Complex Manifolds in Dimension 1

1. Almost complex structures and Hodge decomposition. Cauchy-Riemann equation via the Hodge decomposition. Integrability of almost complex structures. Existence of holomorphic function as a sufficient condition for integrability (dimension 1).
2. Complex manifolds, sheaves of holomorphic functions, equivalence of definitions of a complex manifold (definition via integrable almost complex structure, definition using sheaves, definition through charts and atlases).
3. Real analytic manifolds. Formal integrability of almost complex structures. Newlander-Nirenberg theorem for real analytic manifolds.
4. Examples of Riemann surfaces: Riemann surface of a function, plane curves, complex tori.
6. Normal families. Montel’s theorem. Arzela-Ascoli theorem. Riemann uniformization theorem.
7. Schwartz lemma. Group of automorphisms of a disk. Poincare metric. Kobayashi metric. Existence of constant curvature metrics on Riemann surfaces.
8. Iterated holomorphic mappings. Fatou and Julia: dynamics on Riemann sphere. Fractal sets. Mandelbrot sets.

Knowledge of geometry (manifolds, tangent bundles, Riemann structures) and complex analysis (Cauchy formula, analytic continuation, complex differentiable and complex analytic functions) is required.