This memoir presents results in three drections. In Riemannian geometry, we prove a generalized Günther inequality on volume, and in dimension 4 an isoperimetric inequality for manifold whose curvature is bounded from above. In the geometry of Wasserstein space from optimal transport, we prove embedding and non-embedding results, we compute isometry groups, and we study the dynamics of expanding circle maps acting on measures. In Chabauty topology, we prove that the space of closed subgroups of $R^n$ is simply connected.
