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

Transcription

1 C R M C E N T R E D E R E C H E R C H E S M A T H É M A T I Q U E S Le Bulletin Automne/Fall 2014 Volume 20, N o 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 cohort 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 simpler algebraic insights and combinatorial constructions. Again the lecturers were asked to bring these to the fore, and did so. Bhargava himself is a fantastic lecturer and enjoyed not only gearing his own talks to a wider audience, but working with some of the other lecturers to appropriately focus their talks. The series by Melanie Wood (on all of the rings involved) and by Arul Shankar (on how to count the arithmetic objects involved) were masterful, and gave the listeners the opportunity to delve deep into this important subject with a great sense of the intuition behind these techniques. Bjorn Poonen managed to explain quite a bit of challenging algebraic geometry in a way that brought everyone along, giving insight into conjectures on ranks, as well as the new ways of exploiting Chabauty s method to determine all of the rational points on a given (high genus) curve. Takashi Taniguchi and Frank Thorne exposed the deeper analytic techniques, all the while respecting that the majority of the audience were algebraists so needed to be carefully led through the meaning behind each step. Dick Gross is one of the great arithmetic geometers of our time, and a fantastic expositor, as he once again showed, using the invariant theory (as explained by Jerry Wang) to attack rational points on hyperelliptic curves. There were also excellent one-off lectures by Wei Ho, Jennifer Park, Eknath Ghate (though actually two one-off lectures) and Michael Stoll, on various exciting parts of the theory. And a rather different perspective explained by the always inspiring and entertaining Jordan Ellenberg. Each day was rounded off by an extensive problem session, led by some of the lecturers, challenging the participants to immerse themselves in the details of the theory of that day. Manjul Bhargava at the CRM in 2006, as Aisenstadt Chair ( Photo Réjean Meloche) The importance of the ideas presented at the summer school was dramatically underscored two months later in Seoul, where Manjul Bhargava was awarded one of the four Fields medals at the ICM for developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptic curves. The CRM summer school will be the object of a proceedings volume to which all of the key participants have enthusiastically agreed to contribute, and which will surely have a broad impact as one of the essential references in an exciting and rapidly evolving part of number theory. For more information on activities related to the thematic year go to IN THIS ISSUE / DANS CE NUMÉRO The Aisenstadt Chair Lectures by MASAKI KASHIWARA.. 3 SABIN CAUTIS, the 2014 André-Aisenstadt Prize recipient, describes his research La Plate-forme d innovation des instituts The Magnificent Kerr Metric, by NIKY KAMRAN, 2014 CRM-Fields-PIMS Prize winner FLORIN DIACU et les incontournables mathématiques.. 11 ALESSIO FIGALLI delivers the first CRM Nirenberg Lectures in Geometric Analysis The CRM ISM Postdoctoral Fellows

2 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 Representations 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 parallèle, 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 complètement 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 Investigación en Matemática) à Guanajuato, Mexique, en décembre 2012, et la seconde à l Université Nicolas Copernic à Torun, Pologne, en septembre La troisième a eu lieu à l UQAM en juin 2014 et la quatrième est prévue à Guanajuato en juin Pour la conférence à l UQAM, le comité scientifique était aidé d un comité 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- CIM et du CRM. Les ARTA portent sur les sujets suivants : structure des algèbres de dimension finie et de leurs catégories de modules, méthodes et conjectures homologiques, algèbres de dimension globale finie et auto-injectives, aspects combinatoires, méthodes de géométrie algébrique, catégories triangulées et théorie de l inclinaison, relations avec la théorie de Lie, la théorie des singularités, les modules de Cohen-Macauley, les groupes quantiques et autres structures algébriques. La conférence a rassemblé 70 participants provenant de 18 pays différents, dont plus de la moitié d étudiants et jeunes chercheurs. Au total, 35 exposés ont été donnés, dont 13 de 50 minutes et le reste de 30. Plusieurs exposés (dont 5 de 50 minutes) ont été confiés à des jeunes chercheurs. Parmi les questions abordées figurent l étude des algèbres auto-injectives et de leurs catégories de modules, et les relations entre théorie des représentations et algèbres amassées. Les participants à l ARTA se