约翰·查尔斯·菲尔兹John Charles Fields

约翰·查尔斯·菲尔兹是苏格兰-爱尔兰血统，尽管他的父母哈丽雅特·鲍斯和菲尔兹都是加拿大人，住在汉密尔顿国王街东150号。老菲尔兹是一位商人，在汉密尔顿国王街西32号拥有一家皮革店，但在英国和欧洲广泛旅行。他在儿子十一岁时去世。菲尔兹就读于汉密尔顿学院，在那里他证明自己是一名出色的学生。他获得了多项奖学金，并于1880年从该学院毕业。

离开汉密尔顿学院后，菲尔兹于1880年进入多伦多大学学习数学。他的本科岁月并不太轻松，因为已经失去了父亲，他的母亲在他大学课程中途去世。尽管如此，他拥有出色的本科记录，并于1884年从多伦多大学获得数学学士学位，赢得数学金质奖章。随后他前往美国，在约翰斯威廉·霍普金斯大学攻读博士学位。1887年，他凭借学位论文Symbolic Finite Solutions, and Solutions by Definite Integrals of the Equation $(d^{n}/dx^{n})y - (x^{m})y = 0$获得学位后，继续在约翰斯霍普金斯大学任教两年。

菲尔兹于1889年被任命为阿勒格尼学院数学教授，这是宾夕法尼亚州最古老的学院之一，但三年后他辞职，以便通过在欧洲学习来推进他的数学研究。从1892年到1900年，菲尔兹在巴黎、哥廷根和柏林跟随拉扎勒斯·福克斯、费迪南德·格奥尔格·弗罗贝尼乌斯、库尔特·亨泽尔、赫尔曼·阿曼杜斯·施瓦茨、卡尔·魏尔斯特拉斯和马克斯·普朗克学习。他在柏林获得了数学金质奖章。他还结识了约斯塔·米塔格-莱弗勒，并与之建立了终生的友谊。这一时期对他作为研究数学家的未来发展显然很重要。约翰·L·辛格在[4]中写道：-

这段漫长的学习时期对他的生活和人生观产生了决定性的影响，这得益于一份微薄的私人收入，加上简朴的生活和节制的习惯。

1902年，菲尔兹被任命为多伦多大学讲师，他在那里一直待到去世，尽管他经常访问欧洲，在那里他结识了许多国家领导人。例如，他参加了1912年瑞典国王举行的晚宴。他于1905年成为副教授，1914年晋升为正教授。1923年，他被提升为多伦多大学研究教授。

他的主要研究课题是代数函数。他写了一本重要的书，于1906年出版。这本书的主要目的是呈现波恩哈德·黎曼-Roch定理、卡尔·魏尔斯特拉斯间隙定理和相关的阿道夫·赫维兹定理，以及Brill和马克斯·诺特的定理。菲尔兹的一位同事和以前的学生在[4]中对他贡献的总结如下：-

菲尔兹教授在代数函数方面的工作必须被视为对他在世纪之交恰好拥有的关于该主题的思想的发展和整理。这些思想与当时流行的方法几乎没有关联。他将其作为毕生工作，首先表明它们可以用来创建一个令人满意的理论，然后赋予由此确保的结构以优雅和普遍性。他的处理方式具有极大的优点，即完全代数化，并且在不为几何直觉所动的情况下应对每一个困难。他为此目的而不得不发明的机器很简单，其各部分协调得十分优美。

然而，菲尔兹更擅长的是作为数学的组织者。国际数学家大会系列始于1897年的苏黎世，但在第一次世界大战（1914-18）期间没有举行任何大会。International Mathematical Union于1920年在斯特拉斯堡的战后第一次大会上成立，以举办未来的大会，但在战后的余波中，德国、奥匈帝国、保加利亚和土耳其被排除在该联盟之外。这是不幸的，在这个问题上情绪高涨。许多人认为数学不应受到政治压力，而应同样欢迎所有国家的数学家。这种观点的主要倡导者是戈弗雷·哈罗德·哈代和约斯塔·米塔格-莱弗勒。其他人则同样真诚地认为联盟的排除规则是正确的。

1922年，在纽约申办1924年大会失败后，菲尔兹提出在国际数学联盟的主持下于多伦多举办该大会，尽管由于排除规则这意味着大会不能真正具有国际性，但它确实阻止了大会的分裂。他是一位熟练的政治家，能够说服许多联盟的反对者参加多伦多大会并使其成功。这并非易事，菲尔兹在欧洲花了几个月的时间不懈努力，以使大会取得成功。他还必须获得财政支持，以便帮助欧洲人支付前往北美的旅费。他在获得财政支持方面如此成功，以至于大会结束时他还有剩余的钱。这给了他一个绝妙主意的机会。

菲尔兹最为人所铭记的是构思并资助了一项国际数学杰出奖章。最初的提案于1931年2月24日提交给曾举办1924年大会的委员会，并打算使用财务上的剩余资金。菲尔兹已经准备好一切，准备前往1932年9月在苏黎世举行的大会，提出他的奖章提案。他已经做了基础工作，到1932年1月，他已经获得了法国、德国、意大利、瑞士和美国的主要数学学会对颁发奖章的支持。

