NOTE: The list is updated periodically.

Also keep in mind that there is considerable overlap between different categories listed below and many academic staff work across areas. We have given cross references in many cases, but you should browse the list carefully and consider talking with more than one member of staff.

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Algebra and Group Theory

Dr Inna Capdeboscq is willing to supervise one (or at most two) research projects on a topic in Group Theory. Prospective students would have to see her to discuss further details.

Professor Derek Holt is willing to supervise one (or at most two) research projects on the topic of Computational Group Theory. The project would involve some programming. Prospective students would have to see him to discuss further details.

Dr Daan Krammer offers project in the areas of representation of groups, low-dimensional topology, and combinatorics. Subareas may include reflection groups, braid groups, knots, mapping class groups, hyperplane arrangements, Garside groups, ordered sets, ...... A precise topic for 4th year projects under his supervision are determined after consultation. The keywords above indicate what sort of topics can be expected. This list is far from being exhaustive.

Dr Diane Maclagan offers a project on Hilbert functions of ideals.

The Hilbert function is a basic invariant of an ideal in a graded ring. When the ideal is generated by r polynomials of degree d in a polynomial ring with n variables, there is a bound B(r,d,n) on the number of different Hilbert functions. The goal of this project is to investigate this bound explicitly for small values of r, d, n, and understand the locus of ideals with a given Hilbert function. This has applications to some long standing conjectures in commutative algebra and related algebraic geometry, but no background beyond Algebra II is required.

Dr John Moody is interested in canonical indeterminacy and Chern classes. Prospective students would have to see him to discuss further details.

Dr Dmitriy Rumynin is interested in Algebra and Representation Theory. Prospective students would have to see him to discuss further details.

Dr Marco Schlichting is interested in Algebraic K-theory and higher Grothendieck-Witt groups of schemes, A^1-Homotopy Theory and Motivic Cohomology, Derived Categories, algebraic topology and algebraic geometry. Prospective students would have to see him to discuss further details.

Algebraic number theory

Dr David Loeffler would be willing to supervise one or possibly two projects next year in the field of algebraic number theory. Possible topics are:

Arithmetic of p-adic number fields The field of p-adic numbers is a very important tool in algebraic number theory, giving a bridge between finite fields (which are nice and simple and easy to work with) and the rational number field (which is much more sophisticated). The aim of this project would be to understand some of the basic properties of the fields of p-adic numbers and their finite extensions (such as Hensel's lemma, ramified and unramified extensions, etc); from there the project could develop in several directions, such as the theory of Witt vectors, higher ramification groups, p-adic analysis, or the beginnings of p-adic Hodge theory.

The Birch-Swinnerton-Dyer conjecture. The BSD conjecture is a well-known conjecture about elliptic curves, relating the rank of an elliptic curve to the order of vanishing of its L function. A successful essay should start with an exposition of the conjecture itself, and go on to cover some of the partial results in the direction of the BSD conjecture, such as the work of the Dokchitser brothers which establishes the conjecture modulo 2 using properties of integral group representations.

Dr Adam Epstein offers projects in Dynamical Systems and Number Theory. For example,

Arithmetic Questions in Holomorphic Dynamics: Consider the polynomials F_n(c) = p_c o ... o p_c(0) (n-fold self-composition) where p_c(z) = z^2 + c. It is known that all roots of F_n(c) are simple. The polynomial F_n splits into factors, some of which arise as F_m for smaller m dividing n: when such factors are divided out, the resulting polynomials are conjectured to be irreducible. Questions of this nature arise for other interesting families of rational maps, and little is known in general. Well-organised computer experimentation would be a good start. This would be an appropriate project for a student who has taken, or will be taking Algebraic Number Theory (or Galois Theory). Familiarity with basics from Complex Analysis and Dynamical Systems would also be useful.

