Problems

Among d+1 equiprobable equal-energy signals in Gaussian noise, is the regular simplex optimal for average error probability? Yes. The underlying comparison is that for any m × m correlation matrix R with R - 11^ T/m succeq 0 and X ~…

In the all-heads coin game a player starts with n coins, each showing heads with probability p; each round all remaining coins are flipped, the player must set aside at least one head (losing if none shows), and wins once all coins are set…

A dimension-independent subgaussian concentration bound for Gaussian vectors under coordinate-wise nonlinear maps, valid for any bounded function under a well-conditioned covariance, which answers a question of Simone Bombari on sign…

The group version of the Matrix Spencer conjecture holds: for every finite group G there are signs ε ∈ ± 1^G with |∑_g ∈ G ε_g ρ(g)| ≤ C√|G|, where ρ is the left regular representation and C is universal.

For an unkilled Levy process ξ drifting to +∞ with all positive exponential moments, let I_ξ = ∫_0^∞ e^-ξ_t dt and X_ξ = 1/I_ξ. Bertoin and Yor proved X_ξ is moment-determinate when ξ has no positive jumps and conjectured that this…

For a multidimensional reflected diffusion, does the basic adjoint relationship uniquely characterize the stationary distribution? The question had stood unresolved for more than thirty-five years since the BAR approach was introduced. For…

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…

Let X = (X_1,…,X_n) be a centered Gaussian vector, not necessarily nondegenerate. Then, for every α_1,…,α_n > 0, E[∏_i=1^n |X_i|^α_i] ≥ ∏_i=1^n E[|X_i|^α_i]. Moreover, if Var(X_i) > 0 for every i, then equality holds if and only if…

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?

Let X_1,…,X_n be independent nonnegative random variables with EX_i ≤ 1, and let S be their sum. Is P(S < ES + 1) ≥ 1/e? Feige proved the constant 1/13 and conjectured the sharp 1/e. Three independent July 2026 proofs settle it, both…

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…