Award: DMS 1105700, Principal Investigator: Michael Usher
This project will use a variety of methods, mostly arising from Floer theory, to study questions relating to Hamiltonian diffeomorphisms of symplectic manifolds. The goals of the project include: expanding the class of manifolds for which the Hamiltonian diffeomorphism group (or its universal cover) is known to admit Calabi quasi-morphisms; investigating global geometric questions about Hofer's metrics on the Hamiltonian diffeomorphism group and on spaces of Lagrangian submanifolds (for instance, when do these spaces have infinite diameter?); and developing a new family of relative symplectic capacities that are constructed by using Floer theory in a novel way. To achieve this, the PI will make use of tools coming from the natural real-valued filtrations on Floer complexes, including the well-established Oh-Schwarz spectral invariants and also a newer invariant, the boundary depth, that was first introduced by the principal investigator in 2009. An auxiliary goal of the project is to obtain a better understanding of the behavior of these two useful invariants.
Hamiltonian diffeomorphisms of symplectic manifolds can be used to mathematically model those classical physical systems in which energy is conserved. Thus they are relevant in the study of a very wide range of phenomena, from the motion of planets and satellites (with potential applications to lower-cost space mission planning) to the flow of turbulent fluids. A remarkable geometric structure on the group of Hamiltonian diffeomorphisms of a symplectic manifold was discovered by Hofer over 20 years ago; despite much effort some basic aspects of this geometry remain poorly understood. Much of this project is aimed at obtaining new results about Hofer's geometry, and at clarifying how the behavior of Hamiltonian diffeomorphisms of a symplectic manifold is related to other properties of the manifold.