**Title:** Billiards in Right Triangles and Dynamics on Moduli Space!

**Speaker:** Paul Apisa

**Speaker Info:** U. Michigan

**Brief Description:** Part of Midwest Dynamics Conference

**Special Note**: **Part of Midwest Dynamics Conference**

**Abstract:**

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 (by work of Eskin and Masur) 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.The proof will build upon new techniques for inductively studying orbit closures introduced by Wright and myself. Other 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.

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