Anton Izosimov (University of Toronto) Lie algebroids and vortex sheets Abstract: In 1966, V.Arnold suggested a grouptheoretic framework for ideal hydrodynamics. In this framework, the motion of an incompressible fluid on a Riemannian manifold is described as the geodesic flow of a rightinvariant metric on the group of volumepreserving diffeomorphisms. While Arnold's approach is, in general, not applicable to fluid flows with discontinuities, it has been recently shown by C.Loeschcke and F.Otto that certain discontinuous fluid motions, known as vortex sheets, can also be viewed as geodesics, but on the space of embedded hypersurfaces. In the talk, I will explain how these two pictures (geodesics on a group and geodesics on the space of hypersurfaces) are related to each other via the theory of Lie groupoids and algebroids. In particular, I will describe a large class of discontinuous fluid motions which can be regarded as trajectories of a Hamiltonian system on the dual of a certain infinitedimensional Lie algebroid. Vortex sheets arise in this setting as solutions lying on a special symplectic leaf. This is joint work with B.Khesin. 
