Emmy Noether

• 1. The Woman Who Was Not A Bathhouse

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  In a joint faculty meeting at the University of Göttingen, one professor protested aloud: "What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?" The mathematician David Hilbert answered with indignation. His exact words have not been preserved, but he is often said to have remarked that the university was "not a bathhouse." The woman whose appointment caused this quarrel was Emmy Noether, a German mathematician born to a Jewish family in 1882. Albert Einstein, Pavel Alexandrov, Jean Dieudonné, Hermann Weyl, and Norbert Wiener would later call her the most important woman in the history of mathematics. Yet for much of her career she taught without pay, lectured under a man's name, and was barred from holding a regular professorship. How does someone excluded from the academy reshape its deepest ideas? Why did a result she proved in physics get compared to the Pythagorean theorem? And what happened to her when a new government in Germany decided that her ancestry made her unwelcome?

• 2. An Unconventional Choice

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  In early 1900, Emmy Noether passed an examination to teach French and English and earned the overall score of sehr gut, meaning very good. That result qualified her to teach languages at schools reserved for girls. She chose instead to study mathematics at the University of Erlangen-Nuremberg, where her father, Max Noether, was a professor. Two years earlier the university's Academic Senate had declared that mixed-sex education would "overthrow all academic order." Noether was one of just two women among 986 students, allowed only to audit classes and needing each professor's permission to attend. In 1903, restrictions on women's enrollment in Bavarian universities were lifted, and she officially reentered the university in October 1904. Under the supervision of Paul Gordan, known as the "king of invariant theory," she completed her doctoral dissertation in 1907 and graduated summa cum laude. Her thesis ended with a list of over 300 explicitly worked-out invariants. She later dismissed that early work, and similar papers, as "crap." All of her later work would be in a completely different field, one that would carry her name.

• 3. Seven Years Without Pay

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  From 1908 to 1915, Noether taught at Erlangen's Mathematical Institute without a salary, sometimes substituting for her father when illness kept him from lecturing. She joined the Circolo Matematico di Palermo in 1908 and the Deutsche Mathematiker-Vereinigung in 1909. When Gordan retired in 1910, she taught under his successors, and one of them, Ernst Fischer, introduced her to the work of David Hilbert. Noether and Fischer found such lively enjoyment in mathematics that they discussed lectures long after they ended. She is known to have mailed Fischer postcards carrying her train of mathematical thought further. At Göttingen, after her invitation there in 1915, she again worked without an official position and without pay. Her lectures were advertised under Hilbert's name, with Noether listed only as providing "assistance." Even after a small salary arrived in 1923, she lived simply and saved half of it to bequeath to her nephew, Gottfried E. Noether.

• 4. Symmetry And Conservation

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  On the 26th of July 1918, Felix Klein presented a paper titled Invariante Variationsprobleme to the Royal Society of Sciences at Göttingen. Its author, Emmy Noether, did not present it herself, because she was not a member of the society. Hilbert and Klein had brought her to Göttingen in 1915 precisely because they needed her command of invariant theory to help them understand Albert Einstein's general relativity. Hilbert had noticed that conservation of energy seemed to be violated in the theory, since gravitational energy could itself gravitate. Noether's paper resolved that paradox and gave physics a fundamental tool. Her result, now called Noether's theorem, shows that a conservation law is tied to any continuous symmetry of a physical system. If physical laws behave the same regardless of orientation in space, the theorem shows angular momentum must be conserved. A jagged asteroid tumbling through space conserves angular momentum despite its asymmetry, because the symmetry lies in the laws, not the object. Einstein wrote to Hilbert after reading her work: "The old guard at Göttingen should take some lessons from Miss Noether! She seems to know her stuff." The American physicists Leon M. Lederman and Christopher T. Hill have called the theorem possibly on a par with the Pythagorean theorem.

