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Le Bulletin du CRM• www.crm.umontreal.ca •

Printemps/Spring 2007 | Volume 13 – No 1 | Le Centre de recherches mathématiques

Fall Thematic Program Semester:

Applied Dynamical Systemsby Tony Humphries (McGill University)

The Fall Semester of 2007 at the CRM will bedevoted to Applied Dynamical Systems. Thesemester will consist of 6 main workshops, with5 taking place at the CRM, and a joint AARMS –CRM workshop in Halifax. Graduate studentand early career scientist participation is en-couraged. Two of the Montréal workshops will

be preceded by graduate level minicourses, and there will alsobe two semester long advanced graduate courses as part of theISM applied mathematics programme.

Dynamical systems theory describesqualitative and quantitative featuresof solutions of systems of nonlineardifferential equations, and has di-verse roots in applications such asPoincaré’s work on planetary mo-tion in the 19th century, the Fermi –Pasta – Ulam simulations of anhar-monic lattices in the 1950s, andthe famous Lorenz equations. Al-though a deep and beautiful theoryof nonlinear dynamical systems hasresulted, dynamical systems whicharise in applications often fall out-side the scope of much of this the-ory, either because they contain gen-eralisations to the dynamical sys-tems paradigm, such as variableor state dependent delays or evenadvanced arguments or noise, orbecause assumptions or conditionswithin the theory do not apply.

Despite the difficulties, significantprogress has been made in recentyears in applying dynamical systems particularly in the areasof mathematical biology and physiology. This has led to an in-creasing interest in new problems such as those with noncon-stant and distributed delays, which has given rise to new ana-

lytical and numerical challenges. Thus, the semester will focuson two themes. Firstly, the use of dynamical systems in applica-tions, principally in physiology, and secondly the developmentof new numerical and dynamical systems tools needed in thestudy of such problems. However, in reality applications, anal-ysis and numerical methods are all interconnected and someaspects of all three will be found in each of the workshops.

The semester will begin with a workshop on Advanced Al-gorithms and Numerical Software for the Bifurcation Analysis ofDynamical Systems (2 – 7 July 2007) organised by E. Doedel

(Concordia) and H. Osinga (Bristol),with a preceding minicourse on 30June – 1 July 2007. This workshopwill address the numerical analysisof discrete and continuous dynam-ical systems that model importantphysical phenomena. Specific top-ics will include algorithms and soft-ware for the computation and vi-sualization of bifurcations, invariantmanifolds, and traveling wave phe-nomena, in nonlinear ordinary dif-ferential equations, delay and func-tional differential equations, and incertain classes of partial differen-tial equation, especially nonlinearparabolic systems. This workshopcan be viewed as being part of a se-ries of workshops that was startedby Yuri Kuznetsov at the CWI inAmsterdam in the early 1990s, andwhich has since been held regularlyin Amsterdam, Utrecht, Bielefeld,Gent, Sevilla, and Bristol. This willbe the first workshop in the series to

be held outside Europe. The series has been influential in thedevelopment of algorithms and software for bifurcation prob-lems. It has also led to many new joint research projects related

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•www.crm.umontreal.ca •

The Workshops of the Combinatorial OptimizationTheme Semester (June – December 2006)

by Odile Marcotte (UQÀM and GERAD)

The theme semester included five workshops whose variedtopics reflected the wide scope of combinatorial optimization,a relatively recent field of mathematics. The first workshop,on Approximation Algorithms, took place from June 12 to June14 and was attended by 70 participants. It was organized byMichel Goemans from MIT and Joseph Cheriyan from the Uni-versity of Waterloo. Many combinatorial optimization prob-lems, such as the famous Traveling Salesman Problem and var-ious network design problems, are NP-hard, which means thatit is unlikely that one will find a polynomial-time algorithm tosolve them. One must then look for efficient approximation al-gorithms, that is, polynomial-time algorithms providing solu-tions whose value is relatively close to the optimal value. Find-ing and proving an upper bound on the difference between theapproximate value and the optimal value is often a challenge,requiring the use of sophisticated techniques from combina-torics and mathematical programming.

One important problem for which approximation algorithmshave been proposed is the Steiner Tree Problem: given an undi-rected graph G, a weight for each edge of G and a subset R ofvertices of G, find a subtree of G that includes all the verticesin R and is of minimum total weight. Michel Goemans andDavid Williamson described an approximation algorithm forthe Steiner Tree Problem and other constrained forest problemsin a famous 1994 article, and out of the 26 lectures of the work-shop, 7 lectures dealt with versions of the Steiner Tree Prob-lem or other network design problems. The other lectures dealtwith approximation algorithms for problems in the followingareas: graph homomorphisms, welfare maximization, cost al-location, inventory control, submodular set functions, routing,cycle packing, lattices, machine scheduling, clustering, feed-back arc sets, and game theory. Generally speaking, the work-shop was very lively. Many of the foremost experts on approx-imation algorithms were present in the audience, the lectureswere very well prepared, and the participants did not hesitateto ask questions when they did not understand the speakers’assertions!

The workshop on Network Design: Optimization and AlgorithmicGame Theory took place from August 14 to August 16 and wasattended by 63 participants. It was organized by Shie Mannorand Adrian Vetta, both from McGill University, and brought to-gether academic and industrial researchers from mathematics,computer science, engineering and economics. The goal wasto discuss the fundamental mathematical issues affecting net-work design and applications. Highlights included: theoreti-cal studies on the evolution of networks such as the Internet;algorithmic approaches for efficient routing under restrictivenetwork protocols; mechanism designs and algorithms for on-

line auctions; a mathematical model explaining that dramaticprice change is an inherent characteristic of dynamic spot elec-tricity markets; investigations into game theoretic considera-tions such as multiple ownership, contracts, limited informa-tion, equilibria, etc.

Overall, the workshop was very successful in highlighting keyareas of research and encouraging cross-disciplinary work. Theworkshop stimulated many new open problems and was thecatalyst for new collaborations. Here is an example of the prob-lems that were discussed. Suppose that Internet bids, say, aremade over time for an object or objects. When the bid is made,the seller must make a decision (accept or reject or announcean alternative price). What is the best strategy for the seller? Anopen question in this area is the following: suppose the set ofbids that can be accepted has a specific structure, for instance,the accepted bids must form an independent set in a given ma-troid M (several types of matroids have nice applications, e.g.,to selling airline seats belonging to several classes). If the bidsarrive in a random order, what is the best online algorithm forthis problem? It is conjectured that a factor e competitive algo-rithm exists.

The workshop on Hybrid Methods and Branching Rules in Combi-natorial Optimization was organized by Vašek Chvátal, CanadaResearch Chair in Combinatorial Optimization at ConcordiaUniversity. It was held during the week of September 18 –22 and was attended by 67 participants. As mentioned be-fore, combinatorial optimization problems, especially thosethat must be solved in practice, tend to be “intractable.” As aresult, diverse heuristic and exact methods to solve them havebeen proposed by researchers belonging to several communi-ties. For instance, the problem Satisfiability or SAT (the archety-pal NP-complete problem) has been studied by researchers inmathematical programming and researchers in constraint sat-isfaction programming. The goal of the workshop was to bringthose two communities of researchers together, in order to fos-ter exchanges on branching rules (used in most algorithms tosolve intractable problems) and hybrid methods (i.e., meth-ods combining heuristic search and branch-and-bound algo-rithms).

The workshop featured 26 lectures, of which 8 dealt withSAT and related problems and algorithms (including resolu-tion search, proposed by Vašek Chvátal). The other lecturesdealt with heuristic search methods, mixed integer program-ming, probabilistic branching rules, and methods combiningconstraint programming and integer programming. Each dayof the workshop ended with a round table, and the workshop


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The André-Aisenstadt Chairs for theCombinatorial Optimization Theme Semester

by Odile Marcotte (UQÀM and GERAD)

The André-Aisenstadt chairs for the Combinatorial Optimization Theme Semester were Noga Alon from Tel Aviv Univer-sity and Paul Seymour from Princeton University. Noga Alon is one of the foremost experts in the world on extremal andprobabilistic combinatorics, and Paul Seymour one of the leading experts on structure in graph theory. In the text below wedescribe some of the questions and results they addressed in their talks.

Noga Alon

The first two lectures by Noga Alondealt with approximation resultsfor combinatorial optimization prob-lems. The topic of the first lecturewas the approximation of a quantityknown as the cut-norm of a matrix.Given a real matrix A = (aij)i∈R,j∈S,the cut-norm ‖A‖C of A is definedas the maximum absolute value of asubmatrix sum, i.e.,

maxI⊂R,J⊂S

∣∣∣∣ ∑i∈I,j∈J

aij

∣∣∣∣.Finding sets I and J for which the maximum is achieved is a dif-ficult combinatorial problem. Indeed, unless the most impor-tant question in complexity theory (“Is P equal to NP?”) has anaffirmative answer, it is impossible to compute efficiently setsI and J such that ∣∣∣∣ ∑

i∈I,j∈Jaij

∣∣∣∣ ≥ ρ‖A‖C

for any ρ greater than 16/17.

Thus one is led to consider a related optimization problem, for-mulated as the following integer program (called IP):

maximize ∑i∈R,j∈S

aijxiyj

subject to xi ∈ −1, 1, yj ∈ −1, 1 for all i, j.

We shall use v(IP) to denote the optimal value of IP. In or-der to solve a difficult integer program such as IP, one oftenstarts by solving a relaxation of this problem, i.e., a mathemat-ical program whose feasible set properly contains the feasibleset of IP. In the present case, the constraint “xi ∈ −1, 1” canbe replaced by the constraint “xi is an m-dimensional vectorof unit norm.” The same transformation is applied to the con-straint “yj ∈ −1, 1,” and to extend the definition of the objec-tive function, we replace the product xiyj by the scalar productxi · yj.

By using the characterization of positive semidefinite matrices,it can be shown that the resulting mathematical program,


aijxi · yj

subject to xi ∈ Sm, yj ∈ Sm for all i, j,

is actually equivalent to a semidefinite program called SDP:


aijzij

subject to Z ∈ M, zii = 1 for all i,

where Z denotes the matrix (zij)i∈R,j∈S and M is the set of allpositive semidefinite matrices. The optimal value of this pro-gram will be denoted by v(SDP).

The relaxation SDP is useful in two ways. First, a near-optimalsolution of SDP (i.e., a solution of value greater than v(SDP)−ε) can be computed in time that is polynomial in the inputlength and log(1/ε). Second, v(SDP) provides an upper boundfor v(IP) since SDP is a relaxation of IP and IP is a maxi-mization problem. In a beautiful article, Noga Alon and AssafNaor show that the matrix version of Grothendieck’s inequal-ity (which is a fundamental tool in functional analysis) impliesthat v(SDP)/v(IP) is bounded by a constant (Grothendieck’s con-stant), whose precise value is not known but is comprised be-tween π/

(2 ln(1 +

√2)

)and π/2. They also describe rounding

techniques that can be used to transform an optimal solution ofSDP into a feasible (integral) solution of IP whose value is nottoo far from v(SDP).

By using these techniques and the relationship between v(IP)and ‖A‖C (‖A‖C ≤ v(IP) ≤ 4‖A‖C), Alon and Naor de-signed an efficient deterministic algorithm for approximatingthe cut-norm, i.e., finding sets I and J such that |∑i∈I,j∈J aij| ≥ρ‖A‖C holds for some fixed ρ > 0. They also designed ef-ficient randomized algorithms producing such approximatesolutions, but with better approximation ratios. In a relatedarticle (Quadratic Forms on Graphs), Noga Alon, KonstantinMakarychev, Yury Makarychev and Assaf Naor generalize theprogram IP as follows: given an undirected graph G = (V, E),

maximize ∑u,v∈E

auvyuyv

subject to yu ∈ −1, 1 for all u ∈ V.

