Asymptotic and nonlinear geometries of Banach spaces and their interactions

TL;DR

This book explores how the nonlinear metric structure of Banach spaces relates to their linear asymptotic properties, aiming to find metric characterizations of these properties through the Kalton and Ribe programs.

Contribution

It provides a comprehensive overview of recent research on the preservation of asymptotic properties of Banach spaces under nonlinear embeddings, advancing the understanding of their geometric and metric structures.

Findings

Characterization of Banach space properties via nonlinear embeddings

Connections between metric graph geometries and Banach space asymptotics

Advancement of the Kalton and Ribe programs in Banach space theory

Abstract

This book discusses the interactions between the (nonlinear) metric structure of Banach spaces and their linear asymptotic behavior. The overarching problem is to understand how the various linear structures of a Banach space are preserved under certain nonlinear maps. The first chapters contain what are by now classical results to study the most basic and fundamental rigidity problems: the Lipschitz or uniform classification of Banach spaces. The other chapters form the main contribution of this book. The intended goal is to cover the work of many researchers, in particular their discoveries from the past 25 years, trying to understand how asymptotic properties of Banach spaces are preserved under several essential notions of nonlinear (bi-Lipschitz, coarse-Lipschitz, coarse or uniform) embeddings. This is part of a broader program called the Kalton program. This program, inspired by the Ribe program, seeks to uncover purely metric characterizations of asymptotic properties of Banach spaces. Many of these charaterizations are closely connected to the geometry of families of metric graphs (trees, Hamming graphs, diamond graphs, interlacing graphs) thus this book is also about the geometric structure of those graphs.