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An Introduction to Symplectic Geometry
 
Rolf Berndt University of Hamburg, Hamburg, Germany
An Introduction to Symplectic Geometry
Hardcover ISBN:  978-0-8218-2056-8
Product Code:  GSM/26
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
Softcover ISBN:  978-1-4704-7688-5
Product Code:  GSM/26.S
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-2081-9
Product Code:  GSM/26.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-0-8218-2056-8
eBook: ISBN:  978-1-4704-2081-9
Product Code:  GSM/26.B
List Price: $184.00 $141.50
MAA Member Price: $165.60 $127.35
AMS Member Price: $147.20 $113.20
Softcover ISBN:  978-1-4704-7688-5
eBook: ISBN:  978-1-4704-2081-9
Product Code:  GSM/26.S.B
List Price: $174.00 $131.50
MAA Member Price: $156.60 $118.35
AMS Member Price: $139.20 $105.20
An Introduction to Symplectic Geometry
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An Introduction to Symplectic Geometry
Rolf Berndt University of Hamburg, Hamburg, Germany
Hardcover ISBN:  978-0-8218-2056-8
Product Code:  GSM/26
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
Softcover ISBN:  978-1-4704-7688-5
Product Code:  GSM/26.S
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-2081-9
Product Code:  GSM/26.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-0-8218-2056-8
eBook ISBN:  978-1-4704-2081-9
Product Code:  GSM/26.B
List Price: $184.00 $141.50
MAA Member Price: $165.60 $127.35
AMS Member Price: $147.20 $113.20
Softcover ISBN:  978-1-4704-7688-5
eBook ISBN:  978-1-4704-2081-9
Product Code:  GSM/26.S.B
List Price: $174.00 $131.50
MAA Member Price: $156.60 $118.35
AMS Member Price: $139.20 $105.20
  • Book Details
     
     
    Graduate Studies in Mathematics
    Volume: 262001; 195 pp
    MSC: Primary 53; 20; 81

    Symplectic geometry is a central topic of current research in mathematics. Indeed, symplectic methods are key ingredients in the study of dynamical systems, differential equations, algebraic geometry, topology, mathematical physics and representations of Lie groups.

    This book is a true introduction to symplectic geometry, assuming only a general background in analysis and familiarity with linear algebra. It starts with the basics of the geometry of symplectic vector spaces. Then, symplectic manifolds are defined and explored. In addition to the essential classic results, such as Darboux's theorem, more recent results and ideas are also included here, such as symplectic capacity and pseudoholomorphic curves. These ideas have revolutionized the subject. The main examples of symplectic manifolds are given, including the cotangent bundle, Kähler manifolds, and coadjoint orbits. Further principal ideas are carefully examined, such as Hamiltonian vector fields, the Poisson bracket, and connections with contact manifolds.

    Berndt describes some of the close connections between symplectic geometry and mathematical physics in the last two chapters of the book. In particular, the moment map is defined and explored, both mathematically and in its relation to physics. He also introduces symplectic reduction, which is an important tool for reducing the number of variables in a physical system and for constructing new symplectic manifolds from old. The final chapter is on quantization, which uses symplectic methods to take classical mechanics to quantum mechanics. This section includes a discussion of the Heisenberg group and the Weil (or metaplectic) representation of the symplectic group.

    Several appendices provide background material on vector bundles, on cohomology, and on Lie groups and Lie algebras and their representations.

    Berndt's presentation of symplectic geometry is a clear and concise introduction to the major methods and applications of the subject, and requires only a minimum of prerequisites. This book would be an excellent text for a graduate course or as a source for anyone who wishes to learn about symplectic geometry.

    Readership

    Graduate students and research mathematicians interested in differential geometry.

  • Table of Contents
     
     
    • Chapters
    • Chapter 0. Some aspects of theoretical mechanics
    • Chapter 1. Symplectic algebra
    • Chapter 2. Symplectic manifolds
    • Chapter 3. Hamiltonian vector fields and the Poisson bracket
    • Chapter 4. The moment map
    • Chapter 5. Quantization
    • Appendix A. Differentiable manifolds and vector bundles
    • Appendix B. Lie groups and Lie algebras
    • Appendix C. A little cohomology theory
    • Appendix D. Representations of groups
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 262001; 195 pp
MSC: Primary 53; 20; 81

Symplectic geometry is a central topic of current research in mathematics. Indeed, symplectic methods are key ingredients in the study of dynamical systems, differential equations, algebraic geometry, topology, mathematical physics and representations of Lie groups.

This book is a true introduction to symplectic geometry, assuming only a general background in analysis and familiarity with linear algebra. It starts with the basics of the geometry of symplectic vector spaces. Then, symplectic manifolds are defined and explored. In addition to the essential classic results, such as Darboux's theorem, more recent results and ideas are also included here, such as symplectic capacity and pseudoholomorphic curves. These ideas have revolutionized the subject. The main examples of symplectic manifolds are given, including the cotangent bundle, Kähler manifolds, and coadjoint orbits. Further principal ideas are carefully examined, such as Hamiltonian vector fields, the Poisson bracket, and connections with contact manifolds.

Berndt describes some of the close connections between symplectic geometry and mathematical physics in the last two chapters of the book. In particular, the moment map is defined and explored, both mathematically and in its relation to physics. He also introduces symplectic reduction, which is an important tool for reducing the number of variables in a physical system and for constructing new symplectic manifolds from old. The final chapter is on quantization, which uses symplectic methods to take classical mechanics to quantum mechanics. This section includes a discussion of the Heisenberg group and the Weil (or metaplectic) representation of the symplectic group.

Several appendices provide background material on vector bundles, on cohomology, and on Lie groups and Lie algebras and their representations.

Berndt's presentation of symplectic geometry is a clear and concise introduction to the major methods and applications of the subject, and requires only a minimum of prerequisites. This book would be an excellent text for a graduate course or as a source for anyone who wishes to learn about symplectic geometry.

Readership

Graduate students and research mathematicians interested in differential geometry.

  • Chapters
  • Chapter 0. Some aspects of theoretical mechanics
  • Chapter 1. Symplectic algebra
  • Chapter 2. Symplectic manifolds
  • Chapter 3. Hamiltonian vector fields and the Poisson bracket
  • Chapter 4. The moment map
  • Chapter 5. Quantization
  • Appendix A. Differentiable manifolds and vector bundles
  • Appendix B. Lie groups and Lie algebras
  • Appendix C. A little cohomology theory
  • Appendix D. Representations of groups
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.