圭多·富比尼Guido Fubini

圭多·富比尼的父亲Lazzaro 富比尼是威尼斯Scuola Macchinisti的数学教师，因此他出身于数学背景，幼年时受父亲影响走向数学。富比尼在威尼斯上中学，在那里他显示出数学上的卓越才能，从这一阶段起就明显可以看出他将以这门学科为职业。

1896年，富比尼进入比萨高等师范学校。在那里，他受教于乌利塞·迪尼和路易吉·比安基，他们很快影响了富比尼从事几何研究。1900年，他提交了学位论文Clifford's parallelism in elliptic spaces。大多数年轻博士生需要几年时间才能在自己的领域内知名。然而，富比尼很幸运，因为他的老师路易吉·比安基即将出版一部关于微分几何的重要著作，并在其1902年出版的论著中讨论了富比尼学位论文的结果。

富比尼留在比萨以取得大学教师资格。大多数数学家在职业生涯的这一阶段会扩展他们在学位论文中已经开始的工作。富比尼则不然。他转向了一个与他学位论文所研究的完全不同的新课题，研究常曲率空间中的harmonic functions理论。

1901年10月，富比尼开始在西西里岛的卡塔尼亚大学任教。在那里，他赢得了一场成为教授的竞争，但不久他又北上，被任命为热那亚大学的讲席。1908年，富比尼迁往都灵，在都灵理工大学和都灵大学任教。

富比尼的兴趣异常广泛，从他早期关于微分几何的工作转向分析。在这一领域，他研究微分方程、解析函数和多复变函数。他在都灵理工大学和都灵大学讲授这些分析课题的课程。第一次世界大战期间，富比尼研究了炮火射击的精度，这些研究引导他转向声学和电学研究。

当意大利的政治局势突然使他陷入异常困难的境地时，他的职业生涯已接近尾声。一切似乎都很好，尽管自1933年以来犹太人在德国遭受了种种问题，但意大利似乎不会走上那条路。直到1934年，墨索里尼仍将法西斯主义视为西方文明内部的一种发展，并且不信任德国，尤其不信任希特勒的国家社会主义，他宣称后者是：-

……百分之百的种族主义：反对一切人和一切事物：昨天反对基督教文明，今天反对拉丁文明，明天，谁知道呢，反对全世界的文明。

然而墨索里尼很快发现，他不得不采取与纳粹相同的路线。1938年7月，他发表了《法西斯种族主义宣言》，此后不久，反犹主义成为意大利法西斯官方政策的一部分。一系列法令将犹太人从政府、银行和教育领域有影响力的职位上清除出去。富比尼被迫从都灵的讲席上退休。

富比尼不想离开意大利，但他有两个当工程师的儿子，而且作为一个始终忠于家庭的人，富比尼认定他的儿子们在一个官方政策是反犹主义的国家没有前途。当他在1939年收到普林斯顿高等爱德华·斯图迪的邀请时，富比尼做出了他认为对家庭最有利的决定。他们立即移民到美国，尽管此时富比尼本人的健康状况已相当不佳。不过，尽管有健康问题，他还是能够在纽约教了几年书，但移民五年后，他因心脏问题去世。

如上所述，富比尼对数学的兴趣广泛。除了上面详述的分析领域外，他还研究了变分法，在其中他研究了将卡尔·魏尔斯特拉斯的积分化为昂利·勒贝格积分，并且他还研究了用两个简单积分来表示面积分。他研究的另一个分析主题是非线性积分方程。

富比尼还研究了theory of groups。特别是他研究了线性群和自守函数群。他的兴趣包括continuous groups，在其中他考察了在群上赋予度量的问题。在non-euclidean spaces中，他推广了保尔·阿佩尔和约斯塔·米塔格-莱弗勒的结果。他最重要的工作是在微分射影几何方面，其中他使用了绝对微分学。

P Speziali在[1]中，将富比尼的成就总结如下：——

富比尼是意大利最多产、最博采众长的数学家之一。他的贡献为分析、几何和数学物理的若干领域开辟了新的研究路径。凭借始终警觉的几何直觉和对所有计算技巧的绝对掌握，他能够沿着那些几乎尚未被瞥见的线索前进。他的技术掌握常常使他能够为诸如Berstein和Pringsheim关于泰勒级数展开的定理发现更简单的证明。

富比尼的儿子们接受的是工程师训练，他对他们正在研究的问题产生了兴趣。这促使他解决了一整系列工程问题，并在晚年将其写成教科书。这本与G Albenge合著的教科书直到1954年才出版，即富比尼去世十一年后。这本最后的教科书是分析领域一批重要教科书中的一本，这些教科书令人印象深刻，其中包括描述他所讲授的分析课程的书籍，以及问题集的书籍。

Speziali [1]用以下措辞描述富比尼：——

一个极有教养、本质上正直善良的人，富比尼 拥有无与伦比的教学才能。他机智的戏谑和社交魅力使他成为令人愉快的伙伴；他身材矮小，声音洪亮而悦耳。

他因其广泛的工作而受到高度评价，并配得上[1]中给予的高度赞扬，其中他被说成是：-

……二十世纪上半叶数学界最杰出、最具原创性的头脑之一。

Guido Fubini's father Lazzaro Fubini was a mathematics teacher at the Scuola Macchinisti in Venice so he came from a mathematical background and was influenced by his father towards mathematics when he was young. Guido attended secondary school in Venice where he showed that he was brilliant at mathematics and it was clear from this stage that he would follow career in the subject.

