The non-linear geometry of Banach spaces after Nigel Kalton
Roughly speaking, ``nonlinear geometry of Banach spaces'' stands for a theory that aims at studying the linear structure of Banach spaces by means of nonlinear maps, e.g., Lipschitz or uniformly continuous maps. The survey paper under review presents a number of the deep results in this area obtained by the late Nigel Kalton, some of which were published in his posthumous papers, e.g., [\textit{N.J.~Kalton}, Fundam.\ Math.\ 212, No. 1, 53--69 (2011; Zbl 1220.46014), Math.\ Ann.\ 354, No. 4, 1247--1288 (2012; Zbl 1268.46018), Trans.\ Am.\ Math.\ Soc.\ 365, No. 2, 1051--1079 (2013; Zbl 1275.46011), Isr.\ J. Math.\ 194, Part A, 151--182 (2013; Zbl 1275.46012)]. Several theorems are explained with complete proofs, while other proofs are just sketched, but in a highly illuminating way. The paper starts with some tables indicating between which classical Banach spaces nonlinear embeddings of a special type are possible or not possible. The third section studies asymptotic uniform smoothness and convexity in connection with nonlinear mappings. The subject matter of Section~4 is the characterisation of Banach space properties by means of the embeddability of certain metric spaces (especially graphs), i.e., to describe the linear structure of a Banach space through its structure as a metric space, which has become known as the Ribe programme after the seminal contributions by M.~Ribe in the 1970s. In Section~5 the authors take a look at nonseparable spaces, while they give an introduction to the Novikov conjecture and its relation to coarse embeddings in Section~6. This section is especially designed for the lay reader in these matters and will be much appreciated by such readers, like myself. Finally, in Section~7 the Lipschitz free space, also known as the Arens-Eells space, of a metric or Banach space is presented; this is the canonical predual of the space of Lipschitz functions on the metric space and can be understood as its linearisation. Throughout the text, several open problems are mentioned. This survey, together with the Benyamini-Lindenstrauss volume [\textit{Y.~Benyamini} and \textit{J. Lindenstrauss}, Geometric nonlinear functional analysis.\ Volume~1.\ Colloquium Publications.\ American Mathematical Society 48.\ Providence, RI:\ American Mathematical Society (2000; Zbl 0946.46002)] and \textit{A. Naor}'s paper [Jpn.\ J. Math.\ (3) 7, No. 2, 167--233 (2012; Zbl 1261.46013)], is an ideal entry point to this very active area of contemporary research and will be welcomed by students and specialists alike.
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