This proposal requests support for 10-15 US participants, particularly graduate students and postdocs, in the NATO Advanced Study Institute "Symmetries and Integrability of Difference Equations" to be hosted by the University of Montreal, July 26 - August 6, 1999. The purpose of the Institute is to review the principle developments that have occurred in this area in recent years, particularly in the quantum domain. Integrable systems have long played a crucial role in mathematical physics. Standard examples include central force motion, harmonic oscillators, and systems with a high degree of symmetry, have been greatly enhanced in the past thirty years, and now include interacting many body systems, coupled spin chains, and integrable systems with an infinity of degrees of freedom. The main results leading to current work in the subject will be presented in a coherent form that will be accessible to graduate students and young investigators.
