Selected topics in Analysis and PDE

Selected topics in Analysis and PDE (V5B2, WS 2016)
— Mathematical General Relativity


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Information regarding the lecture in Basis

Time: Tuesday, 10 ct – 12 AM, 18.10.2016 – 7.2.2017
Place: Seminar room 0.006, Mathematical Institute (Endenicher Allee 60, 53115 Bonn)

Outline: In 1915 Einstein laid the foundations for the general theory of relativity by formulating a system of nonlinear partial differential equations. These so-called Einstein equations describe graviation by relating the geometry of spacetime to the matter and energy distribution of the universe. While many explicit solutions were constructed shortly afterwards, it took a long time until it was known that the Einstein equations even allow for an initial value formulation. This initial value formulation and a proof of local existence of solutions is due to Choquet-Bruhat in 1952. The existence of a unique so-called maximal globally hyperbolic development was established in 1969. Many research problems that are currently in the focus of attention are closely related to the initial value formulation and include issues of stability, predictability and singularities.

In this lecture course we will go through the fundamental developments of mathematical general relativity and prove that given initial data there is a maximal globally hyperbolic development to the Einstein equations. The necessary background in geometry and partial differential equations will be provided. If time permits, we will discuss some global results and recent developments.

The outline of the course is as follows:

  • Hyperbolic systems: Sobolev spaces, local well-posedness for linear and nonlinear wave equations, local stability
  • Semi-Riemannian geometry: Lorentzian metrics, connections, curvature, geodesics, causal structure, global hyperbolicity, uniqueness
  • General relativity: initial value formulation, constraint equations, gauge choices, local existence, Cauchy stability, existence of a maximal globally hyperbolic development
  • Further topics: global existence, stability, low-regularity initial data, singularities etc.

Target audience: Master and PhD students of Mathematics

Prerequisits: basic knowledge in ordinary and partial differential equations, functional analysis and differential geometry

Examination: oral exams at the end of the lecture period

[1] S. Alinhac, Geometric analysis of hyperbolic differential equations: an introduction, London Mathematical Society Lecture Note Series 374, Cambridge University Press, 2010.
[2] B. O’Neill, Semi-Riemannian geometry. With applications to relativity, Pure and Applied Mathematics 103, Academic Press, Inc., 1983.
[3] A.D. Rendall, Partial differential equations in general relativity, Oxford Graduate Texts in Mathematics, Oxford University Press, 2008.
[4] H. Ringström, The Cauchy problem in general relativity, Lectures in Mathematics and Physics, European Mathematical Society, 2009.

Further references will be provided during the lecture course.