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Research Project Topics


Keep in mind that there is considerable overlap between different categories listed below and many academic staff work across areas. We have given cross references in many cases, but you should browse the list carefully and talk with more than one member of staff.

You are also strongly encouraged to consult this list of permanent staff. The page of research areas may also be useful.

Algebra and Group Theory

(see also John Greenlees, Nikos Zygouras)

Adam Thomas can supervise projects in group theory, Lie theory and representation theory. Topics could include algebraic groups, Lie algebras, permutation groups or a related area mutually agreed upon.

Inna Capdeboscq can supervise projects in group theory.

Derek Holt is willing to supervise one (or at most two) research projects on the topic of Computational Group Theory. The project would involve some programming. Prospective students would have to contact him to discuss further details.

Diane Maclagan can supervise projects in commutative algebra. Contact her for details. (See also Algebraic Geometry and Combinatorics.)

Dmitriy Rumynin is interested in Algebra and Representation Theory. Prospective students would have to contact him to discuss further details.

Marco Schlichting can supervise projects in Homological/Homotopical Algebra (algebraic K-theory, homology of classical groups, derived categories, Quillen model categories), Algebra (quadratic forms and central simple algebras, projective modules, Milnor K-theory), Algebraic Geometry (algebraic cycles, (oriented) Chow groups, motivic cohomology, A1-homotopy theory) and Algebraic Topology (homotopy theory, topological K-theory)

Gareth Tracey can supervise projects in both finite and inifnite group theory, with particular focus on projects on finite simple groups; permutation groups; the Burnside problems; and problems in the theory of algebraic groups.

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Algebraic Geometry

(See also Marco Schlichting and Weiyi Zhang)

Christian Boehning is interested in algebraic geometry, representation and invariant theory, derived category methods in birational geometry, birational automorphism groups.

Gavin Brown can supervise projects in algebraic geometry, including the projective geometry of curves and surfaces and applications of Riemann-Roch.

Chunyi Li is interested in Algebraic Geometry.

Martin Lotz can supervise projects in effective, computational and numerical algebraic geometry.

Diane Maclagan can supervise projects in several different areas of algebraic geometry, including tropical geometry. Contact her for details. (See also Algebra and Combinatorics.)

Miles Reid has a variety of possible projects on offer related to modern research in algebraic geometry, and to MA4A5 Algebraic Geometry or MA426 Elliptic Curves. He can also propose problems related to advanced topics in Galois theory, commutative algebra and algebraic number theory for suitably motivated students.

Finite subgroups of SL(2, CC) and SL(3, CC)
Explicit generators, invariant theory. Work of Klein around 1870, but recently developed in many directions. Start from my preliminary chapter on cyclic quotient singularities. See for example my Bourbaki seminar or the references given on my McKay correspondence website.

Quaternions, octonions, special geometric structures and exceptional Lie groups
See any introductory text on quaternions (for example, Balazs Szendroi and Miles Reid, Geometry and Topology, Chapter 8), followed by John C Baez, The Octonions, 56 pages, preprint available from as math.RA/0105155, and John H. Conway and Derek A. Smith, On quaternions and octonions: their geometry, arithmetic, and symmetry, A K Peters, Ltd. 2003 IBSN 1568811349

Topology, projective geometry and enumerative geometry of Grassmann varieties and projective homogeneous spaces.

Applications of Riemann Roch on curves and surfaces
Graded rings, computer algebra calculations. Computer enumeration of graded rings of interest in algebraic geometry. Work of mine and my students. It contains lots of fairly simple minded combinatorial problems that lend themselves to computing. Start from my preliminary chapter on graded rings, then look at Gavin Brown's GRDB website.

Damiano Testa is interested in Algebraic geometry and Number Theory. Prospective students would have to contact him to discuss further details.

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Analysis/Applied Analysis/Partial Differential Equations

(See also Thomas Hudson, Shreyas Mandre, Peter Topping, Tim Sullivan, Simon Myerson, Maxwell Stolarski and Fluid Dynamics)

Keith Ball offers projects on 1) the Haar system in function spaces and 2) the structure of null sets in the plane.

David Bate is interested in geometric measure theory, real analysis and functional analysis.

