School of Mathematical Sciences

1.Water trading in the Murray Darling Basin

In this project we will explore how mathematical optimisation can be used to inform the setting of "exchange rates" in the Murray Darling Basin. Unlike money, water cannot flow uphill and so the physical topology of the river system has to be taken into account during water trading. Currently, "exchange rates" are set to determine how much of the water sold at point A can be extracted by the purchaser at point B. These are set by an attempt to conserve water. But why not set these exchange rates to support the management of the river system as a whole? Once we have agreed on the objectives by which the system should be managed, we can use mathematics to ensure that each trade supports these objectives. Perhaps this is as simple as water conservation, but I do not believe so.

2. Stochastic modelling using structured markov chains

In this project we will investigate the mathematical properties of some very recent stochastic models. These models have been developed from the basic principles used in a field known as "Matrix-analytic Methods" (or MAM) where simple exponentially distributed lifetimes are replaced by lifetimes from more complex distributions. When done carefully, the analysis of the whole model becomes matrix-based, rather than scalar-based, hence the name. Of course, this brings all sorts of challenges (for example, the square root operation no longer makes any sense) and requires a much closer connection to the physical model itself. This, and an associated emphasis on computational algorithms, are the main features of this area of stochastic modelling.

1. Modelling Hirshsprung's disease

Hirschsprung's disease is relatively common, affecting roughly 1 in 5000 newly born babies each year in Australia. The disease occurs when there is an incomplete formation of the nervous system in the gut. This project will explore both discrete and continuous mathematical models, which can help in determining the underlying mechanisms that cause the disease.

1. Nanoscaled oscillating systems

Nonoscaled structures such as carbon nanotubes and fullerenes undergo interactions described by van der Waals forces. At very small scales these interactions can lead to extreme accelerations, velocities and, in the case of oscillating systems, frequencies. By modelling the structures as surfaces with uniform atomic densities and the van der Waals interactions using a 6-12 Lennard-Jones potential, we can make predictions regarding these systems including deriving a formula for the frequency which is in good agreement with molecular dynamics simulations. In this project the student will look at models to calculate the force and predict the behaviour of various oscillating systems.

2. Geometries and geometric issues of nanostructures

It is clear from the various structures seen at the nanoscale that the complex interactions of these structures often lead to symmetric conformations. So in satisfying a minimum energy constraint the system often adopts a symmetric structure that shares the energetic costs of bending and stretching covalent bonds equally to all components in the structure. By assuming a symmetric conformation up front, it is possible to reduce fundamentally complex problems of molecular structure to problems with are more mathematically tractable and thereby derive results which can be confirmed by experiment and simulation and can also be used to predict ideal systems and novel structures in certain extreme cases. In this project the student will study models for nanostructures such as nanotubes, cones and spheres (buckyballs) with the aim to provide more precise predictions of structural parameters like length and radius.

1. Drag reduction

This project will introduce the student to the problem of drag reduction through a study of the topic of turbulence in fluids flows. The project will involve a review of the state of the art in drag reduction technologies and can take a variety of directions depending upon the student's interests and background.

2. Extreme waves

Over the past ten years over 200 super-carriers (cargo ships that are over 200 metres long) have been lost at sea. Many of these losses have been attributed to an encounter with a freak wave. Recent data from the European Space Agency's MaxWave experiment detected ten giant waves (each over 25 metres high) in just a three-week period. These waves are not related to tsunamis. They are thought to result from a self- interaction process between much smaller amplitude waves that occur in the ocean.

1. Chaotic Fluid Mixing

This project will introduce the student to chaotic dynamical systems through the modelling and analysis of a prototype micro-scale fluid mixing device. The project will involve a mixture (pun intended) of complex variable techniques for very viscous fluid flow, numerical solution of nonlinear ordinary differential equations, and possibly some bench-top experimental work.

2. Swimming micro-organisms

This project will introduce the student to the fluid flow modelling and dynamical systems analysis surrounding swimming micro-organisms. There will be a particular focus on how swimmers respond to their environment and to ensembles of other swimmers.

