- Collaboration with the Fields Institute
- Partnerships
- Awards, Distinctions & Landmarks
- Computer Facilities

This year the CRM and the Fields Institute have again enjoyed a very high level of collaboration.

First of all, the CRM and the Fields Institute have jointly invested much effort in the development of the PIms/NNCMS initiative. This has required numerous meetings across the country among the directors of both institutes and other parties. These have allowed the two directors to coordinate efficiently the scientific activities of the institutes and to develop new joint projects. It should also be mentioned that Robert Moody is a member of both the CRM's advisory committee and the Fields Institute's scientific committee and he thus facilitates communication between these two committees.

The CRM and the Fields Institute have jointly sponsored many events. For instance the CAMS meeting at Memorial University, the workshop on Partial Differential Equations and their Applications held at the Fields Institute, the meeting on Nonlinear Dynamics and Time Series that took place at the CRM, the one on Semidefinite Programming & Interior-Point Approaches for Combinatorial Optimization Problems held at the Fields Institute and the workshop on Algebraic Approaches to Quantum Dynamics at the Fields Institute.

A number of CRM's members have taken part in the organization of scientific activities at the Fields Institute. J. Patera (together with R. Moody) has been responsible for organizing a theme-semester on the mathematical foundations of periodic order in the fall of 1995. J. Hurtubise was on the scientific committee of the workshop on Interaction between Topology and Physics that took place in April in the framework of theme activities in Homotopy Theory. Niky Kamran coorganized the meeting on Algebraic Approaches to Quantum Dynamics that was co-sponsored by CRM.

The CRM and the Fields Institute have jointly supported a postdoctoral fellow, Sadok Kallel, who was in residence at the Fields Institute during the program in Homotopy Theory.

Plans are also being made to hold at the Fields Institute parts of the CRM 1997 theme activities in Statistics and of the 1998 program in Number Theory and Algebraic Geometry.

Finally the year has also seen the awarding of the second CRM/Fields Institute prize to George Elliott in recognition of outstanding contributions to the mathematical sciences.

The research group PHYSNUM, led by B. Goulard, joined the CRM during the year 95-96. This event was particularly important for the development of research in applied and industrial mathematics. The PHYSNUM group is presently involved in mathematical imaging and risk management. These activities have emerged from more general research programs based on wavelet theory, neural networks and parallel computing. Beside theoretical work on wavelets (now recognized internationally), the PHYSNUM team has acquired expertise in various aspects of image processing. Recent projects include analysis of images (from satellites and of digital mammograms) based on multiscaling representations given by wavelet transforms. PHYSNUM is pursuing a number of project in collaboration with industry. It enjoys a unique partnership with Atlantic Nuclear Services initiated five years ago. It also has ties with the Hôpital Notre-Dame de Montréal, Noranda (Montréal), and the Canadian Centre for Remote Sensing (Ottawa). The group leader, Bernard Goulard, has received an honourable mention from the NSERC University-Industry Synergy 1995 program for establishing successful partnerships. The group also offers industrial fellowships to young scientists with Ph.D.'s in fundamental domains. So far all the fellows of this group have found positions in industry (ANS, CAE, Hydro-Québec, Lockheed Martin Electronic Systems Canada) at or before the end of their stipend.

Over the last three years, the CRM has enjoyed a
close collaboration with two liaison and research transfer centres, the CERCA (*Centre de recherche en calcul
appliqué*) and the CIRANO (*Centre interuniversitaire de
recherche en analyse des organisations*), and two centres
involved in applied and contractual research, the CRT
(*Centre de recherche sur les transports*) and the GERAD
(*Groupe d'études et de recherche en analyse des décisions*).
Their interests are in applied numerical computations
(CERCA), in management, economy, and finance (CIRANO), in problems related to transportation
(CRT) and in risk and decision (GERAD). The collaboration with these partners takes various forms. In particular, it has allowed the CRM to set up an industrial
postdoctoral fellowship program and to hold meetings that were jointly organized.

