In the mid-18th century, Euler derived hisfamous equations of motion of an incompressible fluid, one of the most studiedequations in hydrodynamics. More than 200 years later, in 1966, Arnold observedthat they are, in fact, geodesic equations on the (infinite dimensional)Lie group of volume-preserving diffeomorphisms of a manifold, endowed with acertain right-invariant Riemannian metric. Since then, many importanthydrodynamical PDEs were recast, similarly, as geodesic equations, and similargeodesic equations, in turn, turned out to be important in imaging problems(also known as "shape analysis"). In the talk I will present thebasic settings of Riemannian geometry on diffeomorphism groups (and similarspaces), and discuss some their geometric properties (e.g., completeness,diameter). I will then show how this interpretation can help us studyingcertain PDEs (like incompressible Euler), and discuss some applications toshape analysis.
Thu, 16/01/2020 - 16:00 to 17:15