Thursday 12 March: Emmy Noether, the Mother of Modern Algebra

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On Thursday 12 March, 2.10 to 3.10, Aditi Kar (Oxford University and Southampton University) and Ellen Powell, president of the Cambridge University Emmy Noether Society, will run a session on “Emmy Noether: the Mother of Modern Algebra”.

Aditi’s and Ellen’s session will be pitched for further maths students only.


The video below includes two talks, by Peter Olver and Ruth Gregory, on Noether’s contribution to mathematical physics.


And here are some notes on Emmy Noether’s life, and her work in algebra

Click here to download notes as pdf


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Emmy Noether

Algebra at school is mostly about rules for doing calculations with x’s and y’s. Algebra in real maths is more conceptual, basically about the patterns connecting and governing the structures within which calculations can be done.

The great turning point was 1930-1, the publication date of Bartel van der Waerden’s Moderne Algebra. As one mathematician puts it, that book “forever changed for the mathematical world the meaning of the word algebra from the theory of equations to the theory of algebraic structures”. In 2B6 you can look at a copy of an English translation of the 7th edition of the book. By that time the title had been changed to just “Algebra”.

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Van der Waerden’s “Moderne Algebra”, which brought Noether’s methods to a wide audience

Van der Waerden had been one of the group of mathematicians at Göttingen university, in Germany, then the world’s leading centre of maths, known as “Noether’s boys”. He described his book as based on the ideas of Emmy Noether and another Göttingen mathematician, Emil Artin.

“The ‘abstract’, ‘formal’, or ‘axiomatic’ direction, to which the fresh impetus in algebra is due, has led to… far-reaching results… The principal objective of this book is to introduce the reader into this entire world of concepts… General concepts and methods stand in the foreground”.

The same reorganisation, away from long calculations and towards conceptual thinking, which Noether had inspired in algebra, was carried through in many other fields of maths in the decades that followed, especially by a group of French mathematicians influenced by the time that one of them, André Weil, had spent in Göttingen working with Noether.

For example, Jean Dieudonné’s book Foundations of Modern Analysis (“analysis” means roughly what in school we call differentiation, integration, and differential equations) starts by declaring: “We have everywhere emphasised the conceptual aspect of every notion, rather than its computational aspect, which was the main concern of classical analysis”. There’s a copy of Dieudonné’s book you can look at in 2B6, too.

Emmy Noether revolutionised the way mathematicians think of algebra, and helped revolutionise the way mathematicians think of many other fields, too. She was to algebra what Einstein was to physics, or Watson and Crick to biology.

She was the daughter of a mathematician, Max Noether. In her teens she was not specially interested in maths, and studied to become a language teacher. Then she decided to do maths.

In 1907, aged 25, she completed her work for a postgraduate degree in maths. Then she worked for seven years as an unpaid teaching assistant to her father.

News of her talent got round, and the leading mathematicians at Göttingen, Felix Klein and David Hilbert, invited her to move there. She moved in 1915.

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Göttingen university – the foremost world centre of mathematics for a century, until the Nazis ended that within a few months, in 1933

Mostly because of Klein’s efforts, Göttingen was more enlightened than most universities at the time. Sofia Kovalevskaya, about whom June Barrow-Green talked to us on 8 January, got her degree there. She’d done all her work for it in Berlin, but there, Karl Weierstrass, the professor of maths, was not even able to get Kovalevskaya allowed into the lectures – he had to teach her privately – let alone get her granted a degree.

Still, even at Göttingen, Noether had difficulty getting a proper job. She was at first officially only a substitute teacher, doing lectures officially listed under David Hilbert’s name, and later only a junior lecturer.

In a faculty meeting, Hilbert protested: “Gentlemen, the faculty is not a swimming-pool changing room!”. Hermann Weyl, Hilbert’s successor as head of the world’s most illustrious maths department, would say: “I was ashamed to occupy such a preferred position beside her [Emmy Noether] whom I knew to be my superior”. But the professors of philosophy, history, etc. stonewalled.

Central in Emmy Noether’s work is the idea of invariants (coined by the English mathematician Arthur Cayley in the middle of the 19th century). You’ve already come across some of these in practice, but not the word. If you have two vectors a and b, their components (a1, a2) and (b1, b2) change if you rotate the x and y axes relative to which you measure them, but a.b always stays the same. It doesn’t vary. It’s invariant.

Another example: if you draw a connected system of vertices (points), edges (lines connecting points), and faces (regions marked out by edges), without the lines crossing, then F+V-E (number of faces plus number of vertices minus number of edges is an invariant of the surface you’re drawing on (it’s the same for every drawing on that surface).
See:
http://en.wikipedia.org/wiki/Wallpaper_group
https://www.math.hmc.edu/funfacts/ffiles/10001.4-7.shtml
http://personal.maths.surrey.ac.uk/st/H.Bruin/MMath/EulerCharacteristics.html

The conceptual, broad-picture, stand-back-from-the-details approach she developed in algebra was learned in Göttingen. Back in 1872 Felix Klein had published a manifesto, the Erlangen programme, proposing that the details of different sorts of geometry could and should be brought into a systematic overview in terms of group theory (about which Marcus du Sautoy spoke at CoLA last year).

David Hilbert had written: “I have tried to avoid long numerical computations, thereby following Riemann’s postulate that proofs should be given through ideas and not voluminous computations”. (The quote is pinned up outside 2B6).

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David Hilbert: one of the last “universal mathematicians”, who could make contributions in every part of mathematics

The contrast at that time was between the Göttingen approach, which Noether would develop, and greater emphasis on working through calculations at Berlin, which was then, under Ferdinand Frobenius, the other main centre of mathematical research. Frobenius rejected the Göttingen approach as one “in which one amuses oneself more with rosy images than hard ideas”.

