Introduction to Derived Differential geometry II: Dg-manifolds and derived C-infinity schemes

Abstract: We will start with a careful introduction to the theory of differential graded manifolds (with functions in non-positive cohomological degrees). We will explain how these objects can be used to model derived intersections of manifolds and how this can be made rigorous through a category of fibrant objects structure due to Behrend-Liao-Xu, which leads to a corresponding infinity-category. We will then describe an appropriate version of derived algebraic geometry for C-infinity rings, and show how the associated infinity-category of affine schemes of finite type is equivalent to the infinity category of dg-manifolds. Time permitting, we may also discuss how to incorporate infinite dimensional manifolds and/or other topics.