2. Topology of the moduli of Higgs bundles

The topology of the moduli of Higgs bundles is only really “wellknown” using Morse theoretic techniques for low ranks (rank $>4$ is where these techniques fall apart). We know many things (e.g. about Betti/Hodge numbers) through motivic/point counting methods, but it would be nice to have a concrete Morse theoretic explanation as well.
Problem 2.1.
Establish Morse theoretic approach to the study of the topology of Hitchin moduli spaces in high rank. In particular, find a geometric reason for the independence of the Poincaré polynomial on the degree (without appealing to the character variety).
Cite this as: AimPL: Singular geometry and Higgs bundles in string theory, available at http://aimpl.org/singularhiggs.