Spring 2015 — Volume 21, No 1 — Le Centre De Recherches Mathématiques

Home , Chauvenet Prize, Henri Darmon, Joachim Lambek, Luc Vinet

C CENTRE R DE RECHERCHES Le Bulletin M MATHÉMATIQUES

Printemps/Spring 2015 — Volume 21, No 1 — Le Centre de recherches mathématiques

A conversation with Marco Bertola (Concordia), Robert Bran- type IIb string theory on AdS 5, and N = 4 Super Yang–Mills denberger (McGill), John Harnad (Concordia) and Johannes theory. Walcher (McGill) on December 8, 2014. Integrability, which is our third theme, has also been playing a role in mathematical investigations of string theory Bulletin: Can you elaborate on the theme of the semester? for a long time: its role in the AdS/CFT correspondence J. Walcher [JW]: Our semester has a 3-pronged theme: it’s emerged around 2004, and has been playing an increasing called AdS/CFT, holography and integrability, and I’ll start role in the quantitative development of the correspondence. trying to explain what AdS/CFT is. AdS/CFT stands for That is what our semester is about: it’s the intersection of the Anti–de Sitter/Conformal Field Theory correspondence. It AdS/CFT correspondence as a holographic duality, and inte- was the last of the major dualities discovered in the wake of grable methods as far as they are relevant to the AdS/CFT the second super string revolution of the mid-1990s, and it correspondence. plays a rather special role in the web of dualities. First of all it’s not a duality between two different string theories or two I will let Robert take it from here. different field theories, but rather it involves an equivalence RB: I will start from the physicist’s point of view, and in par- between, on the one side, a theory of quantum gravity, string ticular from the point of view of someone interested in gravity theory, and on the other side a more standard, although very and cosmology. In gravity and cosmology we had for a 100 interesting, quantum field theory. And the dictionary between years basic problems: our theories predict singularities, and these two has the feature of being holographic. although singularities may be interesting from a mathematical Bulletin: Can you explain what holographic means? point of view, they are unphysical. No physical apparatus will ever measure an infinity—the physical apparatus will break R. Brandenberger [RB]: Holographic means that there is one down. And the reason that our theories predict singularities theory, in d dimensions, which is equivalent to a different the- is that they are not correct. The AdS/CFT correspondence ory which lives on the boundary. It’s like in optics, it’s the provides a way to solve these problems. For example, it pro- same use of the word as in optics. vides a means to understand the physical resolution of the JW: The ideas that later developed into AdS/CFT go back singularity at the center of a black hole that is predicted by to the early 1970s, shortly after the birth of string theory. Einstein’s theory. This question is going to be explored in one The first proposals that gravity should be holographic are of the workshops, the one which Alex Maloney and Patrick due to ’t Hooft, Polyakov, and Susskind, and the Bekenstein– Hayden are coordinating, the one on gravity, in September Hawking entropy of black holes was another important clue. [crm.math.ca/2015/Gravity15/]. But the big breakthrough came in 1998, or late 1997, when I’m a cosmologist and cosmologists have this huge problem Juan Maldacena, who is one of the organizers of the the- of the Big Bang singularity, so again we know that the thematic semester, proposed this exact correspondence between ory breaks down, and we hope that through the AdS/CFT crm.math.ca correspondence we can have a model for what happened in 19th century to the 1970s, when it became a clearly devel- the very early universe, what replaces the Big Bang. That’s oped theory. The fantastic new things, which occurred in the going to be the focal point of the workshop on cosmology, the last 10 or 15 years, were absolutely unexpected connections first workshop in July [crm.math.ca/2015/Extension15/]. between integrable systems and other areas of physics and of There’s the hint that one can take the gravitation theory, mathematics. I always think of integrability as a subject that which is singular, and through the AdS/CFT correspondence, has nine lives, because it keeps running out of steam and then which is this holographic correspondence, map it into a non- finding a new reason to come back into the center of atten- gravitational theory, on the boundary. Then, since nongravi- tion. I’m going to just list some of these things (not in order tational theories are under much better mathematical control, of importance), and everything on my list is covered in this we can hope that we can evolve the theory on the boundary. thematic program. If we have the picture in mind that we are evolving towards One thing that is close to my heart is the topic of tau func- the Big Crunch singularity, and we go to the boundary, then tions, which is more of a mathematical application, but closely we hope that we can let time continue beyond where it would related to the theory of Riemann surfaces, essential to string stop in the gravitational theory. There are already indications theory. It turned out that tau functions also play a completely that this would work for a very simple model of cosmology different role, and are extremely important, in a mathemati- where there is just a homogeneous and isotropic universe, no cal sense, as generating functions for invariants in enumera- fluctuations, no people, nothing. And the goal of this cos- tive geometry and graph theory, which are also very central mology workshop is to extend it to a more realistic situation to the string theory application. Tau functions are generat- where we include fluctuations, where we include the possibil- ing functions not just for the enumerative geometry related ity of having galaxies in this picture. This is really going to to Riemann surfaces, but also for the invariants such as Jones be a workshop in the sense that this has never been achieved and HOMFLY polynomials appearing in knot theory. It was before and we are getting the people who are working on this known for a long time that there’s a close connection between topic to come together. That is a really novel feature of this topological field theory, Chern–Simons theory, and knot in- program. variants, but it turns out that there is also a relation to inte- There are also applications of the AdS/CFT correspondence grable systems through the tau function. That’s a fascinat- in different areas of physics. For example, a big challenge ing aspect that will hopefully also be developed during this is to understand gauge theories beyond perturbation theory, program. because in some gauge theories the coupling constant is ac- Another connection that emerged recently was a very surpris- tually large and we cannot trust existing techniques. So now ing connection between scattering amplitudes in supersym- we can go backwards using the AdS/CFT correspondence, we metric Yang–Mills theory and soliton theory. Two aspects of can map this complicated gauge theory into a higher dimen- it entered, one was explicit on-shell calculations of scattering sional gravitational theory, and it works out that if the gauge amplitudes which turned out to have integral representations theory is complicated, the gravity theory is going to be in the involving the integration of differential forms with singular- more easily tractable regime. This has applications both to ities over Grassmannians. This gave exactly the singularity condensed matter physics and to relativistic heavy ion col- structure that arises. It is characterized by what we call the lisions, and that’s the theme of the workshop that Keshav moduli space of solitons. Solitons are essential elementary in- Dasgupta and Charles Gale and Sangyong Jeon are coordi- gredients in integrable systems. It turns out that the param- nating [crm.math.ca/2015/Applications15/]. eters that characterize solitons are exactly what we integrate J. Harnad [JH]: I want to mainly talk about the aspects of over when we compute these on-shell scattering amplitudes— the semester that connect up to the notion of completely in- a totally unexpected connection. At present, this connec- tegrable systems. The point is that the thematic program tion is still not understood, but I have great hopes for this is trying to build a bridge between very new developments, program, where we’re bringing together people from the dif- which are everything related to AdS/CFT holography, and ferent specialties, those specialized in integrable systems and work that really dates back to the mid or early 19th century, those specialized in scattering amplitudes and other aspects of which is on integrable systems. A big breakthrough in inte- quantum field theory, to try to cross-fertilize the two subjects. grable systems occurred in the 1960s, when it was realized There is also a connection through a very intensely studied that there were in fact real genuine physical systems, which new area of mathematics, which is known as cluster alge- are integrable in the sense that they have the maximum num- bras. The topological cell structure of positroid cells in Grass- ber of conservation laws, but are actually infinite-dimensional: mannians may appear to be a totally unrelated mathemati- they have an infinite number of degrees of freedom. cal subject. On the contrary, it turns out that the struc- At the same time, the fact that this is closely connected to ture of positroid cells, which exactly characterizes, on the one quantum mechanics became apparent and there were many hand, nonsingular solitons, and on the other hand, the domain quantum integrable systems that were studied simultaneously of integration which represents the scattering amplitudes, under the general heading of “quantum inverse scattering are very closely related. And the underlying mathematics, methods.” So this is what’s been going on, say, from the (continued on page 17)

BULLETIN CRM–2 crm.math.ca Prix CRM-SSC Fang Yao (Université de Toronto) Quand les données se font courbes et surfaces

Christian Genest (Université McGill)

Fang Yao, professeur titulaire de statistique à l’Université de Toronto et 16e lauréat du prix CRM–SSC, était à Montréal le jeudi 15 janvier pour présenter un survol de ses travaux dans le cadre du Colloque des sciences mathématiques du Qué- bec. Établi au Canada depuis 2006, Fang est un spécialiste de l’analyse de données fonctionnelles (ADF). Instituée et popularisée par Jim Ramsay, l’ADF est une branche relativement récente de la statistique qui traite de l’analyse d’échantillons de courbes et de surfaces. Si un sujet écrit vingt fois l’acronyme FDA (Functional Data Analysis) au stylo, par exemple, on observera des fluctuations dans le tracé (voir figure). En s’appuyant sur des techniques de lissage, des résultats d’analyse fonctionnelle et la théorie des processus stochastiques, les spécialistes de l’ADF développent Fang Yao des méthodes qui permettent de dégager les principales carac- téristiques des courbes observées, après les avoir étalonnées au La mise en œuvre de cette approche pose des défis d’enver- besoin. Dans un article paru en 2013 dans BMC Medical Re- gure, tant au plan numérique que conceptuel. En plus d’avoir search Methodology, Ullah et Finch répertorient plus de 80 étudié les propriétés asymptotiques des estimateurs, Fang a applications de l’ADF publiées dans des revues de sciences ou conçu l’algorithme PACE (Principal Analysis by Conditional de médecine entre 1995 et 2010. Estimation) et la version 1.0 du logiciel du même nom, qui a largement contribué à disséminer cette méthodologie. Aujour- d’hui, ce logiciel est très répandu et continue de se développer au fil des travaux de Fang et de ses collaborateurs. Au cours de son allocution, Fang Yao a donné un tour d’hori- zon de ses nombreuses contributions méthodologiques à l’ADF et des multiples applications qu’il en a faites, notamment en biostatistique et en finance. Il suffit de consulter son pro- fil Google Scholar pour mesurer l’influence considérable qu’il exerce dans le domaine. Le brillant exposé qu’il a donné au CRM restera longtemps dans les mémoires.

Comme les données fonctionnelles sont des réalisations d’un Octav Cornea (Montréal) and Henri Darmon processus stochastique à trajectoires lisses, elles sont forcé- (McGill) Selected as 2015 Simons Fellows in ment éparses ; leur modélisation est donc confrontée au « ﬂéau Mathematics de la dimension ». Pour contourner cette diﬃculté, on fait gé- néralement appel à une version fonctionnelle de l’analyse en Two CRM members, Octav Cornea (CIRGET) and Henri composantes principales. Cette technique s’appuie sur la re- Darmon (CICMA), have recently been selected as Simons présentation de Karhunen–Loève des processus stochastiques Fellows in Mathematics by the Simons Foundation. The de carré intégrable en termes des éléments de la base orthonor- 40 awardees in Mathematics for 2015 (four from Canadian mée associée aux valeurs propres de l’opérateur de Hilbert– universities and the rest from the United States) are distin- Schmidt induit par la fonction d’autocovariance. guished mathematicians at all levels of their respective careers. Pour que l’espérance du processus sous-jacent puisse être es- timée avec précision, il faut que les courbes échantillonnées According to the Simons Foundation website, “The Fellows soient observées sur tout leur domaine ou du moins en un très Programs provide funds to faculty for up to a semester-long grand nombre de points communs. Or ce n’est pas toujours research leave from classroom teaching and administrative le cas, notamment dans les études longitudinales. Pour pal- obligations. Such leaves can increase creativity and provide lier ce problème, Fang a proposé de recourir à des méthodes intellectual stimulation.” de lissage fondées sur des régressions locales ou des splines. BULLETIN CRM–3 crm.math.ca Number Theory from Arithmetic Statistics to Zeta Elements Aisenstadt Chair: Zéev Rudnick September 16–18, 2014

Chantal David (Concordia University)

