## Available Project Titles

**Dr Stephen Baigent**

- Lotka-Volterra systems
- Coupled oscillator models (mostly using Matlab/ Mathematica)

**Dr Timo Betcke**

- Fasr boundary integral equation methods and their appellations
- Spectral properties of boundary integral operators

Please note that both projects require a good deal of programming in Python.

It is therefore essential that candidates have some programming background and are willing to invest effort into learning Python develpment.

**Dr Robert Bowles**

- Stability of shear flows over compliant surfaces
- Modelling social interactions with differential equations
- Motion of a slender tethered blade beneath a free surface or in a compliant channel

**Dr Christian Boehmer**

- Continuum mechanics and general relativity
- Dynamical systems and cosmology

**Professor Erik Burman**

- Investigations of numerical methods for two dimensional incompressible flow at high Reynolds number
- Different regularization methods applied to finite element methods for inverse problems

**Dr Gavin Esler**

1. Stochastic methods in advection-diffusion problems

A flecible numerical method for solving advection-diffusion problems, relevant to atmospheric transport problems, involves using ensembles of trajectories generated by solving stochastic differential equations. An aspect of this method wil be explored.

2. Statistical mechanics of point vortices

The equations of motion of point vortices are Hamiltonian, and have some interesting properties. The methods of staticstical physics can be used to predict the behaviour of the point vortex system, as a fuction of energy, when the number of vortices N is large. We will investigate these predictions in domains generated using conformal maps to the unit circle.

**Dr Jonny Evans**

- Mapping class groups of surfaces

This project concerns symmetries of two-dimensional surfaces. The group of these symmetries (diffeomorphisms considered up to isotopy) is called the mapping class group of the surface and is one of the most interesting algebraic objects in low-dimensional topology. Special cases include braid groups and SL(2,Z). Thurston proved a classification theorem for elements of mapping class groups (generalising the classification of elements of SL(2,Z) into elliptic, parabolic and hyperbolic) and the goal of this project would be to explain his proof (though you could take it in other various directions).

A good place to start (available in the maths library):

B. Farb and D. Margalit (2012) " A primer on mapping class groups", Princeton University Press

Good places to continue:

A. Fathi, F. Laudenbach, V. Poenaru (2012) "Thurston's Work on Surfaces", Princeton University Press (Engl. transl. by D. Kim and D. Margalit of "Travaux de Thurston"). Though this is not yet in the maths library, I have ordered it.

W. Thurston, "On the geometry an dynamics of diffeomorphisms of surfaces", Bulletin of the AMS (New Series), Volume 19, Number 2, 1988 (Open access online).

**Professor Rod Halburd**

- Differential equations in the complex domain

**Dr Richard Hill**

- Verify the Birch-Swinnerton-Dyer conjecture modulo 2 for a family of elliptic curves (prerequisites 3705 and 3703)
- Calculate some Iwasawa invariants, and make some corresponding deductions about the class groups of algebraic number fields (prerequisites 3704 and 3703)

**Professor Ted Johnson**

- Simple inviscid flows with geophysical applications

**Professor Francis EA Johnson**

- Cohomology groups of finite groups
- Algebraic Topology

Dr Ilia Kamotski

- Spectral problems for periodic graphs
- Topics in Homogenisation Theory

**Professor Yaroslav Kurylev**

- Topics in inverse and ill-posed problems

**Dr Jason Lotay**

- De Rham Cohomology
- Holonomy

**Dr Christian Luebbe**

- Geometrical concepts in general relativity
- Projective and conformal differential geometries

Professor Robb McDonald

1. Exact solution methods for Laplacian growth

2. Vortex motion around a sharp edge

Dr Nick Ovenden

- Biomedical Flows
- Sound transmission and propagation

**Dr Karen Page**

- Trust, reputation and the Ultimatum Game
- Students wishing to do a project in mathematical biology are welcome to come and disscuss potential projects directly with the tutor

**Professor Leonid Parnovski**

- Periodic spectral problems
- Variational approach in spectral theory

**Dr Yiannis Petridis**

- Gaps between prime numbers
- The Hardy-Littlewood circle method
- Other topics in analytic number theory
- Random matrices and moments of L-functions
- Counting problems relating to infinite groups, lattices and graphs

**Dr Mark Roberts**

- Non-commutative unique factorisation domains
- Other projects in algebra

**Dr Felix Schulze**

- Minimal surfaces and Bernstein's Theorem

**Dr Nadia Sidorova**

- Topics in probability

**Professor Michael Singer**

- Riemann surfaces and/or algebraic curves
- Differential topology
- Topics in geometric analysis: Hodge theory

**Professor Frank Smith**

- Industrial modelling problems
- Biomedical flows
- Modelling of social dynamics

**Professor Valery Smyshlyaev**

- High frequency scattering: asymptotic methods and analysis
- Multi-scale problems and homogenisation: asymptotics and analysis

**Professor Alex Sobolev**

- Spectra of compact operators

**Dr Isidoros Strouthos**

- Algebraic K-theory
- Whitehead torsion
- Thurston's eight three-dimensional geometries

**Dr Sergei Timoshin**

- Instabilities in weakly non-homogeneous systems
- Turing Instability

**Dr John Talbot**

1. Cliques in graphs

A good place to start:

V. Nikiforov (2010) The number of cliques in graphs. Please click here to download the article.

C. Reiher (2012) The clique density theorem. Please click here to download the article.

**Professor Jean-Marc Vanden-Broeck**

- Analytical and numerical studies of gravity waves of large amplitude

**Professor Dmitri Vassiliev**

- Topics in spectral theory of partial differential operators and microlocal analysis

**Dr Chris Wendl**

- Morse homology

The general idea of Morse theory is to recover information about the topology of a smooth manifold from the critical points of a smooth real-valued fuction on that manifold. Morse homology is a formalisation of this idea, where one studies the spaces of gradient-flow lines of a generic fuction in order to define algebraic invariants that are independent of the choice of function but depend on the topology of the domain. This idea has been extremely popular among symplectic and differential topologist since the 1980s, as it inspired a powerful new set of geometric invariants known as "Floer homologies", which remain an active area of research.

**Dr Helen Wilson**

- Non-Newtonian Fluid Mechanics

**Dr Henry Wilton**

- Free groups and topology of finite graphs

**Dr Andrei Yafaev**

- Topics in arithmetic algebraic geometry

**Professor Alexey Zaikin **

- Stochastic modelling of coupled repressilators

Recent advances of synthetic biology have made it possible to construct synthetic genetic networks which demonstrate oscillations. One of such classical example is a

repressilator which consists of three mutually repressive genes. Recently it was shown that coupled repressilators can demonstrate very rich dynamics e.g. in-phase or anti-phase synchronized oscillations, oscillatory death or clustering. The project is of computational nature. It is proposed to simulate coupled repressilators with Gillespie algorithm and study the influence of noise on coexisting dynamical attractors.

**Dr Sarah Zerbes**

- Brauer groups

Page last modified on 25 mar 13 10:08