Dynamical Systems Seminar

In 1977, Furstenberg gave a dynamical proof of the theorem of Szemerédi on the existence of arithmetic progressions in dense subsets of integers. In doing so, he initiated the use of ergodic methods to solve problems originating from additive combinatorics and number theory. The main tool in this field are multiple ergodic averages, a class of multilinear operators that generalize classical Birkhoff averages and can be used to count the number of arithmetic patterns in dense sets of integers. This talk will survey recent developments on multiple ergodic averages and ensuing applications to additive combinatorics.