How can an ant determine whether it is sitting on a ball or a bagel? How could the ancient Greeks have known that the earth is not flat? The difficulty with solving problems of this nature lies in the fact that a ball, a sphere with a hole in it, and a flat plane look exactly the same in the neighborhood of the observer.

Topology developed in the 19th century as an offshoot of geometry, but it has become a mathematical discipline in its own right over the years. Topologists study qualitative questions about geometrical objects (surfaces and spheres) in two-, three-, and higher-dimensional spaces. These objects—imagine them made out of clay or Play-Doh—may be transformed into one another by stretching and squeezing but without tearing, piercing, or gluing separate bits together. Spheres or cubes, for example, can be manipulated into egg-shaped forms or pyramids and are therefore considered topologically equivalent. A ball, on the other hand, cannot be transformed into a doughnut without punching a hole in it. Finally, a pretzel is not equivalent to a doughnut, topologically speaking, because it has three holes.

The number of holes that an object sports is an important characteristic in topology. But how does one define holes mathematically? Nothing, surrounded by a border? No, that would not do. Theoretically one can proceed as follows: Sling a rubber band around the object to be investigated. If it is a ball, an egg, or another object without a hole, then no matter where the rubber band was slung, one can always shrink the loop continuously to a single point. But if one takes an object such as a doughnut or pretzel and wraps the rubber band along the surfaces, this does not always work. If the rubber band is

threaded through one of the holes, the loop gets caught upon tightening. This is why, in topology, bodies are classified according to the number of their holes.

Surfaces of three-dimensional bodies, such as balls or doughnuts, are called two-dimensional manifolds. What about three-dimensional manifolds—that is, the surfaces belonging to four-dimensional bodies? To investigate these objects, the French mathematician Henri Poincaré (1854–1912) proceeded in the same manner as with two-dimensional manifolds. And he made a bold claim: Three-dimensional manifolds on which any loop can be shrunk to a single point are topologically equivalent to a sphere. When he attempted to provide a proof for this assertion, however, he got into hot water. His attempt proved to be a failure. So in 1904 the word “claim” was changed to “conjecture.”

In the course of the second half of the 20th century, mathematicians succeeded in proving that Poincaré’s conjecture holds true for four-, five-, six-, and all higher-dimensional manifolds. But the original conjecture for three-dimensional manifolds remained unsolved. This was all the more frustrating, since four-dimensional space, where one finds the three-dimensional manifolds under investigation, represents the space-time continuum in which we live.

In the spring of 2003 it was announced that Grigori Perelman, a Russian mathematician from the Steklov Institute in St. Petersburg, might have been successful in providing a proof for the Poincaré conjecture. Similarly to his famous colleague Andrew Wiles, who had “cracked” Fermat’s last theorem in 1995, Perelman too had been working in complete isolation and solitude for eight years. His efforts culminated in three essays that he posted on the Internet, one in November 2002, one in March 2003, and one in July 2003.

Scientists in the former Soviet Union do not have an easy life, and Perelman was no exception. In one of the papers a footnote mentions that he had been able to scrape by financially only thanks to the money he had made as a graduate fellow at American research institutes. In April

2003 Perelman gave a series of lectures in the United States. The intention was to share his results with his colleagues and get their feedback.

In his proof Perelman relied on two tools, developed in previous work by two colleagues. The first is the so-called geometrization conjecture, which was formulated in the 1980s by William Thurston from the University of California, today at Cornell University. It is a known fact—to mathematicians, that is—that three-dimensional manifolds can be decomposed into basic elements. Thurston’s conjecture says that these elements can only be of eight different shapes. Proving this conjecture was an even more ambitious undertaking than proving the Poincaré conjecture. The latter only aims to identify the manifolds that are equivalent to a sphere. Thurston managed to prove his conjecture, but only after making certain additional assumptions. For this feat he was awarded the highest mathematical award, the Fields Medal, in 1983. As far as the most general version of the conjecture is concerned, the version without the assumptions made by Thurston, the conjecture remained unproven, however.

The second tool on which Perelman relied is the so-called Ricci flow. This concept was introduced to topology by Richard Hamilton of Columbia University. Basically, the Ricci flow is a differential equation that is related to the dispersion of heat in a body. In topology the Ricci flow describes the development of a manifold that changes continuously, at a rate inversely proportional to the manifold’s curvature at every point. This permits a deformed body to find a state of constant curvature. Sometimes the Ricci flow allows manifolds to split into several components. Hamilton proved—albeit also under certain limiting conditions—that these components could only take on the eight shapes predicted by Thurston.

Perelman succeeded in extending the theory of the Ricci flow to a complete proof of the general version of Thurston’s geometrization conjecture. From this it follows, as a corollary, that Poincaré’s conjecture is correct: If a loop around a three-dimensional manifold can be shrunk to a point, the manifold is equivalent to a sphere.

The proof that Perelman presented during his lecture series still requires in-depth verification. This could take several years. It would not be the first time that a proof has been found to be lacking only after it has been presented to the mathematical community. In 2002, for example, a year before Perelman’s publications, the English mathematician Martin Dunwoody posted what he believed was a proof for the Poincaré conjecture on the Internet. (See Chapter 5.) Much to his chagrin, a colleague soon noticed that one claim which Dunwoody had made in the course of his five-page paper was not fully proven.

So far there has been no reason to think that the proof Perelman described in his papers is incorrect. No gaps have been uncovered; no errors have been found. Should his proof pass all future tests, it looks as if the Russian mathematician would be the recipient of the first Clay Prize, to be awarded to mathematicians who manage to solve one of the seven “millennium problems.” With the $1 million prize money, Perelman would no longer have to rely on the meager honorariums that guest lecturers generally receive.

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