Title: Ollivier-Ricci curvature in non-smooth Lorentzian geometry and causal set theory

Speaker: Samuël Borza (University of Vienna)

Time: 2026.07.09 16:00-17:00

Tencent Meeting ID: 784756621

Abstract:

This talk will explore some aspects of non-smooth Lorentzian geometry, the mathematical framework underlying Einstein’s general relativity, which is currently being developed. Just as metric length spaces provide a synthetic generalisation of smooth Riemannian manifolds, the time-separation function plays the role of a “distance” in Lorentzian geometry. The need for a non-smooth Lorentzian framework appeared early on, most famously with Penrose’s singularity theorems. After introducing the basic concepts and some initial results in this synthetic setting, we will turn to causal set theory, a radical approach to quantum gravity in which spacetime is modeled as a discrete causal graph. I will formulate a new notion of curvature, inspired by Ollivier-Ricci curvature on metric graphs, using optimal transport between causal diamonds. We will see that it does recover Ricci curvature on smooth Lorentzian manifolds, and numerical examples will be presented. 

About the speaker:  Samuël Borza is currently a Marie Skłodowska-Curie Postdoctoral Fellow in Mathematics at the University of Vienna. From 2021 to 2024, Samuël was a Postdoctoral Researcher in Mathematics at SISSA in Trieste, Italy. Samuël obtained his PhD in 2021 from Durham University, UK.  Samuël’s research lies at the intersection of differential geometry, geometric analysis, and optimal transport, with some applications to mathematical physics. More recently, he started working on causal set theory, a mathematical approach to quantum gravity in which spacetime is modeled as a discrete causal graph.