Bertram Düring's research interests are in applied and computational partial differential equations, spanning modelling, analysis, numerical analysis and optimal control, with a particular interest in applications from finance and socio-economics.

Charles Elliott’s research is centred around the analysis of nonlinear partial differential equations and computational mathematics with applications (mathematical biology, material science, continuum mechanics, phase transitions etc) including numerical analysis and applied analysis. In particular, finite element methods, free boundary problems, geometric evolution equations and surface growth, two phase flow, cell motility, biomembranes and PDE optimisation. In the year 2022/23 he will be giving a year 4 module Variational Analysis and Evolution Equations. This module is a natural companion to Theory of PDEs, Advanced PDEs, Numerical Analysis of PDEs and Advanced Real Analysis as well as other modules in Analysis and Applied Mathematics. Lecture notes will b e available at the beginning of the academic year. Projects may be in any of applied analysis, numerical analysis, computation and modelling and topics include:- (A) Variational problems with point constraints involving inverse, control or cell biology applications (B) PDE methods applied to networks (C) Functions of bounded variation: applications and computation. (D) Time dependent function spaces and PDEs in evolving domains (E) Information propagation on graphs. Prospective students would have to contact him to discuss further details. Please consult his web page for more details of his research interests.

Josephine Evans is interested in analysis of PDEs, stochastic processes or functional inequalities related to many body systems (kinetic theory) coming from both physics and biology.

Susana Gomes is interested in partial differential equations, in particular modelling, control and inverse problems in applications such as pedestrian dynamics or fluid dynamics

András Máthé is interested in geometric measure theory, fractal geometry and real analysis (of combinatorial nature).

Filip Rindler research concerns singularities in nonlinear PDEs and the modern theory of the calculus of variations. In particular, he is interested in oscillation and concentration phenomena and what can be rigorously proved about their "shape". Applications include elasticity and elasto-plasticity theory.

Jose Rodrigo is interested primarily interested in nonlinear partial differential equations (fluid mechanics, reaction-diffusion systems with mass conservation, fractional diffusion, ...) but is also happy to supervise projects on some areas of harmonic analysis (following on from MA433 or MA4J0). Prospective students should contact him to discuss further details.

James Robinson's interests are in partial differential equations and fluid mechanics; infinite-dimensional dynamical systems; embeddings of finite-dimensional sets into Euclidean spaces. Prospective students should contact him to discuss further details.

Vedran Sohinger  is interested in nonlinear dispersive PDEs and their connections with mathematical physics and probability theory. He is happy to offer projects based on the module MA4J0 (Advanced Real Analysis) and on related modules.

Florian Theil is happy to offer projects with the topics

  1. Partial differential equations approximating particle systems
  2. Mathematical models for Lithium-ion batteries

Marie-Therese Wolfram  is interested in partial differential equations, mathematical modeling in socio-economic applications and the life sciences, numerical analysis.

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(See also Joel Moreira)

Agelos Georgakopoulos will not be offering projects in the academic year 2022/23.

Vadim Lozin offers projects in graph theory, combinatorics, discrete mathematics

Diane Maclagan offers several projects related to matroids. Contact her for details. (See also Algebra and Algebraic Geometry.)

Oleg Pikhurko is interested in combinatorics and graph theory, including their connections to other areas (such as analysis, logic, probability, etc). Another possible direction is to use computers to prove new combinatorial results using Razborov's flag algebras.

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Complexity Science and Mathematical Modelling and Materials

(see also Applied Analysis, Computational mathematics and numerical analysis, Fluid Dynamics, Mathematical Biology)

Ed Brambley works on applied mathematical modeling, particularly in aeroacoustics and metal forming. Details of potential projects can be found here.

Randa Herzallah interests are in complex systems modelling and control, dynamical systems, signal processing, data analytics and machine learning, uncertainty characterisation and consideration, quantum systems modelling and control.

Thomas Hudson is interested in connecting models of physical systems at different scales through asymptotic limits. To do so, he uses mathematical tools from the analysis of PDEs and the Calculus of Variations, such as homogenization, Gamma-convergence and Large Deviations. Some examples of project ideas include stochastic modelling and/or simulation of point defect diffusion from atomistic data, studying mean-field PDE models and/or stochastic models for dislocations, and studying equilibrium distributions of stars in globular clusters. Analysis projects related to all of these topics are also possible, and can be adjusted to suit the interests of the student.

