Alles Gute zum Geburtstag, Emmy !
Today's Google-doodle commemorates Emmy Noether. Noether really is a significant
creative figure in the history of mathematics.
Unlike Ada Lovelace, she wasn't anyone's famulus, but forged her
own way independently. And,
despite the ferocious abstraction of her research, she was a nourishing and
accessible person to her circle of mathematicians. As one leading algebraist wrote in his memoirs:
Emmy Noether jouait le rôle de mère
poule protectrice.
-- André Weil, Souvenirs
d’apprentissage (1991)
~
As an undergraduate math major at Harvard, I had barely
heard of Noether; and this, for
three good reasons, none of which had to do with Noether herself.
(1) Undergrads generally need to learn the subject (i.e., what’s
on the test) and not worry about its history.
(2) The exact sciences generally
hew to “Whig history”: whatever
the present state of the field, is all you really need to know. The pioneers were blunderers.
(3) For math specifically, the
Platonic outlook singularly
minimizes the role of the individual researcher. All mathematical truths are already stored-up in
Platonic heaven; whoso proves this
theorem or that, has simply managed to claw loose a pre-existing nugget from
the hoard.
The closest I came to having heard of her, was that I’d
heard of (but not studied) “Noetherian
rings” (an adjective I still do not know how to pronounce). These figured prominently in the
introductory algebra text by Van der Waerden, one of her pupils.
Now:
Noetherian rings (along with the eponymous modules) are all very well in
their way; but they must take
their place in a menagerie of such constructions.
Thus: If you
stick with it long enough, you will learn such gems as this:
In a short exact sequence of
R-modules, A and C noetherian imply B noetherian, and conversely.
-- Saunders MacLane & Garrett Birkhoff, Algebra
(1967; 3rd edn. 1999), p. 380
Hmm!
Mmyess! Good to know! Must make a note of that!
But that is not why she’s famous.
~ ~ ~
Emmy Noether |
Separated at birth ??
Emma Goldman |
~ ~ ~
The discovery (or invention) of Noetherian rings, was internal to algebra. What inscribed her name on the scrolls of History in letters of gold, was the linking of algebra to something seemingly outside itself: specifically, the linking of each algebraic symmetry with a conservation law -- which puts her work at the center of modern physics. In this it bears comparison with the previous Erlangen Program led by Felix Klein, which characterized the plethora of newly recognized geometries (prior to Gauss & Lobachevsky, there was but one) by their symmetry-groups.
Technically stated:
Noether’s principle:
To any continuous one-parameter group of symmetries of the
Lagrangian, there corresponds a
conservation law for the associated Euler-Lagrange PDE.
-- Timothy Gowers, ed., The Princeton Companion to
Mathematics (2008), p. 479
or (seen from the standpoint of QFT):
Noether’s theorem: If
a Lagrangian has a continuous symmetry, then there exists a current associated
with that symmetry that is
conserved when the equations of motion are satisfied.
-- Matthew Schwartz, Quantum
Field Theory and the Standard Model (2014), p. 34
Noether’s results antedate post-quantum particle physics,
but their reach remains. As, “If there
is a gauge invariance, expect to find a conserved charge,” (Roger
Penrose). Penrose however goes on
to note, that the principle is not unlimited: it does not apply to derive the conservation of the energy-momentum
in General Relativity (The Road to Reality (2004), p. 489-90).
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