Emmy Noether was one of the most brilliant and important mathematicians of the 20th century. She altered the course of modern physics. Einstein called her a genius. Yet today, almost nobody knows who she is.

In 1915, Noether uncovered one of science's most extraordinary ideas, proving that every symmetry found in nature has a corresponding law of conservation. So, for example, the fact that physical laws work the same today as they did yesterday turns out to be related to the notion that energy can neither be created nor destroyed. Noether's theorem is a deep insight that underpins much of modern-day physics and things like the search for the Higgs boson.

And yet, as one of the very few female mathematicians working in Germany in her day, Noether faced rampant sexism. As a young woman, she wasn't allowed to formally attend university. Long after she proved herself a first-rate mathematician, male faculties were still reluctant to hire her. If that wasn't enough, in 1933, the Nazis ousted her for being Jewish. Even today, she remains all too obscure.

That should change. And what better time to celebrate her work than on her birthday? (In 2015, Google ushered in March 23 with an Emmy Noether Google Doodle.) So here's an introduction to the life and work of a woman Albert Einstein once called "the most significant creative mathematical genius thus far produced."

Noether was brilliant — yet universities wouldn't hire her

Amalie Emmy Noether was born in 1882 in Erlangen, Germany, to a family of mathematicians. Her father, Max Noether, was a professor at the University of Erlangen. Her brother Fritz later proved worthy in the field of applied math.

Despite this fertile background, it wasn't obvious that Emmy could become a mathematician, too. German universities rarely accepted female students at the time. She had to beg the faculty at Erlangen to let her audit math courses. It was only after she dominated her exams that the school relented, giving her a degree and letting her pursue graduate studies.

Her early work focused on invariants in algebra, looking at which aspects of mathematical functions stay unchanged if you apply certain transformations to them. (To give a very basic example of an invariant, the ratio of a circle's circumference to its diameter is always the same — it's always pi — no matter how big or small you make the circle.) Noether studied invariants for polynomial functions and made some impressive advances.*

"What will our soldiers think... to learn at the feet of a woman?"

Her work got noticed, and in 1915, the renowned mathematician David Hilbert lobbied for the University of Göttingen to hire her. But other male faculty members blocked the move, with one arguing: "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?" So Hilbert had to take Noether on as a guest lecturer for four years. She wasn't paid, and her lectures were often billed under Hilbert's name. She didn't get a full-time position until 1919.

That didn't stop Noether from doing trailblazing work in a number of areas, especially abstract algebra. Rather than focusing on real numbers and polynomials — the algebraic equations we learn in high school — Noether was interested in abstract structures, like rings or groups, that obey certain rules. Abstract algebra was one of the big mathematical innovations of the 20th century, and Noether was hugely influential in shaping it.

But perhaps her most consequential work came in another field: physics. In 1915, Einstein published his general theory of relativity, showing that gravity was a property of space and time, and the University of Göttingen was all abuzz with the the discovery. Hilbert asked Noether to apply her work on algebraic invariants to the equations in Einstein's theory.

In the process, Noether made a startling discovery of her own.

Noether’s theorem: How symmetry explains the world

To put it very simply, what Noether's theorems show is that anytime there’s a continuous symmetry in a physical system, there’s a related law of conservation.**

Here's an example: Let's say we conduct a scientific experiment today. If we then conduct the exact same experiment tomorrow, we'd expect the laws of physics to behave in exactly the same way. This is "time symmetry." Noether showed that if a system has time symmetry, then energy can't be created or destroyed in that system — we get the law of conservation of energy.

Noether had linked together concepts as different as energy and time

Likewise, if we do an experiment, and then do the exact same experiment again 20 miles to the east, that shouldn't make any difference — the laws of physics should work the exact same way in both places. This is known as "translation symmetry." Noether showed that translation symmetry leads to the law of conservation of momentum.

Finally, if we put our experiment on a table and rotate the table 90 degrees, that shouldn't affect the laws of physics, either. This is known as "rotational symmetry." But if rotational symmetry holds in a system, then angular momentum is always conserved. (That is, if you have a spinning bicycle wheel, it should spin in the same direction forever unless friction slows it down.)

This was a stunning revelation. Noether had linked concepts as different as time and energy. What's more, she had showed there was a deep connection between certain abstract algebraic structures — those that deal with symmetry — and physics. As David Goldberg details in his book The Universe in the Rearview Mirror, physicists soon began hunting for yet more symmetries.

In 1954, Chen Ning Yang and Robert Mills showed that other types of symmetries could describe the behavior of a vast array of particles and forces. In 1962, physicist Murray Gell-Mann was able to predict the existence of a new particle after simply studying symmetries written on a blackboard. (That particle was later confirmed by a particle accelerator.) In 1964, Peter Higgs (among others) used symmetries to predict the existence of the Higgs boson — a particle that was found in 2012 by the Large Hadron Collider.

The idea that purely mathematical structures could help find new particles in the physical world is astonishing, when you think about it. And it traces back to a discovery Emmy Noether made in 1915.

Noether fled Germany after the Nazis came to power

Noether continued doing vital mathematical work in abstract algebra and topology all through the 1920s and 1930s. But her career at at Göttingen was cut short when the Nazis came to power in 1932.

As a Jewish academic — and a woman at that — Noether didn't stand much of a chance in Nazi Germany. She was fired from her post, and in 1933 she fled to the United States to teach at Bryn Mawr College. Unfortunately, her life was cut short. Less than two years later, she died at the age of 53, following surgery for an ovarian cyst.

Shortly after Noether's death, in 1935, Albert Einstein wrote a beautiful letter to the New York Times praising her genius and recalling fondly her time at Bryn Mawr:

In the judgment of the most competent living mathematicians, Fraeulein Noether was the most significant creative mathematical genius thus far produced since 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. Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulae are discovered necessary for the deeper penetration into the laws of nature. ... Her unselfish, significant work over a period of many years was rewarded by he new rulers of Germany with a dismissal, which cost her the means of maintaining her simple life and the opportunity to carry on her mathematical studies. Farsighted friends of science in this country were fortunately able to make such arrangements at Bryn Mawr College and at Princeton that she found in America up to the day of her death not only colleagues who esteemed her friendship but grateful pupils who enthusiasm made her last years the happiest and perhaps the most fruitful of her entire career.

Today, Emmy Noether remains relatively unknown outside of math circles. In 2012, physicist David Goldberg told the New York Times that most of his colleagues and students had never heard of her: "Surprisingly few could say exactly who she was or why she was important."

It's about time we fixed that.

Footnotes

* In her 1907 dissertation, for instance, Noether studied degree-four polynomials with three variables. She found that these polynomials had 331 independent invariants, and all other invariants depended on these. This was a mind-numbing feat of calculation — she later described it as "a jungle of formulas." She soon moved on to bigger, conceptual insights.

** A more precise statement of Noether's theorem might go something like: "If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time." Many physicists would put it like this: "Whenever a system exhibits a continuous symmetry, there is an associated conserved charge."

Further reading

— In 2012, Natalie Angier wrote a beautiful profile of Noether for The New York Times. She's got some great additional biographical details.

— This paper by UCLA's Nina Byers offers an excellent history of Noether's conservation theorems and their importance to physics.

— This post at the blog Gravity and Levity offers a wonderful illustration of how Noether's theorem is useful in everyday physics.

— Back in 2014, Evelyn Lamb created a fascinating list of other unjustly forgotten women in mathematics that's very much worth checking out.