Algebraic Geometry

(See also Dr Marco Schlichting)

Professor Miles Reid has a variety of possible projects on offer related to modern research in algebraic geometry, and to MA4A5 Algebraic Geometry or MA426 Elliptic Curves. He can also propose problems related to advanced topics in Galois theory, commutative algebra and algebraic number theory for suitably motivated students.

Finite subgroups of SL(2, CC) and SL(3, CC)
Explicit generators, invariant theory. Work of Klein around 1870, but recently developed in many directions. Start from my preliminary chapter on cyclic quotient singularities. See for example my Bourbaki seminar or the references given on my McKay correspondence website.

Quaternions, octonions, special geometric structures and exceptional Lie groups
See any introductory text on quaternions (for example, Balazs Szendroi and Miles Reid, Geometry and Topology, Chapter 8), followed by John C Baez, The Octonions, 56 pages, preprint available from uk.arXiv.org as math.RA/0105155, and John H. Conway and Derek A. Smith, On quaternions and octonions: their geometry, arithmetic, and symmetry, A K Peters, Ltd. 2003 IBSN 1568811349

Topology, projective geometry and enumerative geometry of Grassmann varieties and projective homogeneous spaces.

Applications of Riemann Roch on curves and surfaces
Graded rings, computer algebra calculations. Computer enumeration of graded rings of interest in algebraic geometry. Work of mine and my students. It contains lots of fairly simple minded combinatorial problems that lend themselves to computing. Start from my preliminary chapter on graded rings, then look at Gavin Brown's GRDB website.

Dr Diane Maclagan offers projects in toric or tropical geometry. Other areas of algebraic geometry or computational algebra may be available by arrangement. (See also Combinatorics section below)

1. Classification of low dimensional toric varieties. Toric varieties are algebraic varieties whose geometry is determined by some combinatorics (a lattice polytope). These are (partially) classified in low dimension and low Picard rank. The goal of this project would be to implement this classification to get a database for use in computer experimentation. This project can be approached from a geometric or a purely combinatorial direction, but anyone interested should plan to take MA 4A5 (Algebraic Geometry). No previous programming experience is required, as programming will be in the computer algebra system Macaulay

2. Tropical geometry. Tropical Geometry is an emerging area at the intersection of algebraic geometry and polyhedral combinatorics with applications to other areas. At its most basic, it is geometry where addition and multiplication are replaced by minimum and addition respectively. This turns familiar geometric objects, such as circles, into piecewise linear objects, which can be studied using combinatorial methods. See here for some pictures, and an idea of what this topic would be like. As for the previous project, willingness to work across several areas is required.

Professor David Mond’s research interests lie in the area of singularity theory and algebraic geometry. Prospective students would have to see him to discuss further details.

Dr Damiano Testa is interested in Algebraic geometry and Number Theory. Prospective students would have to see him to discuss further details.

Analysis

Dr John Erik Andersson Prospective students would have to see him to discuss further details.

Professor Keith Ball offers a project on

Restricted Invertibility: About 20 years ago Bourgain and Tzafriri showed that given $n$ unit vectors $u_1,u_2,...,u_n$ in Euclidean space satisfying an inequality a bit like an orthonormal basis \[ \| \sum \lambda_i u_i \|^2 \leq M^{^2} \sum \lambda_i^{^2} \] it is possible to select a constant proportion (depending upon $M$ but independent of $n$) that behave almost exactly like an orthonormal basis. Recently two computer scientists, Spielman and Srivastava, found a remarkable new proof of this which has a wide range of applications to algorithmic problems on graphs. The proof is motivated by a kind of discrete potential theory and is completely constructive. The aim of the project will be to study and compare the different approaches to the problem. The main prerequisite is Functional Analysis I but Combinatorics or Matrix Analysis might be helpful.

