Dynamics of Semiflexible Biopolymers
Dr. Oliver Harlen
Using techniques from
polymer physics and nonNewtonian fluid mechanics, the dynamics of
semiflexible biopolymers like DNA and Factin 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 twocelled 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
largescale 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
meanfield theory, the foundations of meanfield theory, and solar and
stellar dynamo theory.

Mathematical modelling: chemotaxis in T cells and tumour cells
Dr. Carmen MolinaParís
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 (reactiondiffusion). 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
onedimensional) 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 blowout bifurcations. PhD projects would study these phenomena in
cases where symmetry requires the use of higherdimensional maps to
describe the dynamics.

Semiclassical approximation to cosmology: preheating in inflation
Dr. Carmen MolinaParís
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 ndimensional 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>superintegrable</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 ndimensions. Further details
can be found here.

Star Formation
Prof. Sam Falle
It is known that stars are formed from the Interstellar Gas and that
selfgravity plays a crucial role. However, selfgravity 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 subcellular level.
Examples are mixtures of molecular motors and protein filaments in the
cell cytoplasm. In addition to finding new nonequilibrium 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
twodimensional 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 halogenbased 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
Freesurface viscous fluid mechanics problems
arise naturally in many important areas
of engineering an physical sciences. In
recent decades, the boundaryintegral 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 surfacemesh 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 MolinaParís
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 MonteCarlo 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 diffusiondriven instabilities (similar to
Turing instabilities) and through buoyant (or flowdriven)
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.

