Sommer 2013-2014: Algebraische Kurven und Riemannsche Flächen

This is the website for the course "Algebraische Kurven und Riemannsche Flächen" given at the Institut für Mathematik in Mainz during the Sommersemester of 2013-2014.

Practical information

Teacher: Duco van Straten

Course: Tuesday and Friday, 14-16h.

EXAM: Monday 15th of September, 12h-14h, Room 04-432

Übungsleiter: Ariyan Javanpeykar

Übungsgruppe: Friday 16-18h.

Schedule:

  1. Tuesday 22 April, p. 5-10
  2. Friday 25 April, p. 10-19
  3. Tuesday 29 April, p. 19-24. (1.6) Satz von Mittag-Leffler; (1.7) Funktion zu vorgegebene Divisor; (1.8) Biholomorphe Abbildungen.
  4. Friday 2 May, p. 25-35. (1.8) Fortsetzung, Riemannsche Abbildungssatz; (1.9) Cauchy-Riemann, harmonische Funktionen; (1.10) Dirichlet Prinzip.
  5. Tuesday 6 May, p.36-42.  Riemannsche Flaeche.
  6. Friday 9 May,  p.39-40, p.43-45. Holomorphe und Meromorphe Fkt.
  7. Tuesday 13 May, p.46-p.5. Analytische Funktionen im Grossen.
  8. Friday 16 May, p. 51-57 etwa. Riemannsche Flaeche einer algebraische Kurve.
  9. Tuesday 20 May, p. 58-60. Wenn F irreduzibel, dann X\setminus \Sigma zusammenhaengend. Differentiale auf Riemannsche Flächen.
  10. Friday 23 May, p. 61-65. Divisoren von Differentiale; Lokale Struktur von Abbildungen zwischen Riemannsche Flächen. 
  11. Tuesday 27 May, p. 65-68 Überlagerungen, Holomorphe Abbildungen, Riemann-Hurwitz.
  12. Friday 30 May, p. 71-75. Riemann-Hurwitz Beispiele, Monodromiegruppe einer verzweigte Überlagerung.
  13. Tuesday 3 June
  14. Friday 6 June 
  15. Tuesday 10 June
  16. Friday 13 June
  17. Tuesday  17 June
  18. Friday 20 June, Satz I, Loesung der Poisson gleichung, p.111-116.
  19. Tuesday 24 June, Hodge Zerlegung und Riemannsche Ungleichung, p.116-12.
  20. Friday 27 June, Satz von Roch, p.120-p.124 oben.
  21. Tuesday 1 July
  22. Friday 4 July
  23. Tuesday 8 July
  24. Friday 11 July
  25. Tuesday 15 July
  26. Friday 18 July
  27. Tuesday 22 July
  28. Friday 25 July
  29. Tuesday 29 July
  30. Friday 1 August,

 

Course material

  • We will follow closely the course notes by Duco van Straten. The schedule above refers to these notes.
  • There will be a homework assignment every two weeks. You will find the homework sheets below.
  • There is a lot of literature available; see the course notes of Duco van Straten for a list of some of the books you can consult.
  • The exercises for the Übungsgruppe can be found here. (They are updated every week.)

 

Homework

Below you will find all the Homework sheets. We recommend you write your solutions in LateX. This will increase the quality of your homework, and it's a great way to practice writing mathematics with LateX. To use LateX, download MikTeX and an editor (like TeXmaker or TeXworks). A good tutorial for using LateX can be found under this link.

 

 

  1. To be handed in on May 9      : Homework 1
  2. To be handed in on May 23    : Homework 2
  3. To be handed in on June 6     : Homework 3
  4. To be handed in on June 20   : Homework 4
  5. To be handed in on July 4      : Homework 5
  6. To be handed in on July 18    : Homework 6
  7. To be handed in on August 1 : Homework 7