“Many people are unaware that there are math questions we don’t know the answers to,” says Melanie Matchett Wood from Harvard University and Radcliffe Institute for Advanced Study. For her efforts in finding solutions to these open problems, she was awarded a MacArthur Fellowship (or “genius grants”) recently. The award honors “extraordinarily talented and creative individuals” with an $800,000 “no strings attached” prize.
Wood was honored for her research “addressing foundational issues in number theory”, which focuses more on whole numbers (-1, 2, 3, and so forth) than 1.5 or 3/8. Her fascination extends to prime numbers, which are whole numbers greater than 1 but only divisible by one (such as 2 or 7). Her work is largely based on arithmetic statistical, which focuses on patterns in the behavior and other types numbers. She has dealt with questions about the nature and behavior of primes in systems that include integers (which are zero, whole numbers, and negative multiples thereof), but that also include other numbers. This extension is illustrated by the system a b2, where a and b represent integers. When she needs to solve difficult questions, she uses a variety of tools from other areas of mathematics.
” The nature of the work is “Here’s an issue that we don’t know how to solve.” Wood says, “So come up with a way to solve this problem.” This is very different from the experience most people have with mathematics in school. It’s similar to the difference between reading and writing a book
Wood talked to Scientific American about her recent win, her favorite mathematical tools, and how she tackles “high risk, high reward” problems.
[ An edited transcript of the interview is available . ]
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 are the foundation of mathematics. These are difficult questions, but it is definitely 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 key tools include being able to look at many concrete examples and try out to see which phenomena are emerging–bringing other areas of mathematics in. Although I may be working on a question about prime numbers in number theory, I use tools from all areas of mathematics, including probability and geometry. Another is the ability not to succeed but to learn from my failures.
What’s your favorite prime number?
Two has always been my favorite number. It’s also my favorite prime number.
It seems so easy. It is possible to make such rich mathematics from just the number 2. For example, 2 is responsible for the idea of whether things are odd or even. It is possible to see the richness in simple situations and consider whether numbers are odd or even. It’s small but very powerful.
Also here’s a funny story: I was an undergraduate student at Duke University, and I was part of our [team for] the William Lowell Putnam Mathematical Competition. We wear shirts with numbers on our backs for the math team. Many people have numbers such as pi or 5-fun irrational number. My number was 2. My Duke math jersey had the number 2 on it when I graduated.
Have you always approached your number theory research from the perspective of arithmetic statistics?
Since my graduate school training, I have always been interested in arithmetic statistics, and in particular, the statistical patterns of numbers [including primes] and how they behave within larger number systems.
A major shift in my thinking, especially recently, has been to incorporate more probability theory into the methods I use to answer these questions. Probability theory is a classical theory that deals with the distributions of numbers. You could measure the length and performance of students in a standard test or the length of fish in an ocean. You will get a distribution of numbers. Then you can try to understand the spread out.
For the type of work I do, we need something more like a probability theory. This means that you are not simply measuring a number for each point. There might be a more complex structure, such as a shape. You might get numbers from a shape, such as “How many side does it have?”. But a shape is more than just numbers or a few numbers. It contains more information than that.
What does winning this MacArthur prize mean to you?
It’s an incredible honor. It’s especially exciting for me because the MacArthur Fellowship celebrates creativity, which most people associate more with the arts. It takes a lot creativity to solve math questions that no one can answer. It is a joy to see this 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 is a three-dimensional space. If you look in a small area it looks like the kind we are used to. It might even have surprising connections if you take a long walk through that space. You might find yourself walking in one direction, and then returning to where you started.
That might sound crazy. Think of two different spaces in two dimensions. You can walk straight in any direction on a flat plane. But you won’t be able to go back to the beginning. Then there is the surface of a sphere. If you walk in a certain direction, you will eventually come around. Because we live in three-dimensional space, it is easy to imagine these two types of two-dimensional spaces. There are actually three-dimensional spaces with strange properties that are not like the three-dimensional space we are used to interfacing with.
What is the essence of the work you’re doing on these spaces?
We find that there are certain types of three-dimensional spaces with certain properties. These properties include how you can move around and return to the place you started. These spaces are not shown, described or constructed by us. They are shown using the probabilistic method.
We show that if you arrange a random space in a particular way, there is a positive probability that you will get a specific kind of space. This is how mathematicians can know that something exists, even if they don’t find it. If you can prove that you can do something randomly and that there is a positive chance that you can get it using random constructions, then it must exist.
These tools are used to show that three-dimensional spaces have certain types of properties. We have examples of these properties, even though we don’t know any.
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 is to develop probability theory for more complex structures than numbers. It is high-risk because it is not clear if it will work or if it won’t be as useful as I hope. It’s not clear where it will lead. It could be very powerful if it works out.