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For the math connoisseur
Posted on Tue, Dec. 27, 2005
Peers hail solving of problem
MU math professor finds proof is positive
By MARÁ ROSE WILLIAMS
The Kansas City Star
As a boy, Steven Hofmann dreamed of becoming a professional baseball player. But odds were against that happening, and arithmetic came easier, so he stuck with math.
Unlike athletes, mathematicians rarely garner national or international praise.
But Hofmann is an exception.
The 47-year-old math professor at the University of Missouri-Columbia is in line for applause from around the world for solving a math problem that had baffled his peers for more than 40 years.
Solving the problem got Hofmann an invitation to speak next spring in Madrid, Spain, at the 2006 International Congress of Mathematicians, which is held every four years.
For a mathematician, the event is “a really big deal,” Hofmann said.
“It is like a baseball player being picked for the all-star team.”
Theodore Slaman, chairman of the Department of Mathematics at the University of California-Berkeley, says an opportunity to speak at the international congress is a career achievement. And solving a problem as old as the one Hofmann solved “is like finding the Holy Grail,” Slaman said.
“Once you have solved it, people believe you have an understanding of an entirely new area. The longer a problem has been around, the more cachet associated with solving it.”
For three years, starting in 1996, Hofmann worked on the problem for two to eight hours every day.
“I think I must have been the last person in the world still working on it,” Hofmann said.
The problem that he finally solved in 2000 had been posed in research papers first written in 1953 and again in 1961 by Tosio Kato, a now-deceased Cal-Berkeley mathematician. It became known in the world of mathematics as Kato’s Conjecture.
The one-dimensional version of the problem was solved about 20 years ago. That was a breakthrough, but not a complete solution. Hofmann, working periodically with several colleagues — Pascal Auscher, Michael Lacey, John Lewis, Alan McIntosh and Philippe Tchamitchian — solved the problem in all its dimensions.
This problem was no simple equation. It applies to the theory of waves moving through different media, such as seismic waves traveling through different types of rock.
“It’s complicated,” Hofmann said. It’s more than just a blackboard full of chalk-etched equations. The solution paper he wrote with mathematicians Lacey and McIntosh, is titled “The solution of the Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds.” It is 120 pages of hieroglyphiclike equations and text.
Hofmann said the solution could allow mathematicians to better describe the behavior of waves traveling through a medium that changes over time. But beyond that, he said, it is impossible for him to explain all the real-world applications.
“Philosophically, the reason research in math matters is that by pursuing math ideas that are deep and interesting for their own sake, you will get real-world applications in the future,” Hofmann said.
“It is like making investments.”
In the case of Kato’s Conjecture, Hofmann had been curious about the problem since his undergraduate days when one of his professors, who had tried to solve it, introduced him to the problem.
It was several years after that, though, before Hofmann got serious enough to concentrate on it.
Hofmann majored in math, he said, “because it was the path of least resistance.” While his friends were writing history papers that were many pages long or spending hours in a computer lab, “all I had to do was solve math problems, and it was something that came to me naturally,” he said.
“By the time you get to graduate school, even if it comes naturally, it gets hard, and that is when you begin to develop a skill to go with the ability.”
It took all of Hofmann’s skill, plus a healthy dose of stick-to-itiveness, he said, to get a breakthrough on Kato’s Conjecture.
He went to bed each night and woke each morning thinking about it.
“I could be out for a bike ride, and I would be thinking about it,” Hofmann said. “Sometimes I would be doing something, get an idea and have to stop … and write it down.”
Once he got two-dimensional results, it took another 12 months closeted in an upstairs room of his Columbia home to solve the problem fully.
He remembers the moment.
“I came downstairs and announced to my family, ‘I’ve got it,’ ” Hofmann said. “They were happy but cautious. I had said that several times before. But this time I really did have it.”
In fact, Hofmann said, there were times when he wasn’t sure there was a solution. But he’d worked too hard for too long to just give up. So, he kept trying.
Hofmann said working day and night on a math problem requires a great deal of faith — faith that there is indeed a solution.
Unlike someone who trains for years to climb, say Mount Everest, and then endures the torturous climb to the top, a mathematician trying to solve an equation works with no certainty that a solution truly exists.
To reach Mará Rose Williams, call (816) 234-4419 or send e-mail to [email protected].
