### Billiards

My main research is in polygonal billiards. Consider a single billiard ball, bouncing around inside a polygon. (We make all the standard ridiculous mathematical assumptions of course—it has radius 0, there is no friction, and collisions are perfectly elastic.) Will the ball end up repeating the same pattern over and over, or will it continue to trace out slightly different paths forever?

It turns out that for almost every starting condition, the pattern never repeats—in fact, with the right tools (namely countability) this is easy to prove. But are there any paths which repeat? This turns out to be much harder. Even the case of triangles has been open since 1775.

Below is a basic simulation for exploring these dynamics in triangles. The simulation comes with a few presets to explore important phenomena, such as periodic trajectories, aperiodic trajectories, and how very similar trajectories will eventually diverge.

The most important tool in the study of polygonal billiards is called *unfolding*.
It stems from the observation that when the billiard ball hits the edge of a polygon,
instead of imagining the billiard ball reflecting off that edge, you can imagine the
polygon reflecting across that edge while the ball continues in a straight line.
This is illustrated below whenever you run the demonstration above for a periodic trajectory.

Taking this further, you can imagine the surface built by gluing together copies
of a polygon along their edges. Billiard trajectories are exactly the straight
lines (or *geodesics*) on this surface, and the question of periodic trajectories
becomes: are there any closed geodesics on this surface? We can further note that any
two polygons with the same orientation are identical for the purposes of trajectories,
so we can glue them together, which in some cases gives us a finite surface.

It is not immediately obvious that this is an easier question. But while nobody
has come up with any good techniques for characterizing billiard trajectories
directly, there are a number of advanced techniques known for characterizing
geodesics on surfaces, especially finite surfaces. Of these, the most fruitful is known as
*Teichmüller theory*. However, my research has focused on
*algebraic geometry*, specifically the *fundamental group*.