April 21, 2009

A team of mathematicians has cracked a 45-year-old problem in mathematics with a solution that was once considered so upsetting to mathematicians' expectations that it has been nicknamed "The Doomsday Theorem."

The team, consisting of researchers from the University of Rochester, Harvard University, and the University of Virginia, solved the Kervaire invariant problem, which is one of the great unanswered problems in algebraic topology—the study of shape relationships. They presented their findings on April 21 at a conference celebrating the 80th birthday of Sir Michael Atiyah, one of the most influential mathematicians of the twentieth century.

Douglas Ravenel, professor of mathematics at the University of Rochester and co-discoverer of the solution, said that the excitement surrounding the discovery is due to the fact that the answer to the Kervaire problem essentially had to boil down to a "yes" or a "no"—and since the 1970s, everyone expected the answer to be "yes."

"We surprised ourselves with this when we realized the answer was 'no'," says Ravenel. "We didn't set out to do this. We were working on something else—sort of knocking on the door of another house down the street. It's as if, when we figured out how to get in, we discovered a tunnel from that house to the Kervaire house. While everyone else was pounding on Kervaire's front door, we were crawling in through the tunnel."

It took another three months of work to overcome another obstacle "in the tunnel," but once the team devised a way to solve that obstacle, they were ready to reveal their solution to the world.

The team, which includes Mike Hopkins of Harvard University and Mike Hill of the University of Virginia, had to wait another three months for the right conference, however, never telling anyone what they'd discovered. They even gave a fake name to the title of their talk at the conference to throw other mathematicians off the scent.

"The title of our talk was so bland that people knew it wasn't real and rumors started flying around," says Ravenel. "Some people asked if we'd solved it, and I had to come up with some creative ways to keep it a secret."

Algebraic topology is a way of describing the commonalities among certain shapes, says Ravenel. The classic example is a coffee mug and a donut, each of which has just one hole. If they were made of wet clay, one could be molded into the other without adding or losing the hole. Neither shape could be molded into a ball, however, because a ball has no holes.

The Kervaire invariant is a number that topologists use to define shapes in certain dimensions. Topologists had expected that specific kinds of shapes occur in infinitely many dimensions, but Ravenel's team has shown that they exist in only one more dimension beyond the few already known. Ravenel says the team's solution gives mathematicians new tools with which to tackle other problems in new ways.

"When this problem was articulated nearly 50 years ago, people tried to solve it with the techniques they had on hand at the time," says Ravenel. "We took two giant steps beyond that. One was a sort of tool our team had in our toolkit, and the other was a new tool we dreamed up six months ago when we realized we might be onto the Kervaire invariant solution."

Ravenel says the solution gives mathematicians an entirely new methodology for tackling problems such as some aspects of quantum theory and string theory, both of which might someday describe the very nature of time and space itself.

The team is now writing up their findings in an official paper they plan to submit to the mathematics community in the coming months.

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