Female Contributions to the Development of Mathematics Throughout History

CHAPTER 5 Emmy Amalie Noether – 20th Century Mathematics


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Emmy Amalie Noether – 20th Century Mathematics

Sonya Kovalevskaya had many self-doubts and insecurities, but nine years before her death another great female mathematician was born who lacked these traits to distract her from her goals, but unfortunately had very different obstacles that were out of her power to control.

In 1882 Emmy Amalie Noether was born in the university town of Erlangen, Germany. Her father, Max Noether, was a professor at the University of Erlangen. He was already a famed mathematician, aiding the development of the theory of algebraic functions, so it is not a surprise that Emmy was also very gifted in mathematics. Her mother, Ida Kaufmann came from a wealthy cologne family. Her parents were both Jewish, and therefore so was Emmy.
From 1889 to 1897 Emmy attended the Höhere Töchter Schule in Erlangen. She studied such subjects as arithmetic, languages and music (learning how to play the piano). Typically she mastered the tasks every young girl should – cleaning, cooking, shopping etc. She loved to dance, and when she got older she went to dances with the university boys, flirting and socialising. None of the Noether family considered that a young girl would need any further education than finishing school after completing the basic elementary education. Emmy’s upbringing – although sounding pretty standard – had a very different quality, which meant Emmy was not satisfied with just attending finishing school.

Max Noether was a strong influence on his young children. Naturally he would have colleagues and associates visit the house for social gatherings, and obviously mathematics would be a strong talking point. Emmy would attend his colleague’s children’s parties – something she looked forward to. Therefore the university and mathematical talk was a common focus of her home life. This was an unusual aspect in comparison to the average household, but both Emmy and her brother Fritz took advantage of this and followed in their father’s footsteps.

Typically Emmy went to finishing school, but after this she took charge of her education. She went on to spend three years in a teaching training programme – one of the only available options for girls for further education. By eighteen years of age she was a certified teacher of English and French in Bavarian girls schools.
Emmy was still not satisfied and wanted to take her education even further, she wanted to study mathematics at university. This was a difficult route to take for any woman in Germany in 1900, even in the more liberal country of America only nine women had received doctorates. Emmy was fortunate that one of the three ‘free’ universities of Germany was located in her hometown of Erlangen (a ‘free’ university was one independent of religion).
The University of Erlangen did not allow females to undertake degrees officially, but after appealing to sympathetic lecturers she was allowed to sit in on their classes as an unofficial student (or ‘auditor’). Emmy was lucky to have such supportive teachers as many lecturers were still very opposed to women students in Germany. (Although it must have helped that her father worked at the university!) She continued to be an auditor for two years, then in 1903 it is believed she took the entrance exam in Nürnberg and went to study at the University of Göttingen. From 1903 to 1904 she attended lectures by Blumenthal, Hilbert, Klein and Minkowski.

Finally in 1904, after five years of being an auditor Emmy was officially allowed to enrol at Erlangen. She became half of the entire female population of the one thousand pupils! [34] She was tutored by Paul Gordon, who was described by Hilbert as ‘King of the invariants’. She produced her doctorate dissertation working under Gordon, entitled ‘On Complete Systems of Invariants for Tertiary Biquadratic Forms’. Gordon took a more constructive approach/methods to Hilbert’s 1888 basis theorem which gave an existence result for finiteness of invariants in ‘n’ variables. Emmy’s dissertation followed this constructive approach of Gordon’s and listed systems of three hundred and thirty-one covariant forms. She was awarded the highest honours for this piece of work, and later it was even described as ‘an awe-inspiring piece of work’ by Hermann Weyl the famous mathematician and relativist who was a colleague and friend of Emmys. Although later on Emmy dismissed the accomplishment of her work, commenting that it was just a ‘jungle of formulas’. [2] [41]

Emmy always had a lot of respect for Gordon, according to Weyl she had a photograph of him in her study for many years. [2]
After being granted a doctorate the normal progression for a male would be to acquire an academic post in research/training, but this option was not open to Emmy as such professional opportunities were very scarce. Therefore Emmy stayed in Erlangen helping her father. She would take his lectures when he was ill, supervise doctoral students and give talks etc. all without any acknowledgement or pay (as only men could be employed). She did continue her own research though, initially with Gordon, then after his retirement in 1911 with his predecessor Ernst Fischer. Emmy began to move away from Gordon’s formalist approach, and in doing so she began to demonstrate her exceptional talent for conceptual axiomatic thinking. Her studies focussed primarily on finite rationals and integral bases. Another algebraist Erhard Schmidt also tutored her during this time. She published many papers detailing her results in this research, which are now considered classics in their field. [2] [41]
Emmy’s reputation grew as her papers were published, as a result she was elected to the Circolo Matematico di Palermo in 1908. In 1909 she became a member of the Deutsche Mathematiker Vereinigung, and was invited to address the annual meeting of the Society in Salzburg. During 1913 she lectured in Vienna.
As her father’s health deteriorated over the years she took on more of his lectures and responsibilities. Still no recognition was granted from the university, even though her reputation was beginning to exceed her.