connaissant tous de longue date, des échanges mathématiques ont eu lieu en marge de la conférence tout au long de celle-ci. Plusieurs travaux entamés de recherche conjointe ont avancé et de nouvelles collaborations de recherche ont émergé. Dans certains cas, de nouveaux résultats ont été prouvés. Ceci laisse prévoir à brève échéance de nouvelles publications. Frédéric Gourdeau lauréat du prix Adrien-Pouliot 2014 de la SMC Le professeur Frédéric Gourdeau de l Université Laval est le lauréat 2014 du prix Adrien-Pouliot pour sa contribution exceptionnelle à l enseignement des mathématiques au Canada. Le prix sera remis à l occasion de la Réunion d hiver 2014 de la SMC à Ottawa. «Frédéric Gourdeau a marqué l enseignement des mathématiques tant dans les classes primaires que les cours aux étudiants diplômés», précise Peter Taylor, président du Comité d éducation de la SMC. «Il a de plus fait sa marque à la fois au sein de la très active communauté qui s intéresse à l enseignement des mathématiques au Canada et sur la scène mondiale, comme en témoigne son invitation à prononcer une prestigieuse conférence au Congrès international 2012 à Séoul.» Dynamique et passionné, Frédéric Gourdeau a joué un rôle important dans plusieurs projets d envergure au pays, dont la création en 2006 de la revue Accromαth. Il a fondé en 1997 l Association québécoise des jeux mathématiques (AQJM), qu il préside depuis. L AQJM organise annuellement le concours mathématique attirant la plus forte participation au Québec. Extraits de l annonce faite par la Société mathématique du Canada. BULLETIN CRM 2

3 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 Mathematical Sciences in Kyoto (RIMS) gave a series of three lectures as Aisenstadt chairholder during the workshop on Combinatorial Representation Theory which was part of the thematic semester New Directions in Lie Theory. Professor Kashiwara has received many honours throughout his long career, to recognize the fundamental contributions he made to several fields of mathematics, including algebraic analysis, microlocal analysis, D-module theory, Hodge theory and representation theory. Of particular relevance to the topics discussed during the workshop is his work developing the theory of crystal bases for integrable modules of quantum groups. These crystal bases contain most of the information of the module and can be represented by a directed graph with edges labelled by simple roots, hence directly relating the study of quantum groups to combinatorial ideas. Masaki Kashiwara Professor Kashiwara s first lecture was based on recent work that he did with Professors S. Kang of the Seoul National University and M. Kim of the Korea Institute for Advanced Study, which is the latest in a series of results modelled after the famous representation theoretic result called Schur Weyl duality. This duality is a beautiful classical result concerning the representation theory of the general linear group GL N and the symmetric group S l, which both act on the space (C N ) l. The group S l permutes the tensor factors, and C N is the natural representation for GL N, which acts diagonally on the l factors by simultaneous multiplication. Schur Weyl duality says that the actions of the two groups commute, and that if we decompose the tensor products in terms of simple modules for one group, that will also determine the decomposition for the other group. More precisely, we have a decomposition as a (S l, GL N )-bimodule (C N ) l (U λ W λ ), λ where U λ (respectively W λ ) are the simple modules of S l (resp. GL N ) indexed by partitions λ of l with at most N parts. The assignment M M (C N ) l for an S l -module M is an exact functor, and induces an equivalence of categories between finitedimensional S l -modules and GL N -modules when l N. One would like to understand this as a statement about the actions of two associative algebras. We recall that the representation theory of S l and its group algebra are the same. Similarly, the finitedimensional representation theory of GL N is the same as that of the Lie algebra sl N, which is equivalent to that of its universal enveloping algebra U(sl N ), which is an associative algebra. In this case, we can understand U(sl N ) as acting on the tensor product by taking advantage of its comultiplication, which is cocommutative. Schur Weyl duality can then be interpreted as saying that the images of these two associative algebras in End((C N ) l ) are each other s centralizers. Jimbo generalized the above construction in 1986 to a result concerning the action of the quantum group U q (sl N ) on V l where V is the natural representation of U q (sl N ). The quantum group U q (sl N ) can be considered as a deformation of the universal enveloping algebra U(sl N ) by a parameter q, so that if we set q = 1 we recover the original algebra. This deformation results in noncocommutative 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 H q (l), a deformation of the group algebra of S l, on V l. 