Celebrated Math Problem Solved, Russian Reports

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April 15, 2003
Celebrated Math Problem Solved, Russian Reports
By SARA ROBINSON


Russian mathematician is reporting that he has proved the Poincar� 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 Poincar� in 1904, the Poincar� 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?

Can someone post the Poincar� conjecture in its entirety ??

There have been quite a few claims of having a solution. There is, incidentally, a million-dollar prize. Some background and links here: www.ams.org...

Last year, a guy at Arizona State University called Nikitin claimed to have solved it, for example (a search should find him)
It�s heap big medicine mathematics-wise and up there with the Riemann conjecture. The conjecture isn�t too difficult to understand but the mathematics behind the various claims to having proved it isway beyond me.
It is very much as posted above and, in a nutshell, the hypothesis that every simply connected closed 3-manifold is homeomorphic to the 3-sphere has been known as the Poincar� conjecture. It has inspired topologists ever since, leading to many false proofs, but also to many advances in our understanding of the topology of manifolds.

If your maths is up to it, there�s a good commentary here (albeit in an appalling .pdf format)
www.claymath.org...

and some excellent further background here:
www.math.unl.edu...