Oxford

Let $G $ be a compact Lie group, and consider the variety $\text {Hom} (\bb Z^k,G)$

of representations of the rank $k$ abelian free group $\bb Z^k$ into $G$. We prove

that the fundamental group of $\text {Hom} (\bb Z^k,G) $ is naturally isomorphic to direct

product of $k$ copies of the fundamental group of $G$. This is joint work with

Jose Manuel Gomez and Juan Souto.

I will describe recent work on motivic DT invariants for 3-manifolds, which are expected to be a refinement of Chern-Simons theory. The conclusion will be that these should be possible to define and work with, but there will be some interesting problems along the way. There will be a discussion of the problem of upgrading the description of the moduli space of flat connections as a critical locus to the problem of describing the fundamental group algebra of a 3-fold as a "noncommutative critical locus," including a recent topological result on obstructions for this problem. I will also address the question of how a motivic DT invariant may be expected to pick up a finer invariant of 3-manifolds than just the fundamental group.