I'm researching ways to plan paths for spacecraft so that they can travel throughout the solar system for extremely reduced fuel requirements.

The analogy I like to use is that on earth we have sailing ships which are able to travel the oceans without fuel because they are using natural energy sources - the winds and the currents. In space we could design spacecraft that use the gravity and movement of planets as their natural energy sources. The trouble is that these paths that use the gravity and movement of the planets are extremely difficult to find and also very hard for a spacecraft to execute. So what I did was that I developed a navigation system that would run on the spacecraft that would allow it to both find these low energy paths and fly these paths.

Science in school I almost don't like that much. My favorite subject right now is probably the philosophy class I'm taking. What I like about science is that outside of school - when you're doing it on your own or with a team - you have a choice of what you want to do.

For me, I always go through these [science] magazines and I look for things that interest me. I read different papers and I kind of get a feel for what a particular area has to offer and then I get to decide, well, I want to research this and this hasn't been done before and then I get to do something totally new.

That's what I like about science, you get to have a lot of freedom to discover things for yourself. That's why I do so many research projects. I just have a lot of fun with it.

 

To do my project, I went to a program called RSI last summer at MIT. It was a lot of fun; I recommend it to anyone eligible. I had two mentors. One, a graduate student, was named Dustin Clausen; the other, a professor, was Pavel Etingof.

I definitely had to learn a lot of new mathematics for my project. I knew next to nothing about representation theory (the main field) when I got there; I wasn't even aware of the extent of my ignorance. I had vague notions of category theory, without which my project makes no sense, but I wasn't fluent in it. Now, I still am essentially a novice in these fields, but I've absorbed a fair bit over the past year, largely thanks to my mentors' patience

My project is on representation theory. Basically, this is a branch of mathematics that studies the representations of an object, in other words how it can act on something else. These “something elses” are called vector spaces, and normally their dimension is a positive integer. For instance, the plane is a two-dimensional vector space and is easy to visualize. However, it is no more difficult mathematically to talk about the seventeenth dimension—the algebraic techniques are the same.

In my project, however, the dimension becomes a complex number. There is a certain sense in which a non-integral dimension does make sense, though I will not define it precisely here. An object with non-integral dimension in this sense is not a vector space; it's an object in a certain category.

My project sings and dances to the refrain that “representation theory in transcendental rank resembles classical representation theory.” To explain that technically would go beyond the scope of this article, but the essential idea is that the properties of representation theory can be parametrized by the vanishing of certain polynomials in one variable with rational coefficients. If these polynomials do not vanish on the positive integers, then they are not identically zero, and consequently do not vanish at transcendental numbers. This is the fundamental reason for the whole approach, and it begins Etingof's program.

In the future, it'd be cool to figure out what happens at non-transcendental parameters. But I'll try now to write this into a paper.