See below a list of supervisors, key words and research areas.
In addition to the supervisors listed here, you could contact any member of the Department of Mathematics whose research area interests you.
Keywords: Geometric analysis, calculus of variations, geometric measure theory, elliptic PDEs
Research area: I work in geometric analysis, specifically on regularity questions arising in geometric problems, using tools from geometric measure theory and elliptic PDEs.
Heather Battey with Chris Hallsworth
Statistical theory ensures that methods of inference, crucial tools for applied statisticians and scientists, are calibrated i.e., suitable for purpose. Specifically, in hypothetical repeated application, proposed methods are required to give an answer within a small neighbourhood of truth with quantifiable high probability. The construction of such methods becomes more difficult when there are a large number of nuisance parameters, or equivalently a high-dimensional nuisance parameter. These are aspects that are not of direct concern, but that interfere with inference on aspects of interest, such as the treatment effect of a new drug. Much of my current research concerns principled inference for interest parameters in the presence of a high-dimensional nuisance parameter. This is most successful when the nuisance parameter is evaded completely or approximately by problem-specific manoeuvres that are often difficult to identify. I am working on a unified treatment.
Non-equilibrium statistical mechanics, stochastic dynamics in life sciences, biomathematics, theoretical soft matter physics, fluid dynamics
Our research group is interested in understanding collective and emergent behaviours in out-of-equilibrium and disordered systems. Employing analytical and computational tools from non-equilibrium statistical mechanics, soft condensed matter physics and stochastic processes, our work finds applications in a variety of settings in physical, life and social sciences. Examples include statistical mechanics of active matter, collective dynamics in tissues, jamming in disordered systems, random walks in complex environments, stochastic models of opinion formation, swelling and drying dynamics of gels. Combining theory and modelling, we try to work closely with experimentalists.
Formal proof verification
Me and a small but growing team of mathematicians and computer scientists from across the world are developing a mathematical database of formally proved theorems. One goal is to formalise the entire content of the pure maths part of a standard undergraduate degree, but we are also beginning on formalising the statements of research level mathematical theorems and conjectures. We have been developing teaching materials targeted at undergraduate mathematicians, and over the next few months I will be developing more. We are making a database which AI will be able to understand and analyse; in the future this will enable mathematicians to search for theorems and hopefully will enable computers to start searching for simple proofs to help mathematicians work more efficiently. After that-- who knows?
My research interests span probability, stochastic analysis, stochastic geometry and their applications, especially to mathematical finance and to data science. A ground-breaking tool in stochastic analysis over the past two decades has been the theory of rough paths, and a cornerstone of this theory is the so-called signature transform. I will be offering both theoretical and applied projects based around the rough paths and the signature.
Algebraic and birational geometry, mirror symmetry
I am interested in algebraic geometry, particularly higher dimensional birational geometry, minimal model theory, Fano varieties, and mirror symmetry
Data science, uncertainty quantification, computational mathematics, mathematics of Planet Earth
I'm interested in algorithms that help us to blend data and physical models to produce statistical forecasts of future evolution or estimates of unobserved quantities. I like to work on problems where the models come from discretisations of partial differential equations that require solution on high performance parallel computers, especially models arising in the study of Planet Earth. I'm the (co-)author of an introductory book "Probabilistic Forecasting and Bayesian Data Assimilation", and a lot of my student projects come from the challenges set out in that book.
Metamaterials, Waves, Acoustics, Scientific Computation
We aim to model, design and understand the next generation of designer materials and surfaces. In particular we study how to send light, or sound, to places of our choosing using small devices. Our research uses pieces of pure mathematics (group theory and topology), applied mathematics and scientific computation (modelling, efficient numerical simulation) to generate fast and accurate results. There are many real-world applications to cloaking, vibration control, energy harvesting, soundproofing and antennas.
Applied Statistics, Machine learning, statistical genetics, genetic epidemiology
I work in the field of statistical genetics that aims to understand how the genome is related to the development of complex diseases. Through my research I develop and apply statistical approaches for the analysis of the large datasets that come from different genomics experiments.
Machine learning, applied statistics, public health, deep learning
My research is on scalable methods and flexible models for spatiotemporal statistics and Bayesian machine learning, applied to public policy and social science. I’ve worked on application areas that include public health, crime, voting patterns, filter bubbles / echo chambers in media, the regulation of machine learning algorithms, and emotion.
