NPR 访谈

http://www.npr.org/templates/story/story.php?storyId=5591652

Scott Simon:

今年8月将在西班牙召开国际数学家大会，届时很可能宣布一个著名数学难题被解决
的消息。这个难题被称为庞卡莱猜想，但是对于它的解决，现在还有很多争论。来
自斯坦福大学的Keith Devlin教授，将为大家作一介绍。

Keith Devlin:

庞卡莱猜想是数学中最大的难题之一，也是克雷数学研究所的七道千禧数学难题之
一。它是1904年由伟大的法国数学家亨利·庞卡莱提出的。许多大数学家都对这个
问题作过研究。他们有的还给出了证明，但不久都发现有错。

这个问题就是要确定我们所生活的空间的形状。这是庞卡莱很感兴趣的问题。他是
一位伟大的数学家，甚至比爱因斯坦还要更早地发现了狭义相对论。那么空间的形
状是指什么呢？对数学家来说，就是拓扑形状。在拓扑学中，我们对距离和弯曲程
度不感兴趣，所以用拓扑学术语来说，一个网球，高尔夫球或者足球都是同样的东
西。而甜麦圈就不同了，因为中间有个洞。所以如果问我们所居住的空间是什么形
状，困难就在于我们生活在空间的内部。

你能区分球面和甜麦圈，也就是数学家所称的环面吗？从外部来看，我们很容易看
到它们是不同的，因为环面有一个洞。但是对于生活在曲面上的二维生物来说，它
们怎样来区分球面和环面呢？庞卡莱猜想的挑战就在于我们要从空间内部来决定空
间的形状。这个问题长期以来进展缓慢。重要的进展发生在上个世纪80年代，一位
美国数学家理查德·哈密尔顿引入了Ricci流的方法，给出了用Ricci流证明庞卡莱
猜测的框架。但是他自己没有完全给出证明。到了2003年，一位受人尊敬的俄国数
学家，普莱尔曼在网上公布了三篇文章，声称这三篇文章给出了证明庞卡莱猜想的
一个描述。世界各地的数学家都很兴奋，开始仔细阅读这些文章，试图给出普莱尔
曼所声称的证明。没有数学家发现普莱尔曼文章中的任何错误。但是仍然有一些缺
口需要弥补，所以没人敢说这个证明已经很完整了。于是就出现了这样的情况，人
们经常互相打听，"（庞卡莱猜想）已经证明了没有？"

直到最近，两位中国数学家，其中一位在美国工作。他们写了一篇300多页的文章，
宣称补上了普莱尔曼文章中的所有缺口，从而给出了庞卡莱猜测的完整证明。

哈密尔顿等权威人物，也高度评价这两位中国数学家的工作。

普莱尔曼是一个非常不愿抛头露面的人。他最初公开文章时，曾到美国访问，给了
一系列演讲。许多著名数学家参加了他的讲座。当他回到俄国以后，当人们试图联
系他，请他解释证明中的某些步骤时，却再也找不到他了。他又开始了隐居生活。
他对百万美元奖金也没有兴趣。他把文章放到网上后就不管了。这令许多数学家都
非常困扰。

Scott Simon:

那么解决这个问题究竟有什么好处呢？

Keith Devlin:

数学家思考这个问题已经有很多年了。永远不要低估其重要性。为了解决这个了不
起的问题，已经推动了许多崭新的数学分支的诞生。我经常想，这种情况就如同山
顶发生的雪崩，你并不知道这些雪块会奔向何方，但是有一点可以肯定。就是雪块
所及之处，都会产生巨大的冲击。

Mathematician May Have Solved 100-Year-Old Problem

NPR (National Public Radio) Programm

Scott Simon:

The international congress of mathematicians on August will announce tha
t a famous high complicated math problem has been solved. It's called Po
incare conjecture, but there is much to debate about how and whether it
was ever solved. Keith Devlin, a math guy from Stanford university. Than
k 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 m
ost famous mathematicians of all Frenchman, Henri Poincare. Many great m
athematicians are have worked on it. They proved it and found their proo
fs shut down in a few weeks later.

It was determing the shape of the space that we lived in. This was Poinc
are interested in. He almost invented Relativity before Einstein. The qu
estion 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. B
ut the donut will be a little different, there it got a hole in the midd
le. So the question is, what is the topological shape of the space we li
ve in. What makes us difficult to answer for physicist is trying to answ
er from the inside. It has a three dimensional analogue, it's like a sph
ere or more like a donut shape.

Can you distinguish a surface, like a surface of the sphere from a surfa
ce of a donut, which mathematicians call a torus. From the outside, we c
an see that they are different, because the torus has a hole in the midd
le. But the two dimensional creature living on the surface, a creature f
or what the whole surface is the world. How could that creature determin
e whether it actually live on the surface of the sphere or the surface o
f the donut. Poincare's challenge was to find what clues nature offered
in order to determine the shape from inside. People make progress, but n
o 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 s
howed 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 mat
hematician, Grigori Perelman, put up three papers on the internet, claim
ing that those threee papers outlined a proof of Poincare conjecture. Ma
thematicians around the world were very excited and started to look at t
hese preprints on the web and try to figure out what the proof was Perel
man 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 Uni
ted States, wrote a paper, a three hundred paper. In that paper, they cl
aim to have actually filled in all the gaps in Perelman's proof, correct
ed everything, provided the missing steps and have now nailed a complete
proof.

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

Perelman is a very reclusive guy. When he first posted his papers, he di
d visite United States, give a series of lectures. Many famous mathemati
cians attended the lectures. When he come back to Russia, when people tr
y to contact him and say that there is a step on this paper I don't unde
rstand, can you explain out. He didn't respond, he just went into reclus
ive life in Russia. He has no interest in the million dollar prize. He j
ust 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 t
he world?

Keith Devlin:

Mathematicians have asked themselves for many years "what will be the ca
se?", "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 di
rection these snow are going to flow, but one thing you do know is, a lo
t of snow is going to flow in a lot of different directions that engende
r a huge impact below.