Lecture Notes On Differential Calculus on RCD Spaces

These notes are intended to be an invitation to differential calculus on {RCD} spaces. We start by introducing the concept of an " L^2 -normed L^∞ -module" and show how it can be used to develop a first-order (Sobolev) differential calculus on general metric measure spaces. In the second part of the manuscript we see how, on spaces with Ricci curvature bounded from below, a second-order calculus can also be built: objects like the Hessian, covariant and exterior derivatives and Ricci curvature are all well defined and have many of the properties they have in the smooth category.