Le Bulletin M MATHÉMATIQUES

C CENTRE R DE RECHERCHES Le Bulletin M MATHÉMATIQUES

Automne/Fall 2014 — Volume 20, No 2 — Le Centre de recherches mathématiques

Thematic Year on Number Theory from Arithmetic Statistics to Zeta Elements SMS 2014: Counting Arithmetic Objects

Organizers: Henri Darmon (McGill); Andrew Granville (Montréal); Benedict Gross (Harvard)

The Séminaire de Mathématiques Supérieures (summer school) on Counting Arithmetic Objects was held at the CRM from June 23 to July 4, 2014, and was by all accounts a resounding success. It was the first summer school to focus on the new approach to arithmetic geometry which has been led by Manjul Bhargava, and resulted in several extraordinary results, including that almost 2/3rds of elliptic curves have p-Selmer rank 0 or 1, and hence (thanks to an impressive panoply of deep results in the arithmetic of elliptic curves obtained over the last 30 years) satisfy the Birch–Swinnerton-Dyer conjecture (up to the 2-part of the Tate– Safarevic group). Much of the work in this area has come from Princeton and Boston, and one goal was to draw a much wider co- hort into the subject. The lecturers were specifically asked to help the listeners develop an intuition for the area, and not worry about all of the details. One difficulty in understanding the subject is that the published papers often focus on the more geometric intuition, though much of the (key) ring theory can be developed through Manjul Bhargava at the CRM in 2006, as Aisenstadt Chair (© Photo Réjean Meloche) simpler algebraic insights and combinatorial constructions. Again the lecturers were asked to bring these to the fore, and did so. The importance of the ideas presented at the summer school was dramatically underscored two months later in Seoul, where Man- Bhargava himself is a fantastic lecturer and enjoyed not only gear- jul Bhargava was awarded one of the four Fields medals at the ICM ing his own talks to a wider audience, but working with some of “for developing powerful new methods in the geometry of num- the other lecturers to appropriately focus their talks. The series by bers, which he applied to count rings of small rank and to bound Melanie Wood (on all of the rings involved) and by Arul Shankar the average rank of elliptic curves.” The CRM summer school will (on how to count the arithmetic objects involved) were master- be the object of a proceedings volume to which all of the key par- ful, and gave the listeners the opportunity to delve deep into this ticipants have enthusiastically agreed to contribute, and which important subject with a great sense of the intuition behind these will surely have a broad impact as one of the essential references techniques. Bjorn Poonen managed to explain quite a bit of chal- in an exciting and rapidly evolving part of number theory. lenging algebraic geometry in a way that brought everyone along, giving insight into conjectures on ranks, as well as the new ways For more information on activities related to the thematic year go of exploiting Chabauty’s method to determine all of the ratio- to http://www.crm.umontreal.ca/Number2014/. nal points on a given (high genus) curve. Takashi Taniguchi and Frank Thorne exposed the deeper analytic techniques, all the while IN THIS ISSUE / DANS CE NUMÉRO respecting that the majority of the audience were algebraists so needed to be carefully led through the meaning behind each step. The Aisenstadt Chair Lectures by MASAKI KASHIWARA. . 3 Dick Gross is one of the great arithmetic geometers of our time, SABIN CAUTIS, the 2014 André-Aisenstadt Prize recipient, and a fantastic expositor, as he once again showed, using the in- describes his research ...... 5 variant theory (as explained by Jerry Wang) to attack rational La Plate-forme d’innovation des instituts ...... 6 points on hyperelliptic curves. There were also excellent “one-off” The Magnificent Kerr Metric, by NIKY KAMRAN, 2014 lectures by Wei Ho, Jennifer Park, Eknath Ghate (though actually two “one-off” lectures) and Michael Stoll, on various exciting parts CRM-Fields-PIMS Prize winner ...... 7 of the theory. And a rather different perspective explained by the FLORIN DIACU et les incontournables mathématiques . . 11 always inspiring and entertaining Jordan Ellenberg. ALESSIO FIGALLI delivers the first CRM Nirenberg Each day was rounded off by an extensive problem session, led Lectures in Geometric Analysis...... 12 by some of the lecturers, challenging the participants to immerse The 2014–2015 CRM–ISM Postdoctoral Fellows...... 18 themselves in the details of the theory of that day. crm.math.ca ARTA III

Ibrahim Assem (Sherbrooke)

La troisième conférence ARTA (Advances in Representation Theory of Algebras) a eu lieu du 16 au 20 juin 2014 à l’UQAM. La théorie des représentations des algèbres est un des domaines les plus actifs et dynamiques des mathématiques, se développant en symbiose avec l’algèbre commutative, la géométrie algébrique, l’algèbre homologique, la théorie de Lie et depuis peu les algèbres amassées. Tous les deux ans a lieu quelque part dans le monde une conférence internationale (ICRA International Conference on Rep- resentations of Algebras). Toutefois, ni l’ICRA, ni les ateliers qui ont lieu plusieurs fois par an, ne suffisent pour dominer le sujet. En outre, à cause du manque de temps, certains points de vue sont moins bien traités que d’autres. Ainsi, le nombre considérable de participants aux ICRA oblige à tenir plusieurs sessions en paral- lèle, ce qui est frustrant tant pour les participants que pour les conférenciers, surtout les jeunes, dont les résultats n’obtiennent pas le retentissement souhaité. Le projet ARTA est une initiative de 4 chercheurs (A. Skowronski, de Pologne, J. A. de la Peña du Mexique, I. Assem de Sherbrooke et le regretté D. Happel d’Allemagne) afin de traiter plus complè- tement ces thématiques et de faire parler des jeunes chercheurs dans les meilleures conditions possibles. L’objectif est de tenir une conférence par an, la première a eu lieu au CIMAT (Centro de In- conjointe ont avancé et de nouvelles collaborations de recherche vestigación en Matemática) à Guanajuato, Mexique, en décembre ont émergé. Dans certains cas, de nouveaux résultats ont été prou- 2012, et la seconde à l’Université Nicolas Copernic à Torun, Po- vés. Ceci laisse prévoir à brève échéance de nouvelles publications. logne, en septembre 2013. La troisième a eu lieu à l’UQAM en juin 2014 et la quatrième est prévue à Guanajuato en juin 2015. Pour la conférence à l’UQAM, le comité scientifique était aidé d’un comité Frédéric Gourdeau lauréat du prix local formé de F. Saliola et C. Hohlweg de l’UQAM et de T. Brüstle et I. Assem de Sherbrooke, sans oublier l’appui logistique du La- Adrien-Pouliot 2014 de la SMC CIM et du CRM. Le professeur Frédéric Gourdeau de l’Université Laval est le lau- Les ARTA portent sur les sujets suivants : structure des algèbres réat 2014 du prix Adrien-Pouliot pour sa contribution exception- de dimension finie et de leurs catégories de modules, méthodes nelle à l’enseignement des mathématiques au Canada. Le prix sera et conjectures homologiques, algèbres de dimension globale finie remis à l’occasion de la Réunion d’hiver 2014 de la SMC à Ottawa. et auto-injectives, aspects combinatoires, méthodes de géométrie « Frédéric Gourdeau a marqué l’enseignement des mathématiques algébrique, catégories triangulées et théorie de l’inclinaison, rela- tant dans les classes primaires que les cours aux étudiants diplô- tions avec la théorie de Lie, la théorie des singularités, les modules més », précise Peter Taylor, président du Comité d’éducation de la de Cohen-Macauley, les groupes quantiques et autres structures SMC. « Il a de plus fait sa marque à la fois au sein de la très active algébriques. communauté qui s’intéresse à l’enseignement des mathématiques La conférence a rassemblé 70 participants provenant de 18 pays au Canada et sur la scène mondiale, comme en témoigne son invi- différents, dont plus de la moitié d’étudiants et jeunes chercheurs. tation à prononcer une prestigieuse conférence au Congrès inter- Au total, 35 exposés ont été donnés, dont 13 de 50 minutes et national 2012 à Séoul. » le reste de 30. Plusieurs exposés (dont 5 de 50 minutes) ont été Dynamique et passionné, Frédéric Gourdeau a joué un rôle im- confiés à des jeunes chercheurs. Parmi les questions abordées fi- portant dans plusieurs projets d’envergure au pays, dont la créa- gurent l’étude des algèbres auto-injectives et de leurs catégories tion en 2006 de la revue Accromαth. Il a fondé en 1997 l’Asso- de modules, et les relations entre théorie des représentations et ciation québécoise des jeux mathématiques (AQJM), qu’il préside algèbres amassées. depuis. L’AQJM organise annuellement le concours mathématique Les participants à l’ARTA se connaissant tous de longue date, des attirant la plus forte participation au Québec. échanges mathématiques ont eu lieu en marge de la conférence tout au long de celle-ci. Plusieurs travaux entamés de recherche Extraits de l’annonce faite par la Société mathématique du Canada.

BULLETIN CRM–2 crm.math.ca

Thematic Semester on New Directions in Lie Theory Aisenstadt Chair: Masaki Kashiwara (RIMS, Kyoto)

Matthew Bennett (UE Campinas); Daniele Rosso (UC Riverside)

Professor Masaki Kashiwara, of the Research Institute for Math- GLN ) indexed by partitions λ of ` with at most N parts. The as- N ⊗` ematical Sciences in Kyoto (RIMS) gave a series of three lectures signment M 7→ M ⊗ (C ) for an S`-module M is an exact as Aisenstadt chairholder during the workshop on Combinatorial functor, and induces an equivalence of categories between finite- Representation Theory which was part of the thematic semester dimensional S`-modules and GLN -modules when ` ≤ N. One New Directions in Lie Theory. Professor Kashiwara has received would like to understand this as a statement about the actions of many honours throughout his long career, to recognize the fun- two associative algebras. We recall that the representation theory damental contributions he made to several fields of mathematics, of S` and its group algebra are the same. Similarly, the finite- including algebraic analysis, microlocal analysis, D-module the- dimensional representation theory of GLN is the same as that of ory, Hodge theory and representation theory. Of particular rel- the Lie algebra slN , which is equivalent to that of its universal en- evance to the topics discussed during the workshop is his work veloping algebra U(slN ), which is an associative algebra. In this developing the theory of crystal bases for integrable modules of case, we can understand U(slN ) as acting on the tensor product quantum groups. These crystal bases contain most of the infor- by taking advantage of its comultiplication, which is cocommuta- mation of the module and can be represented by a directed graph tive. Schur–Weyl duality can then be interpreted as saying that with edges labelled by simple roots, hence directly relating the the images of these two associative algebras in End((CN )⊗`) are study of quantum groups to combinatorial ideas. each other’s centralizers. Jimbo generalized the above construction in 1986 to a result con- ⊗` cerning the action of the quantum group U q(slN ) on V where V is the natural representation of U q(slN ). The quantum group U q(slN ) can be considered as a deformation of the universal en- veloping algebra U(slN ) by a parameter q, so that if we set q = 1 we recover the original algebra. This deformation results in non- cocommutative comultiplication. This means that if M and N are representations, then the isomorphism M⊗N → N⊗M is not induced by simply permuting factors, but instead comes from an important transformation called the universal R-matrix. Using this R-matrix, Jimbo defined an action of the Hecke algebra Hq(`), ⊗` a deformation of the group algebra of S`, on V . His theorem says that the two actions commute and that a decomposition into simple modules with respect to one action determines the decomposition for the other action. The assignment M 7→ M ⊗ V ⊗`, Masaki Kashiwara for an Hq(`)-module M, is exact, and, when ` ≤ N, it induces an Professor Kashiwara’s first lecture was based on recent work that equivalence of categories between finite-dimensional Hq(`) mod- he did with Professors S. Kang of the Seoul National University ules and U q(slN ) modules of level `. We can consider this result and M. Kim of the Korea Institute for Advanced Study, which is as a quantized Schur–Weyl duality. the latest in a series of results modelled after the famous representation theoretic result called Schur–Weyl duality. This duality is a There are affine versions for the previous results provided by beautiful classical result concerning the representation theory of Chari–Pressley, Cherednik and Ginzburg–Reshetikhin–Vasserot at various times in the mid 1990s. The affine quantum group the general linear group GLN and the symmetric group S`, which N ⊗` U (sl ) can be considered as a deformation of the universal en- both act on the space (C ) . The group S` permutes the tensor q dN N ±1 ⊗` factors, and C is the natural representation for GLN , which acts veloping algebra for the loop algebra slN ⊗C[t ]. It acts on Vb , diagonally on the ` factors by simultaneous multiplication. Schur– where Vb is the natural representation, and one is interested in Weyl duality says that the actions of the two groups commute, and a universal R-matrix to intertwine tensor products of represen- that if we decompose the tensor products in terms of simple mod- tations. An action by the affine Hecke algebra Hˆq(`) (which is ±1 ±1 ules for one group, that will also determine the decomposition for a semidirect product of Hq(`) with C[t1 , . . . , tn ]) is defined on the other group. More precisely, we have a decomposition as a Vb ⊗` using the R-matrix, and it was proved that these actions com- (S`, GLN )-bimodule mute and that the algebras satisfy the double centralizer property. N ⊗` M ⊗` (C ) ' (Uλ ⊗ Wλ), Once again the assignment M 7→ M ⊗ Vb for a Hbq(`)-module λ M is exact, and when ` ≤ N it induces an equivalence of cat- where Uλ (respectively Wλ) are the simple modules of S` (resp.

