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Explore the third lecture in a mini-course series examining the geometric aspects of diffeomorphism groups and their connection to information geometry. Delve into the mathematical foundations of geometric hydrodynamics, pioneered by V. Arnold in the 1960s, which views ideal fluid flow as geodesic motion on infinite-dimensional groups of volume-preserving diffeomorphisms. Learn about the geometric interpretation of optimal mass transport, Kantorovich-Wasserstein metric, and the information geometry associated with Fisher-Rao metric and Hellinger distance. Examine various Riemannian metrics on diffeomorphism groups and their relationships to information geometry, based on forthcoming research with B. Khesin and G. Misiolek. Gain insights into the infinite-dimensional geometry underlying these mathematical concepts as presented at the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" at the Erwin Schrödinger International Institute.