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 M V l, for an H q (l)-module M, is exact, and, when l N, it induces an equivalence of categories between finite-dimensional H q (l) modules and U q (sl N ) modules of level l. We can consider this result as a quantized Schur Weyl duality. There are affine versions for the previous results provided by Chari Pressley, Cherednik and Ginzburg Reshetikhin Vasserot at various times in the mid 1990s. The affine quantum group U q (ŝl N) can be considered as a deformation of the universal enveloping algebra for the loop algebra sl N C[t ±1 ]. It acts on V l, where V is the natural representation, and one is interested in a universal R-matrix to intertwine tensor products of representations. An action by the affine Hecke algebra Ĥq(l) (which is a semidirect product of H q (l) with C[t ±1 1,..., t±1 n ]) is defined on V l using the R-matrix, and it was proved that these actions commute and that the algebras satisfy the double centralizer property. Once again the assignment M M V l for a Ĥq(l)-module M is exact, and when l N it induces an equivalence of cat- BULLETIN CRM 3

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

5 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 Mathematics 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 he was a recipient of the highly coveted Alfred P. Sloan Fellowship. Sabin Cautis works at the crossroads of algebraic geometry, representation theory, and lowdimensional 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 algebraic geometry to questions in representation theory. For example, the Kazhdan Lusztig conjectures, proven in the early 1980s by Beilinson Bernstein and Brylinski Kashiwara, involved translating the conjectures into more geometric language consisting of D-modules on flag varieties. In much of my work we invert this perspective and also introduce a categorical ingredient. Instead of using geometric tools, such as D-modules, to obtain results in representation theory we apply techniques in (categorical) representation theory to understand geometric situations. One could call this categorical geometric representation theory. For example, in a project with Kamnitzer and Licata we constructed a natural derived equivalence DCoh ( T G(k, N) ) ( DCoh T G(N k, N) ). (1) Here T G(k, N) denotes the cotangent bundle to the Grassmannian of k-planes in C N while DCoh(X) denotes the (derived) category of coherent sheaves on X. Notice that we have T G(k, N) = T G(N k, N) but this isomorphism is not canonical since it requires choosing a bilinear form on C N. Constructing a natural equivalence as in (1) is surprisingly difficult. Earlier work of Namikawa had shown that an obvious guess does not induce an equivalence while subsequent work of Kawamata had conjectured a detailed form of these equivalences when k = 2. To construct the equivalence in (1) we first had to define a categorical sl 2 action E DCoh ( T G(k 1, N) ) E DCoh ( T G(k, N) ) F F E DCoh ( T G(k + 1, N) ) E F F involving the categories DCoh(T G(k, N)) for all 0 k N. Here E and F are functors analogous to the usual generators e = ( ) and f = ( ) of sl 2. The word categorical refers to the fact that the weight spaces are categories rather than vector spaces. Now consider a representation V = l V (l) of sl 2 where V (l) are the weight spaces. A fundamental result in representation theory tells us the element ( ) SL2 (C), which can be expressed in terms of es and fs, induces a natural isomorphism V (l) V ( l). It turns out this result has a categorical analogue. Consider an sl 2 action on categories C = l C(l) (in our example above C(l) = DCoh ( T G(k, N) ) where l = 2k N). One can define a complex of functors built from compositions of Es and Fs which lifts the isomorphism V (l) V ( l) to an equivalence C(l) C( l). A version of this result had been used by Chuang Rouquier to prove that certain blocks of symmetric groups in positive characteristic are derived equivalent. Adapting their method to our geometric situation above gave us the equivalence in (1). In subsequent work with Kamnitzer we showed that a categorical sl n action gives rise to an action of the braid group. This is another result in categorical (also known as higher) representation theory 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 can also be used to construct homological knot invariants, such as Khovanov homology. In more recent work with Licata we tried to develop the categorical representation theory of Heisenberg algebras and vertex (continued on page 16) BULLETIN CRM 5

