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 the PhD Admissions Tutor Dr Gianne Derks.
Normal diffusion in compact group extensions of chaotic dynamical
systems (Supervisors: Dr Henk Bruin and Prof Ian Melbourne)
Many classes of dynamical systems (uniformly hyperbolic, nonuniformly hyperbolic, etc) exhibit strong statistical laws such as the central limit theorem and convergence to Brownian motion.
An important class of partially hyperbolic systems occur in systems with symmetry, where the neutral direction along the symmetry group is an added complication.
Previous work (Melbourne & Nicol; Bruin, Holland & Melbourne) indicates that under certain reasonable conditions, strong statistical laws hold for systems with symmetry despite the presence of the neutral direction. This is a topic for further investigation, but perhaps more interesting is the discovery that in some examples the symmetry helps. That is, for certain classes of observables, the system with symmetry might obey the central limit theorem and so on even though the nonsymmetric system does not. (Nonanomalous anomalous diffusion, so to speak.)
Further details: Dr Henk Bruin, Prof Ian Melbourne
Statistical properties of nonuniformly hyperbolic systems (Supervisor: Prof Ian Melbourne)
Many classical dynamical systems such as the Lorenz attractor, Hènon-like attractors, dispersing billiards and Lorentz gases, are nonuniformly hyperbolic and modelled by a symbolic system known as a Young tower. Recent work of Melbourne & Terhesiu introduces a new technique for studying the statistical behaviour (central limit theorems with error estimates, stable laws, local limit theorems, large deviations etc) of such systems. The possibilities seem to be endless so it would be a good entrance to a PhD project.
Lyapunov exponents for systems with symmetry (Supervisor: Dr Philip Aston, Prof Ian Melbourne)
For uniformly hyperbolic (Axiom A) dynamical systems, it is expected that typically the Lyapunov exponents are distinct. Definitive results are unavailable, but Bonatti & Viana (2004) proved that distinctness of Lyapunov exponents is typical in a related but simpler setting (exponents of linear cocycles).
In systems with symmetry, there are constraints that can force multiple Lyapunov exponents. Results of Aston & Melbourne (2006) on such symmetry-induced multiplicities
are probably optimal, but this has not been pinned down due to the above difficulties of proving distinctness.
This discussion leads naturally to the following problems:
- Extend the multiplicity results of Aston & Melbourne to the setting of linear cocycles (satisfying an appropriate symmetry constraint).
- In the Axiom A case, prove that the results in (i) are sharp using the techniques of Bonatti & Viana.
Further detail: Dr Philip Aston, Prof Ian Melbourne
Deterministic random walks in random environments (Supervisor: Prof Ian Melbourne)
The one-dimensional simple symmetric random walk (SSRW) has diffusive √n growth but the introduction of randomness into the environment leads to trapping regions and a much slower subdiffusive (log n)2 growth rate (Sinai, 1972).
These results have a deterministic analogue in systems with a noncompact group of symmetries (the word symmetry here has almost nothing to do with the symmetry in SSRW) where chaotic dynamics in phase space lead to random-like behaviour in the group directions. In particular, √n growth has been proved in this context. In applications (including dynamics of spiral waves in excitable media) the symmetry is not exact and it is reasonable to assume random perturbations in the group directions. In certain situations, it is reasonable to anticipate subdiffusive growth rates.
The project branches from the outset into various distinct directions including:
- Probabilistic: Analyse Sinai diffusion in analogues of the SSRW in random environments with a group of symmetries.
- Deterministic: Analyse Sinai diffusion for deterministic systems in random environments with a group of symmetries.
Testing for chaos, and Wiener sausages (Supervisor: Prof Ian Melbourne)
Gottwald & Melbourne (2004) introduced a new technique for distinguishing chaotic from regular dynamics for deterministic time series data. Let x1, x2, ... ∈ R be a sequence of data from a dynamical system. Choosec>0 and define p(n)=∑nj=1 eijc xj. The claim is that p(n) is bounded if the dynamics is regular and that p(n)∼W(n) if the dynamics is chaotic. Hence it remains to distinguish bounded behaviour from Brownian-motion behaviour.
The method used so far is to use the fact that the growth rate of p(n) is 0 in the regular case and 1 in the chaotic case, but it is important to study other distinguishing features since they may have better convergence properties in practice.
For example, one could study the sausage associated to the process p(n), namely the ε-envelope. There are precise scaling laws for the volume of such sausages, and it would be interesting to (i) establish laws rigorously in the context of the test for chaos, (ii) see how this works for actual time series data.
Numerical weather prediction: multi-scale variational data assimilation and high-resolution models (Supervisor: Prof. Ian Roulstone)
Assessing risk from extremes of weather and climate, evaluating sustainable energy resources, and generally improving our ability to adapt to a changing climate, all require detailed knowledge of how weather patterns will affect our lives in the future. Reliable weather forecasts are currently issued for 3 to 4 days ahead at continental and global scales, though more detailed local/regional information from higher-resolution weather prediction models requires a radical new approach (because relatively rapid, unstable nonlinear processes become more important).
This project will contribute to the development of high-resolution numerical weather prediction systems (which are recognized as an important part of evaluating the impact of climate change at local or regional scales) and their use to generate meaningful information for end users e.g. where forecasts are used as part of a flood warning system. Success will lead to immediate benefits for all users of forecasts (e.g. Environment Agencies, Aviation Authorities (e.g. CAA), etc.).
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)
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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.
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.
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:
- 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.
- Use the calculus of variations to find solutions of multi-symplectic Hamiltonian PDEs, using the Mountain Pass Theorem and related functional analytical tools.
- Apply numerical methods to either part (I) or part (II).