Analytic Continuation and Riemann Surfaces

Analytic continuation and monodromy theorem, global analytic functions, Riemann surfaces, Poincaré–Volterra theorem on the countability of Riemann surfaces sheets, Puiseux series, notions of Euler characteristic and genus of connected compact 2-dimensional orientable differential real manifolds, affine plane curves, algebraic curves in complex projective space, holomorphic and meromorphic functions in Riemann surfaces, conformal Riemann surfaces, algebraic characterization of conformal Riemann surfaces by the Bers and the Iss’sa theorems, Poincaré or hyperbolic metric, Schwarz–Pick lemma on holomorphic functions of the unit disk with center 0 in itself being contractions in the Poincaré distance, Gauss curvature of a metric conformal to Euclidian metric in a region, ultrahyperbolic metrics, Schwarz–Pick–Ahlfors lemma on ultrahyperbolic metrics, holomorphic functions ranges (Bloch–Ahlfors, Schottky–Ahlfors, little and great Picard theorems), Montel–Carathéodory and Carathéodory–Landau theorems. Exercises, including on domains of holomorphy, M.Riesz bounding theorem, Fatou and M. Riesz convergence theorem, Ostrowski convergence theorem, theta series, Schottky theorem.