MATH669-0101: Selected Topics in Riemann Surfaces-Fall 2025 raw

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This course is an introduction to Riemann surface theory. An emphasis will be given to developing a familiarity with the main concepts and results that often appear in current research.

Prerequisites for this course include basic complex analysis (MATH 463 or 660), some familiarity with the topology of manifolds (e.g. MATH 437 or 740), and real analysis with point-set topology (MATH 410). 

Here is some more basic information about this course:

  • Time & Place: TuTh 2:00-3:15 pm, MTH B0431.

  • Instructor: Professor Richard A. Wentworth

  • Office Hours: TuTh 10-11, or by appointment.

  • Texts: The recommended text for this course is Compact Riemann Surfaces, by R. Narasimhan. Other classic texts that are useful are Lectures on Riemann Surfaces, by R.C. Gunning,  Lectures on Riemann Surfaces, by O. Forster, Riemann Surfaces, by Simon Donaldson, and Riemann Surfaces, by H. Farkas and I. Kra.

  • Grading: The final grade will depend on your participation in class. I will give reading assignments.  Each of you should give at least one presentation on a topic or problem that has come up during lectures. 

(above material dated Sep 2, 2025)

Below is a more detailed description of the topics I hope to cover (this will be updated throughout the semester):

A. Review of complex analysis (here are some notes from Math 660)

  • Holomorphic and meromorphic functions
  • Cauchy formula
  • Runge's theorem
  • Solving LaTeX: \bar\partial

B. Riemann surfaces (notes)

  • Basic definitions
  • Examples, branched covers
  • The Riemann surface of an algebraic function

C. Analysis on Riemann surfaces (notes)

  • Differential forms, PoincarĂ© Lemma, de Rham theorem
  • Harmonic differentials and the Hodge theorem
  • Bilinear relations
  • Existence of meromorphic functions

D. Sheaves and cohomology (notes)

  • Germs of holomorphic functions
  • Cohomology
  • Dolbeault isomorphism

E. Riemann-Roch Theorem (notes)

  • Line bundles and divisors
  • Serre duality
  • Riemann-Roch theorem
  • Holomorphic vector bundles
  • Applications

F. Jacobians and abelian varieties

  • Period matrices
  • Abel embedding
  • Theta functions

G. Uniformization theorem

  • Fuchsian groups
  • Hyperbolic metrics
  • Complex projective structures
  • Differential equations

H. Moduli and Teichmueller space

  • Deformations of Riemann surfaces
  • Beltrami differentials
  • The Kodaira-Spencer class
  • Riemann's moduli

Course Summary:

Course Summary
Date Details Due