Character varieties of closed surfaces have a natural Poisson structure whose quantization may be constructed in terms of the corresponding quantum group. When the quantum parameter is a root of unity, this quantization carries a central subalgebra isomorphic to the algebra of functions on the classical character variety. In this talk I will describe a procedure which allows one to obtain Azumaya algebras via quantum Hamiltonian reduction. As an application, I will show that quantizations of character varieties at roots of unity are Azumaya over the corresponding classical character varieties.
This is a report on joint work with Iordan Ganev and David Jordan.[-]

Reidemeister torsion defines an element in the determinant line of a finite CW complex. I will explain its family version which allows one to define a volume form on a mapping stack whose source has a simple homotopy type. One family of examples is given by character stacks of finite CW complexes: for surfaces one recovers the symplectic volume form while for 3-manifolds one obtains orientation data necessary to define cohomological DT invariants. Another family of examples is given by the volume form on the derived loop space related to the Todd class. This is a report on work joint with Florian Naef.[-]