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Emmy Noether in Erlangen and G¨ottingen Ravi A. Rao School of Mathematics, Tata Institute of Fundamental Research Ravi A. Rao (TIFR) Emmy Noether in Erlangen and G¨ottingen 1 / 57 Abstract We give an overview of Emmy Noether's seminal works; and her contribution to the birth of modern algebra. We also give an indication of its influence in present day research. Ravi A. Rao (TIFR) Emmy Noether in Erlangen and G¨ottingen 2 / 57 Her name In 1809, the Tolerance Edict of Baden stated that the male head of every Jewish family who did not have a distinguishing heriditary surname was to assume one for himself and his children. Elias Samuel, the grandfather of Max Noether, was given the name N¨other.Max and his children used the spelling Noether, though his marriage documents still showed the official spelling as N¨other. Ravi A. Rao (TIFR) Emmy Noether in Erlangen and G¨ottingen 3 / 57 Erlangen Felix Klein (1849-1925) brought world fame to Erlangen with his inaugural lecture of 1872 explaining the significance of the concept of a group in geometry, now known as the \Erlangen Program". One of Felix Klein's friends and colleagues, Paul Gordan (1837-1912) was a colleague of Max Noether (1844-1921), whose daughter Emmy was to become Gordan's only doctoral candidate. It was only in 1904 that the University of Erlangen decided that women should be provided full access; and Emmy was accepted by Gordan (the King of Invariant Theory) to be his first student (at the age of 61). She finished her doctorate (no BA or MA in Math!) in the subject of Computational classical Invariant Theory in 1907. Her thesis ended with 331 formulas on invariants of ternary biquadratic forms! Her degree was awarded \summa cum laude" the highest distinction possible. Ravi A. Rao (TIFR) Emmy Noether in Erlangen and G¨ottingen 4 / 57 Noether's thesis Her thesis was on the construction of the system of forms for the ternary biquadratic form: Uber¨ der Bildung des Formensystems der tern¨aren biquadratischen Form (registered in Erlanger Universit¨atsschriften1907/08 number 202, dated June 2, 1908). This later appeared in Journal f¨urdie reine und angewandte Mathematik, 134, 1908, pg. 23-90. Emmy Noether herself later referred to her thesis, as well as to several consecutive paper on the theory of invariants, as \crap". She said it was a jungle of formulas { routine calculations. As a lecturer (unpaid) in Erlangen she began to study the work of Hilbert. Two students of Hilbert: E. Fischer (1875-1954) and Erhard Schmidt, were the successor of Gordan (who retired in 1910) at Erlangen, It was Fischer's influence that Emmy Noether made the definitive change from the purely computational method distinctly algorithmic approach represented by Gordan to the mode of thinking characteristic of Hilbert. Ravi A. Rao (TIFR) Emmy Noether in Erlangen and G¨ottingen 5 / 57 Noether invited to G¨ottingen In 1913 and 1914 Noether exchanged letters with David Hilbert and his G¨ottingensenior colleague Felix Klein discussing Einstein's Relativity Theory. Hilbert invited her in 1915 to come to G¨ottingenas a Lecturer; but this was opposed by the Humanities department. (What will the soldiers think if they come back from the war and be taught at the feet of a woman? Hilbert argued that: \Gentleman, the Senate is not a bathhouse; so I do not see why a woman cannot enter it. However, he did not succeed in convincing his colleagues. Ravi A. Rao (TIFR) Emmy Noether in Erlangen and G¨ottingen 6 / 57 Privatdozent in G¨ottingen Noether was so eager to join Hilbert's department in G¨ottingenthat, to overcome Hilbert's opponents, she agreed not to be formally appointed as a lecturer and to receive no pay. Her father continued supporting her financially (sadly her mother died in 1915) and the lectures she gave were advertised as lectures by Professor Hilbert, with assistance from Dr. E. Noether. It was only in 1922 that she was given a small salary, as a lecturer in algebra. She received the appointment as honorary Professor, as a result of Courant's efforts she received a so-called Lehraufirag, i.e. a small salary (200-400 marks per month) for her lectures; which required confirmation every year by the Ministry. Thus, after the death of Gordan, and with her brother Fritz, and E. Fischer drafted in the military, and the sudden death of her mother, Emmy moved to G¨ottingenat the end of April, 1915, when she received an invitation from Klein and HIlbert to substitute for the Privatdozenten- someone who had the right to teach without being on the paid staff of the university. Ravi A. Rao (TIFR) Emmy Noether in Erlangen and G¨ottingen 7 / 57 Noether's work Noether's Theorem had