伊斯拉埃尔·盖尔范德Israel Gelfand

伊斯拉埃尔·盖尔范德于1930年16岁时前往莫斯科，当时他尚未完成中学教育。在那里，他做过各种不同的工作，例如列宁图书馆的门卫，但他也开始教授数学。

莫斯科有许多不同的学院在夜校教授数学，盖尔范德在这些学院中的多所教授初等数学，稍后则进阶教授更高等的数学。在进行夜校教学的同时，他还在莫斯科大学听课，他参加的第一门课程是米哈伊尔·阿列克谢耶维奇·拉夫连季耶夫讲授的复变函数论。

1932年，盖尔范德被录取为安德雷·柯尔莫哥洛夫指导下的研究生。他的工作属于泛函分析，他很幸运地身处一个强大的泛函分析学派，因此得到了A E Plessner和L A Lyusternik等其他数学家的许多支持。盖尔范德于1935年提交了他的学位论文Abstract functions and linear operators，其中包含重要结果，但或许更重要的是他所使用的方法，即通过将线性泛函应用于赋范空间上的函数，并利用经典分析来研究由此产生的函数。

盖尔范德的下一个重大成就是交换赋范环理论，这是他在1938年提交的学位论文中创立并研究的。这项工作的重要性在[20]和[21]中得到阐明：-

……正是盖尔范德揭示了极大理想这一基本概念，使得统一先前不协调的事实并创建一门有趣的新理论成为可能。盖尔范德的赋范环理论揭示了斯特凡·巴拿赫的一般泛函分析与经典分析之间的密切联系。

在盖尔范德进行这项学位论文工作期间，他在苏联科学院任教。他从1935年起在科学院任职，直到1941年被任命为莫斯科国立大学教授。

在20世纪40年代初与马克·奈马克的合作中，盖尔范德研究了带有对合的非交换赋范环。他们证明了这些环总可以表示为希尔伯特空间上的线性算子环。

在这篇短文中不可能公正地评价盖尔范德所涵盖的工作范围。然而，我们应当提及他工作的几个主要方向。他在20世纪40年代初开始研究的一个重要领域是非紧群的表示论。这项工作接续了费迪南德·格奥尔格·弗罗贝尼乌斯和伊赛·舒尔关于有限群的表示论，以及赫尔曼·外尔关于紧群的表示论。

他工作的另一个重要领域是在微分方程上，他研究了逆雅克·夏尔·弗朗索瓦·施图姆-约瑟夫·刘维尔问题。他看到了舍盖·索伯列夫和洛朗·施瓦茨关于广义函数与分布理论工作的重要性，并在一系列专著中发展了这一理论。他从事计算数学研究，发展了用数值方法求解数学物理方程的通用方法。在这一领域，他还研究了差分算子。

盖尔范德关于群表示的工作引导他研究积分几何（这一术语由威海姆·布拉希开提出），而积分几何又是通过研究约翰·拉东变换引导他进入的。1968年至1972年间，盖尔范德发表了一系列关于无穷维李代数的上同调的重要论文。

从1958年起，盖尔范德开始对生物学和医学问题感兴趣。1960年，他与Fomin及其他科学家一起建立了苏联科学院的生物物理研究所。他尤其对细胞生物学产生了兴趣，并且除了最初感兴趣的理论工作外，还对实验工作产生了兴趣。他在生物学方面的工作在[9]和[10]中描述如下：-

在实际生物学成果的基础上，他提出了复杂多细胞系统中控制组织的重要一般原理。这些思想除了具有生物学意义外，还成为创建新的求极值方法的起点，这些方法成功地应用于X射线结构分析问题、识别问题……

尽管盖尔范德有着在数学、应用数学和生物学领域发表500多篇论文的惊人记录，但他的兴趣当然不限于研究。他创办了一所数学函授学校，[4]：-

……有助于把丰富的数学经验带给全苏联的学生。

他的教学风格在[20]和[21]中有描述：-

盖尔范德 Moiseevic活动的一个典型特征是他的研究工作与教学工作之间极其紧密的联系。提出新问题和意想不到的问题，倾向于从新的观点看待甚至众所周知的事物，这些特点表明盖尔范德是一位教师，无论他在某一时刻是在与学童交谈还是与自己的同事交谈。

