Eight written versions of four-hour minicourses focus on using techniques from optimal transport to define geometric notions on metric measure spaces; the notion of weak gradient and Sobolev spaces of functions in metric measure spaces; and functional and geometric inequalities ranging from the concentration of measure, isoperimetry
, and minimizers to heat kernel estimates on non-compact manifolds, and Sobolev inequalities in the Sierpinsky carpet.
Here they also describe Poincare-type inequalities, entropy and Orlicz spaces, LSq and Hardy-type inequalities on the line, probability measures satisfying LSq inequalities on the real line, exponential integrability and perturbation of measures, LSq inequalities for Gibbs measures and super Gaussian tails, LSq inequalities and Markov subgroups, isoperimetry
, the localization argument, proofs of theorems, uniformly convex bodies, from isoperimetry
to LSq inequalities and isoperimetric functional inequalities.