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Open covers and complexity (à la Lusternik--Schnirelmann) SemesterWiSe 2022 / 23 Lecturer Kevin Li, Clara Löh, Matthias Uschold Type of course (Veranstaltungsart) Seminar German title Offene Überdeckungen und Komplexität (nach Lusternik--Schnirelmann) Contents 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. Recommended previous knowledge 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. Time/Date Wed 8:30--10:00 Location M 101 Course homepage https://loeh.app.uni-regensburg.de/teaching/catsem_ws2223/ (Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern) Registration
BSem, MV, MSem ECTS 4.5 ECTS |