I will discuss some of Mazur's work about congruences between Eisenstein series and cusp forms, and then end with an application to class groups of fields $\mathbb{Q}(N^{1/p})$, where $N$ and $p$ are primes.  I will only assume some algebraic number theory.  In particular, nothing about modular forms will be assumed.
 

Given an extension L/K of number fields, we say that the Hasse norm principle (HNP) holds if every non-zero element of K which is a norm everywhere locally is in fact a global norm from L. If L/K is cyclic, the original Hasse norm theorem states that the HNP holds. More generally, there is a cohomological description (due to Tate) of the obstruction to the HNP for Galois extensions. In this talk, I will present work (joint with Rachel Newton) developing explicit methods to study this principle for non-Galois extensions. As a key application, I will describe how these methods can be used to characterize the HNP for extensions whose normal closure has Galois group A_n or S_n. I will additionally discuss some recent generalizations of these methods to study the Hasse principle and weak approximation for multinorm equations as well as consequences in the statistics of these local-global principles.

We discuss the problem in representation theory of decomposing restricted representations. We start classically with the symmetric groups via Young diagrams and Young tableaux, and then move into the world of Lie groups. These problems have connections with both physics and number theory, and if there is time I will discuss the Gan-Gross-Prasad conjectures which predict results on restrictions for algebraic groups over both local and global fields. The pre-requisites will build throughout the talk, but it should be accessible to anyone with some knowedge of both finite groups and Lie groups.

In this talk I will give an introduction to random multiplicative functions, and cover the recent developments in this area. I will also explain how RMF's are connected to some of the important open problems in Analytic Number Theory.

In this session we will discuss how interviewing and being interviewed has changed now that interviews are conducted online. We will have a panel comprising Marya Bazzi, Mohit Dalwadi, Sam Cohen, Ian Griffiths and Frances Kirwan who have either experienced being interviewed online and have interviewed online and we will compare experiences with in-person interviews. 

Differential equations and neural networks are two of the most widespread modelling paradigms. I will talk about how to combine the best of both worlds through neural differential equations. These treat differential equations as a learnt component of a differentiable computation graph, and as such integrates tightly with current machine learning practice. Applications are widespread. I will begin with an introduction to the theory of neural ordinary differential equations, which may for example be used to model unknown physics. I will then move on to discussing recent work on neural controlled differential equations, which are state-of-the-art models for (arbitrarily irregular) time series. Next will be some discussion of neural stochastic differential equations: we will see that the mathematics of SDEs is precisely aligned with the machine learning of GANs, and thus NSDEs may be used as generative models. If time allows I will then discuss other recent work, such as how the training of neural differential equations may be sped up by ~40% by tweaking standard numerical solvers to respect the particular nature of the differential equations. This is joint work with Ricky T. Q. Chen, Xuechen Li, James Foster, and James Morrill.