Department of Mathematics
Our goal is a balanced program of pure and applied mathematics, covering the three major aspects of modern mathematics – analysis, modelling and computation. Our department is active in research, focusing on our strengths in:
- Australian category theory
- number theory
- applied mathematics
while recently we have taken a new direction of research in waves and non-linear dynamics.
We offer one PACE (Professional and Community Engagement) unit in Mathematics, and students can select from one of two Planet units in Mathematics available for students outside of the Faculty of Science and Engineering. More information on these units can be found in the Macquarie Course Handbook.
Vacation scholarships are also available during the summer holidays and are typically awarded for five weeks in January and February.
Our department is awarding up to six scholarships in 2017. Please refer to the list of projects below for more details.
For more information
Telephone: (02) 9850 8947 E-mail: firstname.lastname@example.org
Projects on offer in 2017
Up to six (6) scholarships will be awarded to suitable candidates, with details of possible projects on offer listed below. Please list your top four preferences (in order) on the application form.
1. Linear Programming Approach to Optimal Control of Nonlinear Dynamical Systems
The linear programming approach to problems of control of nonlinear dynamical systems is a relatively new technique based on the fact that certain aspects of system’s dynamics can be captured by a system of linear equations. The work on this project will involve learning theoretical foundation of the approach as well as its practical application for finding a numerical solution to a problem of optimal control of a predator-prey model
2. Numerical Modelling of Stratified Free Surface Flow
This project focuses on studying water waves caused by potential flow over a submerged obstacle, such as an ocean shelf or trench. This problem is complicated, as the position of the surface (and therefore the shape of the flow domain) is not known, and must be obtained as part of the solution itself. These problems are often studied using iterative numerical methods. In this project, you will apply iterative methods to study flows in which two fluids with different densities separated by a free interface. This can be thought of as a very simple representation of density stratification seen in the ocean. In particular, you will investigate how stratification affects waves on the surface of the flow.
3. Simulation of Travelling Waves through Particle Chains
Particle chains consist of a string of particles which interact only with their neighbours. These chains can be used to propagate energy or information; however, they are often subject to wave decay due to small imperfections in the chain. This project first involves simulating the behaviour of travelling wave signals sent through ideal particle chains. The second stage of the project is to introduce small imperfections into the system, and study the effect of these imperfections on the propagated waveform.
4. Symmetry and the Hydrogen atom:
The mathematical framework of quantum theory uses techniques from many diverse domains of the mathematical landscape, from the very pure to the highly applied. Group theory in particular, the theory that underlies symmetry, forms a very important and remarkably diverse tool in physical theory. In this project we will explore the non-trivial role that group representation theory plays in constructing physical theories and solving physical problems in general, and in elucidating the spectrum of the Hydrogen atom in particular.
5. The algebra of sets
You know about various functions involving numbers m and n, such as:
How can you adapt these and similar functions so that they make sense when m and n are sets?
6. The geometry of graphs
In vector calculus there is a higher-dimensional analogue of the fundamental theorem of calculus called Stokes’ theorem.
Calculus lies within the “continuous” part of mathematics, which involves things like real numbers, measurement, and integrals.
On the other hand “discrete” mathematics involves things like integers, counting, and summation. An important structure in discrete mathematics is that of a graph: these consist of “vertices” which are connected by “edges” (as you might use to represent a network).
In this project you will look at a version of Stokes’ theorem (and other theorems of calculus and geometry) which apply to graphs.
This project is best suited to someone who has done MATH236, where you learn about the usual Stokes’ theorem, but it would also be possible to do the version for graphs without having done the usual one.
7. Turing degrees
The famous "halting problem" implies the existence of sets of natural numbers which cannot be computed by any algorithm. The Turing degree of a set of natural numbers is a measure of the extent of its computability or non-computability. It turns out that there are infinitely many Turing degrees, of which only two have an easy description; going beyond this requires subtle and sophisticated mathematical arguments. The goal of this project is to understand some of these arguments.
8. Integral Equations
Just as a differential equation is a mathematical equation that relates some function with its derivatives, an integral equation is one in which the unknown function appears under an integral sign. Many interesting physical problems can be expressed in terms of integral equations: for example, the determination of the distribution of electric charge on a conducting surface from a knowledge of the potential, or the determination of the shape of a surface from a knowledge of the energy scattered by some illuminating field. A linear integral equation can be seen as the continuous analogue of a system of linear algebraic equations. This project will explore ways and means of solving integral equations and their interpretation in physical contexts.
