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Mirror Symmetry and Algebraic Geometry

David A. Cox Amherst College, MA
Sheldon Katz Oklahoma State University, Stillwater
Available Formats:
Softcover ISBN: 978-0-8218-2127-5
Product Code: SURV/68.S
List Price: $64.00 MAA Member Price:$57.60
AMS Member Price: $51.20 Electronic ISBN: 978-1-4704-1295-1 Product Code: SURV/68.S.E List Price:$60.00
MAA Member Price: $54.00 AMS Member Price:$48.00
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List Price: $96.00 MAA Member Price:$86.40
AMS Member Price: $76.80 Click above image for expanded view Mirror Symmetry and Algebraic Geometry David A. Cox Amherst College, MA Sheldon Katz Oklahoma State University, Stillwater Available Formats:  Softcover ISBN: 978-0-8218-2127-5 Product Code: SURV/68.S  List Price:$64.00 MAA Member Price: $57.60 AMS Member Price:$51.20
 Electronic ISBN: 978-1-4704-1295-1 Product Code: SURV/68.S.E
 List Price: $60.00 MAA Member Price:$54.00 AMS Member Price: $48.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$96.00 MAA Member Price: $86.40 AMS Member Price:$76.80
• Book Details

Mathematical Surveys and Monographs
Volume: 681999; 469 pp
MSC: Primary 14; Secondary 81;

Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in four-dimensional projective space. Understanding the mathematics behind these predictions has been a substantial challenge. This book is the first completely comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made to date. Subjects discussed include toric varieties, Hodge theory, Kähler geometry, moduli of stable maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the mirror theorem.

Features:

• Numerous examples worked out in detail
• An appendix on mathematical physics
• An exposition of the algebraic theory of Gromov-Witten invariants and quantum cohomology
• A proof of the mirror theorem for the quintic threefold

Graduate students, research mathematicians interested in the relations between mathematics and physics; algebraic geometers, symplectic geometers, and theoretical physicists.

• Chapters
• 1. Introduction
• 2. The quintic threefold
• 3. Toric geometry
• 4. Mirror symmetry constructions
• 5. Hodge theory and Yukawa couplings
• 6. Moduli spaces
• 7. Gromov-Witten invariants
• 8. Quantum cohomology
• 9. Localization
• 10. Quantum differential equations
• 11. The mirror theorem
• 12. Conclusion

• Reviews

• As the authors observed, the greatest obstacle facing a mathematician who wants to learn about mirror symmetry is knowing where to start. Another problem is the scattering of many mathematical ideas throughout the physics literature, which is difficult for mathematicians to read. The present book seems to be a successful attempt to collect all these ideas. It could also be used as a starting reference for mathematicians interested in learning about mirror symmetry. It is especially very helpful for the reader that the authors have summarized in Appendix B some of the key points of physical theories mentioned in the book.

Bulletin of the AMS
• Mathematicians wanting to get into the field will find it an essential book. They will find a very well written and encyclopaedic account of the mathematics which was needed in, and was developed from, what now might be termed classical mirror symmetry. We can be grateful to the authors for a book which not only summarizes current knowledge, but also points to the future.

Bulletin of the LMS
• Featured Review:

The book is highly recommended for everyone who wants to learn about the fascinating recent interplay between physics and mathematics. It can serve as an introduction both for a mathematician who wants to learn about mirror symmetry, and for a physicist who knows about mirror symmetry and wants to understand the mathematics behind it. It even contains enough details to also be useful for the mathematician who actively wants to do research in the subject.

Mathematical Reviews
• Mathematicians wanting to get into the field … will find a very well written and encyclopaedic account of the mathematics which was needed in, and was developed from, what now might be termed classical mirror symmetry.

Bulletin of the LMS
• The book is highly recommended for everyone who wants to learn about the fascinating recent interplay between physics and mathematics.

Mathematical Reviews
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 681999; 469 pp
MSC: Primary 14; Secondary 81;

Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in four-dimensional projective space. Understanding the mathematics behind these predictions has been a substantial challenge. This book is the first completely comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made to date. Subjects discussed include toric varieties, Hodge theory, Kähler geometry, moduli of stable maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the mirror theorem.

Features:

• Numerous examples worked out in detail
• An appendix on mathematical physics
• An exposition of the algebraic theory of Gromov-Witten invariants and quantum cohomology
• A proof of the mirror theorem for the quintic threefold

Graduate students, research mathematicians interested in the relations between mathematics and physics; algebraic geometers, symplectic geometers, and theoretical physicists.

• Chapters
• 1. Introduction
• 2. The quintic threefold
• 3. Toric geometry
• 4. Mirror symmetry constructions
• 5. Hodge theory and Yukawa couplings
• 6. Moduli spaces
• 7. Gromov-Witten invariants
• 8. Quantum cohomology
• 9. Localization
• 10. Quantum differential equations
• 11. The mirror theorem
• 12. Conclusion
• As the authors observed, the greatest obstacle facing a mathematician who wants to learn about mirror symmetry is knowing where to start. Another problem is the scattering of many mathematical ideas throughout the physics literature, which is difficult for mathematicians to read. The present book seems to be a successful attempt to collect all these ideas. It could also be used as a starting reference for mathematicians interested in learning about mirror symmetry. It is especially very helpful for the reader that the authors have summarized in Appendix B some of the key points of physical theories mentioned in the book.

Bulletin of the AMS
• Mathematicians wanting to get into the field will find it an essential book. They will find a very well written and encyclopaedic account of the mathematics which was needed in, and was developed from, what now might be termed classical mirror symmetry. We can be grateful to the authors for a book which not only summarizes current knowledge, but also points to the future.

Bulletin of the LMS
• Featured Review:

The book is highly recommended for everyone who wants to learn about the fascinating recent interplay between physics and mathematics. It can serve as an introduction both for a mathematician who wants to learn about mirror symmetry, and for a physicist who knows about mirror symmetry and wants to understand the mathematics behind it. It even contains enough details to also be useful for the mathematician who actively wants to do research in the subject.

Mathematical Reviews
• Mathematicians wanting to get into the field … will find a very well written and encyclopaedic account of the mathematics which was needed in, and was developed from, what now might be termed classical mirror symmetry.

Bulletin of the LMS
• The book is highly recommended for everyone who wants to learn about the fascinating recent interplay between physics and mathematics.

Mathematical Reviews
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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