Topological symplectic manifolds and bi-Lipschitz structures

Abstract: A symplectic homeomorphism is a C^0 limit of symplectic diffeomorphisms; a topological symplectic manifold is a manifold with an atlas whose transition maps are symplectic homeomorphisms. I will explain recent joint work showing that all such manifolds are bi-Lipschitz. As a consequence, Donaldson theory extends to topological symplectic 4-manifolds. This gives the first examples of manifolds that do not admit topological symplectic structures. Our arguments work for any manifold whose transition maps are C^0 limits of bi-Lipschitz homeomorphisms.