Speaker
Description
One of the strengths of the theory of metric measure spaces comes from the application of the tools of functional analysis to the study of the geometry of the spaces, made possible by the existence of a robust theory of Sobolev spaces. The two main definitions are due to Cheeger -- via relaxation -- and Shanmugalingam -- via the statistical behaviour along curves -- and were developed and unified by Ambrosio, Gigli and Savaré.
In this talk, I will discuss the extension of this framework to the synthetic Lorentzian setting and describe a new definition of Sobolev spaces of causal functions on metric measure spacetimes in the spirit of Cheeger. The proof of the equivalence with the approach à la Shanmugalingam -- adopted in the "octet" paper -- passes through the definition of a Lorentzian analogue of a nonlinear heat flow and the study of the curve induced in the Lorentz-Wasserstein space. The estimates obtained along the way suggest that this trajectory might itself be a Lorentz-Wasserstein gradient flow of a power-type entropy.
This is joint work with Mathias Braun, Nicola Gigli and Robert McCann.