Advanced Complex Analysis
Math 213b / 1011:30 Th Th / Science Center 507
Harvard University  Spring 2007
Instructor:
Curtis T McMullen
(ctm@math.harvard.edu)
Required Texts
 Forster,
Lectures on Riemann Surfaces,
SpringerVerlag, 1981
 Buser,
Geometry and Spectra of Compact Riemann Surfaces,
Birkhauser, 1992
Additional references
 Griffiths and Harris,
Principles of Algebraic Geometry,
Wiley Interscience, 1978
 Farkas and Kra,
Riemann Surfaces,
SpringerVerlag
Prerequisites.
Intended for graduate students.
Prerequesites include algebraic topology, complex analysis
and differential geometry on manifolds.
Topics.
This course will cover fundamentals of the theory of
compact Riemann surfaces from an analytic and
topological perspective.
Topics may include:
 Algebraic functions and branched coverings of P^{1}
 Sheaves and analytic continuation
 Curves in projective space; resultants
 Holomorphic differentials
 Sheaf cohomology
 Line bundles and projective embeddings; canonical curves
 RiemannRoch and Serre duality via distributions
 Jacobian variety
Reading and Lectures.
Students are responsible for all topics covered in
the readings and lectures. Lectures may go beyond the
reading, and not every topic in the reading will be
covered in class.
Grades.
Graduate students who have passed their
quals are excused from a grade for this course.
Grades for other students will be based on homework.
Homework.
Homework will be assigned every few weeks.
Collaboration between students is encouraged, but you
must write your own solutions, understand them and
give credit to your collaborators.
Calendar.
1 Feb (Th)  First class 
2630 Mar (MF)  Spring recess 
3 May (Th)  Last class 
