Dimension-Cruncher: Exotic Spheres Earn Mathematician John Milnor an Abel Prize

His discovery that some seven-dimensional spheres look different under the lens of calculus spurred decades of research in topology
Abel Prize winner John Milnor of Stony Brook University

MFO via Creative Commons License

John Milnor, an American mathematician best known for the discovery of exotic hyperspheres, was awarded the 2011 Abel Prize, the Norwegian Academy of Science and Letters announced March 23.

Milnor, a professor at Stony Brook University in New York State, got a call at his Long Island home at 6 A.M. informing him he was receiving the $1-million prize—an honor first awarded in 2003 as mathematics' answer to the Nobel Prizes. "I knew I was a possible candidate, but I certainly didn't expect it," says Milnor, 80, who had already earned numerous awards during his career, including a Fields Medal in 1962 and a Wolf Prize in 1989. Milnor is the second consecutive American-born Abel laureate; the 2010 prize went to John Tate of the University of Texas at Austin for his contributions to number theory.

Milnor was born in 1931 in Orange, N.J., and graduated from Princeton University in 1951. He earned his PhD in mathematics at Princeton three years later. Milnor worked primarily at Princeton and at the unaffiliated Institute for Advanced Study in Princeton, N.J., before joining the faculty at Stony Brook in 1989, where he co-directs the Institute for Mathematical Sciences. His work spans the fields of topology, geometry and algebra, and displays "profound insights, vivid imagination, striking surprises and supreme beauty," the Norwegian academy said. His first important discovery, the solution to a conjecture on mathematical knots, came when he was a Princeton undergraduate. But the Milnor finding that really shook the mathematical world was his identification of a seven-dimensional sphere with exotic mathematical properties.

Mathematicians have dissected and analyzed the possible shapes, or topologies, of space—whether in two, three or any number of dimensions—ever since the late 1800s, when the field of topology was established.

Among possible shapes, spheres look simplest, although deceptively so. Just like all the points at a given distance from a point in 3-D space form a 2-D surface—the ordinary sphere—the points at a given distance from the origin of, say, eight-dimensional Euclidean space form a seven-dimensional space, or "manifold," called a (standard) seven-dimensional sphere.

In 1956 Milnor discovered a manifold that had the shape of a seven-dimensional sphere—what mathematicians call a "topological" sphere. But the sphere was not equivalent to the standard sphere in a subtle but fundamental way: it differed from the point of view of calculus.

Manifolds are a natural place to do calculus, and thus all sorts of science. Calculus makes it possible to set up scientific problems such as calculating the propagation of waves or the distribution of heat along the manifold. Milnor's discovery implied that the solutions to any such problem on his exotic 7-D sphere could not smoothly translate to solutions on an ordinary 7-D sphere. Instead, any such translation would produce singularities, or kinks. From the point of view of calculus, the two spheres were different animals.

The discovery was somewhat accidental. "Using one argument I could prove one manifold existed and using another I could not," Milnor recalls. The reason why he thought there was a contradiction was that he was assuming that all topologically equivalent spheres would also be smoothly equivalent. "It contradicts people's intuition," Milnor says.

And indeed, for topologists the discovery came like a bolt out of the blue. "Initially, they didn't even believe it," says James Milgram, an algebraic topologist at Stanford University. Later, in collaboration with the late Polish-born mathematician Michel Kervaire, Milnor classified all possible 7-D spheres, showing that there are exactly 27, in addition to the standard one.

Beginning with Milnor's work, the study of spheres has dominated many topologists' work and led to the awarding of at least four Fields Medals. (The Fields Medal, with a smaller cash award than the Abel Prize, is given every four years to a small number of mathematicians up to age 40.) "Spheres have been the central theme of topology for the last 60 years," says Robion Kirby, a topologist at the University of California, Berkeley; the culmination of that line of research was Grigory Perelman's 2004 solution of the Poincaré conjecture, a problem concerning the 3-D sphere.

"For me, Milnor's discovery of the exotic spheres was important for establishing the foundations of topology, towards understanding the role of various structures in that subject," says Stephen Smale, a mathematician at the City University of Hong Kong who won a 1966 Fields Medal for proving the Poincaré conjecture in dimensions higher than four. "That in turn played a big role in my own work on Poincaré's conjecture," Smale adds.

Mathematicians added the last touches to the classification of exotic spheres in any dimension higher than four only in 2009, but the 4-D case is still open. In other words, it is still unknown if there is more than one way to do calculus on the 4-D sphere.

The entire field was motivated by Milnor's discovery in the 1950s. "It certainly opened up a whole range of new questions you can ask," Milnor says.

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