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They introduce a new parameter called the Grothendieck con-stant of the graph G, and give upper and lower estimates for theratio v(IP)/v(SDP).

In his second lecture, Noga Alon considered edge-deletion prob-lems. These are a special type of graph modification problems,which have applications in numerical linear algebra and thephysical mapping of DNA. Let G be an undirected graph, con-sisting of a finite set of vertices and a finite collection of pairsof vertices (called edges). A graph property is monotone if theremoval of a vertex or an edge from a graph G preserves theproperty. For instance, the property of being triangle-free ismonotone. Note that the obvious algorithm for verifying thatG is triangle-free or transforming G into a triangle-free graphtakes time proportional to n3, since it enumerates all subsetsof three vertices. Noga Alon, Asaf Shapira and Benny Sudakovhave given an O(n2) algorithm for the latter problem, not onlyfor the property of being triangle-free but for any monotoneproperty.

More precisely, for any graph G and any monotone propertyP , let E′P (G) denote the smallest number of edges that must bedeleted from G to transform it into a graph satisfying P . Alon,Shapira and Sudakov have shown that for any fixed ε > 0 andany monotone property P , there is an O(n2) deterministic al-gorithm that, given a graph G on n vertices, computes a realnumber r satisfying |r − E′P (G)/n2| ≤ ε. Such an algorithmwas not even known for the property of being triangle-free.Alon and his coauthors have also proved that if all bipartitegraphs satisfy P , then it is NP-hard to approximate E′P (G)/n2

to within an additive error of n−δ for any fixed δ > 0. Thisis also a startling result, since prior to their article it was noteven known that computing E′P (G) precisely was NP-hard,even when P is the property of being triangle-free!

The topic of Noga Alon’s third lecture was graph property test-ing, a topic related to the graph modification problems we havejust described. The study of combinatorial property testing wasfirst motivated by issues arising in program checking. A graphG with n vertices is said to be ε-far from satisfying the propertyP if the number of edge insertions or deletions needed to trans-form G into a graph satisfying P is at least εn2. A graph prop-erty P is testable if there exists a randomized algorithm makingat most q(ε) edge queries that can distinguish with high prob-ability between graphs satisfying P and graphs that are ε-farfrom satisfying P . Note that q(ε) does not depend upon n orthe size of the input. In his third lecture, Noga Alon surveyedthe main results on graph property testing, especially the re-cent ones, and outlined the relationship between these resultsand Szemerédi’s regularity lemma.

Biography

Noga Alon is a Baumritter Professor of Mathematics and Com-puter Science at Tel Aviv University in Israel. He received hisPh.D. in Mathematics at the Hebrew University of Jerusalem in1983 and held visiting positions in various research institutes

including MIT, The Institute for Advanced Study in Princeton,IBM Almaden Research Center, Bell Laboratories, Bellcore andMicrosoft Research. He serves on the editorial boards of morethan a dozen international technical journals and has giveninvited lectures at many conferences, including plenary ad-dresses at the 1996 European Congress of Mathematics and the2002 International Congress of Mathematicians, and an invitedlecture at the 1990 International Congress of Mathematicians.He has published one book and more than three hundred re-search articles, mostly in combinatorics and theoretical com-puter science. He has been a member of the Israel NationalAcademy of Sciences since 1997 and received the Erdos prizein 1989, the Feher prize in 1991, the George Pólya Prize in 2000,the Bruno Memorial Award in 2001, the Landau Prize in 2005and the Gödel Prize in 2005.

Paul Seymour

In his first lecture, geared towards abroad audience, Paul Seymour gave anoverview of some of the most importantstructure theorems in graph theory. Togive the flavour of this lecture, much ap-preciated by the audience, we first recallthe notions of graph and induced sub-graph. Let G = (V, E) be a simple graph,where V is a finite set of vertices and E aset of edges, i.e., pairs of vertices. A sub-

graph H of G is obtained from G by deleting edges or vertices(along with the edges incident to them); H is an induced sub-graph of G if it is of the form (U, F), where U is a subset of Vand F is the set of all edges with endpoints in U. Given twographs G and H, the phrase “G does not contain H” may haveseveral meanings. The first meaning, “G does not contain H asa subgraph,” has received comparatively little attention fromgraph theorists investigating the structure of graphs.

The second meaning, “G does not contain H as an induced sub-graph,” has received much more attention. For instance, Bergegraphs are the graphs that do not contain any odd hole or an-tihole (where a hole is a cycle with at least four vertices andan antihole is the complement of a hole). The Berge graphs arefamous because of a conjecture posed by Claude Berge in 1961,the Strong Perfect Graph Conjecture, which was one of the mostimportant problems in graph theory until its solution in 2002.This conjecture states that a graph G is Berge if and only if G isperfect, i.e., the chromatic number of any induced subgraph Hof G is equal to the clique number of H. In 2002, Paul Seymourand his collaborators (Maria Chudnovsky, Neil Robertson andRobin Thomas) announced a proof of the Strong Perfect GraphConjecture, an achievement that is comparable to the proof ofthe four-colour theorem!

The third meaning of the phrase “G does not contain H” isthat G does not contain H as a minor. We say that H is a minor


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Prix CRM-Fields-PIMS 2007 : Joel Feldmanpar David Brydges (University of British Columbia)

C’est au cours des années 70 qu’étudiant gradué,je rencontrai pour la première fois les travaux deJoel Feldman. À cette époque-là, le programme deconstruction de théories quantiques des champs(TQC) battait son plein. La construction de TQCen deux dimensions d’espace-temps avait été cou-ronnée de succès et même les redoutables diver-

gences de la TQC à trois dimensions subissaient les assautsd’une machinerie inventée par Glimm et Jaffe [14], et connuesous le nom de « développement en cellules dans l’espace desphases ». Joel Feldman faisait partie des rares personnes à com-prendre clairement les principes de cette technique et j’aimaispresque mieux lire ses articles que ceux des maîtres, parcequ’ils constituaient, alors comme maintenant, des modèles declarté. En 1976 Feldman et Osterwalder [13] réussirent fina-lement à prouver qu’une candidate au rang de TQC à troisdimensions, la théorie dite en φ4

3, était bel et bien une théoriequantique des champs, puisqu’elle obéissait aux axiomes deWightman. Quant à quatre dimensions d’espace-temps, mondirecteur de thèse me dit : « Il se pourrait bien que nous devionsattendre cent ans avant de comprendrele cas à quatre dimensions ». Depuis lors,j’ai suivi avec étonnement la façon dontJoel Feldman et ses collaborateurs ontgraduellement étendu la portée du déve-loppement en cellules dans l’espace desphases, au point où leurs travaux récentssur le problème de la surface de Fermi sur-montent un problème bien au-delà des dif-ficultés particulières qui rendaient mon di-recteur de thèse si pessimiste il y a 30 ans.Malheureusement, il pourrait avoir encoreraison en ce que l’existence d’une théorieen quatre dimensions demeure probléma-tique. Mais l’obstruction ne vient plus des« divergences aux courtes distances » quisemblaient si difficiles en 1976.

Pour situer Joel Feldman dans tout cela et donner une idée desréalisations que couronne le Prix CRM-Fields-PIMS, je devraid’abord brosser rapidement la toile de fond constituée par laTQC et le développement en cellules dans l’espace des phases.Quelqu’un d’autre pourrait dans cet article mettre l’accent surdes aspects différents de l’oeuvre de Feldman, comme son tra-vail sur les surfaces de Riemann de genre infini [1], mais c’està ses travaux sur la TQC et la matière condensée que je prendsle plus grand plaisir. À un autre égard au moins cette descrip-tion ne rend pas justice au sujet : le souci de brièveté ne permetpas de décrire convenablement les contributions de la physiquethéorique.

Les axiomes de la théorie quantique des champs (TQC) sontétonnamment simples. À chaque sous-ensemble ouvert bornéde l’espace-temps est associée une algèbre d’opérateurs sur unespace de Hilbert. Ces opérateurs représentent des quantitésmesurables, comme l’énergie du champ à l’intérieur de l’en-semble. La structure de l’espace-temps est inscrite fonctorielle-ment en exigeant que si deux ouverts de l’espace-temps sont re-liés par un morphisme, les algèbres correspondantes d’opéra-teurs soient reliées par un morphisme correspondant.Et pour-tant, cette simplicité recèle l’une des structures les plus belleset les plus difficiles qu’on ait jamais vues en mathématiques.On en entend maintenant constamment parler en géométriedifférentielle et en topologie, mais c’est aux analystes, peut-être, qu’elle offre le plus grand défi. Tant qu’il n’aura pas étérelevé, tous les arguments étranges reposant sur l’« intégralefonctionnelle » continueront de n’être qu’une intuition margi-nale pour des conjectures à prouver d’une autre façon. Le mou-vement brownien, qui fait le bonheur et la fierté des proba-bilistes, est une théorie quantique des champs sur un espace-temps euclidien à une dimension.On découvrira d’autres

trésors parmi les autres théories quan-tiques des champs. On en voit un exemplese dérouler sous nos yeux avec la théo-rie des champs conformes en deux dimen-sions et l’évolution stochastique de Loew-ner (SLE).

L’espace des phases est le produit carté-sien de l’espace Rd avec l’espace des im-pulsions, Rd. Comme dans la théorie desopérateurs pseudo-différentiels, une cel-lule dans l’espace des phases est une boîtecompatible avec l’analyse de Fourier et larelation d’incertitude en ce sens que les cô-tés ∆x dans le Rd spatial et ∆p dans leRd des impulsions obéissent à la relation∆x∆p = 1. Les divergences en TQC dé-coulent du fait que le champ a des fluctua-

tions à toutes les échelles. Si les fluctuations sont décomposéesen fluctuations à l’intérieur des cellules de l’espace des phases,alors il y a en gros un seul degré de liberté qui fluctue à l’inté-rieur de chaque cellule. Les divergences de la TQC se cachentmaintenant dans l’infinité des cellules de l’espace des phases,mais cette réorganisation révèle une propriété d’indépendanceapproximative qui est à l’origine de compensations. Le déve-loppement en cellules de l’espace des phases réalise l’idée quel’intégrale fonctionnelle de la théorie quantique des champs estdonnée approximativement par le produit sur les cellules desintégrales des fluctuations à l’intérieur de chaque cellule. Se-lon le développement en cellules dans l’espace des phases, une

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intégrale fonctionnelle en théorie quantique des champs peutêtre considérée comme une somme convergente de correctionsà cette description très simplifiée. Dans ce développement, lesdivergences de la TQC se combinent de façon à s’annuler. Leterme « divergent » signifie qu’un paramètre est introduit, le« paramètre de coupure ultraviolet », qui spécifie une échellepetite à laquelle sont artificiellement supprimées les fluctua-tions. Cette suppression détruit l’un au moins des axiomes dela TQC, mais le développement en cellules est uniforme en ceparamètre, si bien qu’il devient possible de prendre la limiteoù le paramètre de coupure ultraviolet tend vers zéro et queles axiomes sont rétablis dans cette limite.

