Le Bulletin du CRM · Le Bulletin du CRM • ... (Concordia University) ... (Stanford) and Terence Tao (UCLA). The thematic year be-gan with the SMS–NATO

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

Automne/Fall 2006 | Volume 12 – No 2 | Le Centre de recherches mathématiques

A Review of CRM’s 2005 – 2006 Thematic ProgrammeAn Exciting Year on Analysis in Number Theory

by Chantal David (Concordia University)

The thematic year “Analysis in Number The-ory” that was held at the CRM in 2005 –2006 consisted of two semesters with differ-ent foci, both exploring the fruitful interac-tions between analysis and number theory.The first semester focused on p-adic analy-sis and arithmetic geometry, and the second

semester on classical analysis and analytic number theory. Inboth themes, several workshops, schools and focus periodsconcentrated on the new and exciting developments of the re-cent years that have emerged from the interplay between anal-ysis and number theory. The thematic year was funded by theCRM, NSF, NSERC, FQRNT, the Clay Institute, NATO, andthe Dimatia Institute from Prague. In addition to the partici-pants of the six workshops and two schools held during thethematic year, more than forty mathematicians visited Mon-tréal for periods varying from two weeks to six months. Thethree André-Aisenstadt Chairs were Manjul Bhargava (Prince-ton), K. Soundararajan (Stanford) and Terence Tao (UCLA).

The thematic year be-gan with the SMS – NATOSummer school “Equidis-tribution in Number The-ory,” organised by An-drew Granville (Mon-tréal) and Zeev Rudnick(Tel Aviv), which washeld from July 11 to 22,2005. The summer schoolwas primarily targetedat senior graduate stu-dents, postdoctoral fel-lows and junior facultymembers. It was attendedby more that 100 partic-ipants from around theworld. The main subjectsof the school were uni-

form distribution (applications to cryptographic protocols, dis-

tribution of integers, and level statistics), integer and rationalpoints on varieties (geometry of numbers, the circle method,homogeneous varieties via spectral theory and ergodic theory),the André – Oort conjectures (equidistribution of CM-pointsand Hecke points, and points of small height) and quantumergodicity (quantum maps and modular surfaces) The mainspeakers were Yuri Bilu (Bordeaux I), Bill Duke (UCLA), JohnFriedlander (Toronto), Andrew Granville (Montréal), RogerHeath-Brown (Oxford), Elon Lindenstrauss (New York), JensMarklof (Bristol), Zeev Rudnick (Tel Aviv), Wolfgang Schmidt(Colorado, Boulder and Vienna), K. Soundararajan (Michigan),Yuri Tschinkel (Göttingen), Emmanuel Ullmo (Paris-Sud), andAkshay Venkatesh (MIT).

Terence Tao

The workshop on “p-adic repre-sentations,” organised by HenriDarmon (McGill) and AdrianIovita (Concordia) in September2006, had two major themes: a p-adic Langlands correspondenceand p-adic families of motives.These two themes have manycommon features and they are at astage of development where theycould influence each other if onlygiven an opportunity. The goal of

the workshop was to create such an opportunity by bring-ing together researchers from both fields, and the organisersare happy to report that their hopes were amply fulfilled.They felt that the workshop marked a decisive step in theevolution of the two themes through the important resultsreported at the workshop, but mostly through the interest-ing ideas and questions offered here for the first time andthe large number of collaborations that were started at theworkshop. The workshop began with expository talks on themain ideas involved in the two main themes, by ChristopheBreuil (IHES) and Matthew Emerton (Northwestern). On thetheme of p-adic Langlands correspondance, the main speak-ers were Pierre Colmez (Jussieu), Matthias Strauch (Münster),

(continued on page 10)

•www.crm.umontreal.ca •

André-Aisenstadt Chairsby Chantal David (Concordia University)

THE André-Aisenstadt chairs for the Thematic Year “Analysis in Number Theory” held at the CRM in 2005-2006 were threeexceptional young mathematicians working in different areas of number theory, that have been influenced and developed

by their scientific achievements: Manjul Bhargava (Princeton University), K. Soundararajan (Stanford University) and TerenceTao (UCLA). The organizers are specially proud that they managed to convince the three André-Aisenstadt chairs to stay inresidence at the CRM for extended periods, from one week to several months. The quality of the many (many!) lectures given bythe three André-Aisenstadt chairs, and the inspiration provided by their presence was one of the reasons for the great successof the special year.

Manjul Bhargava

Manjul Bhargava

The work of Manjul Bhargavaon higher composition lawshas revolutionized the field,and created a new and excitingline of research on a topic thathad seen little activity sincethe time of Gauss! His workis appearing in a series of fun-damental papers in Annals ofMathematics that are reshapingseveral areas of number the-ory. The series of lectures thatManjul Bhargava gave on thattopic during his stay in Mon-treal was a magical moment ofthe Thematic Year.

The first example of a composition law is due to Gauss, whoshowed in Disquisitiones Arithmeticae that the set of primitiveSL2(Z)-orbits of binary quadratic forms of discriminant D hasa natural group structure (the class group). To this day, Gausscomposition is still one of the best way to understand the classgroup of quadratic fields. Are there composition laws for other,non-quadratic, fields? This is the question that was answeredby Bhargava. When rephrasing the problem as an orbit prob-lem, this means understanding the integer points on the orbitsof a prehomogeneous vector space, and finding a one-to-onecorrespondence between those points and interesting algebraicstructures. In the case of Gauss, the interesting algebraic struc-ture is of course the group of classes of binary quadratic formsof discriminant D. There are 29 prehomogeneous vector spaces,their classification was an amazing achievement of representa-tion theory, and their dimensions go up to 50. The new insightof Bhargava is to understand those spaces from purely arith-metic methods. Looking again at Gauss composition law, Bhar-gava rephrases it as a “cube law”, corresponding to the pre-homogeneous vector space Z2 ⊗ Z2 ⊗ Z2/ SL2(Z)3. The inte-ger points on that space are in one-to-one correspondence withtriplet of ideals in quadratic rings such that the product is triv-ial. By taking one component of the vector space, one retrievesGauss composition. The cube law also leads to other interesting

algebraic structures, as 3-torsion in the class group of quadraticfields. The next step is cubic fields: are there any cubic compo-sition laws similar to the cube law? This is the magic of Bhar-gava’s work, cubic (and quartic and quintic) fields can be seenas the integer points of some prehomogeneous vector spaces.

A spectacular application of this new point of view is an oldquestion of algebraic number theory, namely counting numberfields by discriminant. Gauss showed that the order of the classgroup of the imaginary quadratic fields of discriminant D is, onaverage, about πD1/2/18ζ(3). The real quadratic analogue isdue to Siegel. Gauss proved his result by constructing a funda-mental domain for the action of SL2(Z) on Sym2(Z2), and bycounting lattice points in that region. This can be done easilyas one can cut out the cusp (an infinite tentacle) which containsno lattice points. One has to wait more than 150 years for a re-sult on cubic fields, which is due to Davenport and Heilbronnin 1970. They showed that the number of cubic fields of abso-lute discriminant at most X is ∼ X/3ζ(3), again counting inte-ger points on a fundamental domain for the action of GL2(Z)on Sym3(Z2). In this case, the region is four-dimensional, thecusps contain many points, and the computations are signif-icantly harder. How can one count integer points in the pre-homogeneous spaces associated to quartic and quintic rings?Those spaces have dimension 12 and 40 respectively, and thereare no known techniques to separate reducible and irreduciblepoints (and we only wish to count the irreducible points). De-veloping hybrids methods between the geometry of numbersand the analytic methods, those cases were also treated byBhargava. An unexpected consequence of his work is that thereis a positive proportion of quartic fields with Galois group D4.Of course, a random polynomial of degree 4 has Galois groupS4, but counting number fields by discriminant yields a totallydifferent story.

The André-Aisenstadt lecture of Manjul Bhargava, which tookplace during the Workshop on Additive Combinatorics, wasconcerned with the representation of positive integers byquadratic forms. The classical Four Squares Theorem of La-grange asserts that any positive integer can be expressed as thesum of four squares; that is, the quadratic form a2 + b2 + c2 + d2

represents all positive integers. The (amazing!) fifteen theo-rem of Conway and Schneeberger (1993) states that if a posi-tive definite quadratic form whose associated matrix has inte-

BULLETIN CRM–2


ger values represents all positive integers ≤ 15, then it rep-resents all positive integers. This result is best possible, asa2 + 2b2 + 5c2 + 5d2 represents all positive integers other than15. The original proof was never published, and no proof ex-isted in the literature before Bhargava presented a simplifiedproof of Conway and Schneeberger’s Theorem. What if theform has integer coefficients, but the associated matrix doesnot necessarily have integer entries? Conway then conjecturedthat if such a quadratic form represents all integers up to 290,then it represents all integers. The computational complexityof this problem is such that Conway predicted that it will notbe solved during his lifetime. This was solved in 2005 by Man-jul Bhargava, in collaboration with Jonathan Hanke, one of themost extraordinary theorems in the theory of representationsof integers by sums of squares.

Terence Tao

The series of lectures of Terence Tao took place during theSchool and Workshop on Additive Combinatorics held at theCRM in April 2006. In the André-Aisenstadt lecture, he de-scribed his work with Ben Green on Long Arithmetic Progres-sions in the Primes which was announced in 2004, stunningthe mathematical community. This is part of the work thatwas recognized by the Fields medal awarded to Terence Taoin Madrid in August 2006, for his contributions to partial dif-ferential equations, combinatorics, harmonic analysis and ad-ditive number theory.

