There were 6 CMA National Research Symposia held in 1999.

• Plane Algebraic Curves and the Jacobian Conjecture
• Differential Geometry and Geometric Analysis
• Algebraic Geometry and Applications
• The fractional quantum Hall effect on the eve of the new millennium
• Numerical Analysis of Boundary Integral Methods and Applications
• CANT'99 -- Group Theory and Computation

Plane Algebraic Curves and the Jacobian Conjecture

Organisers: Professor W Neumann, Dr P Norbury and Ms P Wightwick (Melbourne)
February 8--12, 1999 held at the University of Melbourne
Participants: 20 (8 participants from 3 Australian Universities; 12 participants from overseas institutions)

The conference brought together mathematicians from different parts of the world. Many new results were presented, most notably the anticipated work of Stein giving an algebraic generalisation of the Jacobian conjecture, together with some progress on the problem, and the work of Russell on normal crossing systems. Brown's demonstration of Magma's algebraic geometry capabilities was received with interest. Miyanishi led a problem session at the closing of the conference in which many open problems were put forward by the participants.

The schedule ran as follows: (February 8) R Gurjar: On a generalisation of the Jacobian Conjecture; P Russell: Normal crossing systems of plane one-place curves; T Kuo: Newton polygon relative to an arc; A Assi: Jacobian of meromorphic curves. (February 9) T Kambayashi: Pro-affine algebras, ind-affine schemes and the Jacobian problem; J Yu: Embeddings of hypersurfaces in euclidean spaces; K-M Fan: Fundamental group of the complement of a union of projective lines; A Di Pasquale: Arrangements of curves in the projective plane. (February 10) I Sigray: Jacobian trees and their applications; G Brown: demonstrated the new capabilities of Magma for working with singularities and plane curves. (February 11) M Suzuki: Semigroup theorem on the affine plane curves with one place at infinity; W Neumann: Unfolding polynomial maps at infinity; P Cassou-Nogues: Examples and constructions of affine curves; Y Stein: Weakly nilpotent and weakly semisimple polynomials. (February 12) M Miyanishi: The generalised Jacobian Conjecture; K Masuda: The equivariant Jacobian Conjecture; P Norbury: Rational polynomials.

Differential Geometry and Geometric Analysis

Organisers: Dr B Andrews and Professor K Ecker (Monash)
July 5--9, 1999 held at the University of Melbourne
Participants: 37 (8 participants from the Australian National University; 15 participants from 5 other Australian Universities and 14 participants from overseas institutions)

The symposium was held to coincide with visits to Australia by a number of mathematicians in these areas for the joint meeting of the Australian and American Mathematical Societies the following week. The symposium involved 12 one-hour talks, and had an attendance of approximately 35 including approximately 15 from outside Melbourne. It was a very successful workshop and provided a valuable prelude to the larger conference. In particular, the workshop heard reports of new work in areas ranging from a new construction of invariant connections on tractor bundles in conformal and projective differential geometry by Gover (Auckland), to analysis of isolated singularities in fully nonlinear elliptic partial differential equations by Labutin. A talk by Nikolayevsky (Latrobe) on a family of interesting new problems in conformal integral geometry generated much interest and continuing further research. A talk by Huisken (Tubingen/Princeton) introduced new methods in complex geometry, in particular an evolution equation which deforms pseudoconvex 3-surfaces to become strictly pseudoconvex. In the ensuing discussion a number of further applications and generalisations were suggested, and research on some of these is continuing. Capogna (Pennsylvania) presented a quite general existence and regularity theory for subelliptic operators, including a natural geometric criterion for boundary regularity in the Dirichlet problem. This is naturally related to problems which arise in the work mentioned by Huisken, and participants suggested that some of the geometric ideas might yield an alternative proof of short-time existence for Huisken's evolution equation and similar nonlinear geometric flows with a subelliptic symbol. Michael Gruter (Saarbrucken) presented regularity results for minimal surfaces with corners, and explained in detail the applications of these ideas to prove the existence of multiple solutions to the Plateau problem for certain boundary curves. Hong presented some new partial regularity results for weak solutions of a system arising in electromagnetism, and Hutchinson spoke on new work on phase transitions. Norbury (Melbourne) gave applications of heat flow methods in constructing and hyperbolic monopoles. Ecker (Monash) presented new Logarithmic Sobolev inequalities for submanifolds of Euclidean space, with applications to motion by mean curvature and related problems. Considerable discussion was devoted to possible applications of these ideas, and their implications for the singularity behaviour of the mean curvature flow. Andrews presented two talks on evolution equations for surfaces, describing new methods which yield sharp new results for both surfaces in spheres and surfaces in hyperbolic spaces. The latter proved to be of interest to participants in the concurrent workshop on group actions and low-dimensional topology, who suggested several possible extensions and conjectures with applications in hyperbolic geometry.

