Instructor: Mark Rudelson

Description: This course is an introduction to modern geometry of
high-dimensional convex bodies. We are going to study the distances
between convex bodies, sections and projections, approximation of
systems of functions etc. In these problems the precise structure of
a convex body is usually unknown, so the probabilistic methods will
play a crucial role. In particular, to find a section of a convex
body with certain nice properties, one can consider a random section
and show that the property is satisfied with positive probability.
This approach allows to prove the celebrated Dvoretzky theorem: any
high-dimensional convex body possesses a section of a large
dimension, which is close to an ellipsoid. The existence of an
ellipsoidal section is a surprising result even for simplest convex
bodies, such as cubes or cross-polytopes.

We shall consider the properties of random convex bodies, the
connection between the volume of a body and the structure of its
sections and different applications of convex geometry techniques to
Analysis.