- Scientific Activities

Org.: Michel Delfour (chair, CRM & Université de Montréal), John Chadam (Fields Institute & McMaster University), Michel Fortin (Université Laval), Augusto Visintin (Università degli Studi di Trento, Italie), Jean-Paul Zolésio (Institut Non Linéaire de Nice, France)

The themes of the CRM Summer School at Banff were shape and structural optimization (including microstructures), phase transition and moving boundary problems, numerical methods for the above types of problems. We were not able to cover everything and had to make choices. The choice of the topic was motivated by earlier cooperation between John Chadam and Delfour in the Summer 1990. Chadam was co-organizer of the International Meeting on Free Boundaries, and Delfour was the organizer of the NATO-ASI Séminaire de Mathématiques Supérieures on Shape Optimization and Free Boundaries. It was an attempt to bring together researchers from both communities and favour exchanges and collaboration in areas where the geometry, boundaries and interfaces play a central role. We feel that the experience was extremely successful and that the School (5 years later) confirmed this sentiment.

The use of the geometry in the design, identification and control of technological processes has been steadily increasing. Shape and optimal design have been concerned with the improvement and the design of mechanical parts (N. Kikuchi for the automotive industry). However other original applications appeared in image processing (Geman and Geman, Ambrosio, Richardson, Morel, Lions, etc.), composite materials, aerospace engineering (reservoirs with flexible membranes, shape control of parabolic antennas), etc. In other applications the geometry appears as a control variable as in "optimal swimming" and stabilization of membranes and plates by periodic variations of the boundary.

Geometric measure theory which has been extremely successful in the theory of minimal surfaces is now widely used in Shape optimization. In the same direction the recent work on Motion by Mean Curvature, Hysteresis problems, etc., clearly provides fundamental tools which can be used in other contexts.

The School was primarily aimed at Ph.D. students in their final years and recent Ph.D.'s. Its objective was to give a broad introduction to contemporary problems where the geometry or the structure is a key variable in the understanding, modelling and control of physical and technological systems and problems, and to expose the participants to some of the latest developments in the associated Mathematics, Mechanics, and Physics.

It covered selected aspects of shape optimization and optimal design, mathematical models in material sciences, hysteresis, superconductivity, phase transition, moving boundary problems, and some of the associated numerical issues.

In addition to the 9 main speakers, we had 5 young, outstanding Canadian speakers (3 are women)
to complement and expand the lectures of the main speakers. Six special talks were also given in the afternoon seminar. The main speakers (6 hours) and their
title were: John Chadam (McMaster Univ. and Fields
Institute), *Reactive Flows in Porous Media*; Alain
Damlamian (École Polytechnique, France),
*Variational approach to the Stefan problem and extensions to the phase
field model with constraints*; Michel Delfour (CRM &
Univ. de Montréal), *Introduction to shape and geometric
optimization*; Ingo Mueller (Technische Universität
Berlin), *Mechanics and thermodynamics of phase transitions in shape memory alloys*; Jacob Rubinstein (Technion,
Israel), *Mathematical models in superconductivity*; Mete Soner (Carnegie Mellon, USA), *Front propagation and
phase field theory*; Claudio Verdi (Università di Milano,
Italy), *Numerical analysis of geometric motion of fronts*; Augusto Visintin (Università degli Studi di Trento,
Italy), *Models of Hysteresis*; Jian-Jun Xu (McGill), *Interfacial Instabilities, Pattern Formation and Selection*.

The other speakers (2 hours) and their title were:
Anne Bourlioux (CERCA & Université de Montréal),
*Detonation and propagation of shocks*; Lia Bronsard (McMaster University), *Interface dynamics as singular
limits of Ginzburg-Landau equations*; Katie Coughlin
(CERCA & Université de Montréal), *Transition to turbulence*; Robert Guénette (Université Laval),
*Numerical analysis of viscoelastic fluids and liquid crystals*; Michael Ward (Univ. of British Columbia),
*Dynamical Metastability and Singular Perturbations*.

The six special talks (1 hour) were: Toyohiko Aiki
(Gifu University, Japan), *One-phase Stefan problems for
semilinear parabolic equations*; Changfeng Gui (Univ. of
British Columbia), *A three layered minimizer in triple
phase transition*; William D. Kalies (Georgia Institute
of Technology), *On the asymptotic behaviour of a phase-field model for elastic phase transitions*; Nobuyuki
Kenmochi (Chiba University, Japan), *Attractors for non-isothermal models of phase transitions*; Robert E. O'Malley,
Jr (Chiba University, Japan), *Supersensitivity of shocks
and transition layers for certain singularly perturbed boundary value problems*; Hong-Ming Yin (University of
Notre-Dame, USA), *A free boundary problem describing
a chemical diffusion process with localized reaction*.

The lecture notes of the CRM School will appear in the CRM Proceedings and Lecture Notes Series published by the AMS. In addition a book by C. Verdi has been accepted in our Monograph Series.

