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Melanie Matchett Wood is a professor of mathematics at Harvard University and a Radcliffe Alumnae Professor at the Radcliffe Institute for Advanced Study. In October, she was awarded the MacArthur Fellowship “Genius Grant” for her creative work in number theory.
This interview has been edited for length and clarity.
FM: In online articles profiling you and your research, one of the first questions they always ask is: “What is number theory and arithmetic and algebraic geometry?” It’s an important one for me and for our readers, so could you pare it down from the high level at which you’re working to middle school or high school math?
MMW: Number theory is the study of the whole numbers, and in those, especially the prime numbers, how other whole numbers can be constructed by multiplying together primes and the patterns in which that happens. And it turns out almost all of math is very richly interconnected, and certainly number theory is.
Part of my research and how I try to answer questions about numbers is by using other tools. Algebraic geometry refers to understanding the geometry of shapes that are cut out by polynomial equations, so a very familiar one of these might be a parabola — it’s cut out by this polynomial equation y equals x squared, y and x polynomials. You could graph some shape that equation described, and algebraic geometry is the study of the geometry of the shapes that can be cut out by polynomials. It turns out that these things are very related even though at first it doesn’t sound that way.
FM: I’ve taken advanced math classes most of my life, up through calculus and linear algebra. But how come I’ve never heard of this before? Where does it fit in within the math chronology that people grow up knowing in school? Or if it doesn’t fit in, why not?
MMW: The math chronology in schools is quite different from the lay of the land of math research. It doesn’t fit very well. It’s not that they’re unrelated. The math chronology in schools is not a path through the math that is out there, probably because it’s focused on teaching people certain practical skills. Where you would have seen some bits of something like number theory is probably you learned, at some point, to be able to take an integer and factor it into primes, especially if you were trying to put a fraction in lowest terms.
You see little touches of number theory in the traditional school education but not so much, which is interesting because it’s one of the oldest topics of mathematics that has been studied. A lot of the questions about the whole numbers have been studied for thousands of years.
FM: What are the real-world implications of your work? Like, if we discover answers to these unanswered or open problems, how would that translate into you know, like, real-world applications?
MMW: My work is in what some people call “pure math,” but the basic science of mathematics. I’m trying to understand the foundational things that seem most important to know about numbers, not with some particular application in mind. There’s lots of wonderful applied mathematics, but that's not what I’m doing. However, throughout history, we see that much of science has been built on mathematics that was developed before that science or those applications were even imagined.
For example, today’s entire digital economy and world relies on the fact that we can encrypt information, that we have some way to reasonably easily encode things so that only people with a certain key can decode them. All of today’s encryption systems are built on number theory, on properties of prime numbers in particular, and on mathematics that was developed before there were computers. That’s the picture of how things are. I think we develop basic mathematics, we try to understand the most foundational things behind numbers, shapes, patterns, and it has paid off over time that the mathematics that we’ve developed has been used for things that we might not have imagined when we developed it.
FM: You’ve earned the title “first woman” to do something in your field quite a few times in your career, starting with being the first female competitor on the U.S. Math Olympiad Team. What was it like to come of age in the math competition circuit as one of the only women performing at that level?
MMW: It was difficult in a lot of ways, in different ways. Ways in which I felt like I didn’t belong. It was hard to be the only girl at a month-long summer training camp of 30-something students. It was also hard in ways in which I felt that there was a tremendous amount of attention and focus and pressure, because I was the only girl — as if, how I performed, I was representing girls as a whole. That was hard.
FM: How has women’s role in mathematics changed over the last two decades or so that you’ve been involved?
MMW: Frustratingly slowly. There has been some progress but the percentage, say, of math Ph.D.s that are women has been frustratingly stubborn. There are still many times when I’m the only woman in the room. I don’t want to forget that there has been some progress made. I think in general, there’s a lot more attention now to how we can make the field welcoming to everyone, to men and women and people from all sorts of backgrounds, and I think that’s an improvement. But still, the progress is slow.
FM: What initially drew you to math?
MMW: It was actually math competitions. In seventh grade, I got involved in this Mathcounts competition. And that was the first time I had seen problems where I hadn’t been taught how to do that problem. For me, math in school previously had been: they teach you how to do a problem, and you do the problems in the way that they taught you. What I really loved about the Mathcounts problems for me at that stage was that I hadn’t been taught how to do that particular problem. It was like a puzzle. You had to think and try different things, and then something would work. That was the hook for me into math, the opportunity to figure something out on my own.
FM: What continues to hold your interest?
MMW: Still that same thing. When I started college, I didn’t realize that there was still math that people had to figure out and that was something you could do as part of your job. I didn’t realize until later in college that that was something that my math professors were doing and that was a job. That’s still what I love. I love figuring things out that we didn’t know.
FM: Your work and your research has been lauded for its creativity. Where do you find the creativity? When one approach might fail, how do you find the next thing to turn to?
