The project organiser with be Prof S Majid. Information on projects will move to his webpage shortly. For the moment this page contains preliminary information which will be useful if you want to start thinking about possible project topics over the summer. Confirmation of the availablity of particular projects will appear on this page when known.
This page is maintained by Robert Johnson. Suggestions on the content of this page are welcome.
The report must involve the study of some mathematical topic at the 4th year level and must be the student's own work in the sense that it gives an original account of the material, but it need not contain new mathematical results.
The MSci project length should be about 8000 words: if it is less than 6000 or more than 10000 you may lose marks. For the Third Year Project the suggested length is 4000 words and if the project is less than 3500 or more than 5000 then you may lose marks.
The report can be written in a single semester or the work can be spread over two semesters, depending on the other units taken: the submission deadline will be the same in all cases.
Students should base their 30 minute seminar-style presentation (which usually takes place in the second half of the second semester) on their MSci project work.
The purpose of the project is largely to test the student's ability to work independently. The supervisor will provide a reading list and give general advice on the work for the project and the writing of the report. The supervisor cannot be expected to provide a list of all the individual results that should go into the report, although a few major items will probably be mentioned. The student is strongly advised to pass a first draft to the supervisor, who will then comment on the format and the English style and point out any major mathematical errors. Advice on the suitability or otherwise of particular sections of the report cannot be expected.
The report should be on A4 size paper, typed on a computer or a type writer or else very neatly written. It should be in a simple binding e.g. a ring or springback binder. Each report should include
A list of possible projects is given below. By following the links, you'll find brief details of what is involved in the project, prerequisites and reading material. Please speak to the supervisors of any projects that sound interesting, and let me know your final choice of project by the end of October. Please note that this list is not authoritative – some of the projects might not be available (if the supervisor has withdrawn it without telling me). Some are marked as unavailable, because the supervisor will be unavailable. In this case, it may be possible to do the project with another supervisor.
Another possible source of projects is from the list of possible MSc projects. Please consult with the relevant member of staff and see if they are willing to offer the project as an MSci project. (Owing to differences in the courses this may not be possible.)
Also, most staff are willing to offer projects in subjects related to their research interests, and so if there's something you want to do a project on which is not listed here, then try to find the most appropriate member of staff and suggest a project. If you're not sure who would be best, consult me.
Supervisor: D. Arrowsmith
Intermittency is characteristic of orbital structure in dynamical systems which exhibits small changes in orbital movement. The phenomenon can occur in various ways and it has been used in a wide variety of applications. The report would aim to give an overview of the theory and the most recent applications of intermittency.
The student would need to have computing and numerical skills and have obtained at least a 2.1 in Chaos and Fractals.
Supervisor: D. Arrowsmith
This is a well-founded area which uses control theory techniques allied to chaotic situations. The basic technique should be described mathematically, developed as a computer program and applied to various applications.
The student would need to have computing and numerical skills and have obtained at least a 2.1 in Chaos and Fractals.
Availability: Third year and MSci
Supervisor: R. Bailey
Latin squares are easy to define (see Chapter 6 of 'Combinatorics' by P. J. Cameron) but many problems flow from them which are easy to state but hard to solve. The project could focus on orthogonal Latin squares, on complete Latin squares, or on any one of many other topics.
Algebraic Structures I
Combinatorics
J. Dénes & A. D. Keedwell, Latin squares and their applications, English Universities Press, 1974.
Availability: MSci
Supervisor: R. Bailey
The combinatorial object called a Latin square (see Chapters 3 and 10 of 'Planning of Experiments' by D. R. Cox) can be used to construct a design for an experiment in many different ways: in a square array; as a fractional factorial design; as a neighbour-balanced sequence, etc. The project could explore some of these.
Design of experiments
Combinatorics
Availability: MSci
Supervisor: R. Bailey
Incomplete-block designs arise in both Design of Experiments and
Combinatorics. In the experimental setting, we might have seven
people willing to taste and rate cheeses. There are seven different
cheeses, but each person can taste only four before their palate gets
jaded. How should we allocate the cheeses to the people? A
combinatorialist would regard this as a problem of having a set of
seven cheeses and finding seven subsets of size four, with certain
desirable properties.
This project can be approached from either perspective. For the first,
a relevant starting point is Chapter 11 of Cox's `Planning of
Experiments'. The aim is to find out about the designs, optimality
properties, some constructions. For the second, a relevant starting
point is Chapter 16 of Cameron's `Combinatorics'. This would focus
more on infinite families of designs and their mathematical properties.
In either case, the project is to find out about the topic and
write a coherent account of it.
either Design of Experiments or Combinatorics
Availability: MSci
Supervisor: R. Bailey
Row-column designs are used for experiments in spatial settings when
it is thought that both rows and columns should be used as blocks. A
recent by R. A. Bailey and a co-author gives a recipe for constructing
row-column designs with good properties.
