University of Oxford

The large sieve inequality for Dirichlet characters is a central result in analytic number theory, which encodes a strong orthogonality property between primitive characters of varying conductors. This can be viewed as a statement about $\mathrm{GL}_1$ automorphic representations, and it is a key open problem to prove similar results in the higher $\mathrm{GL}_m$ setting; for $m \ge 2$, our best bounds are far from optimal. We'll outline two approaches to such results (sketching them first in the elementary case of Dirichlet characters), and discuss work-in-progress of Thorner and the author on an improved $\mathrm{GL}_m$ large sieve. No prior knowledge of automorphic representations will be assumed.

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of $\mathbb{Q}_p^{ab}$, the maximal extension of the $p$-adic numbers $\mathbb{Q}_p$ with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of $\mathbb{Q}_p$) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method yields short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).

The security of many widely used communication systems hinges on the presumed difficulty of factoring integers or computing discrete logarithms. However, Shor's celebrated algorithm from 1994 demonstrated that quantum computers can perform these tasks in polynomial time. In 2023, Regev proposed an even faster quantum algorithm for factoring integers. Unfortunately, the correctness of his new method is conditional on an ad hoc number-theoretic conjecture. Using tools from analytic number theory, we establish a result in the direction of Regev's conjecture. This enables us to design a provably correct quantum algorithm for factoring and solving the discrete logarithm problem, whose efficiency is comparable to Regev's approach. In this talk, we will give an accessible account of these developments.

In this talk, I will introduce fusion categories as categorical versions of finite rings. We will discuss some examples which may already be familiar, like the category of representations of a finite group and the category of vector spaces graded over a finite group. Then, we will define Tambara-Yamagami categories, which are a certain type of fusion categories which have one simple object which is non-invertible. I will provide the classification results of Tambara and Yamagami on these categories and give some small examples. Time permitting, I will discuss current work in progress on how to generalize Tambara-Yamagami fusion categories to locally compact groups. 

This talk will not assume familiarity with category theory further than the definition of a category and a functor.

In 1911, Otto Toeplitz posed the intriguing "Square Peg Problem," asking whether every Jordan curve admits an inscribed square. Despite over a century of study, the problem remains unsolved in its full generality. However, significant progress has been made over the years. In this talk, we explore recent advancements by Andrew Lobb and Joshua Greene, who approach the problem through the lens of Lagrangian Floer homology. Specifically, we outline a proof of their result: every smooth Jordan curve inscribes every rectangle up to similarity.

A non-nilpotent graph Γ(G) of a finite group G has elements of G as vertices, with x and y joined by an edge iff a subgroup generated by these two elements is non-nilpotent. During the talk we will prove several (often unrelated) properties of this construction; for instance, any simple graph can be found as an induced subgraph of Γ(G) for some (solvable) group G. The talk is based on my article "A few remarks on the theory of non-nilpotent graphs" (May 2023).

Bernstein–Gelfand–Gelfand (BGG) resolutions and the Grothendieck–Cousin complex both play central roles in modern algebraic geometry and representation theory. The BGG approach provides elegant, combinatorial resolutions for important classes of modules especially those arising in Lie theory; while Grothendieck–Cousin complexes furnish a powerful framework for computing local cohomology via filtrations by support. In this talk, we will give an overview of these two constructions and illustrate how they arise from the same categorical consideration.

We begin with the Fontaine--Wintenberger isomorphism, which gives an example of an extension of Q_p and of F_p((t)) with isomorphic absolute Galois groups. We explain how by trying to lift maps on mod p reductions one encounters Witt vectors. Next, by trying to apply the theory of Witt vectors to the two extensions, we encounter the idea of tilting. Perfectoid fields are then defined more-or-less so that tilting may be reversed. We indicate the proof of the tilting correspondence for perfectoid fields following the Witt vectors approach, classifying the untilts of a given characteristic p perfectoid field along the way. To end, we touch upon the Fargues--Fontaine curve and the geometrization of l-adic local Langlands as motivation for globalizing the tilting correspondence to perfectoid spaces.