Institut de recherche mathématique avancée

Résumé : I will talk about a joint project with Y. Baris Kartal whose purpose is to incorporate Morse--Bott methods into Floer homotopy theory. Morse--Bott methods have the prospect of being useful for two reasons. First, they simplify equivariant constructions: for example, we used these methods to define a lift of circle equivariant symplectic homology to equivariant spectra. Second, they enable new computations. As a step in this direction, we used Morse--Bott methods to compute the (equivariant) local Floer homology of the orbit of an autonomous Hamiltonian, but I hope and expect that one can push such computations significantly further.

Résumé : Although hyperplane arrangement complements are rationally formal, we note that they have rational models which are topologically and combinatorially significant. We construct a combinatorial version of the classical Leray model for arrangement complements that interpolates between the Orlik-Solomon algebra and the Chow ring of a matroid. This gives us the opportunity to present and reconcile much of the Orlik-Solomon and nested set combinatorics that have proven to be rather powerful structural tools in the subject area.

Résumé : En 1990, Mess a démontré le théorème du tremblement de terre de Thurston en utilisant la géométrie anti-de Sitter. Depuis lors, plusieurs idées de Mess ont été utilisées pour étudier la correspondance entre les surfaces dans l'espace anti-de Sitter tridimensionnel et la théorie de Teichmüller. Dans cet exposé, je parlerai du problème d'extension de champs vecteurs sur le cercle au plan hyperbolique, en utilisant la géométrie dite "Half-pipe", qui est duale à la géométrie de Minkowski. Cette construction suggère un lien entre les champs de vecteurs sur le plan hyperbolique et les surfaces dans l'espace Half-pipe. En particulier, nous retrouvons le théorème de Gardiner, qui affirme que tout champ de vecteur Zygmund sur le cercle peut être représenté comme un tremblement de terre infinitésimal. J’expliquerai également un résultat analogue pour les champs de vecteurs harmoniques.

Résumé : Smooth partial quotients form a wide class of smooth compact manifolds with a torus action. They were defined in the framework of toric topology by Buchstaber and Panov in 2002 as orbit spaces of moment-angle manifolds w.r.t. certain freely and smoothly acting compact tori. Apart from moment-angle manifolds, the most well-studied example of a smooth partial quotient is given by a quasitoric manifold. This notion was introduced by Davis and Januszkiewicz in 1991 as a topological counterpart of a projective toric variety and generalizes that of a symplectic toric manifold. However, apart from the general description of cohomology rings due to Baskakov, Buchstaber, Franz and Panov, little is known about the geometry and topology of an arbitrary smooth partial quotient so far. In this talk, we shall discuss the construction of a smooth partial quotient, its basic cohomology ring properties and some applications in bordism theory. The talk is based in part on a joint work with Grigory Solomadin.