Zéev Rudnick is the Cissie and Aaron Beare Chair in Num- hold over function fields, at the q limit (when the size of the ber Theory at Tel Aviv University. He got his Ph.D. from finite field Fq tends to infinity). The proofs involve two main Yale, and was an Assistant Professor at Stanford University steps: first rephrase the original question as a problem about and Princeton University before joining Tel Aviv University zeros of certain L-functions, and then use the powerful theo- in 1995. Rudnick is a recognized leader in analytic num- rems about equidistribution of Frobenius matrices originating ber theory, mathematical physics (especially quantum chaos) in the work of Deligne and Katz–Sarnak. They obtain results and arithmetic statistics, and his work in all those areas is which go much further than one can hope to do in the num- deep and influential. Among the many honours that he re- ber field setting in the foreseeable future, shed light on several ceived for his research, we mention the Erdős Prize of the conjectures in number fields which are currently intractable, Israel Mathematical Union, an Annales Henri Poincaré Dis- and also lead to generate refined conjectures in the number tinguished Paper Award, and a ERC Advanced Grant. He is a field case. Fellow of the American Mathematical Society since 2012, and In his second talk, Rudnick presented his work with his stu- was an invited speaker at the 2014 International Congress dents Entin and Roditty-Gershon, where they were able to use of Mathematicians and at the 2012 European Congress of their results in the function field setting to actually prove the Mathematics. equivalent theorem in the number field setting, namely that the n-level density of low-lying zeroes for classical Dirichet L- functions is given by the scaling density of the unitary sym- plectic group USP(2g) as g → ∞, when the support of the Fourier transform is limited to a certain compact region. This fascinating and new twist on the interactions between number theory over number fields and function fields was achieved by showing that a certain combinatorial factor, which is obtained in the number field setting, and which has resisted several efforts to show that it coincided with the USP factor, is the same in the function field setting (i.e., for the family of hy- perelliptic curves). The authors then use the Katz–Sarnak equidistribution theorem to match the combinatorial factor with the factor coming from random matrix theory. The third talk was entitled Sums of Three Squares and Spa- tial Statistics on the Sphere, and Rudnick presented his recent Zéev Rudnick Photo: Jeff Mozzochi work with Bourgain and Sarnak on the distribution of “Linnik His visit coincided with the workshop Statistics and Num- points” on the sphere. ber Theory, which he coorganized with C. David (Concordia University) and P. Kurlberg (KTH, Stockholm), and he gave three talks during this time. In line with the theme of the workshop, the first two were concerned with the analytic number theory of polynomials over a finite field, and presented the exploration of traditional problems of analytic number theory in the context of function fields over a finite field. In the first talk, Rudnick presented his recent work with Keat- ing where they look at classical conjectures of analytic number √ theory, such as the distribution of primes in short and in long The Linnik points L(n) are the projections (x, y, z)/ n to the unit sphere intervals, and the sum of the Möbius function in short inter- of the various representations of n as a sum of three integer squares vals. In each case, they were able to show that the analogues n = x 2 + y2 + z2. Pictured are the 4,032 Linnik points corresponding to of the classical conjectures (due to Goldston–Montgomery, n = 104,729, the 10,000th prime. Friedlander–Goldston and Good–Churchhouse, respectively) (continued on page 10)

BULLETIN CRM–4 crm.math.ca Number Theory from Arithmetic Statistics to Zeta Elements Aisenstadt Chair: Carl Pomerance December 8–12, 2014

Dimitris Koukoulopoulos (Université de Montréal)

Carl Pomerance is the John G. Kemeny Parents Professor (Z/nZ)×, which denotes the group of reduced residues mod n; at Dartmouth College. He obtained his Ph.D. from Harvard the sum-of-divisors function σ(n); the function s(n) B σ(n)−n, University in 1972. Upon finishing his doctorate studies, he which gives the sum of the proper divisors of n; and the became a professor at the University of Georgia, where he Carmichael function λ(n), defined to be the exponent of the remained till 1999. He then moved to the Bell Labs, where group (Z/nZ)×. All of these functions arise frequently in num- he stayed for four years, before joining Dartmouth College ber theory. In fact, the second and third ones are directly in 2003. Pomerance is a well-recognized leader in number related to one of the oldest problems in mathematics, con- theory, focusing mainly on the analytic, computational and cerning perfect numbers. combinatorial aspects of this vast field. He is also a lead- In his first talk, entitled The Ranges of some Familiar Func- ing expert in applications of number theory to cryptography, tions, Pomerance discussed old and new results concerning the holding for many years the record for the fastest integer fac- value distribution of the aforementioned arithmetic functions. torisation algorithm based on the so-called quadratic sieve. Whereas, by now, we understand relatively well the range of Among the many honors that he has received for his research, ϕ and of σ, as well as their intersection, thanks to work of var- we mention that he is the recipient of the Chauvenet Prize ious authors, such as Erdős, Ford, Luca and Pomerance, the and of the Deborah and Franklin Tepper Haimo Award by range of the other two functions is more mysterious. For ex- the Mathematical Association of America (MAA), and of the ample, up until recently, we did not know the frequency of the Levi L. Conant Prize by the American Mathematical Society integers that are λ-values. In his talk, Pomerance presented (AMS). Moreover, he is a fellow of the American Association some very recent important progress towards this problem for the Advancement of Science (AAAS) and of the AMS, and due to Ford, Luca and himself. Moreover, he talked about he was an invited speaker at the 1994 International Congress another very recent piece of work by himself and Luca that of Mathematicians. resolved affirmatively an old question about whether a positive proportion of even integers are s-values. (The case of odd s-values is easier and well understood.) Pomerance’s second talk, entitled The First Function, had a more historical flavour and was targeted to a broad mathematical audience. His talk took us on a journey through the centuries, starting from Antiquity and the study of the function s and its second iteration s◦s = s2 by Pythagoras. In fact, Pomerance postulated that this is the very first occurence of a mathematical function in history, let alone of a dynamical system, hence the title of his talk. The study of the dynamics of s leaves several tantalizing questions, some still open, despite efforts spanning thousands of years. The fixed points of s are the perfect numbers such as 6 and 28. The question of whether there are infinitely many perfect numbers, dating Carl Pomerance from the time of Pythagoras and Euclid, remains open till today. The 2-cycles of s give rise to amicable pairs of inte- Pomerance visited Montréal for the occasion of two of the gers, such as (220, 284). We still don’t know whether there workshops organized within the framework of the 2014–2015 are infinitely many such pairs either, though it is certainly thematic year: Statistics and Number Theory and New ap- believed so. Pomerance then went on to present several re- proaches in Probabilistic and Multiplicative Number Theory. sults and conjectures on the more general dynamics of the His Aisenstadt lectures were delivered as part of the sec- function s. Some of the conjectures are even mutually exclu- ond of these workshops. In accordance with the theme of sive, thus giving a hint of how little we know about s. This the workshop, his lectures examined the statistical behaviour was a fascinating talk on questions almost as old as mathe- of various important arithmetic functions. More precisely, matics itself and the avenues of modern research they have Pomerance’s talk focused mainly on four functions: Eu- (continued on page 10) ler’s totient function ϕ(n), defined to be the cardinality of

BULLETIN CRM–5 crm.math.ca Counting Arithmetic Objects (Ranks of Elliptic Curves) November 10–14, 2014

Organizers: Henri Darmon (McGill); Jordan S. Ellenberg (Wisconsin–Madison); Andrew Granville (Montréal); Benedict Gross (Harvard) Melanie Matchett Wood (Wisconsin–Madison); Frank Thorne (South Carolina) role played by the Iwasawa main conjectures and by various classical and p-adic Gross–Zagier formulae for counting elliptic curves of rank one). The plan of the workshop reflected the almost “multidisci- plinary” nature of the topic being discussed, and a real effort was made to group speakers together according to the different unifying themes. The first day opened with two lecture series of two hours each, by Manjul Bhargava and Eric Urban, covering the “geometry of numbers” and “Iwasawa main conjecture” aspects of the main argument. (Chris Skinner had been originally sched- uled to give this lecture series but had to cancel, and it was fortunate that Eric Urban, one of Skinner’s key collaborators in the proof of the Main Conjecture, was able to step in to take his place.) Both lecture series were a great success in The workshop Counting Arithmetic Objects (Ranks of Elliptic conveying the broad outline and key ideas in the proof. Curves ) revolved around ranks of elliptic curves, focussing on The second day was dedicated largely to topics growing out what is expected of the “typical behaviour” of this mysteri- of Bhargava’s revolutionary techniques, with lectures devoted ous and subtle invariant. The workshop was unusual in that to counting elliptic curves with various extra structures, or to it concentrated largely on a single breakthrough: the recent suggesting heuristics for the behaviour of Selmer groups and theorem of Manjul Bhargava, Chris Skinner and Wei Zhang Shafarevich Tate groups modelled on the remarkably success- that a positive proportion of elliptic curves over the field of ful Cohen–Lenstra heuristics. rational numbers have rank zero and a positive proportion have rank one (and furthermore, that the Shafarevich–Tate The third day was devoted to the Euler system techniques groups of these elliptic curves are finite, and that their ranks used to relate Heegner points with L-functions, both classical agree with the order of vanishing of the associated Hasse– and p-adic, and Selmer groups, growing out of the fundamen- Weil L-function at the central point, as predicted by the tal work of Gross–Zagier and Kolyvagin. Some of the more re- Birch and Swinnerton-Dyer conjecture). This theorem rep- cent themes covered included the p-adic Gross–Zagier formula resents a symbolic landmark, and its proof combines many of Bertolini–Darmon–Prasanna and the “Jochnowitz congru- of the fundamental advances on elliptic curves and the Birch ences” explored in the previous decade by Bertolini–Darmon, and Swinnerton-Dyer conjecture achieved over the last several which are an important ingredient for exhibiting a positive decades, growing out of the fundamental work of Gross–Zagier proportion of elliptic curves of rank one independently of the and Kolyvagin on Heegner points, and, more recently: Birch–Swinnerton-Dyer and Shafarevich–Tate conjectures. (a) Bhargava’s revolutionary program for counting arithmetic The fourth day focussed on the fundamental recent results on objects, notably, his work with Arul Shankar counting the Iwasawa Main Conjecture growing out of the work of Skin- small order elements in Selmer groups; ner and Urban. The techniques involved here are rather differ- (b) the breakthroughs towards the Iwasawa main conjecture ent, relying on p-adic families of automorphic forms on unitary growing out of the work of Chris Skinner and Eric Urban groups of signature (2, 2) and (3, 1). The day concluded with and its extension by Xin Wan, and the resulting “con- a CRM–ISM Colloquium lecture by Kartik Prasanna which verse of Kolyvagin”-type theorems described in work of aimed to present the overall themes of the entire workshop to Skinner, Venerucci, and Zhang. colleagues in the city from other mathematical areas. The ideas at play in the work of Bhargava, Skinner and Zang The last day returned somewhat to the themes of the sec- involve an appealing mix of techniques, from the very ana- ond day, with pivotal lectures by Arul Shankar on the count- lytic (Bhargava’s profound new geometry of numbers argu- ing of the size of 5-Selmer groups of elliptic curves, and of ments and counting methods) to the algebraic (his methods Vladimir Dokchitser on the important parity conjecture for for parametrizing arithmetic objects of interest via ideas from elliptic curves. The workshop ended on a high note, with a “invariant theory over Z”) and the arithmetic (in the central (continued on page 10)

BULLETIN CRM–6 crm.math.ca Conférence de Théorie des nombres Québec-Maine 2014

Hugo Chapdelaine et Claude Levesque (Université Laval) Les 27 et 28 septembre 2014, la 16e Conférence de Théorie bien ne pas avoir à investir une semaine complète. Le prix à des nombres Québec-Maine se tenait à l’Université Laval en payer, c’est qu’il y a des sessions en parallèle. Bien que les l’honneur de Hershy Kisilevsky (Concordia) et Manfred Kol- exposés en parallèle portent sur des domaines différents, les ster (McMaster) pour souligner leur 70e anniversaire de nais- choix à faire sont souvent cornéliens. Le premier orateur, Mi- sance. Hershy et Manfred sont des mathématiciens bien re- chel Waldschmidt, a donné un joli exposé accessible à tous sur connus dans la communauté internationale qui ont beaucoup les multizêtas ; (Pierre Cartier préfère utiliser le mot « pluri- contribué au développement des mathématiques au Canada. zêtas »). Le dernier orateur, Ram Murty, égal à lui-même, a Tous les deux sont très humbles et leur modestie a certai- donné un merveilleux exposé sur une généralisation du déter- nement un effet d’osmose sur les membres de la jeune géné- minant de Dedekind. Au total, il y a eu 47 exposés et il est ration. Mentionnons d’entrée de jeu que ce fut très difficile possible de lire les résumés de tous les exposés sur la toile de les convaincre d’accepter d’être honorés et nous avons dû en tapant : Québec–Maine Number Theory Conference. Le promettre que nous serions « low profile ». but de ces réunions est d’offrir aux théoriciens des nombres Hershy s’est tout l’occasion de discu- ter et d’échanger avec d’abord impliqué en tous et chacun à pro- théorie d’Iwasawa et depuis une quin- pos de leurs résultats zaine d’années il se et de leurs problèmes, concentre sur les et ce, dans une atmo- courbes elliptiques. sphère très décontrac- tée, pour ne pas dire Il s’est impliqué ludique. dans l’organisation de plusieurs congrès à Ces congrès Québec- Montréal. Parmi ses Maine attirent aussi grandes réalisations, de bons étudiants il a démarré le Sé- sous-gradués. Cette minaire de Théorie année, sous le tutoriat des nombres Québec- de Steven J. Miller Vermont (QVNTS) et (Williams College), on lui doit la création sept étudiants sous- du Centre interuni- gradués ont présenté versitaire en calcul mathématique algébrique (CICMA). Au leurs résultats. Pour la plupart, c’était des calculs effectués début du congrès, on a d’ailleurs annoncé aux nombreux post- sur l’ordinateur. Ils présentaient leurs résultats en équipes. docs et gradués présents que sans l’existence du CICMA, ils On y a vu une équipe de trois étudiants qui, tous les trois à ne seraient probablement pas ici à Québec. l’avant, parlaient à tour de rôle. La formule est intéressante. Quant à Manfred, il a contribué au rayonnement du Ca- Ces étudiants ont été initiés au monde de la communication et se sont enrichis de leur expérience sans que les organisateurs nada en K-théorie algébrique et en cohomologie en organisant aient à offrir à chacun une période de 20 minutes (ou soient quelques conférences, en invitant plusieurs excellents mathé- maticiens et en supervisant des étudiants postdoctoraux. Si tout simplement forcés de refuser de le faire par manque de vous voulez des motifs pour vous motiver à faire l’étude de la temps). Quel charme que d’entendre un étudiant sous-gradué cohomologie motivique, écoutez religieusement les exposés de qui n’a pas encore son B.Sc. dire : « Let us assume the Sato– Manfred et vous verrez que son charisme est contagieux. Il a Tate conjecture ! » écrit un bel opuscule (un bijou !) sur la cohomologie étale. Son Suivant la tradition, le samedi soir nous avons eu droit à un sourire est figé en permanence sur ses lèvres et veuillez nous banquet japonais et nous avons bien échangé en compagnie croire, il a plusieurs belles histoires (aventures et mésaven- de Bacchus. tures) à raconter. Lors de l’un des congrès Québec-Maine, il a Le congrès a bénéficié de soutiens financiers de la part du dû forcer (casser ?) une fenêtre pour entrer dans sa chambre CRM, du Fields, de la NTF (Number Theory Foundation), d’hôtel car il dépassait l’heure du couvre-feu. du CICMA et du Département de mathématiques et de sta- La conférence Québec-Maine a attiré près de 70 mathémati- tistique de l’Université Laval. ciens (professeurs, postdocs, étudiants gradués et même sous- gradués). Les séances sont très intenses et les gens aiment

