Centre de recherches mathématiques, Annual Report 1996-1997

Scientific Activities

The core of each year's scientific program at the CRM is its thematic program. The topic is chosen by the Advisory Committee for its scientific importance, its timeliness, and its impact on the Canadian scientific community. Preceeding years' topics include: Probability and Stochastic Control (1992-93); Dynamical Systems (1993-94); Geometry and Topology (1994-95); and Applied and Numerical Analysis (1995-96). A year's activities can combine a good number of workshops and conferences, one or two Aisenstadt chairs, a certain number of visiting scientists in residence, and some post-doctoral fellowships. Typically, there is some coordination with Montréal universities to offer appropriate graduate courses in order to help graduate students participate in the activities.

Theme Year 1996-1997: Combinatorics and Group Theory


The theme of the 1996 -1997 academic year at the CRM was Combinatorics and Group Theory. Combinatorics is a subject whose importance has grown tremendously in recent years, reflecting its enormous importance in many concrete problems either of computer science and operations research. The thematic program covered a wide variety of areas, including graph theory, with a special session on colouring problems, combinatorial designs, algebraic combinatorics, and "experimental mathematics." The activities in combinatorics were combined with a program in related subjects in group theory such as hyperbolic and automatic groups, actions on trees, and combinatorial group theory.

The holders of the Chaire Aisenstadt for the year were László Babai of Eötvos University and the University of Chicago, and Efim Zelmanov of Yale University.

The scientific events are described below. They took place at the CRM unless specified otherwise.

Pseudorandom Number Generation

3-21 June 1996
Org.:G. Brassard (Montréal), C. Crépeau (Montréal), R. Couture (Montréal), P. L'Écuyer (Montréal), H. Niederreiter (Austrian Academy of Sciences)

The main objective of this workshop was to bring together researchers studying the generation of pseudorandom numbers from various backgrounds: number theorists, physicists who use billions of random numbers in their stochastic simulations and who need fast and reliable generators, cryptologists for whom unpredictability is crucial, statisticians interested in faithful simulations of probabilistic laws of stochastic processes in order to estimate their properties, hackers who discover faults in certain generators, and finally computer scientists who try to implement fast and reliable generators that can satisfy everybody.

Participants came from Asia, Europe and North America. The presence of internationally renowned researchers, like H. Niederreiter and I. Sobol', attracted many first class scientists.

The program was divided into 3 topics, 1 per week:

  1. Characteristics required from good generators. Criteria of selection and statistical tests.
  2. Linear methods.
  3. Nonlinear and cryptographic methods. Software implementation.

An average of two one-hour lectures were given each day, followed by long periods of questions, discussions and comments. This allowed for informal discussions and more collaborations were started here than in usual conferences. Participants reacted very positively to this new format and said that they learned a lot from the wide spectrum of specialists present.

No proceedings were supposed to be published. However a special issue of the quarterly journal ACM Transactions on Modelling and Computer Simulation will be devoted to the subjects covered at the workshop. The invited editors are R. Couture and P. L'Écuyer, two of the organizers.

CRM Summer School on Group Theory

Banff Centre of Arts, Banff, Alberta
11-23 August 1996
Org.:G. Baumslag (CUNY), D. Gildenhuys (McGill), O. Kharlampovich (McGill), E. Zelmanov (Yale)

The summer school was primarily aimed at PhD students in their final years and recent Ph.D.'s. Its main objective was to prepare people for the workshops that were to follow. About 35 students were selected from over 70 applicants. They came from Canada, USA and France.

The following short courses were given:

The proceedings of the Summer School are to be published in a volume of the CRM Conference Proceedings. Invited speakers have submitted short surveys for publication. All submissions have been refereed. O. Kharlampovich is editing the Proceedings.

Cayley Graphs

16-20 September 1996
Org.: G. Hahn and G. Sabidussi (U. de Montréal)
Invited speakers: B. Alspach, L. Babai, P. Cameron, D. J. Dunwoody, W. Imrich, S. Klavzar , D. Marusic, N. Seifter, J. Sirán, V. Trofimov, M. E. Watkins

There were thirty-three participants ranging from invited speakers (fifteen) to students. They came from all over the world Austria, Canada, Great Britain, Slovakia, Slovenia, United States, to name but the countries of the invited speakers. The group spent the week listening to lectures on recent advances in the theory of Cayley graphs, lectures given by recognized

experts as well as by doctoral students. Both finite and infinite graphs and groups were covered in presentations on various subjects. One of the attractions of the workshop was a series of lectures by László Babai on Combinatorial Properties of Vertex-Transitive Graphs.

The invited talks included topics such as:

The presence of computer scientists should be noted. Indeed, Cayley graphs are studied more and more in computer science and so old results are of interest there, while new problems interesting to mathematicians are suggested.

Combinatorics of Words

1 - 5 October 1996, UQAM (Université du Québec à Montréal)
Org.: Sreãko Brlek

The goal of this meeting was twofold. First it aimed at providing an overview of algorithmic problems arising naturally in the study of words based on a finite alphabet. These cover both statistical aspects (computation of complexity measurements) and structural problems (the search for patterns, motives, particular structures). The second aim was to fill the pressing need for the development of formal computational tools adapted to word combinatorics.

The meeting drew around twenty participants, as well as ten students and professors of the LaCIM (UQAM). The speakers were Jean Paul Allouche (LRI, Orsay), Valérie Berthé (IML, Marseille), Julien Cassaigne (IML, Marseille), Michel Koskas (LARIA, Amiens) as well as Dominique Bernardi (TNAD, Paris VI), Sreãko Brlek, Annie Ladouceur and Christophe Reutenauer (LaCIM).

All the lectures were tied to the theme of word complexity. Some combinatorial problems relating the measurements of recursion with complexity were first discussed. Then the particular case of infinite words obtained by rotations of the unit circle was considered. A third lecture developed Cassaigne's fine techniques for fixing bounds on complexity measurements. Finally a review of the links between automatic sequences and the questions of transcendence in number theory was presented.

On October 3rd, a long-term cooperation plan was drawn up by Dominique Bernardi, Sreãko Brlek, Julien Cassaigne, Michel Koskas and Annie Ladouceur to develop a formal computational tool adapted for word combinatorics. Several questions were addressed concerning this project:

(i) Scientific design: algorithms for the construction of infinite words, computation on words, identification of combinatorial properties, functions related to factors of infinite words;

(ii) Technical design: development language (C++), data bases for primitives, interfaces with other existing tools;

(iii) Administrative structure: constitution of the development group CABAC, computer tools, sharing of the central directory at LaCIM. The persons responsible were unanimously chosen as Sreãko Brlek (scientific and administrative management) and Annie Ladouceur (development management). Funding will be sought from the Programmes France-Québec. First contacts have already been established with the funding agencies.

(iv) Future ventures: the meeting succeeded in creating a network of researchers sharing the goal of developing a formal computational tool in word combinatorics. The teams involved are the LRI (Orsay), the IML (Marseille), the TNAD (Paris), the LARIA (Amiens) and the LaCIM (Montréal). The network hopes to have a working prototype available by June 1997. Visits are already scheduled: Koskas (February 1997) and Cassaigne (May 1997) at the LaCIM, Brlek at the LRI (Allouche) and at the IML (Cassaigne and Berthe) and Ladouceur at the LARIA and the LRI. Other meetings have been planned but no dates have been fixed as of now.

Hyperbolic and Automatic Groups; Groups Acting on R-Trees

13-19 October 1996
Org.: G. Baumslag (CUNY), D. Gildenhuys (McGill), O. Kharlampovich (McGill), E. Zelmanov (Yale)

The purpose of the workshop was to bring together specialists approaching group theory from different angles.

