List of possible PhD projects
Below are some examples of PhD projects that we offer in our department. Alternative projects might be formulated following discussions with individual staff members, just contact the staff member or PhD admissions tutor Dr Gianne Derks.
Granular matter (Supervisor: Dr Anne Skeldon)
The equations that govern the motion of fluids are well-established. Less well understood is the motion of particulate materials, such as sand, soil or powders: sometimes these flow like fluids--you can pour gravel out of a bucket; and sometimes they behave like a solid--you can make a castle out of sand but not out of water. This project is joint with Prof Ugur Tuzun in Chemical and Process Engineering at Surrey to develop data analysis techniques/understand the physics of granular media.
Faraday waves (Supervisor: Dr Anne Skeldon)
Patterns can be made to form on the surface of a container of fluid by shaking the container up and down. The shaking has to be at the right frequency and right amplitude for any patterns to form--if you move a container of fluid up and down very slowly then the surface of the fluid will remain flat. There are a wealth of experimental results showing a variety of different possible patterns. On the theoretical front, arguments based on symmetries and the way different patterns interact has led to some knowledge of the mechanisms that cause particular patterns to become dominant. Although these have been tested on mathematical models describing the fluid, there are still significant gaps in our understanding. The aim of this project is to investigate further the theory underlying pattern selection in the Faraday and related problems. This will require using a variety of different dynamical systems techniques both theoretical and computational.
Attractors for hydrodynamical problems in unbounded domains (Supervisor: Dr Sergey Zelik)
The concept of a global attractor plays a very important role in the modern theory of dissipative systems generated by PDEs. For the case of equations in bounded domains (like a square, disk or ball), this attractor is usually finite-dimensional. Thus, despite the infinite-dimensionality of the initial phase space, the limit dynamics reduced to the attractor occurs finite-dimensional and can be studied by the methods of the classical dynamical systems theory. The situation is more difficult for the case where the underlying domain is unbounded (like R2 or R3) since the dimension of the attractor is typically infinite and the finite-dimensional reduction is no more possible. Nevertheless, a reasonable theory has been recently developed for large class of such systems. The aim of this project is to extend/apply this theory to hydrodynamical problems in unbounded (cylindrical) domains. Such extension was problematic during a long time since even the well-posedness of the Navier-Stokes equation in 2D unbounded domains in the proper classes of spatially non-decaying solutions was not known. The situation is changed now due to recent results on the solvability and dissipativity of 2D and 3D Navier-Stokes equations in cylindrical domains.
Construction of invariant sets for nonautonomous ODEs (Supervisor: Dr Jonathan H.B. Deane)
Invariant sets (subsets of the phase plane in which solutions of an ODE remain for all time) are important because they delineate qualitatively different sorts of behaviour displayed by solutions of an ODE, for instance separating solutions that remain bounded for all time from ones which blow up in finite time. Proving that a given set is invariant generally requires the proof of an inequality on the boundary of the set. Even in the case of a second-order non-autonomous ODE, this essentially planar method yields a subset of the actual invariant set. An investigation into how to optimise this procedure to obtain best possible constructions, and possibly also how to automate the construction using computer algebra or a low-level computer language, is the purpose of this PhD project.
Fast ODE solvers using analytical continuation (Supervisor: Dr Jonathan H.B. Deane)
Fast, accurate methods for solving nonlinear ODEs with polynomial nonlinearity are important in many applications. One suitable method is based on Taylor series, and is also known as the cell-to-cell mapping technique. Roughly speaking, the solution of the ODE is expanded in a power series around a point t=t0, and a suitably modified ratio test applied to the high-order coefficients of the series. The test gives an estimate of the radius of convergence of the series, R (among other things), and so we can compute the solution accurately at, say, t=t0+½ R. This is effectively numerically-implemented analytical continuation. In practice, R can be quite large and so the ODE can be solved in correspondingly large time steps. In some recent work on the varactor equation, an increase in speed by a factor of about 10-50 was obtained using this method compared to, for instance, Runge-Kutta methods. The method appears to be promising but a great deal remains unknown about the assumptions on which it is based, its performance and the circumstances under which it fails.
