On two open problems related to optimal transport

In this talk, we present two open problems on which we have partial results. The first problem is related to Wasserstein gradient flows of interaction energy functionals associated with the Riesz kernels. The second problem concerns spaces of so-called nonnegative cross-curvature introduced by Kim and McCann. Very few spaces are known to satisfy this condition related to smoothness of optimal transport and we present some (maybe not so) new examples.