Research within the Department
Our research is recognised nationally and internationally, with groupings of mathematicians and statisticians working in analysis, biostatistics, category theory, fluid mechanics, mathematical and computational statistics, nonlinear dynamics, optimisation and waves. The Department attracts a substantial amount of external research funding and hosts a steady stream of international visitors.
Our mathematicians have many international collaborations. Members of the Department work with colleagues at many of the other leading Australian Mathematics Departments as well as from many high profile international mathematics departments, including Université de recherche Paris Sciences et Lettres, Université de Bordeaux, Sun Yat-sen University, Central China Normal University, University of Birmingham, University of Wroclaw, INRIA (France), University of Bundeswehr, University of Bayreuth, University of Brest, Penn State University, University of Manchester amongst many others.
Our statisticians collaborate widely with statistical colleagues nationally and internationally at institutions such as University College London, Harvard University and the Chinese University of Hong Kong. Our applied research is conducted in collaboration with colleagues from diverse areas across Macquarie campus, for example the Departments of Clinical Medicine, Biological Sciences, Computing and Applied Finance and Actuarial Studies, and externally with collaborators in for example the Sydney Adventist Hospital, the Brain and Mind Institute (University of Sydney) and the NSW Office of Environment and Heritage.
Our research is supported by a number of vibrant seminars series, attracting both domestic and international speakers. In addition the Department's Colloquium, aimed at a general mathematical sciences audience, features talks by visitors to the Department, both international and domestic, as well as the occasional talk by members of the department.
Our major programs of research are detailed below.
Research in Applied Mathematics
Deterministic and Stochastic Modelling
Research in Fluid Mechanics is covers many broad aspects of modern fluid mechanics research, from the fundamental question of how, and when, fluid flows transition from a laminar to a turbulent state through to modern approach to potential flow theory and on into the behaviour of fluids at small scales. Members of the Department working in this area bring the bear the three threads of scientific discovery, theory, computation and experiment, to understand the complex dynamics of a wide range of fluid flows.
Waves and Nonlinear Dynamics
Research in Waves and Nonlinear Dynamics is focussed on wave scattering and imaging, nonlinear dynamics, mechanics and geometry, and the work spans the areas of theoretical modelling, computational modelling and applications, including industrial applications as well as the theoretical development of the mathematical framework of large classes of models. Particular interests include
- the scattering and the detection of acoustic and electromagnetic waves from complex objects with applications to antenna design and to object identification or imaging (inverse scattering);
- the inversion of static field data for the creation of magnetic or gravity profiles for geological purposes; and
- the modelling of nonlinear effects in electronic systems leading to chaos or other dysfunctional behaviour.
Research in Pure Mathematics
The research interests of the Analysis Group span functional analysis, harmonic analysis, Lie groups, symmetric spaces, partial differential equations and control theory. Some of the problems studied are mathematical formulations of physical situations and the solutions provided have applications to real world problems. For example, wavelet analysis is recognized as a significant advance in signal processing and compression. Similarly, answers to control theoretic questions at the quantum level are essential to realize the potential of the coming generation of computing devices. Members of the Analysis Group enjoy extensive research interaction and collaboration, nationally and internationally, and attract many overseas visitors to Macquarie University. They also receive many invitations to be speakers and main speakers at national and international conferences and workshops.
Category theory is a branch of mathematics which seeks to identify patterns and structures across diverse areas of mathematical discourse, and to distil them down to their essential nature. The Centre of Australian Category Theory (CoACT) is one of the world's oldest and most distinguished research groups in category theory. CoACT members are pioneers of monoidal, enriched, and higher-dimensional category theory, areas which are increasingly fundamental to modern algebraic geometry, representation theory, and mathematical physics. Other fields in which CoACT's research has been influential include algebraic topology and homotopy theory, programming language semantics and quantum groups.
A key focus of the group's scientific activities is the Australian Category Seminar, which has met weekly since its inception in 1971. The seminar is frequently visited by major international figures in category theory and cognate disciplines.
Another focus is the recently-launched journal Higher Structures, which is published by Macquarie University; CoACT member Batanin is a founding editor, while Garner, Lack and Street are all on its editorial board.
Research in Statistics
The members of the Biostatistics Research Group use modern statistical methods to analyse epidemiological data, and they carry out epidemiological studies in collaboration with medical researchers. Current research includes menopause studies, hearing studies and survival and quality of life studies of persons with breast cancer.
The group has also been working on competing risks survival analysis, risk factors for HIV/AIDS, age-period-cohort modelling, the assessment of functional independence among the disabled, and the economic costs and benefits of strategies for health improvement.
The group has several full-time PhD students, maintains strong links with NHMRC centres in Sydney, and collaborates closely with epidemiologists in South East Asia.
- Dr Ken Beath
- Associate Professor Ayse Bilgin
- Dr Petra Graham
- Professor Gillian Heller
- Dr Kehui Luo
- Associate Professor Jun Ma
- Professor Ian Marschner
- Dr Maurizio Manuguerra
Mathematical and Computational Statistics
Members of the Mathematical and Computational Statistics group work broadly within the areas of nonparametric statistics, image processing, signal processing, stochastic finance, time series analysis and optimisation. Applications of their work are in finance, animal nutrition, medicine, defence and quantum computing.