- Scientific Activities (continued)

**
The Chaire Aisenstadt was endowed by Montréal philanthropist Dr. André Aisenstadt. Under these
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 on 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 and Blaine Lawson. This
year the CRM offered two Chaire Aisenstadt lectureships, both in close relationship with the semester
on applied analysis; they were held by Y. Meyer and I. Karatzas.
**

CEREMADE, Université de Paris-Dauphine

5 lectures, 11-15 mars 1996, CRM

Prof. Yves Meyer's lectures entitled "Time-scale and Time-frequency Analysis in Signal or Image Processing" were given during a semester focusing on spline functions and the theory of wavelets. An expanded version of these lectures will appear in the CRM Monograph Series published by the American Mathematical Society.

Professor Meyer occupies the position of "Professeur de classe exceptionnelle" at the Université Paris-Dauphine. Since 1991 he has been detached to the Institut Universitaire de France. He is also Membre de l'Institut (Académie de Sciences de Paris) and a foreign honorary member of the American Academy of Arts and Sciences. He has been awarded the following scientific prizes: Peccot (1969), Salem (1970), Carrière (1972), and the Grand Prix de l'Académie des Sciences (1984). He has given invited lectures at the International Congress of Mathematicians in Nice (1970), Warsaw (1983), and Kyoto (1990) as well as the International Congress of Mathematical Physics, Swansea (1988), and the International Congress of Applied Mathematics, Washington (1991).

Following his studies Professor Meyer worked at several universities in France, among them Strasbourg, Université de Paris-Sud, and the École Polytechnique, before accepting his current position. His recent research interests centre around the various aspects of the theory of wavelets, including the construction of orthonormal wavelet bases, image analysis, signal processing, and the equations of Navier-Stokes. In addition to numerous articles in these fields he has published a series of books on the diverse aspects of wavelet theory. He has also made significant contributions to the fields of algebraic numbers, harmonic analysis, and pseudo-differential operators.

Professor Meyer has directed some 30 Ph.D. students, organized several conferences, and edited various proceedings.

Columbia University

9 lectures, 6-27 May 1996, CRM

Ioannis Karatzas received his Diploma from the National Technical University of Athens in 1975 and then went to Columbia University for his M.Sc. and Ph.D. degrees, received in 1975 and 1980 respectively. After a postdoctoral year in Applied Mathematics at Brown University, he returned to Columbia where he is currently Eugene Higgins Professor of Applied Probability. His research interests have been in Probability and Mathematical Statistics, Random Processes, Stochastic Calculus, Stochastic Control and Optimization, and, most recently, Mathematical Economics and Finance.

Professor Karatzas has written 56 articles, several sets of lecture notes, and the well-known book "Brownian Motion and Stochastic Calculus," written with S.E. Shreve. Another book, entitled "Methods of Mathematical Finance," and coauthored by Shreve, is scheduled for publication in 1997. He is a fellow of the Institute of Mathematical Statistics and serves on the editorial board of several distinguished journals.

Professor Karatzas' Aisenstadt lectures were attended by an enthusiastic audience of mathematicians, economists, and workers in the field of finance. They will be published this fall by the AMS in the CRM Monograph Series under the title "Lectures on the Mathematics of Finance."

31 May - 2 June 1995

Sponsors: CRM, Fields Institute, Memorial Univ. (Vice-President/Research, Dean of Science, Dean of Graduate Studies, Dept. of Mathematics)

The well attended meeting was dominated by two special sessions reflecting current trends and activities in their respective fields, namely the special session "Numerical analysis of nonlinear differential and integral equations" (organized by Hermann Brunner) and the special session "Climate, meteorology, environment" (organized by Sam Shen, Univ. of Alberta).

The first of these special sessions involved seven invited speakers: Alastair Spence (Univ. of Bath, UK), Chris Budd (Univ. of Bristol, UK; now at Bath), Andrew Stuart (Stanford Univ.), David Sloan (Univ. of Strathclyde, Glasgow), Sue Campbell (Univ. of Waterloo), Yanping Lin (Univ. of Alberta), and Uri Ascher (Univ. of British Columbia). The topics of their talks ranged from differential-algebraic integral equations arising in the modelling of catalytic combustion, blow-up in semilinear parabolic PDEs, pseudo-spectral methods for singular problems to delay equations, Volterra integro-differential equations, and various aspects of deterministic and probabilistic computations for ODEs and PDEs. Judging from the reaction of the participants, these talks illuminated many of the current research activities in numerical nonlinear analysis, and they were well received because of their uniformly clear and thoughtful presentation. Moreover, they stimulated a series of follow-up discussions, both formal and informal, among a good number of the participants who benefited from the opportunity of having access to the wide variety of expertise of the speakers.

The second special session involved five invited speakers: Bryant Moodie (Univ. of Alberta), Paul Sullivan (Univ. of Western Ontario), Richard Greatbach (Memorial Univ. of Newfoundland), Gerald North (Texas A & M Univ.), and Sam Shen (Univ. of Alberta). Their lectures seemed an ideal complement to the previous ones and dealt with topics such as gravity currents, stability analysis for simple climate models, optimal estimation of global change in climate, the monitoring of dilution of contaminant concentration values, and the question "why is the North Pacific so different from the North Atlantic."

