Speaker: Joe Rosenblatt (IUPUI)
Abstract: A coboundary is a difference $H – H \circ \tau$. The function $H$ is the transfer function. It is well-known that for ergodic transformations, a mean-zero function can be approximated by coboundaries. How well one can do this is constrained mostly by bounds on the norm of the transfer function. Similarly, even ergodic maps have almost invariant functions, with the primary constraint being the degree of almost invariance and how long the almost invariance should last. These two issues are related to the ongoing effort to construct (many) pairs of maps without common coboundaries
Organised in collaboration with the OSU Logic seminar
Speaker: Anush Tserunyan (UIUC)
Title: Probability groups as an alternative to Furstenberg’s correspondence
Abstract: Multiple recurrence results for amenable groups are most commonly proven via the Furstenberg correspondence principle, which allows for switching from the hard-to-work-with translation action of \Gamma on (\Gamma, d), where d is an upper density function, to a more friendly measure-preserving action of \Gamma on a (genuine) probability space (X, \mu). I suggest an alternative correspondence principle, where we switch the group itself to a more infinitary group G, which, nevertheless, is equipped with a translation-invariant (genuine) probability measure \mu, and the corresponding action is simply the translation action of G on itself. The object (G, \mu) is an example of, what I call, a probability group, the class of which includes all compact groups and is closed under ultraproducts. One of the advantages of studying measure-preserving actions of probability groups over those of amenable groups is that we can integrate over the group, which makes many statements, such as the mean ergodic theorem, boil down to Fubini’s theorem. As an example, we will give a short proof of a triple recurrence result proved by Bergelson–Tao for quasirandom groups.
Speaker: Yair Hartman (Weizmann Institute, Israel)
Title: The Furstenberg entropy realization problem
Abstract: Stationary actions are a generalization of measure preserving actions, in the context of a random walk on a group. The Furstenberg entropy of a stationary action is an important invariant which measures its “distance from invariance”. The realization problem is to determine the possible entropy values realizable for a given random walk. In this talk we will see a new characterization of Kazhdan’s property (T) in terms of this problem (for non-singular actions) and will use Invariant Random Subgroups (IRSs) in order to describe a full solution of the problem for lamplighter groups.
Based of several joint works with Lewis Bowen, Omer Tamuz and Ariel Yadin