恩斯特·爱德华·库默尔Ernst Kummer

恩斯特·爱德华·库默尔的父亲卡尔Gotthelf Kummer是一名医生。然而他在库默尔年仅三岁时就去世了，库默尔和他的哥哥由母亲抚养长大。库默尔在九岁进入索劳的文理中学之前接受过私人辅导。1828年，库默尔进入哈勒大学，打算学习新教神学。对数学的幸运之处在于，库默尔在攻读学位期间接受了数学教学，这是为了给哲学研究提供适当的基础而设置的。库默尔的数学讲师H F 海因里希·谢尔克激发了他对数学的兴趣，库默尔很快就把数学作为自己的主修科目，尽管在这个阶段他仍然认为数学会导向日后对哲学的研究。

1831年，库默尔因就谢尔克所出题目撰写的一篇数学论文而获奖。同年，他获得了可以在学校任教的证书，并且凭借那篇获奖论文获得了博士学位。在1831-32学年，库默尔在他曾受教育的索劳文理中学进行了一年的试用期教学。随后，他被任命到利格尼茨的文理中学任教，该地现为波兰的莱格尼察。

库默尔在利格尼茨担任这一教职达10年。在那里他教授数学和物理，有时也教其他科目。他的一些学生能力出众，反过来，他们也极其幸运地遇到了一位像库默尔这样素质高、有能力激发学生的学校教师。他最著名的两个学生是利奥波德·克罗内克和费迪南德·约阿希姆斯塔尔，在库默尔的指导下，他们在上学期间就开始了数学研究。

库默尔本人在利格尼茨任教期间也从事数学研究。1836年，他在Crelle's Journal上发表了一篇关于hypergeometric series的论文，并把论文副本寄给了卡尔·古斯塔夫·雅各布·雅可比。这导致卡尔·古斯塔夫·雅各布·雅可比以及后来的约翰·彼得·古斯塔夫·勒热纳·狄利克雷就数学课题与库默尔通信，他们很快意识到库默尔在最高水平的数学方面具有巨大潜力。1839年，尽管库默尔仍是一名学校教师，但经约翰·彼得·古斯塔夫·勒热纳·狄利克雷推荐，他被选入柏林科学院。卡尔·古斯塔夫·雅各布·雅可比现在意识到，他必须为库默尔找到一个大学教授职位。

1840年，库默尔与约翰·彼得·古斯塔夫·勒热纳·狄利克雷妻子的表亲结婚。这段婚姻只持续了八年，因为他的妻子于1848年不幸去世。然而这些年对库默尔来说是取得巨大成就的时期。1842年，在卡尔·古斯塔夫·雅各布·雅可比和约翰·彼得·古斯塔夫·勒热纳·狄利克雷的大力支持下，他被任命为布雷斯劳大学的正教授，该地现为波兰的弗罗茨瓦夫。在那里，他很快确立了自己作为杰出大学数学教师的地位，并且从迁往布雷斯劳开始，他着手从事数论方面的研究。1848年妻子去世后，库默尔很快就再婚了。

1848年2月22日至24日巴黎起义期间，国王路易-菲利普被推翻。这引发了一系列针对德意志邦联政府的同情性革命。其中大多数是温和的事件，但柏林发生的战斗却激烈而血腥。在这个时期，没有人能够没有强烈的政治观点，事实上库默尔持有强烈的右翼政治观点。他在1848年革命中支持君主立宪制，反对共和制。当1848年3月13日，体制的象征梅特涅被迫辞去奥地利内阁职务时，诸侯们迅速与反对派和解，以防止像法国那样的共和主义和社会主义实验。

