Problems

What is the minimax optimal error rate for density estimation when observations are perturbed by Wasserstein-bounded contaminations? Chao and Dobriban's 2023 preprint left a gap between upper and lower bounds; the sharp rate is now…

Does the finite signed basic adjoint relation determine the invariant signed measure uniquely, and how far beyond the Harrison-Reiman class can uniqueness extend?

For every zonotope Z ⊂ R^d and vectors v_1,…,v_n ∈ Z, there are signs with ∑_i x_i v_i ∈ C√d Z for a universal constant C. This resolves a 2002 conjecture on vector balancing in zonotopes.

The Courtade-Kumar conjecture (2014) posits that dictatorship functions maximize mutual information between a Boolean function's output and a noisy input. The paper resolves an open question posed by Courtade and Kumar themselves - a sharp…

Is the density of Chernoff's distribution - the law of argmax_t W(t) - t^2 for two-sided Brownian motion W - strongly log-concave, as conjectured by Balabdaoui and Wellner in 2014?

For every smooth positive density f on R^d, must the Fisher information t ↦ I(f * γ_t) be log-convex along the heat flow?

Given n independent standard Gaussian vectors in R^d, an ellipsoid fit is a positive semidefinite S with x_i' S x_i = d for every i. Saunderson, Parrilo and Willsky conjectured that this semidefinite feasibility problem has a sharp…

Does the Benjamini-Hochberg procedure always control the false-discovery rate at its nominal level for correlated two-sided Gaussian p-values? A factor model gives FDR > 0.0104 at nominal level α = 0.01.

The discrete-time Kac walk on S^n-1 started from a coordinate vector exhibits total variation cutoff at C_BRW n log n, where C_BRW ≈ 3.8916 is set by the speed of the leftmost particle in a branching random walk. The cutoff is therefore…