Workshop

The talk will give a nontechnical summary of some aspects of my lecture course, concentrating on the geometric Langlands-type structures that can now be defined in a mixed-characteristic setting.

: In this talk, I hope to describe the state-of-the-art of the program of applying the theory of motivic integration (which is based on model theory, and will be defined in the talk) to harmonic analysis on p-adic groups. This program was started by T.C. Hales around 1999, and in particular, led to one of the proofs of transfer of the Fundamental Lemma to local fields of characteristic zero, by Cluckers, Hales and Loeser. I will describe their proof, and also talk about the transfer principles of a more analytic nature that were recently proved by Cluckers, Halupczok and myself. These new transfer principles allow one to transfer statements about integrability and boundedness of functions between characteristic zero and (large) positive  characteristic, and in particular, give a proof of local integrability of Harish-Chandra characters in the large positive characteristic case.

Finally, the same model theory methods give some uniform bounds for orbital integrals, normalized by the square root of the discriminant,  on the groups G(K_v),  where G is a connected reductive group defined over a global field K, and K_v varies over almost all completions of K.

This talk is based on the work of several people, and especially on the joint project with R. Cluckers and I. Halupczok.

Rational Cherednik algebras are (a degenerate version of) generalizations of Hecke algebras. We will construct finite-dimensional representations of these algebras from the cohomology of affine Springer fibers and Hitchin fibers with extra symmetry. We will also give a dimension formula for these representations. This is joint work with A.Oblomkov.

Certain part of Langlands's beyond endoscopy proposal can be formulated in terms of geometry of certain fibration, similar to Hitchin's. In my talk, I will discuss about its construction and new geometric problem it offers. 

We will survey them. The relation to analysis lies in the fact that stratifications are related to differential equations, and to categories in the fact that the fundamental groups in question are those Tannakian fundamental groups of those categories. Else one should honestly say that the deep questions which will be mentioned are rather about the connection between Étale topology  and various crystalline sites.

Locally analytic representations of p-adic Lie groups are of interest in several branches of arithmetic algebraic geometry, notably the p-adic local Langlands program. I will discuss some work in progress towards a Beilinson-Bernstein style localisation theorem for admissible locally analytic representations of semisimple compact p-adic Lie groups using equivariant formal models of rigid analytic flag varieties. 

As the title indicates, this talk will be about classical, function-theoretic harmonic analysis. However, I will discuss some recently discovered structures in harmonic analysis which may have sheaf-theoretic analogs. More precisely, I will discuss the theory of asymptotics on spherical varieties as it appears in the work of Bezrukavnikov-Kazhdan (for the group) and my joint work with Venkatesh (in general), as well as the scattering theory which allows to formulate a Paley-Wiener theorem (joint work with Delorme and Harinck). There will be some overlap with my talk at the NGM Research Seminar at MSRI in October, but the emphasis will be different.

I will explain a new approach to the commutativity constraints of the Satake category. This approach is based on an equality between certain Kazhdan-Lusztig polynomial and Lusztig-Vogan polynomial (conjectured by Lusztig and proved by Lusztig-Yun). It eventually allows us to prove the geometric Satake isomorphism for p-adic groups, without using fusion products.