**Industrial postdoctoral fellowship program.**
This program was established three years ago to support
research in industrial mathematics, contribute to the
training of specialists in this area and foster collaborations between the university and industry. Four fellowships, jointly funded by the CRM and industry, were awarded in 1995-96. The CRM's partners in this
program include Ad Opt and Cognologic for one postdoctoral fellow (pdf) dealing with monthly scheduling of airline flight personnel, Environment Canada
for one pdf dealing with regional-atmosphere modelling, Bombardier Canada, Pratt & Whitney and Environment Canada for one dealing with wing icing during flights, and Bombardier Canada, GE Canada,
Hydro-Québec, and Environment Canada for one
dealing with large-scale turbulence.

**University-industry workshops. **The first semester of
the current thematic year was devoted to computational hydrodynamics and was organized jointly with
CERCA. It included one large conference and two workshops, one with several mini-courses. It welcomed scientists from both industry and academia. A
week-long meeting on the mathematics of finance was
held in April 96, funded by both the CIRANO and the
CRM, with support from AMI Partners Inc., the Bourse
de Montréal, the Caisse de dépôt et placement du
Québec, and Hydro-Québec. The audience was made
up of economists, bankers and mathematicians.

The industrial associate member (E. Shahbazian) and three visiting members of the CRM (L. Gagnon, M. Mayrand and P. Valin) are scientists working for Lockheed Martin Electronic Systems. Two Ph.D. students are co-supervised by J. Patera (Univ. de Montréal), L. Gagnon and E. Shahbazian, with part of their thesis work being done at Lockheed Martin.

The centres CERCA, CIRANO, CRM, CRT and GERAD are currently writing a grant proposal to be submitted to NSERC in its new research network program. If awarded, this grant would allow the CRM to sustain and expand its research efforts in applied and industrial mathematics.

Michel Delfour, regular member at CRM and professor in the Département de mathématiques et de statistique at the Université de Montréal, was awarded in 1995 the Prix Urgel-Archambault by the Association canadienne-française pour l'avancement des sciences (ACFAS). This prize was created in 1953 in memory of Urgel Archambault, director and founder of the École Polytechnique de Montréal. It rewards scientists working in the physical sciences, mathematics and engineering. It is sponsored by Alcan.

The ACFAS citation is as follows: "Michel Delfour's career has deeply marked mathematical life in Quebec and Canada. Born in Paris, he moved to Montréal and studied electrical engineering at McGill University where he won several awards. He chose to specialize in mathematics and obtained a Ph.D. from the Case Western University in Cleveland. Coming back to Canada, he was successively invited researcher at the Université de Montréal, member of the Centre de recherches mathématiques, and professor in the Département de mathématiques et de statistique at this institution.

"Michel Delfour is a brilliant scientist, original and with extremely diversified gifts. His research, concentrated in applied mathematics, has at least five parts. The first two, developed in the context of the Canadian space program, deal with control and stabilization of systems as well as shape and structure optimization. The applications of this work are numerous, ranging from flexible structure modelling to satellite design.

"Michel Delfour has also worked on the modelling and control of systems with delay, on discontinuous approximation methods for ordinary differential equations and on wavelength assignment, a problem with obvious financial consequences. The results of his last project were extremely well received by the federal Ministry of Communications.

"Besides his accomplishments as a researcher, Michel Delfour has two remarkable qualities as a mathematician: conciseness and elegance. His fifty (or so) articles are models of clarity and mathematical taste. He is an exception to the rule that mathematicians prefer to work alone; all his colleagues praise his team spirit and his enthusiasm is contagious.

"Michel Delfour is currently finishing his third monograph. His international reputation is significant, as his invitation to be a plenary speaker at a SIAM conference indicates. (SIAM is the most prestigious society of applied mathematics in the world.) He won a Killam fellowship in 1989-1991 and is the president since 1991 of the Canadian Mathematical Society."

(The mandate of Prof. Delfour as President of the Canadian Mathematical Society ended in 1994.)

In November 1995, the Wolf Foundation of Israel announced the award of the Wolf Prize jointly to two mathematicians. Robert Langlands and Andrew Wiles (Princeton University) shared the $100,000 prize in mathematics for their ground-breaking research in number theory and related areas. The award was presented to them in Jerusalem in March by Ezer Weizman, President of Israel.

