卡米耶·若尔当Camille Jordan

卡米耶·若尔当的父亲埃斯普里-亚历山大若尔当（1800-1888）是一位工程师，曾在巴黎综合理工学院接受教育。若尔当的母亲约瑟芬·皮维·德·夏凡纳是著名画家皮埃尔·皮维·德·夏凡纳的妹妹，后者是19世纪下半叶法国最重要的壁画家。若尔当父亲的家族也相当有名；一位叔祖父也叫埃内蒙-若尔当（1771-1821），曾担任高级政治职位，而一位表亲Alexis Jordan（1814-1897）是著名的植物学家。

若尔当曾在里昂中学和乌兰学院学习。1855年，他进入巴黎综合理工学院学习数学。这所学校培养工程师，而若尔当像他同时代的许多法国数学家一样，取得了工程师资格并从事这一职业。特别是奥古斯丁·路易·柯西曾走过这条路，而像奥古斯丁·路易·柯西一样，若尔当能够作为工程师工作，同时仍投入大量时间进行数学研究。若尔当的学位论文分为两部分，第一部分Sur le nombre des valeurs des fonctions Ⓣ（论函数值的个数）是关于代数的。第二部分题为Sur des periodes des fonctions inverses des intégrales des différentielles algebriques Ⓣ（论代数微分积分的反函数的周期），是关于形式为$\int   u dz$的积分，其中$u$是满足代数方程$f (u, z) = 0$的函数。若尔当于1861年1月14日由让-马里·迪阿梅尔、约瑟夫·阿尔弗雷德·塞雷和维克托·皮瑟进行答辩。事实上，若尔当学位论文第二部分的题目是由维克托·皮瑟提出的，而考官们更偏爱这第二部分。考试结束后，他继续作为工程师工作，先在普里瓦，然后在索恩河畔沙隆，最后在巴黎。

若尔当于1862年与里昂副市长之女玛丽-伊莎贝尔·穆内结婚。他们育有八个孩子，两个女儿和六个儿子。

从1873年起，他在巴黎综合理工学院担任考官，并于1876年11月25日成为分析学教授。从1883年起，他还在法兰西学院担任教授，尽管直到1885年，他至少在理论上仍以工程师为职业。然而，值得注意的是，他在担任工程师时反而有更多时间进行研究。他的大部分原创研究都出自这一时期。

若尔当是一位数学家，他在极其广泛的领域中工作，基本上对当时所研究的每一个数学主题都有贡献。参考文献[3]、[4]、[5]、[6]指的是他全集的四卷，从这些内容可以看出主题的范围。第1卷和第2卷包含若尔当关于有限群的论文，第3卷包含他关于线性与多重线性代数以及数论的论文，而第4卷包含关于多面体拓扑学、微分方程和力学的论文。

拓扑学（当时称为analysis situs）在他最初的一些出版物中发挥了重要作用，这些出版物是关于对称性的组合方法。他在1866年引入了重要的拓扑学概念，这些概念建立在他对波恩哈德·黎曼在拓扑学方面工作的了解之上，但没有基于奥古斯特·费迪南德·莫比乌斯的工作，因为他不知道后者。若尔当引入了路径的同伦概念，研究路径之间的形变。他定义了一个曲面的同伦群，但没有明确使用群术语。

若尔当对有限群论特别感兴趣。事实上，这种说法并不十分准确，因为有理由认为，在若尔当开始这一领域的研究之前，并不存在有限群论。是若尔当首先发展了该主题的系统方法。直到1846年约瑟夫·刘维尔重新发表埃瓦里斯特·伽罗瓦的原始工作，其重要性才被注意到。约瑟夫·阿尔弗雷德·塞雷、约瑟·伯特兰和夏尔·埃尔米特曾听过约瑟夫·刘维尔关于伽罗瓦理论的讲座，并开始为该主题做出贡献，但若尔当是第一个阐明该学科发展方向的人。

