This is a list of indicative projects and supervisors for the MATH3602 module.

## Applied Mathematics

### Wave Trapping Around Islands

Islands are subject to tides, seismic waves as well as wind generated waves. If any of these are trapped, that is the energy of the wave goes continuously around the island without escaping then there are practical consequences in terms of flood prediction as well as the detection of signals of say earthquakes or the effects of distant storms such as hurricanes. The effect of the earth’s rotation is also important if the island is large. This project examines models of wave trapping and derives a condition that governs where it occurs dependant on the geometry of the island.

Supervisor: Prof Phil Dyke

### Natural Ventilation

The study of how to ventilate a room without recourse to air conditioning has practical application to theatres and libraries where all but the quietest noise has to be avoided. This project examines how classical fluid dynamics aids this practical application. You could be using complex analysis through mapping and solving partial differential equations in simple geometry but with mixed boundary conditions.

Supervisor Prof Phil Dyke

### Wind Driven Circulation

In the ocean the main gyres responsible for how the sea circulates are generated by the climatic winds. In this project you will derive the equations that govern this process from first principles. These equations will then be solved in order to understand why there are strong currents on the western side of every world ocean. This project will appeal to those who enjoy using idealized mathematics to understand how the ocean moves. There are some opportunities for using numerical techniques, but primarily this is a theoretical project.

Supervisor: Prof Phil Dyke

### The Numerical Solution of the p-Laplace Equation in a Geometry with Sharp Corners

Certain types of fluids do not respond in a linear way to external forces – they are called non-Newtonian fluids. A simple model of a non-Newtonian fluid which is widely used in industrial applications (e.g. food processing) is the so-called power-law model. In very slow flows, the governing equation turns out to be the p-Laplace equation (a generalisation of Laplace’s equation).

In this project, numerical solutions of the p-Laplace equation will be found for the flow in simple geometries and a geometry that includes a singularity arising from a sharp corner. The project will involve analytical work and a substantial amount of programming.

The contour plot below shows the variation, calculated numerically, of a quantity called the shear rate in a geometry that has two singular points – one at each of the sharp corners at the lower end of the white region. If you look closely you can see two red areas where the shear rate is ~20 (the units are 1/sec) whereas the shear rate is smaller everywhere else. A theoretical analysis indicates that the shear rates should be infinite at these locations. The singular behaviour of the solution at these points affects the convergence of numerical solutions and the project would look at ways to overcome this problem.

Supervisor: Dr Jason Hughes

### Perturbation Methods

Perturbation methods are used to investigate solutions to mathematical problems in which exact solutions can’t be found but in which there is a parameter whose size (usually very small) allows approximate solutions to be found. In this project the aim would be to look at a range of perturbation methods and the types of problems to which they can be applied.

The project would be mainly analytical although it could involve some numerical work.

Supervisor: Dr Jason Hughes

### Non-linear Waves

This project concerns deriving equations that govern the propagation of waves on water, then solving them. These equations have two well-known solutions, the solitary wave (or soliton) and cnoidal waves. Some of the mathematics is quite involved but the satisfaction comes in being able to describe known observable wave phenomena. There is some opportunity for using numerical techniques but this is primarily a theoretical project

Supervisor: Prof Phil Dyke

### Waves

Wave motion is found everywhere: sound waves, water waves, electromagnetic waves, traffic waves and Mexican waves are just some examples. The mathematical description of wave motion can be fairly straightforward, as with the oscillations of a guitar string for example, but as more physical effects are allowed for the theory becomes more complicated.

In this project the aim would be to look at some well-known mathematical equations that describe different aspects of wave motion, to find out how they are solved and to look at different types of solutions. Two examples are Burgers’ equation and the Korteweg de Vries (KdV) equation.

This project would be almost entirely analytical but could include illustrations using Maple. Take a look at the animation at the link below – it shows two solitons (possible solutions of the KdV equation) moving at different speeds, interacting, then emerging as though one had passed through the other.

http://www.math.uwaterloo.ca/~kglamb/course_animations/KdV_inter.gif

Supervisor: Dr David Graham

### Chebychev Polynomials

In this project the aim is to study the mathematical properties of Chebychev polynomials and their role in numerical approximation.