Prof. Robert P. Langlands, 59, did his undergraduate work at the University of
British Columbia and his Ph.D. at Yale University. He was subsequently instructor
and assistant professor at Princeton University and professor at Yale.
Since 1972, he is a member
of the Faculty of the Institute for Advanced Study's
School of Mathematics (Princeton). According to the
Wolf Foundation's citation, Langlands received the
Wolf Prize for his "path-blazing work and extraordinary insights in
the fields of number theory, automorphic forms, and group representation."
His foundational work on Eisenstein series, group representations,
*L*-functions and the Artin conjectures, the principle of functoriality,
and the formulation of the far-reaching Langlands program shaped the modern
theory of automorphic forms and influenced much subsequent research.

Prof. Langlands has been a regular member of the CRM for the past few years. He spends a part of each summer at the CRM. He initiated the CRM summer school, first held in 1990, and has organized a workshop on "Zeta Functions of Picard Modular Surfaces" (spring 88) that still yields scientific results. He has also codirected several students with Y. Saint-Aubin.

Professor Carolyne Van Vliet has retired in 1995 after more than 25 years at the Centre de recherches mathématiques and at the Département de physique at the Université de Montréal.

Prof.Van Vliet obtained her Ph.D. from the Free University of Amsterdam in 1956. From 1956 to 1970,
she was first postdoctoral fellow and subsequently
professor of electrical engineering at the University of
Minnesota. She was one of the first researchers to be
hired at the Centre de recherches mathématiques in
1969. Her interests, since the beginning of her career,
lie in mathematical, statistical and solid state physics,
more specifically in out-of-equilibrium statistical mechanics (in particular linear
response theory and the description of *N*-body correlation and relaxation processes),
quantum transport in condensed matter, fluctuations and stochastic processes, and
mesoscopic phenomena in quantum mechanics and electrodynamics.

Following some severe but valid criticism of Kubo's linear response theory, made for example by
N. van Kampen, Prof. Van Vliet started, in 1978, a profound revision of this theory
consistent with fundamental principles of statistical mechanics such as entropy
production and irreversibility in transport processes. She derived, using Zwanzig's
projection operator technique and within the famous Van Hove limit,
a generalized master equation, with external fields
present and thus more general than that of Pauli or
van Hove, general many-body linear response formulas for the conductivity, and two quantum Boltzmann
equations. Her series of four articles *Linear Response
Theory Revisited I-IV*, published between 1978 and 1984
in the Journal of Mathematical Physics, constitutes a
remarkable contribution to the field. This mathematical theory has been applied to magnetotransport and
hopping conduction in disordered materials. Her graduate students also gave other important applications.

Another of her important results is related to the
long-standing controversy over Handel's quantum theory of the *1/f* noise.
Prof. Van Vliet resolved this debate by giving a rigorous treatment based on quantum
electrodynamics. A further extension of the theory,
involving the very pertinent electron-phonon interaction, was recently completed by one of her students.

Prof. Van Vliet is extremely prolific, with some 200 articles, and is still very active: she pursues her teaching and research career at Florida International University (Miami) while still maintaining her ties to the CRM.

NSERC established the highly competitive Collaborative Grant Program to foster and support
multi-disciplinary research programs. A grant was awarded during the fall of 1995 for
a proposal entitled *Algebraic Combinatorics and Quantum Integrable Models*.
The team consists of: L.Vinet (CRM, team leader), F.Bergeron (UQAM), N.Bergeron (York),
R. Floreanini (INFN, Trieste), A.Garsia (UC at San Diego), J. Harnad
(Concordia), D. Levi (Univ. di Roma III), C. Reutenauer
(UQAM), Y. Saint-Aubin (CRM), Pavel Winternitz (CRM).

The project lies at the interface of theoretical physics and combinatorics,
and focuses on integrable systems and special function theory. Integrable systems
are very important in physics since they often allow
us to find fundamental laws. A paradigm is the Calogero-Sutherland (CS) model which describes
*N* particles on a circle with long-range interactions. It is
known to be relevant to the physics of the fractional
quantum Hall effect and of high-*Tc* superconductivity. It was found recently that the wave functions of
the CS model can be expressed in terms of the multivariate Jack polynomials. The CS model also admits a
relativistic generalization; in this case the quantum
dynamics are governed by a q-difference equation whose solutions are expressed in terms of Macdonald
symmetric polynomials.