The project is to read parts of the paper, explain it in the student's
own words, and construct the designs for smallish values of the
parameters.
either Design of Experiments or Combinatorics
Availability: MSci
Supervisor: R. Bailey
Classical factorial designs, either in blocks or as fractional
replicates, are constructed as (cosets of) subgroups of the group
of all possible treatment combinations. Therefore, there are strong
links both to the theory of linear codes and to the theory of Abelian
groups.
The project is to find out about the connections between these
designs, linear codes and Abelian groups, and write a coherent
account of them.
Design of Experiments
Algebraic Structures I, Coding Theory
Availability: MSci
It is very common to conduct an experiment for n varieties, each
occurring r times, in an r by n rectangle, in such a way that each
variety occurs once per row. Should we take any account of the
pattern of varieties in the columns?
A recent paper by R. A. Bailey proposes two different solutions.
The project is to do the following, for either one of those
solutions: (i) read that part of the paper and reproduce it in the
student's own words (ii) verify the construction of the designs given
in the table, and extend the table a little (iii) make the designs in
the table available in an electronic format suitable for later
conversion for the web server designtheory.org.
Part (i) will be supervised by RAB, part (iii) by LHS, part (ii) by
both, as some computer searching may be involved.
Either Combinatorics or Design of Experiments, in addition to computing skills.
Availability: Third year and MSci
Supervisor: C. Beck
Multifractals are fractals with a probability measure on their support. A good statistical description is achieved by generalizing the concept of a fractal dimension (capacity) to that of a dimension function D(q) depending on a real parameter q. The project is to write a short survey on fractals, multifractals, and the dimension function D(q).
Chaos and Fractals
C.Beck & F. Schloegl, Thermodynamics of Chaotic Systems, Cambridge University Press (1995).
Availability: Third year and MSci
Supervisor: C. Beck
Path integrals yield a very important tool for modern formulations of quantum theory and quantum field theories. The concept was originally introduced by Feynman in his PhD thesis. There is an interesting connection to the theory of stochastic processes and Brownian motion. The project is to write a short survey on path integrals, concentrating on the general definition of a path integral and to apply it to nonrelativistic quantum mechanical systems.
Probability Models
Applied Stochastic Processes (useful)
R. Feynman & A. Hibbs, 'Quantum Mechanics and Path Integrals' (McGraw–Hill 1965).
L. Schulrnan, 'Techniques and Applications of Path Integration' (Wiley 1981).
Supervisor: S. Bullett
On 30th May 1799 Gauss discovered a remarkable relationship between the 'arithmetic–geometric mean and the value of certain 'elliptic integrals' which he described as certain to 'open up an entirely new field of analysis'. He was right. His discoveries are now central to efficient calculations of constants such as π, and his methods fit very well into modern 'chaos theory'.
Geometry
Complex Functions
(maybe also Number Theory and/or Chaos and Fractals)
J. Borwein & P. Borwein, 'Pi and the AGM', (John Wiley 1987).
D. Cox, 'The arithmetic–geometric mean of Gauss' L'Enseignement Mathématique 30 (1984) 275–330.
S. Bullett, 'Dynamics of the arithmetic–geometric mean', Topology 30 (1991) 171–190.
Supervisor: S. Bullett
Various possible topics, including
In 1989 Devaney proposed three criteria for a dynamical system to be regarded as 'chaotic'. Over the years since various authors have shown that these criteria are related to one another in various ways. The project is to write of survey of what is known.
Chaos and Fractals
Topology
J. Banks, J. Brookes, G. Cairns, G. Davis & P. Stacey, 'On Devaney's definition of chaos', Amer. Math. Monthly 99 (1992) 332–334.
M. Vellekoop & R. Berglund, 'On intervals, transitivity = chaos', Amer. Math. Monthly 101 (1994) 353–355.
A. Crannell, 'The role of transitivity in Devaney's definition of chaos', Amer. Math. Monthly 102 (1995) 788–793.
P. Touhey, 'Yet another definition of chaos' Amer. Math. Monthly 104 (1997), 411–414.
Sarkovskii's Theorem is a key theorem in 'Chaos Theory'. It first appeared in the Ukrainian Maths Journal in 1964 (Vol. 16, pages 61–71), but various proofs and generalizations have appeared since.
Chaos and Fractals
Topology
A. Sarkovskii, 'Coexistence of cycles of a continuous map of the line into itself' (translated from Russian), International Journal of Bifurcation and Chaos 5 (1995) 1263–1273.
T. Li & J. Yorke, 'Period three implies chaos', Amer. Math. Monthly 82 (1975) 985–992.
P. Glendinning, 'Stability, Instability and Chaos' (CUP 1994).
C. Ho & C. Morris, 'A graph-theoretic proof of Sarkovskii's Theorem on the periodic points of continuous functions' Pacific J. Math 96 (1981) 361–370.