BULLETIN CRM–7 crm.math.ca Connecting Women in Mathematics across Canada October 3–5, 2014

Organizers: Galia Dafni (Concordia); Sara Faridi (Dalhousie); Shannon Fitzpatrick (Prince Edward Island); Megumi Harada (McMaster); Malabika Pramanik (UBC) Goals Activities This is an exciting time for women and visible minorities in The program made an effort to create a valuable experience the basic sciences. More and more women from diverse back- for the participants within a limited timeframe. The high- grounds continue to beat formidable odds and come to the lights included: forefront of their professions with spectacular achievements. • a collaborative and encouraging environment that facili- For the first time in its distinguished history, the Fields medal tated interaction, as well as one-on-one mentorship; has been awarded to a woman, Professor Maryam Mirzakhani. • expert advice, general and personalized, related to re- The presidents at the helm of the International Mathematical search, teaching, giving talks and other facets of academic Union (IMU) and the Canadian Mathematical Society (CMS) life; are two other highly acclaimed female mathematicians, re- • the presence of many role models. spectively Professors Ingrid Daubechies and Lia Bronsard. For the last decade or so, considerable effort has been in- We wanted the participating students to leave with the un- vested in researching and ensuring gender equity, and we are derstanding that they are not alone (even though they may beginning to see the gratifying repercussions of this height- be among a mere handful of females in their respective pro- ened awareness. At the same time, this increase in diversity grams), that women are being increasingly successful in chal- has not yet resulted in a representation of women and minori- lenging and rewarding careers and having a positive impact ties in graduate and postgraduate programs or as university on their communities. faculty at the rate or proportion one might expect. We are Given the limited amount of time we had, we broke the pro- still looking for ways to support this pipeline. gram into three parts. We made sure all junior participants The aim of the 2-day BIRS workshop was to target top female were allowed time to present their work to the other attendees, junior mathematicians from across the country, who are close as if they were giving a short job talk. We then had general to obtaining a Ph.D. or have just graduated. This is a time discussion times built into the program as well as one on one when researchers are faced with the challenges of job search meetings with assigned mentors so that the junior mathemati- in an increasingly tough academic environment. For gradu- cians could receive feedback on their presentations, as well as ate students, this may involve landing a coveted postdoctoral general career advice. position. Postdocs on the other hand would be looking for Finally, we had three (of the originally planned four) main in- tenure-track and/or instructorship positions after their term. vited lectures focusing on some major aspects of a mathemati- Different job searches come with varying, often nonexplicit, cal career: Research, University Teaching, and Presenting Re- criteria. Preparing oneself for a largely unknown hiring pro- search in Conferences. These fantastic talks were interactive, cess can be stressful. This is especially true for women, who and followed by passionate discussions. are often a small minority within their respective programs, and hence lack as extensive a support system as men. Also, We all came away from the meeting wishing it had been women role models are much rarer, making it difficult for longer! This year we will apply for a 5-day workshop at BIRS the junior women researchers to visualize themselves in se- which will allow us to cover much more ground. nior academic roles.

Contemporary Mathematics, Volume 630 three of the workshops organized during these programs: Ge- Geometric and Spectral Analysis ometry of Eigenvalues and Eigenfunctions, held from June P. Albin, D. Jakobson & F. Rochon (eds.) 4–8, 2012; Manifolds of Metrics and Probabilistic Methods in Geometry and Analysis, held from July 2–6, 2012; and Spectral Invariants on Non-compact and Singular Spaces, held In 2012, the Centre de recherches mathématiques was at the from July 23–27, 2012. center of many interesting developments in geometric and spectral analysis, with a thematic program on Geometric The topics covered in this volume include Fourier integral op- Analysis and Spectral Theory followed by a thematic year erators, eigenfunctions, probability and analysis on singular on Moduli Spaces, Extremality and Global Invariants. spaces, complex geometry, Kähler–Einstein metrics, analytic torsion, and Strichartz estimates. This volume contains original contributions as well as useful survey articles of recent developments by participants from www.ams.org/bookstore/

BULLETIN CRM–8 crm.math.ca Grande Conférence publique du CRM Chris Danforth: Measuring Emotional States in Real-Time

Christiane Rousseau (Université de Montréal)

Chris Danforth from the University of Vermont came to Mont- Moreover, it is observed in many languages. Of course, the réal to give a CRM Grande Conférence (public lecture) on instrument is not perfect, especially on the unhappiness side, November 20, 2014. Chris Danforth is a specialist on chaotic and Chris Danforth’s team is working on improving it. For behaviour in meteorological systems and a member of the instance, the Hedonometer only does a partial coverage of Mathematics Climate Research Network. He has diversified words. The instrument also ignores the context and cannot his research interests in the past years. The common thread detect sarcasm. of his research is that he works on large-scale systems in many The Hedonometer also measures the long-term trends in the fields including sociology, nonlinear dynamics, networks, ecol- world. Compiling data since 1960 shows that the average va- ogy, and physics. For many years now he has been working lence2 of lyrics is decreasing over time. In music, this comes on a fascinating subject, namely how mathematics can help from the appearance of new styles, like Rap/Hip-Hop, Punk in “measuring emotions in real-time,” and that was the title and Metal/Industrial that rate lower than classical music for of his talk. instance. Moreover, the frequencies of some happy words are moving down, while unhappy words are moving up. The Hedonometer further facilitates the analysis of seasonal trends, the correlation of happiness with age: it reveals that the maximum happiness occurs between 45 and 50. On Twit- ter, finer movements are analyzed during the day, such as peaks of food-related words around lunchtime. Using the Hedonometer, we can draw happiness maps with fine detail inside cities, and classify cities in the U.S. from the happi- est (Napa in California) to the saddest (Beaumont in Texas). One can learn a lot about a city from the words the local people use: for instance, the obesity rate is positively correlated with McDonald’s and other “obese” words, and negatively correlated with “skinny” words such as tofu or brunch. More Chris Danforth generally, happiness is positively correlated with income and Why measuring emotional states? Alan Greenspan was con- negatively correlated with the obesity rate. vinced that such data would be very helpful for forecasting the economy: “I’ve been dealing with these big mathematical models of forecasting the economy . . . If I could figure out a way to determine whether or not people are more fearful or changing to more euphoric, I don’t need any of this other stuff. I could forecast the economy better than any way I know.”1 Chris Danforth and his team have built a tool, the Hedonome- ter, allowing them to measure the emotions. It relies on how people feel about individual words. How can we measure happiness? The idea of the Hedonome- ter is to measure the happiness of words. Individual words Daily and weekly cycles are observed with the happiness start- are rated on a 9-point unhappy–happy scale by many people. ing high every morning and decreasing during the day. Hap- But, where do we measure the happiness? The media are a piness is higher on Saturdays and lowest on Tuesdays, and daily source of words. And, now, social media provide a large catastrophes are recorded on the Hedonometer. source of data, already in digital form, that can be analyzed The lecturer ended by presenting some ongoing projects. To everyday. name a few: what is the largest outbreak of happiness? How A first surprising result is that there is a bias towards happi- does happiness vary with proximity to nature? Or to Farmers’ ness: more words seem to be on the happy side than on the (continued on page 10) unhappy side. This frequency-independent happiness bias is 1From an interview with Jon Stewart on the Daily Show (September a very general phenomenon: it can be observed in Twitter, 18, 2007) in books, in the New York Times, as well as in music lyrics. 2A term associated with attractiveness and positive emotions

BULLETIN CRM–9 crm.math.ca

James Maynard Awarded the SASTRA spacing. For this last statistic, they showed that the Linnik Ramanujan Prize and a Clay Research points exhibit random-like behaviour for almost all n. Fellowship This fascinating talk on some exciting breakthroughs on a classical question of analytic number theory was very much In December 2014, James Maynard, a 2013–2014 CRM–ISM appreciated by the broad audience of mathematicians in at- Postdoctoral Fellow (currently at Oxford University), was tendance. awarded the top world prize for a young number theorist, the SASTRA Ramanujan Prize, by SASTRA University in India, citing his “spectacular contributions to number theory, Carl Pomerance (continued from page 5) especially on some famous problems on prime numbers.” In January 2015 it was announced that Maynard has been ap- created. It was very much appreciated by the broad audience pointed as a Clay Research Fellow for a term of three years of mathematicians in attendance. beginning July 1, 2015. The third talk was entitled Amicable Numbers, and its main In November 2013, Maynard proved a remarkable result on focus was some of the most recent developments on the saga of small gaps between primes, which eventually showed that the function s and its iterates, initiated by Pythagoras, as ex- there are infinitely many pairs of consecutive primes whose plained above. More precisely, the presentation was dedicated difference is at most 246. This result has now appeared in the mainly to two recent results. The first one, due to Pomerance Annals of Mathematics. himself, concerns upper bounds for the counting function of Then in August 2014, while still in Montréal, Maynard proved amicable numbers (that is to say, how many there are up to another beautiful result, this time about large gaps between a certain bound x). The second one concerns a result jointly primes, namely how far apart can two consecutive primes get. proven by Luca and Pomerance on the even s-values, also re- This longstanding problem, an improvement on a lower bound ferred to in the first talk. The talk elucidated the details known as Rankin’s formula, was made famous by Paul Erdős, behind the proofs of these two very recent and important ad- who offered his highest award ($10,000) for the solution. A vancements and ended with some potential avenues for future similar result had been proved independently by Ford, Konya- research. gin, Green, and Tao, but Maynard’s proof rests on a different idea, which is simpler and more flexible. A vast improvement Counting Arithmetic Object will appear as a joint paper of all five authors. (continued from page 6)

stimulating and thought-provoking lecture by Bjorn Poonen Zéev Rudnick presenting heuristics that support the belief that the rank of (continued from page 4) elliptic curves might well be uniformly bounded: a prediction The question arises from a celebrated result of Legendre and that runs against a lot of conventional wisdom, although An- Gauss which determines which integers n can be represented drew Granville has arrived at strikingly similar conclusions (primitively) as a sum of three squares, i.e., n = x2 + y2 + z2 based on a very different set of considerations. for some integers x,y, z such that gcd(x,y, z) = 1. This occurs All in all this was an extremely successful, stimulating, and if and only if n . 0, 4, 7 mod 8, and for those there are many thought-provoking workshop on some of the most fundamen- ways of writing n as a sum of three squares. These different√ tal and rapidly evolving questions in number theory. representations give collections of points L(n) = {(x,y, z)/ n} on the unit sphere, and a fundamental result, conjectured by Chris Danforth Linnik, is that the sets L(n) become uniformly distributed on (continued from page 9) the sphere as n → ∞. Linnik proved in 1940 that this is true under the Generalized Riemann Hypothesis, and it was proved markets? Or to a Walmart? Or to one’s partner? In the near unconditionally by Duke and by Golubeva and Fomenko in future dozens of new languages are to be added, together with 1988. In the lecture, Rudnick explored what happens be- geographic mapping of happiness in other parts of the world yond uniform distribution, giving evidence of randomness on and other sources (e.g., Google Trends) thus allowing further smaller scales. This is in sharp contrast to what happens with use in comparative studies. The Hedonometer team is work- sums of two or four or more squares. ing on moving from words to phrases, and to enlarging the The results presented have interesting connection to physics, spectrum of emotions studied, to include, for example, amaze- and configurations of charges on the sphere which minimize ment. This will allow quantifying cultures with the number energy (stable configurations). Finding stable configurations of narratives, metaphors and stories they have. is notoriously difficult. Rudnick and his collaborators have The lecture left no one in the audience indifferent and a pas- shown that the energy of the Linnik points L(n) is close to sionate debate started during the period of questions and con- minimal. They also studied other statistics for the Linnik tinued over for the whole vin d’honneur. points, such as the nearest neighbour distance and the least BULLETIN CRM–10 crm.math.ca SCHOLAR—A Scientific Celebration Highlighting Open Lines of Arithmetic Research

