Forschungsseminar Differentialgeometrie

Christian Bär

Wintersemester 2009/2010

In diesem Seminar werden aktuelle Themen aus der Forschung in der Differentialgeometrie und ihren Nachbargebieten besprochen. Interessenten sind herzlich willkommen.

Wann:

Montag, 15:15-16:45

Wo:

Haus 8, Raum 0.58

Seminarplan (Vorträge):

Datum Vortrag Referent Inhalt
26.10.09Höhere Spinorfelder auf gekrümmten Raumzeiten IRainer MühlhoffBericht über die Diplomarbeit
02.11.09Höhere Spinorfelder auf gekrümmten Raumzeiten IIRainer MühlhoffFortsetzung des Vortrags vom 26.10.2009
09.11.09Vergleich verschiedener Energiedefinitionen für KnotenChristian KarschEs werden verschiedene Konzepte von Knotenenergie dargestellt und miteinander verglichen.
16.11.09Kurzzeitexistenz für Dirac-harmonische FlüsseIn diesem Vortrag soll es um die Kurzzeitexistenz von Dirac-harmonischen Flussgleichungen gehen. Ferner soll für den Fall einer Kurve skizziert werden, wie das Problem der Langzeitexistenz zu behandeln ist.
23.11.09Funktorielle Aspekte von Supergeometrie unter besonderer Berücksichtigung superharmonischer AbbildungenFlorian Hanischtba
30.11.09Conformally covariant differential operators IAndreas JuhlIn a series of 3 lectures we will discuss recent developments around conformally invariant differential operators and Branson's Q-curvature. In the first lecture, we plan to cover the following topics:
  • Conformal geometry and the ambient metric construction
  • Yamabe and Paneitz operator and their role in geometric analysis
  • GJMS-operators and their relation to scattering theory
07.12.09Conformally covariant differential operators IIAndreas JuhlSecond lecture. We will cover the following topics:
  • Branson's Q-curvature: basic properties, its role in geometric analysis
  • Q-curvature and scattering theory
  • holographic formula and its connection with AdS/CFT
14.12.09Conformally covariant differential operators IIIAndreas JuhlThird lecture in this series. We will cover the following topics:
  • Conformally covariant differential operators associated to submanifolds
  • Residue families: definition and basic properties
  • Recursive structure of GJMS-operators and Q-curvatures
04.01.10Introduction to functional renormalizationTim KoslowskiI will give a brief introduction to the ideas of Wilsonian renormalization and then focus on the functional renormalization group. I will apply the methods to scalar field theories and mention generalizations realistic matter content.
11.01.10Magnetische GeodätischeImmanuel AsmusIn diesem Vortrag sollen Abschätzungen für die Energie sowie die kinetische Energie magnetischer Geodätischer vorgestellt werden. Außerdem werden Beispiele für derartige Kurven auf T^2 und T^3 diskutiert.
18.01.10Local Boundary Problems for Dirac Type OperatorsIbrahim LyTo each Dirac type operator one assigns a formally exact elliptic complex of length two. We study the Neumann problem for this complex in a bounded domain with smooth boundary in {tex}$\mathbb{R}^n${/tex}, as it was formulated by Spencer in 1957.
25.01.10Algebraic structures in classical and quantum field theoryKlaus FredenhagenQuantum field theory is governed by the principles of locality and covariance. It will be discussed how these principles can be formulated in the presence of gravitational fields and how they are encoded in various algebraic structures, among them the Peierls bracket of classical field theory, the operator product and the time ordered product of quantum field theory, up to the algebraic properties of the renormalization group. The leading mathematical concept is that of a natural transformation between suitable functors.
01.02.10Monotonicity and entropy formulas for geometric evolution equationsKlaus EckerIn this talk, we discuss monotone quantities which arise naturally in a variety of geometric heat flows such as mean curvature flow and Ricci flow. The monotonicity formulas feature scaling invariant integral quantities involving backward heat kernels which behave monotonically in time and 'stop changing' in the re-scaling limit. They can therefore be used to describe the structure of a solution near a singularity.
Entropy formulas in nonlinear evolution equations were first introduced by Perelman for the case of Ricci flow and played a crucial role in the resolution of the Poincare conjecture. We shall explain their significance and indicate how an analogue for mean curvature flow might look like and how this would solve a number of important open problems.

Erforderliche Vorkenntnisse:

Differentialgeometrie