In 1896 Fubini entered the Scuola Normale Superiore di Pisa. There he was taught by Dini and Bianchi who quickly influenced Fubini to undertake research in geometry. He presented his doctoral thesis Clifford's parallelism in elliptic spaces in 1900. Most young doctoral students take a few years to make themselves well known in their area. However, Fubini was lucky for his teacher Bianchi was about to publish an important work on differential geometry and he discussed the results of Fubini's thesis in his treatise which appeared in 1902.

Fubini remained at Pisa to qualify as a university teacher. Most mathematicians at this stage in their careers extend the work they have begun in their doctoral thesis. Not so Fubini. He attacked a completely new topic to the one he had studied for his doctoral thesis studying the theory of harmonic functions in spaces of constant curvature.

In October 1901 Fubini began teaching at the University of Catania in Sicily. There he won a competition to become a professor but soon he was heading north again when he was appointed to a chair at the University of Genoa. In 1908 Fubini moved to Turin where he taught both at the Politecnico and at the University of Turin.

Fubini's interests were exceptionally wide moving from his early work on differential geometry towards analysis. In this area he worked on differential equations, analytic functions and functions of several complex variables. He taught courses on these analysis topics at both the Politecnico and the University in Turin. During World War I Fubini studied the accuracy of artillery fire and these investigations led him on to work on acoustics and electricity.

He was nearing the end of his career when the political situation in Italy suddenly put him in an exceptionally difficult position. All had seemed well and, despite the problems suffered by Jewish people in Germany from 1933, it seemed as though Italy would not follow that route. Up to 1934 Mussolini regarded Fascism as a development within Western civilisation and distrusted Germany and especially Hitler's National Socialism, which he declared to be:-

... one hundred percent racism: Against everything and everyone: Yesterday against Christian civilisation, today against Latin civilisation, tomorrow, who knows, against the civilisation of the whole world.

However Mussolini soon found that he had to take the same line as the Nazis. In July 1938 he published the Manifesto of Fascist Racism and shortly after this anti-Semitism became part of the official Fascist policy of Italy. A series of decrees removed Jews from positions of influence in government, banking and education. Fubini was forced to retire from his chair in Turin.

Certainly Fubini had no wish to leave Italy but he had two sons who were engineers and, always a man who was devoted to his family, Fubini decided that his sons had no future in a country whose official policy was anti-Semitism. When he received an invitation from the Institute for Advanced Study in Princeton in 1939 Fubini made the decision which he believed was best for his family. They emigrated to the United States immediately, although Fubini himself was in rather poor health by this time. Still, despite his health problem, he was able to teach for a few years in New York but, 5 years after emigrating he died of heart problems.

As remarked above, Fubini's interests in mathematics were wide. In addition to the areas of analysis detailed above, he worked on the calculus of variations where he studied reducing Weierstrass's integral to a Lebesgue integral and also he worked on the expression of surface integrals in terms of two simple integrations. Another analysis topic he studied was non-linear integral equations.

Fubini also worked on the theory of groups. In particular he studied linear groups and groups of automorphic functions. His interests included continuous groups where he looked at the question of putting a metric on the group. In non-euclidean spaces he extended results due to Appell and Mittag-Leffler. His most important work was on differential projective geometry where he used the absolute differential calculus.

P Speziali, writing in [1], sums up Fubini's achievements as follows:-

Fubini was one of Italy's most fecund and eclectic mathematicians. His contributions opened new paths for research in several areas of analysis, geometry, and mathematical physics. Guided by an ever-alert geometric intuition and possessed of an absolute mastery of all the techniques of calculation, he was able to follow leads that had barely been glimpsed. His technical mastery often permitted him to discover simpler demonstrations of such theorems as those of Berstein and Pringsheim on the development of Taylor series.

Since Fubini's sons trained as engineers he took an interest in the problems they were studying. This led him to solve a whole series of engineering problems which he was writing up as a textbook towards the end of his life. The textbook, jointly authored with G Albenge, only appeared in 1954, eleven years after Fubini's death. This last textbook was one of an impressive collection of important textbooks on analysis which included books which described analysis courses which he had given and also books which were collections of problems.

Speziali [1] describes Fubini in these terms:-

A man of great cultivation, fundamentally honourable and kind, Fubini possessed unequalled pedagogic talents. His witty banter and social charm made him delightful company; he was small in stature and his voice was vigorous and pleasant.

He is rated highly for his wide ranging work and merits the high praise given in [1] where he is said to be:-

... one of the most luminous and original minds in mathematics during the first half of the twentieth century.