# Supervisors

## 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. Students with an interest in Geometry and Number Theory can also contact anyone on the LSGNT academic staff list at Imperial, UCL, or King's.**

### Barnett, Dr Ryan

Keywords:

Theoretical Physics, Quantum Superfluids, Topological Insulators

Research Area:

Condensed Matter Theory, Quantum Mechanics

### Battey, Dr Heather

Keywords:

Statistical theory and applied probability motivated by the medical and physical sciences. Particularly: conditional and high-dimensional inference; inferential separation and evasion of nuisance parameters; comparisons and contradictions between alternative modes of inference; probabilistic behaviour of eigenvectors and eigenvalues of structured random matrices.

Research Area:

Much of my research is about calibrated inference for key quantities of interest, like the effect of a drug or treatment, in the presence of a large number of nuisance parameters. The latter are aspects of no direct subject-matter concern, but that are needed to complete the idealised representation of the physical, biological or sociological system. Large numbers of them arise naturally when one wishes to limit the strength of modelling assumptions in the equations describing the data generating process.

### Bertrand, Dr Thibault

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.

### Bodenham, Dr Dean

Keywords:

Changepoint detection; anomaly detection; outlier detection; two-sample testing; computational statistics

Research Area:

My main research interests are in the areas of two-sample testing and changepoint detection. Primarily, I am interested in developing methods that are both nonparametric and computationally efficient, and so can be deployed on large-scale datasets or in a streaming data scenario. A related research interest is outlier detection, which is sometimes called anomaly detection, and involves identifying individual, or groups of individual, anomalous data points. Several real-world applications for these topics include finance, cybersecurity, meteorology and biology, but my main motivation is to develop methods that can be deployed for medical applications.

### Bravi, Dr Barbara

Biomathematics, Stochastic processes, Statistical and Machine Learning

Research Area:

My research concerns methods to reconstruct the dynamics of biological networks and to predict

### Bryn, Dr Davies

Applied and Numerical Analysis, Waves, PDEs, Metamaterials, Aymptotics

Research Area:

I can supervise a range of projects related to metamaterials and wave propagation in complex media. The overarching aim of our work is to design efficient and robust waveguides for use in wave control devices (such as antennas, sound proofing, energy harvesting, sensing). In practice, any such work will involve analysing solutions of differential equations (most likely using asymptotic techniques), performing simulations in Matlab or Python and developing theory to explain our observations (e.g. using ideas from topology or group theory). I have an assortment of projects in mind, which can be adjusted according to a student's interests and expertise. These projects are all related to current lines of research in our group, so the student would be contributing to active research problems.

### Buzzard, Prof Kevin

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. I am particularly interested in formalising algebra, algebraic geometry and number theory. We are gamifying mathematics.

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.

Cheraghi, Dr Davoud

Keywords:

Complex Analysis, Holomorphic Dynamics, Mandelbrot and Julia sets, Renormalisation

Research Area:

I am interested in the interface of analysis, geometry, and combinatorics. Specifically, I work in the area of the dynamics of holomorphic maps, related to the Mandelbrot, Julia and Fatou sets. In general one aims to explain the orbits of points generate by successively applying a holomorphic map. Explaining come in many forms, such as topological, geometric, or probabilistic features of orbits. There is a wide range of interesting problems, and techniques, such as renormalisation methods employed in the area.

### Cohen, Dr Ed

Keywords:

Statistical signal and image processing, event data, point processes, bio-imaging, network monitoring

Research Area:

I am interested in developing new statistical methods for analysing signals and images, with a particular interest in event data (temporal, spatial or spatio-temporal). I am motivated by questions arising in engineering and the natural sciences, for example, detecting if a pair of neurons are firing coherently, looking for anomalous correlations in computer networks, or determining the distribution of molecules on 3D cell membranes.

### Cotter, Prof Colin & Hamdan, Dr Abdalaziz

Keywords:

Prof Colin Cotter: Numerical analysis, scientific computing, numerical weather prediction, finite element methods, time parallel algorithms, data assimilation.