Finding arithmetic progressions in the primes is an old andclassical problem of additive number theory, together with theTwin Prime conjecture (are there infinitely many primes p suchthat p + 2 is also prime?) and Goldbach’s conjecture (is everyeven number ≥ 6 the sum of two primes?). An arithmetic pro-gression of length k is a set of integers of the form a, a + b,a + 2b,. . . , a + (k − 1)b. It is easy to see that an arithmetic pro-gression of length 3 in the primes is equivalent to three primesnumbers p, q, r such that p + r = 2q. It was proven by Vander Corput in 1929 that the primes contain infinitely manyarithmetic progressions of length 3. This uses the same tech-nique (the Circle Method) as Vinogradov’s proof that everysufficiently large odd n is the sum of three primes. The first re-sult for arithmetic progressions of length 4 is the achievementof Heath-Brown, who showed that there are infinitely manyarithmetic progressions in the integers of length 4, with 3 en-tries prime and the fourth entry almost prime (a product of atmost 2 primes). This is the same limitation as in Chen’s theo-rem, who showed that there are infinitely many pairs p, p + 2with p prime and p + 2 almost prime, failing short of provingthe Twin Prime conjecture. Extending Heath-Brown’s result toan arithmetic progression of length 4 in the primes was thenconsidered a fundamental and very difficult problem, and onewhere new ideas and techniques would be needed. The num-ber theory community was thus stunned when news of the the-orem of Green and Tao began to circulate: they had proven that

there are infinitely many arithmetic progressions of length k inthe primes, for any positive integer k (and not just k = 4)! Howcan one prove such a theorem? According to Tao, what is beingused is not new knowledge about the primes, but new knowl-edge about arithmetic progressions. At the heart of the work ofGreen and Tao is Szemerédi’s Theorem (1975) which says thatif a subset of the integers has positive density, then it containsinfinitely many arithmetic progressions of length k for any k. Ofcourse, the primes do not have positive density, and one can-not apply Szemerédi’s theorem directly to them. All the proofsof Szemerédi’s theorem rely on the dichotomy between struc-ture and (pseudo)randomness. Both types of sets contain arith-metic progressions (for completely different reasons). Splittingthe primes into a “structured part” and a “random part,” onehas to deal with each part separately, and the magic of Sze-merédi’s Theorem can be applied to one or the other.

In a series of lectures entitled “Combinatorial and ergodic tech-niques for proving Szemerédi-type theorems” given duringthe CRM-Clay School of Additive Combinatorics, Terence Taoshared with the audience his fascination with Szemerédi’s The-orem, and with the many existing proofs involving seeminglydrastically different ideas, all of them influential, importantand deep: the original proof of Szemerédi using combinato-rial graph theory, the proof of Furstenberg using ergodic the-ory, the proof of Gowers using Fourier analysis, and more re-cently the proof using hypergraph regularity due to Gowers,and to Nagle, Rödl and Schacht. He also presented many newcombinations and developments of the above that are appear-ing in his work, explaining beautifully the analogies betweenthe different approaches, and presenting new points of views,such as his development of an analytic analogy to the Ruzsa –Szemerédi triangle removal lemma.

Finally, during the Workshop on Additive Combinatorics, Ter-ence Tao presented a new approach to hypergraph regularityand removal, which has led to the latest proof of Szemerédi’stheorem. The proofs in this subject, while finite and elemen-tary, involve complicated and lengthy estimates. Tao explainedhow he uses the constructions of ergodic theory to avoid suchcomplicated estimates, though still obtaining explicit results.

During his marathon of brilliant lectures (6 lectures in 7 days!),Terence Tao managed to convey his fascination with additivecombinatorics, explaining his ideas and insights in a very per-sonal and vivid way to a delighted audience.

K. Soundararajan

K. Soundararajan was in Montréal for an extended stay dur-ing the thematic year, and gave two series of lectures, in the“Workshop on L-functions,” and the “Workshop on Anatomyof Integers.” His lectures were a fascinating guided tourthrough the ideas, techniques and challenges of modern ana-lytic number theory, given by one of the leaders in the field.


BULLETIN CRM–3


School on Additive Combinatoricsby Andrew Granville (Université de Montréal)

The CRM – Clay School on Additive Combi-natorics took place at Université de Montréalfrom March 30 to April 5, 2006, with morethan 100 participants from around the world.The school started with two introductory mini-courses given by the two co-organizers, to pre-pare the students for the main lecture series:

Jozsef Solymosi (UBC) gave an introduction to Hungarian stylecombinatorial geometry, and in particular the connection withRoth’s theorem and related results; Andrew Granville (Univer-sité de Montréal) developed an historical view of combinato-rial number theory, then gave complete proofs of the Freiman –Ruzsa structure theorem, and of Roth’s theorem in the spirit ofGowers.

There were then four main lecture series which formed the coreof the School:

Ben Green (University of Bristol) explained in his series of lec-tures “Quadratic Fourier analysis” the developments of theideas from Gowers’ Fourier analytic proof of Szemerédi’s theo-rem that were essential to his work with Terence Tao on primesin arithmetic progressions. He explained beautifully the essen-tial ideas that go into the new multidimensional Fourier analy-sis by expounding on the simplest new case, namely quadraticFourier analysis. The key is in breaking functions up into apart that reflects structure and a part that reflects random be-haviour, something that can now be done very efficiently.

Photo: Antal BalogBen Green (left) and Bryna Kra (right)

Bryna Kra (Northwestern University) gave us insights into “Er-godic methods in combinatorial number theory.” Starting witha detailed explanation of Furstenberg’s key ideas for provingSzemerédi’s theorem, she showed how thinking in ergodic the-ory is developing today (as in her work with Host), and espe-cially how it is effecting the development of methods in com-binatorial enumeration and in modern harmonic analysis.

The lecture series of Terence Tao, entitled “Combinatorial andergodic techniques for proving Szemerédi-type theorems.” isdescribed in another article of this Bulletin.

In “Structure of sumsets and applications,” Van Vu (RutgersUniversity) gave details of the proof of his result with Sze-merédi on the existence of long arithmetic progressions in sum-sets, in which they recently obtained the best possible resultsconjectured by Erdos.

The four principal lecturers delivered well-taught lectures, andeach of them was able to give remarkable insight into their cho-sen subject.

Among the other talks that occured during the School, AntalBalog (Alfréd Rényi Institute of Mathematics) gave a speciallecture on the Balog – Szemerédi – Gowers Theorem, giving acomplete proof.

On the final day of the School, the three senior mathematicianswho did the most to create this field were invited to give pre-sentations. The students found this a very exciting event towrap up the school and, indeed, these “celebrities” were givenan ovation before and after their lectures! In the morning, ImreRuzsa (Alfréd Rényi Institute of Mathematics) gave a beauti-ful presentation of consequences of “Plunnecke’s theorem” aswell as his recent thoughts on non-abelian analogues. In the af-ternoon, Endre Szemerédi (Rutgers, and Alfréd Rényi Instituteof Mathematics) gave a proof of Roth’s theorem he first devel-oped 25 years ago but had never published; and finally GregoriFreiman (Tel Aviv University) in “Inverse additive number the-ory: Results and problems” discussed how he was led to hisstructural understanding, and his belief as to what are the es-sential questions for the future of this subject. The workshopended up with a panel discussion involving all of the speakersat the school, and eventually with all of them speculating as towhat is truly the best possible form of Roth’s theorem.

PublicationsConvexity Properties of

Hamiltonian Group ActionsPerspectives in Riemannian

Geometry

V. Guillemin & R. Sjamaar V. Apostolov et al. (eds.)2005; US$35, AMS members US$28 2006; US$85, AMS members US$68

cf. www.crm.umontreal.ca/pub/BULLETIN CRM–4


Workshop on Additive Combinatoricsby Andrew Granville (Université de Montréal)

THE Workshop on Additive Combinatorics was held at Uni-versité de Montréal from April 6 to April 12, 2006. There

were about 150 participants from about 30 countries. The meet-ing was a high profile event with many of the world’s lead-ing analysts participating, including Jean Bourgain, Tim Gow-ers, Terence Tao, Ben Green, Sergei Konyagin,. . . reflecting thegreat interest in this meeting (the third ever in the sub ject). Theworkshop was funded from the CRM, NSERC, NSF, the ClayFoundation, and the DIMATIA institute in the Czech Republic.The plenary lecturers were:

Terence Tao (UCLA) who discussed “An infinitary approachto (hyper)graph regularity and removal” in which he explainshow he uses the constructions of ergodic theory to avoid com-plicated estimates yet still obtains explicit results.

Harald Helfgott (Université de Montréal) who discussedhis surprising breakthrough on “Growth and generation inSL2(Z/pZ)”; and then Jean Bourgain (Institute for AdvancedStudy) who discussed his work with Gamburd and Sarnak de-veloping Helfgott’s result to other groups and some stunningconsequences in “Sum-product and expanders,” particularlywith reference to certain sieve questions.

Jean Bourgain gave a second plenary talk on “More applica-tions of the sum-product theorem and the quantum cat map” inwhich he solves a well-known question of Kurlberg and Rud-nick. Then Mei-Chu Chang (UC Riverside) gave more detailsof her work with Bourgain on “Sum-product theorems, expo-nential sum bounds and applications.” results that are havingan enormous impact on analytic number theory.