Supervisor: Dr Matt Finn

1. Sea water desalination using carbon nanotubes

Global warming, the rise in the population and increasing expectations in living standards has dramatically raised the demand for water. Australia is one of the driest continents on the planet and the recent surge in population makes it increasingly harder to meet water demands. Thirteen large-scale Reverse Osmosis desalination plants are being proposed and planned nationwide for this purpose. Membrane processes, especially for reverse osmosis desalination, require a lower energy for seawater desalination, and a successful low cost desalination technique is a vital objective for Australia. Nanotubes can be embedded into a polymer matrix to fabricate nanotube membranes that may be used as reverse osmosis membranes to generate rapid water transport. Ultra-small nanotubes are found to repel heavy ions, such as salts and allow only water molecules passing through the nanotube membranes. In addition, the rapid transport of molecules through molecular pores and hydrophobic nanotube channels has been found to be essential for several biological and technological processes, such as bio-catalysis, hydroelectric power converters, seawater desalination and drug delivery for killing tumour cells. In this project, we will consider the use of different types of nanotube membranes for seawater desalination. Feasible topics include the modelling of water configurations inside nanotube membranes, the molecular effects of bulk solution and membrane matrix on water/ion permeation and the connection between the flow arsing from nanofluids and macrofluids.

2. Nano memory device design

The trend for modern computer devices is towards decreasing in size to improve speed and capacity, by reducing the energy and the heat generated. Nanotechnology has brought many revolutionary advanced materials, for which the physical size of the smallest components have reduced from the micrometer scale to the nanometer scale. Many materials at the nanometer scale display exceptional physical characteristics such as their mechanical and electronic properties, and these properties can be quite different compared to those at the micro-scale. Accordingly, nanotechnological components might be one possible solution for future computer design. In the present era of data storage increasing geometrically with a large exponent in the past 20 years, a high capacity data storage device becomes an important issue, which are required to be small in size and posses a high data transfer rate. A two-state memory device could store in the binary system, the information of state zero and state one, and it could be easily to charge the state for rewrite data by applied a voltage or a magnetic field. The normal text could be stored in the binary system by a character-encoding scheme which familiar scheme is 8-bit ASCII code (American Standard Code for Information Interchange). For example, the binary code for the character `A' is 0100 0001 (decimalise 65) and for the character `a' is 0110 0001 (decimalise 97). This project will design memory device using mathematical modelling and continuum approximation on nano-materials such as matellofullerenes, fullerenes, nanotubes and nanocone.

Supervisor: Professor Jim Hill

1. Finite simple groups, trees and complex polynomials

It is known that every (nonabelian) finite simple group can be generated by two elements. Thus there is a way to encode it in a finite graph with some extra structure, known as a dessin. Sometimes the dessin is simply a tree. Then the group is realised as the monodromy group of a complex polynomial of special kind, a so-called Shabat polynomial. You will familiarise yourself with the underlying theory, survey the literature, which is scattered and sketchy, and, for some small simple groups, in particular the smallest sporadic group, produce a corresponding tree and, with the help of a computer algebra system, attempt to calculate a Shabat polynomial for the group. The project requires third-year complex analysis, group theory, and topology.

Supervisor: Associate Professor Finnur Larusson

1. The classical groups

The topic of the project are the classical groups defined as invariance groups of bilinear and Hermitian forms over real, complex and quaternionic vector spaces. The aim is to study them as submanifolds in the vector space of n-by-n matrices and to establish some of the isomorphisms and covering maps for small dimensions n. Further exploration can be related to topological properties of the classical groups or to the quotient spaces arising from them.

2. Quaternion’s and Octonions

In a similar way how the complex numbers are constructed from the reals, the quaternions are constructed from the complex numbers and the octonions from the quaternions. Thus, both can be considered as generalisations of complex numbers to higher dimensions. Many interesting algebraic and geometric phenomena are related to the quaternions and octonions and some interesting groups are related to them. To explore these features and relations is the aim of the project.

1. Fire behaviour models

High-fidelity numerical simulations of bushfires have the potential to increase our knowledge and understanding of bushfire behaviour, and to provide operational forecasts of bushfire progress. The aim of this project is to examine existing fire behaviour models and to use them to run coupled simulations of bushfire/atmosphere interactions.

1. Differential geometry
2. Mathematical physics

I am happy to discuss possible topics in the areas of differential geometry and mathematical physics. Interested students should email me and arrange a meeting.

1. Analysing animal movements using Markov chain methods

Where do animals spend most of their time? This project will investigate a novel method for addressing this question using data on an animal's position at discrete time points.

2. Realistic infectivity profiles in disease dynamics

We will extend a recently developed model of household disease dynamics to incorporate a realistic infectivity profile. The consequences of this novel modification will be explored.

1. Sea-shell modelling

It has been known since the 19th century that the shape of mollusc shells can be modelled by a simple 3 parameter family of surfaces based on logarithmic spirals. More recently effort has been devoted to understanding the formation of the beautiful patterns on such shells. However, most of the historical work has concentrated on the exterior surface of the shell. The goal of this project will be to study these models with a view to creating an animated fly-through of the shells based on mathematical modelling of the internal and external structure. Ultimately, such a model could be used as input to our 3D printer, to print physical replicas of sea shells.