Asymptotic Geometric Analysis: Matrices, Operators and Noncommutative Phenomena

Case Western Reserve University, Cleveland OH

Investigators

Abstract

This project involves continued research on the geometric, probabilistic and combinatorial aspects of functional analysis and convexity theory, with particular attention to noncommutative objects and phenomena. While most of the problems considered in the project have motivation in other fields of mathematical sciences, they are typically expressed in the language of local or finite dimensional geometry of Banach spaces and/or analyzed using the methods prevalent in that area. Sample research topics include: entropy of linear operators and duality of such entropy, structural properties of high-dimensional convex bodies, various analytic and geometric questions related to random matrices, some problems in the "local" operator theory, Gaussian correlation of convex sets, concentration phenomenon in the noncommutative context and some geometric questions related to free and quantum information theories. 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. The number of free parameters in the underlying problem can often be related to the dimension of sets in the corresponding mathematical model. Since real-life problems usually do depend on very many parameters, the high-dimensional setting is of particular interest. This is exactly the domain of Asymptotic Geometric Analysis, which studies quantitative properties of convex sets as the dimension goes to infinity. It constitutes a fertile middle ground between the classical Functional Analysis and the classical Geometry. Functional Analysis is usually concerned with the infinite-dimensional setting (which frequently is an idealization of a very large dimension), but it often provides only qualitative information. On the other hand, Geometry typically yields very precise information for a specific not-too-large dimension. For the last two decades or so the asymptotic theory has been quite successful in identifying and exploiting "approximate" symmetries of various problems that escaped the earlier "too qualitative" or "too rigid" methods. (This led, among others, to the discovery of many links to Computer Science.) Finally, to explain our interest in noncommutativity we point out that it simply reflects the fact that the final outcome of a process may depend on the order of operations involved; the best known, but by far not the only manifestation of that principle is quantum mechanics.