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Cover; Contents; PART I: PRELIMINARIES; 1 Holomorphic functions; 1.1 Simple examples: algebraic functions; 1.2 Analytic continuation: differential equations; Exercises; 2 Surface topology; 2.1 Classification of surfaces; 2.2 Discussion: the mapping class group; Exercises; PART II: BASIC THEORY; 3 Basic definitions; 3.1 Riemann surfaces and holomorphic maps; 3.2 Examples; Exercises; 4 Maps between Riemann surfaces; 4.1 General properties; 4.2 Monodromy and the Riemann Existence Theorem; Exercises; 5 Calculus on surfaces; 5.1 Smooth surfaces; 5.2 de Rham cohomology. 5.3 Calculus on Riemann surfacesExercises; 6 Elliptic functions and integrals; 6.1 Elliptic integrals; 6.2 The Weierstrass [Omitted] function; 6.3 Further topics; Exercises; 7 Applications of the Euler characteristic; 7.1 The Euler characteristic and meromorphic forms; 7.2 Applications; Exercises; PART III: DEEPER THEORY; 8 Meromorphic functions and the Main Theorem for compact Riemann surfaces; 8.1 Consequences of the Main Theorem; 8.2 The Riemann-Roch formula; Exercises; 9 Proof of the Main Theorem; 9.1 Discussion and motivation; 9.2 The Riesz Representation Theorem. 9.3 The heart of the proof9.4 Weyl's Lemma; Exercises; 10 The Uniformisation Theorem; 10.1 Statement; 10.2 Proof of the analogue of the Main Theorem; Exercises; PART IV: FURTHER DEVELOPMENTS; 11 Contrasts in Riemann surface theory; 11.1 Algebraic aspects; 11.2 Hyperbolic surfaces; Exercises; 12 Divisors, line bundles and Jacobians; 12.1 Cohomology and line bundles; 12.2 Jacobians of Riemann surfaces; Exercises; 13 Moduli and deformations; 13.1 Almost-complex structures, Beltrami differentials and the integrability theorem; 13.2 Deformations and cohomology; 13.3 Appendix; Exercises. 14 Mappings and moduli14.1 Diffeomorphisms of the plane; 14.2 Braids, Dehn twists and quadratic singularities; 14.3 Hyperbolic geometry; 14.4 Compactification of the moduli space; Exercises; 15 Ordinary differential equations; 15.1 Conformal mapping; 15.2 Periods of holomorphic forms and ordinary differential equations; Exercises; References; Index; A; B; C; D; E; F; G; H; I; J; K; L; M; N; O; P; Q; R; S; T; U; V; W