Influential Mathematicians: Emmy Amalie Noether
Today we’ll cover Emmy Amalie Noether, a very special mathematician that made important contributions that extended far beyond mathematics despite the unfair discrimination she had to endure in academia for being a woman.
Who was Emmy Noether?
The School of Mathematics and Statistics University of St Andrew has a pretty comprehensive bio on Emmy Noether that I've summarised for you below.
Emmy was the eldest of her parents' four children, the three younger children being boys. Alfred Noether (1883-1918) studied chemistry and was awarded a doctorate from Erlangen in 1909. After elementary school, Emmy Noether attended the Städtische Höhere Töchter Schule on Friedrichstrasse in Erlangen from 1889 until 1897. She had been born in the family home at Hauptstrasse 23 and lived there until, in the middle of her time at high school, in 1892, the family moved to a larger apartment at Nürnberger Strasse 32. At the high school she studied German, English, French, arithmetic and was given piano lessons. She loved dancing and looked forward to parties with children of her father's university colleagues. At this stage her aim was to become a language teacher and after further study of English and French she took the examinations of the State of Bavaria and, in 1900, became a certificated teacher of English and French in Bavarian girls schools. She was awarded the grade of "very good" in the examinations, the weakest part being her classroom teaching.
However Noether never became a language teacher. Instead she decided to take the difficult route for a woman of that time and study mathematics at university. Women were allowed to study at German universities unofficially and each professor had to give permission for his course. Noether obtained permission to sit in on courses at the University of Erlangen during 1900 to 1902. She was one of only two female students sitting in on courses at Erlangen and, in addition to mathematics courses, she continued her interest in languages being taught by the professor of Roman Studies and by an historian. At the same time she was preparing to take the examinations which allowed a student to enter any university. Having taken and passed this matriculation examination in Nürnberg on 14 July 1903, she went to the University of Göttingen. During 1903-04 she attended lectures by Karl Schwarzschild, Otto Blumenthal, David Hilbert, Felix Klein and Hermann Minkowski. Again she was not allowed to be a properly matriculated student but was only allowed to sit in on lectures. After one semester at Göttingen she returned to Erlangen.
At this point the rules were changed and women students were allowed to matriculate on an equal basis to the men. On 24 October 1904 Noether matriculated at Erlangen where she now studied only mathematics. In 1907 she was granted a doctorate after working under Paul Gordan. The oral examination took place on Friday 13 December and she was awarded the degree 'summa cum laude'. Hilbert's basis theorem of 1888 had given an existence result for finiteness of invariants in nn variables. Gordan, however, took a constructive approach and looked at constructive methods to arrive at the same results. Noether's doctoral thesis followed this constructive approach of Gordan and listed systems of 331 covariant forms.
Having completed her doctorate the normal progression to an academic post would have been the habilitation. However this route was not open to women so Noether remained at Erlangen, helping her father who, particularly because of his own disabilities, was grateful for his daughter's help. Noether also worked on her own research, in particular she was influenced by Ernst Fischer who had succeeded Gordan to the chair of mathematics when he retired in 1911. Fischer's influence took Noether towards Hilbert's abstract approach to the subject and away from the constructive approach of Gordan. Now this was very important to her development as a mathematician for Gordan, despite his remarkable achievements, had his limitations.
Noether's reputation grew quickly as her publications appeared. In 1908 she was elected to the Circolo Matematico di Palermo, then in 1909 she was invited to become a member of the Deutsche Mathematiker-Vereinigung and in the same year she was invited to address the annual meeting of the Society in Salzburg. She gave the lecture ‘Zur Invariantentheorie der Formen von n Variabeln’. In 1913 she lectured in Vienna, again to a meeting of the Deutsche Mathematiker-Vereinigung. Her lecture on this occasion was ‘Über rationale Funktionenkörper’. While in Vienna she visited Franz Mertens and discussed mathematics with him.
In 1915 Hilbert and Klein invited Noether to return to Göttingen. The reason for this was that Hilbert was working on physics, in particular on ideas on the theory of relativity close to those of Albert Einstein. He decided that he needed the help of an expert on invariant theory and, after discussions with Klein, they issued the invitation. This result in theoretical physics is sometimes referred to as Noether's Theorem, and proves a relationship between symmetries in physics and conservation principles. This basic result in the theory of relativity was praised by Einstein in a letter to Hilbert when he referred to Noether's penetrating mathematical thinking. Of course, she arrived in Göttingen during World War I. This was a time of extreme difficulty and she lived in poverty during these years and politically she became a radical socialist. However, they were extraordinarily rich years for her mathematically.