1855年，约翰·彼得·古斯塔夫·勒热纳·狄利克雷离开柏林前往哥廷根接替卡尔·弗里德里希·高斯。他推荐柏林将空缺的讲席提供给库默尔，柏林确实这样做了。然后库默尔玩了一个巧妙的政治把戏。他希望卡尔·魏尔斯特拉斯成为柏林大学的同事，但他意识到卡尔·魏尔斯特拉斯是他在布雷斯劳留下的空缺讲席的有力候选人。因此，他推荐布雷斯劳任命他以前的学生费迪南德·约阿希姆斯塔尔。一切按库默尔的计划进行，1856年，卡尔·魏尔斯特拉斯被任命到柏林。利奥波德·克罗内克也在1855年被任命到柏林，因此柏林成为世界领先的数学中心之一。

1861年，库默尔和卡尔·魏尔斯特拉斯在柏林建立了德国第一个纯数学讨论班。K-R Biermann在[1]中写到库默尔在柏林的教学：-

库默尔的柏林讲座总是精心准备，涵盖解析几何、力学、曲面理论和数论。他讲解清晰生动，吸引了大量学生——他的讲座曾多达250人听讲。卡尔·魏尔斯特拉斯和利奥波德·克罗内克在讲座中介绍他们最新的研究成果，而库默尔在创办讨论班后，则局限于奠定坚实的基础。另一方面，在讨论班中，他讨论自己的研究，以鼓励参与者进行独立研究。

Biermann写到库默尔的受欢迎程度：-

库默尔作为教授的受欢迎程度不仅基于他讲座的清晰性，还基于他的魅力和幽默感。此外，他关心学生的福祉，并在他们遇到物质困难时乐意帮助他们……

令人惊讶的是，鉴于库默尔作为数学教师的杰出品质，他从未写过任何教科书。他确实发表了数学讲座，当然，还有许多极具影响力的数学论文。

库默尔在还是学校教师时首次当选为柏林科学院院士，最终在科学院担任高级职务。他从1863年到1878年担任科学院数学/物理部的秘书。他还在柏林大学担任高级职务，1857-58年和1865-66年担任院长。1868-69年担任大学校长。

在柏林期间，库默尔指导了大量博士生，其中许多人后来在大学担任数学讲席，包括保罗·巴赫曼、格奥尔格·康托尔、du Bois-Reymond、保罗·哥尔丹、阿图尔·舍恩弗利斯和赫尔曼·阿曼杜斯·施瓦茨。事实上，赫尔曼·阿曼杜斯·施瓦茨娶了他的一个女儿，从而与库默尔成为亲戚。库默尔还任命了许多有才华的年轻讲师，包括阿尔弗雷德·克莱布什、埃尔温·布鲁诺·克里斯托费尔和拉扎勒斯·福克斯。当库默尔在1883年因记忆力衰退而决定退休时，是拉扎勒斯·福克斯接替了他，尽管除了库默尔本人之外，没有人察觉到这一点。

库默尔的第一个数学时期研究函数论。他扩展了卡尔·弗里德里希·高斯关于超几何级数的工作，给出了在微分方程理论中有用的发展。他是第一个计算这些级数的单值群的人。

1843年，库默尔意识到证明费马大定理的尝试失败，是因为整数的唯一因子分解不能推广到其他复数环，于是试图通过引入“理想”数来恢复因子分解的唯一性。他的工作不仅在关于皮埃尔·德·费马大定理的工作中最为基础，因为此后许多年的所有工作都以此为基础，而且理想的概念使得环论以及大部分抽象代数得以发展。

Paris Academy of Sciences因这项工作于1857年授予库默尔大奖。事实上，3000法郎的奖金是为解决费马大定理而设，但当没有解决方案出现时，即使延长了日期，奖金还是给了库默尔，尽管他并未提交参赛作品。

库默尔获得大奖后不久，他当选为巴黎科学院成员，然后在1863年，他当选为伦敦皇家学会会士。在他漫长的职业生涯中，他还获得了许多其他荣誉。

库默尔的几何时期是他致力于研究威廉·哈密顿曾考察过的射线系统的时期，但库默尔以代数方式处理这些问题。他还发现了以他命名的四阶曲面，基于二次线丛的奇异曲面。库默尔曲面有16个孤立的锥形二重点和16个奇异切线平面，于1864年发表。

