Problems
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Curvature is expected to smooth flame-front wrinkles and so reduce turbulent flame speed, and in two-dimensional shear flows this was proved. In three dimensions it fails: there is a smooth periodic shear flow for which introducing…
Is the sequence W_0, W_1, …, W_n counting the flats of each rank of a matroid always unimodal? Rota conjectured yes in 1970.
The classical problem of maximizing the Shannon entropy of a sum of independent random variables supported on a finite alphabet, settled in the ternary case. For independent X_1, …, X_n taking values in 0,1,2, the entropy of S_n = X_1 + ……
Bondal and Polishchuk conjectured in 1993 that the braid group acts transitively on the set of full exceptional collections in a triangulated category. Chang, Haiden and Schroll disproved it for partially wrapped Fukaya categories, but no…
Mason conjectured the following: let M be a matroid of rank r, and let W_i denote the number of flats of M of rank i. Is it true that for all 1 ≤ i ≤ r - 1, we have W_i^2 ≥ W_i + 1W_i - 1? This is false; a counterexample is given by a…
An n-divisor set contains a multiple of every integer from 1 to n. Umans and Wang proposed, as the arithmetic-progression form of their Strong (α,β)-Divisor Conjecture, that such a progression exists with few terms of bounded magnitude,…
The directed five-dimensional torus D_5(m) has a Hamilton decomposition for every odd m ≥ 3, extending the decomposition program for directed tori beyond the three-dimensional case.
Nikolov and Ullman asked, as Open Problem 1 on DifferentialPrivacy.org, whether k statistical queries over a universe of size T can be released under pure differential privacy at the square-root error rate that the known lower bounds…
The Benjamini-Hochberg procedure is known not to control the false discovery rate at its nominal level under arbitrary dependence. A folklore conjecture in the FDR literature held that it must at least control the FDR up to a universal…
Monical, Tokcan and Yong conjectured that every fixed positive power of the Vandermonde determinant fails to have saturated Newton polytope in sufficiently many variables. For every even power k ≥ 4 there is an explicit lattice point of…
How large can a Bruhat interval in S_n that is a poset hypercube be? Using a permutation pattern suggested by AlphaEvolve, the authors exhibit hypercube intervals of dimension O(n log n) for n a power of 2, matching the largest possible…
Subbarao and Verma asked in 1999 (Problem 5.7, first part) whether the complementary Bell numbers f(n) = B_n(-1) take any given value only finitely many times. Campbell proves they do: for every fixed integer the fiber is finite, a result…
Let n_k be the least n > 2k such that (n-k)(n-k+1)…(n-1) has no prime factor in (k, 2k). Erdos conjectured a superpolynomial lower bound; for all large k, n_k > e^log^2 k / (20 loglog k).
Erdos and Szemeredi conjectured that every finite set of reals satisfies max(|A+A|,|AA|) ≥ |A|^2-o(1). False: there are arbitrarily large A ⊂ R, of algebraic integers in a number field of degree asymp log|A|, with max(|A+A|,|AA|) ≤ |A|^2-c…
A collection of open problems drawn from published lists, including Cahen, Fontana, Frisch and Glaz's Open Problems in Commutative Ring Theory and Erman and Sam's survey of Boij-Soderberg theory, each proved or disproved by one automated…
For every δ > 0 and infinitely many n there is a set of n lines in the plane with no intersecting quadruple such that every subset of size at least n^4/5+δ contains three concurrent lines. This improves the bound for a dual form of a…
A problem from Fajtlowicz's Graffiti program, studied by Erdős and Staton, on the Havel-Hakimi residue of common-divisor graphs. The paper resolves the problem and extends it, determining the residue's first-order scale and its nontrivial…
Two degree inequalities for circle-valued Sobolev maps have constants that degenerate as p → 1^+ or δ → 0^+. Brezis posed the problem of sharpening them; both are now sharpened, by the same power trick with elementary estimates.
For every n≥2 the paper exhibits an n-dimensional K-polystable toric Q-Fano variety whose alpha invariant is exactly 2/2n+1, answering a question of Liu and Zhuang on whether a K-semistable example exists with alpha invariant between 1/n+1…
For A ⊂ F_p of density 1/2, call A almost affine invariant under φ(x) = ax+b if |A triangle φ(A)| = o(p). Problem 90 asks for the threshold K below which A can be almost affine invariant simultaneously under all such φ with |a|, |b| ≤ K…