Riemann Surfaces, Dynamics and Geometry

Math 275 - TuTh 10:00-11:30 pm - 111 Science Center
Harvard University - Spring 1998

Instructor: Curtis T McMullen (; Office hours 12-1 Tu-Th

  • 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

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