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Possible topics for Mathematics and Statistics honours thesis

The topics listed below are just some of the possible topics from which students can choose. Most supervisors are happy to discuss other possible topics with you personally. If you have a particular area of research that you wish to pursue and you do not know who to discuss this with, then contact the Honours Coordinator who can arrange a meeting with a potential supervisor.

Mathematics honours thesis projects

Symbolic Dynamics and Formal Languages

Supervisor: Dr John Banks

The branch of dynamical systems theory known as symbolic dynamics interacts in many ways with the theory of formal languages, a theory of fundamental importance in many areas of computer science. This project would explore some of the many connections between the two theories. Some examples of specific areas that might be investigated include the connection between sofic shifts and regular languages, the application of the theory of context free languages to symbolic dynamics, the application of symbolic dynamics to coding problems such as the coding of data for hard disk storage.

Entropy measures for nonlinear partial differential equations

Supervisor: Professor Philip Broadbridge

Just as some dynamical systems have conservation laws, some have monotonicity laws such as dissipation of energy or increase of entropy. In this project, entropy laws will be found systematically for dynamical systems that are governed by practical nonlinear partial differential equations. These will be used to prove stability of states of maximum entropy and to prove stability of solutions that are invariant under some symmetry. A connection will be made between smoothing properties or positivity properties and decrease of Shannon information. Applications might include heat and mass transfer, metal surface evolution and quantum mechanics.

Quantum mechanics of a scalar field in an accelerating universe

Supervisor: Professor Philip Broadbridge

Since the late 1990s, it has been observed that the universe is not only expanding but accelerating, and that most of the energy is in the form of “dark energy” not associated with known types of waves or particles. If we allow a scalar field to be minimally coupled in an invariant way to the simplest consistent expanding accelerating universe, then the field has energy eigenstates that become unstable at discrete times. These non-oscillatory states have a classical analogue in an oscillating spring on an accelerating platform – eventually the spring ceases to oscillate. The quantized non-oscillatory states have a continuous spectrum (like that of an ionized atom), unlike the quantum particle states that  have a well defined particle number and discrete spectrum (like the bound states of a neutral atom).  This project will predict the partitioning of the scalar field energy into the particle states and the continuous “jelly” states, and will estimate how this non-particulate energy increases in time.

Applying an exactly solvable nonlinear convection-diffusion equation to soil-water flow

Supervisor: Professor Philip Broadbridge

Recently, a formal series solution in terms of Kummer functions, has been constructed for an integrable nonlinear convection-diffusion equation. Up to 180 series coefficients have been calculated explicitly. The series converges in practical terms but its radius of convergence  has not been estimated in any way. This solution leads to an exact infiltration series, a power series in square root time, for the total depth of surface water having infiltrated a soil. The coefficients of this series have been in dispute for some time. These exact results may help to explain pre-existing partial results that seem to be in mutual contradiction. At a practical level, the integrable equation seems to well model water infiltration in soil but the goodness of fit with experimental data needs to be quantified and compared with that of more popular phenomenological models.

Harry Dym equation and its simpler relative

Supervisor: Professor Philip Broadbridge

The Harry Dym equation is a well known integrable nonlinear wave equation. There is another less well known nonlinear  water wave equation that appears to be very similar but it can be solved more simply because it becomes linear after a change of variables. This will allow an investigation of how special is the soliton behaviour of the Harry-Dym equation. The Harry Dym equation has a multi-soliton solution for which the finite number of peaks remains constant. What does the other similar integrable equation predict in similar circumstances? Is there a trace of soliton behaviour? If not, how is the soliton energy dissipated?

Graph theory and Number theory

Supervisor: Associate Professor Grant Cairns

I have a range of possible thesis topics in Combinatorial Game Theory and Number Theory. Some are classical, some are open problems. For further information, come and see me to discuss topics that might interest you.