Le problème de la surface de Fermi se présente en physiquede la matière condensée, quand on essaie de comprendre lecomportement collectif d’un nombre élevé d’électrons en mou-vement dans un cristal. L’expression « comportement collec-tif » signale qu’il s’agit là d’un problème analogue à ceux de laTQC : les nombreux degrés de liberté des électrons engendrentde nouveaux phénomènes comme la supraconductivité, quisont du domaine de la TQC. Si on néglige les interactions entreles électrons, on peut se concentrer en premier lieu sur le mou-vement d’un seul électron dans le cristal. Conformément auxmanuels de mécanique quantique, il peut prendre des énergiesqui sont fonction de son impulsion. Dans ce cas, le fondamen-tal de N électrons sans interaction est l’état où les énergies àun électron les plus basses sont toutes occupées jusqu’à uneénergie maximale bien définie, l’énergie de Fermi, tous les ni-veaux d’énergie supérieure étant vides. Ce remplissage résultede ce que les électrons sont des fermions : la fonction d’onde deN fermions doit être antisymétrique dans l’échange des coor-données de ces particules, ce qui interdit de placer plus d’uneparticule dans chacun des états à une particule, de telle sorteque les élecrons remplissent les niveaux comme l’eau dans unebaignoire. On peut se représenter la surface de Fermi commecelle d’un volume complètement occupé dans l’espace des im-pulsions. Que devient cette description si l’on cesse de négligerles interactions entre électrons ? La densité des états occupésen fonction de l’impulsion présente-t-elle toujours une discon-tinuité ou s’étale-t-elle ? Les travaux de Feldman, Knörrer etTrubowitz répondent à cette question, pour une certaine classede modèles, dans la série d’articles [2,4–12] affichés sur sa pagepersonnelle. Ils montrent que si la surface de Fermi d’un sys-tème d’électrons sans interactions en deux dimensions obéit àcertaines conditions, alors le fait de réintroduire une faible in-teraction déforme la surface sans interaction, mais ne la détruitpas. De plus, la série perturbative renormalisée ne se contentepas d’être finie terme par terme, mais converge.

Parmi les idées nouvelles majeures de cet article se trouvecelle de généraliser le développement en cellules dans l’espacedes phases de façon à introduire des cellules curvilignes quisuivent la surface de Fermi et se raffinent en approchant dela surface. Un électron isolé, sans interaction, se propage selonune équation aux dérivées partielles pour laquelle la surface deFermi est caractéristique. Dans la dynamique de N électrons en

interaction, les cellules au voisinage de cette surface caractéris-tique sont la scène de grandes fluctuations. Cependant, aveccette géométrie des cellules dans l’espace des phases, les inter-actions entre fluctuations jouissent des propriétés d’indépen-dance nécessaires quand on approche de la surface.

Tout comme en TQC il y a des divergences en théorie naïve desperturbations. Elles surviennent parce que la surface de Fermidu système d’électrons libres n’est pas la même que celle dusystème d’électrons en interaction. Renormaliser revient à ef-fectuer le développement autour d’un nouveau système sansinteraction dont la surface de Fermi est la même que celle dusystème en interaction.

Une paire de Cooper est constituée de deux particules dont lesimpulsions sont sur la surface de Fermi et dont l’impulsiontotale est très petite. Si la forme de la surface permet l’exis-tence de telles paires, l’interaction entre elles devient si énormequand leur impulsion totale tend vers zéro que de nouvellesdivergences apparaissent en théorie des perturbations, ce quisignale, une fois de plus, qu’on est en train de prendre commeapproximation du modèle le mauvais modèle sans interaction.Dans ces articles sur la surface de Fermi, ce phénomène estprécisément ce que les hypothèses interdisent. Par conjecture,si elles ne sont pas interdites, les paires de Cooper dominentle comportement à longue distance de la théorie, la surface deFermi est détruite et le système est un supraconducteur.

Au cours des années 80, il est devenu clair, grâce aux travauxde Balaban entre autres, qu’on peut prendre la limite où la cou-pure ultraviolette tend vers zéro, mais qu’une autre obstruc-tion s’oppose à l’existence d’une TQC en quatre dimensions ausens spécifié par le problème de l’Institut Clay. On devrait aussicomprendre le rôle des fluctuations aux très grandes échelles.Le phénomène de la supraconductivité dans le problème àN électrons a beaucoup de pertinence pour les conjecturesformulées sur le comportement des théories quantiques dechamps de jauge en quatre dimensions : les supraconducteursexpulsent un champ magnétique en créant un champ magné-tique opposé grâce à un flux d’électrons. C’est l’effet Meissner.Dans certaines versions de cet effet, au lieu de l’expulsion com-plète qui demande trop d’énergie, un tube mince retourne à laphase non supraconductrice de façon à permettre la présencedu champ magnétique, mais à l’intérieur du tube seulement. Sil’on introduisait deux monopôles magnétiques dans pareil sys-tème, il y aurait de l’un à l’autre un tube de flux dont le coût enénergie croîtrait linéairement avec la longueur. Les monopôlesseraient donc confinés l’un près de l’autre par le tube de flux.On s’attend à ce qu’un mécanisme semblable confine les quarksdans une TQC en quatre dimensions. On a besoin de mieuxsaisir les détails de ce genre de mécanisme pour comprendreles théories quantiques de champs de jauge en quatre dimen-sions. Le travail sur la surface de Fermi a construit une partiede l’infrastructure nécessaire à ce problème. En fait, pour prou-ver l’existence de la surface de Fermi les auteurs ont dû érigerune énorme infrastructure qui inclut des définitions, propriétés

(suite à la page 15)

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Thematic Program Semester:

Dynamical Systems and Evolution EquationsJanuary – June 2008

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

The theme of the Winter Semester of 2008 will be DynamicalSystems, interpreted in a broad sense so as to include appli-cations to fundamental problems in differential geometry aswell as in mathematical physics. Topics of focus will include: (1) interplay between dynamical systems and PDE, in par-ticular in the context of Hamiltonian systems, (2) geometricevolution equations such as Ricci flows and extrinsic curva-ture flows, (3) spectral theory and Hamiltonian dynamics, and(4) Floer theory and Hamiltonian flows. All of these topics arevery active areas of research at the CRM. In the past severalyears there have been dramatic achievements in these four ar-eas, representing progress on a number of the most basic anddifficult questions in mathematics. The purpose of this pro-gram semester is to bring to Montréal representatives of theinternational community of researchers interested in these top-ics, to present a series of advanced-level courses on the subjectso as to make it available to new researchers in the field, andto discuss the perspectives and general directions for the nextadvances and directions of progress in the area.

The subject matter of the program, dynamical systems and evo-lution equations, includes the fundamental theory of Hamilto-nian systems, the mutual cross-fertilization of dynamical sys-tems and PDE, as well as applications to geometry and math-ematical physics. Very recently there has been progress on anumber of central questions in geometry, including proofs ofthe Poincaré conjecture and Thurston’s geometrization conjec-ture for three-manifolds, and in Hamiltonian PDE, where ana-lytic methods of Hamiltonian mechanics are making an impacton the study of the evolution of many of the principal non-linear evolution equations of mathematical physics. These ad-vances have had a broad impact on recent progress in geome-try and topology, and they also shed light on the basic physicalprocesses that are modeled by ordinary and partial differen-tial equations. This is consistent with the vision of Poincaré,recognized as the founder of the modern theory of dynami-cal systems, who foresaw many of the modern advances in theanalysis of differential equations; particularly striking is theway he articulates his vision of the role of differential equa-tions within mathematics in this 1890 paper in the AmericanJournal of Mathematics: Cette revue rapide des diverses parties de laPhysique Mathématique nous a convaincus que tous ces problèmes,malgré l’extrême variété des conditions aux limites et même des équa-tions différentielles, ont, pour ainsi dire, un certain air de famille qu’ilest impossible de méconnaître. In this article Poincaré considersproblems in potential theory, fluid dynamics, heat flow, spec-tral theory and wave propagation, making his point quite clearthat he appreciates both the value to physics of a mathemati-cally rigorous theory, as well as its value to pure mathematics.He goes on to describe the extreme difficulty of the problems at

hand, concluding his introduction to the paper with this justi-fication of his focus on the analytic issues of partial differentialequations and his views as to their rôle in mathematics; Néan-moins toutes les fois que je le pourrai, je viserai à la rigueur absolueet cela pour deux raisons ; en premier lieu, il est toujours dur pourun géomètre d’aborder un problème sans le résoudre complètement ;et en second lieu, les équations que j’étudierai sont susceptibles, nonseulement d’applications physiques, mais encore d’applications ana-lytiques.. . .Et qui nous dit que les autres problèmes de la PhysiqueMathématique ne seront pas un jour, comme l’a déjà été le plus simpled’entre eux, appelés à jouer en Analyse un rôle considérable ? GerardHuisken and Jean-Christophe Yoccoz will hold the AisenstadtChair during the winter of 2008.

The semester will start with two short workshops, providingthe initial momentum for the semester. The first one, the Ini-tial Conditions Workshop (organizers: W. Craig, P. Guan and C.Rousseau), will take place in mid-January 2008. The lecturesof this three day workshop wil be given by a few invited lec-tures together with the organizers of the core workshops andthe CRM postdocs associated with the semester. The point isto start off the thematic semester with the right atmosphere,and to get the participants, in particular the associated post-doctoral fellows, to know each other. Hence the ‘joke’ of thetitle. Several of the lectures, having a broad scope, will intro-duce to the themes of the semester, while the postdocs will beasked to give talks on their own research programs. The secondJanuary workshop is the Young Mathematicians’ Conference (or-ganizers: W. Craig, A, Nachman, D. Pelinovsky, M. Pugh andC. Sulem) January 18 – 19, 2008. This will be the fifth editionof this annual conference, and will take place at the CRM, theprevious ones having been held at McMaster University andthe Fields Institute. The idea is to have a regular meeting ofmathematicians interested in PDE and/or dynamical systems,which would give junior mathematicians the opportunity toknow each other and each other’s work. The program consistsof two senior ‘plenary’ speakers and 8 to 12 regular speakers. Inaddition, young mathematicians from around the country whoare interested in PDE and dynamical systems are invited to at-tend, with support for their participation from the conference(hopefully shared with their doctoral or postdoctoral advisors).Our informal rule is that but for the two ‘plenary’ speakers, allother talks should be given by people within ε either way oftheir Ph.D., (where ε = a few years). More senior mathemati-cians are of course welcome to attend. Young people from allover Canada will be invited to this coming event.

The four areas of concentration, and the subsequent workshopsthat are central to our program are spread through the monthsof March, April and May. The first will be on Spectrum and

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Dynamics (organizers: D. Jakobson et I. Polterovich), proposeddates: April 7 – 11, 2008. The central theme of the meeting is theinterplay between the theory of dynamical systems and spec-tral geometry. In particular, the workshop focuses on emerg-ing applications of dynamics to the study of various spectralasymptotics. Main topics include: spectral asymptotics on neg-atively curved manifolds and hyperbolic dynamics, quantumcompletely integrable systems in the high energy limit, dynam-ical methods in scattering theory, elliptic operators on vectorbundles and partially hyperbolic flows. Just before this work-shop, we plan to run several mini-courses on topics related toits subject.

The next workshop will be focused on Geometric EvolutionEquations (organizers: V. Apostolov, P. Guan, A. Stancu), April16 – 27, 2008. In view of the impressive progress made inthe area of geometric evolution equations, the workshop isplanned as a stimulating meeting between specialists present-ing the state of recent research in the field and mathemati-cians, young or well-established in other areas, for whom cur-vature flows prompt further investigations in diverse direc-tions. It will emphasize successively the theme of intrinsic cur-vature flows with a focus on Ricci and Kähler – Ricci flows, im-plications on the Poincaré conjecture, the geometrization con-jecture, and the existence and uniqueness of special Kählermetrics on projective varieties; and that of extrinsic curvatureflows — mean and inverse mean curvature flows, homogenousand nonhomogenous Gauss curvature flows, general relativity.There will be mini-courses related to each of the two main top-ics during the period of the workshop.

The next workshop Singularities, Hamiltonian and gradient flows(organizers: W. Craig, C. Rousseau, A. Shnirelman), May 12 –16 2008 will focus around the general theme of singularities ofdynamical systems and the rôle they play in ODE and PDE.Among the participants there will be specialists of geometricalmethods in differential equations, geometric analysis, the anal-ysis of Hamiltonian systems and Hamiltonian PDE, summabil-ity techniques in complex dynamics, small divisors and KAMtheory, and applications of dynamical systems to PDE. The sub-jects covered will include normal forms, small divisors, reso-nances, and variational problems in dynamical systems. It willfollow the Aisenstadt lectures by Jean-Christophe Yoccoz andother mini-courses during the week May 5 – 9.

