如何評價數學家John Milnor在數學領域的成就？

謝@小王子與玫瑰邀。

Milnor 是我的偶像。但評價他對我來說是一件非常不自量力的事情。我會儘力介紹一下我了解的他的八卦與工作。

1，把一條繩子打上很多結，再將首尾連接在一起，在什麼情況下我們不剪斷繩子就可以解開所有的結把繩子變成一個圓？

1949年，普林斯頓大學一節微分幾何課上，Tucker 教授為了激發孩子們的興趣提出了這個問題，同時介紹了兩年前波蘭數學家Borsuk的猜想：考察打好結的繩子的彎曲程度，將每個點的曲率在整個繩子上積分起來，如果小於等於4π，那麼所有的結都可以打開。

Tucker沒想到的是，幾天後，它課上的一個18歲的學生像交作業一樣將一份完整地證明交給了他。

On the Total Curvature of Knots on JSTOR

2，大學期間這個智商溢出的孩子並沒有被數學所滿足，還對很多智力項目也都很感興趣，比如圍棋還有一個叫 Nash 的遊戲。認識了遊戲的發明人之後，二人成了好朋友，並發表了關於遊戲理論（博弈論）的一系列文章。最近發現，Nash的成名作好短，加上參考文獻還不到一頁。

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063129/

3，1954年，23歲的Milnor 帶著橫跨幾個方向的近十篇文章結束了博士生涯，成為了普林斯頓的助理教授，開始了一次次顛覆所有拓撲學家價值觀的旅程。

4，成名作：三觀盡失的怪球。1956年數學史上里程碑的構造由25歲的Milnor給出，六頁半的文章足以讓他有資格拿到世界上所有的數學獎項。事實上也確實如此，目前他是僅有的四位包攬 Fields, Wolf,Abel 大滿貫選手之一。1962年，31歲的Milnor拿到了Fields:

"Proved that a7-dimensional sphere can have several differential structures; this led to the creation of the field of differential topology."

5，由Wolf獎委員會發行的獲獎者文集中，收錄了Milnor的四篇文章，第一篇當然是怪球，第二篇是1958年可除代數只有實，復，四元，八元 四種的新證明，其中引用了吳文俊先生的五篇文章，並在文中寫道：「鑒於吳的文章為中文，我們將所用定理的證明在附錄中給出」。

6，第二次顛覆：Hauptvermutung的反例。Hauptvermutung 為德語，直譯為 主要猜想，直指組合拓撲的根本，即同一拓撲空間的三角剖分是否唯一？更數學一點，即是否任一兩個三角剖分都有相同的加細？1961年，30歲的Milnor 直接了當地構造了一個反例。和怪球一樣，這次顛覆也終結了無數人的研究計劃但又為更多的人開啟了新的方向。

Triangulation and the Hauptvermutung

7，再一次顛覆：能否從鼓的聲音中判斷鼓的形狀？這個問題很有影響力，Kac 憑藉這個問題就拿下了兩項大獎：http://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/Kac68chv.pdf

我們耳朵聽到的聲音是由頻率決定的，而頻率則是鼓面的Laplace運算元的特徵值。所以問題化為：如果兩個空間Laplace運算元的譜相同，這兩個空間是否相同？但問題轉瞬間就變成了一個目前還有很多人研究的方向：如果譜相同，什麼條件下空間相同。這是因為 「Almost immediately,」 Milnor 一言不合就拋出了兩個譜相同但空間不同的16維環面的例子。

8，再再一次顛覆：同胚的光滑流形居然可以有不同構的切叢。。。再顛覆我們就審美疲勞了。

9, 龐加萊猜想是他從接觸數學開始一生最大的追求，能在有生之年看到猜想被Perelman解決，不知道他是什麼心情。

10，一般來說高智商的學霸是很難理解學渣在學習中的掙扎的，所以很多天才的教授講課寫書都一塌糊塗。但Milnor 又是個反例，他有一個天賦，可以將一團亂麻的複雜的前沿研究解釋的非常簡單。「Usually when Milnor explains, it is easier.」 Milnor 寫書的水平是古往今來數學界一絕，微分拓撲，示性類，h協邊，Morse理論。。。這些當時莫測高深的前沿在Milnor寫好書後就成了研究生課程。他是唯一一個包攬了美國數學會發給優秀數學著作的Leroy P Steele Prize的人，這是Serge Lang都達不到的成就。

11，我是在讀了數學家McDuff的書之後才知道他居然是Milnor的老婆，她當年為了追Milnor放棄了自己學校的終身職位去做非終身的助理教授只為了離他更近。。。

