Seminars in 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.

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.

Miguel Walsh: Fourier Uniformity of Multiplicative Functions

25th March, 2024

The Fourier uniformity conjecture seeks to understand what multiplicative functions can have large Fourier coefficients on many short intervals. We will discuss recent progress on this problem and explain its connection with the distribution of prime numbers and with other central problems about the behaviour of multiplicative functions, such as the Chowla and Sarnak conjectures.

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.