TA TIBOR ANTAL
TA.IND.1
Number of accessible paths in the hypercube
20 point individual project
Single honours mathematics or any joint honours degree
Motivated by an evolutionary biology question, we will study the following problem: we consider the hypercube {0,1}L where each vertex carries an independent random variable uniformly distributed on [0,1], except (1,1,1,...,1) which carries the value 1 and (0,0,0,...,0) which carries the value x∈[0,1]. We'll study the number of paths from vertex (0,0,0,...,0) to the opposite vertex (1,1,1,...,1) along which the values on the vertex form an increasing sequence. The project is about understanding and extending proofs and theorems and comparing the results to computer simulations.
References:
  1. The number of accessible paths in the hypercube by Julien Berestycki, Eric Brunet, Zhan Shi;
  2. Accessibility percolation with backsteps by Julien Berestycki, Eric Brunet, Zhan Shi.
Requirements: Probability, Stochastic Modelling; Essentials in Analysis and Probability would be useful.
Second marker: Nikola Popovic.
TA.DBL.1
TBA
40 point double project
Joint honours degree
TBA.
References: TBA.
Requirements: TBA.
Second marker: TBA.

JA JONAS AZZAM
JA.GRP.1
Geometric Measure Theory
20 point group project
Single honours mathematics
We will study some topics in geometric measure theory.
References:
  1. Fractal Geometry by Falconer, Wiley 1990;
  2. The Geometry of Fractal Sets by Falconer, Cambridge 1986.
Requirements: Honours Analysis and Essentials in Analysis and Probability.
Second marker: Martin Dindos.
JA.DISS.1
Embeddings of metric spaces
40 point dissertation
Single honours mathematics
When can we fit a metric space into Euclidean space? That is, when can we find a function that maps an abstract metric space X into a finite dimensional Euclidean space but doesn't change the distances between points too much? What's the lowest dimensional Euclidean space we can do this for a given X? This is an old and difficult problem in metric geometry that uses techniques from analysis, probability, combinatorics, and graph theory (and has gained a lot of interest in the last few decades due to applications to computer science). In this project, we will read about famous and classical embedding results such as Bourgain's theorem, Assouad's embedding theorem, and the Johnson-Lindenstrauss Lemma, and also study some modern problems.
References:
  1. Lectures on Discrete Geometry by Matousek, Springer Graduate Texts in Mathematics, 2002;
  2. Union of Euclidean Metric Spaces is Euclidean by Makarychev and Makarychev, Discrete Analysis 2016:14.
Requirements: Honours Analysis and Essentials in Analysis and Probability.
Second marker: Jim Wright or Tony Carbery.

PB PIETER BLUE
PB.GRP.1
Singularity theorems in general relativity
20 point group project
Single honours mathematics and Mathematics and Physics
General relativity describes space-time as a curved, four-dimensional object, and the Einstein equation governs the curvature of this object. Many of the most important, known solutions to the Einstein equation have singularities. These include black holes and cosmological solutions. This project will start with the essentials of general relativity and the differential geometry of manifolds. Students should also be taking the Geometry of General Relativity course. There is a wide range of possible directions to go from there, allowing students to work divide work between them. This could include the singularity theorems of Hawking and Penrose, singularities without divergences including Cauchy horizons, chaos in the Bianchi IX cosmological model, topology of the future and choosing a time function, and cosmological models with less than total isotropy.
References:
  1. An introduction to general relativity: Spacetime and Geometry, S.M. Carroll (2004), Addison-Wesley;
  2. The large scale structure of space-time, S.W. Hawking and G.F.R. Ellis (1973), Cambridge University Press;
  3. Gravitation, C.W. Misnor, K.S. Thorne, J.A. Wheeler (1973), Freeman;
  4. General Relativity, R.M. Wald (1984), University of Chicago Press.
Requirements: Geometry of General Relativity (corequisite). An interest in physics would be useful.
Second marker: James Lucietti.
PB.GRP.2
Solving nonlinear partial differential equations using the Nash-Moser iteration scheme
20 point group project
Single honours mathematics or Maths and physics
Proving the existence of solutions to partial differential equations is an important topic in analysis. Typically, this is very different from the methods for finding explicit solutions, which were used in Honours Differential Equations. The existence of solutions to ordinary differential equations was proved in Honours Analysis using the contraction mapping theorem in the space of continous functions. The contraction mapping theorem can be used to solve some nonlinear elliptic problems, which might describe the static configuration of an elastic body. However, the contraction mappingtheorem is not sufficiently strong to treat some nonlinear evolution problems, such as nonlinear wave equations. The purpose of this project is to explore the relevant spaces of function and to study an improved version of the contraction mapping theorem, developed by Nash and Moser. X. Saint Raymond (2) has written a very clear and concise description of the method, which will be the basis for this project.
References:
  1. Partial Differential Equations, C. Evans (1998), American Mathematical Society;
  2. A simple Nash-Moser implicit function theorem, X. Saint Raymond, L'Enseignement Mathematique, 35 (1989) 217-226.
Requirements: Honours Analysis, Linear analysis, Honours Differential Equations.
Second marker: Nikolaos Bournaveas or Aram Karakhanyan.

HB HARRY BRADEN
HB.GRP.1
Geodesics of the Atiyah-Hitchin Manifold
20 point group project
Mathematics (BSc, MA, MMath-Y4), Maths and physics
The Atiyah-Hitchin manifold is an important 4 dimensional hyperkahler manifold for both mathematics and mathematical physics. This project will explore its properties, including its geodesics.
References: The geometry and dynamics of magnetic monopoles by Michael Atiyah and Nigel Hitchin, Princeton University Press, 1988.
Requirements: An A in Geometry. A corequisite is Geometry of General Relativity. At most two students are allowed in the group.
Second marker: James Lucietti.
HB.GRP.2
Shuffle, Stuffle and Hopf algebras, and the polylogarithm
20 point group project
Mathematics (BSc, MA, MMath-Y4), Maths and physics
This is an algebra project that will explore a number of different algebraic structures and relate them to a class of functions with number theoretic import.
References: Shuffle Algebra.
Requirements: Honours Algebra. A corequisite is Group Theory. At most two students are allowed in the group.
Second marker: David Jordan.
HB.DBL.1
Computational Riemann Surfaces
40 point double project
CS and Maths
Riemann surfaces are important meeting ground of analysis, algebra, geometry and topology; they also underlie many applications in the mathematical sciences. In recent years computational and algorithmic techniques have been brought to bear upon them, constructing for example homology bases and computing the associated period matrix of a surface. Very recently much of this work has been written in Python to be part of the Sage project. The project will explore Riemann surfaces with symmetries and develop appropriate computational tools.
References:
  1. Riemann Surfaces by Farkas and Kra, Springer-Verlag;
  2. Tutorials on Abelfunctions.
Requirements: Strong Python skills.
Second marker: Johan Martens.

MDC MIGUEL DE CARVALHO
MDC.GRP.1
Statistical Learning
20 point group project
Joint degree suitability: Maths with Management, Maths and Stats, Maths and physics
Statistical Learning is a fast-evolving field bridging ideas, concepts, and methods from Statistics and Data Science with those of Machine Learning. The goal of this project is to revise selected methodologies and use these to conduct data analysis in R.
References:
  1. An Introduction to Statistical Learning - with Applications in R by G. James, W. Daniela, T. Hastie and R. Tibshirani, New York: Springer, 2014;
  2. The Elements of Statistical Learning by T. Hastie, R. Tibshirani and J. Friedman, New York: Springer, 2009;
  3. The R Book by M. J. Crawley, New York: Wiley, 2012.
Requirements: At least two Honours statistics courses.
Second marker: Jonathan Gair or Ruth King.
MDC.GRP.2
Computer-Age Statistical Science
20 point group project
Joint degree suitability: Maths with Management, Maths and Stats, Maths and physics
Statistical inference is concerned with the rigorous modelling of uncertainty. As posed by R. A. Fisher: We may at once admit that any inference from the particular to the general must be attended with some degree of uncertainty, but this is not the same as to admit that such inference cannot be absolutely rigorous, for the nature and degree of the uncertainty may itself be capable of rigorous expression. The goal of this project is on surveying modern developments on statistical inference, especially those taking advantage of today's available computing power.
References:
  1. Computer-Age Statistical Inference: Algorithms, Evidence, and Data Science, by B. Efron and T. Hastie, Cambridge University Press, 2016;
  2. The R Book by M. J. Crawley, New York: Wiley, 2012.
Requirements: At least two Honours statistics courses.
Second marker: Vanda Inacio De Carvalho or Bruce Worton.
MDC.IND.1
Bayesian Modelling of Financial Time Series
20 point individual project
Joint degree suitability: Maths with Management, Maths and Stats, Maths and physics
Time series are a topic of wide interest in theoretical and applied statistics. The goal of this project is to survey and apply some selected models for time series, and to use Bayesian inference to learn about parameters from financial data.
References:
  1. Time Series: Modeling, Computation, and Inference by R. Prado and M. West, CRC/Chapman & Hall, 2001;
  2. Bayesian Ideas and Data Analysis: An Introduction for Scientists and Statisticians by R. Christensen, W. O. Johnson, A. Branscum and T. E. Hanson, CRC/Chapman & Hall, 2011;
  3. The R Book by M. J. Crawley, New York: Wiley, 2012.
Requirements: At least two Honours statistics courses.
Second marker: Natalia Bochkina or Ioannis Papastathopoulos.

VDC VANDA DE CARVALHO
VDC.GRP.1
Functional data analysis
20 point group project
Particularly suitable for Mathematics and Statistics. Joint degree suitability: Economics and Statistics
In the last years, technological developments have led to the production of data that are increasingly complex, high-dimensional and structured. A large fraction of these data can be characterised as functional data. Broadly, functional data analysis is about the analysis of the information on curves, spectra, images, and shapes. It finds application in many different areas, such as, medical, environmental, and financial sciences, image analysis, business, and engineering, to name a few. The goal of this project is to survey some functional data analysis techniques and apply them to real data using R.
References:
  1. Functional Data Analysis by J.O. Ramsay and B.W. Silverman, 2005, Springer;
  2. Functional Data Analysis with R and Matlab by J.O. Ramsay, G. Hooker and S. Graver, 2010, Springer;
  3. An introduction with medical applications to functional data analysis by H. Sorensen, J. Goldsmith, L. Sangalli, Statistics in Medicine 32 (2013), 5222-5240.
Requirements: At least two Honours statistics courses. Linear Statistical Modelling is desirable.
Second marker: Miguel de Carvalho or Ruth King.
VDC.IND.1
Quantile regression models
20 point individual project
Particularly suitable for Mathematics and Statistics. Joint degree suitability: Economics and Statistics
One of the top ten reasons to become a statistician (according to Friedman, Friedman, and Amoo): statisticians are mean lovers. The majority of regression models focus on the problem of how a set of covariates are related to the mean of a response variable. Obviously, this results in an inherent information loss. In contrast, quantile regression models, provide a more complete picture, by considering how the covariates relate to the quantiles of the response variable. The goal of this project is to survey quantile regression models, both from a classical and Bayesian perspectives, and apply them to real data using R.
References: Regression: Models, Methods, and Applications by L. Fahrmeir, T. Kneib, S. Lang and B. Marx, 2013, Springer.
Requirements: At least two Honours statistics courses. Linear Statistical Modelling, Likelihood, and Bayesian Theory are desirable.
Second marker: Finn Lindgren or Bruce Worton.
VDC.IND.2
Principal component analysis
20 point group project
Joint degree suitability: Economics and Statistics, AI and Mathematics
Principal components analysis is a widely used multivariate statistical technique. It is a procedure for identifying a smaller number of uncorrelated variables, called principal components, from a large set of data. This project aims to survey the literature and apply the methods to real data using R.
References:
  1. Principal Component Analysis by I.T. Jollife, 2002, Springer;
  2. Analysis of Multivariate and High-Dimensional Data by I. Koch, 2013, Cambridge.
Requirements: At least two Honours statistics courses.
Second marker: Jonathan Gair or Ioannis Papastathopoulos.

