Groups, rings, representations, and algebraic structures, including their connections to geometry and topology.
16 records · Includes focused interests · How we rate progress
Alpöge and Howell report elliptic curves over the rationals with at least 30 and at least 31 independent rational points of infinite order. Dujella’s specialist table records the rank-at-least-31 example as the current record, and the news account attributes assistance to an internal Claude model.
Read original source Original rank-at-least-31 record, point data and attributionelliptic-rank.icarm.cloud
Original report: 2026-08-23The Alpöge–Claude manuscript claims a compact complex threefold, obtained by completing a modular family of complex two-tori, whose underlying smooth manifold is diffeomorphic to S⁶. Philip Engel supplies a separate geometric exposition and self-contained proof of the construction.
Read original source Original Alpöge–Claude constructionalpo.ge
Original report: 2026-08-23Huang, Jackson, Lee, Poonen, Pries and Zhang construct an explicit degree-23 polynomial whose splitting field over the rationals has Galois group M₂₃. They also construct a regular M₂₃ extension of Q(t), completing the realization program for all 26 sporadic finite simple groups.
Read original source Original paper v2: theorems and human/AI attributionarxiv.org
Original report: 2026-08-09Akhil Mathew’s contribution gives a noncommutative finite free group scheme of order four over a nonreduced base that is not killed by four.
Read original source Mathew’s statement and AI disclosuregithub.com
Original report: 2026-07-14OpenAI reports that the rational Grothendieck–Teichmüller Lie algebra is free on one generator in each odd weight starting at three, with the corresponding weight-completed isomorphism.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports a finitely presented torsion-free group with a finite two-dimensional classifying space whose group algebra over the two-element field contains nonzero elements with product zero.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports a finitely presented torsion-free group with a finite two-dimensional classifying complex and elements a,b in its characteristic-two group algebra satisfying ab=1 but ba≠1.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports one fixed finite-dimensional complex algebra with finite-dimensional modules of unbounded finite projective dimension, refuting universal finiteness of little finitistic dimension.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports an infinite discrete group with an ordinary finite presentation that is residually finite and has finite two-power order for every element, with unbounded element orders.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports that Thompson’s group F, viewed as a discrete group, is nonamenable.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06Flake, Gruber and Heidersdorf report that the universal rigid monoidal category on one object has an abelian envelope without the quotient property.
Original report: 2026-09-15The Ulam writeup reports a polynomial map C³→C³ with constant nonzero Jacobian determinant −2 and three distinct points sharing one image.
Original report: 2026-07-20OpenAI claims Lie-group structure for any locally compact second-countable Hausdorff group with a faithful jointly continuous action on a connected finite-dimensional manifold under the stated topological conventions.
Read original source Pinned primary manuscript · family 304github.com
Original report: 2026-10-06OpenAI claims that a hyperbolic group with a two-sphere boundary has a proper cocompact action by isometries on hyperbolic three-space, allowing a finite kernel.
Read original source Pinned primary manuscript · family 246github.com
Original report: 2026-10-06OpenAI claims strict positivity of the Tor Euler characteristic for nonzero finite modules of complementary dimension over a regular local ring, when their tensor product has finite length.
Read original source Pinned primary manuscript · family 193github.com
Original report: 2026-10-06OpenAI claims that rational Hodge classes on any complex CM abelian variety admit algebraic-cycle representatives with rational coefficients, in all codimensions.
Read original source Pinned primary manuscript · family 032github.com
Original report: 2026-10-06Groups, rings, representations, and algebraic structures, including their connections to geometry and topology.
Related focused topics and problems retain their own statements and baselines. A contribution may belong to more than one field; it is counted once within each topic.
Background review: 2026-10-11. Selected sources:
Publication dates and editorial changes.
Alpöge–Howell–Claude: elliptic curves of rank at least 30 and 31
Alpöge–Claude reports: a complex structure on the six-sphere
AI-assisted realization of the Mathieu group M₂₃ over the rationals
AI-assisted counterexample to Grothendieck’s group-scheme question
OpenAI reports: An infinite finitely presented residually finite 2-group
OpenAI reports: A torsion-free direct-finiteness counterexample
AI-assisted counterexample to the abelian-envelope quotient property
OpenAI reports: The rational Hodge conjecture for CM abelian varieties
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