Mathematics

Abstract. Lorentzian geometry is the mathematical foundation of Einstein's theory of general relativity, which explains gravity as a manifestation of spacetime curvature (unlike Newton’s theory, which treats gravity as a force). This course has two objectives. First, we give an overview of classical concepts from Lorentzian geometry, encompassing e.g. length, geodesics, causality theory, curvature, Einstein's equations, black hole models, Hawking-Penrose incompleteness theorems, the splitting theorem, etc. Second, we complement this exposition with an introduction to recent and vibrant research on these aspects through the lens of metric geometry and optimal transport, including e.g. metric measure spacetimes, Lorentz-Wasserstein distance, timelike curvature-dimension condition, needle decomposition, Sobolev calculus, comparison theory, Lorentz-Gromov-Hausdorff convergence, etc. Where appropriate, we will highlight new methods to establish (and generalize) classical results.

 

The precise focus will be adapted to the background and interest of the audience.

 

 

The following standard textbooks will be the basis for the classical theory.

 

To get an inspiration about recent research, the following survey papers are recommended as a starter.