Colloquium Mathematics - Dr. N. Martynchuk

Title: Geometry in the theory of integrable Hamiltonian systems

Abstract:

Various dynamical systems appearing in physics and classical mechanics are integrable in the sense of Liouville, that is, they admit a set of independent commuting integrals of motion (conservation laws), where n is the number of degrees of freedom. The list of examples of such systems includes the Kepler problem, the integrable tops, the Calogero-Moser systems, and certain geodesic flows, to name just a few.

The existence of commuting integrals of motion allows one to approach a given integrable system geometrically, by looking at the projection map induced by these integrals. This viewpoint turned out to be particularly useful for studying concrete integrable models and opened up new connections between quite different fields of mathematics and mathematical physics, such as algebra, topology, symplectic geometry, and quantum mechanics.

In this talk, we shall discuss several aspects of integrable systems, emphasising the interplay between integrability and geometry. In particular, we shall discuss the Morse theory of integrable systems and classical and quantum monodromy. The presentation will consist of two parts, the first part being more introductory and the second part focused on the more recent advances in the field. The exposition will be complemented with a number of examples and possible applications beyond the integrable systems framework.