Dr Davoud Cheraghi offers the following project:

Analytically natural notions of length in the complex plane:The purpose of this project is to study the notions of "extremal length" and "extremal width" in the complex plane. Naturally paused questions of geometry and analysis on the complex plane are invariant under the (local or global) holomorphic maps, for example rescalings. A simple example is the distance between two ball in the complex plane. However, the usual notions of length, for example Euclidian distance, are not invariant under such maps. The project is about introducing more natural notions, and their applications. This is related to other areas of mathematics like probably (random walks), but need not be discussed here.

Professor David Preiss’s research interest lie in the area of mathematical analysis. Prospective students would have to see him to discuss further details.

Applied Analysis

Dr Jose Rodrigo’s research interests lie in the area of partial differential equations and fluid mechanics. Prospective students would have to see him to discuss further details.

Professor James Robinson's interests are in rigorous fluid dynamics and turbulence; infinite-dimensional dynamical systems; random dynamical systems; non-autonomous dynamical systems; embeddings of finite-dimensional sets into Euclidean spaces. Prospective students would have to see him to discuss further details.

Combinatorics

Dr Vadim Lozin offers a project on Structural graph theory. Prospective students would have to see him to discuss further details.

Dr Diane Maclagan

The counter-example to the Hirsch conjecture: The Hirsch conjecture was a conjecture about the complexity of linear programming that was open for fifty years until disproved two years ago. The goal of this project will be to understand this conjecture, why it was important, and most importantly to understand the counter-example. See: http://personales.unican.es/santosf/Hirsch/ for the main paper and some background surveys.

Dr Oleg Pikhurko’s research interests lie in the area of combinatorics and graph theory, in particular in applying tools from probability, analysis, and algebra to discrete problems. This is a deep and thriving area, with many possible projects that can help you to prepare for PhD studies. Prospective students should contact Dr Pikhurko by email.

Dr Bruce Westbury offers the following two topics:

1. Dilogarithm: This is an interesting special function with several elementary definitions. The aim of the essay is to present applications to volumes of hyperbolic tetrahedra. For example, the dilogarithm satisfies a functional equation which has a straightforward interpretation in terms of volumes of ideal tetrahedra. The dilogarithm is also related to volumes of orthoschemes (analogues in 3D of right-angled triangles). This is discussed in:
   Vinberg, È. B. The volume of polyhedra on a sphere and in Lobachevsky space. Algebra and analysis (Kemerovo, 1988), 15--27, Amer. Math. Soc. Transl. Ser. 2, 148, Amer. Math. Soc., Providence, RI, 1991.

2. Cyclic Sieving Phenomenon: This is a branch of combinatorics which studies finite sets with an action of a cyclic group. The orbit structure is encoded by a polynomial. This only requires the character theory of cyclic groups. However studying interesting examples seems to lead to deep mathematics.
   There is an excellent survey by Bruce Sagan.

3. Circle packing (Note this is *not* related to the problem of fitting as many circles into a region as possible.) An excellent introduction to this topic with plenty of pictures is Introduction to circle packing. The theory of discrete analytic functions by Stephenson, Kenneth. Cambridge University Press, Cambridge, 2005. Since this book was published there have much clearer proofs of the main theorem. The theory of circle packings gives a discete theory of analytic functions. The aim of the essay would be to present a proof of the main theorem or an account of an application of circle packing to analytic functions. For example, the uniformisation theorem can be deduced from the main theorem of circle packing.
   
   Stephenson, Kenneth . Introduction to circle packing. The theory of discrete analytic functions. Cambridge University Press, Cambridge, 2005
   Bobenko, Alexander I. ; Springborn, Boris A. Variational principles for circle patterns and Koebe's theorem. Trans. Amer. Math. Soc. 356 (2004), no. 2, 659--689.
   Rivin, Igor . Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. of Math. (2) 139 (1994), no. 3, 553--580.
   Stephenson, Kenneth . Circle packing: a mathematical tale. Notices Amer. Math. Soc. 50 (2003), no. 11, 1376--1388.

4. Rolling Balls. There has recently been a spate of papers explaining how the exceptional simple Lie group G_2 appears in the mechanics of two balls rolling without slipping or spinning on each other but only if the radii are in the ration 1:3.

