Recordings

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.

Pattern Formation Through Spatial Segregation

From Order to Chaos and Chaos to Order in Fluid Flows

The eleven year solar activity cycle is a remarkable example of regular behaviour emerging from an extremely turbulent system. The jets on Jupiter sit unmoving on a sea of turbulent eddies. Astrophysical phenomena often display organisation on spatial and temporal scales much larger than the turbulent processes that drive them. An outstanding problem of astrophysics (and indeed other branches of nonlinear physics) is the mathematical description of such systems that can capture systematic behaviour emerging from the underlying chaos, given that Direct Numerical Simulation of these objects is simply impossible. These fascinating phenomena introduced and and it is discussed how methods from non-equilibrium statistical mechanics may be developed to give some insight into their behaviour.

Finite Shadows of Infinite Groups, Finiteness Properties, and Geometry

There are many situations in geometry and group theory where it is natural, convenient or necessary to explore infinite groups via their actions on finite objects. But how much understanding can one really gain about an infinite group by examining its finite images? Sometimes little, sometimes a lot. In this colloquium talk, it is sketched the rich history of this problem and describe how input from hyperbolic geometry and low-dimensional topology have transformed the subject in recent years.

Data Augmentation and Imagination

The technique of data augmentation, whereby observed data are effectively augmented by additional quantities not actually observed in an experiment, has proved to be extremely powerful in modern statistics generally, and in Bayesian parametric inference in particular.  This talk will describe its application to Bayesian inference for epidemic models where many challenges arise from the typically incomplete observations of epidemic processes.

Artistic Mathematics: Truth and Beauty

This is about Henry’s work in mathematical visualisation: making accurate, effective, and beautiful pictures, models, and experiences of mathematical concepts. He discusses what it is that makes a visualisation compelling, and show many examples in the medium of 3D printing, as well as some work in virtual reality and spherical video. He also discusses his experiences in teaching a project-based class on 3D printing for mathematics students.

Postcards from Network Theory

I will show some postcards of the use of network theory for solving real-world problems. After a very brief introduction I will start by illustrating such postcards. The postcard 1 is about the study of protein-protein interaction (PPI) networks. The postcard 2 is about the use of networks to describe the 3-dimensional structure of biomacromolecules. In particular, I will develop the concept of network communicability and its relation to matrix functions. Postcard 3 is about the study of navigation routes on networks and its implication to traffic in cities. I will show how such diffusive processes on graphs can explain the routes followed by drivers at rush hour in several cities across the world. The final postcard is about the development of mathematical strategies to implement nonlocal interactions in networks.