One of these is the geometric approach. Much of combinatorial group theory has its roots in geometrical and topological ideas. For example, the notion of non-positive curvature, implicitly underlying small-cancellation theory, has been abstracted and exploited by Gromov, Cannon and Rips, among others, resulting in a sizable body of work on hyperbolic (or negatively curved) groups. This work has many links to other areas like hyperbolic metric spaces, hyperbolic manifolds and group actions on R-trees. In this context, we should mention as well the geometric methods based upon Van Kampen diagrams. These methods were substantially developed by Ol'shanskii in his work on the Burnside, and related, problems.

A second approach is via logical methods. The area of algorithmic solvability of problems in group theory fall under this heading. Razborov (1990 Nevanlinna prize winner) has constructed an algorithm which recognizes the solvability, or otherwise, of a system of equations over a free group. His construction uses ideas due to Makanin. The same ideas turned out to be very fruitful in many other investigations, in particular in Rips' solution of the Morgan-Shalen conjecture about groups acting on R-trees.

A third approach to combinatorial group theory has its roots in the Bass-Serre theory of groups acting on trees.

The list of invited speakers included M. Bestvina, M. Bridson, N. Brady, T. Delzant, S. Gersten, G. Levitt, O. Kharlampovich, L. Mosher, A. Myasnikov, A. Ol'shanskii, E. Rips, Z. Sela, M. Staretz.

Distance-Regular Graphs

18-23 November 1996
Org.: G. Hahn and G. Sabidussi (U. de Montréal)
Invited speakers: E. Bannai, N. Biggs, A. E. Brouwer, P. Cameron, A. Cohen, A. D. Gardiner, C. Godsil, A. A. Ivanov, L. K. Jørgensen, M. Mulder, P. Terwilliger

The workshop attracted thirty-nine participants, including many students. As with the workshop on Cayley graphs, the pleasure of being with like-minded souls showed. László Babai continued his lectures, this time on Algorithms in Finite Groups, while the other participants (again, from students to well-known experts) discussed the various aspects of classifying distance-regular and distance-transitive graphs.

The invited talks included topics such as:

Both the workshop on Cayley graphs and the present one included a problem session, and a list of problems from these was sent, after editing, to all participants.

General Combinatorial Group Theory

5-11 April 1997
Org.: G. Baumslag (CUNY), D. Gildenhuys (McGill), O. Kharlampovich (McGill), E. Zelmanov (Yale)

The workshop included a wide variety of topics. The list of invited talks included the following.

There were also some contributed talks.

Transversal Designs and Orthogonal Arrays

21-25 April 1997, Four Points Hotel, Kitchener, Ont.
Org.: Charles Colbourn (Waterloo), Ron Mullin (Waterloo), Alex Rosa (McMaster), Doug Stinson (Nebraska at Lincoln)

The workshop on transversal designs and orthogonal arrays was designed to bring together researchers on combinatorial, algebraic, statistical, and coding theoretic aspects of orthogonal arrays; to focus on applications; and to involve graduate students and beginning researchers. The format of the workshop involved one plenary (one hour) speaker each day, providing a quite comprehensive overview of the main research directions. Kathy Heinrich and Charlie Colbourn presented plenary talks focusing on the use of orthogonal arrays in construction of block designs and related designs. Alex Pott introduced the connections with geometry, number theory, and algebra in the context of difference sets. Neil Sloane described connections with coding theory, particularly with an important new application in numerical finance. Debbie Street described connections with experimental design in statistics.

The plenary talks were well supplemented by both invited and contributed talks. Topics of the invited speakers (fifteen in all) were well distributed among the main research topics on orthogonal arrays. Applications of orthogonal arrays to the construction of (t,m,s)-nets for numerical integration problems, primarily in finance, arose as a theme in four of the talks. Other concentrations were on computational methods for construction of orthogonal arrays, covering arrays and their application in software testing, and to applications in coding theory and experimental design. The workshop's original plan to bring together researchers from a variety of backgrounds paid excellent dividends, particularly in promoting applicable research.

The participation of younger researchers and graduate students through contributed talks, and informal discussions, was also most encouraging. Twelve graduate students participated, among a total of sixty participants (approximately as expected).

The program of talks was very well received. In addition, participants were very positive about the idea of organizing a conference around the theme of a central combinatorial structure but including statisticians, coding theorists, computer scientists, and others. While all of the participants spoke about the same underlying mathematical object, their particular emphases provided a valuable synergy.

A number of research directions were advanced as a result of the interaction at the workshop, including research on (t,m,s)-nets, on covering arrays and software testing, on difference sets, on pairwise balanced designs, and on computational methods. Collaborations begun at the workshop are already bearing fruit.

The workshop was both productive and enjoyable for the participants, and fulfilled its main objectives in bringing together researchers from a variety of backgrounds, and in exposing graduate students and young researchers to a challenging and diverse topic.

The keynote speakers were: Charlie Colbourn (Univ. of Vermont), Katherine Heinrich (Simon Fraser Univ.), Alexander Pott (Univ. Augsburg), Neil Sloane (AT&T Labs, Murray Hill), Deborah Street (Univ. of Technology, Sydney).

The invited speakers were: Julian Abel (Univ. New South Wales), Frank Bennett (Mount St. Vincent Univ.), Jurgen Bierbrauer (Michigan Technological Univ.), Art Drisko (U. S. Dept. of Defense), Dieter Gronau (Univ. Rostock), Donald Kreher (Michigan Technological Univ.), Chuck Laywine (Brock Univ.), Gary Mullen (Pennsylvania State Univ.), Kevin Phelps (Auburn Univ.), Douglas Stinson (Univ. of Manitoba), Luc Teirlinck (Auburn Univ.), Vladimir Tonchev (Michigan Technological Univ.), John van Rees (Univ. of Manitoba), Mieczyslaw Wojtas (Technical Univ. of Wroclaw), Jeff Wu (Univ. of Michigan).

Graph Colouring and Applications

5-9 May 1997, CRM
Sponsors: CRM, GERAD (Groupe d'études et de recherche en analyse des décisions)
Org.: Pierre Hansen, Odile Marcotte

This workshop brought together outstanding researchers in the fields of combinatorial optimization and graph theory. It consisted of eleven lectures by invited speakers and seventeen contributed talks. The opening lecture was given by Paul Seymour, from the mathematics department of Princeton University, who presented the latest and most elegant proof of the famous four-colour theorem. This proof is the result of work carried out by Paul Seymour himself and his collaborators, several of whom were also attending the workshop. The lecture of Professor Carsten Thomassen, from the Technical University of Denmark, had a similar topic, i.e., algorithms for colouring graphs embedded in specific surfaces. Professors Claude Berge (from Paris) and Vasek Chvátal (from Rutgers University) gave lectures on perfect graphs, a topic closely related to the vertex-colouring problem. Professor Anthony Hilton (from Reading) lectured on total colourings of graphs, and Professor Adrian Bondy (from Lyon) lectured on the relationship between colourings and orientations of graphs.

Graph colouring has many applications, some of which arise from practical concerns and others from the natural or social sciences. Professor Horst Sachs (from the Technical University of Ilmenau, in Germany) presented results on invariant polynomials of polyhedra and their application to chemistry. Dominique de Werra (from the École Polytechnique Fédérale de Lausanne) gave a lecture on the application of graph colouring to the problem of register allocation, which arises during the compilation of computer programs. Fred Roberts (from Rutgers University) presented a variant of graph colouring used in the modelling of social roles, and related this concept to other important concepts in the mathematical models of the social sciences. Finally, Mike Carter (from the University of Toronto) and Bjarne Toft (from Odense University, in Denmark) lectured on applications of graph colouring to timetabling. The last main lecture was given by Bjarne Toft, who educated and entertained his audience by discussing the life and correspondence of Julius Petersen, who gave his name to the famous Petersen graph!

