Amalie (Emmy) Noether .5cm [width=2cm]QRNoethersusanka.org/MathPhysics/NoetherHandout.pdfAmalie (Emmy) Noether TOC Intro G¨ottingen 1917-1920 Gottingen 1920-1933¨ Exile References

Amalie (Emmy) Noether TOC Intro Gottingen 1917-1920 Gottingen 1920-1933 Exile References

Amalie (Emmy) Noether

Larry Susanka

October 3, 2019


Amalie (Emmy) Noether

Table of Contents

Introduction

Gottingen and Hilbert andEinstein 1917-1920

Gottingen and Algebra1920-1933

German Nationalism: Exilefrom Germany

References


INTRODUCTION

Emmy Noether (pronouncedNER-ter) was born to aprosperous Jewish family inthe Bavarian university townof Erlangen on March 23, 1882.She died unexpectedly inanother university town, BrynMawr Pennsylvania, on April14, 1935 at the age of 53.

The aim of this talk is to tellpart of the story of thisremarkable woman andprovide an outline of herscientific legacy.

—————


She was the eldest of four children. Little is known about herchildhood beyond what can be inferred from her generalsituation. There are no anecdotes that reveal that she was“destined” to be a great mathematician. Girls at that time andplace and social class simply did not study the sciences, and anacademic career was out of the question.

According to Wikipedia,As a girl, Noether was well liked. She did not stand outacademically although she was known for being clever andfriendly. She was near-sighted and talked with a minor lispduring her childhood.

—————


Her father Max Noether, though, was a Professor ofMathematics at the University of Erlangen and studied whatwe would call algebraic geometry. He was a contemporary ofFelix Klein, and the ideas of the “Erlangen Program” must havebeen in the air.

(One finds variant spellings of the family name. Apparentlyher father’s family name was originally Samuel but waschanged by fiat of the local authorities under a “ToleranceEdict” to Nother sometime after 1809. During her father’sgeneration that was changed again to Noether.)—————


She went to the standard localgirl’s school and trained tobecome a High School teacherin modern languages, Frenchand English. She graduated in1890 with good grades,though apparently “classmanagement” was not herstrong suit.

—————


The oldest of her younger brothers was Alfred, born in 1883,who was awarded a PhD in chemistry in 1909 but died in 1918.

Fritz was born in 1884 and was a good applied mathematician.After the rise of the Nazis made work at German Universitiesimpossible he got a job at Tomsk State University in the SovietUnion. During the Great Purge of 1937 he was arrested andsentenced as a German spy and eventually shot in 1941.

Gustav Robert, was born in 1889 and died in 1928, aftersuffering “chronic illnesses.”—————


—————


In 1900 the law was changed to allow women to audit classes atErlangen, though only with the explicit permission of theinstructor, and Emmy became one of two women (out of athousand students) to attend classes there.

She studied languages and history and mathematics andpresumably at some point it became obvious how good shewas at mathematics, and how much she enjoyed it, and sheswitched to mathematics entirely.

She passed the graduation exam in 1903 and attended lectures(Schwarzschild, Minkowski, Hilbert) for a semester atGottingen in 1903-04.

Her Father’s colleague Paul Gordan agreed to be her PhDthesis supervisor (when that became possible) and she finishedher degree in Mathematics at Erlangen in 1907.—————


Her work with Gordan was in an area called Invariant Theory,and involved finding finite sets of polynomials that generatedinfinite sets of polynomials which leave “invariant” objects wewould now call algebraic varieties.

David Hilbert, at the center of the scientific world in Gottingen,had achieved his first fame by outlining conditions underwhich this was possible.

But actually finding generators for a given situation was adifferent matter. In her work with Gordan based on her thesis,and published in a major journal and widely known, sheprovided sets of generators for more than 300 of these.

Once found these finite sets can easily be shown to generate.

But apparently there were, and are, no explicit, algorithmic,methods for finding even one of these, let alone 300, and thework was regarded as a computational marvel.—————


From 1908 to 1915 Noether continued her work in this area andlearned of the techniques used by David Hilbert, up the road inGottingen.

She expanded her work to study invariants on fields of rationalfunctions and finite groups and worked with several othermathematicians on these topics. (Erhard Schmidt, Ernst Fisher)

She lectured during this time but never in her own name, andnever for pay—it was forbidden by law for a woman to teach atuniversity.—————


GOTTINGEN AND HILBERT AND EINSTEIN 1917-1920In 1907 Einstein created the Special Theory of Relativity, butsoon realized it was missing critical elements (gravitation, forinstance) and turned to Mathematicians to help with theproblems he encountered. In particular he began a livelycorrespondence with David Hilbert, who had becomefascinated by Mathematical Physics, and the two were workingout together the key elements of the General Theory Einsteinproposed for this.

There were major issues: for instance it seemed that energy wasnot conserved in the theory.

Hilbert recognized the issue as being related to invarianttheory, a subject he had not looked at in a while, and realizedhe needed a current expert on the subject to help him here. Heknew of Fraulein Noether from Erlangen who had beenworking in this area.—————


Hilbert and Felix Klein (who had since moved to Gottingen)invited her to visit Gottingen and attempted to recruit her in1915—but an appointment was blocked by the Philosophydepartment, which refused to consider an application forhabilitation by a woman.

