伊萨多·辛格Isadore Singer

伊萨多·辛格 被称为 Is 或 Iz。他的父母 Simon Singer 和 Freda Rose 是波兰人。他们于 1917 年移民到加拿大多伦多，并在该市结婚，之后搬到密歇根州底特律。辛格 在学校时既喜欢科学也喜欢阅读，但当他于 1941 年 9 月进入密歇根大学时，他决定学习物理而不是英国文学。由于第二次世界大战，课程被压缩，他于 1944 年 1 月获得学士学位。随后他在美国陆军通信兵团服役三年。经过初步训练后，辛格 被派往菲律宾管理一所通信兵团学校。在那里期间，他利用晚上时间为自己在芝加哥大学学习物理做准备。他特别学习了群论和 微分几何，意识到拥有扎实数学背景的重要性。

辛格于1947年1月进入芝加哥大学，当时仍打算从事物理学研究。然而，他仍然认为如果自己有扎实的数学背景，就能在物理学中取得更好的进展，于是他攻读了数学硕士学位。他于1948年获得理学硕士学位，但此时他对数学如此兴奋，以至于决定继续从事该领域的研究。他于1950年获得博士学位。他在芝加哥的学位论文导师是尔文·席格，他的学位论文题为Lie Algebras of Unbounded Operators。R S Doran写道：

当时，[芝加哥]系在Marshall Stone的远见卓识领导下，他汇集了世界上最优秀的数学教师队伍之一。这个群体中的资深成员包括陈省身、桑德斯·麦克兰恩、安德烈·韦伊和A Zygmund。厄文·卡普兰斯基、尔文·席格和保罗·哈尔莫斯是系里活跃的年轻研究人员，他们的兴趣包括算子理论和算子代数等。此外，大量杰出的访问学者和非凡的研究生来到芝加哥，与这些杰出的教师一起学习。这就是[辛格所开展]工作所处的令人兴奋的初始环境。

获得博士学位后，辛格于1950年被任命为麻省理工学院的C L E Moore讲师。在麻省理工学院两年后，他被任命为加州大学洛杉矶分校的数学助理教授。1954年，他前往哥伦比亚大学，在1954-55学年担任访问助理教授。接下来的学年也是以访问职位度过的，这次是在普林斯顿的高等爱德华·斯图迪研究所。他职业生涯早期这些年的出版物包括：（与理查德·卡迪森合作）Some remarks on representations of connected groups（1952）；Uniformly continuous representations of Lie groups（1952）；（与Warren Ambrose合作）A theorem on holonomy（1953）；（与Richard Arens合作）Function values as boundary integrals（1954）。

1953 年，厄文·卡普兰斯基 问 辛格，紧 费利克斯·豪斯多夫 空间上的连续复值函数代数具有哪些导子。辛格 在第二天就回答了这个问题，表明答案是 0，这引发了关于该主题的大量工作，包括 辛格 与 J Wermer 合作发表的进一步工作，发表在论文 Derivations on commutative normed algebras（1955 年）中，他们在其中证明了半单交换赋范代数上的所有有界导子都是 0。

辛格 于 1954 年回到麻省理工学院，被任命为助理教授。他于 1958 年晋升为副教授，然后于 1959 年晋升为正教授。1961 年，他与 Sheila Ruff 结婚，他们有一个女儿 Natascha 辛格（她是《纽约时报》的记者）。这段婚姻在 70 年代中期破裂，最终他与 Rosemarie 结婚，并在 乔治·伯克利 有了两个女儿，Emily 和 Annabelle。他于 1970 年被任命为麻省理工学院的 诺伯特·维纳 数学教授。1977 年，辛格 前往加州大学 伯克利 分校，在那里作为访问教授度过了两年。在获得 伯克利 的永久职位后，他于 1979 年从麻省理工学院辞职，接受了 伯克利 的教授职位。他于 1982 年被任命为 伯克利 的 Miller 教授，但在次年离开，回到麻省理工学院，在那里被任命为 弗瑞兹·约翰 D MacArthur 数学教授。他于 1987 年被任命为学院教授。

Doran写道：

辛格因其在几何、分析和拓扑学方面的深刻而壮观的工作而在数学家中当之无愧地闻名，这些工作最终导致了迈克尔·阿蒂亚-辛格指数定理及其在现代数学和量子物理学中的许多分支。

在辛格于2001年获得的凯瑟琳·斯蒂尔终身成就奖的引文中，特别提到了他在迈克尔·阿蒂亚-辛格指标定理方面的工作[15]：-

辛格与迈克尔·阿蒂亚关于椭圆算子指标定理的一系列五篇论文（发表于1968年至1971年），以及他与迈克尔·阿蒂亚和V K Patodi关于带边流形指标定理的三篇论文（发表于1975年至1976年），都是全局分析的伟大经典之作。它们催生了微分几何、微分拓扑和分析中的许多发展……