• 5. Changing The Face Of Algebra

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  In 1921, Noether published Idealtheorie in Ringbereichen, or Theory of Ideals in Ring Domains, a paper the algebraist Irving Kaplansky called "revolutionary." In it she proved that in a ring satisfying the ascending chain condition on ideals, every ideal is finitely generated. The condition became so central that objects satisfying it are called Noetherian in her honor, a term coined by Claude Chevalley in 1943. The same paper established the Lasker-Noether theorem, which can be seen as a generalization of the fundamental theorem of arithmetic, the statement that any positive integer factors uniquely into primes. Her 1927 work characterized the rings now known as Dedekind domains through five conditions and introduced what are now called the isomorphism theorems. Nathan Jacobson wrote that abstract algebra is "largely due to her." Unlike most mathematicians, she did not generalize from known examples but worked directly with the abstractions. Van der Waerden recorded her guiding maxim: relationships between numbers, functions, and operations become "fully productive only after they have been isolated from their particular objects and been formulated as universally valid concepts." She called this begriffliche Mathematik, purely conceptual mathematics.

• 6. The Noether Boys

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  In Göttingen, Noether's students were sometimes called the "Noether Boys," and she supervised more than a dozen doctoral students, though most were registered formally with Edmund Landau because she was not permitted to supervise dissertations on her own. Her first, Grete Hermann, defended in February 1925 and later spoke reverently of her "dissertation-mother." In 1924 the Dutch mathematician Bartel Leendert van der Waerden arrived and soon became the leading expositor of her ideas. The second volume of his 1931 textbook Moderne Algebra borrowed heavily from her work, and he later said her originality was "absolute beyond comparison." Her devotion ran past the academic day. Once, when the building was closed for a state holiday, she gathered her class on the steps, led them through the woods, and lectured at a local coffee house. She did not follow a lesson plan, spoke quickly, and was hard to follow. Outsiders who visited often left within half an hour, and one regular student joked of such a departure: "The enemy has been defeated; he has cleared out." Olga Taussky-Todd recalled a luncheon where Noether, lost in mathematics, "gesticulated wildly" and spilled food on her dress, "completely unperturbed."

• 7. Recognition And Expulsion

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  In 1932, Noether and Emil Artin shared the Ackermann-Teubner Memorial Award, seen as a long-overdue recognition of her work. That September she delivered a plenary address at the International Congress of Mathematicians in Zürich, an event attended by 800 people with only twenty-one plenary addresses given. The 1932 congress is sometimes described as the high point of her career. Yet her colleagues remained frustrated that she was never elected to the Göttingen academy of sciences and never promoted to full professor. When Adolf Hitler became Reichskanzler in January 1933, the attack came quickly. One young protester reportedly demanded: "Aryan students want Aryan mathematics and not Jewish mathematics." In April 1933 Noether received a notice withdrawing her right to teach, citing the Civil Service Code of the 7th of April 1933. She accepted the decision calmly. Hermann Weyl later wrote that she was "in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace." When a student appeared at her apartment in the uniform of the Sturmabteilung, she showed no agitation and reportedly laughed about it afterward.

• 8. America And A Sudden End

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  After negotiations with the Rockefeller Foundation, a grant brought Noether to Bryn Mawr College in Pennsylvania, where she began in late 1933. There she gathered a group sometimes called the Noether girls, among them Grace Shover Quinn, Marie Johanna Weiss, Olga Taussky-Todd, and her only American doctoral student, Ruth Stauffer. In 1934 she also began lecturing at the Institute for Advanced Study in Princeton, though she remarked of nearby Princeton University that she was unwelcome at "the men's university, where nothing female is admitted." In April 1935 doctors found a tumor in her pelvis and discovered during surgery an ovarian cyst "the size of a large cantaloupe." For three days she seemed to recover normally. On the 14th of April she fell unconscious, her temperature soared to 109 degrees Fahrenheit, and she died. She was 53. Her body was cremated and the ashes interred under the walkway around the cloisters of the Old Library at Bryn Mawr. In a letter to a newspaper, Einstein called her the most significant creative mathematical genius produced since the higher education of women began. The historian Jeremy Gray put her legacy plainly: "Mathematicians simply do ring theory her way."