## Module Handbook

• Dynamischer Default-Fachbereich geändert auf MAT

# Module MAT-40-26-M-6

## Module Identification

Module Number Module Name CP (Effort)
MAT-40-26-M-6 Riemannian Surfaces 3.0 CP (90 h)

## Basedata

CP, Effort 3.0 CP = 90 h 1 Sem. irreg. [6] Master (General) [EN] English Decker, Wolfram, Prof. Dr. (PROF | DEPT: MAT) Decker, Wolfram, Prof. Dr. (PROF | DEPT: MAT) Zintl, Jörg, PD Dr. (EXT | DEPT: MAT) [MAT-AGCA] Algebra, Geometry and Computer Algebra [MAT-88.105-SG] M.Sc. Mathematics [NORM] Active

## Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
2V MAT-40-26-K-6
Riemann Surfaces
P 28 h 62 h - - PL1 3.0 irreg.
• About [MAT-40-26-K-6]: Title: "Riemann Surfaces"; Presence-Time: 28 h; Self-Study: 62 h

## Examination achievement PL1

• Form of examination: oral examination (20-30 Min.)
• Examination Frequency: irregular (by arrangement)
• Examination number: 84154 ("Riemannian Surfaces")

## Evaluation of grades

The grade of the module examination is also the module grade.

## Contents

• Riemann surfaces and holomorphic images,
• topological properties, fundamental groups, coverings,
• sheaves, differential forms, integration,
• cohomology and exact sequences,
• Riemann-Roch theorem and Serre duality.

## Competencies / intended learning achievements

Upon successful completion of this module, the students are familiar with the classical theory of Riemann surfaces which has had an influence on subsequent mathematical theories, such as the ambiguous theory of functions (e.g. radical and logarithm), differential geometry or the geometry of algebraic curves. Vice versa, by a specific example which is close to intuition (the Riemann surfaces), the students have learnt to apply general concepts related to complex analysis and geometry. They are able to name and to prove the essential statements of the lecture as well as to classify and to explain the connections.

## Literature

• S. Donaldson: Riemann Surfaces,
• R. Miranda: Algebraic Curves and Riemann Surfaces,
• O. Forster: Lectures on Riemann Surfaces,
• R.C. Gunning: Lectures on Riemann Surfaces.

None

## References to Module / Module Number [MAT-40-26-M-6]

Course of Study Section Choice/Obligation
[MAT-88.105-SG] M.Sc. Mathematics [Core Modules (non specialised)] Pure Mathematics [WP] Compulsory Elective
[MAT-88.706-SG] M.Sc. Mathematics International [Core Modules (non specialised)] Pure Mathematics [WP] Compulsory Elective
Module-Pool Name
[MAT-GM-MPOOL-5] General Mathematics (Introductory Modules M.Sc.)