Emmy Noether

Biography

Early life

Amalie Emmy Noether was born on 23 March 1882 in Erlangen, Bavaria. She was the first of four children of mathematician Max Noether and Ida Amalia Kaufmann, both from wealthy Jewish merchant families. Her first name was "Amalie", but she began using her middle name at a young age and continued to do so in her adult life and her publications.

In her youth, Noether did not stand out academically, but she was known for being clever and friendly. She was nearsighted and had a minor lisp during her childhood. A family friend recounted a story years later about young Noether quickly solving a brain teaser at a children's party, showing logical acumen at an early age. She was taught to cook and clean, as were most girls of the time, and took piano lessons. She pursued none of these activities with passion, but loved to dance.

Noether had three younger brothers. The eldest, Alfred Noether, was born in 1883 and was awarded a doctorate in chemistry from Erlangen in 1909, but died nine years later. Fritz Noether was born in 1884, studied at the Ludwig-Maximilians-Universität München, and made contributions to applied mathematics. He was likely executed in the Soviet Union in 1941 during the Second World War. The youngest, Gustav Robert Noether, was born in 1889. Very little is known about his life; he suffered from chronic illness and died in 1928.

Education

Noether showed early proficiency in French and English. In 1900, she took the examination for teachers of these languages and received an overall score of sehr gut (very good). Her performance qualified her to teach languages at schools reserved for girls, but she chose instead to continue her studies at the University of Erlangen–Nuremberg, where her father was a professor.

This was an unconventional decision; two years earlier, the Academic Senate of the university had declared that allowing mixed-sex education would "overthrow all academic order". One of just two women in a university of 986 students, Noether was allowed only to audit classes rather than participate fully, and she required the permission of individual professors whose lectures she wished to attend. Despite these obstacles, on 14 July 1903 she passed the graduation exam at a Realgymnasium in Nuremberg.

During the 1903–04 winter semester, Noether studied at the University of Göttingen, attending lectures given by astronomer Karl Schwarzschild and mathematicians Hermann Minkowski, Otto Blumenthal, Felix Klein, and David Hilbert.

In 1903, restrictions on women's full enrollment in Bavarian universities were rescinded. Noether returned to Erlangen and officially reentered the university in October 1904, declaring her intention to focus solely on mathematics. She was one of six women in her year (two auditors) and the only woman in her chosen school. Under the supervision of Paul Gordan, she wrote her dissertation, Über die Bildung des Formensystems der ternären biquadratischen Form (On Complete Systems of Invariants for Ternary Biquadratic Forms), in 1907, graduating summa cum laude that year. Gordan was a member of the "computational" school of invariant researchers, and Noether's thesis ended with a list of over 300 explicitly worked-out invariants. This approach to invariants was later superseded by the more abstract and general approach pioneered by Hilbert. It was well received, but Noether later called her thesis and some subsequent similar papers of hers "crap". All her later work was in a completely different field.

University of Erlangen–Nuremberg

From 1908 to 1915, Noether taught at Erlangen's Mathematical Institute without pay, occasionally substituting for her father, Max Noether, when he was too ill to lecture. She joined the Circolo Matematico di Palermo in 1908 and the Deutsche Mathematiker-Vereinigung in 1909. In 1910 and 1911, she published an extension of her thesis work from three variables to n variables.

Gordan retired in 1910, and Noether taught under his successors, Erhard Schmidt and Ernst Fischer, who took over from Schmidt in 1911. According to her colleague Hermann Weyl and her biographer Auguste Dick, Fischer was an important influence on Noether, in particular by introducing her to David Hilbert's work. Noether and Fischer shared lively enjoyment of mathematics and often discussed lectures long after they were over; Noether is known to have sent Fischer postcards continuing her train of mathematical thoughts.

From 1913 to 1916, Noether published several papers extending and applying Hilbert's methods to mathematical objects such as fields of rational functions and the invariants of finite groups. This phase marked Noether's first exposure to abstract algebra, a field to which she made groundbreaking contributions.

In Erlangen, Noether advised two doctoral students: Hans Falckenberg and Fritz Seidelmann, who defended their theses in 1911 and 1916. Despite Noether's significant role, they were both officially under her father's supervision. After completing his doctorate, Falckenberg spent time in Braunschweig and Königsberg before becoming a professor at the University of Giessen. Seidelmann became a professor at the Ludwig-Maximilians-Universität München.

