Topics in Asymptotic Geometric Analysis and its Applications

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AbstractSzarekThis project involves continued research on the geometric, probabilistic and combinatorial aspects of functional analysis and convexity theory, the loosely defined area that has been lately referred to as "asymptotic geometric analysis." Particular attention will be paid to non-commutative objects and phenomena on one hand and, on the other hand, to links with other areas of mathematics and other mathematical sciences, which motivate most of the problems that are being considered. Sample research topics that are proposed to be studied include: structural properties of high-dimensional convex bodies and of high-dimensional normed spaces, metric entropy (of convex sets or of linear operators) and duality of such entropy and some geometric questions related to quantum information theory and quantum. The questions are typically expressed in the language of local or finite dimensional geometry of Banach spaces and are to be analyzed using mainly the diverse methods that originated or were developed in that context.On an elementary level, Analysis is a study of functions, or relationships between quantities and the parameters on which they depend. Since very many naturally appearing relationships are linear or at least convex, 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 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 (or other geometric structures) as the dimension goes to infinity. 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 of classical functional analysis and classical geometry. (This led, among others, to the discovery of many links to computer science.) Finally, to explain our emphasis on non-commutativity 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.