阿道夫·赫维兹Adolf Hurwitz

阿道夫·赫维兹出生于一个犹太家庭。他的父亲Salomon Hurwitz从事制造业，但并非特别富裕。萨洛蒙有三个儿子，马克斯、尤利乌斯和赫维兹，但他们的女儿珍妮在一岁时夭折。令人悲伤的是，赫维兹的母亲埃莉斯·韦特海默在他只有三岁时就去世了。Salomon Hurwitz为儿子们提供了良好的教育，鼓励他们参与音乐、体操、犹太传统和吸烟，“因为他几乎无法想象一位体面的绅士没有雪茄，甚至更好的烟斗”[6]。三兄弟都对数学有特别的才能。

1868年，赫维兹进入位于希尔德斯海姆的Realgymnasium安德烈亚努姆中学。他在那里由赫曼·舒伯特[2]教授数学：-

Schubert每个星期日都抽出部分时间与身为学生的赫维兹一起研究几何，而后者最早的论文之一——写于他仍在安德烈亚努姆中学期间——是一篇合著论文。也正是赫曼·舒伯特说服了赫维兹的父亲允许他上大学，并带着热情的推荐信把他送到慕尼黑的菲利克斯·克莱因那里。

我们注意到，赫维兹的第一篇论文是与赫曼·舒伯特合写的，内容是关于米歇尔·沙勒定理的。Salomon Hurwitz无力送儿子上大学，但他的朋友Edwards先生同意提供经济帮助，从而使赫维兹的大学之路成为可能。他于1877年进入慕尼黑大学，当时还不满十八岁，在那里花了一年时间听菲利克斯·克莱因的讲座。尽管他深受菲利克斯·克莱因的影响，并已开始与其一起从事高级工作，但他在1877-78学年还是前往柏林大学继续学业，在那里听了恩斯特·爱德华·库默尔、卡尔·魏尔斯特拉斯和利奥波德·克罗内克的课。特别是他听了卡尔·魏尔斯特拉斯Introduction to the theory of analytic functions的一学期课程，赫维兹当时所做的笔记被整理成书[2]出版。这些讲座包含了卡尔·魏尔斯特拉斯版本的分析算术化，包括他对实数、分析的ε-δ方法以及基于幂级数的复函数理论的“构造”。

在柏林期间，赫维兹继续与菲利克斯·克莱因保持联系，并协助他撰写一篇关于椭圆模函数的论文。在柏林大学度过三个学期后，赫维兹于1879年回到慕尼黑大学继续与菲利克斯·克莱因合作，因此当菲利克斯·克莱因于1880年10月转到莱比锡大学时，赫维兹随他而去。他的博士学位由菲利克斯·克莱因指导，并于1881年凭借关于椭圆模函数的学位论文Grundlagen einer independenten Theorie der elliptischen Modulfunktionen und Theorie der Multiplikatorgleichungen erster StufeⓉ（独立椭圆模函数理论的基本原理及一阶乘子方程理论）获得学位。

赫维兹成为莱比锡大学的Privatdozent本是自然而然的事，因为他是那里数学教授菲利克斯·克莱因的学生。然而有一个困难——赫维兹的希腊语知识不足以满足院系要求！幸运的是哥廷根没有这样的要求，赫维兹于1882年在那里提交了教授资格论文（Habilitation）论文后成为哥廷根大学的编外讲师。赫维兹在1881-82年期间并不在慕尼黑，而是回到了柏林，在那里听了卡尔·魏尔斯特拉斯和利奥波德·克罗内克的进一步讲座课程。

1884年，赫维兹接受了费迪南德·冯·林德曼的邀请，成为柯尼斯堡的编外教授，并在那里待了八年。在这里，他教过大卫·希尔伯特和赫尔曼·闵可夫斯基，并与大卫·希尔伯特结成了终生的朋友。即使赫尔曼·闵可夫斯基离开柯尼斯堡大学前往波恩后，他仍然每个假期都回到柯尼斯堡，加入赫维兹和大卫·希尔伯特几乎每天的散步[4]：-

在赫维兹居住柯尼斯堡的整个八年期间，这些散步持续不断，几乎探索了当时已知数学世界的每一个角落。

赫维兹遇到了Ida Samuel，她是医学院一位教授的女儿，他们结了婚。这对夫妇有三个孩子：Lisbeth（生于1894年）、
Eva（生于1896年）和Otto（生于1898年）。Eva于1915年开始在苏黎世的ETH学习数学，但几年后她退学了，转向“极端革命”政治，令她的父母非常沮丧。Lisbeth从事社会福利工作者的职业，Otto学习化学。

