The Mighty Mathematician You’ve Never Heard Of

If anything, the article probably understates her importance (and doesn't really explain why she is so important). You do get an idea when you start at her bibliography; not many scientists have their papers subdivided into epochs [1].

To begin with, Einstein didn't describe Emmy Noether (in his obituary of her, sent to the New York Times) as "the most 'significant and 'creative' female mathematician of all time". The exact quote is instead: "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." [2]

In essence, he is describing her as the greatest mathematician of the past decades, male or female. That may have been a slight exaggeration (she was a friend of his and visited frequently at Princeton during her time at Bryn Mawr), but not much: she was definitely one of the most influential mathematicians of the 1920s and early 1930s. Many mathematicians and physicists visited Göttingen specifically to interact with her.

What made her so important were not just her contributions to abstract algebra (groundbreaking as they were, essentially redefining the field), but probably most of all her axiomatic "method of abstraction", pioneered in several papers starting with 1921's "Idealtheorie in Ringbereichen". For non-mathematicians, it is roughly the same concept as abstract data types; instead of proving theorems about very specific mathematical constructs (such as integers and polynomials), you prove them for more general constructs, abstractly defined through operations over them and axioms connecting those operations. It is one of those things that seem blindingly obvious in retrospect, but the approach was actually hotly debated in the 1920s; these days, it's foundational for much of mathematics.

A good example of this is the so-called Lasker-Noether theorem, which was initially proven by Emanuel Lasker for a couple of specific rings, a result that Emmy Noether later generalized for a wide class of rings (rings that we now call, for other reasons, Noetherian rings).

Another reason why she is little known these days even among mathematicians is probably that she didn't care much about getting credit for her ideas (even though she was painstakingly careful about giving others credit in her own papers). Her own publications, numerous as they were [1], are probably only the tip of the iceberg; it is known, for example, that volume 2 of van der Waerden's "Modern Algebra" (the standard textbook for abstract algebra for decades) was essentially her work (van der Waerden never made a secret of it). A lot of her other work was likewise published under the name of her students and colleagues. (Unlike what the article states, she never published under a man's name.)

This is the final way in which she influenced mathematics -- as a teacher. She never was a very good teacher when it came to actual lecturing (being too much of an absent-minded professor), but she was generous with sharing her ideas and an inspiring influence for her students on colleagues (her first Ph.D. student, Grete Hermann, went on to lay the foundations for Computer Algebra, for example).

As van der Waerden wrote, "Each of her lecture series was a paper. And nobody was happier than herself when such a paper was completed by her students. Completely unegotistical and free of vanity, she never claimed anything for herself, but promoted the works of her students above all."

[1] http://en.wikipedia.org/wiki/Emmy_Noether_bibliography

[2] http://www-history.mcs.st-andrews.ac.uk/Obits2/Noether_Emmy_...

The simplest way I've seen to explain Noether's theorem comes from Feynman's Messenger Lectures.

Suppose -- as we have in modern physics -- that the laws of physics can be phrased in a certain way. Namely:

(1) There is some number we can calculate for all of the possible paths that an object could take from point A to point B. We call this number the "action" of the path.

(2) Of all of the paths that an object can take, the one which it does has a stationary action -- meaning that if you wiggle the path a little bit, the action remains the same. This can happen for example if the action is at a minimum for that path -- if you're at a minimum then you can't go any lower, and so there cannot be a proportionate response to any slight change in path.

This work was already done before Noether -- the observation, I mean, that there is a definition of "action" for all known laws in physics such that you can phrase those laws as a stationary-action principle. An example is light. All of these complicated rules of lenses and light bending when it enters water can be phrased as the much simpler rule that "light always takes the least-time path." The action of the path is then the time it takes for light to travel along that path from A to B.

Now, suppose the action has a continuous symmetry. For example, it might be the case that if A and B were moved a centimeter to the right, to two new points A' and B', then the actions on the corresponding paths between them would remain the same.

Now construct a new path: We go from A to B like so: we go from A to A', follow the physical path to B', then go to B, rather than following the physical path from A to B.

Call these two paths:

(1) A → B

(2) A : A' → B' : B

You have two competing facts. First off:

(1) the symmetry: action(A' → B') = action(A → B)

(2) stationary action: action(A : A' → B' : B) = action(A → B).

In other words, as long as the little wiggles described by ":" are very small, the stationary action principle holds and we don't change the action when we move from a physical path.

It must then be the case that action(A : A') = - action(B : B') in some sense. In other words, there is some quantity local to both A and B which is exactly the same -- it is the susceptibility of the action to changes if you shifted in one direction or another. The big symmetry also forces some little number to be constant while this thing is taking whatever path it takes.

