Random structures in high dimensions: Matrices, polynomials and point processes

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The main focus of this project is to identify predictable behavior in large random structures using tools from probability theory, combinatorics, analysis, and theoretical computer science. Specific examples of models considered consist of those used to model noisy data sets, gases, and other types of spatial data. Mathematically, the three classes of models that are the primary focus of the project are random matrices, random polynomials, and Gibbs point processes. The project includes collaboration with graduate students on core problems concerning these three classes of models and mentoring of undergraduate research. A common theme to this research on random polynomials and random matrices is the understanding of universality phenomena. Results of this form are of mathematical interest as well as practical interest, as they show that certain behavior does not depend on the particulars of a given model but only on a small number of general properties of the model. The work on Gibbs point processes consists of understanding and expanding the uniqueness regime of these models, with results and proposed problems both theoretical as well as algorithmic.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.