C R M. Le Bulletin. Printemps/Spring 2015 Volume 21, N o 1 Le Centre de recherches mathématiques
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1 C R M C E N T R E D E R E C H E R C H E S M A T H É M A T I Q U E S Le Bulletin Printemps/Spring 2015 Volume 21, N o 1 Le Centre de recherches mathématiques A conversation with Marco Bertola (Concordia), Robert Brandenberger (McGill), John Harnad (Concordia) and Johannes Walcher (McGill) on December 8, Bulletin: Can you elaborate on the theme of the semester? J. Walcher [ JW]: Our semester has a 3-pronged theme: it s called AdS/CFT, holography and integrability, and I ll start trying to explain what AdS/CFT is. AdS/CFT stands for Anti de Sitter/Conformal Field Theory correspondence. It was the last of the major dualities discovered in the wake of the second super string revolution of the mid-1990s, and it 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 different field theories, but rather it involves an equivalence between, on the one side, a theory of quantum gravity, string theory, and on the other side a more standard, although very interesting, quantum field theory. And the dictionary between these two has the feature of being holographic. Bulletin: Can you explain what holographic means? R. Brandenberger [RB]: Holographic means that there is one theory, in d dimensions, which is equivalent to a different theory which lives on the boundary. It s like in optics, it s the same use of the word as in optics. JW: The ideas that later developed into AdS/CFT go back to the early 1970s, shortly after the birth of string theory. The first proposals that gravity should be holographic are due to t Hooft, Polyakov, and Susskind, and the Bekenstein Hawking entropy of black holes was another important clue. But the big breakthrough came in 1998, or late 1997, when Juan Maldacena, who is one of the organizers of the thematic semester, proposed this exact correspondence between type IIb string theory on AdS 5, and N = 4 Super Yang Mills theory. Integrability, which is our third theme, has also been playing a role in mathematical investigations of string theory for a long time: its role in the AdS/CFT correspondence emerged around 2004, and has been playing an increasing role in the quantitative development of the correspondence. That is what our semester is about: it s the intersection of the AdS/CFT correspondence as a holographic duality, and integrable methods as far as they are relevant to the AdS/CFT correspondence. I will let Robert take it from here. RB: I will start from the physicist s point of view, and in particular from the point of view of someone interested in gravity and cosmology. In gravity and cosmology we had for a 100 years basic problems: our theories predict singularities, and although singularities may be interesting from a mathematical point of view, they are unphysical. No physical apparatus will ever measure an infinity the physical apparatus will break down. And the reason that our theories predict singularities is that they are not correct. The AdS/CFT correspondence provides a way to solve these problems. For example, it provides a means to understand the physical resolution of the singularity at the center of a black hole that is predicted by Einstein s theory. This question is going to be explored in one of the workshops, the one which Alex Maloney and Patrick Hayden are coordinating, the one on gravity, in September [crm.math.ca/2015/gravity15/]. I m a cosmologist and cosmologists have this huge problem of the Big Bang singularity, so again we know that the theory breaks down, and we hope that through the AdS/CFT
2 correspondence we can have a model for what happened in the very early universe, what replaces the Big Bang. That s going to be the focal point of the workshop on cosmology, the first workshop in July [crm.math.ca/2015/extension15/]. There s the hint that one can take the gravitation theory, which is singular, and through the AdS/CFT correspondence, which is this holographic correspondence, map it into a nongravitational theory, on the boundary. Then, since nongravitational theories are under much better mathematical control, we can hope that we can evolve the theory on the boundary. If we have the picture in mind that we are evolving towards the Big Crunch singularity, and we go to the boundary, then we hope that we can let time continue beyond where it would stop in the gravitational theory. There are already indications that this would work for a very simple model of cosmology where there is just a homogeneous and isotropic universe, no fluctuations, no people, nothing. And the goal of this cosmology workshop is to extend it to a more realistic situation where we include fluctuations, where we include the possibility of having galaxies in this picture. This is really going to be a workshop in the sense that this has never been achieved before and we are getting the people who are working on this topic to come together. That is a really novel feature of this program. There are also applications of the AdS/CFT correspondence in different areas of physics. For example, a big challenge is to understand gauge theories beyond perturbation theory, because in some gauge theories the coupling constant is actually large and we cannot trust existing techniques. So now we can go backwards using the AdS/CFT correspondence, we can map this complicated gauge theory into a higher dimensional gravitational theory, and it works out that if the gauge theory is complicated, the gravity theory is going to be in the more easily tractable regime. This has applications both to condensed matter physics and to relativistic heavy ion collisions, and that s the theme of the workshop that Keshav Dasgupta and Charles Gale and Sangyong Jeon are coordinating [crm.math.ca/2015/applications15/]. J. Harnad [JH]: I want to mainly talk about the aspects of the semester that connect up to the notion of completely integrable systems. The point is that the thematic program is trying to build a bridge between very new developments, which are everything related to AdS/CFT holography, and work that really dates back to the mid or early 19th century, which is on integrable systems. A big breakthrough in integrable systems occurred in the 1960s, when it was realized that there were in fact real genuine physical systems, which are integrable in the sense that they have the maximum number of conservation laws, but are actually infinite-dimensional: they have an infinite number of degrees of freedom. At the same time, the fact that this is closely connected to quantum mechanics became apparent and there were many quantum integrable systems that were studied simultaneously under the general heading of quantum inverse scattering methods. So this is what s been going on, say, from the 19th century to the 1970s, when it became a clearly developed theory. The fantastic new things, which occurred in the last 10 or 15 years, were absolutely unexpected connections between integrable systems and other areas of physics and of mathematics. I always think of integrability as a subject that has nine lives, because it keeps running out of steam and then finding a new reason to come back into the center of attention. I m going to just list some of these things (not in order of importance), and everything on my list is covered in this thematic program. One thing that is close to my heart is the topic of tau functions, which is more of a mathematical application, but closely related to the theory of Riemann surfaces, essential to string theory. It turned out that tau functions also play a completely different role, and are extremely important, in a mathematical sense, as generating functions for invariants in enumerative geometry and graph theory, which are also very central to the string theory application. Tau functions are generating functions not just for the enumerative geometry related to Riemann surfaces, but also for the invariants such as Jones and HOMFLY polynomials appearing in knot theory. It was known for a long time that there s a close connection between topological field theory, Chern Simons theory, and knot invariants, but it turns out that there is also a relation to integrable systems through the tau function. That s a fascinating aspect that will hopefully also be developed during this program. Another connection that emerged recently was a very surprising connection between scattering amplitudes in supersymmetric Yang Mills theory and soliton theory. Two aspects of it entered, one was explicit on-shell calculations of scattering amplitudes which turned out to have integral representations involving the integration of differential forms with singularities over Grassmannians. This gave exactly the singularity structure that arises. It is characterized by what we call the moduli space of solitons. Solitons are essential elementary ingredients in integrable systems. It turns out that the parameters that characterize solitons are exactly what we integrate over when we compute these on-shell scattering amplitudes a totally unexpected connection. At present, this connection is still not understood, but I have great hopes for this program, where we re bringing together people from the different specialties, those specialized in integrable systems and those specialized in scattering amplitudes and other aspects of quantum field theory, to try to cross-fertilize the two subjects. There is also a connection through a very intensely studied new area of mathematics, which is known as cluster algebras. The topological cell structure of positroid cells in Grassmannians may appear to be a totally unrelated mathematical subject. On the contrary, it turns out that the structure of positroid cells, which exactly characterizes, on the one hand, nonsingular solitons, and on the other hand, the domain of integration which represents the scattering amplitudes, are very closely related. And the underlying mathematics, (continued on page 17) BULLETIN CRM 2
