乔治·戴维·伯克霍夫George David Birkhoff

乔治·戴维·伯克霍夫 的父母是 伯克霍夫，他是一名医生，以及 Jane Gertrude Droppers。伯克霍夫 在芝加哥接受教育，1896年至1902年间是刘易斯学院的学生。1901年，他开始与 哈里·范迪弗 就数论中的一个问题通信。他们一起研究了 $a^{n} - b^{n}$ 的素因子，后来将他们的工作写成论文发表。1902年从刘易斯学院毕业后，伯克霍夫 开始了他的大学教育。

他于1902年进入芝加哥大学，在那里学习了一年，然后转到哈佛大学，1903年至1905年在那里学习。在哈佛期间，他于1904年将他与 哈里·范迪弗 共同获得的结果提交给 Annals of Mathematics，这篇联合数论论文成为他的第一篇出版物。同样在1904年，他向 美国数学会 提交了一篇关于分析的论文。在哈佛的这两年里，对他影响最大的老师是 马克希莫·博谢，他教他代数和经典分析。伯克霍夫 于1905年被哈佛授予学士学位，1906年获得硕士学位。

伯克霍夫于1905年回到芝加哥大学攻读博士学位。他的研究集中在渐近展开、边值问题以及雅克·夏尔·弗朗索瓦·施图姆-约瑟夫·刘维尔型问题上，但他的学位论文导师伊莱基姆·黑斯廷斯·穆尔对伯克霍夫的指导似乎不如儒勒·昂利·庞加莱有影响力。伯克霍夫阅读了儒勒·昂利·庞加莱关于微分方程和天体力学的工作，他从儒勒·昂利·庞加莱那里学到更多，并且在研究方向上也受到儒勒·昂利·庞加莱更强烈的影响，而非来自他的导师。Archibald在[5]中写道：-

在哈佛的两年间，一年在本科，一年在研究生院，马克希莫·博谢和威廉·佛格·奥斯古德正值鼎盛时期；但在接下来的两年里[在芝加哥]，他在伊莱基姆·黑斯廷斯·穆尔、Bolza和海因里希·马什克手下工作，当时那是美国最鼓舞人心的数学中心。然而，他的学位论文完全是独立完成的。

伯克霍夫提交的博士论文题为Asymptotic Properties of Certain Ordinary Differential Equations with Applications to Boundary Value and Expansion Problems，这使他在1907年获得了博士学位。这是一篇重要的论文，不仅因为它所包含的结果，还因为这项工作具有自然的重要推广。伯克霍夫本人在随后的几年里进一步发展了这些思想，他的两个学生Rudolph Langer和Marshall Stone也是如此。例如，伯克霍夫和Langer在1923年发表了一个重要的推广。伯克霍夫关于线性微分方程、差分方程和广义波恩哈德·黎曼问题的工作，大多都源于他在学位论文中奠定的基础。

伯克霍夫 从1907年到1909年在威斯康星大学麦迪逊分校担任讲师。在此期间，实际上在1908年，他与 Margaret Elizabeth Grafius 结婚；他们有三个孩子。在威斯康星麦迪逊之后，他去了普林斯顿担任数学导师，1911年成为那里的教授。第二年，他转到哈佛担任助理教授，1919年晋升为正教授，余生都在哈佛度过。他于1932年被任命为哈佛的珀金斯教授，然后在1936年成为文理学院院长。

伯克霍夫研究过如此多不同的数学课题，以至于在这样篇幅的传记中很难公正地评价他贡献的范围。然而，我们应当注意到，他的主要工作是动力学和遍历理论。他的遍历定理通过使用昂利·勒贝格测度，将詹姆斯·克拉克·麦克斯韦-路德维希·玻尔兹曼气体动理论转变为一个严格的原理。Butler在[6]中写道：-

伯克霍夫在1931-32年发现了后来被称为“遍历定理”的东西，这是他对动力学最著名的贡献。这个理论原则上解决了气体理论和统计力学中产生的一个基本问题，不仅在动力学本身，而且在概率论、群论和泛函分析中都有影响。

当然，使伯克霍夫成为他那个时代美国最著名数学家的不仅仅是遍历定理。许多年前，当他在1913年证明了儒勒·昂利·庞加莱最后几何定理——三体问题的一个特例时，他已经在大多数数学家眼中获得了这一殊荣。儒勒·昂利·庞加莱于1912年在Sur un théorème de géométrie中陈述了他的定理，但只能在某些特殊情况下给出证明。伯克霍夫在1913年的证明是[1]：-