Photo: Antal Balog

Sergei Konyagin (left) and Imre Ruzsa (right)

Tamara Ziegler (Institute for Ad-vanced Study) proved a wonderfulold conjecture of Erdos in “Configu-rations in sets of positive upper den-sity in Rm” by ergodic theory meth-ods. One of the key difficulties in un-derstanding Gowers’ norms is to un-derstand the structures on which theyare large. At this stage of developmentof this sub ject, it seems vital to givea more accessible description that canbe useful for Fourier analysts, and thiswas the focus of Bryna Kra (North-western University) in “Gowers norms in ergodic theory andadditive combinatorics.” In “Linear equations in primes.” BenGreen (University of Bristol) discussed his latest work withTerry Tao in trying to generalize their methods as much aspossible in the direction of the prime k-tuplets conjecture. Itis shocking to see how much they have succeeded in doing! Inparticular they outlined plans to show that for any admissible

set of k linear forms, no two of which are linearly dependantover the integers, there are infinitely many integer values forthe variables such that the forms all simultaneously take onprime values.

One key issue in trying to gain a complete understanding ofSzemerédi’s theorem is to understand circumstances in whichone has significantly fewer arithmetic progressions of length4 than expected, despite having all of the Fourier coefficientssmall. Timothy Gowers (University of Cambridge) describedhis new elegant construction in “A uniform set with few pro-gressions of length 4”.

Photo: Antal Balog

Timothy Gowers

Gowers’ norms have provedvery fruitful in the hands ofthe analysts and it is inter-esting to see how they ef-fect the more classical meth-ods of analytic number the-ory. Trevor Wooley (Universityof Michigan) and Antal Ba-log (Alfréd Rényi Institute ofMathematics) have been devel-oping “Variants of the CircleMethod after the work of Gow-ers”, and explained how onecan now do much better in the key Diophantine questions inthe circle method. The final plenary talk, by Imre Ruzsa (AlfrédRényi Institute of Mathematics), entitled “2A and 3A” gave de-tails of his new fundamental results on sumsets, particularly inthe non-abelian setting, for bounds on the size of A + B + C.

There were many other exciting talks,among them: Ernie Croot (Georgia In-stitute of Technology) gave a power-ful structure theorem for “Subsets ofFq with the minimal number of three-term arithmetic progressions” whichbeautifully complements the struc-tural theorems of Green and Tao;Javier Cilleruelo (Universidad Au-tonoma de Madrid), in work withMelvyn Nathanson (Lehman College,New York), showed how to constructprescribed “Dense perfect difference

sets of integers”; Ilya Shkredov (Moscow State University) con-tinued his extraordinary work on understanding the structureof “Sets of large exponential sums”; Sergei Konyagin (MoscowState University) gave new results on “Additive propertiesof product sets in fields of prime order”. After his lecture,Helfgott made a few observations that allowed Konyagin to


BULLETIN CRM–5


Research Contributions of the2006 CAP/CRM Award Winner

by John Harnad (Concordia University)

THE 2006 CAP/CRM Prize in Theoretical and Mathematical Physics was awarded to John Harnad (Concordia Universityand CRM) for his remarkable work on the theory of integrable systems. After obtaining a B.Sc. from McGill University

in 1967 and a M.Sc. from the University of Illinois in 1968, John Harnad did a D. Phil. in theoretical physics at the Universityof Oxford under the direction of Professor J. C. Taylor. After postdoctoral years at the Eötvös Institute in Budapest andCarleton University in Ottawa (1972 – 1975), he was a research associate at the CRM (1975 – 1984), and an associate professorat the Stevens Institute of Technology (1985 – 1986) and at the École Polytechnique in Montréal. He became a member of thefaculty at Concordia in 1989 and is now the director of the Mathematical Physics Group at the CRM.

SOMETIMES it is difficult for a mathematical physicist to steera course between the Scylla of mathematical techniques

that appear to most physicists as too abstract to be of gen-uine use, and the Charybdis of imprecise logic that may lurkbehind the guesswork and analogies characteristic of imagi-native work in theoretical physics. It is gratifying therefore tohave work recognized in this award that is based on precise useof mathematical constructions, whether concrete or abstract, inthe resolution of problems of real interest to physics.

John Harnad

When I began my doctoralstudies in Oxford in 1968, rel-ativistic quantum field theory,which had proved so success-ful in the context of quan-tum electrodynamics, seemedvirtually cast aside as a vi-able approach to the subnu-clear forces. This was mainlydue to the seemingly insur-mountable obstacles in imple-menting the renormalizationscheme for short range inter-actions involving vector (andaxial-vector) currents, and the

apparent inconsistency of trying to use a perturbative approachin the case of strong interactions. Both these obstacles weredramatically overcome with two great developments that oc-curred at the beginning of the 1970s. The first of these wasthe discovery that a consistent, renormalizable quantum fieldtheory existed, unifying the electro-weak interactions, basedon a nonabelian extension of the gauge symmetry of quan-tum electrodynamics, combined with the mechanism of spon-taneous symmetry breaking. The second was the demonstra-tion that this framework could be consistently extended to in-clude the strong interactions, since considerations valid to allorders in perturbation theory implied that the effective cou-pling strengths became weak in the short distance limit. Overa dozen Nobel prizes were awarded in the following years inrecognition of the various theoretical and experimental devel-opments confirming these discoveries.

My own thesis work, done at the time of this transition, underthe supervision of one of the pioneers of the unified electro-weak theory, J.C. Taylor, was unfortunately still set in themold of the analyticity-symmetry approach to hadronic scat-tering theory characteristic of the preceding period. The im-portance of nonabelian gauge theory however became quicklyclear to everyone. Approaches based on nonperturbative tech-niques also gained prominence, and there was a period of ac-tivity in which energy minimizing classical configurations inpure gauge theory, such as instantons and monopoles, were in-tensely studied. I was able to make some contributions in thisdirection, together with my collaborators [1], concerning clas-sical solutions to field equations having a high degree of sym-metry. This led to a general geometric framework, somewhatanalogous to the Kaluza-Klein approach to the unification ofgravitation and electrodynamics, which became known as “di-mensional reduction” [2]. We were able to show that the Higgsfield, essential to spontaneous symmetry breaking, could bededuced naturally through the process of symmetry reductionfrom a higher dimensional space. Another result to which Icontributed at this time, together with my students and col-leagues [3, 4] was a proof that the field equations of supersym-metric Yang – Mills theory could, as suggested by Edward Wit-ten, be given a purely geometrical formulation as integrabil-ity conditions of an associated linear system, analogous to thetwistor approach to the self-dual Yang – Mills equations whichhad been so successfully applied to the study of instantons andnonabelian monopoles.

The experience this work provided in the use of geometri-cal techniques, complex analyticity and symmetry in the res-olution of relativistic field equations gave a very useful back-ground when I subsequently changed orientation towards thestudy of completely integrable systems. This area, althoughmuch more modest in its goals than elementary particle the-ory, had some important successes of its own. A fairly generalframework was developed, based on inverse spectral meth-ods, that gave a complete understanding of the mechanism un-derlying long term coherent phenomena characteristic of suchsystems, such as solitons and nonlinear quasi-periodic modes.


BULLETIN CRM–6


Research Contributions of the2006 CRM – SSC Award Winner

by Jeffrey S. Rosenthal (University of Toronto)

THE CRM – SSC Prize in Statistics was awarded this year to Jeffrey S. Rosenthal (Department of Statistics, University ofToronto). Professor Rosenthal received his B.Sc. from the University of Toronto in 1988, and a M.A. (1990) and Ph.D.

(1992) from Harvard University. It was there that his interest in applications and practical issues was piqued. After spendingone year at the University of Minnesota, he joined the staff at the University of Toronto (1993). Within 15 years of his Ph.D.,Jeffrey Rosenthal has made outstanding contributions to asymptotic theory related to Markov processes and, with greatinsight and ingenuity, to clarifying the practical implications of theory in this area. He has received many honours andawards over his career, including being elected IMS Fellow in 2005. He discusses here his main research interests.

I completed my Ph.D. in 1992 at Harvard University underthe direction of Persi Diaconis. The first part of my the-

sis (e.g., [12]) studied the mathematics of random walks oncompact Lie groups, and bounded their rates of convergenceto normalised Haar measure, using the theory of group repre-sentations. The second part (e.g., [13]) concerned the theory ofMarkov chain Monte Carlo (MCMC) algorithms. I soon discov-ered that of the two areas, MCMC was less technical and had amuch wider audience, so I gradually let the Lie groups slide.

Jeffrey S. Rosenthal

MCMC algorithms have a longhistory in physics (e.g., [6]),and became popular in statis-tics with the publication of [3]in 1990. They are used pri-marily to sample from proba-bility distributions π(·) whichare too complicated and high-dimensional to be sampledfrom (or integrated against)analytically or numerically. Insuch cases, one instead defines

a Markov chain {Xn} on a measurable space (X ,F ) which con-verges to π(·) in distribution, i.e., such that the total variationdistance

D(n) ≡ supA∈F

|Prob[Xn ∈ A]− π(A)|

tends to 0 as n → ∞.

One particular interest of mine is quantitative bounds on D(n).Such bounds allow applied users of MCMC algorithms to de-termine a number of iterations n∗ such that the distance D(n∗)is sufficiently small, say D(n∗) < 0.01. However, for compli-cated Markov chains this is not easy. One approach is coupling(e.g., [5]), which says that if {Xn} and {X′

n} are two copies of aMarkov chain, and there is a random time T such that Xn = X′

nwhenever n ≥ T, then D(n) ≤ Prob[T > n].