IC IVAN CHELTSOV
IC.GRP.1
Mathematical question 11851
20 point group project
Single honours mathematics or joint honours mathematics and physics
Let S be a finite set in R2 that consists of N>2 points. Suppose, in addition, that not all points in S are collinear. A line in R2 that contains exactly two points in S is called an ordinary line. By the famous Sylvester–Gallai theorem (posed as a problem by Sylvester in 1893, and proved by Gallai in 1944), ordinary lines always exist. However, this theorem does not say how many ordinary lines exist. Of course, the answer depends on the set S. Denote by t2(N) the minimum number of ordinary lines when S runs through all finite subsets in R2 consisting of N non-collinear points. Then t2(N)>0 by the Sylvester–Gallai theorem. In 1951, Motzkin proved that t2(N)≥ N½. Dirac conjectured that t2(N)≥ N/2 for all N. This project is about this conjecture, which is often referred to as the Dirac-Motzkin conjecture. In 2013, Ben Green and Terence Tao showed that it holds for all sufficiently large N. The goal is to understand their proof, to investigate the Dirac-Motzkin conjecture for small N, and to explore a similar problem in C2.
References:
  1. Mathematical question 11851 by J. J. Sylvester, The Educational Times 46 (1893), 156;
  2. On sets defining few ordinary lines, by Ben Green and Terence Tao, Discrete & Computational Geometry 50, 409-468.
Requirements: It helps if students take Algebraic Geometry in parallel.
Second marker: Harry Braden.
IC.GRP.2
Finite collineation groups
20 point group project
Single honours mathematics
Tetrahedron, cube, octahedron, dodecahedron, and icosahedron are represented by their groups of symmetries. Thus, the famous classification of Platonic solids can be considered as a problem of finding all finite subgroups in SO3(R). Since SO3(R) is a quotient of the group SL2(C) by its center, this gives the list of finite subgroups in SL2(C) as well. This project is about finite subgroups in SL3(C), which were classified by Blichfeldt exactly 100 years ago.
References:
  1. Finite Collineation Groups by Hans Frederik Blichfeldt, 1917;
  2. Gorenstein quotient singularities, Stephen Yau and Yung Yu, Memoirs of the American Mathematical Society, volume 505, 1993.
Requirements: It helps if students have taken Honours Algebra and Algebraic Geometry.
Second marker: Milena Hering or Johan Martens.
IC.DISS.1
Resolution of singularities
40 point dissertation
Single honours mathematics
Resolution of singularities is one of the oldest and prettiest topics of algebraic geometry. In low dimensions, it is also completely explored. This project is about resolving singularities of algebraic curves and algebraic surfaces.
References: Lectures on resolution of singularities by Janos Kollar, Annals of Mathematics Studies 166, Princeton University Press.
Requirements: Commutative Algebra and Algebraic Geometry.
Second marker: Milena Hering or Johan Martens.
IC.DISS.2
Finite subgroups of Cremona group
40 point dissertation
Single honours mathematics
Birational selfmaps of the plane are maps given by rational functions that have inverses of the same kind. The simplest example of such maps is the involution that send (x,y) to (1/x, 1/y), which is usually called the standard Cremona involution. The oldest example of such maps is the famous inversion in a circle that was introduced by Apollonius of Perga two thousand years ago. The Cremona group is the group of all birational selfmaps of the plane. The goal of this project is to study finite subgroups of this group.
References:
  1. Undergraduate Algebraic Geometry, Miles Reid, London Mathematical Society Student Texts (1988);
  2. Rational and nearly rational varieties, Alessio Corti, Janos Kollar, Karen Smith, Cambridge University Press (2003);
  3. Classical Algebraic Geometry: A Modern View by Igor Dolgachev, Cambridge University Press (2012);
  4. Finite abelian subgroups of the Cremona group of the plane, Jeremy Blanc, PHD Thesis, 189 pages (2006);
  5. Finite subgroups of the plane Cremona group, Igor Dolgachev and Vasily Iskovskikh, Birkhauser Boston, Progress in Mathematics 269 (2009), 443-548.
Requirements: Commutative Algebra and Algebraic Geometry.
Second marker: Milena Hering or Johan Martens.

CD CHRIS DENT
CD.DISS.1
Risk assessment of electricity generating capacity shortages
40 point dissertation project
Any degree in mathematics. Suitable for Maths with Management and Maths and Stats
Ensuring an appropriately high level of security of supply is one of the key issues in management of electric power systems and markets, and thus associated risk assessment is a major topic in electric power system analysis. There has been renewed research interest in recent years due to the advent of high capacities of renewable generation in many systems, whose availability has very different statistical properties from that of conventional fossil fuel powered generation (as its availability is primarily determined by the wind or solar resource, rather than by mechanical availability). Within such risk assessments it is necessary to build a joint statistical model of demand and available renewable capacity (a statistical association between these is naturally expected, as in most power systems temperature has an influence on demand through heating or air conditioning load, and available renewable capacity is clearly driven by the weather). This project will take methods developed for modelling wind and demand in the Great Britain system, and apply them to data from other systems supplied by overseas collaborators (for instance in India and in North America). The models developed will then be incorporated into risk assessments for these other systems.
References:
  1. Bayesian Logical Data Analysis for the Physical Sciences by P. Gregory, Cambridge University Press, 2010;
  2. What is an electricity blackout? Durham Energy Institute briefing note in association with the Maxwell Institute for Mathematical Sciences;
  3. Statistical modelling for inclusion of wind generation in industrial adequacy studies by Amy Wilson;
  4. Accounting for wind-demand dependence when estimating LoLE by Amy Wilson, Stan Zachary, Chris Dent.
Requirements: Familiarity with applied statistical modelling in R.
Second marker: Natalia Bochkina, Jonathan Gair, Vanda Inacio De Carvalho, Ruth King, Ioannis Papastathopoulos, Finn Lindgren or Miguel de Carvalho.

MD MARTIN DINDOS
MD.GRP.1
Derivatives, Dini derivatives, approximate derivatives
20 point group project
Single honours mathematics
Everyone is familiar with the basic concept of derivative as well as the fact the there are continuous nowhere differentiable functions. There are however further generalizations of the notion of derivative such such the concept of Dini derivatives, approximate derivatives, etc. The goal of the project is to read about these concept and study the relations between them.
References: Differentiation of real functions by Bruckner.
Requirements: Honours Analysis. At most 3 students should be in the group.
Second marker: Aram Karakhanyan.
MD.DISS.1
Solvability methods for elliptic PDEs
40 point dissertation
Single honours mathematics (not recommend for joint degree students)
Two solvability methods of elliptic PDE will be looked at namely the classical Lax-Milgram theorem and the Peron's method. The student will acquire familiarity with Sobolev spaces, notions of a weak and classical solution.
References: Second order elliptic PDEs by Gilbarg and Trudinger.
Requirements: Honours Analysis, and Essentials in Analysis and Probability is a good bonus.
Second marker: Aram Karakhanyan or Pieter Blue.

GDR GONCALO DOS REIS
GDR.GRP.1
Credit Risk Modelling
20 point group project
Single honours mathematics
The aims of this project are
  • to introduce students to the models used in the management of portfolio credit risk;
  • to explore the mathematical underpinnings of widely-used industry models, such as the Moody's KMV model, CreditMetrics and CreditRisk+, and learn how the critical phenomenon of default dependence is modelled;
  • to show how these portfolio credit models are used to determine capital adequacy and reveal how they have shaped regulation and led to the Basel II capital formula.
The project will cover the following topics:
  • introduction to credit risk (credit-risky instruments, defaults, ratings);
  • Merton's model of the default of a firm;
  • common industry models (KMV, CreditMetrics, CreditRisk+);
  • Modelling dependence between defaults with factor models;
  • latent variable or threshold models of default;
  • mixture models of default;
  • the Basel II regulatory capital formula;
  • calculating the portfolio credit loss distribution;
  • large portfolio behaviour of the credit loss distribution;
  • introduction to credit derivatives.
Learning outcomes of the project:
  • to demonstrate an understanding of the nature of credit risk;
  • to describe the theoretical underpinnings of models used in the financial industry;
  • to show a knowledge of the regulatory framework and, in particular, the Basel II regulatory capital formula;
  • to explain how dependence is modelled in credit portfolios;
  • to explain how latent variable or threshold models are constructed;
  • to describe mixture models of default and derive their mathematical properties;
  • to describe methods for calculating the portfolio loss distribution, including asymptotic approximations;
  • to describe the cash flows of common single-name and basket credit derivatives and have some idea of how they are valued.
References:
  1. Quantitative Risk Management: Concepts, Techniques and Tools by A. McNeil, R. Frey and P. Embrechts, Princeton University Press, 2005,
  2. An Introduction to Credit Risk Modeling by C. Bluhm, L. Overbeck and C. Wagner, Chapman & Hall/CRC, 2002.
Requirements: None.
Second marker: TBA

JRG JONATHAN REAVLEY GAIR
JRG.GRP.1
Calculating Bayesian evidence for model selection
20 point group project
Particularly suitable for Mathematics and Statistics degree students
The ratio of Bayesian evidences (the odds ratio) is commonly used in the physical sciences to compare two different models for some observed data. The computation of the evidence requires the evaluation of a multi-dimensional integral over the parameter space of possible signals within a given model. Difficulties arise in the evaluation of this integral when the posterior distribution has been discretely sampled across parameter space, particularly when the dimensionality of the parameter space is large. Various methods have been proposed - Markov Chain Monte Carlo techniques, Kernel density estimators and nested sampling algorithms of different types. This project will explore these various techniques and identify when each approach is most applicable. The techniques will be illustrated using blind challenges in which the students have to identify the model used to generate various data sets.
References:
  1. Bayesian Logical Data Analysis for the Physical Sciences by P. Gregory, Cambridge University Press, 2010;
  2. Nested sampling for general Bayesian computation by J. Skilling, Bayesian Anal. 1 833 (2006);
  3. Computing the Bayesian Factor from a Markov chain Monte Carlo Simulation of the Posterior Distribution by M.D. Weinberg, Bayesian Anal. 7 737 (2012);
  4. MultiNest: an efficient and robust Bayesian inference tool for cosmology and particle physics by F. Feroz, M.P. Hobson, and M. Bridges, Mon. Not. Roy. Astron. Soc. 398 1601 (2012);
  5. PolyChord: nested sampling for cosmology by W.J. Handley, M.P. Hobson, and A.N. Lasenby, to appear in Mon. Not. Roy. Astron. Soc.
Requirements: Second year statistics and probability modules. Third year statistics desirable. Familiarity with running codes and programming in a language such as C/C++ or Python will be useful. Several evidence computation packages are freely available to download online.
Second marker: Natalia Bochkina or Ioannis Papastathopoulos.
JRG.IND.1
Approximate Bayesian computation
20 point individual project
Particularly suitable for Mathematics and Statistics degree students
In many inference problems, particularly those that rely on numerical simulations of a physical process, the likelihood is intractable, i.e., it cannot be directly computed. Approximate Bayesian computation methods have been developed to handle such scenarios. They rely on performing simulations of the computational model and asking whether they are sufficiently close to the observed data or not. Various methods have been developed for efficient ABC sampling, in order to minimise the number of simulations of the model that are discarded because they are too far from the observations. The aim of this project is to explore different ABC techniques and compare their performance when applied to an illustrative toy model.
References:
  1. A tutorial on approximate Bayesian computation, B. M. Turner and T. Van Zandt, J. Math. Psych. 56, (2012), 69-85;
  2. Sequential Monte Carlo without likelihoods by S. A. Sisson, Y. Fan, and M. M. Tanaka, Proceedings of the National Academy of Sciences 104 (2007), 1760–1765;
  3. Approximate Bayesian computation in evolution and ecology by M. A. Beaumont, Annual Review of Ecology, Evolution, and Systematics 41 (2006), 379–406.
Requirements: Statistics 2 or Statistics 3. Useful: Bayesian Theory and Bayesian Data Analysis. Experience with computer programming also necessary, e.g., R or Python or C/C++.
Second marker: Natalia Bochkina or Ruth King.
JRG.DBL.1
Inference using emulation of computationally intensive models
40 point double project
Joint honours mathematics. Suitable for Applied Maths or Informatics and Math
It is now possible to simulate complex physical and biological processes using computers and it is natural to want to use data in conjunction with those simulations to make inferences about the parameters characterising the underlying physical process. However, these simulations are typically very expensive and so it is not practical to evaluate them for the many thousands of parameter values that will typically be required in an inference calculation. One approach to resolve this problem is to use emulators to interpolate between points in parameter space at which simulations have been performed to other parameter values. These emulators should be sufficiently inexpensive to evaluate that they can then be used in inference computations. The purpose of this project is to explore different approaches to emulation. One popular and promising approach involves representing the model output as a Gaussian process. This project will explore this approach, but could also compare it to other approaches and identify the relative advantages and disadvantages of each.
References:
  1. Introduction to Gaussian Processes by D.J.C. MacKay;
  2. Improving gravitational-wave parameter estimation using Gaussian process regression by C. J. Moore, C. P. L. Berry, A. J. K. Chua and J. R. Gair, Phys. Rev. D 93 (2016), 064001.
Requirements: Statistics 2 or Statistics 3. Useful: Bayesian Theory and Bayesian Data Analysis.
Second marker: Bruce Worton or Ruth King.

