Berlin-Hamburg-Seminar am 2.12.2019

Marco Golla (Nantes) Singular symplectic curves: constructions, isotopy, fillings

I will discuss a construction, existence and uniqueness (up to isotopy) of symplectic curves (mostly in the projective plane) whose singularity are modelled over complex singularities. The focus will be on rational curves with irreducible singularities and their relationship with fillings. This is mostly based on joint projects with Laura Starkston (partly in progress) and with John Etnyre (in progress).

Kathrin Helmsauer (Augsburg) The Euler equation on manifolds with a stable Hamiltonian structure

According to Ebin and Marsden, solving the Euler equation on a manifold is equivalent to finding geodesics on its diffeomorphism group. Using this method, one can prove the local existence of solutions for the Euler equation for volume-presering vector fields, on symplectic manifolds and on certain contact manifolds. We extend this result to manifolds with a stable Hamiltonian strucure. To that end, we have to find conditions so that the diffeomorphism group preserving the stable Hamiltonian structure (SHS diffeomorphisms) is a smooth Hilbert submanifold of the full diffeomorphism group. We also compute the orthogonal projection of the full tangent bundle to the tangent bundle of the SHS diffeomorphisms and show that it is a smooth bundle map. Following Ebin and Marsden, this implies the local existence of solutions to the Euler equation on manifolds with a stable Hamiltonian structure.