The first part of a course is mainly devoted to classical integrable systems. Main algebraic structures will be described such as the Lax equations, the classical r-matrices and related Poisson structures on Lie (co)algebras, coadjoint orbits and Lie groups. We will review different kind of models including examples of many-body systems, integrable tops, the classical (spin) chains as well as their continuous limit leading to integrable field theories having soliton type solutions. The phenomenon of integrability is often related to existence of symmetries, generated by the action of groups, and generating the conservation laws. Using the moment map technique, we will see that integrable systems can be obtained by performing the Hamiltonian reduction starting from a free motion. In the end we come to the main idea of R-matrix quantization underlying the quantum inverse scattering method and the Bethe ansatz. We will explain how the quantum groups appear naturally in this way.

Financial support. The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2022-265).

Fall Semester Schedule of 2022/2023:

Time: Tuesday 10:00 – 11:25

First lecture: September 13

Lecturers
Zotov Andrei Vladimirovich
Matushko Mariya Georgievna

Organizations
Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center