In 1915 Emmy’s father retired, and rather sadly her mother died. These tremendous changes in her home life made her more welcoming to an offer made by David Hilbert and Felix Klein. Hilbert is considered as one of the greatest mathematicians, he and Klein were fully immersed at the time aiding Einstein to try and decipher a relativistic theory of gravity. Emmy came to the attention of Hilbert in 1914 as she had the exact area of expertise on invariants that they were looking for to aid research. Hilbert and Klein invited Emmy to return to the University of Göttingen to join their team. Göttingen was allegedly the centre of the World for mathematical and physics research at that time.

On one of Emmy’s visits to the university Hilbert persuaded her to stay and in 1915 she moved there permanently, joining one of the most creative circles of research in post-war Göttingen. Over the next few years she helped develop elegant mathematical formulations for a number of important concepts in general relativity. Weyl also commented on this work –
“..for two of the most significant sides of the theory of relativity, she gave at that time the genuine and universal mathematical formulation.” [39]
It was at Göttingen that she also became interested in establishing, on an axiomatic basis, a completely general theory of ideals.
Even though Emmy’s contribution was vast she was still not offered an official academic position. Both Hilbert and Klein thought this treatment was appalling and began to fight her case for her! (Klein was actually a strong force behind Göttingen’s decision to grant doctorates to several women in the 1890’s, so he was already accustomed to backing such causes!) Hilbert went to the philosophical faculty (which included philosophers, philologists, historians, natural scientists and mathematicians) to argue Emmy’s case requesting that she was appointed the position of ‘Privatdozent’ (a junior position), but unfortunately most professors agreed that a woman’s place was in the home raising her children. A non-mathematical member of the faculty argued –
“How can it be allowed that a woman become a Privatdozent? Having become a Privatdozent, she can then become a professor and a member of the University Senate…What will our soldiers think when they return to the university and find that they are expected to learn at the feet of a woman?” [40]
This statement annoyed Hilbert, and he angrily retaliated (which probably did more harm than good) by saying –

“Meine Herren, I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, the Senate is not a bathhouse.” [40]

Unfortunately the narrow-minded members of the faculty got their way and Emmy continued to have no official position and no pay, surviving only on a small trust fund set up for her by her mother’s brothers.
To allow Emmy to lecture she would advertise her courses under Hilbert’s name, for example one of the courses she taught in 1916 was advertised in the university’s catalogue as ‘Mathematical Physics Seminar: Professor Hilbert, with the assistance of Dr. E. Noether, Mondays from 4-6, no tuition.’ [41]
The first piece of work that she produced at Göttingen in 1915 is referred to by physicists as ‘Noether’s Theorem’. This theorem relates physical laws of conservation (such as conservation of energy and momentum) to the mathematical property of symmetries. It was of great importance, as demonstrated here –
“Before Noether’s Theorem the principle of conservation of energy was shrouded in mystery…Noether’s simple and profound mathematical formulation did much to demystify physics.” [38]
Her theorem became of central importance to physicists who were searching for a unified theory of forces and particles – a single theory to unite relativity and quantum mechanics into an all-encompassing package.

It was the intervention of World events that provided a small breakthrough for Emmy’s career. After World War I ended the status of women improved a little, with the declaration of the German Republic the social climate of Germany changed. (Although Emmy was not concerned with social or political events of the day, the German revolution did confirm Emmy’s status as a pacifist, an attitude that she felt for the rest of her life). This meant that in 1922 Emmy was officially granted the position of ‘Unofficial Associate Professor’, but she still received no pay. It was only when she was honoured with a ‘Lehrauftrag’ in algebra that she received a very small salary for the first time in her life.

Göttingen became a very popular place to study in the 1920’s – it was the place to be for a budding mathematician. Therefore Emmy enjoyed teaching numerous eager pupils from all over the World. Her lectures were less formal than what was tradition, but she was thought of as an innovative, stimulating, effective and original teacher. Her relationship with her students was legendary, Norbert Wiener once described this relationship as like a swarm of ducklings flocking around a kind, motherly hen. Her most famed students were a group called the ‘Noether Boys’ they were from all over the World, including Germany, Russia, Holland, Israel, China and Japan. Emmy looked upon this group like a family, and any insults made about them she would take far more seriously than she would insults to herself.
Emmy’s students also used to pick up on her informal dress code when she taught, imitating it sometimes, for example, by just wearing shirtsleeves! This was unheard of in comparison to their usual mode of dress for lectures. This was branded the ‘Noether-guard uniform’. Emmy did not care for her clothing or appearance, like her lectures, she cared about substance not form. Unfortunately that meant she sometimes did look quite messy, she has been documented as looking like a washerwoman!
Emmy was particularly close to a Czechoslovakian student named Olga Taussky Todd. Olga described a particular lunchtime that completely summed up Emmy’s persona –
“When lunch came I sat down next to Emmy, to her left. She was very busy discussing mathematics with the man on her right and several people across the table. She was having a very good time. She ate her lunch, but gesticulated violently when eating. This kept her left hand busy too, for she spilled her food constantly and wiped it off from her dress, completely unperturbed.” [34]