6 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 (le Fields Institute et le PIMS) ont reçu cet été une nouvelle subvention du CRSNG, appelée Plate-forme d innovation des instituts. L objectif du CRSNG (pleinement partagé par les trois instituts eux-mêmes) est que les instituts utilisent cette subvention pour susciter de nouvelles collaborations entre des chercheurs en sciences mathématiques et des entreprises, notamment grâce à des ateliers de résolution de problèmes, des ateliers de modélisation mathématique pour les entreprises, des ateliers de maillage universités/entreprises centrés sur des domaines précis des mathématiques, des formations de courte durée et des forums de l emploi. Pour l aider à atteindre les objectifs de la PII, le CRM a constitué un comité de scientifiques occupant des postes de gestion dans les entreprises. Ce comité, le Comité consultatif du CRM pour la Plateforme d innovation des instituts (CCPII), donnera des conseils à la direction du CRM sur les meilleures façons de développer les collaborations universités/entreprises dans le domaine des sciences mathématiques. Il sera impliqué dans la planification des activités mentionnées ci-dessus et jouera un rôle de premier plan dans le lancement de nouvelles activités. Il participera à des échanges avec les comités semblables créés par les autres instituts canadiens. Son président fera un rapport annuel au Conseil d administration du CRM. Voici la liste des membres du CCPII. M. Michel Carreau directeur des énergies renouvelables, Hatch M. Carreau obtint son doctorat en physique du Massachusetts Institute of Technology en 1991 et effectua un stage postdoctoral à la Boston University de 1991 à Après avoir travaillé pour Synexus Global, M. Carreau accepta un poste chez Hatch, une compagnie fournissant (entre autres) des services-conseils en procédés et en affaires et des services à l exploitation pour l industrie des mines et métaux, de l énergie et des infrastructures. Chez Hatch M. Carreau dirige une équipe d ingénieurs performante qui conçoit et implante des projets d énergie renouvelable (éolienne, solaire, géothermique ou hydro-électrique), incluant aussi des solutions hybrides. M. Carreau collabore régulièrement avec des chercheurs universitaires. M. Denis Faubert président-directeur général, CRIAQ M. Faubert détient une maîtrise et un doctorat en physique de l Université Laval. Après avoir travaillé pour plusieurs organismes, dont le Centre de recherche pour la défense, M. Faubert fut nommé directeur général de l IREQ, poste qu il occupa de 2007 à Il est maintenant le président-directeur général du Consortium de recherche et d innovation en aérospatiale au Québec (CRIAQ), un organisme sans but lucratif dont la mission est d accroître la compétitivité de l industrie aérospatiale et d améliorer la base des connaissances collectives dans ce secteur grâce à une meilleure formation des étudiants. M. Pierre Trudeau vice-président, développement des affaires, GIRO Après des études de recherche opérationnelle, M. Trudeau fonda, avec quatre autres personnes, une compagnie spécialisée dans la planification des horaires d équipages aériens (la compagnie AD OPT). Il travailla ensuite pendant plusieurs années pour la compagnie ILOG, comme directeur de la commercialisation de ses produits d optimisation. M. Trudeau est maintenant vice-président de GIRO, une compagnie fournissant aux sociétés de transport public et aux services postaux des logiciels pour planifier, optimiser et gérer les activités des véhicules et de leur personnel. GIRO a des liens étroits avec les chercheurs en optimisation et a présenté plusieurs problèmes aux ateliers de résolution de problèmes organisés par le CRM. M me Roxana Zangor directrice de la sécurité et de la fiabilité des produits, Pratt & Whitney Canada Après des études de mathématiques en Roumanie, M me Zangor reçut une bourse Fulbright et obtint une maîtrise en enseignement des mathématiques et un doctorat en mathématiques appliquées de l Université de Syracuse aux États-Unis. Elle travailla d abord comme scientifique senior au United Technologies Research Center (UTRC), dans l état du Connecticut, où elle fit profiter le centre de son expertise en méthodes numériques, forage de données et traitement d images (entre autres domaines). Depuis 2006 elle travaille chez Pratt & Whitney Canada, une entreprise de classe mondiale en conception, manufacture et entretien d aéronefs. De plus, le CRM a embauché M. Stéphane Rouillon comme agent de développement de partenariats pour soutenir ses efforts en mathématiques industrielles, particulièrement les efforts reliés à la PII. M. Rouillon a étudié à l École Polytechnique de Montréal et détient un baccalauréat en génie physique, une maîtrise en mécanique des fluides et un doctorat en recherche opérationnelle de cette institution. Il a acquis une vaste expérience industrielle en travaillant pour diverses entreprises, en particulier SNC Lavalin, et a réalisé plusieurs projets de vulgarisation scientifique et mathématique. M. Rouillon travaillera au CRM en étroite collaboration avec Odile Marcotte, directrice adjointe aux partenariats. BULLETIN CRM 6