The themes of the contributed talks were similar to those of the main talks, and the workshop was attended by 46 researchers from eight countries (Canada, United States, France, England, Germany, Denmark, Switzerland and Israel). The workshop met with great success, and the conference room was full until the end of the workshop, on Friday afternoon!

Workshop on Experimental Mathematics and Combinatorics

19-30 May 1997
Org.:F. Bergeron (UQAM), G. Labelle (UQAM), P. Leroux (UQAM)

Methods of formal calculus play an ever growing role in mathematics. This is particularly true in domains, like combinatorics, where mathematical experimentation is an integral part of the research process. The aim of this workshop was to do an overview of the development of the computerized research tools for mathematics and to work on the applications of the most recent methods, mainly in combinatorics.

The workshop was held over a period of two weeks, the first at the CRM, the second at the Laboratoire de combinatoire et d'informatique mathématique (LACIM, UQAM). The program consisted of invited lectures, communications, problem periods, a tutorial on the formal calculus software MAPLE and workshops on combinatorial modelisations with formal calculus.

The main themes of the workshop were:




Workshop on Algebraic Combinatorics

9-20 June 1997
Org.: F. Bergeron (UQAM), N. Bergeron (CRM & York Univ.), C. Reutenauer (UQAM)

The purpose of this workshop was to study interactions between algebraic combinatorics and symmetric functions,with a special emphasis on:

  1. descent algebra of Coxeter groups in relation to quasi-symmetric functions and noncommutative symmetric functions, and
  2. doubly parametrised Macdonald (q,t)-symmetric functions, in relation to harmonics of reflection groups.

The first part of the workshop (9-13 June) was devoted to a few conferences by well-known specialists in the area. The format of the first part, one conference in the morning, one in the afternoon, was ideal for discussions and exchange of ideas. Indeed, many junior mathematicians took advantage of this time to discuss problems with the speakers and with the other participants of the conference. The second part (16-20 June) was devoted to communications, problem sessions, software demonstrations and small work groups. Both weeks were a great success in terms of participation, discussion and collaboration. There were 56 official participants from more than ten countries.

Here is a list of the invited speakers for the first week with the title of their talks:

Most of these conferences triggered long discussions afterwards. We certainly missed the presence of I. G. Macdonald and C. Procesi who had to cancel their trip to Montréal at the last minute.

In the second week, communications were given by: G. Bhowmik (Univ. de Valenciennes), P. Cellini (Univ. di Padova), S. Hamel (UQAM), A. Joellenbeck (Univ. Kiel), C. Lenart (MIT), F. Patras (Nice), M. Schocker (Univ. Kiel), F. Sottile (Univ. of Toronto), G. Tesler (UCSD), S. van Willigenburg (St. Andrews). We had two problem sessions. During the first one, the whole period was dedicated to problems suggested by many participants. During the second period, we also had a software demonstration of ACE given by A. Lascoux. The rest of the time was devoted to work in small groups.

Chaire Aisenstadt

The Chaire Aisenstadt was endowed by Montréal philanthropist Dr. André Aisenstadt. Under its auspices, one or two distinguished mathematicians are invited each year for a period of at least one week, ideally one or two months. During their stay the lecturers present a series of courses <> a specialized subject. They are also invited to prepare a monograph. At the request of Dr. Aisenstadt, the first of their lectures should be accessible to a wide audience. Previous holders of the Chaire Aisenstadt are: Marc Kac, Eduardo Zarantonello, Robert Hermann, Marcos Moshinsky, Sybren de Groot, Donald Knuth, Jacques-Louis Lions, R. Tyrell Rockafellar, Yuval Ne'eman, Gian-Carlo Rota, Laurent Schwartz, Gérard Debreu, Philip Holmes, Ronald Graham, Robert Langlands, Yuri Manin, Jerrold Marsden, Dan Voiculescu, James Arthur, Eugene B. Dynkin, David P. Ruelle, Robert Bryant, Blaine Lawson, Yves Meyer and Ioannis Karatzas. This year the Chaire was awarded to Professors László Babai and Efim I. Zelmanov.

Professor László Babai
University of Chicago

László Babai, Professor in the Departments of Mathematics and Computer Science of the University of Chicago, held the Aisenstadt Chair in the fall semester of 1996.

Professor Babai received his Diploma in Mathematics from Eötvos University in Budapest, Hungary and his Ph.D. from the Hungarian Academy of Sciences in 1984. Since then he has been a faculty member at the University of Chicago.

Professor Babai's research is in the fields of combinatorics, graph theory and groups, especially automorphism groups. He is the author of some 140 papers and has given numerous invited addresses. Among his honours are the Gödel Prize, the T. Szele Prize, awarded by the J. Bolyai Mathematical Society, and the Mathematical Prize of the Hungarian Mathematical Society.

During his visit to the CRM in the fall of 1996, Professor Babai gave a series of lectures on Algorithms in Finite Groups. In addition, he gave a public lecture entitled "Surprise Methods in Combinatorics." A book based on these lectures will be published by the American Mathematical Society in the CRM Monograph Series.

Surprise Methods in Combinatorics
Lecture and reception: 22 November 1996, CRM
Algorithms in Finite Groups
Three lectures: 18, 19 and 21 November 1996, CRM, in conjunction with the workshop on "Distance-Regular Graphs"

Professor Efim I. Zelmanov
Yale University

The Aisenstadt Chair was held during the winter semester of the 1996-1997 academic year by Professor Efim I. Zelmanov. Professor Zelmanov received the degrees of M.S. (1977) and Ph.D. (1980) at the Novosibirsk State University, and his Dr. at the Leningrad State University in 1985. After holding various positions at Institutes of Mathematics in the former Soviet Union, he emigrated to the United States in 1990 and was Professor at the University of Wisconsin, the University of Chicago, and Yale University where he currently teaches.

Professor Zelmanov has written more than 60 papers, given numerous invited addresses, and is the editor of 7 major journals in Algebra. His awards include the Fields Medal, awarded in 1994 for the solution of the Burnside Problem, and the Medal of the Collège de France.

During his visit to the CRM in the spring of 1997, Professor Zelmanov gave a public lecture entitled "An Overview of Abstract Algebra in the 20th Century." The lecture was video-taped and is currently available from Les Publications CRM. He also gave a series of lectures covering aspects of the Burnside Problem, groups with the Golod-Shafarevich property, and growth in groups and Lie algebras. An expanded version of these lectures will appear in the CRM Monograph Series published by the American Mathematical Society.

An overview of Abstract Algebra in the 20th Century
Lecture and reception: 9 May 1997, CRM
On the Burnside Problem
Two lectures: 12 and 13 May 1997, CRM

In 1902 W. Burnside formulated his famous problem. Let G be a finitely generated group. Suppose that there exists n > 1 such that xn = 1 for an arbitrary element x from G. Does it make the group G finite? Professor Zelmanov discussed the history of this problem and its relations with ring theory, Lie algebras, profinite groups, combinatorics, logic.

Variations on the theme of Burnside
14 May 1997, CRM

Professor Zelmanov discussed some open problems from different areas of mathematics (algebraic geometry, topology, Galois theory, geometric group theory) that are related to the Burnside problem.

On groups with Golod-Shafarevich property
20 May 1997, CRM

In 1964 Golod and Shafarevich found a sufficient condition for a group presented by generators and relations to be infinite. In the same volume of Izvestia Shafarevich applied this criterion to the construction of an infinite tower of class fields while Golod constructed a counter example to the General Burnside Problem. Professor Zelmanov discussed some recent developments related to groups satisfying the Golod-Shafarevich criterion.

On Growth in Groups and Lie Algebras
21 May 1997, CRM

Professor Zelmanov discussed the notions of growth in Groups and Algebras (the latter known as Gelfand-Kirillov dimension) focusing on Lie algebras and superalgebras of growth 1. Such are, for example, affine Kac-Moody algebras and superconformal algebras.

