Tuesday, April 15, 2003

The Poincare Conjecture The New York Times is reporting that a Russian mathematician named Grigori Perelman is announcing a proof of the Poincare conjecture. The article can be found here.

At the turn of the milennium, a panel of mathematicians was asked to identify the most important unsolved problems in mathematics. The Poincare conjecture made the list. A one million dollar prize has been offered by the Clay Mathematics Institute in Cambridge, Massachusetts to anyone who could solve it. If Perelman's work holds up (a very big if, given the number of false solutions that have been offered over the years), he will receive the prize (together with another mathematician, Richard Hamilton, a professor at Columbia University, on whose work Perelman has based his proof).

To understand what the Poincare conjecture asks, begin by considering the surface of a sphere. Draw a loop on this surface, and imagine that you gradually shrink the loop over the sphere's surface. The loop can be shrunk indefinitely, until it gets down to a single point. Contrast this with the surface of a doughnut. Here it is possible to draw a closed loop on the surface for which it is impossible to shrink it down to a point. For example, imagine that you grab the doughnut with your thumb and index finger, so that your fingers meet through the hole of the doughnut. The loop formed by your thumb and index finger can not be continuously deformed to a single point. Not if we're confined to moving around on the doughnut's surface, anyway. Similarly, if you tie a loop of string around the doughnut like a belt, then this loop also can not be deformed.

The surface of a sphere and the surface of a torus are examples of two-dimensional surfaces. It turns out that the sphere is essentially the only two-dimensional surface on which it is possible to shrink any loop to a point. But wait! What about a cube, or a tetrahedron? Well, it is indeed possible to shrink any loop to a point on the surface of a cube or a tetrahedron. However, mathematicians regard such surfaces as equivalent to a sphere. Imagine that your cube was made of rubber. It would then be possible to smooth out the corners of the cube, and gradually mold the whole surface until a sphere emerged. Any two surfaces that can be deformed into one another by such a "rubber-sheet" transformation are viewed as equivalent. The rubber can be pulled and stretched and molded at will, with the sole restriction being that the rubber must never be torn or cut. A sphere can not be deformed into a torus by such a transformation, so they are not equivalent.

But why should we stop at two-dimensions? Though they are rather hard to picture, it is a simple matter to define higher-dimensional surfaces, and in each dimension there is a surface that is clearly analagous to a sphere. Call that analog the n-dimensional sphere. The question is, if you are given an n-dimensional surface on which it is possible to shrink any closed loop to a point, is your surface necessarily equivalent to the n-dimensional sphere?

If the dimension is five or higher the answer is yes. That was shown by Stephen Smale in 1960. In 1981, Michael Freedman demonstrated that the answer is still yes in dimension four. Smale and Freedman both received Field's Medals (the mathematical equivalent of the Nobel Prize) for their troubles.

All of which should make the attentive reader wonder about dimension three. That's the Poincare conjecture. Specifically, the conjecture is that the answer remains yes in the incredibly complicated case of three-dimensional manifolds, and that is what Dr. Perelman believes he has shown.

Many proposed proofs of the conjecture have come and gone, all of them suffering from subtle flaws. The history of the conjecture is replete with people who devoted huge chunks of their lives to it, only to wind up defeated and insane. Hopefully Dr. Perelman will avoid that fate.


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