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.

Computational Representation Theory

I have done research on the representation theory of the cohomology of the pure configuration space on $\mathbb C^n$. Church, Ellenburg and Farb prove in this preprint that the $S_n$-representation of $H^i(PConf_n(\mathbb C))$ stabilizes as a function of $n$, in a sense they define precisely. I've written a library for computing the stable value of this decomposition, which is available on my GitHub. The project of actually performing these computations is chronicled on my programming page.

Good Problems

These are a few good problems I have collected over the years. Click to show/hide the solutions.

• Consider the sequence $a_n=2^n$. As $n\to\infty$, what fraction of the terms begin with a $1$?

  Note that $2^n$ begins with $1$ iff $\log_{10}2^n\,\mathrm{mod} 1 < \log_{10}2$. Since $\log_{10}2$ is irrational, $\log_{10}2^n=n\log_{10}2$ is equidistributed $\!\mathrm{mod} 1$, so the asymptotic fraction of terms with $\log_{10}2^n \,\mathrm{mod} 1 < \log_{10}2$ is $\log_{10}2$.

• Let $X$ be a metric space. Show that $X$ is compact iff every continuous function $f:X\to \mathbb R$ is bounded.

  If $X$ is compact, then the open cover $\{f^{-1}((-n,n)):n\in\mathbb N\}$ has a finite subcover, hence $f^{-1}((-n,n))=X$ for some $n$, so $f$ is bounded between $-n$ and $n$. If $X$ is not compact, we have some sequence $x_n$ with no convergent subsequence, hence $\{x_n:n\in \mathbb N\}$ is a closed, discrete set. Define $f:\{x_n:n\in \mathbb N\}\to \mathbb R$ by $f(x_n)=n$, which is continuous by discreteness. By the Tietze Extension Theorem, this extends to a (clearly unbounded) function on $X$.

• Take an $8\times 8$ chessboard and remove two opposite corner squares. Can the resulting board be covered by dominoes, which each cover 2 adjacent squares?

  No. Each domino covers one square of each color, but we removed 2 squares of the same color.

The Quadratic Equation and Beyond