2010-10-29

Yuri Berest, Cornell University

Calogero-Moser Spaces and Applications

The Calogero-Moser spaces are complex symplectic algebraic varieties, which play a role in several areas of mathematics (including algebraic geometry, representation theory and integrable systems). After a brief introduction, I will describe a `universal construction' of these spaces as representation varieties and consider two applications. First, for a smooth algebraic curve X, I will explain how to identify the Calogero-Moser spaces over X with moduli spaces of projective D-modules (noncommutative D-bundles) on X. One implication of this construction is a classification of algebras Morita equivalent to the algebra D of global differential operators on X, in terms of Hochschild homology. In the second part of the talk, I will describe the results of my recent joint work with A.Eshmatov and F.Eshmatov on automorphism groups of algebras of differential operators on singular curves. Our method for computing such groups is a mixture of techniques from geometric group theory (Bass-Serre theory of groups acting on graphs) and geometric invariant theory. The results obtained generalize some classic theorems of J. Dixmier and L. Makar-Limanov.