Emmy ignored all the feminine conventions of the day. She was overweight, enthusiastic, opinionated, messy, unfashionable and comfortable, but she was also loving, utterly unselfish, and friendly. This was hard for the conservative professors of the university to deal with, they didn’t know how to cope with a woman that was completely the opposite to what they perceived as ‘female’. Unfortunately due to this non-conformist attitude she was often the butt of jokes. It is interesting to note though that the male equivalent of Emmy, i.e. Einstein, was respected for such traits, and never was humiliated in such a way.

Emmy tried to avoid teaching undergraduate classes, as they could not keep up with the speed of her thought, and even speedier talk! Her audience needed to be concentrating fully and at all times to keep up with her. It is believed that a lot of today’s algebra originates from these lectures – since a lot of her students would pick up ideas and concepts from these and then write them down clearly and painstakingly so that people who could not keep up with Emmy could understand them. Her stimulus and imagination was thought to spark creativity in others. A Dutch student of hers, Bartel Van der Waerden, wrote many classic texts originating from her ideas in the lectures she gave. When he left Göttingen he produced a two-volume book entitled ‘Modern Algebra’, it is believed that the majority of the second volume consists of Emmy’s work. [2] [34]
Emmy produced most of her best work in the 1920’s, this was when her true genius was recognised. Her first piece, and the turning point of her work, was a paper she co-authored on differential operators. This demonstrated her strong interest in the conceptual axiomatic approach. It established Emmy as a force in altering the face of algebra.
She became part of the development of abstract algebra, where instead of studying the results of algebraic operations and solving equations which had been done in the past, they studied the properties of the algebraic operations, such as commutativity, associativitiy, and distributivity. They wanted to investigate what happened if one of these properties was not assumed. They also generalised the number system to other systems of fields, rings, groups, near rings etc.

A ‘ring’ is an abstract structure in which the objects are subject to two operations e.g. addition and multiplication, and satisfy a number of rules. The rules require the existence of certain laws, e.g. the associative law, which must be satisfied by these operations. The ring must include a zero element. An ‘ideal’ of a ring is a subset of the ring (so it’s a ring itself). A ‘chain’ is a relationship in which ideals are linked by the subset relation.

Emmy developed her father’s residual theorem so that it fitted into her general theorem of ideals in arbitrary rings. This furthered the axiomatic and integrative tendencies of abstract algebras.
In 1927 she began work collaborating with Helmut Hasse and Richard Brauer. They produced papers on non-commutative algebras, hypercomplex quantities and the theory of class fields, norm rests, and the principal genus theorem. One of their joint papers proving that every simple algebra over an ordinary algebraic number field is cyclic, was called a classic of it’s kind. Hasse also published a paper on the theory of cyclic algebras, this paper demonstrated Emmy’s theory of cross products.
Emmy also conducted lectures in Moscow and Frankfurt, which heightened her recognition in the European Centres of learning.
In addition to Emmy’s usual responsibilities she also helped to edit the international mathematics journal. Which incidentally she never got credited for, her name was missing from the masthead of it.
Thankfully, there were some recognitions of Emmy’s accomplishments. She was invited to address the International Mathematical Congress twice, once at Bologna in 1928, and once at Zurich in 1932. In the same year she also jointly received the Alfred-Teubner Memorial Prize for the Advancement of Mathematical Knowledge with Artin.

Emmy had established herself as a central figure in the research and teaching bodies of Göttingen. She was wedded to her work, and led a quiet life outside of it, spending even her leisure hours discussing mathematics! This happy existence all changed in 1933 when World events once again completely shook her career. The Nazi’s came to power and all Jews were denied to partake in any academic activity. Therefore she lost both her pain-stakingly-earned position and salary at the university. Her reputation as a great mathematician did not even slightly overshadow the fact that she was a Jew, a woman and a liberal. Even if she hadn’t been a Jew, the fact that she was a woman meant she would have been dismissed from her university post in 1934 anyway. She couldn’t win either way! Weyl wrote of Emmy during this period –

