Seminar: Open covers and complexity (à la LusternikSchnirelmann); 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 LScategory cat(X).
(Despite the name, there is no connection to category theory).
This integervalued 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
counterexamples to the longstanding 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, 8:3010:00, 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 CWcomplexes, 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.