柏林的三位伟大数学家库默尔、卡尔·魏尔斯特拉斯和利奥波德·克罗内克在二十年里是亲密的朋友，他们密切而有效地合作。然而，大约在1875年，卡尔·魏尔斯特拉斯和利奥波德·克罗内克闹翻了。库默尔继续与利奥波德·克罗内克保持友谊，但这使他与卡尔·魏尔斯特拉斯的关系变得紧张。也许这三位伟大的数学天才发生这种事并不太令人惊讶，特别是考虑到利奥波德·克罗内克会猛烈人身攻击任何与他有数学分歧的人。

Eduard Kummer's father, Carl Gotthelf Kummer was a physician. However he died when Eduard was only three years old and Eduard and his elder brother were brought up by their mother. Eduard received private coaching before entering the Gymnasium in Sorau when he was nine years old. In 1828 Kummer entered the University of Halle with the intention of studying Protestant theology. Fortunately for the good of mathematics, Kummer received mathematics teaching as part of his degree, designed to provide a proper foundation to the study of philosophy. Kummer's mathematics lecturer H F Scherk inspired his interest in mathematics and Kummer soon was studying mathematics as his main subject, although at this stage he still saw it as leading to a later study of philosophy.

In 1831 Kummer was awarded a prize for a mathematical essay he wrote on a topic set by Scherk. In the same year he was awarded his certificate to enable him to teach in schools and, on the strength of his prize winning essay, he was awarded a doctorate. During session 1831-32 Kummer taught a probationary year at the Gymnasium in Sorau where he had been educated. Then he was appointed to a teaching post at the Gymnasium in Liegnitz, now Legnica in Poland.

Kummer held this teaching post in Liegnitz for 10 years. There he taught mathematics and physics and sometimes other topics. Some of his pupils had great ability, and conversely they were extremely fortunate to find a school teacher of Kummer's quality and ability to inspire. His two most famous pupils were Kronecker and Joachimsthal and, under Kummer's guidance, they were undertaking mathematical research while at school.

Kummer himself was undertaking mathematics research while teaching at Liegnitz. He published a paper on hypergeometric series in Crelle's Journal in 1836 and he sent a copy of the paper to Jacobi. This led to Jacobi, and later Dirichlet, corresponding with Kummer on mathematical topics and they soon realised the great potential for the highest level of mathematics that Kummer possessed. In 1839, although still a school teacher, Kummer was elected to the Berlin Academy on Dirichlet's recommendation. Jacobi now realised that he had to find Kummer a university professorship.

In 1840 Kummer married a cousin of Dirichlet's wife. This marriage would only last eight years as his wife sadly died in 1848. However these where years of great achievement for Kummer. In 1842, with strong support from Jacobi and Dirichlet, he was appointed a full professor at the University of Breslau, now Wrocław in Poland. There he quickly established himself as an outstanding university teacher of mathematics and, starting with his move to Breslau, he began to undertake research in number theory. After the death of his wife in 1848, Kummer remarried fairly soon.

During the 22 to 24 February 1848 insurrection in Paris, king Louis-Philippe was overthrown. This resulted in a series of sympathetic revolutions against the governments of the German Confederation. Most of them were tame affairs but in the case of the fighting in Berlin it was bitter and bloody. Nobody could fail to have strong political views at this time and indeed Kummer had strong right wing political views. He supported the constitutional monarchy in the 1848 revolution and was against a republic. When on 13 March 1848 Metternich, a symbol of the establishment, was forced to resign his position in the Austrian Cabinet, the princes quickly made peace with the opposition to prevent republican and socialist experiments like those in France.

In 1855 Dirichlet left Berlin to succeed Gauss at Göttingen. He recommended to Berlin that they offer the vacant chair to Kummer, which indeed they did. Then Kummer played a clever political trick. He wanted Weierstrass as a colleague at Berlin, yet he realised that Weierstrass was a strong candidate for the chair he was leaving vacant in Breslau. Hence he recommended to Breslau that they appoint his former student Joachimsthal. All went according to plan for Kummer and, in 1856, Weierstrass was appointed to Berlin. Kronecker had also been appointed to Berlin in 1855 so Berlin became one of the leading mathematical centres in the world.