It appeared, judging from numerous comments received during and after the meeting, that the format of this year's CAMS meeting (where the emphasis was on carefully selected invited talks intended to introduce important current research and to stimulate discussion and possibly collaboration between researchers from different, and yet related, fields) was very successful: this format resembled more a workshop-like meeting than the "usual" conference with less focused talks spread over a wide area.

11-15 June 1995, Université Laval, Québec

Org.: F. Khanna (Univ. of Alberta), L. Vinet (CRM & Univ. de Montréal)

This conference took place during the Congress of the Canadian Association of Physicists (CAP). This Congress was a very important event: CAP celebrated there its 50th anniversary and invited important delegations of the American Physical Society and of the Sociedad Mexicana de Física so as to make this celebration a North-American event. The session sponsored by the CRM on Theoretical and Mathematical Physics reflected this desire to build on the ties that the Canadian community of theoretical physicists has with the United States and Mexico. It was also at the banquet of the Congress that the first CAP/CRM Award in Theoretical and Mathematical Physics was presented to Prof. Werner Israel of the University of Alberta. (See the CRM Prizes.)

The several topics covered at the conference are found among the most active disciplines in theoretical and mathematical physics in Canada: classical and quantum gravitation, quantum field theory, classical and quantum integrable systems and the use of symmetry in physics.

**Classical and quantum gravitation. **This section covered the whole spectrum of the modern problems in this field: the theoretical predictions of general relativity, its geometrical content, quantum gravity and
measurements. The lecturers were: G. Kunstatter (Winnipeg), R. Laflamme (Los Alamos), A. Macias (Iztapalapa), T. Matos (IPN, San Pedro), L.O. Pimentel
(Iztapalapa), E. Poisson (Washington), G.F. Torres del
Castillo (Puebla), J.D. Vergera (UNA de México), H.
Waelbroeck (UNA de México).
**Quantum Field Theory. **This section covered several
aspects of quantum field theory: quantization schemes,
conformal field theory, applications to nonlinear optics and to impurity problems in solid state physics,
lattice models, relationship with statistical mechanics, etc. The speakers were: I. Affleck (UBC),
D. Caenepeel (Montréal), A. Das (Rochester), R. Jackiw
(MIT), A. Leclair (Cornell), P. Lepage (Cornell), E. Lieb
(Princeton), R. Mackenzie (Montréal), P. Nelson (U.
Penn.), P. Ramond (Florida), B. Sakita (City Coll. of
CUNY), M. Shifman (U. Minnesota), T. Steele (Saskatchewan), J. Tuszynski (Alberta).

**Classical and quantum integrable systems.**
Integrability is a rather recent subject that originated with
the discovery of nonlinear pde's with some remarkable properties like superposition principles and elastic
diffusion properties of some of their localized solutions. In less than thirty years it led to outstanding
developments in one- and two-dimensional theoretical physics. The most active areas in this field were
represented: spin chains and their algebraic structures,
solution through algebraic methods, new algebraic structures like quantum groups, parafermionic
algebras and Yangian structure, etc. The speakers were:
H. Bougourzi (Montréal and Stony Brook), F.D.
Haldane (Princeton), V. Korepin (Stony Brook), Hoong-Chin Lee (National Chung Hsing Univ.), P.
Mathieu (Laval), Y. Saint-Aubin (Montréal), L. Vinet
(Montréal), P. Wiegman (Chicago), Yong-Shi Wu
(Utah).

**Symmetry in physics. **Symmetry and the underlying
mathematical structures of groups and Lie algebras
have been one of the most powerful tools of mathematical physics in this century. Several avenues that
are currently being explored were reported on: the
symmetries of differential and difference equations,
calculatory aspects of representation theory of Lie
algebras, aperiodic structures, etc. The lecturers were:
W.E. Baylis (Windsor), R. Floreanini (Trieste), S.
Hacyan (UNA de México), N. Kamran (McGill), M.
Légaré (Alberta), L. Marchildon (UQTR), B. Mielnik
(México and Warsaw), J. Patera (Montréal), D. Provost (Laurentian), M. Thoma (McGill), P. Winternitz (Montréal), K.B. Wolf (UNA de México).

Org.: P. Greiner (Univ. of Toronto), V. Ivrii (Univ. of Toronto), L. Seco (Univ. of Toronto), C. Sulem (Univ. of Toronto)

Sponsors: Canadian Mathematical Society (CMS), CRM, Fields Institute, NSERC and the Department of Mathematics of the University of Toronto

This conference was the 1995 Annual Seminar of the Canadian Mathematical Society and had partial differential equations as its main theme. Its goal was to enhance interaction between PDE and a large number of different areas, such as arithmetic groups, spectral asymptotics, differential geometry, fluid dynamics and quantum physics. Its format consisted of five minicourses and a large number of lectures given by distinguished mathematicians, all of them world leaders in their field. The minicourse given by Charles Fefferman was part of the Fields Institute's Distinguished Lecture Series.

There were also two sessions of contributed talks, given by younger mathematicians. Graduate students from universities in Canada and elsewhere had the exceptional opportunity to attend courses and lectures given by scientists of outstanding stature and well-known for their original work as well as for their excellent presentations. Graduate students took a very active part in the conference. Indeed, some of them agreed to take on the challenging work of writing the notes for the lectures of the minicourse speakers. It is these notes, revised by the lecturer, that will be published in the Seminar Proceedings. The Proceedings are to be published by CRM.