BULLETIN CRM–3 crm.math.ca egories between finite-dimensional Hbq(`) modules and U q(sldN ) polynomial parameters this corresponds to the upper global basis modules of level `. These results can be considered affinizations of U −(g)∨, which was introduced by professor Kashiwara him- Z of the classical Schur–Weyl duality and its quantum version. self. In general, one cannot hope that this correspondence of bases will always hold, since there are known counterexamples. How- There is a generalization of the affine Hecke algebra, calleda ever, the lecture concluded with a generalization of this theorem. quiver Hecke algebra, which was introduced by Khovanov–Lauda The new result presented by Professor Kashiwara was that, aslong and Rouquier to categorify (the lower part of) a quantum group. as g is given by a symmetric generalized Cartan matrix, for any After these authors’ initials, they are also called KLR algebras. generic choice of parameters Q = {Q } (any choice within an They are defined for each positive root β in an affine root system i,i i,j open dense subset of the space of parameters) the basis of simple and hence usually denoted by R(β). In his talk, Prof. Kashiwara modules does correspond to the upper global basis. described a version of Schur–Weyl duality in which R(β) takes the role of the affine Hecke algebra above. The other algebra he Conférence de la chaire Aisenstadt sur la used was a generalization of the affine quantum group, which has its own version of an R-matrix. Prof. Kashiwara described how to correspondance de Riemann–Hilbert pour les use this R-matrix on tensor products of representations and de- D-modules holonomes non nécessairement fined a module Vbβ which admits actions by both the quiver Hecke réguliers algebra and the quantum group, and concluded that the functor David Hernandez (Paris-Diderot) M 7→ M ⊗ Vbβ induces an equivalence of categories. Dans le cadre de sa chaire Aisenstadt, le professeur Masaki Ka- The second lecture by Professor Kashiwara, like his first one, dis- shiwara (Kyoto) a donné une troisième conférence destinée à un cussed Khovanov–Lauda–Rouquier algebras. In this instance the auditoire plus large que celui des seuls participants du programme focus was on categorification. In general, the philosophy of cate- «Nouvelles avenues en théorie de Lie». En présentant sa preuve gorification is to replace elements in a set and functions with ob- (avec Andrea D’Agnolo) de la correspondance de Riemann–Hilbert jects in a category and functors. This leads to a sort of ‘higher- pour les D-modules holonomes non nécessairement réguliers, il a level mathematics’ that sees classical concepts as projections or montré la très grande étendue des domaines des mathématiques ‘shadows’ of a richer structure. For example, the main motivation dans lesquels il a apporté des contributions importantes, mais for introducing KLR algebras was a famous categorification the- aussi l’unité de ses travaux. Si le théorème principal pour cette cor- orem of Khovanov–Lauda and Rouquier that relates the category respondance de Riemann–Hilbert ne semble pas avoir aujourd’hui of modules over these algebras with quantum groups associated d’application directe en théorie des représentations et de Lie, il est to Lie algebras. For a positive root β in a Kac–Moody root system, fort possible que ce soit le cas à l’avenir. On sait en effet l’impor- the algebra R(β) is defined explicitly in terms of three families tance des D-modules dans ce contexte (notamment pour la preuve of generators xk, τj and eν where 1 ≤ k ≤ n, 1 ≤ j ≤ n − 1 de conjectures de Kazdhan–Lusztig). and ν ∈ Iβ (here Iβ is an index set that keeps track of how many e ways there are to write β as a sum of simple roots). These gen- Le problème original de Riemann–Hilbert (généralisation du 21 problème de Hilbert) concerne des équations différentielles li- erators satisfy certain relations, in particular the xk’s commute néaires ordinaires with each other, the eν’s are orthogonal idempotents and the τj’s behave similarly to the generators of the Hecke algebra Hq(`). d m a (z) + ··· + a (z) f(z) = 0 The exact form of these relations depends on the choice of afam- m dz 0 ily of polynomials in two variables Q = {Qi,j(u, v)}i,j, so it is more precise to denote the algebra by R(Q)β. By assigning ap- d’inconnue f(z) définie sur un ouvert complexe. Une telle équa- propriate degrees to the generators we even get a graded algebra tion peut avoir des singularités régulières, c’est-à-dire des points β γ structure. Given two positive roots and , there is an inclusion z0 tels que am(z0) = 0 et R(Q)β ⊗ R(Q)γ ,→ R(Q)β+γ, hence by taking induced modules deg aj(z) ≥ deg am(z) − (m − j) pour 0 ≤ j ≤ m. we get a functor z=z0 z=z0 On suppose que toutes les singularités de l’équation sont régu- R(Q) -gmod × R(Q) -gmod → R(Q) -gmod, β γ β+γ lières, c’est-à-dire qu’il n’y a pas d’autres points d’annulation de where -gmod denotes the category of finite-dimensional graded am. Les solutions de l’équation différentielle produisent alors un modules. By exactness, this functor induces a convolution système local sur le plan complexe privé des singularités régu- K(R(Q) ) := operation on the Grothendieck group -gmod lières (c’est la théorie de la monodromie). La correspondance de L K(R(Q) ) β β-gmod . The categorification theorem then says Riemann–Hilbert peut être vue comme l’étude d’une construction K(R(Q) ) that there is an isomorphism of algebras between -gmod réciproque : pour S un sous-ensemble fini d’une courbe complexe with this convolution operation and the dual of the integral form X, il s’agit d’établir une correspondance entre les systèmes locaux of the lower part of the quantum group U −(g)∨. Z sur X\S et certaines équations différentielles dont les singularités Now, the Grothendieck group K(R(Q)-gmod) has a distinguished régulières sont dans S. basis given by the classes of the simple modules, hence a natural Plus généralement, considérons en langage moderne une va- question to ask is what this basis corresponds to under the iso- riété complexe X. Les équations différentielles avec singularités morphism. In the symmetric case, it is a theorem of Varagnolo– (suite à la page 10) Vasserot and Rouquier that for a specific choice of the family Q of BULLETIN CRM–4 crm.math.ca

2014 André Aisenstadt Prize in Mathematics Sabin Cautis (University of British Columbia)

The International Scientific Advisory Committee of the Centre de recherches mathématiques (CRM) is happy to announce that Sabin Cautis from the University of British Columbia is the 2014 André Aisenstadt Prize recipient. As an undergraduate student at the University of Waterloo, Sabin Cautis was a Putnam Fellow and a member of the winning team in the 1999 Putnam competition. After receiving a Bachelor of Mathe- matics degree in 2001, he continued his education at Harvard University, where he obtained his Ph.D. in 2006 under the supervision of Joe Harris. Before joining the Mathematics Department at UBC, Dr. Cautis held postdoctoral positions at Rice University and MSRI and was Assistant Professor at Columbia University and the University of Southern California. In 2011–2013 he was a recipient of the highly coveted Alfred P. Sloan Fellowship. Sabin Cautis works at the crossroads of algebraic geometry, representation theory, and low- dimensional topology. In his earlier work (joint with the 2011 André Aisenstadt Prize recipient, Joel Sabin Cautis Kamnitzer), he developed a new approach to Khovanov’s knot invariants, which uses algebraic geometry and is inspired by mirror symmetry. Dr. Cautis is a world leader in the area of categorification. His results are expected to have a lasting impact on the field and lead to important developments in low-dimensional topology, the geometric Langlands program, and the mathematical aspects of quantum physics. In particular, his recent work with Anthony Licata on categorification of Heisenberg algebras and vertex operators is a major step in the direction outlined by Igor Frenkel towards categorification of conformal field theory. Dr. Cautis contributed the following description of his work.

In geometric representation theory one often applies tools from involving the categories DCoh(T ?G(k, N)) for all 0 ≤ k ≤ N. algebraic geometry to questions in representation theory. For ex- Here E and F are functors analogous to the usual generators 0 1 0 0 ample, the Kazhdan–Lusztig conjectures, proven in the early 1980s e = ( 0 0 ) and f = ( 1 0 ) of sl2. The word “categorical” refers by Beilinson–Bernstein and Brylinski–Kashiwara, involved trans- to the fact that the weight spaces are categories rather than vector lating the conjectures into more geometric language consisting of spaces. D-modules on flag varieties. L Now consider a representation V = ` V (`) of sl2 where V (`) In much of my work we invert this perspective and also intro- are the weight spaces. A fundamental result in representation 0 1 duce a categorical ingredient. Instead of using geometric tools, theory tells us the element −1 0 ∈ SL2(C), which can be ex- such as D-modules, to obtain results in representation theory we pressed in terms of es and fs, induces a natural isomorphism ∼ apply techniques in (categorical) representation theory to under- V (`) −→ V (−`). stand geometric situations. One could call this categorical geo- It turns out this result has a categorical analogue. Consider an metric representation theory. L sl2 action on categories C = ` C(`) (in our example above ? For example, in a project with Kamnitzer and Licata we con- C(`) = DCoh T G(k, N) where ` = 2k − N). One can de- structed a natural derived equivalence fine a complex of functors built from compositions of Es and ∼ ? ∼ ? Fs which lifts the isomorphism V (`) −→ V (−`) to an equiva- DCoh T G(k, N) −−→ DCoh T G(N − k, N) . (1) ∼ lence C(`) −→ C(−`). A version of this result had been used Here T ? (k, N) denotes the cotangent bundle to the Grassman- G by Chuang–Rouquier to prove that certain blocks of symmetric nian of k-planes in N while DCoh(X) denotes the (derived) cate- C groups in positive characteristic are derived equivalent. Adapting gory of coherent sheaves on X. Notice that we have T ? (k, N) =∼ G their method to our geometric situation above gave us the equiv- T ? (N − k, N) but this isomorphism is not canonical since it G alence in (1). requires choosing a bilinear form on CN . Constructing a natural equivalence as in (1) is surprisingly difficult. Earlier work of In subsequent work with Kamnitzer we showed that a categorical Namikawa had shown that an obvious guess does not induce an sln action gives rise to an action of the braid group. This is another equivalence while subsequent work of Kawamata had conjectured result in categorical (also known as higher) representation theory a detailed form of these equivalences when k = 2. which can be used in geometry to obtain, for instance, an action of the braid group on partial flag varieties. Such a braid group action To construct the equivalence in (1) we first had to define a cate- can also be used to construct homological knot invariants, such as gorical sl action 2 Khovanov homology. E E ? ? ··· o / DCoh T G(k − 1,N) o / DCoh T G(k, N) In more recent work with Licata we tried to develop the cate- F F E E gorical representation theory of Heisenberg algebras and vertex ? o / DCoh T G(k + 1,N) o / ··· (continued on page 16) F F

BULLETIN CRM–5 crm.math.ca La Plate-forme d’innovation des instituts (PII) et le Comité consultatif du CRM pour la PII

Odile Marcotte, directrice adjointe – partenariats

Le CRM et les deux autres instituts de mathématiques canadiens Consortium de recherche et d’innovation en aérospatiale au Qué- (le Fields Institute et le PIMS) ont reçu cet été une nouvelle sub- bec (CRIAQ), un organisme sans but lucratif dont la mission est vention du CRSNG, appelée Plate-forme d’innovation des insti- d’accroître la compétitivité de l’industrie aérospatiale et d’amélio- tuts. L’objectif du CRSNG (pleinement partagé par les trois ins- rer la base des connaissances collectives dans ce secteur grâce à tituts eux-mêmes) est que les instituts utilisent cette subvention une meilleure formation des étudiants. pour susciter de nouvelles collaborations entre des chercheurs en M. Pierre Trudeau vice-président, développement des affaires, sciences mathématiques et des entreprises, notamment grâce à des GIRO ateliers de résolution de problèmes, des ateliers de modélisation mathématique pour les entreprises, des ateliers de maillage uni- Après des études de recherche opérationnelle, M. Trudeau fonda, versités/entreprises centrés sur des domaines précis des mathéma- avec quatre autres personnes, une compagnie spécialisée dans la tiques, des formations de courte durée et des forums de l’emploi. planification des horaires d’équipages aériens (la compagnie AD OPT). Il travailla ensuite pendant plusieurs années pour la compa- Pour l’aider à atteindre les objectifs de la PII, le CRM a constitué gnie ILOG, comme directeur de la commercialisation de ses pro- un comité de scientifiques occupant des postes de gestion dans les duits d’optimisation. M. Trudeau est maintenant vice-président de entreprises. Ce comité, le Comité consultatif du CRM pour la Plate- GIRO, une compagnie fournissant aux sociétés de transport public forme d’innovation des instituts (CCPII), donnera des conseils à la et aux services postaux des logiciels pour planifier, optimiser et gé- direction du CRM sur les meilleures façons de développer les col- rer les activités des véhicules et de leur personnel. GIRO a des liens laborations universités/entreprises dans le domaine des sciences étroits avec les chercheurs en optimisation et a présenté plusieurs mathématiques. Il sera impliqué dans la planification des activités problèmes aux ateliers de résolution de problèmes organisés par mentionnées ci-dessus et jouera un rôle de premier plan dans le le CRM. lancement de nouvelles activités. Il participera à des échanges avec les comités semblables créés par les autres instituts canadiens. Son Mme Roxana Zangor directrice de la sécurité et de la fiabilité des président fera un rapport annuel au Conseil d’administration du produits, Pratt & Whitney Canada CRM. Voici la liste des membres du CCPII. Après des études de mathématiques en Roumanie, Mme Zangor re- çut une bourse Fulbright et obtint une maîtrise en enseignement M. Michel Carreau directeur des énergies renouvelables, Hatch des mathématiques et un doctorat en mathématiques appliquées M. Carreau obtint son doctorat en physique du Massachusetts Ins- de l’Université de Syracuse aux États-Unis. Elle travailla d’abord titute of Technology en 1991 et effectua un stage postdoctoral à comme scientifique senior au United Technologies Research Cen- la Boston University de 1991 à 1993. Après avoir travaillé pour ter (UTRC), dans l’état du Connecticut, où elle fit profiter le centre Synexus Global, M. Carreau accepta un poste chez Hatch, une de son expertise en méthodes numériques, forage de données et compagnie fournissant (entre autres) des services-conseils en pro- traitement d’images (entre autres domaines). Depuis 2006 elle tra- cédés et en affaires et des services à l’exploitation pour l’indus- vaille chez Pratt & Whitney Canada, une entreprise de classe mon- trie des mines et métaux, de l’énergie et des infrastructures. Chez diale en conception, manufacture et entretien d’aéronefs. Hatch M. Carreau dirige une équipe d’ingénieurs performante qui conçoit et implante des projets d’énergie renouvelable (éolienne, De plus, le CRM a embauché M. Stéphane Rouillon comme agent solaire, géothermique ou hydro-électrique), incluant aussi des so- de développement de partenariats pour soutenir ses efforts en ma- lutions hybrides. M. Carreau collabore régulièrement avec des thématiques industrielles, particulièrement les efforts reliés à la chercheurs universitaires. PII. M. Rouillon a étudié à l’École Polytechnique de Montréal et détient un baccalauréat en génie physique, une maîtrise en mé- M. Denis Faubert président-directeur général, CRIAQ canique des fluides et un doctorat en recherche opérationnelle de M. Faubert détient une maîtrise et un doctorat en physique cette institution. Il a acquis une vaste expérience industrielle en de l’Université Laval. Après avoir travaillé pour plusieurs orga- travaillant pour diverses entreprises, en particulier SNC Lavalin, nismes, dont le Centre de recherche pour la défense, M. Faubert et a réalisé plusieurs projets de vulgarisation scientifique et mathé- fut nommé directeur général de l’IREQ, poste qu’il occupa de matique. M. Rouillon travaillera au CRM en étroite collaboration 2007 à 2014. Il est maintenant le président-directeur général du avec Odile Marcotte, directrice adjointe aux partenariats.

