Dr. David Bate (University of Warwick)

Bi-Lipschitz embeddings and optimal partial transport

A bi-Lipschitz embedding of a metric space X into another, Y, preserves relative distances between points, up to a multiplicative error. Usually, one seeks a bi-Lipschitz embedding when X is a metric space of interest, and Y has good geometric properties, since the embedding allows X to inherit these properties.

A classical example is Almgren’s embedding of the space of unordered k-tuples in Euclidean space, equipped with the Wasserstein metric, into a (higher dimensional) Euclidean space. This was the starting point to his treatise on the regularity of area minimising surfaces.

The Wasserstein metric is an example of a transportation metric that compares two probability measures on a space. Recently, there has been significant interest in transportation metrics that compare measures with different total masses. One such metric was introduced by Figalli and Gigli to describe solutions to the heat equation with Dirichlet boundary conditions. It compares two measures in a similar way to the Wasserstein metric, but allows excess mass to be transported to or from the boundary of a domain. This transportation metric also describes the space of persistence barcodes in topological data analysis.

In this talk we will first give a gentle introduction to the transportation metric of Figalli and Gigli. It will then be shown that the space of unordered k-tuples, equipped with this metric, does not bi-Lipschitz embedded into Euclidean space. However, we will then proceed to show how this space does in fact embed into Hilbert space. This talk will be accessible to all, is of a combinatorial nature, and no knowledge of measure theory is required.

This talk is based on joint work with A.L. Garcia-Pulido (Stirling).