Problems

Is the exact nonreal spectral region of the four-cycle family of row-stochastic nonnegative matrices determined by the Karpelevich constraint, as Ran and Teng conjectured in 2024?

Douglas and Yang attach to each nonzero vector x of a quasinilpotent operator T a local resolvent-growth exponent k_x, giving the power set Λ(T) = k_x : x ≠ 0. Ji and Zhang asked whether 1 always belongs to Λ(T). It does, for every…

Cornulier asked, in a MathOverflow discussion, whether amenability of a module over an associative algebra depends on the ground field. It does not: the notion is invariant under change of base field.

Aluffi, Chen and Marcolli conjectured that the Poincare polynomial of the Deligne-Mumford moduli space overlineM_0,n of stable n-pointed rational curves has only real roots. True, with simple roots and strict interlacing between…

Can one construct a plain-model, efficient, information-theoretically secure one-time unclonable-encryption scheme for one classical bit with exponentially small adversarial advantage?

Maz'ya and Shaposhnikova introduced a non-classical maximal operator M^diamond, the maximal convolution with the vector-valued signum kernel truncated to centered balls. One of Maz'ya's 75 open problems in analysis asks whether it can be…

For the adjacent-transposition chain on S_n with a regular parameter vector, Fill's spectral gap conjecture (recently resolved) leaves open the characterization of the equality cases. The paper settles them, constructing the additional…

Whether the real Kalton-Peck space Z_2 is isomorphic to its hyperplanes. It is not: no hyperplane of Z_2 is isomorphic to Z_2, proved through a rank parity theorem for symplectic spaces applied to the Kalton-Swanson symplectic structure.

A question of Averkov, Hofscheier and Nill on whether the Ehrhart h^*-polynomial of a lattice polytope of large lattice width is real-rooted. Proved in fixed dimension for sufficiently large lattice width, giving strict log-concavity and…

A convex body is in Faber-Krahn position if it minimizes the first Dirichlet eigenvalue within its volume-preserving linear orbit. The paper proves this position is unique up to orthogonal transformations, answering a question of…

How many pairwise non-overlapping infinite circular cylinders of unit radius can simultaneously touch a unit ball? Kuperberg conjectured in 1990 that the maximum is six.

Zhu, Gyori, He, Lv, Salia and Xiao conjectured the maximum number of copies of a fixed cycle in an n-vertex graph of bounded circumference, attained by the join of a clique with an independent set. For every fixed s ≥ 3 and L ≥ 2s+2 and…

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,…

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…

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…

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…

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,…