Universität Wien
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250130 VO Metric Geometry (2023W)

6.00 ECTS (4.00 SWS), SPL 25 - Mathematik
ON-SITE
Moodle

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Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

  • Tuesday 03.10. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 05.10. 15:00 - 16:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 10.10. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 12.10. 15:00 - 16:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 17.10. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 19.10. 15:00 - 16:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 24.10. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 31.10. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 07.11. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 09.11. 15:00 - 16:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 14.11. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 16.11. 15:00 - 16:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 21.11. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 23.11. 15:00 - 16:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 28.11. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 30.11. 15:00 - 16:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 05.12. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 07.12. 15:00 - 16:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 12.12. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 14.12. 15:00 - 16:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 09.01. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 11.01. 15:00 - 16:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 16.01. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 18.01. 15:00 - 16:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 23.01. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 25.01. 15:00 - 16:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 30.01. 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.

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, available under

https://www.mat.univie.ac.at/~stein/teaching/skripten/as.pdf

It 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: Tu 08.10.2024 12:06