Alternatively have a look at the program.

## Higgs bundles on Riemann surfaces, I

The relation between representations of the fundamental group of a Riemann surface into the unitary group, flat connections on Hermitian vector bundles, and stable holomorphic bundles on the surface, goes back to the celebrated theorem of Narasimhan and Seshadri. The case of representations into a non-compact reductive Lie group required the introduction of new holomorphic objects on the Riemann surface, called Higgs bundles.

## Higgs bundles on Riemann surfaces, II

On a Riemann Surface $\Sigma$, the moduli space of polystable $\mathrm{SL}_n(\mathbb{C}$)-Higgs bundles can be identified with the space of reductive representations $\pi _1 (\Sigma) \to \mathrm{SL}_n(\mathbb{C})$. In this talk, we discuss a proof of this so called non-abelian Hodge correspondence. Our goal is to understand how

to construct a Higgs bundle from a given representation and how this construction relates to the theory of harmonic maps.

## The Hitchin System

Using the Dolbeault picture of the moduli space of Higgs bundles, we will construct a function from the moduli space to an affine space.This function will be the Hamiltonian of an integrable system called the Hitchin system. This by definition gives a foliation of the moduli space generically by open subsets of abelian varieties. Time permitting, we will see some applications and how this leads to Higgs bundles offering new, and some times unexpected, geometric insight.

© MPI f. Mathematik, Bonn | Impressum & Datenschutz |