1892年，费迪南德·格奥尔格·弗罗贝尼乌斯 离开他在苏黎世联邦理工学院的讲席，返回柏林，而 赫维兹 被任命填补苏黎世空缺的讲席。赫维兹 余生都留在苏黎世，不幸的是他一直饱受健康问题困扰。他的健康问题始于他在慕尼黑求学时染上的伤寒。当时这种疾病在该市广泛传播。事实上，他两次感染伤寒，此后便严重遭受偏头痛的折磨。

尽管正如我们已指出的，赫维兹 余生都留在苏黎世，但这并不是因为德国没有向他提供讲席。赫尔曼·阿曼杜斯·施瓦茨 当时是哥廷根大学的教授，于1892年接受柏林大学的教授职位，接替了 卡尔·魏尔斯特拉斯。哥廷根大学在 赫维兹 接受苏黎世讲席仅几周后便联系他，向他提供空缺的讲席，但他拒绝了这一邀请。对 赫维兹 来说，这一定是一个极其艰难的决定，因为在当时，对于任何德国人来说，像哥廷根这样的德国顶尖大学的讲席都比瑞士的讲席有声望得多。然而，赫维兹 是一个极其忠诚的人，既然他已经承诺接受苏黎世的职位，就不会背弃自己的诺言。

赫维兹的许多数学工作可以看作深受菲利克斯·克莱因的影响（也受波恩哈德·黎曼的影响，其思想通过菲利克斯·克莱因传给了赫维兹）。事实上，赫维兹和菲利克斯·克莱因互补得极好，原因正如威廉·亨利·扬在[4]中所指出的：-

菲利克斯·克莱因的力量……有时被认为更多地在于他思想的丰富性和[天才]，而不是在于发展这些思想的能力。

那么这就是赫维兹的长处——发展菲利克斯·克莱因的思想[1]：-

菲利克斯·克莱因关于模函数的新观点，将基本域等几何方面与同余子群等群论工具以及波恩哈德·黎曼曲面的亏格等拓扑概念结合起来，被赫维兹充分利用。

赫维兹研究了波恩哈德·黎曼曲面的亏格。他致力于如何从模方程推导类数关系。他研究了亏格大于1的代数波恩哈德·黎曼曲面的自守群，证明它们是有限的。他还研究了$SO(n, \mathbb{R})$和$SL(n, \mathbb{R})$的不变积分，Slodowy在[12]中描述了这项工作，连同伊赛·舒尔关于正交关系的工作以及正交群的特征标公式，如何导致了赫尔曼·外尔关于半单李群的表示论的论文。

赫维兹研究的进一步课题包括复变函数论、弗里德里希·威廉·贝塞尔函数的根以及差分方程。他还写了几篇关于Fourier series的论文。在他去苏黎世后不久，他的同事Aurel Stodola向他提出了一个问题，关于一个具有实系数的$n$次多项式何时

$f (x) = a_{0}x^{n} + a_{1}x^{n-1} + ... + a_{n}$

具有正首项系数$a_{0} > 0$的方程只有负实部的根。赫维兹完全解决了这个问题，表明该条件成立当且仅当某个行列式序列全部为正。他于1895年在论文Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt Ⓣ（《方程只有负实部根的条件》）中发表了这一结果，该文于1895年发表在Mathematische Annalen上。这篇极具影响力的论文在100年后，即1995年，被重新收录于赫维兹稳定理论研讨会在阿斯科纳举行的会议录中。优秀的综述[7]出现在同一研讨会的会议录中，而在论文[5]中，详细描述了著名的稳定性判据的赫维兹版本的起源。

赫维兹 在 代数数论 中做出了杰出的工作。例如，他于1896年发表了一篇关于整数 四元数 的因子分解理论的论文，并将其应用于将一个整数表示为四个平方和的问题。赫维兹 思想的完整证明出现在他去世那年出版的一本小册子中。这涉及研究整数四元数环，其中有24个单位。他证明单边理想是主理想，并引入了素四元数和准素四元数。

