04 October 2024: Kevin Buzzard (Imperial) Interactive theorem provers and mathematics

The technology which powers computer theorem provers has existed for decades, but for many years the mainstream mathematics community has not really engaged with it. The last few years have seen a step change here though, with work of several Fields Medallists now having been digitised. Will this technology soon begin to change the way mathematicians work? Where is it going? I will give an overview of the area suitable for mathematicians, assuming no background in computer science.

11 October 2024: Mark Peletier (Eindhoven) In search of structure: Gradient flows, GENERIC systems, and the role of noise

Many ODEs and PDEs describe real-world phenomena. Often it is useful to know that these equations have, more "structure" than that of the bare PDE. For instance, many ODEs and PDEs are Hamiltonian systems, which provides a wealth of additional information about their solutions.

Other ODEs and PDEs are _gradient_flows_, and this is the structure that I will concentrate on. Many evolutionary PDEs are known to be gradient flows in an appropriate sense, and again this property gives deep insight and provides many tools for analysis.

Despite the importance of such gradient structures, it is only relatively recently that we have discovered the reason _why_ many evolutionary PDEs are gradient flows, in particular why there are so many gradient flows based on the _Wasserstein_metric_. It turns out that this is intimately connected to randomness.

In this talk I will discuss gradient flows and their cousin "GENERIC systems", and show how one can understand how these deterministic, geometric structures have their roots in randomness.

18 October 2024: Liz Fearon (UCL) Epidemiological modelling of testing, contact tracing and isolation interventions in epidemic response: experiences from COVID-19

This talk will explore mathematical modelling used to support design and deployment decisions for one of our key epidemic control interventions: testing, tracing and isolation or quarantine (TTI). I will review the types of questions that were posed at different stages of the COVID-19 pandemic in the UK and how we responded to them, highlighting the types of modelling tools that were used. Thinking more broadly and looking forward, I will consider new challenges, including for other types of infections and learnings for how we develop models and engage with communities and other disciplines in doing so.

25 October 2024: Mike Whittaker (Glasgow) Self-similar groups and their limit spaces

A self-similar group (G,X) consists of a group G acting faithfully on a homogeneous rooted tree such that the action satisfies a self-similarity condition. In this talk I will introduce a beautiful theorem of Nekrashevych: there is a self-similar group associated to every post critically finite rational function whose limit space recovers the Julia set of the function. I'll then show how limit spaces of self-similar actions appear in other contexts and reveal dynamical systems in a group theoretic context.

1 November 2024: Harry Schmidt (Warwick) Canonical heights and equidistribution

Algebraic numbers are roots of (non-zero) polynomials with coefficients in the integers. We can measure their size with a height function. There are various height functions associated to geometric and dynamical objects, and I will give some examples coming from dynamical systems induced by polynomial maps. I will present some joint work with Philipp Habegger in which we prove lower bounds for such heights and give some applications of our work. Time permitting, I will also present some joint work with Myrto Mavraki in which we study points that are small with respect to two distinct heights.

8 November 2024: Stefan Güttel (Manchester) Randomized algorithms in numerical linear algebra

Randomization is an established technique to speed up the numerical solution of very large-scale linear algebra problems that have some form of redundancy, with overdetermined least-squares problems and low-rank matrix approximation being the most prominent examples. Until recently, it has been less clear how to apply randomization to problems that do not have inherent redundancy, including linear systems of equations, matrix functions, and (non)linear eigenvalue problems. I will discuss some new ideas to speed up computational methods for these problems.

15 November 2024: Nina Snaith (Bristol) Random matrices, number theory and the derivative of the characteristic polynomial

For over 50 years the connection between random matrix theory and the Riemann zeta function has been studied, allowing calculations on average values of the characteristic polynomial of random unitary matrices to inform studies of number theoretical functions. I will give some stories and history of this connection and then look at some results on the derivative of the characteristic polynomial, in analogy with the derivative of the Riemann zeta function.

22 November 2024: Rob Neel (Lehigh) Coupling in geometry and sub-Riemannian diffusions

We recall the notion of a coupling of two stochastic processes (and more generally two probability measures) and its application to showing convergence to equilibrium. We then describe the classical applications to Riemannian geometry, and how these natural constructions fail in sub-Riemannian geometry, even for the simplest case of the Heisenberg group. After reviewing the situation, we describe an improvement and extension of constructions of non-Markovian reflection couplings on sub-Riemannian model spaces by Banerjee-Gordina-Mariano and Bénéfice. Moreover, this construction is relatively simple and geometrically appealing, being based on global symmetries of the underlying spaces. This talk is based on joint work with Liangbing Luo.

29 November 2024: Robert Kropholler (Warwick) Folding-like techniques for CAT(0) cube complexes

In a seminal paper, Stallings introduced folding of morphisms of graphs. Stallings's methods give effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani--Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. We extend their techniques to fundamental groups of non-positively curved cube complexes. This is joint work with Robbie Lyman and Michael Ben-Zvi.

6 December 2024: Silke Weinfurtner (Nottingham) Exploring Classical and Quantum Fields in Curved Spacetimes: Lab-Based Investigations into Black Holes and Early Universe Physics

Exploring the dynamics of the early universe and black holes unveils profound insights into the interplay between general relativity and classical/quantum fields. Important phenomena emerge when gravitational and/or field interactions are strong, and/or when quantum effects become prominent. Notable examples include Hawking's proposal on the evaporation of black holes, Penrose's conjecture on the spin-down of rotating black holes, and Kofman's proposal on particle production during preheating. Despite their significance, observing these phenomena directly remains elusive. In this presentation, I will report on recent advancements in investigating these processes in laboratory experiments involving normal and quantum liquids.