Alina Carmen Cojocaru (University of Illinois at Chicago) There seem to be at least three layers of understanding: the College); Dinakar Ramakrishnan (Caltech); Michael I. Rosen first is acquaintance with syntax, words, and symbols; the (Brown); N. Saradha (TIFR); Joseph Silverman (Brown); second is meaning; and the third can be called the meaning Cam Stewart (Waterloo); Dinesh S. Thakur (Rochester); Yi- of meaning. The means of entering the deeper layers is to tang Zhang (New Hampshire). Their presentations illus- cogitate, research, reflect what we know at the first layer.1 trated research of the highest caliber on a broad spectrum This is how M. Ram Murty, Professor and Head of the De- of arithmetic topics: abelian varieties; function field arith- partment of Mathematics and Statistics at Queen’s University metic; Galois representations; L-functions; modular forms; in Kingston, Canada, might sound when talking to a friend, sieve methods; transcendental number theory. A special lec- a colleague, or a student. ture was given by Yitang Zhang, who had just been awarded the Cole Prize for his contributions to the twin prime conjecture. Zhang’s talk alone, titled A new method applying to small gaps between primes, attracted more than 130 attendees (conference participants and others from the Montréal mathematical community). Associated with the meeting was also a distinguished lecture, The Sato–Tate Conjecture, given by M. Ram Murty himself within the CRM–ISM Colloquium series on October 18. During the conference, a celebratory dinner was held for all participants. A special guest was Cynthia Fekken, Profes- sor of Psychology and Associate Vice-Principal (Research) at Queen’s University, who travelled to the conference venue solely for conveying Queen’s University’s respect and admira- tion for Murty’s scholarly legacy. Heartfelt stories were shared by several conference participants who lauded Murty’s excep- M. Ram Murty tional contributions and commitment to research, education, A refined scholar, an intellectual with an unquenchable thirst and service, both at his home institution and worldwide. The for all facets of truth, scientific and philosophic, M. Ram evening was filled with deep feelings of gratitude, respect and Murty has had a profound impact on the development of affection for M. Ram Murty and inspired the participants to- number theory throughout the world. To honour his math- wards great pursuits. ematical legacy, a conference focused on new research di- The conference was generously supported by the Centre de rections in number theory inspired by Murty’s most signif- recherches mathématiques, the Fields Institute for Research icant achievements was organized by Alina Carmen Cojocaru in Mathematical Sciences, the Number Theory Foundation, (UIC), Chantal David (Concordia), Hershy Kisilevsky (Con- and the Université de Montréal. It will culminate with a vol- cordia) and Francesco Pappalardi (Roma Tre) at the Centre ume in the CRM Proceedings series, published together with de recherches mathématiques in Montréal. The conference the American Mathematical Society as part of the Contem- was titled SCHOLAR—A Scientific Celebration Highlighting porary Mathematics series. Open Lines of Arithmetic Research and took place October 1 15–17, 2013. M. Ram Murty, The Art of Research, Colloquium Lecture, Queen’s University, Kingston, ON, 2001. A remarkably successful event, the conference had about 100 participants representing several generations of research ABONNEMENT. Pour vous abonner au Bulletin du CRM mathematicians, from beginning graduate students to emeri- veuillez compléter un bref formulaire à la page web : crm. tus professors. The speakers included some of the most promi- math.ca/bulletin/abonnement. Vous avez le choix entre nent researchers in number theory from around the world: les éditions imprimée et électronique. (Pour annuler votre Henri Darmon (McGill); Etienne Fouvry (Paris-Sud); John abonnement, veuillez aller à la même page web.) Friedlander (Toronto); Dorian Goldfeld (Columbia); Henryk SUBSCRIPTIONS. You can subscribe to the printed or elec- Iwaniec (Rutgers); Ernst Kani (Queen’s); Wen-Ching Winnie tronic versions of the Bulletin by completing a brief form Li (Penn State); Kumar Murty (Toronto); Yiannis Petridis at this webpage: crm.math.ca/bulletin/abonnement. (To (University College London); Carl Pomerance (Dartmouth unsubscribe, please go to the same webpage.)

BULLETIN CRM–11 crm.math.ca Joachim Lambek, FRSC December 5, 1922–June 23, 2014

Michael Barr (McGill University), Philip Scott (University of Ottawa) and Robert Seely (McGill University) Professor Joachim (Jim) He was immediately hired as a lecturer in mathematics at Lambek was a prominent McGill. In those postwar days, McGill had to expand rapidly member of the Montréal to make room for returning veterans and one way it dealt with mathematics community the problem was by opening a satellite campus in St. Jean. throughout the second half Getting there was his responsibility. One of Jim’s stories was of the twentieth century, ac- that once he missed the train for reasons beyond his control tive until his death last sum- and skipped the class. The chairman told him he should have mer. During his career he taken a taxi—at his own expense of course. This would have was an inspiration, a men- cost $10 (probably equivalent to at least $150 today), consid- tor, and a dear friend to erably more than his pay for the class. many generations of math- One of Jim’s former students, Prof. Robert Raphael of Con- ematicians, linguists, and cordia University, recalls an amusing anecdote from Jim’s stu- other scholars, and his loss dent years at McGill in the 1940s. Jim faced a problem finish- has left a huge gap in our ing his undergraduate degree, since taking a chemistry course community. was a requirement to graduate. But the director of the chem- The following account of istry building denied Lambek entrance, as Germans were not Joachim Lambek Jim’s life and work is partly allowed because of sensitive experiments taking place in the based on a talk given on the occasion of his 75th birthday, by building. The McGill Senate acted and replaced the chemistry Michael Barr. course with an arts requirement, so Jim graduated. After the Jim was born in Leipzig, Germany, in 1922. In the late ’30s, war, the building director was arrested on charges of spying he was among the last “children” (he was in his mid-teens) for the Soviet Union. to escape Nazi Germany on the Kindertransport to England, In 1950, he completed his Ph.D. under Hans Zassenhaus (in where he worked on a farm. Shortly after, along with other fact, he wrote two theses, one in mathematical physics, then male German Jews, he was deported (as an “enemy alien”) to one in pure mathematics) and was promoted to assistant pro- Canada and interned in a prison work camp. In later years, fessor. In those days, assistant professors taught at least 12 Jim often joked that they were not told until after the ship hours a week and were not expected to do much, if any, re- set sail whether they were headed to Canada or to Australia: search. Yet, between 1951 and 1959 he had 14 publications. being on a ship bound for Canada saved him from having to Half of those papers were joint with Leo Moser and appear to do 2-category theory. After he had spent about two years in be in combinatorics and elementary number theory. However, a camp, mainly in New Brunswick (but including a week on even in those early days he had begun research on several of Île Sainte-Hélène, later the site of Expo 67), the authorities the areas of mathematics that would occupy the rest of his decided that people like Jim should not have been impris- career. In 1958, he published his first paper in linguistics, on oned and he was released. During his time in the work camp the syntactic calculus, as well as his first paper on rings of he had begun studies in mathematics and logic, taught in quotients. the camps by Fritz Rothberger and other detained European Although there were two more publications in mathematical mathematicians, and even entered into correspondences with linguistics in the following three years, he appears to have other mathematicians and logicians, including Willard V.O. abandoned the subject for over a dozen years. But the world Quine. On release, those prisoners who found Canadian spon- did not abandon this work, and a small but lively group of sors were allowed to stay in Canada; fortunately, a Montréal researchers, mainly in Europe, developed the subject of “Lam- businessman agreed to sponsor him. So he settled in Montréal bek grammars.” He returned to the area several times in his and entered McGill, earning an honours degree in mathemat- subsequent career, and it was a dominant theme in his last ics in 1945 and an M.Sc. a year later. One result of all this is decade or so. During the same period, Jim published a short that he spent the war years safely in Canada, while his mother paper in the Bulletin of the Canadian Mathematics Society and sister, who were allowed to stay in England, endured the (1961) entitled How to program an infinite abacus. It turns blitz, and came to North America only at the end of the war. out he co-discovered an important notion of abstract comput- ing device (an alternative to Turing’s machines), today called

BULLETIN CRM–12 crm.math.ca

Register machines. The idea was simultaneously (and appar- In a quite extraordinary series of works late in his career, ently independently) developed by several leading researchers, Jim developed an entirely new approach to his earlier work notably M. Minsky. in linguistics, which he called “pregroup grammars,” a formal From the mid-50s to the mid-60s, Jim put most of his efforts typing system for natural language processing, which actually arose from his growing interests in higher category theory. He on ring theory, particularly rings of quotients. He published From word to sentence many papers on the subject, culminating in the very successful wrote a book for linguists (Polimet- and influential book, Lectures on Rings and Modules (1966). rica, 2008), as well as numerous articles for mathematicians on his new theories of linguistics. Jim spent his sabbatical year 1965-66 in Zürich at the Forschungsintitut für Mathematik der Eidgenössische Tech- Math Reviews lists more than 130 publications at nische Hochschule where Beno Eckmann had gathered together a group of people interested in algebraic topology, ho- present, and that does not From mological algebra, and, incidentally, category theory. That include his last book Rules of Grammar to Laws year in Zürich reoriented his research into category theory. of Nature, which he wrote Two mathematicians who subsequently became pivotal members of the Montréal Category Theory group also were in and saw to publication dur- Zürich around the same time, and met Jim then, or shortly ing the last months of his life. after his year there. Michael Barr remembers being given Jim’s monograph Completions of Categories (SLNM 24, 1966) Jim Lambek’s reputation as soon as he arrived in Zürich; he first met Jim in Chicago does not depend only on the shortly afterwards. Marta Bunge was in Zürich the same year quantity and quality of his as Jim (along with Bill Lawvere, among others), and remem- research, however. Many bers being struck by his quick grasp of what was important of his past students, post- in the subject. Jim arranged to bring both of these future doctoral fellows, colleagues colleagues to Montréal. and friends have commented During the 1970s, his research combined ring theory, torsion on his generosity (and insis- tence on high standards). He helped launch many a career theory and category theory, much of the latter in collaboration with Basil Rattray. During this decade, he renewed his in mathematics by putting innumerable students on the track interest in mathematical linguistics, including formal studies of promising research projects, and providing beginning researchers with postdoctoral fellowships and opportunities to of verb conjugations in French and Latin. As well, he began develop and present their results. He was also a good friend working on applications of category theory to logic. This last 1 interest resulted in a fruitful collaboration with Phil Scott and colleague, a generous host, and excellent company. He will be deeply missed, personally and professionally. (initially a postdoc of Jim’s, later a professor at U. Ottawa), culminating in the now-classic book, Introduction to Higher 1We cannot resist mentioning one of Jim’s favourite stories. Jim Order Categorical Logic, in 1986. Lambek’s works in cate- would often organize workshops and conferences in Montréal, to which both mathematicians and linguists were invited. Jim, always the con- gorical proof theory were of seminal influence. They ranged vivial host, would lead the group to some downtown restaurant. The from his early applications of Gentzen’s methods to coher- problem, he observed, was that the mathematicians could never figure ence theorems in category theory, to his later introduction out the bill, and the linguists couldn’t speak French! and promotion of multicategories, substructural logics, and [. . . ] But it may be Jim Lambek who had the biggest impact bicategories in widely varying areas, from algebra to linguis- on my career of any McGill professor, through the kind of tics to physics. chain of accidents that so often affects a life course. Jim Over the years, Jim’s research work never slacked off. He taught a course on the Theory of Computation and Mathe- regularly published papers in categorical algebra and logic, matical Linguistics. That came in handy when, in my first in linguistics, in philosophy, and even in quantum physics, week of graduate school at Harvard, we were assigned a tech- returning to a topic he addressed in his first Ph.D. thesis. In- nical paper in that field on the problem of language acquisi- deed, he was an enthusiastic supporter of using Hamilton’s tion. Thanks to his class, I was able to understand it, and quaternions in physics; a popular account of his views and of wrote a course paper which became my first major publica- his meeting with Dirac is contained in his article If Hamil- tion. That led to my being hired to teach language acquisi- ton had prevailed: quaternions in physics in the Mathemati- tion in my first job, which led to an encompassing interest cal Intelligencer (1995). During this period he also wrote an in all aspects of language, from how it evolved to how it is undergraduate text on the history of mathematics with Bill used at its best in good writing. Anglin, The Heritage of Thales (1995), which is unique in its Steven Pinker (Harvard University) deep coverage of a wide range of topics. To professor, with love, The McGill News, Fall–Winter 2014

