MATH 7350: Topics in Algebra: Introduction to Integrable Systems (Spring 2012)

Instructor: Yuri Berest

In many areas, from physics to geometry, one encounters a remarkable phenomenon of integrability: when a system of nonlinear differential equations, which looks at first glance complicated and inaccessible, has a deep hidden symmetry. This symmetry is usually linked to algebraic geometry and manifests itself in the existence of nontrivial conserved quantities and explicit solutions (solitons, instantons, etc.). Recognizing integrability, and learning the techniques for exploiting it, has become a necessity for a broad band of mathematicians.

The course is intended to be a general introduction to the theory of (classical) integrable systems. We will try to give a balanced overview of the subject, though in the end we will focus on algebraic and algebro-geometric aspects and some recent results. Tentative syllabus:

• Integrable systems in classical mechanics (Liouville's Theorem, action-angle variables, Lax pairs. Examples: spinning tops (Euler, Lagrange, Kowalevski); Neumann's problem and geodesics on ellipsoids; Calogero-Moser systems.)
• Examples of infinite-dimensional integrable systems: the KP and the KdV hierarchies. Solitons and rational solutions. The inverse scattering transform.
• Lie-theoretic methods (Lax equations with spectral parameters, coadjoint orbits and the factorization problem, the classical Yang-Baxter equations, the Adler-Kostant-Symes construction, dressing transformations. Bi-hamiltonian structures. The Drinfeld-Sokolov reduction and classification of integrable systems.)
• Algebro-geometric methods (Baker-Akhiezer functions, spectral curves and eigenvector bundles, Lax equations associated with vector bundles on a Riemann surface. Example: Hitchin systems.)
• Infinite-dimensional Grassmannians and noncommutative geometry (Loop groups and the Sato-Segal-Wilson Grassmannian, the boson-fermion correspondence and infinite wedge representations, Hirota bilinear equations. Wilson's adelic Grassmannian and the Calogero-Moser correspondence. The ADHM construction and the twistor transform in noncommutative geometry.)