JG JACEK GONDZIO
JG.DISS.1
Column Generation for Combinatorial Optimization Problems
40 point dissertation
Any degree in mathematics or joint degree
This project will require to study and implement a column generation formulation of a particular combinatorial optimization problem. The approach relies on an existing primal-dual column generation algorithm which has already been used in a number of applications:
  • the multicommodity network flow problem (MCNF),
  • the cutting stock problem (CSP),
  • the vehicle routing problem with time windows (VRPTW), and
  • the capacitated lot sizing problem with setup times (CLSPST).
The student may choose their class of combinatorial optimization problem, such as for example, travelling salesman problem (TSP), Hamiltonian cycle problem (HCP), quadratic assignment problem (QAP), or some other relevant OR problem. Students who took the OR in Telecommunication course may suggest an interesting application which arises in telecom industry. Objectives of the projects:
  • study the theoretical background of the chosen problem;
  • study its formulations, especially those eligible for column generation;
  • implement the approach within the context of PDCGM, Primal-Dual Column Generation Method based on HOPDM software;
  • apply this implementation to solve real-life public domain instances of problems.
References:
  1. New Developments in the Primal-Dual Column Generation Technique, J. Gondzio, P. Gonzalez-Brevis, P. Munari, European Journal of Operational Research 224 (2013) 41-51.
  2. Large-Scale Optimization with the Primal-Dual Column Generation Method, J. Gondzio, P. Gonzalez-Brevis, P. Munari, Mathematical Programming Computation 8 (2016), 47-82.
Requirements: Fluency in the C programming language. Very good grasp of FuO (mark above 70%) and ODS and CO.
Second marker: TBA.

RG RICHARD GRATWICK
RG.GRP.1
Convexity in the calculus of variations
20 point group project
Any degree in mathematics or joint degree
The calculus of variations is the search for minimal objects: shapes with the smallest surface area, curves defining a shortest path. Convexity of the underlying quantity turns out to be important in establishing the existence of solutions. But exactly what type of convexity? For multi-dimensional vector-valued problems, quasiconvexity is the key notion. But this is understood woefully poorly: we really struggle to identify quasiconvex functions. More easily spotted are rank-1 convex functions, and polyconvex functions. But these are all, in general, different things. This project will look at the direct method of the calculus of variations, and the importance of convexity to this. The main emphasis will then be on considering the different generalized notions of convexity, and what we know and do not know about them. There are plenty of open questions here for the ambitious and the inspired.
References:
  1. Direct Methods in the Calculus of Variations, B. Dacorogna, Springer, 2008;
  2. Variational models for microstructure and phase transitions, S. Muller, Max Planck Institute Lecture notes, 1998;
  3. The calculus of variations and materials science, J. M. Ball, Quart. Appl. Math. 56 (1998), 719-740.
Requirements: Several Variable Calculus and Differential Equations and Honours Analysis. General Topology and Linear Analysis might be useful at points (but we can discuss what exactly is needed).
Second marker: Aram Karakhanyan, Martin Dindos or Jonas Azzam.
RG.GRP.2
The Lavrentiev Phenomenon
20 point group project
Any degree in mathematics or joint degree
Just under a hundred years ago, a rather unnerving phenomenon was discovered: it is not always possible to approximate the minimal value of a problem in the calculus of variations using smooth competing objects. Not only might the solution be nonsmooth, but the minimal value really is a definite jump away. This is still a relatively little-studied problem, despite it having serious consequences, e.g. for attempts to solve optimization problems numerically. This project will examine the well-known examples of this (the Lavrentiev phenomenon), and look at when it can and cannot occur.
References:
  1. One-dimensional Variatonal Problems, G. Buttazzo, M. Giaquinta, and S. Hildebrandt, 1998, OUP.
  2. A survey on old and recent results about the gap phenomenon in the calculus of variations, G. Buttazzo and M. Belloni, in Recent developments in well-posed variational problems, 1–27, Math. Appl. 331, 1995.
  3. One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, J. M. Ball and V. Mizel, Arch. Rational Mech. Anal. 90 (1985), no. 4, 325–388.
Requirements: Several Variable Calculus and Differential Equations and Honours Analysis. General Topology might be useful in certain directions of the project.
Second marker: Aram Karakhanyan, Martin Dindos or Jonas Azzam.
RG.DBL.1
Differentiability of Lipschitz Functions
40 point double project
Joint honours mathematics. Suitable for Applied Maths or Informatics and Maths
A function between metric spaces is Lipschitz if there is a constant L so that the distance between the image points is at most L times the distance between the points themselves. Thus: a differentiable real-valued function on the real line with bounded derivative is Lipschitz. But must a Lipschitz function be differentiable? If so, where? This turns into a surprisingly complicated question, particularly once you consider functions between higher dimensional Euclidean spaces. This is a project in classical real analysis, beginning with work of Lebesgue and Rademacher, and possibly coming right up to date with results published only in the last couple of years.
References:
  1. Measure Theory and Fine Properties of Functions, L. C. Evans and R. F. Gariepy, CRC, 1992;
  2. Geometry of Sets and Measures in Euclidean Spaces, P. Mattila, CUP, 1995;
  3. A simple proof of Zahorski's description of non-differentiability sets of Lipschitz functions, T. Fowler and D. Preiss, Real Analysis Exchange 34 (2009), 127-138.
  4. Differentiability of Lipschitz functions in Lebesgue null sets, D.Preiss and G. Speight, Invent. Math 199 (2015), 517-559.
Requirements: Honours Analysis and Essentials in Analysis and Probability (for measure theory) would be useful.
Second marker: Martin Dindos or Jonas Azzam.

BG BEN GODDARD
BG.GRP.1
Pseudospectral methods
20 point group project
Any degree in mathematics or joint degree
Pseudospectral methods are a powerful class of widely-used techniques for the numerical solution of boundary value problems, eigenvalue problems and partial differential equations. If one wants to solve such a problem to high accuracy on a simple domain and if the resulting solution is smooth (infinitely differentiable) then pseudeospectral methods are generally the best tool. Typically, one would expect ten digits of accuracy, compared to two or three digits for finite difference. Relatedly, pseudospectral methods demand less computer memory and processor time than the alternatives for a given accuracy. This project concerns both the analytical, underlying mathematical properties of these methods (e.g. exponential convergence rate for smooth solutions, aliasing, computational complexity) and their numerical implementation. The method will be implemented in Matlab for a range of simple problems, allowing the theoretical aspects to be demonstrated numerically.
References:
  1. Spectral methods in MATLAB, L. N. Trefethen, SIAM, 2000;
  2. Chebyshev and Fourier Spectral Methods, J.P. Boyd, Springer-Verlag, 2001.
Requirements: Familiarity with Matlab, Numerical Ordinary Differential Equations and Applications, knowledge of Fourier series would be helpful.
Second marker: Jacques Vanneste or James Maddison.
BG.GRP.2
Image processing
20 point group project
Any degree in mathematics or joint degree
Image processing is the digital manipulation of an image. It has many applications in areas including engineering, computer science, medicine, agriculture, law enforcement and astronomy. The main areas are image enhancement (processing such that the information relevant for the desired application is enhanced), image segmentation (subdividing and image or isolating particular parts), and image restoration (undoing damage). This project will explore the theoretical and practical aspects of image processing. Mathematical tools involved include matrix transformations, Fourier transforms and discrete derivatives. The project will also involve practical implementation via Matlab. There are a wide range of possible topics, ranging from the mathematics behind image capture by the eye or a camera, through the digital manipulation of images, to the storage and compression of image files. More advanced topics could include steganography (hiding messages in images) and fractal image compression.
References:
  1. Fundamentals of Image Processing, Chris Solomon and Toby Breckon, Wiley-Blackwell, 2011;
  2. Digital image processing, Rafael C. Gonzalez, Richard E. Woods, PHI, 2008;
  3. Introduction to digital image processing with Matlab, Alasdair McAndrew (digital copy of similar lecture notes available online).
Requirements: Basic familiarity with Matlab and Introduction to Linear Algebra.
Second marker: Tibor Antal, Tom MacKay, James Maddison, Ben Leimkuhler, Kostas Zygalakis.
BG.DISS.1
Mathematics of the Periodic Table
40 point dissertation
Any degree in mathematics or joint degree
The periodic table of elements was first published in 1869 by Dmitri Mendeleev. However, mathematicians had to wait until 1926, when Erwin Schroedinger published his famous wave equation, before they could begin to explain its features. This lead, three years later, to Paul Dirac's statement that The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. This still remains the case today. However, many fundamental results can be derived mathematically from the Schroedinger equation. Examples of such results that could be investigated in this project include the quantization of the energy levels of hydrogen, the existence of molecules, the Pauli exclusion principle and Heisenberg's uncertainty principle. It should be noted that the choice of topics is flexible, and can be tailored to the background and interests of the students.
References:
  1. Lecture notes on the hydrogen atom;
  2. Chapters 1-3, 19 and 20 in The Feynman Lectures on Physics, Vol III, available online here.
  3. Quantum Mechanics: Non-Relativistic Theory, L. D. Landau and E.M. Lifshitz, Elsevier, 1981.
Requirements: An interest in chemistry/physics, Honours Differential Equations and Probability with Applications.
Second marker: Ben Leimkuhler or Harry Braden.

IG IAIN GORDON
IG.GRP.1
Symmetric Group and Probability
20 point group project
Single honours mathematics or any joint honours degree
This project intertwines probability theory, combinatorics and advanced linear algebra (a.k.a. representation theory). How about the following? First, picture a riffle shuffle: it is the shuffle when you hold half the deck of cards in each hand with the thumbs inward, then released the cards by yours thumbs so that they fall to the table interleaved. It's the technique that's typically used in casinos. Now suppose you want to know how often you'd have to do this to make sure your cards have been shuffled well. This project can tell you how to pose this question effectively, and how to answer it. (The answer is 7.) But you don't need to stick to professional shuffles: take the one you like best and work it out! This is just one of the applications of a theory that involves beautiful mathematics, important open problems and numerous applications. This is not the only thing that I'd suggest that you study: there are directions that follow all sorts of paths.
References:
  1. Chapter 28 in Proofs from the Book by Aigner and Ziegler;
  2. Group Representations in Probability and Statistics by Diaconis.
Requirements: Probability, Honours Algebra and Fundamentals of Pure Mathematics.
Second marker: Harry Braden, Tibor Antal or David Jordan.

