Universität Regensburg   IMPRESSUM   DATENSCHUTZ
Fakultät für Mathematik Universität Regensburg

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de Rham cohomology
Semester
WiSe 2020 / 21

Lecturer
Marc Hoyois

Type of course (Veranstaltungsart)
Seminar

Contents

The goal of this seminar is to introduce and study de Rham cohomology, which is an invariant of smooth manifolds defined using differential forms. It is related both to analysis (via the fundamental theorem of calculus and Stokes' theorem) and algebraic topology (as de Rham cohomology turns out to be isomorphic to singular cohomology). Moreover, the ideas underlying the definition of de Rham cohomology are quite versatile and can be applied in many other geometric contexts, for example in algebraic geometry.

The seminar will closely follow Chapter I of the book "Differential Forms in Algebraic Topology" by R. Bott and L. W. Tu and cover most of the results therein. The main theorem is Poincaré duality, which is a surprising symmetry in the cohomology of smooth compact manifolds. We will discuss it for both oriented and nonorientable manifolds.

See the seminar webpage below for a detailed program of the seminar.



Literature
Raoul Bott, Loring W. Tu, "Differential Forms in Algebraic Topology", Springer 1982

Recommended previous knowledge
Linear algebra, integration in several variables

Time/Date
Di 8-10

Location
online

Additional question session
Time/Date: Mo 10-13
Location: online

Course homepage
http://www.mathematik.ur.de/hoyois/WS21/derham/index.html
(Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)

Registration
  • Organisational meeting/distribution of topics: July 29 2020 at 16:15 online. Please contact me
    by email to get access to the meeting or if you have any questions.
  • Registration for course work/examination/ECTS: FlexNow
Course work (Studienleistungen)
  • Presentation: Giving a seminar talk of roughly 90 minutes
Examination (Prüfungsleistungen)
  • Detailed written report of the seminar talk
Modules
BSem, MV, MSem

ECTS
Siehe Modulkatalog. MV und Nebenfach: 4,5 LP bei Studienbeginn ab WS 15/16, 6 LP bei Studienbeginn
vor WS 15/16