Description: This course will introduce Riemann surfaces and Teichmüller theory with an emphasis on the geometric and complex analytic parts of the theory. It will cover the definition and basic examples of Riemann surfaces, analytic and meromorphic functions and differential forms, covering spaces, the uniformization theorem, quasiconformal mappings in the plane, Teichmüller space, analytic coordinate systems, metrics, and applications. The aim is to develop a detailed understanding of the the basic concepts and a general appreciation of some more advanced topics.
Application and future course: Given time and the interests of the students, we will introduce the topic of conformal field theory (a special 2 dimensional quantum field theory) and its connection to Teichmuller theory. There is the possibility that I will give a course on conformal field in period V.
Motivation: Riemann surfaces are two-dimensional surfaces which are locally modeled by the complex plane, that is, they are 1-dimensional complex manifolds. They form the basis for the global study of analytic functions, and appear in a wide variety of settings in pure and applied mathematics from number theory to quantum field theory.
It is then natural to look at the space of all Riemann surfaces and to put metrics on this space so as to be able to compare surfaces in a systematic way. This leads to the notion of Teichmüller space and moduli space. These spaces that appear in many modern research areas such as: algebraic, arithmetic and hyperbolic geometry, 3-manifold theory, complex dynamics, conformal field theory, and image processing/recognition.
The foundations of quasiconformal mappings and Teichmuller theory was largely developed by the Finnish mathematical community and the extension of these concepts plays an important role in current research here.
Assessment: Homework and in class presentations.
Prerequisites: An undergraduate course in complex analysis is recommended. If you don't have this but are still interested in the course then contact me, as other background may suffice.
Farkas and Kra, Riemann surfaces.
Nag, The complex analytic theory of Teichmüller spaces.
Lehto, Univalent Functions and Teichmüller spaces
Ahlfors, Lectures on quasiconfomal mappings
Papadopoulos, Introduction to Teichmüller theory, old and new. (from the introduction to the Handbook of Teichmuller Theroy, Vol I - IV).
Mumford and Sharon, 2D-Shape Analysis using Conformal Mapping (http://www.cis.upenn.edu/~cis610/sharon-mumford.pdf)
Please contact me at email@example.com (Otakaari 1, Y242b) if you have any questions.