AG ANDREAS GROTHEY
AG.GRP.1
Automatic Differentiation
20 point group project
Single honours mathematic, Maths with Management, Maths and Stats, Maths and physics
Many algorithms in applied mathematics require the evaluation of derivatives of functions. Two possibilities are well know: symbolic differentiation by a package like Maple, or numerically estimating the derivative by finite differences. Automatic differentiation (AD) is a third (in some sense intermediate) techniques that is able to obtain an exact numerical evaluation of a function gradient at the cost of no more than 5 function evaluations (compared to n+1 evaluations for an approximate gradient with finite differences). AD relies on recursive application of the chain rule and clever organisation of computations. The purpose of the project is get familiar with the concept of AD, understand and be able to present the theoretical results regarding computation complexity and if possible be able to program a simple AD engine.
References: None.
Requirements: At most three students should be in the group.
Second marker: Julian Hall, Ken McKinnon or Jacek Gondzio.
AG.GRP.2
Solution approaches for Optimal Power Flow
20 point group project
Single honours mathematics, Maths with Management, Maths and Stats, Maths and physics
The Optimal Power Flow problem is an important problem in Power Systems Engineering which is at the base of many more sophisticated optimization problems relevant to power systems operations. The problem is that, given an electricity network with given demands, to find the cost optimal generation levels of the available generators. The main difficulty of the problem is that power flows are not switchable: the flow follows the Laws of Physics. Simply using the cheapest generators may well violate network constraints (such as line flow limits, voltage levels, reactive power limits). Indeed power flows are AC and the applicable Laws of Physics are the AC versions of Kirchhoffs Laws. At this point approaches vary, but power flows can be described either by complex numbers, or, if staying in the reels, Kirchhoff Laws require trigonometric functions. In any case the resulting constraints lead to a very challenging nonlinear, non-convex optimization problem. The purpose of the project is to explore models and solution approaches that have been proposed for this problem (which cover quite a few areas of nonlinear optimization) and if possible compare different models and implementations. Students would learn quite a bit about modelling and solving challenging real life optimization problems.
References: None.
Requirements: Linear Programming, Modelling and Solution. At least know some basic optimization problem and not be scared to read up on material. Students should be familiar with a modelling system such as XPress (taught in LPMS) or if not be prepared to learn. No previous knowledge of the underlying physics is needed, although students should of course expect to pick up some of this during the course of the project. To some extend the difficulty of the project can be taylored to the background of the student.
Second marker: Chris Dent, Julian Hall or Ken McKinnon.
AG.DISS.1
Extensions to Optimal Power Flow
40 point dissertation
Single honours mathematic
This project will look at possible extensions to the Optimal Power Flow problem. This may look at more sophisticated solution methods such as involving holomorphic embedding or adding binary decisions (network topology, unit commitment) and exploring ways to obtain good relaxations/bounds that can be used in a Branch & Bound framework.
References: None.
Requirements: None.
Second marker: Chris Dent, Julian Hall, Ken McKinnon.

JH JULIAN HALL
JH.DISS.1
Support vector machines
40 point dissertation
Single honours mathematics. Suitable for Applied Maths or Informatics and Maths
A support vector machine (SVM) enables a functional separation of data points into classes. In its simplest form, an optimal separating hyperplane in Rd separates data points into two distinct classes. Refinements include the case where the data points cannot (quite) be separated into two classes; where the separation is nonlinear and where there are more than two classes. This project will study the theory underlying SVMs, study their practical behaviour in Matlab and, possibly, implement a SVM in C++ in order to handle very large data sets
References: Support vector machines explained by T. Fletcher, 2009.
Requirements: None.
Second marker: Jacek Gondzio.
JH.DISS.2
Quadratic programming problems: solution methods and their implementation
40 point dissertation
Single honours mathematics
Although convex quadratic programming (QP) problems are technically nonlinear, the linearity of the gradient function allows them to be solved by a direct method. They are, thus, an important class of optimization problem, whether formulated directly in applications (such as mean-variance models for portfolio optimization) or as sub-problems when solving (genuinely!) nonlinear programming problems by solving a sequence of quadratic programming problems. This project will consider both the theory underlying methods for solving QPs and their practical implementation in C++. Theoretically, the major difference between linear programming (LP) problems and convex QP problems is that there is not usually a vertex solution of a QP problem. A standard solution technique for QP problems is the active set method which can be viewed as a generalisation of the simplex method. The difference between the two methods is that active set methods determine a sequence of points which are generally not vertices of the feasible region. The project will also develop a practical implementation of the active set method in C++. This will be based on the LAPACK numerical linear algebra routines. There is also scope for developing a parallel implementation using the ScaLAPACK numerical linear algebra routines and OpemMP directives for the algorithm-specific computational components.
References: Stable reduced Hessian updates for indefinite quadratic programming by T. Fletcher, Mathematical Programming 87 (2000), 251-264.
Requirements: Numerical Linear Algebra and Applications, Linear Programming, Modelling and Solution, and significant prior programming experience.
Second marker: TBA.
JH.DBL.1
Solving Mixed-integer Linear Programming problems
40 point double project
Joint honours mathematics. Suitable for Applied Maths or Informatics and Maths
Mixed-integer Linear Programming (MILP) problems are the most important model used in optimal decision making. As a consequence, there is a considerable body of theory on how they may be solved efficiently. This project will explore some of this and develop an implementation of a subset enabling large practical MILP problems to be solved. Specifically, the project will consider the fundamental branch-and-bound methodology and its implementation. Prior to applying branch-and-bound there is a considerable performance gain if the MILP problem has been preprocessed via a procedure known as presolve, for which, after the MILP has been solved, there is a corresponding postsolve. Presolve consists of applying a range of logical rules to the problem with the aim of reducing its size and modifying the bounds on variables and constraints. There are many such rules and this project will study the most elementary. Mindful of the fact that presolve rules are relatively hard to implement - particularly if the corresponding postsolve is also implemented - only some will be implemented in practice and postsolve may not be implemented. Within the elementary branch-and-bound procedure there are many further techniques which improve its efficiency considerably. Some of these will be studied and a subset will be implemented. Implementation will be in C++, making use of an existing high performance linear programming solver and the necessary interface required to use it within a MILP solver.
References:
  1. Experiments in mixed-integer linear programming by M. Benichou, J. M. Gauthier, P. Girodet, G. Hentges, G. Ribiere, O. Vincent, Mathematical Programming 1 (1971), 76–94;
  2. Noncommercial Software for Mixed-Integer Linear Programming by J. T. Linderoth and T. K. Ralphs, 2004.
Requirements: None.
Second marker: TBA.

MH MILENA HERING
MH.DISS.1
Polytopes and matroids.
40 point dissertation
Single honours mathematics.
To a lattice polytope one can associate a matroid, and the project will be to learn about matroids and to study what sort of matroids can arise in this way. This matroid is related to the study of the defining equations of the toric variety associated to the polytope.
References: None.
Requirements: None.
Second marker: Ivan Cheltsov.

AK ARAM KARAKHANYAN
AK.GRP.1
The Sturm-Liouville problem
20 point group project
Single honours mathematics or any joint honours degree
The boundary value problems (BVP) for linear second order ODE's have a number of important applications in mathematical physics. For instance, to solve partial differential equations (PDE) in some planar domains one may employ the method of separation of variables. It reduces the PDE to BVP for an eigenvalue problem for second order ODE. The Sturm-Liouville BVP is an important example of this sort. The span of questions closely related to this problem includes generalised Fourier series, orthogonality of eigenfunctions, Bessel series expansion etc.
References:
  1. Differential Equations With Applications and Historical Notes by G. Simmons;
  2. Elementary differential equations by W. Boyce and R. DiPrima.
Requirements: Basic calculus, some functional analysis and ordinary differential equations.
Second marker: Martin Dindos.
AK.DISS.1
The Gauss-Codazzi equations and Bonnet's theorem
40 point dissertation
Single honours mathematics or any joint honours degree
The Gauss-Codazzi equations are fundamental equation in differential geometry, in particular for the embedding of surfaces in a Euclidean space. It is a system of partial differential equations: the first equation relates the intrinsic curvature with the Gauss map and hence involves the second fundamental form of surface. The second equation is compatibility condition on the second derivatives of the Gauss map. Using these equations one can solve the problem of recovering the surface from its two fundamental forms modulo an affine transformation. This is contained in Bonnet's theorem.
References: Differential Geometry by A. Pogorelov, 1954.
Requirements: Basic calculus, good knowledge of differential equations and basic differential geometry.
Second marker: James Lucietti.

RK RUTH KING
RK.GRP.1
Estimating population sizes
20 point group project
Particularly suitable for Mathematics and Statistics. Joint degree suitability: Economics and Statistics
Within many areas of statistics, the estimation of population sizes is of primary interest. Applications range from the number of webpages on the Internet on a given topic or the number of bugs in a computer code to the number of drug addicts in a given area or the number of tigers in a conservation area. Typically, total enumeration is infeasible. Thus, methods have been developed to estimate the total population size from a partial enumeration from a variety of sources. The students will be expected to familiarise themselves with the problem and the different approaches that have been used to estimate the population size. It is anticipated that some programming skills will be used in order to fit some of the models to real data and discuss the interpretation. A review will then be written discussing the issues and associated data analyses conducted.
References:
  1. Discrete Multivariate Analysis: Theory and Practice by Y.M. Bishop, S.E. Fienberg and P.W. Holland, M.I.T. Press, Cambridge, MA, 1975;
  2. Estimating Animal Abundance: Closed Populations by D.L. Borchers, S.T. Buckland and W. Zucchini, Springer Verlag, London, 2002.
Requirements: At least two Honours statistics courses.
Second marker: Bruce Worton or Vanda Inacio De Carvalho.
RK.GRP.2
Assessing the importance of assumptions
20 point group project
Particularly suitable for Mathematics and Statistics. Joint degree suitability: Economics and Statistics
In any statistical analysis, the results obtained are conditional on the underlying assumptions that are made. This project will focus on the analysis of incomplete contingency tables and investigate the impact that the underlying assumptions have on the statistical analysis and corresponding results obtained. The students will initially familiarise themselves with incomplete contingency tables and the associated log-linear models often fitted to such data. They will then investigate the sensitivity of the results to the different underlying assumptions. A substantial simulation study will be undertaken in R investigating the impact of assumptions by relaxing a given assumption, simulating a number of datasets, before fitting the standard log-linear models to the data. The results will be discussed in detail and the general conclusions that can be drawn from the simulation study summarised. This project will involve substantial coding using R.
References:
  1. Discrete Multivariate Analysis: Theory and Practice by Bishop, Fienberg and Holland, M.I.T. Press, 1975;
  2. Categorical Data Analysis by Agresti, Wiley, 2013;
  3. Incomplete Contingency Tables with Censored Cells with Application to Estimating the Number of People who Inject Drugs in Scotland by Overstall, King, Bird, Hutchinson and Hay, Statistics in Medicine 33 (2014), 1564-1579.
Requirements: At least two Honours statistics courses.
Second marker: Miguel de Carvalho or Jonathan Gair.
RK.DISS.1
Dealing with Censoring in Incomplete Contingency Tables
40 point dissertation
Single honours mathematics
Within epidemiology, estimating the size of a population is often of interest, such as the number of individuals with a given disease or the number of homeless individuals within an area. However, complete enumeration of such populations is often infeasible (or even simply impossible). Consequently, capture-recapture data are often used, where several sources observe individuals in the given population. Assuming that individuals are uniquely identifiable, this leads to the data being able to be displayed in a contingency table, where each cell corresponds to the number of individuals that are observed by each combination of sources. Log-linear models are commonly fitted to such data to estimate the number of individuals that are not observed by any source. However, for some studies one (or more) sources used to observe individuals may observe and record individuals that are not members of the population of interest (i.e. non-target individuals). This leads to inflated cell entries (i.e. censored cells). This project will investigate the bias that an inflated cell has on the estimation of the total population size before developing a (classical) model-fitting approach for dealing with the problem. Additional modeling extensions that can be considered (time permitting) include the extension to the case where multiple sources may observe non-target individuals and/or the presence of (discrete) covariates.
References:
  1. Discrete Multivariate Analysis: Theory and Practice by Bishop, Fienberg and Holland, M.I.T. Press, Cambridge, MA, 1975;
  2. Incomplete Contingency Tables with Censored Cells with Application to Estimating the Number of People who Inject Drugs in Scotland by Overstall, King, Bird, Hutchinson and Hay, Statistics in Medicine 33 (2014), 1564-1579.
Requirements: At least three Honours statistics courses.
Second marker: Bruce Worton or Vanda Inacio De Carvalho.

