菲利克斯·克莱因Felix Klein

菲利克斯·克莱因最著名的是他在non-euclidean geometry方面的工作、关于几何与group theory之间联系的工作，以及函数论中的结果。他出生于1849年4月25日，并喜欢指出日（$5^{2}$）、月（$2^{2}$）和年（$43^{2}$）都是素数的平方。

克莱因的父亲是政府首脑的秘书。在Proceedings of the Royal Society:-中他的讣告里对克莱因的出生有一段生动的描述。

外面，大炮在莱茵兰起义者为对抗他们憎恨的普鲁士统治者而筑起的街垒上轰鸣。里面，尽管一切都已为逃亡准备就绪，却没有人想到离开；就在那一夜，一位严厉的普鲁士秘书得了一个儿子。那个儿子就是克莱因。

这场反对普鲁士人的革命，导致了克莱因如此戏剧性的出生，到1849年夏天被彻底镇压。

克莱因在杜塞尔多夫就读于文理中学。毕业后，他进入波恩大学，并在1865-1866年间在那里学习数学和物理。他最初开始职业生涯时打算成为一名物理学家。还在波恩大学学习期间，他于1866年被任命为尤里乌斯·普吕克的实验室助理。尤里乌斯·普吕克在波恩拥有数学和实验物理的讲席，但当克莱因成为他的助理时，尤里乌斯·普吕克的兴趣已经非常牢固地扎根于几何学。克莱因于1868年在波恩大学获得博士学位，其学位论文由尤里乌斯·普吕克指导，题为Über die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien-Koordinaten auf eine kanonische Form Ⓣ（论线坐标间一般二次方程化为标准形式的变换），内容涉及线几何及其在力学中的应用。在他的学位论文中，克莱因利用卡尔·魏尔斯特拉斯的初等因子理论对二次线丛进行了分类。

然而，在克莱因获得博士学位的那一年，尤里乌斯·普吕克去世了，留下了他关于线几何基础的主要著作未完成。克莱因显然是完成尤里乌斯·普吕克的Neue Géometrie des Raumes Ⓣ（新空间几何）第二部分的不二人选，这项工作使他结识了阿尔弗雷德·克莱布什。阿尔弗雷德·克莱布什已于1868年迁往哥廷根，而在1869年期间，克莱因访问了柏林、巴黎和哥廷根。1870年7月，克莱因正在巴黎，当时普鲁士首相俾斯麦发表了一份旨在激怒法国政府的挑衅性消息。法国于7月19日对普鲁士宣战，克莱因觉得自己不能再留在巴黎，于是返回了。随后，在短时间内，他作为医疗勤务兵服了兵役，之后于1871年初被任命为哥廷根的讲师。

1872年，克莱因被任命为德国南部巴伐利亚埃尔朗根大学的教授。他得到了阿尔弗雷德·克莱布什的大力支持，后者认为他很可能成为他那个时代最杰出的数学家，因此克莱因在年仅23岁时就获得了讲席。然而，克莱因并没有在埃尔朗根建立一个学派，因为那里只有少数学生，所以1875年当他被授予慕尼黑工业高等学校的讲席时，他感到很高兴。在那里，他和他的同事Brill向大量优秀学生教授高级课程，克莱因的教学天赋得到了充分展现。克莱因在慕尼黑期间教过的学生包括阿道夫·赫维兹、瓦尔特·冯·戴克、Rohn、卡尔·龙格、马克斯·普朗克、路易吉·比安基和格雷戈里奥·里奇-库尔巴斯托罗。同样在1875年，克莱因娶了哲学家格奥尔格·威廉·弗里德里希·黑格尔的孙女安妮·黑格尔。格蕾丝格蕾丝·奇斯霍姆·杨在[3]中回忆道：-

……那些阳光明媚的日子，当时那位高大英俊的年轻教授追求并赢得了哲学家黑格尔可爱的孙女。

在慕尼黑工业高等学校任教五年后，克莱因被任命为莱比锡大学的几何学讲席。在那里，他有一些才华横溢的年轻讲师作为同事，包括瓦尔特·冯·戴克、Rohn、爱德华·斯图迪和Engel。克莱因在莱比锡度过的1880年至1886年，在许多方面从根本上改变了他的生活。正如D·E·罗在[13]中所写：-

