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OpenAI Aftermath is a guide to mathematical research published by OpenAI and to publicly available follow-up work by others: extensions, applications, formal verifications, corrections and critiques.

How entries are written

• Each result page summarizes the published claim and cites its public source: the paper, the repository commit or the preprint version.
• Follow-up work is listed with its relation to the original result (extension, use, verification, correction or critique), its public date and its public status.
• The publication of a paper, code or a formal proof is reported as such. It is not presented as independent verification.

Browse

• All result pages
• Number theory:
  • Milne's rationality conjecture
  • Irrationality of Catalan's constant
  • The irrationality exponent of π is 2
  • Modularity of elliptic curves over imaginary quadratic fields
  • The quasi-Riemann hypothesis
  • Prime-factor statistics of p − 1
  • Primitive roots for every admissible integer base
  • Bounded gaps between primes
  • Long gaps between primes
• Algebraic geometry:
  • The rational Hodge conjecture for CM abelian varieties
• Convex geometry:
  • The Mahler conjectures
• Algebraic complexity:
  • Matrix multiplication with exponent at most 9/4
• Computational complexity:
  • The Unique Games Conjecture
• Group theory:
  • Classifying spaces and geometric obstructions for Artin groups
  • Thompson's group F is nonamenable
  • Kaplansky's zero-divisor conjecture
• Algebra:
  • The Deligne–Drinfeld conjecture
• Representation theory of algebras:
  • The little finitistic dimension conjecture
• Mathematical logic:
  • Rigidity of the Turing degrees
• Algorithms:
  • A factor-two approximation for shortest common superstring
  • Integer multiplication below n log n
• Combinatorics:
  • Hadwiger's conjecture
  • Monochromatic finite sums and products in the positive integers
  • The hypercube Ramsey number has linear order
  • The Erdős–Gallai cycle-decomposition conjecture
  • Power-saving bounds for square-difference-free sets
  • Polynomial removal for ordered binary matrices
• Discrete geometry:
  • The periodic tiling conjecture in dimension three
  • The higher-dimensional Erdős distinct-distances conjecture
  • The Euclidean plane is not five-colorable
  • Universal optimality of the triangular lattice
  • Classification of finite Euclidean Ramsey configurations
• Real analysis:
  • The Erdős similarity conjecture
• Harmonic analysis:
  • The triangular Hilbert transform at the symmetric point
• Partial differential equations:
  • The hot spots conjecture for simply connected planar domains
  • The critical dimension for one-phase Bernoulli minimizers
• Differential geometry:
  • Smooth isometric immersions of surfaces into four-space
• Mathematical physics:
  • Localization and delocalization in the Anderson model
  • The spin-one Haldane gap
  • The ionization and generalized ionization conjectures
• Quantum information:
  • Classical capacity of generalized amplitude damping
• Probability:
  • Free energies of random perceptrons
  • Optimal mixing of the Thorp shuffle
  • An exact Hausdorff gauge for SLE
  • Conformal invariance of square-lattice random-cluster interfaces
• Dynamical systems:
  • Permanence of weakly reversible mass-action systems

Sources

• OpenAI mathematics collection: github.com/openai/math
• OpenAI's abridged reasoning summaries for ten results: reasoning_traces (list in the repository's README)