Dr Abdalaziz Hamdan: Computational applied mathematics, Parallel-in-time methods, Advanced finite-element methods, Saddle point systems.

Research Area:

Prof Colin Cotter: 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.

Dr Abdalaziz Hamdan: I am interested in developing finite-element methods for PDEs and preconditioners for the arising systems. These days, I am working on parallel-in-time algorithms for PDEs that simulate the formation of sedimentary rocks over geological time.

### Craster, Prof Richard

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.

### Davies, Bryn

Keywords:

Metamaterials, Waves, Acoustics, Scientific Computation

Research Area:

We are a vibrant group of applied mathematicians working on modelling wave propagation in complex and exotic media. We are particularly interested in electronic and acoustic systems which have extraordinary properties such as negative effective mass or the ability to act as cloaking devices. We will work on a project that will give the student an opportunity to learn new techniques while contributing to cutting-edge research. The work will involve analysing solutions of differential equations (most likely using asymptotic techniques), performing simulations in Matlab or Python and developing theory to explain our observations (often using ideas from topology or group theory).

### Ghazouani, Dr Selim

Keywords:

Geometry, dynamical systems

Research Area:

I work at the crossroads between geometry and dynamical systems. A question of particular interest to me is the following: if I pick a physical system "at random", will it be chaotic (it is when its long-time evolution is very hard to predict, like the weather) or well-behaved (things are very accurately predictable, like the motion of planets in the Solar System)?

This is a very difficult question, and I'm working on low-dimensional models where methods from Riemannian geometry and algebraic geometry can be applied to get some valuable insight into the mechanisms driving chaotic or well-behaved dynamics.

### Graefe, Dr Eva-Maria& Eastman, Dr Jessica

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.

### Heard, Prof Nick

Keywords:

Dynamic networks, Bayesian changepoint analysis, Bayesian cluster analysis, meta-analysis, statistical cyber-security.

Research Area:

My main research interests are in finding structure in large, dynamic graphs such as enterprise computer networks, typically through latent factor models. Inference tasks include identifying clusters of similar nodes and performing outlier/anomaly detection. These ideas have clear applications in cyber-security contexts but can also be applied in a wide variety of other domains. I’m also interested in finding temporal changes in network data streams, using Bayesian changepoint models or anomaly detection techniques. I also have interests in meta-analysis, combining evidence from typically weak signals from a large number of data streams to assess overall significance levels; this is also of direct importance in cyber-security problems.

### Holm, Prof Darryl

Keywords:

Geometric Mechanics, Applied Mathematics, Geophysical Fluid Dynamics, Shape Analysis, Integrable Systems

Research Area:

Stochastic PDEs for GFD and Nonlinear waves

### Jacquier, Dr Jack

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.

### Jones, Prof. Nick

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.

### Kestner, Dr Charlotte

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.

### Monod, Dr Anthea

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.

### Olver, Dr Sheehan

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.

### Papageorgiou, Prof Demetrios

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 physical problems and in particular multi fluid flows that support waves at liquid-air and liquid-liquid interfaces. Such mathematical problems known as free-boundary problems since the wave shape evolves spatiotemporally and must be determined as part of the complete solution (whether analytically, computationally or a combination of the two). I study such problems “holistically”, i.e. starting from their physical origin using mathematical modelling to extract reduced-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 many modern 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 waves and coherent structures, and singularity formation and blow-up in PDEs; these are addressed using tools from asymptotic analysis, applied dynamical systems, numerical analysis and scientific computation.

### Papatsouma, Dr Ioanna

Keywords:

Clustering, mixed-type data, robust clustering, clustering validation

Research Area:

My research interests lie broadly in the area of clustering. Many statistical applications are carried out on mixed-type data, that is, data consisting of both continuous and categorical variables. In medical research, for instance, data sets typically include gender, and smoking status (usually categorical variables) together with other measures, such as blood pressure and blood glucose (usually continuous variables). Such heterogeneity demands ways to guide researchers and practitioners in choosing appropriate clustering approaches that will identify distinct groups of individuals and generate hypotheses. Common problems in clustering mixed-type data concern the choice of validation indices for estimating the quality of partitions produced by clustering algorithms and for determining the number of clusters, the use of appropriate distances for mixed-type data, and complexities introduced by high dimensionality, noise, outliers and missing values.

