Statistics:

Applied Mathematics:

Members of the CMA participate with staff from the Department of Mathematics in these programs to form the focal point of research interaction in the School, (refer to SMS Research Programs).

Each program has a large number of visitors and extensive international contacts. As shown later in this report, there is a wealth of powerful international collaboration. A large number of leading mathematicians and statisticians visit us from overseas as well as from within Australia. Our members regularly visit their colleagues overseas, and are invited to participate in international conferences. Indeed, the CMA has an exceptional international profile for its size.

A selection of significant achievements by members of the CMA during the year follows. In collaborative work with the French mathematicians, Auscher and Tchamitchian, and the Americans, Hofmann and Lacey, Alan McIntosh solved a problem which has aroused a lot of interest since it was posed by Kato forty years ago. Min-Chun Hong resolved a long-standing conjecture about energy-minimizing deformations of a ball. Amnon Neeman obtained results with Ranicki about higher-dimensional knot invariants by applyi ng Thomason's localisation theorem to the K-theory of non-commutative rings. Rodney Baxter has given a new formulation of the combinatorial meander problem in terms of the well known Temperley-Lieb algebra.

The CMA conducts several seminar series as well as a weekly colloquium. It publishes a series of research monographs and has extensive series of mathematical and statistical reports.

In its national role, the CMA organises and sponsors programs and conferences involving Australian and overseas researchers. In particular it supports a scheme of National Research Symposia. In 2000, there were 5 such symposia, individual reports of whi ch appear later. The program was overseen by a national committee, while the organisers come from the Australian mathematical and statistical community. The program will continue in 2001, though on a reduced scale. It will be a challenge for the CMA to find support to maintain its national role in the coming years, but every effort will be made to do so. However, its national and international profile remains exceptional, and is expected to continue.

This year Derek Robinson, a long-standing and highly valued member of the CMA, retired; he will continue his association with the CMA as a Visiting Fellow and holds an ARC grant. The next few years will see several more retirements. This presents us wit h the opportunity and the challenge of assessing our role and determining the areas in which to concentrate our resources and build our future strength. We are already committed to development of the applied and computational areas, and have recently app ointed Markus Hegland who has strong connections with other sections of the University and with the Australian Partnership in Advanced Computation. Sue Wilson developed substantial links between our School and the John Curtin School of Medical Research t hrough the establishment of the Centre in Bioinformation Science. The theoretical core of activities must also be kept at its current high level of strength. In this way, the CMA will remain the strongest mathematical and statistical research group in t his country and maintain its international reputation.

Joe Gani was awarded Membership of the Order of Australia (AM) for his contribution to mathematical research, in particular applied probability and mathematical biology. He was also awarded the Moyal Medal when he gave the inaugural Moyal Lecture at Macq uarie University. Peter Hall was elected to Fellowship of the Royal Society of London.

The Mathematics Department is a part of the School of Mathematical Sciences as well as a member of the Faculty of Science. It has both teaching and research roles and carries the primary responsibility for undergraduate teaching in Mathematics in the Uni versity. Much of this year was taken up with digesting the many changes flowing from decisions made by the University in 1999. Overall the year was a successful but difficult one for the Department.

Undergraduate student numbers continued the strong growth of the previous year. Especially pleasing is the growth in numbers of students in our first and second year honours units - they are growing even more strongly than our overall numbers. We contin ue to attract a large number of National Undergraduate Scholars and Distinguished Scholars. Two students were awarded University Medals. The restructuring of our units to fit in with the new 8:8:8 structure has been well received by students. Some unit s were offered by videoconferencing with Monash and Sydney universities. The individual units went well despite the technical inadequacies of present facilities for teaching at this level. However overall there are a number of problems that make it unli kely that this method of teaching will be used extensively. Brian Davies' book "Exploring chaos: Theory and Experiment" has continued to receive critical acclaim - as well as several enthusiastic reviews, Choice (Current Reviews for Academic Libraries) w ill include the book in their annual Outstanding Academic Titles list, published in January 2001. This is a list of 600 books chosen from more that 25,000 titles in all academic disciplines and reviewed by them in the year 2000.

Throughout the year we continued our series of school visits (in collaboration with the University of Canberra) to promote the study of mathematics at the secondary level.

In addition, we have completed the groundwork for several new initiatives which will start next year. A new unit, Poetry of the Universe, will be offered jointly with Philosophy. The Department will receive support from the Vice-Chancellor's Plan for Gr owth to make an appointment in bioinformatics. This will, with the Centre for Bioinformation Science, promote the study of bioinformatics at the ANU. A major initiative is the establishment of a new degree in Computational Science. This recognises the increasingly important role of computational modelling in a wide range of applications and gives students the opportunity to combine depth in mathematics and computer science with the study of an area of application. This initiative is supported by the e ducation program of APAC. The development of an honours school in mathematics as an SMS wide initiative made substantial progress. The research programs of the SMS will offer honours level courses and the SMS has provided funding for the offer of honour s year scholarships from 2001. Postgraduate enrolments remain steady but low; honours graduates in mathematics are in high demand from employers and it is becoming increasingly difficult to recruit postgraduate students in mathematics.

The heavy demands of maintaining our teaching program and adapting to all the changes of the previous year have made research time a precious commodity. The majority of staff have continued to pursue research vigorously and productively. Our output was maintained at a high level, as demonstrated by the number of publications and visitors to the Department and the number of visits by staff to collaborators in other universities. The grant support for mathematics in Australia is very low, with the ARC be ing the only external funding agency to support research in theoretical areas. We had no new large grants in 2000, but received a number of small grants. John Hutchinson has been awarded a large grant for 3 years from 2001 as well as a grant from the ne w Australian-German Joint Research Co-operation Scheme. Several members of staff were successful in obtaining small grants for 2001.

This year saw the retirement of the two longest serving members of the Department (Haralds Petersons and Martin Ward); they continue their association with the Department as Visiting Fellows. A mini-conference was held to celebrate the sixtieth birthday of Bob Bryce. We welcomed back Steve Roberts from his secondment to ACSys. Steve has been one of the driving forces behind the establishment of the new Computational Science degree. We also welcomed the (long overdue) promotion of Rick Loy to Reader.

Maintaining our teaching program has placed considerable strains on the Department. The growth in student numbers together with an infusion of special funds will enable us to make new appointments in the coming year. New staff, growing students numbers and a number of new initiatives mean that we look forward to a year which will be difficult and full of challenges but also full of promise.

In June 2000 academic staff of the School of Mathematics were interviewed by the 7.30 Report to discuss the Navier-Stokes Equations - a riddle that has stumped the world's leading mathematicians for more than 200 years. An offer of big money by the Ameri can Clay Mathematics Institute to solve that and six other extremely complex mathematical problems had provoked the media's interest in the topic.

This occasion provided the School with the opportunity to explain the relevance and importance of mathematics to solving everyday problems and to encourage study in mathematics to a national audience.

By all accounts it was a huge success.

Solution of 40 year old mathematical problem

About 1960 Professor Kato of the University of California Berkeley, during his investigation of the evolution of physical systems, was led to pose a key question about the square roots of operators. The square root of an operator is an operation which wh en performed twice gives the original operator. The question asks whether the square root is stable under small perturbations of the original operator, this being useful in computing mathematical solutions of the physical evolution equations.