University of Göttingen

In early 1915, Noether was invited to return to the University of Göttingen by Hilbert and Felix Klein. Their effort to recruit her was initially blocked by the philologists and historians among the philosophical faculty, who insisted that women should not become privatdozenten. In a joint department meeting on the matter, one faculty member protested: "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?" Hilbert, who believed Noether's qualifications were the only important issue and that gender was irrelevant, objected with indignation and scolded those protesting her habilitation. His exact words have not been preserved, but his objection is often said to have included the remark that the university was "not a bathhouse". Pavel Alexandrov recalled that faculty members' opposition to Noether was based not just in sexism, but also in their objections to her social-democratic political beliefs and Jewish ancestry.

Noether left for Göttingen in late April; two weeks later her mother died suddenly in Erlangen. She had previously received medical care for an eye condition, but its nature and relation to her death is unknown. Around the same time, Noether's father retired and her brother joined the German Army to serve in World War I. She returned to Erlangen for several weeks, mostly to care for her aging father.

During her first years teaching at Göttingen, she had no official position and was not paid. Her lectures often were advertised under Hilbert's name, with Noether providing "assistance".

Soon after arriving at Göttingen, she demonstrated her capabilities by proving the theorem now known as Noether's theorem, which shows that a conservation law is associated with any differentiable symmetry of a physical system. The paper, Invariante Variationsprobleme, was presented by Felix Klein on 26 July 1918 at a meeting of the Royal Society of Sciences at Göttingen. Noether presumably did not present it herself because she was not a member of the society. In their book Symmetry and the Beautiful Universe, physicists Leon M. Lederman and Christopher T. Hill write that Noether's theorem is "certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem".

When World War I ended, the German Revolution of 1918–19 brought a significant change in social attitudes, including more rights for women. In 1919, the University of Göttingen allowed Noether to proceed with her habilitation (eligibility for tenure). Her oral examination was in May, and she successfully delivered her habilitation lecture in June. Noether became a privatdozent, and that fall delivered the first lectures listed under her name. She was still not paid for her work.

Moscow State University

In 1928–1929, Noether accepted an invitation to Moscow State University, where she continued working with P. S. Alexandrov. In addition to carrying on with her research, she taught classes in abstract algebra and algebraic geometry. She worked with the topologists Lev Pontryagin and Nikolai Chebotaryov, who later praised her contributions to the development of Galois theory.

Politics was not central to her life, but Noether took a keen interest in political matters and, according to Alexandrov, showed considerable support for the Russian Revolution. She was especially happy to see Soviet advances in the fields of science and mathematics, which she considered indicative of new opportunities made possible by the Bolshevik project. This attitude caused her problems in Germany, culminating in her eviction from a pension lodging building, after student leaders complained of living with "a Marxist-leaning Jewess". Hermann Weyl recalled that "During the wild times after the Revolution of 1918," Noether "sided more or less with the Social Democrats". She was a member of the Independent Social Democrats from 1919 to 1922, a short-lived splinter party. In the words of logician and historian Colin McLarty, "she was not a Bolshevist, but was not afraid to be called one."

Noether planned to return to Moscow, an effort for which she received support from Alexandrov. After she left Germany in 1933, he tried to help her gain a chair at Moscow State University through the Soviet Education Ministry. This proved unsuccessful, but they corresponded frequently during the 1930s, and in 1935 she made plans for a return to the Soviet Union.

Recognition

In 1932, Emmy Noether and Emil Artin received the Ackermann–Teubner Memorial Award for their contributions to mathematics. The prize included a monetary reward of 500 ℛ︁ℳ︁ and was seen as a long-overdue official recognition of her considerable work in the field. Nevertheless, her colleagues expressed frustration at the fact that she was not elected to the Göttingen Gesellschaft der Wissenschaften (academy of sciences) and was never promoted to the position of Ordentlicher Professor (full professor).

Noether's colleagues celebrated her fiftieth birthday, in 1932, in typical mathematicians' style. Helmut Hasse dedicated an article to her in the Mathematische Annalen, wherein he confirmed her suspicion that some aspects of noncommutative algebra are simpler than those of commutative algebra, by proving a noncommutative reciprocity law. This pleased her immensely. He also sent her a mathematical riddle, which he called the "mμν-riddle of syllables". She solved it immediately, but the riddle has been lost.

