The symmetries of the standard contact structure of a sphere generate families of contact structures. There exists a Serre fibration relating the space of contact structures and the group of contactomorphisms. The homotopy exact sequence for this fibration is studied and the non--triviality of certain elements in the homotopy groups of the contactomorphism group is concluded. Part of the argument applies to $3$--Sasakian manifolds due to their quaternionic symmetries. We comment on an alternative approach to the detection of non--triviality through the definition of a series of indices generalizing the Maslov index in the symplectic case.

Dans cet exposé je présenterai un modèle de type Hartree pour un gaz comprenant une infinité de particules quantiques, et je discuterai du phénomène de dispersion dans ce système infini. Les outils principaux sont 1) une nouvelle inégalité de Strichartz pour des familles de fonctions orthonormées et 2) la conservation de l'énergie (libre) relative. Travaux en collaboration avec Rupert Frank (Caltech), Elliott H. Lieb (Princeton), Julien Sabin (Cergy) et Robert Seiringer (Vienne).

We will talk about our contribution to a large project with the goal of a self-consistent numerical simulation of the evolution of the universe beginning soon after the Big Bang and ending with the formation of realistic stellar systems like the Milky Way. This is a multi-scale problem of vast proportions. It requires the development of new numerical methods that excel in accuracy, parallel scalability, and physical fidelity to the processes relevant in galaxy formation. These numerical methods themselves require the development of mathematical theory in order to guarantee the above mentioned requirements.

In particular we introduce a finite volume code for ideal magnetohydrodynamics (MHD), which possesses excellent stability properties. Ingredients are: an approximate Riemann solver, extension to multidimensions via a Powell term, second order preserving positivity. The scheme’s robustness is due to entropy stability, positivity and properly discretised Powell terms. This can be shown due to a mathematical theory inspired by a kinetic description of the phenomena.

The lecture will be illustrated by movies that show numerical simulations related to the evolution of the universe. This will go beyond our own contributions and also show simulations of my collaborators Volker Springel and Fritz Röpke.

This talk will serve to cover some of the background material for my talk on Friday. We will define symplectic cohomology, wrapped Fukaya category and the closed-open string maps that relates the two.