May 11, 2009
‘Doomsday Theorem’ cracks 45-year-old math problem
jonathan.sherwood@rochester.edu
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 Rochester, Harvard University, and the University of Virginia, solved the Kervaire invariant problem, which was one of the great unanswered problems in algebraic topology—the study of shape relationships. The team presented its findings April 21 at a conference celebrating the 80th birthday of Sir Michael Atiyah, one of the most influential mathematicians of the 20th century.
Douglas Ravenel, professor of and co-discoverer of the solution, said 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’,” Ravenel says. “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 Michael Hopkins of Harvard and Michael Hill of Virginia, had to wait another three months for the right conference while not 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.
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.
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