PhD Projects

I am very happy to consider PhD applications. Projects in a number of areas are available, centred around my interests in applied dynamical systems, both for out-and-out applied mathematicians and for those intrigued by something (even) more interdisciplinary. The exact details of a project are usually best left to be worked out in discussion with potential students. Do contact me to fix a time to have an informal discussion.

Applied mathematics

Two examples of topics I'd be happy to supervise are:
1. Localised pattern formation
Many physical systems which spontaneously form structure do so in a way that allows localised patches of 'pattern' to appear instead of stable (almost) periodic spatial structures. Examples include traditional fluid mechanical problems such as Rayleigh--Benard convection (a plane layer of viscous fluid confined between horizontal plates with the lower plate hotter than the upper plate), reaction-diffusion systems describing the 'mass action' dynamics of chemical kinetics, and the buckling of elastic structures.

Questions that are largely open here include the bifurcation structure of such states in 2D and 3D, the effects of additional large-scale modes on the dynamics, the relation with existing reports of experimental work and the connections with other families of coherent structures ('pulses', 'fronts') constructed in other parameter regimes, far from the initial bifurcation (Turing instability) in which these localised states are 'born'.
2. Robust heteroclinic cycling
Systems in which equilibrium states exist but are always unstable in time to a different equilibrium state often produce 'switching' dynamics in which trajectories in phase state move between neighbourhoods of these saddle-type equilibria without converging to any one of them. The object to which the dynamics as a whole converges can be described as a heteroclinic cycle and in cases where additional constraints on the dynamics exist, such as symmetry or the existence of permanent states (such as population death), these heteroclinic cycles (HCs) are termed 'robust' (RHCs). Many mathematical properties of these dynamical objects remain to be explored, including their response to external driving and the much more complex dynamics that can result if RHCs are coupled together.

Interdisciplinary topics - University Studentships available!

More details of the Unversity Research Studentships scheme are available here.
The four projects I've proposed under this scheme are:
1. Localised states in nonlinear optics
Supervisors: J.H.P. Dawes (Mathematical Sciences); D. Skryabin (Physics)

Project objectives.
The project objective is to enhance our understanding of the generation and control of nonlinear localised states of electromagnetic field in optical systems. The project provides an opportunity for a graduate student to learn a range of widely-applicable methods both in applied dynamical systems theory and in computational and theoretical electromagnetism, and to develop both of these directions further, initially through the study of specific physically-motivated problems. This research area is very well aligned with the established strengths of the mathematics and physics departments: the combination of challenges within the project will naturally produce interdisciplinary research within the faculty. Moreover it is the kind of synergetic collaboration that may well lead to further joint funding applications.

Project description.
Localised states in externally driven and internally lossy systems are sustained through the complex balance of nonlinearity, dissipation, dispersion and geometry of the system. In one spatial dimension, localised patches of energy arise generically near pattern-forming instabilities which underpin the spontaneous formation of spatial structure in many physical situations. The mathematical theory for localised states in one spatial dimension is well understood, and reasonably complete. In two and three spatial dimensions, the theory is much less well developed and there is a clear lack of theoretical results to explain in detail the range of numerical and experimental results obtained over the last two decades or so.
This project will develop analytical and numerical theories of the formation and dynamics of localised states in the real world optical systems. The particular focus of the research will be the investigation, from first principles, of localised states within spatially periodically modulated optical media (photonic crystals) by solving Maxwell's equations in this complicated geometry. On the basis of the detailed data so acquired, it will then be possible to derive simplified mathematical models which can be rigorously analysed through the methods of nonlinear dynamics. Relying on the combination of the above approaches, the project will aim to understand and interpret effects that have been observed previously (both numerically and experimentally) and to predict new physical properties.

2. Neural coding and decision making: theoretical and experimental aspects of winnerless competition
Supervisors: J.H.P. Dawes (Mathematical Sciences); A.R. Nogaret (Physics)

Project objectives.
The project objective is to enhance our understanding of the neural processes involved in information processing and decision making by combining experimental construction and theoretical modelling of simple neural networks that possess complex dynamics. The research project will synthesize the experimental realization of specific neural networks using semiconductor wires with the mathematical analysis of their dynamics using tools from dynamical systems theory. There is thus the opportunity for a graduate student to learn a range of widely-applicable methods both in modern condensed matter physics and in applied dynamical systems theory.

