A Short Course on Non Linear Geometry of Banach Spaces

In this course we show how some linear properties of Banach spaces, in particular properties related to the asymptotic structure of Banach spaces, are stable under coarse-Lipschitz embedddings or uniform homeomorphisms. We will focus on the recent use of some fundamental metric graphs or trees in the subject. Foreword. These notes are based on a series of five lectures given at the Winter school on “Functional and Harmonic Analysis” organized in Lens (France) in December 2010. I am glad to thank S. Grivaux and D. Li for the excellent organization of this school and for giving me the opportunity to present this series of lectures. I also would like to thank the participants and especially the students and young researchers for the very friendly and mathematically intense atmosphere they helped to create during this week. A great part of this course will be based on Nigel Kalton’s work. Unfortunately, Nigel Kalton passed away in August 2010. This is a terrible loss for his family, his friends and for Mathematics. One of the goals of this course is to give a flavour of some of the wonderful ideas he has left for the researchers in this field.