The study of diffeomorphism groups and their applications to problems in analysis and geometry has a long history. Geometric hydrodynamics, pioneered by V. Arnold in the 1960s, considers an ideal fluid flow as the geodesic motion on the infinite-dimensional group of volume-preserving diffeomorphisms of a flow domain in the energy metric. A similar consideration on the space of densities led to a geometric description of the optimal mass transport and the Kantorovich-Wasserstein metric. Likewise, the information geometry associated with the Fisher-Rao metric and Hellinger distance has an equally beautiful infinite-dimensional geometry and can be regarded as a higher Sobolev analogue of optimal transportation.
In this mini-course, I shall describe various Riemannian metrics on diffeomorphism groups and describe the connection to information geometry. The material is based on a forthcoming essay with B. Khesin and G. Misiolek, which shall make up the course literature.