你可以在THIS LINK看到菲尔兹阐述其提案的信件。

然而，他的健康在1932年5月开始恶化，当时他出现了心脏问题。在他去世前几天，他在约翰·L·辛格的床边立下遗嘱，其中包括将47,000美元添加到奖章基金中。他没有活到参加大会，但他的计划仍被提出。1932年在苏黎世举行的国际数学家大会上通过，第一批菲尔兹奖章于1936年在奥斯陆大会上颁发。请注意，尽管他希望奖章不带有任何人的名字，但它们仍被命名为“菲尔兹奖章”。

菲尔兹奖章每四年在国际数学家大会上颁发给两位40岁以下的数学家。这些条件是为了承认菲尔兹的愿望，即奖项既认可已完成的工作，又指向未来成就的潜力。请注意，40岁的年龄限制并非明确由于菲尔兹。第一批菲尔兹奖章于1936年颁发给拉尔斯·阿尔福斯和杰西·道格拉斯。第二次世界大战期间没有颁发奖项，然后从1950年开始，奖章每四年颁发一次。1966年决定，每届大会至少颁发两枚奖章，最多四枚。

每枚菲尔兹奖章都附有15,000加元的奖金，奖章由黄金制成，上面有阿基米德的头像。上面刻有一句据信出自阿基米德的话：-

Transire suum pectus mundoque potiri.
（超越自我，把握世界。）

铭文：-

Congregati ex toto orbe mathematici ob scripta insignia tribuere
（来自世界各地的数学家齐聚于此，为杰出著作致敬）

出现在背面。

菲尔兹获得了多项重要荣誉。他于1907年当选为加拿大皇家学会会士，并于1913年当选为伦敦皇家学会会士。1924年，国际数学家大会在多伦多举行，菲尔兹荣幸地担任了大会主席。他是1928年在博洛尼亚举行的下一届国际数学家大会的副主席（在那届大会上，被排除在外的国家被重新接纳）。他还于1919年至1925年间担任加拿大皇家研究所所长。1924年，他同时担任国际数学联盟主席、British Association for the Advancement of Science副主席以及美国科学促进会副主席。他还当选为俄罗斯科学院和科英布拉研究所的成员。一个有趣的插曲是，意大利政府想授予菲尔兹“意大利王冠司令”的称号，但加拿大政府有法律禁止加拿大公民持有头衔，因此菲尔兹不得不拒绝意大利政府的提议。

让我们以菲尔兹在数学世界之外生活的若干细节来结束这篇传记。他从未结婚。年轻时他热爱运动，打棒球、足球和冰球。他也喜欢散步。音乐对他是一大乐事，他拉小提琴，还喜欢跳舞[4]：-

在生活习惯上，他饮食有节，不喝茶、咖啡、酒，不用调味品，也不吸烟……他很有幽默感……

John Charles Fields was of Scots-Irish extraction although his parents, Harriet Bowes and John Charles Fields, were both Canadians and lived at 150 King Street East, Hamilton. John Charles Fields Senior was a merchant who owned a leather shop at 32 King Street West, Hamilton, but travelled extensively in the United Kingdom and Europe. He died when his son was eleven years old. Fields attended Hamilton Collegiate Institute where he proved himself to be an outstanding pupil. He was awarded several scholarships and graduated from the Institute in 1880.

After leaving Hamilton Collegiate Institute, Fields entered the University of Toronto in 1880 to study mathematics. His undergraduate years were not too easy, for having already lost his father, his mother died while he was in the middle of his university course. Despite this he had an outstanding undergraduate record and he received his B.A. in mathematics from the University of Toronto in 1884, winning the gold medal for mathematics. He then went to the United States to study for his Ph.D. at Johns Hopkins University. After the award of the degree in 1887 for his thesis Symbolic Finite Solutions, and Solutions by Definite Integrals of the Equation $(d^{n}/dx^{n})y - (x^{m})y = 0$, he remained teaching at Johns Hopkins for a further two years.

Fields was appointed in 1889 Professor of Mathematics at Allegheny College, one of the oldest colleges in Pennsylvania, but resigned after three years so that he might further his mathematical researches by studying in Europe. From 1892 to 1900 Fields studied in Paris, Göttingen and Berlin with Fuchs, Frobenius, Hensel, Schwarz, Weierstrass, and Planck. He was awarded the gold medal for mathematics at Berlin. He also met and formed a life-long friendship with Mittag-Leffler. This period was clearly important for his future development as a research mathematician. Synge writes in [4]:-

This long period of study, which exercised a decisive influence on his life and outlook, was rendered possible by a modest private income, combined with simple living and abstemious habits.

In 1902 Fields was appointed to the position of lecturer at the University of Toronto where he remained until his death, although he frequently visited Europe where he was acquainted with many national leaders. For example he attended a dinner party given by the King of Sweden in 1912. He became an Associate Professor in 1905 before being promoted top full Professor in 1914. In 1923 he was promoted to Research Professor at the University of Toronto.

