VI.76 Emmy Noether
b. Erlangen, Germany, 1882; d. Bryn Mawr, Pennsylvania, 1935
Algebra; mathematical physics; topology
Noether began her career with a feat of classical algebra, which she transmuted into the NOETHER CONSERVATION THEOREMS [IV.12 §4.1] for physics. She became a founder of modern abstract algebra and the leader in spreading that algebra all across mathematics.
Her father Max Noether and family friend Paul Gordan were Erlangen mathematicians and favored educating women. Gordan made heroic calculations of invariants in algebra. A quadratic polynomial Ax2 + Bx+ C has essentially just one invariant, the discriminant 
used in the quadratic formula. As Gordan’s student, Noether found 331 independent invariants of degree-four polynomials in three variables, and proved that all others depend on them. It was impressive, though not, as it turned out, groundbreaking.
HILBERT [VI.63] brought her to Göttingen in 1915 to work on invariants for differential equations in general relativity by reducing them to algebra. That year she found her conservation theorems, which show that the conserved quantities of a physical system correspond to its symmetries. For example, if a system has laws unchanging with time, so that a time shift is a symmetry of the system, then energy is conserved in the system (Feynman 1965, chapter 4). These theorems became fundamental in Newtonian physics and especially quantum mechanics. They also showed that general relativity admits conservation laws only in special cases.
Noether saw the creation of general abstract algebra as her life’s work. Instead of classical algebra with real numbers, or complex numbers, and polynomials using them, she would study any system satisfying abstract rules such as the RING AXIOMS [III.81] or the GROUP AXIOMS [I.3 §2.1]. Concrete examples include the ring of all algebraic functions defined on a space (such as a sphere), and the group of all symmetries of a given space. She largely created the now-standard style of abstract algebra. Her ideas were also adopted in ALGEBRAIC GEOMETRY [IV.4] where every abstract ring appears as the ring of functions on a corresponding space called a SCHEME [IV.5 §3].
She turned her attention away from operations on elements of a system, like plus and times, and focused instead on relating whole systems to each other, such as rings R, R′ related by ring HOMOMORPHISMS [I.3 §4.1] from R to R′. She organized all algebra around her homomorphism and isomorphism theorems. Her aim was to show how IDEALS [III.81 §2] and their corresponding homomorphisms could replace equations between elements as the basic tools for stating and proving theorems. (This approach was to come to fruition in the 1950s, with the advent of Grothendieck-style homological algebra.)
Topologists studied TOPOLOGICAL SPACES [III.90] by looking at continuous functions from one space to another. Noether saw how her algebraic methods could apply here, and convinced young topologists in the 1920s to use them in algebraic topology. Each topological space S has HOMOLOGY GROUPS [IV.6 §4] HnS with the property that continuous functions from S to S′ induce group homomorphisms from HnS to HnS′. Theorems of topology follow by abstract algebra. This relationship between homomorphisms and continuous functions is what inspired CATEGORY THEORY [III.8].
In the 1930s Noether pursued the algebra of GALOIS THEORY [V.21] through a radically simplified abstract theory of groups acting on rings. The applications are quite arcane, beginning with CLASS FIELD THEORY [V.28] and eventually growing into group cohomology and many other algebraic and topological methods used in ARITHMETIC GEOMETRY [IV.5].
Exiled from Germany by the Nazis in 1933, she died following surgery in the United States, at the height of her creative power.
Further Reading
Brewer, J., and M. Smith, eds. 1981. Emmy Noether: A Tribute to Her Life and Work. New York: Marcel Dekker.
Feynman, R. 1965. The Character of Physical Law. Cambridge, MA: MIT Press.
Colin McLarty