Sharing AI progress on mathematics (amazing); and my lecture plans

OpenAI released 722 mathematical manuscripts grouped into 372 families, produced by an internal AI model, according to a blog post by mathematician Gil Kalai. Kalai called the release "an amazing milestone for mathematics" but stressed the results still need to be verified and digested by human mathematicians, noting that even Lean verification may have issues. Kalai, midway through a lecture tour visiting Brown University, MIT, Yale, IAS, Rutgers and Princeton University, said he plans to feed an AI tool his talk abstracts and outlines to draft lecture notes, including for a Friday lecture on Borsuk's conjecture at Jeff-Fest.

With Danny and Sharon Kleitman and Michel Goemans at the MIT Endicott house. Update: There are many reactions and emerging details on the new 700 or so papers released by Open AI. On Persiflage there is a very interesting post https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/ with analysis of eight results related to number theory and an earlier post https://scottaaronson.blog/?p=10169 with a list of expectations from AI. On SO ten results https://scottaaronson.blog/?p=10169 in computer science and computability are described. Sharing AI progress on mathematics OpenAI A few hours ago, Open AI shared solution to a few hundred mathematical problems https://openai.com/index/sharing-ai-progress-in-mathematics/ . Certainly this is an amazing milestone for mathematics, and the results will needs to be verified and digested by human mathematicians in the months to come. Several of the problems were discussed here on the blog over the past two decades and I will try to give a more detailed update regarding these problems and other “Math for AI” achievements. This is an amazing development At the end of the post I described early blog posts with some background on 26 of the new results. Both the Open AI list and some news from colleagues from the last days are relevant to my lecture around Borsuk’s conjecture on Friday in Jeff-Fest. My Lecture Tour I am currently in the middle of a nostalgic and hectic tour, visiting Brown University, MIT, Yale, IAS, Rutgers for Jeff Kahn’s conference celebration , and Princeton University. It is a pleasure to meet old friends and both young and old mathematicians that I did not meet before. I truly love both mathematics and the community of mathematicians, and I hope my lectures and posts reflect this feeling. Right now I am writing from Avi and Edna’s home at the IAS, having just returned from a lovely mathematics department event welcoming the new academic year. Tomorrow, I will be speaking here about intersection homology and combinatorics. Later in the evening, I am giving a short lecture in the “discussion series” about my work on quantum computation, which I hope will spark an engaging conversation. It will be quite a challenge to present quantum computation and my two decades of research on the subject in a 20-to-30-minute lecture tailored for a general audience. I also have a rather intense blogging plans: I am planning three posts based on my lectures: one on algebraic shifting at MIT, one on intersection homology and combinatorics at the IAS, and one on problems around Borsuk’s conjecture at “Jeff Fest.” Here is how I plan to go about it: I will feed an AI tool my abstract and a rough outline of the talk, and let it prepare detailed lecture notes that I can use for the presentation itself. For the MIT talk, I only thought of this approach after the fact Later on, I will edit those lecture notes into blog posts. It is quite interesting—and a little amusing—to note that the AI writes not only about the technical matters but also attempts to capture “my” feelings and hopes. I also plan a post with a candidate for “the most outrageous conjecture,” alongside an entertaining double-feature “test your intuition” post. OpenAI’s New Mathematical Results: Connections to Earlier Posts As I mentioned above, OpenAI released an impressive collection of mathematical results https://openai.com/index/sharing-ai-progress-in-mathematics/ produced by an internal AI model: 722 manuscripts grouped into 372 families https://github.com/openai/math . Many of these concern problems that have appeared on this blog over the years. Three disclaimers: first, the collection includes results with different stages of verification; their proofs will need to be checked and digested by the mathematical community. Second, even Lean verification may have issues, and third, there were a variety of other AI based results, also related to earlier blog posts that I did not collected; among those let me mention the Irrationality of , the Navier Stoke problem, The Komlos conjecture, Chvatal’s conjecture, and the KLS conjecture. Here are twenty connections to earlier posts, with related results occasionally grouped together. 1. The Mahler conjectures. I discussed the symmetric Mahler conjecture and its connections to symplectic geometry and my conjectures on faces and flags https://gilkalai.wordpress.com/2024/09/23/viterbos-conjecture-was-refuted-by-pazit-haim-kislev-and-yaron-ostrover/ . The release announces proofs of both the symmetric https://github.com/openai/math/blob/main/preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/paper.pdf and general https://github.com/openai/math/blob/main/preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026/paper.pdf Mahler conjectures in every dimension, including the equality cases of Hanner polytopes and simplices, respectively. 