ulamai/UnsolvedMath

1{2 "schema_version": "1.0",3 "generated_at": "2026-08-14T00:00:00Z",4 "canonical_domains": 26,5 "canonical_records": 3359,6 "status_definitions": {7  "exact": "Canonical problem text retained as the clean formulation; no statement repair was required.",8  "corrected_verified": "Clean formulation repairs the canonical extraction using checked source evidence or an explicit mechanically checkable typo correction.",9  "reconstructed_unverified": "A useful clean reading is recorded, but available source evidence does not uniquely verify it.",10  "unrecoverable": "No defensible single clean statement can be recovered; clean_statement is null and the original is preserved."11 },12 "counts": {13  "corrected_verified": 61,14  "exact": 2886,15  "reconstructed_unverified": 385,16  "unrecoverable": 2717 },18 "clean_statement_sources": {19  "canonical_problem_field": 2886,20  "explicit_typographical_substitution": 23,21  "labeled_recovery_in_report_section_1": 115,22  "no_clean_statement": 27,23  "no_safe_clean_extraction": 30824 },25 "records": {26  "AIM-ALGEBRAIC_GEOMETRY-0001": {27   "statement_status": "exact",28   "original_statement": "How to find (algebraically, say) a mirror construction for fake projective planes (certain rigid surface of general type with same Hodge numbers as projective space, otherwise very different).\n\n\\begin{itemize}\n\\item $\\bullet$ There exist 100 of them, but only 22 have known explicit equations.\n\\item $\\bullet$ Not complete intersections\n\\end{itemize}",29   "clean_statement": "How to find (algebraically, say) a mirror construction for fake projective planes (certain rigid surface of general type with same Hodge numbers as projective space, otherwise very different).\n\n\\begin{itemize}\n\\item $\\bullet$ There exist 100 of them, but only 22 have known explicit equations.\n\\item $\\bullet$ Not complete intersections\n\\end{itemize}",30   "public_statement": "How to find (algebraically, say) a mirror construction for fake projective planes (certain rigid surface of general type with same Hodge numbers as projective space, otherwise very different).\n\n\\begin{itemize}\n\\item $\\bullet$ There exist 100 of them, but only 22 have known explicit equations.\n\\item $\\bullet$ Not complete intersections\n\\end{itemize}",31   "evidence": "The canonical record is item 1.1, “Mirror constructions,” from the AIM workshop *Syzygies and mirror symmetry*. Its text is:",32   "classification_method": "canonical_text_retained_no_recovery_claim",33   "clean_statement_source": "canonical_problem_field",34   "source_file": "aim-algebraic-geometry-notes.json",35   "source_index": 0,36   "attempt": 137  },38  "AIM-ALGEBRAIC_GEOMETRY-0002": {39   "statement_status": "exact",40   "original_statement": "Given a singular variety in char. p, $X$, $\\mathcal{F}:=Frobenius$, then $\\mathcal{F}^{>>0}(Perf(X))=Coh(X)$ (as dg enhanced categories of complexes) [B-I-L-M-P].\n\nCan we see the mirror to this fact explicitly in case e.g. of mirrors to $X=\\{xy=0\\}\\subset \\mathbb{A}_{\\mathbb{F}_p}$? More generally, what is the symplectic mirror to Frobenius for each $p$?",41   "clean_statement": "Given a singular variety in char. p, $X$, $\\mathcal{F}:=Frobenius$, then $\\mathcal{F}^{>>0}(Perf(X))=Coh(X)$ (as dg enhanced categories of complexes) [B-I-L-M-P].\n\nCan we see the mirror to this fact explicitly in case e.g. of mirrors to $X=\\{xy=0\\}\\subset \\mathbb{A}_{\\mathbb{F}_p}$? More generally, what is the symplectic mirror to Frobenius for each $p$?",42   "public_statement": "Given a singular variety in char. p, $X$, $\\mathcal{F}:=Frobenius$, then $\\mathcal{F}^{>>0}(Perf(X))=Coh(X)$ (as dg enhanced categories of complexes) [B-I-L-M-P].\n\nCan we see the mirror to this fact explicitly in case e.g. of mirrors to $X=\\{xy=0\\}\\subset \\mathbb{A}_{\\mathbb{F}_p}$? More generally, what is the symplectic mirror to Frobenius for each $p$?",43   "evidence": "The canonical record is problem 1.2 in the AIM workshop list *Syzygies and mirror symmetry*, section “Mirror constructions.” Its text is:",44   "classification_method": "canonical_text_retained_no_recovery_claim",45   "clean_statement_source": "canonical_problem_field",46   "source_file": "aim-algebraic-geometry-notes.json",47   "source_index": 1,48   "attempt": 149  },50  "AIM-ALGEBRAIC_GEOMETRY-0003": {51   "statement_status": "exact",52   "original_statement": "Probe what is know about symp. mirrors to algebraic geometric operations (not necessary toric) such as blow-ups?",53   "clean_statement": "Probe what is know about symp. mirrors to algebraic geometric operations (not necessary toric) such as blow-ups?",54   "public_statement": "Probe what is know about symp. mirrors to algebraic geometric operations (not necessary toric) such as blow-ups?",55   "evidence": "The canonical record is problem 1.3 in the AIM workshop *Syzygies and mirror symmetry*, section “Mirror constructions.” Its exact extracted text is:",56   "classification_method": "canonical_text_retained_no_recovery_claim",57   "clean_statement_source": "canonical_problem_field",58   "source_file": "aim-algebraic-geometry-notes.json",59   "source_index": 2,60   "attempt": 161  },62  "AIM-ALGEBRAIC_GEOMETRY-0004": {63   "statement_status": "exact",64   "original_statement": "Explore the mirror symmetry correspondence between Morse-theoretic decompositions and algebraic/algebro-geometric decompositions",65   "clean_statement": "Explore the mirror symmetry correspondence between Morse-theoretic decompositions and algebraic/algebro-geometric decompositions",66   "public_statement": "Explore the mirror symmetry correspondence between Morse-theoretic decompositions and algebraic/algebro-geometric decompositions",67   "evidence": "The exact canonical record is problem 1.4 in the AIM workshop *Syzygies and mirror symmetry*, section “Mirror constructions”:",68   "classification_method": "canonical_text_retained_no_recovery_claim",69   "clean_statement_source": "canonical_problem_field",70   "source_file": "aim-algebraic-geometry-notes.json",71   "source_index": 3,72   "attempt": 173  },74  "AIM-ALGEBRAIC_GEOMETRY-0005": {75   "statement_status": "exact",76   "original_statement": "HMS for ATVs (e.g. $x^2+y^2+z^2=0/\\mathbb{R}$)\n\nDoes the derived category of an arithmetic toric variety (i.e., one defined over a non-algebraically closed field) admits an exceptional collection?\n(e.g., $\\{x^2+y^2+z^2=0\\}$ in $\\mathbb{P}^2_{\\mathbb{R}}$)\n\\begin{itemize}\n\\item $\\bullet$ Over $\\mathbb{C}$ is just $\\mathbb{P}^1$ but has no $\\mathbb{R}$-points\n\\item $\\bullet$ has exceptional collection $$ with $End(\\mathcal{E})=\\mathbb{H}$.\n\\item $\\bullet$ has an action of $S^1$ which is the restriction of the $S^1\\subset \\mathbb{C}^*$ action on $\\mathbb{C}-points)\n\\item $\\bullet$ toric data here is roughly data of a fan + data of a $Gal(\\overline{k}/k)$ action on the fan + ...\n\\item $\\bullet$ One source: An exceptional collection where fan automorphism permutes objects; e.g. powers of toric Frobenius applied to $\\mathcal{O}_X$ (gives generating objects acted on by $Gal$, sometimes exceptional collection).\n\\end{itemize}",77   "clean_statement": "HMS for ATVs (e.g. $x^2+y^2+z^2=0/\\mathbb{R}$)\n\nDoes the derived category of an arithmetic toric variety (i.e., one defined over a non-algebraically closed field) admits an exceptional collection?\n(e.g., $\\{x^2+y^2+z^2=0\\}$ in $\\mathbb{P}^2_{\\mathbb{R}}$)\n\\begin{itemize}\n\\item $\\bullet$ Over $\\mathbb{C}$ is just $\\mathbb{P}^1$ but has no $\\mathbb{R}$-points\n\\item $\\bullet$ has exceptional collection $$ with $End(\\mathcal{E})=\\mathbb{H}$.\n\\item $\\bullet$ has an action of $S^1$ which is the restriction of the $S^1\\subset \\mathbb{C}^*$ action on $\\mathbb{C}-points)\n\\item $\\bullet$ toric data here is roughly data of a fan + data of a $Gal(\\overline{k}/k)$ action on the fan + ...\n\\item $\\bullet$ One source: An exceptional collection where fan automorphism permutes objects; e.g. powers of toric Frobenius applied to $\\mathcal{O}_X$ (gives generating objects acted on by $Gal$, sometimes exceptional collection).\n\\end{itemize}",78   "public_statement": "HMS for ATVs (e.g. $x^2+y^2+z^2=0/\\mathbb{R}$)\n\nDoes the derived category of an arithmetic toric variety (i.e., one defined over a non-algebraically closed field) admits an exceptional collection?\n(e.g., $\\{x^2+y^2+z^2=0\\}$ in $\\mathbb{P}^2_{\\mathbb{R}}$)\n\\begin{itemize}\n\\item $\\bullet$ Over $\\mathbb{C}$ is just $\\mathbb{P}^1$ but has no $\\mathbb{R}$-points\n\\item $\\bullet$ has exceptional collection $$ with $End(\\mathcal{E})=\\mathbb{H}$.\n\\item $\\bullet$ has an action of $S^1$ which is the restriction of the $S^1\\subset \\mathbb{C}^*$ action on $\\mathbb{C}-points)\n\\item $\\bullet$ toric data here is roughly data of a fan + data of a $Gal(\\overline{k}/k)$ action on the fan + ...\n\\item $\\bullet$ One source: An exceptional collection where fan automorphism permutes objects; e.g. powers of toric Frobenius applied to $\\mathcal{O}_X$ (gives generating objects acted on by $Gal$, sometimes exceptional collection).\n\\end{itemize}",79   "evidence": "The canonical record asks:",80   "classification_method": "canonical_text_retained_no_recovery_claim",81   "clean_statement_source": "canonical_problem_field",82   "source_file": "aim-algebraic-geometry-notes.json",83   "source_index": 4,84   "attempt": 185  },86  "AIM-ALGEBRAIC_GEOMETRY-0006": {87   "statement_status": "exact",88   "original_statement": "Does every smooth toric variety admits an exceptional collection of sheaves\n\\begin{itemize}\n\\item $\\bullet$ if replace sheaves (good objects of $D^b(X)$, answer is yes)\n\\item $\\bullet$ First open case is the case of toric 3-folds\n\\item $\\bullet$ if replace sheaves with line bundles, answer is no [Etimov '14] counterexample smooth Fano toric with $dim\\geq 0$ and $Pic=3$.\n\\item $\\bullet$ For surfaces can do with line bundles, read off explicitly from fan (exposition of this in [Hacking-Keating])\n\\item $\\bullet$ More specifically, is it possible to find an exceptional collection from (push-forwards of) line bundles on toric strata? (This case can be analyzed, perhaps).\n\\end{itemize}",89   "clean_statement": "Does every smooth toric variety admits an exceptional collection of sheaves\n\\begin{itemize}\n\\item $\\bullet$ if replace sheaves (good objects of $D^b(X)$, answer is yes)\n\\item $\\bullet$ First open case is the case of toric 3-folds\n\\item $\\bullet$ if replace sheaves with line bundles, answer is no [Etimov '14] counterexample smooth Fano toric with $dim\\geq 0$ and $Pic=3$.\n\\item $\\bullet$ For surfaces can do with line bundles, read off explicitly from fan (exposition of this in [Hacking-Keating])\n\\item $\\bullet$ More specifically, is it possible to find an exceptional collection from (push-forwards of) line bundles on toric strata? (This case can be analyzed, perhaps).\n\\end{itemize}",90   "public_statement": "Does every smooth toric variety admits an exceptional collection of sheaves\n\\begin{itemize}\n\\item $\\bullet$ if replace sheaves (good objects of $D^b(X)$, answer is yes)\n\\item $\\bullet$ First open case is the case of toric 3-folds\n\\item $\\bullet$ if replace sheaves with line bundles, answer is no [Etimov '14] counterexample smooth Fano toric with $dim\\geq 0$ and $Pic=3$.\n\\item $\\bullet$ For surfaces can do with line bundles, read off explicitly from fan (exposition of this in [Hacking-Keating])\n\\item $\\bullet$ More specifically, is it possible to find an exceptional collection from (push-forwards of) line bundles on toric strata? (This case can be analyzed, perhaps).\n\\end{itemize}",91   "evidence": "The source is AIM Problem 2.2 in the “Syzygies and mirror symmetry” list, section “Exceptional collections.” It reads:",92   "classification_method": "canonical_text_retained_no_recovery_claim",93   "clean_statement_source": "canonical_problem_field",94   "source_file": "aim-algebraic-geometry-notes.json",95   "source_index": 5,96   "attempt": 197  },98  "AIM-ALGEBRAIC_GEOMETRY-0007": {99   "statement_status": "exact",100   "original_statement": "Can the \"looking glass\" parallel/analogy between rational homotopy theory (= symplectic geometry of cotangent bundles) and local ring theory be generalized/ further exploited to other symplectic manifolds (e.g. Weinstein manifolds)?\n\nA)\n\\begin{itemize}\n\\item $\\bullet$ can \"attach handles\" to Weinstein manifolds\n\\item $\\bullet$ there exists a natural \"categorical compactification\" of Fukaya categories via \"stop\" or \"Lefschetz fibrations\"\n\\end{itemize}\nB)\n\\begin{itemize}\n\\item $\\bullet$ Can rational homotopy theory e.g. Sullivan minimal models can be used to e.g. construct resolutions?\n\\end{itemize}\nComment: The initial parallel comes from\n$$C_*(\\Omega_q M)=Ext_{C^{\\bullet}M}(k,k)=A$$\nwhere\n$$C_*(\\Omega_q M)\\simeq End_{Fuk(T^*M)}(T^*_q M, T^*_q M)$$\n(Here $M$ is assumed to be simply connected).",101   "clean_statement": "Can the \"looking glass\" parallel/analogy between rational homotopy theory (= symplectic geometry of cotangent bundles) and local ring theory be generalized/ further exploited to other symplectic manifolds (e.g. Weinstein manifolds)?\n\nA)\n\\begin{itemize}\n\\item $\\bullet$ can \"attach handles\" to Weinstein manifolds\n\\item $\\bullet$ there exists a natural \"categorical compactification\" of Fukaya categories via \"stop\" or \"Lefschetz fibrations\"\n\\end{itemize}\nB)\n\\begin{itemize}\n\\item $\\bullet$ Can rational homotopy theory e.g. Sullivan minimal models can be used to e.g. construct resolutions?\n\\end{itemize}\nComment: The initial parallel comes from\n$$C_*(\\Omega_q M)=Ext_{C^{\\bullet}M}(k,k)=A$$\nwhere\n$$C_*(\\Omega_q M)\\simeq End_{Fuk(T^*M)}(T^*_q M, T^*_q M)$$\n(Here $M$ is assumed to be simply connected).",102   "public_statement": "Can the \"looking glass\" parallel/analogy between rational homotopy theory (= symplectic geometry of cotangent bundles) and local ring theory be generalized/ further exploited to other symplectic manifolds (e.g. Weinstein manifolds)?\n\nA)\n\\begin{itemize}\n\\item $\\bullet$ can \"attach handles\" to Weinstein manifolds\n\\item $\\bullet$ there exists a natural \"categorical compactification\" of Fukaya categories via \"stop\" or \"Lefschetz fibrations\"\n\\end{itemize}\nB)\n\\begin{itemize}\n\\item $\\bullet$ Can rational homotopy theory e.g. Sullivan minimal models can be used to e.g. construct resolutions?\n\\end{itemize}\nComment: The initial parallel comes from\n$$C_*(\\Omega_q M)=Ext_{C^{\\bullet}M}(k,k)=A$$\nwhere\n$$C_*(\\Omega_q M)\\simeq End_{Fuk(T^*M)}(T^*_q M, T^*_q M)$$\n(Here $M$ is assumed to be simply connected).",103   "evidence": "This is Problem 3.1, “Symplectic geometry,” from the AIM workshop list *Syzygies and mirror symmetry*. The canonical record asks:",104   "classification_method": "canonical_text_retained_no_recovery_claim",105   "clean_statement_source": "canonical_problem_field",106   "source_file": "aim-algebraic-geometry-notes.json",107   "source_index": 6,108   "attempt": 1109  },110  "AIM-ALGEBRAIC_GEOMETRY-0008": {111   "statement_status": "exact",112   "original_statement": "Compare $\\Delta$-resolutions\n\nCan we directly compare the Hanlon-Hicks-Lazarev and Brown-Erman construction of resolutions?\n\n$\\bullet$ Potential mechanism: (starting in degree 0)\n\n{vertices/points in stratification of $T^n$ arising in HHL}\nis in one to one correspondence with\n{additional generators for normalization of the semi-group ring, i.e. elements coming from saturation of semigroup}",113   "clean_statement": "Compare $\\Delta$-resolutions\n\nCan we directly compare the Hanlon-Hicks-Lazarev and Brown-Erman construction of resolutions?\n\n$\\bullet$ Potential mechanism: (starting in degree 0)\n\n{vertices/points in stratification of $T^n$ arising in HHL}\nis in one to one correspondence with\n{additional generators for normalization of the semi-group ring, i.e. elements coming from saturation of semigroup}",114   "public_statement": "Compare $\\Delta$-resolutions\n\nCan we directly compare the Hanlon-Hicks-Lazarev and Brown-Erman construction of resolutions?