She was professionally harassed for being a woman, and then persecuted for her Judaism. In the meantime, Emmy Noether positioned herself as one of the most important mathematicians of the 20th century and developed some of the most important concepts in algebra and physics

"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". These were the words that the great physicist Albert Einstein used to describe mathematician Emmy Noether in response to her death in 1935. Einstein's eulogy was part of a long line of written tributes by mathematicians and physicists that underline Noether as the most important woman in the history of mathematics.

## Auditing student

Amalie Noether was born on March 23rd 1882 in Erlangen, Germany to a traditional Jewish family. Her father, Max Noether was a world-renowned professor of mathematics at the local university, and also the middle of her three younger brothers became a mathematician.

In school, Noether did not stand out as a particularly bright student. Her peers and teachers described her as a four-eyed, smart, likeable and very friendly girl. In high school she began going by her middle name, Emmy, which is how everyone called her. In high school she excelled in English and French and decided to become a language teacher. She also learned to play the piano and enjoyed dancing in parties.

In 1897 she graduated from high school and went on to pursue the path she chose for herself. In 1900 she graduated and received an overall score of "very good" in the examination for teachers of English and French of the state of Bavaria.

Noether may have graduated with excellence, but never taught one day in a classroom. At some point during her studies she decided to change direction and chose to study mathematics. However, in those days, women were not allowed to study in German universities. Nevertheless, they had the option to attend courses as auditors, namely, without receiving credit or a grade. Even for this option, each course they would want to attend required permission from the professor. This was not a problem for Noether, as many were her father's colleagues, and some even allowed her to eventually take the exam.

In the years to come, Noether took many courses at the University of Erlangen, in mathematics and in other fields, such as classical studies, and was one of the only two women to study at the university, among 984 men. In 1903, she successfully passed the entrance exam to the university, but was still not allowed to enroll officially. Following the exam, she took one semester at the University of Göttingen, which was a world-class center for excellence in mathematics and physics at the time. As an auditor, she took courses given by some of the most prominent scientists of the time, including physicist Karl Schwarzschild and mathematician David Hilbert.

## Mathematics without salary

When she completed her studies, she returned to Erlangen, and then her luck changed for the better: Germany changed its law and stated that women are allowed to study in the university. Noether seized the opportunity and went to study only mathematics this time. She received credit for her previous studies, and in 1907 she completed her PhD with highest honors, under the supervision of Paul Gordan.

Noether's dissertation was in the field of differential and algebraic invariants, which is a branch of algebra dealing with actions of groups or vector spaces. Hilbert has already identified the fundamentals of these invariants in the late 19th century, but Noether developed more efficient and practical methods to solve the problems that Hilbert solved only in theory. Though her dissertation was very well received, she described it as "crap" in later years.

After completing her PhD, the natural course of action was to pursue an academic position, but PhD or not, she was still a woman. Therefore, she could not receive a position like this in Erlangen. To be exact, there was a position at hand, but it was unpaid, uncredited and not formally recognized. Noether taught courses at the university, but was officially only a "teaching assistant". She even mentored graduate students, with her father registered as the official supervisor, but she did not receive pay for doing so. Nevertheless, she stayed in Erlangen, also to assist her father, who was in poor health.

Along with teaching and mentoring, Noether continued her research. She was very much influenced by Ernest Fischer, who replaced Gordan after his retirement. She said that he was the one who encouraged her to study abstract algebra from an arithmetic point of view, which shaped her entire future body of work. She continued to publish research papers, building a reputation for herself in the mathematics departments, and was invited to participate and speak at conferences throughout Europe.

A proof of great importance for physics. The title of the paper that contains the Noether's theorem, which ties conservation laws with symmetry in physical systems

## Relations and relativity

In 1915, Hilbert and the world-renowned mathematician Felix Klein offered Noether a position in Göttingen. Hilbert has then begun working on the mathematical aspects of Einstein's theory of relativity, and needed an expert on invariants. Noether could not pass on the opportunity to work with the best mathematicians in the world on problems that were at the cutting edge of science.

Almost as soon as she arrived in Göttingen, Noether solved two seminal problems that Hilbert had been working on. One dealt with Riemannian manifolds, which are used to mathematically describe multi-dimensional structures and were an important tool in Einstein's breakthrough theories. The second problem dealt with the law of energy conservation, as part of the general theory of relativity. Her solution deciphered the connection between symmetry in a physical system and the conservation laws that apply to it. Namely, Noether provided the mathematical explanation for conservation laws, such as conservation of energy, momentum or electric charge, which are pivotal both in theoretical physics and in its applications – from vehicle engineering to calculating the routes of celestial objects. This proof is now called Noether's theorem, and it's publication in 1918 cemented Noether's status as one of the most prominent mathematicians of her time.