Now, if there is δ > 0 and a probability measure ν(·) whichsatisfy the minorisation condition

Prob[Xn ∈ A |, Xn−1 = x] ≥ δ ν(A), x ∈ C, A ∈ F , (1)

for some subset C (called a small set), then for x, x′ ∈ C, we can

ensure that Prob[Xn = X′n | Xn−1 = x, X′

n−1 = x′] ≥ δ. Further-more, we can control the return of the chains to the subset Cusing the drift condition

Ph(x, x′) ≤ h(x, x′)/α, (x, x′) /∈ C × C (2)

for some function h ≥ 1 and some α > 1, where Ph(x, x′) is theconditional expected value

Ph(x, x′) ≡ E[h(Xn, X′n) | Xn−1 = x, X′

n−1 = x′].

Putting this together, and letting Ex be the expected value ofh(x, Z) with respect to Z ∼ π(·), and writing Rh(x, y) for theexpected value of h with respect to the residual measure µ(·)defined for A, A′ ∈ F by

µ(A, A′) =(Prob[Xn ∈ A | Xn−1 = x]− δ ν(A)

)(1− δ)

×(Prob[X′

n ∈ A′ | X′n−1 = x′]− δ ν(A′)

)(1− δ)

we obtain, e.g., the following ( [2, 14, 16]):

Theorem. Assume (1) and (2). Then for any integers 1 ≤ j ≤ n,

supA∈F

|Prob[Xn ∈ A | X0 = x]− π(A)| ≤ (1− δ)j + α−nBj−1Ex,

with B = max[1, α(1− δ) supC×C Rh

].

This result reduces the difficult problem of bounding the dis-tance D(n), to the simpler problem of verifying (1) and (2). Ithas been applied to a number of applied MCMC algorithms(e.g., [4, 15]), providing concrete numbers (like “140”) for thenumber of iterations n∗ required to make D(n∗) < 0.01. Thus,MCMC theory not only involves interesting mathematics, italso sometimes provides genuine insights into statistical appli-cations.

I have since extended theoretical MCMC ideas in various ways,largely joint with G. O. Roberts (see the reviews [8,9]). My latestresearch interest is adaptive MCMC algorithms ( [10,11]), whichimprove convergence by modifying their transition probabili-ties as they run. I have also branched out into other applica-tions of probability, collaborating with economists, computer


BULLETIN CRM–7


Décès de Maurice L’Abbé, fondateur du CRMMaurice L’Abbé (1920-2006)

Maurice L’Abbé

Né à Ottawa le 20 mai 1920, Mau-rice L’Abbé obtient une licence ensciences mathématiques de l’Uni-versité de Montréal en 1945 avantde poursuivre des études à la pres-tigieuse Université Princeton, auNew Jersey. Il y obtient un Ph.D.grâce à une thèse en logique mathé-matique rédigée sous la directionde Alonzo Church et de Leon Hen-kin.

Dès l’automne 1948, M. L’Abbé sejoint au Département de mathéma-

tiques de l’Université de Montréal, où il accédera au rang deprofesseur titulaire. Directeur du Département de 1957 à 1968,il occupe les fonctions de vice-doyen de la Faculté des sciences,de 1964 à 1968, et de vice-recteur à la recherche de 1968 à 1978.

En 1962, il fonde le Séminaire de mathématiques supérieures :encore aujourd’hui, la tenue de ce séminaire constitue un évé-nement important au sein de la communauté scientifique in-ternationale. En 1968, il crée le Centre de recherches mathéma-tiques de l’Université de Montréal, qui deviendra par la suiteun centre de recherches national.

M. L’Abbé a été président de l’Association canadienne-française pour l’avancement des sciences (ACFAS) en 1964 etprésident de la Société mathématique du Canada de 1967 à1969.

En 1980, M. L’Abbé quitte l’Université de Montréal pour de-venir directeur général du Conseil des sciences du Canada. En1983, le gouvernement du Québec crée le Conseil de la scienceet de la technologie et invite M. L’Abbé à en devenir le premierprésident pour un mandat de six ans. En 1991, il est appelé àprésider la Commission de vérification de l’évaluation des pro-grammes, organisme nouvellement crée pour l’ensemble desuniversités du Québec.

Au cours de sa carrière, M. L’Abbé s’est vu décerner plu-sieurs distinctions académiques. Son apport au développe-ment des sciences au Québec et au Canada a notammentété récompensé du prix Armand-Frappier, de l’Ordre natio-nal du Québec et d’un doctorat honorifique ès sciences del’Université McGill. En 1990, il devenait le premier lauréat duprix Walter-Hitschfeld décerné par l’Association canadienned’administration de la recherche universitaire. En 1995, la So-ciété mathématique du Canada lui décernait la distinction pourservice méritoire. Membre de l’Académie des Grands Montréa-lais, Maurice L’Abbé compte également parmi la quarantainede diplômés honorés de l’Ordre du mérite de l’Association desdiplômés de l’Université de Montréal.

Le dimanche 10 septembre une cérémonie commémoratives’est déroulée dans le hall d’honneur de l’Université de Mon-tréal en hommage à monsieur Maurice L’Abbé, décédé le 21juillet 2006 à l’âge de 86 ans. Nour reproduisons ci-dessousdeux des allocutions prononcées à cette occasion, celle deHarold Kuhn, lié à lui par une amitié constante depuis les an-nées où ils firent leur doctorat à Princeton, et celle de CamilleLimoges, qui le connut particulièrement bien, puisqu’il étaitsous-ministre de la Science et de la Technologie au momentoù Maurice L’Abbé devint président du Conseil de la scienceet de la technologie.

Allocution de Harold KuhnAfter receiving his diploma from the University of Montréal, Mauricewent to Princeton University to do graduate work in Mathematics inthe fall of 1945. I began my studies there in 1947, one year beforeMaurice returned to Montréal in 1948. His return to Montréal hadimportant consequences for two of his friends, Leon Henkin and my-self.

The late 40s saw the birth of long playing records and of high fidelitysound equipment. As a student, Maurice indulged his love of classicalmusic by acquiring a large collection of both, which posed a problemfor his return to Canada. Leon Henkin volunteered to borrow his fa-ther’s car to drive Maurice, his records and his sound equipment backto Montréal. Leon, accompanied by his sister Estelle, would spend avacation in Montréal, then return to New York. On the way, theywould pick up a common friend, Kai Lai Chung, at Cornell Univer-sity. Kai Lai’s visa required him to leave the United States before re-turning with permanent status. Thus the trip had three segments:New York – Princeton, Princeton – Ithaca and Ithaca – Montréal.

I knew this itinerary (but did not know that Leon Henkin had a sis-ter). When I went to the Henkin apartment in New York to beg a rideback to Princeton I discovered that indeed he had a sister. We willcelebrate our 57th wedding anniversary in December.

Estelle played a crucial role later in the trip, distracting the custom’sofficer at the border so that Maurice paid no duty on his records andsound equipment.

In Montréal, Maurice introduced Leon, Estelle and Kai Lai to his cir-cle of friends. This is how Leon Henkin met Ginette Potvin. They havejust celebrated their 56th wedding anniversary.

Whatever profound effect Maurice had on the development of mathe-matics in Canada, his return to Montréal from his studies in Prince-ton had an important influence on the lives of two American mathe-maticians.

In the past few weeks I have been in contact with almost everyone,who is still alive, who knew Maurice in Princeton: Leon Henkin, KaiLai Chung, Gil Hunt, Felix Browder. We are unanimous in character-izing Maurice as “easy to talk to,” “non-competitive in the dog-eat-dog atmosphere of Fine Hall,” “someone who enjoyed the closeness ofPrinceton to New York and who exploited the cultural riches of NewYork.”

BULLETIN CRM–8


The introduction by vice-recteur Berthiaume said that I would speakof Maurice L’Abbé the mathematician. His field was symbolic logic, asubject in which I am not an expert. However, his Ph.D. thesis on thesubject of systems of transfinite types involving lambda conversioncontinued to have an impact in the 60s when P.B. Andrews publisheda book on transfinite type theory that built on Maurice’s thesis as afoundation. Maurice’s final oral exam was administered by AlonsoChurch, Roger Lyndon and Father de Baggis, a logician visitor fromNotre Dame. They graded the thesis with the highest grade: EXCEL-LENT. This is a word that describes every thing that Maurice everdid: EXCELLENT!

Allocution de Camille Limoges

« En 1983, quand la Loi pour le développement scientifique du Québecinstitue le Conseil de la science et de la technologie, Maurice L’Abbéapparaît rapidement comme tout désigné pour en assumer la mise surpied et la présidence.

Pendant dix ans, de 1968 à 1978, il a comme premier vice-recteur à larecherche engagé avec un incontestable succès l’Université de Mont-réal dans sa transformation en un établissement voué intensivementà la recherche et à la formation par la recherche. Puis, après un pas-sage à la Division des politiques de la science de l’OCDE, à Paris,il a pendant quatre ans rempli les fonctions de directeur-général duConseil des sciences du Canada. Personne n’était mieux préparé queMaurice pour donner forme au nouvel organisme conseil québécois etpour en piloter le développement.

De fait, au cours des huit années de sa présidence, jusqu’à 1990, Mau-rice a su impulser cette organisation et lui imprimer beaucoup destraits qui sont toujours les siens.

Pour remplir son rôle de définisseur de voies nouvelles, mais ausside critique, Maurice a engagé avec son Conseil une vaste gamme detravaux qui ont notablement influencé le cours des politiques de larecherche au Québec.

