Department Colloquium: Tim Austin
Speaker: Professor Tim Austin, the University of Warwick, recipient of the Ostrowski Prize, 2021.
Title: New developments in entropy and ergodic theory
Abstract: Entropy was brought into ergodic theory by Kolmogorov and Sinai in the 1950s, not long after Shannon used it to lay the foundations of information theory. By now it has been shown to have a remarkable range of consequences for the structure and behaviour of measure-preserving dynamical systems, such as Ornstein's celebrated result that two stationary stochastic processes with independent coordinates are ergodic-theoretically isomorphic if and only if they have the same entropy.
This talk will survey some recent developments in this story, emphasizing the ergodic theory of actions of non-amenable groups such as free groups. In that setting, work by Lewis Bowen and others has revealed whole new vistas in the last twenty years, but large gaps in our understanding remain, and many new phenomena seem to be waiting for us. If time allows, I will build up to a description of a simple, single new example in this setting which shows that several of the main older results for single measure-preserving transformations are false once one considers actions of a `large enough' group. This part of the talk is based on a current joint project with Lewis Bowen and Christopher Shriver.
This talk will assume familiarity with measure theory and the basic language of probability theory (such as `events' and `random variables'). Prior experience with ergodic theory is not essential.