The list will be updated, please come back later if you want to see more topics.

Pure Mathematics

Algebraic number theory

Professor John Cremona (is likely to be already oversubscribed) offers a number of projects in number theory or elliptic curves, including the following:

The Class Number One problem for imaginary quadratic number fields is to prove that the only such fields with class number one (i.e., which have unique factorization) are Q(sqrt(-d)) for d=1,2,3,7,11,19,43,67 and 163. This is related to amusing facts such as the primality of n2+n+41 for all n between 0 and 39. More generally, it is known that there are only finitely many imaginary quadratic fields with any given class number h, and these have all been determined for all h up to 100. The project would be to investigate the theory behind these problems and their solution; someone with a computational bent can also list all negative discriminants with class number up to 100 (the known results only cover fundamental discriminants), which has not been done before. Prerequisite: to have taken MA3A6 — 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:

1. Iwasawa theory. This is a major branch of number theory that deals with relations between arithmetic objects (such as the ideal class groups of number fields) and L-functions. A good source would be the book of Coates and Sujatha, "Cyclotomic Fields and Zeta Values". It would probably be necessary to study MA3A6 (Algebraic Number Theory) beforehand.

2. 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-organized 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

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 dimensionsional 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.

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 this for some pictures, and an idea of what this topic would be like. As for the previous project, willingness to merge algebraic, combinatorics, and computer experiments are required.

Differential Geometry and PDE

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.

Dynamical Systems

(See also Adam Epstein)

Dr Alastair Fletcher is happy to offer a 4th year project.

Quasiregular dynamics: Quasiregular mappings are natural generalizations of holomorphic mappings, sharing many similar properties. Recently, there has been interest in the iteration of quasiregular mappings, in analogy to complex dynamics, and there are many open questions. One possible research project could study aspects of the iteration of compositions of polynomials and affine mappings in the plane. Further topics are possible, and there could be a programming aspect to the project.

See http://www.warwick.ac.uk/~masias/quadqr.pdf for preliminary reading.

Group Theory

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.

Combinatorics

Dr Vadim Lozin offers a project on

Structural graph theory: With the proof of Perfect Graph Theorem and Wagner's Conjecture on graph minors Structural Graph Theory has recently developed into an independent branch of combinatorics with its own problems and methods. This project offers a variety of options to study special graph properties or special graph techniques. Sample topics include: "The structure of fork-free graphs", "Locally bounded coverings of graphs", "Canonical antichains of graphs and permutations", "Local transformations of graphs preserving the independence number.

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 fify years until disproved last year. 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 Bruce Westbury offers the following two topics:

Dilogarithm: This is an interesting special function with several elementary definitions. The applications come from a five term functional equation. For example, this is the volume of an ideal hyperbolic tetrahedron.

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.

He is offering a further topic:

Circle packing based on the book "An Introduction to Circle Packing" by Ken Stephenson.

Applied Mathematics

Complexity

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.

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 Stefan Grosskinsky is offering projects in the area of stochastic processes and applications in complex systems/statistical mechanics. Topics include

1. Urn models and connections to phase transitions in stochastic particle systems,

2. metastability phenomena and connections to quasi-stationary distributions (e.g. in birth-death chains),

3. entropies and convergence of point processes under scaling limits,

4. applications to models of wealth condensation, biodiversity, population growth, traffic modelling, etc.

These projects fit well with the module CO905 - Stochastic models of complex systems in Term 2, which can be taken as an unusual option. In case of interest please contact me via email. For further details on past projects you can also have a look at http://www.warwick.ac.uk/~masgav/.

Computational Mathematics

Professor Dwight Barkley

Macroscopic wave-particle duality.

There has been interest recently in macroscopic systems exhibiting wave-particle duality. See this video. While there is superficial evidence of a deep connection between disparate physical systems, at present we do not know whether the agreement is quantitative. The goal of this project is to provide quantitative measurements of reflection of spiral waves from boundaries (as seen in the video) as a first step in comparing with other physical systems.

This project is largely computational. The code you will be working with is written in C and can be found here. You must be able to modify this code as necessary for the project. You should also be able to use software tools, such as Matlab, as required to process data.

Epidemiology, Ecology and Evolution

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”.

Mathematics of Fluids

Dr Xinyu He and Professor Robert Kerr

Title: Geometry of the 3d Euler equations

Abstract: The Euler equations govern the motion of an ideal, incompressible fluid in three space dimensions. It is an open question as to whether the velocity field $ v(x,t) $ remains regular for all time, starting from sufficiently smooth, compactly supported initial data. An answer to this question is central to our understanding of Navier-Stokes fluid turbulence and whether these equations might also have a singularity. Until recently, most of the mathematical analysis of potential finite-time Euler singularities has used continuum norms, for example showing that the vorticity, defined as $ curl v(x,t) $, controls all singular growth.

A more recent approach is to look at geometrical structures of the system. For example, it is now known that the curvature of the vortex also controls singular growth. However visualisations of Euler calculations have only given us a few hints as to how this can occur.

In this project, the student’s first goal will be to report on what is known mathematically about how curvature controls the growth of vorticity. First by looking at papers from the 1980s using vortex filament approximations of the Euler equations, then more recent direct analysis of the full continuum equations.

The second goal would be to write this analysis in terms of Serret-Frenet equations for how the direction of vorticity, the curvature of vortex lines, and their torsion change along vortex lines as well as the quaternion formulation based on the direction in which vorticity changes [GHKR].