His main research topic was on algebraic functions. He wrote an important book which was published in 1906. The main purpose of the book was to present the Riemann-Roch Theorem, the Weierstrass Gap Theorem and the related Hurwitz Theorem, and theorems of Brill and Max Noether. A colleague and former pupil of Fields gives this summary of his contributions in [4]:-

The work of Professor Fields on algebraic functions must be regarded as the development and organisation of ideas on the subject which he happened to have at the turn of the century. These had very little reference to prevailing methods. He made it his life work, first to show that they could be used to create a satisfactory theory and then to give to the structure thus secured both elegance and generality. His treatment has the great merit of being completely algebraic in character and of meeting every difficulty without an appeal to geometric intuition. The machinery, which he had to invent for the purpose, is simple, and its parts are beautifully coordinated.

It was, however, rather as an organiser of mathematics that Fields excelled. The series of International Congresses of Mathematicians began in Zürich in 1897 but no congress was held during World War I (1914-18). The International Mathematical Union was set up in 1920 at the first post war Congress in Strasbourg to run future congresses but, in the aftermath of war, Germany, Austria-Hungary, Bulgaria, and Turkey were excluded from the Union. This was unfortunate and feelings ran high on the issue. Many felt that mathematics should not be subjected to political pressures but should welcome equally mathematicians of all nations. Leading advocates of that view were Hardy and Mittag-Leffler. Others held equally sincere views that the Union's exclusion rules were right.

In 1922, following the collapse of a bid from New York to hold the 1924 Congress, Fields made a bid to hold it in Toronto under the auspices of the International Mathematical Union, and although this meant that the congress could not be truly international because of the exclusion rules, it did prevent a break-up of the congress. He was a skilled politician and was able to persuade many opponents of the Union to attend the Toronto Congress and have it succeed. This was not easily achieved and Fields spend several months in Europe working relentlessly to make the Congress a success. He also had to get financial support so that Europeans could be helped with their travel costs to North America. So successful was he in getting financial support that he had money left at the end of the Congress. It gave him the opportunity to come up with a wonderful idea.

Fields is best remembered for conceiving the idea of, and for providing funds for, an international medal for mathematical distinction. The original proposals were put on 24 February 1931 to the committee who had run the 1924 Congress, and money left over from the finances were to be used. Fields had everything in place to travel to the September 1932 Congress in Zürich to put forward his proposal for Medals. He had already done the ground work and by January 1932 he had support for awarding Medals from the leading mathematical societies in France, Germany, Italy, Switzerland and the United States.

You can see Fields's Letter setting out his proposals at THIS LINK.

However, his health began to fail in May of 1932 when he suffered heart problems. A few days before his death he drew up a will, with Synge at his bedside, including an amount of $47,000 to be added to the funds for the medals. He did not live to attend the Congress but his plans were still put forward. Adopted at the International Congress of Mathematicians at Zürich in 1932, the first Fields Medals were awarded at the Oslo Congress of 1936. Notice that they were named "Fields Medals" despite his wish that they should not bear anyone's name.

Fields Medals were to be awarded to two mathematicians under 40 years of age every four years at the International Congress of Mathematicians. These conditions were set down to recognise Fields's wish that the awards recognise both work completed and point to the potential for future achievement. Notice that the 40 age limit was not explicitly due to Fields. The first Fields Medals were awarded to Lars Ahlfors and Jesse Douglas in 1936. No awards were made during World War II, then beginning from 1950 the Medals have been awarded every four years. In 1966 it was decided that no fewer than two Medals would be awarded at each Congress, and no more than four.

A prize of 15,000 Canadian dollars is awarded with each Fields Medal, which is made of gold, and shows the head of Archimedes. A quotation attributed to Archimedes is inscribed:-

Transire suum pectus mundoque potiri.
(Rise above oneself and grasp the world.)

The inscription:-

Congregati ex toto orbe mathematici ob scripta insignia tribuere
(The mathematicians assembled here from all over the world pay tribute for outstanding work)

appears on the reverse.

Fields received several important honours. He was elected a fellow of the Royal Society of Canada in 1907 and, in 1913, he was elected a fellow of the Royal Society of London. In 1924 the International Congress of Mathematicians was held at Toronto and Fields was honoured by being President of the Congress. He was vice-president of the next International Congress of Mathematicians at Bologna in 1928 (at which the excluded nations were readmitted). He also held the position of President of the Royal Canadian Institute from 1919 to 1925. He was president of the International Mathematical Union, vice-president of the British Association for the Advancement of Science, and vice-president of the American Association for the Advancement of Science all in the year 1924. He was also elected a member of the Russian Academy of Sciences and of the Institute of Coimbra. An interesting aside is that the Italian Government wanted to honour Fields with the title "Commander of the Crown of Italy", but the Canadian Government had a law forbidding Canadian citizens from holding titles, so Fields had to refuse the Italian government's offer.

Let us end this biography by giving a few details of Fields's life outside the world of mathematics. He never married. As a young man he loved sport, playing baseball, football and hockey. He also loved walking. Music was a great joy to him and he played the violin, and loved dancing [4]:-

In his habits of life he was abstemious, avoiding tea, coffee, alcohol, and condiments, and he did not smoke ... he possessed a good sense of humour ...