": "The canonical AIM record (workshop *Quantum invariants and low-dimensional topology*, problem 2.15) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0042": {
   "statement_status": "exact",
   "original_statement": "State Skein algebra of bigons\n\nFind skeins representing the basis for the isomorphism\n\"Stated Skein Algebra of bigon $\\cong$ $O_q(SL_n)$ for $n\\geq 3$\" where $O_q(SL_n)$ has Kashiwara-Lusztig canonical basis",
   "clean_statement": "State Skein algebra of bigons\n\nFind skeins representing the basis for the isomorphism\n\"Stated Skein Algebra of bigon $\\cong$ $O_q(SL_n)$ for $n\\geq 3$\" where $O_q(SL_n)$ has Kashiwara-Lusztig canonical basis",
   "public_statement": "State Skein algebra of bigons\n\nFind skeins representing the basis for the isomorphism\n\"Stated Skein Algebra of bigon $\\cong$ $O_q(SL_n)$ for $n\\geq 3$\" where $O_q(SL_n)$ has Kashiwara-Lusztig canonical basis",
   "evidence": "The source record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0043": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "KBSM of connected sum of Lens space\n\nCompute the Kauffman Bracket Skein Module of $#$ Lens space (Haken manifold of finite type) over $\\mathbb{Z} [A^{\\pm}]$",
   "clean_statement": null,
   "public_statement": "KBSM of connected sum of Lens space\n\nCompute the Kauffman Bracket Skein Module of $#$ Lens space (Haken manifold of finite type) over $\\mathbb{Z} [A^{\\pm}]$",
   "evidence": "I use the following explicit reconstruction.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0044": {
   "statement_status": "exact",
   "original_statement": "$SL_n$ skein modules\n\nExplore $SL_n$ skein modules for $n\\geq 3$.\nFor example, the dimension $< \\infty$ is proved by Gunningham-Jordan-Safranov.",
   "clean_statement": "$SL_n$ skein modules\n\nExplore $SL_n$ skein modules for $n\\geq 3$.\nFor example, the dimension $< \\infty$ is proved by Gunningham-Jordan-Safranov.",
   "public_statement": "$SL_n$ skein modules\n\nExplore $SL_n$ skein modules for $n\\geq 3$.\nFor example, the dimension $< \\infty$ is proved by Gunningham-Jordan-Safranov.",
   "evidence": "The exact AIM record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0045": {
   "statement_status": "exact",
   "original_statement": "$SL_n$ skein modules definition equivalence\n\nCompare the definitions of $SL_n$ skein modules/algebra for either a generic q or q being a root of unity.",
   "clean_statement": "$SL_n$ skein modules definition equivalence\n\nCompare the definitions of $SL_n$ skein modules/algebra for either a generic q or q being a root of unity.",
   "public_statement": "$SL_n$ skein modules definition equivalence\n\nCompare the definitions of $SL_n$ skein modules/algebra for either a generic q or q being a root of unity.",
   "evidence": "The exact canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0046": {
   "statement_status": "exact",
   "original_statement": "Quantum trace for closed surfaces\n\nFor Bloomquist-Karuo-Le quantum trace for closed surface, what happens when we change the pants decomposition? (Hatcher-Thurston move)",
   "clean_statement": "Quantum trace for closed surfaces\n\nFor Bloomquist-Karuo-Le quantum trace for closed surface, what happens when we change the pants decomposition? (Hatcher-Thurston move)",
   "public_statement": "Quantum trace for closed surfaces\n\nFor Bloomquist-Karuo-Le quantum trace for closed surface, what happens when we change the pants decomposition? (Hatcher-Thurston move)",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0047": {
   "statement_status": "exact",
   "original_statement": "Skein module for 3 manifold with $S(\\Sigma)$ action\n\nLet $S(\\Sigma)$ act on $N$ be a module of the Kauffman Bracket Skein Algebra. Can we determine if N is the skein module of a 3 manifold M with $S(\\Sigma)$ action given by $\\Sigma: \\partial M \\hookrightarrow M$",
   "clean_statement": "Skein module for 3 manifold with $S(\\Sigma)$ action\n\nLet $S(\\Sigma)$ act on $N$ be a module of the Kauffman Bracket Skein Algebra. Can we determine if N is the skein module of a 3 manifold M with $S(\\Sigma)$ action given by $\\Sigma: \\partial M \\hookrightarrow M$",
   "public_statement": "Skein module for 3 manifold with $S(\\Sigma)$ action\n\nLet $S(\\Sigma)$ act on $N$ be a module of the Kauffman Bracket Skein Algebra. Can we determine if N is the skein module of a 3 manifold M with $S(\\Sigma)$ action given by $\\Sigma: \\partial M \\hookrightarrow M$",
   "evidence": "The canonical record (AIM Problem Lists, workshop *Quantum invariants and low-dimensional topology*, problem 2.45) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0048": {
   "statement_status": "exact",
   "original_statement": "Gluing Stated skein algebra\n\nIs there a formula for gluing stated skein algebra along circles?",
   "clean_statement": "Gluing Stated skein algebra\n\nIs there a formula for gluing stated skein algebra along circles?",
   "public_statement": "Gluing Stated skein algebra\n\nIs there a formula for gluing stated skein algebra along circles?",
   "evidence": "There is no apparent OCR error. The wording does omit data which the 2023 AIM workshop report supplies: the intended surface has two boundary circles, each with one marked point, and the circles are identified. The workshop group expected the glued algebra to be the $U_q(\\mathfrak{sl}_2)$-invariants in a relative tensor product over two actions of the once-marked annulus. It constructed a cutting map and a concrete candidate map, but explicitly described well-definedness and bijectivity as conjectural.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0049": {
   "statement_status": "exact",
   "original_statement": "Versions of Stated skeins for non semi-simple categories\n\nCompare different definitions of the Stated Skeins for non semi-simple categories",
   "clean_statement": "Versions of Stated skeins for non semi-simple categories\n\nCompare different definitions of the Stated Skeins for non semi-simple categories",
   "public_statement": "Versions of Stated skeins for non semi-simple categories\n\nCompare different definitions of the Stated Skeins for non semi-simple categories",
   "evidence": "The canonical record is number 3.1 in the AIM workshop list *Quantum invariants and low-dimensional topology*, section “Non semi-simple categories”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0050": {
   "statement_status": "exact",
   "original_statement": "Classification of non-semi simple modular categories\n\nClassify non semi-simple modular categories for low rank (rank $\\geq 2$) where the rank denotes the number of simple objects",
   "clean_statement": "Classification of non-semi simple modular categories\n\nClassify non semi-simple modular categories for low rank (rank $\\geq 2$) where the rank denotes the number of simple objects",
   "public_statement": "Classification of non-semi simple modular categories\n\nClassify non semi-simple modular categories for low rank (rank $\\geq 2$) where the rank denotes the number of simple objects",
   "evidence": "The exact repository record (AIM Problem Lists, workshop *Quantum invariants and low-dimensional topology*, section \"Non semi-simple categories\", Problem 3.2) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0051": {
   "statement_status": "exact",
   "original_statement": "Non semi-simple categories to TQFTS\n\nConstruct non semi-simple categories that give rise to certain TQFTS",
   "clean_statement": "Non semi-simple categories to TQFTS\n\nConstruct non semi-simple categories that give rise to certain TQFTS",
   "public_statement": "Non semi-simple categories to TQFTS\n\nConstruct non semi-simple categories that give rise to certain TQFTS",
   "evidence": "The canonical AIM record says, verbatim:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0052": {
   "statement_status": "exact",
   "original_statement": "Categorification of CJP\n\nCategorify Colored Jones Polynomial to study the volume conjecture.",
   "clean_statement": "Categorification of CJP\n\nCategorify Colored Jones Polynomial to study the volume conjecture.",
   "public_statement": "Categorification of CJP\n\nCategorify Colored Jones Polynomial to study the volume conjecture.",
   "evidence": "The exact repository record (AIM Problem Lists, workshop *Quantum invariants and low-dimensional topology*, section \"Categorification\", Problem 4.2) is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0053": {
   "statement_status": "exact",
   "original_statement": "Categorification of Kauffman Bracket Skein Module\n\nCategorify Kauffman Bracket Skein Module of Lens space L(p,1)",
   "clean_statement": "Categorification of Kauffman Bracket Skein Module\n\nCategorify Kauffman Bracket Skein Module of Lens space L(p,1)",
   "public_statement": "Categorification of Kauffman Bracket Skein Module\n\nCategorify Kauffman Bracket Skein Module of Lens space L(p,1)",
   "evidence": "The canonical record is AIM-TOPOLOGY-0053, source file `aim-topology-notes.json`, record index 52. Its problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0054": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Lift the recursion relation of the colored Jones Polynomial to categorified CJP.",
   "clean_statement": null,
   "public_statement": "Lift the recursion relation of the colored Jones Polynomial to categorified CJP.",
   "evidence": "There are two mathematically plausible readings.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0055": {
   "statement_status": "exact",
   "original_statement": "Make $gl(1|1)$-homology concrete and relate it to knot/tangle Floer homology.",
   "clean_statement": "Make $gl(1|1)$-homology concrete and relate it to knot/tangle Floer homology.",
   "public_statement": "Make $gl(1|1)$-homology concrete and relate it to knot/tangle Floer homology.",
   "evidence": "The exact canonical statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0056": {
   "statement_status": "exact",
   "original_statement": "Does the superspace coinvariant ring defined as $$\\mathbb{C}[x_1,\\ldots,x_n,\\theta_1,\\ldots,\\theta_n]\\,\\big/\\,(\\mathbb{C}[x_1,\\ldots,x_n,\\theta_1,\\ldots,\\theta_n]^{S_n}_+)$$ relate to link homology? What object does $$\\mathbb{C}[x_1,\\ldots,x_n,\\theta_1,\\ldots,\\theta_n]\\,\\big/\\,(\\mathbb{C}[x_1,\\ldots,x_n,\\theta_1,\\ldots,\\theta_n]^{S_n}_+)$$ define in the trace of Hecke category?",
   "clean_statement": "Does the superspace coinvariant ring defined as $$\\mathbb{C}[x_1,\\ldots,x_n,\\theta_1,\\ldots,\\theta_n]\\,\\big/\\,(\\mathbb{C}[x_1,\\ldots,x_n,\\theta_1,\\ldots,\\theta_n]^{S_n}_+)$$ relate to link homology? What object does $$\\mathbb{C}[x_1,\\ldots,x_n,\\theta_1,\\ldots,\\theta_n]\\,\\big/\\,(\\mathbb{C}[x_1,\\ldots,x_n,\\theta_1,\\ldots,\\theta_n]^{S_n}_+)$$ define in the trace of Hecke category?",
   "public_statement": "Does the superspace coinvariant ring defined as $$\\mathbb{C}[x_1,\\ldots,x_n,\\theta_1,\\ldots,\\theta_n]\\,\\big/\\,(\\mathbb{C}[x_1,\\ldots,x_n,\\theta_1,\\ldots,\\theta_n]^{S_n}_+)$$ relate to link homology? What object does $$\\mathbb{C}[x_1,\\ldots,x_n,\\theta_1,\\ldots,\\theta_n]\\,\\big/\\,(\\mathbb{C}[x_1,\\ldots,x_n,\\theta_1,\\ldots,\\theta_n]^{S_n}_+)$$ define in the trace of Hecke category?",
   "evidence": "The canonical record is AIM-TOPOLOGY-0056, source file `aim-topology-notes.json`, zero-based index 55. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0057": {
   "statement_status": "exact",
   "original_statement": "Do $d$-agonal coinvariants relate to link homology?",
   "clean_statement": "Do $d$-agonal coinvariants relate to link homology?",
   "public_statement": "Do $d$-agonal coinvariants relate to link homology?",
   "evidence": "The canonical record from the 2023 AIM workshop *Algebra, geometry, and combinatorics of link homology* says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0058": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Do Haiman's polygraph rings relate to link homology? How do they relate to cables of Hopf link?",
   "clean_statement": null,
   "public_statement": "Do Haiman's polygraph rings relate to link homology? How do they relate to cables of Hopf link?",
   "evidence": "The canonical record is AIM Problem Lists, workshop *Algebra, geometry, and combinatorics of link homology*, section “Link homology,” Problem 1.4:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 57,
   "attempt": 3
  },
  "AIM-TOPOLOGY-0059": {
   "statement_status": "exact",
   "original_statement": "Do Catalanimal operators appear naturally in link homology?",
   "clean_statement": "Do Catalanimal operators appear naturally in link homology?",
   "public_statement": "Do Catalanimal operators appear naturally in link homology?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0060": {
   "statement_status": "exact",
   "original_statement": "Is the Khovanov-Rozansky homology functorial?",
   "clean_statement": "Is the Khovanov-Rozansky homology functorial?",
   "public_statement": "Is the Khovanov-Rozansky homology functorial?",
   "evidence": "The exact canonical record is AIM-TOPOLOGY-0060, record 59 (zero-based) of `aim-topology-notes.json`, from the 2023 AIM workshop *Algebra, geometry, and combinatorics of link homology*. Its entire problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0061": {
   "statement_status": "exact",
   "original_statement": "If $w^2=1$, we have a map $B_w\\rightarrow R$. Is the induced map $HH(B_w)\\rightarrow HH(R)$ injective? Given that the Hochschild homology is defined as $HH(\\beta):=\\text{Ext}_{R-\\text{ bimod}}(R,B)$.",
   "clean_statement": "If $w^2=1$, we have a map $B_w\\rightarrow R$. Is the induced map $HH(B_w)\\rightarrow HH(R)$ injective? Given that the Hochschild homology is defined as $HH(\\beta):=\\text{Ext}_{R-\\text{ bimod}}(R,B)$.",
   "public_statement": "If $w^2=1$, we have a map $B_w\\rightarrow R$. Is the induced map $HH(B_w)\\rightarrow HH(R)$ injective? Given that the Hochschild homology is defined as $HH(\\beta):=\\text{Ext}_{R-\\text{ bimod}}(R,B)$.",
   "evidence": "The exact AIM record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0062": {
   "statement_status": "exact",
   "original_statement": "Compute the Khovanov-Rozansky homology $HHH$ of cables of torus knots and develop a recursion for this class of knots.",
   "clean_statement": "Compute the Khovanov-Rozansky homology $HHH$ of cables of torus knots and develop a recursion for this class of knots.",
   "public_statement": "Compute the Khovanov-Rozansky homology $HHH$ of cables of torus knots and develop a recursion for this class of knots.",
   "evidence": "This text was checked on 2026-08-13 against the live AIM Problem Lists page for Problem 2.3. It agrees verbatim, so there is no OCR repair to make. The statement is nevertheless underspecified: a cable depends on a longitude/framing convention, and $HHH$ may mean reduced or unreduced triply graded HOMFLY--PT/Khovanov--Rozansky homology. Below, $K(p,q)$ means the raw $(p,q)$ satellite using the Seifert (zero) framing. It is not a projector-colored component. The Poincaré series convention is the unreduced convention of Caprau--González--Hogancamp--Mazin (CGHM), in which a factor $(1-q)^{-1}$ occurs.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0063": {
   "statement_status": "exact",
   "original_statement": "Compute $HHH(JM_1^{t_1}\\dots JM_n^{t_n})$ \"for as many $t$'s as possible\" is it parity? Describe it as an $R$-module.",
   "clean_statement": "Compute $HHH(JM_1^{t_1}\\dots JM_n^{t_n})$ \"for as many $t$'s as possible\" is it parity? Describe it as an $R$-module.",
   "public_statement": "Compute $HHH(JM_1^{t_1}\\dots JM_n^{t_n})$ \"for as many $t$'s as possible\" is it parity? Describe it as an $R$-module.",
   "evidence": "**Artifact metadata.** Source file aim-topology-notes.json, zero-based source index \\(62\\), attempt \\(2\\).",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 62,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0064": {
   "statement_status": "exact",
   "original_statement": "Describe annulus maps in Khovanov-Rozansky homology $HHH$.",
   "clean_statement": "Describe annulus maps in Khovanov-Rozansky homology $HHH$.",
   "public_statement": "Describe annulus maps in Khovanov-Rozansky homology $HHH$.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0065": {
   "statement_status": "exact",
   "original_statement": "Compute the Khovanov homology of torus links.",
   "clean_statement": "Compute the Khovanov homology of torus links.",
   "public_statement": "Compute the Khovanov homology of torus links.",
   "evidence": "The live AIM page agrees verbatim with the JSON record. There is no visible OCR error. The sentence is nevertheless underspecified: it does not say reduced or unreduced, integral or field coefficients, ordinary \\(\\mathfrak{sl}_2\\) Khovanov homology or a Khovanov--Rozansky theory, positive or negative torus links, nor a grading normalization.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0066": {
   "statement_status": "exact",
   "original_statement": "Compute $JM_1^{t_1}\\dots JM_n^{t_n}$ for Khovanov homology.",
   "clean_statement": "Compute $JM_1^{t_1}\\dots JM_n^{t_n}$ for Khovanov homology.",
   "public_statement": "Compute $JM_1^{t_1}\\dots JM_n^{t_n}$ for Khovanov homology.",
   "evidence": "The exact canonical AIM record is Problem 3.2 in the “Khovanov homology” section of the 2023 workshop *Algebra, geometry, and combinatorics of link homology*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0067": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "For Khovanov homology develop analogues of\n \\begin{enumerate}[label=\\alph*)]\n \\item The Oblomkov-Rasmussen-Shende conjecture.\n \\item Braid varieties.\n \\item Hilb$^n(\\mathbb{C}^2)$ should be Hilb$^n(x^2=0)$.\n\\end{enumerate}",
   "clean_statement": null,
   "public_statement": "For Khovanov homology develop analogues of\n \\begin{enumerate}[label=\\alph*)]\n \\item The Oblomkov-Rasmussen-Shende conjecture.\n \\item Braid varieties.\n \\item Hilb$^n(\\mathbb{C}^2)$ should be Hilb$^n(x^2=0)$.\n\\end{enumerate}",
   "evidence": "We use the following conservative reconstruction. The established geometric objects in the prompt concern triply graded HOMFLY/Khovanov--Rozansky homology (abbreviated HHH):",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0068": {
   "statement_status": "exact",
   "original_statement": "Fixing a slope, what corresponds to a stable basis in $K$-theory of Hilbert schemes in the cocenter of the Hecke algebra?",
   "clean_statement": "Fixing a slope, what corresponds to a stable basis in $K$-theory of Hilbert schemes in the cocenter of the Hecke algebra?",
   "public_statement": "Fixing a slope, what corresponds to a stable basis in $K$-theory of Hilbert schemes in the cocenter of the Hecke algebra?",
   "evidence": "**Artifact metadata.** Source file aim-topology-notes.json, zero-based source index \\(67\\), attempt \\(1\\).",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0069": {
   "statement_status": "exact",
   "original_statement": "Compute endomorphism algebra (in the homotopy category) of projectors, and relate it to coordinate ring and of charts on Hilb$^n(\\mathbb{C}^2)$",
   "clean_statement": "Compute endomorphism algebra (in the homotopy category) of projectors, and relate it to coordinate ring and of charts on Hilb$^n(\\mathbb{C}^2)$",
   "public_statement": "Compute endomorphism algebra (in the homotopy category) of projectors, and relate it to coordinate ring and of charts on Hilb$^n(\\mathbb{C}^2)$",
   "evidence": "The exact canonical record, AIM Problem 4.2 in the section “Hilbert schemes,” reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0070": {
   "statement_status": "exact",
   "original_statement": "Relate $tr(T(m,n))$ to rings of quasi-invariants and to A. Wilson's conjecture. Describe $H^*(\\text{Hilb}(x^{nd}=y^n))$ as a module over rational Double Affine Hecke algebra (DAHA) and relate it to the above.",
   "clean_statement": "Relate $tr(T(m,n))$ to rings of quasi-invariants and to A. Wilson's conjecture. Describe $H^*(\\text{Hilb}(x^{nd}=y^n))$ as a module over rational Double Affine Hecke algebra (DAHA) and relate it to the above.",
   "public_statement": "Relate $tr(T(m,n))$ to rings of quasi-invariants and to A. Wilson's conjecture. Describe $H^*(\\text{Hilb}(x^{nd}=y^n))$ as a module over rational Double Affine Hecke algebra (DAHA) and relate it to the above.",
   "evidence": "The exact canonical AIM record is Problem 4.3 in the “Hilbert schemes” section of the 2023 workshop *Algebra, geometry, and combinatorics of link homology*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 69,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0071": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "We define Hilb$^n(x^2=0)=\\{\\text{codiminsional }n \\text{ ideals in }\\mathbb{C}[x,y]/(x^2=0)\\}\\subseteq$ Hilb$^n(\\mathbb{C}^2)$\n\nRelate Hilb$^n(x^2=0)$ to $HH$ of the arc algebra.",
   "clean_statement": null,
   "public_statement": "We define Hilb$^n(x^2=0)=\\{\\text{codiminsional }n \\text{ ideals in }\\mathbb{C}[x,y]/(x^2=0)\\}\\subseteq$ Hilb$^n(\\mathbb{C}^2)$\n\nRelate Hilb$^n(x^2=0)$ to $HH$ of the arc algebra.",
   "evidence": "The exact source record (AIM workshop “Algebra, geometry, and combinatorics of link homology,” section “Hilbert schemes,” problem 4.4) reads:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 70,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0072": {
   "statement_status": "exact",
   "original_statement": "Let $\\mathcal{F}(\\beta)$ be the sheaf on Hilb$^n(\\mathbb{C}^2)$ associated to $\\beta$. How does $\\mathcal{F}(\\beta)|_{\\text{Hilb}^n(x^2=0)}$ relate to Khovanov homology of $\\beta$?",
   "clean_statement": "Let $\\mathcal{F}(\\beta)$ be the sheaf on Hilb$^n(\\mathbb{C}^2)$ associated to $\\beta$. How does $\\mathcal{F}(\\beta)|_{\\text{Hilb}^n(x^2=0)}$ relate to Khovanov homology of $\\beta$?",
   "public_statement": "Let $\\mathcal{F}(\\beta)$ be the sheaf on Hilb$^n(\\mathbb{C}^2)$ associated to $\\beta$. How does $\\mathcal{F}(\\beta)|_{\\text{Hilb}^n(x^2=0)}$ relate to Khovanov homology of $\\beta$?",
   "evidence": "The archived AIM page agrees verbatim with the extracted record. Here \\(\\operatorname{Hilb}^n(x^2=0)\\) means the Hilbert scheme of length-\\(n\\) subschemes of the **scheme-theoretic** double line \\(D=\\operatorname{Spec}\\mathbb C[x,y]/(x^2)\\), embedded in \\(H_n=\\operatorname{Hilb}^n(\\mathbb C^2)\\).",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 71,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0073": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Construct link invariants for links in lens spaces using Hilb$([\\mathbb{C}^2/(\\mathbb{Z}/l\\mathbb{Z}])$ and relate to wreath Macdonald polynomials.",
   "clean_statement": null,
   "public_statement": "Construct link invariants for links in lens spaces using Hilb$([\\mathbb{C}^2/(\\mathbb{Z}/l\\mathbb{Z}])$ and relate to wreath Macdonald polynomials.",
   "evidence": "The exact canonical record is Problem 4.6 in the “Hilbert schemes” section:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 72,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0074": {
   "statement_status": "exact",
   "original_statement": "What are Schur expansions of projector closures at the category level and what are their connections to Macdonald theory?",
   "clean_statement": "What are Schur expansions of projector closures at the category level and what are their connections to Macdonald theory?",
   "public_statement": "What are Schur expansions of projector closures at the category level and what are their connections to Macdonald theory?",
   "evidence": "The canonical record is problem 5.1 in the AIM workshop list *Algebra, geometry, and combinatorics of link homology*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 73,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0075": {
   "statement_status": "exact",
   "original_statement": "Categorify the modified Macdonald polynomials $\\tilde{H_\\mu}$.",
   "clean_statement": "Categorify the modified Macdonald polynomials $\\tilde{H_\\mu}$.",
   "public_statement": "Categorify the modified Macdonald polynomials $\\tilde{H_\\mu}$.",
   "evidence": "The canonical AIM record is problem 5.2 in the workshop *Algebra, geometry, and combinatorics of link homology*, section “Macdonald polynomials”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0076": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Let $P\\in SYT(\\lambda)$, $Q\\in SYT(\\mu)$ relate $Hom(tr(P),tr(Q))$ to the Macdonald inner product.",
   "clean_statement": null,
   "public_statement": "Let $P\\in SYT(\\lambda)$, $Q\\in SYT(\\mu)$ relate $Hom(tr(P),tr(Q))$ to the Macdonald inner product.",
   "evidence": "The source sentence is terse and lacks punctuation, but it is mathematically coherent rather than visibly corrupted OCR. The live AIM URL returned an HTTP 502 during this run, so no wording beyond the canonical record could be verified there. Nearby problems ask for Schur expansions of projector closures and a categorification of the modified Macdonald polynomials. In that context the most conservative reconstruction is:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0077": {
   "statement_status": "exact",
   "original_statement": "Find the Macdonald/Schur expansion of the $a$-graded Frobenius character of the parking function space. How does the $tr(T(m,n)$ relate to the parking function module?",
   "clean_statement": "Find the Macdonald/Schur expansion of the $a$-graded Frobenius character of the parking function space. How does the $tr(T(m,n)$ relate to the parking function module?",
   "public_statement": "Find the Macdonald/Schur expansion of the $a$-graded Frobenius character of the parking function space. How does the $tr(T(m,n)$ relate to the parking function module?",
   "evidence": "The exact corpus record (AIM workshop *Algebra, geometry, and combinatorics of link homology*, section 5, item 5.4) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0078": {
   "statement_status": "exact",
   "original_statement": "Develop singular braid varieties and relate them to Richardson varieties in $G/P$ and to colored homology.",
   "clean_statement": "Develop singular braid varieties and relate them to Richardson varieties in $G/P$ and to colored homology.",
   "public_statement": "Develop singular braid varieties and relate them to Richardson varieties in $G/P$ and to colored homology.",
   "evidence": "The AIM record (workshop *Algebra, geometry, and combinatorics of link homology*, section “Braid varieties,” Problem 6.1) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0079": {
   "statement_status": "exact",
   "original_statement": "What properties of braid varieties in $G/B$ should survive in $G/P$? In other words, what properties of full flag varieties remain in partial flag varieties.",
   "clean_statement": "What properties of braid varieties in $G/B$ should survive in $G/P$? In other words, what properties of full flag varieties remain in partial flag varieties.",
   "public_statement": "What properties of braid varieties in $G/B$ should survive in $G/P$? In other words, what properties of full flag varieties remain in partial flag varieties.",
   "evidence": "The exact canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 78,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0080": {
   "statement_status": "exact",
   "original_statement": "What is the relation between braid varieties and singular braid varieties?",
   "clean_statement": "What is the relation between braid varieties and singular braid varieties?",
   "public_statement": "What is the relation between braid varieties and singular braid varieties?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 79,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0081": {
   "statement_status": "exact",
   "original_statement": "Compute $H^*_T(X(\\sigma_1\\dots\\sigma_{n-1})^m\\Delta)$.",
   "clean_statement": "Compute $H^*_T(X(\\sigma_1\\dots\\sigma_{n-1})^m\\Delta)$.",
   "public_statement": "Compute $H^*_T(X(\\sigma_1\\dots\\sigma_{n-1})^m\\Delta)$.",
   "evidence": "The exact AIM problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 80,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0082": {
   "statement_status": "exact",
   "original_statement": "Are there \"braid variety analogues\" of projectors?",
   "clean_statement": "Are there \"braid variety analogues\" of projectors?",
   "public_statement": "Are there \"braid variety analogues\" of projectors?",
   "evidence": "The canonical record is problem 6.5 in the “Braid varieties” section of the AIM workshop *Algebra, geometry, and combinatorics of link homology*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0083": {
   "statement_status": "exact",
   "original_statement": "Can we determine $gl(m|n)$ / Khovanov homology using braid-like varieties?",
   "clean_statement": "Can we determine $gl(m|n)$ / Khovanov homology using braid-like varieties?",
   "public_statement": "Can we determine $gl(m|n)$ / Khovanov homology using braid-like varieties?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 82,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0084": {
   "statement_status": "exact",
   "original_statement": "Define the analogues of braid varieties for $sl_N$ homology, and study their connection with representation varieties.",
   "clean_statement": "Define the analogues of braid varieties for $sl_N$ homology, and study their connection with representation varieties.",
   "public_statement": "Define the analogues of braid varieties for $sl_N$ homology, and study their connection with representation varieties.",
   "evidence": "The extracted statement is intact; there is no apparent OCR error. The official workshop report makes its meaning substantially more precise [AIM]. Ordinary braid varieties for positive braids are smooth complex flag-configuration varieties whose cohomology describes a lowest \\(a\\)-degree part of triply graded Khovanov–Rozansky homology. The report contrasts them with compact real spaces of \\(SU(N)\\)-representations of link groups with meridians in prescribed conjugacy classes. The latter are configurations of lines (or, for exterior-power labels, subspaces) in \\(\\mathbb C^N\\). The working group observed that planar braid-like webs should bring the two constructions closer, but did not produce a satisfactory general definition.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 83,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0085": {
   "statement_status": "exact",
   "original_statement": "What is the action of the full twist on the trace of the Hecke category? Does it relate to $\\nabla$? How does it act on Schur objects?",
   "clean_statement": "What is the action of the full twist on the trace of the Hecke category? Does it relate to $\\nabla$? How does it act on Schur objects?",
   "public_statement": "What is the action of the full twist on the trace of the Hecke category? Does it relate to $\\nabla$? How does it act on Schur objects?",
   "evidence": "There is no apparent OCR corruption. The surrounding workshop material and Gorsky--Hogancamp--Wedrich (GHW) fix the intended setting: the finite type-$A$ Hecke category, modeled by complexes of Soergel bimodules $\\mathrm{SBim}_n$, and its **derived horizontal trace** (the annular trace), completed under cones and homotopy summands. This is not the vertical Hochschild homology vector space, although the latter is the endomorphism algebra of the traced unit, and it is not the trace of the affine Hecke category.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 84,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0086": {
   "statement_status": "exact",
   "original_statement": "Let $B_w\\in$ SBim$_n$ be the indecomposable Soergel bimodule associated to $w\\in S_n$. Compute Ext$_{R-R \\text{ bimod}}(B_v,B_w)$ and develop its diagrammatics.",
   "clean_statement": "Let $B_w\\in$ SBim$_n$ be the indecomposable Soergel bimodule associated to $w\\in S_n$. Compute Ext$_{R-R \\text{ bimod}}(B_v,B_w)$ and develop its diagrammatics.",
   "public_statement": "Let $B_w\\in$ SBim$_n$ be the indecomposable Soergel bimodule associated to $w\\in S_n$. Compute Ext$_{R-R \\text{ bimod}}(B_v,B_w)$ and develop its diagrammatics.",
   "evidence": "The exact canonical question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 85,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0087": {
   "statement_status": "exact",
   "original_statement": "Find functors from derived horizontal trace computing $\\mathcal{gl}(m|n)$-homology.",
   "clean_statement": "Find functors from derived horizontal trace computing $\\mathcal{gl}(m|n)$-homology.",
   "public_statement": "Find functors from derived horizontal trace computing $\\mathcal{gl}(m|n)$-homology.",
   "evidence": "The exact canonical AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 86,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0088": {
   "statement_status": "exact",
   "original_statement": "Find a basis of the coinvariant ring of $$\\mathbb{C}[x_1,\\ldots,x_n,y_1,\\ldots,y_n,\\theta_1,\\ldots,\\theta_n,\\phi_1,\\ldots,\\phi_n]$$ and a combinatorial model for the Frobenius characteristic in the $$\\mathbb{C}[x_1,\\ldots,x_n,y_1,\\ldots,y_n,\\theta_1,\\ldots,\\theta_n]$$ case, does the basis proposed by Haglund-Sergel work?",
   "clean_statement": "Find a basis of the coinvariant ring of $$\\mathbb{C}[x_1,\\ldots,x_n,y_1,\\ldots,y_n,\\theta_1,\\ldots,\\theta_n,\\phi_1,\\ldots,\\phi_n]$$ and a combinatorial model for the Frobenius characteristic in the $$\\mathbb{C}[x_1,\\ldots,x_n,y_1,\\ldots,y_n,\\theta_1,\\ldots,\\theta_n]$$ case, does the basis proposed by Haglund-Sergel work?",
   "public_statement": "Find a basis of the coinvariant ring of $$\\mathbb{C}[x_1,\\ldots,x_n,y_1,\\ldots,y_n,\\theta_1,\\ldots,\\theta_n,\\phi_1,\\ldots,\\phi_n]$$ and a combinatorial model for the Frobenius characteristic in the $$\\mathbb{C}[x_1,\\ldots,x_n,y_1,\\ldots,y_n,\\theta_1,\\ldots,\\theta_n]$$ case, does the basis proposed by Haglund-Sergel work?",
   "evidence": "The canonical AIM record (Topology, workshop *Algebra, geometry, and combinatorics of link homology*, section 9, Problem 9.4) literally asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 87,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0089": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Take $P,Q\\in SYT(\\lambda)$, how do $tr(P)$ and $tr(G)$ relate?",
   "clean_statement": null,
   "public_statement": "Take $P,Q\\in SYT(\\lambda)$, how do $tr(P)$ and $tr(G)$ relate?",
   "evidence": "The evidence is that another record from the same workshop, AIM-TOPOLOGY-0076, asks how $\\operatorname{Hom}(\\operatorname{tr}(P),\\operatorname{tr}(Q))$ relates to the Macdonald inner product. The workshop report also discusses the derived horizontal trace of the type-A Soergel category, Schur objects, and closures of categorified projectors. Consequently the most natural reading of $\\operatorname{tr}$ is a categorical horizontal/derived trace. A second plausible reading is the scalar Markov trace of a Young idempotent. A third reading, in which $G$ denotes some omitted object, cannot be analyzed without a definition of $G$.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 88,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0090": {
   "statement_status": "exact",
   "original_statement": "Compare recursions for $q,t$-Catalan numbers $C_n(q,t)$ with the recursions for $HHH$.",
   "clean_statement": "Compare recursions for $q,t$-Catalan numbers $C_n(q,t)$ with the recursions for $HHH$.",
   "public_statement": "Compare recursions for $q,t$-Catalan numbers $C_n(q,t)$ with the recursions for $HHH$.",
   "evidence": "The exact canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 89,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0091": {
   "statement_status": "exact",
   "original_statement": "\\begin{enumerate}[label=\\alph*)]\n \\item Relate $tr(T(m,n))$ to rings of quasi-invariants and to A. Wilson's conjecture on $\\nabla_{p_1}^n$.\n \\item Describe $H^*(\\text{Hilb}(x^{nd}=y^n))$ as a module over rational Double Affine Hecke algebra (DAHA) and relate it to the above.\n \\end{enumerate}",
   "clean_statement": "\\begin{enumerate}[label=\\alph*)]\n \\item Relate $tr(T(m,n))$ to rings of quasi-invariants and to A. Wilson's conjecture on $\\nabla_{p_1}^n$.\n \\item Describe $H^*(\\text{Hilb}(x^{nd}=y^n))$ as a module over rational Double Affine Hecke algebra (DAHA) and relate it to the above.\n \\end{enumerate}",
   "public_statement": "\\begin{enumerate}[label=\\alph*)]\n \\item Relate $tr(T(m,n))$ to rings of quasi-invariants and to A. Wilson's conjecture on $\\nabla_{p_1}^n$.\n \\item Describe $H^*(\\text{Hilb}(x^{nd}=y^n))$ as a module over rational Double Affine Hecke algebra (DAHA) and relate it to the above.\n \\end{enumerate}",
   "evidence": "The exact canonical record, AIM Problem 9.6 in the “Miscellaneous” section of the 2023 workshop *Algebra, geometry, and combinatorics of link homology*, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 90,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0092": {
   "statement_status": "exact",
   "original_statement": "Higher homotopy groups\n\nDefine higher homotopy groups in digital topology. As a test case, one should verify that $\\pi_2(S^2, *) = \\mathbb{Z}$, where $S^2$ is the digital 2-sphere, i.e., a graph with six vertices appropriately embedded in $\\mathbb{Z}^3$.",
   "clean_statement": "Higher homotopy groups\n\nDefine higher homotopy groups in digital topology. As a test case, one should verify that $\\pi_2(S^2, *) = \\mathbb{Z}$, where $S^2$ is the digital 2-sphere, i.e., a graph with six vertices appropriately embedded in $\\mathbb{Z}^3$.",
   "public_statement": "Higher homotopy groups\n\nDefine higher homotopy groups in digital topology. As a test case, one should verify that $\\pi_2(S^2, *) = \\mathbb{Z}$, where $S^2$ is the digital 2-sphere, i.e., a graph with six vertices appropriately embedded in $\\mathbb{Z}^3$.",
   "evidence": "The canonical AIM record, problem 1.1 in the “Digital topology” section of the 2023 workshop *Discrete and combinatorial homotopy theory*, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 91,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0093": {
   "statement_status": "exact",
   "original_statement": "Loop spaces\n\nIs there a good notion of a loop space of a digital space?",
   "clean_statement": "Loop spaces\n\nIs there a good notion of a loop space of a digital space?",
   "public_statement": "Loop spaces\n\nIs there a good notion of a loop space of a digital space?",
   "evidence": "The canonical AIM record is from the 2023 workshop *Discrete and combinatorial homotopy theory*, section “Digital topology,” Problem 1.2. Its entire question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0094": {
   "statement_status": "exact",
   "original_statement": "Topological realization\n\nThis problem assumes that higher homotopy groups of digital spaces have been constructed.\n\nIs there a functor $F$ from the category of digital images to that of topological spaces such that the digital invariants of a digital image $X$ agree with the corresponding classical invariants of $FX$, e.g., $\\pi_n(X, *) \\cong \\pi_n(FX, *)$?",
   "clean_statement": "Topological realization\n\nThis problem assumes that higher homotopy groups of digital spaces have been constructed.\n\nIs there a functor $F$ from the category of digital images to that of topological spaces such that the digital invariants of a digital image $X$ agree with the corresponding classical invariants of $FX$, e.g., $\\pi_n(X, *) \\cong \\pi_n(FX, *)$?",
   "public_statement": "Topological realization\n\nThis problem assumes that higher homotopy groups of digital spaces have been constructed.\n\nIs there a functor $F$ from the category of digital images to that of topological spaces such that the digital invariants of a digital image $X$ agree with the corresponding classical invariants of $FX$, e.g., $\\pi_n(X, *) \\cong \\pi_n(FX, *)$?",
   "evidence": "The repository text agrees with the [live AIM page](http://aimpl.org/combhomotop/1/) checked on 2026-08-13. No OCR correction or reconstruction is needed. The page still labels the problem “Open,” but that label does not reflect several recent results.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0095": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Digital topology v $x$-homotopy theory\n\nWhat is the relation between digital topology and $\\times$-homotopy theory of reflexive graphs?",
   "clean_statement": "**Digital topology v \\(x\\)-homotopy theory.** What is the relation between digital topology and \\(\\times\\)-homotopy theory of reflexive graphs?",
   "public_statement": "Digital topology v $x$-homotopy theory\n\nWhat is the relation between digital topology and $\\times$-homotopy theory of reflexive graphs?",
   "evidence": "The plain \\(x\\) in the extracted title is almost certainly a rendering/OCR loss: the mathematical question itself contains the unambiguous LaTeX command `\\times`. I preserve the record but recover the intended title as “Digital topology versus \\(\\times\\)-homotopy theory.” The original AIM problem-list URL returned an error during this run. The canonical record, nearby problems, AIM workshop page, and workshop report were available.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-topology-notes.json",
   "source_index": 94,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0096": {
   "statement_status": "exact",
   "original_statement": "Homotopy colimits in A-homotopy theory\n\nThe category of graphs with weak equivalences given by the maps inducing isomorphisms on all A-homotopy groups admits (small) homotopy colimits.",
   "clean_statement": "Homotopy colimits in A-homotopy theory\n\nThe category of graphs with weak equivalences given by the maps inducing isomorphisms on all A-homotopy groups admits (small) homotopy colimits.",
   "public_statement": "Homotopy colimits in A-homotopy theory\n\nThe category of graphs with weak equivalences given by the maps inducing isomorphisms on all A-homotopy groups admits (small) homotopy colimits.",
   "evidence": "Thus the repetition is in the source itself, not an extraction error. The most natural repair in context is “allow \\(n\\) to depend on \\(m\\),” i.e. ask whether, for every desired connectivity \\(m\\), sufficiently long suspensions have an \\(m\\)-connected collapse. The alternative repair “allow \\(m\\) to depend on \\(n\\)” asks for a connectivity estimate as a function of length. Both are mathematically sensible, and neither is silently substituted for the source.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0097": {
   "statement_status": "exact",
   "original_statement": "Applications to subspace arrangements\n\nCan earlier results of Barcelo on the relation between A-theory and subspace arrangements be extended beyond $A_1$ by showing that the higher $A$-groups vanish?",
   "clean_statement": "Applications to subspace arrangements\n\nCan earlier results of Barcelo on the relation between A-theory and subspace arrangements be extended beyond $A_1$ by showing that the higher $A$-groups vanish?",
   "public_statement": "Applications to subspace arrangements\n\nCan earlier results of Barcelo on the relation between A-theory and subspace arrangements be extended beyond $A_1$ by showing that the higher $A$-groups vanish?",
   "evidence": "The canonical AIM record (source file `aim-topology-notes.json`, zero-based index 96) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 96,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0098": {
   "statement_status": "exact",
   "original_statement": "Computation of cubical homology\n\nWhen computing cubical homology of a simple graph $X$, one builds a cubical set $MX$ whose $n$-cubes are maps $I_1^{\\square n} \\to X$ and then takes the usual cubical homology of this cubical set.\n\nCan cubical homology of a graph be computed using only injective maps $I_1^{\\square n} \\to X$? Should this restriction be made at the cubical set or chain complex level?",
   "clean_statement": "Computation of cubical homology\n\nWhen computing cubical homology of a simple graph $X$, one builds a cubical set $MX$ whose $n$-cubes are maps $I_1^{\\square n} \\to X$ and then takes the usual cubical homology of this cubical set.\n\nCan cubical homology of a graph be computed using only injective maps $I_1^{\\square n} \\to X$? Should this restriction be made at the cubical set or chain complex level?",
   "public_statement": "Computation of cubical homology\n\nWhen computing cubical homology of a simple graph $X$, one builds a cubical set $MX$ whose $n$-cubes are maps $I_1^{\\square n} \\to X$ and then takes the usual cubical homology of this cubical set.\n\nCan cubical homology of a graph be computed using only injective maps $I_1^{\\square n} \\to X$? Should this restriction be made at the cubical set or chain complex level?",
   "evidence": "The canonical record is AIM-TOPOLOGY-0098, source file \\(\\texttt{aim-topology-notes.json}\\), zero-based index \\(97\\), from the 2023 AIM workshop *Discrete and combinatorial homotopy theory*, section “A-homotopy theory,” Problem 2.3. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 97,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0099": {
   "statement_status": "exact",
   "original_statement": "Classifying spaces\n\nDo there exist classifying graphs for fibrations with a fixed fiber (up to weak equivalence) in A-theory? Here, by fibrations we mean maps satisfying the graph analogue of the sharp map condition from simplicial sets.",
   "clean_statement": "Classifying spaces\n\nDo there exist classifying graphs for fibrations with a fixed fiber (up to weak equivalence) in A-theory? Here, by fibrations we mean maps satisfying the graph analogue of the sharp map condition from simplicial sets.",
   "public_statement": "Classifying spaces\n\nDo there exist classifying graphs for fibrations with a fixed fiber (up to weak equivalence) in A-theory? Here, by fibrations we mean maps satisfying the graph analogue of the sharp map condition from simplicial sets.",
   "evidence": "The live AIM page was checked on 13 August 2026. It has exactly this wording, attributes the problem to Eric Babson, and contains no status note or later remark. There is no apparent extraction error.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 98,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0100": {
   "statement_status": "exact",
   "original_statement": "Algebraic K-theory of digraphs\n\nBased on the work of Carranza-Doherty-Kapulkin-Opie-Sarazola-Wong, there is a cofibration category structure on the category of digraphs whose weak equivalences are digraph maps that induce isomorphisms on all path homology groups. One can use it to define a Waldhausen category on (pointed) digraphs, which can then be used to take algebraic K-theory.\n\nCompute $K_0(*)$ and check if it agrees with what one might expect from spaces, i.e., an isomorphism $K_0(*) \\cong \\mathbb{Z}$, given by taking a class $[X] \\in K_0(*)$ to the Euler characteristic of $X$.\n\nCan other K-groups be computed?",
   "clean_statement": "Algebraic K-theory of digraphs\n\nBased on the work of Carranza-Doherty-Kapulkin-Opie-Sarazola-Wong, there is a cofibration category structure on the category of digraphs whose weak equivalences are digraph maps that induce isomorphisms on all path homology groups. One can use it to define a Waldhausen category on (pointed) digraphs, which can then be used to take algebraic K-theory.\n\nCompute $K_0(*)$ and check if it agrees with what one might expect from spaces, i.e., an isomorphism $K_0(*) \\cong \\mathbb{Z}$, given by taking a class $[X] \\in K_0(*)$ to the Euler characteristic of $X$.\n\nCan other K-groups be computed?",
   "public_statement": "Algebraic K-theory of digraphs\n\nBased on the work of Carranza-Doherty-Kapulkin-Opie-Sarazola-Wong, there is a cofibration category structure on the category of digraphs whose weak equivalences are digraph maps that induce isomorphisms on all path homology groups. One can use it to define a Waldhausen category on (pointed) digraphs, which can then be used to take algebraic K-theory.\n\nCompute $K_0(*)$ and check if it agrees with what one might expect from spaces, i.e., an isomorphism $K_0(*) \\cong \\mathbb{Z}$, given by taking a class $[X] \\in K_0(*)$ to the Euler characteristic of $X$.\n\nCan other K-groups be computed?",
   "evidence": "The source record itself has no visible OCR corruption. It does leave the finiteness model implicit. The results below use the explicit finite/bounded model in Section 3; changing that model can change its algebraic $K$-theory.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 99,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0101": {
   "statement_status": "exact",
   "original_statement": "Eilenberg-Steenrod axioms\n\nDo the Eilenberg-Steenrod axioms or a variant thereof determine the homology theory uniquely for closure spaces (or graphs)?",
   "clean_statement": "Eilenberg-Steenrod axioms\n\nDo the Eilenberg-Steenrod axioms or a variant thereof determine the homology theory uniquely for closure spaces (or graphs)?",
   "public_statement": "Eilenberg-Steenrod axioms\n\nDo the Eilenberg-Steenrod axioms or a variant thereof determine the homology theory uniquely for closure spaces (or graphs)?",
   "evidence": "The canonical record is `aim-topology-notes.json`, zero-based index 100, workshop *Discrete and combinatorial homotopy theory*, section *Cech closure spaces*, problem 4.1. Its exact mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 100,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0102": {
   "statement_status": "exact",
   "original_statement": "Cubical homology\n\nIs the Excision Axiom true for cubical homology of Cech closure spaces with respect to interior covers?",
   "clean_statement": "Cubical homology\n\nIs the Excision Axiom true for cubical homology of Cech closure spaces with respect to interior covers?",
   "public_statement": "Cubical homology\n\nIs the Excision Axiom true for cubical homology of Cech closure spaces with respect to interior covers?",
   "evidence": "The canonical record is aim-topology-notes.json, zero-based index 101, workshop *Discrete and combinatorial homotopy theory*, section *Cech closure spaces*, problem 4.2. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 101,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0103": {
   "statement_status": "exact",
   "original_statement": "Manifolds/cobordism\n\nWhat are manifolds in Cech closure spaces? What is the notion of cobordism in Cech closure spaces?",
   "clean_statement": "Manifolds/cobordism\n\nWhat are manifolds in Cech closure spaces? What is the notion of cobordism in Cech closure spaces?",
   "public_statement": "Manifolds/cobordism\n\nWhat are manifolds in Cech closure spaces? What is the notion of cobordism in Cech closure spaces?",
   "evidence": "The record has no remarks or literature. There is no apparent OCR corruption beyond omission of the diacritic in “Cech.” The statement is deliberately open-ended: it asks for definitions, not for a theorem with fixed hypotheses. In particular, it does not specify smooth, topological, PL, or homology manifolds; finite versus arbitrary closure spaces; oriented versus unoriented cobordism; or a choice among the several products, intervals, and homology theories now known for closure spaces. Those choices cannot be silently supplied.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 102,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0104": {
   "statement_status": "corrected_verified",
   "original_statement": "Singular homology of graphs\n\nEvery simple graph can be viewed as a closure space as follows: given a graph $X = (V, E)$, we define a closure space by taking its underlying set to be $V$ and $c(A) = \\bigcup_{v \\in A} c(v)$, where $c(v) = \\{ w \\in V \\ | \\ \\{ v, w \\} \\in E\\}$.\nFor $n , k \\in \\mathbb{Z}$, define the graph $(\\mathbb{Z}/n, c_k)$ to have the set of vertices $\\mathbb{Z}/n = \\{ 0, 1, \\ldots, n-1\\}$ and an edge between $i$ and $j$ whenever $i$ and $j$ are no more than $k$ away.\n\nCompute $H^{sing}_*(\\mathbb{Z}/n, c_k)$. Is it isomorphic to the homomology of the clique complex of $(\\mathbb{Z}/n, c_k)$?",
   "clean_statement": "for \\(n\\ge3\\), \\(k\\ge0\\), compute the \\(I\\)-singular homology of the finite closure space associated with the reflexive symmetric graph \\(C_n^k\\), and compare it with \\(H_*(\\operatorname{Cl}(C_n^k);\\mathbb Z)\\). The degenerate cases \\(n=1,2\\) are recorded separately below. Negative \\(k\\) does not produce an extensive closure by this distance prescription.",
   "public_statement": "for \\(n\\ge3\\), \\(k\\ge0\\), compute the \\(I\\)-singular homology of the finite closure space associated with the reflexive symmetric graph \\(C_n^k\\), and compare it with \\(H_*(\\operatorname{Cl}(C_n^k);\\mathbb Z)\\). The degenerate cases \\(n=1,2\\) are recorded separately below. Negative \\(k\\) does not produce an extensive closure by this distance prescription.",
   "evidence": "Three substantive repairs are necessary for a literal well-posed reading. The final word “homomology” in the source is also an evident typographical error for “homology.” The recovered problem is therefore: for \\(n\\ge3\\), \\(k\\ge0\\), compute the \\(I\\)-singular homology of the finite closure space associated with the reflexive symmetric graph \\(C_n^k\\), and compare it with \\(H_*(\\operatorname{Cl}(C_n^k);\\mathbb Z)\\). The degenerate cases \\(n=1,2\\) are recorded separately below. Negative \\(k\\) does not produce an extensive closure by this distance prescription.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-topology-notes.json",
   "source_index": 103,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0105": {
   "statement_status": "exact",
   "original_statement": "Higher homotopy groups\n\nIn $\\times$-homotopy theory, what are the higher homotopy groups?",
   "clean_statement": "Higher homotopy groups\n\nIn $\\times$-homotopy theory, what are the higher homotopy groups?",
   "public_statement": "Higher homotopy groups\n\nIn $\\times$-homotopy theory, what are the higher homotopy groups?",
   "evidence": "The canonical AIM record is Problem 5.1 in the section “×-homotopy theory” of the workshop list *Discrete and combinatorial homotopy theory*. Its mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 104,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0106": {
   "statement_status": "exact",
   "original_statement": "Homology theory for graphs with loops\n\nIs there a good homology theory for graphs with loops? How does it interact with homotopy groups of graphs with loops (which also need to be defined). Does it satisfy a variant Eilenberg-Steenrod axioms?",
   "clean_statement": "Homology theory for graphs with loops\n\nIs there a good homology theory for graphs with loops? How does it interact with homotopy groups of graphs with loops (which also need to be defined). Does it satisfy a variant Eilenberg-Steenrod axioms?",
   "public_statement": "Homology theory for graphs with loops\n\nIs there a good homology theory for graphs with loops? How does it interact with homotopy groups of graphs with loops (which also need to be defined). Does it satisfy a variant Eilenberg-Steenrod axioms?",
   "evidence": "The canonical AIM record is number 5.2 in the section “\\(\\times\\)-homotopy theory” of the workshop *Discrete and combinatorial homotopy theory*. Its exact **problem** field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 105,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0107": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Equivariant discrete homotopy theory\n\nDevelop equivariant discrete homotopy theory.",
   "clean_statement": null,
   "public_statement": "Equivariant discrete homotopy theory\n\nDevelop equivariant discrete homotopy theory.",
   "evidence": "This choice is not asserted to be the intended unique reading of the AIM prompt. It is useful because finite $T_0$ spaces are equivalent to finite posets, order complexes give finite simplicial models, and all subgroup fixed-point data can be retained exactly.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 106,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0108": {
   "statement_status": "exact",
   "original_statement": "Geometric group theory\n\nFor the free group $F_k$ on $k$ generators, consider the generating set given by the closure of the standard generating set under conjugation. When considered with the word metric, are $F_2$ and $F_3$ quasi-isometric?",
   "clean_statement": "Geometric group theory\n\nFor the free group $F_k$ on $k$ generators, consider the generating set given by the closure of the standard generating set under conjugation. When considered with the word metric, are $F_2$ and $F_3$ quasi-isometric?",
   "public_statement": "Geometric group theory\n\nFor the free group $F_k$ on $k$ generators, consider the generating set given by the closure of the standard generating set under conjugation. When considered with the word metric, are $F_2$ and $F_3$ quasi-isometric?",
   "evidence": "The canonical record is Problem 6.2 in the “Other” section of the AIM workshop list *Discrete and combinatorial homotopy theory*. Its exact mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 107,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0109": {
   "statement_status": "exact",
   "original_statement": "Reversal and the bipolar filtration\n\nFor each $n \\in \\mathbb{N}$, find an $n$-bipolar topologically slice knot $K$ such that $K \\# -K^r$ is not smoothly slice.",
   "clean_statement": "Reversal and the bipolar filtration\n\nFor each $n \\in \\mathbb{N}$, find an $n$-bipolar topologically slice knot $K$ such that $K \\# -K^r$ is not smoothly slice.",
   "public_statement": "Reversal and the bipolar filtration\n\nFor each $n \\in \\mathbb{N}$, find an $n$-bipolar topologically slice knot $K$ such that $K \\# -K^r$ is not smoothly slice.",
   "evidence": "The record has no remarks or supplied literature. The current AIM workshop page and problem-list index were checked; the legacy direct `aimpl.org` problem URL returned an HTTP 502 response on 2026-08-13, so the exact problem text is preserved from the canonical record rather than silently reconstructed from that page. The mathematical text has no visible OCR error. The only ambiguity is conventional: some authors use “inverse” for reversal, whereas the modern concordance convention used below is that $K^r$ is string reversal and $-K$ is the group inverse. This convention agrees with Kim--Livingston and Kim. Also, the bipolar filtration is indexed by $n\\geq 0$ in its defining literature; thus $\\mathbb N$ is interpreted as including the nonnegative filtration levels. If the source intended $\\mathbb N=\\{1,2,\\ldots\\}$, the statements below simply omit level zero.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 108,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0110": {
   "statement_status": "exact",
   "original_statement": "4-genus and the bipolar filtration\n\nFor $n \\in \\mathbb{N}$, are there knots in $\\mathcal{T}_n$ with arbitrarily large smooth 4-genus?",
   "clean_statement": "4-genus and the bipolar filtration\n\nFor $n \\in \\mathbb{N}$, are there knots in $\\mathcal{T}_n$ with arbitrarily large smooth 4-genus?",
   "public_statement": "4-genus and the bipolar filtration\n\nFor $n \\in \\mathbb{N}$, are there knots in $\\mathcal{T}_n$ with arbitrarily large smooth 4-genus?",
   "evidence": "The record has no remarks and its literature field is empty. The canonical source URL is <http://aimpl.org/concordsliceknot/1/>. That page was unavailable during this run, but the exact question is independently reproduced as Question 4.18 in Ray's lecture notes [Ray]. There is no visible corruption in the canonical record.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 109,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0111": {
   "statement_status": "exact",
   "original_statement": "Characterization of 0-bipolar knots\n\nCan one characterize 0-positive, 0-negative, or 0-bipolar knots, either via the vanishing of invariants or geometrically?",
   "clean_statement": "Characterization of 0-bipolar knots\n\nCan one characterize 0-positive, 0-negative, or 0-bipolar knots, either via the vanishing of invariants or geometrically?",
   "public_statement": "Characterization of 0-bipolar knots\n\nCan one characterize 0-positive, 0-negative, or 0-bipolar knots, either via the vanishing of invariants or geometrically?",
   "evidence": "The canonical record is AIM Problem Lists, workshop *Smooth concordance classes of topologically slice knots*, section “Filtrations,” Problem 1.4:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 110,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0112": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Torsion and the bipolar filtration\n\nAre all 2-torsion knots 0-bipolar?",
   "clean_statement": null,
   "public_statement": "Torsion and the bipolar filtration\n\nAre all 2-torsion knots 0-bipolar?",
   "evidence": "There is a genuine scope ambiguity. Because the workshop concerns topologically slice knots and the surrounding questions use the induced filtration, the most plausible reading is",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 111,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0113": {
   "statement_status": "exact",
   "original_statement": "Bipolar quotients\n\nDoes $\\mathcal{T}_n/ \\mathcal{T}_{n+1}$ contain a $\\mathbb{Z}^{\\infty}$-summand?\nWhat about a $(\\mathbb{Z}/2\\mathbb{Z})^{\\infty}$ subgroup?",
   "clean_statement": "Bipolar quotients\n\nDoes $\\mathcal{T}_n/ \\mathcal{T}_{n+1}$ contain a $\\mathbb{Z}^{\\infty}$-summand?\nWhat about a $(\\mathbb{Z}/2\\mathbb{Z})^{\\infty}$ subgroup?",
   "public_statement": "Bipolar quotients\n\nDoes $\\mathcal{T}_n/ \\mathcal{T}_{n+1}$ contain a $\\mathbb{Z}^{\\infty}$-summand?\nWhat about a $(\\mathbb{Z}/2\\mathbb{Z})^{\\infty}$ subgroup?",
   "evidence": "The canonical record is zero-based record 112 of `aim-topology-notes.json`, from the AIM workshop *Smooth concordance classes of topologically slice knots*, section “Filtrations,” Problem 1.5. Its exact problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 112,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0114": {
   "statement_status": "exact",
   "original_statement": "Highly solvable knots with large 4-genera.\n\nFor arbitrary $n>2$ and $g>0$ prove that there exist $n$-solvable knots $K$ such that the topological 4-genus of $K$ is strictly more than $g$.",
   "clean_statement": "Highly solvable knots with large 4-genera.\n\nFor arbitrary $n>2$ and $g>0$ prove that there exist $n$-solvable knots $K$ such that the topological 4-genus of $K$ is strictly more than $g$.",
   "public_statement": "Highly solvable knots with large 4-genera.\n\nFor arbitrary $n>2$ and $g>0$ prove that there exist $n$-solvable knots $K$ such that the topological 4-genus of $K$ is strictly more than $g$.",
   "evidence": "No correction to the source statement is needed. The one convention that must not be silently changed is **integral** solvability: producing a merely rationally $n$-solvable knot would not by itself answer this record.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 113,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0115": {
   "statement_status": "exact",
   "original_statement": "Is homology slice the same as slice?\n\nIs there a non-smoothly slice knot $K$ in $S^3$ which is slice in a homology $B^4$?\n(i.e., is $\\phi_{smooth}$ non-injective?)",
   "clean_statement": "Is homology slice the same as slice?\n\nIs there a non-smoothly slice knot $K$ in $S^3$ which is slice in a homology $B^4$?\n(i.e., is $\\phi_{smooth}$ non-injective?)",
   "public_statement": "Is homology slice the same as slice?\n\nIs there a non-smoothly slice knot $K$ in $S^3$ which is slice in a homology $B^4$?\n(i.e., is $\\phi_{smooth}$ non-injective?)",
   "evidence": "The canonical AIM record, from the workshop *Smooth concordance classes of topologically slice knots*, section “Knots in homology spheres,” asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 114,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0116": {
   "statement_status": "exact",
   "original_statement": "Homotopy to slice knots\n\nLet $K$ be a knot in an integer homology sphere $Y$, and suppose that $Y$ bounds an integer homology ball $W$ such that $K$ is null-homotopic in $W$.\nMust $K$ be homotopic in $Y$ to a knot $K'$ which is smoothly slice in $W$?",
   "clean_statement": "Homotopy to slice knots\n\nLet $K$ be a knot in an integer homology sphere $Y$, and suppose that $Y$ bounds an integer homology ball $W$ such that $K$ is null-homotopic in $W$.\nMust $K$ be homotopic in $Y$ to a knot $K'$ which is smoothly slice in $W$?",
   "public_statement": "Homotopy to slice knots\n\nLet $K$ be a knot in an integer homology sphere $Y$, and suppose that $Y$ bounds an integer homology ball $W$ such that $K$ is null-homotopic in $W$.\nMust $K$ be homotopic in $Y$ to a knot $K'$ which is smoothly slice in $W$?",
   "evidence": "No corruption of the canonical statement was detected. The original AIM problem-list URL was unavailable during this run, but the wording and its two stated partial results are corroborated by Davis's paper [Dav20].",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 115,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0117": {
   "statement_status": "exact",
   "original_statement": "Concordance of knots in homology spheres.\n\nIs every knot in every integer homology sphere topologically concordant in some integer homology cobordism to a knot in $S^3$? (i.e. is $\\phi_{top}$ surjective?)",
   "clean_statement": "Concordance of knots in homology spheres.\n\nIs every knot in every integer homology sphere topologically concordant in some integer homology cobordism to a knot in $S^3$? (i.e. is $\\phi_{top}$ surjective?)",
   "public_statement": "Concordance of knots in homology spheres.\n\nIs every knot in every integer homology sphere topologically concordant in some integer homology cobordism to a knot in $S^3$? (i.e. is $\\phi_{top}$ surjective?)",
   "evidence": "The canonical input is zero-based record 116 of `aim-topology-notes.json`, from the AIM workshop *Smooth concordance classes of topologically slice knots*, section “Knots in homology spheres,” Problem 2.4. Its exact problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 116,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0118": {
   "statement_status": "exact",
   "original_statement": "Homology slice vs. contractibly slice\n\nFix an integer homology sphere $Y$ which bounds a contractible 4-manifold $W$.\nIs there a knot $K$ in $Y$ such that $K$ bounds a disc in some integer homology ball $W'$, but $K$ does not bound a disc in any contractible 4-manifold?",
   "clean_statement": "Homology slice vs. contractibly slice\n\nFix an integer homology sphere $Y$ which bounds a contractible 4-manifold $W$.\nIs there a knot $K$ in $Y$ such that $K$ bounds a disc in some integer homology ball $W'$, but $K$ does not bound a disc in any contractible 4-manifold?",
   "public_statement": "Homology slice vs. contractibly slice\n\nFix an integer homology sphere $Y$ which bounds a contractible 4-manifold $W$.\nIs there a knot $K$ in $Y$ such that $K$ bounds a disc in some integer homology ball $W'$, but $K$ does not bound a disc in any contractible 4-manifold?",
   "evidence": "The canonical AIM record (`aim-topology-notes.json`, record 117) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 117,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0119": {
   "statement_status": "exact",
   "original_statement": "Torsion in the cokernel\n\nDoes $coker(\\phi_{smooth})$ have torsion?",
   "clean_statement": "Torsion in the cokernel\n\nDoes $coker(\\phi_{smooth})$ have torsion?",
   "public_statement": "Torsion in the cokernel\n\nDoes $coker(\\phi_{smooth})$ have torsion?",
   "evidence": "The canonical AIM record, Problem 2.3 in the section “Knots in homology spheres” of the workshop list *Smooth concordance classes of topologically slice knots*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 118,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0120": {
   "statement_status": "exact",
   "original_statement": "Does the homotopy class of the infection curve determine the action of a pattern on topological concordance?\n\nDoes the Mazur pattern act by the identity on the topological concordance group?\nMore generally, if $\\eta_1$ and $\\eta_2$ are two homotopic curves in the complement of a slice knot $R$, must the induced maps $R_{\\eta_1}$ and $R_{\\eta_2}$ on the topological concordance group agree?",
   "clean_statement": "Does the homotopy class of the infection curve determine the action of a pattern on topological concordance?\n\nDoes the Mazur pattern act by the identity on the topological concordance group?\nMore generally, if $\\eta_1$ and $\\eta_2$ are two homotopic curves in the complement of a slice knot $R$, must the induced maps $R_{\\eta_1}$ and $R_{\\eta_2}$ on the topological concordance group agree?",
   "public_statement": "Does the homotopy class of the infection curve determine the action of a pattern on topological concordance?\n\nDoes the Mazur pattern act by the identity on the topological concordance group?\nMore generally, if $\\eta_1$ and $\\eta_2$ are two homotopic curves in the complement of a slice knot $R$, must the induced maps $R_{\\eta_1}$ and $R_{\\eta_2}$ on the topological concordance group agree?",
   "evidence": "The canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 119,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0121": {
   "statement_status": "exact",
   "original_statement": "2-torsion in $\\mathcal{C}$\n\nCan one detect whether $K$ is concordant to a negative amphichiral knot?",
   "clean_statement": "2-torsion in $\\mathcal{C}$\n\nCan one detect whether $K$ is concordant to a negative amphichiral knot?",
   "public_statement": "2-torsion in $\\mathcal{C}$\n\nCan one detect whether $K$ is concordant to a negative amphichiral knot?",
   "evidence": "The canonical record is AIM Problem Lists, workshop *Smooth concordance classes of topologically slice knots*, section “Structure and operators,” Problem 3.1:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 120,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0122": {
   "statement_status": "exact",
   "original_statement": "Injectivity of winding number 0 satellite operators\n\nIs there a pattern $P$ with winding number 0 such that $P(K)$ is slice if and only if $K$ is slice, in either category?",
   "clean_statement": "Injectivity of winding number 0 satellite operators\n\nIs there a pattern $P$ with winding number 0 such that $P(K)$ is slice if and only if $K$ is slice, in either category?",
   "public_statement": "Injectivity of winding number 0 satellite operators\n\nIs there a pattern $P$ with winding number 0 such that $P(K)$ is slice if and only if $K$ is slice, in either category?",
   "evidence": "The canonical AIM record is problem 3.4 in the workshop list *Smooth concordance classes of topologically slice knots*, section “Structure and operators.” Its mathematical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 121,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0123": {
   "statement_status": "exact",
   "original_statement": "Small stable 4-genera\n\nThe stable 4-genus of a knot $K$, in either category, is defined as $g_4^{st}(K)= \\lim_{n \\to \\infty} \\frac{ g_4(nK)}{n}$, see Livingston \\cite{MR2745668}.\n\nFind a knot $K$ with $0< g_4^{st}(K)< 1/2$.",
   "clean_statement": "Small stable 4-genera\n\nThe stable 4-genus of a knot $K$, in either category, is defined as $g_4^{st}(K)= \\lim_{n \\to \\infty} \\frac{ g_4(nK)}{n}$, see Livingston \\cite{MR2745668}.\n\nFind a knot $K$ with $0< g_4^{st}(K)< 1/2$.",
   "public_statement": "Small stable 4-genera\n\nThe stable 4-genus of a knot $K$, in either category, is defined as $g_4^{st}(K)= \\lim_{n \\to \\infty} \\frac{ g_4(nK)}{n}$, see Livingston \\cite{MR2745668}.\n\nFind a knot $K$ with $0< g_4^{st}(K)< 1/2$.",
   "evidence": "The canonical record is `aim-topology-notes.json`, zero-based index 122, from the AIM workshop *Smooth concordance classes of topologically slice knots*, section “Structure and operators,” Problem 3.2. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 122,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0124": {
   "statement_status": "exact",
   "original_statement": "Stable 4-genus and 2-torsion\n\nFind a knot $K$ which has infinite order in $\\mathcal{C}$ and yet which has $g_4^{st}(K)=0$.",
   "clean_statement": "Stable 4-genus and 2-torsion\n\nFind a knot $K$ which has infinite order in $\\mathcal{C}$ and yet which has $g_4^{st}(K)=0$.",
   "public_statement": "Stable 4-genus and 2-torsion\n\nFind a knot $K$ which has infinite order in $\\mathcal{C}$ and yet which has $g_4^{st}(K)=0$.",
   "evidence": "The canonical record is `aim-topology-notes.json`, record 123 (zero based), from the AIM workshop *Smooth concordance classes of topologically slice knots*, Section 3.3, “Structure and operators.” Its problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 123,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0125": {
   "statement_status": "exact",
   "original_statement": "Genus $g$ traces and concordance\n\nFor a knot $K$ in $S^3$, let $X^g(K)$ denote the 4-ball union $\\Sigma_g \\times D^2$, where $\\Sigma_g$ is a genus $g$ surface with one boundary component, attached to $B^4$ along a 0-framed neighborhood of $K$.\nAre there nonconcordant knots $K$ and $J$ such that $\\partial X^g(K)$ and $\\partial X^g(J)$ are integrally homology cobordant?",
   "clean_statement": "Genus $g$ traces and concordance\n\nFor a knot $K$ in $S^3$, let $X^g(K)$ denote the 4-ball union $\\Sigma_g \\times D^2$, where $\\Sigma_g$ is a genus $g$ surface with one boundary component, attached to $B^4$ along a 0-framed neighborhood of $K$.\nAre there nonconcordant knots $K$ and $J$ such that $\\partial X^g(K)$ and $\\partial X^g(J)$ are integrally homology cobordant?",
   "public_statement": "Genus $g$ traces and concordance\n\nFor a knot $K$ in $S^3$, let $X^g(K)$ denote the 4-ball union $\\Sigma_g \\times D^2$, where $\\Sigma_g$ is a genus $g$ surface with one boundary component, attached to $B^4$ along a 0-framed neighborhood of $K$.\nAre there nonconcordant knots $K$ and $J$ such that $\\partial X^g(K)$ and $\\partial X^g(J)$ are integrally homology cobordant?",
   "evidence": "The record adds: “homeomorphism of $\\partial X^g(K)$ and $\\partial X^g(J)$ is enough to imply that the knots $K$ and $J$ are isotopic.” The source page was unavailable during this run, but the wording has no visible OCR corruption. It does omit an important quantifier: is $g$ allowed to be zero, or is the intended question for a fixed positive $g$? The construction in Hayden--Piccirillo explicitly permits $g\\geq0$ [HP25], but their rigidity theorem assumes $g>0$. These cases must therefore be separated.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 124,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0126": {
   "statement_status": "exact",
   "original_statement": "Branched covers and concordance\n\nSuppose $K$ and $J$ are knots in $S^3$ such that for every prime power $q$, the $q$th cyclic branched covers $\\Sigma_q(K)$ and $\\Sigma_q(J)$ are rationally homology cobordant. Must $K$ and $J$ be concordant?",
   "clean_statement": "Branched covers and concordance\n\nSuppose $K$ and $J$ are knots in $S^3$ such that for every prime power $q$, the $q$th cyclic branched covers $\\Sigma_q(K)$ and $\\Sigma_q(J)$ are rationally homology cobordant. Must $K$ and $J$ be concordant?",
   "public_statement": "Branched covers and concordance\n\nSuppose $K$ and $J$ are knots in $S^3$ such that for every prime power $q$, the $q$th cyclic branched covers $\\Sigma_q(K)$ and $\\Sigma_q(J)$ are rationally homology cobordant. Must $K$ and $J$ be concordant?",
   "evidence": "The canonical record is `aim-topology-notes.json`, zero-based index 125, from the AIM workshop *Smooth concordance classes of topologically slice knots*, section “Determining concordance,” Problem 4.2. Its exact mathematical question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 125,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0127": {
   "statement_status": "exact",
   "original_statement": "Surgeries and concordance\n\nSuppose $K$ and $J$ are knots in $S^3$ such that for all $p/q \\in \\mathbb{Q}$, the Dehn surgeries $S^3_{p/q}(K)$ and $S^3_{p/q}(J)$ are integer homology cobordant. Must $K$ and $J$ be concordant?",
   "clean_statement": "Surgeries and concordance\n\nSuppose $K$ and $J$ are knots in $S^3$ such that for all $p/q \\in \\mathbb{Q}$, the Dehn surgeries $S^3_{p/q}(K)$ and $S^3_{p/q}(J)$ are integer homology cobordant. Must $K$ and $J$ be concordant?",
   "public_statement": "Surgeries and concordance\n\nSuppose $K$ and $J$ are knots in $S^3$ such that for all $p/q \\in \\mathbb{Q}$, the Dehn surgeries $S^3_{p/q}(K)$ and $S^3_{p/q}(J)$ are integer homology cobordant. Must $K$ and $J$ be concordant?",
   "evidence": "The exact canonical record is aim-topology-notes.json, zero-based record 126, from the 2019 AIM workshop *Smooth concordance classes of topologically slice knots*, Section 4.3:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 126,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0128": {
   "statement_status": "exact",
   "original_statement": "0-surgeries and concordance\n\nSuppose $K$ and $J$ are knots in $S^3$ such that there is an integer homology cobordism $W$ between $S^3_0(K)$ and $S^3_0(J)$ in which the positively oriented meridians $\\mu_K$ and $\\mu_J$ are freely homotopic. Must $K$ and $J$ be concordant?",
