Perfect Rigour

by Masha Gessen

  What makes one manifold different from another is its having a hole, or more than one hole. To a topologist, a ball, a box, a bun, and a blob are essentially the same. But a bagel is different. The key to this is the rubber band, an instrument as important to the topological imagination as the atlas. The imaginary rubber band is placed around the imaginary object and allowed to do its rubber-band thing, which is contract. If a rubber band—a very tight rubber band—is placed around a ball, it will find a way to contract and slip off the ball. It is significant that this will happen no matter where on the ball the band is placed. A bagel, however, is different: if one end of your imaginary rubber band has been threaded through the hole in the bagel and then reconnected to itself, it will stay around the bagel, never slipping off no matter how tight it is. A rubber band can be slipped off any place on a ball, a box, a bun, or a blob without a hole, which makes them all essentially similar or, in the language of topology, diffeomorphic to one another. This means you can reshape any one of them into any other and then back again.

  This more or less brings us to the point where we can understand the Poincaré Conjecture. A bit more than a hundred years ago, Poincaré posed an innocent-sounding question: if a three-dimensional manifold is smooth and simply connected, then is it diffeomorphic to a three-dimensional sphere? Smooth means that the manifold is not twisted (you can imagine that twisting something would cause some problems with the papering-over map project). Simply connected means that it has no holes. And we know what diffeomorphic means. We also know what three-dimensional means: a three-dimensional manifold is the surface of a four-dimensional object. Let us also pause to consider what a sphere is. A sphere is a collection of points that are all equally far from a given point—the center. A one-dimensional sphere (a circumference in regular school geometry) is all of these points in a two-dimensional space (a plane). A two-dimensional sphere (the surface of a ball) is all of these points in a three-dimensional space. What makes spheres particularly interesting to topologists is that they belong to a category called hypersurfaces—objects that have as many dimensions as is possible in a given space (one dimension in a two-dimensional space, two dimensions in a three-dimensional space, and so on). The three-dimensional sphere that so interested Poincaré was the surface of a four-dimensional ball. We cannot imagine this thing, but we just might inhabit it.

  Topologists often tackle problems by trying to solve them for a different number of dimensions. The equivalent of the Poincaré Conjecture for two dimensions—the understanding that the surfaces of a ball, a box, a bun, and a blob without a hole are essentially the same—is basic to topology. But in three dimensions—when we actually get to the conjecture itself—it gets tricky. Mathematicians struggled with the Poincaré Conjecture in its original three dimensions for the better part of a century, but the first breakthroughs came from a different place—or, rather, in higher dimensions.

  At the dawn of the 1960s, several mathematicians—exactly how many and under what circumstances is still a matter of some dispute—proved the Poincaré Conjecture for dimensions five and higher. One was the American John Stallings, who in 1960 published a proof of the conjecture for seven dimensions or more13 just a year after he received his PhD from Princeton.14 Next was the American Stephen Smale, who probably completed his proof earlier than Stallings but published it several months later; he, however, proved the conjecture for dimensions five and higher.15 Then the British mathematician Christopher Zeeman extended Stallings’s proof to dimensions five and six.16 A fourth man in the mix was Andrew Wallace, an American mathematician who in 1961 published a proof essentially similar to Smale’s.17 There was also a Japanese mathematician18 named Hiroshi Yamasuge who published his own proof for dimensions five and higher in 1961.

  So, more than fifty years after it was originally posed, the Poincaré Conjecture started to give—ever so slightly. All of these mathematicians, like countless others who were far less successful, had hoped to prove the conjecture itself—for the three dimensions for which it was stated. And while they will probably be remembered for their groundbreaking contributions to the cause of cracking the conjecture, at least one of them seemed to think himself most remarkable for the contribution he did not make. John Stallings,19 a professor emeritus at Berkeley, listed only a few of his papers on his personal website. The first published paper he mentioned dated back to 1966, and it was called “How Not to Solve the Poincaré Conjecture.”

  “I have committed—the sin20 of falsely proving Poincaré’s Conjecture,” Stallings began. “Now, in hope of deterring others from making similar mistakes, I shall describe my mistaken proof. Who knows but that somehow a small change, a new interpretation, and this line of proof may be rectified!” That is the spirit of hope against hope, at once conscious of the futility of efforts and obsessively incapable of giving up, that characterized the nearly hundred-year battle against the conjecture.

  It was twenty years before the conjecture yielded slightly once again. In 1982 the young—he was thirty-one at the time—American mathematician Michael Freedman published a proof of the conjecture for dimension four.21 The accomplishment was hailed as a breakthrough;22 Freedman received the Fields Medal. But the conjecture for dimension three remained unproven. None of the methods used in the higher dimensions worked for dimension three; there was not enough room in this dimension to allow topologists to wield the tools they used in higher dimensions. It seemed to call for a revolutionary approach, something Poincaré himself could not have envisioned or even suspected.