\n\n$\\bullet$ Potential mechanism: (starting in degree 0)\n\n{vertices/points in stratification of $T^n$ arising in HHL}\nis in one to one correspondence with\n{additional generators for normalization of the semi-group ring, i.e. elements coming from saturation of semigroup}",115   "evidence": "The canonical AIM record reads:",116   "classification_method": "canonical_text_retained_no_recovery_claim",117   "clean_statement_source": "canonical_problem_field",118   "source_file": "aim-algebraic-geometry-notes.json",119   "source_index": 7,120   "attempt": 2121  },122  "AIM-ALGEBRAIC_GEOMETRY-0009": {123   "statement_status": "exact",124   "original_statement": "Topology/Morse theory of cellular resolutions\n\nCan we interpret resolutions via Fukaya categories? (e.g. minimizing resolutions through \"cellular\" operations)",125   "clean_statement": "Topology/Morse theory of cellular resolutions\n\nCan we interpret resolutions via Fukaya categories? (e.g. minimizing resolutions through \"cellular\" operations)",126   "public_statement": "Topology/Morse theory of cellular resolutions\n\nCan we interpret resolutions via Fukaya categories? (e.g. minimizing resolutions through \"cellular\" operations)",127   "evidence": "The canonical record is Problem 4.2 in the “Resolutions” section of the AIM problem list *Syzygies and mirror symmetry*:",128   "classification_method": "canonical_text_retained_no_recovery_claim",129   "clean_statement_source": "canonical_problem_field",130   "source_file": "aim-algebraic-geometry-notes.json",131   "source_index": 8,132   "attempt": 1133  },134  "AIM-ALGEBRAIC_GEOMETRY-0010": {135   "statement_status": "exact",136   "original_statement": "Topology/Morse theory of cellular resolutions\n\nThe Morse-theoretic description of $H_*(M)$ can be refined (at e.g. level of \"framed flow categories\"). can this refined decomposition tell us something on the algebraic side?",137   "clean_statement": "Topology/Morse theory of cellular resolutions\n\nThe Morse-theoretic description of $H_*(M)$ can be refined (at e.g. level of \"framed flow categories\"). can this refined decomposition tell us something on the algebraic side?",138   "public_statement": "Topology/Morse theory of cellular resolutions\n\nThe Morse-theoretic description of $H_*(M)$ can be refined (at e.g. level of \"framed flow categories\"). can this refined decomposition tell us something on the algebraic side?",139   "evidence": "The canonical record is Problem 4.3 in the “Resolutions” section of the AIM workshop list *Syzygies and mirror symmetry*. The source page was checked directly on 2026-07-22 and agrees with the JSON record. Its title and text are:",140   "classification_method": "canonical_text_retained_no_recovery_claim",141   "clean_statement_source": "canonical_problem_field",142   "source_file": "aim-algebraic-geometry-notes.json",143   "source_index": 9,144   "attempt": 1145  },146  "AIM-ALGEBRAIC_GEOMETRY-0011": {147   "statement_status": "exact",148   "original_statement": "$T^n=K(\\pi,1)$\n\nGiven a cellular resolution, is there a meaningful way to algebraically interpret topology of the cell complex beyond homology e.g. $\\pi_1$?",149   "clean_statement": "$T^n=K(\\pi,1)$\n\nGiven a cellular resolution, is there a meaningful way to algebraically interpret topology of the cell complex beyond homology e.g. $\\pi_1$?",150   "public_statement": "$T^n=K(\\pi,1)$\n\nGiven a cellular resolution, is there a meaningful way to algebraically interpret topology of the cell complex beyond homology e.g. $\\pi_1$?",151   "evidence": "The assigned record is Problem 4.4 in the AIM problem list *Syzygies and mirror symmetry*, section “Resolutions”:",152   "classification_method": "canonical_text_retained_no_recovery_claim",153   "clean_statement_source": "canonical_problem_field",154   "source_file": "aim-algebraic-geometry-notes.json",155   "source_index": 10,156   "attempt": 1157  },158  "AIM-ALGEBRAIC_GEOMETRY-0012": {159   "statement_status": "exact",160   "original_statement": "Topology/Morse theory of resolutions\n\nIs there an analogue of \"Morse theory for cellular resolutions\" for virtual resolutions?",161   "clean_statement": "Topology/Morse theory of resolutions\n\nIs there an analogue of \"Morse theory for cellular resolutions\" for virtual resolutions?",162   "public_statement": "Topology/Morse theory of resolutions\n\nIs there an analogue of \"Morse theory for cellular resolutions\" for virtual resolutions?",163   "evidence": "The canonical record is Problem 4.5 in the AIM list *Syzygies and mirror symmetry*, section “Resolutions”:",164   "classification_method": "canonical_text_retained_no_recovery_claim",165   "clean_statement_source": "canonical_problem_field",166   "source_file": "aim-algebraic-geometry-notes.json",167   "source_index": 11,168   "attempt": 1169  },170  "AIM-ALGEBRAIC_GEOMETRY-0013": {171   "statement_status": "exact",172   "original_statement": "Observation: The current definition of virtual resolution is so broad and it is not meaningful to parameterize, then.\n\nIs there a well-behaved moduli theory of virtual resolutions equipped with extra structure? e.g. condition under which there exists a finite set? or space/ stacks/...?\n(c.f. [Boij-Sölderberg] theory constructs a meaningful cone of Betti tables of all free complexes. Look at subcone of virtual resolutions of a given $M$)",173   "clean_statement": "Observation: The current definition of virtual resolution is so broad and it is not meaningful to parameterize, then.\n\nIs there a well-behaved moduli theory of virtual resolutions equipped with extra structure? e.g. condition under which there exists a finite set? or space/ stacks/...?\n(c.f. [Boij-Sölderberg] theory constructs a meaningful cone of Betti tables of all free complexes. Look at subcone of virtual resolutions of a given $M$)",174   "public_statement": "Observation: The current definition of virtual resolution is so broad and it is not meaningful to parameterize, then.\n\nIs there a well-behaved moduli theory of virtual resolutions equipped with extra structure? e.g. condition under which there exists a finite set? or space/ stacks/...?\n(c.f. [Boij-Sölderberg] theory constructs a meaningful cone of Betti tables of all free complexes. Look at subcone of virtual resolutions of a given $M$)",175   "evidence": "This is Problem 4.6 in the **Resolutions** section of the AIM list *Syzygies and mirror symmetry*. The canonical record and the live AIM page agree. The live page was checked on 2026-07-22 and contains no status update or attached remark. Its statement is:",176   "classification_method": "canonical_text_retained_no_recovery_claim",177   "clean_statement_source": "canonical_problem_field",178   "source_file": "aim-algebraic-geometry-notes.json",179   "source_index": 12,180   "attempt": 1181  },182  "AIM-ALGEBRAIC_GEOMETRY-0014": {183   "statement_status": "exact",184   "original_statement": "What does the Fourier-Mukai transform by either resolution of $\\Delta$ applied to $M$ tell us about $M$? More generally what does applying Fourier-Mukai transforms of resolutions of toric subvarieties tell us about the modules apply them to? [e.g. in the spirit of the way we obtain Eagon-Northcott-type complexes for modules]",185   "clean_statement": "What does the Fourier-Mukai transform by either resolution of $\\Delta$ applied to $M$ tell us about $M$? More generally what does applying Fourier-Mukai transforms of resolutions of toric subvarieties tell us about the modules apply them to? [e.g. in the spirit of the way we obtain Eagon-Northcott-type complexes for modules]",186   "public_statement": "What does the Fourier-Mukai transform by either resolution of $\\Delta$ applied to $M$ tell us about $M$? More generally what does applying Fourier-Mukai transforms of resolutions of toric subvarieties tell us about the modules apply them to? [e.g. in the spirit of the way we obtain Eagon-Northcott-type complexes for modules]",187   "evidence": "The exact canonical record is AIM Problem 5.1 from the 2023 workshop *Syzygies and mirror symmetry*:",188   "classification_method": "canonical_text_retained_no_recovery_claim",189   "clean_statement_source": "canonical_problem_field",190   "source_file": "aim-algebraic-geometry-notes.json",191   "source_index": 13,192   "attempt": 1193  },194  "AIM-ALGEBRAIC_GEOMETRY-0015": {195   "statement_status": "exact",196   "original_statement": "Given $M$ an $S$-module, $S=Cox(X)$,\n\\begin{enumerate}\n\\item can we always find another $S$-module $M'$ satisfying\n\\begin{itemize}\n\\item $\\bullet$ $\\tilde{M}=\\tilde{M}'$\n\\item $\\bullet$ $pdim(M')\\geq dim(X)$\n\\end{itemize}\nThis is known to be true if $X=\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$.\n\n\\item What is the relation of this to examples of [Chardin-D'Cruz] where adding embedded components drops regularity?\n\n\\item Potential mechanism by truncating $M$ to $(M)_{\\geq d}$? (This works for $\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$'s)\n\\item Alternatively, could there be an invariant of $\\tilde{M}$ which obstructs existence of such an $M'$?\n\\end{enumerate}",197   "clean_statement": "Given $M$ an $S$-module, $S=Cox(X)$,\n\\begin{enumerate}\n\\item can we always find another $S$-module $M'$ satisfying\n\\begin{itemize}\n\\item $\\bullet$ $\\tilde{M}=\\tilde{M}'$\n\\item $\\bullet$ $pdim(M')\\geq dim(X)$\n\\end{itemize}\nThis is known to be true if $X=\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$.\n\n\\item What is the relation of this to examples of [Chardin-D'Cruz] where adding embedded components drops regularity?\n\n\\item Potential mechanism by truncating $M$ to $(M)_{\\geq d}$? (This works for $\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$'s)\n\\item Alternatively, could there be an invariant of $\\tilde{M}$ which obstructs existence of such an $M'$?\n\\end{enumerate}",198   "public_statement": "Given $M$ an $S$-module, $S=Cox(X)$,\n\\begin{enumerate}\n\\item can we always find another $S$-module $M'$ satisfying\n\\begin{itemize}\n\\item $\\bullet$ $\\tilde{M}=\\tilde{M}'$\n\\item $\\bullet$ $pdim(M')\\geq dim(X)$\n\\end{itemize}\nThis is known to be true if $X=\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$.\n\n\\item What is the relation of this to examples of [Chardin-D'Cruz] where adding embedded components drops regularity?\n\n\\item Potential mechanism by truncating $M$ to $(M)_{\\geq d}$? (This works for $\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$'s)\n\\item Alternatively, could there be an invariant of $\\tilde{M}$ which obstructs existence of such an $M'$?\n\\end{enumerate}",199   "evidence": "The assigned record is Problem 6.1 in the “Modules over the Cox ring” section of the AIM list *Syzygies and mirror symmetry*. The canonical record says:",200   "classification_method": "canonical_text_retained_no_recovery_claim",201   "clean_statement_source": "canonical_problem_field",202   "source_file": "aim-algebraic-geometry-notes.json",203   "source_index": 14,204   "attempt": 1205  },206  "AIM-ALGEBRAIC_GEOMETRY-0016": {207   "statement_status": "exact",208   "original_statement": "What can be said about the Orlov spectrum of a toric variety (or any $X$)? (e.g. about gaps, sequences, etc.)\n\nKnown:\n\\begin{itemize}\n\\item $OrSpec(\\mathbb{P}^1)=\\{1,2\\}$\n\\item On a toric surface, there exists full arbitrary large consecutive sequence.\n\\item More genrally, by replacing $D^b(X)$ with $\\mathcal{C}$, there exists a notion of $OrSpec(\\mathcal{C})$ and $OrSpec(Rep (Q))$ ($Q$ is a quiver), computed when $Q$ is of A/D types.\n\\end{itemize}",209   "clean_statement": "What can be said about the Orlov spectrum of a toric variety (or any $X$)? (e.g. about gaps, sequences, etc.)\n\nKnown:\n\\begin{itemize}\n\\item $OrSpec(\\mathbb{P}^1)=\\{1,2\\}$\n\\item On a toric surface, there exists full arbitrary large consecutive sequence.\n\\item More genrally, by replacing $D^b(X)$ with $\\mathcal{C}$, there exists a notion of $OrSpec(\\mathcal{C})$ and $OrSpec(Rep (Q))$ ($Q$ is a quiver), computed when $Q$ is of A/D types.\n\\end{itemize}",210   "public_statement": "What can be said about the Orlov spectrum of a toric variety (or any $X$)? (e.g. about gaps, sequences, etc.)\n\nKnown:\n\\begin{itemize}\n\\item $OrSpec(\\mathbb{P}^1)=\\{1,2\\}$\n\\item On a toric surface, there exists full arbitrary large consecutive sequence.\n\\item More genrally, by replacing $D^b(X)$ with $\\mathcal{C}$, there exists a notion of $OrSpec(\\mathcal{C})$ and $OrSpec(Rep (Q))$ ($Q$ is a quiver), computed when $Q$ is of A/D types.\n\\end{itemize}",211   "evidence": "The canonical record is Problem 7.1 in the AIM list *Syzygies and mirror symmetry*, section \"Orlov spectrum and Rouquier dimension\":",212   "classification_method": "canonical_text_retained_no_recovery_claim",213   "clean_statement_source": "canonical_problem_field",214   "source_file": "aim-algebraic-geometry-notes.json",215   "source_index": 15,216   "attempt": 1217  },218  "AIM-ALGEBRAIC_GEOMETRY-0017": {219   "statement_status": "exact",220   "original_statement": "Rdim of $D^b Sing$\n\nWhat about Rouquier dimension or orlov spectrum of $D^b(Sing(X))=D^b(Coh(X))/D^b(Perf(X))$?",221   "clean_statement": "Rdim of $D^b Sing$\n\nWhat about Rouquier dimension or orlov spectrum of $D^b(Sing(X))=D^b(Coh(X))/D^b(Perf(X))$?",222   "public_statement": "Rdim of $D^b Sing$\n\nWhat about Rouquier dimension or orlov spectrum of $D^b(Sing(X))=D^b(Coh(X))/D^b(Perf(X))$?",223   "evidence": "The exact canonical record is problem 7.2 in the AIM workshop *Syzygies and mirror symmetry*, section “Orlov spectrum and Rouquier dimension”; it is zero-based record 16 of `aim-algebraic-geometry-notes.json`:",224   "classification_method": "canonical_text_retained_no_recovery_claim",225   "clean_statement_source": "canonical_problem_field",226   "source_file": "aim-algebraic-geometry-notes.json",227   "source_index": 16,228   "attempt": 1229  },230  "AIM-ALGEBRAIC_GEOMETRY-0018": {231   "statement_status": "exact",232   "original_statement": "Same questions as above for $Fuk(X)$ where $X$ is compact symplectic.",233   "clean_statement": "Same questions as above for $Fuk(X)$ where $X$ is compact symplectic.",234   "public_statement": "Same questions as above for $Fuk(X)$ where $X$ is compact symplectic.",235   "evidence": "The canonical record is AIM problem 7.3 from the workshop *Syzygies and mirror symmetry*:",236   "classification_method": "canonical_text_retained_no_recovery_claim",237   "clean_statement_source": "canonical_problem_field",238   "source_file": "aim-algebraic-geometry-notes.json",239   "source_index": 17,240   "attempt": 1241  },242  "AIM-ALGEBRAIC_GEOMETRY-0019": {243   "statement_status": "exact",244   "original_statement": "There are two known ways to bound Rouquier dimension of a category\n\\begin{enumerate}\n\\item bound length of a resolution of a diagonal\n\\item the minimal depth of a presentation of $\\mathcal{C}$ as hocolim (cats with Rouquier dimension $Q$). Meaning if $F:I^{\\leq}\\to Cat$ diagram with $Rdim F(i)=0$, then $Rdim(hocolim F i)\\leq depth(I)$\n\\end{enumerate}\n\nCan (1) be used to bound (2) or vice-versa?",245   "clean_statement": "There are two known ways to bound Rouquier dimension of a category\n\\begin{enumerate}\n\\item bound length of a resolution of a diagonal\n\\item the minimal depth of a presentation of $\\mathcal{C}$ as hocolim (cats with Rouquier dimension $Q$). Meaning if $F:I^{\\leq}\\to Cat$ diagram with $Rdim F(i)=0$, then $Rdim(hocolim F i)\\leq depth(I)$\n\\end{enumerate}\n\nCan (1) be used to bound (2) or vice-versa?",246   "public_statement": "There are two known ways to bound Rouquier dimension of a category\n\\begin{enumerate}\n\\item bound length of a resolution of a diagonal\n\\item the minimal depth of a presentation of $\\mathcal{C}$ as hocolim (cats with Rouquier dimension $Q$). Meaning if $F:I^{\\leq}\\to Cat$ diagram with $Rdim F(i)=0$, then $Rdim(hocolim F i)\\leq depth(I)$\n\\end{enumerate}\n\nCan (1) be used to bound (2) or vice-versa?",247   "evidence": "The displayed text really does contain \\(Q\\), so this is not an extraction error. Because the very next clause says \\(\\operatorname{Rdim}F(i)=0\\), the only coherent reconstruction is that “\\(Q\\)” is a typographical substitution for “\\(0\\).” The notation \\(I^{\\leq}\\) is not defined on the page. I interpret it as a finite partially ordered set \\((I,\\leq)\\), regarded as a category. This agrees with the precise finite-poset formulation subsequently used by Bai--Côté.",248   "classification_method": "explicit_no_change_evidence",249   "clean_statement_source": "canonical_problem_field",250   "source_file": "aim-algebraic-geometry-notes.json",251   "source_index": 18,252   "attempt": 1253  },254  "AIM-ALGEBRAIC_GEOMETRY-0020": {255   "statement_status": "exact",256   "original_statement": "What can be said about the Rouquier dimension of non-toric varieties and/for their symplectic mirrors?\n\\begin{itemize}\n\\item Toric varieties of dim n $\\longleftrightarrow$(mirror) ($T^*T^n$,some stop)\n\\item What known symplecticly for $(T^*M,stops)$ can be find in (Bai-Cote, Hanlon-Hicks-Lazarev, Favero-Huang)\n\\end{itemize}\ne.g. is there a symplectic $(X^{2n},f)$ such that $Rdim \\ Fuk(X,f)\\lneq n$? (for instance by finding $(X,f)$ mirror to a singular $Y$ with $Rdim(D^b(Y))>dim(Y)$)",257   "clean_statement": "What can be said about the Rouquier dimension of non-toric varieties and/for their symplectic mirrors?