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Topologie en meetkunde
Cursusdoel
Zie onder vakinhoud.
Vakinhoudelijk
This course is optional for mathematics students. The course is recommended to students interested in pure mathematics, such as differential geometry, algebraic geometry, algebraic topology, algebra, logic. Please find more information about the study advisory paths in the bachelor at the.
Leerdoelen:
We will start with some questions outside and inside of topology that we will solve during the course using methods of homotopy theory and algebraic topology. In particular, we will define homotopies, homotopy equivalences and the fundamental groups. The fundamental group of a topological space will occupy a major part of this course: The first computation (with already some applications) is the fundamental group of the circle. The key computational tool for more complicated spaces is the van Kampen theorem. We will show that the fundamental group is closely related to free group actions and covering space theory.
An important test ground for these ideas are two-dimensional manifolds, i.e. surfaces. We will give a classification of these, both in the oriented and non-oriented case.
The last part of the course will deal with homology. Homology is an algebraic invariant that is much more suitable to high dimensional phenomena than the fundamental group. While it takes a bit to prove the key computational tools (like homotopy invariance and the Mayer-Vietoris sequence), there will be a significant payoff.
At the end of this course, a student is able to:
Onderwijsvormen:
Two times per week two hours of lectures and two times per week tutorials.
Toetsing:
There are two exams for this class, the midterm will count for 30% and the final exam will count for 50% of the final grade.Furthermore, there is homework which counts for 20% of the final grade.
In case of a retake exam, the retake exam counts for 80% and the homework for 20% of the final grade, while the original grades of midterm and final exam are deleted.
Herkansing en inspanningsverplichting:
Students with a final grade lower than 4 are eligible to do the retake exam only if they have handed in solutions to all hand-in homework problems, either before or after the final exam.
Taal van het vak:
The language of instruction is English.
Leerdoelen:
We will start with some questions outside and inside of topology that we will solve during the course using methods of homotopy theory and algebraic topology. In particular, we will define homotopies, homotopy equivalences and the fundamental groups. The fundamental group of a topological space will occupy a major part of this course: The first computation (with already some applications) is the fundamental group of the circle. The key computational tool for more complicated spaces is the van Kampen theorem. We will show that the fundamental group is closely related to free group actions and covering space theory.
An important test ground for these ideas are two-dimensional manifolds, i.e. surfaces. We will give a classification of these, both in the oriented and non-oriented case.
The last part of the course will deal with homology. Homology is an algebraic invariant that is much more suitable to high dimensional phenomena than the fundamental group. While it takes a bit to prove the key computational tools (like homotopy invariance and the Mayer-Vietoris sequence), there will be a significant payoff.
At the end of this course, a student is able to:
- Define homotopies between maps, homotopy equivalences between topological spaces and illustrate these notions by some examples.
- Define the fundamental group and compute it for a large class of spaces.
- State the classification of surfaces.
- Explain the terms "connected sum", "orientable" and "Euler characteristic".
- Define (universal) covers and know the standard theorems and examples of them
- Define the notion of homology, and compute it for some small familiar examples.
- Decide for many pairs of spaces whether they are homotopy equivalent (or even homeomorphic) or not.
- Provide applications (like the Brouwer fixed point theorem).
Onderwijsvormen:
Two times per week two hours of lectures and two times per week tutorials.
Toetsing:
There are two exams for this class, the midterm will count for 30% and the final exam will count for 50% of the final grade.Furthermore, there is homework which counts for 20% of the final grade.
In case of a retake exam, the retake exam counts for 80% and the homework for 20% of the final grade, while the original grades of midterm and final exam are deleted.
Herkansing en inspanningsverplichting:
Students with a final grade lower than 4 are eligible to do the retake exam only if they have handed in solutions to all hand-in homework problems, either before or after the final exam.
Taal van het vak:
The language of instruction is English.
Werkvormen
Hoorcollege
Werkcollege
Werkcollege
Toetsing
Eindresultaat
Verplicht | Weging 100% | ECTS 7,5
Ingangseisen en voorkennis
Ingangseisen
Er is geen informatie over benodigde voorkennis bekend.
Voorkennis
Inleiding Topologie, WISB243.
Voertalen
- Engels
Cursusmomenten
Gerelateerde studies
- Informatica en wiskunde vanaf 2016-2017
- Informatica en wiskunde vanaf 2019-2020
- Informatica en wiskunde vanaf 2022-2023
- Minor Wiskunde
- Natuurkunde en wiskunde vanaf 2017-2018
- Natuurkunde en wiskunde vanaf 2019-2020
- Natuurkunde en wiskunde vanaf 2020-2021
- Theoretical Physics vanaf 2020-2021
- Wiskunde vanaf 2016-2017
- Wiskunde vanaf 2019-2020
- Wiskunde vanaf 2020-2021
- Wiskunde vanaf 2022-2023
Tentamens
| Type | Datum | Tijd | Locatie |
|---|---|---|---|
| Eindresultaat | dinsdag 12 april 2022 | 09:00 - 12:30 | EDUC THEATRON |
| Eindresultaat | dinsdag 12 april 2022 | 09:00 - 12:00 | EDUC THEATRON |
| Eindresultaat (hertoets) | donderdag 7 juli 2022 | 13:30 - 17:00 | BBG 001 |
| Eindresultaat (hertoets) | donderdag 7 juli 2022 | 13:30 - 16:30 | BBG 001 |
Verplicht materiaal
Er is geen informatie over de verplichte literatuur bekend
Aanbevolen materiaal
| Materiaal | Omschrijving |
|---|---|
| ARTIKELEN | We will start by following these notes on the classification of surfaces http://new.math.uiuc.edu/zipproof/zipproof.pdf |
| BOEK | We will follow the first few chapters of http://www.math.cornell.edu/~hatcher/AT/ATchapters.html(Hatcher's book). |
| SOFTWARE | pdf reader, webbrowser |
Coördinator
| dr. A. del Pino Gomez | a.delpinogomez@uu.nl |
Docenten
| dr. A. del Pino Gomez | a.delpinogomez@uu.nl |
Inschrijving
Deze cursus is open voor bijvakkers. Controleer wel of er aanvullende ingangseisen gelden.
Inschrijving
Van maandag 1 november 2021 tot en met zondag 28 november 2021
Na-inschrijving
Van maandag 24 januari 2022 tot en met dinsdag 25 januari 2022
Inschrijving niet geopend
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