3 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 Québec. 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 des méthodes qui permettent de dégager les principales caractéristiques des courbes observées, après les avoir étalonnées au besoin. Dans un article paru en 2013 dans BMC Medical Research Methodology, Ullah et Finch répertorient plus de 80 applications de l ADF publiées dans des revues de sciences ou de médecine entre 1995 et Fang Yao La mise en œuvre de cette approche pose des défis d envergure, tant au plan numérique que conceptuel. En plus d avoir étudié les propriétés asymptotiques des estimateurs, Fang a conçu l algorithme PACE (Principal Analysis by Conditional Estimation) et la version 1.0 du logiciel du même nom, qui a largement contribué à disséminer cette méthodologie. Aujourd 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 horizon 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 profil 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 processus stochastique à trajectoires lisses, elles sont forcément éparses ; leur modélisation est donc confrontée au «fléau de la dimension». Pour contourner cette difficulté, on fait généralement appel à une version fonctionnelle de l analyse en composantes principales. Cette technique s appuie sur la représentation de Karhunen Loève des processus stochastiques de carré intégrable en termes des éléments de la base orthonormée associée aux valeurs propres de l opérateur de Hilbert Schmidt induit par la fonction d autocovariance. Pour que l espérance du processus sous-jacent puisse être estimée avec précision, il faut que les courbes échantillonnées soient observées sur tout leur domaine ou du moins en un très grand nombre de points communs. Or ce n est pas toujours le cas, notamment dans les études longitudinales. Pour pallier ce problème, Fang a proposé de recourir à des méthodes de lissage fondées sur des régressions locales ou des splines. Octav Cornea (Montréal) and Henri Darmon (McGill) Selected as 2015 Simons Fellows in Mathematics Two CRM members, Octav Cornea (CIRGET) and Henri Darmon (CICMA), have recently been selected as Simons Fellows in Mathematics by the Simons Foundation. The 40 awardees in Mathematics for 2015 (four from Canadian universities and the rest from the United States) are distinguished mathematicians at all levels of their respective careers. According to the Simons Foundation website, The Fellows Programs provide funds to faculty for up to a semester-long research leave from classroom teaching and administrative obligations. Such leaves can increase creativity and provide intellectual stimulation. BULLETIN CRM 3
4 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 Number Theory at Tel Aviv University. He got his Ph.D. from Yale, and was an Assistant Professor at Stanford University and Princeton University before joining Tel Aviv University in Rudnick is a recognized leader in analytic number theory, mathematical physics (especially quantum chaos) and arithmetic statistics, and his work in all those areas is deep and influential. Among the many honours that he received for his research, we mention the Erdős Prize of the Israel Mathematical Union, an Annales Henri Poincaré Distinguished Paper Award, and a ERC Advanced Grant. He is a Fellow of the American Mathematical Society since 2012, and was an invited speaker at the 2014 International Congress of Mathematicians and at the 2012 European Congress of Mathematics. Zéev Rudnick Photo: Jeff Mozzochi His visit coincided with the workshop Statistics and Number 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 Keating where they look at classical conjectures of analytic number theory, such as the distribution of primes in short and in long intervals, and the sum of the Möbius function in short intervals. In each case, they were able to show that the analogues of the classical conjectures (due to Goldston Montgomery, Friedlander Goldston and Good Churchhouse, respectively) hold over function fields, at the q limit (when the size of the finite field F q tends to infinity). The proofs involve two main steps: first rephrase the original question as a problem about zeros of certain L-functions, and then use the powerful theorems about equidistribution of Frobenius matrices originating in the work of Deligne and Katz Sarnak. They obtain results which go much further than one can hope to do in the number field setting in the foreseeable future, shed light on several conjectures in number fields which are currently intractable, and also lead to generate refined conjectures in the number field case. In his second talk, Rudnick presented his work with his students Entin and Roditty-Gershon, where they were able to use their results in the function field setting to actually prove the 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 symplectic group USP(2д) as д, 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 hyperelliptic 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 Spatial Statistics on the Sphere, and Rudnick presented his recent work with Bourgain and Sarnak on the distribution of Linnik points on the sphere. The Linnik points L(n) are the projections (x, y, z)/ n to the unit sphere of the various representations of n as a sum of three integer squares n = x 2 + y 2 + z 2. Pictured are the 4,032 Linnik points corresponding to n = 104,729, the 10,000th prime. (continued on page 10) BULLETIN CRM 4
5 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 at Dartmouth College. He obtained his Ph.D. from Harvard University in Upon finishing his doctorate studies, he became a professor at the University of Georgia, where he remained till He then moved to the Bell Labs, where he stayed for four years, before joining Dartmouth College in Pomerance is a well-recognized leader in number theory, focusing mainly on the analytic, computational and combinatorial aspects of this vast field. He is also a leading expert in applications of number theory to cryptography, holding for many years the record for the fastest integer factorisation algorithm based on the so-called quadratic sieve. Among the many honors that he has received for his research, we mention that he is the recipient of the Chauvenet Prize and of the Deborah and Franklin Tepper Haimo Award by the Mathematical Association of America (MAA), and of the Levi L. Conant Prize by the American Mathematical Society (AMS). Moreover, he is a fellow of the American Association for the Advancement of Science (AAAS) and of the AMS, and he was an invited speaker at the 1994 International Congress of Mathematicians. Carl Pomerance Pomerance visited Montréal for the occasion of two of the workshops organized within the framework of the thematic year: Statistics and Number Theory and New approaches in Probabilistic and Multiplicative Number Theory. His Aisenstadt lectures were delivered as part of the second of these workshops. In accordance with the theme of the workshop, his lectures examined the statistical behaviour of various important arithmetic functions. More precisely, Pomerance s talk focused mainly on four functions: Euler s totient function ϕ(n), defined to be the cardinality of (Z/nZ), which denotes the group of reduced residues mod n; the sum-of-divisors function σ(n); the function s(n) σ(n) n, which gives the sum of the proper divisors of n; and the Carmichael function λ(n), defined to be the exponent of the group (Z/nZ). All of these functions arise frequently in number theory. In fact, the second and third ones are directly related to one of the oldest problems in mathematics, concerning perfect numbers. In his first talk, entitled The Ranges of some Familiar Functions, Pomerance discussed old and new results concerning the value distribution of the aforementioned arithmetic functions. Whereas, by now, we understand relatively well the range of ϕ and of σ, as well as their intersection, thanks to work of various authors, such as Erdős, Ford, Luca and Pomerance, the range of the other two functions is more mysterious. For example, up until recently, we did not know the frequency of the integers that are λ-values. In his talk, Pomerance presented some very recent important progress towards this problem due to Ford, Luca and himself. Moreover, he talked about another very recent piece of work by himself and Luca that 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 = s 2 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 from the time of Pythagoras and Euclid, remains open till today. The 2-cycles of s give rise to amicable pairs of integers, such as (220, 284). We still don t know whether there are infinitely many such pairs either, though it is certainly believed so. Pomerance then went on to present several results and conjectures on the more general dynamics of the function s. Some of the conjectures are even mutually exclusive, thus giving a hint of how little we know about s. This was a fascinating talk on questions almost as old as mathematics itself and the avenues of modern research they have (continued on page 10) BULLETIN CRM 5
6 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 workshop Counting Arithmetic Objects (Ranks of Elliptic Curves) revolved around ranks of elliptic curves, focussing on what is expected of the typical behaviour of this mysterious and subtle invariant. The workshop was unusual in that it concentrated largely on a single breakthrough: the recent theorem of Manjul Bhargava, Chris Skinner and Wei Zhang that a positive proportion of elliptic curves over the field of rational numbers have rank zero and a positive proportion have rank one (and furthermore, that the Shafarevich Tate groups of these elliptic curves are finite, and that their ranks agree with the order of vanishing of the associated Hasse Weil L-function at the central point, as predicted by the Birch and Swinnerton-Dyer conjecture). This theorem represents a symbolic landmark, and its proof combines many of the fundamental advances on elliptic curves and the Birch and Swinnerton-Dyer conjecture achieved over the last several decades, growing out of the fundamental work of Gross Zagier and Kolyvagin on Heegner points, and, more recently: (a) Bhargava s revolutionary program for counting arithmetic objects, notably, his work with Arul Shankar counting small order elements in Selmer groups; (b) the breakthroughs towards the Iwasawa main conjecture growing out of the work of Chris Skinner and Eric Urban and its extension by Xin Wan, and the resulting converse of Kolyvagin -type theorems described in work of Skinner, Venerucci, and Zhang. The ideas at play in the work of Bhargava, Skinner and Zang involve an appealing mix of techniques, from the very analytic (Bhargava s profound new geometry of numbers arguments and counting methods) to the algebraic (his methods for parametrizing arithmetic objects of interest via ideas from invariant theory over Z ) and the arithmetic (in the central The plan of the workshop reflected the almost multidisciplinary 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 scheduled 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 conveying the broad outline and key ideas in the proof. The second day was dedicated largely to topics growing out of Bhargava s revolutionary techniques, with lectures devoted to counting elliptic curves with various extra structures, or to suggesting heuristics for the behaviour of Selmer groups and Shafarevich Tate groups modelled on the remarkably successful Cohen Lenstra heuristics. The third day was devoted to the Euler system techniques used to relate Heegner points with L-functions, both classical and p-adic, and Selmer groups, growing out of the fundamental work of Gross Zagier and Kolyvagin. Some of the more recent themes covered included the p-adic Gross Zagier formula of Bertolini Darmon Prasanna and the Jochnowitz congruences explored in the previous decade by Bertolini Darmon, which are an important ingredient for exhibiting a positive proportion of elliptic curves of rank one independently of the Birch Swinnerton-Dyer and Shafarevich Tate conjectures. The fourth day focussed on the fundamental recent results on the Iwasawa Main Conjecture growing out of the work of Skinner and Urban. The techniques involved here are rather different, relying on p-adic families of automorphic forms on unitary groups of signature (2, 2) and (3, 1). The day concluded with a CRM ISM Colloquium lecture by Kartik Prasanna which aimed to present the overall themes of the entire workshop to colleagues in the city from other mathematical areas. The last day returned somewhat to the themes of the second day, with pivotal lectures by Arul Shankar on the counting of the size of 5-Selmer groups of elliptic curves, and of Vladimir Dokchitser on the important parity conjecture for elliptic curves. The workshop ended on a high note, with a (continued on page 10) BULLETIN CRM 6
