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# Ecole de printemps 2018 du GdR AFHP

## Première annonce

Le Laboratoire de Mathématiques de Besançon, avec le soutien de GdR “Analyse Fonctionnelle, Harmonique et Probabilités” organise du 29 mai au 1 juin 2018 une école de printemps. Trois cours d’une durée 3x1h30 seront donnés à l’intention des doctorants, jeunes chercheurs ou chercheurs confirmés. Ces cours seront donnés par :

 Guillaume Aubrun - Université Lyon 1 Geometry of quantum entanglement Marek Cúth - Université Charles de Prague Lipschitz-free spaces over finite-dimensional spaces Sophie Grivaux - CNRS et Université de Lille Baire Category methods in linear dynamics

Comité d’organisation :
Gilles Lancien et Tony Prochazka, Université Bourgogne Franche-Comté.

Contacts :
Gilles Lancien : gilles.lancien@univ-fcomte.fr
Tony Prochazka : antonin.prochazka@univ-fcomte.fr

## Résumés

 Guillaume Aubrun - Université Lyon 1 Geometry of quantum entanglement We will study the phenomenon of quantum entanglement with the point of view of high-dimensional convex geometry. We will show consequences of classical results from local theory of Banach spaces (Dvoretzky’s theorem, MM* estimate). Marek Cúth - Université Charles de Prague Lipschitz-free spaces over finite-dimensional spaces Given a metric space $M$, it is possible to construct a Banach space $\mathcal{F}(M)$ in such a way that the metric structure of $M$ somehow reflects the linear structure of $\mathcal{F}(M)$. This space $\mathcal{F}(M)$ is called the Lipschitz-free space over M (also known as the Arens-Eells space). The study of Banach space theoretical properties of Lipschitz-free spaces was initiated by a paper by Godefroy and Kalton [GK], where the authors proved, using this notion, e.g. that if a separable Banach space Y is isometric (not necessarily linearly) to a subset of a Banach space X, then Y is already linearly isometric to a subspace of X. Soon after, the study of Lipschitz-free Banach spaces became an active field of study. However, the structure of these spaces is still very poorly understood to this day. For example, it is not known whether $\mathcal{F}(\mathbb{R}^2)$ is linearly isomorphic with $\mathcal{F}(\mathbb{R}^3)$ or whether $\mathcal{F}(E)$ has a monotone Schauder basis for every finite-dimensional space $E$. The aim of this short series of lectures is to describe some of the recent developments concerning the structure of Lipschitz-free spaces over finite-dimensional spaces and their subsets. We will recall the basic definitions and properties of Lipschitz-free spaces. Next, we will sketch some of the results concerning the structure of Lipschitz-free spaces over finite-dimensional spaces and their subsets. We hope to save most of our time to describe the characterization of $\mathcal{F}(\mathbb{R}^d)$ as the quotient of $L_1(\mathbb{R}^d,\mathbb{R}^d)$ by the vector fields with zero divergence in the distributional sense (obtained independently in [CKK], [F] and [GL]) and present the result from [CKK2] that $\mathcal{F}(\mathbb{R}^d)$ is complemented in its bidual, which is based on this characterization. [CKK] M. Cúth, O. F. K. Kalenda, P. Kaplický : Isometric representation of Lipschitz-free spaces over convex domains in finite-dimensional spaces, Mathematika, 63 (2017), 538–552. [CKK2] M. Cúth, O. F. K. Kalenda, P. Kaplický : Finitely additive measures and complementability of Lipschitz-free spaces, preprint available at arxiv.org [F] G. Flores, Estudio de los espacios Lipschitz-libres y una caracterizacin para el caso finitodimensional, Master Thesis, 2016. [GK] G. Godefroy and N. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (2003), 121—141. [GL] G. Godefory, N. Lerner : Some natural subspaces and quotient spaces of $L^1$, Adv. Oper. Theory 3 (2018), no. 1, 61–74. Sophie Grivaux - CNRS et Université de Lille Baire Category methods in linear dynamics Given a separable Hilbert space of infinite dimension $H$, one can consider on the space $\mathcal{B}(H)$ of bounded linear operators on $H$ several natural topologies which turn the closed balls $B_M(H)=\{T\in\mathcal{B}(H) ; \|\|T\|\|\le M\}$ into Polish spaces. The aim of this course will be to present - on the one hand, some results concerning typical properties in the Baire Category sense of operators of $\mathcal{B}_M(H)$ for these topologies ; - on the other hand, some applications of these typicality results to Hilbertian linear dynamics, i.e. to the study of dynamical systems given by the action of an operator $T\in\mathcal{B}(H)$ on $H$.

## Inscriptions

Pour les 30 premiers inscrits parmi les chercheurs non permanents membres du GdR AFHP (doctorants, ATER, post-docs..) le logement sera pris en charge par l’organisation. Les autres participants peuvent faire une demande pour la même prise en charge, qui sera étudiée en fonction des crédits disponibles.

Pour s’inscrire, veuillez copier les lignes suivantes dans un mail, complétez-les et envoyez le tout à Tony Prochazka. La date limite est le 27 avril 2018.
Prénom et Nom :
Université :
Date d’arrivée :
Date de départ :
Souhaitez-vous qu’on prenne en charge votre logement : oui / non

## Info pratique

Travelling to Besancon

The easiest way to get to Besancon is to fly to Paris, Lyon, Basel, Strasbourg, Geneva or Zurich and then take a train to Besancon Viotte train station. (Attention ! Not to be confused with Besancon Franche-Comte TGV train station which is in the middle of the woods about 16 km from Besancon !)
The tickets for the train can be booked here.

From the train station you can

• either catch bus No. 3 going to "Temis" (timetable for going, timetable for returning) to go to the math department (get off at the stop "CROUS Université", it’s about 10 minutes of bus ride + 5 minutes of walking in campus)
• or take the tram No. 2 (timetable) to go to the hotels downtown (for the hotel Zenitude la City, get off at the stop "Canot").

A map of the bus/tram stops near the train station Viotte.

## Participants

 Vidal AGNIEL Université Lille 1/ ENS de Rennes Marcu-Antone ORSONI Université Bordeaux