By Tommy Tomlinson
The Charlotte Observer
| November 16, 2003

QUESTION ONE: You decide you want to solve a math problem that’s so hard, no one’s come close in 25 years. How do you begin?

John Swallow began four and a half years ago, 6,000 miles from home, staring out the window of a bus.

He’s the guy you want next to you on the bus seat. Friendly but quiet. You might not remember him later, unless you glanced over when he’d just thought of something, and you saw his left eyebrow rise over the rim of his glasses.

Swallow started writing computer code when he was 7. He aced college calculus when he was 13. He entered grad school at Yale when he was 19.

But here he was at 28, in Haifa, Israel, with a problem that clogged his mind like kitchen sludge.

Swallow teaches at Davidson College. He went to Israel on a working sabbatical to trade ideas with a professor named Jack Sonn. Swallow and Sonn are two of maybe 100 people in the world who are experts in their particular side street of math.

They study algebra at its highest levels. They work with sets of numbers called Brauer groups, named for a Jewish mathematician who left Germany when Hitler took over. They apply ideas based on Galois theory, named for a 19th-century French mathematician who died in a duel.

The work has practical uses, such as cryptography — the making and breaking of codes. But to Swallow, it combines the things he loves about math: the beautiful patterns in numbers, and the challenge of seeing how far his skill and imagination can stretch.

In Israel, Swallow and Sonn spent a semester warming up with some minor theorems. Then one day Sonn suggested a problem that other experts in their field had thought of back in the ’70s. Many mathematicians worked on it into the ’80s — Sonn among them — but no one ever came up with an answer.

In every branch of math there are problems no one has ever solved. They are numerical shipwrecks. If you dive deep enough you could find treasure. But you might spend years and come out with nothing.

The problem Sonn suggested involves analyzing two Brauer groups — huge algebraic structures, whole fields of numbers — and trying to show that they’re the same.

The numbers in Brauer groups aren’t just the ones you use to balance the checkbook. They’re irrational numbers (like the square root of 2) that can’t be reduced to a fraction. They’re even imaginary numbers (like the square root of -1) that don’t show up on a calculator.

Swallow came to think of his problem as comparing two forests. They look exactly alike. The heights of the trees match. But to prove that they’re identical, you have to get down to every needle and every hunk of bark.

He had never worked on a problem that required so many techniques, so many new ideas, so much brainpower.

For months he sat in Sonn’s office every afternoon, the two of them staring at the blackboard, sometimes for so long that Sonn would doze off.

At night Swallow rode home on a city bus. The other passengers chatted in Hebrew or Arabic, languages he didn’t understand. Swallow thought about all the rest he didn't understand, the equations on the blackboard, the numbers skittering out of reach.

He wondered if he had come this far only to find something he had never run into: a problem that was stronger than his mind.

QUESTION TWO: You’re struggling to solve a math problem that’s so hard, no one’s come close in 25 years. You also have a normal life. How do you balance the world inside your head with the one outside?

Cameron Swallow calls it “the math-problem expression.” She describes it as “an abstracted gazing into the middle distance.”

She first saw it in her husband half their lives ago. They met in choir practice at the University of the South in Tennessee. She was a freshman at 17. He was already a sophomore at 16.

They both loved math and music and English literature. They had long romantic talks about quadratic equations. He had enough credits to graduate early, but he stayed an extra year — partly to finish off a double major, partly to be with her. They got married in 1991, when John was at Yale.

In 1994 they came to Davidson. Soon they had a daughter, Ruth, and then another, Sophie. The talk shifted to whose turn it was to change diapers and when to buy the minivan. John became, as Cameron puts it, the Kitchen Spouse. He makes a mean Reuben sandwich in the Crock-Pot.

In Israel their kids were still small, so John and Cameron had time to talk about the math problem. But he and Sonn weren’t getting far. They had spent six months digging and hadn't hit anything solid. And it was time for the Swallows to go back to Davidson.

Swallow resumed his regular life — teaching during the day, spending time with family at night. He worked on the problem in spare hours, between classes and church services and oil changes. He filled sheets of paper with equations next to phone numbers for the DMV.

Sonn had gone on to other projects. Swallow worked by himself for months. But he kept getting stuck. He thought of it like trying to lay carpet that was too small for the room. Every time he got one corner to fit, another would pop loose.

He worried that he had lost his confidence, lost his aggressiveness, lost his faith.

He put down the problem for nearly a year.

He taught, traveled, read to his daughters. He got in touch with a Canadian collaborator. They worked on a smaller problem that they wrapped up in a few months.

When they were done Swallow went back to the folder in his file cabinet, the one marked ‘Current’ Research.

It was filled with copied pages from textbooks, scribbles on graph paper, half-finished thoughts on index cards. The Brauer groups, those two huge fields of numbers, ran all over the pages. He was sure they were the same. But he had to prove it.

He read the notes over and dug in again.

He spread out his work on a table at Summit Coffee across from the Davidson campus. He tried out theories in his head as he drove back from family visits, Cameron and the kids asleep in the minivan, a band called String Cheese Incident playing on the stereo.

Sometimes he forgot what he’d already done and repeated the mistakes he and Sonn made in Israel. Sometimes he worked for days and ended up back at the same wrong place.

But then he thought about the smaller problem he’d already finished. He realized that some of that work overlapped.

He still wasn’t getting far. He wasn’t even doing enough to call his progress slow and steady. Slow and unsteady, maybe.

Still, after two and a half years, the stubborn numbers in Swallow’s head began to shift a little.

His eyebrow rose.

QUESTION THREE: You’ve spent countless hours trying to solve a math problem that’s so hard, no one’s come close in 25 years. What will it take to finally break through?

