Algebraic structures in Integrable systems

Annotation

The theory of integrable systems is a modern field of scientific research at the intersection of dynamical systems theory, discrete mathematics, quantum physics, low-dimensional topology and statistical mechanics.

The course is a first introduction to the concept of integrability. This semester is devoted to the concept of Liouville integrability, some algebraic and geometric constructions underlying the phenomenon of integrability, including: the Lax representation, the Bihamiltonian structure, r-matrix Poisson structures and Lie-Poisson groups, the Adler-Kostant-Symes scheme.

The course is accompanied by an intensive analysis of examples and applied problems reflecting the property of integrability. The main examples are the harmonic oscillator, the Kepler problem, the Euler top, the open Toda chain, the full symmetric Toda system, the KdV equation, the nonlinear Schrodinger equation.

At the beginning of the course, we will pay attention to the basic elements of the theory of Lie groups and Lie algebras and differential geometry, which are necessary to understand the main part of the course.

Course plan

1. Introductory lectures on the theory of Lie groups and Lie algebras. 
2. Introductory lectures on tensor calculus on manifolds 
3. Introduction of the concept of integrability, Liouville's theorem 
4. Adler-Kostant-Symes scheme (Toda systems) 
5. Bi-Hamiltonian formalism, Lenard-Magry scheme, argument shift method (generalized tops) 
6. R-matrix structures, Lie-Poisson groups  
7. Integrability of field-theory systems (KdV, NS)

Literature