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UNIVERSITY OF EDINBURGH SCHOOL OF MATHEMATICS Year 4 Project Topics, 2011/12
4 April 2011
The following Individual, Combined-degree and Group project topics are available during 2011/12. In addition, a student or group of students may suggest a topic, but requires the agreement of a member of academic staff to supervise it. Under Type, ‘Ind’, ‘Grp’ and ‘Com’ indicate Individual, Group and Combined-honours projects. Individual projects are worth 20 credit points, and Group projects are worth 20 points for each member of the group. Combined-honours students must enrol for a project worth 40 points, except for Mathematics & Physics students, who may choose between a 20-point and a 40-point project. Clicking on the title of a topic takes you to the page on which it is described.
Supervisor Box Title Type
C G G Aitken 1 Interpreting DNA evidence Ind
C G G Aitken 1 Sampling problems in forensic science Ind
C G G Aitken 1 The evaluation of transfer evidence Ind
C G G Aitken 1 Bayesian networks in forensic science Ind
T Antal 1 Stochastic models of DNA evolution Ind
T Antal 1 Combinatorial games Ind
T Antal 1 Urn models Ind
T Antal 1 Tiling a square by unit squares Ind
P Blue 2 Divergence of nonlinear waves Ind
P Blue 3 Measure theory: nondifferentiable solutions to thenonlinear Schrodinger equation
Ind
P Blue 4 Formation of singularities in general relativity Ind
A M Davie Complex dynamics Ind/Com
A M Davie Random graphs Ind
A M Davie High-precision calculation of elementary functions Ind/Com
A M Davie Dynamics of circle mappings Ind
A M Davie The BMV conjecture Ind/Com
A M Davie Almost periodic functions Ind
A M Davie Computational complexity of matrix multiplication Ind/Com
N Bournaveas The Cauchy-Kowalevski Theorem and Lewy's Example. Ind
N Bournaveas Introduction to the Mathematical Theory of Waves Ind
N Bournaveas The mathematics of Medical Imaging Ind
J Figueroa-O’Farrill Tiling a rectangle Ind
J Figueroa-O’Farrill Kuratowski’s 14-set theorem and its generalisations Ind
J Figueroa-O’Farrill 3 Topics in Riemannian geometry Grp
J Figueroa-O’Farrill 3 Lie theory and separation of variables Com/Grp/Ind
J Figueroa-O’Farrill Black hole uniqueness theorems Com/Ind
J Figueroa-O’Farrill Cosmological models Com/Ind
J Hall Modelling languages for Mathematical Programming Com
DC Heggie 4 Japanese Temple Mathematics Grp
DC Heggie 1 Henon's Solution Com
DC Heggie 2 Mathematical Astronomy and Climate Change Ind
A Karakhanyan Sturm-Liouville boundary value problems Ind/Grp
A Karakhanyan Second quadratic form of surface Ind
T Lenagan Project in Algebra Ind
T Lenagan Totally positive matrices Ind
T Lenagan Hilbert Series of Graded Algebras Ind
T Lenagan Growth Com/Grp/Ind
T Lenagan Noetherian rings Grp/Ind
T Lenagan The Weyl Algebra Ind
A Maciocia Unifying Mathematics Com/Ind
A Maciocia Paradoxes in Mathematics Com/Ind
A Maciocia Exotic Logics Ind
A Maciocia Transfinite Arithmetic Ind
A Maciocia Non-standard Arithmetic Ind
A Maciocia Finite projective spaces and -tone scales Com/Ind
A Maciocia The Mathematics of Synthetic Music Com/Ind
A Maciocia Symmetry Groups in Music Com/Ind
TG Mackay 1 Causality and the Kramers-Kronig Relations Com/Ind
TG Mackay 1 Voigt wave propagation in anisotropic materials Com/ind
TG Mackay 1 Depolarization dyadics in electromagnetichomogenization studies
Com/Ind
A Olde Daalhuis The computation of special functions in the complexplane
Grp/Ind
A Olde Daalhuis Elliptic functions Elliptic functions Elliptic functions Ind
A Olde Daalhuis Inner and Outer Conformal Maps Ind
N Papovic 3 Geometric singular perturbation theory Grp/Ind
N Papovic 3 Social aggression in pigs Grp/Ind
A Ranicki The Farey sequences Grp/Ind
A Ranicki Quaternions and rotations Grp/Ind
A Ranicki Simplicial complexes and the Euler characteristic Grp/Ind
M Rasonyi 1 Martingales and stochastic games Com/Ind
M Rasonyi 1 Monte Carlo simulations Com/Ind
M Rasonyi 1 Arbitrage Pricing Theory Com/Ind
A Smoktunowicz Nil and Jacobson radical rings Ind
A Smoktunowicz Polynomial identity (PI) rings Ind
A Smoktunowicz Some famous matrices Ind
C Smyth Bounding and finding roots of polynomials Grp/Ind
C Smyth Covering congruences and their applications Ind
C Smyth Chromatic polynomials of graphs Ind
C Smyth Continued fractions of quadratic irrationals Ind
C Smyth Eigenvalues of graphs Ind
C Theobald 3 Home ground advantage in team sports Grp/Ind
C Theobald 1 Modelling numbers of Olympic medals Ind
C Theobald 1 Statistical modelling of data on the unit disk Ind
M Wemyss 4 McKay Quivers Grp/Ind
M Wemyss 2 Power Series Rings versus Polynomial Rings Ind
M Wemyss 3 Singularities in Algebraic Geometry Grp/Ind
B Worton 1 Statistical modelling of data from controlled laboratory tests Ind
B Worton 1 Finite mixture statistical modelling Ind
B Worton 1 Using empirical likelihood for statistical analysis Ind
B Worton 1 Identifying the number of modes of a density Ind
B Worton 1 Statistical modelling of circular data Ind
Interpreting DNA evidence Type: Individual Supervisor: Professor C.G.G. Aitken ([email protected]) Requirements: Second year statistics and probability modules.
DNA profiling is a very powerful tool in forensic science and paternity cases. Various issues in its interpretation will be investigated including consideration of the weight to be attached to (a) the possibility a relative of the suspect may be the criminal, (b) sub-population structure, (c) the information that a suspect was identified through a database search and (d) evidence of a relative of the alleged father in paternity (in the absence of the DNA profile of the alleged father). Gastwirth, J L (Editor) (2000) Statistical science in the courtroom. Springer Verlag. K5485 Sta Balding, D J (2005) Weight-of-evidence for forensic DNA profiles, John Wiley and Sons Ltd., RA1057.5 Bal.
Sampling problems in forensic science Type: Individual Supervisor: Professor C.G.G. Aitken ([email protected]) Requirements: Second year statistics and probability modules.
It is often impractical for an entire consignment of items, some or all of which may be illegal, to be inspected. Sampling raises an extra element of uncertainty for consideration in the assessment of the guilt or otherwise of a suspect. Issues surrounding sampling in a forensic context and Bayesian inference will be investigated. A related issue is that of quantity. In some jurisdictions, the quantity of an illicit substance which is associated with a criminal act is a factor in the sentencing of the person convicted of the act. The quantity may only be an estimate and it is of interest to determine the estimate which is the best in this context. Gastwirth, J L (Editor) (2000) Statistical science in the courtroom. Springer Verlag. K5485 Sta Aitken, C G G, and Taroni, F, Statistics and the evaluation of evidence for forensic scientists (2nd edition), John Wiley and Sons Ltd. (2004), HV8073 Ait.
The evaluation of transfer evidence Type: Individual Supervisor: Professor C.G.G. Aitken ([email protected]) Requirements: Second year statistics and probability modules. Third year statistics desirable. Knowledge of R very useful. Some analysis of data is required.
Evidence, such as glass fragments, which is transferred to or from a crime scene and a criminal and for which there are measurable characteristics, such as the refractive index or elemen- tal compositions, gives rise to interesting issues of methodology using multivariate Bayesian techniques to estimate the value of the evidence. Gastwirth, J L (Editor) (2000) Statistical science in the courtroom. Springer Verlag. K5485 Sta Aitken, C G G and Taroni, F, Statistics and the evaluation of evidence for forensic scientists (2nd edition), John Wiley and Sons Ltd. (2004), HV8073 Ait.
Bayesian networks in forensic science Type: Individual Supervisor: Professor C.G.G. Aitken ([email protected])
Requirements: Second year statistics and probability modules. Third year statistics desirable. Knowledge of R very useful. The practical application of probabilistic reasoning in forensic science can be assisted and its rationale clarified if it is conducted in a graphical environment. A graph in this context is a set of nodes and edges. Nodes represent pieces of evidence. Edges are lines connecting nodes and represent links between pieces of evidence. A probabilistic table is also required which provides the conditional probabilities associated with pieces of evidence joined by edges. The project will review the role of Bayesian networks in forensic science. Taroni, F., Aitken,C.G.G., Garbolino,P., Biedermann,A. (2006) Bayesian networks and proba- bilistic inference in forensic science. QA279.5 Bay
Stochastic models of DNA evolutionType of project: Individual (20-point)Supervisor: Tibor Antal, ([email protected])Requirements: Probability; Stochastic Modelling could be useful
What can we learn about the mechanisms of evolution from sequencing the DNA of some individuals from a population? How long ago did Adam and Eve live? To start answering these sorts of questions we need to understand some basic probabilistic models of evolution, like the Wright-Fisher process, the coalescent, or the stepping stone model. We are interested in the chance of
finding certain patterns in the DNA, which can be derived by combinatorial means. Or, the same results can be derived more easily by following the evolution backward in time. The project can be purely analytic, or also mostly computational.
Durrett R, Probability Models for DNA Sequence Evolution, Springer 2002; (Darwin Library QH438.4.S73 Dur)
Combinatorial gamesType of project: Individual (20-point)Supervisor: Tibor Antal, ([email protected])Requirements: Discrete Mathematics could be useful
In the old French movie "L'année dernière à Marienbad", the following game is often played by the bored guests of a hotel. There are several piles of chips on the table, and the two players take turns choosing a pile and removing one or more chips from it. The goal is to be the player that takes the last chip. In the movie a scary waiter can always win. Can you? The object of this project is to learn how to win this and similar combinatorial games.
http://mathworld.wolfram.com/Nim.htmlhttp://www.stat.berkeley.edu/~peres/155.html
Urn modelsType of project: Individual (20-point)Supervisor: Tibor Antal, ([email protected])Requirements: Probability; Stochastic Modelling would be useful
There is an urn with a black and a white ball in it. We draw a ball randomly, and return this ball to the urn with an extra ball of the same color. We repeat this process until we have 100 balls in the urn. What is more likely: a single white ball and 99 black balls, or 50 black and 50 white balls? What is the fraction of white balls if we draw long enough? What if we return not one but 1000 extra balls of the same color each time? Would it matter then if we returned 1000 balls of the same color and also one extra ball of the opposite color each time? Apart from having surprising features, urn models have applications in a wide variety of fields ranging from genetics to informatics. The object of the project is to solve explicitly some simple urn schemes, and prove results in the limit of large number of draws. The project then can develop into an analytic or a computational exploration of these models, according to taste.
