This course is an introduction to the theory of length spaces, which covers a broad variety of geometric topics related to the notion of distance. It has experienced a very fast development in the last few decades, and now is regarded as one of the fundamental branches of geometry. It has close links with a number of other disciplines, such as Group Theory, Dynamical Systems, and Partial Differential Equations, and is also suitable for students specialising in these subjects.
The course is oriented on students in Mathematics and Physics, and is one of the modules in the Mathematics Master Programme as well as the Master Programme in Theoretical and Mathematical Physics (TMP); it is worth 9 ECTS points.
The core module "Differenzierbare Mannigfaltigkeiten/Differential geometry"; basic modules on analysis and measure theory.
Lectures will be given in English twice a week; 10.00-12.00 Mon, Room B046, and 10.00-12.00 Wed, Room B 045.
The course starts with a detailed exposition of basic notions, techniques, and constructions of length spaces. Later we also plan to discuss Alexandrov spaces of bounded curvature, Gromov-Hausdorff convergence, and Gromov-Hausdorff limits of metric spaces.
1. Burago, D., Burago, Y., Ivanov, S. A course in metric geometry. Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001.
2. Bridson, M. R., Haefliger, A. Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften, 319. Springer-Verlag, Berlin, 1999.
3. Federer, H. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969.
Exercise classes (Übungen):
The exercise classes will hold at 10.00-12.00 Fri, Room B046.
The problem sets for the exercise classes are typed and posted by Asma Hassannezhad on this page on Mondays and are due a week later. Students are encouraged to do as many problems in these sets as possible, since they comprise an important part of the course and similar problems are likely to appear on the exam.
There will be an oral exam at the end of July. If you would like to take the exam please send an email to register and arrange the date and time. The registration deadline is 12 July.
1. Introduction and Background material
1.1. Metric spaces and Lipschitz maps, examples; compact metric spaces and their isometries.
1.2. Hausdorff measure and dimension; Caratheodory's construction.
1.3. Lipschitz maps between Euclidean spaces: Rademacher's theorem, Sard's theorem, Area and Co-Area formulas.
1.4. Subsets in the Euclidean spaces: Brunn-Minkowski inequality and its consequences (isoperimetric and isodiametric inequalities).
2. Length spaces
2.1. Length structures: basic definitions and examples; length structures defined by metrics, 1-dimensional Hausdorff measure, and Lipschitz speed.
2.2. Existence of shortest paths and Hopf-Rinow theorem on length spaces.
2.3. Locality, metric quotients, and gluing; polyhedral spaces, metric graphs.
2.4. Covering spaces and group actions on length spaces; presentations of groups acting on geodesic spaces, metrics on groups.
2.5. Products, cones, and spherical joins.
2.6. Riemannian volumes: definition via Hausdorff measure, Gromov's theorem, and Besikovitch inequality.
3. Convergence of metric spaces
3.1. Basic distances on the spaces of metric spaces: uniform distance, Lipschitz distance, and Hausdorff distance between subsets.
3.2. Hausdorff distance on subsets in the Euclidean space and Steiner symmetrization
3.3. Gromov-Hausdorff convergence I: four definitions of the Gromov-Hausdorff distance, characterization via \epsilon-isometries, other properties and examples.
3.4. Gromov-Hausdorff convergence II: characterization in terms of convergence of \epsilon-nets, compactness theorems, applications to Riemannian geometry.
3.5. Gromov-Hausdorff convergence III: convergence of length spaces, approximation by metric graphs; convergence of pointed metrics spaces, tangent and asymptotic cones.
3.6. Convergence of metric measure spaces: weak convergence of measures (Levy-Prokhorov metric), Wasserstein spaces, measures Gromov-Hausdorff convergence, other metrics and topologies.