……那个时代最激动人心的数学事件之一。

相对论和量子力学的基础也是伯克霍夫研究的课题。1923年，他与R E Langer合作出版了专著Relativity and Modern Physics。晚年，他在1938年的Electricity as a Fluid中出版了一部更具思辨性的著作，将他关于哲学和科学的思想结合起来。他还在四色定理上做出了重要工作。他发展了一种数学美学理论，并将其应用于艺术、音乐和诗歌。在撰写Aesthetic measure之前，他花了一年时间环游世界，研究艺术、音乐、诗歌以及各个国家。这在[5]中有所提及：-

他告诉我们，西方音乐的形式结构，即旋律之谜，在他本科时期就开始引起他的兴趣；对其中涉及的数学要素的相当深入的思考，促使他也将他的理论应用于多边形、镶嵌图案、花瓶，甚至诗歌等审美对象。

在他的著作中，其中一些我们已在上文提及，包括Relativity and Modern Physics（1923年）、Dynamical Systems（1928年）、Aesthetic Measure（1933年）和Basic Geometry（1941年）。

1923 年，American Mathematical Society 将首届 马克希莫·博谢 纪念奖授予 伯克霍夫，以表彰他的论文 Dynamical systems with two degrees of freedom，该论文他于 1917 年在 Transactions of the American Mathematical Society 上发表。他与 美国数学会 有着长期的合作关系，1919 年担任副主席，1920 年担任学术讨论会讲师，当时他就 动力系统 进行演讲，他于 1921 年至 1924 年编辑 Transactions of the American Mathematical Society，并于 1925 年至 1926 年担任主席。

也许这种对 美国数学会 的高度参与已经表明 伯克霍夫 为推进美国的数学而不懈努力。事实上，奥斯瓦尔德·维布伦 将他在这一方面的努力描述为“一种宗教般的奉献”。当然，他意识到推进美国的数学意味着与世界其他地区的进展保持密切联系。他与欧洲数学家的密切联系使他成为为国际教育委员会撰写关于欧洲数学状况报告的自然人选。在他的欧洲数学家中，他最亲密的朋友是 雅克·阿达马、Niels Nörlund、图利奥·列维-齐维塔 和 埃德蒙·泰勒·惠特克。

在圣安德鲁斯当地引人关注的是，伯克霍夫 是 1926 年圣安德鲁斯学术讨论会的主要演讲者之一，并于 1927 年当选为 爱丁堡数学会 的荣誉会员。他在 1938 年 7 月由 爱丁堡数学会 举办的 詹姆斯·格雷果里 三百周年庆祝活动上做了四次讲座，并在 1938 年访问期间被圣安德鲁斯大学授予荣誉法学博士学位。事实上，这是他获得的十三个荣誉学位之一。

伯克霍夫 获得的其他荣誉包括 1917 年因他 1915 年发表的论文 The restricted problem of three bodies 而获得的皇家威尼斯科学研究所 Querini-Stampalia 奖，1926 年美国科学促进会年度奖，以及 1935 年在罗马的 Accademia dei Lincei 颁发的双年奖。他当选为 国家科学院、美国哲学学会、American Academy of Arts and Sciences、巴黎的 Académie des Sciences、Pontifical Academy of Sciences、Circolo Matematico di Palermo、Royal Danish Academy of Sciences 与文学、Göttingen Academy、博洛尼亚皇家研究所、Edinburgh Mathematical Society、London Mathematical Society 以及 埃隆·拉格斯·利马、秘鲁的国家科学院院士。这是对 伯克霍夫 杰出贡献的非凡世界性认可。

在他去世前不久撰写的文章[5]中，他被描述为：-

……天性极为喜好社交；通过多次前往欧洲以及环球航行，他与世界各地的学者有着广泛的接触；他经常出席和参加各种会议和大会，并经常被邀请作为作家和演讲者就热门主题发表演讲。在最近的行政聚会上，他观点的独创性和广度引人注目。

然而，伯克霍夫的性格中也有消极的一面，我们应当加以评论。阿尔伯特·爱因斯坦说：-

伯克霍夫是世界上最大的反犹主义者之一。

同样地，Chandler Davis在私人通信中写道：-

伯克霍夫是我早期的老师，他的儿子加勒特·伯克霍夫是我（非常感激的）学位论文导师。G D（但不是加勒特·伯克霍夫）一贯反犹，多年来的通信可以证明；参见R Phillips的文章[15]和桑德斯·麦克兰恩为伯克霍夫辩护[11]。他有系统地阻止犹太人进入他的系，但显然晚年有所缓和，到20世纪40年代倾向于任命一位犹太人。他还在20世纪30年代帮助一些犹太难民找到工作，但不是在哈佛，同时总体上阻碍他们入境。尽管他的记录好坏参半，有些人比他更顽固地反犹，但他在这方面的行为很重要，因为他有极大的影响力。然而，他反对任命奥斯卡·扎里斯基到他的系，这似乎不是真的（尽管当时有可信的传言）。正如我提到的，加勒特·伯克霍夫一点也不反犹。