莱比锡似乎是建立他现在所设想的那种学派的绝佳前哨：这种学派将大量借鉴波恩哈德·黎曼的函数论几何方法所提供的丰富资源。但不可预见的事件和他一向虚弱的身体共同阻碍了这一计划。……[在他身上有]两个灵魂……一个渴望宁静的学者生活，另一个渴望作为编辑、教师和科学组织者的活跃生活。……正是在1882年秋天，这两个世界中的第一个向他轰然倒塌……他的健康彻底崩溃，在整个1883-1884年间，他饱受抑郁之苦。

作为研究数学家的生涯基本结束，克莱因于1886年接受了哥廷根大学的讲席。他在哥廷根任教直至1913年退休，但他此时力图重新确立哥廷根作为世界首要数学研究中心的地位。他在莱比锡作为几何学派领袖的角色从未转移到哥廷根。在哥廷根，他教授各种各样的课程，主要是数学与物理之间的交叉领域，如力学和potential theory。

克莱因在哥廷根建立了一个研究中心，该中心后来成为世界各地最佳数学研究中心的典范。他引入了每周讨论会、一个带有数学图书馆的数学阅览室。克莱因于1895年将大卫·希尔伯特从柯尼斯堡带到哥廷根加入他的研究团队。

期刊Mathematische Annalen的声誉建立在克莱因的数学和管理能力之上。该期刊最初由阿尔弗雷德·克莱布什创办，但只有在克莱因的管理下，它才首先与奥古斯都·利奥波德·克雷勒的期刊匹敌，然后在重要性上超越它。从某种意义上说，这些期刊代表了柏林数学学派中经营奥古斯都·利奥波德·克雷勒期刊的竞争团队和支持Mathematische Annalen的阿尔弗雷德·克莱布什的追随者。克莱因建立了一个小型编辑团队，定期开会并做出民主决定。该期刊专注于复分析、代数几何和不变量理论。它也为实分析和群论这一新领域提供了重要的发表渠道。

克莱因因健康不佳于1913年退休。然而，在第一次世界大战期间，他继续在家中教授数学。

理解克莱因对几何学的贡献的重要性有点困难。这并不是因为它今天对我们来说很陌生，恰恰相反，它已经成为我们当前数学思维中如此重要的一部分，以至于我们很难意识到他的结果的新颖性，以及它们并未被他的所有同时代人普遍接受这一事实。

克莱因的第一个重要数学发现是在1870年与索菲斯·李合作做出的。他们发现了恩斯特·爱德华·库默尔曲面上渐近线的基本性质。随后又与索菲斯·李合作，他们研究W曲线，即在射影变换群下不变的曲线。事实上，索菲斯·李在克莱因的发展中起了重要作用，向他介绍了群的概念，这在他后来的工作中起了主要作用。公平地说，卡米耶·若尔当也在教克莱因关于群的知识方面起了一定作用。

1871年在哥廷根期间，克莱因在几何学方面做出了重大发现。他发表了两篇论文On the So-called Non-Euclidean Geometry，其中他表明可以将欧几里得几何和非欧几里得几何视为附加了特定conic section的射影曲面的特殊情况。这有一个显著的推论：非欧几里得几何是相容的当且仅当欧几里得几何是相容的。非欧几里得几何在当时仍是一个有争议的话题这一事实现在消失了。它的地位被置于与欧几里得几何完全相同的基础上。阿瑟·凯莱从未接受克莱因的想法，认为他的论证是循环论证。

克莱因将几何学综合为对在给定变换群下不变的空间性质的研究，即所谓的Erlanger Programm（1872年），深刻影响了数学的发展。这是为克莱因1872年被任命为埃尔朗根教授时的就职演讲而写的，尽管它实际上并不是他在那个场合所做的演讲。埃尔朗根纲领为几何学提供了一种统一的方法，这现在是标准公认的观点。

