Problems

Does the natural trace estimate hold for kinetic energy spaces in the unrestricted Gaussian velocity model on bounded domains (Question 1.8 of Albritton, Armstrong, Mourrat and Novack)? No: for each 1 ≤ p < 2 there are counterexamples on…

If a finite graph has girth at least five, must its minimum dual degree satisfy δ^*(G) ≤ -∂_n(G), where ∂_n(G) is the smallest eigenvalue of its distance matrix? The Hoffman-Singleton graph violates it: dual degree 7 against eigenvalue…

Erdős and Graham asked whether binomnk with 1 ≤ k ≤ n/2 must always have a divisor ≤ n that is close to n, meaning bigger than a fixed constant times n. Settled in both directions: true when k is large enough as a function of n, but false…

Monical, Tokcan and Yong conjectured that Schubitopes, the generalized permutahedra arising as Newton polytopes of Schubert polynomials and of Demazure characters of GL_n, are Ehrhart positive. Disproved by an explicit Schubitope whose…

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…

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?

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…

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…

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…

Aldroubi, Cabrelli, Krishtal and Molter conjectured that for a bounded normal operator T and any vector g, the normalized orbit T^k g / |T^k g| : k ≥ 0 is never a frame. It can be: an explicit construction produces a normalized orbit that…

An ℓ-Oddtown is a family of subsets of an n-element set whose set sizes are not divisible by ℓ while all pairwise intersection sizes are. Berlekamp and Graver showed the maximum size is n for prime ℓ, Babai and Frankl extended this to…

Do n point charges whose electrostatic potential has only non-degenerate critical points always have at most (n-1)^2 of them? A configuration of five charges - three at the vertices of an equilateral triangle plus two small central charges…

The paper constructs an exact cluster F⊆Z^2 of cardinality 8 with full affine span and an F-tiling whose orbit closure contains no 1-periodic F-tiling, giving a non-degenerate counterexample to Nivat's conjecture for non-convex windows.…

For an odd prime p, do Sun's normalized trigonometric permanents satisfy s_p < 0 ⇔ p ≡ 5 pmod12 and s'_p < 0 ⇔ p ≡ 7 pmod 8? Exact computation at p = 29 refutes both sign laws.

For a projective variety X with at worst Gorenstein canonical singularities whose stringy E-function E_st(X; u, v) is a polynomial, all stringy Hodge numbers h^p,q_st(X) are non-negative. (Batyrev 1998, Conjecture 3.10.)

Does exhaustive AdaBoost always converge to a finite cycle of weak classifiers and weight vectors on every finite training set? A finite instance whose orbit never becomes periodic answers no.

For semifree noncommutative differential graded algebras over a nontrivial computable unital commutative ring, are stable tame isomorphism, quasi-isomorphism, or derived Morita equivalence algorithmically decidable? All three are…