7 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 des nombres Québec-Maine se tenait à l Université Laval en l honneur de Hershy Kisilevsky (Concordia) et Manfred Kolster (McMaster) pour souligner leur 70e anniversaire de naissance. Hershy et Manfred sont des mathématiciens bien reconnus dans la communauté internationale qui ont beaucoup contribué au développement des mathématiques au Canada. Tous les deux sont très humbles et leur modestie a certainement un effet d osmose sur les membres de la jeune génération. Mentionnons d entrée de jeu que ce fut très difficile de les convaincre d accepter d être honorés et nous avons dû promettre que nous serions «low profile». Hershy s est tout d abord impliqué en théorie d Iwasawa et depuis une quinzaine d années il se concentre sur les courbes elliptiques. Il s est impliqué dans l organisation de plusieurs congrès à Montréal. Parmi ses grandes réalisations, il a démarré le Séminaire de Théorie des nombres QuébecVermont (QVNTS) et on lui doit la création du Centre interuniversitaire en calcul mathématique algébrique (CICMA). Au début du congrès, on a d ailleurs annoncé aux nombreux postdocs et gradués présents que sans l existence du CICMA, ils ne seraient probablement pas ici à Québec. bien ne pas avoir à investir une semaine complète. Le prix à payer, c est qu il y a des sessions en parallèle. Bien que les exposés en parallèle portent sur des domaines différents, les choix à faire sont souvent cornéliens. Le premier orateur, Michel Waldschmidt, a donné un joli exposé accessible à tous sur les multizêtas ; (Pierre Cartier préfère utiliser le mot «plurizêtas»). Le dernier orateur, Ram Murty, égal à lui-même, a donné un merveilleux exposé sur une généralisation du déterminant de Dedekind. Au total, il y a eu 47 exposés et il est possible de lire les résumés de tous les exposés sur la toile en tapant : Québec Maine Number Theory Conference. Le but de ces réunions est d offrir aux théoriciens des nombres l occasion de discuter et d échanger avec tous et chacun à propos de leurs résultats et de leurs problèmes, et ce, dans une atmosphère très décontractée, pour ne pas dire ludique. Ces congrès QuébecMaine attirent aussi de bons étudiants sous-gradués. Cette année, sous le tutoriat de Steven J. Miller (Williams College), sept étudiants sousgradués ont présenté leurs résultats. Pour la plupart, c était des calculs effectués sur l ordinateur. Ils présentaient leurs résultats en équipes. On y a vu une équipe de trois étudiants qui, tous les trois à 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 mathématiciens 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 stala conférence Québec-Maine a attiré près de 70 mathémati- tistique de l Université Laval. ciens (professeurs, postdocs, étudiants gradués et même sousgradués). Les séances sont très intenses et les gens aiment BULLETIN CRM 7
8 Goals 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) This is an exciting time for women and visible minorities in the basic sciences. More and more women from diverse backgrounds continue to beat formidable odds and come to the forefront of their professions with spectacular achievements. For the first time in its distinguished history, the Fields medal has been awarded to a woman, Professor Maryam Mirzakhani. The presidents at the helm of the International Mathematical Union (IMU) and the Canadian Mathematical Society (CMS) are two other highly acclaimed female mathematicians, respectively Professors Ingrid Daubechies and Lia Bronsard. For the last decade or so, considerable effort has been invested in researching and ensuring gender equity, and we are beginning to see the gratifying repercussions of this heightened awareness. At the same time, this increase in diversity has not yet resulted in a representation of women and minorities in graduate and postgraduate programs or as university faculty at the rate or proportion one might expect. We are still looking for ways to support this pipeline. The aim of the 2-day BIRS workshop was to target top female junior mathematicians from across the country, who are close to obtaining a Ph.D. or have just graduated. This is a time when researchers are faced with the challenges of job search in an increasingly tough academic environment. For graduate students, this may involve landing a coveted postdoctoral position. Postdocs on the other hand would be looking for tenure-track and/or instructorship positions after their term. Different job searches come with varying, often nonexplicit, criteria. Preparing oneself for a largely unknown hiring process can be stressful. This is especially true for women, who are often a small minority within their respective programs, and hence lack as extensive a support system as men. Also, women role models are much rarer, making it difficult for the junior women researchers to visualize themselves in senior academic roles. Activities The program made an effort to create a valuable experience for the participants within a limited timeframe. The highlights included: a collaborative and encouraging environment that facilitated interaction, as well as one-on-one mentorship; expert advice, general and personalized, related to research, teaching, giving talks and other facets of academic life; the presence of many role models. We wanted the participating students to leave with the understanding that they are not alone (even though they may be among a mere handful of females in their respective programs), that women are being increasingly successful in challenging and rewarding careers and having a positive impact on their communities. Given the limited amount of time we had, we broke the program into three parts. We made sure all junior participants were allowed time to present their work to the other attendees, as if they were giving a short job talk. We then had general discussion times built into the program as well as one on one meetings with assigned mentors so that the junior mathematicians could receive feedback on their presentations, as well as general career advice. Finally, we had three (of the originally planned four) main invited lectures focusing on some major aspects of a mathematical career: Research, University Teaching, and Presenting Research in Conferences. These fantastic talks were interactive, and followed by passionate discussions. We all came away from the meeting wishing it had been longer! This year we will apply for a 5-day workshop at BIRS which will allow us to cover much more ground. Contemporary Mathematics, Volume 630 Geometric and Spectral Analysis P. Albin, D. Jakobson & F. Rochon (eds.) In 2012, the Centre de recherches mathématiques was at the center of many interesting developments in geometric and spectral analysis, with a thematic program on Geometric Analysis and Spectral Theory followed by a thematic year on Moduli Spaces, Extremality and Global Invariants. This volume contains original contributions as well as useful survey articles of recent developments by participants from three of the workshops organized during these programs: Geometry of Eigenvalues and Eigenfunctions, held from June 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 from July 23 27, The topics covered in this volume include Fourier integral operators, eigenfunctions, probability and analysis on singular spaces, complex geometry, Kähler Einstein metrics, analytic torsion, and Strichartz estimates. BULLETIN CRM 8
9 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 Montréal to give a CRM Grande Conférence (public lecture) on November 20, Chris Danforth is a specialist on chaotic behaviour in meteorological systems and a member of the Mathematics Climate Research Network. He has diversified his research interests in the past years. The common thread of his research is that he works on large-scale systems in many fields including sociology, nonlinear dynamics, networks, ecology, and physics. For many years now he has been working on a fascinating subject, namely how mathematics can help in measuring emotions in real-time, and that was the title of his talk. Chris Danforth Why measuring emotional states? Alan Greenspan was convinced 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 Hedonometer, allowing them to measure the emotions. It relies on how people feel about individual words. How can we measure happiness? The idea of the Hedonometer is to measure the happiness of words. Individual words are rated on a 9-point unhappy happy scale by many people. But, where do we measure the happiness? The media are a daily source of words. And, now, social media provide a large source of data, already in digital form, that can be analyzed everyday. A first surprising result is that there is a bias towards happiness: more words seem to be on the happy side than on the unhappy side. This frequency-independent happiness bias is a very general phenomenon: it can be observed in Twitter, in books, in the New York Times, as well as in music lyrics. Moreover, it is observed in many languages. Of course, the instrument is not perfect, especially on the unhappiness side, and Chris Danforth s team is working on improving it. For instance, the Hedonometer only does a partial coverage of words. The instrument also ignores the context and cannot detect sarcasm. The Hedonometer also measures the long-term trends in the world. Compiling data since 1960 shows that the average valence2 of lyrics is decreasing over time. In music, this comes from the appearance of new styles, like Rap/Hip-Hop, Punk and Metal/Industrial that rate lower than classical music for 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 Twitter, 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 happiest (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 generally, happiness is positively correlated with income and negatively correlated with the obesity rate. Daily and weekly cycles are observed with the happiness starting high every morning and decreasing during the day. Happiness is higher on Saturdays and lowest on Tuesdays, and catastrophes are recorded on the Hedonometer. The lecturer ended by presenting some ongoing projects. To name a few: what is the largest outbreak of happiness? How does happiness vary with proximity to nature? Or to Farmers (continued on page 10) 1 From an interview with Jon Stewart on the Daily Show (September 18, 2007) 2 A term associated with attractiveness and positive emotions BULLETIN CRM 9
