Adele Jackson is a second-year student at the Australian National University, in the Bachelor of Philosophy (Science) program. She has yet to find an area of mathematics she’s not interested in. In particular, Adele is fascinated by fields on the border of maths and theoretical computer science, which let her combine her algorithms background with her university studies.

As well as maths, she has studied environmental science and computer science. Outside of academics she enjoys singing in choirs, all kinds of sport, and helping out with the Australian informatics olympiad training program.

The unknotting problem is to algorithmically recognise whether a given presentation of a knot is in fact ambient isotopic to the trivial knot.The unknotting problem is known to lie in both NP and co-NP, so is a good candidate to additionally be solvable in polynomial time (that is, to lie in P) in the number of crossings. One complexity concept halfway between NP and P is fixed-parameter tractability, where a problem is solvable in polynomial time in the input when another parameter of the input is fixed. I attempt to show that unknot recognition is fixed-parameter tractable using a mix of topological, algebraic, geometric and combinatorial techniques.

Nicholas Liu is a Bachelor of Science graduate from the University of Melbourne majoring in Physics, with a Diploma in Applied Mathematics. He completed his Bachelors degree under the Chancellors Scholars Program due to his 99.95 ATAR. He is interested in a variety of fields, from theoretical physics to applied mathematics and even aspects of chemistry. This interest was developed through various small research assignments during his degree in chemistry (2015), physics (2016) and maths (2016). The chemistry project focused on the effect of various solvents on photochemical upconversion, the physics project focused on the behaviour of molybdenum disulfide during exfoliation and the maths project focused on modifying the Navier Stokes equations to include a non-constant viscosity. His current research focuses on using novel computational methods to solve the Boltzmann equations for rarefied gas flow.

When outside the world of academics, Nicholas enjoys ice skating and playing the saxophone.

The main aim of the project is to solve the Boltzmann equations for rarefied gas flow, and will use newly-proposed computational and mathematical methods tailor-made for this problem. This is to describe fluid flow on the nano scale to enable the development of technologies like nanoelectromechanical systems. Then, first order approximation of the distribution function will provide useful physical insight.

William Troiani is a Master of Science (Mathematics and Statistics) student at the School of Mathematics and Statistics at the University of Melbourne. In particular, William is part of the Pure Mathematics stream. His interests lie in the interplay between Pure Mathematics and Computer Science. Specifically, his research interests revolve around select areas of Category Theory (such as Symmetric Closed Monoidal Categories), as well as Linear Logic and its connections to Linear Algebra. By studying such areas, William hopes to contribute towards our understanding of computation; what exactly is computation? Why is it worth thinking about?

One possible approach to this, is to examine already existing connections between Mathematical Logic, and programming languages such as the λ-Calculus (the Curry-Howard correspondence provides a concrete example of such a connection). It is through these areas of Mathematics, Logic, and Computer Science, that William hopes to push our Mathematical understanding of computer programs as formal objects, which in a world where the sophistication of modern technology is rapidly advancing, is an important task.

What is computation? The pioneering work of Godel, Turing and Church in the 1930s organised the theory of computation around several different, but equivalent, definitions of the mathematical objects that we now call “computer programs”. Recent developments in mathematics, computer science and physics are leading to a re-imagining of the nature these objects. For example: programs and their execution are often presented using string diagrams and transformations of diagrams, either for λ-terms in Church’s approach to computation or in related approaches like linear logic, an abstract functional programming language introduced by Girard.

Wern Shing (Nicole) Ng is a third year Bachelor of Mathematics student majoring in Applied Statistics at the University of Wollongong. Her academic interests involve econometrics, financial modelling and biostatistics. Academics aside, she likes playing the piano and violin, and also enjoys swimming. Nicole’s accomplishments at UOW include being awarded both the Undergraduate Excellence and Faculty Merit Scholarships, as well as being awarded a place on the Dean’s List in 2016.

In recognition of her achievements, she has been employed as a Peer Academic Study Sessions (PASS) leader in the Peer Learning unit, and as a Math & Stats resource development partner in the Learning Development Unit at UOW. Off campus, she enjoys working casually as an assistant instructor at KUMON, where she shares her interest in Mathematics with children of all ages.

After enrolling in an Applied Mathematical Modelling subject taught by Dr Mark Nelson at UOW in 2016, she gradually developed a strong interest in the use of mathematical modelling in environmental engineering. Due to her father’s long-lived passion in aquaculture, she has chosen to undertake a project involving micro-algae harvesting and wastewater treatments. Upon completing the AMSI Vacation Research Scholarship program, she hopes to graduate in July 2017 before continuing onto pursuing an Honours Degree.

The world has a serious, and possible fatal, addiction to petrochemicals. Almost 80% of energy consumed across the world comes from fossil fuels. Biofuels are a promising solution to reduce the world’s dependence upon a diminishing resource. Where do biofuels come from? One way to obtain biofuels is to use microalgae to remove the organic pollutants that are present in municipal wastewater. This serves a dual purpose. Firstly, contaminated wastewaters are cleaned before their discharge into the environment. Secondly, as they grow the algal biomass can be harvested and used for a variety of applications including the production of biofuels and for producing valuable substances used as feed, food and in the nutraceutical/pharmaceutical industries. Interest in the production of biofuels has gained momentum in recent years as they provide a ‘green’ alternative to exhaustible and environmentally unsafe fossil fuels.

The research component of this project is to extend an engineering model for the biological treatment of wastewater to include terms representing both light `shading’ and harvesting. (Light shading is the phenomenon by which the presence of biomass at the top of a bioreactor reduces the rate of growth in other regions of the bioreactor due to their reducing the intensity of light penetrating into them). The new model will be analysed to identify the strategy that maximises the harvesting of biomass whilst ensuring that the concentration of organic pollutants in the effluent stream of the treatment plant is below a specified level.

