MA 309: Teichmuller theory
Credits: 3:0
Prerequisite courses: MA 224 (Complex Analysis), MA 232 (Introduction to Algebraic Topology)
This course offers an introduction to Teichmüller theory, the study of the deformation spaces of Riemann surfaces (or hyperbolic surfaces). Occupying a central position at the crossroads of complex analysis, hyperbolic geometry, low-dimensional topology, and algebraic geometry, Teichmüller theory provides fundamental tools for understanding geometric structures and their moduli.
Topics will include:
- A brief review of Riemann surfaces and the Uniformization Theorem.
- Hyperbolic geometry in two dimensions.
- Quasiconformal mappings and the Beltrami equation.
- Various definitions of Teichmüller space: analytic, geometric, algebraic.
- Fenchel-Nielsen coordinates and geodesic length functions.
- Holomorphic quadratic differentials and tangent spaces.
- The Teichmüller metric and its geometric properties.
- The Mapping Class Group and its action on Teichmüller space. -Schwarzian derivatives and the Bers embedding.
- The complex structure on Teichmüller space.
- Measured foliations and Thurston’s compactification of Teichmüller space.
- (If time permits) The Weil-Petersson metric and Wolpert’s formula.
Suggested books and references:
- Y. Imayoshi and M. Taniguchi, An Introduction to Teichmüller spaces, Springer-Verlag 1992.
- John H. Hubbard, Teichmuller Theory And Applications To Geometry, Topology, And Dynamics, Volume 1, Matrix editions, 2006.