The workshop on geometric analysis and dynamics will be fol-lowed by one on topology and dynamics, namely Floer Theoryand Symplectic Dynamics (organizers: O. Cornea, L. Polterovich,F. Schlenk), May 19 – 24 2008. In the past two decades therehas been an important interaction between symplectic topol-ogy and theory of dynamical systems. One of the main expo-nents of this interaction is the Floer theory, a powerful ma-chinery which combines ingredients from infinite-dimensionalMorse theory, complex algebraic geometry and elliptic PDEs.In spite of the substantial progress achieved, a number of majorproblems are still open and the foundations are still under con-struction. Nowadays the subject is speedily developing and at-

tracting the best researchers all over the world. The main topicsof the conference include recent exciting developments in thefoundations of the Floer theory, various aspects of symplecticand contact dynamics, symplectic capacities and packing prob-lems, topology of Lagrangian submanifolds and algebraic andgeometric properties of groups of symplectic and contact trans-formations. A special attention will be paid to recent develop-ments in Floer theory related to mathematical physics such asmirror symmetry and symplectic field theory. The meeting willbring together the top world experts in the field as well as ju-nior researchers.

The scientific organization committee includes Vestislav Apos-tolov (Université du Québec à Montréal), Hermann Brun-ner (Memorial University), Octav Cornea (Université de Mon-tréal), Walter Craig (McMaster University), Pengfei Guan(McGill University), Dmitry Jakobson (McGill University),Sergei Kuksin (Herriot-Watt University – Edinburgh), IosifPolterovich (Université de Montréal), Christiane Rousseau(Université de Montréal), Alina Stancu (Concordia Univer-sity), Alexander Schnirelman (Concordia University), EugeneWayne (Boston University), and Maciej Zworski (University ofCalifornia – Berkeley)

Vient de paraître !L’Algèbre et le Groupe de Virasoropar Laurent Guieu et Claude Roger

L’algèbre et le groupe de Virasorojouent un rôle important dans un grandnombre de branches des mathématiqueset de la physique théorique, comme lessystèmes intégrables, les classes caracté-ristiques, la quantification, les systèmesdynamiques, les théories de cordes et lathéorie des champs conformes.

Ce livre constitue un exposé mathéma-tique détaillé de leurs multiples aspects,accompagné des notions algébriques etgéométriques indispensables à l’autono-

mie de l’ouvrage : théorie des groupes de Lie, éléments de géo-métrie symplectique et algèbre homologique. Les trois dernierschapitres sont consacrés à diverses généralisations, comme lesalgèbres-W et les algèbres superconformes. Une notice his-torique illustrant les développements successifs de la théo-rie ainsi qu’un appendice de Vlad Sergiescu sur le groupe deThompson font office d’épilogue temporaire à l’un des do-maines de recherche les plus féconds de la physique théoriqueactuelle, domaine qui est encore loin d’avoir livré tous ses se-crets.

Prix : 88,00 $CAN

cf. www.crm.math.ca/pub/Ventes/Catalogue/

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Workshop on Singularities in PDE andthe Calculus of Variations

by Stan Alama (McMaster University), Lia Bronsard (McMaster University), and Peter Sternberg (Indiana University)

The Workshop on Singular-ities in PDE and the Cal-culus of Variations held atthe CRM from July 17 toJuly 21, 2006 was a greatsuccess. Participants in-cluded researchers andstudents from severalcountries in North andSouth America, Europe,

and Asia, many of whom met for the first time. The partici-pants were extremely impressed by the high level of the talks,the range and depth of the results presented, and the remark-ably high level of congeniality and camaraderie in the meeting.The main theme of the meeting was the formation of geomet-rical singularities in PDE problems with variational formula-tion. These equations typically arise in some applications — tophysics, engineering, or biology, for example — and their res-olution requires combining methods coming from functionaland harmonic analysis, differential geometry, geometric mea-sure theory, for instance.

A recurrent theme in the meeting involved the Cahn – Hilliardmodel for phase transitions, which is ubiquitous in the de-scription of domain walls (codimension-one singularities). BobKohn (NYU), who has played a major role in the developmentof this area, spoke on dynamics of crystal facets as a steep-est descent problem for a nonsmooth energy functional. M.Del Pino (Chile) spoke about the existence of multiple transi-tion layers, while M. Reznikoff (Georgia Tech) presented a newapproach to studying stochastic effects in switching betweenlocal minima in such problems. X. Ren (Utah State) showedhow modifications to Cahn – Hilliard can stabilize stationaryproblems. Radu Ignat (Paris VI) spoke about a related prob-lem from micromagnetism, in which nonlocal effects are intro-duced. R. Choksi (SFU) presented striking results, obtained incollaboration with P. Sternberg (Indiana), on some new nonlo-cal isoperimetric problems arising from the Ohna – Kawasakimodel for diblock copolymers.

A second major theme of the workshop was the family ofGinzburg – Landau models for superconductors, superfluids,and Bose-Einstein condensates. In these models, the singulari-ties are typically “vortices” which are quantized codimension-two defects. F. Bethuel (Paris VI) presented impressive resultson the dynamics and interactions of vortex solutions. Dynam-ical results on the difficult Schrodinger map problem were de-scribed by S. Gustafson (UBC). L. Berlyand (Penn State) dis-cussed the role of the capacity in minimizing the Ginzburg-Landau functional with prescribed degrees. J.A. Montero

(Toronto) presented results, some together with S. Alama andL. Bronsard (McMaster), on local minimizers with vortices in aBose – Einstein condensate.

In the setting of superconductivity, S. Serfaty (NYU) gave asweeping overview of the vorticity of minimizers in increasingmagnetic fields, tying the problem to the methods of div-curlsystems introduced by DiPerna-Majda. S. Jimbo (Hokkaido)and Y. Morita (Ryukoku) described method producing solu-tions in narrow three dimensional superconductors. Y. Almog(LSU) analyzed the normal/superconducting transition in thepresence of electric currents. D. Phillips (Purdue) presented amodel for high-temperature superconductors with results onthe unusual pattern of vortices observed. There were two talkson the Chern – Simons – Higgs functional from gauge-field the-ory: G. Tarantello (Rome “Tor Vergata”) discussed uniquenessproblems in the self-dual regime, while D. pirn (Minnesota)showed gamma-convergence of the functional in the large-coupling limit.

There were also many fascinating talks outside these broadcategories, illustrating the breadth of the field. A. Malchiodi(SISSA) gave a beautiful talk on singular perturbation prob-lems with concentrations along curves or surfaces. P. Padilla(UNAM) gave an enlightening closing talk on the varietyof variational problems in nature. B. Kawohl (Cologne) con-trasted several variational (such as Mumford – Shah) and ge-ometrical (Perona – Malik) models used in image process-ing. C. Liu (Penn State) gave results on difficult problemsof well-posedness in viscoelastic materials. C. Garcia-Cervera(UCSB) presented numerical and analytical results on den-sity functional theory from quantum chemistry. David Kinder-lehrer (CMU) gave an excellent opening talk, about the prob-lem of transport in cells which combines modelling ques-tions with optimization and diffusion. On the analytic foun-dations behind these models, R. Jerrard (Toronto) presented anew measure-theoretic approach to Monge-Ampère functionswhich doesn’t require convexity assumptions. V. Millot (CMU)discussed the lifting of H1/2 maps with values in the circle.W. Ziemer (Indiana) spoke, with the collaboration of his sonW.K. Ziemer (Long Beach State), about a very general form ofthe Gauss – Green theorem, which turned out to be related tomethods described by Serfaty in her talk.

The organisers are grateful to the CRM for its generous fundingof the meeting and the outstanding organization and facilitiesprovided, and also to the NSF for providing a rather generousgrant to support young mathematicians coming from the US.

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The 2007 André-Aisenstadt Prizeby Alexander E. Holroyd (University of British Columbia)

Astonishing Cellular Automata

Cellular automata arise naturally in thestudy of physical systems, and exhibit aseemingly limitless range of intriguing be-haviour. Such models lend themselves natu-rally to simulation, yet rigorous analysis canbe notoriously difficult, and can yield highlyunexpected results.

Bootstrap percolation is a very simple model, originally intro-duced in [13], which turns out to hold many surprises. Cellsarranged in an L by L square grid can be either infected orhealthy. Initially, we flip a biased coin for each cell, declaringit infected with probability p. At each subsequent time step,any healthy cell with 2 or more infected neighbours becomesinfected, while infected cells remain infected forever. (A cell’sneighbours are the cells immediately to its North, South, Eastand West — interior cells have 4 neighbours while boundarycells have fewer). See Figures 1 and 2.

0 1 2 3 4 5 6FIGURE 1. Bootstrap percolation on a square of size 5 at timesteps 0, . . . , 6. Infected cells are shown by Xs

How does the probability

I(L, p) := P(the entire square is eventually infected)

behave for large L? It is easy to guess that for p sufficientlylarge we have I → 1 as L → ∞, since the infection can al-ways invade finite healthy islands. More surprisingly, it turnsout that for any fixed p > 0 we have I → 1 as L → ∞. This wasthe first rigorous result in the subject, proved in [31]. The intu-ition is that certain rare local configurations act as “nucleationcentres” which can spread to infect any region (given a back-ground of randomly infected cells), so any square large enoughto contain a nucleation centre will become entirely infected.

Clearly for fixed L we have I → 0 as p → 0, so it is natu-ral to ask what happens when L and p are varied simultane-ously — how large must a square be, as a function of p, to havehigh probability of becoming entirely infected? An early break-through was the following.

Theorem 1 ([3]). There exist positive constants C± such that, as(L, p) → (∞, 0),

• I(L, p) → 1 if p log L > C+;• I(L, p) → 0 if p log L < C−.

Other aspects of the model were subsequently studied in de-tail (e.g., [4, 9, 28, 30]), but the question of whether C± could

FIGURE 2. Bootstrap percolation on a square of size L = 300.Cells were initially infected with probability p = 0.05. In thisexample all cells eventually became infected, and the coloursindicate time of infection: red (initial infections), black (earli-est), followed by dark blue, light blue, green, yellow (latest)

be replaced by a single sharp constant remained open for 15years until the following result in 2003, which furthermore es-tablishes the constant’s value.

Theorem 2 ([21]). Theorem 1 holds for all C− < π2/18 < C+.

The above is an example of a phase transition — a suddenchange in behaviour as a parameter (here p log L) crosses a crit-ical value (π2/18). The proof involves finding accurate boundson the density of nucleation centres.

Bootstrap percolation seems ideally suited to computer simu-lation. Indeed this can be a valuable tool — Figure 2 immedi-ately suggests the idea of nucleation (a vital ingredient of theproofs). Amazingly however, the resulting numerical predic-tions differ greatly from the above rigorous result. On the basisof simulation of squares up to size L = 28800, the critical valuewas estimated earlier in [2] to be 0.245 ± 0.015. (This conclu-sion was natural given the data, and subsequent larger simula-tions appear at first sight to confirm it.) However the rigorousvalue π2/18 = 0.548311 . . . is larger by more than a factor oftwo! The discrepancy seemingly arises because convergence to

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the limit π2/18 is extremely slow. In fact, combining simula-tion data with Theorem 2 suggests that a square of size at leastL = 1020 would be needed in order to get close. This points tofurther intriguing questions which we shall discuss later.

