Tag - Langlands programme

Tom Gannon: Quantization of the universal centralizer and central D-modules

14th October, 2024

We will discuss joint work with Victor Ginzburg that proves a conjecture of Nadler on the existence of a quantization, or non-commutative deformation, of the Knop-Ngô morphism, a morphism of group schemes used in particular by Ngô in his proof of the fundamental lemma in the Langlands programme. We will first explain the representation-theoretic background, give an extended example of this morphism for the group GLn(ℂ), and then present a precise statement of our theorem.

Time permitting, we will also discuss how the tools used to construct this quantization can also be used to prove conjectures of Ben-Zvi and Gunningham, which predict a relationship between the quantization of the Knop-Ngô morphism and the parabolic induction functor, as well as an "exactness" conjecture of Braverman and Kazhdan in the D-module setting.

Charlotte Chan: Generic character sheaves on parahoric subgroups

27th May, 2024

Lusztig’s theory of character sheaves for connected reductive groups is one of the most important developments in representation theory in the last few decades. I will give an overview of this theory and explain the need, from the perspective of the representation theory of p-adic groups, of a theory of character sheaves on jet schemes. Recently, R. Bezrukavnikov and I have developed the 'generic' part of this desired theory. In the simplest non-trivial case, this resolves a conjecture of Lusztig and produces perverse sheaves on jet schemes compatible with parahoric Deligne-Lusztig induction. This talk is intended to describe in broad strokes what we know about these generic character sheaves, especially within the context of the Langlands programme.

Ana Caraiani: A glimpse into the Langlands programme

20th May, 2024

The goal of this lecture series is to give you a glimpse into the Langlands programme, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. In the first lecture, we will look at a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. I will try to give you a sense of the different meanings of modularity and of the multitude of ingredients that go into establishing such a result. In the following lectures, I will focus on the more geometric ingredients, first in the special case of the modular curve and then for higher-dimensional Shimura varieties.

Sam Raskin: The Geometric Langlands Conjecture

6th May, 2024

I will describe the main ideas that go into the proof of the (unramified, global) geometric Langlands conjecture. All of this work is joint with Gaitsgory and some parts are joint with Arinkin, Beraldo, Chen, Faergeman, Lin, and Rozenblyum. I will also describe recent work on understanding the structure of Hecke eigensheaves (where the attributions are varied and too complicated for an abstract).

Jessica Fintzen: An introduction to representations of p-adic groups

25th March, 2024

An explicit understanding of the category of all (smooth, complex) representations of p-adic groups provides an important tool not just within representation theory, but also for the construction of an explicit and a categorical local Langlands correspondence, and has applications to the study of automorphic forms, for example. In my talk I will introduce p-adic groups and explain that the category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks. I will then provide an overview of what we know about the structure of these Bernstein blocks. In particular, I will sketch how to use a joint project in progress with Jeffrey Adler, Manish Mishra and Kazuma Ohara to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or easier to achieve.

Peter Scholze: Real local Langlands as geometric Langlands on the twistor-ℙ1

4th March, 2024

In 2014, Fargues realized that one can formulate the local Langlands correspondence over p-adic fields as a geometric Langlands correspondence on the Fargues-Fontaine curve. This raises the question of a similar realization of the local Langlands correspondence over the real numbers. The goal of these lectures is to explain a possible formulation. As part of this, we will give a new perspective on the theory of variations of twistor structure, a generalization of the theory of variations of Hodge structure. This uses the theory of analytic stacks developed in our joint work with Clausen, of which we will give a brief overview.

Jack Thorne: Symmetric Power Functoriality For Hilbert Modular Forms

18th May, 2023

Symmetric power functoriality is one of the basic cases of Langlands's functoriality conjectures and is the route to the proof of the Sato-Tate conjecture (concerning the distribution of the modulo p point counts of an elliptic curve over ℚ, as the prime p varies). I will discuss the proof of the existence of the symmetric power liftings of Hilbert modular forms of regular weight. The proof uses automorphy lifting theorems, automorphic forms on unitary groups, and the geometry of Shimura varieties, as well as the fact that Spec(ℤ) is simply connected.

Shai Evra: Optimal strong approximation and the Sarnak-Xue density hypothesis

3rd November, 2022

It is a classical result that the modulo map from SL2(ℤ) to SL2(ℤ/qℤ), is surjective for any integer q. The generalization of this phenomenon to other arithmetic groups goes under the name of strong approximation, and it is well understood. The following natural question was recently raised in a letter of Sarnak: What is the minimal exponent e, such that for any large q, almost any element of SL2(ℤ/qℤ) has a lift in SL2(ℤ) with coefficients of size at most qe? A simple pigeonhole principle shows that e is strictly greater than 3/2. In his letter Sarnak proved that this is in fact tight, namely e = 3/2, and call this optimal strong approximation for SL2(ℤ). The proof relies on a density theorem of the Ramanujan conjecture for SL2(ℤ). In this talk we will give a brief overview of the strong approximation, a quantitative strengthening of it called super strong approximation, and the above mentioned optimal strong approximation phenomena, for arithmetic groups. We highlight the special case of p-arithmetic subgroups of classical definite matrix groups and the connection between the optimal strong approximation and optimal almost diameter for Ramanujan complexes. Finally, we will present the Sarnak-Xue density hypothesis and describe recent ongoing works on it relying on deep results coming from the Langlands programme.

Ngô Bảo Châu: On the kernel of the non-abelian Fourier transform

20th October, 2022

Tate reformulated the theory of the Riemann zeta function and its functional equation as the Mellin shadow of the Fourier transform on a certain space of function on the adeles. Conjecturally, Langlands' general automorphic L-functions and their functional equation can be interpreted in the same way following a framework due to Braverman and Kazhdan with the case of standard L-function associated with automorphic representations of GLn and the standard representation of the dual GLn being well known and due to Godement and Jacquet. This talk is based on a work in progress jointly with Zhilin Luo in which we propose an explicit conjectural construction for the kernel of the non abelian Fourier transform for G=GLn and arbitrary representation of the dual GLn.