Seminar: Open covers and complexity (à la Lusternik--Schnirelmann); WS 2022/23
Prof. Dr. Clara Löh
There are still free slots!
If you are interested in participating: please send me an email.
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
Details: seminar schedule
Wednesday, 8:30--10:00, M 101
Overview of all talks
(and some organisatorial hints).
Hints on writing mathematical texts (in German)
on slide talks
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,
See the commented list of courses.
Last Change: August 4, 2022.