Who Was Emmy Noether, Mathematician Whom Einstein Called A 'Genius'?

If we tell you that her discoveries have had a considerable impact on theoretical physics, that they have allowed us to profoundly reconsider our understanding of the intimate relationships between space, time and energy, who do you think of? Another clue: Einstein – no, it’s not him! – spoke of her – her? – as a “considerable creative mathematical genius” and of her theorems as a “monument of mathematical thought”. This mathematician who discovered fundamental structures and theorems in several areas of mathematics is Emmy Noether. Despite the many obstacles she had to overcome, her astonishing theorems make her a crucial player in the construction of the theoretical foundations of modern physics .

A path strewn with pitfalls

Emmy Noether was born in 1882 in Erlangen, Germany. From childhood, she revealed her exceptional gifts for mathematics, but her career path was not a straight line! Completing her thesis in 1907, she had to work as a volunteer at the Erlangen Institute of Mathematics for 7 years while waiting to obtain a position. Her work then began to be very visible in the mathematics research community, and she was invited by two illustrious mathematicians of the time, Felix Klein and David Hilbert, to join the prestigious mathematics department of the University of Göttingen.

At the beginning of the 20th century  , however, it was extremely difficult for a woman to make her way in scientific society, and there was strong opposition to her being allowed to teach as a professor. For this reason, she had to give her lectures under the name Hilbert! In 1918, she published her first major article: "Variational Invariant Problems" and finally obtained a position as Privatdozent , which allowed her to give lectures, but did not provide any remuneration.

In the early 1920s, she developed her research in algebra, creating the foundations of the theory of rings, fundamental structures with many applications, particularly in cryptography. In 1921, she published a paper that significantly advanced these theories, and exhibited a particular class of these objects, later named in her honor "Noetherian rings". Her contributions to algebra made her famous within the mathematical community. Things then finally seemed to be going well for her, until the Nazis came to power in 1933: as a Jew, she was expelled from the University. She emigrated to the United States, to Pennsylvania, where she obtained a position. A few years later, however, in 1935, aged only fifty-three, she died from an ovarian cyst.

His first major paper in 1918, on the theory of invariants, almost fell into oblivion shortly after its publication. But from the 1950s onwards, he reappeared, this time in the spotlight, attracting the attention of the theoretical physics community, because he enriched with a powerful analytical framework one of the basic building blocks used in the construction of physical theories, namely the principle of least action.

The principle of least action

There is a way to describe the evolution of some physical systems, considering that they try to make a certain quantity as small as possible. For example, if we assume that "light tries to travel its path as quickly as possible", as Pierre de Fermat did in 1657, this allows us to find all the laws of Descartes' geometric optics. Since light propagates at different speeds in air and in glass, the shape of the light rays passing through a lens is the one that minimizes the travel time.

In other words, a ray of light takes the form of a sort of "rubber band", the shortest possible. It is possible to go further, and to consider that this rubber band is stretched in space and time, to describe more general physical systems, as Maupertuis did in 1746, a theory later refined by Euler and Lagrange. Something that would resemble the "length of the rubber band" in space and time is called "action", and this type of reasoning is called a "principle of least action".

Emmy Noether, invariance and conservation laws

The theorems discovered by Emmy Noether in 1918 reveal very profound truths about physical systems governed by a principle of least action. They establish the link between two notions: on the one hand invariance, namely the conditions of an experiment that can be modified without consequences on the result of the experiment, and on the other hand conservation, namely, the existence of physical quantities whose value does not vary during the experiment.

For example, if we perform the same physics experiment today or tomorrow under the same conditions, we should observe the same thing. Noether's theorem tells us that this invariance with respect to time translates into the existence of a conserved physical quantity. If we do the calculation, we find that this quantity corresponds to something well known, namely energy. This is interesting because it gives a better understanding of what energy is, which emerges here as a mathematical property of the equations. Other invariances exist, this time with respect to space, as well as their associated conserved quantities, which allow us, among other things, to explain the principle of inertia and the behavior of a gyroscope.

These conservation laws had been known for a long time by Emmy Noether's time, but through her theorems, she provided for the first time a completely abstract explanation of these laws, and above all, a way to discover new ones. While Einstein had just published his famous theory of general relativity in 1915, Emmy Noether's contribution made it possible to abstract the structure of the reasoning behind certain aspects of relativity, and to transport this reasoning to other areas of physics. For example, by very subtly changing the definition of invariance, namely what can be modified in an experiment without changing the result, we find another famous formula for energy: E = mc 2 .

In search of the laws of nature: a new investigation tool

At the time Emmy Noether published her theorem, it allowed to reveal deep truths in the laws of physics already known or under construction at the time, and to better link them together within a coherent edifice. But it allows to do much more, still today: by analyzing the subtle links existing between invariances and conservation laws, her theorem is a real guide to try to discover new laws, by exploiting the structure of the equations. It plays a role in particular in quantum physics, where we are interested in other invariances called "gauge symmetries", and the associated conserved quantities.

During her short life, while overcoming many obstacles, Emmy Noether revolutionized two different areas of mathematics, leaving us two different legacies, namely the theory of invariants, which is still at the heart of modern physics, and the theory of rings, notably used in cryptography today.

This article was written by Bruno Levy, Inria research director, researcher in digital physics, Inria for The Conversation.