Tau Functions and their Applications

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.