14-16 September 1995, Hotel Bonaventure/Hilton, Montréal

Org.: Paul Arminjon (chair, CRM & Univ. de Montréal), Alain Dervieux (INRIA, France), Michel Fortin (Univ. Laval, Québec), Wagdi G. Habashi (Concordia Univ. and CERCA)

The aim of the conference was to bring together some of the leading specialists in the field of numerical methods for the Euler and Navier-Stokes equations, and to give a comprehensive survey of recent developments in high resolution methods (finite differences, finite elements, finite volumes) for compressible and incompressible flows. A wide range of numerical experiments with numerous applications to engineering problems was presented and discussed.

The total number of registered participants was 75 and the high quality of the speakers led to a clear and interesting exposition of many of the most important aspects of recent research in computational fluid dynamics, with a strong emphasis on compressible flows and applications to aerodynamics.

A large number of advanced Ph.D. students (35) participated in the conference, and several of them mentioned they were very pleased to attend lectures of the highest standard which were at the same time well structured, clearly presented and included interesting recent developments in numerical methods for the Euler and Navier-Stokes equations as well as a large number of numerical applications to real engineering problems. Among those many applications, we should mention: A. Jameson (aerodynamics consultant for major American aircraft manufacturers), A. Dervieux, O. Pironneau, B. Stoufflet (aerodynamics consultants for Dassault Aviation and Aerospatiale, France), B. van Leer, E. Tadmor, E. Turkel, H. Yee (consultants for the NASA), S.K. Godunov (formerly a main scientific consultant for computational aerodynamics to the Russian airplane and space programs), M. Fortin and W.G. Habashi (consultants for Hydro-Quebec and Pratt and Whitney, respectively).

The series of lectures addressed five fundamental aspects of research in Computational Methods for the Euler and Navier-Stokes Equations.

**Numerical methods for the Euler equations. **R. Abgrall (INRIA) presented his work with A. Harten
on Multiresolution Representation in Unstructured Meshes, a technique for representing data which originate from
discretization of functions in unstructured
meshes in terms of their local *scale* components, by
means of a nested sequence of discretizations. A.
Dervieux (INRIA) described two strategies, based on
Mixed Finite Element Finite Volume methods, to at
tenuate or completely avoid spurious numerical diffusion (derivation of a higher-order dissipation or con
struction of flow dependent finite volume partitions).
M. Hafez (U. of California, Davis) presented some anomalies associated with the non-uniqueness of the
numerical solution of the Euler equations. B. van Leer
(University of Michigan) discussed interesting issues
associated with operator splitting and staggered grids,
and presented his joint work with E. Turkel on a comparison of preconditioning methods (for both for Euler
and Navier-Stokes). E. Tadmor (Tel Aviv University
and U.C.L.A.) described his one-dimensional non-oscillatory central differencing scheme where Riemann
problems at the cell interfaces are bypassed by using
two alternate grid systems (joint work with H. Nessyahu). X.D. Liu (Courant Institute for Mathematical Sciences, NYU),
in joint work with P.D. Lax, introduced a new positivity principle for numerical schemes for hyperbolic systems,
and presented a family of space and time - second order accurate schemes
with a very simple structure using characteristic decomposition. Comparison with well-established methods showed that the new schemes are competitive.

**Numerical methods for the Navier-Stokes equations.** M.J. Ivanov (Central Institute of Aviation Motors,
Moscow) described recent CFD developments and applications for steady and transient flows in different types of turboengines and their components. O. Pironneau (Université Paris VI and INRIA) spoke on
wall laws for turbulent flows, which are extensively
used to eliminate from the domains of computation regions of strong gradients or regions where the geometry is complex. He gave an interpretation of wall
laws by domain decomposition, of error estimates for
approximations on a simple potential flow with complex boundaries, and presented several test cases for
turbulent flows. B. Stoufflet (Dassault Aviation) outlined recent efforts and improvements to remedy the
deficiencies of numerical methods when implemented
in industrial codes, and to insert those methods as efficient tools and analysis algorithms in the process
of numerical shape optimization. H. Yee (NASA-Ames) reported on super-stable implicit methods and time-marching approaches, showing how the use of dynamical system theory can contribute to reliability,
efficiency, stability and convergence properties of time-dependent approaches for obtaining steady-state numerical solutions (joint work with P.K. Sweby).

