Lectures on Riemann Surfaces

1 Covering Spaces. - 1. The Definition of Riemann Surfaces. - 2. Elementary Properties of Holomorphic Mappings. - 3. Homotopy of Curves. The Fundamental Group. - 4. Branched and Unbranched Coverings. - 5. The Universal Covering and Covering Transformations. - 6. Sheaves. - 7. Analytic Continuation. - 8. Algebraic Functions. - 9. Differential Forms. - 10. The Integration of Differential Forms. - 11. Linear Differential Equations. - 2 Compact Riemann Surfaces. - 12. Cohomology Groups. - 13. Dolbeault's Lemma. - 14. A Finiteness Theorem. - 15. The Exact Cohomology Sequence. - 16. The Riemann-Roch Theorem. - 17. The Serre Duality Theorem. - 18. Functions and Differential Forms with Prescribed Principal Parts. - 19. Harmonic Differential Forms. - 20. Abel's Theorem. - 21. The Jacobi Inversion Problem. - 3 Non-compact Riemann Surfaces. - 22. The Dirichlet Boundary Value Problem. - 23. Countable Topology. - 24. Weyl's Lemma. - 25. The Runge Approximation Theorem. - 26. The Theorems of Mittag-Leffler and Weierstrass. - 27. The Riemann Mapping Theorem. - 28. Functions with Prescribed Summands of Automorphy. - 29. Line and Vector Bundles. - 30. The Triviality of Vector Bundles. - 31. The Riemann-Hilbert Problem. - A. Partitions of Unity. - B. Topological Vector Spaces. - References. - Symbol Index. - Author and Subject Index. Language: English