Tagged openai-math
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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 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 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 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 Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures
Proves a dimension-free logarithmic Sobolev inequality for centered log-concave densities with uniformly subgaussian linear marginals, with constant bounded by a universal multiple of the squared linear subgaussian…
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Paper The Deligne–Drinfeld conjecture
Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture. () Manuscript September…
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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 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 Bounded-distortion L_{1} embeddings of planar and bounded-treewidth graphs
Resolves the planar and bounded-treewidth cases of the Gupta–Newman–Rabinovich–Sinclair conjecture. Shortest-path metrics of finite connected graphs with arbitrary positive edge lengths embed into real L1 with universal…
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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 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 Counterexamples to Baum–Connes and Kadison–Kaplansky
Disproves the coefficient-free reduced Baum–Connes conjecture through a failure of rational injectivity and a separate failure of surjectivity of assembly. A finitely generated torsion-free example witnesses the…
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Paper Rigidity and arithmetic of lattice von Neumann algebras
Classifies finite-index bimodules between scalar-twisted group factors of ICC groups commensurable with property-(T) lattices over characteristic-zero local fields and arbitrary ICC group factors. Every such…
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Paper Kadison's similarity conjecture
Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra to operators on a Hilbert space becomes a -homomorphism after conjugation by a…
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Paper Relative bicentralizers and modular spectral recovery
Proves Connes' bicentralizer conjecture for every type III1 factor with separable predual and every faithful normal state. More generally, for every inclusion N\subset M of von Neumann algebras with separable…
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Paper The generator problem for finite factors
Proves that every type II1 factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion…
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Paper Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness
Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra admitting normalized 2-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite…
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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 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 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 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 A counterexample to Kurosh’s division-ring problem
Constructs a countable characteristic-zero division ring that is algebraic over its center and generated by two elements over that center, but has infinite dimension over it. This answers Kurosh's division-ring problem…
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Paper The Hilbert–Smith conjecture in every dimension
Every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected finite-dimensional topological manifold is a Lie group. This proves the Hilbert–Smith conjecture in all…
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Paper The purely cosmetic surgery conjecture
Distinct Dehn surgery slopes on a nontrivial smooth knot in S3 never produce orientation-preservingly homeomorphic manifolds, proving the purely cosmetic surgery conjecture. The statement includes the meridional slope.…
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Paper Serre’s intersection-multiplicity conjecture
Proves strict positivity of Serre's intersection multiplicity \chi^R(M,N) for nonzero finitely generated modules over any regular local ring, provided M\otimesR N has finite length and \dim M+\dim N=\dim R.…
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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 Kobayashi’s canonical-ampleness conjecture
Every compact connected Kähler manifold of positive complex dimension with no nonconstant entire curve has ample canonical bundle and is therefore projective. This proves Kobayashi's canonical-ampleness conjecture in…
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Paper Zariski cancellation and affine fibrations over the complex numbers
Constructs an integral complex affine fourfold X\not\cong\mathbb A^4 with X\times\mathbb A^1\cong\mathbb A^5, disproving affine-space cancellation over ℂ in dimension four. It also disproves the…
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Paper The double-dimer loop ensemble converges to \mathrm{CLE}_{4}
Resolves the half-plane Temperleyan form of the double-dimer scaling-limit conjecture: the complete loop ensemble formed by two independent dimer coverings of the Temperleyan square lattice converges to nested CLE4.…
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Paper The Falconer distance conjecture
Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set E\subset\mathbb R^d with Hausdorff dimension greater than d/2 determines a set of Euclidean distances of positive Lebesgue…
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