"Integrable Systems"

1. Billiards, symplectic structure on lines, Crofton's formula
2. Integrability of the geodesic flow on an ellipsoid
3. Hamiltonian systems, first integrals, the Arnold-Liouville theorem, tori
4. Lie algebras, the Lie-Poisson bracket; example: a rigid body
5. Bihamiltonian systems, the Lenard-Magri scheme, the Lax form; example: the Toda lattice
6. Two Hamiltonian structures of the Korteweg-de Vries (KdV) equation
7. The NLS and filament equations
8. Introduction to the inverse scattering; example: KdV
9. Introduction to solitons
10. Near-integrability and non-integrability: glimps of the KAM and Arnold's diffusion
11. Topics and generalizations (time permitted):
    • pentagram maps
    • the Kadomtsev-Petviashvili (KP) hierarchy on PDE's
    • spectral parameter and spectral curve
    • R-matrix

• S. Tabachnikov "Geometry and billiards" (Chapters 3 and 4),
       Amer. Math. Soc., 2005.
• S. Tabachnikov "Introduction to symplectic topology" (Chapter 2),
       Lecture notes (PennState Univ.).
• B. Khesin and R. Wendt "Geometry of infinite-dimensional groups" (Chapters I and II),
       Springer-Verlag, 2008.
• R.S. Palais "The symmetries of solitons"
      Bull. Am. Math. Soc., New Ser. 34, No.4, 339-403 (1997).
• G. Segal, in the book "Integrable Systems."
      Oxford Sci. Publ. 1999
• J. Moser "Various aspects of integrable Hamiltonian systems."
      Progr. Math. 8, Birkhauser, Boston, Mass.,1980, pp. 233-289.

Some familiarity with the main notions of classical mechanics or symplectic geometry will be useful, but not required