Monday, March 23, 2015

Emmy Noether zum Geburtstag

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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