Le Bulletin du CRM• crm.math.ca •

Automne/Fall 2008 | Volume 14 – No 2 | Le Centre de recherches mathématiques

The Fall 2008 Aisenstadt ChairsFour Aisenstadt Chair lecturers will visit the CRM during the 2008-2009 thematic year “Probabilistic Methods in Mathemat-ical Physics.” We report here on the series of lectures of Wendelin Werner (Université Paris-Sud 11) and Andrei Okounkov(Princeton University), both of whom are Fields Medalists, who visited the CRM in August and September 2008 respectively.The other Aisenstadt Chairs will be held by Svante Janson (Uppsala University) and Craig Tracy (University of Californiaat Davis).

Wendelin Wernerde Yvan Saint-Aubin (Université de Montréal)

Wendelin Werner est unspécialiste de la théo-rie des probabilités. Il aobtenu son doctorat en1993 sous la direction deJean-François Le Gall. Ilest professeur au labo-ratoire de mathématiquesà l’Université Paris-SudXI à Orsay depuis 1997,ainsi qu’à l’École nor-

male supérieure depuis 2005. Avec ses collaborateurs Greg Lawleret Oded Schramm, il a montré que la probabilité de deux marchesaléatoires planaires qui s’évitent décroît comme n−5/8 où n est la lon-gueur des marches et a déterminé la dimension de Hausdorff (= 4

3 )de la frontière extérieure du mouvement brownien planaire. Ses tra-vaux sur l’équation de Loewner stochastique et sur les ensembles deboucles conformes ont déjà eu un impact profond sur la descriptionmathématique des phénomènes critiques en deux dimensions.

Werner a reçu le prix de la Société Européenne de Mathématiquesen 2000, le prix Fermat en 2001, le prix Loève en 2005, le prix Pó-lya l’année suivante. En 2006 il devenait le premier probabiliste àrecevoir la médaille Fields. Il a détenu une des chaires Aisenstadt del’année thématique 2008-2009 sur les Méthodes probabilistes en phy-sique mathématique.

Les phénomènes critiques

L’étude des transitions de phase en physique a commencé il ya plus d’un siècle, mais ses progrès importants ont été accom-plis depuis les années quarante du XXe siècle. Son objet touchedes phénomènes physiques variés, dont certains tombent sousl’expérience de tous les jours, comme l’ébullition de l’eau oula miscibilité de gas et de liquide, par exemple dans les bois-sons gaseuses. Dans pratiquement tous ces phénomènes, unequantité macroscopique possède un comportement singulier

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Andrei Okounkovby John Harnad (Concordia University)

A metaphor from an ancient fragment by Archilochus “The Foxknows many things but the Hedgehog knows one big thing”was used by Isaiah Berlin as title and theme of his essay on Tol-stoy’s view of history (“. . . by nature a fox, but believed in be-ing a hedgehog”) [1]. It is also very suitably applied to styles inscience. In his presentation of the work of Andrei Okounkovwhen he was awarded the Fields medal at the 2006 Interna-tional Congress of Mathematicians in Madrid, Giovanni Feldersaid: “Andrei Okounkov’s initial area of research was grouprepresentation theory, with particular emphasis on combinato-rial and asymptotic aspects. He used this subject as a start-ing point to obtain spectacular results in many different areasof mathematics andmathematical physics,from complex and realalgebraic geometry tostatistical mechanics,dynamical systems,probability theory andtopological string the-ory. The research ofOkounkov has its rootsin very basic notionssuch as partitions, which form a recurrent theme in hiswork. . . at its core (is) the idea that partitions and other no-tions of representation theory should be considered as randomobjects with respect to natural probability measures. This ideawas further developed by Okounkov, who showed that, to-gether with insights from geometry and ideas of high energyphysics, it can be applied to the most diverse areas of mathe-matics.”

This would appear to describe the work of a fox but, as withTolstoy, things are not necessarily as they seem. The factthat Okounkov’s work intersects so many fields of mathe-matics and mathematical physics may appear astonishing to

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Wendelin Werner(suite de la page 1)

en fonction des paramètres du système. Voici un exemple ma-thématique célèbre, celui de la percolation.

(a) (b)Fig. 1. Une interface γ12 est tracée en (a), une trajectoire du processusSLE6 en (b).

D’un réseau triangulaire planaire de maille unité, considéronsune région de la forme d’un triangle équilatéral dont les côtésont longueur N, avec N pair. Le site médian de la base est l’ori-gine. Les sites à la frontière de la région sont peints en blancsi leur coordonnée horizontale est positive, en noir autrement.Pour les autres sites (intérieurs), leur couleur sera noir avecprobabilité p ∈ [0, 1], blanc avec probabilité 1− p. Une interfaceγN , séparant les sites noirs des blancs, commence (presque) àl’origine, suit le réseau dual hexagonal et quitte la région tri-angulaire (presque) au sommet. Soit d(p, N) la probabilité quecette interface, issue de l’origine, touche le côté droit avant lecôté gauche. Il est possible de montrer que la limite de d(p, N)lorsque N → ∞ est 0 lorsque p < 1

2 et 1 lorsque p > 12 . Donc

l’« observable » limN→∞ d(p, N) possède un comportement sin-gulier en p = 1

2 . (Pour cette raison pc = 12 est nommé la pro-

babilité critique pour cet exemple.) Cette observable prend lavaleur 1

2 en p = pc. D’autres observables peuvent aussi êtreétudiées, par exemple le nombre moyen de sites dans la plageconnexe de sites noirs contenant l’origine.

Deux conjectures

Cet exemple se prête à de nombreuses généralisations. Il estpossible de changer le réseau triangulaire pour un autre ré-seau périodique ou d’étudier la percolation en d’autres dimen-sions, par exemple sur le réseau Zd ou sur une région finiede ce réseau. D’autres modèles peuvent être aussi proposéspour lesquels la couleur d’un site (ou l’état de son spin) dé-pend de celles de ses voisins, comme pour les modèles d’Isinget de Potts. Pour tous ces modèles, l’existence d’un point cri-tique et les propriétés en ce point lorsqu’il existe dépendent dela dimension du réseau. Mais les physiciens sont convaincus,par des arguments théoriques et des simulations numériques,que pour une dimension donnée les propriétés critiques sontindépendantes du réseau choisi. Cette indépendance constituel’hypothèse d’universalité. Cette conjecture demeure un des pro-blèmes majeurs du sujet.

Une autre hypothèse importante est celle de l’invarianceconforme. Elle fut introduite dans le cadre des phénomènes cri-tiques au début des années 80 par Belavin, Polyakov et Za-molodchikov. Une forme de cette hypothèse, suggérée par Ai-zenman, est particulièrement simple à formuler. Changeonsl’échelle de sorte que le côté de la région triangulaire conserveune longueur unité lorsque N varie et notons par ΓN l’interfaceainsi obtenue. Fixons p = pc et considérons maintenant les in-terfaces commençant au sommet. Divisons la base de la régionen deux segments, le gauche de longueur x ∈ [0, 1], l’autre delongueur 1− x. Une seconde observable est la probabilité π(x)que, dans la limite N → ∞, l’interface ΓN issue du sommettouche le segment gauche avant le droit. Soit maintenant unetransformation conforme φ sur un ouvert contenant la régiontriangulaire. L’image du triangle par cette transformation estmaintenant tracée sur le réseau triangulaire qui, lui, demeureinchangé. L’hypothèse d’invariance conforme stipule que, dansla limite de la maille du réseau allant vers zéro, la probabilitéqu’une interface issue de l’image du sommet atteigne l’imagedu segment gauche avant celle du droit demeure π(x).

SLE

Des énoncés rigoureux existent maintenant pour certains as-pects de ce domaine ; deux résultats particuliers en sont les pré-curseurs. Tout d’abord, Smirnov (1999) a montré qu’une cer-taine observable pour la percolation sur le réseau triangulaireest effectivement conformément invariante lorsque la maille duréseau tend vers zéro. Et Schramm (2000) a lié la loi limite desinterfaces ΓN à celle du mouvement brownien unidimension-nel. Cette idée allait être développée par lui et ses collèguesLawler et Werner en un champ fertile en idées et outils nou-veaux. Il est maintenant connu sous le nom d’équation de Loew-ner stochastique ou équation de Loewner-Schramm (SLE).

(a) La courbe c (b) L’image de la courbe c pargt, t > 0

Fig. 2. Une courbe c est progressivement retirée du demi-plan par lafamille de transformations conformes gt de Loewner.

Le lien entre la loi limite de ΓN et le mouvement brow-nien est faite à l’aide d’un résultat classique de la géo-métrie conforme. Soit une courbe c : [0, T] → H ∪ ∂H

dans le demi-plan supérieur H et l’axe réel dont le pointinitial est en l’origine. (Voir Figure 2(a).) Supposons quecette courbe ne touche ni l’axe réel (sauf l’origine) ni elle-même. (Ces hypothèse sont trop restrictives, mais elles sim-plifient la présentation.) Alors le complément C de cettecourbe dans H est homéomorphe au demi-plan H. Parle théorème de Riemann il existe donc une transformationconforme envoyant C sur H. Il est même possible de trouver

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Prizes

The 2008 CAP – CRM Prize in Theoretical andMathematical PhysicsThe CAP – CRM Prize is given conjointly by the CanadianAssociation of Physicists (CAP) and the CRM every yearsince 1995. It was awarded this year to Richard Cleve fromthe University of Waterloo for fundamental results in quan-tum information theory, including the structure of quantumalgorithms and the foundations of quantum communicationcomplexity. Richard Cleve received the 2008 Prize during theCAP’s awards banquet that was held at Laval University onJune 10th, 2008.

Richard Cleve is an outstanding computer scientist who hasworked at the boundary of physics, mathematics and computerscience. His work has transcended the area of computer scienceto have a broad impact on the physics of quantum information.

Richard Cleve gives the Prize Lecture at Université Laval.

Richard Cleve has made seminal contributions to quantum in-formation science. He has shown how generic quantum algo-rithms can be broken into the fundamental physical buildingblocks, the so called one and two qubits gates. Today mostphysical implementation of quantum computers use this re-sult. He has also created the field of quantum communicationcomplexity which assesses the resources required to performtasks using quantum systems. He has demonstrated how aquantum version of quantum walks gives an exponential al-gorithmic speed up compared to its classical counterpart. Hehas also been involved in the first experiment on order-findingon a quantum computer prototype.

Besides being a first class researcher, Richard Cleve has beeninstrumental in building quantum information processing inCanada. He founded and was the impetus behind the quan-tum group at the University of Calgary before he moved to theInstitute for Quantum Computing at the University of Water-loo, where he holds the Quantum Information Chair. He is alsoa research associate at the Perimeter Institute for TheoreticalPhysics, a founding fellow of the Quantum Information Pro-cessing program at the Canadian Institute for Advanced Re-search, and Team Leader at QuantumWorks, a NSERC innova-tion platform bringing together academia, government and in-dustry to develop quantum information processing in Canada.

The 2008 CRM – SSC Prize in Statistics

The CRM – SSC Prize is given conjointly by the Statisti-cal Society of Canada (SSC) and the CRM every year since1999. It was awarded this year to Paul Gustafson, Professorin the Department of Statistics at the University of BritishColumbia (UBC), for his contributions to Bayesian statisticalmethodology and its application to epidemiology that havehad an immense impact in statistics, biostatistics and publichealth. Paul Gustafson received the 2008 Prize at the jointmeeting of the SSC and the Société française de Statistiqueheld in Ottawa in May 2008.

Within 15 years of his Ph.D., Paul Gustafson has made out-standing contributions to the understanding of Bayesian statis-tical inference, to the implementation of the Bayesian paradigmin the health sciences, and to the development of computa-tional algorithms for Bayesian inference. His work displays adeep knowledge of the foundation of statistical reasoning anda true ability to make substantial contributions to diverse do-mains of application. He has written key papers in several ar-eas of statistics such as survival analysis, the analysis of countdata, computational methods, and disease mapping. He hasmade solid methodological contributions through his collab-orative work with epidemiologists, medical researchers, andpsychologists.

Paul Gustafson receives the CRM – SSC Prize in Statistics fromLouis-Paul Rivest (Université Laval).