若尔当 群就是我们今天所说的 置换群；抽象群的概念要到后来才被研究。为了说明他试图建立群论的方式，我们将稍微谈谈他对有限可解群的贡献。今天定义这类群的标准方式是，它们是合成因子为阿贝尔群的群。事实上，若尔当 引入了合成列的概念（一列子群，每个子群在前一个中正规，并且具有这样的性质：不能再向该列添加更多的项而保持该性质）。群 $G$ 的合成因子是通过计算合成列中相邻群的商群而得到的群。若尔当 证明了 若尔当-奥托·赫尔德 定理，即尽管群可以有不同的合成列，但合成因子的集合是群的不变量。

虽然有限阿贝尔群的分类是直接的，但有限可解群的分类远远超出了今天以及可预见的未来数学家的能力。然而，若尔当 显然将此视为该学科的一个目标，即使它可能永远无法解决。他对这种分类可能如何进行做出了若干显著贡献，建立了一种递归方法来确定给定 $n$ 的所有阶为 $n$ 的可解群。

关于有限群的第二项主要工作是对具有 $p$ 个元素的域上的通用线性群的研究，其中 $p$ 为素数。他应用他在典型群上的工作来确定根被选为与某些几何构型相关联的方程的 Galois group 的结构。

他在1860年至1870年间所做的群论工作被写成了一部重要著作 Traité des substitutions et des équations algebraique Ⓣ（《论替换与代数方程》），于1870年出版。这部论著对 伽罗瓦理论 进行了全面研究，并提供了第一本群论书籍。由于这项工作，他被授予 Académie des Sciences 的 让-维克托·彭赛列 奖。该论著包含了矩阵的“若尔当 标准形”定理，不是在复数上而是在有限域上。他似乎不知道 卡尔·魏尔斯特拉斯 早先关于此类结果的工作。他的书将置换群带入了数学的核心角色，并且直到 威廉·伯恩赛德 在大约30年后写下他著名的群论教科书之前，这项工作为整个学科奠定了基础。也可以公平地说，在 若尔当 的基础性出版物之后的100年里，群论是数学研究的主要领域之一。

若尔当 在1869年将群概念用于几何是受到晶体结构研究的启发。他考虑了欧几里得运动群的分类。他的工作为他赢得了广泛的国际声誉，索菲斯·李 和 菲利克斯·克莱因 都于1870年在巴黎拜访他，与他一起学习。若尔当 对三维空间中欧几里得变换群的兴趣影响了 索菲斯·李 和 菲利克斯·克莱因 关于连续群和不连续群的理论。

Traité des substitutions et des équations algebraique Ⓣ（《论替换与代数方程》）的出版并未标志着 若尔当 对群论贡献的结束。在接下来的十年里，他继续产生了具有根本重要性的进一步结果。他研究了本原置换群并证明了一个有限性定理。他将对称群的子群的类定义为 $c > 1$，如果 $c$ 是该子群有一个移动 $c$ 个点的元素的最小数目。他的有限性定理表明，对于给定的 $c$，除了对称群和交错群之外，只有有限多个类为 $c$ 的本原群。

将拉扎勒斯·福克斯关于线性微分方程的一个结果加以推广，若尔当被引导去研究复数域上$n \times n$matrices的一般线性群的有限子群。尽管这样的有限子群有无穷多个族，若尔当发现它们具有一种非常特殊的群论结构，而他能够对此加以描述。

另一次推广，这次是将夏尔·埃尔米特关于整系数二次型的工作加以推广，使若尔当去考虑复数域上行列式为1的$n \times n$矩阵的特殊线性群作用在$n$个未定元的次数为$m$的复多项式向量空间上。

今天，若尔当在分析学家和拓扑学家中最被铭记的是他证明了简单闭曲线把平面恰好分成两个区域，现在称为Jordan curve theorem。正是他对数学严格性理解的加深，才使他意识到证明这样一个结果是必要的。他还开创了有界变差函数的概念，尤其以他对曲线长度的定义而闻名。这些概念出现在他的Cours d'analyse de l'École Polytechnique中，该书于1882年至1887年间分三卷首次出版。第二版于1893年问世，而若尔当曲线定理则出现在1909年至1915年间出版的该文本第三版中。

