  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.
