FALL 2009

The main hero of this class will be a * Riemann surface *.
This notion classically appeared as a geometric way of understanding multi-valued functions,
but eventually developed in a central concept in mathematics.
The main topics covered are:

The *Uniformization Theorem* that provides a complete classification of Riemann surfaces.

* Function theory * on compact Riemann surfaces: Riemann-Roch and Abel Theorems, Jacobian.

Elements of the * Teichmuller theory * and theory of quasiconformal maps
that describes the deformation space of Riemann surfaces of a given topological type.

The theory will be illustrated with numerous examples from Holomorphic Dynamics, Differential Equations, Hyperbolic Geometry, and Number Theory, which will be partly developed as bi-weekly homework assignments.

** Textbooks:**
We will use the books ``Riemann surfaces'' by H. Farkas and I. Kra and
``Teichmuller theory'' by J.H. Hubbard,
along with a number of additional sources on various supplementary topics.

** Final grade ** will be based upon attendance and homework

** Office Hours:** Tue 2:30 -- 4:00 and by appointment

** Note **
If you have any condition, such as physical or mental
disability, that will make it difficult to complete the course
or that will require extra time on examinations, notify your lecturer
during the first two weeks of class so that appropriate arrangements can be
made.