Tagged openai-math
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Paper Cannon's conjecture
Every word-hyperbolic group with boundary homeomorphic to S2 admits a proper cocompact isometric action on hyperbolic three-space with finite kernel, proving Cannon's conjecture. Every torsion-free such group is…
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Paper Thompson's group F is nonamenable
Proves that Thompson's group F, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem. () Manuscript September 23, 2026 · · Lean…
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Paper A torsion-free hyperbolic group that is neither residually finite nor linear over any field
Constructs a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question negatively. One fixed nonidentity element is killed by every finite-dimensional linear…
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Paper Borsuk's conjecture fails in dimension nine
Constructs a compact subset of ℝ9 that cannot be covered by ten sets of strictly smaller diameter, disproving Borsuk's covering assertion already in dimension nine. The example consists of rank-one orthogonal projectors…
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Paper Isomorphism of the free group factors
Resolves the free group factor isomorphism problem: L(\mathbb F2)\cong L(\mathbb F3), and hence all interpolated free group factors, including L(\mathbb F\infty), are isomorphic. Their common factor has…
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Paper Kirchberg's \mathcal O_2 norm-ultrapower embedding problem
Constructs an explicit separable unital full group C∗-algebra that cannot embed unitally into the norm ultrapower of any fixed nonzero unital nuclear C∗-algebra, for any free ultrafilter on the natural numbers. Taking…
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Paper The Kadison–Ringrose cohomology conjecture
Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive. Equivalently, all higher bounded Hochschild…
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Paper Finite Smith–Toda complexes at every height
For every n ≥ 0, constructs a finite Smith–Toda spectrum at a prime p depending on n, with Brown–Peterson homology BP/(p,v1,\ldots,vn) and the canonical comodule structure. Thus Smith–Toda complexes exist at every…
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Paper Independence of the separable quotient problem
Establishes, relative to the consistency of a measurable cardinal, that the separable quotient problem is independent of ZFC. The assertion that every infinite-dimensional Banach space has a separable…
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Paper Markov type characterizes superreflexivity
Proves that every real Banach space with Markov type p for some p > 1 admits an equivalent uniformly convex norm, answering Naor's renorming question. Together with the known converse, this characterizes…
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Paper A counterexample to the nearby Lagrangian conjecture
Disproves the unrestricted nearby Lagrangian conjecture. For some sufficiently large even N, constructs a closed exact embedded Lagrangian in T^(S^9\times S^{N-1}) that is diffeomorphic to the base but not…
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Paper The metric Blaschke conjecture
Proves the metric Blaschke conjecture: every closed connected Riemannian manifold of positive dimension whose injectivity radius equals its diameter is, up to scaling, a standard compact rank-one symmetric space.…
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Paper Yau’s nodal bounds: surfaces and higher dimensions
Proves the sharp C\sqrt\lambda upper bound for nodal length on every fixed smooth closed surface, completing Yau's conjecture there. The upper bound fails for fixed smooth metrics in dimensions three and four,…
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Paper Kinetic limits and fluctuations over the Boltzmann lifespan
Derives the nonlinear Boltzmann equation from three-dimensional grand-canonical Newtonian gases throughout every regular kinetic interval with uniform Gaussian decay. Stable finite-range radial potentials may have…
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Paper Gaussian free field limits throughout the balanced six-vertex regime
The balanced square-lattice six-vertex height field with a=b=1 and 0\lt c\le2 converges to a Gaussian free field, including at the endpoint c = 2. The plane state is defined by balanced-torus limits. For unit…
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Paper Sharp Cartan–Hadamard isoperimetry and rigidity
Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most κ ≤ 0, every finite-volume finite-perimeter set satisfies the sharp comparison…
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Paper Positive metric entropy for the standard map
Proves that the standard sine map on the two-dimensional torus has positive metric entropy with respect to area for every sufficiently large positive parameter. This establishes Sinai's positive-parameter-measure…
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Paper Fujita’s freeness conjecture
Proves Fujita's freeness conjecture at its sharp bound in every dimension: for a smooth projective complex variety X of dimension n and an ample line bundle L, the adjoint KX+mL is globally generated for every…
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Paper Hyperkähler SYZ and projective-space bases
Proves the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample. It also proves that…
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Paper Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII
Proves that every complex K3 surface is Oka, including nonprojective surfaces. More generally, every connected minimal compact complex surface of Kodaira dimension zero is Oka; a connected minimal compact complex…
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Paper A characteristic-zero counterexample to Lipman–Zariski
Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture. Manuscript September 23, 2026 · · cite as…
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Paper Koebe’s circle-domain conjecture
Resolves the existence part of Koebe's circle-domain conjecture: every domain in the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points. It also proves that…
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Paper The L\log L Fourier-convergence conjecture
Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in L\log L(\mathbb T) converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture…
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Paper The optimal order of convex-body covering density
Determines the optimal worst-case covering density as \Theta(n\log n), for both lattice and unrestricted translative coverings. Every convex body in ℝn, n ≥ 2, admits a lattice covering of density at most $`Cn\log…
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Paper Exact derandomization of logarithmic space: \mathsf L=\mathsf{RL}=\mathsf{BPL}
Proves \mathsf L=\mathsf{RL}=\mathsf{BPL}, resolving derandomization for bounded-error logarithmic-space computation. An effective compiler converts each randomized polynomial-time logarithmic-space machine deciding…
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Paper Perfect completeness for 2-to-1 games
Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational \delta\in(0,1), it is NP-hard to distinguish satisfiable games from games whose optimum is at most δ, on explicit unweighted…
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Paper Integer multiplication below n\log n
Multiplies two n-bit integers exactly at every input length in deterministic worst-case time O(n(\log n)^{1-\kappa}), with \kappa=2^{-182}, on one fixed finite-alphabet Turing machine with finitely many…
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Paper One-sample matroid prophet inequalities against an almighty adversary
For every finite matroid known in advance, gives a distribution-independent online rule using one independent sample per element and earning a universal constant fraction of the expected offline optimum. Values are…
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Paper Beyond the square-root exponent for depth-three circuits
Constructs a single language in deterministic polynomial time whose n-bit membership function requires 2^{\omega(\sqrt n)} total gates in unbounded-fan-in OR–AND–OR circuits, at every sufficiently large input…
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Paper Banach’s simple Lebesgue-spectrum problem
Resolves the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics. A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on…
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