\n\\begin{itemize}\n\\item Toric varieties of dim n $\\longleftrightarrow$(mirror) ($T^*T^n$,some stop)\n\\item What known symplecticly for $(T^*M,stops)$ can be find in (Bai-Cote, Hanlon-Hicks-Lazarev, Favero-Huang)\n\\end{itemize}\ne.g. is there a symplectic $(X^{2n},f)$ such that $Rdim \\ Fuk(X,f)\\lneq n$? (for instance by finding $(X,f)$ mirror to a singular $Y$ with $Rdim(D^b(Y))>dim(Y)$)",258   "public_statement": "What can be said about the Rouquier dimension of non-toric varieties and/for their symplectic mirrors?\n\\begin{itemize}\n\\item Toric varieties of dim n $\\longleftrightarrow$(mirror) ($T^*T^n$,some stop)\n\\item What known symplecticly for $(T^*M,stops)$ can be find in (Bai-Cote, Hanlon-Hicks-Lazarev, Favero-Huang)\n\\end{itemize}\ne.g. is there a symplectic $(X^{2n},f)$ such that $Rdim \\ Fuk(X,f)\\lneq n$? (for instance by finding $(X,f)$ mirror to a singular $Y$ with $Rdim(D^b(Y))>dim(Y)$)",259   "evidence": "This record is Problem 7.5 in the AIM workshop list *Syzygies and mirror symmetry*, section “Orlov spectrum and Rouquier dimension.” The canonical record and the live AIM page were both checked. The page really contains <code>\\lneq</code>; the apparent conflict below is therefore present in the source, rather than being introduced by JSON extraction.",260   "classification_method": "canonical_text_retained_no_recovery_claim",261   "clean_statement_source": "canonical_problem_field",262   "source_file": "aim-algebraic-geometry-notes.json",263   "source_index": 19,264   "attempt": 1265  },266  "AIM-ALGEBRAIC_GEOMETRY-0021": {267   "statement_status": "reconstructed_unverified",268   "original_statement": "What operations on Fukaya categories have meaningful commutative algebra translations? e.g. maybe a spherical twist (via mirror symmetry Dehn twist) around a spherical object supported on irrelevant virtual resolution, or more generally induces an aut (space of virtual resolutions)?",269   "clean_statement": null,270   "public_statement": "What operations on Fukaya categories have meaningful commutative algebra translations? e.g. maybe a spherical twist (via mirror symmetry Dehn twist) around a spherical object supported on irrelevant virtual resolution, or more generally induces an aut (space of virtual resolutions)?",271   "evidence": "* Section 4 asks for Morse theory and a moduli theory for virtual resolutions; Problem 4.6 explicitly asks for a “space/stacks” of virtual resolutions. * Section 6 concerns modules over a Cox ring. * Thus “induces an aut (space of virtual resolutions)” most plausibly abbreviates “induces an automorphism of the space of virtual resolutions”. This is a reconstruction, not verified source text. * “A spherical object supported on irrelevant virtual resolution” has two possible meanings: (i) the center itself has Cox homology supported on the irrelevant locus, or (ii) a nonzero spherical object on the toric variety is represented by a virtual resolution that is allowed irrelevant higher homology. The distinction is decisive. Under (i) the center is zero after sheafification; under (ii) it can define a genuine twist, but only in the quotient by irrelevant homology.",272   "classification_method": "explicit_uncertainty_evidence",273   "clean_statement_source": "no_safe_clean_extraction",274   "source_file": "aim-algebraic-geometry-notes.json",275   "source_index": 20,276   "attempt": 1277  },278  "AIM-ALGEBRAIC_GEOMETRY-0022": {279   "statement_status": "exact",280   "original_statement": "[Wang, Borisov-Wang]: Found a condition on underlying fan for a toric stack to admit infinitely many acyclic line bundles (meaning $\\mathcal{L}$ with $H^*(\\mathcal{L})=0$ for all $*$, e.g. $\\mathcal{O}(-1)$ on $\\mathbb{P}^n$) (e.g. in $2D$ if fan has collinear rays).\n\n\\begin{enumerate}\n\\item Understand symplectic mirror of these criteria.\n\\item Understand consequences of generalizations of these critera (e.g. if there exists a toric subvariety whose fans is contained in a linear subspace) or if generalized.\n\\end{enumerate}",281   "clean_statement": "[Wang, Borisov-Wang]: Found a condition on underlying fan for a toric stack to admit infinitely many acyclic line bundles (meaning $\\mathcal{L}$ with $H^*(\\mathcal{L})=0$ for all $*$, e.g. $\\mathcal{O}(-1)$ on $\\mathbb{P}^n$) (e.g. in $2D$ if fan has collinear rays).\n\n\\begin{enumerate}\n\\item Understand symplectic mirror of these criteria.\n\\item Understand consequences of generalizations of these critera (e.g. if there exists a toric subvariety whose fans is contained in a linear subspace) or if generalized.\n\\end{enumerate}",282   "public_statement": "[Wang, Borisov-Wang]: Found a condition on underlying fan for a toric stack to admit infinitely many acyclic line bundles (meaning $\\mathcal{L}$ with $H^*(\\mathcal{L})=0$ for all $*$, e.g. $\\mathcal{O}(-1)$ on $\\mathbb{P}^n$) (e.g. in $2D$ if fan has collinear rays).\n\n\\begin{enumerate}\n\\item Understand symplectic mirror of these criteria.\n\\item Understand consequences of generalizations of these critera (e.g. if there exists a toric subvariety whose fans is contained in a linear subspace) or if generalized.\n\\end{enumerate}",283   "evidence": "The canonical record is AIM Problem List 9.1 in the section “Line bundles over toric stacks” of *Syzygies and mirror symmetry*. The live AIM page was checked on 2026-07-22 and agrees with the record in `input.json`. Its mathematical content is:",284   "classification_method": "canonical_text_retained_no_recovery_claim",285   "clean_statement_source": "canonical_problem_field",286   "source_file": "aim-algebraic-geometry-notes.json",287   "source_index": 21,288   "attempt": 1289  },290  "AIM-ALGEBRAIC_GEOMETRY-0023": {291   "statement_status": "exact",292   "original_statement": "The motivic filtration on the algebraic $K$-theory of the sphere spectrum\n\nDue to work of Rognes, Blumberg--Mandell, and others, we understand the algebraic K-theory of the sphere spectrum fairly well. Recent work of Hahn--Raksit--Wilson \\cite{arXiv:2206.11208} gives a motivic filtration on TC of $\\mathbb{E}_{\\infty}$-rings which is closely related to the motivic filtration on algebraic K-theory in motivic homotopy theory.\n\nCan one understand the motivic filtration (or motivic spectral sequence) on the $\\mathrm{TC}$ of the sphere spectrum? What about for the motivic filtration on related invariants such as $\\mathrm{THH}$, $\\mathrm{TP}$?",293   "clean_statement": "The motivic filtration on the algebraic $K$-theory of the sphere spectrum\n\nDue to work of Rognes, Blumberg--Mandell, and others, we understand the algebraic K-theory of the sphere spectrum fairly well. Recent work of Hahn--Raksit--Wilson \\cite{arXiv:2206.11208} gives a motivic filtration on TC of $\\mathbb{E}_{\\infty}$-rings which is closely related to the motivic filtration on algebraic K-theory in motivic homotopy theory.\n\nCan one understand the motivic filtration (or motivic spectral sequence) on the $\\mathrm{TC}$ of the sphere spectrum? What about for the motivic filtration on related invariants such as $\\mathrm{THH}$, $\\mathrm{TP}$?",294   "public_statement": "The motivic filtration on the algebraic $K$-theory of the sphere spectrum\n\nDue to work of Rognes, Blumberg--Mandell, and others, we understand the algebraic K-theory of the sphere spectrum fairly well. Recent work of Hahn--Raksit--Wilson \\cite{arXiv:2206.11208} gives a motivic filtration on TC of $\\mathbb{E}_{\\infty}$-rings which is closely related to the motivic filtration on algebraic K-theory in motivic homotopy theory.\n\nCan one understand the motivic filtration (or motivic spectral sequence) on the $\\mathrm{TC}$ of the sphere spectrum? What about for the motivic filtration on related invariants such as $\\mathrm{THH}$, $\\mathrm{TP}$?",295   "evidence": "This is problem 2.1 in the AIM workshop list *Equivariant techniques in stable homotopy theory*, section *Computations in algebraic K-theory*. The repository record and the live AIM page were compared on 2026-07-22. The mathematical question is uncorrupted:",296   "classification_method": "canonical_text_retained_no_recovery_claim",297   "clean_statement_source": "canonical_problem_field",298   "source_file": "aim-algebraic-geometry-notes.json",299   "source_index": 22,300   "attempt": 1301  },302  "AIM-ALGEBRAIC_GEOMETRY-0024": {303   "statement_status": "exact",304   "original_statement": "The $K$-theory of Johnson--Wilson theory\n\nThe algebraic $K$-theory of Johnson--Wilson theory $E(n)$ is closely related to the topological cyclic homology of the connective cover of the two periodic Johnson--Wilson theory $E'(n)$.\n\nCompute something about $\\mathrm{TC}$ (and related invariants such as $\\mathrm{TP}, \\mathrm{THH}$) of $\\tau_{\\geq0}E'(n)$, where $E'(n)$ is the $2$-periodic Johnson--Wilson theory.",305   "clean_statement": "The $K$-theory of Johnson--Wilson theory\n\nThe algebraic $K$-theory of Johnson--Wilson theory $E(n)$ is closely related to the topological cyclic homology of the connective cover of the two periodic Johnson--Wilson theory $E'(n)$.\n\nCompute something about $\\mathrm{TC}$ (and related invariants such as $\\mathrm{TP}, \\mathrm{THH}$) of $\\tau_{\\geq0}E'(n)$, where $E'(n)$ is the $2$-periodic Johnson--Wilson theory.",306   "public_statement": "The $K$-theory of Johnson--Wilson theory\n\nThe algebraic $K$-theory of Johnson--Wilson theory $E(n)$ is closely related to the topological cyclic homology of the connective cover of the two periodic Johnson--Wilson theory $E'(n)$.\n\nCompute something about $\\mathrm{TC}$ (and related invariants such as $\\mathrm{TP}, \\mathrm{THH}$) of $\\tau_{\\geq0}E'(n)$, where $E'(n)$ is the $2$-periodic Johnson--Wilson theory.",307   "evidence": "The canonical record is Problem 2.2 in the AIM list *Equivariant techniques in stable homotopy theory*, in the section \"Computations in algebraic K-theory.\" Its exact mathematical request is:",308   "classification_method": "canonical_text_retained_no_recovery_claim",309   "clean_statement_source": "canonical_problem_field",310   "source_file": "aim-algebraic-geometry-notes.json",311   "source_index": 23,312   "attempt": 2313  },314  "AIM-ALGEBRAIC_GEOMETRY-0025": {315   "statement_status": "exact",316   "original_statement": "Detecting Greek letter families in algebraic $K$-theory\n\nProve Conjecture 1.1 of \\cite{arXiv:1810.10088}. This is a version of redshift for the Greek letter family in the homotopy groups of spheres: it asks that the $K$-theory of an $\\mathbb{E}_{\\infty}$-ring detecting the $\\alpha^{(n)}$-family detects the $\\alpha^{(n+1)}$-family.",317   "clean_statement": "Detecting Greek letter families in algebraic $K$-theory\n\nProve Conjecture 1.1 of \\cite{arXiv:1810.10088}. This is a version of redshift for the Greek letter family in the homotopy groups of spheres: it asks that the $K$-theory of an $\\mathbb{E}_{\\infty}$-ring detecting the $\\alpha^{(n)}$-family detects the $\\alpha^{(n+1)}$-family.",318   "public_statement": "Detecting Greek letter families in algebraic $K$-theory\n\nProve Conjecture 1.1 of \\cite{arXiv:1810.10088}. This is a version of redshift for the Greek letter family in the homotopy groups of spheres: it asks that the $K$-theory of an $\\mathbb{E}_{\\infty}$-ring detecting the $\\alpha^{(n)}$-family detects the $\\alpha^{(n+1)}$-family.",319   "evidence": "The canonical AIM record asks:",320   "classification_method": "canonical_text_retained_no_recovery_claim",321   "clean_statement_source": "canonical_problem_field",322   "source_file": "aim-algebraic-geometry-notes.json",323   "source_index": 24,324   "attempt": 1325  },326  "AIM-ALGEBRAIC_GEOMETRY-0026": {327   "statement_status": "exact",328   "original_statement": "Atiyah--Segal completion thoerem for iterated $K$-theory\n\nCan one prove an Atiyah--Segal completion theorem for the algebraic $K$-theory of topological $K$-theory?",329   "clean_statement": "Atiyah--Segal completion thoerem for iterated $K$-theory\n\nCan one prove an Atiyah--Segal completion theorem for the algebraic $K$-theory of topological $K$-theory?",330   "public_statement": "Atiyah--Segal completion thoerem for iterated $K$-theory\n\nCan one prove an Atiyah--Segal completion theorem for the algebraic $K$-theory of topological $K$-theory?",331   "evidence": "The canonical AIM record (workshop *Equivariant techniques in stable homotopy theory*, section “Computations in algebraic K-theory,” Problem 2.4) reads:",332   "classification_method": "canonical_text_retained_no_recovery_claim",333   "clean_statement_source": "canonical_problem_field",334   "source_file": "aim-algebraic-geometry-notes.json",335   "source_index": 25,336   "attempt": 1337  },338  "AIM-ALGEBRAIC_GEOMETRY-0027": {339   "statement_status": "exact",340   "original_statement": "Motivic filtered Poitou-Tate duality\n\nIs there a duality at the level of motivic filtered spectra $\\mathrm{fil}_{\\text{mot}}\\mathrm{TC}(\\mathcal{O}_K)$ that recovers Poitou--Tate duality at the level of associated graded?",341   "clean_statement": "Motivic filtered Poitou-Tate duality\n\nIs there a duality at the level of motivic filtered spectra $\\mathrm{fil}_{\\text{mot}}\\mathrm{TC}(\\mathcal{O}_K)$ that recovers Poitou--Tate duality at the level of associated graded?",342   "public_statement": "Motivic filtered Poitou-Tate duality\n\nIs there a duality at the level of motivic filtered spectra $\\mathrm{fil}_{\\text{mot}}\\mathrm{TC}(\\mathcal{O}_K)$ that recovers Poitou--Tate duality at the level of associated graded?",343   "evidence": "The canonical record is Problem 2.5 in the AIM workshop list *Equivariant techniques in stable homotopy theory*, section “Computations in algebraic K-theory.” Its statement is:",344   "classification_method": "canonical_text_retained_no_recovery_claim",345   "clean_statement_source": "canonical_problem_field",346   "source_file": "aim-algebraic-geometry-notes.json",347   "source_index": 26,348   "attempt": 1349  },350  "AIM-ALGEBRAIC_GEOMETRY-0028": {351   "statement_status": "exact",352   "original_statement": "Motivic filtration on $\\mathrm{TR}$\n\n\\cite{arXiv:2206.11208} gives a motivic filtration on $\\mathrm{THH}, \\mathrm{TC}, \\mathrm{TC^{-}}, \\mathrm{TP}$ of ring spectra generalizing previously defined motivic filtrations such as in \\cite{arXiv:1802.03261}. In remark 1.13 of loc. cit. it is mentioned that their techniques can be made to additionally define a motivic filtration on $\\mathrm{TR}$ of discrete rings.\n\nCan one define a well behaved motivic filtration on the $\\mathrm{TR}$ of ring spectra? Can one compute it in good cases?",353   "clean_statement": "Motivic filtration on $\\mathrm{TR}$\n\n\\cite{arXiv:2206.11208} gives a motivic filtration on $\\mathrm{THH}, \\mathrm{TC}, \\mathrm{TC^{-}}, \\mathrm{TP}$ of ring spectra generalizing previously defined motivic filtrations such as in \\cite{arXiv:1802.03261}. In remark 1.13 of loc. cit. it is mentioned that their techniques can be made to additionally define a motivic filtration on $\\mathrm{TR}$ of discrete rings.\n\nCan one define a well behaved motivic filtration on the $\\mathrm{TR}$ of ring spectra? Can one compute it in good cases?",354   "public_statement": "Motivic filtration on $\\mathrm{TR}$\n\n\\cite{arXiv:2206.11208} gives a motivic filtration on $\\mathrm{THH}, \\mathrm{TC}, \\mathrm{TC^{-}}, \\mathrm{TP}$ of ring spectra generalizing previously defined motivic filtrations such as in \\cite{arXiv:1802.03261}. In remark 1.13 of loc. cit. it is mentioned that their techniques can be made to additionally define a motivic filtration on $\\mathrm{TR}$ of discrete rings.\n\nCan one define a well behaved motivic filtration on the $\\mathrm{TR}$ of ring spectra? Can one compute it in good cases?",355   "evidence": "The canonical record is Problem 2.7 in the AIM workshop list *Equivariant techniques in stable homotopy theory*, section “Computations in algebraic K-theory.” It is record 27 (zero-based) of `aim-algebraic-geometry-notes.json`. Its statement is intact:",356   "classification_method": "canonical_text_retained_no_recovery_claim",357   "clean_statement_source": "canonical_problem_field",358   "source_file": "aim-algebraic-geometry-notes.json",359   "source_index": 27,360   "attempt": 1361  },362  "AIM-ALGEBRAIC_GEOMETRY-0029": {363   "statement_status": "exact",364   "original_statement": "Calculation of $\\mathrm{TP}(\\mathbb{Z})$\n\nBokstedt and Madsen computed $\\mathrm{TC}(\\mathbb{Z})$, without calculating $\\mathrm{TP}(\\mathbb{Z})$. However, Hesselholt and Madsen calculated (in \\cite{MR1317575}) $(L_{K(1)} \\mathbb{S})^{tS^1}$. Their results suggest the following:\n\nFix a prime $p$. Let $j = \\tau_{\\geq 0} L_{K(1)} \\mathbb{S}$. Then, $\\mathrm{TP}(\\mathbb{Z}_p) \\simeq j^{tS^1}$, for the trivial $S^1$-action on $j$.",365   "clean_statement": "Calculation of $\\mathrm{TP}(\\mathbb{Z})$\n\nBokstedt and Madsen computed $\\mathrm{TC}(\\mathbb{Z})$, without calculating $\\mathrm{TP}(\\mathbb{Z})$. However, Hesselholt and Madsen calculated (in \\cite{MR1317575}) $(L_{K(1)} \\mathbb{S})^{tS^1}$. Their results suggest the following:\n\nFix a prime $p$. Let $j = \\tau_{\\geq 0} L_{K(1)} \\mathbb{S}$. Then, $\\mathrm{TP}(\\mathbb{Z}_p) \\simeq j^{tS^1}$, for the trivial $S^1$-action on $j$.",366   "public_statement": "Calculation of $\\mathrm{TP}(\\mathbb{Z})$\n\nBokstedt and Madsen computed $\\mathrm{TC}(\\mathbb{Z})$, without calculating $\\mathrm{TP}(\\mathbb{Z})$. However, Hesselholt and Madsen calculated (in \\cite{MR1317575}) $(L_{K(1)} \\mathbb{S})^{tS^1}$. Their results suggest the following:\n\nFix a prime $p$. Let $j = \\tau_{\\geq 0} L_{K(1)} \\mathbb{S}$. Then, $\\mathrm{TP}(\\mathbb{Z}_p) \\simeq j^{tS^1}$, for the trivial $S^1$-action on $j$.",367   "evidence": "The record adds that this would follow, for example, from \\(\\operatorname{TR}(\\mathbb Z_p)\\simeq j_p\\). There is no visible corruption in the record. The citation `MR1317575` is Hesselholt--Madsen, *The \\(S^1\\)-Tate spectrum for \\(J\\)*.",368   "classification_method": "explicit_no_change_evidence",369   "clean_statement_source": "canonical_problem_field",370   "source_file": "aim-algebraic-geometry-notes.json",371   "source_index": 28,372   "attempt": 1373  },374  "AIM-ALGEBRAIC_GEOMETRY-0030": {375   "statement_status": "exact",376   "original_statement": "$L$-theory of integers\n\nThe symmetric and normal $L$-theory of the integers was computed in \\cite{arXiv:2004.06889}, and the Tate cohomology of $Q_8$ was computed by Atiyah in \\cite{MR0148722}. The normal $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $Q_8$; similarly, the symmetric $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $\\mathrm{Pin}(2)$.