The seminar drew the attention of graduate students and professional mathematicians from all over the world: there were participants from Austria, Israel, USA, France, England, Russia, Scotland, Spain, Italy, Sweden, Mexico, and of course Canada.

The CMS 1995 Summer Seminar resulted in a significant level of on-going collaborative activity. For example, Papanicolau and Sulem will be writing a book based upon their talks at the Seminar and Ron Howard, a student of Charles Fefferman (Princeton), is now working with Peter Constantin (Chicago). In addition, a remarkable amount of productive research took place during the actual Seminar.

It is also worth noting that participants included not only PDE experts, but algebraic geometers, number theorists, engineers, physicists, and some non-specialists. This helped to give those attending and the public in general a better view of Canadian mathematics and an area of significant research activity.

This workshop had 81 registered participants.

Org.: Colleen D. Cutler (Univ. of Waterloo), Daniel T. Kaplan (McGill Univ.)

Sponsors: Fields Institute and CRM

This workshop was a highly successful effort to bring together leading statisticians and physical/biological scientists working in the area of analysis of data from nonlinear dynamical systems and time series. The primary purpose of the workshop was to promote an exchange between these two groups, in the hope of encouraging further interdisciplinary research and communication. The new perspectives and methodology of time series analysis inspired by recent developments in nonlinear dynamics and "chaos" theory provide new viewpoints and open problems for statisticians; in return, applied scientists have much to gain from the expertise and long experience of statisticians in time series analysis and related areas.

The idea for the workshop originated with Dr. John Chadam, then director of the Fields Institute for Research in the Mathematical Sciences. Colleen Cutler and Danny Kaplan were approached to be co-organizers representing, respectively, statistics and dynamics, and the CRM agreed to co-sponsor, as well as host, the resulting workshop. The workshop was designed to follow some international statistical meetings being held in Montréal in mid-July.

The program consisted of 22 speakers from various countries, with approximately half being statisticians, and the remaining half being scientists working in nonlinear dynamics in mathematics, physics, or biology. Opening overview lectures were given by Professors Henry Abarbanel and Howell Tong, representing scientists and statisticians respectively. Topics covered at the workshop include problems and methodology related to embedding and reconstruction of dynamical systems from observed time series data, forecasting and prediction of nonlinear systems, error bounds and estimation of local Lyapunov exponents, performance of surrogate data techniques, separating deterministic and stochastic components in time series, nonlinearity and estimation for time series with long-range dependence, and ideas and techniques of chaos control.

Approximately 80 additional people attended and participated in the workshop; these participants represented a wide range of scientific disciplines, including mathematics, statistics, economics, physics, biology, geology, and engineering. Discussions and exchange at the workshop were lively and fruitful, and the program was deemed a success by speakers and participants alike.

Formal proceedings from the workshop are being published as a Fields Institute Communications volume, to appear in late 1996.

Org.: David Sankoff (CRM & Univ. de Montréal)

About fifty researchers participated in this meeting, including about equal numbers of "Fellows" and "Scholars" of the programme such as David Sankoff, Robert J. Cedergren and B. Franz Lang of the Université de Montréal, their students and postdoctoral fellows, "Associés" of the programme from Québec, Canada, the United States and overseas, international advisors of the programme and invited speakers.

The colloquium's themes were mathematical analysis of genome rearrangements and algorithms for phylogeny. The invited speakers included the following individuals: M. S. Waterman (USC), J Felsenstein (U. Washington), P. Pevzner and S. Hannanhalli (Penn State and USC), J. Kececioglu (U. Georgia), M. Steel (New Zealand), T. Warnow (Penn), V. King (Victoria), E. Myers (Arizona), S. O'Brien (NIH), G. Olsen (Illinois).

Org.: Jonathan Borwein (SFU)

Sponsors: CECM and CRM

The intention of the workshop was to explore the
possibilities for activating mathematical papers by allowing for computation and
other real-time enhancements. The workshop has led to a CMS Proceedings:
an in-press volume which is the hardcopy version of
the intrinsically electronic "Proceedings of the Organic
Mathematics Workshop." The electronic version is
available at:
** www.cecm.sfu.ca/organics**.

The more precise "raison d'être" of the Conference and the exact nature of the electronic proceedings are described in the accompanying articles:

- schedule of the meeting;
- list of speakers;
- abstracts of talk;
- budget summary;
- a detailed description of the project: What is organic mathematics?;

as are the many issues raised by such a project.

The conference was in purely scientific terms a huge success, both in the calibre of the science and the level of exposition. In addition, the technical staff at CECM provided a superb level of computational assistance with many of the talks being given "on-line."

The Web version of the Proceedings was released in April and has since been visited more than 2,200 times. It has been awarded several web recognitions (including "site of the week" in the Chronicle of Higher Education, three star status by Magellan and inclusion in the Scout Report).

The final question that begs to be asked is, why produce a "hardcopy" version at all. The primary answer is that conventional books still have a far larger potential audience and a clearer archival role. The centre piece of the Organic Mathematics Workshop is the content of the mathematical papers and is primarily text based. This book allows us to make these papers easily and comfortably available to a wide readership, not just those with fast internet access. Additionally this volume provides a fixed, and easily referenced, permanent version of what is otherwise an evolving document.

