Séminaire de topologie, géométrie et algèbre (archives)

The notion of Quillen cohomology, first defined by Quillen in his seminal book and later extended to include coefficients in spectra, is a universal form of cohomology which enjoys a wide range of applications: it features in obstruction theories and spectral sequences abutting to mapping spaces, controls deformations of objects in the relevant contexts and tends to appear in moduli problems classifying objects realizing a given cohomological invariant. In this talk we will describe an approach to the computation of Quillen cohomology of categories enriched in a well behaved model category S. When S is the category of simplicial sets one recovers a model for the theory of oo-categories, in which case our computation identifies the Quillen cohomology of an oo-category C as the cohomology of spectrum valued functors on the twisted arrow category of C. When S is the category of marked simplicial sets one recovers a model for the theory of (oo,2)-categories, in which case our computation identifies the Quillen cohomology of an (oo,2)-category C with the cohomology of spectrum valued functors indexed by what we call the twisted 2-cell category of C. These results can be used to give explicit and computable obstruction theories for problems such as lifting homotopy commutative diagrams (in either a 1-categorical or a 2-categorical setting). This is joint work with Joost Nuiten and Matan Prasma.

We shall discuss the enumerative problem of counting the number of complex curves (in complex projective space of dimension 2) which pass through the requisite number of generic points and has a prescribed singularity at one point. Our exposition will be from a topological point of view via the ubiquitous Euler class.

Demailly (2012) showed that the Hodge conjecture is equivalent to the statement that any (p,p)-dimensional closed current with rational cohomology class can be approximated by linear combinations of integration currents; Moreover, the statement that any (p,p)-dimensional closed current with rational cohomology class can be approximated by linear combinations of integration currents; Moreover, the statement that all strongly positive currents with rational cohomology class can be approximated by positive linear combinations of integration currents, can be viewed as a strong version of the Hodge conjecture (1982). In this talk, I will explain the construction of a current which does not verify the latter statement on a toric variety, where the Hodge conjecture is known to hold. The example belongs to the family of `complex tropical currents', which we extend their framework to toric varieties, discuss their extremality properties, and express their cohomology classes as recession fans of their underlying tropical varieties. Finally, the counter-example will be the tropical current associated to a 2-dimensional balanced subfan of a 4-dimensional toric variety, whose intersection form does not have the right signature in terms of the Hodge index theorem. This is a joint work with June Huh.

https://www.lebesgue.fr/content/sem2017

La conjecture de Weinstein affirme que tout champ de Reeb sur une variété close possède une orbite périodique. Dans cet exposé, on présentera une stratégie de preuve pour le cas des variétés de contact portées par un livre ouvert de forme particulière : la page est obtenue en attachant à un domaine de Weinstein une anse critique le long d'une sphère legendrienne lâche et homologiquement triviale. Cette approche s'appuie sur des travaux récents de Bourgeois, Ekholm et Eliashberg sur les anses critiques

In this talk, I will review the notion of modular functor and explain how one use it to defines bundles over the compactified moduli space of Riemann surfaces. The Chern classes of these bundles turn out to define Cohomological field theories. I will explain how this implies that they can be computed by an inductive procedure called topological recursion. One of the motivating example is the study of the so-called Verlinde bundle associated to Wess-Zmino-Witten Conformal field theories.

Based on a joint work with Andersen and Borot

(travail en commun avec Jean Fasel) Voevodsky a défini la cohomologie motivique en commençant par construire les correspondances finies, une version algébrique des fonctions multivaluées. Les groupes de Chow n’y apparaissent pas explicitement, mais ils sont sous-jacents, et j’expliquerai une façon de les révéler. Cela permettra de les remplacer par les groupes de Chow-Witt, en lien avec les formes quadratiques, et de poursuivre la construction de Voevodsky, afin d’obtenir de nouveaux groupes de cohomologie motivique, dits « généralisés » fournissant des informations plus fines sur la catégorie homotopique des schémas, pendant algébro-géométrique de la catégorie homotopique des espaces topologiques.

Je citerai les quelques calculs non triviaux que nous pouvons faire, notamment une généralisation d’un théorème de Suslin-Nesterenko-Voevodsky-Totaro qui identifie la cohomologie motivique d’un corps à la K-théorie de Milnor.

Originally defined by Nadler, an arboreal singularity of a Lagrangian is a robust kind of singularity; notable in that any Lagrangian singularity can be deformed into one with only arboreal singularities. They are also interesting algebraically, because an arboreal singularity is determined by a quiver (a rooted tree), and the space of constructable sheaves on the arboreal singularity coincides with the space of representations of the quiver. However, often times an arboreal Lagrangian skeleton will have free boundary components, so the space of sheaves is considerable smaller, even locally. Geometrically this corresponds to removing a number of the top dimensional strata, which we call pruning. We prove that the link of a pruned arboreal singularity is loose, if and only if the singularity admits no non-constant constructable sheaves. We'll also discuss how this fits into a larger program of using the wrapped Fukaya category to detect flexibility of Weinstein manifol

Une théorie quantique des champs topologiques (TQFT) en dimension (1+1) associe à chaque cercle une algèbre de Frobenius et à une surface en pantalon le morphisme de co/multiplication de l'algèbre de Frobenius . En particulier elle associe à une surface fermée un nombre.

On s'attachera dans cette exposé à expliquer une construction similaire où on considère certains graphes plantaires à la place des cercles et aussi des mousses qui sont des cobordismes naturelles entre ces graphes. On présentera en particulier une formule permettant de calculer cette TQFT trivalente sur les mousses fermées et on verra comment celle ci-permet de reconstruire la TQFT entièrement grâce à un procédé de construction universelle.

Une bonne partie de l'exposé s'attachera à motiver cette construction et à expliquer ses liens avec la théorie des représentations du groupe quantique de type A et les invariants d'entrelacs associées. On esquissera aussi le lien avec l'anneau de cohomologie des drapeaux partiels et conjecturalement la cohomologie de certains autres espaces de modules.