The lecture will discuss several new existence and non-existence results for bi-Lipschitz embeddings in Banach spaces. One approach to non-existence theorems is based on generalized differentiation theorems in the spirit of Rademacher's theorem on the almost everywhere differentiability of Lipschitz functions on R^n. We first show that earlier differentiation based results of Pansu and Cheeger, which proved non-existence of embeddings into R^k, generalize to many Banach space targets, such as L^p, for 1 < p < infinity. We then focus on the case when the target is L^1, where differentiation theory is known to fail, and the embedding questions are of particular interest in computer science. When the domain is the Heisenberg group with its Carnot-Caratheodory metric, we show that a modified form of differentiation still holds for Lipschitz maps into L^1, by exploiting a new connection with functions of bounded variation, and some very recent advances in geometric measure theory. This lead to a proof of a conjecture of Assaf Naor.

This is joint work with Jeff Cheeger.