# Farkhod Eshmatov

### First Position

Postdoc at the University of Michigan### Dissertation

*The Calogero-Moser Correspondence for Noncommutative Deformations of Kleinian Singularities*

### Advisor

### Research Area

### Abstract

Associated to each finite subgroup Γ of SL_{2}(C) ther is a family of noncommutative algebras *O*_{τ}(Γ), which is a deformation of the coordinate ring of the Kleinian singularity C² //Γ. We study finitely generated projective modules over these algebras.

First, we establish a bijective correspondence between the set of isomorphism classes of rank one projective modules over *O*_{τ}(Γ) and a certain class of quiver varieties associated to Γ. We show that this bijection is naturally equivariant under the action of a "large" Dixmier-type automorphism group *G*. This construction leads to a completely explicit description of ideals of the algebras *O*_{τ}(Γ).

Second, we construct functors between the category of *O*_{τ}(Γ)-modules and that of deformed preprojective algebras of a certain quiver associated to Γ. We show that the restriction of these functors to simple modules over preprojective algebras give another description of *O*_{τ}(Γ)-ideals.