Abstracts

Cours de Patrick Gérard:

Dans ce mini-cours, je discuterai le cas de deux EDP hamiltoniennes intégrables sur le cercle : l'équation de Szegö cubique et l'équation de Benjamin-Ono. Dans une présentation conjointe, je discuterai les structures de paires de Lax pour ces deux équations, et je montrerai comment les bonnes propriétés des opérateurs de Lax par rapport à la structure d'espace de Hardy permettent d'établir des expressions explicites pour les solutions, et de définir un flot sur des espaces de régularité optimale. Le contenu de ce cours est inspiré de travaux en collaboration avec Sandrine Grellier, Alexander Pushnitski, Thomas Kappeler (+) et Peter Topalov.

Dario Bambusi:

I will present a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schr\"odinger equation with a convolution potential and a beam equation. The abstract theorem can also be used to ensure effective stability of the plane waves in NLS. The main technical novelty is that we deal with general tori, including irrational tori, in which the differences of the eigenvalues of the Laplacian are dense on the real axe.

Fabricio Macia:

We address the problem of reconstructing a scalar conductivity from the Dirichlet-to-Neumann map on the boundary of a domain in Euclidean space (the reconstruction aspect of the Calderon problem). It is well-known that, under suitable assumptions on the conductivity, this problem can be reduced to the analysis and reconstruction of the potential of a Schrödinger operator $-\Delta + V$ on the sphere. This problem is rather involved in general, from both the analytical and numerical points of view. Here we introduce an object that is obtained in terms of certain matrix elements of the Dirichlet-to-Neumann map -- the Born approximation -- which is reminiscent of an approximation for the potential that has been extensively studied in the context of inverse scattering theory. We will show a number of interesting analytical properties of the Born approximation, in particular how it can approximate in a suitable sense the potential in the Calderón problem. If time permits, we will also present a novel algorithm for numerical reconstruction based on this object.

Fethi Mahmoudi:

Given an n-dimensional compact riemannian manifold without boundary $(M,g)$ we establish existence of positive solutions for the nonlocal equation \varepsilon^{2s} P_g^s u + u = u^{p}\ \hbox{in} \ (M,g) where $P_g^s$ stands for the fractional Paneitz operator with principal symbol $(-\Delta_g)^s$, $s \in (0,1)$, $ p \in (1,2_{s}^*-1)$ with $2_s^* := \frac{2n}{n-2s}$, $n>2s$, represents the critical Sobolev exponent and $\varepsilon > 0$ is a small real parameter. We construct a family of solutions $u_\varepsilon$ that concentrate as $\varepsilon $ goes to zero near critical points of the mean curvature $H$ if $0 <s< \frac{1}{2}$, and near critical points of a reduced function involving the scalar curvature of the manifold $M$ if $ \frac{1}{2} \leq s < 1$.

AnnaLaura Stingo:

The Kaluza-Klein theories represent the classical mathematical approach to the unification of general relativity with electromagnetism and more generally with gauge fields. In these theories, general relativity is considered in 1+3+d dimensions and in the simplest case d=1 dimensional gravity is compactified on a circle to obtain at low energies a (3+1)-dimensional Einstein-Maxwell-Scalar systems. In this talk I will discuss the problem of the classical global stability of Kaluza-Klein theories when d=1. This is a joint with C. Huneau and Z. Wyatt.

Jeremie Szeftel:

I will introduce the celebrated black hole stability conjecture according to which the Kerr family of metrics are stable as solutions to the Einstein vacuum equations of general relativity. I will then discuss the history of this problem, including a recent work on the resolution of the black hole stability conjecture for small angular momentum.