Billiards and the Arithmetic of Non-Arithmetic Groups

Curtis Mcmullen

What are the slopes of periodic billiard paths in a regular polygon?We will connect this question and others to:        - cusps of thin groups,        - curves on Hilbert modular varieties,        - heights from Jacobians with real multiplication, and         - a spectral gap for the Galois orbits of triangles.Along the way we will encounter issues of chaos and decidability, first appearing in polygons with 7 and 12 sides.

https://www.ias.edu/video/billiards-and-arithmetic-non-arithmetic-groups