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.
Algebra and Group Theory
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 see him to discuss further details.
Dmitriy Rumynin is interested in Algebra and Representation Theory. Prospective students would have to see 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
Algebraic Topology: Homotopy theory, topological K-theory
(See also Adam Epstein)
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.
David Loeffler would be willing to supervise one or possibly two projects next year in the field of algebraic number theory. Possible topics could include:
Arithmetic of p-adic number fields The field of p-adic numbers is a very important tool in algebraic number theory, giving a bridge between finite fields (which are nice and simple and easy to work with) and the rational number field (which is much more sophisticated). The aim of this project would be to understand some of the basic properties of the fields of p-adic numbers and their finite extensions (such as Hensel's lemma, ramified and unramified extensions, etc); from there the project could develop in several directions, such as the theory of Witt vectors, higher ramification groups, p-adic analysis, or the beginnings of p-adic Hodge theory.
Hilbert modular forms. Hilbert modular forms are a generalisation of the classical theory of modular forms, in which functions of one complex variable are replaced by functions of several variables satisfying a transformation property under a larger group. These objects resemble classical modular forms in many ways, but some aspects of the theory are interestingly different, such as the theory of Hecke operators. (MA4H9 Modular Forms would be a prerequisite for this project.)
Samir Siksek is interested in number theory and diophantine equations. Prospective students would have to see him to discuss further details.
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.
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 uk.arXiv.org 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.
Chunyi Li is interested in Algebraic Geometry.
David Mond’s research interests lie in the area of singularity theory and algebraic geometry. Prospective students would have to see him to discuss further details.
Damiano Testa is interested in Algebraic geometry and Number Theory. Prospective students would have to see him to discuss further details.
András Máthé is interested in geometric measure theory, fractal geometry, combinatorics, real analysis, ergodic theory.
Antoine Choffrut's research is in the Analysis of Partial Differential Equations and Mathematical Fluid Mechanics. He is also interested in a new technique called Convex Integration, which has shown surprising connections between geometry and fluid mechanics. Prospective students would have to see him to discuss further details.
Charles Elliott’s research is centred around 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 2015/165 he will be giving the fourth year module MA4K2 Optimisation and Fixed Point Theory in the Autumn Term. 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. 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. Prospective students would have to contact him to discuss further details. Please consult his web page where more details will appear.
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’s research interests lie in the area of partial differential equations and fluid mechanics. He is happy to offer a project in Harmonic Analysis, following on from topics covered in MA433 (Fourier Analysis) and MA4J0 (Advanced Real Analysis). Prospective students would have to see 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 would have to see 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.
Marie-Therese Wolfram is interested in partial differential equations, mathematical modeling
in socio-economic applications and the life sciences, numerical analysis.
(see also András Máthé)
Keith Ball offers offer a project in probabilistic combinatorics. Students should come to discuss it with him.
Agelos Georgakopoulos offers projects combining probability and graph theory, sometimes involving computer simulations.
Vadim Lozin offers projects in graph theory, combinatorics, discrete mathematics
Daniel Kral works in combinatorics and related areas of theoretical computer science. He is open to advising student projects in particular related to limits of combinatorial objects, linear programming methods in combinatorics, classical structural graph theory, and logic-based approaches in algorithm design. There are many problems of different difficulties in these areas, so it would be best if prospective students come to see him and discuss their own interests to choose a suitable topic for the project.
Jonathan Noel offers projects in extremal and probabilistic combinatorics with connections to statistical physics (in particular, bootstrap percolation) and analysis (combinatorial limits). Other projects related to graph colourings and/or computational complexity may be offered.
Oleg Pikhurko’s main research interests lie in combinatorics and graph theory, including their connections to analysis, descriptive set theory, measured group theory, ergodic theory, etc. This is a deep and thriving area, with many possible projects that can help you to prepare for PhD studies. Prospective students should contact Prof Pikhurko by email.
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 would have to see him to discuss further details.
Andrea Mondino works at the interface of Analysis and Geometry. Specifically, variational problems in differential geometry and metric measure spaces with Ricci curvature.
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 8 of term 3.
Ergodic Theory and Dynamical Systems
Claude Baesens's interests are in dynamical systems and applications to physics, and in exponential asymptotics. She is offering projects on Frenkel-Kontorova models and on canard solutions in slow-fast dynamical systems. Prospective students would have to see her to discuss further details.
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 would have to see him to discuss further details.
