Tau Functions and their Applications

by John Harnad, Ferenc Balogh

Overview

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.

Product Details

ISBN-13:	9781108637213
Publisher:	Cambridge University Press
Publication date:	02/04/2021
Series:	Cambridge Monographs on Mathematical Physics
Sold by:	Barnes & Noble
Format:	eBook
File size:	23 MB
Note:	This product may take a few minutes to download.

About the Author

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.
Ferenc Balogh is Professor of Mathematics at John Abbott College, Montréal. He completed his Ph.D. in Mathematics under the supervision of John Harnad, with whom he has since collaborated on a number of research projects. His doctoral thesis was awarded the 2011 Distinguished Doctoral Dissertation Prize in Engineering and Natural Sciences.

Table of Contents

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.

Editorial Reviews

‘This book is a magnificent handbook on the applications of the τ -functions.’ Dimitar A. Kolev, zbMATH

From the Publisher