Riemann Surfaces
Math 213b / MWF 12:001:00 / Science Center 112
Harvard University  Spring 2001
Instructor:
Curtis T McMullen
(ctm@math.harvard.edu)
Course Assistant:
Laura DeMarco
(demarco@math.harvard.edu)
Required Texts
 Forster,
Lectures on Riemann Surfaces,
SpringerVerlag, 1981
 Griffiths and Harris,
Principles of Algebraic Geometry,
Wiley Interscience, 1978
 Buser,
Geometry and Spectra of Compact Riemann Surfaces,
Birkhauser, 1992
Additional references
 Barth, Peters and van de Ven,
Compact Complex Surfaces,
Springer, 1984
 Gardiner,
Teichmueller Theory and Quadratic Differentials,
Wiley, 1987
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.
Possible topics include:

Complex Riemann surfaces
 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
 Torelli theorem

Hyperbolic Riemann surfaces
 Uniformization theorem
 Hyperbolic geometry and trigonometry
 Closed and simple geodesics
 Thickthin decomposition
 Short geodesics and eigenvalues of the Laplacian
 Teichmueller space via pairs of pants
 Mumford's compactness theorem on moduli space

Additional topics (as time allows)
 Automorphic forms
 Univalent functions, Bers embedding, WeilPetersson metric
 Selberg trace formula
 Complex tori and K3 surfaces
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 and
a final project.
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.
31 Jan (W)  First class 
19 Feb (M)  President's day 
2630 Mar (MF)  Spring recess 
4 May (F)  Last class 
11 May (F)  Final papers due 
