29 KAM Mathematical Colloquium
Prof. DAVID PREISS
MARCH 27, 1997
Lecture Room S6, Charles University, Malostranske nam. 25,
The infinite dimensional problem most intimately connected to the main topic
of the talk is the Lipschitz isomorphism problem: Is the linear structure of a
separable Banach space uniquely determined by its Lipschitz isomorphism class?
The `injective' and `surjective' parts of the problem lead to similar
questions for Lipschitz embeddings and Lipschitz quotients. The case of
embeddings is the only one where satisfactory answers are known: Aharoni's
example shows that the existence of a Lipschitz embedding does not imply
that of a linear one if the target is $c_0$, and the results on G\^ateaux
differentiability show that it does if the target is, e.g., reflexive.
The existence of derivative, or at least of a suitable linear approximant,
to a Lipschitz mapping turned out to be one of the main tools in all partial
results found so far. The talk will give a partial overview of this
rapidly developing research area and will describe some very recent
surprising results. The infinite dimensional problems are connected,
directly or indirectly, to a number of finite dimensional ones such as Gromov's
attempt to define quasi-regularity of mappings between spaces of different
dimensions, contractions of sets onto balls, or questions about the structure
of Lebesgue null sets.