Look around at the world, and the objects in it–buildings, trees, people, birds, insects–appear to come in an endless variety of shapes. At first, cataloging these diverse shapes may seem impossible. But on closer inspection, relationships emerge. The bumpy surface of a starfish, for example, is simply a stretched and distorted version of a sphere. The same goes for the surface of a table or a telephone pole. In contrast, a coffee cup is not a sphere but instead a distorted version of a doughnut, and a pretzel can be considered a doughnut with three holes instead of one.
What about more complicated shapes like a fishnet or a bicycle wheel? Amazingly, more than a hundred years ago, mathematicians proved that every closed surface in space is simply some version of a sphere, a doughnut surface–which they call a torus–or a torus with extra holes.
Even though spheres and tori sit in three-dimensional space, mathematicians focus on their surfaces and so view them as two-dimensional, unlike solid balls and filled-in doughnuts, which are three-dimensional. A small patch of a sphere or torus surface looks almost like a piece of a flat plane and has area rather than volume.
Mathematicians also study an analogous collection of what they call closed three-dimensional shapes. Unlike ordinary three-dimensional objects, these shapes live in four-dimensional–or higher–space and curve in on themselves as the sphere and torus do in three-dimensional space. Although such shapes are difficult to visualize, some cosmologists speculate that our own universe may be of that form, rather than the infinitely extending space that most people envision.
For a century, mathematicians have wondered whether there’s a classification of three-dimensional shapes like the simple breakdown of two-dimensional shapes into spheres and tori. Now, a Russian mathematician may finally have proved that the answer is yes (SN: 4/26/03, p. 259: Available to subscribers at Spheres in Disguise: Solid proof offered for famous conjecture). Details are starting to emerge of his work, which gives a way to distort a three-dimensional object, little by little, to make its shape more uniform.
A few years ago, the Clay Mathematics Institute in Cambridge, Mass., offered a $1 million bounty to anyone who could settle the Poincaré conjecture, a 99-year-old question about three-dimensional shapes that’s one of the most famous problems in mathematics. After working for years in near seclusion and supporting himself largely on personal savings, Grigory Perelman of the Steklov Institute of Mathematics in St. Petersburg, Russia, announced that he has proved the conjecture, which gives a way to identify whether a complicated shape is a distorted version of a sphere. He also claims to have proved the much broader Thurston geometrization conjecture, which considers all closed three-dimensional shapes.
Over the years, dozens of mathematicians have mistakenly claimed to have proved the Poincaré conjecture. For this reason, mathematicians–including Perelman himself–are not rushing to judgment. Perelman has declined to talk to the press until colleagues verify his proof.
It will take months, some mathematicians say, to dissect the details of Perelman’s densely written papers. But Perelman’s track record makes many optimistic that his work will stand up to scrutiny. “He’s singularly brilliant,” says Jeff Cheeger of the Courant Institute of Mathematical Sciences at New York University. What’s more, Perelman’s colleagues note, the portions of his work that have already been verified are full of groundbreaking ideas.
“Whether or not he has a complete proof, he has clearly made very important contributions to mathematics,” says John Milnor, a mathematician at the State University of New York at Stony Brook who attended a series of lectures Perelman gave there in April and May.
Many past attempts to prove the Poincaré conjecture have involved intricate, hard-to-check arguments. “This one feels like a much more natural, very promising approach,” Milnor says. “It seems like the right way to handle the problem.”
Recognizing the hypersphere
Even though a sphere and a torus are two-dimensional to mathematicians, there’s no way to fit them into a flat plane without squashing them. Similarly, some three-dimensional shapes can’t fit comfortably into ordinary three-dimensional space.
For instance, just as the sphere is the two-dimensional boundary of the three-dimensional ball, mathematicians have defined the hypersphere as the three-dimensional boundary of the four-dimensional ball–a space that’s hard to visualize but that can nevertheless be analyzed mathematically. Researchers have also discovered a three-dimensional analog of the torus, as well as an infinitely large family of more exotic three-dimensional spaces.