\n\nEquip $\\mathrm{Pin}(2)$ (resp. $Q_8$) with the canonical $\\mathbb{Z}/2$-action (resp. the action given by conjugation by $i$). Then, the $E_\\infty$-map $L^s(\\mathbb{Z})_{(2)} \\to L^n(\\mathbb{Z})$ can be identified by taking strict $\\mathbb{Z}/2$-fixed points of the map $\\mathbb{Z}_{(2)}^{t\\mathrm{Pin}(2)} \\to \\mathbb{Z}^{tQ_8}$. (This map is induced by the inclusion $Q_8\\subseteq \\mathrm{Pin}(2)$.)",377   "clean_statement": "$L$-theory of integers\n\nThe symmetric and normal $L$-theory of the integers was computed in \\cite{arXiv:2004.06889}, and the Tate cohomology of $Q_8$ was computed by Atiyah in \\cite{MR0148722}. The normal $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $Q_8$; similarly, the symmetric $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $\\mathrm{Pin}(2)$.\n\nEquip $\\mathrm{Pin}(2)$ (resp. $Q_8$) with the canonical $\\mathbb{Z}/2$-action (resp. the action given by conjugation by $i$). Then, the $E_\\infty$-map $L^s(\\mathbb{Z})_{(2)} \\to L^n(\\mathbb{Z})$ can be identified by taking strict $\\mathbb{Z}/2$-fixed points of the map $\\mathbb{Z}_{(2)}^{t\\mathrm{Pin}(2)} \\to \\mathbb{Z}^{tQ_8}$. (This map is induced by the inclusion $Q_8\\subseteq \\mathrm{Pin}(2)$.)",378   "public_statement": "$L$-theory of integers\n\nThe symmetric and normal $L$-theory of the integers was computed in \\cite{arXiv:2004.06889}, and the Tate cohomology of $Q_8$ was computed by Atiyah in \\cite{MR0148722}. The normal $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $Q_8$; similarly, the symmetric $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $\\mathrm{Pin}(2)$.\n\nEquip $\\mathrm{Pin}(2)$ (resp. $Q_8$) with the canonical $\\mathbb{Z}/2$-action (resp. the action given by conjugation by $i$). Then, the $E_\\infty$-map $L^s(\\mathbb{Z})_{(2)} \\to L^n(\\mathbb{Z})$ can be identified by taking strict $\\mathbb{Z}/2$-fixed points of the map $\\mathbb{Z}_{(2)}^{t\\mathrm{Pin}(2)} \\to \\mathbb{Z}^{tQ_8}$. (This map is induced by the inclusion $Q_8\\subseteq \\mathrm{Pin}(2)$.)",379   "evidence": "This is Conjecture 2.9, “\\(L\\)-theory of integers,” from the AIM workshop *Equivariant techniques in stable homotopy theory* (October 2022). The canonical record says:",380   "classification_method": "canonical_text_retained_no_recovery_claim",381   "clean_statement_source": "canonical_problem_field",382   "source_file": "aim-algebraic-geometry-notes.json",383   "source_index": 29,384   "attempt": 1385  },386  "AIM-ALGEBRAIC_GEOMETRY-0031": {387   "statement_status": "exact",388   "original_statement": "Devissage for equivariant/hermitian $K$-theory\n\nWe understand devissage for the $K$-theory of stable categories fairly well now due to work of Quillen, Barwick, and Burklund--Levy (see \\cite{arXiv:2112.14723}). See section 2.2 of \\cite{arXiv:2009.07225} for what is known for hermitian $K$-theory.\n\nWhat is the optimal devissage result for equivariant/hermitian $K$-theory?",389   "clean_statement": "Devissage for equivariant/hermitian $K$-theory\n\nWe understand devissage for the $K$-theory of stable categories fairly well now due to work of Quillen, Barwick, and Burklund--Levy (see \\cite{arXiv:2112.14723}). See section 2.2 of \\cite{arXiv:2009.07225} for what is known for hermitian $K$-theory.\n\nWhat is the optimal devissage result for equivariant/hermitian $K$-theory?",390   "public_statement": "Devissage for equivariant/hermitian $K$-theory\n\nWe understand devissage for the $K$-theory of stable categories fairly well now due to work of Quillen, Barwick, and Burklund--Levy (see \\cite{arXiv:2112.14723}). See section 2.2 of \\cite{arXiv:2009.07225} for what is known for hermitian $K$-theory.\n\nWhat is the optimal devissage result for equivariant/hermitian $K$-theory?",391   "evidence": "The source record has no remarks or supplied bibliography beyond those two identifiers. Both were checked: they are Burklund--Levy, *On the \\(K\\)-theory of regular coconnective rings*, and Calmès--Dotto--Harpaz--Hebestreit--Land--Moi--Nardin--Nikolaus--Steimle, *Hermitian \\(K\\)-theory for stable \\(\\infty\\)-categories III: Grothendieck--Witt groups of rings*. The latter was revised as arXiv v4 on 27 April 2026 and accepted by the *Annals of Mathematics*. There is no apparent corruption in the statement.",392   "classification_method": "explicit_no_change_evidence",393   "clean_statement_source": "canonical_problem_field",394   "source_file": "aim-algebraic-geometry-notes.json",395   "source_index": 30,396   "attempt": 1397  },398  "AIM-ALGEBRAIC_GEOMETRY-0032": {399   "statement_status": "exact",400   "original_statement": "a inverted Artin--Tate $\\mathbb{R}$-motives\n\nThe category of Artin--Tate $\\mathbb{R}$-motives has a topological model as a $1$-parameter deformation of $C_2$-equivariant homotopy theory \\cite{arXiv:2010.10325}. Let $\\Sp_{\\mathbb{R}}^{AT}$ denote this category.\n\nWhat category do you get when you invert the class $a$ in $\\Sp_{\\mathbb{R}}^{AT}$?",401   "clean_statement": "a inverted Artin--Tate $\\mathbb{R}$-motives\n\nThe category of Artin--Tate $\\mathbb{R}$-motives has a topological model as a $1$-parameter deformation of $C_2$-equivariant homotopy theory \\cite{arXiv:2010.10325}. Let $\\Sp_{\\mathbb{R}}^{AT}$ denote this category.\n\nWhat category do you get when you invert the class $a$ in $\\Sp_{\\mathbb{R}}^{AT}$?",402   "public_statement": "a inverted Artin--Tate $\\mathbb{R}$-motives\n\nThe category of Artin--Tate $\\mathbb{R}$-motives has a topological model as a $1$-parameter deformation of $C_2$-equivariant homotopy theory \\cite{arXiv:2010.10325}. Let $\\Sp_{\\mathbb{R}}^{AT}$ denote this category.\n\nWhat category do you get when you invert the class $a$ in $\\Sp_{\\mathbb{R}}^{AT}$?",403   "evidence": "The canonical record asks:",404   "classification_method": "canonical_text_retained_no_recovery_claim",405   "clean_statement_source": "canonical_problem_field",406   "source_file": "aim-algebraic-geometry-notes.json",407   "source_index": 31,408   "attempt": 1409  },410  "AIM-ALGEBRAIC_GEOMETRY-0033": {411   "statement_status": "exact",412   "original_statement": "Mahowald invariants increase chromatic height\n\nGiven a $v_n$-periodic family in the stable homotopy groups of spheres, does its Mahowald invariant give rise to an (eventually) $v_{n+1}$-periodic family?",413   "clean_statement": "Mahowald invariants increase chromatic height\n\nGiven a $v_n$-periodic family in the stable homotopy groups of spheres, does its Mahowald invariant give rise to an (eventually) $v_{n+1}$-periodic family?",414   "public_statement": "Mahowald invariants increase chromatic height\n\nGiven a $v_n$-periodic family in the stable homotopy groups of spheres, does its Mahowald invariant give rise to an (eventually) $v_{n+1}$-periodic family?",415   "evidence": "The exact canonical record is Problem 3.3 from the AIM workshop list “Equivariant techniques in stable homotopy theory,” section “Motivic, equivariant, and synthetic spectra”:",416   "classification_method": "canonical_text_retained_no_recovery_claim",417   "clean_statement_source": "canonical_problem_field",418   "source_file": "aim-algebraic-geometry-notes.json",419   "source_index": 32,420   "attempt": 1421  },422  "AIM-ALGEBRAIC_GEOMETRY-0034": {423   "statement_status": "exact",424   "original_statement": "Equivariant and synthetic\n\nCan one profitably mix synthetic spectra with equivariant homotopy theory?",425   "clean_statement": "Equivariant and synthetic\n\nCan one profitably mix synthetic spectra with equivariant homotopy theory?",426   "public_statement": "Equivariant and synthetic\n\nCan one profitably mix synthetic spectra with equivariant homotopy theory?",427   "evidence": "The exact canonical record is AIM Problem List 3.4 from the workshop *Equivariant techniques in stable homotopy theory*:",428   "classification_method": "canonical_text_retained_no_recovery_claim",429   "clean_statement_source": "canonical_problem_field",430   "source_file": "aim-algebraic-geometry-notes.json",431   "source_index": 33,432   "attempt": 1433  },434  "AIM-ALGEBRAIC_GEOMETRY-0035": {435   "statement_status": "exact",436   "original_statement": "Odd primary analog of Wood cofibre sequences\n\nThe Wood cofibre sequence is the cofibre sequence $\\Sigma \\mathrm{KO} \\xrightarrow{\\eta} \\mathrm{KO} \\to \\mathrm{KU}$ (and similarly for the connective versions). There is also a version relating $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle^{C_2}$ to $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle$.\n\nWhat are odd primary analogs of the Wood cofibre sequence? For example, one might expect a $p$-stage filtration on $E_{p-1}$ with associated graded $E_{p-1}^{hC_p}$. Is there an analog for $E_{(p-1)p^{n-1}m}$?",437   "clean_statement": "Odd primary analog of Wood cofibre sequences\n\nThe Wood cofibre sequence is the cofibre sequence $\\Sigma \\mathrm{KO} \\xrightarrow{\\eta} \\mathrm{KO} \\to \\mathrm{KU}$ (and similarly for the connective versions). There is also a version relating $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle^{C_2}$ to $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle$.\n\nWhat are odd primary analogs of the Wood cofibre sequence? For example, one might expect a $p$-stage filtration on $E_{p-1}$ with associated graded $E_{p-1}^{hC_p}$. Is there an analog for $E_{(p-1)p^{n-1}m}$?",438   "public_statement": "Odd primary analog of Wood cofibre sequences\n\nThe Wood cofibre sequence is the cofibre sequence $\\Sigma \\mathrm{KO} \\xrightarrow{\\eta} \\mathrm{KO} \\to \\mathrm{KU}$ (and similarly for the connective versions). There is also a version relating $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle^{C_2}$ to $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle$.\n\nWhat are odd primary analogs of the Wood cofibre sequence? For example, one might expect a $p$-stage filtration on $E_{p-1}$ with associated graded $E_{p-1}^{hC_p}$. Is there an analog for $E_{(p-1)p^{n-1}m}$?",439   "evidence": "The local JSON record agrees with the AIM statement. The AIM web page timed out during this run, but there is no visible corruption or missing notation in the repository copy. I interpret “associated graded \\(E_{p-1}^{hC_p}\\)” in the standard stable sense: the graded pieces are suspensions of \\(E_{p-1}^{hC_p}\\). This convention is necessary even classically, since the two graded pieces of \\(KO\\wedge C(\\eta)\\simeq KU\\) are \\(KO\\) and \\(\\Sigma^2KO\\).",440   "classification_method": "explicit_no_change_evidence",441   "clean_statement_source": "canonical_problem_field",442   "source_file": "aim-algebraic-geometry-notes.json",443   "source_index": 34,444   "attempt": 1445  },446  "AIM-ALGEBRAIC_GEOMETRY-0036": {447   "statement_status": "exact",448   "original_statement": "Motivic modular forms (mmf)\n\nIs there a connective version of motivic modular forms, which should be named $\\mathrm{mmf}$, and is a lift of $\\mathrm{tmf}$?",449   "clean_statement": "Motivic modular forms (mmf)\n\nIs there a connective version of motivic modular forms, which should be named $\\mathrm{mmf}$, and is a lift of $\\mathrm{tmf}$?",450   "public_statement": "Motivic modular forms (mmf)\n\nIs there a connective version of motivic modular forms, which should be named $\\mathrm{mmf}$, and is a lift of $\\mathrm{tmf}$?",451   "evidence": "The exact canonical record is Problem 3.5 in the section “Motivic, equivariant, and synthetic spectra” from the October 24–28, 2022 AIM workshop *Equivariant techniques in stable homotopy theory*:",452   "classification_method": "canonical_text_retained_no_recovery_claim",453   "clean_statement_source": "canonical_problem_field",454   "source_file": "aim-algebraic-geometry-notes.json",455   "source_index": 35,456   "attempt": 1457  },458  "AIM-ALGEBRAIC_GEOMETRY-0037": {459   "statement_status": "exact",460   "original_statement": "The associated graded of the localized slice spectral sequence tower\n\nThe localized slice spectral sequence gives a way to break up the slice spectral sequence into pieces that see phenomena only on certain geometric fixed points. For the group $C_{2^n}$, there is a localized slice spectral sequence for each subgroup.\n\nFor the group $C_{2^n}$ one understand more easily the fibres between the localized slice spectral sequence for two adjacent subgroups $H$ and $H'$?",461   "clean_statement": "The associated graded of the localized slice spectral sequence tower\n\nThe localized slice spectral sequence gives a way to break up the slice spectral sequence into pieces that see phenomena only on certain geometric fixed points. For the group $C_{2^n}$, there is a localized slice spectral sequence for each subgroup.\n\nFor the group $C_{2^n}$ one understand more easily the fibres between the localized slice spectral sequence for two adjacent subgroups $H$ and $H'$?",462   "public_statement": "The associated graded of the localized slice spectral sequence tower\n\nThe localized slice spectral sequence gives a way to break up the slice spectral sequence into pieces that see phenomena only on certain geometric fixed points. For the group $C_{2^n}$, there is a localized slice spectral sequence for each subgroup.\n\nFor the group $C_{2^n}$ one understand more easily the fibres between the localized slice spectral sequence for two adjacent subgroups $H$ and $H'$?",463   "evidence": "The canonical record is Problem 4.1, “The associated graded of the localized slice spectral sequence tower,” in the AIM list *Equivariant techniques in stable homotopy theory*, section “Norms and the slice filtration.” The source record says:",464   "classification_method": "canonical_text_retained_no_recovery_claim",465   "clean_statement_source": "canonical_problem_field",466   "source_file": "aim-algebraic-geometry-notes.json",467   "source_index": 36,468   "attempt": 1469  },470  "AIM-ALGEBRAIC_GEOMETRY-0038": {471   "statement_status": "exact",472   "original_statement": "The image of $J$ in equivariant truncated Brown--Peterson spectra\n\nWhat is the image of the $J$ homomorphism in the fixed points of $\\mathrm{BP}^{((G))}\\langle m\\rangle$?",473   "clean_statement": "The image of $J$ in equivariant truncated Brown--Peterson spectra\n\nWhat is the image of the $J$ homomorphism in the fixed points of $\\mathrm{BP}^{((G))}\\langle m\\rangle$?",474   "public_statement": "The image of $J$ in equivariant truncated Brown--Peterson spectra\n\nWhat is the image of the $J$ homomorphism in the fixed points of $\\mathrm{BP}^{((G))}\\langle m\\rangle$?",475   "evidence": "The canonical record is Problem 4.2 in the AIM workshop list *Equivariant techniques in stable homotopy theory*, section “Norms and the slice filtration”:",476   "classification_method": "canonical_text_retained_no_recovery_claim",477   "clean_statement_source": "canonical_problem_field",478   "source_file": "aim-algebraic-geometry-notes.json",479   "source_index": 37,480   "attempt": 1481  },482  "AIM-ALGEBRAIC_GEOMETRY-0039": {483   "statement_status": "reconstructed_unverified",484   "original_statement": "The Hurewicz image of equivariant truncated Brown-Peterson spectra\n\nWhat is the Hurewicz image of the truncated Brown--Peterson spectrum $\\mathrm{BP}^{((G))}\\langle m \\rangle$?",485   "clean_statement": null,486   "public_statement": "The Hurewicz image of equivariant truncated Brown-Peterson spectra\n\nWhat is the Hurewicz image of the truncated Brown--Peterson spectrum $\\mathrm{BP}^{((G))}\\langle m \\rangle$?",487   "evidence": "Other plausible readings are genuinely different problems:",488   "classification_method": "explicit_uncertainty_evidence",489   "clean_statement_source": "no_safe_clean_extraction",490   "source_file": "aim-algebraic-geometry-notes.json",491   "source_index": 38,492   "attempt": 1493  },494  "AIM-ALGEBRAIC_GEOMETRY-0040": {495   "statement_status": "exact",496   "original_statement": "Understanding $n$-slices\n\n\\cite{arXiv:1711.03472} gives a description of $n$-slices, the objects that appear in the associated graded of the slice filtration.