**Multidimensional algorithms. **P. Arminjon (Université de Montréal) presented a two-dimensional finite
volume extension of the Lax-Friedrichs and Nessyahu-Tadmor schemes (joint work with M.C. Viallon, Univ.
de St-Etienne), with several applications to typical 2-dimensional problems (on unstructured grids).
H. Deconinck (Vrije Universiteit Brussel) reported on compact multidimensional upwind schemes for the
Euler equations on unstructured meshes, showing the relation with finite element methods and other distribution
schemes by reuniting them as Fluctuation-Splitting schemes (joint work with H. Paillère). C. Hirsch
(Vrije Universiteit Brussel) presented some multidimensional upwind algorithms for the Euler and Navier-Stokes equations,
using a cell-centred finite volume approach on structured grids, leading to first
and second order accurate schemes with respectively
minimum and zero cross diffusion (joint work with P.
van Ransbeeck). R.J. LeVeque (University of Washington) described a three-dimensional algorithm for general
hyperbolic systems of conservation laws, consisting of an unsplit finite volume method based on solving one-dimensional Riemann problems on piecewise constant states rather than interpolated values, and
then using the waves arising from this solution to define second order correction terms; the waves are further
split by solving Riemann problems in the transverse direction to model the cross-derivative terms (joint work with J.O. Langseth).

**Grid generation and mesh adaptation. **W.G. Habashi
(Concordia University/Pratt and Whitney) presented
an edge-based mesh adaptation procedure using mesh
orientation and stretching, and based on error estimates and mechanisms to recast the mesh according to these error estimates.
Numerous numerical tests gave good evidence of the efficiency of the method
(joint work with M. Fortin and the two authors' student teams at Concordia and Laval University). S.K.
Godunov (Russian Academy of Sciences) one of the pioneers in Computational Fluid Dynamics, described
his joint work with V.M. Gordienko and G.A. Chumakov on the theory of two-dimensional quasi-isometric grid construction.

**Present status, Challenges and Future in CFD. **Finally, A. Jameson (Princeton University), in the last
lecture, gave a remarkable survey of the role of computational fluid dynamics as a tool for aircraft design.
Addressing the requirements for effective industrial
use, and the trade-offs between modelling accuracy
and computational costs, he discussed the main issues in algorithm design, together with a unified approach to the design of shock capturing schemes. He
then described his use of techniques drawn from control theory to determine optimal aerodynamics shapes,
concluding that, in the future, multidisciplinary analysis and optimization should be combined to provide
an integrated numerical design environment.

2-6 October 1995, CRM

Org.: Katie Coughlin (CERCA & Univ. de Montréal)

This workshop was held as part of the CRM thematic year on Numerical Analysis, Approximation Theory, and Applications, and was one of three held in the fall on numerical methods in fluid dynamics. Thirty-one participants attended, coming from Canada, the U.S., Europe and Asia. Being a somewhat specialized subject, the workshop was organized to be small-scale and informal, with each invited speaker giving two hours of lectures, leaving plenty of time for interaction. The attendance was divided about fifty-fifty between mathematicians and physicists. An effort was made to encourage attendance by researchers and students in Montréal.

The goal of the workshop was to bring together researchers from different disciplines, who share an interest in developing innovative methods for the solutions of partial differential equations, primarily (but not exclusively) the equations describing fluid motion. The particular issue that arises in this context is the coexistence of coherent, structured flow on large scales with apparently random fluctuations on small scales. For computational methods to resolve exactly all the important scales is generally impractical, and so one wants to develop methods which treat the small scales by a statistical model and retain the more-or-less exact dynamics at large scales. This idea has proven to be quite difficult to implement in a well-defined way, suggesting that much of the physics and mathematics of the "apparently random fluctuations on small scales," and how they affect the large scale motion, is not well understood. Each of the invited speakers contributed a different perspective to this basic problem of interaction between large and small scales.

The topics selected were divided into three general themes: physical fluid dynamics and physical modelling; statistical approaches and modelling of data; and the theory of inertial manifolds applied to computational techniques.

In the first category, Phil Marcus (U.C. Berkeley) and Jeff Weiss (U. Colorado) spoke about geophysical fluid dynamics, in which one commonly sees the formation of large "coherent structures" (such as the Red Spot of Jupiter, or the Gulf Stream) in a turbulent background. They emphasized that often the physical processes leading to structure formation can be understood with relatively simple models, even though a complete mathematical description is lacking. Tom Warn (McGill) spoke about the difficulties of developing rigorously consistent models of turbulence. Katie Coughlin (U. de Montréal) discussed the particular problems associated with transitional flows, for which there is a temporal as well as spatial intermittency to the turbulence. Charles Meneveau (Johns Hopkins) discussed large eddy simulations of turbulent flows, which model the effect of small scales by an average taken along a fluid particle path line.

In the second category, Emily Ching (CUHK) presented a theoretical analysis of the statistics of passive scalar fluctuations, with comparison to experiment. S. Balachandar (U. Illinois) discussed a statistical technique for extracting coherent features from velocity data, which can be used for example to compare structures in different physical situations. Henry Greenside (Duke University) discussed various attempts to quantify pattern order and disorder in convection calculations, using ideas taken from statistical and condensed matter physics. Michael Kirby (Colorado State) presented an innovative technique for constructing low dimensional models of PDE's by training a neural net with numerical data. Nadine Aubry (CUNY) presented an extension of the Karhunen-Loève decomposition, a technique which decomposes a random field into a set of modes which is optimal in the energy norm, to space and time dependent fields.

