Nonlinear retracts and the geometry of Banach spaces

Abstract

In the nonlinear geometry of Banach spaces where the objects in the category are Banach spaces as in the linear case, the morphisms in the new setting are taken to comprise of certain nonlinear maps involving say, Lipschitz maps and, in some cases, uniformly continuous, coarse or coarse Lipschitz mappings arising from the underlying metric or the uniform structure attached to the norm of the Banach space. The question as to what extent the Lipschitz or the uniform structure may be used to capture the full linear structure of a Banach space has been one of the most fundamental problems pertaining to the nonlinear structure of Banach spaces since this line of investigation was undertaken by Lindenstrauss in the late sixties. This line of research which is subsumed under the so called Ribe program broadly underscores the view that metric spaces encode a much deeper and hidden structure than is apparent. It is truly surprising how the linear structure of a Banach space gets captured to a considerable extent by its metric space structure. This point of view has led to deep insights into Banach space theory that has paved the way for these ideas being employed in seemingly unrelated disciplines including harmonic analysis, geometric group theory and many other domains of mathematics.