## Engage with us

## Vacation scholarships for undergraduates

A scholarship of a tax-free stipend of up to $400 per week is available for undergraduate students in Australia to be involved in cutting-edge research during the summer vacation.It is typically awarded for four weeks in January to February.

### How to apply

Applications are now closed. Next round will open in Dec 2017 for Jan 2018

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 two academic referees.

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

**For more
information **

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

## 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.

### Projects

Project title | Project description | 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 |