The schedule ran as follows: (July 6) L Capogna: Dirichlet problem for sublaplacians; D Labutin: Isolated singularities for fully nonlinear elliptic equations; Y Nikolayevsky: Conformal integral geometry of curves and surfaces. (July 7) B Andrews: Evolving surfaces, part I: Surfaces in spheres; M-C Hong: Partial regularity of weak solutions to Maxwell's equations in a quasi-static electromagnetic field. (July 8) G Huisken: Flow of hypersurfaces by the trace of the Levi form; P Norbury: Asymptotic values of hyperbolic monopoles; M Gruter: Minimal surfaces with corners; K Ecker: Logarithmic Sobolev Inequalities. (July 9) J Hutchinson: Phase Transitions; R Gover: Fundamental invariant operators and tractor calculus; B Andrews: Evolving surfaces, part II: Surfaces in hyperbolic space.

Algebraic Geometry and Applications

Organisers: Professor A Neeman and Dr B Martin (Sydney)
July 19--23, 1999 held at the Australian National University
Participants: 34 (2 participants from the Australian National University; 13 participants from 4 other Australian Universities and 19 participants from overseas institutions)

The themes of the conference were algebraic geometry and its applications. Many Australian mathematicians study algebraic geometry and use it in their work, but there are seldom opportunities for them to meet with each other and with experts from overseas. The meeting brought together people from Melbourne, Sydney, Canberra, Adelaide and Armidale.

One of the aims was to let graduate students learn about this important branch of mathematics. We were pleased, therefore, that 5 graduate students came to the conference. One student gave a talk.

The talks covered many topics including homotopy theory, number theory, Hecke algebras, reductive algebraic groups, low-dimensional topology and statistical mechanics. Mainly the talks on topology were in the first half of the meeting, while the the talks on representation theory were in the second half. Thus people could go to the talks that they were most interested in without having to stay for the whole meeting. Many attended all of the talks, for even people working in very different areas often study the same basic problems. For example, speakers at the meeting described various approaches to the representation theory of reflection groups and associated algebras: their methods ranged from the algebraic and the combinatorial to the geometric and the topological.

The meeting was a great success; comments from those who took part, both Australians and visitors, were favourable. If we are fortunate enough again to have such a gathering of algebraic geometers in Australia, then a similar meeting would be very worthwhile.

The schedule ran as follows: (July 19) D Christensen: The ghost-length of projective spaces; B Davies: The chiral Potts curves; B Wang: Surgery formulas for monopole equations. (July 20) P Norbury: Unfolding polynomial maps at infinity; E Getzler: Hopf operads; S Tillmann: On character varieties of mutative 3-manifolds. (July 21) B Keller: On the cyclic homology of algebraic varieties; M Kisin: Unit L-functions and a conjecture of Katz. (July 22) L Morris: The role of affine Hecke algebras in the representation theory of reductive p-adic groups; G Lehrer: Regular semisimple varieties and iterated loop spaces; A Mathas: The centre of the Iwahori-Hecke algebra of type A. (July 23) G Roehrle: On abelian normal subgroups in parabolic groups; B Martin: The compactification of a representation variety; M Vazirani: A strong multiplicity one result for Hecke algebra modules.

The fractional quantum Hall effect on the eve of the new millennium

Organisers: Drs P Bouwknegt and M Varghese (Adelaide)
August 16--20, 1999 held at the University of Adelaide
Participants: 36 (3 participants from the Australian National University; 24 participants from 5 other Australian Universities and 9 participants from overseas institutions)

The organization of the workshop was directly motivated by the 1998 Nobel Prize in Physics awarded to Stormer, Tsui and Laughlin for "for their discovery of a new form of quantum fluid with fractionally charged excitations". The purpose of the workshop was to discuss recent advances and future directions of the fractional quantum Hall effect (fqHe) and related topics and to stimulate interaction and explore common interests between experimentalists, theoretical physicists and mathematicians.

An effort was made to make the program interesting and accessible to postgraduate students (and other non-experts) by scheduling a number of pedagogical review talks in addition to the more technical expositions discussing the recent advances. Recent advances discussed in this workshop included a better understanding of the non-Abelian fractional quantum Hall states, the recently discovered metal-insulator transition in the absence of a magnetic field and the non-commutative geometry approach to the fqHe.

