A Lorentzian Analog for Hausdorff Dimension and Measure

Explore a groundbreaking mathematical concept in this 36-minute conference talk from the Workshop on "Non-regular Spacetime Geometry" at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into the development of a one-parameter family of canonical volume measures on Lorentzian length spaces, introducing a geometric dimension analogous to the Hausdorff dimension for metric spaces. Discover how this new approach distinguishes between spacelike and null subspaces of Minkowski spacetime, and learn about its applications in defining natural reference measures for synthetic or limiting spacetimes. Examine the concept of collapsed spacetimes and its parallels with metric measure geometry and Riemannian Ricci limit spaces. Explore crucial tools such as the doubling condition for causal diamonds and causal doubling measures. Gain insights into applications for continuous spacetimes and connections to synthetic timelike curvature bounds. The talk is based on joint work with Robert McCann, with the possibility of discussing recent collaborations with Andrea Mondino if time allows.