University of Oxford

Modular curves play a key role in the Langlands programme, being the simplest example of so-called Shimura varieties.  Their less famous cousins, Shimura curves, are also very interesting, and very concrete. 
In this talk I will give a gentle introduction to the arithmetic of Shimura curves, with lots of explicit examples. Time permitting, I will say something about recent work about intersection numbers of geodesics on Shimura curves.

Roughly speaking, class field theory for a number field K describes the abelianization of its absolute Galois group in terms of the idele class group of K. Geometric class field theory is what we get when K is instead the function field of a smooth projective geometrically connected curve X over a finite field. In this talk, I give a precise statement of geometric class field theory in the unramified case and describe how one can prove it by showing the Picard stack of X is the “free dualizable commutative group stack on X”. A key part is to show that the usual “divisor class group exact sequence“ can be done in families to give the adelic uniformization of the Picard stack by the moduli space of Cartier divisors on X. 

The Hecke algebra is an algebraic gadget for studying the smooth complex representations of locally profinite groups. We demonstrate the spherical Hecke algebra of GL(n,F) is commutative and present a combinatorial proof of the Satake isomorphism. We apply this to the classification of spherical representations of GL(2,F).

Richard Hamilton introduced the Ricci flow as a way to study the Poincaré conjecture, which says that every simply connected, compact three-manifold is homeomorphic to the three-sphere. In this talk, we will introduce the Ricci flow in a way that is accessible to anyone with basic knowledge of Riemannian geometry. We will give some examples, discuss finite time singularities, and give an application to a theorem of Hamilton which says that every compact Riemannian 3-manifold with positive Ricci curvature admits a metric of constant positive sectional curvature.

Two areas of current research in Mathematical Gauge Theory are the study of higher dimensional instantons on manifolds with special holonomy (for example, Calabi-Yau three folds, G₂ and Spin(7) manifolds), and low dimensional gauge theories (for example the Kapustin-Witten, Haydys-Witten and ADHM Seiberg-Witten equations). A common feature of these two sets of theories is that the moduli spaces of solutions are in general not compact. In both cases, compactness issues arise because of solutions to a certain non-linear equation called the Fueter equation. In this talk, I'll explain how this non compactness gives a relationship between these high and low dimensional gauge theories.

In this talk I will give a brief introduction to the field of post-quantum (PQ) cryptography, introducing a few of the most popular computational hardness assumptions. Second, I will give an overview of a recent work of mine on PQ electronic voting. I’ll finish by presenting a short selection of ‘exotic’ cryptographic constructions that I think are particularly hot at the moment (no, not blockchain). The talk will be definitionally light since I expect the area will be quite new to many and I hope this will make for a more engaging introduction.

For a complete theory T, Lascar associated with it a Galois group which we call the Lacsar group. We will talk about some of my work on recovering the Lascar group as the fundamental group of Mod(T) and some recent progress in understanding the higher homotopy groups.

This is a pre-seminar meeting for Margaret Bilu's talk "A motivic circle method", which takes place later in the day at 5PM in L3.

The local Kronecker-Weber theorem states that the maximal abelian extension of p-adic numbers Q_p is obtained from this field by adjoining all roots of unity. In 2018, Koenigsmann conjectured that the maximal abelian extension of Q_p is decidable. In my talk, we will discuss Koenigsmann's proposed axiomatisation. In contrast, the maximal unramified extension of Q_p is known to be decidable, admitting a complete axiomatisation by an informed but simple set of axioms (this is due to Kochen). We explain how the question of completeness can be reduced to an Ax-Kochen-Ershov result in residue characteristic 0 by the method of coarsening.