In the third category, Edriss Titi (U.C. Irvine) presented an overview of the theory of inertial manifolds and the related nonlinear Galerkin numerical methods, and discussed results which suggest that they may provide a framework for understanding finite dimensional behaviour in the Navier-Stokes Equations. Martine Marion (Lyon) presented a finite element algorithm for the equations on an approximate inertial manifold, along with error estimates and different implementations. John Heywood (UBC) pointed out that improved error estimates for nonlinear Galerkin methods may be attributed to a reduction in the severity of the Gibbs phenomena, and that one should be careful in giving physical interpretations of the theory. He also presented a new adaptive Fourier spectral method developed for 2D forced flows.

Overall, the response of the participants (most of whom were unfamiliar with the work of about one half of the other attendees) to the broad range of disciplines represented was very enthusiastic. The mathematicians were interested to see how much could be learned by physical reasoning, and the physicists found the discussion of basic mathematical problems to be very illuminating. The two-hour format was also well-received, as it allowed a thorough presentation of each speaker's material. Thus, as a forum for the exchange of scientific ideas, the workshop was very successful.

13-22 November 1995, CRM

Org.: Alain Vincent (CERCA & Univ. de Montréal)

The goal of this workshop was to bring together during two weeks the top world experts on current numerical methods in fluid mechanics. The lectures were presented at the level of graduate students, researchers and engineers. The 60 or so participants were from universities in Montreal but also from Canada, Europe and North America.

The following themes were covered.

**New numerical techniques for environment study.
**Jean Cote (RPN, Montreal, Canada) gave a lecture on
the computer code "Global Finite Elements" currently
in testing for numerical weather forecasting at Environment Canada. The code uses the semi-Lagrangian method that leads to better time stability. Sylvie Gravel
(RPN, Montreal, Canada) presented the principal problems related to semi-implicit and semi-Lagrangian schemes. David Dritschel (Cambridge University,
UK) gave two lectures on contour dynamics. This numerical method leads to a very precise resolution of
vorticity filaments and can be used for simulations of
the stratospheric polar vortex for example. A new method for continuous remeshing using a calculation
of the coordinate system metrics was used to study
the dynamics of tornadoes by Brian Fiedler (University of Oklahoma). This method offers an interesting
alternative to industrial automatic remeshing codes.
Ue-Li Pen (Princeton) uses a very similar method to
describe astrophysical flows. Ue-Li and Brian were
able to compare their algorithms during the workshop
and the technique developed independently for astrophysical problems appeared to be more powerful.

**Closure techniques for simulation of turbulent fluids **are divided into two categories:
large eddy simulation (LES) and Reynolds stress tensor modelling. The
state of the art for LES was described by Joel Fertziger
(Stanford). In particular he mentioned a LES model
for flows around buildings. The modelling of the Reynolds stress tensor was detailed in a series of 5
lectures by Brian Launder (Manchester). These methods, more complex but also more precise than the "k-epsilon" method, are starting to be used in industry.

**Methods to deal with discontinuities. **Charles Hirsch
(Vrije Universiteit, Bruxelles) spoke about shock capturing schemes and of new multidimensional upwind
schemes that seem to be very powerful. Stanley Osher
(UCLA, Los Angeles) described essentially non-oscillatory schemes (ENO) for the combustion case.
Maurice Meneguzzi (IDRIS, Paris) uses these methods for the simulation of two-phase fluids (e.g. water
and oil) that are important for the oil industry. The
water-oil interface is treated as an internal boundary.
The internal boundary techniques are also active topics of mathematical research.
Alfio Quarteroni (Cagliari, Italy) gave a series of lectures on domain decomposition methods.

**Finite difference methods **can also be used for high
precision calculations by using compact schemes. Sanjiva Lele (Stanford) presented us some of their applications in engineering.

Finally, Claude Basdevant (Ecole Normale Superieure,
Paris) talked about **wavelet methods for the resolution of PDE's**. These questions are still open and, for
the time being, the applications are limited to the one-dimensional Burger equation.

The audience was larger than foreseen, forcing us to change the conference hall at the last minute. The courses were most appreciated. Even the Masters students were able to learn something about these advanced subjects. Not only did it provide an opportunity for young researchers to meet the leaders of the subject, but it also allowed time to develop a professional relationship with them.