变换在现代数学中起着主要作用，克莱因展示了给定几何的本质性质如何可以由保持这些性质的变换群来表示。通过这种方式，埃尔朗根纲领定义了几何学，使其既包括欧几里得几何也包括非欧几里得几何。

克莱因本人将其在函数论方面的工作视为他对数学的主要贡献。正如W Burau和B Schoenberg在[1]中所写：-

克莱因认为他在函数论方面的工作是他数学工作的顶峰。他的一些最伟大的成功归功于他对波恩哈德·黎曼思想的发展，以及他在后期与不变理论、数论和代数、群论、多维几何和微分方程理论的概念之间建立的密切联系，特别是在他自己的领域——椭圆模函数和自守函数中。

通过考虑模群在复平面上的作用，克莱因表明基本区域被移动以铺砌平面。1879年，他研究了$PSL(2,7)$的作用，将其视为模群的像，并获得了波恩哈德·黎曼曲面的显式表示。他表明它在射影空间中具有方程$x^{3}y + y^{3}z + z^{3}x = 0$作为一条曲线，其对称群是阶为168的$PSL(2,7)$。他在1882年写了Riemanns Theorie der algebraischen Funktionen und ihre IntegralsⓉ（黎曼的代数函数及其积分理论），以几何方式处理函数论，连接了位势论和共形映射。他在这项工作中还使用了物理思想，特别是流体动力学的思想。

克莱因考虑了次数大于4的方程，并且特别感兴趣于使用超越方法求解一般的五次方程。在基于夏尔·埃尔米特和利奥波德·克罗内克的方法之上，产生了与弗朗切斯科·布廖斯基类似的结果后，他继续使用二十面体的群完全解决了这个问题。这项工作使他考虑椭圆模函数，他在一系列论文中研究了这些函数。

他发展了自守函数理论，在他1884年关于二十面体的重要著作中连接了代数和几何结果。然而，儒勒·昂利·庞加莱于1881年开始发表他的自守函数理论纲要，正如[13]中所解释的，这导致了两人之间的竞争：-

克莱因与儒勒·昂利·庞加莱开始了通信，很快友好的竞争随之而来，因为两人都试图制定并证明一个宏大的单值化定理，作为这一理论的顶峰。在巨大压力下工作，克莱因成功制定了这样一个定理，并勾勒了证明它的策略。

然而，正是在这项工作期间，如上所述，克莱因的健康状况崩溃了。与1884年来到莱比锡的Robert Fricke一起，克莱因在接下来的20年里撰写了一部关于自守函数和椭圆模函数的四卷本经典著作。

我们还应该提到克莱因瓶，一种以克莱因命名的单侧闭曲面。克莱因瓶无法在欧几里得空间中构造。它最好被想象为一个圆柱体自身回环，与另一端相连。然而，这不是3空间中的连续曲面，因为曲面无法在不产生不连续的情况下穿过自身。可以在非欧几里得空间中构造克莱因瓶。

在1890年代，克莱因对数学物理产生了兴趣，尽管在他的整个职业生涯中，他表现出在态度上从未远离这一领域。出于这种兴趣，他与阿诺·索末菲合写了一部关于陀螺仪的重要著作。

在职业生涯后期，克莱因开始对中学水平的教学产生兴趣。W Burau和B Schoenberg在[1]中写道：-

从1900年开始，他在继续履行学术职责的同时，对大学水平以下的数学教学产生了浓厚的兴趣。作为德国数学教学现代化的倡导者，1905年他在制定“Ⓣ”（梅拉纳教学计划）中发挥了决定性作用。建议的根本性变革是在中学引入微积分和函数概念的初步知识。

在1908年罗马国际数学家大会上，克莱因当选为国际数学教育委员会主席。在他的指导下，该委员会的德国分会出版了许多关于德国各级数学教学的卷册。

他在世纪之交从事的另一个项目是Encyklopädie der Mathematischen WissenschaftenⓉ（数学科学百科全书）。他积极参与这个项目，与K 埃米尔·穆勒共同编辑了关于力学的四卷部分。

