SoSe 2017

Vortragsreihe - Deformation of metrics towards constant scalar curvature

Dozent: Prof. Dr. Simon Brendle

Beschreibung

The classical uniformization theorem asserts that any Riemannian metric on a closed two-
dimensional surface is conformal to a metric of constant Gaussian curvature. A higher dimensio-
nal analogue of this statement is given by the solution of the Yamabe problem: Any metric on
an n-dimensional manifold is conformal to a metric of constant scalar curvature. This problem
is equivalent to the existence of a positive solution of the nonlinear elliptic equation of the form

In this lectures, I will describe the background of this problem, and its variational formulation
in terms of the Yamabe functional. The gradient flow associated with the Yamabe functional
leads to an curvature flow, and I will discuss why this flow converges to a metric of constant
scalar curvature for any initial metric.

Introduction to Numerical Relativity

Dozent: Dr. Leon Escobar
Beginn: Freitag, 21. April 2017
Zeit: Freitags, 10 Uhr c. t. bis 12 Uhr,
Ort: N14

Beschreibung

The cornerstone of the general relativity theory is the Einstein’s field equation which describes the gravitational field as result of the interaction between the curvature of the space-time and its energy content. In general, this equation consists of ten coupled non-linear partial differential equations which are, with exception of some few cases, hard to be solved simultaneously. The numerical relativity emerges as a response for dealing with such complexity by introducing analytical and numerical techniques that makes possible solving the equations in a large variety of situations. The purpose of this lecture is to provide a basic introduction to this subject. In the first part, the geometric 3 + 1-decomposition of the space-time will be introduced as well as the most used approaches for solving the resulting equations. In the second part, it will be presented the standard approaches for the construction of initial data and the choice of suitable coordinates. The course finishes by discussing, in the 3 + 1-decomposition context, the equations that model the most common energy sources, namely; dust, a perfect gas, scalar and electromagnetic fields.

Voraussetzungen

One lecture in differential geometry and numerical methods as well as a very basic knowledge of some programming language.

Literatur

• M. Alcubierre, Introduction to 3 + 1 Numerical Relativity, Oxford Science Publications, 2008.
• T. W. Baumgarte and S. L. Shapiro, Numerical Relativity: Solving Einstein’s Equations on the Computer, Cambridge University Press, 2010.
• E. Gourgoulhon, 3+1 Formalism in General Relativity: Bases of Numerical Relativity, vol 846, Springer Science and Business Media, 2012.

Modulhandbuch

ECTS Punkte: 3
Prüfungsgebiet: Angewandte Mathematik

Studien- und Prüfungsleistungen

There will be an individual assignment which will be worth half of the final mark. The other half will be determined by an oral exam.

Lineare Partielle Differentialgleichungen

Dozent: Dr. Martin Kell
Beginn: Montag, 19. April 2017
Zeit: Montags und Mittwochs, 14:15 Uhr  bis 15:45 Uhr;
Ort: Hörsaal N15

Beschreibung

In dieser Vorlesung werden Grundlagen zur Theorie partieller Differentialgleichungen erarbeitet. Dazu wird die Lösungs- und Regularitätstheorie linearer partieller Differentialgleichungen entwickelt, der Fokus wird auf elliptischen Differentialgleichungen liegen, je nach Zeit werden auch parabolische Gleichungen betrachtet. Unter anderem werden folgende Themen behandelt: Harmonische Funktionen, Maximumprinzipien, Sobolev-Räume, L²-Theorie, Schauder-Abschätzungen, Harnack-Ungleichungen, Hölder-Regularität.

The exercise sessions will be held in English. Depending on the preference of all participants there is a possibility to have English-only lectures. Participants who prefer English may contact me via e-mail before the first lecture.

Voraussetzungen

Grundvorlesungen in Analysis und Linearer Algebra

Hilfreich aber nicht notwendig: Funktionalanalysis

Literatur (Beispiele)

• D. Gilbarg und N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1998.
• L.C. Evans, Partial Differential Equations, American Mathematical Society, 1998.

Modulhandbuch

ECTS Punkte: 10
Prüfungsgebiet: Reine Mathematik

Studien- und Prüfungsleistungen

Für die Zulassung zur Prüfung werden 50% der Übungspunkte benötigt. Je nach Größe der Veranstaltung gibt es eine Klausur oder mündliche Prüfung.

Übungen (werden auf Englisch abgehandelt)

Assistent: Jason Ledwidge
Zeit: TBA
Ort: TBA