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 nonlinear difference equations
 SecondOrder Difference Equations (MATH235 only)
In MATH111 we investigated firstorder difference equations.
In this project we investigate secondorder difference equations.
Topics covered could include:
 Analytical solutions
 Steadystate solutions and stability
 Modelling
 Chaos
 01 test for chaos
 harvesting in predatorprey models
 piecewisesmooth dynamical systems
The assessment will contain `tutorialtype 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 cosupervised 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),
499513.
 Business and politics

 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 `cakecutting' 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 (476550) compute a table of
sines for singles from 3^{o} to 45^{o}?
This project is suitable for secondyear 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 socalled
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),
339, 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 wellknown 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
 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), 599653.
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
 As an ODEs project (perhaps best as a secondyear project).
 As a PDEs project (best as a thirdyear 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 diseasecausing
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
secondyear 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 icecream 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 manydimensions 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 costarring actors.
Because pairwise affinity data can be stored
in a twodimensional 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 diseasecausing pathogen. You will
analyse a prototype mathematical model of IR promoted by a
chemical elicitor.
 Mathematical Medicine.
 Mathematical modelling of Hirschsprungs disease
The prerequisite for this project is that you have passed MATH305.
This project has two components:
 Learning how to solve 1d reactiondiffusion equations
numerically using the method of lines.
 Learning how to analyse 1d reactiondiffusion equations
using phaseplane 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 4variable HIV antioxidanttherapy 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),
501542, 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.
 Nonlinear 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
magnetoelastic. 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), 517528.
Guckenheimer and Holmes.
Moon F.C. and Li G.X. (1985). Fractal Basin Boundaries and Homoclinic
Orbits for Periodic Motion in a TwoWelled 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, SpringerVerlag 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), 669677, 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 welldefined
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
smokefilled 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!
 Snakes and Ladders.
 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 sixsided 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
Markovchain models. We might need the help of a friendly statistician
as I don't know anything about Markovchain 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),
187193.
Notices of the American Mathematical Society,
Volume 51, Number 3,
pages 312319, 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 5050, 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
 Quadratic and cubic reactiondiffusion fronts.
 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 nonhonours level is my list of
potential honours projects.