The concept of ‘good’ and ‘best’ numerical approximations and the Remes algorithm will be investigated and, if time permits, further applications of Chebychev polynomials could be considered.

This project is mainly analytical but would involve some programming (Maple would be fine).

Supervisor: Dr David Graham

### Particle methods for fluid dynamics

Particle-based numerical methods have been used extensively by Hollywood in modelling dramatic events such as floods or sinking ships. This project would investigate smoothed particle hydrodynamics (SPH) models or, possibly, discrete vortex models (DVM) in modelling relatively simple flows. More SPH info can be found here.

Supervisor: Dr David Graham

### Numerical methods for PDE’s

Partial differential equations describe how the three/four dimensional world works. Solving such equations is the key to understanding the behaviour of many physical, chemical, biological or financial systems. A large range of numerical methods is available to solve such problems (for example finite difference, finite element, finite volume, boundary element or pseudo-spectral methods to name a few). This project would investigate the use of one or more of these methods for solving a well-known PDE.

Supervisor: Dr David Graham

### Pricing of American Options

An option is the right (but not the obligation) to buy, at or before an agreed expiry date, a share at a ‘strike price’ (agreed now). An American option allows the option buyer to cash in the option at any time between now and expiry (for European options, the buyer must wait until expiry). Because of the flexibility, pricing such options is mathematically tricky. A project would investigate possible methods for pricing American options.

http://financial-dictionary.thefreedictionary.com/American+Option

Supervisor: Dr David Graham

### Multifractals in Finance

The standard mathematical model of random asset price variation is Geometric Brownian Motion – where the log of the asset price ‘return’ is normally distributed. However, this under-predicts the frequency of dramatic and large changes in the price. In real data, ‘fat-tails’ are observed, meaning that large deviations from mean behaviour are seen more often than in GBM – see here or here . A multifractal approach promises more realistic behaviour. This project would investigate possible uses of multifractal models in finance.

Supervisor: Dr David Graham

### Mathematical Finance

Many topics in mathematical finance can make good project topics. The key to understanding lies in the modelling of the random behaviour seen in prices of shares, oil, foreign exchange rates and the like. Projects in this area usually require solving either stochastic differential equations (SDE’s) or partial differential equations (PDE’s) analytically and/or numerically. A large variety of methods can be used. The numerical methods can be coded in Excel or Maple. Alternatively, for those wishing to develop programming skills, R or fortran enable industrial-scale computations to be carried out.

Supervisor: Dr David Graham

### Theory and Application of Evolutionary Algorithms

Evolutionary algorithms are iterative stochastic optimisation algorithms which achieve approximate solutions of a problem through use of naturally inspired ‘random’ operations such as crossover, mutation and natural selection. They act on a population of individuals, each one an approximation to a solution of the problem. Analysis of the algorithm is based upon analysis of how the population evolves. Classically they are used as optimisers for real-world problems or sets of parameters, but can also be used in such areas as algebra or combinatorics where some optimal solution is sought. Historically, evolutionary algorithms have been applied to ‘complicated’ problems where there either exists no deterministic algorithm or conventional deterministic algorithms are very slow, and have achieved sometimes startling results. This project is of the exploitation of one of the following avenues: a. Implementation and application of a simple evolutionary algorithm to a mathematical problem. In this case, interested students should have good programming skills (e.g., in MATLAB); b. A survey of a small amount of published research material on a particular evolutionary algorithm. For instance: historical foundations, application to industry and the various advantages and disadvantages of genetic algorithms at large (e.g. looping, information about the fitness landscape). The student should aim for a mathematical type of exposition.

References:

- Eiben, A., Smith, J., Introduction to Evolutionary Computing, Natural Computing Series, Springer (2003).
- Goldberg, D., Genetic Algorithms in Search, Optimisation and Machine Learning, Addison-Wesley (1989).
- Michalewicz, Z., Genetic Algorithms + Data Structures = Evolution Programs, Springer (1996).

