University of Oxford

It is a standard result that mapping class groups of high genus do not surject the integers. This is easily shown by computing the abelianization of the mapping class group using a presentation. Once we pass to finite index subgroups, this becomes a conjecture of Ivanov. More generally, we can ask which groups admit epimorphisms from finite index subgroups of the mapping class group. In this talk, I will present a geometric approach to this question, using harmonic maps, and explain some recent results.

In 1735, Euler observed that $ζ(2) = 1 + \frac{1}{2²} + \frac{1}{3²} + ⋯ = \frac{π²}{6}$. This is related to the famous identity $ζ(−1) "=" 1 + 2 + 3 + ⋯ "=" \frac{−1}{12}$. In general, values of the Riemann zeta function at positive even integers are equal to rational numbers multiplied by a power of $π$. The values at positive odd integers are much more mysterious; for example, Apéry proved that $ζ(3) = 1 + \frac{1}{2³} + \frac{1}{3³} + ⋯$ is irrational, but we still don't know if $ζ(5) = 1 + \frac{1}{2⁵} + \frac{1}{3⁵} + ⋯$ is rational or not! In this talk, we will explain the arithmetic significance of these values, their generalizations to Dirichlet/Dedekind L−functions, and to L−functions of elliptic curves. We will also present a new formula for $ζ(3) = 1 + \frac{1}{2³} + \frac{1}{3³} + ...$ in terms of higher algebraic cycles which came out of an ongoing project with Lambert A'Campo.

Functoriality is a key feature in Langlands’ conjectured relationship between automorphic representations and Galois representations; it predicts that certain Galois representations are automorphic, i.e. should come from automorphic representations. We discuss the idea of $p$-adic propagation of automorphy, which seeks to establish the automorphy of everything in a “neighborhood” given the automorphy of something in that neighborhood. The “neighborhoods” that we consider will be the irreducible components of a $p$-adic analytic space called the eigenvariety, which parameterizes $p$-adic automorphic representations. This technique was introduced by Newton and Thorne in their proof of symmetric power functoriality, and can be adapted to investigate similar problems.

I'll give an introduction to a recent theme in the Langlands program over number fields and mixed characteristic local fields (with a much older history over function fields). This is enhancing the traditional 'set-theoretic' Langlands correspondence into something with a more geometric flavour. For example, relating (categories of) representations of p-adic groups to sheaves on moduli spaces of Galois representations. No number theory or 'Langlands' background will be assumed!

The talk is based on joint work with Lasse Grimmelt. We prove a theorem that allows one to count solutions to determinant equations twisted by a periodic weight with high uniformity in the modulus. It is obtained by using spectral methods of $\operatorname{SL}_2(\mathbb{R})$ automorphic forms to study Poincaré series over congruence subgroups while keeping track of interactions between multiple orbits. This approach offers increased flexibility over the widely used sums of Kloosterman sums techniques. We give applications to correlations of the divisor function twisted by periodic functions and the fourth moment of Dirichlet $L$-functions on the critical line.

Introducing an inhomogeneous shift allows for generalisations of many multiplicative results in diophantine approximation. In this talk, we discuss an inhomogeneous version of Gallagher's theorem, established by Chow and Technau, which describes the rates for which we can approximate a typical product of fractional parts. We will sketch the methods used to prove an earlier version of this result due to Chow, using continued fraction expansions and geometry of numbers to analyse the structure of Bohr sets and bound sums of reciprocals of fractional parts.

Kashiwara’s theory of crystal bases provides a powerful tool for studying representations of quantum groups. Crystal bases retain much of the structural information of their corresponding representations, whilst being far more straightforward and ‘stripped-back’ objects (coloured digraphs). Their combinatorial description often enables us to obtain concrete realizations which shed light on the representations, and moreover turn challenging questions in representation theory into far more tractable problems.

After reviewing the construction and basic theory regarding quantum groups, I will introduce and motivate crystal bases as ‘nice q=0 bases’ for their representations. I shall then present (in both finite and affine types) the construction of Young wall models in the important case of highest weight representations. Time permitting, I will finish by discussing some applications across algebra and geometry.

In this talk, we will study the problem in additive combinatorics of determining for a finite Abelian group $G$ the size of its smallest subset $A\subset G$ that has no unique sum, meaning that for every two $a_1,a_2\in A$ we can write $a_1+a_2=a’_1+a’_2$ for different $a’_1,a’_2\in A$. We begin by using classical rectification methods to obtain the previous best lower bounds of the form $|A|\gg \log p(G)$, which stood for 50 years. Our main aim is to outline the proof of a recent improvement and discuss some of its key notions such as additive dimension and the density increment method. This talk is based on Bedert, B. On Unique Sums in Abelian Groups. Combinatorica (2023).

Modular curves are moduli spaces of elliptic curves equipped with certain level structures. This talk will be concerned with how the attendant theory has been used to answer questions about the modularity of elliptic curves over $\mathbb{Q}$ and over quadratic fields. In particular, we will outline two instances of the modularity switching technique over totally real fields: the 3-5 trick of Wiles and the 3-7 trick of Freitas, Le Hung and Siksek. The recent work of Caraiani and Newton over imaginary quadratic fields naturally leads one to consider the descent theory of 'twisted' modular curves, and this will be the focus of the final part of the talk.