   "clean_statement": "0-surgeries and concordance\n\nSuppose $K$ and $J$ are knots in $S^3$ such that there is an integer homology cobordism $W$ between $S^3_0(K)$ and $S^3_0(J)$ in which the positively oriented meridians $\\mu_K$ and $\\mu_J$ are freely homotopic. Must $K$ and $J$ be concordant?",
   "public_statement": "0-surgeries and concordance\n\nSuppose $K$ and $J$ are knots in $S^3$ such that there is an integer homology cobordism $W$ between $S^3_0(K)$ and $S^3_0(J)$ in which the positively oriented meridians $\\mu_K$ and $\\mu_J$ are freely homotopic. Must $K$ and $J$ be concordant?",
   "evidence": "The canonical record has no visible corruption. The original AIM URL timed out during this run, so the wording was checked against the repository record and against Cha--Powell's Question 1.4. There is a category ambiguity that should not be erased: the workshop is about smooth concordance of topologically slice knots, while the remarks distinguish smooth from locally flat topological conclusions. Here CAT means either category when the argument works in both.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 127,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0129": {
   "statement_status": "exact",
   "original_statement": "Rational sliceness\n\nIs every knot in $S^3$ which is slice in some rational homology ball actually slice in some $\\mathbb{Z}[1/2]$-homology ball?",
   "clean_statement": "Rational sliceness\n\nIs every knot in $S^3$ which is slice in some rational homology ball actually slice in some $\\mathbb{Z}[1/2]$-homology ball?",
   "public_statement": "Rational sliceness\n\nIs every knot in $S^3$ which is slice in some rational homology ball actually slice in some $\\mathbb{Z}[1/2]$-homology ball?",
   "evidence": "The record comes from the 2019 AIM workshop *Smooth concordance classes of topologically slice knots*, Section 5, Problem 5.1. The statement has no visible OCR corruption. The legacy source URL was unavailable during this run.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 128,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0130": {
   "statement_status": "exact",
   "original_statement": "Difficult sliceness problems\n\nAre any of the following slice?\n\\begin{enumerate}\n\\item The $(2,1)$ cable of the figure-eight knot.\n\\item The positive Whitehead double of any left-handed torus knot.\n\\item The positive Whitehead double of the figure-eight knot.\n\\item The $(+,+,-)$ Whitehead double of the Borromean rings.\n\\end{enumerate}",
   "clean_statement": "Difficult sliceness problems\n\nAre any of the following slice?\n\\begin{enumerate}\n\\item The $(2,1)$ cable of the figure-eight knot.\n\\item The positive Whitehead double of any left-handed torus knot.\n\\item The positive Whitehead double of the figure-eight knot.\n\\item The $(+,+,-)$ Whitehead double of the Borromean rings.\n\\end{enumerate}",
   "public_statement": "Difficult sliceness problems\n\nAre any of the following slice?\n\\begin{enumerate}\n\\item The $(2,1)$ cable of the figure-eight knot.\n\\item The positive Whitehead double of any left-handed torus knot.\n\\item The positive Whitehead double of the figure-eight knot.\n\\item The $(+,+,-)$ Whitehead double of the Borromean rings.\n\\end{enumerate}",
   "evidence": "The source record is `aim-topology-notes.json`, record 129 (zero based), from the AIM workshop *Smooth concordance classes of topologically slice knots*, section “Miscellaneous,” Problem 5.2. Its exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 129,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0131": {
   "statement_status": "exact",
   "original_statement": "Attacks on slice-ribbon\n\nFind new ways to build knots which are by construction slice but not by construction ribbon.",
   "clean_statement": "Attacks on slice-ribbon\n\nFind new ways to build knots which are by construction slice but not by construction ribbon.",
   "public_statement": "Attacks on slice-ribbon\n\nFind new ways to build knots which are by construction slice but not by construction ribbon.",
   "evidence": "The canonical record is problem 5.3, “Attacks on slice-ribbon,” from the AIM workshop *Smooth concordance classes of topologically slice knots*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 130,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0132": {
   "statement_status": "exact",
   "original_statement": "Understanding topological slice discs\n\nExplicitly describe some non-smooth topologically slice disc.",
   "clean_statement": "Understanding topological slice discs\n\nExplicitly describe some non-smooth topologically slice disc.",
   "public_statement": "Understanding topological slice discs\n\nExplicitly describe some non-smooth topologically slice disc.",
   "evidence": "The canonical record is AIM Problem List 5.4 from the workshop *Smooth concordance classes of topologically slice knots*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 131,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0133": {
   "statement_status": "exact",
   "original_statement": "Topological ribbon concordance\n\nGordon asked whether two knots which are mutually ribbon concordant must be isotopic. This problem offers a topological version.\n\nDefine knots $K$ and $J$ to be topologically homotopy ribbon concordant, and write $K \\leq_{thrc} J$, if $K$ and $J$ are topologically concordant via an annulus $A$ such that the inclusion-induced map $\\pi_1(X_K) \\to \\pi_1(X_A)$ is a surjection and the inclusion-induced map $\\pi_1(X_J) \\to \\pi_1(X_A)$ is an injection.\nIs $\\leq_{thrc}$ a partial order on the collection of knots in $S^3$?",
   "clean_statement": "Topological ribbon concordance\n\nGordon asked whether two knots which are mutually ribbon concordant must be isotopic. This problem offers a topological version.\n\nDefine knots $K$ and $J$ to be topologically homotopy ribbon concordant, and write $K \\leq_{thrc} J$, if $K$ and $J$ are topologically concordant via an annulus $A$ such that the inclusion-induced map $\\pi_1(X_K) \\to \\pi_1(X_A)$ is a surjection and the inclusion-induced map $\\pi_1(X_J) \\to \\pi_1(X_A)$ is an injection.\nIs $\\leq_{thrc}$ a partial order on the collection of knots in $S^3$?",
   "public_statement": "Topological ribbon concordance\n\nGordon asked whether two knots which are mutually ribbon concordant must be isotopic. This problem offers a topological version.\n\nDefine knots $K$ and $J$ to be topologically homotopy ribbon concordant, and write $K \\leq_{thrc} J$, if $K$ and $J$ are topologically concordant via an annulus $A$ such that the inclusion-induced map $\\pi_1(X_K) \\to \\pi_1(X_A)$ is a surjection and the inclusion-induced map $\\pi_1(X_J) \\to \\pi_1(X_A)$ is an injection.\nIs $\\leq_{thrc}$ a partial order on the collection of knots in $S^3$?",
   "evidence": "This is Problem 5.5 in the “Miscellaneous” section of the 2019 AIM workshop *Smooth concordance classes of topologically slice knots*. The canonical text and nearby records show no OCR corruption. The legacy AIM URL timed out during this run, so the statement above is reproduced from the repository record rather than reverified on that page.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 132,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0134": {
   "statement_status": "exact",
   "original_statement": "Grope metric\n\nUnderstand the grope metric $d_1$ defined by Cochran-Harvey-Powell \\cite{MR3665407}.",
   "clean_statement": "Grope metric\n\nUnderstand the grope metric $d_1$ defined by Cochran-Harvey-Powell \\cite{MR3665407}.",
   "public_statement": "Grope metric\n\nUnderstand the grope metric $d_1$ defined by Cochran-Harvey-Powell \\cite{MR3665407}.",
   "evidence": "The exact AIM record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 133,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0135": {
   "statement_status": "exact",
   "original_statement": "Round Handle Problem\n\nGiven a linking number 0 link $L$, construct a 4-manifold $W$ by attaching a round handle to $B^4$ along each component of $L$. Must $L$ be topologically slice in $W$?",
   "clean_statement": "Round Handle Problem\n\nGiven a linking number 0 link $L$, construct a 4-manifold $W$ by attaching a round handle to $B^4$ along each component of $L$. Must $L$ be topologically slice in $W$?",
   "public_statement": "Round Handle Problem\n\nGiven a linking number 0 link $L$, construct a 4-manifold $W$ by attaching a round handle to $B^4$ along each component of $L$. Must $L$ be topologically slice in $W$?",
   "evidence": "The exact AIM record, problem 5.8 from the workshop *Smooth concordance classes of topologically slice knots*, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 134,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0136": {
   "statement_status": "exact",
   "original_statement": "Metrics on $\\mathcal{T}$\n\nConstruct an interesting (i.e. non-discrete, perhaps nice with respect to the bipolar filtration) metric on the set of topologically slice knots.",
   "clean_statement": "Metrics on $\\mathcal{T}$\n\nConstruct an interesting (i.e. non-discrete, perhaps nice with respect to the bipolar filtration) metric on the set of topologically slice knots.",
   "public_statement": "Metrics on $\\mathcal{T}$\n\nConstruct an interesting (i.e. non-discrete, perhaps nice with respect to the bipolar filtration) metric on the set of topologically slice knots.",
   "evidence": "The exact canonical record is AIM Problem List 5.7 from the 2019 workshop *Smooth concordance classes of topologically slice knots*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 135,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0137": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Give a Nielsen-Thurston classification type theorem for big mapping classes.",
   "clean_statement": null,
   "public_statement": "Give a Nielsen-Thurston classification type theorem for big mapping classes.",
   "evidence": "The source page was unavailable during this run, but the canonical JSON record and its nearby section records are internally coherent; no reconstruction of damaged mathematical notation was needed.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 136,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0138": {
   "statement_status": "exact",
   "original_statement": "Does every big mapping class either preserve a hyperbolic metric or preserve a lamination?",
   "clean_statement": "Does every big mapping class either preserve a hyperbolic metric or preserve a lamination?",
   "public_statement": "Does every big mapping class either preserve a hyperbolic metric or preserve a lamination?",
   "evidence": "The canonical AIM record (workshop *Surfaces of infinite type*, section “Classification of elements of big mapping class groups,” Problem 1.2) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 137,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0139": {
   "statement_status": "exact",
   "original_statement": "This was proposed as an \"easier version\" of the previous problem.\n\nGiven an (irreducible) big mapping class $f$ and a simple closed curve $\\alpha$, do $f^n(\\alpha)$ and $f^{-n}(\\alpha)$ converge to a lamination?",
   "clean_statement": "This was proposed as an \"easier version\" of the previous problem.\n\nGiven an (irreducible) big mapping class $f$ and a simple closed curve $\\alpha$, do $f^n(\\alpha)$ and $f^{-n}(\\alpha)$ converge to a lamination?",
   "public_statement": "This was proposed as an \"easier version\" of the previous problem.\n\nGiven an (irreducible) big mapping class $f$ and a simple closed curve $\\alpha$, do $f^n(\\alpha)$ and $f^{-n}(\\alpha)$ converge to a lamination?",
   "evidence": "The preceding record, Problem 1.1, supplies the workshop's provisional meaning of reducibility: a big mapping class is reducible when it preserves a possibly infinite discrete collection of pairwise disjoint essential simple closed curves and proper arcs, where discrete means no accumulation inside the surface. The source contains no OCR error in this record, but it does leave three mathematically consequential choices unstated:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 138,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0140": {
   "statement_status": "exact",
   "original_statement": "Describe all big mapping classes that preserve a train track on the surface.",
   "clean_statement": "Describe all big mapping classes that preserve a train track on the surface.",
   "public_statement": "Describe all big mapping classes that preserve a train track on the surface.",
   "evidence": "The canonical record from the 2019 AIM workshop *Surfaces of infinite type*, in the section “Classification of elements of big mapping class groups,” asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 139,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0141": {
   "statement_status": "exact",
   "original_statement": "Can we characterize big mapping classes whose mapping tori admit a complete hyperbolic metric?",
   "clean_statement": "Can we characterize big mapping classes whose mapping tori admit a complete hyperbolic metric?",
   "public_statement": "Can we characterize big mapping classes whose mapping tori admit a complete hyperbolic metric?",
   "evidence": "The exact AIM problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 140,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0142": {
   "statement_status": "exact",
   "original_statement": "Given a pseudo-Anosov acting on a translation surface, what can be said about the measured laminations in the stable and in the unstable direction? In general, \"How much of Thurston's notes goes through?\"",
   "clean_statement": "Given a pseudo-Anosov acting on a translation surface, what can be said about the measured laminations in the stable and in the unstable direction? In general, \"How much of Thurston's notes goes through?\"",
   "public_statement": "Given a pseudo-Anosov acting on a translation surface, what can be said about the measured laminations in the stable and in the unstable direction? In general, \"How much of Thurston's notes goes through?\"",
   "evidence": "This is Problem 1.6 in the section “Classification of elements of big mapping class groups” from the AIM workshop *Surfaces of infinite type* (source file aim-topology-notes.json, zero-based source index 141). There is no visible OCR corruption.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 141,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0143": {
   "statement_status": "corrected_verified",
   "original_statement": "Is there a dynamical description of an irreducible mapping class? For instance, given two simple closed curves $\\alpha$ and $\\beta$, what can be said about the asymptotics of $i(f^n(\\alpha,\\beta))$?",
   "clean_statement": "For a mapping class of an infinite-type surface that preserves no discrete system of disjoint essential curves and proper arcs, describe the dynamics of curve iterates and, in particular, the asymptotics of $I_n(\\alpha,\\beta)$.",
   "public_statement": "For a mapping class of an infinite-type surface that preserves no discrete system of disjoint essential curves and proper arcs, describe the dynamics of curve iterates and, in particular, the asymptotics of $I_n(\\alpha,\\beta)$.",
   "evidence": "The displayed expression is not well formed: geometric intersection number is a binary function, whereas the parentheses make $f^n$ appear to take the ordered pair $(\\alpha,\\beta)$. The archived AIM page contains the same malformed expression, so this is not an error introduced by the JSON extraction. I use the reconstructed quantity This reconstruction is forced by three checks. First, it is the standard two-curve intersection-growth sequence. Second, the alternative $i(f^n(\\alpha),f^n(\\beta))$ is identically $i(\\alpha,\\beta)$ because homeomorphisms preserve geometric intersection, contradicting the note about growth. Third, Hooper's paper discussed at the workshop proves exactly an asymptotic for $i(\\phi^n(\\alpha),\\beta)$. Thus the correction is documented here but the source record in `input.json` is left unchanged.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-topology-notes.json",
   "source_index": 142,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0144": {
   "statement_status": "exact",
   "original_statement": "Let $R$ be a hyperbolic metric on $S$ and $\\operatorname{Mod}(R)$ the Teichmüller modular group of $R$.\n\nIf $f \\in \\operatorname{Map}(S)$ satisfies that there exists $\\lambda$ such that $\\forall \\alpha, \\beta$ and $n \\in \\mathbb{N}$, $i(f^n(\\alpha),\\beta)< c \\cdot \\lambda^n$, is it true then that $f \\in \\operatorname{Mod}(R)$ for some $R$?",
   "clean_statement": "Let $R$ be a hyperbolic metric on $S$ and $\\operatorname{Mod}(R)$ the Teichmüller modular group of $R$.\n\nIf $f \\in \\operatorname{Map}(S)$ satisfies that there exists $\\lambda$ such that $\\forall \\alpha, \\beta$ and $n \\in \\mathbb{N}$, $i(f^n(\\alpha),\\beta)< c \\cdot \\lambda^n$, is it true then that $f \\in \\operatorname{Mod}(R)$ for some $R$?",
   "public_statement": "Let $R$ be a hyperbolic metric on $S$ and $\\operatorname{Mod}(R)$ the Teichmüller modular group of $R$.\n\nIf $f \\in \\operatorname{Map}(S)$ satisfies that there exists $\\lambda$ such that $\\forall \\alpha, \\beta$ and $n \\in \\mathbb{N}$, $i(f^n(\\alpha),\\beta)< c \\cdot \\lambda^n$, is it true then that $f \\in \\operatorname{Mod}(R)$ for some $R$?",
   "evidence": "The canonical record (AIM Problem Lists, *Surfaces of infinite type*, section “Teichmüller theory and other tools,” Problem 3.2) reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 143,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0145": {
   "statement_status": "exact",
   "original_statement": "Is there a natural bordification/boundary of Teichmüller space for big surfaces?",
   "clean_statement": "Is there a natural bordification/boundary of Teichmüller space for big surfaces?",
   "public_statement": "Is there a natural bordification/boundary of Teichmüller space for big surfaces?",
   "evidence": "The canonical AIM record (Topology, *Surfaces of infinite type*, §3.3, source index 144; source page <http://aimpl.org/genusinfinity/3/>) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 144,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0146": {
   "statement_status": "exact",
   "original_statement": "Are there general methods to take limits, for going from finite-type to infinite-type. What properties hold under limits?",
   "clean_statement": "Are there general methods to take limits, for going from finite-type to infinite-type. What properties hold under limits?",
   "public_statement": "Are there general methods to take limits, for going from finite-type to infinite-type. What properties hold under limits?",
   "evidence": "The canonical AIM record is problem 3.4 in the section “Teichmüller theory and other tools” of the 2019 workshop *Surfaces of infinite type*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 145,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0147": {
   "statement_status": "exact",
   "original_statement": "Consider a geometric invariant for which we know the asymptotics in genus. This describes the generic shape of a surface of high genus. Can we use this information to obtain information about the generic shape of an infinite-genus surface?",
   "clean_statement": "Consider a geometric invariant for which we know the asymptotics in genus. This describes the generic shape of a surface of high genus. Can we use this information to obtain information about the generic shape of an infinite-genus surface?",
   "public_statement": "Consider a geometric invariant for which we know the asymptotics in genus. This describes the generic shape of a surface of high genus. Can we use this information to obtain information about the generic shape of an infinite-genus surface?",
   "evidence": "The canonical AIM record, from *Surfaces of infinite type*, section “Teichmüller theory and other tools,” Problem 3.5, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 146,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0148": {
   "statement_status": "exact",
   "original_statement": "In the finite-type setting, a mapping class has a representative that's a hyperbolic isometry for some hyperbolic metric exactly when it is periodic.\n\nCharacterize the big mapping classes that can be realized by hyperbolic isometries of some hyperbolic metric on the surface.",
   "clean_statement": "In the finite-type setting, a mapping class has a representative that's a hyperbolic isometry for some hyperbolic metric exactly when it is periodic.\n\nCharacterize the big mapping classes that can be realized by hyperbolic isometries of some hyperbolic metric on the surface.",
   "public_statement": "In the finite-type setting, a mapping class has a representative that's a hyperbolic isometry for some hyperbolic metric exactly when it is periodic.\n\nCharacterize the big mapping classes that can be realized by hyperbolic isometries of some hyperbolic metric on the surface.",
   "evidence": "The canonical AIM record (Topology, *Surfaces of infinite type*, §3.1, source index 147; source page <http://aimpl.org/genusinfinity/3/>) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 147,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0149": {
   "statement_status": "exact",
   "original_statement": "Come up with reasonable counting problems and analogues of geodesic currents.\n\nA possible counting problem is as follows. Fix a hyperbolic metric $R$ on the blooming $3$--pod surface. Fix a curve $\\alpha$ and a basepoint $x \\in R$, then does there exists $p(L)$ such that\n \\[\n \\frac{ | \\phi(\\alpha) : \\ell_R(\\phi(\\alpha)) < L | }{p(L)} \\simeq\n \\text{Vol}(B_L(x))?\n \\]",
   "clean_statement": "Come up with reasonable counting problems and analogues of geodesic currents.\n\nA possible counting problem is as follows. Fix a hyperbolic metric $R$ on the blooming $3$--pod surface. Fix a curve $\\alpha$ and a basepoint $x \\in R$, then does there exists $p(L)$ such that\n \\[\n \\frac{ | \\phi(\\alpha) : \\ell_R(\\phi(\\alpha)) < L | }{p(L)} \\simeq\n \\text{Vol}(B_L(x))?\n \\]",
   "public_statement": "Come up with reasonable counting problems and analogues of geodesic currents.\n\nA possible counting problem is as follows. Fix a hyperbolic metric $R$ on the blooming $3$--pod surface. Fix a curve $\\alpha$ and a basepoint $x \\in R$, then does there exists $p(L)$ such that\n \\[\n \\frac{ | \\phi(\\alpha) : \\ell_R(\\phi(\\alpha)) < L | }{p(L)} \\simeq\n \\text{Vol}(B_L(x))?\n \\]",
   "evidence": "The canonical AIM record, problem 3.6 in the section “Teichmüller theory and other tools” of the 2019 workshop *Surfaces of infinite type*, says exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 148,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0150": {
   "statement_status": "exact",
   "original_statement": "Give a complete quasi-isometry classification (in the sense of Rosendal) of big mapping class groups. For instance, let $S$ be the ladder surface and let $S'$ be the $3$--pod blooming surface. Is $\\operatorname{Map}(S)$ quasi-isometric to $\\operatorname{Map}(S')$?",
   "clean_statement": "Give a complete quasi-isometry classification (in the sense of Rosendal) of big mapping class groups. For instance, let $S$ be the ladder surface and let $S'$ be the $3$--pod blooming surface. Is $\\operatorname{Map}(S)$ quasi-isometric to $\\operatorname{Map}(S')$?",
   "public_statement": "Give a complete quasi-isometry classification (in the sense of Rosendal) of big mapping class groups. For instance, let $S$ be the ladder surface and let $S'$ be the $3$--pod blooming surface. Is $\\operatorname{Map}(S)$ quasi-isometric to $\\operatorname{Map}(S')$?",
   "evidence": "The canonical record (`aim-topology-notes.json`, index 149) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 149,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0151": {
   "statement_status": "exact",
   "original_statement": "Give an example of a pair $(S, G)$ of a surface and a countable group such that $G$ is not a subgroup of $\\operatorname{Map}(S)$. What obstructions exist barring countable groups from being subgroups of families of big mapping class groups?",
   "clean_statement": "Give an example of a pair $(S, G)$ of a surface and a countable group such that $G$ is not a subgroup of $\\operatorname{Map}(S)$. What obstructions exist barring countable groups from being subgroups of families of big mapping class groups?",
   "public_statement": "Give an example of a pair $(S, G)$ of a surface and a countable group such that $G$ is not a subgroup of $\\operatorname{Map}(S)$. What obstructions exist barring countable groups from being subgroups of families of big mapping class groups?",
   "evidence": "The canonical record (`aim-topology-notes.json`, zero-based index 150) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 150,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0152": {
   "statement_status": "exact",
   "original_statement": "For every $S$ does there exist a countable group $G$ such that $G$ is not a subgroup of $\\operatorname{Map}(S)$?",
   "clean_statement": "For every $S$ does there exist a countable group $G$ such that $G$ is not a subgroup of $\\operatorname{Map}(S)$?",
   "public_statement": "For every $S$ does there exist a countable group $G$ such that $G$ is not a subgroup of $\\operatorname{Map}(S)$?",
   "evidence": "The exact AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 151,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0153": {
   "statement_status": "exact",
   "original_statement": "Describe the compact subsets/subgroups of $\\operatorname{Map}(S)$. Also describe all bounded subsets/subgroups of $\\operatorname{Map}(S)$.",
   "clean_statement": "Describe the compact subsets/subgroups of $\\operatorname{Map}(S)$. Also describe all bounded subsets/subgroups of $\\operatorname{Map}(S)$.",
   "public_statement": "Describe the compact subsets/subgroups of $\\operatorname{Map}(S)$. Also describe all bounded subsets/subgroups of $\\operatorname{Map}(S)$.",
   "evidence": "The canonical record is AIM-TOPOLOGY-0153, Problem 4.15 from the AIM workshop *Surfaces of infinite type*, section “Algebraic and topological properties of big mapping class groups.” Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 152,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0154": {
   "statement_status": "exact",
   "original_statement": "Are there \"nice\" or \"natural\" generators for $\\operatorname{Map}(S)$?",
   "clean_statement": "Are there \"nice\" or \"natural\" generators for $\\operatorname{Map}(S)$?",
   "public_statement": "Are there \"nice\" or \"natural\" generators for $\\operatorname{Map}(S)$?",
   "evidence": "The canonical record is problem 4.2 in the AIM workshop list *Surfaces of infinite type*, section “Algebraic and topological properties of big mapping class groups”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 153,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0155": {
   "statement_status": "exact",
   "original_statement": "Can we see the topology of $S$ in the algebraic structure of $\\operatorname{Map}(S)$?",
   "clean_statement": "Can we see the topology of $S$ in the algebraic structure of $\\operatorname{Map}(S)$?",
   "public_statement": "Can we see the topology of $S$ in the algebraic structure of $\\operatorname{Map}(S)$?",
   "evidence": "The exact canonical AIM record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 154,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0156": {
   "statement_status": "exact",
   "original_statement": "Does $\\operatorname{Map}(S)$ have a canonical word metric?",
   "clean_statement": "Does $\\operatorname{Map}(S)$ have a canonical word metric?",
   "public_statement": "Does $\\operatorname{Map}(S)$ have a canonical word metric?",
   "evidence": "The repository text is syntactically intact, so no OCR correction is needed. The original AIM page was unavailable during this run. There is, however, a substantive ambiguity: a **word metric** requires an algebraic generating set, while big mapping class groups are typically discussed using topological generation and the quotient compact-open topology. Also, “canonical” may mean a distinguished metric, a metric independent up to bi-Lipschitz equivalence, or only a canonical quasi-isometry class. These readings have different answers.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 155,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0157": {
   "statement_status": "exact",
   "original_statement": "Are there any finite non-abelian quotients of $\\operatorname{PMap}(S)$ or $\\operatorname{Map}(S)$?",
   "clean_statement": "Are there any finite non-abelian quotients of $\\operatorname{PMap}(S)$ or $\\operatorname{Map}(S)$?",
   "public_statement": "Are there any finite non-abelian quotients of $\\operatorname{PMap}(S)$ or $\\operatorname{Map}(S)$?",
   "evidence": "There is no apparent OCR corruption. There is, however, an important quantifier ambiguity. The record does not specify the infinite-type surface $S$, whether the question is existential in $S$ or is meant for a fixed/arbitrary $S$, or whether a quotient homomorphism must be continuous for the usual quotient compact--open topology. These distinctions change the answer. This report treats quotients as abstract group quotients unless “continuous” is stated. Surfaces are connected, orientable, and second countable; homeomorphisms fix the boundary pointwise. Boundaryless hypotheses are stated where used.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 156,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0158": {
   "statement_status": "exact",
   "original_statement": "When $S$ is a finite genus surface, there are forgetful homomorphisms from $\\operatorname{PMap}(S)$ to the mapping class groups of finite-type surfaces.\nThose finite-type mapping class groups are residually finite, so there are many further quotients. What other finite quotients can we have for $\\operatorname{Map}(S)$ and $\\operatorname{PMap}(S)$?",
   "clean_statement": "When $S$ is a finite genus surface, there are forgetful homomorphisms from $\\operatorname{PMap}(S)$ to the mapping class groups of finite-type surfaces.\nThose finite-type mapping class groups are residually finite, so there are many further quotients. What other finite quotients can we have for $\\operatorname{Map}(S)$ and $\\operatorname{PMap}(S)$?",
   "public_statement": "When $S$ is a finite genus surface, there are forgetful homomorphisms from $\\operatorname{PMap}(S)$ to the mapping class groups of finite-type surfaces.\nThose finite-type mapping class groups are residually finite, so there are many further quotients. What other finite quotients can we have for $\\operatorname{Map}(S)$ and $\\operatorname{PMap}(S)$?",
   "evidence": "The canonical record is AIM Problem List 4.4 from the 2019 workshop *Surfaces of infinite type* (source file `aim-topology-notes.json`, record 157). Its exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 157,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0159": {
   "statement_status": "exact",
   "original_statement": "Let $S$ have infinite genus and no punctures, with finitely many ends accumulated by genus. Must every homomorphism factor through an abelian subgroup? Can you forget ends accumulated by genus?\n\nAre surfaces with infinite genus and no punctures, with finitely many ends accumulated by genus quasi-isometric?",
   "clean_statement": "Let $S$ have infinite genus and no punctures, with finitely many ends accumulated by genus. Must every homomorphism factor through an abelian subgroup? Can you forget ends accumulated by genus?\n\nAre surfaces with infinite genus and no punctures, with finitely many ends accumulated by genus quasi-isometric?",
   "public_statement": "Let $S$ have infinite genus and no punctures, with finitely many ends accumulated by genus. Must every homomorphism factor through an abelian subgroup? Can you forget ends accumulated by genus?\n\nAre surfaces with infinite genus and no punctures, with finitely many ends accumulated by genus quasi-isometric?",
   "evidence": "The canonical record (`aim-topology-notes.json`, zero-based index 158) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 158,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0160": {
   "statement_status": "exact",
   "original_statement": "Is every injective self-homomorphism from $\\operatorname{Map}(S)$ an isomorphism? That is to say, for which $S$ is $\\operatorname{Map}(S)$ co-Hopfian (or Hopfian)?",
   "clean_statement": "Is every injective self-homomorphism from $\\operatorname{Map}(S)$ an isomorphism? That is to say, for which $S$ is $\\operatorname{Map}(S)$ co-Hopfian (or Hopfian)?",
   "public_statement": "Is every injective self-homomorphism from $\\operatorname{Map}(S)$ an isomorphism? That is to say, for which $S$ is $\\operatorname{Map}(S)$ co-Hopfian (or Hopfian)?",
   "evidence": "The canonical record (`aim-topology-notes.json`, zero-based index 159) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 159,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0161": {
   "statement_status": "exact",
   "original_statement": "Lattices don't map into $\\operatorname{Mod}(S_g)$ for a compact surface $S_g$ of genus $g$.\n\nDo lattices map into $\\operatorname{Map}(S)$ or $\\overline {\\operatorname{PMap}_c(S)}$? In particular, can higher-rank lattices map into these groups?",
   "clean_statement": "Lattices don't map into $\\operatorname{Mod}(S_g)$ for a compact surface $S_g$ of genus $g$.\n\nDo lattices map into $\\operatorname{Map}(S)$ or $\\overline {\\operatorname{PMap}_c(S)}$? In particular, can higher-rank lattices map into these groups?",
   "public_statement": "Lattices don't map into $\\operatorname{Mod}(S_g)$ for a compact surface $S_g$ of genus $g$.\n\nDo lattices map into $\\operatorname{Map}(S)$ or $\\overline {\\operatorname{PMap}_c(S)}$? In particular, can higher-rank lattices map into these groups?",
   "evidence": "There is no apparent OCR corruption, but “map” is ambiguous. Every group has a trivial homomorphism into every mapping class group. The meaningful alternatives are:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 160,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0162": {
   "statement_status": "exact",
   "original_statement": "For surfaces for which this is not known, can we produce quasimorphisms using actions of $\\operatorname{Map}(S)$?",
   "clean_statement": "For surfaces for which this is not known, can we produce quasimorphisms using actions of $\\operatorname{Map}(S)$?",
   "public_statement": "For surfaces for which this is not known, can we produce quasimorphisms using actions of $\\operatorname{Map}(S)$?",
   "evidence": "The canonical record is AIM-TOPOLOGY-0162, record 161 (zero-based) of `aim-topology-notes.json`, from the AIM workshop *Surfaces of infinite type*, Section 4, item 4.6. Its problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 161,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0163": {
   "statement_status": "exact",
   "original_statement": "No big mapping class groups satisfies the strong Tits alternative, but it is not known about the classical Tits Alternatives.\n\nAre there any big mapping class groups that satisfy the classical Tits alternative?",
   "clean_statement": "No big mapping class groups satisfies the strong Tits alternative, but it is not known about the classical Tits Alternatives.\n\nAre there any big mapping class groups that satisfy the classical Tits alternative?",
   "public_statement": "No big mapping class groups satisfies the strong Tits alternative, but it is not known about the classical Tits Alternatives.\n\nAre there any big mapping class groups that satisfy the classical Tits alternative?",
   "evidence": "The canonical record (`aim-topology-notes.json`, zero-based index 162) says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 162,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0164": {
   "statement_status": "exact",
   "original_statement": "What about bounded cohomology? In $\\mathbb{R}^2-C$, is it true that every subgroup of $\\operatorname{Map}(S)$ has either infinite-dimensional space of quasimorphisms or is amenable?",
   "clean_statement": "What about bounded cohomology? In $\\mathbb{R}^2-C$, is it true that every subgroup of $\\operatorname{Map}(S)$ has either infinite-dimensional space of quasimorphisms or is amenable?",
   "public_statement": "What about bounded cohomology? In $\\mathbb{R}^2-C$, is it true that every subgroup of $\\operatorname{Map}(S)$ has either infinite-dimensional space of quasimorphisms or is amenable?",
   "evidence": "The canonical record is AIM Problem List 4.7 from the 2019 workshop *Surfaces of infinite type*, section “Algebraic and topological properties of big mapping class groups.” Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 163,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0165": {
   "statement_status": "exact",
   "original_statement": "Given a homomorphism $f: \\operatorname{Map}(S) \\to \\operatorname{Map}(S')$, does $f$ preserve the notion of being compactly supported? That is, does $f$ send compactly supported elements to compactly supported elements.",
   "clean_statement": "Given a homomorphism $f: \\operatorname{Map}(S) \\to \\operatorname{Map}(S')$, does $f$ preserve the notion of being compactly supported? That is, does $f$ send compactly supported elements to compactly supported elements.",
   "public_statement": "Given a homomorphism $f: \\operatorname{Map}(S) \\to \\operatorname{Map}(S')$, does $f$ preserve the notion of being compactly supported? That is, does $f$ send compactly supported elements to compactly supported elements.",
   "evidence": "The text is grammatically and mathematically coherent; no OCR correction is needed. There is, however, a terminology issue that matters. In the literature cited by the record, **finite support** means support on a finite-type domain. Literal **compact support** means that a representative is the identity outside a compact subset. These notions agree for pure finitely-supported mapping classes, but can differ in the full mapping class group: a half-twist interchanging two isolated punctures has finite support but is not compactly supported, since a compactly supported homeomorphism fixes every end.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 164,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0166": {
   "statement_status": "exact",
   "original_statement": "For any homomorphism from $\\operatorname{Map}(S)$ to a separable topological group $G$, is this automatically continuous?",
   "clean_statement": "For any homomorphism from $\\operatorname{Map}(S)$ to a separable topological group $G$, is this automatically continuous?",
   "public_statement": "For any homomorphism from $\\operatorname{Map}(S)$ to a separable topological group $G$, is this automatically continuous?",
   "evidence": "The canonical AIM record is Problem 4.8 in the workshop *Surfaces of infinite type*, section “Algebraic and topological properties of big mapping class groups.” Its exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 165,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0167": {
   "statement_status": "exact",
   "original_statement": "Which mapping classes are realized by affine automorphisms on some translation surface?",
   "clean_statement": "Which mapping classes are realized by affine automorphisms on some translation surface?",
   "public_statement": "Which mapping classes are realized by affine automorphisms on some translation surface?",
   "evidence": "The repository record is internally legible and shows no OCR corruption. The listed AIM page was unavailable during this run (HTTP 502), so the wording was checked against the exact repository record and the AIM workshop report rather than silently reconstructed. Nearby records confirm that this is an infinite-type question, followed by questions about flat versus hyperbolic geometry and Veech groups.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 166,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0168": {
   "statement_status": "exact",
   "original_statement": "Can you relate the flat and hyperbolic geometry of a given surface? That is, show how to uniformize the flat structure. For instance, describe (up to quasi-isometry) the hyperbolic structure on the Loch Ness monster corresponding to the flat structure given by the Chamanara surface.",
   "clean_statement": "Can you relate the flat and hyperbolic geometry of a given surface? That is, show how to uniformize the flat structure. For instance, describe (up to quasi-isometry) the hyperbolic structure on the Loch Ness monster corresponding to the flat structure given by the Chamanara surface.",
   "public_statement": "Can you relate the flat and hyperbolic geometry of a given surface? That is, show how to uniformize the flat structure. For instance, describe (up to quasi-isometry) the hyperbolic structure on the Loch Ness monster corresponding to the flat structure given by the Chamanara surface.",
   "evidence": "The canonical AIM record is Problem 5.2 in the section *Infinite translation surfaces* of the workshop list *Surfaces of infinite type*. Its exact problem text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 167,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0169": {
   "statement_status": "exact",
   "original_statement": "In which cases do we have a Veech dichotomy on infinite translation surfaces? Are there any cases at all?",
   "clean_statement": "In which cases do we have a Veech dichotomy on infinite translation surfaces? Are there any cases at all?",
   "public_statement": "In which cases do we have a Veech dichotomy on infinite translation surfaces? Are there any cases at all?",
   "evidence": "The accompanying literature note says: “We know that there are cases in which the Veech dichotomy does not hold.” The source is the AIM problem-list page <http://aimpl.org/genusinfinity/5/>. There is no apparent OCR corruption, but there is an important mathematical ambiguity: on an infinite-area surface there is no normalized area probability measure, so the phrase “uniquely ergodic” is not canonical.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 168,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0170": {
   "statement_status": "exact",
   "original_statement": "Which Veech groups arise from translation structures on the ladder surface?",
   "clean_statement": "Which Veech groups arise from translation structures on the ladder surface?",
   "public_statement": "Which Veech groups arise from translation structures on the ladder surface?",
   "evidence": "The canonical AIM record (`aim-topology-notes.json`, zero-based index 169) asks exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 169,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0171": {
   "statement_status": "exact",
   "original_statement": "Is the billiard flow on the triangle with side lengths $3$, $4$, and $5$ ergodic?",
   "clean_statement": "Is the billiard flow on the triangle with side lengths $3$, $4$, and $5$ ergodic?",
   "public_statement": "Is the billiard flow on the triangle with side lengths $3$, $4$, and $5$ ergodic?",
   "evidence": "There are no remarks or literature entries in the record. The text is legible and has no apparent OCR error. The listed problem page http://aimpl.org/genusinfinity/5/ returned HTTP 502 during this run. The repository wording was therefore preserved exactly and checked against nearby records and the AIM workshop report. The report confirms that the section grew out of discussions of infinite translation surfaces, but does not specify the measure-theoretic convention for this problem.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 170,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0172": {
   "statement_status": "exact",
   "original_statement": "What conditions can we put on an infinite translation surface to ensure that it contains a closed geodesic?",
   "clean_statement": "What conditions can we put on an infinite translation surface to ensure that it contains a closed geodesic?",
   "public_statement": "What conditions can we put on an infinite translation surface to ensure that it contains a closed geodesic?",
   "evidence": "The canonical AIM record is problem 5.6 in the section “Infinite translation surfaces” of the workshop *Surfaces of infinite type*. Its exact text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 171,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0173": {
   "statement_status": "exact",
   "original_statement": "Problem 1.1 (D. Dumas, K. Rafi). Does the Thurston norm at X ∈ T (S)determines X?",
   "clean_statement": "Problem 1.1 (D. Dumas, K. Rafi). Does the Thurston norm at X ∈ T (S)determines X?",
   "public_statement": "Problem 1.1 (D. Dumas, K. Rafi). Does the Thurston norm at X ∈ T (S)determines X?",
   "evidence": "The canonical record (`aim-topology-notes.json`, zero-based index 172) preserves the workshop text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 172,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0174": {
   "statement_status": "exact",
   "original_statement": "Problem 1.1 is motivated by Royden's proof of his celebrated theorem that (roughly speaking) the isometry group of Teichm¨ uller space endowed with the Teichm¨ uller metric is the mapping class group.........................................................................",
   "clean_statement": "Problem 1.1 is motivated by Royden's proof of his celebrated theorem that (roughly speaking) the isometry group of Teichm¨ uller space endowed with the Teichm¨ uller metric is the mapping class group.........................................................................",
   "public_statement": "Problem 1.1 is motivated by Royden's proof of his celebrated theorem that (roughly speaking) the isometry group of Teichm¨ uller space endowed with the Teichm¨ uller metric is the mapping class group.........................................................................",
   "evidence": "The canonical record is preserved exactly in `input.json`. Its `problem` field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 173,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0175": {
   "statement_status": "exact",
   "original_statement": "Problem 1.2 (K. Rafi). What properties of the hyperbolic surface are de-termined by the Thurston infinitesimal norm?........................................................................",
   "clean_statement": "Problem 1.2 (K. Rafi). What properties of the hyperbolic surface are de-termined by the Thurston infinitesimal norm?........................................................................",
   "public_statement": "Problem 1.2 (K. Rafi). What properties of the hyperbolic surface are de-termined by the Thurston infinitesimal norm?........................................................................",
   "evidence": "The canonical record (`aim-topology-notes.json`, zero-based index 174) contains:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 174,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0176": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 1.3 (F. Gu´ ertitaud). Does the unit sphere in T ∗ \n\n> X\n\nT (S) endowed with the Thurston norm determine the global behaviour of stretch lines or geodesics. As shown by Thurston, the behavior of lengths of laminations infinites-imally in measured lamination space gives good global coordinates for Te-ichm¨ uller space.........................................................................",
   "clean_statement": null,
   "public_statement": "Problem 1.3 (F. Gu´ ertitaud). Does the unit sphere in T ∗\n\n> X\n\nT (S) endowed with the Thurston norm determine the global behaviour of stretch lines or geodesics. As shown by Thurston, the behavior of lengths of laminations infinites-imally in measured lamination space gives good global coordinates for Te-ichm¨ uller space.........................................................................",
   "evidence": "The canonical JSON record reads, including its extraction errors:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 175,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0177": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 1.4. Is each local isometry of Teichm¨ uller space with the Thurston metric induced by an element of the extended mapping class group? It was recently observed by Walsh that the horofunction compactification of Teichm¨ uller space with the Thurston metric is naturally identified with Thurston's compactification.........................................................................",
   "clean_statement": "**Problem 1.4.** Is each local isometry of Teichmüller space with the\nThurston metric induced by an element of the extended mapping class group?",
   "public_statement": "Problem 1.4. Is each local isometry of Teichm¨ uller space with the Thurston metric induced by an element of the extended mapping class group? It was recently observed by Walsh that the horofunction compactification of Teichm¨ uller space with the Thurston metric is naturally identified with Thurston's compactification.........................................................................",
   "evidence": "The canonical record is Problem 1.4 of the 2014 AIM list *Problems on Thurston Metric*. Direct inspection of the source PDF gives:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-topology-notes.json",
   "source_index": 176,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0178": {
   "statement_status": "exact",
   "original_statement": "Problem 1.5 (D. Dumas). Is there any sense in which the Thurston metric is compatible with the complex structure of Teichm¨ uller space? We may ask the same question for symplectic structure. \n\n> Date: April 21, 2014.\n> 12WEIXU SU\n\n2. Geodesics of Thurston metric \n\nAny two points in Teichm¨ uller space can be joined by a geodesic path that is a concatenation of stretch segments.",
   "clean_statement": "Problem 1.5 (D. Dumas). Is there any sense in which the Thurston metric is compatible with the complex structure of Teichm¨ uller space? We may ask the same question for symplectic structure.\n\n> Date: April 21, 2014.\n> 12WEIXU SU\n\n2. Geodesics of Thurston metric\n\nAny two points in Teichm¨ uller space can be joined by a geodesic path that is a concatenation of stretch segments.",
   "public_statement": "Problem 1.5 (D. Dumas). Is there any sense in which the Thurston metric is compatible with the complex structure of Teichm¨ uller space? We may ask the same question for symplectic structure.\n\n> Date: April 21, 2014.\n> 12WEIXU SU\n\n2. Geodesics of Thurston metric\n\nAny two points in Teichm¨ uller space can be joined by a geodesic path that is a concatenation of stretch segments.",
   "evidence": "The canonical JSON record contains page-boundary spillover. Inspection of page 1 of the original AIM PDF gives the complete problem as exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 177,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0179": {
   "statement_status": "exact",
   "original_statement": "Problem 2.1. Is there an algorithm to find the maximally stretch lamina-tion?........................................................................ Generically, a Thurston geodesic connecting two points in Teichm¨ uller space is not unique.",
   "clean_statement": "Problem 2.1. Is there an algorithm to find the maximally stretch lamina-tion?........................................................................ Generically, a Thurston geodesic connecting two points in Teichm¨ uller space is not unique.",
   "public_statement": "Problem 2.1. Is there an algorithm to find the maximally stretch lamina-tion?........................................................................ Generically, a Thurston geodesic connecting two points in Teichm¨ uller space is not unique.",
   "evidence": "The source is the AIM problem list *Problems on Thurston metric*, dated April 21, 2014, from the workshop “Lipschitz metric on Teichmüller space.” The PDF prints",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 178,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0180": {
   "statement_status": "exact",
   "original_statement": "Problem 2.2 (F. Gu´ eritau). Given X, Y ∈ T (S), describe the set Env( X, Y ):= ∪{G},\n\nwhere G denotes a Thurston geodesic connecting X to Y.........................................................................",
   "clean_statement": "Problem 2.2 (F. Gu´ eritau). Given X, Y ∈ T (S), describe the set Env( X, Y ):= ∪{G},\n\nwhere G denotes a Thurston geodesic connecting X to Y.........................................................................",
   "public_statement": "Problem 2.2 (F. Gu´ eritau). Given X, Y ∈ T (S), describe the set Env( X, Y ):= ∪{G},\n\nwhere G denotes a Thurston geodesic connecting X to Y.........................................................................",
   "evidence": "The record is Problem 2.2 in Weixu Su's AIM list *Problems on Thurston Metric*, dated April 21, 2014. Direct inspection of the PDF gives:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 179,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0181": {
   "statement_status": "exact",
   "original_statement": "Problem 2.3 (K. Rafi). Does Env( X, Y ) depend continuously on X, Y?........................................................................",
   "clean_statement": "Problem 2.3 (K. Rafi). Does Env( X, Y ) depend continuously on X, Y?........................................................................",
   "public_statement": "Problem 2.3 (K. Rafi). Does Env( X, Y ) depend continuously on X, Y?........................................................................",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 180,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0182": {
   "statement_status": "exact",
   "original_statement": "Problem 2.4 (K. Rafi). Identify curves that are short along the preferred path GX,Y.F. Kassel and K. Rafi prove that, for any X, Y ∈ T (S), there is a Thurston geodesic GX,Y, parametrized linearly in Thurston's shear (cataclysm) coordi-nates associated with a canonical lamination λ(X, Y ) such that lengths of all simple closed curves along GX,Y are convex functions (up to reparametriza-tion).........................................................................",
   "clean_statement": "Problem 2.4 (K. Rafi). Identify curves that are short along the preferred path GX,Y.F. Kassel and K. Rafi prove that, for any X, Y ∈ T (S), there is a Thurston geodesic GX,Y, parametrized linearly in Thurston's shear (cataclysm) coordi-nates associated with a canonical lamination λ(X, Y ) such that lengths of all simple closed curves along GX,Y are convex functions (up to reparametriza-tion).........................................................................",
   "public_statement": "Problem 2.4 (K. Rafi). Identify curves that are short along the preferred path GX,Y.F. Kassel and K. Rafi prove that, for any X, Y ∈ T (S), there is a Thurston geodesic GX,Y, parametrized linearly in Thurston's shear (cataclysm) coordi-nates associated with a canonical lamination λ(X, Y ) such that lengths of all simple closed curves along GX,Y are convex functions (up to reparametriza-tion).........................................................................",
   "evidence": "The source is the AIM workshop list *Problems on Thurston metric*, dated April 21, 2014. Direct inspection of page 2 of the PDF gives the following text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 181,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0183": {
   "statement_status": "exact",
   "original_statement": "Problem 2.5 (K. Rafi). Assume that G(λ) is a stretch line directed by a maximal geodesic lamination λ, then how does the geodesic G(λ) look like? Moreover, if the projection of G(λ) to the arc and curve graph of a subsurface Y is large, can we expect that an interval of time where each boundary component β of Y has bounded length and β is close to a geodesic on Y?........................................................................",
   "clean_statement": "Problem 2.5 (K. Rafi). Assume that G(λ) is a stretch line directed by a maximal geodesic lamination λ, then how does the geodesic G(λ) look like? Moreover, if the projection of G(λ) to the arc and curve graph of a subsurface Y is large, can we expect that an interval of time where each boundary component β of Y has bounded length and β is close to a geodesic on Y?........................................................................",
   "public_statement": "Problem 2.5 (K. Rafi). Assume that G(λ) is a stretch line directed by a maximal geodesic lamination λ, then how does the geodesic G(λ) look like? Moreover, if the projection of G(λ) to the arc and curve graph of a subsurface Y is large, can we expect that an interval of time where each boundary component β of Y has bounded length and β is close to a geodesic on Y?........................................................................",
   "evidence": "The dotted separator appended to the corpus record is page layout, not part of the problem. More importantly, the grammatical defect (“can we expect that an interval”) and the final occurrence of \\(\\beta\\) both occur in the PDF. They are therefore source defects or ambiguities, not OCR errors introduced by the corpus. A natural grammatical repair inserts “there is.” The mathematical repair of the final clause requires more care and is made explicit below; it is not silently substituted into the source.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 182,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0184": {
   "statement_status": "exact",
   "original_statement": "Problem 2.6 (K. Rafi). Can we understand the behavior of Thurston's geodesics inductively, going from surfaces to subsurfaces?........................................................................ There have also been a great deal of study on the dynamical properties of the Teichm¨ uller geodesic flow on moduli space or tangent bundle of muduli space, with application to billiards and flat structures.",
   "clean_statement": "Problem 2.6 (K. Rafi). Can we understand the behavior of Thurston's geodesics inductively, going from surfaces to subsurfaces?........................................................................ There have also been a great deal of study on the dynamical properties of the Teichm¨ uller geodesic flow on moduli space or tangent bundle of muduli space, with application to billiards and flat structures.",
   "public_statement": "Problem 2.6 (K. Rafi). Can we understand the behavior of Thurston's geodesics inductively, going from surfaces to subsurfaces?........................................................................ There have also been a great deal of study on the dynamical properties of the Teichm¨ uller geodesic flow on moduli space or tangent bundle of muduli space, with application to billiards and flat structures.",
   "evidence": "The canonical JSON record merges a question with a later contextual paragraph. Inspection of the original AIM PDF verifies the following page order:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 183,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0185": {
   "statement_status": "exact",
   "original_statement": "Problem 2.7 (K. Rafi). Define the geodesic flow for the Thurston metric and study its properties, such as ergodic or mixing......................................................................... Inspired by the work of Eskin and Mirzakhani on Teichm¨ uller metric, we ask PROBLEMS ON THURSTON METRIC 3",
   "clean_statement": "Problem 2.7 (K. Rafi). Define the geodesic flow for the Thurston metric and study its properties, such as ergodic or mixing......................................................................... Inspired by the work of Eskin and Mirzakhani on Teichm¨ uller metric, we ask PROBLEMS ON THURSTON METRIC 3",
   "public_statement": "Problem 2.7 (K. Rafi). Define the geodesic flow for the Thurston metric and study its properties, such as ergodic or mixing......................................................................... Inspired by the work of Eskin and Mirzakhani on Teichm¨ uller metric, we ask PROBLEMS ON THURSTON METRIC 3",
   "evidence": "The canonical record is Problem 2.7 from the AIM workshop list *Problems on the Thurston metric* (workshop: “Lipschitz metric on Teichmueller space”). The exact mathematical question in the source PDF is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 184,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0186": {
   "statement_status": "exact",
   "original_statement": "Problem 2.8 (K. Rafi). Find the asymptotic behavior for the number of conjugacy classes of pseudo-Anosov elements of the mapping class group whose translation length in the Thurston metric is less than R.........................................................................",
   "clean_statement": "Problem 2.8 (K. Rafi). Find the asymptotic behavior for the number of conjugacy classes of pseudo-Anosov elements of the mapping class group whose translation length in the Thurston metric is less than R.........................................................................",
   "public_statement": "Problem 2.8 (K. Rafi). Find the asymptotic behavior for the number of conjugacy classes of pseudo-Anosov elements of the mapping class group whose translation length in the Thurston metric is less than R.........................................................................",
   "evidence": "The canonical record is Problem 2.8 from the AIM workshop list *Problems on the Thurston metric* (workshop: “Lipschitz metric on Teichmüller space,” dated April 21, 2014). The source fixes a finite-type hyperbolic surface $S$, of genus $g$ with $n$ punctures, and its Teichmüller space $\\mathcal T(S)$. The statement in the original PDF is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 185,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0187": {
   "statement_status": "exact",
   "original_statement": "Problem 2.9. Is there a dense Thurston geodesic in moduli space? Masur used closed Teichm¨ uller geodesics to approximate a Teichm¨ uller geodesic which is dense in moduli space.........................................................................",
   "clean_statement": "Problem 2.9. Is there a dense Thurston geodesic in moduli space? Masur used closed Teichm¨ uller geodesics to approximate a Teichm¨ uller geodesic which is dense in moduli space.........................................................................",
   "public_statement": "Problem 2.9. Is there a dense Thurston geodesic in moduli space? Masur used closed Teichm¨ uller geodesics to approximate a Teichm¨ uller geodesic which is dense in moduli space.........................................................................",
   "evidence": "The canonical record is Problem 2.9 in the AIM list *Problems on Thurston Metric*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 186,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0188": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 2.10 (K. Rafi). Is a stretch line typically recurrent/ dense/ e-quidistributed in moduli space?........................................................................ The question of when two Teichm¨ uller geodesic rays stay bounded dis-tance apart has been answered completely by Lenzhen and Masur.",
   "clean_statement": null,
   "public_statement": "Problem 2.10 (K. Rafi). Is a stretch line typically recurrent/ dense/ e-quidistributed in moduli space?........................................................................ The question of when two Teichm¨ uller geodesic rays stay bounded dis-tance apart has been answered completely by Lenzhen and Masur.",
   "evidence": "The canonical record comes from Weixu Su's 2014 AIM list *Problems on Thurston Metric*, Problem 2.10, attributed to K. Rafi. Inspection of the original PDF gives the question",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 187,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0189": {
   "statement_status": "exact",
   "original_statement": "Problem 2.11. Determine when two Thurston geodesic rays stay bounded distance apart.........................................................................",
   "clean_statement": "Problem 2.11. Determine when two Thurston geodesic rays stay bounded distance apart.........................................................................",
   "public_statement": "Problem 2.11. Determine when two Thurston geodesic rays stay bounded distance apart.........................................................................",
   "evidence": "The canonical record is Problem 2.11 from the AIM workshop list *Lipschitz metric on Teichmueller space*. Inspection of page 2 of the original AIM PDF recovers the complete statement as:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 188,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0190": {
   "statement_status": "exact",
   "original_statement": "Problem 2.12 (K. Rafi). Given two points in Teichm¨ uller space, do we have a comparison between the maximal stretch laminations (which define the concatenation of stretch segments) and the vertical (horizontal) foliations of quadratic differential for the Teichm¨ uller geodesic connecting them? Minsky has studied the approximate behavior of high-energy harmonic maps when the domain surface is varied, and compared the harmonic maps (which are stretched along the Hopf foliation) with Teichm¨ uller maps and stretch maps.........................................................................",
   "clean_statement": "Problem 2.12 (K. Rafi). Given two points in Teichm¨ uller space, do we have a comparison between the maximal stretch laminations (which define the concatenation of stretch segments) and the vertical (horizontal) foliations of quadratic differential for the Teichm¨ uller geodesic connecting them? Minsky has studied the approximate behavior of high-energy harmonic maps when the domain surface is varied, and compared the harmonic maps (which are stretched along the Hopf foliation) with Teichm¨ uller maps and stretch maps.........................................................................",
   "public_statement": "Problem 2.12 (K. Rafi). Given two points in Teichm¨ uller space, do we have a comparison between the maximal stretch laminations (which define the concatenation of stretch segments) and the vertical (horizontal) foliations of quadratic differential for the Teichm¨ uller geodesic connecting them? Minsky has studied the approximate behavior of high-energy harmonic maps when the domain surface is varied, and compared the harmonic maps (which are stretched along the Hopf foliation) with Teichm¨ uller maps and stretch maps.........................................................................",
   "evidence": "The canonical record is Problem 2.12, attributed to K. Rafi, from the AIM workshop *Lipschitz metric on Teichmüller space*. The official AIM PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 189,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0191": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 2.13. What is the quasi-isometric group of Teichm¨ uller space equipped with the Thurston metric? 3. Symmetrization of the Thurston metric",
   "clean_statement": null,
   "public_statement": "Problem 2.13. What is the quasi-isometric group of Teichm¨ uller space equipped with the Thurston metric? 3. Symmetrization of the Thurston metric",
   "evidence": "The canonical record comes from Weixu Su's AIM list *Problems on Thurston Metric*. Inspection of the original PDF shows that the complete statement is",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 190,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0192": {
   "statement_status": "exact",
   "original_statement": "Problem 3.1 (A. Papadopoulos). What is a good way to symmetrize the Thurston metric? The length-spectrum metric by Sorvali is a symmetrization of the Thurston metric.........................................................................",
   "clean_statement": "Problem 3.1 (A. Papadopoulos). What is a good way to symmetrize the Thurston metric? The length-spectrum metric by Sorvali is a symmetrization of the Thurston metric.........................................................................",
   "public_statement": "Problem 3.1 (A. Papadopoulos). What is a good way to symmetrize the Thurston metric? The length-spectrum metric by Sorvali is a symmetrization of the Thurston metric.........................................................................",
   "evidence": "The canonical record is `AIM-TOPOLOGY-0192`, record 191 (zero-based) of `aim-topology-notes.json`. The official AIM problem list, *Problems on Thurston Metric*, Section 3, says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 191,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0193": {
   "statement_status": "exact",
   "original_statement": "Problem 3.2 (A. Papadopoulos). Is the length-spectrum metric Finsler? If yes, give a formula for the infinitesimal norm of a vector on Teichm¨ uller space with respect to this Finsler structure.........................................................................",
   "clean_statement": "Problem 3.2 (A. Papadopoulos). Is the length-spectrum metric Finsler? If yes, give a formula for the infinitesimal norm of a vector on Teichm¨ uller space with respect to this Finsler structure.........................................................................",
   "public_statement": "Problem 3.2 (A. Papadopoulos). Is the length-spectrum metric Finsler? If yes, give a formula for the infinitesimal norm of a vector on Teichm¨ uller space with respect to this Finsler structure.........................................................................",
   "evidence": "The canonical record is Problem 3.2 from the 2012 AIM workshop *Lipschitz metric on Teichmueller space*. The AIM PDF gives:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 192,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0194": {
   "statement_status": "exact",
   "original_statement": "Problem 3.3 (A. Papadopoulos). Is Isom( T (S), d ls ) = Mod( S)?........................................................................ 4 WEIXU SU",
   "clean_statement": "Problem 3.3 (A. Papadopoulos). Is Isom( T (S), d ls ) = Mod( S)?........................................................................ 4 WEIXU SU",
   "public_statement": "Problem 3.3 (A. Papadopoulos). Is Isom( T (S), d ls ) = Mod( S)?........................................................................ 4 WEIXU SU",
   "evidence": "The canonical record is source index 193 of `aim-topology-notes.json`. Its OCR text is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 193,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0195": {
   "statement_status": "exact",
   "original_statement": "Problem 3.4 (K. Rafi). Is the Thurston metric a degeneration of some better vector-valued symmetric distance on Teichm¨ uller space?........................................................................ The reversed Thurston metric d∗ is defined by \n\nd∗(X, Y ) = dTh (Y, X ).",
   "clean_statement": "Problem 3.4 (K. Rafi). Is the Thurston metric a degeneration of some better vector-valued symmetric distance on Teichm¨ uller space?........................................................................ The reversed Thurston metric d∗ is defined by\n\nd∗(X, Y ) = dTh (Y, X ).",
   "public_statement": "Problem 3.4 (K. Rafi). Is the Thurston metric a degeneration of some better vector-valued symmetric distance on Teichm¨ uller space?........................................................................ The reversed Thurston metric d∗ is defined by\n\nd∗(X, Y ) = dTh (Y, X ).",
   "evidence": "The canonical record is zero-based entry 194 of `aim-topology-notes.json`. The mathematical question in the official AIM problem-list PDF is exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 194,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0196": {
   "statement_status": "exact",
   "original_statement": "Problem 3.5 (C. Walsh). What is the horofunction boundary of the re-versed Thurston metric?........................................................................ Let S be a surface of finite type with nonempty boundary. The reduced Teichm¨ uller space T (S) is the set of equivalence classes of marked bordered hypebolic structures on S.",
   "clean_statement": "Problem 3.5 (C. Walsh). What is the horofunction boundary of the re-versed Thurston metric?........................................................................ Let S be a surface of finite type with nonempty boundary. The reduced Teichm¨ uller space T (S) is the set of equivalence classes of marked bordered hypebolic structures on S.",
   "public_statement": "Problem 3.5 (C. Walsh). What is the horofunction boundary of the re-versed Thurston metric?........................................................................ Let S be a surface of finite type with nonempty boundary. The reduced Teichm¨ uller space T (S) is the set of equivalence classes of marked bordered hypebolic structures on S.",
   "evidence": "The canonical input is source index 195 of `aim-topology-notes.json`. Its extracted `problem` field says",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 195,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0197": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 3.6 (F. Gu´ eritaud). Describe the cone of directions in the tangent space T X T (S) which shorten the lengths of all simple closed curves on S......................................................................... The arc metric dA on T (S) is a natural generalization of the Thurston metric By doubling, there is a natural isometric embedding from ( T (S), d A)to ( T (Sd), d Th ). We shall identify T (S) with its image in T (Sd).",
   "clean_statement": "**Problem 3.6 (F. Guéritaud).** Describe the cone of directions in the tangent space \\(T_X\\mathcal T(S)\\) which shorten the lengths of all simple closed curves on \\(S\\).",
   "public_statement": "**Problem 3.6 (F. Guéritaud).** Describe the cone of directions in the tangent space \\(T_X\\mathcal T(S)\\) which shorten the lengths of all simple closed curves on \\(S\\).",
   "evidence": "The official AIM PDF, *Problems on the Lipschitz metric on Teichmüller space*, resolves the extraction boundary. On page 3, lines 123--125 introduce reduced Teichmüller space for a finite-type surface with nonempty boundary. Lines 126--127 contain Problem 3.6. A dotted separator follows on line 128. The arc-metric and doubling paragraph is on lines 129--135, immediately before Problem 3.7 on line 136. It is therefore adjacent context for Problem 3.7, not part of Problem 3.6. The recovered statement is:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-topology-notes.json",
   "source_index": 196,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0198": {
   "statement_status": "exact",
   "original_statement": "Problem 3.7. Is the metric space ( T (S), d A) a length space? Given X, Y ∈T (S), is there a geodesic Γ of ( T (Sd), d Th ) connecting X and Y such that Γ ⊂ T (S)?........................................................................",
   "clean_statement": "Problem 3.7. Is the metric space ( T (S), d A) a length space? Given X, Y ∈T (S), is there a geodesic Γ of ( T (Sd), d Th ) connecting X and Y such that Γ ⊂ T (S)?........................................................................",
   "public_statement": "Problem 3.7. Is the metric space ( T (S), d A) a length space? Given X, Y ∈T (S), is there a geodesic Γ of ( T (Sd), d Th ) connecting X and Y such that Γ ⊂ T (S)?........................................................................",
   "evidence": "The canonical record comes from Weixu Su's 2014 AIM workshop list *Problems on Thurston Metric*, Problem 3.7. The immediately preceding text fixes the notation:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 197,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0199": {
   "statement_status": "exact",
   "original_statement": "Problem 3.8. Let S be a hyperbolic surface with ideal boundary. Can we define the Thurston metric on the non-reduced Teichm¨ uller space of S?........................................................................",
   "clean_statement": "Problem 3.8. Let S be a hyperbolic surface with ideal boundary. Can we define the Thurston metric on the non-reduced Teichm¨ uller space of S?........................................................................",
   "public_statement": "Problem 3.8. Let S be a hyperbolic surface with ideal boundary. Can we define the Thurston metric on the non-reduced Teichm¨ uller space of S?........................................................................",
   "evidence": "The canonical record is zero-based entry 198 of `aim-topology-notes.json`. The official AIM PDF contains the following exact question:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 198,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0200": {
   "statement_status": "exact",
   "original_statement": "Problem 3.9 (D. Dumas). Can we define the Thurston metric on the uni-versal Teichm¨ uller space? 4. Infinitely-generated Fuchsian group of the first kind \n\nLet Γ 0 be an infinite-generated Fuchsian group of the first kind.",
   "clean_statement": "Problem 3.9 (D. Dumas). Can we define the Thurston metric on the uni-versal Teichm¨ uller space? 4. Infinitely-generated Fuchsian group of the first kind\n\nLet Γ 0 be an infinite-generated Fuchsian group of the first kind.",
   "public_statement": "Problem 3.9 (D. Dumas). Can we define the Thurston metric on the uni-versal Teichm¨ uller space? 4. Infinitely-generated Fuchsian group of the first kind\n\nLet Γ 0 be an infinite-generated Fuchsian group of the first kind.",
