The following list does not aim to be exhaustive, but a list of things that seem like they might be interesting to investigate. Feel free to suggest your own ideas.

Applied non-linear difference equations
Second-Order Difference Equations (MATH235 only)
In MATH111 we investigated first-order difference equations. In this project we investigate second-order difference equations. Topics covered could include:
• Analytical solutions
• Modelling
• Chaos
• 0-1 test for chaos
• harvesting in predator-prey models
• piecewise-smooth dynamical systems
The assessment will contain `tutorial-type questions' and Maple projects.
This project is will be jointly supervised with Dr Annette Worthy.
Symmetry in Chaos
This project is based upon the book Symmetry in Chaos and will be based upon a combination of reproducing some of the fantastic colour figures in this book and understanding the mathematics. (Consult the book to see the pictures!). You will need to be proficient in computer programming (not MAPLE) to take this project.
Analysis (applications in image science)
This project is for someone who is interested in analysis. The detection of discontinuities are important in many fields. For instance, in image processing it is often important to detect the boundaries of the items in the image. An important application of edge detection is in the medical sciences. Another area of application is in the movie industry: edge detection is important in the automatic restoration of old film prints and the automatic colouring of black and white film. In this project you will learn how to use Fourier series to detect "edges". Mathematically the problem is this: given the Fourier coefficients for a function how can we determine the location and size of the jump discontinuities of the function?

This project will be co-supervised by Associate Professor Rod Nillsen and will be based around the following article.
Engelberg, S. (2008). Edge Detection Using Fourier Coefficients. The American Mathematical Monthly, 115(6), 499-513.

• Modelling alliances.
• Collective influence and social change. How does public opinion effect issues such as voting choice? What's the relationship between crime and punishment? Marriage as an economic institution.
Electoral Systems.
Is it possible to design a `fair' electoral system? What do we mean by a fair electoral system? How can mathematics be used to analyse and improve electoral systems? By the end of this project you will know!
Cake Cutting
Many problems in economics can be caricatured mathematically as a `cake-cutting' problem. One of the simplest such problems might be written as follows.

Suppose you and I are to cut a cake, so that each of us gets half of the cake, where `half' is interpreted according to our preferences. Suppose you like the chocolate and prefer a piece with at least half of all the chocolate decoration. I, on the other hand, like marzipan and prefer a piece with at least half of all the marizpan decoration. How do we split the cake so that we are equally happy? How do we split the cake between n people with their own preferences?

Calculators: How do they calculate?
How do calculators calculate? What is sin(1)? You might think they calculators use Taylor polynomials to approximate trigonometric functions. This is a good guess, but it is not correct! In this project we will investigate the mathematics used by calculators to determine the values of trigonometric functions and other types of functions. We might also look into how methods used by mathematicians over 1000 years ago to approximate the values of trigonometric functions. For instance, how did the Indian mathematician Aryabhata (476--550) compute a table of sines for singles from 3o to 45o?

This project is suitable for second-year students only.

Chemistry
Wine Fermentation.
We investigate a physical and mathematical model for the kinetics of wine fermentation. The model predicts sugar utilisation curves based on experimental data from wine fermentations with various initial nitrogen and sugar concentrations in the juice.
Climate Change
Climate scientists have hypothesized that an increase in temperature could cause the world's peatlands to overheat, like a compost pile. This project investigates a (very) simple model for global warming in which the average atmospheric temperature is taken as a slowly varying parameter. The model exhibits a `tipping point', leading to spontaneous combustion (of the earth!) if the rate of change of atmopsheric temperature is too large.
Combustion
Spontaneous Combustion of Coal.
When coal reacts with oxygen it releases energy, which is why it is used as a fuel. However, when too much coal is stockpiled it can spontaneously ignite. In this project we will investigate the spontaneous combustion of coal.
see cube.dir for transient problem of estimating parameter values. based upon malow:2004. Cylindrical problem.
Differential equations
Identification of ODE Models.
OK. Many of the projects on this page are "applied des" projects. This is a little different.