Supervisor: Dr Matthew Craven

### Cryptography

Cryptography is an eminently applicable area of mathematics. The problems come in several different flavours and utilise distinct areas of mathematics, giving available projects in at least the following two areas.

a. Number-theoretic cryptography: Number-theoretic cryptosystems depend upon numerical base problems such as the discrete logarithm problem and integer factorisation problem. RSA is one example of such a cryptosystem. The aim of this project would be to conduct a survey of available number theoretic cryptographic schemes and the number theory behind them. You should then cover some attacks on one of the schemes. A discussion of sneaky attacks is expected, as well as a discussion of available mathematical techniques for breaking the applicable base problem. The student should aim to learn some in-depth number theory for this project;

b. Group-based cryptography: There are several available cryptosystems which rely upon group theoretic problems to protect the private key. Moreover, there are several distinct types of group over which the cryptosystems operate, and also some groups are known to have “difficult” base problems. The aim of this project is to briefly write about some distinct types of groups and associated proposed cryptographic schemes. Then a specific family of groups will be chosen and attacks upon the proposed schemes and base problems will be surveyed. A healthy command of algebra, and in particular, group theory, is assumed.

References:

- Koblitz, N., A Course in Number Theory and Cryptography, Graduate Texts in Mathematics 114, Springer (1994).
- Koblitz, N., Algebraic Aspects of Cryptography, Algorithms and Computation in Mathematics 3, Springer (1999).
- Menezes, A., Handbook of Applied Cryptography, downloadable from http://www.cacr.math.uwaterloo.ca/hac/ (1997).
- Schneier, B., Applied Cryptography, Wiley 1995.

Supervisor: Dr Matthew Craven

### Game theory

Game theory, popularised by von Neumann and Morgenstern in 1944, is used to study a large variety of real-life situations concerning strategic decision making. It models conflict and cooperative strategies in business, economics, politics and cybersecurity, to name but a few, and may be used to predict short-term and long-term behaviours in a competitive environment. Simple games may be simulated by hand but larger games are often run as computer simulation. In particular, Nash equilibria may be located for a player, meaning that (with some conditions) the player is playing the best possible strategy for a given game. In large scale simulations, the evolution of societies through agents may be performed, making the topic particularly timely.

References:

- Myerson, R. B., Game Theory: Analysis of Conflict, Harvard University Press (1991).
- Nash, J., Equilibrium Points in n-Person Games, PNAS 36 (1) (1950), 48-49.
- Axelrod, R., The Evolution of Cooperation (Revised ed.), Perseus Books Group (2006).

Supervisor: Dr Matthew Craven

## Pure Mathematics

### The Inverted Pendulum

Why is it that a solid pendulum can be made to stand on its end when it’s other end is made to oscillate? This project will investigate the underlying mathematics behind this very strange phenomena. You could also try modelling this in Maple or even building your own control system to do it.

Supervisor: Dr Colin Christopher

### Mondromy of Abelian Integrals

This project is related to the change in an complex integral as its defining parameters change. You will need to learn some theory about Riemann surfaces and integrals on them, but I would like this project to have more of a research nature with plenty of experiments (and pictures) in Maple to find out what happens in some particular systems.

Supervisor: Dr Colin Christopher

### Hilbert’s 10th Problem

How complicated can it be to find all the integer solutions to a set of polynomial equations? In fact, as complicated as you like! Given any output from a computer program (a list of all prime numbers, for example), you can find a set of equations whose only solutions occur when one of the variables takes any value in the list but no others. This project will look at the mathematics behind trying to make this statement precise. Along the way, it might well morph into some other aspect of mathematical logic (Goedel’s incompleteness theorem, the halting problem, etc.) as it takes your interest or not.

