奥尔加·陶斯基-托德Olga Taussky-Todd

奥尔加·陶斯基-托德结婚前的名字是陶斯基-托德 Taussky，她出生在一个犹太家庭。她的父亲Julius David Taussky是一位工业化学家，也为报纸撰写文章。她的母亲Ida Pollach除了做家庭主妇和照顾孩子外，从未接受过任何训练。陶斯基-托德是三个女孩中的中间孩子，与姐姐和妹妹都相差三岁。关于她的父母，她写道[1]：-

我的父亲是一个非常有趣的人，非常活跃，非常有创造力。……我的母亲是一个乡下姑娘……一位相当安静的女士……被教育成为家庭主妇……

当 陶斯基-托德 三岁时，全家搬到了维也纳。在那里，陶斯基-托德 上了小学，尽管她表现不错，但她的姐姐才是明星学生。陶斯基-托德 在算术上获得了最高或第二高的成绩，但她最喜欢的科目是作文和语法。除了上学，她还上私人音乐课。当第一次世界大战爆发时，维也纳的条件变得非常困难，食物严重短缺，全家几乎饿死。

1916年，Taussky一家再次搬家，这次搬到了林茨，陶斯基-托德的父亲在那里担任一家醋厂的厂长，该厂除了生产醋外还制作果酱和软饮料。Julius Taussky非常胜任这份工作，因为他在1903年与自己的父亲皮埃尔·萨米埃尔 Taussky合著了一本关于醋的著作。在第一次世界大战中期抵达林茨，他们的生活仍然艰难，但食物虽然仍然稀缺，却比维也纳稍微丰富一些。14岁时，陶斯基-托德进入高中，在那里学习了一年，然后在文理中学学习。在这个阶段，她仍然对写作和诗歌感兴趣，最初她在文理中学开始学习的拉丁语让她着迷，尤其是拉丁语语法。然而，突然间，她的兴趣转向了数学[1]：-

当我试着阅读任何到手的书籍时，我突然意识到，最伟大的智慧并非通过读书获得。我觉得科学实验提供了几乎无限的洞察力……数学，在当时对我来说也是一门实验学科……渐渐地，我清楚地认识到[数学]是我的学科。

从大约十五岁起，她开始做家教，起初是给一位同学教化学，但随着时间的推移，她做的越来越多。陶斯基-托德的父亲在她上学的最后一年去世，因此她增加了家教工作，试图帮助家庭解决经济困难。知道很难为她的大学学习提供资金，她整个夏天都在醋厂工作。

Taussky面临很大的家庭压力要学习化学，事实上她也喜欢这门科目，因此她进入维也纳大学时打算攻读化学学位，但她的姐姐获得了化学学位并接管了父亲的工作。家庭压力从Taussky身上消失了，她能够将数学作为她的主要科目。

在维也纳，陶斯基-托德师从菲利浦·富特文勒、汉斯·哈恩、威廉·维廷格、卡尔·门格尔、爱德华·赫吕等人，同时她是库尔特·弗雷德里希·哥德尔的同学。当时，这些人中最著名的是菲利浦·富特文勒，但他健康状况不佳，只能在别人的帮助下行走，讲课时需要一名学生为他写板书。菲利浦·富特文勒成为她的学位论文导师，在享受了他第一年的数论课程和第二年的代数数论讨论班之后，她问他是否可以写关于数论的学位论文。他说她将研究类域论，她写道[1]：-

……是一项巨大的荣誉。

她写了关于代数数域的学位论文，当时类域论正在发展。然而事情并不容易[1]：-

……作为学位论文学生，我度过了一段非常艰难的时光，我没有任何同事，几乎见不到我的导师，他有相当长一段时间没有指导我研究具体问题。

她于1930年获得博士学位，其学位论文于1932年发表在奥古斯都·利奥波德·克雷勒的期刊上。获得博士学位后，她通过辅导赚了一些钱，但也继续在数学上刻苦钻研，继续发展其学位论文中的思想。