在收到斯蒂尔奖的答复中，辛格谈到了他与Michael 迈克尔·阿蒂亚的合作：-

我与迈克尔·阿蒂亚二十多年的合作非常令人兴奋，我们的工作继续产生巨大影响。迈克尔爵士是一位杰出的人，除了数学之外，他还投入了大量时间和精力支持全世界的科学事业。我很幸运在数学和物理领域有许多合作者，与他们共事愉快，并已成为亲密的朋友。我很高兴感谢他们，超过三十人——太多无法在此一一列出。

引文还提到了辛格的其他杰出贡献：-

然而，[指标定理]仅代表他对几何和分析贡献的一小部分。他对几何的其他重要贡献包括与D B Ray关于解析挠率的工作，这是现代几何中“行列式”不变量许多工作的先驱，以及与J A Thorpe合著的一本有影响力的教科书《初等拓扑学与几何讲义》……此外，除了纯数学的工作，辛格二十年来一直致力于将数学家和理论物理学家聚集在一起。这不仅仅是人际关系和研讨会演讲的问题，而是需要长期努力去理解、重新加工，并使现代理论物理学最深刻的结果为数学家所用。这种数学家和物理学家之间严肃互动的复兴，始于20世纪70年代中期，对数学产生了戏剧性的影响，而辛格在这一发展中发挥了重要作用。

辛格荣幸地当选为国家科学院和美国艺术与科学院的成员。在辛格获得的众多奖项中，除了斯蒂尔奖之外，我们还提到来自美国数学会的马克希莫·博谢奖纪念奖（1969年）、尤金·维格纳奖章（1988年）、国家科学奖章（1983年）、来自美国数学会的杰出公共服务奖（1992年）、尼尔斯·阿贝尔奖（2004年）以及James Rhyne Killian教职成就奖（2005年）。他曾在国家科学院理事会、国家研究委员会管理委员会和白宫科学委员会任职。辛格在1970-72年间担任美国数学会的副主席。

2004年的尼尔斯·阿贝尔奖是联合授予迈克尔·阿蒂亚和辛格的：-

……因他们发现并证明了指标定理，将拓扑学、几何学和分析学结合在一起，以及他们在数学与理论物理学之间搭建新桥梁方面所发挥的杰出作用。

教学在 辛格 的生活中也扮演了重要角色。他说[15]：-

对我来说，课堂是研究的重要对应物。我喜欢教各个层次的本科生，我还有许多研究生，其中许多人最终教给我的东西比我教给他们的还要多。

他对“各个层次”教学的热爱在基利安奖的颁奖词中得到了强调[4]：-

他也许是唯一一位拥有杰出大学教授职位却定期教授普通（而非荣誉）第一学期微积分的美国数学家。

至于数学之外的兴趣，辛格列出了文学、徒步旅行和网球。他说[13]：-

我喜欢打网球，我尽量每周打两三次。这让我精神焕发，我认为这帮助我这些年来在数学上努力工作。

Isadore Singer is known as Is or Iz. His parents, Simon Singer and Freda Rose, were Polish. They emigrated to Toronto, Canada, in 1917 and married in that city before moving to Detroit, Michigan. Isadore enjoyed both science and reading while at school but when he entered the University of Michigan in September 1941 he decided to study physics rather than English literature. Because of World War II, courses were compressed and he graduated with a B.S. in January 1944. He then spent three years in the U.S. Army Signal Corps. After initial training, Singer was sent to the Philippines to run a Signal Corps school. While there he spent the evenings preparing himself for studying physics at the University of Chicago. In particular he studied group theory and differential geometry realising the importance of having a strong background in mathematics.

Singer entered the University of Chicago in January 1947, still intending to undertake research in physics. However, still thinking that he would be in a better position to make advances in physics if he had a strong mathematical background, he took a Master's degree in mathematics. He was awarded an M.S. in 1948 but by this time was so excited by mathematics that he decided to continue with research in that topic. He was awarded a Ph.D. in 1950. His thesis advisor at Chicago was Irving Segal and his thesis was entitled Lie Algebras of Unbounded Operators. R S Doran writes:-

At the time the department [at Chicago] was under the visionary leadership of Marshall Stone, who had assembled one of the finest mathematical faculties in the world. Senior members among this group included S S Chern, S Mac Lane, A Weil, and A Zygmund. Irving Kaplansky, Irving Segal, and Paul Halmos were active young researchers in the department with interests, among many other things, in operator theory and operator algebras. In addition, a large number of distinguished visitors and extraordinary graduate students came to Chicago to study with this illustrious faculty. This is the exciting initial environment underlying the work [Singer carried out].

After the award of his doctorate, Singer was appointed C L E Moore Instructor at the Massachusetts Institute of Technology in 1950. After two years at MIT he was appointed as an Assistant Professor of Mathematics at the University of California, Los Angeles. In 1954 he went to the Columbia University where he spent the academic year 1954-55 as a Visiting Assistant Professor. The next academic year was also sent in a visiting position, this time at the Institute for Advanced Study at Princeton. His publications in these early years of his career include: (with Richard V Kadison) Some remarks on representations of connected groups (1952); Uniformly continuous representations of Lie groups (1952); ( with Warren Ambrose) A theorem on holonomy (1953); (with Richard Arens) Function values as boundary integrals (1954).

