Andrea Mondino (Oxford)
Optimal transport tools have been extremely powerful to study Ricci curvature, in particular Ricci lower bounds in the non-smooth setting of metric measure spaces (which can be been as a non-smooth extension of Riemannian manifolds). Since the geometric framework of general relativity is the one of Lorentzian manifolds (or space-times), and the Ricci curvature plays a prominent role in Einstein’s theory of gravity, a natural question is whether optimal transport tools can be useful also in this setting. The goal of the talk is to introduce the topic and to report on recent progress. More precisely: After recalling some basics of optimal transport, we will define "timelike Ricci curvature and dimension bounds" for a possibly non-smooth Lorentzian space in terms of displacement convexity of suitable entropy functions and discuss applications.
Some cases of such bounds have remarkable physical interpretations (like the attractive nature of gravity) and can be used to give a characterisation of the Einstein's equations for a non-smooth space, extend classical singularity theorems to settings of low regularity, and prove new results even for smooth Lorentzian manifolds (such as new isoperimetric-type inequalities).
Based partly on joint work with S. Suhr and partly on joint work with F. Cavalletti.
