Mathematics 272: Riemann Surfaces (Fall 2005)
Description
An introduction to Riemann surfaces:
Abstract Riemann surfaces. Examples: algebraic curves, elliptic curves
and functions on them. Normalization of singular algebraic curves.
Holomorphic and meromorphic functions and differential forms,
divisors and the Mittag-Lefler problem. The analytic genus. Bezout's
theorem and applications.
Introduction to sheaf theory, with applications to constructing
linear series of meromorphic functions.
Serre duality, the existence of meromorphic functions on Riemann
surfaces, the equality of the topological and analytic genera,
the equivalence of algebraic curves and compact Riemann
surfaces, the Riemann-Roch theorem.
Period matrices and the Abel-Jacobi mapping, Jacobi inversion,
the Torelli theorem.
Applications: The canonical curve, classification of
curves of low genus, hyperelliptic curves and integrable systems
(time permitting).
Instructor
Robert L. Bryant
Schedule
Time: Tuesdays and Thursdays, 11:40–12:55
Room: Physics Building, Room 120
Prerequisites
A course in one complex variable, Math 181 at least,
though Math 245 would be preferable. At least one course
in topology, such as Math 261. In special cases, consent
of the instructor may be granted in lieu of these courses.
Text(s)
- Main Text: Introduction to Algebraic Curves,
by Phillip A. Griffiths, Volume 76, Translations of Mathematical
Monographs, American Mathematical Society, 1989. (Required)
- Secondary Text: Lectures on Riemann Surfaces,
by Otto Forster, Graduate Texts in Mathematics, Volume 81,
Springer, 1981. (Not Required)
- Tertiary Text: Riemann Surfaces,
by Robert Bryant, Informal lecture notes. (Supplied in class)
Course Website
For more information see Math 272
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