Introduction
The 1997 IAS/Park City Mathematics Institute on Symplectic Geometry and
Topology took place in Park City, Utah, from June 29 to July 19. Symplectic
geometry has its origins as a geometric language for classical mechanics but has
recently exploded into an independent field of mathematics interconnected with
many other areas of mathematics and physics. The goal of the Graduate Summer
School was to give an intensive introduction to many of these exciting. areas of
current research.
Approximately 85 participants from the graduate program plus many of the 90
research program participants attended portions of the Graduate Summer School.
The School was centered around the following courses, listed as their notes will
appear in this volume:- Introduction to Symplectic Topology by Dusa McDuff;
- Holomorphic Curves and Dynamics in Dimension Three by Helmut Hofer;
- An Introduction to the Seiberg-Witten Equations on Symplectic Manifolds by
Clifford Taubes; - Lectures on Floer Homology by Dietmar Salamon;
- A Tutorial on Quantum Cohomology by Alexander Givental;
- Euler Characteristics and Lagrangian Intersections by Robert MacPherson:
- Hamiltonian Group Actions and Symplectic Reduction by Lisa Jeffrey;
- Mechanics, Dynamics, and Symmetry by Jerrold Marsden.
Each course consisted of five lectures and had accompanying problem sessions. For
all of these courses, lecture notes that closely follow the content of the lectures are
included in this volume. Often these notes have been written jointly by the lecturer
and his or her teaching assistant.
The existence of symplectic topology as a separate field was predicted in the
1960's by V.I. Arnold in a famous series of conjectures. A large body of interrelated
questions emerged with a focus on issues such as the existence and uniqueness
of symplectic structures, invariants of symplectic manifolds, and the existence of
fixed points of Hamiltonian transformations, or, more generally, the existence of a
Morse theory analog for the problem of Lagrangian intersections. The existence of
symplectic topology was confirmed by several authors in the beginning of the 1980's,
but the real breakthrough in this program occurred after M. Gromov introduced
the technique of pseudo holomorphic curves in symplectic manifolds. This led to an
explosion of research in symplectic topology. The courses of McDuff, Hofer, Tau bes,
Salamon, and Givental all involve the theory of pseudo holomorphic curves.
McDuff's course is an introduction to some of the key concepts and results in
symplectic topology. A fascinating aspect of symplectic geometry is the coexistence
of flexible and rigid behavior. McDuff's lectures explore both aspects of the theory
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