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### Geometry of Wasserstein spaces: Hadamard spaces

**Geometry/Topology**

Speaker: | Benoit Kloeckner, University of Grenoble |

Location: | 2112 MSB |

Start time: | Tue, Feb 14 2012, 3:10PM |

Optimal transport enables one to define a metric on the set of sufficiently concentrated probability measures of a metric space X: the squared distance between two measures is defined by the minimal cost of a transport plan betwen them, given that the cost for moving a unit of mass is the square of the distance it is moved by.

The goal of the talk is to study some geometric properties of this metric space of measures, called the Wasserstein space of X, when X is non-positively curved and locally compact. We shall focus on the following result: if X is CAT(-1), then the Euclidean plane cannot be isometrically embedded in the Wasserstein space of X, while there is a wealth of bi-Lipschitz embeddings.

(Joint work with Jerome Bertrand.)