Hardcover ISBN:  9780821887462 
Product Code:  COLL/52.R 
List Price:  $109.00 
MAA Member Price:  $98.10 
AMS Member Price:  $87.20 
Electronic ISBN:  9780821890967 
Product Code:  COLL/52.R.E 
List Price:  $109.00 
MAA Member Price:  $98.10 
AMS Member Price:  $87.20 

Book DetailsColloquium PublicationsVolume: 52; 2012; 726 ppMSC: Primary 53; 57; 37; 32; Secondary 58; 14;
The theory of \(J\)holomorphic curves has been of great importance since its introduction by Gromov in 1985. In mathematics, its applications include many key results in symplectic topology. It was also one of the main inspirations for the creation of Floer homology. In mathematical physics, it provides a natural context in which to define Gromov–Witten invariants and quantum cohomology, two important ingredients of the mirror symmetry conjecture.
The main goal of this book is to establish the fundamental theorems of the subject in full and rigorous detail. In particular, the book contains complete proofs of Gromov's compactness theorem for spheres, of the gluing theorem for spheres, and of the associativity of quantum multiplication in the semipositive case. The book can also serve as an introduction to current work in symplectic topology: there are two long chapters on applications, one concentrating on classical results in symplectic topology and the other concerned with quantum cohomology. The last chapter sketches some recent developments in Floer theory. The five appendices of the book provide necessary background related to the classical theory of linear elliptic operators, Fredholm theory, Sobolev spaces, as well as a discussion of the moduli space of genus zero stable curves and a proof of the positivity of intersections of \(J\)holomorphic curves in fourdimensional manifolds. The second edition clarifies various arguments, corrects several mistakes in the first edition, includes some additional results in Chapter 10 and Appendices C and D, and updates the references to recent developments.ReadershipGraduate students and research mathematicians interested in symplectic topology and geometry.

Table of Contents

Cover

Title page

Contents

Preface to the second edition

Preface

Introduction

𝐽holomorphic curves

Moduli spaces and transversality

Compactness

Stable maps

Moduli spaces of stable maps

Gromov–Witten invariants

Hamiltonian perturbations

Applications in symplectic topology

Gluing

Quantum cohomology

Floer homology

Fredholm theory

Elliptic regularity

The Riemann–Roch theorem

Stable curves of genus zero

Singularities and intersections

Bibliography

List of symbols

Index

Back Cover


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The theory of \(J\)holomorphic curves has been of great importance since its introduction by Gromov in 1985. In mathematics, its applications include many key results in symplectic topology. It was also one of the main inspirations for the creation of Floer homology. In mathematical physics, it provides a natural context in which to define Gromov–Witten invariants and quantum cohomology, two important ingredients of the mirror symmetry conjecture.
The main goal of this book is to establish the fundamental theorems of the subject in full and rigorous detail. In particular, the book contains complete proofs of Gromov's compactness theorem for spheres, of the gluing theorem for spheres, and of the associativity of quantum multiplication in the semipositive case. The book can also serve as an introduction to current work in symplectic topology: there are two long chapters on applications, one concentrating on classical results in symplectic topology and the other concerned with quantum cohomology. The last chapter sketches some recent developments in Floer theory. The five appendices of the book provide necessary background related to the classical theory of linear elliptic operators, Fredholm theory, Sobolev spaces, as well as a discussion of the moduli space of genus zero stable curves and a proof of the positivity of intersections of \(J\)holomorphic curves in fourdimensional manifolds. The second edition clarifies various arguments, corrects several mistakes in the first edition, includes some additional results in Chapter 10 and Appendices C and D, and updates the references to recent developments.
Graduate students and research mathematicians interested in symplectic topology and geometry.

Cover

Title page

Contents

Preface to the second edition

Preface

Introduction

𝐽holomorphic curves

Moduli spaces and transversality

Compactness

Stable maps

Moduli spaces of stable maps

Gromov–Witten invariants

Hamiltonian perturbations

Applications in symplectic topology

Gluing

Quantum cohomology

Floer homology

Fredholm theory

Elliptic regularity

The Riemann–Roch theorem

Stable curves of genus zero

Singularities and intersections

Bibliography

List of symbols

Index

Back Cover