当然，到1882年第一卷出版时，若尔当正在巴黎综合理工学院授课，这本书是作为那里学生的教材而写的。在某些方面这有点奇怪，因为这是一部严格的分析教材，建立在奥古斯丁·路易·柯西开始的、并由卡尔·魏尔斯特拉斯给予相当大推动的为该主题奠定坚实基础的努力之上。然而，巴黎综合理工学院的课程本应培养学生成为土木和军事工程师，而这似乎并不是人们试图向工程师教授微积分应用时会采取的方法。巴黎综合理工学院有着严格分析的传统，当然是由奥古斯丁·路易·柯西本人开创的。若尔当意识到他的工作处于一种对工程学生来说有些不太合适的水平，因为他曾对昂利·勒贝格说，他把它称为“巴黎综合理工学院分析教程”，因为：-

……人们在封面上写上这个是为了取悦出版商……

Gispert-Chambaz在[7]中对比了若尔当在该书第一版和第二版中处理拓扑概念的方式。在第一版中，大多数拓扑概念是在第3卷的一个附录中处理的。然而，在两版之间，若尔当在法兰西学院讲授了更高级的分析课程，这可能影响他在第二版中把集合拓扑学放在最前面。在这方面，人们可以把第二版看作是为分析教材定下了延续至今的基调。

在若尔当对分析的许多贡献中，我们还应当提到他对Fourier series收敛判别法的推广。

Journal de Mathématiques Pure et Appliquées是一份领先的数学期刊，在19世纪数学的发展中发挥了非常重要的作用。它通常被称为Journal de Liouville，因为约瑟夫·刘维尔于1836年创办了该期刊。约瑟夫·刘维尔于1882年去世，1885年若尔当成为该期刊的编辑，他担任这一角色超过35年，直到去世。

1912年，若尔当从职位上退休。然而，他生命的最后几年因1914年开始的第一次世界大战而悲伤。1914年至1916年间，他的六个儿子中有三个在战争中阵亡。在剩下的三个儿子中，若尔当是政府部长，爱德华是索邦大学的历史教授，第三个儿子是工程师。

授予若尔当的荣誉包括他于1881年4月4日当选为Académie des Sciences。1890年7月12日，他成为荣誉军团军官。1920年9月，他在斯特拉斯堡举行的国际数学家大会上担任名誉主席。

最后，我们应该注意到一些相当令人困惑的事实。尽管若尔当在矩阵方面的工作以及Jordan normal form以他的名字命名，但用于求解矩阵方程$\bf{A}\bf{x}= \bf{b}$的卡尔·弗里德里希·高斯-若尔当主元消去法并非如此。卡尔·弗里德里希·高斯-若尔当的若尔当是Wilhelm Jordan（1842年至1899年），他将该方法应用于测量中求平方误差。Jordan algebras以德国物理学家和数学家帕斯库尔·约尔丹（1902年至1980年）命名。

Camille Jordan's father, Esprit-Alexandre Jordan (1800-1888), was an engineer who had been educated at the École Polytechnique. Camille's mother, Joséphine Puvis de Chavannes, was the sister of the famous painter Pierre Puvis de Chavannes who was the foremost French mural painter of the second half of the 19th century. Camille's father's family were also quite well known; a grand-uncle also called Ennemond-Camille Jordan (1771-1821) achieved a high political position while a cousin Alexis Jordan (1814-1897) was a famous botanist.

Jordan studied at the Lycée de Lyon and at the Collège d'Oullins. He entered the École Polytechnique to study mathematics in 1855. This establishment provided training to be an engineer and Jordan, like many other French mathematicians of his time, qualified as an engineer and took up that profession. Cauchy in particular had been one to take this route and, like Cauchy, Jordan was able to work as an engineer and still devote considerable time to mathematical research. Jordan's doctoral thesis was in two parts with the first part Sur le nombre des valeurs des fonctions Ⓣ being on algebra. The second part entitled Sur des periodes des fonctions inverses des intégrales des différentielles algebriques Ⓣ was on integrals of the form $\int   u dz$ where $u$ is a function satisfying an algebraic equation $f (u, z) = 0$. Jordan was examined on 14 January 1861 by Duhamel, Serret and Puiseux. In fact the topic of the second part of Jordan's thesis had been proposed by Puiseux and it was this second part which the examiners preferred. After the examination he continued to work as an engineer, first at Privas, then at Chalon-sur-Saône, and finally in Paris.