In 1953 Kaplansky asked Singer what derivations the algebra of continuous complex-valued functions on a compact Hausdorff space possessed. Singer had answered the question by the following day, showing the answer was 0, and this led to considerable work on the topic including further work by Singer, jointly with J Wermer, published in the paper Derivations on commutative normed algebras (1955) in which they showed that all bounded derivations on a semisimple commutative normed algebra are 0.

Singer returned to the Massachusetts Institute of Technology in 1954 where he was appointed as an Assistant Professor. He was promoted to Associate Professor in 1958, and then to full professor in 1959. In 1961 he married Sheila Ruff and they had a daughter Natascha Singer (who is a reporter for the NYTimes). That marriage broke up in the mid 70's and eventually he married Rosemarie and they had two daughters, Emily and Annabelle, in Berkeley. He was named Norbert Wiener Professor of Mathematics at MIT in 1970. In 1977 Singer went to the University of California at Berkeley where he spent two years as a Visiting Professor. Offered a permanent post at Berkeley, he resigned from MIT in 1979 taking up the professorship at Berkeley. He was named Miller Professor at Berkeley in 1982 but left in the following year to return to MIT where he was named John D MacArthur Professor of Mathematics. He was appointed as an Institute Professor in 1987.

Doran writes:-

Singer is justifiably famous among mathematicians for his deep and spectacular work in geometry, analysis, and topology, culminating in the Atiyah-Singer Index theorem and its many ramifications in modern mathematics and quantum physics.

In the citation for the Steele Prize for Lifetime Achievement which Singer received in 2001, his work on the Atiyah-Singer Index theorem is highlighted [15]:-

Singer's series of five papers with Michael F Atiyah on the Index Theorem for elliptic operators (which appeared in 1968 - 71) and his three papers with Atiyah and V K Patodi on the Index Theorem for manifolds with boundary (which appeared in 1975 - 76) are among the great classics of global analysis. They have spawned many developments in differential geometry, differential topology, and analysis ...

In his reply to receiving the Steele Prize, Singer spoke of his collaboration with Michael Atiyah:-

My collaboration with Sir Michael Atiyah for more than twenty years has been very exciting, and our work continues to have great impact. Sir Michael is a remarkable human being who - mathematics aside - has devoted much time and energy in the support of science throughout the world. I've been fortunate in having many collaborators in mathematics and physics with whom I have enjoyed working and who have become close friends. It is a pleasure to acknowledge them, over thirty in number - too many to list here.

The citation also mentions other outstanding contributions by Singer:-

However, [the Index Theorem] represents only a small part of his contributions to geometry and analysis. Other significant contributions to geometry were his work with D B Ray on analytic torsion, the precursor of much modern work on "determinant" invariants in geometry, and an influential textbook joint with J A Thorpe, Lecture Notes on Elementary Topology and Geometry .... Moreover, in addition to his work in pure mathematics, Singer has laboured for two decades to bring together mathematicians and theoretical physicists. This has been not simply a matter of interpersonal relations and seminar talks, but has entailed a long effort to understand, rework, and make available to mathematicians the deepest results of modern theoretical physics. This renaissance of serious interaction between mathematicians and physicists, which dates from the mid-1970s, has had a dramatic effect on mathematics, and Singer has played a major role in this development.

Singer has been honoured with election to the National Academy of Sciences and to the American Academy of Arts and Sciences. Among the many awards which Singer has received, in addition to the Steele Prize, we mention the Bôcher Prize Memorial Prize from the American Mathematical Society (1969), the Eugene Wigner Medal (1988), the National Medal of Science (1983), the Award for Distinguished Public Service from the American Mathematical Society (1992), the Abel Prize (2004), and the James Rhyne Killian Faculty Achievement Award (2005). He served on the Council of the National Academy of Sciences, the Governing Board of the National Research Council, and the White House Science Council. Singer was vice president of the American Mathematical Society during 1970-72.

The 2004 Abel Prize was a joint award to Atiyah and Singer:-

... for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.

Teaching has also played a large role in Singer's life. He said [15]:-

For me the classroom is an important counterpart to research. I enjoy teaching undergraduates at all levels, and I have a host of graduate students, many of whom have ended up teaching me more than I have taught them.

His liking for teaching "at all levels" was emphasised in the citation for the Killian Award [4]:-

He is perhaps the only American mathematician to hold a Distinguished University Professorship who regularly teaches ordinary (as opposed to Honours) first semester calculus.

As to interests outside mathematics, Singer list literature, hiking and tennis. He said [13]:-

I love to play tennis, and I try to do so two or three times a week. That refreshes me, and I think that it has helped me work hard in mathematics all these years.