Saturday, November 07, 2015

Emmy Noether

The Mighty Mathematician You’ve Never Heard Of
Albert Einstein called her the most “significant” and “creative” female mathematician of all time, and others of her contemporaries were inclined to drop the modification by sex. She invented a theorem that united with magisterial concision two conceptual pillars of physics: symmetry in nature and the universal laws of conservation. Some consider Noether’s theorem, as it is now called, as important as Einstein’s theory of relativity; it undergirds much of today’s vanguard research in physics, including the hunt for the almighty Higgs boson. ....... a brilliant theorist whose unshakable number love and irrationally robust sense of humor helped her overcome severe handicaps — first, being female in Germany at a time when most German universities didn’t accept female students or hire female professors, and then being a Jewish pacifist in the midst of the Nazis’ rise to power. ...... Through it all, Noether was a highly prolific mathematician, publishing groundbreaking papers, sometimes under a man’s name, in rarefied fields of abstract algebra and ring theory. And when she applied her equations to the universe around her, she discovered some of its basic rules,

like how time and energy are related

, and why it is, as the physicist Lee Smolin of the Perimeter Institute put it, “that riding a bicycle is safe.” ........ “You can make a strong case that her theorem is the backbone on which all of modern physics is built.” .... Noether came from a mathematical family. Her father was a distinguished math professor at the universities of Heidelberg and Erlangen, and her brother Fritz won some renown as an applied mathematician. Emmy, as she was known throughout her life, started out studying English, French and piano — subjects more socially acceptable for a girl — but her interests soon turned to math. Barred from matriculating formally at the University of Erlangen, Emmy simply audited all the courses, and she ended up doing so well on her final exams that she was granted the equivalent of a bachelor’s degree. ....... earned her doctorate summa cum laude ..... In 1915 Einstein published his general theory of relativity. The Göttingen math department fell “head over ear” with it, in the words of one observer, and Noether began applying her invariance work to some of the complexities of the theory. That exercise eventually inspired her to formulate what is now called Noether’s theorem, an expression of the deep tie between the underlying geometry of the universe and the behavior of the mass and energy that call the universe home. ........ Wherever you find some sort of symmetry in nature, some predictability or homogeneity of parts, you’ll find lurking in the background a corresponding conservation — of momentum, electric charge, energy or the like. If a bicycle wheel is radially symmetric, if you can spin it on its axis and it still looks the same in all directions, well, then, that symmetric translation must yield a corresponding conservation. By applying the principles and calculations embodied in Noether’s theorem, you’ll see that it is angular momentum, the Newtonian impulse that keeps bicyclists upright and on the move. ..........

Some of the relationships to pop out of the theorem are startling, the most profound one linking time and energy.

Noether’s theorem shows that a symmetry of time — like the fact that whether you throw a ball in the air tomorrow or make the same toss next week will have no effect on the ball’s trajectory — is directly related to the conservation of energy, our old homily that energy can be neither created nor destroyed but merely changes form. ........ “Energy, momentum and other quantities we take for granted gain meaning and even greater value when we understand how these quantities follow from symmetry in time and space.” ...... After meeting the young Czech math star Olga Taussky in 1930, Noether told friends how happy she was that women were finally gaining acceptance in the field, but she herself had so few female students that her many devoted pupils were known around town as Noether’s boys. ...... Noether lived for math and cared nothing for housework or possessions, and if her long, unruly hair began falling from its pins as she talked excitedly about math, she let it fall. She laughed often and in photos is always smiling. .......

When a couple of students started showing up to class wearing Hitler’s brownshirts, she laughed at that, too.

..... In 1933, with the help of Einstein, she was given a job

at Bryn Mawr College, where she said she felt deeply appreciated as she never had been in Germany.

Emmy Noether Google Doodle: Why Einstein called her a ‘creative mathematical genius’
Noether had risen against wall after wall of obstacles to work on such areas as ring theory; now she was counted among those in a most rarefied academic circle...... “Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature.” –A.E. ........ Noether studied French and English as a girl growing up in Bavaria, but upon reaching adulthood, she followed her father (Max Noether) and a brother (Fritz) into math, and it was there she discovered and gave her full expression to the poetry of logical ideas. ........ Einstein called her two years at Pennsylvania’s Bryn Mawr “the happiest and perhaps the most fruitful of her entire career.” ..... “There weren’t any obstacles that would stop Noether from her studies. In this doodle, each circle symbolizes a branch of math or physics that Noether devoted her illustrious career to. From left to right, you can see topology (the donut and coffee mug), ascending/descending chains, Noetherian rings (represented in the doodle by the Lasker-Noether theorem), time, group theory, conservation of angular momentum, and continuous symmetries — and the list keeps going on and on from there! Noether’s advancements not only reflect her brilliance but also her determination in the face of adversity.”
The female mathematician who changed the course of physics—but couldn’t get a job
Noether's Theorem may be the most important theoretical result in modern physics. ...... Göttingen served as the center of mathematics for the Western world by this point, and Hilbert stood as one of its most notorious thinkers. He was a prominent leader for the minority of mathematicians who preferred a symbolic, axiomatic development in contrast to a more concrete style that emphasized the construction of particular solutions. Many of his peers recoiled from these modern methods, one even calling them “theology.” But Hilbert eventually won over most critics through the power and fruitfulness of his research. ......... Her father, Max, was a fairly prominent mathematician, and one of her brothers eventually attained a doctorate in math. In retrospect, perhaps the Noethers may be another historical example of a family with a math gene. ..... she had a facility with languages and was allowed to become certified as a language teacher. But Noether recognized her passion was in mathematics, and she decided to chase her dream and find a way to study the subject at the university level. ...... She also vigorously attacked her own research, forging a personal and original path through abstract algebra. Just a year after her doctorate, Noether's papers and the doctoral research that she was unofficially supervising gained her election to several academic societies, which prompted invitations to speak around Europe. Among those wanting her around, Hilbert reached out to bring Noether to Göttingen in order to tackle Einstein’s theory. ........