“…her courage, her frankness, her unconcern about her own fate, her conciliatory spirit - was in the midst of all hatred and meaness, despair and sorrow surrounding us, a moral solace.” [38]
Emmy and her brother, Fritz, were two of the lucky intellectuals that managed to leave Germany. Fritz managed to get a position in Siberia at the Research Institute for Mathematics and Mechanics in Tomsk, and Emmy accepted a visiting professorship at the women’s college Bryn Mawr in Pennsylvania, America. Emmy’s arrival caused much excitement in the American mathematical fraternity. She also gave weekly lectures at the Institute for Advanced Study in Princeton, New Jersey. Einstein and Weyl were also transferred to Princeton. Emmy found that the respect, recognition, and friendship that she had lacked at Göttingen were heartily bestowed on her at Bryn Mawr and Princeton. She enjoyed a pleasant work life in America, doing what she loved without a struggle. Her English was useable, and she found it fun discovering such a new culture. Although happy, Emmy would sometimes demonstrate signs of anguish, concerning the situation in Germany, and her health. As a result she would expose bouts of anger towards her closest friends – for which she would humbly apologise for later. Unfortunately after only a year and a half of what seemed an idyllic life for Emmy she died very suddenly on 4th April 1935 after a routine operation to remove an ovarian cyst resulted in complications.
Obviously Emmy’s death was a great shock to her family, friends and colleagues, her studious courage in times of such little hope had not prepared them for such an event. At fifty-three years of age she was at her prime, such a late development in a mathematicians creative talent is rare, but Emmy was a ‘slow-burner’ and as Weyl recalled in her obituary she was still at the peak of

“…the native productive power of her mathematical genius”. [35]

The World not only lost a great mathematician, but as Weyl also said ‘a great woman’.
Although Emmy’s life was short she accomplished a great deal. Her name officially appears on 37 publications, but should probably appear on numerous others as her concept of algebra has roots in many students and colleagues’ works. Noether’s theorem led to formulations for several concepts of Einstein’s general theory of relativity. Emmy’s surname is used to designate many concepts specific to abstract algebra, for example, a ring is called Noetherian if each ideal has a finite basis, a group is called Noetherian if each subgroup can be generated by a finite basis. Mathematicians also speak of Noetherian equations, modules, factor systems etc. She was one of the founders of the practice of abstract algebra, which can now be found in every school in America as the ‘New Math’ in a diluted format. So why was it that the Royal Göttingen Academy of Science denied her membership? And why for the eighteen years that she devoted her life to the University of Göttingen was she never granted a proper professorship or a proper wage, never receiving benefits or a pension? She had to endeavour a life of financial hardship, just because she chose an unorthodox career as a woman.
Even her fellow great mathematicians thought of her with great respect and admiration, Einstein said of her –
“In the judgement of the most competent living mathematicians, Fraulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present day younger generation of mathematicians.” [43]

Emmy Noether was a World-class mathematician working at the highest level of mathematics and in the most abstract and ‘coldest’ parts of it. Although she remained, as once described by Weyl as –

“…warm like a loaf of bread…There irradiated from her broad, comforting, vital warmth.” [39]
Which is a definite contrast to some other great mathematicians working in such a field. She possessed the unusual talent to be able to visualise remote, extremely complex connections without resorting to examples. She also had the fantastic gift of then being able to convey these amazingly complex concepts to others so they were able to understand them fully.
Erlangen, her place of birth, has done much to commemorate Emmy’s life, in 1958 a conference was held at the university to commemorate the fiftieth anniversary of Emmy’s well-earned doctorate being granted. Then in 1982, to commemorate the one-hundredth anniversary of Emmy’s birth the Emmy Noether Gymnasium was opened – a co-educational school emphasising mathematics, natural sciences and language.
More recently in 1992 the Emmy Noether Institute for Mathematical Research was opened in Bar Ilan University in Tel Aviv, Israel. Also the Australian Mathematics Trust commemorated Emmy on their 1999 T-shirt, which lists each of the eight chains which commence with 18Z.


Olga Taussky Todd

Olga Taussky Todd, a student of Emmy Noether’s (as mentioned in chapter five), also developed into a great female mathematician. Considering Olga was only a generation in front of Emmy, she did not come across as many obstacles as Emmy did for being a woman.

Olga was born on 30th August 1906 in Olmutz in the Austro-Hungarian Empire, which is now Olomouc in the Czech Republic. Olga was the middle child of three girls, with three years separating each of them. The oldest was Heta, and the youngest Ilona. Her father, Julius David Taussky, was an industrial chemist, who also occasionally wrote newspaper articles. Olga once described her father –

“My father was a very interesting man, very active, very creative…”. [50]
He wished for his daughters to have a good education, and to specialise in the arts. Unfortunately he was to be disappointed, all three of his daughters went into the fields of mathematics and science! Olga’s mother, Ida Pollach, was also very supportive of her daughter’s education’s (although she was never formally educated herself).
When Olga was three years old the family moved to Vienna. She attended primary school there, where she enjoyed essay writing, poetry and music. She also received music lessons to learn to play the piano. At this age Olga looked like she may have fulfilled her father’s aspirations for her.
World War I began in 1914, food shortages in Vienna were extremely harsh and the family was nearly starved. As a result they moved to Linz in Austria where her father had secured a job, food was still scarce, but they were much better off than they were in Vienna. Her father was appointed the position of director at a vinegar factory, a substance that he knew much about after co-authoring a text on it with his father in 1903.
When Olga was fourteen she enrolled at the local high school (mittelschule). She stayed there for a year and then was able to enrol at the gymnasium, the only school girls could attend. She continued to write poetry about events that affected her life, and compose music. She especially liked the Latin that she learnt at the Gymnasium - it fascinated her. Suddenly her interests changed though, and she noted later –