The process of turning a conference into a "virtual book" into a conventional manuscript has been interesting and has necessitated addressing a variety of additional issues. How does one include links? These we have included primarily as footnotes and appendices. A "book" has firm space limits and so some ancilliary material had perforce to be dropped. Colour pictures are expensive so most of these were also excluded. Some of the electronic features like video or interactive Maple sessions simply can't be reduced to text. The above exclusions in some ways diminish the collection but in other ways enhance it; principally by focussing on what is central, the mathematical content.

Also in this period of rapidly emerging evolving network technologies most of us are still most comfortable reading books not screens. Perhaps it will stay this way for quite a while.

15-19 April 1996, McGill University

Org.: J. Choksi (McGill), S. Drury (McGill), R. Gundy (Rutgers), R. Murty (McGill), N. Varopoulos (Univ. Paris VI)

Sponsors: NSERC, Fonds FCAR, CICMA, CMS, CRM

Carl Herz made fundamental contributions to mathematics, especially in analysis.
Among these are results on spectral synthesis, the theory of A_{p} spaces,
the H_{p} theory of martingales, atomic decompositions
and more recently, analysis on Lie groups. His mathematical interests
extended well beyond analysis into number theory,
probability theory and representation theory.

This conference brought together mathematicians of international repute from Canada, the United States, France, Italy, Russia and Australia to talk about work which would have interested Herz. There were 20 talks in all, each of about one hour: there were 4 each on the first three days, 5 on the fourth day and 3 on the last day. The first talk, by Varopoulos, was on the work of Herz. The talks of Stein, Kenig, Lohoue, Cowling, Gundy and Figa-Talamanca dealt with topics heavily influenced by Herz. Of the other talks, those of Christ, Stroock, Havin, Kahane, Koosis, Malliavin and Toth all fell in the domain of analysis, many of them with at least some connection to harmonic analysis, those of Arthur, Sarnak, Boyd and Murty in the domain of Number Theory, that of Kamran on Lie groups and geometry, and that of Langlands on percolation and lattice systems.

We plan to publish the proceedings of the conference in a volume of the CMS Conference Proceedings. Participants as well as invited speakers may submit papers for publication. All submissions will be peer refereed. S. Drury and R. Murty will edit the Proceedings. N. Kamran and R. Murty are editors of the series.

Org.: Joseph Cheriyan, Bill Cunningham, Levent Tunçel, Tony Vannelli, Henry Wolkowicz (Univ. of Waterloo), and Panos Pardalos (Univ. of Florida).

Sponsors: Fields Institute and CRM

Semidefinite Programming (SDP) is a generalization of Linear Programming (LP) in that the nonnegativity constraints on the variables are replaced by a positive semidefinite constraint on matrix variables. Many of the elegant theoretical properties and powerful solution techniques follow through from LP to SDP. In particular, the primal-dual interior-point methods, which are currently so successful for LP, can be used to efficiently solve SDP problems.

In addition to the interesting theoretical and algorithmic questions, SDP has found many important applications in Combinatorial Optimization, Control Theory, Statistics, and other areas of Mathematical Programming. SDP is currently a very hot area of research. This can be seen by the number of talks on SDP at various recent optimization conferences and the number of recent publications.

The workshop attracted roughly 100 researchers. Participants from Australia, Austria, Brazil, Belgium,
Canada, Israel, France, Hungary, Italy, The Netherlands, Puerto Rico, and USA gave the workshop an important international component. There were 39
talks during the three days of the workshop. A list of
participants, speakers, titles, abstracts, and this article, can be found at:
** orion.uwaterloo.ca/~hwolkowi/fields.d/readme.html**.

Following is a short, far from comprehensive, outline of several of the talks.

The workshop started with two talks on Matrix Completion Problems, i.e. under what circumstances does a partial matrix have a completion of a desired type. These problems have been studied extensively beginning in the early 80's and exemplify one of the early instances of SDP.

Monique Laurent spoke on "A Connection Between Positive Semidefinite and Euclidean Distance Matrix Completion Problems." Although there is a strong relationship between positive semidefinite matrices and Euclidean distance matrices, it was not clear how to link the two completion problems. Monique showed how the results for the Euclidean distance matrix completion problem can be derived from the corresponding results for the positive semidefinite completion problem, using a functional transform introduced by Schoenberg.

Charlie Johnson spoke on "Recent Progress on Matrix Completion Problems." After discussing the state of the art on positive definite completion problems, he continued with various other completion problems including those for totally positive matrices, P-matrices, inverse M-matrices, completely positive and doubly nonnegative matrices.

There were many talks on applications to Combinatorial Optimization. One of the main reasons for the current interest in SDP is the success in finding good approximations for the max cut problem. However, SDP has applications to many other combinatorial problems.

Stefan Karisch (with Franz Rendl) spoke on "Semidefinite Programming and Graph Equipartition." Stefan showed how SDP can be used to approximate the problem of partitioning a graph into equally sized components. Improvements were shown on previous eigenvalue approaches.

Christoph Helmberg (with Franz Rendl and R. Weismantel) spoke on "Quadratic Knapsack Relaxations Using Cutting Planes and Semidefinite Programming." Though the quadratic knapsack problem is extremely difficult to solve by linear programming alone, it was shown that SDP is very useful for quadratic knapsack problems.

Hsueh-I Lu (with Philip Klein) spoke on "Approximation Algorithms for Semidefinite Programs arising from Max Cut and Coloring."

