Joseph D. Harris ’72 is a mathematician in the field of algebraic geometry and Harvard’s Math 55 professor.
Joseph D. Harris ’72 is a mathematician in the field of algebraic geometry and Harvard’s Math 55 professor. By Jennifer Z. Liang

Fifteen Questions: Joe Harris ’72 on Math 55, the Dudley Co-op, and Failure

The mathematician sat down with Fifteen Minutes to discuss Math 55’s notorious reputation and his own experience at Harvard. “In math, it’s rare that you would decide to fix on a specific concrete goal, and then either achieve it or not,” he says. “Usually, it’s a matter of exploration.”
By Michal Goldstein

Joseph D. Harris ’72 is a mathematician in the field of algebraic geometry and Harvard’s Math 55 professor.

This interview has been edited for length and clarity.

FM: You’ve been studying math for the majority of your life. What do you think initially drew you to the subject?

JDH: I don’t know. I’ve wondered about that myself. I think it might have been that my father wanted to be a mathematician, but he was growing up during the Depression and that wasn't an option. Among other things, he was Jewish, and at that time, universities, by and large, still had quotas on the number of Jews that they would hire. In any case, he wound up going into medicine, but I suspect he always maintained that interest in math and somehow communicated it to me before I knew any better.

FM: What part of your research into algebraic geometry is most exciting to you? And can you explain it in a way that would make sense to a high school math student?

JDH: Algebraic geometry, it’s something that you’ve encountered already, I suspect. When you describe a geometric object like a circle as a locus of points in the plane xy, whose coordinates satisfy a polynomial equation. So in this case, x squared plus y squared equals 1. And the question of algebraic geometry in general is what’s the relationship between the algebra of the polynomials and the geometry of the figure that they describe. The subject has undergone a lot of revisions, but that’s fundamentally the question.

FM: In the past, you’ve said that nothing you think about could not have been imagined by the Italian geometers who were foundational to the field of algebraic geometry, and that if you enjoy greater success than them, it’s because you have access to better tools. Could you elaborate on this and how technological development has been helpful?

JDH: I think in very rough terms, in the 19th century, when the subject really got underway, people still had a relatively informal style. And ultimately, that reached its limits, and a large part of the 20th century was devoted to putting the subject on a more rigorous foundation. And the result of that was the introduction of a lot of new ideas and techniques that made it possible to revisit some of the problems that they had talked about or thought about in the 19th century, but were unable to resolve then. In some sense, that’s how I make my living.

FM: More so than ever before, today many students opt for what they consider “practical” concentrations that can easily be applied to careers. Why do you believe that pure math should still hold a place in higher education, despite its abstractness?

JDH: There is, certainly, applications of mathematics to adjacent fields — computer science, physics, even economics — and I can get out of the question by citing those. But in fact, I think there’s also a framework of mind that mathematics teaches that I think is valuable, no matter what you go into.

FM: And what do you mean by that — “the framework that it teaches?”

JDH: It’s a very, very strict logical structure, that things are either true or not true, and they’re either provable or not provable. And occasionally, by dint of effort, we can move something from the first column to the second column, or rather from the second column to the first. But, the basic idea that there is a logical structure that is determinative — is fundamental — to the subject, and I think it’s a useful framework to have in other respects.

FM: Math 55 is known as one of the hardest undergraduate classes at Harvard — and in the country. Do you think that Math 55’s notorious reputation is accurate?

JDH: No. I became aware last year that Math 55 was having a moment on social media, and I looked at some of the videos and things, and they’re wildly inaccurate. The fact is, it is a very hard, challenging course. We cover a lot of material, and we do so in what I think is pretty rigorous fashion. But we have the students. The students who show up in 55, they can handle it. They can do it. So I don’t think it's that much of a killer course as it’s made out to be.

FM: How would you describe the students that are drawn to the course — aside from their passion for math? Do you worry that capable students who don’t think they fit that mold turn themselves away from the course?

JDH: Well, we try to make it accessible to everybody. Especially if you haven't seen a lot of the material already in one form or another, I suspect it probably would be pretty time consuming. But I think basically, we’ve encouraged everybody to just give it a try, come to a few lectures. The math department — I don’t know how it is in other departments — but we were all unanimously very upset at the end of shopping period.

Because that was what we would tell students — if you’re not sure if you want to take 55, come to the first week of lectures and then decide. But I think the students who do take it and succeed, they’re mostly kids who were sort of outliers in their high school. And I think for them, what's really a wonderful experience is being in a community of peers, for the first time for most of them.

FM: In 2017, The Crimson published an article about the Math Department at Harvard and the gender divide within it, and it stated that less than 7 percent of Math 55 students were women between the years of 2012 and 2017. How does that statistic compare to today, as far as you know? And how do you think the math department should or is expanding its reach to female students?

JDH: Wow, 7 percent, you said?

FM: Less than 7 percent, in those five years.

JDH: That’s fairly upsetting. But I have to say that it’s not like that. We’re not up to 50 percent, but we’re getting there. I think we’ve just basically tried to make the course as accessible to everyone as we possibly can. And I don’t know. I think we’re succeeding. But I’m not the one to judge.

FM: Despite the difficulty of Math 55, many students laud the wonderful community they are able to build through the class. How have you built, or seen students build, that community in your own class?

JDH: I think the idea is that — and this is something we really encourage the students to do — if the students will talk to each other about basically the material and specifically about the problem sets that are due every week. The course is based on weekly problem sets, and students can talk about them in section and office hours and with each other.

It really doesn’t take long in my experience for the students to get to know each other, or at least a subset of the rest of the class that they can talk to and work with. That’s, in fact, I think that's the great thing about the course is exactly building that community.

FM: You went to Harvard as an undergraduate. Did you take Math 55 when you were here?

JDH: I did, but it was a completely different course back then. It didn’t serve the same purpose either. One thing that’s changed since back then is that students coming out of high school have a much more solid foundation in abstract mathematics than they did back then. So we can do very different things in 55 now than we did back then.

FM: What house were you in?

JDH: I was in Adams House for a year and then I moved off campus so I was officially Dudley. And I wound up my last semester at the Co-op house.

FM: Did you like it?

JDH: Oh, I loved it. That was where I felt I belonged, somehow.

FM: So you’ve been here for a long time. Do you have a favorite hidden gem location at Harvard or in the Square?

JDH: I used to like to hang out in the stacks in Widener, but I haven’t been there for decades now.

You just wander around and pick up random books and see what they’re about.

FM: Often, our 15 Questions series focuses on the successes of the people that we profile. Can you share one of your failures?

JDH: Well, it’s funny because in math, it’s rare that you would decide to fix on a specific concrete goal, and then either achieve it or not. Usually, it’s a matter of exploration. And if you think of it in those terms, you know, you’re trying to explore new territory. There really isn’t a notion of success or failure. If you’re trying to climb a mountain then you either get to the top or you don’t, right. But here it’s more a matter of just wandering around and seeing what you can find.

FM: If you weren’t a mathematician, what do you think you would be doing instead?

JDH: I’ve fantasized about being a cook. But that’s, you know, that’s a fantasy.

— Magazine Editor-at-Large Michal Goldstein can be reached at Follow her on Twitter @bymgoldstein.

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