Paul Gustafson obtained his B.Sc. in Mathematics in 1990 andhis M.Sc. in Statistics in 1991 from the University of BritishColumbia. He completed his Ph.D. in 1994 at Carnegie-Mellon University. He shares Canadian bonds with his super-visor Larry Wasserman, the 2002 recipient of the CRM – SSCaward. Paul is a third generation CRM – SSC winner sinceLarry Wasserman’s adviser, Rob Tibshirani, was the 1999 re-cipient of this award. He held a Postdoctoral fellowship atthe University of British Columbia in 1994, where he was ap-pointed Assistant Professor in 1995, Associate Professor in 2000and Full Professor in 2005. He obtained a UBC Killam FacultyResearch Fellowship in 2001.

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Andrei Okounkov(continued from page 1)

those who encounter his results separately in the context ei-ther of group representation theory or combinatorics, proba-bility, algebraic geometry, algebraic topology, integrable sys-tems, statistical physics, growth phenomena, topological grav-ity, etc. However, for those working in related fields, most es-pecially, Integrable Systems, which is well represented at theCRM, these topics and Okounkov’s contributions to them donot seem quite as diverse as they might otherwise appear.

To begin with, when looked at closely, there really are a num-ber of common elements that point to an underlying unity ofideas and economy of methods. The representation theory ofLie groups is well known to be closely related to that of finiteand discrete groups, in particular, through the essential roleplayed by the Weyl group acting on the root lattice. The ir-reducible group characters for the unitary and general lineargroups, known as Schur functions, were originally introducedand studied by Jacobi in the context of symmetric polynomi-als and their combinatorial properties long before the notion ofa Lie group or its representions was known. There is a well-known “duality” between the unitary (or general linear) groupin N dimensions, and the symmetric group SN of permutationof N elements. In particular, the irreducible characters are de-termined by the same essentially combinatorial objects, parti-tions, in both cases. Partitions, moreover, have their own pla-nar geometrical significance when represented by Young dia-grams, which are formed by stacking aligned rows of adjacentsquare boxes. This construction extends in fact very naturallyto three dimensions, with the squares replaced by cubes. Whena three dimensional partition is represented through projectionto a plane from a suitable perspective, we obtain what appearsas a rhombus tiling of a hexagon, which can be split into oneby equilateral triangles. This leads to natural variations, wherethe boundary is replaced by more general polygons. Con-sidering the tiling as a graph, we also have a dual tiling byhexagons or, after splitting the rhombi into pairs of equilateraltriangles, we may “forget” the three dimensional origins, andorder the pairs in some way, identifying an adjacency matrix(the Kasteleyn matrix) which recovers the three dimensionaldata. This also provides the key tool in computing the detailedcombinatorial/probabilistic quantities characterizing configu-rations sharing certain given boundary requirements throughdeterminantal formulae.

Connecting the centers of the triangles pairwise gives a “dimercovering” (or “perfect matching” of the associated set of ver-tices). Moreover, the three-dimensional interpretation canbe used to associate a non-intersecting set of “horizontalpaths”and, if the original 3-D partition was random, these, toomay be viewed as a set of random paths in the plane, having acertain probabilistic weight, thus defining a random process.

It is also natural to consider the limit where the number oftiles, or cubes is infinite, and all jaggedness is eliminated bytaking a sufficiently distant perspective. We then arrive at“universal shapes” bounding the regions of “disorder.” Theseturn out, very generally, to be real algebraic curves, leadingus into the realm of algebraic geometry. The “large number oftilings” corresponds to large partitions, which in the 2-D caseis related also to the asymptotics of group representations andcharacters. Viewed as random processes consisting of non-intersecting paths, the large number limit may be interpretedas determining a diffusion process, which is also analogous tothe processes that govern the eigenvalues of chains of randommatrices in the continuous time limit.

The connections between random partitions, randomtilings and random processes was the theme of oneof the largest workshops of the CRM 2008 – 2009 The-matic Program on Probabilistic Methods in Mathemati-cal Physics , which took place Sept. 1 – 6, 2008 (URL:crm.umontreal.ca/Tilings08/index_e.shtml) and over-lapped, both in timing and content with the series of Aisen-stadt Chair Lectures “The Algebra and Geometry of RandomSurfaces” given by Andrei Okounkov . The first two of hislectures were included as part of the program of the workshop,while the remaining four lectures systematically developednew approaches to problems of random tilings and asymp-totic bounding curves that Okounkov has pioneered, featuringespecially the link with noncommutative rings and algebraicgeometry.

To obtain concrete results in this direction requires enumera-tive computations of probabilities, statistical correlations be-tween the various random objects involved, and characteriza-tion of their asymptotic properties. Making sense of the rela-tions between the various linked interpretations and applica-tions requires a deep understanding of the underlying struc-ture. Okounkov’s contributions to this very rapidly evolvingfield has been characterized by his introduction of distinctiveapproaches that help both to compute concretely and makesense of the underlying rather subtle structure by bringing newconcepts and insight onto the scene. Okounkov excels at neatlycharacterizing the combinatorial objects and rules that are in-volved with the help of new approaches, such as noncommuta-tive geometry and operator methods, that seem quite unusualand surprising in this context in comparison with more stan-dard combinatorial and probabilistic methods. The construc-tions are ingeniously devised to reduce the complexity of thecomputations at hand, while giving fresh new insights into thethe objects under study.

One such approach consists in expressing a certain probabil-ity weight, or the partition function on a statistical ensemble,or a combinatorial generating function, as matrix elements ofsome infinite dimensional operator on a Fermionic Fock space,

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SMS Summer School 2009: Computational andExperimental Approaches to Automorphic Forms

by Michael Rubinstein (University of Waterloo)

The year 2010 marks the 50th anniversary of the publicationof Eugene Wigner’s famous essay on the “unreasonable effec-tiveness of mathematics in the natural sciences.” The interven-ing five decades have witnessed an explosion in the varietyand scope of the applications of mathematics, to the extent thatone can now speak of an ongoing “mathematization” of manybranches of science and indeed of society as a whole. Numbertheory, traditionally viewed as far removed from the sphere ofapplications, now plays a central role in questions pertaining tothe design of efficient networks as well as in areas like robotics,computer vision, statistics, coding theory, computer security,and cryptography.

The 2010 Winter theme semester Number theory as experimentaland applied science, organized by Henri Darmon and Eyal Goren(McGill) will be devoted to recent developments in numbertheory with a specific focus on significant practical applica-tions, as well as on the many ways in which the field standsto be affected by the emergence of new software and technolo-gies.

The first activity of the theme semester will take place sev-eral months ahead of the semester, during the summer of 2009:the SMS Summer school on Computational and Experimental Ap-proaches to Automorphic Forms, organized by Michael Rubin-stein (Waterloo) and Andreas Strömbergsson (Uppsala) will beheld for two weeks from Monday June 22 to Friday July 3, 2009.

The explosive development of the theory of automorphic formson GL(2) in the second half of the twentieth century allowed anumber of spectacular recent advances in number theory. Themost visible of these was the proof of the Shimura – Taniyama –Weil conjecture and of Fermat’s Last Theorem almost 15 yearsago, but one should also mention other highlights like the sub-sequent proof of the Serre conjectures relating two-dimensionalmod p Galois representations to modular forms, and the im-portant progress on the Fontaine – Mazur conjectures concern-ing p-adic Galois representations. Spurred in part by some ofthese breakthroughs, the last decade has witnessed significantarithmetic applications of the theory of automorphic forms onhigher rank groups, such as the spectacular recent proof ofthe Sato-Tate conjecture and the ongoing work on the Iwasawamain conjecture exploiting Eisenstein series on unitary groups.

Automorphic forms on GL(2) are very accessible to machinecalculation, and there has been a tremendous amount of ac-tivity devoted to developping efficient algorithms for comput-ing with modular forms, and making tables of data related tomodular forms widely accessible. For example, we now haveextensive on-line tables of all elliptic curves over Q of conduc-tor up to 130,000, a database which contains several million

elliptic curves and whose calculation, based on the truth ofthe Shimura – Taniyama conjecture and on the modular symbolmethod for computing holomorphic cusp forms, is both a com-putational tour de force and a real treasure trove for researchersinterested in the arithmetic of elliptic curves.

By contrast, the realm of automorphic forms on groups ofhigher rank has been much less explored computationally, andthis presents a great number of interesting challenges. It wouldbe of great interest, both for theoretical as well as experimentalpurposes, to bring the theory of automorphic forms as much aspossible within reach of practical machine calculations.

The purpose of the SMS is to bring together a number of ex-perts who have been leaders in both the theoretical and morecomputational aspects of the theory of automorphic forms andL-functions, who will offer introductory-level courses aimedat presenting graduate students and postdoctoral fellows withthe state of the art in the subject and report on new advanceswhich have not yet been covered in a forum of this sort.

The lectures of the school will include:Henri Darmon (McGill) and Eyal Goren (McGill), Backgroundand prerequisites.Andrew Booker (Bristol), Maass forms and L-functions.Noam Elkies (Harvard), K3 surfaces of high Picard number andtheir moduli.Dorian Goldfeld (Columbia), Automorphic forms and L-functions in higher rank.Erez Lapid, (Hebrew University), The Arthur – Selberg traceformula and applications.Kamal Khuri-Makdisi (American University of Beirut), Modu-lar interpretation and equations for modular varieties.Michael Rubinstein (Waterloo), Algorithms for the computa-tion of L-functions.Harold Stark (UCSD), Artin L-functions and special values ats = 0.Fredrik Strömberg (TU Clausthal), Maass waveforms forSL(2, Z) and subgroups, from a computational point of view.Andreas Strömbergsson (Uppsala), Computations using theSelberg Trace Formula.Audrey Terras (UCSD), L-functions of graphs.Akshay Venkatesh (Stanford, Aisenstadt Chair), Eisenstein se-ries.John Voight (Vermont), The algorithmic theory of quaternionalgebras.

For more information about the SMS 2009, in particular forgraduate students and postdoctoral fellows who would like toattend and apply for funding, please contact Sakina Benhimaat benhima@CRM.UMontreal.CA.

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Wendelin Werner(suite de la page 2)

une famille de transformations conformes gt qui retire lacourbe c jusqu’au point t < T. (Voir Figure 2(b).) L’équation deLoewner est une équation différentielle donnant cette famillegt. Des conditions supplémentaires permettent de fixer la fa-mille gt uniquement. Le mouvement du point gt(ct) sur l’axeréel dépend de la courbe c. Schramm a montré que, pour la loilimite de ΓN , la loi de ce mouvement est

√κBt où κ = 6 et Bt est

le mouvement brownien. La même construction, pour d’autresvaleurs de κ, a pu être liée à d’autres modèles critiques et leprocessus est donc nommé SLEκ.

L’approche de la criticalité

Les premiers travaux qui ont suivi l’introduction des tech-niques liées à SLE ont été limités aux limites d’échelle à la pro-babilité (ou température) critique. C’est en ces valeurs critiquesque les lois limites sont décrites par SLEκ. Ces travaux lais-saient donc intouché un chapitre important de la théorie destransitions de phase, celui du voisinage du point critique etdes outils développés par les physiciens pour l’étudier : les loisd’échelle et le groupe de renormalisation. Dans les années 60et 70, des physiciens, dont Fisher, Widom, Kadanoff et Wilson,avaient proposé que le comportement de certaines observablesauraient, proche du point critique, un comportement en loi depuissance et que leurs exposants seraient reliés à ceux appa-raissant au point critique. Ce lien entre le modèle critique et lemodèle près du point critique avait été exploré par Harry Kes-ten dans les années 80. Mais depuis quelques mois les probabi-listes ont franchi des étapes importantes pour révéler la finessedes phénomènes proches du point critique. Le dernier exposéde Werner était consacré à décrire ses résultats récents, obte-nus avec Pierre Nolin, en percolation bidimensionnelle quasicritique.

Revenons aux interfaces ΓN partant du point médian de la basede la région triangulaire. Leur loi converge vers le processusSLE6. Nous avons énoncé que la probabilité d(p, N) de toucherle côté droit de la région avant le gauche avait un comporte-ment limite singulier : lim d(p, N) est 0 si p < 1

2 et 1 si p > 12 .

Un processus limite plus fin est donc nécessaire pour étudier larégion proche du point critique. À cette fin, Nolin et Werner dé-finissent p∗(N, ε) = inf {p : d(p, N) > 1

2 + ε} pour ε ∈ (0, 12 ).