It took until 1981 for the one-dimensional version of this problem to be solved in the fundamental work of the Frenchman Meyer, the American Coifman, and the Australian McIntosh, then at Macquarie University. This solution depended on the development of techniques in the deep field of harmonic analysis, and indeed helped contribute to further innovations in this subject. It was only in 2000 that the two dimensional version of this problem fell to the efforts of Hofmann and McIntosh, to be followed some months later by the full three dimensional solution, in a joint attack by the two Americans Hofmann and Lacey, the two Frenchmen Auscher and Tchamitchian, and McIntosh who is now at the ANU.

World Mathematical Year 2000 Conference
Why Mathematics Matters - Exploring ideas and their applications
24-25 November 2000

This conference was organized by the School of Mathematical Sciences, and sponsored by the Australian Mathematical Society and the Australian Academy of Science. The Organizing Committee consisted of Professor Joe Gani (ANU), Professor Cheryl Praeger (U of Western Australia) and Professor Ian Sloan (U of New South Wales).

The Organising Committee canvassed colleagues in all Australian states, and selected speakers from the many names suggested to them. The intention was to cover the widest possible range of mathematical topics in non-technical talks, designed to be ac cessible to school leavers and anyone interested in the role which mathematics plays in everyday life.

The conference was widely advertised in newspapers and on 2 radio stations and gave every impression of being a success. A wide range of people expressed their interest in the topics discussed and commented favourably on the choice of speakers and subjec ts. The talks were clearly enjoyable and informative. However, it was a pity that more school leavers did not attend; one must conclude that alternative strategies for reaching the schools and schoolteachers will need to be devised if further progress w ith outreach programmes is to be made in future.

Public lecture by Vaughan Jones
"A personal and very biased history of the mathematical theory of knots"

On 15 February the School hosted a public lecture by Vaughan Jones, Professor of Mathematics at the University of California at Berkeley, on knots and their mathematical properties. Jones is not your average mathematician. For one thing he is a winner, in 1990, of the Fields Medal (the Mathematics equivalent of a Nobel Prize) for his discovery of quite astonishing relationships between geometric topology and other areas of mathematics. A New Zealander, he delivered his Fields Medal lecture wearing an " All Blacks" rugby jersey. The capacity crowd that attended his ANU lecture in February was not disappointed either. They were captivated by smoke-ring displays and details of how to calculate their own invariant polynomials, as well as hearing a particu larly lucid account of the history of mathematical knot theory and its many practical applications.

Algebra

K-theory

For various knot-theoretic applications, it is very useful to study localisation theorems in the K-theory of non-commutative rings. The results by Schofield (Cambridge) are particularly tantalising, since they yield an exact localisation sequence, which so far has only six terms; it involves K₀ and K₁, and the question whether it can be continued to an infinite sequence remains open.

About 12 years ago, Thomason proved a very nice localisation theorem in K-theory. As originally stated, the theorem was about vector bundles on algebraic varieties, objects which are somehow very like what one studies in commutative ring theory. In the early 90's, Neeman generalised Thomason's result to the topological context. Ranicki (Edinburgh) recently asked whether either Thomason's old theorem, or Neeman's generalisation of it, apply in the context of Schofield's results. The answer seems to be that they do; Neeman and Ranicki are currently writing a joint paper on this.

Neeman also worked with Nenashev (Singapore) on defining the lambda operations in K-theory directly in the derived category.

Finite groups

Bryce and Hawkes (Warwick) have been studying the conjugacy class poset associated with a group. There is a natural partial order defined on the frame, the set of all conjugacy classes of subgroups, of a group. This project is directed at understanding connections between the properties of a group and those of its frame. For example, when is a class of groups characterised by the class of the frames of its members? It can be shown that a finite group whose frame is isomorphic to the frame of the alternating group of degree 5 is isomorphic to the alternating group of degree 5. On the other hand, the class of cyclic groups is not characterised by their frames. The present emphasis of this project is on the class of finite soluble groups. Rather restrictive conditions have been found on the frame of a finite group necessary to ensure the group's solubility. Work is continuing on the question of the sufficiency of these conditions.

Bryce and Serena (Firenze) have continued work on covering groups with subgroups. It has long been known that if a group is the irredundant union of finitely many, say n, proper subgroups, then the intersection of these subgroups has finite index bounded above by some function of n. Let f(n) denote the best such bound. There are known upper and lower estimates for f(n). However, the gap between these increases exponentially with n, so one at least is seriously imprecise. The suspicion is that the known lower estimate is close to f(n); indeed, for some small values of n the known lower estimate is f(n). The calculations of these estimates are elementary, using very little of the plethora of theory potentially available. One part of this project is an attempt to reduce the problem of finding f(n) to one where the theory may be applied. Bryce and Serena have extended somewhat the range of n for which f(n) is known precisely. Another aspect of this research concerns so called minimal covers of a group, that is collections of n subgroups of the group whose union is the whole group, and with n as small as possible. In particular, it is of interest to characterise those groups with a minimal cover of subgroups enjoying certain restrictions. As one step in this direction, Bryce and Serena have characterised all groups with a minimal cover consisting of abelian subgroups.

Cossey and Stonehewer (Warwick) continued their investigation of the normal closure of a cyclic quasi-normal subgroup in a p-group. They have completed the odd order case. The even order case seems much more complicated and is still incomplete.

Cossey and Alejandre (Elche, Spain) began an investigation of the mutually permutable products of finite supersoluble groups; they have obtained a good structure theorem. This work is currently being written up for publication.

Cossey and Hawkes (Warwick) continued their investigation into the relation between the number of distinct irreducible character degrees and the number of distinct conjugacy class sizes, concentrating on the case of two class sizes or two irreducible character degrees. Results so far indicate that in this case there is a relation between the two invariants, and results obtained so far confirm their conjecture about the relation.

Havas (Queensland), Newman and O'Brien (Auckland) have improved methods for finding presentations with deficiency zero for finite groups with trivial multiplicator.

Updating earlier work, Kovács completed a paper with Praeger (UWA) on minimal faithful permutation representations of finite groups: this was published in the course of the year.

Ormerod has continued to work with finite p-groups and has obtained a set of relations which must hold for any finite p-group in which every cyclic subgroup has defect at most 2, as long as the prime involved is greater than 3. A closely related class of groups is the class in which every subgroup has defect at most 2. She has found several families of groups with this property, again when the prime is larger than 3.

Lie algebras

Jurman worked on modular graded Lie algebras of maximal class and thin Lie algebras; in particular he studied a family of simple finite-dimensional Lie algebras in characteristic 2 called bi-Zassenhaus algebras. He investigated the structure, the presentation and the cohomology of those algebras and of their infinite-dimensional loop algebras.
Young has also been studying thin Lie algebras, attempting to classify those algebras where the distance between the second and third diamonds is greater than the distance between the first and second diamonds.

