Piotr Hajlasz - Convex Functions I

I will present a modern theory of convex and monotone functions. The aim is to present proofs of the following results:
(1) Rectifiability of the singular set of a convex function,
(2) Aleksandrov's theorem about second order differentiabilty of convex functions,
(3) [C^{1,1}] approximation by inf-convolution.
The talk will cover quite a lot of material. The proofs will include all details. The presentation will be elementary and accessible to everyone who knows measure theory.
Most likely my talks will have to be split into two seminars.