# First hired for his brilliant mathematical work, he’s now also recognized as an exceptional educator.

By MIT OpenCourseWare

When Professor Haynes Miller conducts office hours for his undergraduate course in differential equations, the room usually resembles a college sit-in, filled with twenty or more students settled across the floor and quietly doing their homework. “Inevitably,” he relates, “a few groups self-organize to start working out problems together on the chalkboard.”

Miller is soft-spoken, and a strong believer in allowing his students to work through challenges unassisted, so he often silently observes their progress from his desk. “Of course, they keep looking back at me for approval, trying to judge how they’re doing,” he chuckles. “One day, one of my students drew a hyperbola on the board after solving a tough equation, which showed the relationship between my face and their mastery of the problem. My expression moves from smiling amusement at their struggles—learning is underway!—to a calm kind of serenity when they finally get the answer.”

The dry, academic wit behind Miller’s story conveys some of the depth and warmth that he brings to a domain that is traditionally considered quite chilly. “The Hollywood perception of mathematics is that it’s very cold and austere. But it’s really one of the most human of all sciences. Mathematics just wouldn’t exist without people, and most of its progress has only occurred through human collaboration. Take chemistry and biology—you could reasonably say that they would exist whether humans studied them or not. But mathematics, because it’s purely conceptual, depends entirely on the thought of human beings.”

“The Hollywood perception of mathematics is that it’s very cold and austere. But it’s really one of the most human of all sciences. Mathematics just wouldn’t exist without people, and most of its progress has only occurred through human collaboration.”

Such a philosophical defense of mathematics makes perfect sense coming from the son of an English professor who’s currently wending his way through Shakespeare’s histories (he’s on *Henry IV, Pt. 2*) and is an avid birdwatcher (the elusive Connecticut Warbler is on the top of his list). His love for teaching has won him the respect of both colleagues and students: Miller has served as Associate Department Head of MIT’s Department of Mathematics, and in 2005 he received the prestigious appointment of MacVicar Faculty Fellow for his “exemplary and sustained contributions to teaching.”

## “I Guess I Feel Lucky”

Growing up in Port Washington, New York, Miller showed an early interest in both math and science, and fondly recalls several NSF-funded programs that allowed him to visit both Columbia University and Notre Dame to study advanced mathematics while still in high school. Upon arriving at Harvard as an undergraduate, he still felt undecided about his major until he fell into the orbit of Raoul Bott, a legendary mathematics professor whose work profoundly changed the landscape of geometry and topology. Later, while earning his doctorate at Princeton, he studied under another mathematics luminary, John Moore. While Moore ran a highly influential seminar that attracted many of the brightest students in his generation, Miller remembers that “a lot of really exciting topology work was also going on at MIT under people like Dan Quillen. There was one postdoc at Princeton who came from MIT, and he brought some truly fascinating ideas along with him.”

Miller spent a decade teaching at the University of Washington and Notre Dame before being invited to join MIT as a professor in 1986. His elegant 1984 proof for the Sullivan Conjecture undoubtedly played a major role in that decision. For those of us who have not yet earned our Ph.D. in algebraic topology, Miller describes its relevance: “Traditional topology exists in higher dimensions, where things get more complicated and diverse. Algebra becomes a bookkeeping device to keep track of all the space in those different dimensions. What the Sullivan Conjecture proved is that at these higher dimensions there are some classes of space that cannot be compared, because they are too radically different.”

Although his proof opened up entirely new fields of mathematical research, Miller views his work with characteristic modesty. “I guess I feel lucky that this proof was useful to other people. For me, I think Andrew Wiles (who solved Fermat’s Last Theorem in 1995) probably said it best. He said that when you’re solving a complex problem, you end up stumbling around a room in the dark for a long while, maybe months, until you find a light switch that illuminates the entire room all at once. That’s the real thrill—that emotional sense of breakthrough. But to be honest, there are new wings to this mansion opening up all the time. The rate of advance is pretty constant.”

## A Dedication to Teaching

Despite his achievements, Miller clearly prefers to focus on teaching. He speaks with admiration for both the graduate and undergraduate students he instructs each year. “One of the first things I noticed when I came to MIT was the quality of the students,” he remarks, “I was lucky enough at the time to work with a professor named Frank Peterson, who ran a seminar here, and he handed the keys over to me, which meant that I had all the fun of talking with the students, with none of the responsibility.”

Miller’s dedication to teaching—while advancing the field of mathematics at MIT and beyond—comes through best in one of his recent endeavors. In collaboration with Karen Willcox from the Department of Aeronautics & Astronautics, he developed a tool called Crosslinks, that is designed to promote deeper cross-domain awareness of how certain concepts are taught and used in different departments. “As professors,” he explains, “we don’t always do a good job stepping outside our bubbles, to make sure that we are teaching in a way that helps students apply that knowledge to other domains, so I’ve been working with other faculty to find new ways to bridge that gap.”

Miller’s students have bestowed their own dubious recognition upon him in typical MIT fashion. He was a recent runner-up for the annual “Big Screw” award, which usually goes to professors whose courses are both popular and outrageously difficult. Miller, of course, is savvy enough to play right along. He shakes the jar of wood screws that he received as a consolation prize with mock astonishment: “The winner gets a giant, four-foot-long, aluminum, left-handed wood screw, but all I get are these defective screws,” he pauses deadpan, “—they’re all right-handed!”

Jokes aside, Miller knows that the jar symbolizes his students’ respect for his dedication to teaching. Indeed, it would be hard to find a better proponent of the joys of mathematics: “One of the true pleasures of teaching is helping students achieve small breakthroughs in their thinking. By creating mathematical exercises, you stimulate that kind of experience for them, and you start to open their eyes to the possibilities for new discoveries in mathematics. That’s precisely why I’m such a big fan of OCW as well, because it allows you to expand the experience to an even broader audience.”