Differential Geometry and Geometric Analysis

Zoll surface
This picture shows the famous Zoll surface: all geodesics are closed and of equal length.
© C. Böhm

Differential manifolds provide higher dimensional generalizations of surfaces. They appear in a very natural manner in many areas of mathematics and physics. On a differential manifold or more generally on a geodesic metric space, one investigates geometric and analytic quantities and concepts, such as geodesics, the curvature of Riemannian metrics or potential theory, and also group actions on such spaces. Of particular interest are connections between such quantities and global, topological properties of the underlying manifold.


The focus topic Differential Geometry and Geometric Analysis is closely related to topology, analysis, stochastics, group theory and to physic, e.g. Einstein's general relativity. A good background in algebra is helpful.

Prerequisites

Prerequisites for the specialization in differential geometry are the lecture courses ''Differential geometry I'' and ''Foundations of analysis, topology and geometry'' (or equivalent courses), with the following contents:

  • Differential geometry I: Riemannian manifolds, geodesics, Levi-Civita-connection, Lemma of Gauß, Theorem of Hopf-Rinow, curvature tensor, first and second variation formula, Lemma of Synge, Theorem of Bonnet-Myers, submanifolds, Gauß equation, Theorema egregium, Jacobi fields, Theorem of Hadamard-Cartan.
  • Foundations of analysis, topology and geometry: notions of point set topology, fundamental group and covering spaces, smooth manifolds.

 

Courses for the specialisation in Differential geometry (DG) and Geometric structures (GS)

Summer semester 2021

Prof. Dr. Hans-Joachim Hein: Regularity theory of elliptic and parabolic PDEs (Type I, II, DG)
Prof. Dr. Gustav Holzegel: Non-linear Wave Equations (Type I, GS)
Prof. Dr. Anna Siffert: Differential geometry on fiber bundles (Type II, DG)
PD Dr. Michael Wiemeler: Transformation groups and equivariant cohomology, in German (Type II, DG)

Winter semester 2021/22

Prof. Dr. Gustav Holzegel: General Relativity and the Analysis of Black Hole Spacetimes (Type I II, GS)
Prof. Dr. Linus Kramer: Geometric Group Theory I, possibly in German (Type I, II, GS)
Prof. Dr. Joachim Lohkamp: Hyperbolic geometry, groups and analysis (Type II, DG)
Prof. Dr. Anna Siffert: Differential geometry II (Type I, II, DG)

Summer semester 2022

Prof. Dr. Linus Kramer: Geometric group theory II (Type II, GS)
Prof. Dr. Joachim Lohkamp: Geometry of Scalar Curvature (Type I, II, DG, GS)
Dr. Bakul Sathaye: Geometric Group Theory II (Typ I, II, GS)
Prof. Dr. Anna Siffert: Spectral Theory (Type II, DG)
apl. Prof. Dr. Wend Werner: Lie Theory and Quantization (Type I, II, DG, GS)
Prof. Dr. Burkhard Wilking: Lie groups (Type II, DG)

Winter semester 2022/2023

Dr. Martin Bays: Geometric Group Theory I (Type I, II, GS)
Prof. Dr. Burkhard Wilking: Differential geometry II (Type I, DG)

Summer semester 2023

Prof. Dr. Anna Siffert: Harmonic maps (Type I, DG, GS)
Prof. Dr. Joachim Lohkamp: Topics in Differential geometry (Type I, DG, GS)
Prof. Dr. Burkhard Wilking: Differential geometry III (Type II, DG)

Seminars

Summer semester 2021

Prof. Dr. Hans-Joachim Hein: Monge-Ampère equations in geometry (DG)
Prof. Dr. Linus Kramer: The Geometry and Topology of Coxeter Groups (GS)
Prof. D. Joachim Lohkamp: Geometry, in German (GS)

Summer semester 2022

Prof. Dr. Joachim Lohkamp: Nash embedding theorems (DG)
Prof. Dr. Anna Siffert: Steklov problems (DG)