a journal of a researcher

Tuesday, August 01, 2006

Mathematician May Have Solved 100-Year-Old Problem

NPR (National Public Radio) Programm

Scott Simon:
The international congress of mathematicians on August will announce that a famous high complicated math problem has been solved. It’s called Poincare conjecture, but there is much to debate about how and whether it was ever solved. Keith Devlin, a math guy from Stanford university. Thank you for being with us.

Keith Devlin:
The biggest unsolved problem in mathematics, is one of the seven-million dollar millennium prize problems. It was posted in 1904 by one of the most famous mathematicians of all Frenchman, Henri Poincare. Many great mathematicians are have worked on it. They proved it and found their proofs shut down in a few weeks later.

It was determing the shape of the space that we lived in. This was Poincare interested in. He almost invented Relativity before Einstein. The question is what is the shape of the space we are living in? by shape, we mean what a mathematician call a topological shape. We don’t worry about the distance and how much things curve exactly, so in topological terms, a tennis ball, a golf ball, a football, a soccer ball, all are same. But the donut will be a little different, there it got a hole in the middle. So the question is, what is the topological shape of the space we live in. What makes us difficult to answer for physicist is trying to answer from the inside. It has a three dimensional analogue, it’s like a sphere or more like a donut shape.

Can you distinguish a surface, like a surface of the sphere from a surface of a donut, which mathematicians call a torus. From the outside, we can see that they are different, because the torus has a hole in the middle. But the two dimensional creature living on the surface, a creature for what the whole surface is the world. How could that creature determine whether it actually live on the surface of the sphere or the surface of the donut. Poincare’s challenge was to find what clues nature offered in order to determine the shape from inside. People make progress, but no one seems to come close to proving it.

Significant progress made in 1980’s was an American mathematician called Richard Hamilton, who took up ideas about fluid flows essentially and showed how you could use his ideas to prove the Poincare conjecture, but he couldn’t push it through. Then in 2003, a very respected Russsian mathematician, Grigori Perelman, put up three papers on the internet, claiming that those threee papers outlined a proof of Poincare conjecture. Mathematicians around the world were very excited and started to look at these preprints on the web and try to figure out what the proof was Perelman claims behind. No mathematician has found any mistakes or any major errors in what Perelman was doing. And yet there was still some gaps and nobody was prepared to say for certain this proof is correct. So we got this bizarre situation where there were thoughts of “is it a proof?”, “isn’t it a proof?”.

Then very recently, two Chinese mathematicians, one of them based in United States, wrote a paper, a three hundred paper. In that paper, they claim to have actually filled in all the gaps in Perelman’s proof, corrected everything, provided the missing steps and have now nailed a complete proof.

People like Richard Hamilton, one of the grandad in the field have started to say that these guys have got it.

Perelman is a very reclusive guy. When he first posted his papers, he did visite United States, give a series of lectures. Many famous mathematicians attended the lectures. When he come back to Russia, when people try to contact him and say that there is a step on this paper I don’t understand, can you explain out. He didn’t respond, he just went into reclusive life in Russia. He has no interest in the million dollar prize. He just put the paper on the internet and then it has nothing to do with it. It’s very frustrating for western mathematicians of course.

Scott Simon:
Besides the intellectural satisfaction of finally resolving this problem, what will happen for resolving this problem? What the importance for the world?

Keith Devlin:
Mathematicians have asked themselves for many years “what will be the case?”, “if the Poincare conjecture is true?”. You can’t overestimate just how much could flow from new mathematics that has brought into solving this magnitude problem. I often think this kind of steps as to starting an avalanche on the top of the mountain, you are not quite sure which direction these snow are going to flow, but one thing you do know is, a lot of snow is going to flow in a lot of different directions that engender a huge impact below.


  • Here's the Lemire conjecture: we have reached a point where it will become increasingly difficult to do any meaningful mathematics by hand (as opposed to using computers).

    By Anonymous Daniel Lemire, at 8:55 AM  

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