1973年，盖尔范德第三次被授予列宁勋章（[9]和[10]）：-

因在发展数学、培养科学专门人才方面的贡献，以及为庆祝他六十岁生日……

这只是盖尔范德多年来因杰出贡献而获得的众多荣誉之一。他在1968-70年间担任莫斯科数学会主席。他当选为美国国家科学院、American Academy of Arts and Sciences、Royal Irish Academy、American Mathematical Society、London Mathematical Society的名誉会员。他获得了许多名誉博士学位，包括牛津大学的名誉博士学位。

1989至1990年，盖尔范德在哈佛大学任教，1990年他还在麻省理工学院任教。同年，即1990年，他移居美国，成为罗格斯大学的杰出访问教授。他还在罗格斯大学离散数学与计算机科学研究所的数学、科学与计算机教育中心的数学系和生物学系担任讲席。

1992年，盖尔范德在美国设立了一个项目，类似于他曾在俄罗斯开办的数学函授学校。盖尔范德外展项目[4]：-

……培养高中生的数学卓越能力。

1994年，盖尔范德获得了弗瑞兹·约翰 T and Catherine D MacArthur基金会颁发的麦克阿瑟 fellowship。麦克阿瑟 fellowship 是[4]：-

……不附带任何条件的奖项，旨在促进广泛人类活动中的创造力。授予盖尔范德的奖金为375,000美元，分五年支付。

让我们提一下授予盖尔范德的另外两项荣誉。1989年，他获得了稻盛财团颁发的京都奖，这是一个国际奖项，旨在表彰那些为人类科学、文化和精神进步做出重大贡献的人。公告中写道：-

盖尔范德是现代数学科学领域的最高权威之一。通过他在数学科学领域的开创性和里程碑式的工作，特别是在泛函分析方面——该领域在本世纪经历了巨大的发展，不仅影响了数学的其他领域，而且为基本粒子物理学和量子力学提供了不可或缺的数学工具——他在其创造性的职业生涯中培养和启发了许多杰出的数学家。他为整个数学科学提供了关键思想和深刻见解，并为该领域的进步做出了杰出贡献。

2005年，盖尔范德因终身成就获得了美国数学会的Leroy P 凯瑟琳·斯蒂尔奖：-

……通过他自己的工作以及与其他数学家和学生的互动，深刻影响了许多研究领域。

Israil Gelfand went to Moscow at the age of 16, in 1930, before completing his secondary education. There he took on a variety of different jobs such as door keeper at the Lenin library, but he also began to teach mathematics.

There were many different institutes in Moscow where mathematics was taught in evening classes and Gelfand taught elementary mathematics in various of these institutes, then a little later progressing to teach more advanced mathematics. While he did this evening teaching he also attended lectures at Moscow University, the first course he attended being the theory of functions of a complex variable by Lavrent'ev.

In 1932 Gelfand was admitted as a research student under Kolmogorov's supervision. His work was in functional analysis and he was fortunate to be in a strong school of functional analysis so he received much support from other mathematicians such as A E Plessner and L A Lyusternik. Gelfand presented his thesis Abstract functions and linear operators in 1935 which contains important results, but is perhaps even more important for the methods that he used, studying functions on normed spaces by applying linear functionals to them and using classical analysis to study the resulting functions.

Gelfand's next major achievement was the theory of commutative normed rings which he created and studied in his D.Sc. thesis submitted in 1938. The importance of this work is brought out in [20] and [21]:-

... it was [Gelfand] who brought to light the fundamental concept of a maximal ideal which made it possible to unite previously uncoordinated facts and to create an interesting new theory. Gelfand's theory of normed rings revealed close connections between Banach's general functional analysis and classical analysis.

During the time that he was carrying out this work for his D.Sc., Gelfand taught at the USSR Academy of Sciences. He held a post at the Academy from 1935 until 1941 when he was appointed as professor at Moscow State University.

In joint work with Naimark in the early 1940s, Gelfand worked on noncommutative normed rings with an involution. They showed that these rings could always be represented as a ring of linear operators on a Hilbert space.

It is impossible to do any justice to the range of work covered by Gelfand in this short article. However, we should mention some of the main strands of his work. One important area which he started work on in the early 1940s was the theory of representations of non-compact groups. This work followed on from the representation theory of finite groups by Frobenius and Schur and the representation theory of compact groups by Weyl.