Supervisor: D. Burgess
Understanding fluid flow in realistic situations is vital in many fields, from weather forecasting to making plastic boxes. This project is to review some of the computational methods used, and their applicability to different types of fluid, and to different systems. Depending on programming background, it may also be possible to write a computer simulation for a fluid.
Calculus III
Dynamics of Physical Systems
Introduction to Numerical Computing (with a good result)
Fluid Dynamics (may be taken simultaneously with the project)
Supervisor: P. Cameron
Finite projective planes are simple to define and entirely combinatorial in nature, and yet there are wide gaps in our knowledge about their existence. There are a number of topics which are not covered in any of our courses but the material can be found without too much difficulty, such as the Bruck–Ryser theorem, the construction of finite non-Desarguesian planes, the relationship between configuration theorems and automorphisms, the connection with orthogonal Latin squares, and the connection with error-correcting codes which underlies the computer proof of the non-existence of a projective plane of order 10.
Combinatorics
D. Hughes & F. Piper, 'Projective Planes' (Springer 1973).
Supervisor: P. Cameron
Ten years ago, Hammons et al. discovered that certain families of non-linear binary codes could be 'linearised' by regarding them as images under a naturally defined 'Gray map' of linear quaternary codes (codes over the integers mod 4). This gave rise to simpler constructions of these codes and to connections with many other parts of mathematics including symplectic and orthogonal geometry over finite fields and systems of lines in real or complex space. The project could involve constructing the codes, explaining the connections, and maybe thinking about generalisations.
Supervisor: P. Cameron
One way to understand a class C of groups (say, abelian groups, nilpotent groups, soluble groups) is to study the minimal non-C groups (those for which every proper subgroup belongs to the class C). There are several classifications of such groups. Moreover, the strategy used to classify the finite simple groups was to study minimal non-K groups, where K is the class of groups all of whose composition factors are "known" simple groups.
The classification of minimal non-nilpotent groups is maybe the earliest such result and is a bit inaccessible: O. J. Schmidt, Mat. Sbornik 1924 (in German), but is referred to in many places. Minimal non-soluble groups are in J. G. Thompson, Bull. Amer. Math. Soc. 1968.
Supervisor: P. Cameron
John McKay pointed out the coincidence that the smallest size of matrices which represent the "Monster" sporadic simple group is 196883, while the smallest non-trivial coefficient in the Laurent series for the classical modular function is 196884. Investigation of this led to the discovery of further coincidences, and then to the formulation of a conjecture by Conway and Norton, which was proved by Borcherds. The aim of the project is to describe the mathematical objects on the two sides of this unexpected connection and, if possible, to say something about how the connection works.
A starting reference can be found at http://www.maths.qmul.ac.uk/~pjc/csgnotes/moon.pdf There are many accessible references.
Supervisor: P. Cameron
A graph is homogeneous if any isomorphism between finite induced subgraphs can be extended to an automorphism of the graph. This is the strongest symmetry condition we can impose on a graph. Finite homogeneous graphs were determined by Gardiner, and countably infinite ones by Lachlan and Woodrow by completely different methods. One can weaken the notion of homogeneity: in some cases we have classifications of the resulting graphs, but in others these are not known. One can also ask about other kinds of structures.
Gardiner's paper is in J. Combinatorial Theory 1976; Lachlan and Woodrow in Trans. Amer. Math. Soc. 1980.
Supervisor: P. Cameron
Various possible topics are described in this document. Contact the supervisor for more details.
Supervisor: C.–H. Chu
Series of various types are widely used in mathematics. However surprising it may be at first glance, the sum of a series can change as a result of a rearrangement of its terms. Two questions arise here: for what series the commutative law is valid; how can the sum change upon rearrangement of the terms? These problems were posed and solved by Riemann for real series. We begin the project by studying Riemann's solution and then explore various generalizations, for instance, to series of vectors.
A good knowledge of Linear Algebra, Calculus and Analysis.
Supervisor: C.–H. Chu
Euclidean geometry can be viewed as a study of properties of objects invariant under isometric transformations of the Euclidean spaces. This project studies the isometries of an n-dimensional Euclidean space and related geometric properties. A main objective, among others, is to describe these isometries completely.
A good knowledge of Linear Algebra and Analysis.
Availability: Third Year, MSci and MSc.
Supervisor: S. Coad
This project involves a detailed study of multivariate analysis of variance (MANOVA), the main statistical technique used to analyse data from a designed experiment in which data consist of observations on more then one variable. In its simplest form, MANOVA can be used to compare several mean vectors, given random samples from multivariate normal distributions each with the same covariance matrix. This is known as one-way MANOVA. The technique extends naturally to more complicated experimental designs, such as randomised block and factorial designs. Other possible topics include repeated measures designs, profile analysis, simultaneous confidence intervals and methods for assessing multivariate normality.