這裡我只提到了我熟悉的容易吸引眼球的八卦，大家千萬不要造成Milnor只是個構造反例專家的印象。事實上他做了無數基礎性的工作：定義連通和並證明三維流形素分解的唯一性，在奇點理論（singularity theory）中定義Milnor 數和Milnor 纖維（http://www.abelprize.no/c53720/binfil/download.php?tid=53745，一個科普，說明巴黎凱旋門附近的Milnor數是25），對Hopf代數的研究，定義環的代數K2群（Quillen 繼續定義高次K群拿到了Fields,Voevodsky通過證明K2群中的Milnor猜想拿到了Fields）還有好多我看都看不懂的東西。研究方向也遍及拓撲，博弈論，群論（Hyperbolic group）,代數和低維動力系統等。

最後以他對Abel獎委員會的回復為結尾，裡面簡略闡述了他自己的數學思想。

The field of mathematics is a marvellous mosaic built up out of contributions by people from many different cultures, speaking many different languages, and stretching back over many hundreds of years. From the beginning mathematics has had a dual nature, partly abstract and self contained, but intimately concerned with understanding of the physical world around us. Much important mathematics was first inspired by real world problems; and mathematics has often contributed in totally unexpected ways. No one could have guessed that Riemann s study of curvature would form the basis for Einstein s theory of gravity; or that Hilbert s theory of infinite dimensional vector spaces would provide the foundations for quantum mechanics. The British mathematician G H Hardy proudly bragged that his work in number theory would never be sullied by applications. He would have been horrified to learn that it is now the basis for methods in cryptography which are fundamentally important in commercial applications and also in military applications. The connections between mathematics and other sciences work in both directions. Claude Shannon s work on communication theory was inspired by the work of physicists on statistical mechanics, and now has important applications not only in computer science but also in the mathematical theory of dynamical systems. The mathematical theory of Riemannn surfaces is now very important to mathematical physicists. Conversely, tools developed by mathematical physicists play a very important role in topology. I have been very lucky to have been able to enjoy this magnificent mosaic of mathematics for more than sixty years, and to make some contributions to it. But of course the contributions of any one person must depend in a very essential way on the cumulative contributions of older generations of mathematicians. The work of Niels Henrik Abel provides a necessary background for a great deal of present day mathematics. The groups studied by Sophus Lie are important in many branches of mathematics. We learn not only from our mathematical ancestors, but also to a great extent from our contemporaries. I have personally benefited from the work of a number of the previous Abel Prize winners. The thesis of Jean-Pierre Serre provided a foundation for nearly all subsequent work on homotopy groups. His beautifulCours d Arithmétiquetaught me about the quadratic forms which play an important role in understanding the topology of manifolds. In fact, this study of quadratic forms was so addictive that I spent some years studying problems in algebra for their own sake. Michael Atiyah s work on K-theory provided the inspiration for my own work on algebraic K-theory; while John Tate helped me to understand the relationship between algebraic K-theory, quadratic forms and Galois cohomology. One great advantage of the long mathematical life which I have enjoyed is that it has enabled me to see amazing progress by others on problems which I had helped to formulate. Thus the work on quadratic forms which I just mentioned led to conjectures which were later verified in very deep work by Vladimir Voevodsky. Similarly, Misha Gromov s work on the growth of finitely generated groups went far beyond anything which I had been able to achieve.

請您繼續閱讀更多來自 知乎 的精彩文章:

※《NPJ-Molecular Phenomics》：聚焦表型組研究前沿，誠邀領域專家投稿
※Google 推出 Gradient Ventures，後者專註於人工智慧領域的風險投資
※在AR領域搶佔先機的Google Android，為何如今卻成為跟隨者？
※CA Technologies被IDC MarketScape評為敏捷項目及組合管理領域的領導者
※Running man成員近況：Gary結婚，其他人拓展領域
※Spotify 收購 Niland 以加強發展 AI 領域
※行業 Industry┃電子商務領域 E-Commerce
※發展AlphaGo的最終目的，是幫助人類在科學領域取得進步 I DeepMind創始人訪談
※試水藍牙耳機領域 Fitbit Flyer會是Ionic的最佳拍檔？
※數據科學和 ML 領域常用的 Python 庫
※George Takei進軍移動遊戲領域，加入Fifth Journey擔任創意總監
※構建加密貨幣和遊戲消費領域的橋樑，GameCredits 發行的 MobileGo 代幣開始在 Bittrex 平台交易
※IBM Watson是AI 領域的「笑話」嗎？
※再次涉足棒球領域！Michael Jordan 成為美職棒球隊股東
※洞悉神祕領域 卡地亞Rotonde de Cartier神祕小時鏤空腕錶
※系統標配View Mixed Reality 微軟在虛擬/增強現實領域發力
※中興通訊Carrier DevOps Builder拿下SDN NFV領域「奧斯卡」大獎
※首個！Illumina子公司推出DNA APP Store，打造「基因測序領域的蘋果」
※常用Twitter和Facebook，那你知道他們在體育領域的布局嗎？