The last 50 years have seen tremendous activity at the
interface between ergodic theory, combinatorics and number theory that
started with Furstenbergâ€™s dynamical proof of the Szemerédi theorem
from the 1970s. The goal of this line of research has been to prove
new multiple recurrence results and then deduce combinatorial
corollaries. To achieve this, one wants to understand the limiting
behaviour of relevant multiple ergodic averages. Of particular
interest are averages of commuting transformations with polynomial
iterates: they play a central role in the polynomial Szemerédi theorem
of Bergelson and Leibman. While their norm convergence has been
established in a celebrated paper of Walsh, little more has been known
for a long time about the form of the limit. In this talk, I will
present some recent results on the limits of such averages obtained
jointly with Nikos Frantzikinakis and explain how they can be used to
answer a number of previously intractable problems at the intersection
between ergodic theory and combinatorics.

Meeting ID: 852 4277 3200
Passcode: 103121