10 James Maynard Awarded the SASTRA Ramanujan Prize and a Clay Research Fellowship In December 2014, James Maynard, a CRM ISM Postdoctoral Fellow (currently at Oxford University), was 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, especially on some famous problems on prime numbers. In January 2015 it was announced that Maynard has been appointed as a Clay Research Fellow for a term of three years beginning July 1, In November 2013, Maynard proved a remarkable result on small gaps between primes, which eventually showed that there are infinitely many pairs of consecutive primes whose difference is at most 246. This result has now appeared in the Annals of Mathematics. Then in August 2014, while still in Montréal, Maynard proved another beautiful result, this time about large gaps between primes, namely how far apart can two consecutive primes get. This longstanding problem, an improvement on a lower bound known as Rankin s formula, was made famous by Paul Erdős, who offered his highest award ($10,000) for the solution. A similar result had been proved independently by Ford, Konyagin, Green, and Tao, but Maynard s proof rests on a different idea, which is simpler and more flexible. A vast improvement will appear as a joint paper of all five authors. crm.math.ca spacing. For this last statistic, they showed that the Linnik points exhibit random-like behaviour for almost all n. This fascinating talk on some exciting breakthroughs on a classical question of analytic number theory was very much appreciated by the broad audience of mathematicians in attendance. Carl Pomerance (continued from page 5) created. It was very much appreciated by the broad audience of mathematicians in attendance. The third talk was entitled Amicable Numbers, and its main focus was some of the most recent developments on the saga of the function s and its iterates, initiated by Pythagoras, as explained above. More precisely, the presentation was dedicated mainly to two recent results. The first one, due to Pomerance himself, concerns upper bounds for the counting function of amicable numbers (that is to say, how many there are up to a certain bound x). The second one concerns a result jointly proven by Luca and Pomerance on the even s-values, also referred to in the first talk. The talk elucidated the details behind the proofs of these two very recent and important advancements and ended with some potential avenues for future research. Counting Arithmetic Object (continued from page 6) Zéev Rudnick (continued from page 4) The question arises from a celebrated result of Legendre and Gauss which determines which integers n can be represented (primitively) as a sum of three squares, i.e., n = x 2 + y 2 + z 2 for some integers x,y, z such that gcd(x,y, z) = 1. This occurs if and only if n 0, 4, 7 mod 8, and for those there are many ways of writing n as a sum of three squares. These different representations give collections of points L(n) = {(x,y, z)/ n} on the unit sphere, and a fundamental result, conjectured by Linnik, is that the sets L(n) become uniformly distributed on the sphere as n. Linnik proved in 1940 that this is true under the Generalized Riemann Hypothesis, and it was proved unconditionally by Duke and by Golubeva and Fomenko in In the lecture, Rudnick explored what happens beyond uniform distribution, giving evidence of randomness on smaller scales. This is in sharp contrast to what happens with sums of two or four or more squares. The results presented have interesting connection to physics, and configurations of charges on the sphere which minimize energy (stable configurations). Finding stable configurations is notoriously difficult. Rudnick and his collaborators have shown that the energy of the Linnik points L(n) is close to minimal. They also studied other statistics for the Linnik points, such as the nearest neighbour distance and the least stimulating and thought-provoking lecture by Bjorn Poonen presenting heuristics that support the belief that the rank of elliptic curves might well be uniformly bounded: a prediction that runs against a lot of conventional wisdom, although Andrew Granville has arrived at strikingly similar conclusions based on a very different set of considerations. All in all this was an extremely successful, stimulating, and thought-provoking workshop on some of the most fundamental and rapidly evolving questions in number theory. Chris Danforth (continued from page 9) markets? Or to a Walmart? Or to one s partner? In the near future dozens of new languages are to be added, together with geographic mapping of happiness in other parts of the world and other sources (e.g., Google Trends) thus allowing further use in comparative studies. The Hedonometer team is working on moving from words to phrases, and to enlarging the spectrum of emotions studied, to include, for example, amazement. This will allow quantifying cultures with the number of narratives, metaphors and stories they have. The lecture left no one in the audience indifferent and a passionate debate started during the period of questions and continued over for the whole vin d honneur. BULLETIN CRM 10
11 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 first is acquaintance with syntax, words, and symbols; the second is meaning; and the third can be called the meaning of meaning. The means of entering the deeper layers is to cogitate, research, reflect what we know at the first layer. 1 This is how M. Ram Murty, Professor and Head of the Department of Mathematics and Statistics at Queen s University in Kingston, Canada, might sound when talking to a friend, a colleague, or a student. M. Ram Murty A refined scholar, an intellectual with an unquenchable thirst for all facets of truth, scientific and philosophic, M. Ram Murty has had a profound impact on the development of number theory throughout the world. To honour his mathematical legacy, a conference focused on new research directions in number theory inspired by Murty s most significant achievements was organized by Alina Carmen Cojocaru (UIC), Chantal David (Concordia), Hershy Kisilevsky (Concordia) and Francesco Pappalardi (Roma Tre) at the Centre de recherches mathématiques in Montréal. The conference was titled SCHOLAR A Scientific Celebration Highlighting Open Lines of Arithmetic Research and took place October 15 17, A remarkably successful event, the conference had about 100 participants representing several generations of research mathematicians, from beginning graduate students to emeritus professors. The speakers included some of the most prominent researchers in number theory from around the world: Henri Darmon (McGill); Etienne Fouvry (Paris-Sud); John Friedlander (Toronto); Dorian Goldfeld (Columbia); Henryk Iwaniec (Rutgers); Ernst Kani (Queen s); Wen-Ching Winnie Li (Penn State); Kumar Murty (Toronto); Yiannis Petridis (University College London); Carl Pomerance (Dartmouth College); Dinakar Ramakrishnan (Caltech); Michael I. Rosen (Brown); N. Saradha (TIFR); Joseph Silverman (Brown); Cam Stewart (Waterloo); Dinesh S. Thakur (Rochester); Yitang Zhang (New Hampshire). Their presentations illustrated research of the highest caliber on a broad spectrum of arithmetic topics: abelian varieties; function field arithmetic; Galois representations; L-functions; modular forms; sieve methods; transcendental number theory. A special lecture 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, Professor 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 admiration for Murty s scholarly legacy. Heartfelt stories were shared by several conference participants who lauded Murty s exceptional contributions and commitment to research, education, and service, both at his home institution and worldwide. The evening was filled with deep feelings of gratitude, respect and affection for M. Ram Murty and inspired the participants towards great pursuits. The conference was generously supported by the Centre de recherches mathématiques, the Fields Institute for Research in Mathematical Sciences, the Number Theory Foundation, and the Université de Montréal. It will culminate with a volume in the CRM Proceedings series, published together with the American Mathematical Society as part of the Contemporary Mathematics series. 1 M. Ram Murty, The Art of Research, Colloquium Lecture, Queen s University, Kingston, ON, ABONNEMENT. Pour vous abonner au Bulletin du CRM veuillez compléter un bref formulaire à la page web : crm. math.ca/bulletin/abonnement. Vous avez le choix entre les éditions imprimée et électronique. (Pour annuler votre abonnement, veuillez aller à la même page web.) SUBSCRIPTIONS. You can subscribe to the printed or electronic versions of the Bulletin by completing a brief form at this webpage: crm.math.ca/bulletin/abonnement. (To unsubscribe, please go to the same webpage.) BULLETIN CRM 11
