Complex Variables II
Mathematics 619: Fall 2005, REF # 339885

Professor: Rachel Pries, e-mail: pries "at" math.colostate "dot" edu; web page: www.math.colostate.edu/~pries; office: Weber 221.

Lecture: MWF 3:10-4:00, Engineering E203.

Prerequisite: Math 566 & 519 or permission of professor.

Course description: Riemann surfaces are ubiquitous in many diverse areas of math, including complex analysis, algebraic geometry, topology, number theory, physics, and differential equations. In this course, we will study Riemann surfaces from various points of view. A Riemann surface looks locally like the complex plane, but its topology and geometry are quite different. One example of a Riemann surface is an elliptic curve or torus (shaped like a doughnut).

Riemann's motivation for introducing these surfaces was to make sense of multi-valued functions, like the square root or logarithm. His idea was to take copies of an open set of the complex plane and to glue them together above the branch cuts of the function. This yields a one-dimensional complex manifold covering the complex plane whose sheets correspond to the possible values of the function. The original multi-valued function on the complex plane can now be thought of as a well-defined single valued function on this Riemann surface.

In the first half of the course we will cover basic topics on Riemann surfaces such as the following: what are Riemann surfaces and where are they found; invariants of Riemann surfaces, such as the genus (the number of holes); functions, differentials, and integration on Riemann surfaces; group actions on Riemann surfaces; parameter spaces for Riemann surfaces.

In the second half of the course, we will cover topics about Riemann surfaces that are the most interesting and motivating for the class.
Topology: universal covers, fundamental groups, monodromy.
Group theory: Riemann surfaces can be viewed as quotients of the upper half plane by the action of a Fuchsian group.
Geometry: the Jacobian, line bundles and the Picard group.
Number theory: An interesting type of Riemann surface comes up when one studies moduli spaces for elliptic curves and modular forms.
Applied Math: fluid flow, potential theory, integrable models, spectral curves, solitons, statistical mechanics, conformal field theory.

Texts: Many excellent textbooks contain this material. I recommend "Algebraic curves and Riemann surfaces" by R. Miranda.