Morse-Kirwan theory on singular spaces

Kirwan's extension of Morse theory has led to a well-developed understanding of the topological structure of moduli spaces defiend as symplectic quotients of smooth spaces. There are many important moduli spaces (such as quiver varieties o r moduli spaces of Higgs bundles) that are defined as symplectic quotients of singular spaces; for example affine algebraic varieties. The goal of this project is to extend Kirwan's theory to these spaces.