Jordan married Marie-Isabelle Munet, the daughter of the deputy mayor of Lyon, in 1862. They had eight children, two daughters and six sons.

From 1873 he was an examiner at the École Polytechnique where he became professor of analysis on 25 November 1876. He was also a professor at the Collège de France from 1883 although until 1885 he was at least theoretically still an engineer by profession. It is significant, however, that he found more time to undertake research when he was an engineer. Most of his original research dates from this period.

Jordan was a mathematician who worked in a wide variety of different areas essentially contributing to every mathematical topic which was studied at that time. The references [3], [4], [5], [6] are to the four volumes of his complete works and the range of topics is seen from the contents of these. Volumes 1 and 2 contain Jordan's papers on finite groups, Volume 3 contains his papers on linear and multilinear algebra and on the theory of numbers, while Volume 4 contains papers on the topology of polyhedra, differential equations, and mechanics.

Topology (called analysis situs at that time) played a major role in some of his first publications which were a combinatorial approach to symmetries. He introduced important topological concepts in 1866 built on his knowledge of Riemann's work in topology but not the work by Möbius for he was unaware of it. Jordan introduced the notion of homotopy of paths looking at the deformation of paths one into the other. He defined a homotopy group of a surface without explicitly using group terminology.

Jordan was particularly interested in the theory of finite groups. In fact this is not really an accurate statement, for it would be reasonable to argue that before Jordan began his research in this area there was no theory of finite groups. It was Jordan who was the first to develop a systematic approach to the topic. It was not until Liouville republished Galois's original work in 1846 that its significance was noticed at all. Serret, Bertrand and Hermite had attended Liouville's lectures on Galois theory and had begun to contribute to the topic but it was Jordan who was the first to formulate the direction the subject would take.

To Jordan a group was what we would call today a permutation group; the concept of an abstract group would only be studied later. To give an illustration of the way he tried to build up groups theory we will say a little about his contributions to finite soluble groups. The standard way to define such groups today would be to say that they are groups whose composition factors are abelian groups. Indeed Jordan introduced the concept of a composition series (a series of subgroups each normal in the preceding with the property that no further terms could be added to the series so that it retains that property). The composition factors of a group $G$ are the groups obtained by computing the factor groups of adjacent groups in the composition series. Jordan proved the Jordan-Hölder theorem, namely that although groups can have different composition series, the set of composition factors is an invariant of the group.

Although the classification of finite abelian groups is straightforward, the classification of finite soluble groups is well beyond mathematicians today and for the foreseeable future. Jordan, however, clearly saw this as an aim of the subject, even if it was not one which might ever be solved. He made some remarkable contributions to how such a classification might proceed setting up a recursive method to determine all soluble groups of order $n$ for a given $n$.

A second major piece of work on finite groups was the study of the general linear group over the field with $p$ elements, $p$ prime. He applied his work on classical groups to determine the structure of the Galois group of equations whose roots were chosen to be associated with certain geometrical configurations.

His work on group theory done between 1860 and 1870 was written up into a major text Traité des substitutions et des équations algebraique Ⓣ which he published in 1870. This treatise gave a comprehensive study of Galois theory as well as providing the first ever group theory book. For this work he was awarded the Poncelet Prize of the Académie des Sciences. The treatise contains the 'Jordan normal form' theorem for matrices, not over the complex numbers but over a finite field. He appears not to have known of earlier results of this type by Weierstrass. His book brought permutation groups into a central role in mathematics and, until Burnside wrote his famous group theory text nearly 30 years later, this work provided the foundation on which the whole subject was built. It would also be fair to say that group theory was one of the major areas of mathematical research for 100 years following Jordan's fundamental publication.

Jordan's use of the group concept in geometry in 1869 was motivated by studies of crystal structure. He considered the classification of groups of Euclidean motions. His work had gained him a wide international reputation and both Sophus Lie and Felix Klein visited him in Paris in 1870 to study with him. Jordan's interest in groups of Euclidean transformations in three dimensional space influenced Lie and Klein in their own theories of continuous and discontinuous groups.