General Program

The CRM's general program funds a wide variety of scientific events, both on-site and around the country. The program is quite flexible, to allow for opportunities as they arise. One new feature is that the CRM, along with the other Canadian Mathematics Institutes, has taken over the responsibility for funding conferences across Canada, following the cancellation of NSERC's conference grant program.

The year 1996-97 saw a remarkable variety of activities being funded, in areas such as number theory, mathematical physics, statistics, complex analysis, dynamical systems, algorithms in CFD, Hopf algebras, and operator theory. Two general meetings were also funded, as well as a summer school in Nonlinear Dynamics in Physiology and Medicine.

5th Conference of the Canadian Number Theory Association (CNTA-5)

17-22 August 1996, Carleton University, Ottawa
Org.: Henri Darmon (McGill Univ.), John B. Friedlander (Univ. of Toronto), Rajiv Gupta (UBC), James G. Huard (Canisius Coll.), Damien Roy (Univ. of Ottawa), Kenneth S. Williams (Carleton Univ.)

The Canadian Number Theory Association (CNTA) is an informalorganization of Canadian number theorists, which was founded in 1987 with the expressed intention of enhancing and promoting both learning and research in number theory and its applications. It has organized four conferences (Banff 1988, UBC 1989, Queen's 1991 and Dalhousie 1994). All of these conferences have been acclaimed by the international number theory community as being among the most successfuland fruitful of such meetings. The number theory community relies onthese meetings to keep abreast of the latest developments in the field. The present conference continued this strong tradition. It focused on the following four areas of number theory: algebraic/computational number theory, analytic number theory, arithmetic algebraic geometry and elliptic curves, and diophantine problems. All of these areas have seen rapid and very deep development in recent years, and all have had a significant impact on real-life applications, for example, computational number theory on public-key cryptography. The scientific program consisted of 1 general public lecture (D. Boyd), 4 one-hour plenary lectures (W.D. Duke, C. Pomerance, K.A. Ribet, M. Waldschmidt), 27 forty-minute special session invited talks, and 71 twenty-minute contributedtalks. There were 180 registered participants: 60 from Canada, 81 from USA, 29 from Europe, 7 from Asia, and 1 each from Africa, Australia andSouth America.

Extended and Quantum Algebras and their Applications to Physics

Sponsors: Nankai Institute of Mathematics (Tianjin) and the CRM
19-24 August 96, Tianjin (China)
Org.: Ge Mo-Lin (Nankai Institute), Yvan Saint-Aubin (CRM), Luc Vinet (CRM)

The goal of this meeting was to foster collaborations between mathematical physicists in China and in Canada. As both the Nankai Institute for Mathematics (Tianjin, China) and the Centre de recherches mathématiques have very active mathematical physics groups, they took the lead in organizing this joint meeting. This collaboration was part of a larger agreement, the 3x3 Consortium, between three Chinese universities (Beijing University, Tianjin University and Tsiang Hua University) and four Canadian ones (McGill University, University of British Columbia, Université de Montréal and University of Toronto) that aims at developing collaborations between the two countries in seven strategic domains, one of them being mathematics.

The meeting brought together physicists and mathematicians from China and Canada, but also from the Pacific Rim (mainly Japan) and from Europe.

The main lectures were:

46e Colloque des sciences mathématiques du Québec et
20e Conférence en mathématiques et statistique de l'Atlantique

18 - 19 October 1996, Université de Moncton, New-Brunswick
Organising Committee: Donald Violette, president; Thu Pham-Gia, director of Dép. de math. et de stat., Univ. de Moncton; Paul Deguire, in charge of the mathematics contest
Sponsors: CIPAS (Conseil des provinces Atlantiques pour les sciences), Coopération N-B/Québec, CRM, Fac. des sciences de l'Univ. de Moncton, Univ. de Moncton, ACFAS-Acadie, Conseil étudiant de la Fac. des sciences de l'Univ. de Moncton

More than 125 participants gathered at this joint meeting of mathematicians from Québec and the Maritimes. Three plenary speakers were invited: G. Brassard (Université de Montréal), L. Glass (McGill University) and D. Dupuis (Technical University of Nova Scotia). Thirty communications completed the scientific program.

The meeting was also the host of a mathematical competition for undergraduates. There were 15 teams and 29 participants. The winning teams have been:

1st prize: Stephen Finbow, Qiyan Li (St. Mary's Univ.);

2nd prize: Guario Laverty, Fai Tam (Univ. of New-Brunswick);

3rd prize: Mark Lewis, Keith Fordham (Dalhousie Univ.);

4th prize: Alex Fraser, James Worrall (Dalhousie Univ.).

Complex Analysis & Geometry

Special Session at the Winter Meeting of the Canadian Mathematical Society
7-9 December 1996, London, Ontario
Org.: Finnur Larusson, (Univ. of Western Ontario)

Starting from the notion of a holomorphic function of one complex variable, complex analysis and its alter ego analytic geometry now encompass the multiple aspects of function theory and geometry of complex analytic spaces, with or without singularities, in one and higher dimensions. It is a major branch of modern mathematics with many important connections to other areas. There were twelve talks at the session, describing recent developments in the field.

Approximately 25 people attended the session. The speakers were: John S. Bland (Univ. of Toronto), Daniel M. Burns (Univ. of Michigan), Frederic Campana (Univ. de Nancy), Bruce Gilligan (Univ. of Regina), Christer Kiselman (Uppsala Univ.), Laszlo Lempert (Purdue Univ.), Steven Shin-Yi Lu (Univ. of Waterloo), Evgeny A. Poletsky (Syracuse Univ.), Mohan Ramachandran (SUNY at Buffalo), Thomas Ransford (Univ. Laval), Ragnar Sigurdsson (Univ. of Iceland), Berit Stensones (Univ. of Michigan).

Symposium for the 60th Birthday of Jií Patera and Pavel Winternitz

9-11 January 1997, CRM
Org.: Y. Saint-Aubin, L. Vinet (CRM)

Jií Patera and Pavel Winternitz were both born in 1936 and their 60th anniversaries provided a wonderful occasion to pay tribute to their remarkable scientific achievements, the enormous role that they have played in the life of the CRM, and what they have done for mathematical physics at the Université de Montréal, and in Canada for that matter.

The theme of this symposium was "Algebraic Methods and Theoretical Physics" and encompassed most of Jií's and Pavel's work. About 60 friends gathered to celebrate the event. Among the group were collaborators, ex- and current postdoctoral fellows and students, and scientists whose career have been influenced by their work.

The titles of the main lectures give an insight into the diversity of our two colleagues' contributions: The Fibonacci deformed harmonic oscillator (J.P. Gazeau, Paris 7), Linearizable continuous and discrete systems: the Riccati saga (B. Grammaticos, École Polytechnique, Paris), Isotropic geometry, Clifford modules and integrable systems (J. Harnad, Concordia), On the abstract structure of Lie pseudogroups of infinite type (N. Kamran, McGill), Lie modules with bounded multiplicities (F.W. Lemire, Windsor), Conditions for the existence of higher symmetries and nonlinear evolution equations on the lattice (D. Levi, Roma), Superintegrability on the two dimensional hyperboloid (W. Miller Jr., Minnesota), The relativistic oscillator and the mass spectra of baryons (M. Moshinsky, UNAM, México), Moving coframes (P.J. Olver, Minnesota), Seiberg-Witten theory without tears (L. O'Raifeartaigh, Dublin Inst. Adv. Studies), Contraction of Lie algebras and separation of variables (G. Pogosyan, Dubna),

Self-dual bilinear forms for discrete Painlevé equations: the grand scheme (A. Ramani, École Polytechnique), Bargman representation revisited for a deformed harmonic oscillator (G. Rideau, Paris 7), A vector-coherent-state inducing construction for coupling coefficients (D.J. Rowe, Toronto), Symmetry Operations in the Brain: Music and Reasoning (G.L. Shaw, UC at Irvine), On the higher-dimensional Laplace transformations and applications (K. Tenenblat, Brasilia), Graded contractions of Lie algebras of physical interest (J. Tolar, Czech Tech. Univ.).