[answers: same; anything; ...; a lot, the fraction tends to 1/2]
Johnson N L and Kotz S, 1977 Urn Models and their Applications (JCM Library QA273 Joh)Freedman D, 1965 Ann. Math. Stat. 36, 956
Tiling a square by unit squaresType of project: Individual (20-point)Supervisor: Tibor Antal, ([email protected])
How many non-overlapping unit squares can you pack into a large square of one million times one
million? Of course at most 1012 . What if we increase the side of the large square by only 1/10: can you pack one more unit square into a square of 1,000,000.1×1,000,000.1 ? There is a lot of extra space, but it's so narrow. Yet, you can pack a lot more then just one extra unit square in. The scope of the project is to learn how to do this, and prove that it is possible. Then one would continue to explore this or similar problems further by analytical or computational means, according to taste.
http://www2.stetson.edu/~efriedma/packing.htmlhttp://www2.stetson.edu/~efriedma/papers/squares/squares.htmlP Erdos and RL Graham, Journal of Combinatorial Theory (A) 19, 119-123 (1975)
Divergence of nonlinear wavesType: Individual Supervisor: Pieter Blue ([email protected])Requirements: PAA, CVD.Degree Programme: Any.
An example of Fritz John shows that even arbitrarily small,smooth initial data to a quadratic nonlinear wave equation mustgenerate a solution that diverges to infinity in finite time. Roughlyspeaking, solutions to the linear wave equation will, like ripples ona pond, spread out getting smaller and smaller. Thus, one mightnaturally expect that if the initial data to a nonlinear wave equationis small, then the influence of the nonlinearity would start small,and only become smaller as time goes on. In some cases, this idea iscorrect, but, perhaps surprisingly, in the example considered in thisproject, it is completely wrong. The starting point of this projectwill be understanding the linear wave equation in 3 spatialdimensions using the Green's function. A wide range of similarexamples of divergence can also be studied, as can criteriadistinguishing between divergent and globally bounded solutions.
This project is suitable for students of all abilities, including weakand strong students.
L.C. Evans (1998), "Partial Differential Equations", AMS.F. John (1989) "Nonlinear wave equations: Formation of Singularities", AMS.
Measure theory: nondifferentiable solutions to the nonlinear Schrodinger equationType: Individual Supervisor: Pieter Blue ([email protected])
Requirements: Metric spaces and PAA. It might be helpful, but is not necessary to be taking some of the following classes: Analysis of Nonlinear Waves, Essentials of Analysis and Probability, Fourier Analysis, Hilbert Spaces, or (the first half of) Topology.Degree Programme: any
This project will focus on the nonlinear Schrodinger equation. This isa partial differential equation. Surprisingly, it is possible to findfunctions which are, in an appropriate sense, solutions of thispartial differential equation, even though the solutions do not have
derivatives. For example, the 1/1∣x∣ has no derivative at 0 ,and the function which is 0 for ∣x∣1 and 0 otherwise has noderivative for ∣x∣=1. Despite the lack of derivatives, both of thesefunctions are valid initial data for the nonlinear Schrodingerequation. The start of this project will focus on understandingLebesgue measure, measurable functions, and how to use these toconstruct vector spaces of functions, in which we can find solutionsto differential equations.
This project would be well suited for moderate to strong students.
L.C. Evans (1998), Partial Differential Equations, AMS.H.L. Royden (1988), Real Analysis, Prentice Hall.R.E. Showalter (1977), Hilbert Space Methods for Partial Differential Equations, Pitman.T. Tao (2006), Nonlinear Dispersive Equations: Local and global Analysis, AMS.
Formation of singularities in general relativityType: Individual Supervisor: Pieter Blue ([email protected]) Requirements: Differential Geometry. An interest in physics would be useful. Degree Programme: Most suited to Mathematics and Physics students.
General relativity describes space-time as a curved, four-dimensionalobject, and the Einstein equation governs the curvature of thisobject. Many of the most important, known solutions to the Einsteinequation have singularities. These include black holes andcosmological solutions. This project will start with the essentials ofgeneral relativity and the differential geometry of manifolds. Thereis a wide range of possible directions to go from there, depending onthe interest of the student. This could include the singularitytheorems of Hawking and Penrose, singularities without divergencesincluding Cauchy horizons, chaos in the Bianchi IX cosmological model,topology of the future and choosing a time function, and cosmologicalmodels with less than total isotropy.
S.M. Carroll (2004), An introduction to general relativity: Spacetime and Geometry, Addison-Wesley.S.W. Hawking & G.F.R. Ellis (1973), The large scale structure of space-time, Cambridge University Press.C.W. Misnor, K.S. Thorne, J.A. Wheeler (1973), Gravitation, Freeman.R.M. Wald (1984), General Relativity, University of Chicago Press.
The Cauchy-Kowalevski Theorem and Lewy's Example.Type: Individual.Supervisor: N. Bournaveas ([email protected])Requirements: PAA and CVD. Metric Spaces (Y3) and Hilbert Spaces (Y4) would be helpfulbut not absolutely necessary.
The Cauchy-Kowalevski Theorem states that the Cauchy problem for any real analytic
PDE with real analytic initial data always has a local real analytic solution. Discoveredat a time when all functions were thought to be analytic it was considered to be the ultimateexistence theorem. In effect, it said that all PDE's always had solutions!As the notion of function was later extended to admit non-analytic functions, newtheorems were discovered which guaranteed the existence of solutions of largeclasses of PDE's. Actually these theorems seemed to suggest that PDEs always had many solutions.Additional conditions had to be imposed (initial conditions, boundaryconditions etc) to uniquely determine a solution.It came as a surprise when H.Lewy, in 1956, discoveredan example of a linear PDE without solutions! His proof uses two very important theorems:Baire's category theorem and the Ascoli-Arzela compactness theorem.This project is intended for a student with a strong liking of the first part of PAA andthe Complex Analysis part of CVD.
F. John, Partial Differential Equations, QA377 JohG.B.Folland: Introduction to PDE, QA374 FolH.L.Royden: Real Analysis, 51.26 Roy
Introduction to the Mathematical Theory of WavesType: IndividualSupervisor: N. Bournaveas ([email protected])Requirements: PAA and CVD.
The broad use of the term wave in every day life makes it difficultto state a single mathematical definition of a wave. Intuitively,a wave is a disturbance moving through a medium such as water, airor a crowd of people at a football game. Mathematically, a wave isrepresented by a function ut , x of a time variable t and a spacevariable x where ut , x is the value of some measurement atposition x at time t . Examining ut , x at times t=t 0t 1t 2...gives an indication of how the wave is propagating through the medium.Partial derivatives of u have important physical meaning as rates ofchange and the way u changes can be expressed as an equation relating
u and its derivatives. In this project we study the wave equation, theKorteweg-de Vries equation and waves in conservation laws.
R.Knobel: An Introduction to the Mathematical Theory of Waves, QA927 KnoJ. Smoller: Shock Waves and Reaction-Diffusion Equations, QA927 Smo
The mathematics of Medical ImagingType: IndividualSupervisor: N. Bournaveas ([email protected])Requirements: PAA and CVD.
Tomography has applications not only in medical science but alsoin radioastronomy, geophysics, the search for oil and other buried treasuresand many other disciplines. Usually a large number of beams are projected
through a body and the attenuation of the x-rays is measured. Theattenuation density is then reconstructed from that data. In mathematical termsthe problem is to reconstruct a function, say on the plane, from itsintegrals along all straight lines. The tool for this is the Radon transformintroduced and studied by Radon in 1917. The aim of the project is tostudy the basic properties of the Radon transform and some of its simplestapplications.
T.Feeman, Mathematics of medical imaging : a beginner's guide RC78.7.D53 Fee.L.A.Shepp; J.B.Kruskal, Computerized Tomography, American Mathematical Monthly, 1978Lawrence Zalcman, Offbeat Integral Geometry, American Mathematical Monthly, 1980R. Strichartz, Radon Inversion-Variations on a Theme, American Mathematical Monthly, 1982G. Folland, Introduction to Partial Differential Equations, QA374 Fol
Complex dynamics Type: Individual or Combined Supervisor: Sandy Davie ([email protected]) Requirements: complex variable part of CVD. Some programming experience could be useful but is not essential.
This project is concerned with the behaviour of iterations of an analytic function of a complex variable. Even for quite simple functions, such as quadratic polynomials, the study of such it- erations can lead to very intricate and spectacular patterns, such as the well-known ‘Julia sets’ and ‘Mandelbrot set’. These sets have been the subject of active research over the last 25 years, combining computational work with deep theoretical results. The object of the project would be first to explore the theoretical aspects, which mainly concern the study of fixed points and periodic points of analytic functions, which are central to understanding the dynamics. This can then be applied to explore the quadratic and possibly other families of functions. This can be done analytically or computationally, or (preferably) using a combination of the two, by exploring the dynamics computationally and interpreting the results theoretically. The emphasis of the project could range from entirely theoretical to mainly computational.
R.L. Devaney, ‘An Introduction to Chaotic Dynamical Systems’ (QA614.8 in JCM Lib), Chapter 3.
Random graphs Type: Individual Supervisor: Sandy Davie ([email protected]) Requirements: some knowledge of probability, such as the second year Probability course. Some programming experience could be useful but is not essential.
A theory has been developed of ‘random graphs’ (where by ‘graph’ is meant a set on n vertices, some pairs of which are joined by edges). One way of constructing such a random graph is to start with n vertices, where n is given, and for each pair of vertices join them with an edge with probability p (where p is given). One can then ask questions such as, what is the probability that the graph is connected? The object of the project would be first to explore the existing theory, which is largely concerned with ‘asymptotic’ results for large n, and second to do calculations of probabilities, which could be exact calculations and/or Monte-Carlo simulations, and compare the results with the theory. The emphasis could be mainly theoretical or mainly computational, according to taste. References: There are two books both entitled ‘Random Graphs’, one by B. Bollobas and one by
V. Kolchin, shelved under QA166.17 in the JCM Library.
High-precision calculation of elementary functions Type: Individual or Combined Degree (Computer Science) Supervisor: Sandy Davie ([email protected]) Requirements: some programming experience desirable (and essential for combined degree ver- sion).
This project is concerned with methods for evaluating constants such as pi and elementary func- tions of given numbers (e.g. log 14) to very high accuracy (millions of digits) in a computationally efficient way. The methods involve series or iterations requiring a relatively small number of al- gebraic calculations. The derivation of these methods require exploration of interesting topics such as the Arithmetic-Geometric Mean (AGM), theta functions and modular functions. Their efficient implementation requires methods for fast multiplication and division of numbers with many digits, and this can be done using the Fast Fourier Transform. The object of the project would be first to explore the theory behind (some of) these methods and see how they can be applied to a variety of functions, and to study their computational efficiency. There is scope for testing the methods by implementing them on a computer. The emphasis of the project could be mainly theoretical or mainly computational or a combination of the two, according to taste. For the combined degree version, it would be appropriate to spend much more time on the details of implementation with attention to matters such as operation counts, design of optimal algorithms and most efficient coding for the calculation of particular functions, etc. ‘Pi and the AGM...’ by J. Borwein and P. Borwein (QA 241 Bor), ‘Pi: a sourcebook’ edited by L. Beggren, J. Borwein and P. Borwein (QA 484 Pi), both in JCM Library. The second of these is a collection of reprinted articles - articles 56 and 64 are particularly relevant.