Supervisor: Dr Colin Christopher

### Algebraic Curves

We know that x

^{2}+y^{2}=1 is the equation of a circle, but what happens when the degree of the equation is greater than two? In this project we investigate some of the geometry of algebraic curves – that is, solutions of polynomial equations P(x,y)=0. In particular, when P is a cubic polynomial, the resulting curve is called an elliptic curve and has some interesting properties (in the complex domain it looks like a doughnut), but higher degree curves are even more interesting.Supervisor: Dr Colin Christopher

### Symmetry Groups and Space

This project will look at the interplay between group theory and symmetry in different spaces. It is deliberately left open for you to explore the areas that you like best – but you will be expected to understand the mathematics behind the pretty pictures too!

Supervisor: Dr Colin Christopher

### Polynomial invariants of graphs and matroids

There are some simple and amazing polynomial invariants of graphs, such as the chromatic polynomial, which are special cases of the Tutte-Grothendieck polynomial invariants of matroids. References:

- Harary, F 1969
*Graph Theory*Addison-Wesley. - Peter Cameron Polynomial aspects of codes, matroids and permutation groups
- Welsh, DJA 1976
*Matroid Theory*Academic Press, London. - Bollobás, B 1998
*Modern Graph Theory*Springer- Verlag, New York.

Supervisor: Professor Stephen Huggett

- Harary, F 1969
### Braid Groups

There are many intriguing invariants of knots, some only discovered recently. Some of these come from studying the related braids, which lead to extremely interesting groups, relatively simple to define but with open questions. References:

- Wikipedia Braid group
- Epple, M 1998
*Orbits of Asteroids, a Braid, and the First Link Invariant*Mathematical Intelligencer, volume 20, number 1, pages 45-52. - Prasolov, VV and Sossinsky, AB 1997
*Knots, Links, Braids, and 3-Manifolds*American Mathematical Society.

Supervisor: Professor Stephen Huggett

### Aperiodic tessellations

It was discovered only fairly recently that there is a pair of tiles with which the entire plane can only be tessellated in a non-periodic way. These tiles are very simple, but have remarkably surprising and deep mathematical properties, as well as giving beautiful tessellations.

References:

- Austin, D Penrose tiles
- Penrose, R 1989
*Tilings and quasicrystals: a nonlocal growth problem?*pages 53-80 in Introduction to the Mathematics of Quasicrystals, edited by Marko Jaric, Academic Press.

Supervisor: Professor Stephen Huggett

### Cartographic Projections

Cartographers use a wide variety of projections from the sphere to the plane, which have a very rich structure studied in

*differential geometry*. Particularly interesting examples are*conformal maps*, in which angles are preserved but not lengths. References:- Feeman, TG 2002
*Portraits of the Earth: A Mathematician Looks at Maps*American Mathematical Society. - McCleary, J 1994
*Geometry from a Differentiable Viewpoint Cambridge University Press.*

Supervisor: Professor Stephen Huggett

- Feeman, TG 2002
### Riemann surfaces

Instead of thinking of complex numbers as lying on a plane, it is much better use

*stereographic projection*to put them on a sphere, called the Riemann sphere. The Möbius group acts on this sphere, giving a fascinating set of transformations, crucially important in many branches of mathematics. A natural development is to consider multi-valued complex functions, such as the logarithm, leading to the amazing concept of a Riemann surface. The more advanced ideas here are at the level of a masters project.References:

- Wikipedia Riemann sphere
- Needham, T 1997
*Visual Complex Analysis*Oxford University Press. - Jones, G and Singerman, D 1987
*Complex Functions*Cambridge University Press.

Supervisor: Professor Stephen Huggett

### Relativity

The spacetime described in Einstein’s special theory of relativity is a very beautiful geometry. In particular, there is a deep relationship between the Lorentz group and the Möbius group, which is an important hint that complex geometry is somehow fundamental. One can see more of this in the projective geometry of twistor space, which is related to spacetime by the Klein correspondence, a piece of classical algebraic geometry itself worthy of independent study. The more advanced ideas here are at the level of a masters project.