Org.: Martin Bilodeau (Université de Montréal), Gilles Deslauriers (École Polytechnique, Montréal), Serge Dubuc (Université de Montréal), Véronique Hussin (Université de Montréal), Jean-Marc Lina (CRM & Université de Montréal), Brenda MacGibbon (UQAM), Marc Moore (École Polytechnique, Montréal), Sherman Riemenschneider (University of Alberta)

Ten workshops on spline functions and the theory of wavelets took place at the CRM during the winter semester. The emphasis was on recent developments in both theoretical aspects and applications, particularly on curve and surface modeling. Among the theoretical aspects was an up-to-date presentation of splines in one and several variables. The applications, beside numerical imaging and interpolation, were signal processing, numerical solutions of differential equations, fractal geometry, and other applications to physics and statistics. Work done in Edmonton and Montreal was thoroughly discussed. The workshops brought together mathematicians, computer scientists, engineers, physicists and statisticians. Unusual for a mathematical meeting were the presence of professionals from the film and animation industry, optometrists, meteorologists, telecommunication scientists, and even one economist. The fact that the subject of wavelets is particularly popular in France, the United States, and Canada was an asset for the organization of the workshops and a factor in their scientific success. The ties among Canada, the United States, and France were reinforced through these meetings.

There were, in total, 89 one-hour lectures and 39 half-hour communications. The average size of the audience was around 30 participants. The Aisenstadt lectureship was held by Prof. Yves Meyer, and a very faithful audience of over one hundred attended his 5 lectures. (See description in the section Chaire Aisenstadt.) Three discussion sessions were organized on geometric modeling, multifractal applications, and wavelets in physics. A video of a conference by Daubechies was presented and two participants gave software demonstrations. Finally two of the participants gave a lecture for undergraduates in the weekly series of the Département de mathématiques et de statistique of the Université de Montréal.

The countries represented were Canada (31), the United States (38), France (21), Germany (7), Argentina (3), Switzerland (3), Australia (2), Belgium (2), Chile (2), Croatia (2), the Netherlands (2), Japan (2), England (1), Cyprus (1), Scotland (1), Israel (1), Poland (1) and Russia (1). To these one must add fifty-eight local participants.

In total the workshop represented 47 days of activity with, on average, 3 lectures per day. This relaxed format allowed for a high level of interaction and met with unanimous approval.

A particular effort was made to maximize the benefit of the workshops for Canadian students. During the fall of 1995, a course on splines and wavelets was given at the graduate level. Around twenty Canadian students received a stipend to attend the workshops and to give lectures on their work.

The one-hour lectures will be collected in a proceedings if the material proves sufficiently original. The refereeing committee will be made up of Alain

Arnéodo, John Benedetto, Hermann Brunner, Stéphane Jaffard, Alain Le Méhauté, Brenda MacGibbon, Sherman Riemenschneider, and Philippe Tchamitchian. Publication is currently set for the end of 1997. The Aisenstadt lectures given by Prof. Meyer will be pub lished separately by the AMS in the CRM Proceedings and Lecture Notes.

**Geometric modeling of splines (22-26 January). **
Surface modeling using spline functions was the main
subject of this workshop. Several applications were
presented: fabric texture representation (J.P. Dussault,
Sherbrooke), cornea modeling for diagnostic purposes
in optometry (B. Barsky, UC at Berkeley), rigid structures and surface intersections in architecture (T.
Grandine, Boeing Comp. Services). A roundtable brought together people from various horizons:
mathematicians interested in surfaces, computer scientists
working in the film industry (more precisely at Softimage),
engineers working on airplane modeling at Boeing or on image processing through telecommunications.

**Splines for approximations and for differential equations (29 January - 2 February). **Two approaches were
discussed here. The first one stressed the use of one- and two-variable functions in approximation problems.
The second one presented the efficient application of splines to the solution of differential equations.
In particular, H. Brunner (Memorial) presented an excellent overview of the latter, covering the main
results of the last twenty years or so. Two researchers from Croatia also gave relevant lectures.

**Splines and wavelets (12-16 February). **This workshop acted as a bridge between the first two and the
one that followed, on the theory of wavelets and their
applications. Six leaders in the field of two-variable
wavelet theory and applications and multivariate splines (Micchelli, Goodman, Riemenschneider, Chui,
Ron, and Ward) were among the lecturers. Prof. Chui
also gave the CRM-ISM Colloquium.

**Wavelets and approximation (19-23 February). **Nonlinear approximations through wavelets was the dominant
theme during this workshop which was directed primarily by De Vore. As was pointed out by B. Lucier (Purdue),
this subject is extremely popular, in particular due to the
decision of the FBI to use wavelets in fingerprint compression. The importance of Besov spaces in
approximation theory was also stressed. D. Hardin (Vanderbilt) presented
a recent construction of "intertwined" multiresolution analysis that enhanced the reputation of the group
at Georgia Tech that discovered it. Other conferences covered the resolution of hyperbolic
equations and multivariate wavelet approximation.

**Multiresolution analysis and subdivision operators
(26 February - 1 March). **Multiresolution analysis is
at the heart of the theory of wavelets. It was discussed
by, among others, Cohen (Univ. Pierre et Marie Curie), who won the Popov Prize for his work
in approximation theory. Dyn (Tel Aviv) presented subdivision
schemes that have important applications in curve and
surface generation by computer. This workshop was very popular with graduate students; five of them also
presented their recent results.