Ordinary differential equation (ODE) models have been widely used to model physical phenomena, engineering systems, economic behaviour, and biomedical processes. In particular, ODE models have recently played a prominent role in describing both the within host dynamics and epidemics of infectious diseases and other complex biochemical processes. Great attention has been paid to the so-called forward problem or simulation problem, i.e.\ predicting and simulating the results of measurements or output variables for a given system with given parameters. However, less effort has been devoted to the inverse problem, i.e., using the measurements of some state or output variables to estimate the parameters that characterises the system, especially for nonlinear ODE models without closed form solutions.
H. Miao et al. SIAM Review 53(1), 3-39, 2011.

Suppose that we have a mathematical that contains n ODEs. We can measure m (m≤n) output variables. The model contains p parameters. Can we estimate the p parameters from measurements on the m output variables?

Drying of particles
Drying of particles.
The drying of small particles is an important industrial process with applications in many industries. A well-known example is the spray drying of milk to produce powdered milk. In this project we will examine some simple models for this process.
Economics
• Can we predict the rhythms of the financial market? Do the assumptions of classical economics ever hold?
• How does the interaction of individual traders lead to flucturations in markets? Do traders act rationally?
• Can we predict market crashes?
• Modelling the growth of firms.
• Globalisation. How does cultural dissemination work? Is diversity eliminated by globalisation?
• Game theory
• How do people/organisations learn how to cooperate?
• Is reciprocity good for us?
• A set of n applicants is to be assigned among m colleges, where qi is the quota of the ith college. Each applicant ranks the colleges in the order of their preference, omitting only those colleges which they would never accept under any circumstances. Each college similarly ranks the students who have applied to it in order of preference, having first eliminated those students that it would not accept under any circumstances. How should the students be allocated to the colleges?

The solution of the above problem was cited by the Royal Swedish Academy of Sciences for the 2012 Nobel Prize in Economics.

• The Gini Index and Measures of Inequality.

"You hear anecdotes all the time: The poorest 20% of the people on Earth earn only 1% of the income. A mere 20% of the people on Earth consume 86% of the consumer goods. Only 3% of the US population owns 95% of the privately held land."

"The Gini index offers a consistent way to talk about statistics like these. A single number that measures how equitably a resource is distributed in a population, the Gini index gives a simple, if blunt, tool for summarizing economic data. It allows us to illustrate how equity has changed in a given situation over time, such as how U.S. family income changed over the 20th century. We can also compare income or wealth across societies, and even analyze salary structures of organisations."

Epidemiology & virus dynamics
We will investigate the spread of infectious diseases through a population and through an individual. We will be interested in questions such as:
• Will a disease spread through a population?
• If so, how many individuals will be affected?
• If the disease is endemic (i.e. habitually present), what is the prevalence of the infection?
• Suppose an infectious disease starts in one city. How quickly will it spread to other cities? What can we do to stop a disease spreading throughout the whole country?
• Seasonal SIR models are interesting.
H.W. Hethcote. 2000. The Mathematics of Infectious Diseases. SIAM Review, Volume 42(4), 599-653.

For virus dynamics we will consider the modelling of HIV.
M.A. Nowak and R.M. May. 2000. Virus Dynamics: Mathematical Principles of Immunology and Virology. Oxford University Press.

good source of ideas and projects. A.B. Gumel and S. Lenhard (editors). Modeling Paradigms and Analysis of Diease Transmission Models. 614.4015118/2.

Recently there have been an interest in modelling how an `infection' of zombies might spread through a population of humans. Can humankind survive or are we all doomed to be converted into zombies? So, if you like watching zombie movies we can investigate this in additional to more traditional models!

Did you watch Contagion? I liked it! If you liked the movie then an interesting project would be to learn Mathematical Epidemiology 101. Then watch the movie carefully to extract "parameter values" for the disease. Then develop a mathematical model for "Australia". Then see how many people might die in Australia before a cure is developed!

Chlamydia is the most common and treatable sexually transmitted diease. You will investigate models for the spread of Chlamydia trachomatis along the genital tract. This project is available in two forms

1. As an ODEs project (perhaps best as a second-year project).
2. As a PDEs project (best as a third-year project, you will learn how to solve PDEs numerically using MATLAB).

Dengue fever in Australia.

