Archive for Wednesday, December 28, 2005

Decades-old math problem solved

Professor’s solution in theory of waves is ‘like finding the Holy Grail’

December 28, 2005

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— A professor at the University of Missouri-Columbia is being recognized for solving a math problem that had stumped his peers for more than 40 years.

The achievement has landed Steven Hofmann an invitation to speak at the 2006 International Congress of Mathematicians in Madrid, Spain, in August.

"It is like a baseball player being picked for the all-star team," Hofmann said of the invitation to the event, which is held every four years.

Hofmann became curious about the problem as an undergraduate when a professor introduced him to it.

The professor was unable to solve the problem. Hofmann, 47, would have more success when the problem began to take over his life in 1996. Until he solved it in 2000, it was the last thing he thought about before he went to bed and the first thing he thought about when he woke.

He spent two to eight hours each day on the problem, working periodically with several colleagues.

"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."

The problem, known as Kato's Conjecture, applies to the theory of waves moving through different media, such as seismic waves traveling through different types of rock. It bears the name of Tosio Kato, a now-deceased mathematician at the University of California-Berkeley, who posed the problem in research papers first written in 1953 and again in 1961.

Part of the problem, called the one-dimensional version, was solved about 20 years ago. Though it was a breakthrough, work remained.

Hofmann solved the problem in all its dimensions in a 120-page paper that he wrote with several colleagues - Pascal Auscher, Michael Lacey, John Lewis, Alan McIntosh and Philippe Tchamitchian.

"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."

Theodore Slaman, chairman of the Department of Mathematics at the University of California-Berkeley, said solving a problem as old as Kato's Conjecture "is like finding the Holy Grail."

"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."

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