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Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether Gottingen¨ and Algebra Amalie (Emmy) Noether 1920-1933 Table of Contents German Nationalism: Exile Introduction from Germany Gottingen¨ and Hilbert and Einstein 1917-1920 References Larry Susanka October 3, 2019 Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References INTRODUCTION She was the eldest of four children. Little is known about her Emmy Noether (pronounced childhood beyond what can be inferred from her general NER-ter) was born to a situation. There are no anecdotes that reveal that she was prosperous Jewish family in “destined” to be a great mathematician. Girls at that time and the Bavarian university town place and social class simply did not study the sciences, and an of Erlangen on March 23, 1882. academic career was out of the question. She died unexpectedly in another university town, Bryn Mawr Pennsylvania, on April According to Wikipedia, 14, 1935 at the age of 53. As a girl, Noether was well liked. She did not stand out academically although she was known for being clever and The aim of this talk is to tell friendly. She was near-sighted and talked with a minor lisp part of the story of this during her childhood. remarkable woman and provide an outline of her ————— scientific legacy. ————— Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Her father Max Noether, though, was a Professor of Mathematics at the University of Erlangen and studied what She went to the standard local we would call algebraic geometry. He was a contemporary of girl’s school and trained to Felix Klein, and the ideas of the “Erlangen Program” must have become a High School teacher been in the air. in modern languages, French and English. She graduated in 1890 with good grades, though apparently “class management” was not her strong suit. (One finds variant spellings of the family name. Apparently her father’s family name was originally Samuel but was changed by fiat of the local authorities under a “Tolerance Edict” to Nother¨ sometime after 1809. During her father’s generation that was changed again to Noether.) ————— ————— Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References 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 Universities impossible he got a job at Tomsk State University in the Soviet Union. During the Great Purge of 1937 he was arrested and sentenced as a German spy and eventually shot in 1941. Gustav Robert, was born in 1889 and died in 1928, after suffering “chronic illnesses.” ————— ————— Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Her work with Gordan was in an area called Invariant Theory, In 1900 the law was changed to allow women to audit classes at and involved finding finite sets of polynomials that generated Erlangen, though only with the explicit permission of the infinite sets of polynomials which leave “invariant” objects we instructor, and Emmy became one of two women (out of a would now call algebraic varieties. thousand students) to attend classes there. David Hilbert, at the center of the scientific world in Gottingen,¨ She studied languages and history and mathematics and had achieved his first fame by outlining conditions under presumably at some point it became obvious how good she which this was possible. was at mathematics, and how much she enjoyed it, and she switched to mathematics entirely. But actually finding generators for a given situation was a different matter. In her work with Gordan based on her thesis, She passed the graduation exam in 1903 and attended lectures and published in a major journal and widely known, she (Schwarzschild, Minkowski, Hilbert) for a semester at provided sets of generators for more than 300 of these. Gottingen¨ in 1903-04. Once found these finite sets can easily be shown to generate. Her Father’s colleague Paul Gordan agreed to be her PhD thesis supervisor (when that became possible) and she finished But apparently there were, and are, no explicit, algorithmic, her degree in Mathematics at Erlangen in 1907. methods for finding even one of these, let alone 300, and the ————— work was regarded as a computational marvel. ————— Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References GOTTINGEN¨ AND HILBERT AND EINSTEIN 1917-1920 In 1907 Einstein created the Special Theory of Relativity, but From 1908 to 1915 Noether continued her work in this area and soon realized it was missing critical elements (gravitation, for learned of the techniques used by David Hilbert, up the road in instance) and turned to Mathematicians to help with the Gottingen.