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S'authentifier
Colloque 2014 du GDR 2875, Topologie Algébrique et Applications
de
mercredi 22 octobre 2014 (08:30)
à
vendredi 24 octobre 2014 (14:20)
lundi 20 octobre 2014
mardi 21 octobre 2014
mercredi 22 octobre 2014
08:30
Accueil
Accueil
08:30 - 09:00
09:00
Hochschild (co)homology, deformation theory and Caldararu's conjecture
-
Michel Van den Bergh
(
Hasselt et NFWO
)
Hochschild (co)homology, deformation theory and Caldararu's conjecture
(TopAlg)
Michel Van den Bergh
(
Hasselt et NFWO
)
09:00 - 10:15
It is well known that the Hochschild (co)homology of a smooth algebraic variety (or real/complex manifold) may be additively identified with its tangent (co)homology through the Hochschild-Kostant-Rosenberg isomorphism. In 2003 Caldararu published an intriguing conjecture (which he attributes to Kontsevich) that in order to make the HKR morphism compatible with the multiplicative structures one has to twist it with the square root of the Todd class, yielding an unexpected connection with the Riemann-Roch theorem. Surprisingly this conjecture has so far resisted all attempts to prove it by elementary means. Calderaru's conjecture was proved by Damien Calaque, Carlo Rossi and myself in 2009 (see arXiv:0904.4890) using techniques from modern deformation theory. The aim of the course is to give an introduction to this theory and to explain the proof of the conjecture. A rough outline will be as follows. (1) Kontsevich and Tsygan formality results. (2) Globalization methods. (3) Application to the proof of Caldararu's conjectury. If time permits then I will discuss some more applications to deformation theory.
10:15
Pause café
Pause café
10:15 - 10:50
10:50
The codimension-three conjecture for holonomic DQ-modules
-
Francois Petit
(
Université du Luxembourg
)
The codimension-three conjecture for holonomic DQ-modules
(TopAlg)
Francois Petit
(
Université du Luxembourg
)
10:50 - 11:30
The codimension 3 conjecture for micro-differential modules was formulated at the end of the seventies by M. Kashiwara and was recently solved by M. Kashiwara and K. Vilonen. It is related to the following problem of extending analytic objects: a holonomic microdifferential module defined outside of a codimension three analytic subset of a Lagrangian submanifold of an open subset of the cotangent bundle extends in a unique way to an holonomic system. This provides informations on the category of perverse sheaves with micro-support in a given conical Lagrangian subvariety of the cotangent bundle. Since DQ-modules provide a generalization of microdifferential modules to arbitrary symplectic manifolds, it is natural to extend the codimension-three conjecture to holonomic DQ-modules. In this talk, I will explain how to obtain a similar result for holonomic DQ-modules on a complex symplectic manifold.
11:40
3-dimensional HQFTs
-
Alexis Virelizier
3-dimensional HQFTs
(TopAlg)
Alexis Virelizier
11:40 - 12:30
Homotopy quantum field theory (HQFT) is a branch of quantum topology concerned with maps from manifolds to a fixed target space. The aim is to define and to study homotopy invariants of such maps using methods of quantum topology. I will focus on 3-dimensional HQFTs with target an Eilenberg-MacLane space K(G,1) where G is a discrete group. (The case G=1 corresponds to more familiar 3-dimensional TQFTs.) These HQFTs provide numerical invariants of principal G-bundles over closed 3-manifolds which can be viewed as "quantum" characteristic numbers. To construct such HQFTs, the relevant algebraic ingredients are G-graded categories, which are monoidal categories whose objects have a multiplicative G-grading.
12:30
Déjeuner
Déjeuner
12:30 - 14:20
14:20
Gröbner methods for generic representation theory and the artinian conjecture
-
Steven Sam
(
University of California, Berkeley
)
Gröbner methods for generic representation theory and the artinian conjecture
(TopAlg)
Steven Sam
(
University of California, Berkeley
)
14:20 - 15:10
Let k be a finite field, let V(k) be the category of finite-dimensional vector spaces over k, and let F(k) be the category of endofunctors of V(k). F(k) is closely related to the category of unstable modules over the Steenrod algebra. A dual version of Schwartz's artinian conjecture states that every finitely generated object in F(k) is noetherian, i.e., satisfies the ascending chain condition for subobjects. I will present the ideas of a proof of a more general version of this conjecture based on Gröbner methods for functor categories developed jointly with Andrew Snowden. Time permitting, I will explain how some related categories are useful for the study of homology of congruence subgroups of automorphism groups of free groups and mapping class groups of surfaces (joint with Andrew Putman).
