Analysis and probability

The analysis and probability theme researches many areas of pure and applied analysis and probability. We are also interested in the applications of our research in economics, engineering, epidemics, communications, continuum mechanics, physics, social networks and other areas. The group fosters industrial partnerships, bringing together rigorous mathematical analysis, scientific computing, and real-world applications, and striving towards impactful research across the pure and applied spectrum of science to deliver solutions to real-world problems.

Limit theorems and approximations

We specialise in limit theorems and approximations, often with explicit error bounds, for a wide range of models and processes with a variety of dependence structures. These allow us to better understand statistics of interest arising from complex stochastic systems. Applications on which we have recently worked include communications and queueing systems, random networks, statistical testing and insurance models, among others. We have also worked on the theoretical underpinnings of several useful and widely applied approximation techniques.

Random graphs

Random graphs are an essential tool in modelling communications, power and computer systems, social media contacts, the internet, and in many other areas. Our work in this area includes the study of both exact and asymptotic properties of random graphs, including limit theorems and approximations. We have worked on understanding the structure and behaviour of different random graph models, including, for example, various measures of connectivity. We also study stochastic processes evolving on random graphs. Applications include the modelling of the spread of computer viruses through computer networks, the spread of information through a social network, and the spread of infection through population.

Ergodicity of stochastic processes

We are interested in research problems related to the question of ergodicity of stochastic processes, with a focus on convergence of Markov processes to equilibrium. Examples of processes under consideration include solutions to stochastic differential equations, Markov chains and birth-death processes. Our research has multiple applications in fields such as computational statistics and machine learning, in problems of sampling (e.g. via Markov Chain Monte Carlo methods) and optimisation (e.g. via stochastic gradient algorithms).

Stochastic differential equations

We study various stability properties of stochastic processes defined as solutions to stochastic differential equations (SDEs), as well as their discrete-time counterparts. This includes classical SDEs driven by Brownian motion, SDEs with jump noise and SDEs of McKean-Vlasov type, as well as interacting particle systems that are used to approximate them. In the context of discretisations of SDEs, we are interested in the numerical analysis of Euler-Maruyama schemes and the theory of the corresponding Markov Chain Monte Carlo methods.

Probabilistic foundations of machine learning

We apply probabilistic techniques to develop the mathematical theory of algorithms used in machine learning for training neural networks, such as stochastic gradient algorithms and other finite- and infinite-dimensional optimisation algorithms. This line of research is strongly connected to the theory of optimal transport and gradient flows. We study problems in infinite-dimensional game theory motivated by the task of training Generative Adversarial Networks (GANs). Operating learning is used for solving problems in scientific computing, partial differential equations (PDEs), and physics-informed machine learning. Probabilistic graphical models and uncertainty quantification provide powerful frameworks for modelling complex dependencies in high-dimensional data.

Interacting-particle systems

Many interesting systems in physics and applied sciences consist of a large number of particles, or agents (e.g. individuals, animals, cells, robots), that interact with each other. The established methodology in statistical mechanics and kinetic theory is to look for simplified models that retain relevant characteristics of the original system by letting the number of particles grow to infinity. Such models are intended to efficiently direct human traffic, to optimise evacuation times, to study rating systems, opinion formation, and animal navigation strategies. Our research activity revolves around the study of systems modelled by stochastic dynamics whose limiting behaviour is described by either a deterministic PDE or by a stochastic PDE (SPDE).

Large deviations

In addition to studying the expected behaviour of a system, one is often interested in the so-called rare events which have a vanishing probability but significant (sometimes disastrous) consequences (bankruptcy of a company, congestion in a network, blackout, etc.) It is of great importance to understand how likely such events are to occur, and also what is the most likely scenario for them to occur. We work on the theory of large deviations as well as its applications in energy, insurance, and queueing networks.

Stochastic networks

We work on various stochastic networks arising from real-world applications including biology, communication, data storage and processing, energy, healthcare, social networks. This is a multifaceted area of research where we study design, modelling, stability, control, performance, approximations and rare events in networks. The research involves development and use of techniques and tools from probability theory, stochastic processes, analysis, optimisation, PDEs, combinatorics and graph theory.

Stochastic modelling of biological systems

Work in this area includes: modelling endemic infections, in particular looking at the time until fade-out of infection, the endemic state prior to extinction, and the effects of different model assumptions upon these; the effects of population heterogeneities, e.g. pens within a pig farm, management groups within a dairy herd, or the existence of a group of "superspreaders"; applications of mathematical control theory in epidemic modelling. The research combines new and existing approaches and techniques from stochastic networks and contact processes.

Spectral theory and analysis of PDEs

Research in this area ranges from the very pure to the more applied side. The interdisciplinary side of this work is reflected through the use of techniques from classical mathematical analysis, applied numerical analysis, approximation theory and differential equations, as well as a new approach to the study of spectral asymmetry via microlocal analysis. Spectral theory also enables the analysis of PDEs ruling the propagation of waves in composites. The global in-time behaviour of solutions of nonlinear dispersive Hamiltonian PDEs is also studied, establishing global well-posedness related to wave turbulence, existence and stability of solitons, and soliton resolution.

Optimal transport theory

Research in optimal transport theory has focused on its applications in materials science, weather modelling, and pattern formation, as well as its connections to machine learning, especially in the context of generative models.