Information Geometry of Diffeomorphism Groups - Part 1

Explore the first lecture in a mini-course series examining the geometric foundations of diffeomorphism groups and their connections to information geometry. Delve into the historical development of diffeomorphism groups and their analytical and geometric applications, starting with Arnold's pioneering work in geometric hydrodynamics from the 1960s. Learn how ideal fluid flow can be understood as geodesic motion on infinite-dimensional groups of volume-preserving diffeomorphisms, and discover the geometric principles underlying optimal mass transport and the Kantorovich-Wasserstein metric. Examine the information geometry associated with the Fisher-Rao metric and Hellinger distance, understanding their relationship to infinite-dimensional geometry and higher Sobolev optimal transportation. Based on forthcoming research with B. Khesin and G. Misiolek, investigate various Riemannian metrics on diffeomorphism groups and their fundamental connection to information geometry in this 52-minute presentation from the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" at the Erwin Schrödinger International Institute for Mathematics and Physics.