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. Information for our distributors: Titles in this series are copublished 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.
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.