桑德斯·麦克兰恩 [11]为伯克霍夫辩护说，无论伯克霍夫可能有过什么样的“反犹主义的不同版本”，这些无疑是他许多同时代人所共有的。

George Birkhoff's parents were David Birkhoff, who was a medical doctor, and Jane Gertrude Droppers. George was educated in Chicago where he was a student at the Lewis Institute from 1896 to 1902. In 1901 he began a correspondence with Harry Vandiver on a problem in number theory. Together they studied the prime factors of $a^{n} - b^{n}$ and later wrote up their work for publication. Graduating from the Lewis Institute in 1902, Birkhoff began his university education.

He entered the University of Chicago in 1902, spending a year there, then moved to Harvard University where he studied from 1903 to 1905. While at Harvard he submitted the results he had obtained with Vandiver to the Annals of Mathematics in 1904 and this joint number theory paper became his first publication. Also in 1904 he submitted a paper on analysis to the American Mathematical Society. During these two years at Harvard the teacher who influenced him most was Bôcher who taught him algebra and classical analysis. Birkhoff was awarded his A.B. by Harvard in 1905 and his A.M. in 1906.

Birkhoff returned to the University of Chicago in 1905 to study for his doctorate. His research concentrated on asymptotic expansions, boundary value problems, and Sturm-Liouville type problems but his thesis advisor Eliakim Moore appears to have been a less influential guide to Birkhoff than was Poincaré. Birkhoff read Poincaré's works on differential equations and celestial mechanics and he learnt more, and was more strongly influenced in the direction his research was taking, by Poincaré than from his supervisor. Archibald writes in [5]:-

During his two years at Harvard, one in college and one in the graduate school, Bôcher and Osgood were in their prime; but during the next two years [at Chicago] he worked under Eliakim Moore, Bolza and Maschke, in the most inspiring mathematical center in the United States at that time. His dissertation was done quite independently however.

The doctoral thesis which Birkhoff submitted was entitled Asymptotic Properties of Certain Ordinary Differential Equations with Applications to Boundary Value and Expansion Problems and it led to the award of his Ph.D. in 1907. It was an important thesis, not just for the results which it contained but also for the fact that the work had natural important extensions. Birkhoff himself developed the ideas further in the following years, as did two of his students, Rudolph Langer and Marshall Stone. For example Birkhoff and Langer published an important extension in 1923. Birkhoff's work on linear differential equations, difference equations and the generalised Riemann problem mostly all arose from the basis he laid in his thesis.

Birkhoff taught at the University of Wisconsin at Madison as an instructor from 1907 to 1909. During this period, in fact in 1908, he married Margaret Elizabeth Grafius; they had three children. After Wisconsin Madison, he went to Princeton as a preceptor in mathematics, becoming a professor there in 1911. In the following year he moved to Harvard as an assistant professor, he was promoted to full professor there in 1919, remaining at Harvard for the rest of his life. He was named Perkins Professor at Harvard in 1932, then in 1936 he became Dean of the Faculty of Arts and Science.

Because Birkhoff worked on so many different mathematical topics it is difficult to do justice to the range of his contributions in a biography of this length. However we should note that his main work was on dynamics and ergodic theory. His ergodic theorem transformed the Maxwell-Boltzmann kinetic theory of gases into a rigorous principle through the use of Lebesgue measure. Butler writes in [6]:-

Birkhoff's discovery of what has come to be known as the "ergodic theorem" in 1931 - 32 is his most well-known contribution to dynamics. This theory, which resolved in principle one of the fundamental problems arising in the theory of gases and statistical mechanics, has been influential not only in dynamics itself but also in probability theory, group theory, and functional analysis.

It is, of course, not only the ergodic theorem that made Birkhoff the most famous mathematician in America in his day. He had already achieved this distinction in most mathematicians eyes many years earlier when he proved Poincaré's Last Geometric Theorem, a special case of the 3-body problem, in 1913. Poincaré had stated his theorem in Sur un théorème de géométrie in 1912 but could only give a proof in certain special cases. Birkhoff's proof in 1913 was [1]:-

.. one of the most exciting mathematical events of the era.