Probability II
Statistical Modelling II
C. Chatfield & A. Collins, 'Introduction to multivariate analysis' (Chapman & Hall 1980).
R. Johnson & D. Wichern, 'Applied multivariate statistical analysis' (4th Ed, Prentice Hall 1997).
Availability: Third Year, MSci and MSc.
Supervisor: S. Coad
When data consist of observations on more than one variable, it is often assumed that they follow a multivariate normal distribution. One of the reasons for making this assumption is that the underlying theory is usually tractable, though complicated. The first part of the project involves the study of some properties of the multivariate normal distribution. Other possible topics include the non-central χ2 and F distributions, distributions of quadratic forms, spherical and elliptical distributions, and distributions of correlation coefficients. A study of some of these topics leads to a better theoretical understanding of certain results met in earlier Statistics and Probability courses. A good knowledge of matrix algebra is required for the project.
Linear Algebra I
Probability II
K. Mardia, J. Kent & J. Bibby, 'Multivariate analysis' (Academic Press 1979).
R. Muirhead, 'Aspects of multivariate distribution theory' (Wiley 1982).
Availability: Third Year, MSci and MSc.
Supervisor: S. Coad
The sequential probability ratio test (SPRT) is designed to decide between two simple hypotheses. The test decides in favour of one of these hypotheses on the basis of observations of the random variable under consideration, subject to certain error probabilities. When the number of observations is fixed, the Neyman–Pearson lemma shows that the most powerful test depends on the likelihood ratio. When observations become available sequentially, it was shown by Wald in the 1940s that the optimum test is the SPRT, which is a sequential analogue of the fixed-sample test. The project involves the study of some properties of the SPRT, such as the operating characteristic function, the average sample number and sample size distribution. Although the SPRT was developed in the context of industrial control, it has also been applied in clinical trials.
Probability II
Statistical Theory
S. Silvey, 'Statistical inference' (Chapman & Hall 1975).
A. Wald, 'Sequential analysis' (Wiley 1947).
Availability: MSci
Supervisor: M. Fayers
A reflection group is a subgroup of the group of symmetries of Euclidean space which is generated by the reflections it contains. Reflection groups have applications to Lie algebras and regular polytopes, but are interesting in their own right. The finite reflection groups have been classified, and remarkably can be characterised as the finite groups with a presentation of a particular kind: the Coxeter groups. The project would give an account of some parts of the book by Humphreys, before perhaps moving on to more specialised topics.
A student doing this project should be very comfortable with groups (though there is no need to have seen presentations of groups before), so Algebraic Structures I is a must and Algebraic Structures II is helpful. Aspects of Geometry I and Linear Algebra I will be needed.
Availability: MSci
Supervisor: M. Fayers
A polytope is like a polygon or polyhedron, but in any number of dimensions. A polytope is regular if it has a sufficiently large symmetry group, where the definition of large is not completely straightforward. A student would given an account of the classification of the regular polytopes, starting from Coxeter's classic (though rather old-fashioned) account. Then there is potential to explore some more of Coxeter's study of the geometry of these polytopes, or to look at some types of semi-regular polytopes, or perhaps (for the more combinatorially inclined) abstract regular polytopes.
Algebraic Structures I is essential, Algebraic Structures II helpful; in particular, the idea of the action of a group will be important. Some aspects of Geometry I will appear.
Availability: MSci
Supervisor: M. Fayers
A partition is a just a weakly decreasing finite sequence of positive integers. A partition is often represented by its Young diagram, which is an array of boxes in the plane. When these boxes are filled with numbers, the resulting diagram is a Young tableau. These have inherent combinatotial interest, as well as applications to areas such as symmetric functions representation theory of the symmetric groups. A student doing this project would start with Fulton's book, and give an account of some of the ideas there, perhaps expanding from other references. Particular sub-topics that might be covered are the hook-length formula, and Littlewood–Richardson coefficients.
None.
Supervisor: S. Gilmour
Continuous data from complex experiments (e.g. split-plot designs) are usually analysed by fitting linear models or linear mixed models. Discrete data are often analysed using generalized linear models (GLMs). However, when a complex design has been used, there should be some additional random effects. This project would involve a review of generalized linear mixed models (GLMMs), and alternatives, and a reanalysis of some published data sets.
Statistical Modelling III, Design of Experiments
Availability MSci and perhaps MSc.
Supervisor: I. Goldsheid
Unlike in the classical case of homogeneous environment, random walks in random environment exhibit unusual features. Read several papers concerned with the subject and explain them.
Knowledge of Linear Algebra and Markov Chains is desirable.
