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The newspaper of The Johns Hopkins University September 20, 2004 | Vol. 34 No. 4
Drumming Up Answer to Math Mystery

Christopher Sogge and Steven Zelditch at the blackboard. "It's pure thought research," Sogge says. "We do it by thinking and brainstorming."

Profs aim to solve 200-year-old puzzle about vibrating shapes

By Lisa De Nike

A $975,398 National Science Foundation grant is allowing a team including mathematicians at Johns Hopkins' Krieger School of Arts and Sciences to tackle a 200-year-old question: How does the shape of drums influence the frequency and geometry of their vibrations when they are struck?

The $402,000 portion of the grant coming to Johns Hopkins is the largest ever awarded to faculty in the Department of Mathematics, according to Christopher Sogge, department chair, who has teamed up with Professor Steven Zelditch for the three-year investigation.

The Johns Hopkins team is sharing the grant with fellow mathematicians Daniel Tataru and Maciej Zworski at University of California, Berkeley, and Hart Smith at the University of Washington, who are taking on the same challenge. According to Sogge, the purpose of the grant is to get like-minded researchers collaborating.

"At the heart of our project is a 200-year-old question that remains largely unanswered today," Sogge says. "Mathematicians are interested in what can be known, and even better, proven, about vibrating shapes. This is important in mathematics and physics because it deals with what we call 'modes of vibration' of objects ranging from drums to atoms and molecules to the whole universe."

The question dates back to the early 19th century, when a German scientist named Ernst Chladni impressed Emperor Napoleon Bonaparte with his ability to make sound waves "visible" using a simple experiment. When Chaldni sprinkled sand onto metal plates and drew a violin bow across the edges, the grains scattered and settled into intricate geometric designs on the portions that were not shaking with sound. Intrigued by those results, the emperor offered a reward to anyone who could explain how the sand's patterns--and the invisible sound waves that produced them--related to the shape of the metal surface upon which they settled.

Sogge and Zelditch have chosen to study various drum shapes because their contours provide a simple model for any vibrating object, including atoms and molecules. The general question of how the frequencies and shapes of vibration reflect the shape of the object is the same, whether one is talking about drums or atoms, they say.

"The shapes we are dealing with range from the kind a drummer would use in a rock band to drums of any dimension and shape," Zelditch says. "We are especially interested in what we call 'extreme drums': those capable of making extreme sounds at a given frequency. Extreme to most people would mean 'loud.' But we think geometrically."

It's important to note that no actual drums are involved in the team's inquiry. Sogge and Zelditch are operating entirely in the realm of the theoretical: Their only hands-on work involves putting chalk to blackboard or pen to paper in the quest for the formula or proof that solves the question at hand. And though some mathematicians and physicists are tackling this challenge with the help of computers, the Johns Hopkins pair is using old-fashioned brainpower.

"It's pure thought research; we do it by thinking and brainstorming," Sogge says. "The best way to visualize what we are doing is to imagine playing billiards on a drumhead and using math to predict and describe the trajectory of the ball when you hit it. For instance, the usual billiard table is rectangular, and the path of a ball hit upon it is, therefore, predictable. But if one makes a pool table in the shape of a football stadium, it turns out that the ball, no matter how you hit it, will move around in a chaotic way. It's as if we are playing an imaginary game of billiards atop drums of various shapes to determine what their vibrations will look like."

Though the Johns Hopkins team anticipates having some answers by the end of the three-year grant, Sogge and Zelditch say they expect to be grappling with similar problems and theories for the rest of their careers.

"What we're doing is of interest to the math world because the properties we are talking about are very basic to lots of areas of mathematics, from number theory to partial differential equations to differential geometry," Sogge says. "But it also is of interest to physicists who study dynamical systems and nanostructures, such as quantum dots and electron corals. In both cases, the concentration patterns of waves and excited states play a fundamental role in the systems they study."


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