Geometric structures on surfaces

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Hours - Lab: 
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Typical Scheduling: 
Not typically scheduled

Differential topology (Math 6452) or permission of the instructor. Familiarity with some basics (Riemannian metrics and geodesics) from Differential Geometry I (Math 6455) is also recommended.


Course Text: 
  • Farb-Margalit’s “A Primer on Mapping Class Groups”. 
  • relevant research articles
Topic Outline: 
The course is intended to be a tour around Mirzakhani’s Fields Medal winning work:
  • hyperbolic (curvature -1) and flat (curvature 0) structures, curves, measured laminations and train tracks on surfaces
  • Teichmuller and moduli spaces
  • McShane’s remarkable identity on the length of all simple closed geodesics on the once punctured torus. Generalization to higher genus surfaces by Mirzakhani. Applying the identity to compute the volume of moduli spaces.
  • Mirzakhani’s theorem counting simple closed geodesics on hyperbolic surfaces.
  • ergodicity properties of geodesic flows on hyperbolic and flat surfaces and on the space of flat surfaces 
  • Billiards on polygonal tables. Masur’s Theorem stating that every polygon with rational angles has a periodic trajectory.