2. Borsuk’s problem in dimension nine. The question of the smallest dimension admitting a counterexample to Borsuk’s conjecture has appeared here repeatedly, including in this 2023 update https://gilkalai.wordpress.com/2023/12/04/progress-around-borsuks-problem/ . The new manuscript https://github.com/openai/math/blob/main/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf gives a compact set in nine-dimensional Euclidean space that cannot be covered by ten sets of smaller diameter. 3. Hadwiger’s conjecture. In Combinatorics News https://gilkalai.wordpress.com/2026/01/01/combinatorics-news/ I discussed Hadwiger’s conjecture, which relates graph coloring to complete minors, while reporting the disproof of its odd-minor strengthening. The new announcement https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf gives counterexamples to the original conjecture, even among graphs with independence number at most two. 4. The chromatic number of the plane. In 2018 I reported Aubrey de Grey’s breakthrough https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ showing that five colors are necessary to avoid monochromatic pairs at distance one. The new manuscript https://github.com/openai/math/blob/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf raises the lower bound to six, leaving six and seven as the two possible answers. 5. The Kakeya problem. My post on Hong Wang and Joshua Zahl’s breakthrough https://gilkalai.wordpress.com/2025/04/04/hong-wang-and-joshua-zahls-solution-for-kakeyas-problem-in-three-dimensions-reflections-and-links/ discussed the three-dimensional Kakeya set conjecture and the stronger maximal-function problem. The release announces the maximal conjecture in three dimensions https://github.com/openai/math/blob/main/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf and the Hausdorff-dimension conjecture in four dimensions https://github.com/openai/math/blob/main/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/paper.pdf . 6. Periodic tiling in dimension three. In An Aperiodic Monotile https://gilkalai.wordpress.com/2023/03/25/an-aperiodic-monotile/ I also mentioned Greenfeld and Tao’s disproof of the periodic translational tiling conjecture. The new construction https://github.com/openai/math/blob/main/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf gives a finite translational tile in that admits no fully periodic tiling, reaching the smallest possible dimension for the lattice problem. 7. Covering density of convex bodies. I described the breakthrough of Ordentlich, Regev, and Weiss on efficient lattice coverings in this post https://gilkalai.wordpress.com/2020/03/18/or-ordentlich-oded-regev-and-barak-weiss-new-bounds-for-covering-density/ . OpenAI announces a universal lattice-covering bound https://github.com/openai/math/blob/main/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf and matching examples even for unrestricted translative coverings https://github.com/openai/math/blob/main/preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026/paper.pdf , determining the worst-case order over convex bodies. 8. Distinct distances and unit distances. These problems featured in János Pach’s guest post on Guth and Katz https://gilkalai.wordpress.com/2010/11/20/janos-pach-guth-and-katzs-solution-of-erdos-distinct-distances-problem/ and in my earlier extremal-combinatorics post https://gilkalai.wordpress.com/2008/07/17/extermal-combinatorics-ii-some-geometry-and-number-theory/ . The release includes the sharp-order bound for distinct distances in every fixed dimension https://github.com/openai/math/blob/main/preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026/paper.pdf and an improved planar unit-distance bound https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf for some absolute . 9. Erdős’s arithmetic-progression conjecture. My post on Bloom and Sisask’s breakthrough https://gilkalai.wordpress.com/2020/07/08/to-cheer-you-up-in-difficult-times-7-bloom-and-sisask-just-broke-the-logarithm-barrier-for-roths-theorem/ discussed the three-term case of the assertion that divergent reciprocal sums force arithmetic progressions. The new result https://github.com/openai/math/blob/main/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf covers every finite length: any set of positive integers with divergent reciprocal sum contains arbitrarily long arithmetic progressions, with quasipolynomial quantitative bounds in Szemerédi’s theorem. 10. Sidorenko’s conjecture. I described Sidorenko’s conjecture https://gilkalai.wordpress.com/2019/08/19/equiangular-lines-with-a-fixed-angle-and-other-breakthroughs-from-yufei-zhaos-blog/ , which predicts that a fixed bipartite graph has at least the density it would have in a random graph of the same edge density. The announced counterexample https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf has 35 vertices and 66 edges and also disproves the related forcing conjecture. 