BULLETIN CRM–6 crm.math.ca The magniﬁcent Kerr metric

Niky Kamran (2014 CRM–Fields–PIMS Prize Winner) Niky Kamran, James McGill Professor at McGill University, is the recipient of the 2014 CRM–Fields–PIMS Prize. Over the past months he has delivered his prize lectures at PIMS, CRM and Fields, and has contributed the following expository article on topics related to his work. Introduction Space-time in Einstein’s General Theory of Relativity is represented by a 4-dimensional smooth manifold, endowed with a metric of Lorentzian signature which satisfies a set of field equations describing the gravitational field through space-time curvature. These equations, which were obtained in their definitive formby Einstein in 1915, are given in the absence of sources by

Rij = 0, (1) where the Rij denote the components of the Ricci curvature of space-time. The equations (1) constitute a system of ten coupled second-order non-linear hyperbolic partial differential equations whose study poses major challenges from both the geometrical and analytical perspectives. It is thus a remarkable fact that there exist families of exact closed-form solutions of (1) which have at the same time considerable physical significance and deep mathe- Niky Kamran (on the left) and Luc Vinet (on the right) matical properties. An example of central importance is given by The Kerr metric enjoys important symmetry properties that we the two-parameter family of Kerr metrics, which describe the ex- will review in this and the next section. It is first of all mani- terior gravitational field of the final state of an isolated object that fest that it admits the two commuting Killing vector fields ∂φ and has collapsed to form a rotating black hole. This metric, which was ∂t corresponding to the invariance of the space-time geometry discovered in 1963 by Roy Kerr, starting from geometrical rather under rotations around the symmetry axis of the black hole and than physical assumptions, is given by time translations. It is also easy to see that the generator of time ∆ dr2 translations is time-like at sufficiently large distances, but that it ds2 = (dt − a sin2 θ dφ)2 − U + dθ2 U ∆ changes causal type and becomes space-like as one gets closer to the event horizon r = r in a region with compact closure, the sin2 θ 1 − (a dt − (r2 + a2) dφ)2, (2) ergosphere, which is defined by the inequality U where r2 − 2Mr + a2 cos2 θ < 0. (6) U := r2 + a2 cos2 θ, ∆ = r2 − 2Mr + a2. (3) This metric depends on two real parameters M and a which we One of the consequences of the presence of the ergosphere is that assume to satisfy the inequality any conserved energy arising from the invariance of the geome- M 2 > a2. (4) try under time translations may change sign and become negative 2 2 within the ergosphere. This makes the analysis of wave equations The limiting case a = M defines the so-called class of extreme and in particular the proof of energy decay results in Kerr geom- Kerr solutions. The physical interpretation of the parameters M etry challenging since the standard approaches to these problems and a is that they correspond respectively to the mass and angu- typically require the presence of a positive energy, giving rise to lar momentum per unit mass of the black hole, measured from a Hilbert space setting in which to solve the initial value prob- infinity. The coordinates in the metric (2) are defined in the range 2 lem and study energy propagation. In the limiting case of the r > r1, −∞ < t < +∞, (θ, φ) ∈ S , where Schwarzschild metric, that is when a = 0, the ergosphere is empty p 2 2 r1 := M + M − a , (5) and the conserved energy is positive outside the event horizon. but the coordinate r can be extended across the locus r = r1, The physical importance of the family of Kerr metrics is under- which becomes a null hypersurface corresponding to the global scored by the celebrated black hole uniqueness theorems of Carter past event√ horizon for the Kerr metric. The hypersurface r = and Israel [5, 23], which characterize this metric as the unique 2 2 r0 := M − M − a is the Cauchy horizon for the metric. In the set of solutions of the boundary value problem corresponding to limit a = 0, the Kerr solution reduces to the well-known spheri- a black hole equilibrium state. It should be remarked that the cally symmetric Schwarzschild solution, discovered in 1915. proof of the original version of the black hole uniqueness theorem depends critically on analyticity assumptions on the space- See http://crm.math.ca/prix/prixCRMFieldsPIMS/ time geometry. These analyticity assumptions have been lifted prixCFP14_an.shtml for the full citation. and significantly weakened in important recent work of Alexakis,

BULLETIN CRM–7 crm.math.ca Ionescu and Klainerman [1], where uniqueness theorems have Decay of Dirac spinors been proved using very different methods. The long term dynamics of Dirac spinors in the Kerr metric were investigated in a series of joint papers with Felix Finster, Joel Separability properties Smoller and Shing-Tung Yau [13, 14, 19]. Our approach used the The Kerr metrics enjoy an extraordinary set of separability proper- separability of the Dirac equation into ordinary differential equa- ties, that led Chandrasekhar to write in [7] that “they have the aura tions to construct an integral spectral representation for the solu- of the miraculous.” These separability results were first discovered tion of the Cauchy problem. We state a couple of theorems in some for the Hamilton-Jacobi equation for the geodesic flow and for the detail, beginning with the solution of the Cauchy problem [19]. Klein–Gordon equation by Carter in 1968, and were subsequently ∞ 2 4 Theorem 1. For every Cauchy data ψ0 ∈ C0 (R × S ) there is a shown to hold by Teukolsky in 1972 for the equations govern- unique solution to the Cauchy problem for the Dirac equation ing the components of maximal spin weight of linearized grav- j itational fields, test Maxwell fields, and test Dirac fields of zero (iγ Dj − m)ψ(t, x) = 0, ψ(0, x) = ψ0(x), (11) rest-mass (Weyl neutrinos). The separability of the massive Dirac given by a (convergent) Fourier decomposition of the form equation in Kerr was proved in 1976 in a remarkable paper by Chandrasekhar, using a different approach from Teukolsky’s. We ψ(x, t) refer to Chandrasekhar’s treatise [8] for a detailed account of all Z +∞ 2 1 X −iωt X kωn kωn kωn these results, which show that every wave equation of physical = e tab ψa (x)hψb | ψ0i dω. (12) π −∞ interest in Kerr geometry can be completely separated into a set k,n∈Z a,b=1 of ordinary differential equations governing individual modes for The integer k is the quantum number corresponding to the projec- the waves being described. It is important to remember that the tion of the angular momentum on the axis of symmetry of the Kerr complete separability of these wave equations cannot be simply metric and n is a generalized total angular momentum quantum be accounted for by the isometries of the Kerr metric. This can number arising from the separation of variables. The coefficients kωn kωn be easily understood when thinking of the complete separabil- tab are generalized transmission coefficients and the ψa are ity in the Staeckel sense of the Hamilton–Jacobi equation for the solutions of the Dirac equation which arise from the separation of geodesic flow, which leads in turn to the complete integrability of variables and behave for |ω| > m like incoming spherical waves the geodesic flow itself. Indeed, there exists, in addition to the con- for a = 1 and outgoing spherical waves for a = 2. For |ω| < m, kωn served linear momenta arising from the commuting Killing vector the ψa are linear combinations of both incoming and outgoing fields ∂φ and ∂t, a quadratic first integral of the geodesic flow, spherical waves near the event horizon. which was discovered by Carter. It is of the form We now consider the long-term dynamics of the solution to the i j Cauchy problem [19]. K = Kijp p . (7) Theorem 2. Consider the Cauchy problem for the Dirac equation in K 2 2 4 where the symmetric tensor ij appearing in the above expres- the Kerr geometry, where the initial data ψ0 is in L ((r1, ∞)×S ) sion is therefore a Killing tensor. The first integral K is deeply ∞ and in Lloc near the horizon, i.e., |ψ0(x)| < c for x ∈ (r1, r1 + ) × 2 connected to all the separability properties of the Kerr metric. It S . Let δ > 0 be given, let R > r1 + δ and consider the compact has a very special structure, as it can be factored into the square space-like hypersurface Kδ,R of E given by of a skew-symmetric Killing–Yano tensor fij, Kδ,R := {(t, r, θ, φ) | r1 + δ ≤ r ≤ R, t = const.}. (13) k Kij = fikf j, (8) Then the probability for the Dirac particle to be inside Kδ,R tends to satisfying zero as t → ∞, that is Z ∇ifjk + ∇jfik = 0. (9) ¯ j lim (ψγ ψ)(t, x) νj dµ = 0, (14) t→∞ While it can be seen quite directly from the separation of vari- Kδ,R ables that the symmetry operator K for the Klein–Gordon opera- where νj denotes the future-pointing normal to Kδ,R and dµ denotes tor given by the induced volume element on Kδ,R. ij K = ∇iK ∇j (10) Theorem 2 implies that the Dirac spinor ψ tends to zero in L∞ admits the separated solutions of the Klein–Gordon equation in loc as t → ∞, or equivalently that the Dirac particle must eventu- Kerr as eigenfunctions, the relationship between the separability ally either disappear into the black hole, or escape to infinity. One of the Dirac equation and the Killing tensor (8) is more hidden. It is may like to compute the probability of these outcomes in terms of Carter and McLenaghan [6] who proved a remarkable result stat- the Cauchy data, as well as the actual rate of decay of the Dirac ing that the Killing–Yano tensor admitted by the Kerr geometry spinor in a compact spatial. These questions were studied in[14], gives rise to an operator that commutes with the Dirac operator assuming that only finitely many angular modes are present in the in Kerr and admits the separated solutions as eigenfunctions. In initial data, meaning that the following section we show how the separability leads to decay results for solutions of the Dirac equation in the Kerr metric. |k| ≤ k0, |l| ≤ l0, (15) BULLETIN CRM–8 crm.math.ca in the Fourier expansion (12). Note that this finiteness assumption everywhere within the ergosphere, making it possible to define is not required in Theorem 2. a positive-definite inner product that is compatible with the sep- We now summarize the main results for the rates of decay, and aration of variables and conserved throughout the evolution. For refer to [14] for the probability estimates. the massless scalar wave equation, no such inner product is available and one has to deal with the non-positivity of the conserved Theorem 3. Consider the Cauchy problem as in Theorem 2, with energy. These issues were addressed in[15,16]. In [15], we proved initial data normalized by hψ0 | ψ0i = 1. Suppose that the bound- that there exists a convergent integral spectral representation for edness condition (15) on the angularmodes of the Cauchy data are the solution of the Cauchy problem for the massless scalar wave satisfied. Then we have: equation with C∞ compactly supported data. The proof used the spectral theory of definitizable operators on Pontrjagin spaces, (i) If for any k and n, together with resolvent estimates needed to deform contours to kωn within an arbitrarily small neighbourhood of the real axis. The lim sup|hψ2 | ψ0i| 6= 0, (16) ω&m deformation to the real axis and the corresponding decay results were established in [16] using improved estimates, but at the cost or kωn of dealing with modes of fixed azimuthal frequency lim inf|hψ2 | ψ0i| 6= 0, (17) ω%−m ψ(t, r, θ, φ) = eikφu(t, r, θ). (19) then, as t → ∞, The very important mode stability theorem of Whiting [31] played |ψ(x, t)| = ct−5/6 + O(t−5/6−), (18) a crucial role in our analysis. The proof of decay results for general where c = c(x) 6= 0, and < 1/30. data rather than the special ones satisfying (19) requires Morawetz multiplier techniques, which are more robust that the ones based kωn (ii) If for all k, n and a = 1, 2, hψa | ψ0i = 0 for all ω in a on separation of variables. For slowly rotating black holes de- neighborhood of ±m, then for any fixed x, ψ(x, t) decays rapidly scribed by Kerr metrics with |a| M, a proof of decay was given in t. by Dafermos and Rodnianski [9]. Recently, Dafermos, Rodnian- ski and Shlapentokh-Rothman [10] achieved a breakthrough by It should be noted that the rate of decay obtained in Theorem 3 is proving decay for scalar waves in the full range a2 < M 2. Other −3/2 slower than the rate of decay of t for the solutions of the Dirac important contributions to this problem are due to Tataru [28], equation in Minkowski space, corresponding to compactly sup- Tataru and Tohaneanu [29], and Andersson and Blue [2] (who ported initial data, [14]. We also briefly mention a result [13, 14] made use of the symmetry associated to the Killing tensor Kij on the nonexistence of time-periodic solutions of the Dirac equa- in their approach to decay). Finally, we mention that the very tion: interesting problem of superradiance for scalar wave packets in Kerr geometry can be studied rigorously using the previous the- Theorem 4. The Dirac equation does not admit any time-periodic orems [17, 18]. A key result that makes this analysis possible L2 solutions in any Kerr solution with a2 < M 2. is the time-independent energy estimate proved by Finster and Smoller [20] for outgoing massless scalar waves in Kerr. Theorem 4 is of a different nature from the preceding ones, in the sense that is considers solutions which are supported across the event horizon, and thus defined in a weak sense. It is also notewor- Some perspectives thy that the conclusion of Theorem 4 does not hold in the extreme Kerr case a2 = M 2. Indeed, it was shown in [26] that the Dirac We conclude this survey by listing a few perspectives. This list is equation admits bound state solutions in the extreme Kerr met- definitely nonexhaustive: ric. We conclude this section on Dirac spinors by mentioning the • The decay problem for solutions of the Klein–Gordon equation important results of Häfner [21] on fermion creation in the Kerr governing massive scalar fields in Kerr geometry, as opposed to black hole, and Häfner and Nicolas on the scattering of massless massless ones, can lead to different conclusions. For example, Dirac fields [22]. there exist exponentially growing finite-energy solutions of the Klein–Gordon equation in Kerr; see [27]. Earlier results on the Decay of massless scalar waves Klein–Gordon equation in Kerr can also be found in [4]. It would Even though the Kerr metric admits an ergosphere where the gen- be of definite interest to further investigate the behaviour of mas- erator of time translations becomes space-like, it turns out that sive fields in Kerr. we were nevertheless able to obtain the convergent Fourier inte- • The Kerr metric admits higher-dimensional generalizations gral representation for the solutions and the corresponding de- known as the Myers–Perry metrics. One can prove decay results cay results that we just reviewed in the previous section for Dirac for Dirac spinors in the 5-dimensional Myers–Perry metric [11], spinors in a Hilbert space setting. This is because the Dirac equa- but there are still many open problems that deserve further inves- tion admits a conserved current of order zero that is positive tigation in this higher-dimensional setting, especially the study of throughout the exterior of the event horizon, and in particular wave equations.