Cellular automaton models may be loosely defined as uniformlocal rules for updating the states of an array of cells. Asidefrom their intrinsic mathematical interest, such models havebeen applied to wide variety of physical problems. In the caseof bootstrap percolation these include magnetic alloys, hydro-gen mixtures, computer storage arrays, and crack formation(see, e.g., the references in [1]). For the last example, imaginea lattice of atoms, with a small fraction p “missing” or “defec-tive,” and where any atom with two or more defective neigh-bours becomes defective. Rather than accurately approximat-ing reality, the aim is to capture phenomena of interest (suchas nucleation), and study them in the simplest setting possible.Interestingly, a relevant scale for some applications might beL ≈ 1010, where the most accurate predictions currently arisefrom interpolation between simulation and theoretical results.A further motivation for studying simple models is for use astools for the rigorous analysis of more complicated systems.See [10, 15] for recent examples in the case of bootstrap per-colation. The model also arose independently in the following(see [32] for background).

Puzzle. Prove that the entire square cannot become infectedwith fewer than L cells initially infected. (There is a one-wordsolution!)

Theorem 1 has been extended to several variant models, some-times with other functional forms replacing the parameterp log L. These include: p log · · · log L for the d-dimensionalmodel [11, 12]; pα log L for 2-dimensional models with varioussymmetric neighbourhoods [16, 17]; p(− log p)β log L for cer-tain asymmetric versions [17,29]; pn24

√n for the hypercube [8];

p√

q for a “q-polluted” lattice [20]. The existence of a sharp con-stant as in Theorem 2 is conjectured in all the cases above, butproved only for two other models below. In the modified model,a healthy cell becomes infected if it has at least one infectedneighbour in each dimension.

Theorem 3 ([22]). For the modified model in d ≥ 2 dimensions, The-orem 2 holds with parameter p log · · · log︸︷︷︸

d−1

L and critical value π2/6.

Theorem 4 ([23]). For the cross-shaped neighbourhood

k−1

in dimension 2 with an infection threshold of k (≥ 2) neigh-bours, Theorem 2 holds with parameter p log L and critical valueπ2/

(3k(k + 1)

).

It is an open problem to extend Theorem 3 to the standardmodel in dimensions ≥ 3, and more generally to find a uni-fied method for proving the existence of sharp critical val-ues. (The latter is reminiscent of the famous random k-SATproblem.) The combinatorial structure behind the constants

π2/(3k(k + 1)

)is somewhat subtle and mysterious. A by-

product of Theorem 4 was the following, which stimulated fur-ther investigations in [5].

Theorem 5 ([23]). The number ak(n) of integer partitions of n notcontaining any k distinct consecutive parts satisfies

log ak(n) ∼ π√

23

(1− 2/

(k(k + 1)

))n as n → ∞.

Adding randomness to the evolution of cellular automata oftenmakes them more tractable. Interacting particle systems (spa-tial models involving Markovian evolution in continuous time)have been studied with great success (see [27]). In contrast,models involving deterministic evolution from random or de-terministic initial conditions, while physically natural, can bevery challenging, and their rigorous study is in its infancy. Be-sides bootstrap percolation, Figure 3 gives tastes of a few recentadvances. Traffic in the BML model (i) is proved to jam at highdensities, but nothing is known about low densities. The nthstage of the rotor aggregation (ii) is known to be within dis-tance n1/4+ε of a disk, but believed to be within distance 2. Cer-tain Packard snowflake models (iii) fill only a proper fraction ofthe plane, yet would completely fill any simulation up to size109. In a different direction, limiting trajectories in the randomsorting network (iv) are believed to be random sine curves, andknown to be Hölder(1/2).

Returning to bootstrap percolation, it is of interest to explainthe apparent discrepancy between simulations and limiting be-haviour. These issues have been investigated partly nonrigor-ously in [14], and some rigorous answers are proved in [19].

FIGURE 3. (i) Biham – Middleton – Levine traffic model [6];(ii) Propp’s rotor-router aggregation [25, 26] (also see [24]);(iii) Packard’s snowflake model [18]; (iv) random sorting net-work [7]

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Theorem 6 ([19]). For bootstrap percolation in 2 dimensions, thereexists a positive constant c such that, as (L, p) → (∞, 0),

I(L, p) → 1 if p log L >π2

18− c(log L)−1/2.

Suppose one tries to estimate the critical value π2/18 usingp1/2 log L, where pα(L) is defined via I(L, pα) = α. In order tohalve the error term c(log L)−1/2, one must raise L to the power4. In the case of the modified model in 2 dimensions, explicit es-timates are available, and imply for example that p1/2 log L isnot within 1% of π2/6 even when L = 103000 (see [19]).

While Theorem 6 shows that the distance from the critical pa-rameter p1/2 log L to its asymptotic value is Ω

((log L)−1/2),

the width of the critical window p1−ε log L − pε log L is muchsmaller, O(log log L/ log L) (or θ(1/ log L) if we fix p ratherthan L) — see [3, 8, 19]. This contrasts with models such as theErdos – Renyi random graph, where the window shrinks moreslowly than its centre converges.

Bootstrap percolation also seems to exhibit crossover — the ap-proximate critical value 0.245 estimated in [2] appears accurateover a wide (but bounded) interval of L. It is a fascinating chal-lenge to understand this phenomenon rigorously. One possi-ble approach is to prove a limiting result for some sequence ofmodels in which this interval becomes longer and longer.

1. J. Adler, Bootstrap percolation, Phys. A 171 (1991), 453 – 470.

2. J. Adler, D. Stauffer, and A. Aharony, Comparison of bootstrappercolation models, J. Phys. A 22 (1989), L297 – L301.

3. M. Aizenman and J. L. Lebowitz, Metastability effects in boot-strap percolation, J. Phys. A 21 (1988), no. 19, 3801 – 3813.

4. E. D. Andjel, Characteristic exponents for two-dimensionalbootstrap percolation, Ann. Probab. 21 (1993), no. 2, 926 – 935.

5. G. E. Andrews, Partitions with short sequences and mock thetafunctions, Proc. Natl. Acad. Sci. USA 102 (2005), no. 13,4666 – 4671 (electronic).

6. O. Angel, A. E. Holroyd, and J. B. Martin, The jammedphase of the Biham – Middleton – Levine traffic model, Electron.Comm. Probab. 10 (2005), 167 – 178 (electronic).

7. O. Angel, A. E. Holroyd, D. Romik, and B. Virag, Randomsorting networks (preprint).

8. J. Balogh and B. Bollobás, Bootstrap percolation on the hyper-cube, Probab. Theory Related Fields 134 (2006), no. 4, 624 –648.

9. J. Balogh, Y. Peres, and G. Pete, Bootstrap percolation on in-finite trees and non-amenable groups, Combin. Probab. Com-put. 15 (2006), no. 5, 715 – 730.

10. N. Cancrini, F. Martinelli, C. Roberto, and C. Toninelli, Ki-netically constrained spin models (preprint).

11. R. Cerf and E. N. M. Cirillo, Finite size scaling in three-dimensional bootstrap percolation, Ann. Probab. 27 (1999),no. 4, 1837 – 1850.

12. R. Cerf and F. Manzo, The threshold regime of finite volumebootstrap percolation, Stochastic Process. Appl. 101 (2002),no. 1, 69 – 82.

13. J. Chalupa, P. L. Leath, and G. R. Reich, Bootstrap percolationon a bethe lattice, J. Phys. C, 12 (1979), L31 – L35.

14. P. De Gregorio, A. Lawlor, P. Bradley, and K. A. Dawson,Exact solution of a jamming transition: closed equations for abootstrap percolation problem, Proc. Natl. Acad. Sci. USA 102(2005), no. 16, 5669 – 5673 (electronic).

15. L. R. Fontes, R. H. Schonmann, and V. Sidoravicius,Stretched exponential fixation in stochastic Ising models at zerotemperature, Comm. Math. Phys. 228 (2002), no. 3, 495 – 518.

16. J. Gravner and D. Griffeath, First passage times for thresholdgrowth dynamics on Z2, Ann. Probab. 24 (1996), no. 4, 1752 –1778.

17. , Scaling laws for a class of critical cellular automatongrowth rules, Random Walks (Budapest, 1998), János BolyaiMath. Soc., Budapest, 1999, pp. 167 – 186.

18. , Modeling snow crystal growth. I: Rigorous results forPackard’s snowflakes (preprint).

19. J. Gravner and A. E. Holroyd, Slow convergence and scalingwindow in bootstrap percolation (in preparation).

20. J. Gravner and E. McDonald, Bootstrap percolation in a pol-luted environment, J. Statist. Phys. 87 (1997), no. 3-4, 915 –927.

21. A. E. Holroyd, Sharp metastability threshold for two-dimensional bootstrap percolation, Probab. Theory RelatedFields 125 (2003), no. 2, 195 – 224.

22. , The metastability threshold for modified bootstrap per-colation in d dimensions, Electron. J. Probab. 11 (2006), no. 17,418 – 433 (electronic).

23. A. E. Holroyd, T. M. Liggett, and D. Romik, Integrals, par-titions, and cellular automata, Trans. Amer. Math. Soc. 356(2004), no. 8, 3349 – 3368 (electronic).

24. A. E. Holroyd and J. Propp, Rotor-router walks (in prepara-tion).

25. L. Levine and Y. Peres, Spherical asymptotics for the rotor-router model in Zd (preprint).

26. , Sharp circularity of the rotor-router and sandpile mod-els (in preparation).

27. T. M. Liggett, Stochastic interacting systems: contact, voterand exclusion processes, Grundlehren Math. Wiss., vol. 324,Springer, Berlin, 1999.

28. T. S. Mountford, Rates for the probability of large cubes be-ing non-internally spanned in modified bootstrap percolation,Probab. Theory Related Fields 9 (1992), no. 2, 159 – 167.

29. , Critical length for semi-oriented bootstrap percolation,Stochastic Process. Appl. 56 (1995), no. 2, 185 – 205.

30. R. H. Schonmann, On the behavior of some cellular automatarelated to bootstrap percolation, Ann. Probab. 20 (1992), no. 1,174 – 193.

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31. A. C. D. van Enter, Proof of Straley’s argument for bootstrappercolation, J. Statist. Phys. 48 (1987), no. 3-4, 943 – 945.

32. P. Winkler, Mathematical puzzles: a connoisseur’s collection, AK Peters Ltd., Natick, MA, 2004.

Applied Dynamical Systems(continued from page 1)

to algorithms and software and their use in the study of a di-verse collection of mathematical models in areas ranging frompopulation dynamics to the design of laser systems, as justsome examples. The meetings have also inspired purely the-oretical work, ranging from the classification of singularitiesto the study of bifurcations in Hamiltonian and reversible sys-tems.

The workshop will also include a celebration of EusebiusDoedel’s 60th birthday. The preceding minicourse aimed atgraduate students and postdoctoral fellows who have a needfor advanced numerical techniques in the study of applied dy-namical systems that arise in their research, will consist of lec-tures and computer work sessions.

A workshop on Mathematical Neuroscience (16 – 19 September2007) organised by S. Coombes (Nottingham), A. Longtin (Ot-tawa), and J. Rubin (Pittsburgh), will provide an overview ofthe current state of research in mathematical approaches toneuroscience. This vibrant area, seeded by successes in under-standing nerve action potentials, dendritic processing, and theneural basis of EEG, has moved on to encompass increasinglysophisticated tools of modern applied mathematics. Includedamong these are Evans function techniques for studying wavestability and bifurcation in tissue level models of synaptic andEEG activity, heteroclinic cycling in theories of olfactory cod-ing, the use of geometric singular perturbation theory in under-standing rhythmogenesis, and stochastic differential equationsdescribing inherent sources of neuronal noise. The novel appli-cation of such techniques in Audition and Parkinsonian tremorand deep brain stimulation will also be addressed in two half-daysessions. The Computational Neuroscience Summer School in Ot-tawa in June 2007 would provide a good introduction to thisfield.

