3rd Year Candidacy Requirement: Unfolding a Billiard Table

Abstract: A common technique in the study of polygonal billiards is that of unfolding. The idea is to straighten the trajectory of the billiard by mirroring the table when the billiard encounters the edge of the table and letting the billiard pass into this new copy. From this process, we can convert the standard "zigzag" billiard flow into a geodesic flow on a translation surface called the unfolding. We consider a new type of polygonal billiard that features "portals", which are pairs of sides that teleport the billiard rather than reflect it. A technique analogous to unfolding can be done for portals, and we again obtain a geodesic flow on a translation surface. We will discuss the problem of computing the genus of this translation surface, first in the case of rational billiards, then in the case of irrational billiards. 

Advisor: Renato Feres