(Emmy) Noether

Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References

Amalie (Emmy) Noether Gottingen¨ and Algebra Amalie (Emmy) Noether 1920-1933 Table of Contents German Nationalism: Exile Introduction from Germany Gottingen¨ and Hilbert and Einstein 1917-1920 References Larry Susanka

October 3, 2019

Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References

INTRODUCTION She was the eldest of four children. Little is known about her Emmy Noether (pronounced childhood beyond what can be inferred from her general NER-ter) was born to a situation. There are no anecdotes that reveal that she was prosperous Jewish family in “destined” to be a great mathematician. Girls at that time and the Bavarian university town place and social class simply did not study the sciences, and an of Erlangen on March 23, 1882. academic career was out of the question. She died unexpectedly in another university town, Bryn Mawr Pennsylvania, on April According to Wikipedia, 14, 1935 at the age of 53. As a girl, Noether was well liked. She did not stand out academically although she was known for being clever and The aim of this talk is to tell friendly. She was near-sighted and talked with a minor lisp part of the story of this during her childhood. remarkable woman and provide an outline of her ————— scientiﬁc legacy.

————— Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References

Her father Max Noether, though, was a Professor of Mathematics at the University of Erlangen and studied what She went to the standard local we would call algebraic geometry. He was a contemporary of girl’s school and trained to Felix Klein, and the ideas of the “Erlangen Program” must have become a High School teacher been in the air. in modern languages, French and English. She graduated in 1890 with good grades, though apparently “class management” was not her strong suit.

(One ﬁnds variant spellings of the family name. Apparently her father’s family name was originally Samuel but was changed by ﬁat of the local authorities under a “Tolerance Edict” to Nother¨ sometime after 1809. During her father’s generation that was changed again to Noether.) ————— —————

Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References

The oldest of her younger brothers was Alfred, born in 1883, who was awarded a PhD in chemistry in 1909 but died in 1918. Fritz was born in 1884 and was a good applied mathematician. After the rise of the Nazis made work at German Universities impossible he got a job at Tomsk State University in the Soviet Union. During the Great Purge of 1937 he was arrested and sentenced as a German spy and eventually shot in 1941. Gustav Robert, was born in 1889 and died in 1928, after suffering “chronic illnesses.” —————

Her work with Gordan was in an area called Invariant Theory, In 1900 the law was changed to allow women to audit classes at and involved finding finite sets of polynomials that generated Erlangen, though only with the explicit permission of the infinite sets of polynomials which leave “invariant” objects we instructor, and Emmy became one of two women (out of a would now call algebraic varieties. thousand students) to attend classes there. David Hilbert, at the center of the scientific world in Gottingen,¨ She studied languages and history and mathematics and had achieved his first fame by outlining conditions under presumably at some point it became obvious how good she which this was possible. was at mathematics, and how much she enjoyed it, and she switched to mathematics entirely. But actually finding generators for a given situation was a different matter. In her work with Gordan based on her thesis, She passed the graduation exam in 1903 and attended lectures and published in a major journal and widely known, she (Schwarzschild, Minkowski, Hilbert) for a semester at provided sets of generators for more than 300 of these. Gottingen¨ in 1903-04. Once found these finite sets can easily be shown to generate. Her Father’s colleague Paul Gordan agreed to be her PhD thesis supervisor (when that became possible) and she finished But apparently there were, and are, no explicit, algorithmic, her degree in Mathematics at Erlangen in 1907. methods for finding even one of these, let alone 300, and the ————— work was regarded as a computational marvel. —————

Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References

GOTTINGENAND¨ HILBERTAND EINSTEIN 1917-1920 In 1907 Einstein created the Special Theory of Relativity, but From 1908 to 1915 Noether continued her work in this area and soon realized it was missing critical elements (gravitation, for learned of the techniques used by David Hilbert, up the road in instance) and turned to Mathematicians to help with the Gottingen.¨ problems he encountered. In particular he began a lively correspondence with David Hilbert, who had become She expanded her work to study invariants on ﬁelds of rational fascinated by Mathematical Physics, and the two were working functions and ﬁnite groups and worked with several other out together the key elements of the General Theory Einstein mathematicians on these topics. (Erhard Schmidt, Ernst Fisher) proposed for this. She lectured during this time but never in her own name, and There were major issues: for instance it seemed that energy was never for pay—it was forbidden by law for a woman to teach at not conserved in the theory. university. Hilbert recognized the issue as being related to invariant ————— theory, a subject he had not looked at in a while, and realized he needed a current expert on the subject to help him here. He knew of Fraulein¨ Noether from Erlangen who had been working in this area. ————— Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References