12 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) Lambek was a prominent member of the Montréal mathematics community throughout the second half of the twentieth century, active until his death last summer. During his career he was an inspiration, a mentor, and a dear friend to many generations of mathematicians, linguists, and other scholars, and his loss has left a huge gap in our community. The following account of Joachim Lambek Jim s life and work is partly based on a talk given on the occasion of his 75th birthday, by Michael Barr. Jim was born in Leipzig, Germany, in In the late 30s, he was among the last children (he was in his mid-teens) to escape Nazi Germany on the Kindertransport to England, where he worked on a farm. Shortly after, along with other male German Jews, he was deported (as an enemy alien ) to Canada and interned in a prison work camp. In later years, Jim often joked that they were not told until after the ship set sail whether they were headed to Canada or to Australia: being on a ship bound for Canada saved him from having to do 2-category theory. After he had spent about two years in a camp, mainly in New Brunswick (but including a week on Île Sainte-Hélène, later the site of Expo 67), the authorities decided that people like Jim should not have been imprisoned and he was released. During his time in the work camp he had begun studies in mathematics and logic, taught in the camps by Fritz Rothberger and other detained European mathematicians, and even entered into correspondences with other mathematicians and logicians, including Willard V.O. Quine. On release, those prisoners who found Canadian sponsors were allowed to stay in Canada; fortunately, a Montréal businessman agreed to sponsor him. So he settled in Montréal and entered McGill, earning an honours degree in mathematics in 1945 and an M.Sc. a year later. One result of all this is that he spent the war years safely in Canada, while his mother and sister, who were allowed to stay in England, endured the blitz, and came to North America only at the end of the war. He was immediately hired as a lecturer in mathematics at McGill. In those postwar days, McGill had to expand rapidly to make room for returning veterans and one way it dealt with the problem was by opening a satellite campus in St. Jean. Getting there was his responsibility. One of Jim s stories was that once he missed the train for reasons beyond his control and skipped the class. The chairman told him he should have taken a taxi at his own expense of course. This would have cost $10 (probably equivalent to at least $150 today), considerably more than his pay for the class. One of Jim s former students, Prof. Robert Raphael of Concordia University, recalls an amusing anecdote from Jim s student years at McGill in the 1940s. Jim faced a problem finishing his undergraduate degree, since taking a chemistry course was a requirement to graduate. But the director of the chemistry building denied Lambek entrance, as Germans were not allowed because of sensitive experiments taking place in the building. The McGill Senate acted and replaced the chemistry course with an arts requirement, so Jim graduated. After the war, the building director was arrested on charges of spying for the Soviet Union. In 1950, he completed his Ph.D. under Hans Zassenhaus (in fact, he wrote two theses, one in mathematical physics, then one in pure mathematics) and was promoted to assistant professor. In those days, assistant professors taught at least 12 hours a week and were not expected to do much, if any, research. Yet, between 1951 and 1959 he had 14 publications. Half of those papers were joint with Leo Moser and appear to be in combinatorics and elementary number theory. However, even in those early days he had begun research on several of the areas of mathematics that would occupy the rest of his career. In 1958, he published his first paper in linguistics, on the syntactic calculus, as well as his first paper on rings of quotients. Although there were two more publications in mathematical linguistics in the following three years, he appears to have abandoned the subject for over a dozen years. But the world did not abandon this work, and a small but lively group of researchers, mainly in Europe, developed the subject of Lambek grammars. He returned to the area several times in his subsequent career, and it was a dominant theme in his last decade or so. During the same period, Jim published a short paper in the Bulletin of the Canadian Mathematics Society (1961) entitled How to program an infinite abacus. It turns out he co-discovered an important notion of abstract computing device (an alternative to Turing s machines), today called BULLETIN CRM 12
13 Register machines. The idea was simultaneously (and apparently independently) developed by several leading researchers, notably M. Minsky. From the mid-50s to the mid-60s, Jim put most of his efforts on ring theory, particularly rings of quotients. He published many papers on the subject, culminating in the very successful and influential book, Lectures on Rings and Modules (1966). Jim spent his sabbatical year in Zürich at the Forschungsintitut für Mathematik der Eidgenössische Technische Hochschule where Beno Eckmann had gathered together a group of people interested in algebraic topology, homological algebra, and, incidentally, category theory. That year in Zürich reoriented his research into category theory. Two mathematicians who subsequently became pivotal members of the Montréal Category Theory group also were in Zürich around the same time, and met Jim then, or shortly after his year there. Michael Barr remembers being given Jim s monograph Completions of Categories (SLNM 24, 1966) as soon as he arrived in Zürich; he first met Jim in Chicago shortly afterwards. Marta Bunge was in Zürich the same year as Jim (along with Bill Lawvere, among others), and remembers being struck by his quick grasp of what was important in the subject. Jim arranged to bring both of these future colleagues to Montréal. During the 1970s, his research combined ring theory, torsion theory and category theory, much of the latter in collaboration with Basil Rattray. During this decade, he renewed his interest in mathematical linguistics, including formal studies of verb conjugations in French and Latin. As well, he began working on applications of category theory to logic. This last interest resulted in a fruitful collaboration with Phil Scott (initially a postdoc of Jim s, later a professor at U. Ottawa), culminating in the now-classic book, Introduction to Higher Order Categorical Logic, in Lambek s works in categorical proof theory were of seminal influence. They ranged from his early applications of Gentzen s methods to coherence theorems in category theory, to his later introduction and promotion of multicategories, substructural logics, and bicategories in widely varying areas, from algebra to linguistics to physics. Over the years, Jim s research work never slacked off. He regularly published papers in categorical algebra and logic, in linguistics, in philosophy, and even in quantum physics, returning to a topic he addressed in his first Ph.D. thesis. Indeed, he was an enthusiastic supporter of using Hamilton s quaternions in physics; a popular account of his views and of his meeting with Dirac is contained in his article If Hamilton had prevailed: quaternions in physics in the Mathematical Intelligencer (1995). During this period he also wrote an undergraduate text on the history of mathematics with Bill Anglin, The Heritage of Thales (1995), which is unique in its deep coverage of a wide range of topics. In a quite extraordinary series of works late in his career, Jim developed an entirely new approach to his earlier work in linguistics, which he called pregroup grammars, a formal typing system for natural language processing, which actually arose from his growing interests in higher category theory. He wrote a book for linguists From word to sentence (Polimetrica, 2008), as well as numerous articles for mathematicians on his new theories of linguistics. Math Reviews lists more than 130 publications at present, and that does not include his last book From Rules of Grammar to Laws of Nature, which he wrote and saw to publication during the last months of his life. Jim Lambek s reputation does not depend only on the quantity and quality of his research, however. Many of his past students, postdoctoral fellows, colleagues and friends have commented on his generosity (and insistence on high standards). He helped launch many a career in mathematics by putting innumerable students on the track of promising research projects, and providing beginning researchers with postdoctoral fellowships and opportunities to develop and present their results. He was also a good friend and colleague, a generous host, and excellent company. 1 He will be deeply missed, personally and professionally. 1 We cannot resist mentioning one of Jim s favourite stories. Jim would often organize workshops and conferences in Montréal, to which both mathematicians and linguists were invited. Jim, always the convivial host, would lead the group to some downtown restaurant. The problem, he observed, was that the mathematicians could never figure out the bill, and the linguists couldn t speak French! [... ] But it may be Jim Lambek who had the biggest impact on my career of any McGill professor, through the kind of chain of accidents that so often affects a life course. Jim taught a course on the Theory of Computation and Mathematical Linguistics. That came in handy when, in my first week of graduate school at Harvard, we were assigned a technical paper in that field on the problem of language acquisition. Thanks to his class, I was able to understand it, and wrote a course paper which became my first major publication. That led to my being hired to teach language acquisition in my first job, which led to an encompassing interest in all aspects of language, from how it evolved to how it is used at its best in good writing. Steven Pinker (Harvard University) To professor, with love, The McGill News, Fall Winter 2014 BULLETIN CRM 13