Differential Geometry and PDE

(See Applied Analysis above and areas such a Fluid Dynamics and Computational Mathematics below for further topics in PDEs).

Dr Mario Micallef’s research interest lie in the area of partial differential equations and differential geometry. Prospective students would have to see him to discuss further details.

Professor Peter Topping offers various options in geometric analysis. There will be many different challenging projects possible in this active area. The suitable student will have interest both in differential geometry and in PDE theory (and should normally be taking Advanced PDE). It might be some effort to understand enough to start the project, but it should leave you in a great position to start PhD research in the area.

Ergodic Theory and Dynamical Systems

(See also Dr Adam Epstein, Professor James Robinson, and Dr David Wood)

Dr Claude Baesens's interests are in dynamical systems and applications to physics, and in exponential asymptotics. See is offering a projects on Frenkel-Kontorova models canard solutions in slow-fast dynamical systems. Prospective students would have to see her to discuss further details.

Dr Davoud Cheraghi offers the following projects:

1. Renormalization through toy models:The renormalization is an important tool in dynamical systems to describe the fine scale geometry of dynamically defined objects. Roughly speaking, recurrence is the main source of complication in dynamics, and renormalization is an approach to understand this obstruction. That is, one defines a new dynamical system by only considering the return map to a small domain (and taking it out of the large picture) and studying it on its own as a new dynamical system. Quite often one repeats this process and defines an operator on the space of dynamical systems, that associates the rerun map to a given system. However, this usually leads to very complicated problems of analysis and distortion. The main purpose of the project is to understand the renromalization technique by recursively replacing the maps obtained from renormlization operator by toy models (say by linear, power, or exponential maps) that closely resemble the behaviour of the actual maps.

2. Visualisation in Dynamics: Visualisation of dynamically defined objects by computers. Requires basic knowledge in C++.

Professor Vassili Gelfreich’s research interests lie in the area of dynamical systems. Prospective students would have to see him to discuss further details.

Dr Oleg Kozlovski is interested in Dynamical systems, ergodic theory, mathematical physics, financial mathematics. Prospective students would have to see him to discuss further details.

Professor Mark Pollicott is interested in Ergodic theory, with applications to analysis, number theory, and geometry. Prospective students would have to see him to discuss further details.

Geometry, Topology and Geometric Group Theory

Professor Brian Bowditch is interested in hyperbolic geometry, low-dimensional topology, geometric group theory. Prospective students would have to see him to discuss further details.

Professor John Jones is interested in geometry topology, gauge theory, K-theory, cyclic homology, index theorems. Prospective students would have to see him to discuss further details.

Professor John Rawnsley is interested in symplectic geometry, geometrical methods of quantisation and the study of Lie groups. Prospective students would have to see him to discuss further details.

Dr Saul Schleimer is interested in geometric topology, group theory, and computation. Prospective students would have to see him to discuss further details.

Complexity

Dr Colm Connaughton welcomes interested students just come and talk to him. Below are some possible titles:

1. Self-similar solutions of nonlinear diffusion equations with spatially varying diffusivity.
2. Low dimensional dynamical system models of Rossby wave interactions in atmospheric dynamics.
3. Parameter estimation for resonant wave interactions in the presence of external noise.
4. Modelling the interaction between zonal flows and drift waves in fusion plasmas.
5. Mean field theory of agglomeration-fragmentation processes in clouds.
6. Stochastic models of consensus formation in online social networks.

7. Lot's of problems to do with fluid dynamics and/or turbulence if someone is particularly interested in these topics.

Dr Leon Danon offer projects in the following areas:

1. The evolution of musician collaboration networks
2. Spread of disease and influence informed by mobile phone data.
3. Methods for detecting communities in dynamic complex networks
4. Models of respondent driven sampling

5. Commuting patterns in the UK

Dr Stefan Grosskinsky's interests are in applied probability theory, stochastic processes and complex systems, statistical mechanics. Prospective students would have to see him to discuss further details.

