Math 7610: Topics in Geometry (Fall 2026)

Teichmuller theory and translation surfaces

Lecture: Tues, Thur 1:25pm-2:40pm, Malott 230

Office hours: Wed 11:30-12:30, Fri 2:30-3:30, Malott 507

Course description

This course will cover topics related to Riemann surfaces, translation surfaces, and hyperbolic geometry. There will be a focus on connections between the areas. Themes include:

Counting problems (for geodesics, and related objects) and their relation to dynamics,
Connections between the theory of translation surfaces and the topology of surfaces, in particular pseudo-Anosov maps.

Topics may include: connections to homogeneous dynamics and lattices, Siegel-Veech theory, Margulis' count of closed geodesics and analogy with graphs, counting simple closed geodesics (including work of Mirzakhani)

Prerequisites

Hyperbolic geometry, complex analysis, and topology, at least at an advanced undergraduate level.

Goals

Lean about Teichmuller theory and translation surfaces
Practice person-to-person communication
Have fun!

Student presentations/projects

Students enrolled in the course will give an in-class presentation on a topic related to the course. Your presentation will be preceded by a shorter "preview" presentation, in which you introduce the topic.

Useful texts

Athreya-Masur "Translation surfaces"

Farb-Margalit "A primer on mapping class groups"

Notes on Riemann surfaces from a previous topics course I taught

Animations

Shape of right-angled hyperbolic hexagons

Finding the hexagon with alternating side lengths specified

Info on presentations

Presentation topics are due Oct 19 (send me an email). "Preview" presentations (5-10 minutes) will be held on Oct 20 and 22. Students should come to office hours before and after the topic is chosen to discuss with presentation with me.

Potential topics: Nielsen realization, Deligne-Mumford compactification, Thurston metric, Royden's Theorem, Length and Laplace spectrum, Bers Embedding, Geometric Shafarevich theorem, Volumes of strata of translation surfaces, Compactifications of strata of translation surfaces, Siegel-Veech theory, Connections between Teichmuller dynamics and homogeneous dynamics, Margulis count of closed geodesics, Mirzakhani count of simple closed geodesics.