Ils peuvent alors montrer que, si une suite {p(N)}N∈N est choi-sie de façon à ce que p(N) soit proche de p∗(N, ε), alors uneloi limite pour les interfaces ΓN peut être obtenue. Cette nou-velle limite est singulière par rapport à SLE6, même si elle enpartage les exposants critiques ; en particulier la dimension deHausdorff des interfaces est presque sûrement 7

4 , comme pourles courbes de SLE6. Voici donc un autre chapitre de l’étudedes phénomènes critiques où les outils proposés par SLE se ré-vèlent fondamentaux.

La communauté du CRM remercie Wendelin de lui avoir pré-senté de façon pédagogique les idées centrales de l’équation deLoewner stochastique en en révélant certains des développe-ments récents.

Andrei Okounkov(continued from page 4)

evaluated between given pairs of states. This brings the under-lying calculations into direct contact with methods and ideasthat are current in the theory of Integrable Systems and Quan-tum Field Theory. Another is based on noting that the adja-cency matrices for on associated set of graphs correspondingto a dimer covering can be realized by multiplication in a non-cummutative version of a polynomial ring. Such constructions,though not at all standard in probability theory or combina-torics, make use of tools that are quite familiar to mathematicalphysicists, who use them in a variety of settings, from quantumfield theory to integrable systems to string theory.

Like Witten, who made remarkable use of ideas and meth-ods from quantum field theory and other physics inspiredtools to arrive at deep new results of a topological and alge-braic geometric nature, Andrei Okounkov draws very prof-itably from the toolbox of mathematical physics to arrive atfresh new perspectives and results relating to problems incombinatorics, probability theory, group representation the-ory, topological invariants and algebraic geometry. Besidesthe fact that this involves some basic constructions in thetheory of integrable systems and noncommutative geome-try, there are also close connections appearing with resultsand methods in the spectral theory of Random Matrices,(which formed the subject of the preceding research workshopthe 2008 – 2009 Thematic Program (Aug. 25 – 30, 2008. URL:crm.umontreal.ca/Matrices08/index_e.shtml). Building onsuch interconnections provides some of the “secret weapons”that have been so innovatively applied by Okounkov, enablinghim to overcome hurdles that might otherwise have stumpedspecialists familiar only with the conventional methods of theirindividual domains.

The successful application of such methods to a wide rangeof problems of considerable current interest in a variety of do-mains not only points to deep underlying connections betweenthem, but suggests that there must be much more along thelines pioneered by Andrei Okounkov that remains worthy offurther investigation. His presence at the CRM during this veryexciting thematic concentration period and his Aisenstadt lec-tures provided a great stimulus to those present, and an inspi-ration to some to carry on the investigation of the challengingrange of ideas involved, deepening our understanding of thelinks between these diverse fields and the beautiful intrinsicstructure that underlies them.

1. I. Berlin, The hedgehog and the fox: an essay on Tolstoy’s view ofhistory, Weidenfeld and Nicolson, London, 1953; Simon andSchuster, New York, 1953.

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Workshop “Mathematical Aspects of Quantum Chaos”by N. Anantharaman, S. Nonnenmacher, Z. Rudnick, and S. Zelditch

The workshop entitled “Mathematical Aspects of QuantumChaos” was held at the CRM on June 2 – 7, 2008. Organized byN. Anantharaman (École Polytechnique, France), S. Nonnen-macher (CEA, France), Z. Rudnick (Tel Aviv University, Israel)and S. Zelditch (Johns Hopkins, USA), it was the first event ofthe thematic year on “Probabilistic Methods in MathematicalPhysics.” It was attended by about 40 participants, a mixtureof pure and applied mathematicians and theoretical physicists.

Participants to the workshop

It is clear from the title that “quantum chaos” is a multidisci-plinary topic. The term was coined by theoretical physicistsand chemists in the 1970s, who were interested in the descrip-tion of highly excited nuclear, atomic or molecular energy lev-els; in mathematical terms, the aim is the spectral analysis ofcertain Schrödinger operators. For the systems under scrutiny,no analytical formulas exist for eigenvalues or eigenfunctions,so one has to resort to semiclassical (high-frequency), numer-ical and/or statistical methods. The common denominator ofthese quantum systems is the fact that their limiting classicaldynamics is (at least partly) chaotic. Soon, theoreticians pre-ferred to study simpler systems, like 2-dimensional Euclideanbilliards, or Riemannian manifolds of negative curvature, thechaotic properties of which had been investigated by the dy-namical systems community. Some (less realistic, but even sim-pler) quantum models were then considered: quantum mapsand quantum graphs. These systems allow “cheaper” nu-merical studies, but they also often allow a deeper analyticalstudy. Some of these systems have strong arithmetical prop-erties: their study forms a proper subfield called “arithmeticquantum chaos.”

These various “quantum chaotic systems” were mentionedduring the workshop, which allowed the audience to appreci-ate their similarities and differences. A “metaconjecture” in thedomain consists in identifying, from a statistical point of view,the eigenvalues and eigenfunctions of individual quantum sys-tems, with those of a class of “random” systems, typically large

random matrices. The Random Matrix Theory (RMT) con-jecture which studies the short-distance correlations betweeneigenvalues (formulated by Bohigas – Giannoni – Schmidt in1984) has recently received a renewed impetus in the theoreti-cal physics community, due to a refined combinatorial analysisof long periodic orbits. These results, presented in the talksof S. Müller (Bristol) and P. Braun (Essen), are still far frommaking a full mathematical proof; the simple quantum graphsstudied by G. Tanner (Nottingham) in this view already showthe difficulty of the question.

On the other hand, the mathematical analysis of eigenfunc-tions has made significant progresses in the recent years. Anow standard result in the domain is the Quantum Ergodic-ity (QE) theorem, which applies to most of the systems men-tioned above. This theorem (first showed by Schnirelman,Zelditch and Colin de Verdière in the 70s and 80s) states thatalmost all the high-frequency eigenfunctions are “macroscopi-cally equidistributed” across the full energy hypersurface, pro-vided the corresponding classical dynamics is ergodic. J. Keat-ing (Bristol) and A. Strohmaier (Loughborough) explained thesubtleties of extending QE to quantum graphs, respectively tomulti-component wavefunctions. Systems with a mixed dy-namics (featuring regions of regular motion surrounded by a“chaotic sea”) were described in the talks of A. Bäcker (Dres-den) and A. Barnett (Dartmouth); so far these systems resist arigorous analysis.

As soon as QE was established, people conjectured that ac-tually all high-frequency eigenfunctions are equidistributed, astatement referred to as Quantum Unique Ergodicity (QUE).This conjecture is still pending, except for the case of arith-metic systems. Lindenstrauss’s recent proof of QUE for Heckeeigenstates on arithmetic hyperbolic surfaces has been re-viewed by L. Silberman (UBC), who also discussed possi-ble extensions to some higher-rank symmetric spaces. AnArithmetic QUE result had been obtained before for quan-tum automorphisms of the 2-dimensional torus (or “quan-tum cat maps”) by Rudnick – Kurlberg; further advances inthe statistics of quantum matrix elements were presentedby P. Kurlberg (Stockholm). On the other hand, quan-tum cat maps also lead to explicit counterexamples to QUE,namely (nonarithmetic) eigenstates which are not equidis-tributed, but have a large probability weight (a “strong scar”)around a single periodic orbit. For automorphisms of higher-dimensional tori, D. Kelmer (Princeton) identified a family ofeigenstates which are both “arithmetic” and localized on astrict submanifold of the phase space, thus disproving Arith-metic QUE for this system. Away from arithmetic systems,the use of the Kolmogorov – Sinai entropy was put forwardby N. Anantharaman a few years ago, when she showedthat high-frequency eigenfunctions on manifolds of (varying)

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Multidisciplinary and Industrial ProgramProgramme multidisciplinaire et industriel

May 20 – 30, 2008. Summer School “SystemsBiology Dynamics: from Genes toOrganisms”by Peter Swain (McGill)

The summer school was organised by the Centre for Non-linearDynamics in Physiology and Medicine, and brought together55 students, postdoctoral fellows, physicians, professors, andindustry representatives from 16 countries for a two-week in-tensive course. The director of the school was Dr. Peter Swainof McGill University. The instructors were primarily membersof the Centre and students associated with the Centre acted asteaching assistants. The mornings consisted of two 1.5 hourlectures and the afternoons of computer laboratory exercisesusing specifically designed and developed material.

The first week focused on non-linear modeling of physiologi-cal dynamics. Following an introductory day, subsequent top-ics included dynamics of normal and pathological behaviorsin both neural and cardiac systems. The second week focusedon modeling single cells and particularly their physiology andthe genetic networks that create it. Applications included mod-eling cell replication and stochastic gene expression, inferringmodel structure from time series data, and synthetic biology,which was taught by Dr. Mads Kaern, a guest lecturer from theUniversity of Ottawa.

The school was organised and supported partially by the CRM,MITACS and INRIA.

Deuxième atelier de résolution de problèmesindustriels de Montréal, du 18 au 22 août 2008de Odile Marcotte (UQÀM et GERAD)

Dans le cadre de son programme multidisciplinaire et indus-triel, le CRM organise des ateliers où des chercheurs, des re-présentants de l’industrie, des étudiants et des stagiaires post-doctoraux tentent de modéliser et de résoudre des problèmesproposés par des partenaires industriels. L’atelier qui s’est tenudu 18 au 22 août 2008 était le deuxième atelier de ce genre orga-nisé par le CRM, et 14 chercheurs, 10 représentants de l’indus-trie et 22 étudiants ou stagiaires postdoctoraux y ont participé.

L’atelier fut organisé conjointement par le CRM, le GERAD, lercm2 (Réseau de calcul et de modélisation mathématique), leCIRANO, MITACS et le CIRRELT, et financé par MITACS et lercm2. Le comité organisateur était formé de Jean-Marc Rous-seau (CIRANO et rcm2, président du comité), Eric Bosco (MI-TACS), Michel Gendreau (CIRRELT et Université de Montréal),Bernard Gendron (Université de Montréal), Alexandra Hae-

drich (ISM), François Lalonde (Université de Montréal), Ro-land Malhamé (École Polytechnique), Odile Marcotte (UQÀMet CRM), Dominique Orban (École Polytechnique) et Louis-Martin Rousseau (École Polytechnique).

Les cinq problèmes étudiés pendant l’atelier provenaient desorganismes suivants : Hydro-Québec, Ressources naturellesCanada, FPInnovations (division Feric), Exact Modus et lacompagnie Les Bois Francs L’Islet-Sud, et la compagnie GIRO.Les organisateurs de l’atelier sont très reconnaissants à ces or-ganismes de leur avoir fourni des problèmes et à Mme SophieD’Amours, professeur à l’Université Laval et codirectrice duCIRRELT, à M. Christian Rouleau, du consortium FORAC, età M. Paul Stuart, titulaire de la chaire CRSNG en génie deconception environnementale à l’École Polytechnique, de lesavoir aidés à trouver des problèmes.

Les cinq problèmes auxquels ont travaillé les équipes rele-vaient de l’optimisation et de la recherche opérationnelle,spécialités auxquelles se consacrent les chercheurs du GE-RAD (www.gerad.ca) et du CIRRELT (www.cirrelt.ca), deuxcentres avec lesquels le CRM a une fructueuse collaboration.Le premier problème, fourni par l’Hydro-Québec, consistait àdéterminer l’emplacement optimal des nouveaux instrumentsde mesure du couvert nival dans le grand nord québécois. Ledeuxième problème, fourni par Ressources naturelles Canada,visait à améliorer le réseau de récupération de la chaleur etle dessin des conduites d’eau dans une usine de pâtes et pa-pier générique. Le troisième problème, fourni par FPInnova-tions, concernait la planification opérationnelle d’une chaîned’approvisionnement forestière. Le quatrième problème rele-vait également du domaine de la foresterie : il consistait à pré-dire la répartition qualitative des planches produites par uneusine en fonction des différents types de billes transforméesen planches par l’usine. Finalement, le cinquième problème,fourni par la compagnie GIRO, consistait à examiner diversesméthodes pour insérer des plaques tournantes dans des tour-nées de véhicules.