Hilbert and Klein persuaded her to remain at Göttingen while they fought a battle to have her officially on the Faculty. In a long battle with the university authorities to allow Noether to obtain her habilitation there were many setbacks and it was not until 1919 that permission was granted and she was given the position of Privatdozent. During this time Hilbert had allowed Noether to lecture by advertising her courses under his own name.
At Göttingen, after 1919, Noether moved away from invariant theory to work on ideal theory, producing an abstract theory which helped develop ring theory into a major mathematical topic. ‘Idealtheorie in Ringbereichen’ (1921) was of fundamental importance in the development of modern algebra. In this paper she gave the decomposition of ideals into intersections of primary ideals in any commutative ring with ascending chain condition. Emanuel Lasker (who became the world chess champion) had already proved this result for a polynomial ring over a field. Noether published ‘Abstrakter Aufbau der Idealtheorie in algebraischen Zahlkorpern’ in 1924. In this paper she gave five conditions on a ring which allowed her to deduce that in such commutative rings every ideal is the unique product of prime ideals.
In the same year of 1924 B L van der Waerden came to Göttingen and spent a year studying with Noether. After returning to Amsterdam van der Waerden wrote his book Moderne Algebra in two volumes. The major part of the second volume consists of Noether's work. From 1927 onwards Noether collaborated with Helmut Hasse and Richard Brauer in work on non-commutative algebras. They wrote a beautiful paper joint paper ‘Beweis eines Hauptsatzes in der Theorie der Algebren’ which was published in 1932. In addition to teaching and research, Noether helped edit Mathematische Annalen. Much of her work appears in papers written by colleagues and students, rather than under her own name.
Further recognition of her outstanding mathematical contributions came with invitations to address the International Congress of Mathematicians at Bologna in September 1928 and again at Zürich in September 1932. Her address to the 1932 Congress was entitled ‘Hyperkomplexe Systeme in ihren Beziehungen zur kommutativen Algebra und zur Zahlentheorie’. In 1932 she also received, jointly with Emil Artin, the Alfred Ackermann-Teubner Memorial Prize for the Advancement of Mathematical Knowledge. In April 1933 her mathematical achievements counted for nothing when the Nazis caused her dismissal from the University of Göttingen because she was Jewish. She received no pension or any other form of compensation but, nevertheless, she considered herself more fortunate than others.
She accepted a one-year visiting professorship at Bryn Mawr College in the USA and in October 1933 sailed to the United States on the ship Bremen to take up the appointment. She had hoped to delay accepting the invitation since she would have liked to have gone to Oxford in England but it soon became clear that she had to leave quickly. At Bryn Mawr she was made very welcome by Anna Johnson Pell Wheeler who was head of mathematics. Noether ran a seminar during the winter semester of 1933-34 for three students and one member of staff. They worked through the first volume of van der Waerden's Moderne Algebra. In February 1934 she began giving weekly lectures at the Institute for Advanced Study, Princeton.
Noether returned to Germany in the summer of 1934. There see saw her brother Fritz for what would be the last time, and visited Artin in Hamburg before going on to Göttingen. She returned to the United States where her visiting professorship at Bryn Mawr had been extended for a further year. She continued her weekly lectures at Princeton where Richard Brauer had now arrived. After her lectures she enjoyed talking about mathematics with Weyl, Veblen and Brauer.
Quantum Formalism Influence
Although Noether did not directly contribute to the actual quantum formalism, her work in invariant theory, abstract algebra and her collaboration with the likes Hermann Weyl, had a great influence in the development of algebraic toolkits used in the most advanced treatments of the theory. Noether was also the supervisor of another great female mathematician that we'll cover later, Grete Hermann, who made important contributions to the foundations of quantum mechanics.
That’s it for today! I hope this inspiring biography of one of the greatest mathematicians of the 20th century was useful. I’ll end with a recommendation of some reading materials related to Noether:
‘Emmy Noether and Hermann Weyl’ (free download)
‘The Life and Times of Emmy Noether’ (free download)
Author: Bambordé, co-founder at Zaiku Group. Please take our upcoming quantum lectures survey if you haven’t already done so - we’ll keep it up until the end of this month!