In 1861 Kummer and Weierstrass established Germany's first pure mathematics seminar in Berlin. K-R Biermann writes of Kummer's teaching in Berlin in [1]:-

Kummer's Berlin lectures, always carefully prepared, covered analytic geometry, mechanics, the theory of surfaces, and number theory. The clarity and vividness of his presentations brought him great numbers of students - as many as 250 were counted at his lectures. While Weierstrass and Kronecker offered the most recent results of their research in their lectures, Kummer in his restricted himself, after instituting the seminar, to laying firm foundations. In the seminar, on the other hand, he discussed his own research in order to encourage the participants to undertake independent investigations.

Explaining Kummer's popularity Biermann writes:-

Kummer's popularity as a professor was based mot only on the clarity of his lectures but on his charm and sense of humour as well. Moreover, he was concerned for the well-being of his students and willingly aided them when material difficulties arose...

Surprisingly, given Kummer's outstanding qualities as a teacher of mathematics, he never wrote any textbooks. He did publish mathematical lectures and, of course, many extremely influential mathematical papers.

Having first been elected to the Berlin Academy while still a school teacher, Kummer ended up with high office in the Academy. He was secretary of the Mathematics/ Physics Section of the Academy from 1863 to 1878. He also held high office in the University of Berlin, being Dean in 1857-58 and again in 1865-66. He was rector of the university in 1868-69.

While at Berlin, Kummer supervised a large number of doctoral students including many who went on to hold mathematics chairs at universities, including Bachmann, Cantor, du Bois-Reymond, Gordan, Schönflies and Schwarz. In fact Schwarz became related to Kummer when he married one of his daughters. Kummer also appointed many talented young lecturers including Clebsch, Christoffel and Fuchs. It was Fuchs who succeeded Kummer when he decided to retire in 1883 on the grounds that his memory was failing, although nobody other than Kummer himself ever detected this.

During Kummer's first period of mathematics he worked on function theory. He extended Gauss's work on hypergeometric series, giving developments that are useful in the theory of differential equations. He was the first to compute the monodromy groups of these series.

In 1843 Kummer, realising that attempts to prove Fermat's Last Theorem broke down because the unique factorisation of integers did not extend to other rings of complex numbers, attempted to restore the uniqueness of factorisation by introducing 'ideal' numbers. Not only has his work been most fundamental in work relating to Fermat's Last Theorem, since all later work was based on it for many years, but the concept of an ideal allowed ring theory, and much of abstract algebra, to develop.

The Paris Academy of Sciences awarded Kummer the Grand Prize in 1857 for this work. In fact the prize of 3000 francs was offered for a solution to Fermat's Last Theorem but when no solution was forthcoming, even after extending the date, the Prize was given to Kummer even though he had not submitted an entry for the Prize.

Soon after Kummer was awarded the Grand Prix he was elected to membership of the Paris Academy of Sciences and then, in 1863, he was elected a Fellow of the Royal Society of London. He received numerous other honours in his long career.

Kummer's geometric period was one when he devoted himself to the study of the ray systems that Hamilton had examined, but Kummer treated these problems algebraically. He also discovered the fourth order surface, now named after him, based on the singular surface of the quadratic line complex. The Kummer surface has 16 isolated conical double points and 16 singular tangent planes and was published in 1864.

The three great mathematicians of Berlin, Kummer, Weierstrass and Kronecker were close friends for twenty years as they worked closely and effectively together, However, around 1875 Weierstrass and Kronecker fell out. Kummer continued his friendship with Kronecker but this put a strain on his relation with Weierstrass. Perhaps it is not too surprising that this should happen to these three great mathematical talents, particularly given that Kronecker vigorously attacked personally anyone with whom he had a mathematical difference.