Whenever you have a stationary-action principle -- which we have for gravity and electromagnetism -- then Noether's principle holds and symmetries in the action become conserved quantities of the physics.

We can also calculate some of these susceptibilities for the most common action principles: we find that symmetry with respect to space-translation (moving 1cm to the side) conserves momentum in that direction. We find that symmetry with respect to rotations about a point conserves angular momentum about that point. We find that if the action remains the same from from one second to another, then the energy is conserved -- literally if someone tells you they've got a "free energy device," they're saying "the laws of physics that this thing uses won't be the same tomorrow as they are today."

But it can get even more interesting. For example, wavefunctions have a phase, but shifting the phase does not change any observables. This shift turns out to have relativistic consequences, and can be associated with conservation of charge. (That's as I understand it, but I only saw a half-sketch of the proof and I haven't worked it out myself, as I have for these other examples.)

Her contributions are core foundations of a theory of commutative rings -- one of the most important properties we usually want from a ring we study is for it to be Noetherian, which means that it satisfies certain finiteness condition, which lets one apply finite and computational methods to study it. Many important theorems in abstract algebra state properties of Noetherian rings, for instance famous Hilbert' basis theorem states that the ring of polynomials over Noetherian rings is Noetherian itself. Other important theorem, Hilbert's Nullstellensatz is frequently proved using a tool called Noether normalisation lemma. Because of that, Noetherian rings, algebras and modules are crucial to many subfields of algebra and algebraic geometry.

One of her other insights, which is especially important in the field I'm interested in, was to notice that homology forms a group -- this led to explosion of new results and marked the beginning of one of the most important and influential field in XX century math, algebraic topology, as we see it today.

The title "mighty mathematician you've never heard of" is odd. Aren't most "mighty mathematicians" people we've never heard of?

It isn't surprising people in the mainstream have never heard of her. Virtually everything she did was abstract algebra and the only connection her work had to real life was studying some invariants of relativity (read: very high level, very specialized). Similarly, how many people have heard of Sierpinski, Voronoy or even Hilbert?

While important, the sort of work she did doesn't lead you to fame. As an applied mathematician I never directly used any of her theorems. The word "ring" (let alone "noetherian") doesn't even appear in my thesis.

There is only one famous mathematician in the world. His name is "that crazy guy played by Russel Crowe".

Interesting: "being female in Germany at a time when most German universities didn’t accept female students or hire female professors"

I'm currently reading "The Poincare Conjecture", which suggested that Erlangen and Gottingen were very open to women, but reading the Wikipedia article it's clear that the quote is correct and the book (or my memory of it) is incorrect; she was one of only 2 women in a class of 986 at Erlangen, and she was actively blocked entrance to Gottingen. It sounds like if it wasn't for Gordan, Klein and Hilbert she might not have been allowed to attend at all.

the interesting thing is that i knew who noether was because of her theorem, but i got all the way through grad school without knowing she was a woman

There's an Emmy Noether Programme from the German Research Foundation [0] promoting young researchers:

The Emmy Noether Programme supports young researchers in achieving independence at an early stage of their scientific careers. Young postdocs gain the qualifications required for a university teaching career during a DFG-funded period, usually lasting five years, in which they lead their own independent junior research group.

[0] http://www.dfg.de/en/research_funding/programmes/individual/...

I will blow my own horn, here, and say that every one of my students has heard of Emmy Noether, and a bit about her story.

I made it a point- following something like Eugene Hecht's lead in his (brilliant) Physics Algebra/Trig text.

Noether is quite famous. My university in New Mexico has a giant picture of her in the "math TA room" ( this is the room where I spent my years as a grad student solving the homeworks of undergraduates ..I mean, helping them solve it :) For the non-math guys, the simplest example of a Noetherian is a Galois Field GFpn - take p prime, n integer, then GFpn is a Galois Field with elements from 0 to pow(p,n)-1. Example let p = 2, n = 2, you get the Galois Field with elements {0,1,2,3}, with the Cayley addition table given by the usual

       def add(a,b) = (a+b)%4

and the Cayley multiplication table given by the regular multiplication except for

         2 times 2 is 3
         2 times 3 is 1
         3 times 3 is 2
         3 times 2 is 2 times 3

That's a trivial Noetherian.

"Prime Obsession" and "Unknown Quantity", both by John Derbyshire, contain significant sections on her (more so in "Unknown Quantity", as it is a popular history of algebra). Both are excellent books, though I'd have to give "Prime Obsession" the edge.

The shadow-sister of Paul Erdos I guess