MA 355: Topics in Geometric Topology: Geometric structures

Credits: 3:0

Pre-requisites :

1. Topology (MA 231)
2. Introduction to Algebraic Topology (MA 232) or equivalent
3. 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:

Part I

1. A review of hyperbolic structures on surfaces
2. A review of Teichmüller spaces and mapping class groups
3. The topology of the PSL(2,R) representation variety
4. The notion of a geometric structure or (G,X)-structure on a manifold

Part II

5. Translation structures on a surface
6. Holomorphic 1-forms and their periods
7. An introduction to Teichmüller dynamics

Part III

8. Complex projective structures on a surface
9. Surface group representations into PSL(2,C)
10. The Schwarzian derivative and holomorphic quadratic differentials
11. Measured laminations and Thurston’s grafting theorem

Part IV

12. Other geometric structures, including affine structures and real projective structures
13. The case of open surfaces.

Suggested books and references:

1. W. P. Thurston, Three-dimensional Geometry and Topology, Princeton University Press, 1997.
2. B. Martelli, An Introduction to Geometric Topology, CreateSpace Publishing, 2016.

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