Shapes, distances, spaces, and their structure, including packings, metric embeddings, manifolds, and algebraic geometry.
30 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: 2026-08-23Math, Inc. reports Lean formalizations of sphere-packing optimality for the E₈ lattice in dimension eight and the Leech lattice in dimension 24, with uniqueness among periodic packings in dimension 24. The human maintainers report checking definitions, axiom dependencies and the supplied code.
Read original source Math, Inc.: reported formalization and scopemath.inc
Original report: 2026-03-02The 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: 2026-08-23For every convex body K in R^n, n≥1, the manuscript reports volume product at its Santaló point at least (n+1)^(n+1)/(n!)², with equality exactly for simplices.
Original report: 2026-10-06Akhil 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 K_X+mL is globally generated on every smooth complex projective n-dimensional variety for every ample line bundle L and every integer m at least n+1.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports that, at r at least ten very general complex plane points, the total multiplicity of every nonzero effective degree-d curve is strictly less than d times the square root of r.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports that every origin-symmetric convex body in n-dimensional real space has volume product at least 4^n/n!, with equality precisely for invertible linear images of Hanner polytopes.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports that the sum of squared Gaussian first-moment lengths of the cells of every finite measurable Euclidean partition is at most 9/(8π), with the three-sector planar construction attaining the bound in the stated dimensions.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports vanishing of universal-cover L²-Betti numbers outside degree two for closed connected aspherical topological four-manifolds, and more generally finite aspherical integral Poincaré complexes of formal dimension four.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports the generalized Cartan–Hadamard isoperimetric inequality in every dimension for complete simply connected smooth manifolds with sectional curvature at most κ≤0, comparing finite-volume finite-perimeter sets with equal-volume model balls.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports that every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to complex Euclidean space.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06Coester, Koutsoupias and Zbysiński report that the work function algorithm is k-competitive on every metric space.
Original report: 2026-09-14The artifact reports exact optimal radii and classifications of optimal packings for one through seven freely rotated unit squares in a disk.
Read original source Original researchgithub.com
Original report: 2026-10-02The Ulam writeup reports a polynomial map C³→C³ with constant nonzero Jacobian determinant −2 and three distinct points sharing one image.
Original report: 2026-07-20The human-verified writeup gives a fixed ε>0 and growing planar point sets with at least n^(1+ε) unit distances.
Original report: 2026-05-20OpenAI claims that every compact boundaryless smooth Riemannian surface has a smooth length-preserving immersion into R⁴, including nonorientable surfaces.
Read original source Pinned primary manuscript · family 333github.com
Original report: 2026-10-06OpenAI 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 reports the sharp invariant-mass lower bound in terms of minimum enclosing area for one-ended spacetime initial-data exteriors in every dimension n≥3, under the paper’s energy, trapping and decay assumptions.
Read original source Pinned primary manuscript · family 260github.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 a conformal circle-domain model for an arbitrary planar domain, allowing uncountably many complementary components.
Read original source Pinned primary manuscript · family 071github.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: 2026-10-06The manuscript reports an L₁ embedding with a universal distortion bound for every finite connected planar graph with positive real edge lengths.
Read original source Original manuscript · family 089github.com
Original report: 2026-10-06The manuscript reports density at most C n log n for a single-lattice covering by translates of every convex body in dimension n ≥ 2.
Read original source Original manuscript · family 092github.com
Original report: 2026-10-06The manuscript reports matching bounds exp(Θ(√(log d · log log d))) for the least ℓ₁ distortion of unit-cost edit distance on strings of length at most d.
Read original source Original manuscript · family 099github.com
Original report: 2026-10-06The manuscript reports a nine-dimensional counterexample using rank-one orthogonal projectors on R⁴ with the Frobenius metric.
Read original source Original manuscript · family 156github.com
Original report: 2026-10-06The manuscript reports that five colors cannot suffice, implying that the plane’s chromatic number is six or seven if the result holds.
Read original source Original manuscript · family 158github.com
Original report: 2026-10-06The authors report configurations packing 11 and 12 unit regular hexagons inside regular hexagons of approximate side lengths 3.931 and 3.942, respectively.
Read original source Georgiev, Gómez-Serrano, Tao, and Wagner paperarxiv.org
Original report: 2025-11-03The September 2026 preprint reports three exact 604-point configurations in eleven dimensions.
Original report: 2026-08-24AlphaEvolve’s authors report an eleven-dimensional kissing configuration with 593 outer spheres.
Original report: 2025-05-14Shapes, distances, spaces, and their structure, including packings, metric embeddings, manifolds, and algebraic geometry.
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
Gauss: AI-assisted sphere-packing formalization in dimensions 8 and 24
Alpöge–Claude reports: a complex structure on the six-sphere
OpenAI reports: Mahler’s conjecture for general convex bodies
AI-assisted counterexample to Grothendieck’s group-scheme question
OpenAI reports: The all-dimensional Gaussian propeller bound
OpenAI reports: Smooth isometric immersion of surfaces into four-space
OpenAI reports: The enclosing-area spacetime Penrose inequality
OpenAI reports: The rational Hodge conjecture for CM abelian varieties
OpenAI reports: Borsuk’s covering question in dimension nine
AlphaEvolve: improved hexagon-packing configurations reported
OpenAI reports: Borsuk’s covering question in dimension nine
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