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April 15, 2003, Tuesday


Russian Reports He Has Solved a Celebrated Math Problem

By SARA ROBINSON (NYT) 1310 words
A Russian mathematician is reporting that he has proved the Poincaré Conjecture, one of the most famous unsolved problems in mathematics.

The mathematician, Dr. Grigori 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 accepted for publication in a refereed research journal and survives two years of scrutiny, Dr. Perelman could be eligible for a $1 million prize sponsored by the Clay Mathematics Institute in Cambridge, Mass., for solving what the institute identifies as one of the seven most important unsolved mathematics problems of the millennium.

Rumors about Dr. 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. Dr. Perelman will give another lecture series at the State University of New York at Stony Brook starting on Monday.

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

For two months, Dr. Tomasz S. Mrowka, a mathematician at M.I.T., has been attending a seminar on Dr. Perelman's work, which relies on ideas pioneered by another mathematician, Richard Hamilton. So far, Dr. Mrowka said, every time someone brings up an issue or objection, Dr. Perelman has a clear and succinct response.

''It's not certain, but we're taking it very seriously,'' Dr. Mrowka said. ''He's obviously thought about this stuff very hard for a long time, and it will be very hard to find any mistakes.''

Formulated by the French mathematician Henri Poincaré in 1904, the Poincaré 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 Poincaré Conjecture makes a corresponding statement about the three-dimensional sphere, a concept that is a stretch for the nonmathematician to visualize. It says, essentially, that the three-dimensional sphere is the only bounded three-dimensional space with no holes.

''The hard part is how to tell globally what a space looks like when you can only see a little piece of it at a time,'' said Dr. Benson Farb, a professor of mathematics at the University of Chicago. ''It was pretty reasonable to think the earth was flat.''

That conjecture is notorious for the many ''solutions'' that later proved false. Indeed, Poincaré himself demonstrated that his earliest version of his conjecture was wrong. Since then, dozens of mathematicians have asserted that they had proofs until experts found fatal flaws.

Although many experts say they are excited and hopeful about Dr. 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.

That was the case in 1993 with Dr. Andrew J. Wiles, the Princeton professor whose celebrated proof for Fermat's Last Theorem turned out to have a serious gap that was repaired after months of effort by Dr. Wiles and a former student, Dr. Richard Taylor.

Dr. Perelman's results go well beyond a solution to the problem at hand, as did those of Dr. Wiles. Dr. Perelman's results say he has proved a much broader conjecture about the geometry of three-dimensional spaces made in the 1970's. The Poincaré Conjecture is but a small part of that.

Dr. Perelman's personal story has parallels to that of Dr. Wiles, who, without confiding in his colleagues, worked alone in his attic on Fermat's Last Theorem. Though his early work has earned him a reputation as a brilliant mathematician, Dr. Perelman spent the last eight years sequestered in Russia, not publishing.

In his paper posted in November, Dr. Perelman, now in his late 30's, 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 that he has proved what is known as the Geometrization Conjecture, a complete characterization of the geometry of three-dimensional spaces.

Since the 19th century, mathematicians have known that a type of two-dimensional space called a manifold can be given a rigid geometric structure that looks the same everywhere. Mathematicians could list all the possible shapes for two-dimensional manifolds and explain how a creature living on the surface of one can tell what kind of space he is on.

In the 1950's, however, a Russian mathematician proved that the problem was impossible to resolve in four dimensions and that even for three dimensions, the question looked hopelessly complex.

In the early 1970's, Dr. William P. Thurston, a professor at the University of California at Davis, conjectured that three-dimensional manifolds are composed of many homogeneous pieces that can be put together only in prescribed ways and proved that in many cases his conjecture was correct. Dr. Thurston won a Fields Medal, the highest honor in mathematics, for his work.

Dr. 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 Poincaré's famous question. Dr. Perelman's approach uses a technique known as the Ricci flow, devised by Dr. 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. Dr. 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,'' Dr. Mrowka said. ''His work relies heavily on Hamilton's work but makes amazing new contributions to that program.''

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

Even if Dr. 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,'' Dr. Mrowka said. ''Either he's done it or he's made some really significant progress, and we're going to learn from it.''

CAPTIONS: Photo: The French mathematician Henri Poincaré lived from 1854 to 1912. (Hulton-Deutsch Collection/Corbis)

Copyright 2003 The New York Times Company