I do not see that the sex of the candidate is an argu-ment against her. After all, we are a university, not abathhouse. (David Hilbert)

Though these efforts did not prevail at this time she did stay atGottingen with financial support from her family and workedwith Hilbert and lectured under his name—without pay.

During the next year she finished the work for which Hilbertasked her to come, the results we now know as Noether’sTheorems.—————


People understood that there were things like linearmomentum, angular momentum, energy and charge that wereconserved in physical theories. But no one really asked why thisshould be true. It wasn’t really necessary in NewtonianMechanics: it was all pretty obvious and fell out of minimizingthe action (an integral involving the Lagrangian) using basiccalculus techniques, calculations of the type we saw in LarryCurnutt’s talk last year.

But GR was a deeper theory, and nothing was really obvious.Why is anything conserved in this theory?

Her work cleared up, incidentally, the issue of conservation ofenergy by showing that, though energy in GR might not appearto be conserved locally it was conserved globally.

But it was about much more than that.

She also proved a converse: whenever you have a conservedquantity there is a corresponding symmetry in physical law.—————


The symmetries of physical law we are talking about arecontinuous symmetries: those of you who attended last year’stalk on group theory and symmetry might recall thesymmetries which I called SquareSym and CircleSym.

SquareSym has a discrete symmetry group. Not the kind towhich Noether’s Theorems could apply.

CircleSym has a continuous symmetry group—homomorphicto the unit circle in the complex plane with complexmultiplication. This is the kind to which Noether’s Theoremscould apply.—————


Einstein and Hilbert needed to understand what, in this newgeometry, stayed the same. She helped them understand howto describe things that stay the same even if you changecoordinates in GR and how to talk about symmetry in physicallaw generally.

By this time Hilbert and Einstein both knew they were workingwith a profound intellect and supported and encouraged herhowever they could.

The easiest way to resolve this is to have Noetherexplain it to me. (comment by Albert Einstein)

—————


Klein presented this work (she could not herself, because shewas not a member) in 1918 at a meeting of the Royal Society ofSciences at Gottingen and in 1919 the University of Gottingenallowed her to use this work for her habilitationsschrift andthey granted her a non-civil-service and unpaid “extraordinaryprofessorship” which allowed her to lecture under her ownname.

In 1920 she was given the position of “Lecturer in Algebra”(Lehrbeauftragte fur Algebra) which was a paid position at last.—————


GOTTINGEN AND ALGEBRA 1920-1933

After her habilitation Noether completely gave up work in thearea of her habilitationsschrift and devoted the rest of hercareer to her first love, the subject we now call AbstractAlgebra, whose form and ways of thinking are largely due toher influence.

This is how Mathematicians know of her, most of whom havenever heard of Noether’s Theorem (except in the context ofvarious important results in algebraic geometry of her fatherMax: Brill-Noether Theory, Noether’s formula,Noether-Lefschetz theorem.)—————


In her classic 1921 paper Theory of Ideals in Ring Domains(Idealtheorie in Ringbereichen) Noether invented the idea ofideals in commutative rings and an entirely new way ofthinking of mathematical objects, sets of elements, as a “whole”with their own properties in relation to other such sets.

For instance there is something called an “ascending chaincondition” in ring theory (and in many other categories as well)and objects satisfying this condition are called Noetherian.

She went on to expand this work to noncommutative algebrasand rephrased the representation theory of groups in terms ofthe theory of modules and ideals.—————


In the winter of 1928-1929 Noether taught and visited atMoscow State University and worked with P. S. Alexandrov,Lev Pontryagin and Nikolai Chebotaryov there who laterpraised her contributions to Galois Theory.

In a 1928 paper, Heinz Hopf credits a conversation withNoether to simplification of his approach to Betti numbers,thereby inventing the concept of homology group, at the verydawn of the subject of Algebraic Topology.—————


As impressive and powerful as these contributions were themost remarkable thing about Noether’s work was the new wayof thinking about these objects. This wholistic approachredefined how mathematicians in many areas worked and isher most lasting mathematical legacy.

This really cannot be attributed to her in the citations of worksthat use it: once you learn to think this way you just do and thepowerful results you can then prove are yours.

Yet her hand is there.—————


During this period students and post-docs came in droves tostudy with her at Gottingen—they were known as “Noether’sboys” and are a “who’s who” of famous algebraists. She wassupervisor to a dozen PhD students in this time.

van der Waerden’s Modern Algebra (1931) looks like acompletely modern graduate algebra text.

Completely unegotistical and free of vanity, she neverclaimed anything for herself, but promoted the works of herstudents above all.

Birkhoff and Mac Lane’s A Survey of Modern Algebra (1941)brought her methods into English

Emil Artin, for whom is named the “descending chaincondition” (Artinian) was one of “Noether’s boys”

Jean Dieudonne praised the work of Noether for liberatingLinear Algebra “from the plague of matrices and determinantsfrom which it had been suffering for a century.”—————


By all accounts Noether was a powerful, incredibly creativeand steadfastly independent mathematician: a completelyconfident personality.

