Tag - Number theory

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.

Charlotte Chan: Deligne-Lusztig theory: examples and applications

20th May, 2024

Geometry and representation theory are intertwined in deep and foundational ways. One of the most important instances of this relationship was uncovered in the 1970s by Deligne and Lusztig: the representation theory of matrix groups over finite fields is encoded in the geometry of a natural 'partition' of flag varieties. Recent developments have revealed rich connections between Deligne-Lusztig varieties and geometry studied in number-theoretic contexts. In this lecture series, we give an example-based tour of these ideas, focusing on how to extract concrete information from theory.

Gyujin Oh: Derived Hecke action for weight 1 modular forms via lassicality

9th May, 2024

It is known that a p-adic family of modular forms does not necessarily specialize into a classical modular form at weight 1, unlike the modular forms of weight 2 or higher. We will explain how this obstruction to classicality leads to a 'derived' action on modular forms of weight 1, which can be understood as the so-called derived Hecke operator at p. We will also investigate the role of the derived action in the study of p-adic periods of the adjoint of the weight 1 modular forms.

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).

Harald Helfgott: Expansion, Divisibility and Parity

5th April, 2024

We will discuss a graph that encodes the divisibility properties of integers by primes. We will prove that this graph has a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier, by combining the main result with Matomaki-Radziwill. (This is joint work with M. Radziwill.) For instance: for λ the Liouville function (that is, the completely multiplicative function with λ(p) = −1 for every prime),

(1/log x) ∑n ≤ x λ(n) λ(n+1)/n=O(1/√(log log x)),

which is stronger than well-known results by Tao and Tao-Teravainen. We also manage to prove, for example, that λ(n+1) averages to 0 at almost all scales when n restricted to have a specific number of prime divisors Ω(n)=k, for any "popular" value of k (that is, k = log log N+O(√(log log N)) for n ≤ N).

James Rickards: The Not-So-Local-Global Conjecture

4th April, 2024

I will introduce Apollonian circle packings, and describe the local-global conjecture, which predicts the set of curvatures of circles occurring in a packing. I will then describe reciprocity obstructions, a phenomenon rooted in reciprocity laws (for instance, quadratic reciprocity), that disproves the conjecture in most cases. I will also describe follow-up work, where we obtain a similar result in a situation related to Zaremba's conjecture on continued fraction expansions, disproving a conjecture of Kontorovich.

David Loeffler: Euler systems and the Bloch-Kato conjecture

27th March, 2024

The Bloch-Kato conjecture, relating special values of L-functions to algebraic data, is one of the most important open problems in number theory; it includes the Birch-Swinnerton-Dyer conjecture for elliptic curves as a special case. I will describe some recent breakthroughs establishing special cases of this conjecture (and related problems such as the Iwasawa
main conjecture) using the method of Euler systems.