Modelling induced resistance to plant diseases. Mathematical modelling is used to investigate the dynamics of induced resistance in a given plant cohort against a selected disease-causing pathogen. You need to be profficient in solving ODEs numerically to work on this project.

Food Engineering
Extraction of nutrients in the intestines
One of the big challenges in the field of biotechnology is the study of food metabolism upon its consumption. The chemical components making up a food product undergo a chain of reactions in the gastro intestinal tract, which can be modelled as a chemical reactor. An understanding of the processes that occur within the intestines will have a major impact in the areas of nutrition science and food engineering and will lead to new advances in targeted drug delivery.

This is a huge problem, and would be a good topic for a PhD. However, in this project we will examine some highly simplified models for the extraction of nutrients in chemical reactors.

The History of Mathematics and Elementary Number Theory
This project might be called "The History of Mathematics through Elementary Number Theory" or alternatively "Elementary Number Theory and the History of Mathematics". The aim of the project is to investigate developments in elementary number theory from approximately 2500 BC to 1500AD. This project will involve a considerable amount of reading leading to a detailed report on the history of mathematics as seen through progress on solving problems in number theory. There will be some mathematics involved, but based upon the perspective of the 21st century the mathematics is straightforward. Consequently, I think that this is a project best suited to a second-year student. <-- S.G. Krantz. An Episodic History of mathematics. 510/395 -->
Industrial Mathematics
Cleaning Industrial Effluent.
Many industrial processes produce effluents that must be cleaned before they can be discharged into rivers. In this project we will analyse simple mathematical models for the degradation of contaminated wastewater in order to maximise the efficiency of the process.

Examples of processes where such models have been used in the past include: the aerobic degradation of waste water originating in the industrial treatment of black olives, the anaerobic treatment of dairy manure, the anaerobic digestion of ice-cream wastewater, the anaerobic treatment of textile wastewater and the aerobic biodegradation of solid municipal organic waste.

Laplace Transforms: Solving Linear PDEs (MATH345)
In MATH202 Laplace transforms were introduced as a technique to solve ordinary differential equations. In this project we extend the method to solve partial differential equations. This project will involve
• extending the theory of Laplace transforms learnt in MATH202
• applying your new technique to solve PDEs arising in mathematical modelling.
This project is will be jointly supervised with Dr Annette Worthy.
Linear Algebra.
Social scientists use adjacency tables to discover indluence networks within and among groups. The singular value decomposition (SVD) is a tool from linear algebra that can be used to distinguish between groups in a mathematically precise manner. The SVD can also be used to `project' large datasets with many-dimensions onto much smaller datasets where, in some sense, all the `action' is. This project is an introduction to the SVD.

The emerging field of network science deals with the tasks of modelling, comparing and summarazing large data sets that describe complex interactions. Examples of such networks include:
• in the cell, connecting genes, proteins, or other genomic quantities;
• in the brain, connecting neurological regions;
• in epidemiology, connecting individuals that come into contact;
• in zoology, connecting animals that interact socially;
• in energy, connecting power suppliers or users;
• in telecommunications, connecting mobile phone users;
• in the World Wide Web, connecting web pages, and
• in the Internet Movie Database, connecting co-starring actors.
Because pairwise affinity data can be stored in a two-dimensional array, graph theory and applied linear algebra provide extremely useful tools. This project is suitable for a student that enjoyed Linear Algebra and is confident in using matlab for matrix calculations.

Mathematical Biology.
There are many potential projects in this area. But here's one idea. Read Ian Stewart's excellent book The Mathematics of Life. Find a couple of chapters that you are interested in and we'll take it from there!

Mathematical Ecology.
Look at how connectivity of habits effects biodiversity.
Modelling induced resistance to plant disease. Plant disease control has traditionally relied heavily on the use of agrochemicals despite their potentially negative impact on the environment. An alternative strategy is that of induced resistance (IR). However, while IR has proven effective in controlled environments, it has shown variable field efficacy, thus raising questions about its potential for disease management in a given crop. Mathematical modelling of IR assists researchers with understanding the dynamics of the phenomenon in a given plant cohort against a selected disease-causing pathogen. You will analyse a prototype mathematical model of IR promoted by a chemical elicitor.

