Spring/Summer 2007

Vol. 4, No. 2

Vol. 4, No. 2

An international standout in geometric analysis, Bill Minicozzi sees math where other people don’t—in staircases, the parking garage. “Like an artist,” says one colleague, “he needs to visualize things.”

On a Friday afternoon in early January, **Bill Minicozzi** holds a plastic baggie full of ice cubes against his left hand. Minicozzi has just returned to his fourth-floor office in Krieger Hall from the gym, where he plays basketball with a group of faculty and staff at lunchtime twice a week. He glances down at his homemade ice pack and grins. “An astrophysicist whacked my hand,” he explains.

Classes haven’t started yet for the spring term, but the J.J. Sylvester Professor of Mathematics still needs basketball to give him a mental break from his work. The board next to his desk is filled with equations and geometric figures drawn in colored chalk. Already today, before he headed to the gym, he and his research partner, Tobias Colding from NYU and MIT, have been on the phone, working through some problems.

Since joining the Krieger School’s math department in 1995, Minicozzi has made an international name for himself in geometric analysis. This field is the intersection of geometry and differential equations; it uses calculations to explain and describe shapes, curves, and surfaces. Minicozzi and Colding have focused much of their research on minimal surfaces, or those surfaces whose smallest pieces have the least area for their boundaries. “There’s nothing that happens in the field of minimal surfaces that doesn’t in some way take into account their work. That’s worldwide,” says Krieger School Math Department Chair **Richard Wentworth**.

It’s not something one would know to meet Minicozzi. He’s a clean-cut, unassuming guy who could probably pass for another graduate student when he isn’t standing in front of the class, holding the chalk. He commutes 55 minutes each way between his home in Washington, D.C., and the Homewood campus. He and his wife, Colleen Doherty-Minicozzi, have two kids, 4-year-old Tim and 2-year-old Nina. He also likes to read, but says these days he has time only for *Curious George* and *Winnie the Pooh*. “The thing that strikes people about Bill is how nice he is,” says Wentworth. “That doesn’t always go hand-in-hand with mathematicians. But he’s a regular guy. He’s the sort of person you grew up with.”

And he’s an enormously popular teacher. Three years ago, he was a finalist for Arts & Sciences’ Excellence in Teaching Award, and in 2004 and 2006, he got the Professor of the Year award given by the graduate students within the math department. “He can take a complicated thing and make it sound so easy for the moment, and then you go home later and realize how complicated it really was,” says Christine Breiner, a third-year graduate student.In addition to the chalkboard, Minicozzi also works on paper. He holds up a white legal pad. “It looks like doodling,” he says. He goes through one of these tablets every couple of days. Most of the work is still done the old-fashioned way: with pen and paper, chalk and brains.

Math is still a discipline where the work is done without much aid from technology. In addition to the chalkboard, Minicozzi works on paper. He holds up a white legal pad. “It looks like doodling,” he says, opening it to a piece of paper covered in equations. He goes through one of these tablets every couple of days. He and Colding do use software that draws shapes, but most of the work is still done the old-fashioned way: with pen and paper, chalk and brains.

Say the word math, and many of us think of numbers and arithmetic, and the way math progressed through our school years: first arithmetic, then algebra and geometry, and then—if we made it that far—calculus. But numbers are just one part of math, and math exists well beyond the classroom. Mathematics is a broader science; it’s the science of patterns. These patterns are everywhere in nature: in the symmetry of a snowflake, in the orbits of planets. It is the job of mathematicians to explain them. As Stanford mathematician and NPR “Math Guy” Keith Devlin has written, math can make the invisible visible. It took Newton’s equations to illustrate gravity and show us why an apple falls to the ground. It takes mathematics to explain what keeps a Boeing 767 in the air. Like poetry, math can make us see.

Which is why doing math is a creative act. Because they are working with what isn’t visible, not even with a microscope, mathematicians are left to imagine. “It takes a lot of lying on the sofa dreaming, with pictures in your head,” says Wentworth. “That’s especially true in Bill’s field. He does a lot of calculations, but he’s also got a huge geometric component. Like an artist, he needs to visualize things.”

When he’s wrestling with a problem, it can be hard to let it go. Minicozzi says he’s always thinking about problems: on his commute, in the shower, when he wakes up at night.

“I remember once, he solved a problem he’d been working on for years when we were on a walk,” says Colleen. “He said, ‘You know, I think I solved it.’ I said, ‘Have you been listening to me at all for the last 45 minutes?’ Actually, he had.”

Even getting out of his office does not get him away from the math, because Minicozzi can see math where other people don’t: “I see helicoids all over the place—parking garages, staircases.” (A helicoid is a twisted spiral shape that resembles a corkscrew.)

Often, the only way to stop thinking about it is to go play basketball. “And it doesn’t even work if I just go shoot around. It has to be a competitive game,” he says. “That’s the only way to get that stuff out of my head: hard physical exercise.”

For Minicozzi, the hardest thing about being a math guy is the uncertainty of trying to solve a problem that might not even be solvable. “The best feeling in math is when you finally see why something is so,” he says. “There really is no room for almost understanding or having a hunch. And once you solve it, everything seems simple.”For Minicozzi, the hardest thing about being a math guy is the uncertainty of trying to solve a problem that might not be solvable. Mathematicians can work for years on something that might not have an attainable answer. That’s also one reason solving a problem is so satisfying—that, and the glorious clarity a solution brings.

“The best feeling in math is when you finally see why something is so,” he says. “There really is no room for almost understanding or having a hunch. And once you solve it, everything seems simple.”

