Séminaire de topologie, géométrie et algèbre (archives)

Soient A un anneau et K un corps, commutatifs. On nomme, d'après Kuhn et en raison d'analogies avec la théorie classique des représentations K-linéaires des groupes, représentation générique sur K des groupes linéaires sur A tout foncteur des A-modules libres de rang fini vers les K-espaces vectoriels, ici supposés de dimension finie. De tels foncteurs, ainsi que des analogues (comme les FI-modules, dont la structure est plus facile à étudier) apparaissent naturellement dans plusieurs problèmes d'algèbre, de topologie ou de K-théorie. Nous présenterons des résultats de structure pour les représentations génériques ayant assez de propriétés de finitude (notamment celles qui possèdent une suite de composition finie) qui impliquent que leur série de Poincaré (dont les coefficients sont les dimensions prises par le foncteur considéré) est très contrainte - c'est une fonction rationnelle.

Cet exposé repose sur plusieurs travaux récents avec Touzé, Vespa et Gaujal.

Given a lagrangian link with k components in the disc, it is possible to define an associated Hofer norm on the braid group with k strands. In this talk we are going to detail this definition, and explain how it is possible to prove non degeneracy if k=2 and certain area conditions on the lagrangian link are met. The proof is based on the construction, using Quantitative Heegaard-Floer Homology, of a family of quasimorphisms which detect linking numbers of braids. Time permitting, we are also going to see how to extend the results to any compact surface with boundary. This talk is based on work of mine, and an ongoing project with Ibrahim Trifa (for the higher genus case).

In a recent joint work with Joan Bellier-Millès, we set up a homotopy theory of curved algebras over curved operads in which one can implement (with the appropriate modifications) usual constructions such as the bar-cobar adjunction and André-Quillen cohomology. The occurrence of curved structures in various research topics (symplectic topology, deformation theory, derived geometry, mathematical physics...) motivate the development of their homotopy theory, cohomology theory and deformation theory. Two important examples fitting in our framework are the homotopy unital curved A-infinity algebras encoding for instance the algebraic structure of Fukaya categories (as in the work of Fukaya-Oh-Ota-Ono) in symplectic topology, and formal integrable almost complex structures in complex geometry. The motivation behind the second example is to get an analogue of the Newlander-Nirenberg integrability theorem in derived complex analytic geometry. At the formal neighbourhood of a point, an integrable almost complex manifold can be described as an algebra over a certain curved operad. Our idea is to consider homotopy curved algebras over this curved operad and to glue these local data up to homotopy. Our results generalize to homotopy sheaves of curved algebras over a curved operad, so in particular they provide a framework to perform such a construction. During this talk, I will explain the various key notions and ideas of our work. If time permits, I will say a few words about the comparison between our model of derived complex analytic spaces and those developped by Lurie, Porta and Pridham.

Weierstrass points on algebraic curves are special points of high importance in algebraic geometry and arithmetic geometry. In this talk, we study how those special points behave when the algebraic curve degenerates to a nodal curve. To this end, we first explain why tropical geometry is a relevant formalism for studying degeneration questions. We then define a tropical analogue on metric graphs (seen as tropical curves) for Weierstrass points, and explore the properties of the so-called “tropical Weierstrass locus”. We also associate intrinsic weights to the connected components of this locus, and show that their total sum for a given metric graph and divisor is a function of few combinatorial parameters (degree and rank of the divisor, genus of the metric graph). Finally, in the case the divisor on the metric graph comes from the tropicalization of a divisor on an algebraic curve, we specify the compatibility between the Weierstrass loci.

In 2000, Eliashberg and Polterovich introduced the notion of orderability to investigate the structure of the group of contact diffeomorphisms and the structure of isotopy classes of Legendrian submanifolds. Roughly speaking, a group of contact diffeomorphisms is orderable if the relation induced by the partial order on contact hamiltonian maps induces a partial order on the associated time-one flows. In this talk, we will explain why orderability is equivalent to the existence of spectral selectors and how these selectors can be used to derive multiple geometric properties in the orderable situation. This is a joint work with Pierre-Alexandre Arlove.

In this talk, I will present recent results concerning special values of certain combinatorial zeta functions counting geodesic paths in triangulations. I will show that those values are related to some topological invariants. As such, we recover the first Betti number or L^2-Betti number of a compact manifold, as well as the linking number of knots in a 3-manifold. This is a joint work with Léo Bénard, Viet Dang and Thomas Schick.

(The talk will be in French.) Je décrirai la première construction, due à Hyde et Lodha, de groupes simples de présentation finie agissant fidèlement sur la droite. Il s'agit de groupes d'homéomorphismes affines par morceaux convenablement définis. L'exposé ne nécessite de prérequis ni en dynamique, ni en théorie des groupes. Il s'agit d'une variante d'une construction due à R. Thompson dans les années 60, qui donnait alors des groupes simples infinis de présentation finie agissant fidèlement sur le cercle - qui étaient alors les premiers exemples connus de groupes simples infinis de présentation finie.