TL TOM LEINSTER
TL.GRP.1
Advanced linear algebra
20 point group project
Single honours mathematics or any joint honours degree
Your first exposure to linear algebra was probably quite computational, involving matrices, row operations, determinants, etc. But there is a different approach to linear algebra, an abstract approach, which is much closer in feel to group and ring theory. Here, the roles of vector space and linear map are emphasized, and as much as possible is done without matrices or determinants. The contrast between the two approaches can be illustrated by two different proofs of the fact that a square matrix of complex numbers (or, more abstractly, an operator on a finite-dimensional vector space over C) always has at least one eigenvalue. The proof you probably know goes via the notions of determinant and characteristic polynomial, and uses the fact that a matrix is non-invertible if and only if its determinant is zero. But there is also a more direct, abstract proof, which you will learn in this project. The core material on eigenvalues and eigenvectors leads on to more advanced topics such as Jordan normal form and operators on inner product spaces.
References:
  1. Linear Algebra Done Right, Sheldon Axler, Springer, 1996;
  2. Down with determinants!, Sheldon Axler, American Mathematical Monthly 102 (1995), 139-154.
Requirements: You should have done well in Honours Algebra and have a taste for abstract algebra in general.
Second marker: Sue Sierra, Antony Maciocia, David Jordan, Arend Bayer, Ivan Cheltsov or Harry Braden.
TL.GRP.2
Information Theory
20 point group project
Single honours mathematics or any joint honours degree
Thsi sentnece demontrates taht, even whn errgors ocucr, relaible commjnication cna oftenbe achheved. Lecture notes are another example: errors are made when the lecturer copies from their notes onto the board, and again when the student copies from the board onto paper, yet usually (we hope) it is possible to reconstruct what was intended. A third example, taken from the book of Jones and Jones cited below, is that space exploration has created a demand for accurate transmission of very weak signals through an extremely noisy channel: there is no point in sending a probe to Mars if one cannot hear and decode the messages it sends back. In 1948, the American engineer Claude Shannon gave the world a new mathematical theory of communication, from which present-day information theory is descended. Among other things, Shannon put forward a precise answer to the question what is information? For example, suppose someone is dictating a word to you down the phone, and the first two letters are Q and U. There is a sense in which the U does not give you very much information, because almost all words starting in Q have U as their second letter. But if the second letter had been A, that would have conveyed more information, because there are relatively few words beginning with QA. The fundamental quantity in information theory is entropy. This is closely related to the entropy that occurs in physics, and is also used in theoretical ecology (although you will not need to know about either of those subjects). For example, the entropy per letter of ordinary English prose is about 4.03, which means that we could get away with about 24=16 letters (instead of 26) if we used them more efficiently. Shannon also proved the surprising and fundamental result that even when the environment is terribly noisy, communication can be achieved with arbitrarily high accuracy and arbitrarily little waste. This is Shannon's Fundamental (or Second) Theorem. Your task is to read, understand and present a significant amount of information theory, including at least one of Shannon's major theorems. Information theory is used in many other branches of science (from physics to engineering to biology), and there may be time at the end of the project for you to address some of the aspects that interest you most.
References:
  1. Elements of Information Theory, Thomas M. Cover and Joy A. Thomas, Wiley, 1991;
  2. Information and Coding Theory, Gareth A. Jones and J. Mary Jones, Springer, 2000.
Requirements: The basic vocabulary of probability theory, e.g. as in the second-year courses Probability or Probability with Applications.
Second marker: Michal Branicki or Joan Simon.
TL.IND.1
Sheaf theory
20 point individual project
Single honours mathematics or any joint honours degree
Sheaves are a central concept of modern topology and geometry. Roughly speaking, a sheaf on a space is a collection of functions that may be defined only locally (i.e. on a small patch of the space) and may be required to satisfy some locally-defined property. For instance, continuous real-valued functions on any given space form a sheaf, because continuity is a local property, but bounded real-valued functions do not, because boundedness is not determined locally. The totality of all sheaves on a given space forms a category, and categories that arise in this way are called toposes. So, any space gives rise to a topos. This topos contains a great deal of information about the original space; indeed, almost everything about the space is determined by its topos of sheaves. From this viewpoint, topology can be understood as the study of toposes. But toposes are not only deeply connected to geometry and topology; they are also deeply connected to logic and set theory. Indeed, a topos can be regarded as a generalized universe of sets as well as a generalized space. It is a universe of sets in which the usual laws of logic do not all apply; for instance, proof by contradiction is not usually valid, and in general we can only use constructive methods. Your role in this project would be to learn and explain some (definitely not all!) of this story, taking a direction that suits your background and taste.
References: Sheaves in Geometry and Logic by Saunders Mac Lane and Ieke Moerdijk, Springer, 1994.
Requirements: You should be taking General Topology or General and Algebraic Topology, and have a taste for abstract algebra.
Second marker: Antony Maciocia, David Jordan, Arend Bayer or Jon Pridham.
TL.DISS.1
Integral geometry
40 point dissertation project
Single honours mathematics
What ways are there of "measuring" a convex subset of Rn? For n=2, there are two obvious ways: take the perimeter or the area. For n = 3, there are also two obvious ways: take the surface area or the volume. But there are also some ways that are useful but less obvious: e.g. for a convex subset X of R3, take the length of the shadow that X casts on a line oriented at random. These measures are called intrinsic volumes, and form part of a subject called integral geometry (a kind of partner to differential geometry). It turns out that using the intrinsic volumes, you can answer many very concrete geometric questions. For example, a 20p piece has the property that no matter how you orient it, it always has the same width w (an important property for vending machines), and by the end of this project you'll understand why this implies that it must have the same perimeter as a disc of diameter w. Some other similar questions are listed here: What is integral geometry? Your task will be to learn some of the theory and its applications.
References: Introduction to Geometric Probability, Daniel A. Klain and Gian-Carlo Rota, Lezioni Lincee, Cambridge University Press, 1997.
Requirements: You will need some measure theory, which means that you should be taking either Essentials in Analysis and Probability or Probability, Measure and Finance.
Second marker: Jonas Azzam.

JL JAMES LUCIETTI
JL.IND.1
Lie groups, Lie algebras and their applications
20 point individual project
Single honours mathematics and also Maths and Physics
The purpose of this project is to become conversant with Lie groups (e.g., matrix groups) and their Lie algebras. Lie groups are groups which can be described by continuous parameters (such as the angle parametrising rotations in the plane). Lie groups are largely determined by their Lie algebra, which is a vector space with additional structure. Hence the project has a very strong linear algebraic flavour. The project can be taken in many directions depending on the interest of the students. For students in the Maths & Physics joint degree, the project will also feature applications to physics.
References: Lie groups: an introduction through linear groups, Wulf Rossmann, Oxford University Press, 2002.
Requirements: Honours Algebra and Several Variable Calculus and Differential Equations.
Second marker: Jose Figueroa-O'Farrill, Johan Martens, Harry Braden or Joan Simon.
JL.DISS.1
Topics in mathematical physics
40 point dissertation
The project is especially suitable for Maths & Physics students.
Content to be designed by discussion with interested student.
References: None.
Requirements: None.
Second marker: Jose Figueroa-O'Farrill or Joan Simon.

AM ANTONY MACIOCIA
AM.GRP.1
Paradoxes in Mathematics
20 point group project
Single honours mathematics, Maths with Management, Maths and Stats, Maths and physics
At the end of the nineteenth century some rather alarming paradoxes concerning the foundations of mathematics were coming to light. For example, consider the collection of all sets. Is this a set? One would like it to be, after all, a set is just a collection, is it not? On the other hand, is the set of all sets a member of itself? There are many other related paradoxes which are usually called Russell's Paradox after Bertrand Russell. There have been several attempts to reconstruct the foundations of mathematics to avoid such problems. Three schools of thought developed: the formalists headed by David Hilbert, the logicists headed by Russell and the intuitionists headed by L.E.J. Brouwer. These led to the development of modern set theory. This project should aim to look at these three schools an to explain how they got around the problems posed by the paradoxes.
References:
  1. Collected Works, L.E.J. Brower, North-Holland;
  2. Foundations of Constructive Mathematics, M.J. Beeson, Springer-Verlag.
Requirements: No prerequisites are required.
Second marker: Tom Leinster or Martin Dindos.
AM.GRP.2
The Mathematics of Synthesised Music
20 point group project
Single honours mathematics or any joint honours degree
There are several sorts of ways to synthesize music. One of the simplest is additive synthesis in which the sound spectrum is built up by adding pure sine waves at various frequencies and amplitudes. More sophisticated techniques are subtractive synthesis in which one or more waveforms are generated and then filters are used to remove or mould the spectrum. The parameters of such waveforms are often varied through time. Another approach is to use one waveform to modulate another. This is called frequency modulation and is especially suited to digital music production: it is responsible for some of those (ghastly) mobile phone ringtones. Frequency modulation is particularly interesting from a mathematical perspective because it introduces Bessel's functions. In this project, which could be a 40 point combined project or a 20 point individual projects, you could take one of these methods (or a combination) and produce a mathematical model to describe them. You might base your analysis on the specification of specific hardware or software synthesizers. You might look at how particular forms of mathematical descriptions could sound better (or worse) than others. You might look at how a mathematical model of real sound (say, from a classical musical instrument) can be synthesized using the type of synthesis you are studying. If you have programming experience, you might look into using software such as csound to experiment with the results of your modeling.
References:
  1. A Mathematical Offering, D. Benson, CUP, 2007;
  2. Google analog synthesis music and digital synthesis music.
Requirements: None.
Second marker: Richard Gratwick, Ben Goddard or Tibor Antal.
AM.DBL.1
Transfinite Arithmetic
40 point double project
Comp Sci and Maths, Economics and Maths, AI and Maths, Philosophy and Maths
Use Set Theory to define transfinite cardinal and ordinal numbers, and to understand their arithmetic. For example, the first infinite ordinal number is just the set of all finite numbers. It is possible to define addition, multiplication and exponentiation for ordinal numbers. It turns out that neither addition nor multiplication is commutative but they are both associative. There is a peculiar theorem which states that any strictly decreasing sequence of ordinal numbers is of finite length. One can also define transfinite cardinal numbers. They can also be added,multiplied and exponentiated. Cardinal numbers measure the size of sets. One important question about cardinal numbers is: is there a cardinal number strictly bigger than the cardinality of the natural numbers but strictly less than the cardinality of the real numbers? The statement that there does not exist such a cardinal number is called the Continuum Hypothesis. This project aims to explore the basic concepts of transfinite arithmetic and study a few applications.
References:
  1. On Numbers and Games, J.H. Conway, Academic Press;
  2. Surreal Numbers, D.E. Knuth, Addison-Wesley;
  3. Fundamentals of Contemporary Set Theory, K. Devlin, Springer-Verlag;
  4. Contributions to the Founding of the Theory of Transfinite Numbers, G. Cantor, Dover;
  5. Notes on Set Theory and Logic, P.T. Johnstone, Cambridge Univ. Press.
Requirements: No prerequisites are required.
Second marker: Tom Leinster or Martin Dindos.
AM.DBL.2
TBA
40 point double project
Joint honours mathematics
TBA
References: TBA.
Requirements: TBA.
Second marker: TBA.
AM.DISS.1
Exotic Logics
40 point dissertation
Single honours mathematics, Maths with Management, Maths and Stats
Suggested topics (one of): Ordinary logic uses negation, and, or, implication and such like to provide a reasoning system for mathematics and philosophy. We also have truth values for such logics which make the logic decidable. But occasionally in mathematics and physics we need more exotic logics. For example, it may be appropriate to have more than just 2 truth values. This leads to notions of fuzzy logic which have important applications in engineering. Or we can introduce time varying logic (so called temporal logics). Or we can include notions of necessity and possibility. This leads to modal logics. The mathematics of such exotic logics is remarkably complex and leads to a variety of axiomatic systems. In this project, you would pick one of the exotic logics and study its axiomatization and look at applications in mathematics and beyond.
References:
  1. An Introduction to Modal Logic, G.E. Hughes and M.J. Cresswell, Methuen (1982);
  2. Introduction to fuzzy arithmetic: theory and applications, A. Kaufmann, Van Nostrand Reinhold (1991).
Requirements: Logic 1.
Second marker: Tom Leinster or Martin Dindos.
AM.DISS.2
Categorical Logic
40 point dissertation
Single honours mathematics
Mathematical theories can be expressed in formal languages, are governed by a specific logic (usually classical predicate logic) and come with a set of axioms. We can use the laws of deduction to then prove theorems in the theory. Models provide us with set realisations of such theories together with a notion of when statements are true. Categories can also provide concrete realisations of theories. Such suitable categories are called elementary toposes. It turns out that these always include an interpretation of a logical system which, in this case, is intuitionistic (so the law of excluded middle or reductio ad absurdum, does not hold). This project will explore how toposes are able to do this and to describe in detail several examples.
References: None.
Requirements: None.
Second marker: Tom Leinster.