BULLETIN CRM–9 crm.math.ca

• The family of Kerr metrics admits generalizations that include a [21] D. Häfner, Creation of fermions by rotating charged black holes, Mém. Soc. cosmological constant, the so-called Kerr–de Sitter corresponding Math. Fr. (N.S.) 117 (2009). to a positive cosmological constant and Kerr–anti-de-Sitter met- [22] D. Häfner and J.-P. Nicolas, Scattering of massless Dirac fields by a Kerr black rics corresponding to a negative one. There are interesting recent hole, Rev. Math. Phys. 16 (2004), no. 1, 29–123. [23] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cam- developments in the study of energy decay for waves in the Kerr– bridge Monogr. Math. Phys., vol. 1, Cambridge Univ. Press, London–New AdS metric [24, 25], but there remain questions worthy of further York, 1973. investigation. [24] G. Holzegel and J. Smulevici, Quasimodes and a lower bound on the uniform energy decay rate for Kerr–AdS spacetimes, available at arXiv:1303.5944. • More generally, the well-posedness of the singular initial- [25] G. Holzegel and C. M. Warnick, Boundedness and growth for the massive wave boundary value problems wave equations, both linear and non- equation on asymptotically anti-de Sitter black holes, J. Funct. Anal. 266 (2014), linear, that arise in asymptotically AdS manifolds, is a subject that no. 4, 2436–2485. needs further study in view of its relation to the AdS/CFT corre- [26] H. Schmid, Bound state solutions of the Dirac equation in the extreme Kerr ge- spondence. Recent results can be found in [12, 30]. ometry, Math. Nachr. 274/275 (2004), 117–129. [27] Y. Shlapentokh-Rothman, Exponentially growing finite energy solutions for the [1] S. Alexakis, A. D. Ionescu, and S. Klainerman, Uniqueness of smooth station- Klein–Gordon equation on sub-extremal Kerr spacetimes, Comm. Math. Phys. ary black holes in vacuum: small perturbations of the Kerr spaces, Comm. Math. 329 (2014), no. 3, 859–891. Phys. 299 (2010), no. 1, 89–127. [28] D. Tataru, Local decay of waves on asymptotically flat stationary space-times, [2] L. Andersson and P. Blue, Hidden symmetries and decay for the wave equation Amer. J. Math. 135 (2013), no. 2, 361–401. on the Kerr spacetime, available at arXiv:0908.2265. [29] D. Tataru and M. Tohaneanu, A local energy estimate on Kerr black hole back- [3] S. Aretakis, Decay of axisymmetric solutions of the wave equation on extreme grounds, Int. Math. Res. Not. IMRN 2 (2011), 248–292. Kerr backgrounds, J. Funct. Anal. 263 (2012), no. 9, 2770–2831. [30] C. M. Warnick, The massive wave equation in asymptotically AdS spacetimes, [4] H. R. Beyer, On the stability of the Kerr metric, Comm. Math. Phys. 221 (2001), Comm. Math. Phys. 321 (2013), no. 1, 85–111. no. 3, 659–676. [31] B. F. Whiting, Mode stability of the Kerr black hole, J. Math. Phys. 30 (1989), [5] B. Carter, Black hole equilibrium states, Black Holes/Les astres occlus (Les no. 6, 1301–1305. Houches, 1972), Gordon and Breach, New York, 1973, pp. 57–214. [6] B. Carter and R. G. McLenaghan, Generalized total angular momentum operator for the Dirac equation in curved space-time, Phys. Rev. D (3) 19 (1979), no. 4, 1093–1097. Masaki Kashiwara (suite de la page 4) [7] S. Chandrasekhar, An introduction to the theory of the Kerr metric and its perturbations, General Relativity: An Introduction Survey (S. W. Hawking and W. Israel, eds.), Cambridge Univ. Press, Cambridge, 1979, pp. 370–453. régulières sont remplacées par les DX -modules holonomes régu- [8] , The mathematical theory of black holes, Internat. Ser. Monogr. Phys., liers sur X, où DX est l’anneau des opérateurs différentiels sur X. vol. 69, Oxford Univ. Press, New York, 1992. La correspondance de Riemann–Hilbert s’exprime alors comme [9] M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on une équivalence de catégorie entre celle de ces DX -modules et Kerr exterior spacetimes. I–II: The cases |a| M or axisymmetry, available at celle des faisceaux pervers sur X, voir [2,3]. Ces faisceaux pervers arXiv:1010.5132. sont certains objets de la catégorie dérivées de faisceaux construc- [10] M. Dafermos, I. Rodnianski, and M. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes. III: The full subextremal tibles sur X, qui ont justement été introduits dans ce contexte. case |a| < M, available at arXiv:1402.7034. Le cas des équations différentielles avec des singularités éventuel- [11] T. Daudé and N. Kamran, Local energy decay of massive Dirac fields in the 5D lement non régulières est ouvert depuis une trentaine d’années. Myers–Perry metric, Classical Quantum Gravity 29 (2012), no. 14, 145007. [12] A. Enciso and N. Kamran, A singular initial-boundary value problem for non- Kashiwara et D’Agnolo [1] ont récemment résolu ce problème diffi- linear wave equations and holography in asymptotically anti-de Sitter spaces, J. cile en établissant une correspondance de Riemann–Hilbert dans Math. Pures Appl. (9), to appear, available at arXiv:1310.0158. ce contexte. Dans une étude remarquable, ils remplacent les fais- [13] F. Finster, N. Kamran, J. Smoller, and S.-T. Yau, Nonexistence of time-periodic ceaux constructibles par des faisceaux sous-analytiques au sens de solutions of the Dirac equation in an axisymmetric black hole geometry, Comm. Kashiwara–Shapira [4]. Ceux-ci se comportent comme des fais- Pure Appl. Math. 53 (2000), no. 7, 902–929. ceaux classiques grâce au formalisme des 6-opérations de Gro- [14] , Decay rates and probability estimates for massive Dirac particles in the Kerr–Newman black hole geometry, Comm. Math. Phys. 230 (2002), no. 2, thendieck. Notons pour finir que le phénomène de Stokes (sur les 201–244. différences de comportements asymptotiques selon les régions [15] , An integral spectral representation of the propagator for the wave de X) est encodé topologiquement dans la correspondance de equation in the Kerr geometry, Comm. Math. Phys. 260 (2005), no. 2, 257–298. D’Agnolo–Kashiwara, ce qui est aussi un point fondamental de [16] , Decay of solutions of the wave equation in the Kerr geometry, Comm. la construction. Math. Phys. 264 (2006), no. 2, 465–503. [17] , A rigorous treatment of energy extraction from a rotating black hole, [1] A. D’Agnolo and M. Kashiwara, Riemann–Hilbert correspondence for holonomic Comm. Math. Phys. 287 (2009), no. 3, 829–847. D-modules, available at arXiv:1311.2374. [18] , Linear waves in the Kerr geometry: a mathematical voyage to black [2] A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and hole physics, Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 4, 635–659. Topology on Singular Spaces. I (Luminy, 1981), Astérisque, vol. 100, Soc. Math. [19] , The long-time dynamics of Dirac particles in the Kerr–Newman black France, Paris, 1982, pp. 5–171. hole geometry, preprint. [3] M. Kashiwara, The Riemann–Hilbert problem for holonomic systems, Publ. Res. [20] F. Finster and J. Smoller, A time-independent energy estimate for outgoing Inst. Math. Sci. 20 (1984), no. 2, 319–365. scalar waves in the Kerr geometry, J. Hyperbolic Differ. Equ. 5 (2008), no. 1, [4] M. Kashiwara and P. Schapira, Ind-sheaves, Astérisque 271 (2001). 221–255.

BULLETIN CRM–10 crm.math.ca

Les 24 heures de science 2014 Incontournables mathématiques

Christiane Rousseau (Montréal)

Le Réseau de calcul et de modélisation mathématique (rcm2) com- dans un canal, sans modification de forme ou diminution de vi- prenant les centres de recherche en sciences mathématiques, dont tesse. Des signes annonciateurs d’un tsunami comme le recul de le CRM, s’est mobilisé pour organiser une demi-journée d’activi- la mer, son bouillonnement, son odeur de pétrole ou d’oeuf pourri, tés sur le thème «Incontournables mathématiques» au pavillon ou encore un boom suivi d’un sifflement, permettent à la popula- André-Aisenstadt le 9 mai dernier. tion de courir se mettre à l’abri sur les hauteurs. Mathématique- L’après-midi a commencé par une conférence Apprivoiser, se mé- ment on peut, suite à un tremblement de terre, estimer la vitesse de propagation potentielle d’un tsunami selon la profondeur de la fier, et travailler avec l’ordinateur omniprésent de Gilles Capaorossi mer et planifier l’évacuation des zones à risque. On sait par contre du Groupe d’études et de recherche en analyse des décisions (GE- que les vagues solitaires ont des chocs élastiques et que l’envoi RAD), un des centres du rcm2. Elle a été suivie par une conférence Comment ne pas modérer ses transports ? de Bernard Gendron dans d’une contre-vague ne pourrait contrer un tsunami. laquelle il a présenté les travaux ayant cours au Centre interunive- Le conférencier a ensuite parlé des tremblements de terre. Une sitaire de recherche sur les réseaux d’entreprise, la logistique et le prévision idéale déterminerait où, quand et de quelle magni- transport (CIRRELT), un troisième centre du rcm2. Les visiteurs tude seraient les prochains tremblements de terre. On sait que ont été accueillis avec des animations mathématiques. Un vin- c’est impossible, mais cela ne signifie pas que l’on ne puisse rien fromage a permis aux participants de se sustenter avant la grande faire. L’évaluation du risque indépendamment du temps permet de conférence publique du CRM donnée par Florin Diacu de l’Uni- concevoir de meilleurs codes du bâtiment et également d’éviter la versité de Victoria. Cette conférence avait pour titre : La prévision construction dans les zones à risque. On peut aussi identifier des des grandes catastrophes et s’inspirait du livre Megadisasters: the plages temporelles pendant lesquelles un tremblement de terre est science of predicting the next catastrophy écrit par le conférencier. plus susceptible de se produire. Aussi, il y a un intervalle de temps de 10 à 30 secondes entre le moment où un tremblement de terre est déclenché et le moment où il frappe, lequel intervalle peut être mis à profit pour arrêter des trains à grande vitesse, couper la four- niture de gaz (pour empêcher des incendies et les explosions), ou lancer un avis d’évacuation. Florin Diacu a ensuite parlé des ouragans. Les météorologues es- saient de prévoir leur trajectoire et leur intensité, ce qui permet aux populations menacées de se protéger ou encore de quitter la zone touchée. Cependant, prévoir la trajectoire peut jouer des tours : l’ouragan Elena en 1985 a suivi une trajectoire imprévi- sible en faisait presque demi-tour sur lui-même. Le conférencier a montré qu’on pouvait expliquer le phénomène par la sensibilité aux conditions initiales en présence d’un point de selle. Une petite erreur dans les conditions initiales peut faire passer la trajectoire de l’autre côté d’une séparatrice avec de très grands effets. Florin Diacu Florin Diacu a ensuite abordé les changements climatiques. L’effet Le livre et la conférence couvrent ce que l’on peut et ne peut pas de serre a déjà été identifié par Fourier au 19e siècle. On sait que prévoir pour huit types de catastrophes : tsunamis, tremblements la Terre a eu des climats chauds et des glaciations dans le passé. de terre, éruptions volcaniques, ouragans, changement climatique Les variations cycliques des paramètres de l’orbite terrestre, ex- rapide, collision avec des astéroïdes ou des comètes, krachs bour- centricité, inclinaison de l’axe de la Terre, etc., dues à l’attraction siers et pandémies. Florin Diacu a expliqué qu’il s’est intéressé au des autres planètes sont appelées cycles de Milankovitch et ex- sujet lorsqu’il a vu la dévastation suite au tsunami meurtrier de pliquent en bonne part ces cycles passés. On procède maintenant 2004 en Indonésie et il a fasciné son auditoire en racontant ce qu’il à des simulations numériques permettant de prévoir les climats avait appris sur le sujet. (suite à la page 16) Lorsqu’on se trouve au bord de la mer, il y a des signes annon- Mme Christiane Rousseau, professeure au Département de ma- ciateurs de l’arrivée d’un tsunami. Un tsunami est une vague so- thématiques et de statistique de l’Université de Montréal, a été litaire, aussi appelée soliton. Le phénomène des vagues solitaires élue membre du comité exécutif de l’Union mathématique in- a été découvert par hasard par John Scott Russell, ingénieur écos- ternationale pour la période 2015-2018. sais, lorsque il a suivi sur des kilomètres une vague se propageant

BULLETIN CRM–11 crm.math.ca

The 2014 CRM Nirenberg Lectures in Geometric Analysis Alessio Figalli (University of Texas at Austin)

Iosif Polterovich (Montréal); Alina Stancu (Concordia)

