Activities This Week

The Gowers uniformity k-norm on a finite abelian group measures the averages of complex functions on such groups over k-dimensional arithmetic cubes. The inverse question about these norms asks if a large norm implies correlation with a function of an algebraic origin. The analogue of the Gowers uniformity norms for measure-preserving abelian actions are the Host-Kra-Gowers seminorms, which are intimately connected to the Host-Kra-Ziegler factors of such systems. The corresponding inverse question, in the dynamical setting, asks for a description of such factors in terms of systems of an algebraic origin. In this talk, we survey recent results about the inverse question in the dynamical and combinatorial settings, and in particular how an answer in the former setting can imply one in the latter. This talk is based on joint works with Asgar Jamneshan and Terence Tao. This talk is aimed at a general audience. In particular, no prior knowledge in ergodic theory or additive combinatorics is required.

Let G be an algebraic group over a local field. We will show that the image of G under an arbitrary continuous homomorphism into any (Hausdorff) topological group is closed if and only if the center of G is compact. We will show how mixing properties for unitary representations follow from this topological property.

In 2003, a big bang occurred when the famous manuscript by Kechris, Pestov, and Todorcevic was uploaded to Arxiv, in which occurred a collusion between topological dynamics, Ramsey theory, and model theory, that left these fields not as they were before - the paper creates a strong link between properties of the automorphism group of a structure (as a topological group) and combinatorial properties of colourings of the same structure.

In this talk, I will survey the history leading to that bang, from Ramsey himself, through Ehrenfeucht and Mostowsky (and from the other side from Mitchell, through Herer, Christensen, Gromov, Milman, Frustenberg, Weiss, Glasner, up to Pestov) and finally Kechris, Pestov, and Todorcevic.

I will also survey applications in modern model theory, and, as time allows, a non-empty set of recent contributions to which I’m proud to be part of.