Geometry Seminar

On a rational right triangle, i.e. one whose angles are all rational multiples of pi, how many (bands of) periodic billiard trajectories of length at most L are there? Amazingly, this question is related to dynamics on the moduli space of Riemann surfaces. Each rational polygon P may be unfolded to a closed surface tiled by copies of P. I will begin by describing how GL(2,R) acts on the collection of such flat surfaces and how the GL(2, R) orbit closure of the unfolding of P controls many dynamical properties of billiard flow on P. I will then explain how to compute the orbit closure of the unfolding of every rational right triangle and describe the consequences it has for billiards.

Key ingredients in the proof include variational formulas in Teichmuller theory, the work of Eskin and Mirzakhani on orbit closures, and the work of Eskin, Kontsevich, and Zorich on sums of Lyapunov exponents.
Speaker(s): Paul Apisa (U Michigan)