格蕾丝·奇斯霍姆·杨在[3]中写到克莱因为数学领域的女性所做的努力：-

当他在[1893年]和其他人成功地为女性打开哥廷根大学的大门时，我认为，对他来说，第一位获得哲学博士学位的女性应该在他的主持下取得学位，并且是一位曾师从于他所尊敬的友人阿瑟·凯莱的格顿学院女生，这确实是一件令他高兴的事。

克莱因于1885年当选为皇家学会成员，并于1912年获得该学会的科普利奖章。London Mathematical Society于1893年授予他奥古斯塔斯·德摩根奖章。

格蕾丝·奇斯霍姆·杨在[3]中写道：-

[他的头脑]充满了想法和辉煌的思考，但确实他的工作缺乏数学精确性所要求的严格方面。通过个人接触，这一点得到了纠正，至少就他的学生而言是如此。他最喜欢的格言是：“永远不要沉闷”。

Felix Klein is best known for his work in non-euclidean geometry, for his work on the connections between geometry and group theory, and for results in function theory. He was born on 25/4/1849 and delighted in pointing out that each of the day ($5^{2}$), month ($2^{2}$), and year ($43^{2}$) was the square of a prime.

Klein's father was secretary to the head of the government. There is a colourful description of Felix's birth in his obituary in the Proceedings of the Royal Society:-

Without, the cannon thundered on the barricades raised by the insurgent Rhinelanders against their hated Prussian rulers. Within, although all had been prepared for flight, there was no thought of departure; on that night was born a son to the stern Prussian secretary. That son was Felix Klein.

The revolution against the Prussians, which resulted in such a dramatic birth for Felix Klein, was completely crushed by the summer of 1849.

Klein attended the Gymnasium in Düsseldorf. After graduating, he entered the University of Bonn and studied mathematics and physics there during 1865-1866. He started out on his career with the intention of becoming a physicist. While still studying at the University of Bonn, he was appointed to the post of laboratory assistant to Plücker in 1866. Plücker held a chair of mathematics and experimental physics at Bonn but, by the time Klein became his assistant, Plücker's interests had become very firmly rooted in geometry. Klein received his doctorate, which was supervised by Plücker, from the University of Bonn in 1868, with a dissertation Über die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien-Koordinaten auf eine kanonische Form Ⓣ on line geometry and its applications to mechanics. In his dissertation Klein classified second degree line complexes using Weierstrass's theory of elementary divisors.

However in the year Klein received his doctorate Plücker died leaving his major work on the foundations of line geometry incomplete. Klein was the obvious person to complete the second part of Plücker's Neue Géometrie des Raumes Ⓣ and this work led him to become acquainted with Clebsch. Clebsch had moved to Göttingen in 1868 and, during 1869, Klein made visits to Berlin and Paris and Göttingen. In July 1870 Klein was in Paris when Bismarck, the Prussian chancellor, published a provocative message aimed at infuriating the French government. France declared war on Prussia on the 19th of July and Klein felt he could no longer remain in Paris and returned. Then, for a short period, he did military service as a medical orderly before being appointed as a lecturer at Göttingen in early 1871.

Klein was appointed professor at Erlangen, in Bavaria in southern Germany, in 1872. He was strongly supported by Clebsch, who regarded him as likely to become the leading mathematician of his day, and so Klein held a chair from the remarkably early age of 23. However Klein did not build a school at Erlangen where there were only a few students, so he was pleased to be offered a chair at the Technische Hochschule at Munich in 1875. There he, and his colleague Brill, taught advanced courses to large numbers of excellent students and Klein's great talent at teaching was fully expressed. Among the students that Klein taught while at Munich were Hurwitz, von Dyck, Rohn, Runge, Planck, Bianchi and Ricci-Curbastro. Also in 1875 Klein married Anne Hegel, the granddaughter of the philosopher Georg Wilhelm Friedrich Hegel. Grace Chisholm Young recalls in [3]:-

... the sunny days when the tall handsome young professor wooed and won the lovely granddaughter of the philosopher Hegel.

