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Hamiltonian Group Actions and Equivariant Cohomology

  • Book
  • © 2019

Overview

Authors:
  1. Shubham Dwivedi

    1. Department of Pure Mathematics, University of Waterloo, Waterloo, Canada

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  2. Jonathan Herman

    1. Department of Mathematical and Computational Sciences, University of Toronto at Mississauga, Mississauga, Canada

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  3. Lisa C. Jeffrey

    1. Department of Mathematics, University of Toronto, Toronto, Canada

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  4. Theo van den Hurk

    1. Department of Mathematics, University of Toronto, Toronto, Canada

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  • Self-contained treatment of equivariant cohomology
  • Treatment of moduli spaces of flat connections (a topic of considerable current interest)
  • The only background required is a course on differential manifolds (a standard offering at the advanced undergraduate or introductory graduate level)

Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xi
    Download chapter PDF

  2. Symplectic Vector Spaces

    • Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk
    Pages 1-7

  3. Hamiltonian Group Actions

    • Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk
    Pages 9-15

  4. The Darboux–Weinstein Theorem

    • Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk
    Pages 17-20

  5. Elementary Properties of Moment Maps

    • Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk
    Pages 21-26

  6. The Symplectic Structure on Coadjoint Orbits

    • Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk
    Pages 27-29

  7. Symplectic Reduction

    • Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk
    Pages 31-46

  8. Convexity

    • Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk
    Pages 47-60

  9. Toric Manifolds

    • Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk
    Pages 61-70

  10. Equivariant Cohomology

    • Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk
    Pages 71-88

  11. The Duistermaat–Heckman Theorem

    • Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk
    Pages 89-102

  12. Geometric Quantization

    • Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk
    Pages 103-112

  13. Flat Connections on 2-Manifolds

    • Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk
    Pages 113-122

  14. Back Matter

    Pages 123-132
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Keywords

About this book

This monograph could be used for a graduate course on symplectic geometry as well as for independent study.

The monograph starts with an introduction of symplectic vector spaces, followed by symplectic manifolds and then Hamiltonian group actions and the Darboux theorem. After discussing moment maps and orbits of the coadjoint action, symplectic quotients are studied. The convexity theorem and toric manifolds come next and we give a comprehensive treatment of Equivariant cohomology. The monograph also contains detailed treatment of the Duistermaat-Heckman Theorem, geometric quantization, and flat connections on 2-manifolds. Finally, there is an appendix which provides background material on Lie groups. A course on differential topology is an essential prerequisite for this course. Some of the later material will be more accessible to readers who have had a basic course on algebraic topology. For some of the later chapters, it would be helpful to have some background on representation theory and complex geometry.

Reviews

“The target audience is graduate students; ... this monograph could easily be used by researchers interested in learning the subject at a fast pace. It is a perfect text for a seminar course. ... the book's material is presented in a crisp and abridged manner. ... This makes the presentation short and highly valuable.” (Eduardo A. Gonzalez, Mathematical Reviews, December, 2020)

Authors and Affiliations

  • Department of Pure Mathematics, University of Waterloo, Waterloo, Canada

    Shubham Dwivedi

  • Department of Mathematical and Computational Sciences, University of Toronto at Mississauga, Mississauga, Canada

    Jonathan Herman

  • Department of Mathematics, University of Toronto, Toronto, Canada

    Lisa C. Jeffrey, Theo van den Hurk

Bibliographic Information

  • Book Title: Hamiltonian Group Actions and Equivariant Cohomology

  • Authors: Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk

  • Series Title: SpringerBriefs in Mathematics

  • DOI: https://doi.org/10.1007/978-3-030-27227-2

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Author(s), under exclusive licence to Springer Nature Switzerland AG 2019

  • Softcover ISBN: 978-3-030-27226-5Published: 07 October 2019

  • eBook ISBN: 978-3-030-27227-2Published: 23 September 2019

  • Series ISSN: 2191-8198

  • Series E-ISSN: 2191-8201

  • Edition Number: 1

  • Number of Pages: XI, 132

  • Number of Illustrations: 2 b/w illustrations, 1 illustrations in colour

  • Topics: Topology, Geometry

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