TM TOM MACKAY
TM.GRP.1
Causality and the Kronig-Kramers Relations
20 point group project
Single honours mathematics or joint honours mathematics and physics
Effect cannot precede cause conveys the general meaning of the principle of causality. For linear dielectric materials, the Kronig-Kramers relations arise as a direct consequence of the principle of causality. The Kronig-Kramers relations are given as a pair of Hilbert transforms, usually expressed in terms of the real and imaginary parts of a complex-valued refractive index. These relations are of considerable practical value in the experimental determination of optical constants of materials. The scope of this project may include:
  • derivations of the Kramers-Kronig relations (using Titchmarsh's Theorem and/or more physically-based methods);
  • applications of the Kramers-Kronig relations in optical measurements;
  • generalisations of the Kramers-Kronig relations (known as dispersion relations) for causal linear systems.
References:
  1. Classical Electrodynamics, J.D. Jackson, 2nd Edition, John Wiley & Sons (1975);
  2. Mathematical Methods for Physicists, G.B. Arfken and H.J. Weber, 4th Edition, Academic Press (1995);
  3. Dispersion Relations and Causal Description, J. Hilgevoord, North Holland Publishing Company (1962).
Requirements: 3rd year core mathematics courses; some physics background would be helpful but not essential.
Second marker: Ben Goddard or Tibor Antal.
TM.GRP.2
Depolarization dyadics in electromagnetic homogenization
20 point group project
Single honours mathematics or joint honours mathematics and physics
This project concerns depolarization dyadics. In effect, a depolarization dyadic represents the electromagnetic scattering response of a small inclusion particle embedded in a homogeneous ambient material. These are important mathematical constructions used in the estimation of the effective electromagnetic properties of composite materials. Explicit forms for depolarization dyadics are available for certain relatively simple inclusion particles (e.g., spherical particles made from isotropic materials), but this is not the case for more complex inclusion particles. The aim of this project is to explore the nature of depolarization for spherical and nonspherical particles made from isotropic and anisotropic materials. This project builds upon mathematical concepts covered in the core 3rd year mathematics courses.
References:
  1. Electromagnetic mixing formulas and applications, A.H. Sihvola, IEE (1999);
  2. Electromagnetic anisotropy and bianisotropy, T.G. Mackay and A. Lakhtakia, World Scientific (2010).
Requirements: 3rd year core mathematics courses; some physics background would be helpful but not essential.
Second marker: Ben Goddard or Tibor Antal.
TM.GRP.3
Voigt wave propagation in anisotropic materials
20 point group project
Single honours mathematics or joint honours mathematics and physics
This project relates to materials that have electromagnetic properties which depend upon direction; these are called anisotropic materials. The propagation of electromagnetic plane waves in anisotropic materials, such as biaxial crystals, is to be investigated. In general, two plane waves with mutually orthogonal polarizations can propagate along any particular direction in such materials. This phenomenon--which is known as birefringence--is commonly described in introductory courses on electromagnetics and/or optics. However, what is not so well-known is that there are instances when the two plane waves are non-orthogonal. Furthermore, the two plane waves then coalesce into a single plane wave which propagates with an amplitude proportional to the propagation distance. This composite wave is called a Voigt wave. This aim of this project is to explore the link between the symmetry of the anisotropic material and the propagation of Voigt waves. The mathematics to be applied in this project will include eigenvector/eigenvalue analysis, Fourier transforms, differential equations, and vector/matrix manipulations.
References:
  1. Electromagnetic Fields and Waves, P. Lorrain, D.R. Corson and F. Lorrain, 3rd Edition, Freeman (1988);
  2. Principles of Optics, M. Born and E. Wolf, 6th Edition, Pergamon (1980);
  3. Physical Properties of Crystals, J.F. Nye, OUP (1985).
Requirements: 3rd year core mathematics courses; some physics background would be helpful but not essential.
Second marker: Ben Goddard or Tibor Antal.

JRM JAMES MADDISON
JRM.IND.1
Adjoint models
20 point individual project
Single honours mathematics
Numerical models are generally given a set of parameter inputs, compute a numerical solution, and then compute a series of diagnostic outputs. For example a model for a linear pendulum requires an initial displacement, initial velocity, and angular frequency. These are used to compute later displacements and velocities. Useful diagnostics for this problem could consist, for example, of the final kinetic energy of the pendulum, or the final displacement. An important question in numerical modelling concerns the sensitivity of a diagnostic to a change in an input: for example, how does the final displacement of a pendulum depend upon the angular frequency? That is, what is the derivative of a given output with respect to a given input? A so-called tangent-linear model enables the derivative of many outputs with respect to a single input to be computed very efficiently. A so-called adjoint model enables the derivative of a single output with respect to many inputs to be computed very efficiently. This project will study the principles of model sensitivity calculation via the derivation of tangent-linear and adjoint models.
References:
  1. Mathematical background: adjoints and their applications;
  2. Perspectives in Flow Control and Optimization by M. D. Gunzburger (SIAM, 2003).
Requirements: Computing and Numerics.
Second marker: Michal Branicki, Ben Goddard or Jacques Vanneste.
JRM.DISS.1
The Finite Element Method
40 point dissertation
Single honours mathematics
The finite element method is an elegant approach for discretising partial differential equations (PDEs) on complex unstructured numerical grids. The finite element method constructs numerical approximations for solutions through approximations of weak integral forms of the PDEs. This leads to a close relationship between the study of PDEs, and the study of their numerical discretisation through the finite element method. This project will introduce the key principles involved in the development and application of the finite element method. Particular emphasis will be placed on the construction of working examples which illustrate the implementation of the approach, and study the properties of the discretisation.
References: Automated solution of differential equations by the finite element method: The FEniCS book by A. Logg, K.-A. Mardal, and G. N. Wells (Editors), Springer, 2012.
Requirements: Previous experience with partial differential equations is strongly recommended. This project will involve programming in Python or MATLAB.
Second marker: Michal Branicki, Ben Goddard or Jacques Vanneste.

JM JOHAN MARTENS
JM.GRP.1
Kovalevskaya's top
20 point group project
Single honours mathematics or Maths and physics
The Kovalevskaya top was the last of a short list of integrable systems, whose discovery came as a surprise to mathematicians of the late 19th century and won Sofia Kovalevskaya a major price of the French Academy of Sciences. This project has two aims: first to introduce students to the topic of integrable systems, and discuss the tops of Euler, Lagrange and Kovalevskaya from a modern point of view. The second aim concerns the history of mathematics: on many fronts Kovalevskaya was a very unusual person, who lived in turbulent times. The second aspect of the project aim to place her mathematics in its historical context.
References:
  1. Spinning Tops: A Course on Integrable Systems, M. Audin, Cambridge Studies in Advanced Mathematics, 1996;
  2. Remembering Sofya Kovalevskaya, M. Audin, Springer, 2011.
Requirements: Differentiable Manifolds.
Second marker: Harry Braden, James Lucietti or Jose Figueroa-O'Farrill.
JM.GRP.2
Convexity theorems for moment maps
20 point group project
Single honours mathematics or Maths and physics
Moment maps in symplectic geometry are the mathematical formulation of a basic principle of physics: symmetry of a physical system results in conservation laws. In the 1980s some surprising general aspects of moment maps for compact symmetry groups were discovered by Atiyah, Guillemin and Sternberg, and Kirwan: their images are convex, giving rise to polytopes. In this project various aspects of these theorems will be investigated, in particular some of the pre-cursors to the general theorem in terms of the distribution of eigenvalues of Hermitian matrices (Schur-Horn theorem).
References: Lectures on Symplectic Geometry by Canas da Silva.
Requirements: Introduction to Lie Groups and Differentiable Manifolds.
Second marker: James Lucietti, Jose Figueroa-O'Farrill or Antony Maciocia.
JM.IND.1
Borel-Weil theorem
20 point individual project
Single honours mathematics or Maths and physics
It is one of the triumphs of representation theory in the middle of the 20th century to give a complete classification of all finite-dimensional representations of compact Lie groups (or complex reductive groups). The Borel-Weil theorem in particular covers the existence part of this classification, by giving an explicit geometric construction of irreducible finite dimensional representations, in terms of sections of a line bundle over an associated flag variety. In this project the theorem and all its prerequisites will be developed. Depending on the preferences of the student this project could be either focussed on algebraic geometry, of on differential/symplectic geometry aspects of the theorem (in terms of geometric quantisation).
References:
  1. Representation Theory of Semisimple Groups: An Overview Based on Examples by Knapp;
  2. Lie Groups, by Duistermaat and Kolk.
Requirements: Introduction to Lie Groups, Algebraic Geometry and/or Differentiable Manifolds.
Second marker: David Jordan, Arend Bayer, Ivan Cheltsov, Milena Hering or Jon Pridham.

AOD ADRI OLDE DAALHUIS
AOD.GRP.1
Computation of special functions in the complex plane
20 point group project
Any degree in mathematics or Maths and physics
Special functions (Bessel functions, orthogonal polynomials, incomplete gamma and beta functions, etc.) are often functions of several (complex) variables, say functions on Cn, n>1. Methods to compute these functions are: differential equations, recurrence relations, integral representations, Taylor series expansions, asymptotic expansions. None of these methods will give satisfying results in the whole of Cn, and a combination of the methods is needed. In this project we will take one special function and try to find a complete set of methods to approximate the function in the whole of Cn. A comparison with the methods that MAPLE uses might also be interesting.
References:
  1. NIST Digital Library of Mathematical Functions;
  2. Special Functions: An Introduction to the Classical Functions of Mathematical Physics, N.M. Temme, Wiley, 1996.
Requirements: Honours Differential Equations and Honours Complex Variables.
Second marker: Ben Goddard, Tibor Antal, Tom MacKay, Jacques Vanneste, Ben Leimkuhler, Michal Branicki, Nikola Popovic, Maximilian Ruffert or Noel Smyth.
AOD.IND.1
Elliptic functions
20 point individual project
Any degree in mathematics or Maths and physics
Elliptic functions are analytic functions in the complex plane that are doubly-periodic. These functions have many nice properties. For example, the most basic elliptic function is the Weierstrass function, and it is possible to express any elliptic function in terms of a rational combination of the Weierstrass function and its derivative. Another example is that when a meromorphic function f(z) is a solution of a first order algebraic differential equation then f(z) is either rational, singly-periodic, or elliptic. In this project you are expected to reconstruct many of the proofs, and apply the results to non-trivial examples.
References:
  1. A Course of Modern Analysis, E. T. Whittaker and G. N. Watson, 1927;
  2. Elliptic Functions and Applications, D. F. Lawden, 1989;
  3. Elements of the theory of elliptic functions, N. I. Akhiezer, 1990.
Requirements: Honours Differential Equations and Honours Complex Variables.
Second marker: Ben Goddard, Tibor Antal, Tom MacKay, Jacques Vanneste, Ben Leimkuhler, Michal Branicki, Nikola Popovic, Maximilian Ruffert, Noel Smyth, Harry Braden, James Lucietti, Jose Figueroa-O'Farrill or Joan Simon.
AOD.IND.2
Inner and Outer Conformal Maps
20 point individual project
Any degree in mathematics or Maths and physics
A conformal map is simply a 1-1 map produced by an analytic function. Given a simple, closed contour C, call such a map from the interior of the unit circle onto the interior of C an inner conformal map, and such a map from the exterior of the unit circle onto the exterior of C (that maps infinity to infinity) an outer conformal map. The project will seek examples of such pairs of maps that can be expressed in terms of known functions.
References:
  1. Handbook of Conformal Mapping with Computer- Aided Visualization, V.I. Ivanov and M.K. Trubetskov, CRC Press, 1995;
  2. Complex variables: introduction and applications, M.J. Ablowitz and A.S. Fokas, 2003;
  3. Theoretical hydrodynamics, L.M. Milne-Thomson, 1968;
  4. Conformal mappings and boundary value problems, Guo-Chun Wen, 1992;
  5. Complex variables and applications, J.W. Brown, 1996.
Requirements: Honours Differential Equations and Honours Complex Variables.
Second marker: Ben Goddard, Tibor Antal, Tom MacKay, Jacques Vanneste, Ben Leimkuhler, Michal Branicki, Nikola Popovic, Maximilian Ruffert or Noel Smyth.

IP IOANNIS PAPASTATHOPOULOS
IP.IND.1
Topics in extremes of time series
20 point individual project
Maths and Stats
A range of different methods used for analysing extremes of time series will be studied. Emphasis will placed be on time series models that exhibit temporal clustering at extreme levels. The students will explore, construct and implement model fitting routines in order to estimate and forecast key quantities of interest such as return levels and mean sojourn times. The models developed will be implemented to environmental and/or financial data.
References: An Introduction to the Statistical Analysis of Extreme values by Stuart Coles, Springer-Verlag, 2001.
Requirements: Statistics Year 2 and knowledge of R/Python.
Second marker: Miguel de Carvalho.