BULLETIN CRM–13 crm.math.ca Vision de l’espace : hommage à Alexander Grothendieck

Jean-Pierre Marquis (Université de Montréal) Les 20 et 21 février dernier, se tenait à l’Université de Mont- versité McGill et un des bâtisseurs de la logique catégorique, réal un colloque intitulé « Vision de l’espace : hommage à s’est attardé, pour sa part à la notion de prétopos. Sa confé- Alexander Grothendieck ». Comme le sous-titre l’indique, ce rence, intitulée « Grothendieck’s concept of pretopos : its role colloque se voulait un hommage au mathématicien Alexan- in algebraic geometry and in logic », a permis à l’auditoire der Grothendieck qui s’est éteint le 13 novembre dernier à de mieux comprendre comment la notion de catégorie cohé- Saint-Lizier en France à l’âge de 86 ans. Récipiendaire de la rente et la notion de prétopos étaient liées non seulement dans médaille Fields en 1966, Grothendieck est considéré à juste l’œuvre de Grothendieck en géométrie algébrique, mais éga- titre comme l’un des plus importants mathématiciens du xxe lement en logique catégorique. Le philosophe des mathéma- siècle. Entre 1955 et 1970, il a complètement réécrit les fonde- tiques Jean-Jacques Szczeciniarz, professeur des universités à ments de la géométrie algébrique en introduisant les concepts l’Université Paris-Diderot et directeur du département d’his- de schéma, de topos, de motifs et toute une panoplie de théo- toire et de philosophie des sciences à la même université, s’est ries cohomologiques dans le but de démontrer les conjectures concentré sur les aspects ontologiques et épistémologiques in- de Weil sur les fonctions zêta locales. La dernière des conjec- hérents à la notion de schéma. Sa conférence, intitulée « Ré- tures de Weil fut démontrée par un étudiant de Grothendieck, flexions sur la révolution théorique opérée par Grothendieck Pierre Deligne, en 1974. Cette refonte en profondeur de la en géométrie algébrique », a mis en lumière comment dans géométrie algébrique repose sur une utilisation intensive, ori- l’approche de ce dernier la connaissance des objets introduits ginale et essentielle de la théorie des catégories, théorie que devenait indirecte et dépendante des outils développés en pa- l’on croyait à l’époque limitée au statut de langage utile et ul- rallèle aux objets eux-mêmes. La notion de schéma et les nou- timement accessoire. On peut affirmer sans sourciller que les velles théories cohomologiques créées en parallèle illustraient travaux de Grothendieck constituent un nouveau paradigme parfaitement ses propos. La première journée s’est terminée mathématique au sein duquel les concepts catégoriques sont par la conférence de l’éminent mathématicien français Pierre inextricables. L’impact de ses travaux a inspiré plusieurs ma- Cartier, membre du groupe Bourbaki, professeur à l’IHES et thématiciens de par le monde ; et Montréal, plaque tournante ancien collègue de Grothendieck, qui a fait un survol éclairé de la théorie des catégories depuis la fin des années 1960 jus- et éclairant des notions de topos et de champs. qu’à aujourd’hui, y a largement puisé. C’est dans cet esprit Au cours de la seconde journée, les conférenciers se sont tour- que les organisateurs de ce colloque ont invité mathématiciens nés vers les mathématiques développées par Grothendieck au et philosophes des mathématiques à venir rendre un hommage cours des années 1980 et 1990. Ainsi, Mathieu Anel, cher- posthume à l’œuvre de Grothendieck. cheur au CNRS et associé au Laboratoire Sphère de l’Univer- Les notions de schéma et sité Paris-Diderot, a mené les participants « au-delà des idées de topos ont constitué les de Grothendieck », car c’était bien là le titre de sa contri- principaux pôles des confé- bution. Mathieu Anel a montré de manière saisissante com- rences de la première jour- ment les notions de catégories dérivées et de dérivation en née. La conférence de Colin général menaient directement à la théorie des catégories su- McLarty, qui occupe le poste périeures et comment cette dernière, non seulement simpli- de Truman P. Handy Profes- fiait plusieurs définitions classiques, mais comment plusieurs sor of Philosophy à la Case constructions qui semblent artificielles et ad hoc dans un Western Reserve University, contexte catégorique traditionnel deviennent naturelles dans s’intitulait « How (and why) le nouveau contexte. Finalement, André Joyal, professeur ho- Grothendieck simplified co- noraire à l’UQAM, a enchaîné en récapitulant l’histoire des homology theory ». Après quasi-catégories et des catégories supérieures, développement avoir présenté un survol his- dans l’esprit des travaux de Grothendieck sur les champs et torique des théories cohomo- la notion de catégorie test. Alexander Grothendieck en 1970 logiques jusqu’à l’époque de Photo : Konrad Jacobs ; © MFO Centré sur les concepts plutôt que sur les résultats, les outils Grothendieck, le professeur et les développements techniques, ce fut l’occasion d’évaluer McLarty a montré clairement comment, par le biais de la la richesse des contributions de Grothendieck sur les mathé- théorie des schémas et de la théorie des topos ce dernier a matiques de la dernière moitié du xxe siècle et sur les mathé- bel et bien simplifié la théorie de cohomologie, et ce, même matiques de demain. si le prix à payer semble pour certains trop marqué par la généralité et l’abstraction. Michael Makkai, logicien à l’Uni-

BULLETIN CRM–14 crm.math.ca Les mille et une façons de partager sa passion pour les mathématiques

Jean-Marie De Koninck (Université Laval) Pour arriver à convaincre jeunes et moins jeunes que les ma- SMATH habitant sur cette planète est à bord d’un vaisseau thématiques sont partout autour de nous, les mathématiciens spatial en direction de la planète Terre. Il s’y rend puisqu’il utilisent toutes sortes d’astuces : des conférences grand public, a entendu dire qu’on utilise les nombres depuis des milliers des livres de vulgarisation très accessibles, etc. C’est dans ce d’années sur la Terre et que cela s’est avéré fort utile. Son contexte et avec l’appui de l’Université Laval et de Mitacs que vaisseau atterrit sur la scène et voilà que SMATH demande si nous avons créé, en 2005, le programme SMAC (Sciences et quelqu’un dans la salle connaît les mathématiques. C’est à ce mathématiques en action), dont les principaux objectifs sont moment que j’interviens et que j’invite SMATH et les jeunes de susciter chez les jeunes un intérêt pour les mathématiques dans la salle à explorer avec moi l’univers fascinant des ma- et les sciences et de démystifier les mathématiques auprès de thématiques. Nous parcourons alors ensemble trois modules : la population en général. La voie que nous avons choisie est (1) l’histoire des nombres en remontant à leur invention il y a celle d’offrir des activités ludiques et amusantes tels des jeux 30 000 ans ; (2) le phénomène du son : sa vitesse, son intensité et des spectacles. Je présente ici celles relevant du domaine et sa fréquence ; et (3) l’exploration spatiale : notre système du spectacle. solaire, ses planètes et ses comètes, ainsi que la taille relative de tous ces objets. À la fin du spectacle, SMATH quitte la Un spectacle multimédia sur les mathématiques Terre avec Léa, une petite fille qu’il a rencontrée au cours du spectacle, tout en déclarant qu’il va partager avec ses amis En 2005, nous avons créé pour les écoles secondaires une tout ce qu’il a appris sur les nombres. conférence-spectacle multimédia appelée Show Math. Le spectacle fait intervenir sur scène des comédiens professionnels. L’objectif de cette activité est de faire rire les jeunes dans un environnement mathématique et de leur faire réaliser que les maths sont partout. Les sujets abordés dans Show Math in- cluent : (1) comment Ératosthène s’y est pris pour mesurer la circonférence de la Terre en utilisant seulement des notions rudimentaires de géométrie ; (2) les mathématiques qui nous permettent de stocker des centaines de chansons sur un petit appareil MP3 ; et (3) le paradoxe des anniversaires selon lequel dans une salle où il y a 23 personnes, la probabilité qu’au moins deux d’entre elles aient le même anniversaire est de 50%, et que dans une salle où il y a 57 personnes, la pro- babilité grimpe à 99%. Comme la plupart des écoles qui ayant accueilli Show Math souhaitaient avoir une suite, nous avons développé Jean-Marie De Koninck Show Math 2 en 2009. Les sujets abordés comprennent (1) comment la notion de fractal a vu le jour et comment elle Pluton va en appel! est utilisée par les producteurs de films pour créer des pay- sages fantastiques comme dans Star Trek et Avatar ; (2) pour- Notre dernière création est Pluton va en appel ! Il s’agit d’une quoi les méthodes de chiffrement les plus sécuritaires reposent pièce de théâtre dans laquelle l’ancienne planète, Pluton, va sur la difficulté à factoriser des grands nombres ; et (3) le fait en appel devant le Soleil afin de pouvoir réintégrer le rang que le GPS utilise l’intersection de trois sphères pour localiser des planètes. Rappelons d’abord que dans les livres d’astro- notre emplacement. nomie publiés avant 2006, on peut lire que notre système solaire est composé de neuf planètes : les quatre planètes ro- Petit Show Math cheuses (Mercure, Vénus, la Terre et Mars), les quatre pla- nètes gazeuses (Jupiter, Saturne, Uranus et Neptune) et enfin Pour répondre au souhait exprimé par les directeurs d’écoles Pluton, que l’on croit faite de roche et de glace. Cependant, primaires, nous avons créé Petit Show Math. Il s’agit d’un à compter de 2003, les astronomes se sont mis à découvrir autre spectacle à saveur mathématique, mais adapté aux plus que plusieurs autres corps en orbite autour du Soleil étaient jeunes. Le fil conducteur est comme suit : Dans une ga- beaucoup plus gros et plus massifs que Pluton et ont ainsi laxie très lointaine, il existe une petite planète où les habi- commencé à se demander s’il n’était pas temps d’établir les tants ignorent la notion de nombre. Un petit garçon nommé critères de ce qui constituait véritablement une planète. C’est BULLETIN CRM–15 crm.math.ca ainsi qu’en 2006, l’Union astronomique internationale a intro- Kai Behrend received a Ph.D. in 1991 at the University of duit une nouvelle classe d’objets appelés planètes naines et California, Berkeley. He joined the faculty of the University décida d’y inclure Pluton. La décision fut loin d’être unanime of British Columbia in 1994. Professor Behrend has received et créa du coup toute une controverse. Plusieurs prétendaient numerous recognitions for his research, including the 2001 que pour des raisons historiques, Pluton devrait conserver son Coxeter–James Prize and the 2011 Jeffery–Williams Prize of titre de planète. En mars 2007, la Chambre des députés de the Canadian Mathematical Society, as well as an invitation l’état du Nouveau-Mexique a même fait adopter une résolu- to speak at the International Congress of Mathematicians in tion stipulant que Pluton conserverait son titre de neuvième Seoul in 2014. planète. J’ai eu l’idée de créer une pièce de théâtre basée sur cette controverse en lisant le livre The Case for Pluto d’Alan Boyle. Dans la pièce, Mercure, parlant au nom des planètes Christiane Rousseau nommée membre du rocheuses, soutient que Pluton doit retrouver son titre, alors comité scientifique de l’IBSP de l’UNESCO que Neptune, qui parle au nom des planètes gazeuses, ar- gumente contre, craignant que si Pluton redevient une pla- Christiane Rousseau, membre du Centre de recherches mathé- nète, cela aura pour effet de redonner la majorité aux pla- matiques et professeure au Département de mathématiques nètes rocheuses. Le spectacle est un prétexte pour expliquer et de statistique de l’Université de Montréal, a été nommée diverses notions mathématiques telles que celle des points de membre du prestigieux conseil scientifique de l’International Lagrange et pour utiliser les mathématiques qui permettent Basic Sciences Programme (IBSP) de l’UNESCO pour un de décrire le phénomène d’assistance gravitationnelle permet- mandat de trois ans de 2015 à 2017. tant aux sondes spatiales de se déplacer dans le système solaire Christiane Rousseau a fait ses études en mathématiques à en utilisant la force gravitationnelle exercée par les différentes l’Université de Montréal où elle a obtenu son doctorat en 1977. planètes. Pluton va en appel ! sera certes encore plus popu- Après des études postdoctorales à McGill, elle est revenue à laire dans l’année qui vient parce que les médias seront alors très intéressés par les péripéties de la sonde New Horizons l’Université de Montréal comme professeure. qui a quitté la Terre en 2006 et qui rejoindra Pluton le 15 Elle a été directrice de son département de 1993 à 1997 et juillet 2015. présidente de la Société mathématique du Canada de 2002 à 2004. Lorsqu’elle dirigeait le Centre de recherches mathéma- tiques (CRM) en 2008–2009, elle a lancé l’année internationale Kai Behrend « Mathématiques de la planète Terre 2013 » sous le patro- 2015 CRM–Fields–PIMS Recipient nage de l’UNESCO. De 2011 à 2015, elle a été vice-présidente Professor Behrend is an internationally recognized leader in de l’Union mathématique internationale (UMI), et elle conti- the field of algebraic geometry, whose contributions to the nue de siéger au comité exécutif de l’UMI durant la période subject are noted both for their depth and scope. He has ob- 2015–2018. Pendant toute sa carrière, elle a mené en parallèle tained fundamental results in the theory of algebraic stacks, des activités de recherche et d’encadrement d’une part et des Gromov–Witten theory and the study of Donaldson–Thomas activités de vulgarisation et de sensibilisation aux mathéma- invariants. In particular, his pioneering works on the con- tiques : conférences dans les cégeps, organisation de camps struction of a “virtual fundamental class” played a key role mathématiques, articles dans des magazines mathématiques. in laying the algebraic foundations of the Gromov–Witten Le directeur du IBSP a souligné la réputation de Christiane theory. Later, he made a breakthrough in the study of the Rousseau en sciences fondamentales et son engagement sou- Donaldson–Thomas invariants by showing that, for certain tenu envers la coopération internationale. Les membres du spaces, the degree of the virtual fundamental class could be conseil scientifique, au nombre de 30, sont chargés de suivre, expressed as the topological Euler characteristic weighted by d’orienter et de superviser du point de vue scientifique, les a natural constructible function, depending only on the intrin- projets proposés par l’IBSP. sic properties of the space. This function is now widely known Créé en 2005, l’IBSP met l’accent sur la promotion de la for- as Behrend’s function. It allowed the use of motivic methods mation et de la recherche en sciences fondamentales et en to compute Donaldson–Thomas invariants, and made it pos- enseignement des sciences, ainsi que l’utilisation pratique des sible to obtain their categorified and motivic versions, which percées scientifiques pour répondre aux problèmes environne- is currently among the hottest trends in the subject. In his mentaux et aux besoins humains, et contribuer à l’améliora- earlier work, Professor Behrend obtained an important gen- tion de la qualité de vie et de l’éducation. eralization of the Lefschetz trace formula for algebraic stacks, presently known as Behrend’s trace formula. The ideas put forward by Kai Behrend have already proven to be immensely influential and will undoubtedly have a lasting impact on this area of mathematics.