受到鼓励参加德国数学会的两次会议，她在那里讲授了她的结果，并开始与伊赛·舒尔的学生A Scholzy在group theory上合作。在其中一次会议上，Hahn将她推荐给理查·科朗特，1931年她被任命为哥廷根的助理。理查·科朗特一直在寻找与威廉·马格努斯和Helmut Ulm一起编辑大卫·希尔伯特关于数论的全集第一卷的人，Taussky完全符合要求。

在哥廷根期间，Taussky还编辑了埃米尔·阿廷关于类域论的讲座（1932年），协助埃米·诺特研究她的类域论，并协助理查·科朗特研究他的微分方程课程。1932年夏天离开哥廷根后，在新学年开始前，她收到理查·科朗特的一封信，建议她不要返回哥廷根，因为政治局势恶化导致大学动荡。

1932至1933年间，Taussky在维也纳担任辅导教师，之后她在布林莫尔度过了一年，随后于1935年获得了剑桥大学格顿学院的研究奖学金。戈弗雷·哈罗德·哈代帮助她于1937年在伦敦的一所学院获得了教职，在那里她很快遇到了一位爱尔兰人Jack（John Todd），他正在伦敦的另一所学院教授分析。他们开始一起研究数学，并于次年9月29日结婚。Taussky在50年后说[14]：-

如果我的爱尔兰人没有出现，我的生活和事业将会截然不同。

第二次世界大战期间，陶斯基-托德和Jack辗转各地。他们曾在贝尔法斯特与Jack的母亲共度了一段时间，也曾在牛津待过一段时间，当时陶斯基-托德的学院为了躲避伦敦的空袭而迁到了那里。此时，陶斯基-托德写了一些关于群论的论文，例如研究按今天的术语来说每个次正规子群都是正规子群的群。她证明的结论之一是，具有循环彼得·卢德维格·梅德尔·西罗子群的群满足这一性质。她还写了几篇关于有限阶matrices的论文，如1940年的Matrices with finite period和次年的Matrices of finite period，这两篇都是与她的丈夫合写的。

在牛津附近任教期间，她在1942年和1943年指导了Hanna Neumann关于组合群论的D.Phil.学位论文。陶斯基-托德获准离开学院从事战时工作，于1943年至1946年间在特丁顿的国家物理实验室工作。这对她的数学来说是一次有用的经历[1]：-

我在空气动力学工作中的职责非常繁重。……我第一次意识到微分方程研究的美妙——这是我以前的老板理查·科朗特未能灌输给我的。其次，我学到了很多矩阵理论。

此处一段未译出，以下为英文原文 Just to illustrate this let us note that during this time Taussky-Todd wrote several papers which were published by the Ministry of Aircraft Production in London. For example she published in 1944 A note on skew-symmetric matrices in which she discussed the form of the matrices $X'X$ and $XX'$, where $X'$ is the transpose of the matrix $X$ whose columns are the eigenvectors of a skew-symmetric matrix with distinct eigenvalues. In the following year she published On some boundary value problems in the theory of the non-uniform supersonic motion of an aerofoil in which she gives rigorous proofs of methods to find the velocity potential due to a two-dimensional airfoil in a supersonic stream whose shape and motion are given. These papers she published under the name Olga Todd, while most of her other papers she wrote under the name Olga Taussky, with a few latter papers written under the name Olga Taussky-Todd.

1947年，陶斯基-托德和Jack去了美国，原打算在那里待一年，他们在普林斯顿高等爱德华·斯图迪研究院工作了一段时间，参与冯·诺伊曼的计算机项目后，又在美国国家标准局的国家应用数学实验室工作。她在国家标准局担任数学顾问，由于她在那里从事计算机工作，她被描述为：-

……计算机先驱……为与计算机应用相关的问题的解决做出了重大贡献。

她继续撰写关于矩阵论、群论、代数数论的论文，但也撰写数值分析方面的论文。

1955年，陶斯基-托德和她的丈夫在纽约的理查·科朗特研究所度过了一年的休假，在那里她教授矩阵理论课程，她的丈夫教授数值分析课程。尽管陶斯基-托德喜欢在国家标准局工作，但她怀念教学[1]：-