NP NIKOLA POPOVIC
NP.GRP.1
Geometric singular perturbation theory
20 point group project
Mathematics (BSc, MA & MMath-Y4)
Singular perturbation problems feature prominently both in the theory of differential equations and in their applications. Singularly perturbed equations are characterised by the presence of at least two fundamentally different scales, and have traditionally been studied using a variety of (often formal) techniques. More recently, a unified geometric approach has been developed, which is based on dynamical systems theory and, in particular, on invariant manifold methods. While the approach is complete in the hyperbolic setting, it breaks down at non-hyperbolic points; this loss of hyperbolicity can often be remedied by geometric desingularisation (blow-up). In this project, you will familiarise yourself with geometric singular perturbation theory and blow-up. A combination of the two techniques can frequently yield a fairly complete global picture of the system dynamics; examples include neuronal spiking in models of Hodgkin-Huxley type and the propagation of front solutions in degenerate reaction-diffusion systems. You will explore these and similar sample applications both analytically and numerically.
References:
  1. Matched asymptotic expansions-ideas and techniques, P.A. Lagerstrom, Springer-Verlag, 1988;
  2. Methods and applications of singular perturbations: boundary layers and multiple timescale dynamics, F. Verhulst, Springer-Verlag, 2005;
  3. Geometric singular perturbation theory, C.K.R.T. Jones in Dynamical Systems, Montecatini Terme, 1994 (available on request);
  4. A survey on the blow-up technique, M.J. Alvarez et al., preprint, 2010 (available on request).
Requirements: Honours Differential Equations and Applied Dynamical Systems. Some knowledge of Maple/Matlab/Mathematica is desirable.
Second marker: Tom MacKay, Adri Olde Daalhuis, Noel Smyth or Jacques Vanneste.
NP.DBL.1
Mixed-mode oscillatory dynamics
40 point double course
Applied Mathematics or Maths and Biology
Mixed-mode dynamics is a type of complex oscillatory behaviour that is characterised by an alternation of oscillations of small and large amplitudes. Mixed-mode oscillations (MMOs) frequently occur in fast-slow systems of ordinary differential equations; the resulting scale separation allows for the application of perturbation techniques. Mixed-mode behaviour is routinely observed in models from mathematical neuroscience that are based on the classical Hodgkin-Huxley formalism; there, MMOs correspond to complicated firing patterns that are oftentimes seen in experimental recordings of neural activity. In this project, we will consider examples of three-dimensional Hodgkin-Huxley type models that may exhibit mixed-mode dynamics - as well as systems that can be reduced effectively to three dimensions - both analytically and numerically. On the basis of the well-developed geometric theory of mixed-mode dynamics, we will characterise the admissible mixed-mode patterns in these examples, and we will identify relevant control parameters. In particular, we will formulate simplified normal form equations which still capture the essential dynamics of the corresponding full models, and we will describe the canard phenomena that often underlie mixed-mode dynamics in fast-slow dynamical systems. (Canards are solutions that can stay close to strongly repelling manifolds for substantial amounts of time.)
References:
  1. Geometric singular perturbation theory, C.K.R.T. Jones in Dynamical Systems, Montecatini Terme, 1994 (available on request);
  2. A survey on the blow-up technique, M.J. Alvarez et al., preprint, 2010 (available on request);
  3. Canards in R3, P. Szmolyan and M. Wechselberger, J. Differential Equations 177(2), 419-453, 2011;
  4. Mixed-mode oscillations with multiple time scales, M. Desroches et al.; ; SIAM Rev. 54(2), 211-288, 2012.
Requirements: Honours Differential Equations and Applied Dynamical Systems. Some knowledge of Maple/Matlab/Mathematica is desirable.
Second marker: Tom MacKay, Adri Olde Daalhuis, Noel Smyth or Jacques Vanneste.

DQ DAVID QUINN
DQ.GRP.1
Groebner bases and convex polytopes
20 point group project
Any degree in mathematics or joint degree
Groebner bases are an important tool in commutative algebra and their algorithmic nature quickly leads to applications of commutative algebra in other areas. The project will intially focus on Strumfel's text before moving on to use the tools which are developed there. The ultimate direction of the project will depend on the interests of the students but can range from algebraic geometry to algebraic statistics and other areas not necessarily in-between.
References: Groebner Bases and Convex Polytopes by Bernd Strumfels.
Requirements: Commutative Algebra (at least in parallel).
Second marker: Milena Hering or Ivan Cheltsov.
DQ.GRP.2
Toric varieties
20 point group project
Any degree in mathematics or joint degree
The geometry of toric varieties can be studied through combinatorial objects such as polytopes and polyhedral fans. One can initially work using elementary linear algebra before moving into a richer theory within algebraic geometry.
References:
  1. Lectures on toric varieties by David Cox;
  2. Introduction to toric varieties by Bill Fulton.
Requirements: Commutative Algebra and Algebraic Geometry (can be taken in parallel).
Second marker: Ivan Cheltsov or Iain Gordon.

MR MAX RUFFERT
MR.DISS.1
Unstable Closed Orbits - Choreographies
40 point dissertation
Single honours mathematics
The orbits of two gravitationally interacting bodies are well known and stable. However, already for three equal-mass interacting bodies, described as the three-body problem, only a very small number of special cases have stable orbits. Simo and others have found periodic but unstable many-body solutions called choreographies. This project would include writing several computer programmes to reproduce some of the basic 2-dimensional choreographies with 3-5 bodies and extend this to include orbits with more than 6 bodies and possibly orbits in 3-dimensions. This project is very numerical in nature, with all calculations done by computer code. The student will have to write this code to suit the needs of the problem. No specific text book will be adhered to. If required, a bibliography can be provided from previous student's project work.
References:
  1. The choreographies of the N-body problem;
  2. Periodic orbits (video) for the n-body problem;
  3. Strange Orbits;
  4. A new solution to the three body problem - and more;
  5. Figure Eight Orbits.
Requirements: Solid experience in computer programming (preferably Fortran or C or Matlab).
Second marker: Michal Branicki, Tibor Antal, Ben Goddard, Ben Leimkuhler, Tom MacKay, Nikola Popovic, Adri Olde Daalhuis or Noel Smyth.

CS CHRIS SANGWIN
CS.GRP.1
The "big three": integrate, factor and solve
20 point group project
Single honours mathematics
Computer algebra systems have three significant algorithms: integrate, factor and solve. The mathematical techniques used to implement the algorithms differ significantly from the methods typically taught to undergraduate mathematicians. The algorithms often make use of research mathematics undertaken in the late 20th centurary, rather than traditional 19th centruary calculus and algebra. This project will examine at least two of these three algorithms, and will develop a curricula sequence which would enable undergraduate students to study these algorithms. That is to say, the project will give an elementary account of the algorithms and identify prerequisite knowledge and critical examples. This project takes place at the intersection of mathematics, education and computer science.
References:
  1. The Computer Modelling of Mathematical Reasoning, A. Bundy, 1983, Academic Press;
  2. Computer algebra: systems and algorithms for algebraic computation, J. H. Davenport, Y. Siret, E. Tournier, 1993, Academic Press Professional.
Requirements: Honours Algebra.
Second marker: Antony Maciocia or Toby Bailey.
CS.IND.1
The mathematics of the rattleback
20 point individual project
Single honours mathematics or Mathematics and physics
This project reviews the mathematical models developed for the rattleback/CELT, a curious mechanical device.
References: Rattleback
Requirements: Honours Differential Equations and Numerical Ordinary Differential Equations and Applications.
Second marker: Ben Goddard.
CS.IND.2
The organica of Van Schooten
20 point individual project
Single honours mathematics
When people say they have "read Descartes" the usually mean they have read Van Schooten's 1649 Latin translation of and commentary on Descartes' Geometrie. This project will write a commentary on Van Schooten's little-known work De organica conicarum sectionum in plano descriptione of 1649. The project will also place the organica in the context of the mathematics of the time, and within contemporary mathematics. Note, this work does not yet exist in English.
References: De organica conicarum sectionum in plano descriptione by Franciscus van Schooten, Lugd. Batavor.: Ex Officina Elzeviriorum, 1646.
Requirements: Latin 1B
Second marker: TBA

SS SOTIRIOS SABANIS
SS.DISS.1
Pricing Vanilla and Exotic options using the 3/2 Stochastic Volatility model
40 point dissertation
Single honours mathematics
Using a new generation of explicit Euler-type numerical schemes for SDEs, one can show that the pricing of Exotic options under the assumption of the 3/2 Stochastic Volatility model can be performed in a more efficient way than before. (Multi-level) Monte Carlo simulations can be used to illustrate the implementation of this new methodology (and its agreement with the theoretical results).
References:
  1. Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients by S. Sabanis, Annals of Applied Probability 26 (2016), 2083-2105;
  2. Numerical Solution of Stochastic Differential Equations by P. Kloeden and E. Platen, Applications of Mathematics, 1999, Springer, Berlin;
  3. Stochastic volatility models and the pricing of VIX options by J. Goard and M. Mazur, Mathematical Finance 23 (2013), 439-458.
Requirements: Probability, Measure and Finance. Knowledge of some programming language (such as C++, Java, etc) or relevant mathematical package (MAPLE, Matlab, etc) is considered essential.
Second marker: TBA.

SJS SUE SIERRA
SJS.GRP.1
Characters and representations of finite groups
20 point group project
Single honours mathematics
A representation of a group is a homomorphism from the group to a matrix group; equivalently, a representation is a vector space on which the group acts. Representations in fact control the structure and properties of the group; even more powerfully, in many cases all one needs to know about the representation are the traces of the matrices involved. In this project we'll explore the representation theory of finite groups. The core material includes semisimplicity, orthogonality relations, and the character table. From there students will have options to move in various directions, from computing character tables of various interesting groups to proving Burnside's famous theorem about the structure of groups of order paqb.
References:
  1. Representations and Characters of Groups by James and Liebeck;
  2. Introduction to Group Characters by Ledermann.
Requirements: Fundmentals of Pure Mathematics and Honours Algebra. Students should be simultaneously enrolled in Group Theory.
Second marker: Harry Braden, David Jordan, Ivan Cheltsov or Iain Gordon.
SJS.DISS.1
Noncommutative algebraic geometry
40 point dissertation
Single honours mathematics
It's well-known that commutative rings are closely associated to algebraic geometry. Is there geometry associated to noncommutative rings? The answer is yes but the connections are much more subtle; this field has only been developing for about 25 years. In this project you'll learn noncommutative ring theory and noncommutative algebraic geometry. There will be some general theory, but also many concrete examples. You'll compute geometric spaces associated to free algebras, quantum n-space and Sklyanin algebras, learn how to twist a commutative ring, and will learn why noncommutative projective curves are actually commutative.
References: An Introduction to Noncommutative Projective Geometry by Dan Rogalski.
Requirements: Commutative Algebra and Algebraic Geometry.
Second marker: David Jordan or Iain Gordon.

JS JOAN SIMON
JS.GRP.1
Cosmological Models
20 point group project
Single honours mathematics or Maths and Physics
The universe we live in is currently accelerating. Thus, it changes with time. It contains different sources of energy : regular matter, dark matter, radiation and dark energy. Different sources dominated the expansion of the universe at different stages in the history of its evolution. The purpose of this project is to use Einstein's equations to study the properties of the universe at these different stages.
References:
  1. Gravitation and Cosmology, S. Weinberg (1972);
  2. Introduction to Cosmology, B. Ryden (2003).
Requirements: None, though depending on the motivation learning differential geometry and having some computer skills can help.
Second marker: Harry Braden, Pieter Blue or Jose Figueroa-O'Farrill.
JS.GRP.2
Topics in mathematical physics
20 point group project
Single honours mathematics or Maths and Physics
Any suitable topic of interest in mathematical physics agreed between students and supervisor.
References: Depends on the topic.
Requirements: Depend on topic.
Second marker: Harry Braden, James Lucietti or Jose Figueroa-O'Farrill.
JS.DBL.1
Topics in Quantum Information
40 point double project
Joint honours mathematics
Any suitable topic of interest in quantum information agreed between students and supervisor.
References: Quantum computation and Quantum information by M.A. Nielsen and I.L. Chuang, Cambridge University Press, 2010.
Requirements: Abstract Linear Algebra and some Quantum mechanics and statistical mechanics.
Second marker: Jose Figueroa-O'Farrill.
JS.DISS.1
TBA
40 point dissertation
Single honours mathematics or Maths and Physics
In this project we will explore the basic tools developed in statistical mechanics to study how to describe complex systems in terms of coarse grained variables by the introduction of entropy. Depending on the evolution of our goals, we will consider relations to information theory and/or quantum mechanics.
References: TBA
Requirements: TBA
Second marker: TBA.

