Asymptotic Geometric Analysis: Matrices, Operators and Noncommutative Phenomena

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This project involves continued research on the geometric,probabilistic and combinatorial aspects of functional analysis andconvexity theory, with particular attention to noncommutative objects andphenomena. While most of the problems considered in the project havemotivation in other fields of mathematical sciences, they are typicallyexpressed in the language of local or finite dimensional geometry of Banachspaces and/or analyzed using the methods prevalent in that area. Sampleresearch topics include: entropy of linear operators and duality of suchentropy, structural properties of high-dimensional convex bodies, variousanalytic and geometric questions related to random matrices, some problemsin the "local" operator theory, Gaussian correlation of convex sets,concentration phenomenon in the noncommutative context and some geometricquestions related to free and quantum information theories. On an elementary level, Analysis is a study of functions, orrelationships between quantities and the parameters on which they depend.As it happens, very many naturally appearing relationships are linear orat least convex. Thus, a good understanding of convex functions and,consequently, of convex sets is a prerequisite for understanding thoserelationships. The number of free parameters in the underlying problem canoften be related to the dimension of sets in the corresponding mathematicalmodel. Since real-life problems usually do depend on very many parameters,the high-dimensional setting is of particular interest. This is exactlythe domain of Asymptotic Geometric Analysis, which studies quantitativeproperties of convex sets as the dimension goes to infinity. Itconstitutes a fertile middle ground between the classical FunctionalAnalysis and the classical Geometry. Functional Analysis is usuallyconcerned with the infinite-dimensional setting (which frequently is anidealization of a very large dimension), but it often provides onlyqualitative information. On the other hand, Geometry typically yieldsvery precise information for a specific not-too-large dimension. For thelast two decades or so the asymptotic theory has been quite successful inidentifying and exploiting "approximate" symmetries of various problemsthat escaped the earlier "too qualitative" or "too rigid" methods. (Thisled, among others, to the discovery of many links to Computer Science.)Finally, to explain our interest in noncommutativity we point out that itsimply reflects the fact that the final outcome of a process may depend onthe order of operations involved; the best known, but by far not the onlymanifestation of that principle is quantum mechanics.