Oleg Kozlovski is interested in Dynamical systems, ergodic theory, mathematical physics, financial mathematics. Prospective students would have to see him to discuss further details.
Ian Melbourne is interested in Ergodic theory and Dynamical systems, including probabilistic or stochastic aspects of deterministic dynamical systems. Prospective students would have to see him to discuss further details.
Mark Pollicott is interested in Ergodic theory, with applications to analysis, number theory, and geometry. Prospective students would have to see him to discuss further details.
Richard Sharp is willing to supervise one (or at most two) research projects on a topic in Ergodic Theory or 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.
Geometry, Topology and Geometric Group Theory
(see also Marco Schlichting)
Brian Bowditch offers project in hyperbolic geometry, low-dimensional topology, geometric group theory
Saul Schleimer is interested in geometric topology, group theory, and computation. Prospective students would have to see him to discuss further details.
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.
Complexity Science and Mathematical Modelling and Materials
(see also Applied Analysis, Fluid Dynamics, Mathematical Biology)
Colm Connaughton offers project in non-equilibrium statistical mechanics, fluid dynamics and turbulence, nonlinear waves, interacting particle systems
Stefan Grosskinsky's interests are in applied probability theory, stochastic processes and complex systems, statistical mechanics. Prospective students would have to see him to discuss further details.
Thomas Hudson is interested in micromechanics of materials, analysis of PDE and the Calculus of Variations, and caling limits, thermodynamic limits, and discrete-to-continuum limit processes.
Samuel Johnson is interested in complex networks, dynamical systems, nonequilibrium physics, ecology, computational neuroscience, cooperation.
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 would have to see him to discuss further details.
Robert MacKay offers projects in a range of applications of mathematics: Hierarchical Aggregation of Markov processes, Metrics on probability distributions for probabilistic cellular automata, Global Positioning Systems, Triple Linkage coding, A mixing volume-preserving vector field in a figure-eight knot complement, Optimal design of assessment graphs, Stroke treatment, Pledge game, Stellarator design, Integrable optics.
See here for further details.
Computational Mathematics and Numerical Analysis
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 a high-level language C/C++/Fortran. 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.
Andread 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 see him to discuss further details.
Christoph Ortner offers 4th year project(s) on the analysis and numerics of molecular mechanics and multiscale models. Possible projects may involve any combination of the model formulation, numerical computations or the analysis of a mathematical model. Prospective students would have to see him to discuss further details.
Björn Stinner is interested in free-boundary problems, PDEs on manifolds, finite element methods and their numerical analysis, asymptotic methods for analysing non-linear partial differential equations, and continuum modelling, particularly based on the phase field methodology, with applications in materials science, fluids, and cell biology. Projects may involve modelling, analysis, and computing or any combination of these. Possible topics are:
- Controlling geometric PDEs for applications in biology: Geometric equations with forcing terms are simple models for moving cells. Using image data and introducing a cost functional for the forcing yields an optimal control problem. Objectives could be of analytical nature (well-posedness, possibly of extensions) or of computational nature (programming the problem and experimenting with the cost functional).
- Computational stochastic PDEs: Objectives could be to learn some principles underlying numerical approaches to approximate solutions to stochastic (parabolic) PDE. There are possible applications in cell biology.
- R-adaptivity: Meshes are underlying many computational methods for PDEs. The typical goal is to establish convergence when the mesh is refined. Adaptivity refers to optimising this refinement, i.e, reducing the error as fast as possible. For some problem, particularly those involving transport, just moving some mesh points already is beneficial, which is referred to as r-adaptivity. Objectives could be to understand these techniques and apply them to phase field equations, for instance, for crystal growth.
Matteo Icardi offers the following projects in applied maths:
Colloid transport in subsurface flows: development and simulation of stochastic Langevin transport models in complex geometries
Granular media: simulation and statistical analysis of synthetic random packings of convex-shaped grains
Fluid-particle systems: development of novel momentum transfer closures based on accurate statistical estimation of random particle configurations in turbulent flows
Carbon dioxide storage and sequestration: parameter estimation and inverse problems in capillary dominated porous media flows
Pore-scale simulations: mesh adaptivity and a-posteriori error estimation for 3D Stokes equation in complex geometries
All the projects require basic numerical analysis and fluid mechanics knowledge and motivation towards C++ or Python programming in Unix environment. The code developments are to be integrated in existing open-source libraries (e.g., OpenFOAM or other in-house codes)
Robert Kerr's current interests include the mathematics of the turbulent energy cascade and conditions for singular behaviour in fluid and related equations, including the three-dimensional nonlinear Schroedinger equations for quantum fluids and atmospheric wave equations.
