LOGOS Seminar

I will talk about Gromov-hyperbolic Lorentzian length spaces, a generalization of negative upper curvature bounds which is independent of changing the time separation function in bi-Lipschitz and finite ways. We will cover some equivalent formulations, and if time permits, will briefly touch on the timelike ideal boundary.

Abstract: We compare two formulations of lower Ricci curvature bounds on metric measure spaces: the Lott–Sturm–Villani condition, based on entropy convexity along Wasserstein geodesics, and the Bakry–Émery condition, based on \Gamma-calculus for diffusion semigroups. After a brief introduction to Dirichlet forms and heat flow, we discuss the implications between these conditions and the role of infinitesimal Hilbertianity.

Penrose built impulsive pp-waves by cutting Minkowski space along a null hyperplane and regluing the two halves with an x-dependent shift f. In my talk I will treat this construction as a gluing of Lorentzian length spaces. Using the notion of synthetic timelike sectional curvature bounds, we show that the curvature bounds depend on the sign of the Hessian of f.

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.

Null geodesics play an important role in Lorentzian geometry, especially in Penrose’s singularity theorem, which predicts null geodesic incompleteness. While the affine parametrization of a timelike geodesic has a geometrical and a physical interpretation as length and as proper time, this is not the case for a null geodesic. The affine parametrization of a null geodesic is usually defined in terms of the geodesic equation, which is problematic for low-regularity and synthetic approaches. One might be led to interpret null geodesics as limits of timelike ones. In this talk we will discuss how this approach may lead to notions of null geodesic (in)completeness through a series of results and examples which involve gluing along null boundaries.