Timothy Collier is a postgraduate student at the University of Sydney who has recently completed a Bachelor of Science (Adv. Maths) majoring in Mathematics and Physics. Timothy has previously been involved in projects at the University of Sydney in harmonic function theory and the perturbation theory of eigenvalues as well as research in quantum error correction and quantum Hall effect circulators with Prof. Andrew Doherty of the University’s quantum science research group. He intends to continue with Honours in Mathematics, focusing on the Navier-Stokes equations under the supervision of Dr. Daniel Hauer.

Consider a container filled with some fluid like water or oil. Every fluid admits a viscosity depending on its physical properties. For a given initial state and external force, the motion of the fluid in the container is described by the celebrated Navier-Stokes equations. The aim of this project is to establish global uniform bounds on the velocities inside the container for weak solutions to these equations with zero external force.

This involves three stages: An introduction into the theory of hydrodynamics with the aim to derive the equations of Navier-Stokes. Second, learning De Georgi’s method for regularity of solutions of elliptic equations. Finally, the research component will involve adapting these techniques to establish new regularity results and to prove the bounds on weak solutions mentioned.

Terry Shang is a third-year student at the University of Sydney, currently studying a degree in Physics and Maths. He is interested in a variety of mathematical fields, in particular differential equations and fluid mechanics, and is hoping to pursue an Honours degree in Physics in the near future.

Hill’s equation in the real is a classical subject with many beautiful results. An ongoing study of stability of stationary solutions of the Euler fluid equations on the torus leads to a Hill’s equation with a complex parameter. The goal of the project is to study such a complex Hill’s equation, in particular it’s spectrum when viewed as a differential operator.

Stephen is a third-year Mathematics student at the University of Western Australia, interested in the fields of differential geometry and variational calculus. He hopes to do an honours degree next year, researching into optimal control theory and extensions of variational calculus into non-homogeneous spaces. Stephen wishes to gain entrance into medicine so that he may put his problem-solving skills to use in real-life situations where people are in need of help.

The aim of my project is to develop significant expertise in techniques used to study abnormal extremals in the calculus of variations and optimal control theory. The first part of the project will be to understand and outline selected variational problems and generate the differential equations for their solutions. These classical problems pose very interesting questions that require investigation. The second part of this project will be to investigate methods of solving these differential equations, in the attempt to obtain numerical solutions to the problems.

The research component is to explore extensions of classical problems to variational problems for curves in Riemannian, then sub-Riemannian, geometry. This project will help build a strong framework for further research at a postgraduate level.

Sean is currently a third year student at the Queensland University of Technology, studying a Bachelor of Mathematics, majoring in Applied and Computational Mathematics. He is interested in many area of mathematics, in particular, the biomedical applications of differential equations, calculus and linear algebra.

A recent collaboration between applied mathematicians and medical researchers at Queensland University of Technology has successfully applied an ordinary differential equation-based model of heat conduction to a series of animal experiments where heat conduction in living porcine tissue is investigated (Andrews et al. 2016). This initial study provides reliable estimates of the thermal conductivity of living porcine skin. This vacation scholar project will build on these preliminary results by analysing an existing, but as yet unpublished data set that explores the question of how to optimally cool a burn injury. This new data set describes temperature data from an in vivo porcine (pig) model of heat conduction where several burn injuries are subjected to different kinds of treatments such as applying cool water of varying temperatures over varying durations. The outcome of this project will be to identify the best strategy for cooling a burn injury. For example, we will explore how cool should the water be, and how long should it be applied for.

This project will train the vacation scholar in several key areas of applied mathematics. Skills developed will include:

• Mathematical modelling – how to idealise a physical heat conduction problem into a suitable model based on ordinary and partial differential equations;

• Mathematical analysis – exact solutions of the differential equations will be obtained using a variety of techniques (direct integration, integral transforms); and

• Model calibration – gradient-based methods for calibrating the mathematical model to experimental data will be used.

This project is sufficiently open-ended that the vacation research student could continue to study an extension of this topic for their Honours project in 2017.

Seamus Albion is currently completing his third year of a Bachelor of Mathematics at the University of Queensland, specialising in pure mathematics. His research interests lie mainly in algebra and number theory. Throughout his degree, Seamus has gained valuable research experience through several projects in mathematical analysis, and wishes to broaden his mathematical knowledge through his vacation research scholarship. In the future, he wishes to undertake an honours project in an area closely related to his project for the summer.

Along with his studies, Seamus is involved in the UQ Mathematics Students Society as president, which organises social events for both undergraduate and postgraduate mathematics students. He is also an avid musician, being a member of the UQ Symphony Orchestra as well as other ensembles both inside and outside UQ.

This is a project in pure mathematics. Its aim is to explore methods from algebraic combinatorics, number theory and representation theory to give a combinatorial description of the characters of affine Lie algebras of type A and their underlying Rogers-Ramanujan type q-series.

Sam is a fourth year student at Monash University, studying dual bachelor degrees in Computer Science and Science. His interests lie in data science and big data analytics.

The goal of this project is to study and improve the optimisation algorithms that are used in deep neural network implementations, and test the improved algorithms on several real-life problems. We will investigate the stochastic gradient descent (SGD) method and will consider methods of variance reduction and second-order versions to accelerate convergence. This will be considered in the TensorFlow framework [1], which is an open source software library that was originally developed by Google researchers and engineers for the purposes of conducting deep neural networks research. We will test the improved algorithms on applications such as the classification of electron microscope images of biological molecules, natural language processing for artificial intelligence, and other related areas.