**Wavelets and differential equations (4-8 March). **The importance of wavelets in the solution of
pde's is being increasingly recognized and this workshop provided an opportunity to survey their various uses.
France and Germany were well represented. Several discussions on approaches to numerical solution of
pde's took place between the American school (represented by G. Beylkin (Colorado)) and the French one
(P. Tchamitchian (Aix-Marseille), V. Perrier (Lab. de
Météorologie Dynamique-ENS, Paris), J. Liandrat
(Marseille)). Elliptic, hyperbolic, and parabolic pde's
were covered as well as nonlinear equations such as
those of Burger and Kuramoto-Sivashinsky. Turbulence was discussed by M. Farge (ENS, Paris) and M.
Wickerhauser (Washington Univ.).

**Wavelets in signal processing and image analysis (11-15 March). **With the explosion of telecommunications
and numerical image analysis, wavelets became powerful tools in the field of signal processing,
even before the word "wavelet" was coined by Meyer. This
workshop attracted the largest audience. It coincided
with the Chaire Aisenstadt held by Meyer himself. His
main topics were multifractal analysis, microlocal
analysis, Mallat's algorithm (often used in image analysis),
and the Marseille algorithm for the detection of chirps and fractionnary brownian motion. Due
to the wide spectrum of applications to image analysis and to physics, a large audience attended the entire series of lectures.
The workshop also offered other exceptional lectures. D. Donoho (Stanford) presented
schemes of nonlinear refinement for signal statistical
analysis. G. Strang (MIT), Kovasevic (AT&T-Bell Labs),
Benedetto (Maryland) and S. Myers (IBM) also made

important contributions. The topics ranged from voice recognition and detection of early signs of epileptic seizures to filters and image analysis. The speakers also included engineers form Toronto, scientists from the Centre for Research in Communications located in Ottawa, and scientists from Australia.

**Wavelets and fractals (18-21 March). **Very often in
fractal geometry can one witness a multiresolution
analysis intimately tied to wavelet theory. One of the
leaders in this field, S. Jaffard (Créteil) has shown that
the coefficients of the wavelet expansion give the
fractal dimension of the set of singular points of a function. These fundamental results show the importance
of wavelet multifractal analysis. This was the central
theme of the workshop. A. Arnéodo (CNRS-Bordeaux)
gave an interesting application of wavelets to DNA
chains.

**Wavelets in physics (25-29 March). **Quantum mechanics had an impact early
in the development of wavelets, and it was natural that this workshop should
start with a detailed presentation of this subject (T. Ali
(Concordia)). G. Battle (Texas A&M) explored this
theme further in quantum field theory. Other subjects
in physics were also touched upon: multifractal formalism applied to turbulence and to cluster growth may be among the most exciting.
A. Arnéodo, J. Muzy (CNRS-Bordeaux) and S. Jaffard gave a detailed account of the formalism with its thermodynamical cor respondence.
J.P. Antoine (Louvain-la-Neuve) presented recent applications in solid state and atomic physics.

**Splines and wavelets in statistics (8-12 April). **Statisticians often use powerful tools based on spline functions and wavelets.
Statistics, on the other hand, also raises new questions about these tools. Johnstone (Stanford), Houdré (Atlanta) and
von Sachs (Kaiserlautern) discussed several uses of wavelets for data
analysis with noise. A. Anroniadis (IMAG, France) and
N. Heckman (UBC), among others, showed that splines often lead to better estimation of parameters.
J. Ramsay (Florida) expanded on several statistical
problems related to emission tomography.

Org.: Yoshua Bengio (Univ. de Montréal), Renato de Mori (McGill Univ.), Bertrand Giraud (CEA, France), Bernard Goulard (CRM & Univ. de Montréal)

The workshops on artificial neural networks (ANN) were aimed at providing a state-of-the-art survey of the field. These activities were devoted to the theoretical aspects (statistics and learning) during the first week, and then to the structure and applications to signal processing (in one and two dimensions). The present workshop was a natural complement to the several workshops on wavelets of the thematic semester. Its program was intended to clarify the relationship between ANN-statistics (early workshops) and the wavelets-statistics relations that were discussed in the final workshop on wavelets. Discussions of ANN's in finance (last two days) produced a natural transition to the workshop on the Mathematics of Finance (see below).

Several topics were vigorously debated. On the theoretical side, the question of how to control the problem of over-generalization was raised. After an introduction of the subject by Yoshua Bengio, Frederico Girosi and Vladimir Vapnik, two viewpoints were presented, namely that of regularization and that of automatic control of capacity. Another theoretical idea (though close to applications) appeared repeatedly: the use of a set of models for the reduction of the variance of the generalization error (Yoshua Bengio, Jean-Pierre Nadal, Nathan Intrator). It was stimulating to hear rather different viewpoints expressed by physicists, computer scientists, and statisticians during the question periods.

