Let $X$ be a metric measure space with a positive Borel measure. For  $1\leq p\leq\infty$, the (homogeneous)  Hajlasz Sobolev space $\dot{M}^1_p$ is defined  as  the set of all functions $u\in L_{1,loc}$ such that there exists a measurable function $g\geq 0$, $g\in L_p$, satisfying
 \begin{equation*}
 |u(x)-u(y)|\leq d(x,y)(g(x)+g(y))\quad\mu-a.e.
\end{equation*}
Assume now that $X$ is a Riemannian manifold satisfying the doubling volume property and  admitting a Poincaré inequality $(P_p)$. Then  the Hajlasz Sobolev space and the usual Sobolev space coincide if $1<p \leq +\infty$. This is not the case for $p=1$: $\dot{M}^1_1$ is a strict subspace of $\dot{W}^1_1$. In this talk, we prove that $\dot{M}^1_1$ is identified with a Hardy-Sobolev space defined in terms of atoms. We also give a maximal characterization of this space. This is a joint work with G. Dafni. 

Given a compact manifold $N$,   we consider the space $W^{k,p}(\mathbb{B}^m,N)$ of those Sobolev maps from the unit ball in $\mathbb{R}^m$ with values into $N.$   It may happen that the set of smooth maps $C^{\infty}(\mathbb{B}^m,N)$ is not dense in $W^{k,p}(\mathbb{B}^m,N).$  This is the main aim of the talk to present the topological obstructions involving such a phenomenon.  Special emphasis will be placed on the case when  $N$ is a  sphere. This is a joint work with Augusto Ponce and Jean Van Schaftingen.

It is not difficult to prove that the drift-diffusion equation $\partial \mu_t=\nabla\cdot(\mu_t(\nabla \log\mu_t+\nabla \psi))$ converges weakly to the steady state $e^{-\psi}$. Moreover, under the convexity condition $\nabla^2\geq\lambda Id$ with $\lambda>0$ (also called Bakry-Emery condition) the solution converges to the steady state with an exponential rate. Our main purpose is to study the exponential convergence in Wasserstein distance of such equation and to exhibit more examples. Work written in collaboration with F. Bolley and A. Guillin.

We consider the simplified Ginzburg-Landau energy $\displaystyle\dfrac 12\int_\Omega|\nabla u|^2+\dfrac 1{4\varepsilon^2}(1-|u|^2)^2$. Here, $\Omega$ is a domain in ${\mathbb R}^2$ and $u$ is complex-valued. On $\partial\Omega$, we prescribe $|u|=1$ and the winding numbers of $u$. This is the simplest model leading to non scalar bubbles. I will discuss existence/non existence results for minimizers/critical points. The talk is based on results of Berlyand, Dos Santos, Farina, Golovaty, Rybalko and the lecturer.

Multiplicative chaos theory has been introduced by J.P. Kahane to construct more realistic models than the celebrated B. Mandelbrot's cascades in many fields of applications: turbulence, finance,... The first part of the talk will be devoted to introducing the model as well as emphasizing its main properties. In particular, I will insist on the notion of stochastic scale invariance which can be thought of as a natural extension of the notion of self-similarity for stochastic processes. In the second part of the talk, I will set forth related research fields and new open problems.

Let $1<p<+\infty$. It is well-known that, for all functions $f\in {\mathcal D}({\mathbb R}^n)$,
\begin{equation} \tag{1}
\left\Vert (-\Delta)^{1/2}f\right\Vert_{L^p(\mathbb R^n)}\sim \sum_{j=1}^n \left\Vert \partial_jf\right\Vert_{L^p(\mathbb R^n)}.
\end{equation}
When $p=1$, a version of (1) holds, where $L^1(\mathbb R^n)$ has to be replaced by the real Hardy space $H^1(\mathbb R^n)$.
In this talk, we will discuss different generalizations of (1) in various contexts, replacing $\mathbb R^n$ by domains of $\mathbb R^n$, Riemannian manifolds or graphs, and $\Delta$ by more general second order uniformly elliptic operators. We will also present related open problems. The results presented in this talk were obtained in collaboration with Pascal Auscher, Nadine Badr, Alan McIntosh and Philippe Tchamitchian.

In this talk we deal with the existence and structure of solutions of overdetermined elliptic problems. In particular, we show that many results and ideas coming from the isoperimetric problem, and more generally from the geometric theory of constant mean curvature surfaces, have their corresponding results or ideas in the context of overdetermined elliptic problems, and represent hence some interesting tools of analysis.

I will prove pointwise algebra properties of Besov spaces defined on Lie groups. In a first time, I will introduce a Littlewood-Paley decomposition on Lie groups of polynomial volume growth and then define Besov spaces in this context. Then I will prove that Besov spaces are algebras under pointwise multiplication. I will also particularize the results to groups of Heisenberg type, where a group Fourier transform is available. I will explain how to develop a para-product and a pseudo-differential calculus on such groups. This is joint work with Isabelle Gallagher.

We study the fluid limit of a kinetic equation involving a small parameter perturbed by a smooth random term (which also involves the small parameter). This is a first attempt in a program of analysis of fluid limits of random kinetic equations. We will show here the convergence to a stochastic diffusion equation. This is a joint work with A. Debussche.