NS NOEL SMYTH
NS.IND.1
Nonlinear Wave Equations and Inverse Scattering
20 point individual project
Suitable for Mathematics and Mathematics and Physics degrees
While the wave equation's name suggests that it may describe all wave phenomena, there are many waves in nature which are not described by the wave equation. The most familiar example is waves on the surface of the ocean. This project will look at equations which describe such physical processes such as waves on the surface of a shallow fluid and light in optical telecommunication fibres. While the underlying physics of these waves id different, it turns out that a large array of waves in nature are governed by generic equations such as the Korteweg-de Vries (KdV) equation and the nonlinear Schrodinger (NLS) equation. These equations also have wave solutions which are non-oscillatory, but take the form of humps, termed solitary waves. The first example of a solitary wave was seen on the Union Canal near Heriot-Watt University in 1834. Remarkably, the KdV and NLS equations have exact solutions via the method of inverse scattering. This method arose in geophysical exploration and the need to determine the structure of the Earth beneath the ground. The project will look at where equations such as the KdV and NLS equations arise and how, their solitary wave solutions and the method of inverse scattering.
References:
  1. Linear and Nonlinear Waves, G.B. Whitham, J. Wiley and Sons, New York (1974);
  2. Nonlinear Dispersive Waves. Asymptotic Analysis and Solitons, M.J. Ablowitz, Cambridge University Press (2011);
  3. Optical Solitons. From Fibers to Photonic Crystals, Y.S. Kivshar and G.P. Agrawal, Academic Press, San Diego (2003).
Requirements: Several Variable Calculus and Differential Equations and Honours Differential Equations.
Second marker: Jacques Vanneste or Stuart King.
NS.IND.2
Asymptotic Solution of Differential Equations
20 point individual project
Suitable for Mathematics and Mathematics and Physics degrees
While courses give a number of techniques for solving ordinary differential equations, these apply to only a very small number of equation types. In particular, if the ordinary differential equation is nonlinear, then it is highly unlikely that an exact solution can be found. This project will look at the asymptotic (approximate in some sense) solution methods for ordinary differential equations termed the Poincare-Lindstedt method (or Stokes' expansion) and the method of multiple scales. These techniques rely on there being a small parameter in the equations. Both techniques can be applied to problems in wave theory, water waves, mechanical oscillators, celestial mechanics and general relativity, among many areas.
References:
  1. Multiple Scale and Singular Perturbation Methods, J. Kevorkian and J.D. Cole, Springer-Verlag (1996);
  2. Advanced Mathematical Methods for Scientists and Engineers, C.M. Bender and S.A. Orszag, McGraw-Hill, New York (1978).
Requirements: Several Variable Calculus and Differential Equations and Honours Differential Equations.
Second marker: James Maddison or Adri Olde Daalhuis.
NS.IND.3
Dispersive Waves
20 point individual project
Suitable for Mathematics and Mathematics and Physics degrees
Due to its name, the wave equation is assumed to govern common wave phenomena in nature. However, this is not the case, an example being waves on the surface of the ocean. Surface water waves are termed dispersive waves and are not governed by the wave equation. This project will look at what dispersive waves are and how they are modelled and analysed. It will then study specific examples of waves in fluids and in optics, for instance the waves generated by a moving ship and the use of light in telecommunication optical fibres.
References:
  1. Linear and Nonlinear Waves, G.B. Whitham, J. Wiley and Sons, New York (1974);
  2. Nonlinear Dispersive Waves. Asymptotic Analysis and Solitons, M.J. Ablowitz, Cambridge University Press (2011);
  3. Optical Solitons. From Fibers to Photonic Crystals, Y.S. Kivshar and G.P. Agrawal, Academic Press, San Diego (2003).
Requirements: Several Variable Calculus and Differential Equations and Honours Differential Equations.
Second marker: Ben Goddard, Jacques Vanneste or Stuart King.

JV JACQUES VANNESTE
JV.GRP.1
Sound generation by fluid flow
20 point group project
Single honours mathematics, Maths and Stats, Maths and physics
The dynamics of many fluids is often separated conceptually into flow (characterised by large particle motion) and waves (characterised by oscillatory motion). This separation is not exact, however: flows often generate waves and vice versa. In the case of compressible fluids, the waves are sound waves, and their generation by flows is familiar to anyone who has heard planes fly by. An elegant theory, originally formulated by Lighthill, provides a way of computing the sound waves radiated by a given flow or by moving surfaces (such as turbine blades). This project will study this theory, starting from the necessary fluid-dynamics background, examine some of its applications, and consider some extensions (such as the feedback of the sound wave emitted on the flow). It will also be possible to explore the analogy between sound radiation in fluids and the radiation of gravitational waves as predicted from general relativity.
References: The theory of vortex sound by Howe, Cambridge University Press, 2002.
Requirements: Several Variable Calculus and Differential Equations and Honours Differential Equations. Some background in physics would be useful.
Second marker: Michal Branicki, Ben Goddard, James Maddison, Maximilian Ruffert or Noel Smyth.
JV.GRP.2
Dispersion in disordered media
20 point group project
Single honours mathematics, Maths with Management, Maths and Stats, Maths and physics
The dispersion of particles experiencing random motion has many applications including to the spreading of pollutants or the movement of molecules inside biological cells. In homogeneous environments, this dispersion is described by classical diffusion and governed by the heat equation, leading to the linear growth of mean-square displacement and to Gaussian statistics. The situation is much richer in heterogeneous environments, and especially in random environments: anomalous diffusion (with mean-square displacements growing nonlinearly) and non-Gaussian statistics are then typical, and their description requires more advanced mathematics. This project will examine simple models of dispersion in heterogeneous environments and show how their properties can be quantified, in many cases explicitly, using tools of probability theory. Examples of models include dispersion with random obstacles, with random traps, on graphs or on fractals. Numerical computations (using matlab) will be used to illustrate and test the theoretical results.
References:
  1. Anomalous diffusion in disordered media by Bouchaud and George, Phys. Rep. 195 (1990), 127-293;
  2. Diffusion and reaction in fractals and disordered systems by Ben-Avraham and Havlin, Cambridge University Press, 2000.
Requirements: Probability or Probability with Applications. Stochastic Modelling may be useful.
Second marker: Michal Branicki, Tibor Antal, Ben Leimkuhler, Nikola Popovic, or Kostas Zygalakis

BW BRUCE WORTON
BW.GRP.1
Statistical modelling of circular data
20 point group project
Particularly suitable for Mathematics and Statistics degree students
Interest in developing statistical methods to analyze directional data dates back as far as Gauss. Such data include: wind directions, vanishing angles of homing pigeons - measured in range (0,π) or (0,360), times of birth over the day in hours - convert by multiplying by 360/24, times of death from a single cause over years. This project would study the theory and application of circular data. Topics would include:
  • descriptive directional statistics,
  • common parametric models,
  • inference problems on the circle.
If time permits it may also be possible to study correlation and regression for directions. Application to meteorological data from JCMB weather station will provide an insight into the times of the year when KB is most windy, and if this relates to other variables.
References:
  1. Statistical Analysis of Circular Data, N.I. Fisher (1993);
  2. Statistical Analysis of Spherical Data, N.I. Fisher, T. Lewis and B. J. J. Embleton (1987);
  3. Statistics of Directional Data, 2nd edition, K.V. Mardia and P.E. Jupp (2000).
Requirements: Linear Statistical Modelling, Likelihood and Statistical Communication Skills.
Second marker: Ioannis Papastathopoulos and Ruth King
BW.DBL.1
The Laplace Distribution and Generalizations
40 point double project
Joint degree suitability: Economics and Statistics
This project involves studying the Laplace distribution, and its numerous generalizations and extensions, for statistical modelling. There will be the opportunity to investigate the theory as well as apply the methods to data sets from areas such as Communications, Economics, Engineering, and Finance.
References: The Laplace Distribution and Generalizations by S. Kotz, T. Kozubowski, and K. Podgorski (2001).
Requirements: Linear Statistical Modelling, Likelihood and Statistical Communication Skills.
Second marker: Ioannis Papastathopoulos and Ruth King
BW.DISS.1
Using empirical likelihood for statistical analysis
40 point dissertation
Single honours mathematics and MMATH
Empirical likelihood methods use empirical distributions to define a likelihood function for a parameter of interest, e.g., a population mean. This approach provides an alternative way of obtaining a likelihood to conventional parametric modelling. This statistical project will study ways of computing empirical likelihoods for location parameters, and regression parameters. Iterative numerical methods are required, as empirical likelihoods cannot be written in closed form.
References:
  1. Elements of Statistical Computing, R.A. Thisted, Chapman and Hall, 1988;
  2. Empirical likelihood ratio confidence intervals for a single functional, A. B. Owen, Biometrika 75 (1988), 237-249;
  3. Empirical Likelihood, A.B. Owen, Chapman and Hall/CRC, 2001.
Requirements: Linear Statistical Modelling, Likelihood and Statistical Communication Skills.
Second marker: Miguel de Carvalho and Vanda Inacio De Carvalho.

KZ KOSTAS ZYGALAKIS
KZ.GRP.1
Chaotic Dynamical Systems
20 point group project
Single honours mathematics or joint honours mathematics and physics, Maths and Stats
Chaos theory is the field of study in mathematics that studies the behavior of dynamical systems that are highly sensitive to initial conditions, a response popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future behaviour is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behaviour is known as deterministic chaos, or simply chaos. Two simple examples of such chaotic systems are the logistic map and the Lorenz equations. In this project you will be asked to investigate (depending on your preferences) either numerically or analytically (where possible) the properties of those equation and the transitions of their solutions in the chaotic regime [1,2]. Other possible topics in the area, includes the calculation of the Liapunov exponent (a measurement of the sensitivity of the system with respect to its initial conditions) as well as control of chaotic trajectories (for the logistic map).
References:
  1. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, S. Strogatz, Westview Press, 2001;
  2. Discrete Chaos, S. Elaydi, Chapman and Hall/CRC, 2000;
  3. Likelihood and Bayesian Prediction of Chaotic Systems, L. Mark Berliner, Journal of the American Statistical Association 86 (1991), 416;
  4. Chaos, and what to do about it?, specialized open online course.
Requirements: Honours Differential Equations, knowledge of Matlab or another programming language.
Second marker: Tibor Antal.
KZ.GRP.2
When will a large complex system be stable?
20 point group project
Single honours mathematics or Maths and Stats
The notion of stability of stationary solutions is central to the study of dynamical systems. For many applied questions, it is pivotal to know that the solution will return to the stationary values/orbits upon perturbation. A prime example is the one coming from ecology and from stability of food webs [1,2,3,4,5]. In the simplest possible case Lord May [1], asked the question of what is the probability of a system being stable around an equilibirum point as the dimension of the system grows, and showed that this probability becomes vanishing small. However, if one imposes extra conditions this doesn't have to be the case [5,7]. In this group project, you would be asked to investigate the stability properties of equilibrium points of dynamical systems using a combination of numerical investigations and theory. In doing so, you will need to study element of Random Matrix theory [6].
References:
  1. Will a Large Complex System be Stable, R. May, Nature 238, 1972;
  2. Stability and diversity of ecological communities, S. McNaughton, Nature 274, 1978;
  3. The stability of real ecosystems, P. Yodzis, Nature 289, 1981;
  4. The diversity–stability debate, K. S. McCann, Nature 405, 2000;
  5. Stability criteria for complex ecosystems, S. Allesina, S. Tang, Nature 483, 2012;
  6. Random matrix theory and its innovative applications, A. Edelman, Y. Wang, Advances in Applied Mathematics, Modeling, and Computational Science, 2013'
  7. Conditional random matrix ensembles and the stability of dynamical systems, P. Kirk, D. M. Y. Rolando, A. L. MacLean and M. P. H. Stumpf, New J. Phys. 17, 2015.
Requirements: Honours Differential Equations, knowledge of Matlab or another programming language.
Second marker: Tibor Antal.