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### Sunday Morning Video (in french): Les travaux de Grothendieck.sur les espaces de Banach, Gilles. Pisier (Lectures grothendieckiennes)
La thèse de Grothendieck et son article ultérieur intitulé “Résumé de la théorie métrique des produits tensoriels topologiques” (1956) a eu un énorme impact sur le développement de la géométrie des espaces de Banach pendant les 60 dernières années. Nous passerons en revue ce “Résumé” en nous concentrant sur le résultat que Grothendieck lui-même a appelé le théorème fondamental de la théorie métrique des produits tensoriels, maintenant devenu “l’inégalité de Grothendieck” ou “le théorème de Grothendieck”. Ce résultat a récemment fait une apparition pour le moins inattendue dans plusieurs domaines a priori fort éloignés des préoccupations de Grothendieck. L’une a trait aux C ∗ -algèbres et aux espaces d’opérateurs (ou “espaces de Banach non-commutatifs”), une autre aux inégalités de Bell et à leur “violation” en mécanique quantique, une dernière relie la constante de Grothendieck au problème P=NP et à la théorie des graphes.
Here is a review that covers some of what is mentioned in the video:
Probably the most famous of Grothendieck’s contributions to Banach space theory is the result that he himself described as “the fundamental theorem in the metric theory of tensor products”. That is now commonly referred to as “Grothendieck’s theorem” (GT in short), or sometimes as “Grothendieck’s inequality”. This had a major impact first in Banach space theory (roughly after 1968), then, later on, in C∗-algebra theory, (roughly after 1978). More recently, in this millennium, a new version of GT has been successfully developed in the framework of “operator spaces” or non-commutative Banach spaces. In addition, GT independently surfaced in several quite unrelated fields:\ in connection with Bell’s inequality in quantum mechanics, in graph theory where the Grothendieck constant of a graph has been introduced and in computer science where the Grothendieck inequality is invoked to replace certain NP hard problems by others that can be treated by “semidefinite programming” and hence solved in polynomial time. In this expository paper, we present a review of all these topics, starting from the original GT. We concentrate on the more recent developments and merely outline those of the first Banach space period since detailed accounts of that are already available, for instance the author’s 1986 CBMS notes.
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