# 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.