Topological dynamics of the plane

Supervisor: Associate Professor Grant Cairns

Mort Brown discovered an unusual periodic homeomorphism of the plane that has sparked interest in several fields. This project investigates this and related examples.

General algebra

Supervisor: Professor Brian Davey

The 'classical' kinds of algebras include groups, rings and vector spaces. I am mainly interested in more 'modern' kinds of algebras, like lattices and Boolean algebras, which arise in logic and theoretical computer science. Possible topics in general algebra include:

1. Primal and quasi-primal algebras. Primal and quasi-primal algebras are important generalisations of the two-element Boolean algebra, which is familiar from discrete mathematics, and arise in many parts of general algebra. The aim of this project will be to find simple and effective tests for quasi-primality based on some old and some recent tests for primality.
2. Graph algebras. Every graph (in the sense of discrete mathematics) can be converted in a natural way into an algebra with a single binary operation. This project will review the many applications that graph algebras have found and will include some recent papers illustrating their use on the fascinating interface between complexity theory and algebra.
3. NU algebras. Algebras with a near-unanimity term are an important generalisation of lattices. This project will review the various properties that these algebras possess and study some very recent papers, including the proof that it is decidable whether or not a finite algebra actually has a near-unanimity term.

Algebras of programs

Supervisor: Dr Marcel Jackson

There are a number of logical and algebraic approaches to analysing the correctness of computer programs. This project would involve an exposition of some of these approaches and their interrelationships, and lead to an investigation into the relationship between these and algebras of functions and relations. This project is best suited to students with some extra background in logic, or with a strong background in algebra with a computer science interest. No knowledge of programming is necessary, since the focus is on algorithms, rather than actual programming languages.

Algebras of functions and relations

Supervisor: Dr Marcel Jackson

This topic concerns the problem of classifying various algebras of functions and binary relations.  For example: the Cayley representation for groups shows that groups are precisely the algebras arising as systems of permutations of a set under the operations of composition and inverse. What about algebras of binary relations under the operation of composition and intersection? There are a plethora of such questions, many motivated from the kind of issue considered in the previous topic "Algebras of programs".

Computational complexity in algebra

Supervisor: Dr Marcel Jackson

How hard is it to decide if two arbitrary finite graphs are isomorphic? What about two finite groups? This project would involve a study of computational questions such as these: the focus would be on general algebraic structures, and connections between computational problems and logical properties of algebras. This project could be directed to computational problems on finite structures (a popular current trend) or for infinite structures (more classical, but still active and important).

Simon's Theorem (automata and semigroups)

Supervisor: Dr Marcel Jackson

Finite state automata (as encountered in, say MAT1DM) and algebraic structures known as "semigroups" have a close relationship: finite semigroups all arise as the algebra of input transitions of a FSA under composition, and every finite semigroup gives rise to a finite automata for which it is the algebra of transitions.  Simon's Theorem relates languages recognised by machines that read a fixed multiple number of bits at a time to an elementary structural property of finite semigroups. It has a number of different proofs.  This topic is suitable for a student with an algebra background.

Note: Dr Marcel Jackson will also consider other proposals in semigroup theory or universal algebra.

Constructions with straightedge and compass

Supervisor: Dr Yuri Nikolayevsky

This is a classical geometric topic dating back to antiquity. The thesis will contain two main parts: an overview of some well known and less known constructions (examples: construct a triangle given three altitudes; construct a regular pentagon) and the proof of non-constructibility for some classical problems (like trisection of angle and duplication of cube). While the first part only requires some general plane geometry knowledge, in the second part we will need more sophisticated algebraic structures (so called quadratic field extensions).

