After discussing the basic facts and examples of Riemann surfaces,
we will proceed to more advanced topics: the Riemann surface of an
algebraic function, cohomology of line bundles, divisors and the
Riemann-Roch theorem, the canonical bundle and Serre duality. Our
proofs will be analytic, for example Serre duality will be proved
using a regularity theorem for the d-bar operator. We will then
discuss Abel's theorem and the Jacobian, and continue with a
treatment of theta functions, the theta divisor, and Riemann's
theorem about meromorphic functions and theta. If
time permits we will prove Torelli's theorem and discuss
the Dirichlet problem on Riemann surfaces and the uniformization
theorem.
Assignment 2 (Due 10/4/18): tex, ps, pdf
Assignment 3 (Due 10/18/18): tex, ps, pdf
Assignment 4 (Due 11/8/18): tex, ps, pdf
Assignment 5 (Due 12/6/18): tex, ps, pdf