Galois theory for Hopf algebras has its roots in works of Chase, Harrison and Rosenberg in 1965 where they wanted to extend the classical Galois theory of fields. Fifteen years later, Kreimer and Takeuchi defined Hopf Galois extensions which are noncommutative analogue of torsors and principal bundles.

In this talk we explain the basics of Hopf Galois extensions and introduce several examples. Furthermore we explain the homology theories related to a Hopf Galois extension and explain some new results.

The automorphism group of a group $G$ comes endowed with a natural filtration: an automorphism belongs to the $k$-th term of this ``Johnson filtration" if it has the same $k$-jet as the identity, with respect to the lower central series of $G$. In this talk, I will discuss the Johnson filtration of the automorphism group of a finitely generated free group, and that of the mapping class group of a surface, with emphasis on the homological finiteness properties of the first few terms in these filtrations.

A key ingredient in this approach is a rather surprising relationship between the classical representation theory of a complex, semisimple Lie algebra $\mathfrak{g}$ and the resonance varieties $R(V,K)\subset V^*$ attached to irreducible $\mathfrak{g}$-modules $V$ and submodules $K\subset V\wedge V$. In the case when $\mathfrak{g}= \mathfrak{sl}_2(\mathbb{C})$, this relationship sheds new light on certain modules studied by Weyman and Eisenbud in the context of Green's conjecture on free resolutions of canonical curves.

This is joint work with Stefan Papadima (arXiv:1011.5292, 1207.2038).