But Emmy Noether moved on from the more Berlin-influenced methods of her degree dissertation, which she now described as “crap” and “a mess of formulas”.

The main structures in algebra are groups, rings, and fields. In groups, very simple calculations are possible. For example, the symmetries of an object (the ways you can transform it and end back where you started) form a group, and you can “multiply” one symmetry by another by doing one transformation and then the other one. The permutations (shufflings) of a list also form a group.

In rings, both something like addition and something like multiplication can be done. For example, square matrices of a particular size form a ring. Noether’s name is especially associated with ring theory.

And then there are fields – like the real numbers or the complex numbers – which have even more structure, so for example division can be done (which it can’t always in rings).

Before her work on algebra, Emmy Noether had discovered, in 1914, a theorem which is now considered the “workhorse of modern theoretical physics”.

It states that for any symmetry of a physical system, there is a conservation law of an associated observable quantity:

1. The symmetry of time translation, i.e. the fact that Newton’s laws are the same if time is measured from an earlier or later starting point, gives conservation of energy
2. The symmetries of spatial translation, i.e. the fact that Newton’s laws are the same if displacements are measured from a different origin, give conservation of momentum
3. Rotational symmetry gives conservation of angular momentum
4. Phase transformation symmetry gives conservation of electrical charge

The theorem sheds a new light on classical mechanics, and solves difficult problems in relativistic physics. In 2B6 you can look at a book discussing this theorem, Emmy Noether’s Wonderful Theorem, by Dwight Neuenschwander. Or click here to read an informative article on the theorem and its context: http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html.

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In 1933 the Nazis came to power. Emmy Noether represented all that they hated. She was Jewish; and since World War One (which she opposed) and the German revolution of 1918-9, she had been a left-wing socialist, though not particularly politically active. The Nazi government got her sacked within six weeks of its coming to power.

Other mathematicians were sacked and fled, too, and within months Göttingen abruptly ceased to be the world centre of maths. David Hilbert, by now elderly and retired, did not emigrate. Isolated and bitter, he was asked at some formal occasion by the Nazi minister of education how things were going now “the Jewish element” had been removed. “Mathematics at Göttingen?”, replied Hilbert. “There really is none any more”.

In a line which foreshadows the rubbish talked by the current government about “British values”, the few mathematicians who remained in Germany were reduced to boasting that their mathematics was good because it showed “German intuition” as opposed to “French logic”. Values, intuition, or logic which are worth anything can no more be German or Alaskan, British or New-Zealander, French or Tierra-del-Fuegan, than logarithms can be yellow.

It was not easy for Jews fleeing the Nazis. The USA and Britain had dropped the policy on which they prided themselves in the 19th century of offering refuge to the persecuted and dissident. (The USA did that after World War One; Britain started to move to a meaner and narrower-spirited approach with the anti-Jewish Aliens Act of 1905). It was difficult for Jews to get in anywhere, unless they could smuggle or negotiate their way round the British-imposed restrictions on migration to Palestine.

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Jewish refugee ship

Emmy Noether’s fame helped her, however. Her friend Albert Einstein, himself a refugee, got her a job at Bryn Mawr college in the USA.

Noether is described as having always been cheerful, kind, and buoyant. As Hermann Weyl wrote after her death: “Without regard for your own fate, open-hearted and without fear, always conciliatory, you went your own way… You were a great woman mathematician – I have no reservations in calling you the greatest that history has known”.

She liked it at Bryn Mawr, though she commented wryly that even in the USA she never really felt accepted at what she called “the men’s university”, Princeton. Einstein and other refugees from Nazism worked at Princeton, and it was becoming for post-1933 maths something like what Göttingen had been for pre-1933 maths.

In 1935 Emmy Noether died, aged 53, as the result of a medical operation going wrong.

Albert Einstein wrote: “Emmy Noether was the most significant creative mathematical genius thus far produced since the higher education of women began… In spite of the efforts of the great Göttingen mathematician, Hilbert, she never reached the academic standing due her in her own country, but none the less surrounded herself with a group of students and investigators at Göttingen, who have already become distinguished as teachers and investigators. Her unselfish, significant work over a period of many years was rewarded by the new rulers of Germany [the Nazis] with a dismissal, which cost her the means of maintaining her simple life and the opportunity to carry on her mathematical studies…”

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Some writers recently have drawn a new distinction between two strands in maths, criticising the approach of authors like Dieudonné as too academic and prissy and contrasting to it more practically-oriented, rough-and-ready, “street-fighting” maths. They quote Benoit Mandelbrot: “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line”.

There is a real issue, I think, with different attitudes to the use of diagrams and pictures in maths. Dieudonné and others like him rejected diagrams and pictures, saying that they confused things by making them too specific and obscuring the more important abstract patterns. There are many parts of maths for which pictures are very important.

But fundamentally rough-and-ready maths and precise maths are two parts of the same whole, not two rival approaches. Laurent Schwartz, one of the group of French mathematicians led by André Weil and Jean Dieudonné who developed Noether’s abstract, conceptual approach in other fields of maths, made his biggest research contribution by developing the idea of distributions as a further generalisation of the idea of functions. He did that by starting off from very rough-and-ready maths used by the physicists Oliver Heaviside and Paul Dirac. Converting Heaviside’s and Dirac’s rough “I-don’t-know-why-it-works-but-it-does” calculations into precise mathematical concepts was not a bit of prissiness: it enabled mathematicians to define the exact limits to when that method would work, and where it could be generalised.

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André Weil, who had studied with Noether in Göttingen in the 1920s, started the “Bourbaki” group of French mathematicians, who tried to reconstruct other branches of mathematics on the same lines as Noether