Tamas Terlaky with J.P. Warners, C. Roos, B. Jansen) spoke on "Potential Reduction Algorithms for Structured Combinatorial Optimization Problems." Tamas presented a modified potential function, for binary feasibility problems that is computationally more attractive than the existing ones. A special class of binary feasibility problems were reformulated as nonconvex quadratic optimization problems. The reformulation is very compact. Computational results on several instances of the graph coloring and frequency assignment problem were presented. These results compared three different potential functions.

Qing Zhao (with Stefan Karisch, Franz Rendl and Henry Wolkowicz) spoke on "Semidefinite Programming Relaxations for the Quadratic Assignment Problem." Qing used the special structure of QAP to construct a gangster operator which enabled him to work in the minimal face of the feasible set. By exploiting this special structure and also using a conjugate gradient method, he was able to get strong bounds for the Nugent test set.

Many talks concentrated on Primal-Dual Interior-Point Methods for SDP. One of the reasons for the success of SDP is that interior-point approaches from LP can be extended to SDP; though the extension is not completely straightforward. Interesting complications can arise such as duality gaps, lack of strict complementary slackness, and confusing choices in the complementarity equations.

Kees Roos (with Tamas Terlaky and Etienne deKlerk) spoke on "Initialization in Semidefinite Programming via a Self-Dual Embedding." In this way the initialization problem for semidefinite problems can be solved nicely. The method also provides a solution for the initialization of quadratic programs and it is applicable as well to more general convex problems with conic formulation.

Zhi-Quan Luo (with Jos F. Sturm and Shuzhong Zhang) spoke on "Superlinear Convergence of a Symmetric Primal-Dual Path Following Algorithm for Semidefinite Programming."

Michael J. Todd (with Kim Chuan Toh and Reha H. Tutuncu) spoke on "The Nesterov-Todd direction in semidefinite programming." Mike showed how to compute the direction efficiently and how to view it as a Newton direction.

Romesh Saigal (with Chih-Jen Lin) spoke on "An Infeasible Start Predictor Corrector Method for Semidefinite Linear Programming."

Shuzhong Zhang (with Jos F. Sturm) spoke on "Symmetric Primal-Dual Path Following Algorithms for SDP."

Yin Zhang spoke on "Infeasible Primal-Dual Interior-Point Methods for Semidefinite Programming." Yin presented formulations, or symmetrization schemes, for the complementarity condition to obtain square optimality systems so that Newton-type methods can be applied. He also discussed a complexity theorem for an infeasible, long-step, path-following algorithm and several computational issues with preliminary numerical results.

Several talks discussed the Geometry, Duality and Complexity of solving SDP:

Franz Renkl (with C. Helmberg) spoke on "Large Scale SDP using Eigenvalues." The advantage of this approach lies in the fact that extreme eigenvalues of symmetric matrices can be computed without having the matrix explicitly available.

Gabor Pataki spoke on "Cone-LP's and Semidefinite Programs: Geometry and a Simplex-type Method."

Motakuri Ramana (with Levent Tunçel and Henry Wolkowicz) spoke on "Strong Duality for Semidefinite Programming." Unlike LP, a duality gap can occur in SDP. Two approaches for closing the gap are compared and shown to have a common basis, though one is of polynomial size while the other is not.

Jos F. Sturm (with Zhi-Quan Luo and Shuzhong Zhang) spoke on "Duality and Self-Duality for semidefinite and Conic Convex Programming."

Laurent Porkolab (with Leoni Khachiyan) spoke on "Bounds on Feasible Solutions of Semidefinite Programs."

Lleonide Faybusovich spoke on "Infinite-dimensional Semidefinite Programming: Self-Concordant Barriers and Path-Following Algorithms for Semidefinite Programming."

Alexander Shapiro spoke on "Second Order Optimality Conditions and Stability Analysis of Semidefinite Programs."

Katya Scheinberg (with D. Goldfarb) spoke on "Interior Point Trajectories in Semidefinite Programming."

Manuel A. Nunez (with Robert M. Freund) spoke on "Condition Measures and Properties of the Central Trajectory of a Semidefinite Program." Manuel used Renegar's condition number to provide bounds on solution size and rates of change of solution. This shows that Renegar's condition number can greatly simplify sensitivity analysis for SDP.

Other applications of SDP were introduced and discussed. The wide ranging applicability, as well as fascinating and intriguing theoretical qualities, shows that SDP promises to be an active discipline of mathematical programming for many years to come.

Lieven Vandenberghe (with Stephen Boyd and Shao-Po Wu) spoke on "Determinant Maximization with Linear Matrix Inequality Constraints." This problem has many interesting applications including finding the ellipsoid of minimal volume that contains a given polytope or given points.

Arjan Berkelaar (with Shuzhong Zhang) spoke on "Convergence Issues and Path-following Algorithms for Semidefinite Programming." Arjan discussed convergence issues related to recent primal-dual interior-point algorithms for the monotone semidefinite linear complementarity problem.

Renato Monteiro "On Two Interior-Point Mappings for Nonlinear Semidefinite Complementarity Problems." Renato presented properties of two fundamental mappings associated with the family of interior-point methods for solving monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices. The first of these maps lead to a family of new continuous trajectories which include the central trajectory as a special case. The trajectories completely "fill up" the set of interior feasible points of the problem in the same way as the weighted central paths do the interior of the feasible region of a linear program.