BULLETIN CRM–16 crm.math.ca

AdS/CFT, Homolography, Integrability that came out of the AdS/CFT correspondence, the so-called (continued from page 2) viscosity bound: the lower bound on the ratio of the viscosity over the entropy density, that was conjectured to hold for all which has been developed independently by combinatorists, physical systems, all hydrodynamical systems, as a result of and by group theorists, having to do with directed networks, certain calculations done in gravity, that η/s should always be positroid cells, and plabic graphs, just happened to do the job bigger than 1/(4π). First of all it’s satisfied for all physical for these two different areas. Obviously one wants to bring systems known in nature, but it also created interest when the people together who are familiar with the different aspects viscosity of the quark-gluon plasma in heavy ion collisions (at so that they can interact and find if there are some deeper RHIC and LHC) was measured to be very close to this bound. reasons for these connections. JH: The Nobel prize winner, Ketterle, from MIT, once said Bulletin: What will be the theme of the Aisenstadt Chair lec- there were two frontiers in theoretical and experimental tures? physics: one is the high energy frontier and the other is the JH: Bertrand Eynard [CEA Saclay], one of the semester’s low temperature frontier. This workshop is unique in that it Aisenstadt Chairs, and a member of our Mathematical is using mathematics to connect these two different frontiers. Physics Laboratory, has been a pioneer in developing what Very surprisingly, AdS/CFT, which was conceived purely for goes under the general heading of “topological recursion” in relativistic four-dimensional quantum field theory, originally, moduli spaces. The idea is, within the general framework of in fact seems to have applications also to strongly correlated moduli spaces of Riemann surfaces, or more generally these low temperature phenomena. And the methods of integrable topological invariants that I mentioned, and geometric enu- systems, which were thought of as something very idealized, merative invariants, there is a unified, common structure turn out to have real world applications. It’s really a beauti- which has to do with solving a sequence of recursive relations ful unifying aspect that’s bringing together the two frontiers which may be interpreted in the context of matrix models as of both experimental and theoretical physics. being correlators or multi-trace invariants, with data that are essentially algebro-geometric in nature. The key ingredient is We forgot one thing which I hope Marco will talk about and that for each one of these (basically it’s a tau function that that’s the role of asymptotics. controls either the Gromov–Witten invariants, Hurwitz sys- M. Bertola: I would like to describe the workshop Asymp- tems, knot invariants, etc.) there is some kind of a recursive totics in Integrable Systems, Random Matrices and Random system of equations, which from the algebro-geometric data, Processes and Universality [crm.math.ca/2015/Deift15/] relating to what’s called a spectral curve, generates all of the Integrable systems are one of the main themes of the semester. higher invariants in a completely finite algorithmic way (by As John said, although the mathematics of nature is not, in recursions). general, “integrable,” there is a sense that methods and re- It’s also closely related to the very last workshop sults of integrable systems keep recurring. A sort of “strange [crm.math.ca/2016/Moduli16/]. That workshop is rather attraction.” late (early January 2016) and Eynard will be here in both One example is the notion of “universality” in the behaviour October–November 2015 and early January 2016. or large systems: under certain asymptotic regimes some fea- Bulletin: Can you say something about the other Aisenstadt tures of a large system behave independently of the minute chairholder? details of the system itself, and they do behave like a given JH: Before I do that let me just comment on what Robert prototypical system that turns out to be integrable. This re- mentioned about applications to physics: some condensed curring theme appears in the statistics of eigenvalues of large matter applications, the ideas of AdS/CFT, and also cosmo- matrices, the small-scale behaviour of nonlinear waves, and logical applications. Another workshop, which is being orga- also in combinatorics, representation theory of groups and nized by Jean-Sébastien Caux, Masudul Haque and Robert string theory. Konik, is also very much about experimental physics. That The workshop itself fits perfectly in the general theme and workshop, which is early in the program, is about near is expected to attract many outstanding world-wide experts. integrability [crm.math.ca/2015/Quantum15/]. In real life Quite serendipitously, in the early stages of the organization physics exact integrability is very rare, but systems that are we realized that the workshop coincides also with the celebra- nearly integrable can very often be understood by methods tions for the 70th anniversary of one of the key figures in the coming from integrable systems. area of asymptotics of integrable systems. I am referring to This approach has two main applications: condensed matter Percy Deift, who, amongst several outstanding contributions, physics in the very low temperature regime, and strongly cor- introduced and perfected the “steepest descent” method for related phenomena. I can’t speak in detail about that but it noncommutative Riemann–Hilbert problems (a certain form should be also a very important connection with experimental of matrix-valued boundary value problems in the complex do- physics. main). This turns out to be one fantastic gift that keeps on giving because this very type of boundary value problems im- JW: Since we mention this connection to experiments, via the pinges on so many types of integrable systems. heavy ion collisions, there is a really very concrete application BULLETIN CRM–17 crm.math.ca

The workshop was also recommended for support from the on to physical applications, we have AdS/CFT and quantum NSF, which will help in attracting and funding a large cohort gravity, which Robert mentioned, about the conundrums as- of young researchers. sociated with black holes. And then there’s the workshop JH: Now about the theme of the other Aisenstadt Chair, on applications to cosmology and the Big Bang singularity Nikita Nekrasov [IHES/SCGP]: this is also a beautiful sur- that Robert talked about, and the workshop on applications prising connection between fully relativistic four dimensional to QCD (quantum chromo-dynamics) and condensed matter quantum field theory and integrable systems, but it’s through physics (probably it will be mostly applications to QCD, the another route. There’s a big class of classical and quantum strong interactions), by Keshav. We also have another work- integrable systems which play a huge role, called the Hitchin shop called Beyond Integrability where condensed matter sys- systems. The Nekrasov approach, and also others which con- tems will be featured. Finally, sometime in September, there tributed to this, connects up the supersymmetric N = 2 will be the Aisenstadt Chair lectures by Nikita Nekrasov. Yang–Mills theory with Hitchin systems, which in turn are JH: Just to elaborate on the three-week session from the very programmed into some of the mathematical structures that last week of July to the second week of August, which is a were mentioned before, namely cluster algebras and Poisson sequence of two different workshops (one of one week and structures. This is a very ongoing, very active, development, one of two weeks) about two different aspects: one is posi- and that’s what Nekrasov and others helped to launch, a few tive Grassmannians and cluster algebras and related topics years ago. [crm.math.ca/2015/Amplitudes15/], and the other is called JW: As John was mentioning, the integrable systems were Hidden Symmetries and Integrability Methods [crm.math.ca/ first noticed in the N = 2 context, which is the Seiberg– 2015/Symmetries15/]. That’s going to be a monstrous two- Witten theory, for certain physical quantities, and then they week session where almost everyone in the subject will be appeared in N = 4 super Yang–Mills, for certain other physi- present, and the idea is to have some people from the first cal quantities, and it’s not exactly understood, one should say, workshop stay over for the second, and some people from how these two things are related. And Nikita Nekrasov will the second arrive early so that there can be some cross- speak about that. One of the ideas of this particular semester fertilization. is to bring together these new developments for N = 4 with Bulletin: Are these going to be people working in mathematics the somewhat older story relating to N = 2 super Yang–Mills or physics? and these generating functions: the tau functions, enumera- RB: In the workshops which I mentioned it’s going to be tive generating functions, and so on. mainly physicists, but specifically the one I’m organizing is JH: Let me clarify that N means the extent of the supersym- going to be a mixture between mathematics and physics, be- metry. Supersymmetry has to do with identifying commuting cause we’re actually using mathematical tools in order to and anti-commuting degrees of freedom, in a symmetric way, make progress on the physical problems. And here the math- and the amount, that is the dimension, of the anti-commuting ematical tools come mostly from the areas of partial differen- part, which relates to Grassmann algebras, is what is being tial equations and dynamical systems. So that’s where Walter labelled by this N . If N is zero then there’s no super symme- Craig and Niky Kamran fit in. try, if N is 1 it’s the minimal extension. For super Yang–Mills JH: I would say globally, it’s hard to say who’s a physicist theory N = 4 is the maximum, I think for gravity N = 8. So and who’s a mathematician in this area, but it’s very nicely this is a sort of code word for specialists in supersymmetry. balanced between the physics aspects and the mathematics Bulletin: Will Juan Maldecena be here? aspects. JW: Yes, he will. To summarize, let me just give a brief JW: Let me conclude by mentioning something which is re- overview of the activities. The semester empiète (encroaches/ ally special about our program and different from others. One spills over): it starts in June (2015) and ends in January of the more “meta” motivations for this entire semester is to (2016). The full schedule is available on the CRM’s website give a forum for the AdS/CFT correspondence at a mathe- [crm.math.ca/Holography2015] Instead of going in chronolog- matical institute. There have been, of course, programs and ical order, I would like to list them starting from the most workshops organized at math institutes featuring integrability mathematical and going to the physical applications: we have and quantum field theory, and also the applications to enu- topological recursions, which connect with very abstract alge- merative geometry that John has mentioned, but as far as we braic geometry—that is associated with the Aisenstadt Chair, know there has been no semester where AdS/CFT and the Bertrand Eynard, and is actually the last workshop chrono- holographic correspondence was put at the center. We want logically, in January 2016. Then we have asymptotics, which to raise awareness of the AdS/CFT correspondence among Marco talked about, which is the very first workshop, in June, mathematicians, how special it is, how different it is from the in honor of Percy Deift. Next there is the positivity, and the other dualities. The other dualities that we have in physics are core of this program is the 3-week focus period in August: dualities that mathematicians are aware of, there are mathe- one week on positivity, positive Grassmannians and the role matical counterparts. But AdS/CFT is really different from they play in integrable systems, and two weeks on integrabil- (continued on page 19) ity in AdS 4 and AdS 5, that’s the pièce de résistance. Moving

BULLETIN CRM–18 crm.math.ca Explorer de nouveaux horizons en modélisation par copules

