MAT 543 Syllabus Complex Analysis II Fall 2017

The schedule for topics covered in lecture is as follows.

• Tuesday, August 29
  Overview of Riemann surface, holomorphic line bundles, and meromorphic sections.
  
  
• Thursday, August 31
  Overview of cohomology, Riemann-Roch, and Riemann-Hurwitz.
  
  
• Tuesday, September 5
  Labor Day. No class.
  
  
• Thursday, September 7
  Section 1. The definition of Riemann surfaces.
  
  
• Tuesday, September 12
  Sections 2 and 3. Elementary properties of holomorphic mappings. Homotopy of Curves. The fundamental group.
  
  
• Thursday, September 14
  Section 4. Branched and unbranched coverings.
  
  
• Tuesday, September 19
  Section 5. The universal covering and covering transformations.
  
  
• Thursday, September 21
  Sections 6 and 7. Sheaves. Analytic continuation.
  Problem Set 1 due in lecture.
  
  
• Tuesday, September 26
  Section 8. Algebraic functions.
  
  
• Thursday, September 28
  Section 9. Differential forms.
  Problem Set 2 due in lecture.
  
  
• Tuesday, October 3
  Section 10. The integration of differential forms.
  
  
• Thursday, October 5
  Section 12. Cohomology groups.
  Problem Set 3 due in lecture.
  
  
• Tuesday, October 10
  Section 13. Dolbeault's lemma.
  
  
• Thursday, October 12
  Section 14. A finiteness theorem.
  Problem Set 4 due in lecture.
  
  
• Tuesday, October 17
  Section 15. The exact cohomology sequence.
  
  
• Thursday, October 19
  Section 16. The Riemann-Roch theorem.
  Problem Set 5 due in lecture.
  
  
• Tuesday, October 24
  Section 17. The Serre duality theorem.
  
  
• Thursday, October 26
  Section 18. Functions and differential forms with prescribed principal parts.
  Problem Set 6 due in lecture.
  
  
• Tuesday, October 31
  Section 19. Harmonic differential forms.
  
  
• Thursday, November 2
  Section 20. Abel's theorem.
  Problem Set 7 due in lecture.
  
  
• Tuesday, November 7
  Section 21. The Jacobi inversion problem.
  
  
• Thursday, November 9
  Section 22. The Dirichlet boundary value problem.
  Problem Set 8 due in lecture.
  
  
• Tuesday, November 14
  Section 24. Weyl's Lemma.
  
  
• Thursday, November 16
  Section 25. The Runge approximation theorem.
  Problem Set 9 due in lecture.
  
  
• Tuesday, November 21
  Section 26. The theorems of Mittag-Leffler and Weierstrass.
  Deadline to confirm oral presentation topic with instructor.
  
  
• Thursday, November 23
  Thanksgiving. No class.
  
  
• Tuesday, November 28
  Section 27. The Riemann mapping theorem.
  
  
• Thursday, November 30
  Section 29. Line and vector bundles.
  Problem Set 10 due in lecture.
  
  
• Tuesday, December 5
  Sections 30 and 31. The triviality of vector bundles. The Riemann-Hilbert problem.
  
  
• Thursday, December 7
  Additional topics.
  
  

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