Kovács continued work with Bryant and Stöhr (UMIST) on the module structure of free Lie rings and free Lie algebras. A substantial paper was completed and submitted (and issued as a Manchester preprint), and two other papers have been drafted and are close to completion. One of the latter brings to a very satisfying conclusion a line of research began nearly thirty years ago, settling the GL(2,p)-module structure of all homogeneous components of the free Lie algebra of rank 2 over an arbitrary field of characteristic p. The isomorphism types of the indecomposable direct summands are determined and a recursive method given for calculating their multiplicities. In particular, this shows that the indecomposables involved are either simple or projective or of dimension p - 1 and composition length 2.

Newman continued his work with Caranti (Trento) on graded Lie algebras with maximal class. They are now turning their attention to finite-dimensional algebras of this type which cannot occur as quotients of infinite-dimensional algebras of this kind. They have developed new methods for constructing such algebras.

Miscellaneous

Jurman began the study of cryptology, mainly focussing on recent results about successful cryptanalytic attacks to public-key encryption methods, like RSA.

Neumann continued to work in Universal Algebra and in Group Theory.

Newman continued work on certain families of fundamental groups of three-manifolds. The groups in each family are parameterised by an integer parameter. He obtained lower bounds for the number of generators for these groups which are linear in the parameter thus refuting the suggestion that all these groups can be generated by two elements.

Analysis and Geometry

Analysis on Lie Groups

Robinson and Dungey, with ter Elst (Eindhoven), studied Gaussian bounds for the kernel of a semigroup generated by a sum of the 2m^th powers of invariant vector fields satisfying the Hörmander condition on groups of polynomial growth. They showed that the decay of the kernel for large times is direction dependent: in some directions the decay is like an operator of order 2m, in others only like a second order operator. They related this in a precise way to the Lie algebraic properties of the invariant vector fields. The analysis was closely related to analysis of almost periodic operators on Euclidean space.

Robinson, Dungey and ter Elst analysed second order operators in divergence form with almost periodic coefficients. It was established that the corresponding kernels are Hölder continuous and the kernels and their Hölder derivatives satisfy global Gaussian bounds. Moreover the Riesz transforms are bounded. This work builds upon prior research with Sikora on operators with periodic coefficients.

Kato square root problem

The Kato problem asks whether the square root of an operator L, a second order elliptic operator on a bounded domain in divergence form with bounded measurable complex coefficients, is a well behaved operator in the sense that it is bounded from the Sobolev space H¹ to L². This had been a celebrated open problem since the early 1960's. McIntosh and Hofmann (Columbia, Missouri) solved the problem in two dimensions, and showed that the square root property is retained under small perturbations. The problem in full generality was then solved by McIntosh, Hofmann, Auscher (Amiens), Lacey (Georgia Tech) and Tchamitchian (Marseilles).

Clifford analysis on Lipschitz domains

McIntosh and Axelsson are using Clifford analysis and Dirac operators to study electromagnetic waves of a fixed frequency propagating in a three dimensional space which contains homogeneous objects which are opaque to or penetrable by the radiation. In joint work with Hogan (Arkansas) and Grognard (University of NSW) they used Rellich inequalities and Hardy spaces to show the solvability of boundary value problems with square integrable boundary data.

In continuing work, they are looking at how to match piecewise defined electromagnetic fields across interfaces in order to study the problem of transmission of electromagnetic waves in the presence of penetrable Lipschitz-like objects with distinct characteristics.

Spectral multiplier theorems

A spectral multiplier theorem is a statement about functions F(L) of an operator L, for example that F(L) is bounded on L^p (given conditions on F and p). Sikora and Cowling (University of NSW) studied spectral multiplier theorems for sublaplacians on SU(2). In standard spectral multiplier theorems for elliptic operators the critical index p is determined by the dimension of the underlying space. Sikora and Cowling proved a Hörmander type spectral multiplier theorem for a sublaplacian on SU(2) with the critical exponent equal to half the Euclidean dimension of the group. This is an analogue of the result obtained by Hebisch, Müller and Stein for the Heisenberg group. They also developed techniques of proving very precise multiplier results for operators whose corresponding wave equation has the finite propagation speed property.

Sikora, Duong (Macquarie) and Ouhabaz (Marne-da-Vallée) studied general spectral multiplier theorems for self-adjoint positive definite operators on L²(X), where X is any open subset of a space of homogeneous type. They showed that sharp Hörmander-type spectral multiplier theorems follow from the appropriate estimates of the L² norm of the kernel of spectral multipliers and the Gaussian bounds for the corresponding heat kernel. Examples include sharp spectral multiplier theorems for a class of scattering operators and new spectral multiplier theorems for Laguerre and Hermite expansions.

Hardy spaces

McIntosh and Lou showed that divergence-free Hardy spaces can be represented using divergence-free atoms. They are now working to prove a similar result for spaces on bounded domains.

Banach algebras

A Banach algebra is said to be amenable if every derivation into any dual bimodule is inner. Loy and Ghahramani (Manitoba) are investigating several different generalisations of this notion. They have shown that in some classical situations the generalised notions coincide with the standard one. In particular this has yielded completely new proofs of several known results.

Loy and Willis (Newcastle) are investigating the sequence of norms of powers of a quasinilpotent element of a Banach algebra. The idea is to use this sequence as a tool in place of the spectrum which gives no information in this case. As a first step they are examining the case of compact multiplication, where recent work with Read (Leeds) and Runde (Alberta) produced examples having hitherto unexpected properties. They have obtained a characterization of which sequences can occur as the norms of powers of a compact quasinilpotent element.

Right topological groups

Loy and Lau (Alberta) are studying compact right topological groups, which are structures which arise as compactifications of general locally compact groups, as well as in topological dynamics. These groups are in general not topological groups in that left multiplication and inversion fail to be continuous. Loy and Lau have developed a new approach to convolution on such objects, allowing them to bring the methods of abstract harmonic analysis to bear on the study. They have showed that the resulting algebras agree with the usual ones if and only if the group is a compact topological group.

Spectral and scattering theory

Hassell and Vasy (MIT) studied the resolvent kernel for the Laplacian on an asymptotically conic space (that is, a complete manifold locally Euclidean at infinity). Such spaces have a natural compactification to a manifold X with boundary. They showed that when the spectral parameter h lies in the continuous spectrum, the resolvent kernels R(h ± i0) are Legendrian distributions on a manifold with corners which is a blowup of the space X × X.

Hassell, Melrose (MIT) and Vasy have been studying scattering theory for nondecaying potentials on asymptotically conic spaces. A typical example of the operators considered is delta + V on Euclidean space, where V(x) is a smooth bounded function which is homogeneous of degree zero for |x| > 1. They have characterized the generalized eigenfunctions and analysed the scattering matrix for such operators at finite energy. The scattering matrix is completely different to the case of decaying potentials; it lives only at the critical points of the potential restricted to infinity.

Hassell and Vasy also analysed Legendrian distributions on manifolds with boundary, and specified the precise relationship between two different classes of distributions which have been used in microlocal analysis on manifolds with boundary.

Functional calculus

McIntosh, Auscher and Duong showed that operators with heat kernel bounds which satisfy L² quadratic estimates also have L^p quadratic estimates. They are now working on the problem of determining the correct definition of a Hardy space associated with operators which have a bounded holomorphic functional calculus in an L² space.