Dynamics of circle mappings Type: Individual Supervisor: Sandy Davie ([email protected]) Requirements: no specific requirements; some programming experience could be useful but is not essential. This project is concerned with the behaviour of iterations of a function which map a circle to itself. Such circle mappings arise in a natural way in the study of dynamical systems in two or more dimensions. In the case of invertible circle mappings the behaviour depends on a quantity called the ‘rotation number’ of the mapping, there being two types of behaviour according to whether this number is rational or not. The object of the project would be first to explore the theory of the dynamics of invertible mappings and the rotation number, then to apply it to some typical families of circle mappings. This can be done analytically or computationally, or using a combination of the two. If time permits the (more complicated) non-invertible case could be looked at. The emphasis of the project could range from entirely theoretical to mainly computational. R.L. Devaney, ‘An Introduction to Chaotic Dynamical Systems’ (QA614.8 in JCM Lib), Section 1.14 J. Hale and H. Kocak, ‘Dynamics and Bifurcations’ (QA372), Chapter 6.
The BMV conjecture Type: Individual or Combined
Supervisor: Sandy Davie ([email protected]) Requirements: PAA and Algebra desirable. Some programming experience could be useful but is not essential. The Bessis-Moussa-Villani conjecture, which arises from mathematical Physics, was first formu- lated in 1975, and asserts the following: suppose A and B are positive definite matrices of the same size, let n be a positive integer, and define f (t) to be the trace of the matrix (A + tB)n . Then f is a polynomial of degree n and the assertion of the conjecture is that its coefficients are all positive. This has been proved for 2 × 2 matrices (for any n) and for any matrices when n is less than 14. But whether it is always true is still a wide open problem, on which there has been much research in the last few years (the result for n less than 14 was proved in 2008). It is an example of an open problem for which recent research is accessible without a lot of specialist knowledge. Possible topics for the project include: exploring the relations between various equivalent for- mulations of the conjecture (the original version looks rather different from that given above) - this would involve Fourier and Laplace transforms; studying some of the recent research on the conjecture, which involves algebraic methods and approaches using optimisation theory; and computational work, for example experimentation with random matrices. References: a link to some notes on the BMV conjecture can be found on my home page (don’t worry if you don’t understand everything in the notes!): http://www.maths.ed.ac.uk/ adavie
Almost periodic functions Type: Individual Supervisor: Sandy Davie ([email protected]) Requirements: Pure & Applied Analysis
Periodic functions arise in many applications. Often however some system may exhibit ’periodic- like’ behaviour which is too complicated to describe by a standard periodic function with a single period. There may be two or more different periods involved. An example is the motion of the earth, where the rotation of the earth on its axis and the revolution round the sun have different periods, so the combined motion is not periodic. The notion of ’almost periodic function’ was introduced to cover more complicated behaviour of this sort. There are different but equivalent ways of defining ’almost periodic’, and the equivalence is one of the main results of the basic theory. The other main result is the existence of a Fourier series expansion analogous to that for periodic functions. The methods used are largely those of the first half of PAA: uniform convergence, subsequences, Fourier series. The project would involve studying the theory and then investigating how it applies to some examples. References: C. Corduneanu, ’Almost periodic functions’ (QA404) gives a good intrduction. H. A. Bohr, ’Almost periodic functions’ (QA403) is an early work by a founder of the theory; the type-setting is somewhat hard on the eye. B. M. Levitan, ’Almost periodic functions and differential equations’ (QA353.P4) is a more ad- vanced treatment. All three books are in JCM Library.
Computational complexity of matrix multiplication Type: Individual or Combined Supervisor: Sandy Davie ([email protected]) Requirements: some programming experience would be useful (especially for combined degree)
The standard algorithm for multiplying two n × n matrices requires approximately 2 times n3 operations. In 1968 Strassen showed that for large n this number of operations can be reduced: starting from a method for multiplying two 2-by-2 matrices using 7 (rather than the usual 8) multiplications, he showed that two n × n matrices can be multiplied in less than a constant times n2.81 operations. Since then a variety of ingenious techniques have been developed to achieve successive reductions in the exponent 2.81, reaching 2.3755 in 1990. That remained the best known result, until a further reduction to 2.3737 was attained in late 2009 by a PhD student in this School. The object of the project would be to understand the algebraic theory behind these developments, and to develop algorithms implementing some of them. D. E. Knuth, ‘The Art of Computer Programming’, Vol 2, 3rd Ed (QA76.6, in Robertson and Main Libraries), Section 4.6.4, gives an introduction to this topic and its connection to related problems. Burgisser, Clausen & Shokrollahi, ‘Algebraic complexity theory’ (QA3 Gru.v.315, JCM Lib), Chapter 15, gives a more extensive and advanced account (with rather forbidding notation, un- fortunately).
Tiling a rectangle Type: Individual Supervisor: Jose Figueroa-O’Farrill ([email protected]) Requirements: None, but an affinity for discrete mathematics is helpful Degree Programme: Any
I heard Alain Connes once say that one does not really understand the integers unless one under- stands the following Theorem. Whenever a rectangle is tiled by rectangles each of which has at least one integer side, then the tiled rectangle has at least one integer side. By now a number of proofs have appeared, ranging from the elementary to the sophisticated, each one suggesting the proof of some generalisation of the theorem. The point of this project is to explore some of these proofs (there are proofs for all mathematical tastes!) and to study some of the generalisations of this result. The project can be extended to consider more general kinds of tilings. Resources: S Wagon, Fourteen proofs of a result about tiling a rectangle, The American Mathematical Monthly (1987) vol. 94 (7) pp. 601-617 R Kenyon, A note on tiling with integer-sided rectangles, J. Combin. Theory Ser. A (1996) vol. 74 (2) pp. 321-332 NG de Bruijn, Filling Boxes with Bricks, Amer. Math. Monthly (1969) vol. 76 (1) pp. 37-40 DG Mead, Dissection of the Hypercube into Simplexes, Proc. Amer. Math. Soc. (1979) vol. 76 (2)pp. 302-304
Kuratowski’s 14-set theorem and its generalisations Type: Individual Supervisor: Jose Figueroa-O’Farrill ([email protected]) Requirements: None, but again an affinity for discrete mathematical structures is helpful Degree Programme: Any
The Kuratowski 14-set theorem is the following neat fact:
Theorem. Let E⊂ X be a subset of a topological space. The number of distinct sets which can be obtained from E by successively taking closures and complements (in any order) is at most 14. Moreover, there are subsets of the reals for which 14 is attained. Despite being phrased in the language of topology, this theorem is algebraic in nature. It uses the language of “topological calculus” and “closure algebras”. The purpose of the project is to understand the proof of the theorem and explore some of its generalisations. In the process the student will learn about posets, lattices, Hasse diagrams, semigroups,...
Resources: D Sherman, Variations on Kuratowski’s 14-set theorem, The American Mathematical Monthly, February 2010 BJ Gardner and M Jackson, The Kuratowski Closure-Complement Theorem, New Zealand J. Math., to appear, 34 pp, preprint JCC McKinsey, A Tarski, The Algebra of Topology, Ann. Math. (1944) vol. 45 (1) pp. 141-191 Topics in Riemannian geometry Type: Individual or Combined or Group Supervisor: Jose Figueroa-O’Farrill ([email protected]) Requirements: Third year Differential Geometry Degree Programme: Any
This project is a continuation of the third year Differential Geometry module and aims to explore the intrinsic differential geometry of higher-dimensional analogues of surfaces called “manifolds”. The differential geometry of manifolds is a vast subject, but whose foundations can be quickly covered to allow students to take the project in different directions, depending on their interests. Foundational topics: differentiable manifolds, riemannian manifolds, the Levi-Civita connection, the Riemann curvature tensor, geodesics, immersions, spaces of constant curvature... Possible continuations: symplectic geometry, complex and Kahler geometry, Hodge theory, pseudoriemannian geometry, <your favourite geometrical topic goes here>,...
Resources: M do Carmo, Riemannian Geometry, Birkhauser, QA649 Car. A Besse, Einstein manifolds, QA649 Bes. M Berger, A panoramic view of Riemannian geometry, QA649 Ber.
Lie theory and separation of variables Type: Individual or Combined or Group Only OK for strong students Supervisor: Jose Figueroa-O’Farrill ([email protected]) Requirements: None, but mastery of the JH core modules is helpful Degree Programme: Any
Solving partial differential equations is not easy. However in many situations it is possible to reduce PDEs to systems of ODEs, which makes the problem much more tractable. This method goes by the name of separation of variables and students have seen this in third year in the context of the wave, heat and Laplace’s equations. The existence of variables in which a given PDE separates can seem rather magical at first, but is actually underpinned by the existence of symmetries of the partial differential equation. The purpose of this project is to understand this more systematic approach to separation of variables and to apply it to derive separating variables for some famous partial differential equations. Resources:
A long series of papers Lie theory and separation of variables by Willard Miller and collaborators. (Find them using MathSciNet.)
Black hole uniqueness theorems Type: Individual or Combined Supervisor: Jose Figueroa-O’Farrill ([email protected]) Requirements: Third year Differential Geometry Degree Programme: Any
Black holes are time-independent solutions to Einstein’s equations. They describe regions of spacetime with such strong gravitational interactions that light can not escape from them. In this project, we will study the uniqueness theorems in four-dimensional General Relativity which state that the only solutions to Einstein’s equations, under some conditions, are the Schwarzschild and the Kerr black holes. Resources: S Chandrasekhar, The Mathematical Theory of Black Holes, QB843.B55 Cha. S Carroll, Spacetime and geometry: an introduction to general relativity, QC173.6 Car. JD Walecka, Introduction to General Relativity, QC173.6 Wal. PK Townsend, Black Holes, arXiv:gr-qc/9707012
Cosmological models Type: Individual or Combined Only OK for strong students Supervisor: Jose Figueroa-O’Farrill ([email protected]) Requirements: Maths 3 Differential geometry Degree Programme: Any
Our universe is currently accelerating. Thus, it changes with time. It contains different sources of energy : regular matter, dark matter, radiation and dark energy. Different sources dominated the expansion of the universe at different stages in the history of its evolution. The purpose of this project is to use Einstein’s equations to study the properties of the universe at these different stages. Resources: Chapters 2-3, and section 11.4.2 in EW Kolb and MS Turner, The Early Universe, 1994 S Carroll, Spacetime and geometry: an introduction to general relativity, QC173.6 Car. S Weinberg, Gravitation and Cosmology, QC6 Wei. JD Walecka, Introduction to General Relativity, QC173.6 Wal. S Chandrasekhar, The Mathematical Theory of Black Holes, QB843.B55 Cha. BS Ryden, Introduction to Cosmology, QB981 Ryd.
Modelling languages for Mathematical ProgrammingType: Combined-degree 40-pointSupervisor: Julian Hall ([email protected]) Requirements: None.Degree programme: Mathematics and Business Studies
The formulation of non-trivial optimal decision-making problems asmathematical programming problems is greatly aided by the use of
modelling languages. There are many academic and commercial modellinglanguages such as Xpress, GAMS, AIMMS, AMPL and MPL. In thisproject the student would study Xpress and one other language, usingthem to model a set of case studies. These would guide the studentthrough a range of classes of mathematical programming problems (LP,MIP and QP or stochastic LP) and allow the facilities of the twolanguages to be compared. See
Xpress: http://www.maths.ed.ac.uk/hall/Xpress-MP/index.htmlGAMS: http://www.gams.comAIMMS: http://www.aimms.comAMPL: http://www.ampl.comMPL: http://www.maximal-usa.com/mpl
Japanese Temple MathematicsType: GroupSupervisor: D.C. Heggie ([email protected])Requirements: No knowledge of Japanese requiredDegree Programme: any single honours
In the nineteenth century, Japanese mathematicians hung solutions ofmathematical problems in temples. Though many have disappeared, manywere collected into books which have survived. At first sight theyhave the appearance of geometrical brain teasers. But many of themare hard, and some of them benefit from a relatively sophisticatedmethod of approach. This project will consider a sample of a few ofthe more interesting problems, and their connections with Westernapproaches to similar problems.
