The following is a list of current opportunities (with contacts) for PhD study within
the Applied and Computational Mathematics group:
High performance computers are increasingly based on complicated interconnect and memory access architectures. Where the traditional "von Neumann machine" processes data in a sequential fashion, these new, evolving computer systems benefit from concurrent number crunching in order to speed up large simulations. At least that is how things should work; in practice the design of numerical methods and software has not kept pace with the new hardware technology, so much of the current fleet of CPUs sits idle for much of its useful life or is performing a lot of redundant busy-work. The aim of the centre is to develop new numerical methods, computer algorithms, along with their mathematical analysis and their implementation in computer software.
NAIS has a budget of over £ 8M dedicated to finding new, innovative approaches for these challenging tasks. A large team of mathematical scientists will be assembled and focussed on all aspects of the problem. Mathematics research students (with basic training in computer programming and exposure to numerical analysis concepts through undergraduate courses) are needed to join our team, to help design, implement and analyse new methods and algorithms.
A very innovative aspect is the linking of the work on the mathematical side with work by computer scientists on new types of advanced compilers.
See
Contacts: Ben Leimkuhler (
The idea is to look at applying geometric methods to control non-linear dynamical systems; to observe non-linear systems subjected to random noise; and to optimally control non-linear systems, especially in the presence of noise.
Contacts: Ben Leimkuhler (
Due to recent experimental progress, the field of mathematical biology is rapidly growing. There
are plenty of biological systems where mathematical models and analysis are needed. Closely
interacting with experimentalists, the PhD candidate would formulate and analyze models of
cancer progression, virus dynamics, bacterial evolution, and possibly other systems related to
molecular motor motion or the origins of life. The project starts with first building and exploring
simple model systems, and continues with their study by computer simulations and analytical
methods. Knowledge of the biological background is not necessary at the beginning, but one will
eventually learn some biology in order to do relevant research.
See also http://www.maths.ed.ac.uk/~antal
Contact: Tibor Antal (tibor.antal@ed.ac.uk)
Biology is a fast growing area for applications of probability. Since still in its infancy, there are many
unexplored areas and open problems. In particular, there is a great interest in stochastic models of
cancers. This PhD project would focus on understanding the most basic and fundamental models
of tumor progression. These models include branching processes, other models borrowed from
population genetics, or spatial Poisson processes. The work has a light numerical aspect to it, but
would focus more on finding exact solutions, and establishing limit theorems. No knowledge of
biology required.
See also http://www.maths.ed.ac.uk/~antal
Contact: Tibor Antal (tibor.antal@ed.ac.uk)
There are several topics available for PhD research: The first topic
is Exponential Asymptotics. Usually asymptotic expansions are divergent, and
that means that exponentially small phenomena are hidden in the tail
of the divergent series. In exponential asymptotics we make these
exponentially small contributions visible. In this way we obtain much
more accurate approximations and increase the regions of validity, we
obtain the most powerful method to compute the so-called Stokes
multipliers, and in many applications explain (and compute) the
appearance of oscillatory behaviour when certain Stokes lines are
crossed. Topics available are exponential
asymptotics for linear or nonlinear ODEs and PDEs with a small parameter.
The second topic is Uniform Asymptotics for Difference Equations. This
field of research is relatively new. For differential equations and
integrals it is well-known how to obtain uniform asymptotic
expansions. These expansions are valid in large regions, and especially near
critical points where the behaviour of the solutions changes dramatically.
There are some results in the literature for difference equations,
but none of them is as simple or as powerful as what is known for
differential equations and integrals. Hence, any new result is worth
publishing.
Contact: Adri Olde Daalhuis (a.oldedaalhuis@ed.ac.uk)
This project deals with a recently discovered oscillatory mode in the evolution of rich stellar systems, like the globular cluster NGC6397.
The aim is to identify the mechanism which drives this mode, using both N-body computer simulation (using a Graphics Processing Unit for high- performance computation) and simplified dynamical systems based on models of chaotically driven oscillators.
Contact: Douglas Heggie (d.c.heggie@ed.ac.uk)
Molecular dynamics is the modelling of the motion and structure of materials, biomolecules (e.g. DNA fragments or protein chains) using a relatively simple set of physical laws such as potential energies of interaction between atoms. Molecular dynamics only yields useful information when the boundary conditions and environmental parameters like temperature and pressure are correctly maintained. Current methods often perform erratically. The goal of this project is to design better molecular models that provide more useful outputs, as well as better numerical methods and tools for using high performance computers for molecular simulation. The project is funded and jointly supervised by chemist Carole Morrison.
Contact: Ben Leimkuhler (b.leimkuhler@ed.ac.uk)
STFs are nanostructured materials with unidirectionally varying properties
that can be engineered to exhibit exotic electromagnetic properties. To date, the chief applications of STFs are in optics as polarization filters, Bragg filters, and spectral hole filters. Applications such as biosensors, tissue scaffolds, drug-delivery platforms, virus traps, and labs-on-a-chip are also envisaged.