7 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 form by Einstein in 1915, are given in the absence of sources by R ij = 0, (1) where the R ij 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 mathematical properties. An example of central importance is given by the two-parameter family of Kerr metrics, which describe the exterior gravitational field of the final state of an isolated object that has collapsed to form a rotating black hole. This metric, which was discovered in 1963 by Roy Kerr, starting from geometrical rather than physical assumptions, is given by ds 2 = ( ) dr U (dt a sin2 θ dφ) 2 2 U + dθ2 where sin2 θ U (a dt (r2 + a 2 ) dφ) 2, (2) U := r 2 + a 2 cos 2 θ, = r 2 2Mr + a 2. (3) This metric depends on two real parameters M and a which we assume to satisfy the inequality M 2 > a 2. (4) The limiting case a 2 = M 2 defines the so-called class of extreme Kerr solutions. The physical interpretation of the parameters M and a is that they correspond respectively to the mass and angular momentum per unit mass of the black hole, measured from infinity. The coordinates in the metric (2) are defined in the range r > r 1, < t < +, (θ, φ) S 2, where r 1 := M + M 2 a 2, (5) but the coordinate r can be extended across the locus r = r 1, which becomes a null hypersurface corresponding to the global past event horizon for the Kerr metric. The hypersurface r = r 0 := M M 2 a 2 is the Cauchy horizon for the metric. In the limit a = 0, the Kerr solution reduces to the well-known spherically symmetric Schwarzschild solution, discovered in See prixcfp14_an.shtml for the full citation. Niky Kamran (on the left) and Luc Vinet (on the right) The Kerr metric enjoys important symmetry properties that we will review in this and the next section. It is first of all manifest that it admits the two commuting Killing vector fields φ and t corresponding to the invariance of the space-time geometry under rotations around the symmetry axis of the black hole and time translations. It is also easy to see that the generator of time translations is time-like at sufficiently large distances, but that it changes causal type and becomes space-like as one gets closer to the event horizon r = r 1 in a region with compact closure, the ergosphere, which is defined by the inequality r 2 2Mr + a 2 cos 2 θ < 0. (6) One of the consequences of the presence of the ergosphere is that any conserved energy arising from the invariance of the geometry under time translations may change sign and become negative within the ergosphere. This makes the analysis of wave equations and in particular the proof of energy decay results in Kerr geometry challenging since the standard approaches to these problems typically require the presence of a positive energy, giving rise to a Hilbert space setting in which to solve the initial value problem and study energy propagation. In the limiting case of the Schwarzschild metric, that is when a = 0, the ergosphere is empty and the conserved energy is positive outside the event horizon. The physical importance of the family of Kerr metrics is underscored by the celebrated black hole uniqueness theorems of Carter and Israel [5, 23], which characterize this metric as the unique set of solutions of the boundary value problem corresponding to a black hole equilibrium state. It should be remarked that the proof of the original version of the black hole uniqueness theorem depends critically on analyticity assumptions on the spacetime geometry. These analyticity assumptions have been lifted and significantly weakened in important recent work of Alexakis, BULLETIN CRM 7

8 Ionescu and Klainerman [1], where uniqueness theorems have been proved using very different methods. Separability properties The Kerr metrics enjoy an extraordinary set of separability properties, that led Chandrasekhar to write in [7] that they have the aura of the miraculous. These separability results were first discovered for the Hamilton-Jacobi equation for the geodesic flow and for the Klein Gordon equation by Carter in 1968, and were subsequently shown to hold by Teukolsky in 1972 for the equations governing the components of maximal spin weight of linearized gravitational fields, test Maxwell fields, and test Dirac fields of zero rest-mass (Weyl neutrinos). The separability of the massive Dirac equation in Kerr was proved in 1976 in a remarkable paper by Chandrasekhar, using a different approach from Teukolsky s. We refer to Chandrasekhar s treatise [8] for a detailed account of all these results, which show that every wave equation of physical interest in Kerr geometry can be completely separated into a set of ordinary differential