Des travaux sur la reconnaissance du secteur des technologies del’information comme prioritaire, sur l’utilisation des marchés publicspour le développement de la technologie québécoise, sur la mise enoeuvre de la politique du Virage technologique, sur le développementindustriel des biotechnologies, sur la fiscalité de la R-D, mais aussisur l’emploi des diplômés des sciences humaines et sociales ou encore,dès 1986, sur la participation des femmes en science et en technologie.

Tablant sur le caractère public des travaux du Conseil, Maurice s’estnon seulement imposé comme conseiller des ministres, mais aussi ila joué plus largement un rôle moteur de sensibilisation, d’éducation,de mobilisation.

Ainsi, si certains dans la belle ville de Québec, pour ne rien dire desrégions, avaient pu s’inquiéter de ce que le président du Conseil soitun Montréalais — donc automatiquement soupçonné de « montréa-lisme » — c’est néanmoins à Maurice que l’on doit le lancement d’unegrande série de bilans régionaux, étalés sur une décennie, qui ontdressé l’état de situation pour chacune des régions du Québec et pourla première fois esquissé, avec la contribution des acteurs régionaux,

des pistes de développement. Des exercices associés souvent aussi àun sain rappel au réalisme.

Maurice n’avait pas été seulement choisi pour ses réalisations anté-rieures, si impressionnantes fussent-elles. Il l’avait été aussi pour desqualités tout à fait personnelles. Expert, pédagogue, fin dialecticien etdiplomate, Maurice avait la capacité de susciter l’écoute des ministresavec lesquels il eut à travailler, pourtant bien différents les uns desautres : Gilbert Paquette, Yves Bérubé, Claude Ryan, par exemple.

D’une totale probité, respectueux mais sans complaisance pour l’au-torité politique, Maurice était vraiment courageux. À certains mo-ments, j’en puis témoigner, il était vraiment dérangeant. En privé, illui arrivait d’avoir la critique rude.

Esprit aigu, ce rationaliste sceptique était aussi un homme de convic-tion. Ses critiques portaient parce qu’il n’était guère soupçonnable departi pris, si ce n’est d’un parti pris continu et affiché pour la re-cherche et pour l’excellence universitaire.

L’homme Maurice L’Abbé attachait par un style unique, nourri parune intarissable curiosité intellectuelle, un style fait de rigueur, maisaussi d’humour, amalgamés à une dose marquée de discrétion et deretenue. Cet homme avait son quant-à-soi.

Même pour qui, sans être de ses familiers, l’avait fréquenté beaucoup,il demeurait un fond, quelque chose d’énigmatique chez lui. D’où lesurnom de « Chinois » que lui avait conféré l’une de nos amies. Il yavait en effet chez notre ami, sans rien de hautain ni de figé ou decompassé, quelque chose de la sérénité disciplinée que l’on prête aumandarin, lettré raffiné bien que rompu aux joutes des corridors dupouvoir.

Je l’imagine donc volontiers aujourd’hui, dans l’un de ces mondes pa-rallèles issus des audaces de théoriciens de la physique, devisant avecles ombres de certains de ceux auxquels il vouait une admiration forte,comme Georges Perec, Kurt Gödel ou Gottlob Frege et, pourquoi pas,avec l’ombre de Confucius aussi.

Mais, à l’aise dans l’intemporel littéraire ou mathématique, Mauricene quittait jamais longtemps le souci du temps présent.

Aussi, par un matin aussi éclatant que celui-ci, je l’imagine encore,engageant la conversation avec un couple inattendu, survenu au dé-tour de sa promenade, deux de ses anciens ministres, Yves Bérubé etClaude Ryan. Et il les fait aisément tomber d’accord1 : la situation dufinancement des universités au Québec appelle un énergique redres-sement et la politique de la recherche et de l’innovation a décidémentun sérieux besoin d’oxygène. C’est que Maurice persiste en effet dansses partis pris de naguère, ceux-là même pour lesquels nous lui de-meurons si intensément redevables.

Comme citoyens, pour les services qu’il a rendus, nous avons tousen effet une dette à l’endroit de Maurice L’Abbé. C’est d’ailleurs sonexceptionnelle contribution qu’est venue reconnaître en 1994 l’attri-bution à Maurice du Prix du Québec. Mais nous n’en sommes pasquittes pour autant.

Merci et salut, Maurice !

Et si ce n’était du respect que l’on doit aux grandes ombres, j’auraisenvie de conclure : Merci et salut, Chinois ! »

1Ces deux-là, il est vrai qu’ils paraissent plus souvent proches l’un de l’autre maintenant dans leur monde parallèle qu’ils n’avaient su l’être dans lenôtre.

BULLETIN CRM–9


Thematic Year(continued from page 1)

Vytautas Paskunas (Bielefeld), Marie-France Vignéras(Jussieu), Jeremy Teitelbaum (Illinois at Chicago), MatthewEmerton (Northwestern) and Mark Kisin (Chicago). On thetheme of p-adic variation of motives, the main speakers wereFrank Calegari (Harvard), Glenn Stevens (Boston), Eric Ur-ban (Columbia), Joël Bellaiche (Columbia), Abdellah Mokrane(Paris 13) and Haruzo Hida (UCLA).

The workshop “Intersection of Arithmetic Cycles and Auto-morphic Forms,” organised by Henri Darmon (McGill) andEyal Goren (McGill), took place in December 2006. The skele-ton of the workshop consisted of five lecture series. One lec-ture series was given by Jan Bruinier (Koln) who discussedarithmetic intersection theory on Hilbert modular surfaces andvalues of functions at CM points. His lectures explained theessential use of Borcherds theory and Arakelov theory in ob-taining the results. Another lecture series was given by BrianConrad (Michigan, Ann Arbor) who explained the local in-tersection numbers calculations in the work of Gross and Za-gier. His lectures provided a solid and conceptual foundationto such calculations and one expects that a similar approachwill be indispensable in higher-dimensional cases that are yetto be studied. Steve Kudla (Maryland) explained some of hisresearch, spanning almost a decade, focusing on joint workwith Michael Rapoport and Tonghai Yang that relates a par-ticular generating series of zero cycles on an arithmetic surfaceand (derivatives of) an Eisenstein series. His lectures stressedagain Arakelov theory and Borcherds theory techniques. Thefourth lecture series was given by Shou-Wu Zhang who dis-cussed two topics. The first was a research joint with Johan deJong concerning sub Shimura varieties of the moduli space ofcurves; the second was a survey, and discussion of some recentresults, connecting period integrals and values of (derivativesof) L-functions. The last lecture series was given by Jose Igan-cio Burgos Gil (Barcelona), Jurg Kramer and Ulf Kuhn (Hum-boldt University) who described a joint project of themselves(partially joint with Bruinier) on extending Arakelov theory. Inaddition to these lecture series, there were 14 additional talkscovering a spectrum of topics related to the theme of the work-shop. The concluding talk for the workshop was given as a col-loquium talk by James Cogdell (Ohio State) who talked on “L-functions, modularity and functoriality.”

There was an intense blitz of activities during the Wintersemester, with four workshops, a school and the lecture seriesof the three André-Aisenstadt Chairs.

The first workshop on “L-functions,” organized by ChantalDavid (Concordia) and Ram Murty (Queen’s), took place inFebruary 2006. It was focusing on the recent developments inthe study of L-functions, with driving themes: vanishing ofL-functions, zero-free regions, size of L-functions, moments.There was a large number of new and exciting results on those

subjects in the last few years, and there are very tantalizingquestions which are just at the limit of the current techniques.The skeleton of the workshop was three series of lecturesby K. Soundararajan (one of the André-Aisenstadt Chairs),Philippe Michel (Montpellier) and Kumar Murty (Toronto).The lectures of Soundararajan began with the general audiencetalk “What are L-functions and what are they good for?" fromthe Aisenstadt lecture series. In his subsequent lectures, hegave an introduction to the classical and related techniques ofamplification, mollification and resonance. The leading threadsof the lectures of Philippe Michel on “Periods, equidistribu-tion, subconvexity and non-vanishing of L-functions” werethree classical problems of equidistribution of integral pointson spheres solved by Linnik and his school in the sixties. Thischoice allowed him to present very recent techniques usedin the proofs of new equidistribution problems about specialpoints on homogeneous varieties under the action of an alge-braic group. Those recent techniques rely primarily on pow-erful tools from harmonic analysis, as Langlands functorial-ity and approximations towards the Ramanujan – Peterson –Selberg conjecture, and on tools from ergodic theory, as Rat-ner’s theorem on the classification of invariant ergodic mea-sures under the action of a unipotent subgroup. Kumar Murtydiscussed in “Families of L-functions” the general space of L-functions and various ways of metrizing this space. He alsotouched upon the new theme of limits of L-functions and theChebotarev density theorem for infinite extensions. In additionto these lecture series, there were 15 invited talks on variousaspects of L-functions. The workshop ended with a lively dis-cussions among all the participants, speculating on the rank ofelliptic curves in families of quadratic twists, following the talkof Andrew Granville (Montréal) about his work in progresswith Mark Watkins (Bristol) on that topic.

