Differential Geometry and Geometric Analysis

Zoll surface
This picture shows the famous Zoll surface: all geodesics are closed and of equal length.
© Mario B. Schulz

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.

 

This page presents the plan at the time of writing for the courses in future semesters. Please note that this plan is subject to change, and courses may be dropped, added, or modified in reaction to currently unforeseen events.

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

Winter semester 2025/2026

Prof. Dr. Christoph Böhm: Differential geometry I (Type I, II, GS)
Prof. Dr. Hans-Joachim Hein: Geometric Analysis I (Type I, II, GS)
Prof. Dr. Joachim Lohkamp: General Relativity (Type I, II, DG, GS)
Dr. Lucas Mann: Bruhat-Tits theory (Type II, GS)
Prof. Dr. Dr. Katrin Tent: Stable Groups (Type I, II, GS)
Dr. Michael Wiemeler: Differential geometry II (Type I, DG, GS)

Summer semester 2025

Prof. Dr. Urs Hartl: Linear Algebraic Groups (Type I, II, GS)
Prof. Dr. Shirly Geffen: Advanced Topics in Functional Analysis (Type I, II, GS)
Prof. Dr. Katrin Tent: Geometric Group Theory (Type I, II, GS)

Winter semester 2024/2025

Prof. Dr. Joachim Lohkamp:  Differential Geometry 2 (Type I, DG; Type I, II, GS)
apl.Prof. Dr. Michael Joachim: Topology 3 (Type I, II, GS)
Prof. Dr. Linus Kramer: Locally compact groups (Type I, II, GS)
Dr. Bianca Santoro:  Kähler Geometry (Type I, DG, GS)
Prof. Dr. Burkhard Wilking: Differential Geometry 1 (Type I, II, GS)

Summer semester 2024

Prof. Dr. Christoph Böhm: Ricci flow (Type II, DG)
Prof. Dr. Ursula Ludwig: Invariants in Global Analysis (Type II, GS)
Prof. Dr. Anna Siffert: Geometric Variational Problems (Type I, DG)

Winter semester 2023/2024

Prof. Dr. Christoph Böhm: Differential Geometry II  (Type I, DG)
Prof. Dr. Ursula Ludwig: Global Analysis II (Index Theory) (Type I, II, DG, GS)
Prof. Dr. Anna Siffert: Harmonic Maps 2 (Type I, II, 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)

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)

 

Seminars

Winter semester 2025/2026

Prof. Dr. Urs Hartl: Reductive Groups (GS)
Prof. Dr. Joachim Lohkamp: Advanced Seminar Geometric Analysis (DG, GS)
Prof. Dr. Joachim Lohkamp: Seminar on Hyperbolic Geometry (DG, GS)
Dr. Bianca Santoro: Complex Geometry (DG, GS)
Prof. Dr. Anna Siffert: Harmonic Maps (GS)

Summer semester 2025

Prof. Dr. Franziska Jahnke, Prof. Dr. Aleksandra Kwiatkowska: An Introduction to Stability Theory (GS)
Prof. Dr. Joachim Lohkamp: Mass and Energy in Relativity (DG, GS)
Dr. Bianca Santoro: Complex Geometry Seminar  (DG)
Prof. Dr. Anna Siffert: New Trends in Spectral Geometry (DG, GS)
Prof. Dr. Katrin Tent: Boone-Higman Conjecture and twisted Brin-Thompson groups (GS)
Dr. Michael Wiemeler: Morse theory (GS)
Prof. Dr. Eva Viehmann: Coxeter Groups  (GS)
Prof. Dr. Benedikt Wirth: Shape Space Theory  (DG)

Winter semester 2024/2025

Prof. Dr. Shirly Geffen, Prof. Dr. David Kerr, Prof. Dr. Aleksandra Kwiatkowska, Prof. Dr. Dr. Katrin Tent: Topological and Measurable Full Groups (GS)
Prof. Dr. Joachim Lohkamp:  Geometric Analysis (GS)
Prof. Dr. Joachim Lohkamp: Minimal Surfaces (GS)
Prof. Dr. Joachim Lohkamp: Yamabe Problem (GS)
Prof. Dr. Anna Siffert: Tonics in spectral geometry (GS)
Prof. Dr. Benedikt Wirth: Master Seminar on Geometry of shape spaces (DG)

Summer semester 2024

Prof. Dr. Christoph Böhm: Seminar on Ricci flow
Prof. Dr. Ursula Ludwig: Global Analysis
Prof. Dr. Hans-Joachim Hein: Spectral theory of hyperbolic surfaces 
Prof. Dr. Anna Siffert: Harmonic Maps 

Winter semester 2023/2024

Prof. Dr. Ursula Ludwig: Selectic topics in Global Analysis