Markus Kirkilionis's interests are in complex systems, mathematical biology, dynamic network models, numerical analysis, pattern formation, physiologically structured Population models, (monotone) dynamical systems. Prospective students should contact him to discuss further details.

Robert MacKay is on sabbatical for the 2022/23 academic year and will therefore not be offering projects except by special arrangement.

Ferran Brosa Planella research interests are in the broad area of industrial and applied mathematics, in particular heat and mass transfer, continuum mechanics, moving boundary problems, and dynamical systems. In his work, he combines analytical techniques and numerical techniques to develop and study mathematical models.

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Computational Mathematics and Numerical Analysis

(See also Bertram Düring, Charles Elliott, Tobias Grafke, Radu Cimpeanu, Tim Sullivan, and Marie-Therese Wolfram)

Dwight Barkley offers projects on modern approaches to turbulence and on nonlinear waves. The projects involve numerical simulations and concepts from dynamical systems and complexity science. Students must be comfortable with numerical computations and must able to program in C/C++/Fortran/Python. Please see the following to get a flavour of the work: here, here, here, here, here, and here. Several past projects have resulted in publications in scientific journals.

Andreas Dedner's interests are in numerical analysis and scientific computing, with particular emphasis in high-order methods for non-linear equations and applications in geophysical flows, radiation magnetohydrodynamics, and reaction-diffusion equations. Prospective students would have to contact him to discuss further details.

Martin Lotz can supervise projects on numerical linear and non-linear algebra, compressed sensing, and deep learning (theory and applications).

Björn Stinner works in the analysis and numerical analysis of partial differential equations. Specifically, he is interested in free-boundary problems, partial differential equations on manifolds, finite element methods, and phase field modelling in materials science, fluids, and cell biology. Projects may involve modelling, analysis, and computing or any combination of these.

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Differential Geometry and PDE

(See Applied Analysis above and areas such a Fluid Dynamics and Computational Mathematics below for further topics in PDEs).

Mario Micallef’s research interest lie in the area of partial differential equations and differential geometry. Prospective students should contact him to discuss further details.

Felix Schulze’s research interests lie in Geometric Analysis, more specifically geometric flows, minimal surfaces and their applications to geometric problems such as for example isoperimetric inequalities. Interested students should have a strong background in Differential Geometry and PDE as well as interest in Geometric Measure Theory. Interested students should contact Prof. Schulze by the end of week 11 of term 3. Interested MASt students are welcome to contact Prof. Schulze also after this date.

Maxwell Stolarski, joining the Mathematics Institute in September 2022, research interests lie in geometric analysis with an emphasis on singularity analysis of geometric flows. Interested students should have a background in differential geometry and PDE. Prospective students should contact him to discuss further details.

Peter Topping offers various options in geometric analysis. There will be many different challenging projects possible in this active area. The suitable student will have interest both in differential geometry and in PDE theory (and should normally be taking Advanced PDE). It might be some effort to understand enough to start the project, but it should leave you in a great position to start PhD research in the area. Interested students must contact Prof. Topping by the end of week 9 of term 3.

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Ergodic Theory and Dynamical Systems

(See also Robert MacKay, James Robinson, and David Wood)

Adam Epstein offers projects in Complex and Arithmetic Dynamical Systems, for example:

Arithmetic Questions in Holomorphic Dynamics: Consider the polynomials F_n(c) = p_c o ... o p_c(0) (n-fold self-composition) where p_c(z) = z^2 + c. It is known that all roots of F_n(c) are simple. The polynomial F_n splits into factors, some of which arise as F_m for smaller m dividing n: when such factors are divided out, the resulting polynomials are conjectured to be irreducible. Questions of this nature arise for other interesting families of rational maps, and little is known in general. Well-organised computer experimentation would be a good start. This would be an appropriate project for a student who has taken, or will be taking Algebraic Number Theory (or Galois Theory). Familiarity with basics from Complex Analysis and Dynamical Systems would also be useful.

He is also willing to supervise appropriate mutually agreed projects in set theory and logic.

