Fifteen Questions: Curtis T. McMullen on Shared Truths, Unsolved Problems, and How to Illustrate Infinity

Professor Curtis T. McMullen received the 1998 Fields Medal, deemed the most prestigious award in mathematics, for his work on complex dynamics. McMullen is also the creator of the Illustrating Infinity exhibit that hangs in the Science Center atrium. This interview has been edited for length and clarity.

FM: Explain it to me like I’m five. What have you primarily been studying the last few years?

CTM: My mathematics is as much about questions as it is about results, formulas, images. I feel that finding the right kind of question to attack is a big part of crafting a beneficial and interesting and deep research program.

It’s easy to understand a lot of the questions that I’m trying to address, but to answer them brings in some of the deepest issues in modern mathematics. There’s lots of problems which are unsolved, and there’s lots of problems which we may never be able to solve. It’s an area which has an easy entry point, but it involves all sorts of branches of mathematics, algebra, number theory, analysis, dynamical systems, and especially the complex numbers — things like the square root of minus one.

FM: The way you talk about mathematics sounds pretty different from what I heard in high school algebra and intro calculus. Given that most people’s experience with math is limited to those high school and early college courses, what are the biggest misconceptions that you run into when you talk about being a professional mathematician?

CTM: We’ve been trained from an early age to think of math as something we have to learn, as opposed to something we might want to learn.

I think that’s the major misconception — that math is some sort of computational grind. Especially these days, with access to computers, I certainly don’t do difficult calculations by hand. Formulating questions? Developing mental imagery for things? This is much more important in research.

FM: Speaking of developing imagery, I walk past the Negatively Curved Crystals exhibit almost every day between classes in the Science Center, and I think they are beautiful and fascinating. What has the reaction to this art been over the last five years?

CTM: I felt that there was no presence of the math department, or really of any department, in the Science Center — on the ground floor, in the lobby, in the atrium.

I talked to the people who ran the Science Center, and I said, “Who’s in charge of this space?” They said “No one is in charge,” so I just got together some materials and guerilla-style took over the space. Every now and then, I see people standing in front of them or taking pictures, but I also think I just love having them in the space and seeing what it brings to the sense that you’re in a science center.

FM: Can you tell me more about the process that you actually used to generate the images?

CTM: I’d wanted to try this out for a long time. Of course, I had drawn computer pictures on my computer screen and printed them out on a laser printer. What I discovered right away is there’s something really different about working at a scale of, say, five feet versus a sheet of notebook paper, and seeing it from 20 feet away versus five feet. Very luckily, I got access through a friend to a plotter at Mass Art. This is something that’s used in architecture, and it allows you to make very large prints very inexpensively.

I used this plotter to get a feel for how things would look and to develop a lot of the parameters about cropping, about resolution, and things like that.

I decided to limit my options because there’s so many pictures one can possibly make. So, I decided to only make black and white prints and to only use circles in the images. That was the first iteration, and that was called negatively curved crystals because these circular patterns have to do with shapes in negatively curved space — shapes that form a repetitive pattern in the same way a salt crystal does.

FM: You mentioned that that was the first iteration, and I’ve also heard you describe the exhibit as ever-evolving. How has it changed in the last several years?

CTM: Every year I make some new images. I swap some images out. My own interests shift — I often like to try to introduce something that’s related to my current research.

All of the imagery aims to evoke what I think is one of the most interesting notions in mathematics, which is the idea of infinity. You see a pattern which is very rich, but in your imagination, you imagine it going on forever. Of course, I can’t have microscopic detail in my prints, but they’re meant to evoke this — it’s like an introduction to the concept of infinity. I hope the viewer then takes over in their own imagination and can, in their own way, feel the infinite complexity that I’m trying to allude to in these images.

FM: You’ve mentioned that the creation of these images reflects your current research. I’m curious how your perspective on mathematics, or on the concept of infinity, has changed as you’ve created artwork that illustrates these geometric rules?

CTM: First let me say, I usually don’t refer to this as artwork. I refer to it as graphics or as illustration. I think that illustration is, in fact, a very important facet of mathematical communication and even of mathematical research.

What’s great is that once you’ve developed the computer skills and the computational tools to render some of these images, you can then tweak them in a minor way and produce infinitely many different images. There were many things I wanted to see when I wrote my first program that generated these patterns of circles. It was in 1995, I wrote it to make one illustration that no one had ever seen before, and I was just tremendously excited when I finally saw this illustration. But I’ve now used that precise program for 30 years, and it’s extremely versatile.

FM: When do you think you first fell in love with mathematics?

CTM: I’m not in love with mathematics — I remember very well in fifth grade talking to an adult and being asked what was the subject I liked the least in school, and I said mathematics. And then they asked what I was the best at, and it was also mathematics.

What really got me excited was not mathematics, per se, but computers.

I was allowed to go visit the high school and try programming on their 10-character-per-second teletype, and that was just tremendously exciting. So I got very interested in computation, I think, before I knew any advanced mathematics.

FM: Do you think you were destined to be a mathematician?

CTM: No. I think I’m very lucky to have become a mathematician. There were several moments in my life where if luck had gone a different way, I don’t know where I would be, but not here.