   "evidence": "The canonical JSON record contains material from the next section. Inspection of the original AIM PDF shows the exact record boundary (page 4 of the PDF):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 199,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0201": {
   "statement_status": "exact",
   "original_statement": "Problem 4.1. Can we deform a infinite-generated Fuchsian group of the first kind (or, a closed hyperbolic surface of infinite area) by quasiconformal maps such that the lengths of all simple closed curves are not increased? Alessandrini and Gu´ eritaud have recent results on this question.........................................................................",
   "clean_statement": "Problem 4.1. Can we deform a infinite-generated Fuchsian group of the first kind (or, a closed hyperbolic surface of infinite area) by quasiconformal maps such that the lengths of all simple closed curves are not increased? Alessandrini and Gu´ eritaud have recent results on this question.........................................................................",
   "public_statement": "Problem 4.1. Can we deform a infinite-generated Fuchsian group of the first kind (or, a closed hyperbolic surface of infinite area) by quasiconformal maps such that the lengths of all simple closed curves are not increased? Alessandrini and Gu´ eritaud have recent results on this question.........................................................................",
   "evidence": "The canonical record comes from Problem 4.1 of the official AIM list *Problems on the Lipschitz metric on Teichmüller space*. Its mathematical content can be recovered unambiguously:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 200,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0202": {
   "statement_status": "exact",
   "original_statement": "Problem 4.2. Can we defining Thurston metric on Tqc (Γ 0)?........................................................................",
   "clean_statement": "Problem 4.2. Can we defining Thurston metric on Tqc (Γ 0)?........................................................................",
   "public_statement": "Problem 4.2. Can we defining Thurston metric on Tqc (Γ 0)?........................................................................",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 201,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0203": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 4.3 (M. Kapovich). Can we construct example of Γ 0 (infinite-generated and of the first kind) where the critical exponent of some elements in Tqc (Γ 0) are distinct? PROBLEMS ON THURSTON METRIC 5\n\nM. Kapovich suggested that the above question maybe related to",
   "clean_statement": "**Problem 4.3.** Construct an infinitely generated Fuchsian group\n\\(\\Gamma _0\\) of the first kind for which the critical exponent function is\nnonconstant on \\(\\mathcal T_{qc}(\\Gamma _0)\\).",
   "public_statement": "**Problem 4.3.** Construct an infinitely generated Fuchsian group\n\\(\\Gamma _0\\) of the first kind for which the critical exponent function is\nnonconstant on \\(\\mathcal T_{qc}(\\Gamma _0)\\).",
   "evidence": "The exact canonical record is visibly truncated at a PDF page break: The official AIM PDF verifies the continuation:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-topology-notes.json",
   "source_index": 202,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0204": {
   "statement_status": "exact",
   "original_statement": "Problem 4.1. He guessed that the construction depends on the ergodic properties of Γ0......................................................................... One idea to understand surfaces of infinite type is approximate it by subsurfaces of finite type. Let Γ 0 be a infinite-generated Fuchsian group of the first kind. Denote by Γ n the fundamental group carried by the n-ball in \n\nH/Γ0.",
   "clean_statement": "Problem 4.1. He guessed that the construction depends on the ergodic properties of Γ0......................................................................... One idea to understand surfaces of infinite type is approximate it by subsurfaces of finite type. Let Γ 0 be a infinite-generated Fuchsian group of the first kind. Denote by Γ n the fundamental group carried by the n-ball in\n\nH/Γ0.",
   "public_statement": "Problem 4.1. He guessed that the construction depends on the ergodic properties of Γ0......................................................................... One idea to understand surfaces of infinite type is approximate it by subsurfaces of finite type. Let Γ 0 be a infinite-generated Fuchsian group of the first kind. Denote by Γ n the fundamental group carried by the n-ball in\n\nH/Γ0.",
   "evidence": "This canonical record is not an autonomous problem. The official AIM PDF shows that it splices two paragraphs on page 5 and adds a false “Problem 4.1” label.",
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  "AIM-TOPOLOGY-0205": {
   "statement_status": "exact",
   "original_statement": "Problem 4.4 (F. Gu´ ertitaud). Does δ(Γ n) → δ(Γ 0) as n → ∞?........................................................................",
   "clean_statement": "Problem 4.4 (F. Gu´ ertitaud). Does δ(Γ n) → δ(Γ 0) as n → ∞?........................................................................",
   "public_statement": "Problem 4.4 (F. Gu´ ertitaud). Does δ(Γ n) → δ(Γ 0) as n → ∞?........................................................................",
   "evidence": "The canonical JSON contains only the displayed question, so the immediately preceding paragraph in the official AIM PDF is essential. It says, with minor grammatical errors:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
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  "AIM-TOPOLOGY-0206": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 4.5 (F. Gu´ ertitaud). Can we have a quasiconformal deformation of Γ 0 which bend a variation of δ(Γ n) independent of n?5. Other questions",
   "clean_statement": "Problem 4.5 (F. Gu´ ertitaud). Can we have a quasiconformal deformation of Γ 0 which bend a variation of δ(Γ n) independent of n?which bend a variation",
   "public_statement": "Problem 4.5 (F. Gu´ ertitaud). Can we have a quasiconformal deformation of Γ 0 which bend a variation of δ(Γ n) independent of n?5. Other questions",
   "evidence": "I therefore keep the source wording visible and separate three plausible readings:",
   "classification_method": "explicit_uncertainty_evidence",
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   "source_file": "aim-topology-notes.json",
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  "AIM-TOPOLOGY-0207": {
   "statement_status": "exact",
   "original_statement": "Problem 5.1. What is Thurston metric on H2, viewed as the Teichm¨ uller space of punctured tori? Belkhirat, Papadopoulos and Troyanov have studied the Thurston metric on Teichm¨ uller space of flat tori.........................................................................",
   "clean_statement": "Problem 5.1. What is Thurston metric on H2, viewed as the Teichm¨ uller space of punctured tori? Belkhirat, Papadopoulos and Troyanov have studied the Thurston metric on Teichm¨ uller space of flat tori.........................................................................",
   "public_statement": "Problem 5.1. What is Thurston metric on H2, viewed as the Teichm¨ uller space of punctured tori? Belkhirat, Papadopoulos and Troyanov have studied the Thurston metric on Teichm¨ uller space of flat tori.........................................................................",
   "evidence": "The canonical OCR record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
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   "source_file": "aim-topology-notes.json",
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  "AIM-TOPOLOGY-0208": {
   "statement_status": "exact",
   "original_statement": "Problem 5.2 (D. Dumas). Does Thurston's metric have any relation with the following unsolved problem: Consider all conformal metrics ρ on a Rie-mann surface which satisfy \n\ninf \n\n> γ\n\n`ρ(γ) ≥ 1.\n\nIs there always one of such metrics with least area? \n\nSee the work of Wolf and Zwiebach for study of the above question and application to string theory.........................................................................",
   "clean_statement": "Problem 5.2 (D. Dumas). Does Thurston's metric have any relation with the following unsolved problem: Consider all conformal metrics ρ on a Rie-mann surface which satisfy\n\ninf\n\n> γ\n\n`ρ(γ) ≥ 1.\n\nIs there always one of such metrics with least area?\n\nSee the work of Wolf and Zwiebach for study of the above question and application to string theory.........................................................................",
   "public_statement": "Problem 5.2 (D. Dumas). Does Thurston's metric have any relation with the following unsolved problem: Consider all conformal metrics ρ on a Rie-mann surface which satisfy\n\ninf\n\n> γ\n\n`ρ(γ) ≥ 1.\n\nIs there always one of such metrics with least area?\n\nSee the work of Wolf and Zwiebach for study of the above question and application to string theory.........................................................................",
   "evidence": "The canonical JSON record is visibly damaged by PDF extraction: it contains the fragments `inf`, `> gamma`, and `` `rho(gamma) >= 1``. The official AIM PDF (Weixu Su, *Problems on Thurston metric*, Problem 5.2) displays the intended formula as",
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   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 207,
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  "AIM-TOPOLOGY-0209": {
   "statement_status": "exact",
   "original_statement": "Problem 5.3. Define and study the Thurston metric on the space of flat \n\nn-tori SL n(R)/SL n(Z).........................................................................",
   "clean_statement": "Problem 5.3. Define and study the Thurston metric on the space of flat\n\nn-tori SL n(R)/SL n(Z).........................................................................",
   "public_statement": "Problem 5.3. Define and study the Thurston metric on the space of flat\n\nn-tori SL n(R)/SL n(Z).........................................................................",
   "evidence": "The official AIM PDF states, without OCR ambiguity:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 208,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0210": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 5.4 (D. Dumas). How does the \"Lipschitz constant function\" (g, h ) 7 → inf \n\n> φ:g→h\n\nLip( φ)behave on a class of metrics larger than the set of hyperbolic metrics, for example, the set of negatively curved Riemannian metrics or its closure?",
   "clean_statement": null,
   "public_statement": "Problem 5.4 (D. Dumas). How does the \"Lipschitz constant function\" (g, h ) 7 → inf\n\n> φ:g→h\n\nLip( φ)behave on a class of metrics larger than the set of hyperbolic metrics, for example, the set of negatively curved Riemannian metrics or its closure?",
   "evidence": "The PDF contains no condition below \\(\\phi:g\\to h\\). Taken literally, the infimum is zero because constant maps are allowed. The surrounding subject is marked Teichmüller space, so the conservative reconstruction is:",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 209,
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  "AIM-TOPOLOGY-0211": {
   "statement_status": "unrecoverable",
   "original_statement": "1 Monday \n\n1.1 Flexible contact structures \n\nE. Murphy: Is there a class of contact structures on closed manifolds abiding to an h-principle?Meaning a class of contact structure for which homotopy through almost contact structure implies isotopy. Such a class would provide a generalization of overtwisted contact 3-manifolds to higher dimensions [12]. \n\n1.2 Test cases for flexibility \n\nSince the preceding question may be very hard, one can rather try to prove by ad hoc methods that some operations do not change contact structures once they look flexible. \n\nC. Wendl: Given a contact manifold (V, ξ ) there exists an operation L called Lutz-Mori twist \n\n(see [28, 31]) that takes ξ to another contact structure L(ξ) on V in the same homotopy class of almost contact structures. (It can be performed on any contact 5-manifold and at least on some examples in higher dimensions). It is likely to kill the contact homology of V [5, 6]. If you apply the twist twice you get another contact structure L2(ξ) on V, is it contactomorphic to L(ξ)?\n\nJ. Etnyre: There is another kind of twist L′ due to Etnyre and Pancholi [16] which applies to any dimensions. \n\nK. Niederkrüger: To complete the list, there is also the negative stabilization process L′′ by E. Giroux (see [22, 7]). It starts with a supporting open book for a given contact manifold (V, ξ ),adds a critical Weinstein handle to the page along a Legendrian sphere bounding a Lagrangian disk and composes the monodromy with a left-handed Dehn twist along the Lagrangian sphere obtained by capping the disk with the handle core. In dimension 5, this preserves the homotopy class of the almost contact structure. Somebody mentioned that in dimension greater than 5,applying it twice also preserves the homotopy class of the almost contact structure. In the same vein as C. Wendl's question, denoting N the process of applying L′′ twice, do we have N 2 = N?\n\nMany people: What is the relationship between L, L′ and N? J. Etnyre and P. Massot remark that they all produce contact structures that are non fillable, have vanishing contact homology and all their Reeb vector fields have a contractible closed orbit. 1P. Massot: How to find bLobs or other remarkable n+1 -dimensional submanifolds in negatively stabilized contact manifolds? \n\n1.3 Convex hypersurface theory \n\nA. Mori: What would be a useful theory of convex hypersurfaces in high dimension? There is a definition [20] as a hypersurface that is transverse to a contact vector field but E. Giroux points out that in opposition to the 3-dimensional case, this is not generic in higher dimensions. Indeed, E. Giroux says one should be able to construct examples of hypersurfaces whose characteristic foliation admits a closed orbit which is neither repelling nor attracting but rather has a hyperbolic type dynamic. This will remain after perturbation and is an obstruction to convexity. A. Mori mentioned that [32] gives an explicit example of this phenomenon. \n\nE. Murphy: Are there other obstructions for perturbing a given hypersurface to a convex one and are there conditions that guarantee the existence of such perturbations? \n\nJ. Etnyre: Is every hypersurface at least homologous to a convex hypersurface? This is a potentially easier question and is relevant to the Thurston-Bennequin question below. \n\nK. Honda: What is a good notion of a bypass in high dimension [23]? P. Massot remarks that there is a natural definition involving topologically canceling contact handles [20, 38] (but non trivial regarding contact topology), and asks if this is a good one? \n\n1.4 Thurston-Bennequin inequality \n\nA. Mori: formulated a Thurston-Bennequin type inequality [13] in any dimension that gen-eralizes the 3-dimensional case (see [32]) to some hypersurfaces whose boundary is a contact submanifold. Does it hold for the standard contact sphere S2n+1? He points out that the Lutz-Mori twist L produces contact structure that violate this inequality. He also says there is an absolute version of this inequality that trivially holds for the standard contact sphere. \n\nP. Massot: Is there a Thurston-Bennequin inequality which holds for closed hypersurfaces in fillable or tight contact manifolds? The first case to look at would be: is there any constraint on the Chern class of a 5-dimensional fillable contact manifold? \n\n1.5 Characterization of the standard contact sphere \n\nK. Niederkrüger: Are there properties that uniquely determine the standard contact S2n+1?For example, are there other contact structures on S2n+1 that are filled by a symplectic manifolds diffeomorphic to the ball D2n+2? M. Abouzaid and M. McLean answer yes: there are examples constructed by McLean in [30] (see also [2]) of Stein manifolds diffeomorphic to Cn with non standard contact boundary. \n\nM. McLean: What constraint on the symplectic structure of D2n+2 would imply that its boundary is the standard contact S2n+1? He suggests symplectic balls D2n+2 that are symplec-tomorphic to a smooth affine variety of negative log-Kodaira dimension. \n\nP. Massot: If ξ is a contact structure on S2n+1 with CH ∗(S2n+1, ξ ) ' CH ∗(S2n+1, ξ std ), are ξ\n\nand ξstd contactomorphic? \n\n1.6 Contact structure on M × S2\n\nF. Presas: Given a contact manifold (M, ξ ), can you build a contact structure on M × S2? We know there is no homotopy obstruction to this. If yes can you require in addition that for some \n\np ∈ S2, M × { p} is a contact submanifold contactomorphic to (M, ξ )?21.7 Contact fibration and orderability \n\nE. Giroux: From the paper of Eliashberg and Polterovich introducing the notion of orderability of a contact manifold [15] we can get the following statement: (M, ξ ) is non-orderable if and only if there exists a contact fibration M × S2 → S2 with fiber contactomorphic to (M, ξ ). What about topologically non trivial bundles? For 3-manifold, the problem of constructing a contact structure transverse to a given circle bundle is related to the Milnor-Wood inequality and to quasimorphisms on Diff( S1) (see [21]). Analogously, the higher dimensional question is probably related to the existence of quasimorphisms on the group of contactomorphisms of (M, ξ ).\n\n1.8 Generalized Giroux torsion \n\nWhat is the generalization of Giroux torsion for higher dimensional contact manifolds? In the morning, P. Massot proposed the notion of Giroux domain introduced recently in [28]. There is a model case S1 × M × D1 with M × D1 a Liouville manifold, which allows to produce non fillable manifolds, yet not flexible and having Reeb vector fields without contractible Reeb orbits. There is some definition of algebraic 1-torsion in SFT [25]. It is conjectured that geometric torsion implies algebraic torsion. \n\n1.9 Fillability and cobordisms \n\nRecall the general picture about fillable contact manifolds: \n\n{Exact }⊂⊂{Stein=Weinstein } {Strong } ⊂ {Weak }⊂⊂{Holomorphic }\n\nJ. Latschev: Are there obstructions to exact or Weinstein cobordisms between contact mani-folds? For example, are there strongly fillable manifolds with no exact filling? \n\nY. Eliashberg: (RP 2n+1, ξ std ) is holomorphically fillable but not Stein fillable ([14, 39]). They probably do not have exact fillings. Are (T 2n+1, ξ Bourgeois ) exactly fillable? He says they are not Stein fillable ([14]) but according to P. Massot they are weakly fillable ([28]). Y. Eliashberg expects these manifolds not to be strongly fillable. \n\nC. Wendl: In dimension 3 and 5, there are examples of weakly but not strongly fillable manifolds (see [28]). What about dimension greater than 5?\n\n1.10 Lefschetz fibration \n\nO. Plamenevskaya: Does every Weinstein domain admit a Lefschetz fibration over the disc D2?E. Giroux says he has a proof in mind using Donaldson's approximately holomorphic techniques [11], it should also work in the case of Stein domains (requiring the projection to be holomorphic) thanks to Hörmander's L2-theory, but it is not yet written. \n\nC. Wendl: Is there a hyperplane pencil decomposition of contact manifolds? By this we mean a kind of open book decomposition but with 2-dimensional pages (instead of codimension 2 pages). One interest for this comes from making these pages holomorphic while applying holomorphic curves techniques. For instance, it would probably allow to prove the Weinstein conjecture in some cases. F. Presas says there exists a notion for this and they can be constructed using approximately holomorphic techniques but resulting maps will have singularities modeled on singularities of maps from Cn+1 to Cn which are very complicated. 31.11 Open book decomposition \n\nO. Plamenevskaya: Can you say anything about contact structure using monodromy data? For example, do we have Stein fillable ⇒ there is a supporting open book whose monodromy is a product of positive Dehn twists along Lagrangian spheres? (it would be a corollary of the existence of Lefschetz fibration on Stein domains). K. Honda remarks that in low dimension there are examples of open book decomposition of Stein fillable manifolds whose monodromy is not a product of Dehn twists so one cannot hope for the stronger result that any supporting open book has such kind of monodromy. \n\nY. Eliashberg: How to read strong fillability on monodromy data? P. Massot says that it is not even clear in dimension 3.\n\nF. Presas: For (M, ξ ) an exact fillable contact manifold, does there exist an open book decom-position whose monodromy is a product of positive and negative Dehn twists? He remarks that it is a non trivial condition since there are symplectomorphisms which are not products of Dehn twists. \n\nM. Abouzaid: For example take T ∗CP n with the associated \"Dehn twist\" (not to confuse with a Dehn twist along a Lagrangian sphere, indeed it is not a product of Dehn twists because there are no Lagrangian spheres), is the contact manifold with the corresponding open book exactly fillable? E. Giroux remarks that there are other examples of symplectomorphisms which are not products of Dehn twists, the so-called fibered Dehn twists. For example, take T ∗RP n =\n\nCP n \\ Quadric, the corresponding fibered Dehn twist is not a product of Dehn twists since there are no Lagrangian spheres. \n\nC. Wendl: Take the negative \"Dehn twist\" on T ∗CP n, the associated contact manifold has probably zero contact homology (this should follow from the strategy of [7]). How does this relate to notions of overtwistedness? \n\n1.12 Fillings \n\nE. Murphy: What can you say about contact structures that are filled by a subcritical or flexible Weinstein manifold? Can you classify their fillings? Y. Eliashberg underlines that we have to study the topological type but also the symplectic type of fillings. A naive question would be: are they all symplectomorphic? \n\nY. Eliashberg: To give a concrete example, take the standard contact sphere S2n+1, we know that the fillings are all diffeomorphic to the ball D2n+2 [29], but are they all symplectomorphic? \n\n1.13 Contact manifolds with a lot of symmetries \n\nY. Karshon: Two families of contact manifolds with a lot of symmetry are \n\n• Contact toric manifolds [26]; \n\n• Prequantization circle bundles of coadjoint orbits of Lie Groups. Every compact contact manifold that admits a transitive action of a compact Lie group by coorientation preserving contactomorphisms lies in the second family above [4]. These families of manifolds constitute a good playground for contact topology. It can be interesting to compute their contact topological invariants. \n\nY. Eliashberg: Take a complex line bundle L over an integral symplectic manifold (M, ω ) with first Chern class c1(L) = n ω. The associated circle bundle V is a contact manifold. Suppose n\n\nis large, is V Stein fillable? F. Presas asks why n is supposed to be large in this problem, and Y. Eliashberg explains that it corresponds somehow to the fact that we need very ample divisor instead of just ample. Without n being large, it might be only symplectically fillable but not 4Stein fillable. E. Giroux suggests that in the case where M is a torus, the cohomology ring of V\n\ncan be an obstruction to fillability (compare [39]). M. Abouzaid asks if there are obstruction for an algebraic variety to be realized as a divisor. After some discussion, it turns out that the question has already been considered at least for hyperplane sections in [46]. \n\n1.14 Symplectization \n\nF. Presas: Suppose two contact manifolds have symplectomorphic symplectizations, are they contactomorphic? Maybe, assume in addition that the manifolds are simply connected. \n\n1.15 Sasakian manifolds \n\nK. Honda: Is there anything symplectic geometry can say about Sasakian manifolds [8]? R. Komendarczyk says that many things are known in dimension 3. In all dimensions, Sasakian manifolds are fillable (see [40, 35]). A. Mori points out that there is also a theorem of D. Martínez Torres [27] that every Sasakian manifold M 2n+1 admits a contact immersion into (S4n+3, ξ std )\n\nwhich pulls back the standard open book of S4n+3 to a supporting open book of M.\n\nF. Presas: A simply-connected closed manifold is formal over rational (resp. real) numbers if its rational (resp. real) homotopy type can be recovered from its cohomology ring. Simply-connected closed orientable manifolds of dimension ≤ 6 are formal, and there are examples of non-formal simply-connected manifolds in any dimension ≥ 7 (see [17] and the references therein). It is known [42] that Sasakian manifolds are formal (for real homotopy type). This leads us to the question of a producing non-formal contact manifolds. For example, are there non-formal simply-connected closed contact manifolds (of dim necessarily ≥ 7)? (see [18, 3] for related work). \n\n1.16 Exotic spheres and contact geometry \n\nP. Massot: Let Σn be an exotic sphere. Is (ST ∗Σn, ξ std ) contactomorphic to (ST ∗Sn, ξ std )?Compare this also to the related result for cotangent bundles [1].",
   "clean_statement": null,
   "public_statement": "1 Monday\n\n1.1 Flexible contact structures\n\nE. Murphy: Is there a class of contact structures on closed manifolds abiding to an h-principle?Meaning a class of contact structure for which homotopy through almost contact structure implies isotopy. Such a class would provide a generalization of overtwisted contact 3-manifolds to higher dimensions [12].\n\n1.2 Test cases for flexibility\n\nSince the preceding question may be very hard, one can rather try to prove by ad hoc methods that some operations do not change contact structures once they look flexible.\n\nC. Wendl: Given a contact manifold (V, ξ ) there exists an operation L called Lutz-Mori twist\n\n(see [28, 31]) that takes ξ to another contact structure L(ξ) on V in the same homotopy class of almost contact structures. (It can be performed on any contact 5-manifold and at least on some examples in higher dimensions). It is likely to kill the contact homology of V [5, 6]. If you apply the twist twice you get another contact structure L2(ξ) on V, is it contactomorphic to L(ξ)?\n\nJ. Etnyre: There is another kind of twist L′ due to Etnyre and Pancholi [16] which applies to any dimensions.\n\nK. Niederkrüger: To complete the list, there is also the negative stabilization process L′′ by E. Giroux (see [22, 7]). It starts with a supporting open book for a given contact manifold (V, ξ ),adds a critical Weinstein handle to the page along a Legendrian sphere bounding a Lagrangian disk and composes the monodromy with a left-handed Dehn twist along the Lagrangian sphere obtained by capping the disk with the handle core. In dimension 5, this preserves the homotopy class of the almost contact structure. Somebody mentioned that in dimension greater than 5,applying it twice also preserves the homotopy class of the almost contact structure. In the same vein as C. Wendl's question, denoting N the process of applying L′′ twice, do we have N 2 = N?\n\nMany people: What is the relationship between L, L′ and N? J. Etnyre and P. Massot remark that they all produce contact structures that are non fillable, have vanishing contact homology and all their Reeb vector fields have a contractible closed orbit. 1P. Massot: How to find bLobs or other remarkable n+1 -dimensional submanifolds in negatively stabilized contact manifolds?\n\n1.3 Convex hypersurface theory\n\nA. Mori: What would be a useful theory of convex hypersurfaces in high dimension? There is a definition [20] as a hypersurface that is transverse to a contact vector field but E. Giroux points out that in opposition to the 3-dimensional case, this is not generic in higher dimensions. Indeed, E. Giroux says one should be able to construct examples of hypersurfaces whose characteristic foliation admits a closed orbit which is neither repelling nor attracting but rather has a hyperbolic type dynamic. This will remain after perturbation and is an obstruction to convexity. A. Mori mentioned that [32] gives an explicit example of this phenomenon.\n\nE. Murphy: Are there other obstructions for perturbing a given hypersurface to a convex one and are there conditions that guarantee the existence of such perturbations?\n\nJ. Etnyre: Is every hypersurface at least homologous to a convex hypersurface? This is a potentially easier question and is relevant to the Thurston-Bennequin question below.\n\nK. Honda: What is a good notion of a bypass in high dimension [23]? P. Massot remarks that there is a natural definition involving topologically canceling contact handles [20, 38] (but non trivial regarding contact topology), and asks if this is a good one?\n\n1.4 Thurston-Bennequin inequality\n\nA. Mori: formulated a Thurston-Bennequin type inequality [13] in any dimension that gen-eralizes the 3-dimensional case (see [32]) to some hypersurfaces whose boundary is a contact submanifold. Does it hold for the standard contact sphere S2n+1? He points out that the Lutz-Mori twist L produces contact structure that violate this inequality. He also says there is an absolute version of this inequality that trivially holds for the standard contact sphere.\n\nP. Massot: Is there a Thurston-Bennequin inequality which holds for closed hypersurfaces in fillable or tight contact manifolds? The first case to look at would be: is there any constraint on the Chern class of a 5-dimensional fillable contact manifold?\n\n1.5 Characterization of the standard contact sphere\n\nK. Niederkrüger: Are there properties that uniquely determine the standard contact S2n+1?For example, are there other contact structures on S2n+1 that are filled by a symplectic manifolds diffeomorphic to the ball D2n+2? M. Abouzaid and M. McLean answer yes: there are examples constructed by McLean in [30] (see also [2]) of Stein manifolds diffeomorphic to Cn with non standard contact boundary.\n\nM. McLean: What constraint on the symplectic structure of D2n+2 would imply that its boundary is the standard contact S2n+1? He suggests symplectic balls D2n+2 that are symplec-tomorphic to a smooth affine variety of negative log-Kodaira dimension.\n\nP. Massot: If ξ is a contact structure on S2n+1 with CH ∗(S2n+1, ξ ) ' CH ∗(S2n+1, ξ std ), are ξ\n\nand ξstd contactomorphic?\n\n1.6 Contact structure on M × S2\n\nF. Presas: Given a contact manifold (M, ξ ), can you build a contact structure on M × S2? We know there is no homotopy obstruction to this. If yes can you require in addition that for some\n\np ∈ S2, M × { p} is a contact submanifold contactomorphic to (M, ξ )?21.7 Contact fibration and orderability\n\nE. Giroux: From the paper of Eliashberg and Polterovich introducing the notion of orderability of a contact manifold [15] we can get the following statement: (M, ξ ) is non-orderable if and only if there exists a contact fibration M × S2 → S2 with fiber contactomorphic to (M, ξ ). What about topologically non trivial bundles? For 3-manifold, the problem of constructing a contact structure transverse to a given circle bundle is related to the Milnor-Wood inequality and to quasimorphisms on Diff( S1) (see [21]). Analogously, the higher dimensional question is probably related to the existence of quasimorphisms on the group of contactomorphisms of (M, ξ ).\n\n1.8 Generalized Giroux torsion\n\nWhat is the generalization of Giroux torsion for higher dimensional contact manifolds? In the morning, P. Massot proposed the notion of Giroux domain introduced recently in [28]. There is a model case S1 × M × D1 with M × D1 a Liouville manifold, which allows to produce non fillable manifolds, yet not flexible and having Reeb vector fields without contractible Reeb orbits. There is some definition of algebraic 1-torsion in SFT [25]. It is conjectured that geometric torsion implies algebraic torsion.\n\n1.9 Fillability and cobordisms\n\nRecall the general picture about fillable contact manifolds:\n\n{Exact }⊂⊂{Stein=Weinstein } {Strong } ⊂ {Weak }⊂⊂{Holomorphic }\n\nJ. Latschev: Are there obstructions to exact or Weinstein cobordisms between contact mani-folds? For example, are there strongly fillable manifolds with no exact filling?\n\nY. Eliashberg: (RP 2n+1, ξ std ) is holomorphically fillable but not Stein fillable ([14, 39]). They probably do not have exact fillings. Are (T 2n+1, ξ Bourgeois ) exactly fillable? He says they are not Stein fillable ([14]) but according to P. Massot they are weakly fillable ([28]). Y. Eliashberg expects these manifolds not to be strongly fillable.\n\nC. Wendl: In dimension 3 and 5, there are examples of weakly but not strongly fillable manifolds (see [28]). What about dimension greater than 5?\n\n1.10 Lefschetz fibration\n\nO. Plamenevskaya: Does every Weinstein domain admit a Lefschetz fibration over the disc D2?E. Giroux says he has a proof in mind using Donaldson's approximately holomorphic techniques [11], it should also work in the case of Stein domains (requiring the projection to be holomorphic) thanks to Hörmander's L2-theory, but it is not yet written.\n\nC. Wendl: Is there a hyperplane pencil decomposition of contact manifolds? By this we mean a kind of open book decomposition but with 2-dimensional pages (instead of codimension 2 pages). One interest for this comes from making these pages holomorphic while applying holomorphic curves techniques. For instance, it would probably allow to prove the Weinstein conjecture in some cases. F. Presas says there exists a notion for this and they can be constructed using approximately holomorphic techniques but resulting maps will have singularities modeled on singularities of maps from Cn+1 to Cn which are very complicated. 31.11 Open book decomposition\n\nO. Plamenevskaya: Can you say anything about contact structure using monodromy data? For example, do we have Stein fillable ⇒ there is a supporting open book whose monodromy is a product of positive Dehn twists along Lagrangian spheres? (it would be a corollary of the existence of Lefschetz fibration on Stein domains). K. Honda remarks that in low dimension there are examples of open book decomposition of Stein fillable manifolds whose monodromy is not a product of Dehn twists so one cannot hope for the stronger result that any supporting open book has such kind of monodromy.\n\nY. Eliashberg: How to read strong fillability on monodromy data? P. Massot says that it is not even clear in dimension 3.\n\nF. Presas: For (M, ξ ) an exact fillable contact manifold, does there exist an open book decom-position whose monodromy is a product of positive and negative Dehn twists? He remarks that it is a non trivial condition since there are symplectomorphisms which are not products of Dehn twists.\n\nM. Abouzaid: For example take T ∗CP n with the associated \"Dehn twist\" (not to confuse with a Dehn twist along a Lagrangian sphere, indeed it is not a product of Dehn twists because there are no Lagrangian spheres), is the contact manifold with the corresponding open book exactly fillable? E. Giroux remarks that there are other examples of symplectomorphisms which are not products of Dehn twists, the so-called fibered Dehn twists. For example, take T ∗RP n =\n\nCP n \\ Quadric, the corresponding fibered Dehn twist is not a product of Dehn twists since there are no Lagrangian spheres.\n\nC. Wendl: Take the negative \"Dehn twist\" on T ∗CP n, the associated contact manifold has probably zero contact homology (this should follow from the strategy of [7]). How does this relate to notions of overtwistedness?\n\n1.12 Fillings\n\nE. Murphy: What can you say about contact structures that are filled by a subcritical or flexible Weinstein manifold? Can you classify their fillings? Y. Eliashberg underlines that we have to study the topological type but also the symplectic type of fillings. A naive question would be: are they all symplectomorphic?\n\nY. Eliashberg: To give a concrete example, take the standard contact sphere S2n+1, we know that the fillings are all diffeomorphic to the ball D2n+2 [29], but are they all symplectomorphic?\n\n1.13 Contact manifolds with a lot of symmetries\n\nY. Karshon: Two families of contact manifolds with a lot of symmetry are\n\n• Contact toric manifolds [26];\n\n• Prequantization circle bundles of coadjoint orbits of Lie Groups. Every compact contact manifold that admits a transitive action of a compact Lie group by coorientation preserving contactomorphisms lies in the second family above [4]. These families of manifolds constitute a good playground for contact topology. It can be interesting to compute their contact topological invariants.\n\nY. Eliashberg: Take a complex line bundle L over an integral symplectic manifold (M, ω ) with first Chern class c1(L) = n ω. The associated circle bundle V is a contact manifold. Suppose n\n\nis large, is V Stein fillable? F. Presas asks why n is supposed to be large in this problem, and Y. Eliashberg explains that it corresponds somehow to the fact that we need very ample divisor instead of just ample. Without n being large, it might be only symplectically fillable but not 4Stein fillable. E. Giroux suggests that in the case where M is a torus, the cohomology ring of V\n\ncan be an obstruction to fillability (compare [39]). M. Abouzaid asks if there are obstruction for an algebraic variety to be realized as a divisor. After some discussion, it turns out that the question has already been considered at least for hyperplane sections in [46].\n\n1.14 Symplectization\n\nF. Presas: Suppose two contact manifolds have symplectomorphic symplectizations, are they contactomorphic? Maybe, assume in addition that the manifolds are simply connected.\n\n1.15 Sasakian manifolds\n\nK. Honda: Is there anything symplectic geometry can say about Sasakian manifolds [8]? R. Komendarczyk says that many things are known in dimension 3. In all dimensions, Sasakian manifolds are fillable (see [40, 35]). A. Mori points out that there is also a theorem of D. Martínez Torres [27] that every Sasakian manifold M 2n+1 admits a contact immersion into (S4n+3, ξ std )\n\nwhich pulls back the standard open book of S4n+3 to a supporting open book of M.\n\nF. Presas: A simply-connected closed manifold is formal over rational (resp. real) numbers if its rational (resp. real) homotopy type can be recovered from its cohomology ring. Simply-connected closed orientable manifolds of dimension ≤ 6 are formal, and there are examples of non-formal simply-connected manifolds in any dimension ≥ 7 (see [17] and the references therein). It is known [42] that Sasakian manifolds are formal (for real homotopy type). This leads us to the question of a producing non-formal contact manifolds. For example, are there non-formal simply-connected closed contact manifolds (of dim necessarily ≥ 7)? (see [18, 3] for related work).\n\n1.16 Exotic spheres and contact geometry\n\nP. Massot: Let Σn be an exotic sphere. Is (ST ∗Σn, ξ std ) contactomorphic to (ST ∗Sn, ξ std )?Compare this also to the related result for cotangent bundles [1].",
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   "original_statement": "2 Tuesday \n\n2.1 Metrics on contactomorphism group \n\nM. Sandon: There is an integer-valued biinvariant metric on the universal cover of the contac-tomorphism group of any contact manifold, which was recently constructed by M. Sandon and V. Colin in [10]. It is called the discriminant metric. Can you find examples of contact manifolds for which this metric is unbounded? We already know that it is bounded for standard S2n+1 and \n\nR2n+1 and unbounded for RP 2n+1 and R2n × S1. Are there necessary or sufficient conditions for this metric to be unbounded? Having a 1-periodic Reeb flow is not sufficient, but maybe one only needs to add the hypothesis that Reeb orbits are non contractible. Is it compatible with the partial order constructed in [15]? \n\nV. Colin: If there are no contractible Reeb orbits, are Reeb flows geodesics in the contactomor-phism group with respect to the discriminant metric (meaning length-minimizing path)? There is also a metric for Legendrian isotopies (in fact the metric on contactomorphism group comes from this.) In the case of T 2 × [− π \n\n> 2, π \n\n> 2\n\n], with contact structure ker (cos( t)d x − sin( t)d y),take the Legendrian circle {y = 0 } in T 2 × { 0} and the isotopy that rotates this in the y direction \n\nn times. The length of this Legendrian isotopy with respect to the discriminant metric is exactly 5n (see [10]). What happens for the length of this isotopy if we replace T 2 × [− π \n\n> 2, π \n\n> 2\n\n] by T 2 × R?Intuitively, it should be the same result but there is no proof at present. E. Giroux asks if we know something in the overtwisted case, for example if T 2 is the boundary of a Lutz tube. Again, it is not known. P. Massot and C. Wendl discuss also that there may be higher dimensional analogues of this question. \n\nM. Fraser: There is also a metric constructed by M. Fraser and L. Polterovich, and yet another one by F. Zapolsky [45]. How do they relate to each other? Are they quasi-isometric? \n\n2.2 Lagrangian concordance \n\nY. Eliashberg: Let L ⊂ (M, ξ ) a Legendrian submanifold. Take an exact Lagrangian concor-dance Λ in the symplectization SM of M between L at the top and another Legendrian at the bottom. Then the Liouville form restricted to Λ writes df for some function f: Λ → R which is constant on L and uniquely defined by imposing that this constant is zero. How large can f\n\nbe at the bottom? Is there a bound? Is it always unbounded? M. Abouzaid remarks that if the Reeb flow on M is complete (for example if M is a closed manifold), then flowing L along the Reeb flow while moving down in the symplectization yields f as large as we want at the bot-tom. However the question is interesting for manifolds with non complete Reeb flow, typically the complement of a Legendrian submanifold in a contact manifold. V. Colin points out that this might be related to the previous question about length of contact isotopies with respect to metrics on the contactomorphism group. \n\n2.3 Loose Legendrians \n\nE. Murphy: We know that the space of loose Legendrians is C0-dense in the space of Legen-drians (see [33]). Is it also C0-open? That is, if we take a loose Legendrian and we C0-perturb it, is it still loose? V. Colin remarks that in dimension 3, all knots C0-close to a stabilized one are also stabilized, which is the 3-dimensional analogue of the question. \n\nA. Mori: He explains that Lutz-Mori tubes can be deformed into foliations [31], and asks whether this may give restrictions on loose knots. \n\n2.4 Open book decompositions \n\nC. Wendl: Given a contact manifold (M, ξ ), what are the constraints on open books supporting \n\nξ? For example, M. Abouzaid formulates something vague like, given an abstract open book decomposition of a manifold with pages admitting Weinstein structure, can it support a given contact structure? F. Presas points out that it is related to the following problem: given a diffeomorphism of a Weinstein manifold which is the identity near the boundary, can it be deformed into a symplectic diffeomorphism among diffeomorphism that are the identity on the boundary? Of course a positive answer is rather unlikely and would lead to existence of contact structures in higher dimensions. \n\nE. Giroux: gives an example of this problem: the group π0 Diff( D6, ∂D 6) has 28 connected components [24, 9], each of which gives an open book for the corresponding exotic 7sphere. Can these diffeomorphisms be deformed to symplectic diffeomorphisms? We can also ask the question for higher dimensional balls. \n\nO. van Koert: Take T ∗S2 with even multiples of the right-handed Dehn twist τ, it gives an infinite family of contact manifolds Mk = OB (T ∗S2, τ 2k) for all k ≥ 1. These manifolds are diffeomorphic to S2 × S3, the contact structures are homotopic as almost contact structures and all have the same contact homology. Are they contactomorphic? E. Giroux also asks the question for negative k.62.5 Contact structures on S5\n\nE. Giroux: How many different contact structures do we know on S5? O. van Koert explains that Brieskorn spheres provide an infinite family [43]. Then it was shown that a connected sum of Brieskorn spheres is no longer a Brieskorn sphere [44], so this produces new ones. M. McLean also points out that there are infinitely many different symplectic balls D6, and their boundaries are very likely to be non contactomorphic (but it is not yet proved). We know that there is at least one non-standard S5 in this family because it has exponential growth of periodic Reeb orbits, and therefore cannot be the standard contact sphere. There is also a non-fillable contact structure on any sphere constructed in [37].",
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   "public_statement": "2 Tuesday\n\n2.1 Metrics on contactomorphism group\n\nM. Sandon: There is an integer-valued biinvariant metric on the universal cover of the contac-tomorphism group of any contact manifold, which was recently constructed by M. Sandon and V. Colin in [10]. It is called the discriminant metric. Can you find examples of contact manifolds for which this metric is unbounded? We already know that it is bounded for standard S2n+1 and\n\nR2n+1 and unbounded for RP 2n+1 and R2n × S1. Are there necessary or sufficient conditions for this metric to be unbounded? Having a 1-periodic Reeb flow is not sufficient, but maybe one only needs to add the hypothesis that Reeb orbits are non contractible. Is it compatible with the partial order constructed in [15]?\n\nV. Colin: If there are no contractible Reeb orbits, are Reeb flows geodesics in the contactomor-phism group with respect to the discriminant metric (meaning length-minimizing path)? There is also a metric for Legendrian isotopies (in fact the metric on contactomorphism group comes from this.) In the case of T 2 × [− π\n\n> 2, π\n\n> 2\n\n], with contact structure ker (cos( t)d x − sin( t)d y),take the Legendrian circle {y = 0 } in T 2 × { 0} and the isotopy that rotates this in the y direction\n\nn times. The length of this Legendrian isotopy with respect to the discriminant metric is exactly 5n (see [10]). What happens for the length of this isotopy if we replace T 2 × [− π\n\n> 2, π\n\n> 2\n\n] by T 2 × R?Intuitively, it should be the same result but there is no proof at present. E. Giroux asks if we know something in the overtwisted case, for example if T 2 is the boundary of a Lutz tube. Again, it is not known. P. Massot and C. Wendl discuss also that there may be higher dimensional analogues of this question.\n\nM. Fraser: There is also a metric constructed by M. Fraser and L. Polterovich, and yet another one by F. Zapolsky [45]. How do they relate to each other? Are they quasi-isometric?\n\n2.2 Lagrangian concordance\n\nY. Eliashberg: Let L ⊂ (M, ξ ) a Legendrian submanifold. Take an exact Lagrangian concor-dance Λ in the symplectization SM of M between L at the top and another Legendrian at the bottom. Then the Liouville form restricted to Λ writes df for some function f: Λ → R which is constant on L and uniquely defined by imposing that this constant is zero. How large can f\n\nbe at the bottom? Is there a bound? Is it always unbounded? M. Abouzaid remarks that if the Reeb flow on M is complete (for example if M is a closed manifold), then flowing L along the Reeb flow while moving down in the symplectization yields f as large as we want at the bot-tom. However the question is interesting for manifolds with non complete Reeb flow, typically the complement of a Legendrian submanifold in a contact manifold. V. Colin points out that this might be related to the previous question about length of contact isotopies with respect to metrics on the contactomorphism group.\n\n2.3 Loose Legendrians\n\nE. Murphy: We know that the space of loose Legendrians is C0-dense in the space of Legen-drians (see [33]). Is it also C0-open? That is, if we take a loose Legendrian and we C0-perturb it, is it still loose? V. Colin remarks that in dimension 3, all knots C0-close to a stabilized one are also stabilized, which is the 3-dimensional analogue of the question.\n\nA. Mori: He explains that Lutz-Mori tubes can be deformed into foliations [31], and asks whether this may give restrictions on loose knots.\n\n2.4 Open book decompositions\n\nC. Wendl: Given a contact manifold (M, ξ ), what are the constraints on open books supporting\n\nξ? For example, M. Abouzaid formulates something vague like, given an abstract open book decomposition of a manifold with pages admitting Weinstein structure, can it support a given contact structure? F. Presas points out that it is related to the following problem: given a diffeomorphism of a Weinstein manifold which is the identity near the boundary, can it be deformed into a symplectic diffeomorphism among diffeomorphism that are the identity on the boundary? Of course a positive answer is rather unlikely and would lead to existence of contact structures in higher dimensions.\n\nE. Giroux: gives an example of this problem: the group π0 Diff( D6, ∂D 6) has 28 connected components [24, 9], each of which gives an open book for the corresponding exotic 7sphere. Can these diffeomorphisms be deformed to symplectic diffeomorphisms? We can also ask the question for higher dimensional balls.\n\nO. van Koert: Take T ∗S2 with even multiples of the right-handed Dehn twist τ, it gives an infinite family of contact manifolds Mk = OB (T ∗S2, τ 2k) for all k ≥ 1. These manifolds are diffeomorphic to S2 × S3, the contact structures are homotopic as almost contact structures and all have the same contact homology. Are they contactomorphic? E. Giroux also asks the question for negative k.62.5 Contact structures on S5\n\nE. Giroux: How many different contact structures do we know on S5? O. van Koert explains that Brieskorn spheres provide an infinite family [43]. Then it was shown that a connected sum of Brieskorn spheres is no longer a Brieskorn sphere [44], so this produces new ones. M. McLean also points out that there are infinitely many different symplectic balls D6, and their boundaries are very likely to be non contactomorphic (but it is not yet proved). We know that there is at least one non-standard S5 in this family because it has exponential growth of periodic Reeb orbits, and therefore cannot be the standard contact sphere. There is also a non-fillable contact structure on any sphere constructed in [37].",
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   "original_statement": "3 Friday \n\n3.1 Lagrangian caps \n\nY. Eliashberg: Explore relation between Lagrangian caps and closed immersed Lagrangians. See the work of D. Sauvaget on closed immersed Lagrangians [41]. \n\n3.2 Plastikstufe \n\nJ. Etnyre: Suppose (M 2n−1, ξ ) contains a Plastikstufe [34] with some core Bn−1, can we also find a Plastikstufe with core T n−1? Can any manifold B′ be realized as the core of a Plastikstufe in M?\n\n3.3 Loose knots \n\nK. Niederkrüger: Take M = Not ×D2 \n\n> R\n\nwith contact form αot +r2 dθ where αot is an overtwisted contact form on N and (r, θ ) are polar coordinates on the disc DR of radius R. What is the influence of R on the looseness of knots in M? (see [36]) \n\n3.4 Contact bundles and contactomorphism group \n\nE. Giroux: Let (N, ξ ) be a contact manifold and S a surface. We denote by G the contactomor-phism group of N and by ˜G its universal cover. Given a contact bundle M → S with fiber (N, ξ ),can we construct a contact structure on M inducing the given contact structure on the fibers? Assuming triviality of the bundle on the 1-skeleton, the bundle is described by an element of \n\nγ ∈ π1(G). Then the question becomes: can we find a product of commutators ∏2gi=1 [ϕi, ψ i] in ˜G\n\nbigger (maybe smaller depending on conventions) than γ? L. Polterovich says it seems possible for S3 and Y. Eliashberg says it should be true for orderable contact manifold. E. Giroux asks for explicit constructions of big products of commutators in ˜G. Is there a bound on the length of such a product of commutators coming from a quasimorphism on ˜G?\n\nL. Polterovich: There is a related question in the symplectic case. Can we find a commutator in Ham( M, ω ) with arbitrary large Hofer norm? \n\n3.5 Convex hypersurfaces \n\nK. Honda: Let N 2n−1 be a contact submanifold of (M 2n+1, ξ ) with trivial normal bundle. Can the boundary of a tubular neighborhood of N be perturbed to a convex hypersurface? Y. Eliashberg suggests to try on S3 ⊂ S5 and in higher codimension, for example for S1 ⊂\n\n(M 5, ξ ).73.6 Submanifolds with Legendrian foliations \n\nK. Niederkrüger: Let (M 2n+1, ξ ) be a contact manifold, develop tools to find submanifolds \n\nN n+1 in M with Legendrian foliation in view of applying holomorphic techniques. Y. Eliashberg adds that the foliation has to be given by a closed 1-form to control the behavior of holomorphic curves. \n\n3.7 Liouville domain with disconnected boundary \n\nC. Wendl: A Stein domain of (real) dimension 2n admits a handle decomposition with handles only of index n and lower. This is why such a manifold will always have connected boundary if its dimension is at least 4. Liouville manifolds with disconnected boundary however do exist. Examples have been constructed in dimension 4 [29], 6 [19] and then in any dimension [28] but they are still rare. Can we develop methods for finding more Liouville domains with disconnected boundary? \n\n3.8 Contact structures on exotic spheres \n\nY. Eliashberg: Let (ft)t∈S1 be a loop of diffeomorphisms of S2n−1 based at the identity, and \n\nF the diffeomorphism of U = S2n−1 × [0, 1] given by: \n\nF (t, x ) = ( ft(x), t )\n\nLet UF = U × [0, 1] /(x,t, 1) ∼(F (x,t ),0) be the associated mapping torus, it has two boundary component diffeomorphic to S2n−1 ×S1, fill in one of these by attaching S2n−1 ×D2 to get M 2n+1.Can we construct a contact structure on M? Denoting by ξ the standard contact structure on \n\nS2n−1, f ∗ \n\n> t\n\nξ is a loop of contact structure on S2n−1. If this loop is contractible then ft is isotopic to a loop in Aut( S2n−1, ξ ) and you get a contact open book on M. A related question is how to construct contact structures on homotopy spheres? Do these homotopy sphere bound manifold with half-dimensional homotopy type? (it is an obvious necessary condition to admit a Stein fillable contact structure).",
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   "public_statement": "3 Friday\n\n3.1 Lagrangian caps\n\nY. Eliashberg: Explore relation between Lagrangian caps and closed immersed Lagrangians. See the work of D. Sauvaget on closed immersed Lagrangians [41].\n\n3.2 Plastikstufe\n\nJ. Etnyre: Suppose (M 2n−1, ξ ) contains a Plastikstufe [34] with some core Bn−1, can we also find a Plastikstufe with core T n−1? Can any manifold B′ be realized as the core of a Plastikstufe in M?\n\n3.3 Loose knots\n\nK. Niederkrüger: Take M = Not ×D2\n\n> R\n\nwith contact form αot +r2 dθ where αot is an overtwisted contact form on N and (r, θ ) are polar coordinates on the disc DR of radius R. What is the influence of R on the looseness of knots in M? (see [36])\n\n3.4 Contact bundles and contactomorphism group\n\nE. Giroux: Let (N, ξ ) be a contact manifold and S a surface. We denote by G the contactomor-phism group of N and by ˜G its universal cover. Given a contact bundle M → S with fiber (N, ξ ),can we construct a contact structure on M inducing the given contact structure on the fibers? Assuming triviality of the bundle on the 1-skeleton, the bundle is described by an element of\n\nγ ∈ π1(G). Then the question becomes: can we find a product of commutators ∏2gi=1 [ϕi, ψ i] in ˜G\n\nbigger (maybe smaller depending on conventions) than γ? L. Polterovich says it seems possible for S3 and Y. Eliashberg says it should be true for orderable contact manifold. E. Giroux asks for explicit constructions of big products of commutators in ˜G. Is there a bound on the length of such a product of commutators coming from a quasimorphism on ˜G?\n\nL. Polterovich: There is a related question in the symplectic case. Can we find a commutator in Ham( M, ω ) with arbitrary large Hofer norm?\n\n3.5 Convex hypersurfaces\n\nK. Honda: Let N 2n−1 be a contact submanifold of (M 2n+1, ξ ) with trivial normal bundle. Can the boundary of a tubular neighborhood of N be perturbed to a convex hypersurface? Y. Eliashberg suggests to try on S3 ⊂ S5 and in higher codimension, for example for S1 ⊂\n\n(M 5, ξ ).73.6 Submanifolds with Legendrian foliations\n\nK. Niederkrüger: Let (M 2n+1, ξ ) be a contact manifold, develop tools to find submanifolds\n\nN n+1 in M with Legendrian foliation in view of applying holomorphic techniques. Y. Eliashberg adds that the foliation has to be given by a closed 1-form to control the behavior of holomorphic curves.\n\n3.7 Liouville domain with disconnected boundary\n\nC. Wendl: A Stein domain of (real) dimension 2n admits a handle decomposition with handles only of index n and lower. This is why such a manifold will always have connected boundary if its dimension is at least 4. Liouville manifolds with disconnected boundary however do exist. Examples have been constructed in dimension 4 [29], 6 [19] and then in any dimension [28] but they are still rare. Can we develop methods for finding more Liouville domains with disconnected boundary?\n\n3.8 Contact structures on exotic spheres\n\nY. Eliashberg: Let (ft)t∈S1 be a loop of diffeomorphisms of S2n−1 based at the identity, and\n\nF the diffeomorphism of U = S2n−1 × [0, 1] given by:\n\nF (t, x ) = ( ft(x), t )\n\nLet UF = U × [0, 1] /(x,t, 1) ∼(F (x,t ),0) be the associated mapping torus, it has two boundary component diffeomorphic to S2n−1 ×S1, fill in one of these by attaching S2n−1 ×D2 to get M 2n+1.Can we construct a contact structure on M? Denoting by ξ the standard contact structure on\n\nS2n−1, f ∗\n\n> t\n\nξ is a loop of contact structure on S2n−1. If this loop is contractible then ft is isotopic to a loop in Aut( S2n−1, ξ ) and you get a contact open book on M. A related question is how to construct contact structures on homotopy spheres? Do these homotopy sphere bound manifold with half-dimensional homotopy type? (it is an obvious necessary condition to admit a Stein fillable contact structure).",
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   "original_statement": "Problem 1. Does the Grove-Petersen-Wu Finiteness Theorem hold in dimension 4, i.e. are there only finitely many diffeomorphism types in the class of 4-dimensional Riemannian manifolds satisfying sec M ≥ − Λ2, diam( M ) ≤ D, Vol( M ) ≥ V?",
   "clean_statement": "Problem 1. Does the Grove-Petersen-Wu Finiteness Theorem hold in dimension 4, i.e. are there only finitely many diffeomorphism types in the class of 4-dimensional Riemannian manifolds satisfying sec M ≥ − Λ2, diam( M ) ≤ D, Vol( M ) ≥ V?",
   "public_statement": "Problem 1. Does the Grove-Petersen-Wu Finiteness Theorem hold in dimension 4, i.e. are there only finitely many diffeomorphism types in the class of 4-dimensional Riemannian manifolds satisfying sec M ≥ − Λ2, diam( M ) ≤ D, Vol( M ) ≥ V?",
   "evidence": "The official AIM PDF, *Open Problems in Non-Negative Sectional Curvature*, was compiled by M. Kerin after the September 2007 AIM workshop. Its first problem reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 213,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0215": {
   "statement_status": "exact",
   "original_statement": "Problem 2. If sec M ≥ 1 and diam( M ) > π \n\n> 2, must M be diffeomorphic to Sn?",
   "clean_statement": "Problem 2. If sec M ≥ 1 and diam( M ) > π\n\n> 2, must M be diffeomorphic to Sn?",
   "public_statement": "Problem 2. If sec M ≥ 1 and diam( M ) > π\n\n> 2, must M be diffeomorphic to Sn?",
   "evidence": "The canonical JSON record contains an OCR line break, `diam(M) > π > 2`. The original AIM PDF, *Open Problems in Non-negative Sectional Curvature*, compiled by M. Kerin after the September 2007 AIM workshop, gives the unambiguous statement in its section “Diameter Pinching”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 214,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0216": {
   "statement_status": "exact",
   "original_statement": "Problem 3. Let M n be a manifold for which sec M ≥ 1 and diam( M ) ≥ π \n\n> 2\n\n− ε(n). Is M\n\nhomeomorphic to a manifold M ′, where sec M ′ ≥ 1, diam( M ′) ≥ π \n\n> 2? This problem may be easier to address if we also assume that Vol( M ) ≥ V and ε = ε(n, V ). 2. Collapse and Alexandrov Geometry",
   "clean_statement": "Problem 3. Let M n be a manifold for which sec M ≥ 1 and diam( M ) ≥ π\n\n> 2\n\n− ε(n). Is M\n\nhomeomorphic to a manifold M ′, where sec M ′ ≥ 1, diam( M ′) ≥ π\n\n> 2? This problem may be easier to address if we also assume that Vol( M ) ≥ V and ε = ε(n, V ). 2. Collapse and Alexandrov Geometry",
   "public_statement": "Problem 3. Let M n be a manifold for which sec M ≥ 1 and diam( M ) ≥ π\n\n> 2\n\n− ε(n). Is M\n\nhomeomorphic to a manifold M ′, where sec M ′ ≥ 1, diam( M ′) ≥ π\n\n> 2? This problem may be easier to address if we also assume that Vol( M ) ≥ V and ε = ε(n, V ). 2. Collapse and Alexandrov Geometry",
   "evidence": "The canonical JSON record has line-break OCR damage in the fractions and has accidentally appended the next section heading. Page 1 of the official AIM workshop PDF gives the following text:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 215,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0217": {
   "statement_status": "exact",
   "original_statement": "Problem 4. Perelman's Stability Theorem yields that manifolds in a given sequence of non-collapsing manifolds are eventually pairwise homeomorphic. Are they also PL-homeomorphic or diffeomorphic?",
   "clean_statement": "Problem 4. Perelman's Stability Theorem yields that manifolds in a given sequence of non-collapsing manifolds are eventually pairwise homeomorphic. Are they also PL-homeomorphic or diffeomorphic?",
   "public_statement": "Problem 4. Perelman's Stability Theorem yields that manifolds in a given sequence of non-collapsing manifolds are eventually pairwise homeomorphic. Are they also PL-homeomorphic or diffeomorphic?",
   "evidence": "The AIM source states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 216,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0218": {
   "statement_status": "exact",
   "original_statement": "Problem 5. Understand DC-structures on manifolds. In particular, does Perelman's Sta-bility Theorem hold in the DC-category. Is PL = DC always?",
   "clean_statement": "Problem 5. Understand DC-structures on manifolds. In particular, does Perelman's Sta-bility Theorem hold in the DC-category. Is PL = DC always?",
   "public_statement": "Problem 5. Understand DC-structures on manifolds. In particular, does Perelman's Sta-bility Theorem hold in the DC-category. Is PL = DC always?",
   "evidence": "The canonical record is Problem 5 in the AIM workshop list *Manifolds with nonnegative sectional curvature*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 217,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0219": {
   "statement_status": "exact",
   "original_statement": "Problem 6. Extend the Wilking Connectivity Theorem to Alexandrov spaces, i.e. if X is a positively curved Alexandrov space and Y ⊂ X a totally geodesic subspace of codimension \n\nk, is it true that X − Y has homology only up to dimension 2 k − 2?",
   "clean_statement": "Problem 6. Extend the Wilking Connectivity Theorem to Alexandrov spaces, i.e. if X is a positively curved Alexandrov space and Y ⊂ X a totally geodesic subspace of codimension\n\nk, is it true that X − Y has homology only up to dimension 2 k − 2?",
   "public_statement": "Problem 6. Extend the Wilking Connectivity Theorem to Alexandrov spaces, i.e. if X is a positively curved Alexandrov space and Y ⊂ X a totally geodesic subspace of codimension\n\nk, is it true that X − Y has homology only up to dimension 2 k − 2?",
   "evidence": "The official AIM list states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 218,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0220": {
   "statement_status": "exact",
   "original_statement": "Problem 7. Suppose X is the non-collapsed Gromov-Hausdorff limit of ( M ni, g i), where \n\n| sec Mi | ≤ 1, and that along every geodesic on Mi one hits a conjugate point before \n\nt = π + 1 \n\n> i. Is X rigid in any sense?",
   "clean_statement": "Problem 7. Suppose X is the non-collapsed Gromov-Hausdorff limit of ( M ni, g i), where\n\n| sec Mi | ≤ 1, and that along every geodesic on Mi one hits a conjugate point before\n\nt = π + 1\n\n> i. Is X rigid in any sense?",
   "public_statement": "Problem 7. Suppose X is the non-collapsed Gromov-Hausdorff limit of ( M ni, g i), where\n\n| sec Mi | ≤ 1, and that along every geodesic on Mi one hits a conjugate point before\n\nt = π + 1\n\n> i. Is X rigid in any sense?",
   "evidence": "The canonical JSON has two OCR errors. The official 2007 AIM problem list reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 219,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0221": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 8. Is there a sequence of simply-connected, pointwise strictly 14 -pinched manifolds \n\nM ni, n > 2, that collapse?",
   "clean_statement": "**Problem 8.** Is there a sequence of simply-connected, pointwise strictly \\(\\frac14\\)-pinched manifolds \\(M_i^n\\), \\(n>2\\), that collapse?",
   "public_statement": "**Problem 8.** Is there a sequence of simply-connected, pointwise strictly \\(\\frac14\\)-pinched manifolds \\(M_i^n\\), \\(n>2\\), that collapse?",
   "evidence": "The official AIM PDF, in Section 2 (“Collapse and Alexandrov Geometry”), reads: Thus `14 -pinched` in the extracted record is an OCR loss of the fraction \\(\\frac14\\), and `M ni` is \\(M_i^n\\). The correction is verified from the PDF and does not modify the canonical input.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-topology-notes.json",
   "source_index": 220,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0222": {
   "statement_status": "exact",
   "original_statement": "Problem 9. Find an appropriate definition of Morse functions on Alexandrov spaces and construct examples. \n\n> Date: October 26, 2007.\n> 12COMPILED BY M. KERIN",
   "clean_statement": "Problem 9. Find an appropriate definition of Morse functions on Alexandrov spaces and construct examples.\n\n> Date: October 26, 2007.\n> 12COMPILED BY M. KERIN",
   "public_statement": "Problem 9. Find an appropriate definition of Morse functions on Alexandrov spaces and construct examples.\n\n> Date: October 26, 2007.\n> 12COMPILED BY M. KERIN",
   "evidence": "The official AIM PDF contains exactly the mathematical sentence",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 221,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0223": {
   "statement_status": "exact",
   "original_statement": "Problem 10. Study the collapse of Alexandrov spaces.",
   "clean_statement": "Problem 10. Study the collapse of Alexandrov spaces.",
   "public_statement": "Problem 10. Study the collapse of Alexandrov spaces.",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 222,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0224": {
   "statement_status": "exact",
   "original_statement": "Problem 11. Consider finite towers \n\nM0 \n\n> F1//\n\nM1 \n\n> F2//... Fk / / Mk\n\nof fiber bundles, where the fibers {F1,..., F k} and Mk are fixed topological manifolds. Loosen this notion to get a \"brotherhood\" on the manifolds M0 and a property of such \n\nM0 not known to be possessed by all manifolds of sec ≥ K, diam ≤ 1.",
   "clean_statement": "Problem 11. Consider finite towers\n\nM0\n\n> F1//\n\nM1\n\n> F2//... Fk / / Mk\n\nof fiber bundles, where the fibers {F1,..., F k} and Mk are fixed topological manifolds. Loosen this notion to get a \"brotherhood\" on the manifolds M0 and a property of such\n\nM0 not known to be possessed by all manifolds of sec ≥ K, diam ≤ 1.",
   "public_statement": "Problem 11. Consider finite towers\n\nM0\n\n> F1//\n\nM1\n\n> F2//... Fk / / Mk\n\nof fiber bundles, where the fibers {F1,..., F k} and Mk are fixed topological manifolds. Loosen this notion to get a \"brotherhood\" on the manifolds M0 and a property of such\n\nM0 not known to be possessed by all manifolds of sec ≥ K, diam ≤ 1.",
   "evidence": "The canonical record is Problem 11 from the AIM workshop list *Open Problems in Non-negative Sectional Curvature*. The PDF extraction has broken the diagram across lines. Inspection of the official PDF recovers it as a tower",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 223,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0225": {
   "statement_status": "exact",
   "original_statement": "Problem 12. Give an Alexandrov analogue of rational ellipticity. In particular, are mani-folds with almost non-negative sectional curvature rationally elliptic?",
   "clean_statement": "Problem 12. Give an Alexandrov analogue of rational ellipticity. In particular, are mani-folds with almost non-negative sectional curvature rationally elliptic?",
   "public_statement": "Problem 12. Give an Alexandrov analogue of rational ellipticity. In particular, are mani-folds with almost non-negative sectional curvature rationally elliptic?",
   "evidence": "The official AIM PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 224,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0226": {
   "statement_status": "exact",
   "original_statement": "Problem 13. Given a non-collapsing Gromov-Hausdorff convergence Mi −→ X, can one find a \"tangent bundle\" structure on X that is sensitive to the diffeomorphism class of the Mi?",
   "clean_statement": "Problem 13. Given a non-collapsing Gromov-Hausdorff convergence Mi −→ X, can one find a \"tangent bundle\" structure on X that is sensitive to the diffeomorphism class of the Mi?",
   "public_statement": "Problem 13. Given a non-collapsing Gromov-Hausdorff convergence Mi −→ X, can one find a \"tangent bundle\" structure on X that is sensitive to the diffeomorphism class of the Mi?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 225,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0227": {
   "statement_status": "exact",
   "original_statement": "Problem 14. Is there a Gauss formula for Alexandrov spaces, i.e. must a convex hyper-surface Y of an Alexandrov space X have sec Y ≥ sec X?",
   "clean_statement": "Problem 14. Is there a Gauss formula for Alexandrov spaces, i.e. must a convex hyper-surface Y of an Alexandrov space X have sec Y ≥ sec X?",
   "public_statement": "Problem 14. Is there a Gauss formula for Alexandrov spaces, i.e. must a convex hyper-surface Y of an Alexandrov space X have sec Y ≥ sec X?",
   "evidence": "The official AIM PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 226,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0228": {
   "statement_status": "exact",
   "original_statement": "Problem 15. Is every finite dimensional Alexandrov space a limit of Riemannian manifolds with sec ≥ K?",
   "clean_statement": "Problem 15. Is every finite dimensional Alexandrov space a limit of Riemannian manifolds with sec ≥ K?",
   "public_statement": "Problem 15. Is every finite dimensional Alexandrov space a limit of Riemannian manifolds with sec ≥ K?",
   "evidence": "The official AIM PDF states, without OCR ambiguity:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 227,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0229": {
   "statement_status": "exact",
   "original_statement": "Problem 16. Study Alexandrov (almost) submetries.",
   "clean_statement": "Problem 16. Study Alexandrov (almost) submetries.",
   "public_statement": "Problem 16. Study Alexandrov (almost) submetries.",
   "evidence": "The same sentence appears in the official AIM PDF. Thus there is no apparent OCR error to repair. The wording is nevertheless deliberately broad: it specifies neither a theorem to prove nor a definition of “almost submetry.” Here “Alexandrov” is read as referring to maps involving Alexandrov spaces, while the metric lemmas below are stated for arbitrary metric spaces and hence apply to that setting.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 228,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0230": {
   "statement_status": "exact",
   "original_statement": "Problem 17. Is there an alternate approach to homotopy groups that is adapted to Alexan-drov spaces?",
   "clean_statement": "Problem 17. Is there an alternate approach to homotopy groups that is adapted to Alexan-drov spaces?",
   "public_statement": "Problem 17. Is there an alternate approach to homotopy groups that is adapted to Alexan-drov spaces?",
   "evidence": "The official AIM PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 229,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0231": {
   "statement_status": "exact",
   "original_statement": "Problem 18. Study collapse to a ray.",
   "clean_statement": "Problem 18. Study collapse to a ray.",
   "public_statement": "Problem 18. Study collapse to a ray.",
   "evidence": "The source is M. Kerin's compilation *Open Problems in Non-negative Sectional Curvature*, assembled from the September 2007 AIM workshop *Manifolds with Non-negative Sectional Curvature*. The exact source text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 230,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0232": {
   "statement_status": "exact",
   "original_statement": "Problem 19. Can an n-dimensional torus collapse to an interval? (The answer to this question is \"No\", essentially settled at the workshop.)",
   "clean_statement": "Problem 19. Can an n-dimensional torus collapse to an interval? (The answer to this question is \"No\", essentially settled at the workshop.)",
   "public_statement": "Problem 19. Can an n-dimensional torus collapse to an interval? (The answer to this question is \"No\", essentially settled at the workshop.)",
   "evidence": "The official PDF contains the same sentence. There is no OCR error to correct. There is, however, suppressed mathematical context: “collapse” means Gromov--Hausdorff convergence of closed smooth Riemannian tori under a uniform lower sectional-curvature bound. The precise form subsequently proved is:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 231,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0233": {
   "statement_status": "exact",
   "original_statement": "Problem 20. Study the collapse of Riemannian manifolds with boundary which have sec ≥\n\nK on the interior and controlled boundary concavity.",
   "clean_statement": "Problem 20. Study the collapse of Riemannian manifolds with boundary which have sec ≥\n\nK on the interior and controlled boundary concavity.",
   "public_statement": "Problem 20. Study the collapse of Riemannian manifolds with boundary which have sec ≥\n\nK on the interior and controlled boundary concavity.",
   "evidence": "The canonical JSON record reads",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 232,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0234": {
   "statement_status": "exact",
   "original_statement": "Problem 21. Find an application where infinite-dimensional Alexandrov spaces appear as limits of manifolds of increasing dimension. 3. Group Actions and Submersions",
   "clean_statement": "Problem 21. Find an application where infinite-dimensional Alexandrov spaces appear as limits of manifolds of increasing dimension. 3. Group Actions and Submersions",
   "public_statement": "Problem 21. Find an application where infinite-dimensional Alexandrov spaces appear as limits of manifolds of increasing dimension. 3. Group Actions and Submersions",
   "evidence": "The exact extracted record in input.json reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 233,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0235": {
   "statement_status": "exact",
   "original_statement": "Problem 22. Let M n be a manifold with sec > 0 or sec ≥ 0 or almost non-negative curvature. Does M n have a positive symmetry degree, i.e. is there an S1 ⊂ Diff( M n)?",
   "clean_statement": "Problem 22. Let M n be a manifold with sec > 0 or sec ≥ 0 or almost non-negative curvature. Does M n have a positive symmetry degree, i.e. is there an S1 ⊂ Diff( M n)?",
   "public_statement": "Problem 22. Let M n be a manifold with sec > 0 or sec ≥ 0 or almost non-negative curvature. Does M n have a positive symmetry degree, i.e. is there an S1 ⊂ Diff( M n)?",
   "evidence": "The canonical AIM record, from Problem 22 of *Open Problems in Non-Negative Sectional Curvature*, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 234,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0236": {
   "statement_status": "exact",
   "original_statement": "Problem 23. Is there a principal T 2-bundle whose total space admits sec > 0?",
   "clean_statement": "Problem 23. Is there a principal T 2-bundle whose total space admits sec > 0?",
   "public_statement": "Problem 23. Is there a principal T 2-bundle whose total space admits sec > 0?",
   "evidence": "The extracted record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 235,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0237": {
   "statement_status": "exact",
   "original_statement": "Problem 24. Given a fat G-principal bundle, must G be S1, S3 or SO (3)?",
   "clean_statement": "Problem 24. Given a fat G-principal bundle, must G be S1, S3 or SO (3)?",
   "public_statement": "Problem 24. Given a fat G-principal bundle, must G be S1, S3 or SO (3)?",
   "evidence": "The exact extracted record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 236,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0238": {
   "statement_status": "exact",
   "original_statement": "Problem 25. Can one reduce the structure group of a fat principal G-bundle?",
   "clean_statement": "Problem 25. Can one reduce the structure group of a fat principal G-bundle?",
