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: Machine learning, applied statistics, public health, deep learning
Research area: 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.
Keywords: Algebraic geometry, moduli problems, derived categories, mirror symmetry
Research area: I work in bits of algebraic geometry related to theoretical physics, such as enumerative algebraic geometry, mirror symmetry, and categories of branes.
Keywords: Bayesian statistics, public health, HIV, phylogenetics
Research area: Biostatistical methods to estimate how infectious diseases spread.
Keywords: Non-equilibrium statistical mechanics, stochastic dynamics in life sciences, biomathematics, theoretical soft matter physics, fluid dynamics
Research area: 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.
Keywords: General relativity, black holes, kinetic theory, geometric analysis, partial differential equations.
Research area: 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.
Keywords: Biomathematics, Computational Biology, Systems Biology, Bioinformatics.
Research area: 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. The goal of my research is to identify the principles that govern the design and robust function of networks of interacting genes and proteins. Such design principles are ones that have been consistently chosen by evolution. 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. Experimental advances in high-throughput methods in molecular biology provide systems-level data, while single cell studies quantify variations in biomolecular levels and their corresponding phenotypic effects. Together, these data provide an excellent opportunity for system-level and quantitative studying of mechanism of robust cellular function and also phenotypic variability.
Keywords: Algebraic geometry, birational geometry.
Research area: My research area is birational geometry. More specifically, I am interested in the Minimal Model Program, in algebraic geometry in positive characteristic and in the birational geometry of foliations on complex varieties.
Keywords: stochastic differential equations, statistical mechanics, Markov Chain Monte Carlo, agent based modelling, optimal control for PDEs
Research Area: Mathematical statistical mechanics, computational statistical mechanics, analysis of algorithms for sampling and for optimisation, analysis of dynamical models exhibiting phase transitions, mathematical modelling in the social sciences, in particular models for opinion formation, optimal control for linear and nonlinear Fokker-Planck equations, numerical methods for linear and nonlinear Fokker-Planck equations
Keywords: Numerical analysis, scientific computing, numerical weather prediction, finite element methods, time parallel algorithms, data assimilation.
Research Area: My research in numerical analysis and scientific computing focusses on the design, analysis and implementation of numerical methods for weather forecasting, ocean modelling and climate simulation. Recently, I have been developing compatible finite element methods and time parallel algorithms for these applications. I am motivated by the geometric variational structure of fluid equations, which are also relevant for algorithms for exploring the geometric structure of shapes and images. I also develop numerical algorithms for probabilistic forecasting and data assimilation.
Keywords: Geometric Mechanics, Applied Mathematics, Geophysical Fluid Dynamics, Shape Analysis, Integrable Systems
Research Area: Stochastic PDEs for GFD and Nonlinear waves
Keywords: Computational mathematics, simulation science, finite element, ocean, atmosphere, climate
Research Area: 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.
Keywords: Asymptotic analysis and singular perturbation theory, wave propagation, microhydrodynamics.
Research Area: 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.
Keywords: Quantum theory, mathematical physics, chaos, dynamics
Research Area: 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.
Key words: Nonautonomous and random dynamical systems, bifurcation theory
Research Area: In contrast to classical dynamical systems, the time evolution of a nonautonomous or random dynamical system is influenced by an independent process (which reflects either changes in the rules governing the system or randomness available in the system). The importance of nonautonomous and random dynamical systems is illustrated by the fact that a significant number of real-world applications, ranging from climate, ecology to finance, are governed by time-dependent inputs or subjected to random perturbations, and the traditional mathematical theory fails to address dynamical changes in these contexts. My research group develops tools to understand qualitative properties of these systems, with main focus on bifurcation theory, in order to understand and characterise changes in the behaviour of the system when external parameters are varied.
Keywords: Formal proof verification, Lean theorem prover
Research Area: Me and a small but growing team of mathematicians and computer scientists from across the world are developing a mathematical database of computer-checked theorems using a computer program called the Lean Formal Proof Verification System. To get a feeling for what it's like try the [natural number game](http://wwwf.imperial.ac.uk/~buzzard/xena/natural_number_game/). I am particularly interested in formalising algebra, algebraic geometry and number theory. [here is a list](https://github.com/kbuzzard/xena/blob/master/many_maths_challenges.txt) of various things I would like to formalise in Lean. We are gamifying mathematics.
Keywords: Theoretical Physics, Quantum Superfluids, Topological Insulators
Research Area: Condensed Matter Theory, Quantum Mechanics
Keywords: Statistical cyber-security, changepoint analysis, dynamic networks, computational Bayesian inference, meta-analysis
Research Area: Development of statistical methods for anomaly detection in security applications, in collaboration with government, industry and US National labs. Particular focus on methods using changepoint detection, latent factor models for networks, combining evidence from weak signals.
Keywords: Biostatistics, network analysis, mitochondrial systems biology
Research Area: The `Systems & Signals’ group works on a broad range of topics by applying mathematics to the natural world, especially the biological sciences. For example, we use statistical methods to investigate single-cell RNA sequencing data and so uncover the biological origins of disease and ageing. We also use tools from network science to gain an understanding of biological pathways, specifically in mitochondria. Research projects will have a strong computational component but depending on the student’s interest they can be complemented by some analytical work.