After five years at the Technische Hochschule at Munich, Klein was appointed to a chair of geometry at Leipzig. There he had as colleagues a number of talented young lecturers, including von Dyck, Rohn, Study and Engel. The years 1880 to 1886 that Klein spent at Leipzig were in many ways to fundamentally change his life. As D E Rowe writes in [13]:-

Leipzig seemed to be a superb outpost for building the kind of school he now had in mind: one that would draw heavily on the abundant riches offered by Riemann's geometric approach to function theory. But unforeseen events and his always delicate health conspired against this plan. .. [In him were] two souls ... one longing for the tranquil scholar's life, the other for the active life of an editor, teacher, and scientific organiser. ... It was during the autumn of 1882 that the first of these two worlds came crashing down upon him ... his health collapsed completely, and throughout the years 1883-1884 he was plagued by depression.

His career as a research mathematician essentially over, Klein accepted a chair at the University of Göttingen in 1886. He taught at Göttingen until he retired in 1913 but he now sought to re-establish Göttingen as the foremost mathematics research centre in the world. His own role as the leader of a geometrical school at Leipzig was never transferred to Göttingen. At Göttingen he taught a wide variety of courses, mainly on the interface between mathematics and physics, such as mechanics and potential theory.

Klein established a research centre at Göttingen which was to serve as a model for the best mathematical research centres throughout the world. He introduced weekly discussion meetings, a mathematical reading room with a mathematical library. Klein brought Hilbert from Königsberg to join his research team at Göttingen in 1895.

The fame of the journal Mathematische Annalen is based on Klein's mathematical and management abilities. The journal was originally founded by Clebsch but only under Klein's management did it first rival, and then surpass in importance, Crelle's journal. In a sense these journals represented the rival teams of the Berlin school of mathematics who ran Crelle's journal and the followers of Clebsch who supported the Mathematische Annalen. Klein set up a small team of editors who met regularly and made democratic decisions. The journal specialised in complex analysis, algebraic geometry and invariant theory. It also provided an important outlet for real analysis and the new area of group theory.

Klein retired due to ill health in 1913. However he continued to teach mathematics at his home during the years of World War I.

It is a little hard to understand the significance of Klein's contributions to geometry. This is not because it is strange to us today, quite the reverse, it has become so much a part of our present mathematical thinking that it is hard for us to realise the novelty of his results and also the fact that they were not universally accepted by all his contemporaries.

Klein's first important mathematical discoveries were made in 1870 in collaboration with Lie. They discovered the fundamental properties of the asymptotic lines on the Kummer surface. Further collaboration with Lie followed and they worked on an investigation of W-curves, curves invariant under a group of projective transformations. In fact Lie played an important role in Klein's development, introducing him to the group concept which played a major role in his later work. It is fair to add that Camille Jordan also played a part in teaching Klein about groups.

During his time at Göttingen in 1871 Klein made major discoveries regarding geometry. He published two papers On the So-called Non-Euclidean Geometry in which he showed that it was possible to consider euclidean geometry and non-euclidean geometry as special cases a projective surface with a specific conic section adjoined. This had the remarkable corollary that non-euclidean geometry was consistent if and only if euclidean geometry was consistent. The fact that non-euclidean geometry was at the time still a controversial topic now vanished. Its status was put on an identical footing to euclidean geometry. Cayley never accepted Klein's ideas believing his arguments to be circular.

Klein's synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programm (1872), profoundly influenced mathematical development. This was written for the occasion of Klein's inaugural address when he was appointed professor at Erlangen in 1872 although it was not actually the speech he gave on that occasion. The Erlanger Programm gave a unified approach to geometry which is now the standard accepted view.

Transformations play a major role in modern mathematics and Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. In this way the Erlanger Programm defined geometry so that it included both Euclidean geometry and non-Euclidean geometry.

