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Emmy Noether revolutionized mathematics — and still faced sexism all her life

(The hunt for the Higgs boson can be traced back to Noether's insight on symmetries. https://cdn1.vox-cdn.com/thumbor/rmcM8i9waOXBFdweDzHgY9AHjDs=/800x0/filters:no_upscale()/cdn0.vox-cdn.com/uploads/chorus_asset/file/3529274/147756618.0.jpg)

Emmy Noether revolutionized mathematics — and still faced sexism all her life

Painting of Emmy Noether by Jennifer Mondfrans from her series, "At Least I Have You, To Remember Me" (Maia Weinstock/Flickr)

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.

"Despite her brilliance, universities didn't want to hire a woman"

Even so, 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. Even after she proved herself a first-rate mathematician, male faculties were 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. So it’s welcome news that Google is honoring Noether today with a Google Doodle on her 133rd birthday. To celebrate, 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

(<a href="http://en.wikipedia.org/wiki/Noether%27s_theorem#/media/File:Noether.jpg">Wikimedia Commons</a>

. . . .

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.
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The hunt for the Higgs boson can be traced back to Noether's insight on symmetries. (Fabrice Coffrini/AFP/Getty Images)

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.

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Born: Erlangen, Germany, March 23, 1882
Died: Bryn Mawr, Pennsylvania, April 14, 1935
Creative Mathematical Genius

. . . .

Noether worked at the Mathematical Institute of Erlangen, without pay or title, from 1908 to 1915. It was during this time that she collaborated with the algebraist Ernst Otto Fischer and started work on the more general, theoretical algebra for which she would later be recognized. She also worked with the prominent mathematicians Hermann Minkowski, Felix Klein, and David Hilbert, whom she had met at Göttingen. In 1915 she joined the Mathematical Institute in Göttingen and started working with Klein and Hilbert on Einstein's general relativity theory. In 1918 she proved two theorems that were basic for both general relativity and elementary particle physics. One is still known as "Noether's Theorem."

But she still could not join the faculty at Göttingen University because of her gender. Noether was only allowed to lecture under Hilbert's name, as his assistant. Hilbert and Albert Einstein interceded for her, and in 1919 she obtained her permission to lecture, although still without a salary. In 1922 she became an "associate professor without tenure" and began to receive a small salary. Her status did not change while she remained at Göttingen, owing not only to prejudices against women, but also because she was a Jew, a Social Democrat, and a pacifist.*

During the 1920s Noether did foundational work on abstract algebra, working in group theory, ring theory, group representations, and number theory. Her mathematics would be very useful for physicists and crystallographers, but it was controversial then. There was debate whether mathematics should be conceptual and abstract (intuitionist) or more physically based and applied (constructionist). Noether's conceptual approach to algebra led to a body of principles unifying algebra, geometry, linear algebra, topology, and logic.

In 1928-29 she was a visiting professor at the University of Moscow. In 1930, she taught at Frankfurt. The International Mathematical Congress in Zurich asked her to give a plenary lecture in 1932, and in the same year she was awarded the prestigious Ackermann-Teubner Memorial Prize in mathematics.

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. . . .

The Mighty Mathematician You’ve Never Heard Of
.. . .
In 1915 Einstein published his general theory of relativity. The Göttingen math department fell “head over ear” with it, in the words of one observer, and Noether began applying her invariance work to some of the complexities of the theory. That exercise eventually inspired her to formulate what is now called Noether’s theorem, an expression of the deep tie between the underlying geometry of the universe and the behavior of the mass and energy that call the universe home.

What the revolutionary theorem says, in cartoon essence, is the following: Wherever you find some sort of symmetry in nature, some predictability or homogeneity of parts, you’ll find lurking in the background a corresponding conservation — of momentum, electric charge, energy or the like. If a bicycle wheel is radially symmetric, if you can spin it on its axis and it still looks the same in all directions, well, then, that symmetric translation must yield a corresponding conservation. By applying the principles and calculations embodied in Noether’s theorem, you’ll see that it is angular momentum, the Newtonian impulse that keeps bicyclists upright and on the move.

Some of the relationships to pop out of the theorem are startling, the most profound one linking time and energy. Noether’s theorem shows that a symmetry of time — like the fact that whether you throw a ball in the air tomorrow or make the same toss next week will have no effect on the ball’s trajectory — is directly related to the conservation of energy, our old homily that energy can be neither created nor destroyed but merely changes form.

The connections that Noether forged are “critical” to modern physics, said Lisa Randall, a professor of theoretical particle physics and cosmology at Harvard. “Energy, momentum and other quantities we take for granted gain meaning and even greater value when we understand how these quantities follow from symmetry in time and space.”

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