Oxford University

A universal D-module of dimension n is a rule assigning to every family of smooth $n$-dimensional varieties a family of D-modules, in a compatible way. This seems like a huge amount of data, but it turns out to be entirely determined by its value over a single formal disc. We begin by recalling (or perhaps introducing) the notion of a D-module, and proceed to define the category $M_n$ of universal D-modules. Following Beilinson and Drinfeld we define the Gelfand-Kazhdan structure over a smooth variety (or family of varieties) of dimension $n$, and use it to build examples of universal D-modules and to exhibit a correspondence between $M_n$ and the category of modules over the group-scheme of continuous automorphisms of formal power series in $n$ variables

In this talk I will explain the basic motivation behind the trace formula and give some simple examples. I will then discuss how it can be used to prove things about automorphic representations on general reductive groups.

A finite set of elements in a connected real Lie group is "Diophantine" if non-identity short words in the set all lie far away from the identity. It has long been understood that in abelian groups, such sets are abundant. In this talk I will discuss recent work of Aka; Breuillard; Rosenzweig and de Saxce concerning this phenomenon (and its limitations) in the more general setting of nilpotent groups. 

In the first part of the talk we are going to build up some intuition about limit-periodic functions and I will explain why they are the 'simplest' class of arithmetic functions appearing in analytic number theory. In the second part, I will give an equivalent description of 'limit-periodicity' by using exponential sums and explain how this property allows us to solve 'twin-prime'-like problems by the circle method.

A distinct covering system of congruences is a collection

\[

(a_i \bmod m_i), \qquad 1\ \textless\ m_1\ \textless\ m_2\ \textless\ \ldots\ \textless\ m_k

\]

whose union is the integers. Erd\"os asked whether there are covering systems for which $m_1$ is arbitrarily large. I will describe my negative answer to this problem, which involves the Lov\'{a}sz Local Lemma and the theory of smooth numbers.

This talk will discuss the discovery of Heegner points from a historic perspective. They are a beautiful application of analytic techniques to the study of rational points on elliptic curves, which is now a ubiquitous theme in number theory. We will start with a historical account of elliptic curves in the 60's and 70's, and a correspondence between Birch and Gross, culminating in the Gross-Zagier formula in the 80's. Time permitting, we will discuss certain applications and ramifications of these ideas in modern number theory.