A mapping between metric spaces is L-bi-Lipschitz if it stretches distances by a factor of at most L, and compresses them by a factor no worse than 1/L. A basic problem in geometric analysis is to determine when one metric space can be bi-Lipschitz embedded in another, and if so, to estimate the optimal bi-Lipschitz constant. In recent years this question has generated great interest in Computer Science, since many data sets can be represented as metric spaces, and associated algorithms can be simplified, improved, or estimated, provided one knows that the metric space space in question can be bi-Lipschitz embedded (with controlled bi-Lipschitz constant) in a nice space, such as L^2 or L^1.
The lecture will discuss several new existence and non-existence results for bi-Lipschitz embeddings into Banach spaces. The second part of the lecture will focus on the case when the target it L^1. Here a crucial role is played by a new connection with functions of bounded variation, and some very recent advances in geometric measure theory.
This is joint work with Jeff Cheeger.