Heat kernel for 1-forms on RCD spaces
On a smooth Riemannian manifold, the heat kernel on 1-forms encodes much information about the interplay between analysis, geometry, and topology, e.g. via the Gauß-Bonnet-Chern formula. We outline the construction of a heat kernel on 1-forms on general RCD spaces. A key ingredient is the Hess-Schrader-Uhlenbrock inequality, through which the heat flow on 1-forms can be controlled by the heat flow on functions. Eventually, this inequality will transfer to the level of integral kernels, which entails immediate first estimates for the heat kernel on 1-forms from known ones for its functional counterpart.