“Many people don’t realize that there are math questions that we don’t know how to answer,” says mathematician Melanie Matchett Wood of Harvard University and the Radcliffe Institute for Advanced Study at Harvard. She recently won a MacArthur Fellowship (or “genius grant”) for her work seeking solutions to some of those open problems. The award honors “extraordinarily talented and creative individuals” with an $800,000 “no strings attached” prize.
Wood was recognized for her research “addressing foundational questions in number theory,” which focuses on the whole numbers—1, 2, 3, and so on, rather than 1.5 or 3/8, for instance. Prime numbers, whole numbers that are greater than 1 and only divisible by 1 and themselves (such as 2 and 7), also fascinate her. Much of her work uses arithmetic statistics, a field that focuses on discovering patterns in the behavior of primes and other types of numbers. She has tackled questions about the nature of primes in systems of numbers that include the integers (these are zero, the whole numbers and negative multiples of the whole numbers) but that are “extended” to include some other numbers as well. For example, the system a + b√2 (where a and b are integers) is such an extension. She also uses a smorgasbord of tools from other areas of math when invoking those ideas could help solve challenging questions.
“The nature of the work is ‘Here’s a question that we have no method to solve. So come up with a method,’” Wood says. “That’s very different from most people’s experience of mathematics in school. It’s like the difference between reading a book and writing a book.”
Wood spoke to Scientific American about her recent win, her favorite mathematical tools and tackling “high risk, high reward” problems.
[An edited transcript of the interview follows.]
What makes a mathematical question intriguing?
I’m drawn in by questions about foundational structures, such as the whole numbers, that we don’t really have any tools to answer. [These] structures of numbers underpin everything in mathematics. Those are hard questions, but that is certainly exciting to me.
If you were to build an imaginary tool belt with some of the mathematical instruments and ideas you find most useful in research, what would you put in it?
Some of the key tools are being willing to look at a lot of concrete examples and try to see what phenomena are emerging—bringing in other areas of math. Even though, maybe, I work on a question in number theory about something like prime numbers, I use tools from across mathematics, from probability, from geometry. Another is the ability to try things that don't work but learn from those failures.
What’s your favorite prime number?
Two is my favorite number, so it’s definitely my favorite prime number.
It seems so simple. Yet such rich mathematics can come out of just the number 2. For example, 2 is kind of responsible for the concept of whether things are even or odd. There is a tremendous richness that can come from just considering things in complicated situations, about whether numbers are even or odd. I like it because even though it’s small, it’s very powerful.
Also, here’s a fun story: I was an undergraduate at Duke [University], and I was on our [team for the William Lowell Putnam Mathematical Competition. For the math team, we have shirts with numbers on the back. Many people have numbers like pi or √5—fun irrational numbers. But my number was 2. When I graduated from Duke, they retired my math jersey with the number 2 on it.
Have you always approached your number theory research from the perspective of arithmetic statistics?
Starting with my training in graduate school, I have always come from this arithmetic statistics perspective, in terms of wanting to understand the statistical patterns of numbers, [including] primes and how they behave in larger number systems.
A big shift for me, especially lately, has been [bringing] more probability theory into the methods for working on these questions. Probability theory, classically, is about distributions of numbers. You could measure the length of fish in the ocean or performance of students on a standardized test. You get a distribution of numbers and try to understand how those numbers are [spread out].
For the kind of work that I’m doing, we need something that is more like a probability theory, where you’re not just measuring a number for each data point. You have some more complex structure—for example, maybe it’s a shape. From a shape, you might get numbers, such as “How many sides does it have?” But a shape is not just a number or a couple of numbers; it has more information than that.
What does winning this MacArthur prize mean to you?
It's a tremendous honor. It is, in particular, exciting to me because the MacArthur Fellowship really celebrates creativity, and most people associate that more with the arts. But to make progress on math questions that no one knows how to answer also requires a lot of creativity. It makes me happy to see that recognized in mathematics.
Harvard mathematician Michael Hopkins described your work on three-dimensional manifolds as “a dazzling combination of geometry and algebra.” What is a three-dimensional manifold?
It’s a three-dimensional space that, if you just look around in a small area, looks like the kind of three-dimensional space that we’re used to. But if you go on a long walk in that space, it might have surprising connections. Like, you walk in one direction and end up back where you started.
That might sound kind of crazy. But think about two different two-dimensional spaces. There's a flat plane, where you can walk straight in every direction, and you’ll never come back to where you start. Then there’s the surface of the sphere. If you walk in some direction, you’ll eventually come back around. We can picture those two different kinds of two-dimensional spaces because we live in three-dimensional space. Well, there are in fact three-dimensional spaces that have these funny properties that are different than the three-dimensional space that we're used to interacting with.
What is the essence of the work you’re doing on these spaces?
We find that certain kinds of three-dimensional spaces exist with certain properties having to do with how you can walk around and come back to where you started in them. We don’t exhibit, construct or describe those spaces. We show that they exist using the probabilistic method.
We show that if you take a random space in a certain way, there is some positive probability that you’ll get a certain kind of space. This is a beautiful way that mathematicians know something exists without finding it. If you prove that you can do something randomly, and there’s some positive chance, no matter how small, that you can get it from some random construction, then it must exist.
We use these tools to show that there exist three-dimensional spaces that have certain kinds of properties. Even though we don’t know of any examples, we prove they exist.
Last year you won a $1-million Alan T. Waterman Award from the U.S. National Science Foundation. The Harvard Gazette noted that you planned to use that funding to tackle “high-risk, high-reward projects.” What are some examples?
This direction of developing probability theory for more complicated structures than numbers is an example. It’s high-risk, because it’s not clear that it’s going to work, or maybe it won’t turn out to be as useful as I hope. There’s no clear blueprint for where it will go. But if it does work out, it could be very powerful.