Bien que la majorité des participants fussent des chercheurs etétudiants de Montréal et de Québec, le CRM a accueilli desparticipants venant d’Oxford, de Toronto, de Waterloo et duMaryland, ainsi que des étudiants français qui avaient passéquelques mois à Montréal dans le cadre de leur stage. L’ateliers’est déroulé dans une ambiance conviviale et chaleureuse etles participants ont trouvé leur expérience très enrichissante.Le lecteur trouvera des détails supplémentaires sur l’atelier enconsultant le site crm.math.ca/probindustriels2008, et unedescription générale du fonctionnement des ateliers de réso-lution de problèmes industriels dans le bulletin du CRM del’automne 2007.

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The Winter 2008 Aisenstadt Chairsby Walter Craig (McMaster), Christiane Rousseau (Université de Montréal), and Alina Stancu (Concordia)

The Aisenstadt Chairs for the theme semester “Dynamical Systems and Evolution Equations” held at the CRM from Januaryto May 2008 were Gerhard Huisken (Max-Planck-Institut für Gravitationsphysik), who visited the CRM in April, and Jean-Christophe Yoccoz (Collège de France) who visited the CRM in May.

Gerhard Huisken

Gerhard Huisken pioneered thestudy of the mean curvature flowin the 80s and continued to beat the forefront of research inthe area of curvature flows. Inthe 90s, in his work with Ilma-nen, Huisken developed the the-ory of weak solutions to the in-verse mean curvature flow andapplied it to prove the Rieman-nian Penrose Inequality: the to-

tal mass of an asymptotically flat three-manifold of nonneg-ative scalar curvature is bounded below in terms of the areaof each smooth, compact, connected “outermost” minimal sur-face (i.e., not separated from infinity by any other compact min-imal surface) in the three-manifold. A 1998 ICM speaker, Ger-hard Huisken is also the recipient of the Medal of AustralianMathematical Society (1991) and the Leibniz Prize (2003). Since2002, Huisken is Director and Scientific Member at the MaxPlanck Institute for Gravitational Physics.

Gerhard Huisken gave a series of four lectures, three of thempart of the workshop “Geometric Evolution Equations” orga-nized in April 2008 by V. Apostolov (UQÀM), P. Guan (McGillUniversity) and A. Stancu (Concordia University).

The first lecture, Geometric Variational Problems in General Rel-ativity, open to the general public, focused on the interac-tion between geometric analysis and physical concepts in thestudy of isolated gravitational systems. Huisken showedhow the Eisenstein equations governing the gravitational phe-nomena in General Relativity can be derived variationally onLorentzian manifolds, thus other concepts related to gravi-tation are also best formulated in within the framework ofLorentzian metric. The second lecture focused on the use of themean curvature flow to derive isoperimetric inequalities. Atthe core of this argument lie monotonicity results which werereconsidered in Huisken’s third talk for the inverse mean cur-vature flow. In the latter context, the monotonicity argumentsled to applications in conformal geometry and General Relativ-ity. Gerhard Huisken’s last talk, An Isoperimetric Concept for theMass in General Relativity, brought together techniques from thethree previous lectures to make precise and give the proof ofthe lower bound on the total mass of an isolated gravitationalsystem.

Jean-Christophe Yoccoz

The winter semester 2008 “Dynamical Systems and EvolutionEquations” at the CRM was focused on dynamical systems,and accordingly the Aisenstadt Lectures of May 9 – 12 were de-livered by Jean-Christophe Yoccoz, one of the leading mathe-maticians working in this area. He delivered four in-depth lec-tures in the form of a series dedicated to the theory of intervalexchange maps, the first lecture being of colloquium style. Healso gave a lecture on affine interval exchange maps during thefollowing week, which coincided with one of the major work-shops of the thematic semester on dynamical systems and evo-lution equations. The lectures were given at a high level, butnonetheless were accessible, and many members of the CRM,as well as the visitors for the thematic semester, were able tobenefit from them.

The central theme of this lectureseries was the analysis of inter-val exchange maps. The classof diffeomorphisms of the circlehas been a classical topic of studyin dynamical systems since thedays of Poincaré, Denjoy, andtheir contemporaries. At thispoint, the ideas of V. Arnold,J. Moser, M. Herman and J.-Ch. Yoccoz have given quite a refined picture of their conju-gacy classes and their dependence upon rotation number andsmoothness. The category of (generalized) interval exchangemaps can be seen as the extension of this study to cases ofpiecewise-increasing discontinuous maps of the interval withmore than one discontinuity. More specifically, a (standard)interval exchange map (IEM) is a one:one map of the interval[0, 1] which is locally a translation except at finitely many dis-continuities. For example, if the number d of discontinuities isd = 1, then an IEM is a rigid rotation. New and fascinatingdynamical questions arise for d ≥ 2. Up to a point, the dynam-ics of IEMs is similar to the dynamics of rotations on the circle:there is a notion of generalized rotation number, and, when itis irrational, all orbits of the transformation are dense.

A major part of the lecture series was devoted to the detaileddevelopment of the theory of this generalized rotation numberin the setting of interval exchange maps.

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Thematic program semester

Dynamical Systems and Evolution EquationsCentre de recherches mathématiques, Montréal — January – June, 2008

by Walter Craig (McMaster University) and Christiane Rousseau (Université de Montréal)

The winter semester 2008 “Dynamical Systems and EvolutionEquations" at CRM has focused on dynamical systems, inter-preted in a broad sense so as to include applications to funda-mental problems in differential geometry as well as in mathe-matical physics. The four principal topics of the semester in-volve, as a common theme, the interplay between classical dy-namical systems and the qualitative properties of solutions ofpartial differential equations. They had been chosen becauseof the recent dramatic achievements, representing progress ona number of the most basic and difficult questions in this field.Among these, we find the proofs of the Poincaré conjecture andof Thurston’s geometrization conjecture for three manifolds.Also in Hamiltonian PDE, the analytic methods of Hamilto-nian mechanics are making an impact on the study of the evo-lution of many of the principal nonlinear evolution equationsof mathematical physics. These advances have had a broad im-pact in geometry and topology, and they shed light on the basicphysical processes that are modeled by ODE and PDE.

Acknowledgements. The semester was supported by the NSFand the Clay foundation, in addition to CRM. We are verygrateful to Eugene Wayne who piloted the NSF application.

There were six workshops during the semester and severaladvanced-level courses, including two Aisenstadt series of lec-tures. The semester started in January with two smaller events.

“Young Mathematicians’ Conference,” January 18 – 19, 2008Organizers: Walter Craig (McMaster), Adrian Nachman(Toronto), Nilima Nigam (McGill), Dmitry Pelinovsky (Mc-Master), Mary Pugh (Toronto), Catherine Sulem (Toronto)

This workshop was the fifth annual Young Mathematicians’Conferences and the first in Montréal. This is a conference ofjunior researchers in the areas of partial differential equations(PDEs) and Dynamical Systems, whose goal is to encouragescientific exchange, to create an opportunity for mathemati-cians in an early stage of their career to speak at a major or-ganized conference, and to get to know each other and eachother’s work. And this most recent one in Montréal definitelylived up to the high standards set by the previous meetings.

The two-day conference this January featured two plenarytalks given by Ivar Ekeland (UBC) and Walter Craig (McMas-ter), and eleven twenty five minute talks given by junior math-ematicians (advanced graduate students and postdocs). I. Eke-land spoke on the foundations of micro-economic theory andthe role that the calculus of differential forms plays in the the-ory. W. Craig spoke on his recent work with A. Biryuk (aformer junior YMC speaker) on new estimates for weak solu-

tions of the Navier – Stokes equations, and their on the Kol-mogorov – Richardson picture of the energy cascade and thepower law behavior of the energy spectral function in fully de-veloped turbulence. The talks given by the young mathemati-cians ranged from numerical modeling of fingering in porousmedia flows (M. Chapwanya, SFU), to complex dynamical sys-tems (R. Roeder, Toronto), to a survey of recent results relatedto the Minkowski problem for convex domains and their gen-eralizations (J.-F. Li, McGill).

“Initial condition workshop,” January 24 – 25, 2008Organizers: Walter Craig (McMaster), Pengfei Guan (McGill)and Christiane Rousseau (Montréal)

This workshop took place right at the beginning of the semesterand allowed the organizers and the participants, in particularthe associated postdoctoral fellows, to know each other and tohave an introduction to each other’s research. The workshopstarted by the invited lecture of P. Mardešic (Dijon). The otherlectures have been given by the organizers of the core work-shops and the CRM postdocs associated to the semester.

These two workshops were followed by a two months break.Then started a blitz of four sparkling workshops.

Workshop “Spectrum and Dynamics,” April 7 – 11, 2008Organizers: Dmitry Jakobson (McGill) and Iosif Polterovich(Montréal)

The central theme was the interplay between the theory of dy-namical systems and spectral geometry, and the conferencebrought together the leading experts in these two fields. Inparticular, the workshop focused on applications of dynam-ics to the study of spectral asymptotics. Main topics included:spectral asymptotics on negatively curved manifolds and hy-perbolic dynamics, Laplace and length spectra, elliptic opera-tors on vector bundles and partially hyperbolic flows, billiardsand spectral theory on manifolds with boundary, ergodic the-ory of group actions, and theory of thermodynamic formalism.A poster session was organized.

Mini-courses: The program of the workshop included twomini-courses, which continued during the week after the work-shop.

S. Nonnenmacher (CEA-Saclay) presented a mini-course onrecent spectacular results on the entropy of chaotic eigen-states, obtained by N. Anantharaman, himself and H. Koch.They show that the entropy of semiclassical measures (limitmeasures associated to a sequence of chaotic eigenstates) isbounded from below by approximately half the maximal en-

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tropy. This implies that high-energy chaotic eigenstates are “atleast half-delocalized.” This partially answers a long-standingquestion in the theory of quantum chaos.

R. Schubert (Bristol) presented an introduction to his work onlong-time behavior of wave propagation on negatively-curvedmanifolds. Until recently, wave propagation in chaotic sys-tems has been well understood up to logarithmic times only:this is the so called Ehrenfest time barrier in quantum mechan-ics. Schubert’s results can be viewed as the first step towardsunderstanding the wave propagation up to polynomial times,which is an outstanding problem in mathematical physics.

The proceedings of the conference will be published in theCRM Proceedings and Lecture Notes, edited by D. Jakobson,S. Nonnenmacher and I. Polterovich.

Workshop on “Geometric Evolution Equations,” April 16 –27, 2008Organizers: Vestislav Apostolov (UQÀM), Pengfei Guan(McGill) and Alina Stancu (Concordia)

This 10-days workshop, held during the spring of 2008, was fo-cused on the study of various evolution equations in geometryand general relativity, including intrinsic curvature flows witha focus on Ricci and Kähler – Ricci flows, and extrinsic curva-ture flows — mean and inverse mean curvature flows, homoge-nous and non-homogenous Gauss curvature flows, relativity.

The workshop was attended by over 40 participants, rangingfrom physicists to geometers to analysts. Most participantswere speakers. Many new results were reported for the firsttime during the workshop and many participants left Montrealready to work on new projects. We highlight below the fourlectures series in which the lecturers did a wonderful work insurveying vibrant fields of current research.

Gerhard Huisken gave a cycle of 4 lectures, including the Ain-senstadt distinguished lecture, thus providing a comprehen-sive survey on geometric variational problems in General Rel-ativity, including aspects of the mean and inverse mean cur-vature flows, different concepts of mass and their relationshipwith isoperimetric inequalities.

Burkhard Wilking lectured on Ricci flow and its spectacular ap-plications to the recent resolution of classical curvature pinch-ing conjectures in Riemannian geometry, including the 1/4pinching sphere theorem.

Lei Ni gave a far-reaching introduction to Ricci flow with aspecial attention to solitons. After explaining the Li – Yau –Hamilton Harnack inequalities for Kähler – Ricci flows, he pre-sented a detailed study of gradient Ricci solitons and theircanonical forms. Xiuxiong Chen surveyed the exciting recentbreakthrough in the theory of extremal Kähler metrics that heand Tian made by proving their uniqueness within a fixed Käh-ler class, as well as the boundedness of the (relative) K-energy,by using suitable approximations of the geodesics in the space

of Kähler metrics. He linked these results to the convergenceof the Kähler – Ricci and Calabi flows.

Thus, in 10 intensive days, the program succeeded in tying to-gether most of the new results in the subject and a variety ofnew projects were born. The participants affirmed frequentlyand spontaneously that the program was a great success.