Reference:Sacred Mathematics: Japanese Temple GeometryFukagawa Hidetoshi & Tony Rothman2008 (Princeton)
Henon's SolutionType: CombinedSupervisor: D.C. Heggie ([email protected])Requirements: Some numerically efficient programming language (e.g. matlab)Degree Programme: BSc Applied, or BSc Mathematics and Physics
The gravitational N-body problem underpins many problems inmathematical astronomy, but is intractible for N>2. When N is largeenough, however, the problem can be modelled statistically, by aFokker-Planck equation (which is essentially like a heat equation).If one seeks an important special kind of solution (homologous, orscale-invariant solutions), it reduces to a coupled fourth-ordersystem, first solved numerically by Michel Henon in 1965. Thisambitious project aims to bring Henon's solution up to date, with a
view to future developments.
Reference:Hénon, M, 1965, Sur l'évolution dynamique des amas globulaires II. Amasisol\'e, Annales d'Astrophysique, 28, 62 (a translation of this article also exists).
Background: Binney J., Tremaine S., 2008, Galactic Dynamics, 2ndedition, Princeton, especially Sec.7.4.
Mathematical Astronomy and Climate ChangeType: IndividualSupervisor: D.C. Heggie ([email protected])Requirements: Some programming languageDegree Programme: Any, but students without applied/physics interests would be less suited
It is well accepted that, besides anthropogenic factors, variations inthe orientation and orbit of the Earth, caused by gravitationalinteractions between the planets, are a major cause of climaticvariations. This project first reviews the evidence, and themechanisms which cause these orbital changes, and in the course of theproject toy models which illustrate them will be constructed.
Reference:Hays J.D., Imbrie J., Shackleton N.J. (1976) "Variations in theEarth's Orbit: Pacemaker of the Ice Ages", Science 194 (4270):1121 1132.
Background: Murray C.D., Dermott S.F., 1999, Solar System Dynamics,Cambridge, especially Sec.7.8.
Sturm-Liouville boundary value problems
Type: Individual/GroupSupervisor: Aram Karakhanyan ([email protected])Requirements: Basic calculus, some functional analysis and ordinary differential equations.Degree Programme: Any
The boundary value problems (BVP) for linear second order ODE's have a number of important applications in mathematical physics. For instance one of the basic techniques for solving partial differential equations (PDE) in some planar domains is the method of separation of variables. It reduces the PDE to BVP for a second order ODE with eigenvalues. Sturm-Liouville BVP is an important example of this sort. The class of questions closely related to this problem includes the generalised Fourier series, orthogonality of eigenfunctions, Bessel series expansion etc.
References
G.Simmons, Differential Equations With Applications and Historical Notes, JCM Library, QA372 Sim.
W. Boyce and DiPrima, Elementary differential equations, Robertson Libarary, QA371 Boy.
Second quadratic form of surfaceType: IndividualSupervisor: Aram Karakhanyan ([email protected])Requirements: Basic calculus, GCV.Degree Programme: Any
The second fundamental form of surface defines the extrinsic invariants of the surface, its principle curvatures. One cannot obtain this information from the first fundamental form. In other words by considering the second fundamental form we move away from the particular curves in the surface and focus on the surface itself. Two important characteristics of a surface in R3 are the mean and Gaussian curvatures. One of the aims of this project is to calculate these curvatures for ruled surfaces and surfaces of revolution.
References:A. Pogorelov Differential Geometry, Darwin Library, QA641 Pog.D. Struik, Lectures on classical differential geometry, Darwin Library, QA641 Str.H.S.M. Coxeter, Introduction to Geometry, JCM Library, QA445 Cox.
Project in AlgebraType: IndividualSupervisor: Tom Lenagan ([email protected])
I have several possible projects in Algebra for a student that hopes to get a good first class degree and is willing to work hard on a project to ensure this aim. If this sounds like you, then please come along to discuss possible projects with me.
In addition, here are some topics that you could Google:
(i) `Quantum Calculus' Kac Cheung (or consult book in Library)
(ii) `Integer partitions' Andrews Eriksson (or consult book in Library)
(iii) `Young Tableaux' Fulton (or consult book in Library)
Totally positive matricesType: Individual Supervisor: Tom Lenagan ([email protected]) Requirements: no special requirements other than to know what a determinant is.
A real matrix is totally positive if all of the determinants of each of its square submatrices are positive. More generally, a matrix is totally non-negative if all of the determinants of each of its square submatrices are non-negative. Totally positive/non-negative matrices arise in many areas; for
example, oscillations in mechanical systems, stochastic processes and approximation theory, planar resistor networks, ... .
However, this project will be a pure mathematical project, suited to students that like abstract mathematics. A typical question to investigate might be the following. The total number of square submatrices of a matrix grows exponentially so it would be very unwieldy to have to check every square submatrix to establish whether or not a matrix is totally positive/non-negative. Is there any more efficient way to do this? (The answer turns out to be yes, with a number of submatrices to check which only grows quadratically.)
There is much current research interest in this area.
S. Fomin and A. Zelevinsky, Total positivity: tests and parametrizations, Math. Intelligencer 22 (2000), 23-33.
Hilbert Series of Graded AlgebrasType: Individual Supervisor: Tom Lenagan ([email protected])
Let R be the ring of polynomials in, say, 3 variables, x , y , zover R . The set Rn of polynomials all of whose terms have total degree n , together with zero, forms a finite dimensional vector subspace of R , and R=R0⊕R1⊕R2⊕... Denote the dimension of the vector space Rn
by f n. The aim is to understand the sequence f n for this and other interesting rings; it turns out that knowledge of this sequence can tell you a lot about the ring. In this example you can check that the sequence is (starting at f 0 ): 1,3, 6, 10,... Can you spot the pattern, and prove your guess is correct?
One useful way of studying sequences such as these is via the generating function, or Hilbert series
HR t :=∑n=0
∞
f n tn.
In our example, we have HR t =13t6 t2
10 t 3... ,
which can be shown to be equal to 1−t −3 (can you check this?)
This project can be treated at several different levels. It can be computational or theoretical and computing can be incorporated or not depending on the preferences of the student.
GrowthType: Individual, Group or Combined Supervisor: Tom Lenagan ([email protected])
Degree programme: Mathematics or Computer Science and Mathematics.
In many algebraic objects there is a notion of length, which takes positive integer values, and it is important to study how the number of objects of length n grows with n. Here are three examples.
(i) Let be a directed graph. For each positive integer n , let a n be the number of paths of length n in
(ii) Let G be a group generated by a set of elements S ; so that every group element can be written as a product of elements from Sand the inverses of such elements. Every group element g can be written in many different ways, but there will be a least number nsuch that we can write g=s1... sn , for some si∈S∪S−1 . This number is called the length of g. Define
a n to be the number of group elements that have length n.
(iii) Let F be a field and R=F [a1,. .., ad] be a finitely generated F -algebra; that is, R is a ring that is generated over
F by the elements a1,. .., ad. Let V be the F -vector space spanned by the elements 1,a1, ... , ad ; so that
F⊆V⊆V 2⊆...⊆¿V n
=R.The function f n=dim V n is the growth function of R .
Such functions are known as Growth Functions, and, in each of these cases, knowledge of the rate of growth of a n with n reveals important information about the object being studied.
The aim of the project is to study growth functions.
The project will start by calculating growth functions for elementary examples.
Next, the combinatorics of generating functions will be studied. This is because the study of growth functions is facilitated by using the Poincare Series (or generating function) of a n;that is, the power series whose coefficient of the n -th power of
t is a n.
Finally, Poincare series will be calculated for several examples involving one or more of (i), (ii) and (iii) above.
The project will involve graph theory, combinatorics and algebra.
For those that like to, computing can be used to obtain experimental evidence and also to save time doing computations (some of the computing can be done in Maple).
Noetherian rings Type: Individual or Group
Supervisor: Tom Lenagan ([email protected])
In Numbers and Rings, you have seen that each ideal of Z , the integers, or F [x ] , polynomials over a field, can be generated by a single element. In contrast, the ideal of Z [ x ] generated by 2 and x cannot be generated by a single element. However, all is not lost: it can be shown that each ideal of Z [ x ] can be generated by a finite number of elements. A ring with this property is called a Noetherian Ring. Many important rings are Noetherian rings, and an extensive theory exists about them. This project will study the basic properties of Noetherian rings and then illustrate them in several important examples.
This project is suitable for students that enjoy abstract algebra.
For a flavour of the material involved, look at `Introductory lectures on rings and modules' by John A. Beachy (JCMB Library Shelfmark QA251.4 Bea).
The Weyl AlgebraType: Individual Supervisor: Tom Lenagan ([email protected])
The Weyl Algebra is a fundamental example of an algebraic structure that has a noncommutative multiplication. It arises in many different contexts. Perhaps the simplest way to introduce it is the following. Consider C [x ] , the ring of polynomials over the field of complex numbers, C . Let
X be the operation that multiplies a polynomial f in C [x ] by x , so that X f =xf. Let D be the operation that differentiates the
polynomial f , so that D f =f ' . A short computation reveals that DX−XD f =f , so that DX−XD=1 is the identity operation on
C [X ]. The algebraic structure C [X ,D ] is known as the Weyl Algebra, and is a noncommutative structure, by virtue of the above relation. As examples of its utility, many questions about differential equations can be studied from the point of view of this algebra. In addition, the relation DX−XD=1 is a scaled version of the equation qp−pq=i /¿ which is the algebraic manifestation of the Heisenberg Uncertainty Principle in quantum mechanics.
This project will concentrate on the algebraic properties of the Weyl Algebra. It is a suitable project for a student that likes abstract algebra.
Unifying Mathematics Type: Individual or Combined Supervisor: Dr Antony Maciocia ([email protected]) Requirements: None
Degree Programme: Any
Categories can be viewed as foundational tool for mathematics. They provide an abstraction of a whole range of mathematical concepts and their use has become widespread in many areas (including Mathematical Physics and Computer Science). They allow us to extract systematically those parts of a theory which relate “things” (called objects) such as groups, topological spaces, rings, even categories themselves to their natural maps. It turns out that many constructions that occur in such fields can be expressed purely in these terms and the resulting simplicity often shortens proofs and separates the abstraction from the essentials. This project can take various forms depending on the interests of the student. For example, we can explore the currently trendy notion of abelian categories, either from the point of view of algebra, or from the point of view of geometry. Another possibility is to study applications of categories in logic. We can also look at applications of the adjoint functor theorem in several areas of mathematics (one of which was to lead to the construction of a knot polynomial). Basic aims: 1. Understand the definition of a category and some basic examples 2. Explore the constructions of limits and universal objects 3. Describe how these arise in various areas of mathematics 4. (for 40 point version) Explore the construction of adjoints and the Yoneda lemma. Texts: P. Freyd, ‘Abelian Categories’, Harper & Row, QA614.5 Abe. S. Maclane, ‘Categories for the Working Mathematician’, Springer-Verlag, QA3 Gra. v.5. T. S. Blyth, ‘Categories’, Longman, QA169 Bly.
