I will give a short introduction into spectral zeta functions for manifolds and domains and explain the relation between heat kernel bounds, finite propagation speed bounds, and bounds for the spectral zeta function. In principle these results can be used to compute the spectral determinant rather accurately with rigorous error estimates. I will discuss briefly some questions about extremal properties of spectral zeta functions.

Intersection cohomology, or IH*, is a topological tool defined in the 1980s to investigate a class of singular spaces called pseudo manifolds. Every point on a pseudo manifold, X, has a neighbourhood that looks like a disk cross the cone on a lower dimensional pseudo manifold called the link at that point. IH*(X) arises from a local upper truncation of the link cohomology. Several Hodge theorems have been proved relating IH(X) to harmonic forms for finite volume metrics on the regular part of X.

HI cohomology is a new cohomology theory for pseudo manifolds defined by M. Banagl based on the idea of co-truncation in the links, that is, using a lower truncation in the link cohomology. This talk will describe ongoing work relating this cohomology to harmonic forms for infinite volume metrics on the regular part of X.

(Joint work with M. Banagl)

Dans cet exposé nous considérons les questions de stabilité et d’instabilité dans le problème inverse de diffusion pour l’équation de Schrödinger et l’équation acoustique en dimension d>=2. En particulier, nous allons présenter des nouvelles estimations de stabilité globale qui dépendent explicitement de la régularité du coefficient et/ou de l’énergie. De plus, en utilisant des idées remontant à Kolmogorov, Tikhomirov et Vitushkin nous montrons l'optimalité des estimations de stabilité de ce type.

Is it possible to find an embedded Lagrangian disk in \C^n - B^2n, so that the boundary is a Legendrian in S^{2n-1}? When n=2 the answer is no, but in all higher dimensions such disks exist in abundance. This follows from a more general existence theorem for Lagrangian embeddings with loose concave boundary; in this two-part talk we precisely state and prove this theorem. The proof has two main components: an action-balancing lemma for Lagrangian immersions, and a Lagrangian Whitney trick. We discuss the proof of both, in particular discussing how they both rely on the classification theorem for loose Legendrians. This project is joint work with Yakov Eliashberg.

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.