14 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 Montréal un colloque intitulé «Vision de l espace : hommage à Alexander Grothendieck». Comme le sous-titre l indique, ce colloque se voulait un hommage au mathématicien Alexander Grothendieck qui s est éteint le 13 novembre dernier à Saint-Lizier en France à l âge de 86 ans. Récipiendaire de la médaille Fields en 1966, Grothendieck est considéré à juste titre comme l un des plus importants mathématiciens du xx e siècle. Entre 1955 et 1970, il a complètement réécrit les fondements de la géométrie algébrique en introduisant les concepts de schéma, de topos, de motifs et toute une panoplie de théories cohomologiques dans le but de démontrer les conjectures de Weil sur les fonctions zêta locales. La dernière des conjectures de Weil fut démontrée par un étudiant de Grothendieck, Pierre Deligne, en Cette refonte en profondeur de la géométrie algébrique repose sur une utilisation intensive, originale et essentielle de la théorie des catégories, théorie que l on croyait à l époque limitée au statut de langage utile et ultimement accessoire. On peut affirmer sans sourciller que les travaux de Grothendieck constituent un nouveau paradigme mathématique au sein duquel les concepts catégoriques sont inextricables. L impact de ses travaux a inspiré plusieurs mathématiciens de par le monde ; et Montréal, plaque tournante de la théorie des catégories depuis la fin des années 1960 jusqu à aujourd hui, y a largement puisé. C est dans cet esprit que les organisateurs de ce colloque ont invité mathématiciens et philosophes des mathématiques à venir rendre un hommage posthume à l œuvre de Grothendieck. Alexander Grothendieck en 1970 Photo : Konrad Jacobs ; MFO Les notions de schéma et de topos ont constitué les principaux pôles des conférences de la première journée. La conférence de Colin McLarty, qui occupe le poste de Truman P. Handy Professor of Philosophy à la Case Western Reserve University, s intitulait «How (and why) Grothendieck simplified cohomology theory». Après avoir présenté un survol historique des théories cohomologiques jusqu à l époque de Grothendieck, le professeur McLarty a montré clairement comment, par le biais de la théorie des schémas et de la théorie des topos ce dernier a bel et bien simplifié la théorie de cohomologie, et ce, même si le prix à payer semble pour certains trop marqué par la généralité et l abstraction. Michael Makkai, logicien à l Université McGill et un des bâtisseurs de la logique catégorique, s est attardé, pour sa part à la notion de prétopos. Sa conférence, intitulée «Grothendieck s concept of pretopos : its role in algebraic geometry and in logic», a permis à l auditoire de mieux comprendre comment la notion de catégorie cohérente et la notion de prétopos étaient liées non seulement dans l œuvre de Grothendieck en géométrie algébrique, mais également en logique catégorique. Le philosophe des mathématiques Jean-Jacques Szczeciniarz, professeur des universités à l Université Paris-Diderot et directeur du département d histoire et de philosophie des sciences à la même université, s est concentré sur les aspects ontologiques et épistémologiques inhérents à la notion de schéma. Sa conférence, intitulée «Réflexions sur la révolution théorique opérée par Grothendieck en géométrie algébrique», a mis en lumière comment dans l approche de ce dernier la connaissance des objets introduits devenait indirecte et dépendante des outils développés en parallèle aux objets eux-mêmes. La notion de schéma et les nouvelles théories cohomologiques créées en parallèle illustraient parfaitement ses propos. La première journée s est terminée par la conférence de l éminent mathématicien français Pierre Cartier, membre du groupe Bourbaki, professeur à l IHES et ancien collègue de Grothendieck, qui a fait un survol éclairé et éclairant des notions de topos et de champs. Au cours de la seconde journée, les conférenciers se sont tournés vers les mathématiques développées par Grothendieck au cours des années 1980 et Ainsi, Mathieu Anel, chercheur au CNRS et associé au Laboratoire Sphère de l Université Paris-Diderot, a mené les participants «au-delà des idées de Grothendieck», car c était bien là le titre de sa contribution. Mathieu Anel a montré de manière saisissante comment les notions de catégories dérivées et de dérivation en général menaient directement à la théorie des catégories supérieures et comment cette dernière, non seulement simplifiait plusieurs définitions classiques, mais comment plusieurs constructions qui semblent artificielles et ad hoc dans un contexte catégorique traditionnel deviennent naturelles dans le nouveau contexte. Finalement, André Joyal, professeur honoraire à l UQAM, a enchaîné en récapitulant l histoire des quasi-catégories et des catégories supérieures, développement dans l esprit des travaux de Grothendieck sur les champs et la notion de catégorie test. Centré sur les concepts plutôt que sur les résultats, les outils et les développements techniques, ce fut l occasion d évaluer la richesse des contributions de Grothendieck sur les mathématiques de la dernière moitié du xx e siècle et sur les mathématiques de demain. BULLETIN CRM 14
15 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 mathématiques sont partout autour de nous, les mathématiciens utilisent toutes sortes d astuces : des conférences grand public, des livres de vulgarisation très accessibles, etc. C est dans ce contexte et avec l appui de l Université Laval et de Mitacs que nous avons créé, en 2005, le programme SMAC (Sciences et mathématiques en action), dont les principaux objectifs sont de susciter chez les jeunes un intérêt pour les mathématiques et les sciences et de démystifier les mathématiques auprès de la population en général. La voie que nous avons choisie est celle d offrir des activités ludiques et amusantes tels des jeux et des spectacles. Je présente ici celles relevant du domaine du spectacle. Un spectacle multimédia sur les mathématiques En 2005, nous avons créé pour les écoles secondaires une 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 incluent : (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 probabilité grimpe à 99%. Comme la plupart des écoles qui ayant accueilli Show Math souhaitaient avoir une suite, nous avons développé Show Math 2 en Les sujets abordés comprennent (1) comment la notion de fractal a vu le jour et comment elle est utilisée par les producteurs de films pour créer des paysages fantastiques comme dans Star Trek et Avatar ; (2) pourquoi les méthodes de chiffrement les plus sécuritaires reposent sur la difficulté à factoriser des grands nombres ; et (3) le fait que le GPS utilise l intersection de trois sphères pour localiser notre emplacement. Petit Show Math Pour répondre au souhait exprimé par les directeurs d écoles primaires, nous avons créé Petit Show Math. Il s agit d un autre spectacle à saveur mathématique, mais adapté aux plus jeunes. Le fil conducteur est comme suit : Dans une galaxie très lointaine, il existe une petite planète où les habitants ignorent la notion de nombre. Un petit garçon nommé SMATH habitant sur cette planète est à bord d un vaisseau spatial en direction de la planète Terre. Il s y rend puisqu il a entendu dire qu on utilise les nombres depuis des milliers d années sur la Terre et que cela s est avéré fort utile. Son vaisseau atterrit sur la scène et voilà que SMATH demande si quelqu un dans la salle connaît les mathématiques. C est à ce moment que j interviens et que j invite SMATH et les jeunes dans la salle à explorer avec moi l univers fascinant des mathématiques. Nous parcourons alors ensemble trois modules : (1) l histoire des nombres en remontant à leur invention il y a ans ; (2) le phénomène du son : sa vitesse, son intensité et sa fréquence ; et (3) l exploration spatiale : notre système solaire, ses planètes et ses comètes, ainsi que la taille relative de tous ces objets. À la fin du spectacle, SMATH quitte la Terre avec Léa, une petite fille qu il a rencontrée au cours du spectacle, tout en déclarant qu il va partager avec ses amis tout ce qu il a appris sur les nombres. Jean-Marie De Koninck Pluton va en appel! Notre dernière création est Pluton va en appel! Il s agit d une pièce de théâtre dans laquelle l ancienne planète, Pluton, va en appel devant le Soleil afin de pouvoir réintégrer le rang des planètes. Rappelons d abord que dans les livres d astronomie publiés avant 2006, on peut lire que notre système solaire est composé de neuf planètes : les quatre planètes rocheuses (Mercure, Vénus, la Terre et Mars), les quatre planètes gazeuses (Jupiter, Saturne, Uranus et Neptune) et enfin Pluton, que l on croit faite de roche et de glace. Cependant, à compter de 2003, les astronomes se sont mis à découvrir que plusieurs autres corps en orbite autour du Soleil étaient beaucoup plus gros et plus massifs que Pluton et ont ainsi commencé à se demander s il n était pas temps d établir les critères de ce qui constituait véritablement une planète. C est BULLETIN CRM 15
16 ainsi qu en 2006, l Union astronomique internationale a introduit une nouvelle classe d objets appelés planètes naines et décida d y inclure Pluton. La décision fut loin d être unanime et créa du coup toute une controverse. Plusieurs prétendaient que pour des raisons historiques, Pluton devrait conserver son titre de planète. En mars 2007, la Chambre des députés de l état du Nouveau-Mexique a même fait adopter une résolution stipulant que Pluton conserverait son titre de neuvième 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 rocheuses, soutient que Pluton doit retrouver son titre, alors que Neptune, qui parle au nom des planètes gazeuses, argumente contre, craignant que si Pluton redevient une planète, cela aura pour effet de redonner la majorité aux planètes rocheuses. Le spectacle est un prétexte pour expliquer diverses notions mathématiques telles que celle des points de Lagrange et pour utiliser les mathématiques qui permettent de décrire le phénomène d assistance gravitationnelle permettant aux sondes spatiales de se déplacer dans le système solaire en utilisant la force gravitationnelle exercée par les différentes planètes. Pluton va en appel! sera certes encore plus populaire dans l année qui vient parce que les médias seront alors très intéressés par les péripéties de la sonde New Horizons qui a quitté la Terre en 2006 et qui rejoindra Pluton le 15 juillet Kai Behrend 2015 CRM Fields PIMS Recipient Professor Behrend is an internationally recognized leader in the field of algebraic geometry, whose contributions to the subject are noted both for their depth and scope. He has obtained fundamental results in the theory of algebraic stacks, Gromov Witten theory and the study of Donaldson Thomas invariants. In particular, his pioneering works on the construction of a virtual fundamental class played a key role in laying the algebraic foundations of the Gromov Witten theory. Later, he made a breakthrough in the study of the Donaldson Thomas invariants by showing that, for certain spaces, the degree of the virtual fundamental class could be expressed as the topological Euler characteristic weighted by a natural constructible function, depending only on the intrinsic properties of the space. This function is now widely known as Behrend s function. It allowed the use of motivic methods to compute Donaldson Thomas invariants, and made it possible to obtain their categorified and motivic versions, which is currently among the hottest trends in the subject. In his earlier work, Professor Behrend obtained an important generalization 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. Kai Behrend received a Ph.D. in 1991 at the University of California, Berkeley. He