Indiana University  Research & Creative Activity Spring 2002 • Volume XXIV Number 3


A
BEAUTIFUL
EQUATION

by Lauren J. Bryant

Peter Sternberg and Michael Jolly hate telling people what they do. “You just …” Sternberg stops. “Even my wife just doesn’t ask.”

“At parties,” says Jolly with a big shrug, “when you tell someone what you do, they just say, ‘Well that was my worst subject,’ and that’s the end of that.”

Jolly and Sternberg are mathematicians. They specialize in partial differential equations. If you type “partial differential equations made easy” into a Web search engine like Yahoo! or Google, you’ll find a “Guide to Math Needed to Study Physics,” where you’ll learn this: “For doing physics in more than one dimension, it becomes necessary to use partial derivatives and hence, partial differential equations.”

photo of Peter Sternberg photo of Michael Jolly
Peter Sternberg, left, and Michael Jolly study problems in applied mathematics as part of the Indiana University Bloomington Department of Mathematics faculty. photos © 2001 Tyagan Miller

Reading that, most of us would probably think, “Well, that’s the end of that.” But catch Sternberg and Jolly after class or over a bagel, and you’ll learn some fascinating things about mathematics.

PLAYING AROUND

Peter Sternberg is about to give a quiz in M119, Indiana University Bloomington’s broadly required and often feared course in introductory calculus. In his blue jeans, high-top tennis shoes, and plaid shirt, the bearded Sternberg looks close in age to his students, but he’s been a member of IUB’s math department since 1988. Over the years, he’s taught M119 many times. It’s a huge course with hundreds of students, so he’s learned “to be a little bit of a disciplinarian,” he says, “especially in the first weeks.”

But Sternberg’s hardly a tough guy. As he preps students for the quiz, he’s enthusiastic and encouraging (“Don’t be embarrassed, come ask me!”), lending out his own calculator and allowing some students (“just for today, I’ll make an exception”) to share theirs. As the quiz begins, lip-chewing silence fills the room, and Sternberg lopes up and down the aisles helping stuck students with a smile or a quick hand to a shoulder.

Sternberg sympathizes with being stuck. “Math is a very tough profession in that regard,” he says. “I may spend a year on something, get nowhere, and then, in 20 minutes while I’m helping my kid learn to ride a bicycle, I suddenly hit on it.
“You have to go down all these dead ends,” he continues, “until finally, hopefully, things work out. But I would say, in math, you’re frustrated by a problem most of the time, which may be when you’re working hardest of all.”

Sternberg is an applied mathematician, so physics and engineering furnish most of the problems he considers. One of those is superconductivity. When chilled to extraordinarily cold temperatures, certain materials conduct an electrical current in a “super” way—it never dies or dissipates. But achieving the extreme cold needed for superconductivity requires a huge energy output, and that’s a major drawback. Still, never-ending currents have some monster applications, including biomedical technologies, high-energy physics, transportation (trains that can “float”), and high-speed data communications.

Estimates put the worldwide market for superconductor products at $90 billion or more by 2010. But first, scientists have to figure out how to achieve superconductivity at much warmer temperatures.

Which is where the math comes in.

Sternberg uses what are called the Ginzburg-Landau equations, a model constructed by physicists to describe the properties of superconductors. Working through these equations can reveal, for instance, just how cold you have to make a material to achieve superconductivity.

It may surprise non-mathematicians to learn that Sternberg never really “solves” the equations he deploys.

“Mathematical researchers don’t usually solve an equation,” he explains. “You don’t say ‘the answer is 1.375.’ Instead, you say, for example, ‘There is a solution’ or ‘There isn’t.’”

What kind of answer is “there-is-no-answer”? A useful one, according to Sternberg: “If you say ‘there isn’t,’ you send a message to the physicist, ‘You might have messed up here, it’s a bad model.’ Because we know it’s supposed to be a superconductor, but there’s no solution.”

Sternberg also uses mathematical theory to estimate the general qualities a solution might have, such as its size (“it gets really big”) or its shape (“it oscillates a lot”). He looks for the optimal shape of a superconducting material. In some of his research, supported by the National Science Foundation, Sternberg has found that lots of wiggles or ridges in a material’s boundary favor superconductivity—the current makes its way through a material at a warmer temperature.

For all of its high-tech applications, however, Sternberg’s research is decidedly low-tech. “I need a pencil, a napkin, and three numbers,” he says. “A zero, a one, and occasionally a two. Those are the only numbers I ever use.”

But use them he does, as his pathway to a world he likens to a vast playground for the mind. Sitting beneath a poster of a Jackson Pollock painting hung on his office wall, Sternberg talks, somewhat shyly, about the creative side of math.

“It sounds very pompous, but being a mathematician may be closer to being an artist than it is to, say, being a chemist or a biologist,” he says. “Yes, you’re given rules of how things can be added and subtracted and whatnot, but then you can say, ‘OK, given these rules, let’s play with this. Let’s play around and see where it leads.’”