In 2014, at the initiative of P. Guan (McGill), D. Jakobson (McGill), In order to answer this question, one needs to define what being I. Polterovich (Montréal) and A. Stancu (Concordia), the Centre de “close to a ball” actually means. When dealing with general non- recherches mathématiques launched a new annual series of dis- convex sets, or disconnected sets, the Hausdorff distance from E tinguished lectures — the CRM Nirenberg Lectures in Geomet- to a ball is not useful. Take, for example, the union of a unit ball ric Analysis. The series is named in honour of Louis Nirenberg, and a far away tiny one, or, if n ≥ 3, a unit ball with a very long, an eminent Canadian-born mathematician who lived in Montreal thin tube attached to it. In both cases, the isoperimetric deficit of during his youth and studied at McGill University. the set is small while the Hausdorff distance to a ball is large. A more natural notion to use is the Fraenkel asymmetry of E, denoted by A(E), which is defined as the infimum of the volume of the symmetric difference of E and all the translates of a ball of the same volume as E. If E is itself a ball, then, clearly, A(E) = 0. We say that a domain is close to a ball if A(E) is small. Let P (E) δ(E) = − 1 (2) n|B|1/n|E|1−1/n

be the isoperimetric deficit of the set E. The question about the stability of the isoperimetric inequality can be then formulated as follows: does A(E) → 0 as δ(E) → 0? Moreover, one may ask a quantitative version of this question: do there exist constants C, α > 0 such that A(E) ≤ Cδ(E)α, and, if so, what is the optimal power α? Louis Nirenberg (on the left) and Alessio Figalli (on the right) Figalli presented here a new approach to the stability problems The first Nirenberg Lectures took place on May 13–16, 2014 and based on the Monge–Kantorovich optimal transportation theory. were given by Alessio Figalli, professor and R. L. Moore Chair at This approach was developed in his joint work with F. Maggi and the University of Texas at Austin. He delivered three talks on Sta- A. Pratelli (2010), using some ideas of M. Gromov, Y. Brenier and bility results for geometric and functional inequalities. There have R. McCann. In particular, the speaker sketched a proof of the in- been a number of important recent developments in this subject, equality A(E) ≤ C(n)δ(E)1/2. One should note that the expo- 1 many of them due to the speaker and his collaborators. nent α = 2 is optimal in any dimension n. This result was first ob- To give a flavour of the results discussed by Alessio Figalli, contained by N. Fusco, F. Maggi and A. Pratelli (2008) using a different sider one of the most well-known inequalities in mathematics — method. The advantage of the optimal transportation approach the isoperimetric inequality. In its full generality, the isoperimet- is its simultaneous applicability to the anisotropic isoperimetric n inequality, which is a far-reaching generalization of its classical ric inequality states that, for any set E in R whose closure has finite Lebesgue measure |E|, its (n−1)-Minkowski content P (E) counterpart. is minimized by the (n − 1)-Minkowski content of the ball of the The idea to use optimal transportation in studying the stability same measure, or, equivalently of the isoperimetric inequality came from one of its proofs. Al- though many different proofs of this inequality were given over the years, the tools from optimal transport were better suited for P (E) ≥ n|B|1/n|E|1−1/n (1) a quantitative version of the isoperimetric inequality. We present below a sketch of the argument. E |E| where B is the unit ball. For a nice enough set , while is the Let T be the optimal transport map from E to B with respect n P (E) (n − 1) -dimensional volume, is the -dimensional volume to the cost given by the distance squared. A classical theorem of the boundary ∂E. of Brenier says that T is a gradient of a convex function and, Equality in (1) is obtained if and only if E is a Euclidean ball (pos- putting aside any possible differentiability issues here and there- sibly up to a set of measure zero). It is then natural to ask whether after, ∇T (which is the Hessian of that convex function) is a sym- a set that almost realizes the equality, must be, in an appropriate metric matrix with non-negative eigenvalues. On the other hand, sense, close to a ball. If this is the case, we say that the isoperi- the change of variables formula applied to T implies that the de- metric inequality is stable. terminant of the gradient of T is |B|/|E|, while the arithmetic-

BULLETIN CRM–12 crm.math.ca geometric mean inequality applied to the eigenvalues of ∇T , im- and thus due to (4), we control the Fraenkel asymmetry of E. plies div T ≥ n det(∇T )1/n on E. Thus, by the divergence theo- However, in general, E may be disconnected, or could lack the rem, necessary boundary regularity for the Sobolev-Poincaré and trace inequalities to hold. Figalli, Maggi and Pratelli proved that, pro- n|B|1/n|E|1−1/n Z Z vided δ(E) ≤ δ(n) for some small δ(n), one can cut off from 1/n = n det(∇T ) dx ≤ div T (x) dx E some sets of very small volume and/or “long, thin tentacles,” E E Z in order to obtain a maximal subset of E, which has almost the = hT (x), ν(x)i dHn−1(x) ≤ P (E), (3) same isoperimetric deficit, and for which the Sobolev–Poincaré ∂E and trace inequalities hold. The argument above could be then where ν : ∂E → n is the outer normal field along ∂E and applied to this maximal “regular” subset, which ultimately leads R 1/2 dHn−1(x) is the (n − 1)-dimensional Hausdorff measure. We to the estimate A(E) ≤ C(n)δ(E) for the set E. used here that hT (x), ν(x)i ≤ 1 since T takes values in B, and The last topic of the lecture series was the stability of theBrunn– thus the norm of T (x) is at most one. The proof of the isoperi- Minkowski inequality. This celebrated inequality states that metric inequality is therefore completed and, if δ(E) = 0, modulo given two sets A and B in the n-dimensional Euclidean space, translations, dilations and modifications up to a set of measure |A + B|1/n ≥ |A|1/n + |B|1/n, with equality if and only if zero, we can deduce that E is B. both sets are convex and homothetic. Here A + B denotes the The proof of the stability of the isoperimetric inequality is substan- Minkowski sum of the two sets, and |·| stands again for the tially more technical, particularly due to the complications arising n-dimensional volume. It is not difficult to see that the Brunn– in the use of the Sobolev–Poincaré and trace inequalities, which Minkowski inequality implies the isoperimetric inequality. are discussed below. However, the outline of the argument is still The following result discussed during the talk was particularly n quite transparent. For a nice enough set E of R , we have the stunning. Let A = B and n = 1. Denote by δBM(A) = following estimate on its Fraenkel asymmetry |A + A| − 2|A| the deficit in the Brunn–Minkowski inequality. Z Does δBM(A) control, in some sense, to which extent the set A is C(n) A(E) ≤ |kxk − 1|dHn−1(x). (4) close to a convex set? In other words, can one bound |co(A) \ A| ∂E from above in terms of δBM(A), where co(A) denotes the convex As mentioned before, the stability problem uses the isoperimet- hull of A? ric deficit δ(E), for δ(E) small enough, to estimate the distance Without loss of generality, suppose that |A| = 1. Taking AL = from E to the optimal set B, given in this case by the Fraenkel 1 1 [0, 2 ] ∪ [L, L + 2 ] ⊂ R, one immediately gets that δBM(AL) = 1 asymmetry. To do so, assuming |E| = |B|, we have 1 for all L > 0, while |co(AL) \ AL| = L − 2 → ∞ as L → ∞, Z and hence no control seems to be possible. However, if in addi- n−1 n|B|δ(E) ≥ (1 − kT (x)k) dH (x). (5) tion to the hypothesis |A| = 1 one assumes that δBM(A) < 1, ∂E then |co(A) \ A| ≤ δBM(A)! This inequality, as well as its higher- Indeed, by (3), dimensional generalization, was obtained in a joint work of the Z Z speaker and D. Jerison (2014). Quite unexpectedly, the proof is n|B| ≤ hT (x), ν(x)i dHn−1(x) ≤ kT (x)k dHn−1(x), based on a deep theorem due to G. Freiman (1959) in additive com- ∂E ∂E binatorics — an area of mathematics which is quite distant from and (5) then immediately follows from (2). geometric analysis. On the other hand, for δ(E) = 0, we have that ∇T = Id almost The first Nirenberg Lectures attracted significant attention from everywhere on E. Thus, it is somewhat natural that, for small the Montreal mathematical community. The lecture hall was full enough δ(E), we can obtain an estimate of the form during all three talks. The credit for that goes, of course, to Alessio Figalli who spoke about fascinating modern mathematics in a su- q Z Z perbly clear and accessible way. It was particularly rewarding C(n) δ(E) ≥ |∇T − Id| dx = |∇(T (x) − x)| dx. (6) E E for the organizers to see many graduate students and young re- p searchers in the audience. Here we use the notation |M| = trace(M >M) for a given matrix M and, for the sake of simplicity, we denote all constants de- Last, but not least, it was a special pleasure and honour to have pending on n by C(n). Furthermore, by the Sobolev–Poincaré and Louis Nirenberg join us from New York to inaugurate the series. trace inequalities applied to the last term, one can obtain His presence, marked by a number of valuable comments and observations, as well as his characteristic humour, contributed in an q Z important way to the success of the event. C(n) δ(E) ≥ kT (x) − xk dHn−1(x). (7) ∂E By adding (5) and (7), we have, for small enough δ(E), q Z C(n) δ(E) ≥ |kxk − 1| dHn−1(x), (8) ∂E BULLETIN CRM–13 crm.math.ca Computational Methods for Surveys and Census Data in the Social Sciences

Organizers: Mary Thompson (Waterloo); David Haziza (Montréal); Louis-Paul Rivest and Anne-Sophie Charest (Laval); Mike Hidiroglou (Statistics Canada); Jean Poirier (Centre interuniversitaire québécois de statistiques sociales) The workshop was co-sponsored by the Canadian Statistical Sci- Australia and New Zealand. Some topics, such as record linkage ences Institute (CANSSI), the Statistical and Applied Mathematical and weighting in surveys generated debate between theorists and Sciences Institute (SAMSI) and the CRM. It grew out of Canadian practitioners and between proponents of Bayesian and frequen- participation in the Computational Methods in the Social Sciences tist approaches. It is felt that these kinds of discussions could lead (CMSS) theme year at SAMSI. A major purpose for the workshop to important syntheses of methodology for both official statistics was to increase opportunities for statisticians and social scientists and the social sciences in the next few years. to communicate about problems and new directions. In the lead-off session, Chris Skinner (London School of Eco- Mathematical Methods and Models nomics) spoke on using paradata to correct for measurement er- ror in social and economic survey data. The second session laid in Laser Filamentation some groundwork on various aspects of combining data from multiple sources with talks by social and health scientists Claire Organizers: André D. Bandrauk (Sherbrooke), Emmanuel Lorin Durand (Université de Montréal), France Labrèche (Institut de (Carleton), Jerome V. Moloney (Arizona) recherche Robert-Sauvé en santé et en sécurité du travail), and Hélène Vézina (Université du Québec à Chicoutimi). Topics in- From March 10 to 14, 2014, applied mathematicians, theoreti- cluded: analysis techniques, administrative and ethical challenges, cal physicists, and specialists of nonlinear optics and laser prop- and record linkage. The afternoon focused on the practice and agation in matter at high intensities gathered at a workshop theory of record linkage, beginning with an overview by Mauri- held at the CRM. Their purpose was to rigorously create, derive, cio Sadinle (Carnegie Mellon University). Jimmy Baulne (Insti- and discuss models and numerical algorithms based on quantum tut de la statistique du Québec) and Abdelnasser Saidi (Statistics Schrödinger, Dirac and classical Maxwell equations for describ- Canada) spoke about record linkage techniques and collaborations ing the nonlinear, nonperturbative regime of ultrashort intense with researchers. Roee Gutman (Brown University) and Rebecca laser interaction with gases (such as the atmosphere) and solids Steorts (Carnegie Mellon University) talked about applications of (for laser writing, etc). New phenomena, such as multiple “soli- Bayesian record linkage approaches to problems in medical record tons” and “rogue” waves which occur in refocusing of laser ener- mining and estimation of human rights violations, respectively. gies, were a main subject of discussion due to their instability to After dinner, Lisa Y. Dillon (Université de Montréal) spoke about ionization of electrons, thus creating plasmas. Reduction of the the census Mining Microdata project. relevant quantum mechanical equations, Schrödinger and Dirac, the latter for relativistic effects, to quasi-classical equations was Most of the second day was devoted to survey data analysis. The another main subject of interest, as was input of nonlinear matter morning sessions, with speakers Rod Little (University of Michi- response in Maxwell’s equations for laser propagation in the at- gan), Jean-François Beaumont (Statistics Canada), Changbao Wu mosphere, liquids, and solids. (University of Waterloo), Qixuan Chen (Columbia University) and Suojin Wang (Texas A&M University) — and discussion by Jean The year 2015 was declared as the “Year of Light” by UNESCO Opsomer (Colorado State University) and Michael Elliott (Univer- and the workshop was integrated into an international network sity of Michigan) — discussed the evolving role of weighting in sanctioned by UNESCO-“MPE-Mathematics of Planet Earth.” The survey data analysis, one of the major themes of the SAMSI CMSS CRM will be active in this network through the continuing work program. In the afternoon, there were talks on the use of network of Christiane Rousseau. The interaction of ultrafast intense laser data in sampling and estimation, by Ray Chambers of Wollongong pulses with the atmosphere has become an important research University and Pierre Lavallée (Statistics Canada), with discussion subject in this international program and the CRM workshop was by Phil Kott (Research Triangle Institute). The workshop wrapped the first Canadian activity in this new research direction involving up with a session on survey data exploration, visualization, and applied mathematicians. mapping, as well as subjects of increasing importance in the era The success of this workshop was recognized last month bythe of “big data.” The speakers were statisticians Thomas Lumley (Uni- international committee of COFIL, International Symposium on versity of Auckland) and Wayne Oldford (University of Waterloo), Filamentation, by an invitation to A. D. Bandrauk (Sherbrooke), and epidemiologist Yan Kestens (Université de Montréal). S. L. Chin (Laval), J.-C. Kiefer (INRS-ÉMT), and F. Théberge (Dept. The discussion was very lively throughout the workshop. There of National Defence, Valcartier) to coorganize the 5th such sympo- were 46 participants, forming a good mix of relatively new and sium, COFIL2016 in Québec. Finally, the organizers of the March more seasoned researchers. They were mainly from Canada and 2014 CRM workshop will be editing a volume of proceedings based the US, but some also came from as far away as Saudi Arabia, on the invited lectures presented, to be published by the CRM and Springer in 2015.

