Recordings

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.

The emergence of hydrodynamics in many-body systems

One of the most important problems of modern science is that of emergence. How do laws of motion emerge at large scales of space and time, from much different laws at small scales? Hydrodynamics offers a basic but very relevant example. Molecules in air simply go along their journey following Newton’s equations. But when there are very many of them, the solution to these equations becomes extremely complicated; seeking the knowledge of each molecule’s individual trajectory is completely impractical. Happily it is also unnecessary. At our human scale, new, different equations emerge for aggregate quantities: those of hydrodynamics. And these are apparently all we need to know in order to understand the weather! Despite its conceptual significance, the passage from microscopic dynamics to hydrodynamics remains a notorious open problem of mathematical physics. This goes much beyond molecules in air: similar principles hold very generally, including in other physical systems such as quantum gases and spin lattices, where the resulting equations themselves can be very different. In particular, integrable models, where an extensive mathematical structure allows us to make progress, admit an entirely new universality class of hydrodynamic equations. In this talk, I will discuss in a pedagogical fashion the general problem and principles, and some recent advances in our understanding, including those obtained in integrable models.

Category Theory in Epidemiology

“Stock and flow diagrams” are widely used for modeling in epidemiology. Modelers often regard these diagrams as an informal step toward a mathematically rigorous formulation of a model in terms of ordinary differential equations. However, these diagrams have a precise syntax, which can be explicated using category theory. Although commercial tools already exist for drawing these diagrams and solving the differential equations they describe, my collaborators and I have created new software that overcomes some limitations of existing tools. Basing this software on categories has many advantages, but I will explain three: functorial semantics, model composition, and model stratification. This is joint work with Xiaoyan Li, Sophie Libkind, Nathaniel Osgood, Evan Patterson and Eric Redekopp.

Navigating the Unknown: Harnessing Uncertainty in Renewable Energy and Heart Health

Uncertainty Quantification (UQ) emerges as a guiding force in the turbulent sea of data-driven domains, from energy to health. This talk presents a methodology that harnesses UQ for robust renewable energy forecasting, employing a stochastic differential equation model that sails beyond the challenges of wind and solar predictability. Shifting the focus to biomedical imaging, we unveil a novel UQ approach that dives into the depths of MRI analysis, providing more precise insights into cardiac health without being anchored to specific segmentation techniques. Embark on a voyage that highlights how navigating the uncertainties of today with UQ can steer us toward the precision of tomorrow’s applications.