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 outandout 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
RayleighBenard convection (a plane layer of viscous fluid confined between horizontal
plates with the lower plate hotter than the upper plate), reactiondiffusion 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 largescale 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 saddletype 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 widelyapplicable 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 physicallymotivated 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 patternforming 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 widelyapplicable methods both in modern condensed matter physics and in applied dynamical systems theory.
Project description.
The experimentallymeasured characteristics of biological neurons have been successfully reproduced by artificial solidstate `neurons' fabricated by growing semiconductor pn 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 smallscale 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 `winnertakesall' dynamics where a single neural state outcompetes 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 lowdimensional 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. Eventdriven 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. Buoyancydriven 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 nonNewtonian, or
`viscoelastic' 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.
NonNewtonian 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 wellknown in the Newtonian case become much harder for
a nonNewtonian 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
`doublydiffusive' 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 largescale
ocean circulations that are generated by these buoyancy differences, and this project will,
through a combination of theoretical work and laboratoryscale experiments,
develop robust methods of analysis that provide insights into the dynamics of planetaryscale
fluid flows. This project therefore will contribute to our understanding of the Earth's climate
and the linkages between the different subsystems within it.