Emmy Noether: Mathematician Trailblaer (Part VII)

First epoch (1908–1919): Physics

Noether was brought to Göttingen in 1915 by David Hilbert and Felix Klein, who wanted her expertise in invariant theory to help them in understanding general relativity, a geometrical theory of gravitation developed mainly by Albert Einstein. Hilbert had observed that the conservation of energy seemed to be violated in general relativity, because gravitational energy could itself gravitate. Noether provided the resolution of this paradox, and a fundamental tool of modern theoretical physics, with Noether's first theorem, which she proved in 1915, but did not publish until 1918.  She not only solved the problem for general relativity, but also determined the conserved quantities for every system of physical laws that possesses some continuous symmetry. Upon receiving her work, Einstein wrote to Hilbert:

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.

For illustration, if a physical system behaves the same, regardless of how it is oriented in space, the physical laws that govern it are rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved.  The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. Rather, the symmetry of the physical laws governing the system is responsible for the conservation law. As another example, if a physical experiment has the same outcome at any place and at any time, then its laws are symmetric under continuous translations in space and time; by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.

Noether's theorem has become a fundamental tool of modern theoretical physics, both because of the insight it gives into conservation laws, and also, as a practical calculation tool.  Her theorem allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it facilitates the description of a physical system based on classes of hypothetical physical laws. For illustration, suppose that a new physical phenomenon is discovered. Noether's theorem provides a test for theoretical models of the phenomenon:

If the theory has a continuous symmetry, then Noether's theorem guarantees that the theory has a conserved quantity, and for the theory to be correct, this conservation must be observable in experiments.

Second epoch (1920–1926): Ascending and descending chain conditions

In this epoch, Noether became famous for her deft use of ascending (Teilerkettensatz) or descending (Vielfachenkettensatz) chain conditions. A sequence of non-empty subsets A1, A2, A3, etc. of a set S is usually said to be ascending, if each is a subset of the next:

A 1 ⊂ A 2 ⊂ A 3 ⊂ ⋯ . {\displaystyle A_{1}\subset A_{2}\subset A_{3}\subset \cdots .} A_{1} \subset A_{2} \subset A_{3} \subset \cdots.

Conversely, a sequence of subsets of S is called descending if each contains the next subset:

A 1 ⊃ A 2 ⊃ A 3 ⊃ ⋯ . {\displaystyle A_{1}\supset A_{2}\supset A_{3}\supset \cdots .} A_{1} \supset A_{2} \supset A_{3} \supset \cdots.

A chain becomes constant after a finite number of steps if there is an n such that A n = A m {\displaystyle A_{n}=A_{m}} {\displaystyle A_{n}=A_{m}} for all m ≥ n. A collection of subsets of a given set satisfies the ascending chain condition if any ascending sequence becomes constant after a finite number of steps. It satisfies the descending chain condition if any descending sequence becomes constant after a finite number of steps.

Ascending and descending chain conditions are general, meaning that they can be applied to many types of mathematical objects—and, on the surface, they might not seem very powerful. Noether showed how to exploit such conditions, however, to maximum advantage.

For example: How to use chain conditions to show that every set of sub-objects has a maximal/minimal element or that a complex object can be generated by a smaller number of elements. These conclusions often are crucial steps in a proof.

Many types of objects in abstract algebra can satisfy chain conditions, and usually if they satisfy an ascending chain condition, they are called Noetherian in her honor. By definition, a Noetherian ring satisfies an ascending chain condition on its left and right ideals, whereas a Noetherian group is defined as a group in which every strictly ascending chain of subgroups is finite. A Noetherian module is a module in which every strictly ascending chain of submodules becomes constant after a finite number of steps. A Noetherian space is a topological space in which every strictly ascending chain of open subspaces becomes constant after a finite number of steps; this definition makes the spectrum of a Noetherian ring a Noetherian topological space.

The chain condition often is "inherited" by sub-objects. For example, all subspaces of a Noetherian space, are Noetherian themselves; all subgroups and quotient groups of a Noetherian group are likewise, Noetherian; and, mutatis mutandis, the same holds for submodules and quotient modules of a Noetherian module. All quotient rings of a Noetherian ring are Noetherian, but that does not necessarily hold for its subrings. The chain condition also may be inherited by combinations or extensions of a Noetherian object. For example, finite direct sums of Noetherian rings are Noetherian, as is the ring of formal power series over a Noetherian ring.

Another application of such chain conditions is in Noetherian induction—also known as well-founded induction—which is a generalization of mathematical induction. It frequently is used to reduce general statements about collections of objects to statements about specific objects in that collection. Suppose that S is a partially ordered set. One way of proving a statement about the objects of S is to assume the existence of a counterexample and deduce a contradiction, thereby proving the contrapositive of the original statement. The basic premise of Noetherian induction is that every non-empty subset of S contains a minimal element. In particular, the set of all counterexamples contains a minimal element, the minimal counterexample. In order to prove the original statement, therefore, it suffices to prove something seemingly much weaker: For any counter-example, there is a smaller counter-example.