Talk
Geometry and Hydrodynamics
Abstract
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We will begin with the geometric approach to ideal hydrodynamics introduced by Arnold. In particular, we will derive the incompressible Euler equations as a reduced geodesic equation on the group of volume-preserving diffeomorphisms of the fluid. This will be shown to be a specific instance of a more general notion of Euler-Arnold equation; a reduced geodesic equation on a Lie group equipped with a right-invariant metric. We will demonstrate features of the equation inherited by both the Riemannian and Lie group structure. Other potential topics include: Fredholmness of the associated exponential maps, conjugate points and pathologies arising in the study of general infinite-dimensional manifolds.
Keywords
Geometric Hydrodynamics, Infinite-Dimensional Geometry, Incompressible Euler Equations
Prerequisites
Useful, but not necessary: Riemannian and Lie geometry, PDEs, Functional Analysis.
