This morning on NPR's Morning Edition, I heard a story about how a British mathematician claims to have solved one of math's great unanswered questions, the Poincare Conjecture. You can listen to the story here. They spoke to Arthur Jaffe of the Clay Mathematics Institute, which is offering a $1 million prize for each of seven great unsolved problems, about this problem and its possible solution.
Like many great unsolved problems, the Poincare Conjecture can be broken down into a bunch of smaller problems. With some of these conjectures, if one solves a smaller, more focused question, one gets the desired larger result. That was the case with the so-called Last Theorem of Fermat (it can now be properly called a Theorem since it has been proven; before that it was merely a conjecture), which was solved by Andrew Wiles by proving the Taniyama-Shimura conjecture.
Poincare's Conjecture differs from Fermat's in that most of it has already been solved. The Conjecture has to do with classifying geometric things known as "manifolds". One talks about manifolds of a certain number of dimensions. For example, a two-dimensional sphere is just what you think it is - something that looks like a beach ball. The object itself is represented in three dimensions, which is how we see it, but is called a two-dimensional object because its surface is similar to the two-dimensional plane. (Mathematicians say that any neighborhood on the two-dimensional sphere is like the two-dimensional plane. This should be apparent to you because as far as you can see, the surface of the Earth on which you now sit looks flat.)
Anyway, the Conjecture attempts to state when two manifolds are mathematically the same thing and when they are not. A two-dimensional sphere is mathematically the same thing as a two-dimensional box because you can transform one to the other without poking holes or tearing the surface. You can't change a beach ball into an inner tube by squishing or stretching or flattening it, so they're considered different mathematical structures. Of course, mathematicians need more exactness than this, so Poincare developed a way of identifying each manifold with a "group", which is a set of numbers and an operation (such as addition) that follows certain rules. (That's an oversimplification, but it'll do.) The benefit here is that it's easier mathematically to deal with groups. Two manifolds are said to be "isomorphic", which is a fancy way of saying "identical", if they have the same "fundamental group" associated with them.
Poincare himself proved the Conjecture for all two-dimensional manifolds. They are spheres (beach balls), toruses (inner tubes) and projective planes, which is what you get when you take a Moebius strip and glue its edges together. Unlike its other two-dimensional siblings, you cannot represent the projective plane in three dimensions, so I can't give you a better picture of what it looks like than that.
The Conjecture is known to be true in all dimensions other than three. Six dimensions and higher were fairly easy to solve. Five dimensions and four dimensions were harder to solve and were accomplished fairly recently, in 1960 by Stephen Smale and in 1980 by Michael Freedman (for which he won a MacArthur Genius Grant), respectively.
The three-dimensional case is the really tricky one. One thing that makes it tricky is that there's more than one kind of "fundamental group" which can be associated with manifolds. In the three-dimensional case, Poincare found an object which has one kind of fundamental group identical to the three-dimensional sphere, but not another. For whatever the reason, the three-dimensional case is more complex than that of other dimensions.
You may ask what the point of all this is. Why do we care about these silly things? I can (as Albert Jaffe did) point out that an awful lot of abstract math has turned out to have applications in unexpected places, like particle physics and cryptography, but I believe there is value in learning for its own sake. The mathematician G.H. Hardy wrote a book called A Mathematician's Apology in which he expressed regret for not doing anything useful but was proud that he added to the world's knowledge. Turns out Hardy spoke too soon - his work in number theory has had wide application in cryptography. So who can say where Poincare may eventually lead us?
As a math major and math geek (as if you couldn't tell), I'm always happy to hear about my favorite subject in the news. Math doesn't get a lot of mainstream respect. I'd love to see a TV show make math look sexy in the way CSI glamorizes science, but even I'm hard-pressed to imagine how it could be done. For now, I'll settle for an NPR segment on one of the subject's enduring challenges, and the thought that some now-obscure professor may be on his way to claiming a million bucks.Posted by Charles Kuffner on April 16, 2002 to Technology, science, and math