SM Exposés 2014-2015
No. SM-139 27/06/2015 14:00 ~ 16:00 Salle U/V
An introduction to the homotopy theory of Berkovich analytic spaces
John WELLIAVEETIL, Université Paris 6
Abstract: Ever since K. Hensel introduced the field of p-adic numbers, there have been several attempts at developing a suitable theory of geometry over non-Archimedean valued fields analogous to the theory over the complex numbers. In this talk, we discuss Berkovich's approach to this problem, using which one can associate a Hausdorff topological space with favourable properties to a variety of finite type over a non-Archimedean real valued field. In 2010, Hrushovski and Loeser using Model theoretic techniques were able to prove profound results concerning the homotopy type of the Berkovich analytification of a variety. We discuss these results and if possible give a rough outline of their constructions.
No. SM-138 20/06/2015 14:00 ~ 16:00 IHES (Amphithéatre Léon Motchane)
(II) Spectral side of Arthur-Selberg Trace formula
YU Hongjie (余红杰), Université Paris-Sud
Abstract: Continuing with last presentation, I will explain briefly the proof by Lafforgue for the geometric side of trace formula over a function field. Then I give a statement of Langlands spectral decomposition on which the spectral side of trace formula bases and a statement of trace formula itself.
No. SM-137 13/06/2015 14:00 ~ 16:00 Salle U/V
Deformations of Kac-Moody algebras
Alexandre BOUAYAD, University of Cambridge
Abstract: I will present a new approach to deformations of a (symmetrizable) Kac-Moody algebra -- of it universal enveloping algebra more precisely -- and of its modules. I will focus on the case of the Lie algebra sl2, and I will briefly report on ongoing work concerning the general case. The construction is meant to be both elementary and systematic. I will also explain how such deformations can give a positive answer to conjectures of Frenkel-Hernandez on Langlands duality in quantum groups.
No. SM-136 06/06/2015 14:00 ~ 16:00 Salle U/V
Hall algebras and quantum groups
Alexander MINETS, ENS
Abstract: I will give the definition of the Hall algebra of an abelian category together with a few motivations, and prove the classical theorem of Ringel, which realizes positive half of a quantum group as the Hall algebra of the category of representations of the corresponding quiver. If time permits, I will also say a few words about canonical bases and/or several "derived" versions of the notion of Hall algebra.
No. SM-135 30/05/2015 14:00 ~ 16:00 IHES (Amphithéatre Léon Motchane)
(I) Truncated kernel in Arthur-Selberg Trace formula over a function field
YU Hongjie (余红杰), Université Paris-Sud
Abstract: Arthur-Selberg Trace formula is an important tool in Langlands Program. For a uniform lattice in a locally compact abelian group, the formula reduces to the Poisson summation formula. For general cases, Arthur use a truncation process to obtain a "trace formula". I will explain in this talk which kind of trace formula we expect to have, and give the truncated kernel for the group GL(n) with Adèle coefficients over a function field. In such case, the truncation process can be done by the correspondence between Adèles and vector bundles on a projective smooth curve.
No. SM-134 02/05/2015 14:00 ~ 16:00 Salle U/V
Formes modulaires surconvergentes et prolongement analytique
Valentin HERNANDEZ, Université Paris 6
Abstract: Après avoir rappelé les définitions de courbes modulaires, et certains outils d'étude de celles-ci, on introduira la notion de formes modulaires surconvergentes et on démontrera par la méthode dite de prolongement analytique, due à Buzzard-Kassaei, le théorème de Coleman, selon lequel toute forme surconvergente de poids $k$, propre pour les opérateurs de Hecke, de pente strictement plus petite que $k-1$ est une forme modulaire classique.
No. SM-133 25/04/2015 14:00 ~ 16:00 Salle U/V
Classes de Hodge absolues
LIN Hsueh-Yung(林學庸), École Polytechnique
Résumé: Grace au théorème de comparaison entre les cohomologies de de Rham et de Betti à coefficients complexes du à Grothendieck, une piste arithmétique est ouverte pour comprendre une partie de la topologie des variétés algésbriques complexes, indépendante de la topologie de C. Un développement majeur dans cette direction était du à Deligne qui a introduit la notion de classe de Hodge absolue et a démontré que toute classe de Hodge dans une variété abélienne l'est absolument. Outre les propriétés de bases de ces derniers objets, je présenterai l'idée motivique qui mène à leur introduction et les cycles motivés d'Y. André, puis la preuve du théorème de Deligne ci-dessus, améliorée par Y. André et C. Voisin, et rédigée par F. Charles et C. Schnell dans un article de survol.
No. SM-132 11/04/2015 14:00 ~ 16:00 Salle U/V
An introduction to infinity categories
Tony Yue YU (余越), Université Paris 7
Abstract: I will give an informal introduction to the theory of infinity categories following Lurie. Motivations and key ideas will be presented. Time permitting, I will explain how infinity categories are used in my recent joint work arXiv:1412.5166 with M. Porta, on the GAGA theorem for analytic stacks.
No. SM-131 04/04/2015 14:00 ~ 16:00 Salle U/V
Autour des conjectures de Hodge, Tate et Mumford-Tate sur les variétés abéliennes
Victoria CANTORAL-FARFAN, Université Paris 7
Cliquez ici pour les notes.
No. SM-130 07/02/2015 14:00 ~ 16:00 Salle U/V
KLR algebras
Ruslan MAKSIMAU, Université Paris 7
Abstract: The KLR algebras are invented by Khovanov-Lauda and Rouquier to categorify the negative part of a quantum group. These algebras appear naturally as Borel-Moore homology of a version of the Steinberg variety. On the other hand, they have a nice diagrammatic description.
