We shall discuss convex functions and their properties, the Legendre transform and other transforms. 

We will investigate the concept of measure transportation with respect to various costs, the Brenier map, Monge Kantorovich duality. 

We will dig deep into concentration of measure, for classical spaces such as Gaussian and the sphere, Sobolev inequality, Poincare, log-Sobolev, the isoperimetric inequality, Prekopa Leindler and its applications for different measures, as well as the Santalo inequality and its applications for concentration. 

We shall discuss inequalities for transportation which imply concentration, Marton-Talagrand, and will investigate hypercontractivity.