  Perhaps one of the problems with four-dimensional spaces is that, unlike higher-dimensional ones, they are not quite abstractions; it seems that we humans may very well inhabit a three-dimensional space embedded in four dimensions, even if most of us cannot wrap our minds around it. But experts say there is one living man, the American geometer William Thurston, who can imagine four dimensions. Thurston, they say, is possessed of a geometric intuition unlike that of any other human. “When you see him or talk to him, he is often staring out into space and you can see that he sees these pictures,” said John Morgan,23 a professor at Columbia University, a friend of Thurston’s, and a coauthor of one of several books written about Perelman’s proof of the Poincaré Conjecture. “His geometric insight is unlike anyone I’ve ever met. So can there be a type of mathematician like Bill Thurston? How can someone have that kind of geometric insight? You know, I’ve got a fair amount of mathematical talent myself but I don’t approach the human conclusions he does.”

  Thurston talked of three-dimensional manifolds in four-dimensional spaces as though he could see and manipulate them. He described the ways they could be cut up, and what would happen if they were. To a topologist, this was a very important exercise; complex objects are usually studied through their simpler composite parts, and understanding the nature of these parts and their relationships is essential to understanding the larger object. Thurston suggested that all three-dimensional manifolds could be carved up in particular ways that yielded objects that belonged to one of eight specific varieties of three-dimensional manifolds. It would not be quite right to call Thurston’s conjecture a step toward proving Poincaré’s. Indeed, it was even more ambitious, if a bit less famous. If Thurston had proved his conjecture, Poincaré’s would automatically have followed. But he could not prove it.

  “I watched Bill make progress,” Morgan recalled. “And when he didn’t get it, I thought, ‘I’m not going to get it, nobody is going to get it.’ Just as Jeff [Cheeger] said one time, ‘It just gets too complicated to keep practicing the Poincaré Conjecture.’”

  While other mathematicians wisely chose to direct their energies elsewhere, a Berkeley professor named Richard Hamilton persisted in tackling the Poincaré and then the Thurston conjectures. The standard journalistic description of Hamilton usually contains the word flamboyant, which seems to mean, basically, that he is interested not only in mathematics but also in surfing a
nd in women. He is sociable, charming, and absolutely brilliant—for it was he who devised the way to prove both of the conjectures.

  In the early 1980s Hamilton proposed something that can sound deceptively obvious. The surface of a sphere in any dimension has a constant positive curvature; this is a basic quality of the object. So if one could find a way to measure the curvature of an unidentifiable, unimaginable three-dimensional blob and then start reshaping the blob, all the while measuring its curvature, then one might eventually get to the point where the curvature was both positive and constant, whereby the blob would definitively be proven to be a three-dimensional sphere. That would mean that the blob had been a sphere all along, since reshaping does not actually change the topological qualities of objects—it just makes them more recognizable.

  Hamilton devised a way of placing a metric on the blob to measure the curvature, and he wrote an equation that showed the way the blob, and the metric, would change over time. He proved that as the blob was molded, its curvature would not decrease but would necessarily grow—and this helped him demonstrate that the curvature would indeed be positive. But how to ensure that it would be constant? Hamilton got stuck.

  Think about a simple function of the sort you studied in high school. Say, 1/x. A graph of this function would look like a smooth line until it got to the point where x = 0. Then things would get crazy, because you cannot divide by zero. The line of your graph would suddenly soar toward eternity. This is called a singularity.

  The process of transforming the metric described by the equation devised by Hamilton is called the Ricci flow. As the flow worked its theoretical magic on the imaginary metric on the unimaginable blob, every so often, a singularity would develop. Hamilton suggested that the singularities could be predicted and disarmed by stopping the function—the Ricci flow—fixing the problem by hand, and resuming the flow. When a mathematician says that he has fixed something “by hand,” he actually means that he has devised a different function for the problem piece. An example is something that often happens in computer programming, where different functions are used depending on the conditions. When, say, your function is equal to x for all cases where x is equal to or greater than 0, and equal to –x for all cases where x is less than 0. In topology, where imaginary hands intervene in the imaginary transformation of an object, this intervention is called surgery. So the process that Hamilton envisioned was Ricci flow with surgery.

