The homotopy groups of CW complexes and of the mapping spaces between them are notoriously difficult to compute. However, if one disregards torsion, rational homotopy theory becomes very effective and can easily solve such problems. Moreover, it produces efficient invariants of homotopy classes of maps, called Maurer–Cartan elements, which encode the rational type of path components. I will give a couple of examples and then explain how this extends to embedding spaces.

La marche aléatoire "bootstrap" (A. Collevecchio, K. Hamza et M. Shi 2016) désigne la marche aléatoire Y dont les incréments sont les produits cumulés des incréments d'une marche aléatoire simple symétrique X. Bien que Y soit fonction déterministe de X, elle a néanmoins la propriété que le processus (X,Y) est récurrent et se renormalise vers un mouvement brownien à composantes indépendantes. L'objectif de l'exposé est de s'intéresser à des extensions de ce résultat lorsque la "fonction déterministe" en question est choisie génériquement. Travail en collaboration avec Kais Hamza. 

The Zakharov-Kuznetsov equation is a model of propagation of acoustic waves in a plasma. Mathematically, it corresponds to a multi-dimensional generalisation of the Korteweg-de-Vries equation, but it is anisotropic. In particular, its linear part enjoys dispersive properties and one expects that it satisfies a property of scattering: if the initial data is small enough, the solution should behave asymptotically as a linear solution. 
In this talk, I will introduce this equation and two results of scattering, in 2D and 3D, and will explain how to adapt the method of space-time resonances introduced by Germain-Masmoudi-Shatah to overcome the difficulties caused by anisotropy. 

We consider the Navier-Stokes equations with constant/variable density in a 2D slab domain, horizontally periodic.  We will study those two nonlinear perturbed problems around special equilibrium, under which Navier-slip boundary conditions are imposed, with constant slip coefficients. In this talk, the influence of slip coefficients on linear instability will be presented to obtain infinitely many normal mode solutions, following the operator method of Lafitte and Nguyen. Hence, we prove the nonlinear instability with a wide class of initial data, thanks to the abstract frameworks of Guo-Strauss and of Desjardins-Grenier. 

We study the propagation of coherent states in self-interacting bosonic quantum field theories in the semi-classical (mean-field) regime. Our work focuses in particular on the spatially cutoff P(\phi)_2 model, under standard assumptions ensuring essential self-adjointness of the Hamiltonian. For this model, the classical field equation is given by a non-linear Klein-Gordon equation, whose well-posedness constitutes the first step of our analysis. We then apply Hepp’s method, combined with a detailed study of both the classical and quantum dynamics, in order to construct an asymptotic expansion of arbitrary order for the quantum evolution of coherent states.