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Harmonic Maps, Loop Groups, and Integrable Systems

Harmonic Maps, Loop Groups, and Integrable Systems

Part of London Mathematical Society Student Texts

  • Date Published: April 1997
  • availability: Available
  • format: Paperback
  • isbn: 9780521589321


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About the Authors
  • Harmonic maps are generalisations of the concept of geodesics. They encompass many fundamental examples in differential geometry and have recently become of widespread use in many areas of mathematics and mathematical physics. This is an accessible introduction to some of the fundamental connections between differential geometry, Lie groups, and integrable Hamiltonian systems. The specific goal of the book is to show how the theory of loop groups can be used to study harmonic maps. By concentrating on the main ideas and examples, the author leads up to topics of current research. The book is suitable for students who are beginning to study manifolds and Lie groups, and should be of interest both to mathematicians and to theoretical physicists.

    • Deals with the intersection of three important areas: harmonic maps, loop groups, and integrable systems
    • Guides the reader from elementary topics to current research
    • Accessible: emphasises main ideas and examples
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    Reviews & endorsements

    '… an accessible introduction to some of the fundamental connections beween differental geometry, Lie groups and integrable Hamiltonian systems.' L'Enseignement Mathématique

    'It is very rare to find a book that can take a student from the very basics of a subject to the frontiers of active research. The author is to be congratulated on having produced just such a rarity!' Bulletin of the London Mathematics Society

    'The book will certainly be appreciated by mathematicians as well as theoretical physics interested in the subject.' European Mathematical Society

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    Product details

    • Date Published: April 1997
    • format: Paperback
    • isbn: 9780521589321
    • length: 212 pages
    • dimensions: 229 x 153 x 13 mm
    • weight: 0.29kg
    • availability: Available
  • Table of Contents

    Part I. One-Dimensional Integrable Systems:
    1. Lie groups
    2. Lie algebras
    3. Factorizations and homogeneous spaces
    4. Hamilton's equations and Hamiltonian systems
    5. Lax equations
    6. Adler-Kostant-Symes
    7. Adler-Kostant-Symes (continued)
    8. Concluding remarks on one-dimensional Lax equations
    Part II. Two-Dimensional Integrable Systems:
    9. Zero-curvature equations
    10. Some solutions of zero-curvature equations
    11. Loop groups and loop algebras
    12. Factorizations and homogeneous spaces
    13. The two-dimensional Toda lattice
    14. T-functions and the Bruhat decomposition
    15. Solutions of the two-dimensional Toda lattice
    16. Harmonic maps from C to a Lie group G
    17. Harmonic maps from C to a Lie group (continued)
    18. Harmonic maps from C to a symmetric space
    19. Harmonic maps from C to a symmetric space (continued)
    20. Application: harmonic maps from S2 to CPn
    21. Primitive maps
    22. Weierstrass formulae for harmonic maps
    Part III. One-Dimensional and Two-Dimensional Integrable Systems:
    23. From 2 Lax equations to 1 zero-curvature equation
    24. Harmonic maps of finite type
    25. Application: harmonic maps from T2 to S2
    26. Epilogue

  • Author

    Martin A. Guest, Tokyo Metropolitan University

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