“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.” [50]

Unfortunately her beloved Latin was taught instead of mathematics and science at the Gymnasium. Her only access to such material came from a lecture series put on by the local high school. There was no local university so the teachers of the high school, who had doctorates and continued their research, would host these lectures to share their results with the general public.
Olga’s father recognised her mathematical talent and gave her problems to solve concerning the vinegar factory, for example the proportion of water necessary to achieve specific pH levels in the vinegar. She also developed a chronological ordering system for her father’s magazines, which was similar to systems included in computer programming. [46]
Olga was asked to tutor her father’s boss’s daughter, which she did, but her father would not accept any money for this service, therefore the mother of the girl gave Olga expensive books instead. Olga continued tutoring more and more pupils. Although her father still would not allow payment - he would have been embarrassed if one of his daughters was earning money. [46] [50]
In her last year of school she had to do a creative research project, it could be on any topic they wished. Olga chose to do hers on Pascal pyramids of all dimensions, instead of the Pascal triangle, and other aspects connected with binomial coefficients. It was entitled ‘From the binomial to the polynomial theorem’. This was quite an impressive title for a school project, but was also quite an unusual topic for a girl at the time.
When World War I ended in 1918 the Taussky family became citizens of Czechoslovakia, but the country was in poverty.

Before the war career options for girls were limited to teaching in girl’s schools, secretaries, shop assistants, domestic service, nurses, dress makers etc. The social climate changed with the war and nurses began to receive an education. Female students felt pressure to eradicate all female conventions though, such as appearance – hairdos and cosmetics were frowned upon.

Olga decided that after all her hours of tutoring, teaching would be the best career for her. Before she could start such a career though she had to help with unavoidable family problems. In her last year of school her father had died, leaving the family with no income. To help with finances Olga increased her tutoring (this time charging for her services!) and took on work at the vinegar factory. Consequently though her career was not progressing, and it was only when she had a particular conversation that her career got kick started. She was talking to an elderly lady that was expressing how she would have liked to have gone into mathematics, Olga realised this could be her telling the exact same thing to another young girl fifty years on, so she decided to do something about it and enrol at university.
There was an immense pressure from her family to study chemistry – to follow in her father’s footsteps. Therefore she applied to the University of Vienna to study a mathematics and chemistry degree. Her older sister, Heta, qualified as an industrial chemist at this point and took over her father’s work, so the pressure to study chemistry was not so intense for Olga and she dropped the chemistry part of her degree to just study purely mathematics. (Heta actually pioneered the exploitation of the Jojoba plant, which has been utilised in the production of medicine and food. The unique oil contained within the plant has also aided the development of cosmetics).

In 1925 Olga enrolled at the University of Vienna. Lecturers such as Furtwängler, Hahn, Wirtinger, Menger, Helly and others taught her. Experiencing mathematics such as finite group theory, algebraic functions, and topology and abstract algebra. Philip Furtwängler was a number theoretician from Germany, he was the most famous of her lecturers, but most probably the one with the worst health – he was only able to walk when aided and he had to have a student write the notes for him when he lectured. In her final year Furtwängler became Olga’s thesis supervisor, she had enjoyed his past lectures on number theory and algebraic number theory so she requested that she could write her thesis on such a topic. Furtwängler agreed and specified that she should focus on class field theory.

Doing her thesis on such a topic consequently helped her career as few people worked in this field giving her an advantage. She found it hard work though, and very lonesome, she had no colleagues due to the newness of the topic, and she hardly ever saw her supervisor, as he was always ill, he didn’t even direct her towards a specific problem for quite a while. She finally based her research on algebraic number fields.
Olga went to Zurich for her last semester where she conducted weekly lectures at the colloquium. She finally received her doctorate in Vienna in 1930. Her thesis was published in Crelle’s journal in 1932.
After receiving her doctorate she continued to tutor to earn some money, but she also continued her research (unpaid), extending her thesis in the topic of class field theory. She attended two meetings held by the German Mathematical Society to lecture on the results she had found from her research. At one of these meetings she met A.Scholz and began collaborating with him on group theory. This resulted in Olga and Scholz solving the class field tower problem. She also came across a lot of criticism for her work, which ultimately was criticism for Furtwängler’s work, as this was what her research was based on. [46] [50]
An excellent opportunity for Olga arose from these meetings. After being introduced and recommended (by Hahn) to Courant who was looking for a candidate to work with Wilhelm Magnus and Helmut Ulm editing the first volume of Hilbert’s complete works on number theory, she was offered the position. With Olga’s experience in this field she was the perfect person for the job. Therefore in 1931 she was appointed as assistant at the University of Göttingen.

