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Peter Bouwknegt, Alex Molev and Hugh Possingham share the Australian Mathematical Society 2001 Medal

The Medal of the Australian Mathematical Society for 2001 is shared between Dr Peter Bouwknegt and Dr Alex Molev and Professor Hugh Possingham. This is the first time that the Medal has been shared between three recipients.

Dr Peter Bouwknegt

Dr Peter Bouwknegt is an outstanding expert in many aspects of mathematical physics most notably conformal field theory. He was one of the principal developers of the theory of W-algebras which are extensions of the two-dimensional conformal algebra and are relevant in the classification of all (rational) 2-dimensional conformal field theories. Bouwknegt co-authored with K. Schoutens two of the early papers in the area which strongly influenced the development of the subject. Bouwknegt's monograph and review of work in the area (`W-symmetry in conformal field theory') have been extensively cited, and he is considered to be `one of the best-known and internationally most highly-regarded researchers in the mathematical sciences in Australia'.

Bouwknegt's work on W-algebras is regarded as a classic that has had a huge influence on the development of algebraic quantum field theory, especially on the theory of vertex algebras. His work revealed a close connection between W-algebras and affine Lie algebras or Kac-Moody algebras. W-algebras are still an important and active area of research, both in physics and mathematics. Bouwknegt and his collaborators have also established very significant connections between conformal field theory and the theory of quantum groups.

With his collaborators, J. McCarthy and K. Pilch, Bouwknegt has made important contributions to the use of free field techniques in conformal field theory. In particular they pioneered the use of homological algebra techniques in conformal field theory. The most influential contributions are the construction of two-sided resolutions for irreducible affine Lie algebra modules in terms of free field Fock spaces and its underlying quantum group structure and the determination of the spectra of physical states through the computation of the cohomology of a Becchi-Rouet-Stora-Tyutin operator using spectral sequence techniques. This line of research culminated in a highly regarded research monograph.

With K. Schoutens and A. Ludwig, Bouwknegt recognized the importance of quasiparticles in conformal field theories. Their work contributed strongly to the development of this subject. Applications of this work range from condensed matter physics, where it is applied to the fractional quantum Hall effect, to pure mathematics, where it finds application in the geometry of infinite dimensional projective varieties, combinatorics and the representation theory of infinite dimensional algebras. This is still a very active area of research in mathematical physics.

Most recently, with V. Mathai, Bouwknegt pointed out the relevance of Dixmier-Douady theory and the twisted K-theory of certain infinite dimensional C*-algebras in the classification of D-brane charges. This work has applications to string theory. In the opinion of many physicists string theory continues to offer the best chance of developing a successful unified theory of matter incorporating a quantum theory of gravity. Bouwknegt's work with Mathai is presently having a very big impact on the development of string theory and will inevitably feed back into significant new developments in K-theory.

Dr Bouwknegt is an Australian Senior Research Fellow at Adelaide University.

Dr Alexander Molev

Dr Alexander Molev is an expert in the area of representation theory of groups and algebras. In his early work Molev proved deep results on the algebraic structure of the infinite dimensional Lie algebras of vector fields. This was followed by his solution of a difficult problem concerning the unitarizability of the Enright-Varadarajan modules for the Lie algebra u(p,q). As a result, Molev obtained, for the first time, a rigorous derivation of the Gelfand-Graev formulas for u(p,q), which go back to 1965.

A systematic theory of so-called `Yangians' and their finite dimensional representations was created by Drinfeld. Molev developed a highly nontrivial analogue of Drinfeld's theory for the the so-called twisted Yangians and succeeded in providing a complete classificationof irreducible representations of all twisted Yangians.

In 1950, Gelfand and Tsetlin introduced special bases for irreducible finite dimensional representations of the classical Lie algebras of type A, B, D and found explicit formulas for the action of the generators in these bases. However, they were unable to handle the series C. During the next 50 years, the problem of finding an analogue of the Gelfand-Tsetlin explicit formulas for the series C remained one of the most difficult open problems in representation theory. Many people attacked it but no substantial progress was achieved until Molev applied his Yangian techniques to completely solve the problem.

The Sklyanin determinant is a `quantum' (noncommutative) determinant. It is closely connected with the so-called reflection equation and provides a generating series for central elements in twisted Yangians. Molev discovered a remarkably simple formula for the Sklyanin determinant which has a number of applications to explicit constructions of generators in the centre of the universal enveloping algebras of the classical series B, C, D and to the so-called characteristic identities.

The Capelli identity is one of the central results in the classical invariant theory. In joint work with M. Nazarov Molev obtained Capelli-type identities for the first time in the framework of Howe's dual reductive pairs, when one of the two commuting group actions is not linear. This work on the Capelli identities led Molev to introduce a new `super' generalization of the classical Schur functions. In joint work with B. Sagan Molev used an original new approach based on interpolation properties of symmetric functions to obtain a multiplication rule for the factorial Schur functions. This remarkably unifies several versions of the classical Littlewood-Richardson rule. The new `Molev-Sagan' rule has already found applications in the geometry of Grassmannians (Knutson, Tao).

While the classification of irreducible finite-dimensional modules for the Yangian Y(gl(n)) has been given by Drinfeld, the structure of a general representation still remains unknown. Molev has obtained a strong result providing an explicit combinatorial criterion for irreducibility of such tensor products of evaluation modules. This provides an explicit description of a large class of Yangian representations.

Dr Molev is a Lecturer in the School of Mathematics and Statistics at the University of Sydney.

Professor Hugh Possingham

Professor Hugh Possingham is an applied mathematician with a particular interest in the application of mathematical techniques to ecology and conservation. He ranks internationally amongst the leaders of those applied mathematicians who are bringing mathematical tools to coherent analytic planning of effective conservation action. In doing this, he combines a keen appreciation of the practical biological realities with highly original mathematical and computational tools focussed on understanding the population dynamics of target species in relation to their interactions with other species in the community and environment of which they are a part. He is regarded as a "most notable figure among the younger generation of conservation biologists in the world".

Possingham has introduced a wide variety of novel mathematical tools to problems in ecology. For example he has used stochastic mathematical programming and simulated annealing to solve problems in fisheries, population harvesting, fire management, nature reserve design and threatened species management. He has used deterministic methods, mainly differential equations, in general ecological theory as well as stochastic methods (Markov chains) in the study of foraging theory and population dynamics. These applications of advanced mathematical tools have resulted in major advances in ecological theory.

Practical benefits have flowed from his work through computational approaches that he and his group have pioneered. Using specific computer software tools of his own design, he has been highly successful in the modelling of vulnerable populations for conservation management. For example Possingham's population viability analysis package ALEX is a unique and widely used tool for researchers in this field. Other statistical methods that he has applied successfully include bootstrapping analyses for assessing the interpretation of sparse data, and maximum likelihood methods for constructing and testing new models for ecological systems as disparate as mound springs metapopulations and fisheries populations.

Possingham's work on integrating spatial analysis of endangered species into the framework of optimal decision making for acquiring reserve areas makes him "probably the most important person working in conservation anywhere". He brings to these difficult but fascinating problems unrivalled mathematical and analytical skills, together with an outstanding ability to glean and comprehend the major ecological issues involved.

Possingham's expertise has made him highly sought after as a conservation spokesperson and a consultant to government on ecological planning issues. This activity has not diminished his remarkable research productivity or his dedication to training the next generation of mathematical ecologists and applied mathematicians. Possingham has become a leader and a spokesperson for applied mathematics at a remarkably early age.

Professor Possingham holds a Chair in both Mathematics and Zoology at the University of Queensland.

Past Medal Winners

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Last update: 02/10/01