PART I: Complex analysis in the complex plane.
We will follow in part Complex Analysis in One Variable by Narasimhan, and there will also be course notes.
23 August. Narasimhan, 3-9. Course notes 3-9.
25 August. Course notes 10-19.
30 August. Exercise session.
1 September. Course notes 20-26.
6 September. Exercise session.
8 September. Course notes 20-26. Schwarz' Lemma in Narasimhan.
13 September. Proof of Riemann mapping theorem in Narasimhan. Meromorphic functions, course notes 29-30
15 September. Singular functions, course notes 31-34. Runges approximation theorem, course notes 35-40.
20 September. Runge's Approximation Theorem, course notes 35-40.
22 September. Partitions of Unity. Narasimhan 97-99. A note on C^1-smooth functions. Course notes, appendix B.
27 September. The dibar-equation and the Mittag-Leffler Theorem. Course notes 41-44.
29 September. Mittag-Leffler Theorem and a preview of cohomology. Course notes 43-45.
4 October. No teaching - work on the mandatory assignment.
6 October. No teaching - work on the mandatory assignment.
PART II: Riemann Surfaces.
We will mainly follow O. Forster's book "Lectures on Riemann Surfaces". The course notes are partly supplementary and overlapping, but here and there proofs are different.
11 October. The definition of Riemann Surfaces. Forster 1-9.
Elementary Properties. Forster 10-12.
13 October. Sheaves. Forster 40-44
18 October. Sheaf cohomology. Forster 96-104
20 October. Exact sequences. Forster 118-125.
22 October. Differential forms. Dolbeault's Lemma on charts on
Riemann Surfaces. Forster 59-63. Course notes 49-51.
- Finite dimensionality of H^1(X,\mathcal O) for compact Riemann Surfaces. Course
notes 51-55. Forster, 109-118.
- The Riemann-Roch Theorem. Forster, 126-131. Course notes, 59-60.
- Integration of differential forms. Course notes 57-59. Forster, 68-80.
- The Serre Duality Theorem. Forster, 126-132. Notes.
- Hodge-Decomposition. Notes.