In September of the same year, Noether delivered a plenary address (großer Vortrag) on "Hyper-complex systems in their relations to commutative algebra and to number theory" at the International Congress of Mathematicians in Zürich. The congress was attended by 800 people, including Noether's colleagues Hermann Weyl, Edmund Landau, and Wolfgang Krull. There were 420 official participants and twenty-one plenary addresses presented. Apparently, Noether's prominent speaking position was a recognition of the importance of her contributions to mathematics. The 1932 congress is sometimes described as the high point of her career.

Expulsion from Göttingen by Nazi Germany

When Adolf Hitler became the German Reichskanzler in January 1933, Nazi activity around the country increased dramatically. At the University of Göttingen, the German Student Association led the attack on the "un-German spirit" attributed to Jews and was aided by privatdozent and Noether's former student Werner Weber. Antisemitic attitudes created a climate hostile to Jewish professors. One young protester reportedly demanded: "Aryan students want Aryan mathematics and not Jewish mathematics."

One of the first actions of Hitler's administration was the Law for the Restoration of the Professional Civil Service which removed Jews and politically suspect government employees (including university professors) from their jobs unless they had "demonstrated their loyalty to Germany" by serving in World War I. In April 1933, Noether received a notice from the Prussian Ministry for Sciences, Art, and Public Education which read: "On the basis of paragraph 3 of the Civil Service Code of 7 April 1933, I hereby withdraw from you the right to teach at the University of Göttingen." Several of Noether's colleagues, including Max Born and Richard Courant, also had their positions revoked.

Noether accepted the decision calmly, providing support for others during this difficult time. Hermann Weyl later 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 surrounding us, a moral solace." Typically, Noether remained focused on mathematics, gathering students in her apartment to discuss class field theory. When one of her students appeared in the uniform of the Nazi paramilitary organization Sturmabteilung (SA), she showed no sign of agitation and, reportedly, even laughed about it later.

Refuge at Bryn Mawr and Princeton

As dozens of newly unemployed professors began searching for positions outside Germany, their colleagues in the United States provided assistance and job opportunities for them. Albert Einstein and Hermann Weyl were appointed by the Institute for Advanced Study in Princeton, while others worked to find a sponsor required for legal immigration. Noether was contacted by representatives of two educational institutions: Bryn Mawr College, in the U.S., and Somerville College at the University of Oxford, in England. After a series of negotiations with the Rockefeller Foundation, a grant to Bryn Mawr was approved for Noether and she took a position there in late 1933.

At Bryn Mawr, Noether met and befriended Anna Wheeler, who had studied at Göttingen just before Noether arrived there. Another source of support at the college was Bryn Mawr's president, Marion Edwards Park, who enthusiastically invited mathematicians in the area to "see Dr. Noether in action!"

At Bryn Mawr, Noether formed a group, sometimes called the Noether girls, of four post-doctoral (Grace Shover Quinn, Marie Johanna Weiss, Olga Taussky-Todd) and doctoral (Ruth Stauffer) students. They enthusiastically worked through van der Waerden's Moderne Algebra I and parts of Erich Hecke's Theorie der algebraischen Zahlen (Theory of algebraic numbers). Stauffer was Noether's only doctoral student in the U.S.; Noether died shortly before she graduated. She took her examination with Richard Brauer and received her degree in June 1935, with a thesis on separable normal extensions. After her doctorate, Stauffer worked as a teacher for a short period and as a statistician for over 30 years.

In 1934, Noether began lecturing at the Institute for Advanced Study in Princeton upon the invitation of Abraham Flexner and Oswald Veblen. She also worked with Abraham Albert and Harry Vandiver. She said of Princeton University that she was not welcome at "the men's university, where nothing female is admitted".

Her time in the U.S. was pleasant; she was surrounded by supportive colleagues and absorbed in her favorite subjects. In mid-1934, she briefly returned to Germany to see Emil Artin and her brother Fritz. The latter, after having been forced out of his job at the Technische Hochschule Breslau, had accepted a position at the Research Institute for Mathematics and Mechanics in Tomsk, in the Siberian Federal District of Russia.

