Mathematical Billiards is a study of the dynamical system that results from letting a point mass travel around in a domain with optical reflection when it hits the walls (angle of incidence equals angle of reflection). There’s a lot of math there!
This research cluster will be focused on the special case that the “billiard table” is a polygon in the plane. If the angles of the polygon are rational multiples of π, the billiard system can be studied by considering curves on an associated surface, so the techniques of topology also come to bear.
images: Chris O’Donnell
Billiards are also amenable to exploration by computer. For instance, Rich Schwartz and Pat Hooper built the computer system McBilliards and Rich used it in his proof of the very cool One Hundred Degree Theorem: every triangular table whose largest angle is <100° has a closed orbit (i.e., there’s a trajectory that repeats periodically forever).
Some expository sources
(This isn’t prerequisite reading, but just for reference.)
Marianne Freiberger, Chaos on the Billiard Table — interview with Corinna Ulcigrai about billiards
Serge Tabachnikov, Geometry and Billiards — beautiful undergraduate-level introduction; not totally focused on polygonal case
Alex Wright, From Billiards to Dynamics on Moduli Spaces — short but high-level note touching on recent breakthroughs