Mathematical Medicine.
Mathematical modelling of Hirschsprungs disease
The pre-requisite for this project is that you have passed MATH305. This project has two components:
1. Learning how to solve 1-d reaction-diffusion equations numerically using the method of lines.
2. Learning how to analyse 1-d reaction-diffusion equations using phase-plane methods.
This project is based upon research carried out by Professor Kerry Landman. For more information see her web page http://www.ms.unimelb.edu.au/~kal.
Tumor Dynamics.
Why do some tumors grow during treatment and others shrink after the treatment stops?

To investigate this we consider a model with three types of cells (immune cells, tumor cells, and normal cells) that takes into account the competition for resources between tumor and normal cells. This model has multiple equilibrium points, including a region in which there is one unstable and two stable equilibrium points: this represents a time during treatment at which the patient's tumor could be driven to either to steadily shrink or to grow.

This model can be extended to include treatment, in the form of chemotherapy.

SIAM News 42(8), October 2009.

Modeling Growth in Biological Materials

The simplest way to model growth at the cell level is to use cellular automaton models. Such models have been used for many years on biological problems. Models in this class divide the region of space being modewlled into a number of lattice sites. For models of cell population, the main property to be tracked is whether the lattice site is occupied by a cell or not. One of the main biological applications of cellular automata is in models of tumour growth.

The basic structure is to allow each cell to make a decision on how it will behave at each time step. These decisions include whether to prliferate, die, move, or mutate. The decision may be deterministic or stochastic, and may depend on external cues such as the number of neighbors or the concentration of messenger chemicals. The ability to study mutation and natural selection is one of the attractions of such models.

To take this project you need to be proficient in programming or willing to become proficient in programming very quickly.

Recurrent infection

Recurrent infection is characterised by short episodes of high viral reproduction, separated by long periods of relative quiescence. This recurrent pattern is observed in many persistent infections, including ``viral blips'' observed during chronic infection with the HIV virus.

The aim of this project is to investigate analytically and numerically a 4-variable HIV antioxidant-therapy model which exhibits viral blips.

Identifying the conditions under which antibodies protect against infection

The ability to predict the conditions under which antibodies protect against viral infection would transform our approach to vaccine development. The aim of this project is to develop a more complete understanding of antibody protection against lentivirus infection, as well as the role of mutation in resistance to an antiboyd vaccine.

Mathematical Physics
I am not very enthusiastic about supervising a project in mathematical physics... but if you have enjoyed MATH203 and you are a physics/mathematics students then I can offer you a project on vortex lattice theory. A vortex lattice is an arrangement of magnetic flux tubes that appear in superconducting materials. These tubes can be visualised by imaging the location of their centres when they are arranged perpendicular to the image plane, and can be seen to form a lattice pattern. This project uses linear algebra to examine the stability of some simple lattice patterns.

This project is based on
P.K. Newton and G. Chamoun. Vortex Lattice Theory: A Particle Interaction Perspective. SIMA Review, 51(3), 501-542, 2009.
This paper contains some very nice images of experimentally observed vortex lattices.

Multiple Scales
One of the challenges in modelling the climate is that it involves the interaction of processes that occur on widely differing physical scales. In this project we will investigate what mathematics can offer to overcome some of these problems. To do this we will learn about the technique of `multiple scales' and apply to the following `toy' problem: an externally driven oscillator with small mass and small damping.
Non-linear oscillators
Duffings Equation (3rd year students only really).
Duffing's equation is an example of a nonlinear oscillator. It can be used to model a variety of mechanical problems including the dynamics of a buckled beam or plate, the forced vibrations of a cantilever beam in a nonuniform field of two permanent magnets

The plan would be to analyse the steady states and stability of the damped Duffing eqn with no forcing (zero or positive damping), then the Duffing equation with negative stiffness - which acts as a model for the magneto-elastic. Apply increasing amounts of forcing to the system to investigate how the basins of attraction change. See how computing power has changed - write a matlab program to repeat on your own PC the calculated that Moon and Li (1985) did on a supercomputer!

