Random matrices and geometric functional analysis

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This mathematics research project belongs to the interface of probability, convex geometry, and functional analysis. One of the main areas of investigation is the non-asymptotic theory of random matrices, a new and rapidly developing area of research, which analyzes spectral characteristics of a random matrix of large, but fixed size, and strives to obtain bounds, which are valid with high probability. The requirements for such bounds arise naturally in convex geometry and geometric functional analysis, where random matrices are used to study typical sections of a high-dimensional convex body. Another direction of this research is the geometry of non-symmetric convex bodies, where the probabilistic methods also play a crucial role.Several components of this mathematics research project are motivated by problems in computer science and engineering. These include signal reconstruction problems arising in computer tomography, as well as questions related to protecting privacy while releasing statistical information. Rudelson and his collaborators will establish new connections between probability and geometric functional analysis with the objective to study the spectral properties of random matrices. Random matrices have become the main model for signal reconstruction in wireless communication and can lead to a better understanding of the convergence of many computer science algorithms.