Riemann surfaces Riemann surface is a complex manifold of dimension 1. Equivalently, Riemann surface is a 2-dimensional smooth manifold with a fixed conformal structure. The "conformal structure" is a Riemannian structure, defined up to a multiplication by a positive function, that is, a way of defining "angles" between tangent vectors. Equivalence of these two definitions is a non-trivial theorem, and one of the main results of the course. Most of the time, we will discuss complex geometry, with a special emphasis on dimension 1, but there is a bunch of results which are special for Riemannian surfaces alone. These are "uniformization theorems", which can be understood as existence of canonical (and very symmetric) complete metric on any Riemann surface. Using this metric, all Riemann surfaces except three are obtained as quotients of the Poincare plane by a discrete group of isometries. I will try to explain the relevant complex geometry and some of the uniformization theorems. Program 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. Frobenius theorem. Newlander-Nirenberg theorem for real analytic almost complex structure. 4. Examples of Riemann surfaces: Riemann surface of a function, plane curves, complex tori. 5.(*) Weierstrass p-function. Elliptic curve and its equation. j-invariant of an elliptic curve. 6.(*) Classification of compact 2-dimensional manifolds. Genus of a curve and its Euler characteristic. Hurwitz formula. Hyperelliptic curves and their genus. Plane curves and their genus. 7. Normal families. Montel's theorem. Arzela-Ascoli theorem. Riemann uniformization theorem. 8. Schwartz lemma. Group of automorphisms of a disk. Poincare metric. Kobayashi metric. Existence of constant curvature metrics on Riemann surfaces. (*) Brody's lemma. Poincare-Koebe uniformization theorem. 9.(*) 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. Parts marked with (*) would be possibly omitted for the lack of time. Literature: John W Milnor, "Dynamics in one complex variable", Annals of mathematics studies, no. 160, Princeton University Press Lars V. Ahlfors, "Lectures on Quasiconformal Mappings", University Lecture Series 38, American Mathematical Society, 2006. Simon Donaldson, "Riemann Surfaces", Oxford Graduate Texts in Mathematics 22, Oxford University Press, 2011. C Herbert Clemens, "A scrapbook of complex curve theory" University series in mathematics (Plenum Press) Plenum Press, New York, 1980.