15:10
Pause café
Pause café
15:10 - 15:40
15:40
Topology of complex projective varieties with isolated singularities
-
David Chataur
(
Laboratoire Paul Painlevé
)
Topology of complex projective varieties with isolated singularities
(TopAlg)
David Chataur
(
Laboratoire Paul Painlevé
)
15:40 - 16:30
The cohomology of smooth complex projective varieties comes endowed with its Hodge decomposition. This structure imposes very drastic conditions on the topology of such varieties. For example, the rational homotopy type is formal (Deligne-Griffiths-Morgan-Sullivan). - In the case of singular varieties, we know after works of Deligne, Hain, Morgan, Navarro-Aznar... that we can endow the cohomology and the rational homotopy groups with a mixed Hodge structure. These Mixed Hodge structures carry interesting geometric information. - Moreover Goresky and MacPherson introduced Intersection cohomology in order to have a "good" cohomology for singular spaces, where we still have Poincaré duality. In this talk I will survey a homotopical treatment of Intersection cohomology (developped in collaboration with Martin Saralegui and Daniel Tanré), where we associate to each singular space its perverse homotopy type. I will discuss how the theory works in the case of complex projective varieties with isolated singularities where we can endow the perverse homotopy type with a mixed Hodge structure and how we get some formality results (work in progress with Joana Cirici).
jeudi 23 octobre 2014
09:00
Hochschild (co)homology, deformation theory and Caldararu's conjecture
Hochschild (co)homology, deformation theory and Caldararu's conjecture
09:00 - 10:15
Lecture 2
10:15
Pause café
Pause café
10:15 - 10:40
10:40
Profinite completion of operads and the Grothendieck-Teichmüller group
-
Geoffroy Horel
(
Université de Münster
)
Profinite completion of operads and the Grothendieck-Teichmüller group
(TopAlg)
Geoffroy Horel
(
Université de Münster
)
10:40 - 11:30
I will define the category of operads in profinite spaces and construct a profinite completion functor from the category of operads in spaces to the category of operads in profinite spaces. I will then try to explain how one can compute the group of homotopy automorphisms of the profinite completion of the little 2-disks operad and show that this group is isomorphic to the Grothendieck-Teichmüller group (up to an extension by a group of order 2). A prounipotent version of this theorem is due to Benoit Fresse. Je définirai la catégorie des opérades en espaces profinis et un foncteur de complétion de la catégorie des opérades en espaces topologiques vers la catégorie des opérades en espaces profinis. Je montrerai ensuite comment on peut calculer le groupe des automorphismes homotopiques de la complétion de l'opérade des petits disques. Ce groupe s'identifie au groupe de Grothendieck-Teichmüller à une extension par un groupe d'ordre 2 près. Une version prounipotente de ce résultat a été prouvée par Benoit Fresse.
11:40
Cohomologie équivariante orientée des variétés de drapeaux et restriction aux points fixes
-
Baptiste Calmes
Cohomologie équivariante orientée des variétés de drapeaux et restriction aux points fixes
(TopAlg)
Baptiste Calmes
11:40 - 12:30
(Travail en commun avec C. Zhong et K. Zainoulline) Soit G un groupe algébrique linéaire semi-simple déployé sur un corps k, soit T un tore maximal déployé de G et soit B un sous-groupe de Borel contenant T. Notre objet principal d’étude est la structure d’anneau de h_T(G/B), où h_T désigne une théorie cohomologique orientée T-équivariante sur les variétés lisses sur k, munies d’une action de T. Cela pourrait être l’anneau de Chow , le groupe de Grothendieck ou une théorie plus compliquée comme le cobordisme algébrique. J’expliquerai comment la méthode de restriction aux points fixes utilisée par plusieurs auteurs (Brion, Atiyah-Bott, Goretsky-Kottwitz-MacPherson) peut être appliquée au cas d’une cohomologie équivariante oritentée quelconque, en utilisant un formalisme algébrique qui généralise des constructions de Demazure, Kostant-Kumar en tenant compte de la loi de groupe formel de la théorie. Je tâcherai de mettre en évidence les différences entre les cas classiques de ces derniers auteurs (anneau de Chow, groupe de Grothendieck) et le cas général. Je mentionnerai également le cas d’une variété G/P, quotient par un parabolique.
