This webpage gives detailed information and a list of possible projects for the modules MTH717U MSci Project and MTH6138 Third-Year Project. The organiser for these mathematics project modules is me Prof Leonard Soicher. My office is B52 and please feel free to see me, either by appointment or during my office hours.
For students thinking of taking the third-year project module: Taking the third-year project module is dependent on the module organiser (me), your advisor and a project supervisor agreeing. Normally we advise against taking a project module unless you have a 2:1 average over the first 2 years (this is because the project is not an easy option and weaker students often end up getting poor reward for their effort).
Before the start of a project module, the most important thing is to find a supervisor. Look at the list of suggested projects on this webpage, choose 2 or 3 which appeal and go and speak to the supervisors. If you are interested in an area of mathematics which doesn't seem to be represented on the list of projects then contact the lecturer of any relevant module and see if they can suggest some project related to their module. The decision on registering for the project is made in September at the same time as you choose your other modules.
The project must involve the study of some mathematical topic, not covered by a lecture module, at least at the 4th year level for MSci projects and at least at the 3rd year level for third-year projects. The project report must be the student's own work in the sense that it gives an original account of the material, including the student's selection and structuring of the material and the selection of bibliographic references, but the report need not contain new mathematical results. You should, however, try to make some original contribution (this is especially important for an MSci Project). For example, you might do one or more of the following: give a new interpretation or view of the mathematics; give new examples; do your own detailed calculations or data analysis; write, test and run your own computer program. You must make it completely clear where you are providing your own original contribution rather than paraphrasing published information. In addition, the project must not include material which has been used for other assessment purposes.
The MSci project report 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 report length is 4000 words, and if a third-year project report is less than 3500 or more than 5000 words long then you may lose marks.
The MSci project report can be written in a single semester or the work can be spread over two semesters, depending on the other modules taken: the submission deadline will be the same in all cases.
Although the third-year project is a one semester module, the work may be done in either semester. In all cases the submission deadline will be the same. You are strongly recommended to do the third-year project in the term you register for it. You do not want your other modules to suffer.
One purpose of the project is 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 typewriter or else very neatly written. It should be in a simple binding, e.g. a ring or springback binder. The report must be written in good English and precise mathematics, and include:
Two copies should be handed in, and the candidate will usually want to keep a third copy for himself/herself.
Your lecture notes for the module MTH5117 Mathematical Writing (if you took it) and the course materials for that module will be useful.
Here is some very useful advice and guidance from a previous project module organiser (updated 12 March 2010).
If you plan to typeset your project in LaTeX then you may find one of the following introductions useful:
In the second-half of the second semester, you will be required to give a short seminar-style presentation on your project/its topic. This should be 30 minutes long for MSci projects and 20 minutes long for third-year projects. The intention is to give you an opportunity to practise mathematical speaking. It is not intended to be intimidating. Thus, provided any moderate attempt is made, the presentation will only increase your mark for the project.
Presentations will be held at the end of March 2012. You will be contacted by email regarding these.
The deadline for submission for both MSci and third-year project reports is 4pm Wednesday 2nd May 2012. However, students are strongly advised to complete their reports before the end of Semester B, so that the vacation can be used for revision for the written examinations in May and June.
Please hand in two copies of your project report to the Mathematics Office during its normal opening hours. The staff there will note the time and date of submission. Direct submission to your supervisor WILL NOT COUNT as a submission. Submission directly to me will also not count except under very exceptional circumstances and if agreed in advance and in writing with me.
A list of possible projects is given below. By following the relevant link, you'll find whether versions of the project are available for the MSci project, the third-year project or both, 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.
Another possible source of MSci 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 programmes 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.
Availability: Third-year, MSci and MSc.
Supervisor: A. Baule
The random behaviour of share prices is usually modelled in terms of geometric Brownian motion (gBm), which underlies the celebrated Black-Scholes pricing model for options. However, realistic market data often exhibits considerable deviations from gBm, manifest, e.g., as wide tails in the probability distribution of log returns.
In this project, the student will become familiar with stochastic models for share price dynamics beyond gBm. A particular emphasis will be put on Lévy stable processes and related models. Apart from a literature review and some analytical work, the student will also have the opportunity to analyze real world financial data sets.
MTH6121 Introduction to Mathematical Finance.
MTH5121 Probability Models and MTH6141 Random Processes
(desirable).
Availability: Third-year, MSci and MSc.
Supervisor: A. Baule
Motor proteins such as Myosin or Kinesin are nanoscale machines that transform chemical energy into useful mechanical work (e.g. transport of cargo in cellular processes). On these length scales, random fluctuations (noise) in the thermal environment as well as in the chemical energy input are significant, leading to an essentially random motion of the motors, which can be described by stochastic differential equations.
