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.