|Emmy Noether, linocut, 11" x 14", Ele Willoughby, 2018|
The three examples I give are probably the best known, but just give a hint of the power of this theorem. If you do an experiment and then move three steps to the right and repeat it, you usually expect the same results. In general, a lot of things will have this translational symmetry. Noether's Theorem shows that if you get the same result in two reference frames which are shifted from one another, your system conserves momentum (p with an arrow, as a vector quantity). Thus, we have conservation of momentum in any inertial frame of reference. That means that any place where we don't have to worry about any significant differences from acceleration or gravity, we can solve physics problems by simply knowing that the total momentum never changes. In my print I show a set of x, y, z axes moved (translated) to get a new set of axes x', y' and z' and then the quantity p. Similarly, if your system doesn't care if you rotate it or how it's oriented in space, the conserved quantity is angular momentum (L with an arrow, as a vector quantity); hence in my print, I show a set of x, y, z axes rotated x', y' and z' along with conserved quantity L. Your system itself doesn't need to be symmetric. A lumpy asteroid conserves angular momentum every bit as much as a planetary system made of perfect spheres. If it's irrelevant to results whether you do your experiment at 3:00 or 6:25 then your system has a time symmetry and conserves energy (E). This method of using observed symmetries of something and then finding things which are invariant allows us to easily solve all sorts of problems in physics. Further, using observed symmetries of the Universe allows us to know which things are invariant, know more about the nature of reality and assess any new theories by checking whether they also produce the same conserved quantities.*
Here's a nice video which talks about Noether's Thereom.
Her male colleagues Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl, and Norbert Wiener described Noether as the most important woman in the history of mathematics - a compliment which betrays the biases of the times in comparing her only to those of the same sex. She was quite simply, one of the most important mathematicians period, and her impact on physics was tremendous. (My portrait betrays my own biases, focusing on the physics of Noether's Theorem, rather than her contributions to mathematics... but there you are. I'm a physicist by training, not a mathematician).
Born in Erlangen, Germany, Emmy Noether initially planned to teach girls English and French, rather than follow in her father's footsteps and become a professor of mathematics. But ultimately, she choose to study mathematics at the University of Erlangen, where he was a lecturer. Pursuing mathematics was unconventional for a woman; the university had recently declared that mixed-sex education would "overthrow all academic order" and as one of 2 female students (out of 986) she was only able to audit classes at the discretion of professors. She nonetheless managed to pass the graduation exam in 1903 and was granted a degree. She spent the winter semester at the University of Göttingen attending lectures from astronomer Karl Schwarzschild and mathematicians Hermann Minkowski, Otto Blumenthal, Felix Klein, and David Hilbert, before returning to Erlanger. She completed a dissertation supervised by Paul Gordan, On Complete Systems of Invariants for Ternary Biquadratic Forms (1907) using the "computational" approach to invariants, later superseded by Hilbert's more abstract and general approach. She later referred to this well-received thesis and the first few similar papers as "crap". She continued to work at the university for 7 years, but as a woman she was excluded from an academic position and in fact forced to worked without pay.
In 1915 she was recruited to come to the renown University of Göttingen and work with famed mathematicians David Hilbert and Felix Klein. However, some philologists and historians in the philosophical department protested that a woman must not become a Privatdozent, an additional post-doctoral rank required in Germany and certain other European nations to act as a university professor. Famously, a 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?" but Hilbert defended her indignantly, with one of my favourite lines in response to such entrenched academic sexism: "I do not see that the sex of the candidate is an argument against her admission as privatdozent. After all, we are a university, not a bath house." There she still faced hurdles and had to rely on her family to support her financially, as she was unpaid and could only lecture under Hilbert's name until 1919 despite already having published her eponymous Noether's Theorem in 1918! After Einstein published his theory of general relativity in 1915 and Noether responded by applying her invariance work to some of its complexities and this eventually lead her to prove her famous theorem. As Einstein wrote when he read her paper, "Yesterday I received from Miss Noether a very interesting paper on invariants. I'm impressed that such things can be understood in such a general way. The old guard at Göttingen should take some lessons from Miss Noether! She seems to know her stuff."