References:

- Rindler, W 1991
*Introduction to Special Relativity*Oxford University Press. - Penrose, R 1959
*The apparent shape of a relativistically moving sphere*Proc. Camb. Phil. Soc. volume 55, pages 137-139. - Huggett, S and Keast, S
*The Penrose Transform*unpublished

Supervisor: Professor Stephen Huggett

- Rindler, W 1991
### Studying varieties with computer algebra

A variety is the solution set of a system of polynomial equations, usually in several unknowns. For instance, some famous plane curves are described by one polynomial equation in x and y. A possible project is to discuss how computer algebra, e.g., Maple, can be applied to work with varieties. Interesting operations are, e.g., taking the union of two varieties, projecting a variety onto a coordinate subspace, finding the subvarieties of a variety which are not unions of two proper subvarieties, etc. (The picture shows a conic section.)

Reference:

- D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms, Springer, 3rd ed. 2007.

Supervisor: Dr Daniel Robertz

## Statistics

### Bayesian Methods for Analysing Historic Data

This project will concern (Bayesian) models for the survival time of historic populations. Such models can provide interesting historical insights. For example, we can understand how survival times have increased across the centuries and how they differ between sexes. In this project will will develop survival analysis methodology for populations for which good data are available, such as popes, monarchs and presidents. Results will be compared with actuarial tables, wherever possible.

Supervisor: Dr Julian Stander### Dynamic Social Media Information Extraction

This project concerns the extraction of information from Social Media such as Facebook and Twitter. It will develop methodology to provide an understanding of how sentiments expressed on social media change over time. A Shiny app that will, for example, provide a user who inputs a topic and a time frame with a detailed understanding of sentiments about that topic will be developed.Supervisor: Dr Julian Stander### Regression modelling of competing risks

Motivation:

Survival analysis takes into account of whether an event occurs as well as the time to such an event. Right censoring can be due to the occurrence of another event, which is called competing risks. An example of competing risk is disease progression due to a local relapse or a distant relapse. Analysis assuming censoring due to the other event can introduce bias.Methods:

In competing risks models, the probability of a local relapse is not only a hazard function of local relapses but also a hazard function of distant relapses.Aims:

This Bsc project will review the methodology for regression modelling of competing risks and develops an application of the method, and involves programming in R or other statistical software.References:

- Fine J.P., Gray R.J.(1999) A proportional hazards model for the subdistribution of a competing risk. Journal of the American Statistical Association, 94(446): 496-509.
- Satagopan J.M. et al (2004) A note on competing risks in survival analysis. British Journal of Cancer, 91, 1229-1235.

Supervisor: Dr Yinghui Wei

### Network meta-analysis of randomized controlled trials

Motivation:

A number of treatments may be available to patients with the same health condition. Policy-maker, clinicians and patients may want to know what the optimal treatment is. This requires comparing the benefits and harms of all available treatments. However, not all the direct comparisons are available from randomized trials. This makes the multiple comparisons challenging, so statistical advance is well motivated.Methods:

Network meta-analysis is a statistical method to compare multiple treatments simultaneously within a single framework. It can be implemented within either a frequentist or a Bayesian framework.Aims:

This Bsc project will review the network meta-analysis methodology and develop an application of the method, and involves programming in WinBUGS and/or R.References:

- Lu, G., Ades, A.E. (2006) Assessing evidence consistency in mixed treatment comparisons. Journal of the American Statistical Association, 101, 447-459.
- Ciprinani et. al. (2013) Conceptual and technical challenges in network meta-analysis. Annals of Internal Medicine, 159(2):130-197.

Supervisor: Dr Yinghui Wei

## Theoretical Physics

### Divergent Series

This project is for a student who finds the following result shocking but intriguing:

1 + 2 + 3 + . . . = -1/12

This looks wrong in so many ways what with the sum of positive terms being negative and an infinite sum being finite. The aim of the project is to understand why this and other similar results are not completely crazy and, in fact, hint at a deeper role for divergent series in various areas of mathematics and its applications. The background for this project is covered in MATH2405 (Real and Complex Analysis).