Y. Sinai, 'Probability Theory – An Introductory Course' (Springer 1992).
F. Solomon, 'Random walks in random environment', Annals of Probability 3 (1975), 1–31.
H. Kesten, M. Kozlov, F. Spitzer, 'Limit law for random walks in random environment', Compositio Mathematica 30 (1975), 145–168.
Availability: MSci only.
Supervisor: R. Harris
By comparing the probability of observing a given stochastic trajectory to the probability of seeing the time-reversed trajectory, fluctuation theorems make predictions about entropy production in systems far from equilibrium. The project is to survey some results in this area using different mathematical tools including large deviation theory. The fluctuation relations discussed should be illustrated by application to simple models, e.g., random walks or interacting many-particle systems. A computationally-minded student could also explore various algorithms which have recently been proposed to efficiently measure large deviations in simulations of such systems.
Probability III
Availability: MSci or MSc.
Supervisor: R. Harris
The asymmetric simple exclusion process (ASEP) is a continuous-time Markov process which plays an important role in non-equilibrium statistical physics. Generalizations of the ASEP have been used as toy models to describe vehicular traffic and this project will focus on understanding and reproducing some work in this area. A particular aim will be to extend some recent results for a simple model of a road intersection [3]. The emphasis here may be computational or analytical depending on the interests and background of the student.
Probability III
Availability: Third Year, MSci and MSc.
Supervisor: B. Jackson
A bar-and-joint framework is a straight line realization of a graph in d-dimensional Euclidean space. We think of the framework as a collection of bars and joints where vertices correspond to joints and each edge to a bar joining its end-vertices. The framework is rigid if it has no non-trivial continuous deformations. Thus a triangle in the plane is rigid but a square is not since it can be continuously deformed into a parallelogram. The project is to give a survey of results on the rigidity of frameworks.
Knowledge of graph theory and Linear Algebra I.
Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures, by Jack E. Graver. Mathematical Association of America, Dolciani Mathematical Expositions number 25, 2001. ISBN: 0-88385-331-0.
Availability: Third year and MSci.
The sequence 00010111 has the amusing property that every 0–1 sequence of length 3 occurs as a consecutive block in it exactly once, provided that the sequence is regarded cyclically (that is with 'wrap around'). Such a sequence is called a de Bruijn cycle of order 3. The notion of a universal cycle extends this idea to other combinatorial families such as permutations, subsets and set partitions. The project would be to survey results and/or applications of universal cycles in general or to focus in more detail on a particular aspect of the theory. The paper [1] is a readable and entertaining introduction.
[1] F. Chung, P. Diaconis & R. Graham, 'Universal cycles for combinatorial structures', Discrete Mathematics 110 (1992), 43–59.
Availability: Third year and MSci.
How large can a family of r-subsets of an n-set be if any two of them have non-empty intersection? This question is answered by the Erdős–Ko–Rado theorem which says that we cannot do better than taking all r-sets containing a fixed element of our n-set. The project would be to survey one of the numerous extensions of this result. Possible examples being to look at what happens if we demand that our sets intersect in many points, or the intersection size satisfies some modular condition, or our n-set has some arithmetric or graph-theoretic structure.
MAS219 Combinatorics would be desirable.
Availability: Third Year, MSci and MSc.
Supervisor: W. Just
Various projects are possible in the fields of Intermittency, Dynamics with Time Delay, and Complex Coupled Systems. See this document or contact the supervisor for details.Related Lecture Courses: Chaos and Fractals, Complex Functions, Calculus III
(Textbook-based, but requires learning about some more complicated concepts.) The Baker map is one of the simplest dynamical systems exhibiting chaotic behavior. Based on different tetxbooks, summarize important dynamical systems properties of this paradigmatic model such as its dynamical instability, ergodicity, mixing behavior, being a K-system, and being Bernoulli. This requires to calculate analytically quantities like Lyapunov exponents, topological and metric entropies. If you like, also discuss Arnold's cat map.
Chaos and Fractals and/or Introduction to Dynamical Systems
J.R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge University Press, 1999)
V.I. Arnold, A. Avez, Ergodic problems of classical mechanics
T. Tel, M. Gruiz, Chaotic Dynamics (Cambridge, 2006)
Briefly outline the idea of chaotic diffusion and define the concept of a diffusion coefficient. Then derive the Taylor-Green-Kubo formula for diffusion in maps. By using this formula, calculate the diffusion coefficient for a lifted Bernoulli shift. It is obtained in terms of the famous Takagi function, which is a continuous but nowhere differentiable fractal that can be calculated by solving a functional recursion relation. Consider a generalized, parameter-dependent version of the lifted Bernoulli shift and try to calculate the diffusion coefficient again by the same method (this is new, I have not done this). Calculate the fractal dimension of all (generalized) Takagi functions involved (also new).
