MAT 1341, Currents and counting of curves, Winter 2018
Office: BA 6180 (Bahen Centre)
MeetingsTuesdays 11 - 12 pm BA 6180
Thursdays 10 - 12 pm BA 6183
Office hoursThursdays 2 - 4 pm or by appointment.
We study the geometry and dynamical properties of Riemann surfarces and examine the set of closed curves on a surface. We start with background material, reviewing measured laminations, currents and the Weil-Peterson metric on Teichmüller space. We then cover some results of Mirzakhani, their applications and generalizations.
Below are some references for the topics covered in class:
- Curves and laminations.
- Thurston's Work on Surfaces, by Fathi, Laudenbach, Poénaru; translated by Djun Kim and Dan Margalit.
- Combinatorics of Train Tracks, by Pener and Harer.
- The Geometry and Topology of 3-manifold, by William Thurston [pdf]
- The Fenchel-Nielsen deformation, by Scott Wolpert.
- Simple geodesics and a series constant over Teichmuller space, by Greg McShane.
- Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, by Maryam Mirzakhani
- The geometry of Teichmüller space via geodesic currents, by F. Bonahon.
- Growth of the number of simple closed geodesics on hyperbolic surfaces, by Maryam Mirzakhani.
- Counting Mapping Class group orbits on hyperbolic surfaces, by Maryam Mirzakhani.
- Ergodic theory of the space of measured laminations, by Lindenstrauss and Mirzakhani.
- Geodesics Currents and Counting Problems, by Kasra Rafi and Juan Souto<./li>
- Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus, by Maryam Mirzakhani.
- Lengths of closed geodesics on random surfaces of large genus, by Maryam Mirzakhani, Bram Petri.
- Mixing, counting, and equidistribution in Lie groups, by Alex Eskin and Curt McMullen
- Distribution des orbites des réseaux sur le plan relé by F. Ledrappier
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