Enumerative geometry, wall-crossing and Virasoro constraints

Given a moduli space of either sheaves on a smooth projective variety or a moduli space of representations of a quiver, there are several invariants that we can extract. One of the ways to get numbers out of a moduli space is to integrate (possibly against a virtual fundamental class) certain tautological classes. Such numbers often have interesting structures behind, and I will talk about two: how they change when one changes a stability condition (wall-crossing formulas) and some universal and explicit linear relations that those invariants always seem to satisfy (Virasoro constraints). Both of these phenomena are related to a vertex algebra found by D. Joyce. For simplicity I will mostly focus on the case of representations of a quiver. The talk is based on joint work with A. Bojko and W. Lim.