Vassili Gelfreich’s research interests lie in the area of dynamical systems. Prospective students should contact him to discuss potential projects in more detail.

Oleg Kozlovski is interested in Dynamical systems, ergodic theory, mathematical physics, financial mathematics. Prospective students should contact him to discuss potential projects in more detail.

Ian Melbourne is interested in Ergodic theory and Dynamical systems, including probabilistic or stochastic aspects of deterministic dynamical systems. Prospective students should contact him to discuss potential projects in more detail.

Joel Moreira is interested in Ergodic theory and applications to combinatorics and number theory and is willing to supervise projects in these areas.

Richard Sharp is interested in Ergodic Theory and its applications to other areas of pure mathematics. Prospective students would have to see him to discuss further details.

John Smillie is interested in translation surfaces and complex dynamics in higher dimensions.

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Fluid Dynamics

(See also Dwight Barkley, Tobias Grafke and the Applied Analysis and Computational Mathematics sections above)

Thomasina Ball is interested in mathematical modelling of fluid dynamical phenomena motivated by observations of the natural world. The applications predominantly originate from geophysical flows at both large- and small-scales but can also have significant relevance to many flows found in industrial processes. Specific topics include gravity-driven flows, non-Newtonian rheologies, Hele-Shaw flows, instabilities (Saffman-Taylor instability, fracture instability of soft gels, buckling instabilities due to rheology contrasts). Projects are likely to include a range of theoretical and computational modelling of PDEs. An interest in modules such as MA3D1 Fluid Dynamics and MA4L0 Advanced Topics in Fluids is useful.

Radu Cimpeanu supervises projects at the intersection between mathematical modelling, asymptotic analysis, solutions for ordinary/partial differential equations and high performance computing (including computational linear algebra and large scale system solvers). Specific topics in fluid mechanics include interfacial flows (modelling, analysis, simulation and applications of drops, bubbles and liquid films), rheological flows (from chocolate to de-icing fluids) and novel mathematical models for the cultivated meat industry. Some projects are likely to include multi-physics aspects such as acoustics, heat transfer and electromagnetism. From a mathematical standpoint, there are exciting opportunities to combine classical (continuum) modelling with discrete and data scientific streams, as well as integrating approaches such as control theory or hybrid modelling into areas previously unexplored with such tools. More generally, work on any of the above will likely involve a mixture of analytical and computational techniques (which can be tailored depending on your own interests and what skillset you wish to develop), as well as the interplay between them.

Shreyas Mandre's research topics are based on application of PDEs (generally arising from continuum considerations) to mathematical modelling of real-world problems. The applications usually arise from curiosity about the natural world and range from biomechanics (e.g. human foot, fish fins and embryos, bird flight and fish swimming) to interfacial fluid mechanics (e.g. surfactant boats, soap films and bubbles), environment (e.g. life of submerged vegetation). More recently, I have been dabbling in understanding how natural selection leads to form and function in biology. If you have a curiousity-driven mathematical modelling problem that you wish to investigate, I welcome them. For inquiring about specific projects, I encourage students to get in touch over email with a brief background and expression of the topic that most interests them. Having taken MA3D1: Fluid Dynamics is an additional advantage. For more information, see my website.

James Sprittles offers projects on the mathematical modelling and computational simulation of fluid mechanical phenomena, particularly those driven by complex interfacial effects which are prevalent in the emerging field of micro/nanofluidics. These microflows often require new modelling approaches, involving both continuum and particle-based methods, coupled to efficient computational techniques and are currently an area of intensive research interest. Possible projects include (but are not limited to):

Gas Dynamics in Free-Surface Flows: Thin films of gas often have a huge influence on the dynamics of liquid volumes, e.g. when drops impact solids, but at present are lacking an accurate theoretical description (in fact, often their influence is ignored). The problem is that their dimension is such that classical continuum mechanics fails and kinetic theory governed by the Boltzmann equation is required. This project will involve developing new models for this class of flows, with particular attention applied to the coupling of kinetic theory with continuum mechanics, and exploiting scale-separation to make these models computationally tractable.