Dr Markus Kirkilionis's interests are in complex systems, mathematical biology, dynamic network models, numerical analysis, pattern formation, physiologically structured Population models, (monotone) dynamical systems. Prospective students would have to see him to discuss further details.

Professor Robert MacKay offers projects in a range of topics on applications of mathematics:

1. Statistics of foams for bullet-proof vests: Mark Williams (WMG) has an X-ray tomography machine with which he can produce a data-structure representing any object he puts in it. One example he showed me is a shear-stiffening foam used in the latest generation of bullet-proof vests. The question the manufacturer asks is what features of the distribution of the holes lead to good or poor performance. So first we need a good way of extracting relevant statistics from the images. One that would be neat is to apply "persistent homology": construct a Cech complex from the data for the interface between material and air and determine the Betti numbers as a function of precision. The zeroth Betti number should boil down to the number of holes of size bigger than epsilon. Google any words you don't recognise. You'll probably get to a nice review paper by Carlsson.

2. Calibration method for selection of Erasmus Mundus applicants: When a panel of assessors evaluates a set of objects they often divide the labour but then end up with a calibration problem. I propose a method (you can find a draft paper on it on my website) which should provide automatic calibration if the bipartite graph of assessors and objects is sufficiently well connected. It allows the assessors to give confidence intervals not just scores. The project is to write a computer package to implement the method and if you have it ready in time I'd like to try it out as a support tool for this year's applications to our Erasmus Mundus Masters in Complex Systems Science.

3. Hierarchical aggregation method for fast computation of shortest paths in a graph: CAMVIT, a sat nav route finding company in Cambridge, is keen to interact with us on developing faster ways to find shortest routes, which could cope with real-time traffic updates. I propose a hierarchical aggregation method but there are many features which remain to be determined (plus there is some literature on this already). The idea is that a divide and conquer strategy could lead to significant speed-up and also save recalculating everything when only one part changes. This is pretty open-ended but probably someone from DIMAP could help point to shortcuts and dead-ends to avoid, and CAMVIT can point to data sources to try any resulting method on.

4. Singularities of a robot arm: Nikon Metrology (Derby) produce a robotic arm for measurement in complex environments (e.g. through the windows of car frames) . It consists of a 7-axis arm inside a 6-axis exoskeleton, to separate the measurement from the driving, with three constraining rings to couple the two. The map from configurations of the whole system to configurations of the exoskeleton has singularities, however (points where the derivative has less than maximal rank), and driving into these produces large stresses. For a 2-axis in 2-axis exoskeleton version we computed the typical singularity set (paper on my website), suing an astute trick to reduce it to a 3D problem which Mathematica could plot. The project consists in extending this to 4-axis in 4-axis exoskeleton, for which the trick is not valid and thus notionally plotting must be done in 8D either by writing your own method to build the 4D configuration manifold in 8D, or perhaps by figuring out how to work Multifario, a package produced by Henderson that should be able to do this. Of course, you may end up projecting to 3D to visualise the result but I think the computation is going to have to be done in 8D.

5. Crowd psychology of biomolecular motors: Rob Cross (Centre for Mechanochemical cell biology) takes fantastic videos of kinesins running along microtubule networks, jamming each other and falling off. Build a virtual world of automata interacting with a track network and each other based on rules that come from the experiments, and try to gain some understanding of what dynamical regimes can arise and their dependence on the rules. There is scope to develop some theory here too (cf. the forthcoming PhD thesis of Paul Chleboun on circular tracks), but we'd have to start with relatively simple networks and rules first.

