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
Iteration on Teichmueller space
  • 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
Hyperbolic 3-manifolds
  • 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
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 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
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 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.

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