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.