Math 7580 (Spring 2023)

Topics in Topology: Riemann Surfaces and their Moduli

Lecture: Mon, Wed 11:25am-12:40pm, Malott 206

Office hours: Mon 1-2pm, Wed 4-5pm, Malott 582

Course notes

Student presentations

Students enrolled in the course are expected to give a 45 minute presentation (40 min talk + 5 min questions) during one of the lecture slots about some topic related to the course. Preferably, this would be after the first month of classes. Please pick a date and let me know by Feb 7. A topic can be chosen a bit later.

Possible topics include: Quasiconformal maps in full generality, Bers Embedding, Geometric Shafarevich,

Scheduled presentations:

April 10 - Thurston metric
April 12 - Kobayashi hyperbolicity
April 17 - NO PRESENTATION
April 19 - Royden's Theorem
April 24 - Teichmuller disks
April 26 - Properties of Teichmuller geodesic flow
May 1- no presentation (rescheduled to May 8)
May 3 - Strebel differentials
May 8 - Laplace and length spectrum

Useful texts

Farb and Margalit "A primer on mapping class groups"

Hubbard "Teichmuller theory: Volume 1"

McMullen Notes on Complex Analysis on Riemann Surfaces

Course description

We will study Riemann surfaces and their moduli, primarily from the perspective of complex analysis and hyperbolic geometry. We will begin with various constructions of Riemann surfaces. Further topics: Teichmuller space, character variety, the mapping class group and Nielsen-Thurston classification, moduli space and the Deligne-Mumford compactification, Kobayashi hyperbolicity, the Weil-Petersson symplectic form (and metric and measure), extremal length, quadratic differentials and the Teichmuller metric, and dynamics of the Teichmuller geodesic flow.