Boris Mirkin "A Mine of Semidefinite Programming Problems"

In addition, Related Problems presented included:

Ding-zhu Du spoke on "On Floorplan Design and Optimization."

Jun Gu spoke on "Parallel Algorithms for Satisfiability (SAT) Problem."

Linas Mockus (with J. Mockus and A. Mockus) spoke on "Bayesian Approach to Combinatorial Optimization."

Mauricio Resende (with Panos Pardalos) spoke on "Using linear programming to help solve quadratic assignment problems." The linear programming problems used to obtain lower bounds for quadratic assignment problems are very large and highly degenerate. For such problems simplex type algorithms or interior point methods based on direct factorizations, can only handle small instances. Large instance can be successfully solved with an interior point algorithm that uses a preconditioned conjugate gradient approach to approximately compute interior point directions.

M.R. Emamy-K "How efficient can we maximize threshold pseudo-Boolean functions?" The class of threshold pseudo-boolean functions was introduced by P. L. Hammer et al. about 10 years ago. Existence of a polynomial algorithm to maximize these functions has been an open problem since then.

Pham Dinh Tao (with Le Thi Hoai An) "D.c. (difference of convex functions) Optimization: Theory, Algorithms and Applications." A general discussion of d.c. optimization was presented. Its importance lies in the fact that d.c. programming marks the passage from convex optimization to nonconvex optimization.

Laaura Palagi (with Stefano Lucidi) "Trust region Problems: Theoretic Results and New Algorithmic Developments." The trust region subproblem is important in nonlinear programming as well as other applications in combinatorial optimization. The special structure of the problem allows one to solve an equivalent unconstrained minimization of a piecewise quartic merit function.

Le Thi Hoai An (with Pham Dinh Tao) "An efficient adapted DCA and Branch-and-Bound algorithm for globally solving large-scale 0-1 quadratic programming problems."

A proceedings of the workshop will be published by the American Mathematical Society in the Fields Series.

Org.: Niky Kamran (McGill), Peter J. Olver (Univ. of Minnesota), Josef Paldus (Univ. of Waterloo), Mikhail Shifman (Univ. of Minnesota), Alexander Turbiner (UNAM, Mexico)

Sponsors: CRM, Fields Institute and The Theoretical Physics Institute at the Univ. of Minnesota

Since the early days of quantum mechanics, the representation theory of Lie algebras and Lie groups has been successfully employed to compute and analyze the spectra of complicated physical systems which have a great degree of geometrical or dynamical symmetry. Over the past decade, algebraic methods have also started to play a significant role for problems of quantum and statistical mechanics where no such symmetries are present. A simple example which comes to mind is the theory of quasi-exactly solvable systems in quantum mechanics, where the spectral problem associated with the Schrödinger operator may not be exactly solvable, but for which at least part of the spectrum can be computed algebraically owing to the existence of a "hidden" symmetry algebra. Other significant examples of these recent developments include the applications of spectrum generating algebras to molecular dynamics and the role played by quantum algebras in some exactly solvable models in statistical mechanics.

The goal of this workshop was to bring together mathematicians, theoretical physicists and chemists who are actively involved in the development of these new algebraic approaches. There were 29 speakers altogether, whose lectures were organized around three main subthemes:

Quasi-exact solvability and spectrum generating algebras: Alhassid (Yale), Brihaye (Mons), Iachello (Yale), Gonzaalez-Lopez (Madrid), Kamran (McGill), Levine (Hebrew), Lipkin (Weizmann), Matsen (Austin), Milson (Minnesota), Novikov (Steklov/Maryland), Olver (Minnesota), Paldus (Waterloo), Turbiner (UNAM), Ushveridze (Lodz), Wulfman (Pacific), Zaslavskii (Kharkov).

Algebraic methods in statistical mechanics, including quantum groups, the Bethe ansatz and the Yang-Baxter equation: Cizek (Waterloo), Gates (Gainesville), Gould (Brisbane), Kauffmann (Chicago), Kibler (Lyon), Oerhn (Gainesville), Saint-Aubin (CRM), Vinet (CRM), Wiegmann (Chicago).

Other mathematical topics: Gerstenhaber (Pennsylvania), Patera (CRM), Rowe (Toronto), Winternitz (CRM).

Because of the somewhat multi-disciplinary nature of the subject, it was of particular concern to the organizers that there be a true dialogue among the mathematicians, physicists and chemists participating in the conference. The expectations of the organizing committee were largely met in this regard. For example, the precise nature of the relationship between the hidden symmetry algebras studied by the mathematicians and the spectrum-generating algebras developed by theoretical physicists and quantum chemists was elucidated during the conference. Likewise, a much better understanding was gained of the elusive problem of quasi-exact solvability in higher-dimensions.

The proceedings of this conference will be published. The organizing committee is presently considering offers from several publishers.

Org.: Michael C. Mackey and the Centre for Nonlinear Dynamics in Physiology and Medicine.

Sponsors: CRM, Fields Institute, Department of Physiology of McGill University, Addison-Wesley, Math Works Inc., Springer Verlag

A summer school on Nonlinear Dynamics in Physiology and Medicine ("Montréal 96") was organized and run for the first time by the Montréal based Centre for Nonlinear Dynamics in Physiology and Medicine (CNLD) in the facilities of the Department of Physiology, McGill University. The more than 60 students were selected on a first come, first served basis from over 100 applicants. They came from 16 countries, ranging in subject specialization from biology, medicine, psychology, physiology, and theoretical physics through applied mathematics. Career levels varied from advanced undergraduates through graduate students, postdoctoral fellows, university faculty members and physicians.

