Applied Mathematics is an interdisciplinary area, encompassing modelling, analysis, algorithm development and simulation for problems arising throughout science, engineering and commerce. Research in this area explores the mathematical properties of systems in physics, chemistry, biology, geology, astronomy, materials science, fluid mechanics, the environment, finance, management and many other disciplines.

The Department of Applied Mathematics at UNSW is one of the largest and most successful in Australia. It is an active centre for research, both fundamental and applied. The research areas of the Department are currently grouped as follows:

The research of the Department of Applied Mathematics has been recognised by substantial support from the Australian Research Council, with a total of over $2.8M awarded in Large Grants for the last five years.

Department members have worked on a wide range of research projects in applied mathematics, and have participated in collaborative research activities with numerous distinguished researchers. The national and international standing of the Department in its chosen areas of research has been maintained by a distinguished publication record, with over 300 peer-reviewed research papers appearing over the last five years. Department members have received prestigious awards and international recognition for outstanding research in recent years. Recent highlights are Professor Michael Banner’s award of the Sverdrup Gold Medal of the American Meteorology Society, Professor Ian Sloan’s award of both the Thomas Ranken Lyle Medal of the Australian Academy of Science and the George Szekeres Medal of the Australian Mathematical Society, and Professor Lance Leslie’s Australian Meteorological and Oceanographic Society Medal for 2003.

The Department maintains an active visitors program for international research collaboration, with over 150 distinguished researchers visiting over the last five years in all research areas of the Department.

The Climate and Environmental Dynamics Laboratory (CEDL) is a Research Centre in the School of Mathematics and Statistics, consisting of members from the Department of Applied Mathematics. It is one of Australia's largest and most successful teaching and research groups in meteorology and oceanography, providing high quality postgraduate teaching programmes in environmental modelling and prediction. The Laboratory produces state-ofthe-art environmental prediction models, including coupled ocean-atmosphere-land surface models and predictive systems for the weather, ocean and land surface climate, with a focus on extreme conditions. The center also produces publications in quality journals and monographs, documentation, software, as well as associated analysis tools for end-users.

The Department of Applied Mathematics is also a key participant in the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems. In December 2002 the Federal Government announced funding of $11M over five years for the centre. Its mission is to stimulate research in mathematical and statistical modelling of complex systems, and to encourage cross-fertilization of ideas and techniques. Special areas of focus for the center include dynamical systems, Monte Carlo methods, and scientific computation.

The department also has close links with, and is a heavy user of, regional (ac3) and national (APAC) centres for high performance computing and communications.

The Department of Applied Mathematics provides a vigorous postgraduate program and has a substantial enrolment of high-quality postgraduate students. Over the last five years, 34 students were awarded a PhD in Applied Mathematics.

The fifth International Congress in Industrial and Applied Mathematics, held in Sydney in July 2003, was be the largest mathematical event ever staged in Australia. Members of the Department are involved in organizing the scientific program and are holding a 2-day satellite conference in the School of Mathematics and Statistics on Mathematics of Computation and Approximation. In October 2003 Professor Ian Sloan became president of the International Council on Industrial and Applied Mathematics.

Nonlinear equations describe fundamental physical phenomena in nature ranging from chaotic behaviour in biological systems, plasma containment in tokomaks and stellarators for energy generation, to solitonic fibre optical communication devices. The Nonlinear Phenomena Group at UNSW is world-renowned for its work in soliton theory and dynamical systems and attracts visitors of international repute in these areas on a regular basis. Its activities have been recognised by Chief Investigator representation in the recently awarded Centre for Excellence in Mathematical and Statistical Modelling of Complex Systems. This will provide a base for a major research programme involving the analysis of complex physical and biological systems and the systematic investigation of potential new areas of application of modern soliton theory in magneto hydrodynamics, the fabrication of fibre-reinforced composites and elastic shell structure design.

The design and analysis of computational algorithms, an area of strength at an internationally distinguished level at UNSW, underpins computational modelling in every area of technology and science, and nowadays in finance, medicine and the environment. Some areas of special focus include advanced methods for differential and integral equations, and algorithms for high dimensional problems. Developments in quantum computing are expected to become of increasing importance. Academic and research staff in this area make extensive use of high performance computing and advanced visualisation, in combination with modern analysis. They have strong involvement in the Centre of Excellence in Mathematical and Statistical Modelling of Complex Systems.

Optimisation is the science that integrates information into mathematical models whose solution yields optimal decisions. Optimisation is the most widely used branch of Applied Mathematics in commerce and industry. Research areas of special focus include the variational analysis of nonsmooth optimisation problems, global optimisation and the development of computational methods for structured constrained optimisation problems. These areas are both of intrinsic interest and have wide applications including mathematical finance, medicine and constructive approximation. Extensive use is made of modern nonlinear functional analysis and high performance computing. Applied Mathematics at UNSW aims to expand its activities to develop discrete optimisation.

Stochastic methods are essential to develop higher levels of understanding of systems in which there are uncertainties. Uncertainties are ubiquitous in ocean-atmosphere systems, financial systems and biomedical systems. For example, two recently developed tools that have proven particularly useful for analysing aperiodic and noisy data are the multifractal spectrum and the wavelet transform. The incorporation of stochastic methods in Applied Mathematics is seen as one of the key areas for future development, and will benefit greatly from collaboration with the Department of Statistics.

This area covers meteorology, oceanography, theoretical and applied fluid dynamics, and climate science. Strengths include a high publication rate, success in external grant earnings, and strong postgraduate student numbers. The Geophysical Fluid Dynamics group at UNSW leads the Australian University sector in these three measures of success. Opportunities for the future include linkages within UNSW (BEES, CMCS, Engineering), participation in national and international research programmes, and strengthened undergraduate training.

The discipline of biology is changing from a mostly descriptive science to a quantitative science. Biomathematics is facilitating this transition by developing new techniques to convert the logical but qualitative models of biology into formulae that can be used for quantitative prediction and improvement as well as for extracting patterns from complex data. The recently formed Biomathematics group at UNSW already has internationally competitive research projects in HIV, epidemiology, bioelectric phenomena and tumour growth. It has significant links with biomedical research centers both overseas and within Australia, and is well positioned to expand and make major contributions to the field.

Quantitative finance and risk management have become increasingly mathematical, using sophisticated techniques from stochastic calculus and related numerical methods. This area will continue to grow, involving collaboration with the Department of Statistics within the School and the School of Banking and Finance and the Department of Actuarial Studies from the Faculty of Commerce and Economics.

While engineering and physics are traditional drivers of applied mathematics, they are also the spur for many contemporary activities. An example of current interest to the Nonlinear group is elastic shell structure design and fibre reinforced composites. For the Computational Mathematics group current interests include boundary element analysis, forward and inverse acoustic and electromagnetic scattering.