11. Ryser’s covering conjecture. I discussed Ryser’s conjecture https://gilkalai.wordpress.com/2012/11/25/happy-birthday-ron-aharoni/ in my birthday post for Ron Aharoni, including his proof of the tripartite case. The new counterexamples https://github.com/openai/math/blob/main/preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026/paper.pdf are intersecting -partite, -uniform hypergraphs requiring vertices to meet every edge, whereas the conjecture predicts . 12. Crossing numbers of complete and complete bipartite graphs. These problems appeared in my discussion of Turán’s mathematics https://gilkalai.wordpress.com/2008/05/20/local-events-turans-problem-and-limits-of-graphs-and-hypergraphs/ . The release reports proofs of the Harary–Hill https://github.com/openai/math/blob/main/preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf and Zarankiewicz https://github.com/openai/math/blob/main/preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026/paper.pdf formulas, determining the minimum number of crossings in plane drawings of complete and complete bipartite graphs. 13. Barnette’s Hamiltonian-cycle conjecture. In Coloring Simple Polytopes and Triangulations https://gilkalai.wordpress.com/2014/12/06/coloring-simple-polytopes-and-triangulations/ I discussed Barnette’s conjecture that every cubic, bipartite, 3-connected planar graph has a Hamiltonian cycle. The new manuscript https://github.com/openai/math/blob/main/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf announces a proof of this conjecture. 14. Combinatorial invariance of Kazhdan–Lusztig polynomials. This was Problem 11 in my discussions of the G-programme, and I returned to it in a 2021 post about DeepMind’s mathematical work https://gilkalai.wordpress.com/2021/12/04/to-cheer-you-up-in-difficult-times-33-deep-learning-leads-to-progress-in-knot-theory-and-on-the-conjecture-that-kazhdan-lustig-polynomials-are-combinatorial/ . The announced theorem https://github.com/openai/math/blob/main/preprints/Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026/paper.pdf says that isomorphic Bruhat intervals have the same equal-parameter Kazhdan–Lusztig polynomial, even when the intervals come from different Coxeter systems. 15. Ramanujan graphs and higher-dimensional expanders. The subject of New Ramanujan Graphs https://gilkalai.wordpress.com/2013/05/01/new-ramanujan-graphs/ returns with a deterministic polynomial-time construction https://github.com/openai/math/blob/main/preprints/Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026/paper.pdf of nonbipartite Ramanujan graphs, for every fixed degree at least three and every sufficiently large even number of vertices. In higher dimensions, the release announces bounded-vertex-degree coboundary expanders https://github.com/openai/math/blob/main/preprints/Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026/paper.pdf over in every dimension at least three, continuing the theme of my introductory posts https://gilkalai.wordpress.com/2011/11/12/high-dimensional-expanders-introduction-i/ . 16. The second Kahn–Kalai conjecture and Talagrand’s conjectures. The release reports proofs of the second Kahn–Kalai conjecture https://github.com/openai/math/blob/main/preprints/The-second-Kahn-Kalai-conjecture-September-24-2026/paper.pdf , which bounds graph-containment thresholds using subgraph expectations, and of Talagrand’s discrete convexity conjecture https://github.com/openai/math/blob/main/preprints/Talagrands-discrete-convexity-conjecture-September-23-2026/paper.pdf , both discussed in my account of Jinyoung Park’s ICM lecture https://gilkalai.wordpress.com/2026/08/01/icm-2026-a-magical-saturday-afternoon-with-park-braden-and-proudfoot-and-meka/ . A companion result makes integral and fractional expectation thresholds equivalent up to a universal constant https://github.com/openai/math/blob/main/preprints/Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026/paper.pdf . 17. The Friedgut–Kalai sharp-threshold conjecture. Sharp thresholds and the role of symmetry were central themes of Boolean Functions: Influence, Threshold, and Noise https://gilkalai.wordpress.com/2017/01/24/boolean-functions-influence-threshold-and-noise/ . The release announces the threshold-width bound for monotone graph properties https://github.com/openai/math/blob/main/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf between fixed intermediate probabilities, where counts vertices, together with the conjectured extension to uniform hypergraphs https://github.com/openai/math/blob/main/preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026/hypergraph-influences.pdf . 18. Unique Games and 2-to-1 Games. My 2018 post on the 2-to-2 Games Theorem https://gilkalai.wordpress.com/2018/01/10/subhash-khot-dor-minzer-and-muli-safra-proved-the-2-to-2-games-conjecture/ discussed progress toward Khot’s conjectures and the issue of perfect completeness. The release announces a proof of the Unique Games Conjecture https://github.com/openai/math/blob/main/preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf and, separately, the 2-to-1 Games Conjecture with perfect completeness https://github.com/openai/math/blob/main/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/paper.pdf . 19. Matrix multiplication. My post Cap Sets, Sunflowers, and Matrix Multiplication https://gilkalai.wordpress.com/2011/12/09/cup-sets-sunflowers-and-matrix-multiplication/ explored the connections between these problems. The new bound is over the complex numbers https://github.com/openai/math/blob/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf , a substantial advance toward the conjecture . 