Tau functions and their applications

著者

書誌事項

Tau functions and their applications

John Harnad, Ferenc Balogh

（Cambridge monographs on mathematical physics）

Cambridge University Press, 2021

  • : hardback

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注記

Includes bibliographical references (p. [494]-509) and index

内容説明・目次

内容説明

Tau functions are a central tool in the modern theory of integrable systems. This volume provides a thorough introduction, starting from the basics and extending to recent research results. It covers a wide range of applications, including generating functions for solutions of integrable hierarchies, correlation functions in the spectral theory of random matrices and combinatorial generating functions for enumerative geometrical and topological invariants. A self-contained summary of more advanced topics needed to understand the material is provided, as are solutions and hints for the various exercises and problems that are included throughout the text to enrich the subject matter and engage the reader. Building on knowledge of standard topics in undergraduate mathematics and basic concepts and methods of classical and quantum mechanics, this monograph is ideal for graduate students and researchers who wish to become acquainted with the full range of applications of the theory of tau functions.

目次

  • Preface
  • List of symbols
  • 1. Examples
  • 2. KP flows and the Sato-Segal-Wilson Grassmannian
  • 3. The KP hierarchy and its standard reductions
  • 4. Infinite dimensional Grassmannians
  • 5. Fermionic representation of tau functions and Baker functions
  • 6. Finite dimensional reductions of the infinite Grassmannian and their associated tau functions
  • 7. Other related integrable hierarchies
  • 8. Convolution symmetries
  • 9. Isomonodromic deformations
  • 10. Integrable integral operators and dual isomonodromic deformations
  • 11. Random matrix models I. Partition functions and correlators
  • 12. Random matrix models II. Level spacings
  • 13. Generating functions for characters, intersection indices and Brezin-Hikami matrix models
  • 14. Generating functions for weighted Hurwitz numbers: enumeration of branched coverings
  • Appendix A. Integer partitions
  • Appendix B. Determinantal and Pfaffian identities
  • Appendix C. Grassmann manifolds and flag manifolds
  • Appendix D. Symmetric functions
  • Appendix E. Finite dimensional fermions: Clifford and Grassmann algebras, spinors, isotropic Grassmannians
  • Appendix F. Riemann surfaces, holomorphic differentials and theta functions
  • Appendix G. Orthogonal polynomials
  • Appendix H. Solutions of selected exercises
  • References
  • Alphabetical Index.

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