Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces
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 we recognize an integrable system? His own contribution then develops connections with algebraic geometry, and includes an introduction to Riemann surfaces, sheaves, and line bundles. Graeme Segal takes the Kortewegde Vries and nonlinear Schrödinger equations as central examples, and explores the mathematical structures underlying the inverse scattering transform. He explains the roles of loop groups, the Grassmannian, and algebraic curves. In the final part of the book, Richard Ward explores the connection between integrability and the self-dual Yang-Mills equations, and describes the correspondence between solutions to integrable equations and holomorphic vector bundles over twistor space.
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algebraic basis called closed coefficients compact support complex condition connection conserved consider consists constant construction coordinate corresponding curvature curve defined Definition denote depends described determined differential dim H'(M dimension direct element equation exact sequence example existence fact finite flow follows functions gauge geometry given gives half-plane Hamiltonian holomorphic holomorphic sections independent integrable integrable systems inverse isomorphism KdV equation known line bundle linear look loop manifold matrix means meromorphic multiplication non-linear Note operator pair periodic plane poles polynomial positive problem proof Proposition rank rational Riemann surfaces satisfies scattering sheaf simple smooth solitons solutions space spectrum structure subspace Suppose symplectic theorem theory tion transform trivial unique values vanishes vector bundle vector field vector space write zero