Though she was blatantly discriminated against as a woman(and because of her Jewish heritage) she was reportedly agood-humored and happy person, warm and helpful toeveryone around her, especially her students and juniorcolleagues, who were devoted to her.

She had the recognition of the male colleagues around her andwas an acknowledged leader. She was the first female plenaryspeaker at the International Congress of Mathematicians(Zurich 1932) and at that point was at the height of her powers,influencing everyone around her.—————


In 1932 Emmy Noether (together with Emil Artin) received theAckermann-Teubner Memorial Award for their contributionsto mathematics.

In the judgment of the most competent living mathemati-cians, Fraulein Noether was the most significant creativemathematical genius thus far produced since the higher ed-ucation of women began. In the realm of algebra, in whichthe most gifted mathematicians have been busy for centuries,she discovered methods which have proved of enormous im-portance in the development of the present-day younger gen-eration of mathematicians. Albert Einstein

—————


Miss Noether is the greatest woman mathematician who hasever lived and the greatest woman scientist of any sort nowliving, and a scholar at least on the plane of Madame Curie.

Norbert Wiener

During her lifetime, and even until today, Noether has beencharacterized as the greatest woman mathematician inrecorded history by mathematicians such as Pavel Alexandrov,Hermann Weyl and Jean Dieudonne. Wikipedia—————


Noether . . . taught us to think in terms of simple and gen-eral algebraic concepts—homomorphic mappings, groupsand rings with operators, ideals—and not in cumbersomealgebraic computations; and she thereby opened up the pathto finding algebraic principles in places where such princi-ples had been obscured by some complicated special situa-tion. Pavel Alexandrov


My methods are really methods of working and thinking;this is why they have crept in everywhere anonymously.

Emmy Noether

—————


GERMAN NATIONALISM: HER EXILE FROM GERMANY

In 1933 Hitler rose to power and within months the Law for theRestoration of the Professional Civil Service was passed, whichremoved Jews from their jobs.

Hermann Weyl wrote thatEmmy Noether—her courage, her frankness, her unconcernabout her own fate, her conciliatory spirit—was in the midstof all the hatred and meanness, despair and sorrow sur-rounding us, a moral solace.

—————


Even after her firing, when she was trying to decide what to donext, she held seminars and worked with students in herapartment.

Dozens of unemployed professors were looking for work:Albert Einstein and Hermann Weyl were appointed by theInstitute for Advanced Study and used connections to get her ajob (still 1933) at Bryn Mawr College.—————


This position was not really suitable for someone of her stature,though Bryn Mawr was very welcoming and proud to have herthere. She went weekly to talk and work with people atPrinceton, but (apart from a few) did not find the faculty therevery welcoming to her as a female mathematician.

In 1934 at the invitation of Oswald Veblen she began lecturingat the Institute for Advanced Study and she supervised twoPhD students in this time. She continued a collaboration(begun in 1930 in Europe) with Richard Brauer, who had alsomoved to the United States.—————


She was reported to be in good spirits and focussed on herwork, but she was still finding her place in this new world. Apermanent position was in the works when (18 months afterher move) she developed an ovarian cyst and went in for whatwas expected to be relatively minor surgery.

She recovered well for several days but a high fever hitsuddenly and she died on the fourth day, April 14, 1935 at theage of 53 and at the height of her powers.—————


Her work, however, does live on. In Mathematics her influenceis so thorough that people don’t even understand that it camefrom her: it is just the air that grad students andmathematicians breathe, even in areas barely related to herwork.

In applications areas the Noether Theorems remain pervasiveand are critical in such places as Fluid Mechanics (potentialvorticity) and Numerical Analysis and even ComputerGraphics, models of the world.

—————


But particle Physicists use the work everywhere: there haslikely not been a particle physics paper in fifty years in whichsymmetry has not been mentioned.

Forty years after her death, the Standard Model of particlephysics was invented using explicit reference to her work. Shewas the inventor and first user, in her theorems, of whatPhysicists now call Gauge Theory. Five Nobel prizes have beengiven for work in modern Particle Physics alone that would notexist without her theorems.

—————


REFERENCES

Dwight Neuenschwander Emmy Noether’s Wonderful Theorem

Steve Nadis Discover June 2019 How Mathematician EmmyNoether’s Theorem Changed Physics

Numerous Wikipedia Entries

Saunders Mac Lane Journal of Pure and Applied Algebra(1986) Topology becomes Algebraic with Vietoris and Noether

Yvette Kosmann-Schwarzbach The Noether Theorems

Gennadi Sardanashvily Noether’s Theorems

Emily Conover Science News June 2018 In her short life,mathematician Emmy Noether changed the face of physics

BBC Radio In Our Time January 24, 2019 Podcast Emmy Noether—————

Documents

Amalie (Emmy) Noether .5cm [width=2cm]QRNoethersusanka.org/MathPhysics/NoetherHandout.pdfAmalie (Emmy) Noether TOC Intro G¨ottingen 1917-1920 Gottingen 1920-1933¨ Exile References