On the applications side, several practical uses of ANN's were discussed by Marco Gori, Bertrand Giraud, Simon Haykin, Hervé Bourlard, Michael Mozer, Patrice Simard, Yann Le Cun and Paul Refenes. Marco Gori gave a presentation on a laptop computer of a recognition system for license plates. Many new algorithms were presented, such as those of Geoff Hinton and Peter Dayan (Helmholtz' machine), Samy Bengio (Markov models of input-output), Michael Jordan (graphical models), to name only a few. More specialized communications were also presented: on cognitive psychology and reinforcement algorithms (Jordan Pollak, Sue Becker, Geoff Hinton, Fernando Pineda, Michael Mozer), implementation on chips (Hans Peter Graf, Jocelyn Cloutier), and financial applications (Yoshua Bengio, Paul Refenes, René Garcia).

The number of participants during the two weeks was ninety. The audience was larger than expected and required a last minute change of conference hall. Many of the participants commented on the high level of the conference. Several discussions led to new collaborations, in particular with P. Refenes (London School of Economics), K. Muller (GMD-FIRST-Berlin) and B. Giraud (Physique théorique-Saclay), and probably many others.

30 April - 3 May 1996, CRM

Org.: Jérôme Detemple (CIRANO & McGill Univ. ), Daniel Dufresne (CRM & Univ. de Montréal), Éric Ghysels (CIRANO & Univ. de Montréal), Martin Goldstein (CRM & Univ. de Montréal)

Advisory Committee: Luis Caffarelli (I.A.S.), Darrell Duffie (Stanford), Ioannis Karatzas (Columbia), Stanley Pliska (Univ. of Illinois at Chicago)

Sponsors: AMI Associés Inc., Bourse de Montréal, Caisse de Dépôt et Placement du Québec, Hydro-Québec

Org.: L. Vinet and P. Winternitz (CRM & Univ. de Montréal)

This workshop was the first of a "tandem" of two,
both devoted to quite recent developments in special
function theory. The second, (reviewed separately),
concerned *q*-special functions, satisfying linear equations. All speakers were invited to participate in both
meetings; quite a few of them did.

The Painlevé transcendents were introduced at the
turn of the century, specifically to solve a certain class of
second order nonlinear ordinary differential equations.
More specifically, P. Painlevé and B. Gambier identified
all equations of the form
where the right hand side is rational in *y* and
and analytical in *x*, having what is now called the "Painlevé property."
This means that the general solution of the equation
is single valued in the neighbourhood of any one of
its "movable" (i.e. depending on initial conditions)
singularities. Fifty classes of such equations were identified and six of them turned out to be irreducible: their
general solution cannot be expressed in terms of previously known functions, like elliptic functions,
or solutions of linear equations.

During the last 30 years or so, since the develop ment of soliton theory and more generally the modern theory of infinite-dimensional integrable systems, the Painlevé transcendents have become extremely important. They occur as solutions of a very large class of physical problems, coming from nonlinear optics, wave propagation in fluids and plasmas, quantum field theory, statistical mechanics and many others. The problems are described by integrable partial differential equations. The Painlevé transcendents occur as special solutions, usually particularly stable ones, providing the asymptotic behaviour for solutions of large classes of Cauchy problems.

The aim of the workshop was to bring together experts in a booming field of research, either to give review talks or to present new and important results. The speakers succeeded in doing this to a remarkable degree. New results were presented in the context of a review of the field. This brings us to the second goal of the workshop: i.e., to introduce participating students, postdoctoral fellows, and interested scientists from related fields to the subject and to produce a book based on the talks presented. Such a book is in preparation and should serve as a multiauthored monograph on a well focused branch of applied mathematics and theoretical physics.

The Workshop on Painlevé Transcendents had 36 registered participants. They came from 12 countries (13 from Canada, 4 each from the USA and Japan, 3 from France, 2 each from Australia, Italy, Poland and the United Kingdom, 1 each from Belgium, Mexico, Taiwan and Russia). The speakers were, in alphabetical order, Yu. Berest (Canada), R. Conte (France), B. Dubrovin (Italy), A.S. Fokas (U.K.), J. Harnad (Canada), A. Its (USA), N. Joshi (Australia), A.V. Kitaev (Russia), M.D. Kruskal (USA), V. Matveev (France), M. Musette (Belgium), F.W. Nijhoff (U.K.), K. Okamoto (Japan), A. Ramani (France), C. Rogers (Australia), V. Spiridonov (Canada), H. Umemura (Japan), P. Wiegmann (USA) and P. Winternitz (Canada).

A.S. Fokas presented a series of four lectures on the isomonodromy method for the solution of Painlevé equations and on specific physical problems leading to these equations. All other talks were one hour, leaving a lot of time for discussions.

