Integrable systems : twistors, loop groups, and Riemann surfaces : based on lectures given at a conference on integrable systems organized by N.M.J. Woodhouse and held at the Mathematical Institute, University of Oxford, in September 1997 / N.J. Hitchin, Savilian, professor of Geometry, University of Oxford, G.B. Segal, Lowndean, professor of Astronomy and Geometry, University of Cambridge, R.S. Ward, professor of Mathematics, University of Durham.

Print version record.

Includes bibliographical references.

Cover; Contents; List of contributors; 1 Introduction; Bibliography; 2 Riemann surfaces and integrable systems; 1 Riemann surfaces; 2 Line bundles and sheaves; 3 Vector bundles; 4 Direct images of line bundles; 5 Matrix polynomials and Lax pairs; 6 Completely integrable Hamiltonian systems; Bibliography; 3 Integrable systems and inverse scattering; 1 Solitons and the KdV equation; 2 Classical dynamical systems and integrability; 3 Some classical integrable systems; 4 Formal pseudo-differential operators; 5 Scattering theory; 6 The non-linear Schrodinger equation and its scattering.

7 Families of flat connections and harmonic maps8 The KdV equation as an Euler equation; 9 Determinants and holonomy; 10 Local conservation laws; 11 The classical moment problem; 12 Inverse scattering; 13 Loop groups and the restricted Grassmannian; 14 Integrable systems and the restricted Grassmannian; 15 Algebraic curves and the Grassmannian; Bibliography; 4 Integrable Systems and Twisters; 1 General comments on integrable systems; 2 Some elementary geometry; 3 First example: self-dual Yang-Mills; 4 Twistor space and holomorphic vector bundles.

5 Yang-Mills-Higgs solitons and minitwistor space6 Generalizations; Bibliography; Index; B; C; D; E; F; G; H; I; J; K; L; M; N; O; P; R; S; T; V; W; Y; Z.

This textbook is designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors. The book has its origins in a series of lecture courses given by the authors, all of whom are internationally known mathematicians and renowned expositors. It is written in an accessible and informal style, and fills a gap in the existing literature. The introduction by Nigel Hitchin addresses the meaning of integrability: how do werecognize an integrable system? His own contribution then develops connections with algebraic geometry, and inclu.

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