Project description.
The experimentally-measured characteristics of biological neurons have been successfully reproduced by artificial solid-state `neurons' fabricated by growing semiconductor p-n layers on a GaAs substrate. The artificial neurons have 2 micron wide nerve fibres that converge to the decision centre of the neuron that regenerates above threshold pulses. Such small-scale devices lend themselves naturally to the construction of artificial neural networks. Such networks, even with only very small numbers of neurons, for example six, are known mathematically to be capable of generating very complex dynamical responses to external stimuli: these allow even very small neural networks to be able to perform subtle information coding and decision making tasks. Theoretical research into information coding in networks in general has emerged recently as a research area that looks to be both important and timely.
Mathematically an intriguing recent description of neural dynamics is as `winnerless competition' (in contrast to `winner-takes-all' dynamics where a single neural state out-competes others). Winnerless competition corresponds to the existence of mathematical objects known as `robust heteroclinic cycles' (RHCs) in the system phase space. Much remains to be understood about the dynamics near RHCs, even in low-dimensional cases.The student will (i) develop models for information coding by spike trains (i.e. periodic oscillations) in such winnerless networks and will (ii) work towards the interpretation of experimental results using the models they develop.

3. Event-driven contact networks
Supervisors: J.H.P. Dawes (Mathematical Sciences); R. James (Physics); M. Patterson (Arch and Civil Eng)

Project objectives.
The project has two objectives. Firstly, to formulate a new theoretical framework for the analysis of networks defined by contacts at specific times between individuals. Secondly, to apply these theoretical ideas to capture and quantify the behaviour of insect colony feeding strategies, motived by the behaviour of ant colonies studied over recent years by James and collaborators in the School of Biological Sciences, University of Bristol.
The project lends itself naturally to an interdisciplinary proposal. It provides opportunities for a graduate student to learn an extremely wide range of mathematical ideas: graph theory, statistical mechanics, dynamical systems and recent concepts in the statistical analysis of networks as well as being exposed to the challenges involved in dealing with real biological data.

Project description.
Most network problems (for example for energy distribution, transport or communication) rely on an almost static infrastructure, or at least a connectivity that evolves much more slowly than the dynamics that takes place on the network. Most current 'network theory' is oriented towards these kinds of network: as the connectivity is fixed, attention can be concentrated on (i) producing quantitative characterisations of the network (e.g. clustering coefficients, centrality) and (ii) describing (optimal) flows and routes through the network.
In other settings, including this project, by contrast, there is no such separation between the network connectivity and the network dynamics because the network connectivity is defined entirely by the dynamics, i.e. the contacts at specific times between specific individuals. For example, this kind of contact dynamics is closer to the real situation arising in epidemiological problems. Moreover, it is exactly the case that arises in the `famine relief' food distribution problems studied currently by R. James in collaboration with the University of Bristol 'AntLab'. This project seeks to develop our understanding of dynamical networks problems such as this, where there is no `network structure' except for the feeding contacts between individuals. Very little is known about the characterisation of these networks, for example the relation between strategies adopted by individual ants and the overall proportion of the colony that is fed. The use of real biological data brings further challenges, and M. Patterson has recently worked on the feasibility of extracting contact information from video clips of the ant motion.

4. Buoyancy-driven fluid flows.
Supervisors: J.H.P. Dawes (Mathematical Sciences); M. Patterson (Arch and Civil Engineering)

Project objectives.
Fluid mechanics is a broad subject: the objective in this project is to focus on developing an understanding of fundamental processes that are essential to the dynamics of oceans and the Earth's mantle. M. Patterson has recently substantially built up the capacity of Arch and Civil Engineering to carry out detailed experimental work on fluid flow problems, including the construction of a Particle Imaging Velocimetry (PIV) setup which has not previously existed. This project proposes a collaboration that will bring theoretical and experimental work together in ways that have not been possible before in Bath.

Project description.
The overwhelming majority of fluid flows in geophysical problems are driven by buoyancy effects: less dense fluid rises and more dense fluid falls. For example the motion of the Earth's mantle is driven (primarily) by temperature effects: the material at the base of the mantle is substantially hotter than than at the upper boundary with the Earth's crust. This convective process is well known, but in the mantle it is made much more complicated by the non-Newtonian, or `visco-elastic' nature of the material. Newtonian materials are those such as water, for which it is reasonable to propose that the stress (force per unit area) is proportional to the strain rate. Non-Newtonian fluids exhibit a more complex response when stressed: examples within our everyday experience include toothpaste, blood, and solutions of cornflour (custard powder) in water. Even the analysis of flow instabilities that are well-known in the Newtonian case become much harder for a non-Newtonian fluid.
The second geophysical motivation is the study of flows that are driven by two components that differentially affect buoyancy, for example the diffusion of temperature and salinity. Such a `doubly-diffusive' flow arises in the Earth's oceans since evaporation at the equator naturally produces hotter and saltier fluid that then comes into contact with colder but fresher water near the poles. There remains plenty of work (both experimental and theoretical) to be done investigating the dynamics of large-scale ocean circulations that are generated by these buoyancy differences, and this project will, through a combination of theoretical work and laboratory-scale experiments, develop robust methods of analysis that provide insights into the dynamics of planetary-scale fluid flows. This project therefore will contribute to our understanding of the Earth's climate and the linkages between the different sub-systems within it.