Manchester Geometry Seminar 2007/2008

At first I am going to discuss integrability of quantum Calogero-Moser system and of its generalization when interacting particles have two different masses. Then I'll focus on the usual Calogero-Moser system at integer parameter of interaction. In this case the ring of quantum integrals (the ring of quasi-invariants) has a remarkable structure. Namely, the quasi-invariant polynomials are defined by simple conditions of vanishing of certain derivatives; the ring of quasi-invariants is a free module over its subring consisting of symmetric polynomials. When interaction parameter is zero the quasi-invariants coincide with all polynomials and can be thought of as cohomology ring of the tautological bundle BTⁿ→ BUⁿ. In general we do not know a natural generalization of this picture to a non-zero interaction parameter. I'll discuss one-dimensional case where a natural space was constructed jointly with Konstantin Feldman with the help of Arnold-Maxwell topological theorem.