Problems

For a bipartite graph without isolated vertices and with a unique minimum dominating set, does the proposed function m(n, γ) bound the number of edges whenever γ ≥ 2 and n ≥ 3γ? A 13-vertex bipartite graph with 22 edges exceeds the…

Assuming the Unique Games Conjecture, it is NP-hard to approximate MAX-3-CUT better than the Frieze-Jerrum semidefinite program does, and similarly for Quantum MAX-CUT: the sharpness question in the Khot-Kindler-Mossel-O'Donnell line,…

Let A(n) be the least positive integer not dividing binom2nn. Erdos asked for the behaviour of A(n) for reasonable n. Under an explicit dyadic-regularity formalization of reasonable, the distribution is determined on dyadic intervals…

Krauth and Mezard predicted in 1989 that the storage capacity of the Ising perceptron at zero margin is an explicit constant α_⋆ ≈ 0.8330786. Ding and Sun proved the matching lower bound and Huang the upper bound, but each was conditional…

Huybrechts conjectured that for every Brauer class alpha on a hyperkahler variety X, the index divides the period raised to the power dim(X)/2, strengthening the usual period-index conjecture. Disproved on certain hyperkahler fourfolds, in…

For p ≥ 2, does Carbery's proposed many-function almost-orthogonality inequality hold with the pairwise overlap coefficients raised to the power 2 - and if not, what is the largest possible exponent?

Talagrand's convexity problem asks whether a universal number of Minkowski sum operations turns any set of large Gaussian measure into one containing a convex body of comparable measure. It is equivalent to a question about subgaussian…

For a perfect field k and a representation-infinite finite-dimensional k-algebra A, the Auslander–Reiten quiver of A has infinitely many connected components. This establishes a conjecture of Auslander, Reiten and Smalø, for…

Mihail and Vazirani conjectured that the graph of every 0/1-polytope has edge expansion at least one. Disproved by a family of 0/1-polytopes whose edge expansion decreases exponentially in the dimension.

Friedland and coauthors proposed a quantum analogue of the p-Wasserstein distance and conjectured that, though only a semidistance in general, it is a true distance for a particular quantum cost matrix and for cost matrices near it. The…

Sabok asked whether the compact convex set S'(X) attached to a separable metric space of diameter at most one is always a simplex, and whether S'(U_1) is the Poulsen simplex. Both answers are negative, with obstructions already visible for…

Wu and Santhanam asked whether one can determine, from an increasing i.i.d. sample of binary random matrices, whether the unknown mean matrix is diagonalizable, while making only finitely many errors almost surely. Answered affirmatively…

For a continuous bounded-variation path with signature g, logarithmic signature l and increment v, the modified Lyons–Sidorova conjecture predicts the structure of g when R(l)=∞. The paper proves it: g=1 when v=0, and otherwise a prefix α…

Kusner conjectured in 1983 that the maximum number of points in R^n that are pairwise at ℓ_p-distance one is exactly n+1 for every 2 < p < ∞, as in the Euclidean case. False: an explicit configuration of n+2 equilateral points exists for…

A paper torus is an embedded polyhedral torus isometric to a flat torus. Schwartz proves no paper torus with 7 vertices exists and constructs one with 8, settling the minimum-vertex question in the flat-torus embedding tradition of…

Kac's walk on the rotation group, introduced by Hastings in 1970, is a central high-dimensional Markov chain in statistical physics and computational science. The paper proves it mixes in n^2 log n steps, the conjectured optimal rate,…

Given a binary matrix M, decide whether its columns can be permuted so that every row contains at most two blocks of 1s and, if it contains two blocks, they are separated by at most one 0. The claimed theorem proves that this (2,1)-Gapped…

Zhi-Wei Sun conjectured a closed evaluation of a truncated Legendre-symbol determinant. For every prime p ≡ 3 pmod 4 it equals ⌊ (p-2)/3 ⌋^2 x, proved by reducing to inverse data for Chapman's full Legendre-symbol matrix and evaluating…

For an irreducible crystallographic root system of rank r with Coxeter number h, the paper proves that Au's normalized Witten zeta function has a simple pole at 2/h and evaluates its residue in closed form in terms of the Cartan…

If a chromatic symmetric function is Schur positive, must every finite-variable specialization X_G(x_1, …, x_k) have a saturated Newton polytope? A 12-vertex bipartite graph realizes weights (6,6,0) and (8,2,2) but omits their midpoint…