DOCUMENTA MATHEMATICA, Extra Volume ICM I (1998), 255-280

Helmut H. W. Hofer

Title: Dynamics, Topology, and Holomorphic Curves

In this paper we describe the intimate interplay between certain classes of dynamical systems and a holomorphic curve theory. There are many aspects touching areas like Gromov-Witten invariants, quantum cohomology, symplectic homology, Seiberg-Witten invariants, Hamiltonian dynamics and more. Emphasized is this interplay in real dimension three. In this case the methods give a tool to construct global surfaces of section and generalizations thereof for the large class of Reeb vector fields. This class of vector fields, includes, in particular, all geodesic flows on surfaces.

1991 Mathematics Subject Classification: 32, 34, 35, 49, 58, 70

Keywords and Phrases: Hamiltonian dynamics, contact forms, Reeb vector fields, quantum cohomology, Gromov-Witten invariants, Arnold conjecture, Weinstein conjecture, holomorphic curves, symplectic homology, surfaces of section.

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