We’ve just reached the halfway point of the Dictionary of Mathematical Eponymy project, and it’s time for a fairly famous one (and again, one I’ve been meaning to understand better).

What is Noether’s Theorem?

Emmy Noether has several theorems named for her, but the first (and probably most important) can be loosely stated as:

If a system has some sort of symmetry, it has some corresponding quantity that’s conserved.

For example, if I spin a bike wheel around its axis, it doesn’t matter (in principle) whether I’m pointing the bike north, east or sideways - the spinning doesn’t depend on the orientation. This rotational symmetry corresponds to the conservation of angular momentum.

If my bike behaves the same today as it did yesterday, there’s a time symmetry - this corresponds to the conservation of energy.

If it behaces the same in Weymouth as it does in Bournemouth, there’s a positional symmetry - that corresponds to the conservation of linear momentum.

Why is it important?

Noether’s theorem allows the inference of a system’s invariants - the conserved quantities - from observations of how the system behaves, and vice versa. It narrows down the possible functions (Lagrangians) that can describe the system.

Noether’s Theorem pulled together several differing approaches to finding invariants, and opened the door to extensions into the quantum arena.

The idea of the Noether charge, the conserved quantity, is also used in the study of the entropy of black holes.

Who was Emmy Noether?

Amalie Emmy Noether (1882-1935), one of the world’s finest mathematicians, suffered the double-whammy of being a Jewish woman in early 20th century Germany.

As a woman, she was not originally allowed to teach officially at the University of Gottingen (she and David Hilbert conspired to get around this by her teaching under his name); she eventually became a Privatdozent in 1919, and continued her research at Gottingen until 1933, when - as a Jew - she was dismissed from her university post. Noether escaped Germany to take up a position at Bryn Mawr in Pennsylvania, where she died two years later.

As well as her remarkable theorems, she developed an amazing array of mathematical objects (fields, rings and algebras, among others), revolutionising modern algebra - helping to move it from the computational to the abstract. (Notably, she spent her undergraduate studies laboriously computing invariants; she would later call her thesis and related papers “crap”.)

She was a stellar mathematician, and it is shameful that the prejudices of her day held her back. Even with the disadvantages forced upon her, she stands as one of the greatest-ever mathematicians.

* See Katie Mack’s article on Emmy Noether for a much better ending. The content is excellent, too.