In this two-part talk, we will discuss the classification of loose Legendrians in high dimensional contact manifolds. The full classification of Legendrian embeddings up to isotopy is likely an intractable problem. Loose Legendrians are special, in that they are determined up to isotopy by their topological (rather than geometric) invariants. They are also very common, in that any Legendrian can be made loose by altering it in a small neighborhood of a point. To construct isotopies between loose Legendrians we use tools coming from the world of h-principles. In particular we use theorems about directed embeddings, holonomic approximation, and wrinkled embeddings to prove an isotopy theorem for Legendrians with unfurled swallowtail singularities, and finally show how the loose condition allows us to surger away those singularities. No prior knowledge of h-principles or contact geometry will be assumed, other than basics such as Darboux's theorem.

In this two-part talk, we will discuss the classification of loose Legendrians in high dimensional contact manifolds. The full classification of Legendrian embeddings up to isotopy is likely an intractable problem. Loose Legendrians are special, in that they are determined up to isotopy by their topological (rather than geometric) invariants. They are also very common, in that any Legendrian can be made loose by altering it in a small neighborhood of a point. To construct isotopies between loose Legendrians we use tools coming from the world of h-principles. In particular we use theorems about directed embeddings, holonomic approximation, and wrinkled embeddings to prove an isotopy theorem for Legendrians with unfurled swallowtail singularities, and finally show how the loose condition allows us to surger away those singularities. No prior knowledge of h-principles or contact geometry will be assumed, other than basics such as Darboux's theorem.

Si Y est une 3-variété topologique compacte, connexe, fermée et orientée, on peut lui associer son homologie de Heegaard Floer HF(Y) et son homologie de contact plongée ECH(Y); par un théorème de Colin, Ghiggini et Honda ces deux homologies sont isomorphes (comme groupes). D'un autre coté, si K est un noeud dans Y on peut aussi définir des homologies de Heegaard Floer HFK(K,Y) et de contact plongée ECK(K,Y) pour K. Conjecture: HFK(K,Y) est isomorphe à ECK(K,Y). Dans cet exposé on rappellera les definitions des homologies ci-dessus dans le cas où K est un noeud fibré et on donnera des indices de la véracité de la conjecture.

Nous parlerons d'un travail joint avec Roman Golovko et Georgios Rizell où l'on démontre des restrictions fortes sur la topologie des cobordismes lagrangiens d'une variété legendrienne vers elle-même lorsque les variété legendriennes considérées ont des augmentations.

Let G be a torsion-free, discrete subgroup of either the isometry group of quaternionic or Cayley hyperbolic space; that is, up to isogeny, $G < Sp(n;1)$ for $n \geq 2$ or $G < F4^{-20}$. We prove that if G contains no parabolics then there is a gap in the possible homological dimension $hd(G)$ of $G$. Namely, if $G < Sp(n;1)$, either $hd(G) = 4n$ or $hd(G)\leq 4n-2$, and if $G < F4^{-20}$, then either $hd(G) = 16$ or $hd(G)\leq 12$. This result does not hold in the real or complex hyperbolic cases, or if $G$ is allowed to have parabolics (even for subgroups of lattices). Our method requires a generalization of work of Besson–Courtois–Gallot on estimates of p–Jacobians of natural maps. We also generalize an inequality of M. Kapovich between the homological dimension and critical exponent for discrete subgroups of the isometry group of real hyperbolic n–space. This is joint work with Benson Farb and Ben McReynolds.

Après avoir rappelé quelques résultats classiques sur les spineurs de Killing et les spineurs de Killing généralisés, nous donnerons des résultats récents. Tout d'abord sur les spineurs de Killing généralisés antisymétriques en dimension 2 et 3 et ensuite sur les spineurs de Killing généralisés sur un fibré des spineurs tordu. Ces derniers apparaissent dans le cas limite d'une minoration de la première valeur propre de l'opérateur de Dirac d'une sous-variété.

L. Thomann présentera la strucutre générale et les prérequis de l'article de Hani, Tzvetkov, Pausader et Visciglia.

Ce premier exposé sera l'occasion de présenter quelques notions sur les martingales et processus à temps discret.