  Hamilton was not the first mathematician who thought he knew how to prove the Poincaré Conjecture. He was also not the first to encounter insurmountable obstacles on his way to a proof. In order for his program—as mathematicians call it—to work, several things had to be true. First, the curvature he was attempting to measure had to have a constant limit, a sort of uniform boundary; if he assumed this was true, the proof would probably work—but how could he know that his assumption was correct? Second, while Hamilton devised Ricci flow with surgery and could show that it would be effective in some cases, he could not prove that it could be used effectively no matter what kind of singularity developed. He could theorize about the sorts of singularities that would appear, but he could not find a way to tame all of them or even claim to have identified all of them. Here was another man who “made progress and then didn’t get it.” Here was another man for whom, as Morgan quoted Jeff Cheeger as saying, it got “too complicated to keep practicing the Poincaré Conjecture.”

  Twenty-five years later, two things are perfectly clear. First, Hamilton did indeed create the blueprint for proving both the Poincaré and the Geometrization conjectures. Second, his personal tragedy was as great as his professional achievement: at the age of forty, Hamilton became stuck and, apparently, remained stuck.

  The point at which Hamilton got stuck is roughly the point at which Perelman began to engage the Poincaré Conjecture. It was also the point at which Perelman began to disappear; he went to fewer seminars, gradually reduced his hours at the Steklov so that he really only appeared when it was time to pick up his monthly pay. He slowed his e-mail correspondence to such a degree that most acquaintances assumed he had become yet another mathematician who had once shown promise but then met a problem and was crushed by it, reduced to mathematical nonexistence.

  We know now that this was not the case. Rather, Perelman had completed his mathematical education and began to apply it. As it happened, the process of being educated—or, perhaps more precisely, the desire he had for mathematical knowledge that could be imparted by others—was what had kept him connected to the outside world. Now that world was more or less used up; its utility was negligible, and its demands therefore incomprehensible and even more irritating than before. Perelman, naturally, turned his back on the world and faced the problem.

  What the world had given Perelman was the habit of honing the power of his incomparable mind on a single problem. What Hamilton had essentially done was turn the Poincaré Conjecture into a super mathematical-olympiad problem. He had, in a sense, taken it down a notch. In the world of top mathematicians, the intellectual elite are people who open new horizons by posing questions no one else has thought to ask. A step down are the people who devise ways to answer those questions; often these are members of the elite at earlier stages in their career—a few years after obtaining their PhDs, for example, when they are proving other people’s theorems before they start formulating their own. And finally, there are the rare birds, those who take the last steps in completing proofs. These are the persistent, exacting, patient mathematicians who finally lay down the paths others have dreamed up and marked out. In our story, Poincaré and Thurston represent the first group, Hamilton the second group, and Perelman the one who finished the job.

  So who was he? He was the man who had never met a problem he could not solve. Whatever he had been trying to do with Alexandrov spaces at Berkeley might have been an exception—he might indeed have gotten stuck—but then it might also have been the only time he tried to do something that fell into the second or even the first category of mathematical work rather than the third. The third category is essentially similar to solving a mathematical-olympiad problem: it has been clearly stated, and restrictions have been placed on its solution—the path to proof had been marked out by Hamilton. This was a very, very complicated olympiad problem; it could not be solved in hours, or weeks, or even months. Indeed, it was a problem that perhaps could not be solved in any amount of time by anyone—except Perelman. And Perelman was a man in search of just such a problem, one that would finally utilize the full capacity of the supercompactor that was his mind.

  Perelman managed to prove two main things. First, he showed that Hamilton did not need to assume that the curvature would always be uniformly bound; in the imaginary space in which the proof unfolded, this simply would always be the case. Second, he showed that all the singularities that could develop stemmed from the same root; they would appear when the curvature began to “blow up,” to grow unmanageable. Since all the singularities had the same nature, a single tool would be effective against all of them—and the surgery originally envisioned by Hamilton would do the job. Moreover, Perelman proved that some of the singularities Hamilton had hypothesized would never occur at all.

  There is something peculiar and slightly ironic in the logic of Perelman’s proof. He succeeded because he used the unfathomable power of his mind to grasp the entire scope of possibilities: he was ultimately able to claim that he knew all that could happen as the matrix grew and the object reshaped itself. Knowing it all, he was able to exclude some of the topological developments as impossible. Speaking of the imaginary four-dimensional space, he referred to things that could and could not occur “in nature.” In essence, he was able to do in mathematics what he had tried to do in life: grasp at once all the possibilities of nature and annihilate everything that fell outside that realm—castrati voices, cars, anti-Semitism, and any other uncomfortable singularity.

  PE
RFECT RIGOR

  THE PROOF EMERGES

  9

  The Proof Emerges

  Date: Tue, 12 Nov 2002 05:09:02 -0500 (EST)

  From: Grigori Perelman

  To: [multiple recipients]

  Subject: new preprint

  Dear [Name],

  may I bring to your attention my paper in arXiv math.DG 0211159.

  Abstract:

  We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton’s program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.