BULLETIN CRM–14 crm.math.ca Exact Solvability and Symmetry Avatars Conference held on the occasion of Luc Vinet’s 60th birthday

Organizers: Decio Levi (Roma Tre); Pavel Winternitz (Montréal); Willard Miller Jr. (Minnesota); Yvan Saint-Aubin (Montréal)

The conference Exact Solvability and Symmetry Avatars celebrated K. Teneblat), cellular automata (H. Ujino), “integrable” combi- the sixtieth birthday of Professor Luc Vinet. Luc’s scientific contri- natorics (P. Di Francesco, J. Harnad), integrable lattice models butions are in several different areas of mathematics and physics. (A. Morin-Duchesne), isomonodromy and conformal field the- They share a common theme, that of symmetries in nature andsci- ory (A.R. Its), quantum and string theory (E. D’Hoker, R. Jackiw, ence. The conference thus brought together scientists who would D. H. Phong), quantum walks (F. A. Grünbaum) , and fundamental usually not attend the same event. It turned out that they wereall results in algebras and representation theory (L. Snobl, P. M. Ter- able to communicate easily and fruitfully. Indeed all participants williger). use symmetries, groups, algebras and their representation theory as everyday tools. One of Luc’s earliest research topics was the use of symmetries to study the dynamics of classical and quantum systems. This includes the problems of separation of variables, integrability and superintegrability, in both continuous and discrete realms. This is currently a very active field of research and several talks were devoted to new developments in this area (A. M. Escobar Ruiz, D. Levi, W. Miller Jr., G. Pogosyan, S. Post, A. Turbiner and W. Za- krzewski). The study and classification of orthogonal polynomials are an important part of Luc’s current interests and this From left to right: Keti Tennenblat (Brasilia), Philippe Di Francesco (Illinois, Urbana- field was well-represented in the talks (P. Desrosiers, C. F. Dunkl, Champaign), Eric D’Hoker (UCLA), Luc Vinet and D. H. Phong (Columbia). E. Emsiz, V. Genest, Y. Grandati, M. Ismail, E. Koelink, T. Koorn- Photo: Letitia Muresan winder, L. Lapointe, F. Marcellán, P. Mathieu, D. W. Stanton, The celebration provided an ideal occasion for many young sci- J. F. van Diejen, O. Zhedanov). Quantum information theory entists to thank Luc for his mentoring. Several were among the also makes use of the idea of symmetries. While Luc has not speakers: Luc Lapointe, Sarah Post, Jan Felipe van Diejen and published many papers in this field, his work there also had an Vincent Genest who is currently finishing his PhD under Luc’s important impact. Many leaders of this recent field presented supervision. Others acknowledged gratefully the influence of their results (M. Christandl, G. Coutinho, E. Farhi, R. Floreanini). Luc’s seminal ideas on their work, his pregnant questions during Several other fields were represented, relating various aspects talks and the pleasure of working with him on scientific projects. of symmetry and integrability: coherent states (A. Strasburger), differential and spectral geometry (N. Kamran, A. V. Penskoi,

Iosif Polterovich nouveau directeur adjoint – programmes scientiﬁques au CRM

M. Iosif Polterovich a succédé à M. Octav Cornea au poste de di- M. Polterovich travaille en théorie spectrale géométrique, au car- recteur adjoint – programmes scientifiques. Son mandat a débuté refour de la géométrie différentielle, de la théorie des équations le 1er juin 2014. M. Polterovich se joint au Comité de direction du aux dérivées partielles et de la physique mathématique. Ses contri- CRM, qui inclut aussi M. Luc Vinet, directeur du CRM et Mmes Ga- butions à la recherche ont été largement reconnues et il a reçu lia Dafni (Concordia), directrice adjointe – publications et Odile plusieurs prix, en particulier le prix Coxeter-James (2011), le prix Marcotte (UQAM et GERAD), directrice adjointe – partenariats. G. de B. Robinson (2008) de la Société mathématique du Canada et le prix André-Aisenstadt décerné par le CRM (2006). M. Iosif Polterovich est actuellement professeur titulaire au Dé- partement de mathématiques et de statistique de l’Université de Lors de l’annonce de la nomination de M. Polterovich, M. Luc Montréal, où il détient une chaire de recherche du Canada en géo- Vinet, a déclaré : «Je suis ravi que Iosif Polterovich, qui est un métrie et théorie spectrale. Il a obtenu une maîtrise de l’Université éminent chercheur et un organisateur très efficace qui a un ex- d’État de Moscou en 1995 et un doctorat de l’Institut Weizmann cellent jugement scientifique, ait accepté de se joindre à l’équipe des sciences en l’an 2000. de direction du CRM et de contribuer avec ses multiples talents au Avant de se joindre à l’Université de Montréal en 2002, il a été développement d’une des plus remarquables organisations cana- boursier postdoctoral et scientifique invité au CRM, au MSRI et à diennes». l’Institut Max-Planck de mathématiques à Bonn. BULLETIN CRM–15 crm.math.ca Dmitry Jakobson nommé rédacteur en chef des Annales mathématiques du Québec

La direction de l’ISM est heureuse d’annoncer que M. Dmitry Ja- chercheur dynamique, dont les travaux sont à la croisée de l’ana- kobson, professeur à l’Université McGill, vient d’être nommé ré- lyse géométrique, de la théorie des équations aux dérivées par- dacteur en chef des Annales mathématiques du Québec (AMQ) pour tielles et de la physique mathématique. Dmitry Jakobson a large- un mandat de trois ans. Il entrera en fonction le 1er janvier 2015. ment contribué, entre autres, à l’étude du spectre et des fonctions Dmitry Jakobson occupe la chaire propres des opérateurs de type Laplace et Dirac. Il s’intéresse aussi Peter Redpath au Département de à la théorie ergodique, à la théorie des probabilités, à la théorie des nombres et à la théorie des graphes. Il a été boursier de la fonda- mathématiques et de statistique tion Alfred P. Sloan en 2001 et a été nommé chercheur-boursier de l’Université McGill. Il a ter- miné ses études de 1er cycle à William Dawson par l’Université McGill en 2003. De plus, la So- l’Institut de technologie du Mas- ciété mathématique du Canada lui a décerné le prix G. de B. Ro- sachusetts en 1991 et a obtenu son binson en 2008. doctorat de l’Université de Prin- Il y a tout lieu de croire que les AMQ continueront de croître et ceton en 1995. Après avoir été de prospérer sous la direction de Dmitry Jakobson. Nos vœux de Dmitry Jakobson boursier postdoctoral à l’Institut succès l’accompagnent dans l’exercice de ses nouvelles responsa- de technologie de Californie et à l’Institute for Advanced Study de bilités. Nous exprimons en outre toute notre gratitude envers le Princeton, il a été professeur adjoint à l’Université de Chicago pen- rédacteur en chef sortant, M. Claude Levesque, pour le travail re- dant un an avant d’être recruté par l’Université McGill en 2000. marquable qu’il a accompli depuis 2008.

En recommandant la nomination de Dmitry Jakobson à la tête des Christian Genest, PhD, PStat AMQ, le Comité de sélection présidé par François Lalonde s’est Directeur de l’ISM dit convaincu que la revue tirerait grand proﬁt de l’expertise de ce

Daniel Wise elected to Florin Diacu the Royal Society of Canada (suite de la page 11) futurs. Modéliser le climat revient à éliminer les effets aléatoires Daniel Wise, Professor at McGill University and a member of the des prévisions météorologiques pour retenir les tendances et le CRM, has been elected to the Royal Society of Canada. He is one conférencier a montré comment les observations et simulations of the two mathematicians in the 2014 class of fellows. Profes- actuelles confortent la thèse du réchauffement climatique. sor Wise is cited for his fundamental contributions to geometric group theory and low-dimensional topology that “… stand at the Florin Diacu, spécialiste de la mécanique céleste, s’est attardé sur core of the most important development in geometry and topol- la prédiction ou prévention de collisions avec des comètes et asté- ogy since the proof of the Poincaré Conjecture, namely the proof roïdes, et il a expliqué comment le programme Space Watch suit of Thurston’s virtually fibered conjecture for hyperbolic three- à la trace les objets célestes qui passent au voisinage de la Terre. manifolds.” Ce même programme tente d’imaginer des moyens dignes de la science fiction pour protéger notre planète de ces objets : utiliser des dispositifs nucléaires pour détruire l’astéroïde, colorier l’asté- Sabin Cautis roïde, ce qui change légèrement son absorption de l’énergie solaire (continued from page 5) et donc sa trajectoire, encastrer des fusées dans l’astéroïde, etc. operators. In the process we were able to lift (categorify) the La conférence s’est terminée sur la prévention des krachs bour- Frenkel–Kac–Segal construction of the basic representation. Our siers. Certains indicateurs existent, dont un conçu par Robert Shil- results can be applied, for instance, to study Hilbert schemes of ler, lauréat du prix Nobel d’économie, et qui avait prévu le krach points on surfaces. de 2000 quelques mois avant qu’il ne se produise. L’un des signes précurseurs est le trop grand écart entre les dividendes et le prix The categorical representation theory of quantum groups, Heisen- des actions. berg algebras, vertex operator algebras, etc. is in its infancy. We hope to develop this area further and to apply it to categories of sheaves on a growing number of geometric spaces. Ultimately one would like to completely describe such categories in terms of structures from categorical representation theory.

BULLETIN CRM–16 crm.math.ca La statistique au service de la collectivité Hommage à Louis-Paul Rivest pour ses 60 ans

Thierry Duchesne (Laval); Christian Genest (McGill)

Le 29 août a eu lieu à Québec un colloque en l’honneur de Louis- de données familiales. Après la pause, François Pageau a brossé Paul Rivest, professeur à l’Université Laval et titulaire de la chaire un tableau des aspects statistiques de la maîtrise de la qualité dans de recherche du Canada en échantillonnage statistique et en ana- l’industrie de la défense, telle qu’elle se pratique entre autres chez lyse de données. L’événement visait en outre à faire la promo- General Dynamics Produits de défense et systèmes tactiques – Ca- tion de la statistique comme champ d’étude et de recherche. Les nada. Eve Belmonte, directrice de l’actuariat pour le Régime de re- quelque 85 participants réunis au pavillon La Laurentienne de la traite de l’Université du Québec, a ensuite expliqué comment la cité universitaire ont eu droit à un programme riche et diversi- statistique pourrait être mise à profit pour trouver des solutions fié, à l’image de la carrière de Louis-Paul. Dix exposés de 30 mi- viables à l’actuelle crise de financement des régimes de retraite. nutes, conçus pour susciter l’intérêt d’un large public, ont dressé En après-midi, Jean-François Beaumont a donné un tour d’hori- un vaste et étonnant panorama des applications de la «science des zon des méthodes utilisées par Statistique Canada pour l’estima- données» dans les différentes sphères de l’activité humaine. tion de paramètres de populations finies en présence de données Tous les conférenciers étaient aberrantes. Rébecca Tremblay, du ministère de l’Emploi et de la So- d’anciens étudiants ou lidarité sociale du Québec, a par la suite montré comment les tech- d’étroits collaborateurs de niques d’enquête et de modélisation statistique permettent d’ins- Louis-Paul. Cette brochette de trumenter la reddition de comptes et d’améliorer les services ren- professionnels talentueux et dus à la population, notamment par la mesure des perceptions de bien formés, qui mènent de la clientèle à l’égard des services gouvernementaux. Puis, parlant belles carrières dans des mi- au nom de l’Institut de la statistique du Québec, Éric Gagnon a lieux variés, constituait en soi enchaîné avec un survol des contributions de Louis-Paul aux tra- un hommage à celui qui a si vaux menés par cet organisme au cours des 30 dernières années. largement contribué au déve- Après une courte pause, Nathalie Vandal a illustré l’usage répandu loppement de notre discipline, de la statistique à l’Institut national de santé publique du Québec tant par ses écrits que par l’en- en mettant l’accent sur son aide précieuse dans l’élaboration du cadrement d’une soixantaine portrait de santé des Québécois et des relations entre les détermi- d’étudiants de 2e cycle, une nants de la santé. Enfin, Lajmi Lakhal-Chaïeb, professeur à l’Uni- demi-douzaine de doctorants versité Laval, a donné un aperçu de travaux qu’il a effectués en et plusieurs stagiaires post- collaboration avec Louis-Paul et une doctorante, Héla Romdhani, doctoraux. L’influence et l’am- maintenant stagiaire postdoctorale à McGill. Son exposé portait Louis-Paul Rivest pleur des travaux de Louis- sur la façon d’adapter le tau de Kendall pour mesurer la dépen- Paul lui ont d’ailleurs valu en 2010 la plus haute distinction en dance entre des variables hiérarchiques groupées. statistique au pays, la médaille d’or de la Société statistique du À la fin du colloque, Louis-Paul et une cinquantaine de collègues Canada. et amis ont été conviés à l’une des meilleures tables de Québec, Le Pour refléter l’étendue de l’expertise et des champs d’intérêt de Galopin. Un ami et collaborateur de longue date, Pierre Lavallée, Louis-Paul, les exposés avaient été regroupés en divers thèmes : la Directeur adjoint à la Division des méthodes d’enquêtes auprès statistique au service de l’Homme et de son milieu, de l’industrie, des entreprises à Statistique Canada, a livré un vibrant hommage de l’État, de la gouvernance et de la santé. Le colloque a été ouvert au jubilaire, rappelant les principaux jalons de sa carrière et cer- par le Doyen de la Faculté des sciences et de génie de l’Université taines de ses réalisations les plus marquantes, dont la fondation du Laval, M. André Darveau, qui a souligné l’importance de la statis- Service de consultation statistique de l’Université Laval en 1984. tique et le rôle de premier plan que Louis-Paul joue depuis plus de En fin de soirée, une affiche du colloque a été remise à Louis-Paul, 30 ans au sein du groupe de statistique de l’établissement. après qu’elle ait été signée par toutes les personnes présentes. En matinée, Nicolas Bousquet, affilié à Électricité de France et à Dans les jours qui ont suivi, de multiples marques d’appréciation l’Université Paul-Sabatier, a mis en lumière quelques apports de nous ont été adressées, ainsi qu’à Louis-Paul. Au nom du Labora- la statistique dans la gestion des ressources environnementales toire de statistique du CRM, promoteur de l’événement, nous lui et industrielles. Les participants ont ensuite pu entendre Bastien réitérons ici tous nos vœux et notre admiration. Qu’il nous soit Ferland-Raymond, du ministère des Forêts, de la Faune et des Parcs aussi permis de souligner avec gratitude l’appui logistique et fi- du Québec, et Karim Oualkacha, professeur à l’Université duQué- nancier des parrains de l’événement, à savoir l’Institut des sciences bec à Montréal, qui ont décrit l’emploi de méthodes statistiques mathématiques du Québec, l’Université Laval, ainsi que l’Associa- dans les inventaires forestiers et l’étude de l’héritabilité au moyen tion des statisticiennes et statisticiens du Québec.

BULLETIN CRM–17 crm.math.ca The 2014–2015 CRM–ISM Postdoctoral Fellows

In the academic year 2014–2015, the members of the CRM are supervising around 30 postdoctoral fellows in all ﬁelds of mathematics, including 10 which are involved in the activities of the thematic year on Algebra and Number Theory. Presented here are this year’s recipients of a CRM–ISM postdoctoral fellowship (see below for further information).