The workshop on “Anatomy of Integers,” organised by An-drew Granville (Montréal) and Jean-Marie De Koninck (Laval)took place in March, 2006. There has never really been a meet-ing dedicated to these questions before (even though this sub-ject includes some of the most important questions in analyticnumber theory, they have always been seen as part of othertopics). Two of the biggest breakthroughs in the area in the lastfew years have been the following. Firstly, the proof of Gold-ston, Pintz and Yildirim that there are “small” gaps betweenprimes infinitely often (this uses classical sieve theory). DanGoldston (San Jose State) gave a three hour lecture series en-titled “Small Prime Gaps: From the Riemann Zeta Functionand Pair Correlation to the Circle Method” in which he gavea broad perspective on such questions and indeed his ownfailed attempts over the last fifteen years! It was a masterfulaccount of many related ideas and the development of thoughtin the subject. Secondly, the work of Kevin Ford on many fa-mous questions on the distribution of divisors of integers, asubject that long intrigued Paul Erdos. The question of howoften an integer has a pair of divisors a < b < 2a has notbeen well understood until very recently, in a highly insight-

BULLETIN CRM–10


ful paper of Kevin Ford (Illinois at Urbana-Champaign), whogave a three hour lecture series on his work, in which he re-solves several conjectures of Erdos, and indeed to an extentthat it is necessary to rewrite what are the main questions tobe studied in this area. Most intriguing perhaps is how Fordconverts the problem into questions about random walks andanswers very similar questions to some of those that appearin recent developments of percolation theory. The AisenstadtChair K. Soundararajan (Stanford) gave two lectures on someof his exciting recent results. The first on his improvement(with Andrew Granville) of the Polya – Vinogradov inequality,the second on his work with J. Lagarias on smooth solutionsto a + b = c. This result was improved, during the meeting bySoundararajan in collaboration with Konyagin.

The next event was a concentration period on additive com-binatorics, organised by Andrew Granville (Montréal), JozsefSolymosi (UBC) and David Ellwood (Clay). Those intense twoweeks of activities are described on pages 4 – 5 of the presentBulletin.

The last event of the theme year was the workshop “Analyticmethods for Diophantine equations” held at the Banff Inter-national Research Station (BIRS) in May, 2006. This event wasorganised jointly by the CRM and the MSRI, as both instituteswere hosting semester long programs on different aspects onthese questions. It was decided to get together at the end of theacademic year for a joint meeting to discuss issues that arise atthe thematic programs at each institute. Thus the participantswere primarily people who had been at one special year orthe other, though perhaps a third were other researchers whoare expert in Diophantine equations. Perhaps the most consis-tent theme of this meeting was the topic of counting pointson higher dimensional varieties, particularly Manin’s conjec-ture. We heard a highly motivating survey by Yuri Tschinkel(Gauss Chair at Göttingen), exciting new research from a geo-metric perspective by Par Salberger (Chalmers, Sweden), froma perspective of automorphic forms by Ramin Takloo-Bighash(Princeton) and from a perspective closer to Diophantine ap-proximations by Jeff Thunder (Northern Illinois). To countpoints on higher dimensional varieties one can also proceedby the classical circle method. Roger Heath-Brown (Oxford)told us about his recent major breakthrough on counting pointson cubic hypersurfaces (reducing the number of variables inDavenport’s famous result), and the extension to quartic va-rieties was discussed by Tim Browning (Bristol). Trevor Woo-ley (Michigan) explained his idea to prove that the local-globalprinciple works almost always and discussed what he hasshown to date. All participants seemed to have greatly enjoyedthe meeting. It was an interesting “coming together” of dif-ferent approaches to important questions, and most speakerstried to be accessible, so a lot was learned. There were severalnew collaborations formed during the meeting, and even someresults proved, while in Banff.

André-Aisenstadt Chairs(continued from page 3)

K. Soundararajan

The lectures of Soundarara-jan began with the André-Aisenstadt lecture “What areL-functions and what arethey good for?”. The first L-functions appeared in 1837in the work of Dirichlet, whoused L-functions associatedwith characters of the multi-

plicative group modulo q (now called Dirichlet characters) toshow that when (a, q) = 1, there are infinitely many primesin the arithmetic progression a, a + q, a + 2q,. . . . In this con-text, the Riemann zeta function is the L-function with triv-ial character. It was the insight of Riemann to consider ζ(s)from a complex analytic analytic point of view, and to linkthe understanding of the primes to the understanding of ζ(s).A prototypical example is the explicit formula, which relatesthe primes to the zeroes of ζ(s), and links the distributionof the primes to the famous Riemann hypothesis. Other L-functions include the Dedekind zeta functions associated tonumber fields, Artin L-functions associated to representationsof Galois groups, L-functions of elliptic curves, the Ramanu-jan’s ∆ function, or in general L-functions attached to modularforms. . . This visit through the zoo of L-functions was moti-vated by the applications: counting primes in arithmetic pro-gressions leads to Dirichlet’s L-functions, the congruent num-ber problem leads to L-function of elliptic curves, etc. The talkended with one fascinating illustration of the applications ofL-functions to arithmetic questions, namely the work of Onoand Soundararajan on integers represented by Ramanujan’sternary quadratic form x2 + y2 + 10z2. For odd integers, thereare no congruence obstructions, and we want to know if theform represents all odd integers with finitely many excep-tions. Ramanujan found 16 of those exceptions, and Ono andSoundararajan showed that under the Generalized RiemannHypothesis (GRH), there are exactly 18 exceptions. The proofuses Dirichlet L-functions and L-functions of elliptic curves,which are related to coefficients of weight 3

2 modular formsby Waldspurger’s theorem. Assuming that the Riemann Hy-pothesis holds for Dirichlet’s L-functions and elliptic curvesL-functions, Ono and Soundararajan showed that the only oddintegers not represented by x2 + y2 + 10z2 are 3, 7, 21, 31, 33,43, 67, 79, 87, 133, 217, 219, 223, 253, 307, 391, 679, 2719.

In his subsequent lectures during the L-functions workshop,Soundararajan gave an introduction to the classical and re-lated techniques of “amplification,” “mollification,” and “res-onance” (or “soundification” . . . ) Mollification was introducedby Selberg to show that a positive proportion of the zeroes ofζ(s) lie on the critical line Re(s) = 1

2 . Since then, it was exten-sively used to prove non-vanishing in families of L-functions,as in the work of Soundararajan, who showed that at least 7

8 of

BULLETIN CRM–11


the Dirichlet quadratic twists do not vanish at s = 12 . It is strik-

ing that the proportion of non-vanishing in this case is twiceas good as the proportion obtained when χ is not quadratic(the best proportion there is due to Iwaniec and Sarnak, andis about 1

3 ). An explanation for that high proportion is pro-vided by the Katz-Sarnak models of random matrix theory. Inall those applications, the role of the mollifier is to tame the os-cillations in the size of L( 1

2 , f ), allowing a good bound on thesecond moment.

The technique of amplification works in the opposite way: in-stead of taming the oscillations of the L-values, one wants toamplify them, or more precisely amplify one L-value such thatupper bounds for the first moment can be translated into upperbounds for this L-value. This was first used by Duke, Friedlan-der and Iwaniec in their work on subconvexity. The techniqueof resonance leads to results on extreme values of L-functions,lower bounds of the correct order of magnitude for momentsof L-functions, and results on large character sums.

Soundararajan also gave a series of lectures in the WorkshopAnatomy of Integers where he presented his work (in collab-oration with Andrew Granville) on pretentious characters andthe Pólya – Vinogradov theorem. The best bound for charactersums of non-principal Dirichlet characters was given indepen-dently by Pólya and Vinogradov in 1918, and there has beenno subsequent improvements in their bound other than in theimplicit constant. Surprisingly, Granville and Soundararajanrecently proved that the bound of Pólya – Vinogradov can besubstantially improved for characters of odd, bounded order.Their proof relies on a characterization of the characters withlarge sum, which reveals a hidden structure among such char-acters: the character sum can be large only when the values ofthe character χ are close to the values of a character ξ of smallconductor, for many small primes. As χ “pretends” to be ξ, itis called a pretentious character.

Workshop on Additive Combinatorics(continued from page 5)

prove the remaining major conjectures in this area! Jean-MarcDeshouillers (Université de Bordeaux) gave a beautiful sur-vey on the classical theory of “Bases : structure”, where hehighlighted the connections with some of the extremal casesin the Gowers theory; Gyorgy Elekes (Eotvos University) dis-cussed his recent work with Ruzsa “On the structure of setswith many ‘medium’size’ arithmetic progressions.” After thelecture, Granville pointed out that this should easily lead toa general structure theorem for almost all arithmetic progres-sions inside a set. One of the most extraordinary talks was byBalázs Szegedy (University of Toronto) who in joint work withLászló Lovász has been developing a theory of “Graph lim-its,” allowing them to prove many of the deep theorems of Sze-merédi, Ruzsa and others by passing to limits and using topo-logical ideas on the set of such graphs. This work comes close

to some the ideas of Tao. Ram Murty (Queen’s University) dis-cussed his recent work, with Kumar Murty, on understandinghow often coefficients of a given higher weight modular formtake on a given odd value, in analogy with “The Lang – TrotterConjecture.” In the final lecture of the meeting, Vsevolod Lev(University of Haifa at Oranim) explained how he has recentlycompletely understood “The rectifiability threshold in abeliangroups” over finite fields of prime order.

There were also several lectures of Ramsey theory, which isan important and related subject. We heard Ron Graham (UCSan Diego) explaining “Some of my favorite Ramsey theoryproblems” and Jaroslav Nešetril (DIMATIA Charles Univer-sity Prague) giving a broad survey on “Ramsey classes of finitestructure”. A graduate student, David Conlon from Universityof Cambridge, has made an enormous advance on “New Up-per Bounds for Ramsey Numbers,” the biggest increase in thebounds since the 1930s! An undergraduate student, Jacob Foxfrom MIT, described his many beautiful contributions to infi-nite Ramsey theory in “Partition regularity of linear equations”in which many answers depend on which axioms of arithmeticone assumes!

