Perfect complexes are finite rank graded vector spaces with a differential of degree +1, and more generally, twisted complexes of vector bundles over a manifold. There is a generalization of the theory of characteristic classes to this setting - a major complication, however, is that the holonomy of a connection is no longer necessarily invertible, but only quasi-invertible.
In the first talk, we introduce the derived stack associated to a differential graded category: our approach is more explicit than the original approach of Toën, Vaquié and Vezzosi, and is based on joint work with Kai Behrend. We also introduce the de Rham complex of a derived stack.
In the second talk (Tuesday 18 Dec 3:30pm), we will give an explicit formula for the Chern character of a perfect complex. Our formulas are explicit, unlike in the original construction of Toën and Vezzosi, who used the cobordism hypothesis to prove existence.