Week 1 - 30th September

Martí Oller Riera (University of Cambridge) - Arithmetic Statistics and Dynkin Diagrams

Arithmetic Statistics is an area of Number Theory that has massively grown in the past decades. One of the most impressive recent results in the area is the work of Bhargava and Shankar, which proved that the average rank of elliptic curves is bounded, a result that was later generalised to Jacobians of hyperelliptic curves. We will begin the talk by giving an overview of the proof of these results, and later in the talk we will explain how such results can be interpreted and generalised using Dynkin diagrams.

Week 2 - 7th October

Ned Carmichael (Kings College London) - Sums of Arithmetical Functions

In this talk, we introduce the Dirichlet divisor problem, overviewing some classical questions and results. A well-known problem in analytic number theory, this asks for the best possible asymptotic approximations to sums of the divisor function. Finally, we consider an analogous problem where the divisor function is replaced by coefficients of cusp forms, and explain some results in this new setting (with comparison to the classical case).

Week 3 - 14th October

Julie Tavernier (University of Bath) - Counting number fields whose conductor is the sum of two squares

A conjecture by Malle proposes an asymptotic formula for the number of number fields of bounded discriminant and given Galois group. In this talk we will consider abelian extensions of fixed Galois group G whose conductor is bounded and the sum of two squares, and show how one can employ techniques from harmonic analysis and class field theory to count such extensions. We then explain how this can be extended to a more general class of field extensions whose conductor satisfies some frobenian condition and conclude by mentioning how some of these results could be interpreted in terms of geometric objects.