   "public_statement": "Problem 25. Can one reduce the structure group of a fat principal G-bundle?",
   "evidence": "Inspection of the official PDF confirms this exact sentence; there is no OCR corruption. It occurs in the section “Group Actions and Submersions,” immediately after Problem 24, which asks whether the group of a fat principal bundle must be \\(S^1\\), \\(S^3\\), or \\(SO(3)\\).",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 237,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0239": {
   "statement_status": "exact",
   "original_statement": "Problem 26. Given a homogeneous space G/H, with G compact, classify all homogeneous metrics with sec ≥ 0. 3",
   "clean_statement": "Problem 26. Given a homogeneous space G/H, with G compact, classify all homogeneous metrics with sec ≥ 0. 3",
   "public_statement": "Problem 26. Given a homogeneous space G/H, with G compact, classify all homogeneous metrics with sec ≥ 0. 3",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 238,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0240": {
   "statement_status": "exact",
   "original_statement": "Problem 27. Is there a positively curved 5-manifold with a free isometric S3 or SO (3) action?",
   "clean_statement": "Problem 27. Is there a positively curved 5-manifold with a free isometric S3 or SO (3) action?",
   "public_statement": "Problem 27. Is there a positively curved 5-manifold with a free isometric S3 or SO (3) action?",
   "evidence": "The canonical record is Problem 27 in the AIM list *Open Problems in Non-negative Sectional Curvature* (dated October 26, 2007):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 239,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0241": {
   "statement_status": "exact",
   "original_statement": "Problem 28. Given a Riemannian submersion with positively curved total space, is the dimension of the fiber less than the dimension of the base? It is perhaps simpler to decide if there is a bound on the dimension of the fiber in terms of the dimension of the base. Is the image of the A-tensor large at some point?",
   "clean_statement": "Problem 28. Given a Riemannian submersion with positively curved total space, is the dimension of the fiber less than the dimension of the base? It is perhaps simpler to decide if there is a bound on the dimension of the fiber in terms of the dimension of the base. Is the image of the A-tensor large at some point?",
   "public_statement": "Problem 28. Given a Riemannian submersion with positively curved total space, is the dimension of the fiber less than the dimension of the base? It is perhaps simpler to decide if there is a bound on the dimension of the fiber in terms of the dimension of the base. Is the image of the A-tensor large at some point?",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 240,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0242": {
   "statement_status": "exact",
   "original_statement": "Problem 29. Classify Riemannian submersions from a Lie group with a bi-invariant metric. (As was observed at the workshop, they are not necessarily biquotient submersions.)",
   "clean_statement": "Problem 29. Classify Riemannian submersions from a Lie group with a bi-invariant metric. (As was observed at the workshop, they are not necessarily biquotient submersions.)",
   "public_statement": "Problem 29. Classify Riemannian submersions from a Lie group with a bi-invariant metric. (As was observed at the workshop, they are not necessarily biquotient submersions.)",
   "evidence": "Source: [AIM workshop problem list](https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf), `aim-topology-notes.json`, record 241 (zero-based). The repository text agrees with the source; no OCR correction is needed. The statement is deliberately broad: it does not specify compactness, connectedness, connected fibers, or an equivalence relation.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 241,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0243": {
   "statement_status": "exact",
   "original_statement": "Problem 30. Suppose ( M n, g ) is simply-connected and sec M ≥ 0. Is rank(Iso( M n, g )) ≤ \n\n> 23\n\nn?",
   "clean_statement": "Problem 30. Suppose ( M n, g ) is simply-connected and sec M ≥ 0. Is rank(Iso( M n, g )) ≤\n\n> 23\n\nn?",
   "public_statement": "Problem 30. Suppose ( M n, g ) is simply-connected and sec M ≥ 0. Is rank(Iso( M n, g )) ≤\n\n> 23\n\nn?",
   "evidence": "The canonical extraction is visibly corrupted:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 242,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0244": {
   "statement_status": "exact",
   "original_statement": "Problem 31. Suppose we have an isometric group action on M. If one changes the metric on the orbit space, does it lift to an invariant metric on M?",
   "clean_statement": "Problem 31. Suppose we have an isometric group action on M. If one changes the metric on the orbit space, does it lift to an invariant metric on M?",
   "public_statement": "Problem 31. Suppose we have an isometric group action on M. If one changes the metric on the orbit space, does it lift to an invariant metric on M?",
   "evidence": "The source is Problem 31 in the AIM workshop list *Manifolds with nonnegative sectional curvature*. The exact database text agrees with the wording in the source PDF:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 243,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0245": {
   "statement_status": "exact",
   "original_statement": "Problem 32. If a group acts isometrically on a manifold of sec ≥ 1 and the fixed point set is a circle, is it's length ≤ 2π?4. Manifolds of Cohomogeneity-one and Polar Actions",
   "clean_statement": "Problem 32. If a group acts isometrically on a manifold of sec ≥ 1 and the fixed point set is a circle, is it's length ≤ 2π?4. Manifolds of Cohomogeneity-one and Polar Actions",
   "public_statement": "Problem 32. If a group acts isometrically on a manifold of sec ≥ 1 and the fixed point set is a circle, is it's length ≤ 2π?4. Manifolds of Cohomogeneity-one and Polar Actions",
   "evidence": "The canonical record is Problem 32 in M. Kerin's compilation of questions from the 2007 AIM workshop *Manifolds with Non-negative Sectional Curvature*. The official PDF prints:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 244,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0246": {
   "statement_status": "exact",
   "original_statement": "Problem 33. Study the existence and non-existence of metrics with non-negative curvature on manifolds of cohomogeneity-one.",
   "clean_statement": "Problem 33. Study the existence and non-existence of metrics with non-negative curvature on manifolds of cohomogeneity-one.",
   "public_statement": "Problem 33. Study the existence and non-existence of metrics with non-negative curvature on manifolds of cohomogeneity-one.",
   "evidence": "The wording was checked against the official five-page PDF. There is no apparent OCR loss or ambiguity. There is an important scope distinction: if one starts with a smooth manifold carrying a cohomogeneity-one action, existence of an arbitrary non-invariant metric with sectional curvature at least zero is different from existence of an invariant one. Averaging does not preserve sectional curvature. This report concerns invariant metrics, as do the principal existence and obstruction results cited below.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 245,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0247": {
   "statement_status": "exact",
   "original_statement": "Problem 34. Find cohomogeneity-one manifolds with \"interesting\" topology, in particular not homeomorphic to a symmetric space. Study curvature properties of these manifolds.",
   "clean_statement": "Problem 34. Find cohomogeneity-one manifolds with \"interesting\" topology, in particular not homeomorphic to a symmetric space. Study curvature properties of these manifolds.",
   "public_statement": "Problem 34. Find cohomogeneity-one manifolds with \"interesting\" topology, in particular not homeomorphic to a symmetric space. Study curvature properties of these manifolds.",
   "evidence": "The canonical AIM record is Problem 34 from the workshop *Manifolds with nonnegative sectional curvature*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 246,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0248": {
   "statement_status": "exact",
   "original_statement": "Problem 35. Classify cohomogeneity-one manifolds with sec ≥ 0 and at least one totally geodesic principal orbit.",
   "clean_statement": "Problem 35. Classify cohomogeneity-one manifolds with sec ≥ 0 and at least one totally geodesic principal orbit.",
   "public_statement": "Problem 35. Classify cohomogeneity-one manifolds with sec ≥ 0 and at least one totally geodesic principal orbit.",
   "evidence": "The official AIM PDF, in Section 4 (*Manifolds of Cohomogeneity-one and Polar Actions*), states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 247,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0249": {
   "statement_status": "exact",
   "original_statement": "Problem 36. Compute topological invariants of cohomogeneity-one manifolds. Classify topologically the new candidates for positive curvature.",
   "clean_statement": "Problem 36. Compute topological invariants of cohomogeneity-one manifolds. Classify topologically the new candidates for positive curvature.",
   "public_statement": "Problem 36. Compute topological invariants of cohomogeneity-one manifolds. Classify topologically the new candidates for positive curvature.",
   "evidence": "This wording was checked against the official five-page PDF compiled by M. Kerin after the September 2007 AIM workshop. The extraction is exact; no OCR correction or reconstruction is needed.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 248,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0250": {
   "statement_status": "exact",
   "original_statement": "Problem 37. Study the existence of Einstein metrics on manifolds of cohomogeneity-one.",
   "clean_statement": "Problem 37. Study the existence of Einstein metrics on manifolds of cohomogeneity-one.",
   "public_statement": "Problem 37. Study the existence of Einstein metrics on manifolds of cohomogeneity-one.",
   "evidence": "The AIM list *Manifolds with nonnegative sectional curvature* states, as Problem 37:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 249,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0251": {
   "statement_status": "exact",
   "original_statement": "Problem 38. Suppose M is a polar manifold with sec > 0. Must M be diffeomorphic to a compact rank-one symmetric space?",
   "clean_statement": "Problem 38. Suppose M is a polar manifold with sec > 0. Must M be diffeomorphic to a compact rank-one symmetric space?",
   "public_statement": "Problem 38. Suppose M is a polar manifold with sec > 0. Must M be diffeomorphic to a compact rank-one symmetric space?",
   "evidence": "The canonical record is Problem 38 in the AIM workshop list *Manifolds with nonnegative sectional curvature*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 250,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0252": {
   "statement_status": "exact",
   "original_statement": "Problem 39. Suppose Σ ⊂ M n is the section of a polar action. If Σ is rationally elliptic, must M n be rationally elliptic? 5. Vector Bundles",
   "clean_statement": "Problem 39. Suppose Σ ⊂ M n is the section of a polar action. If Σ is rationally elliptic, must M n be rationally elliptic? 5. Vector Bundles",
   "public_statement": "Problem 39. Suppose Σ ⊂ M n is the section of a polar action. If Σ is rationally elliptic, must M n be rationally elliptic? 5. Vector Bundles",
   "evidence": "The canonical JSON record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 251,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0253": {
   "statement_status": "exact",
   "original_statement": "Problem 40. Is there a metric with sec ≥ 0 on R6-bundles over S3 × S3 with non-trivial Euler class? Wilking has shown that the answer is \"No\" if the soul is S3 × S3 with the product metric.",
   "clean_statement": "Problem 40. Is there a metric with sec ≥ 0 on R6-bundles over S3 × S3 with non-trivial Euler class? Wilking has shown that the answer is \"No\" if the soul is S3 × S3 with the product metric.",
   "public_statement": "Problem 40. Is there a metric with sec ≥ 0 on R6-bundles over S3 × S3 with non-trivial Euler class? Wilking has shown that the answer is \"No\" if the soul is S3 × S3 with the product metric.",
   "evidence": "The canonical record is Problem 40 in M. Kerin's compilation *Open Problems in Non-negative Sectional Curvature*, arising from the 2007 AIM workshop “Manifolds with nonnegative sectional curvature.” The official PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 252,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0254": {
   "statement_status": "exact",
   "original_statement": "Problem 41. Which vector bundles over S2 × S2 or CP 2± CP 2 where the structure group does not reduce to a torus admit sec ≥ 0? 4 COMPILED BY M. KERIN",
   "clean_statement": "Problem 41. Which vector bundles over S2 × S2 or CP 2± CP 2 where the structure group does not reduce to a torus admit sec ≥ 0? 4 COMPILED BY M. KERIN",
   "public_statement": "Problem 41. Which vector bundles over S2 × S2 or CP 2± CP 2 where the structure group does not reduce to a torus admit sec ≥ 0? 4 COMPILED BY M. KERIN",
   "evidence": "The canonical extraction reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 253,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0255": {
   "statement_status": "exact",
   "original_statement": "Problem 42. Classify metrics with sec ≥ 0 on S2 × R4 (or, more generally, on Sn × Rk).",
   "clean_statement": "Problem 42. Classify metrics with sec ≥ 0 on S2 × R4 (or, more generally, on Sn × Rk).",
   "public_statement": "Problem 42. Classify metrics with sec ≥ 0 on S2 × R4 (or, more generally, on Sn × Rk).",
   "evidence": "The official AIM PDF was checked directly. On page 3 it states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 254,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0256": {
   "statement_status": "exact",
   "original_statement": "Problem 43. Suppose E −→ M is a vector bundle over a compact, simply-connected manifold M for which sec M ≥ 0. Does E ⊕ Rk −→ M have sec ≥ 0 for k large? 6. Quasi-positive Curvature and Positive Curvature on an Open Dense Set",
   "clean_statement": "Problem 43. Suppose E −→ M is a vector bundle over a compact, simply-connected manifold M for which sec M ≥ 0. Does E ⊕ Rk −→ M have sec ≥ 0 for k large? 6. Quasi-positive Curvature and Positive Curvature on an Open Dense Set",
   "public_statement": "Problem 43. Suppose E −→ M is a vector bundle over a compact, simply-connected manifold M for which sec M ≥ 0. Does E ⊕ Rk −→ M have sec ≥ 0 for k large? 6. Quasi-positive Curvature and Positive Curvature on an Open Dense Set",
   "evidence": "The exact corpus record ends with text that does not belong to Problem 43:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 255,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0257": {
   "statement_status": "exact",
   "original_statement": "Problem 44. Which theorems from sec > 0 carry over to positive curvature on an open dense set?",
   "clean_statement": "Problem 44. Which theorems from sec > 0 carry over to positive curvature on an open dense set?",
   "public_statement": "Problem 44. Which theorems from sec > 0 carry over to positive curvature on an open dense set?",
   "evidence": "The canonical record is Problem 44 in M. Kerin's compilation *Open Problems in Non-negative Sectional Curvature*, produced from the 2007 AIM workshop “Manifolds with nonnegative sectional curvature.” The official source reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 256,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0258": {
   "statement_status": "exact",
   "original_statement": "Problem 45. Suppose G is a compact Lie group with a left-invariant metric. Are there any new examples H\\G with quasi-positive curvature?",
   "clean_statement": "Problem 45. Suppose G is a compact Lie group with a left-invariant metric. Are there any new examples H\\G with quasi-positive curvature?",
   "public_statement": "Problem 45. Suppose G is a compact Lie group with a left-invariant metric. Are there any new examples H\\G with quasi-positive curvature?",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 257,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0259": {
   "statement_status": "exact",
   "original_statement": "Problem 46. Find new examples of fundamental groups in quasi-positive curvature or positive curvature on an open dense set.",
   "clean_statement": "Problem 46. Find new examples of fundamental groups in quasi-positive curvature or positive curvature on an open dense set.",
   "public_statement": "Problem 46. Find new examples of fundamental groups in quasi-positive curvature or positive curvature on an open dense set.",
   "evidence": "The corpus record is uncorrupted:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 258,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0260": {
   "statement_status": "exact",
   "original_statement": "Problem 47. Fix k ∈ N. Is there a n0 = n0(k) such that for any quasi-positively curved manifold ( M n, g ) with n ≥ n0 and cohom( M n, g ) ≤ k, there exists a chain \n\nM0 = M n ⊂ M n+k \n\n> 1\n\n⊂ M n+2 k \n\n> 2\n\n⊂ · · · \n\nsuch that all inclusions are totally geodesic, the manifolds Mi are quasi-positively curved, \n\n∪Mi is the classifying space of a Lie group, and Mi/ Iso( Mi, g ) is isometric to M/ Iso( M, g )? 7. Ricci Flow",
   "clean_statement": "Problem 47. Fix k ∈ N. Is there a n0 = n0(k) such that for any quasi-positively curved manifold ( M n, g ) with n ≥ n0 and cohom( M n, g ) ≤ k, there exists a chain\n\nM0 = M n ⊂ M n+k\n\n> 1\n\n⊂ M n+2 k\n\n> 2\n\n⊂ · · ·\n\nsuch that all inclusions are totally geodesic, the manifolds Mi are quasi-positively curved,\n\n∪Mi is the classifying space of a Lie group, and Mi/ Iso( Mi, g ) is isometric to M/ Iso( M, g )? 7. Ricci Flow",
   "public_statement": "Problem 47. Fix k ∈ N. Is there a n0 = n0(k) such that for any quasi-positively curved manifold ( M n, g ) with n ≥ n0 and cohom( M n, g ) ≤ k, there exists a chain\n\nM0 = M n ⊂ M n+k\n\n> 1\n\n⊂ M n+2 k\n\n> 2\n\n⊂ · · ·\n\nsuch that all inclusions are totally geodesic, the manifolds Mi are quasi-positively curved,\n\n∪Mi is the classifying space of a Lie group, and Mi/ Iso( Mi, g ) is isometric to M/ Iso( M, g )? 7. Ricci Flow",
   "evidence": "The canonical JSON is corrupted at the displayed chain: it detaches the digits $1,2$, inserts stray greater-than signs, and appends the next section heading, “7. Ricci Flow.” I checked page 4 of the official AIM PDF, including the font sizes and vertical coordinates in its PDF content stream. The verified statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 259,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0261": {
   "statement_status": "exact",
   "original_statement": "Problem 48. Let M n be a compact manifold with positive isotropic curvature. Does any blow-up limit of the Ricci flow have non-negative curvature operator?",
   "clean_statement": "Problem 48. Let M n be a compact manifold with positive isotropic curvature. Does any blow-up limit of the Ricci flow have non-negative curvature operator?",
   "public_statement": "Problem 48. Let M n be a compact manifold with positive isotropic curvature. Does any blow-up limit of the Ricci flow have non-negative curvature operator?",
   "evidence": "The corpus text reads “\\(M n\\)” because the superscript was lost in extraction. Comparison with the official PDF confirms that the intended notation is \\(M^n\\); no other correction is needed.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 260,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0262": {
   "statement_status": "exact",
   "original_statement": "Problem 49. Suppose G acts freely and isometrically on M. What kind of flows on M/G \n\nare induced by the Ricci flow on M?",
   "clean_statement": "Problem 49. Suppose G acts freely and isometrically on M. What kind of flows on M/G\n\nare induced by the Ricci flow on M?",
   "public_statement": "Problem 49. Suppose G acts freely and isometrically on M. What kind of flows on M/G\n\nare induced by the Ricci flow on M?",
   "evidence": "The canonical record is Problem 49 in the 2007 AIM list *Open Problems in Non-negative Sectional Curvature*, in Section 7, “Ricci Flow”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 261,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0263": {
   "statement_status": "exact",
   "original_statement": "Problem 50. Can one improve the Hsiang-Kleiner theorem on positively curved 4-manifolds with symmetry from homeomorphism to diffeomorphism by using the Ricci flow? 8. Miscellaneous Problems",
   "clean_statement": "Problem 50. Can one improve the Hsiang-Kleiner theorem on positively curved 4-manifolds with symmetry from homeomorphism to diffeomorphism by using the Ricci flow? 8. Miscellaneous Problems",
   "public_statement": "Problem 50. Can one improve the Hsiang-Kleiner theorem on positively curved 4-manifolds with symmetry from homeomorphism to diffeomorphism by using the Ricci flow? 8. Miscellaneous Problems",
   "evidence": "The exact canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 262,
   "attempt": 2
  },
  "AIM-TOPOLOGY-0264": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 51. Is a \"generic\" manifold a K(π, 1)-space (where \"generic\" is to be deter-mined)?",
   "clean_statement": "**Problem 51.** Is a “generic” manifold a $K(\\pi,1)$-space (where\n“generic” is to be determined)?",
   "public_statement": "**Problem 51.** Is a “generic” manifold a $K(\\pi,1)$-space (where\n“generic” is to be determined)?",
   "evidence": "The official AIM PDF places this on page 4 as the first question in “8. Miscellaneous Problems.” The hyphen in “deter-mined” is only a line-break hyphen. The recovered statement is therefore:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-topology-notes.json",
   "source_index": 263,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0265": {
   "statement_status": "exact",
   "original_statement": "Problem 52. Does positive sectional curvature imply that the manifold is formal (in the sense of Sullivan's minimal model)?",
   "clean_statement": "Problem 52. Does positive sectional curvature imply that the manifold is formal (in the sense of Sullivan's minimal model)?",
   "public_statement": "Problem 52. Does positive sectional curvature imply that the manifold is formal (in the sense of Sullivan's minimal model)?",
   "evidence": "The canonical record is Problem 52 in the AIM list *Open Problems in Non-negative Sectional Curvature*, compiled by M. Kerin. The official PDF gives the following wording:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 264,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0266": {
   "statement_status": "exact",
   "original_statement": "Problem 53. For compact, odd dimensional, positively curved manifolds is there a cyclic subgroup of the fundamental group whose index is bounded only in terms of the dimen-sion?",
   "clean_statement": "Problem 53. For compact, odd dimensional, positively curved manifolds is there a cyclic subgroup of the fundamental group whose index is bounded only in terms of the dimen-sion?",
   "public_statement": "Problem 53. For compact, odd dimensional, positively curved manifolds is there a cyclic subgroup of the fundamental group whose index is bounded only in terms of the dimen-sion?",
   "evidence": "The record is Problem 53 in Section 8, “Miscellaneous Problems,” of the AIM workshop list *Open Problems in Non-negative Sectional Curvature*. The official PDF reads across a line break:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 265,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0267": {
   "statement_status": "exact",
   "original_statement": "Problem 54. Is there a finiteness result for n-dimensional, positively curved manifolds with π1 = π2 = 0? 5",
   "clean_statement": "Problem 54. Is there a finiteness result for n-dimensional, positively curved manifolds with π1 = π2 = 0? 5",
   "public_statement": "Problem 54. Is there a finiteness result for n-dimensional, positively curved manifolds with π1 = π2 = 0? 5",
   "evidence": "The canonical record reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 266,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0268": {
   "statement_status": "exact",
   "original_statement": "Problem 55. Is there a δ(n) > 0 such that any n-dimensional, positively curved manifold carries a δ(n)-pinched metric?",
   "clean_statement": "Problem 55. Is there a δ(n) > 0 such that any n-dimensional, positively curved manifold carries a δ(n)-pinched metric?",
   "public_statement": "Problem 55. Is there a δ(n) > 0 such that any n-dimensional, positively curved manifold carries a δ(n)-pinched metric?",
   "evidence": "The official AIM workshop PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 267,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0269": {
   "statement_status": "exact",
   "original_statement": "Question 1.1 (Bill Goldman). Can every maximal representation in Sp(4, R) be deformed to a proper Zariski closed subgroup? \n\nComment 1.2 (Bill Goldman). Every maximal representation of the unitary group can be deformed in this way. \n\nComment 1.3 (Anna Wienhard). The problem should be easier for Sp(2 n, R) with n > 2 (it is still not known though). For n = 2 progress was made at the conference on many com-ponents, however there exist 2( g − 2) \"exotic\" components for which nothing is currently known. \n\n2 Geometry of the Hitchin component",
   "clean_statement": "Question 1.1 (Bill Goldman). Can every maximal representation in Sp(4, R) be deformed to a proper Zariski closed subgroup?\n\nComment 1.2 (Bill Goldman). Every maximal representation of the unitary group can be deformed in this way.\n\nComment 1.3 (Anna Wienhard). The problem should be easier for Sp(2 n, R) with n > 2 (it is still not known though). For n = 2 progress was made at the conference on many com-ponents, however there exist 2( g − 2) \"exotic\" components for which nothing is currently known.\n\n2 Geometry of the Hitchin component",
   "public_statement": "Question 1.1 (Bill Goldman). Can every maximal representation in Sp(4, R) be deformed to a proper Zariski closed subgroup?\n\nComment 1.2 (Bill Goldman). Every maximal representation of the unitary group can be deformed in this way.\n\nComment 1.3 (Anna Wienhard). The problem should be easier for Sp(2 n, R) with n > 2 (it is still not known though). For n = 2 progress was made at the conference on many com-ponents, however there exist 2( g − 2) \"exotic\" components for which nothing is currently known.\n\n2 Geometry of the Hitchin component",
   "evidence": "The source is the AIM workshop list *Representations of surface groups*, from the workshop held March 19--23, 2007. The relevant text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 268,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0270": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 2.1 (John Loftin). Does there exist a mapping class group-invariant K¨ ahler structure on the Hitchin component for SL(3, R)?\n\nComment 2.2 (Richard Wentworth). Theorem 1.0.2 of [Lab06] shows that the Hitchin com-ponent for SL(3, R) is given by the bundle of cubic differentials on Teichm ¨ uller space. There is a mapping class group-invariant complex structure on this space. 1Comment 2.3 (John Loftin). There is evidence for a K¨ ahler structure, since transverse to the fibres there is a K¨ ahler metric (Weil-Petersson), and on the fibres there is a K¨ ahler metric.",
   "clean_statement": null,
   "public_statement": "Question 2.1 (John Loftin). Does there exist a mapping class group-invariant K¨ ahler structure on the Hitchin component for SL(3, R)?\n\nComment 2.2 (Richard Wentworth). Theorem 1.0.2 of [Lab06] shows that the Hitchin com-ponent for SL(3, R) is given by the bundle of cubic differentials on Teichm ¨ uller space. There is a mapping class group-invariant complex structure on this space. 1Comment 2.3 (John Loftin). There is evidence for a K¨ ahler structure, since transverse to the fibres there is a K¨ ahler metric (Weil-Petersson), and on the fibres there is a K¨ ahler metric.",
   "evidence": "The canonical record is Question 2.1 from the 2007 AIM workshop *Representations of Surface Groups*. The official four-page PDF was checked directly. Its intended text is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-topology-notes.json",
   "source_index": 269,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0271": {
   "statement_status": "exact",
   "original_statement": "Question 2.4 (Richard Wentworth, Francois Labourie). Let S be a closed surface, ρ:\n\nπ1(S) → SL( n, R) a representation in the Hitchin component. Given a complex structure J\n\non S there exists a unique ρ-equivariant harmonic map u: ( ˜S, J ) → X = SL( n, R)/ SO( n)\n\n(Corollary 3.5 of [Cor88]). Is there a unique complex structure J such that u is also con-formal? \n\nComment 2.5 (Richard Wentworth). This is true for n = 2, 3 (see Theorem 9.3.2 of [Lab06]). \n\nComment 2.6 (Anna Wienhard). One can ask the same question for maximal representa-tions. \n\n3 Surface Bundles",
   "clean_statement": "Question 2.4 (Richard Wentworth, Francois Labourie). Let S be a closed surface, ρ:\n\nπ1(S) → SL( n, R) a representation in the Hitchin component. Given a complex structure J\n\non S there exists a unique ρ-equivariant harmonic map u: ( ˜S, J ) → X = SL( n, R)/ SO( n)\n\n(Corollary 3.5 of [Cor88]). Is there a unique complex structure J such that u is also con-formal?\n\nComment 2.5 (Richard Wentworth). This is true for n = 2, 3 (see Theorem 9.3.2 of [Lab06]).\n\nComment 2.6 (Anna Wienhard). One can ask the same question for maximal representa-tions.\n\n3 Surface Bundles",
   "public_statement": "Question 2.4 (Richard Wentworth, Francois Labourie). Let S be a closed surface, ρ:\n\nπ1(S) → SL( n, R) a representation in the Hitchin component. Given a complex structure J\n\non S there exists a unique ρ-equivariant harmonic map u: ( ˜S, J ) → X = SL( n, R)/ SO( n)\n\n(Corollary 3.5 of [Cor88]). Is there a unique complex structure J such that u is also con-formal?\n\nComment 2.5 (Richard Wentworth). This is true for n = 2, 3 (see Theorem 9.3.2 of [Lab06]).\n\nComment 2.6 (Anna Wienhard). One can ask the same question for maximal representa-tions.\n\n3 Surface Bundles",
   "evidence": "The official AIM PDF, from the 2007 workshop *Representations of Surface Groups*, reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 270,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0272": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 3.1 (Dieter Kotschick). Fix a closed Riemann surface B of genus g ≥ 3, and fix \n\nh ≥ 2. There exist at most finitely many non-isotrivial holomorphic genus h fibrations over B, Fh → X → B.This gives a conjugacy class of representations ρ: π1(B) → MCG( Fh). How to charac-terise the representations which are the holonomy of a holomorphic fibration? \n\nComment 3.2 (Dieter Kotschick). When the fibration is holomorphic there is a K¨ ahler struc-ture on the surface bundle X, and so the cohomology of X satisfies certain constraints from Hodge theory (e.g. h1(X) is even). These constraints give some restrictions on the representations, but they are not enough to give an \"if and only if\" statement.",
   "clean_statement": "**Question 3.1 (Dieter Kotschick).** Fix a closed Riemann surface \\(B\\) of genus \\(g\\geq 3\\), and fix \\(h\\geq 2\\). There exist at most finitely many non-isotrivial holomorphic genus \\(h\\) fibrations over \\(B\\),\n\\[\nF_h\\longrightarrow X\\longrightarrow B.\n\\]\nThis gives a conjugacy class of representations\n\\[\n\\rho:\\pi_1(B)\\longrightarrow \\operatorname{MCG}(F_h).\n\\]\nHow to characterise the representations which are the holonomy of a holomorphic fibration?\n\n**Comment 3.2 (Dieter Kotschick).** When the fibration is holomorphic there is a Kähler structure on the surface bundle \\(X\\), and so the cohomology of \\(X\\) satisfies certain constraints from Hodge theory (e.g. \\(h_1(X)\\) is even). These constraints give some restrictions on the representations, but they are not enough to give an “if and only if” statement.",
   "public_statement": "Question 3.1 (Dieter Kotschick). Fix a closed Riemann surface B of genus g ≥ 3, and fix\n\nh ≥ 2. There exist at most finitely many non-isotrivial holomorphic genus h fibrations over B, Fh → X → B.This gives a conjugacy class of representations ρ: π1(B) → MCG( Fh). How to charac-terise the representations which are the holonomy of a holomorphic fibration?\n\nComment 3.2 (Dieter Kotschick). When the fibration is holomorphic there is a K¨ ahler struc-ture on the surface bundle X, and so the cohomology of X satisfies certain constraints from Hodge theory (e.g. h1(X) is even). These constraints give some restrictions on the representations, but they are not enough to give an \"if and only if\" statement.",
   "evidence": "The source is the AIM workshop list *Representations of surface groups*, produced at the March 19--23, 2007 workshop. The PDF reads:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-topology-notes.json",
   "source_index": 271,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0273": {
   "statement_status": "exact",
   "original_statement": "Question 3.3 (Dieter Kotschick). Fix g, h with h ≥ 2. Does there exist a locally trivial fibration of surfaces Fh → X → Bg such that X admits a metric of strictly negative curvature? \n\nComment 3.4 (Dieter Kotschick). A necessary condition is that the monodromy is pseudo-Anosov. If the curvature is also constant then the signature must be zero and so the Toledo invariant vanishes. \n\n4 Invariants of representations",
   "clean_statement": "Question 3.3 (Dieter Kotschick). Fix g, h with h ≥ 2. Does there exist a locally trivial fibration of surfaces Fh → X → Bg such that X admits a metric of strictly negative curvature?\n\nComment 3.4 (Dieter Kotschick). A necessary condition is that the monodromy is pseudo-Anosov. If the curvature is also constant then the signature must be zero and so the Toledo invariant vanishes.\n\n4 Invariants of representations",
   "public_statement": "Question 3.3 (Dieter Kotschick). Fix g, h with h ≥ 2. Does there exist a locally trivial fibration of surfaces Fh → X → Bg such that X admits a metric of strictly negative curvature?\n\nComment 3.4 (Dieter Kotschick). A necessary condition is that the monodromy is pseudo-Anosov. If the curvature is also constant then the signature must be zero and so the Toledo invariant vanishes.\n\n4 Invariants of representations",
   "evidence": "The official AIM problem-list PDF gives the following question of Dieter Kotschick:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 272,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0274": {
   "statement_status": "exact",
   "original_statement": "Question 4.1 (Marc Burger). Let M be a compact manifold foliated by surfaces, with transverse measure μ. Denote the foliation by F. Let X be a Hermitian symmetric space, \n\nG = Isom( X)0 and ρ: π1(M ) → G. Given a ρ-equivariant map f: ˜M → X we have \n\nf ∗ω ∈ H2(F), and let C be a Ruelle-Sullivan cycle. Find bounds on 〈f ∗ω, C 〉 ∈ R and study the maximal representations.",
   "clean_statement": "Question 4.1 (Marc Burger). Let M be a compact manifold foliated by surfaces, with transverse measure μ. Denote the foliation by F. Let X be a Hermitian symmetric space,\n\nG = Isom( X)0 and ρ: π1(M ) → G. Given a ρ-equivariant map f: ˜M → X we have\n\nf ∗ω ∈ H2(F), and let C be a Ruelle-Sullivan cycle. Find bounds on 〈f ∗ω, C 〉 ∈ R and study the maximal representations.",
   "public_statement": "Question 4.1 (Marc Burger). Let M be a compact manifold foliated by surfaces, with transverse measure μ. Denote the foliation by F. Let X be a Hermitian symmetric space,\n\nG = Isom( X)0 and ρ: π1(M ) → G. Given a ρ-equivariant map f: ˜M → X we have\n\nf ∗ω ∈ H2(F), and let C be a Ruelle-Sullivan cycle. Find bounds on 〈f ∗ω, C 〉 ∈ R and study the maximal representations.",
   "evidence": "The official four-page PDF of the 2007 AIM workshop *Representations of Surface Groups* was checked at Question 4.1. With notation restored, it asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 273,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0275": {
   "statement_status": "exact",
   "original_statement": "Question 4.2 (Oscar Garcia-Prada). Consider a real group G with a symmetric space X =\n\nG/K of quaternion-K¨ ahler type. Let Ω be a 4-form on X, M a 4-manifold and ρ: π1(M ) →\n\nG. Study the Toledo invariant 〈f ∗Ω, [M ]〉. An interesting special case is when M is a K¨ ahler surface. 2Comment 4.3 (Dieter Kotschick). It might be interesting to study the case where M is as-pherical. \n\nComment 4.4 (Domingo Toledo). Other interesting special cases are: When M is a complex hyperbolic surface, when M has constant curvature and when M is the product of two surfaces.",
   "clean_statement": "Question 4.2 (Oscar Garcia-Prada). Consider a real group G with a symmetric space X =\n\nG/K of quaternion-K¨ ahler type. Let Ω be a 4-form on X, M a 4-manifold and ρ: π1(M ) →\n\nG. Study the Toledo invariant 〈f ∗Ω, [M ]〉. An interesting special case is when M is a K¨ ahler surface. 2Comment 4.3 (Dieter Kotschick). It might be interesting to study the case where M is as-pherical.\n\nComment 4.4 (Domingo Toledo). Other interesting special cases are: When M is a complex hyperbolic surface, when M has constant curvature and when M is the product of two surfaces.",
   "public_statement": "Question 4.2 (Oscar Garcia-Prada). Consider a real group G with a symmetric space X =\n\nG/K of quaternion-K¨ ahler type. Let Ω be a 4-form on X, M a 4-manifold and ρ: π1(M ) →\n\nG. Study the Toledo invariant 〈f ∗Ω, [M ]〉. An interesting special case is when M is a K¨ ahler surface. 2Comment 4.3 (Dieter Kotschick). It might be interesting to study the case where M is as-pherical.\n\nComment 4.4 (Domingo Toledo). Other interesting special cases are: When M is a complex hyperbolic surface, when M has constant curvature and when M is the product of two surfaces.",
   "evidence": "The official AIM PDF gives the following question.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 274,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0276": {
   "statement_status": "exact",
   "original_statement": "Question 4.5 (Marc Burger). Let k be a field and V a symplectic vector space over k. There exists a central extension \n\n1 → W (k) → ˜Sp( V ) → Sp( V ) → 1\n\nwhere W (k) is the Witt group. Given a representation ρ: π1(S) → Sp( V ), the Toledo invariant τ (ρ) is an element of W (k). Let A be an algebraic set and consider a map u: A →\n\nRep( π1(S), Sp(2 n, R)). To this map associate the tautological representation ρ: π1(S) →\n\nSp(2 n, k (A)), and define the invariant τ (ρ) ∈ W (k(A)). Study this. \n\nComment 4.6 (Marc Burger). When A is a point this is the classical Toledo invariant.",
   "clean_statement": "Question 4.5 (Marc Burger). Let k be a field and V a symplectic vector space over k. There exists a central extension\n\n1 → W (k) → ˜Sp( V ) → Sp( V ) → 1\n\nwhere W (k) is the Witt group. Given a representation ρ: π1(S) → Sp( V ), the Toledo invariant τ (ρ) is an element of W (k). Let A be an algebraic set and consider a map u: A →\n\nRep( π1(S), Sp(2 n, R)). To this map associate the tautological representation ρ: π1(S) →\n\nSp(2 n, k (A)), and define the invariant τ (ρ) ∈ W (k(A)). Study this.\n\nComment 4.6 (Marc Burger). When A is a point this is the classical Toledo invariant.",
   "public_statement": "Question 4.5 (Marc Burger). Let k be a field and V a symplectic vector space over k. There exists a central extension\n\n1 → W (k) → ˜Sp( V ) → Sp( V ) → 1\n\nwhere W (k) is the Witt group. Given a representation ρ: π1(S) → Sp( V ), the Toledo invariant τ (ρ) is an element of W (k). Let A be an algebraic set and consider a map u: A →\n\nRep( π1(S), Sp(2 n, R)). To this map associate the tautological representation ρ: π1(S) →\n\nSp(2 n, k (A)), and define the invariant τ (ρ) ∈ W (k(A)). Study this.\n\nComment 4.6 (Marc Burger). When A is a point this is the classical Toledo invariant.",
   "evidence": "The official AIM workshop PDF, *Representations of surface groups*, Question 4.5 (Marc Burger), reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 275,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0277": {
   "statement_status": "corrected_verified",
   "original_statement": "Question 4.7 (Olivier Guichard). Following on from the previous question, there is also a central extension for SL( n)1 → K2(A) → E(n, A ) → SL( n, A ) → 1\n\nLet Hn be the Hitchin component, and let A = Q(Hn). Given a representation ρ: π1(S) →\n\nSL( n, A ) we obtain q ∈ K2(A). There is a map d log: K2(A) → Ω2(Hn). Is d log( q) the Weil-Petersson form? \n\n5 Other questions",
   "clean_statement": "Following on from the previous question, there is also a central extension for \\(\\mathrm{SL}(n)\\)\n\n\\[\n1\\longrightarrow K_2(A)\\longrightarrow E(n,A)\\longrightarrow\n\\mathrm{SL}(n,A)\\longrightarrow 1.\n\\]\nLet \\(H_n\\) be the Hitchin component, and let \\(A=\\mathbb Q(H_n)\\). Given\n\\(\\rho:\\pi_1(S)\\to\\mathrm{SL}(n,A)\\), we obtain \\(q\\in K_2(A)\\).\nThere is \\(d\\log:K_2(A)\\to\\Omega^2(H_n)\\). Is \\(d\\log(q)\\) the\nWeil--Petersson form?",
   "public_statement": "Following on from the previous question, there is also a central extension for \\(\\mathrm{SL}(n)\\)\n\n\\[\n1\\longrightarrow K_2(A)\\longrightarrow E(n,A)\\longrightarrow\n\\mathrm{SL}(n,A)\\longrightarrow 1.\n\\]\nLet \\(H_n\\) be the Hitchin component, and let \\(A=\\mathbb Q(H_n)\\). Given\n\\(\\rho:\\pi_1(S)\\to\\mathrm{SL}(n,A)\\), we obtain \\(q\\in K_2(A)\\).\nThere is \\(d\\log:K_2(A)\\to\\Omega^2(H_n)\\). Is \\(d\\log(q)\\) the\nWeil--Petersson form?",
   "evidence": "The source is Question 4.7, attributed to Olivier Guichard, in the AIM workshop list *Representations of surface groups*. Inspection of the source PDF recovers: The string “5 Other questions” in the extracted JSON is the next section heading, not part of Question 4.7. The missing separator between \\(\\mathrm{SL}(n)\\) and the displayed \\(1\\) is also an extraction error.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-topology-notes.json",
   "source_index": 276,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0278": {
   "statement_status": "exact",
   "original_statement": "Question 5.1 (Oscar Garcia-Prada). The cyclic group of order n acts on the Hitchin com-ponent of Rep( π1(S), SL( n, R)) (multiply the Higgs field by roots of unity). Call the sub-space of Z/n Z-invariant Higgs fields \"cyclotomic Higgs fields\", which are in one-to-one correspondence with H0(S, K n). Is this space independent of the complex structure on S?",
   "clean_statement": "Question 5.1 (Oscar Garcia-Prada). The cyclic group of order n acts on the Hitchin com-ponent of Rep( π1(S), SL( n, R)) (multiply the Higgs field by roots of unity). Call the sub-space of Z/n Z-invariant Higgs fields \"cyclotomic Higgs fields\", which are in one-to-one correspondence with H0(S, K n). Is this space independent of the complex structure on S?",
   "public_statement": "Question 5.1 (Oscar Garcia-Prada). The cyclic group of order n acts on the Hitchin com-ponent of Rep( π1(S), SL( n, R)) (multiply the Higgs field by roots of unity). Call the sub-space of Z/n Z-invariant Higgs fields \"cyclotomic Higgs fields\", which are in one-to-one correspondence with H0(S, K n). Is this space independent of the complex structure on S?",
   "evidence": "The official PDF of the 2007 AIM workshop *Representations of Surface Groups* was checked at page 3, Question 5.1. With only line-break hyphenation and mathematical superscripts restored, it asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 277,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0279": {
   "statement_status": "exact",
   "original_statement": "Question 5.2 (Anna Wienhard). Do there exist embeddings SL(2, R) ↪→ G such that the component in Rep( π1(S), G ) containing the representation \n\nπ1(S) / / \n\n> F uchsian %%JJJJJJJJJ\n\nG\n\nSL(2, R)\n\n> ϕ\n> O\n> O\n\nconsists entirely of discrete and faithful representations? \n\nComment 5.3 (Anna Wienhard). For G being a split real form and SL(2, R) → G an irre-ducible representation this gives the Hitchin component. For G being of Hermitian type and SL(2, R) → G a tight representation, then this gives one connected component in the space of maximal representations. 3",
   "clean_statement": "Question 5.2 (Anna Wienhard). Do there exist embeddings SL(2, R) ↪→ G such that the component in Rep( π1(S), G ) containing the representation\n\nπ1(S) / /\n\n> F uchsian %%JJJJJJJJJ\n\nG\n\nSL(2, R)\n\n> ϕ\n> O\n> O\n\nconsists entirely of discrete and faithful representations?\n\nComment 5.3 (Anna Wienhard). For G being a split real form and SL(2, R) → G an irre-ducible representation this gives the Hitchin component. For G being of Hermitian type and SL(2, R) → G a tight representation, then this gives one connected component in the space of maximal representations. 3",
   "public_statement": "Question 5.2 (Anna Wienhard). Do there exist embeddings SL(2, R) ↪→ G such that the component in Rep( π1(S), G ) containing the representation\n\nπ1(S) / /\n\n> F uchsian %%JJJJJJJJJ\n\nG\n\nSL(2, R)\n\n> ϕ\n> O\n> O\n\nconsists entirely of discrete and faithful representations?\n\nComment 5.3 (Anna Wienhard). For G being a split real form and SL(2, R) → G an irre-ducible representation this gives the Hitchin component. For G being of Hermitian type and SL(2, R) → G a tight representation, then this gives one connected component in the space of maximal representations. 3",
   "evidence": "The source is Question 5.2 in the AIM workshop list *Representations of surface groups*. The JSON extraction damaged a commutative diagram and attached the next page number to Comment 5.3. Inspection of the official four-page PDF gives the following reconstruction. Here $S$ is a closed, connected, oriented surface of genus $g\\geq 2$, $\\Gamma=\\pi_1(S)$, and $\\rho_0$ is Fuchsian:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 278,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0280": {
   "statement_status": "exact",
   "original_statement": "Question 5.4 (Marc Burger). Count rational representations whose image lies in the set of integer points of the symplectic group.",
   "clean_statement": "Question 5.4 (Marc Burger). Count rational representations whose image lies in the set of integer points of the symplectic group.",
   "public_statement": "Question 5.4 (Marc Burger). Count rational representations whose image lies in the set of integer points of the symplectic group.",
   "evidence": "The AIM PDF was checked against the extracted record. The wording above is faithful; there is no substantive OCR error. Nearby questions confirm the surface-group setting, but Question 5.4 itself does not specify:",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 279,
   "attempt": 1
  },
  "AIM-TOPOLOGY-0281": {
   "statement_status": "exact",
   "original_statement": "Question 5.5 (Bill Goldman). Let S be a closed orientable surface with χ(S) < 0, and G\n\na semisimple Lie group. Define Eρ: TS → R to be the energy function associated to a representation ρ: π1(S) → G (see [GW05] for more details). What are the most general conditions on ρ for which Eρ is proper? \n\nComment 5.6 (Bill Goldman). If G is split and ρ is a Hitchin representation or maximal symplectic representation then Eρ is proper (see [Lab05]). If ρ is convex co-compact then Eρ is proper (see [GW05]).",
   "clean_statement": "Question 5.5 (Bill Goldman). Let S be a closed orientable surface with χ(S) < 0, and G\n\na semisimple Lie group. Define Eρ: TS → R to be the energy function associated to a representation ρ: π1(S) → G (see [GW05] for more details). What are the most general conditions on ρ for which Eρ is proper?\n\nComment 5.6 (Bill Goldman). If G is split and ρ is a Hitchin representation or maximal symplectic representation then Eρ is proper (see [Lab05]). If ρ is convex co-compact then Eρ is proper (see [GW05]).",
   "public_statement": "Question 5.5 (Bill Goldman). Let S be a closed orientable surface with χ(S) < 0, and G\n\na semisimple Lie group. Define Eρ: TS → R to be the energy function associated to a representation ρ: π1(S) → G (see [GW05] for more details). What are the most general conditions on ρ for which Eρ is proper?\n\nComment 5.6 (Bill Goldman). If G is split and ρ is a Hitchin representation or maximal symplectic representation then Eρ is proper (see [Lab05]). If ρ is convex co-compact then Eρ is proper (see [GW05]).",
   "evidence": "The canonical JSON record reads “Define \\(E_\\rho:T S\\to\\mathbb R\\).” Inspection of the original AIM workshop PDF shows that the intended domain is the Teichmüller space \\(\\mathcal T_S\\), not the tangent bundle \\(TS\\). This is an OCR/typesetting-loss correction; the canonical input has not been altered.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-topology-notes.json",
   "source_index": 280,
   "attempt": 1
  },
  "AIM-OTHER-0001": {
   "statement_status": "exact",
   "original_statement": "Problem 1.1. What is the dimension of V0, and what is a natural basis for it? \n\nWe know the answer when P is a product of two chains and T is rowmotion or promotion, but not for rowmotion and promotion of other posets. One can ask the same question with antichains instead of order ideals. Again, we know the answer when P is a product of two chains and T is rowmotion or promotion, but not for the rowmotion and promotion actions on other posets. \n\n1.2 Telescoping \n\nLet T be an action on a set X all of whose orbits are finite, and let f be some real-valued 0-mesy of (X, T ) (that is, a real-valued function on X whose average value on each T -orbit is 0, or equivalently, whose sum on each T -orbit is 0), The 2most straightforward way to prove that the sum of f over every T -orbit is zero is to find a way to exhibit a function g on X for which f (x) = g(x) − g(T (x)) for all x ∈ X; for in that case 0-mesy is just a matter of telescoping. And indeed, some of the known proofs of homomesy rely upon similar tricks. This prompts one to ask: To what extent can proofs of homomesy results (especially 0-mesy results) be simplified by finding constructions of such \"integrals\" g?Note that for any 0-mesic function f there are typically many functions g\n\nsatisfying the relation f (x) = g(x) − g(T (x)); the problem is the following.",
   "clean_statement": "Problem 1.1. What is the dimension of V0, and what is a natural basis for it?\n\nWe know the answer when P is a product of two chains and T is rowmotion or promotion, but not for rowmotion and promotion of other posets. One can ask the same question with antichains instead of order ideals. Again, we know the answer when P is a product of two chains and T is rowmotion or promotion, but not for the rowmotion and promotion actions on other posets.\n\n1.2 Telescoping\n\nLet T be an action on a set X all of whose orbits are finite, and let f be some real-valued 0-mesy of (X, T ) (that is, a real-valued function on X whose average value on each T -orbit is 0, or equivalently, whose sum on each T -orbit is 0), The 2most straightforward way to prove that the sum of f over every T -orbit is zero is to find a way to exhibit a function g on X for which f (x) = g(x) − g(T (x)) for all x ∈ X; for in that case 0-mesy is just a matter of telescoping. And indeed, some of the known proofs of homomesy rely upon similar tricks. This prompts one to ask: To what extent can proofs of homomesy results (especially 0-mesy results) be simplified by finding constructions of such \"integrals\" g?Note that for any 0-mesic function f there are typically many functions g\n\nsatisfying the relation f (x) = g(x) − g(T (x)); the problem is the following.",
   "public_statement": "Problem 1.1. What is the dimension of V0, and what is a natural basis for it?\n\nWe know the answer when P is a product of two chains and T is rowmotion or promotion, but not for rowmotion and promotion of other posets. One can ask the same question with antichains instead of order ideals. Again, we know the answer when P is a product of two chains and T is rowmotion or promotion, but not for the rowmotion and promotion actions on other posets.\n\n1.2 Telescoping\n\nLet T be an action on a set X all of whose orbits are finite, and let f be some real-valued 0-mesy of (X, T ) (that is, a real-valued function on X whose average value on each T -orbit is 0, or equivalently, whose sum on each T -orbit is 0), The 2most straightforward way to prove that the sum of f over every T -orbit is zero is to find a way to exhibit a function g on X for which f (x) = g(x) − g(T (x)) for all x ∈ X; for in that case 0-mesy is just a matter of telescoping. And indeed, some of the known proofs of homomesy rely upon similar tricks. This prompts one to ask: To what extent can proofs of homomesy results (especially 0-mesy results) be simplified by finding constructions of such \"integrals\" g?Note that for any 0-mesic function f there are typically many functions g\n\nsatisfying the relation f (x) = g(x) − g(T (x)); the problem is the following.",
   "evidence": "The assigned record comes from the AIM pre-workshop problem list for *Dynamical Algebraic Combinatorics*. The PDF first fixes a finite poset \\(P\\), an invertible map \\[ T:J(P)\\longrightarrow J(P), \\] and, for \\(x\\in P\\), the membership indicator \\(\\mathbf 1_x(I)=1\\) if \\(x\\in I\\) and \\(0\\) otherwise. It defines \\[ V=\\operatorname{span}_{\\mathbb R}\\{\\mathbf 1_x:x\\in P\\} \\] and lets \\(V_0\\) be the subspace of functions whose sum on every \\(T\\)-orbit is zero. The exact question in Problem 1.1 is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 0,
   "attempt": 1
  },
  "AIM-OTHER-0002": {
   "statement_status": "exact",
   "original_statement": "Problem 1.2. Find simple and uniform constructions for such functions g for a wide range of 0-mesies f.\n\n1.3 Dynamical closures \n\nIn most cases of combinatorial interest, the \"feature space\" V is not preserved by the map T; that is, for f: X → R a function in V, the time-shifted function \n\nf ◦T: X → R typically is not in V. It therefore seems natural, when T is of finite order (order n, say), to replace V by a larger but still finite-dimensional space \n\nV T (the \"dynamical closure\" of V ) defined as the smallest set of functions that contains V and contains f ◦ T whenever it contains f. Indeed, many important dynamical properties of (X, T, V ) reduce to linear (or more generally affine) relations satisfied in V T: invariance asserts that for some particular f ∈ V, f\n\nequals f ◦ T; homomesy asserts that for some particular f ∈ V, f + f ◦ T + f ◦\n\nT 2 + · · · + f ◦ T n−1 equals a constant function; and reciprocity asserts that for some particular f, g ∈ V and some particular k, f + g ◦ T k is the zero function. Also (to give an example that seems to be very narrow but whose seeming narrowness may be merely a reflection of our ignorance of analogous behavior in other dynamical systems), we may note that the combinatorial fact that underlies the existence of the Armstrong-Stanley way of looking at Panyushev complementation of antichains in [a] × [b], namely, the fact that A contains an element in the ith fiber if and only if the Panyushev complement of A contains an element in the ( i + 1 )st fiber (as long as i < a ), can also be expressed as a linear relation in V T.Here is an outline for how to approach V T systematically. Letting f1,..., f N\n\ndenote a basis for V T, define V T (X) to be the set of N -tuples (f1(x),..., f N (x)) \n\nas x varies over X. Figuring out what linear relations are satisfied by the functions f1,..., f N when they are restricted to V T (X) is equivalent to:",
   "clean_statement": "Problem 1.2. Find simple and uniform constructions for such functions g for a wide range of 0-mesies f.\n\n1.3 Dynamical closures\n\nIn most cases of combinatorial interest, the \"feature space\" V is not preserved by the map T; that is, for f: X → R a function in V, the time-shifted function\n\nf ◦T: X → R typically is not in V. It therefore seems natural, when T is of finite order (order n, say), to replace V by a larger but still finite-dimensional space\n\nV T (the \"dynamical closure\" of V ) defined as the smallest set of functions that contains V and contains f ◦ T whenever it contains f. Indeed, many important dynamical properties of (X, T, V ) reduce to linear (or more generally affine) relations satisfied in V T: invariance asserts that for some particular f ∈ V, f\n\nequals f ◦ T; homomesy asserts that for some particular f ∈ V, f + f ◦ T + f ◦\n\nT 2 + · · · + f ◦ T n−1 equals a constant function; and reciprocity asserts that for some particular f, g ∈ V and some particular k, f + g ◦ T k is the zero function. Also (to give an example that seems to be very narrow but whose seeming narrowness may be merely a reflection of our ignorance of analogous behavior in other dynamical systems), we may note that the combinatorial fact that underlies the existence of the Armstrong-Stanley way of looking at Panyushev complementation of antichains in [a] × [b], namely, the fact that A contains an element in the ith fiber if and only if the Panyushev complement of A contains an element in the ( i + 1 )st fiber (as long as i < a ), can also be expressed as a linear relation in V T.Here is an outline for how to approach V T systematically. Letting f1,..., f N\n\ndenote a basis for V T, define V T (X) to be the set of N -tuples (f1(x),..., f N (x))\n\nas x varies over X. Figuring out what linear relations are satisfied by the functions f1,..., f N when they are restricted to V T (X) is equivalent to:",
   "public_statement": "Problem 1.2. Find simple and uniform constructions for such functions g for a wide range of 0-mesies f.\n\n1.3 Dynamical closures\n\nIn most cases of combinatorial interest, the \"feature space\" V is not preserved by the map T; that is, for f: X → R a function in V, the time-shifted function\n\nf ◦T: X → R typically is not in V. It therefore seems natural, when T is of finite order (order n, say), to replace V by a larger but still finite-dimensional space\n\nV T (the \"dynamical closure\" of V ) defined as the smallest set of functions that contains V and contains f ◦ T whenever it contains f. Indeed, many important dynamical properties of (X, T, V ) reduce to linear (or more generally affine) relations satisfied in V T: invariance asserts that for some particular f ∈ V, f\n\nequals f ◦ T; homomesy asserts that for some particular f ∈ V, f + f ◦ T + f ◦\n\nT 2 + · · · + f ◦ T n−1 equals a constant function; and reciprocity asserts that for some particular f, g ∈ V and some particular k, f + g ◦ T k is the zero function. Also (to give an example that seems to be very narrow but whose seeming narrowness may be merely a reflection of our ignorance of analogous behavior in other dynamical systems), we may note that the combinatorial fact that underlies the existence of the Armstrong-Stanley way of looking at Panyushev complementation of antichains in [a] × [b], namely, the fact that A contains an element in the ith fiber if and only if the Panyushev complement of A contains an element in the ( i + 1 )st fiber (as long as i < a ), can also be expressed as a linear relation in V T.Here is an outline for how to approach V T systematically. Letting f1,..., f N\n\ndenote a basis for V T, define V T (X) to be the set of N -tuples (f1(x),..., f N (x))\n\nas x varies over X. Figuring out what linear relations are satisfied by the functions f1,..., f N when they are restricted to V T (X) is equivalent to:",
   "evidence": "The canonical record has a boundary error: after the one-sentence Problem 1.2, it appends most of Section 1.3, “Dynamical closures.” The official AIM PDF and the preceding canonical record recover the intended setup:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 1,
   "attempt": 1
  },
  "AIM-OTHER-0003": {
   "statement_status": "exact",
   "original_statement": "Problem 1.3. Characterize the affine closure of V T (X) in V T.\n\nOne could explore this computationally for specific dynamical systems (X, T ),and infer patterns that would lead to conjectures. The cases to start with are rowmotion and promotion on [a] × [b], since a wealth of homomesy and reci-procity relations are known. Are there invariance relations as well, or other sorts of affine relations satisfied by V T (X) that don't follow from known homo-mesy and reciprocity relations? 32 Combinatorial Problems \n\n2.1 The middle runner problem \n\nImagine n runners on a circular track, moving at various non-zero speeds (for simplicity, assume that the speeds are commensurable). The track has a starting line for counting laps (though runners are not required to start at the starting line), and the position of each runner at every instant is written as a number between 0 and 1 (representing how much of his/her current lap the runner has completed). At any instant t, let \n\np1(t) ≤ p2(t) ≤ · · · ≤ pn(t)\n\nbe the sorted positions of the runners, and for 1 ≤ i ≤ n let pi denote the average value of pi(t) over the course of one full period (the time it takes for all the runners to return to where they respectively started, which is just the lcm of the periods of all the runners).",
   "clean_statement": "Problem 1.3. Characterize the affine closure of V T (X) in V T.\n\nOne could explore this computationally for specific dynamical systems (X, T ),and infer patterns that would lead to conjectures. The cases to start with are rowmotion and promotion on [a] × [b], since a wealth of homomesy and reci-procity relations are known. Are there invariance relations as well, or other sorts of affine relations satisfied by V T (X) that don't follow from known homo-mesy and reciprocity relations? 32 Combinatorial Problems\n\n2.1 The middle runner problem\n\nImagine n runners on a circular track, moving at various non-zero speeds (for simplicity, assume that the speeds are commensurable). The track has a starting line for counting laps (though runners are not required to start at the starting line), and the position of each runner at every instant is written as a number between 0 and 1 (representing how much of his/her current lap the runner has completed). At any instant t, let\n\np1(t) ≤ p2(t) ≤ · · · ≤ pn(t)\n\nbe the sorted positions of the runners, and for 1 ≤ i ≤ n let pi denote the average value of pi(t) over the course of one full period (the time it takes for all the runners to return to where they respectively started, which is just the lcm of the periods of all the runners).",
   "public_statement": "Problem 1.3. Characterize the affine closure of V T (X) in V T.\n\nOne could explore this computationally for specific dynamical systems (X, T ),and infer patterns that would lead to conjectures. The cases to start with are rowmotion and promotion on [a] × [b], since a wealth of homomesy and reci-procity relations are known. Are there invariance relations as well, or other sorts of affine relations satisfied by V T (X) that don't follow from known homo-mesy and reciprocity relations? 32 Combinatorial Problems\n\n2.1 The middle runner problem\n\nImagine n runners on a circular track, moving at various non-zero speeds (for simplicity, assume that the speeds are commensurable). The track has a starting line for counting laps (though runners are not required to start at the starting line), and the position of each runner at every instant is written as a number between 0 and 1 (representing how much of his/her current lap the runner has completed). At any instant t, let\n\np1(t) ≤ p2(t) ≤ · · · ≤ pn(t)\n\nbe the sorted positions of the runners, and for 1 ≤ i ≤ n let pi denote the average value of pi(t) over the course of one full period (the time it takes for all the runners to return to where they respectively started, which is just the lcm of the periods of all the runners).",
   "evidence": "The canonical record is corrupted in two independent ways:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 2,
   "attempt": 1
  },
  "AIM-OTHER-0004": {
   "statement_status": "exact",
   "original_statement": "Conjecture 2.1. For i + j = n + 1, pi + pj = 1.\n\nThe middle runner problem takes its name from the special case in which n\n\nis odd and i = j = ( n + 1) /2: it says that the position of the middle runner is 1/2 on average. The case in which all runners have the same speed has been solved, but the general case remains open. \n\n2.2 Cores \n\nThere are bijection between simultaneous (a, b )-cores, lattice paths in the tri-angular region with vertices (0, 0), (a, 0), (a, b ), and certain (a, b )-noncrossing partitions [AHJ14, ARW13]. There are 1\n\n> a+b\n\n(a+bb\n\n) such objects. Armstrong conjectured that the average size |c| of an (a, b )-core c was \n\n(a − 1)( b − 1)( a + b − 1) 24.\n\nThis was proved in the Catalan case in [SZ13], the Fuss-Catalan case in [Agg14], and (very recently!) in full generality (using the polynomial method and Er-hart theory-cores are naturally points in the root lattice in a dilation of the fundamental alcove in affine type A) in [Joh15]. Thus, the number and av-erage size correspond to the 0th and 1st moments of the generating function ∑ \n\n> can (a,b )−core\n\nq|c|.",
   "clean_statement": "Conjecture 2.1. For i + j = n + 1, pi + pj = 1.\n\nThe middle runner problem takes its name from the special case in which n\n\nis odd and i = j = ( n + 1) /2: it says that the position of the middle runner is 1/2 on average. The case in which all runners have the same speed has been solved, but the general case remains open.\n\n2.2 Cores\n\nThere are bijection between simultaneous (a, b )-cores, lattice paths in the tri-angular region with vertices (0, 0), (a, 0), (a, b ), and certain (a, b )-noncrossing partitions [AHJ14, ARW13]. There are 1\n\n> a+b\n\n(a+bb\n\n) such objects. Armstrong conjectured that the average size |c| of an (a, b )-core c was\n\n(a − 1)( b − 1)( a + b − 1) 24.\n\nThis was proved in the Catalan case in [SZ13], the Fuss-Catalan case in [Agg14], and (very recently!) in full generality (using the polynomial method and Er-hart theory-cores are naturally points in the root lattice in a dilation of the fundamental alcove in affine type A) in [Joh15]. Thus, the number and av-erage size correspond to the 0th and 1st moments of the generating function ∑\n\n> can (a,b )−core\n\nq|c|.",
   "public_statement": "Conjecture 2.1. For i + j = n + 1, pi + pj = 1.\n\nThe middle runner problem takes its name from the special case in which n\n\nis odd and i = j = ( n + 1) /2: it says that the position of the middle runner is 1/2 on average. The case in which all runners have the same speed has been solved, but the general case remains open.\n\n2.2 Cores\n\nThere are bijection between simultaneous (a, b )-cores, lattice paths in the tri-angular region with vertices (0, 0), (a, 0), (a, b ), and certain (a, b )-noncrossing partitions [AHJ14, ARW13]. There are 1\n\n> a+b\n\n(a+bb\n\n) such objects. Armstrong conjectured that the average size |c| of an (a, b )-core c was\n\n(a − 1)( b − 1)( a + b − 1) 24.\n\nThis was proved in the Catalan case in [SZ13], the Fuss-Catalan case in [Agg14], and (very recently!) in full generality (using the polynomial method and Er-hart theory-cores are naturally points in the root lattice in a dilation of the fundamental alcove in affine type A) in [Joh15]. Thus, the number and av-erage size correspond to the 0th and 1st moments of the generating function ∑\n\n> can (a,b )−core\n\nq|c|.",
   "evidence": "The canonical record has two extraction defects.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 3,
   "attempt": 1
  },
  "AIM-OTHER-0005": {
   "statement_status": "exact",
   "original_statement": "Problem 2.2. Find and prove formulas for higher moments of cores: \n\n∑ \n\n> can (a,b )−core\n\n|c|i.\n\nOne can rotate an (a, b )-noncrossing partition, which can be modeled using toggles on rational slope lattice paths. 4",
   "clean_statement": "Problem 2.2. Find and prove formulas for higher moments of cores:\n\n∑\n\n> can (a,b )−core\n\n|c|i.\n\nOne can rotate an (a, b )-noncrossing partition, which can be modeled using toggles on rational slope lattice paths. 4",
   "public_statement": "Problem 2.2. Find and prove formulas for higher moments of cores:\n\n∑\n\n> can (a,b )−core\n\n|c|i.\n\nOne can rotate an (a, b )-noncrossing partition, which can be modeled using toggles on rational slope lattice paths. 4",
   "evidence": "The assigned record is Problem 2.2 in the official AIM pre-workshop list *Dynamical Algebraic Combinatorics*, dated May 29, 2015. Direct inspection of the PDF text around the damaged extraction recovers the display as \\[ \\boxed{\\quad \\sum_{\\substack{c\\ \\mathrm{an}\\ (a,b)\\text{-core}}}|c|^i . \\quad} \\] Thus the exact problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 4,
   "attempt": 1
  },
  "AIM-OTHER-0006": {
   "statement_status": "exact",
   "original_statement": "Conjecture 2.3. The set of (a, b )-noncrossing partitions under rotation exhibits the CSP using the polynomial 1[a+b]q\n\n(a+bb\n\n) \n\n> q.\n\n2.3 Perfect matchings \n\nThe Aztec diamond graph of order n has 2n(n+1) /2 perfect matchings (see e.g. [EKLP92a, EKLP92b]); fix n, and let X be the set of all such perfect matchings. for each edge e and each perfect matching M of the graph, let \n\n1e(M ) be 1 if e belongs to M and 0 otherwise. Then the average value of 1e(M )\n\nas M ranges over X can be interpreted as the probability that, if one chooses uniformly at random from the set X, the perfect matching one chooses will contain the edge e. These probabilities (as e ranges over the set of edges of the Aztec diamond graph of order n) are all rational numbers with denominators di-viding 2n(n+1) /2, and there is a priori no reason to think that the edge-inclusion probabilities (that is, the probabilities that specific edges will appear in a uni-formly random perfect matching) should be expressible as fractions with a much smaller denominator. However, it is known (though possibly not mentioned in published articles) that all such probabilities can be written as fractions with denominators dividing 2n (which is on the order of the square root of 2n(n+1) /2).",
   "clean_statement": "Conjecture 2.3. The set of (a, b )-noncrossing partitions under rotation exhibits the CSP using the polynomial 1[a+b]q\n\n(a+bb\n\n)\n\n> q.\n\n2.3 Perfect matchings\n\nThe Aztec diamond graph of order n has 2n(n+1) /2 perfect matchings (see e.g. [EKLP92a, EKLP92b]); fix n, and let X be the set of all such perfect matchings. for each edge e and each perfect matching M of the graph, let\n\n1e(M ) be 1 if e belongs to M and 0 otherwise. Then the average value of 1e(M )\n\nas M ranges over X can be interpreted as the probability that, if one chooses uniformly at random from the set X, the perfect matching one chooses will contain the edge e. These probabilities (as e ranges over the set of edges of the Aztec diamond graph of order n) are all rational numbers with denominators di-viding 2n(n+1) /2, and there is a priori no reason to think that the edge-inclusion probabilities (that is, the probabilities that specific edges will appear in a uni-formly random perfect matching) should be expressible as fractions with a much smaller denominator. However, it is known (though possibly not mentioned in published articles) that all such probabilities can be written as fractions with denominators dividing 2n (which is on the order of the square root of 2n(n+1) /2).",
   "public_statement": "Conjecture 2.3. The set of (a, b )-noncrossing partitions under rotation exhibits the CSP using the polynomial 1[a+b]q\n\n(a+bb\n\n)\n\n> q.\n\n2.3 Perfect matchings\n\nThe Aztec diamond graph of order n has 2n(n+1) /2 perfect matchings (see e.g. [EKLP92a, EKLP92b]); fix n, and let X be the set of all such perfect matchings. for each edge e and each perfect matching M of the graph, let\n\n1e(M ) be 1 if e belongs to M and 0 otherwise. Then the average value of 1e(M )\n\nas M ranges over X can be interpreted as the probability that, if one chooses uniformly at random from the set X, the perfect matching one chooses will contain the edge e. These probabilities (as e ranges over the set of edges of the Aztec diamond graph of order n) are all rational numbers with denominators di-viding 2n(n+1) /2, and there is a priori no reason to think that the edge-inclusion probabilities (that is, the probabilities that specific edges will appear in a uni-formly random perfect matching) should be expressible as fractions with a much smaller denominator. However, it is known (though possibly not mentioned in published articles) that all such probabilities can be written as fractions with denominators dividing 2n (which is on the order of the square root of 2n(n+1) /2).",
   "evidence": "The stored OCR record is corrupted in two independent ways. First, the displayed Gaussian binomial coefficient was flattened. Second, the record continues through the heading “2.3 Perfect matchings” and part of the Aztec-diamond discussion. The official AIM PDF shows that Conjecture 2.3 ends before that heading, on page 5 of the PDF (printed page 4). The perfect-matching text belongs to the next subsection and is not part of this problem.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 5,
   "attempt": 1
  },
  "AIM-OTHER-0007": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 2.4. Is there a cyclic action of order 2n on the set of perfect matchings of the Aztec diamond graph of order n, such that the edge-inclusion indicator functions associated with all the edges of the Aztec diamond graph are all ho-momesic? \n\nThis line of thinking is inspired by Sam Hopkins' succinct formulation of the homomesy enterprise via the slogan \"Small denominators are explained by group actions.\" A possible avenue to pursue in solving",
   "clean_statement": "**Problem 2.4.** Is there a cyclic action of order \\(2^n\\) on the set of perfect\nmatchings of the Aztec diamond graph of order \\(n\\), such that the edge-inclusion\nindicator functions associated with all the edges of the Aztec diamond graph are all\nhomomesic?",
   "public_statement": "Problem 2.4. Is there a cyclic action of order 2n on the set of perfect matchings of the Aztec diamond graph of order n, such that the edge-inclusion indicator functions associated with all the edges of the Aztec diamond graph are all ho-momesic?\n\nThis line of thinking is inspired by Sam Hopkins' succinct formulation of the homomesy enterprise via the slogan \"Small denominators are explained by group actions.\" A possible avenue to pursue in solving",
   "evidence": "The canonical JSON record is both truncated and affected by lost-superscript OCR. The official AIM preworkshop problem list, page 4, gives the following mathematical data:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-other-notes.json",
   "source_index": 6,
   "attempt": 1
  },
  "AIM-OTHER-0008": {
   "statement_status": "exact",
   "original_statement": "Problem 2.4 may be an analysis of domino shuffling, since the set of Aztec diamonds is naturally divided into \"equivalence classes\" of size 2n by the shuffling algorithm. \n\n2.4 Resonance \n\nSome researchers have recently been studying combinatorial actions that are not strictly speaking of finite order, or at least not uniformly finite as some size parameter n varies, but still exhibit some forms of periodicity. An important example is Wieland's gyration operation on Alternating Sign Matrices (ASMs). When n is small, the 2nth power of the gyration operation on n-by-n ASMs is the identity map, so that all orbits have size dividing 2n, but as n gets larger this ceases to be the case. Instead one finds orbits whose sizes are \"mostly\" multiple of 2n, or submultiples k(2 n)/m where m is a small divisor of 2n. This is a fairly squishy notion, for what do \"mostly\" and \"small\" mean? Without having answers to these questions, we have charged forward and dubbed this \"resonance\"; part of the challenge here is making a good definition of the phenomenon being studied.",
   "clean_statement": "Problem 2.4 may be an analysis of domino shuffling, since the set of Aztec diamonds is naturally divided into \"equivalence classes\" of size 2n by the shuffling algorithm.\n\n2.4 Resonance\n\nSome researchers have recently been studying combinatorial actions that are not strictly speaking of finite order, or at least not uniformly finite as some size parameter n varies, but still exhibit some forms of periodicity. An important example is Wieland's gyration operation on Alternating Sign Matrices (ASMs). When n is small, the 2nth power of the gyration operation on n-by-n ASMs is the identity map, so that all orbits have size dividing 2n, but as n gets larger this ceases to be the case. Instead one finds orbits whose sizes are \"mostly\" multiple of 2n, or submultiples k(2 n)/m where m is a small divisor of 2n. This is a fairly squishy notion, for what do \"mostly\" and \"small\" mean? Without having answers to these questions, we have charged forward and dubbed this \"resonance\"; part of the challenge here is making a good definition of the phenomenon being studied.",
