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James Brody January 20th, 2011 12:13 PM

Ken Ono: Fractal Partition Numbers
 
"On Friday, Emory mathematician Ken Ono will unveil new theories that answer these famous old questions.

"Ono and his research team have discovered that partition numbers behave like fractals. They have unlocked the divisibility properties of partitions, and developed a mathematical theory for "seeing" their infinitely repeating superstructure. And they have devised the first finite formula to calculate the partitions of any number.

"Our work brings completely new ideas to the problems," says Ono, who will explain the findings in a public lecture at 8 p.m. Friday on the Emory campus. "We prove that partition numbers are 'fractal' for every prime. These numbers, in a way we make precise, are self-similar in a shocking way. Our 'zooming' procedure resolves several open conjectures, and it will change how mathematicians study partitions."

"The work was funded by the American Institute of Mathematics (AIM) and the National Science Foundation. Last year, AIM assembled the world's leading experts on partitions, including Ono, to attack some of the remaining big questions in the field. Ono, who is a chaired professor at both Emory and the University of Wisconsin at Madison, led a team consisting of Jan Bruinier, from the Technical University of Darmstadt in Germany; Amanda Folsom, from Yale; and Zach Kent, a post-doctoral fellow at Emory.

"'Ken Ono has achieved absolutely breathtaking breakthroughs in the theory of partitions,' says George Andrews, professor at Pennsylvania State University and president of the American Mathematical Society. 'He proved divisibility properties of the basic partition function that are astounding. He went on to provide a superstructure that no one anticipated just a few years ago. He is a phenomenon….'

“Child's play
“On the surface, partition numbers seem like mathematical child's play. A partition of a number is a sequence of positive integers that add up to that number. For example, 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1. So we say there are 5 partitions of the number 4.

“It sounds simple, and yet the partition numbers grow at an incredible rate. The amount of partitions for the number 10 is 42. For the number 100, the partitions explode to more than 190,000,000.

"‘Partition numbers are a crazy sequence of integers which race rapidly off to infinity,’ Ono says. ‘This provocative sequence evokes wonder, and has long fascinated mathematicians.’

“By definition, partition numbers are tantalizingly simple. But until the breakthroughs by Ono's team, no one was unable to unlock the secret of the complex pattern underlying this rapid growth.”

http://www.eurekalert.org/pub_releas...-nmt011911.php


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