Non-Asymptotic Random Matrix Theory and Geometric Functional Analysis

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The research is intended to provide new connections between two areas of mathematics, probability and functional analysis. One of the main objects of investigation is a random matrix, a large rectangular array of random data. The PI strives to understand the properties of such arrays which hold with high probability and the dependence of those properties on the nature of random entries and the structure of the matrix. This study will have potential applications beyond the realm of pure mathematics, as random matrices are used in statistics, computer algorithms, and wireless communication. Another direction of the proposed research is the study of high dimensional convex sets with the emphasis on their complexity and approximation. This research will also have computer science applications including rate estimates for various high-dimensional algorithms.One of the main directions of this research is the non-asymptotic theory of random matrices, a new and rapidly developing area of research analyzing spectral characteristics of a random matrix of a large but fixed size and striving to obtain bounds valid with high probability. The PI intends to study singular values, eigenvalues, and eigenvectors of different ensembles of random matrices of a large size. The results obtained in this direction would have important applications within the random matrix theory in proving limit laws for the spectral characteristics of random matrices. Another group of problems comes from geometric functional analysis, an area of mathematics concerned with the study of high-dimensional convex bodies and normed spaces. A progress in this direction would lead to better understanding of the structure of sections and projections of such bodies, as well as possibility of approximation of a general convex body by a body with certain nice properties. Both directions have a significant component related to the theoretical computer science.