Mathematical Modelling of 3D Printing:  This project will focus on the creation of ‘printed electronics’, where metallic nanoparticles are suspended within a liquid microdrop, deposited upon a (also printed) polymer substrate and then form (e.g.) conductive tracks when the liquid evaporates. This project will develop a mathematical model that captures the drop dynamics and offers unique insight and understanding that cannot be obtained from experiments alone. Classical applied mathematics techniques will be deployed to provide simple analyses alongside more direct computational approaches.

Thermal Fluctuations in Nanoscale Flows: This project will focus on the dynamics of liquids at the nanoscale, where thermal fluctuations drive nanowaves on interfaces (e.g. a water-air one). From a mathematical perspective, this means solving stochastic PDEs to understand the dynamics of small liquid volumes, bringing with it new and exciting computational/theoretical challenges.
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Geometry, Topology and Geometric Group Theory

(see also Marco Schlichting)

Brian Bowditch offers project in hyperbolic geometry, low-dimensional topology, geometric group theory

John Greenlees is interested in algebraic topology, homotopy theory, commutative algebra and representation theory.

Robert Kropholler would be interested in supervising projects in geometric group theory and low-dimensional topology. Students should contact him to discuss potential projects in more detail.

Martin Lotz can supervise projects in topological data analysis and persistent homology.

Saul Schleimer is interested in geometric topology, group theory, and computation. Students should contact him to discuss potential projects in more detail.

Karen Vogtmann is interested in geometric group theory, low-dimensional topology, cohomology of groups.

Weiyi Zhang is interested in symplectic topology, complex geometry and low dimensional topology.

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Mathematical Biology, Epidemiology, Ecology and Evolution

(see also Charles Elliott, Shreyas Mandre and Björn Stinner)

Nigel Burroughs applies mathematical and statistical methods to biological systems. He is primarily interested in understanding the mechanics and mechanisms of cells, in effect how they work and achieve the spectacular range of behaviours that are observed. He uses a combination of model development, mathematical analysis (eg dynamical systems methods), simulation and statistical computation (Markov chain Monte Carlo methods), with both deterministic and stochastic models. The projects below should give you some idea of what projects I can suggest, but these are not definitive. 1,2 examine the mechanics of cell division, from the duplication and separation of the chromosomes to the mechanical separation of cells into two daughter cells, 3,4 are on microtubule modelling and 5 is on cancer modelling.

1. Modelling chromosome dynamics. Chromosomes are duplicated but then have to be divided so that each daughter cell gets one and only one copy. This is achieved by 'holding' the pairs at the cell equator until all pairs are in position. Mathematically we can think of this system as two particles in a 1D box connected by a spring and pushed/pulled separately from the ends. The surprising observation is that chromosomes oscillate from side to side across the cell equator. How this occurs is poorly understood, but this is believed to involve a tension sensor that effectively acts as a means of communication between the two chromosomes. This project would involve analysis of deterministic and stochastic models of oscillation of paired chromosomes (dynamical systems techniques and simulation), examining different mechanisms of feedback/communication that give rise to oscillations. Deterministic models are non-smooth dynamical systems, whilst stochastic models are based on the Ornstein-Uhlenbeck process (or more generally switching diffusions).

2. Statistical computation (Markov chain Monte Carlo, MCMC) analysis of chromosome oscillations. See above (1) for system description. Here you would use our large database of chromosome trajectories (1000s of trajectories) to understand the statistical structure of chromosome oscillations, fitting saw-tooth like oscillatory profiles. Experience with/exposure to MCMC, hidden Markov chains and probability theory is essential.

3. Modelling microtubule bending. Microtubules are biological polymers that polymerise into tubes and exhibit elastic bending properties. Here you will examine models and simulations of microtubule bending in fluid flow and binding of proteins that enhance the curvature.

4. Reverse engineering microtubules. Microtubules are biological polymers; they polymerise into tubes and exhibit what is called dynamical instability, switching from periods of growth to decay. What causes this (stochastic) switching is unknown but likely an emergent property of the microtubule lattice. The idea of this project would be to construct models and an MCMC algorithm to fit those models of lattice dynamics to data to examine the degree to which such data can inform on the underlying processes. Experience with MCMC and probability theory is essential.