6. Subversive group formation and fragmentation: Neil Johnson (Miami) has a model for how terrorist and other groups who want to avoid detection but achieve some coordinated actions grow and if spotted fragment to limit the damage. Google to find his papers. Devise an experiment to test this. Gordon Brown (Psychology dept, not the past-PM!) can advise on design of the protocol and Decision Research @ Warwick may be able to provide use of an experimental facility, though you may prefer to do it all on the web. Your results may suggest refinements to the model, in which case it would be good to analyse their effects.

7. Stability of electricity pricing systems: With the growth of distributed generation and of smart-metered consumers there is going to be a need for a real-time (and spatially dependent) pricing signal. Michael Caramanis (Boston University, a partner of Warwick) has some papers on this (google him), The question is how real producers and consumers will respond to the price signal and what instabilities the resulting behaviour might produce. Ultimately we'd want to design a robust pricing system, but demonstrations of what can go wrong are vital at this stage. If we have something to show we will try it out on National Grid or E.ON.

8. Protein configurations: The protein backbone can be idealised as having rigid bond lengths and angles, trans peptide bonds and two torsion angles per amino-acid (or just one for a proline). Given a sequence of N amino-acids between clamped ends the configuration manifold has dimension 2N-6. You can describe it succinctly with quaternions. The question is what you can say about it qualitatively and about its typical bifurcations as one end is moved relative to the other. If you get something interesting then David Wild (Systems Biology) will be interested in having you use your knowledge to design localised Monte Carlo moves to improve on his current "crankshaft" moves for determination of protein folding structure.

Computational Mathematics and Numerical Analysis

Professor Dwight Barkley is interested in many areas of applied and computational mathematics and can offer computational projects in pattern formation and fluid dynamics. Students must be able to program in a high-level language C/C++/Fortran. Prospective students would have to see him to discuss further details.

Dr Andread Dedner's interests are in numerical analysis and scientific computing, with particular emphasis in high-order methods for non-linear equations and applications in geophysical flows, radiation magnetohydrodynamics, and reaction-diffusion equations. Prospective students would have to see him to discuss further details.

Professor Charles Elliott research is centred around nonlinear partial differential equations and computational mathematics with applications(mathematical biology, material science, continuum mechanics, phase transitions etc) including numerical analysis and applied analysis. In particular, finite element methods, free boundary problems, geometric evolution equations and surface growth, two phase flow, cell motility, biomembranes and PDE optimisation. Prospective students would have to see him to discuss further details.

Dr Björn Stinner is interested in free-boundary problems and PDEs on manifolds, applied analysis of nonlinear partial differential equations,
finite element methods and their numerical analysis, continuum modelling, particularly based on the phase field methodology. Prospective students would have to see him to discuss further details.

Professor Andrew Stuart works in applied and computational mathematics, with particular interests in Inverse Problems for Differential Equation (both Deterministic and Stochastic), mainly arising from applications in the physical sciences. Prospective students should contact him for further details.

Fluid Dynamics

(See also Dr Colm Connaughton and the Applied Analysis and Computational Mathematics sections above)

Dr Xinyu He offers the following project on the Geometry of the 3d Euler equations

Let v(x,t) be a solution to the Euler equation, and denote the vorticity by curl v(x,t) whose direction is shown to control singular growth. It has been really challenging to understand how the curvature and the torsion of vortex lines affect the smoothness of solutions. The project is to analyse geometrical structures of the system in terms of Serret-Frenet frame, for this purpose there are also complementary numerical data available.

Professor Sergey Nazarenko is interested in turbulence; vortex and wave dynamics; Navier-Stokes; MHD and NLS equations; plasma; superfluids. Prospective students would have to see him to discuss further details.

Professor Robert Kerr's current interests include the mathematics of the turbulent energy cascade and conditions for singular behaviour in fluid and related equations, including the three-dimensional nonlinear Schroedinger equations for quantum fluids and atmospheric wave equations.

