Title: Structured extensions and characteristic factors in ergodic theory
Abstract:
In 2005--2007, Host--Kra and Ziegler showed that for an ergodic system $(X,T)$, the multiple ergodic averages
of the form
$$\mathbb{E}_{n \in [N]} f_1(T^n x)\, f_2(T^{2n} x)\cdots f_k(T^{kn} x)$$
are controlled by a single factor: the maximal $(k-1)$-step pronil factor of $(X,T)$. In particular, the
limits of these averages depend only on the conditional expectations of the functions $f_i$ onto this factor,
so that this pronil system is the characteristic factor for these averages.
In the early 2010s, Austin studied analogous characteristic factors for systems $(X,T_1,\dots,T_k)$ of commuting
measure-preserving transformations. He discovered that in order to obtain suitably structured characteristic
factors in this more general setting, one must first pass to an extension of the original system.
In this talk, I will survey a lot of the motivation and ideas behind these works, and then discuss more recent developments
constructing new structured extensions that admit good characteristic factors for a wide class of multiple
ergodic averages arising from commuting transformations. If time permits, I will also explain how these results imply that every multi-correlation sequence can be decomposed as the sum of a generalized nilsequence and a null-sequence.