Riemann Surfaces(D)

RIEMANN SURFACES (D)
Complex Analysis is essential and Analysis II desirable.

The complex logarithm. Analytic continuation in the plane; natural boundaries of power series. Informal examples of Riemann surfaces of simple functions (via analytic continuation). Examples of Riemann surfaces, including the Riemann sphere, and the torus as a quotient surface. [4]
Analytic, meromorphic and harmonic functions on a Riemann surface; analytic maps between two Riemann surfaces. The open mapping theorem, the local representation of an analytic function as z 7→ zk. Complex-valued analytic and harmonic functions on a compact surface are constant. [2]
Germs of an analytic map between two Riemann surfaces; the space of germs as a covering surface (in the sense of complex analysis). The monodromy theorem (statement only). The analytic continuation of a germ over a simply connected domain is single-valued. [3]
The degree of a map between compact Riemann surfaces; Branched covering maps and the Riemann Hurwitz relation (assuming the existence of a triangulation). The fundamental theorem of algebra. Rational functions as meromorphic functions from the sphere to the sphere. [3]
Meromorphic periodic functions; elliptic functions as functions from a torus to the sphere. The Weierstrass P-function. [3]
Statement of the Uniformization Theorem; applications to conformal structure on the sphere, ∗to tori, and the hyperbolic geometry of Riemann surfaces∗. [1]

Appropriate books

L.V.Ahlfors Complex Analysis. McGraw-Hill, 1979
A.F.Beardon A Primer on Riemann Surfaces. Cambridge University Press, 2004
G.A.Jones and D.Singerman Complex functions: an algebraic and geometric viewpoint. Cambridge University Press, 1987
E.T.Whittaker and G.N.Watson A Course of Modern Analysis Chapters XX and XXI, 4th Edition. Cambridge University Press, 1996