¨ problems he encountered. In particular he began a lively correspondence with David Hilbert, who had become She expanded her work to study invariants on fields of rational fascinated by Mathematical Physics, and the two were working functions and finite groups and worked with several other out together the key elements of the General Theory Einstein mathematicians on these topics. (Erhard Schmidt, Ernst Fisher) proposed for this. She lectured during this time but never in her own name, and There were major issues: for instance it seemed that energy was never for pay—it was forbidden by law for a woman to teach at not conserved in the theory. university. Hilbert recognized the issue as being related to invariant ————— theory, a subject he had not looked at in a while, and realized he needed a current expert on the subject to help him here. He knew of Fraulein¨ Noether from Erlangen who had been working in this area. ————— Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References People understood that there were things like linear Hilbert and Felix Klein (who had since moved to Gottingen)¨ momentum, angular momentum, energy and charge that were invited her to visit Gottingen¨ and attempted to recruit her in conserved in physical theories. But no one really asked why this 1915—but an appointment was blocked by the Philosophy should be true. It wasn’t really necessary in Newtonian department, which refused to consider an application for Mechanics: it was all pretty obvious and fell out of minimizing habilitation by a woman. the action (an integral involving the Lagrangian) using basic I do not see that the sex of the candidate is an argu- calculus techniques, calculations of the type we saw in Larry ment against her. After all, we are a university, not a Curnutt’s talk last year. bathhouse. (David Hilbert) But GR was a deeper theory, and nothing was really obvious. Why is anything conserved in this theory? Though these efforts did not prevail at this time she did stay at Gottingen¨ with financial support from her family and worked Her work cleared up, incidentally, the issue of conservation of with Hilbert and lectured under his name—without pay. energy by showing that, though energy in GR might not appear to be conserved locally it was conserved globally. During the next year she finished the work for which Hilbert asked her to come, the results we now know as Noether’s But it was about much more than that. Theorems. She also proved a converse: whenever you have a conserved ————— quantity there is a corresponding symmetry in physical law. ————— Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References The symmetries of physical law we are talking about are continuous symmetries: those of you who attended last year’s talk on group theory and symmetry might recall the Einstein and Hilbert needed to understand what, in this new symmetries which I called SquareSym and CircleSym. geometry, stayed the same. She helped them understand how to describe things that stay the same even if you change coordinates in GR and how to talk about symmetry in physical law generally. By this time Hilbert and Einstein both knew they were working with a profound intellect and supported and encouraged her however they could. SquareSym has a discrete symmetry group. Not the kind to which Noether’s Theorems could apply. The easiest way to resolve this is to have Noether CircleSym has a continuous symmetry group—homomorphic explain it to me. (comment by Albert Einstein) to the unit circle in the complex plane with complex ————— multiplication. This is the kind to which Noether’s Theorems could apply. ————— Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References GOTTINGEN¨ AND ALGEBRA 1920-1933 Klein presented this work (she could not herself, because she After her habilitation Noether completely gave up work in the was not a member) in 1918 at a meeting of the Royal Society of area of her habilitationsschrift and devoted the rest of her Sciences at Gottingen¨ and in 1919 the University of Gottingen¨ career to her first love, the subject we now call Abstract allowed her to use this work for her habilitationsschrift and Algebra, whose form and ways of thinking are largely due to they granted her a non-civil-service and unpaid “extraordinary her influence.
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  • Arxiv:1604.08579V1 [Math.MG] 28 Apr 2016 ..Caatrztoso I Groups Lie of Characterizations 5.1

    Arxiv:1604.08579V1 [Math.MG] 28 Apr 2016 ..Caatrztoso I Groups Lie of Characterizations 5.1

    A PRIMER ON CARNOT GROUPS: HOMOGENOUS GROUPS, CC SPACES, AND REGULARITY OF THEIR ISOMETRIES ENRICO LE DONNE Abstract. Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks. We consider them as special cases of graded groups and as homogeneous metric spaces. We discuss the regularity of isometries in the general case of Carnot-Carathéodory spaces and of nilpotent metric Lie groups. Contents 0. Prototypical examples 4 1. Stratifications 6 1.1. Definitions 6 1.2. Uniqueness of stratifications 10 2. Metric groups 11 2.1. Abelian groups 12 2.2. Heisenberg group 12 2.3. Carnot groups 12 2.4. Carnot-Carathéodory spaces 12 2.5. Continuity of homogeneous distances 13 3. Limits of Riemannian manifolds 13 arXiv:1604.08579v1 [math.MG] 28 Apr 2016 3.1. Tangents to CC-spaces: Mitchell Theorem 14 3.2. Asymptotic cones of nilmanifolds 14 4. Isometrically homogeneous geodesic manifolds 15 4.1. Homogeneous metric spaces 15 4.2. Berestovskii’s characterization 15 5. A metric characterization of Carnot groups 16 5.1. Characterizations of Lie groups 16 Date: April 29, 2016. 2010 Mathematics Subject Classification. 53C17, 43A80, 22E25, 22F30, 14M17. This essay is a survey written for a summer school on metric spaces. 1 2 ENRICO LE DONNE 5.2. A metric characterization of Carnot groups 17 6. Isometries of metric groups 17 6.1.