After two weeks of additive combinatorics with the schooland workshop devoted to that subject, the organizers thoughtit would be fun to break things up by having the André-Aisenstadt Lecture of Manjul Bhargava (Princeton) on the rep-resentation of integers by quadratic forms. This was a stun-ningly beautiful lecture that evidently delighted the audience.

John Harnad(continued from page 6)

From previous familiarity with twistor transform methods, itwas clear to me that the inverse spectral transform, based uponthe Riemann – Hilbert factorization problem of analytic func-tion theory, had an analogous rôle and, indeed, the two couldbe usefully related through dimensional reduction.

My work on soliton theory began with the recognition that lin-earization of flows could be achieved by casting the equationsof inverse spectral theory into a special form, which geometri-cally describe flows induced by linear group actions on Grass-mann manifolds [5]. This related the “dressing method” of Za-kharov and Shabat, based on the Riemann – Hilbert problem,to the transformation group approach of the Kyoto school. Italso became clear that the abstract notion of symmetry reduc-tion could be viewed, within a Hamiltonian context, as the keystructural feature underlying the factorization method.

In subsequent work (again together with a number of close col-laborators), I developed a method for canonically embeddingfinite dimensional integrable systems into a sort of “univer-sal phase space” provided by loop algebras and loop groups,through a “dual pair” of moment mappings [6]. The notionof “duality” arose naturally in this context, providing two al-ternative representations of the same dynamics as isospectral

BULLETIN CRM–12


flows in loop algebras. These involved a pair of rational “Laxmatrices” sharing the same invariant “spectral curve,” whichencoded the same conserved quantities. Our approach madeclear that the linearization of flows on the complex tori associ-ated to the spectral curves was simply an instance of classicalcanonical transformation theory [7].

In 1991, I attended some lectures by Witten at the Institute forAdvanced Study in Princeton relating the flows of the KP in-tegrable hierarchy to a matrix integral shown by Maxim Kont-sevich to be the generating function for intersection indices onmoduli spaces of Riemann surfaces, a quantity that entered intothe theory of 2D topological gravity. Subsequently, at the 1992Scheveningen summer meeting, while giving some lectures onlinearization of isospectral flows in loop algebras, I met CraigTracy, who was lecturing about his new results, with HaroldWidom, on edge spacing distributions in random matrix en-sembles. He introduced a system of nonlinear Hamiltonianequations that underlay the deformation dynamics, and notedthat these appeared almost exactly identical to the isospectralones. On closer examination, it became clear that the key dif-ference lay in the fact that parameters that were fixed in theisospectral equations became identified as times in the Tracy –Widom system, making them nonautonomous, and changingthem from isospectral flows to deformations of associated ra-tional covariant derivative operators that preserved their mon-odromy — so-called “isomonodromic deformations.”

We wrote up a brief paper together explaining this fact [8], andthis became the starting point of much of my subsequent re-search. I was able afterwards to develop a general Hamiltoniantheory of such isomonodromic deformation equations [9], aswell as their applications to the deformation approach to thespectral statistics of random matrices [10]. Amongst the usefulresults that emerged from this work was the recognition thatthe same canonical framework (R-matrix theory) that under-lies integrable isospectral flows in loop algebras was applica-ble in this nonautonomous setting to isomonodromic systems.Furthermore, the notion of “duality” turned out to be at leastas pertinent to isomonodromic systems, providing alternativeimplementations of the same deformation dynamics [9, 11].

This also turned out to have importance in applications ofisomonodromic deformations to the study of two-matrix mod-els, since the duality previously introduced for quite generalisomonodromic systems had in this setting very immediate im-plications for the spectral correlation functions. A key devel-opment of recent years, obtained through collaborative workwith my colleagues Marco Bertola (Concordia and CRM) andBertrand Eynard (CEA, Saclay) was the derivation of a Rie-mann – Hilbert framework characterizing the integral kernelsentering in the computation of the correlators [12]. The largeN asymptotics of such correlators are the main objects of in-terest, and further development of the Riemann – Hilbert tech-nique, applied to the computation of their asymptotic limits,is the goal of an ongoing program of research on multi-matrixmodels.

Amongst the many other colleagues and students with whomI have had the pleasure of working, and sharing ideas, I wouldlike to specially mention: Steve Shnider, with whom I had along and rewarding collaboration over several years, JacquesHurtubise, who contributed enormously to our many jointprojects, and remains a close friend and colleague, Luc Vinet,whom I had the immeasurable good luck to have had asmy first, unofficial graduate student, and Alexander Its, fromwhom I learned a great deal, and with whom I have had thepleasure of an ongoing scientific collaboration together with ashared friendship.

1. J. Harnad, S. Shnider, and L. Vinet, Group actions on princi-pal bundles and invariance conditions for gauge fields, J. Math.Phys. 21 (1980), 2719- – 2724.

2. J. Harnad, S. Shnider, and J. Tafel, Group actions on principalbundles and dimensional reduction, Lett. Math. Phys. 4 (1980),107 – 113.

3. J. Harnad, J. Hurtubise, M. Légaré, and S. Shnider, Con-straint equations and field equations for supersymmetric Yang –Mills theory, Nucl. Phys. B256 (1985), 609 – 620.

4. J. Harnad and S. Shnider, Constraint equations and field equa-tions for ten-dimensional super Yang – Mills theory, Commun.Math. Phys. 106 (1986), 183 – 199.

5. J. Harnad, Y. Saint-Aubin, and S. Shnider, The soliton corre-lation matrix and the reduction problem for integrable systems,Commun. Math. Phys. 93 (1984), 33 – 56.

6. M. Adams, J. Harnad, and J. Hurtubise, Dual moment mapsinto loop algebras, Lett. Math. Phys. 20 (1990), 299 – 308.

7. M. R. Adams, J. Harnad, and J. Hurtubise, Darboux coordi-nates and Liouville – Arnold integration in loop algebras, Com-mun. Math. Phys. 155 (1993), 385 – 413.

8. J. Harnad, C. A. Tracy, and H. Widom, Hamiltonian structureof equations appearing in random matrices, Low-DimensionalTopology and Quantum Field Theory (H. Osborn, ed.),Plenum, New York, 1993, pp. 231 – 245.

9. J. Harnad, Dual isomonodromic deformations and momentmaps into loop algebras, Commun. Math. Phys. 166 (1994),337 – 365.

10. M. Bertola, B. Eynard, and J. Harnad, Partition functions formatrix models and isomonodromic tau functions, J. Phys. A:Math. Gen. 36 (2003), 3067 – 3983.

11. J. Harnad and Alexander R. Its Integrable Fredholm operatorsand dual isomonodromic deformations, Commun. Math. Phys.226 (2002), 497 – 530.

12. M. Bertola, B. Eynard, and J. Harnad Differential systems forbiorthogonal polynomials appearing in 2-matrix models, and theassociated Riemann – Hilbert problem, Commun. Math. Phys.243 (2003), 193 – 240.

BULLETIN CRM–13


Jeffrey S. Rosenthal(continued from page 7)

scientists, finance researchers, and psychologists (e.g., [1, 7]).

Recently, I took an unexpected detour, writing and publicisinga book for the general public about probability and uncertaintyin our everyday lives [17]. This in turn led to extensive mediainterviews and presentations to non-mathematical audiences,which has been an interesting and different experience. By theway, did I mention that the book makes a great gift?

1. A. Borodin, G. O. Roberts, J. S. Rosenthal, and P. Tsaparas,Link analysis ranking: algorithms, theory, and experiments.ACM Transactions on Internet Technology 5 (2005), 231 –297.

2. R. Douc, E. Moulines, and J. S. Rosenthal, Quantitativebounds on convergence of time-inhomogeneous Markov Chains.Ann. Appl. Prob. 14 (2004), 1643 – 1665.

3. A. E. Gelfand and A. F. M. Smith, Sampling based approachesto calculating marginal densities, J. Amer. Stat. Assoc. 85(1990), 398 – 409.

4. G. L. Jones and J. P. Hobert, Honest exploration of intractableprobability distributions via Markov chain Monte Carlo, Statis-tical Science 16 (2001), 312 – 334.

5. T. Lindvall, Lectures on the coupling method, Wiley & Sons,New York, 1992.

6. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller,and E. Teller, Equations of state calculations by fast computingmachines, J. Chem. Phys. 21 (1953), 1087 – 1091.

7. M. J. Osborne, J. S. Rosenthal, and M. A. Turner, Meet-ings with costly participation, American Economic Review 90(2000), 927 – 943.

8. G. O. Roberts and J. S. Rosenthal, Optimal scaling for vari-ous Metropolis – Hastings algorithms, Stat. Sci. 16 (2001), 351 –367.

9. G. O. Roberts and J. S. Rosenthal, General state space Markovchains and MCMC algorithms, Probability Surveys 1 (2004),20 – 71.

10. G. O. Roberts and J. S. Rosenthal, Coupling and ergodicity ofadaptive MCMC, revised for publication.

11. G. O. Roberts and J. S. Rosenthal, Simulation examples ofadaptive MCMC, in preparation.

12. J. S. Rosenthal, Random rotations: characters and random walkson SO(N), Ann. Prob. 22 (1994), 398 – 423.

13. J. S. Rosenthal, Rates of convergence for Gibbs sampling forvariance components models, Ann. Stat. 23 (1995), 740 – 761.

14. J. S. Rosenthal, Minorization conditions and convergence ratesfor Markov chain Monte Carlo, J. Amer. Stat. Assoc. 90 (1995),558 – 566.

