Some recent convergence results for nonconventional ergodic averages

Timothy Austin, UCLA
Fine Hall 314

The phenomenon of multiple recurrence in ergodic theory occurs when several of the images of one fixed positive-measure set under some probability-preserving transformations, indexed by the points of some finite configuration in the acting group, all overlap in a set of positive measure. Instances of this were first investigated very generally by Furstenberg, who related them to Szemeredi's Theorem in additive combinatorics and so gave a new proof of that theorem. His analysis focuses on certain `nonconventional' ergodic averages, and these have gone on to attract considerable further interest. In this talk we will discuss some recent progress in their analysis, showing how an extension of an initially-given system of commuting probability-preserving transformations can be used in a proof of the norm convergence of some such averages.