Differential calculus on RCD spaces - part 1

In the lecture I will survey recent advances on calculus on RCD spaces. I shall start recalling how the concept of `L2-normed L1 module' allows to give a reasonable definition of
(co)tangent space and to build a first-order theory on abstract metric measure spaces. The Bochner inequality, which properly interpreted defines RCD spaces, will then allow
to build a second order calculus in such setting: we shall review it and, if time permits, see some applications to the study of geometry of RCD spaces.