Research

January 21, 2011

New theories reveal the nature of numbers


For centuries, some of the greatest names in math have tried to make sense of partition numbers, the basis for adding and counting. Many mathematicians added major pieces to the puzzle, but all of them fell short of a full theory to explain partitions. Instead, their work raised more questions about this fundamental area of math.

Emory mathematician Ken Ono is unveiling 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 . "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 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 postdoctoral fellow at Emory.

"Ken Ono has achieved absolutely breathtaking breakthroughs in the theory of partitions," says George Andrews, 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."

How did a hike to Georgia's Tallulah Falls lead to a eureka moment for the researchers? Read more about this major math discovery at eScienceCommons. »

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