   "public_statement": "Problem 2.4 may be an analysis of domino shuffling, since the set of Aztec diamonds is naturally divided into \"equivalence classes\" of size 2n by the shuffling algorithm.\n\n2.4 Resonance\n\nSome researchers have recently been studying combinatorial actions that are not strictly speaking of finite order, or at least not uniformly finite as some size parameter n varies, but still exhibit some forms of periodicity. An important example is Wieland's gyration operation on Alternating Sign Matrices (ASMs). When n is small, the 2nth power of the gyration operation on n-by-n ASMs is the identity map, so that all orbits have size dividing 2n, but as n gets larger this ceases to be the case. Instead one finds orbits whose sizes are \"mostly\" multiple of 2n, or submultiples k(2 n)/m where m is a small divisor of 2n. This is a fairly squishy notion, for what do \"mostly\" and \"small\" mean? Without having answers to these questions, we have charged forward and dubbed this \"resonance\"; part of the challenge here is making a good definition of the phenomenon being studied.",
   "evidence": "The canonical record is not a self-contained numbered problem. It is a splice of the end of the discussion following Problem 2.4 with the opening prose of Section 2.4, “Resonance.” The canonical extraction begins",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 7,
   "attempt": 1
  },
  "AIM-OTHER-0009": {
   "statement_status": "exact",
   "original_statement": "Problem 2.5. What is resonance? And why are there systems that exhibit it? \n\n5Another example of resonance appears with regard to rowmotion on order ideals of posets of the form [a] × [b] × [c] once a, b, and c are all large enough. One attempt to understand resonance has looked at the piecewise-linear (PL) \n\nlifts of maps that at the combinatorial level exhibit resonance. In many cases, we find that the PL dynamical system exhibits a whole spectrum of periods, each associated with a positive-measure subset of the polytope on which the map is acting. (For instance, for ASMs of order 4, the PL lift of gyration has lots of orbits of size 8, and lots of orbits of size 24, and many of much larger size, though curiously there are essentially none of size 16.) It would be good to understand this phenomenon better, and to find ways to relate it to the original, vaguely defined notion of resonance which pertains to orbits involving vertices of these polytopes, rather than interior points. This leads to the following problem.",
   "clean_statement": "Problem 2.5. What is resonance? And why are there systems that exhibit it?\n\n5Another example of resonance appears with regard to rowmotion on order ideals of posets of the form [a] × [b] × [c] once a, b, and c are all large enough. One attempt to understand resonance has looked at the piecewise-linear (PL)\n\nlifts of maps that at the combinatorial level exhibit resonance. In many cases, we find that the PL dynamical system exhibits a whole spectrum of periods, each associated with a positive-measure subset of the polytope on which the map is acting. (For instance, for ASMs of order 4, the PL lift of gyration has lots of orbits of size 8, and lots of orbits of size 24, and many of much larger size, though curiously there are essentially none of size 16.) It would be good to understand this phenomenon better, and to find ways to relate it to the original, vaguely defined notion of resonance which pertains to orbits involving vertices of these polytopes, rather than interior points. This leads to the following problem.",
   "public_statement": "Problem 2.5. What is resonance? And why are there systems that exhibit it?\n\n5Another example of resonance appears with regard to rowmotion on order ideals of posets of the form [a] × [b] × [c] once a, b, and c are all large enough. One attempt to understand resonance has looked at the piecewise-linear (PL)\n\nlifts of maps that at the combinatorial level exhibit resonance. In many cases, we find that the PL dynamical system exhibits a whole spectrum of periods, each associated with a positive-measure subset of the polytope on which the map is acting. (For instance, for ASMs of order 4, the PL lift of gyration has lots of orbits of size 8, and lots of orbits of size 24, and many of much larger size, though curiously there are essentially none of size 16.) It would be good to understand this phenomenon better, and to find ways to relate it to the original, vaguely defined notion of resonance which pertains to orbits involving vertices of these polytopes, rather than interior points. This leads to the following problem.",
   "evidence": "The official AIM PDF places this record in subsection 2.4, “Resonance.” The question itself is numbered 2.5:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 8,
   "attempt": 1
  },
  "AIM-OTHER-0010": {
   "statement_status": "exact",
   "original_statement": "Problem 2.6. Find/define analogues of resonance phenomena for the PL dy-namical systems mentioned above and their birational lifts. \n\nFor background on piecewise-linear and birational lifts of toggle-group ac-tions, see [EPb] and the more detailed article-in-progress [EPa]. \n\n2.5 Undiscovered combinatorial models \n\n2.5.1 The 3n − 2 Problem \n\nThe fact that Wieland's gyration operation T on n-by-n ASMs \"resonates with \n\n2n\" (even though it is not periodic with period 2n) can be readily understood in terms of the Fully Packed Loops model and the link pattern associated with an ASM (see [Pro01]); these link patterns admit a natural rotation action of order \n\n2n, and this action is compatible with gyration in the sense that turning an ASM into a link pattern and then applying rotation gives the same outcome as first applying gyration and then turning the resulting ASM into a link pattern. Striker and Williams found another action on ASMs, called superpromotion \n\nin [SW12], that resonates with 3n − 2. This suggests that there may be a map from n-by-n ASMs to some other class of combinatorial objects that admits a natural cyclic group action of order 3n − 2.",
   "clean_statement": "Problem 2.6. Find/define analogues of resonance phenomena for the PL dy-namical systems mentioned above and their birational lifts.\n\nFor background on piecewise-linear and birational lifts of toggle-group ac-tions, see [EPb] and the more detailed article-in-progress [EPa].\n\n2.5 Undiscovered combinatorial models\n\n2.5.1 The 3n − 2 Problem\n\nThe fact that Wieland's gyration operation T on n-by-n ASMs \"resonates with\n\n2n\" (even though it is not periodic with period 2n) can be readily understood in terms of the Fully Packed Loops model and the link pattern associated with an ASM (see [Pro01]); these link patterns admit a natural rotation action of order\n\n2n, and this action is compatible with gyration in the sense that turning an ASM into a link pattern and then applying rotation gives the same outcome as first applying gyration and then turning the resulting ASM into a link pattern. Striker and Williams found another action on ASMs, called superpromotion\n\nin [SW12], that resonates with 3n − 2. This suggests that there may be a map from n-by-n ASMs to some other class of combinatorial objects that admits a natural cyclic group action of order 3n − 2.",
   "public_statement": "Problem 2.6. Find/define analogues of resonance phenomena for the PL dy-namical systems mentioned above and their birational lifts.\n\nFor background on piecewise-linear and birational lifts of toggle-group ac-tions, see [EPb] and the more detailed article-in-progress [EPa].\n\n2.5 Undiscovered combinatorial models\n\n2.5.1 The 3n − 2 Problem\n\nThe fact that Wieland's gyration operation T on n-by-n ASMs \"resonates with\n\n2n\" (even though it is not periodic with period 2n) can be readily understood in terms of the Fully Packed Loops model and the link pattern associated with an ASM (see [Pro01]); these link patterns admit a natural rotation action of order\n\n2n, and this action is compatible with gyration in the sense that turning an ASM into a link pattern and then applying rotation gives the same outcome as first applying gyration and then turning the resulting ASM into a link pattern. Striker and Williams found another action on ASMs, called superpromotion\n\nin [SW12], that resonates with 3n − 2. This suggests that there may be a map from n-by-n ASMs to some other class of combinatorial objects that admits a natural cyclic group action of order 3n − 2.",
   "evidence": "The canonical record contains the genuine Problem 2.6 followed by text from the next subsection. The official AIM preworkshop PDF, page 5, has this exact boundary:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 9,
   "attempt": 1
  },
  "AIM-OTHER-0011": {
   "statement_status": "exact",
   "original_statement": "Problem 2.7. What action of order 3n − 2, on combinatorial objects of some unspecified kind, plays the role that rotation of link-patterns does in the case of gyration? \n\nOne possible approach would be to encode ASMs of order n as certain points in kO(Φ +(An)). The vertices kO(Φ +(An)) are naturally labeled by noncrossing partitions: on such an encoding, is the link pattern uncovered by projecting an ASM to a vertex of kO(Φ +(An))? If so, the vertices are also labeled by triangulations. Nathan Williams has pointed out that n + 2 (the order of rotation of a triangulation), 2n (the order of the Kreweras complement), and 3n − 2 (the order of this conjectural mystery action) are in arithmetic progression. 62.5.2 Multi-noncrossing models \n\nOne can construct multi-noncrossing objects using a subword construction due to Ceballos, Labbé, Stump [CLS14]. These have a well-known triangulation model, which was studied, for exam-ple, by Pilaud in [PP12]. In types A, B, H 3, I 2(m), these also have a nonnesting model as P -partitions in the root poset.",
   "clean_statement": "Problem 2.7. What action of order 3n − 2, on combinatorial objects of some unspecified kind, plays the role that rotation of link-patterns does in the case of gyration?\n\nOne possible approach would be to encode ASMs of order n as certain points in kO(Φ +(An)). The vertices kO(Φ +(An)) are naturally labeled by noncrossing partitions: on such an encoding, is the link pattern uncovered by projecting an ASM to a vertex of kO(Φ +(An))? If so, the vertices are also labeled by triangulations. Nathan Williams has pointed out that n + 2 (the order of rotation of a triangulation), 2n (the order of the Kreweras complement), and 3n − 2 (the order of this conjectural mystery action) are in arithmetic progression. 62.5.2 Multi-noncrossing models\n\nOne can construct multi-noncrossing objects using a subword construction due to Ceballos, Labbé, Stump [CLS14]. These have a well-known triangulation model, which was studied, for exam-ple, by Pilaud in [PP12]. In types A, B, H 3, I 2(m), these also have a nonnesting model as P -partitions in the root poset.",
   "public_statement": "Problem 2.7. What action of order 3n − 2, on combinatorial objects of some unspecified kind, plays the role that rotation of link-patterns does in the case of gyration?\n\nOne possible approach would be to encode ASMs of order n as certain points in kO(Φ +(An)). The vertices kO(Φ +(An)) are naturally labeled by noncrossing partitions: on such an encoding, is the link pattern uncovered by projecting an ASM to a vertex of kO(Φ +(An))? If so, the vertices are also labeled by triangulations. Nathan Williams has pointed out that n + 2 (the order of rotation of a triangulation), 2n (the order of the Kreweras complement), and 3n − 2 (the order of this conjectural mystery action) are in arithmetic progression. 62.5.2 Multi-noncrossing models\n\nOne can construct multi-noncrossing objects using a subword construction due to Ceballos, Labbé, Stump [CLS14]. These have a well-known triangulation model, which was studied, for exam-ple, by Pilaud in [PP12]. In types A, B, H 3, I 2(m), these also have a nonnesting model as P -partitions in the root poset.",
   "evidence": "The canonical record is Problem 2.7 in *Problems for 2015 AIM workshop on Dynamical Algebraic Combinatorics* (May 29, 2015), prepared by Jim Propp, Tom Roby, Jessica Striker, and Nathan Williams. The source asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 10,
   "attempt": 1
  },
  "AIM-OTHER-0012": {
   "statement_status": "exact",
   "original_statement": "Problem 2.8 ([CLS14]). Prove a cyclic sieving phenomenon for multitriangu-lations, using the analogue of Cambrian rotation and the polynomial \n\n∏\n\n> 0≤j<k\n\n∏\n\n> 1≤i≤n\n\n[di + h + 2 j]q\n\n[di + 2 j]q.",
   "clean_statement": "Problem 2.8 ([CLS14]). Prove a cyclic sieving phenomenon for multitriangu-lations, using the analogue of Cambrian rotation and the polynomial\n\n∏\n\n> 0≤j<k\n\n∏\n\n> 1≤i≤n\n\n[di + h + 2 j]q\n\n[di + 2 j]q.",
   "public_statement": "Problem 2.8 ([CLS14]). Prove a cyclic sieving phenomenon for multitriangu-lations, using the analogue of Cambrian rotation and the polynomial\n\n∏\n\n> 0≤j<k\n\n∏\n\n> 1≤i≤n\n\n[di + h + 2 j]q\n\n[di + 2 j]q.",
   "evidence": "The official AIM PDF gives the following question in subsection 2.5.2, “Multi-noncrossing models”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 11,
   "attempt": 1
  },
  "AIM-OTHER-0013": {
   "statement_status": "exact",
   "original_statement": "Problem 2.9. Is there a corresponding multi-noncrossing partition model? \n\nOne would expect that this has something to do with the root configuration, and the case k = 1 is well-known in classical types. Using the correspondence between the type Bn root poset and [n] × [n], the case k = 2 for the type Bn\n\nroot poset corresponds to certain Narayana numbers [SW12]. Similarly, the case \n\nk = 3 corresponds to Baxter numbers [Dil14]. Both of these have noncrossing models: Narayana numbers are well-known to be noncrossing partitions with a specified number of blocks; N. Reading has given a combinatorial description of Baxter permutations as noncrossing diagrams on a horizontal line of points using arcs that stay above or below the points [Rea14]. \n\n2.6 Products of chains \n\nBloom et al. [BPS13] proved a homomesy result for rectangular semistandard tableaux under promotion. But we also know that promotion of semistandard tableaux is a special case of PL promotion in the order polytope of a prod-uct of two chains (see http://jamespropp.org/gtt-promotion.txt), and we also know homomesy results for rowmotion and promotion in products of two chains (Propp and Roby).",
   "clean_statement": "Problem 2.9. Is there a corresponding multi-noncrossing partition model?\n\nOne would expect that this has something to do with the root configuration, and the case k = 1 is well-known in classical types. Using the correspondence between the type Bn root poset and [n] × [n], the case k = 2 for the type Bn\n\nroot poset corresponds to certain Narayana numbers [SW12]. Similarly, the case\n\nk = 3 corresponds to Baxter numbers [Dil14]. Both of these have noncrossing models: Narayana numbers are well-known to be noncrossing partitions with a specified number of blocks; N. Reading has given a combinatorial description of Baxter permutations as noncrossing diagrams on a horizontal line of points using arcs that stay above or below the points [Rea14].\n\n2.6 Products of chains\n\nBloom et al. [BPS13] proved a homomesy result for rectangular semistandard tableaux under promotion. But we also know that promotion of semistandard tableaux is a special case of PL promotion in the order polytope of a prod-uct of two chains (see http://jamespropp.org/gtt-promotion.txt), and we also know homomesy results for rowmotion and promotion in products of two chains (Propp and Roby).",
   "public_statement": "Problem 2.9. Is there a corresponding multi-noncrossing partition model?\n\nOne would expect that this has something to do with the root configuration, and the case k = 1 is well-known in classical types. Using the correspondence between the type Bn root poset and [n] × [n], the case k = 2 for the type Bn\n\nroot poset corresponds to certain Narayana numbers [SW12]. Similarly, the case\n\nk = 3 corresponds to Baxter numbers [Dil14]. Both of these have noncrossing models: Narayana numbers are well-known to be noncrossing partitions with a specified number of blocks; N. Reading has given a combinatorial description of Baxter permutations as noncrossing diagrams on a horizontal line of points using arcs that stay above or below the points [Rea14].\n\n2.6 Products of chains\n\nBloom et al. [BPS13] proved a homomesy result for rectangular semistandard tableaux under promotion. But we also know that promotion of semistandard tableaux is a special case of PL promotion in the order polytope of a prod-uct of two chains (see http://jamespropp.org/gtt-promotion.txt), and we also know homomesy results for rowmotion and promotion in products of two chains (Propp and Roby).",
   "evidence": "The canonical JSON record has merged the end of Problem 2.9 with the beginning of Section 2.6. Inspection of the official AIM preworkshop PDF gives the following clean record (notation normalized only by writing subscripts and Cartesian products in LaTeX):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 12,
   "attempt": 1
  },
  "AIM-OTHER-0014": {
   "statement_status": "exact",
   "original_statement": "Problem 2.10. How do the results of Bloom, Pechenik, and Saracino relate to the results of Propp and Roby? \n\nAlso, there is work to be done regarding rowmotion on products of three chains. The obvious cardinality statistic is not in general homomesic under rowmotion in general, but various other statistics are (or at least appear to be, experimentally).",
   "clean_statement": "Problem 2.10. How do the results of Bloom, Pechenik, and Saracino relate to the results of Propp and Roby?\n\nAlso, there is work to be done regarding rowmotion on products of three chains. The obvious cardinality statistic is not in general homomesic under rowmotion in general, but various other statistics are (or at least appear to be, experimentally).",
   "public_statement": "Problem 2.10. How do the results of Bloom, Pechenik, and Saracino relate to the results of Propp and Roby?\n\nAlso, there is work to be done regarding rowmotion on products of three chains. The obvious cardinality statistic is not in general homomesic under rowmotion in general, but various other statistics are (or at least appear to be, experimentally).",
   "evidence": "The canonical record comes from Section 2.6, “Products of chains,” of *Problems for 2015 AIM workshop on Dynamical Algebraic Combinatorics*. The official PDF has the following sequence on page 6:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 13,
   "attempt": 1
  },
  "AIM-OTHER-0015": {
   "statement_status": "exact",
   "original_statement": "Problem 2.11. What are the homomesies of rowmotion acting order ideals in a product of three chains? Likewise, what are the homomesies of Panyushev complementation acting on antichains in a product of three chains? \n\n73 Coxeter-theoretic Problems \n\n3.1 Bijactions in Cataland \n\nThe following problem is given in greater detail in [Wil14] and in much greater detail in [Wil13]. Let W be a finite Weyl group (or type H3 or I2(m)) and let c be a Coxeter element. Fix the word in simple reflections Q = ( Q1, Q2,..., QN +n):= cw o(c)\n\n(here wo(c) is the c-sorting word for wo). The (W, c )-subwords are the elements of the set \n\nAsoc (W, c ):= {(i1 ≤ i2 ≤ · · · ≤ iN ): Qi1 Qi2 · · · QiN = wo}.\n\nWe find it convenient to complete the word Q to the word QQ = ch+2 (up to commutations), and to think of a subword in Asoc (W, c ) as a doubled subword of this doubled word. Define the nonnesting c-Cambrian rotation Camb c: NN (W ) → NN (W ) by \n\nCamb c:= Tog inv( wo(c)) Tog +inv( wo(c)),\n\nwhere Tog + \n\n> α\n\n(x):= \n\n{ Tog α(x) if α 6 ∈ ∆( W ); \n\nx otherwise, and Tog + \n\n> α1α2··· αi:= Tog + \n\n> αi\n\n· · · Tog + \n\n> α1.",
   "clean_statement": "Problem 2.11. What are the homomesies of rowmotion acting order ideals in a product of three chains? Likewise, what are the homomesies of Panyushev complementation acting on antichains in a product of three chains?\n\n73 Coxeter-theoretic Problems\n\n3.1 Bijactions in Cataland\n\nThe following problem is given in greater detail in [Wil14] and in much greater detail in [Wil13]. Let W be a finite Weyl group (or type H3 or I2(m)) and let c be a Coxeter element. Fix the word in simple reflections Q = ( Q1, Q2,..., QN +n):= cw o(c)\n\n(here wo(c) is the c-sorting word for wo). The (W, c )-subwords are the elements of the set\n\nAsoc (W, c ):= {(i1 ≤ i2 ≤ · · · ≤ iN ): Qi1 Qi2 · · · QiN = wo}.\n\nWe find it convenient to complete the word Q to the word QQ = ch+2 (up to commutations), and to think of a subword in Asoc (W, c ) as a doubled subword of this doubled word. Define the nonnesting c-Cambrian rotation Camb c: NN (W ) → NN (W ) by\n\nCamb c:= Tog inv( wo(c)) Tog +inv( wo(c)),\n\nwhere Tog +\n\n> α\n\n(x):=\n\n{ Tog α(x) if α 6 ∈ ∆( W );\n\nx otherwise, and Tog +\n\n> α1α2··· αi:= Tog +\n\n> αi\n\n· · · Tog +\n\n> α1.",
   "public_statement": "Problem 2.11. What are the homomesies of rowmotion acting order ideals in a product of three chains? Likewise, what are the homomesies of Panyushev complementation acting on antichains in a product of three chains?\n\n73 Coxeter-theoretic Problems\n\n3.1 Bijactions in Cataland\n\nThe following problem is given in greater detail in [Wil14] and in much greater detail in [Wil13]. Let W be a finite Weyl group (or type H3 or I2(m)) and let c be a Coxeter element. Fix the word in simple reflections Q = ( Q1, Q2,..., QN +n):= cw o(c)\n\n(here wo(c) is the c-sorting word for wo). The (W, c )-subwords are the elements of the set\n\nAsoc (W, c ):= {(i1 ≤ i2 ≤ · · · ≤ iN ): Qi1 Qi2 · · · QiN = wo}.\n\nWe find it convenient to complete the word Q to the word QQ = ch+2 (up to commutations), and to think of a subword in Asoc (W, c ) as a doubled subword of this doubled word. Define the nonnesting c-Cambrian rotation Camb c: NN (W ) → NN (W ) by\n\nCamb c:= Tog inv( wo(c)) Tog +inv( wo(c)),\n\nwhere Tog +\n\n> α\n\n(x):=\n\n{ Tog α(x) if α 6 ∈ ∆( W );\n\nx otherwise, and Tog +\n\n> α1α2··· αi:= Tog +\n\n> αi\n\n· · · Tog +\n\n> α1.",
   "evidence": "The canonical JSON record contains a page/section-boundary merge. The mathematical problem ends after its second question; the following text beginning `73 Coxeter-theoretic Problems` is the printed page number 7 followed by the heading `3 Coxeter-theoretic Problems`, and belongs to the next section and next canonical record. The recovered statement is therefore:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 14,
   "attempt": 1
  },
  "AIM-OTHER-0016": {
   "statement_status": "exact",
   "original_statement": "Conjecture 3.1. A bijaction 1 from J(Φ +(W )) under Camb c to Asoc (W, c )\n\n(under noncrossing Cambrian rotation) is given as follows. Beginning with a nonnesting partition x, compute the orbit \n\n(\n\nx, Camb c(x), Camb 2 \n\n> c\n\n(x),..., Camb h+1 \n\n> c\n\n(x)\n\n).\n\nThe subword of ch+2 is given by replacing each nonnesting partition Camb kc (x)\n\nin this sequence by a copy of c, adding to the subword those simple reflections whose corresponding roots are in Camb kc (x).\n\nThis has immediate homomesy implications-for example, in type An, this corresponds to homomesies of rotation of a triangulation. There is so much more to say on this: there is a similar and intimately related map for the Kreweras complement to noncrossing partitions; one can walk on the Cambrian lattice in this order to realize the noncrossing versions. \n\n3.2 Nonnesting Cataland Lifts \n\nBirational toggles in wo(c) root orders appear to continue to have order 2h in type An. This fails, for example, in type D4 (as does birational rowmotion).",
   "clean_statement": "Conjecture 3.1. A bijaction 1 from J(Φ +(W )) under Camb c to Asoc (W, c )\n\n(under noncrossing Cambrian rotation) is given as follows. Beginning with a nonnesting partition x, compute the orbit\n\n(\n\nx, Camb c(x), Camb 2\n\n> c\n\n(x),..., Camb h+1\n\n> c\n\n(x)\n\n).\n\nThe subword of ch+2 is given by replacing each nonnesting partition Camb kc (x)\n\nin this sequence by a copy of c, adding to the subword those simple reflections whose corresponding roots are in Camb kc (x).\n\nThis has immediate homomesy implications-for example, in type An, this corresponds to homomesies of rotation of a triangulation. There is so much more to say on this: there is a similar and intimately related map for the Kreweras complement to noncrossing partitions; one can walk on the Cambrian lattice in this order to realize the noncrossing versions.\n\n3.2 Nonnesting Cataland Lifts\n\nBirational toggles in wo(c) root orders appear to continue to have order 2h in type An. This fails, for example, in type D4 (as does birational rowmotion).",
   "public_statement": "Conjecture 3.1. A bijaction 1 from J(Φ +(W )) under Camb c to Asoc (W, c )\n\n(under noncrossing Cambrian rotation) is given as follows. Beginning with a nonnesting partition x, compute the orbit\n\n(\n\nx, Camb c(x), Camb 2\n\n> c\n\n(x),..., Camb h+1\n\n> c\n\n(x)\n\n).\n\nThe subword of ch+2 is given by replacing each nonnesting partition Camb kc (x)\n\nin this sequence by a copy of c, adding to the subword those simple reflections whose corresponding roots are in Camb kc (x).\n\nThis has immediate homomesy implications-for example, in type An, this corresponds to homomesies of rotation of a triangulation. There is so much more to say on this: there is a similar and intimately related map for the Kreweras complement to noncrossing partitions; one can walk on the Cambrian lattice in this order to realize the noncrossing versions.\n\n3.2 Nonnesting Cataland Lifts\n\nBirational toggles in wo(c) root orders appear to continue to have order 2h in type An. This fails, for example, in type D4 (as does birational rowmotion).",
   "evidence": "The canonical input is Conjecture 3.1 of the AIM pre-workshop list *Dynamical algebraic combinatorics*. Its notation is introduced at the end of the preceding canonical record. Let \\(W\\) be a finite Weyl group of rank \\(n\\), let \\(c=s_1\\cdots s_n\\) be a Coxeter element, let \\(h\\) be the Coxeter number, and put \\(N=|\\Phi^+(W)|=nh/2\\). If \\(w_0(c)\\) is the \\(c\\)-sorting word for the longest element, set",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 15,
   "attempt": 1
  },
  "AIM-OTHER-0017": {
   "statement_status": "exact",
   "original_statement": "Conjecture 3.2. Birational toggles in wo(c) root orders have order 2h in the coincidental types A, B, H 3, I 2(m).\n\n> 1Bijaction: a bijection induced by an action.\n\n8Can we give piecewise-linear and birational analogues of Armstrong, Stump, and Thomas' proof (see [AST13]) of Panyushev's",
   "clean_statement": "Conjecture 3.2. Birational toggles in wo(c) root orders have order 2h in the coincidental types A, B, H 3, I 2(m).\n\n> 1Bijaction: a bijection induced by an action.\n\n8Can we give piecewise-linear and birational analogues of Armstrong, Stump, and Thomas' proof (see [AST13]) of Panyushev's",
   "public_statement": "Conjecture 3.2. Birational toggles in wo(c) root orders have order 2h in the coincidental types A, B, H 3, I 2(m).\n\n> 1Bijaction: a bijection induced by an action.\n\n8Can we give piecewise-linear and birational analogues of Armstrong, Stump, and Thomas' proof (see [AST13]) of Panyushev's",
   "evidence": "The official AIM pre-workshop PDF has the following complete text in §3.2:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 16,
   "attempt": 1
  },
  "AIM-OTHER-0018": {
   "statement_status": "exact",
   "original_statement": "Conjecture 2.1 (see [Pan09]) asserting homomesy of antichain cardinality under rowmotion? To clarify the meaning of this question we provide a bit of background. Let P be the root poset of type An, with order polytope O(P ) and chain polytope C(P ). Let ρ: O(P ) → O (P ) be PL rowmotion and φ: O(P ) →C(P ) be Stanley's transfer map [Sta86]. Then the map ρ′:= φ ◦ ρ ◦ φ−1:\n\nC(P ) → C (P ) may be viewed as a PL analogue of the Panyushev complement (since its restriction to the vertices of C(P ), that is, to the antichains of P, is Panyushev complementation). Let f: C(P ) → R be the function that adds all the coordinates of a point in C(P ) (the PL analogue of the cardinality of an antichain).",
   "clean_statement": "Conjecture 2.1 (see [Pan09]) asserting homomesy of antichain cardinality under rowmotion? To clarify the meaning of this question we provide a bit of background. Let P be the root poset of type An, with order polytope O(P ) and chain polytope C(P ). Let ρ: O(P ) → O (P ) be PL rowmotion and φ: O(P ) →C(P ) be Stanley's transfer map [Sta86]. Then the map ρ′:= φ ◦ ρ ◦ φ−1:\n\nC(P ) → C (P ) may be viewed as a PL analogue of the Panyushev complement (since its restriction to the vertices of C(P ), that is, to the antichains of P, is Panyushev complementation). Let f: C(P ) → R be the function that adds all the coordinates of a point in C(P ) (the PL analogue of the cardinality of an antichain).",
   "public_statement": "Conjecture 2.1 (see [Pan09]) asserting homomesy of antichain cardinality under rowmotion? To clarify the meaning of this question we provide a bit of background. Let P be the root poset of type An, with order polytope O(P ) and chain polytope C(P ). Let ρ: O(P ) → O (P ) be PL rowmotion and φ: O(P ) →C(P ) be Stanley's transfer map [Sta86]. Then the map ρ′:= φ ◦ ρ ◦ φ−1:\n\nC(P ) → C (P ) may be viewed as a PL analogue of the Panyushev complement (since its restriction to the vertices of C(P ), that is, to the antichains of P, is Panyushev complementation). Let f: C(P ) → R be the function that adds all the coordinates of a point in C(P ) (the PL analogue of the cardinality of an antichain).",
   "evidence": "The canonical record is cut at both ends. Inspection of the official AIM PDF, pages 8--9 of the document (PDF pages 7--8), gives the missing beginning:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 17,
   "attempt": 1
  },
  "AIM-OTHER-0019": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 3.3. f is homomesic under the action of ρ′, with average value \n\nn/ 2.\n\nIt appears that a similar homomesy holds for the natural birational lift of ρ′.Experiments in Mathematica show (by brute force) that ̂ f is 0-mesic under the action of ̂ ρ′ for the cases n ≤ 3, where ̂ f is the logarithm of the product of the entries of a triangular array of formal indeterminates, and ̂ ρ′ is the birational lift of ρ′ defined in the most straightforward fashion. A birational Armstrong-Stump-Thomas theorem would yield the \"classical\" Armstrong-Stump-Thomas result as a corollary in the usual way (first tropicalize to obtain the PL version, then specialize to the vertices of the order polytope). It should also be noted that Panyushev's article contains other conjectures about homomesy for cardinality of antichains, which apparently have not been proved. \n\n3.3 Coincidental Types \n\nThis problem is taken from [Wil13, Wil14]. Define the posets n:= [ n]×[n], n = J ([2] ×[n]), and n:= J n([2] ×[2]).\n\nThese are the (Gaussian/minuscule) root posets for certain maximal parabolic quotients W J [Ste96]. \n\nTheorem 3.4. We have the following equalities: \n\n|L (Φ +(An)) | = 2 n(n−1) /2|L ( n)|, 2n|J (Φ +(An) × [k]) | = |J ( n × [2 k + 1]) |;\n\n|L (Φ +(Bn)) | = |L ( n)|, |J (Φ +(Bn) × [k]) | = |J ( n × [k]) |;\n\n|L (Φ +(H3)) | = |L ( 5)|, |J (Φ +(H3) × [k]) | = |J ( 5 × [k]) |; and \n\n|L (Φ +(I2(2 m))) | = |L ( m−2)|, |J (Φ +(I2(2 m)) × [k]) | = |J ( m−2 × [k]) | for m ≥ 2.",
   "clean_statement": "**Conjecture 3.3.** The statistic \\(F\\) is homomesic under \\(\\rho'\\), with average \\(n/2\\).",
   "public_statement": "Conjecture 3.3. f is homomesic under the action of ρ′, with average value\n\nn/ 2.\n\nIt appears that a similar homomesy holds for the natural birational lift of ρ′.Experiments in Mathematica show (by brute force) that ̂ f is 0-mesic under the action of ̂ ρ′ for the cases n ≤ 3, where ̂ f is the logarithm of the product of the entries of a triangular array of formal indeterminates, and ̂ ρ′ is the birational lift of ρ′ defined in the most straightforward fashion. A birational Armstrong-Stump-Thomas theorem would yield the \"classical\" Armstrong-Stump-Thomas result as a corollary in the usual way (first tropicalize to obtain the PL version, then specialize to the vertices of the order polytope). It should also be noted that Panyushev's article contains other conjectures about homomesy for cardinality of antichains, which apparently have not been proved.\n\n3.3 Coincidental Types\n\nThis problem is taken from [Wil13, Wil14]. Define the posets n:= [ n]×[n], n = J ([2] ×[n]), and n:= J n([2] ×[2]).\n\nThese are the (Gaussian/minuscule) root posets for certain maximal parabolic quotients W J [Ste96].\n\nTheorem 3.4. We have the following equalities:\n\n|L (Φ +(An)) | = 2 n(n−1) /2|L ( n)|, 2n|J (Φ +(An) × [k]) | = |J ( n × [2 k + 1]) |;\n\n|L (Φ +(Bn)) | = |L ( n)|, |J (Φ +(Bn) × [k]) | = |J ( n × [k]) |;\n\n|L (Φ +(H3)) | = |L ( 5)|, |J (Φ +(H3) × [k]) | = |J ( 5 × [k]) |; and\n\n|L (Φ +(I2(2 m))) | = |L ( m−2)|, |J (Φ +(I2(2 m)) × [k]) | = |J ( m−2 × [k]) | for m ≥ 2.",
   "evidence": "The canonical record is extracted from the 2015 AIM pre-workshop document *Problems for 2015 AIM workshop on Dynamical Algebraic Combinatorics*, Section 3.2, “Nonnesting Cataland Lifts.” The extraction accidentally continues past Conjecture 3.3 into the heading “3.3 Coincidental Types” and Theorem 3.4. That later material is not part of this problem.",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-other-notes.json",
   "source_index": 18,
   "attempt": 1
  },
  "AIM-OTHER-0020": {
   "statement_status": "exact",
   "original_statement": "Problem 3.5. Give combinatorial proofs of the equalities above. \n\nWe will refer to an equation in Theorem 3.4 by its row ( A, B, H, or I) and its column ( L or J ). Note that AJ is already interesting for k = 1.9",
   "clean_statement": "Problem 3.5. Give combinatorial proofs of the equalities above.\n\nWe will refer to an equation in Theorem 3.4 by its row ( A, B, H, or I) and its column ( L or J ). Note that AJ is already interesting for k = 1.9",
   "public_statement": "Problem 3.5. Give combinatorial proofs of the equalities above.\n\nWe will refer to an equation in Theorem 3.4 by its row ( A, B, H, or I) and its column ( L or J ). Note that AJ is already interesting for k = 1.9",
   "evidence": "The canonical record comes from page 9 of the official 2015 AIM pre-workshop problem list, Section 3.3, “Coincidental Types.” Its exact problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 19,
   "attempt": 1
  },
  "AIM-OTHER-0021": {
   "statement_status": "exact",
   "original_statement": "Problem 3.6. Relate both sides of the L identities under promotion, and both sides of the J identities under (birational) promotion/rowmotion. \n\nThe hook-length and shifted hook-length formulas prove AL and BL. R. Proctor simultaneously established BL and BJ with a representation-theoretic proof of BJ [Pro83], while K. Purbhoo in unpublished work and M. Haiman in [Hai92] found beautiful jeu-de-taquin bijections for AL and BL, respectively. I believe that the remaining equalities are new or trivial. 2\n\n3.4 Hurwitz Actions on Factorizations of c\n\nThis problem comes from [Wil13]; D. Bessis made a reference to the possible existence of such a problem in Bielefeld ( http://www.math.uni-bielefeld. de/birep/meetings/ncp2014/ ). For W a Coxeter group with degrees d1, d 2,..., d n and Coxeter number h,the factorizations of a Coxeter element c are counted by the uniform formula \n\n|Red T (c)| = n!hn\n\n|W | =\n\n> n\n\n∏\n\n> i=1\n\nih di.\n\nIn type An, these are equinumerous with parking functions, so it is possible to rephrase this problem in type A in terms of parking functions; there is a simple bijection for linear c due to Stanley (a similar idea also works in types B and \n\nD). We act on T -words (words using reflections T ) using the dual braid move \n\nTi: Red T (w) → Red T (w) by \n\nTi\n\n(\n\nt1,..., t i, t i+1,..., t `\n\n)\n\n=\n\n(\n\nt1,..., t i+1, (ti+1 titi+1 ),..., t `\n\n).\n\nFix W and a reduced word w = si1 · · · sik for w in An−1. Define the action \n\nTw:= Ti1 · · · Tik.\n\nIt is easy to see Tw does not depend on the choice of reduced word for w.One can compute that Two has order 2h on Red T (c), and that Tc has order nh \n\non Red T (c).",
   "clean_statement": "Problem 3.6. Relate both sides of the L identities under promotion, and both sides of the J identities under (birational) promotion/rowmotion.\n\nThe hook-length and shifted hook-length formulas prove AL and BL. R. Proctor simultaneously established BL and BJ with a representation-theoretic proof of BJ [Pro83], while K. Purbhoo in unpublished work and M. Haiman in [Hai92] found beautiful jeu-de-taquin bijections for AL and BL, respectively. I believe that the remaining equalities are new or trivial. 2\n\n3.4 Hurwitz Actions on Factorizations of c\n\nThis problem comes from [Wil13]; D. Bessis made a reference to the possible existence of such a problem in Bielefeld ( http://www.math.uni-bielefeld. de/birep/meetings/ncp2014/ ). For W a Coxeter group with degrees d1, d 2,..., d n and Coxeter number h,the factorizations of a Coxeter element c are counted by the uniform formula\n\n|Red T (c)| = n!hn\n\n|W | =\n\n> n\n\n∏\n\n> i=1\n\nih di.\n\nIn type An, these are equinumerous with parking functions, so it is possible to rephrase this problem in type A in terms of parking functions; there is a simple bijection for linear c due to Stanley (a similar idea also works in types B and\n\nD). We act on T -words (words using reflections T ) using the dual braid move\n\nTi: Red T (w) → Red T (w) by\n\nTi\n\n(\n\nt1,..., t i, t i+1,..., t `\n\n)\n\n=\n\n(\n\nt1,..., t i+1, (ti+1 titi+1 ),..., t `\n\n).\n\nFix W and a reduced word w = si1 · · · sik for w in An−1. Define the action\n\nTw:= Ti1 · · · Tik.\n\nIt is easy to see Tw does not depend on the choice of reduced word for w.One can compute that Two has order 2h on Red T (c), and that Tc has order nh\n\non Red T (c).",
   "public_statement": "Problem 3.6. Relate both sides of the L identities under promotion, and both sides of the J identities under (birational) promotion/rowmotion.\n\nThe hook-length and shifted hook-length formulas prove AL and BL. R. Proctor simultaneously established BL and BJ with a representation-theoretic proof of BJ [Pro83], while K. Purbhoo in unpublished work and M. Haiman in [Hai92] found beautiful jeu-de-taquin bijections for AL and BL, respectively. I believe that the remaining equalities are new or trivial. 2\n\n3.4 Hurwitz Actions on Factorizations of c\n\nThis problem comes from [Wil13]; D. Bessis made a reference to the possible existence of such a problem in Bielefeld ( http://www.math.uni-bielefeld. de/birep/meetings/ncp2014/ ). For W a Coxeter group with degrees d1, d 2,..., d n and Coxeter number h,the factorizations of a Coxeter element c are counted by the uniform formula\n\n|Red T (c)| = n!hn\n\n|W | =\n\n> n\n\n∏\n\n> i=1\n\nih di.\n\nIn type An, these are equinumerous with parking functions, so it is possible to rephrase this problem in type A in terms of parking functions; there is a simple bijection for linear c due to Stanley (a similar idea also works in types B and\n\nD). We act on T -words (words using reflections T ) using the dual braid move\n\nTi: Red T (w) → Red T (w) by\n\nTi\n\n(\n\nt1,..., t i, t i+1,..., t `\n\n)\n\n=\n\n(\n\nt1,..., t i+1, (ti+1 titi+1 ),..., t `\n\n).\n\nFix W and a reduced word w = si1 · · · sik for w in An−1. Define the action\n\nTw:= Ti1 · · · Tik.\n\nIt is easy to see Tw does not depend on the choice of reduced word for w.One can compute that Two has order 2h on Red T (c), and that Tc has order nh\n\non Red T (c).",
   "evidence": "The source is the AIM pre-workshop list *Problems in Dynamical Algebraic Combinatorics* (2015), §3.3, pp. 8--9. Its exact question is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 20,
   "attempt": 1
  },
  "AIM-OTHER-0022": {
   "statement_status": "exact",
   "original_statement": "Conjecture 3.7. For w = wo, c,\n\n(\n\nRed T (c),\n\n> n\n\n∏\n\n> i=1\n\n[ih ]q\n\n[di]q, Tw\n\n)\n\nexhibits the CSP. \n\n> 2There is a simple representation-theoretic proof of AJ, but I don't know of any such proof for HJ.\n\n10 This can probably be proved pretty easily for w = wo using a combinatorial construction that associates pairs of factors in the orbit under Two to certain bicolored quadrangulations. There is an action of order h on Red T (c), which is quite simply conjugation by c. It is easy to see that this gives all orbits of size h.",
   "clean_statement": "Conjecture 3.7. For w = wo, c,\n\n(\n\nRed T (c),\n\n> n\n\n∏\n\n> i=1\n\n[ih ]q\n\n[di]q, Tw\n\n)\n\nexhibits the CSP.\n\n> 2There is a simple representation-theoretic proof of AJ, but I don't know of any such proof for HJ.\n\n10 This can probably be proved pretty easily for w = wo using a combinatorial construction that associates pairs of factors in the orbit under Two to certain bicolored quadrangulations. There is an action of order h on Red T (c), which is quite simply conjugation by c. It is easy to see that this gives all orbits of size h.",
   "public_statement": "Conjecture 3.7. For w = wo, c,\n\n(\n\nRed T (c),\n\n> n\n\n∏\n\n> i=1\n\n[ih ]q\n\n[di]q, Tw\n\n)\n\nexhibits the CSP.\n\n> 2There is a simple representation-theoretic proof of AJ, but I don't know of any such proof for HJ.\n\n10 This can probably be proved pretty easily for w = wo using a combinatorial construction that associates pairs of factors in the orbit under Two to certain bicolored quadrangulations. There is an action of order h on Red T (c), which is quite simply conjugation by c. It is easy to see that this gives all orbits of size h.",
   "evidence": "The canonical record is OCR-damaged. The surrounding subsection of the AIM pre-workshop document supplies the notation. Let \\(W\\) be a finite irreducible Coxeter group of rank \\(n\\), with reflection set \\(T\\), degrees \\(d_1,\\ldots,d_n\\), Coxeter number \\(h\\), and a fixed Coxeter element \\(c_W\\in W\\). Write",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 21,
   "attempt": 1
  },
  "AIM-OTHER-0023": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 3.8. The polynomial ∏ni=1 [ih ]q \n\n> [di]q\n\nappears to propose orbit sizes for other multiples of h between h and nh when corresponding roots of unity are plugged in-can the conjecture above be generalized by describing the corresponding ac-tions Tw?\n\nThese elements are presumably related to solving wp = cn in the braid group of type An−1, where n is the rank of W. For example, (wo)p = cn for p = 2, so \n\nTwo gives an order ph = 2 h action; similarly, cp = cn for p = n, so Tc gives an order ph = nh action. \n\n4 Piecewise-Linear and Birational Toggles \n\n4.1 Order polytope promotion and rowmotion",
   "clean_statement": "**Problem 3.8 (recovered).** The polynomial \\(F_W(q)\\) appears, when evaluated at roots of unity, to prescribe orbit sizes for other multiples \\(ph\\) of \\(h\\) between \\(h\\) and \\(nh\\). Can Conjecture 3.7 be generalized by describing corresponding Hurwitz actions? Such elements should be related to solutions of \\(\\boldsymbol w^p=\\boldsymbol c^n\\) in the braid group \\(B_n\\). The endpoint examples are \\(\\boldsymbol w_0^2=\\boldsymbol c^n\\) and \\(\\boldsymbol c^p=\\boldsymbol c^n\\) for \\(p=n\\).",
   "public_statement": "Problem 3.8. The polynomial ∏ni=1 [ih ]q\n\n> [di]q\n\nappears to propose orbit sizes for other multiples of h between h and nh when corresponding roots of unity are plugged in-can the conjecture above be generalized by describing the corresponding ac-tions Tw?\n\nThese elements are presumably related to solving wp = cn in the braid group of type An−1, where n is the rank of W. For example, (wo)p = cn for p = 2, so\n\nTwo gives an order ph = 2 h action; similarly, cp = cn for p = n, so Tc gives an order ph = nh action.\n\n4 Piecewise-Linear and Birational Toggles\n\n4.1 Order polytope promotion and rowmotion",
   "evidence": "The canonical record is an OCR-damaged extraction of page 11 of the official 2015 AIM pre-workshop problem list. The preceding page defines the notation. Let \\(W\\) be a finite irreducible Coxeter group of rank \\(n\\), with reflection set \\(T\\), degrees \\(d_1,\\ldots,d_n\\), Coxeter number \\(h\\), and Coxeter element \\(c_W\\). Put",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-other-notes.json",
   "source_index": 22,
   "attempt": 1
  },
  "AIM-OTHER-0024": {
   "statement_status": "exact",
   "original_statement": "Problem 4.1. Explicitly describe the decomposition of O(P ) under piecewise linear promotion/rowmotion. \n\nOf particular interest would be to do this for [a] × [b] and the type Bn root poset [Pro83, Ste86]. The case P = [2] ×[n] has been settled (the polytope is divided into \"Catalan-many\" simplices such that every power of the map, restricted to any particular simplex, is a linear map), but the case P = [3] × [n] is more complicated and has not been resolved. \n\n4.2 Birational rowmotion on G/P",
   "clean_statement": "Problem 4.1. Explicitly describe the decomposition of O(P ) under piecewise linear promotion/rowmotion.\n\nOf particular interest would be to do this for [a] × [b] and the type Bn root poset [Pro83, Ste86]. The case P = [2] ×[n] has been settled (the polytope is divided into \"Catalan-many\" simplices such that every power of the map, restricted to any particular simplex, is a linear map), but the case P = [3] × [n] is more complicated and has not been resolved.\n\n4.2 Birational rowmotion on G/P",
   "public_statement": "Problem 4.1. Explicitly describe the decomposition of O(P ) under piecewise linear promotion/rowmotion.\n\nOf particular interest would be to do this for [a] × [b] and the type Bn root poset [Pro83, Ste86]. The case P = [2] ×[n] has been settled (the polytope is divided into \"Catalan-many\" simplices such that every power of the map, restricted to any particular simplex, is a linear map), but the case P = [3] × [n] is more complicated and has not been resolved.\n\n4.2 Birational rowmotion on G/P",
   "evidence": "The canonical JSON record has two extraction defects: ordinary `O(P )` is the order-polytope notation \\(\\mathcal O(P)\\), and the final line, “4.2 Birational rowmotion on \\(G/P\\),” is the heading of the next problem rather than part of Problem 4.1. The official AIM PDF gives the following recovered statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 23,
   "attempt": 1
  },
  "AIM-OTHER-0025": {
   "statement_status": "exact",
   "original_statement": "Problem 4.2. Generalize Grinberg and Roby's proof of periodicity of birational rowmotion on rectangles uniformly to G/P.\n\nGrinberg and Roby's proof corresponds to the case when G = GL (n) and \n\nP is a maximal parabolic subgroup-their coordinates appear to be related to its Plücker embedding, which has generalizations for other quotients (see, for example, [Hil82, page 184] or [FZ00, Section 3.1]. When P is minuscule, this would be a birational generalization of [RS13]. See also [RSW04].",
   "clean_statement": "Problem 4.2. Generalize Grinberg and Roby's proof of periodicity of birational rowmotion on rectangles uniformly to G/P.\n\nGrinberg and Roby's proof corresponds to the case when G = GL (n) and\n\nP is a maximal parabolic subgroup-their coordinates appear to be related to its Plücker embedding, which has generalizations for other quotients (see, for example, [Hil82, page 184] or [FZ00, Section 3.1]. When P is minuscule, this would be a birational generalization of [RS13]. See also [RSW04].",
   "public_statement": "Problem 4.2. Generalize Grinberg and Roby's proof of periodicity of birational rowmotion on rectangles uniformly to G/P.\n\nGrinberg and Roby's proof corresponds to the case when G = GL (n) and\n\nP is a maximal parabolic subgroup-their coordinates appear to be related to its Plücker embedding, which has generalizations for other quotients (see, for example, [Hil82, page 184] or [FZ00, Section 3.1]. When P is minuscule, this would be a birational generalization of [RS13]. See also [RSW04].",
   "evidence": "The canonical record is aim-other-notes.json, index 24, Problem 4.2 of the AIM preworkshop notes *Dynamical algebraic combinatorics*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 24,
   "attempt": 1
  },
  "AIM-OTHER-0026": {
   "statement_status": "exact",
   "original_statement": "Problem 4.3. Study birational rowmotion on Proctor's d-complete posets. \n\n4.3 When is birational rowmotion periodic? \n\nGrinberg and Roby show that birational rowmotion has finite order (i.e., is periodic) for a variety of graded posets of interest, but in general this appears 11 to be the exception rather than the rule. When it holds, periodicity also follows (by tropicalization) for piecewise-linear rowmotion on the corresponding order polytopes, and generally one finds that the order of combinatorial rowmotion on the poset itself has much smaller order than one might naively expect. We currently know that birational rowmotion is periodic for the following posets: \n\n• the poset [p] × [q] which is the product of two chains, with order p + q.\n\n• triangular posets created by cutting [p]×[p] square in half either vertically or horizontally, generally getting order 2p.\n\n• the class of skeletal posets, which generalize the class of graded forests. These are built up inductively by successively \"grafting\" multiple an-tichains above or below an existing poset, or by taking disjoint unions of graded skeletal posets of the same rank. The order can be easily bounded and computed algorithmically. On the other hand, birational rowmotion (over fields of characteristic zero) has infinite order for the following simple examples: \n\n• If P is the poset {x1, x 2, x 3, x 4, x 5} with relations x1 < x 3, x1 < x 4,\n\nx1 < x 5, x2 < x 4 and x2 < x 5 (this is a 5-element 2-graded poset), then \n\nord ( RP ) = ∞.\n\n• If P is the \"chain-link fence\" \n\n• If P is the Boolean lattice [2] × [2] × [2], then ord ( RP ) = ∞.We conjecture that birational rowmotion has order p for triangular posets made by cutting a [p] × [p] square into quarters, the two distinct cases being the \"northeast\" and \"southeast\" corners. We can show it holds for p odd, but the case of even p remains open (though all the evidence suggests that it's true). A generalization of this for \"trapezoids\" due to N. Williams is as follows:",
   "clean_statement": "Problem 4.3. Study birational rowmotion on Proctor's d-complete posets.\n\n4.3 When is birational rowmotion periodic?\n\nGrinberg and Roby show that birational rowmotion has finite order (i.e., is periodic) for a variety of graded posets of interest, but in general this appears 11 to be the exception rather than the rule. When it holds, periodicity also follows (by tropicalization) for piecewise-linear rowmotion on the corresponding order polytopes, and generally one finds that the order of combinatorial rowmotion on the poset itself has much smaller order than one might naively expect. We currently know that birational rowmotion is periodic for the following posets:\n\n• the poset [p] × [q] which is the product of two chains, with order p + q.\n\n• triangular posets created by cutting [p]×[p] square in half either vertically or horizontally, generally getting order 2p.\n\n• the class of skeletal posets, which generalize the class of graded forests. These are built up inductively by successively \"grafting\" multiple an-tichains above or below an existing poset, or by taking disjoint unions of graded skeletal posets of the same rank. The order can be easily bounded and computed algorithmically. On the other hand, birational rowmotion (over fields of characteristic zero) has infinite order for the following simple examples:\n\n• If P is the poset {x1, x 2, x 3, x 4, x 5} with relations x1 < x 3, x1 < x 4,\n\nx1 < x 5, x2 < x 4 and x2 < x 5 (this is a 5-element 2-graded poset), then\n\nord ( RP ) = ∞.\n\n• If P is the \"chain-link fence\"\n\n• If P is the Boolean lattice [2] × [2] × [2], then ord ( RP ) = ∞.We conjecture that birational rowmotion has order p for triangular posets made by cutting a [p] × [p] square into quarters, the two distinct cases being the \"northeast\" and \"southeast\" corners. We can show it holds for p odd, but the case of even p remains open (though all the evidence suggests that it's true). A generalization of this for \"trapezoids\" due to N. Williams is as follows:",
   "public_statement": "Problem 4.3. Study birational rowmotion on Proctor's d-complete posets.\n\n4.3 When is birational rowmotion periodic?\n\nGrinberg and Roby show that birational rowmotion has finite order (i.e., is periodic) for a variety of graded posets of interest, but in general this appears 11 to be the exception rather than the rule. When it holds, periodicity also follows (by tropicalization) for piecewise-linear rowmotion on the corresponding order polytopes, and generally one finds that the order of combinatorial rowmotion on the poset itself has much smaller order than one might naively expect. We currently know that birational rowmotion is periodic for the following posets:\n\n• the poset [p] × [q] which is the product of two chains, with order p + q.\n\n• triangular posets created by cutting [p]×[p] square in half either vertically or horizontally, generally getting order 2p.\n\n• the class of skeletal posets, which generalize the class of graded forests. These are built up inductively by successively \"grafting\" multiple an-tichains above or below an existing poset, or by taking disjoint unions of graded skeletal posets of the same rank. The order can be easily bounded and computed algorithmically. On the other hand, birational rowmotion (over fields of characteristic zero) has infinite order for the following simple examples:\n\n• If P is the poset {x1, x 2, x 3, x 4, x 5} with relations x1 < x 3, x1 < x 4,\n\nx1 < x 5, x2 < x 4 and x2 < x 5 (this is a 5-element 2-graded poset), then\n\nord ( RP ) = ∞.\n\n• If P is the \"chain-link fence\"\n\n• If P is the Boolean lattice [2] × [2] × [2], then ord ( RP ) = ∞.We conjecture that birational rowmotion has order p for triangular posets made by cutting a [p] × [p] square into quarters, the two distinct cases being the \"northeast\" and \"southeast\" corners. We can show it holds for p odd, but the case of even p remains open (though all the evidence suggests that it's true). A generalization of this for \"trapezoids\" due to N. Williams is as follows:",
   "evidence": "The canonical JSON record is contaminated by text from the following subsection. The official AIM PDF is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 25,
   "attempt": 2
  },
  "AIM-OTHER-0027": {
   "statement_status": "corrected_verified",
   "original_statement": "Conjecture 4.4. Let p be an integer > 1, and s ∈ N. Let NEtri ′ (p) be the poset \n\n{(i, k ) ∈ [p] × [p] | i ≤ k; i + k > p + 1; and k ≥ s} Then, ord (RNEtri ′(p)\n\n) | p.\n\nIn general it seems that birational rowmotion has finite order for posets related to root systems, so there are several general classes that could be studied separately, or perhaps treated in a uniform way. For pictures and further details about all of this, the most complete and up-to-date source to consult is § 18- 21 of http://web.mit.edu/~darij/www/algebra/skeletal.pdf. A concise sketch of the ideas involved is available in the twelve-page extended abstract for FPSAC 2014 [GR14]. 12 4.4 Order polytopes and P -partitions \n\nIf we dilate the order polytope O(P ) of a poset P by a factor of k, then the integer points of kO(P ) are in bijection with P -partitions of height k, or- equivalently- J(P ×[k]). The usual piecewise linear toggles on O(P ) now induce a toggle operation on these P -partitions. (For P of tableaux shape with boxes p ∈ P, we record the number of elements \n\n(p, j ) in the box p, and we may then add i to the boxes in the ith row to get column-strict tableaux.) For certain posets (minuscule, types A, B, H 3, I 2(m)), there are very nice for-mulas for the number of these plane partitions. (Since they have hook-length for-mulas, we expect that there must also be nice formulas for P -partitions of height \n\nk in d-complete posets). For example, minuscule posets P have P -partitions of height k counted by \n\nJ(P × [k]) = ∏\n\n> x∈P\n\n[ht (x) + k]q\n\n[ht (x)] q,\n\nwhile types W = A, B, H 3, I 2(m) have the \"uniform\" formula [CLS14] \n\nJ(Φ +(W ) × [k]) = ∏\n\n> 0≤j<k\n\n∏\n\n> 1≤i≤n\n\n[di + h + 2 j]q\n\n[di + 2 j]q.",
   "clean_statement": "**Conjecture 4.4 (Williams, as recorded by AIM).** Let \\(p>1\\) be an integer and \\(s\\in\\mathbb N\\). Then the order of birational rowmotion on \\(P_{p,s}\\) divides \\(p\\).",
   "public_statement": "**Conjecture 4.4 (Williams, as recorded by AIM).** Let \\(p>1\\) be an integer and \\(s\\in\\mathbb N\\). Then the order of birational rowmotion on \\(P_{p,s}\\) divides \\(p\\).",
   "evidence": "The canonical JSON record contains two extraction defects. First, it appends the end of Section 4.3 and the beginning of Section 4.4 to Conjecture 4.4. Inspection of page 12 of the official AIM PDF shows that the conjecture ends immediately after the divisibility assertion. Second, the PDF itself calls the poset `NEtri' (p)` although the definition depends on \\(s\\); thus the missing \\(s\\) in the name is a source-level typographical inconsistency, not merely OCR. The defining predicate is visually clear. To avoid silently repairing the source's name, write",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-other-notes.json",
   "source_index": 26,
   "attempt": 1
  },
  "AIM-OTHER-0028": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 4.5. When P is minuscule or coincidental, is there a cyclic sieving phenomenon the integer points of kO(P ) under the induced actions of promo-tion/rowmotion? \n\nIn the case of the root poset of An, there is a statistic-generalizing the major index-such that J(Φ +(An) × [k]) is the weight-generating function for this statistic. Specifically, given a P -partition (where P is the root poset for An)whose entries lie between 0 and k, create a larger triangular array by sticking a row of k's at the bottom, then apply Stanley's transfer map to turn this into a point x in the chain polytope, with coordinates x1 through xp; the weight of the original P -partition can then be defined as q to the power of λ(x), where λ\n\nis the linear form that weights entries in the jth column of the triangular array by j − n − 1 (for 1 ≤ j ≤ 2n + 1 ). This weight doesn't just give a nice formula for the sum of the weights of the P -partitions of ceiling k, for each individual k;it does so in a uniform way (as in Chapoton's q-Ehrhart theory). Perhaps we should not be separating into cases according to k, but should be treating all \n\nk's together, by letting the cyclic group act on the a cone containing infinitely many points? \n\n4.5 Cluster algebras and birational toggling \n\nCluster algebras have flips that change variables by acting on the Dynkin dia-gram (of simple roots). Birational toggles change variables by acting on the root 13 poset (of all positive roots). For finite Weyl groups, there are wonderful duali-ties between the set of simple roots S and the set of all roots T. For example, \n\n2|T | = h|S|-for more information, see [Bes03].",
   "clean_statement": null,
   "public_statement": "Problem 4.5. When P is minuscule or coincidental, is there a cyclic sieving phenomenon the integer points of kO(P ) under the induced actions of promo-tion/rowmotion?\n\nIn the case of the root poset of An, there is a statistic-generalizing the major index-such that J(Φ +(An) × [k]) is the weight-generating function for this statistic. Specifically, given a P -partition (where P is the root poset for An)whose entries lie between 0 and k, create a larger triangular array by sticking a row of k's at the bottom, then apply Stanley's transfer map to turn this into a point x in the chain polytope, with coordinates x1 through xp; the weight of the original P -partition can then be defined as q to the power of λ(x), where λ\n\nis the linear form that weights entries in the jth column of the triangular array by j − n − 1 (for 1 ≤ j ≤ 2n + 1 ). This weight doesn't just give a nice formula for the sum of the weights of the P -partitions of ceiling k, for each individual k;it does so in a uniform way (as in Chapoton's q-Ehrhart theory). Perhaps we should not be separating into cases according to k, but should be treating all\n\nk's together, by letting the cyclic group act on the a cone containing infinitely many points?\n\n4.5 Cluster algebras and birational toggling\n\nCluster algebras have flips that change variables by acting on the Dynkin dia-gram (of simple roots). Birational toggles change variables by acting on the root 13 poset (of all positive roots). For finite Weyl groups, there are wonderful duali-ties between the set of simple roots S and the set of all roots T. For example,\n\n2|T | = h|S|-for more information, see [Bes03].",
   "evidence": "This record is Problem 4.5 in the pre-workshop document for the AIM workshop *Dynamical algebraic combinatorics*. The source text, with line-break hyphenation removed and the visibly omitted word “for” supplied in brackets, asks:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 27,
   "attempt": 1
  },
  "AIM-OTHER-0029": {
   "statement_status": "exact",
   "original_statement": "Problem 4.6. Make the analogy between birational toggles and cluster flips explicit. \n\nPresumably, this should fit into the S vs. T duality mentioned above. Astarting point is the question of whether there is an analogue of the bijection between cluster variables and almost positive roots on iterates of rowmotion. Perhaps frieze patterns (of both the PL and birational sort) would be a fruitful place to start. In the simplest non-trivial case, the dynamics of shifting the frieze pattern is essentially the dynamics of Lyness 5-cycles, as is briefly described in section 2.6 of (the November 2014 version of) Propp and Roby's \"Homomesy in products of two chains\" (arXiv:1310.5201v5). \n\n4.6 Gelfand-Tsetlin triangles \n\nKirillov and Berenstein (see math.uoregon.edu/ ∼arkadiy/bk1.pdf ) describe what in modern parlance would be called a toggle-action on Gelfand-Tsetlin tri-angles. Such triangles may be viewed as lattice points in the order polytope of a certain poset, and Kirillov and Berenstein's involutions were the prototypical examples of fiber-flipping. As those authors noted, one can define an operation equivalent to Schützenberger promotion by taking appropriate products of these involutions; this gives rise to a cyclic group action on Gelfand-Tsetlin triangles whose properties deserve study (and indeed Grinberg has already proved a ho-momesy property of this action). However, a bigger finite group is only slightly offstage: the full symmetric group. (Joel Kamnitzer summarizes the construc-tion of this action by saying \"you take the cactus group of the root system and then quotient by the braid relations\".)",
   "clean_statement": "Problem 4.6. Make the analogy between birational toggles and cluster flips explicit.\n\nPresumably, this should fit into the S vs. T duality mentioned above. Astarting point is the question of whether there is an analogue of the bijection between cluster variables and almost positive roots on iterates of rowmotion. Perhaps frieze patterns (of both the PL and birational sort) would be a fruitful place to start. In the simplest non-trivial case, the dynamics of shifting the frieze pattern is essentially the dynamics of Lyness 5-cycles, as is briefly described in section 2.6 of (the November 2014 version of) Propp and Roby's \"Homomesy in products of two chains\" (arXiv:1310.5201v5).\n\n4.6 Gelfand-Tsetlin triangles\n\nKirillov and Berenstein (see math.uoregon.edu/ ∼arkadiy/bk1.pdf ) describe what in modern parlance would be called a toggle-action on Gelfand-Tsetlin tri-angles. Such triangles may be viewed as lattice points in the order polytope of a certain poset, and Kirillov and Berenstein's involutions were the prototypical examples of fiber-flipping. As those authors noted, one can define an operation equivalent to Schützenberger promotion by taking appropriate products of these involutions; this gives rise to a cyclic group action on Gelfand-Tsetlin triangles whose properties deserve study (and indeed Grinberg has already proved a ho-momesy property of this action). However, a bigger finite group is only slightly offstage: the full symmetric group. (Joel Kamnitzer summarizes the construc-tion of this action by saying \"you take the cactus group of the root system and then quotient by the braid relations\".)",
   "public_statement": "Problem 4.6. Make the analogy between birational toggles and cluster flips explicit.\n\nPresumably, this should fit into the S vs. T duality mentioned above. Astarting point is the question of whether there is an analogue of the bijection between cluster variables and almost positive roots on iterates of rowmotion. Perhaps frieze patterns (of both the PL and birational sort) would be a fruitful place to start. In the simplest non-trivial case, the dynamics of shifting the frieze pattern is essentially the dynamics of Lyness 5-cycles, as is briefly described in section 2.6 of (the November 2014 version of) Propp and Roby's \"Homomesy in products of two chains\" (arXiv:1310.5201v5).\n\n4.6 Gelfand-Tsetlin triangles\n\nKirillov and Berenstein (see math.uoregon.edu/ ∼arkadiy/bk1.pdf ) describe what in modern parlance would be called a toggle-action on Gelfand-Tsetlin tri-angles. Such triangles may be viewed as lattice points in the order polytope of a certain poset, and Kirillov and Berenstein's involutions were the prototypical examples of fiber-flipping. As those authors noted, one can define an operation equivalent to Schützenberger promotion by taking appropriate products of these involutions; this gives rise to a cyclic group action on Gelfand-Tsetlin triangles whose properties deserve study (and indeed Grinberg has already proved a ho-momesy property of this action). However, a bigger finite group is only slightly offstage: the full symmetric group. (Joel Kamnitzer summarizes the construc-tion of this action by saying \"you take the cactus group of the root system and then quotient by the braid relations\".)",
   "evidence": "The official workshop PDF was checked directly. The canonical record contains OCR spill from the next subsection. Its actual problem ends immediately before the heading “4.6 Gelfand–Tsetlin triangles.” The recovered text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 28,
   "attempt": 1
  },
  "AIM-OTHER-0030": {
   "statement_status": "exact",
   "original_statement": "Problem 4.7. What are the homomesies of the symmetric group action on Gelfand-Tsetlin triangles? \n\nOne might expect (at least naively) that there are more homomesies for the symmetric group action than for its cyclic subactions: making the group bigger means merging orbits, and this merging permits more averaging to take place. Note that there is significant overlap with the question raised in Section??.\n\n4.7 The birational toggle group \n\nLet P be a finite poset. All the birational toggle operations taken together generate a group.",
   "clean_statement": "Problem 4.7. What are the homomesies of the symmetric group action on Gelfand-Tsetlin triangles?\n\nOne might expect (at least naively) that there are more homomesies for the symmetric group action than for its cyclic subactions: making the group bigger means merging orbits, and this merging permits more averaging to take place. Note that there is significant overlap with the question raised in Section??.\n\n4.7 The birational toggle group\n\nLet P be a finite poset. All the birational toggle operations taken together generate a group.",
   "public_statement": "Problem 4.7. What are the homomesies of the symmetric group action on Gelfand-Tsetlin triangles?\n\nOne might expect (at least naively) that there are more homomesies for the symmetric group action than for its cyclic subactions: making the group bigger means merging orbits, and this merging permits more averaging to take place. Note that there is significant overlap with the question raised in Section??.\n\n4.7 The birational toggle group\n\nLet P be a finite poset. All the birational toggle operations taken together generate a group.",
   "evidence": "The canonical JSON record joins two different pieces of the source PDF. The recoverable problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 29,
   "attempt": 1
  },
  "AIM-OTHER-0031": {
   "statement_status": "exact",
   "original_statement": "Problem 4.8. When is the birational toggle group finitely presented? \n\nWe know that this is the case when P is just a chain, for then the PL toggles are actual linear maps, and their birational lifts are just monomial maps. But even for as simple a poset as [2] × [2], we don't know the answer to this question. 14 Perhaps it is better to approach the birational toggle group from above. Cer-tain algebraic combinations of the variables are invariant under all the birational toggles.",
   "clean_statement": "Problem 4.8. When is the birational toggle group finitely presented?\n\nWe know that this is the case when P is just a chain, for then the PL toggles are actual linear maps, and their birational lifts are just monomial maps. But even for as simple a poset as [2] × [2], we don't know the answer to this question. 14 Perhaps it is better to approach the birational toggle group from above. Cer-tain algebraic combinations of the variables are invariant under all the birational toggles.",
   "public_statement": "Problem 4.8. When is the birational toggle group finitely presented?\n\nWe know that this is the case when P is just a chain, for then the PL toggles are actual linear maps, and their birational lifts are just monomial maps. But even for as simple a poset as [2] × [2], we don't know the answer to this question. 14 Perhaps it is better to approach the birational toggle group from above. Cer-tain algebraic combinations of the variables are invariant under all the birational toggles.",
   "evidence": "The canonical record is Problem 4.8 in Section 4.7, “The birational toggle group,” of the AIM pre-workshop document *Dynamical algebraic combinatorics*. The question and its explanatory paragraph are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 30,
   "attempt": 1
  },
  "AIM-OTHER-0032": {
   "statement_status": "corrected_verified",
   "original_statement": "Problem 4.9. Can we say what those combinations are? Can we then char-acterize the birational toggle group as being precisely the group of birational transformations that preserve those algebraic combinations? \n\nApropos of invariance under toggling, we mention the case of the full toggle group on the poset [2] × [2]. Of the 24 orders in which one can compose all 4 toggles, 8 are not conjugate to rowmotion or promotion or their inverses; indeed, the PL and birational versions of these compositions were shown by Einstein to be of infinite order. Nevertheless, there are things to be proved about the map (called \"locomotion\" for present purposes). For instance, the pictures at http://jamespropp.org/locomotion.pdf, which shows some projections of an or-bit of locomotion strongly suggests that the orbit lies in a 2-dimensional surface in R4; that is, there are two conserved quantities, of which we know only one. Likewise, experimental studies suggest that other quantities are homomesic un-der locomotion, in an appropriately asymptotic sense of the word \"average\". (Similar, polyhedral pictures appear when one replaces birational locomotion by PL locomotion.)",
   "clean_statement": "**Problem 4.9.** Can we say what those combinations are? Can we then characterize the birational toggle group as being precisely the group of birational transformations that preserve those algebraic combinations?\n\nApropos of invariance under toggling, we mention the case of the full toggle group on the poset \\([2]\\times[2]\\). Of the 24 orders in which one can compose all 4 toggles, 8 are not conjugate to rowmotion or promotion or their inverses; indeed, the PL and birational versions of these compositions were shown by Einstein to be of infinite order. Nevertheless, there are things to be proved about the map (called “locomotion” for present purposes). For instance, the pictures at http://jamespropp.org/locomotion.pdf, which shows some projections of an orbit of locomotion strongly suggests that the orbit lies in a 2-dimensional surface in \\(\\mathbb R^4\\); that is, there are two conserved quantities, of which we know only one. Likewise, experimental studies suggest that other quantities are homomesic under locomotion, in an appropriately asymptotic sense of the word “average.” (Similar, polyhedral pictures appear when one replaces birational locomotion by PL locomotion.)",
   "public_statement": "**Problem 4.9.** Can we say what those combinations are? Can we then characterize the birational toggle group as being precisely the group of birational transformations that preserve those algebraic combinations?\n\nApropos of invariance under toggling, we mention the case of the full toggle group on the poset \\([2]\\times[2]\\). Of the 24 orders in which one can compose all 4 toggles, 8 are not conjugate to rowmotion or promotion or their inverses; indeed, the PL and birational versions of these compositions were shown by Einstein to be of infinite order. Nevertheless, there are things to be proved about the map (called “locomotion” for present purposes). For instance, the pictures at http://jamespropp.org/locomotion.pdf, which shows some projections of an orbit of locomotion strongly suggests that the orbit lies in a 2-dimensional surface in \\(\\mathbb R^4\\); that is, there are two conserved quantities, of which we know only one. Likewise, experimental studies suggest that other quantities are homomesic under locomotion, in an appropriately asymptotic sense of the word “average.” (Similar, polyhedral pictures appear when one replaces birational locomotion by PL locomotion.)",
   "evidence": "The canonical record was checked against the official AIM workshop PDF. The only repairs made below are line-break hyphenations: “char-acterize,” “or-bit,” and “un-der” become “characterize,” “orbit,” and “under.” The recovered problem is:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-other-notes.json",
   "source_index": 31,
   "attempt": 1
  },
  "AIM-OTHER-0033": {
   "statement_status": "exact",
   "original_statement": "Problem 4.10. Is the PL toggle group isomorphic to the birational toggle group? Or is it a proper quotient group? \n\n5 Generalized Toggling \n\nThe following problems are associated to [Str15]. The main observation is that the toggle group need not be restricted to order ideals of a poset P. The particular structure of order ideals in a poset is unnecessary in the definition of the toggle group; the essential structure is merely that an order ideal is a subset of poset elements. Thus, given a finite ground set E, we can define a toggle group T (L) on any set of subsets L ⊆ 2E.\n\nDefinition 5.1. Let E be a finite set and L ⊆ 2E. For each element e ∈ E\n\ndefine its toggle te: L → L as follows. \n\nte(X) = \n\n\n\nX ∪ { e} if e / ∈ X and X ∪ { e} ∈ L \n\nX \\ { e} if e ∈ X and X \\ { e} ∈ L \n\nX otherwise Note that t2 \n\n> e\n\n= 1 for all e ∈ E. We define the generalized toggle group as the group generated by these toggles. \n\nDefinition 5.2. Let T (L) be the subgroup of the symmetric group SL, gener-ated by {te | e ∈ E}. Call T (L) the toggle group on L.15 Therefore, if we isolate any set of subsets L that has combinatorial meaning, we can use the toggle group to gain insight on these objects in ways similar to order ideals. Several examples of potentially interesting toggle groups are: \n\n• Poset structures: chains, antichains, or interval-closed sets; \n\n• More than one partial order on the same ground set; \n\n• Graph structures: independent sets, acyclic subgraphs, vertex covers, edge covers, connected subgraphs; \n\n• Matroids; \n\n• Antimatroids. \n\n5.1 Generalized toggling from the bottom up",
   "clean_statement": "Problem 4.10. Is the PL toggle group isomorphic to the birational toggle group? Or is it a proper quotient group?\n\n5 Generalized Toggling\n\nThe following problems are associated to [Str15]. The main observation is that the toggle group need not be restricted to order ideals of a poset P. The particular structure of order ideals in a poset is unnecessary in the definition of the toggle group; the essential structure is merely that an order ideal is a subset of poset elements. Thus, given a finite ground set E, we can define a toggle group T (L) on any set of subsets L ⊆ 2E.\n\nDefinition 5.1. Let E be a finite set and L ⊆ 2E. For each element e ∈ E\n\ndefine its toggle te: L → L as follows.\n\nte(X) =\n\n\n\nX ∪ { e} if e / ∈ X and X ∪ { e} ∈ L\n\nX \\ { e} if e ∈ X and X \\ { e} ∈ L\n\nX otherwise Note that t2\n\n> e\n\n= 1 for all e ∈ E. We define the generalized toggle group as the group generated by these toggles.\n\nDefinition 5.2. Let T (L) be the subgroup of the symmetric group SL, gener-ated by {te | e ∈ E}. Call T (L) the toggle group on L.15 Therefore, if we isolate any set of subsets L that has combinatorial meaning, we can use the toggle group to gain insight on these objects in ways similar to order ideals. Several examples of potentially interesting toggle groups are:\n\n• Poset structures: chains, antichains, or interval-closed sets;\n\n• More than one partial order on the same ground set;\n\n• Graph structures: independent sets, acyclic subgraphs, vertex covers, edge covers, connected subgraphs;\n\n• Matroids;\n\n• Antimatroids.\n\n5.1 Generalized toggling from the bottom up",
