Contact and Symplectic Structures and Holomorphic Curves

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DMS-0102298Helmut H. HoferThe main tool for most of the proposed projects is a particularly adapted theory of holomorphic curves, which has been developed by Dr. Hofer and his co-workers in recent years. The fact, that there is such a close, but completely unexpected, relationship between some important class of dynamical systems and a suitable theory of holomorphic curves has deep implications. Immediately it makes it, in principle, feasible to relate questions about large classes of dynamical systems to recent mathematical theories like quantum cohomology, Gromov-Witten invariants, and in low dimensions even to Seiberg-Witten invariants. One of the main goals of this project will be the development of a mathematical machinery for constructing global surfaces of section for three-dimensional Reeb flows and generalizations thereof. In practice this means that the study of certain three-dimensional dynamical system can effectively be reduced to two dimensions. The other goal is the development of a symplectic field theory. Particular cases of such a theory are Gromov-Witten invariants and Floer Homology. Many physical systems like the flow of an incompressibleideal fluid, the movement of a satellite under the gravitational forces of celestial bodies, or the movement of charged particles in a magnetic field, to name a few, are examples of so called dynamical systems. The mathematical theory of dynamical systems provides tools to understand their complex behavior and allowsto make predictions. The particular examples mentioned above are of so-called Hamiltonian nature. For such systems, beginning with the work of Lagrange and Hamilton, geometric tools have been developed leading to the modern theory of contact and symplectic geometry. These geometries play a central role in connecting mathematical areas like dynamical systems, algebraic geometry and smooth topology. This makes it possible to employ powerful tools from different mathematical areas in the study of important classes of dynamical systems and also to use ideas from dynamical systems to study important intrinsic questions in other fields by new methods.