Around 1900, French mathematician Henri Poincaré wondered whether there’s an easy way to tell when a given closed three-dimensional space is a distorted version of the hypersphere. Poincaré made a daring conjecture. To recognize a hypersphere, he guessed, all that’s needed is information about one-dimensional curves in the space. If every closed loop of thread in the space can be drawn in to a single point, then the space is a hypersphere in disguise, he hypothesized. On a torus, by contrast, a loop that goes around the hole can’t be pulled tight to a single point.
Poincaré’s conjecture is one of the simplest possible questions to ask about three-dimensional spaces, yet it has stumped mathematicians from Poincaré’s time to the present. Surprisingly, higher-dimensional spheres turn out to be more amenable to analysis. Decades ago, mathematicians proved the corresponding conjectures for spheres of four dimensions and higher.
Geometric building blocks
In the late 1970s, mathematician William Thurston, now at the University of California, Davis, envisioned a way to tame the menagerie of three-dimensional spaces–an idea that gave mathematicians a roadmap for proving the Poincaré conjecture. The key, Thurston suspected, was in an analogy between the geometry of three-dimensional spaces and that of two-dimensional surfaces.
Every closed surface can be distorted into a particular shape with an especially uniform geometry. For starfish, tables, and telephone poles, that most uniform shape is simply the sphere, which looks the same at every point.
Among tori, the doughnut surface is more homogeneous than the coffee cup, but it is not perfectly uniform. Points on the outer ring are positively curved, like a sphere, while points on the inner ring are negatively curved, like a saddle’s central point. However, mathematicians have found a way to conceptualize a completely uniform torus, in which each small patch of the torus has the same geometric structure as a flat piece of paper.
All other two-dimensional surfaces–the tori with multiple holes–can be given what’s called hyperbolic geometry, which makes the surfaces negatively curved at all points.
Among closed surfaces, spherical, flat, and hyperbolic geometry are mutually exclusive. Breaking down these surfaces into geometric types thus gives a way to distinguish two-dimensional spheres, for example, from other surfaces. A similar breakdown for three-dimensional spaces, Thurston realized, would give mathematicians a useful tool for distinguishing hyperspheres from other shapes, the goal of the Poincaré conjecture.
Mathematicians have known for decades that three-dimensional spaces can’t be categorized as neatly as two-dimensional surfaces can. Some spaces, for instance, consist of a hyperbolic chunk and a flat chunk sewn together. Other spaces have geometric structures that don’t match any of spherical, flat, or hyperbolic geometry.
In pioneering work, Thurston proposed that there is nevertheless a precise way to classify the geometry of three-dimensional spaces. Each closed space, he conjectured, can be given a special geometric structure built from components selected from eight geometric types. Three of the eight are spherical, flat, and hyperbolic geometry; the other five are slightly more complicated but still uniform geometries. Thurston, who proved large portions of his conjecture, was awarded a Fields Medal–mathematics’ version of a Nobel prize–in large part for this body of work.
“What Thurston proposed was a revolutionary idea that went well beyond the Poincaré conjecture,” Cheeger says.
Erasing the bar
If Thurston’s conjecture can be proved, the Poincaré conjecture will follow automatically. The logic goes more or less like this: In a closed three-dimensional space, if all loops of thread can be pulled tight to a point, mathematicians know that the only one of the eight geometries that can fit the space is spherical geometry. That means that no matter how convoluted the space appears, it must simply be a distorted version of the hypersphere.
After Thurston’s work, mathematicians who wanted to prove the Poincaré conjecture could focus on demonstrating that Thurston’s vision of three-dimensional spaces is correct. By the early 1990s, Richard Hamilton of Columbia University had proposed a technique that he hoped would do just that–show that each three-dimensional space can be smoothed out into Thurston’s special pieces. He defined a method, called the Ricci flow, for changing the shape gradually at each point to make the space more uniform. His equation resembles the physics equation that describes how heat spreads through a material.
“If you take a body where parts are hot and parts are cold and you let it stand, heat tends to flow by itself until the temperature is even,” Milnor says. “In Hamilton’s process, you have a manifold that is very curved in some places, maybe flat or negatively curved in other places, and you just let the curvature flow and try to even itself out.”
For instance, the Ricci flow would make an egg-shaped surface gradually flatten out on the ends and bulge even more in the middle, getting closer and closer to a perfect sphere.