20. Percolation: criticality and nonuniqueness. The release includes absence of infinite critical clusters https://github.com/openai/math/blob/main/preprints/No-percolation-at-criticality-on-quasi-transitive-graphs-September-24-2026/paper.pdf for bond percolation on infinite connected locally finite quasi-transitive graphs with , extending the setting of my recent post on an AI proof of critical nonpercolation https://gilkalai.wordpress.com/2026/09/03/amazing-there-is-no-percolation-at-the-critical-probability-in-all-dimensions-solved-by-ai-via-a-conjecture-of-gady-kozma-and-shahaf-nitzan/ . It also announces for nonamenable graphs in this class https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf , resolving the Benjamini–Schramm nonuniqueness conjecture discussed here in 2009 https://gilkalai.wordpress.com/2009/04/25/a-problem-on-planar-percolation/ . A few more related posts 1. The logarithmic Brunn–Minkowski conjecture. My post The Logarithmic Minkowski Problem https://gilkalai.wordpress.com/2021/11/22/the-logarithmic-minkowski-problem/ 2021 discussed the closely related existence problem for convex bodies with prescribed cone-volume data. The release announces a proof of the logarithmic Brunn–Minkowski conjecture https://github.com/openai/math/blob/main/preprints/The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026/paper.pdf for origin-symmetric convex bodies in every dimension. 2. The two-point Chowla conjecture. In EDP Reflections and Celebrations https://gilkalai.wordpress.com/2015/10/22/edp-reflections-and-celebrations/ 2015 , I discussed Tao’s logarithmically averaged Chowla and Elliott results and their role in solving the Erdős discrepancy problem. The new manuscript https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf announces the ordinary two-point Chowla conjecture, giving cancellation in the Liouville correlations \sum {n\le X}\lambda n \lambda n+h =o X for every fixed h\ge1 . 3. High-dimensional expanders. In High Dimensional Expanders: Introduction I https://gilkalai.wordpress.com/2011/11/12/high-dimensional-expanders-introduction-i/ 2011 , I discussed several notions of expansion for simplicial complexes, including coboundary expansion. The new construction https://github.com/openai/math/blob/main/preprints/Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026/paper.pdf gives arbitrarily large complexes in every dimension d\ge3 , with bounded vertex degrees and uniform \mathbb F 2 coboundary expansion in every degree below d . 4. Amenability of Thompson’s group F . My post The Thompson Group https://gilkalai.wordpress.com/2009/05/01/the-thompson-group/ 2009 introduced the group of dyadic piecewise-linear homeomorphisms of the interval and highlighted the question of whether it is amenable. The new manuscript https://github.com/openai/math/blob/main/preprints/Thompsons-group-F-is-nonamenable-September-23-2026/paper.pdf announces that Thompson’s group F is nonamenable. 5. Flat and ultraflat Littlewood polynomials. In Flat polynomials exist https://gilkalai.wordpress.com/2019/09/09/paul-balister-bela-bollobas-robert-morris-julian-sahasrabudhe-and-marius-tiba-flat-polynomials-exist/ 2019 , I described the breakthrough of Balister, Bollobás, Morris, Sahasrabudhe, and Tiba and emphasized the remaining ultraflat problem for coefficients \pm1 . The new manuscript https://github.com/openai/math/blob/main/preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/ultraflat-real-littlewood-polynomials.pdf announces polynomials with N such coefficients whose modulus is 1+o 1 \sqrt N uniformly on the entire unit circle, answering that stronger question. 6. First-passage percolation. In Analysis of Boolean Functions week 5 and 6 https://gilkalai.wordpress.com/2013/10/07/analysis-of-boolean-functions-week-5-and-6/ 2013 , I discussed passage-time fluctuations and the challenge of improving variance bounds in first-passage percolation. The release announces absence of doubly infinite geodesics https://github.com/openai/math/blob/main/preprints/No-bigeodesics-in-planar-first-passage-percolation-September-24-2026/main.pdf in the planar model for i.i.d. nonnegative nonatomic weights whose minimum over four independent samples has finite second moment, and strict convexity and a continuously differentiable limit-shape boundary https://github.com/openai/math/blob/main/preprints/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026/main.pdf for exponential weights. 7. Stochastic games and the backgammon question. added Oct. 22 In Is Backgammon in P? https://gilkalai.wordpress.com/2011/01/14/is-backgammon-in-p/ 2011 , I discussed the problem of finding polynomial-time algorithms for two-player zero-sum stochastic games with perfect information. The new manuscript https://github.com/openai/math/blob/main/preprints/Turn-Based-Stochastic-Mean-Payoff-Games-in-Deterministic-Quasipolynomial-Time-October-5-2026/stochastic-mean-payoff-games.pdf announces a deterministic quasipolynomial-time algorithm for the exact value-threshold problem in turn-based stochastic mean-payoff games, making progress toward the polynomial-time goal.