When Albert Einstein wrote an obituary for Emmy Noether in 1935, he described her as a "creative mathematical genius" who - despite "selfless, important work over a period of many years" - did not receive the recognition she deserved.
Noether made groundbreaking contributions to mathematics at a time when women were excluded from academia and when Jewish people like her were persecuted in Nazi Germany, where she lived.
2021 marked the hundredth anniversary of Noether's seminal paper on ring theory, a branch of theoretical mathematics that continues to fascinate and challenge mathematicians like me today.
I remember the first time I heard about Noether and the surprise I felt when my professor called the brilliant ring theorist "she." Even though I am a woman who does math, I had assumed Noether would be a man. I was surprised at how touched I was when I learned that she was also a woman.
Her inspiring story is one that not many people know.
A rare woman in mathematics
Noether was born in 1882 in Erlangen, Germany. Her father was a mathematics professor, but it must have seemed unlikely to the young Noether that she would follow in his footsteps. At the time, few women took classes at German universities, and if they did, they could only audition them. Teaching at a university was out of the question.
But in 1903 - a few years after Noether graduated from an all-girls high school - Erlangen University began allowing women to enroll. Noether applied and eventually received her doctorate in mathematics there.
That PhD should have meant the end of her mathematical career. In Germany at the time, women were still not allowed to teach at universities. But Noether stuck with mathematics anyway, staying in Erlangen and unofficially supervising PhD students without pay. In 1915 she applied for a position at the prestigious University of Göttingen. The dean of the university, also a mathematician, was in favor of hiring Noether, although his argument was far from feminist.
"I think the female brain is unsuitable for mathematical production," he wrote, but Noether stood out as "one of the rare exceptions."
Unfortunately for Noether, the Prussian Ministry of Education would not allow the university to have a woman on their faculty, no matter how talented. Noether nevertheless remained in Göttingen and taught courses under the name of a male faculty member.
During those years she continued to conduct research. While she was still an unofficial teacher, Noether made important contributions to theoretical physics and Einstein's theory of relativity. The university finally awarded her lecturer status in 1919 - four years after she enrolled.
A revolution in ring theory
In 1921, just two years after becoming an official lecturer, Noether published revolutionary discoveries in ring theory that mathematicians continue to ponder and build on today. Noether's work on ring theory is the main reason I, like many contemporary mathematicians, know her name.
Ring theory is the study of mathematical objects called rings. Despite the name, these rings have nothing to do with circles or ring-shaped objects - theoretical or otherwise. In mathematics, a ring is a collection of items that you can add, subtract, and multiply, so you always get another object in the set.
A classic example is the ring known as Z. It is made of all integers - positive and negative integers such as 0, 1, 2, 3, -1, -2, -3 and so on - and it is a ring because if you add, subtract, or multiply two integers, you always get another integer.
There are infinite rings, and each ring is different. A ring can consist of numbers, functions, matrices, polynomials or other abstract objects - as long as there is a way to add, subtract and multiply them.
One reason why rings are so interesting to mathematicians is that it is often possible to say that something is a ring, but it is difficult to know much about the specifics of that particular ring. It's like seeing a croissant in a fancy bakery. You know you're looking at a croissant, but you might not know if it's filled with almond paste, chocolate, or something else.
Rather than focusing on one ring at a time, Noether showed that an entire class of easily identified rings all share a common internal structure, like a row of houses with the same floor plan. These rings are now called Noetherian rings, and the structure they share is like a map that guides the mathematicians who study them.
Noetherian rings appear constantly in modern mathematics. Mathematicians still use Noether's map today, not only in ring theory, but also in other areas such as number theory and algebraic geometry.
Escape from Nazi Germany
Noether published her famous paper on ring theory and other important results in mathematics while she was a teacher in Göttingen from 1919 to 1933. But in the spring of 1933, the University of Göttingen received a telegram: six faculty members - including Noether - had to quit teaching immediately. The Nazis had passed a law banning Jews from professorships.
It seems that Noether's response was calm. "This thing is far less terrible for me than for many others," she wrote in a letter to a fellow mathematician. But she no longer had a job and no university in Germany could hire her.
Help came from the United States. Bryn Mawr, a women's college in Pennsylvania, offered Noether a professorship through a special fund for refugee German scholars. Noether accepted the offer, and as a professor at Bryn Mawr, she mentored four younger women-one doctoral student and three postdoctoral fellows-in advanced mathematics.
Noether's time at Bryn Mawr was, tragically, short-lived. In 1935 she had surgery to remove a tumor and died unexpectedly four days later.
At Noether's funeral, mathematician Hermann Weyl compared her sudden death to 'the echo of a clap of thunder'. In her short life, Noether shook up mathematics. She continued to teach and learn even when women and Jews were not welcome. A hundred years later, her mathematical genius and "unbreakable optimism" are qualities to be admired.
This article is republished from The Conversation, an independent nonprofit organization providing facts and trusted analysis to help you understand our complex world. It was written by: Tamar Lichter Blanks, Rutgers University Read more: Tamar Lichter Blanks received funding from the National Science Foundation through the Graduate Research Fellowship Program. She is an American Association for the Advancement of Science (AAAS) Mass Media Science & Engineering Fellow at The Conversation US, sponsored by the American Mathematical Society.