Christian Genest (Université McGill) et Louis-Paul Rivest (Université Laval) en présence de divers risques environnementaux ou quand on veut intégrer des données locales dans un modèle régional. Richard Cook (Waterloo) et Lajmi Lakhal Chaïeb (Laval) ont recensé des applications des copules en sciences de la santé, notamment aux fins de calcul de tailles d’échantillon, de prévi- sion de la progression de maladies et de détection de variantes génétiques rares. Matthias Scherer (Munich), auteur de deux livres récents sur les copules, a présenté diverses méthodes de construction de modèles de copules échangeables et un autre chercheur prolifique intéressé aux questions de dépen- dance, Peter Song (Michigan), a proposé de nouveaux tests Un atelier de l’INCASS sur les nouveaux horizons en modéli- d’ajustement pour les modèles basés sur des copules semi- sation par copules a eu lieu au CRM du 15 au 18 décembre paramétriques. 2014. Le Laboratoire de statistique du CRM et la Chaire de recherche du Canada en modélisation de la dépendance sto- La dernière matinée mettait en vedette Gal Elidan, un spé- chastique, basée à McGill, parrainaient cet événement qui de cialiste de l’apprentissage machine actuellement en stage chez l’avis de tous a été un franc succès. Trois jours et demi riches Google, et Harry Joe (UBC), auteur de livres influents sur en exposés, en séances d’affiches et en échanges informels ont la modélisation de la dépendance. Elidan a expliqué que les permis aux 65 participants venant de 10 pays et trois conti- chercheurs en apprentissage machine s’intéressent davantage nents de faire le point sur les nouvelles techniques de mo- aux algorithmes et aux mesures de performance qu’aux mo- délisation de données multivariées à l’aide de copules. Une dèles probabilistes. Il s’est dit convaincu que des échanges large place était aussi faite aux applications, tant dans des plus nourris avec les statisticiens seraient bénéfiques aux deux domaines où l’approche est déjà bien implantée – tel l’actua- communautés et à titre d’exemple, il a montré comment riat, la finance et l’hydrologie – que dans ceux où sa popula- les copules peuvent accélérer les algorithmes d’apprentissage. rité croît rapidement, dont la biostatistique et l’apprentissage Quant à Joe, il a fait valoir la nécessité d’aller au-delà des mo- machine. dèles de vignes pour mieux refléter les structures de données échangeables. Il a expliqué comment incorporer des variables Plusieurs exposés ont vanté les mérites des vignes pour la latentes à une vigne et déduit quelques propriétés de la classe modélisation de données de grande dimension. L’approche a de modèles résultante. été décrite par une spécialiste du sujet, Claudia Czado (Mu- nich), et de nombreuses applications intéressantes en ont été L’atelier comportait en tout 18 exposés. Deux séances d’af- données tout au long du congrès. Tobias Erhardt (Munich) fiches ont aussi permis à 15 chercheurs de présenter un large et Benedikt Gräler (Münster) ont notamment montré com- éventail d’exemples d’application des modèles de copules, ment se servir de vignes pour analyser des données spatiales ainsi que des travaux méthodologiques portant sur les pro- ou pour modéliser des champs aléatoires spatio-temporels. priétés structurelles de ce type de modèle. La qualité des pré- sentations a été unanimement saluée par les participants, qui L’utilisation des copules continue de se répandre en écono- ont semblé particulièrement séduits par les nouvelles possibili- mique, en finance et en actuariat. Dans leurs exposés, An- tés qu’offre l’utilisation des copules en apprentissage machine. drew Patton (Duke), Valérie Chavez-Demoulin (Lausanne) et On pourrait envisager d’organiser un atelier sur ce thème dans Brendan Beare (San Diego) ont présenté de nouvelles façons quelques années. de bâtir des modèles économétriques complexes au moyen de copules. Des applications de haut calibre en assurance et en gestion ont aussi motivé et servi à illustrer les travaux rappor- AdS/CFT, Homolography, Integrability tés par Jed Frees (Wisconsin-Madison) et Giovanni Puccetti (continued from page 18) (Florence). this, and we want to emphasize how different it is. There is L’usage grandissant des copules en hydrologie et en sciences one special version of AdS/CFT which has a very concrete de l’environnement a été souligné par Philippe Naveau (Ver- mathematical counterpart, which is the correspondence be- sailles), Luis Mediero (Madrid), Gianfausto Salvadori (Sa- tween Chern–Simons theory and knot invariants, and enu- lento) et Salvatore Grimaldi (Tuscia), ex-président de la Com- merative geometry, so that can be viewed as a special case mission internationale d’hydrologie statistique. Les partici- of the holographic correspondence between gauge theory and pants ont pu apprécier l’utilité des copules, quand on cherche gravity. But it goes far beyond, and we want to raise aware- par exemple à estimer le temps de retour de catastrophes ness of its special status for mathematical physics of the 21st century.

BULLETIN CRM–19 crm.math.ca Bilan de l’UMI CRM

Laurent Habsieger (codirecteur de l’UMI CRM) L’UMI CRM est actuellement à un tournant de son histoire. • Pour 2013, l’UMI a présenté deux projets, et un a été re- Ce 7 avril, un comité la visitera en vue de son renouvellement, tenu : CAESAR, dont je suis le coordinateur local. Cela et un nouveau directeur est annoncé dès septembre 2015. Il représente 22 200 e hors frais de gestion, pour la période est donc temps de présenter un bilan de ses premières années 2013-2015. d’existence. • Pour 2014, l’UMI a présenté quatre projets et deux ont Fonctionnement été retenus : GeRaSic, dont Dmitry Jakobson (McGill) est le coordinateur local, et HR-CEM, dont Yves Bourgault La direction de l’UMI est assurée par deux codirecteurs, le (Ottawa) est le coordinateur local. Les deux projets rete- directeur du CRM et moi-même. J’ai ainsi pu travailler en nus couvrent la période 2014-2017. Hors frais de gestion, étroite concertation avec François Lalonde puis Luc Vinet, cela représente 40 000 e pour GeRaSic et 34 800 e pour et mener une politique scientifique qui reflète celle du CRM. HR-CEM. L’année est marquée par deux temps forts : l’automne, avec • Pour 2015, l’UMI a présenté trois pré-projets, mais aucun les appels à projets de l’INSMI pour les postes rouges et les n’a été retenu. séjours longue durée, et avec les appels à projets de l’ANR, • Pour 2016, l’UMI a présenté un pré-projet, et le résultat et le début d’année où je lance un appel à projets, pour éta- n’est pas encore connu. blir le budget de l’UMI. Pour les appels à projets de l’INSMI, j’accompagne les candidats en leur fournissant toutes les in- Avec trois programmes ANR en activité, l’UMI CRM fait par- formations pertinentes et en leur rédigeant une lettre d’ap- tie des meilleures UMI dans le monde : aucune UMI n’a ac- pui personnalisée. Pour les projets ANR, je tiens le rôle de tuellement réussi à obtenir quatre ANR, selon les statistiques consultant, et assure la jonction avec le service Partenariat et présentées en mai dernier à la réunion des directeurs d’UMI Valorisation de la délégation Paris Michel Ange. Une fois re- d’Amérique du Nord. cueillis les résultats de mon appel à projet de début d’année, Visites au sein de l’UMI je convoque le conseil d’unité, et il est procédé à la répartition des fonds disponibles. Les visites longues sont celles qui durent plus de trois mois et que l’UMI ne peut donc financer intégralement. Il faut donc J’assure également la gestion administrative de l’UMI, en par- présenter des demandes au CNRS, de divers types : déléga- tenariat avec la délégation Paris Michel Ange. Afin de facili- tion (pour permettre à l’universitaire de dépendre du CNRS), ter la compréhension du système français, j’ai rédigé une note soutien financier direct, affectation. La formule la plus avan- explicative des procédures à suivre au sein du CNRS pour la tageuse est celle de l’affectation, et l’UMI a été particulière- prise en charge de missions, avec un lexique des termes admi- ment bien servie lors des deux dernières années : sur un total nistratifs, document que j’ai diffusé aux coordinateurs locaux de 38 affectations possibles au CNRS sur l’ensemble des dis- de projets ANR. Aussi bien pour la subvention d’état que ciplines et des 35 UMI, l’UMI CRM en a obtenu 3,5 en 2014 pour les programmes de l’ANR, j’assure le suivi des finances, et 2015. Au total, ce sont 17 visiteurs qui ont pu ainsi venir, accompagne les missionnaires tout au long du processus de pour des durées majoritairement comprises entre six mois et prise en charge de leur mission, depuis la demande d’ordre de un an. Il faut remarquer que l’UMI rayonne bien au-delà de mission jusqu’à la rédaction de l’état de frais. l’INSMI : des membres d’autres Instituts (INP pour les physi- Suzette Paradis, webmestre du CRM, a mis en place un site ciens théoriciens, INS2I pour les combinatoristes) souhaitent web pour l’UMI, dont je suis administrateur, et que je mets visiter l’UMI. à jour périodiquement. Son URL est crm.math.ca/UMI/. Pour les visites de plus courte durée, les (co)financements sont Financement gérés par l’UMI. Ainsi, au premier janvier 2015, l’UMI a fi- nancé directement (sur ses fonds propres) ou indirectement L’UMI a reçu une subvention d’état annuelle de 15 000 e en (sur projet ANR) vingt-deux visites à l’UMI. 2012, 2013 et 2014. Chaque année, dans la demande de moyens que je présente au CNRS, je sollicite une augmentation de Visites de membres de l’UMI en France cette dotation, pour tenir compte à la fois de l’éloignement La plupart des membres de l’UMI ont déjà des subventions géographique de l’UMI (par rapport à une UMI européenne) individuelles qui leur permettent de voyager. L’UMI fait donc et du dynamisme de celle-ci. J’ai été entendu et la subvention face à une demande moins forte dans le sens Canada-France d’état annuelle est passée à 25 000 e en 2015. que dans le sens France-Canada. Notons que l’obtention de Dès mon arrivée à l’UMI, j’ai insisté sur les possibilités offertes programmes de l’ANR permet d’atténuer cette tendance. par les programmes de l’ANR. Ainsi l’UMI a financé sept visites de courte durée, directe- BULLETIN CRM–20 crm.math.ca ment ou indirectement. L’UMI a aussi transmis au CNRS des partement de mathématiques et de statistique de l’Université candidatures de très bon niveau pour les postes rouges (3 mois de Montréal et du CRM, les participants au colloque ont eu au CNRS). accès à des installations conviviales pour échanger pendant • En 2014, Sabin Lessard (Montréal) a présenté une demande une fin de semaine des idées, communiquer des problèmes ma- thématiques et présenter leurs résultats de recherches. Ce fut et a été sélectionné. • En 2013, Javad Mashreghi (Laval) a présenté une demande, l’occasion pour plusieurs de rencontrer ou revoir leurs col- puis s’est retiré, ayant été sélectionné sur une autre de- lègues provenant des diverses universités de la métropole et de l’extérieur. De constater autant de cohésion au sein de la mande antérieure présentée par son laboratoire d’accueil français (Lille). communauté mathématique étudiante est pour nous fort ré- • En 2012, Dmitry Jakobson (McGill) et Christophe Hol- jouissant. weg (UQAM) ont présenté une demande. Dmitry Jakob- son a été sélectionné. Il faut aussi noter qu’Adrian Iovita (Concordia) a aussi bénéficié d’un poste rouge, présenté par son laboratoire d’accueil français. Colloques L’UMI subventionne des rencontres, soit directement en leur versant des fonds, comme pour les Entretiens Jacques Cartier, soit indirectement en prenant en charge la venue de conféren- ciers par exemple. D’autre part l’UMI finance deux types de colloques : ceux organisés dans le cadre des activités théma- tiques du CRM, et ceux organisés indépendamment. L’UMI a ainsi subventionné sept programmes thématiques du CRM, comptant au total plus d’une douzaine d’ateliers, et dix-neuf autres colloques à travers le monde. Photo : Gida Hussami Actions en cours Vingt-et-une conférences étudiantes sur une palette variée de Depuis septembre 2013, Luc Vinet et moi sommes en contact sujets ont ainsi été données en anglais et en français, en plus avec Emmanuel Ullmo, directeur de l’IHÉS, en vue de créer des traditionnelles conférences plénières des quatre profes- un site miroir de l’UMI au sein de l’IHÉS. La principale diffi- seurs invités des différentes universités de Montréal. Ces der- rs culté à surmonter est d’ordre juridique : il faut que l’IHÉS soit nières conférences, des P Yvan Saint-Aubin, Léa Popovic, doté d’une structure CNRS. Ce problème est résolu depuis le Niky Kamran et Olivier Collin, ont été hautement appréciées. 1er janvier 2015, lorsque l’IHÉS est devenue une équipe de Cette année, l’événement était subventionné et soutenu par les recherche labellisée (ERL). Une étape importante a été fran- départements de mathématiques et de statistique des univer- chie début mars 2015, lorsqu’un accord de principe fut signé sités montréalaises, le Département de physique et la Faculté à l’occasion de la visite de Philippe Couillard en France. des arts et des sciences de l’Université de Montréal, l’ISM, le CRM, Maplesoft ainsi que par les associations étudiantes de Alain Fuchs a signé en personne la convention créant l’UMI, mathématiques et de statistique de la métropole. lorsqu’il a été invité à Montréal pour recevoir un doctorat ho- noris causa dans le cadre des Entretiens Jacques Cartier. Le délégué général du Centre Jacques Cartier, Alain Bideau, m’a alors contacté. Il souhaitait réintroduire les mathématiques au sein des Entretiens. L’UMI a ainsi soutenu quatre colloques dans ce cadre. Suite au changement de direction et de politique du Centre en 2014, l’UMI n’a pas proposé d’Entretien pour 2015, sans préjuger de 2016. SUMM 2015

Marc-André Miron (Université de Montréal) Une centaine d’étudiants et d’étudiantes en mathématique du Québec et de l’Ontario se sont rassemblés en janvier à l’occasion de la sixième édition des Séminaires universitaires en Niky Kamran Photo : Gida Hussami mathématiques à Montréal. Grâce à l’accueil généreux du Dé- BULLETIN CRM–21 crm.math.ca

