CAREER: Non-Asymptotic Random Matrix Theory and Connections

Random matrices are ubiquitous throughout mathematics, science, and engineering. For instance, they can model physical systems in mathematical and statistical physics, sample covariance matrices in statistics, benchmarks in numerical analysis, algorithmic tools in signal processing; they also play a role in the probabilistic method in geometric functional analysis. This project aims to develop novel methods in random matrix theory with a view towards resolving fundamental problems in a wide range of areas, including classical and combinatorial random matrix theory, numerical analysis, Markov chain Monte Carlo, statistical physics, discrepancy theory, and exact algorithms. The educational component of this project includes training, research, and mentoring opportunities for students at all levels. A key element will be interdisciplinary workshops targeted at graduate students and early-career mathematicians, with a focus on professional development of participants and fostering research collaborations. The research program is broadly composed of three components. The first component concerns the non-asymptotic study of the extreme singular values of random matrices, motivated by fundamental problems including the singularity problem for discrete random matrices and the matrix Spencer conjecture. The second component involves the application of non-asymptotic random matrix theory to the mixing time of several classical Markov chains in statistics and statistical mechanics. The third component focuses on the study of the anti-concentration phenomenon and its applications to combinatorics and computer science. Progress in each of these components is anticipated to lead to the development of new probabilistic and combinatorial techniques. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.