We give an introduction to a rigorous renormalization group analysis of the sine-Gordon model with a focus on deriving the lowest-order beta function.

The representation theory of Lie algebras over fields of positive characteristic behaves quite differently to the characteristic zero case. For example, in positive characteristic, the dimension of all simple modules is finite and bounded. In this talk, we’ll begin by recalling the classification of finite simple representations of sl_2, and then explore how this changes when we move to the positive characteristic setting. Along the way, we’ll discuss the additional structures that appear in positive characteristic, such as restricted Lie algebras, the p-centre, and reduced enveloping algebras.

In this talk, I'm going to give an introduction to my area of research, which concerns automorphic L-functions. We're going to start by introducing the ring of adeles and how it leads us to an integral representation of the Riemann zeta function. We'll see how this can be generalised for an arbitrary automorphic representation and pose general conjectures which resemble the Riemann Hypothesis. I'll finish by presenting the statement and an idea behind my recent result related to those conjectures.

Bring interesting problems (relating to your research or otherwise) for a unique brainstorming session

Fusion systems are a generalisation of finite groups designed in a way to capture local structure at a prime motivated by the existence of "exotic" fusion systems; local structures that do not appear in any finite group. In this talk I will give a brief introduction to fusion systems with emphasis on how they relate to groups. I will then discuss recent work done on fusion invariant character theory, concluding with a short excursion into biset functor theory to state a character value formula for "induction" between fusion systems and a Frobenius reciprocity analogue.

A (saturated) fusion system on a p-group P contains data about conjugacy within P, the typical case being the system induced by a group on its Sylow p-subgroup. Fusion systems are completely determined by looking at their essential subgroups, which must admit an automorphism of order coprime to p. For p=2, we describe two new methods that address the question: given an essential subgroup $E<P$ of a fusion system on P, what can we say about P? In particular, one method gives us sufficient conditions to deduce that $E\triangleleft P$, while the other explores cases where we have strong control over the normaliser tower of E in P.

• as holomorphic functions on certain symplectic spaces
• as matrix coefficients of the Heisenberg (and metaplectic) representation,
• as sections of line bundles on abelian varieties.