Barycenters are defined to average probability measures on metric spaces. For Wasserstein spaces, (Wasserstein) barycenter is a direct generalization of the celebrated McCann interpolation, which corresponds to the barycenters of measures $\lambda \delta_{\mu_1} + (1 - \lambda) \delta_{\mu_2}$.
In the talk, we consider Wasserstein barycenters on Riemannian manifolds, and discuss the displacement functional used by the author in arXiv:2310.13832 to prove their absolute continuity with lower Ricci curvature bound assumptions. It is different from the widely used displacement convexity property combined with gradient flow, but still manifests an intriguing connection with the curvature-dimension condition. If time allowed, we will also explain how the Souslin space theory is applied in the proof, which is an unexpected technique for optimal transport.
Maxime Laborde
