I'm in Erlangen, where the great German mathematician Emmy Noether was born in 1882. She was the daughter of the well-known mathematician Max Noether - but as a woman, she was only allowed to audit courses at the university here. Somehow she finished a PhD thesis in 1907. She then worked here without pay for 7 years, since women were excluded from academic jobs.
Her thesis advisor, Paul Gordan, specialized in doing complicated calculations to find all the polynomials that were unchanged by certain symmetries. Around this time David Hilbert proved a powerful general theorem that said all these polynomials could be gotten by adding, subtracting and multiplying a finite set of them, called 'generators'. But he didn't say how to find these generators! Gordan said "this is not mathematics; this is theology."
Noether did her thesis, On Complete Systems of Invariants for Ternary Biquadratic Forms, in the style of Gordan's work. It was well received, but she later said it was "crap". While working without pay, she learned Hilbert's ideas and started revolutionizing the subject of algebra.
In 1915 she was invited to the University of Göttingen by David Hilbert and Felix Klein. Their attempt to recruit her was fought by the philologists and historians, who didn't want a woman on the faculty. Hilbert fought back, saying "After all, we are a university, not a bath house."
It took years for her to actually get paid, but she started working at Göttingen and soon proved the theorem physicists remember her for, relating symmetries and conservation laws. They call it Noether's Theorem.
In fact she proved two important theorems on this subject, but the easier one is more famous: Leon Lederman said it's "certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem."
Her theorem applies to classical mechanics and classical field theory, but there's also a quantum version, and more recently Brendan Fong and I proved a 'stochastic' version, which applies to random processes. The stochastic version is weirdly different from the quantum version, but +Ville Bergholm has just written a nice article discussing this issue, and some results he discovered with +Jacob Biamonte and Mauro Faccin:
http://johncarlosbaez.wordpress.com/2014/05/03/noethers-theorem-quantum-vs-stochastic/
Check it out!
Emmy Noether finally started getting a salary in 1923, sixteen years after finishing her thesis. If anyone asks why there are fewer famous women mathematicians than men, consider pointing this out!
Noether did extraordinary work until 1933, when the Nazis kicked her out of the University of Göttingen. She wound up in Bryn Mawr College, a women's college near Philadelphia. She died of complications from surgery in 1935.
But here are some of the wonderful things she did:
In 1921 she stated the general definition of 'ring' and 'ideal', and proved that in a ring where every increasing sequence of ideals stops growing after finitely many steps, every ideal has finitely many generators. Such rings are now called Noetherian.
In 1927 she gave a massive generalization of the fundamental theorem of arithmetic, about unique factorization into primes. She characterized commutative rings in which the ideals have unique factorization into prime ideals as the integral domains that are Noetherian, 0- or 1-dimensional, and integrally closed in their quotient fields. Sorry - this sounds technical, and it is! But everyone who studies modern number theory takes this result as basic: such rings are now called Dedekind domains, but Noether discovered them.
Even more important than either of these massive results are the beautifully simple 'Noether isomorphism theorems' that everyone learns near the start of a course on group theory.
And perhaps even more important was her discovery of 'homology groups' while attending lectures by the famous topologists Alexandrov and Hopf. Other people would have made a whole career out of this discovery, which utterly revolutionized topology. But she only gave it a tiny mention in one of her works on group theory! She was truly a fountain of new ideas.
I now have an office in the Emmy-Noether-Zentrum für Algebra at the university in Erlangen.
For more, try:
https://en.wikipedia.org/wiki/Emmy_Noether
Her thesis advisor, Paul Gordan, specialized in doing complicated calculations to find all the polynomials that were unchanged by certain symmetries. Around this time David Hilbert proved a powerful general theorem that said all these polynomials could be gotten by adding, subtracting and multiplying a finite set of them, called 'generators'. But he didn't say how to find these generators! Gordan said "this is not mathematics; this is theology."
Noether did her thesis, On Complete Systems of Invariants for Ternary Biquadratic Forms, in the style of Gordan's work. It was well received, but she later said it was "crap". While working without pay, she learned Hilbert's ideas and started revolutionizing the subject of algebra.
In 1915 she was invited to the University of Göttingen by David Hilbert and Felix Klein. Their attempt to recruit her was fought by the philologists and historians, who didn't want a woman on the faculty. Hilbert fought back, saying "After all, we are a university, not a bath house."
It took years for her to actually get paid, but she started working at Göttingen and soon proved the theorem physicists remember her for, relating symmetries and conservation laws. They call it Noether's Theorem.
In fact she proved two important theorems on this subject, but the easier one is more famous: Leon Lederman said it's "certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem."
Her theorem applies to classical mechanics and classical field theory, but there's also a quantum version, and more recently Brendan Fong and I proved a 'stochastic' version, which applies to random processes. The stochastic version is weirdly different from the quantum version, but +Ville Bergholm has just written a nice article discussing this issue, and some results he discovered with +Jacob Biamonte and Mauro Faccin:
http://johncarlosbaez.wordpress.com/2014/05/03/noethers-theorem-quantum-vs-stochastic/
Check it out!
Emmy Noether finally started getting a salary in 1923, sixteen years after finishing her thesis. If anyone asks why there are fewer famous women mathematicians than men, consider pointing this out!
Noether did extraordinary work until 1933, when the Nazis kicked her out of the University of Göttingen. She wound up in Bryn Mawr College, a women's college near Philadelphia. She died of complications from surgery in 1935.
But here are some of the wonderful things she did:
In 1921 she stated the general definition of 'ring' and 'ideal', and proved that in a ring where every increasing sequence of ideals stops growing after finitely many steps, every ideal has finitely many generators. Such rings are now called Noetherian.
In 1927 she gave a massive generalization of the fundamental theorem of arithmetic, about unique factorization into primes. She characterized commutative rings in which the ideals have unique factorization into prime ideals as the integral domains that are Noetherian, 0- or 1-dimensional, and integrally closed in their quotient fields. Sorry - this sounds technical, and it is! But everyone who studies modern number theory takes this result as basic: such rings are now called Dedekind domains, but Noether discovered them.
Even more important than either of these massive results are the beautifully simple 'Noether isomorphism theorems' that everyone learns near the start of a course on group theory.
And perhaps even more important was her discovery of 'homology groups' while attending lectures by the famous topologists Alexandrov and Hopf. Other people would have made a whole career out of this discovery, which utterly revolutionized topology. But she only gave it a tiny mention in one of her works on group theory! She was truly a fountain of new ideas.
I now have an office in the Emmy-Noether-Zentrum für Algebra at the university in Erlangen.
For more, try:
https://en.wikipedia.org/wiki/Emmy_Noether
