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German original

Holomorphic Functions and Abelian Varieties
(dt. Holomorphe Funktionen und Abelsche Varietäten)

Level, degree of commitment in original study programme Advanced module, compulsory elective module
Forms of teaching and learning,
workload
Lecture (4 SWS), recitation class (2 SWS),
270 hours (90 h attendance, 180 h private study)
Credit points,
formal requirements
9 CP
Course requirement: Successful completion of at least 50 percent of the points from the weekly exercises.
Examination type: Written or oral examination
Language,
Grading
German,
The grading is done with 0 to 15 points according to the examination regulations for study course M.Sc. Mathematics.
Original study programme M.Sc. Mathematik / Vertiefungsbereich Mathematik
Duration,
frequency
One semester,
Im Wechsel mit anderen advanced moduleen
Person in charge of the module's outline Prof. Dr. Thomas Bauer

Contents

Holomorphic functions: Deepening knowledge about the theory of holomorphic functions of one variable (Theorem of Mittag-Leffler, Weierstraßsch product theorem, elliptic functions)

Analytic functions of several variables: Holomorphic functions, Weierstraß preparation theorem, algebraic properties of the ring of power series

Abelian varieties: Complex tori and Abelian varieties, theta functions, divisors, Néron-Severi group, Riemann-Roch theorem, projective embeddings


Qualification Goals

The students shall

  • learn classical results of the advanced function theory of one variable,
  • learn to deal with holomorphic functions in several variables required in complex and algebraic geometry,
  • get to know Abelian varieties as an important class of complex manifolds,
  • understand the study of divisors on these manifolds as an essential tool for understanding geometry and possible projective embeddings,
  • be introduced to current research questions,
  • practice mathematical working methods (development of mathematical intuition and its formal justification, training of the ability to abstract, proof techniques),
  • improve their oral communication skills in the exercises by practicing free speech in front of an audience and during discussion.

Prerequisites

None. The competences taught in the following modules are recommended: either Foundations of Mathematics and Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis, either Complex Analysis and Vector Analysis or Complex Analysis.


Recommended Reading

  • Fischer/Lieb: Funktionentheorie. Vieweg-Verlag.
  • S. Lang: Introduction to Algebraic and Abelian Functions. Springer-Verlag.



Please note:

This page describes a module according to the latest valid module guide in Wintersemester 2021/22. Most rules valid for a module are not covered by the examination regulations and can therefore be updated on a semesterly basis. The following versions are available in the online module guide:

The module guide contains all modules, independent of the current event offer. Please compare the current course catalogue in Marvin.

The information in this online module guide was created automatically. Legally binding is only the information in the examination regulations (Prüfungsordnung). If you notice any discrepancies or errors, we would be grateful for any advice.