In addition to her role as editor at the university she also assisted Courant in his differential equations course (although she had no formal training on this topic). She also edited Emil Aritn’s 1932 lecture notes in class field theory and principal ideal theorem and translated it into a statement on finite non-abelian groups. Whilst at Göttingen she also crossed paths with Emmy Noether (as mentioned in chapter five), Olga assisted Emmy with her lectures on class field theory also and in return Emmy introduced Olga to the notion of algebraic systems. They became good friends.

Maybe as women mathematicians they had a common ground that they could both relate to. Olga definitely showed her loyalty to Emmy when a top man in the mathematics department degraded and criticised Emmy, Olga went straight up to him and declared that his behaviour offended her, he then apologised to the whole department. Therefore even though Olga did not have as many obstacles as Emmy did because she was a woman, the obstacles were still present, but Olga demonstrated that maybe female academics were beginning to find their feet – and their voice! – in what had been such alienating surroundings.
Before the new academic year began in 1932 Courant wrote to Olga warning her of unrest at the university due to the deteriorating political situation in Germany at that time. He advised her not to return to the university (being Jewish the Nazi’s would have eventually segregated her). She readily took this kind advice and returned to Vienna to begin tutoring again, she also acquired a small fixed appointment in the mathematics department there. Olga continued her work with Scholz, but for her own research she started to investigate topological algebra, where an algebraic and geometric point of view is studied simultaneously.
Whilst at Vienna Olga worked with the professors, Hahn, Menger and Furtwängler. They exposed her to topics such as functional analysis, abstract spaces on which a Euclidean-type distance is introduced, and the sum of squares.
Olga applied for a science fellowship to Girton College in Cambridge, England, but before it had been processed she received an invite to attend an appointment at Bryn Mawr College in Pennsylvania, America, so she accepted this position.

In 1934 Olga began her journey to Pennsylvania. She amazingly learnt the English language on her trip across the continent. She entered Bryn Mawr as a graduate student on fellowship. Here she met up again with Emmy Noether who also had been exiled from Göttingen. Olga did not enjoy her stay at Bryn Mawr, she was forced to live in a dormitory, and the university system was very different from the European universities she was used to. She was able to travel with Emmy to the Institute for Advanced Studies in Princeton though, where she got to meet Einstein.

Olga only spent a year at Bryn Mawr before she accepted a research fellowship (the one she originally applied for) from Girton College. Here she was given the title of a ‘fellow’ (or a ‘don’). This title enabled her to pursue more activities in academia.
The actual job interview Olga attended for the position was amusing as she was asked about her jointly written papers. More specifically she was tactfully asked if she was the senior or junior author. Could they not believe that a woman might have equal input into a published paper? On another job interview she was asked why she had only collaborated with men, she retaliated by saying that that was why she was applying for a position in a women’s college! Obviously the gender issue was beginning to become a hot topic in academia!
Olga experienced at Girton College quite opposing views on women, but ridiculously from another woman! The female head of the college insisted that students should not work under Olga when constructing their dissertation, as she believed it would be damaging to their career if they had had a woman supervisor!
Olga felt lonely at Girton College, as once again no one was involved in her field of research. Therefore in 1937 Hardy helped her to obtain a teaching position at Westfield College of the London University, a women’s college. She had to teach nine courses that were out of her field.

At this point she met her husband-to-be John (Jack) Todd on intercollegiate seminars where she lectured. Apparently Jack approached Olga for her help on a technical mathematics question, which initially she could not solve! Jack taught analysis at one of the other London colleges, he was a northern Irish Presbyterian. His scientific background as well as his personal background was also very different to Olga’s, he was trained in classical analysis. Not deterred by the differences they married on 29 September 1938, and had an extraordinarily fruitful marriage for over fifty-seven years. They even found many mathematical topics that they could both have an interest in (considering the diversity of their fields), for example the Hilbert matrix. They collaborated on merely a few joint papers, but they discussed everything and influenced each other greatly. Olga said some fifty years later –

“My life and my career would have been so different if my Irishman had not come along.” [50]
During World War II it is claimed that Olga and Jack had to move eighteen times due to bombing in London. They moved between Belfast, where they stayed to be with Jack’s mother, and Oxford, where Westfield College had been transferred. During this period Olga wrote quite a few papers, some being on group theory. These papers discussed particular groups where every subnormal subgroup is normal, she proved as a result, along with other things, that a group with cyclic Sylow subgroups satisfied this property. She also wrote several papers on matrices of finite order – she eventually made important contributions to matrix theory, even though it was originally not in her field of work. She jointly wrote with her husband on this topic in 1940, they produced a paper entitled ‘Matrices with finite period’ and in 1941 another paper on ‘Matrices of finite period’. Olga also supervised thesis work, during 1942 and 1943 she supervised Hanna Neumann’s thesis on combinatorial group theory.
Between 1943 and 1946 Olga was given leave of absence from Westfield College to help with the war effort. She and Jack both began work on aerodynamics at the National Physical Laboratory at Teddington, in conjunction with the Ministry of Aircraft Production. Olga worked with a group investigating an aerodynamic phenomenon called ‘flutter’.

