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.

10 January 2025: Jerome Neufeld (Cambridge) Flow and flexure: Subglacial hydrology and the transient response of ice sheets

The response of the Greenland and Antarctic ice sheets to a changing climate is one of the largest sources of uncertainty in future sea level predictions. The behaviour of the subglacial environment, where ice meets hard rock or soft sediment, is a key determinant in the flux of ice towards the ocean, and hence the loss of ice over time. Predicting how ice sheets respond on a range of timescales brings together mathematical models of the elastic and viscous response of the ice, subglacial sediment and water and is a rich playground where the simplified models of the contact between ice, rock and ocean can shed light on very large scale questions. In this talk we’ll see how these simplified models can make sense of a variety of field and laboratory data in order to understand the dynamical phenomena controlling the transient response of large ice sheets.

17 January 2025: Lukas Eigentler (Warwick) Modelling dryland vegetation patterns

Vegetation patterns are a ubiquitous feature of semi-arid regions and are a prime example of a self-organisation principle in ecology. In this talk, I present a broad overview of how I have used mathematical modelling and analysis to develop new insights into vegetation pattern dynamics, for example on how mathematical stability properties relate to ecosystem degradation in changing environments, and a possible resolution for contradicting field data on the long-term migration dynamics of vegetation patterns.

24 January 2025: Max Stolarski (Warwick) Singularity Analysis of Geometric Flows

Geometric flows, like the mean curvature flow and Ricci flow, prescribe the time evolution of some geometric object according to its geometry. These flows have found applications in geometry and topology as well as mathematical models of physical systems that include moving boundaries. Due to nonlinear effects, geometric flows often form singularities in finite time. Analysis of these singularities is then essential for applications. We'll discuss geometric flow singularities where the evolving manifold looks like a cone near the singularity. These conical singularities give a rich source of examples of interesting dynamics, including bounded curvature quantities near singularities and non-unique continuations through singularities. We'll explore developments here and discuss implications for the general dynamics of weak geometric flow solutions.

31 January 2025: Michael Hochmann (Hebrew University Jerusalem) Expansion of numbers in different bases and equidistribution

Every number in [0,1] can be expanded in different integer bases (binary, decimal,...). The conversion between bases is algorithmically trivial; but the relation between the expansions is quite mysterious. For instance, although Lebesgue-a.e. point has maximally complex expansions in every base, there is still no explicit construction of a point with this property. Much of what is known about this problem comes from the study of randomly generated numbers, an approach that goes back to the work of Cassels and Schmidt in the late 1950s, They showed that numbers that are generated using biased independent digits in one base, still appear uniformly random in other bases (turning this around, uniform randomness in one base does not guarantee it in other bases!). This has been generalized in many ways, and is closely related to the phenomenon of measure rigidity in ergodic theory, and also to other problems, primarily Furstenberg's transversality conjecture. In this introductory talk I'll focus on the many things we do not understand in this area, and describe several old and new results. All the relevant background will be explained in the talk.

7 February 2025: Ben Green (Oxford) Intersective sets

A set S of natural numbers is intersective if every set of positive (upper) density contains two elements differing by an element of S. Classical examples are S = squares (Furstenberg-Sarkozy) and S = {p - 1: p prime}. I will survey this topic and then discuss some recent progress on quantitative aspects of the theory (partially in joint work with Mehtaab Sawhney).

14 February 2025: Maciej Zworski (Berkeley) Mathematics of magic angles in 2D structures

Magic angles refer to a remarkable theoretical (Bistritzer–MacDonald, 2011) and experimental (Cao et al 2018) discovery, that two sheets of graphene twisted by certain (magic) angles display unusual electronic properties such as superconductivity.

Mathematically, this is related to having flat bands of nontrivial topology for the corresponding periodic Hamiltonian and their existence was shown in the chiral model of twisted bilayer graphene (Tarnopolsky-Kruchkov-Vishwanath, 2019). A spectral characterization of magic angles (Becker–Embree–Wittsten–Z, 2021, Galkowski–Z, 2023) also produces complex values and the distribution of their reciprocals looks remarkably like a distribution of scattering or Pollicott–Ruelle resonances, with the real magic angles corresponding to anti-bound states. I will provide a gentle introduction to the subject and highlight some recent results.