Death

In April 1935, doctors discovered a tumor in Noether's pelvis. Worried about complications from surgery, they ordered two days of bed rest first. During the operation they discovered an ovarian cyst "the size of a large cantaloupe". Two smaller tumors in her uterus appeared to be benign and were not removed to avoid prolonging surgery. For three days she appeared to convalesce normally, and she recovered quickly from a circulatory collapse on the fourth. On 14 April, Noether fell unconscious, her temperature soared to 109 °F (42.8 °C), and she died. "[I]t is not easy to say what had occurred in Dr. Noether", one of the physicians wrote. "It is possible that there was some form of unusual and virulent infection, which struck the base of the brain where the heat centers are supposed to be located." She was 53.

A few days after Noether's death, her friends and associates at Bryn Mawr held a small memorial service at College President Park's house. Hermann Weyl and Richard Brauer both traveled from Princeton and delivered eulogies. In the months that followed, written tributes began to appear: Albert Einstein joined van der Waerden, Weyl, and Pavel Alexandrov in paying their respects. Her body was cremated and the ashes interred under the walkway around the cloisters of the Old Library at Bryn Mawr.

Contributions to mathematics and physics

Noether's work in abstract algebra and topology was influential in mathematics, while Noether's theorem has widespread consequences for theoretical physics and dynamical systems. Noether showed an acute propensity for abstract thought, which allowed her to approach problems of mathematics in fresh and original ways. Her friend and colleague Hermann Weyl described her scholarly output in three epochs:

(1) the period of relative dependence, 1907–1919

(2) the investigations grouped around the general theory of ideals 1920–1926

(3) the study of the non-commutative algebras, their representations by linear transformations, and their application to the study of commutative number fields and their arithmetics

In the first epoch (1907–1919), Noether dealt primarily with differential and algebraic invariants, beginning with her dissertation under Paul Gordan. Her mathematical horizons broadened, and her work became more general and abstract, as she became acquainted with the work of David Hilbert, through close interactions with a successor to Gordan, Ernst Sigismund Fischer. Shortly after moving to Göttingen in 1915, she proved the two Noether's theorems, "one of the most important mathematical theorems ever proved in guiding the development of modern physics".

In the second epoch (1920–1926), Noether devoted herself to developing the theory of mathematical rings. In the third epoch (1927–1935), Noether focused on noncommutative algebra, linear transformations, and commutative number fields. The results of Noether's first epoch were impressive and useful, but her fame among mathematicians rests more on the groundbreaking work she did in her second and third epochs, as noted by Hermann Weyl and B. L. van der Waerden in their obituaries of her.

In these epochs, she was not merely applying ideas and methods of the earlier mathematicians; rather, she was crafting new systems of mathematical definitions that would be used by future mathematicians. In particular, she developed a completely new theory of ideals in rings, generalizing the earlier work of Richard Dedekind. She is also renowned for developing ascending chain conditions – a simple finiteness condition that yielded powerful results in her hands. Such conditions and the theory of ideals enabled Noether to generalize many older results and to treat old problems from a new perspective, such as the topics of algebraic invariants that had been studied by her father and elimination theory, discussed below.

First epoch (1908–1919)

Much of Noether's work in the first epoch of her career was associated with invariant theory, principally algebraic invariant theory. Invariant theory is concerned with expressions that remain constant (invariant) under a group of transformations. As an everyday example, if a rigid metre-stick is rotated, the coordinates of its endpoints change, but its length remains the same. A more sophisticated example of an invariant is the discriminant B2 − 4AC of a homogeneous quadratic polynomial Ax2 + Bxy + Cy2, where x and y are indeterminates. The discriminant is called "invariant" because it is not changed by linear substitutions x → ax + by and y → cx + dy with determinant ad − bc = 1. These substitutions form the special linear group SL2.

One can ask for all polynomials in A, B, and C that are unchanged by the action of SL2; these turn out to be the polynomials in the discriminant. More generally, one can ask for the invariants of homogeneous polynomials A0xry0 + ... + Arx0yr of higher degree, which will be certain polynomials in the coefficients A0, ..., Ar, and more generally still, one can ask the similar question for homogeneous polynomials in more than two variables.

One of the main goals of invariant theory was to solve the "finite basis problem". The sum or product of any two invariants is invariant, and the finite basis problem asked whether it was possible to get all the invariants by starting with a finite list of invariants, called generators, and then, adding or multiplying the generators together. For example, the discriminant gives a finite basis (with one element) for the invariants of a quadratic polynomial.

