L'utilisation de schémas Lagrangiens pour l'approximation de la dynamique des gaz est un outil puissant. En particulier, en présence de plusieurs matériaux non miscibles, on évite l'utilisation de modèles de mélanges puisque les discontinuités de contact sont automatiquement préservées.

Pour des écoulements multidimensionnels et choqués, on obtient naturellement un glissement à la discontinuité de contact séparant les matériaux. Comme dans le cas Eulérien, il s'avère nécessaire d'effectuer un traitement particulier pour prendre en compte ce phénomène.

On présentera une méthode de glissement conservative en énergie totale, en impulsion et en masse pour des schémas volumes finis lagrangiens. La méthode, établie dans un cadre abstrait, est de type joints. Elle s'appuie sur une formulation variationnelle qui permet le calcul de la vitesse des grilles et des flux. On décrira et analysera un cas pratique de discrétisation P1-P0. Finalement, on montrera des résultats numériques représentatifs justifiant la démarche choisie.

Il s'agit d'un travail en collaboration avec Silvia Bertoluzza de l'Université de Pavie et avec Emmanuel Labourasse du CEA.

We study a damped semi-linear wave equation in a bounded domain of R^3 with smooth boundary. It is proved that any H^2-smooth solution can be stabilised locally by a finite-dimensional feedback control supported by a given open subset satisfying a geometric condition. The proof is based on an investigation of the linearised equation, for which we construct a stabilising control satisfying the required properties. We next prove that the same control stabilises locally the non-linear problem. This is a joint work with Thomas Duyckaerts and Armen Shirikyan.

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.