William Thurston (1946-2012): the man who could see the 3-folds


下午在Wiki上传出William Thurston去世的消息。他的儿子,Dylan Thurston,确认了这个不幸的传闻:

From: Dylan Thurston <dpt@math.columbia.edu <mailto:dpt@math.columbia.edu>>

Subject: Bill Thurston, 1946-2012

Date: August 22, 2012 5:38:27 AM GMT+02:00

Dear all, Bill passed away at 8PM on August 21 in Rochester, NY. He was surrounded by family, and went very peacefully, after a fight with melanoma since spring 2011. Please pass this on as appropriate.

Sincerely,

Dylan Thurston

William Thurston (1946-2012)

2007年,Princeton为庆祝Thurston的60大寿举行了一场学术会议,这张照片出现在会议海报上。严格地说,Thurston的工作领域并非扭结理论(而是与之紧密相关的3维流形的几何和拓扑),但我还是很喜欢这张照片。在我看来(冒着过度诠释的风险),它模仿了著名雕塑Gruppo del Laocoonte,也继承了原作的肃穆感:祭司与天意搏斗,数学家则与扭结、量子群和模形式搏斗,这同样是, fight with infinity.

高中的时候读Atiyah的访问记(这场访问因为Atiyah对有限单群分类的苛刻评论闻名),诧异于居然有人能“看到”3维流形——这是Poincaré本人尝试过却承认自己做不到的事情,自此就对Thurston这个名字留下了印象。后来知道他和Yau同年获得Fields奖,在低维拓扑学界极受尊崇(例如王诗宬就是他的崇拜者),正式接触他的工作却要从读他的名著Three-Dimensional Geometry and Topology, Vol. 1算起。这本书用初等且直观的方式讨论了几何化猜想,以“Thurston的八正道”命名的系列posts是我的读书笔记。当时我对Lie群的了解很少,回头看有不少没有写透的地方,希望有机会改写。

Remark  The second volume of this book was expected for a long time. Here are some unpublished notes. It is a pity that Thurston could no longer work on it.

后来在MathOverflow上见到名为Bill Thurston的帐号,才知道Thurston更愿意熟悉的人称呼他为Bill. 他在MO上的自我介绍如是说:

Mathematics is a process of staring hard enough with enough perseverenceat at the fog of muddle and confusion to eventually break through to improved clarity. I’m happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarassment that I might reveal ignorance or confusion. Over the years, this has helped me develop clarity in some things, but I remain muddled in many others. I enjoy questions that seem honest, even when they admit or reveal confusion, in preference to questions that appear designed to project sophistication.

And some comments are ready. Thurston研究数学的方式是非常直观的,他惊人的视觉化能力常常使他能够“看出”某个关于3维流形的结论是对的 (却难以写下Bourbaki式的严格证明)。这种状况在他研究的后期越发严重,人们往往分不清在他的论文里哪些是已获得证明的定理而哪些又是基于几何直观的猜想。作为德高望重的前辈,Serre曾经不点名地批评过“此类数学家” (当然所有人都知道Serre针对的是谁,因为这种风格在当今数学界几乎是独一无二的)。上面这段自白,可以算是对Serre,也是对所有质疑者的回应。一个更正式的陈述参见

Thurston  On proof and progress in mathematics

这种层次的争论中显然没有我置喙的余地。我只想指出,Thurston的研究风格其实可以上溯到3维拓扑学的祖师爷Poincaré:

“Habitué à négliger les détails et à ne regarder que les cimes, il passait de l’une à l’autre avec une promptitude surprenante et les faits qu’il découvrait se groupant d’eux-mêmes autour de leur centre étaient instantanément et automatiquement classés dans sa mémoire.”                                                   ——Believe (1956)                                                                     

除了极少数早夭的天才,数学家多半很长寿,尚未到66周岁的Thurston中道崩阻,当然令人惋惜。不过,也可以换一个角度看:随着Perelman对几何化猜想的证明,3维流形的几何与拓扑俨然已臻于成熟之境,而回顾40年来的历程,奠基性的工作几乎是由Thurston一个人完成的——他已进入“立言”的不朽之域,又有什么可遗憾的呢。更何况,Dylan也是从事几何拓扑研究的数学家,薪尽火传,Thurston的生命已在双重意义上得以延续。

May you rest in peace, Bill.

[2012.8.23: For those who are not “mathematically educated”( in its most restricted sense) but want to learn more about the life and work of W.Thurston, I recommend the following post: the mathematical legacy of William Thurston, Scientific American]

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