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.

Week 4 - 21st October

Yicheng Yang (LSGNT) - Higher Hida theory on the modular curve

In the 80's, Hida introduced an ordinary projector on modular forms and constructed p-adic families of ordinary modular forms. In the talk we will give a construction of integral Hecke operators and explain the idea of higher Hida theory which works for all cohomological degrees and can be generalised to higher dimensional.

Week 5 - 28th October

Besfort Shala (Bristol) - Chowla's Conjecture for Random Multiplicative Functions

Let f be a Steinhaus or Rademacher random multiplicative function. Even though a lot is known about partial sums of f over positive integers due to work of Harper, the exact distribution is still elusive. However, partial sums of f over some restricted sets of positive integers are better understood and are expected/known to mimic a sum of independent random variables, in the sense that they satisfy a central limit theorem and/or law of iterated logarithm. There are several results in this direction such as summing over: (i) integers with a restricted number of prime factors (Hough, Harper), (ii) integers in a short interval (Chatterjee-Sound, Sound-Xu), (iii) values of integer polynomials in the Steinhaus case (Klurman-Shkredov-Xu). The latter may be viewed as a random analogue for Chowla's conjecture or the related Elliott's conjecture. In this talk, I will give a survey of the above results and discuss joint work with Jake Chinis, where we address the random analogue of Chowla's conjecture for Rademacher multiplicative functions. If time permits, I will also discuss some open questions in this area, some of which we address in ongoing joint work with Christopher Atherfold.