\n\nIs there a way to get a good computational grasp of $n$-slices? For example understanding their Postnikov towers, homotopy groups etc.",497   "clean_statement": "Understanding $n$-slices\n\n\\cite{arXiv:1711.03472} gives a description of $n$-slices, the objects that appear in the associated graded of the slice filtration.\n\nIs there a way to get a good computational grasp of $n$-slices? For example understanding their Postnikov towers, homotopy groups etc.",498   "public_statement": "Understanding $n$-slices\n\n\\cite{arXiv:1711.03472} gives a description of $n$-slices, the objects that appear in the associated graded of the slice filtration.\n\nIs there a way to get a good computational grasp of $n$-slices? For example understanding their Postnikov towers, homotopy groups etc.",499   "evidence": "This agrees with the JSON record. There is no corruption to repair. The question is intentionally broad rather than a yes/no conjecture.",500   "classification_method": "explicit_no_change_evidence",501   "clean_statement_source": "canonical_problem_field",502   "source_file": "aim-algebraic-geometry-notes.json",503   "source_index": 39,504   "attempt": 1505  },506  "AIM-ALGEBRAIC_GEOMETRY-0041": {507   "statement_status": "exact",508   "original_statement": "Heights of slice differentials\n\nThe differentials in the slice spectral sequence for $\\mathrm{BP}^{((G))}\\langle n \\rangle$ look like they are stratified by height.\n\nIs it possible to make sense of the height of a differential in the slice spectral sequence of $\\mathrm{BP}^{((G))}\\langle n \\rangle$ so that it has a well defined height?",509   "clean_statement": "Heights of slice differentials\n\nThe differentials in the slice spectral sequence for $\\mathrm{BP}^{((G))}\\langle n \\rangle$ look like they are stratified by height.\n\nIs it possible to make sense of the height of a differential in the slice spectral sequence of $\\mathrm{BP}^{((G))}\\langle n \\rangle$ so that it has a well defined height?",510   "public_statement": "Heights of slice differentials\n\nThe differentials in the slice spectral sequence for $\\mathrm{BP}^{((G))}\\langle n \\rangle$ look like they are stratified by height.\n\nIs it possible to make sense of the height of a differential in the slice spectral sequence of $\\mathrm{BP}^{((G))}\\langle n \\rangle$ so that it has a well defined height?",511   "evidence": "The exact AIM record (workshop *Equivariant techniques in stable homotopy theory*, section “Norms and the slice filtration,” Problem 4.5) asks:",512   "classification_method": "canonical_text_retained_no_recovery_claim",513   "clean_statement_source": "canonical_problem_field",514   "source_file": "aim-algebraic-geometry-notes.json",515   "source_index": 40,516   "attempt": 1517  },518  "AIM-ALGEBRAIC_GEOMETRY-0042": {519   "statement_status": "exact",520   "original_statement": "The $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for $\\mathrm{BP}^{((G))}\\langle n\\rangle$\n\n$\\mathrm{BP}^{((G))}\\langle n-1\\rangle$ is the quotient of $\\mathrm{BP}^{((G))}\\langle n\\rangle$ by $G\\cdot \\overline{v}_n$.\n\nCan one understand the $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for this quotient map?",521   "clean_statement": "The $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for $\\mathrm{BP}^{((G))}\\langle n\\rangle$\n\n$\\mathrm{BP}^{((G))}\\langle n-1\\rangle$ is the quotient of $\\mathrm{BP}^{((G))}\\langle n\\rangle$ by $G\\cdot \\overline{v}_n$.\n\nCan one understand the $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for this quotient map?",522   "public_statement": "The $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for $\\mathrm{BP}^{((G))}\\langle n\\rangle$\n\n$\\mathrm{BP}^{((G))}\\langle n-1\\rangle$ is the quotient of $\\mathrm{BP}^{((G))}\\langle n\\rangle$ by $G\\cdot \\overline{v}_n$.\n\nCan one understand the $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for this quotient map?",523   "evidence": "The canonical AIM record (source file `aim-algebraic-geometry-notes.json`, zero-based index 41, problem 4.6 in the section “Norms and the slice filtration”) asks:",524   "classification_method": "canonical_text_retained_no_recovery_claim",525   "clean_statement_source": "canonical_problem_field",526   "source_file": "aim-algebraic-geometry-notes.json",527   "source_index": 41,528   "attempt": 1529  },530  "AIM-ALGEBRAIC_GEOMETRY-0043": {531   "statement_status": "exact",532   "original_statement": "Connecting slice differentials via power operations\n\nWe would like a systematic way of understanding slice differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$.\n\nAre there power operations connecting differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$",533   "clean_statement": "Connecting slice differentials via power operations\n\nWe would like a systematic way of understanding slice differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$.\n\nAre there power operations connecting differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$",534   "public_statement": "Connecting slice differentials via power operations\n\nWe would like a systematic way of understanding slice differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$.\n\nAre there power operations connecting differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$",535   "evidence": "The assigned record is problem 4.7 in the section “Norms and the slice filtration” of the AIM workshop list *Equivariant techniques in stable homotopy theory*. The exact stored text is:",536   "classification_method": "canonical_text_retained_no_recovery_claim",537   "clean_statement_source": "canonical_problem_field",538   "source_file": "aim-algebraic-geometry-notes.json",539   "source_index": 42,540   "attempt": 1541  },542  "AIM-ALGEBRAIC_GEOMETRY-0044": {543   "statement_status": "exact",544   "original_statement": "Relations in $\\mathrm{BP}^{((G))}$\n\nIs $8\\sigma = 0$ in the fixed points of $\\mathrm{BP}^{((G))}$?",545   "clean_statement": "Relations in $\\mathrm{BP}^{((G))}$\n\nIs $8\\sigma = 0$ in the fixed points of $\\mathrm{BP}^{((G))}$?",546   "public_statement": "Relations in $\\mathrm{BP}^{((G))}$\n\nIs $8\\sigma = 0$ in the fixed points of $\\mathrm{BP}^{((G))}$?",547   "evidence": "The exact corpus record is:",548   "classification_method": "canonical_text_retained_no_recovery_claim",549   "clean_statement_source": "canonical_problem_field",550   "source_file": "aim-algebraic-geometry-notes.json",551   "source_index": 43,552   "attempt": 1553  },554  "AIM-ALGEBRAIC_GEOMETRY-0045": {555   "statement_status": "exact",556   "original_statement": "Problem 1\n\nFind more integers $d$ and $n$ such that a general hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.\nSimilarly,\nfind more integers $d$ and $n$ such that every smooth hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.",557   "clean_statement": "Problem 1\n\nFind more integers $d$ and $n$ such that a general hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.\nSimilarly,\nfind more integers $d$ and $n$ such that every smooth hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.",558   "public_statement": "Problem 1\n\nFind more integers $d$ and $n$ such that a general hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.\nSimilarly,\nfind more integers $d$ and $n$ such that every smooth hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.",559   "evidence": "The canonical source is aim-algebraic-geometry-notes.json, zero-based index 44, problem 1.02 from the AIM workshop *Rationality problems in algebraic geometry*. The source record states:",560   "classification_method": "canonical_text_retained_no_recovery_claim",561   "clean_statement_source": "canonical_problem_field",562   "source_file": "aim-algebraic-geometry-notes.json",563   "source_index": 44,564   "attempt": 1565  },566  "AIM-ALGEBRAIC_GEOMETRY-0046": {567   "statement_status": "exact",568   "original_statement": "Given $d$, what is the smallest $n$ such that there is some smooth unirational hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$?",569   "clean_statement": "Given $d$, what is the smallest $n$ such that there is some smooth unirational hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$?",570   "public_statement": "Given $d$, what is the smallest $n$ such that there is some smooth unirational hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$?",571   "evidence": "The canonical record is problem 1.04 from the AIM workshop *Rationality problems in algebraic geometry*, posed by Stefan Schreieder. Its exact text is:",572   "classification_method": "canonical_text_retained_no_recovery_claim",573   "clean_statement_source": "canonical_problem_field",574   "source_file": "aim-algebraic-geometry-notes.json",575   "source_index": 45,576   "attempt": 1577  },578  "AIM-ALGEBRAIC_GEOMETRY-0047": {579   "statement_status": "exact",580   "original_statement": "Does unirationality specialize in smooth projective families?",581   "clean_statement": "Does unirationality specialize in smooth projective families?",582   "public_statement": "Does unirationality specialize in smooth projective families?",583   "evidence": "The exact AIM record (problem 1.06 in the 2019 workshop *Rationality problems in algebraic geometry*) asks:",584   "classification_method": "canonical_text_retained_no_recovery_claim",585   "clean_statement_source": "canonical_problem_field",586   "source_file": "aim-algebraic-geometry-notes.json",587   "source_index": 46,588   "attempt": 1589  },590  "AIM-ALGEBRAIC_GEOMETRY-0048": {591   "statement_status": "exact",592   "original_statement": "Does there exist an irrational unirational variety with unirational parametrizations of coprime degrees?",593   "clean_statement": "Does there exist an irrational unirational variety with unirational parametrizations of coprime degrees?",594   "public_statement": "Does there exist an irrational unirational variety with unirational parametrizations of coprime degrees?",595   "evidence": "The canonical record is problem 1.08 from the AIM workshop *Rationality problems in algebraic geometry*:",596   "classification_method": "canonical_text_retained_no_recovery_claim",597   "clean_statement_source": "canonical_problem_field",598   "source_file": "aim-algebraic-geometry-notes.json",599   "source_index": 47,600   "attempt": 1601  },602  "AIM-ALGEBRAIC_GEOMETRY-0049": {603   "statement_status": "exact",604   "original_statement": "Does there exist a rationally connected smooth complex projective variety, not stably rational, that has an integral decomposition of the diagonal?",605   "clean_statement": "Does there exist a rationally connected smooth complex projective variety, not stably rational, that has an integral decomposition of the diagonal?",606   "public_statement": "Does there exist a rationally connected smooth complex projective variety, not stably rational, that has an integral decomposition of the diagonal?",607   "evidence": "The canonical AIM record is problem 1.1 from the workshop *Rationality problems in algebraic geometry*. Its question is:",608   "classification_method": "canonical_text_retained_no_recovery_claim",609   "clean_statement_source": "canonical_problem_field",610   "source_file": "aim-algebraic-geometry-notes.json",611   "source_index": 48,612   "attempt": 1613  },614  "AIM-ALGEBRAIC_GEOMETRY-0050": {615   "statement_status": "exact",616   "original_statement": "Does there exist a unirational variety $X$ of dimension at least 4 such that\n\\begin{align*}\n\\frac{H^4(X, \\mathbb Z) \\cap H^{2,2}(X)}{(H^4(X, \\mathbb Z) \\cap H^{2,2}(X))_{tors}}\n\\end{align*}\nis non-algebraic? Here, the subscript tors denotes torsion classes.\nDoes there exist such a variety of dimension exactly 4?",617   "clean_statement": "Does there exist a unirational variety $X$ of dimension at least 4 such that\n\\begin{align*}\n\\frac{H^4(X, \\mathbb Z) \\cap H^{2,2}(X)}{(H^4(X, \\mathbb Z) \\cap H^{2,2}(X))_{tors}}\n\\end{align*}\nis non-algebraic? Here, the subscript tors denotes torsion classes.\nDoes there exist such a variety of dimension exactly 4?",618   "public_statement": "Does there exist a unirational variety $X$ of dimension at least 4 such that\n\\begin{align*}\n\\frac{H^4(X, \\mathbb Z) \\cap H^{2,2}(X)}{(H^4(X, \\mathbb Z) \\cap H^{2,2}(X))_{tors}}\n\\end{align*}\nis non-algebraic? Here, the subscript tors denotes torsion classes.\nDoes there exist such a variety of dimension exactly 4?",619   "evidence": "The canonical AIM record, Problem 1.12 from the workshop *Rationality problems in algebraic geometry*, asks:",620   "classification_method": "canonical_text_retained_no_recovery_claim",621   "clean_statement_source": "canonical_problem_field",622   "source_file": "aim-algebraic-geometry-notes.json",623   "source_index": 49,624   "attempt": 1625  },626  "AIM-ALGEBRAIC_GEOMETRY-0051": {627   "statement_status": "exact",628   "original_statement": "Is there a smooth projective $3$-fold $X$ over a finite field for which $H^3_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z(2))$ is not zero?",629   "clean_statement": "Is there a smooth projective $3$-fold $X$ over a finite field for which $H^3_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z(2))$ is not zero?",630   "public_statement": "Is there a smooth projective $3$-fold $X$ over a finite field for which $H^3_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z(2))$ is not zero?",631   "evidence": "The canonical record is Problem 1.14 from the AIM workshop *Rationality problems in algebraic geometry*:",632   "classification_method": "canonical_text_retained_no_recovery_claim",633   "clean_statement_source": "canonical_problem_field",634   "source_file": "aim-algebraic-geometry-notes.json",635   "source_index": 50,636   "attempt": 1637  },638  "AIM-ALGEBRAIC_GEOMETRY-0052": {639   "statement_status": "exact",640   "original_statement": "Is there a smooth rational cubic hypersurface of odd dimension?",641   "clean_statement": "Is there a smooth rational cubic hypersurface of odd dimension?",642   "public_statement": "Is there a smooth rational cubic hypersurface of odd dimension?",643   "evidence": "The canonical AIM record is Problem 1.16 from the workshop “Rationality problems in algebraic geometry”:",644   "classification_method": "canonical_text_retained_no_recovery_claim",645   "clean_statement_source": "canonical_problem_field",646   "source_file": "aim-algebraic-geometry-notes.json",647   "source_index": 51,648   "attempt": 1649  },650  "AIM-ALGEBRAIC_GEOMETRY-0053": {651   "statement_status": "exact",652   "original_statement": "Is there a smooth rational quartic hypersurface of some dimension?",653   "clean_statement": "Is there a smooth rational quartic hypersurface of some dimension?",654   "public_statement": "Is there a smooth rational quartic hypersurface of some dimension?",655   "evidence": "The canonical record is problem 1.18 from the AIM workshop “Rationality problems in algebraic geometry,” in the section “The problems.” Its exact text is:",656   "classification_method": "canonical_text_retained_no_recovery_claim",657   "clean_statement_source": "canonical_problem_field",658   "source_file": "aim-algebraic-geometry-notes.json",659   "source_index": 52,660   "attempt": 1661  },662  "AIM-ALGEBRAIC_GEOMETRY-0054": {663   "statement_status": "exact",664   "original_statement": "Can we use the degeneration method to obstruct stable rationality for $4$-folds fibered by $3$-folds over $\\mathbb P^1$?",665   "clean_statement": "Can we use the degeneration method to obstruct stable rationality for $4$-folds fibered by $3$-folds over $\\mathbb P^1$?",666   "public_statement": "Can we use the degeneration method to obstruct stable rationality for $4$-folds fibered by $3$-folds over $\\mathbb P^1$?",667   "evidence": "The canonical record is Problem 1.2 from the AIM workshop list “Rationality problems in algebraic geometry,” posed in the list by Asher Auel:",668   "classification_method": "canonical_text_retained_no_recovery_claim",669   "clean_statement_source": "canonical_problem_field",670   "source_file": "aim-algebraic-geometry-notes.json",671   "source_index": 53,672   "attempt": 1673  },674  "AIM-ALGEBRAIC_GEOMETRY-0055": {675   "statement_status": "exact",676   "original_statement": "If $X$ is a smooth rational variety of dimension $n$ and $p \\in X$ is a point, is there a Zariski open $U \\subset X$ with $p \\in U$ such that $U$ is isomorphic to a Zariski open subset of $\\mathbb P^n$?",677   "clean_statement": "If $X$ is a smooth rational variety of dimension $n$ and $p \\in X$ is a point, is there a Zariski open $U \\subset X$ with $p \\in U$ such that $U$ is isomorphic to a Zariski open subset of $\\mathbb P^n$?",678   "public_statement": "If $X$ is a smooth rational variety of dimension $n$ and $p \\in X$ is a point, is there a Zariski open $U \\subset X$ with $p \\in U$ such that $U$ is isomorphic to a Zariski open subset of $\\mathbb P^n$?",679   "evidence": "The canonical AIM record is Problem 1.22 from the workshop “Rationality problems in algebraic geometry”:",680   "classification_method": "canonical_text_retained_no_recovery_claim",681   "clean_statement_source": "canonical_problem_field",682   "source_file": "aim-algebraic-geometry-notes.json",683   "source_index": 54,684   "attempt": 1685  },686  "AIM-ALGEBRAIC_GEOMETRY-0056": {687   "statement_status": "exact",688   "original_statement": "Let $X$ be a rationally connected $3$-fold which is a del Pezzo fibration over $\\mathbb P^1$.