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Softcover ISBN:  9780821803325 
Product Code:  ULECT/6 
List Price:  $69.00 
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eBook ISBN:  9780821834398 
Product Code:  ULECT/6.E 
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Softcover ISBN:  9780821803325 
eBook ISBN:  9780821834398 
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Book DetailsUniversity Lecture SeriesVolume: 6; 1994; 209 ppMSC: Primary 53; Secondary 58; 57;
\(J\)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.
ReadershipAdvanced graduate students, research mathematicians, and mathematical physicists.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. Local behaviour

Chapter 3. Moduli spaces and transversality

Chapter 4. Compactness

Chapter 5. Compactification of moduli spaces

Chapter 6. Evaluation maps and transversality

Chapter 7. GromovWitten invariants

Chapter 8. Quantum cohomology

Chapter 9. Novikov rings and CalabiYau manifolds

Chapter 10. Floer homology

Appendix A. Gluing

Appendix B. Elliptic regularity


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


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\(J\)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.
Advanced graduate students, research mathematicians, and mathematical physicists.

Chapters

Chapter 1. Introduction

Chapter 2. Local behaviour

Chapter 3. Moduli spaces and transversality

Chapter 4. Compactness

Chapter 5. Compactification of moduli spaces

Chapter 6. Evaluation maps and transversality

Chapter 7. GromovWitten invariants

Chapter 8. Quantum cohomology

Chapter 9. Novikov rings and CalabiYau manifolds

Chapter 10. Floer homology

Appendix A. Gluing

Appendix B. Elliptic regularity

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