That was the case in his recent discovery about minimal surfaces. The most common example of a minimal surface is the soap film in a kids’ toy bubble wand. This simple kind of minimal surface, one that doesn’t curve too much and isn’t too large, is one mathematicians already understood. But the math behind complex minimal surfaces had largely eluded mathematicians for 250 years—until last July. That’s when Minicozzi and Colding published a paper in the Proceedings of the National Academy of Sciences showing that all embedded (not self-intersecting) minimal surfaces are built of simple shapes: pieces of planes, helicoids, and catenoids, another simple shape. “If you had suggested this to me when we were starting, I would have said it was too good to be true,” he says. “Good because it’s simple. Surfaces can be extremely complicated. Knowing you can break all of them down into these few simple pieces tells you a lot.”

Their discovery has huge implications for nanotechnologists, physicists, and materials scientists. Minicozzi is glad about that, but the application of his work is not what drives him. This is a guy who solves problems for their own sake. He’s hardly unique. “When that ‘aha’ moment arrives, every mathematician believes she is discovering a new truth just as physicists discover new particles or biologists a new enzyme,” says Wentworth.

When Minicozzi and Colding are trying to solve a big problem, they begin by breaking it down into steps. “We like to start with what we think is the weakest link, where there is the most chance of something going awry,” Minicozzi says.

Some weak links don’t take long to eliminate, but some do. He remembers a piece of one problem that took them about a year to solve. They were trying to prove something called the one-sided curvature estimate, and the key was to rule out a shape that looked like a helicoid but instead of spiraling straight up, it spiraled up and then back down. “We even gave it a name. We called it the string of pearls,” he says and draws on the board what looks like a series of cursive e’s. “We showed that this could not happen, and that in fact any embedded minimal disk would have to spiral vertically like the helicoid,” he says. “Toby and I figured it out together. The solution wasn’t simple, but we definitely had a sense of understanding.”

Minicozzi and math go way back. By third grade, he’d figured out that math was much easier for him than for other kids. As a freshman at Gonzaga College High School in Washington, D.C., he tied with a senior to win a school-wide math competition. “Math was unusually easy,” he says. “I would wile away time in religion class cooking up math problems to solve.”

In 1986, he entered Princeton as a declared math major. There, he met Colleen. They both went on to Stanford, with Colleen at the law school and Bill in the math department studying under Richard Schoen, a pre-eminent mathematician in the field of geometric analysis. “After a year or so, Bill was teaching me as much math as I was teaching him,” remembers Schoen. “He became more like a faculty colleague than a PhD student.”

When Minicozzi came to Hopkins in 1995, his hiring was part of a shift in the Hopkins Math Department, which had traditionally been focused on algebra and number theory. In the early 1990s, as geometric analysis underwent a renaissance, largely due to the work of Schoen and his doctoral advisor, Fields Medalist Shing-Tung Yau of Harvard, the department began to hire more people in that field. Today the group includes Minicozzi, Christopher Sogge, Joel Spruck, Steve Zelditch, Bernard Shiffman, Chicako Mese, and Wentworth. “That group is among the top two or three in the country, in my opinion,” says Schoen.

By the time he got to Hopkins, Minicozzi had also acquired his other life partner, so to speak: Colding. Between Stanford and coming to Hopkins, Minicozzi spent a year as a post-doc at the Courant Institute of Mathematical Sciences at New York University. Colding was there, too. Like many good relationships, theirs had a rocky start. When Colding arrived in New York, he didn’t have an apartment yet. The department secretary lent Colding the keys to Minicozzi’s apartment since he wasn’t due to arrive for a couple of days—or so she thought. “Bill found me sleeping on the floor,” remembers Colding, chuckling. “He probably thought I was a homeless person. He closed the door immediately, and then a couple of days later we met.”

That’s all it took. Their collaboration was successful from the get-go. In the mid-1990s, Colding and Minicozzi solved a conjecture of Yau, a polynomial growth conjecture of harmonic functions. After that, they started focusing on minimal surfaces. They have written more than 35 papers together. One of the biggies was related to Grigori Perelman’s proof of the Poincaré Conjecture, one of the seven “Millennium Problems” identified in 2000 by the Clay Mathematics Institute as the most important unsolved mathematical problems of the 21st century. Last year, Perelman was awarded the Fields Medal for solving the century-old conjecture. This conjecture is a topological question that, to oversimplify, seeks to mathematically distinguish an apple from a doughnut. Perelman posted his proof on the web in three separate papers. The third paper, posted in 2003, was a key step about something called finite extinction time. A couple of years earlier, at a dinner in New York, Perelman had asked Colding about the finite extinction time problem. Colding and Minicozzi got to work on it. Shortly after Perelman posted his own proof, the duo posted an alternative argument, which they later published in the April 2005 issue of the *Journal of the American Mathematical Society*.

Working together is common in mathematics, but the length and intensity of their collaboration is unusual. They e-mail back and forth constantly and talk several times a day. “They’re always on the phone,” says Minicozzi’s friend and colleague Chris Sogge. “I think they switched to the same cell phone company to save money.”

“We’ve been working together so long,” Minicozzi says, “we have the same thoughts now.”

Nowadays, he and Colding are sharing thoughts to see if what they learned about minimal surfaces in three-dimensional space, “the one we think we live in,” also applies to curved spaces. They already have a couple of papers in the works.

Minicozzi is excited about the discoveries that lie ahead. “My hope,” he says, “is that it’s like anything in basic science—you ask the simplest questions, then the answer has to be useful. It’s just gotta be.”

Kristi Birch is a science writer at Hopkins’ Center for Talented Youth.