2005 Fall MATH 272-01
Bulletin Course Description Compact Riemann Surfaces, maps to projective space, Riemann-Roch Theorem, Serre duality, Hurwitz formula, Hodge theory in dimension one, Jacobians, the Abel-Jacobi map, sheaves, Cech cohomology. Instructor: Staff
(Instructor named in bulletin description above may not be current. For current instructor, see listing below.)
Title RIEMANN SURFACES Department MATH Course Number 2005 Fall 272 Section Number 01 Primary Instructor Bryant,Robert L Prerequisites Prerequisite: Mathematics 245 and Mathematics 261 or consent of instructor. Course Homepage www.math.duke.edu/~bryant/267/
Prerequisites
Math 181 or Math 245 (preferred) and Math 261 or consent of instructor.
Synopsis of course content
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).
Textbooks
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)
Assignments
Regular homework exercises will be assigned during the lectures
Exams
No exams
Term Papers
No term papers
Grade to be based on
Homework assignments completed and class participation