This spontaneous pursuit of new ideas sustains Sternberg in a world where few people understand what he does.

“Often the answer to the question ‘Why am I doing this?’ is that it looks fun, it looks interesting,” he says. “A lot of times, mathematicians just play around and they’re happy, even if nobody else in the world cares.
“Mostly, we don’t do this work so that we can ‘build a better bridge.’ Thirty years later, someone might realize, ‘Oh, this helps describe genetics,’ but that’s almost by accident.”

AN OBJECT OF DESIRE

Mike Jolly builds a really big bagel sandwich. His roast beef-Swiss cheese-lettuce-cucumber-carrots-green peppers-black olives construction gets topped by a small spoonful of capers. (“The capers are what make it,” he says.)

Jolly is a lean, outdoorsy type who wears Birkenstocks and cites the Beatles during classes on differential equations, and his research is as unusual as his bagel sandwiches.

No pencils and napkins for Jolly, who describes himself as “one of the first people hired [in IUB’s math department] who would admit to using a computer as part of his research.” Today, he uses high-speed computers “in an intimate way,” he says, to construct intricate, sinuous visualizations of possible solutions to certain mathematical equations. In one recent article, he even ventured a bit of imagery rare to academic math; he named a visualization after a Viking ship.

“I was working with a colleague at Princeton on a project that had gone on a long time,” Jolly explains. “He kept describing the computer images we were coming up with as ‘a cup’ and ‘a boat.’ We needed some motivation to finish the paper so I got on the Web and started looking around for pictures. Up came this image of a Viking ship, and I thought, that’s it!”
The ship was one recovered in an archaeological dig in Oseberg, Norway, in 1903. The title of the co-authored paper became “The Oseberg Transition.”

Jolly discovered his math-and-machine niche in his final year of a mathematics doctorate at University of Minnesota. During a summer project at the Los Alamos Laboratory in New Mexico, he found he liked “being very computational.”

“I discovered right away that there was a lot of computational stuff I could do to support rigorous applied mathematics,” he says.

Jolly notes that since he joined IUB in 1989, computer use in applied mathematics research has steadily increased. “It’s a growing area called scientific computing,” he says. “Because computers have become so much more powerful, we can now consider a lot of problems—much bigger problems like how air flows around an airplane—that we couldn’t fully consider before.”

Sometimes, Jolly adds, his computational side leads him back to the theoretical.

“I take various equations as a given and try to simulate them on the computer,” he says, “but I also use the results from computational experiments to raise new questions. Sometimes the computer can guide you toward a purely mathematical result, which is always the goal in mathematics.”

To begin to grasp what Jolly does, he suggests imagining the earth’s atmosphere as seen from space. Each point in the volume of that swirling mass is described by variables such as temperature, velocity, and air pressure, adding up to an incalculable amount of information. “One simple but critical fact,” Jolly says, “is that there are an infinite number of points in that volume, so the system (the atmosphere) can be described only by an infinite amount of information.”

Computer simulations of partial differential equations are typically based on approximations, using a large but finite amount of information. The approach Jolly uses reduces a problem to a small amount of information (at the cost of complexity). This allows for certain phenomena to be visualized.

In particular, Jolly works on developing alternative approaches to systems that change over time. With support from the National Science Foundation, his research has focused on combustion, fluid flow, and turbulence.

Turbulence—the chaotic motion of particles which most of us recognize as the agitated air flowing around airplanes—is Jolly’s newest area of research. He is collaborating with Ciprian Foias, a Distinguished Professor of mathematics at IUB whose studies of turbulence have had a notable impact in the field. Foias has established mathematical backing for some of the theories physicists have come up with to describe turbulent flow.

“Foias has had some really good insights lately,” Jolly says. “The proof is much more complete than he ever thought it would be. My contribution is to help write up the mathematical results he generates and carry out simulations, which may lead to more results.”

As he describes his research, Jolly sounds like a poet, a philosopher, and sometimes, a secret admirer. Talking about how variables in a system change (such as temperature and air pressure in the atmosphere), he says the variables trace out trajectories “like trails of smoke in the sky of the three-dimensional space we call home.” The trajectories eventually settle on something mathematicians call a global attractor, “a collection of all long-time behavior, that which persists,” says Jolly. He calls the global attractor “the main object of my desire.” Although the global attractor can be located by following trajectories forward in time, Jolly says he is “trying to find the attractor by some other means than just following trajectories and lurching toward it.”

It can be an unrequited search. Like his colleague and friend Peter Sternberg, Jolly has pursued projects that simply don’t work. “I leave them and try to come back to them, but it’s so painful trying to get back to where I left off,” he says.
But for research mathematicians, the right answer isn’t the prize. What motivates Jolly, and Sternberg too, is the thrill of the chase, the addiction to the unexpected, the pleasures of the new.

“It’s gratifying to visualize complicated mathematical phenomena on a computer,” Jolly says. “I see a lot of exploration I can do.”

As Sternberg puts it, “A lot of it is just the sheer intellectual pursuit of seeing where you can go.”

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