The majority of my graduate research has been into "Billiards with Bombs", a variant of billiards loosely inspired by the classic arcade game Breakout. In my variant, the plane is covered by a square grid and a ball bounces around, erasing some pattern of walls (the bomb) every time it hits. Depending on the starting conditions and the shape of the bomb, it's possible to wind up with the ball quickly clearing a periodic or almost-periodic tunnel; with a lengthy period of unpredictable behaviour which eventually turns into a periodic tunnel; or with the ball continuing to clear a giant blob around its starting place for as long as human simulation can run.

People wanting to find out more about this problem are encouraged to read the arxiv preprint of my Experimental Mathematics paper.

I've also coded several Java simulators to help explore the problem's behavior given various starting points. Each simulator can be used by unzipping the linked file to a separate folder and double-clicking the appropriate .jar. They all assume that the ball starts out somewhere in the square [0,1) x [0,1) traveling to the right and either up or down in the direction given by the associated slope; full instructions are included in the zip files.

A periodic tunnel,The ball will keep clearing this tunnel forever . . .

delayed tunnels,. . . and this tunnel will also continue forever, though it took a while to set up. The speckled blob on the right extends to twelve times the height shown; the rest was snipped for sanity's sake.

a big blob Sometimes, however, the ball never seems to start tunneling. In this picture, it's cleared six million walls and is still going strong.

When I'm not working on Breakout, I've recently been playing around with chip-firing and the sandpile problem (some wiser heads' overview of the problem can be found on the arxiv), though at the moment I don't have much to show for it beyond pretty pictures and some interesting speculation about hexagonal graphs.

Pretty pattern . . .