Soliton Switching in Fibres (Supervisor: Dr Gianne Derks)
For the optical transmission of data across a cable, one can use two (or more) coupled fibre cables. Experiments have shown that if a certain type of signal is put at one end of the cable, it will go to the other end of this cable and hardly anything happens in the other cable. However, if one puts other types of signals on the cable, the signal will switch to the other cable. This gives a convenient way of sending data consisting of zeros and ones. In this project we will aim for a better understanding of this experimentally observed process by investigating the family of soliton-like solutions, especially issues like existence, stability, bifurcations and invariant manifolds will be investigated.
Patterns in Surface Chemistry (Supervisor: Dr Rebecca Hoyle)
Regular patterns arise naturally in many physical, chemical and
biological systems - from hexagonal convection cells on the surface of
the sun to stripes on a zebra's back. Constantly changing irregular
patterns of carbon monoxide (CO) and oxygen are seen during CO
oxidation on platinum crystals in the [100] orientation. Recently a
reaction-diffusion model is developed to reproduce this pattern
formation and created numerical simulations that show patterns made up
of moving CO and oxygen fronts. Possible PhD projects in this area
include: extending the model to include the formation of subsurface
oxygen at higher pressures or developing a similar model for the NO +
NH3 reaction on Pt{100}. These interdisciplinary projects are great
opportunities for Maths graduates to apply their skills in a new area,
or for Chemistry graduates with good maths and computing skills to
move into theory.
Mathematics of Storytelling (Supervisor: Dr Rebecca Hoyle)
How do oral histories, tales that encode some part of a community's history or shared culture, spread and persist? Can we model this mathematically, perhaps using an agent-based approach, where we create individuals, give them attributes and behaviours and link them together in an evolving social network? I'd like to find out, perhaps using the evolution of children's nursery rhymes as an example. This project would involve researching the history and geographical distribution of nursery rhymes and attempting to build a model that can reproduce a similar pattern of spread. It would suit a Maths or Computing Science graduate with an interest in social science and some programming skills who is comfortable with an open-ended and exploratory approach in the initial stages.
Molecular Motors (Supervisor: Dr Rebecca Hoyle)
Molecular motors are proteins that transform chemical energy into
mechanical work on a molecular level, generating forces and leading
to motion. We are studying myosin V, a motor involved in intracellular
transport in animal cells. It has two heads that bind to an actin filament
and a long neck that attaches to its cargo, such as vesicles and
organelles. The myosin molecule walks hand-over-hand along the actin
track via the coordinated binding and release of its heads. We have used
energetics to model the interaction of external load and intramolecular
strain with the ATP hydrolysis cycle that drives the stepping action,
and performed a detailed quantitative fit to experimental data.
Possible PhD projects include: applying the same methodology to a variety
of other molecular motors to determine how well the established models
compare with experimental data, and how the evolved physical characteristics
of the motors relate to their biological function. This is interdisciplinary
work in an exciting and fast-moving area of biophysics. An enthusiasm
for learning about biophysics and communicating with experimentalists
is essential. This project would suit a graduate in Applied Maths or
Physics, or possibly a Biology graduate with strong quantitative skills.
Programming skills are needed to adapt existing codes.
Chaotic Advection in Large Airways (Supervisor: Prof Peter Hydon)
Babies who are born very prematurely usually need mechanical help to breathe. One particularly effective type of mechanical ventilator uses High Frequency Ventilation (HFV), in which a low volume of air is delivered to the lung at a frequency of 10-15 Hz. However the volume of air used is so small that the conventional theory of breathing does not explain how adequate gas transport can be achieved. It has recently been shown that a type of stirring called`chaotic advection' greatly enhances transport in small airways. The purpose of this project is to find out the extent to which chaotic advection operates in the large airways, and thus to develop strategies for obtaining the best possible transport at various frequencies. This project combines mathematical biology, dynamical systems theory and computational fluid dynamics. It is of direct relevance to medicine, and the student will have the opportunity to work with some of the UK's leading medical specialists in HFV.