The geometry of a Lie algebra

Supervisor: Dr Yuri Nikolayevsky

In this project, starting with the definition and general properties of Lie algebras, we will study the curvature properties of Lie algebras. In the first part of the project, we will classify low-dimensional Lie algebras and compute their curvatures. Depending on the pace of the project, we might then work on a particular case of one long-standing conjecture in Riemannian geometry. No heavy machinery will be used, but the good understanding of the material of MAT3AC (metric and curvature) and MAT2LAL (symmetric and skew-symmetric matrices, eigenvalues and eigenspaces) is desirable. The project may also involve some calculations using computer algebra packages (Maple or Mathematica).

Conformal transformations and conformal geometry

Supervisor: Dr Yuri Nikolayevsky

Conformal transformations are the ones which preserve angles. We will study conformal transformations and conformal geometry from three different viewpoints. First, we will look at the complex analytic functions as conformal mappings (extending the material taught in MAT3CZ). Secondly, we will discuss the conformal transformations in a more general geometric context (this includes the stereographic projection from cartography and the famous Liouville's theorem). Thirdly, we will study the conformal property from the viewpoint of differential geometry (this part of the project will require some material of MAT3AC (metric and curvature) and a little bit of differential equations).

Differential equations for orthogonal polynomials

Supervisor: Dr Chris Ormerod

Orthogonal polynomial systems are sequences of polynomials that are orthonormal with respect to some linear form. The study of non-standard orthogonal polynomials have many applications to classically integrable systems such as the Schrödinger wave equation.
The integrable systems that arise are a ubiquitous part of various quantum mechanical models. This project may be directed towards any number of integrable aspects of orthogonal polynomial systems.

The Inverse Problem in the Calculus of Variations

Supervisor: Professor Geoff Prince

When are the solutions of a system of second order ordinary differential equations (ode’s)



the solution of an Euler-Lagrange equation



This is a famous question in the calculus of variations, important in differential geometry, mechanics and relativity. This project involves an exploration of this inverse problem for some important classes of differential equations using differential geometry and computer algebra.

Numerical integration by arcs

Supervisor: Professor Geoff Prince

Numerical integration of differential equations uses approximation by straight line segments to construct solutions. This project explores what seems to be a completely new idea: approximation by circular arcs. It involves development and implementation of analogues of Euler's method and Runge-Kutta methods and comparison with the traditional algorithms.

Note: Professor Geoff Prince will also consider proposals in differential geometry, differential equations, relativity and mechanics.

To discover and explain features of integrable systems

Supervisor: Dr Peter van der Kamp

Amongst all differential/difference equations, or mappings/correspondences, the integrable ones are the nice ones. They always carry special structures. Examples are a Lax-pair, infinitely many symmetries, sufficiently many integrals of motion, and polynomial growth (as opposed to exponential growth) of degrees of iterates and multivaluedness. Often one is able to calculate the first few terms of an infinite series from which one can guess the general form. At this point you want to write down a proof! What does the identity, which makes it happen, look like?

Statistics honours thesis projects

Competing Risks

Supervisor: Dr Siew Pang Chan

Cox regression is the most established statistical method for analysing time-to-event data, where the objective is to ascertain how identified factors are associated with an event of interest (eg, time to hospitalisation, recovery from an illness, or death). It takes into account the presence of incomplete records (say premature withdrawal, late recruitment to the study), which hinder the observation of the event of interest.  This topic attempts to analyse time-to-event data with an alternative strategy known as competing risk. In general, a competing risk situation arises when an individual can experience more than one type of event and the occurrence of one of such events hinders the occurrence of other types of events. For example, a group of patients with a specific condition are followed up in order to observe an event of interest, say hospitalization. If by the end of the study, each patient is either hospitalized, alive or remain free from disease, then the conventional statistical methods may apply. But if some patients might have died from other causes (e.g., accidents), then a competing risk situation has arisen because death has hindered the occurrence of hospitalization.  Hence, the patients in question should not be considered as censored because their data are not incomplete.  The R and Stata packages are recommended for this topic.

Note: Please contact Dr Siew Pang Chan if you would like to propose other applied topics, such as data mining, medical decision analysis, longitudinal data analysis, structural-equation model and reliability engineering.

