Pete Brooksbank, Mathematics

Pete Brooksbank

photo from bucknell.edu

 Is there any way you could in laymen’s terms structure one of the algebraic problems you are trying to solve using these really advanced and complicated models?
"Sure. Maybe a good thing to think about is the familiar Rubik’s cube. With the cube, there six different faces that you can rotate, and there are many different configurations that you can generate from these face rotations. Rather amusingly, Ideal Toy, the original manufacturers of the cube, wrote on the packaging that there are more than 3 billion configurations. This sounds like an awful lot, but in fact they underestimated the actual number by a factor of about 10 billion! The challenge, if you’ve ever played with the cube, is to take a scrambled configuration and return it to its original solved state. People who are good at this come up with various "macros" that move the cube from more scrambled to less scrambled states, with each successive state being a better approximation of the solved cube. In this way they are able rather quickly to reach the solved state. Of course, when human beings solve the cube, they are really using a lot of geometric insight and visual recognition. However, you can also think of the cube problem in purely algebraic terms: underlying it is an example of the kind of algebraic problem that I’m interested in. Specifically, you have a couple of fixed symmetries of the cube, namely the face rotations, and you are given a scrambled configuration of the cube. Your job is to unscramble this configuration back to the solved one. The catch is that you don't have a physical cube to work with! All you have is the algebraic description of what these different face rotations are doing. Well, you can in fact program a computer to do this, but it’s not going to produce the optimal solution because you have not told it that there's a cube behind the problem. Thus, the computer cannot have the same geometric intuition as human beings. Rather, it works only with the algebraic structure. Despite this apparent handicap, computer algorithms can solve the cube almost effortlessly using a moderate number of moves. So that’s one type of problem that I'm interested in. It’s an example of a "constructive word problem" for algebraic objects called "groups". There are far more general versions of that problem, and highly refined methods for solving them. As these methods go, the Rubik’s cube problem is child’s play: the computer can solve the cube in the blink of an eye. In my work, I deal with vastly larger algebraic objects whose underlying structure is far more complex than that of the cube."