Week 1 of Montréal 96 emphasized application of nonlinear dynamics to study the stability of steady states, oscillation, and chaos in biological systems. Week 2 morning lectures dealt with more advanced topics involving bifurcation theory, stochastic models, case studies of situations modelled with inherent delays (hematological disorders), and partial differential equations. The afternoon stream of Week 2 was devoted to time series analysis and the connection with dynamics. The final Week 3 of Montréal 96 was devoted to further case studies that illustrated the use of the techniques developed in the first two weeks in neurological tremor, stochastic resonance in sensory neurons, cardiac re-entry problems, the pupil light reflex, and chaotic behaviour in the periodically stimulated squid giant axon. The close association of all lecturers with the CNLD allowed a tight integration of lecture material and smooth transitions in difficulty of the topics considered. An extensive set of lecture notes written by the lecturers was compiled for each student.

Two features of Montréal 96 were unique. The first was the inclusion and integration of lectures on time series analysis techniques with concepts from dynamics. The second was the daily computer laboratory designed to illustrate the concepts of the lectures through numerical experiments using software written by the lecturers utilizing either the commercially available MATLAB or the freeware XPPAUT (written by Bard Ermentrout, Univ. of Pittsburgh) which incorporates the AUTO bifurcation analysis package written by the CNLD member Eusebius Doedel. Graduate students and postdoctoral fellows of the CNLD served as laboratory assistants, as did the lecturers, to supplement the aid given in the laboratory manual.

Plans are already in progress for a sequel, Montréal 97, and information
will be available at the website
** www.cnd.mcgill.ca**.

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 first CRM/Fields Prize was given last year to **Professor H. S. M. Coxeter** of the
University of Toronto. Due to scheduling constraints the award ceremony took place only this fall.
On September 22, 1995, Prof. Coxeter gave his lecture "Evolution of Coxeter-Dynkin Diagrams;"
afterwards director Luc Vinet presented the award. Prof. Coxeter very kindly accepted to give a second
talk, this time for undergraduates. In an overcrowded hall, with people standing along the walls and
sitting on the floor, Prof. Coxeter gave a very accessible lecture entitled "Euler's formula
for polyhedra." It is an experience that will be remembered by many young mathematicians.

The CRM-Fields Institute Prize for 1995 was awarded to **Professor George A. Elliott** of the
University of Toronto and the University of Copenhagen. The awards ceremony took place at the CRM
on April 19, 1996, following a lecture by Professor Elliot entitled "*C*-*algebras at the
CRM." He was cited in particular for his classification of *C*-*algebras via invariants
related to ordered *K*-theory. His work has so influenced this field that specialists speak
of "the Elliot Program," and he was an invited speaker at the International Congress of
Mathematicians in 1994.

George Elliott obtained his B.Sc. and M.Sc. at Queens University in Kingston in 1965 and 1966 and his Ph.D. at the University of Toronto in 1969. After postdoctoral fellowships at the University of British Columbia and Queens University and a year at the Institute for Advanced Study in Princeton, he accepted the position of lektor at the University of Copenhagen in 1972. In 1984 he was named adjunct professor at the University of Toronto.

Professor Elliott has published more than 100 papers and given invited talks at some 48 universities and numerous meetings. He served as editor of the Canadian Journal of Mathematics and the Canadian Mathematical Bulletin and currently edits the Mathematical Reports of the Academy of Sciences of Canada.

In addition to the work cited above George Elliott has made important contributions in other
areas of operator algebras: derivations and automorphisms of *C*-*algebras, classification
of *AF-*algebras in terms of their ordered *Ko-*groups, and non-commutative tori.

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 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. Niky Kamran (1991), Ian Putnam (1992), Michael Ward (1994) and Nigel Higson (1994) are the former recipients.

The André Aisenstadt Mathematics Prize for 1995 was awarded to **Professor Adrian S. Lewis **of the
University of Waterloo. Professor Lewis was cited for
his deep contributions in a wide range of mathematical areas: mathematical optimization,
convex and nonsmooth analysis, functional analysis, matrix
theory, and computational optimization. In particular he is world renowned for his work in the field of
convex programming of Hermitian matrices. The prize
was awarded on April 26, 1996 at the CRM by Luc Vinet, director, following a lecture by Professor Lewis
entitled "Convex Analysis and Applications."

Professor Lewis did his undergraduate and graduate studies at Cambridge University in England, obtaining his Ph.D. in 1987. His thesis was entitled "Extreme point methods for infinite linear programming." After research fellowships at Cambridge University and Dalhousie University he moved to the University of Waterloo in 1988 where he is currently Associate Professor in the Department of Combinatorics and Optimization.

Adrian Lewis has published more than 30 articles in prestigious refereed journals and has given numerous invited presentations and colloquia, among them a keynote speech at the SIAM Conference on Optimization in 1996. He has also accepted invitations to Marseilles and Toulouse for joint research and expositions of his work. He is a member of the editorial board of the SIAM Journal on Optimization, and referees and reviews for ten other important journals.

