Problems
Everything in the archive: the problem as it was posed, what has been attempted against it, and who checked each attempt. The mark down the left of the list says who has looked — a person, a machine, or nobody yet. Human reviews and machine checks are counted separately and are never added together.
37 problems
The swap chain flips checkerboard 2×2 blocks to sample 0/1 matrices with fixed row and column sums. Kannan, Tetali and Vempala conjectured in 1997 that it mixes in polynomial time for all feasible margins; the lazy chain is shown to have…
A precise asymptotic formula for the number of n × 4t partial Hadamard matrices in the regimes t/n^3 → ∞ and t/n^3 → Θ, reaching the cubic regime that previous approaches (de Launey-Levin and successors) could not.
Mossel and Peres showed that a single-variable function admits a finite-automata Bernoulli factory exactly when it is rational, and their Theorem 2.9 claimed the same extension to multivariable functions. The multivariable claim is false,…
Along the heat flow, do the successive time derivatives of the entropy of X + √t Z alternate in sign, as conjectured by Cheng and Geng? An explicit measure on R has a fifth derivative with the forbidden sign.
A structured special case of the Matrix Spencer conjecture, reached through the representation theory of finite-dimensional C*-algebras: the conjectured discrepancy bound holds for every family of contractions contained in a suitable…
Hall and Ho conjectured how the zeros of the heat-flow-evolved characteristic polynomial of a random matrix behave in the large-n limit. General cases are proved; in particular, for a complex Ginibre matrix the empirical measure of those…
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…
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…
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 the switch-walk-switch lamplighter walk on Z_2 wr T_d, prove the sharp asymptotic p_2n(e,e) = ρ_d^2n exp[-(π^2 (log(d-1))^2 + o(1)) n/log^2 n] with ρ_d = frac2√d-1d.
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,…
For P_n(z) = ∑_k=0^n ε_k z^k with independent uniform signs, does the number R_n of roots in |z| ≤ 1 satisfy R_n/(n/2) → 1 almost surely? The manuscript proves the strong law with R_n = n/2 + O_ω(n^149/150).
Litvak conjectured in 2018 that for every p > 0 the quantity E[min_i ≤ n |g_i|^p], for g ~ N(0,Σ), is minimized over n × n correlation matrices by the Gram matrix of the regular simplex in R^n-1. False: the matrix Σ^cos_ij = cos(π(i-j)/n)…
For the switch-walk-switch walk on Z_2 wr Z started at (0,0) and (0,2), prove |P_t^x - P_t^y|_TV asymp t^-1/2.
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 + ……
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…
The Gaussian Moments Conjecture asks whether, for complex polynomials P,Q in n independent standard real Gaussian variables, E(P^m)=0 for all m≥ 1 forces E(QP^m)=0 for all large m. Explicit counterexamples with E(P^m)=0 and E(QP^m)=m!≠ 0…
Do the one-species current marginals of type-D ASEP have the predicted Tracy-Widom long-time asymptotics despite the model's two-species interactions?
Chafai, Dadoun and Youssef asked whether the quadratically penalised logarithmic energy of mean empirical spectral distributions is monotone in the dimension, for Wigner matrices and for matrices with i.i.d. entries. Neither holds: a…