The perspective of the project is numerical and would be based on the latest high-resolution calculations. Potential topics include comparing some of the new statistics generated by these simulations to the latest proofs from rigorous mathematics for singularities, or visualising the underlying structural alignments that have either been assumed or shown analytically must exist if these equations have singularities. For example, I would like to understand helicity, a topological measure of the twisting of vortex lines. This can be represented from either in physical space, in which case the Frenet-Serret equations for the curvature of vortex lines would be used, or from a Fourier perspective.

Dr Oleg Zaboronski offers a project related to

Metastability in stochastic Euler dynamics. During this project, the student will learn about:
1. Arnold's algebraic formulation of 2d Euler equation
2. Zeitlin's su(N) approximation of 2d Euler flow
3. Stochastic Euler equation on su(N)
4. Control of stochastic Euler flow on su(N) using gradient dissipation built out of integrals of motion and the associated invariant measures.
5. Dynamics of stochastic Euler flow on su(N) for multimodal invariant measures using Wentzel-Fradkin theory.

Mathematical Biology, Epidemiology, Ecology and Evolution

(see also Dr Leon Danon)

Dr Thomas House would like to offer a project on Modelling Infectious Diseases: Increasingly, mathematical models are used to inform policy and give greater scientific understanding of infectious diseases. These include 'emerging' infections like SARS or pandemic influenza, and also established, 'endemic' diseases like STIs and seasonal influenza. A large variety of mathematical and computational techniques are used in the modelling, and a project in this area would be suitable for a student looking to apply techniques learned during their undergraduate studies to issues of immediate public-health significance.

Professor Matt Keeling offers projects in the areas of Epidemiology,Ecology or Evolution. Ecology (the study of animal/plant populations and their environment), Epidemiology (the study of disease spread and control) and Evolution (the study of the long-term dynamics of populations) present a wide variety of interesting problems that require a mathematical approach. There are too many questions and problems to list here, so it would be simplest if you came to talk to me and then I tried to fit your interests and experiences to the problems available.

Dr Dave Wood would be willing to discuss possible topics given the information below. His main interests for projects include ecological type modelling, investigating systems with symmetry (including but not limited to applications to arthropod locomotion) and applications of mathematics in industry. The industrial applications could cover a broad range of mathematical disciplines and be a survey of some problems that have already been studied or a look at a new problem involving original research. Systems with symmetry could be theoretical or applied, but would concentrate on using equivariant bifurcation theory and so having taken MA240 Modelling Nature's Nonlinearity and MA371 QTODE would be highly desirable. I would also be interested in formulating a more innovative project based on use of Web 2.0 and virtual worlds for teaching mathematics to undergraduates and/or gifted and talented, although the feasibility of this will only be looked into if someone keen on the idea comes forward (this may be of interest to students who have taken IE2A6 Introduction to Secondary School Teaching).

Recent titles he has supervised include: “Coupled cell networks, bifurcations and symmetry”, “The effects of tuna fishing on dolphin populations”, “Symmetry in coupled cells and neuronal networks”, “Applications of maths in industry”, “Discrete maths in industry”, “Modelling the Future of the Hawaiian Honeycreeper: An Ecological and Epidemiological problem”.

Dr Hugo van den Berg offers projects in the area of Mathematical Biology. Prospective students would have to see him to discuss further details. Prospective students would have to see him to discuss further details.

Professor Nigel Burroughs is interested in application of mathematics and statistics to biological and medical problems. Prospective students would have to see him to discuss further details.

Professor David Rand's main research interest is Systems Biology, particularly understanding the design principles of regulatory and signalling systems in cells. Prospective students would have to see him to discuss further details.

Mathematical Physics, Molecular Dynamics, and Statistical Mechanics

Dr Stefan Adams's interests are in large deviation theory, probability theory, Brownian motions, statistical mechanics, gradient models, multiscale systems. Prospective students would have to see him to discuss further details.

Professor Roman Kotecky's interests are in discrete mathematics, probability theory, statistical mechanics, mathematical physics. Prospective students would have to see him to discuss further details.