Paradoxes in Mathematics Type: Individual or Combined Supervisor: Dr Antony Maciocia ([email protected]) Requirements: None Degree programme: Any
At the end of the nineteenth century some rather alarming paradoxes concerning the foundations of mathematics were coming to light. For example, consider the collection of all sets. Is this a set? One would like it to be, after all, a set is just a collection, is it not? On the other hand, is the set of all sets a member of itself? There are many other related paradoxes which are usually called Russell’s Paradox after Bertrand Russell. There have been several attempts to reconstruct the foundations of mathematics to avoid such problems. Three schools of thought developed: the formalists headed by David Hilbert, the logicists headed by Russell and the intuitionists headed by L.E.J. Brouwer. These led to the development of modern set theory. This project should aim to look at these three schools an to explain how they got around the problems posed by the paradoxes. Basic Aims: 1. To explore the historical background to one or more of the fundamental paradoxes. 2. To explore how the paradoxes as resolved. 3. To give a modern account of one of the particular solutions and how it affects modern mathematics. L. E. J. Brower, ‘Collected Works’, North-Holland. M. J. Beeson, ‘Foundations of Constructive Mathematics’, Springer-Verlag. P. Benacerraf & H. Putnam, ‘Philosophy of Mathematics: selected readings’. B. Russell & A. N. Whitehead, ‘Principia Mathematica’, Cambridge University Press.
Exotic Logics Type: Individual Supervisor: Antony Maciocia ([email protected]) Requirements: None (but Logic 1 would be helpful) Degree programme: Any but especially suitable for Philosophy and Maths, or CS and Maths Level of difficulty: Any
Ordinary logic uses negation, and, or, implication and such like to provide a reasoning system for mathematics and philosophy. We also have truth values for such logics which make the logic decidable. But occasionally in mathematics and physics we need more exotic logics. For example, it may be appropriate to have more than just 2 truth values. This leads to notions of fuzzy logic which have important applications in engineering. Or we can introduce time varying logic (so called temporal logics). Or we can include notions of necessity and possibility. This leads to modal logics. The mathematics of such exotic logics is remarkably complex and leads to a variety of axiomatic systems. In this project, you would pick one of the exotic logics and study its axiomatization and look at applications in mathematics and beyond. G. E. Hughes & M. J. Cresswell, An Introduction to Modal Logic, Methuen (1982) A. Kaufmann, Introduction to fuzzy arithmetic : theory and applications, Van Nostrand Reinhold (1991) E. Turunen, Mathematics behind fuzzy logic, Physica-Verlag Heidelberg (1999). G. S. Boolos & R. C. Jeffrey, Computability and Logic, CUP (1980).
Transfinite Arithmetic Type: Individual Supervisor: Antony Maciocia ([email protected]) Degree Programme: Any but particularly suitable for Philosophy and Mathematics
Use Set Theory to define transfinite cardinal and ordinal numbers, and to understand their arith- metic. For example, the first infinite ordinal number is just the set of all finite numbers. It is possible to define addition, multiplication and exponentiation for ordinal numbers. It turns out that neither addition nor multiplication is commutative but they are both associative. There is a peculiar theorem which states that any strictly decreasing sequence of ordinal numbers is of finite length. One can also define transfinite cardinal numbers. They can also be added, multiplied and exponentiated. Cardinal numbers measure the size of sets. One important question about cardinal numbers is: is there a cardinal number strictly bigger than the cardinality of the natural numbers but strictly less than the cardinality of the real numbers? The statement that there does not exist such a cardinal number is called the Continuum Hypothesis. This project aims to explore the basic concepts of transfinite arithmetic and study a few appli- cations. Which topics are covered will depend on whether the Set Theory 4th year course is running. Basic Aims: 1. Understand the definitions of ordinal and cardinal numbers. 2. Understand their arithmetic (including arithmetic rules). 3a. Explore the ramifications in complexity theory, or 3b. Use of ordinals to define Conway’s surreal numbers. Texts: K. Devlin, ‘Fundamentals of Contemporary Set Theory’, Springer-Verlag. G. Cantor, ‘Contributions to the Founding of the Theory of TransfiniteNumbers’, Dover. P. T. Johnstone, ‘Notes on Set Theory and Logic’, Cambridge Univ. Press. J. H. Conway, ‘On Numbers and Games’, Academic Press D. E. Knuth, ‘Surreal Numbers’, Addison-Wesley
Non-standard Arithmetic Type: Individual Supervisor: Antony Maciocia ([email protected])
It is possible to give axioms for integer arithmetic. These are called Peano’s Axioms. The usual natural numbers provide a model for the axioms. The axioms include the order of the numbers and the fact that they are inductive (‘there is always one more number’). But it turns out that this set is not the only countable model. There are other non-isomorphic ones and these include so-called non-standard integers. The aim of this project is to understand these and/or other non-standard number systems and explore their applications. Texts: G. S. Boolos & R. C. Jeffrey, Computability and Logic, CUP (1980). J. H. Conway, On Numbers and Games, Academic Press (1977).
Finite projective spaces and n2 −n1 -tone scales Type: Combined or Individual Degree Programme: Any but especially suitable for Mathematics and Music. Supervisor: Antony Maciocia ([email protected]) Requirements: Some knowledge of music theory. Even tempered scales in which each semitone is exactly the same rise in pitch is generally hard to achieve and does not always sound the most pleasing. Scales in which the frequency of a fifth is exactly half of the third harmonic of the tonic will not be even tempered. There i a lot of interesting mathematical points in studying the possible scales which goes back to the Pythagoreans. One interesting observation is that when there are n2 −n1 tones in a scale then there is a description of equal tempered scales and their symmetries using finite projective planes ( n2 −n1 is the number of points in a finite projective plane in characteristic n−1 ). Such scales are called microtonal and have recently become fashionable in both classical and popular music. This links the geometry of such planes to the music. More details can be found in Chapter 9 in reference (2) below. In this project, you might make a general study of the mathematics of tonal systems and then provide a description of the finite projective plane example in relation to certain equal tempered scales. Alternatively, you could make a more detailed study of finite projective planes and explore what the geometry of such planes can reveal about musical compositions. There is software available to play microtonal compositions on computer (notably scala). 1. D. Benson, ‘Music, A Mathematical Offering’, CUP, 2007, On-line at http://www.maths.abdn.ac.uk/ bensondj/html/maths-music.html2. J. Fauvel, R. Flood, R. Wilson, ‘Music and Mathematics: From Pythagoras to Fractals’, OUP, 2003, JCM Lib: ML3800 Mus. 3. ‘musical scales’ or ‘microtonal’ in google will find numerous useful references.
The Mathematics of Synthetic Music Type: Combined or Individual Supervisor: Antony Maciocia ([email protected]) Requirements: 3rd year Algebra, Fourier Theory Degree Programme: Any but especially suitable for Mathematics and Music.
There are several sorts of ways to synthesise music. One of the simplest is additive synthesis in which the sound spectrum is built up by adding pure sine waves at various frequencies and ampli- tudes. More sophisticated techniques are subtractive synthesis in which one or more waveforms
are generated and then filters are used to remove or mould the spectrum. The parameters of such waveforms are often varied through time. Another approach is to use one waveform to modulate another. This is called frequency modulation and is especially suited to digital music production: it is responsible for some of those (ghastly) mobile phone ringtones. Frequency modulation is particularly interesting from a mathematical perspective because it introduces Bessel’s functions. In this project, which could be a 40 point combined project or a 20 point individual projects, you could take one of these methods (or a combination) and produce a mathematical model to describe them. You might base your analysis on the specification of specific hardware or software synthesisers. You might look at how particular forms of mathematical descriptions could sound better (or worse) than others. You might look at how a mathematical model of real sound (say, from a classical musical instrument) can be synthesised using the type of synthesis you are studying. If you have programming experience, you might look into using software such as csound to experiment with the results of your modelling. 1. D. Benson, ‘Music, A Mathematical Offering’, CUP, 2007, On-line at http://www.maths.abdn.ac.uk/ bensondj/html/maths-music.html2. J. Fauvel, R. Flood, R. Wilson, ‘Music and Mathematics: From Pythagoras to Fractals’, OUP, 2003, JCM Lib: ML3800 Mus. 3. ‘analog synthesis music’ or ‘digital synthesis music’ in google will find numerous useful references.
Symmetry Groups in Music Type: Combined or Individual Supervisor: Antony Maciocia ([email protected]) Requirements: 3rd year Algebra and some knowledge of musical melody and harmony theory. Degree Programme: Any but especially suitable for Mathematics and Music. Many composers have deliberately or subconsciously used symmetries in writing music. Such symmetries allow a small melody or harmony to generate larger coherent pieces of music by a process of translation, rotation and harmonic reflections. It is natural to try to study these using Group Theory. many groups can occur but dihedral groups, infinite cyclic groups and products involving Z12 occur most frequently. In this project, you might look at the literature on group theory in music and study some examples from the great composers (Beethoven, Bartok, Bach, Mozart etc). Alternatively, you might look particularly at using group theory to make numerical estimates on the possible number of genuinely distinct melodic phrases. 1. D. Benson, ‘Music, A Mathematical Offering’, CUP, 2007, On-line at http://www.maths.abdn.ac.uk/ bensondj/html/maths-music.html2. J. Fauvel, R. Flood, R. Wilson, ‘Music and Mathematics: From Pythagoras to Fractals’, OUP, 2003, JCM Lib: ML3800 Mus. 3. ‘analog synthesis music’ or ‘digital synthesis music’ in google will find numerous useful references.
Causality and the Kramers-Kronig RelationsSupervisor: T.G. Mackay [email protected] programme: single honours mathematics or joint honours mathematics and physics
``Effect cannot precede cause'' conveys the general meaning of the principle of causality. For linear dielectric materials, the Kramers-Kronig relations arise as a direct consequence of the principle of causality. The Kramers-Kronig relations are given as a pair of Hilbert transforms, usually expressed in terms of the real and imaginary parts of a complex-valued refractive index. These relations are of considerable practical value in the experimental determination of optical constants of materials.
The scope of this project may include:
(i) derivations of the Kramers-Kronig relations (using Titchmarsh'sTheorem and/or more physically-based methods);
(ii) applications of the Kramers-Kronig relations in opticalmeasurements;
(iii) generalisations of the Kramers-Kronig relations (known asdispersion relations) for causal linear systems.
This project builds upon mathematical concepts covered in the core 3rd year mathematics courses.
References:
Accounts of the Kramers-Kronig relations may be found in most advanced undergraduate or postgraduate textbooks on electromagnetic theory; for example,
1. ``Classical Electrodynamics'', J.D. Jackson, 2nd Edition, JohnWiley & Sons (1975).
A readable account of general dispersion relations can be found in
2. ``Mathematical Methods for Physicists'', G.B. Arfken and H.J.Weber, 4th Edition, Academic Press (1995);
while further technical details are given in
3. ``Dispersion Relations and Causal Description'', J. Hilgevoord,North Holland Publishing Company (1962).
Voigt wave propagation in anisotropic materialsSupervisor: T.G. Mackay [email protected] programme: single honours mathematics or joint honours mathematics and physics
This project relates to materials that have electromagnetic properties which depend upon direction; these are called anisotropic materials. The propagation of electromagnetic plane waves in anisotropic materials, such as biaxial crystals, is to be investigated. In general, two plane waves with mutually orthogonal polarizations can propagate along any particular direction in such materials. This phenomenon--which is known as birefringence--is commonly described in introductory courses on electromagnetics and/or optics. However, what is not so well-known is that there are instances when the two plane waves are non-orthogonal. Furthermore, the two plane waves then coalesce into a single plane wave which propagates with an amplitude proportional to the propagation distance. This composite wave is called a Voigt wave. This aim of this project is to explore the link between the symmetry of the anisotropic material and the propagation of Voigt
waves. The mathematics to be applied in this project will include eigenvector/eigenvalue analysis, Fourier transforms, differential equations, and vector/matrix manipulations.
