Integrable Systems

This informative textbook, Integrable Systems, provides graduate students with insights into integrable systems through Riemann surfaces and loop groups.

The textbook Integrable Systems is crafted to provide graduate students with a comprehensive understanding of integrable systems through the exploration of Riemann surfaces, loop groups, and twistors. Authored by a team of internationally recognized mathematicians, the book is presented in an informal and accessible manner, making complex concepts more approachable for readers. It addresses a significant gap in the existing literature, offering a unique perspective on the subject matter.

The origins of Integrable Systems can be traced back to a series of lecture courses delivered by the authors, who are esteemed figures in the mathematical community. The introduction by Nigel Hitchin delves into the essence of integrability, posing the critical question: how do we identify an integrable system? Hitchin’s contribution also establishes vital connections with algebraic geometry, introducing essential concepts such as Riemann surfaces, sheaves, and line bundles.

In the latter sections, Graeme Segal focuses on the Korteweg-de Vries and nonlinear Schrödinger equations, utilizing them as key examples to investigate the mathematical frameworks that underlie the inverse scattering transform. He elucidates the significance of loop groups, the Grassmannian, and algebraic curves. The final part of the book, led by Richard Ward, investigates the relationship between integrability and self-dual Yang-Mills equations, highlighting the correspondence between solutions to integrable equations and holomorphic vector bundles over twistor space.