Introduction to Dynamics on Moduli Spaces
Introduction to ergodic theory and dynamics on the moduli space of translation surfaces, covering counting problems in rational billiards and flat surfaces.
Summary
Recent recorded grades — Fall 2025: 4.00 GPA, 100.0% A/AB (n=23 letter grades).
What you’ll be able to do
Apply fundamental dynamical techniques and ergodic theorems to counting problems, for instance counting periodic trajectories on a billiard table or cylinders on a translation surface using the Siegel-Veech theorem, Nevo's ergodic theorem, and/or the trapezoid method of Eskin-Masur.
Write clear and well-reasoned mathematical arguments to describe the frequency with which a trajectory will recur to a neighborhood of its starting point using Margulis functions.
State the Eskin-Mirzakhani classification of SL(2,R) ergodic invariant measures on moduli spaces of translation surfaces and describe the use of random walks and the exponential drift argument of Benoist-Quint to prove measure rigidity statements.
Apply results in ergodic theory and dynamics to solve novel problems in geometric topology.
Analyze a range of mathematical proofs in dynamical systems on translation surfaces and their moduli spaces and recognize the underlying principles and techniques.
Source: UW Guide
Grade history
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Prerequisites
Course mapGraduate/professional standing or declared in Mathematics VISP (graduate or dissertator)
- Graduate/professional standing or declared in Mathematics VISP (graduate or dissertator)
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- Graduate/professional standing or declared in Mathematics VISP (graduate or dissertator)
Professors
Spring 2027No instructors recorded for this selection.
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Student experience
the class
Recent recorded grades — Fall 2025: 4.00 GPA, 100.0% A/AB (n=23 letter grades).
difficulty & workload
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Topics
- Ergodic theory and dynamics on the moduli space of translation surfaces
- Counting problems in rational billiards and on flat surfaces
- Quantitative recurrence results using Margulis functions
- Measure rigidity statements using random walks inspired by Benoist and Quint
- Finiteness results using Zimmer's theory of the algebraic hull
Skills
- Apply dynamical techniques and ergodic theorems to counting problems using the Siegel-Veech theorem, Nevo's ergodic theorem, or the trapezoid method.
- Write mathematical arguments describing trajectory recurrence frequencies using Margulis functions.
- State the Eskin-Mirzakhani classification and describe random walk methods for proving measure rigidity statements.
- Apply ergodic theory and dynamics results to solve novel problems in geometric topology.
Grades
Latest available · Fall 2025— not enough history to project Spring 2027.
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Explore recorded gradesRecorded 2026-10-14
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Model & dataset provenance
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