Tagged openai-math
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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 Smooth isometric immersions of surfaces into \mathbb{R}^{4}
Every closed smooth Riemannian surface admits a smooth isometric immersion into ℝ4, resolving the closed-surface form of the four-dimensional isometric-immersion problem. This includes nonorientable surfaces and metrics…
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Paper Global smoothness for relativistic Vlasov–Maxwell
Proves large-data global existence and uniqueness for the three-dimensional, one-species relativistic Vlasov–Maxwell system. Smooth admissible initial data may be arbitrary provided the particle density is compactly…
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Paper Yau's uniformization conjecture
Proves Yau's uniformization conjecture: every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to ℂn. Manuscript September 23, 2026 · · cite as…
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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 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 The circulant Hadamard and Barker-sequence conjectures
Proves that real circulant Hadamard matrices exist exactly in orders 1 and 4, resolving the circulant Hadamard conjecture. Together with classical Barker-sequence results, this shows that binary sequences whose…
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Paper Deterministic nonbipartite Ramanujan graphs in every fixed degree
For every fixed d ≥ 3, constructs a simple d-regular nonbipartite Ramanujan graph on every sufficiently large even number n of vertices, with every nonconstant adjacency eigenvalue strictly between -2\sqrt{d-1} and…
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Paper Deterministic construction of strong thin spanning trees
Resolves the strong thin-tree conjecture constructively. Every finite loopless k-edge-connected multigraph on at least two vertices has a spanning tree containing at most a universal C/k fraction of the edges of…
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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 Seymour’s second-neighborhood conjecture
Proves Seymour's second-neighborhood conjecture: every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one. Oriented graphs may be…
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Paper The Mahler conjectures, functional inequalities and polar-product symplectic width
Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The…
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