The Centre de recherches mathématiques and the Canadian Association of Physicists
jointly created this year the CAP/CRM Medal for outstanding achievement in theoretical and
mathematical physics. The first Medal was presented at the 1995 CAP Annual Congress to
**Professor Werner Israel** of the University of Alberta.

In presenting the Medal to his thesis advisor, mentor, and friend, Dr. Eric Poisson had the following to say regarding Dr. Israel: "Werner was born in Berlin in the early nineteen thirties. Soon after, he and his family moved to Cape Town, South Africa. There he stayed until he moved to Dublin, Ireland, to pursue a graduate degree. Werner obtained his doctorate from Trinity College in 1960. In Dublin, Werner met and married Inge, and the two of them came to live in Edmonton. Werner joined the University of Alberta as an Assistant Professor in 1958, and there he remained to this day.

Werner's field of research is general relativity, most especially black holes. His contributions to this field are numerous and far reaching; throughout his career his role has been that of a leader.

In the late nineteen sixties, Werner formulated a theorem which took everybody working in the field by surprise. Werner showed that nonrotating black holes in isolation must be spherically symmetric, no matter how aspherical the collapsing star initially was. The star could be a cube, and the resulting black hole would still be spherical! This theory created a lot of excitement in the field, and over a period of several years, it was generalized (by Werner as well as other workers) to the case of charged and rotating black holes. This result, now known as the no-hair theorem for black holes, is one of the most powerful and beautiful achievements of gravitation theory.

Most recently, Werner's scientific focus has been on the internal constitution of black holes. His work establishes that the singularity of an aging black hole is lightlike (as opposed to spacelike), and far more ordered than was initially expected.

Werner's work combines deep physical significance with elegant mathematical formulation. Moreover, the vast majority of his work has been original and innovative, which establishes Werner as one of the true leaders in the field of general relativity. In his letter of support for this Prize, Kip S. Thorne writes:

With two exceptions (Stephen Hawking of Cambridge University and Roger Penrose of Oxford University), nobody has contributed more than Werner Israel to our understanding of gravitational theory, during the past three decades."

**
The CRM offers numerous lectures which are part of a regular seminar series. These are
generally arranged by local CRM members and take various forms. In some cases there are
formal lectures, in others working groups on general subjects are led by researchers.
**

- Seminar on "Wavelets"

Org.: Jean-Marc Lina (CRM & Univ. de Montréal) - Seminar on "History of Mathematics"

Org.: Liliane Beaulieu (CRM) - Seminar on "Nonlinear Analysis"

Org.: Marlène Frigon (Univ. de Montréal) - Seminar on "Integrable Systems"

Org.: John Harnad (CRM & Concordia Univ. ) - Seminar on "Mathematical Physics"

Org.: Pavel Winternitz (CRM & Univ. de Montréal)

- Luc Vinet (CRM), 29 September 1995
*"Symétries quantiques: introduction aux groupes quantiques"* - Alessio Corti (University of Chicago), 6 October 1995
*"Minimal models and the classification of algebraic varieties"* - Beno Eckmann (E.T.H. Zurich), 13 October 1995
*"Composition des formes quadratiques et le groupe unitaire infini"* - Harold Stark (University of California, San Diego), 20 October 1995
*"Can an L-function be determined by a single value"* - Nigel Higson (Penn State University), 27 October 1995
*"C*-Algebras and the Topology of Group Representations"* - Victor Snaith (McMaster University), 3 November 1995
*"Some natural occurrences of plus or minus one in number theory"* - Robert Guénette (Université Laval), 10 November 1995
*"Fluides rhéologiquement complexes: problèmes théoriques et numériques"* - Bertram Kostant (M.I.T.), 17 November 1995
*"Harmonic decompositions and deformed Lie algebra cohomology"* - Yakov Eliashberg (Stanford University & Harvard University), 24 November 1995
*"Symplectic geometry and knots"* - François Soumis (GERAD & École Polytechnique), 1 December 1995
- Martin Kruskal (Rutgers University & Princeton University), 8 December 1995
*"How to tell whether a nonlinear differential equation is integrable"* - Michel Fortin (Université Laval), 26 January 1996
*"Méthodes de stabilisation pour les méthodes d'éléments finis mixtes"* - Ed Bierstone (University of Toronto), 2 February 1996
*"Resolution of singularities"* - André Joyal (UQAM), 9 February 1996
*"Jeux, catégories et communications"* - Charles Chui (Texas A&M), 16 February 1996
*"What are wavelets"* - Mark Goresky (I.A.S.), 23 February 1996
*"The geometry behind Arthur's trace formula"* - Barbara Keyfitz (University of Houston), 1 March 1996
*"Multidimensional conversation laws"* - Jérôme Detemple (CIRANO & McGill Univer sity), 8 March 1996
*"American Options and Stochastic Volatility"* - Hector Sussmann (Rutgers University), 22 March 1996
*"Geometry and nonlinear control, 300 years after Johann Bernoulli's brachistochrone"* - Paul Deheuvels (Université de Paris), 29 March 1996
*"Fractals Aléatoires"* - Michael Thaddeus (Harvard University), 12 April 1996
*"Complete Quadrics and Punctured Curves"* - Catherine Sulem (University of Toronto), 3 May 1996
*"Phénomène d'autofocalisation pour les équations de Schrödinger nonlinéaires"*

**Return to the Table of Contents of this Annual Report****Continue with the next part of this Annual Report**

**6 November 1998, webmaster@CRM.UMontreal.CA**