Continuing the biological theme, the workshop on Deconstruct-ing Biochemical Networks (24 – 28 September 2007) organizedP.S. Swain (McGill), B.P. Ingalls (Waterloo) and M.C. Mackey(McGill) will be preceded by a minicourse (22 – 23 September2007). This workshop will bring together both modellers andexperimentalists, and will focus on network design. New tech-nologies have revealed additional complexity in biochemicalnetworks. Proteins and genes were initially thought to exist inlinear pathways, now connections and crosstalk between path-ways is the norm, and it appears that the nonlinear dynami-cal properties of these networks determine not only cell phys-

iology but also organismal development and behaviour. Al-though reproducing cellular behaviour with simulation has in-valuable predictive power, the simulation may become almostas complex as the network itself. An intuitive understandingof network properties will possibly only be reached by decon-structing networks into functional modules as, for example, anelectrical system can be deconstructed into its component tran-sistors and switches. Such an approach has been invaluable inthe physical sciences and the multidisciplinary character of thisworkshop will provide momentum to its application to biol-ogy.

The Joint AARMS – CRM Workshop on Recent Advances inFunctional and Delay Differential Equations will take place atDalhousie University, Halifax (1 – 5 November 2007) organ-ised by J. Appleby (Dublin City), H. Brunner (Memorial), A.R.Humphries (McGill), and D.E. Pelinovsky (McMaster), with lo-cal organisers P. Keast (Dalhousie) and P. Muir (St Mary’s). Thisworkshop will provide a wide perspective on current researchand open problems, covering theory, applications and numeri-cal analysis of these equations. Recent advances across the fieldof DDEs will be covered, with particular concentrations in non-constant delay equations, advance-delay differential equationsand Volterra functional equations, including problems with“differential-algebraic” structure. The workshop will bring to-gether specialists in analysis and numerical computations ofthe functional and delay equations, with particular applica-tions in biological problems and traveling waves in nonlinearlattices.

This is followed by a workshop on Dynamical Systems and Con-tinuum Physics at the CRM (14 – 16 November 2007) organisedby L. Tuckerman (PMMH-ESPCI, France) which will focus onrecent research in laminar and turbulent hydrodynamics andin some more discrete systems, notably granular media andfoams.

The semester concludes with a workshop on Chaos and Ergod-icity of Realistic Hamiltonian Systems (11 – 14 December 2007) or-ganised by H. Broer (Groningen) and P. Tupper (McGill). In sta-tistical physics the dynamical systems defined by the mechan-ics of large system of interacting particles are often assumed tobe ergodic, but, this appears to be contradicted by results in thetheory of Hamiltonian systems that imply that ergodicity is nota generic property. This workshop will investigate what is trueof the ergodic properties of flows arising from these systems

Two advanced graduate courses associated with the semesterwill be included in the ISM applied mathematics program forFall 2007: A Practical Introduction to SDEs, at McGill with P. Tup-per and Numerical Analysis of Nonlinear Equations at Concordiawith E. Doedel.

For more information on the programme and the support avail-able see http://www.crm.math.ca/Dynamics2007/

BULLETIN CRM–13


Combinatorial Optimization Theme Semester(continued from page 2)

fostered interactions between researchers from several commu-nities, as well as generating much interest in the Montréal op-erations research community.

The workshop on Data Mining and Mathematical Programmingwas held during the week of October 10-13 and organized byPierre Hansen (HÉC Montréal) and Panos Pardalos (Florida atGainesville). The goal of the new and active field of data min-ing is to find interesting, useful or profitable information invery large databases. The main tasks of data mining, i.e., dis-crimination, clustering and relation finding, have already beenexplored in other fields, but the challenge resides in the hugesize of the problems considered nowadays. Thus it is necessaryto revise and streamline traditional methods and to devise newones. Mathematical programming plays a key role in this en-deavor, since it leads to rigorous formulations of the problemsconsidered in data mining and provides powerful mathemati-cal tools for solving them.

Most of the 24 workshop lectures were one-hour lectures ona wide variety of topics and applications, including machinelearning, logical analysis of data, clustering, network optimiza-tion, similarity analysis, functional data analysis, and appli-cations to genomics and neuroscience. Professor Steve Smale,a Fields medallist, gave a fascinating lecture on patterns indata, showing how it is possible to “learn” the homology ofa submanifold when data arises from a probability distribu-tion whose support is on or near a submanifold of Euclidean

space. Overall, the workshop was attended by 84 participantsand was a great success.

The workshop on Polyhedral Computation took place from Oc-tober 17 to October 20 and was attended by 39 participants. Itwas organized by David Avis (McGill), David Bremner (NewBrunswick) and Antoine Deza (McMaster). The workshop con-sisted mainly of 40-minute talks by the invited speakers. Theycame from various fields: polyhedral combinatorics, computa-tional algebra, lattice theory, semi-definite programming andapplications of polyhedral computation. The goal of the work-shop was to explore the use of symmetry and algebra to al-low the extension of current computational methods to largescale symmetric polyhedra. All participants were mathemati-cians who actively design, implement or use computer soft-ware in their work, and many of the principal authors of suchsoftware were present at the workshop. For example, MathieuDufour spoke on vertex enumeration under symmetry, and re-ported on work where he used codes developed by six peoplein the audience.

The participants were able to get a sample of the wide vari-ety of mathematical problems for which their software is nowused, its current limitations, and ideas for future extensions.The talks were very well prepared, so that participants outsideof the speaker’s area could understand the contributions tothe theme of the workshop, i.e., polyhedral computation. Theworkshop included an open problems session on the first dayand various free discussion periods throughout the four days.The workshop is thought to be the first ever on the topic ofpolyhedral computation, and a follow-up meeting is plannedfor 2008 in Europe.

André-Aisenstadt Chairs(continued from page 4)

of G if H is obtained from G by deleting edges or vertices orcontracting edges (contracting the edge u, v amounts to iden-tifying the vertices u and v). Minor containment is often en-countered in topological graph theory, because it is closely re-lated to topological containment. The best-known example inthis regard is the strong version of Kuratowski’s theorem givenby Klaus Wagner: a graph G is planar if and only if it does notcontain K5 (the complete graph on five vertices) or K3,3 (thecomplete bipartite graph with three vertices in each part) asa minor. Another example is Tutte’s conjecture (1966), a gen-eralization of the four-colour theorem: the edges of any cubic2-connected graph not containing the Petersen graph as a mi-nor can be coloured with three colours. Tutte’s conjecture wasalso proved by Paul Seymour and Neil Robertson in 1997!

Given a notion of graph containment and a fixed graph H, onemay ask the following question: how does one construct ex-plicitly all the graphs not containing H? After giving examplesof such constructions for the two principal meanings of “con-tainment,” Paul Seymour argued that the phrase “explicit con-

struction” still lacks a clear definition! To illustrate this, he gavethe example of claw-free graphs. Given a graph G and a ver-tex v of G, N(v) denotes the set of vertices adjacent to v. Thegraph G is claw-free if for any v, N(v) does not contain threepairwise nonadjacent vertices. Here is an obvious (inductive)way of constructing a claw-free graph: given a claw-free graphG and a subset X not containing three pairwise nonadjacentvertices, add a vertex to G whose set of neighbours is X. Theresulting graph is claw-free.

This inductive definition yields a polynomial-time algorithm,but it does not seem to be what we want! Indeed, we have to“guess” the set X of neighbours of the new vertex. Fortunately,Maria Chudnovsky and Paul Seymour recently proved a deepstructure theorem for claw-free graphs (its proof is 100-pagelong!), and a likely by-product of this theorem is an explicitconstruction for claw-free graphs that avoids such “guesses.”Paul Seymour concluded his first lecture by raising the possi-bility that one could use mathematical logic to give a precisemeaning to the phrase “explicit construction.”

In his second lecture, Paul Seymour turned to the algorithmicaspect of induced graph containment. Let F be a fixed fam-

BULLETIN CRM–14


ily of graphs, and for any graph G and any subset U of ver-tices of G, let G[U] denote the subgraph induced by U. Con-sider the following problem: given a graph G and a subset Sof vertices of G, does G have an induced subgraph G[U] be-longing to F and such that U contains S? Most problems ofthis sort seem to be NP-complete. Maria Chudnovsky and PaulSeymour, however, gave a polynomial-time algorithm to solvethe above problem when F is the family of trees and S a setof three vertices. They called this special case the “three-in-a-tree” problem, because it consists of deciding whether a graphG has an induced tree containing three specified vertices. Thealgorithm of Chudnovsky and Seymour can be used to decidewhether or not a graph G contains an induced subgraph iso-morphic to a theta, i.e., a subgraph consisting of three vertex-disjoint paths between the vertices a and b.

The topic of Paul Seymour’s third lecture was the Caccetta –Häggkvist conjecture. A digraph G is a couple (V, A), where Vis a finite set and A a finite subset of V × V. The members ofV are the vertices of G and those of A the arcs of G. A loop isa couple of the form (u, u), and a digon is a pair of arcs of theform (u, v), (v, u) for two distinct vertices u and v. A digraphG is simple if it contains no loop or digon. In 1978, Caccettaand Häggkvist made the following conjecture: every simple di-graph with n vertices and minimum out-degree at least equalto r contains a directed cycle of length at most dn/re. Manyspecial cases of this conjecture are known to be true; in par-ticular, Jian Shen proved in 2003 that the Caccetta-Häggkvistconjecture holds if r is at most

√n/2.

The case where r is equal to n/3 (among others) seems to bedifficult, and researchers have tried to determine the smallestconstant c such that G contains a directed triangle if its min-imum out-degree is at least cn. The best result so far, due toJian Shen, is that c is at most 3 −

√7 (it is conjectured that c

is equal to 13 ). In a similar vein, Seymour, de Graaf and Schri-

jver proposed to determine the smallest constant β such thatG contains a directed triangle if its minimum out-degree andits minimum in-degree are at least βn. They proved that β is atmost 0.3487; Jian Shen used their work and one of his previousresults to show that β is at most 0.3477.

Finally, the formulation of the Caccetta-Häggkvist conjecturecan be slightly modified to include the minimum in-degree;for instance, does every simple digraph whose minimum out-degree and minimum in-degree are at least n/3 contain a di-rected triangle? This special case would be implied by the fol-lowing conjecture (Seymour’s Second Neighbourhood Conjecture):any simple digraph with no loop or digon has a vertex v whosesecond neighbourhood is at least as large as its first neighbour-hood. The second (resp. first) neighbourhood of v is the set ofvertices u such that there is a shortest path of length 2 (resp. 1)between v and u.

It is impossible to do justice to Paul Seymour’s work in such ashort space. To conclude, however, let us mention his work onregular matroids, on matroid theory in general, on Hadwiger’sconjecture, and on graph minors. The latter work appears ina series of more than twenty articles in the Journal of Combi-natorial Theory, Series B, and in particular the twentieth articlecontains the proof of a celebrated conjecture of Klaus Wagner(1970) asserting that “for every infinite set of finite graphs, oneof its members is isomorphic to a minor of another.”

Biography

Paul Seymour is Professor of Mathematics at Princeton Univer-sity. After obtaining a D.Phil. from the University of Oxford,he held positions at the Universities of Oxford and Waterlooand at the Ohio State University. He was a Member of Tech-nical Staff and a Senior Scientist at Bellcore from 1984 to 1996and has been a Professor at Princeton University since 1996.He has published more than 150 research articles and is Editor-in-Chief for the Journal of Graph Theory (jointly with CarstenThomassen). He is also Editor for Combinatorica and the Jour-nal of Combinatorial Theory, Series B. He was awarded the mostprestigious prize in combinatorial optimization, the D.R. Fulk-erson Prize, in 1979, 1994 and 2006 (jointly with Neil Robertsonand Robin Thomas in 1994 and jointly with Neil Robertson in2006). He was also awarded the Ostrowski Prize in 2004 and theGeorge Pólya Prize in 1983 and 2004 (jointly with Neil Robert-son in 2004).