5. Cancer dynamics. What is the optimal cancer therapy for a given cancer? By using mathematical models of tumours and drug activity, control theory can be used to optimise the therapy - for instance both dose scheduling and combinations of drugs. Fundamental to this is what should be optimised, in a mathematical framework what objective function should be used. This project would involve examining various growth models of tumours and normal tissue. Both deterministic (ODEs, PDEs) and stochastic (branching processes, stochastic logistic model) can be looked at.

Prospective students should contact him to discuss further details. [Experience with MCMC means acquaintance with Gibbs and Metropolis-Hastings algorithms and their use in simulating posterior probability distributions. Experience coding an algorithm for a simple problem would be an advantage].

Emma Davis uses mathematical models to investigate the emergence and spread of infectious diseases through human populations, with a particular focus on tropical diseases that affect populations with poor healthcare access. Mathematical techniques could feed into a project include stochastic / branching processes, systems of differential equations, statistical analysis and computer simulations. Students would benefit from taking "MA4E7: Population Dynamics" and/or "MA4M1: Epidemiology by Example" either before or during the project, due to a substantial overlap in methodology and theory. Example ideas for projects include: 1. Developing a meta-population model (representing interacting populations) to investigate the impact of cross-border migration on neglected tropical disease transmission in scenarios where neighbouring areas or countries may have contrasting public health strategies or resources. 2. Investigating how assumptions around clustering (or 'super-spreading') of SARS-CoV-2 transmission can impact model forecasts. 3. Using a model of SARS-CoV-2 lateral flow test positivity over time since exposure to derive transmission networks by determining the likely direction of transmission between positive contacts.

Louise Dyson works on mathematical modelling of biological systems, especially the epidemiology of neglected tropical diseases and the analysis of biological systems in which noise plays an important role.

Ed Hill is a mathematical epidemiologist and infectious disease modeller. I like to offer projects in public and veterinary health policy, zoonoses and interdisciplinary problems in epidemiology that involve the dynamics of behaviour. Approaches involve the application of mathematical and computational methods, including the development of models, parameter inference and the evaluation of interventions via computational simulation. Students wishing to undertake a project in these topics would benefit from taking "MA4E7: Population Dynamics" and/or "MA4M1: Epidemiology by Example" either before or during the project. A list of example projects may be found hereLink opens in a new window.

Matt Keeling offers projects in the areas of Epidemiology,Ecology or Evolution. Epidemiology (the study of infectious diseases and their spread in populations), Ecology (the study of animal/plant populations and their environment), Epidemiology (the study of disease spread and control) and Evolution (the study of the long-term dynamics of populations) present a wide variety of interesting problems that require a mathematical approach. There are a vast number of problems and approaches that could be studied, ranging from model-development, to computer simulation, to statistical analysis. If you've attended (or planning to attend) MA4E7: Population Dynamics, that would be a distinct advantage although not essential. I like to offer projects that show how the mathematical techniques you've learnt can be applied to real questions to obtain useful or meaningful insights. If you're interest, its probably easiest to come and talk with me, and together we can determine a specific project that matches your interests and skills.
Recent projects have included the evolution and competition of influenza strains, diffusion approximations to disease spread on networks, optimal control of spatial epidemics, Nicholson-Bailey lattice models.

David Rand. His main research interest is Systems Biology, particularly understanding the design principles of regulatory and signalling systems in cells. Prospective students would have to contact him to discuss further details.

Kat Rock is a mathematical epidemiologist interested in population-level disease dynamics. In particular her focus is on transmission of vector-borne (insect-transmitted) infections, infections in low- or middle-income countries, and bringing together dynamic transmission modelling and health economic evaluation. She teaches the MA4M1 “Epidemiology by Example” course and it would be recommended to have taken this and/or “Population Dynamics” (MA4E7) already or alongside an R-project.

Robin Thompson develops mathematical models of infectious disease outbreaks and uses them to guide control measures. This includes developing stochastic models, parameterising them by applying statistical inference techniques, and then using them to test out different possible interventions. Recent research projects have involved the use of stochastic models early in outbreaks (to estimate the risk that initial cases will lead to a major epidemic as opposed to fading out as a minor outbreak) and late in outbreaks (to determine when outbreaks have finished so that control measures can be removed).