Projects in Applied Statistics

Supervisor: Dr David Farchione

Note: Please contact Dr David Farchione to discuss possible projects in applied statistics.

Confidence intervals for the difference of two normal means without assuming equal population variances

Supervisor: Associate Professor Paul Kabaila

Suppose that X₁;:::;X_m and Y₁;:::;Y_n are independent random variables. Also suppose that X₁;:::;X_m are independent and identically N(¹₁;¾₁²) distributed and Y₁;:::;Y_n are independent and identically N(¹₁;¾₁²) distributed. We do not require the population variances ¾₁² and ¾₂² to be equal. A classic problem in statistics, called the Behrens-Fisher problem, is how one should carry out inference about '1¡ '2. The purpose of the project is to compare several confidence intervals for '1¡ '2 that have coverage probability that never falls below a specified value '1¡ ®. Included in this comparison is the method described in section 3 of Kabaila (2005).

Reference

• Kabaila, P. (2005)  Assessment of a preliminary F-test solution to the Behrens_Fisher problem. Communications in Statistics – Theory and Methods, 34, 291 – 302.

Integrated likelihood methods

Supervisor: Associate Professor Paul Kabaila

Suppose that the distribution of the data is determined by the parameter vector (µ;¸), where µ is the scalar parameter of interest and the vector parameter ¸ is of no direct interest to us. The presence of the parameter ¸ (called a nuisance parameter) causes difficulties for statistical inference about  µ.  One method of carrying out this inference is to use a weighted integration, with respect to ¸ , of the likelihood function. This is called an integrated likelihood method. An introduction to such methods is provided by Berger et al (1999). The purpose of the project is to analyse the properties of such methods by considering some novel integrated likelihoods.

Reference

• Berger, J.O., Liseo, B. & Wolpert, R.L. (1999) Integrated likelihood methods for eliminating nuisance parameters. Statistical Science, 14, 1 – 28.

Stein estimation

Supervisor: Associate Professor Paul Kabaila

Suppose that the random vector X has a multivariate normal distribution with mean θ and, for simplicity, covariance the identity matrix. Also suppose that X has dimension 3 or greater and that the parameter of interest is θ. The usual estimator of  θ is X. However, in 1961, James and Stein found an estimator that is, according to a particular plausible criterion, better than X. This is a very surprising result that is sometimes referred to as Stein's Paradox (see section 10.7 of Casella & Berger, 1990). Furthermore, one can construct a confidence set centred at this estimator that has both smaller expected volume and larger coverage probability than the usual confidence set, which is centred at X (see e.g. Casella & Hwang, 1983). Casella & Hwang (1987) state that these confidence sets provide improved analysis of one-way ANOVA data. The purpose of this project is two-fold:

1. After reading over see section 10.7 of Casella & Berger (1990), to provide some of the details of the derivations of Casella & Hwang (1983).
2. To examine the extent to which the confidence sets of Casella & Hwang (1983) can be used to provide an improvement of the analysis of one-way ANOVA data.

References

• Casella, G. & Berger, R.L. (1990) Statistical Inference. Wadsworth.
• Casella, G. & Hwang, J.T. (1983) Empirical Bayes confidence sets for the mean of a multivariate normal distribution. Journal of the American Statistical Association, 78, 688-698.
• Casella, G. & Hwang, J.T. (1987) Employing vague prior information in the construction of confidence sets. Journal of Multivariate Analysis, 21, 79-104.

WKS sampling reconstructions

Supervisor: Dr Andriy Olenko

WKS sampling is a process of representing continuous signals by sequences of numbers (called samples). A classical approach to this problem is provided by Shannon’s sampling theorem which states that if a signal is band-limited, then it is uniquely determined by its sampled values. This result is one of the basic tools in signal processing. In direct numerical implementations we consider the truncated variant of Shannon’s sampling theorem. Restoring continuous signals with given accuracy from discrete samples or assessing the information lost in the sampling process are fundamental problems.  This project will analyse truncation errors and the information lost for deterministic and stochastic signals.  Applications in signal processing and information compression will be considered.

