We start this short course by  introducing some of the geometric aspects of  the theory of optimal transport that have been developed by Lott, Villani  and Sturm. The theory which establishes a relation between statistical mechanics and geometry leads to  a notion for a measured length space to have Ricci curvature bounded below.  (about 5 sessions) For this part we start by Monge Kantarovich theory and then review the relevant  role of Ricci curvature in differential geometry and continue by   discussing the stablility properties of the notion of lower Ricci curvature bound and if time permits the smoothness of optimal transport in curved spaces.

Next for about two sessions we try to cover some standard mean field equations in random energy models and talk about its generalization in the framework of graphical models and variational Bayesian inference (2 sessions).

At the end we will talk about our results on the theory of mean field variational Bayesian approximation which paves the way for a rigorous treatment of this area of Bayesian inference and its applications.

Familiarity with un undergraduate level of analysis and differential geometry and probability.

[1] Optimal transport and curvature by Alessio Figalli and Cedric Villani.
[2] Information Physics and Computation, Marc Mezard and Andrea Montanari.
[3] Optimal transport, Ricci curvature and mean field variational Bayesian approximation, my preprint.