## Celebrated Math Problem Solved?

- By Sara Robinson
- Apr. 16 2003 00:00

The mathematician, Grigory Perelman of the Steklov Institute of Mathematics of the Russian Academy of Sciences in St. Petersburg, is describing his work in a series of papers, not yet completed.

It will be months before the proof can be thoroughly checked. But if true, it will verify a statement about three-dimensional objects that has haunted mathematicians for nearly a century, and its consequences will reverberate through geometry and physics.

If his proof is published in a refereed research journal and survives two years of scrutiny, Perelman could be eligible for a $1 million prize sponsored by the Clay Mathematics Institute in Cambridge, Massachusetts, for solving what the institute identifies as one of the seven most important unsolved mathematics problems of the millennium.

Rumors about Perelman's work have been circulating since November, when he posted the first of his papers reporting the result on an Internet preprint server.

Last week at the Massachusetts Institute of Technology, he gave his first formal lectures on his work to a packed auditorium. Perelman will give another lecture series at the State University of New York at Stony Brook starting Monday.

Perelman declined to be interviewed, saying publicity would be premature.

Formulated by French mathematician Henri Poincare in 1904, the Poincare Conjecture is a central question in topology, the study of the geometrical properties of objects that do not change when the object is stretched, twisted or shrunk.

The hollow shell of the surface of the earth is what topologists would call a two-dimensional sphere. It has the property that every lasso of string encircling it can be pulled tight to one spot.

On the surface of a doughnut, by contrast, a lasso passing through the hole in the center cannot be shrunk to a point without cutting through the surface.

Since the 19th century, mathematicians have known that the sphere is the only bounded two-dimensional space with this property, but what about higher dimensions?

The Poincare Conjecture makes a corresponding statement about the three-dimensional sphere, a concept that is a stretch for the non-mathematician to visualize. It says, essentially, the three-dimensional sphere is the only bounded three-dimensional space with no holes.

Although many experts say they are excited and hopeful about Perelman's effort, they also urge caution, noting that not all of the proof has been written down and that even the most reliable researchers make mistakes.

Though Perelman's early work has earned him a reputation as a brilliant mathematician, he spent the last eight years sequestered in Russia, not publishing. Without confiding in his colleagues, he worked alone in his attic on Fermat's Last Theorem.

In his paper posted in November, Perelman, now in his late 30s, thanks the Courant Institute at New York University, SUNY Stony Brook and the University of California at Berkeley, because his savings from visiting positions at those institutions helped support him in Russia.

His papers say he has proved what is known as the Geometrization Conjecture, a complete characterization of the geometry of three-dimensional spaces.

Perelman's work, if correct, would provide the final piece of a complete description of the structure of three-dimensional manifolds and, almost as an afterthought, would resolve Poincare's famous question. Perelman's approach uses a technique known as the Ricci flow, devised by Hamilton, who is now at Columbia University.

The Ricci flow is an averaging process used to smooth out the bumps of a manifold and make it look more uniform. Hamilton uses the Ricci flow to prove the Geometrization Conjecture in some cases and outlined a general program of how it could be used to prove the Geometrization Conjecture in all cases. He ran into problems, however, coping with certain types of large lumps that tended to grow uncontrollably under the averaging process.

"What Perelman has done is to figure out some new and interesting ways to tame these singularities," said Tomasz Mrowka, a mathematician at MIT. "His work relies heavily on Hamilton's work but makes amazing new contributions to that program."

If Perelman succeeds in resolving Poincare, he will probably share the Clay Mathematics Institute Award with Hamilton, mathematicians said.

Even if Perelman's work does not prove the Geometrization Conjecture, mathematicians said, it is clear that his work will make a substantial contribution to mathematics.

"This is one of those happy circumstances where it's going to be fun no matter what," Mrowka said. "Either he's done it or he's made some really significant progress, and we're going to learn from it."