爱德华·威滕Edward Witten

爱德华·威滕出生于一个犹太家庭。他的母亲是Lorraine W 威滕，而他的父亲Louis Witten是一位理论物理学家，专攻引力和广义相对论。他的兄弟Matt 威滕作为编剧和电视制片人而闻名。

威滕就读于布兰迪斯大学，并于1971年获得学士学位。之后他前往普林斯顿，于1974年获得硕士学位，1976年获得博士学位。

完成博士学位后，威滕前往哈佛，在1976-77学年担任博士后研究员，随后从1977年到1980年担任初级研究员。1980年9月，威滕被任命为普林斯顿大学物理学教授。他于1982年获得麦克阿瑟 fellowship，并一直担任普林斯顿大学物理学教授，直到1987年被任命为高等爱德华·斯图迪研究院自然科学学院的教授。

威滕是一位数学物理学家，他拥有大量重要出版物，这些出版物完全属于物理学领域。然而，正如迈克尔·阿蒂亚在[3]中所写：-

尽管他无疑是一位物理学家（他的出版物清单清楚地表明了这一点），但很少有数学家在数学造诣上能与他匹敌，而他以数学形式阐释物理思想的能力更是独一无二。他一次又一次地以对物理洞察力的卓越应用引出新的深刻数学定理，令数学界感到惊讶。

在1988年的美国数学会百年纪念研讨会上，威滕解释了几何学与理论物理学之间的关系：-

过去，当人们想到物理学中的几何学时，主要想到的是经典物理学——特别是广义相对论——而不是quantum physics。……当然，量子物理学从一开始就在数学的许多领域产生了显著影响——仅举两例，泛函分析和表示论。……若干重要影响带来了这一状况的改变。主要影响之一是对非阿贝尔规范理论在基本粒子物理学中核心作用的认识——这一认识在20世纪70年代中期已明确确立。另一个主要影响来自对超对称和弦理论的新兴研究。

在对这些理论物理学领域的研究中，威滕达到了很高的数学水平，这使他获得了数学家所能获得的最高荣誉，即菲尔兹奖。他于1990年在日本京都举行的国际数学家大会上领取了该奖章。大会论文集包含两篇文章，描述了使威滕获奖的数学工作。主要的颂词是迈克尔·阿蒂亚的文章[3]，但迈克尔·阿蒂亚未能到京都发表演讲，因此大会上的演讲由路德维希·德米特里耶维奇·法捷耶夫[5]发表，他大量引用了迈克尔·阿蒂亚[3]。

导致威滕获得约翰·查尔斯·菲尔兹奖章的第一个主要贡献是他对正质量猜想的更简单证明，该猜想曾使丘成桐于1982年获得菲尔兹奖章。Gawedzki和Soulé在[9]中描述了威滕于1981年发表的这项工作：-

该证明……以一种微妙的方式运用了超对称的思想。这成为威滕后续许多工作的核心……

威滕的后续工作之一是一篇迈克尔·阿蒂亚在[3]中特别提到的论文，即1984年发表在Journal of 微分几何上的Supersymmetry and Morse theory。迈克尔·阿蒂亚写道，这篇论文是：-

……对于有兴趣理解现代量子场论的几何学家来说是必读的。它还包含了对经典莫尔斯不等式的精彩证明，将临界点与同调联系起来。……威滕解释说，“超对称量子力学”不过是威廉·瓦兰斯·道格拉斯·霍奇-乔治·德拉姆理论。然而，这篇论文的真正目的是为作为无限维流形的威廉·瓦兰斯·道格拉斯·霍奇-乔治·德拉姆理论的超对称量子场论奠定基础。威滕对这一领域的掌握程度，体现在他能够在后续的许多工作中明智而巧妙地运用这一困难的视角。

自从这篇极具影响力的论文发表以来，其中的思想在微分几何研究中已占据核心地位。威滕引入了进一步具有根本重要性的新思想，并在[9]中进行了描述：-

威滕随后给出了椭圆亏格的弦论解释，并为其刚性提供了论证……另一项新数学源于威滕关于整体引力反常的论文。……近年来，威滕将注意力集中在topological量子场论上。这些对应于拉格朗日量……形式地给出流形不变量。威滕用西蒙·唐纳森和安德烈斯·弗洛尔的不变量描述了这些（扩展了迈克尔·阿蒂亚的早期思想），并推广了沃恩·弗雷德里克·兰德尔·琼斯纽结多项式……

