Geometry and Integrability

Based on courses held at the Feza GÜ rsey Institute, this collection of survey articles introduces advanced graduate students to an exciting area on the border of mathematics and mathematical physics. Including articles by key names such as Calogero, Donagi and Mason, it features the algebro-geometric material from Donagi as well as the twistor space methods in Woodhouse's contribution, forming a bridge between the pure mathematics and the more physical approaches.

Most integrable systems owe their origin to problems in geometry and they are best understood in a geometrical context. This is especially true today when the heroic days of KdV-type integrability are over. Problems that can be solved using the inverse scattering transformation have reached the point of diminishing returns. Two major techniques have emerged for dealing with multi-dimensional integrable systems: twistor theory and the d-bar method, both of which form the subject of this book. It is intended to be an introduction, though by no means an elementary one, to current research on integrable systems in the framework of differential geometry and algebraic geometry. This book arose from a seminar, held at the Feza Gursey Institute, to introduce advanced graduate students to this area of research. The articles are all written by leading researchers and are designed to introduce the reader to contemporary research topics.

1. Introduction Lionel Mason; 2. Differential equations featuring many periodic solutions F. Calogero; 3. Geometry and integrability R. Y. Donagi; 4. The anti self-dual Yang-Mills equations and their reductions Lionel Mason; 5. Curvature and integrability for Bianchi-type IX metrics K. P. Tod; 6. Twistor theory for integrable equations N. M. J. Woodhouse; 7. Nonlinear equations and the d-bar problem P. Santini.

Based on a seminar for graduates held at the Feza Gürsey Institute, there are articles by some key researchers. One attractive feature is the inclusion of the algebro-geometric material as well as the twistor space methods, which form a bridge between the pure mathematics and the more physical approaches.