Colloquium

Tuesday, Feb. 26, 2008

4:00 PM, BAC 219

Speaker: Vitali Milman, Tel Aviv University, Israel and Ohio State University, Columbus, Ohio

Host: Barry Turett

Title: Asymptotic Geometric Analysis: The Concept of Polarity and Geometrization of Probability

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.

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