Abstract: In recent decades, the theory of optimal transport has advanced rapidly, finding an ever-growing range of applications. The original problem of Monge is to find the cheapest way to transform one probability distribution into another when the cost is proportional to the distance. The most important metric structure that is related to optimal transport is the so-called p-Wasserstein space [denoted by Wp(X)] over the metric space X.

The pioneering work of Bertrand and Kloeckner started to explore fundamental geometric features of 2-Wasserstein spaces, including the description of complete geodesics and geodesic rays, determining their different types of ranks, and understanding the structure of their isometry group.

In this talk I will focus on isometry groups. A notable and useful property of p-Wasserstein spaces is that X embeds isometrically into Wp(X), moreover an isometry of X induces an isometry of Wp(X) by the push-forward operation. These induced isometries are called trivial isometries, and we say that Wp(X) is isometrically rigid if all its isometries are trivial. The question is: are there non-rigid Wasserstein spaces? What does a non-trivial isometry look like? Until very recently, only a few non-rigid examples were known such as the 2-Wasserstein space over R^n, and the 1-Wasserstein space over [0,1].

In the first part of the talk, I will introduce some key concepts and notation. The main focus will then shift to exploring results concerning both rigidity and non-rigidity in Wasserstein spaces.

For further information please contact elisur.magrini@unibocconi.it