The question of bi-Lipschitz embeddability of Wasserstein spaces into classical Banach spaces has attracted much attention. The importance of such embeddings can be seen, for instance, in the proof of Almgren's partial regularity theorem.

In optimal transport, Wasserstein distances are the prime examples transportation metrics to compare measures of the same total mass. This talk will...

The Lott–Sturm–Villani curvature-dimension condition CD(K,N) provides a synthetic notion for a metric measure space to have curvature bounded from below by K and dimension bounded from above by N. It has been recently proved that this condition does not hold in any sub-Riemannian manifold equipped with a positive smooth measure, for every choice of the parameters K and N. In this talk, we...

The classical Courant nodal domain theorem states that the number of nodal domains of the k-th Dirichlet eigenfunction is bounded above by k. Pleijel later showed that only a finite number of eigenfunctions realised this bound. There have been extensions and improvements of Pleijel's result to the Robin and Neumann boundary conditions requiring boundary regularity that is much stronger than...