Noether's advisor, Paul Gordan, was known as the "king of invariant theory", and his chief contribution to mathematics was his 1870 solution of the finite basis problem for invariants of homogeneous polynomials in two variables. He proved this by giving a constructive method for finding all of the invariants and their generators, but was not able to carry out this constructive approach for invariants in three or more variables. In 1890, David Hilbert proved a similar statement for the invariants of homogeneous polynomials in any number of variables. Furthermore, his method worked, not only for the special linear group, but also for some of its subgroups such as the special orthogonal group.

Noether followed Gordan's lead, writing her doctoral dissertation and several other publications on invariant theory. She extended Gordan's results and also built upon Hilbert's research. Later, she would disparage this work, finding it of little interest and admitting to forgetting the details of it. Hermann Weyl wrote, [A] greater contrast is hardly imaginable than between her first paper, the dissertation, and her works of maturity; for the former is an extreme example of formal computations and the latter constitute an extreme and grandiose example of conceptual axiomatic thinking in mathematics.

Second epoch (1920–1926)

In this epoch, Noether became famous for her deft use of ascending (Teilerkettensatz) or descending (Vielfachenkettensatz) chain conditions. A sequence of non-empty subsets A1, A2, A3, ... of a set S is usually said to be ascending if each is a subset of the next:

A

1

⊆

A

2

⊆

A

3

⊆

⋯

.

{\displaystyle A_{1}\subseteq A_{2}\subseteq A_{3}\subseteq \cdots .}

Conversely, a sequence of subsets of S is called descending if each contains the next subset:

A

1

⊇

A

2

⊇

A

3

⊇

⋯

.

{\displaystyle A_{1}\supseteq A_{2}\supseteq A_{3}\supseteq \cdots .}

A chain becomes constant after a finite number of steps if there is an n such that

Third epoch (1927–1935)

Much work on hypercomplex numbers and group representations was carried out in the nineteenth and early twentieth centuries, but remained disparate. Noether united these earlier results and gave the first general representation theory of groups and algebras. This single work by Noether was said to have ushered in a new period in modern algebra and to have been of fundamental importance for its development.

Briefly, Noether subsumed the structure theory of associative algebras and the representation theory of groups into a single arithmetic theory of modules and ideals in rings satisfying ascending chain conditions.

Noether also was responsible for a number of other advances in the field of algebra. With Emil Artin, Richard Brauer, and Helmut Hasse, she founded the theory of central simple algebras.

A paper by Noether, Hasse, and Brauer pertains to division algebras, which are algebraic systems in which division is possible. They proved two important theorems: a local-global theorem stating that if a finite-dimensional central division algebra over a number field splits locally everywhere then it splits globally (so is trivial), and from this, deduced their Hauptsatz ("main theorem"):Every finite-dimensional central division algebra over an algebraic number field F splits over a cyclic cyclotomic extension.These theorems allow one to classify all finite-dimensional central division algebras over a given number field. A subsequent paper by Noether showed, as a special case of a more general theorem, that all maximal subfields of a division algebra D are splitting fields. This paper also contains the Skolem–Noether theorem, which states that any two embeddings of an extension of a field k into a finite-dimensional central simple algebra over k are conjugate. The Brauer–Noether theorem gives a characterization of the splitting fields of a central division algebra over a field.

Legacy

Noether's work continues to be relevant for the development of theoretical physics and mathematics, and she is considered one of the most important mathematicians of the 20th century. She has been called the greatest woman mathematician in recorded history by mathematicians such as Pavel Alexandrov, Hermann Weyl, and Jean Dieudonné.

In a letter to The New York Times, Albert Einstein wrote:

In the judgment of the most competent living mathematicians, Fräulein 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.

In his obituary, algebraist B. L. van der Waerden wrote that her mathematical originality was "absolute beyond comparison", and Weyl said that Noether "changed the face of [abstract] algebra by her work". Mathematician and historian Jeremy Gray wrote that all textbooks on abstract algebra bear the evidence of Noether's contributions: "Mathematicians simply do ring theory her way." Several things now bear her name, including many mathematical objects, and an asteroid, 7001 Noether. In 2019 Time created 89 new covers to celebrate women of the year starting from 1920; it chose Noether for 1921.