This module is part of a long-term overall curriculum plan for the Master's programme, which ensures that modules with an amount of at least 20 CP are offered in all five fields each year. The module is offered at irregular intervals as part of this overall curriculum planning.
10 Credit points
For information on the duration of the modul, refer to the courses of study in which the module is used.
Non-official translation of the module descriptions. Only the German version is legally binding.
Students master advanced content and methods of the theory of Riemann Surfaces, in particular they can independently carry out very complex proofs in this area requiring a high level of mathematical expertise.
Students grasp the underlying properties of complex manifolds of dimension 1 and the associated theory of algebraic curves. They can transfer basic concepts, fundamental properties and theorems from Complex Analysis. They can use cohomological methods to solve geometric problems. The students know examples of Riemann surfaces such as the complex plane C, the projective space P^1 over C or elliptic curves over C.
Students will be introduced to current research questions in the area of Riemann Surfaces. They are able to recognise and assess further development opportunities and research goals.
Furthermore, students recognise further-reaching connections to mathematical issues that have already been worked out. They can transfer and apply the knowledge and methods they have learnt so far to deeper mathematical problem areas. Students also expand their mathematical intuition as a result of more intensive study.
In combination with other in-depth modules, they will be able to write their own research papers, e.g. a master's thesis in the field of Riemann Surfaces.
In the tutorials, students develop their ability to discuss mathematical topics and thus further prepare themselves for the requirements of the Master's module, in particular for the scientific discussion within the Master's seminar presentation and the defence of their Master's thesis.
The following advanced content of teaching from the area of Riemannian surfaces is compulsory:
Holomorphic and meromorphic functions, atlas, fundamental groups and covering maps, sheaves and sheaf cohomology, divisors, Riemann-Roch theorem
In addition, the following content of teaching can be covered, for example:
Triangulations, Line bundles, Weierstrass points
This module prepares the content of a master's thesis.
Algebra, basic knowledge of complex analysis and Geometry/Topology
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Module structure: 1 SL, 1 bPr 1
| Allocated examiner | Workload | LP2 |
|---|---|---|
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Teaching staff of the course
Tutorials Riemann Surfaces
(exercise)
Regular completion of the exercises, each with a recognisable solution approach, as well as participation in the exercise groups for the module's lecture. As a rule, participation in the exercise group includes presenting solutions to exercises twice after being asked to do so as well as regular contributions to the scientific discussion in the exercise group, for example in the form of comments and questions on the proposed solutions presented. The organiser may replace some of the exercises with face-to-face exercises. |
see above |
see above
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(electronic) written examination in presence of usually 120 minutes, oral examination in presence or remote of usually 40 minutes, A remote electronic written examination is not permitted.
| Degree programme | Profile | Recommended start 3 | Duration | Mandatory option 4 |
|---|---|---|---|---|
| Mathematical Economics / Master of Science [FsB vom 28.02.2025] | Mathematics | 2. o. 3. | one semester | Compulsory optional subject |
| Mathematical Economics / Master of Science [FsB vom 28.02.2025] | Economics | 2. o. 3. | one semester | Compulsory optional subject |
| Mathematics / Master of Science [FsB vom 28.02.2025] | 2. o. 3. | one semester | Compulsory optional subject |
The system can perform an automatic check for completeness for this module.