Workshop on Dynamical Systems

20-25 January 1997, CRM
Org.: Dana Schlomiuk (Univ. de Montréal)

This workshop, in which six lectures were given, was held during the winter semester of the academic year 1996-1997. Two of the speakers (Robert Roussarie and Robert Kooij) spent two weeks at the CRM. Ana Guzman from UNAM-México, who was spending the academic year at the Université de Montréal, also gave a lecture in the workshop. The workshop stimulated scientific discussions, and joint projects were initiated (for example Dana Schlomiuk with Robert Roussarie on the geometry of quadratic vector fields). Robert Kooij obtained a new result during the time he spent at CRM and a CRM report based on this result and entitled "Limit cycles in quadratic systems with a weak focus and a strong focus" was written by him and André Zegeling. Students enrolled in the Master Degree and Ph.D. programs participated at the workshop and some of them benefited from the presence of foreign visitors to advance their research. The following lectures were given at this workshop:

Monday January 20

Wednesday January 22

Thursday January 23

Calogero-Moser-Sutherland models

10-15 March 1997, CRM
Org.: J.F. van Diejen (CRM) and L. Vinet (CRM & Univ. de Montréal)

The Calogero-Moser-Sutherland (CMS) models are certain integrable systems of n interacting particles on the line or circle, which were discovered in the early seventies by F. Calogero, J. Moser, and B. Sutherland. One of the intriguing aspects of these models is their remarkably ubiquitous nature. For instance, at the level of classical mechanics the CMS models are intimately related to the theory of solitons and nonlinear integrable wave equations (e.g. sine-Gordon, KdV, KP), the geodesic motion on simple Lie groups, questions in symplectic and algebraic geometry, , whereas at the level of quantum mechanics there are close connections with the harmonic analysis on symmetric spaces and their quantum (i.e. q-) versions, the combinatorial theory of symmetric functions (Jack and Macdonald polynomials), the theory of special functions in several variables, random matrices, exactly solvable models in condensed matter and quantum field theory (e.g. spin models, quantum sine-Gordon theory), anyon physics, conformal field theory, Recent years have shown an increasing amount of activity in this area of research, stimulated, for example, by the discovery of relativistic (Ruijsenaars-Schneider) and spin (Haldane-Shastry) type versions of the CMS model.

The purpose of the present workshop was to stimulate communication between experts covering a relatively wide spectrum of research in areas of mathematics and physics where the CMS systems are studied from different perspectives, and to provide an overview of the present state of the art regarding the various research activities involving CMS systems. The workshop was attended by 64 registered participants from Canada (23), USA (12) Japan (8), Russia (4), UK (3), France (2), Italy (2), Switzerland (2), Australia (1), Belgium (1), Greece (1), Germany (1), India (1), Mexico (1), The Netherlands (1), Ukraine (1).

Among the participants were two of the three founding fathers of the field: F. Calogero (Italy) and B. Sutherland (USA). In addition to the invited lectures by Professors Calogero (Tricks of the trade: relating and deriving solvable integrable dynamical systems) and Sutherland (Exactly solved many-body problems: old and new results) the program consisted of twenty-five 45-minute invited lectures by J. Avan (France), H. Awata (USA), T. Baker (Australia), R. Bhaduri (Canada), O.

Bogoyavlenskij (Canada), H. Braden (UK), B. Enriquez (France), P. Di Francesco (USA), E. Gutkin (USA), F.D. Haldane (USA), V. Inozemtsev (Russia), I. Krichever (USA), F. Lesage (USA), P. Mathieu (Canada), N. Nekrasov (USA), M. Olshanetsky (Russia), A. Polychronakos (Greece), S. Ruijsenaars (The Netherlands) E. Sklyanin (Japan), T. Shiota (Japan) C. Tracy (USA), A. Varchenko (USA), A. Veselov (UK), M. Wadati (Japan), G. Wilson (UK) and fifteen 30-minute contributed lectures by Y. Berest (Canada), P. Choquard (Switzerland), F. van Diejen (Canada), M. Dijkhuizen (Japan), R. Floreanini (Italy), A. Kasman (Canada), A. N. Kirillov (Canada), K. Hasegawa (Japan), C. Quesne (Belgium) D. Sen (India), K. Taniguchi (Japan), A. Turbiner (Mexico), D. Uglov (Japan), K. Vaninsky (USA), A. Zhedanov (Ukraine).

In the week prior to the CMS workshop, Prof. Ruijsenaars gave an introductory crash course on the Classical Calogero-Moser-Sutherland and Toda type systems aimed mainly at graduate students of the CRM and the four Montréal based universities (Université de Montréal, McGill University, Concordia University and UQAM).

Aside from the workshop program, Prof. Calogero who is also secretary general of the Pugwash conferences on science and world affairs presented a special lecture entitled: A nuclear-weapon-free world: is it desirable? is it feasible? is it likely? In this connection an interview with Prof. Calogero appeared in the March 24, 1997 issue of Forum, the weekly newspaper of the Université de Montréal (vol. 31, no. 25).

Themes addressed by the lecturers include: dynamical R matrices for CMS models; the underlying algebraic structure of CMS models and their relativistic- and spin-type generalizations; Dunkl operators and multivariable Hermite polynomials; the problem of lacunae and analysis on root systems; collective field methods and two-dimensional systems; symmetries of integrable Hamiltonian systems and applications; functional equations and R-matrix constructions associated with CM models; tricks for relating and deriving solvable and integrable dynamical systems; Coulomb system with energy spectrum equivalent to CMS model; meander determinants; quantisation of Poisson brackets on symmetric spaces and multivariable Askey-Wilson polynomials; elliptic quantum groups and quasi-Hopf algebras; conditions for integrability of an n-particle model; the Haldane-Shastry spin chain and the ideal spinon gas model; Ruijsenaars commuting difference operators from Belavin's elliptic R-matrix; integrability and diagonalization of the Haldane-Shastry spin chain with elliptic interaction; bispectrality and linearisation of Calogero-Moser systems; elliptic solutions to difference nonlinear equations and Bethe ansatz; Yangian symmetry in conformal field theory; CMS systems in gauge theories; the Painlevé-Calogero correspondence; multidimensional Calogero models; three-body generalizations of the Sutherland problem; generalized Lamé functions; multispecies CMS models; Calogero-Moser and KP hierarchy; separation of variables in Jack and Macdonald polynomials; exactly solved many body problems, non-diffractive scattering and asymptotic Bethe ansatz; differential operators that commute with the inverse-square Hamiltonian; distribution of largest eigenvalue in Gaussian random matrix ensembles; quasi-exactly solvable many-body problems; Yangian Gelfand-Zetlin bases, gl(n)-Jack polynomials and dynamical correlation functions for the CMS model; a hierarchy of CMS systems with polynomial wave functions; thermodynamics of Calogero-Moser potentials and the Seiberg-Witten solution; algebraic Bethe ansatz for the elliptic quantum group Et,h(sl2); algebraic integrability of generalized Calogero-Moser systems and deformations of root systems; towards an algebraic treatment of Calogero models; the quantum Calogero model: integrability, algebraic structures and orthogonal basis; collisions in Calogero-Moser models in the complex domain; multidimensional factorization method and multivariable Krawtchouk polynomials; quadratic algebras, Dunkl elements and Schubert calculus.