Wavelets for stochastic processes

Supervisor: Dr Andriy Olenko

In various statistical, data compression, signal processing applications and simulation, it could be used to convert the problem of analysing a continuous-time random process to that of analysing a random sequence, which is much simpler. Multiresolution analysis provides an efficient framework for the decomposition of random processes. This approach is widely used in statistics to estimate a curve given observations of the curve plus some noise. Various extensions of the standard statistical methodology were proposed recently. These include curve estimation in the presence of correlated noise. For these purposes the wavelet based expansions have numerous advantages over Fourier series, and often lead to stable computations. However, in many cases numerical simulation results need to be confirmed by theoretical analysis. Recently, a considerable attention was given to the properties of the wavelet transform and of the wavelet orthonormal series representation of random processes. This project will analyse uniform convergence of wavelet expansions for Gaussian random processes and fields. Applications in signal processing and information compression will be considered.

Clustering large financial data

Supervisor: Dr Andriy Olenko

There is a growing need for a more automated system of partitioning data sets into groups, or clusters. Clustering techniques can be used to discover natural groups in data sets and to identify abstract structures that might reside there, without having any background knowledge of the characteristics of the data. Clustering has been used in a variety of areas, including computer vision, data mining, bioinformatics (gene expression analysis), and information retrieval, to name just a few. This project focuses on a few of the most important clustering algorithms.  Applications in stocks clustering with actual data will be considered.

Note: Please see Dr Andriy Olenko if you would like to propose other areas in which to conduct a project. Dr Andriy Olenko would agree to supervise a project in such areas including spatial statistics, limit theorems, actuarial mathematics and approximation theory.   

Using area under the curve to analyse repeated measures data

Supervisor: Dr Luke Prendergast

In many areas of research it is useful to measure individuals at multiple time points to analyse the effect of time on a variable of interest.  For example, suppose that a new treatment is available that is specifically designed to assist in lowering blood pressure for hypertensive patients.  To assess the effectiveness of this new treatment, scientists recruit a number of patients and place them on either the new treatment or an existing treatment.  Patients are then monitored for 24 hours and have their blood pressure recorded every hour.  The data is then used to compare the effectiveness of the treatments.  The analysis of data of this type is not so simple and it is common in many areas of research to focus on ‘area under curve’ (AUC) summary measures.  The data for each patient can be displayed as a line plot with time on the horizontal axis and the variable of interest on the vertical axis.  The AUC for each patient is then simply the total area under this line and can be used as a single summary measure in place of the multiple measures recorded for each patient.  For the example described above, the average AUC for each group can then be used to compare the effectiveness of the treatments.  Whilst such an approach is quite simple and easy to understand, important information specifically related to the variable time is lost.  This project will discuss the limitations of such an approach by looking at both real and simulated data sets.

Fitting linear mixed effects models using R

Supervisor: Dr Luke Prendergast

Early R functionality that allowed the fitting of linear mixed effect (LME) models was designed for nested random effects.  However, crossed random effects arise commonly in practice and newer R functionality allows for the fitting of such models.  A drawback of using this functionality is that p-values associated with the testing of model parameters are not provided in the summary output.  This is not simply an oversight and there are good reasons as to why they are not given.  Douglas Bates, the author of the R functionality that allows for the fitting of models with crossed random effects, provides a detailed discussion as to why this is the case and recommends using a Markov Chain Monte Carlo approach to assess the model parameters.  This project will discuss the fitting of LME’s in R with crossed random effects and consider the effectiveness of this Monte Carlo approach via simulation.

The evidence in non-central chi-squared statistics and applications to goodness-of-fit

Supervisor: Professor Robert Staudte

For details please speak to Professor Robert Staudte.

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