Phd opportunities in Applied Mathematics

The department of Applied Mathematics has an extremely large range of opportunities for postgraduate research -- a selection of which is
included below. More information can be obtained from the research pages of the individual research groups below.

Dynamics of Semi-flexible Bio-polymers
Dr. Oliver Harlen

Using techniques from polymer physics and non-Newtonian fluid mechanics, the dynamics of semi-flexible bio-polymers like DNA and F-actin will be studied. Macroscopic properties of solutions of such polymers will be obtained from microscopic models.

Sensory detection in noisy environments
Dr. Grant Lythe and Dr. Dean Waters (Faculty of Biological Sciences)

Many animals manage to detect a weak signal against a noisy background. The key to understanding this may lie in the concept of stochastic resonance, a mechanism by which noise actually assists, rather than hinders, detection of signals. An example that is being actively investigated at Leeds is found in moths, which have tympanic organs sensitive to ultrasound that enable them to detect the echolocation calls of bat predators and take evasive action. Resolving the puzzle of how the two-celled moth organ is capable of performing this detection will require integrating experiment and theory from beginning to end, including stochastic modelling and statistical analysis of time series.

Dynamo Theory
Dr. Steve Tobias

Projects involve the interaction of small- and large-scale dynamos, fast dynamo theory including the role of the Lorentz force and the suppression of chaos in the underlying flow, the turbulent transport of magnetic fields and the nonlinear dependence of the alpha, beta and gamma coefficients of mean-field theory, the foundations of mean-field theory, and solar and stellar dynamo theory.

Mathematical modelling: chemotaxis in T cells and tumour cells
Dr. Carmen Molina-París

The project is motivated by experimental observations and makes use of a mathematical model of chemotactically directed tumor growth. It will involve both analytical study as well as a numerical one. The student will be able to learn the biology behind the project (tumour growth and invasion and T cell development and maturation) and make use of it to develop a mathematical model (reaction-diffusion). The aim is to link this continuum model to a random walk model and to be able to use experimental data to fit the transition probabilities and parameters.

Nonlinear dynamics: symmetries and chaos
Dr. Alastair Rucklidge

Global bifurcations are often responsible for creating chaotic dynamics in dissipative differential equations. Much of the complicated behaviour exhibited by chaotic systems can be explained by constructing maps (usually one-dimensional) that are valid near such a bifurcation. The presence of symmetry makes the analysis more difficult, and introduces the possibility of new types of phenomena: synchronisation, cycling chaos, and blow-out bifurcations. PhD projects would study these phenomena in cases where symmetry requires the use of higher-dimensional maps to describe the dynamics.

Semiclassical approximation to cosmology: preheating in inflation
Dr. Carmen Molina-París

In all inflationary models of the Early Universe there must be a period after inflation in which a substantial amount of particle production occurs. If the particle production rate is much faster than the thermalization rate then this period is sometimes called preheating. The process of particle production after the rapid expansion phase of inflationary models is crucial to the viability of these models, not just to populate the universe with matter and radiation but to make sure that the universe reheats to a temperature that is not too large or some benefits of inflation such as the effective elimination of magnetic monopoles are lost, and not too small or baryogensis may not occur. The project will involve making use of the semiclassical approximation to obtain an accurate picture of the amount of particle production that occurs and the rate that it occurs at.

Quantum Integrable Systems and Special Functions
Prof. Allan Fordy

An n-dimensional Quantum Integrable System is defined by the existence of n
mutually commuting differential operators.  Exactly Solvable Quantum Systems
have extra structure, such as ladder operators and Darboux transformations,
which enable us to <b>explicitly</b> build the spectrum and eigenfunctions.
Some systems are <b>super-integrable</b>, meaning that they possess
additional commuting operators, giving an alternative way to build
eigenfunctions.  The subject is closely related to the theory of orthogonal
polynomials and other special functions in n-dimensions.  Further details
can be found here.

Star Formation
Prof. Sam Falle

It is known that stars are formed from the Interstellar Gas and that
self-gravity plays a crucial role. However, self-gravity cannot by itself
explain the structure of star forming regions and it is also not able to
account for the observed distribution of stellar masses. At Leeds we have
been exploring the idea that magnetohydrodynamic waves are responsible for
much of the structure in star forming regions and therefore have an
important effect on star formation. Projects in this area involve a
combination of analytic and numerical work on these ideas.