Complex analysis

Isaev and Kruzhilin (Moscow) classified connected n-dimensional complex manifolds that admit an effective action of the unitary group U_n by biholomorphic transformations. One somewhat surprising consequence of this classification is an affirmative answer to the following problem formulated by Krantz (Washington). Let M be a connected complex manifold of dimension n and Aut(M) the group of all biholomorphic automorphisms of M. Assume that Aut(M) is isomorphic as a topological group equipped with the compact-open topology to the automorphism group Aut(C_n) of complex space C_n; does it then follow that M is biholomorphically equivalent to C_n?

Isaev also found an easier proof of the affirmative answer to Krantz's question in the case of Stein manifolds.

Theoretical rheology.

Loy and Anderssen (CSIRO) have worked on the theoretical underpinnings of various models in polymer dynamics. Together with Davies and Newbury they have given a rigorous proof of a recent sampling localization result of Anderssen and Davies, which used a formal argument to determine the support of what was loosely termed a distribution. The proof holds for suitably restricted data functions, the restriction being very mild from a practical point of view. Loy and Anderssen also gave a technically rigorous proof of a result of Anderssen and Mead concerning molecular weight scaling, which is needed in recovering molecular weight information about polymers from experimental data.

Nonlinear and Applied Analysis

Nonlinear Partial Differential Equations and Applications

Andrews established a new regularity theory for parabolic equations in two spatial variables, analogous to that known for elliptic equations in two variables. In higher dimensions he extended regularity theory to treat equations satisfying weaker concavi ty conditions as a function of second derivatives. New lower order methods were developed for proving gradient estimates for quasilinear non-divergence form parabolic and elliptic equations.

Booth studied the existence proof to solutions of the stationary Maxwell-Dirac equations, in the light of the k-monogenic function theory of McIntosh et al. Problems under investigation included: generalizing k-monogenicity to the case wher e k is a vector, and fully understanding the significance of the nonlinearity.

Trudinger and Wang completed a paper on weak continuity and potential theory for quasilinear subelliptic equations. This was an outgrowth of their previous treatment of Hessian operators, including applications to potential theory by Labutin, and include d, for example, a Wiener criterion for regular points.

Computational Mathematics

Hutchinson finished work with Dziuk (Freiburg) on approximating surfaces of prescribed mean curvature. They also studied the problem of convergence estimates for numerical approximations to mean curvature flow for surfaces and to the flow for curves driv en by elastic energy.

Trudinger with Kuo (Taiwan) investigated the discrete Aleksandrov maximum principle and Schauder estimates, with applications to discrete schemes, thus extending prior results.

Geometric Problems

Andrews investigated the use of highly nonlinear parabolic equations with carefully chosen nonlinearity for various specific applications. He applied these methods to prove new results on evolution and classification of surfaces, hypersurfaces and Rieman nian metrics. He also continued his work on higher order diffusion equations arising in affine, conformal and projective geometry.

Fang with Hwang (Academia Sinica), studied minimal surfaces bounded by parallel lines. They constructed new reflective examples of such surfaces and generalized doubly periodic minimal surfaces. They also proved graph theorems for such surfaces and for surfaces of more general type. They are currently working on the uniqueness of solutions to the Dirichlet problem on unbounded domains. With Weihuan Chen (Peking), Fang studied self-theta congruent minimal surfaces.

Hong with Giaquinta (Pisa) proved the partial regularity of minimizers of a Dirichlet p-energy type functional involving both forms and maps.

Hong also showed for any real number p with 1 < p < n - 1 that the map x/|x| is the unique minimizer of the p-energy functional for maps from an n-dimensional ball into an (n-1)-dimensional sphere. This question had been open since the 1989 paper of Coron and Gulliver and was finally completely solved by Hong.

Trudinger and Wang continued previous work on the Chern conjecture, addressing boundary value problems and regularity.

Urbas studied Hessian and curvature equations, proving the existence of globally smooth Weingarten graphs with prescribed Gauss image. This continued a series of papers on boundary value problems where the boundary condition is to prescribe the gradient image of the solution. In previous work this problem was solved for the Monge-Ampčre equation (1997) and for Hessian equations (to appear in 2001). More general boundary conditions of this kind are of interest in connection with mass transfer problems. Some progress was made on this during the year, but further investigation is required. Some progress was also made on the second boundary value problem for two dimensional Monge-Ampčre equations in nonconvex domains.

Urbas also studied the global regularity of two dimensional graphs of prescribed Gauss curvature. A particular case of this work is the following optimal result: any two dimensional surface of constant positive Gauss curvature which is a graph over a smo oth convex domain is globally Hölder continuous with exponent 1/2. Urbas continued his research on interior regularity of solutions of curvature equations and related monotonicity formulae, with the aim of extending his recent results for Hessian equatio ns.

Other Physical Problems

Andrews studied qualitative behaviour of interface evolution with anisotropic diffusion. He proved general results concerning regularity and asymptotic behaviour of hypersurface evolution equations involving explicit dependence of speed on direction.

In previous work on the nonlinear coupled Maxwell-Dirac equations, Booth with Jarvis (Tasmania) and Legg (Tasmania) solved (algebraically) the Dirac equation for the potential in terms of the Dirac spinors and their first derivatives. They then derived s ome consistency conditions which the spinors must satisfy. Using the properties of the Clifford algebra, they are now examining these conditions and generalizing them to higher dimensions. This higher dimensional version corresponds to super-symmetric m atter.

Howe applied estimates for eigenvalues to obtain stability tests for systems of differential equations from feedback control.

Hutchinson with Rüschendorf (Freiburg) worked on the problem of applying techniques from their recent papers on random fractals to the approximation of stochastic processes with self-similarity properties.

Trudinger and Wang wrote a paper providing a short proof of the Monge mass transfer problem. In view of a recently found gap in Sudakov's original proof in 1979, this may be the first published complete proof of the solution of this historic problem. Th ey also found counterexamples showing that solutions are not smooth in general.

Statistical Science

Statistical Methodology and Theory

Data sharpening techniques involve perturbing data values with the aim of enhancing particular features of an estimator. Hall and Kang's research in this area included techniques for reducing bias, reducing variance, enforcing qualitative constraints (su ch as monotonicity or unimodality of curve estimators) and enhancing robustness.

Conventional methods for conducting nonparametric smoothing using kernel smoothing involve using the same bandwidth both inside and outside the kernel. However Hall showed that the two bandwidths can profitably be taken to be different, particularly if b ias is to be reduced, and in particular bias can generally be reduced by an order of magnitude by allowing the inner bandwidth to be a slight perturbation of the outer one.

There is no extant method for consistently, and nonparametrically, estimating a mixture distribution without using training data. However in some cases, for example where the components are multivariate with independent marginals and the number of dimens ions is sufficiently high, Hall showed that root-n consistent estimation is possible.

Hirst, together with Hall, examined some of the practical and theoretical issues involved with the bandwidth test for regression functions, and is developing a method of calibrating this test to improve the accuracy of its results.

Huang collaborated with Y Ogata (Institute of Statistical Mathematics, Tokyo) on spatial statistics, and developed a new class of estimator based on maximizing the generalized pseudolikelihood. Also, he studied MCMC (Markov Chain Monte Carlo) methods for calculating values of the maximum likelihood estimator and for high-dimensional integration, as well as for the spatial model selection problem.

