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250130 VO Metric Geometry (2025S)
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Note: The time of your registration within the registration period has no effect on the allocation of places (no first come, first served).
Details
Language: English
Lecturers
Classes (iCal) - next class is marked with N
Please note that the course only starts on Thu. March 6.
- Thursday 06.03. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
- Monday 10.03. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
- Thursday 13.03. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
- Monday 17.03. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
- Thursday 20.03. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
- Monday 24.03. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
- Thursday 27.03. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
- Monday 31.03. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
- Thursday 03.04. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
- Monday 07.04. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
- Thursday 10.04. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
- Monday 28.04. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
- Monday 05.05. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
- Thursday 08.05. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
- Monday 12.05. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
- Thursday 15.05. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
- Monday 19.05. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
- Thursday 22.05. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
- Monday 26.05. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
- Monday 02.06. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
- Thursday 05.06. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
- Thursday 12.06. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
- Monday 16.06. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
- Monday 23.06. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
- Thursday 26.06. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
- N Monday 30.06. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
Information
Aims, contents and method of the course
This is a first course on metric geometry. The central idea of this field is to describe geometric properties (such as length, angles and curvature) in terms of metric distances alone. As it turns out, many notions familiar from differential geometry can indeed be captured in such "synthetic" terms alone.The foundational notion is that of a length space, i.e., a metric space where the metric distance between two points is given by the infimum of the length of all connecting curves. Key examples are Riemannian manifolds and polyhedra.Curvature bounds in such spaces are based on comparison with triangles in certain model spaces. E.g., the sphere has positive curvature because triangles are fatter than Euclidean triangles of the same sidelengths. Spaces with a curvature bound in this sense from below/above are called Alexandrov/CAT(k) spaces.Metric geometry, and in particular the theory of length spaces, is a vast and very active field of research that has found applications in diverse mathematical disciplines, such as differential geometry, group theory, dynamical systems and partial differential equations. It has led to identifying the ‘metric core’ of many results in differential geometry, to clarifying the interdependence of various concepts, and to generalizations of central notions in the field to low regularity situations.The prerequisites for following this course are mild and I will soley assume konwledge of (metric) topology. To fully cherish the final chapter familarity with Riemannian geometry or elementary differential geometry is, however, useful.Remark: In recent years our research group---driven by the quest to understand low regularity spacetimes in General Relativity, that is Lorentzian manifolds with non-smooth metrics---has developed a Lorentzian version of metric geometry which provides the basis for (sectional and Ricci) curvature bounds. While not directly being the topic of this course the latter may serve as an entry point to a master thesis in our research group. For general information consult https://ef-geometry.univie.ac.at
Assessment and permitted materials
Oral examination.
Minimum requirements and assessment criteria
Examination topics
Reading list
We will follow the lecture notes of Mike Kunzinger and myself. Here is the final version for the course (all 4 chapters)https://ucloud.univie.ac.at/index.php/s/ZzfRzNi23WzdMqdFor reference I also leave the first part of the updated version here https://ucloud.univie.ac.at/index.php/s/RSAy2mArM68CMkc
as well as the previous edition https://www.mat.univie.ac.at/~stein/teaching/skripten/as.pdfIt is based on the following three standard references, mainly the first one:
Dimitri Burago, Yuri Burago, Sergei Ivanov, "A Course in Metric Geometry" (AMS, 2001)
Martin R. Bridson, Andre Häfliger, "Metric Spaces of Non-Positive Curvature" (Springer, 2011)
Athanase Papadopoulos, "Metric Spaces, Convexity and Nonpositive Curvature" (EMS, 2004)
as well as the previous edition https://www.mat.univie.ac.at/~stein/teaching/skripten/as.pdfIt is based on the following three standard references, mainly the first one:
Dimitri Burago, Yuri Burago, Sergei Ivanov, "A Course in Metric Geometry" (AMS, 2001)
Martin R. Bridson, Andre Häfliger, "Metric Spaces of Non-Positive Curvature" (Springer, 2011)
Athanase Papadopoulos, "Metric Spaces, Convexity and Nonpositive Curvature" (EMS, 2004)
Association in the course directory
MGEV
Last modified: Th 08.05.2025 14:06