revolutionized physics. In 1919 the full force of her powerful mind turned towards pure mathematics. In this discipline, she was one of the principle architects of abstract algebra. Her name is remembered in many of its concepts, structures and objects, such as: Noetherian, Noetherian group, Noetherian induction, Noether normalization, Noether problem, Noetherian ring, Noetherian module, Noetherian scheme, Noetherian space, Albert{Brauer{Hasse{Noether theorem, Lasker{Noether theorem, and Skolem{Noether theorem. Ravi A. Rao (TIFR) Emmy Noether in Erlangen and G¨ottingen 8 / 57 Contd. Her work was pivotal in the fields of: mathematical rings she established the modern axiomatic definition of a commutative ring and developed the basis of commutative ring theory commutative number fields, linear transformations, noncommutative algebras { Hermann Weyl credited Noether with representations of noncommutative algebras by linear transformations, and their application to the study of commutative number fields and their arithmetics. Ravi A. Rao (TIFR) Emmy Noether in Erlangen and G¨ottingen 9 / 57 Steinitz paper on field theory In 1910 in Vol. 137 of Crelle, 167{309, there appeared a 134 page paper with the title Algebraische Theorie der K¨orper [Algebraic Theory of Fields] by Ernst Steinitz. Steinitz paper was the first time that an abstract algebraic structure was studied on the basis of its system of axioms. On the one hand the formal algebraic thinking, on the other hand the connection to set theory{these are the characteristics of the modern algebra of structures. Noether carried this program further in her works, Idealtheorie in Ringbereischen (1921) and Abstrakter Aufbau der Idealtheorie in algebraischen Zahl und Funktionenkorpern (1926). Ravi A. Rao (TIFR) Emmy Noether in Erlangen and G¨ottingen 10 / 57 Invariant Theory Three of Hilbert's fundamental contributions to modern algebra, Nullstellensatz, Basis Theorem, Syzygy Theorem, were first proved as lemmas in his Invariant Theory papers (1890, 1893). A polynomial f 2 C[X1;:::; Xn] is said to be symmetric if it is invariant under any permutation of the variables X1;:::; Xn i.e. f (X1;:::; Xn) = f (Xσ(1);:::; Xσ(n)), 8σ 2 Sn, the group of permutations on n-symbols. For instance, X "1 = X1 + ::: + Xn = Xi i "2 = X1X2 + X1X3 + ::: + X2X3 + ::: + Xn−1Xn Y = Xi Xj i<j Y "3 = Xi Xj Xk i<j<k ::: = ::: "n = X1X2 ::: Xn Ravi A. Rao (TIFR) Emmy Noether in Erlangen and G¨ottingen 11 / 57 Contd. are invariant under the action of Sn. These are called the elementary symmetric polynomials. Every symmetric polynomial f can be written uniquely as a polynomial f ("1;:::;"n) 2 C[X1;:::; Xn]. These fundamental invariants are algebraically independent. There is no non-zero syzygy. Let G be a subgroup of GLn(C), the general linear group. GLn(C) acts linearly on the variables X1;:::; Xn: t t If α 2 GLn(C), send (X1;:::; Xn) 7! α(X1;:::; Xn) , and f 7! f ◦ α. 0 1 Example: Let f = X 2 + 2X X , α = , so X 7! X , and 1 1 2 −1 0 1 2 2 X2 7! −X1, and f 7! X2 − 2X1X2. Ravi A. Rao (TIFR) Emmy Noether in Erlangen and G¨ottingen 12 / 57 The ring of invariants G C[X1;:::; Xn] = ff 2 C[X1;:::; Xn] j f = f ◦ γ; 8γ 2 Gg is the subring of C[X1;:::; Xn] consisting of polynomials which are invariant under every element of G. The following are some of the relevant questions about these rings: G • Finite Generation: Is C[X1;:::; Xn] generated by a finite set of generators? • Syzygies: What are (all) the relations between these generators? • Computational Algorithm: Can you give an algorithm to write down an arbitrary invariant as a polynomial in some fundamental invariants? Ravi A. Rao (TIFR) Emmy Noether in Erlangen and G¨ottingen 13 / 57 Noether's degree bound Hilbert's Finiteness Theorem showed that for finite groups G ⊂ GLn(C), G the ring of invariants C[X1;:::; Xn] has transcendence degree n over C and is finitely generated. Nagata (1959) gave examples to show that certain \exotic" linear groups do not have finitely generated invariant rings. (This meant that Hilbert's sixth problem had a negative solution in general.) In her last theorem proved in Erlangen, Noether (1916) gave an efficient version of Hilbert's theorem: Ravi A. Rao (TIFR) Emmy Noether in Erlangen and G¨ottingen 14 / 57 Contd. Theorem (Noether's degree bound) G The invariant ring C[X1;:::; Xn] if a finite natrix group G has a basis of n+jGj at most n invariants whose degrees are bounded above by jGj. This theorem has great impact on computational commutative algebra. More generally, Noether's theorem works if jGj is invertible in the base field; and so this has also led to active research in modular invariant theory. Ravi A.
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