**12/1/2008**

**Alessio Figalli**

**Université de Nice Sophia-Antipolis and Ecole Polytechnique**

**Regularity results for optimal transport maps on Riemannian manifolds**

In this talk I will describe some recent developments of the regularity theory for optimal transport maps on Riemannian manifolds, when the cost function is given by the squared distance. We will see in particular that the regularity theory of optimal maps allows to study the geometry of the cut locus of the manifold.