Le Bulletin du CRM Call for Proposals Volume 21, No 1 Printemps 2015 The CRM (Centre de recherches mathé- favourable decision, the prospective or- Le Bulletin du CRM est une lettre matiques) is soliciting applications for ganizers will be asked to submit a de- d’information à contenu scientifique, scientific activities to take place at the tailed proposal that will be evaluated faisant le point sur les actualités du Centre de recherches mathématiques CRM. The proposals are divided into by the International Scientific Advisory (CRM). three categories: thematic semesters, of Committee of the CRM. a duration of up to six months; work- ISSN 1492-7659 Proposals and further inquiries should shops, conferences or schools, whose du- Le Centre de recherches mathéma- be emailed to the CRM at proposal@ ration can vary from a couple of days to tiques a vu le jour en 1969. Actuelle- crm.umontreal.ca. ment dirigé par Luc Vinet, il a pour up to two weeks; and targeted research objectif de servir de centre natio- periods for small groups of researchers, More information can be found on the nal pour la recherche fondamentale crm.math.ca/Proposals en mathématiques et leurs applica- lasting one to two weeks. website . tions. Le personnel scientifique du Preliminary contact and informal in- CRM regroupe plus d’une centaine A commitment to diversity. Appli- quiries with the Director or the Deputy de membres réguliers et de boursiers cants are encouraged to ensure partici- postdoctoraux. De plus, le CRM ac- Director (Scientific Programs) of the pation of women and underrepresented cueille chaque année entre mille et CRM are encouraged. Budget indi- groups in the proposed activity. They mille cinq cents chercheurs du monde cations and information on sources of entier. should also plan to involve scientists at funding will be provided. Le CRM coordonne des cours de different stages of their career, including cycles supérieurs et joue un rôle All letters of intent will be examined by postdoctoral fellows and students, com- prépondérant (en collaboration avec the CRM’s local and international sci- ing from diverse institutions and loca- l’ISM) dans la formation de jeunes chercheurs. On retrouve partout entific advisory committees. In case of a tions in Canada and abroad. dans le monde de nombreux chercheurs ayant eu l’occasion de parfaire leur formation en recherche au CRM. Le Centre est un lieu privilégié de rencontres où tous les membres béné- Appel à projets ficient de nombreux échanges et col- laborations scientifiques. Le Centre de recherches mathématiques soumettre une proposition détaillée qui Le CRM tient à remercier ses divers partenaires pour leur appui financier (CRM) vous invite à proposer des pro- sera évaluée par le Comité scientifique à sa mission : le Conseil de recherches jets d’activités scientifiques. Les activi- international du CRM. en sciences naturelles et en génie du tés scientifiques se divisent en trois caté- Les lettres d’intention et les demandes Canada, le Fonds de recherche du gories : les semestres thématiques d’une Québec – Nature et technologies, la d’informations doivent être envoyées au durée de six mois ; les conférences, ate- National Science Foundation, l’Uni- CRM par courriel à l’adresse suivante : versité de Montréal, l’Université du liers ou écoles d’une durée de quelques [email protected]. Québec à Montréal, l’Université Mc- jours à 2 semaines ; et les périodes de re- Gill, l’Université Concordia, l’Uni- cherche ciblée d’une durée d’une à deux Pour plus d’informations, visitez le site versité Laval, l’Université d’Ottawa, l’Université de Sherbrooke, le réseau semaines. crm.math.ca/Propositions. Mitacs, ainsi que les fonds de dotation André-Aisenstadt et Serge-Bis- Avant la soumission formelle, nous en- Un engagement à la diversité. Les sonnette. courageons les organisateurs éventuels chercheurs proposant une activité sont Directeur : Luc Vinet à prendre contact avec le directeur du encouragés à s’assurer de la participa- Directrice d’édition : Galia Dafni CRM ou le directeur adjoint responsable tion des femmes et d’autres membres Conception : André Montpetit des programmes. Des indications sur le de groupes traditionnellement peu re- Centre de recherches mathématiques budget et les sources de financement leur présentés. De plus, nous encourageons Université de Montréal seront fournies. les organisateurs à inviter des partici- C.P. 6128, succ. Centre-ville Montréal, QC H3C 3J7 Toutes les lettres d’intention reçues se- pants d’origines et de statuts divers : chercheurs en début de carrière, cher- Téléphone : 514.343.7501 ront examinées par les comités scienti- Courriel : [email protected] fiques du CRM (le comité local et le cheurs chevronnés, boursiers postdoctoraux et étudiants, provenant d’universi- Le Bulletin est disponible à : comité international). Si leur lettre est tés de toutes les régions du Canada et crm.math.ca/docs/docBul_fr.shtml. agréée, les organisateurs seront invités à de l’étranger.

BULLETIN CRM–22 crm.math.ca A Word from the Director

Luc Vinet, Director of the CRM

Twice a year, the publication of the Bulletin offers me the we haven reasons to hope and believe that the outcome will occasion to muse about the achievements of the CRM and its be commensurate to our merits and needs. members. As the present issue illustrates the last semester The CRM (represented by Deputy Director Odile Marcotte) has certainly been quite eventful also. took part in the recent mission of the Premier of Québec in The year-long program in Number Theory, which will con- France and signed two partnership agreements with distin- clude in May with the second series of Aisenstadt Chair lec- guished French research institutes on this occasion: one with tures by Sophie Morel, has been absolutely remarkable. The the Institut Henri Poincaré (IHP) and the other with the In- participation has been overwhelming, the quality of the speak- stitut des Hautes Études Scientifiques (IHES). We are look- ers has been exceptional and scientific breaktroughs have been ing forward with great anticipation to the enactment of these witnessed. There is a very large number of postdoctoral fel- agreements. lows number theory in residence this year at the CRM and I The CRM is privileged to host a Unité Mixte Internationale cannot help but think that their stay with us will be a defining (UMI) of the CNRS. This structure enriches the CRM in experience. many ways and is highly appreciated. In fact the agreement The preparation for the upcoming thematic semester on the with the IHES was developed with an eye to establishing a AdS/CFT correspondence—a first in a “mathematics insti- mirror site in France of the CRM-UMI. Let me mention that tute”—is now being finalized and it promises to be outstand- our UMI is up for renewal this year and will be the object of ing with a substantial number of the leaders in the field, physi- a site visit this April. In concluding and in this connection, cists and mathematicians, having confirmed their participa- I wish to pay a vibrant hommage to the founding co-director tion. You can read about it in the conversation with some of of the UMI, Laurent Habsieger, whose mandate will end this the organizers. summer. The success and popularity of the UMI CRM have Meanwhile we are actively preparing the thematic semesters been noted and are largely due to Laurent who has been a fabulous leader. With passion, altruism, inspiration and ded- for 2016 and beyond. ication, in four years, Laurent will have made the UMI CRM The general program has also been vibrant with events cele- one of the best performing UMIs there are. We thus owe brating some of our distinguished colleagues, liaising activi- much to Laurent and it was a great pleasure to collaborate ties with the industry, conferences for students, talks given by with him. Wishing him well in his future endeavours, I want great prize winners and wonderful presentations for the public to convey the high esteem and friendship in which I hold him, at large. Of note is the restructuring of the CRM–ISM Col- and in the name of every one at the CRM and the UMI CRM, loquium series into the Québec Colloquium in Mathematical to voice our heartfelt thanks. Sciences. This year again, the CRM–ISM postdoctoral fellowship program has attracted exceptional candidates and we are looking forward to seeing the winners of the competition in our midst soon. The last months have also seen the integration into the CRM of our three new laboratories: CAMBAM (mathematical bi- ology), Probability and QUANTACT (financial and actuar- ial mathematics). This exciting development testifies to the CRM’s dynamism, to its leadership and to its commitment to core mathematics and interdisciplinary outreach. A number of representatives of the CRM have met early in the year with the FRQNT expert (sub)committee that will make a recommendation on our Québec funding. Given the Laurent Habsieger (standing, second from right) with then CRM di- importance of the CRM as a driver of the National Policy for rector François Lalonde (standing, first from right) on the occasion of Research and Innovation, its stellar international track record, the signing of the agreement creating the UMI CRM on October 4, the growth in its membership, the creation of three new labs 2011, by Joseph Hubert (Vice-rector Research and International Re- lations at Université de Montréal) and Alain Fuchs (President of the and the increase in support from all its partner universities, CNRS, France).

BULLETIN CRM–23 crm.math.ca Mot du directeur

Luc Vinet, directeur du CRM Deux fois par année, la publication du Bulletin m’offre l’occasion d’offrir une petite chronique du CRM et de ses membres. Comme le présent numéro en fait foi, le semestre dernier est à la hauteur des précédents. L’année thématique en théorie des nombres, qui prendra fin en mai avec la deuxième série de conférences de la chaire Aisenstadt par Sophie Morel, aura été absolument remarquable. Le CRM a accueilli, comme toujours, un grand nombre de participants ; la qualité des confé- renciers fut exceptionnelle et des percées scientifiques impor- tantes ont été réalisées. Un très grand nombre de chercheurs postdoctoraux effectuent leur stage au CRM cette année et tout porte à penser que leur séjour parmi nous sera détermi- nant. La préparation du prochain semestre thématique sur la cor- respondance AdS/TCC - une première dans un institut de Luc Vinet (gauche) avec les conférenciers de la chaire Aisenstadt Pierre sciences mathématiques - sera bientôt finalisée. Le programme Colmez (centre) et Sophie Morel (droite) de ce semestre est extraordinaire puisqu’une proportion im- route internationale étincelante, l’augmentation du nombre portante de chefs de file du domaine - physiciens ou mathé- de ses membres, la création de trois nouveaux laboratoires, maticiens - a confirmé sa participation. Vous en saurez plus l’augmentation de l’appui financier de tous ses partenaires si vous lisez l’entrevue accordée par quelques-uns des organi- universitaires, il nous est permis d’espérer et de croire que la sateurs du semestre thématique. Nous sommes aussi en train subvention du FRQNT sera à la hauteur de nos réalisations d’élaborer les semestres thématiques pour 2016 et les années et de nos besoins. subséquentes. Le CRM (représenté par sa directrice adjointe Odile Mar- Le programme général n’a pas été en reste : il a comporté des cotte) a pris part à la mission récente du premier ministre évènements pour célébrer certains de nos collègues réputés, du Québec en France et en a profité pour signer deux en- des activités de liaison avec l’industrie, des conférences pour tentes de partenariat avec de prestigieux instituts de recherche les étudiants, des exposés donnés par de formidables récipien- français : une entente avec l’Institut Henri Poincaré (IHP) et daires de prix, de merveilleuses présentations pour le grand l’autre avec l’Institut des Hautes Études Scientifiques (IHÉS). public et j’en passe. Je souligne la restructuration de la série C’est avec beaucoup de joie que nous entrevoyons la mise en de colloques pour en faire le Colloque des sciences mathéma- œuvre de ces ententes. Le CRM a le privilège d’héberger une tiques du Québec ; nous nous attendons à ce que la nouvelle Unité Mixte Internationale (UMI) du CNRS. Cette structure formule attire encore plus d’auditeurs et touche le public ma- enrichit considérablement le CRM de plusieurs manières et thématique dans plusieurs localités du Québec. Cette année est hautement appréciée. En fait, l’entente avec l’IHÉS a été encore, le programme de bourses postdoctorales CRM–ISM conclue dans la perspective de la création en France d’un site a attiré des candidatures exceptionnelles et c’est avec beau- miroir de l’UMI CRM. Mentionnons que notre UMI est en re- coup de fierté, d’intérêt et de plaisir que nous accueillerons nouvellement cette année et qu’elle sera l’objet d’une visite de les lauréats à Montréal. site en avril prochain. À ce chapitre et en conclusion, j’aime- Les derniers mois ont vu la création de trois nouveaux labo- rais rendre un vibrant hommage au codirecteur fondateur de ratoires du CRM : le laboratoire de probabilités, CAMBAM l’UMI CRM, Laurent Habsieger, dont le mandat prendra fin (laboratoire de biologie mathématique) et QUANTACT (la- cet été. Le succès et la popularité de l’UMI CRM sont remar- boratoire de mathématiques financières et actuarielles). Cette qués et sont dus très largement à Laurent qui est un leader création témoigne du dynamisme du CRM, de son leadership formidable. Avec passion, altruisme, inspiration et doigté, en et de son engagement envers les mathématiques et l’interdis- quatre ans seulement, Laurent aura su faire de l’UMI CRM ciplinarité. l’une des UMI les plus performantes du CNRS. Nous devons Des représentants du CRM ont rencontré il y a quelques se- beaucoup à Laurent et cela aura été de plus un très grand maines le (sous-)comité du FRQNT chargé de faire les recom- plaisir de collaborer avec lui. En lui souhaitant tout le succès mandations de subventions aux regroupements stratégiques qu’il mérite dans ses entreprises futures, je lui dis toute mon (et au CRM en particulier). Étant donné l’importance du estime et mon amitié et au nom de tous les membres du CRM CRM comme agent de développement de la Politique natio- et de l’UMI CRM je lui exprime du fond du cœur toute notre nale pour la recherche et l’innovation (PNRI), sa feuille de reconnaissance.

BULLETIN CRM–24