Hamilton was aware, however, that the flow would not always produce a uniform geometry. At any point in the space, the flow is determined mainly by the local geometry, not by the overall shape of the space. So, sometimes the geometry of one part of the space might change much faster than that of another part, producing a highly uneven geometry overall.
For example, picture a dumbbell–two weights connected by a thin bar–each portion of which is flowing with a mind of its own. The bar wants to even out its geometry with the weights to turn the whole thing into a nicely rounded sphere. Each weight, on the other hand, wants to make itself as spherical as possible. In the three-dimensional version of the dumbbell, depending on the initial geometry, the weights may predominate, growing rounder and rounder while the bar stretches into a long, thin neck.
Hamilton’s idea for dealing with this difficulty was simply to snip out the neck at some appropriate point, continue the Ricci flow on the pieces, and glue the neck back in at the end. The resulting shape would have the right kinds of building blocks for Thurston’s conjecture. But for more complicated shapes than the dumbbell, he couldn’t show that these necks were the only extreme geometric forms the flow would produce. Other extremities, such as awkward protrusions he called cigars, might result.
What’s more, perhaps every time the flow evened out one portion of the space, that portion’s extreme shape would have moved somewhere else, like bulges in a rug that is being fit into a room too small for it. Extreme geometric features might cycle around and around, without the whole space ever growing uniform.
These questions dogged Hamilton and his followers for more than a decade. Then last November, Perelman sent several mathematicians an e-mail, saying only that he had posted a paper on the Internet that might be of interest to them. In the paper, he writes that his work “removes the major stumbling block in Hamilton’s approach to geometrization.” Although the posted paper makes no reference to the Poincaré conjecture, experts in the field immediately realized what he was driving at.
Music of the spheres
In the early 1990s, working in the United States, Perelman had emerged as a major player in Riemannian geometry, which studies subjects such as curvature. “In that domain he was considered a phenomenon at that time, incredibly brilliant,” recalls Cheeger.
Then abruptly, Perelman all but vanished from the mathematical scene. In 1995, he turned down job offers from several top universities and returned to Russia. When U.S. mathematicians asked Perelman’s colleagues at the Steklov Institute what he was working on, they generally replied that they had no clue.
Some mathematicians speculated that Perelman had quit mathematics. Every now and then, however, one or another mathematician would receive an e-mail from Perelman with probing, insightful questions. “All of a sudden, there would be concrete evidence that he was following certain developments,” Cheeger says.
Once Perelman’s first paper on the Ricci flow appeared on the Internet in November 2002, rumors started flying that he had proven the Poincaré conjecture and Thurston’s geometrization conjecture. On March 10, Perelman posted a second paper that developed the ideas in his first paper and explicitly claimed a proof of the two conjectures. He has promised a third paper with a few remaining details.
This spring, Perelman visited the United States to present lectures on his work in Cambridge, Mass., and Stony Brook. So far, he has answered all the questions raised about his work, several mathematicians told Science News.
To understand the behavior of the Ricci flow, Perelman devised a way to capture a specific characteristic of any three-dimensional space. Roughly, he described what the pitch of a space would be if someone could ring the space like a bell.
Perelman then proved that as the space slowly morphs under the Ricci flow, its pitch gets higher and higher.
Perelman’s result immediately shows that the geometry of a space can’t cycle around under the Ricci flow–if it did, its pitch would be unchanged after each cycle. Perelman claims that the result about pitch, together with other ideas that he develops in his papers, also does away with the possibility of cigars and other potential obstacles to carrying out Hamilton’s program.
“Perelman’s results are as spectacular as the Poincaré conjecture,” says Dennis Sullivan, a mathematician at Stony Brook. “In just a few pages of work, he puts a hand grenade in the brick wall Hamilton had run into and blows a hole through it.
Whether that has enabled him to crawl through to the meadow on the other side remains to be seen.”
Many mathematicians have accepted the correctness of Perelman’s result about the pitch of a space, but they have not finished studying the portions of Perelman’s papers that explore the ramifications of the result. Once Perelman’s papers have been published, if no one exposes a hole in his work within 2 years, he will be eligible for the Clay Institute’s prize.
For many mathematicians, however, the appeal of the Poincaré conjecture lies beyond the million-dollar prize and accompanying fame. “It’s important for the same reason Beethoven’s Ninth Symphony is important,” Sullivan says. “It’s great.”
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