Riemann Surfaces and Hyperbolic Geometry
Math 275 - MWF 12-1 pm - Science Center 216
Harvard University - Fall 2009
Instructor:
Curtis T McMullen
Suggested Texts
- J. H. Hubbard, Teichmüller Theory, vol. 1, Matrix Editions, 2006.
- O. Lehto, Univalent Functions and Teichmüller Spaces, Springer-Verlag, 1987
- Matsuzaki and Taniguchi, Hyperbolic Manifolds and Kleinian Groups, Oxford Science Publications, 1998
- J. Milnor, Dynamics in One Complex Variable , Third Edition. Princeton University Press, 2006.
- J. Ratcliffe, Foundations of Hyperbolic Manifolds, 2nd Edition. Springer, 2006.
- W. P. Thurston, Three-dimensional Geometry and Topology, Princeton University Press, 1997.
Other useful references
- M. B. Bekka and M. Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces, Cambridge University Press, 2000.
- Benedetti and Petronio, Lectures on Hyperbolic Geometry, Springer-Verlag, 1992.
- B. Bollobas, The asymptotic number of unlabelled regular graphs
- E. Ghys, Dynamique des flots unipotents sur les espaces homogènes, Sem. Bourbaki 1991/92; Asterisque 206.
- M. Gromov, Volume and bounded cohomology
- I. Kapovich and N. Benakli, Boundaries of hyperbolic groups
- S. Kerckhoff, The Nielsen Realization Problem
- C. McMullen,
From dynamics on surfaces to rational points on curves.
- C. McMullen, Notes on Teichmüller Theory and Complex Dynamics
- C. McMullen, Notes on Ergodic Theory
- G. Mess, Lorentz spacetimes of constant curvature (see Prop. 22 for the Earthquake Theorem).
- C. Series, Martin boundaries of random walks on Fuchsian groups
- W. P. Thurston, Geometry and Topology of 3-Manifolds.
Prerequisites.
Intended for advanced graduate students.
Description
Fundamental results and topics in Teichmüller theory, hyperbolic
3-manifolds, complex dynamics and the geometry of moduli space. Topics may include:
- Random walks on Riemann surfaces
- Fuchsian and quasifuchsian groups
- Holomorphic quadratic differentials
- Perspectives on Teichmüller theory
- The mapping-class group
- Moduli space and its compactifications
- Curves in moduli spce
- Iterated rational maps
- Rational maps with given combinatorics
- Kleinian groups and hyperbolic 3-manifolds
- Mostow rigidity
- Geometrization of 3-manifolds
Grades.
Enrolled students should attend the course regularly.
Calendar.
| 2 Sept (W) | First class |
| 7 Sept (M) | Labor day - no class |
| 12 Oct (M) | Columbus day -- no class |
| 11 Nov (W) | Veteran's day -- no class |
| 27 Nov (F) | Thanksgiving -- no class |
| 2 Dec (M) | Last class |
| 4-11 Dec (Tu-F) | Reading period |