Consider a weighted homogeneous polynomial $f: \mathbb{C}^k \to \mathbb{C}$. For example, $f = x^2 - y^3 : \mathbb{C}^2 \to \mathbb{C}$. In this talk I will discuss the moduli space of flat connections on $\mathbb{C}^k$ which have logarithmic singularities along $D=f^{-1}(0)$. This naturally has the structure of an infinite-dimensional derived moduli stack, but under a certain condition on $D$ (namely that it is a `free divisor') there is an equivalent finite-dimensional model. In the talk I will describe this finite-dimensional model and discuss the proof of the equivalence.

I will try to make the talk expository, and will not assume any prior knowledge of derived geometry. In particular, I will introduce the notion of `bundles of curved differential graded Lie algebras' (due to Behrend and collaborators) which provides a relatively concrete and down to earth model for derived geometry.

If there is time, I will discuss the case of $f = x^p - y^q : \mathbb{C}^2 \to \mathbb{C}$, which already seems to exhibit a lot of interesting behaviours.