……我在局里的工作对我来说并不完全合适。这确实是一份非常有趣的工作。我在那里学到了很多，贡献了很多，并受到了极大的礼遇。

1957年，陶斯基-托德和Jack都接受了加州理工学院的聘任。她写道：-

在多年主要与应用数学打交道的工作之后，我起初对教学相当没有把握。但同样是学生们来帮助了我[就像在伦敦的那所学院里一样]。他们很清楚我有很多数学知识可以教给他们，并且他们迫使我把它讲出来。

1977年，她退休成为荣休教授，但她继续过着非常活跃的数学人生。特别是，她能够把更多时间投入到与世界各地许多人的大量数学通信中。

陶斯基-托德的荣誉和工作在[13]中描述如下：-

陶斯基-托德是一位杰出而多产的数学家，撰写了约300篇论文。她一生中获得许多荣誉和殊荣，最著名的是荣誉十字勋章，这是她的祖国奥地利授予贡献的最高认可。陶斯基-托德最著名和最有影响的工作是在矩阵理论领域，尽管她也在数论方面做出了重要贡献。

在[13]中也有许多来自她多年来所教学生的致敬。Rober Guralnick写道：-

能与一位对该领域如此精通的人合作，我感到非常幸运。

汤姆·麦克·阿波斯托写道：-

陶斯基-托德是一位极出色的导师，鼓励人、劝诱人，并且总是在那里提供帮助。

但她也并非没有自己的独特习惯，正如Charles Johnson所写：-

……她对数学写作有相当特别的看法。不允许有图片或图表，而且，尽管我想在我的学位论文中用“$n \times n$”来描述矩阵的维数，但必须写成“$n$乘$n$”。我和我的学生至今仍写“$n$乘$n$”。

最后，我们应该提到陶斯基-托德因其数学贡献而获得的许多荣誉。1971年，她因一篇关于平方和的论文而获得了美国数学协会颁发的莱斯特·佛特奖。她当选为多个科学院院士：当选为奥地利科学院（1975年）；当选为巴伐利亚科学院院士（1985年）；当选为美国科学促进会会士（1991年）。1978年，奥地利政府授予她最高奖项——科学与艺术荣誉十字勋章一级，1980年，她获得了维也纳大学的荣誉博士学位。1988年，她还获得了南加州大学的荣誉博士学位。

陶斯基-托德曾在伦敦数学会理事会任职（1946-47年），并曾在美国数学会理事会任职（1972-78年）。她于1985年担任美国数学会副主席。

最后让我们引用她自己的话 [1]：-

……从幼年起，诗歌和写作对我来说就是自然而然的事。但在我看来，无论是在他人的作品中还是在我自己的作品中，我寻求的是美，而不仅仅是成就。……一个以我那种热情开始、并且这种热情并未因艰辛、困难和失望而减退的人，‘是没有选择的’，‘未来的阴影是看不见的’。

Olga Taussky-Todd's name before she married was Olga Taussky and she was born into a Jewish family. Her father, Julius David Taussky, was an industrial chemist who also wrote articles for a newspaper. Her mother, Ida Pollach, was never trained for anything other than being a housewife and looking after her children. Olga was the middle child of three girls, with three years separating her both from her older sister and from her younger sister. Of her parents she wrote [1]:-

My father was a very interesting man, very active, very creative. ... My mother was a country girl ... a rather quiet lady ... educated to be a housewife ...

When Olga was three years old the family moved to Vienna. There Olga attended primary school and although she did well, her older sister was the star pupil. Olga received the top or second top grade for arithmetic but her favourite subjects were essay writing and grammar. In addition to her schooling she had private music lessons. When World War I broke out conditions became very difficult indeed in Vienna, there was severe shortages of food and the family were near to starving.

In 1916 the Taussky family moved again, this time to Linz where Olga's father had a job as director of a vinegar factory which, in addition to vinegar made jams and soft drinks. Julius Taussky was well qualifed for the job for he had coauthored a text on vinegar with his own father Samuel Taussky in 1903. Arriving in Linz in the middle of World War I, life was still hard for them but food, although still scare, was slightly more plentiful than in Vienna. When she was 14 years old Olga entered the high school where she spent a year before studying at the Gymnasium. At this stage she was still interested in writing and poetry, and initially the Latin that she began to study at the Gymnasium fascinated her, particularly Latin grammar. Suddenly, however, her interests moved to mathematics [1]:-

While I tried to read any books that came my way, the realisation that the greatest wisdom was not to be gained by reading books struck me suddenly. I felt that scientific experiments provided almost unlimited insight ... Mathematics, too, came to me at that time as an experimental subject ... Gradually it became clear to me that [mathematics] was my subject.