The end of WWI and German Revolution of 1918-1919 lead to social change and increased rights for women. Her habilitation was approved and she obtained the rank of Privatdozent in 1919. Three years later she was promoted to an untenured professor (nicht beamteter ausserordentlicher Professor) but her work remained unpaid until the next year when she was finally granted a special position (Lehrbeauftragte für Algebra).
Until 1919 she focused on theories of algebraic invariants and number fields. While her incredible contribution to physics had already occurred in 1918, mathematicians remember her for her central role in the 20th century revolution in mathematics, the development of abstract algebra, and her prolific work including Ring Theory from 1920 to 1926, as well as Noetherian rings, Noether groups, Noether equations, Noether modules and more. Her revolutionary 1921 paper Theory of Ideals in Ring Domains is considered a classic and objects which satisfy the ascending chain condition are named Noetherian, in her honour. In the final stage of her career, she focused on noncommutative algebras and hypercomplex numbers and united the representation theory of groups with the theory of modules and ideals. She was a leader in the strong University of Göttingen math department until 1933. Her colleague Dutch mathematician B. L. van der Waerden made her work the foundation of the second volume of his influential 1931 textbook, Moderne Algebra; it was typical of her to allow students and colleagues to receive credit for her ideas. She supervised more than a dozen doctoral students. She was known for her patient guidance but insistence on accuracy. van der Waerden wrote that she was, "Completely unegotistical and free of vanity, she never claimed anything for herself, but promoted the works of her students above all." She learned to live frugally, having gone so long without a salary, and took no concern about her manners, housework or appearance. She used her lecturers as a time for spontaneous discussions of the latest mathematics with students and a place to explore ideas (many of which would become major publications of those students). She spent the winter of 1928–29 at Moscow State University, working with P. S. Alexandrov. She was interested in and supportive of the Russian Revolution and her political opinions got her evicted from her lodging back in Germany when students there complained of living with "a Marxist-leaning Jewess". In 1932, she won the received the Ackermann–Teubner Memorial prize for her contributions to mathematics, which came with 500 Reichsmarks and she gave the plenary address at the 1932 International Congress of Mathematicians in Zürich, a sign of her international stature in the field. Colleagues complained that she was however never elected to the Göttingen Gesellschaft der Wissenschaften (academy of sciences) or promoted to full professor. Within a year Nazi Germany moved to dismiss her and all Jewish academics from university positions. The German Student Association, aided by one of Noether's own former students, a privatdozent named Werner Weber, led the attack on Jews at the University of Göttingen. She merely laughed when students showed up dressed as Hilter's brownshirts. Dedicated to her students, she invited them to her home to discuss math and their plans for the future. Herman Weyl wrote "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." Emmy Noether was able to find a position at Bryn Mawr College in Pennsylvania in 1933, where she finally gained the appreciation she deserved. In 1934 she lectured at the Institute for Advanced Study in Princeton, but remarked that she was not welcome at the "men's university, where nothing female is admitted." Tragically, she died 4 days after surgery to remove an ovarian cyst in 1935 when she was only 53.
Noether's theorem remains fundamental to physics, and has been especially vital to particle physics in the decades since her death. Her originality in mathematics was beyond compare and in Weyl's words she "changed the face of algebra by her work."
Emmy Noether, wikipedia article access June 6, 2018
Noether E (1918). "Invariante Variationsprobleme". Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys. Klasse. 1918: 235–257.
M. A. Tavel's English translation of Noether's Theorems (1918)
Matthew R. Francis, Mathematician to know: Emmy Noether, Symmetry Magazine, June 18, 2015.
Natalie Angier, The Mighty Mathematician You’ve Never Heard Of, The New York Times, March 26, 2012
*Now, if you're interested in the equation itself here's one good online explation (if say, you have most of an undergraduate degree in physics or more). A more intuitive a bit more straightforward explanation is here. The original paper is here and can be found in translation here.