Quiver varieties have been invented
by Nakajima to give a geometric construction of representations
of quantum groups. They have also been used recently to
construct monodial categorifications of cluster algebras. Quiver
varieties associated with a quiver Q are defined by geometric
invariant theory. In my talk, I will explain how they can be
described explicitly as modules of certain mesh categories. We
show that there is a bijection between their strata and the
isomorphism classes of objects in the derived category of the
quiver Q. We will also establish the link between quiver
varieties and the desingularisation map defined in Giovanni
Cerulli-Irelli's talk. This is joint work with Bernhard Keller.