For further details, see
Contact: Tom Mackay (
t.mackay@ed.ac.uk)
A composite medium may viewed as an effectively homogeneous medium
provided that wavelength(s) are sufficiently large. Significantly,
such homogenized composite mediums may exhibit properties not exhibited
by their components. Recently, several interesting results have emerged
relating to metamaterials.
For further details, see
Contact: Tom Mackay (t.mackay@ed.ac.uk)
A range of exotic and potentially useful phenomenons, such as negative
refraction, inverse Doppler effect, inverse Cerenkov radiation, and so
forth, are associated with the propagation of electromagnetic plane
waves with negative phase velocity (NPV). While much research to date
has been directed towards NPV propagation in metamaterials, recent
theoretical results suggest interesting possibilities in astronomical
scenarios.
For further details, see
Contact: Tom Mackay (
t.mackay@ed.ac.uk)
Mixed-mode dynamics is a type of complex behaviour which is frequently encountered in systems of ordinary differential equations that involve multiple scales in space or time. Mixed-mode oscillations are characterised by time series in which small-amplitude (sub-threshold) oscillations and large-amplitude (relaxation-type) excursions alternate. The recently developed geometric approach to multiple-scale problems, which combines geometric singular perturbation theory and the desingularisation technique known as blow-up, is particularly well-suited for the analysis of mixed-mode behaviour. That geometric theory is fairly complete for fast-slow differential equation models from mathematical neuroscience that are based on the classical Hodgkin-Huxley formalism. The irregular firing patterns observed in these models are often of mixed-mode type and can be explained by reducing the models to normal forms which, though analytically simpler, still capture the essential dynamics of the full equations. For a more complete classification of the possible mixed-mode dynamics in Hodgkin-Huxley type models, a "toolbox" of such normal form systems needs to be developed and characterised geometrically. Other relevant questions in this context concern the (potentially chaotic) dynamics of the discrete maps induced by these continuous systems, the geometric characterisation of related firing patterns, such as bursting, and the efficient reduction of high-dimensional models to their low-dimensional normal forms.
Contact: Nikola Popovic
(nikola.popovic@ed.ac.uk)
The mysterious gamma-ray bursts are the largest explosions since the
big bang. They happen on average once a day and last for only a few
seconds. One scenario to explain these cataclysmic events involves the
collisions of neutrons stars and black holes. The detailed numerical
calculations are done on powerful computers to try to simulate the
collisions and then explain these bursts. The project would involve
first to get acquainted with the computer code and the astrophysical
background, and then to extend the code to include new physical
effects and/or explore the parameter space.
See
Contact: Max Ruffert (m.ruffert@ed.ac.uk)
Nonlinear beams in liquid crystals, termed nematicons, hold great promise as basic elements of all-optical devices, such as re-configurable circuits and all-optical logic elements. There has been a great deal of experimental work in this area, but little mathematical theory. Mathematical modelling can play a vital part in the on-going development of optical devices based on liquid crystals as both a predictive tool and a tool for understanding physical mechanisms. The main current research areas are the interaction of beams and the interaction of beams with varying media as a method for trajectory shaping. The mathematics involves the analysis and numerical solution of systems of nonlinear partial differential equations.
Contact: Noel Smyth (n.smyth@ed.ac.uk)
Scalar advection. This project explores the evolution of the concentration of
constituents in complex flows that are modelled by random processes. Interesting
questions concern the statistics of extreme events, such as the persistence of
anomalously high concentrations in spite of the mixing.
Radiation-induced complexity. In many fluids (from relativistic fluids to
superfluids), wave radiation and the energy loss it induces lead to complex, turbulent
dynamics. Conservation laws and asymptotic methods can be used to examine
how wave radiation destabilises some flows and predict their long-term behaviour.
Acoustic mixing. This project considers methods of driving complex, mixing fluid
flows using sound waves.
Contact: Jacques Vanneste (j.vanneste@ed.ac.uk
)
Applications are invited for a 3-year PhD studentship titled "Spatial and stochastic modelling of algal and plant circadian clocks" in the group of Dr. Ramon Grima at the Centre for Systems Biology in Edinburgh (CSBE) starting in October 2009. This focused project has as its main aim, the study of the combined impact of molecular noise and spatial compartmentalization on the dynamics of the circadian clock in individual algal and plant cells. This will be a first in plant systems and computational biology since standard models ignore both factors. The student will work closely with the Millar and Nagy labs, co-located in CSBE, to utilize experimental data of the plant Arabidopsis thaliana and the algae Ostreococcus tauri to identify intracellular compartments central to the clock's function; this data will be used to guide the construction of the initial stochastic models. Subsequently the models will be analyzed using both theoretical methods and computer simulation and their output compared with that of standard non-spatial and deterministic models to assess the impact of noise and spatial organization on the working of the clock. The student will additionally benefit from the excellent interdisciplinary atmosphere of CSBE which is housed in a purpose-built facility on the Kings Building campus of the University of Edinburgh.
Contact: Ramon Grima (School of Biological Sciences,
ramon.grima@ed.ac.uk)
There is also the possibility for a more numerically oriented PhD Project under NAIS involving Ramon Grima and Nikola Popovic on mesoscale modelling of complex enzyme reactions.
