This will be a continuation of Math 503. The course will focus on the foundation of theory of Riemann surfaces and its relationship to other fields (geometry and topology, analysis and algebraic curves).
Topics. This course will cover fundamentals of the theory of compact Riemann surfaces from an analytic and topological perspective. Topics may include:
1. Algebraic functions and branched coverings
2. Sheaves and analytic continuation
3. Elliptic functions
4. Curves in projective space
5. Holomorphic differentials
6. Sheaf cohomology
7. Riemann-Roch, Abel and Jacobi theorems
8. Dirichlet problem and uniformization theorem.
 Forster, Lectures on Riemann Surfaces, Springer-Verlag, 1981
 McMullen, C. Lecture note on Riemann surfaces, Harvard University
 Farkas and Kra, Riemann Surfaces, Springer-Verlag, 1991
 Ahlfors, complex analysis, McGraw-Hill Education; 3rd edition, 1979
Forster, Lectures on Riemann Surfaces, Springer-Verlag, 1981
503, point set topology