## Les événements de septembre 2017

• ### Jeudi 28 septembre 15:00-17:30 - Mathieu Colin (Univ. de Bordeaux) ; Alberto Farina (Univ. d’Amiens)

Séminaires Equations aux Dérivées Partielles

Résumé : Mathieu Colin : "Ondes solitaires et systèmes de Schrödinger"
Alberto Farina : "Un résultat de type Bernstein pour l’équation des surfaces minimales"

Lieu : 316B

• ### Mardi 26 septembre 13:45-14:45 - Colin Petitjean - UFC

The linear structure of some dual Lipschitz free spaces

Résumé : Consider a metric space $M$ with a distinguished point $0_M$. Let $Lip_0(M)$ be the Banach space of Lipschitz functions from $M$ to $\mathbb R$ satisfying $f(0_M) = 0$ (the canonical norm being the best Lipschitz constant). The Lipschitz-free space $\mathcal F(M)$ over $M$ is defined as the closed linear span in $Lip_0(M)^*$ of $\delta(M)$ where $\delta (x)$ denotes the Dirac measure defined by $\langle \delta (x) , f \rangle = f(x)$. The Lipschitz free space $\mathcal F(M)$ is a Banach space such that every Lipschitz function on $M$ admits a canonical linear extension defined on $\mathcal F(M)$. It follows easily from this fundamental linearisation property that the dual of $\mathcal F (M)$ is in fact $Lip_0(M)$. A considerable effort to study the linear structure and geometry of these spaces has been undergone by many researchers in the last two or three decades.
In this talk, we first focus on some classes of metric spaces $M$ for which $\mathcal F(M)$ is isometrically isomorphic to a dual Banach space. After a quick overview of the already known results in this line, we define and study the notion of "natural predual". A natural predual is a Banach space $X$ such that $X^* = \mathcal F(M)$ isometrically and $\delta(M)$ is $\sigma(\mathcal F(M),X)$-closed. As we shall see, $\delta(M)$ is always $\sigma(\mathcal F(M),Lip_0(M))$-closed but it may happened that it is not $\sigma(\mathcal F(M),X)$-closed for some predual $X$. We characterise the existence of a natural predual in some particular classes of metric spaces. Notably, we concentrate on the class of uniformly discrete and bounded (shortened u.d.b.) metric spaces, for which it is well known that $\mathcal F(M)$ is isomorphic to $\ell_1$. In particular, we exhibit an example of a u.d.b. metric space $M$ for which $\mathcal F(M)$ is a dual isometrically but which does not have any natural predual. We also provide a u.d.b. metric space $M$ such that $\mathcal F(M)$ is not a dual isometrically. We finish with the study of the extremal structure of Lipschitz free spaces admitting a natural predual.
This is part of a joint work with L. García-Lirola, A. Procházka and A. Rueda Zoca.

Lieu : 316Bbis

• ### Jeudi 28 septembre 16:30-18:00 -

Commission Informatique (CI)

Lieu : Salle 316Bbis

• ### Vendredi 29 septembre 10:00-12:00 -

Soutenance de thèse de Tianxiang GOU

Lieu : Salle 316B (LMB, UFR ST)

• ### Mardi 26 septembre 14:15-16:00 -

Soutenance d’habilitation d’Antoine PERASSO

Lieu : Salles de Actes (UFR ST, Besançon)