16:640:504 Theory of Functions of a Complex Variable II

Spring 2020

Feng Luo

Course Description:

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.
Main Reference
[1] Forster, Lectures on Riemann Surfaces, Springer-Verlag, 1981
Additional references
[2] McMullen, C. Lecture note on Riemann surfaces, Harvard University
[3] Farkas and Kra, Riemann Surfaces, Springer-Verlag, 1991
[4] Ahlfors, complex, McGraw-Hill Education; 3rd edition, 1979
Prerequisites: math 503 and basic point set topology

Textbook:

Forster, Lectures on Riemann Surfaces, Springer-Verlag, 1981

Prerequisites:

503, point set topology