## Department of Mathematics

## Why Study Maths?

Mathematics is the language of our technological society. It is is used in a wide range of jobs, and as such, is highly sought after by employers. Employers have indicated that they are experiencing a shortage of adequately skilled candidates with the problem solving and logical thinking skills that a mathematics degree provides. In 2015, CareerCast released a report of the top 10 jobs for the year evaluating several variables including income, outlook, environmental factors, stress, and physical demands for 200 professions across a wide variety of industries, salary ranges, and skill levels. Mathematicians shared top spot on the list, recording mid-career income levels well above an average graduate’s salary.

### Careers in Mathematics are Booming!

Interested in a career in Mathematics? Check out the new Math Adds booklet (pdf), highlighting many of the exciting career awaiting you when you study for a degree in Mathematics.

## Vacation Scholarships

- involve cutting-edge research during the summer vacation. Research projects listed below .
- are open to undergraduate mathematics students in Australia
- have a scholarship of a tax-free stipend of up to $400 per week
- are available in areas of research excellence
- are typically awarded for 4 weeks in Jan – Feb

### How to apply

**Applications are now open and close by 25 November 2016.**

The application process comprises:

- a completed application form (PDF - fillable)
- a covering letter
- a copy of your academic transcript
- your CV
- contact details and references from
academic referees.*two*

At the completion of the scholarship students will be required to provide a one-page report detailing their experience.

### For more information

**Telephone: **(02) 9850 8947** E-mail:** sci.maths-adm@mq.edu.au

Terms of the award Selection procedures

#### Projects on offer for 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.

## Project Title | ## Supervisor |
---|---|

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 | Vlad Gaitsgory |

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. | Chris Lustri |

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. | Chris Lustri |

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. | Frank Valckenborgh |

You know about various functions involving numbers m and n, such as: - (i) m+n
- (ii) m times n
- (iii) m^n
- (iv) binomial coefficient C(n,2): “n choose 2”
- (v) e^n
- (vi) 1/(1-n).
How can you adapt these and similar functions so that they make sense when m and n are sets? | Steve Lack |

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. | Steve Lack |

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. | Richard Garner |

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. | Paul Smith |

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. | Sophie Calabretto |

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. | Jim Denier |

We consider one dimensional Schrödinger operator _{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. | Adam Sikora |

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. | Ross Moore |