Institut de recherche mathématique avancée

Résumé : In his foundational work, Margulis demonstrated a superrigidity phenomenon for higher-rank lattices, showing that any “non-elementary” homomorphism from such lattices to a semisimple Lie group extends to the ambient group. Zimmer extended this result to the setting of cocycles, with significant consequences for orbit equivalence rigidity. Since then, many related problems have been explored, extending these ideas to situations where the target group is a more general topological group preserving certain structures. In this talk, we address the case where the target group is the group of isometries of a finite-rank median space, and the source group is a product of locally compact second countable groups. These spaces have attracted interest in geometric group theory due to the unified framework they provide for studying actions on real trees and CAT(0) cube complexes, as well as the characterization they offer of Kazhdan’s property (T).

Résumé : The space of (local) symmetries of a given geometric structure has the natural structure of a Lie (pseudo)group. Conversely, geometric structures admitting a local model can be described via the pseudogroup of symmetries of such local model. The main goal of this talk is to provide several examples and give an intuitive understanding of the slogan above, which can be made precise at various levels of generality (depending on the definition of "geometric structure") and using different tools/methods. Moreover, I will sketch a new framework, which include previous formalisms (e.g. G-structures or Cartan geometries) and allows us to prove integrability theorems. In particular, I will provide intuition on the relevant objects which make this approach work, namely Lie groupoids endowed with a multiplicative "PDE-structure" and their principal actions. Poisson geometry will give us the guiding principles to understand those objects, which are directly inspired from, respectively, symplectic groupoids and principal Hamiltonian bundles. This is based on a forthcoming book written jointly with Luca Accornero, Marius Crainic and María Amelia Salazar.

Résumé : TBA

Résumé : Résumé : J'expliquerai comment on peut approcher le type d'homotopie des espaces de plongements entre variétés différentiables en termes de modules sur l'opérade des petits disques. Cette méthode due à Goodwillie et Weiss devient particulièrement puissante quand on la combine avec des outils de théorie de l'homotopie rationnelle. En particulier, je formulerai un résultat récent obtenu avec Pedro Boavida de Brito donnant une description explicite du type d’homotopie rationnel d'un espace de plongements sous des hypothèses assez générales.

Résumé : Given a topological surface S, there is a plethora of Riemann surface structures (i.e. complex structures) that you can endow it with. In fact, the space of such structures is a smooth manifold called the Teichmüller space. So given a Riemann surface, how can you continuously deform it to obtain a new one? In this talk, I will describe two different ways to do this, namely twisting and grafting. After giving some geometric intuition of these deformations, I will discuss a result of McMullen relating the two through the Kähler geometry of Teichmüller space.