15. J.S. Rosenthal, Analysis of the Gibbs sampler for a model relatedto James – Stein estimators, Stat. and Comput. 6 (1996), 269 –275.

16. J. S. Rosenthal, Quantitative convergence rates of Markovchains: A simple account, Elec. Comm. Prob. 7 (2002), No. 13,123–128.

17. J. S. Rosenthal, Struck by lightning: the curious world of prob-abilities. HarperCollins Canada, 2005 (probability.ca/sbl).

A Year that Set a New Standard for Canadian Mathematics(continued from page 16)

event to bring together all mathematicians developping vari-ous aspects of the “ultimate theory” of Symplectic Field Theory.Other new international agreements have been signed this yearby the CRM, notably with INSERM and the CNRS in France,with Latin America and with Central Europe (Dimatia).

I would like to thank the organizers of the 2005 – 2006 thematicprogramme at the CRM, especially Andrew Granville, ChantalDavid and Henri Darmon (but there are so many other names),as well as the CRM staff for their commitment and dedication.I would also like to thank the VPR’s of the seven large uni-versities involved in the CRM enterprise — their support is ab-solutely essential. And, above all, I would like to emphasizethat the CRM is here to serve the international community of

mathematical scientists in all possible ways. The three directorshave been working in close collaboration on many national andinternational issues, like the applications to the ICM, ICIAM,the Second Canada – France Congress (with Octav Cornea andNassif Ghoussoub as scientific directors), the support to thethree mathematical and statistical societies, to AARMS and tothe Statistical programme NPCDS. It has been, once again thisyear, a pleasure to share our vision and efforts, in which theLiaison Committee led by Richard Kane has played a vital role.

François LalondeDirector, CRM

BULLETIN CRM–14


Nouvelles/NewsPavel Winternitz

Le 3 octobre 2006, le professeur Pavel Winternitz rece-vait un doctorat honoris causa de l’Université techniquetchèque à Prague. Membre du CRM depuis 1972, il estprofesseur au Département de mathématiques et de statis-tiques de l’Université de Montréal depuis 1984. Il s’est faitremarquer, entre autres, par ses travaux sur la théorie desalgèbres et des groupes de Lie ainsi que sur leur applica-tion à l’étude des équations différentielles et des équationsaux différences finies. En 2002, Pavel Winternitz recevait leprix ACP-CRM de physique théorique et mathématique.

Andrew GranvilleAu cours de l’année 2006, Andrew Granville (Universitéde Montréal) a été élu à la Société royale du Canada, et aaussi reçu le prix Jeffrey-Williams de la Société mathéma-tique du Canada (SMC/CMS). Ces deux honneurs récom-pensent une contribution exceptionnelle et soutenue à larecherche.Andrew Granville est titulaire d’une chaire de recherchedu Canada (CRC) à l’Université de Montréal depuis 2002.Son domaine de recherche est la théorie analytique des

nombres. Entre autres contributions exceptionnelles, mentionnons la preuve (en col-laboration avec A. Alford et C. Pomerance) de l’infinité des nombres de Carmichael,ses travaux récents avec K. Soundararajan sur les fonctions multiplicatives, les va-leurs extrêmes des fonctions L et les sommes de caractères, ses travaux avec H. Starkreliant la conjecture abc aux zeros de Siegel, et ses travaux avec J. Friedlander surl’équidistribution des nombres premiers.

Laboratoire informatiqueEn mai dernier, François Lalonde, directeur du CRM, et Daniel Ouimet, administra-teur des systèmes informatiques, inaugurèrent un nouveau laboratoire informatiquedestiné à mieux servir les chercheurs et visiteurs du Centre. Construit en abattant lesmurs entre trois petits laboratoires antérieurs et un ancien bureau, il offre un équipe-ment informatique complètement renouvelé, constitué de quatorze PC sous Linuxet d’une imprimante dernier cri. Une nouvelle porte a été ajoutée pour les cher-cheurs du corridor ouest du Centre, donnant ainsi un accès direct depuis tous lescorridors. Tout le mobilier a également été renouvelé. Ce laboratoire s’adresse sur-tout aux chercheurs de passage qui n’ont pas d’ordinateur dans leur bureau, et auxvisiteurs, qui pourront également brancher directement leur portable sur l’Internet.

Le Bulletin du CRM

Volume 12, No 2Automne 2006

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 : Jean LeTourneuxConception 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–15


A Year that Set a New Standardfor Canadian Mathematics

The year 2005 – 2006 at the CRM has set newstandards: it was both a peak for the numberof Canadians invited to speak at the ICM and aremarkably deep, fruitful year at the CRM. Thebeauty is that both are very much linked: it isas if the CRM’s last thematic year had been arehearsal for the ICM, at least that part of it re-

lating to number theory. Since, like other insitutes, the CRMmust prepare its thematic programmes three years in advanceand send the invitations at least a year prior, just as the ICMdoes, it is striking that both the ICM Programme Committeeand the CRM Scientific Advisory Panel came up with the sameinvitations at roughly the same time, but in totally indepen-dent ways. Another striking relation is the fact that the Chairof the ICM Programme Committee, Noga Alon (Tel Aviv), isalso the CRM’s Aisenstadt Chair holder this semester — this isof course a pure coincidence, but an interesting one. The lastparallel between the ICM and the CRM is the emergence forthe first time at an ICM meeting of a very strong session inprobability, precisely at the moment when the CRM is workinghard, and has been for the past year, to organize two simultane-ous full-year programmes in 2008 – 2009: one on probabilisticmethods in mathematical physics and a second, in close col-laboration with PIMS, on probability. It is not a coincidencethat both the ICM and the CRM have seen the need to developthis field at the same time: probabilistic methods are becomingmuch more important in a large spectrum of problems from al-most all fields of mathematics, computer science, physics and,of course, statistics and biology. A good and simple example ofthis fundamental trend is the recent overwhelming penetrationof probabilistic methods in graph theory, large scale networksand designs where one must draw reliable estimates about anetwork or polytope made of millions of vertices and edges(like the web) by considering a collection of much smaller sam-ples in a very clever way. It is interesting to reflect on the evo-lution of computer science over the last fifty years: from a fielddominated by logic and algebra, it has become one where prob-ability and the quantum world are occupying more and moreplace at the forefront of the most exciting paths of research.

One of the best indicators of the vitality of mathematical re-search in Canada is the competition for the Aisenstadt Prizeawarded every year to a young Canadian mathematician (in-cluding pure and applied mathematics as well as statistics) bythe CRM Scientific Advisory Panel. At the moment of writingthis note, just three days before the meeting that will decide thewinner, I can testify that the new candidates this year, proposedby universities in all regions of Canada, are of such quality thatit will be a torment to make a choice amongst them.

There is, however, one worrying fact about the last ICM:only one woman was invited to give a plenary talk (Michèle

Vergne), and it seems that there were not more than ten ortwelve women invited speakers out of a total of 160 talks inall sessions. I was invited by Rachel Kuske to participate thisFall in a workshop at BIRS on ”Women in Mathematics.” TheCRM is committed to working on this situation. The CRM di-rectorate (directors and deputy directors) now has two womanout of a total of five people; the CRM Scientific panel, whichdid not include a single woman for 35 years (except the ex-officio members), will have three top level women mathemati-cians next year out of a total of twelve people, which meansthat half of the new appointments over the last two years havebeen filled by women mathematicians. The same applies toall other CRM committees and to CRM’s personnel at all lev-els. On a personal level, I like mixity in any working environ-ment — and I am very glad that the field in which I work, sym-plectic geometry and topology, has constantly been one wherewomen have reached the highest levels (Noether, Uhlenbeck,Vergne, McDuff, Kirwan,. . . ) and where the new generation ofyoung researchers is a model of gender mixity. This was obvi-ous at the Meeting in honour of Dusa McDuff’s sixtieth birth-day (which has actually become a tradition since it s now heldevery year !... a very enjoyable thing): the 2006 Dusafest clearlyshowed the persistent and extremely solid impact of women inthese fields.

The CRM is currently setting up two new laboratories, onein Quantum Informatics, called INTRIQ (Institut transdisci-plinaire en informatique quantique) and another one that willtake shape soon. This means that the CRM is going to be homefor ten large laboratories, spread over seven major universi-ties in Quebec and Ontario and jointly funded by the univer-sities and the CRM. We have also worked intensively on ournew web site, with 80,000 dynamic pages updated daily and7,500 hand-written pages, in both languages (French and En-glish). This website is the result of years of effort to give a com-plete and easily accessible view of everything that is of inter-est to mathematicians and mathematical scientists. It is fed by“CRM’s Monster,” one of the largest databases in mathemat-ics in the world. Click on http://www.crm.umontreal.ca/ andchoose the language of your choice (the choice is quite limited!but still, you have a choice).

The CRM was the first Canadian institute to seek funds fromthe NSF, NATO, and the Clay Institute simultaneously, withan enthusiastic response: none of our applications have beenturned down — it is particularly interesting to see that the onlyNATO Advanced Study Institute in the world, in all fields,funded year after year is the CRM’s “Séminaire de mathéma-tiques supérieures,” jointly sponsored by Université de Mon-tréal’s DMS. In 2007, we will innovate in another way: we willorganize an event to be held at Stanford, the first international


BULLETIN CRM–16

Documents

Le Bulletin du CRM · Le Bulletin du CRM • ... (Concordia University) ... (Stanford) and Terence Tao (UCLA). The thematic year be-gan with the SMS–NATO