Mike Tildesley works on the development of mathematical models to simulate the spread of livestock and zoonotic diseases.

Dave Wood would be willing to discuss possible topics given the information below. His main interests for projects include ecological type modelling, investigating systems with symmetry (including but not limited to applications to arthropod locomotion) and applications of mathematics in industry. Systems with symmetry could be theoretical or applied, but would concentrate on using methods from MA3J3 Bifurcations, Catastrophes and Symmetry, so this may interest students who have taken that. The industrial applications could cover a broad range of mathematical disciplines and be a survey of some problems that have already been studied or a look at a new problem involving original research. Ecological modelling I would be happy to consider any suitable application that a student feels passionate about (see below for a couple of past such projects).

Previous titles he has supervised include: “Coupled cell networks, bifurcations and symmetry”, “The effects of tuna fishing on dolphin populations”, “Symmetry in coupled cells and neuronal networks”, “Applications of maths in industry”, “Discrete maths in industry”, “Modelling the Future of the Hawaiian Honeycreeper: An Ecological and Epidemiological problem” (the latter of which led to a successful PhD).

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Mathematical Physics, Molecular Dynamics, and Statistical Mechanics

(see also Vedran Sohinger, Nikos Zygouras, and Stochastic Analysis and Probability)

Stefan Adams offers project on large deviation theory, probability theory, Brownian motions, statistical mechanics, gradient models, multiscale systems.

Siri Chongchitnan offers projects on cosmology and theoretical astrophysics. Potential topics include: cosmological inflation, primordial gravitational waves, large-scale structures and Bayesian methods in cosmology. These projects will involve extensive use of Python. Background knowledge of cosmology or astrophysics, whilst ideal, is not necessary. An ideal student would be someone who is deeply curious about the nature and the origin of the Universe. Further enquiries are welcome.

Daniel Ueltschi's interests are in analysis, probability theory, statistical mechanics, mathematical physics. Prospective students would have to contact him to discuss further details.

Roger Tribe offers projects on determinantal point processes - randomly arranged points whose distribution is characterized via determinants. (See Terrence Tao blog on Other projects available: random matrices/interacting particle systems/large deviations.

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Number theory

(See also Adam Epstein, Joel Moreira)

Sam Chow supervises projects in analytic number theory, diophantine equations, and diophantine approximation.

Adam Harper is interested in analytic and probabilistic number theory, and would be willing to supervise projects in these areas. The basic goal is to use methods of analysis (real, complex, Fourier, probability) to understand the distribution of number theoretic objects, like prime numbers. A project would likely involve working through a couple of papers from the (fairly) recent research literature, and trying to understand and synthesise them.

Pak-Hin Lee is primarily interested in algebraic number theory, particularly topics concerning modular forms, elliptic curves and Iwasawa theory. He can also supervise projects related to the Langlands program, such as automorphic forms, Eisenstein series and L-functions; these are suitable, but not limited, to students with a strong background in analysis.

David Loeffler would be willing to supervise projects in various areas of number theory, with possible topics including aspects of p-adic numbers, modular forms, and elliptic curves. See here for a list of previous topics.

Minhyong Kim works on arithmetic geometry, the study of mathematical structures incorporating a mixture of arithmetic and geometric properties. He is also interested in topology and mathematical physics, especially the foundational aspects of quantum field theory. He is willing to supervise a project in any combination of these areas.

Simon Myerson offers projects in analytic number theory and harmonic analysis. The fourth year course called "analytic number theory" is actually about one rather special part of analytic number theory and not very relevant to what I usually do. An interest in number theory and analysis is a must. There are applications to PDEs which would interest students talking MA4J0, although they are unlikely to be the focus of the project. Projects close to algebraic number theory would also be possible if you have an idea in mind.

Samir Siksek is interested in number theory and diophantine equations. Prospective students would have to contact him to discuss further details.
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Stochastic Analysis and Probability

(see also Josephine Evans, Adam Harper, Ian Melbourne, Oleg Zaboronski, as well as the Complexity Science and Mathematical Physics)

Tobias Grafke is interested in numerical methods and mathematical tools to analyse stochastic systems. Applications include fluid dynamics and turbulence, atmosphere and ocean dynamics, and biological and chemical systems.