For background reading on the propagation of electromagnetic plane waves, a good place to start is:
1. ``Electromagnetic Fields and Waves'', P. Lorrain, D.R. Corson andF. Lorrain, 3rd Edition, Freeman (1988).
For accounts of plane wave propagation in anisotropic materials one should look to books describing `crystal optics'. The classic references are:
2. ``Principles of Optics'', M. Born and E. Wolf, 6th Edition, Pergamon (1980)
and
3. ``Physical Properties of Crystals'', J.F. Nye, OUP (1985).
Depolarization dyadics in electromagnetic homogenization studiesProposed 4th year individual project 2011/2012Supervisor: T.G. Mackay [email protected] programme: single honours mathematics or joint honours mathematics and physics
This project concerns depolarization dyadics. In effect, a depolarization dyadic yields the electromagnetic scattering response of a small inclusion particle embedded in a homogeneous ambient material. These are important mathematical constructions used in the estimation of the effective electromagnetic properties of composite materials. Explicit forms for depolarization dyadics are available for certain relatively simple inclusion particles (e.g., spherical particles made from isotropic materials), but this is not the case for more complex inclusion particles. The aim of this project is to explore the nature of depolarization dyadics for spherical and nonspherical particles made from isotropic and anisotropic materials. This project builds upon mathematical concepts covered in the core 3rd year mathematics courses.
For background reading on electromagnetic homogenization studiessee:
1. ``Electromagnetic mixing formulas and applications," A.H. Sihvola,IEE (1999).
For details on depolarization dyadics see:
2. ``Electromagnetic anisotropy and bianisotropy," T.G. Mackayand A. Lakhtakia, World Scientific (2010).
The computation of special functions in the complex plane Type: Individual or Group
Supervisor: Adri Olde Daalhuis ([email protected]) Requirements: MAT-3-CVD
Special functions (Bessel functions, orthogonal polynomials, incomplete gamma and beta func- tions, etc.) are often functions of several (complex) variables, say functions on Cn , n1 . Methods to compute these functions are: differential equations, recurrence relations, integral representations, Taylor series expansions, asymptotic expansions. None of these methods will give satisfying results in the whole of Cn, and a combination of the methods is needed. In this project we will take one special function and try to find a complete set of methods to approximate the function in the whole of Cn. A comparison with the methods that MAPLE uses might also be interesting. N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, 1996. (QC20.7.F87 Tem)
Elliptic functions Type: Individual Supervisor: Adri Olde Daalhuis ([email protected]) Requirements: MAT-3-CVD
Elliptic functions are analytic functions in the complex plane that are doubly-periodic. These func- tions have many nice properties. For example, the most basic elliptic function is the Weierstrass ℘ function, and it is possible to express any elliptic function in terms of a rational combination
of the Weierstrass ℘ function and its derivative. Another example is that when a meromorphic function f z is a solution of a first order algebraic differential equation then f z is either (1) rational, (2) singly-periodic, or (3) elliptic. In this project you are expected to reconstruct many of the proofs, and apply the results to non-trivial examples. D. F. Lawden, Elliptic Functions and Applications, Springer-Verlag, 1989. (QA343 Law) E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, CUP, 1927. (QA295 Whi)
Martingales and stochastic gamesType: Individual or Combined DegreeSupervisor: Miklos Rasonyi ([email protected]) Requirements: some knowledge of probability and willingness to learn more about it.
The aim of the project is to study an important class ofstochastic processes: martingales. These are closely related toreal-life phenomena such as gambling or the stock exchange.We would apply their theory to the analysis ofwinning strategies in games whose outcome depends on randomness.
References (the book can be found in the JCM library, the two articles are not easily available but I can provide electronic copies of them):Brzezniak, Z. and Zastawniak, T.: Basic stochastic processes.A course through exercises.Springer Undergraduate Mathematics Series. Springer, 1999. T. F. Mori: On favourable stochastic games. Annales Univ. Sci. Budapest. R. Eotvos Nom., Sect. Comput. 3, 99--103, 1982.T. F. M\'ori and G. J. Szekely: How to win if you can? In: Limit Theorems in Probability and Statistics vol.2, Colloq. Math. Soc. J. Bolyai 36, North-Holland, 791--806, 1984.
Monte Carlo simulationsType: Individual or Combined Degree (Computer Science) Supervisor: Miklos Rasonyi ([email protected]) Requirements: basic knowledge of probability; some programming experience is needed, especially for combined degree version.
In many areas of applied probability one needs to calculate theexpectation of complicated random variables (e.g. prices of financialassets or delay of a telecommunication system). Generically, there are no closed-form solutions so simulations are performed. The related techniques arereferred to as ``Monte Carlo methods''. In this project we learn about the most important ideas in this field and run simulations to illustrate them. Our examples will be mainly coming fromfinancial mathematics.
References (both in JCM Library):Hammersley, J. M. and Handscomb, D. C.: Monte Carlo methods.Methuen and Co., 1965.Ross, S. M.: Simulation. Second edition. Academic Press, 1997.
Arbitrage Pricing TheoryType: Individual or Combined Degree (Economics) Supervisor: Miklos Rasonyi ([email protected]) Requirements: a basic knowledge of probability required. In parallel, the student will be required to read the year 4 Hilbert space course.
Arbitrage pricing theory was invented by the economist S. Ross in theseventies. It is based on the following ideas: There exist market indices (such as Dow Jones or Nasdaq) which are supposedto describe the general state of the economy quite well. Let usfix such an index and consider a collection of individual stocks.What is the relationship between the return on these stocks andtheir correlation with the given index ? It was argued by economists that this relationship must be linear. Arbitragepricing theory provides a rigorous mathematical formulationand a proof (under suitable assumptions) that this relationship must be asymptotically linear.We'll explore the economic theory and mathematics behind and see how simple ideas may lead to fascinating conclusions. References (the book is in the Main Library, there is onlineaccess to the paper through the library system):Huang, C. and Litzenberger, R.: Foundations for financial economics. Prentice Hall, 1988.Huberman, G.: A simple approach to arbitrage pricing theory. J. Econom. Theory, 28, 183--191, 1982.
Inner and Outer Conformal Maps Type: Individual Supervisor: Adri Olde Daalhuis ([email protected]) Requirements: The complex variable part of CVD. Some programming experience would be useful.
A conformal map is simply a 1–1 map produced by an analytic function. Given a simple, closed contour C , call such a map from the interior of the unit circle onto the interior of C an inner conformal map, and such a map from the exterior of the unit circle onto the exterior of C (that maps ∞ to ∞) an outer conformal map. The project will seek examples of such pairs of maps that can be expressed in terms of known functions. Reference: V. I. Ivanov & M. K. Trubetskov: Handbook of Conformal Mapping with Computer- Aided Visualization, CRC Press, 1995.
Geometric singular perturbation theoryType: Individual or GroupSupervisor: N. Popovic ([email protected])Requirements: Differential Equations, Dynamical Systems
Singular perturbation problems feature prominently both in the theory of differential equations and in their applications. Singularly perturbed equations are characterised by the presence of at least two fundamentally different scales, and have traditionally been studied using a variety of (often formal) techniques. More recently, a unified geometric approach has been developed, which is based on dynamical systems theory and, in particular, on invariant manifold methods. While the approach is complete in the hyperbolic setting, it breaks down at non-hyperbolic points; this loss of hyperbolicity can often be remedied by geometric desingularisation (blow-up).
In this project, you will familiarise yourself with geometric singular perturbation theory and blow-up. A combination of the two techniques can frequently yield a fairly complete global picture of the system dynamics; examples include neuronal spiking in models of Hodgkin-Huxley type and the propagation of front solutions in degenerate reaction-diffusion systems. You will explore these and similar sample applications both analytically and numerically.
P. A. Lagerstrom; Matched asymptotic expansions--ideas and techniques; Springer-Verlag, 1988C. K. R. T. Jones; Geometric singular perturbation theory; in Dynamical Systems, Montecatini Terme, 1994 (available on request)M. J. Alvarez et al.; A survey on the blow-up technique; preprint, 2010 (available on request)
Social aggression in pigsType: Individual or GroupSupervisor: N. Popovic ([email protected])Requirements: Dynamical Systems, Statistics
Aggressive behaviour in pigs is a major issue in animal welfare that affects the profitability and environmental impact of pig production. Several factors influencing this so-called social aggression have been identified: thus, the group structure and dynamics have been observed to affect fighting and feeding behaviour. However, little is known about the underlying genetic basis and biological pathways and the role of dynamic interactions between individuals in the emergence of stable hierarchies.
Recently, a stochastic agent-based model has been developed to describe aggressive behaviour in time based on pairwise interactions in a group of pigs. In this project, you will validate and extend that modelling framework, drawing on a large dataset of observations of social aggression. By performing a statistical analysis of the data, you will establish characteristics for quantifying aggressive behaviour and hierarchy formation in the group, and you will investigate the key factors that drive aggression.
The project will be supervised jointly with A. Wilson (The Roslin Institute and Royal (Dick) School of Veterinary Studies, [email protected]).
R. B. D?Eath and A. B. Doeschl-Wilson; Modelling the dynamics of aggressive behaviour in groups of pigs; poster presentation, 2009 (available on request)M. Enquist and O. Leimar; Evolution of fighting behavior--decision rules and assessment of relative strength; J. Theor. Biology 102, 387-410, 1983J. J. Bryson, A. Yasushi, H. Lehman; Agent-based modelling as scientific method: a case study analysing primate social behaviour; Phil. Trans. R. Soc. B 263, 1685-1698, 2007
The Farey sequences Type: Individual or Group Supervisor: A.Ranicki ([email protected]) Second marker: Anyone in the Geometry/Topology or Algebra/Number Theory groups Requirements: Suitable for any student, but the stronger the better!
The Farey sequence Fn of order n is the sequence of rational numbers p/q ( p ,qcoprime, with 0≤p≤q≤n ) , ordered by size. For example,
F3={0/1,1/3, 1/2,2/3, 1/1}.The Farey sequences and associated diagrams have many applications in number theory (continued fractions, Riemann hypothesis etc.), geometry, topology, fractals . . . . The Internet is a wonderful source of material about the sequence (sometimes called the Farey series), starting inevitably with the Wikipedia article (1), and including the history (2). There are 35,000 entries to chose from. The object of the project is to select some favourite application(s) of the student(s), describe them, and do some mathematics in that area. Just copying out chunks of material from the Internet is not good enough! For example, doing the problem set (3) would be a good idea if the applications to geometry are being considered. References: (1) Wikipedia http://en.wikipedia.org/wiki/Farey_Sequence (2) History http://www.cut-the-knot.org/blue/FareyHistory.shtml (3) Problem set http://math.stanford.edu/circle/ProblemSetFord.pdf
Quaternions and rotations Type: Individual or Group Supervisor: A. Ranicki ([email protected]) Requirements: Topology (Suitable for all students, provided they are hardworking)
Quaternions are ”hypercomplex numbers” with interesting applications in algebra, topology, physics and engineering. They were discovered in 1843 by William Rowan Hamilton in Dublin, and are still going strong! A quaternion has the form
q=a ib jc kdwith a ,b , c , d real numbers. Quaternions have a noncommutative multiplication according to the
rules i 2= j2=k2=−1 ,i j=− j i=k ,j k=−k j=i ,k i=− i k= j .
