Riemann Surfaces

Math 213b / MWF 12:00-1:00 / 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, Springer-Verlag, 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 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
    • Torelli theorem
  • Hyperbolic Riemann surfaces
    • Uniformization theorem
    • Hyperbolic geometry and trigonometry
    • Closed and simple geodesics
    • Thick-thin 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, Weil-Petersson 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.

31 Jan (W) First class
19 Feb (M) President's day
26-30 Mar (M-F) Spring recess
4 May (F) Last class
11 May (F) Final papers due

Course home page: http://math.harvard.edu/~ctm/math213b