Adrian Dawid – Hofer geometry and spectral invariants

Abstract: On a symplectic manifold M, the group of Hamiltonian diffeomorphisms Ham(M) is a natural object of study. In 1990, Hofer introduced a bi-invariant metric on Ham(M). When M models the phase space of a mechanical system, Ham(M) arises as the group describing all admissible motions. Intuitively, the Hofer distance between a Hamiltonian diffeomorphism and the identity can be thought of as the minimal energy necessary to generate the diffeomorphism. Many properties of the Hofer metric remain unknown. Spectral invariants coming from Floer theory are a powerful tool used to study the Hofer metric. Using spectral invariants, we will show that Ham(M) admits quasi-isometric embeddings of Euclidean space in various settings.