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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 arith- metic 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.
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