Another important area of his work is that on differential equations where he worked on the inverse Sturm-Liouville problem. He saw the importance of the work of Sobolev and Schwartz on the theory of generalised functions and distributions, and he developed this theory in a series of monographs. He worked on computational mathematics, developing general methods for solving the equations of mathematical physics by numerical means. In this area he also worked on difference operators.

Gelfand's work on group representations led him to study integral geometry (a term due to Blaschke) which in turn he was led to by a study of the Radon transform. Between 1968 and 1972 Gelfand produced a series of important papers on the cohomology of infinite dimensional Lie algebras.

From 1958 onwards Gelfand became interested in problems in biology and medicine. In 1960, together with Fomin and other scientists, he set up the Institute of Biological Physics of the USSR Academy of Sciences. In particular he became interested in cell biology and also became interested in experimental work as well as the theoretical work which was his first interest. His work in biology is described in [9] and [10] as follows:-

On the basis of actual biological results, he developed important general principles of the organisation of control in complex multi-cell systems. These ideas, apart from their biological significance, served as a starting point for the creation of new methods of finding an extremum, which were succesfully applied to problems of X-ray structural analysis, problems of recognition, .....

Gelfand's interests were certainly not confined to research despite his incredible record of having published over 500 papers in mathematics, applied mathematics and biology. He established a correspondence school in mathematics which [4]:-

... helped to bring rich mathematical experiences to students all over the Soviet Union.

His style of teaching is described in [20] and [21]:-

One of the characteristic features of Israil Moiseevic's activities has been the extremely close bond between his research work and his teaching. The formulation of new problems and unexpected questions, a tendency to look at even well known things from a new point of view characterises Gelfand as a teacher, regardless of whether at a given moment he is holding a conversation with schoolchildren or with his own colleagues.

In 1973 Gelfand was awarded the Order of Lenin for the third time ([9] and [10]):-

For services in the development of mathematics, the training of scientific specialists and in celebration of his sixtieth birthday ...

This is only one of a very large number of honours which have been given to Gelfand over many years of outstanding contributions. He was president of the Moscow Mathematical Society during 1968-70. He was elected an honorary member of the American National Academy of Science, the American Academy of Arts and Sciences, the Royal Irish Academy, the American Mathematical Society, the London Mathematical Society. He has been awarded many honorary doctorates including one from the University of Oxford.

In 1989-90 Gelfand taught at Harvard University and in 1990 he also taught at Massachusetts Institute of Technology. In that same year, 1990, he emigrated to the United States where he became Distinguished Visiting Professor at Rutgers. He also holds a chair in the departments of mathematics and biology at the Center for Mathematics, Science, and Computer Education in the Institute for Discrete Mathematics and Computer Science at Rutgers University.

In 1992 Gelfand set up a programme in the United States similar to the correspondence school in mathematics which he had run in Russia. The Gelfand Outreach Program [4]:-

... fosters mathematical excellence in high school students.

In 1994 Gelfand was awarded a MacArthur Fellowship from the John T and Catherine D MacArthur Foundation. The MacArthur Fellowships are [4]:-

... no-strings-attached awards that are intended to foster creativity in a wide range of human endeavours. The award to Gelfand is $375,000, to be paid over five years.

Let us mention two other honours given to Gelfand. In 1989 he received the Kyoto Prize from the Inamori Foundation, an international award to honour those who have contributed significantly to the scientific, cultural, and spiritual betterment of mankind. The announcement states:-

Izrail Moiseevich Gelfand is one of the highest authorities in modern mathematical sciences. Through his pioneering and monumental work in mathematical sciences, especially in functional analysis - which has experienced tremendous development this century and not only affected other areas of mathematics but has also provided indispensable mathematical tools for the physics of elementary particles and quantum mechanics - he has brought up and inspired many prominent mathematicians in the course of his creative career. He has provided key ideas and deep insights to the whole of mathematical sciences and made outstanding contributions to the advancement of the field.

In 2005 Gelfand received the Leroy P Steele Prize of the American Mathematical Society for Lifetime Achievement:-

... for profoundly influencing many fields of research through his own work and through his interactions with other mathematicians and students.