Past Meetings: 2023-24

Robin Ammon- Arithmetic Statistics and the Cohen--Lenstra principle.

Arithmetic Statistics is an area of number theory that deals with statistical questions about various objects of number-theoretic relevance, ranging from century old questions about the distribution of prime numbers to more modern problems about number fields or elliptic curves, proofs of which often have far-reaching consequences. There has been a recent surge in work in arithmetic statistics following a seminal paper by H. Cohen and H. Lenstra, in which they observe that the previously mysterious seeming behaviour of ideal class groups of number fields appears to be governed by a fundamental principle about the distribution of random mathematical objects. Their paper demonstrated the power of a statistical approach in number theory and gave a new perspective on how to understand mathematical objects, and afterwards many other occurrences of their principle have been found, in number theory and beyond.

In my talk, I will give an introduction to arithmetic statistics and explain Cohen and Lenstra's principle with many examples. I will discuss their conjecture about the distribution of ideal class groups and briefly talk about my own work, which seeks to generalise the conjecture to ray class groups. I will not assume any knowledge in number theory!

Oli Jones- Fixed Points of Automorphisms of Torus Knot Groups.

Given an automorphism f of a group G, the set of fixed points Fix(f)= {g | f(g) = g} form a subgroup. Fixed point subgroups are a topic of long-term interest in geometric group theory, dating back to the Scott conjecture for free groups in the 1970s. In this talk we will explore how we can use an action of Aut(G) on a space X which extends an action of G on X to understand fixed points of automorphisms. We will illustrate this by framework by studying fixed points of the Torus Knot Groups <x,y | x^n = y^m>, via their actions on trees.

David Cueto Noval - Cluster Structures, Integrable Systems and Gauge Theories.

This talk is meant to be an introduction to the use of cluster structures in the context of integrable systems and gauge theories. We will start by reviewing the notion of cluster variety. We will focus on the relativistic Toda chain and its connections to 5d N =1 SU(N) gauge theory with no matter multiplets.