Prix CRM-Fields-PIMS 2007(suite de la page 6)

et évaluations d’intégrales sur des algèbres de Grassmann per-mettant de décrire les fermions dans le langage de la TQC,l’analyse et l’évaluation de classes de diagrammes en échelle,ainsi que la technologie de cellules dans l’espace des phasesadaptées à la surface de Fermi.

Quand je contemple l’étendue des phénomènes régis par laTQC, il me semble que la Nature vit à la lisière du possible etqu’il nous faut une préparation d’une extraordinaire ampleurpour la suivre jusque-là. En regroupant autour d’un seul su-

jet des preuves complètes permettant de contrôler fortement lathéorie des perturbations pour la matière condensés, l’expédi-tion s’est bel et bien mise en route.

(traduit de l’anglais par Jean LeTourneux)

1. Joel Feldman, Horst Knörrer, and Eugene Trubowitz, Rie-mann surfaces of infinite genus, CRM Monogr. Ser., vol. 20,Amer. Math. Soc., Providence, RI, 2003.

2. , Single scale analysis of many fermion systems. I. Insu-lators, Rev. Math. Phys. 15 (2003), no. 9, 949–993.

(suite à la page 16)

BULLETIN CRM–15


Conference on Geometric Group Theoryby Daniel Wise (McGill University)

A conference on Geometric Group Theorywas held at the CRM from July 3 to July14, 2006. The conference was a resound-ing success with ninety-eight participantsduring the two weeks.

The first week of the conference was ded-icated to five mini-courses which werepitched at an introductory level aimed atstudents entering the field. Chris Hruska(Chicago) gave a mini-course on “Rela-

tive Hyperbolicity” which is a topic of continual interest thathas matured during the last couple of years. Graham Niblo(Southamton) and Michah Sageev (Technion) jointly gave amini-course on “CAT(0) Cube Complexes” which is becom-ing an increasingly central topic in geometric group theoryrekindling in higher dimensions the monumental impact ofgroup actions on trees introduced by Serre in the 1970s. JacekSwiatkowski (Wroclaw) gave a mini-course on “SimplicialNonpositive curvature” which is an exciting emerging area andmany of the student participants now embarking on a PhD areactually planning on working in this topic. Kevin Whyte (UIC)gave a mini-course on “Quasi-isometric Rigidity” which liesat the heart of geometric group theory and has featured manystriking results beginning with Gromov’s virtually nilpotentclassification of groups with polynomial growth. Finally, MikeDavis (Ohio) gave a mini-course on “L2-Betti numbers” whichis an older and more analytic topic that is showing an increas-ingly strong presence in the theory of infinite groups.

These courses were a great success. In addition to a substan-tial student presence, there were a fair number of seasonedmathematicians from neighboring fields who were interested

in learning more about geometric group theory. Their numer-ous informed questions heightened the level of communicationduring the mini-courses for all participants.

During the second week, there were approximately thirty ad-vanced research lectures on a wide variety of topics within ge-ometric group theory. We were able to include a number ofshorter contributed talks by additional participants and manyof these talks built upon the mini-courses from the previ-ous week. For instance: Bruce Kleiner spoke about the quasi-isometric rigidity of right-angled Artin groups, building uponboth the quasi-isometric rigidity, and the CAT(0) cube com-plex mini-courses. Tim Hsu spoke about cubulating graphs ofgroups, building upon CAT(0) cube complexes and relative hy-perbolicity. Roman Sauer spoke about measure theoretic meth-ods in the study of L2-Betti numbers, building upon the intro-duction to L2-Betti numbers in group theory. There were excit-ing new announcements of results in the area, such as Osin’sannouncement of the construction of certain small-cancellationmonsters settling old questions from the Kovourka problembook. Mark Feighn announced a surprising characterizationof constructible subsets of free groups. There were also manynew research directions opened: Jon McCammond outlined anew approach to geometrize Artin groups to resolve notoriousproblems about their algebraic properties.

The participants were able to spend much time together in theevenings and during the breaks, to carry on ongoing collabora-tions. The conference will have a major impact on the next gen-eration of researchers in the area, by introducing them to sev-eral of the most important current topics, and allowing them tointeract with seasoned researchers in the field.

Prix CRM-Fields-PIMS 2007(suite de la page 15)

4. , Single scale analysis of many fermion systems. II. Thefirst scale, Rev. Math. Phys. 15 (2003), no. 9, 995 – 1037.

5. , Single scale analysis of many fermion systems. III. Sec-torized norms, Rev. Math. Phys. 15 (2003), no. 9, 1039 – 1120.

6. , Single scale analysis of many fermion systems. IV. Sec-tor counting, Rev. Math. Phys. 15 (2003), no. 9, 1121 – 1169.

7. , A two dimensional Fermi liquid. I. Overview, Comm.Math. Phys. 247 (2004), no. 1, 1 – 47.

8. A two dimensional Fermi liquid. II. Convergence,Comm. Math. Phys. 247 (2004), no. 1, 49 – 111.

9. , A two dimensional Fermi liquid. III. The Fermi surface,Comm. Math. Phys. 247 (2004), no. 1, 113 – 177.

10. , Particle — hole ladders, Comm. Math. Phys. 247(2004), no. 1, 179 – 194.

11. , Convergence of perturbation expansions in fermionicmodels. I. Nonperturbative bounds, Comm. Math. Phys. 247(2004) no. 1, 195 – 242.

12. , Convergence of perturbation expansions in Fermionicmodels. II. Overlapping loops, Comm. Math. Phys. 247 (2004)no. 1, 243–319.

13. J. S. Feldman and K. Osterwalder, The Wightman axioms andthe mass gap for weakly coupled (Φ4)3 quantum field theories,Ann. Physics 97 (1976), no. 1, 80 – 135 .

14. J. Glimm and A. Jaffe, Positivity of the φ43 Hamiltonian, Fort-

schr. Physik 21 (1973), 327 – 376.

BULLETIN CRM–16


INTRIQby Gilles Brassard (Université de Montréal)

The launch of the Transdisciplinary Institute forQuantum Information (INTRIQ) took place at theCRM on Thursday November 9, 2006.

Quantum information is an emerging field of re-search, at the crossroads of mathematics, physics,computer science and chemistry. It could poten-tially bring a fundamental revolution, not only

in our way of processing information, but also in our way ofunderstanding the world. Ever since the fundamental work ofClaude Shannon and Alan Turing during the first half of the20th century, information theory has been anchored in a clas-sical conception of the physical world inherited from Newtonand Einstein. This conception has prevented us from exploitingthe full potential of the physical world for the processing of in-formation since in reality the world is governed by the laws ofquantum mechanics. These laws are quite different from thoseof classical physics. For instance, quantum mechanics teachesus that elementary particles do not behave like macroscopic ob-jects such as planets and hockey pucks. What influence couldthis reality have on our way of processing information? Sev-eral laboratories have been established worldwide to investi-gate this avenue of research; the members of the INTRIQ areamong the pioneers of this extraordinary scientific adventure.

The day started with welcoming remarks from Gilles Brassard,CRC in Quantum Information Processing in the department ofcomputer science and operations research at the Université deMontréal and director of INTRIQ, and Francois Lalonde, di-rector of the CRM. This was followed by five talks on differ-ent aspects of quantum information: Barry Sanders (Univer-sity of Calgary) lectured on Implementing quantum information,Michele Mosca (University of Waterloo) lectured on Self-testingof quantum circuits, Chip Elliott (BBN) lectured on Architecturesfor quantum networks, John Watrous (University of Waterloo)lectured on Zero-knowledge against quantum attacks, and CharlesH. Bennett (IBM Research) lectured on Privacy, Publicity, andPermanence of Information. The day ended with the first plenarymeeting of the INTRIQ members, and a reception not to be for-gotten soon.

The initial membership of INTRIQ was determined at theplenary meeting. In alphabetical order of institutions andthen of members, we have José Manuel Fernandez and Nico-las Godbout (École Polytechnique de Montréal), David Avis,Aashish Clerk, Claude Crépeau, Guillaume Gervais, PeterGrütter, Patrick Hayden, Michael Hilke, and Prakash Pa-nengaden (McGill University), Michel Boyer, Gilles Brassard,Richard MacKenzie, and Alain Tapp (Université de Montréal)and Alexandre Blais, Patrick Fournier, and André-Marie Trem-blay (Université de Sherbrooke).

During that meeting, Gilles Brassard was confirmed in his po-sition of Director of the Institute. It was decided that director-ships would last for a period of two years and would neverbe twice in a row at the same institution. It was also decidedthat the next Director will be Michael Hilke. A “comité dessages” was formed, which includes the Director (Brassard), theDirector-elect (Hilke), and one additional member per institu-tion: Alexandre Blais, Nicolas Godbout, Patrick Hayden, andRichard MacKenzie.

It was decided that one of the most important activities wouldbe a regular seminar (which has since then been scheduled onTuesday mornings at 11).

For more detail, please consult the INTRIQ website athttp://www.intriq.org.

CRM – ISM PostdoctoralFellows 2006 – 2007

Here is a list of the 9 CRM – ISM Postdoctoral Fellows who arein residence at one (or several) of the universities associatedwith the CRM during the academic year 2006 – 2007. For eachpostdoctoral fellow, the years of the fellowship, the date andinstitution of Ph.D., the field of research and the supervisor(s)are listed.

Félix Carbonell (2006 – 2008). Ph.D. 2006, Universidad de LaHabana. Statistics. Supervisor: Keith Worsley (McGill).

Pierre Charollois (2005 – 2006). Ph.D. 2005, Université de Bor-deaux. Number Theory. Supervisor: Henri Darmon(McGill).

Stefan Friedl (2006 – 2008). Ph.D. 2003, Brandeis University.Geometry and Topology. Supervisors: Olivier Collin andSteven Boyer (UQÀM).

Amy Glen (2006 – 2008). Ph.D. 2006, University of Adelaide.Combinatorics. Supervisors: Srecko Brlek and ChristopheReutenauer (UQÀM).

Basak Gurel (2006 – 2008). Ph.D. 2003, UC Santa Cruz. Geom-etry and Topology. Supervisors: Octav Cornea and Fran-cois Lalonde (Montréal).

Nathan Jones (2005 – 2007). Ph.D. 2005, UC Los Aangeles.Number Theory. Supervisors: Chantal David (Concordia)and Andrew Granville (Montréal).

Emmanuel Lorin (2005 – 2007). Ph.D. 2001, ENS – Cachan. Ap-plied Mathematics. Supervisor: André Bandrauk (Sher-brooke).

Andrew McIntyre (2005 – 2007). Ph.D. 2002, SUNY at StonyBrook. Mathematical Physics. Supervisor: Dimitri Ko-rotkin (Concordia).

Dan Mangoubi (2006). Ph.D. 2006, Technion. Analysis. Super-visors: Dmitry Jakobson (McGill) and Iosif Polterovich(Montréal).

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News from the LaboratoriesNouvelles des laboratoires

Conference on 3-manifold topology in honourof Peter Shalen’s 60th birthdayby Steven Boyer (UQÀM and CIRGET Laboratory)

A conference in honour of Peter Shalen, whose work has beena major force in bringing many different aspect of mathemat-ics to bear on the study of 3-manifolds, and in expanding theinfluence of 3-manifold topology into other areas, was held atthe CRM on July 12 – 15, 2006. The organisers were S. Boyer(UQÀM), R. Canary (Michigan), M. Culler (UIC), N. Dunfield(Caltech) and B. Farb (Chicago), and the main speakers were I.Agol (UIC), C. Gordon (University of Texas), A. Lubotzky (He-brew University), D. Margalit (University of Utah), Y. Minsky(Yale), M. Mirzakhani (Princeton), J. Morgan (Columbia), L. Ng(Stanford) P. Ozsvath (Columbia), J. Rasmussen (Princeton), M.Sageev (Technion) and P. Storm (Stanford).