In this project, the student will analytically investigate simple stochastic models for such noise induced transport and discuss the connection to real biological systems.
Some physical intuition.
MTH5121 Probability Models and MTH6141 Random Processes (desirable).
Availability: Third-year, MSci and MSc.
Supervisor: A. Baule
Stochastic differential equations with Lévy or Poissonian distributed random variables occur in many applications from physics to finance. In this project, the student will implement algorithms to generate sample trajectories of such non-Gaussian stochastic processes and numerically determine the associated statistical quantities, such as the probability density function and correlation functions. As a further application of the code, mean first passage time problems can be considered.
MTH5121 Probability Models and MTH6141 Random Processes (desirable).
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).
Availability: MSci
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 Analysis
(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.
Availability: Third-year and MSci
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: 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.
Availability: Third-Year and MSci.
Supervisor: C.–H. Chu
This project studies the basic structures of Jordan algebras and their connections to the geometry of symmetric domains in finite dimensions. These domains were classified by E. Cartan using Lie groups. The Jordan theory offers an alternative approach which can be extended to infinite dimensions.
A good knowledge of Algebraic Structures, Linear Algebra, Analysis and Geometry.
Availability: Third-Year and MSci.
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 Models
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 Models
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 Models
Statistical Theory
S. Silvey, 'Statistical inference' (Chapman & Hall 1975).
A. Wald, 'Sequential analysis' (Wiley 1947).
Availability: MSci
Supervisor: D. Ellis
Szemerédi's Regularity Lemma is one of the most surprising and important results of modern graph theory, and has led to the resolution of a wide range of problems in combinatorics and number theory over the last thirty years. Roughly speaking, it says that any graph can be partitioned into a small number of blocks, in such a way that the edges between most pairs of blocks are distributed in a highly uniform way.
The aim of this project would be to understand and present a proof of Szemerédi's Regularity Lemma, to give a survey of its applications, and to explain some of these in depth. It is slightly towards the harder end of the spectrum of MSci projects, but should be more rewarding as a result.
Combinatorics would be helpful, but not essential.
Availability: Third-year and MSci.
Subject area: Combinatorics / Discrete Mathematics / Graph Theory
Supervisor: D. Ellis
Ramsey theory is a beautiful part of pure mathematics, which can be described as "finding order in chaos". The recurring theme is that often, "complete disorder is impossible": any sufficiently large structure must always contain a relatively large piece which is highly ordered. For example, whenever there are six people in a room, there are always either three who all know one another, or three who are all strangers to one another. In general, for any number k, there is a number n such that if there are n people in a room, there are always either k people who all know one another, or k people who are all strangers to one another. (This is one way of stating Ramsey's theorem for graphs.) Many results in Ramsey theory have beautiful and ingenious proofs, which use ideas from probability theory and algebra as well as from combinatorics. There are many well-known open problems in the area.
The primary aim of the project would be to look at some of the central results in Ramsey theory, develop a full understanding of their proofs, and explain them in your own words. The secondary aim would be to read about and then discuss open problems in the area.
There are no formal prerequisites, although Algorithmic Graph Theory, Combinatorics, Extremal Combinatorics and Advanced Combinatorics are all related to the project, and helpful in developing the right ways of thinking.
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.
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: Third-year, MSci and MSc.
Supervisor: R. J. Harris
The project is to consider some examples of random walkers with steps correlated in time, starting with simple textbook examples (e.g., the persistent random walk) and extending to more complicated memory dependence such as coupled walkers or the "elephant" and "Alzheimer" random walks considered in recent literature.
Random Processes (or Probability III) and/or Topics in Probability and Stochastic Processes.
Availability: Third-year, MSci and MSc.
Supervisor: R. J. Harris
The zero-range process is a paradigmatic Markov model in the field of interacting particle systems. The project is to understand the model, reproduce known results (analytically and/or numerically) and possibly consider some simple extensions, e.g., to more complex geometries.
Random Processes (or Probability III) and/or Topics in Probability and Stochastic Processes.
Availability: Third-year and MSci.
Supervisor: R. J. Harris
Spitzer's identity is a classical result which connects the probability distribution of the maxima of a random walk with the probability distribution of the walk itself. The project is to understand the result and present one or more proofs of it (at least for the case of right-continuous walks as treated, for example, in Grimmett and Stirzaker). The student could also give an exposition of some generalizations/consequences of Spitzer's identity and/or test it numerically.
Random Processes (or Probability III) and/or Topics in Probability and Stochastic Processes.
Availability: Third-year, MSci and MSc.