   "public_statement": "Problem 4.10. Is the PL toggle group isomorphic to the birational toggle group? Or is it a proper quotient group?\n\n5 Generalized Toggling\n\nThe following problems are associated to [Str15]. The main observation is that the toggle group need not be restricted to order ideals of a poset P. The particular structure of order ideals in a poset is unnecessary in the definition of the toggle group; the essential structure is merely that an order ideal is a subset of poset elements. Thus, given a finite ground set E, we can define a toggle group T (L) on any set of subsets L ⊆ 2E.\n\nDefinition 5.1. Let E be a finite set and L ⊆ 2E. For each element e ∈ E\n\ndefine its toggle te: L → L as follows.\n\nte(X) =\n\n\n\nX ∪ { e} if e / ∈ X and X ∪ { e} ∈ L\n\nX \\ { e} if e ∈ X and X \\ { e} ∈ L\n\nX otherwise Note that t2\n\n> e\n\n= 1 for all e ∈ E. We define the generalized toggle group as the group generated by these toggles.\n\nDefinition 5.2. Let T (L) be the subgroup of the symmetric group SL, gener-ated by {te | e ∈ E}. Call T (L) the toggle group on L.15 Therefore, if we isolate any set of subsets L that has combinatorial meaning, we can use the toggle group to gain insight on these objects in ways similar to order ideals. Several examples of potentially interesting toggle groups are:\n\n• Poset structures: chains, antichains, or interval-closed sets;\n\n• More than one partial order on the same ground set;\n\n• Graph structures: independent sets, acyclic subgraphs, vertex covers, edge covers, connected subgraphs;\n\n• Matroids;\n\n• Antimatroids.\n\n5.1 Generalized toggling from the bottom up",
   "evidence": "The canonical JSON record contains a duplicated word and then continues into the next section. The official AIM PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 32,
   "attempt": 1
  },
  "AIM-OTHER-0034": {
   "statement_status": "exact",
   "original_statement": "Problem 5.3. Explore these (and other) generalized toggle groups, look for homomesy and CSP, look at piecewise-linear and birational liftings. \n\nFor example, one could look at toggling chains, then the piecewise-linear extension should be to the order complex. I have Sage code for generalized toggles and to search for homomesy in generalized toggle groups. Perhaps we could work on adapting Darij's birational code to the generalized lifted toggles at Sage Days. \n\n5.2 Generalized toggling from the top down",
   "clean_statement": "Problem 5.3. Explore these (and other) generalized toggle groups, look for homomesy and CSP, look at piecewise-linear and birational liftings.\n\nFor example, one could look at toggling chains, then the piecewise-linear extension should be to the order complex. I have Sage code for generalized toggles and to search for homomesy in generalized toggle groups. Perhaps we could work on adapting Darij's birational code to the generalized lifted toggles at Sage Days.\n\n5.2 Generalized toggling from the top down",
   "public_statement": "Problem 5.3. Explore these (and other) generalized toggle groups, look for homomesy and CSP, look at piecewise-linear and birational liftings.\n\nFor example, one could look at toggling chains, then the piecewise-linear extension should be to the order complex. I have Sage code for generalized toggles and to search for homomesy in generalized toggle groups. Perhaps we could work on adapting Darij's birational code to the generalized lifted toggles at Sage Days.\n\n5.2 Generalized toggling from the top down",
   "evidence": "The source is Problem 5.3 in the AIM pre-workshop notes for *Dynamical algebraic combinatorics*. The operative text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 33,
   "attempt": 1
  },
  "AIM-OTHER-0035": {
   "statement_status": "exact",
   "original_statement": "Problem 5.4. Start with a known action in the birational (or piecewise-linear) realm and find the corresponding generalized toggle group action in the combi-natorial realm. \n\nOne such example of a birational map is the pentagram map. \n\n5.3 Subset toggling \n\nGeneralized subset-toggle groups are defined below. We could ask the same questions from the previous problem in this further-generalized context. Let E be a countable set and L ⊆ 2E.\n\nDefinition 5.5. For any subset S ⊆ E define its (subset-)toggle tS: L → L as \n\ntS (X) = \n\n{\n\nX4S if X4S ∈ L \n\nX otherwise where X4S denotes the symmetric difference of the sets X and S, that is, \n\nX4S = ( X \\ S) ∪ (S \\ X). We call {tS | S ⊆ E} the set of (subset-)toggles.We define the power set toggle group as the group generated by all the (subset-)toggles on L.16 Definition 5.6. Let T2E (L) be the subgroup of the symmetric group SL, gen-erated by {tS | S ⊆ E}. Call T2E (L) the power set toggle group on L.One could also construct a toggle group using only some of the subset-toggles. \n\nDefinition 5.7. Let K ⊆ 2E. Define TK(L) to be the subgroup of T2E (L) gen-erated by {tS | S ∈ K}. Call TK(L) the K-toggle subgroup on L (or generically, we call any TK(L) a subset-toggle group ). \n\n5.4 Toggling noncrossing partitions \n\nGiven a noncrossing partition π of {1, 2,..., n } ∈ NC (n) and 1 ≤ i < j ≤ n,define τ ′ as follows: \n\n• if i and j are consecutive elements of the same block B, split the block into two blocks (one consisting of all the elements of B that are ≤ i and the other consisting of all the elements of B that are ≥ j) and leave all the other blocks alone; \n\n• if i is the largest element of one block B1 and j is the smallest element of another block B2, merge the two blocks into one block B1 ∪ B2 (as long as this will not violate the noncrossing condition) and leave all the other blocks alone; \n\n• otherwise, do nothing. We write τ ′ = τi,j (π), and call the involution τi,j: NC (n) → NC (n) toggling at (i, j ).Define the composite operation σ obtained by successively toggling at (1, 2),\n\n(2, 3),..., (n − 1, n ), (1, 3),..., (n − 2, n ), (1, 4),..., (1, n − 1), (2, n ), (1, n ).",
   "clean_statement": "Problem 5.4. Start with a known action in the birational (or piecewise-linear) realm and find the corresponding generalized toggle group action in the combi-natorial realm.\n\nOne such example of a birational map is the pentagram map.\n\n5.3 Subset toggling\n\nGeneralized subset-toggle groups are defined below. We could ask the same questions from the previous problem in this further-generalized context. Let E be a countable set and L ⊆ 2E.\n\nDefinition 5.5. For any subset S ⊆ E define its (subset-)toggle tS: L → L as\n\ntS (X) =\n\n{\n\nX4S if X4S ∈ L\n\nX otherwise where X4S denotes the symmetric difference of the sets X and S, that is,\n\nX4S = ( X \\ S) ∪ (S \\ X). We call {tS | S ⊆ E} the set of (subset-)toggles.We define the power set toggle group as the group generated by all the (subset-)toggles on L.16 Definition 5.6. Let T2E (L) be the subgroup of the symmetric group SL, gen-erated by {tS | S ⊆ E}. Call T2E (L) the power set toggle group on L.One could also construct a toggle group using only some of the subset-toggles.\n\nDefinition 5.7. Let K ⊆ 2E. Define TK(L) to be the subgroup of T2E (L) gen-erated by {tS | S ∈ K}. Call TK(L) the K-toggle subgroup on L (or generically, we call any TK(L) a subset-toggle group ).\n\n5.4 Toggling noncrossing partitions\n\nGiven a noncrossing partition π of {1, 2,..., n } ∈ NC (n) and 1 ≤ i < j ≤ n,define τ ′ as follows:\n\n• if i and j are consecutive elements of the same block B, split the block into two blocks (one consisting of all the elements of B that are ≤ i and the other consisting of all the elements of B that are ≥ j) and leave all the other blocks alone;\n\n• if i is the largest element of one block B1 and j is the smallest element of another block B2, merge the two blocks into one block B1 ∪ B2 (as long as this will not violate the noncrossing condition) and leave all the other blocks alone;\n\n• otherwise, do nothing. We write τ ′ = τi,j (π), and call the involution τi,j: NC (n) → NC (n) toggling at (i, j ).Define the composite operation σ obtained by successively toggling at (1, 2),\n\n(2, 3),..., (n − 1, n ), (1, 3),..., (n − 2, n ), (1, 4),..., (1, n − 1), (2, n ), (1, n ).",
   "public_statement": "Problem 5.4. Start with a known action in the birational (or piecewise-linear) realm and find the corresponding generalized toggle group action in the combi-natorial realm.\n\nOne such example of a birational map is the pentagram map.\n\n5.3 Subset toggling\n\nGeneralized subset-toggle groups are defined below. We could ask the same questions from the previous problem in this further-generalized context. Let E be a countable set and L ⊆ 2E.\n\nDefinition 5.5. For any subset S ⊆ E define its (subset-)toggle tS: L → L as\n\ntS (X) =\n\n{\n\nX4S if X4S ∈ L\n\nX otherwise where X4S denotes the symmetric difference of the sets X and S, that is,\n\nX4S = ( X \\ S) ∪ (S \\ X). We call {tS | S ⊆ E} the set of (subset-)toggles.We define the power set toggle group as the group generated by all the (subset-)toggles on L.16 Definition 5.6. Let T2E (L) be the subgroup of the symmetric group SL, gen-erated by {tS | S ⊆ E}. Call T2E (L) the power set toggle group on L.One could also construct a toggle group using only some of the subset-toggles.\n\nDefinition 5.7. Let K ⊆ 2E. Define TK(L) to be the subgroup of T2E (L) gen-erated by {tS | S ∈ K}. Call TK(L) the K-toggle subgroup on L (or generically, we call any TK(L) a subset-toggle group ).\n\n5.4 Toggling noncrossing partitions\n\nGiven a noncrossing partition π of {1, 2,..., n } ∈ NC (n) and 1 ≤ i < j ≤ n,define τ ′ as follows:\n\n• if i and j are consecutive elements of the same block B, split the block into two blocks (one consisting of all the elements of B that are ≤ i and the other consisting of all the elements of B that are ≥ j) and leave all the other blocks alone;\n\n• if i is the largest element of one block B1 and j is the smallest element of another block B2, merge the two blocks into one block B1 ∪ B2 (as long as this will not violate the noncrossing condition) and leave all the other blocks alone;\n\n• otherwise, do nothing. We write τ ′ = τi,j (π), and call the involution τi,j: NC (n) → NC (n) toggling at (i, j ).Define the composite operation σ obtained by successively toggling at (1, 2),\n\n(2, 3),..., (n − 1, n ), (1, 3),..., (n − 2, n ), (1, 4),..., (1, n − 1), (2, n ), (1, n ).",
   "evidence": "The canonical record is index 34 of `aim-other-notes.json`, extracted from the AIM workshop list *Dynamical Algebraic Combinatorics* (2015). Inspection of the official PDF shows that the problem ends immediately before the heading “5.3 Subset toggling.” The text from that heading onward in `input.json` is spillover from later problems and definitions, not part of Problem 5.4.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 34,
   "attempt": 1
  },
  "AIM-OTHER-0036": {
   "statement_status": "exact",
   "original_statement": "Conjecture 5.8. The number of blocks (a statistic on NC (n)) is homomesic under σ with average value (n + 1) /2 on each σ-orbit. \n\nThis conjecture has been verified for n ≤ 8, with help from Striker's gen-eralized toggling code (written in Sage). It should be noted that this action is not conjugate to the Panyushev complement, since for instance NC (4) consists of a single orbit of size 14 (rather than orbits of size 8, 4, and 2). In more detail, the orbit-decompositions for 2 ≤ n ≤ 8 are 2 = 2, 5 = 3 + 2, 14 = 14,\n\n42 = 15 + 13 + 5 + 3 + 3 + 3, 132 = 112 + 20, 429 = 133 + 109 + 39 + 39 + 31 + 15 + 13 + 11 + 9 + 9 + 6 + 3 + 3 + 3 + 3 + 3, and 1430 = 1240 + 144 + 32 + 8 + 6.Note that when n is even, the conjecture implies that all orbits have even cardinality. Also, if the conjecture is true, it's a dramatic illustration of the claim that homomesy can occur even when the orbit structure of a combinatorial dynamical system is \"horrible\" (from the point of view of cyclic sieving, say). Variants of σ that arise from composing the τi,j 's in a different order appear to be related to already-studied maps on Catalan objects. 17 References \n\n[Agg14] Amol Aggarwal, Armstrong's conjecture for (k, mk + 1) -core parti-tions, arXiv preprint arXiv:1407.5134 (2014). [AHJ14] Drew Armstrong, Christopher RH Hanusa, and Brant C Jones, \n\nResults and conjectures on simultaneous core partitions, European Journal of Combinatorics 41 (2014), 205-220. [ARW13] Drew Armstrong, Brendon Rhoades, and Nathan Williams, Ratio-nal associahedra and noncrossing partitions, The Electronic Journal of Combinatorics 20 (2013), no. 3, P54. [AST13] Drew Armstrong, Christian Stump, and Hugh Thomas, A uniform bijection between nonnesting and noncrossing partitions, Transac-tions of the American Mathematical Society 365 (2013), no. 8, 4121-4151. [Bes03] David Bessis, The dual braid monoid, Annales scientifiques de lâĂŹEcole normale supérieure, vol. 36, Elsevier, 2003, pp. 647-683. [BPS13] Jonathan Bloom, Oliver Pechenik, and Dan Saracino, A homomesy conjecture of J. Propp and T. Roby, http://arxiv.org/abs/1308. 0546.[CLS14] Cesar Ceballos, Jean-Philippe Labbé, and Christian Stump, \n\nSubword complexes, cluster complexes, and generalized multi-associahedra, Journal of Algebraic Combinatorics 39 (2014), no. 1, 17-51. [Dil14] Kevin Dilks, Involutions on baxter objects, arXiv preprint arXiv:1402.2961 (2014). [EKLP92a] Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp, \n\nAlternating-sign matrices and domino tilings (Part I), Journal of Algebraic Combinatorics 1 (1992), no. 2, 111-132. [EKLP92b], Alternating-sign matrices and domino tilings (Part II),Journal of Algebraic Combinatorics 1 (1992), no. 3, 219-234. [EPa] David Einstein and James Propp, Combinatorial, piecewise-linear, and birational homomesy for products of two chains, preprint; \n\nhttp://arxiv.org/abs/1310.5294.[EPb], Piecewise-linear and birational toggling, preprint; http: //arxiv-web3.library.cornell.edu/abs/1404.3455.[FZ00] Sergey Fomin and Andrei Zelevinsky, Recognizing Schubert cells,Journal of Algebraic Combinatorics 12 (2000), no. 1, 37-57. 18 [GR14] Darij Grinberg and Tom Roby, The order of birational rowmotion,Disc. Math & Theor. Comp. Sci. FPSAC 2014 (2014), 753-764. [Hai92] Mark D Haiman, Dual equivalence with applications, including a conjecture of Proctor, Discrete Mathematics 99 (1992), no. 1, 79- 113. [Hil82] Howard Hiller, Geometry of Coxeter groups, Pitman Pub., 1982. [Joh15] Paul Johnson, Lattice points and simultaneous core partitions,ArXiv e-prints (2015). [Pan09] Dmitri I Panyushev, On orbits of antichains of positive roots,European Journal of Combinatorics 30 (2009), no. 2, 586-594, \n\nhttp://arxiv.org/abs/0711.3353.[PP12] Vincent Pilaud and Michel Pocchiola, Multitriangulations, pseudo-triangulations and primitive sorting networks, Discrete & Compu-tational Geometry 48 (2012), no. 1, 142-191. [Pro83] Robert A Proctor, Shifted plane partitions of trapezoidal shape, Pro-ceedings of the American Mathematical Society 89 (1983), no. 3, 553-559. [Pro01] James Propp, The many faces of alternating-sign matrices, Discrete Mathematics and Theoretical Computer Science 43 (2001), 58. [Rea14] Nathan Reading, Noncrossing diagrams and canonical join repre-sentations, arXiv preprint arXiv:1405.6904 (2014). [RS13] David B Rush and XiaoLin Shi, On orbits of order ideals of minus-cule posets, Journal of Algebraic Combinatorics 37 (2013), no. 3, 545-569. [RSW04] Victor Reiner, Dennis Stanton, and Dennis White, The cyclic siev-ing phenomenon, Journal of Combinatorial Theory, Series A 108 \n\n(2004), no. 1, 17-50. [Sta86] Richard P Stanley, Two poset polytopes, Discrete & Computa-tional Geometry 1 (1986), no. 1, 9-23, http://dedekind.mit.edu/ ~rstan/pubs/pubfiles/66.pdf.[Ste86] John R Stembridge, Trapezoidal chains and antichains, European Journal of Combinatorics 7 (1986), no. 4, 377-387. [Ste96], On the fully commutative elements of coxeter groups, Jour-nal of Algebraic Combinatorics 5 (1996), no. 4, 353-385. [Str15] Jessica Striker, Rowmotion and the generalized toggle group,preprint (2015). 19 [SW12] Jessica Striker and Nathan Williams, Promotion and rowmotion,European Journal of Combinatorics 33 (2012), no. 8, 1919-1942. [SZ13] Richard P Stanley and Fabrizio Zanello, The Catalan case of armstrong's conjecture on core partitions, arXiv preprint arXiv:1312.4352 (2013). [Wil13] Nathan Williams, Cataland, Ph.D. thesis, 2013. [Wil14], Bijactions in Cataland, DMTCS Proceedings (2014), no. 01, 597-608. 20",
   "clean_statement": "Conjecture 5.8. The number of blocks (a statistic on NC (n)) is homomesic under σ with average value (n + 1) /2 on each σ-orbit.\n\nThis conjecture has been verified for n ≤ 8, with help from Striker's gen-eralized toggling code (written in Sage). It should be noted that this action is not conjugate to the Panyushev complement, since for instance NC (4) consists of a single orbit of size 14 (rather than orbits of size 8, 4, and 2). In more detail, the orbit-decompositions for 2 ≤ n ≤ 8 are 2 = 2, 5 = 3 + 2, 14 = 14,\n\n42 = 15 + 13 + 5 + 3 + 3 + 3, 132 = 112 + 20, 429 = 133 + 109 + 39 + 39 + 31 + 15 + 13 + 11 + 9 + 9 + 6 + 3 + 3 + 3 + 3 + 3, and 1430 = 1240 + 144 + 32 + 8 + 6.Note that when n is even, the conjecture implies that all orbits have even cardinality. Also, if the conjecture is true, it's a dramatic illustration of the claim that homomesy can occur even when the orbit structure of a combinatorial dynamical system is \"horrible\" (from the point of view of cyclic sieving, say). Variants of σ that arise from composing the τi,j 's in a different order appear to be related to already-studied maps on Catalan objects. 17 References\n\n[Agg14] Amol Aggarwal, Armstrong's conjecture for (k, mk + 1) -core parti-tions, arXiv preprint arXiv:1407.5134 (2014). [AHJ14] Drew Armstrong, Christopher RH Hanusa, and Brant C Jones,\n\nResults and conjectures on simultaneous core partitions, European Journal of Combinatorics 41 (2014), 205-220. [ARW13] Drew Armstrong, Brendon Rhoades, and Nathan Williams, Ratio-nal associahedra and noncrossing partitions, The Electronic Journal of Combinatorics 20 (2013), no. 3, P54. [AST13] Drew Armstrong, Christian Stump, and Hugh Thomas, A uniform bijection between nonnesting and noncrossing partitions, Transac-tions of the American Mathematical Society 365 (2013), no. 8, 4121-4151. [Bes03] David Bessis, The dual braid monoid, Annales scientifiques de lâĂŹEcole normale supérieure, vol. 36, Elsevier, 2003, pp. 647-683. [BPS13] Jonathan Bloom, Oliver Pechenik, and Dan Saracino, A homomesy conjecture of J. Propp and T. Roby, http://arxiv.org/abs/1308. 0546.[CLS14] Cesar Ceballos, Jean-Philippe Labbé, and Christian Stump,\n\nSubword complexes, cluster complexes, and generalized multi-associahedra, Journal of Algebraic Combinatorics 39 (2014), no. 1, 17-51. [Dil14] Kevin Dilks, Involutions on baxter objects, arXiv preprint arXiv:1402.2961 (2014). [EKLP92a] Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp,\n\nAlternating-sign matrices and domino tilings (Part I), Journal of Algebraic Combinatorics 1 (1992), no. 2, 111-132. [EKLP92b], Alternating-sign matrices and domino tilings (Part II),Journal of Algebraic Combinatorics 1 (1992), no. 3, 219-234. [EPa] David Einstein and James Propp, Combinatorial, piecewise-linear, and birational homomesy for products of two chains, preprint;\n\nhttp://arxiv.org/abs/1310.5294.[EPb], Piecewise-linear and birational toggling, preprint; http: //arxiv-web3.library.cornell.edu/abs/1404.3455.[FZ00] Sergey Fomin and Andrei Zelevinsky, Recognizing Schubert cells,Journal of Algebraic Combinatorics 12 (2000), no. 1, 37-57. 18 [GR14] Darij Grinberg and Tom Roby, The order of birational rowmotion,Disc. Math & Theor. Comp. Sci. FPSAC 2014 (2014), 753-764. [Hai92] Mark D Haiman, Dual equivalence with applications, including a conjecture of Proctor, Discrete Mathematics 99 (1992), no. 1, 79- 113. [Hil82] Howard Hiller, Geometry of Coxeter groups, Pitman Pub., 1982. [Joh15] Paul Johnson, Lattice points and simultaneous core partitions,ArXiv e-prints (2015). [Pan09] Dmitri I Panyushev, On orbits of antichains of positive roots,European Journal of Combinatorics 30 (2009), no. 2, 586-594,\n\nhttp://arxiv.org/abs/0711.3353.[PP12] Vincent Pilaud and Michel Pocchiola, Multitriangulations, pseudo-triangulations and primitive sorting networks, Discrete & Compu-tational Geometry 48 (2012), no. 1, 142-191. [Pro83] Robert A Proctor, Shifted plane partitions of trapezoidal shape, Pro-ceedings of the American Mathematical Society 89 (1983), no. 3, 553-559. [Pro01] James Propp, The many faces of alternating-sign matrices, Discrete Mathematics and Theoretical Computer Science 43 (2001), 58. [Rea14] Nathan Reading, Noncrossing diagrams and canonical join repre-sentations, arXiv preprint arXiv:1405.6904 (2014). [RS13] David B Rush and XiaoLin Shi, On orbits of order ideals of minus-cule posets, Journal of Algebraic Combinatorics 37 (2013), no. 3, 545-569. [RSW04] Victor Reiner, Dennis Stanton, and Dennis White, The cyclic siev-ing phenomenon, Journal of Combinatorial Theory, Series A 108\n\n(2004), no. 1, 17-50. [Sta86] Richard P Stanley, Two poset polytopes, Discrete & Computa-tional Geometry 1 (1986), no. 1, 9-23, http://dedekind.mit.edu/ ~rstan/pubs/pubfiles/66.pdf.[Ste86] John R Stembridge, Trapezoidal chains and antichains, European Journal of Combinatorics 7 (1986), no. 4, 377-387. [Ste96], On the fully commutative elements of coxeter groups, Jour-nal of Algebraic Combinatorics 5 (1996), no. 4, 353-385. [Str15] Jessica Striker, Rowmotion and the generalized toggle group,preprint (2015). 19 [SW12] Jessica Striker and Nathan Williams, Promotion and rowmotion,European Journal of Combinatorics 33 (2012), no. 8, 1919-1942. [SZ13] Richard P Stanley and Fabrizio Zanello, The Catalan case of armstrong's conjecture on core partitions, arXiv preprint arXiv:1312.4352 (2013). [Wil13] Nathan Williams, Cataland, Ph.D. thesis, 2013. [Wil14], Bijactions in Cataland, DMTCS Proceedings (2014), no. 01, 597-608. 20",
   "public_statement": "Conjecture 5.8. The number of blocks (a statistic on NC (n)) is homomesic under σ with average value (n + 1) /2 on each σ-orbit.\n\nThis conjecture has been verified for n ≤ 8, with help from Striker's gen-eralized toggling code (written in Sage). It should be noted that this action is not conjugate to the Panyushev complement, since for instance NC (4) consists of a single orbit of size 14 (rather than orbits of size 8, 4, and 2). In more detail, the orbit-decompositions for 2 ≤ n ≤ 8 are 2 = 2, 5 = 3 + 2, 14 = 14,\n\n42 = 15 + 13 + 5 + 3 + 3 + 3, 132 = 112 + 20, 429 = 133 + 109 + 39 + 39 + 31 + 15 + 13 + 11 + 9 + 9 + 6 + 3 + 3 + 3 + 3 + 3, and 1430 = 1240 + 144 + 32 + 8 + 6.Note that when n is even, the conjecture implies that all orbits have even cardinality. Also, if the conjecture is true, it's a dramatic illustration of the claim that homomesy can occur even when the orbit structure of a combinatorial dynamical system is \"horrible\" (from the point of view of cyclic sieving, say). Variants of σ that arise from composing the τi,j 's in a different order appear to be related to already-studied maps on Catalan objects. 17 References\n\n[Agg14] Amol Aggarwal, Armstrong's conjecture for (k, mk + 1) -core parti-tions, arXiv preprint arXiv:1407.5134 (2014). [AHJ14] Drew Armstrong, Christopher RH Hanusa, and Brant C Jones,\n\nResults and conjectures on simultaneous core partitions, European Journal of Combinatorics 41 (2014), 205-220. [ARW13] Drew Armstrong, Brendon Rhoades, and Nathan Williams, Ratio-nal associahedra and noncrossing partitions, The Electronic Journal of Combinatorics 20 (2013), no. 3, P54. [AST13] Drew Armstrong, Christian Stump, and Hugh Thomas, A uniform bijection between nonnesting and noncrossing partitions, Transac-tions of the American Mathematical Society 365 (2013), no. 8, 4121-4151. [Bes03] David Bessis, The dual braid monoid, Annales scientifiques de lâĂŹEcole normale supérieure, vol. 36, Elsevier, 2003, pp. 647-683. [BPS13] Jonathan Bloom, Oliver Pechenik, and Dan Saracino, A homomesy conjecture of J. Propp and T. Roby, http://arxiv.org/abs/1308. 0546.[CLS14] Cesar Ceballos, Jean-Philippe Labbé, and Christian Stump,\n\nSubword complexes, cluster complexes, and generalized multi-associahedra, Journal of Algebraic Combinatorics 39 (2014), no. 1, 17-51. [Dil14] Kevin Dilks, Involutions on baxter objects, arXiv preprint arXiv:1402.2961 (2014). [EKLP92a] Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp,\n\nAlternating-sign matrices and domino tilings (Part I), Journal of Algebraic Combinatorics 1 (1992), no. 2, 111-132. [EKLP92b], Alternating-sign matrices and domino tilings (Part II),Journal of Algebraic Combinatorics 1 (1992), no. 3, 219-234. [EPa] David Einstein and James Propp, Combinatorial, piecewise-linear, and birational homomesy for products of two chains, preprint;\n\nhttp://arxiv.org/abs/1310.5294.[EPb], Piecewise-linear and birational toggling, preprint; http: //arxiv-web3.library.cornell.edu/abs/1404.3455.[FZ00] Sergey Fomin and Andrei Zelevinsky, Recognizing Schubert cells,Journal of Algebraic Combinatorics 12 (2000), no. 1, 37-57. 18 [GR14] Darij Grinberg and Tom Roby, The order of birational rowmotion,Disc. Math & Theor. Comp. Sci. FPSAC 2014 (2014), 753-764. [Hai92] Mark D Haiman, Dual equivalence with applications, including a conjecture of Proctor, Discrete Mathematics 99 (1992), no. 1, 79- 113. [Hil82] Howard Hiller, Geometry of Coxeter groups, Pitman Pub., 1982. [Joh15] Paul Johnson, Lattice points and simultaneous core partitions,ArXiv e-prints (2015). [Pan09] Dmitri I Panyushev, On orbits of antichains of positive roots,European Journal of Combinatorics 30 (2009), no. 2, 586-594,\n\nhttp://arxiv.org/abs/0711.3353.[PP12] Vincent Pilaud and Michel Pocchiola, Multitriangulations, pseudo-triangulations and primitive sorting networks, Discrete & Compu-tational Geometry 48 (2012), no. 1, 142-191. [Pro83] Robert A Proctor, Shifted plane partitions of trapezoidal shape, Pro-ceedings of the American Mathematical Society 89 (1983), no. 3, 553-559. [Pro01] James Propp, The many faces of alternating-sign matrices, Discrete Mathematics and Theoretical Computer Science 43 (2001), 58. [Rea14] Nathan Reading, Noncrossing diagrams and canonical join repre-sentations, arXiv preprint arXiv:1405.6904 (2014). [RS13] David B Rush and XiaoLin Shi, On orbits of order ideals of minus-cule posets, Journal of Algebraic Combinatorics 37 (2013), no. 3, 545-569. [RSW04] Victor Reiner, Dennis Stanton, and Dennis White, The cyclic siev-ing phenomenon, Journal of Combinatorial Theory, Series A 108\n\n(2004), no. 1, 17-50. [Sta86] Richard P Stanley, Two poset polytopes, Discrete & Computa-tional Geometry 1 (1986), no. 1, 9-23, http://dedekind.mit.edu/ ~rstan/pubs/pubfiles/66.pdf.[Ste86] John R Stembridge, Trapezoidal chains and antichains, European Journal of Combinatorics 7 (1986), no. 4, 377-387. [Ste96], On the fully commutative elements of coxeter groups, Jour-nal of Algebraic Combinatorics 5 (1996), no. 4, 353-385. [Str15] Jessica Striker, Rowmotion and the generalized toggle group,preprint (2015). 19 [SW12] Jessica Striker and Nathan Williams, Promotion and rowmotion,European Journal of Combinatorics 33 (2012), no. 8, 1919-1942. [SZ13] Richard P Stanley and Fabrizio Zanello, The Catalan case of armstrong's conjecture on core partitions, arXiv preprint arXiv:1312.4352 (2013). [Wil13] Nathan Williams, Cataland, Ph.D. thesis, 2013. [Wil14], Bijactions in Cataland, DMTCS Proceedings (2014), no. 01, 597-608. 20",
   "evidence": "The source is the AIM *Dynamical Algebraic Combinatorics* preworkshop list, Conjecture 5.8. Its mathematical statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 35,
   "attempt": 1
  },
  "AIM-OTHER-0037": {
   "statement_status": "exact",
   "original_statement": "Is any integral fusion category unitarizable?",
   "clean_statement": "Is any integral fusion category unitarizable?",
   "public_statement": "Is any integral fusion category unitarizable?",
   "evidence": "The exact AIM problem is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 36,
   "attempt": 1
  },
  "AIM-OTHER-0038": {
   "statement_status": "exact",
   "original_statement": "Is every integral fusion category weakly group theoretical?",
   "clean_statement": "Is every integral fusion category weakly group theoretical?",
   "public_statement": "Is every integral fusion category weakly group theoretical?",
   "evidence": "The exact AIM question, listed as Problem 1.2 under “General fusion category questions” in the *Classifying fusion categories* problem list, is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 37,
   "attempt": 1
  },
  "AIM-OTHER-0039": {
   "statement_status": "exact",
   "original_statement": "Does pseudo-unitary imply unitarizable?",
   "clean_statement": "Does pseudo-unitary imply unitarizable?",
   "public_statement": "Does pseudo-unitary imply unitarizable?",
   "evidence": "The canonical record is number 1.3 in the AIM workshop list “Classifying fusion categories,” section “General fusion category questions”:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 38,
   "attempt": 1
  },
  "AIM-OTHER-0040": {
   "statement_status": "exact",
   "original_statement": "Are all fusion categories pivotal? Are all fusion categories spherical? Does it depend on the ground field ($k$ v.s. $\\C$)?",
   "clean_statement": "Are all fusion categories pivotal? Are all fusion categories spherical? Does it depend on the ground field ($k$ v.s. $\\C$)?",
   "public_statement": "Are all fusion categories pivotal? Are all fusion categories spherical? Does it depend on the ground field ($k$ v.s. $\\C$)?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 39,
   "attempt": 1
  },
  "AIM-OTHER-0041": {
   "statement_status": "exact",
   "original_statement": "\"Kaplansky's sixth conjecture\" for fusion categories\n\nIs $\\displaystyle \\frac{\\text{FPdim}(\\mathcal{C})}{\\text{FPdim}(X)}$ an algebraic integer for every $X\\in\\text{Irr}(\\mathcal{C})$?",
   "clean_statement": "\"Kaplansky's sixth conjecture\" for fusion categories\n\nIs $\\displaystyle \\frac{\\text{FPdim}(\\mathcal{C})}{\\text{FPdim}(X)}$ an algebraic integer for every $X\\in\\text{Irr}(\\mathcal{C})$?",
   "public_statement": "\"Kaplansky's sixth conjecture\" for fusion categories\n\nIs $\\displaystyle \\frac{\\text{FPdim}(\\mathcal{C})}{\\text{FPdim}(X)}$ an algebraic integer for every $X\\in\\text{Irr}(\\mathcal{C})$?",
   "evidence": "The exact canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 40,
   "attempt": 1
  },
  "AIM-OTHER-0042": {
   "statement_status": "exact",
   "original_statement": "Can we find all fusion categories with a given smallest simple object (which is not invertible)?",
   "clean_statement": "Can we find all fusion categories with a given smallest simple object (which is not invertible)?",
   "public_statement": "Can we find all fusion categories with a given smallest simple object (which is not invertible)?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 41,
   "attempt": 1
  },
  "AIM-OTHER-0043": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Is there an effective version of Ocneanu rigidity? Is there a sub-exponential bound on the number of unitary fusion categories with respect to $N$, the sum of all the fusion multiplicities $N_{i,j}^k$?",
   "clean_statement": null,
   "public_statement": "Is there an effective version of Ocneanu rigidity? Is there a sub-exponential bound on the number of unitary fusion categories with respect to $N$, the sum of all the fusion multiplicities $N_{i,j}^k$?",
   "evidence": "The canonical record is AIM Problem List entry 2.2 from the 2012 workshop *Classifying fusion categories*. Its exact text is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 42,
   "attempt": 1
  },
  "AIM-OTHER-0044": {
   "statement_status": "exact",
   "original_statement": "How many fusion categories have the same given fusion rules?",
   "clean_statement": "How many fusion categories have the same given fusion rules?",
   "public_statement": "How many fusion categories have the same given fusion rules?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 43,
   "attempt": 1
  },
  "AIM-OTHER-0045": {
   "statement_status": "exact",
   "original_statement": "What can you say about all fusion categories $\\mathcal{C}$ for which\n$\\#\\{\\dim(X)|X\\in\\mathcal{C}\\}=2?$",
   "clean_statement": "What can you say about all fusion categories $\\mathcal{C}$ for which\n$\\#\\{\\dim(X)|X\\in\\mathcal{C}\\}=2?$",
   "public_statement": "What can you say about all fusion categories $\\mathcal{C}$ for which\n$\\#\\{\\dim(X)|X\\in\\mathcal{C}\\}=2?$",
   "evidence": "The exact canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 44,
   "attempt": 1
  },
  "AIM-OTHER-0046": {
   "statement_status": "exact",
   "original_statement": "Property F conjecture\n\nIf a fusion category has property F, then can't use braiding alone for a universal quantum computer.\n\nGiven a braided, weakly integral fusion category, is the image of the braid group finite?\n\nIs braided and weakly integral fusion equivalent to finite image of the braid group?",
   "clean_statement": "Property F conjecture\n\nIf a fusion category has property F, then can't use braiding alone for a universal quantum computer.\n\nGiven a braided, weakly integral fusion category, is the image of the braid group finite?\n\nIs braided and weakly integral fusion equivalent to finite image of the braid group?",
   "public_statement": "Property F conjecture\n\nIf a fusion category has property F, then can't use braiding alone for a universal quantum computer.\n\nGiven a braided, weakly integral fusion category, is the image of the braid group finite?\n\nIs braided and weakly integral fusion equivalent to finite image of the braid group?",
   "evidence": "The accessible primary AIM workshop notes contain the same text as **Problem 9.13 (Rowell, Property F conjecture)** [AIM, pp. 14–15]. The canonical numbering “3.1” is a website-section number, whereas the workshop PDF uses 9.13. No OCR correction is needed.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 45,
   "attempt": 1
  },
  "AIM-OTHER-0047": {
   "statement_status": "exact",
   "original_statement": "Is there a physical model which gives infinite image for the braid group?",
   "clean_statement": "Is there a physical model which gives infinite image for the braid group?",
   "public_statement": "Is there a physical model which gives infinite image for the braid group?",
   "evidence": "This is Problem 3.2 in the repository extraction from the AIM workshop *Classifying fusion categories*. The original workshop PDF gives the same text as Problem 9.14, immediately after the discussion of Property F and quantum computation. No OCR correction is needed.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 46,
   "attempt": 1
  },
  "AIM-OTHER-0048": {
   "statement_status": "exact",
   "original_statement": "For unitary theories, we can choose a gauge so the $F$ matrices formed by the $6j$ symbols are unitary, and the braiding matrices are diagonal with respect to a certain basis. Can this happen for some non-unitary fusion category?\n\nIs the unitarity of the $F$ matrices equivalent to unitarity?",
   "clean_statement": "For unitary theories, we can choose a gauge so the $F$ matrices formed by the $6j$ symbols are unitary, and the braiding matrices are diagonal with respect to a certain basis. Can this happen for some non-unitary fusion category?\n\nIs the unitarity of the $F$ matrices equivalent to unitarity?",
   "public_statement": "For unitary theories, we can choose a gauge so the $F$ matrices formed by the $6j$ symbols are unitary, and the braiding matrices are diagonal with respect to a certain basis. Can this happen for some non-unitary fusion category?\n\nIs the unitarity of the $F$ matrices equivalent to unitarity?",
   "evidence": "The exact canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 47,
   "attempt": 1
  },
  "AIM-OTHER-0049": {
   "statement_status": "exact",
   "original_statement": "Is there a sense in which a randomly chosen fusion graph doesn't have cylotomic dimensions?",
   "clean_statement": "Is there a sense in which a randomly chosen fusion graph doesn't have cylotomic dimensions?",
   "public_statement": "Is there a sense in which a randomly chosen fusion graph doesn't have cylotomic dimensions?",
   "evidence": "The canonical record in `aim-other-notes.json` (zero-based index 48) reads exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 48,
   "attempt": 1
  },
  "AIM-OTHER-0050": {
   "statement_status": "exact",
   "original_statement": "Look at all spoke graphs with $N>0$ arms. Are there finitely many $N$-tuples $(\\ell_1,\\dots, \\ell_N)$ such that the spoke graph with $N$ arms of lengths $\\ell_1,\\dots, \\ell_N$ has cyclotomic norm squared?",
   "clean_statement": "Look at all spoke graphs with $N>0$ arms. Are there finitely many $N$-tuples $(\\ell_1,\\dots, \\ell_N)$ such that the spoke graph with $N$ arms of lengths $\\ell_1,\\dots, \\ell_N$ has cyclotomic norm squared?",
   "public_statement": "Look at all spoke graphs with $N>0$ arms. Are there finitely many $N$-tuples $(\\ell_1,\\dots, \\ell_N)$ such that the spoke graph with $N$ arms of lengths $\\ell_1,\\dots, \\ell_N$ has cyclotomic norm squared?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 49,
   "attempt": 1
  },
  "AIM-OTHER-0051": {
   "statement_status": "exact",
   "original_statement": "Is there a positive real number which is a cyclotomic integer and is largest amongst its Galois conjugates, but which is not realized as the dimension of an object in a fusion category?",
   "clean_statement": "Is there a positive real number which is a cyclotomic integer and is largest amongst its Galois conjugates, but which is not realized as the dimension of an object in a fusion category?",
   "public_statement": "Is there a positive real number which is a cyclotomic integer and is largest amongst its Galois conjugates, but which is not realized as the dimension of an object in a fusion category?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 50,
   "attempt": 1
  },
  "AIM-OTHER-0052": {
   "statement_status": "exact",
   "original_statement": "What values can $\\dim(X)$ take in $[2,3]$ for $X\\in\\mathcal{C}$, a braided fusion category?",
   "clean_statement": "What values can $\\dim(X)$ take in $[2,3]$ for $X\\in\\mathcal{C}$, a braided fusion category?",
   "public_statement": "What values can $\\dim(X)$ take in $[2,3]$ for $X\\in\\mathcal{C}$, a braided fusion category?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 51,
   "attempt": 1
  },
  "AIM-OTHER-0053": {
   "statement_status": "exact",
   "original_statement": "What are all the $\\mathcal{C}$ generated by $X$ with $\\text{FPdim}(X)\\leq2$, and $X$ not self-dual? Are they group theoretical if $\\dim(X)=2$?",
   "clean_statement": "What are all the $\\mathcal{C}$ generated by $X$ with $\\text{FPdim}(X)\\leq2$, and $X$ not self-dual? Are they group theoretical if $\\dim(X)=2$?",
   "public_statement": "What are all the $\\mathcal{C}$ generated by $X$ with $\\text{FPdim}(X)\\leq2$, and $X$ not self-dual? Are they group theoretical if $\\dim(X)=2$?",
   "evidence": "The canonical AIM record (Classifying fusion categories, section “Objects in fusion categories,” Problem 5.2) asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 52,
   "attempt": 1
  },
  "AIM-OTHER-0054": {
   "statement_status": "exact",
   "original_statement": "A fusion category version of supertransitivity\n\nIn a fusion category, is there an upper bound on the $N$ such that $X^{\\otimes N}$ is a simple object (where $\\dim(X)>1$)?",
   "clean_statement": "A fusion category version of supertransitivity\n\nIn a fusion category, is there an upper bound on the $N$ such that $X^{\\otimes N}$ is a simple object (where $\\dim(X)>1$)?",
   "public_statement": "A fusion category version of supertransitivity\n\nIn a fusion category, is there an upper bound on the $N$ such that $X^{\\otimes N}$ is a simple object (where $\\dim(X)>1$)?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 53,
   "attempt": 1
  },
  "AIM-OTHER-0055": {
   "statement_status": "exact",
   "original_statement": "There are accumulation points from below for $\\text{FPdim}(X)$ for an object in a fusion category or $[M\\colon N]$ for finite depth subfactors. Are there any accumulation points from above?",
   "clean_statement": "There are accumulation points from below for $\\text{FPdim}(X)$ for an object in a fusion category or $[M\\colon N]$ for finite depth subfactors. Are there any accumulation points from above?",
   "public_statement": "There are accumulation points from below for $\\text{FPdim}(X)$ for an object in a fusion category or $[M\\colon N]$ for finite depth subfactors. Are there any accumulation points from above?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 54,
   "attempt": 1
  },
  "AIM-OTHER-0056": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Is there a way to find the Frobenius-Schur exponent of $\\mathcal{C}$ without computing $Z(\\mathcal{C})$?",
   "clean_statement": null,
   "public_statement": "Is there a way to find the Frobenius-Schur exponent of $\\mathcal{C}$ without computing $Z(\\mathcal{C})$?",
   "evidence": "The canonical record asks:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 55,
   "attempt": 1
  },
  "AIM-OTHER-0057": {
   "statement_status": "exact",
   "original_statement": "Classify module categories and Brauer-Picard groups for known examples.",
   "clean_statement": "Classify module categories and Brauer-Picard groups for known examples.",
   "public_statement": "Classify module categories and Brauer-Picard groups for known examples.",
   "evidence": "The canonical record is AIM-OTHER-0057, source file `aim-other-notes.json`, zero-based index 56. Its statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 56,
   "attempt": 1
  },
  "AIM-OTHER-0058": {
   "statement_status": "exact",
   "original_statement": "The supertransitivity with respect to an object $X$ with $\\dim(X)>2$ is the largest $N$ such that $\\text{Hom}(1,X^{\\otimes n})$ is Temperley-Lieb.\n\nSupertransitivity is the analog of transitivity of group actions.\n\nIs there an upper bound on the supertransitivity of a subfactor planar algebra?",
   "clean_statement": "The supertransitivity with respect to an object $X$ with $\\dim(X)>2$ is the largest $N$ such that $\\text{Hom}(1,X^{\\otimes n})$ is Temperley-Lieb.\n\nSupertransitivity is the analog of transitivity of group actions.\n\nIs there an upper bound on the supertransitivity of a subfactor planar algebra?",
   "public_statement": "The supertransitivity with respect to an object $X$ with $\\dim(X)>2$ is the largest $N$ such that $\\text{Hom}(1,X^{\\otimes n})$ is Temperley-Lieb.\n\nSupertransitivity is the analog of transitivity of group actions.\n\nIs there an upper bound on the supertransitivity of a subfactor planar algebra?",
   "evidence": "The canonical AIM record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 57,
   "attempt": 1
  },
  "AIM-OTHER-0059": {
   "statement_status": "exact",
   "original_statement": "Find a non-number theoretic argument to rule out the rest of the Haagerup family vine.\n\nFor example, is there a diagram that evaluates in two different ways?",
   "clean_statement": "Find a non-number theoretic argument to rule out the rest of the Haagerup family vine.\n\nFor example, is there a diagram that evaluates in two different ways?",
   "public_statement": "Find a non-number theoretic argument to rule out the rest of the Haagerup family vine.\n\nFor example, is there a diagram that evaluates in two different ways?",
   "evidence": "The official AIM workshop compilation contains the same text as Problem 9.29 (Snyder), except that it abbreviates “For example” as “E.g.” There is no OCR error in the canonical record. The extracted record does omit the graph pictures and notation implicit in the workshop discussion, so those must be recovered from the cited subfactor literature.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 58,
   "attempt": 1
  },
  "AIM-OTHER-0060": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Is there a polymer theory of principal graphs? What graphs can appear as subgraphs of principal graphs?",
   "clean_statement": null,
   "public_statement": "Is there a polymer theory of principal graphs? What graphs can appear as subgraphs of principal graphs?",
   "evidence": "The canonical record is AIM-OTHER-0060, from `aim-other-notes.json` at zero-based index 59. Its exact question is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 59,
   "attempt": 1
  },
  "AIM-OTHER-0061": {
   "statement_status": "exact",
   "original_statement": "Is there an extension theory for fusion categories extended by fusion rings? (For example, near group categories)",
   "clean_statement": "Is there an extension theory for fusion categories extended by fusion rings? (For example, near group categories)",
   "public_statement": "Is there an extension theory for fusion categories extended by fusion rings? (For example, near group categories)",
   "evidence": "This agrees exactly with Problem 9.2 in the official notes from the 2011 AIM workshop *Classifying Fusion Categories*. The repository number 9.1 is a local section number, not an OCR error. The immediately preceding workshop section, “New from old,” lists \\(G\\)-extensions—categories \\(\\mathcal D=\\bigoplus_{g\\in G}\\mathcal D_g\\) with \\(\\mathcal D_e=\\mathcal C\\)—among standard constructions. It also lists short exact sequences, equivariantization, de-equivariantization, Hopf monads, and other genuinely categorical constructions. The next open problem asks for “fusion rings” in place of groups.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 60,
   "attempt": 1
  },
  "AIM-OTHER-0062": {
   "statement_status": "exact",
   "original_statement": "Describe functors between group-theoretical categories.\n\nWhat is known for Verlinde categories? (quantum groups at roots of unity)",
   "clean_statement": "Describe functors between group-theoretical categories.\n\nWhat is known for Verlinde categories? (quantum groups at roots of unity)",
   "public_statement": "Describe functors between group-theoretical categories.\n\nWhat is known for Verlinde categories? (quantum groups at roots of unity)",
   "evidence": "The official AIM workshop compilation contains exactly this text as Problem 9.10 (Gelaki). There is no OCR corruption. The source page labels it “Tensor functors,” so “functors” is interpreted as exact \\(k\\)-linear strong monoidal functors between fusion categories over an algebraically closed field \\(k\\) of characteristic zero, considered up to monoidal natural isomorphism.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 61,
   "attempt": 1
  },
  "AIM-OTHER-0063": {
   "statement_status": "exact",
   "original_statement": "Can you classify all algebras in fusion categories with small Frobenius-Perron dimension (e.g., less than $3+\\sqrt{3}$)?",
   "clean_statement": "Can you classify all algebras in fusion categories with small Frobenius-Perron dimension (e.g., less than $3+\\sqrt{3}$)?",
   "public_statement": "Can you classify all algebras in fusion categories with small Frobenius-Perron dimension (e.g., less than $3+\\sqrt{3}$)?",
   "evidence": "The canonical record, from `aim-other-notes.json` at zero-based index 62, asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 62,
   "attempt": 1
  },
  "AIM-OTHER-0064": {
   "statement_status": "exact",
   "original_statement": "Compute the center of the even half of the Asaeda-Haagerup and extended Haagerup subfactors.",
   "clean_statement": "Compute the center of the even half of the Asaeda-Haagerup and extended Haagerup subfactors.",
   "public_statement": "Compute the center of the even half of the Asaeda-Haagerup and extended Haagerup subfactors.",
   "evidence": "The wording is faithful to Problem 9.11 (attributed to Peters) in the official notes of the 2011 AIM workshop *Classifying Fusion Categories*. The repository number 11.2 is a local indexing choice; there is no substantive OCR error.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 63,
   "attempt": 1
  },
  "AIM-OTHER-0065": {
   "statement_status": "exact",
   "original_statement": "Is there a conceptual construction of the even half of 4442?",
   "clean_statement": "Is there a conceptual construction of the even half of 4442?",
   "public_statement": "Is there a conceptual construction of the even half of 4442?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 64,
   "attempt": 1
  },
  "AIM-OTHER-0066": {
   "statement_status": "exact",
   "original_statement": "Suppose $\\mathcal{C}$ is a finite tensor category over $\\C$ with prime Frobenius-Perron dimension. Is $\\mathcal{C}$ fusion?\n\n(Hence, it would be of the form $\\text{Vect}(\\Z/p,\\omega)$. This would be an extension of a result in Hopf algebras.)",
   "clean_statement": "Suppose $\\mathcal{C}$ is a finite tensor category over $\\C$ with prime Frobenius-Perron dimension. Is $\\mathcal{C}$ fusion?\n\n(Hence, it would be of the form $\\text{Vect}(\\Z/p,\\omega)$. This would be an extension of a result in Hopf algebras.)",
   "public_statement": "Suppose $\\mathcal{C}$ is a finite tensor category over $\\C$ with prime Frobenius-Perron dimension. Is $\\mathcal{C}$ fusion?\n\n(Hence, it would be of the form $\\text{Vect}(\\Z/p,\\omega)$. This would be an extension of a result in Hopf algebras.)",
   "evidence": "The canonical record is Problem 12.1 from the AIM workshop *Classifying fusion categories*:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 65,
   "attempt": 1
  },
  "AIM-OTHER-0067": {
   "statement_status": "exact",
   "original_statement": "How much from modular representations of finite groups can be carried to finite tensor categories?",
   "clean_statement": "How much from modular representations of finite groups can be carried to finite tensor categories?",
   "public_statement": "How much from modular representations of finite groups can be carried to finite tensor categories?",
   "evidence": "The canonical record asks:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 66,
   "attempt": 1
  },
  "AIM-OTHER-0068": {
   "statement_status": "unrecoverable",
   "original_statement": "Conjecture 21 10. Arithmeticity 23 11. Symbolic coding 23 References 24 \n\n1. Local rigidity \n\n1.1. It is well-known that an Anosov diffeomorphism is structurally stable: ev-ery C1-diffeomorphism which is sufficiently close in C1-topology to an Anosov diffeomorphism is topologically conjugate to it. However, the conjugation map is not differentiable in general. On the other hand, Anosov 1 actions by higher rank abelian groups exhibit much more rigid behavior (see [93] for the first re-sult of this type). It was shown in [97] that most of known algebraic 2 Anosov \n\nZkand Rkactions, k ≥ 2, are locally C∞-rigid. Recall that a C∞-action of Zk\n\nis called locally C∞-rigid if any C∞-action of Zk which is sufficiently C1-close to this action is conjugate to it by a C∞-map. A C∞-action of Rk is called \n\nlocally C∞-rigid if any C1-small perturbation of this action is C∞-conjugate to it up to an automorphism of Rk. \n\n> Date: November 1, 2004; Scribe: A. Gorodnik.\n> 1An action of a group Gis called Anosov if there is an element g∈Gthat acts normally hyperbolically with respect to the orbit foliation of G.\n> 2That is, the actions on infrahomogeneous spaces of Lie groups induced by either auto-morphisms or translations.\n> 1OPEN PROBLEMS 2\n\nIt was shown in [97] that most natural algebraic Anosov Zkand Rkactions, \n\nk ≥ 2, are locally C∞-rigid provided that they do not reduce to rank one actions via some elementary constructions. We call such actions \"irreducible\".See, for example, [97] for some natural conditions that guarantee that an action is \"irreducible\". Recently, local rigidity was proved in [39] for partially hyperbolic higher rank abelian actions by toral automorphisms. The method of [39] allows to construct C∞-conjugacy only for Cl-perturbations of the original action for some large l. Another interesting example of a partially hyperbolic action is given in the following conjecture, which was communicated by R. Spatzier:",
   "clean_statement": null,
   "public_statement": "Conjecture 21 10. Arithmeticity 23 11. Symbolic coding 23 References 24\n\n1. Local rigidity\n\n1.1. It is well-known that an Anosov diffeomorphism is structurally stable: ev-ery C1-diffeomorphism which is sufficiently close in C1-topology to an Anosov diffeomorphism is topologically conjugate to it. However, the conjugation map is not differentiable in general. On the other hand, Anosov 1 actions by higher rank abelian groups exhibit much more rigid behavior (see [93] for the first re-sult of this type). It was shown in [97] that most of known algebraic 2 Anosov\n\nZkand Rkactions, k ≥ 2, are locally C∞-rigid. Recall that a C∞-action of Zk\n\nis called locally C∞-rigid if any C∞-action of Zk which is sufficiently C1-close to this action is conjugate to it by a C∞-map. A C∞-action of Rk is called\n\nlocally C∞-rigid if any C1-small perturbation of this action is C∞-conjugate to it up to an automorphism of Rk.\n\n> Date: November 1, 2004; Scribe: A. Gorodnik.\n> 1An action of a group Gis called Anosov if there is an element g∈Gthat acts normally hyperbolically with respect to the orbit foliation of G.\n> 2That is, the actions on infrahomogeneous spaces of Lie groups induced by either auto-morphisms or translations.\n> 1OPEN PROBLEMS 2\n\nIt was shown in [97] that most natural algebraic Anosov Zkand Rkactions,\n\nk ≥ 2, are locally C∞-rigid provided that they do not reduce to rank one actions via some elementary constructions. We call such actions \"irreducible\".See, for example, [97] for some natural conditions that guarantee that an action is \"irreducible\". Recently, local rigidity was proved in [39] for partially hyperbolic higher rank abelian actions by toral automorphisms. The method of [39] allows to construct C∞-conjugacy only for Cl-perturbations of the original action for some large l. Another interesting example of a partially hyperbolic action is given in the following conjecture, which was communicated by R. Spatzier:",
   "evidence": "**Recovered-statement verdict:** there is no mathematical assertion to recover for AIM-OTHER-0068. It is a spurious/contextual record and should have status `invalid_statement`. Neither the neighboring Conjecture 1 nor the later genuine Conjecture 21 should be silently substituted for it.",
   "classification_method": "explicit_unrecoverable_evidence",
   "clean_statement_source": "no_clean_statement",
   "source_file": "aim-other-notes.json",
   "source_index": 67,
   "attempt": 1
  },
  "AIM-OTHER-0069": {
   "statement_status": "exact",
   "original_statement": "Conjecture 1. Let G be a connected semisimple Lie group, Γ an irreducible lattice in G, and A a closed subgroup of a split Cartan subgroup of G with \n\ndim A > 1. Then any C1-small perturbation of the action of A on G/ Γ is \n\nC∞-conjugate to the action of A defined by a continuous homomorphism from \n\nA to the centralizer of A in G.\n\nWe also state one of important partial cases of",
   "clean_statement": "Conjecture 1. Let G be a connected semisimple Lie group, Γ an irreducible lattice in G, and A a closed subgroup of a split Cartan subgroup of G with\n\ndim A > 1. Then any C1-small perturbation of the action of A on G/ Γ is\n\nC∞-conjugate to the action of A defined by a continuous homomorphism from\n\nA to the centralizer of A in G.\n\nWe also state one of important partial cases of",
   "public_statement": "Conjecture 1. Let G be a connected semisimple Lie group, Γ an irreducible lattice in G, and A a closed subgroup of a split Cartan subgroup of G with\n\ndim A > 1. Then any C1-small perturbation of the action of A on G/ Γ is\n\nC∞-conjugate to the action of A defined by a continuous homomorphism from\n\nA to the centralizer of A in G.\n\nWe also state one of important partial cases of",
   "evidence": "The primary AIM workshop PDF gives the following complete statement on page 2:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 68,
   "attempt": 1
  },
  "AIM-OTHER-0070": {
   "statement_status": "exact",
   "original_statement": "Conjecture 1 the group A is not contained in a wall of a Weyl chamber of the split Cartan subgroup D, then any C1-small perturbation of the action of A on G/ Γ is C∞-conjugate to the action of A defined by a continuous homomorphism from A to D.",
   "clean_statement": "Conjecture 1 the group A is not contained in a wall of a Weyl chamber of the split Cartan subgroup D, then any C1-small perturbation of the action of A on G/ Γ is C∞-conjugate to the action of A defined by a continuous homomorphism from A to D.",
   "public_statement": "Conjecture 1 the group A is not contained in a wall of a Weyl chamber of the split Cartan subgroup D, then any C1-small perturbation of the action of A on G/ Γ is C∞-conjugate to the action of A defined by a continuous homomorphism from A to D.",
   "evidence": "The canonical text begins:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 69,
   "attempt": 2
  },
  "AIM-OTHER-0071": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 2 was proved in [97] when A is the full split Cartan subgroup. It was pointed out by A. Katok that these conjectures might be possible to solve using the method from [39]. 1.2. Local rigidity for semisimple Lie groups of higher rank and their lattices (motivated by the program of R. Zimmer [209]) has been an active area of research too. First results in this direction were obtained for Anosov actions (see [85, 94, 95, 97]) and for actions with weaker hyperbolicity assumptions (see [131] and references therein). Recently, local rigidity results were established without any hyperbolicity assumptions (see [60]). 2. Global rigidity \n\n2.1. The only known examples of Anosov diffeomorphisms are automorphisms of infranilmanifolds. Moreover, every Anosov diffeomorphism on an infranil-manifold is topologically conjugate to a hyperbolic automorphism (see [62, 127]). This motivates the following \" ¤100,000\" folklore conjecture (stated in [130]):",
   "clean_statement": null,
   "public_statement": "Conjecture 2 was proved in [97] when A is the full split Cartan subgroup. It was pointed out by A. Katok that these conjectures might be possible to solve using the method from [39]. 1.2. Local rigidity for semisimple Lie groups of higher rank and their lattices (motivated by the program of R. Zimmer [209]) has been an active area of research too. First results in this direction were obtained for Anosov actions (see [85, 94, 95, 97]) and for actions with weaker hyperbolicity assumptions (see [131] and references therein). Recently, local rigidity results were established without any hyperbolicity assumptions (see [60]). 2. Global rigidity\n\n2.1. The only known examples of Anosov diffeomorphisms are automorphisms of infranilmanifolds. Moreover, every Anosov diffeomorphism on an infranil-manifold is topologically conjugate to a hyperbolic automorphism (see [62, 127]). This motivates the following \" ¤100,000\" folklore conjecture (stated in [130]):",
   "evidence": "The canonical problem field is the following extraction from the 2004 AIM workshop report *Emerging Applications of Measure Rigidity*:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 70,
   "attempt": 1
  },
  "AIM-OTHER-0072": {
   "statement_status": "exact",
   "original_statement": "Conjecture 3. Every Anosov diffeomorphism is topologically conjugate to a hyperbolic automorphism on an infranilmanifold. OPEN PROBLEMS 3\n\nAlthough there are some partial results on",
   "clean_statement": "Conjecture 3. Every Anosov diffeomorphism is topologically conjugate to a hyperbolic automorphism on an infranilmanifold. OPEN PROBLEMS 3\n\nAlthough there are some partial results on",
   "public_statement": "Conjecture 3. Every Anosov diffeomorphism is topologically conjugate to a hyperbolic automorphism on an infranilmanifold. OPEN PROBLEMS 3\n\nAlthough there are some partial results on",
   "evidence": "The source record in `input.json` has been preserved verbatim; only this report separates the page artifact and records the affine correction.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 71,
   "attempt": 1
  },
  "AIM-OTHER-0073": {
   "statement_status": "exact",
   "original_statement": "Conjecture 3 (see, for example, [61, 14, 69, 90]), it is not even known whether an Anosov automorphism is topologically transitive in general. We also mention that the conjugation map in",
   "clean_statement": "Conjecture 3 (see, for example, [61, 14, 69, 90]), it is not even known whether an Anosov automorphism is topologically transitive in general. We also mention that the conjugation map in",
   "public_statement": "Conjecture 3 (see, for example, [61, 14, 69, 90]), it is not even known whether an Anosov automorphism is topologically transitive in general. We also mention that the conjugation map in",
   "evidence": "The PDF itself says “Anosov automorphism,” so that noun is not an OCR error. It is, however, mathematically ambiguous in context. Under standard algebraic terminology, a toral automorphism is induced by a matrix in $\\mathrm{GL}(d,\\mathbb Z)$, and an Anosov toral automorphism is hyperbolic; such a map is topologically mixing, as proved below. Modern primary literature formulates the unresolved assertion for arbitrary Anosov **diffeomorphisms**. The safest recovery is therefore to preserve the printed word while analyzing both readings explicitly.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 72,
   "attempt": 2
  },
  "AIM-OTHER-0074": {
   "statement_status": "exact",
   "original_statement": "Conjecture 3 is not necessarily smooth. There are examples of Anosov diffeomorphisms on manifolds that are homeomorphic but not diffeomorphic to a torus (see [54]). In contrast, there exist Anosov flows that are not topologically transitive (see [63]), and it is not clear how to state a conjecture regarding classification of general Anosov flows. Such a conjecture is available in the special case when either stable or unstable foliation has dimension one (see [195, 70]). 2.2. One can also state an analog of",
   "clean_statement": "Conjecture 3 is not necessarily smooth. There are examples of Anosov diffeomorphisms on manifolds that are homeomorphic but not diffeomorphic to a torus (see [54]). In contrast, there exist Anosov flows that are not topologically transitive (see [63]), and it is not clear how to state a conjecture regarding classification of general Anosov flows. Such a conjecture is available in the special case when either stable or unstable foliation has dimension one (see [195, 70]). 2.2. One can also state an analog of",
   "public_statement": "Conjecture 3 is not necessarily smooth. There are examples of Anosov diffeomorphisms on manifolds that are homeomorphic but not diffeomorphic to a torus (see [54]). In contrast, there exist Anosov flows that are not topologically transitive (see [63]), and it is not clear how to state a conjecture regarding classification of general Anosov flows. Such a conjecture is available in the special case when either stable or unstable foliation has dimension one (see [195, 70]). 2.2. One can also state an analog of",
   "evidence": "The exact canonical text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 73,
   "attempt": 2
  },
  "AIM-OTHER-0075": {
   "statement_status": "exact",
   "original_statement": "Conjecture 3 for Anosov Zkand Rk-actions, k ≥ 2. In this case, it is usually possible to show that if a continuous conjugation map exists, it is also smooth (see, for example, [85, 93, 97]). R. Spatzier communicated the following conjecture.",
   "clean_statement": "Conjecture 3 for Anosov Zkand Rk-actions, k ≥ 2. In this case, it is usually possible to show that if a continuous conjugation map exists, it is also smooth (see, for example, [85, 93, 97]). R. Spatzier communicated the following conjecture.",
   "public_statement": "Conjecture 3 for Anosov Zkand Rk-actions, k ≥ 2. In this case, it is usually possible to show that if a continuous conjugation map exists, it is also smooth (see, for example, [85, 93, 97]). R. Spatzier communicated the following conjecture.",
   "evidence": "The canonical record is an OCR-fragment:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 74,
   "attempt": 1
  },
  "AIM-OTHER-0076": {
   "statement_status": "exact",
   "original_statement": "Conjecture 4. Every \"irreducible\" Anosov Zkand Rkaction, k ≥ 2, is \n\nC∞-conjugate to an algebraic action. \n\nSome partial results on",
   "clean_statement": "Conjecture 4. Every \"irreducible\" Anosov Zkand Rkaction, k ≥ 2, is\n\nC∞-conjugate to an algebraic action.\n\nSome partial results on",
   "public_statement": "Conjecture 4. Every \"irreducible\" Anosov Zkand Rkaction, k ≥ 2, is\n\nC∞-conjugate to an algebraic action.\n\nSome partial results on",
   "evidence": "The canonical JSON record is visibly damaged by extraction:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 75,
   "attempt": 1
  },
  "AIM-OTHER-0077": {
   "statement_status": "exact",
   "original_statement": "Conjecture 4 were obtained in [93, 142] and, recently, by F. Rodriguez-Hertz [162] and B. Kalinin, R. Spatzier. One may also hope to classify \"irreducible\" partially hyperbolic Zkand Rkaction, k ≥ 2, and more generally, higher rank actions of commuting expanding maps. 2.3. There are also analogous conjectures for actions of connected semisim-ple Lie groups of higher rank and their lattices satisfying some hyperbolicity assumptions (see [85, 131]).",
   "clean_statement": "Conjecture 4 were obtained in [93, 142] and, recently, by F. Rodriguez-Hertz [162] and B. Kalinin, R. Spatzier. One may also hope to classify \"irreducible\" partially hyperbolic Zkand Rkaction, k ≥ 2, and more generally, higher rank actions of commuting expanding maps. 2.3. There are also analogous conjectures for actions of connected semisim-ple Lie groups of higher rank and their lattices satisfying some hyperbolicity assumptions (see [85, 131]).",
   "public_statement": "Conjecture 4 were obtained in [93, 142] and, recently, by F. Rodriguez-Hertz [162] and B. Kalinin, R. Spatzier. One may also hope to classify \"irreducible\" partially hyperbolic Zkand Rkaction, k ≥ 2, and more generally, higher rank actions of commuting expanding maps. 2.3. There are also analogous conjectures for actions of connected semisim-ple Lie groups of higher rank and their lattices satisfying some hyperbolicity assumptions (see [85, 131]).",
   "evidence": "The canonical record is not a self-contained conjecture. Its text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 76,
   "attempt": 1
  },
  "AIM-OTHER-0078": {
   "statement_status": "corrected_verified",
   "original_statement": "Conjecture 5. Every action of a connected semisimple Lie group of higher rank (i.e., all simple factors have real rank at least 2) or its lattice, which that has an element which acts non-trivially uniformly partially hyperbolicly, is C∞-conjugate to an algebraic action. \n\nPartial results on",
   "clean_statement": "Conjecture 5. Every action of a connected semisimple Lie group of higher rank (i.e., all simple factors have real rank at least 2) or its lattice, which that has an element which acts non-trivially uniformly partially hyperbolically,, is C∞-conjugate to an algebraic action.\n\nPartial results on",
   "public_statement": "Conjecture 5. Every action of a connected semisimple Lie group of higher rank (i.e., all simple factors have real rank at least 2) or its lattice, which that has an element which acts non-trivially uniformly partially hyperbolically,, is C∞-conjugate to an algebraic action.\n\nPartial results on",
   "evidence": "The official AIM PDF itself prints (with the displayed line breaks suppressed): Thus **“which that” and “partially hyperbolicly” are source typos, not OCR defects**. A labeled editorial repair is:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "explicit_typographical_substitution",
   "source_file": "aim-other-notes.json",
   "source_index": 77,
   "attempt": 1
  },
  "AIM-OTHER-0079": {
   "statement_status": "exact",
   "original_statement": "Conjecture 5 were obtained in [85, 94, 95, 158, 72, 131]. Note that without partial hyperbolicity assumption, one may only hope to classify the actions when restricted to an open dense subset. See [94, 59] for examples of nonstandard lattice actions. In general, there are conjectures originated from [209] on classification of actions satisfying some transitivity assumptions or preserving a rigid geometric structure in the sense of Gromov (see [117, 130] for up-to-date statements). 2.4. Ratner's measure rigidity theorem has applications to the study of gen-eral properties of continuous volume preserving actions of higher rank semisim-ple Lie groups and their lattices on compact manifolds. In particular, Ratner's theorem plays a key role in the construction of arithmetic quotients of such OPEN PROBLEMS 4\n\nactions (see [123, 124] for connected groups, [56, 57] for lattices, and [58] for a survey). This raises the following question:",
   "clean_statement": "Conjecture 5 were obtained in [85, 94, 95, 158, 72, 131]. Note that without partial hyperbolicity assumption, one may only hope to classify the actions when restricted to an open dense subset. See [94, 59] for examples of nonstandard lattice actions. In general, there are conjectures originated from [209] on classification of actions satisfying some transitivity assumptions or preserving a rigid geometric structure in the sense of Gromov (see [117, 130] for up-to-date statements). 2.4. Ratner's measure rigidity theorem has applications to the study of gen-eral properties of continuous volume preserving actions of higher rank semisim-ple Lie groups and their lattices on compact manifolds. In particular, Ratner's theorem plays a key role in the construction of arithmetic quotients of such OPEN PROBLEMS 4\n\nactions (see [123, 124] for connected groups, [56, 57] for lattices, and [58] for a survey). This raises the following question:",
   "public_statement": "Conjecture 5 were obtained in [85, 94, 95, 158, 72, 131]. Note that without partial hyperbolicity assumption, one may only hope to classify the actions when restricted to an open dense subset. See [94, 59] for examples of nonstandard lattice actions. In general, there are conjectures originated from [209] on classification of actions satisfying some transitivity assumptions or preserving a rigid geometric structure in the sense of Gromov (see [117, 130] for up-to-date statements). 2.4. Ratner's measure rigidity theorem has applications to the study of gen-eral properties of continuous volume preserving actions of higher rank semisim-ple Lie groups and their lattices on compact manifolds. In particular, Ratner's theorem plays a key role in the construction of arithmetic quotients of such OPEN PROBLEMS 4\n\nactions (see [123, 124] for connected groups, [56, 57] for lattices, and [58] for a survey). This raises the following question:",
   "evidence": "The canonical record is not a standalone problem. It is the connective paragraph after Conjecture 5 and before subsection 2.4 and Question 6 in the AIM report *Emerging applications of measure rigidity*.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 78,
   "attempt": 1
  },
  "AIM-OTHER-0080": {
   "statement_status": "exact",
   "original_statement": "Question 6 (D. Fisher). Do the new results on measure rigidity for actions of higher rank abelian groups give rise to obstructions to smooth or continuous actions of a higher rank abelian group on a compact manifold? \n\nSome basic obstructions for smooth volume preserving actions of higher rank abelian groups are already known, see particularly work of H. Hu and A. Katok. The question is whether one can use results on measure rigidity to obtain more information. The results on arithmetic quotients for actions of semisimple groups and their lattices have no straightforward analogues here, since the proofs of those results use not only Ratner's theorem but applications of the cocycle super-rigidity theorems to cocycles which are necessarily only measurable. Though some cocycle superrigidity theorems are known for particular classes of actions of higher rank abelian groups, none of these apply to measurable cocycles be-cause of the Dye theorem [43] and its generalizations [154, 37]. 3. Measure rigidity \n\n3.1. Let G be a Lie group, Γ a discrete subgroup, and H a subgroup of G\n\ngenerated by one-parameter unipotent subgroups. One of the prototypical examples of measure rigidity is the classification of finite ergodic H-invariant measures on G/ Γ (see [159], and [144] for an accessible exposition).",
   "clean_statement": "Question 6 (D. Fisher). Do the new results on measure rigidity for actions of higher rank abelian groups give rise to obstructions to smooth or continuous actions of a higher rank abelian group on a compact manifold?\n\nSome basic obstructions for smooth volume preserving actions of higher rank abelian groups are already known, see particularly work of H. Hu and A. Katok. The question is whether one can use results on measure rigidity to obtain more information. The results on arithmetic quotients for actions of semisimple groups and their lattices have no straightforward analogues here, since the proofs of those results use not only Ratner's theorem but applications of the cocycle super-rigidity theorems to cocycles which are necessarily only measurable. Though some cocycle superrigidity theorems are known for particular classes of actions of higher rank abelian groups, none of these apply to measurable cocycles be-cause of the Dye theorem [43] and its generalizations [154, 37]. 3. Measure rigidity\n\n3.1. Let G be a Lie group, Γ a discrete subgroup, and H a subgroup of G\n\ngenerated by one-parameter unipotent subgroups. One of the prototypical examples of measure rigidity is the classification of finite ergodic H-invariant measures on G/ Γ (see [159], and [144] for an accessible exposition).",
   "public_statement": "Question 6 (D. Fisher). Do the new results on measure rigidity for actions of higher rank abelian groups give rise to obstructions to smooth or continuous actions of a higher rank abelian group on a compact manifold?\n\nSome basic obstructions for smooth volume preserving actions of higher rank abelian groups are already known, see particularly work of H. Hu and A. Katok. The question is whether one can use results on measure rigidity to obtain more information. The results on arithmetic quotients for actions of semisimple groups and their lattices have no straightforward analogues here, since the proofs of those results use not only Ratner's theorem but applications of the cocycle super-rigidity theorems to cocycles which are necessarily only measurable. Though some cocycle superrigidity theorems are known for particular classes of actions of higher rank abelian groups, none of these apply to measurable cocycles be-cause of the Dye theorem [43] and its generalizations [154, 37]. 3. Measure rigidity\n\n3.1. Let G be a Lie group, Γ a discrete subgroup, and H a subgroup of G\n\ngenerated by one-parameter unipotent subgroups. One of the prototypical examples of measure rigidity is the classification of finite ergodic H-invariant measures on G/ Γ (see [159], and [144] for an accessible exposition).",
   "evidence": "The canonical record contains Fisher's Question 6, its two explanatory paragraphs, and then the beginning of the next numbered section. The exact canonical text is preserved in `input.json`. Inspection of page 4 of the official AIM PDF verifies that the question proper is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 79,
   "attempt": 1
  },
  "AIM-OTHER-0081": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 7 (L. Silberman). Extend the results on measure rigidity of unipotent flows to adelic setting. \n\nIt seems natural to expect (and is known in some cases) that the set of finite ergodic invariant measures for other dynamical systems with parabolic behav-ior has a manageable structure, which is possible to described in algebraic terms. Suppose that H is a connected semisimple Lie subgroup of a Lie group G,and let P be a parabolic subgroup of H. One of manifestations of the measure rigidity of unipotent flows is the fact that every finite P -invariant measure on \n\nG/ Γ is H-invariant (see [146]).",
   "clean_statement": null,
   "public_statement": "Problem 7 (L. Silberman). Extend the results on measure rigidity of unipotent flows to adelic setting.\n\nIt seems natural to expect (and is known in some cases) that the set of finite ergodic invariant measures for other dynamical systems with parabolic behav-ior has a manageable structure, which is possible to described in algebraic terms. Suppose that H is a connected semisimple Lie subgroup of a Lie group G,and let P be a parabolic subgroup of H. One of manifestations of the measure rigidity of unipotent flows is the fact that every finite P -invariant measure on\n\nG/ Γ is H-invariant (see [146]).",
   "evidence": "The source itself prints the ungrammatical phrase “which is possible to described.” It is preserved above. An editorial reading is “which can possibly be described” or “which is possible to describe,” but that is reconstruction, not verified source text. By contrast, `behav-ior` is a line-break artifact; `G,and`, `P -invariant`, and `G/ Γ` are extraction-spacing artifacts.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 80,
   "attempt": 1
  },
  "AIM-OTHER-0082": {
   "statement_status": "exact",
   "original_statement": "Question 8 (E. Lindenstrauss). Suppose that H acts on a space X preserv-ing some geometric structure. Under what conditions on X, every finite P -invariant measure is H-invariant? In other words, which H-actions are stiff (see [64] )? \n\nFor example, one may consider an SL(2, R)-action on the moduli space of quadratic differentials over complex structures on a compact surface. There are a lot of similarities between this action and SL(2, R)-actions on homogeneous OPEN PROBLEMS 5\n\nspaces (see [48, 199, 141]). Some partial results on topological and measure rigidity for the latter actions were obtained in [143] and [51]. 3.2. A. Katok constructed an example of a Finsler metric on 2-dimensional sphere such that the corresponding geodesic flow is ergodic and has only two periodic orbits (see [92]).",
