Syllabus
Riemann Surfaces, Dynamics and Geometry
Math 275 - TuTh 10:00-11:30 pm - 111 Science Center
Harvard University - Spring 1998
Instructor:
Curtis T McMullen
(ctm@math.harvard.edu);
Office hours 12-1 Tu-Th
Texts
- Benedetti and Petronio. Lectures on Hyperbolic Geometry. Springer-Verlag, 1992.
- Carleson and Gamelin. Complex Dynamics. Springer-Verlag, 1993.
- Lehto. Univalent Functions and Teichmüller Spaces. Springer-Verlag, 1987.
- Otal. Le théorème d'hyperbolisation pour les variétés fibrées de dimension trois , Astérisque volume 235 (1996). Distributed by AMS.
- Thurston. Three-Dimensional Geometry and Topology. Princeton University Press, 1997.
- Zinsmeister. Formalisme thermodynamique et systèmes dynamiques holomorphes , SMF 1996. Distributed by AMS.
Prerequisites.
Intended for advanced graduate students.
Acquaintance with complex analysis, hyperbolic
geometry, Lie groups and dynamical systems
will be useful.
Topics.
We will discuss iterated rational maps and
hyperbolic 3-manifolds from many points of view,
especially in relation to topology,
Teichmüller theory and ergodic theory.
Some possible topics include:
-
Dynamics of rational maps
- Local holomorphic dynamics
- Normal families and the Julia set
- Classification of stable regions
- Sullivan's no-wandering-domains theorem
- Holomorphic families of rational maps
- The Teichmüller space of n points on a sphere
- Thurston's characterization of critically finite maps
- The Teichmüller space of a rational map
- Invariant line fields and the hyperbolicity conjecture
-
Hyperbolic manifolds: flexibility and rigidity
- Kleinian groups and the limit set
- The thick-thin decomposition
- Convex cocompact groups
- Mostow rigidity
- Patterson-Sullivan measure
- Hausdorff dimension of limit sets
- Dimension 2 for geometrically infinite groups (Bishop-Jones)
- The Ahlfors measure zero conjecture
- Sullivan's no invariant line field theorem
- Teichmüller theory injects into ergodic theory
- The Connes-Sullivan quantum integral
- Thurston's geometrization conjecture
- Universal Teichmüller space
- Amenability and the Theta conjecture
- Geometrization of Haken manifolds
Additional References
- Beardon. The Geometry of Discrete Groups. Springer-Verlag, 1983.
- Bishop and Jones. Hausdorff dimension and Kleinian groups. Preprint.
- Bowen. Hausdorff dimension of quasi-circles. Publ. Math. IHES 50(1978), 11-25.
- Connes. Non Commutative Geometry. Academic Press, 1994.
- Douady and Hubbard. A proof of Thurston's topological characterization of rational maps. Acta Math. 171(1993), 263-297.
- Gardiner. Teichmüller Theory and Quadratic Differentials. Wiley Interscience, 1987.
- Imayoshi and Taniguchi. Introduction to Teichmüller Spaces. Springer-Verlag, 1992.
-
McMullen.
Complex Dynamics and Renormalization .
Annals of Math Studies 135, Princeton University Press, 1994.
- McMullen. Renormalization and 3-Manifolds which Fiber over the Circle. Annals of Math Studies 142, Princeton University Press, 1996.
- Milnor. Dynamics in one complex variable: Introductory lectures. Stony Brook IMS Preprint 1990/5.
- Nicholls. The Ergodic Theory of Discrete Groups. Cambridge University Press, 1989.
- Ratcliffe. Foundations of Hyperbolic Manifolds. Springer-Verlag, 1994.
- Sullivan. On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions. In: Kra and Maskit, editors, Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference. Annals of Math. Studies 97, Princeton, 1981.
- Thurston. Geometry and Topology of Three-Manifolds. Lecture Notes, Princeton University, 1979.