Asymptotic and nonlinear geometries of Banach spaces and their interactions
Florent P. Baudier, Gilles Lancien
TL;DR
The work surveys the nonlinear geometry of Banach spaces, focusing on how metric (nonlinear) structures interact with linear and asymptotic properties. It develops a framework linking Lipschitz, coarse, and uniform embeddings to linear invariants via tools such as Lipschitz-free spaces, ultrapowers, and invariant means, culminating in Ribe and Kalton programs that aim at metric characterizations of asymptotic geometry. Key contributions include detailed treatments of metrical universality, linear reductions of Lipschitz problems, differentiation in infinite dimensions, and the Godefroy–Kalton Lipschitz-free machinery, which together establish rigidity results, stability under nonlinear quotients, and structural decompositions with broad implications for the geometry of Banach spaces. The chapters build a cohesive program for translating nonlinear geometric questions into linear or asymptotic ones, enabling rigorous classification and embedding results across a wide landscape of spaces such as ℓp, Lp, c0, and their free-space analogues. Overall, the text advances the Kalton program by clarifying how asymptotic moduli, renorming, and graph-geometric methods interact with nonlinear mappings to reveal the metric underpinnings of Banach space geometry.
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.