Paula Lins at the LMS Undergraduate Summer School 2024 – Essex University

In July 2024, Paula Lins took part in the LMS Undergraduate Summer School held at the University of Essex, UK. As part of the programme, she gave a colloquium talk titled “Zeta Functions of Groups”, introducing students to intriguing connections between algebra and analytic structures.

It was a joy to be involved in this lively and supportive event, aimed at encouraging undergraduate students to explore mathematical research. The Summer School brought together curious minds from across the UK, and Paula was glad to contribute to the teaching programme and share her enthusiasm for group theory.

Title: Zeta functions of groups.

Abstract

Group Theory is the area in mathematics that studies symmetries through structures known as groups. Although of very abstract nature, groups are everywhere in science. Ranging from chemistry to physical and dynamical systems, and all the way to cryptography and even music theory. In this talk, we will discuss a tool that is very useful to understand infinite groups; the so-called zeta functions of groups. These are generalisations of the famous Riemann zeta function, that allow us to investigate group properties through analytic properties of the function. The idea is the following: we split the elements of a group into finite pieces according to some interesting property. (For instance, we could split the integer numbers accordingly to being even or odd, getting precisely two pieces). Then, we define a zeta function that encodes arithmetic data of these smaller pieces. One can then study properties of the function, and these will reflect properties of the group.