What is your research about?
I work in the field of arithmetic geometry. One type of fundamental problem is finding whole-number solutions to equations such as x⁵ + y⁵ = z⁵, or showing that no such solutions exist. This kind of problem goes back almost 2,000 years to the Greek mathematician Diophantus; hence they are called Diophantine equations.
The idea behind arithmetic geometry is to first consider the space of solutions to the equation in complex numbers x,y,z. This space is geometric in nature; for instance, if you squint hard enough, the space of solutions to x⁵ + y⁵ = z⁵ starts to look like a donut with six holes.
One then makes geometric arguments about this space and uses some very deep theorems to derive the properties of the set of whole-number solutions.
Math is worth doing for its own intrinsic beauty and for the subtle understanding about the world that it gives us. Although pure math is not concerned with practical applications, historically it has proved over and over again to be useful in the most surprising and important ways. A recent example is the use of elliptic curves defined over finite fields (an important player in arithmetic geometry) in some of the most advanced encryption algorithms in use today.
What has been the most exciting time so far in your research life?
I spend about 95% of my research time writing and revising papers, doing straightforward verifications, or just plain being stuck. The other 5% is where the "aha!" moments happen that make it all worth it.
So far the most memorable time was when I solved my Ph.D. thesis problem. For weeks, I had been thinking hard about the same thing. Then one day, just as I was spreading mayonnaise on my sandwich for lunch, I realized what to do to make the final step work. From there, the solution was like a cascade of dominoes, with everything falling exactly into place. That was not only extremely satisfying. It also launched my career: I was one of the first people to use so-called tropical geometry to solve a problem in arithmetic geometry, which strategy has become something of a cottage industry now.
How did you find your way to mathematics research?
My father has a Ph.D. in physics, so I grew up assuming I'd get a Ph.D. as well. I was always interested in thinking about math. For instance, in high school, when I realized I didn't know why the Pythagorean theorem was true, I spent one evening working out a nice geometric proof. Of course this proof had been known since Greek times, but it was satisfying to work something out on my own.
I didn’t get serious about studying math until freshman year of college, when I discovered that I enjoyed my math course more than my physics course.
What advice would you give to a college freshman who wants to be a mathematician?
Being a mathematician is both an extremely solitary and a very social activity. Learning, understanding, or communicating mathematics takes a large amount of care and rigor. It is best done alone, with no distractions and with long periods of concentration. But you should also interact with a community of peers, to chat about the most compelling things you’ve learned or thought about recently and to work together when you get stuck, which happens daily.
Take intellectual risks. Sign up for a graduate course even if you’re not sure you'll get an A in it. Go to seminars and expose yourself to concepts you might not understand. Try undergraduate research programs. Never be afraid to tell someone that you’re confused, and ask them to explain something more slowly.
If you could not be a mathematician, in what line of work would you be now?
I'd probably be a computer programmer. I've always been good with computers. I learned Basic programming when I was around 12.
What is the most exciting thing about being a part of Georgia Tech?
The students. I really enjoy teaching upper-level undergraduate math classes. Some students are extremely hard-working and talented. I derive a lot of pleasure from interactions in class and office hours.
What are you most surprised about in your encounters with Georgia Tech students?
Individual students often surprise me greatly. I've had very good students who participate in activities such as professional cage fighting, EMT work in ambulances, cheerleading for a major professional sports team, and serious bodybuilding. I never know what to expect when a new student walks into my office.
What is an unusual skill, talent, or quality you have now that is not obvious to your colleagues?
I used to be a very good lindy hop dancer. You can find videos on YouTube. Start by searching for the Rock Step Lobstahs.
What is your ideal way of relaxation?
The real answer is a movie and a beer, but I’m going to go with jogging. I run about 4.5 miles almost every day, a great way to clear my head. My wife and I just had a baby, though, so all of my routines are up in the air at the moment.
What three destinations are still in your travel to-do list?
I have to do a lot of traveling for work, 5-10 conferences all over the world each year. But I would always prefer it if the conference were at Georgia Tech and I could stay at home. So instead of listing places I wish I could visit, I'll mention the three most interesting places where I've attended a conference since I came to Georgia Tech: Fukuoka, Japan; Papeete, French Polynesia; Rio de Janeiro, Brazil.
If you won $10 million in a lottery, what would you with it?
I'd put it in low-risk investments and live off the interest. I never particularly wanted to be rich. I'd much rather have stability than wealth, thus my choice to become a tenured professor. That said, with $10 million, I’d have enough income to build an obscenely powerful personal computer, just for kicks.