University Lecture Series 1994; 209 pp; softcover Volume: 6 Reprint/Revision History: reprinted 1995 ISBN10: 0821803328 ISBN13: 9780821803325 List Price: US$30 Member Price: US$24 Order Code: ULECT/6
 \(J\)holomorphic curves revolutionized the study of symplectic geometry when Gromov first introduced them in 1985. Through quantum cohomology, these curves are now linked to many of the most exciting new ideas in mathematical physics. This book presents the first coherent and full account of the theory of \(J\)holomorphic curves, the details of which are presently scattered in various research papers. The first half of the book is an expository account of the field, explaining the main technical aspects. McDuff and Salamon give complete proofs of Gromov's compactness theorem for spheres and of the existence of the GromovWitten invariants. The second half of the book focuses on the definition of quantum cohomology. The authors establish that this multiplication exists, and give a new proof of the RuanTian result that is associative on appropriate manifolds. They then describe the GiventalKim calculation of the quantum cohomology of flag manifolds, leading to quantum Chern classes and Witten's calculation for Grassmannians, which relates to the Verlinde algebra. The Dubrovin connection, GromovWitten potential on quantum cohomology, and curve counting formulas are also discussed. The book closes with an outline of connections to Floer theory. Readership Advanced graduate students, research mathematicians, and mathematical physicists. Reviews "All in all it is rewarding to read this book, as many delicate points are first explained in easytounderstand terms before the authors dive into the proofs ... this book will certainly remain a standard for background on quantum cohomology for many years to come."  Mathematical Reviews Table of Contents  Introduction
 Local behaviour
 Moduli spaces and transversality
 Compactness
 Compactification of moduli spaces
 Evaluation maps and transversality
 GromovWitten invariants
 Quantum cohomology
 Novikov rings and CalabiYau manifolds
 Floer homology
 Gluing
 Elliptic regularity
 Bibliography
 Indexes
