Honor Among Geniuses
An article by Sylvia Nasar and David Gruber in the New Yorker provides a much deeper view into the political opera behind the recent news of Grigory Perelman, the mathematician who, drawing on others’ work, solved the famously intractable Poincaré conjecture then refused the mathematics equivalent of the Nobel Prize. The article illuminates the complexity of the problem and places it within its knotty historical context. More enthralling, however–and the math is pretty enthralling–are the tensions, trashtalk and professional tiger traps that emerge among a global community of brilliant mathematicians.
It’s a fascinating story and a terrific piece of journalism. Come for the math, stay for the bloodfeud:
Proofs of the Poincaré have been announced nearly every year since the conjecture was formulated, by Henri Poincaré, more than a hundred years ago. Poincaré was a cousin of Raymond Poincaré, the President of France during the First World War, and one of the most creative mathematicians of the nineteenth century. Slight, myopic, and notoriously absent-minded, he conceived his famous problem in 1904, eight years before he died, and tucked it as an offhand question into the end of a sixty-five-page paper. Poincaré didn’t make much progress on proving the conjecture. “Cette question nous entraînerait trop loin” (“This question would take us too far”), he wrote.
He was a founder of topology, also known as “rubber-sheet geometry,” for its focus on the intrinsic properties of spaces. From a topologist’s perspective, there is no difference between a bagel and a coffee cup with a handle. Each has a single hole and can be manipulated to resemble the other without being torn or cut. Poincaré used the term “manifold” to describe such an abstract topological space. The simplest possible two-dimensional manifold is the surface of a soccer ball, which, to a topologist, is a sphere—even when it is stomped on, stretched, or crumpled. …
Two-dimensional manifolds were well understood by the mid-nineteenth century. But it remained unclear whether what was true for two dimensions was also true for three. Poincaré proposed that all closed, simply connected, three-dimensional manifolds—those which lack holes and are of finite extent—were spheres. The conjecture was potentially important for scientists studying the largest known three-dimensional manifold: the universe. …
By the nineteen-sixties, topology had become one of the most productive areas of mathematics, and young topologists were launching regular attacks on the Poincaré. To the astonishment of most mathematicians, it turned out that manifolds of the fourth, fifth, and higher dimensions were more tractable than those of the third dimension. By 1982, Poincaré’s conjecture had been proved in all dimensions except the third. In 2000, the Clay Mathematics Institute, a private foundation that promotes mathematical research, named the Poincaré one of the seven most important outstanding problems in mathematics and offered a million dollars to anyone who could prove it.
“My whole life as a mathematician has been dominated by the Poincaré conjecture,” John Morgan, the head of the mathematics department at Columbia University, said. “I never thought I’d see a solution. I thought nobody could touch it.”
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Thurston proposed that every three-dimensional manifold could be broken down into one or more of eight types of component, including a spherical type. Thurston’s theory—which became known as the geometrization conjecture—describes all possible three-dimensional manifolds and is thus a powerful generalization of the Poincaré. If it was confirmed, then Poincaré’s conjecture would be, too. Proving Thurston and Poincaré “definitely swings open doors,” Barry Mazur, a mathematician at Harvard, said. The implications of the conjectures for other disciplines may not be apparent for years, but for mathematicians the problems are fundamental. “This is a kind of twentieth-century Pythagorean theorem,” Mazur added. “It changes the landscape.”
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Barry Mazur, the Harvard mathematician, uses the image of a dented fender to describe Perelman’s achievement: “Suppose your car has a dented fender and you call a mechanic to ask how to smooth it out. The mechanic would have a hard time telling you what to do over the phone. You would have to bring the car into the garage for him to examine. Then he could tell you where to give it a few knocks. What Hamilton introduced and Perelman completed is a procedure that is independent of the particularities of the blemish. If you apply the Ricci flow to a 3-D space, it will begin to undent it and smooth it out. The mechanic would not need to even see the car—just apply the equation.”
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Zhu and Cao’s paper was more than three hundred pages long and filled the A.J.M.’s entire June issue. The bulk of the paper is devoted to reconstructing many of Hamilton’s Ricci-flow results—including results that Perelman had made use of in his proof—and much of Perelman’s proof of the Poincaré. In their introduction, Zhu and Cao credit Perelman with having “brought in fresh new ideas to figure out important steps to overcome the main obstacles that remained in the program of Hamilton.” However, they write, they were obliged to “substitute several key arguments of Perelman by new approaches based on our study, because we were unable to comprehend these original arguments of Perelman which are essential to the completion of the geometrization program.” Mathematicians familiar with Perelman’s proof disputed the idea that Zhu and Cao had contributed significant new approaches to the Poincaré. “Perelman already did it and what he did was complete and correct,” John Morgan said. “I don’t see that they did anything different.”
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The prospect of being awarded a Fields Medal had forced him to make a complete break with his profession. “As long as I was not conspicuous, I had a choice,” Perelman explained. “Either to make some ugly thing”—a fuss about the math community’s lack of integrity—“or, if I didn’t do this kind of thing, to be treated as a pet. Now, when I become a very conspicuous person, I cannot stay a pet and say nothing. That is why I had to quit.”
In retrospect, I think I was probably right to stop at calculus.
