Non-asymptotic theory of random matrices

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This project focuses on spectral properties of random matrices of a large size. A random matrix is used to model a typical behavior in many problems arising in high-dimensional convexity, functional analysis, as well as in physics and computer science. In all these areas the size of a matrix, i.e. the available number of degrees of freedom is very large, but fixed. This project aims to investigate the spectral phenomena which arise with high probability, and obtain explicit probability bounds. One of the central questions here is evaluation of the probability that a large random matrix is non-singular, and obtaining quantitative characteristics of non-singularity. To this end, the project brings into play a wide array of tools, ranging from classical probability, to asymptotic geometric analysis, and additive combinatorics.The theory of random matrices is currently undergoing a period of rapid development. In a large part this is due to the demand for explicit probabilistic estimates arising in various problems of computer science and engineering. This includes, among others, the wireless communication, where random matrices are used to model interactions between antennas and obstacles on the ground. The study of random matrices is also of importance for internet privacy protection, since certain kinds of random matrices became tools of choice in constructing hacking algorithms. The project will develop explicit quantitative spectral bounds, which are required for the aforementioned problems.