Einstein’s Theory of General Relativity was undoubtedly beautiful. It was unlike any theory of nature yet imagined by humankind

...... The mass that determined the strength of the gravitational force was the same mass that appeared in Newton’s second law of motion, F = ma; gravitational mass was the same as the “inertial mass.” There was no apparent reason this had to be true, it simply was. ....... but Hilbert could not overcome the resistance of the humanities professors, who simply could not stomach the idea of a female teacher. ...... Noether immediately grasped the problem with Einstein's theory. Over the course of three years, she not only solved it, but in doing so she proved a theorem that simultaneously reached back to the dawn of physics and pushed forward to the physics of today.

Noether’s Theorem

, as it is now called, lies at the heart of modern physics, unifying everything from the orbits of planets to the theories of elementary particles........ the theorem uncovers a hidden relationship between symmetry and conservation, and that relationship is what came to unify all of physics. ....... If the equations that describe the universe changed as time passed, we could never make sense of anything. ..... Physics should be the same no matter where in space we are, if nothing else changes. The equations of motion need to be the same in New York or Göttingen. ....... Noether’s Theorem relates continuous invariants to conservation laws. A conservation law is a rule that says that some quantity remains numerically constant as the system evolves in time. Conservation of energy, momentum, and angular momentum from classical physics are famous examples. ....... conservation of energy was not discovered for almost 200 years after Newton published his laws of motion. ........

Noether’s Theorem proves that for every invariant, there is a corresponding conservation law. She also proved the converse, meaning that for every conservation law there must be an invariant behind it.

...... The theorem shows that conservation of energy is equivalent to time invariance in classical physics. This hard-won yet essential conservation law is directly implied by, and implies, a fundamental symmetry of nature. It shows that momentum conservation is equivalent to spatial invariance. It establishes the equivalence of other symmetries, more mathematical in flavor, with other conservation laws. For example, the conservation of charge is related to a gauge symmetry, a complex mathematical symmetry in the equations of electrodynamics. ...... It is the theorem’s power to derive new conservation laws from abstract symmetries that has guided physical theory up to the present day. Noether’s result is an important tool in contemporary areas like particle physics, and it’s likely to remain so. ........ Noether’s work helped shed light on the fact that Einstein’s gravity behaves as no theory devised before, in that the energy of matter moving in a gravitational field can not be considered separately from the energy of the field itself. There is a conservation law, but it involves taking all of matter and gravity in a region of space as a unified whole .......... Noether showed that Hilbert was correct­—normal local energy conservation did not hold in Einstein’s work. However, she discovered that this was because of the peculiar kind of symmetry in general relativity. In this radically new model of the universe, gravity altered the very geometry of space and time.

In a Euclidean world, the ratio of the circumference of a circle to its diameter equals π. But in Einstein’s universe, this ratio depends on where in space you happen to be.

.......... energy conservation in general relativity just could not take the form that it had in all previous physics theories. ..... classroom teaching wasn’t her strength, Noether proved to be a superb leader of small research groups. Her advanced students were devoted to her. .......

One-third of the mathematics professors, and three-fourths of the heads of Göttingen’s mathematics and physics institutes, were Jewish despite less than one percent of the German population identifying that way at the time.

....... Noether’s work seemed to unify the most abstract mathematics with the most basic physical intuition, unifying the earliest successful systems of physics with science yet unborn. The circumstances of her life provide a powerful example of the humanizing influence of science and mathematics. It was the exponents of these fields who were eager to welcome her into their fellowship without regard for her sex or ancestry; the men of philosophy, history, politics, and government sought to exclude her for these very reasons. ....... A street and school in her home town have been named after her, as well as a crater on the moon. And for her birthday on March 23, Google dedicated its coveted Doodle real estate to one of history's most under-appreciated minds.

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