Lindström的论文[10]展示了赫维兹工作的另一个方面。以下是Lindström摘要的一部分：-

1893年，瑞典精算师兼数学史家Gustaf Eneström在一篇关于养老保险的论文（以瑞典文发表）中发表了一个关于某些实系数多项式复根的定理。这个结果现在常被称为Eneström-Kakeya定理，因为S Kakeya在1912—1913年发表了类似的结果。但Kakeya的定理含有一个错误，该错误由A 赫维兹于1913年更正。赫维兹将Kakeya（已更正）的结果告知了埃德蒙·朗道；埃德蒙·朗道在证明一个关于无穷幂级数的定理时需要这个结果。……我们提及Eneström定理的一个推广，并给出它在赫维兹一个类似结果上的应用。

第一次国际数学家大会于1897年在苏黎世举行。[6] 赫维兹在1896年7月的初次会议上加入了组织委员会，并成为大会的主要组织者之一。他担任接待委员会主席，该委员会负责与娱乐委员会一起组织大会的社交部分，并负责提供所有大会出版物（邀请函、日程、规章、决议、海报、徽章等）的德文和法文版本。

在四个小组委员会中，接待委员会的工作量似乎最大，赫维兹建议成立一个出版委员会，以减轻他所在委员会的一些职责。最终，组织委员会采纳了Rudio的建议，扩大现有的接待委员会。与Geiser和赫尔曼·闵可夫斯基一起，赫维兹负责（根据Rudio的建议）挑选全会演讲人。他还请菲利克斯·克莱因为筹备工作出力。此外，赫维兹编写了与会者名单，分发给每位大会参加者。他还审看了汽船游览期间的音乐娱乐节目。

8月8日，赫维兹花了一整天“在火车站迎接陆续抵达的数学家，发放大会证件，并按要求为抵达者安排住宿”。晚上，在⟦L1⟧的欢迎会上，他致欢迎辞，强调不同国家数学家之间个人关系的重要性。

在大会本身，他主持了第二分会：函数论与分析。更重要的是，他是四位全会演讲人之一，也是其中唯一代表瑞士的人。他在8月9日第一次全体会议上作了题为Entwicklung der allgemeinen Theorie der analytischen Funktionen in neuerer Zeit Ⓣ（近代解析函数一般理论的发展）的报告，大卫·希尔伯特称其“因清晰简洁的风格以及对这个如此广泛主题的成功选择而堪称典范”。

偏头痛并不是赫维兹健康问题的全部，他的健康状况变得越来越严重。他的肾脏发生病变，并于1905年摘除了一只。只剩下一只肾脏，而且这一只功能也不正常，他的生活质量非常差。Young[2]写道：-

……他的一生[是]一场与消耗性疾病的漫长斗争。这场斗争竟能相对成功地持续这么多年，几乎令人难以置信，只能归因于[他的]妻子始终如一的照料与奉献。

在[1]中记载了大卫·希尔伯特对赫维兹其人的评论：-

大卫·希尔伯特将他描绘为一个和谐的精神；一位睿智的哲学家；一个谦逊、无野心的人；一位音乐爱好者和业余钢琴家；一个友好而不装腔作势的人，他那生动的眼睛透露出他的精神。

Adolf Hurwitz was born into a Jewish family. His father, Salomon Hurwitz, was in the manufacturing business but was not particularly well off. Salomon had three sons, Max, Julius and Adolf, but their daughter Jenny died at the age of one. Sadly, Adolf's mother Elise Wertheimer died when he was only three years old. Salomon Hurwitz offered his sons a good education, encouraging them to engage in music, gymnastics, Jewish traditions and smoking, "as he could scarcely imagine a proper gentleman without a cigar or even better a pipe" [6]. The three brothers all had a particular talent for mathematics.

Hurwitz entered the Realgymnasium Andreanum in Hildesheim in 1868. He was taught mathematics there by Schubert [2]:-

Schubert gave up part of every Sunday to working at geometry with the schoolboy Hurwitz, and the first of the latter's papers, written when he was still at the Andreanum, was a joint paper. It was also Schubert who persuaded Hurwitz's father to allow him to go to university and who sent him with warm recommendations to Klein at Munich.

We note that this first paper by Hurwitz, written jointly with Schubert, was on Chasles's theorem. Salomon Hurwitz could not afford to send his son to university but his friend, Mr Edwards, agreed to help out financially, so making a university career for Hurwitz possible. He entered the University of Munich in 1877, before he was eighteen years old, and spent a year there attending lectures by Klein. Although he was greatly influenced by Klein and had already begun to undertake advanced work with him, he went for academic year 1877-78 to continue his studies at the University of Berlin where he attended classes by Kummer, Weierstrass and Kronecker. In particular he attended the one semester course by Weierstrass Introduction to the theory of analytic functions and the notes taken by Hurwitz at this time are reproduced as the book [2]. The lectures contained Weierstrass's version of the arithmetisation of analysis including his "construction" of the real numbers, the ε, δ approach to analysis and his theory of complex functions based on power series.