The perspective of the project is numerical and would be based on the latest high-resolution calculations. Potential topics include comparing some of the new statistics generated by these simulations to the latest proofs from rigorous mathematics for singularities, or visualising the underlying structural alignments that have either been assumed or shown analytically must exist if these equations have singularities. For example, I would like to understand helicity, a topological measure of the twisting of vortex lines. Helicity can be analysed from either a physical space perspective, in which case the Frenet-Serret equations for the curvature of vortex lines would be used, or from a Fourier perspective.
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.
Universal Behaviour of Drop Formation: The formation of a drop from a column of liquid (as seen from a dripping tap) is a singular flow which, remarkably, is governed by a 'universal' similarity solution (i.e. the final stages of the liquid breaking always look the same). However, recent computational results have revealed, twenty years after the original theory was proposed, that the approach to this solution is not uniform, so that the liquid's surface oscillates in time. This project will involve developing theory and computations to investigate the stability of the similarity solution in order to understand these new surprising results.
Oleg Zaboronski offers a project related to
Metastability in stochastic Euler dynamics. During this project, the student will learn about:
1. Arnold's algebraic formulation of 2d Euler equation
2. Zeitlin's su(N) approximation of 2d Euler flow
3. Stochastic Euler equation on su(N)
4. Control of stochastic Euler flow on su(N) using gradient dissipation built out of integrals of motion and the associated invariant measures.
5. Dynamics of stochastic Euler flow on su(N) for multimodal invariant measures using Wentzel-Fradkin theory.
Mathematical Biology, Epidemiology, Ecology and Evolution
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.
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.
Kat Rock is a mathematical epidemiologist interested in population-level disease dynamics.
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).
Hugo van den Berg offers projects in the area of Mathematical Biology. Prospective students would have to see him to discuss further details. Prospective students would have to see him to discuss further details.
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 (dynamical systems, perturbation theory), simulation and statistical computation (Markov chain Monte Carlo methods), with both deterministic and stochastic models/systems, although most are stochastic given the fact that cells often show stochastic behaviour. Projects are in the mechanics of cell division, from the duplication and separation of the chromosomes to the mechanical separation of cells into two daughter cells, (1, 2 below), microtubule modelling (3, 4) and cancer modelling (5):
1. Modelling chromosome oscillations. 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 communication system 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.
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 paired chromosome trajectories (1000s of trajectories) to understand the statistical structure of chromosome oscillations, fitting saw-tooth like oscillatory profiles. Experience with 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 evolution. Cancer is a Darwinian process whereby the cell functional pathways are mutated to escape growth control processes. There are multiple control processes, each controlled by a large number of genes. Under Darwinian evolution one predicts that only one mutation is required in each of these groups of genes. You will construct an MCMC algorithm to test this on real data to demonstrate this mutual exclusion prediction.
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].
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 see him to discuss further details.
Mike Tildesley works on the development of mathematical models to simulate the spread of livestock and zoonotic diseases.
Mathematical Physics, Molecular Dynamics, and Statistical Mechanics
(see also Sohinger and Stochastic Analysis and Probability)
Stefan Adams offers project on large deviation theory, probability theory, Brownian motions, statistical mechanics, gradient models, multiscale systems.
Roman Kotecky's interests are in discrete mathematics, probability theory, statistical mechanics, mathematical physics. Prospective students would have to see him to discuss further details.
Florian Theil offers project in the area of molecular dynamics which is an active and fast developing field in Applied Mathematics. The objective is to obtain insight into complex molecular systems by means of computer simulations instead of using experimental techniques. Due the high dimensionality the simulations are very costly and consequentially analytical results can provide valuable insights which potentially improve the efficiency of the numerical approaches. The proposed fourth year projects involve the application of tools from Stochastic Analysis and PDE theory to MD systems.
Daniel Ueltschi's interests are in analysis, probability theory, statistical mechanics, mathematical physics. Prospective students would have to see 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 https://terrytao.wordpress.com/2009/08/23/determinantal-processes/). Other projects also available.
Stochastic Analysis and Probability
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.
Sebastian Vollmer is interested in Monte Carlo Methods, Stochastic Gradient Methods, Stochastic Processes, Applications of Data Science.