Department Colloquium
Department of Mathematics
Princeton University
November 22nd, 2000
Billiards in Rational Polygons and Moduli Spaces of Holomorphic
Differentials
A polygon is called rational if all angles are rational
multiples of pi. It turns out that the problem of counting
periodic billiard trajectories on such a polygon can be
reduced to a certain dynamical problem of the moduli space of pairs
(M,w) where M is a Riemann surface, and w is a holomorphic
1-form on M. Even though it is not locally homogeneous,
this moduli space is analogous in many ways to the moduli space
of Euclidean lattices SL(n,R)/SL(n,Z). I will discuss
these constructions and some associated problems in combinatorial
enumeration.