Eva-Maria Graefe and Jessica Eastman
Quantum theory, mathematical physics, chaos, dynamics
The physical laws of the world of our daily experiences ("classical physics") are very different from those of quantum mechanics governing the behaviour of microscopic particles such as electrons and atoms. Given that the world is ultimately made up of these small "quantum" building blocks, this discrepancy in behaviour is a little puzzling. A focus of the work of our group is trying to understand the connections between quantum and classical physics better. We also work on certain open quantum systems, where particles can leave or enter the system of interest. We describe these using different models, such as complex energies and non-Hermitian Hamiltonians, as well as Lindblad equations. We like to think our ideas are useful in cold atoms, optics, materials, and other applications.
Computational mathematics, simulation science, finite element, ocean, atmosphere, climate
My team develops mathematical software which is used around the world to simulate physical phenomena, especially those of the natural world. Our work bridges the gap between Mathematica-style symbolic computation and high performance computing, enabling scientists to write maths, and have the simulation happen automatically. Our system, the Firedrake project, is used to simulate a huge range of processes, with a particular focus on natural phenomena such as the flow in the atmosphere and ocean. Research students working with us have the chance to push the boundaries of what is possible in mathematical simulation, and to make contributions that will have real impact on the work of many simulation scientists.
Geometric mechanics, fluid circulation, stochastic transport by fluid currents
The way we view hydrodynamics changed forever in 1966 when VI Arnold made his revolutionary discovery that the Euler equations for an ideal fluid represent geodesic motion on the manifold of smooth invertible maps acting on reference configuration of a fluid in the domain of flow. This revolutionary viewpoint brought infinite-dimensional geometry into the foundations of fluid dynamics. We are studying the implications of Arnold’s geometric viewpoint and making them stochastic, so we can use this viewpoint to estimate uncertainty in the predictions of climate, weather and ocean circulation. The way we view hydrodynamics changed forever in 1966 when VI Arnold made his revolutionary discovery that the Euler equations for an ideal fluid represent geodesic motion on the manifold of smooth invertible maps acting on reference configuration of a fluid in the domain of flow. This revolutionary viewpoint brought infinite-dimensional geometry into the foundations of fluid dynamics. We are studying the implications of Arnold’s geometric viewpoint and making them stochastic, so we can use this viewpoint to estimate uncertainty in the predictions of climate, weather and ocean circulation.
Gustav Holzegel and Thomas Johnson
General Relativity, Black Holes, Partial Differential Equations
Me and my group are interested in partial differential equations in a geometric context, mostly in connection with Einstein’s theory of general relativity. Topics include the study of the global properties of solutions to wave equations on black hole spacetimes, the stability of black hole spacetimes, unique continuation problems in connection with Einstein’s equations and many more. The research project will be chosen according to the interests of the student.
Key words: Stochastic differential equations, stochastic dynamics, Perturbations, Malliavin Calculus, stochastic processes on manifolds, limit theorems, stochastic flows, averaged dynamics And diffusion creation
Stochastic equations come from modelling randomness, they are related to differential operators, parabolic equations, semi-group theory, they pick up the geometry from its state space and from the interactions of the variables. The basic problems are : smooth dependence on the initial data, large time asymptotic and ergodicity, perturbation problems including complexity reduction and effective dynamics. Probabilistic representations are useful for handling some PDE problems. I have also worked on Malliavin Calculus, analysis of spaces of continuous paths, and on differential forms. I am also interested in techniques arising from regularity structure theory, and work with some rough paths.
Statistics; Bayesian inference; astrophysics
I am interested in both foundational statistical issues, primarily related to Bayesian inference and hypothesis testing, and the application of statistical methodology to data analysis problems in astrophysics.
Monte Carlo, Computational Statistics, Inference and Optimisation
My research focuses on developing Monte Carlo methodology for inference and optimisation problems with emphasis to particle methods. I am also interested in numerical methods and applications for a variety of problems such as: non-linear filtering, Stochastic optimal control, Parameter inference for general state space models, data assimilation and inverse problems.
Non-equilibrium statistical mechanics, biological physics, active matter, reaction-diffusion processes, field-theory.
Most of my group research the statistical mechanics of non-equilibrium phenomena in biological systems. We often use field-theoretic methods to develop a theoretical understanding of the problem at hand. Some problems studied are fairly abstract and general (say first-passage times), others have concrete applications biochemistry (say microtubule polymerisation). Numerical simulations very often guide analytical insight as they help us develop an intuition. UROP projects are normally co-supervised with a PhD student.
