Riemann Surfaces
|
Math 213b / 12:00-1:15 Tu Th / Science Center room 228
Harvard University - Spring 2026
Instructor:
Curtis T McMullen
(ctm@math.harvard.edu)
Required Texts
- Forster, Lectures on Riemann Surfaces, Springer-Verlag, 1981
Additional references
- Buser, Geometry and Spectra of Compact Riemann Surfaces, Birkhauser, 1992
- Griffiths and Harris, Principles of Algebraic Geometry, Wiley Interscience, 1978
- Farkas and Kra, Riemann Surfaces, Springer-Verlag
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 P1
- Sheaves and analytic continuation
- Curves in projective space; resultants
- Holomorphic differentials
- Sheaf cohomology
- Line bundles and projective embeddings; canonical curves
- Riemann-Roch 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.
In-class presence.
All enrolled students are expected to attend lectures regularly, and participate in discussions.
Electronics.
To maximize engagement, the use of laptops and cell phones during class is not
permitted. Notebooks and tablets may be used for taking notes.
Grades.
Graduate students who have passed their quals are excused
from a grade for this course.
Homework.
For others, grades will be based on weekly homework.
Collaboration between students is encouraged, but you must write your own solutions, understand them and
give credit to your collaborators.
There will also be two comprehensive, `free solo' homeworks,
to be completed without collaboration,
and to take the place of a midterm and a final.
Late homework is not accepted, but your lowest score on collaborative homework will be dropped. (The `free solo' homeworks cannot be dropped.)
AI and online resources.
Only materials from this course can be referred to and used for the homework. E.g. the use of AI, searching for solutions on the web, etc, is not permitted. The use of the many available NI resources is encouraged.