### Pavliotis, Prof Grigorios

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

### Pike-Burke, Dr Ciara

Keywords:

Statistical 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.

### Pruessner, Dr Gunnar

Keywords:

Active matter, field theory, entropy production, biological physics

Research Area:

Active matter is a collection of entities that transform some form of fuel from the environment into mechanical action. Typically this action is self-propulsion, but it can be more subtle. The resulting phenomena are often strikingly different to what we are used to from equilibrium. In our group we use field theoretic methods to characterise such active matter. While somewhat cumbersome to get started with, once a phenomenon has been cast in the language of field theory, it can be investigated perturbatively in a systematic way, using diagrams to help our intuition and our calculations. We often use numerics to guide our research. Applications range from fundamental technical questions to morphogenesis and flocking and swarming.

**Rasmussen, Dr Martin**

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.

### Ratmann, Dr Oliver & Blenkinsop, Dr Alexandra

Keywords:

Applied Bayesian statistics, machine learning, public health, phylogenetics, infectious diseases

Research area:

The work in my group focuses on applied Bayesian modelling for Public Good. We develop flexible, robust, and computationally scalable models and methods, and apply these tools in applied fields of phylogenetics, infectious disease dynamics, and social science. I am particularly interested in novel approaches that harness information in viral deep sequence data, mobile phone mobility data, and time-resolved patient data to characterise the spread of infectious diseases, and to guide public health interventions.

### Rodriguez, Dr Pierre-Francois

Keywords:

Probability, Analysis, Critical Phenomena, Renormalization

Research Area:

I work on various problems involving randomness, often concerning models (for instance: percolation) that display phase transitions. For many of these systems, physicists predict a rich phenomenology near the critical point, involving random fractal geometric objects that exhibit a universal scaling behavior.

**Salvi, Dr Cristopher**

Keywords:

Rough Analysis, Deep Learning, Kernel Methods, Signal Processing.

Research Area:

My research interests are in the areas of rough path theory, deep learning, kernel methods and signal processing. Generally speaking, I develop machine learning algorithms for processing irregular, high-dimensional time series in continuous time. I am also interested in the interplay between neural networks and differential equations.

### Sanna Passino, Dr Francesco

Keywords:

dynamic networks, clustering, statistical cyber-security

Research Area:

My main research interests are broadly based on statistical analysis of dynamic networks. In my work, I enjoy exploring an array of different statistical techniques, adapted and extended to dynamic network modelling, such as latent variable models and model-based clustering. In recent years, I have also developed an interest for statistical analysis of event-time data, topic modelling, Bayesian non-parametric methods, and recommender systems. My research has been mainly applied to statistical cyber-security problems, but also to social networks, music streaming services, and bike sharing systems.

### Schedler, Dr Travis

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.

### Schnitzer, Dr Ory

Keywords:

Asymptotic Analysis, Singular Perturbation Theory, Low-Reynolds Number Hydrodynamics, Transport Phenomena, Wave Motion.

Research Area:

I use mathematical modelling and asymptotic methods to illuminate fundamental problems in fluid dynamics, transport phenomena and wave motion. A current focus is problems where the coupling between inertialess (small-scale) flows and other physics gives rise to instabilities and spontaneous symmetry-breaking dynamics. In particular, I'm currently studying the spontaneous mobility of levitated Leidenfrost drops, the memory-mediated spontaneous dynamics of chemically active drops and the electrohydrodynamic Quincke rotation of droplets in strong electric fields.

### Shahrezaei, Dr Vahid

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.

### Taylor, Dr Martin

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.

### Thomas, Prof Richard Thomas

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.

### Zatorska, Dr Ewelina

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.