equations governing individual modes for the waves being described. It is important to remember that the complete separability of these wave equations cannot be simply be accounted for by the isometries of the Kerr metric. This can be easily understood when thinking of the complete separability in the Staeckel sense of the Hamilton Jacobi equation for the geodesic flow, which leads in turn to the complete integrability of the geodesic flow itself. Indeed, there exists, in addition to the conserved linear momenta arising from the commuting Killing vector fields φ and t, a quadratic first integral of the geodesic flow, which was discovered by Carter. It is of the form K = K ij p i p j. (7) where the symmetric tensor K ij appearing in the above expression is therefore a Killing tensor. The first integral K is deeply connected to all the separability properties of the Kerr metric. It has a very special structure, as it can be factored into the square of a skew-symmetric Killing Yano tensor f ij, satisfying K ij = f ik f k j, (8) i f jk + j f ik = 0. (9) While it can be seen quite directly from the separation of variables that the symmetry operator K for the Klein Gordon operator given by K = i K ij j (10) admits the separated solutions of the Klein Gordon equation in Kerr as eigenfunctions, the relationship between the separability of the Dirac equation and the Killing tensor (8) is more hidden. It is Carter and McLenaghan [6] who proved a remarkable result stating that the Killing Yano tensor admitted by the Kerr geometry gives rise to an operator that commutes with the Dirac operator in Kerr and admits the separated solutions as eigenfunctions. In the following section we show how the separability leads to decay results for solutions of the Dirac equation in the Kerr metric. BULLETIN CRM 8 crm.math.ca Decay of Dirac spinors The long term dynamics of Dirac spinors in the Kerr metric were investigated in a series of joint papers with Felix Finster, Joel Smoller and Shing-Tung Yau [13, 14, 19]. Our approach used the separability of the Dirac equation into ordinary differential equations to construct an integral spectral representation for the solution of the Cauchy problem. We state a couple of theorems in some detail, beginning with the solution of the Cauchy problem [19]. Theorem 1. For every Cauchy data ψ 0 C 0 (R S 2 ) 4 there is a unique solution to the Cauchy problem for the Dirac equation (iγ j D j m)ψ(t, x) = 0, ψ(0, x) = ψ 0 (x), (11) given by a (convergent) Fourier decomposition of the form ψ(x, t) = 1 π + 2 e iωt k,n Z a,b=1 t kωn ab ψa kωn (x) ψb kωn ψ 0 dω. (12) The integer k is the quantum number corresponding to the projection of the angular momentum on the axis of symmetry of the Kerr metric and n is a generalized total angular momentum quantum number arising from the separation of variables. The coefficients t kωn ab are generalized transmission coefficients and the ψa kωn are solutions of the Dirac equation which arise from the separation of variables and behave for ω > m like incoming spherical waves for a = 1 and outgoing spherical waves for a = 2. For ω < m, the ψa kωn are linear combinations of both incoming and outgoing spherical waves near the event horizon. We now consider the long-term dynamics of the solution to the Cauchy problem [19]. Theorem 2. Consider the Cauchy problem for the Dirac equation in the Kerr geometry, where the initial data ψ 0 is in L 2 ((r 1, ) S 2 ) 4 and in L loc near the horizon, i.e., ψ 0(x) < c for x (r 1, r 1 + ɛ) S 2. Let δ > 0 be given, let R > r 1 + δ and consider the compact space-like hypersurface K δ,r of E given by K δ,r := {(t, r, θ, φ) r 1 + δ r R, t = const.}. (13) Then the probability for the Dirac particle to be inside K δ,r tends to zero as t, that is lim ( ψγ j ψ)(t, x) ν j dµ = 0, (14) t K δ,r where ν j denotes the future-pointing normal to K δ,r and dµ denotes the induced volume element on K δ,r. Theorem 2 implies that the Dirac spinor ψ tends to zero in L loc as t, or equivalently that the Dirac particle must eventually either disappear into the black hole, or escape to infinity. One may like to compute the probability of these outcomes in terms of the Cauchy data, as well as the actual rate of decay of the Dirac spinor in a compact spatial. These questions were studied in [14], assuming that only finitely many angular modes are present in the initial data, meaning that k k 0, l l 0, (15)