The speaker was boring. Worse yet, he was boring in French. By now it was July 2001. Swallow had come to Lille, France, north of Paris, for a math conference. He knows French. But this guy at the front of the lecture hall was talking so fast that Swallow couldn't understand half the words, and didn't care about the rest.

Eventually he gave up. He reached over and pulled out his notes on the problem he and Jack Sonn had been working on.

All of a sudden a fresh thought flashed in his mind. He grabbed a pen and wrote one word.


He followed that word with a string of equations that set new limits on the number fields.

Maybe if he put just a few restrictions on the problem, narrowed the scope just a bit, it would work. He went back to the idea of trying to lay a carpet that’s too small for the room. Maybe the answer was to make the room a little smaller.

For the next couple of days he did calculations in every spare moment. The numbers were lining up, making graceful curves on his worksheets. But there were still places where the numbers strayed.

One morning Swallow skipped the conference and went looking for coffee. He ended up in a shopping center and found a table in a restaurant called Quick — a European version of McDonald’s.

He doesn’t remember much about the scene around him. The steam coming off the coffee. A woman pushing a baby stroller.

Then, another flash.

All along he had struggled with a few key places where the two number fields could have been different. If they were the same, he could apply an equation to both fields and the two sides would add up to zero. But one side always came up with the wrong result.

This time Swallow tried a new technique, something he’d never thought of before that moment in the fast-food joint in France.

It was as if he had been trying to train a dog for months, and the dog finally came.

Lots of dogs. Whole fields of them. The numbers lined up and sat still. Swallow applied an equation to both Brauer groups. Did the calculations.

They added up to zero. Perfect balance. He had made it down to the needle and the bark. The forests were the same.

Swallow still had to try his new thoughts on other parts of the problem. He still had to recheck his calculations. He still had to trust himself. He went back to Davidson. His wife noticed the old “math-problem expression.“ Swallow ran through the steps of his solution over and over until he felt sure.

In the fall of ’01 he sent Jack Sonn a draft of the solution. For the next six months they e-mailed back and forth, challenging each other’s ideas, getting stuck and starting over. Swallow had to refine his work, make the path to the answer more clear.

The revisions took more than a year. In November 2002, Swallow sent Sonn a draft that contained all the changes. Sonn spent two months looking them over.

And then Sonn e-mailed back with the words Swallow had waited to hear for almost exactly four years: “Looks good.”

QUESTION FOUR: You think you’ve solved a math problem that’s so hard, no one’s come close in 25 years. How do you know when you’re done?

At the highest levels, every math problem is solved twice: once in private, once in public. Swallow and Sonn agreed that they’d found the answer. But now the math world would get to check their work.

They typed up a formal version of the proof: “Brauer Groups of Genus Zero Extensions of Number Fields.” It ran 22 pages. Swallow sent copies to several other experts. He and Sonn posted their work on Web sites devoted to new research papers.

Based on the feedback, they made a few small fixes. Then they got the proof ready for the final step — submitting it to one of the academic journals.

The journals are the hockey goalies of math. If they think a paper is worthy, they send it to referees — other mathematicians who go over every detail. The referees are anonymous. If they agree with the proof, most mathematicians consider the problem solved.

Most journals get more submissions than they can publish. One journal decided not to look at Sonn and Swallow’s proof. They sent their work to a second journal. It was now February 2003. Swallow thought it might be another year before they heard back.

Swallow picks up his office mail at the college union. In summer he goes by every couple of days. In early August he found a letter. It was from an editor of Transactions of the American Mathematical Society.

I am pleased to inform you that your manuscript has been accepted for publication.

One referee suggested two tiny changes. The other didn’t suggest any.

It had been four years and seven months since they started. Now they were officially finished.

Swallow sent Jack Sonn an e-mail. He said it was time for a drink.

BONUS QUESTION: You’ve solved a math problem that was so hard, no one else came close for 25 years. What did you learn?

The first breath of fall is blowing across Main Street in Davidson. The folks behind the counter at Summit Coffee learned long ago what John Swallow wants. Regular latte if it's the morning, decaf latte if it’s the afternoon.

They know what he wants to drink, they might know what he does for a living, but they don’t know what he has accomplished. Not many people do — his family, a few other faculty members, maybe 50 mathematicians worldwide.

The problem he solved won’t win any of the big math prizes or make it into Newsweek. It’s not even necessarily the kind of thing that would earn him a raise.

But there are rewards. He’ll move up in the eyes of those who study top-level algebra. People will ask him to speak at conferences, publish papers, collaborate on new ideas. He’s already got a textbook due in December.

He knows now that figuring out the mysteries of giant number fields isn’t that different from working out the problems of everyday life. You break them down into small steps. You leave them alone now and then so you can come back fresh. Mainly, you trust what your instincts tell you.

These days Swallow is in charge of figuring out supper and hustling the kids to the car pool. Cameron has gone back to work; she teaches algebra at Smith Language Academy, a Charlotte-Mecklenburg magnet school. Ruth is 7, and Sophie’s 5. They’re ahead of their age groups in math.

Swallow is due for another sabbatical in 2005. He’s thinking about taking the family to France. He’s had good luck in France.

Meanwhile he daydreams about the next big problem, wonders what mental turn he'll have to take to solve it.

“There are lots of good ideas, but at first they are only ideas,” he says. “They have this feeling of novelty and newness. But until you sit down and hack it out, look at the details, you’re never sure what you’ve got. The idea can be beautiful. But only the work can make it beautiful.”

And his left eyebrow rises up.


Reprinted with permission of the Charlotte Observer.

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