Abstract:

Convexity is one of the most simple regularity properties outside of smoothness requirements. Its apparent ubiquity makes convex functions an invaluable tool in many settings. For example, the relation between convexity and curvature is a source of deep insights. One of principal aspects, which has concrete consequences in metric measure geometry, is the theory of gradient flows: despite the possible nonsmoothness, gradient flows of convex functions are often well posed.

In this talk I will introduce the notion of causally convex functions, a proposal for a Lorentzian counterpart to convex functions, and some natural notions of gradient flow. I will sketch the basic regularity properties of those functions and the well posedness of their gradient flows, comparing them with what happens in positive signature.
This is a work in collaboration with Mathias Braun, Nicola Gigli, Robert McCann and Matteo Zanardini.