Her success did not translate, however, to an improved financial or employment situation. Actually, it was quite the contrary. Exactly as it was in Erlangen, she was officially only Hilbert's teaching assistant, but in actuality she taught his courses without being paid for it. Noether lived with little means, and the fact that it was a time of war did not improve the situation. Poverty was what most likely drove her to enthusiastically embrace socialist views, but even though she expressed them often, she did not take part in any organized political activity. Noether was also a pacifist her whole life, and opposed war of any kind.

The efforts of Hilbert and Klein to arrange a position for her were not successful, and even their appeal to the German Ministry of Education was rejected. One of the university faculty members, who opposed hiring women, asked what would the soldiers returning from the war would think, when they find out they have to learn from women? Hilbert replied to this question by saying that he does not see what gender has to do with it – "After all, we are a university, not a bath house", he said.

Ironically, it was actually the war that Noether has opposed so much that facilitated the change in her status. Germany's defeat in the war led to many legislative changes, including the law that prevented women from becoming university faculty. In 1919, Noether was finally allowed to habilitate, still with no tenure or social benefits, but at least she could teach courses under her name and mentor graduate students.

Received great professional appreciation, but with no financial welfare. Noether at Göttingen | Source: Bar-Ilan University

## Rings and collaborations

In the following years in Göttingen, Noether would mainly work in the field of abstract algebra, and was one of its founders. She led new developments in the branch of "ring theory", which defines combinations of groups of numbers and operations. Noether accounts for some of the breakthroughs in the field. Among others, Noether found a generalized method of defining rings with certain properties that are now called Noetherian rings.

In 1924 the Dutch mathematician Bartel van der Waerden came to work with Noether, and later published his well-known textbook Moderne Algebra, a large section of which is based on her work. This story shows another side to Noether, who often collaborated with other mathematicians, and allowed them to receive credit for work she has done herself. She also did this many times with the students she was mentoring. Many colleagues, as they describe her impact on mathematics, describe not only her many published papers, but also her prolific collaborations and the inspiration and assistance she gave her to collaborators.

By the 1920's and early 1930's, Noether was considered one of the most important mathematicians in the world. She was invited to give speeches in important conferences throughout Europe, and was even invited to teach one semester at Moscow State University – where her socialist tendencies couldn't hurt. In 1932, she received the Ackermann–Teubner Award given by the University of Leipzig for considerable contribution to mathematics.

The end of the mathematics era in Göttingen. The letter ordering Noether's dismissal in September 1933 | Source: National Library of Israel

## The greatest

In 1933, Nazis rose to power in Germany, and all of Noether's mathematical achievements were of no importance to them, as she was Jewish. Within a short time, the Law for the Restoration of the Professional Civil Service, which enabled the dismissal of Jews from government positions, was passed. Soon enough, University of Göttingen received a letter ordering it to dismiss Noether. Expelling all Jewish mathematicians from Göttingen was detrimental to the university. In 1934, the Reich Minister of Culture met with Hilbert, who opposed expelling the Jews, and asked him how the department of mathematics was doing, now that it has been released from the burden of Jews. Hilbert simply replied that there is no mathematics anymore in Göttingen.

After being expelled from University of Göttingen, Noether crossed the Atlantic Ocean. Albert Einstein, who was already in the US and was quite familiar with her work, assisted her in receiving a faculty position at Bryn Mawr College, a prestigious women college in Pennsylvania. For the first time in her life, she received, aside from professional appreciation, also a nice salary and decent work conditions. In 1934 she also gave a workshop at the Institute for Advanced Study at Princeton University, which is one of the most important centers in the world for mathematics and theoretical physics.

However, Noether's new path was not a long one. In April 1935, after only one year and a half in the US, doctors found she had a pelvic tumor. She underwent surgery to have it removed, and in the first days she seemed to be recovering well, but a number of days later her fever shot up and she died on April 14th, shortly after her 53rd birthday.

The written tributes following Noether's death highlight her indisputable greatness. Van der Waerden wrote that her originality was "absolute beyond comparison", and mathematician Hermann Weyl added that Noether's work "changed the face of algebra". Another mathematician, Norbert Wiener wrote just a few weeks before her death that 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's body was cremated, and her ashes were buried under the walkway around the library at Bryn Mawr, which was her home following Nazi persecution. Many things were named in her honor, including a street at her birth town, schools and academic departments in many places, including the Department of Mathematics at Bar-Ilan University. In addition, a crater on the moon and an asteroid bear her name, in honor of her great contribution to physics.

Translated by Elee Shimshoni