This project involves 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, and to links with other areas of mathematics and other mathematical sciences, which motivate most of the problems that are being studied. Sample research topics 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, derandomization of various probabilistic constructions appearing in functional analysis, some geometric questions related to quantum information theory and quantum computing, and problems motivated by links to mathematical programming. Typically, the questions are (or can be) expressed in the language of geometry of Banach spaces (of high, but finite dimension) and are to be analyzed using the diverse methods that originated or were developed in that context. The approach depends on 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. 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.
On the 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 objects in the corresponding mathematical model. Since real-life systems or processes (say, physical, biological or economic) usually exhibit very many degrees of freedom, the high-dimensional setting is of particular interest. This is exactly the domain of asymptotic geometric analysis, which studies quantitative properties of various geometric structures as the dimension goes to infinity. While investigation of high-dimensional phenomena often suffers from the curse of dimensionality (the complexity of the problem exploding with the increase in dimension and resulting in intractability), we may say that the asymptotic approach exploits the blessing of dimensionality, with symmetries of the problem becoming apparent only when the dimension is large. While this is a project in pure mathematics, many of the research topics arose in, or were motivated by, other fields such as mathematical physics, operations research, control theory, computer science or probability and statistics. Accordingly, any progress has a potential impact cutting across disciplines. On the one hand, this research may help to advance the applied areas by providing a new perspective and ? conceivably ? leading to breakthroughs. On the other hand, the problems and ideas arising in those areas will feed back into the fundamental research, contribute to maintaining the vitality of mathematics and potentially open completely new directions of inquiry. Additionally, the project will involve graduate and undergraduate students in intensive research, thus contributing to the development of scientific base and infrastructure.
