Can we talk about German mathematician Emily Noether? Let’s talk about Emily Noether. Noether’s work spans:
- Algebraic invariant theory, which is concerned with what expressions (for example, measurable properties of spacetime) are invariant — that is, unchanging — under groups of transformations (such as rotations, dilations, reflections, and projections). Noether developed constructively and singlehandedly an extension that allows us to study relationships between invariants in three variables. Though this was a significant computational problem, Noether later called this work Mist (crap) and Formelngestrupp (a jungle of equations).
- Galois theory in abstract algebra, which is concerned with what “splitting” fields provide solutions to equations with a given “ground” field of coefficients — for example, the complex numbers are a splitting field for all polynomials with real coefficients — and what sorts of transformations on the splitting field preserve the roots of an equation. Noether published a groundbreaking paper on the inverse Galois problem, which remains unsolved to this day.
- General relativity, for which she was sought by David Hilbert and Felix Klein (yes, namesakes of Hilbert spaces and the Klein bottle) for her unique knowledge of invariants. The two had noticed a hole in relativity that energy appeared to not be conserved. Noether not only solved the problem for general relativity, she developed Noether’s theorem, “what some call the most beautiful, deepest result in theoretical physics”, which determines the conserved quantities for any physical system with continuous symmetry.
- Chaining conditions on sets and Noetherian induction, powerful tools which she developed (or invented) and made popular as proof techniques.
- Topology, which Noether first suggested studying algebraically.
- And of course, general commutative ring theory, representation theory, and central simple algebras, all of which Noether either founded or co-founded as fields of research.
Noether applied to the University of Nottingen, the premier mathematics research institution in 1915, to become privatdozent (of lecturer status). Citing tradition, the philosophy and history faculties reportedly said, “How can it be allowed that a woman become a Privatdozent? Having become a Privatdozent, she can then become a professor and a member of the University Senate. Is it permitted that a woman enter the Senate? What will our soldiers think when they return to the University and find that they are expected to learn at the feet of a woman?” Hilbert famously replied, “This is a university, not a bath house.” Noether was rejected, but she lectured at the University under Hilbert’s name for four years. After she became even more extensively published, it was only by Hilbert’s threat of resignation that Noether was promoted to privatdozent.
Nevertheless, she was recognized outside the University. She was described by Albert Einstein, who was very much impressed with her seminal work on relativity (“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 Gottingen should take some lessons from Miss Noether! She seems to know her stuff.”) as “the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians.”
Noether’s approach to theoretical mathematics can be described as begriffliche Mathematik (purely conceptual mathematics). In her obituary, her friend B.L. van der Waerden said that her mathematical philosophy could be formulated as “Any relationships between numbers, functions, and operations become transparent, generally applicable, and fully productive only after they have been isolated from their particular objects and been formulated as universally valid concepts.” This approach is essential to modern algebra.
This all being said, the xkcd comic — which was posted outside my physics teacher’s classroom in high school — has a point. We should remember that no one in mathematics is really expected to be a genius, and that the fact that anyone of any gender is seeking to become a mathematician is a blessing in itself to our community.
Source: Wikipedia, University of Evansville [CJH]