Methods for Obtaining Discrete Symmetries (Supervisor: Prof Peter Hydon)
Differential equations are widely used as models of real systems. To understand the behaviour of a system, it is usually necessary to find the symmetries of the model differential equation. Recent work at Surrey has led to the first systematic technique for obtaining discrete symmetries. The purpose of this project is to simplify and extend the technique, and to create a Maple package that will systematically obtain the discrete symmetries of a given differential equation. This project combines differential equation techniques, linear algebra, group theory and Maple programming. It is expected that the resulting package will be distributed with future releases of Maple, and will be used by those working in many areas of mathematics and theoretical physics.
Spiral waves and defect interaction (Supervisor: David Lloyd and Bjorn Sandstede)
Spiral waves have been found in many experiments such as the famous Belousov-Zhabotinsky reaction and the oxidation of carbon-monoxide on platinum surfaces. In the former reaction, period doubling of spiral waves has been observed, and a numerical simulation of a model problem is shown to the left. This bifurcation has been analysed in 2D. Of interest would be an analytic and numerical study of this phenomenon in 3D where the line defect that connects the core to the bottom boundary becomes a two-dimensional surface, which itself may exhibit a interested dynamics. More generally, the various line defects visible in the simulation may interact with each other, and it would be of interest to study the time and length scales involved in this interaction from a analytical and numerical perspective.
Degradation Modelling (Supervisor: Dr Karen Young)
Degradation models can be used in for example medicine and engineering. In situations where you are interested in the time to an event such as failure, but the event occurs where a suitable measure of degradation has reached a threshold, we considered a case where the degradation followed a Wiener process so that time to a threshold had an inverse Gaussian distribution. In this project we would extend this work in a number of different directions eg looking at other processes for degradation, comparison with traditional survival analysis, diagnostics to detect outliers and influential observations.
Theory and numerics of reversing symmetry breaking bifurcations of Hamiltonian relative periodic orbits (Supervisor: Dr Claudia Wulff)
Relative periodic orbits (RPOs) of symmetric dynamical systems are
periodic orbits for the symmetry reduced system. Both the theory and
numerics of periodic orbits of general dynamical systems is
well-developed. But additional structure such as time-reversing and
preserving symmetries and symplecticity changes the generic behaviour
of dynamical systems dramatically. Recently there has been a lot of
progress in the development of a bifurcation theory for symmetric and
Hamiltonian systems, but the theory is far from being complete. In
particular the bifurcation theory of reversible symmetric periodic and
relative periodic orbits, is still in its beginning, as is the numerical
analysis of symmmetry breaking bifurcations of reversible periodic
orbits. The topic of this project is to analyze reversing symmetry breaking
bifurcations of RPOs theoretically and to derive and implement
numerical methods for their detection and computation within the
package SYMPERCON of Wulff, Schebesch, Schilder.
The results will be applied to various symmetric Hamiltonian
systems, in particular to N-body systems.
Multi-symplectic structures and nonlinear PDEs (Supervisor: Prof Thomas J. Bridges)
An understanding of nonlinear PDEs is one of the great challenges for
the twenty-first century. In this project the geometry of nonlinear
PDEs is studied. Multi-symplectic structure, which generalises
classical symplectic geometry, is used as a backbone for the analysis.
The project has three parts:
(I) Develop the geometry of the total exterior algebra bundle for smooth two-dimensional manifolds and construct Dirac operators on this bundle as a basis for canonical hyperbolic and elliptic PDEs.
(II) Use the calculus of variations to find solutions of multi-symplectic Hamiltonian PDEs, using the Mountain Pass Theorem and related functional analytical tools.
(III) Apply numerical methods to either part (I) or part (II).
