**Complex Variables II**

Mathematics 619: Fall 2005, REF # 339885

** 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.

**Grading: ** This is an advanced graduate course and course grades will be computed accordingly.

50% Homework; 10% Short Presentation; 40% Project.

**Homework and short presentation:**
Homework is the most important part of this class.
It should demonstrate your knowledge of the material,
your investigation of open ended questions, and your skill at writing proofs.
Homework is due every week.
Please make sure your homework is neat, legible, and stapled.
I encourage you to brainstorm the problems together and write up your
solutions independently.

**
Detailed information on homework & short presentation**

** Project:**
The project is an opportunity to learn more about a topic in this area that interests you or will be
relevant for your future graduate work.
It gives us a chance to hear about cool ideas which we will not have time to cover in class.
It is also a good opportunity to develop more skill at writing and speaking on mathematics.
**
Detailed information on projects **

**Help:** Help is always available if you have trouble with homework
or lecture material. If your classmates can't answer your question,
come ask me! Office hours will be (TBA) or are available by appointment.