The last part of this project could be numerical if the student is interested. For this, data sets for the vorticity from high resolution simulations of the Euler equations using Matlab could be compared against the analytic predictions, as was done to superfluid data by a student last year.

[GHKR] J. D. Gibbon, D. D. Holm, R. M. Kerr & I. Roulstone, Quaternions and particle dynamics in the Euler fluid equations, Nonlinearity (2006), vol 19, 1969-1983.

Professor Robert Kerr

Topological change in quantum fluids: There are two directions this could go. Each requires some straightforward analytic work first to determine the equations we want to analyse, followed by applying this analysis using Matlab to existing three-dimensional data sets generated using the non-linear Schrödinger equation applicable to quantum fluids. Sample Matlab codes have been written.

The first direction would be to verify, then apply, a new formulation of the energy transfer in Fourier space between the two energy components of a quantum fluid, the kinetic energy and internal energy. The Matlab part would be debug and run an existing code to determine how these energy transfer functions.

The second direction would be to characterise the curvature and torsion of colliding quantum vortex lines using the Frenet-Serret from differential geometry. Two existing theories are: That interactions between these vortices can be modeled using the law of Biot-Savart. Or a reduction of this law known as the local induction approximation or LIA. In my new paper in Physical Review Letters it is demonstrated that these approximations do not apply when the initial vortices are close together, but these approximations might still be valid in other cases that have been simulated.

The long term goal is to understand superfluid experiments where the kinetic energy decays. This phenomena is now called quantum turbulence, because we expect to observe it shortly in Bose-Einstein condensates. The mechanism observed starts with vortex reconnection, followed by the creation of vortex rings and the propagation of the vortex rings out of the interaction region. These observations could explain a longstanding experimental paradox: How can a inviscid, Hamiltonian (energy-conserving) system generate kinetic energy decay that is virtually indistinguishable from the decay of kinetic energy in a classical, viscous, Navier-Stokes fluid. The mechanism observed would confirm a speculation originally made by Feynman in the 1950s. The figure below shows the configuration of the vortices just prior to reconnection. The isosurfaces show density contours. The density is identically zero on the lines representing the vortex cores.

For further information on related turbulence research, see http://www.maths.warwick.ac.uk/research/research_areas/turb.htm

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

Dr Hugo van den Berg offers projects in the area of Mathematical Biology.

Mathematical Physics

Dr Volker Betz offers several 4th year projects:

1. Spatial random permutations: The points of a sqaure (or cubic) grid are connected by arrows, where each point must have exactly one incoming and one outgoing arrow. Taken on its own, each arrow as a probability simple distribution that decays rapidly, e.g. Gaussian. But of course the total probability distribution of all the arrows is more complicated, as neighbouring arrows 'block' possible targets of each other. The resulting model is called spatial random permutation. There are many open questions about these, including connections to self-avoiding random walks, Bose-Einstein condensation and even Schramm-Loewner equations. In a 4th year project you will be following one of these connections. After learning the basic tools, you will either be running careful simulations (a working MCMC code exists) in order to come to conjectures, or try to explore theoretically some of the properties of spatial random permutations.

2. Interacting particle systems: Spatial random permutations are one example of an interesting class of probability models called interacting (particle) systems. In general, these systems contain a large number of simple parts (such as the jumps in the spatial random permutations) that interact and create interesting effects on a macroscopic (i.e. system-wide) scale. Examples include population growth models, phase boundary models or percolation models, where in the latter one is interested in the probability of creating a path from one edge of the system to the other by randomly putting down pieces of path throughout the system. If you are interested in a project on any of these (or any other interacting particle) model, you are welcome to contact me.

3. Singularly perturbed differential equations play an important role in many physics and engeneering problems. The field of exponential asymptotics tries to find very accurate approximate solutions to them, often involving complex analysis and recursive schemes. Much of this work is non-rigorous. In your project, you will apply a known method in order to obtain new rigorous results about nonlinear ODEs such as e.g. the KdV equation.

Molecular Dynamics

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.

Project 1: Analysis of transition state theory.

Project 2: Thermostats.

Probability

Dr Roger Tribe offers 2 probability projects: The first requires only elementary tools (though experience with Markov chains is an advantage), the second requires the student to be interested in Brownian motion and possibly Stochastic differential equations.

1. Negative dependence. For vectors of random variables, there are various notions of negative dependence, various techniques for establishing them, and plenty of open problems. The aim would be to look at some applying these techniques to some simple models for interacting particles.

2. One dimensional diffusions. These, and their behaviour near boundaries, were understood in the 1960/70's but remains one of the most useful set of models for approximation and comparisons in more intricate models. The project might lead in various directions: time dependent models, discrete approximations, ...

Stochastic Analysis

Dr Xue-Mei Li would be happy to supervise a 4th year project in the field of stochastic analysis. The topics could be on one of the following: Levy processes, Brownian motions and stochastic processes on geometric spaces, stochastic differential equations; Malliavin Calculus, etc.

Academic Staff

Please note that any member of academic staff is a potential supervisor, not only those listed here. You can also have a look on the past projects. Some members of staff are NOT available in 2011-2012 academic year: Dr Anthony Manning will retire next Summer, Dr Jose Rodrigo will be on a study leave but willing to discuss a project, and Dr Dmitriy Rumynin will be on a study leave.

Year 1 regs and modules
G100 G103 GL11 G1NC

Year 2 regs and modules
G100 G103 GL11 G1NC

Year 3 regs and modules
G100 G103

Year 4 regs and modules
G103

Exam information
Core module averages