The schedule ran as follows: (August 16) K Schoutens: The fractional quantum Hall effect; D Neilson: Destabilization of the 2D conducting phase by an in-plane magnetic field; M Simmons: Metal-insulator transitions in two dimensions; E Tosatti: How are the zero field 2D metal-insulator and the Quantum Hall-insulator transitions connected? (August 17) P Bouwknegt: The fractional quantum Hall effect and conformal field theory; A Capelli: W-Infinity symmetry in the quantum Hall effect; L Chim: Exclusion statistics and fermionic character formulas; V Bazhanov: Quantum Brownian motion in a periodic potential and conformal field theory. (August 18) J Bellissard: The non-commutative Brillouin zone; A Hassell: On the ground state energy of the fractional quantum Hall effect; D Robinson: On periodic systems; I Raeburn: Modelling graphs in C*-algebras; K Schoutens: Non-abelian quantum Hall states. (August 19) J Bellissard: The four traces way; T Sunada: Long time asymptotics for the transition probability of a random walk on a crystal lattice; K Hannabuss: The quantum Hall effect in non-commutative hyperbolic space; P Forrester: Correlation functions and Jack polynomials; P Jarvis: S-function determinant formulae, tableaux decompositions and the fermion-boson correspondence. (August 20) J Bellissard: Why are the plateaux so flat?

Numerical Analysis of Boundary Integral Methods and Applications

Organisers: Professors I Sloan (New South Wales) and W Wendland (Stuttgart)
September 27 -- October 1 1999, held at the University for New South Wales
Participants: 33 (1 participant from the Australian National University; 12 participants from 6 other Australian Universities and 20 participants from overseas institutions)

The symposium, also advertised as the German-Australian Workshop in the Numerical Analysis of Boundary Integral Methods and Applications.

During the symposium, 35 lectures were presented, covering almost all aspects of contemporary research on boundary integral methods and their applications. Highlights of the lecture program included an authoritative lecture by Wendland on approximation of the Steklov-Poincare operator; one by Costabel giving mathematical justification for a fast convergence method for singular integral equations; and one by Hackbusch (a leader in decomposition methods) on his newly developed concept of hierarchal matrices. The symposium concluded with an open-ended Discussion Forum on "Boundary Integral Methods -- What Next?". At this forum a problem that generated intense discussion was that of closing the gap between mathematicians and engineers who work in the boundary element field. Engineering colleagues emphasized the need for boundary element methods to be robust, and for the existence of software of high quality. The visit to Australia of so many highly ranked German researchers provided a major stimulus to collaborative research projects, for example between Stephan and Tran, Grigorieff and Sloan, Wendland and Sloan, Kress and Ganesh, Steinbach and Ganesh. Many German visitors took the opportunity provided by the visit to Australia to attend the CTAC Conference in Canberra, or (as in the case of Costabel and Stephan) to strengthen links with researchers in the School of Mathematical Sciences at the ANU. By common consent, the symposium was an outstanding success, with the scientific program being at the highest international level. The report of Professor Wendland to the DFG, which is highly enthusiastic in tone, concludes with the remark: "Der workshop war zweifellos ein grosser Erfolg" ("The workshop was undoubtedly a great success.").

The schedule ran as follows: (September 27) W Wendland: On boundary element approximations of the Steklov-Poincare operator; D Clements: Fundamental solutions for second order linear elliptic partial differential equations; C Schneider: Collocation on graded meshes; G Chandler: The fundamental solutions method for moving boundary problems; D Scheen: Layer potential approach to the inversion of discontinuous conductivities; J Elschner: Diffraction in periodic structures and optimal design of binary gratings; V Didenko: On stability of approximation methods for Muskhelishvili equation; A Sandig: Transmission problems: Regularity results and boundary integral formulation. (September 28) M Costabel: An exponentially convergent singular function BEM; Y Jeon: Scalar boundary integral equation formulas for the biharmonic equation Ellipticity results; S Roch: Index calculus for approximation methods with applications to boundary integral equations; R Grigorieff: A superapproximation property for periodic splines with multiple knots and applications; B Silbermann: BEM and topics in asymptotic spectral theory; M Ganesh: Boundary element methods for a DtN map based nonlinear equation; P Junghanns: On the numerical solution of systems of Cauchy singular integral equations in the non-periodic case; D Elliott: Why go beyond the trapezoidal rule? K Diethelm: Analysis of quadrature formulas for strongly singular integrals arising in the BEM. (September 29) G Rodin: Fast boundary element methods for large problems arising in micromechanics of composite materials; H Andra: Application of the Galerkin-type boundary element method for the reconstruction of interface cracks in composite materials; J Watson: Quadratic and singular boundary elements for cracks in three dimensions; R Schneider: Biorthogonal wavelet bases for boundary integral equations; C Bourgeois: Prewavelet analysis of the heat equation; A Rathsfeld: Wavelet collocation for integral equations, Linear basis functions and quadrature. (September 30) E Stephan: Schwarz methods for the h-p version of the boundary element method; T Tran: Overlapping additive Schwarz preconditioners for Galerkin boundary element methods; K Giebermann: A multilevel algorithm for the fast evaluation of boundary element matrices; W Hacksbusch: On hierarchical matrices; W McLean: Boundary element preconditioners for a hypersingular integral equation; M Maischak: Fast solvers for bem and fem-bem coupling with Signorini-type boundary conditions; C Carstensen: Remarks on a posteriori error estimates for BEM; O Steinbach: Adaptive boundary element methods: A posteriori error estimates and preconditioning; T Cao: Adaptive h-, r- and h-r methods for Symm's integral equation. (October 1) P Johnston: Simple transformations for evaluating singular integrals; R Kress: Inverse obstacle scattering with modified or reduced data; I Sloan: Analysis of a spectral method for 3D acoustic scattering; I Sloan and W Wendland: (Discussion Forum) Boundary integral methods -- what next?