MMW: There are a lot of different sources of new ideas when the ones you already had weren’t working. Good collaborators is important, having someone to talk to and bounce ideas off of back and forth is really helpful. I often find it helpful to play devil’s advocate. If I’m trying to show that something is true and the things I’m doing aren’t working, I think, “Well, what if I were trying to show it’s false?” Even if I think it’s true, I try to find a counterexample. Sometimes, you can sort of feel where you’re being blocked to find a counterexample, maybe that’s the hint. For me,also bringing in perspectives from a lot of different kinds of mathematics, sometimes directly tools from other parts of mathematics but also sometimes just even the perspective, the way of looking at things, organizing things, thinking about them, has been useful.
FM: The MacArthur Fellowship ‘Genius Grant’ comes with a $800,000 “no strings attached” prize. How are you planning to use the grant for your research? Or is there anything else you’re planning on doing with it?
MMW: I’ll be honest: At first, I was imagining organizing conferences and workshops that were a little off the beaten path, that got people talking to each other and learning from each other more than traditionally happens. But upon further reflection, I realized that is something I can probably do through more traditional means and that this fellowship is a really special opportunity. Part of “no strings attached” was that there wasn’t a hurry to decide what to do with it. And so I’m now thinking that I don’t know yet, but that it's worth waiting and thinking and trying to find something that I really couldn’t do any other way.
FM: Tackling problems in number theory that have stood unanswered for decades, if not centuries, seems like a daunting task. What’s the day-to-day like of your job?
MMW: The day-to-day involves a lot of things that of course aren’t just tackling those problems. Working with students and mentoring and teaching and going to seminars and working on things with my colleagues and the department.
When I do sit down and research, for me, it’s a lot of filling sheafs and sheafs of paper. I write a lot and I try a lot of things. And if you saw me, I would just be filling pages and pages of different things and trying things, so perhaps not very exciting to watch. It’s slow, many of the things take many years. But the day-to-day is trying different ideas and looking for examples and following different leads and seeing where they go and all of that that physically comprises filling sheafs and sheafs of paper.
FM: Do you have any tangible goals for 2023?
MMW: I have a paper that I’m working on with a colleague at Columbia. It was a paper that we set out to write a couple of years ago but we kept finding that before we could finish that project, we needed to do another project first. So now we have completed three projects that we found in the way of the project we really wanted to finish. Now we’re working on that, and I’m hoping we can get it finished, though there was recently an article about this project in Quantum Magazine and my collaborator was quoted saying, “I don’t think this project is ever going to end.” So we’ll see. The things that we discovered along the way have been fun and exciting as well, so if it keeps happening that we discover surprising and exciting things along the way, I guess I’ll be thankful for that too.
FM: Are there any big open questions that you predict might be answered in the next 10 years? Or is that too short of a time span?
MMW: In terms of famous open questions, I don’t know of any that seem likely to be solved in the next 10 years. But one of the things about math is that sometimes it is hard to predict. You don’t know where the ideas might come from. It’s hard to say. Often, many big developments come as a surprise.
FM: Do you have any hidden talents or interests besides math that people might be surprised to learn?
MMW: The thing that I studied most in college after math was theater, particularly classical theater like Shakespeare.
FM: What's your favorite Shakespeare play? And what roles did you remember loving?
MMW: Perhaps the thing that really stands out to me most: I assistant-directed a production of “Macbeth” that I was very proud to be part of. Now, I only enjoy theater as an audience member.
FM: What was the last show you saw that was really memorable?
MMW: I just, actually, last Friday night took my kids to see “Hamilton.” That was maybe my fourth or fifth time seeing it.
My daughter and I were supposed to see it in 2020. We had gotten tickets, and they got canceled because of the pandemic.
FM: Dean Khurana of the College recently shared a resume of his rejections — projects that never came to fruition, grants, fellowships, and academic appointments that he was turned down for — as an exercise, and others have posted resumes of rejections or failures online as well. What’s one item on your resume of failures, and what advice do you have for students or researchers who are dealing with their own share?
MMW: I first think of many examples in my current life when I submit papers for publication. A lot of the process of submitting papers for publication is you get them rejected, and you try again. But to something closer to the student experience, when I was applying to grad school and applying for fellowships, I applied for a Defense Department graduate fellowship, and I didn’t get in. But I applied again the next year, and I applied for other things.
I think this is a nice idea to share since people share their successes but not their rejections and you don’t realize that everyone has lots of rejections. That’s part of the process of life. If you’re not ever getting rejections or failing, you’re not trying hard or interesting enough things. You have to keep trying and interpret that as a sign that you’re on a good path of challenging yourself and striving when you run into failures and rejection.
FM: There’s a lot on the internet about how mathematicians love their chalk. Do you have a favorite chalk?
MMW: Absolutely. When the news broke that Hagoromo was going out of business, I calculated what I thought would be my lifetime supply and purchased a personal lifetime supply of Hagoromo chalk. It turned out that they sold the formula to another company and so it was perhaps unnecessary, but just in case, I am prepared.
— Associate Magazine Editor Dina R. Zeldin can be reached at dina.zeldin@thecrimson.com.
