Abstract: The Asymptotic Geometric Analysis studies the asymptotic behavior of finite- (but very high-)dimensional normed spaces and convex bodies when dimension tends to infinity. Contrary to common intuition, which anticipates enormous diversity and chaotic behavior, we observe a uniform behavior for the whole family of finite- (but high-)dimensional spaces. In the Introduction to our talk we will demonstrate a couple of different and unexpected phenomena accompanying high dimension. In the second, the main part of the talk we will explain how the geometric theory of convexity is extended to a larger category of log-concave measures which bring inside this class of (probability) measures geometric vision and approach. In particular, this point of view introduces functional versions for many geometric inequalities, and also leads to solutions of some central problems of the theory. It also leads to the discovery of the abstract notion of Duality (Polarity) with many unexpected results outside the particular field we discuss.

Notes: The talk will be understandable to any graduate student in Mathematics.