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ADVISORY BOARD
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Director |
Professor of Mathematics, Mathematics Department, Macquarie University | |
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Associate Director |
Professor Michael Johnson PhD |
Director of the Macquarie ICT Innovations Centre, Division of ICS, Macquarie University |
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External Member |
Professor G. Max Kelly PhD Cambridge, FAA |
Professorial Fellow and Emeritus Professor, School of Mathematics and Statistics, University of Sydney |
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External Member |
Dr Wesley Phoa BSc ANU, PhD Cambridge |
Capital Strategy Research, The Capital Group Companies, Los Angeles |
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Honorary Consultant |
Associate Professor Dominic Verity PhD Cambridge |
Postgraduate Coursework Programs Director, Division of ICS, Macquarie University (Previously Director of Categorical Solutions) |
General Description of the Area: Categories concern transformation and composition in mathematics. They provide an algebra of wide-spread applicability for the synthesis and analysis of systems and processes in fields as diverse as physics and computer science, but also in mathematics itself. They can be used to clarify and simplify the learning, teaching and development of mathematics. The aim of the Centre is basic research on categories, training of high quality mathematics and computer science students, and application to geometry, physics, computing, finance, and other industries.
Research: The Australian Category Seminar began in 1971 and has been held alternately at Macquarie and Sydney Universities nearly all Wednesdays since. The meetings typically last three hours and include two lectures. The germ of the Research Centre has existed since 1986 when it was awarded a six year ARC Program Grant. The output of the Centre and the quality of research work has been rewarded by continued ARC Large Grants: in 1998 the four senior members are the Chief Investigators of a joint initial grant (administered by Macquarie University), while Kelly and Street have renewed individual grants. The international reputation of the Centre can be judged from the following anonymous referee's report on our 1998 initial proposal:
"The applicants modestly estimate their team as 'one of the top four or five' internationally in categorical research. However, although Montréal, Edinburgh, and Louvain may house a comparable number of experts, it is clear that those centres derive a significant part of their vitality from their frequent and continuing contacts with the Macquarie-Sydney group. The weekly half-day seminars evidence a degree of organization and dedication equalled by no other team in the world. Those seminars, with their stream of foreign visitors and frequent collaborative missions abroad, are in fact the centre of world category theory and should be adequately funded in order to so continue, in the interest both of world science and Australia's prestige within it."
Other Grant referees' reports can be supplied on request. This year, senior visitors to the Centre include Professor André Joyal (from Montréal, Canada, for the period June 1998 - June 1999), Professor Robin Cockett (from Calgary, Canada, for the period March-May 1998), Professor George Janelidze (from Georgia, Eastern Europe, for the period November 1997-April 1998). Several junior bright mathematicians will also visit for 2-3 month periods. Centre members are frequently invited to give plenary lectures at, and be on scientific committees for, international conferences.
Teaching: This decade the Director and Associate Director decided especially to attract and train high-quality senior students. Street taught a graduate course on quantum groups in the first half of 1990; the notes later appeared on the Web and this had an influence on prospective students and postdoctoral fellows. Currently Street is supervising 5 excellent PhD students. In 1997, one of Johnson's students received the award for the year's Most Distinguished Australian PhD in Computer Science and another was awarded a tenurable Lecturer B position at UNSW on the basis of his research even before he submitted his PhD thesis. Johnson currently supervises 8 PhD students, including 3 University Medallists. Graduates or research associates from the Centre hold positions as senior lecturers, as a senior research fellow in Edinburgh, and responsible positions at IBM, Swiss Bank, and HSBC James Capel. Furthermore, in 1997, Johnson was awarded an Outstanding Teacher Award by Macquarie University and was one of 3 finalists for the inaugural Australian Award for University Teaching in Science. Other teaching activities that Centre members take extremely seriously with successful results are the supervision of Vacation Scholars (designed to give undergraduates some insight into mathematical research) and the supervision of Honours Degree projects.
Applications: Much of the Centre's work is inspired by other areas of mathematics, by physics, by computing. "The mathematical sciences are vital to economic competitiveness. They are a critical, generic, enabling technology." [Mathematical Sciences: Adding to Australia (Australian Government, January 1996) page 33 quoting a US-sourced review by J.G. Glimm.] In particular, recent PhDs in areas related to this project found mathematical research jobs in the Finance Sector working on Options Pricing. Our colleague Professor André Joyal (U Québec à Montréal) runs courses on the mathematics of finance: the original motivation for him came from Categories. Associate Director Johnson and Associate Professor Kit Dampney engage in industrial consultancies. The most immediate industrial application of some of their current work is in the design and analysis of information systems and other complex systems, including business systems. This work has led to consultancies with Telecom Australia and with Caltex Oil Australia, as well as professional education courses delivered to industry by Johnson and Dampney and tutorial introductions to the method at for example the Australian Software Engineering Conference, at industrially organised presentations, and at a number of universities internationally.
A major project of the Centre is the application of higher dimensional category theory to computer science. Monoidal bicategories are an appropriate algebra for studying processes in computer science: the relevant operations on processes being serial composition, parallel composition and feedback. The higher-dimensional aspect arises in analysing abstraction, refinement and bisimulation of processes. Associated with such an algebra is a geometric picture (a process diagram) of expressions in the algebra. Practical applications currently under study from this point of view are: accounting; concurrent, distributed algorithms; model checking of distributed control structures; asynchronous circuit theory; dynamic aspects of data bases; verification of distributed hybrid systems; and, hierarchical structure of communication protocols. Another higher dimensional approach uses dimension to encapsulate modules in software, and this approach is expected to have significant impact on the Software Engineering of concurrent systems.
The Centre's work includes the implementation of category theoretic methods enabling the above practical applications. Indeed, Johnson and Walters are two of the earliest workers in computational category theory, and Johnson is currently editing a special issue of the prestigious Journal of Symbolic Computation devoted to progress in computational category theory.
As an illustration of the diversity of our applications, another major project is the development and application of the theory of weak higher dimensional categories for theoretical physics. This lies behind Street's interest in quantum groups, and his pioneering work is applied by both physicists and mathematicians.
Research Centre: The Centre is at a high point of productivity with international demand for our latest work. There are many excellent research associates and experienced teachers available to join us; there are many high-calibre students, both undergraduate and postgraduate, strongly attracted to the goals of the Centre; and industry is showing readiness to incorporate Category Theory including the high-profile example of Boeing. These three components will be the focus of the Research Centre.
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