0 An Overview. - 0. 1. Topological Aspects Uniformization and Fuchsian Groups. - 0. 2. Algebraic Functions. - 0. 3. Abelian Varieties. - 0. 4. More Analytic Aspects. - I Riemann Surfaces. - I. 1. Definitions and Examples. - I. 2. Topology of Riemann Surfaces. - I. 3. Differential Forms. - I. 4. Integration Formulae. - II Existence Theorems. - II. 1. Hilbert Space TheoryA Quick Review. - II. 2. Weyl's Lemma. - II. 3. The Hilbert Space of Square Integrable Forms. - II. 4. Harmonic Differentials. - II. 5. Meromorphic Functions and Differentials. - III Compact Riemann Surfaces. - III. 1. Intersection Theory on Compact Surfaces. - III. 2. Harmonic and Analytic Differentials on Compact Surfaces. - III. 3. Bilinear Relations. - III. 4. Divisors and the Riemann-Roch Theorem. - III. 5. Applications of the Riemann-Roch Theorem. - III. 6. Abel's Theorem and the Jacobi Inversion Problem. - III. 7. Hyperelliptic Riemann Surfaces. - III. 8. Special Divisors on Compact Surfaces. - III. 9. Multivalued Functions. - III. 10. Projective Imbeddings. - III. 11. More on the Jacobian Variety. - III. 12. Torelli's Theorem. - IV Uniformization. - IV. 1. More on Harmonic Functions (A Quick Review). - IV. 2. Subharmonic Functions and Perron's Method. - IV. 3. A Classification of Riemann Surfaces. - IV. 4. The Uniformization Theorem for Simply Connected Surfaces. - IV. 5. Uniformization of Arbitrary Riemann Surfaces. - IV. 6. The Exceptional Riemann Surfaces. - IV. 7. Two Problems on Moduli. - IV. 8. Riemannian Metrics. - IV. 9. Discontinuous Groups and Branched Coverings. - IV. 10. Riemann-RochAn Alternate Approach. - IV. 11. Algebraic Function Fields in One Variable. - V Automorphisms of Compact SurfacesElementary Theory. - V. l. Hurwitz's Theorem. - V. 2. Representations of the Automorphism Group on Spaces of Differentials. - V. 3. Representationof Aut M on H1(M). - V. 4. The Exceptional Riemann Surfaces. - VI Theta Functions. - VI. 1. The Riemann Theta Function. - VI. 2. The Theta Functions Associated with a Riemann Surface. - VI. 3. The Theta Divisor. - VII Examples. - VII. 1. Hyperelliptic Surfaces (Once Again). - VII. 2. Relations Among Quadratic Differentials. - VII. 3. Examples of Non-hyperelliptic Surfaces. - VII. 4. Branch Points of Hyperelliptic Surfaces as Holomorphic Functions of the Periods. - VII. 5. Examples of Prym Differentials. - VII. 6. The Trisecant Formula. Language: English