Geometric Analysis and Riemannian Geometry
I am interested in geometric flows, such as mean curvature flow and Ricci flow, as well as minimal surfaces, Willmore surfaces and optimal geometric inequalities.
Asymptotic analysis and singular perturbation theory, wave propagation, microhydrodynamics.
Our goal is to develop asymptotic models in order to address fundamental open problems arising in applied physics and engineering. Our work spans a broad spectrum of scientific interests including wave diffraction in micro-structured media (e.g., acoustic and photonic crystals and metamaterials, plasmonics) and microhydrodynamics (e.g., superhydrophobic surfaces, active suspensions, electrohydrodynamics). We focus our efforts on identifying and exploiting the singular limits in the problem, innovating and adapting classical singular perturbation techniques such as matched asymptotic expansions, WKB and ray methods, as well as homogenisation and other multiple-scale methods. This approach is very powerful when seeking a simple description of phenomena characterised by multiple length and time scales or near-singular effects (e.g., resonance). Such asymptotic analyses often furnish surprisingly simple asymptotic formulae (typically power laws and logarithms) capable of explaining numerical or experimental findings; in other cases the result is a reduced model that is easier to study numerically. Many of the problems we work on involve partial differential equations, with geometry playing a key role. For example, we have an ongoing funded project on optical (“plasmonic”) resonances of metallic nanoparticles; the key mathematical problem here is an eigenvalue problem, derived from a long-wavelength limit of Maxwell’s equations, determining the resonance frequencies of the particle as a function of the particle’s geometry.
Algebraic geometry, homological algebra
I'm interested in interactions between geometry, algebra, and theoretical physics.
Finite element methods, numerical weather prediction, atmospheric and oceanic physics
I develop the Gusto dynamical core toolkit which builds upon the Firedrake finite element library to provide algorithms for solving equations relevant to numerical weather prediction. These algorithms are the basis of the new Met Office weather and climate prediction model. Students working on this project would learn about geophysical fluid dynamics and the algorithms underlying weather and climate prediction, and would be able to contribute to the current effort to design efficient algorithms for the next Met Office model.
Mathematical Biology, Systems Biology
I am broadly interested in understanding what enables cells to grow, divide, transmit signals, process information and make decisions in spite of the inherent stochasticity in the underlying biochemical networks. Biological systems are too complex to be understood by studying their individual components alone. Systems-level study of biological networks demands a theoretical approach. Through the application of tools from physics, mathematics and bioinformatics, I study the temporal, spatial and stochastic dynamics of biochemical network models using analytical and computational methods. In my research, I strive to stay close to experimental data and we collaborate closely with biologists.
Partial differential equations, geometric analysis, general relativity, kinetic theory
I am interested in understanding the asymptotic behaviour of solutions of certain geometric hyperbolic and kinetic equations arising in mathematical physics, such as the Einstein equations of general relativity and the Boltzmann and Vlasov--Maxwell equations of kinetic theory. One current topic of very active research is the famous black hole stability problem in general relativity. A common theme in such problems is to exploit a quantitative understanding of geometric properties of the equations in order to prove analytic results. Research students will have the chance gain experience with the cutting edge techniques currently being employed to address these types of questions.
Algebraic geometry, moduli problems, derived categories, mirror symmetry
I work in bits of algebraic geometry related to theoretical physics, such as enumerative algebraic geometry, mirror symmetry, and categories of branes.
Deep learning, machine learning, dynamical systems, bifurcation theory, functional analysis, statistical learning theory, music information retrieval.
I am interested in the mathematical foundations of machine learning and deep learning algorithms, in particular understanding how these systems learn, and how mathematical tools can be applied to make these systems more efficient, accurate and generalisable. I am actively involved in growing the machine learning activities and capability within the mathematics department, and the course offerings to students. I am particularly interested in learning compressed representations from data, stochastic latent variable modelling and generative models such as the variational autoencoder and normalising flows. A specific area of interest is the application of these methods to problems in audio modelling, particularly music.
Applied mathematics, data science, reliability engineering, sales forecasting, process/condition monitoring
My research involves developing novel and applying existing mathematical methodologies to solve real-world problems. Examples include the application of time series models in forecasting sales units; dimensional reduction and machine learning techniques in fault detection; reliability analysis as a decision-making tool for Unmanned Aerial Vehicles using Graphic models; Maintenance process modelling and optimisations using simulations and the genetic algorithms. Research students will have the chance to leverage their mathematical modelling skills to tackle real-world engineering or market research problems.