Lecture 1: Representations of surface groups and harmonic maps

Classical Hodge theory uses harmonic forms as preferred representatives of cohomology classes. A representation of the fundamental group of a Riemann surface gives rise to a corresponding flat bundle. A Theorem of Donaldson and Corlette shows how to find a harmonic metric in this bundle. A flat bundle corresponds to class in first non-abelian cohomology and the Theorem can be viewed as an analogue of the classical representation of de Rham cohomology classes by harmonic forms.

Lecture 2: Higgs bundles and the Hitchin-Kobayashi correspondence

A Higgs bundle on a Riemann surface is a pair consisting of a holomorphic bundle and a holomorphic one-form, the Higgs field, with values in a certain associated vector bundle. A theorem of Hitchin and Simpson says that a stable Higgs bundle admits a metric satisfying Hitchin's equations. Together with the Theorem of Corlette and Donaldson, the Hitchin-Kobayashi correspondence generalizes the classical Hodge decomposition of the first cohomology of the Riemann surface, providing a correspondence between isomorphism classes of Higgs
bundles and representations of the fundamental group of the surface.

Lecture 3: Higgs bundles and character varieties for surface groups

In the final lecture, we give some applications of Higgs bundle theory to the study of the geometry and topology of character varieties for surface groups, via the identification with moduli of Higgs bundles.