CAREER: Analytic and High-dimensional Methods in Probability

This project focuses on the mathematical area of probability theory, the study of random structures. Random structures are ubiquitous throughout the sciences for their use as models as well as their use in the design of algorithms. The main focus of this project is on random structures in high dimensions, meaning random structures with many degrees of freedom. Some examples are random matrices, random sphere packings and random polynomials. Each of these classes of models has direct application in various other scientific fields such as data science, statistical physics and theoretical computer science. The project includes workshops for early-career researchers and graduate students, with an aim of bringing together disparate mathematical subfields. The project consists of three components, with specific problems chosen with the aim of developing new techniques in high-dimensional probability and the use of analytic approaches in probability theory. The first component of the project concerns universality properties of random polynomials along with their use in optimization and algorithmic problems. The second component focuses on the structure of random sphere packings using connections to more combinatorial objects such as independent sets. The third component studies the non-asymptotic theory of random matrices with a focus on extremal behavior such as understanding the behavior of the least singular value in models without independent entries. 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.