People understood that there were things like linear Hilbert and Felix Klein (who had since moved to Gottingen)¨ momentum, angular momentum, energy and charge that were invited her to visit Gottingen¨ and attempted to recruit her in conserved in physical theories. But no one really asked why this 1915—but an appointment was blocked by the Philosophy should be true. It wasn’t really necessary in Newtonian department, which refused to consider an application for Mechanics: it was all pretty obvious and fell out of minimizing habilitation by a woman. the action (an integral involving the Lagrangian) using basic I do not see that the sex of the candidate is an argu- calculus techniques, calculations of the type we saw in Larry ment against her. After all, we are a university, not a Curnutt’s talk last year. bathhouse. (David Hilbert) But GR was a deeper theory, and nothing was really obvious. Why is anything conserved in this theory? Though these efforts did not prevail at this time she did stay at Gottingen¨ with ﬁnancial support from her family and worked Her work cleared up, incidentally, the issue of conservation of with Hilbert and lectured under his name—without pay. energy by showing that, though energy in GR might not appear to be conserved locally it was conserved globally. During the next year she ﬁnished the work for which Hilbert asked her to come, the results we now know as Noether’s But it was about much more than that. Theorems. She also proved a converse: whenever you have a conserved ————— quantity there is a corresponding symmetry in physical law. —————

Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References

The symmetries of physical law we are talking about are continuous symmetries: those of you who attended last year’s talk on group theory and symmetry might recall the Einstein and Hilbert needed to understand what, in this new symmetries which I called SquareSym and CircleSym. geometry, stayed the same. She helped them understand how to describe things that stay the same even if you change coordinates in GR and how to talk about symmetry in physical law generally. By this time Hilbert and Einstein both knew they were working with a profound intellect and supported and encouraged her however they could. SquareSym has a discrete symmetry group. Not the kind to which Noether’s Theorems could apply. The easiest way to resolve this is to have Noether CircleSym has a continuous symmetry group—homomorphic explain it to me. (comment by Albert Einstein) to the unit circle in the complex plane with complex ————— multiplication. This is the kind to which Noether’s Theorems could apply. ————— Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References

GOTTINGENAND¨ ALGEBRA 1920-1933

Klein presented this work (she could not herself, because she After her habilitation Noether completely gave up work in the was not a member) in 1918 at a meeting of the Royal Society of area of her habilitationsschrift and devoted the rest of her Sciences at Gottingen¨ and in 1919 the University of Gottingen¨ career to her ﬁrst love, the subject we now call Abstract allowed her to use this work for her habilitationsschrift and Algebra, whose form and ways of thinking are largely due to they granted her a non-civil-service and unpaid “extraordinary her inﬂuence. professorship” which allowed her to lecture under her own name. This is how Mathematicians know of her, most of whom have never heard of Noether’s Theorem (except in the context of In 1920 she was given the position of “Lecturer in Algebra” various important results in algebraic geometry of her father (Lehrbeauftragte fur¨ Algebra) which was a paid position at last. Max: Brill-Noether Theory, Noether’s formula, ————— Noether-Lefschetz theorem.) —————

Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References

In her classic 1921 paper Theory of Ideals in Ring Domains In the winter of 1928-1929 Noether taught and visited at (Idealtheorie in Ringbereichen) Noether invented the idea of Moscow State University and worked with P. S. Alexandrov, ideals in commutative rings and an entirely new way of Lev Pontryagin and Nikolai Chebotaryov there who later thinking of mathematical objects, sets of elements, as a “whole” praised her contributions to Galois Theory. with their own properties in relation to other such sets. For instance there is something called an “ascending chain condition” in ring theory (and in many other categories as well) and objects satisfying this condition are called Noetherian. In a 1928 paper, Heinz Hopf credits a conversation with Noether to simpliﬁcation of his approach to Betti numbers, She went on to expand this work to noncommutative algebras thereby inventing the concept of homology group, at the very and rephrased the representation theory of groups in terms of dawn of the subject of Algebraic Topology. the theory of modules and ideals. ————— ————— Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References

During this period students and post-docs came in droves to study with her at Gottingen—they¨ were known as “Noether’s boys” and are a “who’s who” of famous algebraists. She was As impressive and powerful as these contributions were the supervisor to a dozen PhD students in this time. most remarkable thing about Noether’s work was the new way van der Waerden’s Modern Algebra (1931) looks like a of thinking about these objects. This wholistic approach completely modern graduate algebra text. redeﬁned how mathematicians in many areas worked and is Completely unegotistical and free of vanity, she never her most lasting mathematical legacy. claimed anything for herself, but promoted the works of her This really cannot be attributed to her in the citations of works students above all. that use it: once you learn to think this way you just do and the Birkhoff and Mac Lane’s A Survey of Modern Algebra (1941) powerful results you can then prove are yours. brought her methods into English Yet her hand is there. Emil Artin, for whom is named the “descending chain ————— condition” (Artinian) was one of “Noether’s boys” Jean Dieudonne´ praised the work of Noether for liberating Linear Algebra “from the plague of matrices and determinants from which it had been suffering for a century.” —————

Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References

By all accounts Noether was a powerful, incredibly creative In 1932 Emmy Noether (together with Emil Artin) received the and steadfastly independent mathematician: a completely Ackermann-Teubner Memorial Award for their contributions confident personality. to mathematics. Though she was blatantly discriminated against as a woman (and because of her Jewish heritage) she was reportedly a good-humored and happy person, warm and helpful to In the judgment of the most competent living mathemati- everyone around her, especially her students and junior cians, FräuleinNoether was the most significant creative colleagues, who were devoted to her. mathematical genius thus far produced since the higher ed- ucation of women began. In the realm of algebra, in which She had the recognition of the male colleagues around her and the most gifted mathematicians have been busy for centuries, was an acknowledged leader. She was the first female plenary she discovered methods which have proved of enormous im- speaker at the International Congress of Mathematicians portance in the development of the present-day younger gen- (Zurich¨ 1932) and at that point was at the height of her powers, eration of mathematicians. Albert Einstein influencing everyone around her. ————— ————— Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References

Miss Noether is the greatest woman mathematician who has ever lived and the greatest woman scientist of any sort now living, and a scholar at least on the plane of Madame Curie. Noether . . . taught us to think in terms of simple and gen- Norbert Wiener eral algebraic concepts—homomorphic mappings, groups and rings with operators, ideals—and not in cumbersome algebraic computations; and she thereby opened up the path to ﬁnding algebraic principles in places where such princi- During her lifetime, and even until today, Noether has been ples had been obscured by some complicated special situa- characterized as the greatest woman mathematician in tion. Pavel Alexandrov recorded history by mathematicians such as Pavel Alexandrov, Hermann Weyl and Jean Dieudonne.´ Wikipedia —————

Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References

My methods are really methods of working and thinking; this is why they have crept in everywhere anonymously. GERMAN NATIONALISM:HER EXILEFROM GERMANY Emmy Noether

In 1933 Hitler rose to power and within months the Law for the Restoration of the Professional Civil Service was passed, which removed Jews from their jobs. Hermann Weyl wrote that Emmy Noether—her courage, her frankness, her unconcern about her own fate, her conciliatory spirit—was in the midst of all the hatred and meanness, despair and sorrow sur- rounding us, a moral solace. —————

This position was not really suitable for someone of her stature, Even after her ﬁring, when she was trying to decide what to do though Bryn Mawr was very welcoming and proud to have her next, she held seminars and worked with students in her there. She went weekly to talk and work with people at apartment. Princeton, but (apart from a few) did not ﬁnd the faculty there very welcoming to her as a female mathematician. Dozens of unemployed professors were looking for work: Albert Einstein and Hermann Weyl were appointed by the In 1934 at the invitation of Oswald Veblen she began lecturing Institute for Advanced Study and used connections to get her a at the Institute for Advanced Study and she supervised two job (still 1933) at Bryn Mawr College. PhD students in this time. She continued a collaboration ————— (begun in 1930 in Europe) with Richard Brauer, who had also moved to the United States. —————

Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References

Her work, however, does live on. In Mathematics her inﬂuence She was reported to be in good spirits and focussed on her is so thorough that people don’t even understand that it came work, but she was still ﬁnding her place in this new world. A from her: it is just the air that grad students and permanent position was in the works when (18 months after mathematicians breathe, even in areas barely related to her her move) she developed an ovarian cyst and went in for what work. was expected to be relatively minor surgery. In applications areas the Noether Theorems remain pervasive She recovered well for several days but a high fever hit and are critical in such places as Fluid Mechanics (potential suddenly and she died on the fourth day, April 14, 1935 at the vorticity) and Numerical Analysis and even Computer age of 53 and at the height of her powers. Graphics, models of the world. ————— ————— Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References

But particle Physicists use the work everywhere: there has EFERENCES likely not been a particle physics paper in ﬁfty years in which R symmetry has not been mentioned. Dwight Neuenschwander Emmy Noether’s Wonderful Theorem Forty years after her death, the Standard Model of particle Steve Nadis Discover June 2019 How Mathematician Emmy physics was invented using explicit reference to her work. She Noether’s Theorem Changed Physics was the inventor and ﬁrst user, in her theorems, of what Physicists now call Gauge Theory. Five Nobel prizes have been Numerous Wikipedia Entries given for work in modern Particle Physics alone that would not Saunders Mac Lane Journal of Pure and Applied Algebra exist without her theorems. (1986) Topology becomes Algebraic with Vietoris and Noether Yvette Kosmann-Schwarzbach The Noether Theorems Gennadi Sardanashvily Noether’s Theorems Emily Conover Science News June 2018 In her short life, mathematician Emmy Noether changed the face of physics BBC Radio In Our Time January 24, 2019 Podcast Emmy Noether ————— —————