Tim Sullivan offers projects in uncertainty quantification and inverse problems, understood as the meeting point of numerical analysis, applied probability and statistics, and scientific computation. He also has interests in data science and machine learning.

Nikos Zygouras: I am working in the area of probability theory and I am interested in studying phenomena that arise from statistical mechanics
of disordered systems (random polymers, motion in random media, stochastic growth etc) and stochastic analysis (SPDEs). I am also interested in the connections to random matrices, algebraic combinatorics, representation theory.
I am happy to supervise fourth year projects focused on two topics
  1. INTEGRABLE PROBABILITY : this is a rather new area within probability which explores remarkable connections between probability and algebraic structures (representation theory, algebraic combinatorics etc). A motivation for this development has been the understanding of the so called Kardar-Parisi-Zhang universality, which governs fluctuations of stochastic growth model (colonies of bacteria, spread of fluid in porous media etc). Remarkably, these fluctuation are not governed by the standard central limit theorem but rather from distributions related to random matrix theory. Some favour on this topic is provided by these notes
  2. SCALING LIMITS OF DISORDERED SYSTEMS AND SPDEs: Stochastic PDEs (SPDEs) have been proposed as continuum scaling limits of statistical mechanics models. However, both the well posedness of the SPDEs and the convergence of the discrete models to these are very challenging problems. The first problem has been recently largely settled below the so-called "critical dimension”) through the theory of Regularity Structures (by Hairer) or Paracontrollled Districutions (by Gubinelli-Imkelle-Perkowski). At the critical dimension things are much more challenging. This project can spin around the above theories, some exploration of critical dimensional statistical mechanics and/or scaling limits. Some flavour of the topic can be obtained in these notes
Oleg Zaboronski offers the following four projects.

Title: Metastability and large deviations for system of SDE's.

Description: An interesting phenomenon has been observed recently for systems of differential equations inspired by hydrodynamics: the addition of noise to a system of ODE's with a single fixed point leads to the emergence of multiple (quasi) fixed points. The aim of the project is to study classes of such ODE's and possibly apply the findings to the study of metastability in turbulence.

Key words and phrases: metastability; stochastic differential equations; time scale separation; large deviations; Wentzell-Freidlin theory; instanton trajectory; Fredholm determinants; Szego's theorems

Title: Matrix valued Brownian motions

Description: Random matrix theory is a multi-disciplinary subject possessing an intrinsic mathematical beauty and having applications to a range of applied scinces from string theory to data science and statistics. A dynamical version of random matrix theory is matrix valued Brownian motion. For the Hermitian matrices this gives rise to such a classical stochastic process as Dyson Brownian motion. The aim of the project is to study random matrix evolution for the non-Hermitian case.

Key words and phrases: matrix-valued Brownian motion; Berezin caclulus; supersymmetry; determinant; pfaffian; stochastic differential equations; point processes

Title: Markov dualities and interacting particle systems in one dimension

Description: More often than not, Markov interacting particle systems in one dimensions exhibit strong fluctuations which render their approximate description using differential equations ('mean field theory') useless. One of the methods of analysing such systems is based on Markov duality which allows to extract at least partial information about the system from its dual. But how do you find these dualities? The aim of the project is to learn about Markov dualities and investigate the dualities for certain classes of strongly fluctuating interacting particle systems.

Key words and phrases: Markov processes in continuous time, interacting particle systems, Markov duality; Hecke algebras; Yang-Baxter equation; R-matrix; determinatal point processes; pfaffian point processes

Title: Moment factorisation for the stochastic heat equation.

Description: It has been discovered recently, that the exponential moments of the solution to the stochastic partial differential heat equation satisfy classical integrable PDE's such as Kadomtsev-Petviashvili equations. The discovery came from the analysis of exact formulae for the exponential moments available for certain initial conditions. One of the consequences is that certain moments for the stochastic heat equations must factor into the linear combinations of products of the lower order moments. The aim of the project is to investigate the factorisation phenomenon from the point of view of stochastic analysis studying the solutions to the stochastic heat equation itself.

Keywords: Stochastic heat equation: stochastic processes, Ito's calculus; KPZ equation; KP integrable hierarchy; Fredholm determinants; intermittency; moment problem

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