High Dimensional Probability and Combinatorics

This project concerns three related topics concerning high dimensional probability and combinatorics. The first part concerns fundamental questions concerning random matrices. These include the problem of finding sharp bounds on the singularity probabilities for discrete random matrices; and determining the limiting spectral distributions of ``sparse'' matrix models. The second major goal of this project is to make progress on the hypergraph Ramsey numbers and the multi-colour Ramsey numbers. These are two absolutely central objects in combinatorics which remain mysterious in many respects, despite many years of intense activity. Finally, I propose to study the Chowla cosine problem. This is an old problem in the area of harmonic and Fourier analysis which the PI has recently been studying through a probabilistic lens.