12:30
Déjeuner
Déjeuner
12:30 - 14:20
14:20
Derived symplectic geometry and TFTs
-
Damien Calaque
(
Université Montpellier 2
)
Derived symplectic geometry and TFTs
(TopAlg)
Damien Calaque
(
Université Montpellier 2
)
14:20 - 15:10
We give an informal introduction to the new field of derived symplectic geometry (after Pantev-Toën-Vaquié-Vezzosi), and present some applications to topological field theories. We in particular explain that derived symplectic geometry provides a suitable framework for the so-called AKSZ construction (after Alexandrov-Kontsevich-Schwartz-Zaboronski).
15:20
Une application de la théorie des k-invariants algébriques
-
Van Tuan PHAM
(
16/09/2014
)
Une application de la théorie des k-invariants algébriques
(TopAlg)
Van Tuan PHAM
(
16/09/2014
)
15:20 - 16:10
Les k-invariants algébriques sont définis par Dold [Albrecht Dold. Zur Homotopietheorie der Kettenkomplexe. Math. Ann., 140:278--298, 1960]. En utilisant cette théorie, nous obtenons une reformulation plus facilement exploitable de la formalité d'un complexe de foncteurs strictement polynomiaux. Comme une application, nous calculons des groupes d'extensions de foncteurs strictement polynomiaux.
16:10
Pause café
Pause café
16:10 - 16:40
16:40
Des catégories triangulées aux catégories de modules via l'algèbre homotopique.
-
Yann Palu
Des catégories triangulées aux catégories de modules via l'algèbre homotopique.
(TopAlg)
Yann Palu
16:40 - 17:30
La théorie du "basculement" (tilting) est un outil fondamental dans l'étude des algèbres de dimension finie, permettant de caractériser les équivalences dérivées. La catégorification des algèbres amassées a apporté un souffle nouveau à cette théorie en donnant naissance à "l'amas-basculement" (cluster-tilting), motivant ainsi l'étude des algèbres d'endomorphisme d'objets rigides dans certaines catégories triangulées. Soit C une catégorie triangulée linéaire et Hom-finie et soit A l'algèbre d'endomorphisme d'un objet rigide de C. La catégorie des modules sur A possède alors deux descriptions différentes : l'une en terme de sous-quotient de C ; l'autre en terme de localisation de C. On peut penser cette situation comme une réminiscence de la construction de la catégorie homotopique d'une catégorie de modèle. Notre objectif, dans cet exposé, est d'expliquer cette double description en rendant plus précise l'analogie avec les catégories de modèle.
vendredi 24 octobre 2014
09:00
Hochschild (co)homology, deformation theory and Caldararu's conjecture
Hochschild (co)homology, deformation theory and Caldararu's conjecture
09:00 - 10:15
Lecture 3
10:15
Pause café
Pause café
10:15 - 10:40
10:40
Realizing unstable coalgebras
-
Georg Biedermann
(
Nantes
)
Realizing unstable coalgebras
(TopAlg)
Georg Biedermann
(
Nantes
)
10:40 - 11:30
(Joint with G. Raptis and M. Stelzer) Unstable coalgebras form the natural target category for mod p singular homology of spaces. We construct a tower of spaces converging to the moduli space of realizations of an unstable coalgebra C ie spaces whose mod p homology is isomorphic to C. As a consequence we can reformulate, unify and generalize associated obstruction theories by Harper, Bousfield and Blanc. We adapt the approach taken by Blanc/Dwyer/Goerss for realizing \Pi-algebras. One long term goal is to obtain an analogous deformation theory for mod p coefficients as in the rational case given by Schlessinger/Stasheff.
11:40
La structure de Batalin-Vilkovisky et la dualité de Koszul
-
Guodong Zhou
(
Shanghai
)
La structure de Batalin-Vilkovisky et la dualité de Koszul
(TopAlg)
Guodong Zhou
(
Shanghai
)
11:40 - 12:30
Analogue à un résultat récent de N. Kowalzig et U. Kraehmer, on montre que la cohomologie de Hochschild d'une algèbre de Frobenius est une algèbre de Batalin-Vilkovisky, à condition que son automorphisme de Nakayama soit semisimple. Etant donné une algèbre de Koszul qui est une algèbre de Calabi-Yau tordue dont l'automorphisme de Nakayama est semisimple, alors il existe un isomorphisme d'algèbres de Batalin-Vilkovisky entre la cohomologie de Hochschild de cette algèbre et celle de son dual de Koszul.
12:30
Déjeuner
Déjeuner
12:30 - 14:20