Geometry and Topology of moduli spaces of Higgs bundles and quiver varieties

The moduli space of Higgs bundles is a manifold with a very rich geometry structure, with a wide variety of applications to current topics of interest in geometry and physics, such as Geometric Langlands, Mirror Symmetry, Gravitational Instantons and many more. Quiver varieties have many features in common with Higgs bundles, and these have found spectacular applications in Representation Theory through work of Nakajima. This project proposes to use Morse theory to gain a greater understanding of the geometry and topology of these moduli spaces.