Topics in Asymptotic Geometric Analysis and Applications

The theory of Asymptotic Geometric Analysis is a comparably young and lively mathematical discipline that emerged from the local theory of Banach spaces. It bridges, in essence, three areas of mathematics: functional analysis, convex and discrete geometry, and probability theory, studying systems that involve a huge amount of parameters. The theory demonstrates new and unexpected phenomena characteristic of high dimensions. Considering that high- dimensional systems are very frequent in mathematics and applied sciences today, understanding of those phenomena is becoming increasingly important. The research carried out within this project reflects the broad spectrum of the theory in several ways. Some problems have an analytic flavor, while others are more geometric or probabilistic in nature. However, they all share a lively interplay of geometric, analytic and probabilistic ideas as is typical for the theory. We shall study central limit theorems and large deviations for the volume of random projections of lp-balls, large deviations for random projections of product measures, the geometry of tensor products of lp-spaces, entropy and Gelfand numbers for embeddings of Schatten classes, operator norms of structured random matrices, and look at new applications of asymptotic geometric analysis to information-based complexity, especially minimal dispersion of point sets.

No additional funding sources recorded.