The Jack and the Macdonald polynomials are some of the principal objects of study in algebraic combinatorics. Many important and challenging conjectures about these polynomials are still unproved. Very recently, a representation-theoretical setting for the Macdonald basis was discovered by members of the team. This construction led them to discover remarkable properties of these polynomials by skilfully using techniques of algebraic geometry, algebraic combinatorics and computer algebra. Drawing from physical intuition, other members of the team have obtained an operator solution of the CS model that sheds new light on many outstanding physical and mathematical problems.

The main objective of the program is to study the various algebraic and combinatorial aspects of the CS model and its generalizations. This will most likely lead to significant advances in many important mathematical and physical areas (condensed matter physics, special function theory, conformal field theory, representation theory, algebra, integrable systems, etc.) that this model connects in an extraordinary way.

Each year the popular science monthly *Québec Science *presents a group of 10 discoveries done during
the last twelve months in the Province of Québec. The
selection criteria are numerous: among them are the
impact on the field of research, publication in prestigious journals and the improvement of human welfare.
The issue of February 1996 contained this year's winners. A recent work of Luc Vinet, director of the CRM,
and Luc Lapointe, his Ph.D. student (Département de
physique at the Université de Montréal) was cited as
one of the 10 discoveries for 1995.

This recognition came for their proof of a famous conjecture by Macdonald on a class of symmetric functions known as the Jack polynomials. These functions, as was mentioned before, play, on the one hand, a key role in algebraic combinatorics they are deformations of the Schur functions and appear, on the other hand, in the expression of the Calogero Sutherland (CS) model wave functions.

The Macdonald conjecture that Lapointe and Vinet proved can be presented roughly like this. The Jack polynomial depend on one parameter (the coupling constant of the CS model). These functions are usually presented through their expansion on another set of symmetric functions known as the symmetric monomials. The Macdonald conjecture asserted essentially that the expansion coefficients are themselves polynomials in the parameter. This conjecture has resisted for more than seven years the efforts made to prove it.

The proof that Lapointe and Vinet gave makes use of the methods of mathematical physics. They found the creation operators of the CS model whose action on the ground state wave function generates the wave functions of the excited states. They have thus discovered a remarkable formula for the Jack polynomials from which the Macdonald conjecture followed as a simple corollary.

The CRM offers to its members and visitors a Unix environment based on a SUN Sparc 1000 equipped with eight 100-Mhz processors and 256 Mb of memory. This computing power is distributed through the offices and common rooms via Sun stations (Sparc 4, 5 and 10) and X-terminals. The software libraries include compilers (several C and C++, Fortran, etc.), symbolic manipulation programs (Mathematica, Maple and Macaulay), several text editors, web browsers, mail tools, and most utilities common to the mathematical world. Upgrades to TeX and its dialects are uploaded whenever they are released.

The local net is linked to the network of the Université de Montréal that maintains the connections with RISQ (Réseau interordinateurs scientifique québécois) and CA*net (the Canadian internet transit service).

The CRM also has a Silicon Graphics Challenge L with six R4400 processors at 100 Mhz and 128 Mb of memory that was purchased through the NSERC grant of one of its research teams. Access to this server is limited to the members of the team or, upon request, to other members with numerical processing needs.

The support staff works on Sun stations, on X-terminals or on Macintoshes tied to the Sun server for mail services and back-ups.

This year the CRM, the Fields Institute and the PIms submitted a joint grant application to NSERC's Major Facilities Access program. The proposal calls for the establishment of a National Computer Network for Mathematical Research. This network would help maintain and enhance the computing facilities offered at the three institutes, and would allow for the development of better electronic connectivity in the mathematics community. Though the three-year grant was not awarded, NSERC recognized the value of the project by giving $75K as seed money for the network.

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**29 May 1998, webmaster@CRM.UMontreal.CA**