Symplectic geometry has its origins as a geometric language for classical mechanics. But it has recently exploded into an independent field interconnected with many other areas of mathematics and physics. The goal of the IAS/Park City Mathematics Institute Graduate Summer School on Symplectic Geometry and Topology was to give an intensive introduction to these exciting areas of current research. Included in this proceedings are lecture notes from the following courses: Introduction to Symplectic Topology by D. McDuff; Holomorphic Curves and Dynamics in Dimension Three by H. Hofer; An Introduction to the Seiberg-Witten Equations on Symplectic Manifolds by C. Taubes; Lectures on Floer Homology by D. Salamon; A Tutorial on Quantum Cohomology by A. Givental; Euler Characteristics and Lagrangian Intersections by R. MacPherson; Hamiltonian Group Actions and Symplectic Reduction by L. Jeffrey; and Mechanics: Symmetry and Dynamics by J. Marsden.

Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Table of Contents

• Introduction
• Introduction to symplectic topology
• Introduction
• Basics
• Moser's argument
• The linear theory
• The nonsqueezing theorem and capacities
• Sketch proof of the nonsqueezing theorem
• Bibliography
• Holomorphic curves and dynamics in dimension three
• Problems, basic concepts and overview
• Analytical tools
• The Weinstein conjecture in the overtwisted case
• The Weinstein conjecture in the tight case
• Some outlook
• Bibliography
• An introduction to the Seiberg-Witten equations on symplectic manifolds
• Introduction
• Background from differential geometry
• Spin and the Seiberg-Witten equations
• The Seiberg-Witten invariants
• The symplectic case, part I
• The symplectic case, part II
• Bibliography
• Lectures on Floer homology
• Introduction
• Symplectic fixed points and Morse theory
• Fredholm theory
• Floer homology
• Gromov compactness and stable maps
• Multi-valued perturbations
• Bibliography
• A tutorial on quantum cohomology
• Introduction
• Moduli spaces of stable maps
• \(QH^*(G/B)\) and quantum Toda lattices
• Singularity theory
• Toda lattices and the mirror conjecture
• Bibliography
• Euler characteristics and Lagrangian intersections
• Introduction
• Lecture 1
• Lecture 2
• Lecture 3
• Lecture 4
• Lecture 5
• Bibliography
• Hamiltonian group actions and symplectic reduction
• Introduction to Hamiltonian group actions
• The geometry of the moment map
• Equivariant cohomology and the Cartan model
• The Duistermaat-Heckman theorem and applications to the cohomology of symplectic quotients
• Moduli spaces of vector bundles over Riemann surfaces
• Exercises
• Bibliography
• Park City lectures on mechanics, dynamics, and symmetry
• Introduction
• Reduction for mechanical systems with symmetry
• Stability, underwater vehicle dynamics and phases
• Systems with rolling constraints and locomotion
• Optimal control and stabilization of balance systems
• Variational integrators
• Bibliography