Moduli of local systems and connections on smooth varieties

This is a joint work with T. Pantev. I will explain how to construct derived moduli spaces of local systems and flat bundles on smooth but possibly non-proper varieties, and discuss their representability. I will explain why these moduli spaces carry canonical Poisson structures, in the sense of shifted symplectic geometry, and relate their symplectic leaves with the subspaces of local systems with fixed data at infinity.