The topics covered include:

- discrete Painlevé equations, singularity confinement for solutions of nonlinear difference equations, asymptotics of discrete Painlevé equations,
- Hamiltonian,
*R*-matrix, isomonodromy and Riemann-Hilbert methods for studying properties of the Painlevé functions, - detailed asymptotics of the continuous Painlevé equations,
- classical special solutions of the Painlevé equations,
- physical applications of the Painlevé equations,
- other topics, such as the relation to the classical problem of lacunas for linear hyperbolic equations, and the use of algebraic methods to study the Painlevé equations or to generate new equations with the Painlevé property.

Org.: Luc Vinet (CRM & Univ. de Montréal)

The following scientific report was written by Prof. Charles Dunkl and published in the "Newsletter of the SIAM Activity Group on Orthogonal Polynomials and Special Functions" (ed. Wolfram Koepf), June 1996, and posted on the electronic news net of this Activity Group (eds. T.H. Koornwinder and M. Muldoon).

Monday, May 20, was a national holiday in Canada (Victoria Day) and this pushed the start of the
workshop to Tuesday morning. Luc Vinet, co-organizer with Pavel Winternitz and director of the Centre,
opened the session at 9:00, welcomed the participants
and dedicated the workshop to the memory of Waleed
Al-Salam, who passed away April 14 of this year. There
were approximately seventeen invited speakers who gave hourlong talks; there were about fourteen who
contributed half-hour talks. David and Gregory Chudnovsky were invited but were unable to attend and
their time on the program was taken by other events.
Sergei Suslov was invited, could not attend, but Dick
Askey delivered his lecture. Dick also gave his own
lecture (more details later). There was indeed a heavy
emphasis on *q*-special functions, both of one-variable
Askey-Wilson type and of several-variable Macdonald type. The algebraic methods were highly refined
and sophisticated, mostly based on root systems and
associated mathematical objects such as double affine
Hecke algebras.

Here is an alphabetical list of the invited speakers and the titles of their lectures:

George Andrews (Plane partitions and MacMahon's partition analysis), Richard Askey (An inequality of Vietoris and
some related hypergeometric sums),
Ivan Cherednik (Spherical difference Fourier transform), Charles Dunkl (Intertwining operators and
polynomials associated with the symmetric group), Pavel Etingof (Macdonald eigenvalue problem and
representations of quantum *gl(n)*), Roberto Floreanini
(Quantum algebras and generalized hypergeometric functions), Adriano Garsia (Polynomiality of the Macdonald *q,t*-Kostka coefficients: a short proof), Mourad
Ismail (Moment problems and orthogonal polynomials), Tom Koornwinder (The *A1*-tableau of Dunkl-Cherednik operators),
Boris Kupershmidt (The great
powers of *q*-calculus), Ian Macdonald (Symmetric and
non-symmetric orthogonal polynomials), David Masson (Contiguous relations, continued fractions and
orthogonality: a ten year journey up the Askey chart),
Willard Miller, Jr. (Tensor products of *q*-superalgebras
and *q*-series identities), Masatoshi Noumi (Raising
operators for Macdonald polynomials), Eric Opdam (Spectral analysis of Hecke algebras), Siddhartha Sahi
(Recent results on Jack polynomials and Macdonald polynomials), Dennis Stanton (*q*-orthogonal polynomials as moments),
Sergei Suslo ([talk delivered by R.
Askey] Some basic hypergeometric series and *q*-Bessel functions), Luc Vinet (Creation operators for Macdonald polynomials. Simple and simpler proofs).

Contributed talks were given by N. Atakishiyev, R. Chouikha, P. Floris, A. Grunbaum, K. Kadell, M. Kapilevich, J. LeTourneux, K. Mimachi, A. Odzijewicz, V. Spiridonov, A. Strasburger, N. Takayama, F. van Diejen, L. Vinet. Ian Macdonald started his lecture Tuesday morning and finished it on Wednesday. Dick

Askey gave an extra half-hour talk on Wednesday containing an overview of Askey-Wilson polynomials. Tom Koornwinder used another one of the hours originally scheduled for the Chudnovsky's to discuss
René Swarttouw's web site
** www.can.nl/~demo/CAOP/CAOP.html**
which gives access to a vast collection of formulas and
references for the polynomials contained in the Askey tableaux (

There were approximately sixty participants; as well some members of the Centre dropped in on the lectures. The languages spoken at coffee and in the hallways appeared to be English, Russian, French, Japanese, Dutch, Polish. The weather was mostly delightful with a few evening showers. The University is in a scenic location near the Mont Royal; the Pavillon André-Aisenstadt, which houses the Centre, is a beautifully designed and equipped academic building, with a wonderful view of the northwest of the city. Ian Macdonald gave the first lecture on Tuesday, Adriano Garsia gave the last one on Saturday, thus bracketing an intense period of leading-edge mathematics. It was generally agreed that the workshop was excellent both in organization and inspiration to the participants for future work. At the conclusion all applauded and thanked the organizers for this exciting conference.

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