[9]的作者们总结了威滕对数学的贡献：-

尽管大多不是以完整证明的形式出现，威滕的思想凭借其远见和概念清晰性引发了重大的数学发展，他的主要发现很快就成为了定理。他在1990年国际数学家大会上获得的菲尔兹奖章承认了他的工作对当代数学日益增长的影响。

迈克尔·阿蒂亚在[3]中以如下方式表达了同样的思想：-

……他对当代数学产生了深远的影响。在他手中，物理学再次为数学提供了丰富的灵感和洞见。当然，物理洞见并不总能立即导致严格的数学证明，但它常常引导人走向正确的方向，然后有望找到技术上正确的证明。威滕的工作就是这种情况。迄今为止，这种洞见从未让他失望，我们数学家理应期望的那种标准的严格证明也总是随之而来。

除了菲尔兹奖章之外，威滕还获得了令人惊讶的一长串奖项，这表明他的贡献受到了异常高度的重视。他获得了布兰迪斯大学（1988年）、耶路撒冷希伯来大学（1993年）、纽约哥伦比亚大学（1996年）、南加州大学（2004年）、约翰斯威廉·霍普金斯大学（2005年）和哈佛大学（2005年）的名誉学位。他获得了瑞士伯尔尼阿尔伯特·爱因斯坦学会颁发的阿尔伯特·爱因斯坦奖章（1985年）、New York Academy of Sciences颁发的物理与数学科学奖（1985年）、国际理论物理中心颁发的保罗·狄拉克奖章（1985年）、美国国家科学基金会颁发的Alan T Waterman奖（1986年）、普林斯顿大学颁发的麦迪逊奖章（1992年）、新泽西骄傲奖（1996年）、美国成就学院颁发的金盘奖（1997年）、斯德哥尔摩大学颁发的菲利克斯·克莱因奖章（1998年）、美国物理联合会颁发的Dannie Heineman奖（1998年）、西北大学颁发的Nemmers数学奖（2000年）、克莱数学研究所颁发的克莱研究奖（2001年）、美国和平现在组织颁发的Shalom奖（2002年）、国家科学奖章（2003年）、意大利克罗托内颁发的毕达哥拉斯奖（2005年）、以色列理工学院颁发的哈维奖（2006年）、国际数学物理协会颁发的儒勒·昂利·庞加莱奖（2006年）以及瑞典皇家科学院颁发的克拉福德数学奖（2008年）。他当选为众多科学院和学会的会士，如美国艺术与科学院（1984年）、美国物理学会（1984年）、National Academy of Sciences（1988年）、美国哲学学会（1994年）、Royal Society of London（1998年）、Academy of Sciences of Paris（2000年）和宗座科学院（2006年）。

Edward Witten was born into a Jewish family. His mother was Lorraine W Witten while his father, Louis Witten, was a theoretical physicist specializing in gravitation and general relativity. His brother, Matt Witten, has achieved fame as a screenwriter and television producer.

Witten studied at Brandeis University and received his B.A. in 1971. From there he went to Princeton receiving his M.A. in 1974 and his Ph.D. in 1976.

After completing his doctorate, Witten went to Harvard where he was postdoctoral fellow during session 1976-77 and then a Junior Fellow from 1977 to 1980. In September 1980 Witten was appointed professor of Physics at Princeton. He was awarded a MacArthur Fellowship in 1982 and remained as professor of Physics at Princeton until 1987 when he was appointed as a Professor in the School of Natural Sciences at the Institute for Advanced Study.

Basically Witten is a mathematical physicist and he has a wealth of important publications which are properly in physics. However, as Atiyah writes in [3]:-

Although he is definitely a physicist (as his list of publications clearly shows) his command of mathematics is rivalled by few mathematicians, and his ability to interpret physical ideas in mathematical form is quite unique. Time and again he has surprised the mathematical community by his brilliant application of physical insight leading to new and deep mathematical theorems.

Speaking at the American Mathematical Society Centennial Symposium in 1988, Witten explained the relation between geometry and theoretical physics:-

It used to be that when one thought of geometry in physics, one thought chiefly of classical physics - and in particular general relativity - rather than quantum physics. ... Of course, quantum physics had from the beginning a marked influence in many areas of mathematics - functional analysis and representation theory, to mention just two. ... Several important influences have brought about a change in this situation. One of the principal influences was the recognition - clearly established by the middle 1970s - of the central role of nonabelian gauge theory in elementary particle physics. The other main influence came from the emerging study of supersymmetry and string theory.