Supervisor: R. J. Harris
The project is to study some simple mathematical models of ion channels, including those based on Markov chains. For a suitably inclined student it may also be possible to build on previous simulation work (in C++) and make comparisons with experimental results from the group of Mark Baker (QMUL Centre for Neuroscience and Trauma).
Random Processes (or Probability III) and/or Topics in Probability and Stochastic Processes.
Availability: Third-year and MSci.
Supervisor: M. Jerrum
The Tutte polynomial T(G;x,y) is a two-variable polynomial associated with an undirected graph G. Evaluating the polynomial at various points provides a wealth of information about G. For example, T(G;1,1) is the number of spanning trees in G and T(G,-2,0) is the number of 3-colourings of G. The project would need to survey different definitions of the Tutte polynomial (in terms of contraction/deletion, and in terms of a rank function), and survey (with proofs) several of the combinatorial interpretations of the polynomial.
MTH6105 Algorithmic Graph Theory would be an advantage.
Availability: MSci
Supervisor: M. Jerrum
Coupling of random variables is a simple but powerful method introduced by Doeblin in the 1930's. An important application of the technique is to establishing upper bounds on mixing time of Markov chains. (The mixing time of an ergodic Markov chain is the number of steps taken until it is close to its limiting distribution.) The project would cover the connection between mixing time of a Markov chain and coalescence time of a coupling, and would illustrate the method using examples drawn from application areas such as statistical physics. There is scope for novel applications, and for a potential extension of the project to "coupling from the past".
Random Processes (or Probability III).
Availability: Third-year and MSci.
Supervisor: R. Johnson
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.
Supervisor: R. Johnson
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.
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 modules: Chaos and Fractals, Complex Variables, Calculus III
Supervisor: R. Klages
(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)
Supervisor: R. Klages
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
Supervisor: R. Klages
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
Supervisor: R. Klages
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.
Supervisor: R. Klages
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.
Availability: Third-Year, MSci and MSc.
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 an account of one or two such models both in terms of their mathematical structure, including the Poincare quantum group (this will require explaining what a quantum group is), and in terms of a derivation of an experimental prediction. For a physics project you could start by explaining the Planck scale and the problem of the continuum.
Linear algebra
Calculus II
Exposure to applied mathematics modules relating to Relativity, quantum theory and/or gravitation would be helpful.
(Preliminary reading - pop science will not be enough but could try: S. Majid, Chapter 2 of "On Space and Time", Cambridge University Press.(2008))
(Poincare algebra: Chapter 2.4 of S. Weinberg, "The Quantum Theory of Fields. 1", Cambridge University press (1995))
G. Amelino-Camelia and S. Majid, Waves on Noncommutative Spacetime and Gamma-Ray Bursts, Int. J. Mod. Phys. A15 (2000) 4301-4323
S. Majid, Algebraic Approach to Quantum Gravity II: noncommutative spacetimes, in Approaches to Quantum Gravity, ed. D. Oriti. C.U.P. (2009) 466-492
S. Majid, 'Foundations of Quantum Group Theory' (CUP 1995,2000).
Availability: Third-Year, MSci and MSc.
Supervisor: S. Majid
Quantum entanglement is about non-separable states in a tensor product quantum system that can lead to correlations between the parts even if they are spatially separated. The notion is key to analysis of the Einstein-Podansky-Rosenberg paradox and to the Bell inequalities, but has since found application in quantum information and quantum teleportation. Could include a clear mathematical definition, analysis of at least one experiment or thought experiment, and quantum entropy as a measure of entanglement. Possibly also how entanglement is related to decoherence and/or a treatment in terms of pure and mixed states on C* algebras.
Linear algebra
Calculus II
Quantum Mechanics
Exposure to other applied mathematics/mathematical physics modules would be helpful.
(preliminary reading; understanding the pop science would not be enough but try: Lousia Gilder, "The Age of Entanglement")
Horodecki R, Horodecki P, Horodecki M, Horodecki K (2007), "Quantum entanglement", in Reviews of Modern Physics ( http://arxiv.org/pdf/quant-ph/0702225v2 )
Bengtsson I, Zyczkowski K (2006). "Geometry of Quantum States: An Introduction to Quantum Entanglement". Cambridge: Cambridge University Press.
D. Salart, A. Baas, C. Branciard, N. Gisin, H. Zbinden, "Testing spooky action at a distance", Nature 454, 2008, 861-864
D. Bouwmeester, J-W. Pan, K. Mattle, M. Eibl, H. Weinfurter & A. Zeilinger, "Experimental Quantum Teleportation", Nature, 390, 1997, pp.575
Availability: Third-Year, MSci and MSc.