Flutter is caused when in flight, interactions between aerodynamic forces and a flexing airframe induce vibrations. When an aircraft flies at a speed greater than a certain threshold, those self-excited vibrations become unstable, leading to flutter. Therefore when designing an aircraft, it’s crucial to know the flutter speed before the aircraft is built and flown. To estimate the speed approximate solutions of certain differential equations must be found. The best way to achieve this was to determine the eigenvalues of a square matrix containing these equations. Although at the time several methods of determining the eigenvalues were available, they were all quite tricky and time consuming. Usually numerous young women were contracted in to operate hand-cranked calculating machines.

Olga, after research, provided a method that considerably reduced the excessive calculations and computational workload. She utilised the Gershgorin circle theorem to narrow down the possible range of solutions eliminating the time consuming methods that were necessary to compute the exact solutions, i.e. the results that Olga produced were enough to estimate the speed of the flutter, without all the work! Olga remembered Gershgorin circle theorem from her student days, although she didn’t do numerical mathematics at university she came across it as a lemma in the algebraic number theory she learnt from Furtwängler.
In conjunction with her discovery Olga published several papers. In 1944 she published ‘A note on skew-symmetric matrices’ and in 1945 she published ‘On some boundary value problems in the theory of the non-uniform supersonic motion of an aerofoil’.
Mathematically, Olga’s time at the National Physical Laboratory was very beneficial for her. She once commented –
“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.” [50]
In 1946 Olga served on the council of the London Mathematical Society for a year.

In 1947 Olga and Jack moved to America with the intention of staying for just a year. After a short stay at the Institute for Advanced Study at Princeton working on Von Neumann’s computer project, they both obtained positions at the National Bureau of Standards’. Many leading mathematicians of the time either worked here or kept in touch with the activities of the institute. Olga lectured at numerous universities on matrix theory and boundary-value problems for a hyperbolic differential equation. She then did a lecture tour in Los Angeles, and eventually settled in California where a new research building had been constructed.

Olga and Jack returned to London for a short period where she conducted research on bounds for eigenvalues of finite matrix, integral matrices, and eigenvalues of sums and products of finite matrix. Following this they returned on request to America, to Washington and the National Bureau of Standards. Here Olga became a mathematical consultant, gaining responsibilities such as monitoring visiting professors and post doctorates, and dealing with any correspondence received. Meanwhile she also carried out research on the L-property (the concept of a special set of matrix pencils) with Motzkin, and highlighted ideas from the ‘Lyapunov functions’ that led to the creation of a general inertia theory.
In 1951 Olga led a conference on ‘Simultaneous Linear Equations and the Determination of Eigenvalues’. This was the first conference of its kind on numerical aspects of matrix theory.
During this time Olga continued to write papers on matrix theory, group theory, algebraic number theory and numerical analysis. Matrix theory became a major contributor to the computer revolution at this time. Therefore she was particularly credited for such work, once being described as a –
“…computer pioneer…who provided significant contributions to solutions of problems associated with applications of computers.” [50]
In 1955 Olga and Jack took a year out from the Bureau to work at the Courant Institute in New York. She lectured on matrix theory, and Jack lectured on numerical analysis.

When Olga returned to the Bureau she lectured on a course detailing bounds for eigenvalues. She also wrote three chapters of a book entitled the ‘Handbook of Physics’. These chapters covered algebra, operator theory and differential equations. Jack also became editor of a book entitled a ‘Survey of Numerical Analysis’, the book contained Olga’s lecture notes.

After her break from the Bureau Olga realised that her position there was not quite ideal, although stimulating and enjoyable, she missed teaching. Olga always loved lecturing her students, she undeniably brought out the best in them, even her postdoctoral fellows found her enthusiasm for work contagious when she was teaching. It was what she enjoyed most.
Therefore in 1957 when Olga and Jack were offered teaching/research positions at Caltech – the California Institute of Technology, they eagerly accepted and left the Bureau. The offer they received was actually that Jack would become the Professor and Olga would be the research associate. This offer to husband and wife was common at the time, unfortunately the male still had to have the dominant status over the female. Olga was not too fussed about such a stigma attached to the job, as after assessment both their offices were the same size and adjacent to one another, and Olga was permitted to conduct lectures and supervise theses, which was all she cared for, so they accepted.
Although the situation did become problematic in 1969 when a young Assistant Professor of English was advertised by the media as being the first woman faculty member at Caltech. This angered Olga, as she was supposed to be on the faculty already, but obviously due to office politics concerning her title the media or Caltech had dismissed her. Therefore she went straight to the powers at be and demanded a resolution to such an infuriating situation. From 1971 Olga was granted the title of Professor.