Chaos and Fractals and/or Introduction to Dynamical Systems
Anomalous diffusion defines a very active field of current research. What does it mean to say that a system is anomalously diffusive? Outline the basic idea. Introduce a simple subdiffusive map and qualitatively discuss its dynamics. Then calculate the anomalous diffusion coefficient of this model by continuous time random walk theory by explaining what this theory is about. This assumes familiarity with Fourier-Laplace transformations and the like. Explain the idea of fractional derivatives and derive a fractional diffusion equation for this model. - Although the results of this project are known, it poses a very challenging task that connects directly with active research.
Chaos and Fractals and/or Introduction to Dynamical Systems
A ball bouncing on a flat vibrating plate yields a simple but paradigmatic nonlinear dynamical system exhibiting surprisingly complex behavior. Review the literature and summarize what makes this system so interesting. Implement a computer program (see links below) and undermine your summary by simulations.
Chaos and Fractals and/or Introduction to Dynamical Systems
N. Tufillaro, T. Abbott & J. Reilly, 'An experimental approach to nonlinear dynamics and chaos' (Addison Wesley 1992).
J. Guckenheimer & P. Holmes, 'Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields' (Springer 1990).
Pieranski's webpage: http://fizyka.phys.put.poznan.pl/~pieransk/BouncigBall.html.
Tuffillaro's webpage: http://www.drchaos.net/drchaos/bb.html.
A point particle scattering with a rotating disk defines a nonlinear dynamical system exhibiting chaotic behavior. Very recently introduced, this model is simple enough to be analyzed rigorously mathematically for its dynamical systems properties. However, it became also very popular for performing computer simulations of heat conduction. Review the recent literature and find out at which point the model becomes non-trivial. Explain why. If you like, demonstrate your assessment in the form of computer simulations.
Chaos and Fractals and/or Introduction to Dynamical Systems
C. Mejia–Monasterio, H. Larralde & F. Leyvraz, Phys. Rev. Lett. 86 (2001) 5417–5420.
K. Rateitschak, R. Klages & G. Nicolis, J. Stat. Phys. 99 (2000) 1339–1364.
P. Balint & S. Troubetzkoy, 'Rotor interaction in the annulus billiard', http://uk.arxiv.org/abs/math.DS/0401194.
J.–P. Eckmann & L.–S. Young, 'Temperature Profiles in Hamiltonian Heat Conduction', http://uk.arxiv.org/abs/cond-mat/0401320.
Supervisor: B. Khoruzhenko
The aim is to obtain analytically the p.d.f. of spacing between eigenvalues of 2×2 matrices with random entries and 'confirm' the obtained answer using numerical simulations.
Probability I
Statistics I
Matrices and Geometry
Calculus II
Experimental Mathematics
F. Haake, 'The Quantum Signature of Chaos' (Springer 1991).
Supervisor: C Leedham–Green
The aim is to produce a modern translation, with simple annotation, of part of this most famous work. Newton's Latin is very simple, and the vocabulary is very limited, so reading the text is not linguistically very challenging. If anyone would prefer to tackle a classic in French or German, that could be arranged.
A slight knowledge of Latin and a knowledge of mechanics up to A level standard.
Supervisor: C Leedham–Green
Determine the complexity of various algorithms in algebra. For example, we have an algorithm for multiplying elements of finite soluble groups, but have no analysis of how efficient the method is. On a more practical level, write a program (probably in C) to multiply elements of a finite soluble group.
Supervisor: S. Majid
There are several models in the literature of models of spacetime wherein the x,y,z, or t coordinates are operators rather than numbers and, in particular, need not commute with each other. The project would involve a review of such models both from the point of view of their mathematical structure and, possibly, with reference to predictions whereby such models could be tested experimentally.
Linear algebra
Calculus II
Exposure to applied mathematics modules relating to quantum theory and/or gravitation would be helpful.
S. Majid, 'Foundations of Quantum Group Theory' (CUP 1995,2000).
Supervisor: S. Majid
The octonions are a famous 8-dimensional algebra that does not obey the usual associativity law, instead it is variously described as 'alternative' or 'quasiassociative' depending on what replaces the associativity. The project would intensely study the octonions and similar algebras particularly from the point of view as algebras in monoidal categories. Monoidal categories have a tensor product which is nonassociative as controlled by an associator obeying a famous pentagon condition.
Linear algebra
Algebraic Structures I
H. Albuquerque & S. Majid, 'Quasialgebra structure of the octonions', J. Algebra 220 (1999) 188–224.
Texts on division algebras (e.g. book of I. Porteus).
S. Mac Lane, 'Categories for the working mathematician' (Springer–Verlag 1994).
Availability: Third year and MSci.