By definition, a rotation of Euclidean n -space ℝn is a distance-preserving transformation
T :ℝn→ℝn such that T 0=0, and which preserves orientation. There is one rotation for
each n×n orthogonal matrix R with det R=1 , and the group SO n of such matrices has a topology. For n=1 there is only 1 rotation. For n=2 there is one rotation for each angle ∈[0, 2, and SO 2 is homeomorphic to S1. The object of the project is to use
quaternions to give descriptions of SO n for n=3,4, and to describe as many related applications as the student(s) can manage. References: [1] Quaternion Wikipedia article http://en.wikipedia.org/wiki/Quaternions [2] Quaternions and spatial rotation Wikipedia article http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation [3] J. Kuipers, Quaternions and rotation sequences, Princeton (1999) (in JCMB Library, QA196 Kui) [4] J.Weiner and G.Wilkens, Quaternions and rotations in ℝ4, American Mathematical Monthly 112, 69–76 (2005) (in JCMB Library)
Simplicial complexes and the Euler characteristic Type: Individual or Group Supervisor: A. Ranicki ([email protected]) Requirements: Topology (Suitable for all students, provided they are interested in topology)
A simplicial complex X is a topological space which is obtained by glueing together points, line segments, triangles and higher-dimensional analogues called n -simplexes (with a point being a
0 -simplex). The Euler characteristic of a compact X is
χ X=∑n=0
∞
−1n bn∈ℤ
with bn the number of n-simplexes. The Wikipedia articles [1],[2] are excellent introductions! The literature on simplicial complexes and the Euler characteristic is vast, and the challenge for the student (or students) undertaking this topic is to narrow down the possibilities to the scope of a project, while at the same time affording the student(s) the opportunity of showing off their mathematical skills! For the more advanced student(s) the paper [3] of Ghrist and de Silva is a very recent application of the Euler characteristic to target enumeration, which would surely repay study : but it is challenging! Working out simple examples to illustrate the general theory would be one specific task. References: [1] Euler characteristic Wikipedia article http://en.wikipedia.org/wiki/Euler_characteristic [2] Simplicial complex Wikipedia article http://en.wikipedia.org/wiki/Simplicial_complex[3] Coordinate-free coverage in sensor networks with controlled boundaries via homology Robert Ghrist and Vin de Silva http://www.math.uiuc.edu/ ghrist/preprints/controlledboundary.pdf
Nil and Jacobson radical ringsType: Individual Supervisor: Agata Smoktunowicz ([email protected])Requirements:
The aim of this project is to study nil and Jacobson radical rings. Let R be a ring; we say that R is Jacobson radical if for every element r∈ R there is element r ′ ∈ R such that rr ′=rr ′ . Moreover, R is a nil ring if for every element r∈ R there is natural number
n=nr such that rn=0. Note that
every nil ring is Jacobson radical. Possible topics to include in this project may be chosen from the following suggestions: 1. Commutative nil rings and Jacobson radical rings. 2. Finite rings. 3. Noetherian rings. Artinian rings. 4. Nil rings of bounded index. 5. Graded rings. Rings with all homogeneous elements nilpotent. 6. Special types of nil rings ex. semicommutative nil rings, duo nil rings. 7. Examples of nil rings with exotic properties; for example finitely gen- erated Jacobson radical and not nil rings.
It is also possible to base the whole project solely on explaining the proof of Golod-Shafarevich theorem which says that there are finitely generated infinitely dimensional nil rings. Considered rings and associative and non- commutative.
Recommended reading ∙ J.A. Beachy, Introductory lectures on rings and modules, student texts 47, London Mathematical Society 1999. ∙ G. Calugareanu, P Hamburg, Exercises in basic ring theory, Kluwer 1998. ∙ T.Y.Lam, A first course in noncommutative rings, 2001, Graduate Texts in Mathematics, Springler-Verlag, New York, Berlin, Heidelber. ∙ Carl Faith, Rings and things and the fine array of 20th century non- commutative algebra, Mathematical Surveys and Mponographs, 2004. ∙ L H Rowen, Graduate algebra:noncommutative view, Graduate Studies in Mathematics, Volume 91, 2000.
Polynomial identity (PI) rings Type: Individual Supervisor: A Smoktunowicz([email protected])Requirements:
A ring R is a PI ring if there is a noncommutative polynomial which vanishes when we substitute arbitrary elements from ring R for the variables. For example, all commutative rings are PI, satisfying polynomial f x , y =xy− yx. This project will concern mainly matrices over commutative rings. By the Postner Theorem, under mild assumptions, polynomial identity algebras are just subalgebras of matrix algebras with coefficients from commutative rings. The aim of the project is to study some famous results on PI algebras and prove some special cases of such theorems using this fact. All considered rings are associative and noncommutative. Recommended reading ∙ Alexei Kanel-Belov, Louis Halle Rowen, Computational Aspects of poly- nomial identities, Research Notes in Mathematics, Volume 9, 2005.
∙ Edward Formanek, The polynomial identities and invariants of nxn matrices, American Mathematical Society 1992. ∙ A. Giambruno, M. Zaicev, Polynomial identities and asymptotic methods, Mathematical Surveys and Monographs Volume 122, AMS, 2005. ∙ C. Procesi, Rings with Polynomial identities, Dekker, 1973.
Some famous matricesType: Individual Supervisor: A Smoktunowicz ([email protected])Requirements:
The aim of the project is to study basic matrix theory. Topics to investigate in the project may include some of the following: 1. Toeplitz matrices. 2. Householder matrices. 3. Eigenvalues, Gresgorin discs. 4. Pseudoinverses of matrices. 5. Stochastic matrices. Peron-Frobenius theorem. Google matrices. 6. Vandermonde matrices. Magic squares. 7. Fourier matrices. It is recommended to either concentrate more extensively on one topic or choose a few topics. Recommended reading ∙ Roger A. Horn, Charles Johnson, Topics in matrix analysis, Cambridge University Press 1991. ∙ Roger A. Horn, Charles Johnson, Matrix analysis, Cambridge Univer- sity Press 2005. ∙ Adi Ben-Israel, Thomas Nall Eden Grenville, Generalized inverses:theory and applications, Canadian Mathematican Society, 2003. ∙ Richard S.Varga, Gresgorin and His Circles, Springer Series in Com- putational Mathematics 36, 2004. ∙ http://mathworld.wolfram.com
Bounding and finding roots of polynomials Type: Individual or Group Supervisor: Chris Smyth ([email protected]) Requirements:
This project has two related aspects. (a) If you know that the coefficients of a polynomial lie within certain bounds, how does that restrict the position in the complex plane of the roots of the polynomial? If you know further that the polynomial has all real roots, can you say more? In general, the problem of finding inequalities restricting the position of the roots, as a function of the coefficients, is an interesting and challenging one. A starting point could be Section 3.4 of [1]. (b) What are efficient algorithms for finding the roots of a polynomial? Do they work when the polynomial has degree 200? One approach is to take the companion matrix of the polynomial. Then the problem of finding its roots is the same as the problem of finding the eigenvalues of this matrix. Thus methods for finding matrix eigenvalues can be used. A starting point could be Section 9.5 of [2]. Some programming (e.g. in Maple) could be undertaken to compare the various methods on a well-chosen set of example
polynomials. The project could combine both parts, perhaps concentrating on one part more heavily, according to interest. [1 ] D.S. Mitrinovic, Analytic inequalities, Springer Verlag 1970. [2 ] W. Press, B. Flannery, S. Teulosky, W. Vetterling, Numerical recipes, CUP 1986 (or later editions).
Covering congruences and their applications Type: Individual Supervisor: Chris Smyth ([email protected]) Requirements:
A system of covering congruences is a finite set of congruences j1mod m1 , . . ., jn mod mn with the mi all different and greater than 1 such that every integer is congruent to ji mod mi for some i. An example is
0 mod 2 ,0mod 3 ,1mod 4 ,3mod 8 ,7 mod 12,23mod 24 .The aim of this project is to study these systems. For instance, how small can
n be? How large can the smallest mi be? A starting point could be the references [1] and [2] below. These covering systems have applications, and another aim of the project is to study some of these. Possible examples of applications are given in references [3], [4] and [5], but no doubt more can be found in the literature.
1. Choi, S. L. G. Covering the set of integers by congruence classes of distinct moduli. Math. Comp. 25 (1971), 885–895. 2. Churchhouse, R. F. Covering sets and systems of congruences. 1968 Computers in Mathematical Research pp. 20–36 North-Holland, Amsterdam 3. Erd s, P. On integers of the form 2kp and some related problems. Summa Brasil. Math. 2, (1950). 113–123. 4. Filaseta, Michael Coverings of the integers associated with an irreducibility theorem of A.Schinzel. Number theory for the millennium, II (Urbana, IL, 2000), 1–24, A K Peters, Natick, MA, 2002. 5. Schinzel, A. Reducibility of polynomials and covering systems of congruences. Acta Arith. 13 1967/1968 91–101.
Chromatic polynomials of graphs Type: Individual Supervisor: Chris Smyth ([email protected])
Given a graph G and k different colours, in how many ways can you colour the vertices of G in such a way that adjacent (joined by an edge) vertices are coloured differently? It turns out that for each G there is a polynomial PGx ,called the chromatic polynomial of G , such that for every k the number of such colourings is PGk .The aim of this project is to study these chromatic polynomials. For instance, where do their zeros lie? For which values of x are they positive? What special properties do chromatic polynomials of planar graphs have? (The celebrated Four-Colour Theorem tells us that PG40 for all planar graphs G. ) A starting point could be the references [1] and [2] below.
1. Read, R. C. An introduction to chromatic polynomials. J. Combinatorial Theory 4 (1968), 52–71. 2. Read, R. C.; Tutte, W. T. Chromatic polynomials. Selected topics in graph theory, 3, 15–42, Academic Press, San Diego, CA, 1988.
Continued fractions of quadratic irrationals Type: Individual Supervisor: Chris Smyth ([email protected])Requirements:
Given a real number x1 , we can write it as x=⌊ x ⌋x , where ⌊ x ⌋ is the integer part of x (the largest integer not greater than x ) and x is the fractional part of x. Then one can produce another real number x ′1 by defining x ′=1/ x . Repeating this process gives a sequence of real numbers, and their integer parts a a0 , a1 , a2 , a3 . .. , the partial quotients of x , give the partial fraction representation x=a01/a11/ a21/ a31 /. . . of x. The aim of this project is to study the continued fraction expansion of x when
x is a quadratic irrational, and so of the form x=uv d , where u and v are rationals, and d is a nonsquare positive integer. One might also investigate how this expansion is related to the problem of finding integer solutions x , y to Pell’s equation x2− dy2=1.There are many books on number theory that contain sections on continued fractions, as well as some specialised texts. One of each kind is given below.is given below.
1. Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L. An introduction to the theory of numbers. Fifth edition. John Wiley & Sons, Inc., New York, 1991. 2. Rockett, Andrew M.; Szsz, Peter, Continued fractions. World Scientific Publishing Co., Inc., River Edge, NJ, 1992.