\nIs there a component $M$ of the space of sections of the map $X \\rightarrow \\mathbb P^1$\nsuch that the Stein factorization of\n$\\alpha: M \\rightarrow J(X)$\nis the MRC fibration of $M$?\n\nCan we find an example of such a rationally connected $3$-fold $X$ with does not admit a decomposition of the diagonal\nand with the above property for the components of spaces of rational curves on X?",689   "clean_statement": "Let $X$ be a rationally connected $3$-fold which is a del Pezzo fibration over $\\mathbb P^1$.\nIs there a component $M$ of the space of sections of the map $X \\rightarrow \\mathbb P^1$\nsuch that the Stein factorization of\n$\\alpha: M \\rightarrow J(X)$\nis the MRC fibration of $M$?\n\nCan we find an example of such a rationally connected $3$-fold $X$ with does not admit a decomposition of the diagonal\nand with the above property for the components of spaces of rational curves on X?",690   "public_statement": "Let $X$ be a rationally connected $3$-fold which is a del Pezzo fibration over $\\mathbb P^1$.\nIs there a component $M$ of the space of sections of the map $X \\rightarrow \\mathbb P^1$\nsuch that the Stein factorization of\n$\\alpha: M \\rightarrow J(X)$\nis the MRC fibration of $M$?\n\nCan we find an example of such a rationally connected $3$-fold $X$ with does not admit a decomposition of the diagonal\nand with the above property for the components of spaces of rational curves on X?",691   "evidence": "The canonical AIM record (problem 1.24 in the workshop *Rationality problems in algebraic geometry*) reads:",692   "classification_method": "canonical_text_retained_no_recovery_claim",693   "clean_statement_source": "canonical_problem_field",694   "source_file": "aim-algebraic-geometry-notes.json",695   "source_index": 55,696   "attempt": 1697  },698  "AIM-ALGEBRAIC_GEOMETRY-0057": {699   "statement_status": "exact",700   "original_statement": "The following is question 2 on the website announcement for this conference.\n\nColliot-Thélène and Pirutka defined the notion of a $CH_0$-universally trivial resolution of singularities. As a preliminary question, is admitting a $CH_0$-universally trivial resolution\nan analytic local property for complex varieties? Assuming the answer is yes, we pose the following questions.\n\nIn low dimensions, say up to 4, can we give an analytic local classification of $CH_0$-universally trivial singularities?\n\nMore vaguely, is there a local classification of $CH_0$-universally trivial singularities in terms of MMP data?\n\nWe can ask the same questions for $L$-rational singularities (defined by Nicaise and Shinder), and for $B$-rational singularities (defined by Kontsevich and Tschinkel).",701   "clean_statement": "The following is question 2 on the website announcement for this conference.\n\nColliot-Thélène and Pirutka defined the notion of a $CH_0$-universally trivial resolution of singularities. As a preliminary question, is admitting a $CH_0$-universally trivial resolution\nan analytic local property for complex varieties? Assuming the answer is yes, we pose the following questions.\n\nIn low dimensions, say up to 4, can we give an analytic local classification of $CH_0$-universally trivial singularities?\n\nMore vaguely, is there a local classification of $CH_0$-universally trivial singularities in terms of MMP data?\n\nWe can ask the same questions for $L$-rational singularities (defined by Nicaise and Shinder), and for $B$-rational singularities (defined by Kontsevich and Tschinkel).",702   "public_statement": "The following is question 2 on the website announcement for this conference.\n\nColliot-Thélène and Pirutka defined the notion of a $CH_0$-universally trivial resolution of singularities. As a preliminary question, is admitting a $CH_0$-universally trivial resolution\nan analytic local property for complex varieties? Assuming the answer is yes, we pose the following questions.\n\nIn low dimensions, say up to 4, can we give an analytic local classification of $CH_0$-universally trivial singularities?\n\nMore vaguely, is there a local classification of $CH_0$-universally trivial singularities in terms of MMP data?\n\nWe can ask the same questions for $L$-rational singularities (defined by Nicaise and Shinder), and for $B$-rational singularities (defined by Kontsevich and Tschinkel).",703   "evidence": "The canonical record is AIM Problem List item 1.26, attributed on the live AIM page to Asher Auel. Its mathematical text is:",704   "classification_method": "canonical_text_retained_no_recovery_claim",705   "clean_statement_source": "canonical_problem_field",706   "source_file": "aim-algebraic-geometry-notes.json",707   "source_index": 56,708   "attempt": 1709  },710  "AIM-ALGEBRAIC_GEOMETRY-0058": {711   "statement_status": "exact",712   "original_statement": "Compare $CH_0$-universally trivial, $B$-rational, and $L$-rational singularities.",713   "clean_statement": "Compare $CH_0$-universally trivial, $B$-rational, and $L$-rational singularities.",714   "public_statement": "Compare $CH_0$-universally trivial, $B$-rational, and $L$-rational singularities.",715   "evidence": "The canonical record is AIM problem 1.28 from the workshop *Rationality problems in algebraic geometry*:",716   "classification_method": "canonical_text_retained_no_recovery_claim",717   "clean_statement_source": "canonical_problem_field",718   "source_file": "aim-algebraic-geometry-notes.json",719   "source_index": 57,720   "attempt": 1721  },722  "AIM-ALGEBRAIC_GEOMETRY-0059": {723   "statement_status": "exact",724   "original_statement": "Which birational transformations preserve the following property?\n\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$\nFor future reference, we call this property (*).",725   "clean_statement": "Which birational transformations preserve the following property?\n\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$\nFor future reference, we call this property (*).",726   "public_statement": "Which birational transformations preserve the following property?\n\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$\nFor future reference, we call this property (*).",727   "evidence": "The canonical record is problem 1.3 from the AIM workshop *Rationality problems in algebraic geometry*, stored as record 58 (zero-based) of `aim-algebraic-geometry-notes.json`. Its problem text is:",728   "classification_method": "canonical_text_retained_no_recovery_claim",729   "clean_statement_source": "canonical_problem_field",730   "source_file": "aim-algebraic-geometry-notes.json",731   "source_index": 58,732   "attempt": 1733  },734  "AIM-ALGEBRAIC_GEOMETRY-0060": {735   "statement_status": "exact",736   "original_statement": "Is there a rationally connected irrational variety $X$ with property (*)?\n\nRecall Property (*) from a previous question was defined as follows.\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$",737   "clean_statement": "Is there a rationally connected irrational variety $X$ with property (*)?\n\nRecall Property (*) from a previous question was defined as follows.\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$",738   "public_statement": "Is there a rationally connected irrational variety $X$ with property (*)?\n\nRecall Property (*) from a previous question was defined as follows.\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$",739   "evidence": "The canonical record is problem 1.32 from the 2019 AIM workshop *Rationality problems in algebraic geometry*. The preceding canonical record, problem 1.30, introduces property (*). Together they ask:",740   "classification_method": "canonical_text_retained_no_recovery_claim",741   "clean_statement_source": "canonical_problem_field",742   "source_file": "aim-algebraic-geometry-notes.json",743   "source_index": 59,744   "attempt": 1745  },746  "AIM-ALGEBRAIC_GEOMETRY-0061": {747   "statement_status": "exact",748   "original_statement": "There have been new rationality constructions of special cubic fourfolds $X$ due to Russo and Staglianò, for discriminant $26$ or $38$. Using these explicit constructions, write down an explicit K3 surface S and an equivalence $Ku(X) \\simeq D(S)$.",749   "clean_statement": "There have been new rationality constructions of special cubic fourfolds $X$ due to Russo and Staglianò, for discriminant $26$ or $38$. Using these explicit constructions, write down an explicit K3 surface S and an equivalence $Ku(X) \\simeq D(S)$.",750   "public_statement": "There have been new rationality constructions of special cubic fourfolds $X$ due to Russo and Staglianò, for discriminant $26$ or $38$. Using these explicit constructions, write down an explicit K3 surface S and an equivalence $Ku(X) \\simeq D(S)$.",751   "evidence": "The canonical record asks:",752   "classification_method": "canonical_text_retained_no_recovery_claim",753   "clean_statement_source": "canonical_problem_field",754   "source_file": "aim-algebraic-geometry-notes.json",755   "source_index": 60,756   "attempt": 1757  },758  "AIM-ALGEBRAIC_GEOMETRY-0062": {759   "statement_status": "exact",760   "original_statement": "For a very general cubic fourfold $X$, is there a surface $S$ such that the Kuznetsov component $Ku(X) \\subset D(X)$ has an embedding $Ku(X) \\hookrightarrow D(S)$?",761   "clean_statement": "For a very general cubic fourfold $X$, is there a surface $S$ such that the Kuznetsov component $Ku(X) \\subset D(X)$ has an embedding $Ku(X) \\hookrightarrow D(S)$?",762   "public_statement": "For a very general cubic fourfold $X$, is there a surface $S$ such that the Kuznetsov component $Ku(X) \\subset D(X)$ has an embedding $Ku(X) \\hookrightarrow D(S)$?",763   "evidence": "The canonical record is problem 1.36 from the AIM workshop *Rationality problems in algebraic geometry*:",764   "classification_method": "canonical_text_retained_no_recovery_claim",765   "clean_statement_source": "canonical_problem_field",766   "source_file": "aim-algebraic-geometry-notes.json",767   "source_index": 61,768   "attempt": 1769  },770  "AIM-ALGEBRAIC_GEOMETRY-0063": {771   "statement_status": "exact",772   "original_statement": "Let $X$ and $Y$ be cubic fourfolds. If $Ku(X) \\simeq Ku(Y)$, is $X$ birational to $Y$?",773   "clean_statement": "Let $X$ and $Y$ be cubic fourfolds. If $Ku(X) \\simeq Ku(Y)$, is $X$ birational to $Y$?",774   "public_statement": "Let $X$ and $Y$ be cubic fourfolds. If $Ku(X) \\simeq Ku(Y)$, is $X$ birational to $Y$?",775   "evidence": "The canonical record is AIM Problem Lists, workshop *Rationality problems in algebraic geometry*, problem 1.38, source file `aim-algebraic-geometry-notes.json`, zero-based record index 62. Its exact problem text is:",776   "classification_method": "canonical_text_retained_no_recovery_claim",777   "clean_statement_source": "canonical_problem_field",778   "source_file": "aim-algebraic-geometry-notes.json",779   "source_index": 62,780   "attempt": 1781  },782  "AIM-ALGEBRAIC_GEOMETRY-0064": {783   "statement_status": "exact",784   "original_statement": "If two smooth projective varieties $X$ and $Y$ are K-equivalent, is the Grothendieck group $K_0(X)$ isomorphic to $K_0(Y)$?",785   "clean_statement": "If two smooth projective varieties $X$ and $Y$ are K-equivalent, is the Grothendieck group $K_0(X)$ isomorphic to $K_0(Y)$?",786   "public_statement": "If two smooth projective varieties $X$ and $Y$ are K-equivalent, is the Grothendieck group $K_0(X)$ isomorphic to $K_0(Y)$?",787   "evidence": "The exact source record asks:",788   "classification_method": "canonical_text_retained_no_recovery_claim",789   "clean_statement_source": "canonical_problem_field",790   "source_file": "aim-algebraic-geometry-notes.json",791   "source_index": 63,792   "attempt": 1793  },794  "AIM-ALGEBRAIC_GEOMETRY-0065": {795   "statement_status": "exact",796   "original_statement": "Investigate $Ku(X)$ for a Fano hypersurface $X$ in a product of projective spaces. Specifically, investigate this for $X \\subset \\mathbb P^2 \\times \\mathbb P^3$ of bidegree $(d,2)$ for $d \\geq 2$.",797   "clean_statement": "Investigate $Ku(X)$ for a Fano hypersurface $X$ in a product of projective spaces. Specifically, investigate this for $X \\subset \\mathbb P^2 \\times \\mathbb P^3$ of bidegree $(d,2)$ for $d \\geq 2$.",798   "public_statement": "Investigate $Ku(X)$ for a Fano hypersurface $X$ in a product of projective spaces. Specifically, investigate this for $X \\subset \\mathbb P^2 \\times \\mathbb P^3$ of bidegree $(d,2)$ for $d \\geq 2$.",799   "evidence": "The canonical record (AIM Problem List, workshop *Rationality problems in algebraic geometry*, problem 1.42) reads:",800   "classification_method": "canonical_text_retained_no_recovery_claim",801   "clean_statement_source": "canonical_problem_field",802   "source_file": "aim-algebraic-geometry-notes.json",803   "source_index": 64,804   "attempt": 1805  },806  "AIM-ALGEBRAIC_GEOMETRY-0066": {807   "statement_status": "exact",808   "original_statement": "Investigate $Ku(X)$ for $X$ a smooth quartic double 5 fold, that is, a double cover of $\\mathbb P^5$ branched along a quartic.\nThis variety is Calabi-Yau and it is never of the form $D(Y)$ for $Y$ a variety. Further, the Serre functor of $X$ is equal to shift by 3; in other words, the category $Ku(X)$ is 3-Calabi Yau.",809   "clean_statement": "Investigate $Ku(X)$ for $X$ a smooth quartic double 5 fold, that is, a double cover of $\\mathbb P^5$ branched along a quartic.\nThis variety is Calabi-Yau and it is never of the form $D(Y)$ for $Y$ a variety. Further, the Serre functor of $X$ is equal to shift by 3; in other words, the category $Ku(X)$ is 3-Calabi Yau.",810   "public_statement": "Investigate $Ku(X)$ for $X$ a smooth quartic double 5 fold, that is, a double cover of $\\mathbb P^5$ branched along a quartic.\nThis variety is Calabi-Yau and it is never of the form $D(Y)$ for $Y$ a variety. Further, the Serre functor of $X$ is equal to shift by 3; in other words, the category $Ku(X)$ is 3-Calabi Yau.",811   "evidence": "The canonical AIM record (problem 1.44, source index 65) reads:",812   "classification_method": "canonical_text_retained_no_recovery_claim",813   "clean_statement_source": "canonical_problem_field",814   "source_file": "aim-algebraic-geometry-notes.json",815   "source_index": 65,816   "attempt": 1817  },818  "AIM-ALGEBRAIC_GEOMETRY-0067": {819   "statement_status": "exact",820   "original_statement": "Show that smooth quartic double 5-folds are irrational.",821   "clean_statement": "Show that smooth quartic double 5-folds are irrational.",822   "public_statement": "Show that smooth quartic double 5-folds are irrational.",823   "evidence": "The canonical AIM record (source file `aim-algebraic-geometry-notes.json`, zero-based index 66, Problem 1.46 of the workshop *Rationality problems in algebraic geometry*) asks:",824   "classification_method": "canonical_text_retained_no_recovery_claim",825   "clean_statement_source": "canonical_problem_field",826   "source_file": "aim-algebraic-geometry-notes.json",827   "source_index": 66,828   "attempt": 1829  },830  "AIM-ALGEBRAIC_GEOMETRY-0068": {831   "statement_status": "exact",832   "original_statement": "Hassett and Tschinkel showed that a complete intersection of two quadrics in $\\mathbb P^5$ over an arbitrary field $k$ is rational if and only if it contains a line over $k$.\n\nGive criteria for other geometrically rational Fano $3$-folds to be rational over a field $k$.\n\nGive criteria for the complete intersection of two quadrics in $\\mathbb P^n$ to be rational over $k$.",833   "clean_statement": "Hassett and Tschinkel showed that a complete intersection of two quadrics in $\\mathbb P^5$ over an arbitrary field $k$ is rational if and only if it contains a line over $k$.