The foundations of relativity and quantum mechanics were also topics which Birkhoff studied. Jointly with R E Langer, he published the monograph Relativity and Modern Physics in 1923. Near the end of his life he published a more speculative work, combining his ideas on philosophy and science, in Electricity as a Fluid in 1938. He also did important work on the four colour theorem. He developed a mathematical theory of aesthetics which he applied to art, music and poetry. Before writing Aesthetic measure he spent a year travelling round the world studying art, music and poetry and various countries. This is referred to in [5]:-

He has told us that the formal structure of western music, the riddle of melody, began to interest him in undergraduate days; somewhat intense consideration of the mathematical elements here involved led him to apply his theory also to aesthetic objects such as polygons, tilings, vases, and even poetry.

Among his works, some of which we have already mentioned above, are Relativity and Modern Physics (1923), Dynamical Systems (1928), Aesthetic Measure (1933), and Basic Geometry (1941).

In 1923 the American Mathematical Society made the first award of the Bôcher Memorial Prize to Birkhoff for his memoir, Dynamical systems with two degrees of freedom which he had published in the Transactions of the American Mathematical Society in 1917. He had a long association with the American Mathematical Society being Vice-President in 1919, Colloquium lecturer in 1920 when he lectured on Dynamical Systems, he edited the Transactions of the American Mathematical Society from 1921 to 1924 and was President from 1925 to 1926.

Perhaps this high level of involvement with the American Mathematical Society already suggest that Birkhoff worked tirelessly to advance mathematics in America. In fact Veblen described his efforts in this direct as "a sort of religious devotion". He realised, of course, that advancing mathematics in America meant having close contacts with advances in the rest of the world. His close contacts with mathematicians in Europe made him a natural person to write a report for the International Education Board on the state of mathematics in Europe. Among the European mathematicians his closest friends were Hadamard, Niels Nörlund, Levi-Civita, and Whittaker.

Of local interest in St Andrews is the fact that Birkhoff was one of the main speakers at the 1926 St Andrews Colloquium and was elected an honorary member of the Edinburgh Mathematical Society in 1927. He gave four lectures at the Gregory Tercentenary celebrations by the Edinburgh Mathematical Society in July 1938 and was awarded an honorary LL.D. by the University of St Andrews during his 1938 visit. This was, in fact, one of thirteen honorary degrees which he received.

Other honours which Birkhoff received include the Querini-Stampalia prize from the Royal Venice Institute of Science in 1917 for his paper The restricted problem of three bodies published in 1915, the annual prize from the American Association for the Advancement of Science in 1926, and the biennial prize from the Accademia dei Lincei in Rome in 1935. He was elected to the National Academy of Sciences, the American Philosophical Society, the American Academy of Arts and Sciences, the Académie des Sciences in Paris, the Pontifical Academy of Sciences, the Circolo Matematico di Palermo, the Royal Danish Academy of Sciences and Letters, the Göttingen Academy, the Royal Institute of Bologna, the Edinburgh Mathematical Society, the London Mathematical Society, and the National Academy of Sciences of Lima, Peru. This is remarkable world-wide recognition for Birkhoff's outstanding contribution.

The article [5], written shortly before his death, describes him as:-

... by nature intensely social; through many trips to Europe, and circumnavigation of the globe, he has been extensively in contact with scholars throughout the world; he is a constant attendant and participant at meetings and congresses, and has been frequently in demand as a writer and speaker on popular themes. In recent administrative gatherings the originality and breadth of his views have been noteworthy.

There was, however, a negative side to Birkhoff's character which we should comment on. Einstein said:-

G D Birkhoff is one of the world's great anti-Semites.

In a similar vein Chandler Davis, in a personal communication, writes:-

G D Birkhoff was an early teacher of mine, and his son Garrett was my (much appreciated) thesis supervisor. G D (but not Garrett) was consistently anti-Semitic, as shown in correspondence over the years; see R Phillips' article [15] and S Mac Lane coming to Birkhoff's defence [11]. He systematically kept Jews out of his department, but apparently relented late in life and favoured appointing ONE by the 1940s. He also helped some Jewish refugees find jobs NOT at Harvard in the 1930s, while acting generally to hinder their entry. Though his record is mixed and some were more implacably anti-Semitic than he was, his actions in this regard are important because of his very great influence. However, it does not seem to be true (as rumoured credibly at the time) that he opposed the appointment of Oscar Zariski to his department. As I mentioned, Garrett was not anti-Semitic at all.

Mac Lane [11], defends Birkhoff by saying that whatever "diffuse and varied versions of anti-Semitism" Birkhoff may have had, they were undoubtedly shared by many of his contemporaries.