The talk is based on joint works with S Becker, M Embree, J Galkowski, M Hitrik, T Humbert, Z Tao, J Wittsten and H Zeng.

21 February 2025: Jose Carrillo (Oxford) Aggregation-Diffusion Equations for Collective Behaviour in the Sciences

Many phenomena in the life sciences, ranging from the microscopic to macroscopic level, exhibit surprisingly similar structures. Behaviour at the microscopic level, including ion channel transport, chemotaxis, and angiogenesis, and behaviour at the macroscopic level, including herding of animal populations, motion of human crowds, and bacteria orientation, are both largely driven by long-range attractive forces, due to electrical, chemical or social interactions, and short-range repulsion, due to dissipation or finite size effects. Various modelling approaches at the agent-based level, from cellular automata to Brownian particles, have been used to describe these phenomena. An alternative way to pass from microscopic models to continuum descriptions requires the analysis of the mean-field limit, as the number of agents becomes large. All these approaches lead to a continuum kinematic equation for the evolution of the density of individuals known as the aggregation-diffusion equation. This equation models the evolution of the density of individuals of a population, that move driven by the balances of forces: on one hand, the diffusive term models diffusion of the population, where individuals escape high concentration of individuals, and on the other hand, the aggregation forces due to the drifts modelling attraction/repulsion at a distance. The aggregation-diffusion equation can also be understood as the steepest-descent curve (gradient flow) of free energies coming from statistical physics. Significant effort has been devoted to the subtle mechanism of balance between aggregation and diffusion. In some extreme cases, the minimisation of the free energy leads to partial concentration of the mass. Aggregation-diffusion equations are present in a wealth of applications across science and engineering. Of particular relevance is mathematical biology, with an emphasis on cell population models. The aggregation terms, either in scalar or in system form, is often used to model the motion of cells as they concentrate or separate from a target or interact through chemical cues. The diffusion effects described above are consistent with population pressure effects, whereby groups of cells naturally spread away from areas of high concentration. This talk will give an overview of the state of the art in the understanding of aggregation-diffusion equations, and their applications in mathematical biology.

28 February 2025: Carola Bibiane Schönlieb (Cambridge) Mathematical imaging: from PDEs to deep learning for images

Images are a rich source of beautiful mathematical formalism and analysis. Associated mathematical problems arise in functional and non-smooth analysis, the theory and numerical analysis of nonlinear partial differential equations, inverse problems, harmonic, stochastic, and statistical analysis, and optimization, just to name a few. Applications of mathematical imaging are profound and arise in biomedicine, material sciences, astronomy, digital humanities, as well as many technological developments such as autonomous driving, facial screening and many more.
In this talk I will discuss my perspective onto mathematical imaging, share my fascination and vision for the subject. I will then zoom into one research problem that I am currently most excited about and that we helped make first advances on: the mathematical formalisation of machine learned approaches for solving inverse imaging problems.

7 March 2025: David Masser (Basel) Algebraic integrals and transcendence

We know much about the transcendence of algebraic integrals like

$$\int_0^1\sqrt{-{1 \over 2}-x^2+{\sqrt{1+8x^2}\over 2}}{\rm d}x,~~~\int_0^1{{\rm d}x \over (2+x)\sqrt{1+x^3}},~~~\int_0^{\sqrt{2}/2}{4x^4-5x^2+2 \over \sqrt{-4x^6+9x^4-7x^2+2}}{\rm d}x.$$

Here the first arises from the area of a lemniscate considered by Fagnano (who was much more interested in the arc length). The third arises from an arc length of a different lemniscate considered by Huygens (who was much more interested in the area). But how does one actually determine whether given integrals like the above are transcendental or not? In this talk I review some of what was known, and I also sketch an effective (and reasonably practical) method for integrals such as these.

14 March 2025: Rajula Srivastava (Bonn) Counting, Curvature and Convex Duality

How many rational points with denominator of a given size lie within a specified distance from a compact, “non-degenerate” manifold? A precise answer to this question has significant implications for a host of problems including Diophantine approximation on manifolds. In this talk, I will describe how the analytic and geometric properties of the manifold critically influence this count, and provide a heuristic for it. Further, I will talk about recent work which leverages Fourier analytic methods to establish the desired asymptotic for manifolds satisfying a strong curvature condition. At the heart of the proof is a bootstrapping argument that combines Poisson summation, convex duality, and oscillatory integral techniques.