Mathematics Department Colloquium
Spring 2005

Date: Thursday, January 13
Time: 4:00 p.m.
Location: 215 Thomas Building
Name: Helmut Hofer
Affiliation: Courant Institute
Title: Dynamical systems at the interface of symplectic geometry and three-dimensional topology

Abstract: Recently it became clear that there is a large class of vector fields for which a global theory of surfaces of section can be developed. The vector fields in question are the so-called Reeb vector fields, which naturally arise in the study of contact structures (the odd-dimensional analogue of a symplectic structure). A typical example for a Reeb vector field is the vector field generating the geodesic flow restricted to the unit sphere bundle. It can be shown that there is a close relationship between the dynamics of a Reeb vector field $X$ on $M$ and a holomorphic curve theory for a particular almost complex structure on ${\bf R}\times M$. Exploiting this fact one can develop a global PDE-approach for constructing global systems for surfaces of section, which is a concept generalizing the notion of surface of section. Ongoing research seems to indicate that the existence of a global system of surfaces of section is not depending on the particular vector field, but rather on the homotopy class of the underlying contact structure. The global system, however, will in general depend strongly on the vector field, but as already said not its existence. Moreover, its seems feasible, to associate to the geometry of a global system of surfaces section invariants, which only depend on the contact structure, but not on the particular Reeb vector field.