Workshop on “Singularities, and gradient and Hamiltonianflows,” May 5 – 16, 2008Organizers: Walter Craig (McMaster), Christiane Rousseau(Montréal), Alexander Schnirelman (Concordia)

The first week was dedicated to a series of three short coursesby J.-C. Yoccoz (Collège de France), L. Stolovitch (Toulouse),and R. Montgomery (UCSB), and followed by the workshop.The worshop itself was heterogeneous, with specialists talk-ing on a broad set of themes, whose basis consists of ideas andthe philosophy of dynamical systems. The lecture room was ingeneral quite full of CRM visitors, postdoctoral fellows, work-shop speakers and their graduate students, as well as mathe-maticians from the Montréal area. With lively question and an-swer periods, it was a week full of mathematical interactions.

Short courses: J.-C. Yoccoz gave his lectures in the context ofthe semester’s André Aisenstadt Chair (see other article).

Laurent Stolovitch’s lectures dealt with the classification ofPoisson structures, in the holomorphic and the C∞ category.This is the resolution of a program dating from the early 1980sdue to A. Weinstein and J. Marsden, giving the classification ofsingular Poisson structures up to holomorphic (smooth) con-jugacy, under a Diophantine condition on the linearization atthe singular point. It is surprisingly enough a small divisorproblem. Stolovitch went on to discuss the similarity of theproblem with the Siegel problem of conjugacy of holomorphicvector fields, and furthermore developed his ideas on exten-sions of the work to various settings of simultaneous conjugacyof bi-vector fields. R. Montgomery talked on the classificationof distributions, discussing the problem of distributions withlarge growth vector and the construction of prolonging a distri-bution, and giving an introduction to the theory of Legendriancurves. His talks described the extension of the classical ideasof Cartan by an infinitary suspension method that he termedthe ‘friendly monster.’

As for the workshop itself, a number of lectures concerned re-sults on the n-body problem and Arnold diffusion. V. Kaloshin,the Clay lecturer for this workshop, gave two lectures on thetopic. The first one, a colloquium type talk, discussed spe-cial types of solutions of the 3-body problem and their Haus-dorff dimension, while the second one presented an example ofArnold diffusion. Concerning the n-body problem, S. Terracinidiscussed collision solutions via variational methods, and K.-C. Chen the stability of satellite solutions. V. Gelfreich de-scribed an example of Arnold diffusion in a non integrable bil-liard. His method can be applied in the more general contextof slow-fast Hamiltonians.

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P. de Maesschalck gave neat proofs of general theorems forthe existence of invariant manifold in analytic or Gevrey class.Moduli spaces of analytic classifications of unfoldings of codi-mension 1 saddle-node vector fields and parabolic points werediscussed by R. Schäfke and C. Rousseau.

Several talks dealt with normal forms near singularities of ODEand PDE and their applications to describe the qualitative be-haviour of solutions of PDE of small norms. L. Stolovitch dis-cussed normal forms of singular points of vector fields withnon diagonal linear part. Other lectures on the subject weregiven by D. Bambusi (wave equations) and H. Ito.

F. Dumortier explained conditions on singularly perturbedsystmes allowing to conclude to the birth of more than onelimit cycles when several critical values of the slow curvecoallesce. B. Fiedler discussed the global qualitative dynamicsof the scalar reaction-advection-diffusion PDE on the unit in-terval. He showed how the global attractor of the PDE consistsof all equilibria and heteroclinic orbits between them.

Dynamical systems methods applied to PDE was a recurrenttheme. H. Eliasson and M. Keel discussed Schrödinger equa-tions. M. Keel was concerned with the energy transfer and itsgrowth to infinity in the presence of resonances. E. Wayne dis-cussed the existence of travelling wave solutions of the Fermi –Pasta – Ulam model coming from a pair of counterpropagatingwaves and ending in such a pair. M. Bialy discussed the peri-odic solutions of a model of quasi-periodic solutions given byBenney chains, together with a conjecture that all the periodicsolutions are standing waves. W. Craig exploited a dynamicalsystems argument to give new estimates on weak solutions ofNavier – Stokes equations, thus yielding nontrivial bounds onthe Kolmogorov spectrum of solutions.

An alternative machinery to KAM consisting in using renor-malization and universality for Hamiltonian flows was intro-duced by D. Gaydashev.

A geometric approach to controllability and optimal control ofmultiparticle system was discussed by A. Sarychev and illus-trated on a system of N interacting particles on a straight line.

Workshop “Floer Theory and Symplectic Dynamics,” May19 – 24, 2008Organizers: Octav Cornea (Montréal), Leonid Polterovich (TelAviv), Felix Schlenk (ULB)

The semester ended beautifully with this huge workshop (68participants) in May. The main theme of the workshop wassymplectic topology and Hamiltonian dynamics. Specific top-ics covered various homology theories arising in symplecticand contact topology, their algebraic and analytic aspects, ap-plications and ramifications (Abbondandolo, Abouzaid, Bour-geois, Frauenfelder, Ginzburg, Lisi, Oancea, Ostrover, Ono,Salamon, Schwarz), new results on Lagrangian tori (Auroux,Biran, Chekanov, Fukaya), Hofer’s geometry (Karshon, Ker-man), rigidity of Poisson brackets (Buhovsky, Entov), symplec-

tic homogenization (Viterbo), Weinstein conjecture (Albers)and rigidity/flexibility of symplectic embeddings (Hind).

Particularly impressive was Edi Zehnder’s talk about surfacesof section for Reeb flows in dimension three, that started withthe work of Poincaré and ended with recent and strong resultson closed orbits of Reeb flows. Another touching moment wasGuangcun Lu’s lecture on the Conley conjecture in cotangentbundles, that showed that the symplectic community forms abig family over all the world.

The meeting was held in a rather informal and very stimulat-ing atmosphere. The many PhD students helped keeping themeeting lively, and proved that symplectic topology is a veryactive field of research. They were five talks a day, besides onWednesday, where the afternoon was left free for discussions.

The many positive comments indicated that the participantswere very happy with the conference. They felt that they prof-itted from the many interesting and carefully presented lec-tures, as well as from the individual discussions and the op-portunities to meet and to collaborate with their colleagues onan international level.

Jean-Christophe Yoccoz(continued from page 9)

Yoccoz also described the construction of Riemann surfacesbuilt to support a translational flow which can be viewed assuspensions of IEM’s. Their genus plays a major rôle in thecomplexity of the orbit structure of the problem.

At the level of parameter space, there is a continued fraction al-gorithm discovered by Rauzy, Veech and Zorich which gener-alizes the classical Gauss continued fraction algorithm, whichcorresponds to a renormalization process. The dynamics of thisalgorithm is hyperbolic and the corresponding Lyapunov ex-ponents have been investigated by Forni, Avila and Viana. Thisleads to a surprising behavior of the Birkhoff sums of a typicalIEM, known as the Zorich phenomenon. The field is at the con-fluence of complex analysis, dynamical systems, number the-ory and uses a wide range of techniques.

The last lecture dealt with affine interval exchange maps(AEM) and stressed the similarities and differences with IEMs.The rotation number can be defined for affine interval ex-change maps in the same way as it is defined for IEMs. How-ever, while IEMs possess no wandering interval, this is not thecase with affine interval exchange maps as shown by Marmi,Moussa, and Yoccoz in 2005: For combinatorial data of genusg > 1, for almost all rotation numbers, and for most slopes, an AEMpossesses a wandering interval.

Yoccoz’s lectures were dynamic and clear, and the many at-tendees appreciated being shown an expert’s overview of thisemerging area of low dimensional dynamical systems. We arelooking forward to the extension of these results to discontinu-ous, non necessarily piecewise linear maps of the interval.

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Activités externes et programme général

Première rencontre CRM-INRIA-MITACSde Michel Delfour (CRM et Université de Montréal)

Le CRM et l’INRIA entretiennent depuis longtemps des liensde coopération. Le 24 août 1973, un accord de coopération futsigné entre M. L’Abbé (vice recteur à la recherche, Universitéde Montréal), L. Le Cam (directeur du CRM) et l’académi-cien J.-L. Lions (directeur) du LABORIA de l’Institut de Re-cherche d’Informatique et d’Automatique (IRIA - premier nomde l’INRIA). Plus récemment, en 2003, l’équipe INRIA associéeà l’étranger ACE (Aérosols, Cœur, Endoprothèses) fut créée parl’INRIA avec la participation du CRM, du GIREF et de l’Uni-versité d’Ottawa et subséquemment élargie à Taiwan. En mai2005, une école de printemps et un atelier sur les traitementsmini-invasifs en médecine et chirurgie furent organisés à Mont-réal et subventionnés par le CRM, l’INRIA et MITACS. La co-opération entre le CRM et l’INRIA s’est vue confortée en 2005par un accord de coopération avec le Gouvernment du Québec(FQRNT) dans le cadre de sa stratégie éducationnelle et indus-trielle, et par une visite du président de l’INRIA, Michel Cos-nard, aux quartiers généraux de MITACS fin juillet 2007.

La collaboration avec le Canada, et en particulier le Québec, seplace très haut dans les priorités de l’INRIA dans le domainedes relations internationales. Il y a en ce moment pas moins de10 équipes associées entre l’INRIA et des équipes de chercheurscanadiens. Un « Memorandum of Understanding for Collabo-ration » avec MITACS a été signé à Ottawa le 2 juillet 2008 enprésence du Premier ministre français F. Fillon et de son homo-logue canadien S. Harper.

La première rencontre CRM-INRIA-MITACS qui s’est tenue àMontréal du 5 au 9 mai 2008 fut une occasion propice au ren-forcement des liens et à la création de nouveaux, entre l’INRIA,MITACS, et les instituts canadiens, s’appuyant sur le dossier decoopération entre le CRM et l’INRIA.

Les organisateurs de la rencontre étaient Michel Delfour (CRMet Université de Montréal, président), Yves Bourgault (Univer-sité d’Ottawa), Fahima Nekka (MITACS et Université de Mont-réal) et Marc Thiriet (INRIA et CNRS).

La rencontre était organisée autour de sept thèmes principaux :Systèmes biologiques, cellule, et tissus ; Imagerie et fonction-nalité du cerveau ; Milieux composites, modélisation multié-chelle, optimisation de forme et design ; Cœur ; Dispositifs mé-dicaux implantables, acheminement des médicaments ; Ecoule-ments physiologiques ; et Système respiratoire. Un des pointsforts de la rencontre a été la mise en place de bonnes condi-tions pour créér et faciliter des interactions entre les différentsgroupes participant à chacun des thèmes ainsi qu’une forte dé-légation d’étudiants français et canadiens.

La pleine intégration des principes du ICST (Institute for Com-puter Sciences, Social-Informatics and Telecommunications Enginee-

ring) dans les technologies médicales est une des sept prioritésdu Plan Stratégique 2003-2007 de l’INRIA. Cet engagement esttoujours à l’agenda de l’Institut, puisque la Médecine Compu-tationnelle est une des sept priorités du Plan Stratégique 2008-2012. Les activités de l’équipe REO dirigée par Marc Thiriet etl’équipe CFT sont donc au cœur des orientations soutenues parl’INRIA. Parallèlement, il faut rappeler que le comité visiteurdu CRSNG qui a évalué les trois instituts canadiens de ma-thématiques a estimé que les mathématiques de la médecineétaient un point fort et distinctif du CRM.

Colloque d’algèbre non commutativede Ibrahim Assem (Université Sherbrooke)

Cette rencontre s’inscrit dans le cadre d’une série de col-loques d’algèbre non commutative entre mathématiciens fran-çais, sud-américains et québécois. Se tenant d’habitude enFrance ou en Amérique du Sud, elle a eu lieu pour la premièrefois en Amérique du Nord, et les universités de Sherbrooke etBishop’s en ont été les hôtes du 9 au 14 juin 2008. Elle a compté65 participants, dont 24 étudiants ou stagiaires postdoctoraux.

