# Geometric Analysis and Topology Seminar

#### Some Open Problems on Measure Concentration in Abstract Ergodic Theory

**Speaker:**
Tim Austin, NYU

**Location:**
Warren Weaver Hall 517

**Date:**
Friday, October 18, 2013, 11 a.m.

**Synopsis:**

One of the great achievements of structural ergodic theory was Ornstein's proof that any two Bernoulli shifts of equal entropy are isomorphic. In fact Ornstein's argument does much more: it gives a very concrete necessary and sufficient condition for an abitrary probability-preserving transformation to be isomorphic to a Bernoulli shift. Among various equivalent formulations of Ornstein's condition, one of the most natural is in terms of measure concentration for the finite marginals of the process. This version is also easily generalized to other acting groups. In this talk I will give a brief overview of basic entropy theory and Ornstein Theory for probability-preserving transformations in terms of the metric geometry of the finite marginals of processes, viewed as metric measure spaces. Then I will explain some ergodic-theoretic open problems that seem to be related to these metric measure spaces, and sketch their relation to corresponding problems in discrete metric geometry.