前幾天（2022年7月3日）在新聞版《環球大觀》欄目讀到一篇文章：《頂級數學家有多厲害？厲害到你會覺得我在胡說…》，作者署名“知乎”。

俺作了下面的評論：

“該文謊話連篇。

克雷所確實通知佩雷爾曼獲獎，但拒絕的原因卻不是文中所說‘對錢不感興趣’，而是指責克雷所‘不公平’。至於‘面對記者的提問’則純屬杜撰，他獲克雷獎後沒有記者報道過對其的採訪。事實上，佩雷爾曼對錢感興趣，因為他在‘龐加萊猜想’事件前領過獎。

佩雷爾曼共發表了三篇網文(preprint)，第二篇網文敘述了一個定理（7.4)卻沒給出證明，只是說在下一篇preprint中給出證明。前兩篇論文的目標是瑟斯頓猜想（其結果包含了龐加萊猜想）。但是，他的“下一篇”卻沒有給出所預報的證明，而是給出龐加萊猜想所需的一些引理。

也就是說，佩雷爾曼第二篇論文的定理7.4至今仍未有證明。俺認為，這是他避不見人的主因。否則，內行問起‘定理7.4’，他如何回答？”

這裡簡單說說整個爭端的來龍去脈。2002年，佩氏貼出兩篇論文，其中第二篇有個定理7.4，從三個條件推導出一個結論。但佩氏隨後說：“第三個條件可以去掉，具體證明將在下一篇文章中給出”。他隨後到美國講學，說這些方法證明了瑟斯頓猜想（比龐加萊猜想更大的猜想）。回到俄國後，他貼出第三篇論文，並沒有前述定理7.4的證明，只有針對龐加萊猜想的幾個定理。從2002年到2006年，有三個團隊在解讀佩氏的論文：曹-朱、克-洛、摩-田。其中，克-洛在其論文寫道：佩氏的定理7.4，如果用三個條件，證明就簡單得多，但實用性不大（“it is NOT clear this strong condition” can be applied）。最後，丘成桐在2006年4月26日宣布，曹-朱的論文將在6月發表，給出完整證明。另外兩組相繼於5月貼出了他們匆忙趕製的論文。有興趣者可以對比這他們在2006年5月貼的論文與最後定稿版的區別。《科學》雜誌年終文章說道：龐加萊猜想在2006年得到解決，三組數學家填補了佩雷爾曼文章的漏缺（gap)。

上面是基本事實。《紐約客》作者納薩寫了篇故事，指責丘成桐的學生曹-朱搶功。據納薩所寫的故事，佩雷爾曼本人指責“有些人不誠實”。

俺也寫了個故事：2002年佩氏到美國研討後，發現自己的定理7.4（兩條件版）有錯，回國後馬上把大猜想（瑟斯頓猜想）放下，寫出第三篇論文專攻小猜想（龐加萊猜想）。此後就不敢見人，怕人問“定理7.4”。

俺搜到2006年在《紐約客》論壇的辯論，貼到這裡，多一個地方保留。

俺的《致佩雷爾曼博士的公開信》貼出後次日，在《紐約客》論壇讀到一篇文章，作者自報為數學教授，但不在微分幾何方向。他說：“原本到此壇的目的是支持納薩。有人指控她對丘的措辭過嚴，我當時認為還不夠嚴。但來看了一下爭論，看來事實完全與納薩所編的故事相反。”在這位教授貼出網文的第二天，《紐約客》關閉了論壇。很可惜，教授的帖子沒抄下來。

- - - -
下面是俺的兩份公開信（分别致納薩和佩氏）。
大致翻譯給佩氏的信：

親愛的佩博士：
從您朋友納薩的文章《流形的命運》知道，您為了某種不可告人的目的在數學研究中“冒了巨大的風險”。雖然俺很反對這種行為，但仍認為您與漢密爾頓教授對證明龐加萊猜想作出90%的貢獻，而且很讚賞兩位的貢獻。

…繆納教授在美國數學會會刊《通知》上發表文章，說您“宣布了一個解決辦法，並保證會在第四篇論文給出定理7.4的證明”（見《通知》2003年11月號，第1231頁）。如今，瑟斯頓猜想已經被證明，而證明者被您及其支持者包括這位繆納教授攻擊為“不誠實、搶功”。

請問，是否能通過“宣布”和“保證”來獲取數學定理的證明權？



 Letters to Ms. Nasar and Dr. Perelman
Dear Ms. Nasar:

I would like to report two obvious mistakes you made in your article entitled "Manifold Destiny".

1. Your article reads:"Yau added, 'Given the significance of the Poincare, that Chinese mathematicians played a 30% role is by no means easy.'" However, the Chinese official news website attributed the quoted words to Yang. [To verify, you can use google to search "content_4644722" +"2006-06", then click translation]. So, you made an undeniable mistake. By the way, on June 9, Yang publicly denied he said that.

2. Cao-Zhu indeed said that they "substitute several key arguments" for "the completion of the geometrization program." It does not contradict with Morgan's comments about "the Poincare". Note that the geometrization conjecture is not "the Poincare" and your words about "the idea that Zhu and Cao had contributed significant new approaches to the Poincaré" are wrong.

For your info: Prof. Milnor wrote:"Perelman has announced a resolution of these difficulties and promised a proof of the Thurston (geometrization) conjecture based on Hamilton's ideas, with further details to be provided in a fourth preprint." (AMS Notices, Nov. 2003, Page 1231). The "fouth preprint" has not been posted yet. You and Morgan should see the difference now: Perelman "announced" and "promised a proof", but Cao and Zhu provided a proof, although their proof was described by Prof. Milnor as "throwing 'a monkey wrench' into the question of who gets credit" (see Wall Street Journal).

"Politics, power, and control have no legitimate role in our community, and they threaten the integrity of our field,"Phillip Griffiths said and we all should remember these.

3. I would appreciate if you would do me a favor to pass the following message to Dr. Perelman, since you are a friend of this "risk" taker and he refuses to receive mails.
Thank you.

Sincerely,
SK

sk08.blogspot.com
cc: Ms. Rosenberg, Mr. Gruber, Mr. Cooper and Dr. Yau

===========

Open letter to Dr. Perelman.

Dear Dr. Perelman:

From the story “Manifold Destiny” written by your friend Ms. Nasar, I found that you took “a considerable risk” in your math research for some untold purpose. Although I strongly disapprove this kind of action, I think that you and Hamilton contributed at least 90% in the Poincare and admire your contribution.

Prof. Milnor wrote: "Perelman has announced a resolution of these difficulties and promised a proof of the Thurston conjecture (TC) based on Hamilton's ideas, with further details to be provided in a fourth preprint." (AMS Notices, Nov. 2003, Page 1231) . Three years have passed and your “fourth preprint” has not been posted yet. Now, two teams have provided those “further details”, but one of them was blamed as “not honest” by you and “stealing credit from Perelman” by your backers. Do you think that one can claim full credit by “announced” and “promised a proof”? I believe people will know who is the one that is “not honest” when they know the truth.

Many people believe that you can prove TC. Now, since two teams provided proofs that circumvented your Thm. 7.4, you still have two ways to prove that you are able to realize your promise:
(i) prove Thm 7.4 with only the first two conditions; or
(ii) prove Thm 7.4 with all three conditions and use it to complete the proof of TC. Note that Kleiner & Lott said that “it is not clear this strong condition” can be applied. You may give them a lesson.

So, finish your job and don’t let your backers down.

Sincerely yours,
SK