The Bispectral Problem

17-21 March 1997
Org.: J. Harnad, A. Kasman and P. Winternitz

It has been 15 years since F. Alberto Grünbaum introduced the notion of bispectrality of an operator. Originally proposed in the context of medical imaging, the "Bispectral Problem" has since been related to diverse areas of mathematical physics including: representation theory, Darboux transformations, integrable systems of particles, soliton equations, orthogonal polynomials, isomonodromy, and Huygens' principle. Interest in bispectrality continues to grow as new connections are found.

This workshop was the first scientific meeting devoted to the bispectral problem. Many of the 38 registered participants of the workshop were researchers who have made significant contributions to our understanding of bispectrality, and others were mathematicians and physicists who work in related fields. Thus, the workshop fulfilled its goal of bringing together experts from a wide variety of areas to collectively address the bispectral problem.

The scheduled talks included both surveys of some of the research that form the foundation of the investigation of the field and exciting new results. Two topics of particular interest were recent results by Horozov et al. relating bispectrality to the highest weight representations of W1+ algebras and new developments in discrete versions of the bispectral problem. All those present benefited from the superb talks given by the invited and contributing speakers and also from the informal discussion groups that they inspired.

The principal speakers at this meeting were: Y. Berest (Université de Montréal), F. A. Grünbaum (University of California, Berkeley), A. Kasman (Concordia University and CRM), L. Haine (Université Catholique de Louvain), J. Harnad (Concordia University and CRM), E. Horozov (Sofia University), A. Its (Indiana University-Purdue University at Indianapolis), F. Magri (Università di Milano), V. Matveev (Université de Bourgogne), A. Orlov (Université de Montréal), A. Radul (Howard University), M. Rothstein (University of Georgia Athens), T. Shiota (Kyoto University), A. Veselov (Loughborough University), G. Wilson (Imperial College), and Jorge P. Zubelli (IMPA, Brazil). In addition, there were excellent contributed talks by M. Gekhtman (University of Michigan), M. Kovalyov (University of Alberta), F. Levstein (Universidad Nacional de Cordoba), J. McKay (Concordia University) V. Retakh (Harvard University), and J. van de Leur (University of Twente).

The proceedings of this workshop will be published in the CRM-AMS Proceedings Series and should prove to be a valuable resource for those working with bispectrality and related topics.

International Workshop on Parallel Algorithms and Software for CFD

24 - 25 March 1997, at CERCA
Sponsors: CERCA (Centre de Recherche en Calcul Appliqué), CRM, Département d'informatique et de recherche opérationnelle (Univ. de Montréal), Silicon Graphics Inc., Environment Canada
Org.: Andrei Malevsky (CERCA), Maurice Meneguzzi (Saclay), Ahmed Sameh (Univ. of Minnesota)

Computational Fluid Dynamics (CFD) problems such as numerical weather forecasting, modelling of flows around aircraft and vehicles, and oil reservoir simulations are among the major clients of High Performance Computing (HPC) technology. The arrival of distributed-memory massively parallel processors (MPP) had promised a dramatic increase in processing speed and computer memory available for applications, and several researchers have demonstrated that their CFD applications can be ported to MPP's with a significant speed-up. Despite these facts, parallel computing has not yet gained a widespread acceptance in the scientific and engineering communities, in part because of lack of software to support the CFD applications. Until recently, every parallel computer manufacturer had offered his own version of software, often incompatible with the products of other vendors. A few general-purpose problem-solving techniques have been available, but achieving good performance on parallel machines resembled an art more than a science.

About 40 specialists in CFD and parallel computing were invited at CERCA to discuss this issue, within the framework of an international workshop. The main conclusions were as follows:

Researchers apply the HPC technology to model diverse CFD problems. A variety of numerical methods have been used on distributed-memory architectures:

Numerical weather prediction is among the major users of HPC technology and is often referred to as one of the Grand Challenge problems of computing. S. Thomas (Environment Canada) presented the parallel distributed-memory implementation of the Mesoscale Compressible Community Model (MC2). The parallel model allows to increase the resolution of regional weather simulations and can also be used for high-resolution studies of atmospheric turbulence.

All these applications have proved the feasibility of the use of distributed-memory parallel architectures for CFD research and engineering.

Hopf Algebras and Group Rings

5-10 May 1997, Memorial University of Newfoundland (MUN), St. John's, Newfoundland
Org.: M.Beattie (Mount Allison Univ.), E.Jespers (MUN)
Sponsors: Atlantic Association for Research in the Mathematical Sciences (AARMS), CRM, Fields Institute, Dalhousie Univ., Mount Allison Univ., MUN, Univ. of New Brunswick (UNB)

The aim of this workshop was to foster cooperation and collaboration between algebra groups in the Atlantic provinces, including graduate students. The workshop was timed in early May to take advantage of planned visits by several international visitors at that time.

Dr. Yuly Billig (UNB, Fredericton) offered a short course (10 hours) on Kac-Moody algebras in the afternoons; this opportunity alone justified the workshop. In addition to the mini-course on Kac-Moody algebras, eighteen of the twenty-four participants presented their recent work. Of these talks, half were primarily about problems in group rings; the others were concerned with problems in more general Hopf algebras. In the course of the week, several working groups emerged.

Of the twenty-four participants, fourteen came from Canada's Atlantic region, two from other regions of Canada, three from the U.S., four from Europe and one from South America.

The workshop was planned to be as informal as possible, to encourage participants to talk about their latest work, ideas and conjectures, to share problems

and expertise. Judging from comments at the close of the week, participants found the workshop valuable and tentative plans for another "Atlantic Algebra Workshop" are underway.

Joint Meeting:
Canadian Operator Algebra Symposium & Great Plains Operator Theory Seminar

18 - 22 May 1997, Queen's University, Kingston, Ont.
Org.: James A. Mingo (Queen's Univ.), Norberto Salinas (Univ. of Kansas)

This joint meeting of the two symposia celebrated the 25th meeting of the Canadian symposium, the 60th birthday of Peter Fillmore, and the 50th birthday of Alain Connes.

The Canadian symposium first met on March 31, 1972 at the University of Toronto. It has met every year since then (except 1979) to present the work of Canadian and foreign researchers in operator algebras and operator theory. An important feature of the symposium has been the support (both financial and moral) given to graduate students and junior researchers. The success of the symposium may be gauged by its longevity and the number of similar conferences it has spawned: the Great Plains Operator Theory Seminar, the West Coast Operator Algebra Conference, and the annual meetings of a group funded by the European Union.

161 participants from Argentina, Australia, Canada, Denmark, France, Germany, Greece, Ireland, Japan, Mexico, Norway, Poland, Slovenia, Switzerland, Turkey, Venezuela, Uruguay, and the United States came to Kingston to hear talks from seven invited speakers: D. Bisch (Santa Barbara), M. Dadarlat (Purdue), T. Giordano (Ottawa), N. Higson (Penn. State), E.Kirchberg (Humbolt), A. Nica (Michigan), G. Pisier (Paris 6 and Texas A & M). An eighth invited speaker (A. Connes, Collège de France) was unable to attend for health reasons. Besides the invited talks, there were 89 contributed papers.

The main themes of the conference were: classification of C*-algebras, K-theory and E-theory, subfactors, noncommutative probability, non-self-adjoint algebras, completely bounded mappings and operator spaces, C*-algebras and dynamical systems, and operator theory.

Montréal '97 Summer School
Nonlinear Dynamics in Physiology and Medicine: An Intensive Introduction

25 May - 6 June 1997, McGill University
Org.: Leon Glass, Michael Mackey, Daniel Kaplan (McGill)

This second Summer School on Nonlinear Dynamics in Physiology and Medicine, Montréal97, was organized and run based on the highly successful Montréal96 of last year. This year's summer school had lectures held in the Department of Physiology, and computer laboratories held in the Faculty of Arts Computer Laboratory. Montréal97 was partially supported by the Department of Energy (USA) and Math Works Inc. who supplied Matlab software. TheCentre de Recherches Mathématiques partially defrayed student tuition fees, and the McGill University Department of Physiology provided infrastructure and administrative support.