Eigenvalues of graphs Type: Individual Supervisor: Chris Smyth ([email protected]) Requirements:
The eigenvalues of a graph are defined as the eigenvalues of its associated adjaceny matrix. How large can the eigenvalue of a graph be? How are the eigenvalues of a graph related to the eigenvalues of its subgraphs? What do the eigenvalues of a graph tell you about the graph? These are some of the questions you might address in this project.
1. Cvetkovi , D.; Rowlinson, P. The largest eigenvalue of a graph: a survey. Linear and Multilinear Algebra 28 (1990), no. 1-2, 3–33. 2. Godsil, Chris; Royle, Gordon, Algebraic graph theory. Springer-Verlag, New York, 2001. Home ground advantage in team sportsType: Individual or Group Supervisor: Chris Theobald ([email protected]) Requirements: Likelihood; knowledge of an appropriate team sport
Sports teams playing on their home ground are often considered to have an advantage. Clarke and Norman (1995) studied the performance of English football teams over several seasons, and estimated the home advantage of individual teams. An individual project could use data on a team sport of the student's choice, and would take a different approach, modelling the scores in each match using log-linear models with effects including measures of team ability and home advantage. Home advantage might be related to influences such as the distance between club grounds. For a group project, the members of the group would be expected to examine data on different sports, and these could include individual sports. Any student expressing a preference for this topic should state on the Project Registration Form which sports are of interest.
Clarke, S. R. and Norman, J. M. (1995). Home ground advantage of individual clubs in English soccer. Statistician, 44, 509-521.
Modelling numbers of Olympic medalsType: Individual Supervisor: Chris Theobald ([email protected]) Requirements: Likelihood
Populous and wealthy countries tend to head published tables of Olympic medals: tables for medal performance per capita tend to be dominated by small island nations, while those for medals relative to gross domestic product (GDP) are dominated by communist (or formerly communist) nations. The paper below considers log-linear relationships between medal performance, population and GDP, but treats medal numbers as continuous random variables and ignores countries gaining no medals. This project would involve fitting Poisson log-linear models, and generalizations of such models, to the medal numbers in the 2008 Olympic Games, and possibly to those for earlier ones, and identifying nations with especially high or low numbers relative to their population and GDP.
Morton, R. H. (2002). Who won the Sydney 2000 Olympics?: an allometric approach. Statistician 51, 147-155.
Statistical modelling of data on the unit diskType: Individual Supervisor: Chris Theobald ([email protected])Requirements: Likelihood
Most statistical methods are defined for data taking real values or non-negative integer values, but sometimes it is necessary to consider other spaces for data, such as the circle, sphere or simplex. This project considers probability distributions whose support is the unit disk, particularly the M\"{o}bius distribution. This distribution is applied to examine data on the response of insect parasites (such as larvae of the
wheat bulb fly) to chemical constituents of the plants they attack.
Jones, M. C. (2004). The M\"{o}bius distribution on the disc. Annals of the Institute of Statistical Mathematics, 56, 73--742. http://www.ism.ac.jp/editsec/aism/pdf/056_4_0733.pdf http://www.ism.ac.jp/editsec/aism/pdf/056\_4\_0733.pdf
McKay Quivers Supervisor: Michael Wemyss ([email protected]) Type: Individual or Group Requirements: None
Way back in 1979, McKay made the observation that a simple procedure using the character theory of a finite subgroup of SL 2,ℂ is related to the geometry of a well–understood geometric space. McKay’s combinatorial rule gives rise to a directed graph, which is known as a quiver. This is now known as the McKay quiver. The point is that, once you know something called the character table of G , the McKay quiver associated to G is very easy to write down. Once you know this quiver, you have a method to attack some geometric problems, as it gives you a good way of visualizing things. The first part of this project would be an introduction to character theory, closely following the very good book ‘Representations and Characters of Groups’ by James and Liebeck. You would learn the basics of character theory, and be able to compute character tables. The second part of the project involves using this knowledge to draw the McKay quivers associated to some finite subgroups G of GL 3,ℂ. This part is easy, once you know the character table. There are also other options available, depending on the students interests.
James and Liebeck, Representations and Characters of Groups.
Power Series Rings versus Polynomial Rings Supervisor: Michael Wemyss ([email protected]) Type: Individual Requirements: None
The polynomial ring k [x1 ,. . . , xn] by definition consists of polynomials, which by definition are finite sums of products of powers of the x i . Power series rings k [[x1 , .. . , xn]] are the objects that arise when we allow ourselves to take infinite sums. These objects are both very similar and very different. For example, in k [x ] the polynomial
1x does not have an inverse, whereas in k [[x ]] it does. There are others too, some of which are very surprising. This project will focus on investigating these similarities and differences. It will begin by looking at the noetherian property, and will explain ideals in an easy way by using Hilbert Basis Theorem. It will then touch on the idea of completion of commutative rings, before illustrating the dramatically different number of maximal ideals by using the Nullstellensatz. There are other options too, including the production of examples where the behaviour of the ring changes when we pass to the formal power series ring. Some of these calculations could be (but don’t have to be) done using computer algebra. Although the project has a slightly geometric flavour, the project would best suit a student who is interested in abstract algebra.
R Y Sharp, Steps in Commutative Algebra. M F Atiyah and I G Macdonald, Introduction to Commutative Algebra. Miles Reid, Undergraduate Commutative Algebra.
Singularities in Algebraic Geometry Supervisor: Michael Wemyss ([email protected]) Type: Individual or Group Requirements: None.
Algebraic Geometry studies the solutions of a set of polynomial equations. If you draw the solution set, it often becomes clear that some points look much different than others, and this leads to the definition of a singularity. This project will start with a rigorous definition of a singularity, with some calculations to gain some insight into what is (and what is not) singular. However, there are basic questions about singularities that are hard to answer using this definition, so we are led to develop some more abstract machinery. Where the project goes from here is up to the student. On the algebraic side, the project could give some characterizations of singular spaces which are easier to work with, and show that they are equivalent to the original definition. The ambitious could aim for the famous result of Auslander–Buchsbaum and Serre, which characterizes singularities in terms of an invariant called global dimension. On the geometric side (suitable for those taking the Algebraic Geometry course), an alternative project would be to aim to improve singularities via a process called blowing up. This could involve either learning theory or doing calculations (or both), then using this knowledge to link to other areas of mathematics. There is also an option of plotting various singular spaces by using a computer algebra package called Singular.
Miles Reid, Undergraduate Algebraic Geometry. Miles Reid, Undergraduate Commutative Algebra. David Eisenbud, Commutative Algebra with a view towards Algebraic Geometry.
Statistical modelling of data from controlled laboratory teststo provide a greater understanding of earthquakesType: Individual Supervisor: Bruce Worton ([email protected]) Requirements: Linear Statistical Modelling (MAT-3-LSM) and Likelihood (MAT-3-Lik) \\Degree programme: Would be particularly suitable for a Mathematics and Statistics degree student.
This project would use statistical modelling to investigate data obtainedin a series of controlled laboratory tests on rock samples.The objective is to use the laboratory tests to provide a greater understanding of brittle failure events,such as earthquakes.
A model proposed by a geologist (Main, 2000) can be fitted in variousways to data. The data take the form of axial strain as the responseover time for the brittle creep experiments.
Main, I.G. (2000). A damage mechanics model for power-law creep andearthquake aftershock and foreshock sequences, Geophys. J. Int. 142,151-161.
Finite mixture statistical modellingType: Individual Supervisor: Bruce Worton ([email protected])Requirements: Linear Statistical Modelling (MAT-3-LSM) and Likelihood (MAT-3-Lik) Degree programme: Would be particularly suitable for a Mathematics and Statistics degree student.
Finite mixture distributions arise in a variety of application areas.They provide natural models for situations where there are componentpopulations within a population. One particular use of finite mixture modelsis in the identification of clusters.
The project will review methods for finite mixture models. Application of finitenormal mixture distributions to various data sets, including velocities ofgalaxies, will give the opportunity to apply and assess the methods studied.
Everitt, B. S. and Hand, D. J. (1981). Finite Mixture Distributions.Chapman and Hall.
McLachlan, G. J. and Peel, D. (2000). Finite Mixture Models. Wiley.
McLachlan, G. J. and Basford, K. E. (1988). Mixture Models:Inference and Applications to Clustering. Marcel Dekker.
Titterington, D. M., Smith, A. F. M., and Makov, U. E. (1985).Statistical Analysis of Finite Mixture Distributions. Wiley.
Using empirical likelihood for statistical analysisType: Individual Supervisor: Bruce Worton ([email protected]) Requirements: Linear Statistical Modelling (MAT-3-LSM) and Likelihood (MAT-3-Lik) Degree programme: Would be particularly suitable for a Mathematics and Statistics degree student.
Empirical likelihood methods use empirical distributions to define alikelihood function for a parameter of interest, e.g., a populationmean. This approach provides an alternative way of obtaining alikelihood to conventional parametric modelling.
This statistical project will study ways of computing empiricallikelihoods for location parameters, and regression parameters.Iterative numerical methods are required, as empirical likelihoodscannot be written in closed form.
Thisted, R. A. (1988). Elements of Statistical Computing. Chapmanand Hall.
Owen, A. B. (1988). Empirical likelihood ratio confidence intervalsfor a single functional. Biometrika, 75, 237-249.
Owen, A. B. (2001). Empirical Likelihood. Chapman and Hall/CRC.
Identifying the number of modes of a densityType: Individual Supervisor: Bruce Worton ([email protected]) Requirements: Linear Statistical Modelling (MAT-3-LSM) and Likelihood (MAT-3-Lik) Degree programme: Would be particularly suitable for a Mathematics and Statistics degree student.
It is often of interest to identify the number of modes of aprobability density function based on a random sample of data. Forexample, multimodal distributions may arise in a finite mixture distribution (i.e., a distribution which is composed of componentdistributions), and we would like to identify the number of componentdistributions.
In this project a statistical testing procedure will be used toidentify the number of modes, and assess whether an apparent mode in adata set is real or spurious. This procedure is based on a bootstrapmethod which is a way of making statistical inferences withoutnecessarily making strong assumptions about the form of the underlyingdistribution. The project would consider applications of the bootstraptest for modes.
Efron, B. and Tibshirani, R.J. (1993). An Introduction to the Bootstrap.London: Chapman and Hall.
Everitt, B.S. and Hand, D.J. (1981). Finite Mixture Distributions.London: Chapman and Hall.
Titterington, D.M., Smith, A.F.M., and Makov, U.E. (1985).Statistical Analysis of Finite Mixture Distributions.Chichester: Wiley.
Statistical modelling of circular dataType: Individual Supervisor: Bruce Worton ([email protected]) Requirements: Linear Statistical Modelling (MAT-3-LSM) and Likelihood (MAT-3-Lik) Degree programme: Would be particularly suitable for a Mathematics and Statistics degree student.
Interest in developing statistical methods to analyse directional datadates back as far as Gauss. Such data include:
• wind directions• vanishing angles of homing pigeons --- measured in range 0,2 or 0,360• \item times of birth over the day in hours --- convert by multiplying by 360/24• times of death from a single cause over years
This project would study the theory and application of circular data.Topics would include: (i) basic descriptive directional statistics, (ii) common parametric models, (iii) basic inferenceproblems on the circle.
If time permits it may also be possible tostudy correlation and regression for directions.
Fisher, N.I. (1993). Statistical Analysis of Circular Data.
Fisher, N.I., Lewis, T. and Embleton, B J.J. (1987). Statistical Analysis of Spherical Data.
Mardia, K.V. and Jupp, P.E. (2000). Statistics of Directional Data.(2nd edition)