Reducing Coincidence with Mathematics: An Interview with Nets Katz

Raised in Grand Prairie, Texas, Nets Katz began pursuing mathematics at an early age, earning a bachelor's from Rice University in 1990 at the age of 17 and a doctorate from the University of Pennsylvania in 1993 at 20. After completing several postdoctoral fellowships, Katz went on to an assistant professorship at the University of Illinois at Chicago, an associate professorship at Washington University in Saint Louis, and a full professorship at Indiana University before joining the faculty at Caltech in January. Recently, Katz answered a few questions about his move to Southern California and his research in a field of math called combinatorics.

What brought you to Caltech, and why are you excited to be here?

I was offered a job here! It's a great institution; I've always admired it. Tom Wolff, who was a math professor here about 12 years ago, was a major influence in my career, and I flatter myself to think that I'm continuing some of his work.

I'm also really excited about teaching these students. I'm teaching Math 1 in the fall, and I'm really looking forward to the unique opportunity to get across deep and useful ideas to the very best students, including students who aren't in my field. Mathematics has always had a significant impact on the other sciences and engineering, and I think it will continue to do so.

What are your research interests?

I'm interested in showing that you can't have very many coincidences. The problems that I'm interested in are mostly considered to be in combinatorics [a field of math concerned with finding maximum, minimum, and optimum configurations—such as the absolute largest or smallest possible size of an object]. In a problem I worked on a few years ago, called the Erdos distance problem, we wanted to know the minimum number of distinct distances possible between a set number of points in a plane.

For example, say you're playing a game in which you ask an opponent to draw a finite number of n dots on a sheet of paper; the object of the game is for the opponent to position their dots so that the number of distinct distances between the dots is as small as possible. You then determine how well they did by tallying the number of distinct distances between these dots. If the dots are truly positioned randomly, some of the distances between the dots could be the same, but almost all of the distances will be different—meaning your opponent didn't do very well.

Placing the dots in a grid-like lattice pattern would be a relatively good strategy to win the game—this arrangement allows you to position the dots in such a way that a lot of the distances are the same. When working on this problem, we were able to prove that you can't get fewer than n/log n distinct distances—which, surprisingly, means that there isn't a strategy much better than the lattice. If there were a better strategy or design, it would involve a lot of coincidences—and too many coincidences aren't possible. You have to have some really incredibly special design to come close to the lattice arrangement, and what we were able to show is that even the best "incredibly special design" really isn't better than the lattice by very much.

How did you first become interested in math?

My father was a physicist, so we would have conversations about math at the dinner table. Of course, he was very far from a mathematician, but among physicists in his day he was quite well versed in math. And he had criticisms of how math was done, not all of which made sense, so we would have discussions about the foundations of things that were really exciting.

Here we were in Grand Prairie—I, a little kid no one had ever heard of, and he, a physicist largely forgotten—talking about how we might set the foundations of mathematics in a more clever way than all the denizens of the MITs and Caltechs of the world had ever managed to do. It was incredibly empowering. If it weren't constrained by the substantive requirements of mathematics, it might have been megalomaniacal. He made me feel that a person out of nowhere could really change the way people think about things. This was very exciting to me.

How do you like living in Southern California?

Actually, I have to admit I'm much more of an Indiana person—I prefer small towns and rural areas to densely populated cities—so I find I am experiencing a lot of culture shock living in Southern California. But there are nice things about the location; I have a lot of friends at UCLA and being closer is definitely a plus.