In my talk I will explain the geometric construction of KLR algebras, I will give a small rank example (nil-Hecke algebra) and I will finish by categorification of the negative part of a quantum group.
No. SM-129 31/01/2015 14:00 ~ 16:00 Salle U/V
Théorie de Lubin-Tate non abélienne
Arthur-César LE BRAS, Université Paris 6
Résumé: Soit K un corps p-adique, O_K son anneau des entiers. Inspirés par la théorie des courbes elliptiques à multiplication complexe, Lubin et Tate ont introduit et étudié, il y a tout juste cinquante ans, les O_K-modules formels. Les espaces de déformations de ces objets sont appelés espaces de Lubin-Tate. On peut penser aux espaces de Lubin-Tate comme à des analogues locaux de certaines variétés de Shimura. Leur géométrie est encore mystérieuse mais plus accessible que celle de ces dernières. Son étude est motivée par le fait que la correspondance de Langlands locale se réalise (au moins partiellement) dans la cohomologie des espaces de Lubin-Tate. J'essaierai dans cet exposé de présenter quelques résultats sur ce thème.
No. SM-128 03/01/2015 15:30 ~ 17:30 Salle U/V
The vertex representations of quantum toroidal algebras
YANG Yaping (杨亚萍), Northeastern University
Abstract: The quantum toroidal algebras were first introduced by Ginzburg-Kapranov-Vasserot in studying the Langlands reciprocity for algebraic surfaces. They are quantization of the universal enveloping algebra of the universal central extension of the "double loop" Lie algebras (the classical toroidal algebras). Their representation theory seems to be rich and promising. In this talk, I will talk about the geometric approach to the quantum toroidal algebras by Ginzburg-Kapranov-Vasserot. The main point is the algebra of Hecke operators for vector bundles on an elliptic surface is a homomorphic image of the quantum toroidal algebra. I will also talk about the algebraic approach by Saito to construct the vertex representations of the quantum toroidal algebra.
References:
1. Langlands reciprocity for algebraic surfaces, by Ginzburg, Kapranov, Vasserot.
2. Vertex Algebras and Algebraic Curves, by Frenkel and Ben-Zvi.
3. Quantum toroidal algebras and their vertex representations, by Saito.
No. SM-127 03/01/2015 14:00 ~ 15:00 Salle U/V
The rank spectral sequence of Algebraic K-theory
SUN Fei (孙飞), Université Paris 6
Abstract: Bruno Kahn has constructed a rank spectral sequence by using a purely categorical approach. This spectral sequence was derived by using a filtration of the category of torsion-free modules over integral domain by ranks and hence the name: rank spectral sequence. The E^1 terms of this sepctral sequence coincide with E^2 terms of Quillen's spectral sequence used to prove the finite generation of K-groups of ring of integers.
In this talk, we will show how to calculate the d^1-differential of the rank spectral sequence. We will put the differential in certain distinguished triangles of coefficients/functors over some categories, and make these functors explicit in terms of Tits building and Ash-Rudolph's modular symbols. To accomplish this, we shall use Quillen's categorical homotopy theory intensively and introduce the notion of extended (modular) symbols which is equivalent to Ash-Rudolph's via the suspension of Tits buildings.
No. SM-126 13/12/2014 14:00 ~ 16:00 Salle U/V
Algebraic elliptic cohomology theory and flops
ZHAO Gufang (赵顾舫), Institut de Mathématiques de Jussieu
Abstract: In the first part of the talk I will recall the definition of algebraic cobordism and oriented cohomology theory. Then I will talk about algebraic elliptic cohomology theory. In the second part, I will discuss the relation between the elliptic genus and classical flops in birational geometry. The main theorem is that the ideal in the algebraic cobordism ring generated by differences of classical flops coincide the kernel of the elliptic genus. The corresponding statement for complex cobordism theory was proved by Totaro. The second part of the talk is based on my joint work with Marc Levine and Yaping Yang.
No. SM-125 22/11/2014 14:00 ~ 16:00 Salle U/V
Holomorphic Morse inequalities and its applications
CAO Junyan (曹俊彦), Université Paris 6
Abstract: I will explain the holomorphic Morse inequalities, an important tool in complex analytic and algebraic geometry. After sketching a proof of the holomorphic Morse inequalities, we will concentrate on one important application in complex algebraic geometry, namely the duality of some positive cones.
Reference: Holomorphic Morse inequalities, The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension.
Click here for notes.
No. SM-124 08/11/2014 14:00 ~ 16:00 Salle U/V
Characteristic cycle of l-adic sheaves
HU Haoyu (胡昊宇), IHES & Nankai University
Abstract: To any holonomic D-modules on a complex smooth variety, one can associate a characteristic cycle that lives in the cotangent bundle. The Eular-Poincaré characteristic can be computed as the intersection of the characteristic cycle with the zero-section of the cotangent bundle. There have been many attempts to define the analogue notion of characteristic cycle for l-adic sheaves. The first example is the Grothendieck-Ogg-Shafarevich formula that computes the Euler-Poincaré characteristic of an l-adic sheaf on a smooth curve over a field. In this talk, I will discuss some approaches for l-adic sheaves on higher dimension schemes due to Abbes and Saito.
Click here for notes.
No. SM-123 25/10/2014 14:00 ~ 16:00 Salle U/V
Presheaf with transfers, Gersten resolution and motivic cohomology
CAO Yang (曹阳), Université Paris-Sud
Abstract: Firstly, I will give the definition of correspondence, presheaf with transfers and some examples. Secondly I will talk about Gersten resolution and its application to Chow groups, K-groups and unramifie cohomology. Thirdly, I will give the definition of motivic complex, and its application to cohomological invariant (if I have time).