   "clean_statement": "Question 8 (E. Lindenstrauss). Suppose that H acts on a space X preserv-ing some geometric structure. Under what conditions on X, every finite P -invariant measure is H-invariant? In other words, which H-actions are stiff (see [64] )?\n\nFor example, one may consider an SL(2, R)-action on the moduli space of quadratic differentials over complex structures on a compact surface. There are a lot of similarities between this action and SL(2, R)-actions on homogeneous OPEN PROBLEMS 5\n\nspaces (see [48, 199, 141]). Some partial results on topological and measure rigidity for the latter actions were obtained in [143] and [51]. 3.2. A. Katok constructed an example of a Finsler metric on 2-dimensional sphere such that the corresponding geodesic flow is ergodic and has only two periodic orbits (see [92]).",
   "public_statement": "Question 8 (E. Lindenstrauss). Suppose that H acts on a space X preserv-ing some geometric structure. Under what conditions on X, every finite P -invariant measure is H-invariant? In other words, which H-actions are stiff (see [64] )?\n\nFor example, one may consider an SL(2, R)-action on the moduli space of quadratic differentials over complex structures on a compact surface. There are a lot of similarities between this action and SL(2, R)-actions on homogeneous OPEN PROBLEMS 5\n\nspaces (see [48, 199, 141]). Some partial results on topological and measure rigidity for the latter actions were obtained in [143] and [51]. 3.2. A. Katok constructed an example of a Finsler metric on 2-dimensional sphere such that the corresponding geodesic flow is ergodic and has only two periodic orbits (see [92]).",
   "evidence": "The canonical input is record 81 (zero-based) of aim-other-notes.json, from the 2004 AIM workshop *Emerging applications of measure rigidity*. The record must be preserved verbatim; in particular, its extracted problem field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 81,
   "attempt": 1
  },
  "AIM-OTHER-0083": {
   "statement_status": "exact",
   "original_statement": "Conjecture 9 (A. Katok). The only ergodic probability measures for this ex-ample are the smooth measure and the measures supported on periodic orbits. \n\n3.3. Since a polygonal billiard is a parabolic dynamical system, one expects that invariant measures and invariant closed sets should be scarce.",
   "clean_statement": "Conjecture 9 (A. Katok). The only ergodic probability measures for this ex-ample are the smooth measure and the measures supported on periodic orbits.\n\n3.3. Since a polygonal billiard is a parabolic dynamical system, one expects that invariant measures and invariant closed sets should be scarce.",
   "public_statement": "Conjecture 9 (A. Katok). The only ergodic probability measures for this ex-ample are the smooth measure and the measures supported on periodic orbits.\n\n3.3. Since a polygonal billiard is a parabolic dynamical system, one expects that invariant measures and invariant closed sets should be scarce.",
   "evidence": "The exact canonical record is preserved in input.json. The official AIM PDF and neighboring records show that its intended statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 82,
   "attempt": 1
  },
  "AIM-OTHER-0084": {
   "statement_status": "exact",
   "original_statement": "Question 10 (A. Katok). Classify ergodic invariant probability measure and closed invariant subsets for polygonal billiards. \n\nFor rational polygonal billiards, the phase space decomposes into invariant subsets Pθ that correspond to directions θ of the flow. It was shown in [103] that the billiard flow is uniquely ergodic on Pθ for the set of directions θ of full measure. There is an estimate of the Hausdorff dimension of this set, which may be positive (see [140, 139, 34]). It is also known that the restriction of the billiard flow on all but countably many of the subsets Pθ is minimal. 3.4. Let Γ be a discrete subgroup of SL(2, R). If Γ has infinite covolume, then the only finite ergodic invariant measures for the horocyclic flow ut =\n\n( 1 t0 1\n\n)\n\non SL(2, R)/Γ are the the ones supported on periodic orbits. It turns out that there is a large family of infinite ergodic invariant Radon measures. Such mea-sures can be constructed from the minimal positive Γ-invariant eigenfunction of the Laplacian (see, for example, [6]). Recently, F. Ledrappier and O. Sarig proved that if Γ is a normal subgroup of a uniform lattice in SL(2, R), then every ut-ergodic Radon measure on SL(2, R)/Γ is of this form up to a constant (see also [7, 166] for previous classification results).",
   "clean_statement": "Question 10 (A. Katok). Classify ergodic invariant probability measure and closed invariant subsets for polygonal billiards.\n\nFor rational polygonal billiards, the phase space decomposes into invariant subsets Pθ that correspond to directions θ of the flow. It was shown in [103] that the billiard flow is uniquely ergodic on Pθ for the set of directions θ of full measure. There is an estimate of the Hausdorff dimension of this set, which may be positive (see [140, 139, 34]). It is also known that the restriction of the billiard flow on all but countably many of the subsets Pθ is minimal. 3.4. Let Γ be a discrete subgroup of SL(2, R). If Γ has infinite covolume, then the only finite ergodic invariant measures for the horocyclic flow ut =\n\n( 1 t0 1\n\n)\n\non SL(2, R)/Γ are the the ones supported on periodic orbits. It turns out that there is a large family of infinite ergodic invariant Radon measures. Such mea-sures can be constructed from the minimal positive Γ-invariant eigenfunction of the Laplacian (see, for example, [6]). Recently, F. Ledrappier and O. Sarig proved that if Γ is a normal subgroup of a uniform lattice in SL(2, R), then every ut-ergodic Radon measure on SL(2, R)/Γ is of this form up to a constant (see also [7, 166] for previous classification results).",
   "public_statement": "Question 10 (A. Katok). Classify ergodic invariant probability measure and closed invariant subsets for polygonal billiards.\n\nFor rational polygonal billiards, the phase space decomposes into invariant subsets Pθ that correspond to directions θ of the flow. It was shown in [103] that the billiard flow is uniquely ergodic on Pθ for the set of directions θ of full measure. There is an estimate of the Hausdorff dimension of this set, which may be positive (see [140, 139, 34]). It is also known that the restriction of the billiard flow on all but countably many of the subsets Pθ is minimal. 3.4. Let Γ be a discrete subgroup of SL(2, R). If Γ has infinite covolume, then the only finite ergodic invariant measures for the horocyclic flow ut =\n\n( 1 t0 1\n\n)\n\non SL(2, R)/Γ are the the ones supported on periodic orbits. It turns out that there is a large family of infinite ergodic invariant Radon measures. Such mea-sures can be constructed from the minimal positive Γ-invariant eigenfunction of the Laplacian (see, for example, [6]). Recently, F. Ledrappier and O. Sarig proved that if Γ is a normal subgroup of a uniform lattice in SL(2, R), then every ut-ergodic Radon measure on SL(2, R)/Γ is of this form up to a constant (see also [7, 166] for previous classification results).",
   "evidence": "1. The canonical JSON continues with “3.4. Let \\(\\Gamma\\) be a discrete subgroup of \\(\\mathrm{SL}(2,\\mathbb R)\\) ...”. Inspection of the PDF shows that this is the next section, about horocycle flows, and is not part of Question 10. It is therefore excluded from the mathematical problem treated here. 2. The PDF has the singular phrase “probability measure”; the natural grammatical reading is “probability measures,” but the wording above is preserved. 3. The sentence “Hausdorff dimension of this set” is preserved verbatim. Taken literally, “this set” would be the full-measure set of uniquely ergodic directions, whose Hausdorff dimension is already one. The cited papers [Cheung2003], [Masur1992], and [MasurSmillie1991] instead study the exceptional set of nonergodic directions. Thus the exceptional-set reading is an editorial inference from the citations, not a silent correction of the...",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 83,
   "attempt": 1
  },
  "AIM-OTHER-0085": {
   "statement_status": "exact",
   "original_statement": "Question 11 (F. Ledrappier, O. Sarig). Let G be a noncompact semisimple Lie group of rank one, Γ a disctrete subgroup of G, and U a horospherical sub-group of G. What are the U -ergodic Radon measures on G/ Γ? In particular, are they either carried by closed U -orbits or given by the harmonic function construction? \n\nA ut-invariant measure μ is called squashable if the centralizer of ut contains an invertible nonsingular transformation that does not preserve μ. F. Ledrap-pier and O. Sarig showed recently that for a normal coabelian subgroup Γ of a uniform lattice, the only nonsquashable ut-ergodic measure on SL(2, R)/Γis Haar. If Γ is conilpotent, then any ut-ergodic measure on SL(2, R)/Γ is squashable except possibly the Haar measure. OPEN PROBLEMS 6",
   "clean_statement": "Question 11 (F. Ledrappier, O. Sarig). Let G be a noncompact semisimple Lie group of rank one, Γ a disctrete subgroup of G, and U a horospherical sub-group of G. What are the U -ergodic Radon measures on G/ Γ? In particular, are they either carried by closed U -orbits or given by the harmonic function construction?\n\nA ut-invariant measure μ is called squashable if the centralizer of ut contains an invertible nonsingular transformation that does not preserve μ. F. Ledrap-pier and O. Sarig showed recently that for a normal coabelian subgroup Γ of a uniform lattice, the only nonsquashable ut-ergodic measure on SL(2, R)/Γis Haar. If Γ is conilpotent, then any ut-ergodic measure on SL(2, R)/Γ is squashable except possibly the Haar measure. OPEN PROBLEMS 6",
   "public_statement": "Question 11 (F. Ledrappier, O. Sarig). Let G be a noncompact semisimple Lie group of rank one, Γ a disctrete subgroup of G, and U a horospherical sub-group of G. What are the U -ergodic Radon measures on G/ Γ? In particular, are they either carried by closed U -orbits or given by the harmonic function construction?\n\nA ut-invariant measure μ is called squashable if the centralizer of ut contains an invertible nonsingular transformation that does not preserve μ. F. Ledrap-pier and O. Sarig showed recently that for a normal coabelian subgroup Γ of a uniform lattice, the only nonsquashable ut-ergodic measure on SL(2, R)/Γis Haar. If Γ is conilpotent, then any ut-ergodic measure on SL(2, R)/Γ is squashable except possibly the Haar measure. OPEN PROBLEMS 6",
   "evidence": "The record is Question 11 in the AIM list *Emerging applications of measure rigidity* (June 2004), attributed to F. Ledrappier and O. Sarig. The PDF reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 84,
   "attempt": 1
  },
  "AIM-OTHER-0086": {
   "statement_status": "exact",
   "original_statement": "Question 12 (F. Ledrappier, O. Sarig). Let Γ be a normal conilpotent sub-group of a uniform lattice in SL(2, R). Is the Haar measure nonsquashable? \n\nFor general discrete subgroup Γ ⊂ SL(2, R), it is not known whether the Haar measure on SL(2, R)/Γ is nonsquashable, or whether there exist other nonsquashable ut-ergodic Radon measures. 3.5. Another important class of examples with rigid behavior is provided by the algebraic actions of higher rank abelian groups (see [121] for a survey). Although several complementary approaches have been developed for the study of invariant measures in this case (see [121]), all of them require some positive entropy assumptions. Such an assumption is not needed in the adelic setting. Let \n\nA ⊂ ∏\n\n> v-place\n\nQv\n\ndenote the ring of adeles and D the diagonal subgroup in SL(2). E. Linden-strauss showed that the only probability D(A)-invariant measure on SL(2, A)/SL(2, Q)is the Haar measure (see [122]).",
   "clean_statement": "Question 12 (F. Ledrappier, O. Sarig). Let Γ be a normal conilpotent sub-group of a uniform lattice in SL(2, R). Is the Haar measure nonsquashable?\n\nFor general discrete subgroup Γ ⊂ SL(2, R), it is not known whether the Haar measure on SL(2, R)/Γ is nonsquashable, or whether there exist other nonsquashable ut-ergodic Radon measures. 3.5. Another important class of examples with rigid behavior is provided by the algebraic actions of higher rank abelian groups (see [121] for a survey). Although several complementary approaches have been developed for the study of invariant measures in this case (see [121]), all of them require some positive entropy assumptions. Such an assumption is not needed in the adelic setting. Let\n\nA ⊂ ∏\n\n> v-place\n\nQv\n\ndenote the ring of adeles and D the diagonal subgroup in SL(2). E. Linden-strauss showed that the only probability D(A)-invariant measure on SL(2, A)/SL(2, Q)is the Haar measure (see [122]).",
   "public_statement": "Question 12 (F. Ledrappier, O. Sarig). Let Γ be a normal conilpotent sub-group of a uniform lattice in SL(2, R). Is the Haar measure nonsquashable?\n\nFor general discrete subgroup Γ ⊂ SL(2, R), it is not known whether the Haar measure on SL(2, R)/Γ is nonsquashable, or whether there exist other nonsquashable ut-ergodic Radon measures. 3.5. Another important class of examples with rigid behavior is provided by the algebraic actions of higher rank abelian groups (see [121] for a survey). Although several complementary approaches have been developed for the study of invariant measures in this case (see [121]), all of them require some positive entropy assumptions. Such an assumption is not needed in the adelic setting. Let\n\nA ⊂ ∏\n\n> v-place\n\nQv\n\ndenote the ring of adeles and D the diagonal subgroup in SL(2). E. Linden-strauss showed that the only probability D(A)-invariant measure on SL(2, A)/SL(2, Q)is the Haar measure (see [122]).",
   "evidence": "The canonical record comes from the AIM workshop list *Emerging applications of measure rigidity*. The official PDF, not merely the extracted JSON, prints the following wording:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 85,
   "attempt": 1
  },
  "AIM-OTHER-0087": {
   "statement_status": "exact",
   "original_statement": "Question 13 (E. Lindenstrauss). Let A′ be defined as the ring of adeles, but the product is taken over a subset of places of Q and Γ an \"irreducible\" lattice in SL(2, A′). What are the finite ergodic D(A′)-invariant measures on \n\nSL(2, A′)/Γ?\n\n3.6. One may also expect measure rigidity for algebraic actions of \"large\" groups.",
   "clean_statement": "Question 13 (E. Lindenstrauss). Let A′ be defined as the ring of adeles, but the product is taken over a subset of places of Q and Γ an \"irreducible\" lattice in SL(2, A′). What are the finite ergodic D(A′)-invariant measures on\n\nSL(2, A′)/Γ?\n\n3.6. One may also expect measure rigidity for algebraic actions of \"large\" groups.",
   "public_statement": "Question 13 (E. Lindenstrauss). Let A′ be defined as the ring of adeles, but the product is taken over a subset of places of Q and Γ an \"irreducible\" lattice in SL(2, A′). What are the finite ergodic D(A′)-invariant measures on\n\nSL(2, A′)/Γ?\n\n3.6. One may also expect measure rigidity for algebraic actions of \"large\" groups.",
   "evidence": "The official AIM PDF, *Emerging applications of measure rigidity*, gives the following question (there numbered Question 13):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 86,
   "attempt": 1
  },
  "AIM-OTHER-0088": {
   "statement_status": "exact",
   "original_statement": "Conjecture 14 (A. Furman). Consider one of the following actions of a group \n\nΓ:\n\n(1) Γ is a \"large\" subgroup of the group of automorphism of a nilmanifold of finite volume. \n\n(2) Γ is a \"large\" subgroup of a Lie group acting by translations on G/ Λ\n\nwhere Λ is a lattice in G.Then the only ergodic Γ-invariant probability measures are the measures sup-ported on finite Γ-orbits and the Haar measure. \n\nThere are results on the topological analog of this conjecture (see [16, 185, 148, 149, 74]). 4. Equidistribution \n\n4.1. Let G be a Lie group, Γ a lattice in G, and U = {u(t)} ⊂ G a one-parameter Ad-unipotent subgroup. Suppose that for x ∈ G/ Γ, U x is dense in \n\nG/ Γ. OPEN PROBLEMS 7",
   "clean_statement": "Conjecture 14 (A. Furman). Consider one of the following actions of a group\n\nΓ:\n\n(1) Γ is a \"large\" subgroup of the group of automorphism of a nilmanifold of finite volume.\n\n(2) Γ is a \"large\" subgroup of a Lie group acting by translations on G/ Λ\n\nwhere Λ is a lattice in G.Then the only ergodic Γ-invariant probability measures are the measures sup-ported on finite Γ-orbits and the Haar measure.\n\nThere are results on the topological analog of this conjecture (see [16, 185, 148, 149, 74]). 4. Equidistribution\n\n4.1. Let G be a Lie group, Γ a lattice in G, and U = {u(t)} ⊂ G a one-parameter Ad-unipotent subgroup. Suppose that for x ∈ G/ Γ, U x is dense in\n\nG/ Γ. OPEN PROBLEMS 7",
   "public_statement": "Conjecture 14 (A. Furman). Consider one of the following actions of a group\n\nΓ:\n\n(1) Γ is a \"large\" subgroup of the group of automorphism of a nilmanifold of finite volume.\n\n(2) Γ is a \"large\" subgroup of a Lie group acting by translations on G/ Λ\n\nwhere Λ is a lattice in G.Then the only ergodic Γ-invariant probability measures are the measures sup-ported on finite Γ-orbits and the Haar measure.\n\nThere are results on the topological analog of this conjecture (see [16, 185, 148, 149, 74]). 4. Equidistribution\n\n4.1. Let G be a Lie group, Γ a lattice in G, and U = {u(t)} ⊂ G a one-parameter Ad-unipotent subgroup. Suppose that for x ∈ G/ Γ, U x is dense in\n\nG/ Γ. OPEN PROBLEMS 7",
   "evidence": "The record comes from the AIM workshop list *Emerging applications of measure rigidity*, Conjecture 14, attributed to A. Furman. Inspection of the official PDF gives the following statement (typographical spacing normalized, but the scare quotes retained):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 87,
   "attempt": 1
  },
  "AIM-OTHER-0089": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 15 (G. Margulis [130]). Prove equidistribution of the sequence {u(tn)x}\n\nin G/ Γ, where tn is one of the following: \n\n(1) tn = [ nα]3 for α > 1,\n\n(2) tn = [ P (n)], where P (x) is a polynomial, \n\n(3) tn is the n-th prime number. \n\nA. Venkatesh suggested a proof of (1) when α is close to 1. It is known that ˇCesaro averages along sequences as in",
   "clean_statement": null,
   "public_statement": "Question 15 (G. Margulis [130]). Prove equidistribution of the sequence {u(tn)x}\n\nin G/ Γ, where tn is one of the following:\n\n(1) tn = [ nα]3 for α > 1,\n\n(2) tn = [ P (n)], where P (x) is a polynomial,\n\n(3) tn is the n-th prime number.\n\nA. Venkatesh suggested a proof of (1) when α is close to 1. It is known that ˇCesaro averages along sequences as in",
   "evidence": "The canonical JSON record is truncated and contains a misleading OCR/rendering artifact. The original AIM workshop PDF was checked directly. Section 4.1 fixes the following setting:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 88,
   "attempt": 1
  },
  "AIM-OTHER-0090": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 15 converge almost everywhere for functions in Lp, p > 1 (see [22, 23, 24, 25, 202]). Note that there is a subtle difference between sequences tn = [ nα] and tn = nα for \n\nα ∈ Q − Z. In fact, there is no general pointwise ergodic theorem possible for the latter sequence (see [18]). 4.2. Let V be a connected Ad-unipotent subgroup of the Lie group G such that V x is dense in G/ Γ for some x ∈ G/ Γ.",
   "clean_statement": null,
   "public_statement": "Question 15 converge almost everywhere for functions in Lp, p > 1 (see [22, 23, 24, 25, 202]). Note that there is a subtle difference between sequences tn = [ nα] and tn = nα for\n\nα ∈ Q − Z. In fact, there is no general pointwise ergodic theorem possible for the latter sequence (see [18]). 4.2. Let V be a connected Ad-unipotent subgroup of the Lie group G such that V x is dense in G/ Γ for some x ∈ G/ Γ.",
   "evidence": "The apparent superscript “3” after \\(\\lfloor n^\\alpha\\rfloor\\) in the preceding JSON record is footnote 3, not a cube; the footnote says that \\([x]\\) denotes the integer part of \\(x\\). The notation is therefore recovered as \\(\\lfloor n^\\alpha\\rfloor\\). The raw record also loses superscript formatting (`nα`) and spacing (`G/ Γ`); these are restored only in the explicitly labeled reconstruction.",
   "classification_method": "explicit_uncertainty_evidence",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 89,
   "attempt": 1
  },
  "AIM-OTHER-0091": {
   "statement_status": "exact",
   "original_statement": "Question 16 (G. Margulis [130]). Show that for a \"good\" sequence of subsets \n\nAn ⊂ V and every f ∈ Cc(G/ Γ),\n\nlim \n\n> n→∞\n\n1\n\nVol( An)\n\n∫\n\n> An\n\nf (vx ) dv =\n\n∫\n\n> G/ Γ\n\nf dμ \n\nwith effective error term, where dv is a Haar measure on V, and μ is the probability Haar measure on G/ Γ.\n\nSuch equidistribution results were proved by several authors (see [160, 132, 176]), but the methods of the proofs are not effective. In the case when V\n\nis a horospherical subgroup of G (see Section 4.3 below), one can deduce an equidistribution result with explicit error term from decay of matrix coefficient on L2(G/ Γ) (see [108]). 4.3. Let L be Lie group, G a closed subgroup of L, and Λ a lattice in L.For a semisimple element a ∈ G, the expanding horospherical subgroup U of G\n\nassociated to a is defined by \n\nU = {g ∈ G: a−nga n → e as n → ∞}.\n\nSuppose that for x0 ∈ L/ Λ, the orbit Gx 0 is dense in L/ Λ. Let μ be a measure on U x 0 which is the image of a probability measure on \n\nU, absolutely continuous with respect to the Haar measure on U, under the map u 7 → ux 0, u ∈ U. Then it is known that anμ → λ as n → ∞ where λ is the probability Haar measure on L/ Λ (see [177]). One may consider the following refinement of the above result. Take any analytic curve γ: [0, 1] → U, and let ν be the image of the Lebesgue measure on [0, 1] under the map t 7 → γ(t)x0. \n\n> 3Here [ x] denotes the integer part of x.OPEN PROBLEMS 8",
   "clean_statement": "Question 16 (G. Margulis [130]). Show that for a \"good\" sequence of subsets\n\nAn ⊂ V and every f ∈ Cc(G/ Γ),\n\nlim\n\n> n→∞\n\n1\n\nVol( An)\n\n∫\n\n> An\n\nf (vx ) dv =\n\n∫\n\n> G/ Γ\n\nf dμ\n\nwith effective error term, where dv is a Haar measure on V, and μ is the probability Haar measure on G/ Γ.\n\nSuch equidistribution results were proved by several authors (see [160, 132, 176]), but the methods of the proofs are not effective. In the case when V\n\nis a horospherical subgroup of G (see Section 4.3 below), one can deduce an equidistribution result with explicit error term from decay of matrix coefficient on L2(G/ Γ) (see [108]). 4.3. Let L be Lie group, G a closed subgroup of L, and Λ a lattice in L.For a semisimple element a ∈ G, the expanding horospherical subgroup U of G\n\nassociated to a is defined by\n\nU = {g ∈ G: a−nga n → e as n → ∞}.\n\nSuppose that for x0 ∈ L/ Λ, the orbit Gx 0 is dense in L/ Λ. Let μ be a measure on U x 0 which is the image of a probability measure on\n\nU, absolutely continuous with respect to the Haar measure on U, under the map u 7 → ux 0, u ∈ U. Then it is known that anμ → λ as n → ∞ where λ is the probability Haar measure on L/ Λ (see [177]). One may consider the following refinement of the above result. Take any analytic curve γ: [0, 1] → U, and let ν be the image of the Lebesgue measure on [0, 1] under the map t 7 → γ(t)x0.\n\n> 3Here [ x] denotes the integer part of x.OPEN PROBLEMS 8",
   "public_statement": "Question 16 (G. Margulis [130]). Show that for a \"good\" sequence of subsets\n\nAn ⊂ V and every f ∈ Cc(G/ Γ),\n\nlim\n\n> n→∞\n\n1\n\nVol( An)\n\n∫\n\n> An\n\nf (vx ) dv =\n\n∫\n\n> G/ Γ\n\nf dμ\n\nwith effective error term, where dv is a Haar measure on V, and μ is the probability Haar measure on G/ Γ.\n\nSuch equidistribution results were proved by several authors (see [160, 132, 176]), but the methods of the proofs are not effective. In the case when V\n\nis a horospherical subgroup of G (see Section 4.3 below), one can deduce an equidistribution result with explicit error term from decay of matrix coefficient on L2(G/ Γ) (see [108]). 4.3. Let L be Lie group, G a closed subgroup of L, and Λ a lattice in L.For a semisimple element a ∈ G, the expanding horospherical subgroup U of G\n\nassociated to a is defined by\n\nU = {g ∈ G: a−nga n → e as n → ∞}.\n\nSuppose that for x0 ∈ L/ Λ, the orbit Gx 0 is dense in L/ Λ. Let μ be a measure on U x 0 which is the image of a probability measure on\n\nU, absolutely continuous with respect to the Haar measure on U, under the map u 7 → ux 0, u ∈ U. Then it is known that anμ → λ as n → ∞ where λ is the probability Haar measure on L/ Λ (see [177]). One may consider the following refinement of the above result. Take any analytic curve γ: [0, 1] → U, and let ν be the image of the Lebesgue measure on [0, 1] under the map t 7 → γ(t)x0.\n\n> 3Here [ x] denotes the integer part of x.OPEN PROBLEMS 8",
   "evidence": "The canonical JSON begins at Question 16 and then accidentally absorbs the next subsection. Inspection of the official AIM PDF gives the missing setup and the correct boundary:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 90,
   "attempt": 2
  },
  "AIM-OTHER-0092": {
   "statement_status": "corrected_verified",
   "original_statement": "Question 17 (N. Shah). Under what condition on γ, we have that anν → λ\n\nas n → ∞?\n\nRecently",
   "clean_statement": "**Question 17 (N. Shah).** Under what condition on \\(\\gamma\\) do we have\n\\[\na^n\\nu\\overset{w^*}{\\longrightarrow}\\lambda\n\\qquad(n\\to\\infty)?\n\\]",
   "public_statement": "**Question 17 (N. Shah).** Under what condition on \\(\\gamma\\) do we have\n\\[\na^n\\nu\\overset{w^*}{\\longrightarrow}\\lambda\n\\qquad(n\\to\\infty)?\n\\]",
   "evidence": "The canonical extraction is truncated: The official AIM PDF, *Emerging applications of measure rigidity*, Section 4.3, supplies the missing setup and correct typography. Let \\(L\\) be a Lie group, let \\(G\\) be a closed subgroup of \\(L\\), and let \\(\\Lambda\\) be a lattice in \\(L\\). For a semisimple element \\(a\\in G\\), define its expanding horospherical subgroup by",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-other-notes.json",
   "source_index": 91,
   "attempt": 1
  },
  "AIM-OTHER-0093": {
   "statement_status": "exact",
   "original_statement": "Question 17 was solved by N. Shah for L = SO( m, 1) and G =SO( n, 1), m > n. He showed that anν → λ as n → ∞ provided γ([0, 1]) does not lie on an proper affine subspace or an ( n − 2)-dimensional sphere in U.",
   "clean_statement": "Question 17 was solved by N. Shah for L = SO( m, 1) and G =SO( n, 1), m > n. He showed that anν → λ as n → ∞ provided γ([0, 1]) does not lie on an proper affine subspace or an ( n − 2)-dimensional sphere in U.",
   "public_statement": "Question 17 was solved by N. Shah for L = SO( m, 1) and G =SO( n, 1), m > n. He showed that anν → λ as n → ∞ provided γ([0, 1]) does not lie on an proper affine subspace or an ( n − 2)-dimensional sphere in U.",
   "evidence": "This canonical record is not a second Question 17. It is an explanatory paragraph immediately following Question 17 in Section 4.3 of the official AIM workshop PDF. The preceding setup is essential.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 92,
   "attempt": 1
  },
  "AIM-OTHER-0094": {
   "statement_status": "exact",
   "original_statement": "Conjecture 18 (N. Shah). The same result holds for all Lie groups L con-taining G = SO( n, 1).\n\nThe above kind of questions are related to the following more general prob-lem. Consider a representation of a semisimple Lie group G on real vector space V equipped with a norm ‖ · ‖. Take a point p ∈ V, and consider the set \n\nRT = {g ∈ G: ‖gp ‖ < T }\n\nfor T > 0. Suppose that the stabilizer of p is finite, so RT is compact for each \n\nT. Let Γ be be a lattice in G, and let μT denote the image of the normalized Haar measure on RT projected to G/ Γ.",
   "clean_statement": "Conjecture 18 (N. Shah). The same result holds for all Lie groups L con-taining G = SO( n, 1).\n\nThe above kind of questions are related to the following more general prob-lem. Consider a representation of a semisimple Lie group G on real vector space V equipped with a norm ‖ · ‖. Take a point p ∈ V, and consider the set\n\nRT = {g ∈ G: ‖gp ‖ < T }\n\nfor T > 0. Suppose that the stabilizer of p is finite, so RT is compact for each\n\nT. Let Γ be be a lattice in G, and let μT denote the image of the normalized Haar measure on RT projected to G/ Γ.",
   "public_statement": "Conjecture 18 (N. Shah). The same result holds for all Lie groups L con-taining G = SO( n, 1).\n\nThe above kind of questions are related to the following more general prob-lem. Consider a representation of a semisimple Lie group G on real vector space V equipped with a norm ‖ · ‖. Take a point p ∈ V, and consider the set\n\nRT = {g ∈ G: ‖gp ‖ < T }\n\nfor T > 0. Suppose that the stabilizer of p is finite, so RT is compact for each\n\nT. Let Γ be be a lattice in G, and let μT denote the image of the normalized Haar measure on RT projected to G/ Γ.",
   "evidence": "The canonical record comes from Section 4.3 of the AIM workshop list *Open Problems from the Workshop “Emerging Applications of Measure Rigidity”*. Its first sentence is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 93,
   "attempt": 1
  },
  "AIM-OTHER-0095": {
   "statement_status": "exact",
   "original_statement": "Question 19 (N. Shah). What is the limiting distribution of the measure μT\n\nas T → ∞?\n\nIn some examples such results are known (see [53, 73]), but more general answers can be very important for understanding distribution of Γ-orbits on homogeneous spaces G/H where either Γ ∩ H is a lattice in H or Γ H is dense in G.4.4. For irrational α, the sequence {αn 2 (mod 1): n ≥ 1} is equidistributed in [0, 1]. In fact, one expects that if α is badly approximable by rationals, then statistical properties of this sequence are the same as the sequence of independent uniformly distributed random variables. For [ a, b ] ⊂ [0, 1], we define pair correlation: \n\nR2([ a, b ], N, α ) = 1\n\nN #\n\n{\n\n1 ≤ i 6 = j ≤ N: αi 2 − αj 2 ∈ 1\n\nN [a, b ] (mod 1) \n\n}.",
   "clean_statement": "Question 19 (N. Shah). What is the limiting distribution of the measure μT\n\nas T → ∞?\n\nIn some examples such results are known (see [53, 73]), but more general answers can be very important for understanding distribution of Γ-orbits on homogeneous spaces G/H where either Γ ∩ H is a lattice in H or Γ H is dense in G.4.4. For irrational α, the sequence {αn 2 (mod 1): n ≥ 1} is equidistributed in [0, 1]. In fact, one expects that if α is badly approximable by rationals, then statistical properties of this sequence are the same as the sequence of independent uniformly distributed random variables. For [ a, b ] ⊂ [0, 1], we define pair correlation:\n\nR2([ a, b ], N, α ) = 1\n\nN #\n\n{\n\n1 ≤ i 6 = j ≤ N: αi 2 − αj 2 ∈ 1\n\nN [a, b ] (mod 1)\n\n}.",
   "public_statement": "Question 19 (N. Shah). What is the limiting distribution of the measure μT\n\nas T → ∞?\n\nIn some examples such results are known (see [53, 73]), but more general answers can be very important for understanding distribution of Γ-orbits on homogeneous spaces G/H where either Γ ∩ H is a lattice in H or Γ H is dense in G.4.4. For irrational α, the sequence {αn 2 (mod 1): n ≥ 1} is equidistributed in [0, 1]. In fact, one expects that if α is badly approximable by rationals, then statistical properties of this sequence are the same as the sequence of independent uniformly distributed random variables. For [ a, b ] ⊂ [0, 1], we define pair correlation:\n\nR2([ a, b ], N, α ) = 1\n\nN #\n\n{\n\n1 ≤ i 6 = j ≤ N: αi 2 − αj 2 ∈ 1\n\nN [a, b ] (mod 1)\n\n}.",
   "evidence": "The canonical extraction joins two different sections of the AIM workshop document. The exact Question 19 occupies the end of Section 4.3; all text beginning with “4.4. For irrational \\(\\alpha\\), the sequence \\(\\{\\alpha n^2\\pmod 1\\}\\) ...” belongs to the next section and is not part of this record.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 94,
   "attempt": 1
  },
  "AIM-OTHER-0096": {
   "statement_status": "exact",
   "original_statement": "Conjecture 20 (Z. Rudnick, P. Sarnak). If α ∈ R is badly approximable by rationals (see [165] for exact conditions), then \n\n(1) R2([ a, b ], N, α ) → b − a as N → ∞.\n\nAlthough it was shown that (1) holds on the set of α of full measure (see [164]) and on a residual set of α in the sense of Baire category (see [165]), one does not know any explicit α for which it is true. It is expected that (1) holds for algebraic integers, and it is not hard to show that there are well approximable irrational α for which (1) fails. OPEN PROBLEMS 9\n\nIt was discovered in [136] that",
   "clean_statement": "Conjecture 20 (Z. Rudnick, P. Sarnak). If α ∈ R is badly approximable by rationals (see [165] for exact conditions), then\n\n(1) R2([ a, b ], N, α ) → b − a as N → ∞.\n\nAlthough it was shown that (1) holds on the set of α of full measure (see [164]) and on a residual set of α in the sense of Baire category (see [165]), one does not know any explicit α for which it is true. It is expected that (1) holds for algebraic integers, and it is not hard to show that there are well approximable irrational α for which (1) fails. OPEN PROBLEMS 9\n\nIt was discovered in [136] that",
   "public_statement": "Conjecture 20 (Z. Rudnick, P. Sarnak). If α ∈ R is badly approximable by rationals (see [165] for exact conditions), then\n\n(1) R2([ a, b ], N, α ) → b − a as N → ∞.\n\nAlthough it was shown that (1) holds on the set of α of full measure (see [164]) and on a residual set of α in the sense of Baire category (see [165]), one does not know any explicit α for which it is true. It is expected that (1) holds for algebraic integers, and it is not hard to show that there are well approximable irrational α for which (1) fails. OPEN PROBLEMS 9\n\nIt was discovered in [136] that",
   "evidence": "The official AIM workshop PDF defines, for irrational \\(\\alpha\\) and a fixed interval \\([a,b]\\subset[0,1]\\), \\[ R_2([a,b],N,\\alpha) =\\frac1N\\#\\left\\{1\\leq i\\ne j\\leq N: \\alpha i^2-\\alpha j^2\\in \\frac1N[a,b]\\pmod 1\\right\\}. \\] Thus the pairs are ordered, the diagonal is excluded, and membership means \\(\\alpha(i^2-j^2)\\in[a/N,b/N]+\\mathbb Z\\). Conjecture 20 asks for \\[ R_2([a,b],N,\\alpha)\\longrightarrow b-a. \\]",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 95,
   "attempt": 1
  },
  "AIM-OTHER-0097": {
   "statement_status": "exact",
   "original_statement": "Conjecture 20 is related to an equidistribution problem on a hyperbolic surface X = Γ \\H2, Γ a lattice. We assume that Γ ∩ { z 7 → z + a: a ∈ R} = {z 7 → z + a: a ∈ Z}.\n\nThen the curve {x + iy: x ∈ [0, 1] } corresponds to a closed horocycle {uy(t): 0 ≤ t ≤ 1} of length y−1 in in the unit tangent bundle T 1(X), and it is well-known that it becomes equidistributed in T 1(X) as y → 0+. Also, for irrational \n\nα, the sequence {uy(αn ): n ≥ 1} is equidistributed in the horocycle. This motivates the following conjecture:",
   "clean_statement": "Conjecture 20 is related to an equidistribution problem on a hyperbolic surface X = Γ \\H2, Γ a lattice. We assume that Γ ∩ { z 7 → z + a: a ∈ R} = {z 7 → z + a: a ∈ Z}.\n\nThen the curve {x + iy: x ∈ [0, 1] } corresponds to a closed horocycle {uy(t): 0 ≤ t ≤ 1} of length y−1 in in the unit tangent bundle T 1(X), and it is well-known that it becomes equidistributed in T 1(X) as y → 0+. Also, for irrational\n\nα, the sequence {uy(αn ): n ≥ 1} is equidistributed in the horocycle. This motivates the following conjecture:",
   "public_statement": "Conjecture 20 is related to an equidistribution problem on a hyperbolic surface X = Γ \\H2, Γ a lattice. We assume that Γ ∩ { z 7 → z + a: a ∈ R} = {z 7 → z + a: a ∈ Z}.\n\nThen the curve {x + iy: x ∈ [0, 1] } corresponds to a closed horocycle {uy(t): 0 ≤ t ≤ 1} of length y−1 in in the unit tangent bundle T 1(X), and it is well-known that it becomes equidistributed in T 1(X) as y → 0+. Also, for irrational\n\nα, the sequence {uy(αn ): n ≥ 1} is equidistributed in the horocycle. This motivates the following conjecture:",
   "evidence": "The canonical input is a paragraph from Section 4.4 of the AIM workshop list *Open Problems from the Workshop “Emerging Applications of Measure Rigidity”*. It begins after Conjecture 20 of Rudnick and Sarnak and ends with “This motivates the following conjecture:”. The official PDF then starts Conjecture 21 in the next paragraph. Therefore this record is explanatory context, not an independent conjecture despite its inherited tag.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 96,
   "attempt": 1
  },
  "AIM-OTHER-0098": {
   "statement_status": "exact",
   "original_statement": "Conjecture 21 (J. Marklof, A. Str¨ ombergsson). Let f be a continuous func-tion on T 1(X) with given growth condition at the cusps (see [136] ). Then for \n\nα ∈ R which is badly approximable by rationals and 0 < c 1 < c 2,\n\n1\n\nM\n\n> M\n\n∑\n\n> m=1\n\nf (uy(αm )) →\n\n∫\n\n> T1(X)\n\nf dλ \n\nuniformly as M → ∞ and c1M −2 ≤ y ≤ c2M −2, where λ denotes the Liouville measure. \n\nIt was observed in [136] that",
   "clean_statement": "Conjecture 21 (J. Marklof, A. Str¨ ombergsson). Let f be a continuous func-tion on T 1(X) with given growth condition at the cusps (see [136] ). Then for\n\nα ∈ R which is badly approximable by rationals and 0 < c 1 < c 2,\n\n1\n\nM\n\n> M\n\n∑\n\n> m=1\n\nf (uy(αm )) →\n\n∫\n\n> T1(X)\n\nf dλ\n\nuniformly as M → ∞ and c1M −2 ≤ y ≤ c2M −2, where λ denotes the Liouville measure.\n\nIt was observed in [136] that",
   "public_statement": "Conjecture 21 (J. Marklof, A. Str¨ ombergsson). Let f be a continuous func-tion on T 1(X) with given growth condition at the cusps (see [136] ). Then for\n\nα ∈ R which is badly approximable by rationals and 0 < c 1 < c 2,\n\n1\n\nM\n\n> M\n\n∑\n\n> m=1\n\nf (uy(αm )) →\n\n∫\n\n> T1(X)\n\nf dλ\n\nuniformly as M → ∞ and c1M −2 ≤ y ≤ c2M −2, where λ denotes the Liouville measure.\n\nIt was observed in [136] that",
   "evidence": "The assigned record is Conjecture 21 from the AIM workshop list *Emerging applications of measure rigidity*. The PDF first fixes \\[ X=\\Gamma\\backslash\\mathbb H, \\qquad \\Gamma\\cap\\{z\\mapsto z+a:a\\in\\mathbb R\\} =\\{z\\mapsto z+a:a\\in\\mathbb Z\\}, \\] and denotes by \\(u_y(t)\\), \\(0\\leq t\\leq1\\), the resulting closed horocycle of length \\(y^{-1}\\) in \\(T^1X\\). With this notation the recovered conjecture is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 97,
   "attempt": 1
  },
  "AIM-OTHER-0099": {
   "statement_status": "exact",
   "original_statement": "Conjecture 20. Conjec-ture 21 was proved in [136] under the condition that c1M −ν ≤ y ≤ c2M −ν for some ν < 2. Furthermore, the statement of",
   "clean_statement": "Conjecture 20. Conjec-ture 21 was proved in [136] under the condition that c1M −ν ≤ y ≤ c2M −ν for some ν < 2. Furthermore, the statement of",
   "public_statement": "Conjecture 20. Conjec-ture 21 was proved in [136] under the condition that c1M −ν ≤ y ≤ c2M −ν for some ν < 2. Furthermore, the statement of",
   "evidence": "The exact canonical OCR record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 98,
   "attempt": 1
  },
  "AIM-OTHER-0100": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 21 holds for almost all \n\nα with respect to Lebesgue measure [136] for any positive ν, in particular for \n\nν = 2. Hence this gives a new proof of the main result in [164]. 4.5. Let M be a compact Riemannian manifold, and φt: M → M is an \n\nAnosov flow, that is, φt is a C1-flow and there exists a continuous invariant splitting \n\nT M = E0 ⊕ Es ⊕ Eu\n\nwhere E0 is the one-dimensional bundle tangent to the flow direction, and for some C, λ > 0, \n\n‖Dφ tv‖ ≤ Ce −λt ‖v‖, v ∈ Es, t ≥ 0; \n\n‖Dφ tv‖ ≥ C−1eλt ‖v‖, v ∈ Eu, t ≥ 0.\n\nThe distribution Es is tangent to the strong stable manifolds \n\nW ss (x) = {y ∈ M: d(φtx, φ ty) → 0 as t → +∞}.\n\nSuppose that the flow is topologically transitive. Then it is known (see [28]) that the foliation W ss (x), x ∈ X, is uniquely ergodic, i.e., there is a unique holonomy invariant transverse measure. OPEN PROBLEMS 10 \n\nIn addition to the above assumptions, we suppose that there exists a con-tinuous invariant splitting Es = Es \n\n> +\n\n+ Es \n\n> −\n\nsuch that for some C > 0 and \n\nμ+ > μ − > λ,\n\n‖Dφ tv‖ ≤ Ce −μ+t‖v‖, v ∈ Es\n\n> +, t ≥ 0; \n\n‖Dφ tv‖ ≥ C−1e−μ−t‖v‖, v ∈ Eu\n\n> −, t ≥ 0.\n\nA basic example of such splitting is the geodesic flow of CH 2. The distribution \n\nEs \n\n> +\n\nintegrates to the fast stable foliation W s\n\n> +.The following is a nonlinear analog of the Ragunathan's question about classifications of measures invariant under unipotent flows:",
   "clean_statement": null,
   "public_statement": "Conjecture 21 holds for almost all\n\nα with respect to Lebesgue measure [136] for any positive ν, in particular for\n\nν = 2. Hence this gives a new proof of the main result in [164]. 4.5. Let M be a compact Riemannian manifold, and φt: M → M is an\n\nAnosov flow, that is, φt is a C1-flow and there exists a continuous invariant splitting\n\nT M = E0 ⊕ Es ⊕ Eu\n\nwhere E0 is the one-dimensional bundle tangent to the flow direction, and for some C, λ > 0,\n\n‖Dφ tv‖ ≤ Ce −λt ‖v‖, v ∈ Es, t ≥ 0;\n\n‖Dφ tv‖ ≥ C−1eλt ‖v‖, v ∈ Eu, t ≥ 0.\n\nThe distribution Es is tangent to the strong stable manifolds\n\nW ss (x) = {y ∈ M: d(φtx, φ ty) → 0 as t → +∞}.\n\nSuppose that the flow is topologically transitive. Then it is known (see [28]) that the foliation W ss (x), x ∈ X, is uniquely ergodic, i.e., there is a unique holonomy invariant transverse measure. OPEN PROBLEMS 10\n\nIn addition to the above assumptions, we suppose that there exists a con-tinuous invariant splitting Es = Es\n\n> +\n\n+ Es\n\n> −\n\nsuch that for some C > 0 and\n\nμ+ > μ − > λ,\n\n‖Dφ tv‖ ≤ Ce −μ+t‖v‖, v ∈ Es\n\n> +, t ≥ 0;\n\n‖Dφ tv‖ ≥ C−1e−μ−t‖v‖, v ∈ Eu\n\n> −, t ≥ 0.\n\nA basic example of such splitting is the geodesic flow of CH 2. The distribution\n\nEs\n\n> +\n\nintegrates to the fast stable foliation W s\n\n> +.The following is a nonlinear analog of the Ragunathan's question about classifications of measures invariant under unipotent flows:",
   "evidence": "The canonical `problem` field is preserved verbatim in `input.json`. It is not one independent conjecture. Comparison with the official AIM PDF and the adjacent canonical records gives three distinct pieces:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 99,
   "attempt": 1
  },
  "AIM-OTHER-0101": {
   "statement_status": "exact",
   "original_statement": "Question 22 (F. Ledrappier). Describe the invariant ergodic measures for the fast stable foliation W s\n\n> +.\n\nOne may ask the same question for Anosov diffeomorphisms as well. 4.6. Let X be a Riemannian locally symmetric space of noncompact type and of finite volume. A flat in X is a totally geodesic submanifold of sectional curvature zero. Note that X = Γ \\G/K, where G is a connected semisimple real algebraic group, K is a maximal compact subgroups of G, and Γ is a lattices in G, and flats are Γ gAK, g ∈ G, for a Cartan subgroup A of G. It was shown (see [150]) that the number of compact flats with bounded volume is finite. Note that this number is related to the number of totally real number fields of fixed degree with bounded regulator (see [150]).",
   "clean_statement": "Question 22 (F. Ledrappier). Describe the invariant ergodic measures for the fast stable foliation W s\n\n> +.\n\nOne may ask the same question for Anosov diffeomorphisms as well. 4.6. Let X be a Riemannian locally symmetric space of noncompact type and of finite volume. A flat in X is a totally geodesic submanifold of sectional curvature zero. Note that X = Γ \\G/K, where G is a connected semisimple real algebraic group, K is a maximal compact subgroups of G, and Γ is a lattices in G, and flats are Γ gAK, g ∈ G, for a Cartan subgroup A of G. It was shown (see [150]) that the number of compact flats with bounded volume is finite. Note that this number is related to the number of totally real number fields of fixed degree with bounded regulator (see [150]).",
   "public_statement": "Question 22 (F. Ledrappier). Describe the invariant ergodic measures for the fast stable foliation W s\n\n> +.\n\nOne may ask the same question for Anosov diffeomorphisms as well. 4.6. Let X be a Riemannian locally symmetric space of noncompact type and of finite volume. A flat in X is a totally geodesic submanifold of sectional curvature zero. Note that X = Γ \\G/K, where G is a connected semisimple real algebraic group, K is a maximal compact subgroups of G, and Γ is a lattices in G, and flats are Γ gAK, g ∈ G, for a Cartan subgroup A of G. It was shown (see [150]) that the number of compact flats with bounded volume is finite. Note that this number is related to the number of totally real number fields of fixed degree with bounded regulator (see [150]).",
   "evidence": "The canonical record correctly begins with Question 22 but then absorbs the opening of the next section. Inspection of the official AIM PDF gives the source-verified question:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 100,
   "attempt": 1
  },
  "AIM-OTHER-0102": {
   "statement_status": "exact",
   "original_statement": "Question 23 (H. Oh). Determine the asymptotics of the number of compact flats with volume less than T as T → ∞.\n\nThis asymptotics and the rate of convergence has been determined for rank one spaces (see [128, 79, 67, 68, 205, 114, 156, 112]); however, the question about optimal rate of convergence is still open (see [88, 126, 125, 31]). When \n\nX is compact, using techniques developed in [183], one can determine the asymptotics of the sum ∑ \n\n> Fregular,systol( F)<T\n\nVol( F).\n\nHere a flat is called regular if its shortest closed geodesics goes in the regular direction and systol( F) denotes the length of this geodesics. See also [40] for another analog of the prime geodesic theorem for higher rank compact symmetric spaces. Another open question concerns the distribution of compact maximal flats on the unit tangent bundle T 1(X). Since the identity component G of the isometry group does not act transitively on X in the higher rank case, it is more convenient to consider the compact orbits of a Cartan subgroup A on the OPEN PROBLEMS 11 \n\nhomogeneous space Γ \\G, which we also call flats. Denote by μF the Lebesgue measure on a flat F ⊂ Γ\\G and by ¯ μF the normalized Lebesgue measure on \n\nF. Let \n\nνT =\n\n∑ \n\n> F: Vol( F)<T\n\nμF\n\n∑ \n\n> F: Vol( F)<T\n\nVol( F) and ¯νT =\n\n∑ \n\n> F: Vol( F)<T\n\n¯μF\n\n#{F: Vol( F) < T }.",
   "clean_statement": "Question 23 (H. Oh). Determine the asymptotics of the number of compact flats with volume less than T as T → ∞.\n\nThis asymptotics and the rate of convergence has been determined for rank one spaces (see [128, 79, 67, 68, 205, 114, 156, 112]); however, the question about optimal rate of convergence is still open (see [88, 126, 125, 31]). When\n\nX is compact, using techniques developed in [183], one can determine the asymptotics of the sum ∑\n\n> Fregular,systol( F)<T\n\nVol( F).\n\nHere a flat is called regular if its shortest closed geodesics goes in the regular direction and systol( F) denotes the length of this geodesics. See also [40] for another analog of the prime geodesic theorem for higher rank compact symmetric spaces. Another open question concerns the distribution of compact maximal flats on the unit tangent bundle T 1(X). Since the identity component G of the isometry group does not act transitively on X in the higher rank case, it is more convenient to consider the compact orbits of a Cartan subgroup A on the OPEN PROBLEMS 11\n\nhomogeneous space Γ \\G, which we also call flats. Denote by μF the Lebesgue measure on a flat F ⊂ Γ\\G and by ¯ μF the normalized Lebesgue measure on\n\nF. Let\n\nνT =\n\n∑\n\n> F: Vol( F)<T\n\nμF\n\n∑\n\n> F: Vol( F)<T\n\nVol( F) and ¯νT =\n\n∑\n\n> F: Vol( F)<T\n\n¯μF\n\n#{F: Vol( F) < T }.",
   "public_statement": "Question 23 (H. Oh). Determine the asymptotics of the number of compact flats with volume less than T as T → ∞.\n\nThis asymptotics and the rate of convergence has been determined for rank one spaces (see [128, 79, 67, 68, 205, 114, 156, 112]); however, the question about optimal rate of convergence is still open (see [88, 126, 125, 31]). When\n\nX is compact, using techniques developed in [183], one can determine the asymptotics of the sum ∑\n\n> Fregular,systol( F)<T\n\nVol( F).\n\nHere a flat is called regular if its shortest closed geodesics goes in the regular direction and systol( F) denotes the length of this geodesics. See also [40] for another analog of the prime geodesic theorem for higher rank compact symmetric spaces. Another open question concerns the distribution of compact maximal flats on the unit tangent bundle T 1(X). Since the identity component G of the isometry group does not act transitively on X in the higher rank case, it is more convenient to consider the compact orbits of a Cartan subgroup A on the OPEN PROBLEMS 11\n\nhomogeneous space Γ \\G, which we also call flats. Denote by μF the Lebesgue measure on a flat F ⊂ Γ\\G and by ¯ μF the normalized Lebesgue measure on\n\nF. Let\n\nνT =\n\n∑\n\n> F: Vol( F)<T\n\nμF\n\n∑\n\n> F: Vol( F)<T\n\nVol( F) and ¯νT =\n\n∑\n\n> F: Vol( F)<T\n\n¯μF\n\n#{F: Vol( F) < T }.",
   "evidence": "The official AIM workshop PDF, *Emerging applications of measure rigidity* (2004), Section 4.6, takes \\(X=\\Gamma\\backslash G/K\\), a finite-volume locally symmetric space of noncompact type. A flat is a totally geodesic zero-curvature submanifold; maximal flats arise from a Cartan subgroup \\(A\\). Oh had proved that only finitely many compact maximal flats have bounded volume.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 101,
   "attempt": 1
  },
  "AIM-OTHER-0103": {
   "statement_status": "exact",
   "original_statement": "Question 24 (H. Oh). Do the measures νT and ¯νT converge to the normalized Haar measure on Γ\\G?\n\nAccording to [113], the normalized Haar measure on Γ \\G is the unique ergodic measure of maximal entropy for the geodesic flow. Thus, to resolve",
   "clean_statement": "Question 24 (H. Oh). Do the measures νT and ¯νT converge to the normalized Haar measure on Γ\\G?\n\nAccording to [113], the normalized Haar measure on Γ \\G is the unique ergodic measure of maximal entropy for the geodesic flow. Thus, to resolve",
   "public_statement": "Question 24 (H. Oh). Do the measures νT and ¯νT converge to the normalized Haar measure on Γ\\G?\n\nAccording to [113], the normalized Haar measure on Γ \\G is the unique ergodic measure of maximal entropy for the geodesic flow. Thus, to resolve",
   "evidence": "The exact assigned field is preserved in `input.json`. It reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 102,
   "attempt": 1
  },
  "AIM-OTHER-0104": {
   "statement_status": "exact",
   "original_statement": "Question 24, it suffices to estimate the entropy of the weak ∗ limit points of νT\n\nand ¯ νT and show that they do not escape to infinity. For the rank one groups,",
   "clean_statement": "Question 24, it suffices to estimate the entropy of the weak ∗ limit points of νT\n\nand ¯ νT and show that they do not escape to infinity. For the rank one groups,",
   "public_statement": "Question 24, it suffices to estimate the entropy of the weak ∗ limit points of νT\n\nand ¯ νT and show that they do not escape to infinity. For the rank one groups,",
   "evidence": "The exact input is a visibly split fragment:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 103,
   "attempt": 1
  },
  "AIM-OTHER-0105": {
   "statement_status": "exact",
   "original_statement": "Question 24 has been answered positively (see [26, 206, 155, 168, 104]). In higher rank case, the equidistribution of ε-seperated closed geodesics from different homotopy classes was established in [113]. As in [113], one can also prove analog of the",
   "clean_statement": "Question 24 has been answered positively (see [26, 206, 155, 168, 104]). In higher rank case, the equidistribution of ε-seperated closed geodesics from different homotopy classes was established in [113]. As in [113], one can also prove analog of the",
   "public_statement": "Question 24 has been answered positively (see [26, 206, 155, 168, 104]). In higher rank case, the equidistribution of ε-seperated closed geodesics from different homotopy classes was established in [113]. As in [113], one can also prove analog of the",
   "evidence": "The exact database text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 104,
   "attempt": 1
  },
  "AIM-OTHER-0106": {
   "statement_status": "exact",
   "original_statement": "Conjecture 24 for compact Γ \\G and the measure νT as above with summation taken over regular flats F such that systol( F ) < T. Another equidistribution result was obtained in a recent work of Y. Benoist and H. Oh, where the averages along Hecke orbits of maximal compact flats were considered. 5. Diophantine analysis \n\n5.1. A vector y = ( y1,..., y n) ∈ Rn is called v-approximable (for v > 0) if there are infinitely many q = ( q1,..., q n) ∈ Zn and p ∈ Z such that \n\n|y1q1 + · · · + ynqn − p| < ‖q‖−v.\n\nHere ‖ · ‖ denotes the max-norm on Rn. If a vector y ∈ Rn is ( n + ε)-approximable for some ε > 0, it is called very well approximable (VWA). An easy argument (using Borel-Cantelli lemma) implies that the set of VWA vectors has Lebesgue measure zero in Rn. Therefore, it is natural to expect that a generic point on a \"nondegenerate\" submanifold of Rn is not VWA. This is the Sprindˇ zuk conjecture proved in [109] (see also [17]). Similarly, an m × n real matrix A is called VWA if for some ε > 0 there are infinitely many q ∈ Zn and p ∈ Zm such that \n\n‖Aq − p‖m < ‖q‖−n−ε.\n\nIt is easy to see that the set of VWA matrices has measure zero in Rm×n, and one hopes that an analog of the Sprindˇ zuk conjecture holds in this set-up as well. The definition of nondegenerate submanifold of Rn in [109], which is well-suited for the case of vectors, is a manifold with smooth coordinate charts \n\nf: U (⊂ Rk) → Rn such that the spaces spanned by the partial derivatives f at OPEN PROBLEMS 12 \n\npoints of U have dimension n. It is not quite clear what is the right definition of \"nondegenerate\" submanifold for the case of matrices.",
   "clean_statement": "Conjecture 24 for compact Γ \\G and the measure νT as above with summation taken over regular flats F such that systol( F ) < T. Another equidistribution result was obtained in a recent work of Y. Benoist and H. Oh, where the averages along Hecke orbits of maximal compact flats were considered. 5. Diophantine analysis\n\n5.1. A vector y = ( y1,..., y n) ∈ Rn is called v-approximable (for v > 0) if there are infinitely many q = ( q1,..., q n) ∈ Zn and p ∈ Z such that\n\n|y1q1 + · · · + ynqn − p| < ‖q‖−v.\n\nHere ‖ · ‖ denotes the max-norm on Rn. If a vector y ∈ Rn is ( n + ε)-approximable for some ε > 0, it is called very well approximable (VWA). An easy argument (using Borel-Cantelli lemma) implies that the set of VWA vectors has Lebesgue measure zero in Rn. Therefore, it is natural to expect that a generic point on a \"nondegenerate\" submanifold of Rn is not VWA. This is the Sprindˇ zuk conjecture proved in [109] (see also [17]). Similarly, an m × n real matrix A is called VWA if for some ε > 0 there are infinitely many q ∈ Zn and p ∈ Zm such that\n\n‖Aq − p‖m < ‖q‖−n−ε.\n\nIt is easy to see that the set of VWA matrices has measure zero in Rm×n, and one hopes that an analog of the Sprindˇ zuk conjecture holds in this set-up as well. The definition of nondegenerate submanifold of Rn in [109], which is well-suited for the case of vectors, is a manifold with smooth coordinate charts\n\nf: U (⊂ Rk) → Rn such that the spaces spanned by the partial derivatives f at OPEN PROBLEMS 12\n\npoints of U have dimension n. It is not quite clear what is the right definition of \"nondegenerate\" submanifold for the case of matrices.",
   "public_statement": "Conjecture 24 for compact Γ \\G and the measure νT as above with summation taken over regular flats F such that systol( F ) < T. Another equidistribution result was obtained in a recent work of Y. Benoist and H. Oh, where the averages along Hecke orbits of maximal compact flats were considered. 5. Diophantine analysis\n\n5.1. A vector y = ( y1,..., y n) ∈ Rn is called v-approximable (for v > 0) if there are infinitely many q = ( q1,..., q n) ∈ Zn and p ∈ Z such that\n\n|y1q1 + · · · + ynqn − p| < ‖q‖−v.\n\nHere ‖ · ‖ denotes the max-norm on Rn. If a vector y ∈ Rn is ( n + ε)-approximable for some ε > 0, it is called very well approximable (VWA). An easy argument (using Borel-Cantelli lemma) implies that the set of VWA vectors has Lebesgue measure zero in Rn. Therefore, it is natural to expect that a generic point on a \"nondegenerate\" submanifold of Rn is not VWA. This is the Sprindˇ zuk conjecture proved in [109] (see also [17]). Similarly, an m × n real matrix A is called VWA if for some ε > 0 there are infinitely many q ∈ Zn and p ∈ Zm such that\n\n‖Aq − p‖m < ‖q‖−n−ε.\n\nIt is easy to see that the set of VWA matrices has measure zero in Rm×n, and one hopes that an analog of the Sprindˇ zuk conjecture holds in this set-up as well. The definition of nondegenerate submanifold of Rn in [109], which is well-suited for the case of vectors, is a manifold with smooth coordinate charts\n\nf: U (⊂ Rk) → Rn such that the spaces spanned by the partial derivatives f at OPEN PROBLEMS 12\n\npoints of U have dimension n. It is not quite clear what is the right definition of \"nondegenerate\" submanifold for the case of matrices.",
   "evidence": "Assigned metadata: `id` AIM-OTHER-0106; `source_file` `aim-other-notes.json`; zero-based `source_index` 105; `attempt` 1; source URL https://aimath.org/WWN/measrigid/measrigid.pdf. The exact source object is preserved verbatim in `input.json`. Its exact `problem` field is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 105,
   "attempt": 1
  },
  "AIM-OTHER-0107": {
   "statement_status": "exact",
   "original_statement": "Question 25 (D. Kleinbock, G. Margulis (see [109], Sec. 6.2)). Find rea-sonable and checkable conditions for a smooth map f: U (⊂ Rk) → Mm×n(R)\n\nwhich generalizes nondegeneracy of vector-valued maps and implies that almost every point of f (U ) is not VWA. \n\nSuch conditions were obtained in some cases in [115, 116, 105]. 5.2. A far-reaching generalization of the Sprindˇ zuk conjecture was suggested in [106]. Let μ be a measure on Rk and and f: supp( μ) → Rn that sat-isfy some reasonable conditions. What are the Diophantine properties of the generic points in Rn with respect to the measure f∗μ? Several results in this direction were obtained in [198, 107, 111] for locally finite measure. It would be interesting to consider the case of Hausdorff measures:",
   "clean_statement": "Question 25 (D. Kleinbock, G. Margulis (see [109], Sec. 6.2)). Find rea-sonable and checkable conditions for a smooth map f: U (⊂ Rk) → Mm×n(R)\n\nwhich generalizes nondegeneracy of vector-valued maps and implies that almost every point of f (U ) is not VWA.\n\nSuch conditions were obtained in some cases in [115, 116, 105]. 5.2. A far-reaching generalization of the Sprindˇ zuk conjecture was suggested in [106]. Let μ be a measure on Rk and and f: supp( μ) → Rn that sat-isfy some reasonable conditions. What are the Diophantine properties of the generic points in Rn with respect to the measure f∗μ? Several results in this direction were obtained in [198, 107, 111] for locally finite measure. It would be interesting to consider the case of Hausdorff measures:",
   "public_statement": "Question 25 (D. Kleinbock, G. Margulis (see [109], Sec. 6.2)). Find rea-sonable and checkable conditions for a smooth map f: U (⊂ Rk) → Mm×n(R)\n\nwhich generalizes nondegeneracy of vector-valued maps and implies that almost every point of f (U ) is not VWA.\n\nSuch conditions were obtained in some cases in [115, 116, 105]. 5.2. A far-reaching generalization of the Sprindˇ zuk conjecture was suggested in [106]. Let μ be a measure on Rk and and f: supp( μ) → Rn that sat-isfy some reasonable conditions. What are the Diophantine properties of the generic points in Rn with respect to the measure f∗μ? Several results in this direction were obtained in [198, 107, 111] for locally finite measure. It would be interesting to consider the case of Hausdorff measures:",
   "evidence": "The exact database field is preserved in `input.json`. It contains both Question 25 and the beginning of the next subsection. Its OCR includes “rea-sonable,” `Mm×n(R)`, `f (U )`, “Sprindˇ zuk,” a duplicated “and,” “sat-isfy,” and `f∗μ`.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 106,
   "attempt": 1
  },
  "AIM-OTHER-0108": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 26 (D. Kleinbock). Give estimates on Hausdorff dimension of v-approximable vectors in a nondegenerate submanifold of Rn using dynamics. \n\nSee [41] for a discussion of what is currently known about the Hausdorff dimension and for a related result. 5.3. For α ∈ R, let 〈α〉 = dist( α, Z). It is not hard to show that the set of (α, β ) ∈ R2 such that lim inf \n\n> q→∞\n\nq(log q)2+ ε 〈qα 〉 〈 qβ 〉 > 0for every ε > 0 has full Lebesgue measure. In fact, it was shown in [184] that lim \n\n> q→∞\n\nq(log q)2+ ε 〈qα 〉 〈 qβ 〉 = ∞\n\non a set of ( α, β ) ∈ R2 of full measure. On the other hand, the following question remains open:",
   "clean_statement": null,
   "public_statement": "Question 26 (D. Kleinbock). Give estimates on Hausdorff dimension of v-approximable vectors in a nondegenerate submanifold of Rn using dynamics.\n\nSee [41] for a discussion of what is currently known about the Hausdorff dimension and for a related result. 5.3. For α ∈ R, let 〈α〉 = dist( α, Z). It is not hard to show that the set of (α, β ) ∈ R2 such that lim inf\n\n> q→∞\n\nq(log q)2+ ε 〈qα 〉 〈 qβ 〉 > 0for every ε > 0 has full Lebesgue measure. In fact, it was shown in [184] that lim\n\n> q→∞\n\nq(log q)2+ ε 〈qα 〉 〈 qβ 〉 = ∞\n\non a set of ( α, β ) ∈ R2 of full measure. On the other hand, the following question remains open:",
   "evidence": "The canonical record is zero-based record 107 of aim-other-notes.json, from the June 2004 AIM workshop *Emerging applications of measure rigidity*. The source PDF identifies the record as follows (notation repaired but wording unchanged):",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 107,
   "attempt": 1
  },
  "AIM-OTHER-0109": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 27 (A. Pollington). Are there α, β ∈ R such that for every ε > 0,\n\nlim inf \n\n> q→∞\n\nq(log q)2−ε 〈qα 〉 〈 qβ 〉 > 0? It follows from [65] that lim inf \n\n> q→∞\n\nq(log q)2 〈qα 〉 〈 qβ 〉 = 0 for almost all ( α, β ). Thus, the set of ( α, β ) in",
   "clean_statement": null,
   "public_statement": "Question 27 (A. Pollington). Are there α, β ∈ R such that for every ε > 0,\n\nlim inf\n\n> q→∞\n\nq(log q)2−ε 〈qα 〉 〈 qβ 〉 > 0? It follows from [65] that lim inf\n\n> q→∞\n\nq(log q)2 〈qα 〉 〈 qβ 〉 = 0 for almost all ( α, β ). Thus, the set of ( α, β ) in",
   "evidence": "The exact assigned OCR record is preserved in input.json. It stops in the middle of the sentence “Thus, the set of \\((\\alpha,\\beta)\\) in”. The official AIM PDF gives the following unambiguous mathematical text:",
   "classification_method": "invalid_literal_with_repaired_reading",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 108,
   "attempt": 1
  },
  "AIM-OTHER-0110": {
   "statement_status": "exact",
   "original_statement": "Question 27 is related to the well-known conjecture of Littlewood:",
   "clean_statement": "Question 27 is related to the well-known conjecture of Littlewood:",
   "public_statement": "Question 27 is related to the well-known conjecture of Littlewood:",
   "evidence": "The exact canonical record is the sentence",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 109,
   "attempt": 1
  },
  "AIM-OTHER-0111": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 28 (Littlewood). For any α, β ∈ R,\n\nlim inf \n\n> q→∞\n\nq 〈qα 〉 〈 qβ 〉 = 0.OPEN PROBLEMS 13 \n\nThe best result on",
   "clean_statement": null,
   "public_statement": "Conjecture 28 (Littlewood). For any α, β ∈ R,\n\nlim inf\n\n> q→∞\n\nq 〈qα 〉 〈 qβ 〉 = 0.OPEN PROBLEMS 13\n\nThe best result on",
   "evidence": "The canonical JSON record is visibly truncated and contaminated by a page transition:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 110,
   "attempt": 1
  },
  "AIM-OTHER-0112": {
   "statement_status": "exact",
   "original_statement": "Conjecture 28 is [47], which shows that the set of ex-ceptions ( α, β ) for the Littlewood conjecture is a countable union of sets of box dimension zero. The proof in [47] uses dynamics on the homogeneous space SL(3, R)/SL(3, Z). It was observed some time ago that",
   "clean_statement": "Conjecture 28 is [47], which shows that the set of ex-ceptions ( α, β ) for the Littlewood conjecture is a countable union of sets of box dimension zero. The proof in [47] uses dynamics on the homogeneous space SL(3, R)/SL(3, Z). It was observed some time ago that",
   "public_statement": "Conjecture 28 is [47], which shows that the set of ex-ceptions ( α, β ) for the Littlewood conjecture is a countable union of sets of box dimension zero. The proof in [47] uses dynamics on the homogeneous space SL(3, R)/SL(3, Z). It was observed some time ago that",
   "evidence": "The exact assigned OCR record is preserved in input.json:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 111,
   "attempt": 1
  },
  "AIM-OTHER-0113": {
   "statement_status": "exact",
   "original_statement": "Conjecture 28 is implied by the following conjecture:",
   "clean_statement": "Conjecture 28 is implied by the following conjecture:",
   "public_statement": "Conjecture 28 is implied by the following conjecture:",
   "evidence": "The canonical record is not itself a mathematical conjecture. Its complete text is the bridge sentence",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 112,
   "attempt": 1
  },
  "AIM-OTHER-0114": {
   "statement_status": "corrected_verified",
   "original_statement": "Conjecture 29 (G. Margulis [130]). Let A be the group of all diagonal ma-trices in SL(3, R). Then every bounded A-orbit in SL(3, R)/SL(3, Z) is closed.",
   "clean_statement": "**Conjecture 29 (G. Margulis).** Let \\(A\\) be the group of all diagonal matrices in \\(\\mathrm{SL}(3,\\mathbb R)\\). Then every bounded \\(A\\)-orbit in\n\\(\\mathrm{SL}(3,\\mathbb R)/\\mathrm{SL}(3,\\mathbb Z)\\) is closed.",
   "public_statement": "**Conjecture 29 (G. Margulis).** Let \\(A\\) be the group of all diagonal matrices in \\(\\mathrm{SL}(3,\\mathbb R)\\). Then every bounded \\(A\\)-orbit in\n\\(\\mathrm{SL}(3,\\mathbb R)/\\mathrm{SL}(3,\\mathbb Z)\\) is closed.",
   "evidence": "Inspection of the official AIM PDF confirms that the only corruption is the line-break hyphen in “ma-trices.” The recovered statement is:",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-other-notes.json",
   "source_index": 113,
   "attempt": 1
  },
  "AIM-OTHER-0115": {
   "statement_status": "exact",
   "original_statement": "Conjecture 29 is a very special case of the general conjecture describing closed invariant subsets for actions of Cartan subgroups on general homoge-neous spaces (see [130]). G. Margulis suggested the following conjecture, which might be easier to handle than",
   "clean_statement": "Conjecture 29 is a very special case of the general conjecture describing closed invariant subsets for actions of Cartan subgroups on general homoge-neous spaces (see [130]). G. Margulis suggested the following conjecture, which might be easier to handle than",
   "public_statement": "Conjecture 29 is a very special case of the general conjecture describing closed invariant subsets for actions of Cartan subgroups on general homoge-neous spaces (see [130]). G. Margulis suggested the following conjecture, which might be easier to handle than",
   "evidence": "The exact assigned OCR record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 114,
   "attempt": 1
  },
  "AIM-OTHER-0116": {
   "statement_status": "exact",
   "original_statement": "Conjecture 30 (G. Margulis). For every compact set K of SL(3, R)/SL(3, Z),there are only finitely many closed A-orbits contained in K.\n\nThis conjecture can be reformulated in terms of the Markov spectrum of forms \n\nF (x) = \n\n> 3\n\n∏\n\n> i=1\n\n( 3∑\n\n> j=1\n\naij xj\n\n), aij ∈ R.\n\nLet ∆( F ) = det( aij ) and m(F ) = inf \n\n{∣ ∣∣∣\n\nF (x)\n\n∆( F )\n\n∣∣∣∣: x ∈ Z3 − 0\n\n}.\n\nThen",
   "clean_statement": "Conjecture 30 (G. Margulis). For every compact set K of SL(3, R)/SL(3, Z),there are only finitely many closed A-orbits contained in K.\n\nThis conjecture can be reformulated in terms of the Markov spectrum of forms\n\nF (x) =\n\n> 3\n\n∏\n\n> i=1\n\n( 3∑\n\n> j=1\n\naij xj\n\n), aij ∈ R.\n\nLet ∆( F ) = det( aij ) and m(F ) = inf\n\n{∣ ∣∣∣\n\nF (x)\n\n∆( F )\n\n∣∣∣∣: x ∈ Z3 − 0\n\n}.\n\nThen",
   "public_statement": "Conjecture 30 (G. Margulis). For every compact set K of SL(3, R)/SL(3, Z),there are only finitely many closed A-orbits contained in K.\n\nThis conjecture can be reformulated in terms of the Markov spectrum of forms\n\nF (x) =\n\n> 3\n\n∏\n\n> i=1\n\n( 3∑\n\n> j=1\n\naij xj\n\n), aij ∈ R.\n\nLet ∆( F ) = det( aij ) and m(F ) = inf\n\n{∣ ∣∣∣\n\nF (x)\n\n∆( F )\n\n∣∣∣∣: x ∈ Z3 − 0\n\n}.\n\nThen",
   "evidence": "The assigned record is Conjecture 30 from the American Institute of Mathematics workshop *Emerging Applications of Measure Rigidity* (June 2004; scribe A. Gorodnik, document dated November 1, 2004). Its mathematical statement is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 115,
   "attempt": 1
  },
  "AIM-OTHER-0117": {
   "statement_status": "exact",
   "original_statement": "Question 30 is equivalent to the following question:",
   "clean_statement": "Question 30 is equivalent to the following question:",
   "public_statement": "Question 30 is equivalent to the following question:",
   "evidence": "The exact canonical record is the sentence fragment",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 116,
   "attempt": 1
  },
  "AIM-OTHER-0118": {
   "statement_status": "exact",
   "original_statement": "Question 31 (G. Margulis). Show that for every ε > 0, the set [ε, ∞)∩{ m(F )}\n\nis finite. \n\n5.4. For 0 ≤ s ≤ 1, define \n\nCs =\n\n{\n\n(α, β ) ∈ R2: inf \n\n> q≥1\n\nmax {qs 〈qα 〉, q 1−s 〈qβ 〉} > 0\n\n}.\n\nIn particular, C1/2 is the set of badly approximable vectors. Since",
   "clean_statement": "Question 31 (G. Margulis). Show that for every ε > 0, the set [ε, ∞)∩{ m(F )}\n\nis finite.\n\n5.4. For 0 ≤ s ≤ 1, define\n\nCs =\n\n{\n\n(α, β ) ∈ R2: inf\n\n> q≥1\n\nmax {qs 〈qα 〉, q 1−s 〈qβ 〉} > 0\n\n}.\n\nIn particular, C1/2 is the set of badly approximable vectors. Since",
   "public_statement": "Question 31 (G. Margulis). Show that for every ε > 0, the set [ε, ∞)∩{ m(F )}\n\nis finite.\n\n5.4. For 0 ≤ s ≤ 1, define\n\nCs =\n\n{\n\n(α, β ) ∈ R2: inf\n\n> q≥1\n\nmax {qs 〈qα 〉, q 1−s 〈qβ 〉} > 0\n\n}.\n\nIn particular, C1/2 is the set of badly approximable vectors. Since",
   "evidence": "The canonical OCR record begins",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 117,
   "attempt": 1
  },
  "AIM-OTHER-0119": {
   "statement_status": "exact",
   "original_statement": "Conjecture 28 holds for all ( α, β ) /∈ C s, one may naively hope to prove it by showing that intersection of the sets Cs, 0 ≤ s ≤ 1, is empty. In this regard, we mention the following conjecture:",
   "clean_statement": "Conjecture 28 holds for all ( α, β ) /∈ C s, one may naively hope to prove it by showing that intersection of the sets Cs, 0 ≤ s ≤ 1, is empty. In this regard, we mention the following conjecture:",
   "public_statement": "Conjecture 28 holds for all ( α, β ) /∈ C s, one may naively hope to prove it by showing that intersection of the sets Cs, 0 ≤ s ≤ 1, is empty. In this regard, we mention the following conjecture:",
   "evidence": "The exact canonical record is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 118,
   "attempt": 1
  },
  "AIM-OTHER-0120": {
   "statement_status": "exact",
   "original_statement": "Conjecture 32 (W. Schmidt [170]). For any s, t ∈ [0, 1], we have Cs ∩ C t 6 = ∅.\n\nNote that W. Schmidt stated this conjecture in [170] only for s = 1 /3 and \n\nt = 2 /3. It is known that each of the sets Cs has zero measure and full Hausdorff dimension. It was shown that the set Cs ∩ C 0 ∩ C 1 has full Hausdorff dimension as well (see [157]).",
   "clean_statement": "Conjecture 32 (W. Schmidt [170]). For any s, t ∈ [0, 1], we have Cs ∩ C t 6 = ∅.\n\nNote that W. Schmidt stated this conjecture in [170] only for s = 1 /3 and\n\nt = 2 /3. It is known that each of the sets Cs has zero measure and full Hausdorff dimension. It was shown that the set Cs ∩ C 0 ∩ C 1 has full Hausdorff dimension as well (see [157]).",
   "public_statement": "Conjecture 32 (W. Schmidt [170]). For any s, t ∈ [0, 1], we have Cs ∩ C t 6 = ∅.\n\nNote that W. Schmidt stated this conjecture in [170] only for s = 1 /3 and\n\nt = 2 /3. It is known that each of the sets Cs has zero measure and full Hausdorff dimension. It was shown that the set Cs ∩ C 0 ∩ C 1 has full Hausdorff dimension as well (see [157]).",
   "evidence": "The official AIM PDF defines, for \\(0\\le u\\le1\\), \\[ C_u=\\left\\{(\\alpha,\\beta)\\in\\mathbb R^2: \\inf_{q\\ge1}\\max\\left\\{q^u\\langle q\\alpha\\rangle, q^{1-u}\\langle q\\beta\\rangle\\right\\}>0\\right\\}. \\] In this Diophantine-approximation context, \\(\\langle x\\rangle\\) denotes distance to the nearest integer. Below it is written in the now-standard notation \\[ \\|x\\|:=\\min_{p\\in\\mathbb Z}|x-p|. \\]",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 119,
   "attempt": 1
  },
  "AIM-OTHER-0121": {
   "statement_status": "exact",
   "original_statement": "Conjecture 32 is related to the following conjecture: OPEN PROBLEMS 14",
   "clean_statement": "Conjecture 32 is related to the following conjecture: OPEN PROBLEMS 14",
   "public_statement": "Conjecture 32 is related to the following conjecture: OPEN PROBLEMS 14",
   "evidence": "The exact extracted record is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 120,
   "attempt": 1
  },
  "AIM-OTHER-0122": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 33. Let A be the group of all diagonal matrices in SL(3, R), and \n\nA1, A 2 ⊂ A are rays in A. Then there exists x ∈ SL(3, R)/SL(3, Z) such that \n\nA1x and A2x are bounded, but Ax is not bounded in SL(3, R)/SL(3, Z).\n\nNote that for rays A1 and A2 which lie in the cone \n\n{diag( eu, e v, e −u−v): u, v ≥ 0} ⊂ A,",
   "clean_statement": null,
   "public_statement": "Conjecture 33. Let A be the group of all diagonal matrices in SL(3, R), and\n\nA1, A 2 ⊂ A are rays in A. Then there exists x ∈ SL(3, R)/SL(3, Z) such that\n\nA1x and A2x are bounded, but Ax is not bounded in SL(3, R)/SL(3, Z).\n\nNote that for rays A1 and A2 which lie in the cone\n\n{diag( eu, e v, e −u−v): u, v ≥ 0} ⊂ A,",
   "evidence": "The canonical record is an OCR extraction of Conjecture 33 from the AIM list *Emerging applications of measure rigidity*. It ends in the middle of the sentence after the displayed cone. The official AIM PDF gives the following statement (notation normalized only by adding superscripts to the exponentials):",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 121,
   "attempt": 1
  },
  "AIM-OTHER-0123": {
   "statement_status": "exact",
   "original_statement": "Conjecture 33 follows from Conjectures 32. On the other hand, it was pointed out by D. Kleinbock that in the case when A1 and A2 lie in the opposite Weyl chambers,",
   "clean_statement": "Conjecture 33 follows from Conjectures 32. On the other hand, it was pointed out by D. Kleinbock that in the case when A1 and A2 lie in the opposite Weyl chambers,",
   "public_statement": "Conjecture 33 follows from Conjectures 32. On the other hand, it was pointed out by D. Kleinbock that in the case when A1 and A2 lie in the opposite Weyl chambers,",
   "evidence": "The canonical record is the fragment",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 122,
   "attempt": 1
  },
  "AIM-OTHER-0124": {
   "statement_status": "exact",
   "original_statement": "Conjecture 33 can be proved using the argument from [108], and moreover, the set of x which satisfy",
   "clean_statement": "Conjecture 33 can be proved using the argument from [108], and moreover, the set of x which satisfy",
   "public_statement": "Conjecture 33 can be proved using the argument from [108], and moreover, the set of x which satisfy",
   "evidence": "The exact canonical record assigned here is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 123,
   "attempt": 1
  },
  "AIM-OTHER-0125": {
   "statement_status": "exact",
   "original_statement": "Conjecture 33 has full Hausdorff dimension. 5.5. Let Q be a nondegenerate positive definite quadratic form of dimension \n\nd ≥ 3.",
   "clean_statement": "Conjecture 33 has full Hausdorff dimension. 5.5. Let Q be a nondegenerate positive definite quadratic form of dimension\n\nd ≥ 3.",
   "public_statement": "Conjecture 33 has full Hausdorff dimension. 5.5. Let Q be a nondegenerate positive definite quadratic form of dimension\n\nd ≥ 3.",
   "evidence": "The exact canonical input is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 124,
   "attempt": 1
  },
  "AIM-OTHER-0126": {
   "statement_status": "exact",
   "original_statement": "Conjecture 34 (Davenport-Lewis). Suppose that Q is not a multiple of a rational form. Then the gaps between consecutive elements of the set {Q(x): \n\nx ∈ Zd} go to zero as Q(x) → ∞.",
   "clean_statement": "Conjecture 34 (Davenport-Lewis). Suppose that Q is not a multiple of a rational form. Then the gaps between consecutive elements of the set {Q(x):\n\nx ∈ Zd} go to zero as Q(x) → ∞.",
   "public_statement": "Conjecture 34 (Davenport-Lewis). Suppose that Q is not a multiple of a rational form. Then the gaps between consecutive elements of the set {Q(x):\n\nx ∈ Zd} go to zero as Q(x) → ∞.",
   "evidence": "The extracted record says:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 125,
   "attempt": 1
  },