joined the faculty of the University of British Columbia in Professor Behrend has received numerous recognitions for his research, including the 2001 Coxeter James Prize and the 2011 Jeffery Williams Prize of the Canadian Mathematical Society, as well as an invitation to speak at the International Congress of Mathematicians in Seoul in Christiane Rousseau nommée membre du comité scientifique de l IBSP de l UNESCO Christiane Rousseau, membre du Centre de recherches mathématiques et professeure au Département de mathématiques et de statistique de l Université de Montréal, a été nommée membre du prestigieux conseil scientifique de l International Basic Sciences Programme (IBSP) de l UNESCO pour un mandat de trois ans de 2015 à Christiane Rousseau a fait ses études en mathématiques à l Université de Montréal où elle a obtenu son doctorat en Après des études postdoctorales à McGill, elle est revenue à l Université de Montréal comme professeure. Elle a été directrice de son département de 1993 à 1997 et présidente de la Société mathématique du Canada de 2002 à Lorsqu elle dirigeait le Centre de recherches mathématiques (CRM) en , elle a lancé l année internationale «Mathématiques de la planète Terre 2013» sous le patronage de l UNESCO. De 2011 à 2015, elle a été vice-présidente de l Union mathématique internationale (UMI), et elle continue de siéger au comité exécutif de l UMI durant la période Pendant toute sa carrière, elle a mené en parallèle des activités de recherche et d encadrement d une part et des activités de vulgarisation et de sensibilisation aux mathématiques : conférences dans les cégeps, organisation de camps mathématiques, articles dans des magazines mathématiques. Le directeur du IBSP a souligné la réputation de Christiane Rousseau en sciences fondamentales et son engagement soutenu envers la coopération internationale. Les membres du conseil scientifique, au nombre de 30, sont chargés de suivre, d orienter et de superviser du point de vue scientifique, les projets proposés par l IBSP. Créé en 2005, l IBSP met l accent sur la promotion de la formation et de la recherche en sciences fondamentales et en enseignement des sciences, ainsi que l utilisation pratique des percées scientifiques pour répondre aux problèmes environnementaux et aux besoins humains, et contribuer à l amélioration de la qualité de vie et de l éducation. BULLETIN CRM 16
17 AdS/CFT, Homolography, Integrability (continued from page 2) which has been developed independently by combinatorists, and by group theorists, having to do with directed networks, positroid cells, and plabic graphs, just happened to do the job for these two different areas. Obviously one wants to bring people together who are familiar with the different aspects so that they can interact and find if there are some deeper reasons for these connections. Bulletin: What will be the theme of the Aisenstadt Chair lectures? JH: Bertrand Eynard [CEA Saclay], one of the semester s Aisenstadt Chairs, and a member of our Mathematical Physics Laboratory, has been a pioneer in developing what goes under the general heading of topological recursion in moduli spaces. The idea is, within the general framework of moduli spaces of Riemann surfaces, or more generally these topological invariants that I mentioned, and geometric enumerative invariants, there is a unified, common structure which has to do with solving a sequence of recursive relations which may be interpreted in the context of matrix models as being correlators or multi-trace invariants, with data that are essentially algebro-geometric in nature. The key ingredient is that for each one of these (basically it s a tau function that controls either the Gromov Witten invariants, Hurwitz systems, knot invariants, etc.) there is some kind of a recursive system of equations, which from the algebro-geometric data, relating to what s called a spectral curve, generates all of the higher invariants in a completely finite algorithmic way (by recursions). It s also closely related to the very last workshop [crm.math.ca/2016/moduli16/]. That workshop is rather late (early January 2016) and Eynard will be here in both October November 2015 and early January Bulletin: Can you say something about the other Aisenstadt chairholder? JH: Before I do that let me just comment on what Robert mentioned about applications to physics: some condensed matter applications, the ideas of AdS/CFT, and also cosmological applications. Another workshop, which is being organized by Jean-Sébastien Caux, Masudul Haque and Robert Konik, is also very much about experimental physics. That workshop, which is early in the program, is about near integrability [crm.math.ca/2015/quantum15/]. In real life physics exact integrability is very rare, but systems that are nearly integrable can very often be understood by methods coming from integrable systems. This approach has two main applications: condensed matter physics in the very low temperature regime, and strongly correlated phenomena. I can t speak in detail about that but it should be also a very important connection with experimental physics. JW: Since we mention this connection to experiments, via the heavy ion collisions, there is a really very concrete application crm.math.ca that came out of the AdS/CFT correspondence, the so-called viscosity bound: the lower bound on the ratio of the viscosity over the entropy density, that was conjectured to hold for all physical systems, all hydrodynamical systems, as a result of certain calculations done in gravity, that η/s should always be bigger than 1/(4π). First of all it s satisfied for all physical systems known in nature, but it also created interest when the viscosity of the quark-gluon plasma in heavy ion collisions (at RHIC and LHC) was measured to be very close to this bound. JH: The Nobel prize winner, Ketterle, from MIT, once said there were two frontiers in theoretical and experimental physics: one is the high energy frontier and the other is the low temperature frontier. This workshop is unique in that it is using mathematics to connect these two different frontiers. Very surprisingly, AdS/CFT, which was conceived purely for relativistic four-dimensional quantum field theory, originally, in fact seems to have applications also to strongly correlated low temperature phenomena. And the methods of integrable systems, which were thought of as something very idealized, turn out to have real world applications. It s really a beautiful unifying aspect that s bringing together the two frontiers of both experimental and theoretical physics. We forgot one thing which I hope Marco will talk about and that s the role of asymptotics. M. Bertola: I would like to describe the workshop Asymptotics in Integrable Systems, Random Matrices and Random Processes and Universality [crm.math.ca/2015/deift15/] Integrable systems are one of the main themes of the semester. As John said, although the mathematics of nature is not, in general, integrable, there is a sense that methods and results of integrable systems keep recurring. A sort of strange attraction. One example is the notion of universality in the behaviour or large systems: under certain asymptotic regimes some features of a large system behave independently of the minute details of the system itself, and they do behave like a given prototypical system that turns out to be integrable. This recurring theme appears in the statistics of eigenvalues of large matrices, the small-scale behaviour of nonlinear waves, and also in combinatorics, representation theory of groups and string theory. The workshop itself fits perfectly in the general theme and is expected to attract many outstanding world-wide experts. Quite serendipitously, in the early stages of the organization we realized that the workshop coincides also with the celebrations for the 70th anniversary of one of the key figures in the area of asymptotics of integrable systems. I am referring to Percy Deift, who, amongst several outstanding contributions, introduced and perfected the steepest descent method for noncommutative Riemann Hilbert problems (a certain form of matrix-valued boundary value problems in the complex domain). This turns out to be one fantastic gift that keeps on giving because this very type of boundary value problems impinges on so many types of integrable systems. BULLETIN CRM 17
18 The workshop was also recommended for support from the NSF, which will help in attracting and funding a large cohort of young researchers. JH: Now about the theme of the other Aisenstadt Chair, Nikita Nekrasov [IHES/SCGP]: this is also a beautiful surprising connection between fully relativistic four dimensional quantum field theory and integrable systems, but it s through another route. There s a big class of classical and quantum integrable systems which play a huge role, called the Hitchin systems. The Nekrasov approach, and also others which contributed to this, connects up the supersymmetric N = 2 Yang Mills theory with Hitchin systems, which in turn are programmed into some of the mathematical structures that were mentioned before, namely cluster algebras and Poisson structures. This is a very ongoing, very active, development, and that s what Nekrasov and others helped to launch, a few years ago. JW: As John was mentioning, the integrable systems were first noticed in the N = 2 context, which is the Seiberg Witten theory, for certain physical quantities, and then they appeared in N = 4 super Yang Mills, for certain other physical quantities, and it s not exactly understood, one should say, how these two things are related. And Nikita Nekrasov will speak about that. One of the ideas of this particular semester is to bring together these new developments for N = 4 with the somewhat older story relating to N = 2 super Yang Mills and these generating functions: the tau functions, enumerative generating functions, and so on. JH: Let me clarify that N means the extent of the supersymmetry. Supersymmetry has to do with identifying commuting and anti-commuting degrees of freedom, in a symmetric way, and the amount, that is the dimension, of the anti-commuting part, which relates to Grassmann algebras, is what is being labelled by this N. If N is zero then there s no super symmetry, if N is 1 it s the minimal extension. For super Yang Mills theory N = 4 is the maximum, I think for gravity N = 8. So this is a sort of code word for specialists in supersymmetry. Bulletin: Will Juan Maldecena be here? JW: Yes, he will. To summarize, let me just give a brief overview of the activities. The semester empiète (encroaches/ spills over): it starts in June (2015) and ends in January (2016). The full schedule is