L’objectif de ces rencontres est de réunir un groupe varié d’ex-perts de problèmes d’algèbre non commutative qui tous uti-lisent comme outil essentiel l’algèbre homologique. Le colloquea comporté quatre mini-cours, dans des domaines différents del’algèbre non commutative, à savoir la K-théorie (M. Karoubi,Université Paris 7), la théorie d’Auslander-Reiten dans une ca-tégorie dérivée (D. Happel, Université de Chemnitz), les iden-tités polynômiales en géométrie non commutative (C. Kassel,CNRS, Université Louis Pasteur, Strasbourg) et enfin la classi-fication des algèbres auto-injectives (A. Skowronski, UniversitéNicolas Copernic, Torun).

Parmi les exposés (de 45 ou de 30 minutes), deux lignes deforce se sont imposées : d’abord l’étude des relations entre al-gèbre non commutative et topologie, et ensuite celle des al-gèbres amassées (« cluster algebras »). La première, qui est à labase de la K-théorie, s’est vue illustrée de diverses façons dansles exposés de A. Ádem (UBC), D. Castonguay (Université Fé-dérale de Goias), P. Le Meur (ENS Cachan) et J. de la Peña(UNAM). Les algèbres amassées, qui ont acquis une impor-tance exceptionnelle en peu de temps, ont été le sujet des ex-posés de A. Zelevinski (Université Northeastern), un des fon-dateurs de la théorie avec S. Fomin, et de O. Kerner (UniversitéHeinrich Heine, Düsseldorf), D. Labardini-Fragoso (UniversitéNortheastern), R. Schiffler (UMass Amherst) et D. Smith (Uni-versité Bishop’s).

Le reste des exposés a montré la variété des sujets d’intérêtactuel en algèbre non commutative. La théorie d’Auslander-Reiten, abordée par D. Happel dans son mini-cours, a été l’objet

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A Celebration of Raoul Bott’s Legacy in Mathematicsby Robert Kotiuga (Boston University)

This five day conference was held and the CRM on June 9 – 13, 2008. Montreal was a natural venue for such an event sinceRaoul Bott obtained two degrees in Electrical Engineering at McGill University in the 1940s and an Honorary Doctoratefrom McGill in 1987. The conference was cosponsored by the Clay Institure, and partially funded by the NSF. The scientificcommittee consisted of Sir Michael Atiyah (Edinburgh), Octavian Cornea (Montréal) David Ellwood (Clay Institute), JacquesHurtubise (McGill), Francois Lalonde (Montréal), David Mumford (Brown), Graeme Segal (Oxford), Stephen Smale (TTIChicago, UCB Emeritus) Jim Stasheff (Pennsylvania, UNC Emeritus) and Edward Witten (IAS, Princeton). The conferencewas organised by Robert Kotiuga (Boston University) who is reflecting in this text on Bott’s Legacy and on the conference.

The conference focused on the mathematical legacy of RaoulBott and the extraordinary impact he had on both topology andinteractions between mathematics, physics and technology.The forward-looking event brought together some prominentmathematicians with close connections to him, in order to dis-cuss the emerging fields of mathematics and applications. Thefollowing focus topics emerged: Operator algebraic methods,K-theory, Supersymmetry and Quantum Field Theory, Morsetheory, string topology, pseudoholomorphic curves, Floer ho-mology, Equivariant cohomology and localization, General-ized cohomology theories and physics, Networks and ModularForms.

Background

A 1942 picture at the South-facing lookout at the chalet ofMont-Royal.

Raoul Bott was a “citizen of theworld” in that he was born in Bu-dapest Hungary in 1923, movedto Slovakia soon after, and whenhe was 16, spent a year in be-ing schooled in England. Hethen lived in Canada for sixyears where he obtained two de-grees in Electrical Engineeringfrom McGill University, duringwhich time he resolved to be-come a mathematician. Bott fin-ished a ScD in applied mathe-matics under Richard Duffin at

CMU (which at that time was Carnegie-Tech), with a thesisthat solved the most challenging synthesis problems in elec-trical network theory at that time. A humbling encounter withHerman Weyl landed him an invitation to the IAS at Prince-ton where, after a few relatively intense transitional years,the application of Morse theory to the problem of computinggeodesics on Lie Groups, and the Bott periodicity theorem forthe stable homotopy groups of the classical Lie groups, helpedmetamorphose him into one of the most influential topolo-gists of the 20th century. In a period of about a decade, hewas awarded tenure at the University of Michigan and thenat Harvard, and among his graduate students, two (DanielQuillen and Stephen Smale) would eventually be awardedFields medals for their work in topology. After four decades

at Harvard and many more students, he retired to San Diegowhere he succumbed to lung cancer late in 2005.

Raoul Bott with Principal Johnson at Place des Arts, receiving his honorarydegree from McGill in 1987.

Raoul Bott’s collected papers testify to his contributions tomathematics. The conference strived to feature prominent re-searchers who saw further by standing on Bott’s shoulders. Bylooking forward, the conference was not a systematic attemptto summarize the mathematics found in the fourvolumes of hiscollected works, published over a decade ago. Rather, we hada view of how the mathematics Bott mastered is manifested incurrent mathematical research and in emerging applications tomathematical physics.

The Main Activities of the Conference

We were fortunate to have recruited influential speakers andpanelists from three generations in order to cover six decadesof Raoul Bott’s research and collectively identify his endur-ing mathematical legacy. The scientific activities of the confer-ence consisted of two panel sessions, and of twenty-four hoursof lectures given by Sir Michael Atiyah (Edinburgh), PaulBaum (Pennsylvania State), James Bernhard (Puget), Octa-vian Cornea (Montréal), R. Cohen (Stanford), Marco Gualtieri(MIT), James Heitsch (UIC Emitus; NWU), Nancy Hingston(The College of New Jersey), Morris W. Hirsch (Berkeley),John Hubbard (Cornell), Lisa Jeffrey (Toronto), Nitya Kitchloo

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(UC San Diego), Joseph Kohn (Princeton), P. Robert Kotiuga(Boston University), Jim Lambek (McGill), Peter D. Lax (NewYork University), John Morgan (Columbia), Stephen Smale(Toyota Technological Institute at Chicago), Andras Szenes(Geneva), Constantin Teleman (Edinburgh) Susan Tolman(Urbana-Champaign), Loring Tu (Tufts), Cumrun Vafa (Har-vard), Jonathan Weitsman (UC Santa Cruz), Edward Witten,(Institute for Advanced Study).

This provided a unique experience for all involved, and dis-semination of the presentations of this conference should proveto be useful to a wide audience. An extended conference pro-ceedings is being prepared, and is expected to appear in 2009in the series “CRM Proceedings and Lecture Notes” publishedby the AMS. The volume will be attributed to both the CRMand the CMI.

The masterful presentations of Michael Atiyah and many othersenoir colleaues could have been anticipated and even ex-pected. The speakers who talked about localization and sin-gularity theory clearly built on the last two decades of Bott’sresearch. The talks by C. Vafa and E. Witten dwelled on astream of dualities which quantum field theory has been of-fering mathematics in recent decades, and the tantalizing newconnections to number theory. Many other talks rounded outthe conference in other ways. However, there were severalunexpected developments where a “big picture” seemed toevolve magically out of smaller parts, and it is useful to focuson one which was not obvious before the conference. Looselyspeaking, it pertains to Chas – Sullivan “string topology” andits relation to Floer homology via Morse theory. Here, on onehand, the work of N. Hingston and M. Goresky recast stringtopology in terms Morse theory as applied to loop spaces byBott in the 1950s. On the other hand, the works of N. Kitchlooand R. Cohen build on Morse theory in the context of “quan-tum topology,” and refine the use of Morse theory in low-dimensional topology. In the talks given by these speakers, aswell as those of O. Cornea and C. Teleman, one could sensewhere manifold topology was headed in the next few years,and while many of the new results would be new to Bott, theconnection to his mathematical perspective and legacy is in-escapable!

Panel sessions were held on the first and last days. The Mon-day panel session, “Raoul Bott as teacher, mentor, and col-league,” was chaired by Nancy Hingston, and the panelistswere: Paul Baum, Jim Stasheff and Loring Tu. Many wonder-ful anecdotes were shared. The Monday panel session featuredthe film, “A Peek Into the Book,” by Vanessa Scott. The movieis about her grandfather, Raoul Bott, and Vanessa’s accompa-nying remarks were read by Candace Bott. The movie was ahit, it brought back many memories, and it was rerun at theend of the conference. Other memorable anecdotes about Bottwere also related at the banquet in old Montreal, in speechesgiven by Michael Atiyah, Stephen Smale, and Candace Bott.

The Friday panel session “Examining Raoul Bott’s Legacyin Mathematics” was chaired by Michael Atiyah, and the

panelists were Nancy Hingston, Jacques Hurtubise, NityaKitchloo, and Susan Tolman. Unfortunately, it would not dojustice to the many threads of the discussion if they were sum-marized in the space available. However, one aspect can easilybe related.

In addition to being a profound and influential researcher, it iswell known that Bott was a wonderful lecturer. Although thishas been documented in many places, the conference has pro-duced some unexpected posthumous testimony! At the endof the Friday panel session, the conference organizer empha-sized that the conference was not organized around a math-ematical topic but a mathematician, and asked the youngerattendees what they thought of the concept. A student whoidentified himself as a graduate student working in an unre-lated field made what was considered a remarkable comment:He said he learnt more at this conference than at mathemat-ics conferences in his area of expertise since speakers at thisconference seemed to make an extraordinary effort to commu-nicate their ideas in the simplest and most visual terms possi-ble. What was more remarkable was that the “instant consen-sus” in the room was that this was a manifestation of all thespeakers being influenced by Bott’s lecturing style and his in-sistence on understanding deep mathematical concepts in thesimplest terms possible. Hopefully this ease of communicationwill be reflected in the extended conference proceedings whichare currently in the works.

Colloque d’algèbre non commutative(suite de la page 13)

des exposés de C. Chaio (Université Nationale de Mar delPlata), F. Coelho (Université de São Paulo), S. Liu (Universitéde Sherbrooke) et S. Trepode (Université Nationale de Mar delPlata). L’étude des algèbres de Koszul et de Calabi-Yau a oc-cupé les conférences de R. Berger (Université de Saint-Étienne)et de A. Martsinkovsky (Université Northeastern). Des progrèsintéressants sur la fameuse conjecture de la dimension fini-tiste due à Bass ont été exposés par O. Mendoza (UNAM) etM. Lanzilotta (Université de la République, Montevideo). Lastructure multiplicative des anneaux de cohomologie de Hoch-schild a été abordée par M. Suárez-Alvarez (Université de Bue-nos Aires) qui a annoncé la preuve d’une conjecture de Busta-mante, et par S. Sanchez-Flores (Université Montpellier 2), quis’est plutôt intéressée à la structure d’algèbre de Lie de ces an-neaux. Des algèbres de Lie apparaissant en théorie de représen-tations des algèbres ont aussi été l’objet de l’exposé de J. Ko-sakowska (Université Nicolas Copernic, Torun). Si on ajoutel’étude des t-structures dans des catégories dérivées (M. Saorín,Université de Murcie), celle des groupes de Coxeter au moyende modules postprojectifs (M. Kleiner, Université de Syracuse),celle des groupes d’Artin de type fini au moyen de suites ex-ceptionnelles (H. Thomas, Université du Nouveau-Brunswick)et des dégénérations dans les catégories de modules (P. Malicki,Université Nicolas Copernic), on aura un panorama complet dela rencontre.

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Problèmes mathématiques en imagerie:du neurone au monde quantique

de Jean-Marc Lina (École de technologie supérieure) et André D. Bandrauk (Université de Sherbrooke)

Le CRM sera l’hôte en 2009 de trois ateliers en neuro-imagerieauxquels s’ajoutera un quatrième dans un nouveau domainede recherche, l’imagerie dynamique quantique. La modélisa-tion mathématique, le traitement du signal et des images sontdes outils indispensables aux quatre ateliers. Les trois pre-miers seront axés sur l’imagerie de l’activité cérébrale où l’ana-lyse statistique est un élément important. L’imagerie des phé-nomènes quantiques traitera des problèmes pour représenteren image le mouvement des particules quantiques, en parti-culier, l’électron aux dimensions moléculaires, moins que lenanomètre (10−9 m) et l’échelle temporelle de l’attoseconde(10−18 s). Les quatre ateliers feront le point sur les mathéma-tiques des modèles dynamiques intrinsèques aux deux disci-plines : la neuro-imagerie et l’imagerie quantique.