9. Mathematics of the brain
The human brain is composed of individual nerve cells, or neurons, whose main function is to transmit and conduct information through the propagation of an action potential. Several mathematical models exist to reproduce certain behaviours of this electrical signalling (or firing) of neurons in the brain. By performing bifurcation analysis and using numerical continuation methods on these systems, we are able to unlock some of the rich dynamics within.
10. Gentle mixing
Ever tried mixing two different coloured paints together? One easy way is to simply put the two different coloured paints in a pot and give it a good shake. This will work and will make use of the fact that the motion of the paint in the pot becomes turbulent (disordered). But this is a very energetic way to achieve this result! Suppose instead you want to achieve the optimum mixing whilst expending the least amount of energy (and without resorting to turbulent motions to do this). One way is to make use of principles of the chaotic advection in which the dynamical system describing the paint’s motion is forced into a chaotic region of phase-space. This project will explore some of the theory behind chaotic advection and will allow you to develop your programming skills in Matlab.
11. Quantum Rectangular Potential Barrier
We consider one dimensional Schrödinger operator –d2x + bX(–a,a), where X(–a,a) is a characteristic function of the interval (–a,a) and b is a real number. The aim is to compute explicit formulae for the heat kernel and resolvent corresponding to this operator and try to understand Schrödinger mechanic interpretation of such calculations.
12. Detailed exploration of the logistic map to estimate the Feigenbaum number and the limit of period-doubling.
The Bifurcation diagram for the real (quadratic) logistic map exhibits period-doubling as the lambda-parameter increases, before reaching a limit after which chaos ensues. By making high-precision measurements of certain features of the bifurcation diagram, it is possible to estimate just when this limit actually occurs.
In this project, we make measurements from computer-generated images to obtain parameter values for 3 different kinds of phenomena. Each of these sets then provide the values for such an estimate of the limit. However, each of these phenomena have associated practical difficulties in locating the correct values for recording. Part of the project is to appreciate these difficulties and find the best ways to use the software in order to get sufficiently precise measurements, or otherwise understand as much as possible about the practical limits of the techniques being used.
Research in the Department of Mathematics at Macquarie University is recognised nationally and internationally, with groupings of mathematicians working in analysis, category theory, fluid mechanics, nonlinear dynamics, number theory, optimisation and waves. We attract a substantial amount of external research funding and host a steady stream of international visitors.
Our research is supported by a number of vibrant seminars series, attracting both domestic and international speakers. In addition the Department's Colloquium, aimed at a general mathematical audience, features talks by visitors to the department, both international and domestic, as well as the occasional talk by members of the department.
Our international collaborations are many and varied. Members of the department work with colleagues at many of the other leading Australian mathematics departments as well as from many high profile international mathematics departments, including:
- Université de recherche Paris Sciences et Lettres
- Université de Bordeaux
- Sun Yat-sen University
- Central China Normal University
- University of Birmingham
- University of Wroclaw
- INRIA (France)
- University of Bundeswehr
- University of Bayreuth
- University of Brest
- Penn State University
- University of Manchester
amongst many others.
Our major programs of research are:
- category theory
- computational mathematics
- fluid mechanics
- waves and nonlinear dynamics
Higher degree research
Our department offers programs leading to the following postgraduate research degrees:
In Excellence in Research for Australia (ERA) 2015, research in mathematical sciences received a rating of 4 out of 5 – ‘performance above world standard’ – for the discipline overall. Macquarie also received the top rating of 5 – ‘outstanding performance well above world standard’ – for the sub-discipline of pure mathematics.
This places Macquarie among the top six Australian universities for pure mathematics. Discover more about the research work undertaken by researchers within this department.
If you are interested in doing research at Macquarie University, please contact a researcher in your field of interest to discuss your ideas before applying. Contact the Higher Degree Research Office to find out more about applying or look for a supervisor.
We offer a number of community outreach programs and operate a dedicated training and upskilling service called the Numeracy Centre, which offers free maths in-service support, via weekly workshops for some first year Mathematics courses, bridging programs and preparatory courses at the beginning of each semester, along with a range of on-line learning resources.
We run a Drop-in Centre, staffed by University tutors, which is open from week 2 of semester, Monday-Friday, for about 35 hours per week.
We also run a regular seminar series, where both local and international speakers talk on a wide range of topics. They include the Applied Analysis Seminar, weekly Australian Category Seminars, Advanced Computing – Algorithms and Cryptography and Statistics Research Seminar.