The subject of 3-manifold topology has a long history of deepand interesting interactions with other parts of mathematics.The complexity of these connections has increased with time,as the subject developed, and has exploded in recent years. Thecoincidence of this explosion with the recent solution of severalof the major outstanding problems in the area made this a goodtime for reflection on its relationship to the rest of mathematics.

The conference brought together a varied group of leadingresearchers whose work demonstrates deep connections be-tween 3-manifold topology and other areas of mathematics.The areas covered by the lectures included: geometrizationof 3-manifolds; virtual properties of 3-manifolds; combinato-rial group theory and coarse geometric properties of groups;properties of random 3-manifolds and asymptotic propertiesof hyperbolic surfaces; Floer homology, Khovanov – Rozanskyhomology; 3-manifold invariants arising from contact struc-tures; character varieties; and Dehn surgery. The five lecturesdelivered by recent Ph.D.s and the large number of studentsand young mathematicians who attended reflected the robusthealth of the subject.

Nouvelles du laboratoire de mathématiquesappliquéespar Robert Owens (Université de Montréal)

– Les Quatrièmes Journées montréalaises de calcul scientifique au-ront lieu les 16 et 17 avril 2007 au CRM, et ont pour butd’encourager les échanges scientifiques au sein de la com-munauté de calcul scientifique au Québec et ailleurs.

– Le premier Atelier de résolution de problèmes industriels deMontréal aura lieu à la fin du mois d’août 2007, et réunira des

représentants de l’industrie, des chercheurs universitaires,des étudiants des cycles supérieurs et des stagiaires post-doctoraux. Il est organisé par le CRM, en collaboration avecle GERAD, le CRT et le rcm2, et est parrainé par le réseaude centres d’excellence MITACS. Anne Bourlioux (Montréal)sera l’une des organisatrices.

– Peter Bartello (McGill) sera un des organisateurs du pro-gramme Nature of High-Reynolds Number Turbulence qui seratenu a l’automne 2008 a l’Institut Isaac Newton de l’univer-sité Cambridge.

– André Bandrauk (Sherbrooke) vient d’être nommé Fellowpar la Humboldt Foundation à Berlin, Allemagne. Il feradonc 3 stages de 4 mois à l’Université Libre de Berlin et auMax Born Institute en 2007, 2008 et 2009.

Series of lectures by Jean-Pierre Serreby John Labute (McGill and CICMA Laboratory)

From September 4 to October 4, 2006, Jean-Pierre Serre (Col-lège de France) gave a series of 8 lectures on Witt invariants inGalois Cohomology at McGill University.

To describe the main topics treated in the lectures, fix a groundfield k0 and consider field extensions k/k0. Let Etn(k) be the setof isomorphism classes of etale algebras L over k of rank n andlet W(k) be the Witt ring of k. The invariants in question are thenatural transformations of Etn(k) to W(k). One such invariantis given by the trace form qL of an etale algebra L, and one ofthe aims of the lectures was to give the proof that every Wittinvariant can be written in a unique way as

L 7→[n/2]

∑i=0

yiλiqL,

where yi ∈ W(k0) and λiqL is the ith exterior power of qL.

The result can also be stated in terms of Sn-torsors sinceEtn(k) = H1(k, Sn) and the main motivation for the lectureswas to extend the above result to An-torsors. Serre showed that,in this case, the Witt invariants form a free W(k0)-module withbasis λiqL (0 ≤ i ≤ [n/4]).

In the second part of the lectures, Serre gave an application ofthese results to give a criterion for the existence of a self-dualnormal basis (BL basis) for an Sn or An-Galois algebra L. Thecondition for the Sn case is that for the etale algebra E = LSn−1

we have qE = 〈1, . . . , 1〉. For An we have the same conditionwith An−1 replacing Sn−1.

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News/Nouvelles2006 CRM – Fields – PIMS Prize Lecture by NicoleTomczak-Jaegermannby Alina Stancu (Concordia University)

It is always impressive to hear outstanding math-ematicians talk on results that have revolutionizeda field. Nicole Tomczak-Jaegermann’s lecture as re-cipient of the 2006 CRM – Fields – PIMS prize at theCRM on November 22 was no exception. Span-ning a decade of her latest work, she described re-sults in Banach space theory, asymptotic geometricanalysis and the interaction between the two. Thetalk presented many instances where the progressachieved in her recent work was remarkable. Forinstance, let’s recall that Dvoretzky’s theorem was,in the local theory of normed spaces, the first in a

sequence of results to show that any normed space of sufficiently high dimensionadmits a subspace or a quotient (or a subspace of a quotient) which is close, in a cer-tain sense, to being the k-dimensional Euclidean space. Then, in the mid-eighties, Vi-tali Milman asked whether, under some mild conditions, every n-dimensional spaceadmits a subspace or a quotient of some special form. The breakthrough achieved byNicole Tomczak-Jaegermann and her collaborator Stanislaw Szarek consisted in ob-taining a general scheme for constructing spaces saturated with copies of any givenspecific space, with some restrictions on the dimensions. A little harder to state here,but nevertheless equally significant, were her results on the topics of random matri-ces, random polytopes and their relationships to each other.

2007 André-Aisenstadt Prize awarded to Alexander E.Holroyd and Gregory G. SmithThe André-Aisenstadt Mathematics Prize of the CRM is awarded every year forresearch achievement in pure and applied mathematics by young Canadian mathe-maticians. It consists of a $3,000 award and a medal. The CRM is proud to announcethat the recipients of the 2007 André-Aisenstadt Mathematics Prize are Alexander E.Holroyd (UBC) and Gregory G. Smith (Queen’s). An article describing their math-ematical achievements will be published by both recipients in the Bulletin du CRM;the article of Alexander E. Holroyd can be found in the present Bulletin, and thearticle of Gregory G. Smith will appear in the Fall Bulletin.

Le Bulletin du CRM

Volume 13, No 1Printemps 2007

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) de l’Université de Montréal a vule jour en 1969. Présentement dirigé parle professeur François Lalonde, il a pourobjectif de servir de centre national pourla recherche fondamentale en mathéma-tiques et dans leurs applications. Le per-sonnel scientifique du CRM regroupeplus d’une centaine de membres régu-liers et de boursiers postdoctoraux. Deplus, le CRM accueille d’année en annéeun grand nombre de chercheurs invités.

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 denotre mission : le Conseil de recherchesen sciences naturelles et en génie duCanada, le Fonds québécois de la re-cherche sur la nature et les technologies,l’Université de Montréal, l’Université duQuébec à Montréal, l’Université McGill,l’Université Concordia, l’Université La-val, l’Université d’Ottawa, ainsi que lesfonds de dotation André-Aisenstadt etSerge-Bissonnette.

Directeur : François Lalonde

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 : [email protected]

Le Bulletin est disponible auwww.crm.umontreal.ca/rapports/

bulletins.html/

BULLETIN CRM–19


Message from the Directionby Chantal David (Concordia University and Deputy Director of the CRM)

The 2006 – 2007 CRM thematic year is ded-icated to various aspect of combinatorics,with a theme semester on “CombinatorialOptimization” in the Fall, and a themesemester on “Recent advances in Combina-torics” in the Winter. The topics of the yearare as varied as approximation algorithms,

network design, data mining, graph theory, algebraic combi-natorics, combinatorics on words, Macdonald polynomials andHopf algebras, enumerative questions in statistical mechanics,etc. The year will end in beauty with four intensive weeks onlinks between geometry and combinatorics, with concentra-tion periods on “Algebraic Geometry and Algebraic Combina-torics” and “Real, Tropical, and Complex Enumerative Geom-etry”. The André-Aisenstadt chairs of the Fall semester wereNoga Alon (Tel Aviv) and Paul Seymour (Princeton), and theAndré-Aisenstadt chair of the Winter semester is Richard Stan-ley (MIT), who will be visiting the CRM in the next weeks.

The 2006 – 2007 academic year will also be remembered as theyear of the NSERC MRS application, culminating in the SiteVisit which took place at the end of January. Those were hecticweeks, and now that the dust is settling, the first thing to do isto thank once again everybody who participated in this event,the personnel of the CRM, the researchers from Montréal, fromeverywhere in Québec, from Canada and around the world,the graduate students, post-doctoral fellows and administra-tors from the various universities associated with the CRM.The site visit was an occasion to assemble those who workedhard to make the CRM into the center of excellence that it isnow, to look at what was accomplished in the past, and to pre-pare for the future. Presenting to the Site Visit Committee allthe scientific activities taking place at the CRM was a highlynon-trivial challenge: the theme years and theme semesters ofcourse, but also the multidisciplinary program, the industrialprogram, the education and outreach program, the general pro-gram and short thematic programs, the laboratories, etc. Morethan one hundred persons were present to describe those vari-ous scientific activities.

The big blue boards with the names and photographs of theAndré-Aisenstadt Chair holders which have just been installedat the entrance of the CRM, give a glimpse of its activities sincethe seventies. And the impressive list of prestigious speak-ers will continue to get longer in the coming years. For theacademic year 2007-2008, dedicated to “Applied DynamicalSystems” and “Dynamical Systems and Evolution Equations,”the Aisenstadt chairs will be Gerhard Huisken (Max-Planck-Institut), John Rinzel (Courant), John Tyson (Virginia Tech) andJean-Christophe Yoccoz (Collège de France). Finally, during the

academic year 2008 – 2009, dedicated to “Probalistic Methodsin Mathematical Physics” and “Perspectives and Challenges inProbability”, the confirmed Aisenstadt chairs will be AndreiOkounkov (Princeton), Craig Tracy (UC Davis) and WendelinWerner (Orsay).

The prize lecture of Nicole Tomczak-Jaegermann took place atthe CRM last November. Nicole Tomczak-Jaegermann is thefirst recipient of the new CRM – Fields – PIMS prize, fundedequally by the three Canadian institutes of mathematics. Theprize, which was established by the CRM and the Fields Insti-tute as the CRM – Fields Prize in 1994, changed its name to theCRM – Fields – PIMS Prize in 2005 when PIMS became an equalpartner. The prize recognizes exceptional achievement in themathematical sciences. The recipient of the 2007 CRM – Fields –PIMS prize is Joel S. Feldman (UBC), and the present Bulletinpresents an account of his work by his collegue David Brydges(UBC). The 2007 André-Aisenstadt Prize recipients are GregoryG. Smith (Queen’s) and Alexander E. Holroyd (UBC). To dojustice to the wonderful pictures in Holroyd’s article about hiswork, some pages of the present Bulletin appears in colour, apremière for the Bulletin du CRM!

In the series of “Grandes Conférences du CRM,” which startedlast year, scientists with a gift for communication are invitedto present exciting recent developments in mathematics to ageneral public. Bart de Smit (Universiteit Leiden) presented anow famous conference on Escher and the Droste Effect at McGillUniversity last November. This project, an initiative of Hen-drik Lenstra (Leiden and Berkeley) is now run by Bart de Smit.Their joint paper in the Notices of the AMS gives a complete ac-count of the project, about one of the lithographs of the famousDutch artist M.C. Escher. In his “Print Gallery,” a man looks ata print of a city. By a strange twist, both the gallery in whichhe is standing and he himself are part of the print of the city.Escher left a mysterious white hole in the middle. The lectureexplains the mathematical structure of the print, and includeswonderful computer animations showing how the hole in themiddle could be filled. One of the fascinating aspects of thetalk was the way it showed how Escher, without any mathe-matical formation, was led to subtle mathematical concepts asconformal mapping from an aesthetic point of view. The next“Grande Conférence du CRM” will take place at UQÀM inMarch, and will be given by Jean-Paul Delahaye (Laboratoired’informatique fondamentale, Université de Lille) on Les limiteslogiques et mathématiques.

Finally, a new CRM laboratory has just been added to the eightalready existing laboratories: the “Transdisciplinary Institutefor Quantum Information (INTRIQ),” directed by Gilles Bras-sard of Université de Montréal. Welcome aboard!

BULLETIN CRM–20

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