The more than 80 students were selected on a first-come, first-served basis from over 100 applicants. They came from 18 countries, ranging in subject specialization from biology and medicine through theoretical physics and applied mathematics. Student housing was provided in the McGill University dormitories.

Three features of Montréal97 were unique. The first was that the lectures were given by individuals with first-hand knowledge in the practical application of nonlinear dynamics to biological problems. Often in such schools, lecturers have one skill or the other, but rarely both. The second was the inclusion and integration of lectures on time series analysis techniques with concepts from dynamics. The third was the daily computer laboratory designed to illustrate the concepts of the lectures through numerical experiments using software written by the lecturers. Graduate and undergraduate students of the CNLD served as laboratory assistants, as did the lecturers, to supplement the instructions given in the laboratory manual.

Based on the evaluations by the students attending, the response was even more positive that the one held last year. By all accounts, the students found it to be a highly rewarding experience, and we are all justifiably proud of the way in which we have organized and carried off this rather unique event.

Those interested in finding out more details of Montréal97 may consult: http://www.cnd.mcgill.ca/. A sequel, Montréal99, is under consideration.

18th Annual Meeting of the Canadian Applied Mathematics Society

30 May - 1 June 1997, Fields Institute
Co-chairs of org. comm.: A.Lawniczak (Guelph), W.Langford (Fields), P.Sullivan (Waterloo)

The three-day meeting included 10 plenary lectures and 18 mini-symposia on bioinformatics, communication networks, computing technology in mathematics education, cryptography, environmental problems, financial mathematics and risk management, fractal image compression, geophysical fluid dynamics, industrial mathematics, mathematics in the biomedical sciences, mesoscale phenomena in fluids and materials, modelling of polymers, neural networks, numerical analysis of differential equations, parallel computation, topics in pulsatile flow, research partnerships of academia, industry and government, and nonlinear waves.

CRM Prizes

CRM/Fields Prize

The Centre de recherches mathématiques and the Fields Institute announced in early 1994 the creation of a new prize aiming at recognizing exceptional work in the mathematical sciences. The recipient is chosen by the Advisory Committee of the CRM and the Scientific Advisory Panel of the Fields Institute on the basis of outstanding contributions to the advancement of research. The main selection criterion is research excellence. A prize of $5000 is awarded and the recipient presents a lecture at the CRM and the Fields Institute. The previous winners of the CRM/Fields Prize were Prof. H. S. M. Coxeter (University of Toronto) and Prof. G. A. Elliott (University of Toronto).

The 1996 CRM-Fields Institute Prize was awarded to Professor James Arthur of the Department of Mathematics, University of Toronto. Professor Arthur was honoured for a career of research contributions in a variety of mathematical fields including harmonic analysis, number theory, Lie-group theory, representation theory, and geometry. He has combined tools in each of these fields to achieve a deep understanding of basic mathematical phenomena. To quote the letter of nomination: "Arthur is a powerful mathematician whose trace formula is having a profound effect on representation theory and automorphic forms", and "His recent work and conjectures on Arthur's packets and the local trace formula are major milestones that are setting the direction for research in a central part of mathematics."

James Arthur obtained the B.Sc. (1966) and the M.Sc. (1967) in Mathematics from the University of Toronto and his Ph.D. in 1970 from Yale University under the direction of Robert Langlands. He then spent time at Princeton, Duke and Yale Universities before coming to the Department of Mathematics of the University of Toronto in 1978. He currently holds the position of University Professor.

Professor Arthur has written over forty articles and given numerous invited lectures at important meetings. Among the honours he has received are the Sloan Fellowship and the E.W.R. Steacie Memorial Fellowship. He has also been named a Fellow of the Royal Society of London and a Fellow of the Royal Society

of Canada which presented him the Synge Award in 1987.

Professor Arthur visited the CRM on April 4, 1997 to deliver the 1996 CRM-Fields Institute Prize Lecture which was entitled Harmonic Analysis and Trace Formulas. An abstract of this talk follows:

Harmonic analysis could be interpreted broadly as a general principle which relates analytic and geometric objects. Examples occur throughout many areas of mathematics. In group theory, the geometric objects are conjugacy classes, the analytic objects are irreducible characters, and the two can be related by means of trace formulas. We shall give a general introduction to trace formulas, and their applications to group representations and number theory.

André-Aisenstadt Prize

Created in 1991, the André-Aisenstadt Mathematics Prize is intended to recognize and reward talented young Canadian mathematicians. The Prize, which is given for research achievement in pure and applied mathematics, consists of a $3000 award. The recipient is chosen by the CRM Advisory Committee. At the time of nomination, candidates must be Canadian citizens or permanent residents of Canada, and no more than seven years from their Ph.D. The previous winners of the André-Aisenstadt Prize were Niky Kamran(1991), Ian Putnam(1992), Michael Ward(1994), Nigel Higson(1994) and Adrian S. Lewis(1995).

The CRM Advisory Committee recommended the awarding of two Aisenstadt Prizes for the academic year 1996-97. These went to Henri Darmon and Lisa Jeffrey both of McGill University.

Henri Darmon was cited for his remarkable work in the area of elliptic curves, particularly for his refinements of the famous Birch Swinnerton-Dyer conjecture. He has also made significant contributions to research on variants of the Fermat equation. In addition to his superb research contributions, Professor Darmon is a splendid expositor and a recent paper explaining the subtleties of Wiles' work on the Shimura-Taniyama conjecture has been widely celebrated.

Professor Darmon obtained his B.Sc. in Mathematics and Computer Science at McGill University in 1987 and his Ph.D. in Mathematics at Harvard University in 1991. He then spent 4 years at Princeton University before coming to McGill University where he is currently an associate professor.

He has also held various visiting positions at such institutions as ETH in Zurich, the Mathematical Sciences Research Institute in Berkeley, and IHES in Paris.

Henri Darmon has published more than 26 research papers and won numerous awards, among them an Alfred P. Sloan Doctoral Dissertation Fellowship and an Alfred P. Sloan Research Award.

Lisa Jeffrey was awarded the Aisenstadt Prize for her distinguished research contributions in Symplectic Geometry and various aspects of the relation between Topology and Physics. In particular, in joint work with Frances Kirwan, she obtained a complete description of the cohomology ring of the moduli space of vector bundles on a Riemann surface, solving an important conjecture of Witten. Techniques invented in the course of this work have proved useful in solving other significant problems as well.

Professor Jeffrey obtained the A.B. at Princeton University, the M.A. at Cambridge University and the Ph.D. at Oxford University in 1992 under the direction of M. F. Atiyah. She then held positions at the Institute for Advanced Study, Cambridge University, and Princeton University before coming to McGill University in 1995 where she is currently Associate Professor of Mathematics.

Professor Jeffrey has written over 21 research papers and numerous review articles. Among the honours she has received are the Kusaka Memorial Prize in Physics from Princeton University and the Smith Prize from Cambridge University.

The two Aisenstadt prizewinners gave lectures on their work at the CRM on February 28, 1997. Professor Darmon's lecture was entitled "Faltings plus epsilon et l'équation de Fermat généralisée" and Professor Jeffrey's was entitled "Flat connections on Riemann surfaces."


The CRM-CAP Prize is given for outstanding contributions to theoretical and mathematical physics. The first winner (1995) was Werner Israel (University of Alberta).