From about the age of fifteen she tutored, starting by teaching chemistry to a classmate, but doing more and more over time. Olga's father died when she was in her final year at school so she increased her tutoring to try to help out the family finances. Knowing that it would be hard to fund her university studies she worked through the summer at the vinegar factory.

There was a lot of family pressure on Taussky to study chemistry, and indeed she loved the subject, so she entered the University of Vienna with a chemistry degree in mind, but her older sister qualified in chemistry and took over her father's work. The family pressure came off Taussky, and she was able to make mathematics her main subject.

At Vienna Olga was taught by Furtwängler, Hahn, Wirtinger, Menger, Helly and others, while she was a fellow student of Gödel. At that time, the most famous of these was Furtwängler but he was in poor health, only being able to walk with help and a student had to write for him when he lectured. Furtwängler became her thesis supervisor and after enjoying his course on number theory in her first year and his algebraic number theory seminar in her second year she asked him if she could write her thesis on number theory. He said that she would work on class field theory which, she wrote [1]:-

... was a great honour.

She wrote her thesis on algebraic number fields just as class field theory was being developed. However things were not easy [1]:-

... I had a very tough time as a thesis student, I had no colleagues whatsoever and hardly saw my teacher, who for quite a while did not direct me towards a specific problem.

She was awarded a doctorate in 1930 and her thesis was published in Crelle's Journal in 1932. After the award of her doctorate she earned some money by tutoring but also continued to work hard on her mathematics, continuing to develop the ideas from her thesis.

Encouraged to attend two meetings of the German Mathematical Society, she lectured there on her results and began a collaboration on group theory with A Scholzy, a student of Schur. At one of these meetings Hahn recommended her to Courant and, in 1931 she was appointed as assistant at Göttingen. Courant had been looking for someone to work with Wilhelm Magnus and Helmut Ulm editing the first volume of Hilbert's complete works on number theory and Taussky fitted the bill perfectly.

While in Göttingen Taussky also edited Artin's lectures in class field theory (1932), assisted Emmy Noether in her class field theory and Courant with his differential equations course. Leaving Göttingen in the summer of 1932, she received a letter from Courant before the new academic year started advising her not to return to Göttingen due to unrest at the university caused by the deteriotating political situation.

In 1932-1933 Taussky tutored in Vienna, then she spent a year at Bryn Mawr before taking up a research fellowship from Girton College, Cambridge in 1935. Hardy helped her obtain a teaching post in a London college in 1937 where she soon met an Irishman Jack (John Todd) who was teaching analysis at a different London College. They began working on mathematics together and were married on 29 September of the following year. Taussky said 50 years later [14]:-

My life and my career would have been so different if my Irishman had not come along.

During World War II Olga and Jack moved from place to place. They spent some time in Belfast with Jack's mother and some time in Oxford where Taussky-Todd's College had moved to avoid the London air raids. At this time Taussky-Todd wrote some papers on group theory, for example studying groups in which, in today's terminology, every subnormal subgroup is normal. She proved, among other things, that a group with cyclic Sylow subgroups satisfies this property. She also wrote several papers on matrices of finite order such at Matrices with finite period in 1940 and Matrices of finite period in the following year, both written jointly with her husband.

While teaching near Oxford she supervised Hanna Neumann's D.Phil. thesis on combinatorial group theory during 1942 and 1943. Given leave of absence from her College to undertake war work, Taussky-Todd worked at the National Physical Laboratory ay Teddington between 1943 and 1946. It was a useful experience for her mathematically [1]:-

The duties in my aerodynamics job were very heavy. ... For the first time I realised the beauty of research on differential equations - something that my former boss, Professor Courant, had not been able to instil in me. Secondly I learned a lot of matrix theory.

