Activities This Week

A classical Theorem of Delorme-Guichardet states that a group G has property (T) if and only if every continuous affine isometric action of G on a Hilbert space has a fixed point.

There was a conjecture (attributed to Margulis) that for simple higher rank algebraic groups, this result has the following far reaching generalization: For a simple higher rank algebraic group with a finite center G, every continuous affine isometric action of G on a uniformly convex space has a fixed point.

This conjecture was recently settled by the joint works of V. Lafforgue, Liao for the non-Archimedean case, and myself, and de Laat and de la Salle in the real case.

In my lecture, I will discuss the history of the conjecture mentioned above and a further generalization of its solution beyond algebraic groups (namely, for higher rank universal lattices and Steinberg groups).

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.