Rau's research with Hall has been targeting questions of locally-parametric estimation of curves and boundaries in a bivariate regression or density surface. Potential application areas include oceanography and geology, image analysis more generally, and econometrics.

Nonparametric prediction is an underdeveloped field in statistics. Rieck, together with Hall, has been developing an approach for the construction of nonparametric prediction intervals that exhibit beneficial theoretical and numerical properties. While some of the techniques being proposed are computationally intensive, they are convenient to implement.

In nonparametric statistics, Welsh worked with Yee (Auckland), Carroll (Texas A & M) and Lin (Texas) on marginal nonparametric models for correlated data. This work showed a particularly interesting difference between kernel and spline methods which is n ot apparent in the independent case.

In ongoing research in robustness, Welsh worked with Ruckstuhl on robustness issues with binomial-like data and with Peng on robustness with extremes.

The density estimate for a data set should reflect the overall pattern of the data - the peaks and valleys and symmetry of the distribution. Often the boundaries of the distribution (points within which all observations lie) are known, and the density pattern near these boundaries is of particular importance. Whiting, toge ther with Hall, has been exploring some novel proposals for remedying edge effects with the aim of producing a method of density estimation applicable to multivariate data that possesses improved performance at endpoints without suffering from the common faults of many popular remedies.

Data Mining

A range of current problems in data mining involve identifying or filtering documents on the world wide web by measuring their distances to other documents. Current uses of this technique are to identify redundant documents in web searches, and to identi fy forgeries. Hall has been employing related ideas to develop techniques for dating medieval manuscripts, using training data from dated manuscripts. The first step of the project involves developing a large and rich class of metrics among "shingles", or word patterns, on manuscripts. The next consists of using nonparametric regression, and the distance functions, to regress the dates of dated manuscripts onto properties of shingles. Undated manuscripts that are neighbours of dated ones may be identi fied using the metrics, and the regression formula developed for dated manuscripts may then be used to impute the missing dates.

Do patients keep returning to the same doctor, to the same laboratory, to the same pharmacy, or do they shop around? To answer questions such as these, Huang as a member of ACSys Data Mining group developed data mining techniques to analyse two huge data sets from HIC (Health Insurance Commission) and HAC (Department of Health and Aged Care). The two data sets are of 5.8GB and 8.5GB sizes. Both are of Medicare transaction records of patients' payments to hospitals, laboratories or pharmacies. For the HIC data set, for example, Huang proposed the concepts of a loyalty score, an attrition score and departure rate for evaluating doctors, laboratories or pharmacies. He also designed a fast algorithm for calculating those scores and departure rates, which was not an easy task as the data set is huge. Concerning the HAC data set, Huang found the encryption algorithms used for several sub-data sets were different. This is an important discovery as the inconsistency is a serious error making nonsense of an y further analysis. Based on methodology from coding theory and statistics, he calculated some distribution values of the HAC data, which showed the inconsistency.

Applied Statistics

Hall and Rao have been developing methods for the discrimination of remotely-sensed data. In particular they treated the data as replicas of random surfaces, using functional principal components analysis.

In the area of modelling data with extra zeros, Welsh worked with Zhou (Indiana) on developing transformation models for heteroscedastic, nonlinear data with extra zeros. He also continued work with Dobbie on handling correlated data with extra zeros. They have been investigating the possibility of modelling such data using the Neyman type A distribution, and the proposed methods were illustrated by analysis of counts of Leadbeater's possums. These data were also analysed using a conditional Poisson m odel, and to formally compare the models from the two different methods, they extended Cox's test for comparing non-nested hypotheses. If the counts are recorded for the same subjects over time, the readings may be correlated. They extended the conditio nal Poisson model to take account of this possible serial dependence between counts. Their model for correlated zero-inflated count data was illustrated through analysis of counts of Noisy Friarbirds in Canberra. Finally, they investigated models for co rrelated zero-inflated counts where the maximum count is small. An analysis of Golden Whistler counts was used to illustrate these methods.

In estimating abundance of plant and animal populations, Welsh worked with Melville to develop methods that overcome the difficulties he and S Barry had previously found with the standard methods.

Statistical genetics

Wicks' research has been using transmission data for genetic marker loci in families in which one or more members are affected by a disease. The aim is to map disease-susceptibility loci relative to the known marker loci. This involved the development o f complex statistical genetic modelling and the application of likelihood theory.

Wilson worked in the general area of the identification and analysis of candidate genes for complex diseases. For identification of candidate genes bioinformatic methods are being introduced, while for the analysis of candidate genes very general statist ical modelling methods are being developed that can incorporate genetic and environmental effects and their interactions. In particular, she examined what the effect on the analysis of data might be if a single disease gene is assumed when, instead, two (or more) genes are interacting to cause the disease. Using a general genetic model, she developed a global approach to analysing two marker loci. Wilson showed that the problems of reproducibility could indeed arise when multiple genes interact to caus e disease. So, finding that conclusions differ from study to study may be indicative that the "disease gene" under investigation is interacting with other disease genes. This is a general conclusion with broad implications.

Wilson also worked with Huttley (JCSMR) on the development of statistical methodology for a variety of genomic data problems. In particular, they have been developing techniques for the analysis of population genetic variation at the molecular level.

Stochastic Analysis

Epidemics

Daley continued discussions with Isham (U Coll. London) on amalgamation of different mechanisms describing different phases of the evolution of epidemic models.

Gani studied particular models for the transmission of HIV by needle exchanges among intravenous drug users, and the spread of the Ebola virus by the re-use of infected needles.

Examination marks

Interest revived in questions regarding scaling examination marks, both with regard to NSW system and the ACT system. Of interest to both is the simulation of examination mark datasets. For the latter, Daley revisited his 1990 Report to the ACT BSSS and the system now implements another of its recommendations with respect to the use of a reference scale (i.e. SAT) scores. Work has begun on studying the question of significant variability in the error variance between colleges as a possible source of ex cessive numbers of students from larger colleges gaining 'top' scores.

Queuing problems

Daley's work, with Servi (GTE Laboratories Inc., Waltham, USA) on modelling call centre operation and management continued with the study of hand-off traffic in mobile telecommunication systems. The work was largely completed, and pointed to a need for a simple way of demonstrating sample path comparisons.

Daley, in joint work with Glynn (Stanford) made progress on making inferences from limited information in queuing models with balking. This work is ready for final review and submission.

Long-range dependence

Heyde's work with Liu and Anh (QUT) on long-range dependence continues under an ARC Large Grant.

Daley completed and published joint work with Vesilo (Macquarie) and Rolski (Wroclaw). Independent work on the moment index for the busy period of an infinite server queue was extended to establish conditions for the busy period distribution to have a su bexponential density function when the same is true of the service time distribution. Real variable techniques further emphasized the fact that transforms are not needed in such studies.

Stochastic geometry problems

Daley completed to technical report stage a study of germ-grain models that may describe realistically the observed volume fraction of aggregate in concrete.

Point processes

Daley began work in earnest on a revised edition of the treatise (with Vere-Jones (Auckland)) "An Introduction to the Theory of Point Processes" (Springer Verlag, 1988). An amalgam of the two long basic theory chapters was prepared, and over a half of th e remaining old text was proof-read after re-keying.

