# Proposals for Research Topics

The academic staff listed below have indicated their willingness to supervise thesis research. In some cases specific topics are listed, while in other cases a general area of interest is indicated.

Prospective students should feel free to approach any of these staff members to discuss the possibility of alternative topics to those listed here, or indeed any other staff member to discuss possible thesis topics. Further ideas for possible research topics may also be obtained from the Graduate Studies web page.

## Topics and Areas of Interest

### Assoc Prof R.K. Beatson

Multivariate Approximation Theory of an applied variety. Particularly interested in the mathematics and numerical analysis underlying such methods, with a view to a rich variety of applications such as Computer Aided Geometric Design, Image Processing, Natural Resource Modelling and Neural Nets.

### Prof D.S. Bridges

#### Constructive foundations of manifolds and Riemannian geometry

Constructive mathematics is the term applied to mathematics that is based on intuitionistic Zermelo–Fraenkel set theory and uses only intuitionistic logic. This logic was formulated by Heyting in 1930, and captures the essence of an algorithmic approach to mathematical proof. One outstanding gap in the constructive research literature is the subject of differential manifolds and Riemannian geometry; no–one has yet produced a reasonable constructive notion of "differential manifold", let alone tackled the really hard problem of developing the theory, through tangent bundles, vector fields, and Lie groups, towards Riemannian geometry. Since Einstein’s General Theory of Relativity is based on classical Riemannian geometry, a constructive development of the latter is essential before we can hope to provide a constructive mathematical approach to gravitation and cosmology.

The aim of this PhD project is first to produce a constructively useful definition of "differential manifold", and then to develop, constructively, the fundamentals of Riemannian geometry. Ideally, we will produce conditions under which one can construct geodesics on such a manifold; this is likely to be very difficult, since the classical existence proof uses general results about continuous functions that are not applicable constructively and do not even give any clues about how to develop a constructive existence proof.

#### Constructive operator theory

Various topics in the theory of operators and operator algebras on a Hilbert space, investigated constructively.

#### Apartness spaces

The notion of an apartness space seems to bear great promise for a powerful constructive development of general topology. There are several aspects of the theory of apartness spaces that would make good doctoral research, all of which would be a part of a project involving co–workers in Germany, Japan, and Romania.

### Dr J.A. Brown

#### Development of efficient survey designs for environmental monitoring

This project involves looking at optimal survey designs for monitoring changes in animal population abundance, in particular, vertebrates pests species. The project may involve some field work.

#### Incorporating information on spatial pattern into ecological models

Statistics on spatial patterns will be used to model optimal network designs for environmental monitoring. Aspects of adaptive sampling will be used to develop optimal designs.

#### Monitoring of spatial and temporal variation in possum populations

The feasibility of using existing possum–monitoring data for estimating population density surfaces which allow for spatial and temporal variation in abundance will be investigated.

### Dr J. Hannah

#### Mathematics education (Masters or PhD):

- Visualization and the teaching and learning of mathematics.
- Technology in mathematics education.

Background: For all of these topics you should know something about research methods in education (for example, from EDUC320).

#### History of mathematics (Masters):

- I am happy to discuss any proposal which takes advantage of a student’s particular background or interests.

I’m also willing to consider proposals about Masters in algebra (ring theory or linear algebra).

### Dr M.S. Hickman

#### Geometric Theory of Differential Equations

Differential equations may be viewed as a variety (i.e. a surface) in a high dimensional space called the jet bundle. The advantage of this approach is that it allows one to use geometric insights to examine the properties of individual solutions and to investigation connections between different sets of differential equations. The computations that this field requires have only become viable with the development over the last decade of efficient algebraic computing engines like MAPLE and MATHEMATICA. Research in this area that I am prepared to supervise include symmetry investigations of specific equations or classes of equations; differential invariants, their computation and application to problems which include the computer recognition of objects; extensions to differential-difference equations and to the development of efficient MAPLE code to implement the various types of computations that this field requires.

### Dr Dominic Lee

Topics in Computational Statistics and Bayesian Statistics, including applications in functional genomics and gene expression analysis, medical/biological/engineering model verification, and signal and image processing.

### Dr C. Montelle

#### History of Mathematics

- Historical Episode: the historical and mathematical analysis of a significant episode in the history of mathematics.
- Historical Work: the textual and mathematical analysis of a historical work, either in translation or original language (English, Cuneiform, Greek, Latin, Sanskrit, Arabic, French).
- Historical Theme: the study of various mathematical themes from the history of mathematics, including (but by no means limited to) mathematical notation and symbolic reasoning, use of diagrams, mathematical tables, parallel insights, approximation techniques, and mathematical astronomy, for example.
- Ethnomathematics

#### Philosophy of Mathematics

Happy to supervise various topics in philosophy of mathematics jointly with colleagues in philosophy.

### Dr M. Plank

- Weed risk assessment and control.
- Modelling spread of invasive plant species.
- Modelling the early stages of atherosclerosis.

### Dr C. Semple

#### Phylogenetics

Phylogenetics is the reconstruction and analysis of phylogenetic trees and networks based on inherited characteristics.

Possible Masters and PhD projects include

Mathematical phylogenetics associated with reticulate evolution.

Combinatorial problems in supertree construction.

#### Matroid Theory

Matroids are precisely the structures that underlie the solution of many combinatorial optimisation problems. These problems include scheduling and timetabling, and finding the minimum cost of a communications network between cities. Moreover, matroid theory unifies the notions of linear independence in linear algebra and forests in graph theory as well as the notions of duality for graphs and codes.

Possible Masters and PhD projects include

- Matroid connectivity and structure.
- Matroid representation.

### Prof M.A. Steel

#### MSc Projects

- Combinatorial/topological/geometric aspects of the space of evolutionary trees.
- Ordinal methods in phylogeny.
- Determining statistical consistency of minimum evolution methods.
- Other topics: Population genetics; rates of evolution; evolutionary game theory, Modelling directional evolution, Abstract origin–of–life models, genomic analysis.

#### PhD Projects

- Modelling hybridization and representation of evolutionary history by digraphs.
- Combinatorial/topological/geometric aspects of the space of evolutionary trees.

### Dr Günter Steinke

#### Geometry

Interpolating systems are families of curves each of which is uniquely determined by a certain number of its points. Among the many properties every interpolating system has (analytic, geometric and numerical) the focus is on the geometric aspect.

Possible Masters and PhD projects deal with

- Construction of interpolating systems;
- Characterisation of certain interpolating systems by configurational conditions;
- Determination of the automorphism groups of interpolating systems and classification of the most homogeneous ones;
- Characterisation of certain interpolating systems by groups of transitive automorphisms;
- Applications to secrete sharing schemes.

I'm also willing to consider proposals in algebra, combinatorics or topology.

### Assoc Prof D.J. Wall

- Mathematical Wave propagation for inverse and direct problems, both numerical and theoretical aspects.
- Mathematical Biology studying the mathematical aspects of cellular secretion from corticotroph cells in the pituitary.

### Assoc Prof N.A. Watson

At the Masters or PhD levels, I can propose topics on the Potential Theory of partial differential equations. The equations can be elliptic or parabolic, and with constant or variable coefficients. For the Potential Theory of Laplace's equation, which is the prototype for all others, see the books “Introduction to Potential Theory” by L.L.Helms (Wiley, 1969), and “Classical Potential Theory” by D.H.Armitage & S.J.Gardiner (Springer, 2001).

At the Masters level, I'll also consider supervising any topic of an analytic nature, be it in analysis, topology, fuzzy topology, or whatever. So if you've got an idea, run it past me.