Abstract:
Lorentzian geometry is the mathematical language of General Relativity
(GR), Einstein's theory of space, time, and gravity. However, the geometries
arising in GR often exhibit non-smooth features that challenge traditional
differential geometric approaches to Lorentzian geometry. In its sister
theory, Riemannian geometry, synthetic approaches using triangle
comparison and optimal transport methods, have led to defining
curvature bounds in a very general setting, allowing for non-smoothness
or even the absence of a differential structure at all. However, in the
Lorentzian setting, the absence of a metric structure has for a long time
blocked a similar development. Only recently, the foundations for an
analogous synthetic Lorentzian geometry, based on the fundamental
notion of Lorentzian length spaces, have emerged. These spaces capture
the essential causal structure of spacetime without requiring smoothness
or a manifold structure at all. In this talk, we explain the basics of this new
geometry, outline initial results including comparison theorems and
convergence results, and explore potential applications in GR and discrete
approaches to quantum gravity.