Illumination and Blocking in Rational Polygons

Abstract: Suppose that you are in a room with mirrored walls. The classical illumination problem asks: if you put a light source at a given point, will it light up the whole room? Are there rooms which can never be fully lit up? Now suppose that there are other people standing in the room: how does this affect the points which are illuminated? Starting from first concepts, I will explain Lelièvre, Monteil, and Weiss’s solution in the case when the room is a polygon whose angles are rational multiples of π. Along the way, we will get a glimpse at the rich interaction between the dynamics of billiards and the geometry of translation surfaces, as well as the breakthrough results of Eskin, Mirzakhani, and Mohammadi.