Module 24-B-GT Geometry and Topology

Faculty

Person responsible for module

Regular cycle (beginning)

Every summer semester

Credit points and duration

10 Credit points

For information on the duration of the modul, refer to the courses of study in which the module is used.

Competencies

Non-official translation of the module descriptions. Only the German version is legally binding.

Students familiarise themselves with the basic concepts of set-theoretical topology. They develop an understanding of how these concepts provide a connection to spatial visualisation for many initially abstract and vague problems. Students will be able to use spatial visualisation to conduct independent mathematical proofs. They learn how to deal with various geometric objects of central importance and acquire basic knowledge and skills that are required in in-depth courses on algebraic geometry, algebraic topology, differential geometry, global analysis, functional analysis, algebra, number theory and mathematical physics. They are confident in applying the methods of geometry and topology and can successfully transfer them to new problems in geometry and topology.

In the tutorials, students demonstrate the acquisition of competences in the technologies of geometry and topology, the ability to apply the associated concepts, presentation and communication skills as well as perseverance as a basic mathematical competence. An understanding of the concepts and their interrelationships as well as confidence in applying the methods to new problems is demonstrated in the final exam.

Content of teaching

  • Topological spaces, continuous mappings and associated constructions
  • Coherence, separation and compactness properties
  • Fundamental group, theorem of Seifert and van Kampen
  • Superposition theory, lifting theorem and topological Galois theory
  • Manifolds, vector and fibre bundles, vector fields and differential forms

Optionally, the teaching staff can cover further topics:

  • Introduction of categories, functors and sheaves
  • Gaussian curvature, Gauss-Bonnet theorem, Euler characteristic
  • Fibres

Recommended previous knowledge

Knowledge of analysis and linear algebra

Necessary requirements

—

Explanation regarding the elements of the module

The module cannot be studied together with module 24-B-GT-5.

Module structure: 1 SL, 1 bPr 1

Courses

Geometry and Topology
Type lecture
Regular cycle SoSe
Workload5 60 h (60 + 0)
LP 2 [Pr]
Tutorial on Geometry and Topology
Type exercise
Regular cycle SoSe
Workload5 90 h (30 + 60)
LP 3 [SL]

Study requirements

Allocated examiner Workload LP2
Teaching staff of the course Tutorial on Geometry and Topology (exercise)

Regular completion of the exercises on geometry and topology, each with a recognisable solution approach, as well as participation in the exercise groups on geometry and topology (The students regularly contribute to the scientific discussions in the exercise group. In particular, comments and questions on the proposed solutions presented as well as twice-calculating exercises when requested). The organiser may replace some of the exercises with face-to-face exercises.

 see above see above

Examinations

e-portfolio with final oral examination o. e-portfolio with final written examination o. portfolio with final oral examination o. portfolio with final written examination
Allocated examiner Teaching staff of the course Geometry and Topology (lecture)
Weighting 1
Workload 150h
LP2 5

Proof of a sufficient number of correctly solved exercises, which are worked on as part of the study requirements of the module (usually 50% of the points achievable in the semester for solving the exercises) and passing a final exam in the form of a written exam (usually 90 min) or anl oral exam (usually 30 min). The final exam relates to the content of the lecture and the tutorial and is used for assessment.
A remote electronic written examination is not permitted as a final exam.

The module is used in these degree programmes:

Degree programme Version Profile Recom­mended start 3 Duration Manda­tory option 4
Extension subject / Master of Education Erweiterungsfach [Prüfungsordnung vom 21.03.2023 mit Änderungen vom 30.11.2023 und 26.04.2024] Mathematics (Subject-specific Regulations from 2019): Extension subject Master of Education Continuation Minor Subject (Advanced Secondary and Comprehensive Schools ('Gymnasium' and 'Gesamtschule')) 3. o. 4. one semester Compul­sory optional subject
Mathematical and Theoretical Physics / Master of Science [FsB vom 26.04.2024 mit Änderungen vom 29.05.2024 und 10.12.2024 und der Berichtigung vom 01.04.2025] Admission Track Profile B 1. o. 2. one semester Compul­sory optional subject
Mathematics / Bachelor of Science [FsB vom 30.09.2016 mit Änderung vom 10.12.2024] Major Subject (Academic) 3. o. 4. one semester Compul­sory optional subject
Mathematics / Bachelor of Science [FsB vom 30.09.2016 mit Änderung vom 10.12.2024] Major Subject (Academic) Strukturierte Ergänzung des fw Bachelor KF 3. o. 4. o. 5. one semester Compul­sory optional subject
Mathematics / Bachelor [FsB vom 30.09.2016 mit Änderung vom 10.12.2024] Minor Subject (Academic), 60 CPs 3. o. 4. o. 5. one semester Compul­sory optional subject
Mathematics / Bachelor of Science [FsB vom 30.09.2016 mit Änderung vom 10.12.2024] Major Subject (Advanced Secondary and Comprehensive Schools ('Gymnasium' and 'Gesamtschule')) 3. o. 4. one semester Compul­sory optional subject
Mathematics / Master of Education [FsB vom 01.03.2019 mit Änderung vom 10.12.2024] Continuation Major Subject (Advanced Secondary and Comprehensive Schools ('Gymnasium' and 'Gesamtschule')) 3. o. 4. one semester Compul­sory optional subject
Mathematics / Master of Education [FsB vom 01.03.2019 mit Änderung vom 10.12.2024] Continuation Minor Subject (Advanced Secondary and Comprehensive Schools ('Gymnasium' and 'Gesamtschule')) 3. o. 4. one semester Compul­sory optional subject
Mathematical Economics / Bachelor of Science [FsB vom 30.09.2016 mit Berichtigung vom 10.01.2017 und Änderungen vom 15.05.2017, 01.03.2018, 01.03.2019, 16.09.2019, 02.03.2020 und 10.12.2024] Bachelor with One Core Subject (Academic) Business Administration 3. o. 4. o. 5. one semester Compul­sory optional subject
Mathematical Economics / Bachelor of Science [FsB vom 30.09.2016 mit Berichtigung vom 10.01.2017 und Änderungen vom 15.05.2017, 01.03.2018, 01.03.2019, 16.09.2019, 02.03.2020 und 10.12.2024] Bachelor with One Core Subject (Academic) Political Economy 3. o. 4. o. 5. one semester Compul­sory optional subject

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Legend

1
The module structure displays the required number of study requirements and examinations.
2
LP is the short form for credit points.
3
The figures in this column are the specialist semesters in which it is recommended to start the module. Depending on the individual study schedule, entirely different courses of study are possible and advisable.
4
Explanations on mandatory option: "Obligation" means: This module is mandatory for the course of the studies; "Optional obligation" means: This module belongs to a number of modules available for selection under certain circumstances. This is more precisely regulated by the "Subject-related regulations" (see navigation).
5
Workload (contact time + self-study)
SoSe
Summer semester
WiSe
Winter semester
SL
Study requirement
Pr
Examination
bPr
Number of examinations with grades
uPr
Number of examinations without grades
This academic achievement can be reported and recognised.

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Elements of the module

Courses

Study requirements

Examinations

Programme of lectures (eKVV)

Programme of lectures (eKVV)

Show lists of modules

Mathematics (Subject-specific Regulations from 2019) (Extension subject) / Master of Education Erweiterungsfach: Extension subject Master of Education Continuation Minor Subject (Advanced Secondary and Comprehensive Schools ('Gymnasium' and 'Gesamtschule'))

Mathematical and Theoretical Physics / Master of Science // Admission Track Profile B

Mathematics / Bachelor of Science: Major Subject (Academic) [FsB vom 30.09.2016 mit Änderung vom 10.12.2024]

Mathematics / Bachelor of Science: Major Subject (Academic) // Strukturierte Ergänzung des fw Bachelor KF [FsB vom 30.09.2016 mit Änderung vom 10.12.2024]

Mathematics / Bachelor: Minor Subject (Academic), 60 CPs [FsB vom 30.09.2016 mit Änderung vom 10.12.2024]

Mathematics / Bachelor of Science: Major Subject (Advanced Secondary and Comprehensive Schools ('Gymnasium' and 'Gesamtschule')) [FsB vom 30.09.2016 mit Änderung vom 10.12.2024]

Mathematics / Master of Education: Continuation Major Subject (Advanced Secondary and Comprehensive Schools ('Gymnasium' and 'Gesamtschule'))

Mathematics / Master of Education: Continuation Minor Subject (Advanced Secondary and Comprehensive Schools ('Gymnasium' and 'Gesamtschule'))

Mathematical Economics / Bachelor of Science: Bachelor with One Core Subject (Academic) // Business Administration [FsB vom 30.09.2016 mit Berichtigung vom 10.01.2017 und Änderungen vom 15.05.2017, 01.03.2018, 01.03.2019, 16.09.2019, 02.03.2020 und 10.12.2024]

Mathematical Economics / Bachelor of Science: Bachelor with One Core Subject (Academic) // Political Economy [FsB vom 30.09.2016 mit Berichtigung vom 10.01.2017 und Änderungen vom 15.05.2017, 01.03.2018, 01.03.2019, 16.09.2019, 02.03.2020 und 10.12.2024]