Dr Florian Theil offers project in the area of molecular dynamics which is an active and fast developing field in Applied Mathematics. The objective is to obtain insight into complex molecular systems by means of computer simulations instead of using experimental techniques. Due the high dimensionality the simulations are very costly and consequentially analytical results can provide valuable insights which potentially improve the efficiency of the numerical approaches. The proposed fourth year projects involve the application of tools from Stochastic Analysis and PDE theory to MD systems.

Dr Daniel Ueltschi's interests are in analysis, probability theory, statistical mechanics, mathematical physics. Prospective students would have to see him to discuss further details.

Multiscale Methods

Dr Christoph Ortner offers 4th year project(s) on the analysis and numerics of molecular mechanics and multiscale models. Possible projects may involve any combination of the model formulation, numerical computations or the analysis of a mathematical model. Prospective students would have to see him to discuss further details.

Stochastic Analysis and Probability

(see also Dr Oleg Zaboronski, Professor Andrew Stuart, as well as the Complexity and Mathematical Physics)

Professor Neil O'Connell's interests are in Brownian motion, random matrices and related topics. Prospective students would have to see him to discuss further details.

Professor Martin Hairer's interests are in stochastic PDE's, stochastic analysis, functional analysis, homogenisation theory. Prospective students would have to see him to discuss further details.

Dr Xue-Mei Li would be happy to supervise a 4th year project in stochastic analysis. She is interested in Markov processes on Riemannian manifolds, geometry of stochastic processes and of second order differential operators on manifolds, stochastic differential equations, Malliavin Calculus, analysis of path spaces, limit theorems of stochastic processes, averaging, homogenisation, numerical computations of stochastic differential equations and coalescing flow. Specific projects are:

1. Stochastic Differential equations driven by Levy Processes: This topic is ideal for somebody who wishes to take further the study of the subject in Stochastic Analysis (MA482). The aim is to understand Stochastic differential equations driven by Brownian motion plus stochastic process that jumps (discontinuous in time).

2. Stochastic Differential Equations on Heisenberg Group: This topic is ideal for somebody who wish to take the study of the subject in Stochastic Analysis (MA 482 ) further, however it can be taken by somebody who has interests in Lie groups, manifolds, and theory of measure. Heisenberg group is a Lie group that has structural constants 1, 0,0. There is one hypoelliptic operator, the hypoelliptic Laplacian, related to a basis of the Lie algebra. The hypoelliptic Laplacian missing one direction from the usual Laplacian. The associated stochastic differential equation is hypoelliptic, which means that the distribution of the stochastic process at each time is absolutely continuous with respect to the Lebesque measure and hence has a kernel. We investigate this SDE and its solutions, the hypo-elliptic Brownian motion. A route that does not involve stochastic differential equation theory can also be taken: we would look into the Carnot-Caratheordory distances, sub-Riemannian metrics, sub-Riemannian geodesics, Bakry-Emery condition, hypoelliptic `heat kernels'. The study of the joint distribution of a 2-dimensional Brownian motion and the corresponding stochastic area is also an interesting subject of study in conjunction with the space of measures with Wasserstein distance associated to the sub-Riemannian metric.

3. Weak gradients and functional inequalities in the space of probability Measures and Mass Transport Problem (2 projects): Given a complete separable metric space and a real valued function, weak gradient can be defined. There are two interesting metric spaces on which this theory can be applied. The first is the space of all probability measures on Rn or on a more general finite dimensional manifold with Wasserstein distance. The Wasserstein distance may be related to the usual distance function of the Riemannian structure on the finite dimensional manifold or associated to other geometrically constructed distance function, e.g. the Carnot-Caratheordory distance for a sub-elliptic structure. Associated problem includes the gradient flow of the Entropy of a measure relative to a reference measure. The other example is Wiener space: the space of all continuous curves on Rn with time running from 0 to 1 with supremum distance and the study related to Malliavin Calculus. We aim to understand and apply the theory of weak gradients to one of the above mentioned examples.