The publication of Traité des substitutions et des équations algebraique Ⓣ did not mark the end of Jordan's contribution to group theory. He went on over the next decade to produce further results of fundamental importance. He studied primitive permutation groups and proved a finiteness theorem. He defined the class of a subgroup of the symmetric group to be $c > 1$ if $c$ was the smallest number such that the subgroup had an element moving $c$ points. His finiteness theorem showed that for a given $c$ there are only finitely many primitive groups with class $c$ other than the symmetric and alternating groups.

Generalising a result of Fuchs on linear differential equations, Jordan was led to study the finite subgroups of the general linear group of $n \times n$ matrices over the complex numbers. Although there are infinite families of such finite subgroups, Jordan found that they were of a very specific group theoretic structure which he was able to describe.

Another generalisation, this time of work by Hermite on quadratic forms with integral coefficients, led Jordan to consider the special linear group of $n \times n$ matrices of determinant 1 over the complex numbers acting on the vector space of complex polynomials in $n$ indeterminates of degree $m$.

Jordan is best remembered today among analysts and topologists for his proof that a simply closed curve divides a plane into exactly two regions, now called the Jordan curve theorem. It was only his increased understanding of mathematical rigour which made him realise that a proof of such a result was necessary. He also originated the concept of functions of bounded variation and is known especially for his definition of the length of a curve. These concepts appears in his Cours d'analyse de l'École Polytechnique first published in three volumes between 1882 and 1887. The second edition appeared in 1893 while the Jordan curve theorem appeared in the third edition of the text which appeared between 1909 and 1915.

Of course by 1882, when the first volume was published, Jordan was lecturing at the École Polytechnique and the book was written as a text for the students there. In some respects this is a little strange since it is a rigorous analysis text built on top of the attempts to put the topic on a firm foundation begun by Cauchy and given considerable impetus by Weierstrass. However, the courses at the École Polytechnique were supposed to train students to become civil and military engineers and this does not seem to be the approach which one would take trying to teach applications of the calculus to engineers. There had been a tradition of rigorous analysis at the École Polytechnique begun, of course, by Cauchy himself. Jordan was aware that his work was at a level that would be somewhat inappropriate for engineering students for he once said to Lebesgue that he called it "École Polytechnique analysis course" since:-

... one puts that on the cover to please the publisher...

Gispert-Chambaz in [7] contrasts the way that topological concepts are treated by Jordan in the first and second editions of the book. In the first edition most of the topological concepts are dealt with in a supplement to Volume 3. However between the editions Jordan had taught more advanced courses on analysis at the Collège de France and this may have influenced him to put set topology right up front in the second edition. In this respect one can see the second edition as setting a tone for analysis textbooks which continues today.

Among Jordan's many contributions to analysis we should also mention his generalisation of the criteria for the convergence of a Fourier series.

The Journal de Mathématiques Pure et Appliquées was a leading mathematical journal and played a very significant part in the development of mathematics throughout the 19th century. It was usually known as the Journal de Liouville since Liouville had founded the journal in 1836. Liouville died in 1882 and in 1885 Jordan became editor of the Journal, a role he kept for over 35 years until his death.

In 1912 Jordan retired from his positions. The final years of his life were saddened, however, because of World War I which began in 1914. Between 1914 and 1916 three of his six sons were killed in the war. Of his three remaining sons, Camille was a government minister, Édouard was a professor of history at the Sorbonne, and the third son was an engineer.

Among the honours given to Jordan was his election to the Académie des Sciences on 4 April 1881. On 12 July 1890 he became an officer of the Légion d'Honneur. He was the Honorary President of the International Congress of Mathematicians at Strasbourg in September 1920.

Finally we should note some rather confusing facts. Although given Jordan's work on matrices and the fact that the Jordan normal form is named after him, the Gauss-Jordan pivoting elimination method for solving the matrix equation $\bf{A}\bf{x}= \bf{b}$ is not. The Jordan of Gauss-Jordan is Wilhelm Jordan (1842 to 1899) who applied the method to finding squared errors to work on surveying. Jordan algebras are called after the German physicist and mathematician Pascual Jordan (1902 to 1980).