NPR 訪談

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

Scott Simon:

今年8月將在西班牙召開國際數學家大會，屆時很可能宣布一個著名數學難題被解決
的消息。這個難題被稱為龐卡萊猜想，但是對於它的解決，現在還有很多爭論。來
自斯坦福大學的Keith Devlin教授，將為大家作一介紹。

Keith Devlin:

龐卡萊猜想是數學中最大的難題之一，也是克雷數學研究所的七道千禧數學難題之
一。它是1904年由偉大的法國數學家亨利·龐卡萊提出的。許多大數學家都對這個
問題作過研究。他們有的還給出了證明，但不久都發現有錯。

這個問題就是要確定我們所生活的空間的形狀。這是龐卡萊很感興趣的問題。他是
一位偉大的數學家，甚至比愛因斯坦還要更早地發現了狹義相對論。那麼空間的形
狀是指什麼呢？對數學家來說，就是拓撲形狀。在拓撲學中，我們對距離和彎曲程
度不感興趣，所以用拓撲學術語來說，一個網球，高爾夫球或者足球都是同樣的東
西。而甜麥圈就不同了，因為中間有個洞。所以如果問我們所居住的空間是什麼形
狀，困難就在於我們生活在空間的內部。

你能區分球面和甜麥圈，也就是數學家所稱的環面嗎？從外部來看，我們很容易看
到它們是不同的，因為環面有一個洞。但是對於生活在曲面上的二維生物來說，它
們怎樣來區分球面和環面呢？龐卡萊猜想的挑戰就在於我們要從空間內部來決定空
間的形狀。這個問題長期以來進展緩慢。重要的進展發生在上個世紀80年代，一位
美國數學家理查德·哈密爾頓引入了Ricci流的方法，給出了用Ricci流證明龐卡萊
猜測的框架。但是他自己沒有完全給出證明。到了2003年，一位受人尊敬的俄國數
學家，普萊爾曼在網上公布了三篇文章，聲稱這三篇文章給出了證明龐卡萊猜想的
一個描述。世界各地的數學家都很興奮，開始仔細閱讀這些文章，試圖給出普萊爾
曼所聲稱的證明。沒有數學家發現普萊爾曼文章中的任何錯誤。但是仍然有一些缺
口需要彌補，所以沒人敢說這個證明已經很完整了。於是就出現了這樣的情況，人
們經常互相打聽，"（龐卡萊猜想）已經證明了沒有？"

直到最近，兩位中國數學家，其中一位在美國工作。他們寫了一篇300多頁的文章，
宣稱補上了普萊爾曼文章中的所有缺口，從而給出了龐卡萊猜測的完整證明。

哈密爾頓等權威人物，也高度評價這兩位中國數學家的工作。

普萊爾曼是一個非常不願拋頭露面的人。他最初公開文章時，曾到美國訪問，給了
一系列演講。許多著名數學家參加了他的講座。當他回到俄國以後，當人們試圖聯
系他，請他解釋證明中的某些步驟時，卻再也找不到他了。他又開始了隱居生活。
他對百萬美元獎金也沒有興趣。他把文章放到網上後就不管了。這令許多數學家都
非常困擾。

Scott Simon:

那麼解決這個問題究竟有什麼好處呢？

Keith Devlin:

數學家思考這個問題已經有很多年了。永遠不要低估其重要性。為了解決這個了不
起的問題，已經推動了許多嶄新的數學分支的誕生。我經常想，這種情況就如同山
頂發生的雪崩，你並不知道這些雪塊會奔向何方，但是有一點可以肯定。就是雪塊
所及之處，都會產生巨大的衝擊。

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.