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Explore a 57-minute lecture from the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" that delves into the geometric aspects of diffeomorphism groups and their connections to information geometry. Learn about various Riemannian metrics on diffeomorphism groups, building upon Arnold's pioneering work in geometric hydrodynamics from the 1960s. Discover how ideal fluid flow can be viewed as geodesic motion on infinite-dimensional groups of volume-preserving diffeomorphisms, and understand the geometric principles behind optimal mass transport and the Kantorovich-Wasserstein metric. Examine the information geometry associated with the Fisher-Rao metric and Hellinger distance, presented as a higher Sobolev analogue of optimal transportation. Based on forthcoming research with B. Khesin and G. Misiolek, gain insights into this complex mathematical framework that bridges differential geometry, information theory, and fluid dynamics.
