Title: Isometric Rigidity and Flexibility of Wasserstein Spaces
Abstract: The optimal transport theory has been used to model
several phenomena both within mathematics and in other fields and is
currently a very active area of research. In this talk, we will delve
into a geometric aspect of this theory, specifically concerning the
symmetries of Wasserstein spaces.
More precisely, given a real number p ∊ [1,∞) and a metric
space (X,d), the p -Wasserstein space over X is the space
Pp(X) consisting of Borel probability
measures on X with finite p-moment, endowed with the distance
function induced by solving the optimal transport problem with the
cost function c(x,y)=d(x,y)p. We say that X is isometrically
rigid with respect to the p-Wasserstein distance if the group of
isometries of Pp(X) is isomorphic to the group of
isometries of X; otherwise, we say it is isometrically flexible.
In general, determining if X is isometrically rigid with respect to
the p-Wasserstein distance heavily relies on the geometry of X and
the value of p . In this talk, I will give an overview of this topic
and present some recent results about the isometric rigidity and
flexibility for some families of spaces, based on a collaboration with
Fernando Galaz-García, Martin Kerin, and Jaime Santos-Rodríguez.