Geometric analysis of high-dimensional convex bodies and random processes

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The proposed research is on related topics in Functional Analysis, Convex Geometry, Asymptotic Theory of Normed Spaces, and Probability Theory. Developing theory and tools from these areas I intend to investigate quantitative relations among finite-dimensional convex bodies, various parameters and properties of convex bodies, behavior of random matrices and sequences of random variables, as well as to provide applications to various related areas in Pure and Applied Mathematics. The project's significance is to expand the knowledge and to strengthen the bonds between important branches of mathematics, namely, Convex Geometry, Local Theory of Banach Spaces, and Probability Theory, as well as to find new applications to related fields in Pure and Applied Mathematics. It is planed to solve several open problems. Investigations in these directions will enrich the areas with new ideas, methods, and techniques. The results will also have applications in related areas.    The project deals with longstanding open problems of Asymptotic Theory and its non-symmetric counterpart. In particular, I plan to work on such well-known problems as duality of entropy and covering numbers, Banach-Mazur distance between arbitrary convex bodies, Illumination problem, and others. Probabilistic technique plays a crucial role in Asymptotic Theory. The project also deals with the study of the smallest non-trivial eigenvalue of a random matrix and corresponding deviation inequalities, as well as with investigation of the behavior of order statistics of a sequence of random variables in terms of the sequence of their first moments. Order statistics have been intensively studied during last century but almost nothing is known about the general case considered in the proposal.