Just to illustrate this let us note that during this time Taussky-Todd wrote several papers which were published by the Ministry of Aircraft Production in London. For example she published in 1944 A note on skew-symmetric matrices in which she discussed the form of the matrices $X'X$ and $XX'$, where $X'$ is the transpose of the matrix $X$ whose columns are the eigenvectors of a skew-symmetric matrix with distinct eigenvalues. In the following year she published On some boundary value problems in the theory of the non-uniform supersonic motion of an aerofoil in which she gives rigorous proofs of methods to find the velocity potential due to a two-dimensional airfoil in a supersonic stream whose shape and motion are given. These papers she published under the name Olga Todd, while most of her other papers she wrote under the name Olga Taussky, with a few latter papers written under the name Olga Taussky-Todd.

In 1947 Olga and Jack when to the United States, with the original intention of spending a year there, and they worked at the National Bureau of Standards' National Applied Mathematics Laboratory after spending a while at the Institute for Advanced Study at Princeton working on von Neumann's computer project. She worked as a consultant in mathematics at the National Bureau of Standards and because of her work there with computers she has been described as a:-

... computer pioneer ... who provided significant contributions to solutions of problems associated with applications of computers.

She continued to write papers on matrix theory, group theory, algebraic number theory but she also wrote on numerical analysis.

In 1955 Taussky-Todd and her husband spent a year's leave at the Courant Institute in New York where she taught a matrix theory course and her husband taught a numerical analysis course. Although Taussky-Todd enjoyed working at the National Bureau of Standards she missed teaching [1]:-

... my job at the Bureau was not exactly right for me. It was certainly a very interesting job. I learned a lot there, contributed a lot, and was treated with great courtesy.

In 1957 Olga and Jack both accepted appointments at the California Institute of Technology. She wrote:-

After many years of work mostly with applied mathematics, I was in the beginning rather uncertain about the teaching. But again it was the students who came to my assistance [as in the college in London]. It was clear to them that I had much mathematics to give them and they forced it out of me.

In 1977 she retired becoming Professor Emeritus, but she continued to lead a very active mathematical life. Particularly she could devote even more time to the very large mathematical correspondence which she carried on with many throughout the world.

Olga's honours and work is described in [13] as follows:-

Olga Taussky-Todd was a distinguished and prolific mathematician who wrote about 300 papers. Throughout her life she received many honors and distinctions, most notably the Cross of Honor, the highest recognition of contributions given by her native Austria. Olga's best-known and most influential work was in the field of matrix theory, though she also made important contributions to number theory.

In [13] there are also a number of tributes from those she taught over the years. Rober Guralnick writes:-

I felt very fortunate to have this chance to work with someone who had such a command of the field.

Tom Apostol wrote:-

Olga was a superb advisor, encouraging, cajoling, and always there to help.

But she was not without her idiosyncrasies as Charles Johnson writes:-

... she had rather particular ideas about mathematical writing. No pictures or diagrams were allowed, and, much as I wanted to write "$n \times n$ " to describe the dimensions of a matrix in my thesis, it had to be "$n$-by-$n$". I, and my students, still write "$n$-by-$n$".

Finally we should mention some of the many honours Taussky-Todd received for her mathematical contributions. She received the Ford Prize from the Mathematical Association of America in 1971 for a paper on the sums of squares. She was elected to several academies: elected to the Austrian Academy of Sciences (1975); elected to the Bavarian Academy of Sciences (1985); elected a Fellow of the American Association for the Advancement of Science (1991). She was honoured by the Austrian government in 1978 with their highest award, The Cross of Honour in Science and Arts, First Class and, in 1980, received an honorary doctorate from the University of Vienna. She also was awarded an honorary doctorate from the University of Southern California in 1988.

Taussky-Todd served on the Council of the London Mathematical Society (1946-47 and she served on the Council of the American Mathematical Society (1972-78). She was Vice-President of the American Mathematical Society in 1985.

Let us finish by quoting her own words [1]:-

... from early childhood on, poetry and writing came to me in a natural way. But it seems to me that both in the work of others and in my own I look for beauty, and not only for achievement. ... a person who started with the enthusiasm that came to me and did not diminish through hardships, difficulties, and disappointments 'is not given a choice' and 'shadows of the future one does not see'.