Some references
T.H. Fay and S.V. Joubert. (2007). Nonlinear resonance and Duffing's spring equation II. International Journal of Mathematical Education in Science and Technology, 38(4), 517-528. Guckenheimer and Holmes.
Moon F.C. and Li G.-X. (1985). Fractal Basin Boundaries and Homoclinic Orbits for Periodic Motion in a Two-Welled Potential. Physical Review Letters, Volume 55, 14.
Strogatz, S.H. (1994) Nonlinear Dynamics and Chaos. Perseus Books Publishing , LLC. Wiggins S. (1990). Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag New York Inc.

Forced van der Pol equation
T.H. Fay. The forced can der Pol equation. International Journal of Mathematical Education in Science and Technology, 40(5), 669-677, 2009.
Physics.
Problems in classical physics can be solved by applying Newton's laws. Problems in quantum physics can be solved working within the axios of quantum mechanics. In both cases physical problems lead to well-defined mathematical problems. Problems in the social sciences do not lead to well defined mathematical problems - for how can the forces that act between individuals be cast into mathematical form?

HOWEVER, it is indeed possible to model the ways in which humans behave and organise themselves, by applying ideas from statistical physics. Possible problems that may be investigated in this project include:

Dynamics of Growth and Movement.
• how do crowds move? What's the best strategy to escape from a smoke-filled room when you don't know where the exits are? What's the best way to disperse a large crowd?
• How do cities grow?
• The dynamics of traffic. How does traffic move and clog? Can we predict when and where congestion will occur?
Social Networks
• The Kevin Bacon Game, six degrees of freedom, social networks and small worlds.
• What is the shape of the internet?
Population Models.
We investigate population models for single and interacting species. We are interested in question such as
• Under what conditions will a population become extinct?
• What are the conditions for sustainable harvesting of a renewable resource?
• What happens when several species compete for scarce resources?
• Ecosystem modelling, what is the effect of habitat destruction?
the mathematics of Search engines.
How do internet search engines work? They work because the designers understand the basics of linear algebra and matrices. Goggle's success derives in large part from the way in which it's search engine tanks the importance of web pages. project you use basic ideas from MATH203 to understand the principles behind popular internet search engines. Why does linear algebra matter? Do this project to find out!
I am sure that you are familiar with the rules of snakes and ladders. One rule that you might have forgotten. If you are square 98 and the finish square is 100 you have to roll a two to finish the game. Suppose you have a six-sided dice. What is the expected number of turns that it will take to complete the game? How does the expected number of turns change as you change the number of sides on your dice? This project will provide a gentle introduction to Markov-chain models. We might need the help of a friendly statistician as I don't know anything about Markov-chain models!
Synchronisation.
Interactions between individuals - be they fireflies, pendula or people - can often lead to the emergence of coherent actions. Mathematically, such models can be represented by a coupled system of phase oscillators. Simply models can lead to surprising behaviour. See, for example.
Y. Zhou, W. Gall and K. Nabb. (2006). Synchronizing Fireflies. The College Mathematics Journal, 37(2), 187-193.
Notices of the American Mathematical Society, Volume 51, Number 3, pages 312-319, March 2004.
the mathematics of Tennis and similar games
Tennis anyone? In this project we will investigate very simple mathematical models for games such as badminton, squash and tennis.

Simple thoughts. What are your chances of winning a game if you know your chances of winning any given point? Did you know that if your chances of winning a point are 50-50, then a 1% increase in your probability of winning each point yields a 2.5% probability of winning the game: so average players have the most to gain by improving a bit.

Travelling Waves
It would help if you had taken MATH305.

The following list does not aim to be exhaustive, but a list of things that I am particularly interested in investigating NOW. Another good place to look for NEW projects that I might be willing to supervisor at a non-honours level is my list of potential honours projects.

Drying of particles
Drying of particles.
The drying of small particles is an important industrial process with applications in many industries. A well-known example is the spray drying of milk to produce powdered milk. In this project we will examine some simple models for this process.
Reformulated models for PEM fuel cells
The aim of this project is to investigate simplified solutions for models of fuel-cells. (It will help if you have taken MATH305). It might also involving models for optimising the use of batteries.

<< Move to the list of undergraduate projects (non-honours).
<< Move to my start page.

Page Created: 13th July 2005.
Last Updated: 23rd June 2017.