Oggi è il giorno in cui si celebra la festa dell’Immacolata.
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.
Emmy Noether my greatest love
How I Shoot: Snapping the Perfect #Puddlegram with @mortenordstrom
How I Shoot is a series where we ask Instagrammers to tell us about the set-up and process behind their photos and videos. This week, Morten Nordstrøm (@mortenordstrom) shares his tips for capturing and editing the perfect #puddlegram on a Windows phone. Follow him on Instagram for puddle shots of Copenhagen and browse the #puddlegram hashtag for more inspiration!
When Morten Nordstrøm (@mortenordstrom) isn’t working or studying business administration and communications, he’s out showcasing the beauty of Copenhagen, Denmark, as reflected through the city’s many puddles.
Getting the most out of a puddle’s reflection is an art, and Morten offered these tips for capturing and editing a #puddlegram on a Windows phone:
Nokia Lumia 925
"First of all, when shooting puddlegrams you will obviously need a puddle. The bigger the better, but you will be surprised how little water you need to make a big impact.
"My experience is that I get the best effect when I look for a strong central focal point and try to get some depth in the perspective, maybe even a vanishing point. That’s one of the reasons why the majority of my puddlegrams are shot on streets; they often meet these requirements. Moreover they are often full of life, which gives life to the picture and helps to tell a story.
"If you want to do a proper #puddlegram, make sure your lens is as close to the ground as possible—even if it means you have to turn your device upside-down. It’s a little effort and it makes a huge difference. Moreover, mind the weather. Windy puddles won’t serve you well."
"I often think a lot about my perspective before I shoot. Other times I just walk around and look for details and shoot what I find interesting and inspiring.
"I’m always shooting with my Nokia Lumia 925 and I have been experimenting a lot with the different camera possibilities. Normally I just use the native camera, other times I use the Nokia Pro Camera which gives me more adjustability. My best advice when taking puddlegrams is to experiment with the number of shots you’re taking and see the different outcomes you can get. Analyzing strengths and weaknesses in these will help make you a more skilled puddleshooter!
"Note that people will be looking when you sit down and place your phone in a puddle. But remember, that is only because they don’t know about the magical perspectives you’re capturing. I sometimes pretend to tie my shoe laces, until the largest group of people has passed."
"This is where the magic happens. After extensive exploration of Windows Phone options, I ended up always using the same two apps: the Nokia Creative Studio and an app called Fhotoroom. They are very different but complement each other well. I like to keep things simple and often aim for a high sharpness, cold tones and not too strong colors, since I find them disturbing. Finally, I pay a lot of attention in the cropping process. A good crop can change a picture entirely—and so can a bad one.”