CANT'99 -- Group Theory and Computation

Organisers: Drs J Cannon (Sydney) and L Kovacs
November 29 -- December 3 1999, held at the University of Sydney
Participants: 84 (7 participants from the Australian National University; 13 participants from 6 other Australian Universities and 64 participants from overseas institutions)

In the last decade or two, computational group theory has seen a shift towards algorithms which make very substantial use of deep theoretical results about groups, particularly of results of a probabilistic nature. This is reflected especially in recent work on the structure of modules and matrix groups. In the same period, several major theoretical achievements built on insights obtained from highly demanding and imaginative computational investigations. Zelmanov's solution of the classic Restricted Burnside Problem and the formulation and eventual proof of the Coclass Conjectures of Leedham-Green and Newman are just two examples of theoretical breakthroughs that demanded, inspired and exploited computational advances. The aim of this symposium was to extend the range of this synergy by bringing together leading experts from algorithmic and computational aspects of group theory, with experts from various branches of group theory that are already using, or that may in future profitably exploit, computational methods. It was the concensus of the participants that the resulting interchanges have enriched and benefited both research communities. In particular, the meeting informed designers and users of algorithms about recent developments in cognate areas of group theory, informed theoretical mathematicians of the possibilities offered by the computational algebra tools that are now becoming available, and helped to identify key directions for future research and interaction. It was also very significant that the meeting provided top-level exposure, for local researchers and graduate students, to the leading experts in the areas and to the excitement of current developments. There will be no proceedings published, but it should be noted that several of the talks were closely related to papers dedicated to Mike Newman in the last two 1999 issues of the Journal of the Australian Mathematical Society (Series A).

The schedule ran as follows: (November 29) K Gupta: Varieties of groups and group representations; L Kovacs: Module structure of free Lie algebras; A Caranti: Graded Lie algebras of maximal class; A Schneider: Classifying p-groups beyond coclass; C Schneider: On the derived series of finite p-groups; J Groves: Modules over free nilpotent groups; M Vaughan-Lee: Variations on a theme. (November 30) A Mann: Some enumeration problems regarding p-groups; G Havas: Some new results in group theory via computation; S Pride: New invariants for groups; M Elder: Geodesic structure for a class of groups; G Willis: Totally disconnected groups; S Todorovic-Vasiljevic: Bounds on the number of symmetries of a compact non-orientable surface; E Hart: Open questions in combinatorial group theory arising from Nielsen fixed point theory. (December 1) J Carlson: Homological algebra over basic algebras: Programs and examples; A Kelarev: Algorithms for computing the Jacobson radical of group rings; A Steel: Advanced algorithms for matrix echelonization and their application; G Kemper: Invariant theory -- algorithms and questions; M Newman: Some desiderata for group libraries; G Kemper: The invariants data base -- a collection of invariant rings; S Linton: Exploiting information from databases in algebraic computation; B Eick: Small groups. (December 2) D Holt: Computing automorphism groups of finite groups; M Smith & M Slattery: Lifting soluble group automorphisms using cohomology; A Niemeyer: Random element generation in finite groups; A Seress: Recognition algorithms for finite simple groups; B Souvignier: Exploiting structural properties of permutation groups; B Unger: Computing the p-core of a permutation group; D Taylor: A computational approach to algebraic structures associated with exceptional Lie algebras; V Gebhardt: Constructing presentations for permutation groups; D Easdown: General products and ring decompositions. (December 3) C Praeger: Cyclic matrices and the Meataxe; S Murray: Conjugacy classes in parabolic subgroups; K Geissler: Galois group computation for transitive groups of degree up to 15; E O'Brien: Effective computing approaches to large degree matrix groups; M Conder: Finding normal subgroups of low index in finitely-presented groups; P Dobcsanyi: A parallel implementation of the low index subgroups algorithm; A Solomon: Infrastructure and algorithms for semigroups in GAP - a progress report; C Sims: Algorithmic questions in rings of rational matrices.