Alain Valette (University of Neuchatel)

Title: The Wasserstein distance on metric trees

Abstract:

Let (X,d) be a metric space. The Wasserstein distance, or earthmover distance, is a metric on the space of probability measures on X, that originates from optimal transportation: if \mu,\nu are probability measures on X, the Wasserstein distance between \mu and \nu intuitively represents the minimal amount of work necessary to transform \mu into \nu. Since the definition of the Wasserstein distance involves an infimum, it is not expected in general that there is a closed formula for this distance. Such a closed formula however exists for metric trees (i.e. combinatorial trees where the length of an edge can be any positive real number), and this closed formula has an interesting history that makes it suitable for a colloquium talk: it appeared first in computer science papers (Charikar 2002), then surfaced again in bio-mathematics (Evans-Matsen 2012), before catching the interest of pure mathematicians. In joint work with M. Mathey-Prévôt, we advocate that the right framework for this closed formula is real trees (i.e. geodesic metric spaces with the property that any two points are connected by a unique arc); we give two proofs of the closed formula, one algorithmic, the other one connecting with Lipschitz-free spaces from Banach space theory.