Literature: G.H. Hardy, Divergent Series. Clarendon Press, Oxford. 1949 (a free ebook version is on the web)

Supervisor: Prof David McMullan

### Scaling

Scaling is an interesting and powerful approach to mathematics. There are many introductory applications ranging from predicting how the period of a pendulum depends on its length to why almost all animals, large and small, can jump about the same height. More sophisticated topics include fractional scaling laws in physics and biology and fractal dimensions. This project would start with dimensional analysis and can go in various directions.

Literature: See, e.g., G.I. Barenblatt “Scaling”, partly available online at: google books

Supervisor: Dr Martin Lavelle

### Classical Mechanics as a Field Theory

In 2013, a team of university physicists, speaking on behalf of the German Physical Society, criticised the adoption of the Karlsruhe Physics Course (KPK) at some grammar schools in South West Germany. The incriminated course suggests (among other things) to abandon the Newtonian mechanics of point particles in favour of a continuous formulation employing fields, similar to fluid dynamics. In this project you are supposed to critically assess the feasibility of this approach from a theoretical physics point of view.

Supervisor: Dr Tom Heinzl

### Charge Dynamics in Electromagnetic Fields

In strong electromagnetic fields the Lorentz force experienced by charges can become so large that the charge motion becomes relativistic. Newton’s equation of motion, F = ma, thus has to be modified as to include relativistic effects. In other words, one has to use relativistic mechanics in order to solve for the charge dynamics. In this project you are to determine the motion (velocities, trajectories) of charged particles in simple electromagnetic field configurations.

Supervisor: Dr Tom Heinzl

### Hybrid mesons from a potential model

The elementary particle called quarks combine together to form mesons. For heavy quarks the masses of the mesons can be obtained by numerically solving Schrodinger’s equation (from quantum mechanics) with a suitable potential (see http://de.arxiv.org/abs/0805.2704 for example). The potential can be modified to predict the masses of hybrid mesons. The aim of the project is write a code to solve Schrodinger’s equation to compute the masses of heavy mesons and hybrid mesons.

Supervisor: Dr Craig McNeile

### Avalanches

An avalanche is an example of a system displaying “self-organisation”. On a clear day, the great mass of snow on a mountain may appear to be in equilibrium, but the slightest change such as a gentle breeze or the drop of a snowflake can trigger the release a dramatic amount of energy as the system “self-organises” into a new (quasi-) equilibrium. Such self-organisation can be found throughout nature, from the outbreak of natural disasters such as earthquakes, to the spread of internet memes. This project will involve implementation of numerical algorithms whose results will be interpreted using analytic calculation. A particular focus will be understanding the criticality of the 1D BTW sandpile model [1,2].

[1] P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. A 38 364-374 (1988)

[2] K. Christensen and N. R. Moloney, Complexity and Criticality, Imperial College Press (2005)Supervisor: Dr Ben King

### Fireflies

Fireflies provide an example from nature of the phenomenon of synchronisation. When one firefly flashes near another, it can cause another to match the frequency of its flashing. Using analytical and/or numerical methods, the project will investigate models for synchronising oscillators, covering topics such as phase-locking and entrainment.

Supervisor: Dr Ben King

### Shock waves: Burgers’ Equation

Shock waves occur throughout nature: in fluids, gases and plasmas. A classic example is the “sonic boom” produced around an aircraft when it exceeds the speed of sound. This project will involve the study of Burgers’ equation, which is a fundamental partial differential equation that demonstrates shock wave behaviour. Using analytical and/or numerical methods, the mathematics and nature of shocks will be explored in detail.

Supervisor: Dr Ben King

### The Abelian Sandpile Model

The Abelian Sandpile Model is a cellular automaton whose discrete dynamics reaches an out-of-equilibrium steady state resembling avalanches in piles of sand. This system shows, in an apparently unpredictable way, bursts of activity such as avalanches. This kind of model can be eventually driven into an out-of-equilibrium steady state. We propose a numerical study of the “recurrent configurations”, i.e. a set of patterns recreated periodically after each avalanche.

Supervisor: Dr Antonio Rago