Jorge Cely: The fundamental lemma for spherical Hecke algebras and motivic integration
The fundamental lemma was formulated by Langlands and it is about some identities of integrals related with the Arthur-Selberg trace formula. In the first part of the talk I will give an introduction (and motivation) of the fundamental lemma. Then I will explain the main ideas in the proof of the Langlands-Shelstad fundamental lemma for the spherical Hecke algebra for unramified p-adic reductive groups in large positive characteristic (joint work with W. Casselman and T. Hales). The proof is based on the transfer principle for constructible motivic functions. As an important step, we encode the entire spherical Hecke algebra into a single constructible function. If time permits, I will make some comments about possible connections with the local trace formula.
Tomás Ibarlucía: Rigidity results for strongly ergodic actions, from a model-theoretic point of view
I will discuss recent applications of continuous logic to ergodic theory. More precisely, I will give a model-theoretic interpretation of some rigidity phenomena associated with strongly ergodic actions of countable groups. The model-theoretic viewpoint allows us to go one step further and obtain a new result about coalescence of systems with generalized discrete spectrum, with an application to actions of property (T) groups. This is joint work with Todor Tsankov.
Martin Bays: Pseudofiniteness and modularity in fields
Large finite subsets of characteristic zero fields which form combinatorially extreme configurations with respect to solutions to polynomial equations often appear "linear". One example of this is Szemeredi-Trotter, where exceeding a certain bound on the number of intersections in an algebraic incidence system forces the configuration to be essentially degenerate. Another is the Elekes-Szabo theorem which finds an (abelian) algebraic group must explain any configuration which looks algebraically and combinatorially like a group. Elaborating on work of Hrushovski, we discuss a formalism in which these phenomena become instances of modularity of an associated geometry, which is a precise notion of "linearity" with strong consequences. This is part of a project with Emmanuel Breuillard.