9 in the Fourier expansion (12). Note that this finiteness assumption is not required in Theorem 2. We now summarize the main results for the rates of decay, and refer to [14] for the probability estimates. Theorem 3. Consider the Cauchy problem as in Theorem 2, with initial data normalized by ψ 0 ψ 0 = 1. Suppose that the boundedness condition (15) on the angularmodes of the Cauchy data are satisfied. Then we have: (i) If for any k and n, or then, as t, lim sup ψ2 kωn ψ 0 = 0, (16) ω m lim inf ω m ψkωn 2 ψ 0 = 0, (17) ψ(x, t) = ct 5/6 + O(t 5/6 ɛ ), (18) where c = c(x) 0, and ɛ < 1/30. (ii) If for all k, n and a = 1, 2, ψa kωn ψ 0 = 0 for all ω in a neighborhood of ±m, then for any fixed x, ψ(x, t) decays rapidly in t. It should be noted that the rate of decay obtained in Theorem 3 is slower than the rate of decay of t 3/2 for the solutions of the Dirac equation in Minkowski space, corresponding to compactly supported initial data, [14]. We also briefly mention a result [13, 14] on the nonexistence of time-periodic solutions of the Dirac equation: Theorem 4. The Dirac equation does not admit any time-periodic L 2 solutions in any Kerr solution with a 2 < M 2. 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 noteworthy that the conclusion of Theorem 4 does not hold in the extreme Kerr case a 2 = M 2. Indeed, it was shown in [26] that the Dirac equation admits bound state solutions in the extreme Kerr metric. We conclude this section on Dirac spinors by mentioning the important results of Häfner [21] on fermion creation in the Kerr black hole, and Häfner and Nicolas on the scattering of massless Dirac fields [22]. Decay of massless scalar waves Even though the Kerr metric admits an ergosphere where the generator of time translations becomes space-like, it turns out that we were nevertheless able to obtain the convergent Fourier integral representation for the solutions and the corresponding decay results that we just reviewed in the previous section for Dirac spinors in a Hilbert space setting. This is because the Dirac equation admits a conserved current of order zero that is positive throughout the exterior of the event horizon, and in particular everywhere within the ergosphere, making it possible to define a positive-definite inner product that is compatible with the separation of variables and conserved throughout the evolution. For the massless scalar wave equation, no such inner product is available and one has to deal with the non-positivity of the conserved energy. These issues were addressed in [15,16]. In [15], we proved that there exists a convergent integral spectral representation for the solution of the Cauchy problem for the massless scalar wave equation with C compactly supported data. The proof used the spectral theory of definitizable operators on Pontrjagin spaces, together with resolvent estimates needed to deform contours to within an arbitrarily small neighbourhood of the real axis. The deformation to the real axis and the corresponding decay results were established in [16] using improved estimates, but at the cost of dealing with modes of fixed azimuthal frequency ψ(t, r, θ, φ) = e ikφ u(t, r, θ). (19) The very important mode stability theorem of Whiting [31] played a crucial role in our analysis. The proof of decay results for general data rather than the special ones satisfying (19) requires Morawetz multiplier techniques, which are more robust that the ones based on separation of variables. For slowly rotating black holes described by Kerr metrics with a M, a proof of decay was given by Dafermos and Rodnianski [9]. Recently, Dafermos, Rodnianski and Shlapentokh-Rothman [10] achieved a breakthrough by proving decay for scalar waves in the full range a 2 < M 2. Other important contributions to this problem are due to Tataru [28], Tataru and Tohaneanu [29], and Andersson and Blue [2] (who made use of the symmetry associated to the Killing tensor K ij in their approach to decay). Finally, we mention that the very interesting problem of superradiance for scalar wave packets in Kerr geometry can be studied rigorously using the previous theorems [17, 18]. A key result that makes this analysis possible is the time-independent energy estimate proved by Finster and Smoller [20] for outgoing massless scalar waves in Kerr. Some perspectives We conclude this survey by listing a few perspectives. This list is definitely nonexhaustive: The decay problem for solutions of the Klein Gordon equation governing massive scalar fields in Kerr geometry, as opposed to massless ones, can lead to different conclusions. For example, there exist exponentially growing finite-energy solutions of the Klein Gordon equation in Kerr; see [27]. Earlier results on the Klein Gordon equation in Kerr can also be found in [4]. It would be of definite interest to further investigate the behaviour of massive fields in Kerr. The Kerr metric admits higher-dimensional generalizations known as the Myers Perry metrics. One can prove decay results for Dirac spinors in the 5-dimensional Myers Perry metric [11], but there are still many open problems that deserve further investigation in this higher-dimensional setting, especially the study of wave equations. BULLETIN CRM 9

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