My thesis advisor was not at Harvard — it was Dennis Sullivan, who was going between New York and France at the time. I had the great fortune that the chairman here at Harvard called him up on the phone and said, “We have a graduate student who’s interested in your work.” I flew down to meet him, he agreed to be my advisor, and he said, “Well, I guess you’ll have to come to Paris to study with me next fall.” I went from having no thesis advisor and no thesis topic to being on the verge of graduating in like four months.

FM: You are a highly decorated mathematician, from Fields Medal to Cabot Professor of Mathematics and so much more. What do you think has been the most rewarding moment of your career?

CTM: I want to mention a saying I learned from Lipman Bers, who was one of my mentors — a Latvian mathematician. He said, “Mathematics is something we do for the begrudging admiration of a few close friends.” I think I never expected more reward from mathematics than something like that. It’s something you do because you love it.

A truly amazing moment in my career was when my former graduate student Maryam Mirzakhani received the Fields Medal. To have a student be so highly recognized, and especially to become the first woman to receive the Fields Medal, was something I’m very proud to have been part of. I gave the laudation for her in Seoul, and I’m very glad to have had that as part of my life. I’m very sad that she died at such an early age. But I have to say that every time a student graduates or comes to me with a great idea, it’s almost the same as when Miriam won the Fields Medal.

FM: I love that answer, and I actually wanted to ask you more about that. You have this role as a mentor and a teacher to future generations of mathematicians, and I’m curious what it’s like to contribute to the lineage of your field’s community in that way.

CTM: One of the great things about being at the various universities where I’ve taught, including Harvard, is just coming into contact with so many students who have such incredible potential. My role is, I feel, really just to guide them into a fruitful beginning of their own research trajectory.

I’m helping my students cultivate their own creativity, pick challenges that are deep and interesting, learn how to formulate interesting questions, and learn what to do when a question is too hard for you.

Every student is different, and every student is rewarding to have.

FM: When you’re working with these students, or you’re teaching your math classes, what topics tend to be the most difficult to communicate to students about math?

CTM: I think that one of the things that’s difficult to communicate, especially to undergraduates and to people in general, is how alive the subject of mathematics is — that it’s not something that’s finished.

FM: What do you feel like are the biggest questions or unknowns that are guiding research right now?

CTM: There’s a narrative about mathematics which centers famous unsolved problems, extraordinary individual accomplishments, and the idea of genius. That’s a very exclusive narrative. It makes it sound like there’s only a handful of the chosen who can really do mathematics, and that the only thing we’re really concerned about is climbing the next Mount Everest. This is not how mathematics actually works.

It’s about developing a human understanding of an abstract field.

I think one of the most remarkable things about mathematics is that it’s evolved for centuries. Some parts of mathematics are like a symphony that’s been written in part by a reclusive Russian, in part by an elite Frenchman, and in part by an American Quaker. At all at different points in time, they’ve all made contributions to it. The fact that through the miracle of culture and language, we’re able to communicate what we know so far to the next generation, is what makes it this wonderfully cooperative societal project.

It’s really not about just discovering what’s true. It’s about investigating various arenas of intellectual endeavor and finding productive ways of picking up problems in that area and turning them around, seeing them from different perspectives, becoming an expert in them — more so than summiting any particular peak.

FM: How do you stay motivated to continue researching when you’re facing these questions that maybe you don’t even know whether or not it’s possible to solve them?

CTM: Well, you don’t just keep looking at a blank blackboard day after day.

Some people talk about the obstacles to solving math problems as being psychological. I think what they mean by that is a lot of times we have preconceptions about how things work, and we have to let go of those because they may be wrong.

There’s a very simple rule about solving problems, which is that no matter what question you have, there is a simpler question which you can answer. Your job is to find that question and then answer it, and then you can formulate the next question.

FM: What were you up to during your sabbatical at Berkeley?

CTM: I actually spent a lot of time writing a new paper about a certain fractal shape and its relation to an object invented in 1904 called Minkowski’s Question Mark Function.

I love writing, and it’s also very difficult. Having a sabbatical is the ideal situation because you don’t lose the content that you already have in your mind — you’re immersed in the project.

FM: We’ve talked a lot today about how mathematical problems are psychological and about how math is partly finding a human understanding of this abstract field. I'm curious if we can flip it — I’d love to hear from you how math has changed your perspective on the world outside of your research.

CTM: What I have come to feel over many decades of being involved in it, is that mathematics is most real at the moment it’s being communicated from one person to another. I think mathematics has helped me to appreciate the importance of communication and also to be very sensitive to miscommunication.

The second thing that I feel I really came to appreciate the value of in mathematics is the importance of truth. Mathematics is probably the field in which there’s the most universal unanimity about what’s true and what’s not true. It’s miraculous that people from all different cultures — from different centuries — all agree on what is correct in math and what’s not correct.

I feel it’s unfortunate that Harvard dropped mathematics as one of its undergraduate requirements — not that people need the actual material in some particular math class, but they need exposure to the idea that there are domains of thought where there are absolutely correct answers. Especially in an era where truth itself is under attack, it’s very important to have some grounding, somewhere.

— Associate Magazine Editor Kate J. Kaufman can be reached at kate.kaufman@thecrimson.com