Topics in Geometry and Dynamics
Math 275 - Tu Th 10-11:30 - Science Center room 216
Harvard University - Spring 2017
Instructor:
Curtis T McMullen
Description:
A survey of fundamental results and current research. Topics may be chosen
from the several interacting areas described below.
Riemann surfaces and Teichmueller theory
- Hyperbolic surfaces
- The Poincare' metric on a plane region
- Fenchel-Nielsen coordinates
- The Selberg Trace Formula
- Sunada's construction
- Complex projective structures
- Quasifuchsian groups
- Quasiconformal mappings
- Extremal length
- Bers embedding
- Teichmueller's theorem
- The Weil-Petersson metric
- Kaehler hyperbolicity
- Earthquakes
- Geodesic currents
- The mapping-class group
- Chararacterization of rational maps
- 3-manifolds that fiber over the circle
- The Theta conjecture
- Gluing acylindrical manifolds
- Geometrization of Haken manifolds
- The Mordell-Shafarevich conjecture
- Knot complements
- Reflection groups
- Mostow rigidity
- Margulis tubes
- Hyperbolic volume
- Dehn filling
- Ahlfors finiteness theorem
- Bers area theorem
- Sullivan bound on cusps
- Limit sets and Hausdorff dimension
- The bifurcation current
- Ratner-Shah rigidity of immersed planes
- Julia sets
- Montel's theorem
- Classification of stable regions
- No wandering domains
- Holomorphic motions
- Bifurcations and stability
- Hausdorff dimension and measures
- Critically finite rational maps
- Families of rational maps
- Heights and periodic points
- Solvability of the quintic
- Billiards
- Geodesic and horocycle flows over moduli space
- Recurrence and unique ergodicity
- Entropy
- Curves systems and pseudo-Anosov maps
- Teichmueller curves
- Regular polygons
- L-shaped tables
- Jacobians with real multiplication
- Rigidity of VHS (Schmid)
- Real multiplication and torsion packets (Moeller)
- Cubic curves in the plane
- Totally geodesic surfaces
- Coxeter groups
- Lattices
- Glue
- Entropy and homology
- Entropy in holomorphic dynamics
- Manifestations of Lehmer's number
- Dynamics on K3 surfaces
- Dynamics on rational surfaces
- Triangle groups
- Braid groups and Hodge theory
- The clique polynomial
Course Notes and Papers
- C. McMullen, Riemann surfaces, dynamics and geometry
- C. McMullen, Hyperbolic manifolds, discrete groups and ergodic theory
- C. McMullen, Various papers
Suggested Texts
- M. B. Bekka and M. Mayer,
Ergodic theory and topological dynamics of group actions on homogeneous spaces,
Cambridge University Press, 2000.
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), 2007.
- 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
- R. Mañé, Ergodic Theory and Differentiable Dynamics
- 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.
Undergraduate enrollment requires permission of the instructor.
Grades.
Enrolled students should attend the course regularly.
Assignments will be provided for students requiring a letter grade.