Mathematical Sciences: Geometric & Probabilistic Aspects of Convexity and Functional Analysis

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NSF PROPOSAL # 9623984 Dr. Stanislaw J. Szarek Case Western Reserve University ABSTRACT The project involves continued research on the geometric, probabilistic and combinatorial aspects of functional analysis and convexity theory. Many of the problems considered are expressed in the language of the asymptotic theory of finite dimensional normed spaces or, what is essentially equivalent, as "isomorphic" statements about high dimensional convex bodies. Some questions have motivation in the theory of operator algebras, mathematical physics, operations research, computer science or probability and statistics. Typical research topics include: entropy of linear operators, existence of nontrivial linear operators on general Banach spaces, norms of random matrices, some problems in the "local" operator theory, Gaussian correlation of convex sets, balancing vectors in normed spaces and other questions on the border of probability, combinatorics and convexity. On an elementary level, analysis is a study of "functions", or relationships, between quantities and the parameters on which they depend. As it happens, very many naturally appearing relationships are linear or at least convex. Thus a good understanding of convex functions (and consequently of convex sets) is a prerequisite for understanding those relationships. Depending on whether the set of parameters is finite or infinite (the latter may also be an "idealization" of "very large"; real life problems usually do depend on very many parameters), we are dealing with either finite dimensional convexity or infinite dimensional functional analysis. The "isomorphic" asymptotic theory was, over the last twenty years or so, quite successful in identifying and exploiting "approximate" symmetries of various problems (involving high dimensional convex sets) that could not be analyzed with earlier more "rigid" methods. One example of a specific link to applications is a recent advance (in computer science) in the area of efficient computation of volume of high dimensional convex sets, which utilized results and ideas from the asymptotic theory.