A World of Doughnuts and Spheres April 20, 2003 By GEORGE JOHNSON Though you might not guess it from trying to read some of the research papers, the whole point of mathematics is to make things simpler. No one has taken this more seriously than the topologists, a rarefied breed of thinkers who insist that the world, however messy and diverse it may appear, is really made of just two basic shapes, the doughnut and the sphere. Actually it's a bit more complicated than that - the doughnuts can have more than one hole, for example, and the topologists don't limit themselves to the usual three dimensions. Lately, they have been preoccupied with claims that a Russian mathematician has solved a famous century-old problem involving what might be called hyperdoughnuts and hyperspheres existing in an imaginary four-dimensional space. Grappling with such slippery abstractions, Dr. Grigori Perelman of the Steklov Institute of Mathematics in St. Petersburg says he has found a proof of the Poincaré Conjecture, which seeks to explain how some of these airy higher-dimensional objects behave. He outlined his approach earlier this month in a series of lectures at the Massachusetts Institute of Technology. If he is right, it will be the biggest mathematical news since 1995, when Dr. Andrew J. Wiles, a Princeton professor, proved Fermat's Last Theorem. Sweetening the victory, Dr. Perelman would be eligible for a $1 million prize, sponsored by the Clay Mathematics Institute in Cambridge, Mass., for solving what it considers one of the seven most important problems of the millennium. The money is almost beside the point. That grown men and women can make a living pondering such matters is a sign that civilization, as fragile as it may sometimes seem, remains intact. "When you spend years closeted away with these things, they are as real to you as your family," said Dr. Michael Freedman, a mathematician at Microsoft who made his mark 22 years ago proving the Poincaré Conjecture for objects in five-dimensional space. Before that it had been proven for all dimensions beyond five. "Ironically the higher dimensions turned out to be easier than the lower ones," Dr. Freedman said. They offered more wiggle room. Topology is the study of that which remains constant as an object is bent, stretched or squeezed. A coffee cup with a looped handle, a bugle and a garden hose can each be transformed into a doughnut (more formally called a torus). Likewise, anything without a hole through it - a pencil, a brick, a piece of spaghetti (but not rigatoni, which is a very long and skinny doughnut) - can be molded into a sphere. The topological cookbook does not permit tearing an object or joining two unconnected points. That would be cheating and would allow anything to be transformed into anything else. Try as you might, you cannot turn a sphere into a doughnut or a doughnut into a sphere. Topologically they are as immiscible as oil and water. Having cataloged all the possible shapes in this realm, topologists have been reaching further. A sphere can be thought of as the three-dimensional world's version of a circle. So, going one level higher, what would be the four-dimensional equivalent of a sphere? And the five-dimensional version and so on? Seeking to find some order, the French mathematician Henri Poincaré proposed almost a century ago that the world of four dimensions obeys a rule similar to the one that prevails down here: Things without a hole are just different squishings of some canonical four-dimensional answer to the sphere. The technical name for this impossible object is the 3-sphere. Just as an ordinary sphere is a two-dimensional surface curving to form an enclosed object in three-dimensional space, a 3-sphere is a three-dimensional surface curving in on itself in four dimensions. Every few years someone claims to have tamed this monster, coming forth with a proof of the conjecture that is subsequently torn to shreds. "It's a famous problem, and at any time maybe a dozen people are working on it," Dr. Freedman explained. "Statistically one or two of them will be convinced that they almost have it." Hearing that a competitor may be on the verge of a breakthrough, "you work for four nights in a row, and then in some crazed state you claim that you've also proved it," he said. Dr. Perelman has raised the stakes even higher, claiming not only to have finished off the Poincaré Conjecture but to have listed every possible kind of object that can exist in the four-dimensional world - 3-spheres and who knows what else, an atlas of an invisible neighboring realm. His approach is innovative enough to make many topologists optimistic that the answer is finally in sight. If so, when the celebrations are over, the result may be met with a bit of melancholy. "Say you're a graduate student and you're picking a subject that will become your career," Dr. Freedman said. Do you really want to pick an area whose main problem has just been solved? "It's not a tragedy, because these people will go into other things," he said. "But it's a small sorrow for this particular branch of topology. You won't have the brilliant young people you have now." http://www.nytimes.com/2003/04/20/weekinreview/20JOHN.html?ex=1051909579&ei= 1&en=d9bfd625f0906cea