Calculus, heat flow, optimal transport and curvature-dimension bounds in metric measure spaces

In this survey lecture, modelled in large part on the ICM2018 one, I will illustrate the most recent developments on calculus in metric measure spaces and the key role played by the theory of optimal transport in the derivation of synthetic lower bounds on Ricci curvature and upper bounds on dimension, for metric measure structures. If time permits, I will also illustrate the emerging role of optimal transport and gradient flows in the field of machine learning.