## Program

 June 27 June 28 June 29 10h-10h55 Nadine Badr 9h15-10h10 Pieralberto Sicbaldi 9h30-11h Governing board meeting 11h-11h55 Pierre Bousquet 10h15-11h10 Yannick Sire 12h-13h30 Lunch 11h15-12h15 Julien Vovelle 13h30-14h25 Ivan Gentil 12h15-14h Lunch 14h30-15h25 Rémi Rhodes 15h30-16h Break 15h30-17h Poster session 16h-16h55 Petru Mironescu 17h-17h55 Emmanuel Russ

## Titles and abstracts

 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\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 $1 0$ (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. Petru Mironescu Title: Ginzburg-Landau energy with prescribed degrees Abstract: 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. 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 self-similarity 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