Fachbereich Mathematik

Sommersemester 2019

Space-Like Hypersurfaces in Lorentzian Manifolds

Dozent: Prof. Dr. Gerhard Huisken
Beginn: Freitag, 26. April 2019
Zeit: Freitag, 10 Uhr c.t. bis 12 Uhr, N14

Zielgruppe: Master in Mathematik und Mathematical Physics
Prüfungsgebiet: Reine Mathematik

Beschreibung / Description

The course describes analytical and geometric aspects of space-like slices in Lorentzian manifolds in
the context of models in General Relativity. Particular topics to be discussed are “maximal surfaces”,
“constant mean curvature surfaces”, “(3+1)-formulation of the Einstein equations in a given slicing”,
“existence and uniqueness results for space-like slices”.

Die Vorlesung behandelt analytische und geometrische Eigenschaften raumartiger Hyperflächen in Lo-
retzschen Mannigfaltigkeiten, jeweils im Zusammenhang mit physikalischen Motivationen durch Model-
le der Algemeinen Relativitätstheorie. Spezifische Themen sind “Maximalflächen”, “Flächen konstanter
mittlerer Krümmung”, “(3+1)-Formulierung der Einstein-Gleichungen in geeigneten Blätterungen der
Raum-Zeit”, “Existenz und Eindeutigkeit bestimmter Hyperflächen”.

Voraussetzungen / Prerequisites

One course in differential geometry and one course in partial differential equations
Je eine Vorlesung über Partielle Differentialgleichungen und Differentialgeometrie

Literatur

Hawking–Ellis, The Large-Scale-Structure of Space-Time, Cambridge Univ. Press.
O’Neill, Semi-Riemannian Geometry, Academic Press
Gilbarg–Trudinger, Elliptic PDEs of Second Order, Springer Grundlehren
Wald, General Relativity, The University of Chicago Press

Prüfung

Written or oral exam depending on course size

Je nach Größe der Veranstaltung gibt es eine Klausur oder mündliche Prüfung.

Übungsgruppe

Stephen Lynch
Einzeltermin: Dienstag, den 30. April 2019, 10-12 Uhr, S 8
Mittwochs, 10-12 Uhr, S 11; Beginn: 08. Mai 2019

 

Mathematical Relativity

Lecturer: Dr. Melanie Graf
Time: Tuesday and Thursday, 4.15pm to 6pm, starting April 16th 2019
Place: N17
Teaching assistant: Sophia Jahns, [cryptmail:pdlowr-mdkqvCpdwk1xql0wxhelqjhq1gh|H0Pdlo vhqghq]

Description

After a short introduction to Special Relativity and its underlying Minkowskian geometry, we will study
general Lorentzian manifolds and the Einstein equations of General Relativity.

One part of the lecture course will focus on static solutions of the Einstein equation, describing space-
times that are in a state of equilibrium. These solutions are geometrically rather simple and therefore
suitable for a first approach to geometric, analytic, and physical questions about spacetimes and iso-
lated systems. In particular, we will prove the Bunting–Masood-ul-Alaam static black hole uniqueness theorem.

In the second part, we will investigate causality, cosmological models, and the Big Bang, specifically
the Penrose–Hawking singularity theorems.

Requirements

Geometry in Physics or Differential Geometry or Mathematische Physik: Klassische Mechanik
Useful, but not required: Linear PDEs

Literature

R. M. Wald, General Relativity, The University of Chicago Press (1984).
H. Fischer und H. Kaul
, Mathematik fur Physiker, Band 3, Springer Spektrum, 3. Auflage (2013)
B. O’Neill, Semi-Riemannian Geometry With Applications to Relativity, Academic Press, Math. 103
S. W. Hawking und G. F. R. Ellis
, The large scale structure of space-time, Cambridge Monographs
on Mathematical Physics (1973).

Exam

To be admitted to the exam, you will need to get 50% of all points on the exercise sheets (including the
project theses, see below). Depending on the number of participants, the exam will be written or oral

Project theses

In the week of June 24 to 28, the participants will be asked to write little project theses about classicals
result in GR instead of solving exercises. The project theses will count like two exercise sheets.

Exercise classes

Time and place to be determined in the first lecture.

 

Limits of Spaces

Lecturer: PD Dr. Martin Kell
Time: Mondays and Wednesdays, 10:15pm to 12pm, starting April 15th 2019
Place: S09

Description

In this course basic concepts of metric geometry like geodesics, doubling property of measures and Hausdorff measures are introduced and their properties are investigated. Furthermore, generalized curvature conditions in the sense of Alexandrov and Busemann are studied and the convergence concepts of Gromov-Hausdorff and the ultra convergence are presented and a proof of Gromov's Precompactness Theorem and other stability theorems will be developed.

Requirements

Analysis I+II and some measure theory.

Literature

Jeff Cheeger, David Ebin: Comparison Theorems in Riemannian Geometry, AMS 1975.
Dimitri Burago, Yuri Burago, Sergei Ivano: A Course in Metric Geometry, AMS 2001.
Mikhail Gromov: Metric Structures for Riemannian and Non-Riemannian Spaces, Springer 2007.