Beijing Institute of Mathematical Sciences and Applications

This course continues the study of group-theoretical and geometric methods in integrable systems and explores their connections with field theory and representation theory. The course introduces integrable hierarchies such as the KdV and KP systems, their Lax representations, and the geometric structures underlying their integrability. We discuss the Drinfeld–Sokolov reduction and its role in the construction of integrable hierarchies and classical (W)-algebras. The course also covers Poisson–Lie groups and their associated integrable systems, with relation to quantum groups. A further part of the course is devoted to integrable systems arising from field theories and gauge-theoretic constructions. We consider the self-dual Yang–Mills equations and Hitchin integrable systems.