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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

## Program

The lectures will start on Tuesday, May 29, in the afternoon (around 13:30)
and will end on Friday, June 1, in the morning (around 12:30).

Schedule

In both cases the calendar will stay synchronized should there be any changes.

## Info pratique

Conference venue
The conference will take place in Amphi B of the building "Metrologie" (the same building as where the math department (LmB) is located).

To arrive from the hotel to the conference venue, take the bus nr. 10 from the stop "Chamars" until the stop "Beaux Arts". Check the map below for further details.

The map of campus and city center.

Travelling to Besancon

As you might know already, the French trains are planning to exercise their right to strike during several days preceding and following the Spring school. The concrete dates can be checked on this web. The most relevant for our school are of course the May 28 and 29, and June 2 and 3. It is difficult to say to which extent the trains will not be working on these days. The most likely scenarios are "every second train canceled" or "no train on the rails". If it is still possible for you, avoid taking the train on these dates or choose rather some of the alternatives below: Bus: flixbus.fr, ouibus.fr Carpooling: blablacar.fr and others... Please let us know about your planned travel dates, so that we can adjust the accommodation reservation and/or adjust the daily schedule of the Spring school. Do not hesitate to write us if you need any help or advice!

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

 Arafat ABBAR Paris Est Marne-la-Vallée Vidal AGNIEL Université Lille 1/ ENS de Rennes Rim ALHAJJ Université Lille 1 Loris ARNOLD Université Bourgogne Franche-Comté Guillaume AUBRUN Université Lyon 1 Isabelle BARAQUIN Université Bourgogne Franche-Comté Jean-Christophe BOURIN Université Bourgogne Franche-Comté Léonard CADILHAC Université de Caen-Normandie Irène CASSELI Université Aix-Marseille Benjamin CELARIES Université Lyon 1 Marek CUTH Charles University Prague Romuald ERNST Université du Littoral Côte d’Opale Tonie FARES Université d’Artois Valentin FERENCZI Universidade de São Paulo Pedro FERNÁNDEZ MARTÍNEZ Universidad de Murcia Uwe FRANZ Université Bourgogne Franche-Comté Clifford GILMORE University of Helsinki Sophie GRIVAUX CNRS/Université de Lille Rainis HALLER Univeristy of Tartu Yulia KUZNETSOVA Université Bourgogne Franche-Comté Gilles LANCIEN Université Bourgogne Franche-Comté Florence LANCIEN Université Bourgogne Franche-Comté Christian LE MERDY Université Bourgogne Franche-Comté Laurent LOOSVELDT Université de Liège Monia MESTIRI Université de Mons Arman MOLLA Université de Liège Stefan NEUWIRTH Université Bourgogne Franche-Comté Alexandre NOU Université Bourgogne Franche-Comté Marcu-Antone ORSONI Université Bordeaux Andre OSTRAK Univeristy of Tartu Colin PETITJEAN Université Bourgogne Franche-Comté Yoël PERREAU Université Bourgogne Franche-Comté Tony PROCHAZKA Université Bourgogne Franche-Comté Matías RAJA Universidad de Murcia