Recordings

Encoding Fourier multipliers in matrix algebras

Fourier multipliers are among the most important operators in analysis. They act on a given function $f$ by pointwise multiplication of its Fourier transform with a fixed function $f (m f pt )^$. This action can be vastly extended using more general notions of Fourier transform via group representation theory. In this talk, we will explain how these fundamental maps can be encoded as Schur multipliers in matrix algebras, which act simply as $$A ( M(j,k) A_j,k ).$$ In particular, remarkable inequalities for Fourier multipliers extend naturally to general Schur multipliers, even in the absence of a Fourier transform connection. If time permits, we will discuss applications to harmonic analysis on Lie groups.

Bi-Lipschitz embeddings and optimal partial transport

A bi-Lipschitz embedding of a metric space X into another, Y, preserves relative distances between points, up to a multiplicative error. Usually, one seeks a bi-Lipschitz embedding when X is a metric space of interest, and Y has good geometric properties, since the embedding allows X to inherit these properties.

A classical example is Almgren’s embedding of the space of unordered k-tuples in Euclidean space, equipped with the Wasserstein metric, into a (higher dimensional) Euclidean space. This was the starting point to his treatise on the regularity of area minimising surfaces.

The Wasserstein metric is an example of a transportation metric that compares two probability measures on a space. Recently, there has been significant interest in transportation metrics that compare measures with different total masses. One such metric was introduced by Figalli and Gigli to describe solutions to the heat equation with Dirichlet boundary conditions. It compares two measures in a similar way to the Wasserstein metric, but allows excess mass to be transported to or from the boundary of a domain. This transportation metric also describes the space of persistence barcodes in topological data analysis.

In this talk we will first give a gentle introduction to the transportation metric of Figalli and Gigli. It will then be shown that the space of unordered k-tuples, equipped with this metric, does not bi-Lipschitz embedded into Euclidean space. However, we will then proceed to show how this space does in fact embed into Hilbert space. This talk will be accessible to all, is of a combinatorial nature, and no knowledge of measure theory is required.

This talk is based on joint work with A.L. Garcia-Pulido (Stirling).

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.