Universität Wien

250093 VO G-Structures (2021W)

5.00 ECTS (3.00 SWS), SPL 25 - Mathematik
MIXED
Moodle

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Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

  • Wednesday 06.10. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 07.10. 10:45 - 11:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 13.10. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 14.10. 10:45 - 11:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 20.10. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 21.10. 10:45 - 11:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 27.10. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 28.10. 10:45 - 11:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 03.11. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 04.11. 10:45 - 11:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 10.11. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 11.11. 10:45 - 11:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 17.11. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 18.11. 10:45 - 11:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 24.11. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 25.11. 10:45 - 11:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 01.12. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 02.12. 10:45 - 11:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 09.12. 10:45 - 11:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 15.12. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 16.12. 10:45 - 11:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 12.01. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 13.01. 10:45 - 11:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 19.01. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 20.01. 10:45 - 11:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 26.01. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 27.01. 10:45 - 11:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock

Information

Aims, contents and method of the course

This is a topics course in the area of specialization "geometry and topology" of the master program which is also suitable for doctoral students. It can be used as an elective in other areas of the master program, too.

G-structures provide a general approach to the concept of a "geometric structure" in differential geometry. We start by discussing from the examples of Riemannian metrics and (almost) symplectic structures, which look similar, but have very different properties. To formulate the general concept of a G-structure, we have to develop the language of fiber bundles and in particular principal bundles that is fundamental for several areas of differential geometry and theoretical physics. The concept of principal connections then leads to compatible connections on G-structures which are the fundamental tool for studying their properties. It turns out that a surprising amount of information about these properties can be deduces using finite dimensional linear algebra and representation theory. The discussion of a large number of examples will be a substantial part of the course. Depending on the available time, I will also discuss Cartan geometries and/or filtered analogs of G-structures.

The current plan is to teach the course in presence. To be on the safe side, there will be a moodle page for the course which will be the main point of contact for the course, in particular if distance teaching becomes neccessary.

Assessment and permitted materials

oral exam after the end of the course

Minimum requirements and assessment criteria

Students know the fundamental concepts and results of fiber bundles, connections and the theory of G-structures as discussed in the course and can describe the proofs of the central results presented in the course; the usual standards for topics courses in the master program will be employed

Examination topics

the contents of the course

Reading list

Lecture notes for the course will be distributed via the webpage
https://www.mat.univie.ac.at/~cap/lectnotes.html in parts during the semester. Discussions of G-structrues in the literature are available for example in the classical books "Transformation groups in differential geometry" by S. Kobayashi (Springer, 1972) and "Lectures on differential geometry" by S. Sternberg (2nd edition, Chelsea Publishing, 1983).

Association in the course directory

MGEV

Last modified: We 15.11.2023 00:20