Festschrift editing

Daley began a major task of editing and preparing for publication a festschrift for Vere-Jones entitled "Probability, Statistics and Seismology". The third of these topics is a fertile area of both probability modelling and statistical inferential proble ms. The problem of approximating the Kagan distribution was tackled, but limited progress made.

Subexponentiality

As a result of an extended visit to Kluppelberg (Technical University of Munich), Daley began work with Baltrunas (Vilnius University) on large deviation problems involving subexponential distributions. It was developed and refined.

Liu commenced work on matrix calculus, inequalities, regression and ARCH models. Much attention was paid to time series analysis, especially analysing financial time series data with heavy tails, long-range dependence and self-similarity.

Patterns in sequences of random events

Gani continued his work in this area. Interest is heightening with possible applications of the ideas to studying gene sequences.

Risky asset models

Heyde, jointly with Liu and Wong, Gay (Monash) and Kou (Columbia) has done considerable work on the incorporation of new evidence about strong dependence of financial time series and fractal scaling into a minimal market model.

History of Statistics

Heyde with Seneta (Sydney) essentially completed a major project for the International Statistical Institute in editing a volume "Statisticians of the Centuries" containing over 100 commissioned biographical articles on statisticians born before 1901.

Advanced Computation and Modelling

Smoothing methods applied to data mining

Roberts, Hegland (RSISE) and Altas (CSU) have developed a finite element approximation of the 2 dimensional thin plate spline. Roberts and Stal (Old Dominican U) have extended the method to 3 dimensions. The method uses piecewise continuous elements in a formulation which only needs first order derivatives. The resolution of the method is chosen independently of the number of observations which only need to be read from secondary storage once and do not require to be stored in memory. The formulation leads to a saddle point problem. Convergence and solution of the method and its relationship to the standard thin plate splines have been investigated.

Numerical modelling of rapidly varying flood flow

Roberts and Zoppou (ACTEW/AGL) have developed a computer model that can be used to assess the potential damage of rapid floods and quantify the effectiveness of alternative remedial strategies. In particular, they have developed a robust and efficient nu merical algorithm capable of simulating rapidly-varying fluid flow using the two-dimensional shallow water equations. They have demonstrated the usefulness of the model by applying it to a small number of typical water supply reservoir sites in the Canbe rra area. ACTEW/AGL Corporation intends to use this technology in a risk assessment of these water supply reservoirs. Extension of the methods to equations with higher order Boussinesq terms that better model the vertical flow behaviour is to be investi gated.

Geophysical computational methods

Roberts and Matthäi (ETH Zürich) have developed finite element software which has been used to model the reaction of minerals flowing through complicated faulted sedimentary basins with spatial resolution ranging over three orders of magnitude. In additi on, the system has been used to simulate the transient drawdown behaviour of oil wells in geologically-realistic faulted sandstone oil reservoirs. The development of higher order advection methods and three-dimensional models is currently being undertake n.

Tolerant qualocation methods

Qualocation methods, since developed by Sloan (UNSW) more than a decade ago, have been a prolific topic in the boundary element literature to study efficient solution techniques for boundary integral equations. The methods obtain the high order of conver gence of the Galerkin method, while preserving the simple implementation of the collocation method. However, a drawback is a requirement of extra smoothness of the exact solution. Tran and Sloan (UNSW) have focused on developing qualocation methods whic h do not require the extra smoothness mentioned above, thus named "tolerant qualocation methods". Equations with variable coefficients were considered. The same convergence results as for the Galerkin method were obtained.
Inverse scattering

Tran and Kress (Göttingen) studied inverse scattering to determine the shape of a local perturbation of a plate. This work contributed to the study of Newton's method for inverse scattering problems on domains with corners.

A posteriori error estimation for nonlinear parabolic equations

Tran and Duong analysed a-posteriori error estimation with the finite element method of lines for nonlinear parabolic equations which were carried out in previous works by other authors. The improvement reduces the assumptions on the solution thus allows the method to be widely used. This method will be extended to other equations like the Benjamin-Bona-Mahony equation and degenerate equations.

Simplicial algorithms for minimizing polyhedral function

Osborne completed the book "Simplicial Algorithms for Minimizing Polyhedral Functions". This book provides the first general account of the development of simplicial algorithms. These include the ubiquitous simplex method of linear programming widely us ed in industrial optimization and strategic decision making and methods important in data analysis, such as problems involving very large datasets. The theoretical development is based on a new way of representing the underlying geometry of polyhedra fun ctions (functions whose graphs are made up of plane faces), and is capable of resolving problems that occur when combinatorially large numbers of faces intersect at each vertex.

Mathematical Physics

Investigation has continued into solvable models in statistical mechanics and related areas of mathematical physics.

Spin ladders and dilute A_L models

Batchelor, Maslen and de Gier continued work on the exactly solvable quantum spin ladders introduced last year. The general construction of solvable ladders was extended to allow mixed spins on the ladder legs. Calculation of the physical properties, su ch as spin gaps and magnetisation plateaus, via the Bethe Ansatz solutions is underway. There is some hope that the models may describe the magnetic properties of experimentally realisable ladder-like compounds.

Batchelor and Seaton (Latrobe) continued their work on the family of dilute A_L lattice models. Explicit calculations were completed for the transfer matrix excitation spectrum of the dilute A₄ model. The results obtai ned after long and difficult calculations verified their previous conjecture for the connection with the E₇ mass spectrum. The results have been shown to give new universal amplitudes which may be observed in other models in the same un iversality class such as the tricritical Ising model.

Random tilings and traffic

De Gier and Batchelor continued work on solvable random tiling models and traffic models. One specific tiling model under investigation is the square-triangle model with open boundaries. De Gier collaborated with Nienhuis (Amsterdam) on a generalisation of the square-triangle model where a defect tile is introduced while retaining integrability: they found an interesting hierarchy of integrable models corresponding to this generalisation.

De Gier also worked on a diffusion problem strongly related to a model used for traffic flow. In particular, the stationary state for this model was calculated exactly for an important new case where cars move with a high velocity.

Stromatolite morphogenesis and tree rings

Batchelor, Burne (Geology) and Henry (UNSW) continued work on mathematical models for the growth of stromatolites. These are laminated structures produced as a result of the environmental interactions of microbial communities. They are the only macrosco pic evidence of life on Earth prior to the evolution of macroscopic plants and animals. A set of solutions has been found for the deterministic version of the Kardar-Parisi-Zhang equation which are suggestive of conophyton. The origin and growth of thes e pyramid-like structures have posed a number of longstanding and exciting theoretical challenges to geologists which we hope to address.

Some intriguing similarities between solutions of the radial growth equations investigated earlier by Batchelor, Henry and Watt and the annular growth and form of tree rings were raised in the conference proceedings of the 3rd Tohwa International Statisti cal Physics Meeting. This work involved discussions with Banks (Forestry).

Random rooted maps and meanders

Since 1978 there has been a great deal of work in statistical mechanics on models on random lattices. This appears to have overlooked earlier work by W.T. Tutte on dichromatic polynomials, i.e. Potts models, on random rooted planar maps. Baxter has ext ended Tutte's work to non-separable maps to obtain a non-linear recursion relation for the partition function. He has solved three special cases exactly, and have verified from numerical studies that the system has a transition much like the regular latt ice Potts model. One advantage of this model is that it should be possible to calculate the free energy and thermodynamic properties away from the phase transition.