When first starting at Caltech Olga was concerned about the time she had spent absent from teaching, but she soon realised that her students would draw the mathematics out of her, whether she proffered it or not! As the years passed increased numbers of thesis students realised that Olga was a superb supervisor, encouraging, cajoling, and supportive, as a result her popularity increased and students flocked to her – her reputation as a fine teacher was spreading!

During her time at Caltech Olga studied topological algebra. From this developed her research on Sum of Squares, where unusual links between number theory, geometry, topology, partial differential equations, Galois theory and algebras are observed – subjects that spanned her field of knowledge. This research resulted in her paper on ‘Sum of Squares’ winning the Ford Prize from the Mathematical Association of America in 1971.
In 1972 Olga was elected to the council of the American Mathematical Society (AMS) where she served till 1978. During this time she became editor of their AMS bulletin and was elected to vice-president between 1986 and 1987.
In 1977 Olga retired becoming a Professor Emeritus. In honour of her career the ‘Journal of Linear Algebra and Applications’ (for which she was an editor) and the ‘Journal of Linear and Multilinear Algebra’ published issues dedicated to her. Also, the ‘Journal of Number Theory’ published a book, ‘Algebra and Number Theory’ in which they included an autobiographical sketch and technical survey of her work. Although retired, Olga continued with her mathematical research, believing that she just had more time to devote to the correspondence that she received and wrote to colleagues all over the World.

During the last couple of decades of Olga’s life she began to get the recognition she deserved. In 1963 she won the ‘Woman of the Year Award’ from the LA times. In 1965 she received a Fulbright professorship to the University of Vienna. In 1975 she was elected to the Austrian Academy of Sciences. In 1978 the Austrian government, her native home honoured her with their highest award, ‘The Cross of Honour in Science and Arts, First Class’. In 1980 she received an honorary doctorate from the University of Vienna. In 1985 she was elected to the Bavarian Academy of Sciences, and in 1988 she was awarded an honorary doctorate from the University of Southern California.

One of Olga’s last papers, published in 1988, was entitled ‘How I Became a Torch Bearer for Matrix Theory’. It details her development, love for, and devotion to matrix theory. The title of her paper sums it up really. Olga popularised the Gershgorin circle theorem, strengthening the method, and initiating the mathematical research of its finer details. Her papers have cast an influence over the research of hundreds of people over the decades since. The standing ovation Olga received after addressing the second Raleigh Conference in 1982, concerning linear algebra applications through real and complex matrix theory demonstrated her excellence in the subject. As a result of Olga’s work matrix theory became more than just a part of a scientist’s toolkit, but in itself became an important field of mathematical research.
In 1992 colleagues, family and friends created the Olga Taussky-John Todd Lecture Program of the International Linear Algebra Society (ILAS), to honour the contributions made by Olga and Jack to the field of Linear Algebra. This programme endeavours to maintain that every three to four years a junior mathematician in linear algebra is invited to address a one-hour meeting endorsed by ILAS. The first of these lectures was in 1993 and Helene Shapiro – a former student of Olga’s, delivered the lecture at the Pure and Applied Linear Algebra Conference in Pensacola, Florida.
On 7th October 1995 Olga Taussky Todd died in Pasadena, California. During her lifetime she had produced approximately three hundred papers, she was a great mathematician. There was much more to her than this though, she was an inspiring teacher, colleague and friend who was warm and supportive.

Olga especially encouraged women mathematicians, as she was always disappointed by the fact that she never had many females to work with. At a particular AMS meeting in 1962 Marjorie Senechal had to dictate one of her papers for the first time, her recollection of the experience brings fond memories of Olga who made the whole event a lot more bearable. Olga came up to Marjorie smiling and introducing herself, saying, “It’s so nice to have another woman here! Welcome to mathematics!” [49]

In 1981 Olga conducted a ‘Noether Lecture’ for the Association of Women in Mathematics, this commemorated another great woman mathematician, but was special for Olga as Emmy Noether was also a great friend and colleague of hers. The lecture was later published in full in a paper entitled ‘The Many Aspects of Pythagorean Triangles’.
In 1999 a conference was held to honour Olga’s achievements as a woman, it was entitled the ‘Olga Taussky Todd Celebration of Careers in Mathematics for Women’. The Mathematical Sciences Research Institute (MSRI) in Berkeley, California hosted it. It detailed the research of outstanding women in mathematics. It also highlighted various issues of concern of women entering the mathematical research community, something that obviously would have been very important to Olga.

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