# 29 KAM Mathematical Colloquium

## Prof. DAVID PREISS

### LONDON

## LIPSCHITZOVSKA ZOBRAZENI

MARCH 27, 1997

Lecture Room S6, Charles University, Malostranske nam. 25,
Praha 1

11:00 AM

## Abstract

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.