\n\nGive criteria for other geometrically rational Fano $3$-folds to be rational over a field $k$.\n\nGive criteria for the complete intersection of two quadrics in $\\mathbb P^n$ to be rational over $k$.",834   "public_statement": "Hassett and Tschinkel showed that a complete intersection of two quadrics in $\\mathbb P^5$ over an arbitrary field $k$ is rational if and only if it contains a line over $k$.\n\nGive criteria for other geometrically rational Fano $3$-folds to be rational over a field $k$.\n\nGive criteria for the complete intersection of two quadrics in $\\mathbb P^n$ to be rational over $k$.",835   "evidence": "The canonical AIM record asks:",836   "classification_method": "canonical_text_retained_no_recovery_claim",837   "clean_statement_source": "canonical_problem_field",838   "source_file": "aim-algebraic-geometry-notes.json",839   "source_index": 67,840   "attempt": 1841  },842  "AIM-ALGEBRAIC_GEOMETRY-0069": {843   "statement_status": "exact",844   "original_statement": "Let $X$ be a smooth projective complex variety. For a finite abelian group $M$, is unramified cohomology $H^i_{nr}(k(X)/k, M)$ always finite?",845   "clean_statement": "Let $X$ be a smooth projective complex variety. For a finite abelian group $M$, is unramified cohomology $H^i_{nr}(k(X)/k, M)$ always finite?",846   "public_statement": "Let $X$ be a smooth projective complex variety. For a finite abelian group $M$, is unramified cohomology $H^i_{nr}(k(X)/k, M)$ always finite?",847   "evidence": "The canonical AIM record asks:",848   "classification_method": "canonical_text_retained_no_recovery_claim",849   "clean_statement_source": "canonical_problem_field",850   "source_file": "aim-algebraic-geometry-notes.json",851   "source_index": 68,852   "attempt": 1853  },854  "AIM-ALGEBRAIC_GEOMETRY-0070": {855   "statement_status": "exact",856   "original_statement": "Let $X$ be a rationally connected smooth projective variety of dimension $n$.\nIs $H^n_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z) = 0$?",857   "clean_statement": "Let $X$ be a rationally connected smooth projective variety of dimension $n$.\nIs $H^n_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z) = 0$?",858   "public_statement": "Let $X$ be a rationally connected smooth projective variety of dimension $n$.\nIs $H^n_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z) = 0$?",859   "evidence": "The exact AIM record is:",860   "classification_method": "canonical_text_retained_no_recovery_claim",861   "clean_statement_source": "canonical_problem_field",862   "source_file": "aim-algebraic-geometry-notes.json",863   "source_index": 69,864   "attempt": 1865  },866  "AIM-ALGEBRAIC_GEOMETRY-0071": {867   "statement_status": "exact",868   "original_statement": "Let $X$ be a variety over $\\overline{\\mathbb Q}$. Is there an algorithm to determine whether\n$X$ is rational?\n\nSame question for unirationality.",869   "clean_statement": "Let $X$ be a variety over $\\overline{\\mathbb Q}$. Is there an algorithm to determine whether\n$X$ is rational?\n\nSame question for unirationality.",870   "public_statement": "Let $X$ be a variety over $\\overline{\\mathbb Q}$. Is there an algorithm to determine whether\n$X$ is rational?\n\nSame question for unirationality.",871   "evidence": "The canonical record is Problem 1.54 from the AIM workshop *Rationality problems in algebraic geometry*:",872   "classification_method": "canonical_text_retained_no_recovery_claim",873   "clean_statement_source": "canonical_problem_field",874   "source_file": "aim-algebraic-geometry-notes.json",875   "source_index": 70,876   "attempt": 1877  },878  "AIM-ALGEBRAIC_GEOMETRY-0072": {879   "statement_status": "exact",880   "original_statement": "Given a rationally connected variety which is not universally $CH_0$-trivial,\ndetermine a nonzero unramified class in some cycle module (in the sense of Rost).",881   "clean_statement": "Given a rationally connected variety which is not universally $CH_0$-trivial,\ndetermine a nonzero unramified class in some cycle module (in the sense of Rost).",882   "public_statement": "Given a rationally connected variety which is not universally $CH_0$-trivial,\ndetermine a nonzero unramified class in some cycle module (in the sense of Rost).",883   "evidence": "There is no visible corruption or ambiguity in the source text. To make the mathematical hypotheses precise, this report works with a smooth, proper, geometrically integral \\(X/k\\). The main new formulation also assumes that \\(X\\) has a zero-cycle \\(z\\) of degree one. This is automatic when \\(k\\) is algebraically closed and whenever \\(X(k)\\ne\\varnothing\\), which covers the usual geometric setting of the AIM question. In characteristic zero, a smooth projective rationally connected variety is rationally chain connected. Rational chain connectedness is used below only to ensure a finite torsion order.",884   "classification_method": "explicit_no_change_evidence",885   "clean_statement_source": "canonical_problem_field",886   "source_file": "aim-algebraic-geometry-notes.json",887   "source_index": 71,888   "attempt": 1889  },890  "AIM-ALGEBRAIC_GEOMETRY-0073": {891   "statement_status": "exact",892   "original_statement": "Prove that Oblomkov-Rozansky triply graded cohomology (HHH$^{\\text{geom}}$) is isomorphic to Khovanov-Rozansky triply graded cohomology (HHH).",893   "clean_statement": "Prove that Oblomkov-Rozansky triply graded cohomology (HHH$^{\\text{geom}}$) is isomorphic to Khovanov-Rozansky triply graded cohomology (HHH).",894   "public_statement": "Prove that Oblomkov-Rozansky triply graded cohomology (HHH$^{\\text{geom}}$) is isomorphic to Khovanov-Rozansky triply graded cohomology (HHH).",895   "evidence": "The canonical AIM record is Problem 1.1 in the section “Oblomkov-Rozansky link invariant” of the 2018 AIM workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*:",896   "classification_method": "canonical_text_retained_no_recovery_claim",897   "clean_statement_source": "canonical_problem_field",898   "source_file": "aim-algebraic-geometry-notes.json",899   "source_index": 72,900   "attempt": 1901  },902  "AIM-ALGEBRAIC_GEOMETRY-0074": {903   "statement_status": "exact",904   "original_statement": "Prove that MF$((\\mathscr{X}^\\text{big})^\\text{stable},w)$ is equivalent as a monoidal category to K$^b(\\mathbb{S}\\text{Bim}_n)$, where $\\mathscr{X}^\\text{big}=\\mathfrak{g} \\times G \\times \\mathfrak{n} \\times G \\times \\mathfrak{n}$.",905   "clean_statement": "Prove that MF$((\\mathscr{X}^\\text{big})^\\text{stable},w)$ is equivalent as a monoidal category to K$^b(\\mathbb{S}\\text{Bim}_n)$, where $\\mathscr{X}^\\text{big}=\\mathfrak{g} \\times G \\times \\mathfrak{n} \\times G \\times \\mathfrak{n}$.",906   "public_statement": "Prove that MF$((\\mathscr{X}^\\text{big})^\\text{stable},w)$ is equivalent as a monoidal category to K$^b(\\mathbb{S}\\text{Bim}_n)$, where $\\mathscr{X}^\\text{big}=\\mathfrak{g} \\times G \\times \\mathfrak{n} \\times G \\times \\mathfrak{n}$.",907   "evidence": "The exact AIM record is:",908   "classification_method": "canonical_text_retained_no_recovery_claim",909   "clean_statement_source": "canonical_problem_field",910   "source_file": "aim-algebraic-geometry-notes.json",911   "source_index": 73,912   "attempt": 1913  },914  "AIM-ALGEBRAIC_GEOMETRY-0075": {915   "statement_status": "exact",916   "original_statement": "On HHH$^\\text{geom}$ of a braid $\\beta$ there is a mysterious 4th $\\mathbb{Z}$-grading coming from the $\\mathbb{Z}/2$-grading on matrix factorizations. Construct the analog for HHH of $\\beta$.",917   "clean_statement": "On HHH$^\\text{geom}$ of a braid $\\beta$ there is a mysterious 4th $\\mathbb{Z}$-grading coming from the $\\mathbb{Z}/2$-grading on matrix factorizations. Construct the analog for HHH of $\\beta$.",918   "public_statement": "On HHH$^\\text{geom}$ of a braid $\\beta$ there is a mysterious 4th $\\mathbb{Z}$-grading coming from the $\\mathbb{Z}/2$-grading on matrix factorizations. Construct the analog for HHH of $\\beta$.",919   "evidence": "The exact AIM record is:",920   "classification_method": "canonical_text_retained_no_recovery_claim",921   "clean_statement_source": "canonical_problem_field",922   "source_file": "aim-algebraic-geometry-notes.json",923   "source_index": 74,924   "attempt": 1925  },926  "AIM-ALGEBRAIC_GEOMETRY-0076": {927   "statement_status": "reconstructed_unverified",928   "original_statement": "To what extend can computers compute HHH$^\\text{geom}$ directly from the definition?",929   "clean_statement": null,930   "public_statement": "To what extend can computers compute HHH$^\\text{geom}$ directly from the definition?",931   "evidence": "The word “extend” is preserved from the record; it is almost certainly a typographical error for “extent.” The mathematical phrase “directly from the definition” is not defined in the record. The contemporaneous Oblomkov–Rozansky definition gives the following unambiguous strict reading. For a braid \\(\\beta\\in Br_n\\), one:",932   "classification_method": "explicit_uncertainty_evidence",933   "clean_statement_source": "no_safe_clean_extraction",934   "source_file": "aim-algebraic-geometry-notes.json",935   "source_index": 75,936   "attempt": 1937  },938  "AIM-ALGEBRAIC_GEOMETRY-0077": {939   "statement_status": "exact",940   "original_statement": "The space $\\mathscr{X}^{\\text{big}}$ and category MF($\\mathscr{X}^{\\text{big}},w$) makes sense not just in type A. What is the analog of $(\\mathscr{X}^{\\text{big}})^{\\text{stable}}$ in other types? What is the categorical analog of the map Br$_\\text{ext} \\to$ Br$_\\text{fin}$.",941   "clean_statement": "The space $\\mathscr{X}^{\\text{big}}$ and category MF($\\mathscr{X}^{\\text{big}},w$) makes sense not just in type A. What is the analog of $(\\mathscr{X}^{\\text{big}})^{\\text{stable}}$ in other types? What is the categorical analog of the map Br$_\\text{ext} \\to$ Br$_\\text{fin}$.",942   "public_statement": "The space $\\mathscr{X}^{\\text{big}}$ and category MF($\\mathscr{X}^{\\text{big}},w$) makes sense not just in type A. What is the analog of $(\\mathscr{X}^{\\text{big}})^{\\text{stable}}$ in other types? What is the categorical analog of the map Br$_\\text{ext} \\to$ Br$_\\text{fin}$.",943   "evidence": "The preceding AIM record specifies \\[ \\mathscr X^{\\mathrm{big}} =\\mathfrak g\\times G\\times\\mathfrak n\\times G\\times\\mathfrak n. \\] This agrees with the “non-reduced” two-fold space \\(\\mathcal X_2=\\mathfrak g\\times(G\\times\\mathfrak n)^2\\) in Oblomkov--Rozansky. We work over \\(\\mathbb C\\), take \\(G\\) to be a connected reductive group, \\(B\\subset G\\) a Borel subgroup, and \\(\\mathfrak n=\\operatorname{Lie}R_u(B)\\). Fixing a nondegenerate \\(G\\)-invariant bilinear form \\(\\kappa\\) on \\(\\mathfrak g\\), the potential is \\[ w(X,g_1,Y_1,g_2,Y_2) =\\kappa\\!\\left(X,\\operatorname{Ad}_{g_1}Y_1- \\operatorname{Ad}_{g_2}Y_2\\right). \\] No corruption of the source statement was found. “Stable” is interpreted through the type A construction in arXiv:1702.03569, Section 2.6.",944   "classification_method": "explicit_no_change_evidence",945   "clean_statement_source": "canonical_problem_field",946   "source_file": "aim-algebraic-geometry-notes.json",947   "source_index": 76,948   "attempt": 1949  },950  "AIM-ALGEBRAIC_GEOMETRY-0078": {951   "statement_status": "exact",952   "original_statement": "What is the interaction between cell theory and the Hilbert scheme? Matrix factorizations?\n\nIn the affine Steinberg setting this corresponds to the stratification by nilpotent orbits. Does this have an analog for $\\mathscr{X}^\\text{big}$?",953   "clean_statement": "What is the interaction between cell theory and the Hilbert scheme? Matrix factorizations?\n\nIn the affine Steinberg setting this corresponds to the stratification by nilpotent orbits. Does this have an analog for $\\mathscr{X}^\\text{big}$?",954   "public_statement": "What is the interaction between cell theory and the Hilbert scheme? Matrix factorizations?\n\nIn the affine Steinberg setting this corresponds to the stratification by nilpotent orbits. Does this have an analog for $\\mathscr{X}^\\text{big}$?",955   "evidence": "The exact AIM record is:",956   "classification_method": "canonical_text_retained_no_recovery_claim",957   "clean_statement_source": "canonical_problem_field",958   "source_file": "aim-algebraic-geometry-notes.json",959   "source_index": 77,960   "attempt": 1961  },962  "AIM-ALGEBRAIC_GEOMETRY-0079": {963   "statement_status": "exact",964   "original_statement": "Call a braid $\\beta$ parity if HHH$(\\beta)$ is parity. Find necessary and/or sufficient conditions for a braid to be parity.\n\nAre there operations on braids (for example, adding a positive number of full twists) which transform it into a parity braid or which preserve parity braids?",965   "clean_statement": "Call a braid $\\beta$ parity if HHH$(\\beta)$ is parity. Find necessary and/or sufficient conditions for a braid to be parity.\n\nAre there operations on braids (for example, adding a positive number of full twists) which transform it into a parity braid or which preserve parity braids?",966   "public_statement": "Call a braid $\\beta$ parity if HHH$(\\beta)$ is parity. Find necessary and/or sufficient conditions for a braid to be parity.\n\nAre there operations on braids (for example, adding a positive number of full twists) which transform it into a parity braid or which preserve parity braids?",967   "evidence": "The canonical record is problem 2.1, “Parity,” from the AIM workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*:",968   "classification_method": "canonical_text_retained_no_recovery_claim",969   "clean_statement_source": "canonical_problem_field",970   "source_file": "aim-algebraic-geometry-notes.json",971   "source_index": 78,972   "attempt": 1973  },974  "AIM-ALGEBRAIC_GEOMETRY-0080": {975   "statement_status": "exact",976   "original_statement": "Are algebraic braids parity? If so find a (recursive) formula for HHH of an algebraic braid.",977   "clean_statement": "Are algebraic braids parity? If so find a (recursive) formula for HHH of an algebraic braid.",978   "public_statement": "Are algebraic braids parity? If so find a (recursive) formula for HHH of an algebraic braid.",979   "evidence": "The exact AIM record is:",980   "classification_method": "canonical_text_retained_no_recovery_claim",981   "clean_statement_source": "canonical_problem_field",982   "source_file": "aim-algebraic-geometry-notes.json",983   "source_index": 79,984   "attempt": 1985  },986  "AIM-ALGEBRAIC_GEOMETRY-0081": {987   "statement_status": "exact",988   "original_statement": "There is a commutative family of braids built as products of FT$_k$, the full twist on the first $k$ strands, for $k \\leq n$. Are the positive products in this family parity? If so, find a (recursive) formula for HHH.",989   "clean_statement": "There is a commutative family of braids built as products of FT$_k$, the full twist on the first $k$ strands, for $k \\leq n$. Are the positive products in this family parity? If so, find a (recursive) formula for HHH.",990   "public_statement": "There is a commutative family of braids built as products of FT$_k$, the full twist on the first $k$ strands, for $k \\leq n$. Are the positive products in this family parity? If so, find a (recursive) formula for HHH.",991   "evidence": "The archived AIM page timed out during this run, but the repository record is syntactically complete and the notation is fixed by the adjacent records. There is no apparent OCR corruption.",992   "classification_method": "explicit_no_change_evidence",993   "clean_statement_source": "canonical_problem_field",994   "source_file": "aim-algebraic-geometry-notes.json",995   "source_index": 80,996   "attempt": 1997  },998  "AIM-ALGEBRAIC_GEOMETRY-0082": {999   "statement_status": "exact",1000   "original_statement": "In the presence of parity, there are two combinatorial formulas for HHH: one for localization of Hilbert schemes from the GNR conjecture and one from the Elias-Hogencamp-Mellit papers. How are these formulas related? How are they related to Demazure crystals?",1001   "clean_statement": "In the presence of parity, there are two combinatorial formulas for HHH: one for localization of Hilbert schemes from the GNR conjecture and one from the Elias-Hogencamp-Mellit papers. How are these formulas related? How are they related to Demazure crystals?",1002   "public_statement": "In the presence of parity, there are two combinatorial formulas for HHH: one for localization of Hilbert schemes from the GNR conjecture and one from the Elias-Hogencamp-Mellit papers. How are these formulas related? How are they related to Demazure crystals?",1003   "evidence": "The canonical AIM record asks:",1004   "classification_method": "canonical_text_retained_no_recovery_claim",1005   "clean_statement_source": "canonical_problem_field",1006   "source_file": "aim-algebraic-geometry-notes.json",1007   "source_index": 81,1008   "attempt": 11009  },1010  "AIM-ALGEBRAIC_GEOMETRY-0083": {1011   "statement_status": "exact",1012   "original_statement": "To an algebraic link one can associate a Springer fibre and a Milnor fibre. Can one compute HHH from the Milnor fibre?",1013   "clean_statement": "To an algebraic link one can associate a Springer fibre and a Milnor fibre. Can one compute HHH from the Milnor fibre?",1014   "public_statement": "To an algebraic link one can associate a Springer fibre and a Milnor fibre. Can one compute HHH from the Milnor fibre?",1015   "evidence": "The exact AIM record is:",1016   "classification_method": "canonical_text_retained_no_recovery_claim",1017   "clean_statement_source": "canonical_problem_field",1018   "source_file": "aim-algebraic-geometry-notes.json",1019   "source_index": 82,1020   "attempt": 11021  },1022  "AIM-ALGEBRAIC_GEOMETRY-0084": {1023   "statement_status": "exact",1024   "original_statement": "It is known that the Alexander polynomial can be computed from both the Springer fibre and the Milnor fibre. Understand the relationship between these two constructions.",1025   "clean_statement": "It is known that the Alexander polynomial can be computed from both the Springer fibre and the Milnor fibre. Understand the relationship between these two constructions.",1026   "public_statement": "It is known that the Alexander polynomial can be computed from both the Springer fibre and the Milnor fibre. Understand the relationship between these two constructions.",1027   "evidence": "The exact AIM record is:",1028   "classification_method": "canonical_text_retained_no_recovery_claim",1029   "clean_statement_source": "canonical_problem_field",1030   "source_file": "aim-algebraic-geometry-notes.json",1031   "source_index": 83,1032   "attempt": 11033  },1034  "AIM-ALGEBRAIC_GEOMETRY-0085": {1035   "statement_status": "exact",1036   "original_statement": "Outside of type A is there still a relation between Hochschild cohomology of Rouquier complexes and affine Springer fibres?",1037   "clean_statement": "Outside of type A is there still a relation between Hochschild cohomology of Rouquier complexes and affine Springer fibres?",1038   "public_statement": "Outside of type A is there still a relation between Hochschild cohomology of Rouquier complexes and affine Springer fibres?",1039   "evidence": "The canonical AIM record is Problem 3.3 in the “Springer fibres” section of the workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*:",1040   "classification_method": "canonical_text_retained_no_recovery_claim",1041   "clean_statement_source": "canonical_problem_field",1042   "source_file": "aim-algebraic-geometry-notes.json",1043   "source_index": 84,1044   "attempt": 11045  },1046  "AIM-ALGEBRAIC_GEOMETRY-0086": {1047   "statement_status": "exact",1048   "original_statement": "What is the Drinfeld center of K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$? K$^b(\\mathbb{S}\\text{Bim}_{\\text{aff}})$? Are there additional complexes in K$^b(\\mathbb{S}\\text{Bim})$ which commute with $\\mathbb{S}$Bim?",1049   "clean_statement": "What is the Drinfeld center of K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$? K$^b(\\mathbb{S}\\text{Bim}_{\\text{aff}})$? Are there additional complexes in K$^b(\\mathbb{S}\\text{Bim})$ which commute with $\\mathbb{S}$Bim?",1050   "public_statement": "What is the Drinfeld center of K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$? K$^b(\\mathbb{S}\\text{Bim}_{\\text{aff}})$? Are there additional complexes in K$^b(\\mathbb{S}\\text{Bim})$ which commute with $\\mathbb{S}$Bim?",1051   "evidence": "The canonical AIM record is Problem 4.1 in the section “Soergel bimodules” of the workshop list *Categorified Hecke algebras, link homology, and Hilbert schemes*. Its exact text is:",1052   "classification_method": "canonical_text_retained_no_recovery_claim",1053   "clean_statement_source": "canonical_problem_field",1054   "source_file": "aim-algebraic-geometry-notes.json",1055   "source_index": 85,1056   "attempt": 11057  },1058  "AIM-ALGEBRAIC_GEOMETRY-0087": {1059   "statement_status": "exact",1060   "original_statement": "Is Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{ext}})$) generated by Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ generated by images of Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ split generated by powers of FT$_n$?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ symmetric monoidal?",1061   "clean_statement": "Is Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{ext}})$) generated by Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ generated by images of Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ split generated by powers of FT$_n$?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ symmetric monoidal?",1062   "public_statement": "Is Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{ext}})$) generated by Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ generated by images of Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ split generated by powers of FT$_n$?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ symmetric monoidal?",1063   "evidence": "The canonical record is Problem 4.2 in the “Soergel bimodules” section of the AIM workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*. The source page and the neighboring Problems 4.1 and 4.3 confirm the following reading:",1064   "classification_method": "canonical_text_retained_no_recovery_claim",1065   "clean_statement_source": "canonical_problem_field",1066   "source_file": "aim-algebraic-geometry-notes.json",1067   "source_index": 86,1068   "attempt": 11069  },1070  "AIM-ALGEBRAIC_GEOMETRY-0088": {1071   "statement_status": "exact",1072   "original_statement": "Is the flattening of the Gaitsgory complex for the standard representation equal to the tautological sheaf from the GNR conjecture?",1073   "clean_statement": "Is the flattening of the Gaitsgory complex for the standard representation equal to the tautological sheaf from the GNR conjecture?",1074   "public_statement": "Is the flattening of the Gaitsgory complex for the standard representation equal to the tautological sheaf from the GNR conjecture?",1075   "evidence": "The AIM record asks:",1076   "classification_method": "canonical_text_retained_no_recovery_claim",1077   "clean_statement_source": "canonical_problem_field",1078   "source_file": "aim-algebraic-geometry-notes.json",1079   "source_index": 87,1080   "attempt": 11081  },1082  "AIM-ALGEBRAIC_GEOMETRY-0089": {1083   "statement_status": "exact",1084   "original_statement": "A theorem of Morton-Samuelson connected the Elliptic Hall algebra at $q=t$ to the Skein algebra of the torus. Categorify this result.",1085   "clean_statement": "A theorem of Morton-Samuelson connected the Elliptic Hall algebra at $q=t$ to the Skein algebra of the torus. Categorify this result.",1086   "public_statement": "A theorem of Morton-Samuelson connected the Elliptic Hall algebra at $q=t$ to the Skein algebra of the torus. Categorify this result.",1087   "evidence": "The canonical record is Problem 5.1 in the “Others” section of the AIM workshop list *Categorified Hecke algebras, link homology, and Hilbert schemes*. The live AIM page confirms the exact text:",1088   "classification_method": "canonical_text_retained_no_recovery_claim",1089   "clean_statement_source": "canonical_problem_field",1090   "source_file": "aim-algebraic-geometry-notes.json",1091   "source_index": 88,1092   "attempt": 11093  },1094  "AIM-ALGEBRAIC_GEOMETRY-0090": {1095   "statement_status": "exact",1096   "original_statement": "Are there simplifications of the above problem for the $\\mathfrak{sl}_n$-quotient invariant?",1097   "clean_statement": "Are there simplifications of the above problem for the $\\mathfrak{sl}_n$-quotient invariant?",1098   "public_statement": "Are there simplifications of the above problem for the $\\mathfrak{sl}_n$-quotient invariant?",1099   "evidence": "The canonical record is Problem 5.2 in the “Others” section of the 2018 AIM workshop list *Categorified Hecke algebras, link homology, and Hilbert schemes*. The live AIM page confirms the exact wording:",1100   "classification_method": "canonical_text_retained_no_recovery_claim",1101   "clean_statement_source": "canonical_problem_field",1102   "source_file": "aim-algebraic-geometry-notes.json",1103   "source_index": 89,1104   "attempt": 11105  },1106  "AIM-ALGEBRAIC_GEOMETRY-0091": {1107   "statement_status": "exact",1108   "original_statement": "Is there an equivalence of categories between DAHA representations and a suitable extension of the Fukaya category of the character variety of a single punctured torus?",1109   "clean_statement": "Is there an equivalence of categories between DAHA representations and a suitable extension of the Fukaya category of the character variety of a single punctured torus?",1110   "public_statement": "Is there an equivalence of categories between DAHA representations and a suitable extension of the Fukaya category of the character variety of a single punctured torus?",1111   "evidence": "The canonical AIM record, from the workshop *Categorified Hecke algebras, link homology, and Hilbert schemes* (AIM, 1--5 October 2018), asks:",1112   "classification_method": "canonical_text_retained_no_recovery_claim",1113   "clean_statement_source": "canonical_problem_field",1114   "source_file": "aim-algebraic-geometry-notes.json",1115   "source_index": 90,1116   "attempt": 11117  },1118  "AIM-ALGEBRAIC_GEOMETRY-0092": {1119   "statement_status": "exact",1120   "original_statement": "Is there a braid invariant extension of HHH which replaces bigraded vector spaces with bigraded $S_n$-modules suitably compatible with the action of $R=\\mathbb{Q}[x_1, \\dots, x_n]$ and the action of $\\mathbb{Q}[y_1, \\dots, y_n]$ coming from $y$-ification?",1121   "clean_statement": "Is there a braid invariant extension of HHH which replaces bigraded vector spaces with bigraded $S_n$-modules suitably compatible with the action of $R=\\mathbb{Q}[x_1, \\dots, x_n]$ and the action of $\\mathbb{Q}[y_1, \\dots, y_n]$ coming from $y$-ification?",1122   "public_statement": "Is there a braid invariant extension of HHH which replaces bigraded vector spaces with bigraded $S_n$-modules suitably compatible with the action of $R=\\mathbb{Q}[x_1, \\dots, x_n]$ and the action of $\\mathbb{Q}[y_1, \\dots, y_n]$ coming from $y$-ification?",1123   "evidence": "The canonical record is problem 5.4, “Others,” from the AIM workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*. Its exact statement is:",1124   "classification_method": "canonical_text_retained_no_recovery_claim",1125   "clean_statement_source": "canonical_problem_field",1126   "source_file": "aim-algebraic-geometry-notes.json",1127   "source_index": 91,1128   "attempt": 11129  },1130  "AIM-ALGEBRAIC_GEOMETRY-0093": {1131   "statement_status": "exact",1132   "original_statement": "Can one describe HHH$^{a>0}$ using HHH$^{a=0}$ after tensoring with a suitable central complex?",1133   "clean_statement": "Can one describe HHH$^{a>0}$ using HHH$^{a=0}$ after tensoring with a suitable central complex?",1134   "public_statement": "Can one describe HHH$^{a>0}$ using HHH$^{a=0}$ after tensoring with a suitable central complex?",1135   "evidence": "The exact AIM record is Problem 5.5 in the “Others” section of the 2018 workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*:",1136   "classification_method": "canonical_text_retained_no_recovery_claim",1137   "clean_statement_source": "canonical_problem_field",1138   "source_file": "aim-algebraic-geometry-notes.json",1139   "source_index": 92,1140   "attempt": 11141  },1142  "AIM-ALGEBRAIC_GEOMETRY-0094": {1143   "statement_status": "exact",1144   "original_statement": "How does one describe the Procesi bundle in the other settings, like K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$?",1145   "clean_statement": "How does one describe the Procesi bundle in the other settings, like K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$?",1146   "public_statement": "How does one describe the Procesi bundle in the other settings, like K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$?",1147   "evidence": "The exact AIM record is:",1148   "classification_method": "canonical_text_retained_no_recovery_claim",1149   "clean_statement_source": "canonical_problem_field",1150   "source_file": "aim-algebraic-geometry-notes.json",1151   "source_index": 93,1152   "attempt": 11153  },1154  "AIM-ALGEBRAIC_GEOMETRY-0095": {1155   "statement_status": "exact",1156   "original_statement": "Can we give lower bounds on the degree of irrationality / covering gonality.... of hypersurfaces in char $p$?\nHow do we make sense of these notions over countable / finite fields?",1157   "clean_statement": "Can we give lower bounds on the degree of irrationality / covering gonality.... of hypersurfaces in char $p$?\nHow do we make sense of these notions over countable / finite fields?",1158   "public_statement": "Can we give lower bounds on the degree of irrationality / covering gonality.... of hypersurfaces in char $p$?\nHow do we make sense of these notions over countable / finite fields?",1159   "evidence": "The canonical record is AIM-ALGEBRAIC_GEOMETRY-0095, item 2.1 in the section “Degree of irrationality and covering gonality” of the AIM workshop *Rational subvarieties in positive characteristic*. Its exact problem text is:",1160   "classification_method": "canonical_text_retained_no_recovery_claim",1161   "clean_statement_source": "canonical_problem_field",1162   "source_file": "aim-algebraic-geometry-notes.json",1163   "source_index": 94,1164   "attempt": 11165  },1166  "AIM-ALGEBRAIC_GEOMETRY-0096": {1167   "statement_status": "exact",1168   "original_statement": "How does rationality behave in a family over $\\text{Spec} \\mathbb{Z}$ or $\\text{Spec} \\mathbb{Z}[\\frac{1}{n}]$.\nIf $X$ is rational / unirational for almost all $p$, does it follow $X$ is rational / unirational in char $0$?",1169   "clean_statement": "How does rationality behave in a family over $\\text{Spec} \\mathbb{Z}$ or $\\text{Spec} \\mathbb{Z}[\\frac{1}{n}]$.\nIf $X$ is rational / unirational for almost all $p$, does it follow $X$ is rational / unirational in char $0$?",1170   "public_statement": "How does rationality behave in a family over $\\text{Spec} \\mathbb{Z}$ or $\\text{Spec} \\mathbb{Z}[\\frac{1}{n}]$.\nIf $X$ is rational / unirational for almost all $p$, does it follow $X$ is rational / unirational in char $0$?",1171   "evidence": "The record is Problem 3.06 in the section “Rationality in a family” of the 2016 AIM workshop *Rational subvarieties in positive characteristic*. Its exact text is:",1172   "classification_method": "canonical_text_retained_no_recovery_claim",1173   "clean_statement_source": "canonical_problem_field",1174   "source_file": "aim-algebraic-geometry-notes.json",1175   "source_index": 95,1176   "attempt": 11177  },1178  "AIM-ALGEBRAIC_GEOMETRY-0097": {1179   "statement_status": "corrected_verified",1180   "original_statement": "Can we find a cubic fourfold which is conjectually irratioal, but (most of ) its reductions are rational?",1181   "clean_statement": "Can one find a smooth cubic fourfold over a number field which is conjecturally irrational over an algebraic closure of characteristic zero, but for which “most” good reductions are rational?",1182   "public_statement": "Can one find a smooth cubic fourfold over a number field which is conjecturally irrational over an algebraic closure of characteristic zero, but for which “most” good reductions are rational?",1183   "evidence": "The same wording appears on the AIM source page, with no appended remark. I reconstruct only the evident typographical error “irratioal” as “irrational.” The mathematically recovered question is therefore:",1184   "classification_method": "source_or_typo_verified_repair",1185   "clean_statement_source": "labeled_recovery_in_report_section_1",1186   "source_file": "aim-algebraic-geometry-notes.json",1187   "source_index": 96,1188   "attempt": 11189  },1190  "AIM-ALGEBRAIC_GEOMETRY-0098": {1191   "statement_status": "exact",1192   "original_statement": "If the general member of a family of smooth cubic is rational, is the special fiber also rational?",1193   "clean_statement": "If the general member of a family of smooth cubic is rational, is the special fiber also rational?",1194   "public_statement": "If the general member of a family of smooth cubic is rational, is the special fiber also rational?",1195   "evidence": "The canonical record asks:",1196   "classification_method": "canonical_text_retained_no_recovery_claim",1197   "clean_statement_source": "canonical_problem_field",1198   "source_file": "aim-algebraic-geometry-notes.json",1199   "source_index": 97,1200   "attempt": 1

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