June
27 
June
28 
June
29 

10h10h55 
Nadine Badr  9h1510h10 
Pieralberto Sicbaldi  9h3011h 
Governing
board
meeting 
11h11h55 
Pierre Bousquet  10h1511h10 
Yannick Sire  
12h13h30 
Lunch  11h1512h15 
Julien Vovelle  
13h3014h25 
Ivan Gentil  12h1514h  Lunch 

14h3015h25 
Rémi Rhodes  
15h3016h 
Break 
15h3017h  Poster session  
16h16h55 
Petru Mironescu  
17h17h55 
Emmanuel Russ 
Nadine Badr 
Title: An atomic decomposition and maximal characterization of the Hajlasz Sobolev space $\dot{M}^1_1$ on manifolds Abstract: 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\mua.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 HardySobolev space defined in terms of atoms. We also give a maximal characterization of this space. This is a joint work with G. Dafni. 
Pierre Bousquet 
Title: Density and singularities in Sobolev spaces between manifolds Abstract: 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.

Ivan Gentil 
Title:
Exponential convergence of diffusion equation in Wasserstein distance Abstract: It is not difficult to prove
that the driftdiffusion 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 BakryEmery 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.

Petru Mironescu 
Title: GinzburgLandau energy with prescribed degrees Abstract: We consider the simplified
GinzburgLandau energy $\displaystyle\dfrac
12\int_\Omega\nabla u^2+\dfrac 1{4\varepsilon^2}(1u^2)^2$. Here,
$\Omega$ is a domain in ${\mathbb R}^2$ and $u$ is complexvalued. 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.

Rémi Rhodes 
Title: Introduction to multiplicative chaos theory and related open problems Abstract: 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 selfsimilarity for stochastic processes. In the second part
of the talk, I will set forth related research fields and new open
problems.

Emmanuel Russ 
Title: Some problems about the square root of uniformly elliptic operators Abstract: Let $1<p<+\infty$. It is
wellknown 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. 
Pieralberto Sicbaldi 
Title: From constant mean curvature surfaces to overdetermined elliptic problems Abstract: 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.

Yannick Sire 
Title: Besov algebras on Lie groups and paraand pseudodifferential calculus on Htype groups Abstract: I will prove pointwise algebra
properties of Besov spaces
defined on Lie groups. In a first time, I will introduce a
LittlewoodPaley 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
paraproduct and a pseudodifferential calculus on such groups. This is
joint work with Isabelle Gallagher.

Julien Vovelle 
Title: Diffusion limit for a stochastic kinetic problem Abstract: 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.