While at Berlin Hurwitz continued to keep in contact with Klein and assisted him with a paper on elliptic modular functions which he was writing. After three semesters at the University of Berlin, Hurwitz returned to the University of Munich in 1879 to continue working with Klein, so when Klein moved to the University of Leipzig in October 1880, Hurwitz went with him. His Ph. D. was supervised by Klein and he received the degree in 1881 for his dissertation on elliptic modular functions Grundlagen einer independenten Theorie der elliptischen Modulfunktionen und Theorie der Multiplikatorgleichungen erster Stufe Ⓣ.

It would have been natural for Hurwitz to become a Privatdozent at the University of Leipzig since he was a student of Klein, the professor of mathematics there. However there was a difficulty -- Hurwitz did not have sufficient knowledge of Greek to satisfy the Faculty requirements! Luckily Göttingen had no such requirement and Hurwitz became a Privatdozent at the University of Göttingen after submitting his habilitation thesis there in 1882. Hurwitz had not been at Munich during 1881-82, rather he had returned to Berlin where he attended further courses of lectures by Weierstrass and Kronecker.

In 1884 Hurwitz accepted an invitation from Lindemann to become an extraordinary Professor at Königsberg and he was to remain there for eight years. Here he taught Hilbert and Minkowski, becoming a life long friend of Hilbert. Even after Minkowski left the University of Königsberg and went to Bonn, he still returned to Königsberg for every vacation and joined Hurwitz and Hilbert in their almost daily walks [4]:-

During these walks, continued over the whole of the eight year period of Hurwitz's residence in Königsberg, well nigh every corner of the then known mathematical world was explored.

In Königsberg Hurwitz met Ida Samuel, the daughter a professor in the faculty of medicine, and they married. The couple had three children: Lisbeth (born 1894),
Eva (born 1896) and Otto (born 1898). Eva began studying mathematics at the ETH in Zürich in 1915, but she dropped out a few years later and turned to 'extreme-revolutionary' politics, much to her parents' dismay. Lisbeth pursued a career as a social welfare worker and Otto studied chemistry.

In 1892 Frobenius left his chair at Eidgenössische Polytechnikum Zürich to return to Berlin and Hurwitz was appointed to the vacant chair at Zürich. Hurwitz remained at Zürich for the rest of his life, unfortunately continually suffering from ill health. His health problems had begun when he contracted typhoid in Munich when he was a student there. The disease was widely spread though the city at that time. In fact he twice contracted typhoid and from then on suffered badly from migraine headaches.

Although, as we have pointed out, Hurwitz remained at Zürich for the rest of his life, that was not because he had not been offered in chair in Germany. Schwarz, who was professor at Göttingen, succeeded Weierstrass by accepting his professorship in Berlin in 1892. Göttingen approached Hurwitz and offered him the vacant chair only weeks after he had accepted the Zürich chair, but he turned down the offer. This must have been a remarkably hard decision for Hurwitz since at that time a chair at a leading German university such as Göttingen would have been much more prestigious to any German than a chair in Switzerland. However Hurwitz was an extremely loyal person, and having given his word that he would accept the Zürich position he would not renege on his promise.

Much of Hurwitz's mathematics can be seen as being strongly influenced by Klein (and also by Riemann whose ideas where transmitted to Hurwitz via Klein). In fact Hurwitz and Klein complemented each other extremely well for the reasons that Young indicates in [4]:-

Klein's strength ... was sometimes regarded as consisting still more in the fertility and the [genius] of his ideas than in the power of developing them.

Here then was Hurwitz's strength -- in developing Klein's ideas [1]:-

Klein's new view on modular functions, uniting geometrical aspects such as the fundamental domain with group theory tools such as the congruence subgroups and with topological notions such as the genus of the Riemann surface, was fully exploited by Hurwitz.

Hurwitz studied the genus of the Riemann surface. He worked on how to derive class number relations from modular equations. He investigated the automorphic groups of algebraic Riemann surfaces of genus greater than 1, showing that they were finite. He also studied invariant integrals for $SO(n, \mathbb{R})$ and $SL(n, \mathbb{R})$ and Slodowy describes in [12] how this work, together with Schur's work on orthogonality relations and the character formula for the orthogonal groups, led to Weyl's papers on the representation theory of semisimple Lie groups.

