Tau Functions and their Applications

John Harnad; Ferenc Balogh

doi:10.1017/9781108610902

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.

‘This book is a magnificent handbook on the applications of the τ -functions.’

Dimitar A. Kolev Source: zbMATH

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Contents

Frontmatter
pp i-vi
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Contents
pp vii-xvi
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Preface
pp xvii-xxiv
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Acknowledgements
pp xxv-xxvi
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1 - Examples
pp 1-26
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2 - KP flows and the Sato–Segal–Wilson Grassmannian
pp 27-34
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3 - The KP hierarchy and its standard reductions
pp 35-72
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4 - Infinite dimensional Grassmannians
pp 73-104
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5 - Fermionic representation of τ-functions and Baker functions
pp 105-138
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6 - Finite dimensional reductions of the infinite Grassmannian and their associated τ-functions
pp 139-180
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7 - Other related integrable hierarchies
pp 181-235
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8 - Convolution symmetries
pp 236-245
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9 - Isomonodromic deformations
pp 246-285
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10 - Integrable integral operators and dual isomonodromicdeformations
pp 286-313
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11 - Random matrix models I. Partitions functions and correlators
pp 314-340
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12 - Random matrix models II. Level spacings
pp 341-355
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13 - Generating functions for characters, intersection indicesand Brézin–Hikami matrix models
pp 356-368
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14 - Generating functions for weighted Hurwitz numbers:enumeration of branched coverings
pp 369-387
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Appendix A - Integer partitions
pp 388-392
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Appendix B - Determinantal and Pfaffian identities
pp 393-400
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Appendix C - Grassmann manifolds and flag manifolds
pp 401-428
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Appendix D - Symmetric functions
pp 429-457
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Appendix E - Finite dimensional fermions:Clifford and Grassmann algebras,spinors, isotropic Grassmannians
pp 458-473
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Appendix F - Riemann surfaces, holomorphic differentialsand θ functions
pp 474-481
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Appendix G - Orthogonal polynomials
pp 482-486
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Appendix H - Solutions of selected exercises
pp 487-493
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References
pp 494-509
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List of Symbols
pp 510-512
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Index
pp 513-522
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