Atelier 1 – La modélisation de l’activité neuronale : des pe-tites vers les grandes échelles (17 au 22 août 2009) Organisa-teurs : F. Lesage, Polytechnique Montréal et CRM ; T. Huppert,Pittsburgh ; H. Benali, Paris 6 et CRM

L’interprétation des signaux en imagerie fonctionnelle céré-brale (IRMf, spectroscopie RMN, MEG/EEG, imagerie optiquediffuse) ne peut faire l’économie d’une modélisation mathé-matique des processus physiologiques cérébraux. Ces modèlesdu tissu cérébral relient les différents compartiments de l’élec-trophysiologie au métabolisme énergétique et à l’hémodyna-mique. Ils se déclinent sous forme de systèmes d’équationsdifférentielles dont la complexité ne réside pas tant dans leurnombre que leur couplage et leur non-linéarité inhérente. L’ate-lier fera le point sur ces modèles dynamiques en les confrontantavec les différentes modalités de mesure de l’activité cérébrale.

Atelier 2 – La résolution de problèmes inverses : approchesstatistiques, variationnelles et bayésiennes au service dela fusion multimodale (24 au 29 août 2009) Organisateurs :C. Grova, McGill ; S. Baillet, Paris 6 ; J.-M. Lina, ÉTS et CRM

Dans une représentation distribuée de l’activité cérébrale, ladétermination robuste des régions actives et la description de ladynamique de cette activité demeurent un sujet ouvert. L’IRMfest la principale technique d’imagerie qui permet une détectionstatistique de l’activité cérébrale dans tout le volume cérébral,et par le biais de phénomènes hémodynamique avec une réso-lution temporelle de l’ordre de la seconde. D’autres techniquesd’exploration non invasives permettent d’accéder à une bienmeilleure résolution temporelle pour suivre l’activité cérébrale,mais ce au prix de signaux mesurés sur le scalp dont la locali-sation dans le cerveau nécessite la résolution d’un problèmeinverse.

Atelier 3 – Les « décompositions atomiques » en imagerie cé-rébrale : reconstructions tomographiques et analyse des si-

gnaux fonctionnels (14 au 19 septembre 2009) Organisateurs :J.-M. Lina, ÉTS et CRM ; S. Jaffard, Paris 12 ; F. Lesage, Poly-technique Montréal et CRM

Bien que l’analyse multi-échelle en imagerie cérébrale ait étémise de l’avant récemment, elle est implicitement présentedans les travaux précurseurs de Worsley et al. qui sont évoquésdans la session précédente. Dans la session présente, on consi-dère la « décomposition atomique » des signaux en termes defonctions élémentaires (harmoniques, ondelettes, curvelets,. . .)dans un perspective d’analyse des signaux fonctionnels.

Deux problématiques intéressantes : d’une part, l’analyse mi-crolocale des signaux physiologiques et bioélectriques donne-t-elle une signature de certains processus mesurés en IRM, enEEG, en MEG ou en Imagerie optique ? L’estimation par onde-lettes est-elle efficace voire utile ?

La seconde problématique porte sur la « représentation sparse »des signaux a l’aide dictionnaires redondants. Dans cette pro-blématique propre à la théorie de l’approximation, le compres-sive sensing qui permet de relier la question de mesure avecla notion d’information (sujette a la compression) est certaine-ment un secteur des mathématiques appliquées en continuitéavec l’héritage des ondelettes. Son application initiale pourl’acquisition rapide et efficace en neuro-imagerie par IRM anti-cipe certainement des développements et des applications im-portantes.

Pendant cet atelier, Stéphane Mallat (Polytechnique Paris etCourant Institute, NYU) sera le conférencier de la chaire Ai-senstadt.

Atelier 4 – Imagerie dynamique quantique (19 au 23 octobre2009) Organisateurs : A. D. Bandrauk (Sherbrooke) ; M. Y. Iva-nov (CNRC, Ottawa et Imperial College, UK)

L’utilisation de la lumière ou des photons pour représenter enimage, contrôler et transmettre de l’information moléculaire estun des domaines de recherche les plus significatifs à émergerdans les dernières années. Les spécialités les plus actives de cedomaine incluent l’imagerie temporelle de phénomènes quan-tiques, la dynamique moléculaire dans l’échelle de temps desfemtosecondes (10−15) pour le mouvement atomique, ou dansl’échelle de temps des attosecondes (10−18) pour le mouvementdes électrons. L’étude des phénomènes dans l’échelle des atto-secondes est maintenant reconnue internationalement commel’une des innovations scientifiques les plus importantes du 21e

siècle. Un aspect important du dévelopement de l’imagerieultrarapide temporelle en femto et attosecondes est la théo-rie et la simulation numérique des réponses non linéaires etnon perturbatives des atomes et molécules a des pulsions laser

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Last Publications

Data Mining and Mathematical ProgrammingPanos M. Pardalos and Pierre Hansen, editors

CRM Proceedings & Lecture Notes, volume 45

Data mining aims at finding interesting,useful or profitable information in verylarge databases. The enormous increasein the size of available scientific and com-mercial databases (data avalanche) as wellas the continuing and exponential growthin performance of present day comput-ers make data mining a very active field.In many cases, the burgeoning volume ofdata sets has grown so large that it threat-

ens to overwhelm rather than enlighten scientists. There-fore, traditional methods are revised and streamlined, com-plemented by many new methods to address challenging newproblems. Mathematical Programming plays a key role in thisendeavor. It helps us to formulate precise objectives (e.g., aclustering criterion or a measure of discrimination) as well asthe constraints imposed on the solution (e.g., find a partition,a covering or a hierarchy in clustering). It also provides pow-erful mathematical tools to build highly performing exact orapproximate algorithms.

This book is based on lectures presented at the workshop on“Data Mining and Mathematical Programming” (October 10 –13, 2006, Montréal) and will be a valuable scientific source ofinformation to faculty, students, and researchers in optimiza-tion, data analysis and data mining, as well as people workingin computer science, engineering and applied mathematics.

Contents: E. Carrizosa, Support vector machines and distanceminimization; H. Chen and J. Peng, 0 – 1 semidefinite pro-gramming for graph-cut clustering: modelling and approxi-mation; Z. Csizmadia, P.L. Hammer, and B. Vizvári, Artificialattributes in analyzing biomedical databases; Y.-J. Fan, C. Iyi-gun, and W.A. Chaovalitwongse, Recent advances in math-ematical programming for classification and cluster analysis;P.G. Georgiev Nonlinear skeletons of data sets and applica-tions — methods based on subspace clustering; M.R. Guarra-cino, S. Cuciniello, D. Feminiano, G. Toraldo, and P.M. Parda-los, Current classification algorithms for biomedical applica-tions; G. Kunapuli, K.P. Bennett, J. Hu, and J.-S. Pang, Bilevelmodel selection for support vector machines; V. Makarenkov,A. Boc, A.B. Diallo, and A.B. Diallo Algorithms for detect-ing complete and partial horizontal gene transfers: theory andpractice; O.L. Mangasarian and E.W. Wild, Nonlinear knowl-edge in kernel machines; F. Murtagh, Ultrametric embedding:application to data fingerprinting and to fast data clustering;O. Seref, O.E. Kundakcioglu, and P.M. Pardalos, Selective lin-ear and nonlinear classification.

Anatomy of IntegersJean-Marie De Koninck, Andrew Granville, and FlorianLuca, editorsCRM Proceedings & Lecture Notes, volume 46

The book is mostly devoted to the study ofthe prime factors of integers, their size andtheir quantity, to good bounds on the num-ber of integers with different properties (forexample, those with only large prime fac-tors) and to the distribution of divisors ofintegers in a given interval. In particular,various estimates concerning smooth num-bers are developed. A large emphasis is puton the study of additive and multiplicative

functions as well as various arithmetic functions such as thepartition function. More specific topics include the Erdos –Kac theorem, cyclotomic polynomials, combinatorial meth-ods, quadratic forms, zeta functions, Dirichlet series and L-functions. All these create an intimate understanding of theproperties of integers and lead to fascinating and unexpectedconsequences. The volume includes contributions from lead-ing participants in this active area of research, such as KevinFord, Carl Pomerance, K. Soundararajan and Gérald Tenen-baum.

Contents: V. Blomer, Ternary quadratic forms, and sums ofthree squares with restricted variables; R. de la Bretèche, En-tiers ayant exactement r diviseurs dans un intervalle donné;P. Erdos, F. Luca, and C. Pomerance, On the proportion ofnumbers coprime to a given integer; K. Ford, Integers witha divisor in (y, 2y]; H.A. Helfgott; Power-free values, repul-sion between points, differing beliefs and the existence of error;H. Maier, Anatomy of integers and cyclotomic polynomials; J.-L. Nicolas, Parité des valeurs de la fonction de partition p(n)et anatomie des entiers; K. Soundararajan, The distributionof smooth numbers in arithmetic progressions; G. Tenenbaumand J. Wu, Moyennes de certaines fonctions multiplicatives surles entiers friables, 4; A. Akbary and M.R. Murty, Uniformdistribution of zeros of Dirichlet series; S. Baier and L. Zhao,On primes represented by quadratic polynomials; W.D. Banks,A.M. Güloglu, C.W. Nevans, and F. Saidak, Descartes num-bers; E. Croot, A combinatorial method for developing Lucassequence identities; J.-M. De Koninck and F. Luca, On thedifference of arithmetic functions at consecutive arguments;A. Granville and K. Soundararajan, Pretentious multiplicativefunctions and an inequality for the zeta-function; R. Khan, Onthe distribution of ω(n); W. Kuo and Y.-R. Liu, The Erdos – Kactheorem and its generalizations; J. Liu, E. Royer, and J. Wu,On a conjecture of Montgomery – Vaughan on extreme valuesof automorphic L-functions at 1; N. Ng, The Möbius function

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in short intervals; P. Pollack, An explicit approach to hypoth-esis H for polynomials over a finite field; C.L. Stewart, Onprime factors of integers which are sums or shifted products;E.B. Wong, Simultaneous approximation of reals by values ofarithmetic functions.

Biology and Mechanics of Blood FlowsMarc Thiriet

CRM Series in Mathematical Physics

Biology and Mechanics of Blood Flows presentsthe basic knowledge and state-of-the-arttechniques necessary to carry out investi-gations of the cardiovascular system usingmodeling and simulation. Part I of thistwo-volume sequence, Biology, addressesthe nanoscopic and microscopic scales. Thenanoscale corresponds to the scale of bio-chemical reaction cascades involved in celladaptation to mechanical stresses among

other stimuli. The microscale is the scale of stress-inducedtissue remodeling associated with acute or chronic loadings.The cardiovascular system, like any physiological system, has acomplicated three-dimensional structure and composition. Itstime dependent behavior is regulated, and this complex sys-tem has many components. In this authoritative work, the au-thor provides a survey of relevant cell components and pro-cesses, with detailed coverage of the electrical and mechanicalbehaviors of vascular cells, tissues, and organs. Because the be-haviors of vascular cells and tissues are tightly coupled to themechanics of flowing blood, the major features of blood flowsand the Navier – Stokes equations of mass and momentum con-servation are introduced at the conclusion of this volume. Thisbook will appeal to any biologist, chemist, physicist, or appliedmathematician with an interest in the functioning of the cardio-vascular system.

Contents: Volume I. Biology: Introduction; 1. The cell;2. Plasma membrane and cell environment; 3. Ion carries andreceptors; 4. Signaling pathways; 5. Transport and cell mo-tion; 6. Blood; 7. Heart wall; 8. Vessel wall, 9. Endothelium;10. Tissue growth, repair and remodeling; Concluding re-marks; A. Endocrine system and hormones; B. Coagulation fac-tors; C. Basic mechanics; D. Numerical simulations.

Volume II. Mechanics and medical aspects: Introduction;1. Anatomy of the cardiovascular system; 2. Cardiovascu-lar physiology; 3. Images, signals and movements; 4. Rhe-ology. 5. Hemodynamics; 6. Numerical simulations; 7. Car-diovascular diseases; 8. Treatments of cardiovascular diseases;Conclusion; A. Anatomical, biological, medical glossaries;B. Adipocyte; C. Basic aspects in mechanics; D. Numerical sim-ulations.

