MA 355: Topics in Geometric Topology: Geometric structures
Topology (MA 231)
Introduction to Algebraic Topology (MA 232) or equivalent
preferably MA 335 (Introduction to Hyperbolic Manifolds) or equivalent
This course would be a survey of fundamental results as well as current research. Topics will be related to the following areas: geometric structures on surfaces, hyperbolic 3-manifolds, Riemann surfaces and Teichmüller theory, and will focus on the various interactions between these fields. Students will be encouraged to explore open-ended questions and/or write related computer programs. The following is the course plan:
A review of hyperbolic structures on surfaces
A review of Teichmüller spaces and mapping class groups
The topology of the PSL(2,R) representation variety
The notion of a geometric structure or (G,X)-structure on a manifold
Translation structures on a surface
Holomorphic 1-forms and their periods
An introduction to Teichmüller dynamics
Complex projective structures on a surface
Surface group representations into PSL(2,C)
The Schwarzian derivative and holomorphic quadratic differentials
Measured laminations and Thurston’s grafting theorem
Other geometric structures, including affine structures and real projective structures
The case of open surfaces.
Suggested books and references:
W. P. Thurston, Three-dimensional Geometry and Topology, Princeton University Press, 1997.
B. Martelli, An Introduction to Geometric Topology, CreateSpace Publishing, 2016.