Stephan Ehlen, Ph.D. TU Darmstadt, 2013 Boaz Slomka, Ph.D. Tel Aviv, 2014 Supervisors: Henri Darmon and Eyal Goren (McGill) Supervisors: Alina Stancu (Concordia) and Dmitry Jakobson My area of research is number theory, in (McGill) particular modular forms and more re- My research is mainly concerned with cently also arithmetic geometry. I am us- questions rising from the field of asymp- ing regularized theta lifts to study rela- totic convex geometry and which are at tions between spaces of modular forms, the meeting point of geometry and anal- obtain explicit constructions and applica- ysis. Its focus is on the study of the so- tions in arithmetic intersection theory. In called “Geometry of Log-concave Func- my thesis, I studied the values of regular- tions” in which concepts and related in- ized theta lifts at CM points on orthog- equalities from the theory of convex bod- onal Shimura varieties. These values are ies are extended to the realms of log- closely related to coefficients of harmonic weak Maass forms and concave functions. The interplay between intersections of special divisors on modular curves. In a recent convex bodies and log-concave functions project my collaborators and I studied the asymptotic behaviour has proven to be beneficial and has also led to significant progress of the dimension of the space of vector-valued modular forms for in the understanding of some central problems in the field of the Weil representation and some geometric applications related asymptotic convex geometry. Recently, I have been working on to the Borcherds lift. problems related to covering numbers and on their functional ex- tensions and applications. Alastair Irving, D.Phil. Oxford, 2014 Supervisors: Andrew Granville and Dimitrios Koukoulopoulos Weiwei Wu, Ph.D., Michigan State, 2012 (Montréal) Supervisors: Octav Cornea and François Lalonde (Montréal) My research is in analytic number theory. I work on problems related to topology I am interested in the application of analytic and geometry of symplectic manifolds. In techniques, such as sieves and bounds for the past I have investigated problems of exponential sums, to problems in classical isotopy classes of Lagrangian submani- number theory. Recently I have proved a folds, the homotopy type of symplecto- new equidistribution theorem on the divi- morphism groups and the space of em- sor function in arithmetic progressions as bedded Lagrangians, and also problems well as some results on almost-primes in related to symplectic quasi-morphisms various polynomial sequences. by computing Floer cohomology. Most of these works involved the theory of Amy Pang, Ph.D. Stanford, 2014 psuedo-holomorphic curves invented by Gromov, and also Supervisors: François Bergeron, Christophe Hohlweg, Christophe flavours of symplectic field theories developed by Eliashberg– Reutenauer and Franco Saliola (UQAM) Hofer–Givental in the last decade. Recently I have been thinking about (1) homological mirror symmetry, trying to prove some My main research interest is in the in- new cases, and also to extract more refined properties from basic teraction of algebraic combinatorics and examples, and (2) understand and compare birational aspects in probability. Lately I’ve been using com- algebraic geometry with symplectic geometry. binatorial Hopf algebras to study Markov chains which model the breaking and recombining of combinatorial objects. CRM–ISM postdoctoral fellowships in all fields of research in Many models of card-shuffling fit into my which the CRM and the ISM are actively involved are awarded framework, including the popular riffle- to promising researchers who have recently obtained or ex- shuffle and the top-to-random shuffle. pect to obtain a Ph.D. in the mathematical sciences. The dead- The techniques I use have a heavy linear-algebraic flavour: for ex- line for the CRM–ISM postdoctoral fellowships is December ample, calculating eigenvectors for various maps on the Hopf al- 12th, 2014. Cf. http://crm.math.ca/~crmism/index_ gebras gives the stationary distribution of the Markov chains and en.html other information about their long-term behaviour. BULLETIN CRM–18 crm.math.ca CRM Thematic Postdoctoral Fellows Kevin Ventullo, Ph.D. UCLA, 2014 Supervisors: Henri Darmon (McGill) Daniel Barrera, Ph.D. Lille 1, 2013 Supervisors: Adrian Iovita (Concordia), Kassaei Payman (McGill) My research is in the area of algebraic and Andrew Granville (Montréal) number theory, specifically Iwasawa theory, p-adic L-functions, modular forms, My main research area is that of al- and Galois representations. I am mainly gebraic number theory, I have mostly interested in questions surrounding the worked on problems related to the Iwasawa main conjecture and Gross– p-adic arithmetic of geometric ob- Stark conjecture, which respectively deal jects. In particular, I am interested in with the order of vanishing and lead- the construction and study of p-adic ing terms of p-adic L-functions. Lately L-functions attached to Hilbert modular I have been trying to make headway on forms and some potentially automor- the Gross–Stark conjecture when the order of vanishing is greater phic motives. I am also interested in the than one, as well as simplifying (and completing when p = 2) relationship between p-adic families of Wiles’s proof of the main conjecture at the exceptional zeroes. modular symbols and overconvergent modular forms of finite I am also interested in questions of modularity in the residually slope. reducible, non-p-distinguished setting.

Sary Drappeau, Ph.D. Paris-Diderot, 2013 Yongqiang Zhao, Ph.D. Waterloo, 2013 Supervisors: Andrew Granville (Montréal) Supervisors: Henri Darmon (McGill), Chantal David (Concordia) I work on problems in analytic number the- and Andrew Granville (Montréal) ory. My research has been mainly focused My research is in the area of arithmetic on distribution and multiplicative proper- geometry. I am interested in a variety of ties of integers with relatively small prime topics which include arithmetic statistics, factors. This includes equidistribution in Manin’s program on the distribution of ra- arithmetic progressions, sums of multi- tional points on Fano varieties and Vojta’s plicative functions, or exponential sums, conjecture. Currently, I am working with and can be used to study related additive David McKinnon on a problem of counting problems. integral points under ﬁnitely many coprime restrictions.

27e Entretiens Jacques Cartier Nouveaux horizons en combinatoire additive

Organisateurs : David Conlon (Oxford); Ben Green (Cambridge); Laurent Habsieger (UMI CNRS-CRM); Andrew Granville (Montréal); Alain Plagne (École polytechnique) Le colloque Nouveaux horizons en combinatoire additive a connu 4,5 mois), Olivier Fouquet (Orsay, 3 mois), Adam J. Harper (Cam- un franc succès international. Il a attiré 61 participants de 9 pays, bridge, 3 mois), Alastair Irving (Oxford, 1 an), Robert Lemke Oli- répartis en trois tiers bien identiﬁés : 23 du Canada, 20 des USA et ver (Stanford, 5 mois), Kaisa Matomäki (Finlande, 4 mois), Marc 18 d’Europe (dont 7 de France). Il s’est tenu du 6 au 10 octobre 2014 Munsch (Marseille, 1 an), Jennifer Park (MIT, 1 an), Maksym Rad- et il a été programmé en coordination avec l’IMA à Minneapolis, ziwill (IAS Princeton, 1 an), Fernando Xuancheng Shao (Stanford, qui a organisé un colloque intitulé Additive and Analytic Combina- 2 mois), Jesse Thorner (Emory, 3 mois), Yongqiang Zhao (Water- torics du 29 septembre au 3 octobre. Cela a permis à 19 chercheurs loo, 1 an). de participer aux deux colloques. L’aspect social fut très présent pendant le colloque. Outre la céré- Pendant cinq jours, il y a eu 30 exposés, dont un, destiné à une monie d’ouverture des Entretiens Jacques Cartier, la réception du audience plus large, dans le cadre du colloque des sciences mathé- maire de Montréal et le concert d’orgue, une réception fut organi- matiques du Québec, par Alex Kontorovich (Rutgers). L’assistance sée par le Centre de recherches mathématiques le mardi 7 octobre, fut fournie, jusqu’au dernier exposé. et les participants se sont retrouvés le jeudi 9 octobre pour un ban- De nombreux participants séjourneront à Montréal pour une du- quet convivial. rée plus longue : Antal Balog (Hongrie, 1 mois), Daniel Barrera En résumé, le colloque fut très réussi, comme en témoignent les (1 an), Stephan Ehlen (Allemagne, 1 an), Daniel Fiorilli (Ottawa, nombreux remerciements improvisés par les conférenciers.

BULLETIN CRM–19 crm.math.ca New CRM Publications

Function Theory on Symplectic Manifolds famous modular forms with their striking product formula — an Leonid Polterovich and Daniel Rosen idea due to Bruinier–Burgos–Kühn and Kudla. This should be seen as an Arakelov analogue of the classical calculation of volumes of This is a book on symplectic orthogonal locally symmetric spaces by Siegel and Weil. In the topology, a rapidly developing latter theory, the volumes are related to special values of (normal- ﬁeld of mathematics which orig- ized) Siegel Eisenstein series. inated as a geometric tool for In this book, it is proved that the Arakelov analogues are related problems of classical mechanics. to special derivatives of such Eisenstein series. This result gives Since the 1980s, powerful meth- substantial evidence in the direction of Kudla’s conjectures in ar- ods such as Gromov’s pseudo- bitrary dimensions. The validity of the full set of conjectures of holomorphic curves and Morse– Kudla, in turn, would give a conceptual proof and far reaching Floer theory on loop spaces gave generalizations of the work of Gross and Zagier on the Birch and rise to the discovery of unex- Swinnerton-Dyer conjecture. pected symplectic phenomena. The present book focuses on CRM Monograph Series, Volume 35, 2014; 152 pp function spaces associated with ISBN: 978-1-4704-1912-7 a symplectic manifold. A number of recent advances show that these spaces exhibit intriguing Geometric and Spectral Analysis properties and structures, giving rise to an alternative intuition Pierre Albin, Dmitry Jakobson, Frédéric Rochon, eds. and new tools in symplectic topology. The book provides an essen- tially self-contained introduction into these developments along In 2012, the Centre de with applications to symplectic topology, algebra and geometry recherches mathématiques was of symplectomorphism groups, Hamiltonian dynamics and quan- at the center of many interest- tum mechanics. It will appeal to researchers and students from ing developments in geomet- the graduate level onwards. ric and spectral analysis, with a thematic program on Geomet- CRM Monograph Series, Volume 35, 2014; 203 pp ric Analysis and Spectral Theory ISBN: 978-1-4704-1693-5 followed by a thematic year on Moduli Spaces, Extremality and The Geometric and Arithmetic Volume of Global Invariants. Shimura Varieties of Orthogonal Type This volume contains original Fritz Hörmann contributions as well as useful survey articles of recent devel- This book outlines a functo- opments by participants from rial theory of integral models of three of the workshops orga- (mixed) Shimura varieties and nized during these programs: Geometry of Eigenvalues and Eigen- of their toroidal compactiﬁca- functions, held from June 4–8, 2012; Manifolds of Metrics and Prob- tions, for odd primes of good abilistic Methods in Geometry and Analysis, held from July 2–6, reduction. This is the integral 2012; and Spectral Invariants on Non-compact and Singular Spaces, version, developed in the au- held from July 23–27, 2012. thor’s thesis, of the theory in- The topics covered in this volume include Fourier integral opera- vented by Deligne and Pink in tors, eigenfunctions, probability and analysis on singular spaces, the rational case. In addition, complex geometry, Kähler–Einstein metrics, analytic torsion, and the author develops a theory of Strichartz estimates. arithmetic Chern classes of integral automorphic vector bun- Contemporary Mathematics, Volume 630; to appear dles with singular metrics using work of Burgos, Kramer and Kühn. The main application is calculating arithmetic volumes or “heights” of Shimura varieties of orthogonal type using Borcherds’

BULLETIN CRM–20 crm.math.ca

2014 CRM–SSC Prize 2014 CAP–CRM Prize Fang Yao, University of Toronto Mark Van Raamsdonk, UBC

The CRM–SSC Prize in statistics is The 2014 CAP–CRM Prize in awarded annually by the Centre de Theoretical and Mathematical recherches mathématiques and the Physics is awarded to Prof Mark Statistical Society of Canada to rec- Van Raamsdonk, University of ognize a statistical scientist’s profes- British Columbia, for his highly sional accomplishments in research original, influential contribu- during the first fifteen years after tions to several areas of theo- earning a doctorate. This year’s win- retical physics, including string ner is Fang Yao of the University of theory, quantum field theory, Toronto. and quantum gravity. High- lights include advances in the the- Fang Yao Fang Yao has conducted his most im- Mark Van Raamsdonk ory of D-branes and other non- portant research since arriving at the University of Toronto in perturbative objects in string theory, the ultraviolet-infrared mix- 2006; it is both seminal and beautiful. ing in non-commutative quantum field theory, and the decon- Fang received his Bachelors degree from the University of Science finement transition in gauge theory, as well as his novel proposal and Technology in China. He was admitted to the University of that the emergence of spacetime is profoundly connected with California, Davis, for graduate studies, where he completed both quantum entanglement. his MSc and his PhD in three years. His doctoral dissertation was completed under the joint supervision of Hans-Georg Müller Mark Van Raamsdonk is a theoretical physicist whose work lies and Jane-Ling Wang. The dissertation was novel for coupling ad- in the field of string theory and quantum gravity. Through his vanced methodological machinery with a data-type critical for es- work, he attempts to answer some of the grandest open ques- tablishing causal relationships. Subsequent to completing his de- tion in theoretical physics: What is space? What is time? How did our universe begin? How will it end? What is inside a black gree in 2003, Fang assumed a position in the Department of Statis- hole? Some of his most significant contributions include a study tics at Colorado State University. The Department of Statistical Sciences at the University of Toronto successfully recruited Fang of actions governing the interactions of multiple D-branes (funda- in 2006; he was subsequently promoted to the rank of Associate mental degrees of freedom in strongly coupled string theory) with Professor with tenure in 2008. During his first sabbatical, Fang gravitational and other classical fields, a discovery of UV-IR mix- was invited to the Statistical and Applied Mathematical Sciences ing in quantum field theories on noncommutative spacetime, and the first analytic study of deconfinement in a four-dimensional Institute in North Carolina as a Research Fellow, where he gave gauge theory. In 2008 his work led to a breakthrough in which the opening address for a thematic program and subsequently led one of the working groups during the fall term. He then moved the actions governing membranes (fundamental degrees of free- on to UBC for the rest of his sabbatical leave, where he created a dom in M-theory) were explicitly determined for the first time. new advanced graduate course, Topics in Smoothing: Functional More recently, he has been pioneering a new approach to under- Data Analysis. standing the origin of classical spacetime in quantum gravity using entanglement of the underlying degrees of freedom. His inno- Fang’s expertise is in the area of functional data analysis (FDA), a vative thinking manifested itself in his essay for the Gravity Re- relatively new field in statistical science that regards data as a set search Foundation competition, winning him the First Award in of functions. Fang’s contributions to FDA have been fundamen- 2010. Finally, throughout his career, he has made many novel con- tal. His research program, characterized by great ambition and tributions applying string theory to other areas of physics, from breadth, continues to lay the foundations of FDA by framing it in nuclear matter at high densities to properties of black holes. terms of interpretable models with complex correlation structures that improve the efficiency of inference techniques. Methods he Prof. Van Raamsdonk received his BSc from the University of develops are useful for both representation and regression prob- British Columbia and his PhD from Princeton. He is currently a lems, and are accessible through the development of his publicly Professor in the Department of Physics at the University of British available software PACE, which has significantly extended the im- Columbia, where he has been since 2002. His work has been recognized by a Sloan Foundation Fellowship and a Canada Research pact of his research in statistical science and other fields. Fang is Chair. the consummate statistician, having a deep understanding of rigorous mathematical techniques, a broad statistical knowledge and Prof. Mark Van Raamsdonk will give a CRM lecture sometimes the ability to apply both in substantive problems. next year. Professor Yao will give his CRM lecture on January 15, 2015.