An intriguing problem in combinatorics is that of counting "meanders", i.e. the number of distinct non-self-intersecting walks that cross a river 2n times, returning to the starting point. Baxter has shown that this can be expressed as a trace ove r an operator in the Temperley-Lieb algebra that has played such a significant role in the planar six-vertex and Potts model, so is in that sense a (rather simple) partition function. So far the problem has proved intractable: can this algebra provide a route to its solution?

Chiral Potts model and educational software

Davies has continued his research into the algebraic geometry of the chiral Potts model. The long term objective of this work is to tackle outstanding problems which have proved intractable using "classical" mathematics. Davies has also developed softwa re for investigating, experimenting with, and visualising nonlinear dynamical systems. The first application version was released in 2000, following successful extensively testing in TLTSU laboratories.

Zamolodchikov and Calogero-Sutherland models

Mangazeev in a collaboration with Boos (Bonn) continued his work on the three-layer Zamolodchikov model. He studied string-like distributions of zeros of the transfer-matrices. In the thermodynamic limit he obtained an exact solution for the distributio n densities for the ground state.

Also Mangazeev studied a separation of variables for the classical and quantum Calogero-Sutherland model. In particular, for the A₃ case he constructed the kernel of the quantum separating operator which in the quasiclassical limit pro duces a generating function of the separating canonical transform. These results were known only for A₁ and A₂ cases.

Geophysics

Petersons studied the long eleven year period electromagnetic response of the Earth and its implications for the electrical conductivity of the mantle. By applying stringent selection criteria to the geomagnetic data and taking into account the variabili ty in both amplitude and period of the solar cycle a more accurate determination of the response function was determined. This response function when compared to the response functions obtained from some theoretical conductivity models supports increased electrical conductivity near the base of the mantle. In the course of this work he found that a set of overdamped systems of a certain type may be related to sunspot numbers.

Theoretical Astrophysics

Professor Gary Schmidt of the University of Arizona visited the School of Mathematical Sciences for a period of 6 months in the latter part of 2000 funded by a Small ARC grant. Professor Schmidt arranged for the University of Arizona spectropolarimeter t o be shipped to the ANU for the duration of his visit. This unique instrument which was designed by himself had hitherto been used exclusively only in the northern hemisphere. The instrument measures the four Stokes vectors which characterise electromag netic radiation, thus allowing a study of magnetism in astronomical objects. The instrument was used on the 74 inch telescope at Mount Stromlo observatory for about 80 nights. The science that has come out of this project has been spectacular. The mode lling of the polarimetric data is being carried out by the staff and students attached to the Astronomy and Astrophysics Program. It is expected that this set of data will be a source of new scientific results and discoveries for years to come.

The year also saw a visit by Professor Brian Warner of the University of Cape Town. As always, his visit stimulated new directions of research.

Members of the Program were heavily involved in the teaching and supervision of honours students in the Astronomy and Astrophysics honours program which is run jointly by the Department of Mathematics, the RSAA and the Department of Physics. Professor W ickramasinghe and Professor Monaghan of Monash University ran specialised courses using the video-conferencing facilities at the ANU and Monash University for the first time. The supervision of projects and the delivery of courses for honours students wa s shared equally between the SMS and the RSAA.

The research highlights from 2000 are summarised below.

Wickramasinghe in collaboration with Tout and Regos (University of Cambridge) investigated the possible progenitors of Type Ia Supernovae. By synthesising the population of binary stars using monte carlo simulations, they concluded that the binary sy stems which lead to Type Ia supernovae are dominated by edge-lit detonations of sub-Chandrasekhar mass white dwarfs and not by super Chandrasekhar mass explosions as is usually believed. Furthermore, their calculations show that the population of binarie s which give rise to Type Ia supernovae evolves with redshift thus questioning the use of supernovae as standard candles for measuring the acceleration of the Universe.

Vennes studied problems in the formation and evolution of compact stars and implications for the general stellar population and for the population of cataclysmic variables. The research involves intensive numerical calculations as well as observations w ith space-borne and ground-based telescopes such as KPNO4m, Lick3m, Stromlo2m, Hubble Space Telescope, Far Ultraviolet Explorer, Chandra X-ray telescope, and the Extreme Ultraviolet Explorer.

Wickramasinghe and Ferrario have written a comprehensive and extensive (52 pages) invited review article for the Publications of the Astronomical Society of the Pacific on "Magnetism in Isolated and Binary White Dwarfs".

Ferrario, P Maxted, T R Marsh (U of Southampton, England) and Wickramasinghe have constructed Zeeman models to explain the peculiar spectrum of the isolated magnetic white dwarf WD1953-011. This star (which also exhibits a rotational period of hours or days) shows a very complex magnetic field structure. The modelling suggests that the spectra can be explained by a high field region of magnetic field strength 500 kiloGauss covering about 10% of the surface area of the star superimposed on an underly ing dipolar field of mean field strength 70 kG.

Ferrario, G Schmidt (ANU and U of Arizona), P Smith (U of Arizona) and Wickramasinghe have modelled the phase-resolved optical spectropolarimetry of the magnetic cataclysmic variable V884 Her. Their modelling found that the set of narrow, polarised absor ption features matches the Zeeman pattern of hydrogen for a nearly uniform magnetic field of B=150 MegaGauss, indicating that the features are "halo" absorption lines arising in a relatively cool reversing layer above the accretion shock. With this ident ification, the broad polarization humps observed in the spectrum of this object are assigned to cyclotron emission from the fundamental and first harmonic (m=2), respectively. V884 Her is only the second AM Her system known with a field exceeding 100 MegaGauss, and the first case in which the cyclotron fundamental has been directly observed from a magnetic white dwarf.

Ferrario, G Schmidt (ANU and U of Arizona) and Wickramasinghe have constructed models of polarised line emission from magnetised accretion flows and investigated how line circular polarisation can be used to infer the dynamics and the physics of the field channelled flow in AM Herculis systems. Their models have been applied to the high field AM Herculis systems AR UMa and V884 Her.

There were 5 CMA National Research Symposia held in 2000:

• The Baxter Revolution in Mathematical Physics
• Group Theory and Applications
• Non-linear Time Series, Stochastic Networks and Allied Moder Statistical Techniques
• Geometric Analysis and Applications
• Geometric Numerical Integration

Organisers: Professor V Bazhanov, Drs M Batchelor and P Pearce (Melbourne)

February 13-19, 2000 held at the ANU

Participants: 76 (21 participants from the Australian National University, 22 participants from 5 other Australian Universities and 33 participants from overseas institutions)

This conference was held in honour of Professor Rodney Baxter's sixtieth birthday. It highlighted areas where his work has been most influential. These included exactly solved lattice models, integrable quantum field theory, quantum groups, knot the ory, q-series, combinatorics and applications to statistical physics. It brought together international experts in these fields from Australia, China, Korea, Japan, Brazil, France, Germany, The Netherlands, Russia and the USA. The conference was highly successful in bringing the participating scientists up to date with the most recent developments and in encouraging communication and collaboration between geographically separated groups.

