Riemann Surfaces

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 Theory—A 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-Roch—An Alternate Approach.- IV. 11. Algebraic Function Fields in One Variable.- V Automorphisms of Compact Surfaces—Elementary 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.