Polynomial equations define geometric spaces.
5 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: 2026-08-23Akhil 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: 2026-07-14The 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 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: 2026-10-06Polynomial equations define geometric spaces. This topic follows algebraic cycles, cohomology, and complex varieties; different subclasses of the Hodge conjecture remain distinct.
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 counterexample to Grothendieck’s group-scheme question
OpenAI reports: The rational Hodge conjecture for CM abelian varieties
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