Seminar: Open covers and complexity (à la Lusternik--Schnirelmann); WS 2022/23

Prof. Dr. Clara Löh / Kevin Li / Matthias Uschold

News

There are still free slots! If you are interested in participating: please send me an email.
The templates for reports/handouts are online (see below); of course, you can also produce the report/handout by other means.
The schedule is now available.

Seminar: Open covers and complexity

A measure of complexity for a topological space X is given by the smallest number of contractible open subsets needed to cover X, called the Lusternik–Schnirelmann category or LS-category cat(X). (Despite the name, there is no connection to category theory).

This integer-valued homotopy invariant is difficult to compute in general and therefore approximated by lower and upper bounds, e.g., using (co)homology. A highlight of the seminar will be Lusternik–Schnirelmann’s classical result that every smooth function on a smooth manifold M has at least cat(M) many critical points. A second goal will be Iwase’s construction of counter-examples to the long-standing Ganea conjecture, using Hopf invariants.

We conclude with applications of the ideas and techniques developed in this seminar to algorithmic problems.

Details: seminar schedule.

Time/Location

Wednesday, 8:30--10:00, M 101

Material

Overview of all talks (and some organisatorial hints).
LaTeX template for handouts: .tex, .pdf
(The template is in German, but that hopefully won't cause any problems).
LaTeX template for reports: .tex, .pdf
(The template is in German, but that hopefully won't cause any problems).
Hints on writing mathematical texts (in German)
More such hints
Small print on slide talks

Prerequisites

Basic algebraic topology, including CW-complexes, homology, and the fundamental group. Prior exposure to cohomology and basic homotopy theory ((co)fibrations, homotopy groups, etc.) is helpful, but not strictly necessary.

This seminar is suitable for Bachelor/Master students (and ambitious Lehramststudenten) with an interest in groups and geometry, and could be the starting point for a project under my supervision (e.g., bachelor/master thesis, Zulassungsarbeit).

Formalities/Credits

See the commented list of courses.

Last Change: October 14, 2022.