The Department of Mathematics investigates a range of mathematical problems involving differential equations and dynamical systems, hydrodynamic stability theory and optimisation and operations research. Other topics of research include mathematics education, statistical mechanics, molecular dynamics and nanofluidics, fluid dynamics and modelling in material sciences and mathematical biology.
More detail on mathematics research is included below.
Differential equations and dynamical systems
Our group investigates the properties of solutions of ordinary differential equations and employs analytical, computational and numerical methods. Our research is focused on systems as diverse as chemical reactors, DNA regulatory networks, epidemiology and interactions between tumours and the immune system. We are also focused on large scale numerical solution of nonlinear partial differential equations, with applications in modelling material failure, fracture and rupture propagation.
Fluid dynamics modelling and computational mechanics in material science
Our group develops models of material failure and damage evolution in anisotropic and composite materials, with the goal of forecasting risk and damage in a range of applications in oil and gas, mining, construction and manufacturing industries.
We are also interested in range of problems at the frontier of fluid dynamics and material science, including drag reduction in fluids, properties of colloidal suspensions, separation of granular and powder materials, cavitation, electrochemical dynamics and Leidenfrost effects.
Hydrodynamic stability theory
Hydrodynamic stability theory aims at finding solutions of partial differential equations describing fluid motion, for example Navier-Stokes, thermal energy equations and constitutive equations describing fluid's physical properties. Our current interests include flows of complex fluids with unique physical properties found in a wide range of modern technological and physical applications e.g. magnetic nano-fluids, piezo-viscous fluids and fluids that exhibit strong variation of their transport properties with temperature.
Others areas of interest include:
- Ferrofluids, suspensions of magnetic nanoparticles in a carrier liquid such as water, kerosene or mineral oil. Due to their very complicated nature, the physical flow behaviour of ferrofluids in non-uniform magnetic and thermal fields remain poorly understood.
- The Faraday instability of a two-layer liquid film with deformable upper surface.
- Gaining a better understanding of geophysical fluid dynamics phenomena, in particular atmospheric and oceanic flows such as hurricanes, ocean currents, atmospheric and oceanic circulations. Deducing under what conditions Langmuir circulation form, grow and die, is of fundamental interest to oceanography and climatology.
Mathematical biology is an active and fast growing inter-disciplinary area in which mathematical concepts and techniques are being applied to a variety of problems in the biological sciences. Our group is very active in the mathematical modelling of a number of fundamental biological processes including:
- predator-prey models for the control of insect populations and pests
- predictive modelling for the development and spread of infectious diseases (pertussis) and epidemics (influenza, malaria and AIDS)
- stochastic and computational models of bacteria and active particles on thin films.
We are also investigating mathematical immunology, with a focus on tumour-immune interactions, immunotherapy and oncolytic virotherapy.
We use a variety of techniques to investigate the effect of new technology and learning practices in tertiary teaching. Examples include:
- the use of tablets by lecturers and student in classes
- the production, usage and assessment of screencasting technology in mathematics and statistics pedagogy, with a focus on the evaluation of the impact video technology has on tertiary students’ learning
- best practice in captioning videos
- Open Educational Resources and Open Educational Practice
- Students as Partners to explain the relevance of mathematics
We are also developing statistical instruments and procedures for learner analytics, with the aim of improving intervention strategies for student retention rates. The engagement of lecturers in education technologies, the impact of Massive Open Online Courses on the Australian higher education sector and the transition from secondary to tertiary education for national and international students are also being investigated.
Optimisation and operations research
The goal of optimisation in mathematical programming is to obtain "best available" values of an objective function subject to certain constraints. Many real-life applications, including scheduling, outputs and quality-monitoring in manufacturing and resource planning require the optimisation of different types of functions. Our main interests include:
- the design of efficient and reliable computational algorithms for the solutions of complex optimisation problems
- the use of polynomial spline approximation through optimisation
- the limits of approximating methods for a continuous function using a piecewise polynomial. The case of knots joining the polynomials being also variable is still an open problem.
Contact: Dr Nadezda Sukhorukova
Statistical mechanics, molecular dynamics and nanofluidics
Our work in this area is primarily devoted to understanding the behaviour of soft matter, i.e. water, polymers in the liquid phase and mixtures, when confined at nano-scales and driven by external fields (e.g. temperature gradients, mechanical stresses, rotating electrodynamic fields). We are particularly interested in developing simulation algorithms that are fully compatible with the principles of statistical mechanics and have a firm theoretical foundation.
Our group has developed some of the most useful and powerful non-equilibrium molecular dynamics algorithms for simulating both bulk and confined atomistic and molecular fluids. Examples of our current research includes:
- the fundamental problem of how one modifies the highly successful Navier-Stokes equations for fluid flow at the nanometer length scale
- the use of non-equilibrium molecular dynamics methods in systems of alkanes and dense polymer melts to study the foundations of rheology
- the role of microscopic chaos in determining the transport coefficients and the ergodic behaviour of atomic systems at the nano-scale
- the effect of velocity and thermal slip in the design of carbon nano-tubes
- the development and application of response theory (both linear and non-linear) to systems of atoms and molecules driven out of equilibrium by external fields.