Tagged openai-math
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Paper Gigli’s characterization of Alexandrov curvature
Proves Gigli's conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support \mathop{\mathrm{RCD}}\nolimits ((n-1)\kappa,n) condition with reference measure…
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Paper A three-manifold without conjugate points or nonpositive curvature
Constructs a closed connected orientable smooth three-manifold that admits a metric without conjugate points but no metric of nonpositive sectional curvature. This answers negatively, already in dimension three, whether…
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Paper The hot spots conjecture for simply connected planar domains
Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior…
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Paper Cyclic length and chromatic fixed-point loss
Determines the optimal chromatic loss from geometric H-fixed points to geometric G-fixed points for every subgroup H of a finite p-group G. At every nonnegative height, the loss equals the shortest subnormal-chain…
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Paper Nonexpansive fixed points in reflexive Banach spaces
Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without…
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Paper A counterexample to metric-entropy duality
Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and…
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Paper The three-dimensional Ball–Evans approximation problem
Resolves the three-dimensional Ball–Evans approximation problem: every W^{1,p} homeomorphism between arbitrary bounded domains in ℝ3, for 1\le p\lt \infty, is a strong W^{1,p} limit of smooth diffeomorphisms…
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Paper Irrationality of Catalan’s constant
Proves that Catalan's constant G=\sum{j\ge0}(-1)^j/(2j+1)^2 is irrational. () Manuscript September 24, 2026 · · cite as `\cite{OAI:Catalans-constant-is-irrational-September-24-2026}` We prove that Catalan's constant…
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Paper Ordinary two-point correlations and the corrected Elliott conjecture
Proves the ordinary two-point Chowla conjecture, with a bound O(X/(\log X)^c) for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute. More generally, proves the binary…
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Paper Squarefree quartics and power-free polynomial values
Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally,…
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Paper Independent largest prime factors of consecutive integers
Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and n+1 are asymptotically independent in ordinary natural density. In particular, the integers…
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Paper The equivariant cohomological Hikita conjecture
Proves the equivariant cohomological Hikita correspondence for every finite quiver, including loops and multiple arrows, with arbitrary dimension and framing vectors and commuting flavor torus. When every semistable…
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Paper A counterexample to Griffiths’ positivity conjecture
Constructs ample rank-two bundles on \mathbb P^1\times\mathbb P^1 with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.…
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Paper A counterexample to Pixton completeness in Chow
Constructs a tautological relation on a moduli space of stable pointed curves that vanishes in rational Chow, hence in rational cohomology, but lies outside Pixton's original relation span. This disproves the Chow and…
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Paper Gepner symmetry and large-volume stability on threefolds
Proves Toda's Gepner conjecture for every smooth complex quintic threefold, constructing a numerical Bridgeland stability condition with the prescribed phase shift 2/5. Also constructs numerical Bridgeland stability…
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Paper The Global Spherical Shell conjecture
Every connected minimal compact complex surface of class VII with b2\gt 0 contains a global spherical shell, proving the positive-b2 Global Spherical Shell conjecture. Such a shell is a holomorphically embedded…
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Paper The generalized Mukai conjecture
Proves the generalized Mukai conjecture: every positive-dimensional smooth complex projective Fano manifold of dimension n, Picard number ρ and pseudoindex ι satisfies \rho(\iota-1)\le n, with equality exactly for…
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Paper Brennan's conjecture and the integral-means spectrum
Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, |\phi'|^s is area-integrable for 4/3\lt s\lt 4. The sharp universal integral-means identity is…
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Paper The Euclidean Steinitz–Bergström bound
Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within C\sqrt d, independently of length. Consequently every zero-sum family can be reordered with the same…
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Paper Kakeya in three and four dimensions
Resolves the Kakeya maximal conjecture in three dimensions and the Hausdorff-dimension conjecture in four. In three dimensions, the radius-δ tube maximal operator maps L^3(\mathbb R^3) to L^3(S^2) with norm…
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Paper The three-dimensional Bochner–Riesz conjecture
Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on L^p(\mathbb R^3) for every 1\le p\le\infty whenever $`\delta\gt…
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Paper Local smoothing in three dimensions
Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions. The estimate holds throughout the full strict range 2\lt p\lt \infty, with Sobolev regularity above…
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Paper Riesz transforms and rectifiability in higher codimension
Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and 2\le n\le d-2, an n-Ahlfors–David regular Radon measure on ℝd is uniformly n-rectifiable whenever its n-dimensional…
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Paper Sharp projection-body inequalities and a counterexample to simplex maximization
Proves Petty's projection-volume conjecture in the remaining dimensions n ≥ 4: ellipsoids uniquely minimize projection-body volume at fixed body volume. Also establishes the full Lutwak–Petty projection inequalities. In…
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Paper The Gaussian propeller conjecture in every dimension
Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most 9/(8\pi). In dimension at least two, three planar sectors of angle 2\pi/3, extended orthogonally, attain the…
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Paper Finite symmetric tensor categories and the Verlinde tower
Proves that every finite symmetric tensor category over an algebraically closed field k of characteristic p > 0 admits a k-linear exact faithful strong symmetric monoidal fiber functor to a higher Verlinde category…
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Paper Sampling and counting contingency tables with arbitrary margins
For nonnegative integer matrices with prescribed row and column sums, gives exact uniform sampling in expected polynomial bit time and almost-uniform sampling in worst-case polynomial bit time. The dimensions and…
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Paper Bin packing and unbounded configuration-LP gaps
Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant.…
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Paper Almost-linear-time exact matching and prescribed-degree factors in general graphs
Gives a randomized algorithm finding an exact maximum-cardinality matching in any simple undirected graph in (n+m)^{1+o(1)} word time, with success probability at least 2/3. The time bound holds on every computation…
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Paper Almost-linear approximation of edit distance
For every fixed rational \varepsilon\in(0,1), gives a randomized (1+\varepsilon) approximation to unit-cost edit distance in worst-case expected time N^{1+o(1)}, with success probability at least 2/3. The…
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