The workshop was opened by Professor Barry McCoy with a memorable overview of Professor Baxter's many revolutionary accomplishments. This began with his solution of the eight-vertex model in 1971 and the invention of corner transfer matrices in 1976 and continuing through to the creation of the RSOS models in 1984 and his present work on chiral Potts models. Professor Vaughan Jones gave a public lecture on the topic of knots which drew one of the largest ever audiences for a public lecture at the Austra lian National University.

The proceedings of the conference will appear in the Journal of Statistical Physics. A picture archive of the speakers and social activities can be found at http://www.maths.anu.edu.au/conferences/Baxter2000.

The schedule ran as follows: (February 14) B McCoy: The Baxter Revolution; B Nienhuis: Packing order: tiles and colours; C Hamer: Finite-size scaling and effective Lagrangians at quantum phase transitions; A Klumper: Thermodynamics of integrable quantum chains; J Oitmaa: Ladders, combs, bilayers and brushes; Y Stroganov: Ground state of the quantum symmetric finite size XXZ spin chain with anisotropy parameter Delta = 1/2; R Quispel: Piecewise-linear integrable systems; C Rim: Reflection amplitudes and one point function of (affine) Toda field theories. (February 15) R Baxter: Models I have known; P Wiegmann: Arithmetics and Cantor spectra of integrable models: Hofstadter problem; M Wadati: Symmetric and non-symmetric bases of quantum integrable particle systems with long-range interactions; V Bazhanov: The mysterious Q; J-M Maillard: Let's Baxterise; A Owczaarek: The combinatorics of the free fermion condition in vertex models; I Enting: The computational complexity of series expansions for the Potts model using the finite lattice method; V Jones: Knots (Public Lecture). (February 16) T Miwa: Combinatorics of coinvariants; F Smirnov: Dual Baxter's equations and quantum algebraic geometry; O Foda: Combinatorial aspects of Baxter-type models; M Gould: Type II quantum superalgebras and integrable models; A Carey: Calogero-Sutherland systems; P Bouwknegt: The universal chiral partition function in a fractional quantum Hall basis; A Kuniba: Soliton cellular automata from vertex models; S Warnaar: Bailey's lemma and Kostka polynomials; V Jones: The quantum dodecahedron. (February 17) J Perk: Wavevector-dependent susceptibilities in quasiperiodic Ising models; F Wu: Dimer statistics and spanning trees; P Forrester: Random matrices in a bigger picture; T Guttmann: Progress in the study of the susceptibility of 2d Ising model; K Seaton: The inversion relation and the dilute A_3,4,6 eigenspectrum; J de Gier: Fractional magnetization plateaus in exactly solved spin ladders; R Kashaev: Quantum dilogarithm and strongly coupled quantum discrete Liouville model; T Nakanishi: Bethe equation at q = 0 and weight multiplicities; I Jensen: Enumerations of lattice animals and trees; J Suzuki: Functional relations in Stokes multipliers; G Campbell: Transforming q series into new Dirichlet series involving Riemann zeta fu nctions. (February 18) C Tracy: Applications of random matrix theory to combinatorics and growth processes; A Zamolodchikov: Yang-Baxter equations in integrable relativistic field theory; P Pearce: Integrable and conformal boundary conditions; V Ko repin: Solution of inverse scattering problem for eight-vertex model; F Alcaraz: Exact solution of asymmetric diffusion with second-class particles of arbitrary size; M-L Ge: Yangian symmetry in Yang-Baxter systems - applications; Y Pugai: Free fie ld construction for ABF model in regime II; C Ahn: Reflection amplitudes and thermodynamic Bethe Ansatz for simply and non-simply laced affine Toda theories; H Yang: Studies of Baxter's solution for the free energy of the Chiral Potts model; J McGuir e: An extension of the Bethe Ansatz - the quantum three-particle ring. (February 19) B McCoy: The sl₂ loop algebra symmetry of the six-vertex model at roots of unity; J Noh: Symmetry property and incompleteness of the Bethe Ansatz w ave functions for Heisenberg XXZ chain; V Fridkin: Eta-dependence of the ground state energy of the finite XXZ spin chain; J Martinez: The spin-1/2 XXZ quantum chain at finite magnetic fields - crossover phenomena driven by temperature; M Gulacsi: T he exact Schrieffer-Wolff transformation; V Mangazeev: Some exact results for the Zamolodchikov model.

Group Theory and Applications

Organisers: Professor A Neeman and Dr L Kovács

February 23-26, 2000 held at the ANU

Participants: 23 (10 participants from the Australian National University, 10 participants from 4 other Australian Universities and 3 participants from overseas institutions)

Australia in general, and the ANU more particularly, have a long tradition of exceptional strength in group theory, and more especially in the theory of finite groups. The idea of this conference was to focus more on the applications of group theory. Th ese applications include other branches of mathematics, and other sciences (especially physics). We wanted to bring together a reasonably diverse group, to discuss a broad range of applications.

When one asks people to attend a conference that focuses on areas only tangentially related to their immediate expertise, one expects a modest turnout. Only the courageous and open-minded are likely to come. In the circumstances, the list of 23 particip ants is impressive; it is particularly encouraging that so many graduate students came. We heard talks about a range of applications of group theory. The talks ranged from physics to topology to algebraic geometry. The talks were about very current, o ften exciting recent developments. For example, Henning Andersen reported about some developments with one of Lusztig's conjectures. The conjecture in question concerns the multiplicities of irreducible modules in the reduction modulo p of a Weyl module. Until recently, the best result in the subject was joint work of Andersen, Jantzen and Soergel; they proved the conjecture for very large p. But now there may be work substantially improving the results. Assuming the new result is true (it is still work in progress), there will soon be a fairly small, effective bound on p.

The National Symposia are intended to promote friendly interactions between the participants. It is fair to say that we had great success. This symposium was somewhat interdisciplinary, so the participants were not a bunch of old friends getting togethe r yet again. Nevertheless, we got on very well, and the people seemed quite keen on the idea of meeting again, sometime soon.

The schedule ran as follows: (February 23) J Du: Finite dimensional algebras and cellular systems; H Andersen: p-filtrations for modular representations; R Zhang: Quantum superalgebras and Vassiliev invariants; C Mak: Quasi-parabolic subgroup s of generalised symmetric groups. (February 24) S Gadde: Splittings of groups and intersection numbers; J Carlson: Categorical equivalences and orthogonal subcategories over group algebras; J Hillman: Applications of L²-Betti numb ers to low-dimensional topology; I Aitchison: Mr Spock as a toral tangram. (February 25) G Lehrer: Action of a reflection group on the cohomology of its configuration space; A Neeman: Automorphisms of the Yang-Baxter equations, for the Chiral Potts model; B Howlett: Automorphisms of Coxeter groups of rank 3. (February 26) Informal discussions.

Non-linear Time Series, Stochastic Networks and Allied Modern Statistical Techniques

Organisers: Dr R Wolff (QUT) and Dr D Guégan (Reims)

June 28-July 2, 2000 held at the ANU

Participants: 20 (4 participants from the Australian National University, 9 participants from 6 other Australian Universities and 7 participants from overseas institutions)

The following is a list of activities which have followed as a result of the symposium.