ouverture

Sur des variétés asymptotiquement hyperboliques en dimension 4

• Yuxin Ge

Room: Salle Pellos (1R2-203) Etant donnée une variété compacte de dimension 3 $(M^3 , [h])$, quand l’on pourrait remplir par une variété asymptotiquement hyperbolique de dimension 4 $(X^4 , g_+ )$ telle que $r^2 g_+ |_M = h$ sur le bord $M = \partial X$ pour certaine fonction définissante $r$ sur $X^4$ ? Ce problème est motivé par la correspondance AdS/CFT en gravité quantique proposé par Maldacena en 1998 et provient également de l’étude de la structure des variétés asymptotiquement hyperboliques. Dans cet exposé, je discute le problème de la compacité des variétés asymptotiquement hyperboliques en dimension 4, c’est-à-dire, comment la compacité de l’infini conforme entraı̂ne la compacité de la compactification des telles variétés sous des hypothèses convenables sur la topologie et des invariants conformes. En tant qu’applications, on montre quelques résultats sur l’existence de tels remplissages.

Entropies and Aviles-Giga model

• Abdelmajid Moustajab

Room: Salle Pellos (1R2-203) Domain walls are transition layers connecting two limit states in physical models (e.g. ferromagnetic thin films, liquid-crystals...) We study them in a Ginzburg-Landau type model for curl-free vector fields, the so-called Aviles-Giga model, that depends on a small parameter $\varepsilon. $ As $\varepsilon \to 0, $ the limit configurations satisfy the eikonal equation. We develop a theory of entropies/calibrations for this equation and prove their characterization and a Liouville type rigidity result which enable us to obtain sharp lower bounds for domain walls in dimension 3 for the Aviles-Giga model.

Isoperimetric and geometric inequalities in quantitative form: Stein's method approach

• Jordan Serres

Room: Salle Pellos (1R2-203) Stein’s method is a collection of tools designed to quantify the closeness between two probability measures. It was initially developed extensively to measure the distance to the Gaussian distribution and has since been generalized to measures with full support on $\mathbb{R}^d$. The application of this method to uniform distributions over different domains presents additional difficulties due to boundary effects, and has only recently been explored, allowing the introduction of new notions of distance between domains. In this talk, I will present the fundamental ideas of Stein's method for shapes, and then mention an application of the method to derive stability estimates in Steklov's spectral problem.

On the stability of the Brascamp-Lieb inequality

• Aldéric Joulin

Room: Salle Pellos (1R2-203) In this talk, we address the stability problem of the famous Brascamp-Lieb inequality for striclty log-concave probability measures on the Euclidean space. More precisely, if a given function almost satisfies the equality in the Brascamp-Lieb inequality, is it true that it is close in some sense to the underlying extremal functions ? Under some assumptions on the eigenvalues of the Hessian matrix of the associated potential, we prove that the distance to the extremal functions in quadratic norm is of order square root of the deficit parameter and involves the second positive eigenvalue of a convenient diffusion operator we wish to estimate. Our results are illustrated by some examples for which the usual uniform convexity assumption on the potential is relaxed. Joint work with M. Bonnefont (Bordeaux) and J. Serres (IMT).