git & nahc

Here is the current version of the seminar plan!

Update 2024-04-09: Final update, all notes uploaded! Thank you all for participating!

The nonabelian Hodge correspondence is a deep and beautiful theory that identifies three moduli spaces: the Betti moduli space \(\mathcal{M}_{\text{B}}\) of surface group representations, the de Rham moduli space \(\mathcal{M}_{\text{dR}}\) of flat connections and the Dolbeault moduli space \(\mathcal{M}_{\text{Dol}}\) of Higgs bundles, whose objects on the surface appear to arise from very different contexts. The moduli spaces themselves are constructed as GIT quotients based on the geometric invariant theory pioneered by Mumford. The correspondence between these moduli spaces is the cumulative work of Corlette, Donaldson, Hitchin and Simpson built on that of many mathematicians and the machinery involved to prove these results lies in the intersection of algebraic geometry, complex geometry, geometric analysis and gauge theory.

The goal of this reading seminar is two-fold:

• Understand the workings of geometric invariant theory and how they are applied in the construction of the moduli spaces involved in the nonabelian Hodge correspondence.
• Understand the geometric properties of objects in the moduli spaces and without going into too much nasty detail, sketch the correspondence between the moduli spaces for \(G = \mathrm{GL}(n, \mathbb{C})\).

Schedule