available on the CRM s website [crm.math.ca/holography2015] Instead of going in chronological order, I would like to list them starting from the most mathematical and going to the physical applications: we have topological recursions, which connect with very abstract algebraic geometry that is associated with the Aisenstadt Chair, Bertrand Eynard, and is actually the last workshop chronologically, in January Then we have asymptotics, which Marco talked about, which is the very first workshop, in June, in honor of Percy Deift. Next there is the positivity, and the core of this program is the 3-week focus period in August: one week on positivity, positive Grassmannians and the role they play in integrable systems, and two weeks on integrability in AdS 4 and AdS 5, that s the pièce de résistance. Moving on to physical applications, we have AdS/CFT and quantum gravity, which Robert mentioned, about the conundrums associated with black holes. And then there s the workshop on applications to cosmology and the Big Bang singularity that Robert talked about, and the workshop on applications to QCD (quantum chromo-dynamics) and condensed matter physics (probably it will be mostly applications to QCD, the strong interactions), by Keshav. We also have another workshop called Beyond Integrability where condensed matter systems will be featured. Finally, sometime in September, there will be the Aisenstadt Chair lectures by Nikita Nekrasov. JH: Just to elaborate on the three-week session from the very last week of July to the second week of August, which is a sequence of two different workshops (one of one week and one of two weeks) about two different aspects: one is positive Grassmannians and cluster algebras and related topics [crm.math.ca/2015/amplitudes15/], and the other is called Hidden Symmetries and Integrability Methods [crm.math.ca/ 2015/Symmetries15/]. That s going to be a monstrous twoweek session where almost everyone in the subject will be present, and the idea is to have some people from the first workshop stay over for the second, and some people from the second arrive early so that there can be some crossfertilization. Bulletin: Are these going to be people working in mathematics or physics? RB: In the workshops which I mentioned it s going to be mainly physicists, but specifically the one I m organizing is going to be a mixture between mathematics and physics, because we re actually using mathematical tools in order to make progress on the physical problems. And here the mathematical tools come mostly from the areas of partial differential equations and dynamical systems. So that s where Walter Craig and Niky Kamran fit in. JH: I would say globally, it s hard to say who s a physicist and who s a mathematician in this area, but it s very nicely balanced between the physics aspects and the mathematics aspects. JW: Let me conclude by mentioning something which is really special about our program and different from others. One of the more meta motivations for this entire semester is to give a forum for the AdS/CFT correspondence at a mathematical institute. There have been, of course, programs and workshops organized at math institutes featuring integrability and quantum field theory, and also the applications to enumerative geometry that John has mentioned, but as far as we know there has been no semester where AdS/CFT and the holographic correspondence was put at the center. We want to raise awareness of the AdS/CFT correspondence among mathematicians, how special it is, how different it is from the other dualities. The other dualities that we have in physics are dualities that mathematicians are aware of, there are mathematical counterparts. But AdS/CFT is really different from (continued on page 19) BULLETIN CRM 18
19 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 prévision 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 dépendance, Peter Song (Michigan), a proposé de nouveaux tests Un atelier de l INCASS sur les nouveaux horizons en modélid ajustement pour les modèles basés sur des copules semisation par copules a eu lieu au CRM du 15 au 18 décembre paramétriques 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 spéchastique, 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 modélisation 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 modèles de vignes pour mieux refléter les structures de données machine. 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 (Munich), et de nombreuses applications intéressantes en ont été L atelier comportait en tout 18 exposés. Deux séances d afdonnées 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 propriétés structurelles de ce type de modèle. La qualité des préou pour modéliser des champs aléatoires spatio-temporels. sentations a été unanimement saluée par les participants, qui L utilisation des copules continue de se répandre en éconoont semblé particulièrement séduits par les nouvelles possibilimique, en finance et en actuariat. Dans leurs exposés, Antés 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 (Vermathematical counterpart, which is the correspondence besailles), Luis Mediero (Madrid), Gianfausto Salvadori (Satween Chern Simons theory and knot invariants, and enulento) et Salvatore Grimaldi (Tuscia), ex-président de la Commerative geometry, so that can be viewed as a special case mission internationale d hydrologie statistique. Les particiof 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 awarepar exemple à estimer le temps de retour de catastrophes ness of its special status for mathematical physics of the 21st century. BULLETIN CRM 19
20 Bilan de l UMI CRM Laurent Habsieger (codirecteur de l UMI CRM) L UMI CRM est actuellement à un tournant de son histoire. Ce 7 avril, un comité la visitera en vue de son renouvellement, et un nouveau directeur est annoncé dès septembre Il est donc temps de présenter un bilan de ses premières années d existence. Fonctionnement La direction de l UMI est assurée par deux codirecteurs, le directeur du CRM et moi-même. J ai ainsi pu travailler en étroite concertation avec François Lalonde puis Luc Vinet, et mener une politique scientifique qui reflète celle du CRM. L année est marquée par deux temps forts : l automne, avec les appels à projets de l INSMI pour les postes rouges et les séjours longue durée, et avec les appels à projets de l ANR, et le début d année où je lance un appel à projets, pour établir le budget de l UMI. Pour les appels à projets de l INSMI, j accompagne les candidats en leur fournissant toutes les informations pertinentes et en leur rédigeant une lettre d appui personnalisée. Pour les projets ANR, je tiens le rôle de consultant, et assure la jonction avec le service Partenariat et Valorisation de la délégation Paris Michel Ange. Une fois recueillis les résultats de mon appel à projet de début d année, je convoque le conseil d unité, et il est procédé à la répartition des fonds disponibles. J assure également la gestion administrative de l UMI, en partenariat avec la délégation Paris Michel Ange. Afin de faciliter la compréhension du système français, j ai rédigé une note explicative des procédures à suivre au sein du CNRS pour la prise en charge de missions, avec un lexique des termes administratifs, document que j ai diffusé aux coordinateurs locaux de projets ANR. Aussi bien pour la subvention d état que pour les programmes de l ANR, j assure le suivi des finances, accompagne les missionnaires tout au long du processus de prise en charge de leur mission, depuis la demande d ordre de mission jusqu à la rédaction de l état de frais. Suzette Paradis, webmestre du CRM, a mis en place un site web pour l UMI, dont je suis administrateur, et que je mets à jour périodiquement. Son URL est crm.math.ca/umi/. Financement L UMI a reçu une subvention d état annuelle de e en 2012, 2013 et Chaque année, dans la demande de moyens que je présente au CNRS, je sollicite une augmentation de cette dotation, pour tenir compte à la fois de l éloignement géographique de l UMI (par rapport à une UMI européenne) et du dynamisme de celle-ci. J ai été entendu et la subvention d état annuelle est passée à e en Dès mon arrivée à l UMI, j ai insisté sur les possibilités offertes par les programmes de l ANR. Pour 2013, l UMI a présenté deux projets, et un a été retenu : CAESAR, dont je suis le coordinateur local. Cela représente e hors frais de gestion, pour la période Pour 2014, l UMI a présenté quatre projets et deux ont été retenus : GeRaSic, dont Dmitry Jakobson (McGill) est le coordinateur local, et HR-CEM, dont Yves Bourgault (Ottawa) est le coordinateur local. Les deux projets retenus couvrent la période Hors frais de gestion, cela représente e pour GeRaSic et e pour HR-CEM. Pour 2015, l UMI a présenté trois pré-projets, mais aucun n a été retenu. Pour 2016, l UMI a présenté un pré-projet, et le résultat n est pas encore connu. Avec trois programmes ANR en activité, l UMI CRM fait partie des meilleures UMI dans le monde : aucune UMI n a actuellement réussi à obtenir quatre ANR, selon les statistiques présentées en mai dernier à la réunion des directeurs d UMI d Amérique du Nord. Visites au sein de l UMI Les visites longues sont celles qui durent plus de trois mois et que l UMI ne peut donc financer intégralement. Il faut donc présenter des demandes au CNRS, de divers types : délégation (pour permettre à l universitaire de dépendre du CNRS), soutien financier direct, affectation. La formule la plus avantageuse est celle de l affectation, et l UMI a été particulièrement bien servie lors des deux dernières années : sur un total de 38 affectations possibles au CNRS sur l ensemble des disciplines et des 35 UMI, l UMI CRM en a obtenu 3,5 en 2014 et Au total, ce sont 17 visiteurs qui ont pu ainsi venir, pour des durées majoritairement comprises entre six mois et un an. Il faut remarquer que l UMI rayonne bien au-delà de l INSMI : des membres d autres Instituts (INP pour les physiciens théoriciens, INS2I pour les combinatoristes) souhaitent visiter l UMI. Pour les visites de plus courte durée, les (co)financements sont gérés par l UMI. Ainsi, au premier janvier 2015, l UMI a financé directement (sur ses fonds propres) ou indirectement (sur projet ANR) vingt-deux visites à l UMI. Visites de membres de l UMI en France La plupart des membres de l UMI ont déjà des subventions individuelles qui leur permettent de voyager. L UMI fait donc face à une demande moins forte dans le sens Canada-France que dans le sens France-Canada. Notons que l obtention de programmes de l ANR permet d atténuer cette tendance. Ainsi l UMI a financé sept visites de courte durée, directe- BULLETIN CRM 20
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