Abstract
In the last thirty years three a priori very different fields of
mathematics, Optimal Transport Theory, Riemannian Geometry and
Probability Theory, have come together in a remarkable way,
leading to a very substantial improvement of our understanding
of what may look like a very specific question, namely the
analysis of spaces whose Ricci curvature admits a lower bound.
The purpose of these lectures is, starting from the classical context,
to present the basics of the three fields that lead to an interesting
generalisation of the concepts, and to highlight some of the most
striking new developments.