Topics in Geometry and Dynamics
Math 275  Tu Th 1011:30  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
 FenchelNielsen coordinates
 The Selberg Trace Formula
 Sunada's construction
 Complex projective structures
 Quasifuchsian groups
 Quasiconformal mappings
 Extremal length
 Bers embedding
 Teichmueller's theorem
 The WeilPetersson metric
 Kaehler hyperbolicity
 Earthquakes
 Geodesic currents
Iteration on Teichmueller space
 The mappingclass group
 Chararacterization of rational maps
 3manifolds that fiber over the circle
 The Theta conjecture
 Gluing acylindrical manifolds
 Geometrization of Haken manifolds
 The MordellShafarevich conjecture
Hyperbolic 3manifolds
 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
 RatnerShah rigidity of immersed planes
Conformal dynamics
 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
Dynamics on moduli spaces
 Billiards
 Geodesic and horocycle flows over moduli space
 Recurrence and unique ergodicity
 Entropy
 Curves systems and pseudoAnosov maps
 Teichmueller curves
 Regular polygons
 Lshaped tables
 Jacobians with real multiplication
 Rigidity of VHS (Schmid)
 Real multiplication and torsion packets (Moeller)
 Cubic curves in the plane
 Totally geodesic surfaces
Reflections groups, entropy, algebraic dynamics
 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
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,
SpringerVerlag, 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,
Threedimensional 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.
