Hamiltonian Dynamics and Pseudoholomorphic Curves

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An example of a system studied in classical mechanics is the motion of a small satellite under the gravitational influence of planets. Historically, symplectic geometry grew out of studying situations such as this, in which the collection of all possible positions and momenta of the system (phase space) was seen to have the structure of a special space known as a symplectic manifold. Physical laws, like conservation of energy, then gave rise to models of the systems as dynamical systems on submanifolds of phase space corresponding to different energy levels. Modern symplectic methods have established deep connections between global properties of symplectic manifolds and the associated dynamics on a special class of energy levels, namely contact-type hypersurfaces, but have thus far proved to be of rather limited use for a general energy level. This project aims to rectify this deficiency and use global symplectic techniques to study dynamics on arbitrary energy levels by analyzing a generalized class of curves, known as feral pseudoholomorphic curves. This project concerns research at the interface of geometric analysis, symplectic topology, and dynamical systems. The key idea is the use of a certain newly defined class of infinite Hofer-energy pseudoholomorphic curves, namely feral curves, to study Hamiltonian flows on prescribed energy surfaces in symplectic manifolds of arbitrary dimension and certain volume-preserving flows on three-manifolds in general. More specifically, the aim is to study these curves and their properties and to use them to establish non-minimality of a wide range of volume preserving flows in dimension three. A further aim is to study the generic asymptotic limit of such feral curves and to determine whether or not symplectic field theory admits an extension to symplectic manifolds with generic smooth boundary. Successful completion of this research has the potential to solve the volume-preserving Gottschalk conjecture, broadly extend symplectic field theory, and possibly yield key new insights on low dimensional differential topology.