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Differentieerbare variëteiten
Cursusdoel
Vakinhoudelijk
This course is optional for mathematics students. The course is recommended to students interested in pure mathematics, such as differential geometry, topology, algebraic geometry, pure analysis. Please find more information about the study advisory paths in the bachelor at the student website.
Leerdoelen:
This course will cover the following concepts:
- definition and examples of manifolds,
- smooth maps, immersions, submersions, diffeomorphisms
- special submanifolds,
- Lie groups, quotients
- tangent and cotangent spaces/bundles,
- vector fields, Lie derivatives and flows,
- differential forms, exterior derivative and de Rham cohomology,
- integration and Stoke’s theorem.
- implicit and inverse function theorems,
- Cartan identities and Cartan calculus,
- Stoke’s theorem
- the definition of a manifold as well as ways to obtain several examples e.g.,
- by finding parametrizations,
- as regular level sets of functions,
- as quotients of other manifolds by group actions.
- the various equivalent description of tangent vectors.
- the relationship between vector fields and curves (flows).
- Differential forms and the various interpretations/properties of the exterior (DeRham) derivatice.
- Orintations, volume forms and integration of differential forms.
- Stokes’ theorem and the very basics of DeRham cohomology.
- Be fluent in using the regular value theorem in order to obtain (sub)manifolds and compute their tangent spaces.
- Be able to compute flows of vector fields.
- Be able to manipulate with differential forms both locally (in coordinate charts) as well as more globally (e.g. using global formulas for Lie derivarives, DeRham differential, etc). In particular, make use of the Cartan calculus.
- Be able to integrate differential forms and derive consequences of Stoke’s theorem.
- Compute DeRham cohomology of some simple spaces.
Two times per week two hours of lectures and two times per week two hours of tutorials.
Toetsing:
- There are homeworks every week graded from 1 to 10, with a final average called HW.
- There is one final exam with a mark E between 1 and 10. To pass the course, E has to be at least $5$.
- There will be other activities such as quizzes (with a mark Q) or bonus exercises (B).
Herkansing en inspanningsverplichting:
For students who failed for the course, an active participation in the course is required to participate in the retake of this course.
Taal van het vak:
The language of instruction is English.
Werkvormen
Werkcollege
Toetsing
Eindresultaat
Verplicht | Weging 100% | ECTS 7,5
Ingangseisen en voorkennis
Ingangseisen
Er is geen informatie over verplichte ingangseisen bekend.
Voorkennis
Lineaire algebra (WISB107 and WISB108), Analyse in meer variabelen (WISB212), Inleiding topologie (WISB 243) and Inleiding groepen en ringen (WISB124). Zie de cursusplanner (cursusplanner.uu.nl) voor de inhoud van deze vakken: selecteer Faculteit Betawetenschappen en vervolgens het programma van de bachelor Wiskunde van het meest recente jaar.
Voertalen
- Engels
Cursusmomenten
Gerelateerde studies
- Informatica en wiskunde vanaf 2019-2020
- Informatica en wiskunde vanaf 2022-2023
- Minor Wiskunde
- Natuurkunde en Wiskunde 2023-2024
- Natuurkunde en wiskunde vanaf 2017-2018
- Natuurkunde en wiskunde vanaf 2019-2020
- Natuurkunde en wiskunde vanaf 2020-2021
- Theoretical Physics vanaf 2020-2021
- Wiskunde en Economie vanaf 2022-2023
- Wiskunde vanaf 2016-2017
- Wiskunde vanaf 2019-2020
- Wiskunde vanaf 2020-2021
- Wiskunde vanaf 2022-2023
Tentamens
Er is geen tentamenrooster beschikbaar voor deze cursus
Verplicht materiaal
Materiaal | Omschrijving |
---|---|
DICTAAT | There are lecture notes for the course, which will be made available on the webpage of the course: https://webspace.science.uu.nl/~crain101/manifolds-2021/ These will be a revised version of the lecture notes from the previous year- see https://webspace.science.uu.nl/~crain101/manifolds-2020/ |
Aanbevolen materiaal
Materiaal | Omschrijving |
---|---|
BOEK | Lee - Introduction to Smooth Manifolds, gratis verkrijgbaar via de website van Springer. (This is for the students that want a book that does more things for them, e.g. more details. But please be aware that, as good/attractive as that sounds, having (too) many things done for you is not necessarily positive ...). |
BOEK | Guillemin, Pollack - Differential Topology - (This is for the students that want to get some more geometric, intuitive insight, with some nicer stories, not all details worked out but fun to read/consult e.g. during a train ride). |
Coördinator
prof. dr. M.N. Crainic | m.crainic@uu.nl |
Docenten
prof. dr. M.N. Crainic | m.crainic@uu.nl |
Inschrijving
Inschrijving
Van maandag 30 mei 2022 tot en met vrijdag 24 juni 2022
Na-inschrijving
Van maandag 22 augustus 2022 tot en met dinsdag 23 augustus 2022
Inschrijving niet geopend
Permanente link naar de cursuspagina
Laat in de Cursus-Catalogus zien