Emmy Noether

Emmy Noether (born March 23, 1882, Erlangen, Germany—died April 14, 1935, Bryn Mawr, Pennsylvania, U.S.) was a German mathematician whose innovations in higher algebra gained her recognition as the most creative abstract algebraist of modern times.

Noether was certified to teach English and French in schools for girls in 1900, but she instead chose to study mathematics at the University of Erlangen (now University of Erlangen-Nürnberg). At that time, women were only allowed to audit classes with the permission of the instructor. She spent the winter of 1903–04 auditing classes at the University of Göttingen taught by mathematicians David Hilbert, Felix Klein, and Hermann Minkowski and astronomer Karl Schwarzschild. She returned to Erlangen in 1904 when women were allowed to be full students there. She received a Ph.D. degree from Erlangen in 1907, with a dissertation on algebraic invariants. She remained at Erlangen, where she worked without pay on her own research and assisting her father, mathematician Max Noether (1844–1921).

In 1915 Noether was invited to Göttingen by Hilbert and Klein and soon used her knowledge of invariants helping them to explore the mathematics behind Albert Einstein’s recently published theory of general relativity. Hilbert and Klein persuaded her to remain there despite the vehement objections of some faculty members to a woman teaching at the university. Nevertheless, she could only lecture in classes under Hilbert’s name. In 1918 Noether discovered that if the Lagrangian (a quantity that characterizes a physical system; in mechanics, it is kinetic minus potential energy) does not change when the coordinate system changes, then there is a quantity that is conserved. For example, when the Lagrangian is independent of changes in time, then energy is the conserved quantity. This relation between what are known as the symmetries of a physical system and its conservation laws is known as Noether’s theorem and has proven to be a key result in theoretical physics. She won formal admission as an academic lecturer in 1919.

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The appearance of “Moduln in nichtkommutativen Bereichen, insbesondere aus Differential- und Differenzenausdrücken” (1920; “Concerning Moduli in Noncommutative Fields, Particularly in Differential and Difference Terms”), written in collaboration with a Göttingen colleague, Werner Schmeidler, and published in Mathematische Zeitschrift, marked the first notice of Noether as an extraordinary mathematician. For the next six years her investigations centred on the general theory of ideals (special subsets of rings), for which her residual theorem is an important part. On an axiomatic basis she developed a general theory of ideals for all cases. Her abstract theory helped draw together many important mathematical developments.

From 1927 Noether concentrated on noncommutative algebras (algebras in which the order in which numbers are multiplied affects the answer), their linear transformations, and their application to commutative number fields. She built up the theory of noncommutative algebras in a newly unified and purely conceptual way. In collaboration with Helmut Hasse and Richard Brauer, she investigated the structure of noncommutative algebras and their application to commutative fields by means of cross product (a form of multiplication used between two vectors). Important papers from this period are “Hyperkomplexe Grössen und Darstellungstheorie” (1929; “Hypercomplex Number Systems and Their Representation”) and “Nichtkommutative Algebra” (1933; “Noncommutative Algebra”).

In addition to research and teaching, Noether helped edit the Mathematische Annalen. From 1930 to 1933 she was the centre of the strongest mathematical activity at Göttingen. The extent and significance of her work cannot be accurately judged from her papers. Much of her work appeared in the publications of students and colleagues; many times a suggestion or even a casual remark revealed her great insight and stimulated another to complete and perfect some idea.

When the Nazis came to power in Germany in 1933, Noether and many other Jewish professors at Göttingen were dismissed. In October she left for the United States to become a visiting professor of mathematics at Bryn Mawr College and to lecture and conduct research at the Institute for Advanced Study in Princeton, New Jersey. She died suddenly of complications from an operation on an ovarian cyst. Einstein wrote shortly after her death that “Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.”

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