If you are interested in PhD studies,
it would be best to either email me or chat about what aspects you
are interested in. The below are only meant to give a flavour of the
sort of topics that I would be interested in supervising. I also have
some ideas for BSc/ MMath/ MSc projects
below.
Possible PhD projects
Synchronization and pattern formation in
chaotic systems
The brain is
composed of a vast number of almost identical cells each of which has
comparatively simple dynamics. One of the fundamental problems in
neurophysiology is to understand how such a system can organise
itself to permit information processing and storage. This project
aims to look at very simple models of such systems of `coupled
oscillators' in an attempt to understand and classify the possible
types of behaviour of such systems, with a view to applying them to
more physically relevant models studied by researchers in
neurophysiology and physics.
Spiral wave motion
We
normally think of waves as patterns that propagate in space. Spiral
waves are such waves where one end of the wave is pinned at a `spiral
core' and the wave rotates around this. Such waves have been observed
to arise in many systems, from the behaviour of heart muscle during
heart attacks to the oxidation of carbon monoxide on catalytic
converters. This project will aim to develop a better understanding
of the existence, stability and bifurcation of such waves through the
use of dynamical system theory.
Nonlinear dynamics of climate models
Climate
systems or subsystems are often highly nonlinear with a range of
feedbacks present. This project, join with members of Exeter Climate
Systems, will look at some aspects of these models, ranging from
"tipping points" to coupled global circulation models.
Numerical approximation of random
attractors
If a system is
forced by a random noise input, one might think that only statistical
models will be useful. By viewing the noise as coming from a
deterministic dynamical system we can apply a variety of techniques
of `random dynamical systems'. This project will examine the
existence of and aim to develop new theory for the behaviour of
so-called random attractors in numerical approximations of randomly
forced system.
Dynamics in the presence of
discontinuities
The dynamics of
systems where all are equations are smooth is at a high level of
sophistication. By contrast, those of systems with discontinuities
are poorly understood, partly because there are many ways in which
this can happen. However there are very basic problems that remain
unsolved, for example: consider a triangle in which we play
`billiard', i.e. we draw a line inside the triangle and reflect at
each boundary it hits. It is unknown whether all triangles have a
periodic trajectory, i.e. a trajectory that repeats exactly! Similar
problems arise in the mechanics of impacting systems and digital
signal processing. There is plenty of scope in this project to
specialise on applications or to work on theoretical problems. This
project will work with the supervisor and interact with colleagues in
Exeter, San Francisco and Marseille at developing a theory for
understanding such maps.
Binocular rivalry For experiments where people are shown differing images appear in each eye, the brain attempts to make sense of the contradictory information by alternating perception between the two different images rather than necessarily trying to fuse them. This is a simple test system where one can begin to understand decision making processes within the brain and there are a variety of mathematical models available to explain the cognitive processes involved. This project will look at some mathematical models of this process, including one proposed by Chow and Laing in 2002.
Chaotic attractors and riddled basins It is well known that nonlinear iterated mappings can behave in a seemingly unpredictable way; the phenomenon of chaotic attractors. Basins of attraction for chaotic attractors can display fascinating and complicated fractal geometry, including what has been called riddled basins. This project aims to look at some of the theory and numerical examples of riddled basin attractors.
Topic in the History of Mathematics A possible topic would be a biography of a mathematician or group of mathematicians, or the development of a particular concept or method within mathematics. You will be expected to read original (primary) and secondary sources related to the topic.
Evolutionary Game Theory This project will look at the fundamentals and some applications of evolutionary game theory. This has arisen from applications in biology and economics where interactions between different species or agents lead to them changing their strategy according to a number of models. This project will look in particular at the `replicator equations' and issues of whether a strategy will disappear as a result of the evolutionary game.
Bifurcation theory Bifurcation theory is a powerful theory for understanding the behaviour of systems of nonlinear differential equations on varying a parameter. By studying the change in solutions one can better understand fundamental instabilities in many systems, focussing on problems in, for example the Couette-Taylor problem of stability of fluid flow between two rotating cylinders.
Fractal dimension and measure The images of fractals (sets with dimension that is not an integer) are used often in popular culture and for example advertisements; they also have serious uses in science and technology. In fact there are many different types of fractal set that can be characterised in many different ways. This project examines some definitions and fundamentals underlying fractal geometry and dimension before moving on to generate and analyse examples of fractals.
Billiards in polygons We investigate the mathematics of idealized billiards within a polygon. This is a simple model for a one-particle gas in two dimensions where a particle travels in straight lines between bouncing off the walls. The dynamics of the billiard system depends critically on the shape of the polygon and can be surprisingly non-trivial.