For many computational problems involving randomness, intricate geometric features of the
solution space have been used to rigorously rule out powerful classes of algorithms. This is …

We study the critical window of the symmetric binary perceptron, or equivalently, random
combinatorial discrepancy. Consider the problem of finding a±1-valued vector σ satisfying …

The symmetric binary perceptron (SBP) is a random constraint satisfaction problem (CSP)
and a single-layer neural network; it exhibits intriguing features, most notably a sharp phase …

We introduce a general random model of a combinatorial optimization problem with
geometric structure that encapsulates both linear programming and integer linear …

We show that two related classes of algorithms, stable algorithms and Boolean circuits with
bounded depth, cannot produce an approximate sample from the uniform measure over the …

We survey the current state of affairs in the study of thresholds and sharp thresholds in
random structures on the occasion of the recent proof of the Kahn--Kalai Conjecture by Park …

We show that the capacity of the Ising perceptron is with high probability upper bounded by
the constant $\alpha_\star\approx 0.833$ conjectured by Krauth and M\'ezard, under the …

The symmetric binary perceptron (SBP) is a random constraint satisfaction problem (CSP)
and a single-layer neural network; it exhibits intriguing features, most notably a sharp phase …

We study two constrained problems in high dimensions. We study a high dimensional
inequality for the binary entropy. The perceptron is a natural model in high-dimensional …