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

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

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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

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: August 4, 2022.