In this thorough and coherent introduction to tau functions, Harnad and Balogh start with the basics and extend right through to the most recent research results. This monograph is ideal for graduates or researchers in related fields unacquainted with the full range of applications of the theory of tau functions.
In this thorough and coherent introduction to tau functions, Harnad and Balogh start with the basics and extend right through to the most recent research results. This monograph is ideal for graduates or researchers in related fields unacquainted with the full range of applications of the theory of tau functions.
John Harnad is Director of the Mathematical Physics Laboratory at the Centre de Recherches Mathématiques, and Professor of Mathematics at Concordia University in Montréal. Over his career he has made numerous contributions to a variety of fields of mathematical physics, including: gauge field theory, integrable systems, random matrices, isomonodromic deformations and generating functions for graphical enumeration. He was the recipient of the 2006 Canadian Association of Physicists Prize in Theoretical and Mathematical Physics.
Inhaltsangabe
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 Brézin-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.
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 Brézin-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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