  "AIM-OTHER-0127": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 34 was proved in [15] for d ≥ 9, and recently the method in [15] was extended to d ≥ 5 as well. The case d = 3, 4 is still open. When Q is a nondegenerate indefinite definite quadratic form of dimension \n\nd ≥ 3 which is not a multiple of a rational quadratic form, the set {Q(x): x ∈\n\nZd} is dense in R. This is the Oppenheim conjecture proved by Margulis in [129]. However, the proof in [129] is not effective.",
   "clean_statement": null,
   "public_statement": "Conjecture 34 was proved in [15] for d ≥ 9, and recently the method in [15] was extended to d ≥ 5 as well. The case d = 3, 4 is still open. When Q is a nondegenerate indefinite definite quadratic form of dimension\n\nd ≥ 3 which is not a multiple of a rational quadratic form, the set {Q(x): x ∈\n\nZd} is dense in R. This is the Oppenheim conjecture proved by Margulis in [129]. However, the proof in [129] is not effective.",
   "evidence": "This record is from the AIM workshop list *Emerging applications of measure rigidity*, item 34. The exact extracted text is:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 126,
   "attempt": 1
  },
  "AIM-OTHER-0128": {
   "statement_status": "exact",
   "original_statement": "Question 35 (G. Margulis [130]). Give an effective estimate on T = T (ε)\n\nsuch that there exists x ∈ Zd with \n\n0 < |Q(x)| < ε and ‖x‖ < T. \n\nThis question is especially difficult since the estimate on T should depend on the Diophantine properties of coefficients of the quadratic form Q. An easier question with x satisfying conditions \n\n|Q(x)| < ε and ‖x‖ < T \n\nis treated for d ≥ 5 in an upcoming work of G. Margulis and F. G¨ otze. 5.6. Let Q(x) = ax 21 + bx 1x2 + cx 22 be a nondegenerate indefinite quadratic form with rational coefficients that does not represent zero over Q. For x ∈ R2,define \n\nm(Q, x ) = inf \n\n> z∈Z2\n\n|Q(x + z)| and m(Q) = sup \n\n> x∈R2\n\nm(Q, x ).\n\nIf the supremum m(Q) is isolated, we also define \n\nm2(Q) = sup {m(Q, x ): x ∈ R2, m (Q, x ) < m (Q)}.OPEN PROBLEMS 15 \n\nThe interest in the quantity m(Q) was motivated by the study of existence of a Euclidean algorithm in quadratic fields Q(√m), m > 0. If Q represents the norm of Q(√m) computed with respect to an integral basis, then Euclidean algorithm exists iff m(Q) < 1. The following conjecture was communicated by A. Pollington:",
   "clean_statement": "Question 35 (G. Margulis [130]). Give an effective estimate on T = T (ε)\n\nsuch that there exists x ∈ Zd with\n\n0 < |Q(x)| < ε and ‖x‖ < T.\n\nThis question is especially difficult since the estimate on T should depend on the Diophantine properties of coefficients of the quadratic form Q. An easier question with x satisfying conditions\n\n|Q(x)| < ε and ‖x‖ < T\n\nis treated for d ≥ 5 in an upcoming work of G. Margulis and F. G¨ otze. 5.6. Let Q(x) = ax 21 + bx 1x2 + cx 22 be a nondegenerate indefinite quadratic form with rational coefficients that does not represent zero over Q. For x ∈ R2,define\n\nm(Q, x ) = inf\n\n> z∈Z2\n\n|Q(x + z)| and m(Q) = sup\n\n> x∈R2\n\nm(Q, x ).\n\nIf the supremum m(Q) is isolated, we also define\n\nm2(Q) = sup {m(Q, x ): x ∈ R2, m (Q, x ) < m (Q)}.OPEN PROBLEMS 15\n\nThe interest in the quantity m(Q) was motivated by the study of existence of a Euclidean algorithm in quadratic fields Q(√m), m > 0. If Q represents the norm of Q(√m) computed with respect to an integral basis, then Euclidean algorithm exists iff m(Q) < 1. The following conjecture was communicated by A. Pollington:",
   "public_statement": "Question 35 (G. Margulis [130]). Give an effective estimate on T = T (ε)\n\nsuch that there exists x ∈ Zd with\n\n0 < |Q(x)| < ε and ‖x‖ < T.\n\nThis question is especially difficult since the estimate on T should depend on the Diophantine properties of coefficients of the quadratic form Q. An easier question with x satisfying conditions\n\n|Q(x)| < ε and ‖x‖ < T\n\nis treated for d ≥ 5 in an upcoming work of G. Margulis and F. G¨ otze. 5.6. Let Q(x) = ax 21 + bx 1x2 + cx 22 be a nondegenerate indefinite quadratic form with rational coefficients that does not represent zero over Q. For x ∈ R2,define\n\nm(Q, x ) = inf\n\n> z∈Z2\n\n|Q(x + z)| and m(Q) = sup\n\n> x∈R2\n\nm(Q, x ).\n\nIf the supremum m(Q) is isolated, we also define\n\nm2(Q) = sup {m(Q, x ): x ∈ R2, m (Q, x ) < m (Q)}.OPEN PROBLEMS 15\n\nThe interest in the quantity m(Q) was motivated by the study of existence of a Euclidean algorithm in quadratic fields Q(√m), m > 0. If Q represents the norm of Q(√m) computed with respect to an integral basis, then Euclidean algorithm exists iff m(Q) < 1. The following conjecture was communicated by A. Pollington:",
   "evidence": "The canonical record is extracted from Section 5.5 of the 2004 AIM list *Open Problems from the Workshop “Emerging Applications of Measure Rigidity.”* The paragraph immediately before Question 35 supplies the hypotheses. After correcting one duplicated word in the printed PDF, they are:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 127,
   "attempt": 1
  },
  "AIM-OTHER-0129": {
   "statement_status": "corrected_verified",
   "original_statement": "Conjecture 36 (E. Barnes, H. Swinnerton-Dyer [11]). For any quadratic form \n\nQ as above, the supremum m(Q) is rational and isolated. Both m(Q) and \n\nm2(Q) are attained at points with coordinates in the the splitting field of Q.",
   "clean_statement": "Conjecture 36 (E. Barnes, H. Swinnerton-Dyer [11]). For any quadratic form Q as above, the supremum m(Q) is rational and isolated. Both m(Q) and m2(Q) are attained at points with coordinates in the the splitting field of Q.",
   "public_statement": "Conjecture 36 (E. Barnes, H. Swinnerton-Dyer [11]). For any quadratic form Q as above, the supremum m(Q) is rational and isolated. Both m(Q) and m2(Q) are attained at points with coordinates in the the splitting field of Q.",
   "evidence": "Thus the recovered conjecture is that \\(m(Q)\\) is rational and isolated, and that both \\(m(Q)\\) and \\(m_2(Q)\\) are realized by torus classes admitting representatives in \\(K^2\\). The repeated word “the” occurs in the official PDF and is a typographical duplication, not an OCR corruption. The notation \\(m_2(Q)\\) is meaningful here only after isolation of \\(m(Q)\\) has been established. The PDF also warns, citing Godwin, that \\(m_2(Q)\\) itself need not be isolated.",
   "classification_method": "source_or_typo_verified_repair",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-other-notes.json",
   "source_index": 128,
   "attempt": 1
  },
  "AIM-OTHER-0130": {
   "statement_status": "exact",
   "original_statement": "Conjecture 36 is based on numerous computations performed in [10, 11]. The supremum m2(Q) need not be isolated (see [71]). 5.7. The following question was communicated by D. Kleinbock:",
   "clean_statement": "Conjecture 36 is based on numerous computations performed in [10, 11]. The supremum m2(Q) need not be isolated (see [71]). 5.7. The following question was communicated by D. Kleinbock:",
   "public_statement": "Conjecture 36 is based on numerous computations performed in [10, 11]. The supremum m2(Q) need not be isolated (see [71]). 5.7. The following question was communicated by D. Kleinbock:",
   "evidence": "The exact canonical fragment is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 129,
   "attempt": 2
  },
  "AIM-OTHER-0131": {
   "statement_status": "exact",
   "original_statement": "Question 37 (Y. Bugeaud). Let \n\nBs =\n\n{\n\n(α, β ) ∈ R2: inf \n\n> n≥1\n\nn(min {‖ nα ‖, ‖nβ ‖} )s(max {‖ nα ‖, ‖nβ ‖} )2−s > 0\n\n}.\n\nCompute the Hausdorff dimension of the set Bs, 0 < s < 1.\n\nNote that B0 is the set of badly approximable vectors and its Hausdorff dimension is 2. On the other hand, B1 is the set of exceptions of the Littlewood conjecture, and its Hausdorff dimension is 0. 5.8. Some other interesting open problems on Diophantine approximation are stated in [110], Section 13. 6. Quantum Chaos \n\n6.1. The term \"quantum chaos\" refers to the study of quantizations of Hamil-tonian systems whose dynamics is chaotic. We concentrate on the case of the geodesic flow on compact (or, more generally, finite volume) Riemannian man-ifold X possibly with piecewise smooth boundary (e.g. billiards in R2). The geodesic flow on the boundary is defined as elastic reflection. Denote by ∆ the Laplace-Beltrami operator on X and by dV the normalized Riemannian volume on X. Let 0 = λ0 < λ 1 ≤ λ2... be the eigenvalues of −∆ and φi,\n\ni ≥ 0, the corresponding eigenfunctions with the Dirichlet boundary condition such that ‖φi‖2 = 1: \n\n−∆φi = λiφi, φi|∂X = 0.\n\nOne is interested in the semiclassical limit of this system, i.e., in the behavior of the eigenvalues and the eigenfunctions as i → ∞. According to the corre-spondence principle in quantum mechanics, certain properties of the classical dynamical system are inherited by the semiclassical limit of its quantization. OPEN PROBLEMS 16 \n\nConsider the probability measures \n\ndμ i(x) = |φi(x)|2dV (x)on X. One of the fundamental questions is to describe all possible weak ∗\n\nlimits of the sequence {μi} as i → ∞, which are called quantum limits. It was shown (see also [182, 204, 36, 207]) that if the geodesic flow is ergodic on X,then μik → dV in the weak ∗ topology as ik → ∞ along a subsequence {ik} of density one. This property is referred as quantum ergodicity.In general, it might be possible that some of the quantum limits are not absolutely continuous and even assign positive measure to an unstable periodic orbit (this is called a scar ) or to a family of marginally stable periodic orbits (this is called a bouncing ball mode ). However, it seems that no rigorous proof of this phenomena has been given. For example, for the stadium billiard there are substantial numerical and heuristic evidences of the existence of scars and bouncing ball modes (see, for example, [82, 91, 119, 8, 187] and references therein). On the other hand, the numerical data in [12] suggest that no scarring occurs for some dispersive billiards. Some numerical experiments were performed for X = Γ \\H2, where Γ is an arithmetic lattice, and no scars were observed (see [81, 80, 4]). Z. Rudnick and P. Sarnak [163] formulated the following conjecture:",
   "clean_statement": "Question 37 (Y. Bugeaud). Let\n\nBs =\n\n{\n\n(α, β ) ∈ R2: inf\n\n> n≥1\n\nn(min {‖ nα ‖, ‖nβ ‖} )s(max {‖ nα ‖, ‖nβ ‖} )2−s > 0\n\n}.\n\nCompute the Hausdorff dimension of the set Bs, 0 < s < 1.\n\nNote that B0 is the set of badly approximable vectors and its Hausdorff dimension is 2. On the other hand, B1 is the set of exceptions of the Littlewood conjecture, and its Hausdorff dimension is 0. 5.8. Some other interesting open problems on Diophantine approximation are stated in [110], Section 13. 6. Quantum Chaos\n\n6.1. The term \"quantum chaos\" refers to the study of quantizations of Hamil-tonian systems whose dynamics is chaotic. We concentrate on the case of the geodesic flow on compact (or, more generally, finite volume) Riemannian man-ifold X possibly with piecewise smooth boundary (e.g. billiards in R2). The geodesic flow on the boundary is defined as elastic reflection. Denote by ∆ the Laplace-Beltrami operator on X and by dV the normalized Riemannian volume on X. Let 0 = λ0 < λ 1 ≤ λ2... be the eigenvalues of −∆ and φi,\n\ni ≥ 0, the corresponding eigenfunctions with the Dirichlet boundary condition such that ‖φi‖2 = 1:\n\n−∆φi = λiφi, φi|∂X = 0.\n\nOne is interested in the semiclassical limit of this system, i.e., in the behavior of the eigenvalues and the eigenfunctions as i → ∞. According to the corre-spondence principle in quantum mechanics, certain properties of the classical dynamical system are inherited by the semiclassical limit of its quantization. OPEN PROBLEMS 16\n\nConsider the probability measures\n\ndμ i(x) = |φi(x)|2dV (x)on X. One of the fundamental questions is to describe all possible weak ∗\n\nlimits of the sequence {μi} as i → ∞, which are called quantum limits. It was shown (see also [182, 204, 36, 207]) that if the geodesic flow is ergodic on X,then μik → dV in the weak ∗ topology as ik → ∞ along a subsequence {ik} of density one. This property is referred as quantum ergodicity.In general, it might be possible that some of the quantum limits are not absolutely continuous and even assign positive measure to an unstable periodic orbit (this is called a scar ) or to a family of marginally stable periodic orbits (this is called a bouncing ball mode ). However, it seems that no rigorous proof of this phenomena has been given. For example, for the stadium billiard there are substantial numerical and heuristic evidences of the existence of scars and bouncing ball modes (see, for example, [82, 91, 119, 8, 187] and references therein). On the other hand, the numerical data in [12] suggest that no scarring occurs for some dispersive billiards. Some numerical experiments were performed for X = Γ \\H2, where Γ is an arithmetic lattice, and no scars were observed (see [81, 80, 4]). Z. Rudnick and P. Sarnak [163] formulated the following conjecture:",
   "public_statement": "Question 37 (Y. Bugeaud). Let\n\nBs =\n\n{\n\n(α, β ) ∈ R2: inf\n\n> n≥1\n\nn(min {‖ nα ‖, ‖nβ ‖} )s(max {‖ nα ‖, ‖nβ ‖} )2−s > 0\n\n}.\n\nCompute the Hausdorff dimension of the set Bs, 0 < s < 1.\n\nNote that B0 is the set of badly approximable vectors and its Hausdorff dimension is 2. On the other hand, B1 is the set of exceptions of the Littlewood conjecture, and its Hausdorff dimension is 0. 5.8. Some other interesting open problems on Diophantine approximation are stated in [110], Section 13. 6. Quantum Chaos\n\n6.1. The term \"quantum chaos\" refers to the study of quantizations of Hamil-tonian systems whose dynamics is chaotic. We concentrate on the case of the geodesic flow on compact (or, more generally, finite volume) Riemannian man-ifold X possibly with piecewise smooth boundary (e.g. billiards in R2). The geodesic flow on the boundary is defined as elastic reflection. Denote by ∆ the Laplace-Beltrami operator on X and by dV the normalized Riemannian volume on X. Let 0 = λ0 < λ 1 ≤ λ2... be the eigenvalues of −∆ and φi,\n\ni ≥ 0, the corresponding eigenfunctions with the Dirichlet boundary condition such that ‖φi‖2 = 1:\n\n−∆φi = λiφi, φi|∂X = 0.\n\nOne is interested in the semiclassical limit of this system, i.e., in the behavior of the eigenvalues and the eigenfunctions as i → ∞. According to the corre-spondence principle in quantum mechanics, certain properties of the classical dynamical system are inherited by the semiclassical limit of its quantization. OPEN PROBLEMS 16\n\nConsider the probability measures\n\ndμ i(x) = |φi(x)|2dV (x)on X. One of the fundamental questions is to describe all possible weak ∗\n\nlimits of the sequence {μi} as i → ∞, which are called quantum limits. It was shown (see also [182, 204, 36, 207]) that if the geodesic flow is ergodic on X,then μik → dV in the weak ∗ topology as ik → ∞ along a subsequence {ik} of density one. This property is referred as quantum ergodicity.In general, it might be possible that some of the quantum limits are not absolutely continuous and even assign positive measure to an unstable periodic orbit (this is called a scar ) or to a family of marginally stable periodic orbits (this is called a bouncing ball mode ). However, it seems that no rigorous proof of this phenomena has been given. For example, for the stadium billiard there are substantial numerical and heuristic evidences of the existence of scars and bouncing ball modes (see, for example, [82, 91, 119, 8, 187] and references therein). On the other hand, the numerical data in [12] suggest that no scarring occurs for some dispersive billiards. Some numerical experiments were performed for X = Γ \\H2, where Γ is an arithmetic lattice, and no scars were observed (see [81, 80, 4]). Z. Rudnick and P. Sarnak [163] formulated the following conjecture:",
   "evidence": "The source is Question 37, communicated by Yann Bugeaud, in the AIM problem list *Emerging applications of measure rigidity*. Write \\(\\|x\\|\\) for the distance from \\(x\\) to the nearest integer. The statement verified against the source PDF is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 130,
   "attempt": 1
  },
  "AIM-OTHER-0132": {
   "statement_status": "exact",
   "original_statement": "Conjecture 38 (Quantum unique ergodicity). Suppose that X has negative sectional curvature. Then \n\nμi → dV as i → ∞.\n\nThe current research on this conjecture is concentrated on the case of arith-metic manifolds X = Γ \\ ˜X, where ˜X is a symmetric space of noncompact type and Γ an arithmetic lattice. The arithmeticity assumption implies that there is an infinite set of Hecke operators acting on X, which commute with the left invariant differential operators (in rank one the Laplacian is the only such operator). We assume that φi, i ≥ 0, are joint eigenfunctions of the invariant differential operators and Hecke operators. Then the weak ∗ limits of the se-quence of measures {μi} are called arithmetic quantum limits. It is believed (see [32]) that the Laplace-Beltrami operator on X = SL 2(Z)\\H2 has simple cuspidal spectrum; then the assumption on Hecke operators is automatic.",
   "clean_statement": "Conjecture 38 (Quantum unique ergodicity). Suppose that X has negative sectional curvature. Then\n\nμi → dV as i → ∞.\n\nThe current research on this conjecture is concentrated on the case of arith-metic manifolds X = Γ \\ ˜X, where ˜X is a symmetric space of noncompact type and Γ an arithmetic lattice. The arithmeticity assumption implies that there is an infinite set of Hecke operators acting on X, which commute with the left invariant differential operators (in rank one the Laplacian is the only such operator). We assume that φi, i ≥ 0, are joint eigenfunctions of the invariant differential operators and Hecke operators. Then the weak ∗ limits of the se-quence of measures {μi} are called arithmetic quantum limits. It is believed (see [32]) that the Laplace-Beltrami operator on X = SL 2(Z)\\H2 has simple cuspidal spectrum; then the assumption on Hecke operators is automatic.",
   "public_statement": "Conjecture 38 (Quantum unique ergodicity). Suppose that X has negative sectional curvature. Then\n\nμi → dV as i → ∞.\n\nThe current research on this conjecture is concentrated on the case of arith-metic manifolds X = Γ \\ ˜X, where ˜X is a symmetric space of noncompact type and Γ an arithmetic lattice. The arithmeticity assumption implies that there is an infinite set of Hecke operators acting on X, which commute with the left invariant differential operators (in rank one the Laplacian is the only such operator). We assume that φi, i ≥ 0, are joint eigenfunctions of the invariant differential operators and Hecke operators. Then the weak ∗ limits of the se-quence of measures {μi} are called arithmetic quantum limits. It is believed (see [32]) that the Laplace-Beltrami operator on X = SL 2(Z)\\H2 has simple cuspidal spectrum; then the assumption on Hecke operators is automatic.",
   "evidence": "The raw canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 131,
   "attempt": 1
  },
  "AIM-OTHER-0133": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 39 (Arithmetic quantum unique ergodicity). The Riemannian volume is the only arithmetic quantum limit. \n\nPositive results towards this conjecture were obtained for the case X =Γ\\H2, where Γ is a congruence subgroup in either SL 2(Z) or in the group of quaternions of norm one. In this case, T. Watson [197] proved",
   "clean_statement": null,
   "public_statement": "Conjecture 39 (Arithmetic quantum unique ergodicity). The Riemannian volume is the only arithmetic quantum limit.\n\nPositive results towards this conjecture were obtained for the case X =Γ\\H2, where Γ is a congruence subgroup in either SL 2(Z) or in the group of quaternions of norm one. In this case, T. Watson [197] proved",
   "evidence": "The exact canonical record is the truncated fragment",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 132,
   "attempt": 1
  },
  "AIM-OTHER-0134": {
   "statement_status": "exact",
   "original_statement": "Conjecture 39 assuming the generalized Riemann hypothesis. His proof also implies the optimal rate of convergence. Unconditionally,",
   "clean_statement": "Conjecture 39 assuming the generalized Riemann hypothesis. His proof also implies the optimal rate of convergence. Unconditionally,",
   "public_statement": "Conjecture 39 assuming the generalized Riemann hypothesis. His proof also implies the optimal rate of convergence. Unconditionally,",
   "evidence": "The canonical record is not a complete mathematical sentence. It reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 133,
   "attempt": 1
  },
  "AIM-OTHER-0135": {
   "statement_status": "exact",
   "original_statement": "Conjecture 39 for this case was OPEN PROBLEMS 17 \n\nproved by E. Lindenstrauss [120]. The only issue that was not handled in [120] is the escape of the limit measure to the cusp in noncompact case. To handle this difficulty, E. Lindenstrauss suggested the following intermediate problem:",
   "clean_statement": "Conjecture 39 for this case was OPEN PROBLEMS 17\n\nproved by E. Lindenstrauss [120]. The only issue that was not handled in [120] is the escape of the limit measure to the cusp in noncompact case. To handle this difficulty, E. Lindenstrauss suggested the following intermediate problem:",
   "public_statement": "Conjecture 39 for this case was OPEN PROBLEMS 17\n\nproved by E. Lindenstrauss [120]. The only issue that was not handled in [120] is the escape of the limit measure to the cusp in noncompact case. To handle this difficulty, E. Lindenstrauss suggested the following intermediate problem:",
   "evidence": "The exact canonical record is only the following fragment:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 134,
   "attempt": 1
  },
  "AIM-OTHER-0136": {
   "statement_status": "exact",
   "original_statement": "Problem 40 (E. Lindenstrauss). Let X = Γ \\H2 where Γ is a noncocompact arithmetic lattice. Show that for all f, g ∈ Cc(X),\n\n∫ \n\n> X\n\nf dμ i\n\n∫ \n\n> X\n\ng dμ i\n\n→\n\n∫ \n\n> X\n\nf dV \n\n∫ \n\n> X\n\ng dV as i → ∞.\n\nThe analog of",
   "clean_statement": "Problem 40 (E. Lindenstrauss). Let X = Γ \\H2 where Γ is a noncocompact arithmetic lattice. Show that for all f, g ∈ Cc(X),\n\n∫\n\n> X\n\nf dμ i\n\n∫\n\n> X\n\ng dμ i\n\n→\n\n∫\n\n> X\n\nf dV\n\n∫\n\n> X\n\ng dV as i → ∞.\n\nThe analog of",
   "public_statement": "Problem 40 (E. Lindenstrauss). Let X = Γ \\H2 where Γ is a noncocompact arithmetic lattice. Show that for all f, g ∈ Cc(X),\n\n∫\n\n> X\n\nf dμ i\n\n∫\n\n> X\n\ng dμ i\n\n→\n\n∫\n\n> X\n\nf dV\n\n∫\n\n> X\n\ng dV as i → ∞.\n\nThe analog of",
   "evidence": "The canonical record contains severe line-oriented OCR damage:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 135,
   "attempt": 1
  },
  "AIM-OTHER-0137": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Problem 40 for continuous spectrum was proved in [89, 126]. Results toward",
   "clean_statement": null,
   "public_statement": "Problem 40 for continuous spectrum was proved in [89, 126]. Results toward",
   "evidence": "The exact canonical record is the fragment",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 136,
   "attempt": 1
  },
  "AIM-OTHER-0138": {
   "statement_status": "exact",
   "original_statement": "Conjecture 39 for some higher-rank symmetric spaces were recently proved by L. Silberman and A. Venkatesh [178]. 6.2. M. Berry [19] conjectured that eigenfunctions of a typical chaotic sys-tems behave like a superposition of plane waves with random amplitude, phase and direction. This model predicts that the eigenfunctions φi behave like in-dependent Gaussian random variables as i → ∞. In particular, the follow-ing conjecture should hold for generic negatively curved compact Riemannian manifolds:",
   "clean_statement": "Conjecture 39 for some higher-rank symmetric spaces were recently proved by L. Silberman and A. Venkatesh [178]. 6.2. M. Berry [19] conjectured that eigenfunctions of a typical chaotic sys-tems behave like a superposition of plane waves with random amplitude, phase and direction. This model predicts that the eigenfunctions φi behave like in-dependent Gaussian random variables as i → ∞. In particular, the follow-ing conjecture should hold for generic negatively curved compact Riemannian manifolds:",
   "public_statement": "Conjecture 39 for some higher-rank symmetric spaces were recently proved by L. Silberman and A. Venkatesh [178]. 6.2. M. Berry [19] conjectured that eigenfunctions of a typical chaotic sys-tems behave like a superposition of plane waves with random amplitude, phase and direction. This model predicts that the eigenfunctions φi behave like in-dependent Gaussian random variables as i → ∞. In particular, the follow-ing conjecture should hold for generic negatively curved compact Riemannian manifolds:",
   "evidence": "The canonical record combines the end of one section with the beginning of the next. Its exact extracted text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 137,
   "attempt": 1
  },
  "AIM-OTHER-0139": {
   "statement_status": "exact",
   "original_statement": "Conjecture 41 (J. Marklof).\n\nVol( {x ∈ X: a ≤ φi(x) ≤ b}) → 1\n\n√2π\n\n∫ ba\n\ne−t2/2dt \n\nas i → ∞.",
   "clean_statement": "Conjecture 41 (J. Marklof).\n\nVol( {x ∈ X: a ≤ φi(x) ≤ b}) → 1\n\n√2π\n\n∫ ba\n\ne−t2/2dt\n\nas i → ∞.",
   "public_statement": "Conjecture 41 (J. Marklof).\n\nVol( {x ∈ X: a ≤ φi(x) ≤ b}) → 1\n\n√2π\n\n∫ ba\n\ne−t2/2dt\n\nas i → ∞.",
   "evidence": "The record is Conjecture 41, attributed to J. Marklof, in the AIM workshop problem list *Emerging applications of measure rigidity* (dated 1 November 2004). The extracted record has lost fraction bars, limits, and subscripts. The source PDF gives the surrounding conventions: \\(X\\) is a compact or finite-volume Riemannian manifold (possibly with boundary), \\(dV\\) is **normalized** Riemannian volume, and",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 138,
   "attempt": 1
  },
  "AIM-OTHER-0140": {
   "statement_status": "exact",
   "original_statement": "Conjecture 41 is supported by numerical experiments (see [81, 82]). The random wave model also predicts a central limit theorem for the con-vergence in",
   "clean_statement": "Conjecture 41 is supported by numerical experiments (see [81, 82]). The random wave model also predicts a central limit theorem for the con-vergence in",
   "public_statement": "Conjecture 41 is supported by numerical experiments (see [81, 82]). The random wave model also predicts a central limit theorem for the con-vergence in",
   "evidence": "The canonical record is a broken cross-record extraction from the AIM workshop report *Emerging applications of measure rigidity*. It reads:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 139,
   "attempt": 1
  },
  "AIM-OTHER-0141": {
   "statement_status": "exact",
   "original_statement": "Conjecture 38 (see [55, 44]). To formulate this, we need some notations. For a smooth function a on the unit cotangent bundle S∗X of X,we denote by Op( a) a pseododifferential operator of order zero with principal symbol a. For example, when a is a function on X, then Op( a) is a multipli-cation by a. Let λ be the Liouville measure on S∗X and gt is the geodesic flow. It is expected that for generic negatively curved compact Riemannian manifolds,we have the following. (Suppose w.l.o.g. the surface has area 4 π so that Weyl's law reads N (λ) = {i: λi ≤ λ} ∼ λ.)",
   "clean_statement": "Conjecture 38 (see [55, 44]). To formulate this, we need some notations. For a smooth function a on the unit cotangent bundle S∗X of X,we denote by Op( a) a pseododifferential operator of order zero with principal symbol a. For example, when a is a function on X, then Op( a) is a multipli-cation by a. Let λ be the Liouville measure on S∗X and gt is the geodesic flow. It is expected that for generic negatively curved compact Riemannian manifolds,we have the following. (Suppose w.l.o.g. the surface has area 4 π so that Weyl's law reads N (λ) = {i: λi ≤ λ} ∼ λ.)",
   "public_statement": "Conjecture 38 (see [55, 44]). To formulate this, we need some notations. For a smooth function a on the unit cotangent bundle S∗X of X,we denote by Op( a) a pseododifferential operator of order zero with principal symbol a. For example, when a is a function on X, then Op( a) is a multipli-cation by a. Let λ be the Liouville measure on S∗X and gt is the geodesic flow. It is expected that for generic negatively curved compact Riemannian manifolds,we have the following. (Suppose w.l.o.g. the surface has area 4 π so that Weyl's law reads N (λ) = {i: λi ≤ λ} ∼ λ.)",
   "evidence": "The canonical record is a fragment extracted from the AIM workshop list *Emerging applications of measure rigidity*. It starts with “Conjecture 38 (see [55, 44])” and ends immediately before the displayed statement of Conjecture 42. Inspection of the official AIM PDF and the neighboring records shows that the first words complete the preceding sentence. The recovered text is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 140,
   "attempt": 1
  },
  "AIM-OTHER-0142": {
   "statement_status": "exact",
   "original_statement": "Conjecture 42 (J. Marklof (after Feingold-Peres [55])). Suppose X is \"generic\". For a ∈ C∞ \n\n> c\n\n(S∗X, R) with ∫\n\n> S∗X\n\na dλ = 0,\n\nput \n\nξi(a) = λ1/4 \n\n> i\n\n|〈 Op( a)φi, φ i〉|.OPEN PROBLEMS 18 \n\nThen the sequence ξi(a) has a Gaussian limit distribution whose variance is given by the classical autocorrelation function \n\nV (a) def \n\n=\n\n∫ ∞−∞ \n\n∫\n\n> S∗X\n\na(xg t)a(x) dλ (x)dt. \n\nThat is, as λ → ∞,\n\n(1) 1\n\nλ\n\n∑\n\n> λi≤λ\n\nξi(a)2 → V (a),\n\n(2) for any interval I ∈ R\n\n1\n\nλ{λi ≤ λ: V (a)−1/2ξi(a) ∈ I} → 1\n\n√2π\n\n∫\n\n> I\n\ne−t2/2dt. \n\nIt was proved by W. Luo and P. Sarnak that for the modular surface X =SL 2(Z)\\H2, ∑\n\n> λi≤λ\n\n|〈 Op( a)φi, φ i〉| 2 ∼ √λB (a) as λ → ∞,\n\nwhere B is a quadratic form on C∞ \n\n> c\n\n(X) which is closely related to but distinct from the form V defined above (see [169]). In this respect the modular and other arithmetic surfaces are ruled out as \"generic\" examples for the above conjecture. 6.3. A. Katok suggested polygonal billiards as a promising model for quantum chaos.",
   "clean_statement": "Conjecture 42 (J. Marklof (after Feingold-Peres [55])). Suppose X is \"generic\". For a ∈ C∞\n\n> c\n\n(S∗X, R) with ∫\n\n> S∗X\n\na dλ = 0,\n\nput\n\nξi(a) = λ1/4\n\n> i\n\n|〈 Op( a)φi, φ i〉|.OPEN PROBLEMS 18\n\nThen the sequence ξi(a) has a Gaussian limit distribution whose variance is given by the classical autocorrelation function\n\nV (a) def\n\n=\n\n∫ ∞−∞\n\n∫\n\n> S∗X\n\na(xg t)a(x) dλ (x)dt.\n\nThat is, as λ → ∞,\n\n(1) 1\n\nλ\n\n∑\n\n> λi≤λ\n\nξi(a)2 → V (a),\n\n(2) for any interval I ∈ R\n\n1\n\nλ{λi ≤ λ: V (a)−1/2ξi(a) ∈ I} → 1\n\n√2π\n\n∫\n\n> I\n\ne−t2/2dt.\n\nIt was proved by W. Luo and P. Sarnak that for the modular surface X =SL 2(Z)\\H2, ∑\n\n> λi≤λ\n\n|〈 Op( a)φi, φ i〉| 2 ∼ √λB (a) as λ → ∞,\n\nwhere B is a quadratic form on C∞\n\n> c\n\n(X) which is closely related to but distinct from the form V defined above (see [169]). In this respect the modular and other arithmetic surfaces are ruled out as \"generic\" examples for the above conjecture. 6.3. A. Katok suggested polygonal billiards as a promising model for quantum chaos.",
   "public_statement": "Conjecture 42 (J. Marklof (after Feingold-Peres [55])). Suppose X is \"generic\". For a ∈ C∞\n\n> c\n\n(S∗X, R) with ∫\n\n> S∗X\n\na dλ = 0,\n\nput\n\nξi(a) = λ1/4\n\n> i\n\n|〈 Op( a)φi, φ i〉|.OPEN PROBLEMS 18\n\nThen the sequence ξi(a) has a Gaussian limit distribution whose variance is given by the classical autocorrelation function\n\nV (a) def\n\n=\n\n∫ ∞−∞\n\n∫\n\n> S∗X\n\na(xg t)a(x) dλ (x)dt.\n\nThat is, as λ → ∞,\n\n(1) 1\n\nλ\n\n∑\n\n> λi≤λ\n\nξi(a)2 → V (a),\n\n(2) for any interval I ∈ R\n\n1\n\nλ{λi ≤ λ: V (a)−1/2ξi(a) ∈ I} → 1\n\n√2π\n\n∫\n\n> I\n\ne−t2/2dt.\n\nIt was proved by W. Luo and P. Sarnak that for the modular surface X =SL 2(Z)\\H2, ∑\n\n> λi≤λ\n\n|〈 Op( a)φi, φ i〉| 2 ∼ √λB (a) as λ → ∞,\n\nwhere B is a quadratic form on C∞\n\n> c\n\n(X) which is closely related to but distinct from the form V defined above (see [169]). In this respect the modular and other arithmetic surfaces are ruled out as \"generic\" examples for the above conjecture. 6.3. A. Katok suggested polygonal billiards as a promising model for quantum chaos.",
   "evidence": "The source is Conjecture 42, attributed to J. Marklof after Feingold--Peres, in the AIM workshop list *Emerging applications of measure rigidity*. I checked the official AIM PDF, including its underlying PDF text stream, because the extracted record breaks several displayed formulas across lines.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 141,
   "attempt": 1
  },
  "AIM-OTHER-0143": {
   "statement_status": "exact",
   "original_statement": "Question 43 (A. Katok). Do periodic orbits in a triangular billiard corre-spond to scars? More precisely, are there quantum limits supported on periodic orbits? \n\nBased on the investigation [21], it seems likely that the answer to this ques-tion is 'yes' for rational billiards.",
   "clean_statement": "Question 43 (A. Katok). Do periodic orbits in a triangular billiard corre-spond to scars? More precisely, are there quantum limits supported on periodic orbits?\n\nBased on the investigation [21], it seems likely that the answer to this ques-tion is 'yes' for rational billiards.",
   "public_statement": "Question 43 (A. Katok). Do periodic orbits in a triangular billiard corre-spond to scars? More precisely, are there quantum limits supported on periodic orbits?\n\nBased on the investigation [21], it seems likely that the answer to this ques-tion is 'yes' for rational billiards.",
   "evidence": "The official AIM PDF gives the following text. The only repairs below remove line-break hyphenation in “correspond” and “question.”",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 142,
   "attempt": 1
  },
  "AIM-OTHER-0144": {
   "statement_status": "exact",
   "original_statement": "Problem 44 (J. Marklof). Classify all quantum limits of the eigenfunctions of a polygonal billiard. \n\n6.4. According to the Berry-Tabor conjecture (see [133]), the eigenvalues of the Laplacian for generic integrable dynamical system have the same statistical properties as a Poisson process. For 2-dimensional torus, the set of eigenvalues is {Q(x): x ∈ Z2} where Q is a positive definite quadratic form.",
   "clean_statement": "Problem 44 (J. Marklof). Classify all quantum limits of the eigenfunctions of a polygonal billiard.\n\n6.4. According to the Berry-Tabor conjecture (see [133]), the eigenvalues of the Laplacian for generic integrable dynamical system have the same statistical properties as a Poisson process. For 2-dimensional torus, the set of eigenvalues is {Q(x): x ∈ Z2} where Q is a positive definite quadratic form.",
   "public_statement": "Problem 44 (J. Marklof). Classify all quantum limits of the eigenfunctions of a polygonal billiard.\n\n6.4. According to the Berry-Tabor conjecture (see [133]), the eigenvalues of the Laplacian for generic integrable dynamical system have the same statistical properties as a Poisson process. For 2-dimensional torus, the set of eigenvalues is {Q(x): x ∈ Z2} where Q is a positive definite quadratic form.",
   "evidence": "The official AIM PDF gives the complete problem as one sentence:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 143,
   "attempt": 1
  },
  "AIM-OTHER-0145": {
   "statement_status": "exact",
   "original_statement": "Question 45 (J. Marklof). What is the distribution of the set \n\n{(Q(x1) − Q(x2), Q (x2) − Q(x3)): xi ∈ Z2, x i 6 = xj for i 6 = j} ⊂ R2?OPEN PROBLEMS 19 \n\nMore precisely, determine the asymptotics of \n\nNT (( a, b ), (c, d )) def \n\n= \n\n(x1, x 2, x 3) ∈ (Z2)3:\n\na < Q (x1) − Q(x2) < b,\n\nc < Q (x2) − Q(x3) < d,\n\nxi 6 = xj for i 6 = j, ‖xi‖ < T \n\n.\n\nThe distribution of the set {Q(x) − Q(y): x, y ∈ Z2} was studied in [167, 49, 50], and in [134, 135] in the case of rational forms over shifted lattice points. It depends on Diophantine properties of coefficients of the quadratic form. The Berry-Tabor conjecture predicts that the set in",
   "clean_statement": "Question 45 (J. Marklof). What is the distribution of the set\n\n{(Q(x1) − Q(x2), Q (x2) − Q(x3)): xi ∈ Z2, x i 6 = xj for i 6 = j} ⊂ R2?OPEN PROBLEMS 19\n\nMore precisely, determine the asymptotics of\n\nNT (( a, b ), (c, d )) def\n\n=\n\n(x1, x 2, x 3) ∈ (Z2)3:\n\na < Q (x1) − Q(x2) < b,\n\nc < Q (x2) − Q(x3) < d,\n\nxi 6 = xj for i 6 = j, ‖xi‖ < T\n\n.\n\nThe distribution of the set {Q(x) − Q(y): x, y ∈ Z2} was studied in [167, 49, 50], and in [134, 135] in the case of rational forms over shifted lattice points. It depends on Diophantine properties of coefficients of the quadratic form. The Berry-Tabor conjecture predicts that the set in",
   "public_statement": "Question 45 (J. Marklof). What is the distribution of the set\n\n{(Q(x1) − Q(x2), Q (x2) − Q(x3)): xi ∈ Z2, x i 6 = xj for i 6 = j} ⊂ R2?OPEN PROBLEMS 19\n\nMore precisely, determine the asymptotics of\n\nNT (( a, b ), (c, d )) def\n\n=\n\n(x1, x 2, x 3) ∈ (Z2)3:\n\na < Q (x1) − Q(x2) < b,\n\nc < Q (x2) − Q(x3) < d,\n\nxi 6 = xj for i 6 = j, ‖xi‖ < T\n\n.\n\nThe distribution of the set {Q(x) − Q(y): x, y ∈ Z2} was studied in [167, 49, 50], and in [134, 135] in the case of rational forms over shifted lattice points. It depends on Diophantine properties of coefficients of the quadratic form. The Berry-Tabor conjecture predicts that the set in",
   "evidence": "This is Question 45, attributed to J. Marklof, in the AIM workshop list *Emerging applications of measure rigidity*. The corpus record has several OCR errors: `Z2` means \\(\\mathbb Z^2\\), `6 =` means \\(\\ne\\), the string `OPEN PROBLEMS 19` is a page header, and the sentence at the end continues on the next PDF page. Inspection of the original PDF gives the following statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 144,
   "attempt": 1
  },
  "AIM-OTHER-0146": {
   "statement_status": "exact",
   "original_statement": "Question 45 should be equidistributed in R2 for a generic quadratic form Q, i.e, after a suitable normalization, NT (( a, b ), (c, d )) converges to ( b − a)( d − c). However, it is not even known whether the set in",
   "clean_statement": "Question 45 should be equidistributed in R2 for a generic quadratic form Q, i.e, after a suitable normalization, NT (( a, b ), (c, d )) converges to ( b − a)( d − c). However, it is not even known whether the set in",
   "public_statement": "Question 45 should be equidistributed in R2 for a generic quadratic form Q, i.e, after a suitable normalization, NT (( a, b ), (c, d )) converges to ( b − a)( d − c). However, it is not even known whether the set in",
   "evidence": "The source itself says “Conjecture 45” in the last sentence even though the heading is “Question 45”; this is an internal cross-reference typo, not an OCR error. The words after “set in” lie in AIM-OTHER-0147. The running heading “OPEN PROBLEMS 19” is not mathematical text.",
   "classification_method": "explicit_no_change_evidence",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 145,
   "attempt": 1
  },
  "AIM-OTHER-0147": {
   "statement_status": "exact",
   "original_statement": "Conjecture 45 is dense in R2.7. Polygonal billiards",
   "clean_statement": "Conjecture 45 is dense in R2.7. Polygonal billiards",
   "public_statement": "Conjecture 45 is dense in R2.7. Polygonal billiards",
   "evidence": "The canonical record is:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 146,
   "attempt": 1
  },
  "AIM-OTHER-0148": {
   "statement_status": "exact",
   "original_statement": "Question 46 (A. Katok). Construct periodic orbits for triangular billiards. \n\nEvery acute triangle has one obvious periodic orbit, but it is not known whether a general acute triangle has other periodic orbits. It is also not known whether a general obtuse triangle has at least one periodic orbit. Periodic orbits were constructed for some special classes of triangles (see [66, 83, 35, 77, 171]). Recently, a computer aided proof, which uses the program McBilliards [84], was found that shows that every triangle with all angles less than 100 degrees has a periodic orbit (see [172]). The situation is much better for rational triangles and polygons (i.e., if the angles are rational multiples of π). Unfolding the billiard table, one can construct a compact Riemannian surface with a flat structure so that billiard trajectories correspond to geodesics on this surface (see [141] for a survey). Using this technique, it was shown that the number N (T ) of periodic orbits of length at most T is bounded from above and below by quadratic polynomials in T (see [137, 138]). Moreover, for some billiard table this number has qua-dratic asymptotics (see [193, 194, 52, 51]), but it seems unknown whether the quadratic asymptotics holds for rational polygons in general. Note that the convergence N (T ) → ∞ cannot be uniform even on a compact set of triangles. In fact, it was announced by R. E. Schwatz that for any given any ε > 0 there exists a triangle, within ε of the 30-60-90 triangle, which has no periodic paths of length less than 1 /ε.",
   "clean_statement": "Question 46 (A. Katok). Construct periodic orbits for triangular billiards.\n\nEvery acute triangle has one obvious periodic orbit, but it is not known whether a general acute triangle has other periodic orbits. It is also not known whether a general obtuse triangle has at least one periodic orbit. Periodic orbits were constructed for some special classes of triangles (see [66, 83, 35, 77, 171]). Recently, a computer aided proof, which uses the program McBilliards [84], was found that shows that every triangle with all angles less than 100 degrees has a periodic orbit (see [172]). The situation is much better for rational triangles and polygons (i.e., if the angles are rational multiples of π). Unfolding the billiard table, one can construct a compact Riemannian surface with a flat structure so that billiard trajectories correspond to geodesics on this surface (see [141] for a survey). Using this technique, it was shown that the number N (T ) of periodic orbits of length at most T is bounded from above and below by quadratic polynomials in T (see [137, 138]). Moreover, for some billiard table this number has qua-dratic asymptotics (see [193, 194, 52, 51]), but it seems unknown whether the quadratic asymptotics holds for rational polygons in general. Note that the convergence N (T ) → ∞ cannot be uniform even on a compact set of triangles. In fact, it was announced by R. E. Schwatz that for any given any ε > 0 there exists a triangle, within ε of the 30-60-90 triangle, which has no periodic paths of length less than 1 /ε.",
   "public_statement": "Question 46 (A. Katok). Construct periodic orbits for triangular billiards.\n\nEvery acute triangle has one obvious periodic orbit, but it is not known whether a general acute triangle has other periodic orbits. It is also not known whether a general obtuse triangle has at least one periodic orbit. Periodic orbits were constructed for some special classes of triangles (see [66, 83, 35, 77, 171]). Recently, a computer aided proof, which uses the program McBilliards [84], was found that shows that every triangle with all angles less than 100 degrees has a periodic orbit (see [172]). The situation is much better for rational triangles and polygons (i.e., if the angles are rational multiples of π). Unfolding the billiard table, one can construct a compact Riemannian surface with a flat structure so that billiard trajectories correspond to geodesics on this surface (see [141] for a survey). Using this technique, it was shown that the number N (T ) of periodic orbits of length at most T is bounded from above and below by quadratic polynomials in T (see [137, 138]). Moreover, for some billiard table this number has qua-dratic asymptotics (see [193, 194, 52, 51]), but it seems unknown whether the quadratic asymptotics holds for rational polygons in general. Note that the convergence N (T ) → ∞ cannot be uniform even on a compact set of triangles. In fact, it was announced by R. E. Schwatz that for any given any ε > 0 there exists a triangle, within ε of the 30-60-90 triangle, which has no periodic paths of length less than 1 /ε.",
   "evidence": "This record is Question 46, attributed to A. Katok, in the June 2004 AIM workshop list *Emerging applications of measure rigidity*. Inspection of page 19 of the original PDF verifies the following statement.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 147,
   "attempt": 1
  },
  "AIM-OTHER-0149": {
   "statement_status": "exact",
   "original_statement": "Question 47 (A. Katok). Prove ergodicity for triangular billiards with irra-tional angles. \n\nIt was proved that the set of ergodic (topologically transitive) triangular billiard table is residual in the sense of Baire category (see [103, 99]). How-ever, it seems unknown whether the set of ergodic billiard tables has positive OPEN PROBLEMS 20 \n\nmeasure. It is easy to see that rational billiards cannot be ergodic. On the other hand, it was shown in [196] that if an irrational billiard table is very well approximable by a rational one, then it is ergodic. In the case when one of the angles of a triangles is rational, there is a useful unfolding procedure (see [189]) that may lead to a proof of ergodicity. Also, it seems unknown whether there is a weakly mixing polygonal billiard (see [76] for a positive result in this direction). It was shown in [5] that typical interval exchange transformation which is not defined by a cyclic permutation is weakly mixing (see also [98, 192]). 8. Divergent trajectories \n\nLet G be a semisimple real algebraic group, Γ a noncocompact arithmetic lattice in G, and D a closed subgroup of a maximal R-split torus A. An orbit \n\nDx of D in G/ Γ is called divergent is the map d 7 → dx, d ∈ D, is proper. One can construct a divergent orbits using the following observation. Suppose that \n\nD is the union of open subsemigroups D1,..., D l such that for every i there exists a representation ρi: G → GL( Vi), defined over Q, and vi ∈ Vi, such that ρi(dx )vi → 0 as d ∈ Di goes to ∞. Then Dx is divergent. Such divergent orbits are called obvious.",
   "clean_statement": "Question 47 (A. Katok). Prove ergodicity for triangular billiards with irra-tional angles.\n\nIt was proved that the set of ergodic (topologically transitive) triangular billiard table is residual in the sense of Baire category (see [103, 99]). How-ever, it seems unknown whether the set of ergodic billiard tables has positive OPEN PROBLEMS 20\n\nmeasure. It is easy to see that rational billiards cannot be ergodic. On the other hand, it was shown in [196] that if an irrational billiard table is very well approximable by a rational one, then it is ergodic. In the case when one of the angles of a triangles is rational, there is a useful unfolding procedure (see [189]) that may lead to a proof of ergodicity. Also, it seems unknown whether there is a weakly mixing polygonal billiard (see [76] for a positive result in this direction). It was shown in [5] that typical interval exchange transformation which is not defined by a cyclic permutation is weakly mixing (see also [98, 192]). 8. Divergent trajectories\n\nLet G be a semisimple real algebraic group, Γ a noncocompact arithmetic lattice in G, and D a closed subgroup of a maximal R-split torus A. An orbit\n\nDx of D in G/ Γ is called divergent is the map d 7 → dx, d ∈ D, is proper. One can construct a divergent orbits using the following observation. Suppose that\n\nD is the union of open subsemigroups D1,..., D l such that for every i there exists a representation ρi: G → GL( Vi), defined over Q, and vi ∈ Vi, such that ρi(dx )vi → 0 as d ∈ Di goes to ∞. Then Dx is divergent. Such divergent orbits are called obvious.",
   "public_statement": "Question 47 (A. Katok). Prove ergodicity for triangular billiards with irra-tional angles.\n\nIt was proved that the set of ergodic (topologically transitive) triangular billiard table is residual in the sense of Baire category (see [103, 99]). How-ever, it seems unknown whether the set of ergodic billiard tables has positive OPEN PROBLEMS 20\n\nmeasure. It is easy to see that rational billiards cannot be ergodic. On the other hand, it was shown in [196] that if an irrational billiard table is very well approximable by a rational one, then it is ergodic. In the case when one of the angles of a triangles is rational, there is a useful unfolding procedure (see [189]) that may lead to a proof of ergodicity. Also, it seems unknown whether there is a weakly mixing polygonal billiard (see [76] for a positive result in this direction). It was shown in [5] that typical interval exchange transformation which is not defined by a cyclic permutation is weakly mixing (see also [98, 192]). 8. Divergent trajectories\n\nLet G be a semisimple real algebraic group, Γ a noncocompact arithmetic lattice in G, and D a closed subgroup of a maximal R-split torus A. An orbit\n\nDx of D in G/ Γ is called divergent is the map d 7 → dx, d ∈ D, is proper. One can construct a divergent orbits using the following observation. Suppose that\n\nD is the union of open subsemigroups D1,..., D l such that for every i there exists a representation ρi: G → GL( Vi), defined over Q, and vi ∈ Vi, such that ρi(dx )vi → 0 as d ∈ Di goes to ∞. Then Dx is divergent. Such divergent orbits are called obvious.",
   "evidence": "The canonical record is Question 47 from Section 7, “Polygonal billiards,” of the AIM workshop list *Emerging applications of measure rigidity* (June 2004; document dated 1 November 2004). The official PDF gives the question across printed pages 19–20:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 148,
   "attempt": 2
  },
  "AIM-OTHER-0150": {
   "statement_status": "exact",
   "original_statement": "Conjecture 48 (B. Weiss). (1) If dim D > rank QG, then there are no divergent orbits of D.\n\n(2) If dim D = rank QG, then the only divergent trajectories are obvious ones. \n\n(3) If dim D < rank QG, then there are non-obvious divergent trajectories.",
   "clean_statement": "Conjecture 48 (B. Weiss). (1) If dim D > rank QG, then there are no divergent orbits of D.\n\n(2) If dim D = rank QG, then the only divergent trajectories are obvious ones.\n\n(3) If dim D < rank QG, then there are non-obvious divergent trajectories.",
   "public_statement": "Conjecture 48 (B. Weiss). (1) If dim D > rank QG, then there are no divergent orbits of D.\n\n(2) If dim D = rank QG, then the only divergent trajectories are obvious ones.\n\n(3) If dim D < rank QG, then there are non-obvious divergent trajectories.",
   "evidence": "The canonical record is a genuine conjecture, but its definitions occur at the end of the preceding record. The official AIM workshop PDF states:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 149,
   "attempt": 1
  },
  "AIM-OTHER-0151": {
   "statement_status": "exact",
   "original_statement": "Conjecture 48 was formulated in [200] where several special cases of it were checked, and it was shown in particular that",
   "clean_statement": "Conjecture 48 was formulated in [200] where several special cases of it were checked, and it was shown in particular that",
   "public_statement": "Conjecture 48 was formulated in [200] where several special cases of it were checked, and it was shown in particular that",
   "evidence": "The exact canonical record is the fragment",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 150,
   "attempt": 1
  },
  "AIM-OTHER-0152": {
   "statement_status": "exact",
   "original_statement": "Conjecture 48 holds when dim D = 1 (see also [38]). The case D = A was settled in [188]. Recently,",
   "clean_statement": "Conjecture 48 holds when dim D = 1 (see also [38]). The case D = A was settled in [188]. Recently,",
   "public_statement": "Conjecture 48 holds when dim D = 1 (see also [38]). The case D = A was settled in [188]. Recently,",
   "evidence": "The canonical record reads, exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 151,
   "attempt": 1
  },
  "AIM-OTHER-0153": {
   "statement_status": "exact",
   "original_statement": "Conjecture 48(1) was proved in [33] when rank QG = 2 and in [201] in com-plete generality. We also mention that",
   "clean_statement": "Conjecture 48(1) was proved in [33] when rank QG = 2 and in [201] in com-plete generality. We also mention that",
   "public_statement": "Conjecture 48(1) was proved in [33] when rank QG = 2 and in [201] in com-plete generality. We also mention that",
   "evidence": "The canonical text is",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 152,
   "attempt": 1
  },
  "AIM-OTHER-0154": {
   "statement_status": "exact",
   "original_statement": "Conjecture 48(3) was checked in [200] for G = SL(4, R) for all diagonal subgroups D except \n\nD = {(s, s −1, t, t −1): s, t > 0}.\n\nNext, we discuss a similar problem when D is a cone in A. There are ex-amples of cones D that admit non-obvious divergent trajectories (e.g., a Weyl chamber) as well as an example of cones that admit only obvious divergent trajectories (see [200]). The latter example was constructed for G = SL(3, R)and the argument used essentially that dim D = 2.",
   "clean_statement": "Conjecture 48(3) was checked in [200] for G = SL(4, R) for all diagonal subgroups D except\n\nD = {(s, s −1, t, t −1): s, t > 0}.\n\nNext, we discuss a similar problem when D is a cone in A. There are ex-amples of cones D that admit non-obvious divergent trajectories (e.g., a Weyl chamber) as well as an example of cones that admit only obvious divergent trajectories (see [200]). The latter example was constructed for G = SL(3, R)and the argument used essentially that dim D = 2.",
   "public_statement": "Conjecture 48(3) was checked in [200] for G = SL(4, R) for all diagonal subgroups D except\n\nD = {(s, s −1, t, t −1): s, t > 0}.\n\nNext, we discuss a similar problem when D is a cone in A. There are ex-amples of cones D that admit non-obvious divergent trajectories (e.g., a Weyl chamber) as well as an example of cones that admit only obvious divergent trajectories (see [200]). The latter example was constructed for G = SL(3, R)and the argument used essentially that dim D = 2.",
   "evidence": "The canonical record has two roles. First, it records the sole \\(\\mathrm{SL}_4\\) subgroup not covered by a 2004 result of Weiss:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 153,
   "attempt": 2
  },
  "AIM-OTHER-0155": {
   "statement_status": "exact",
   "original_statement": "Question 49 (B. Weiss). Construct examples of cones D in A with dim D ≥ 3\n\nand no non-obvious divergent trajectories. \n\nLet AT denote the ball of radius T in A and λ a Haar measure on A.OPEN PROBLEMS 21",
   "clean_statement": "Question 49 (B. Weiss). Construct examples of cones D in A with dim D ≥ 3\n\nand no non-obvious divergent trajectories.\n\nLet AT denote the ball of radius T in A and λ a Haar measure on A.OPEN PROBLEMS 21",
   "public_statement": "Question 49 (B. Weiss). Construct examples of cones D in A with dim D ≥ 3\n\nand no non-obvious divergent trajectories.\n\nLet AT denote the ball of radius T in A and λ a Haar measure on A.OPEN PROBLEMS 21",
   "evidence": "The canonical OCR record is contaminated by the beginning of the next question and by a page header. Inspection of the official AIM workshop PDF, on printed page 20 (PDF page 21), recovers the complete statement as",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 154,
   "attempt": 1
  },
  "AIM-OTHER-0156": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Question 50 (B. Weiss). Suppose that dim A ≥ 2 and for some x ∈ G/ Γ and every one-parameter subgroup D of A, the orbits Dx is not divergent in G/ Γ.Is it true that there exists a compact set K ⊂ G/ Γ such that \n\nlim sup \n\n> T→∞\n\n1\n\nλ(AT ) λ({a ∈ AT: ax ∈ K}) > 0? 9. Andr´ e-Oort Conjecture \n\nA Shimura datum is a pair ( G, X ) where G is a reductive algebraic group defined over Q and X is a G(R)-conjugacy class of homomorphisms h: C× →\n\nG(R) such that (1) The adjoint action of h(C×) on Lie( Gad (R)) 4 decomposes as a direct sum of eigenspaces with characters z/ ¯z, 1, ¯ z/z.(2) ad h(i) acts as a Cartan involution on Gad (R). (3) Gad (R) has no factors on which the adjoint action of h(C×) is trivial. Morphisms ( ˜G, ˜X) → (G, X ) of Shimura datums are induced by morphisms ˜G → G of algebraic groups in obvious way. Note that X has a natural struc-ture of complex manifold such that its connected components are Hermitian symmetric domains, G(R) acts on X by holomorphic automorphisms, and morphisms are equivariant holomorphic maps. Let Af denote the ring of finite adeles, and K is an open compact subgroup in G(Af ). Define Sh K (G, X ) = G(Q)\\(X × G(Af )) /K \n\nOne can show that Sh K (G, X ) is a finite disjoint union of Hermitian locally symmetric domains. In particular, by the Baily-Borel theorem, Sh K (G, X ) has a natural structure of an algebraic variety. \n\nExample: Let G = GL 2, h(a + bi ) = \n\n( a b\n\n−b a\n\n)\n\nand K = GL 2(ˆZ). Then Sh K (G, X ) ' SL 2(Z)\\H2 parametrizes isomorphism classes of elliptic curves over C.The Shimura variety associated to ( G, X ) is the projective limit of Sh K (G, X )where K runs over open compact subgroups of G(Af ). A point h ∈ X is called \n\nspecial if there exists a torus T of G defined over Q such that h(C×) ⊂ T (R). One can check that in the above example, the special points are imaginary quadratic irrationals that correspond to elliptic curves with complex multipli-cation. \n\n> 4Gad is the adjoint group which is the factor of Gby its center. OPEN PROBLEMS 22\n\nFor g ∈ G(Af ), we have natural projection maps \n\nπ1: Sh K∩gKg −1 (G, X ) → Sh K (G, X )\n\nπ2: Sh K∩gKg −1 (G, X ) → Sh gKg −1 (G, X ).\n\nwith finite fibers. This defines Hecke correspondence \n\nTg(x) = π2(π−11 (x)) g: Sh K (G, X ) → Sh K (G, X ).\n\nLet ( ˜G, ˜X) → (G, X ) be morphism of Shimura datums that induces map Sh ˜K ( ˜G, ˜X) → Sh K (G, X ). The special subvarieties (also called subvarieties of Hodge type) are the irreducible components of the image Sh ˜K ( ˜G, ˜X) → Sh K (G, X ) Tg\n\n−→ Sh K (G, X ).\n\nUsing Hecke correspondences, one shows that the set of special points in a special subvariety is dense with respect to Zariski (or even analytic) topology. The following conjecture is the converse of this fact.",
   "clean_statement": null,
   "public_statement": "Question 50 (B. Weiss). Suppose that dim A ≥ 2 and for some x ∈ G/ Γ and every one-parameter subgroup D of A, the orbits Dx is not divergent in G/ Γ.Is it true that there exists a compact set K ⊂ G/ Γ such that\n\nlim sup\n\n> T→∞\n\n1\n\nλ(AT ) λ({a ∈ AT: ax ∈ K}) > 0? 9. Andr´ e-Oort Conjecture\n\nA Shimura datum is a pair ( G, X ) where G is a reductive algebraic group defined over Q and X is a G(R)-conjugacy class of homomorphisms h: C× →\n\nG(R) such that (1) The adjoint action of h(C×) on Lie( Gad (R)) 4 decomposes as a direct sum of eigenspaces with characters z/ ¯z, 1, ¯ z/z.(2) ad h(i) acts as a Cartan involution on Gad (R). (3) Gad (R) has no factors on which the adjoint action of h(C×) is trivial. Morphisms ( ˜G, ˜X) → (G, X ) of Shimura datums are induced by morphisms ˜G → G of algebraic groups in obvious way. Note that X has a natural struc-ture of complex manifold such that its connected components are Hermitian symmetric domains, G(R) acts on X by holomorphic automorphisms, and morphisms are equivariant holomorphic maps. Let Af denote the ring of finite adeles, and K is an open compact subgroup in G(Af ). Define Sh K (G, X ) = G(Q)\\(X × G(Af )) /K\n\nOne can show that Sh K (G, X ) is a finite disjoint union of Hermitian locally symmetric domains. In particular, by the Baily-Borel theorem, Sh K (G, X ) has a natural structure of an algebraic variety.\n\nExample: Let G = GL 2, h(a + bi ) =\n\n( a b\n\n−b a\n\n)\n\nand K = GL 2(ˆZ). Then Sh K (G, X ) ' SL 2(Z)\\H2 parametrizes isomorphism classes of elliptic curves over C.The Shimura variety associated to ( G, X ) is the projective limit of Sh K (G, X )where K runs over open compact subgroups of G(Af ). A point h ∈ X is called\n\nspecial if there exists a torus T of G defined over Q such that h(C×) ⊂ T (R). One can check that in the above example, the special points are imaginary quadratic irrationals that correspond to elliptic curves with complex multipli-cation.\n\n> 4Gad is the adjoint group which is the factor of Gby its center. OPEN PROBLEMS 22\n\nFor g ∈ G(Af ), we have natural projection maps\n\nπ1: Sh K∩gKg −1 (G, X ) → Sh K (G, X )\n\nπ2: Sh K∩gKg −1 (G, X ) → Sh gKg −1 (G, X ).\n\nwith finite fibers. This defines Hecke correspondence\n\nTg(x) = π2(π−11 (x)) g: Sh K (G, X ) → Sh K (G, X ).\n\nLet ( ˜G, ˜X) → (G, X ) be morphism of Shimura datums that induces map Sh ˜K ( ˜G, ˜X) → Sh K (G, X ). The special subvarieties (also called subvarieties of Hodge type) are the irreducible components of the image Sh ˜K ( ˜G, ˜X) → Sh K (G, X ) Tg\n\n−→ Sh K (G, X ).\n\nUsing Hecke correspondences, one shows that the set of special points in a special subvariety is dense with respect to Zariski (or even analytic) topology. The following conjecture is the converse of this fact.",
   "evidence": "The canonical `problem` field begins with Question 50 but then continues for more than a page through definitions of Shimura data, Hecke correspondences, and special subvarieties. Inspection of the official AIM workshop PDF fixes the boundary exactly. The preceding sentence defines \\(A_T\\) and the question is:",
   "classification_method": "verified_repair_without_extractable_clean_statement",
   "clean_statement_source": "no_safe_clean_extraction",
   "source_file": "aim-other-notes.json",
   "source_index": 155,
   "attempt": 1
  },
  "AIM-OTHER-0157": {
   "statement_status": "exact",
   "original_statement": "Conjecture 51 (Y. Andr´ e-F. Oort [3, 153]). Zariski closure of a set of special point is a finite union of special subvarieties. \n\nSee [45, 46] and references therein for partial results on this conjecture. We also mention that",
   "clean_statement": "Conjecture 51 (Y. Andr´ e-F. Oort [3, 153]). Zariski closure of a set of special point is a finite union of special subvarieties.\n\nSee [45, 46] and references therein for partial results on this conjecture. We also mention that",
   "public_statement": "Conjecture 51 (Y. Andr´ e-F. Oort [3, 153]). Zariski closure of a set of special point is a finite union of special subvarieties.\n\nSee [45, 46] and references therein for partial results on this conjecture. We also mention that",
   "evidence": "The canonical record ends mid-sentence:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 156,
   "attempt": 1
  },
  "AIM-OTHER-0158": {
   "statement_status": "exact",
   "original_statement": "Conjecture 51 was proved in the case when the dimension of the Zariski closure is one assuming generalized Riemannian hypothesis (see [203]).",
   "clean_statement": "Conjecture 51 was proved in the case when the dimension of the Zariski closure is one assuming generalized Riemannian hypothesis (see [203]).",
   "public_statement": "Conjecture 51 was proved in the case when the dimension of the Zariski closure is one assuming generalized Riemannian hypothesis (see [203]).",
   "evidence": "The canonical record is a historical continuation of Conjecture 51, not a self-contained conjecture:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 157,
   "attempt": 1
  },
  "AIM-OTHER-0159": {
   "statement_status": "exact",
   "original_statement": "Conjecture 51 was partially motivated by an analogy with the theory of abelian varieties, according to which special points correspond to torsion points and special subvarieties correspond to translates of abelian subvarieties by torsion points. Analogous conjectures for abelian varieties is due to S. Lang, Yu. Manin, and D. Mumford. These conjectures were settled (see [190] for a survey). One of the proofs (see [191, 208]) is based on equidistribution of Galois orbits of \"generic\" sequences of points, which was established in [186] (see also [20]). This approach may also lead to a proof of",
   "clean_statement": "Conjecture 51 was partially motivated by an analogy with the theory of abelian varieties, according to which special points correspond to torsion points and special subvarieties correspond to translates of abelian subvarieties by torsion points. Analogous conjectures for abelian varieties is due to S. Lang, Yu. Manin, and D. Mumford. These conjectures were settled (see [190] for a survey). One of the proofs (see [191, 208]) is based on equidistribution of Galois orbits of \"generic\" sequences of points, which was established in [186] (see also [20]). This approach may also lead to a proof of",
   "public_statement": "Conjecture 51 was partially motivated by an analogy with the theory of abelian varieties, according to which special points correspond to torsion points and special subvarieties correspond to translates of abelian subvarieties by torsion points. Analogous conjectures for abelian varieties is due to S. Lang, Yu. Manin, and D. Mumford. These conjectures were settled (see [190] for a survey). One of the proofs (see [191, 208]) is based on equidistribution of Galois orbits of \"generic\" sequences of points, which was established in [186] (see also [20]). This approach may also lead to a proof of",
   "evidence": "The canonical record is not itself a new conjecture. It is the explanatory paragraph following Conjecture 51 in the AIM workshop list *Emerging applications of measure rigidity*. The extracted text ends:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 158,
   "attempt": 1
  },
  "AIM-OTHER-0160": {
   "statement_status": "exact",
   "original_statement": "Conjecture 52. Let {xn} be a sequence of special points on a Shimura variety. Suppose that xn lies outside of any special subvariety for sufficiently large n.Then the (finite) Galois orbits of xn become equidistributed as n → ∞ with respect to the normalized Haar measure. \n\nFor some partial results on this conjecture, see [78] and references therein. In particular, for the above example,",
   "clean_statement": "Conjecture 52. Let {xn} be a sequence of special points on a Shimura variety. Suppose that xn lies outside of any special subvariety for sufficiently large n.Then the (finite) Galois orbits of xn become equidistributed as n → ∞ with respect to the normalized Haar measure.\n\nFor some partial results on this conjecture, see [78] and references therein. In particular, for the above example,",
   "public_statement": "Conjecture 52. Let {xn} be a sequence of special points on a Shimura variety. Suppose that xn lies outside of any special subvariety for sufficiently large n.Then the (finite) Galois orbits of xn become equidistributed as n → ∞ with respect to the normalized Haar measure.\n\nFor some partial results on this conjecture, see [78] and references therein. In particular, for the above example,",
   "evidence": "The canonical record stops in the middle of a sentence. Inspection of page 22 of the official AIM problem PDF recovers both the conjecture and its dangling continuation. In unambiguous notation, the source asserts the following.",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 159,
   "attempt": 1
  },
  "AIM-OTHER-0161": {
   "statement_status": "exact",
   "original_statement": "Conjecture 52 was established in [42].",
   "clean_statement": "Conjecture 52 was established in [42].",
   "public_statement": "Conjecture 52 was established in [42].",
   "evidence": "The canonical record is the one-sentence fragment",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 160,
   "attempt": 1
  },
  "AIM-OTHER-0162": {
   "statement_status": "exact",
   "original_statement": "Question 53 (L. Silberman). Give an ergodic-theoretic proof of the equidis-tribution of special points on SL 2(Z)\\H2.OPEN PROBLEMS 23 \n\n10. Arithmeticity \n\nLet G be the direct product of k copies of SL(2, R), k ≥ 2, and U + and U −\n\nthe upper and lower unipotent subgroups G respectively. Let Γ + and Γ − be lattices in U + and U −. We assume that these lattices are \"irreducible\" in the sense that the projection maps from G to its components are injective on Γ +\n\nand Γ −.The following conjecture was communicated by H. Oh:",
   "clean_statement": "Question 53 (L. Silberman). Give an ergodic-theoretic proof of the equidis-tribution of special points on SL 2(Z)\\H2.OPEN PROBLEMS 23\n\n10. Arithmeticity\n\nLet G be the direct product of k copies of SL(2, R), k ≥ 2, and U + and U −\n\nthe upper and lower unipotent subgroups G respectively. Let Γ + and Γ − be lattices in U + and U −. We assume that these lattices are \"irreducible\" in the sense that the projection maps from G to its components are injective on Γ +\n\nand Γ −.The following conjecture was communicated by H. Oh:",
   "public_statement": "Question 53 (L. Silberman). Give an ergodic-theoretic proof of the equidis-tribution of special points on SL 2(Z)\\H2.OPEN PROBLEMS 23\n\n10. Arithmeticity\n\nLet G be the direct product of k copies of SL(2, R), k ≥ 2, and U + and U −\n\nthe upper and lower unipotent subgroups G respectively. Let Γ + and Γ − be lattices in U + and U −. We assume that these lattices are \"irreducible\" in the sense that the projection maps from G to its components are injective on Γ +\n\nand Γ −.The following conjecture was communicated by H. Oh:",
   "evidence": "The canonical OCR record joins the end of page 21 of the AIM list to the page header and the beginning of Section 10 on Arithmeticity. Inspection of the official PDF shows that the clean record ends before the page header. It is exactly:",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 161,
   "attempt": 1
  },
  "AIM-OTHER-0163": {
   "statement_status": "reconstructed_unverified",
   "original_statement": "Conjecture 54 (G. Margulis, A. Selberg). If the group 〈Γ+, Γ−〉 is discrete, then it an arithmetic lattice in G.\n\nIt was observed in [151] that",
   "clean_statement": "Conjecture 54 (G. Margulis, A. Selberg). If the group \\(\\langle\\Gamma^+,\\Gamma^-\\rangle\\) is discrete, then it an arithmetic lattice in \\(G\\). It was observed in [151] that",
   "public_statement": "Conjecture 54 (G. Margulis, A. Selberg). If the group 〈Γ+, Γ−〉 is discrete, then it an arithmetic lattice in G.\n\nIt was observed in [151] that",
   "evidence": "The canonical record is truncated. It reads:",
   "classification_method": "repair_without_verification",
   "clean_statement_source": "labeled_recovery_in_report_section_1",
   "source_file": "aim-other-notes.json",
   "source_index": 162,
   "attempt": 1
  },
  "AIM-OTHER-0164": {
   "statement_status": "exact",
   "original_statement": "Conjecture 54 for k ≥ 3 follows from",
   "clean_statement": "Conjecture 54 for k ≥ 3 follows from",
   "public_statement": "Conjecture 54 for k ≥ 3 follows from",
   "evidence": "The canonical record is not a complete conjecture. It contains only",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 163,
   "attempt": 1
  },
  "AIM-OTHER-0165": {
   "statement_status": "exact",
   "original_statement": "Conjecture 29. In particular, one can show that the Hausdorff dimension of the set of irreducible lattices Γ + ⊂ U + for which 〈Γ+, Γ−〉 is discrete for some irreducible lattice Γ − ⊂ U − is exactly k (see [152]). 11. Symbolic coding \n\nSymbolic dynamics plays important role in the study of Anosov flows (see, for example, [27]). In the case of surfaces of constant negative curvature, a symbolic representation of the geodesic flow in terms of a Markov chain can be given quite explicitly. Such constructions go back to M. Morse, E. Artin, and G. Hedlund. More recently, these constructions were generalized and improved by several authors (see [30, 173, 174, 175], [1, 2], [100, 75, 101, 102]). The \n\ngeometric code of a geodesic is a biinfinite sequence of symbols that obtained by fixing a fundamental domain and recording which sides the geodesic hits along its pass. The arithmetic code of a geodesic is obtained by expanding the coordinates of the endpoints of the geodesic into a continued fraction expansion.",
   "clean_statement": "Conjecture 29. In particular, one can show that the Hausdorff dimension of the set of irreducible lattices Γ + ⊂ U + for which 〈Γ+, Γ−〉 is discrete for some irreducible lattice Γ − ⊂ U − is exactly k (see [152]). 11. Symbolic coding\n\nSymbolic dynamics plays important role in the study of Anosov flows (see, for example, [27]). In the case of surfaces of constant negative curvature, a symbolic representation of the geodesic flow in terms of a Markov chain can be given quite explicitly. Such constructions go back to M. Morse, E. Artin, and G. Hedlund. More recently, these constructions were generalized and improved by several authors (see [30, 173, 174, 175], [1, 2], [100, 75, 101, 102]). The\n\ngeometric code of a geodesic is a biinfinite sequence of symbols that obtained by fixing a fundamental domain and recording which sides the geodesic hits along its pass. The arithmetic code of a geodesic is obtained by expanding the coordinates of the endpoints of the geodesic into a continued fraction expansion.",
   "public_statement": "Conjecture 29. In particular, one can show that the Hausdorff dimension of the set of irreducible lattices Γ + ⊂ U + for which 〈Γ+, Γ−〉 is discrete for some irreducible lattice Γ − ⊂ U − is exactly k (see [152]). 11. Symbolic coding\n\nSymbolic dynamics plays important role in the study of Anosov flows (see, for example, [27]). In the case of surfaces of constant negative curvature, a symbolic representation of the geodesic flow in terms of a Markov chain can be given quite explicitly. Such constructions go back to M. Morse, E. Artin, and G. Hedlund. More recently, these constructions were generalized and improved by several authors (see [30, 173, 174, 175], [1, 2], [100, 75, 101, 102]). The\n\ngeometric code of a geodesic is a biinfinite sequence of symbols that obtained by fixing a fundamental domain and recording which sides the geodesic hits along its pass. The arithmetic code of a geodesic is obtained by expanding the coordinates of the endpoints of the geodesic into a continued fraction expansion.",
   "evidence": "The canonical record is not a faithful standalone conjecture. In the AIM source, the relevant passage occurs at the end of Section 10. Let",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 164,
   "attempt": 1
  },
  "AIM-OTHER-0166": {
   "statement_status": "exact",
   "original_statement": "Question 55 (S. Katok). Construct analogs of the geometric coding and the arithmetic codings for the Weyl chamber flow (i.e., the action the diagonal group) on SL( n, R)/SL( n, Z).\n\nIt was pointed out by B. Weiss that an interesting symbolic coding for the Weyl chamber flow was used in [181], where a (wrong) proof of the Littlewood conjecture was given (see also [179, 180, 9]). One should mention that symbolic representations in higher dimensions are usually quite involved and not explicit. For example, any Markov partition of a hyperbolic toral automorphism on the 3-dimensional torus must consist of fractal sets (see [29]). F. Ledrappier and S. Mozes suggested to look for a convenient symbolic representation of the Weyl chamber flow using fractal tilings. This approach was successfully applied to construct explicit symbolic representations for some automorphisms and shifts on higher-dimensional tori OPEN PROBLEMS 24 \n\n(see [161, 13, 86, 87]) and for the Cartan action on GL(2, Qp) × GL(2, Qq)/Γ, Γ a irreducible lattice (see [145, 147]).",
   "clean_statement": "Question 55 (S. Katok). Construct analogs of the geometric coding and the arithmetic codings for the Weyl chamber flow (i.e., the action the diagonal group) on SL( n, R)/SL( n, Z).\n\nIt was pointed out by B. Weiss that an interesting symbolic coding for the Weyl chamber flow was used in [181], where a (wrong) proof of the Littlewood conjecture was given (see also [179, 180, 9]). One should mention that symbolic representations in higher dimensions are usually quite involved and not explicit. For example, any Markov partition of a hyperbolic toral automorphism on the 3-dimensional torus must consist of fractal sets (see [29]). F. Ledrappier and S. Mozes suggested to look for a convenient symbolic representation of the Weyl chamber flow using fractal tilings. This approach was successfully applied to construct explicit symbolic representations for some automorphisms and shifts on higher-dimensional tori OPEN PROBLEMS 24\n\n(see [161, 13, 86, 87]) and for the Cartan action on GL(2, Qp) × GL(2, Qq)/Γ, Γ a irreducible lattice (see [145, 147]).",
   "public_statement": "Question 55 (S. Katok). Construct analogs of the geometric coding and the arithmetic codings for the Weyl chamber flow (i.e., the action the diagonal group) on SL( n, R)/SL( n, Z).\n\nIt was pointed out by B. Weiss that an interesting symbolic coding for the Weyl chamber flow was used in [181], where a (wrong) proof of the Littlewood conjecture was given (see also [179, 180, 9]). One should mention that symbolic representations in higher dimensions are usually quite involved and not explicit. For example, any Markov partition of a hyperbolic toral automorphism on the 3-dimensional torus must consist of fractal sets (see [29]). F. Ledrappier and S. Mozes suggested to look for a convenient symbolic representation of the Weyl chamber flow using fractal tilings. This approach was successfully applied to construct explicit symbolic representations for some automorphisms and shifts on higher-dimensional tori OPEN PROBLEMS 24\n\n(see [161, 13, 86, 87]) and for the Cartan action on GL(2, Qp) × GL(2, Qq)/Γ, Γ a irreducible lattice (see [145, 147]).",
   "evidence": "The record is Question 55, attributed to S. Katok, in the notes from the June 2004 AIM workshop *Emerging Applications of Measure Rigidity* (scribe A. Gorodnik, dated November 1, 2004):",
   "classification_method": "canonical_text_retained_no_recovery_claim",
   "clean_statement_source": "canonical_problem_field",
   "source_file": "aim-other-notes.json",
   "source_index": 165,
   "attempt": 1
  }
 }
}