Numerical Methods for Stochastic Differential Equations
Dr. Grant Lythe

Solving a stochastic differential equation is akin to solving an ordinary differential equation: exact analytical solutions are seldom available, but paths can be generated in a matter of seconds on a computer.

Dynamics of Soft Active Systems
Dr. Tanniemola Liverpool

Projects include the development of new mathematical techniques to study the behaviour of active biological systems. Energy is not conserved within living systems which use up food (fuel) to function even at the sub-cellular level. Examples are mixtures of molecular motors and protein filaments in the cell cytoplasm. In addition to finding new non-equilibrium steady states, the response of such systems to external forcing will also be studied.

Pattern formation and equivariant bifurcation theory
Dr. Alastair Rucklidge

Many physical, chemical and biological systems spontaneously develop patterns when driven hard enough. Many pattern formation problems can be analysed using equivariant bifurcation theory, but there are still many experimentally observed patterns that cannot yet be explained within this framework. Examples include quasipatterns and spatially modulated two-dimensional patterns. PhD projects would examine these kinds of patterns with an aim to developing new theory.

Extinction of combustion
Prof. John Merkin

Since the Montreal protocol, the use of halogen-based fire extinguishants has been banned and alternative strategies are required. One such strategy is to use a `safe' chemical inhibitor within a water spray. This acts as a `radical scavenger', removing the active combustion radicals and so slowing down and inhibiting combustion. The project is to start by examining relatively simple models for this process, to determine the parameter ranges where flame extinction occurs, and to use this study to develop more realistic models.

Perturbation methods and algebraic manipulators
Dr. Mark Kelmanson

Free-surface viscous fluid mechanics problems arise naturally in many important areas of engineering an physical sciences. In recent decades, the boundary-integral method has proved to be a powerful tool in solving such problems. Dr Kelmanson is interested in improving the accuracy of such methods by appealing to the techniques of spectral methods, particularly for problems in which rapid variations occur in the curvature of the free surface, so that surface-mesh dynamical adaption is also required. The ultimate aim is to develop general algorithms that are not restricted to any particular application.

Stochastic Dynamics of the T Cell Repertoire
Dr. Carmen Molina-París

Projects involve modelling the stochastic dynamics of T cell activation arising from recognition by the T cell receptor of an antigen presented by professional cells. This recognition process is highly stochastic. The project requires modelling the T cell and the professional cell interaction. In particular the calculation of activation probabilities is very well suited to large deviation techniques and Monte-Carlo simulations. Other related projects involve developing a T cell competition model for the antigen, and how the T cell repertoire is maintained by means of this competition. The mathematical tools needed for these projects are based on approximations to solutions of a large (order of a million) system of coupled SDEs, and Markov stochastic processes.

Dynamics of kinks
Dr. Grant Lythe

Stochastic Partial Differential equations describe spatially extended systems with noise. The theory of such equations is still in its infancy, but the rapid increase in the computing power available to scientists has permitted their increasing use. The focus of current research is the statistics and dynamics of coherent structures that maintain their identity as they move and are buffeted by local fluctuations. In their simplest manifestation they are known as kinks and a systematic study is possible, which will serve as a stepping stone towards the study of more complicated structures such as vortices.

Hydrodynamic and Magnetohydrodynamic Stability
Dr. Rainer Hollerbach

A recurring theme in many different areas of fluid dynamics is the
transition to more and more complicated flows as the system is forced
increasingly hard.  Hydrodynamic stability theory seeks to understand the
nature of these transitions, both in terms of the underlying fluid dynamics,
as well as more abstractly, in terms of the symmetries that are broken by
the various bifurcations.  Applications include not just classical fluid
dynamics, such as the flow between differentially rotating cylinders, but
also geophysically and astrophysically motivated problems.  PhD projects
would involve studying one of these applications, and numerically
investigating the resulting instabilities.

Stability of reaction fronts
Prof. John Merkin

Planar reaction fronts can become unstable both through diffusion-driven instabilities (similar to Turing instabilities) and through buoyant (or flow-driven) instabilities arising from the difference in reactant densities ahead and at the rear of the front caused by the reaction. Additional features, such as applying electric fields, can radically alter the the structure of the front and hence the stability characteristics of the front. The project is to examine, through both stability analysis and numerical simulation, prototype models which exhibit these features, to examine the interaction between diffusion and flow driven instabilities and to assess the effects that electric fields can have on the overall stability.