Course Overview
The course gives an introduction to a topic of current interest in Lorentzian geometic analysis and mathematical General Relativity: an approach to nonregular spacetimes based on a “metric” point of view.
Learning Outcomes
Becoming acquainted with Lorentzian length spaces, sectional and Ricci curvature bounds for non-regular Lorentzian spaces and the appropriate techniques.
Course Synopsis
Lecture 1a: Review of Lorentzian geometry, spaces of constant curvature, causality theory, singularity theorems.
Lecture 1b: Introduction to Lorentzian length spaces, timelike sectional curvature bounds.
Lecture 2a: Optimal transport, timelike Ricci curvature bounds
Lecture 2b: Sobolev calculus for time functions. Literature: [O’N83, KS18, CM20].
Reading group: Depending on student’s interest one could discuss the papers [GKS19, AGKS21, ABS22].
References
[ABS22] L. Aké Hau, S. Burgos, and D. A. Solis. Causal completions as Lorentzian pre-length spaces. General Relativity and Gravitation, 54(9), 2022. doi:10.1007/s10714-022-02980-x.
[AGKS21] S. B. Alexander, M. Graf, M. Kunzinger, and C. Sämann. Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems. Comm. Anal. Geom., to appear, 2021. doi:10.48550/arXiv.1909.09575. arXiv:1909.09575 [math.MG].
[CM20] F. Cavalletti and A. Mondino. Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications. Cambridge Journal of Mathematics, to appear, arXiv:2004.08934 [math.MG], 2020. doi:10.48550/arXiv.2004.08934.
[GKS19] J. D. E. Grant, M. Kunzinger, and C. Sämann. Inextendibility of spacetimes and Lorentzian length spaces. Ann. Global Anal. Geom., 55(1):133–147, 2019. doi:10.1007/s10455-018-9637-x.
[KS18] M. Kunzinger and C. Sämann. Lorentzian length spaces. Ann. Glob. Anal. Geom., 54(3):399–447, 2018. doi:10.1007/s10455-018-9633-1.
[O’N83] B. O’Neill. Semi-Riemannian geometry with applications to relativity, volume 103 of Pure and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983.
Should you be interested in taking part in the course, please send an email to @email by 10 May 2024.
Further Information
Sessions led by Dr Clemens Sämann will take place on:
Tuesday, 14 May 10am-12pm C5 (Lecture)
Thursday, 16 May 10am-12pm C5 (Lecture)
Tuesday, 28 May 10am-12pm C5 (Reading group)
Participants should have a good knowledge of differential geometry and metric spaces (basics of Lorentzian geometry will be reviewed). Some knowledge of measure theory, functional analysis (in particular Sobolev spaces) and optimal transport is recommended but we will try to be as self-contained as possible.