Mathematician May Have Solved 100-Year-Old Problem
SCOTT SIMON, host:
It's going to be a mathematics smackdown in Madrid this summer, the rumble of the numbers set. When the International Congress of Mathematicians convenes at the end of August, members are expected to announce that a famous, highly complicated math problem has been solved. It's called Poincare's Conjecture. But there is much debate about how and whether it was ever solved. Keith Devlin is our Math Guy here at WEEKEND EDITION, joins us from the campus of Stanford University to talk about it. Keith, thanks for being with us.
KEITH DEVLIN reporting:
Hi, Scott, Thanks for having me on.
SIMON: And mathematicians are really worked up about this, aren't they?
DEVLIN: They are indeed worked up about it. This is one of the biggest, biggest unsolved problems in mathematics. It's one of the seven million dollar millennium prize problems. It was posed in 1904 by one of the most famous mathematicians of all, the Frenchman Henri Poincare. Many great mathematicians have at various stages thought they have proved it, only to find their proofs shot down in flames a few weeks or months later. It's one of these gold ring problems that any mathematician would love to solve. It's like winning the Tour de France except you don't need testosterone do it.
(Soundbite of laughter)
SIMON: Keith, in 15 million words or less, explain Poincare's Conjecture to us.
DEVLIN: Yeah, it's really about determining the shape of space that we live in. And this is what Poincare's interested in - he was as much a physicist as a mathematician. In fact, he almost invented spatial relativity before Einstein did. He was a contemporary of Einstein. And the question is, what is the shape of the space that we live in. And by shape, we mean what mathematicians call a topological shape. We don't worry about the distance and how much things curve exactly, so in topological terms, a tennis ball, golf ball, a football, a soccer ball, are all the same, but a donut shape, a sort of tire inner tube, would be different because it's got a hole in the middle. And so the question is, what is the topological shape of the space we live in?
What makes that difficult to answer for physicists is trying to answer it from the inside. Is there a way from inside space of determining what its topological space is? Is it like a three-dimensional analog of a sphere? Or is it more like a donut shape?
The best way to get at this is to think of the two-dimensional analog, which is not the thing that Poincare was interested in, but just imagine surfaces. Can you distinguish a surface like a surface of a sphere from the surface of a donut, what mathematicians call a torus? From the outside we can see they're different because the torus has a hole in the middle. But a two-dimensional creature living on the surface, a creature for what that - for whom that surface is the world - how could that creature determine whether it was actually living on the surface of a sphere or the surface of a donut?
Poincare's challenge was to find what clues nature offered in order to determine the shape from inside. And just no one could make - well, people made progress - but no one seemed to come close to proving it.
SIMON: Tell us about this man who has come forward and posted his thinking, if you please, on the Web and says...
DEVLIN: Ahh. Well, now we get into the most tangled part of the entire tale. Progress started to be made - significant progress was made in the early 1980s with an American mathematician called Richard Hamilton, who took some ideas about fluid flows, essentially, and showed how you could use these ideas to start to build toward a proof of the Poincare conjecture, but he couldn't push it through. Then in 2003, a very respected Russian mathematician, Grigori Perelman, posted a series of three papers on the Internet, claiming that those three papers outlined a proof of the Poincare conjecture.
Mathematicians around the world were very excited and started looking at these preprints on the Web and started to try and figure out what the proof was that Perelman claimed he had. And no mathematicians have found any mistakes, or any major errors in what Perelman was doing, and yet there were still some gaps that people were unsure off. And nobody was prepared to say for certain this proof is correct. So we've got this bizarre situation where there's a sort of is it a proof, isn't it a proof, and no one seems to be able to get beyond that.
Then very recently, two Chinese mathematicians - one of them based in the United States - wrote a paper - a 300-page paper, incidentally, it's a huge, mammoth paper. And in that paper they claim to have actually filled in all the gaps of Perelman's proof, corrected everything, provided the missing steps, and have now nailed a complete proof. And people like Richard Hamilton, one of the granddaddies of the field, have actually started saying yup, it looks as though these guys have done it.
SIMON: Now, I understand there's some controversy over a million different things as to why he posted it on the Web and didn't submit it to a professional jury.
DEVLIN: Yeah, you know, Perelman is a very, very - I've never met him - he's a very reclusive guy. When he first published it - when he first posted these papers, he did visit the United States, gave a series of lectures, people - very famous mathematicians attended the lectures and sort of convinced themselves that this looked like it might well be a correct proof. But then he went back to Russia and when people tried to contact him, and saying, you know, there's a step on this page I don't understand and can you explain that, he didn't respond. He just sort of went into recluse - reclusive life in Russia, showed no interest in taking the steps to perform - that you'd have to perform if you wanted to claim this million-dollar prize. He'd simply put it on the Internet and then had nothing more to do with it, which was very frustrating for Western mathematicians, of course.
SIMON: Aside from the intellectual satisfaction of finally resolving this, if that's what's happened, what's the import for the world? I mean, will we be able to make pigs fly?
(Soundbite of laughter)
DEVLIN: You know, mathematicians have asked themselves for many years what would be the case if the Poincare conjecture is true. So, in a sense, there's no surprises that will follow from now knowing that it is true if indeed that's the case. You can't overestimate just how much could flow from the new mathematics that's brought into solving a problem of this magnitude. I often liken these kinds of steps as to starting an avalanche on the top of a mountain, you're not quite sure which direction the snow is going to fall, but one thing you do know is a lot of snow is going to fall in a lot of different directions and have a huge impact on the mountain below. And that's what's going to be the case here, I'm sure.
SIMON: Keith, thanks very much.
DEVLIN: My pleasure, Scott.
SIMON: Our math guy, Keith Devlin, executive director of the Center for the Study of Language Information at Stanford University. His most recent book is The Math Instinct: Why You're a Mathematical Genius Along With Lobsters, Birds, Cats and Dogs.
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