BULLETIN CRM–21 crm.math.ca

Call for Proposals

The CRM (Centre de recherches mathématiques) is soliciting applications for scientiﬁc activities to take place attheCRM.The proposals are divided in three categories: thematic semesters, of a duration of up to six months; workshops, conferences or schools, whose duration can vary from a couple of days up to two weeks; and targeted research periods for small groups of researchers lasting one to two weeks.

A commitment to diversity. Applicants are encouraged to ensure participation of women and underrepresented groups in the proposed activity. They should also plan to involve scientists at diﬀerent stages of their career, including postdoctoral fellowsand students, coming from diverse institutions and locations in Canada and abroad.

Thematic Programs — Letters of Intent All letters of intent will be examined by the CRM’s local and international scientific advisory committees. In case of a favourable The Centre de recherches mathématiques (CRM) in Montréal is decision, the prospective organizers will be asked to submit a de- soliciting applications for thematic programs to be held at the tailed proposal that will be evaluated by the International Scien- CRM during the calendar year 2017. The thematic semesters take tific Advisory Committee of the CRM. place from January to June (for the Winter–Spring Semester), and Proposals and further inquiries should be emailed to the CRM at from July to December (for the Summer–Fall Semester). The duration of each program is either one or two semesters. The scien- [email protected] tific activities of each semester typically include 3–5 workshops; Deadline to submit letters of intent: January 15, 2015 some funds are also allocated to support postdoctoral fellows, short- and long-term visitors and two Aisenstadt Chairs. Pro- Workshop, Schools and Conferences grams to be organized jointly (in whole or in part) with other institutes are welcome. The proposal for workshop, schools or conferences must bere- ceived by the CRM at least one year before the date proposed for The prospective organizers are invited to submit letters of intent the event. In exceptional cases, this deadline can be reduced to including the following information: less than a year. The proposals will be evaluated by the Interna- tional Scientific Advisory Committee of the CRM. • The title and dates of the proposed thematic program. The proposals must include: • The composition of the organizing committee, with the names and affiliations of its members. It is recommended that at least • the title and dates of the proposed event; one member of the organizing committee belongs to the local • the composition of the organizing committee, with the names, mathematical community, and at least half of the organizers affiliations and C.V. of its members; are external. • a detailed scientific description of the event (approximately 3–5 • A brief scientific description of the thematic semester (1–2 pages); pages). The description must emphasize the goals, the back- ground, and the timeliness of the proposed workshops, as well • a tentative list of the principal invited speakers, and their dis- as training activities (schools, mini-courses, etc.) for graduate cipline if other than mathematics or statistics; students and postdoctoral fellows. • a budget showing expected revenues and expenditures. • A tentative list of the principal invited speakers, including the names of the potential Aisenstadt Chairs. The candidates Proposals should be sent to the CRM by email at: proposal@ for the Aisenstadt Chairs should be world-famous mathemati- crm.umontreal.ca. cians. They are expected to spend at least one week attheCRM Targeted Research Periods during the semester, and to give a series of 3–4 lectures within the thematic program. One of these lectures should be acces- Targeted research periods for small groups of researchers are sible to a general mathematical audience. designed to seize research opportunities focussing on ground- breaking topics in the mathematical sciences. Proposals can be Preliminary contact and informal inquiries with the Director and submitted at any time and will be evaluated in a timely fashion. Deputy Director – Scientific Programs of the CRM are encour- Proposals should describe the research to be worked on during aged. Budget indications and information on sources of funding the period as well as justify its need and timeliness. C.V.’s of the will be provided. invited participants need to be submitted with the proposal.

BULLETIN CRM–22 crm.math.ca

Appel à projets

Le Centre de recherches mathématiques (CRM) vous invite à proposer des projets d’activités scientiﬁques. Les activités scientiﬁques se divisent en trois catégories : les semestres thématiques d’une durée de six mois; les conférences, ateliers ou écoles d’une durée de quelques jours à 2 semaines; et les périodes de recherche ciblée d’une durée d’une à deux semaines.

Un engagement à la diversité. Nous encourageons les chercheurs proposant une activité à s’assurer de la participation des femmes et d’autres membres de groupes traditionnellement peu représentés. De plus, nous encourageons les organisateurs à inviter des participants d’origines et de statuts divers : chercheurs en début de carrière, chercheurs chevronnés, boursiers postdoctoraux et étudiants, provenant d’universités de toutes les régions du Canada et de l’étranger.

Programmes thématiques – Lettres d’intention tional). Si leur lettre est agréée, les organisateurs seront invités à soumettre une proposition détaillée qui sera évaluée par leCo- mité scientifique international du CRM. Le Centre de recherches mathématiques (CRM) sollicite des propositions de programmes thématiques qui pourraient se tenir au Les lettres d’intention et les demandes d’informations doivent CRM (à Montréal) en 2017. Les semestres thématiques se dé- être envoyées au CRM par courriel à l’adresse ci-dessous. roulent de janvier à juin (semestre hiver-printemps) ou de juillet à [email protected] décembre (semestre été-automne). Les programmes peuvent du- Date limite : 15 janvier 2015 rer un ou deux semestres. Les activités scientifiques de chaque semestre incluent habituellement 3 à 5 ateliers; des fonds sont Ateliers, écoles et conférences également alloués pour des stages postdoctoraux, deux chaires Aisenstadt et des visiteurs faisant au CRM des séjours courts ou Les propositions pour les ateliers, écoles ou conférences doivent longs. Les programmes peuvent être organisés conjointement (en être reçues par le CRM au moins un an avant la date proposée totalité ou en partie) avec d’autres instituts. pour la tenue de l’événement. Exceptionnellement, l’échéance peut être réduite à six mois. Ces demandes seront évaluées par Les organisateurs éventuels sont invités à soumettre une lettre le Comité scientifique aviseur international du CRM. d’intention qui inclura les informations suivantes : Les demandes doivent inclure : – le titre et les dates du programme thématique proposé, – les membres du comité organisateur, incluant leurs établisse- – le titre et les dates proposées pour la tenue de l’événement; ments d’affiliation (il est recommandé que le comité inclue au – la composition du comité organisateur, avec les noms, les affi- moins un membre de la communauté mathématique locale et liations des membres du comité, et un bref curriculum vitæ; qu’au moins la moitié des organisateurs viennent de l’exté- – une description scientifique détaillée de l’événement sur 3 à 5 rieur), pages qui doit d’abord rappeler le contexte scientifique, l’im- – une description du programme thématique d’une à deux pages, portance du sujet à l’heure actuelle, et les défis, questions et mettant l’accent sur les buts, le contexte scientifique, la perti- conjectures que l’événement se propose d’étudier, en mettant nence et l’actualité des ateliers proposés, ainsi que sur les ac- l’accent sur la pertinence de tenir cet événement maintenant; tivités de formation (écoles et mini-cours) pour étudiants aux – une liste provisoire des principaux conférenciers avec leur affi- cycles supérieurs et stagiaires postdoctoraux, et liation et leur discipline (si autres que mathématiques ou sta- – une liste préliminaire des principaux conférenciers invités, tistique); ainsi que les noms des titulaires potentiels de la chaire Aisens- – un budget des revenus et les dépenses prévus. tadt. Ces titulaires potentiels doivent être des chercheurs de re- Les propositions d’activités doivent être envoyées à : projet@ nommée internationale, passer au moins une semaine au CRM crm.umontreal.ca. et donner une série de 3 à 4 conférences dans le cadre du programme thématique. Une des conférences doit être accessible Périodes de recherche ciblée à un large public de chercheurs et d’étudiants en mathéma- tiques. Les périodes de recherche ciblée sont conçues pour permettre à de petits groupes de chercheurs de saisir une occasion spéciale Nous encourageons les organisateurs éventuels à faire une de- de recherche portant sur des sujets de pointe en sciences mathé- mande de rencontre préliminaire et informelle avec le directeur matiques. Les propositions peuvent être soumises en tout temps du CRM ou le directeur adjoint responsable des programmes. Des et seront évaluées en temps opportun. Les propositions doivent indications sur le budget et les sources de financement leur seront décrire les questions sur lesquelles le groupe travaillera ainsi que fournies. justifier la nécessité et la pertinence d’organiser cette période de Toutes les lettres d’intention reçues seront examinées par les co- travail. Le curriculum vitæ des chercheurs doit être soumis avec mités scientifiques du CRM (le comité local et le comité interna- la proposition.

BULLETIN CRM–23 crm.math.ca

Le Bulletin du CRM A Word from the Director Volume 20, No 2 Automne 2014 Mot du directeur Le Bulletin du CRM est une lettre d’information à contenu scientifique, fai- As the pages of this Bulletin reflect, the sant le point sur les actualités du Centre CRM is as vibrant as always. The highly de recherches mathématiques. successful semester on “New Directions in ISSN 1492-7659 Lie Theory” was barely completed when the annual program aimed at covering Le Centre de recherches mathématiques (CRM) a vu le jour en 1969. Actuelle- Number theory from A to Z kicked off with ment dirigé par Luc Vinet, il a pour ob- the SMS summer school in June, featuring jectif de servir de centre national pour la the (then soon-to-be) 2014 Field medalist recherche fondamentale en mathéma- Manjul Bhargava and other distinguished tiques et leurs applications. Le person- lecturers. Meanwhile, numerous activi- nel scientifique du CRM regroupe plus ties, forming what is referred to as the d’une centaine de membres réguliers et CRM General Program, have taken place in de boursiers postdoctoraux. De plus, le parallel. CRM accueille chaque année entre mille Luc Vinet et mille cinq cents chercheurs du monde Le dynamisme du CRM se manifeste aussi entier. dans l’élargissement remarquable de l’en- universités partenaires ont augmenté leur appui financier au CRM relativement à ce Le CRM coordonne des cours de cycles semble de ses laboratoires. En effet, trois qu’il était dans le cadre de la demande pré- supérieurs et joue un rôle prépondé- nouveaux laboratoires ont été créés et in- rant (en collaboration avec l’ISM) dans tégrés récemment à savoir le Laboratoire cédente; je tiens à les en remercier chaleu- la formation de jeunes chercheurs. On de Probabilités, QUANTACT en mathéma- reusement. Dans le contexte financier diffi- retrouve partout dans le monde de nom- tiques actuarielles et financières et CAM- cile que nous connaissons au Québec, cela breux chercheurs ayant eu l’occasion de BAM en biologie mathématique. Cela té- témoigne d’une reconnaissance de l’excel- parfaire leur formation en recherche au moigne de la très grande vitalité de notre lence et de l’importance stratégique du CRM. Le Centre est un lieu privilégié de institut. Le CRM étend ainsi le spectre de CRM qui est sans équivoque. Sur la base rencontres où tous les membres bénéfi- ses recherches, rassemble encore davan- de ces appuis et avec le développement de cient de nombreux échanges et collabo- nos laboratoires, j’ai la conviction que nous rations scientifiques. tage les forces vives en sciences mathéma- tiques et favorise des interactions diversi- avons présenté une demande très forte qui Le CRM tient à remercier ses divers par- fiées. rend justice à la place prépondérante qu’oc- tenaires pour leur appui financier à sa cupe le CRM au niveau international. J’ose mission : le Conseil de recherches en We have been notified a few months ago espérer que cela nous vaudra un finance- sciences naturelles et en génie du Ca- that NSERC will be providing $1,15M per ment accru même en ces temps de com- nada, le Fonds de recherche du Qué- year to the CRM in the context of the pressions. bec — Nature et technologies, la Na- CTRMS program. In addition, $500K have tional Science Foundation, l’Université been awarded annually by NSERC again, In closing, I want to offer kudos to Chris- de Montréal, l’Université du Québec à tiane (Rousseau) for her election to the Ex- Montréal, l’Université McGill, l’Univer- to the CRM, Fields Institute and PIMS as a 3-year research partnership grant, for the ecutive Committee of the IMU until 2018, sité Concordia, l’Université Laval, l’Uni- to Frédéric (Gourdeau), the 2014 winner versité d’Ottawa, l’Université de Sher- establishment of the Institute Innovation of the Adrien-Pouliot prize of the CMS brooke, le réseau Mitacs, ainsi que les Platform (IIP). We rejoice in this great news fonds de dotation André-Aisenstadt et and are busily at work to meet and exceed and to Dani (Wise) for his election to the Serge-Bissonnette. the related expectations, and in particular Royal Society of Canada. I also wish to cordially thank Octav (Cornea) who Directeur : Luc Vinet to set up the IIP as a vehicle to expand our partnerships with industry. completed his term as Deputy Director Directrice d’édition : Galia Dafni and to warmly welcome Iosif (Polterovich) Conception : André Montpetit Le renouvellement de nos grandes sub- who has been recently appointed Deputy Centre de recherches mathématiques ventions nous a grandement accaparés Director – Scientific Program. Université de Montréal depuis l’an dernier. Après celle présentée C. P. 6128, succ. Centre-ville au CRSNG l’an dernier, nous venons tout Bonne lecture. Montréal, QC H3C 3J7 juste de soumettre au FRQNT la demande Téléphone : 514.343.7501 de renouvellement de notre octroi de re- Courriel : [email protected] groupement stratégique. À cet égard, il me Luc Vinet Le Bulletin est disponible à : fait plaisir de vous informer que toutes nos crm.math.ca/docs/docBul_fr.shtml.

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