Supervisor: S. Majid
Many of the key equations in mathematical physics become quite beautiful in the language of differential forms. The project would describe what these objects are, the associated de Rham complex and the notion of cohomology. This would be followed either by more advanced elements of Hodge-de Rham theory or by applications in mathematical physics (including Maxwell's equations), or both. A more advanced project might include Riemannian geometry in terms of differential forms.
Linear algebra
Calculus II
Exposure to either applied mathematics/mathematical physics modules or to geometry modules such as Geometry II
I. Madsen and J. Tornehave, `From calculus to cohomology', Cambridge Univ. Press, 1997. .
Availability: Third-Year, MSci and MSc.
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: MSci and MSc.
Supervisor: S. Majid
In mathematics we are familiar with the idea of objects with structure and maps between two of them. This notion of `category' has served mathematicians well for some decades but in recent years something new has emerged, the notion of objects with structure, maps between then and maps between the maps. This is a 2-category. The project would be to explain their definition and key examples such as 2-groups and 2-Lie algebras. For MSci or MSc level, braided 2-categories might also be covered.
Linear algebra
Algebraic Structures I
Tom Leinster `Higher Operads, Higher Categories' (Cambridge University Press, 2004)
S. Mac Lane, 'Categories for the working mathematician' (Springer–Verlag 1994).
Availability: Third-year and MSci.
Supervisor: L Pettit
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 Models or Statistical Methods
B. Morgan, 'Elements of simulation' (Chapman–Hall).
B. Ripley, 'Stochastic Simulation' (Wiley).
Availability: MSci.
Supervisor: L Pettit
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 Models or Statistical Methods
PC Fishburn (1986) The Axioms of Subjective Probability, Statistical Science, 1, 335-358
JM Bernardo and AFM Smith (1994) Bayesian Theory. Wiley
Availability: Third-year and MSci.
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.
Combinatorics would be desirable.
H.S.Wilf, Generatingfunctionology, Academic Press, 2004.
P.Flajolet and R.Sedgwick, Analytic Combinatorics, Cambridge University Press, 2009.
Availability: Third-year, MSci and MSc.
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).
Some basic knowledge of programming in C.
[2] T. Prellberg & J. Krawczyk, 'Flat histogram version of the pruned and enriched Rosenbluth method', Phys. Rev. Lett. 92 (2004) 120602.
Availability: MSci and MSc.
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.
Enumerative and Asymptotic Combinatorics would be desirable.
[3] L. Takacs, 'Some asymptotic formulas for lattice paths', J. Stat. Plan. Inf. 14 (1986), 123–142.
Availability: Third-year and MSci.
Supervisor: L. Soicher
For t a non-negative integer, a t-(v,k,λ) design (or simply a t-design) is an ordered pair (V,B), such that V is a finite non-empty set whose elements are called points, B is a finite non-empty collection of non-empty subsets of V called blocks, V has size v, each block has size k, and every t-element subset of V is contained in exactly λ blocks.
For example, if V={1,...,7} and B=[{1,2,4},{2,3,5},{3,4,6},{4,5,7},{1,5,6},{2,6,7},{1,3,7}], then (V,B) is a 2-(7,3,1) design.
The theory of t-designs is extensive and has applications in many areas of mathematics, including experimental design, group theory, coding theory and finite geometry. This project would include the basic theory of t-designs, followed either by a selected application area, or one or more significant results (with proof) in the theory of t-designs.
Combinatorics would be desirable.
Availability: MSci and MSc.
Supervisor: L. Soicher
Recently, Peter Cameron introduced a new class of block designs which generalises the class of t-designs, and also includes orthogonal arrays, resolved 2-designs, and other classes of combinatorial designs. The main idea is that the set of points of the design is structured by an ordered partition of that set, and the blocks of the design and t-subsets of the point-set also have structure with respect to this point-set partition.
The main purpose of this project is to study Cameron's original paper [1], together with a very recent paper [2], which establishes some basic theory of generalised t-designs, generalising some well-known results in the theory of ordinary t-designs. This topic area is very new, and there is certainly scope for a project student to determine new examples, results, or contructions.
Combinatorics would be desirable.
[1] P.J. Cameron, A generalisation of t-designs, Discrete Math.
309 (2009), 4835-4842.
[2] L.H. Soicher, On generalised t-designs and their parameters,
Discrete Math. 311 (2011), 1136-1141.
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).
Availability: 3rd year and MSci project
Supervisor: J.A. Valiente
The purpose of the project is to explore the so-called effect of warp-drive in General Relativity which could allow hyper-fast (but not faster than light!) travel.
A sufficient prerequisite for this project is the material discussed in the course of General Relativity (MTH6132).
M. Alcubierre. The warp drive: hyper-fast travel within General Relativity. Class. Quantum Grav. 11, L73-L77 (1994).
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.
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.
Webpage last updated: 29 September 2011