For integrals which cannot be evaluated analytically one option is to use a numerical quadrature rule, e.g. trapezium rule or Simpson's rule. Another option, which turns out to be more accurate for high dimensional integrals, is to use a Monte Carlo method based on simulating certain random variables. In this project you will compare different Monte Carlo methods which have been proposed for evaluating integrals.
Probability II
B. Morgan, 'Elements of simulation' (Chapman–Hall).
B. Ripley, 'Stochastic Simulation' (Wiley).
Availability: MSci.
Kolmogorov provided a set of axioms which underpin the study of probability. However,
these do not tell us how a probability should be interpreted. The idea of probability
as a subjective degree of belief has a long history. In the twentieth century a number of
axiomatic treatments were suggested which emphasize this subjective idea.
The aim of the project would be to give a discussion of the historical development of such
axiom systems with a detailed critical account of at least one such system.
Probability II
PC Fishburn (1986) The Axioms of Subjective Probability, Statistical Science, 1, 335-358
JM Bernardo and AFM Smith (1994) Bayesian Theory. Wiley
Supervisor: T. Prellberg
Lagrange inversion is a powerful tool in enumerative combinatorics. This project will require you to compare several proofs of the Lagrange Inversion Theorem (combinatorial and analytic) and apply it to several examples.
Supervisor: T. Prellberg
Intermittency is one of the mechanisms by which a dynamical system can change from regular behaviour to chaotic behaviour. A powerful tool for the analysis of such systems is the transfer operator. Using the method described in [1], the spectral properties of various transfer operators for intermittent systems will be analysed numerically.
Supervisor: T. Prellberg
Using existing computer programs which implement the algorithm described in [2], the phase diagram of various polymeric systems can be explored. There is an endless supply of interesting systems (and therefore projects).
Supervisor: T. Prellberg
Counting directed lattice paths with respect to length and area gives rise to Gaussian polynomials. It is known that the resulting distribution is after suitable rescaling described by a normal distribution [3]. The purpose of this project is to refine this result, using methods from probability theory and/or asymptotic analysis.
Supervisor: D. Stark
In a top-to-random card shuffle the top card is removed and placed at a random position in the deck. The goal of this project is to understand how quickly information is lost through repeated performances of this shuffle. Computer simulations will be involved.
Discrete Mathematics and/or Combinatorics
Applied Stochastic Processes
Computational Maths
D. Aldous, 'Random walks on finite groups and rapidly mixing Markov chains', Seminaire de Probabilités XVII, Lecture Notes in Mathematics 986 (Springer 1983).
P. Diaconis, 'Group Representations in Probability and Statistics', IMS Lecture Notes – (1988).
Supervisor: H. Touchette
The Legendre transform is an important construction in mathematics that finds applications in various areas of applied mathematics, physics, and engineering. This project should summarize the construction and main properties of Legendre transforms, as well as their generalization proposed by Fenchel.
Calculus II and Convergence and Continuity
J. van Tiel, Convex Analysis: An Introductory Text. John Wiley, New York, 1984.
R. T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, 1970.
Supervisor: H. Touchette
The theory of large deviations was developed by Srinivas Varadhan (the Abel prize of 2007) to find approximations of probability measures of sums of random variables, as well as other random variables resulting from the interplay of many sub-random variables. The goal of this project is to summarize and exemplify this theory for the simple case of sums of independent and identically distributed random variables.
Probability II and III
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, New York, 1998.
Supervisor: H. Touchette
Random numbers generated by computers are not perfectly random: they are actually the result of recursive sequences based on chaotic dynamical systems. For this project the student should describe a number of techniques used for generating (pseudo) random numbers on computers. The student is also expected to implement these techniques on a computer using either Maple or Mathematica.
Introduction to Mathematical Computing, Introduction to Numerical Computing
W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, 2007.
Supervisor: H. Touchette
The calculus of variations is a classical subject of analysis constantly used in physics and engineering to find functions that minimize certain integrals or functionals, i.e., functions of functions. The goal of this project is to summarize this subject and prove two of its basic results: the Euler-Lagrange equation and the Hamilton-Jacobi equation.
Calculus II and III
M. Mesterton-Gibbons, A Primer on the Calculus of Variations and Optimal Control Theory, AMS, 2009.
S. E. Dreyfus, Dynamic Programming and the Calculus of Variations, Academic Press, 1965.
Availability: MSci and MSc. (Possibly third year but rather challenging.)
Supervisor: F. Vivaldi
When space is discretised, one of the simplest dynamical systems – the planar rotation – becomes extremely complicated. A connection exists between arithmetical properties of the angle of rotation and probabilistic properties of the discretised motion. These are to be studied and could also be explored numerically, with the aim of formulating conjectures.
This project requires the ability to write simple computer programs (in C, say).
