Kapustin and Witten introduced a powerful perspective on the geometric Langlands correspondence as an aspect of electric-magnetic duality in four dimensional gauge theory. While the familiar (de Rham) correspondence is best seen as a statement in conformal field theory, much of the structure can be seen in the simpler (Betti) setting of topological field theory using Lurie's proof of the Cobordism Hypothesis. In these lectures I will explain this perspective and illustrate its applications to representation theory following joint work with Nadler as well as Brochier, Gunningham, Jordan and Preygel.[-]

Let $p$ be a prime number and $F$ be a non-archimedean field with finite residue class field of characteristic $p$. Understanding the category of Iwahori-Hecke modules for $SL_2(F)$ is of great interest in the study of $p$-modular smooth representations of $SL_2(F)$, as these modules naturally show up as spaces of invariant vectors under the action of the standard pro-$p$-Iwahori subgroup. In this talk, we will discuss a work in progress in which we aim to classify all non-trivial extensions between these modules and to compare them with their analogues for $p$-modular smooth representations of $SL_2(F)$ and with their Galois counterpart in the setting of the local Langlands correspondences in natural characteristic.[-]

I will talk about my joint work with Aubert where we prove the Local Langlands Conjecture for $G_2$ (explicitly). This uses our earlier results on Hecke algebras attached to Bernstein components of (arbitrary) reductive $p$-adic groups, as well as an expected property on cuspidal support, along with a list of characterizing properties (including stability). In particular, we obtain 'mixed' L-packets containing F-singular supercuspidals and nonsupercuspidals. Our methods are inspired by the Langlands-Shahidi method, Deligne-Lusztig and Lusztig theories etc. If time permits, I will explain how to characterize our correspondence using stability of L-packets, by computing character formulae in terms of (generalized) Green functions ; one key input is a homogeneity result due to Waldspurger and DeBacker. Moreover, I will mention how to adapt our general strategy to construct LLC for other reductive groups, such as $G S p(4), S p(4)$, etc. The latter parts are based on recent joint work with Suzuki.[-]

Kapustin and Witten introduced a powerful perspective on the geometric Langlands correspondence as an aspect of electric-magnetic duality in four dimensional gauge theory. While the familiar (de Rham) correspondence is best seen as a statement in conformal field theory, much of the structure can be seen in the simpler (Betti) setting of topological field theory using Lurie's proof of the Cobordism Hypothesis. In these lectures I will explain this perspective and illustrate its applications to representation theory following joint work with Nadler as well as Brochier, Gunningham, Jordan and Preygel.[-]

We present a new approach to affine Deligne Lusztig varieties which allows us to study the so called "non-basic" case in a type free manner. The central idea is to translate the question of non-emptiness and the computation of the dimensions of these varieties into geometric questions in the Bruhat-Tits building. All boils down to understand existence of certain positively folded galleries in affine Coxeter complexes. To do so, we explicitly construct such galleries and use, among other techniques, the root operators introduced by Gaussent and Littelmann to manipulate them.[-]

This is a report on joint work with François Digne. Quasisemisimple elements are a generalisation of semisimple elements to disconnected reductive groups (or equivalently, to algebraic automorphisms of reductive groups). In the setting of reductive groups over an algebraically closed field, we discuss the classification of quasisemisimple classes, including isolated and quasi-isolated ones. The talk will start with the basic theory of non-connected reductive groups.[-]

Let $G$ be a semisimple algebraic group of adjoint type. The universal centralizer is the family of centralizers in $G$ of regular elements in Lie$(G)$, parametrized by their conjugacy classes. It has a natural symplectic structure, obtained by Hamiltonian reduction from the cotangent bundle $T^∗ G$. We consider a partial compactification of the universal centralizer, where each centralizer fiber is replaced by its closure inside the wonderful compactification of $G$. The symplectic structure extends to a log-symplectic Poisson structure on this partial compactification, whose fibers are isomorphic to regular Hessenberg varieties.[-]

Zaremba's conjecture belongs to the area of continued fractions. It predicts that for any given positive integer q there is a positive a, a < q, (a,q)=1 such that all partial quotients b_j in its continued fractions expansion a/q = 1/b_1+1/b_2 +... + 1/b_s are bounded by five. At the moment the question is widely open although the area has a rich history of works by Korobov, Hensley, Niederreiter, Bourgain and many others. We survey certain results concerning this hypothesis and show how growth in groups helps to solve different relaxations of Zaremba's conjecture. In particular, we show that a deeper hypothesis of Hensley concerning some Cantor-type set with the Hausdorff dimension >1/2 takes place for the so-called modular form of Zaremba's conjecture.[-]