Further topics studied by Hurwitz include complex function theory, the roots of Bessel functions, and difference equations. He also wrote several papers on Fourier series. Soon after he went to Zürich he was asked a question by Aurel Stodola, one of his colleagues, concerning when an $n$th-degree polynomial with real coefficients

$f (x) = a_{0}x^{n} + a_{1}x^{n-1} + ... + a_{n}$

with positive leading coefficient $a_{0} > 0$ has only roots with negative real parts. Hurwitz solved this problem completely showing that the condition held if and only if a certain sequence of determinants are all positive. He published this in 1895 in the paper Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt Ⓣ which appeared in Mathematische Annalen in 1895. This remarkably influential paper was reprinted 100 years later in the proceedings of the Hurwitz Symposium on Stability theory in Ascona in 1995. The excellent review [7] appears in the proceedings of the same symposium, and in the paper [5] the genesis of Hurwitz's version of the well-known stability criterion is described in detail.

Hurwitz did excellent work in algebraic number theory. For example he published a paper on a factorisation theory for integer quaternions in 1896 and applied it to the problem of representing an integer as the sum of four squares. A full proof of Hurwitz's ideas appears in a booklet published in the year of his death. This involves studying the ring of integer quaternions in which there are 24 units. He shows that one-sided ideals are principal and introduces prime and primary quaternions.

The paper [10] by Lindström shows another aspect of Hurwitz's work. Here is part of Lindström's summary:-

In 1893 the Swedish actuary and mathematics historian Gustaf Eneström published a theorem on the complex roots of certain polynomials with real coefficients in a paper on pension insurance (in Swedish). This result is now often called the Eneström-Kakeya theorem, since S Kakeya published a similar result in 1912-1913. But Kakeya's theorem contained a mistake, which was corrected by A Hurwitz in 1913. Hurwitz informed E Landau about Kakeya's result (corrected); Landau needed the result in a proof of a theorem on infinite power series. ... We mention a generalization of Eneström's theorem and give an application to a similar result by Hurwitz.

The first International Congress of Mathematicians was held in Zürich in 1897. [6] Hurwitz joined the organising committee at the initial meeting in July 1896 and became one of the main organisers of the congress. He was president of the reception committee, which was responsible for organising the social part of the congress together with the amusement committee, and for providing all congress publications (invitations, programmes, regulations, resolutions, posters, badges etc.) in both German and French.

The reception committee seems to have had the highest workload out of the four sub-committees, and Hurwitz suggested that a publications committee be set up in order to relieve his committee of some of its duties. In the end, the organising committee opted for Rudio's suggestion of enlarging the existing reception committee. Together with Geiser and Minkowski, Hurwitz was responsible for choosing the plenary speakers (on Rudio's suggestion). He also asked Klein to contribute to the preparations. Furthermore, Hurwitz wrote up the attendance list, which was given to every congress participant. He also reviewed the musical entertainment during the steamboat excursion.

On 8th August, Hurwitz spent the entire day 'at the train station, welcoming the arriving mathematicians, handing out the congress cards and organising accommodation for the arrivals if desired'. In the evening, at the collation in the Tonhalle, he gave a welcome speech, in which he stressed the importance of personal relations between mathematicians of different countries.

At the congress itself, he chaired section II: Theory of Functions and Analysis. More importantly, he was one of the four plenary speakers, and the only one of them representing Switzerland. He gave his talk Entwicklung der allgemeinen Theorie der analytischen Funktionen in neuerer Zeit Ⓣ (Development of the General Theory of Analytic Functions in Recent Times), which Hilbert describes as "exemplary due to the clear and concise style as well as the successful selection of this so extensive topic", in the first general meeting on 9th August.

Migraine was not the extent of Hurwitz's health problems which became increasingly severe. His kidneys became diseased and he had one removed in 1905. With only one kidney, and that one not functioning properly, the quality of his life was very poor. Young [2] writes that:-

... his life [was] one long struggle with a wasting disease. That this struggle was waged with comparative success for so many years appears almost incredible, and can only be accounted for by the constant care and devotion of [his] wife.

In [1] Hilbert's comments on Hurwitz as a person are recorded:-

Hilbert depicted him as a harmonious spirit; a wise philosopher; a modest, unambitious man; a lover of music and amateur pianist; a friendly unassuming man whose vivid eyes revealed his spirit.