In his study of these areas of theoretical physics, Witten has achieved a level of mathematics which has led him to be awarded the highest honour that a mathematician can receive, namely a Fields Medal. He received the medal at the International Congress of Mathematicians which was held in Kyoto, Japan in 1990. The Proceedings of the Congress contains two articles describing Witten's mathematical work which led to the award. The main tribute is the article [3] by Atiyah, but Atiyah could not be in Kyoto to deliver the address so the address at the Congress was delivered by Faddeev [5] who quotes freely from Atiyah [3].

The first major contribution which led to Witten's Fields Medal was his simpler proof of the positive mass conjecture which had led to a Fields Medal for Yau in 1982. Gawedzki and Soulé describe this work by Witten, which appeared in 1981, in [9]:-

The proof ... employed in a subtle way the idea of supersymmetry. This became the centrepiece of many of Witten's subsequent works...

One of Witten's subsequent works was a paper which Atiyah singles out for special mention in [3], namely Supersymmetry and Morse theory which appeared in the Journal of differential geometry in 1984. Atiyah writes that this paper is:-

... obligatory reading for geometers interested in understanding modern quantum field theory. It also contains a brilliant proof of the classic Morse inequalities, relating critical points to homology. ... Witten explains that "supersymmetric quantum mechanics" is just Hodge-de Rham theory. The real aim of the paper is however to prepare the ground for supersymmetric quantum field theory as the Hodge-de Rham theory of infinite dimensional manifolds. It is a measure of Witten's mastery of the field that he has been able to make intelligent and skilful use of this difficult point of view in much of his subsequent work.

Since this highly influential paper, the ideas in it have become of central importance in the study of differential geometry. Further new ideas of fundamental importance were introduced by Witten and described in [9]:-

Witten subsequently gave a string interpretation of the elliptic genus and provided arguments for its rigidity ... Another piece of new mathematics stemmed from Witten's papers on global gravitational anomalies. ... In recent years, Witten focused his attention on topological quantum field theories. These correspond to Lagrangians ... formally giving manifold invariants. Witten described these in terms of the invariants of Donaldson and Floer (extending the earlier ideas of Atiyah) and generalised the Jones knot polynomial ...

The authors of [9] sum up Witten's contributions to mathematics:-

Although mostly not in the form of completed proofs, Witten's ideas have triggered major mathematical developments by the force of their vision and their conceptual clarity, his main discoveries soon becoming theorems. His Fields Medal at the 1990 International Congress of Mathematicians acknowledged the growing impact of his work on contemporary mathematics.

Atiyah, in [3], expresses the same ideas in the following way:-

... he has made a profound impact on contemporary mathematics. In his hands physics is once again providing a rich source of inspiration and insight in mathematics. Of course physical insight does not always lead to immediately rigorous mathematical proofs but it frequently leads one in the right direction, and technically correct proofs can then hopefully be found. This is the case with Witten's work. So far the insight has never let him down and rigorous proofs, of the standard we mathematicians rightly expect, have always been forthcoming.

Witten has received an amazingly long list of awards, in addition to the Fields Medal, which shows the exceptionally high regard in which his contributions are held. He has been awarded honorary degrees from Brandeis University (1988), the Hebrew University of Jerusalem (1993), Columbia University, New York (1996), the University of Southern California (2004), Johns Hopkins University (2005), and Harvard University (2005). He has received the Einstein Medal from the Einstein Society of Berne, Switzerland (1985), the Award for Physical and Mathematical Sciences from the New York Academy of Sciences (1985), the Dirac Medal from the International Center for Theoretical Physics (1985), the Alan T Waterman Award from the National Science Foundation (1986), the Madison Medal from Princeton University (1992), the New Jersey Pride Award (1996), the Award of the Golden Plate from the American Academy of Achievement (1997), the Klein Medal from Stockholm University (1998), the Dannie Heineman Prize from the American Institute of Physics (1998), the Nemmers Prize in Mathematics from Northwestern University (2000), the Clay Research Award from the Clay Mathematics Institute (2001), the Shalom Award from Americans for Peace Now (2002), the National Medal of Science (2003), the Pythagoras Award from Crotone, Italy (2005), the Harvey Prize from the Technion, Israel (2006), the Poincaré Prize from the International Association Of Mathematical Physics (2006), and the Crafoord Prize in Mathematics from The Royal Swedish Academy of Sciences (2008). He has been elected a Fellow of numerous academies and societies such as American Academy of Arts and Sciences (1984), the American Physical Society (1984), the National Academy of Sciences (1988), the American Philosophical Society (1994), the Royal Society of London (1998), the Academy of Sciences of Paris (2000), and the Pontifical Academy of Sciences (2006).