Number theory (essential)
Probability I (desirable)
Chaos and Fractals (desirable)
F. Vivaldi, 'Periodicity and transport from round-off errors', Experim. Math. 3 (1994) 303–315.
Availability: MSci and MSc. (Possibly third year but rather challenging.)
Supervisor: F. Vivaldi
This project is appropriate for someone interested in dynamics,
computation, and a bit of number theory.
Take a set of n real
numbers, and adjoin to it a new real number, uniquely determined by
the rule that the arithmetic mean of the extended set be equal to the
median of the original set. Repeating this procedure we obtain a
recursive sequence of real numbers, which depends on the initial
set. It is not difficult to show that for n = 1, 2, such a sequence is
eventually fixed. In a recent paper [1] the authors, building on
previous work by Shultz and Shiflett for the case n = 3, conjecture
that this sequence is eventually fixed for any initial set. They prove
some auxiliary results and perform some computations, which reveal a
very interesting dynamics of unexpected complexity.
The main goal of this project is to advance our understanding of this
problem in two significant cases: the case in which the starting set
consists of rational numbers, and of algebraic numbers of degree two.
This is a research project: I do not know the outcome.
Fluency in writing computer programs in Maple, and also in some compiled language (C, Fortran, etc)
[1] M Chamberland and M Martelli, The mean-median map, J. Difference Equ. Appl. 13 (2007) 577–583.
Supervisor: R. Wilson
An elliptic curve is a cubic curve of the form y^2=ax^3+bx^2+cx+d. It can be made into an abelian group in such a way that three points sum to zero if and only if they lie on a straight line. (The associative law is quite tricky to prove!) It is possible to design a public-key cryptosystem analogous to RSA, using elliptic curves over finite fields in place of modular arithmetic. Find out about this and explain it in your own words.
MAS335 Cryptography, MAS201 Algebraic structures I
Availability: Third year and MSci
Supervisor: F. Wright
Below, 'Maple' means a computer algebra system such as Maple, REDUCE or Mathematica; 'Java' means a general-purpose programming language such as Java, C, C++, Pascal, or Fortran.
Develop procedures to integrate along both open and closed contours (curves) in the complex plane automatically by computer, in particular by using residue calculus to evaluate integrals around closed contours. Particular issues to investigate are how to represent general integration contours on a computer, how to determine what singularities lie within a closed contour and how to determine the nature of a singularity. Display integrals (integrands and contours) graphically in the complex plane. Consider applications to real definite integration, integral transforms, etc.
In Java, implement rational arithmetic. Implement Gaussian elimination for matrices over the real and rational numbers (R and Q) and compare the performance. Understand and implement Bareiss elimination over the integers Z, and compare its performance with Gaussian elimination regarding Z as a sub-ring of Q and R. In Maple, generalise to polynomials and rational functions. [Ref. BCL, DST, GCL]
A resultant is essentially the result of eliminating a variable between two polynomials. Discuss the theory and applications of resultants, and implement and test one or more algorithms to compute resultants. Investigate the advantages or otherwise of first performing a squarefree factorisation of the input polynomials (easier in Maple than Java). [Ref. BCL, DST, GCL]
Understand and implement an algorithm using p-adic lifting. Consider possible extensions to systems of polynomials and approximation of real roots. [Loos]
Understand and implement algorithms for expressing a polynomial or rational function as a composition of such functions. This involves some field theory. [Zippel, Proc. ISSAC'92]
[BCL] B. Buchberger, G. Collins, R. Loos, with R. Albrecht (eds.), 'Computer Algebra: symbolic amid algebraic
computation' (2nd ed., Springer 1983).
[DST] J. Davenport, Y. Siret & E. Tournier, 'Computer Algebra: systems and algorithms for algebraic computation' (Academic Press 1988).
[CCL] K. Geddes, S. Czapor & G. Labahn, 'Algorithms for Computer Algebra' (Kluwer Academic Publishers 1992).
[Loos] R. Loos, 'Computing rational zeros of integral polynomials by p-adic expansion', SIAM J. Computing. 12 (1983) 286–293.
Supervisor: R. Nelson
Accretion discs play an important role in many areas of astrophysics, from black holes and active galactic nucleii, to star and planet formation. The project will provide an up to date review of existing work in this area, either by providing a broad description of accretion phenomena in astrophysics, or by focussing on one or more of its applications.
Suitable astronomy courses
Supervisor: R. Nelson
About 110 planets have been discovered orbiting around stars other than the sun. This project will review some of the following themes in this area:
Suitable astronomy courses
Your lecture notes for the module MTH5117 Mathematical Writing (if you took it) and the course materials for that course will be useful.
Some advice (updated 12 March 2010).
If you plan to typeset your project in LaTeX then you may find one of the following introductions useful:
Getting Started with LaTeX, by D.R. Wilkin
LaTeX for Complete Novices, by N.L.C. Talbot