Keywords: Logic, Model theory, Geometric stability theory
Research Area: Geometric stability theory aims to classify mathematical structures, according to logical properties, to create a “geography of mathematical structures." The idea is to show that whole groups of seemingly different mathematical structures share certain basic properties; usually combinatorial properties of sets defined by formulas in the chosen first order language. The real strength of modern model theory is in using what is essentially a combinatorial geography to reach geometric conclusions about areas of the geography (for example the existence of dimension). Research summer students would get the chance to learn about a specific area of this classification. The ultimate aim of a project would be to prove something new about either a specific area, or show where a specific structure fits into the geography.
Keywords: Metamaterials, Waves, Acoustics, Scientific Computation
Research Area: 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.
Keywords: Statistical signal and image processing, event data, point processes, bio-imaging, network monitoring
Research Area: I am broadly interested in the development of statistical methods for the analysis of signals and images. Recently, I have been particularly interested in the construction of statistical machinery for the analysis of event data; be it temporal, spatial or spatiotemporal. This includes change point detection, analysing correlation structures, and characterising self and mutually exciting behaviour. I am motivated by applications in the natural sciences and engineering, including network monitoring for cyber-security, quantitative analysis of super-resolution bio-imaging data, and detecting relationships in neural firing patterns.
Keywords: Topological data analysis; Algebraic statistics; Biomathematics; Applied algebraic geometry; Tropical Geometry
Research Area: I am a mathematical data scientist; I study random algebraic structures and randomness in algebraic settings. I leverage theory from algebraic topology and algebraic geometry to develop methodology to handle complex data structures. I have applied my methods in real biological settings.
Keywords: Spectral methods, orthogonal polynomials, differential equations, singular integral equations, Riemann–Hilbert problems, random matrix theory, applied complex analysis
Research Area: I work on spectral methods for solving differential equations, singular integral equations, and problems in applied complex analysis. These methods use orthogonal polynomials to construct sparse discretisations that are efficiently solvable even when there are millions of degrees of freedom. Lately, I’m particularly interested in constructing new families of orthogonal polynomials on algebraic curves and surfaces in 2D and 3D for solving partial differential equations on complicated geometries, and in equations involving fractional Laplacians.
Keywords: Interface of algebra, geometry, physics, Poisson and symplectic geometry, moduli spaces, homology, deformation theory.
Research Area: I work in various subjects in the interface of algebra, geometry, and physics, specifically: representations of groups and algebras, Poisson and symplectic geometry, moduli spaces, homology and deformation theory.
Keywords: Fluid dynamics, free-boundary problems, nonlinear waves, nonlinear PDEs, scientific computing
Research Area: I am interested in the study of partial differential equations (PDEs) arising from physicalproblems and in particular multi fluid flows that support waves at liquid-air andliquid-liquid interfaces. Such mathematical problems known as free-boundary problems since the wave shape evolves spatiotemporally and must be determinedas part of the complete solution (whether analytically, computationally or a combination of the two). I study suchproblems “holistically”, i.e. starting from their physical origin using mathematical modelling to extractreduced-dimension systems, to the analytical and computational study of such systems, and where possible use mathematical structures to compare with experiments.There is a wide spectrum of research problems that emerge with applications to manymodern technologies such as superhydrophobic surfaces and propulsion and mixing on the micro-scale.
Mathematically issues that emerge include, for example, spatiotemporal chaos and its control, nonlinear wavesand coherent structures, and singularity formation and blowup in PDEs; these are addressed using tools fromasymptotic analysis, applied dynamical systems, numerical analysis and scientific computation.
Keywords: Statisticall machine learning, sequential decision making, reinforcement learning, multi-armed bandits
Research Area: My research is in the field of statistical machine learning with a focus on sequential decision making problems and online learning. I am interested in problems where we receive data sequentially and use this data to learn to make better decisions. Some examples of these problems are multi-armed bandit and reinforcement learning problems which arise naturally in many settings such as web-advertising, product recommendation or healthcare.
Keywords: Applied Analysis, Partial Differential Equations, Collective Behaviour, Fluid Mechanics
Research Area: I am interested in understanding how emergence of swarms, traffic jams, consensuses, or opinions can be described using macroscopic models. This is done through analysis of systems of partial differential equations describing interacting groups of individuals macroscopically. These system are similar in their structure to classical equations of Mathematical Fluid Mechanics (compressible Euler and Navier-Stokes equations) but they also include terms corresponding to specific interactions between the agents like avoidance of collisions, or ability to choose the optimal route to reach a target.
Keywords: Monte Carlo methods, particle filters, Optimisation and Control
Research Area: I work on a variety of problems and numerical methods for challenging inference or optimisation. Some recent work is on parameter estimation in large scale agent based models, estimation and control for Hidden Markov models, control and inference of interacting stochastic differential equations.
Keywords: Data modelling, stochastic analysis, quantum computing, deep learning
Research Area: I am interested in developing algorithms and models to capture the random (some would say erratic) behaviour of financial markets. The tools that I am using range from stochastic analysis to machine learning and quantum computing. Many exciting projects can be developed either theoretically or numerically, depending on students’ preferences.
Keywords: Algebraic Geometry, Moduli Problems, Geometric Invariant Theory
Research Area: I work in algebraic geometry, mainly doing things with moduli spaces. At the moment I'm thinking about moduli spaces of curve singularities, del Pezzos surfaces, and sheaves. The main tool that I use is 'non-reductive geometric invariant theory', which allows us to quotient algebraic varieties by group actions.