1996 CRM-CAP Prize: Professor William J. Unruh
Physics Dept., University of British Columbia

Bill Unruh was born and raised in Winnipeg. He graduated in 1967 with a B.Sc. from the University of Manitoba and received his M.A. and Ph.D. from Princeton University. An NRC Postdoctoral fellowship taken at Birkbeck College, London was followed by a Miller Fellowship at Berkeley. Bill then returned to Canada to teach in the Department of Applied Mathematics at McMaster University. He was recruited by the UBC Physics Department in 1976. Since 1986, he has also been the Director of the Cosmology Program of the Canadian Institute for Advanced Research. Due in large part to his inspirational leadership, this is a very successful program with a high international profile.

Bill is an exceptionally gifted scientist of truly international stature. His contributions to theoretical physics reflect range, versatility and creativity. His work can be categorised under five broad headings: quantum field theory in curved space-times; fundamental basis of quantum mechanics and measurement theory; cosmology; foundations of quantum gravity; and, more recently quantum computers.

He was first internationally recognised for his pioneering work on the quantum radiation from black holes. His 1976 paper on acceleration radiation is universally acknowledged as a landmark in the field. This classic contribution, together with Hawking's simultaneous and complementary discovery that black holes are hot, revealed deep and previously unsuspected interconnections, linking quantum theory, gravity acceleration and thermodynamics. The "Unruh vacuum", the "Unruh temperature" and "Unruh detectors" are phrases used by workers in this field and reflect the importance of his contributions. Currently he has been examining the mechanism of black hole evaporation and particle creation near the horizon by using a hydrodynamic model.

Bill's study of quantum non-demolition measurements have been relevant to the problem of gravitational wave detection, and his explanation (with Zurek) of why "Schrödinger cat states" are not generally observed in macroscopic systems is widely accepted.

He has also been a major influence in the arena of cosmology. In 1985 (with Mazenko and Wald) the "new inflation" theory was shown to be flawed. A paper (with Wald) studied the damping mechanisms for coherent oscillations of axions and put constraints on the observation of these dark-matter candidates. Bill has also made important studies of cosmic strings and made a significant contribution to the debate as to why the cosmological constant is so small.

From the early 1980's to the present, he has thought about quantum gravity and emphasised the crucial role of time. There is a fundamental incompatibility between the current theories of quantum mechanics and general relativity which has to be reconciled in any new theory that seeks to explain the physical world at all levels.

Although he is an expert in theoretical and mathematical physics, Bill is interested in science in general and gets involved with the problems of his experimental colleagues. His fields of interest range from applied subjects such as intercalation batteries to interesting contributions to the American Journal of Physics. Importantly he is also willing to participate in outreach programs from school students. He has served on the Canada Council Killam Selection Committee (1984-88) and was a member of the NSERC Space and Astronomy Committee (1989-92).

His contributions to science have been recognised by many awards and prizes. As an undergraduate he placed in the top 10 of the William Putnam Mathematics Competition (1966) and won First Place in the C.A.P. Undergraduate Examination (1967). He was made a Fellow of the Royal Society of Canada in 1984. Other awards include an Alfred P. Sloan Fellowship (1978-80), the Rutherford Medal from the Royal Society of Canada (1982), the C.A.P. Herzberg Medal (1983), the Steacie Prize (1984), the Steacie Fellowship (1984-86), the B.C. Science Council Distinguished Research (1990). In 1995 he was awarded the C.A.P. Medal of Achievement, and this year he was the recipient of the Canada Council lzaak Walton Killam Memorial Prize in the Natural Sciences.

Bill's many friends and acquaintances in Canada and abroad will share the pleasure of his UBC colleagues in this award. Although he has won other prestigious prizes, this one is special because it represents the appreciation of his talents by his peers in Theoretical and Mathematical Physics.

The above text by Brian G. Turrell (UBC) is taken from Physics in Canada, July/August 1996.

1997 CRM-CAP Prize: Professor Ian Affleck
Physics Dept., University of British Columbia

The CAP-CRM prize in theoretical physics is presented to Dr. Ian Affleck of the University of British Columbia for his very successful work in theoretical condensed matter physics highlighting field-theoretic methods and related mathematical physics.

In the 1970's and the early 1980's, most applications of field theory to critical phenomena were devoted to classical systems, mainly through renormalisation-group methods. Dr. Affleck has been one of the first physicists to systematically apply quantum field theory methods to the more subtle problem of quantum critical phenomena, focusing mainly on one-dimensional systems. On this subject he made important contributions in the period 1985-89. For instance, he proposed that Wess-Zumino-Witten models are the fixed points governing quantum spin chains. He also related this description of spin chains with the large-spin approximation based on the nonlinear sigma model. [Wess-Zumino-Witten models are two-dimensional quantum field theories with Lie group symmetry; they can be regarded as the building blocks of all two-dimensional quantum field theories having conformal invariance.] He also made significant contributions to the physics of quasi-one-dimensional spin-1 antiferromagnets in relation with the Haldane gap. Some of his discoveries in the area of quantum spin chains are now standard material in theoretical physics. The essentials of his results and insights were reported in the remarkable lectures presented at Les Houches in 1988, which became an influential review paper.

In 1990, in collaboration with A. Ludwig, he initiated a very important research program on the applications of the recently discovered boundary conformal field theory (formulated by Cardy) to various one-dimensional quantum impurity problems. These applications are mainly in condensed matter physics (e.g., the Kondo effect and its multi-channel and higher-spin generalizations, quantum wires, etc.) but some are related to particle physics and cosmology (the Callan-Rubakov effect describing baryon scattering off monopoles). Conformal field theory provides numerous examples of exactly solvable models (i.e, of which all correlation functions can be calculated exactly). However, apart from two-point functions from which critical exponents may be extracted, very little can be compared with experiments. In their approach to quantum impurities problems, Affleck and Ludwig have shown that various physical quantities (resistivity, specific heat, etc.) can indeed be calculated and compared with experimental data. These studies, in turn, have stimulated new experimental work, which further demonstrates the impact of Dr. Affleck's work. Moreover, this work often provides striking and original physical realizations of some of the most abstract constructions of conformal field theory, such as fusion rules or conformal embeddings in the treatment of the multi-channel Kondo effect.

Although Dr. Affleck's main focus has been the application of field theory methods to condensed matter physics, he has to his credit two fundamental discoveries in conformal field theory. One is the interpretation of the conformal anomaly the basic parameter of a conformal theory as a finite-size or Casimir effect. (The same result was found independently by Blöte, Cardy and Nightingale.) The second one, obtained in collaboration with Ludwig, is the discovery of the g-function, the "ground-state degeneracy", a measure of the impurity entropy, which has been shown to be always decreasing under renormalisation (in close analogy with the Zamolodchikov ctheorem). The importance of both contributions has been widely acknowledged.

Dr. Affleck joined UBC in 1987 as professor and fellow of the CIAR Cosmology Program. He is now also an associate of the CIAR Superconductivity Program. He graduated from Trent University in 1975 and obtained his Ph.D from Harvard University in 1979. He was junior fellow of the Harvard Society of Fellows from 1979 to 1981 and then associate professor at Princeton University until 1987. He received various distinctions since his arrival at UBC: Steacie Prize in 1988, the CAP Herzberg Medal in 1989, elected Fellow of the Royal Society of Canada in 1991, Rutherford Medal of the Royal Society of Canada in 1991, UBC Senior Killam Research Prize and UBC Jacob Bieley Prize in 1992.

The above text by Pierre Mathieu (Université Laval) is taken from Physics in Canada, July/August 1997.

Members' Seminars and Special Events

The members of the CRM are encouraged to organise seminars and other scientific activities during their stay at the CRM, and the CRM hosted several seminar series during the year 1996-97:
There were also a few special events:

CRM-ISM Colloquium

The CRM, together with the Institut des Sciences Mathématiques, the graduate consortium of the four Montréal Universities, runs the Montréal mathematics colloquium, which, during the university year, organises survey talks by distinguished mathematicians on topics of current interests.

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