One useful strategy for studying moduli spaces of sheaves on projective varieties is to study “local” analogs where the moduli space has more noncompact directions, but also possesses a rich group of symmetries. I will give an introduction to this strategy and explain how these ideas can be applied to control invariants of moduli spaces of stable Higgs sheaves on surfaces. The local analog we use in this case is an affine quiver variety, the so-called “instanton moduli space” of torsion free framed sheaves on P^2. The applications include results in preparation with M. Kool and T. Laarakker.