Riemann surfaces, dynamics and hyperbolic geometry
Harvard University - Spring 2024
Instructor:
Curtis T McMullen
Description:
A survey of fundamental results and current research in several interacting areas.
Topics may include:
- Hyperbolic manifolds in dimensions 2 and 3
- Quadratic differentials
- Measured laminations
- Teichmueller theory
- Fenchel-Nielsen coordinates
- Quasifuchsian groups
- Bers embedding
- Mostow rigditiy
- Patterson-Sullivan measure
- Ahlfors' finiteness theorem
- Bers' area theorem
- No wandering domains
- Ratner rigidity
- Ergodic theory
- Arithmetic of discrete groups
- Quaternion algebras
- Moduli spaces of Riemann surfaces
- Moduli spaces of holomorphic 1-forms
- Conformal dynamical systems
- The Mordell-Shafarevich conjecture for function fields
Suggested Texts
- 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.
- E. Ghys, Dynamique des flots unipotents sur les espaces homogènes, Sem. Bourbaki 1991/92; Asterisque 206.
- M. Gromov, Volume and bounded cohomology
- J. H. Hubbard, Teichmueller Theory (vol. 1-3)
- O. Lehto, Univalent Functions and Teichmueller Spaces
- C. Maclachlan and A. Reid, Arithmetic of Hyperbolic 3-Manifolds, Springer-Verlag, 2003.
- C. McMullen, The evolution of geometric structures on 3-manifolds.
- C. McMullen, Renormalization and 3-Manifolds which Fiber over the Circle
- D. Witte Morris, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Math. Series, 2005.
- J. Ratcliffe, Foundations of Hyperbolic Manifolds, 2nd Edition. Springer, 2006.
- W. P. Thurston, Three-Dimensional Geometry and Topology, Princeton University Press, 1997.
Prerequisites.
Intended for advanced graduate students.
Readings and Lectures.
Students are encouraged to explore the several course notes and references
for more material on topics covered in lectures.
Homework.
Problems to complement the lecture material will be collected in the course notes.
Expectations.
Enrolled students should attend lectures regularly and work on the problems, which are an important part of the course.
Collaboration is encouraged.
Grades.
Letter grades, for those requiring them, will be based on submitted homework.