However Klein himself saw his work on function theory as his major contribution to mathematics. As W Burau and B Schoenberg write in [1]:-

Klein considered his work in function theory to be the summit of his work in mathematics. He owed some of his greatest successes to his development of Riemann's ideas and to the intimate alliance he forged between the later and the conception of invariant theory, of number theory and algebra, of group theory, and of multidimensional geometry and the theory of differential equations, especially in his own fields, elliptic modular functions and automorphic functions.

By considering the action of the modular group on the complex plane, Klein showed that the fundamental region is moved around to tessellate the plane. In 1879 he looked at the action of $PSL(2,7)$, thought of as an image of the modular group, and obtained an explicit representation of a Riemann surface. He showed it had equation $x^{3}y + y^{3}z + z^{3}x = 0$ as a curve in projective space and its group of symmetries was $PSL(2,7)$ of order 168. He wrote Riemanns Theorie der algebraischen Funktionen und ihre Integrals Ⓣ in 1882 which treats function theory in a geometric way connecting potential theory and conformal mappings. He also used physical ideas in this work, especially those of fluid dynamics.

Klein considered equations of degree greater than 4 and was particularly interested in using transcendental methods to solve the general equation of the fifth degree. After building on methods due to Hermite and Kronecker, producing similar results to Brioschi, he went on to completely solve the problem using the group of the icosahedron. This work led him to consider elliptic modular functions which he studied in a series of papers.

He developed a theory of automorphic functions, connecting algebraic and geometric results in his important 1884 book on the icosahedron. However Poincaré began publishing an outline of his theory of automorphic functions in 1881 and, as explained in [13], this led to a competition between the two:-

Klein initiated a correspondence with Poincaré, and soon a friendly rivalry ensued as both sought to formulate and prove a grand uniformization theorem that would serve as a capstone to this theory. Working under great stress, Klein succeeded in formulating such a theorem and in sketching a strategy for proving it.

However it was during this work that Klein's health collapsed as mentioned above. With Robert Fricke who came to Leipzig in 1884, Klein wrote a major four volume classic on automorphic and elliptic modular functions produced over the following 20 years.

We should also mention the Klein bottle, a one-sided closed surface named after Klein. A Klein bottle cannot be constructed in Euclidean space. It is best pictured as a cylinder looped back through itself to join with its other end. However this is not a continuous surface in 3-space as the surface cannot go through itself without a discontinuity. It is possible to construct a Klein bottle in non-Euclidean space.

In the 1890s Klein became interested in mathematical physics, although throughout his career he showed he was never far from this area in attitude. Following from this interest, he wrote an important work on the gyroscope with A Sommerfeld.

Later in his career Klein became interested in teaching at school level. W Burau and B Schoenberg write in [1]:-

Starting in 1900 he began to take a lively interest in mathematical instruction below university level while continuing to pursue his academic functions. An advocate of modernizing mathematics instruction in Germany, in 1905 he played a decisive role in formulating the "Meraner Lehrplanestwürfe" Ⓣ. The essential change recommended was the introduction in secondary schools of the rudiments of differential and integral calculus and the function concept.

Klein was elected chairman of the International Commission on Mathematical Instruction at the Rome International Mathematical Congress of 1908. Under his guidance the German branch of the Commission published many volumes on the teaching of mathematics at all levels in Germany.

Another project he worked on around the turn of the century was the Encyklopädie der Mathematischen Wissenschaften Ⓣ. He took an active part in this project, editing with K Müller the four volume section on mechanics.

Grace Chisholm Young writes in [3] of Klein's efforts on behalf of women in mathematics:-

When in [1893] he and others succeeded in opening the doors of the University of Göttingen to women, it was, I think, a real pleasure to him that the first woman to take the degree of D.Phil. should do so under his auspices, and should be a Girton girl who had sat at the feet of his revered friend Cayley.

Klein was elected a member of the Royal Society in 1885 and received the Copley medal of the Society in 1912. The London Mathematical Society awarded him their De Morgan Medal in 1893.

Chisholm Young writes in [3]:-

[His mind] teemed with ideas and brilliant reflections, but it is true that his work lacks the stern aspects required by mathematical exactitude. It was in personal contact that this was corrected, at least in so far as his students were concerned. His favourite maxim was, "Never be dull".