Geometry and dynamics of hyperbolic surfaces

MATH-541 / 5 credits

Language: English

Summary

This course begins by describing and classifying hyperbolic surfaces before focussing on their geodesic flow. Topics include ergodicity, mixing, the Anosov property and equirepartition of closed geodesics.

Content

In this lecture, I will begin by outlining some key aspects of planar hyperbolic geometry using a metric approach. Next, I will define, construct, and classify closed hyperbolic surfaces using a pair of pants. Then I will delve into the geodesic flow of these surfaces, explaining the Anosov property, ergodicity, and mixing properties using the Howe-Moore theorem and spectral interpretation. Building on this, I will proceed to Margulis' proof of equidistribution. Finally, I will present the Ratner theorem and discuss some applications, including the equidistribution of Hecke points and perhaps touch on some more advanced topics. The course is based on this unfinished set of notes that you can find on my webpage.

Learning Prerequisites

Recommended courses

Differential geometry.

Important concepts to start the course

Metric spaces, projective line.

Assessment methods

Throughout the semester, there will be three short half-hour in-class assignments counting for half the grade. Depending on attendance, a three hour final class assignment or several presentations will conclude the semester.