Combinatorics on WordsJean Berstel, Aaron Lauve, Christophe Reutenauer, andFranco V. SaliolaCRM Monograph Series, volume 27

The two parts of this text are based on twoseries of lectures delivered by Jean Bersteland Christophe Reutenauer in March 2007at the Centre de recherches mathématiques,Montréal, Canada. Part I represents thefirst modern and comprehensive expositionof the theory of Christoffel words. Part IIpresents numerous combinatorial and al-gorithmic aspects of repetition-free wordsstemming from the work of Axel Thue — a

pioneer in the theory of combinatorics on words.

A beginner to the theory of combinatorics on words will bemotivated by the numerous examples, and the large variety ofexercises, which make the book unique at this level of exposi-tion. The clean and streamlined exposition and the extensivebibliography will also be appreciated. After reading this book,beginners should be ready to read modern research papers inthis rapidly growing field and contribute their own research toits development.

Experienced readers will be interested in the finitary approachto Sturmian words that Christoffel words offer, as well as thenovel geometric and algebraic approach chosen for their expo-sition. They will also appreciate the historical presentation ofthe Thue – Morse word and its applications, and the novel re-sults on Abelian repetition-free words.

Contents: Part I. Christoffel words: 1. Christoffel words;2. Christoffel morphisms; 3. Standard factorization; 4. Palin-dromization; 5. Primitive elements in the free group F2; 6. Char-acterizations; 7. Continued fractions; 8. The theory of Markoffnumbers. Part II. Repetitions in words: 1. The Thue – Morseword; 2. Combinatorics of the Thue – Morse word; 3. Square-free words; 4. Squares in words; 5. Repetitions and patterns.

Problèmes mathématiques en imagerie(suite de la page 16)

ultracourtes. L’imagerie dynamique quantique (IDQ) est doncune nouvelle frontière scientifique requérant de nouvellesavancées mathématiques pour analyser et résoudre certainesEDP multidimentionelles spatiales et temporelles comme leséquations de Schrödinger et de Dirac dépendantes du temps etles équations de Maxwell pour les photons dans des effets depropagation collectifs. L’inversion des données expérimentalesd’IDQ présente donc de nouveaux défis mathématiques, et lesimplications pour la recherche et le dévelopement se ferontsentir dans des domaines aussi divers que l’optoélectronique,les sciences des matériaux et même l’information quantique,étant donné la propriété de l’IDQ d’être un outil essentiel decontrôle de la matière au niveau moléculaire.

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• crm.math.ca •

Stagiaires postdoctoraux CRM-ISM 2008-2009

Renouvellements

Bryden Cais (University of Michigan, juin 2007) sous la supervision de Adrian Iovita(Concordia) et Henri Darmon (McGill).

Seung-Yeop Lee (University of Chicago, juin 2007) sous la supervision de Marco Ber-tola et John Harnad (Concordia)

Jeehon Park (Boston University, mai 2007) sous la supervision de Henri Darmon(McGill)

Igor Wigman (Université de Tel Aviv, mai 2006) sous la supervision de Dima Jacob-son (McGill), Andrew Granville et Iosif Polterevich (Montréal)

Nouveaux

Nadine Badr (Université Paris-Sud 11, décembre 2007) sous la supervision de GaliaDafni (Concordia)

Norman Do (University of Melbourne, avril 2008) sous la supervision de JacquesHurtubise (McGill)

Xander Faber (Columbia University, 2008) sous la supervision de Henri Darmon(McGill) et Andrew Granville (Montréal)

Benjamin Young (University of British Columbia, mai 2006) sous la supervision deJacques Hurtubise (McGill)

Associés à l’année thématique 2008-2009

Robert Buckingham (Duke University, 2005) sous la supervision de Marco Bertola,John Harnad et Dimitri Korotkin (Concordia)

Dong Wang (Brandeis University, 2008) sous la supervision de Marco Bertola, JohnHarnad et Dimitri Korotkin (Concordia)

Mathematical Aspects of Quantum Chaos(continued from page 7)

negative curvature necessarily have a positive entropy (hence cannot be completelylocalized around a single orbit). B. Gutkin (Essen) and S. Brooks (Princeton) man-aged to strengthen the constraints on the entropy of semiclassical measures for cer-tain families of quantum maps, and show that the obtained bounds are sharp.

Beyond the macroscopic description of the eigenfunctions embodied in Q(U)E theo-rems, one may focus on the microscopic fluctuations; on such a scale, the eigenfunc-tions of a quantized chaotic systems are believed to resemble random superpositionsof plane waves. One way to measure this “randomness” is to study properties of thenodal sets of the eigenfunctions (that is, the set of points where the functions vanish).The most precise results on this matter require real-analyticity of the system. Oneis then able to locally represent eigenfunctions by polynomials, and study the nodalsets of the latter. While D. Mangoubi (MPI Bonn) studied the volume of thin “tubes”surrounding the nodal sets on real-analytic manifolds, J. Toth (McGill) counted theopen nodal lines for the Dirichlet or Neumann eigenfunctions on real-analytic do-mains of the plane: their number grows like the square-root of the (very general)Courant bound on the number of nodal domains. For the quasi-1-dimensionalcase of a quantum graph, G. Berkolaiko (Texas A&M) showed a much more pre-cise counting of the nodal points. Some specific manifolds (flat sphere or flat torus)enjoy large spectral degeneracies: it then makes sense to equip the eigenspaces witha certain probability measure, and study the properties of a “typical” or “random”eigenfunction, and compare them with standard models of random superpositionsof plane waves. Such a study for the 2-dimensional torus was presented by I. Wig-man (CRM): it involves subtle number-theoretic properties.

Le Bulletin du CRM

Volume 14, No 2Automne 2008

Le Bulletin du CRM est une lettre d’in-formation à contenu scientifique, faisantle point sur les actualités du Centre derecherches mathématiques.

ISSN 1492-7659Le Centre de recherches mathématiques(CRM) a vu le jour en 1969. Présentementdirigé par Mme Christiane Rousseau, il apour objectif de servir de centre nationalpour la recherche fondamentale en ma-thématiques et dans leurs applications.Le personnel scientifique du CRM re-groupe plus d’une centaine de membresréguliers et de boursiers postdoctoraux.De plus, le CRM accueille chaque annéeentre mille et mille cinq cents checheursdu monde entier.Le CRM coordonne des cours graduéset joue un rôle prépondérant en colla-boration avec l’ISM dans la formationde jeunes chercheurs. On retrouve par-tout dans le monde de nombreux cher-cheurs ayant eu l’occasion de perfection-ner leur formation en recherche au CRM.Le Centre est un lieu privilégié de ren-contres où tous les membres bénéficientde nombreux échanges et collaborationsscientifiques.

Le CRM tient à remercier ses divers par-tenaires pour leur appui financier dansnotre mission : le Conseil de recherchesen sciences naturelles et en génie duCanada, le Fonds québécois de la re-cherche sur la nature et les technolo-gies, la National Science Foundation, leClay Mathematics Institute, l’Universitéde Montréal, l’Université du Québec àMontréal, l’Université McGill, l’Univer-sité Concordia, l’Université Laval, l’Uni-versité d’Ottawa, l’Université de Sher-brooke, ainsi que les fonds de dotationAndré-Aisenstadt et Serge-Bissonnette.

Directrice : Christiane Rousseau

Directeur d’édition : Chantal DavidConception et infographie : André Montpetit

Centre de recherches mathématiquesPavillon André-AisenstadtUniversité de MontréalC.P. 6128, succ. Centre-VilleMontréal, QC H3C 3J7Téléphone : 514.343.7501Télecopieur : 514.343.2254Courriel : CRM@CRM.UMontreal.CA

Le Bulletin est disponible aucrm.math.ca/docs/docBul_fr.shtml

BULLETIN CRM–19

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Mot de la directrice Word of the DirectorC’est presqu’au moment de mettre sous presse que j’ai acceptéde prendre la direction intérimaire du CRM jusqu’au 31 mai2009 et ce mot de la directrice sera donc bref. Le CRM est unfleuron de la communauté mathématique canadienne. En té-moignent les grandes années thématiques, mais aussi la parti-cipation aux grands événements nationaux. Il insuffle un dy-namisme à la communauté mathématique québécoise, entreautres par le biais des laboratoires, des ateliers de résolutionde problèmes industriels et des activités pour le public.

Ce succès est attribuable en grande partie au travail de titan desdirecteurs précédents et je profite de ce message pour remer-cier, au nom de la communauté du CRM, François Lalonde dutravail gigantesque qu’il a accompli pendant ses quatre annéesà la direction du CRM. Je remercie également Anne Bourliouxqui a accepté la direction intérimaire pendant l’été 2008. Annea accepté de prolonger son mandat au-delà du 31 août, malgréses nombreux autres engagements pour l’année en cours.

Au-delà du travail des directeurs et directriceprécédents, le succès du CRM est également dûau travail et au dévouement de tous les cher-cheurs, des directeurs et directrices adjoints, etdu personnel du CRM.

J’hérite du CRM en pleine santé. Pour la pre-mière fois depuis des années nous avons uneannée thématique plutôt que deux semestresdisjoints. Cette année thématique en cours estmagnifique. Le thème « Méthodes probabilistesen physique mathématique » rejoint de nom-breux chercheurs montréalais qui passent unepartie importante de leur temps au CRM. La programma-tion thématique comprend également un programme conjointavec le PIMS sur « Les défis et perspectives en probabilités ».Quatre titulaires de la chaire Aisenstadt ont visité ou visiterontle CRM en 2008-2009, soit Svante Janson, Andrei Okounkov,Craig Tracy et Wendelin Werner. Parallèlement, le CRM a tenudeux ateliers de résolution de problèmes industriels en 2007 et2008 et en prépare un pour 2009. Il a participé à l’école d’été« Systems Biology Dynamics : from Genes to Organisms » pen-dant l’été 2008.

Les discussions des derniers mois ont révélé le dynamismede la communauté des chercheurs du CRM et la programma-tion des futures années scientifiques bat son plein. Les projetspan-canadiens ne manquent pas alors que nous sortons d’uncongrès Canada-France qui nous a tous réunis et que la com-munauté mathématique canadienne prépare la candidature deMontréal pour le Congrès international des mathématiciens de2014. Je me réjouis de travailler avec la communauté du CRMet avec la communauté mathématique canadienne à maintenirla tradition d’excellence des activités du CRM et des mathéma-tiques canadiennes.

Christiane Rousseau

It is nearly at the time of sending the bulletin to press that I have ac-cepted to become Acting Director of CRM till May 31 2009. Hencethis “word of the Director” will be short. The CRM is a success storyin Canadian mathematics. This is acknowledged by its thematic yearsor semesters, and by its participation to large national events. It alsolies at the heart of the Québec mathematical community, in partic-ular through its laboratories, industrial problem solving workshopsand activities for the public.

This success is greatly due to the gigantic task accomplished by theprevious directors. I use this opportunity to thank François Lalonde,in the name of the CRM community, for the great work he did duringhis four years as Director of CRM. I am also very grateful to AnneBourlioux who agreed to be Acting Director during the summer andto extend her mandate past August 31, despite numerous other com-mitments.

Beyond the contribution of previous Directors, the CRM’s success isequally due to the work and commitment of all itsresearchers, associate Directors and staff members.

The CRM is in great shape. For the first time inseveral years, we are having a thematic year ratherthan two disjoint thematic semesters. The currentmathematical year is splendid. The theme “Proba-bilistic methods in mathematical physics” is close tothe interests of a number of members from Montrealwho spend a significant portion of their time at theCRM. There is also a joint program with PIMS on“Challenges and perspectives in probability.” FourAisenstadt Chairs lecturers have visited or will visit

the CRM in 2008 – 2009, namely Svante Janson, Andrei Okounkov,Craig Tracy and Wendelin Werner. In parallel, the CRM held two In-dustrial problem solving workshops in 2007 and 2008, and is prepar-ing a third one for 2009. The CRM was a partner in the summerschool “Systems Biology Dynamics: from Genes to Organisms” dur-ing the summer 2008.

The discussions of the last months have revealed how dynamic thecommunity of CRM members is and the planning of future scien-tific thematic years is presently very active. Pan-Canadian venturesare numerous: the Canadian mathematical community just ended theCanada – France congress and is now preparing the Canadian bid tohold ICM 2014 in Montréal.

I am looking forward to work with both the CRM and the Canadianmathematical community to maintain the tradition of excellence inthe CRM activities and Canadian mathematics.

Christiane Rousseau

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