Recordings

Venue	Date	Speaker and Affiliation
University of Stirling	21 Nov 2025	David Bate (University of Warwick)
University of Glasgow	19 Dec 2025	Javier Parcet (ICMAT Madrid)
University of Edinburgh	23 Jan 2026	Alison Etheridge (University of Oxford)
Heriot-Watt University	20 Feb 2026	Ulisse Stefanelli (University of Vienna)
University of St Andrews	27 March 2026	Mikaela Iacobelli (ETH Zürich)
University of Aberdeen	22 May 2026	Heather Harrington (University of Oxford)

The Torelli group: a quick tour

In this talk we will give a gentle introduction to the Torelli group of a surface.  The talk will survey some of its algebraic properties as well as its connection with low-dimensional topology.    Along the way we will highlight some seminal work of Joan Birman and Dennis Johnson, among others.

Note: due to technical issues, the first 10 minutes of the talk were not recorded. We apologise for the inconvenience.

Maths’ Greatest Unsolved Puzzles

While mathematicians are undoubtedly brilliant, and their work is used in all kinds of amazing scientific and technological discoveries, there are still questions they can’t answer. Every mathematical question is a puzzle to be solved, and while there’ll be plenty of puzzles for you to chew on, we’ll also discuss some of the questions that still leave mathematicians stumped – from simple-sounding number and shape problems to some truly mind-bending fundamental questions.

Orthogonal Sobolev polynomials and spectral methods for boundary value problems on the unit ball

Our main objective in this talk is to demonstrate how orthogonal Sobolev polynomials emerge as a useful tool within the framework of spectral methods for boundary-value problems. The solution of a boundary-value problem for a stationary Schrödinger equation on the unit ball can be studied from a variational perspective. In this variational formulation, a Sobolev inner product naturally arises. As test functions, we consider the linear space of polynomials satisfying the boundary conditions on the sphere, and a basis of
mutually orthogonal polynomials with respect to the Sobolev inner product is provided. The basis of the proposed method is provided in terms of spherical harmonics and univariate orthogonal Sobolev polynomials. The connection formula between these orthogonal Sobolev polynomials and classical orthogonal polynomials on the sphere is established. Consequently, the Sobolev Fourier coefficients of a function satisfying the boundary value problem are recursively derived. Finally, numerical experiments were presented.

Exponential asymptotics and applied mathematics

Divergent series are the invention of the devil, and it is shameful to base on them
any demonstration whatsoever.” – N. H. Abel.
The lecture will introduce the concept of an asymptotic series, showing how useful divergent
series can be, despite Abel’s reservations. We will then discuss Stokes’ phenomenon, whereby
the coefficients in the series appear to change discontinuously. We will show how
understanding Stokes’ phenomenon is the key which allows us to determine the qualitative
and quantitative behaviour of the solution in many practical problems. Examples will be
drawn from the areas of surface waves on fluids, crystal growth, dislocation dynamics, HeleShaw flow, thin film rupture, quantum mechanics, and atmospheric dynamics.

Seeing is deceiving: The mathematics of visual illusions

Illusions have been a constant source of amusement but they are also a unique gateway into understanding the way we perceive the world and how the brain processes information. The simplest visual illusions often involve a primary element—be it a line or a circle—that undergoes deformation or displacement due to the influence of surrounding elements, such as additional lines or dots. Mathematically, these deformations can be modelled as transformations in the plane. I will show that that these perceptual interactions between different elements can indeed be modelled through universal visual laws inducing small deformations and explaining a wide range of illusions. Moreover, these laws find their foundations in models of our visual processing system that exhibits a rich geometric structure, thereby forging a direct connection between the processing of geometric information in our brain and the emergence of illusions.

Layer potentials – quadrature error estimates and approximation with error control

When numerically solving PDEs reformulated as integral equations, so-called layer potentials must be evaluated. The quadrature error associated with a regular quadrature rule for evaluation of such integrals increases rapidly when the evaluation point approaches the surface and the integrand becomes sharply peaked. Error estimates are needed to determine when the accuracy becomes insufficient, and then, a sufficiently accurate special quadrature method needs to be employed.
 
In this talk, we discuss how to estimate quadrature errors, building up from simple integrals in one dimension to layer potentials over smooth surfaces in three dimensions. We also discuss a new special quadrature technique for axisymmetric surfaces with error control.  The underlying technique is so-called interpolatory semi-analytical quadrature in conjunction with a singularity swap technique. Here, adaptive discretizations and parameters are set automatically given an error tolerance, utilizing further quadrature and interpolation error estimates derived for this purpose.

The AI Mathematician

We argue how AI can assist mathematics in three ways: theorem-proving,
conjecture formulation, and language processing.
Inspired by initial experiments in geometry and string theory in 2017, we summarize how this
emerging field has grown over the past years, and show how various machine-learning
algorithms can help with pattern detection across disciplines ranging from algebraic
geometry to representation theory, to combinatorics, and to number theory.
At the heart of the programme is the question how does AI help with theoretical discovery,
and the implications for the future of mathematics

A recording of the talk can be found on Youtube.

Chance, luck, and ignorance; how to put our uncertainty into numbers

We all have to live with uncertainty about what is going to happen, what has happened, and why things turned out how they did.  We attribute good and bad events as ‘due to chance’, label people as ‘lucky’, and (sometimes) admit our ignorance.  I will show how to use the theory of probability to take apart all these ideas, and demonstrate how you can put numbers on your ignorance, and then measure how good those numbers are. Along the way we will look at three types of luck, and judge whether Derren Brown was lucky or unlucky when he was filmed flipping ten Heads in a row.

Perfect sampling, old and new

The possibility of obtaining perfect samples efficiently from a complex probability distribution entered the consciousness of the community in the mid-nineties with the invention of `coupling from the past’ by Propp and Wilson.  The study of perfect samplers of course has considerable theoretical appeal.  But, in addition, their ‘self clocking’ aspect may have practical advantages.  For example, to obtain (imperfect) samples by direct Markov chain simulation we need an a priori analytic bound on the mixing time of the Markov chain, which even now may be very weak;  in contrast, a simulation by coupling from the past halts as soon as a perfect sample has been computed.  Coupling may occur much earlier than the analytic mixing time bound might suggest.

In the last decade or so, a remarkable variety of perfect samplers have come to light, too many to survey in one talk.  To cut down the scope to something reasonable, I’ll focus on perfect samplers that have the additional property that they are able to sample perfectly (a finite window onto) a random configuration of infinite spatial extent, such as one finds in statistical mechanics.  Coupling from the past has this property, as does an approach that Konrad Anand (a PhD student at QMUL) and I have been investigating, and which we are calling `lazy depth-first sampling’.  I feel this approach gives a particularly concrete account of what it means for an infinite system to have a unique Gibbs measure.